Cerius2 - ESIvi Cerius2 Property Prediction/December 1998 Applied stress/strain tables 10 Standard Voigt notation 10 Editing the defaults 10 Plot stress/strain profile 10 Choosing

Cerius2Property Prediction

December 1998

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Cerius2 Property Prediction/December 1998 v

How to Use This Book xxvii

Who should use this book . . . . . . . . . . . . . . . . . . . . . . . . xxviiThings to be familiar with xxviiWorkstation requirements xxviii

How to find information . . . . . . . . . . . . . . . . . . . . . . . . . .xxixUsing other Cerius2 books. . . . . . . . . . . . . . . . . . . . . . . . .xxixOnline Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxTypographical conventions . . . . . . . . . . . . . . . . . . . . . . . . xxx

1. Mechanical Properties 1

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1Sections in this chapter 1

General methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2Second derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2Constant-stress minimization . . . . . . . . . . . . . . . . . . . . .3

Default mode 3Custom mode 4

Constant strain minimization . . . . . . . . . . . . . . . . . . . . .4Default and custom mode 4

Advantages and drawbacks of the three methods . . . . .4Calculating mechanical properties . . . . . . . . . . . . . . . . .5

Module design 5Novice or expert 5Methods 5Sweep parameters 6Output 6Minimize Model First 6Accumulate Averages 6Data analysis 7Crystal constraints 7Restoring data 7

To calculate the mechanical properties of a model . .7To determine the effect of crystal constraints . . . . . .9

Specifying the applied stress/strain and mode . . . . . . .9Sweep defaults 9Choosing values 9Choosing the mode 9

vi Cerius2 Property Prediction/December 1998

Applied stress/strain tables 10Standard Voigt notation 10Editing the defaults 10Plot stress/strain profile 10Choosing the mode 10

To specify variables for constant-stress minimization11

Sweep Defaults 11Mode 11Stress profile 11Plot profile 11

To specify the constant strain minimization variables12

Sweep Defaults 12Mode 12Strain profile 12Plot profile 12

Displaying and saving results . . . . . . . . . . . . . . . . . . . 12Output variables . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Text and tables files 13Level of detail 13

To specify the output variables . . . . . . . . . . . . . . . 13Analyzing the data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Data plotting 14Data fitting 15

To analyze constant-stress minimization data . . . . 15Data plotting 15Curve fitting 15Average Y values 16Updating properties 16

To analyze constant-strain minimization data . . . . 16Restoring data for analysis . . . . . . . . . . . . . . . . . . . . . . 16

Reload options 16To reload a mechanical properties calculation file . 17

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Deriving mechanical properties . . . . . . . . . . . . . . . . . . 17

Compliance matrix 17Compressibility 17Bulk modulus 17

Cerius2 Property Prediction/December 1998 vii

Young’s modulus 17Poisson’s ratios 18Velocities of sound 18Lamé constants 18

Model, forcefield, and energy setup . . . . . . . . . . . . . . .19Models 19Force field and energy setup 19

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20

2. Polymer Properties 21Sections in this chapter 21

General Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22Selecting the trajectory file . . . . . . . . . . . . . . . . . . . . . .22

To select the trajectory file to be analyzed . . . . . . .23Selecting the trajectory file frames . . . . . . . . . . . . . . . .23

To specify the trajectory frames to be analyzed . . .23Calculating physical and chain properties . . . . . . . . . .23

To calculate physical or chain properties . . . . . . . .24Select the physical properties 24Select the chain properties 24

To analyze the Voronoi volume. . . . . . . . . . . . . . . . . . .25Calculating dihedral distributions . . . . . . . . . . . . . . . .26

1D or 2D plots, timeevolution 26Where variables are set 26Input 26Defining the dihedrals 26

To obtain a 1D distribution or 2D plot . . . . . . . . . .27Input the model or trajectory data 27Specify the output 27Define the dihedrals 27Do the calculations 27

To plot the time evolution of a single dihedral. . . .28Specifying the torsion selection rules . . . . . . . . . . . . . .28

Defaults 29To specify the torsion selection rules . . . . . . . . . . .29

Obtaining orientation function results . . . . . . . . . . . . .29Input 29Where variables are set 30Reference direction 30

viii Cerius2 Property Prediction/December 1998

Structural unit vector 30Information obtained 30

To calculate the orientation function and angledistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Input the model ortrajectory data 30Specify the output 31Define the reference direction 31Define the structural unit vector 31Do the calculations 31

To plot the time evolution of a single vector . . . . . 32Specifying the bond selection rules . . . . . . . . . . . . . . . 32

Defaults 33To specify the bond selection rules . . . . . . . . . . . . 33

To specify a particular bond pair type 33To specify a particular bond order 33To include hydrogen bonds 33

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Physical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Dipole moment 33Quadrupole tensor 34Molecular weight 34Density 34

Chain properties — radius of gyration and end-to-enddistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Radius of gyrationtensor (S) 35Principal valuesand directions 35Radius of gyration (s) 36End-to-end distance (R) 36

Voronoi volume analysis . . . . . . . . . . . . . . . . . . . . . . . 36Voronoi polygon 36Determining Voronoi volume 36Information obtained 37Cutoff distance 38Order versus disorder 38External effects 38

Cerius2 Property Prediction/December 1998 ix

Calculating the orientation function (order parameter) 38Defining the terms . . . . . . . . . . . . . . . . . . . . . . . . .39

Hermans orientation function, P2 39Order parameter, S 39Average value used, <cos2θ> 39Range of values for P2 39Random orientation 40Average angle 40

3. Blends 41

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41Sections in this chapter 41

Utility and applications . . . . . . . . . . . . . . . . . . . . . . . .42Applications 42

Using Blends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43Calculate functions 43Analyze functions 43Extracting pairs 44

General methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44To calculate pair interaction energies . . . . . . . . . . . . . .44

Specify the packingvariables 44Specify the calculation controls 45Specify the outputcontrols 45To calculate a single Eij 45To calculate all Eijs (E11, E12, E21, and E22) 45

To calculate the coordination numbers . . . . . . . . . . . . .45Specify the packingvariables 46Specify the calculation controls 46Specify the output 46To calculate a single Z 46To calculate all Zs (Z11, Z12, Z21, and Z22) 46

To specify the packing variables . . . . . . . . . . . . . . . . . .46Specify the alignment 46Specify the noncontact atoms 47

To fit the mixing energy data . . . . . . . . . . . . . . . . . . . .47

x Cerius2 Property Prediction/December 1998

To plot the interaction parameter Chi(T) . . . . . . . . . . . 48To calculate a phase diagram . . . . . . . . . . . . . . . . . . . . 48Plotting pair energy distributions . . . . . . . . . . . . . . . . 49

To plot a pair energy distribution . . . . . . . . . . . . . 51Plotting thermodynamic functions . . . . . . . . . . . . . . . 51

Isotherms 52To calculate and plot thermodynamic functions . . 52

Specify the functions to be computed 52Specify the model 52Specify the degree of polymerization 52

Plotting thermodynamic isotherms . . . . . . . . . . . . . . . 52To plot thermodynamic isotherms. . . . . . . . . . . . . 53

Specify the model 53Specify the degree of polymerization 53

Extracting molecular pairs . . . . . . . . . . . . . . . . . . . . . . 53To extract molecular pair configurations . . . . . . . . 54

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Theoretical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Flory-Huggins model 54Other theories 55

Molecular simulations . . . . . . . . . . . . . . . . . . . . . . . . . 56Blends approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Extensions to Flory-Huggins model 56Analytical fit 57Information obtained 57Force field employed 57

Calculating pair interaction energies . . . . . . . . . . . . . . 58Sampling problems 58Molecular dynamics 58

Blends Monte Carlo sampling technique. . . . . . . . 58The pairs method. . . . . . . . . . . . . . . . . . . . . . . . . . 59

Packing variables 60Output 60Where variables are set 61

Calculating coordination numbers. . . . . . . . . . . . . . . . 61Nearest-neighborpacking 62Packing variables 62Averaged Zijs obtained 63Where variables are set 63Averaged Zs used to

Cerius2 Property Prediction/December 1998 xi

calculate χ 63Specifying packing variables . . . . . . . . . . . . . . . . . . . .63

Isotropic versus axial packing . . . . . . . . . . . . . . . .63Excluded atom constraints . . . . . . . . . . . . . . . . . . .64

Fitting the mixing energy and calculating Chi . . . . . . .64Fitting Emix (T) 64Plotting χ(T) 65

Calculating phase diagrams . . . . . . . . . . . . . . . . . . . . .65Calculating ∆G 66

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .67

4. Synthia 69

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69Sections in this chapter 69

Using Synthia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .70A. Building the copolymer . . . . . . . . . . . . . . . . . . .70B. Predicting properties . . . . . . . . . . . . . . . . . . . . .72C. Studying a range of concentrations . . . . . . . . . .74

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75General methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . .76

Calculable properties 76Scope and limitations . . . . . . . . . . . . . . . . . . . . . . . . . .78Computing repeat unit length . . . . . . . . . . . . . . . . . . .79Energy minimization of repeat units . . . . . . . . . . . . . .80Values of cohesive energy and solubility parameter used

in correlations for other properties . . . . . . . . . . . . .80Representation of amide groups . . . . . . . . . . . . . . . . . .81Units of permeability . . . . . . . . . . . . . . . . . . . . . . . . . .81Developing correlations with QSAR. . . . . . . . . . . . . . .81Performing polymer properties calculations . . . . . . . .82

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .85Background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .85Connectivity indices of polymer repeat units . . . . . . . .86Connectivity indices of polymer chains . . . . . . . . . . . .87General forms of the correlations in terms of connectivity

indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .89Backbone and side group connectivity indices . . . . . . .89Actual correlations for various properties and validation

against experimental data . . . . . . . . . . . . . . . . . . .90References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .90

xii Cerius2 Property Prediction/December 1998

5. RMMC (RIS MetropolisMonte Carlo) 91

Using RMMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91General Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

RIS Metropolis Monte Carlo (RMMC) Concepts . . . . . 94Rotatable Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . 94Energy Calculation . . . . . . . . . . . . . . . . . . . . . . . . 95Parameters Controlling the Simulation . . . . . . . . . 96

“RIS” Metropolis Monte Carlo (RMMC) Simulations . 98Computing Dihedral Distribution Functions . . . . 98Output Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99RIS Metropolis Monte Carlo Simulation . . . . . . . . . . . 99

Properties Calculated with RMMC . . . . . . . . . . . 100The RMMC Algorithm . . . . . . . . . . . . . . . . . . . . 100Treatment of Constraints . . . . . . . . . . . . . . . . . . . 101

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6. Crystal Packer 105

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105Sections in this chapter 105

About Crystal Packer . . . . . . . . . . . . . . . . . . . . . . . . . 106Rigid units 106Torsional subrotations 106Quick defaultminimization 106

General methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107Model initialization . . . . . . . . . . . . . . . . . . . . . . . . . . 107

To initialize the model . . . . . . . . . . . . . . . . . . . . . 107To set energy calculation options . . . . . . . . . . . . . . . . 108

External pressure 108van der Waals preferences 108On-diagonal VDW parameters 108Off-diagonal parameters 109van der Waals interaction range 109Coulomb preferences 109H-bond preferences 109Subrotations 111Defining subrotations 111Fourier parameter sets 111

Cerius2 Property Prediction/December 1998 xiii

1–4 interactions 111Minimizer constraints setup . . . . . . . . . . . . . . . . . . . .112

To set up minimizer constraints . . . . . . . . . . . . . .112Variable cell parameters 112Rigid units 113Variable subrotations 113

Minimizer preferences setup. . . . . . . . . . . . . . . . . . . .113Minimization algorithm 113Modified Newton 114Steepest descents 114Maximum increments 114The contact table 114Updating the table 115Minimization termination 115Energy tolerance 115

To set minimization preferences . . . . . . . . . . . . . .115Control parameters 115Maximum increments 115Termination criteria 116

Calculating energy and running the packing calculation .116To calculate model energy . . . . . . . . . . . . . . . . . .116To perform a minimization of model energy . . . .117

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .117Energy expression setup . . . . . . . . . . . . . . . . . . . . . . .117

The van der Waals term . . . . . . . . . . . . . . . . . . . .118Polymer, surface, andnetwork models 118Treatment of long-range interactions 119Close contact check 119The Lennard-Jones functional form 119On-diagonals 119Off-diagonals 120

The Coulomb term . . . . . . . . . . . . . . . . . . . . . . . .120Minimum charge 121The Ewald sum 121

The hydrogen bond term . . . . . . . . . . . . . . . . . . .121Functional form ofH-bond potential 122

xiv Cerius2 Property Prediction/December 1998

H-bond parameters 123van der Waals parameters and H-bonds 123

The torsional energy term . . . . . . . . . . . . . . . . . . 123The external pressure term . . . . . . . . . . . . . . . . . 125

Uses for external pressure 125References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

7. Polymorph Predictor 127


Using the Polymorph Predictor . . . . . . . . . . . . . . . . . . . . 128Step 1: Setup 128Step 2: Monte Carlo packing simulation 128Step 3: Cluster analysis 128Step 4: Energy minimization 128Step 5: Cluster minimized structures 129Trajectory file merging 129Trajectory file analysis 129Reliability checks 129

General Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129Setting up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129Configuring forcefields . . . . . . . . . . . . . . . . . . . . . . . 130

Preparing models. . . . . . . . . . . . . . . . . . . . . . . . . 130Minimize and calculate charges 130Flexible molecules 131

Simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . 132Predicting Polymorphs . . . . . . . . . . . . . . . . . . . . . . . 132

Polymorph Predictor limitations . . . . . . . . . . . . . 132Flexible molecules 132Forcefields 132Processing time 133Intramolecular symmetry 133

Running a prediction . . . . . . . . . . . . . . . . . . . . . . 133Restarting interrupted prediction procedures 133Defining the molecules in the crystal structure 134

To run a complete polymorph prediction sequence .134

To predict polymorphs manually . . . . . . . . . . . . 136Monte Carlo packing simulation . . . . . . . . . . . . . . . . 138

The simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Cerius2 Property Prediction/December 1998 xv

Generating initialstructures 138Heating phase 139Cooling phase 139

Trial steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .139Setting Monte Carlo preferences . . . . . . . . . . . . .140

Search parameters . . . . . . . . . . . . . . . . . . . . .141Effects of changing search parameters . . . . . .141

Heating phase 141Cooling phase 142

Space groups . . . . . . . . . . . . . . . . . . . . . . . . .142Output options . . . . . . . . . . . . . . . . . . . . . . . . . . .144

Cluster analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .144Clustering Monte Carlo output 144Clustering energyminimization output 144

Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .144Clustering algorithm . . . . . . . . . . . . . . . . . . . . . .145Setting Cluster Analysis preferences . . . . . . . . . .145Input file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .145Clustering parameters . . . . . . . . . . . . . . . . . . . . .146Clustering tolerances for Monte Carlo and minimized

output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .146Monte Carlo output 146Minimized output 146Identifying sensible clustering values 146

Output options . . . . . . . . . . . . . . . . . . . . . . . . . . .147Energy minimization . . . . . . . . . . . . . . . . . . . . . . . . .147

Optimizing the current model 147Setting minimization preferences . . . . . . . . . . . . .147

Input file 147Termination criteria 147Rigid Bodies 148Output options 148

Notes on rigid body minimization . . . . . . . . . . . . . . .148Trajectory file analysis and model extraction . . . . . . .149

Select trajectory file 149Properties 149Model extraction 150

Automatically comparing powder spectra . . . . . . . . .151Select trajectory file 151

xvi Cerius2 Property Prediction/December 1998

Select experimental spectrum 151Transform experimental spectrum 152Specify Diffraction-Crystal settings 152Set CMACS intervals 153Specify an identifier for the comparison 153Calculate measures of comparison 153Analyze measures of comparison 153Show settings 154Delete Set 154To automatically compare powder spectra 154

Merging trajectory files . . . . . . . . . . . . . . . . . . . . . . . 155Reliability checking . . . . . . . . . . . . . . . . . . . . . . . . . . 156

Examine structures/properties 156Comparison against experimental data 156Symmetry 157Change the FF dielectric parameters 157

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157Predicting polymorphs . . . . . . . . . . . . . . . . . . . . . . . 157

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158Effect of disregarding TS 158Reliability checks 159Further reading 159

Simulated annealing theory . . . . . . . . . . . . . . . . . . . . 160Packing simulation difficulties 160Solutions provided by simulated annealing 160

Automatically comparing powder spectrad . . . . . . . 161References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

8. Morphology 165


Using Morphology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166Step 1: Setup 166Step 2: Calculations 166Step 3: Visualization 167Step 4: Listing, editing, adding and removing growth faces

Cerius2 Property Prediction/December 1998 xvii

167Step 5: Crystal attributes 167Step 6: Saving themorphology 167

General Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . .168Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .168

Other Cerius2 modules 168Calculating Morphology with the BFDH method . . .168

Growth planes and growth rate list 169Minimum slice thickness 169Displaying themorphology 169

To calculate morphology with the BFDH method 169Calculating morphology with the Attachment or Surface

Energy methods . . . . . . . . . . . . . . . . . . . . . . . . . .170 Correct molecule 170 Listing the growth faces 170 Calculating attachment energies 170Slice positioning 171Calculating surface energies 171Deducing the morphology 171Energy setup 172

To calculate morphology with the Attachment orSurface Energy methods . . . . . . . . . . . . . . . .172

Setting up the energy calculations . . . . . . . . . . . . . . .174 Choosing the force field 174 Auto force field switch 175Non bonded energy terms 175 Lattice energy and interaction radius 175Saving the energy data 176

To set up the energy calculations . . . . . . . . . . . . .176To check the lattice energy . . . . . . . . . . . . . . . . . .177

Slice positioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . .177Finding the most stable slice 177Center slice on allmolecules 178Slice offset 178

To specify the slice positioning variables . . . . . . .178Editing, adding and removing crystal faces . . . . . . . .179

xviii Cerius2 Property Prediction/December 1998

Face list 179Selecting faces 179Editing faces 179Adding faces 179 Removing faces 180Listing faces 180

To edit, add, and remove crystal faces . . . . . . . . . 180Calculating morphology using the Hartman-Perdok

Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Generating the Crystal Graph 181Generating Connected Nets 181Calculating the Morphology 182Creating a model of a connected net 182 Saving the crystal graph and the connected nets 182

To calculate morphology with the Hartman-PerdokMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

Generating and editing the Crystal Graph. . . . . . . . . 183Defining the spatial range 183Generating crystal bonds 183Visualizing crystal bonds 183Editing crystal bonds 184

To Generate and edit the Crystal Graph . . . . . . . 184Generating and editing Periodic Bond Chains and

Connected Nets . . . . . . . . . . . . . . . . . . . . . . . . . . 185Generating connected nets 185 Generating the crystal morphology 186 List of connected nets 186Deleting less stable connected nets 186 Analyzing connected nets 187Create a model of a connected net 187Display style of a connected net 187Removing dummy atoms 187

To Generate and edit Periodic Bond Chains andConnected Nets . . . . . . . . . . . . . . . . . . . . . . . 188

Displaying the morphology . . . . . . . . . . . . . . . . . . . . 188 Visualization 188Scale factor 189Transparency 189 Face label 189

Cerius2 Property Prediction/December 1998 xix

Redisplay morphology 189To specify the display controls . . . . . . . . . . . . . . .189

Analyzing the morphology. . . . . . . . . . . . . . . . . . . . .190Interplanar angles and surface areas 190List Areas by Form option 190 Aspect ratio 190Cleave selected face 190

To analyze the crystal morphology . . . . . . . . . . .190Storing morphologies . . . . . . . . . . . . . . . . . . . . . . . . .191

Saving 191 Loading 191

To save the current morphology. . . . . . . . . . . . . .192To load a morphology . . . . . . . . . . . . . . . . . . . . .192

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .192Bravais Friedel Donnay Harker method . . . . . . . . . . .193

Growth rates - Bravais Friedel rules 193 Growth planes - Donnay Harker rules 193

The Attachment Energy method. . . . . . . . . . . . . . . . .194Calculating Eatt 194Deducing morphology 194 Assumptions 194

The Equilibrium Morphology . . . . . . . . . . . . . . . . . . .194 Calculating Esurf 195Assumptions 195

The Hartman-Perdok method. . . . . . . . . . . . . . . . . . .196 Crystal bonds and the crystal graph 196Periodic bond chains (PBCs) and connected nets 196Deducing morphology 197Assumptions 197

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .197

9. Flexisorb 199

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .199Using Flexisorb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .199Tutorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .200

Physi-sorption II - simulating large hydrocarbons . . .200General Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . .211

Preparing a model. . . . . . . . . . . . . . . . . . . . . . . . . . . .211Selecting a forcefield . . . . . . . . . . . . . . . . . . . . . . . . . .211Atom typing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .212

xx Cerius2 Property Prediction/December 1998

United-atom versus all-atom . . . . . . . . . . . . . . . . . . . 212Calculating energy maps . . . . . . . . . . . . . . . . . . . . . . 212Job control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213Calculating gas phase chemical potential . . . . . . . . . 213Using existing map files . . . . . . . . . . . . . . . . . . . . . . . 214Calculating the isosteric heat . . . . . . . . . . . . . . . . . . . 214Predicting the uptake isotherm . . . . . . . . . . . . . . . . . 215Non-Ideal gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216Output files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216Creating an isosurface . . . . . . . . . . . . . . . . . . . . . . . . 217Mapping a property onto a surface . . . . . . . . . . . . . . 217Creating a slice plane . . . . . . . . . . . . . . . . . . . . . . . . . 218Displaying a cloud point diagram . . . . . . . . . . . . . . . 219

Troubleshooting 219Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

General CB-GCMC Acceptance Rules . . . . . . . . . . . . 220Model details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

Forcefield 224Other simulation details 226

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

10. Sorption 229

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229Information obtained 229Output forms 229Sections in this chapter 229

Using Sorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230Running a sorption simulation . . . . . . . . . . . . . . . . . 231

General methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232Settings for the energy calculation . . . . . . . . . . . . . . . 232

Bump checks 232Excluded volumes 233van der Waals radius reduction 233van der Waals energy 233Minimum image convention 233Coulomb energy 234Ewald summation method 234

Setting energy calculation options for a sorptionsimulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

Forcefield 235

Cerius2 Property Prediction/December 1998 xxi

Atom types 235Bad contacts 235van der Waals energy 236Coulomb energy 236Using Ewald Grids 236

Output during the simulation. . . . . . . . . . . . . . . . . . .237Setting output options for the simulation . . . . . .238

Model window output 238Plotted output 239Trajectory file output 239Snapshot file output 239Text output 239

Setting up and running the simulation . . . . . . . . . . . .239Framework 239Surface 240Surface Setup 240Sorbate 240Move probabilities 240Maximum step sizes 241Rescaling step sizes 241

To set up and run the simulation . . . . . . . . . . . . .242Begin the simulation 244Stopping the simulation 244Restarting the simulation 244

Analysis of sorption trajectory files . . . . . . . . . . . . . .245Trajectory file plots 245Mass distribution plots 246Energy distribution plots 247Loading-curve plots 247Mass cloud plots 248

Plotting sorption trajectory files . . . . . . . . . . . . . .249Plotting trajectory files . . . . . . . . . . . . . . . . . . . . .249Plotting mass distribution of sorbates in framework .

250Plotting energy distribution of sorbates . . . . . . . .250Plotting a loading curve . . . . . . . . . . . . . . . . . . . .251Plotting mass clouds (dot density maps) of sorbate

positions. . . . . . . . . . . . . . . . . . . . . . . . . . . . .252Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .252

Simulation methods . . . . . . . . . . . . . . . . . . . . . . . . . .252

xxii Cerius2 Property Prediction/December 1998

Fixed loading (canonical ensemble) simulation. . 253Fixed pressure (grand canonical ensemble)

simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 254Henry’s constant simulation . . . . . . . . . . . . . . . . 256

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

11. MesoDyn 259Sections in this chapter 259

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260Using MesoDyn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261General methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267Building a molecular ensemble . . . . . . . . . . . . . . . . . 269Specifying the Run parameters for a MesoDyn

simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272Setting the host machine and parallel nodes, job

monitoring and output handling. . . . . . . . . . . . . 274Analyzing the results . . . . . . . . . . . . . . . . . . . . . . . . . 278

Selecting a system 278Plotting the thermodynamics functions 278Exploring the morphology 279

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285Parameterization: Mapping of the atomistic level to the

mesoscale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287Numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

12. Dissipative Particle Dynamics (DPD) 291Sections in this chapter 291

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292Using DPD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293General methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

Building a molecular ensemble . . . . . . . . . . . . . . . . . 300Specifying the Run parameters for a DPD simulation 300Setting the host machine, job monitoring and output

handling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304Analysing the result: Selecting a system, inspecting the

files, plotting the thermodynamic functions, andexploring the morphology . . . . . . . . . . . . . . . . . . 305

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

Cerius2 Property Prediction/December 1998 xxiii

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .309Equations of motion for a DPD system. . . . . . . . . . . .310Integration scheme in DPD . . . . . . . . . . . . . . . . . . . . .312Choosing the dissipation and random noise magnitudes.

313Choosing the repulsion parameters . . . . . . . . . . . . . .313Mapping the interactions onto Flory-Huggins theory 314Calculation of Flory-Huggins χ parameters as input to

DPD simulations . . . . . . . . . . . . . . . . . . . . . . . . .318References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .319

A. File Formats 321

Mechanical Properties files . . . . . . . . . . . . . . . . . . . . . . . .321Output files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .321

File names 321Morphology files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .322

CIF format. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .322Interactions.dat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .324

Polymorph files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .325Trajectory file usage and naming . . . . . . . . . . . . . . . .325

Tip 327

B. Using MSI Online Documentation 329

MSI Hypertext Locations. . . . . . . . . . . . . . . . . . . . . . . . . .329

xxiv Cerius2 Property Prediction/December 1998

Cerius2 Property Prediction/December 1998 xxv

LIST OF FIGURESFigure 1. Voronoi polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37

Figure 2. The Repeat Unit (a), andCorresponding Hydrogen-Suppressed Graph (b)86

Figure 3. The Atomic and Bond Connectivity (a), and Valence (b)Indices for the PVF Repeat Unit . . . . . . . . . . . . .87

Figure 4. Nonbond interactions in RMMC energy calculation .96

Figure 5. The Cerius2 Visualizer window showing theMESOSCALE card and the main MESODYNmenu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .268

Figure 6. The MesoDyn Build panels for defining beads, moleculesand interactions. . . . . . . . . . . . . . . . . . . . . . . . .270

Figure 7. The MesoDyn Constraints panels for defining regionsfrom which the beads are excluded. . . . . . . . . .271

Figure 8. The MesoDyn Run control panel. . . . . . . . . . . . . . . .273

Figure 9. The MesoDyn Job Control panel. . . . . . . . . . . . . . . .275

Figure 10. The MesoDyn Job Control Options subpanel, withnodes and output paths specified for running ontwo nodes of a particular IBM sp2. . . . . . . . . . .276

Figure 11. The MesoDyn Systems Analysis panel. . . . . . . . . . .278

Figure 12. The MesoDyn Thermodynamics panel for plotting thefree energy, entropy, and other time-dependentaverages. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .279

Figure 13. The MesoDyn Profiles panels for creating and analyzingslices through the density fields. . . . . . . . . . . . .280

Figure 14. The MesoDyn Isodensities control panel for creatingand analyzing isosurfaces of the density fields. 281

Figure 15. The Cerius2 Visualizer window showing theMESOSCALE card and the main DPD menu. . .299

Figure 16. The DPD Build panels for defining beads, molecules,interactions and dissipations. . . . . . . . . . . . . . .301

Figure 17. The DPD Run panel. . . . . . . . . . . . . . . . . . . . . . . . .302

Figure 18. The DPD bead and molecule Display panels. . . . . .303

Figure 19. The DPD Job Control panel. . . . . . . . . . . . . . . . . . .304

Figure 20. The DPD System Analysis panel. . . . . . . . . . . . . . .306

xxvi Cerius2 Property Prediction/December 1998

Figure 21. The DPD Plot panel for plotting the bead diffusioncoefficients, and polymer endpoint and bond lengthdistributions. . . . . . . . . . . . . . . . . . . . . . . . . . . 306

Figure 22. The DPD Profiles panel for creating and analyzing slicesthrough the density fields. . . . . . . . . . . . . . . . . 307

Figure 23. The DPD Isodensities panel for creating and analyzingisosurfaces of the density fields. . . . . . . . . . . . . 308

Cerius2 Property Prediction/December 1998 xxvii

How to Use This Book

Cerius2 Property Prediction is a complete guide to the Cerius2 mate-rials science instruments that require force field energy calcula-tions. Several modules are discussed in this book: MechanicalProperties, Polymer Properties, Blends, Synthia, Crystal Packer,Polymorph Predictor, Morphology, and Sorption.

This book describes each instrument in a separate chapter, provid-ing general information on the instruments and step-by-step pro-cedures on using them. Please see the online help for detaileddescriptions of the control panels.

You need not read this book from cover to cover before you startusing Cerius2 — simply turn to the chapter pertaining to theinstrument that you want to use.

Who should use this book

This book is intended for users of any of the Property Predictionmodules; specifically, those users:

♦ Working with polymers, polymer mixtures, and polymer solu-tions (see Chapters 1, 2, 3, 4, and 5).

♦ Working with crystal growth or the packing behavior of crys-tals (see Chapters 6, 7, and 8).

♦ Wanting to perform sorption simulations (see Chapter 9).

Things to be familiar with You will probably want to familiarize yourself with a few thingsbefore working with the Cerius2 Property Prediction modules:

♦ The windowing software on your workstation.

♦ Use of the mouse on your workstation.

♦ Basic UNIX® commands.

xxviii Cerius2 Property Prediction/December 1998


♦ The Cerius2 Modeling Environment book

♦ Chapter 1, “Open Force Field” of the Cerius2 Simulation Toolsbook

Workstation requirements Before you begin, be certain that you have these things availableon your workstation:

♦ An installed and licensed copy of Cerius2

♦ The Open Force Field (OFF) module included in your licensedcopy of Cerius2 (OFF is not needed with Crystal Packer)

♦ A home directory in which you can create subdirectories

How to find information

Cerius2 Property Prediction/December 1998 xxix

How to find information

Using other Cerius2 books

You can find additional information about Cerius2 in several otherbooks:

♦ Cerius2 Modeling Environment — Describes the integrated set oftools for session management and atomistic modeling thatform the core of the Cerius2 modeling environment.

♦ Cerius2 Builders — Discusses the specialized builder modulesthat can be added to supplement the basic model sketchingcapabilities provided by the Cerius2 modeling environment

If you want to know about… Read…

Calculating the mechanical propertiesof materials

Chapter 1, “Mechanical Properties”

Measuring physical and chain proper-ties of macromolecules

Chapter 2, “Polymer Properties”

Calculating the thermodynamics oftwo component solutions and blends

Chapter 3, “Blends”

Estimating polymer properties rapidly Chapter 4, “Synthia”Computing conformational properties

of polymer chains.Chapter 5, “RMMC (RIS Metropolis

Monte Carlo)”Estimating the sublimation energies and

packing of molecular crystalsChapter 6, “Crystal Packer”

Generating a selection of stable pack-ing structures for molecular crystals

Chapter 7, “Polymorph Predictor”

Predicting growth rates and shapes ofbulk crystals

Chapter 8, “Morphology”

Running sorption molecular simulations Chapter 9, “Sorption”Predicting mesoscale structures of soft-

condensed matterChapter 10, “MesoDyn”

Simulating complex fluids such as sur-factant solutions and copolymermelts.

Chapter 11, Dissipative ParticleDynamics (DPD)

xxx Cerius2 Property Prediction/December 1998


(that is, the Analog Builder, Crystal Builder, Surface Builder,Interface Builder, Polymer Builder, and Amorphous Buildermodules).

♦ Cerius2 Simulation Tools — Discusses the Open Force Field,Force Field Editor, Charges, Minimizer, Dynamics Simulationand Analysis, Conformers, and Field Calculation modules.

♦ Cerius2 Analytical Instruments — Discusses the diffraction-related Cerius2 materials science modules: Diffraction-Crystal,Diffraction-Amorphous, Diffraction-Faulted, Distance LeastSquared (DLS), Rietveld, IR-Raman, Powder Indexing, EXAFS,LEED/RHEED, and HRTEM.

♦ Cerius2 Command Script Guide — Shows how to capture andreplay a script of Cerius2 commands, and how to enhance yourcommand scripts with the features of the Tool Command Lan-guage (Tcl). This book is only available online (as part of theSoftware Developer’s Kit (SDK) documentation) athttp://www.msi.com/support/sdk/pub/scripting/macro_language.html. Please contact customer support if you needthis guide and do not have internet access.

♦ Cerius2 Installation and Administration Guide — Provides step-by-step instructions for installing and administering Cerius2 inyour operating environment.

Online Documentation

Some docmentation complimentary to the information in thisbook exists online at HTTP://www.msi.com/doc. Please seeAppendix B. “Using MSI Online Documentation” for more infor-mation about accessing these documents.

Typographical conventions

Unless otherwise noted in the text, this book uses the typographi-cal conventions described below:

Typographical conventions

Cerius2 Property Prediction/December 1998 xxxi

♦ Names of modules and names of things in the Cerius2 interfaceare presented in bold type. For example:

Go to the SYNTHIA card.

♦ UNIX command dialog and file samples are represented in acourier font. If the dialog is something you must type, it isgiven in bold courier font. For example:

> cerius2 -b output scriptfile

xxxii Cerius2 Property Prediction/December 1998


Cerius2 Property Prediction/December 1998 1

1 Mechanical Properties

The Mechanical Properties module is a computational instrumentfor calculating the mechanical properties of materials. Severalelastic properties can be predicted: compliance and stiffness matri-ces; Young’s, shear, and bulk moduli; volume compressibility;Poisson’s ratios; Lamé constants; and velocities of sound in speci-fied directions. You can use one of three methods to make predic-tions:

♦ Second derivative calculations.

♦ Constant stress minimization.

♦ Constant strain minimization.

After the predictions are made, you can:

♦ Analyze the data by creating several types of plots, includingstress-strain and pressure-volume curves.

♦ Fit the data to a range of curve types.

♦ Automatically derive moduli.

♦ Investigate constraints imposed by crystal symmetry.

Difficulties in processing raw materials often make experimentalmeasurement of elastic properties difficult. Mechanical Propertiesis particularly useful in allowing researchers to predict themechanical properties of these substances. The visualization pro-vided by simulation also gives valuable insight into the molecular-level mechanisms that control mechanical properties.

Introduction

Sections in this chapter General methodology

Calculating mechanical properties

2 Cerius2 Property Prediction/December 1998

1. Mechanical Properties

Specifying the applied stress/strain and mode

Displaying and saving results

Analyzing the data

Restoring data for analysis

Theory

References

General methodology

Three basic methods are available for calculating mechanical prop-erties: second derivative, constant stress minimization, and constantstrain minimization. These techniques can all be used to obtain thestiffness matrix as well as its inverse, called the compliance matrix.These two matrices are then used to derive the other properties —Young’s modulus, bulk modulus, volume compressibility, Pois-son’s ratios, Lamé constants, and velocities of sound (see “Deriv-ing mechanical properties” on page 17).

Second derivative

The second derivative method uses a single-point energy calcula-tion to obtain the second derivatives of the lattice energy with

For information about See

Editing and manipulating graphs The Cerius2 Modeling Environmentbook

Loading forcefields and setting upenergy calculations

The Cerius2 Simulation Tools book

Setting minimization variables andminimizing the energy of a struc-ture

The Cerius2 Simulation Tools

Calculating physical characteristicsof amorphous polymers (such asRg)

The “Polymer Properties” chapter inthis book

General methodology


respect to the lattice parameters and the atomic coordinates. Thefollowing energy expression is used:

+ higher order terms Eq. 1

Where:

U0 = Equilibrium energy

ε = Strain

When the structure is at an energy minimum (that is, all first deriv-atives of the lattice energy are zero), the second derivative term

can be used to calculate the components Cij of the stiff-ness matrix:

Eq. 2

The stiffness matrix computed by this method is always symmet-ric; that is, .

The second-derivative method requires no input parameters.

Constant-stress minimization

Constant-stress minimization applies an external stress to a mini-mized periodic system. The structure is then reminimized, allow-ing all the lattice parameters to vary, and the resultant strain ismeasured. This is repeated for a series of stresses. The variation ofthe measured strain as a function of external stress is used toderive the stiffness matrix. Stresses can be applied in the x, y, or zdirections, or shear stresses can be applied. Positive stresses resultin expansion; negative stresses result in compression.

Two modes of operation are provided: default and custom.

Default mode In default mode, the stiffness matrix is calculated in the simplestmanner. Each of the six independent elements of the stress tensoris varied individually while the other five elements are kept atzero. This results in six sweeps being defined. For each sweep, cal-culations are performed for up to 20 values of the variable tensor

U U0∂U∂ε-------εi

12--- ∂U

2

∂εi∂εj---------------εiεj

ij∑+

i∑+=

∂U2 ∂εi∂εj⁄

Cij∂U

2

∂εi∂εj---------------=

Cij Cji=



element. Default values for each sweep can be edited, but eachsweep must remain within elastic limits in order to produce a truestiffness matrix. The sweep parameters are specified using theoptions on the Constant Stress Minimization Prefs control panel(see “Specifying the applied stress/strain and mode” on page 9).

The slopes of the stress-strain graphs correspond to the elementsof the compliance matrix. Both the compliance matrix and itsinverse, the stiffness matrix, are used to derive the other mechani-cal properties.

Custom mode In custom mode, only one sweep is used, allowing all six elementsof the stress tensor to vary independently. You have complete con-trol over the stress profile; that is, you can specify applied stressvalues for all the elements of the stress tensor. Custom mode isspecified by checking the Use Only One Customized Sweep boxon the Constant Stress Minimization Prefs control panel (see theonline help for more information).

Constant strain minimization

This technique is the complement of the constant stress minimiza-tion described above. Small strains are applied to a periodic struc-ture at an energy minimum. The structure is reminimized keepingthe lattice parameters fixed, and the resultant stress in the mini-mized structure is measured. This is repeated for a series of strains.The variation in the measured stress as a function of applied strainis used to derive the stiffness matrix. Strains can be applied in thex, y, or z directions or shear strains can be applied.

Default and custommode

For constant stress minimization, either the default or the custommode can be used. They work as for constant stress minimizationexcept that the strain tensor is varied (instead of the stress tensor)and the sweep parameters are specified using the options on theConstant Strain Minimization Prefs control panel (see the onlinehelp for more information).

Advantages and drawbacks of the three methods

The second derivative method is generally considered the bestmethod of the three; its principal drawback is that for large sys-tems it is both slow and memory-hungry. The constant stress

General methodology


method works for larger systems than the second derivativemethod but is also slow. The constant strain method is the quickestfor large systems but is the least likely to be accurate.

Ideally, all three methods should produce the same results. Inpractice, however, differences between the methods are due toerrors in the calculation (for example, insufficient minimization, ordeformations beyond the elastic limits).

While the second derivative method always yields a symmetricmatrix, this is not necessarily true of the constant stress and con-stant strain minimization methods, where the derivation of ele-ment Cij takes a different route from the derivation of element Cji.The deviation of the stiffness matrix derived by a minimizationmethod can be used to judge the accuracy of the calculation.

Note

Calculating mechanical properties

Module design The mechanical properties of a periodic model can be calculatedby simply clicking the Calculate button on the Mech Props Runcontrol panel (see the online help). The calculation uses the param-eters currently specified.

The method and other general variables are set on this controlpanel, while method-specific variables are set on other panelsaccessed using Preferences... buttons. Analysis functions and out-put are specified on separate control panels accessed using itemson the MECHANICAL PROPERTIES card.

Novice or expert The module is designed so that the occasional user can quickly andeasily predict mechanical properties, but features are also avail-able that allow the expert to fine-tune the calculations and tomodel a custom stress or strain profile.

Methods Three property prediction methods are provided: second deriva-tive, constant-stress minimization, and constant strain minimiza-tion. Select the method you want to use from the Method popup.

The shear modulus cannot be obtained using the second deriva-tive method. To calculate the shear modulus, you must eitheruse the constant stress method or the constant strain method.



Sweep parameters The minimization methods involve applying a series of stresses orstrains to the periodic structure. The direction and value of theexternal stresses or strains and the mode used are specified on theConstant Stress Minimization Prefs and Constant Strain Minimi-zation Prefs control panels, accessed using a Preferences... button.You can use the defaults or you can set up your own custom stressor strain profile (see “Specifying the applied stress/strain andmode” on page 9).

Output The results of the mechanical properties calculations are displayedin the text window and saved in a text file. Alternatively, you cansave the results in table format. The controls that specify outputformat and level of output detail are set on the Mech Props Outputcontrol panel. Mechanical Properties also outputs files containingthe run parameters, model coordinates, and trajectories. Youshould enter a seed name for all the output files before initiating anew calculation. For details, see “Displaying and saving results”on page 12.

Minimize Model First All of the methods rely on minimizing the structure before per-forming the calculations. Thus, the Minimize Model First switchis preset to on. The structure is minimized under zero stress usingthe minimization variables currently specified. The default set-tings use the conjugate gradient algorithm (for both the atoms andunit cell) and termination criteria of 500 steps or rms force of0.1000 kcal/mol/Å. You can specify different termination valuesusing the options in the Minimizer. For information about whenyou should do this, see the note on page 8.

Accumulate Averages An Accumulate Averages switch allows you to calculate cumula-tive averages from successive calculations or reloads. Averagesfrom different calculation methods cannot be done (for example,only data from a constant stress minimization can be averagedwith data from another constant stress minimization). A genera-tion number is inserted into the names of the output files when theAccumulate Averages switch is on; files are simply overwrittenwhen the switch is off.

Typically, you might use the Accumulate Averages function if youhave several structures created by the Amorphous Builder for thesame polymer chain. Averaging the mechanical properties for allof these structures is likely to yield a more accurate set of propertydata than performing the calculation on only one structure.

General methodology


Three separate accumulators are provided. Mechanical Propertiesremembers and separately maintains averages for each methodtype. Thus, you can easily compare results from the different cal-culation methods averaged over a series of models.

Data analysis Analysis functions are provided for plotting and fitting dataobtained from constant stress and constant strain minimization.Several different types of plots can be created including stress-strain and pressure-volume curves. You can fit the plotted data toa range of curve types, automatically derive moduli, and calculateaverage Y values. The derived mechanical properties can also berecalculated using the fitted values. For details, see “Analyzing thedata” on page 14.

Crystal constraints Options are provided that allow you to investigate the effects ofconstraints imposed by crystal symmetry on the mechanical prop-erties of a material. When a material belongs to certain crystalclasses, relationships exist between elements of the stiffnessmatrix. For example, when the crystal class is cubic, the followingrelationships apply within the stiffness matrix C:

C(1,1) = C(2,2) = C(3,3)

C(4,4) = C(5,5) = C(6,6)

C(1,2) = C(1,3) = C(2,3)

All other elements are constrained to be zero. In all cases, the stiff-ness matrix should be symmetric.

The crystal class to be applied is selected from the Crystal Con-straints popup. The Redisplay Results button can then be used torecalculate the mechanical properties given these constraints. Theresults are displayed in the text window and written to an outputfile.

Restoring data The data saved from a mechanical properties calculation can bereloaded and analyzed at a later date (see “Restoring data for anal-ysis” on page 16).

To calculate the mechanical properties of a model

1. Place a periodic structure in the current model space. The struc-ture can be loaded in from file or created using the Cerius2

Builders. (If the structure’s symmetry is not primitive, convert



it to P1 using the Superlattice button on the Crystal Buildingcontrol panel of the Crystal Builder.)

2. If you don’t want to use the default settings of the Open ForceField module, you should set up the energy calculations: youmay want to load a particular forcefield and use its defaults, oryou may want to do a complete setup of the energy expression.For more information on the Open Force Field module, see theCerius2 Simulation Tools book.

3. Select Run on the MECHANICAL PROPERTIES card to bringup the Mech Props Run control panel.

4. Select the method to be used from the Method popup.

5. If using one of the minimization methods (Const Stress Min orConst Strain Min), specify the mode and values for appliedstress/strain in the Preferences control panel (see “Specifyingthe applied stress/strain and mode” on page 9).

6. Enter the file name seed for the output files in the Files Prefixentry box.

7. If the structure has already been minimized, you can uncheckthe Minimize Model First box. Otherwise, leave it checked.

Note

8. If you want to accumulate averages from successive runs, leavethe Accumulate Averages box checked; otherwise, uncheck it.Alternatively, to zero the accumulator, uncheck and recheck theAccumulate Averages box.

9. Specify the level of detail and file(s) used to save the results (see“Displaying and saving results” on page 12).

10.Click the Calculate button.

If you don’t want to use the currently specified minimizationmethod, you can change the method in the Minimizer module(see Cerius2 Simulation Tools). For example, the default rms forcetermination value of 0.1 kcal/mol/Å is generally too large formechanical properties applications and should be reduced tothe order of 10-2 to 10-3 by editing the RMS Force entry box onthe Energy Minimization control panel in the Minimizer.

General methodology


The amount of information sent to the text window and writtento file depends upon the output variables you’ve specified. (See“To specify the output variables” on page 13.)

To determine the effect of crystal constraints

1. Calculate the mechanical properties of the model given no con-straints (the default). Follow the procedure “To calculate themechanical properties of a model” described on page 7.

2. Select the crystal class to be applied from the Crystal Con-straints popup at the bottom of the Mech Props Run controlpanel.

3. Click the Redisplay Results button.

The constraints are applied to the data and the results are dis-played in the text window and written to the .txt file.

Specifying the applied stress/strain and mode

The external stress values and mode used when performing con-stant stress minimization are specified in the Constant Stress Min-imization Prefs control panel (see the online help for more controlpanel information). A similar control panel is available for specify-ing the applied strain and mode for constant-strain minimization.

Sweep defaults Default values for applied stress or strain are easily specified usingthe options under Sweep Defaults. Initial and final values areused to define the range. The total number of points must also beentered (as many as 20 points are allowed for each sweep). Theother values are automatically calculated from the initial and finalvalues so that the intervals between points are equal.

Choosing values You should choose initial and final sweep values that result in onlysmall (elastic) deformations of the initial structure. While thesevalues will depend upon the stiffness of the material, you shouldaim to keep the change in cell parameters to less than 1%. In somecases, if the material has a soft shear modulus, the maximum elas-tic deformation should be even smaller — on the order of 0.1%.

Choosing the mode Either the default or custom mode can be used to perform constantstress and constant strain minimization. These modes aredescribed under “Constant-stress minimization” on page 3. Cus-



tom mode is specified by checking the Use Only One CustomizedSweep box. Default mode is specified by leaving this boxunchecked (default).

Applied stress/straintables

The values shown in the Applied Stress Table determine the stressprofile that will be applied during the mechanical properties cal-culations. Similarly, the Applied Strain Table reflects the strainprofile. The values specified under Sweep Defaults are initiallyassigned to each sweep. In custom mode, these defaults areassigned to only two components of the stress/strain tensor (xxand yy); the values for the other four components are set to zero.

For stresses, the xx values represent the amount of force that isapplied to a surface perpendicular to x in the direction of x. Valuesare in GPa. The same applies to yy and zz, but for the y and z direc-tions, respectively. The other values are shear stresses. For exam-ple, the yz values represent the amount of force that is applied toa surface perpendicular to z in the direction of y. The same appliesto the strain tensor, except that the values represent the strain, thatis, the percentage change in length.

Standard Voigt notation The stress/strain tensor values are ordered and applied followingthe standard Voigt notation. For example, an xy stress value of 0.1corresponds to setting the values of the xy and yx elements of thestress tensor to 0.1.

Editing the defaults If you don’t want to use the default values, you can easily enternew values in the Applied Stress/Strain Table. When using defaultmode, values can only be entered for one sweep at a time. Thesweep to be edited is selected from a popup menu. Each sweep isreferred to by the component of the stress/strain tensor that isallowed to vary. When using custom mode, values for all compo-nents of the stress/strain tensor appear in the table and can beedited.

Plot stress/strain profile You can view the currently specified stress or strain profile byclicking the Plot Stress/Strain Profile button.

Choosing the mode Default or custom mode can be used to perform constant stressand constant strain minimization. Custom mode is specified bychecking the Use Only One Customized Sweep box. Defaultmode is specified by leaving this box unchecked (default). Thesemodes are described under “Constant-stress minimization” onpage 3.

General methodology


To specify variables for constant-stress minimization

1. If necessary, open the Mech Props Run control panel by select-ing Run from the MECHANICAL PROPERTIES card.

2. Make sure the Method popup is set to Const Stress Min.

3. Click the Preferences... button to bring up the Constant StressMinimization Prefs control panel.

Sweep Defaults 4. Enter a value for Number of Points per Sweep (2 to 20).

5. Enter a value for Initial Stress (GPa).

6. Enter a value for Final Stress (GPa).

Mode 7. If using custom mode, check the Use Only One CustomizedSweep box (at the bottom of the control panel); if using defaultmode, leave the box unchecked.

Stress profile 8. If not using the default values for applied stress, edit theApplied Stress Table:

Default mode:

a. Select the stress tensor element whose values are to beedited from the Stress Components Which Vary DuringSweep popup.

b. Enter the new values for this sweep in the entry boxes.

c. Repeat the above two steps for each sweep to be edited.

Custom mode: Enter new values for the stress tensor elements(any or all may be changed).

Plot profile 9. To view a plot of the stress profile:

a. Select the stress tensor element to be plotted from the StressComponents Which Vary During Sweep popup (skip thisstep if using custom mode).

b. Click the Plot Stress Profile button.

In custom mode, all the tensor elements are displayed on the plotin the graph window. In default mode, only the element selectedby the popup is plotted.



To specify the constant strain minimization variables

1. If necessary, open the Mech Props Run control panel (selectRun from the MECHANICAL PROPERTIES menu card).

2. Make sure the Method popup is set to Const Strain Min.

3. Click the Preferences... button to bring up the Constant StrainMinimization Prefs control panel.

Sweep Defaults 4. Enter a value for Number of Points per Sweep (2 to 20).

5. Enter a value for Initial Strain (%).

6. Enter a value for Final Strain (%).

Mode 7. If using custom mode, check the Use Only One CustomizedSweep box (at the bottom of the control panel); if using defaultmode, leave the box unchecked.

Strain profile 8. If not using the default values for applied strain, edit theApplied Strain Table:

Default mode:

a. Select the strain tensor element whose values are to beedited from the Strain Components Which Vary DuringSweep popup.

b. Enter the new values for this sweep in the entry boxes.

c. Repeat the above two steps for each sweep to be edited.

Custom mode: Enter new values for the strain tensor elements(any or all may be changed).

Plot profile 9. To view a plot of the strain profile:

a. Select the strain tensor element to be plotted from the StrainComponents Which Vary During Sweep popup (skip thisstep if using custom mode).

b. Click the Plot Strain Profile button.

Displaying and saving results

Each time a mechanical properties calculation is performed, theresults of the energy calculations are displayed in the text window.

General methodology


When the energy calculations are complete, the mechanical prop-erties are automatically calculated and displayed. These include:

♦ Elastic stiffness constants, Cij (all 36 components).

♦ Elastic compliance constants, Sij (all 36 components).

♦ Bulk modulus.

♦ Compressibility.

♦ Young’s modulus (x, y, and z directions).

♦ Poisson’s ratios.

♦ Velocities of sound (nine directions).

♦ Lamé constants λ and µ (for isotropic materials).

For definitions of these properties, see “Deriving mechanical prop-erties” on page 17.

Output variables

The level of detail and the format used to save the results of themechanical properties calculations can be specified. This is doneusing options on the Mech Props Output control panel (see theonline help for more information).

Text and tables files The results are written to a text file (.txt) by default. You can turnthis off. You can also specify that the results should be saved intable format (.dat file). The two check boxes under Files are pro-vided for this purpose.

Level of detail The Output Level popup is used to specify the detail level of theresults that are displayed in the text window and written to thetext file. Three settings are provided: None, Normal (default), andDetailed.

Please see the “File Formats” appendix for more detailed informa-tion on files generated by the Mechanical Properties module.

To specify the output variables

1. Select Output from the MECHANICAL PROPERTIES card toopen the Mech Props Output control panel.

2. Select the level of detail (None, Normal, or Detailed) from theOutput Level popup.



3. To save the results in a text file (.txt), check the Output Resultsto File box (default); otherwise, uncheck it.

4. To save the results in table format (.dat file), check the OutputResults to File Formatted for Tables Input box; otherwise,leave the box unchecked.

Note

Analyzing the data

The data obtained from constant stress minimization can be ana-lyzed using the options on the Constant Stress Minimization Anal-ysis control panel (see the online help). An identical control panelis provided for analyzing constant strain minimization data (seethe online help for the Constant Strain Minimization Analysis con-trol panel). Two types of analysis are provided: data plotting anddata fitting.

Data plotting Several different types of plots can be generated from the datasaved in the minimization trajectory files. These include plots of:

♦ Measured stress or strain

♦ Applied stress

♦ Stress versus strain

♦ Cell shape matrix

♦ Pressure versus volume

♦ Poisson’s ratios

♦ Various moduli (bulk, shear, and Young’s)

For many of these plots, values for a given direction can be plotted(xx, yy, zz, xy, yz, xz) on the x and/or y axes. The trajectory file tobe used is selected from a popup. The type of plot and data sets tobe used for the x and y axes are specified using the Plot Type, XAxis, and Y Axis options. The Plot From Sweep button is thenused to generate the plot.

The seed name for the output files is specified using the FilesPrefix option on the Mech Props Run control panel (see theonline help).

General methodology


The plots are displayed in the graph window. They can be savedand their appearance can be manipulated using the functions inthe Graphs module (see Cerius2 Modeling Environment).

Data fitting Four data fitting algorithms can be used to fit the plotted data: lin-ear, quadratic, cubic, and quartic. The type is selected from theCurve Type popup. Click the Fit __ Curve to Data Points buttonto initiate the data fitting. The fitted curve is added to the plot andthe correlation coefficient is calculated and displayed in the textwindow. You can exclude certain points from the fit, if appropri-ate. This is done by selecting a subset of data points from the graphwindow. Only these are used in the fit. Average Y values can alsobe calculated.

To analyze constant-stress minimization data

1. Open the Constant Stress Minimization Analysis control panel.Select Analysis from the MECHANICAL PROPERTIES card,then choose Constant Stress from the popup menu.

2. If analyzing saved data from a previous session, reload theappropriate .mp file (see “Restoring data for analysis” onpage 16).

Data plotting 3. To plot data from a sweep:

a. Select the trajectory file to be used from the Sweep Trajec-tory File popup.

b. Select the type of plot from the Plot Type popup.

c. If appropriate, select the data sets that will be used for the xand y axes from the X Axis and Y Axis popups, respectively.

d. Click the Plot From Sweep button.

e. To display additional plots, repeat Steps a through d.

Curve fitting 4. To fit the most recently plotted data:

a. Select the type of curve from the Curve Type popup (Linear,Quadratic, Cubic, or Quartic).

b. If not using all the data points, select from the graph win-dow those to be included in the fit.

c. Click the Fit __ Curve to Data Points button.



Average Y values 5. To calculate the average Y values for the last plotted data:

a. If not using all the data points, use the mouse and Shift keyto select from the graph window those points to be includedin the averaging.

b. Click the Calculate Average Y Value of Data Points button.

Updating properties 6. To update the mechanical properties using the results of the lastfitting or averaging operation, click the Update CalculatedProperties Using Last Fit button.

7. To view the updated values, click the Redisplay Results buttonon the Mech Props Run control panel. (If needed, open thispanel by selecting Run from the MECHANICAL PROPER-TIES card.)

To analyze constant-strain minimization data

1. Open the Constant Strain Minimization Analysis control panel.(Select Analysis from the MECHANICAL PROPERTIES card,then choose Constant Strain from the popup menu.)

2. Follow Steps 2 through 8 under “To analyze constant-stressminimization data” on page 15.

Restoring data for analysis

You can reload a mechanical properties calculation file (.mp) thatwas created from a previous run. Reloading restores the runparameters, recalculates the derived properties, and sets up thetrajectory file(s) for data analysis. This allows you to run the calcu-lation at one time and then reload the results for analysis later.

Reload options Reloading is done using the controls on the Mech Props Reloadcontrol panel (see the online help). The .mp file is selected using abrowser box. Information about the file including the method andtype of applied stress/strain can be viewed using the Show Infor-mation button. The Reload Sweeps button is used to reload thefile.

Theory


To reload a mechanical properties calculation file

1. Select Load from the MECHANICAL PROPERTIES menucard to open the Mech Props Reload control panel.

2. Use the browser box to select the .mp file to be reloaded.

3. To display information about the file, click the Show Informa-tion button.

4. Click the Reload Sweeps button.

Theory

Deriving mechanical properties

Any of the three methods described in this chapter (second deriv-ative, constant-stress minimization, or constant-strain minimiza-tion) can be used to calculate the stiffness matrix. Once thestiffness matrix has been obtained, several related mechanicalproperties can be derived from it.

Compliance matrix The compliance matrix, S, is calculated as the inverse of the stiff-ness matrix:

Eq. 3

Compressibility Volume compressibility is calculated from the compliance matrixas follows:

Eq. 4

Bulk modulus Bulk modulus is the inverse of the compressibility:

Eq. 5

Young’s modulus Young’s modulus is calculated from the compliance matrix. Valuesare given for three directions (x, y, and z) as follows:

S C1–

=

β S 1 1,( ) S 2 2,( ) S 3 3,( ) 2 S 3 1,( ) S 2 1,( ) S 3 2,( )+ +[ ]+ + +=

B1β---=



Eq. 6

Poisson’s ratios Poisson’s ratios are calculated from the compliance matrix for sixcomponents:

Eq. 7

Velocities of sound The velocities of sound, V, are calculated from the eigenvalues ofthe corresponding stiffness submatrix, λsc, and the density, ρ:

Eq. 8

Three submatrices can be derived from the stiffness matrix, C.Each of these submatrices has three eigenvalues, giving a total ofnine eigenvalues and, therefore, nine components for the velocityof sound.

Lamé constants Lamé constants are calculated from the stiffness matrix and applyonly to isotropic materials:

Eq. 9

Eq. 10

Ex1

S 1 1,( )-------------------= Ey

1S 2 2,( )-------------------= Ez

1S 3 3,( )-------------------=

νxyS 2 1,( )S 1 1,( )-------------------–= νxz

S 3 1,( )S 1 1,( )-------------------–=

νyxS 2 1,( )S 2 2,( )-------------------–=

νyzS 3 2,( )S 2 2,( )-------------------–=

νzxS 3 1,( )S 3 3,( )-------------------–=

νzyS 3 2,( )S 3 3,( )-------------------–=

Vλsc

ρ-------=

λ 13--- C 1 1,( ) C 2 2,( ) C 3 3,( )+ +[ ] 2

3--- C 4 4,( ) C 5 5,( ) C 6 6,( )+ +[ ]–=

µ 13--- C 4 4,( ) C 5 5,( ) C 6 6,( )+ +[ ]=

Theory


Model, forcefield, and energy setup

Models The Mechanical Properties module performs calculations on thestructure in the current model space. A primitive periodic struc-ture of any type can be used, polymeric (Fan and Hsu 1992, The-odorou and Suter 1986) or nonpolymeric. Thus, you can modelcrystalline or amorphous polymers, inorganic or molecular crys-tals, alloys, or semiconductors. The model can be derived fromeither of two sources:

♦ Cerius2 Builders — Periodic structures can be built and modi-fied using the build tools provided by the Visualizer or any ofthe more specialized builders (see the Cerius2 Builders book).

♦ Stored files — Models can be loaded from any appropriatemodel structure file. To load a model file, choose Load Model...from the File pulldown menu.

Structures of nonprimitive symmetry can be converted to primi-tive lattices using the Superstructure Generation Superlatticebutton on the Crystal Building control panel of the Crystal Buildermodule. The effect of symmetry on the mechanical properties iscalculated separately according to lattice type (see “Crystal con-straints” on page 7).

No limits are set on the unit cell size or on the number of atoms inthe unit cell. In practice, however, the amount of available memoryand/or the time required for large calculations eventually limitsthe number of atoms you can use. The second derivative methodis more memory-intensive than either of the two minimizationmethods.

Force field and energysetup

All of the property prediction methods perform energy calcula-tions on the model. The energy is calculated using the currentforcefield energy expression. If no energy expression is set up,Mechanical Properties does it automatically using the default set-tings for the Open Force Field. However, you are more likely towant to exercise control over the forcefield and energy expressionsetup. Cerius2 provides a wide range of forcefields and parametersthat allow you to choose the one best suited to your problem. Forinformation about the forcefields available and how to load them,atom typing of models, choosing energy terms, setting atom exclu-sions and restraints, and setting up the energy expression, see the



Cerius2 Simulation Tools book and the Cerius2 Forcefield-Based Simu-lations book.

References

Fan, C. F.; Hsu, S. L. Macromolecules, 25, 266 (1992).

Theodorou, D. N.; Suter, U. W. Macromolecules, 19, 139 (1986).


2 Polymer Properties

The Polymer Properties module is a tool for analyzing someimportant structural features of polymers. These analyses canreveal the characteristic nature of polymer structures, therebyplaying an important role in helping you understand the behaviorof polymer materials in various environments. The resultsobtained can be directly correlated to the applications of polymermaterials.

The structural characteristics analyzed include radius of gyrationtensor, end-to-end distance of a polymer chain, distribution ofselected dihedrals, orientation function (order parameter), dipolemoment and quadrupole tensor, and Voronoi volume distribu-tions. Useful plots can be generated from the analyses.

Sections in this chapter General Methodology

Selecting the trajectory file

Selecting the trajectory file frames

Calculating physical and chain properties

To analyze the Voronoi volume

Calculating dihedral distributions

Specifying the torsion selection rules

Obtaining orientation function results

Specifying the bond selection rules

Theory


2. Polymer Properties

General Methodology

Selecting the trajectory file

Trajectory file data generated from molecular dynamics simula-tions can be used in all the polymer property calculations andanalyses provided by this module. The results obtained are aver-aged over all the snapshots and, thus, often provide better esti-mates than those obtained from a single structure. The use oftrajectory file data also allows you to study the time evolution ofseveral polymer properties. These include density, end-to-end dis-tance, radius of gyration, Voronoi volume, dihedral angle, and ori-entation angle. Plots of these properties as a function of time arecalculated and displayed in the graph window.

The trajectory file is selected and loaded using the browser box onthe Polymer Trajectory control panel (see the online help for moreinformation on control panels).


Editing and manipulating graphs The Cerius2 Modeling Environmentbook

Calculating the compatibility ofpolymer mixtures, analyzing bind-ing energies, or calculating ther-modynamic mixing variables

The “Blends” chapter in this book

Calculating the pair distributionfunction and structure factor ofpolymers


Calculating the velocity autocorre-lation function and predictingvibrational properties, calculatingthe self-diffusion constant, or per-forming statistical analysis of prop-erties


General Methodology


Note

To select the trajectory file to be analyzed

1. Select Trajectory File from the POLYMER PROPERTIES cardto bring up the Polymer Trajectory control panel.

2. Use the browser box to select the trajectory file to be analyzed.Trajectory file names use the .trj, .qtrj, or .atrj extensions.

Selecting the trajectory file frames

You can use the entire trajectory file in the analysis or you can spec-ify that a segment of the file be used. The data to be included isdefined by specifying the first and last frame number and theframe interval. This is done using the options on the TrajectoryFrames control panel (see the online help for more information).The corresponding simulation times are also shown. The defaultsetting specifies that all frames be included in the file.

To specify the trajectory frames to be analyzed

1. Choose Frame Selection from the POLYMER PROPERTIEScard to bring up the Trajectory Frames control panel.

2. Enter the starting and ending frame numbers in the First andLast entry boxes.

3. Specify the frame interval by entering the number of frames inthe Step entry box.

4. To reset to include all frames, click the Reset to Full Trajectorybutton.

Calculating physical and chain properties

The physical and chain properties of polymers can be calculatedusing the options on the Physical & Chain Properties control panel(see the online help). Calculated values are reported in the text

The corresponding .msi or .bgf coordinates file is automaticallyloaded at the same time. However, if an msi or bgf file with thesame file name prefix cannot be found in the current directory,then the current model structure is used.



window. If you are using trajectory file data, the time evolution ofsome properties (that is, density, radius of gyration, and end-to-end distance) is also plotted in the graph window.

Note

To calculate physical or chain properties

1. If using trajectory file data:

a. Choose Trajectory File from the POLYMER PROPERTIEScard and use the browser box to select the trajectory file tobe analyzed.

b. Specify the frames that are to be included in the analysis (see“Selecting the trajectory file frames” on page 23).

Otherwise, read in or build the model that is to be used in theanalysis.

Note

2. Choose Properties from the POLYMER PROPERTIES card,then select Physical & Chain to bring up the Physical & ChainProperties control panel.

Select the physical prop-erties

3. Check the box for each physical property to be calculated:

Dipole Moment

Quadrupole Tensor

Molecular Weight

Density

4. If calculating dipole moment and using all the atoms in themodel, choose All from the popup; otherwise, choose Selectedand select the atoms to be included from the model window.

Select the chain proper-ties

5. Check the box for each chain property to be calculated:

Radius of Gyration

End-to-End Distance

These calculations apply only to single-chain polymers. Themodels used can be either periodic or nonperiodic.

Single-chain periodic or nonperiodic models must be used.

General Methodology


6. Click the Calculate button.

To analyze the Voronoi volume


a. Choose Trajectory File from the POLYMER PROPERTIESmenu card and use the browser box to select the trajectoryfile to be analyzed.


Otherwise, read in or build the model that is to be used in theanalysis. A periodic model must be used.

2. Choose Properties from the POLYMER PROPERTIES card,then select Volume to bring up the Volume Properties controlpanel.

3. To calculate the Voronoi volume:

a. Check the Voronoi Volume box.

b. If using all the atoms in the model, choose All from thepopup; otherwise, choose Selected and select the atoms tobe included.

c. Enter the cutoff distance to be used.

d. Enter the number of bins to be used in the distribution his-togram.

4. To calculate the coordination number, check the CoordinationNumber box.

5. If you want to list additional data in the text window, check theUse Long Output Format box.

6. To calculate the cell volume, check the Cell Volume box.


Please see page 36 for a theoretical discussion of Voronoi volumes.



Calculating dihedral distributions

The most important motion for polymeric materials is dihedralrotation. Combined with a conformational energy plot (1D curveor 2D contour map), the distributions of dihedrals provide veryuseful information about the conformation of the chain, its ener-getic states, mobility, and possible transition paths.

1D or 2D plots, timeevolution

Polymer Properties can be used to calculate and plot the distribu-tion of selected types of rotatable dihedrals; a 1D plot of frequencyversus dihedral angle is generated. The interdependence of twoconsecutive dihedrals can also be illustrated with a 2D plot ofdihedral one versus dihedral two (only the discrete points are plot-ted). If trajectory file data is being used, averaged values for all thesnapshots are used in the plots. The time evolution of a singledihedral in the trajectory file can also be plotted.

Where variables are set You perform the calculations using options on the Dihedral Distri-butions control panel (see the online help). The type of plot gener-ated and the dihedrals used are also specified on this control panel.

Input The calculations can be performed either on a single isolated struc-ture or periodic model, or on data from a trajectory file. The trajec-tory file and frames are specified using both the PolymerTrajectory control panel and the Trajectory Frames control panel(see the online help). The structure must be a single-chain polymerconsisting of well-defined regular repeat units; at least four repeatunits are required.

Defining the dihedrals You define the dihedrals by selecting them from a list. The type ofdihedrals that appear in the list are specified using the options onthe Torsion Selection Rules control panel (see the online help). Youcan have all rotatable dihedrals in the polymer listed, or you canlist only unique dihedrals. You can specify that only dihedralswith a particular central bond pair type, such as C—O, C—N, orC—C, be shown. You can also determine whether dihedrals withterminal hydrogen bonds are listed and, if so, whether those withone or all terminal hydrogens are included.

You can then use the Find button to find and list all the specifieddihedrals (the element label and number are given for the fouratoms). You define the dihedrals to be used in the plots by select-ing them from those listed.

General Methodology


To obtain a 1D distribution or 2D plot

Input the model or trajec-tory data




Otherwise, read in or build the model that is to be used.

Note

2. Choose Properties from the POLYMER PROPERTIES card,then select Dihedral Distributions to bring up the DihedralDistributions control panel.

Specify the output 3. Specify the type of plot by selecting from the Trajectory PlotType popup:

1-D Average to plot a dihedral distribution

2-D Average to plot the interdependence of two dihedrals

4. If doing a distribution plot, enter the bin width to be used.


Define the dihedrals 6. Specify the torsion selection rules (see “Specifying the torsionselection rules” on page 28).

7. Click the Find button to find and list all specified dihedrals.

8. Select the dihedrals to be analyzed from the CorrespondingDihedrals of the Repeat Unit list.

9. To highlight the selected dihedrals on the model, click theShow Torsions button.

10.To deselect all dihedrals, click the DeSelect All button. To rede-fine dihedrals, repeat steps 6 through 9 as needed.

Do the calculations 11. Click the Calculate button.

A single isolated structure or periodic model can be used, but itmust be a single-chain polymer consisting of regular repeatunits (at least four repeat units are required).



To plot the time evolution of a single dihedral

1. Choose Trajectory File from the POLYMER PROPERTIES cardand use the browser box to select the trajectory file to be ana-lyzed.

2. Specify the frames that are to be included in the analysis (see“Selecting the trajectory file frames” on page 23).

Note


4. Select Evolution from the Trajectory Plot Type popup.

5. Enter the repeat unit to be used in the Repeat ID Number entrybox.


7. Define the dihedral to be plotted as described in steps 6 through10 in “To obtain a 1D distribution or 2D plot” on page 27.


Specifying the torsion selection rules

The dihedrals to be used in the plots are defined by selecting themfrom the Corresponding Dihedrals of the Repeat Unit list. Therules used to find and list the dihedrals are set using options on theTorsion Selection Rules control panel (see the online help for moreinformation on control panels).

Only rotatable dihedrals are allowed. You can have all the rotat-able dihedrals listed, or you can restrict the listings to a particulartype; this often makes the selection process easier. For example,you can specify that only unique dihedrals be listed or that onlydihedrals with a particular central bond pair type, such as C—O,C—N, or C—C, be shown. You can also determine whether dihe-

The trajectory file must have been generated from a single-chain polymer model consisting of regular repeat units; at leastfour repeat units are required.

General Methodology


drals with terminal hydrogen bonds are listed and, if so, whetherthose with one or all terminal hydrogens are included.

Defaults The defaults list only unique rotatable dihedrals; all central bondpair types are included, but dihedrals with terminal hydrogens arenot allowed.

To specify the torsion selection rules

Note


2. Click the Rules... button to bring up the Torsion Selection Rulescontrol panel.

3. To list only unique dihedrals, check the Unique Torsions Onlybox; otherwise, uncheck it.

4. Specify the torsion center bond types to be listed:

a. Click the Torsion Center Bond action button to update thetorsion center bond types for the current model

b. Select the type desired from the adjacent popup (All or abond type such as C—C or C—O)

5. Specify the dihedrals with terminal hydrogens that are to beincluded by selecting one of the options from the TerminalHydrogens popup (None, One, or All).

Obtaining orientation function results

Input The calculations can be performed either on a single isolated struc-ture or periodic model, or on data from a trajectory file. The trajec-tory file and frames are specified using both the PolymerTrajectory control panel and the Trajectory Frames control panel(see the online help). The model must be a single-chain polymerconsisting of regular repeat units (at least four repeat units arerequired).

Before you can specify rules, you must first either read in orbuild the model polymer, or input the trajectory file.



Where variables are set The reference direction, structural unit vector, and plot and outputvariables used in the orientation function calculations are specifiedusing the options on the Orientation Function control panel (seethe online help).

Reference direction The reference direction is specified by selecting an axis (x, y, or z),by entering the x, y, and z components of a vector, or by selectingatoms to define a vector.

Structural unit vector The structural unit vector is defined by selecting a particular bondor set of bonds from those that appear in a list.

You first specify the type of bonds that are to be included in the list.This is done using options on the Bond Selection Rules controlpanel (see the online help). You can specify that all bonds in therepeat unit be listed, only single, double, or triple bonds be listed,or only particular bond pairs, such as C::O, C::N, or C::C, beshown. Next, you click the Find button to find and list all specifiedbonds in the repeat unit (the element label and number are given).Finally, you define the structural unit vector by selecting thedesired bonds from those listed.

Information obtained The distribution of θ, the average angle , and the orientationfunction P2 are calculated and displayed in the text window, and ahistogram showing the orientation angle distribution is plotted inthe graph window. If a trajectory file is being used, averages arecalculated and displayed for each snapshot, and the average orien-tation angle distribution for all the snapshots is plotted. Alterna-tively, the time evolution of θ for a single vector in the trajectoryfile can be plotted.

To calculate the orientation function and angle distribution

Input the model ortrajectory data




Otherwise, read in or build the model that is to be used.

θ

General Methodology


Note

2. Choose Properties from the POLYMER PROPERTIES card,then select Orientation Function to bring up the OrientationFunction control panel.

Specify the output 3. Select Average from the Trajectory Plot Type popup.

4. Enter the bin width to be used in the distribution histogram.


Define the referencedirection

6. Select the reference direction method from the Define Bypopup (Axis, Atoms, or Components):

♦ If Axis is chosen, select the coordinate axis to be used from thepopup (X-axis, Y-axis, or Z-axis).

♦ If Atoms is chosen, select the atoms that are to define the vectorfrom the model window, then click the Pick button.

♦ If Components is chosen, enter the x, y, and z components ofthe vector in the Cartesian Direction entry boxes.

Define the structural unitvector

7. Specify the bond selection rules (see “Specifying the bondselection rules” on page 32).

8. Click the Find button to find and list all specified bonds in therepeat unit.

9. Select the bonds that define the structural unit vector from theCorresponding Bonds of the Repeat Unit list.

10.To highlight the selected bonds on the model, click the ShowBonds button.

11. To deselect all bonds, click the DeSelect All button. To redefinethe structural unit vector, repeat steps 7 through 10 as needed.

Do the calculations 12.Click the Calculate button.

A single isolated structure or periodic model can be used; itmust be a single-chain polymer made up of regular repeat units(at least four repeat units are required).



To plot the time evolution of a single vector

1. Choose Trajectory File from the POLYMER PROPERTIES cardand use the browser box to select the trajectory file to be ana-lyzed.

2. Specify the frames that are to be included in the analysis (see“Selecting the trajectory file frames” on page 23).

Note


4. Select Evolution from the Trajectory Plot Type popup.

5. Enter the repeat unit to be used in the Repeat ID Number entrybox.


7. Define the reference direction and structural unit vector asdescribed in steps 6 through 11 in “To calculate the orientationfunction and angle distribution” on page 30.


Specifying the bond selection rules

The structural unit vector is defined by selecting bonds from theCorresponding Bonds of the Repeat Unit list (see above). You canhave all bonds in the repeat unit listed, or you can restrict the list-ings to bonds of a particular type; this often makes the selectionprocess easier. For example, you can specify that only single, dou-ble, or triple bonds be listed, or that only particular bond pairs,such as C::O, C::N, or C::C, be shown. The rules used to find andlist the bonds are specified using the options on the Bond SelectionRules control panel (see the online help).

The trajectory file must have been generated from a single-chain polymer model made up of regular repeat units (at leastfour repeat units are required).

Theory


Defaults The defaults specify that all bond pair types and bond orders (sin-gle, double, and triple), except bonds with terminal hydrogens, beincluded in the list.

To specify the bond selection rules

Note


2. Click the Rules... button to bring up the Bond Selection Rulescontrol panel.

To specify a particularbond pair type

3. Click the Bonding Pair button to update the bonding atom pairtypes listed for the current model, then select the type desiredfrom the adjacent popup.

To specify a particularbond order

4. Select the bond type desired from the Bond Order popup (All,Single, Double, or Triple).

To include hydrogenbonds

5. Check the Include Hydrogens box.

Theory

Physical properties

The physical properties that can be calculated include the dipolemoment, quadrupole tensor, molecular weight, and density.

Dipole moment The dipole moment formed by a pair of equivalent point chargeswith opposite sign is defined as:

Eq. 1

Where:

µ = Dipole moment

Before you can specify rules, you must first either read in orbuild the model polymer, or input the trajectory file.

µ ql=



q = Charge

l = Vector pointing from the negative charge to the positive one

The components of µ are calculated from the atomic coordinatesand atomic charges using the following equations:

Eq. 2

Where:

xi, yi, zi = Coordinates of atom i

Qi = Charge on atom i

Eq. 2 can be used as a more general definition of a dipole when asystem is not neutral.

The dipole moment can be calculated for the entire model or justfor selected atoms. Values reported to the text window are themagnitude of the dipole moment and the three Cartesian compo-nents µx, µy, and µz.

Quadrupole tensor The components of the quadrupole tensor Θ are calculated usingthe following equations and their equivalents:

Diagonal components Eq. 3

Off-diagonal components Eq. 4

The nine components of the quadrupole tensor are reported in thetext window.

Molecular weight Molecular weight is calculated from the atom masses. Currently,calculations include the entire model. However, the ability to usemultiple-chain structures and to select specific molecules will beavailable in the future. The value reported in the text window isgiven in grams.

Density Density of periodic systems is calculated from both the molecularweight and the unit cell parameters. Values listed in the text win-dow are given in g/cc. If trajectory file data is being used, the timeevolution is also plotted in the graph window.

µx xiQi∑= µy yiQi∑= µz ziQi∑=

Θxx xixiQi∑=

Θxy xiyiQi∑=

Theory


Chain properties — radius of gyration and end-to-enddistance

The radius of gyration tensor, S, characterizes the overall shapeand orientation of a polymer chain. It is defined as:

Radius of gyrationtensor (S)

Eq. 5

Where:

r0 (x0, y0, z0) = Position of the center of gravity

ri (xi, yi, zi) = Position of atom i (i = 1,N)

Here the average is over all atoms.

The center of gravity of the polymer can be calculated fromweight-averaged coordinates:

, , Eq. 6

If every atom (or segment) has the same mass, then , andsimplified equations like the following can be used to calculate theelements of the S matrix:

Eq. 7

Eq. 8

These equations are also applicable if the center of geometry isused as a reference point instead of the center of gravity.

Principal valuesand directions

The diagonalization of the S matrix gives its principal values S12,

S22, S3

2 (eigenvalues) and their orientation (eigenvectors) in theCartesian frame.

S

xi x0–( ) 2xi x0–( ) yi y0–( ) xi x0–( ) zi z0–( )

yi y0–( ) xi x0–( ) yi y0–( ) 2yi y0–( ) zi z0–( )

zi z0–( ) xi x0–( ) zi z0–( ) yi y0–( ) zi z0–( ) 2

=

x0

Mixi∑Mi∑

----------------------= y0

Miyi∑Mi∑

----------------------= z0

Mizi∑Mi∑

---------------------=

x0 xi=

xi x0–( ) 2xi

2xi

2–=

xi x0–( ) yi y0–( )– xiyi xiyi–=



Radius of gyration (s) The radius of gyration, s, is the square root of the first invariant ofS:

Eq. 9

End-to-end distance (R) The end-to-end distance, R, is simply calculated between the firstand last skeleton atoms in the polymer chain.

Values reported to the text window are the radius of gyration ten-sor, its principal values and directions, and the scalar value of theradius of gyration. If trajectory file data is used, plots also appearin the graph window, showing the time evolution for the radius ofgyration and end-to-end distance.

Voronoi volume analysis

The degree of order or disorder of a polymer can be determined byanalyzing the volume of the Voronoi polyhedron of atoms.Because this applies to polymers in the condensed state, periodicmodels must be used. The Voronoi polyhedron of an atom (or asegment) is defined as the region of space closer to that atom thanany other.

Voronoi polygon In the two-dimensional case, the Voronoi polyhedron becomes theVoronoi polygon. The construction of a Voronoi polygon is illus-trated in the figure below:

In three dimensions, the edges shown above become faces. Eachface of the polyhedron is a polygon.

Determining Voronoi vol-ume

The volume of a Voronoi polyhedron for an atom is determined asfollows:

1. All the neighboring atoms within a given cutoff distance arefound.

2. The polyhedron for the atom is defined:

a. For each neighboring atom, a plane is defined whose normalis the vector pointing from the center of the atom to thisneighbor. The plane is positioned so that the space betweenthe two atoms is divided equally.

b. Only the closest planes are used.

s I1 S12

S22

S32

+ +

= =

Theory


3. The volume of the polyhedron is calculated:

a. The volume contributed by each face is calculated:

Eq. 10

Where:

Aface = Area of the face

d = Distance from the face to the center atom

b. The volume of the polyhedron is determined by summingthe volumes of all the faces.

Information obtained The information obtained from Voronoi volume analysis includesthe coordination number (that is, the number of faces) for eachatom, the Voronoi volume for each atom, the average Voronoi vol-ume, and the Voronoi volume distribution. These results arereported in the text window, and a histogram of the Voronoi vol-ume distribution is plotted in the graph window. If trajectory filedata is used, the time evolution of the Voronoi volume is also plot-ted.

Figure 1. Voronoi polygon

Neighboring atom

Central atom

Edge of Voronoi polygo

Vface

Aface d×3

----------------------=



Voronoi volume analysis can be performed on all atoms in a peri-odic model or only on selected atoms.

The calculations are performed using options on the Volume Prop-erties control panel (see the online help). The cutoff distance andother variables are also set on this control panel.

Cutoff distance Specifying an appropriate cutoff distance is often critical. The cal-culation time for large systems can be rather long (on the order ofhours). Unnecessarily large cutoff distances significantly increasethe calculation time; this is because the number of neighboringatoms increases with distance. On the other hand, if the cutoff dis-tance is too small, some neighboring atoms may be excluded, lead-ing to incorrect results. The default value, 6.0 Å, is usuallyappropriate for systems with an even distribution of atoms.

Order versus disorder For a completely ordered structure, the shape of the Voronoi poly-hedron is well-defined. The number of types of polyhedrons isdetermined by the number of atom types in the model. For a dis-ordered structure, however, the random nature of the local envi-ronment can cause the shape of the Voronoi polyhedron to bedifferent, even for the same type of atoms. The distribution ofVoronoi volumes is therefore quite different for disordered struc-tures.

External effects External influences such as temperature, strain, and electric fieldcan change the Voronoi volume distribution. An analysis of thesechanges can reveal useful information about the structuralchanges of polymers.

Calculating the orientation function (order parameter)

The orientation of a structural unit with respect to a chosen refer-ence direction plays an important role in determining the proper-ties of polymers and liquid crystals. The structural unit can be asmall molecule, polymer segment, bond vector, or any other struc-tural element. The reference direction can be the fiber axis, thedirection of an external electric or magnetic field, the draw direc-tion in deformation, the director in a liquid crystalline polymer, orany direction defined by a vector.

A measure of alignment can be obtained by calculating the orien-tation function. Useful information can also be revealed by calcu-

Theory


lating the average orientation angle, and by analyzing orientationangle distributions.

Defining the terms

Polymer scientists describe the level of structural alignment withthe term orientation function or, specifically, the Hermans orientationfunction, and designate this quantity P2. It is defined as follows:

Hermans orientation func-tion, P2

Eq. 11

Where:

θ = Angle between the structural unit vector and the referencedirection.

Order parameter, S In liquid crystalline science, the term order parameter is often used,designated by the symbol S. The definition of the order parameteris identical to that given for the orientation function in Eq. 11.Therefore, they are completely equivalent.

Average value used,<cos2θ>

Because a model system can contain a large number of structuralunits, cos2θ in Eq. 11 has a particular distribution. As a result,Polymer Properties uses the average value, <cos2θ>, in calculatingP2:

Eq. 12

Range of values for P2 The orientation functions for three typical situations are shown inthe table below.

The values for P2 vary from 1 (perfect alignment) to -0.50 (perpen-dicular); the P2 for random orientation is 0.

<cos2θ> * P2Perfect alignment 1 0 1Random orientation 1/3 54.73 0In perpendicular plane 0 90 −0.50*(0 ≤ ≤ 90)

P212--- 3 2θ 1–cos( )=

P212--- 3 2θcos⟨ ⟩ 1–( )=

θ

θ



Random orientation In the case of random orientation, it can be shown that the distri-bution function, f(θ), is a sine function:

Eq. 13

The <cos2θ> is obtained from:

Eq. 14

Average angle The average angle is given by:

Eq. 15

The values for used in the calculations and reported in the textwindow are restricted to:

Eq. 16

f θ( ) 12--- θsin=

θcos2⟨ ⟩ θ f θ( ) dθcos2

0

π

∫ 13---= =

θ

θ 2θcos⟨ ⟩( )1–

cos=

θ

0 θ 90≤ ≤


3 Blends

The Blends module combines a modified Flory-Huggins modeland molecular simulation techniques to calculate the compatibil-ity of binary mixtures. These mixtures range from small models tolarge systems, including polymer solutions, polymer blends, andalloys. The information obtained includes phase diagrams (bin-odal and spinodal curves), thermodynamic mixing variables(enthalpy, entropy, change in free energy), the temperature depen-dent interaction parameter χ, binding energy component analysis,and the identification of favorable binding configurations betweenmolecular pairs, molecules and surfaces, additives and bulk mate-rials, and liquids and crystals.

Introduction

Sections in this chapter Using Blends

To calculate pair interaction energies

To calculate the coordination numbers

To specify the packing variables

To fit the mixing energy data

To plot the interaction parameter Chi(T)

To calculate a phase diagram

Plotting pair energy distributions

Plotting thermodynamic functions

Extracting molecular pairs

Theory

References


3. Blends

Utility and applications

The product formulations that are the most successful today tendto be complex mixtures. As a result, the properties affecting misci-bility, mixing, compatibility, and adhesion play a critical role in thedevelopment of new products. Unfortunately, the time required todevelop new formulations in the laboratory can be extensive. It isestimated that the average time to fully test a new binary polymermixture, for example, is three work weeks. A direct method forestimating the free energy of mixing between formulation compo-nents can speed the process considerably.

Blends provides a way to shorten the discovery process by esti-mating the miscibility behavior of binary mixtures. These includesolvent-solvent, polymer-solvent, and polymer-polymer mixtures.Blends predicts the thermodynamics of mixing directly from thechemical structures of the two components and, therefore, requiresonly their molecular structures and a reasonable choice of non-bond forcefield parameters as input.

The information generated by Blends includes:

♦ Phase diagrams (binodal and spinodal curves).

♦ Interaction parameter χ (T).

♦ Free energy of mixing (∆Gmix).

♦ Enthalpic and entropic contributions (∆Hmix and T∆Smix).

♦ Binding energy component analysis (van der Waals, electro-static, and H-bond).

♦ Binding energy distribution functions.

♦ Identification of favorable binding configurations betweenmolecular pairs, molecules and surfaces, additives and bulkmaterials, and liquid-crystalline mixtures.

Applications Blends is targeted to the following industrial areas:

♦ Polymer blend compatibility — For amorphous and semi-crystalline polymers.

♦ Additives — Plasticizers, modifiers, fillers, dyes, antioxidants,UV stabilizers, extrusion agents, surface modifiers.

Using Blends


♦ Solvent effects — Swelling, dissolution, diffusion, mixing, sol-vent casting.

♦ Formulation work — Effects of temperature and concentrationon formulation stability.

♦ Adhesion — Polymer-polymer, polymer-metal, and polymer-ceramic interfaces.

♦ Partition coefficients — To correlate molecular structure topartitioning in various solvents.

♦ Liquid-liquid phase equilibria — Detailed analysis can lead tothe design of better separation equipment.

♦ Separations technology — Design of stereoselective chromato-graphic columns and other separation and analysis media.

Using Blends

The procedures used to determine χ, calculate phase diagrams,and plot thermodynamic functions all start with the samesequence of steps:

Calculate functions 1. Specify the blends molecules.

2. Specify the packing variables.

3. Calculate the pair interaction energies.

4. Calculate the coordination numbers.

5. Fit the energy of mixing data with an analytical model.

This data can then be used to:

♦ Calculate χ(T).

♦ Calculate phase diagrams.

The options used to implement these functions are located on theRun Blends control panel and its subpanels (see the online help formore detailed information on control panels).

Analyze functions After performing the appropriate calculations, the following ana-lytical functions can be performed:

♦ Plot pair-energy distributions.


3. Blends

♦ Plot thermodynamic functions (that is, enthalpy, entropy, andfree energy of mixing).

♦ Plot thermodynamic isotherms.

These analytical functions are Located on the Blends Analysis con-trol panels, which are discussed in detail later in this chapter.

Extracting pairs Lastly, molecular pair configurations can be extracted from Blendsinteraction energy (.enr) files. This is done using the Extract Pairscontrol panel, also discussed later in this chapter.

Note

General methodology

Both theoretical models and molecular simulation techniqueshave been employed in an effort to predict the thermodynamicbehavior of binary mixtures. For more discussion of theoreticalmodels, please see the “Theory” section on page 54.

To calculate pair interaction energies

1. Place the two blend molecules to be used in two model spaces(these can either be loaded from a file or built using one or moreof the builders).

Note

2. Select Calculate on the BLENDS card to bring up the RunBlends control panel.

3. Enter the model number for molecules 1 and 2 in the Modelentry boxes.

Specify the packingvariables

4. See “To specify the packing variables” on page 46.

Please see the online help for more detailed information on thecontrol panels.

The blend molecules used to generate the molecular pairsshould be individually and fully minimized prior toperforming the calculations.

General methodology


5. Click the Preferences... button next to Calculate InteractionEnergies to bring up the Interaction Energies control panel.

Specify the calculationcontrols

6. Enter the number of molecular pairs to be generated.

7. To obtain a different set of molecular pairs, change the numberused as the random seed initializer.

Specify the outputcontrols

8. Enter the file prefix for the energy files to be created.

9. Select the text output level from the popup (None, Default, orVerbose).

10.To view the pairs as they are generated, check the UpdateModel box and enter the Frequency (number of pairs gener-ated between updates).

11. To plot an energy distribution histogram as the pairs are gener-ated:

a. Check the Update Graph box.

b. Enter the frequency (number of pairs between updates).

c. Enter the binwidth (kcal/mol).

To calculate a single Eij 12.Click the Calculate button for the Eij to be calculated (E11, E12,E21, or E22).

To calculate all Eijs (E11,E12, E21, and E22)

13.Return to the Run Blends control panel and click the CalculateInteraction Energies button.

See “Calculating pair interaction energies” on page 58 for theoret-ical background on this method.

To calculate the coordination numbers

1. Place the two blend molecules to be used in two model spaces(these can either be loaded from a file or built using one of thebuilders).

2. Select Calculate on the BLENDS card to bring up the RunBlends control panel.

3. Enter the model number for molecules 1 and 2 in the entryboxes under Model.


3. Blends

Specify the packingvariables

4. See “To specify the packing variables” on page 46.

5. Click the Preferences... button next to Calculate CoordinationNumbers to bring up the Z Number Calculation control panel.

Specify the calculationcontrols

6. Enter a value for the number of clusters to be generated.

7. Enter the number of trials per cluster.

8. To obtain a different set of clusters, change the number used asthe random seed initializer.

Specify the output 9. Select the text output level from the popup (None, Default, orVerbose).

10.To view the clusters as they are generated, check the UpdateModel box and enter the frequency (that is, number of clustersgenerated between updates).

11. To plot the running average of Z as the configurations are gen-erated, check the Update Graph box and enter the frequency(that is, number of clusters between updates).

To calculate a single Z 12.Click the Calculate button for the coordination number to becalculated (Z11, Z12, Z21, or Z22).

To calculate all Zs (Z11, Z12,Z21, and Z22)

Return to the Run Blends control panel and click the CalculateCoordination Numbers button.

Please see “Calculating coordination numbers” on page 61 for the-oretical background on the above method.

To specify the packing variables

1. Select the Calculate item on the BLENDS card to bring up theRun Blends control panel.

2. Click the top Packing... button to bring up the Molecule 1 Pack-ing control panel.

Specify the alignment 3. To reorient the molecules so that the long axis is along the zdirection, check the Align Along Principal Axes box; other-wise, uncheck it.

4. Select Isotropic or Axial Distribution from the popup.

5. If using axial distribution, enter the allowed angular spread(degrees).

General methodology


Specify the noncontactatoms

6. If using excluded atom constraints:

a. Check the Use Non-contact Atom List box (otherwise, leaveit unchecked).

b. Select the atoms to be excluded.

c. Click the Add Selected Atoms to List action button.

d. To clear the noncontact atoms list, click the Clear ContactList action button.

e. To show the noncontact atoms in the model window, clickthe Show Contact List action button.

7. Return to the Run Blends control panel and click the bottomPacking... button to bring up the Molecule 2 Packing controlpanel.

8. Repeat Steps 3 through 6 for molecule 2.

Please see “Specifying packing variables” on page 63 for the theo-retical background on this method.

To fit the mixing energy data

Note

1. Bring up the Run Blends control panel.

2. Click the Preferences... button next to Calculate Emix (T)Model to bring up the Fit Mixing Energy control panel.

3. To list additional data in the text window, check the Print Datain Text Window box.

4. Select the algorithm to be used from those listed in the Emix (T)Model popup.

5. If not using calculated values, enter the coordination numbersto be used in the Z11, Z12, Z21, and Z22 entry boxes.

The interaction energies for the two blend molecules mustalready have been calculated and saved in .enr files (see“Calculating pair interaction energies” on page 58). Also, if youare using calculated values for Z, these calculations mustalready have been performed (see “Calculating coordinationnumbers” on page 61).


3. Blends

6. Enter the minimum temperature, maximum temperature, andnumber of temperature points to be used in the __ K to __ KUsing __ Points entry boxes.

7. If not using the interaction energy files currently listed (theseentry boxes are automatically updated following Eij calcula-tions), enter the file names directly or select them from abrowser box by clicking the adjacent E11..., E12..., E21..., andE22... buttons.

8. Click the Fit Mixing Energy Model button. (Alternatively, theCalculate Emix (T) Model button on the Run Blends controlpanel can be used.)

Please see “Fitting the mixing energy and calculating Chi” onpage 64 for more theoretical background on this method.

To plot the interaction parameter Chi(T)

1. Calculate the interaction energies for the two blend molecules(see “Calculating pair interaction energies” on page 58).

2. Calculate the coordination numbers (see “Calculating coordi-nation numbers” on page 61), unless you are using specifiedvalues for Z.

3. Fit the mixing energy data with an analytical model (see “To fitthe mixing energy data” on page 47).

4. If desired, repeat Step 3 using different Emix (T) model algo-rithms to determine which one gives the lowest standard devi-ation.

5. Click the Plot Interaction Parameter Chi(T) button.

Please see “Fitting the mixing energy and calculating Chi” onpage 64 for more theoretical background on this method.

To calculate a phase diagram

1. Calculate the pairwise interaction energies for the two compo-nents of the binary mixture (see “Calculating pair interactionenergies” on page 58).

General methodology


2. Calculate the coordination number Zij for each of the four paircombinations (see “Calculating coordination numbers” onpage 61). This step can be skipped if using your own Z values.

3. Do an analytical fit of the energy of mixing data (see “To fit themixing energy data” on page 47).

4. If desired, repeat step 3 using different Emix (T) model algo-rithms to determine which one gives the lowest standard devi-ation.

5. Return to the Run Blends control panel and click the Prefer-ences... button next to Calculate Phase Diagram to bring up thePhase Diagram control panel.

6. Select the Emix (T) model to be used from the popup.

7. If not using the model parameters calculated from the energy ofmixing fit, enter your own parameter values in the entry boxesnext to A, B, and C.

8. Enter the degree of polymerization to be used for component 1(X1) and component 2 (X2).

9. Enter the minimum temperature, maximum temperature, andnumber of temperature points to be used in the __ K to __ KUsing __ Points entry boxes.

10.Click the Calculate Phase Diagram button. (Alternatively, clickthe Calculate Phase Diagram button on the Run Blends controlpanel.)

Please see “Calculating phase diagrams” on page 65 for more the-oretical information on this method.

Plotting pair energy distributions

The generation of different orientations using the Pairs Methodleads to configurations of varying energetics. Often, very favor-able energies are found; many actually locate a local minimum onthe potential energy surface. The distribution of the pair energiesobtained can be plotted using the options on the Blends EnergyAnalysis control panel (see the online help for more control panelinformation). The total energy of interaction can be plotted, or justthe electrostatic, van der Waals, or hydrogen bond terms. Inspec-


3. Blends

tion of these distributions can be helpful in noting the differentmodes and frequencies with which two molecules interact.

For example, the pair energy distribution for n-hexane andnitrobenzene is plotted in the figure below.

Fifty thousand pairs were generated and no energy minimizationwas done. The distribution is somewhat asymmetrical and there isa long, low-energy tail. The number of configurations with energyhigher than the peak value decreases dramatically. This phenome-non clearly indicates the merit of the sampling method used; thatis, very few configurations with high energy or bad van der Waalscontacts are generated. The average Eij obtained for this case was0.9214 kcal/mol. The error in the average Eij values decreases asthe inverse square root of the number of configurations.

The histograms can be normalized, if desired. Temperature effectscan also be taken into account by temperature weighting wherethe distribution is multiplied by the Boltzmann factor, exp (-E/RT).

General methodology


To plot a pair energy distribution

The interaction energies for the pair must already have been calcu-lated and saved in a .enr file (see “Calculating pair interactionenergies” on page 58).

1. Choose Analyze on the BLENDS card, then select Energiesfrom the popup to bring up the Blends Energy Analysis controlpanel.

2. Select the .enr file to be analyzed (11, 12, 21, or 22) from thebrowser box at the bottom of the panel, or enter the file namedirectly in the entry box.

3. Choose the energy term to be plotted (Total, VdW, Coulomb,or H-Bond) from the popup next to the Plot button.

4. To normalize the histogram, check the Normalize Histogrambox.

5. To temperature-weight the histogram, check the TemperatureWeight box and enter the temperature to be used (in degreesK).

6. Enter the binwidth (kcal/mol).

7. Click the Plot button.

A plot of the pair energy distribution appears in the graph win-dow.

Plotting thermodynamic functions

Blends analysis options can be used both to calculate thermody-namic functions (enthalpy, entropy, free energy of mixing) for abinary system, and to create plots of these functions versus com-position at a specified temperature. The plots generated reflect thecurrent choice of the interaction parameter model χ(T) and thedegree of polymerization (X1, X2) of the two components. Thesethermodynamic analysis options are provided on the Thermody-namic Analysis control panel (see the online help for more controlpanel information).

You can specify the model and its parameters directly, or Blendscan calculate these values based on molecular simulations. Thestarting point of such simulations is the molecular structures of the


3. Blends

two mixture components. Intermediate steps include the calcula-tion of interaction energies (*ij.enr files) and coordination numbers(Zij) followed by an analytic fit of the resulting energy of mixingversus temperature Emix(T) function. The degree of polymeriza-tion (X1,X2) of the two components is also part of the model. Formore details, see “General methodology” on page 44.

Isotherms Several temperatures (isotherms) for a given thermodynamicfunction can also be calculated and plotted using the options onthe Blends Isotherms control panel (see “Plotting thermodynamicisotherms” on page 52).

To calculate and plot thermodynamic functions

1. Select Analyze on the BLENDS card, then choose Thermody-namics to bring up the Thermodynamic Analysis control panel.

Specify the functions tobe computed

2. To plot enthalpy, check the Plot Enthalpy of Mixing (dH) box.

3. To plot entropy, check the Plot Entropy of Mixing (-TdS) box.

4. To plot the Gibbs free energy of mixing, check the Plot FreeEnergy of Mixing (dG) box.

5. Enter the Temperature (K) value to be used for the plots.

Specify the model 6. Click the Thermodynamic Model... button to bring up theThermodynamic Model control panel.


8. Enter values for the model parameters (A, B, and C) or usethose calculated from a Blends energy of mixing fit (see “Fittingthe mixing energy and calculating Chi” on page 64).

Specify the degree ofpolymerization


10.Return to the Thermodynamic Analysis control panel and clickthe Plot Thermodynamic Functions button.

Plotting thermodynamic isotherms

Several temperatures (isotherms) for a given thermodynamicfunction can be calculated and plotted using the options on theBlends Isotherms control panel (see the online help). The functions

General methodology


that can be plotted are free energy of mixing, enthalpy, andentropy. You can specify the number of curves plotted and therange of temperatures used. The plots generated reflect the currentchoice of the interaction parameter model χ(T) and the degree ofpolymerization (X1, X2) of the two components. You can specifythe model parameters directly, or Blends can calculate these basedon a molecular simulation. For more details, see “Fitting the mix-ing energy and calculating Chi” on page 64.

To plot more than one thermodynamic mixing function on thesame plot at a fixed temperature, use the options on the Thermo-dynamics Analysis control panel (see “Plotting thermodynamicfunctions” on page 51).

To plot thermodynamic isotherms

1. Select Analyze on the BLENDS card, then choose Isotherms tobring up the Blends Isotherms control panel.

2. Select the thermodynamic function to be plotted (Free Energy,Entropy, or Enthalpy).

3. Enter the number of isotherms to be plotted.

4. Enter the temperatures for the first and last isotherms in the __K to __K entry boxes.

Specify the model 5. Click the Thermodynamic Model... button to bring up theThermodynamic Model control panel.


7. Enter values for the model parameters (A, B, and C) or usethose calculated from a Blends energy of mixing fit (see “Fittingthe mixing energy and calculating Chi” on page 64).

Specify the degree ofpolymerization


9. Return to the Blends Isotherms control panel, and click the PlotMixing Isotherms button.

Extracting molecular pairs

Molecular pair configurations can be extracted from blends inter-action energy files and saved in .msi files so that they can be


3. Blends

viewed later. The configurations are sorted according to energy(lowest to highest) using the energy term specified (total energy,van der Waals, electrostatic, or hydrogen bond energy). Theextracted configurations can be restricted to a maximum numberhaving energies within a specified range. Appropriate values canbe determined by viewing the Eij distribution for the correspond-ing pair interaction type (see “Plotting pair energy distributions”on page 49). The options used to extract molecular pairs are foundon the Extract Pairs control panel (see the online help).

To extract molecular pair configurations

1. Choose Extract Pairs on the BLENDS card to open the ExtractPairs control panel.

2. Enter the name of the interaction energy (.enr) file to be used inthe File name entry box or select the file using the browser boxat the bottom of the control panel.

3. Select the energy term to be used from the popup (Total, vdW,Coulomb, or H-Bond).

4. Enter the lowest and highest energy values for the pairs to beextracted (in kcal/mol).

5. Enter the maximum number of pairs to be extracted.

6. Click the Extract Molecular Pairs button.

Note

Theory

Theoretical models

Flory-Huggins model Undoubtedly, the simplest and best-known theory of the thermo-dynamics of mixing and phase separation in binary systems is the

To perform the extraction, Blends requires two model filescontaining the original blends molecules. To ensure accuratereproduction of pair configurations, make sure that you haveloaded the same forcefield as the one used in the original blendsinteraction energy calculation.

Theory


Flory-Huggins lattice theory (Flory 1953). The general expressionfor calculating the free energy of mixing ∆G for a binary system is:

Eq. 1

Where:

∆G = Free energy of mixing per mole.

φ = Volume fraction.

X = Degree of polymerization (chain length).

Each repeating unit is defined as occupying a lattice site. The inter-action parameter χ is defined as:

Eq. 2

Where:

Z = Coordination number of the model lattice.

∆E12 = Differential energy of interaction of an unlike pair.

Eq. 3

Here Eij is the energy of a particular ij pair.

Other theories Certain deficiencies have been noted in the Flory-Huggins model,and more sophisticated theories, such as the reference interactionsite model (RISM) and lattice cluster theory (LCT), have recentlybeen developed to treat the problems beyond Flory-Huggins orig-inal approximations (Schweizer and Curro 1989, Freed 1985). Theapplicability of these theories is limited, however, becausedetailed information on each component required by these theo-ries is often absent. Many parameters characteristic of a binarymixture are thus obtained by fitting a theoretical model with someexperimental data. Prediction of the thermodynamic behavior fora system that is not well known remains a difficult task.

∆GRT--------

φ1

X1------lnφ1

φ2

X2------lnφ2 χφ1φ2+ +=

χZ∆E12

RT----------------=

∆E1212--- E12 E21+( ) 1

2--- E11 E22+( )–=


3. Blends

Molecular simulations

Recent advances in molecular simulation techniques haveimproved this situation. Accurate forcefields can be obtained bydefining parameters using structural and spectral data. Molecularsimulations can be performed on well-characterized systems lead-ing to a better fundamental understanding of atomic level interac-tions. This information can then be used to predict useful physicalproperties of systems that are less well characterized.

Many factors govern mixing processes, including the temperatureand the chemical nature of the individual components. Additionalfactors may be involved for polymers; for example, chain packing,the degree of crystallinity, molecular weight, and chain flexibility.Not all of these factors can be addressed using molecular simula-tion techniques. However, it is possible to obtain structural datafor the individual components in a mixture and to calculate theinteraction energy terms required for the thermodynamic expres-sions.

As with any molecular simulation technique, the results aredependent on the molecular forcefield employed. It has beenshown, however, that even a generic forcefield can provide phasediagram and miscibility information that compares favorably withexperiment (Fan et al. 1992).

Blends approach

Blends combines a modified Flory-Huggins model and molecularsimulation techniques to calculate the compatibility of binary mix-tures. Two important extensions to the Flory-Huggins model areemployed:

Extensions to Flory-Hug-gins model

♦ Blends is an off-lattice calculation, meaning that molecules arenot arranged on a regular lattice as in the original Flory-Hug-gins theory. The coordination number is explicitly calculatedfor each of the possible molecular pairs using molecular simu-lations (see “Calculating coordination numbers” on page 61).

♦ Blends incorporates an explicit temperature dependence on theinteraction parameter χ. This is accomplished by first carryingout a molecular simulation of the differential energy of binding

Theory


between similar and dissimilar pairs, then temperature averag-ing the results using the Boltzmann method (see “Calculatingpair interaction energies” on page 58).

These two extensions to the classical Flory-Huggins theory of mix-ing are documented in recent publications by Blanco (1991) andFan (1992) and coworkers.

Analytical fit The temperature dependence of χ is fitted with one of several ana-lytical models, χ(T), then used in the expression for the free energyof mixing:

Eq. 4

A popular temperature model is the Kamide expression:

Eq. 5

Here, the mixture-dependent parameters A, B, and C are fullydetermined through a Blends molecular simulation (see “Fittingthe mixing energy and calculating Chi” on page 64).

Information obtained Analytical expressions of this type allow the accurate determina-tion of first, second, and higher-order derivatives of the freeenergy of mixing with respect to volume fraction as a function oftemperature. The derivatives are used to locate the coexistingcurve (binodal) and the stability curve (spinodal) in a two-phasediagram. Critical mixing temperatures and volume fractions caneasily be read from such a phase diagram (see “Calculating phasediagrams” on page 65). Thermodynamic functions, such asenthalpy, entropy, and free energy of mixing, can also be plotted(see “Plotting thermodynamic functions” on page 51).

Force field employed As with any molecular simulation, the results obtained depend onthe accuracy of the forcefield. By default, the Universal forcefieldis used. However, you can either load a different forcefield usingthe Open Force Field (OFF) module or specify different parametersusing the Force Field Editor. For information about the OFF andthe Force Field Editor, see the Cerius2 Simulation Tools book.

∆GRT--------

φ1

X1------lnφ1

φ2

X2------lnφ2 χ T( ) φ1φ2+ +=

χ T( ) A BT CTlnT+ +RT

----------------------------------------=


3. Blends

Calculating pair interaction energies

Sampling problems The differential energy of interaction of an unlike pair (∆Eij) can beobtained in a straightforward manner simply by calculating theenergies of the four different pairs as defined by Eq. 3 of the Flory-Huggins model (see page 55). However, a question arises as towhether the ∆Eij calculated from simple pairwise interactions issufficiently accurate to represent interactions in the actual con-densed state. It is important to take into account and to properlyweigh a large number of relative orientations of the two molecules.Simple energy minimization using several selected configurationsis not representative of the interaction energy for binary mixturesexhibiting normal Boltzmann distributions.

Molecular dynamics Molecular dynamics represents an improved sampling technique.However, molecular dynamics has difficulties in sampling signifi-cantly different regions of orientation space. If the temperatureused is low, the two molecules remain trapped in a local energyminimum. If the temperature is too high, the molecules drift apartbecause sufficient kinetic energy is present for them to escape localenergy minima. The use of periodic boundary conditions reducessome of these problems at the cost of longer simulation times. Ingeneral, the use of molecular dynamics for molecular sampling ofbinding energies is rather time consuming because the equationsof motion need to be integrated with short time steps. Many timesteps are needed to generate significantly different molecular ori-entations. These problems are illustrated by the fact that simplemolecular dynamics simulations have shown that an averaged∆E12 is strongly dependent on the initial configurations chosen.

It is therefore critical to establish an efficient algorithm for sam-pling relative orientations of a pair of molecules.

Blends Monte Carlo sampling technique

Blends uses Monte Carlo atomistic simulations both to generatethousands of different molecular orientations and to calculatetheir pair-interaction energies; this is called the pairs method. Thisapproach generates energetically favorable configurations byemploying a Monte Carlo technique that includes excluded-vol-ume constraints and takes temperature effects into account.

Theory


The excluded-volume constraint method is a modified version ofBlanco’s molecular silverware algorithm, which aligns the mole-cules so that their van der Waals surfaces are barely touching. Thismethod saves computing time by ensuring that only meaningfulconfigurations are generated. It can be applied in a variety of situ-ations to sample energetics of molecules embedded in simple orcomplex topological environments (that is, molecular packing orsolvation).

The pairs method

The pairs method results in four Boltzmann-averaged Eij values.These Eij values can be used to calculate both ∆Eij using Eq. 3 (seepage 55) and Emix(T) using Eq. 7 (see page 65).

Proper structures for each of the molecules are constructed andoptimized. The overall shape of each molecule is represented by itsvan der Waals surface.

7. The centers of mass of both molecules are positioned near theorigin of the Cartesian coordinate frame; the coordinates ofmolecule 1 (white in the above illustration) are unchangedthroughout the calculations.

8. A particular orientation of molecule 2 (gray in the above illus-tration) is determined by randomly choosing three Eulerangles.

9. A vector (n) that points from the origin to the surface of a unitsphere is randomly chosen.

10.Molecule 2 is translated along the vector determined by step 4until the van der Waals surfaces of each molecule just toucheach other.

11. The pairwise interaction energy (Eij) of this specific configura-tion is calculated and stored.

12.Steps 3 through 6 are repeated a specified number of times, anda probability distribution P(Eij) is constructed.

13.Temperature effects are taken into account by weighting thedistribution by the Boltzmann factor exp (-Eij/kT). The meanvalue of Eij as a function of temperature is given by:


3. Blends

Eq. 6

14.Steps 2 through 8 are repeated for the other pair interactiontypes, and four Boltzmann-averaged Eij(T) values are obtained:

Note

Packing variables In the pair-generation method illustrated here, the orientation ofmolecule 2 is determined randomly. However, this may not beappropriate for oriented polymers or rigid-rod molecules such asliquid crystals. Connectivity between polymer segments shouldalso be considered. As a result, Blends provides options that placerestrictions on both molecule alignment and atom contacts duringpacking and, thus, allow you to obtain more representative Eij val-ues (see “Specifying packing variables” on page 63).

Output By default, the mean interaction energies for every 100 pairs gen-erated are reported in the text window along with the total bindingenergy, the total nonbond energy, and the energies due to van der

Value Molecule 1 Molecule 2

E11(T) Component 1 Component 1E12(T) Component 1 Component 2E21(T) Component 2 Component 1E22(T) Component 2 Component 2

In addition to the Boltzmann-averaging method described here,two other methods are available for calculating thermally-averaged interaction energies: Metropolis and BiasedMetropolis (see Fan et al. 1992). These methods use a MetropolisMonte Carlo sampling algorithm that introduces temperaturein a different way. They are implemented by entering thefollowing command in the text window:

> BLENDS/AVERAGING (argument)

where argument is either METROPOLIS, BIASED, orBOLTZMANN (the default).

Eij T( )⟨ ⟩dEijP Eij( ) Eijexp

Eij–

kT----------

∫dEijP Eij( ) exp

Eij–

kT----------

∫-------------------------------------------------------------------=

Theory


Waals, electrostatic, and hydrogen bond interactions. Each of thepairs generated is displayed in the model window, and an energydistribution histogram is plotted in the graph window. The resultsof the calculations are saved in interaction energy (.enr) files. Moredetailed analysis of the pair-energy distributions can be obtainedusing the energy analysis functions (see “Plotting pair energy dis-tributions” on page 49).

Where variables are set The number of pairs calculated, the interaction energy file name,and other output variables are specified on the Interaction Ener-gies control panel. By default, 10,000 pairs are calculated, but ahigher number is recommended for large molecules, moleculescapable of hydrogen bonding, and polar molecules. The packingvariables are specified on the Molecule 1 Packing and Molecule 2Packing control panels (see the online help for more control panelinformation).

Calculating coordination numbers

The coordination number Zij is the number of molecules of type jthat can be packed around a single molecule of type i. A singlecoordination number has a definite physical significance onlywhen the two components of the binary mixture have similar vol-umes or surface areas. The difficulty in defining a coordinationnumber for a system in which two components are not similar insize somewhat limits the applicability of the pairs method. For abinary system, at least four different combinations are possible;that is, the central molecule, as well as the surrounding molecule,can be either component 1 or component 2. This leads to four coor-dination numbers.

Number Central molecule Surrounding molecule

Z11 Component 1 Component 1Z12 Component 1 Component 2Z21 Component 2 Component 1Z22 Component 2 Component 2


3. Blends

The Blends module differs from the Flory-Huggins model in thatit uses an off-lattice calculation; that is, molecules are not arrangedon a regular lattice as in the original Flory-Huggins theory. Thecoordination number Z is explicitly calculated for each of the pos-sible molecular pairs using molecular simulations.

Nearest-neighborpacking

The technique involves generating clusters in which nearestneighbors are packed around the central molecule until no morewill fit.

The van der Waals surfaces are used to represent the shapes of themolecules. The central molecule is shown in white. The dark graymolecule represents the one being packed, and the light gray mol-ecules represent existing nearest neighbors. The positions of thenearest neighbors are determined in the same way as in Steps 2through 5 of the pairs method (see page 59). Each nearest neighboradded must just touch the central molecule while avoiding over-lapping the other nearest neighbors. A specified number of pack-ing trials are allowed before determining that no more will fit:

♦ a — With no existing nearest neighbor.

♦ b, c, and d — With one, two, and three existing nearest neigh-bors.

♦ e — With four existing nearest neighbors.

After e, no more nearest neighbors could be packed in the allowednumber of packing trials. The Z obtained in this particular searchwas five.

The Z obtained depends on the number of packing trials allowed.In general, after a certain point, the number of nearest neighborsincreases very slowly with an increase in the number of packingtrials. The default maximum number of trials is 100.

Packing variables In the method illustrated here, the orientation of the surroundingmolecules is determined randomly. However, this may not beappropriate for oriented polymers or rigid-rod molecules such asliquid crystals. Connectivity between polymer segments shouldalso be considered. Blends, therefore, provides options that allowyou to place restrictions on molecule alignment and atom contactsduring packing and, thus, obtain more representative values for Z(see “Specifying packing variables” on page 63).

Theory


Averaged Zijs obtained The Z values are calculated for several clusters and an average Zijis obtained for each of the four pair combinations. Individual andaverage values are reported in the text window, and the runningaverages are plotted in the graph window. The plots show howwell each Zij converged. By default, 100 clusters are generated foreach Zij calculation.

Where variables are set The number of clusters generated, the number of trials per cluster,and the output variables are specified using the options on theZ Number Calculation control panel. The packing variables arespecified on the Molecule 1 Packing and Molecule 2 Packing con-trol panels (see the online help for more control panel informa-tion).

Averaged Zs used tocalculate χ

Averaged coordination numbers are employed in the expressionfor the temperature dependence of the interaction parameter χ:

Eq. 7

The Zij values used to calculate χ are shown on the Fit MixingEnergy control panel (see the online help). These values areupdated when a coordination number calculation is performed. Ifyou do not run a coordination number calculation, default valuesare employed in the χ(T) expression. Alternatively, you can specifythe Z values to be used.

Specifying packing variables

Packing can be affected by placing restrictions on molecule align-ment and atom contacts. The packing variables are set on the Mol-ecule 1 Packing and Molecule 2 Packing control panels (see theonline help).

Isotropic versus axial packing

In the pairs method and the Z calculation method describedabove, pair or cluster generation was illustrated using isotropicpacking; that is, the orientation of the surrounding molecules withrespect to the central molecule was determined randomly; all ori-entations were possible. However, isotropic packing may not be

χ T( )Emix T( )

RT----------------------

Z12E12 Z21E21 Z11E11– Z22E22–+( )2RT

----------------------------------------------------------------------------------------------= =


3. Blends

appropriate for oriented polymers or rigid-rod molecules, such asliquid crystals.

The Blends module, therefore, provides axial packing where thesurrounding molecules are packed only around the principal axesof the central molecule. Some deviation from parallel is allowed.The allowed range of orientations (Angular Spread) is specified indegrees. A value θ different from zero leads to orientations in therange of −θ to +θ. Packing occurs randomly within this range.Axial packing can be used in conjunction with another option thataligns the molecules along their principal axes; the long axis isreoriented along the z-direction.

Excluded atom constraints

For some systems it is important to consider chain connectivityand accessibility in packing. For example, in long-chain polymers,the ends of a polymer segment are normally connected to othersegments, making some positions of the polymer segment inacces-sible to other molecules. Blends allows you to exclude certainatoms from coming into contact with other atoms during packing.Configurations that place surrounding molecules in contact withan excluded atom are rejected.

The white circles in (b) above represent two excluded atoms at thehead and tail of a polymer segment. The radii of the spheres are thevan der Waals radii of the excluded atoms. Because of chain con-nectivity, these two positions are inaccessible to other molecules.Two molecules (dark gray in the illustration above) are in contactwith these excluded atoms, causing them to be rejected.

Excluded atom constraints are particularly useful in the modelingof polymer-solvent and polymer-polymer mixtures. Notice thatthe Flory-Huggins equation, Eq. 1, is written in terms of polymersegments (lattice sites). The polymer segment connectivity renderssome regions inaccessible for binding. Typically, the head and tailatoms in a polymer segment should be designated as excludedatoms.

Fitting the mixing energy and calculating Chi

Fitting Emix (T) The interaction energy data contained in the interaction energyfiles can be plotted as a function of temperature, and fitted with a

Theory


selected curve-fitting model. The energy of mixing is defined fromthe interaction energies and coordination numbers as follows:

Eq. 8

Here the temperature dependence is introduced as a result ofBoltzmann averaging the Eij values at each of the requested tem-peratures. For example, the values at each temperature T for Emixcould be fitted using a model of the form A + BT + CT ln T, which,when divided by RT, gives the Kamide expression for the interac-tion parameter χ(T). Five curve-fitting models are provided.

The model, coordination numbers, temperatures, energy files, andother variables used to perform the fit are specified on the Fit Mix-ing Energy control panel (see the online help). The coordinationnumbers used can be entered directly, or atomistic estimates can becalculated by generating molecular clusters (see “Calculatingcoordination numbers” on page 61).

The scatter plot and curve obtained from the fit are displayed inthe graph window. Values for the fitted model parameters (A, B,and C) and the standard deviation for the curve are calculated anddisplayed in the text window. The model that gives the loweststandard deviation provides the best results.

Plotting χ(T) The interaction parameter χ can then be plotted as a function oftemperature from the Emix (T) model data, using the fitted param-eter values for A, B, and C:

Eq. 9

Calculating phase diagrams

The compatibility of binary mixtures can be illustrated by generat-ing phase diagrams. These diagrams are obtained by calculatingthe free energy of mixing (∆G) as a function of composition at dif-ferent temperatures.

Emix T( )Z12E12 T( ) Z21E21 T( ) Z11E11 T( )– Z22E22 T( )–+[ ]

2-----------------------------------------------------------------------------------------------------------------------------------------=

χ T( )Emix T( )

RT----------------------=


3. Blends

At temperatures above the critical temperature (Tcr), the ∆G (Gibbsfree energy of mixing) versus φ2 (volume fraction of the secondcomponent) has only one minimum. Two components are misciblefor any composition. For temperatures below Tcr, the two points incontact with the straight line define two binodal points, A and D(below Tcr are two minima with ∆G/φ = 0). Two inflection pointswith 2∆G/φ2 = 0, define spinodal points B and C. In the region ofφ2 < φA and φ2 > φD, the two components are miscible. In the regionbetween φB and φC, the system is unstable, separating into twophases with compositions equal to φA and φD. At the critical tem-perature, Tcr, A–D merge into a single point defined by 2∆G/φ2 =3∆G/φ3 = 0.

By connecting the binodal points at different temperatures, thecoexistence curve designated by the dashed line can be obtained.This binodal curve, also known as the phase diagram, describesthe phase change as a function of temperature and composition.

The spinodal curve, designated by the dotted line, indicates theregion of instability, and of metastability for the mixtures. Metasta-ble compositions are stable only at small fluctuations in composi-tion, while unstable compositions spontaneously phase segregateinto two solution components indicated by the binodal curve atany fixed temperature.

If no binodal curve exists (no phase diagram), this implies that thetwo components are miscible at all proportions for the currentselection of volume fraction and temperature ranges.

Calculating ∆G The Blends method for calculating ∆G is based on a modifiedFlory-Huggins model (see “General methodology” on page 44).

The phase diagrams generated reflect the current choice of theEmix (T) model, the degree of polymerization (X1, X2) of the twocomponents, and the temperature points. These variables are spec-ified on the Phase Diagram control panel (see the online help).

The Emix (T) model parameters are calculated using molecular sim-ulations. This involves calculating the interaction energies (*ij.enrfiles) and coordination numbers (Zij) for the two components fol-lowed by an analytic fit of the resulting energy of mixing versustemperature Emix(T) function. The use of analytical expressions infitting the data allows the accurate determination of first, second,and higher order derivatives of the free energy of mixing withrespect to volume fraction as a function of temperature. The deriv-

References


atives are used to locate the coexisting curve (binodal) and the sta-bility curve (spinodal) in the phase diagram. Critical mixingtemperatures and volume fractions can easily be read from such aphase diagram.

References

Bawendi, M. G.; Freed, K. F.; Mohanty, U. J.; Chem. Phys., 84, 7036(1986)

Bawendi, M. G.; Freed, K. F.; Mohanty, U. J. Chem. Phys., 87, 5534(1987).

Bawendi, M. G.; Freed, K. F. J. Chem. Phys., 88, 2741 (1988).

Blanco, M., J. Comput. Chem., 12, No 2, 237 (1991).

Fan, C. F.; Olafson, B. D.; Blanco, M.; Hsu, S. L. Macromolecules, 25,3667 (1992).

Freed, K. F. J. Phys. A,. 18, 871 (1985).

Flory, P. J. Principles of Polymer Chemistry, Cornell University Press,Ithaca, New York (1953).

Jacobson, S. et al, CDA News, November, (1991).

Nemirovsky, A. M.; Bawendi, M. G.; Freed, K. F. J. Chem. Phys., 87,7272 (1987).

Pesci, A. I.; Freed, K. F. J. Chem. Phys. 90, 2003 (1989).

Schweizer, K. S.; Curro, J. G. J. Chem. Phys., 91, 5059 (1989).


3. Blends


4 Synthia

Introduction

The Synthia module allows you to make rapid estimates of poly-mer properties using empirical and semi-empirical methods.Synthia can predict a wide range of thermodynamic, mechanical,and transport properties for bulk amorphous homopolymers andstatistical copolymers. The key advantage of Synthia is that it usesconnectivity indices as opposed to group contributions in its cor-relations; this means that no database of group contributions isrequired, and properties may be predicted for any polymer com-prised of any of the following nine elements: carbon, hydrogen,nitrogen, oxygen, silicon, sulfur, fluorine, chlorine, or bromine.This methodology is based on research conducted by Dr. JozefBicerano (1993) of The Dow Chemical Company.

Sections in this chapter Using Synthia

General methodology

Computing repeat unit length

Energy minimization of repeat units

Values of cohesive energy and solubility parameter used in cor-relations for other properties

Representation of amide groups

Performing polymer properties calculations

Theory

References


4. Synthia

Using Synthia

This section contains a simple example to get you started with theSynthia module. Steps you must perform for the tutorial to workas designed are in boxes. Explanations are printed in italics. Com-mand, popup, and pushbutton names are given in bold.

The specific example described in this section allows you to esti-mate the solubility parameter and the Young’s modulus of an eth-ylene-propylene copolymer over a range of concentrations atroom temperature. Note that the predictions of the Synthia mod-ule apply to an isotropic, atactic, amorphous polymer phase; theeffects of ordering, tacticity, and crystallinity, in particular, are nottaken into account. The results in this example thus apply to theamorphous regions of the copolymer.

This example is divided into sections:

A. Building the copolymer

B. Predicting properties

C. Studying a range of concentrations

A. Building the copolymer

1. Starting Cerius2

2. Accessing Synthia

Open a new UNIX shell and type:

> cerius2

From the Visualizer panel, go to the POLYMER Deck andchoose the SYNTHIA card.

Using Synthia


3. Load the monomer subunits and build the copolymer

You now see a list of monomers under the category polyolefin.

You have now loaded the ethylene monomer.

You have now loaded the propylene monomer. Note that the concen-trations of the two monomers are each equal to 0.5. This is because nei-ther concentration has been marked as fixed; the programautomatically divides unassigned concentrations equally among

Select the Study/Copolymer item to bring up the Copoly-mer control panel.

In the box labeled Polymer enter co_eth_prop as a name foryour new copolymer.

Click the arrow to the right of the first Monomer Name textfield. Select polyolefin from the list of monomer types.

Click once again on the same arrow.

Select PE from the list.

Now click the arrow to the right of the second MonomerName text field. This time select polyolefin and then PPfrom the list.


4. Synthia

unfixed monomers.

4. Load the copolymer into Synthia

This loads your copolymer into Synthia. Note that the two monomersare loaded in as models and that the propylene monomer is displayedin the model window.

5. Display the study table

This brings up a table displaying the name of your copolymer, thestructures of each constituent monomer, their respective concentra-tions, and the default temperature and molecular weight for the study(298 K and 100000 respectively).

B. Predicting properties

1. Open the menu of properties

This brings up the Select Properties panel. The panel should displaythe Thermophysical category by default.

Click the blue button labeled Add Copolymer to NEWStudy.

Select the Study/Show item on the SYNTHIA card.

Go to the SYNTHIA card and select the Properties/Selectitem.

Using Synthia


2. Choose the thermophysical properties

3. Choose the mechanical properties

This displays a list of mechanical properties.

4. Predict the selected properties for the copolymer

The selected properties should be displayed in the table for the copoly-mer made up of ethylene and propylene in equal mole fractions. (Othermole fractions can be entered by checking the box marked Fixed(located to the right of the monomer name) for a monomer and thentyping in a value into the box to its right.)

Scroll through the list of properties on the panel and selectSolubility Parameter (van Krevelen).

Click the yellow popup labeled Thermophysical and selectMechanical.

Select Young’s Modulus from the list.

Now go to the Study Table and click the PREDICT button.


4. Synthia

C. Studying a range of concentrations

1. Initiate a concentration study

This brings up the Concentration Range panel. The name co_eth_prop should be displayed in the text field labeled Polymer.

You will see a list of allowed range monomers for co_eth_prop.

2. Set the concentration step

3. Display the range in the study table

You should see six rows displayed in the table, one corresponding toeach set of compositions in the range just defined. Note that both theYoung’s modulus and the solubility parameter increase monotonicallywith rising ethylene fraction.

Select the Study/Concentration Range item on theSYNTHIA card.

Click the yellow Range popup currently displaying thevalue None.

Set the popup to PE.

Enter a Concentration Step of 0.2.

Check the box labeled Display Range in Table.

Using Synthia


4. Plot the solubility parameter

This graphically displays the variation of the copolymer solubilityparameter with the mole fraction of ethylene.

5. Finishing Up

Summary

In this short example, you:

♦ Used monomers from the built-in monomer library to constructa simple copolymer.

♦ Predicted some of the copolymer properties as a function of itscomposition.

♦ Displayed your results in a Study Table,

♦ Displayed on of your results in a graph.

You can also build and edit monomers using the building capabil-ities of the 3D Sketcher, (accessible via the Build/3D-Sketch itemon the top menu bar).

Click the down arrow next to Over Range and selectSolubility Parameter.

Click the Plot button.

Select the File/Exit command from the top menu bar andclick Exit in the blue dialog box that pops up.

Alternatively, if you wish to continue on with other work,select the File/New Session command and click Confirm inthe blue dialog box.


4. Synthia

For further practice with tutorials in Synthia or other polymermodules, please see the Cerius2 Tutorials—Materials Science book.

General methodology

Calculable properties Table 1 lists the properties that you can calculate with Synthia,together with references to the expressions in Bicerano (1993) thatare evaluated when performing these calculations, and an esti-mate of the standard deviation in the predicted values. In manycases, the standard deviations are values of standard deviations inthe mean that are taken directly from the correlations performedin Bicerano (1993). For some properties, correlations in Bicerano(1993) were performed against values computed from group con-tributions. In these cases, the values of standard deviation listed inTable 1 are a combination of uncertainties in the correlations per-formed in Bicerano (1993) and uncertainties in the original groupcontribution approach (the latter usually being the major contrib-utor). The standard deviations listed in Table 1 are only intendedto serve as a guide for you to assess the relative merit of the largerange of properties that may be computed by the Synthia module.

Table 1. Properties Calculable By the Synthia Module

Property Reference+StandardDeviation

Thermophysical Propertiesvan der Waals volume Eq. 3.10, 3.11; p. 54 1.8%Glass transition temperature Eq. 6.2, 6.3; p. 145, 148 6.7%Temp. of half decomposition Eq. 16.3, 16.4; p. 338, 341 7.0%Coeff. vol. thermal expansion Eq. 3.15, 3.16, 3.17; p. 73 6.7%Molar volumeat 298 K Eq. 3.13, 3.14; p. 65 2.2%at other temps. Eq. 3.15, 3.16, 3.18, 3.19;p.73 2.5%Densityat 298 K Eq. 3.13, 3.14; p. 65 2.2%at other temps. Eq. 3.15, 3.16, 3.18, 3.19;p.73 2.5%

+Bicerano (1993).

General methodology


Molar heat cap. of solidat 298 K Eq. 4.13; p. 86 5.0%at other temps. Eq. 4.10; p. 81 7.0%Molar heat cap. of liquidat 298 K Eq. 4.14; p. 92 4.7%at other temps. Eq. 4.15; p. 92 15%Change in molar heat cap at

TgEq. 4.17; p. 97 17%

Cohesive energy at 298 KFedors Eq. 5.8, 5.9, 5.10; p. 110, 111 3.9%van Krevelen Eq. 5.12, 5.13; p. 115, 116 3.0%Solubility parameter at 298 KFedors Eq. 5.4; p. 106 3.0%van Krevelen Eq. 5.4; p. 106 2.6%Surface TensionMolar parachor Eq. 7.2, 7.9, 7.10; p. 164, 168 4.0%Fedors at 298 K Eq. 7.1; p. 163 5.0%van Krevelen at 298 K Eq. 7.1; p. 163 5.0%Thermal conductivityat 298 K Eq. 14.6, 14.7; p. 315 10.2%at other temps. Eq. 14.2, 14.3; p. 311 12.7%Electrical, Optical, and Magnetic PropertiesRefractive indexat 298 K Eq. 8.5, 8.6; p. 179, 180 1.0%at other temps. Eq. 8.1, p. 176 3.0%Molar refraction at 298 K Eq. 8.3; p. 177 2.6%Dielectric constant at 298 K Eq. 9.11, 9.12; p. 202, 203 3.0%Volume resistivity at 298 K Eq. 9.7; p. 200 20%Diamagnetic susceptibility at

298 KEq. 10.3, 10.4; p. 228 5.0%

Mechanical PropertiesBulk modulus Eq. 11.13; p. 239 15%Shear modulus Eq. 11.7; p. 236 18%Young’s modulus Eq. 11.7; p. 236 18%

Table 1. Properties Calculable By the Synthia Module (Continued)


+Bicerano (1993).


4. Synthia

Scope and limitations

The correlations implemented in the Synthia module were devel-oped for isotropic (unoriented) amorphous atactic homopolymersand alternating and random copolymers constructed from the fol-lowing nine elements: carbon, nitrogen, oxygen, silicon, sulfur, flu-orine, chlorine, and bromine. They are also applicable to theamorphous phase of semi-crystalline polymers. Consequently, thefollowing predictions are beyond the scope of Synthia: effects oftacticity, ordering (crystallinity and liquid crystallinity), and orien-tation of polymers; predictions for cross-linked, ladder, and bio-logical polymers, for block copolymers, and for polymeric systems

Poisson’s ratioat 298 K Eq. 11.10; p. 239 3%at other temps. Eq. 11.12; p. 239 5%Shear yield stress Eq. 11.23; p. 251 20%Brittle fracture stress Eq. 11.22; p. 250 20%Chain Stiffness and Entanglement PropertiesSteric hindrance parameter Eq. 12.13; p. 281 4.5%Characteristic ratio Eq. 12.14, 12.15; p. 285 9.0%Molar stiffness function Eq. 12.19; p. 288 10%Entanglement molecular

weightEq. 11.18; p. 243 30%

Critical molecular weight Eq. 11.17; p. 243 30%Entanglement length 30%Transport PropertiesActivation energy for viscous

flowEq. 13.8, 13.15; p. 298, 302 3%

Permeability of oxygen Eq. 15.13; p. 325 50%Permeability of nitrogen Eq. 15.14; p. 325 50%Permeability of carbon dioxide Eq. 15.15; p. 325 50%Zero-Shear Viscosity Eq. 13.9,13.10;p. 298,299

Table 1. Properties Calculable By the Synthia Module (Continued)


+Bicerano (1993).

General methodology


containing additives and impurities that have a significant effecton the properties of the polymer. Also, Synthia does not predict thedependence of most properties on molecular weight; the estimatescorrespond to typical high molecular weight polymers (the onlycurrent exception is zero-shear viscosity as defined in chapter 13 ofBicerano (1993).

Computing repeat unit length

The repeat unit length is used in correlations for many of themechanical properties. To compute this distance appropriately,you should set all repeat units to their fully extended all-trans con-formations and energy minimize these structures before you sub-mit them to Synthia. For most polymer repeat units, you canadequately perform this minimization using the Run item on theMINIMIZER card in the OFF Methods deck.

All repeat units you submit to Synthia are assumed to be in the all-trans conformation. If you use monomers from the monomerlibrary, the head and tail atom of the unit are predefined; if you arebuilding your own monomers, you must select the head and tailatom before doing a Synthia calculation using the Edit Monomeritem on the POLYMER BUILDER card. Typically you have severalchoices of head and tail atom for the monomer that are topologi-cally equivalent to the selected one; the program identifies the pairT, T’ that is most widely separated.

The bond lengths that connect T and T’ to terminal heavy back-bone atoms H and H’ are next extrapolated to the expected bondlength between monomeric units, that is, between the atom typescorresponding to H and H’. The distances between these extrapo-lated positions of the terminal atoms and the corresponding termi-nal heavy atoms at the opposite end of the unit aremeasured. The repeat unit length is then computed as:

Eq. 1

lTH ′ and lHT ′( )l( )

llTH ′

2---------

lHT ′2

---------+=


4. Synthia

Energy minimization of repeat units

It is advised that you energy minimize all repeat units before sub-mitting them to the Synthia module. The fact that minimizationmay affect the repeat unit length determined by Synthia has beenmentioned in the previous section. In turn, this length affects sev-eral other aspects of geometry used to compute polymer proper-ties. To achieve adequate minimization of structures it is usuallysufficient to follow these steps:

1. Set repeat units to their all-trans conformation.

2. Set all parameters to their default values.

3. Select the Run item from the MINIMIZER card in the OFFMETHODS deck.

Values of cohesive energy and solubility parameterused in correlations for other properties

Two independent values are computed for the cohesive energy ofa polymer, and subsequently two corresponding values are calcu-lated for the solubility parameter. These two sets of values resultfrom correlations to cohesive energies predicted by group contri-butions developed by Fedors (1974a and b) and van Krevelen(1990). Consequently, they are termed Fedors-like and van Krev-elen-like values, and are listed this way in the output file.

Values of cohesive energy and solubility parameters are used incorrelations to predict other properties. Unless explicitly statedotherwise in the output file, any value of cohesive energy that is soused is the Fedors-like quantity, but any value of solubility parameteris the van Krevelen-like quantity.

General methodology


Representation of amide groups

It is recommended that you build amide and similar functionalgroups with the following structure:

That is, try to use only single and double bonds and no partial dou-ble bonds. Synthia uses bond type information to compute severalproperties, and this type of construction is necessary to maintainconsistency with the way in which Synthia was parameterized.

Units of permeability

The units for the permeability values that are given in the Synthiaoutput file are: cm3 mil (day 100 inches2 atm)-1, which are oftenreferred to as “Dow Units” (DU). Here, mil denotes one thou-sandth of an inch.

Developing correlations with QSAR

The Synthia module allows you to develop your own correlationsfor properties if you also have access to the QSAR module, withwhich Synthia is integrated. The Correlate item on the SYNTHIAcard automatically brings up a QSAR table. In addition to thestructural descriptors used internally by Synthia, all the propertiesavailable for prediction in Synthia at 298 K and for a molecularweight of 100000 are available for use as descriptors in QSARstudies.

For a detailed description of the significance of various structuralparameters useful in correlating polymer properties, please seeBicerano (1993).

O

C

H

N


4. Synthia

Performing polymer properties calculations

The method for performing polymer properties calculationsinvolves the following steps:

1. Build or edit monomers by selecting the Build/3D-Sketch itemfrom the top menu bar of the Visualizer panel. Alternatively,you can select monomers from the monomer library: these areaccessible directly within Synthia during homopolymer andcopolymer calculations.

2. If you build your own monomers, go to the POLYMERBUILDER card and select the Edit Monomers item to bring upthe Monomer Editor control panel. Use the icons on this panelto define the head and tail atom of the monomer.

Note

3. To perform a homopolymer study, go to the SYNTHIA cardand select the Study/Homopolymer item. Click the arrow nextto the Monomer label on the Homopolymer panel and selectthe monomer you are interested in from the displayed mono-mer list.

To use a model you have built, select Models from the mono-mer list and pick your monomer. Type a name for yourhomopolymer in the box labeled Polymer and enter a temper-ature for the study.

4. Add the homopolymer to the study by pushing the AddHomopolymer to Study button.

5. For a copolymer study, select the Study/Copolymer item to dis-play the Copolymer control panel. Choose up to five constitu-ent monomers from the monomer choosers under MonomerName.

To access a monomer you have built, select Models from the listand then the pick the specific monomer of interest. To assign afixed concentration to a monomer, check the Fixed box associ-ated with it and enter the concentration in the text field to itsright. Note that concentrations are in either mole or weight frac-

Step 2 is not necessary for structures in the monomer library,since they have predefined head and tail atoms.

General methodology


tions; unassigned fractions are automatically divided equallyamong unfixed monomers to bring the total to one.

When the copolymer is fully defined, click the Add button toadd it to either a new or current study.

6. To choose which properties to predict for your polymer, go tothe SYNTHIA card and select the Properties/Select item toopen the Select Properties panel. Choose the category of prop-erties you are interested in from the yellow popup above theproperty list. Then click on properties within the list displayedto add them to the set of properties to compute in your study.

Note that you can make multiple selections by dragging themouse or by combining mouse clicks with the control and shiftkeys.

7. Display the study table by selecting the Study/Show item.

8. To display the selected properties for the homopolymer orcopolymer of interest, click the PREDICT button on the StudyTable.

9. For homopolymers you can go directly to the Predict panel byselecting the Predict Properties item on the SYNTHIA card.Pick a monomer and select properties for prediction.

You can display and select different property categories usingthe category chooser located above the list of shown properties.You can perform calculations on models chosen from the Mod-els window on the Visualizer panel. To do this:

a. Check the Selected Models box on the Predict panel

b. Click the Predict Properties button to display the results ofthe calculation in the current table.

10.To calculate properties over a temperature range, go to the topmenu bar in the Study Table and select the Ranges/Tempera-ture Range item (or you can select the Study/TemperatureRange item from the SYNTHIA card). This opens the Temper-ature control panel.

a. Select the homopolymer or copolymer to use as the subjectof the study from the list you see when you click thePolymer arrow button.

b. Enter a Start Temperature.


4. Synthia

c. Enter a Final Temperature.

d. Enter a Temperature Step.

e. Check the Display Range in Table box to display the resultsof the temperature range calculation in the Study table. (Oryou can click the Plot button to display the predictions ofselected properties graphically.)

11. To calculate the properties of a copolymer over a concentrationrange, go to the top menu bar in the Study Table and select theRanges/Concentration Range item (or select the Study/Concentration Range item on the SYNTHIA card). This bringsup the Concentration Range panel.

a. Select a copolymer for the study from the Polymer list

b. Select a monomer as the Range variable from the list ofallowed range monomers displayed by clicking the yellowpopup.

c. Pick initial and final concentrations and a concentration stepfor the range monomer.

d. As in the case of temperature ranges, either use the DisplayRange in Table check box to display the results of the calcu-lation in the Study table or click the Plot button to view theresults graphically.

12.To develop your own correlations (if you have access to the QSARmodule), select the Correlate/QSAR Study Table item from theSYNTHIA card. This brings up a QSAR Study Table. Using theitems on the Molecules pulldown in the QSAR Study Table’smenu bar, add any molecules of interest; use the Descriptors/Databases item to identify the descriptors to employ in theQSAR study.

Double-clicking on a column of descriptors already added tothe QSAR Study Table brings up help information on the nameand significance of the descriptor.

Theory


Theory

Background

Group additive methods have been used successfully for manyyears to predict the properties of polymers as well as small mole-cules. These methods are extremely fast and easy to use. Conse-quently, they are of greatest utility when a rapid estimate of aproperty is required without a detailed understanding of the ato-mistic interactions that give rise to it. However, the principal short-coming of these methods is their reliance upon a database of groupcontributions, the contents of which are obtained by fitting toexperimentally observed properties of interest. Thus, if a polymercontains a group for which the group contribution cannot be esti-mated, then the property of that polymer cannot be calculated.

To circumvent this limitation, the method implemented in theSynthia module uses topological information about polymers(instead of group contributions) in the predictive correlations. Spe-cifically, connectivity indices derived from graph theory areemployed. Thus, no database of group contributions is required,and properties may be predicted for any polymer comprised ofany of the nine elements: carbon, hydrogen, nitrogen, oxygen, sil-icon, sulfur, fluorine, chlorine, and bromine.

No attempt is made here to discuss or explain the details of all ofthe correlations that are used within the Synthia module. For thisinformation, refer to Jozef Bicerano’s research monograph (Bicer-ano 1993). The description of the theory underlying this module isrestricted in this document to a brief outline summarized fromBicerano’s work. A complete description is given, however, of allassumptions and modifications made to the original approach(Bicerano 1993) in the course of implementing this computer pro-gram.


4. Synthia

Connectivity indices of polymer repeat units

In this section a brief description is given of how connectivity indi-ces are determined for polymer repeat units. For more detailedinformation see Bicerano (1993), and Kier and Hall (1976, 1986).

Graphical theoretical treatment of molecular properties starts byconstruction of the hydrogen-suppressed graph of the molecule.To represent a repeat unit, special considerations are required totake into account chain continuation in a consistent manner, andnot introduce truncation errors. The procedure is illustrated hereby considering the repeat unit of poly(vinyl fluoride) (PVF). Thisrepeat unit is shown in Figure 2, together with the hydrogen-sup-pressed graph that will be utilized.

The next step is to define two atomic indices ( and ), thatdescribe the bonding and electronic environment of each non-hydrogen atom. The first is the simple connectivity index, ,which equals the number of non-hydrogen atoms to which anatom is bonded. The latter atomic index, , contains electronicconfiguration information for the atom and is given by:

Eq. 2

Figure 2. The Repeat Unit (a), andCorresponding Hydrogen-Suppressed Graph (b)

C — C

H F

H H

(a) (b)

δ δv

δ

δv

δvZv NH–

Z Zv– 1–------------------------=

Theory


where is the number of valence electrons of the atom, is thenumber of hydrogens bonded to it, and is its atomic number.

Bond indices and can also be defined in terms of the atomicindices. These are defined by:

Eq. 3

and:

Eq. 4

The values of the atomic and bond indices that are obtained for thePVF repeat unit are shown in Figure 3. In this figure the connectiv-ity indices and are shown in Figure 3(a), and the valence indi-ces and are shown in Figure 3(b).

Connectivity indices of polymer chains

The connectivity indices for the polymer chain can now bedefined. It is these chain connectivity indices that are later used inthe correlations with polymer properties.

The zeroth-order (atomic) connectivity indices and for thepolymer molecule are defined in terms of the summations:

Figure 3. The Atomic and Bond Connectivity (a), and Valence (b)Indices for the PVF Repeat Unit

Zv NHZ

β βv

βij δi δj•=

βvij δv

i δvj•=

δ βδv βv

(a)

26

63

1

33

(b)

26

621

7

33

χ0 χv0


4. Synthia

Eq. 5

and:

Eq. 6

over the vertices of the hydrogen-suppressed graph.

The first-order (bond) connectivity indices and for thepolymer molecule are defined in terms of the summations:

Eq. 7

and:

Eq. 8

over the edges of the hydrogen-suppressed graph.

Thus, for PVF we have:

Eq. 9

Eq. 10

Eq. 11

Eq. 12

Note that the and values outside of the square brackets inFigure 3 are not included in the summations. They are included inthe figure to take the chain connectivity into account, and to allowthe correct assignment of the and values for all of the atomsand bonds within the brackets.

χ0 1

δ-------∑=

χ0 v 1

δv---------∑=

χ1 χ1 v

χ1 1

β-------∑=

χv1 1

βv----------∑=

χ0 12--- 1

3--- 1

1---+ + 2.2845= =

χ0 v 12--- 1

3--- 1

7---+ + 1.6624= =

χ1 16--- 1

6--- 1

3---+ + 1.3938= =

χv1 16--- 1

6--- 1

21------+ + 1.0347= =

δ δv

δv βv

Theory


General forms of the correlations in terms of connectivityindices

Extensive properties depend upon the size of the system, and theirvalues increase in direct proportion to the amount of materialpresent. Examples of such properties include: molar volume,molar heat capacity, and cohesive energy. Such properties are cor-related directly with the values of .

Intensive properties are essentially independent of the amount ofmaterial present, providing that it is nonzero. These properties arecorrelated with values of scaled by the number of non-hydrogenatoms in the system N. These scaled values are denoted by theGreek letter , with all the usual superscripts and subscripts thatare used with the corresponding .

Thus, two general forms of correlation are used:

Eq. 13

Eq. 14

Here, a and b are sets of linear regression coefficients. There is noadditive constant term in the correlation for an extensive propertybecause the value of constant does not change as a function of theamount of material present. The structural parameters are combi-nations of connectivity indices and geometrical parameters thatare used in correlations for certain properties. The atomic andgroup correction terms are terms that are dependent upon thenumber of certain types of atoms or groups that may be present.

Backbone and side group connectivity indices

In correlations to predict certain properties, it is necessary to sep-arate the contributions of atoms in the backbone and those in side

χ

χ

ξχ

Extensive Property( ) aχ∑ extensive structural parameters, and(+=atomic or group correction terms)

Intensive Property( ) bξ∑ intensive structural parameters, and(+=atomic or group correction terms)

constant+


4. Synthia

chains into two sets of χs. This is not discussed further here—foradditional information please see Bicerano (1993).

Actual correlations for various properties and validationagainst experimental data

All the correlations used in the Synthia module are fully docu-mented in Bicerano (1993), in addition to their validation againstextensive experimental data.

References

Bicerano, J. Prediction of Polymer Properties, Marcel Dekker, Inc.:New York (1993).

Elias, H.G.; Vohwinkel, F. New Commercial Polymers 2, Gordon andBreach Science Publishers: New York (1986).

Fedors, R.F. Polymer Engineering and Science, 14, 147-154 (1974a).

Fedors, R.F. Polymer Engineering and Science, 14, 472 (1974b)

Kier, L.B.; Hall, L.H. Molecular Connectivity in Chemistry and DrugResearch, Academic Press: New York (1976).

Kier, L.B.; Hall, L.H. Molecular Connectivity in Structure-ActivityAnalysis, John Wiley & Sons: New York (1986).

van Krevelen, D.W.; Hoftyzer, P.J. Properties of Polymers, Their Esti-mation and Correlation with Chemical Structure, 2nd Edition,Elsevier: New York (1990).


5 RMMC (RIS MetropolisMonte Carlo)

The RMMC module is used to compute the conformational prop-erties of a polymer chain. RMMC performs an RIS MetropolisMonte Carlo (RMMC) simulation of your polymer, calculates theproperties that you specify, and can then save the conformationsfor later viewing or analysis. You can also choose to modify vari-ous parameters controlling the simulation to enhance the qualityof the results.

Note

Using RMMC

This section contains a simple example to get you started with theRMMC module. Steps you must perform for the tutorial to workas designed are in boxes. Explanations are printed in italics. Carddeck, card, item, and control panel widget names are given inbold.

The specific example described in this section allows you to esti-mate the chain statistics of a linear alkane melt at room tempera-ture.

Additional documentation for RMMC is available at: HTTP://www.msi.com/doc. Please see Appendix B, “Using MSI OnlineDocumentation” for more information about accessing thesedocuments.


5. RMMC (RIS Metropolis Monte Carlo)

1. Starting Cerius2

2. Accessing POLYMER BUILDER

3. Build a decane molecule

You now see a decane molecule in the model window.

4. Setting the run mode to INTERACTIVE

This displays the RMMC Job Control panel.

Open a new UNIX shell and type:

> cerius2

Within the Visualizer, go to the POLYMER1 card deck andchoose the POLYMER BUILDER card.

Select the Homopolymer item on the POLYMER BUILDERcard. When the Homopolymer Builder panel appears, clickthe BUILD button.

Go to the RMMC card and select the Job Control item.

Click the yellow popup labeled Run Mode and change thesetting from BACKGROUND to INTERACTIVE.

Using RMMC


5. Accessing the RMMC Run panel

This displays the RMMC Run panel. The Equilibration andSimulation check boxes are selected, but the Preparation andMinimization check boxes are not.

You now see messages in the text window indicating that the simula-tion has started. The logfile for the calculation, test_decane.rislog, isautomatically brought up in a separate window and is updated as thesimulation proceeds. The calculation should taken only a few minutesto complete. On completion, the displayed logfile contains estimatesfor several properties calculated in the RMMC simulation, includingthe average end-to-end distance of the decane molecule.

6. Display the results of the calculation

This brings up a file browser containing the names of files available for

Now select the Run item from the RMMC card.

Check the Minimization check box.

In the text entry box labeled File Prefix, type the text: test_decane.

Set the number of Equilibration Steps to 1000 and the num-ber of Simulation Steps to 5000.

Click the RUN button to launch the RMMC calculation.

Go back to the RMMC card and select the Analyze item.



analysis from the RMMC simulation run.

Pushing the buttons displays the distribution of dihedral angles aver-aged over the simulation run. Click on the yellow popups labeled Dis-tribution and Trajectory to select other simulation predictions, andclick the appropriate Plot radio button to display the results for theselected property.

Note that this example is an unusually short simulation run. Toobtain reliable average chain properties for a typical polymer system,you may need to conduct runs of a million steps or more to sample alarge-enough subset of the configuration space of the polymer.

General Methodology

RIS Metropolis Monte Carlo (RMMC) Concepts

Rotatable Bonds

In an RMMC simulation, you can potentially change the torsionangle of every rotatable bond. By default, all single and partialdouble bonds in the model are treated as rotatable. If the check boxlabeled Mark All Bonds Rotatable on the RMMC Monte CarloPreferences panel is deselected (access this panel by selecting thePreferences->Monte Carlo item), the RMMC program uses back-bone atom flags to determine which bonds are rotatable for simu-lation purposes. A bond is considered rotatable in this context if itsatisfies all of the following criteria:

♦ Both atoms of the bond are backbone atoms.

♦ Each of the two atoms connected by the bond is attached to atleast one other atom. (Thus, for example, a C–Cl bond is notrotatable, because the Cl is bonded to no atom other than thecarbon atom.)

Select the file test_decane_dih.tbl. Click the radio buttonlabeled Plot DIHEDRAL Distribution.

General Methodology


♦ The bond is a single or partial double bond.

♦ The bond is not in a ring.

By using the commands on the Polymer Editor panel (accessed bygoing to the POLYMER BUILDER card and selecting theEdit-->Polymers item), you can control which atoms are consid-ered backbone atoms, and thus which bonds are considered rotat-able. This allows you to treat branched chains, dendrimers, andpolymers with flexible side groups using RMMC. In all cases, anunbroken backbone path through the molecule must exist forRMMC to work properly.

Energy Calculation

Unlike traditional RIS calculations, RMMC does not use statisticalweights. Instead, it uses the energy as computed from a forcefieldin order to calculate chain conformational properties. The onlyenergy terms considered in a RMMC calculation are:

♦ Torsion terms

♦ Van der Waals terms

♦ Coulombic terms

There are two parameters—Minimum Bonds and MaximumBonds—that determine the interacting pairs of atoms for the pur-pose of calculating nonbond (van der Waals and Coulomb) ener-gies. (Access these parameters on the RMMC Interactions panel,by going to the RMMC card deck and selecting thePreferences->Interactions item.) Nonbond energies are not com-puted for atoms closer than Minimum BONDS bonds away fromeach other. (See Figure 4.) The usual value for Minimum BONDSis 3. Nonbond energies are also neglected for atoms further thanMaximum BONDS bonds away from each other. Reasonable val-ues for Maximum BONDS range from 4 to about 6 for polymerchains in theta conditions. (Larger values of Maximum BONDSmay greatly increase CPU and memory requirements, althoughsome excluded volume effects might, in principle, be treated inthis way.



Parameters Controlling the Simulation

A number of parameters can affect the outcome of a RMMC simu-lation. These include the following:

♦ Minimum bonds, maximum bonds

♦ Temperature

♦ Forcefield (CVFF, CFF91, PCFF or COMPASS)

♦ Charges (whether on or off), dielectric constant

♦ Rigidity or flexibility of articulated side groups

♦ Energy scaling factor

Minimum BONDS and Maximum BONDS have already beenmentioned. The temperature enters the Boltzmann factor that isused to determine whether to accept or reject a conformation. Theforcefield provides the parameters for the torsional and nonbondenergy calculation. The inclusion or exclusion of atomic partialcharges and the dielectric constant determine the calculated Cou-lomb energy.

Whether articulated side groups are treated as flexible in a RMMCsimulation is determined by whether the side group atoms havetheir backbone flags set. (The commands on the Monomer Editorand Polymer Editor panels can be used to set these flags. (Access

Figure 4. Nonbond interactions in RMMC energy calculationBlack atoms interact with the central atom (indicated by thearrow); white atoms do not. In this example, Minimum BONDS is 3and Maximum BONDS is 5.

General Methodology


these panels by going to the POLYMER BUILDER card and select-ing the Edit—>Polymers and Edit—> Monomers items) It is bestto set these flags on the repeat units before building the polymer.)In principle, it is more realistic to treat side groups as flexiblerather than rigid. However, doing so increases the computationtime and might not be necessary if the groups are small (as, forexample, in polypropylene).

The “energy scaling factor” is a number that multiplies all com-puted energies. If experimental data are available, this factor canbe used to refine the properties calculated by RMMC so that theymatch experimental properties as closely as possible. The samefactor can then be used for calculations on related chains for whichno experimental data exist.

The reason for the relatively large number of parameters control-ling the energy calculation is as follows. A RMMC simulation isperformed on an isolated polymer chain. Yet, the propertiesdesired are those for a chain in solution or in the melt. If solventmolecules or other chain molecules were present in the simulation,then a high quality forcefield such as COMPASS should be capableof predicting the correct properties. Because these other moleculesare not explicitly present, their effects must be mimicked by alter-ing the way the energy is computed. (In the language of statisticalmechanics, it is not “bare” interaction energies but potentials ofmean force (PMF) that determine the polymer conformations in asolvent or in the presence of other chains. See McQuarrie (1976) formore on the PMF concept. The goal in a RMMC simulation is tohave the computed energy come as close to the potential of meanforce as possible.)

Using Maximum BONDS to limit the range of nonbond interac-tions is a way of mimicking theta conditions. However, there is noknown a priori way to determine the ideal value of MaximumBONDS for a particular polymer. Thus, calculations with differingvalues of this parameter should be performed and the results eval-uated according to their reasonableness or agreement with estab-lished data.

The dielectric constant provides another way of mimicking thepresence of solvent. But it must be kept in mind that, at a molecu-lar level, a solvent is not a dielectric continuum, so that this too isan approximation. Consequently, the best value for the dielectricconstant in a RMMC calculation might not be the same as the sys-



tem’s bulk dielectric constant. In the event that variation of Maxi-mum BONDS and the dielectric constant within reasonable limitsdoes not give adequate results, the energy scaling parameter isavailable as a last resort.

“RIS” Metropolis Monte Carlo (RMMC) Simulations

If your chain is branched, or if its statistical weights are unknown,RMMC simulation is an alternative to traditional RIS approaches.To calculate the properties of a polymer chain using RMMC, per-form the following steps.

1. Build the chain using the POLYMER BUILDER card.

2. Go to the RMMC card and select the Properties item; choosethe properties you wish to calculate on the control panel thatopens.

3. Select various preferences for the simulation using the panelstriggered by the Preferences item on the RMMC card.

4. On the RMMC Run panel, define the length of each stage toconduct in the simulation and select a temperature. Choose aforce field for use in computing the nonbond and coulombicinteractions in the simulation.

5. Use the Run button (the RUN_RMMC command) to executethe RMMC simulation.

6. When the simulation finishes, use the Analyze item on theRMMC card to plot the computed chain properties.

7. After looking over the results, perform the calculation with adifferent value of Maximum BONDS and/or the dielectricconstant in order to test the sensitivity of the results to the sim-ulation parameters.

Computing Dihedral Distribution Functions

RMMC calculates distribution functions for the dihedral anglethat are averaged over all rotatable bonds in the model. There iscurrently no provision for the estimation of distribution functionsfor specific torsional types, or for the computation of bond pairdistribution functions.

Theory


Output Files

An RMMC calculation may produce a number of output files.Listed below are the extensions and contents of these files.

♦ .rislog file: General information; <r2>, Cn, <s2>, <a>, andother properties; error messages.

♦ _dih.tbl file: Dihedral distribution functions for all rotatablebackbone bonds (averaged together),

♦ _rsq.tbl file: Block-averaged values of <r2> and the energyas functions of the simulation step number. (Plotting this filecan help indicate whether the system is fully equilibrated. Anylong term drift in <r2> or the energy over the course of the sim-ulation is evidence that the chain has not reached a steadystate.)

♦ _pr.tbl file: Vector and scalar distribution functions of theend-to-end distance.

♦ .tab file: General information and properties.

♦ .rismnt file: Internal moments, <rija>.

♦ .arc file: Coordinates of conformations produced during thesimulation. This file can be analyzed using commands in theAnalysis module.

♦ _out.car file: The atomic coordinates at the end of the simu-lation.

Theory

RIS Metropolis Monte Carlo Simulation

Conventional RIS methods require that statistical weights exist forthe polymer of interest. The RIS Metropolis Monte Carlo (RMMC)simulation does not require statistical weights. Instead, RMMCuses a forcefield (such as CVFF, PCFF or COMPASS) to determinethe conformational properties of a chain. Because it does not usestatistical weights (or even discrete torsion angles), RMMC is not,strictly speaking, a rotational isomeric state method. However, it



shares several assumptions with RIS theory, and is used to com-pute the same types of properties as RIS. In this sense, it is closelyrelated to RIS theory.

Properties Calculated with RMMC

The RMMC program can directly calculate the following proper-ties:

♦ Mean squared end-to-end distance (<r2>), characteristic ratio(Cn), and radius of gyration (<s2>).

♦ Probability distribution function of the end-to-end distance(P(r) or P(|r|)).

♦ Chain persistence length (<a>).

♦ Mean squared chain dipole moment (<m2>).

♦ Dihedral (torsion) angle distribution function for all rotatablebonds averaged together.

♦ Various derived properties, including the intrinsic viscosity,molar stiffness function, and critical molar mass.

You can also save the chain conformations in a file for analysis byother programs.

The RMMC Algorithm

The RMMC method was developed at Biosym (now MolecularSimulations). Although most of the ideas underlying the methodare not new, the particular way these ideas have been put togetheris. The closest method to RMMC that has been published wasdeveloped by Dodd and Theodorou (1994). As with all RIS-relatedmethods, RMMC is appropriate for theta chains—whether in solu-tion or in the melt. Long-range excluded volume interactions areneglected.

As in traditional RIS methods, only torsional degrees of freedomare considered in determining a chain’s conformation; bondlengths and angles are fixed. Unlike these methods, RMMC allowstorsion angles to vary continuously; it does not impose theassumption of discrete rotational states.

As its name indicates, RMMC is a Metropolis (Metropolis 1953)Monte Carlo method. (This can be contrasted to the Markovian

Theory


approach used in a RIS Monte Carlo calculation to build indepen-dent chains.) In a Metropolis simulation of a polymer, one beginswith a chain in an arbitrary conformation. A Monte Carlo step con-sists of making a small change to that conformation—e.g., by rotat-ing a bond—and then deciding whether or not to retain thatchange, based on the temperature and the energy of the new con-formation relative to the old one. This process is repeated manytimes in order to yield a set of conformations characteristic of thatchain at the specified temperature.

In outline, a RMMC simulation proceeds as follows:

1. Perform an energy minimization on the molecule (so that bondlengths and angles adopt reasonable values).

2. Randomly select a rotatable backbone bond.

3. Select a random torsion value for this bond between –180 and+180 degrees.

4. Rotate the bond to its new torsion value and compute the newenergy of the chain.

5. Generate a random number, R, between 0 and 1. If exp[–(Enew–Eold)/kT] > R, keep the new torsion value. Otherwise, restorethe old value.

6. After enough steps have been done that the chain is equili-brated: compute the properties of the chain conformation (e.g.,r2, s2), and update the running averages of these properties.

7. Repeat from step (2) until the desired number of iterations hasbeen performed.

Treatment of Constraints

In a RMMC simulation, bond lengths and bond angles are con-strained. (For this reason, “pre-minimization” is recommended inorder that the bonds lengths and angles adopt reasonable values.)The next three paragraphs deal with some of the technical subtle-ties of simulations with constraints. Knowledge of these subtletiesis not required for a basic understanding of the RMMC simulationapproach, but is included here for the sake of completeness.

In a Monte Carlo simulation that incorporates constraints (such asfixed bond lengths and angles) and uses internal coordinates



(rather than Cartesian coordinates) in making its moves, care mustbe taken that the conformations are sampled correctly. Simulationsmay be performed with constraints that are regarded as “rigid” oras “flexible.” In the rigid case, the momenta conjugate to the con-strained coordinates are neglected. In the flexible case, it is imag-ined that the force constants for the constrained coordinates are solarge that the coordinates do not move significantly from theirequilibrium values, but that the conjugate momenta are activated.It is hard to say a priori which is a more reasonable approximationfor a polymer. In a real polymer, bond lengths and angles are, ofcourse, not rigidly constrained. On the other hand, according toquantum mechanics, very stiff degrees of freedom are not acti-vated at ordinary temperatures. Go and Scheraga (1976) arguethat, in practice, flexible constraints should be more quantitativelyaccurate.

It should be emphasized that the distinction between “rigid” and“flexible” constraints is in the assumptions behind the physicalmodel. Thus, even in the “flexible” case, the constrained coordi-nates do not move during the simulation.

A flexible constraint requires no special treatment in a MonteCarlo simulation. This is because the configuration integral, asexpressed in terms of torsional degrees of freedom, contains aJacobian-like term that is independent of these degrees of freedom.Thus, this term can be moved outside the integral, and is dividedout of any conformational averages. In the rigid case, this is nottrue, and in principle the Boltzmann factor used in the Metropolisacceptance criterion needs to be multiplied by an extra term (Fix-man 1974). In practice, this extra term has been found to changethe conformational statistics only slightly (Almarza et al. 1990).For simplicity, the flexible constraint assumption, with no extraterms in the Boltzmann factor, is made for the RMMC algorithm.Any errors introduced by this approach are likely to be muchsmaller than those due to other approximations in the RMMCalgorithm--such as approximating the potential of mean force by aforcefield with a truncated interaction range. For more on the issueof constraints in simulations, see Fixman (1974), Allen and Tildes-ley (1987), and Almarza et al. (1990).

References


References

Allen, M.P.; Tildesley, D.J. Computer Simulation of Liquids, OxfordUniversity Press (1987).

Almarza, N.G.; Enciso, E.; Alonso, J.; Bermejo, F.J.; Alvarez, M.“Monte Carlo simulations of liquid n-butane,” Mol. Phys., 70,485 (1990).

Dodd, L.R.; Theodorou, D.N. “Atomistic Monte-Carlo Simulationand Continuum Mean-Field Theory of the Structure andEquation of State Properties of Alkane and Polymer Melts,”Advances in Polymer Science, 116, 249 (1994).

Fixman, M., “Classical Statistical Mechanics of Constraints: A The-orem and Application to Polymers,” Proc. Nat. Acad. Sci., 71,3050 (1974).

Go, N.; Scheraga, H.A., “On the Use of Classical StatisticalMechanics in the Treatment of Polymer Chain Conformation,”Macromolecules, 9, 535 (1976).

Metropolis, N.; Rosenbluth, A.W.; Rosenbluth, M.N.; Teller, A.H.;Teller, E. J. Chem. Phys., 21, 1087 (1953).

McQuarrie, D.A. Statistical Mechanics, Harper and Rowe (1976).




6 Crystal Packer

Crystal Packer is a computational module that assists in the esti-mation of sublimation energies and the packing of molecular crys-tals.

Energy calculations take into account van der Waals interactions,Coulomb charges, hydrogen bonding, internal rotations, andhydrostatic pressure. The design of Crystal Packer permits greaterflexibility in the use of these terms and in their parameterization.

The current version of Crystal Packer is completely independentof the Cerius2 Open Force Field. Crystal Packer maintains its ownDreiding forcefield parameters, which are available for editingthrough its control panels.

Introduction

Sections in this chapter Model initialization

To set energy calculation options

Minimizer constraints setup

Minimizer preferences setup

Calculating energy and running the packing calculation

Theory

References


6. Crystal Packer

About Crystal Packer

Crystal Packer calculates energies and performs energy minimiza-tions (packing) of molecular crystal structures. These packing andenergy calculations use the same energy expression and forcefieldparameters. As mentioned above, Crystal Packer is not linked tothe Open Force Field, but instead contains its own independentforcefield based upon Dreiding II.

Rigid units The asymmetric unit of the crystal is divided into fragment-basedrigid units. Nonbond (van der Waals, electrostatic, and H-bond)energies are calculated between the rigid units. During energyminimization, the rigid units can be translated and rotated, andthe unit cell parameters varied.

Torsional subrotations Torsional subrotations within rigid units can be defined, thus intro-ducing an internal degree of freedom to the unit and dividing theunit into rigid subunits. Nonbond energies are calculated betweensubunits, but with 1–2 and 1–3 valence exclusion applied.

Quick defaultminimization

You can perform a default minimization on the current model inthree simple steps:

1. Open the Calculate control panel.

2. Click the Initialize Model button.

3. Click the Minimize Energy of Crystal button.

Defaults for all Crystal Packer options and controls have beencarefully chosen so that reasonable results can generally beobtained. However, to take advantage of this module’s potentialand flexibility, you should take time to thoughtfully prepare the


Building crystals The Cerius2 Builders bookNonbond energy calculations The Forcefield-Based Simulations

bookAssigning charges The Cerius2 Simulation Tools book

General methodology


model, forcefield parameters, and energy expression and minimi-zation options as described in this chapter.

General methodology

Model initialization

Before an energy calculation or minimization is performed for thefirst time on a crystal model, the model must be initialized. Duringinitialization, Crystal Packer assigns rigid-unit groups andassesses the dimensionality of the model. For detailed informationabout model dimensionality, see “The van der Waals term” onpage 118.

The rigid units are based on bonding, so that each molecule is arigid unit. Rigid-unit groups can be edited after initialization.

You can control two aspects of the initialization procedure:

♦ Hydrogen bonding — You can request that Crystal Packer findhydrogen bonds automatically, and load them into the H-bondtable during initialization.

♦ Rigid unit display — You can choose to distinguish betweenrigid units by displaying each in a different solid color.

To initialize the model

1. Open the Calculate control panel by selecting the Calculateitem on the CRYSTAL PACKER card.

2. Open the Initialization Preferences control panel from thePreferences… button on the Calculate control panel.

3. Check the Search for Hydrogen Bonds box if you want CrystalPacker to automatically find and display H-bonds when themodel is initialized.

You may want to edit options in the H-bond Calculation Pref-erences control panel to set the distance and angle criteria forhydrogen bonding. For information about setting these prefer-ences, see the Cerius2 Modeling Environment book.


6. Crystal Packer

4. Check the Load Up Hydrogen Bonds box to have hydrogenbonds automatically placed into the list box on the HydrogenBonds control panel when the model is initialized (see theonline help for more control panel information).

This feature need not be used with the Search for HydrogenBonds option. For example, you can draw your own choice ofH-bonds on the model, then select the Load Up HydrogenBonds option.

5. Click the Rigid Unit Display button if you want each type ofrigid unit to be displayed in a different color in the model win-dow when the model is initialized. This option is useful onlywhen there is more than one rigid unit type in the model.

To return to color-by-element display, choose Default from theColor popup on the Display Attributes control panel (seeCerius2 Modeling Environment).

To set energy calculation options

1. Open the Energy Terms control panel by selecting the EnergyOptions/Energy terms item on the CRYSTAL PACKER card.

2. Check those terms that you want to include in the CrystalPacker energy calculation. (The van der Waals term is alwaysincluded in the minimization calculation.)

External pressure 3. If the external pressure term is to be included, enter a value inthe External Pressure entry box.

van der Waals prefer-ences

4. Open the Van der Waals control panel (see the online help)using the Preferences… button on the Energy Terms controlpanel, or the Energy Options/VdW item on the CRYSTALPACKER card.

On-diagonal VDWparameters

5. On-diagonal VDW parameters of the Dreiding II forcefield areloaded automatically. If you want to use an alternative set ofparameters, load them using the Load From File browser boxon the Van der Waals control panel.

6. If you want to alter the on-diagonal parameters for specific ele-ments, open the On-Diagonal VdW Terms control panel (seethe online help) using the Edit On-Diagonal Terms… button.

General methodology


7. Select one of the Radius Combination Rule radio buttons toaverage the on-diagonal van der Waals radius parameters oftwo elements using the arithmetic or geometric mean.

Off-diagonal parameters 8. If you want to specify off-diagonal parameters for specific two-atom interactions, open the Off-Diag VdW (Pair Potentials)control panel (see the online help) using the Edit off-diagonalterms… button.

van der Waals interactionrange

9. Using the Van der Waals control panel, set the interaction rangefor the VDW energy calculation:

a. If you don’t want to use the defaults, edit the nonbond cut-off and close-contact warning distances.

b. By default, the convergence-smearing technique is used toestimate VDW energy of interactions beyond the cutoff;however, you can choose not to use it.

Coulomb preferences 10.If the Coulombic term is included, open the Electrostatic con-trol panel (see the online help) using the Preferences… buttonon the Energy Terms control panel, or the Energy Options/Coulomb item on the Crystal Packer menu card.

If the Coulombic term isn’t included in the energy calculation,skip to Step 12.

11. The Ewald Sum Constant, the Real Sum Limit, and the Recip-rocal Sum Limit control the range and real/reciprocal spacepartitioning of the Ewald summation of electrostatic energy.

All charges with magnitudes less than the minimum charge areset to zero. In some cases, this can shorten the calculation timeby avoiding summation of very small charges.

Generally, you do not need to alter the values on this controlpanel from the defaults.

H-bond preferences 12.If the Hydrogen bond term is included, open the HydrogenBonds control panel (see the online help) from the Prefer-ences… button on the Energy Terms control panel. You can alsoopen it from the Energy Options/Hydrogen Bonds item on theCRYSTAL PACKER card.

If hydrogen bonds are not included in the energy calculation,skip to Step 17.


6. Crystal Packer

13.If hydrogen bonds have been searched for, displayed, andloaded during the model initialization step (see “Model initial-ization” on page 107), examine the H-bonds in the model andin the list to ascertain that they are what you want.

If you want to edit the hydrogen bonding in the model, use theEdit H-bond… button to open the Edit Hydrogen Bonding con-trol panel (see the Cerius2 Modeling Environment book). Onceyou’ve finished editing, update the H-bond list box by choos-ing the Load Hydrogen Bond Table action button on theHydrogen Bonds control panel.

14.If hydrogen bonds were not assigned during the initializationstep, assign them now using the Edit Hydrogen Bonding con-trol panel.

Use the controls on the Edit Hydrogen Bonding control panelto calculate, make, and delete H-bonds. Once you’ve finishedediting, update the H-bond list box by clicking the LoadHydrogen Bond Table action button on the Hydrogen Bondscontrol panel.

15.If necessary, edit the assignment of functional parameters to thehydrogen bonds in the list box. Use the Parameters popup tospecify whether the parameters are displayed as A and B coef-ficients or as well-radius and depth values. By default, all H-bonds are assigned the same parameter set.

If you want to specify your own parameters, enter the values inthe two entry boxes under the Function Editing heading, thenclick the Add Set button. The new set is added to the 12–10Function Parameters list.

By default, the H-bond energy is a function of angle as well asdistance. Uncheck the Use Angle-Dependent Functional Formcheck box to use a function independent of angle for H-bondenergy calculation.

16.If you do not want to calculate the van der Waals energybetween H-bonded atoms, or if you want to modify the VDWparameters between H-bonded atoms, open the Acid Hydro-gens control panel (see the online help) using the Acid HVDW… button at the bottom of the Hydrogen Bonds controlpanel.

General methodology


Subrotations 17.If subrotation terms are included in the energy calculation,open the Subrotation Terms control panel (see the online help)from the Preferences… button on the Energy Terms controlpanel, or from the Energy Options/Subrotations item on theCRYSTAL PACKER card.

If subrotation terms are not included in the energy calculation,this is the final step in the procedure.

Defining subrotations 18.All subrotations must be explicitly defined:

a. Select four bonded atoms in the model window that definethe subrotation. The last atom picked is the one that moveswhen the torsion angle is varied.

b. Click the Define Subrotation From Selected Atoms actionbutton. The newly defined subrotation appears in the Cur-rent Subrotations list box.

If you make a mistake and want to remove a subrotationfrom the list, enter its number in the Remove SubrotationNumber text box, or clear the list completely with theDelete All Subrotations action button.

Fourier parameter sets 19.A set of Fourier parameters must be assigned to each definedsubrotation. Cerius2 always assigns the first parameter set inthe list, but this is often inappropriate.

a. To assign a parameter set in the Fourier Function Parameterslist box to a defined subrotation, enter the parameter setnumber in the Function entry box next to the subrotation inthe Current Subrotations list box.

b. To add a new set of Fourier parameters to the Fourier Func-tion Parameters list box, enter the new values for T0, C1, C2,C3, and C6 into the entry boxes, and click the Add FourierParameters action button. The new set appears in the Fou-rier Function Parameters list box.

c. Should you want to delete a parameter set from the list,enter its number in the Remove parameters entry box.

1–4 interactions 20.If you want to scale or exclude 1-4 valence-atom pairs fromVDW and Coulombic energy calculations, edit the Scaling Fac-tor For 1-4 Nonbonded Interactions entry box.


6. Crystal Packer

For more information about the underlying theory of this method,please see “Energy expression setup” on page 117

Minimizer constraints setup

A Crystal Packer minimization can contain several variableparameters:

♦ Six cell parameters (a, b, c, α, β, γ).

♦ Three translational variables (a, b, c,) per rigid unit.

♦ One rotational variable per rigid unit.

♦ Any number of subrotations, depending on the number of tor-sion angles defined in the rigid units.

It is unlikely that you would choose to vary all of these at once.

Cell parameters must be constrained to maintain the lattice type.For example, in a cubic lattice-cell minimization, only the a dimen-sion is allowed to vary. The b and c dimensions are tied to the alength, and the α, β, and γ cell angles are held at 90˚.

Translational and rotational freedom may need to be constrainedfor various reasons. If the rigid unit contains only one atom, itsrotational degrees of freedom should be fixed. If only one rigidunit is in the cell and no symmetries applied, the unit’s transla-tional degrees of freedom should be fixed.

To set up minimizer constraints

1. Open the Minimization Constraints control panel (see theonline help) from the Minimization Options/Constraints itemon the CRYSTAL PACKER card.

Variable cell parameters 2. Once the model has been initialized, Cerius2 suggests appropri-ate constraints in the Variable Cell Parameters section of thecontrol panel. The boxes checked are those allowed to varyindependently. To impose additional cell constraints, uncheckthe boxes.

General methodology


Note

Rigid units 3. To view and alter allowed rotations and translations for a rigidunit:

a. Select the rigid unit in the model window.

b. Click the Show button. The Translation and Rotation checkboxes are updated to show the current constraints for theselected rigid unit. (The checked directions are those inwhich movement is allowed.)

c. If you want to edit the constraints for the selected rigid unit,check or uncheck boxes as required, then click the Applybutton.

4. Repeat Step 3 for each rigid unit in the model.

Variable subrotations 5. If subrotations have been defined (see page 111), they are listedin the Variable Subrotations list box. Check those subrotationsthat you want to vary during the minimization. Those notchecked are held fixed. Fixed subrotations are marked with anF in the Subrotation Terms control panel (see the online help formore control panel information).

Minimizer preferences setup

During minimization, cell parameters, molecular segment posi-tion, and orientation and subrotation torsion angles may all bevaried in the search for energy minima.

Minimization algorithm Crystal Packer provides a choice of two minimization algorithms:

♦ Modified Newton

♦ Steepest descents

Crystal Packer uses analytical second derivatives for optimal effi-ciency. If the matrix of second derivatives is positive definite,Newton’s method is used to step to the minimum of the local qua-dratic model. If this matrix is not positive definite, no such mini-mum exists, so another strategy must be used.

If the model was built with general symmetry positions ratherthan space groups, the suggested cell constraints may not becorrect (see the Cerius2 Builders book.) Therefore, make sure thatthe cell constraints are consistent with the lattice type.


6. Crystal Packer

Modified Newton One strategy is to construct a related positive-definite matrixthrough Cholesky factorization. This is the modified Newtonmethod.

Steepest descents Another strategy is to use the steepest descents method, in whichcoordinates are displaced in a direction opposite to the gradient ofthe potential energy.

In most situations, the Modified Newton algorithm performs bet-ter and is, in general, preferred to the Steepest Descent method.For more information about the concepts involved in minimiza-tion, see Gill et al. (1981).

Minimizer progress is marked by minimization steps or iterations.Each iteration involves a series of sub-steps:

1. Crystal Packer calculates the energy, as well as the first and sec-ond derivatives of the energy.

2. The derivatives are used to determine the best search directionin the crystal-parameter space.

3. The minimum energy position in this search direction is deter-mined.

4. The crystal structure is then updated to this new lower-energyposition unless it requires a movement exceeding one of themaximum increment values to reach the new structure.

Maximum increments Five user-specified maximum-increment values exist: cell edge,cell angle, molecule translation, molecule rotation, and subrota-tion torsion. Setting a maximum limit on movement allowed inone iteration promotes the investigation of local minima. This isbecause these limits restrict the minimizer to exploring only asmall area of the crystal parameter space at a time.

The contact table Because atom-energy potentials have infinite range, but calculatedenergies are only between atom pairs within the cutoff distance,the crystal energy can vary discontinuously as the crystal spaceparameters change and atoms move in and out of the interactingradius. This sort of discontinuity can make minimization searchesvery difficult. Crystal Packer combats this problem by maintaininga nonbond list of interacting atom pairs that is updated only whennecessary. This nonbond list, called the contact table, is used for thecalculation of VDW, Coulomb, and H-bond energies.

General methodology


Updating the table Instead of updating the contact table at regular step intervals assome modules do, Crystal Packer updates its nonbond list when agiven maximum interatomic displacement is reached. A runningsum of atom displacements is maintained, and the nonbond list isrefreshed when any interatom distance has changed by more thanthe threshold displacement since the last refresh.

Minimization termination The minimization ends when the maximum number of iterationshas been performed or when the overall energy gradient falls tobelow the specified gradient tolerance value. (Energy gradientswith respect to each degree of freedom are first calculated, thenthese components are combined in quadrature to produce theoverall gradient.)

Energy tolerance Associated with the gradient tolerance is the energy tolerance. Theenergy-tolerance check is performed after the gradient-tolerancecondition has been met. The energy-tolerance check is necessary ifthe nonbond list is not updated after each iteration. If the nonbondlist is out of date when the minimization converges, it could be thatthe convergence condition would not have been met if the non-bond list were current.

To check for this, before terminating the minimization CrystalPacker updates the nonbond list and recalculates the energy. If thedifference between the energy calculated using the old nonbondlist and the energy with the new list is less than the energy toler-ance, the minimization is terminated. Otherwise, the minimizationis continued until convergence is achieved again.

To set minimization preferences

1. Open the Minimizer Preferences control panel (see the onlinehelp) from the Minimization Options/Preferences item on theCRYSTAL PACKER card.

Control parameters 2. Change the Algorithm popup from the default (ModifiedNewton) to the Steepest Descent only if the Modified Newtonmethod proves unsatisfactory.

3. If necessary, edit the value in the Non-bond Update Displace-ment Threshold entry box.

Maximum increments 4. If necessary, edit the maximum increments for cell edge, cellangle, translation, rotation, and subrotation.


6. Crystal Packer

Termination criteria 5. If necessary, edit the maximum number of minimization itera-tions.

6. If necessary, edit the Gradient Tolerance value. The units are inkcal/mol/Å and kcal/mol/rad. This is the sole criterion forsuccessful termination of the minimization.

7. If necessary, edit the Energy Tolerance value. This value veri-fies that the nonbond list is not significantly out of date whenthe gradient tolerance condition is met.

Calculating energy and running the packing calculation

The energy calculation and energy minimization functions are thecore of Crystal Packer. All procedures discussed until now havebeen done to prepare the model, the energy expression, and theminimizer for the energy and packing calculations.

Two calculations can be performed:

♦ Single sublimation energy calculation.

♦ Crystal packing minimization.

The energy expression used for each is the same, except where theVDW energy term is not included (see the online help for morecontrol panel information).

Output from both calculations is sent to the text window. Addi-tionally, for the minimization calculation, a plot of energy (kcal/cell) versus minimization step is sent to the graph window.

To calculate model energy

Before doing a single-energy calculation on the model, the initial-ization procedure (see page 107) must be carried out. Also, unlessyou are using default energy terms and not defining subrotations,you should first perform the energy term setup procedure (seepage 108). To calculate model energy:

1. Open the Calculate control panel from the Calculate item onthe CRYSTAL PACKER card.

2. Click the Calculate Energy of Crystal button. The total modelenergy, as well as a breakdown of the terms, is sent to the textwindow.

Theory


To perform a minimization of model energy

Before performing a minimization calculation on the model, theinitialization procedure (see page 107) must be carried out. Also,unless you are using default settings for all options, you shouldfirst perform the energy terms setup procedure (see page 108), theminimization constraints procedure (see page 112), and the mini-mization preferences procedure (see page 115). To perform a min-imization:

1. Open the Calculate control panel from the Calculate item onthe CRYSTAL PACKER card.

2. Click the Minimize button. The minimization proceeds untilthe maximum number of iteration steps is performed or untilconvergence.

Information on the minimization is sent to the text window,and a plot of energy as a function of minimization step is sentto the graph window.

Theory

Energy expression setup

This section describes both the energy expression and calculationsused in Crystal Packer. Although there is much flexibility over theparameters governing the energy expression and forcefieldparameters, care has been taken to set robust default values. As aresult, you should obtain reasonable results with little or no edit-ing.


6. Crystal Packer

The energy expression set up by Crystal Packer can involve up tofive terms:

Each energy term is discussed below.

The van der Waals term

The van der Waals (VDW) term is nearly always included in theenergy calculation and is always included during minimization.

VDW interactions apply to all atom pairs that are:

♦ In different rigid units (or different subunits if subrotationshave been defined).

♦ Closer than the defined nonbond cutoff distance.

Bonds are not allowed between different rigid units, nor betweenrigid units and their symmetry copies. If atoms in different rigidunits are separated by typical covalent bonding distances, thentheir nonbond interaction energy becomes unphysically large, andthe cell energy huge. Thus, correct definition of rigid units is veryimportant for correct results from Crystal Packer calculations.

Polymer, surface, andnetwork models

Nonetheless, some exceptions to the rule prohibiting interunitbonding must be made; that is, for polymer, surface, and 3D net-work models. Since polymers in crystals are covalently bondedthroughout the crystal in the c-axis direction, Crystal Packerignores interactions between a polymer segment and translationalimages of that segment in the c direction. Similarly, a 2D surfacenetwork is assumed to be bonded in the b-c plane, and interactionswith images in cells (0, m, n) are excluded. A 3D network isbonded throughout the crystal, and no self-segment interactionsare included. During the initialization step, Cerius2 automaticallysets the rigid unit dimensionality according to the dimensionalityof the model.

Energy = van der Waals term +Coulomb term +hydrogen bond term +

torsional energy term +

external pressure term

nonbondterms

Theory


Treatment of long-rangeinteractions

Crystal Packer provides an option to use convergence acceleratingsmearing (Pertsin and Kitaigorodsky 1987) to estimate the VDWenergy of interactions between atom pairs further apart than thecutoff distance. This method works by assuming that, outside thetruncation radius, atoms are uniformly smeared over space. Thisassumption leads to an easily calculated expression for estimationof long-range VDW interactions.

Close contact check During the energy calculation, Cerius2 also checks for close contactatoms. For all atoms in different rigid units that are separated byless than the specified close contact distance, a warning message issent to the text window. This warning has no effect on the calcula-tion.

The Lennard-Jones func-tional form

The VDW energy of atom pairs is calculated using the Lennard-Jones functional form (also called the 12–6 form):

Eq. 1

Where:

Evdw = Van der Waals interaction energy between atoms i and j.

R = Interatomic separation between atoms i and j.

Dij0 = Lennard-Jones well-depth parameter for atoms i and j.

Rij0 = Lennard-Jones radius parameter for atoms i and j.

Or an equivalent equation:

Eq. 2

Where:

A and B = Lennard-Jones coefficients.

On-diagonals By default, Dij0 and Rij0 are calculated by combining values for thetwo elements involved in the interaction. The well depth is alwayscombined geometrically, but the well radius can be either the arith-metic or geometric mean of the homo-atom radii:

Eq. 3

Evdw R( ) Dij0

Rij0

R---------

122

Rij0

R---------

6–=

Evdw R( ) AR 12– BR 6––=

Dij0 Di0Dj0=


6. Crystal Packer

and

(arithmetic mean) Eq. 4

or

(geometric mean) Eq. 5

Where:

Di0 = Lennard-Jones well depth for element i.

Rii0 = Lennard-Jones well minimum for element i.

Crystal Packer maintains a set of these on-diagonal well-depth andradius parameters for most elements. This set is based on the Dre-iding II (Mayo et al. 1990) forcefield, but an alternative set basedon the Tripos 5.2 (Clark et al. 1989) forcefield is also provided.Crystal Packer also lets you load your own customized set ofparameters from file.

Your choice of combination rule depends upon the forcefield; theforcefield parameters supplied in Crystal Packer are from the Dre-iding II forcefield, which uses the arithmetic mean.

Off-diagonals Crystal Packer also allows you to specify off-diagonal parametersfor unique pairs of atoms. These pair potentials can be entered aswell depth and radii (Dij0 and Rij0) or as R-12 and R-6 coefficients(A and B). Values defined for specific pairs take precedence overvalues determined by elemental combination.

The off-diagonal parameter lists can be saved to a file and re-loaded in later sessions. The CERIUS2Resources directory con-tains two sample files: pots1 and pots2; the sources for these filesare given in their headers.

The Coulomb term

Electrostatic interaction energy can be included in the CrystalPacker energy expression. Partial charges can be assigned or calcu-lated using the Charges module (see the Cerius2 Simulation Toolsbook). The overall charge of the crystal must be neutral.

Rij0

Rii0 Rjj0+

2-------------------------=

Rij0 Rii0Rjj0=

Theory


Minimum charge The electrostatic interaction is calculated between all chargedatoms residing in different rigid units (or different subunits, if sub-rotations have been defined). To prevent summing over small,essentially random charges, you may want to set a minimum chargevalue, so that any smaller charges are excluded from the Coulombsum.

The Ewald sum The Ewald summation method (Ewald 1921, Karasawa and God-dard 1989) is used to calculate the Coulombic energy. The slowlyconverging real-space Coulomb sum is split into a quickly con-verging modified real-space sum and a summation in reciprocal-space.

The division is specified by three parameters:

♦ Ewald sum constant

♦ Real-space sum cutoff

♦ Reciprocal-space sum cutoff

The Ewald sum constant controls the division of work between thereal- and reciprocal-space sums. The larger the sum constant, themore quickly convergent is the real-space sum, but the moreslowly convergent is the reciprocal-space sum.

The choice of these three parameters can be quite tricky. The largerthe two limits, the more accurate the Coulombic energy, but themore expensive the calculation. Because the reciprocal-space sumis cheaper than the real-space sum, it is better to err on the side ofa large sum constant, small real-space sum limit, and large recip-rocal-space sum limit. However, the default values of 0.40, 10, and0.5 for the Ewald sum constant, the real-space sum limit, and thereciprocal-space sum limit, respectively, are reasonable choicesthat the novice user may never need to alter.

Note

The hydrogen bond term

Hydrogen-bond energy can be included in the Crystal Packerenergy expression. Crystal Packer can be set to automatically find

The Ewald summation technique converges only if the unit cellis electrically neutral. Therefore, Crystal Packer does not permitCoulombic energy calculations on cells with a non-zero netcharge.


6. Crystal Packer

all H-bonds in a model according to specified bond search criteria,or you can assign H-bonds to the model yourself.

H-bond interactions should be included only between rigid unitsor subunits. H-bond interactions that are wholly within fixed unitsare undesirable; that is, they add calculation time, yet add a con-stant term to the energy.

The hydrogen bonds used by Crystal Packer are those displayed inthe model window when the Load Up Hydrogen Bonds option isset (see the online help for the Initialization Preferences controlpanel). Because Crystal Packer is designed for use on structuresthat are not far from a minimum, hydrogen bonds are not updatedduring minimization.

Functional form ofH-bond potential

Either one of two 12–10 functional forms can be used to calculatedthe H-bond energy:

♦ The CHARMm-like, angle-dependent potential:

Eq. 6


Eq. 7

Where:

R = Donor-to-acceptor distance

θA H D = Angle formed by the acceptor, hydrogen, and donoratoms

D0 = Well depth

R0 = Radius of the well minimum

A = Coefficient (=D05R012)

B = Coefficient (=D06R010)

♦ The angle-independent potential:

Eq. 8

Ehb R θA H D,( ) D0 5R0

R------

126

R0

R------

10– θA H Dcos4=

Ehb R θA H D,( ) AR 12– BR 10––( ) θA H Dcos4=

Ehb R( ) D0 5R0

R------

126

R0

R------

10–=

Theory



Eq. 9

Where:

R = Hydrogen-to-acceptor distance

The other variables are as defined for Eq. 6 above.

H-bond parameters Parameters, expressed as R0 and D0, or as A and B, are supplied inthe Hydrogen Bonds control panel; their source is the Dreiding IIforcefield (Mayo et al. 1990). Some of these parameters aredesigned for use when Coulomb energies are included and othersare for use when electrostatic interactions are ignored.

Alternatively, you can specify your own 12–10 parameters for theH-bond potential function.

By default, all H-bonds in the model are assigned the same param-eter set. However, this need not be the case. Crystal Packer allowsyou to assign different parameter sets to different H-bonds in themodel.

van der Waals parametersand H-bonds

In some forcefields, H-bonds are parameterized assuming noexplicit van der Waals interaction between the hydrogen atom andthe acceptor atom. Crystal Packer lets you choose to exclude H-Acceptor van der Waals interactions from the van der Waalsenergy calculation.

However, if you do include H-Acceptor van der Waals interac-tions, you have the option of modifying the magnitude of theinteraction by editing van der Waals (12–6) well-depth and radiusparameters for the acid hydrogen atom, thereby creating a moreshallow well or decreasing the radius.

All 12–10 parameters provided in Crystal Packer are defined suchthat modification of the H__a VDW parameter is unnecessary.

The torsional energy term

During minimization, molecules in a crystal are given transla-tional and rotational freedom. Additionally, internal degrees offreedom may be added by allowing torsional rotation within themolecules. Each rotatable torsion, called a subrotation, must beexplicitly defined before the model is initialized.

Ehb R( ) AR 12– BR 10––( )=


6. Crystal Packer

The subrotation is defined by selecting four atoms that constitutethe chosen torsion. The order of atom selection is important.

The rotation takes place about the bond between atoms B and C;and the atoms A and D define the dihedral angle. The rigid sub-unit attached to atom C moves when the dihedral angle is varied.This is an important point if several subrotations are to be definedin one rigid unit. This is because Crystal Packer requires that eachrigid unit have some portion that is not moved by subrotations.

The potential energy function for a subrotation takes the form of aFourier series:

Eq. 10

Where:

C1, C2, C3, and C6 = Constants.

θt = Torsion angle (as shown in figure above).

θ0 = Reference torsion angle.

Etorsion(θt) = Torsion energy as a function of torsion angle.

Nine sets of function parameters (θt, C1, C2, C3, and C6) from theDreiding forcefield are supplied with Crystal Packer, but these areeasily edited to suit your particular application (Mayo et al. 1990).When a torsion is initially defined, it is arbitrarily assigned func-tion one of the Dreiding set, regardless of the chemical nature of

A

B

C

D

Torsion specification:Four atoms in torsion

Select A with firstclick, B with secondclick, …

The torsion angle, θ,looking down alongthe B-C bond

A

B, C

D

θ

Etorsion θτ( ) C1 θτ θ0–( )cos C2 2 θτ θ0–( ) +cos+=

C3 3 θτ θ0–( )cos C6 6 θτ θ0–( )cos+

Theory


the chosen subrotation. If function one is not suitable to you,assign a different parameter set from the list or enter a new set ofyour own.

Each subrotation that is defined converts a single rigid unit intotwo rigid units. For the purposes of van der Waals, Coulomb, andhydrogen-bond energies, all suitable interactions between atomsin different rigid units are included, thus giving rise to both intra-and inter-molecular terms. Valence exclusion is applied to all 1–2and 1–3 nonbond interactions, but 1–4 nonbond interactions (suchas between atoms A and D above) are included in the energy cal-culation. The strength of 1–4 nonbond interactions can be attenu-ated by scaling factor or, by setting the scaling factor to zero, 1-4interactions can be excluded. For more information about valenceexclusion and 1–4 scaling, see the Forcefield-Based Simulations book.

The external pressure term

The hydrostatic energy term contribution to the energy (Ehyd) is:

Eq. 11

Where:

p = External pressure [kbar]

V = Unit cell volume

Uses for external pressure This external pressure function is primarily intended for use withthe Crystal Packer minimizer. Some sample minimization strate-gies that use the external pressure term are listed below:

♦ Begin a minimization — In minimizing a very low density cell,you may find that the intermolecular distances are greater thanthe nonbond cutoff distance. In this situation, minimization isunimportant because no attractive inter-unit nonbond forcesare calculated. However, by applying a small external pressureat the start of the minimization, you can bring the moleculesinto closer contact, to a distance less than the nonbond cutoffrange.

♦ Perform a quick minimization — Find a rough minimum witha small interaction radius (about 5 Å) and an applied pressure(about 10 kbar), then increase the interaction radius andremove the pressure.

Ehyd pV=


6. Crystal Packer

♦ Find a new local minimum — Apply a high pressure (about100 kbar), forcing atoms past each other and, perhaps, into alower energy structure, then remove the pressure and continueminimization.

References

Pertsin, A. J.; Kitaigorodsky, A. I. The Atom-Atom Potential Method,Springer-Verlag, Berlin (1987).

Mayo, S. L.; Olafson, B. D.; Goddard, W. A. J. Phys. Chem., 94, 8897(1990).

Clark, M.; Cramer, R. D.; Van Opdenbosch, N. J. Comp. Chem., 10,982 (1989).

Ewald, P. P. Ann Phys (Leipzig), 64, 253 (1921).

Karasawa, N.; Goddard, W. A. J.Phys. Chem, 93, 7320 (1989).

Gill, P. E.; Murray W.; Wright, M. H. Practical Optimization,Chapter 4, Academic Press, London (1981).


7 Polymorph Predictor

The Polymorph Predictor module simulates the packing processfor molecular crystals, generating a number of stable crystal struc-tures that have a high probability of being found experimentally.The polymorphs generated are saved in trajectory files; tools areprovided to analyze the properties of structures in these trajectoryfiles, and to extract those required for further study into modelspaces.

Introduction

Sections in this chapter Using the Polymorph Predictor

Setting up

Monte Carlo packing simulation

Cluster analysis

Energy minimization

Trajectory file analysis and model extraction

Merging trajectory files

Reliability checking

Theory

References

Topic Reference

Loading forcefields The Cerius2 Simulation Tools bookPreparing conformers for flexible

moleculesThe Cerius2 Conformational Search

and Analysis book


7. Polymorph Predictor

Using the Polymorph Predictor

Several time-consuming steps are required to generate probablemolecular crystals using Polymorph Predictor. However, once youhave prepared and specified the molecular models to include inthe asymmetric unit, and set up all parameters and options, theprogram can handle the entire prediction sequence automatically.

Step 1: Setup Before you start, you must load and configure an appropriateCerius2 forcefield. You must then prepare molecules to include inthe asymmetric unit of the crystal structure being studied, andappropriately configure prediction procedure parameters andoptions. For details, see “Setting up” on page 129.

Step 2: Monte Carlo pack-ing simulation

The Monte Carlo packing simulation searches each specified spacegroup for the most probable crystal structures. For details, see“Monte Carlo packing simulation” on page 138.

Note

Step 3: Cluster analysis Trajectory files generated by the Monte Carlo simulation for eachspace group contain large numbers of unoptimized structures;those with the lowest energy are the most likely to occur experi-mentally. At the local minima of the energy hyper surface exploredduring the simulation, clusters of many very similar structures (ofwhich only the lowest energy representative is significant) areexpected.

These clusters of similar structures and their lowest energy repre-sentatives can be identified by performing cluster analysis on thetrajectory files output by the Monte Carlo simulation. Low energyrepresentatives identified by clustering are output in trajectory fileformat. For details, see “Cluster analysis” on page 144.

Step 4: Energy minimiza-tion

The low-energy structures output by cluster analysis are still ener-getically unrefined and require optimization with respect to alldegrees of freedom.

A trajectory file containing all accepted structures is output foreach space group searched. You must perform steps 2, 3, 4, and5 of the prediction sequence for each space group, using thetrajectory files output from the previous step as input. SeeAppendix B, “File Formats” for more information abouttrajectory files.

General Methodology


Energy minimization subjects each structure in a trajectory fileoutput by cluster analysis to a full body minimization. The result-ing optimized structures are saved in an output trajectory file. Fordetails, see “Energy minimization” on page 147.

Step 5: Cluster minimizedstructures

Initially distinct structures may become very similar, or even iden-tical, during minimization. You should remove such redundantstructures from the trajectory files produced by energy minimiza-tion by running a final cluster analysis. The remaining structuresare saved in an output trajectory file.

Trajectory file merging Having performed steps 2 through 5 for each of a number of spacegroups, you are left with several output files (one per space group)containing optimized structures ready for final analysis and modelextraction. However, it is possible that structures in the outputfrom different space groups are identical. You may want to mergethe output trajectory files and perform a final cluster analysis toremove such redundant structures. See “Merging trajectory files”on page 155. For more detailed information on trajectory files,please see Appendix B, “File Formats”.

Trajectory file analysis After each polymorph run, the resulting trajectory file can be ana-lyzed using the Analysis item on the POLYMORPH card. Pleasesee “Trajectory file analysis and model extraction” on page 149 formore information.

Reliability checks It is recommended that you confirm the accuracy of the polymor-phs obtained by the prediction procedures by performing one ormore reliability checks on them. See “Reliability checking” onpage 156.

General Methodology

Setting up

Before you start a polymorph prediction run, you must appropri-ately configure a Cerius2 forcefield and specify and prepare mod-els of all molecules in the asymmetric unit of the crystal. Once youhave defined the asymmetric unit, you can initiate further steps inthe prediction sequence (Monte Carlo packing simulation, clusteranalysis, minimization, and final clustering) as a complete, auto-



mated sequence or as individual steps. Either way, you should setparameters and options for each step prior to initiating it.

Configuring forcefields

You should use a forcefield that is appropriate for the atom types(that is, the elements and their hybridization, as well as the localgeometry) in the structure. The latest version of the Dreiding force-field, Dreiding2.21, is the most appropriate for molecular crystalswithout metal atoms and is recommended for Polymorph Predic-tor runs.

You must specify use of the Ewald long-range summation methodfor electrostatic and van der Waals interactions, as other methodsdo not handle interactions between molecules correctly. To findout more about Cerius2 forcefields, how to load them, and setparameters for energy terms, see the chapter on the Open ForceField in Cerius2 Simulation Tools.

Preparing models

You should prepare and specify models that will be included in theasymmetric unit of the crystal structure being studied (includingthe number of times the model should appear in the unit) usingthe Polymorph Run control panel.

Minimize and calculatecharges

Each model you specify for inclusion in the asymmetric unitshould be energetically minimized and have charges calculatedand assigned to its atoms before polymorph prediction begins.

Use the Cerius2•Minimizer to perform energy minimization oneach molecule (see the Cerius2 Simulation Tools). If you have exper-imental data for comparison (for example, a powder pattern),MOPAC ESP charges for atoms in the models obtained using theCerius2•MOPAC module should be sufficiently accurate. If yourequire improved quality predicted structures, particularly for abinitio predictions where no experimental data is available, use ofthe Gaussian or another ab initio quantum mechanics package isrecommended:

1. Perform a Gaussian optimization of the molecules in the asym-metric unit, preferably with the 6-31G** basis set.

2. Obtain Gaussian ESP charges for all atoms.

General Methodology


3. Edit Open Force Field parameters so that the forcefield mini-mized structure is as close as possible to the Gaussian opti-mized structure. When performing this editing, it isrecommended that you start with the simple parameters andwork upward in complexity (for example, edit bond potentialsbefore valence potentials) as the amount of work required torefine increases exponentially.

4. Specify rigid body constraints which you wish to apply duringthe final optimization of the structures. By default, PolymorphPredictor treats molecules as rigid during the initial MonteCarlo/simulated annealing step, but, during the final minimi-zation, the flexibility of the molecules is taken into account andall atomic positions are optimized individually. If you wish cer-tain structural units or whole molecules to remain rigid duringthis final minimization, you must specify these rigid body con-straints before starting the simulation sequence. See the Cerius2

Forcefield Engines manual on how to define rigid body con-straints.

Note

Flexible molecules The Monte Carlo packing simulation treats all molecules as rigidstructures. For flexible molecules, you should prepare conformerscorresponding to local energy minima of the conformational spaceusing Conformer Search and Conformer Analysis. After preparingthe conformers, perform the polymorph prediction procedure oneach one. (See the Cerius2 Conformer Search and Analysis book.)

Where a flexible molecule occurs more than once in the asymmet-ric unit, you must perform prediction runs for each possible com-bination of stable conformers. For example, for a flexible moleculewith three stable conformers, A, B, and C, where you suspect thatthe molecule occurs twice in the asymmetric unit, six predictionsequences are required — for combinations A+A, A+B, A+C, B+B,B+C, and C+C.

With no experimental data available, it is impossible to checkthe results of ab initio predictions. You may find it useful topack similar molecules for which experimental data isavailable. You can then check whether the motifs of the packing(for example, hydrogen bonding patterns) are found usingPolymorph Predictor.



Simulation parameters

The Monte Carlo packing simulation, cluster analysis, and energyminimization processes each have a number of associated optionsand parameters that affect their operation. Unless you want to usedefault values, you should access the preferences panels that con-tain the controls for these parameters, and set them appropriatelyprior to initiating any of these procedures.

Predicting Polymorphs

Polymorph Predictor limitations

Flexible molecules Polymorph Predictor treats all molecules as rigid structures. Forflexible molecules, conformers corresponding to local energy min-ima of the conformational space must be prepared and the poly-morph prediction procedure performed on each one.

Where a flexible molecule occurs more than once in the asymmet-ric unit, prediction runs must be performed for each possible com-bination of stable conformers. This means that the number of runsrequired quickly becomes unmanageable as the number of stableconformers and/or molecules in the asymmetric unit increases.

For practical reasons then, molecules included in the asymmetricunit for prediction runs should have limited flexibility. Sensiblelimits are five or fewer conformers, within approximately twokcal/mol of the global minimum.

Forcefields Polymorph Predictor calculates likely crystal structures withforcefield technology. Many different forcefields are available,each optimized for different elements and structure types. Theaccuracy of a prediction run is therefore related to the suitability ofthe forcefield for the crystal being studied.

You should use a forcefield that works well for the atom types (thatis, the elements and their hybridization) in the structure. The latestversion of the Dreiding forcefield, Dreiding2.21, is optimized formolecular crystals; however, it does not work well for crystalstructures containing metals. Dreiding models carbon, nitrogen,oxygen, and hydrogen most accurately; it models sulphur, phos-

General Methodology


phorous, chlorine, and bromine less accurately, and so on downthe periodic table.

Processing time Prediction time increases in proportion to the number of:

♦ Atoms in each molecule

♦ Molecules in the asymmetric unit

♦ Symmetry operators for the space groups being searched

Doubling any of the listed values roughly doubles processing timefor the Monte Carlo step and quadruples processing time for theminimization. Recommended limits to the number of atoms in heasymmetric unit are 150 as an upper bound and 100 as a reason-able maximum.

Intramolecular symmetry Polymorph Predictor assumes no intramolecular symmetry —there should be no bonds between the asymmetric unit and anysymmetry copies of the asymmetric unit.

Running a prediction

You can initiate a complete polymorph prediction sequence or youcan launch each step manually from the Polymorph Run controlpanel. The following steps are performed for each space group to besearched during the Monte Carlo simulation (please see the onlinehelp for more information on the control panels):

♦ Monte Carlo packing simulation.

♦ Cluster analysis of the Monte Carlo trajectory.

♦ Minimization of the structures produced by cluster analysis.

♦ Cluster analysis of the minimized structures.

You perform each step using the parameters and options on corre-sponding preferences panels. Open the appropriate panels tochange any of their values from their defaults.

Restarting interrupted pre-diction procedures

Each of the polymorph prediction procedures (Monte Carlo simu-lation, cluster analysis, and energy minimization) can take a longtime (up to several hours) to run. To avoid the potential time wast-age and inconvenience that a system crash or unavoidable processinterruption could cause, Polymorph Predictor provides a mecha-nism for restarting such interrupted jobs.



Note

In addition to the trajectory files that are generated during eachprocedure, restart and summary files (with the same names as thetrajectory file, but with extensions of .rst and .summary ratherthan .pmp) are periodically written. If the run completes success-fully the restart file is deleted, but not the summary file.

Restart files contain information necessary to complete the inter-rupted run from the point at which the file was last updated,including the settings of all relevant Polymorph Predictor optionsand parameters for the operation. All such options are restoredand used if a run is restarted. Note that forcefield and other set-tings are not included in restart files and must be restored by alter-native means. The frequency of restart file generation isnecessarily different for each procedure:

♦ Monte Carlo simulation — A restart file is written every 100Monte Carlo trials. A restarted run continues from the trial atwhich the file was written, so a maximum of 99 trials acceptedafter the file was last written can be lost.

♦ Energy minimization — A restart file is written each time min-imization of a new structure begins. A restarted run continuesfrom this point.

Defining the molecules inthe crystal structure

The molecules to include in the asymmetric unit of the crystalstructure you are studying are defined (by model number) in thelower portion of the Polymorph Run control panel. You can alsospecify that multiple copies of a model be incorporated into theasymmetric unit. The content of each model is treated as a rigidunit during the Monte Carlo simulation.

To run a complete polymorph prediction sequence

1. Load and/or sketch molecules that you want to include in theasymmetric unit of the crystal system you want to study.

2. Load an appropriate forcefield (Dreiding2.21 is recommended)and select the Ewald long-range summation method for elec-trostatic interactions using the Open Force Field.

Only the operation in process at the time of the interruption canbe restarted. Automatic prediction sequences cannot beresumed; you must manually complete any operations requiredafter you restart a procedure.

General Methodology


Note

3. Minimize, obtain MOPAC or Gaussian ESP charges, and, if flex-ible, obtain low energy conformers for all molecules in theasymmetric unit.

4. If you wish structural units or whole molecules to remain rigidduring the minimization step of the Polymorph predictionsequence, specify these rigid body constraints for each mole-cule in the asymmetric unit using the options provided on theOPEN FORCE FIELD/Constraints or MINIMIZER/Con-straints control panels. See the Forcefield Based Simulationsmanual for details on how to define rigid body constraints.

5. Open the Polymorph Run control panel by choosing Run fromthe POLYMORPH PREDICTOR card.

6. Specify the model or models containing the molecules youwant to include in the asymmetric unit of the crystal. Specifythe number of copies of any model that should be includedmore than once in the asymmetric unit.

7. If the default settings for the Monte Carlo simulation are notsatisfactory, specify new preferences:

a. Open the Polymorph MC Preference control panel by press-ing the Preferences... button beside the Run Monte Carlobutton.

b. Set the parameters and options on the panel and its two sub-panels: Polymorph MC Parameters and Polymorph MCOutput. See “Setting Monte Carlo preferences” on page 140.

8. If the default settings for the cluster analysis are not satisfac-tory, specify new preferences:

a. Open the Polymorph Cluster Prefs control panel by pressingthe Preferences... button beside the Run Cluster Analysisbutton on the Polymorph Run control panel.

b. Set the parameters and options on the panel and the Poly-morph Cluster Output subpanel. See “Setting Cluster Anal-ysis preferences” on page 145.

If you use the OFF forcefield on a non-periodic model afterselecting the Ewald method, the method will automaticallyswitch to spline.



9. If the default settings for energy minimization are not satisfac-tory, specify new preferences:

a. Open the Polymorph Minimize Prefs control panel by press-ing the Preferences... button beside the Run Energy Mini-mization button on the Polymorph Run control panel.

b. Set the parameters and options on the panel and the Poly-morph Minimize Output subpanel. See “Setting minimiza-tion preferences” on page 147.

10.Initiate the polymorph prediction sequence by pressing thePredict Polymorphs button on the Polymorph Run controlpanel. Polymorph Predictor performs all necessary proceduresautomatically, including Monte Carlo simulation and subse-quent cluster analysis, minimization, and final clustering foreach specified space group.

11. Perform reliability checks to verify the accuracy of the poly-morphs obtained by the prediction procedure. See “Reliabilitychecking” on page 156.

To predict polymorphs manually

1. Load and/or sketch molecules that you want to include in theasymmetric unit of the crystal system you want to study.

2. Load an appropriate forcefield (Dreiding2.21 is recommended)and select the Ewald long-range summation method for elec-trostatic interactions using the Open Force Field.

Note

3. Minimize, obtain MOPAC or Gaussian ESP charges, and, if flex-ible, obtain low energy conformers for all molecules in theasymmetric unit.

4. If you wish structural units or whole molecules to remain rigidduring the minimization step of the Polymorph predictionsequence, specify these rigid body constraints for each mole-cule in the asymmetric unit using the options provided on theOPEN FORCE FIELD/Constraints or MINIMIZER/Con-

If you use the OFF forcefield on a non-periodic model afterselecting the Ewald method, the method will automaticallyswitch to spline.

General Methodology


straints control panels. See the Forcefield Based Simulationsmanual for details on how to define rigid body constraints.

5. Open the Polymorph Run control panel by choosing Run fromthe POLYMORPH PREDICTOR card.

6. Specify the model or models containing the molecules youwant to include in the asymmetric unit of the crystal. Specifythe number of copies of any model that should be includedmore than once in the asymmetric unit.

7. If the default settings for the Monte Carlo simulation are notsatisfactory, specify new preferences:

a. Open the Polymorph MC Preference control panel by press-ing the Preferences... button beside the Run Monte Carlobutton.

b. Set the parameters and options on the panel and its two sub-panels: Polymorph MC Parameters and Polymorph MCOutput. See “Setting Monte Carlo preferences” on page 140.

8. Run the Monte Carlo simulation by pressing the Run MonteCarlo button on the Polymorph Run or Polymorph MC Prefer-ences control panel. A trajectory file is output for each spacegroup searched.

9. If the default settings for cluster analysis are not satisfactory,specify new preferences:

a. Open the Polymorph Cluster Prefs control panel by pressingthe Preferences... button beside the Run Cluster Analysisbutton on the Polymorph Run control panel.

b. Set the parameters and options on the panel and the Poly-morph Cluster Output subpanel. See “Setting Cluster Anal-ysis preferences” on page 145.

10.Perform cluster analysis on the contents of each trajectory file(one per space group searched) output by the Monte Carlo sim-ulation by pressing the Run Cluster Analysis button on thePolymorph Run or Polymorph Cluster Prefs control panel.

11. If the default settings for energy minimization are not satisfac-tory, specify new preferences:



a. Open the Polymorph Minimize Prefs control panel by press-ing the Preferences... button beside the Energy Minimizebutton on the Polymorph Run control panel.

b. Set the parameters and options on the panel and the Poly-morph Minimize Output subpanel. See “Setting minimiza-tion preferences” on page 147.

12.Perform energy minimization on the contents of each trajectoryfile (one per space group searched) output by cluster analysisby pressing the Energy Minimize button on the PolymorphRun control panel or the Minimize button on the PolymorphMinimize Prefs control panel.

13.Specify preferences for and perform a final cluster analysis oneach trajectory file (one per space group searched) output byenergy minimization. See steps 8 and 9 above.

14.Perform reliability checks to verify the accuracy of the poly-morphs obtained by the prediction procedure. See “Reliabilitychecking” on page 156.


Monte Carlo packing simulation

The first step of a polymorph prediction sequence is a Monte Carlosimulation of the thermodynamic movement of the system foreach selected space group. The simulation consists of two phases,heating and cooling, and is thus termed simulated annealing. TheMetropolis algorithm is used to determine whether generated trialstructures are accepted or rejected.

The simulation

Generating initialstructures

An initial structure is generated for each space group from themodels specified on the Polymorph Run control panel. Each of thespecified models is copied into the current model space (whichshould be empty) before starting the simulation. If more than onecopy of any model is required, the fragment is copied the appro-priate number of times.

General Methodology


Heating phase Firstly, each trial crystal is heated. During heating, the temperaturefor each new trial, Tnew, is obtained from that used for the previoustrial, Told, using a user-definable heating factor, Th, by:

Eq. 1

Heating continues until either a preset maximum temperature isreached or a specified number of consecutive trial moves accepted.

If the current trial is rejected, the Monte Carlo move factor ishalved. (Note that the move factor has an initial value of 1.0 duringthe cooling phase, at which all parts of phase space are accessiblewithin one move. During the heating phase, the value is fixed at1.0). If the trial is accepted this factor is doubled, with a maximumallowed value of 1.0.

Cooling phase If a trial is accepted during cooling, the temperature to use for thenext trial Tnew, is obtained from that used for the previous trial,Told, using a user-definable cooling factor, Tc, by:

Eq. 2

The default cooling factor specified is modified according to thenumber of rigid units in the asymmetric unit, N, using the follow-ing algorithm:

Eq. 3

The Monte Carlo move factor is updated with the same algorithmused during the heating phase. The run finishes when either thetemperature reaches a preset minimum, a specified maximumnumber of trials have occurred, or the Monte Carlo move factorbecomes lower than a minimum allowed value.

Trial steps

Each Monte Carlo trial consists of the following steps:

1. Make a random change to the orientation of each rigid unitwithin the asymmetric unit.

Tnew Told 1.0 Th+( )×=

Tnew Told 1.0 Tc–( )×=

Tc Tc 1 N 1–( )N 1+( )

--------------------– ×=



2. If more than one rigid unit is present in the asymmetric unit,make a random change to the relative orientation of each rigidunit within the asymmetric unit.

3. Depending on space group restrictions, make a random changeto the relative orientation of the lattice vectors of the unit cell.

4. If there is more than one rigid unit in the asymmetric unit:

a. Expand the asymmetric unit in free space until no van derWaals contacts can be found within the asymmetric unit.(Relatively large step sizes are used during expansion.)

b. Contract the asymmetric unit in free space while maintain-ing zero van der Waals contacts within the asymmetric unitsuch that the molecules in the asymmetric unit are broughtinto close contact. (Smaller step sizes are used to pack thecrystal as tightly as possible.)

Steps a and b both use a random selection procedure to movethe relevant translational degrees of freedom.

5. Remove van der Waals contacts from the crystal:

a. Increase the magnitude of the lattice vectors until there areno van der Waals contacts. (Relatively large step sizes areused.)

b. Decrease the magnitude of the lattice vectors while main-taining zero van der Waals contacts. (Smaller step sizes areused to pack the crystal as tightly as possible.)

Steps a and b both use a random selection procedure to movethe relevant translational degrees of freedom. When appropri-ate, the translation of the asymmetric unit within the unit cellalso moves randomly during step B.

6. Calculate the energy and density of the crystal and apply astandard Metropolis acceptance/rejection test. In addition, adensity constraint is applied: if the density of the structure isbelow 0.3 g/cm3, the step is always rejected.

Setting Monte Carlo preferences

You specify the parameters and options that affect the Monte Carlopacking simulation using options on (or accessible from) the Poly-

General Methodology


morph MC Preferences control panel (please see the online helpfor control panel information).

Search parameters You can individually define search param-eters such as the heating and cooling factors (Th and Tc) describedin the previous section (see page 138). Alternatively, you can con-veniently adjust the default values of all the parameters necessaryto provide a coarse, medium, or fine search of phase space duringthe simulation using the Search Level popup.

The most frequently used options are available on the PolymorphMC Preference panel. Others are located on the directly accessiblePolymorph MC Parameters control panel (by pressing Prefer-ences...).

Effects of changing search parameters This section dis-cusses the effects and consequences of changing three of the mostsignificant search parameters:

♦ Number of trials to accept before cooling.

♦ Heating factor, Th.

♦ Cooling factor, Tc.

Heating phase The purpose of the heating phase is to find an acceptable startingtemperature for simulated annealing (that is, the cooling phase).Ideally, the cooling phase should start from a temperature atwhich all possible moves are accepted, that is, where phase spacecan be sampled completely randomly. In order not to waste cool-ing steps during which only random moves are made, you shouldmake this temperature as low as possible. It is also important thatthis temperature is determined with the minimum of effort. Ifcooling is started after a certain number of trials are accepted in arow then there is a good likelihood that the desired temperaturehas been reached somewhere early in the heating phase.

Scaling the temperature for each trial step may result in overshoot-ing the desired value. To attempt to avoid this you can reduce thenumber of trials to accept before cooling. However, this increasesthe risk of actually starting at too low a temperature; there is afinite probability of accepting any number of steps at any temper-ature and this probability increases as the required number ofaccepted steps is decreased. The default values are a reasonablecompromise.



Cooling phase The rate of cooling is the most important factor during cooling. Itis useful to think of a particular cooling rate as defining a level ofdetail for the search of phase space. A longer search at a particulartemperature increases the probability of finding the energy min-ima at a particular level of detail.

It is thus apparent that the cooling factor is the fundamentalparameter in determining the quality of the configurations gener-ated during the Monte Carlo search. Decreasing the cooling factorimproves the detail and increases the length of the simulation. Thedefault cooling factor is inversely proportional to the number ofmolecules in the asymmetric unit. You may want to also reduce itfor larger molecules. The default settings provide a good compro-mise between accuracy and length of search.

Space groups You can search up to 20 space groups during thesimulation. However, such a search is very time consuming andmay not yield significantly greater accuracy than a search of justthe few most significant space groups. This is illustrated by the fol-lowing table, which lists the 17 most common space groups (whichappear on the Space Group Selector popup) and the percentage oforganic crystals that occur in them.

Space group Crystal system

Number ofsymbolicoperators

Degreesoffreedom

Occurrence(%)

P 21/c Monoclinic 4 10 35.9P -1 Triclinic 2

41217

13.7

P 21 21 21 Orthorhombic 4 9 11.6P 21 Monoclinic 2 9 6.7C 2/c Monoclinic 8 10 6.6P b c a Orthorhombic 8 9 4.3P n m a Orthorhombic 8 9 1.9P n a 21 Orthorhombic 4 8 1.8P b c n Orthorhombic 8 9 1.2P 1 Triclinic 1

24

91527

1.0

C c Monoclinic 4 8 0.9

General Methodology


♦ All entries assume one rigid unit in the asymmetric unit except:

P -1 and P 1 — The second line refers to two rigid units

P 1 — The third line refers to four rigid units in the asym-metric unit

♦ The fourth column refers to the total number of degrees of free-dom; that is, translational and rotational. Note that rotationaldegrees of freedom are modified and translational degrees offreedom optimized during each Monte Carlo move.

♦ In general, more degrees of freedom require more intensive cal-culation, thereby increasing processing time.

You should define space groups using the standard nomenclaturein International Tables of Crystallography Volume A (1989) or Interna-tional Tables numbers.

You should use brief descriptions, with single spaces placedbetween the descriptor for the lattice type and each of the symme-try directions. You should represent subscripts, as used for screwaxes, by normal numbers (so that a screw dyad is represented by 21,a triad by 31 or 32, and so on). Represent inversion axes withminus signs (for example, -3, -4, -6). Some examples are:

P 21/cP 42/mP -4 21 c

You can define only the first unit cell option for each space group.Alternatively, you can supply International Tables numbers.

C 2 Monoclinic 4 9 0.9P c a 21 Orthorhombic 4 9 0.8P 21/m Monoclinic 4 10 0.8C 2/m Monoclinic 8 10 0.6P 21 21 2 Orthorhombic 4 9 0.6P 2/c Monoclinic 4 10 0.5

Space group Crystal system

Number ofsymbolicoperators

Degreesoffreedom

Occurrence(%)



Output options

Output options, such as trajectory file naming, progress plotting,model viewing, and text window output are set on the PolymorphMC Output control panel, which is accessed from the PolymorphMC Preference panel (by pressing Output...).

Please see the online help for detailed information on the controlpanels.

Cluster analysis

Clustering Monte Carlooutput

The Monte Carlo simulation outputs large numbers of unopti-mized structures for each space group. You can expect these toinclude low-energy structures and clusters of many very similarstructures that are obtained at the local minima of the energy sur-face of the crystal.

Of these structures, those with the lowest energy are the mostlikely to occur experimentally and are, therefore, of interest; youcan disregard the rest.

To obtain the low-energy representatives of each cluster, you per-form a cluster analysis procedure on trajectory output from MonteCarlo simulations.

Clustering energyminimization output

Initially-distinct low-energy structures produced by cluster analy-sis of Monte Carlo output can become very similar during energyminimization. You can remove these redundant structures by per-forming a final cluster analysis procedure on trajectory outputfrom energy minimization procedures.

Procedure

The Polymorph Predictor automated cluster analysis procedureconsists of the following steps:

1. The source trajectory file is traversed and the unclustered struc-ture with the lowest energy is identified and written to the out-put trajectory file.

2. All other unclustered structures are compared to thelow-energy cluster representative (obtained in step 1) by meansof a clustering algorithm. The algorithm makes its determina-

General Methodology


tion by examining the partial radial distribution functionsbetween pairs of forcefield atom types of the two structuresbeing compared.

3. Structures deemed to be similar to the low-energy referencestructure are marked as belonging to the current cluster.

4. Steps 1, 2, and 3 are repeated until there are no further struc-tures in the source trajectory file, or a user-specified number ofclusters has been identified.

When the procedure terminates, the output trajectory file con-tains only the low-energy representative of each cluster identi-fied.

Clustering algorithm

1. Groups atoms in the asymmetric unit cell by element, atomname or forcefield type for the low energy reference structure.

2. For each pair of forcefield types, calculates all squared inter-atomic distances between atoms of those types less than a spec-ified cutoff distance.

3. Calculates partial radial distribution function (.rdf file), identi-fies the peaks, and compares the peaks for two structures.

4. If radial distribution function between two structures is identi-cal to the low energy reference structure, removes duplicatestructure.

5. Repeats steps 1 through 4 for all the clusters.

Setting Cluster Analysis preferences

The parameters and options that affect the cluster analysis proce-dure are specified using options on (or accessible from) the Poly-morph Cluster Prefs control panel (see the online help).

Input file

By default, the trajectory file output by the last run prediction pro-cedure is selected as the input file for cluster analysis. If you do notwant to use this file, choose another using the file selector on thepanel.



Clustering parameters

You can specify the cutoff distance for interatomic relationships tobe examined during clustering such that only sensible interactionsare considered.

You can also specify the tolerance of the clustering algorithm todifferences between structures. The lower the tolerance, the moresimilar a structure must be to the reference structure to be deemeda member of the cluster. Low tolerances generally result in theidentification of a greater number of smaller clusters. You can limitthe number of clusters output to a sensible value; if the specifiednumber of clusters is obtained before all structures in the input tra-jectory file are clustered, the procedure is terminated. However, itis sensible to use a combination of parameters that place the largemajority of structures in one of the identified clusters.

Sensible tolerance and maximum cluster values to provide acoarse, medium, or fine detailed cluster analysis can be conve-niently obtained using the Search Level popup.

Clustering tolerances for Monte Carlo and minimized output

The clustering tolerances required for output from the MonteCarlo and energy minimization phases are significantly different.The clustering level required after the minimization phase needsto be much finer than that required after the Monte Carlo search.

Monte Carlo output The Monte Carlo search outputs a lot of raw structures that differsignificantly. Here you should set a tolerance that performs asmuch data reduction as possible while obtaining as many differentstarting structures for minimization as possible.

Minimized output Minimized structures are fully optimized; a small differencebetween these structures may represent a different polymorph. Atthis stage, you typically have tens of structures and can, therefore,afford to perform a more detailed search. It is strongly recom-mended that you use the defaults associated with Fine clusteringfor all post minimization clustering. You can identify any remain-ing truly similar structures by visual comparison and/or usingother Cerius2 instruments (for example, Diffraction).

Identifying sensible clus-tering values

You can use the Cluster Measure tool available on the PolymorphProperties control panel (see “Trajectory file analysis and modelextraction” on page 149) to investigate sensible clustering values

General Methodology


for an output file. To do this extract the first frame into a model andobtain measures against all other frames.

Output options

Output options, such as output file naming, progress plotting, andtext window output are set on the Polymorph Cluster Output con-trol panel, which is accessed from the Polymorph Cluster Prefscontrol panel (by pressing Output...).

Please see the online help for more information on the controlpanels.

Energy minimization

Cluster analysis outputs low energy cluster representatives thatare still energetically unrefined and require optimization withrespect to all degrees of freedom.

The energy minimization procedure does a full minimization ofeach structure output into a trajectory file by cluster analysis. Theminimization takes into account any rigid body constraintsdefined on the molecules forming the asymmetric unit.

Optimizing the currentmodel

The Polymorph Predictor energy minimization procedure can alsobe performed on individual models. The minimizer used in theprocedure is the same one used by other Cerius2 modules, and cantherefore be used with other structures in other modules.

Setting minimization preferences

You can specify the parameters and options that affect energy min-imization using options on (or accessible from) the PolymorphMinimize Prefs control panel (see the online help).

Input file By default, the trajectory file output by the last run prediction pro-cedure is selected as the input file for energy minimization. If youdo not want to use this file, choose another using the file selectoron the control panel.

Termination criteria Various termination criteria for the minimization in Cerius2 exist:a target root mean squared force or a maximum number of mini-mization steps.



Rigid Bodies If rigid bodies are defined on the molecules forming the asymmet-ric unit, the Minimizer normally recognizes these rigid bodies anddoes not change the relative position of atoms within a rigid body.It is often desirable to compare results with and without rigidbody minimization -- for this purpose, a switch Ignore rigid bod-ies is provided, allowing you to rerun a minimization on a trajec-tory file as if no rigid bodies had been defined.

Output options You can set output options, such as output file naming, progressplotting, model and text window output, on the Polymorph Mini-mize Output control panel (accessed from the Polymorph Mini-mize Prefs control panel by pressing Output...).

Please see the online help for more information about control pan-els.

Notes on rigid body minimization

♦ In many cases, rigid body minimization can save considerablecompute time, since the number of degrees of freedom is usu-ally significantly lower for rigid bodies compared to an all-atom minimization. Also, in cases where the intra-moleculardegrees of freedom are not well-described by the forcefield orthe conformation of the molecule(s) in the crystalline state isknown in advance, rigid body minimization can be useful toprevent the program from making unphysical changes to theconformation of the molecules. On the other hand, rigid bodiesshould be used with considerable caution -- sometimes, even smallchanges in individual atom positions can have a significanteffect on the energies, and by prohibiting such small changes,errors may be introduced.

♦ Other than completely ignoring all rigid bodies, it is currentlynot possible to modify the rigid body definition on an existingtrajectory file - any rigid bodies must be defined before the startof the Polymorph prediction sequence, and if you wish to mod-ify the rigid body definition, you need to rerun the whole Poly-morph sequence.

♦ For efficiency, the Open Force Field disregards any interactionswithin rigid bodies (see the Forcefield Based Simulations manual).This means that you have to exercise care when comparingenergies obtained with and without rigid bodies. The differ-

General Methodology


ence between the energy of two essentially identical models,one with and one without rigid bodies defined on them, will bea constant for a given fixed molecular geometry. This constantenergy is easily determined. First, you calculate the energy ofthe crystal with rigid bodies defined using the options pro-vided on the OFF SETUP/OPEN FORCE FIELD/EnergyExpression/Setup card. Then you delete the rigid bodies andrepeat the energy calculation. The difference between the twocalculations is the constant energy of the intra-rigid-body inter-actions; it should not change even if you vary the lattice param-eters, the position of the molecules in the unit cell or thepositions of the rigid bodies relative to one another.

Trajectory file analysis and model extraction

Polymorph Predictor generates and outputs each structure fromthe Monte Carlo, cluster analysis, and energy minimization proce-dures as a frame in a trajectory file. If you want to study a structurein this form using the full range of Cerius2 visualization tools andcomputational instruments, you must extract it from the trajectoryfile into a model space.

Polymorph Predictor analysis tools allow you to study the proper-ties of the contents of a trajectory file, identify the most interestingstructures, then extract them into model spaces. Three steps areinvolved, the controls for each contained on a separate controlpanel available from the Analysis pullright on the POLYMORPHcard.

Select trajectory file By default, the trajectory file output by the last-run prediction pro-cedure is selected as the input file for analysis and extraction. Ifyou do not want to analyze this file, choose another using the fileselector on the Polymorph Analysis File control panel (see theonline help).

Properties Tools on the Polymorph Properties control panel (see the onlinehelp) allow you to plot profiles and distribution histograms ofstructures in the trajectory file (by frame number) against a num-ber of their properties:

♦ Total energy

♦ van der Waals energy



♦ Coulomb energy

♦ H-bond energy

♦ Temperature

♦ Density

♦ Cell angles

♦ Cell lengths

If you have carried out an automatic comparison of powder spec-tra (see below) for the selected trajectory, the measures of compar-ison (R_P, R_WP, R-INDEX and CMACS) are listed as additionalproperties.

You can also search for structures in the file with the lowest orhighest values (as appropriate) for these properties.

You can compare an existing model to the currently-selected anal-ysis trajectory file using the clustering algorithm. Structures aresorted by similarity to the reference model, and their frame num-bers listed in the text window. A small Cluster Measure Prefer-ences panel allows you to set the cluster parameters used for thiscomparison. Note that changing the settings specified on thispanel does not affect the clustering parameters used during aPolymorph Prediction run. These can only be set on the Poly-morph Cluster Prefs panel, which is available from the main Poly-morph Run panel (please see “Setting Cluster Analysispreferences” on page 145 for more information).

Model extraction Having identified a structure for further study, you can extract itinto the current model space using the controls on the PolymorphExtract control panel (see the online help). You specify the struc-ture to extract using its frame number; information on the selectedstructure can be listed in the text window prior to extraction. Youcan also write frames represented by points selected on a graph toa new trajectory file.

Please see the online help for more information on the control pan-els.

General Methodology


Automatically comparing powder spectra

The Polymorph Predictor identifies a limited number of likelylow-energy crystal structures for a given compound. To determinewhich of these candidate structures is/are experimentallyobserved, a common technique is to compare simulated powderpatterns with experimentally observed powder patterns.

Cerius2 incorporates functionality automating this time-consum-ing step. With the help of the Diffraction-Crystal module, the pro-gram automatically generates simulated powder patterns for eachframe of a polymorph trajectory and computes figures of meritindicating the “closeness” between simulated and experimentalpattern for each frame. These figures of merit can then be analyzedwith the usual tools provided on the Polymorph Analysis/Proper-ties control panel.

Automatically comparing powder patterns is done using the Pow-der Comparison control panel accessible from the PolymorphAnalysis pullright. For a given trajectory file, you can carry out anumber of different comparisons - for example, you may wish tocompare the frames in the trajectory to more than one experimen-tal pattern, or you may wish to change the powder diffraction set-tings used to generate the simulated spectra which are thencompared to the experimental spectrum. You can store each ofthese comparisons with the trajectory file, giving them ResultsIdentifiers.

To set up an automatic powder comparison analysis, you shouldbe familiar with the Diffraction-Crystal module (see the Cerius2

3.8 Analytical Instruments manual), especially with the generationof simulated powder patterns.

Select trajectory file By default, the trajectory file output by the last-run prediction pro-cedure is selected as the input file for the powder comparison. Ifyou do not want to analyze this file, choose another using the fileselector on the Polymorph Analysis File control panel.

Select experimental spec-trum

Next, you need to specify with which experimental you wish tocompare the frames in the trajectory. The Load Powder ... buttonon the Powder Comparison control panel opens the 1-D Experi-mental Data control panel (also accessible from the DIFFRAC-TION-CRYSTAL card). A large variety of different file formats is



supported, the only restriction being that the abscissae of theexperimental spectrum need to be equally spaced.

Transform experimentalspectrum

It is often necessary to subtract the background from the experi-mental spectrum and/or to artificially broaden it, in order to allowa meaningful comparison between the simulated and experimen-tal spectrum. Broadening and background subtraction are per-formed using the Prepare Powder button on the PowderComparison control panel, which opens the 1-D Data Transforma-tions control panel.

You can fit and subtract a background polynomial using the Sub-tract polynomial button. The order of the background polynomialcan be specified using the input field below - the default is 5. Ahigher order usually guarantees a more detailed fit. In most cases,the algorithm employed to subtract the background works satis-factorily -- in a few cases, you may need to try a number of itera-tions with varying order of the background polynomial beforeachieving a satisfactory result.

For the RP, RWP, and R-Index measures, it is often advantageous tobroaden the experimental spectrum before comparison (see thetheory below). The options on the 1-D Data Transformations panelallow you to uniformly broaden the experimental spectrum with aGaussian or Lorentzian. You can also specify the width of thebroadening function.

Specify Diffraction-Crystalsettings

To perform a meaningful comparison, you must ensure that thesettings used to generate the simulated spectra mimic the experi-mental conditions as closely as possible. In particular, the displayrange, the radiation wavelength and instrumental or other peakbroadening factors may need to be adjusted. Usually, the chosendisplay range should be similar to the range on which the experi-mental data is defined.

You adjust all these parameters using the Calculate Crystal Dif-fraction control panel accessible via the Diffraction Prefs ... but-ton, and the Preferences control panels accessible from there (seethe Cerius2 3.8 Analytical Instruments manual). If you wish to testyour settings before carrying out the comparison for the full trajec-tory, you can first extract a frame from the trajectory into a modeland then use CALCULATE diffraction pattern on the CalculateCrystal Diffraction control panel to carry out a simulation (see theCerius2 3.8 Analytical Instruments manual).

General Methodology


Set CMACS intervals The CMACS measure is calculated individually on a number ofequally spaced intervals over the display range, and the results areaveraged over these intervals. The default for the number of inter-vals is 10. In cases where the spectrum contains a very large num-ber of peaks, you may wish to increase this value. You can changethis value using the CMACS Intervals input field on the PowderComparison control panel.

Specify an identifier forthe comparison

Since you may wish to carry out more than one comparison for agiven trajectory file, you need to specify a name for each compari-son. Simply enter the desired name in the Results Identifier field,or use the default name suggested.

Calculate measures ofcomparison

Use the Compare radio button to compare the current trajectorywith the current experimental powder pattern. All four measuresof comparison are evaluated (RP, RWP, R-Index, and CMACS). Thecalculation may take a few minutes, depending on the size of thetrajectory. You can monitor the progress on the status bar whichappears after you start the calculation.

Note

Analyze measures ofcomparison

Once a powder comparison has been done, you can use the stan-dard tools on the Polymorph Analysis control panel to analyze theresults. If you have carried out more than one comparison for thetrajectory file, you must choose which results to analyze, by usingthe pulldown menu in the lower half of the Powder Comparisoncontrol panel. All the analysis tools extract information from thecalculation specified in this field. Using the Property pulldown onthe Polymorph Analysis control panel, you can now choose whichof the four measures of comparison you wish to analyze. You canplot the measures as a function of frame number, find the frameswith the best match (using the Search for Lowest ... Framesoption), and create a new trajectory containing only frames whose

The calculated measures of comparison are not written to thetrajectory file itself, but to a separate file with the same basename as the trajectory file. It can be recognized by the suffix .rfc.If you copy/move trajectory files using UNIX commands andyou want to keep the results of your powder comparison, besure to also copy/move the .rfc files to the same directory as the.pmp files - otherwise, Polymorph will not know about theirexistence.



powder pattern is close to the experimental pattern, indicated bysmall values for the comparison measures.

Show settings If you do not remember the Diffraction Crystal settings and/or theexperimental powder pattern to which a particular comparisoncorresponds, use the Show settings button on the Powder Com-parison control panel. It prints, among other things, a number ofDIFF-CRYSTAL commands which, if executed as a script, willreset the DIFFRACTION-CRYSTAL settings to the values used forthe powder comparison identified by Comparison results to ana-lyze.

Delete Set You can delete the comparison identified by Comparison resultsto analyze using the Delete option on the Powder Comparisoncontrol panel.

To automatically com-pare powder spectra

1. Select a trajectory file using the Polymorph Analysis File con-trol panel.

2. Load an experimental spectrum using the Load Powder ...option.

3. If desired, subtract a background and/or broaden the experi-mental spectrum using the Prepare Powder ... options.

4. Set up appropriate parameters for simulating powder spectrausing the Diffraction Prefs ... options.

5. If required, modify the number of intervals for the CMACSmeasure on the Powder Comparison control panel.

6. Type in an identifier for the calculation in the Results Identi-fier field.

7. Perform the comparison by clicking the Compare using ... but-ton.

8. To extract information about a particular calculation, toggle theComparison results to analyze field to the results identifier forthat calculation.

9. Select the comparison measure you wish to print/visualize/analyze using the Property pulldown on the Polymorph Anal-ysis card.

10. Print/plot/analyze the comparison measures using the stan-dard tools on the Polymorph Analysis control panel.

General Methodology


11. Print information about the calculation identified by Compar-ison results to analyze using the Show settings option.

12. Delete a calculation using the Delete option.

Note

Merging trajectory files

Having clustered, minimized, and re-clustered the output fromthe Monte Carlo search for a number of space groups, you are leftwith several trajectory files (one per space group searched) con-taining structures that are likely to be found experimentally.

Although the structures in each file are unique within that file, it ispossible that structures in the output files derived from MonteCarlo searches of different space groups are, in fact, identical. Toremove such redundant structures and provide a single file forfinal analysis and model extraction, you can merge the final trajec-tory files (using the controls on the Polymorph File Merge controlpanel) and perform a final cluster analysis.

It is important to point out the limitations of this feature. Inmany trials, we have found that if one of the frames in the tra-jectory is reasonably close to an experimental spectrum, theprogram is usually able to identify this frame as one which hasthe lowest or at least one of the lowest CMACS measures. Thereverse is not true -- a small CMACS measure does not guaran-tee that the frame really corresponds to the experimental spec-trum. Additional analysis is required to establish this -- forexample, you can extract a promising frame and try to useRietveld refinement (see the Cerius2 3.8 Analytical Instrumentsmanual) to test the fit between theory and experiment.

If you find a frame which is among the lowest in energy withinthe trajectory and for which the comparison measures consis-tently indicate it is close to the experimental spectrum, there isa good chance that you have identified the correct structure. Butadditional tests (visually inspecting the spectra, Rietveld refine-ment, etc.) always need to be carried out to verify the result.



Note


Reliability checking

It is recommended that you confirm the reliability of the searchprocedures by performing one or more checks on the validity ofthe final output structures. This section describes a few such tests.

Examine structures/properties

You may be able to quickly determine the plausibility of a poly-morph by examining its structure and basic properties to deter-mine whether their values seem reasonable. Properties to examineinclude:

♦ Density

♦ Close contact distances

♦ H-bonding

♦ Interactions between conjugated systems

Comparison againstexperimental data

Where experimental data is available, comparison against proper-ties calculated for a predicted structure using Cerius2 computa-tional instruments can be very effective.

For example, take a case in which an experimentally obtained x-ray powder pattern is available. Using the Powder Comparisonfeature described above, you can compare simulated powderspectra from the predicted structures to the experimental spec-trum and identify structures within the trajectory whose powderspectrum is similar to the experimental spectrum. Powder Com-parison uses the capabilities of the Cerius2 Diffraction-Crystalmodule. Good agreement between peak positions and intensitiesindicates that the predicted structure is accurate. You can thenextract likely candidate structures and use Cerius2 Rietveld toadjust the predicted structure to give a good match between theexperimental and predicted x-ray powder curves.

All the merged trajectory files should contain output derivedfrom Monte Carlo searches of the same asymmetric unit. Inparticular, the number of rigid units in the asymmetric unitinitially searched must be the same.

Theory


Symmetry If a structure is predicted in a particular symmetry then it shouldalso be predicted in a lower symmetry provided that the lowersymmetry has sufficient degrees of freedom (that is, permittedtranslations, cell angles, cell lengths, and types of symmetry oper-ators) to reproduce the higher symmetry.

For example, for a search in P 21 with one molecule in the asym-metric unit (Z=1) there are two molecules in the unit cell. P 1 sym-metry has sufficient degrees of freedom to reproduce a P 21structure if there are two molecules in the asymmetric unit (Z=2),which in this case will also be the number of molecules in the unitcell. Therefore, a P 1 (Z=2) prediction should find any structuresfound in a P 21 (Z=1) prediction. However, the P 1 (Z=2) predictionmay find many better (that is, lower energy) structures.

Remember that the efficiency of the predictor is expected todegrade with increasing numbers of molecules in the asymmetricunit. You should not expect a P 1 (Z=16) prediction to reproduce aP 4/m m m (Z=1) structure.

Note

Change the FF dielectricparameters

Current forcefield technology neglects polarization effects. If thestability ranking of predicted structures is not affected by chang-ing the values of the dielectric constant epsilon (used in electro-static calculations), then the prediction can be assumed reliable. Tofind out more about the dielectric parameters and recalculating theenergies of your structures, see Cerius2 Simulation Tools.

Theory

Predicting polymorphs

Molecules tend to exhibit polymorphism, that is, they crystallize inmany modifications, as a result of the physical conditions and themanner in which the crystals are obtained. Different polymorphspossess different values of the Gibbs free energy (G). However, it

In certain cases, for example, if the molecule is not chiral, P 1(Z=2) should reproduce P -1 (Z=1). However, if the molecule ischiral then P 1 (Z=2) should not reproduce P -1 (Z=1).



does not necessarily follow that a crystallization process yields themodification with the lowest value of G. In fact, the relationshipbetween crystallization conditions and the polymorph obtained isnot generally understood.

However, with certain limitations, Polymorph Predictor can gen-erate crystal structures highly likely to be found experimentally byusing sophisticated simulation algorithms. To find out more aboutPolymorph Predictor’s limitations, see “Polymorph Predictor lim-itations” on page 132.

Theory

As stated earlier, experiments are likely to find crystal structureswith a fairly low value of G. Thermodynamics states that Gdecomposes into with energy E, pressure p, vol-ume V, temperature T, and entropy S.

The product pV is easily calculated, but can be disregarded as itsvalue is too small to have any noticeable influence on G at mean-ingful pressures.

Effect of disregarding TS Different crystal structures have different entropies at finite tem-peratures. TS therefore has a significant influence on G. However,because it is difficult to compute the entropy of a crystal, TS is notconsidered. Neglecting this factor changes the calculated energyslightly from the true value of G at finite temperatures. However,these effects are only expected to reorder the relative stabilities ofthe different low energy structures predicted at zero kelvin — thelowest energy structures Polymorph Predictor obtains should rep-resent those found experimentally, but probably with different rel-ative stabilities.

This leaves the energy, E, as the quantity that must be optimized.For an arrangement of molecules, and thus also of a crystal as aspecial case of a regular arrangement, E is efficiently computedusing a forcefield. Cerius2 provides advanced forcefield technol-ogy through the Open Force Field (OFF) and Force Field Editor(see the Cerius2 Simulation Tools).

The structure of a molecular crystal is efficiently mathematicallydefined by specifying the content of the unit cell and the values ofthe unit cell parameters; that is, the lengths of the unit cell vectors(a, b, and c) and the angles between them (α, β, and γ). The content

G E pV TS–+=

Theory


of the unit cell is specified by the coordinates of the asymmetricunit and by the space group.

Therefore, to find the most probable crystal structures, it may atfirst appear that all possible space groups (230 in total) must besearched with different numbers of molecules in the asymmetricunit, a very time-consuming process. However, the search can bedramatically reduced in scope because of the following factors:

♦ Most organic crystals crystallize in only a few space groups. Forexample, 88.6 percent of the organic molecular crystal struc-tures published between 1929 and 1975 belonged to one of thefollowing nine space groups: P21/c, P212121, P1, P21, C2/c,Pbca, Pna21, Pnma, Pbcn (Belsky and Zorkii 1977).

♦ A connection exists between the symmetry of the moleculesand the possible space groups of a crystal (Kitaigorodsky 1961).

Consequently, one can obtain highly probable molecular crystalstructures by determining the most stable structures in just a fewspace groups and comparing the results in energy and structure. Itmay be possible to find a particular structure in a space groupwithin the results of a sub group.

Reliability checks In global optimization problems there is usually no guarantee thatthe global minimum has been located. In crystal packing prob-lems, these difficulties are worsened since the complete lower partof the energy spectrum must be calculated. Polymorph Predictorprovides many adjustable parameters that allow you to fine tunethe packing simulation and increase search reliability. However,such increased reliability comes at the cost of dramaticallyincreased processing demands, so it is not generally practical toperform the most detailed search possible.

It is advisable, therefore, to confirm the reliability of the searchprocedures by performing one or more checks on the validity ofthe output structures. If the results appear accurate, you know thatyour search accuracy was adequate; if not, you should adjust thesearch parameters accordingly. For some recommended reliabilitychecks, see “Reliability checking” on page 156.

Further reading Please see the following references for a more detailed discussionof the theory and background of the prediction procedures imple-mented in Polymorph Predictor (Gdanitz 1992, Karfunkel and



Gdanitz 1992, Karfunkel and Leusen 1992, Karfunkel et al 1993,Gdanitz et al 1993, Karfunkel et al 1994).

Simulated annealing theory

Polymorph Predictor’s Monte Carlo packing algorithm performsa simulated annealing procedure intended to search for the lowestminima of the energy function E of molecular crystals. Such asearch is too difficult for classical algorithms for numerous rea-sons:

Packing simulation diffi-culties

♦ E is not bounded from above, since it can take arbitrarily highvalues when molecules interpenetrate. This is the case whenlattice vectors become too small.

♦ Local minima of E, which correspond to dense crystals, are inthe direct vicinities of the singularities described above. This isbecause large pressures are required to further compress adensely packed crystal.

♦ If a lattice vector a becomes long enough to prevent contactbetween corresponding layers in the crystal, the slope of E withrespect to change in the length of the lattice vector, dE/da,quickly tends to zero. It is impossible to compute a step to reachreasonable regions of E from these positions.

♦ E possesses a huge number of local minima. When shifting lay-ers of a molecular crystal relative to one another, it is easy tofind stable configurations.

♦ Only a limited number of interactions between atoms can beconsidered in energy calculations. A cutoff distance is used tolimit the number of these interactions. Due to this cutoff of thecrystal fragment, E oscillates very fast with a small but numer-ically noticeable amplitude. This makes it virtually impossibleto obtain the same crystal structure when starting an optimiza-tion at different regions.

Solutions provided by sim-ulated annealing

The simulated annealing method works around these difficultiesby treating the search for the global minimum of E as a thermody-namic problem. At some non-zero temperature, T, the crystalchanges its structure randomly, and its energy fluctuates accord-ingly. To prevent the simulation from becoming trapped in a localminimum, cooling begins at a relatively high temperature; every

Theory


crystal structure (within the constraints of the space group andasymmetric unit contents) can then theoretically be reached, andergodicity therefore ensured.

As the temperature is lowered, the algorithm explores the poten-tial energy hypersurface of the crystal with greater and greaterprecision. At lower temperatures, movement is restricted toregions where the value of E is relatively low. The simulation endswhen T is so low that the crystal is frozen and the global minimumof E is hopefully found. The optimization process discovers otherlow-lying minima of E, and records the corresponding stablestructures.

Automatically comparing powder spectrad

The automatic comparison of powder patterns relies on the avail-ability of a computational measure to indicate “closeness”between powder spectra. Each powder pattern is characterized bya set of intensity values at N observation points k = 1, ..., N, corre-sponding to a set of diffraction angles. Four different figures ofmerit have been implemented to enable you to judge how “close”the simulated powder pattern from a given trajectory frame is tothe experimental pattern. In each case, a small value indicates aclose match and a large value indicates that the powder patternsare very different. The four measures of comparison are:

Eq. 4

Eq. 5

RP

Yk exp( ) Yk calc( )–

k∑

Ik

k∑

----------------------------------------------------------------------=

RWP

wk Yk exp( ) Yk calc( )–2

k∑

wk Ik( ) 2

k∑

--------------------------------------------------------------------------------

12---

=



Eq. 6

Eq. 7

The weights wk at the observation points k are defined via wk = 1/Ik , where Ik = Yk(exp) + Yk(background) is the experimentallyobserved intensity. Ik must be positive everywhere and Yk(back-ground) should be previously calculated and subtracted from theexperimental spectrum using the Load Powder option. The firstthree of these measures are essentially Rietveld-like measures. Ifno background is available, wk = 1 is assumed for RWP and R-INDEX. Yk(calc) is the calculated intensity, rescaled such that

Eq. 8

is minimized.

The CMACS measure has been developed at MSI and has per-formed well in preliminary tests. In contrast to the Rietveld mea-sures, it does not rely on a significant overlap between peaks in thesimulated and experimental spectrum, which for very sharplypeaked spectra exists only if the simulated structure is very closeto the experimental structure. The spectra are not compareddirectly, but instead they are integrated first, and these integratedfunctions are then compared. The comparison is carried out sepa-rately on each of a user-specified number of intervals over the dis-play range, and the values obtained are averaged over theseintervals.

Other measures quoted in the literature, such as the FOLD algo-rithm (Karfunkel et al 1993) and the overlap integral (Lawton and

R INDEX–RWP

R ectedexp-----------------------, where=

R ectedexp

wkIk

k∑

wk Ik( ) 2

k∑-------------------------------

12---

=

CMACS Continuous Measure for the Automatic Comparison of Spectra=

M Yk exp( ) Yk calc( )–[ ] 2for Rp( ) or

k∑=

M wk Yk exp( ) Yk calc( )–[ ] 2for RWP( )

k∑=

References


Bartell 1994), are all based on artificially broadening the experi-mental spectrum before carrying out a least squares comparison.Since RWP and R-INDEX are essentially least-squares measures,the FOLD algorithm can be shown to be essentially equivalent tosimply broadening the experimental powder pattern before apply-ing the RWP or R-INDEX measure

References

Belsky, V.K. and Zorkii, P.M., Acta. Cryst. A33 1004 (1977).

Gdanitz, R.J., Chem. Phys. Letters, 190, 391 (1992)

Gdanitz, R.J., Karfunkel, H.R., and Leusen, F.J.J., J. Mol. Graphics,11, 275 (1993)

Karfunkel, H.R. and Gdanitz, R.J., J. Comput. Chem., 13, 1171 (1992)

Karfunkel, H.R. and Leusen, F.J.J., Speedup J., 6/2, 43 (1992)

Karfunkel, H.R., Rohde, B., Leusen, F.J.J., Gdanitz, R.J., and Rihs,G. J. Comput. Chem., 14, 1125 (1993)

Karfunkel, H.R., Leusen, F.J.J., and Gdanitz, R.J., J. Comput.-AidedMat. Design, 1, 177 (1994).

Kitaigorodsky, A.I., Organic Chemical Crystallography, Consult-ants Bureau, New York (1961).

Lawton, S.L., and Bartell, L.S., Powder Diffraction 9, 124 (1994)




8 Morphology

Introduction

The Morphology module predicts the external morphology ofcrystalline materials from the internal crystal structure. Morphol-ogy provides four methods:

1. The Bravais Friedel Donnay Harker (BFDH) method.

2. The Attachment Energy (AE) method for growth morpholo-gies.

3. The Surface Energy (SE) method for equilibrium morphologies.

4. The Hartman-Perdok (HP) method.

The BFDH method is an approximation based on geometrical con-siderations. The AE method relies on a calculation of the energyreleased when a growth slice is added to a growing plane. Bothpredict the relative growth rates of possible growth faces. In con-trast, the SE method predicts the morphology which minimizesthe total surface energy of the crystal, by determining the surfaceenergy of the relevant faces. Finally, the HP method uses a power-ful algorithm for identifying stable growth planes by generatingconnected chains and nets of strong bonds in the crystal. Attach-ment Energies of these growth planes are then calculated and usedto predict the growth morphology.

For all the above methods, you can also calculate crystal attributessuch as inter planar angles, aspect ratio, surface areas, and volume.

Sections in this chapter Using Morphology

Calculating Morphology with the BFDH method

Calculating morphology with the Attachment or Surface Energymethods


8. Morphology

Setting up the energy calculations

Slice positioning

Editing, adding and removing crystal faces

Calculating morphology using the Hartman-Perdok Method

Generating and editing the Crystal Graph

Generating and editing Periodic Bond Chains and Connected Nets

Displaying the morphology

Analyzing the morphology

Storing morphologies

Theory

References

Using Morphology

Calculating morphologies using the Morphology module isstraightforward and largely automated. The accuracy of the pre-diction depends not only on the system under investigation butalso on the setup of force field parameters. Moreover, you shouldbear in mind that the current methods employed by Morphologydo not take into account factors such as the presence or absence ofsolvents or excipients and possible surface reconstructions, whichcan have a profound influence on experimentally observed mor-phologies.

Step 1: Setup Before you start, you must ensure that the current model spacecontains a crystalline model, loaded from a file or constructedusing the Crystal Builder. For all calculations except BFDH (step2), you will also need to load and configure an appropriate Cerius2force field, unless you wish to use the provided default force field.

Step 2: Calculations You can make predictions using a number of different methods.

The BFDH, Attachment Energy, and Hartman-Perdok methodspredict the relative growth rates of possible growth faces, fromwhich you can deduce the growth morphology.

Using Morphology


The Surface Energy method predicts the equilibrium morphologybased on relative surface energies of possible faces.

You can carry out the calculations using the options on the Calcu-late Morphology control panel and on subpanels accessible fromthere (see Calculating Morphology with the BFDH method, Calculatingmorphology with the Attachment or Surface Energy methods, and Cal-culating morphology using the Hartman-Perdok Method). The theorybehind the calculations is outlined in the Theory section.

Step 3: Visualization The calculated morphology is displayed in the model window.Visualization options allow you to display the external morphol-ogy with or without the atomistic structure, to control the trans-parency of the crystal faces and to label the faces with Millerindices or surface area. You can set these variables using theoptions on the Morphology Display control panel (see Displayingthe morphology).

Step 4: Listing, editing,adding and removinggrowth faces

A conventional way to describe crystal morphology is to list theMiller indices of all present faces and their perpendicular distancefrom the center of the crystal; that is, the center-to-face distance.After a morphology calculation, such a list is available on the Edit/Add Faces control panel. This control panel also allows you to editthe properties of the growth faces, add new faces or removeundesired faces (see Editing, adding and removing crystal faces).

Step 5: Crystal attributes You can compute several properties of the calculated morphologyusing the options on the Morphology Analysis control panel.These include interplanar angles, aspect ratio, and the percentagesurface area accounted for by certain forms (see Analyzing the mor-phology).

Step 6: Saving themorphology

You can save the morphology of the crystal structures to a file andaccess it later (see Storing morphologies).

If you use the Hartman-Perdok method, further analysis capabili-ties for the bonds and connected nets within the crystal are pro-vided (see Generating and editing the Crystal Graph and Generatingand editing Periodic Bond Chains and Connected Nets).


8. Morphology

General Methodology

Applications

Morphology allows researchers both to study particle shape and toconsider the effects of altering the growth rate of particular faceson crystal morphology. In particular, Morphology can look at theeffect of tailor-made additives in modifying growth. Knowledge ofshape and aspect ratio is essential to understanding packing, flowproblems, clogging of filters, and other problems. The Morphol-ogy module can also provide insight into other properties, such asthe texture of powders and polymorphism. In addition, the Hart-man-Perdok module enables you to understand the role of bondednetworks in stabilizing particular faces and to study the effect ofaltering interactions between molecules ("bonds") on the morphol-ogy. Morphology's application areas include pharmaceuticals,agrochemicals, food sciences, petrochemicals, cements, and com-modity and specialty chemicals.

Other Cerius2 modules Morphology is an ideal complement to other Cerius2 modules:

♦ Polymorph, Crystal Packer, and Diffraction-Crystal aid indetermining crystal structures. For more information aboutthese modules, see the corresponding Cerius2 online documen-tation.

♦ Sorption, Surface Builder, Minimizer, and Dynamics Simula-tion study surface reactivity or catalytic behavior of surfacespredicted by Morphology. For more information about thesemodules, see the corresponding Cerius2 online documentation.

Calculating Morphology with the BFDH method

Morphology calculations use the options on the Calculate Mor-phology control panel. Once the crystal structure is in the currentmodel space, you can predict the morphology by simply clickingthe Calculate BFDH Morphology button. The prediction is basedon the crystal cell and any symmetry operators present. It is there-fore important that you build the crystal model with the correctsymmetry.

General Methodology


Growth planes andgrowth rate list

The prediction generates a list of possible growth planes that sat-isfy the Donnay Harker rules for the current symmetry. The calcu-lation loops through all values of h, k, and l choosing faces. Anyface forbidden by the symmetry (Donnay Harker) has its indicesincreased as necessary (for example, 2 0 0 may replace 1 0 0). Next,a center-to-face distance is assigned according to Bravais Friedel asbeing proportional to the reciprocal of the lattice spacing. For the-oretical details, see Bravais Friedel Donnay Harker method.

The list of generated planes is displayed in the Edit/Add Facescontrol panel. The listed center-to-face distances indicate the rela-tive growth rates of the planes. The actual values assigned are arbi-trary and are in fact equal to 100/d, where d is the inter planespacing or slice thickness.

Minimum slice thickness The 1000 planes with the smallest center-to-face distances arelisted; these planes must all have a d value greater than the mini-mum slice thickness. This value controls the number of faces con-sidered. If the number of planes exceeds 1000, a larger value for theminimum slice thickness is automatically used.

Displaying themorphology

From this list of relative growth rates, Cerius2 deduces the crystalmorphology and displays it in the model window. By default, themorphology is shown with the molecular model, and the planesmaking up the displayed morphology are transparent. You canchange this and other visualization variables using the options onthe Morphology Display control panel. You should consider theshape you obtain to be a good first approximation to the morphol-ogy of any crystalline system and an ideal starting point for attach-ment energy calculations in molecular crystals.

To calculate morphology with the BFDH method

1. Place the crystal structure in the currently active model space.You can either load the crystal structure from a file or build itusing the Crystal Builder.

2. Go to the MORPHOLOGY card and select the Calculate itemto bring up the Calculate Morphology control panel.

3. Click the Calculate button in the Bravais Friedel DonnnayHarker subpanel.


8. Morphology

Calculating morphology with the Attachment or SurfaceEnergy methods

As with the BFDH method, you do morphology calculations usingthe options on the Calculate Morphology control panel (see theonline help). Once you set up the energy calculation, you can pre-dict the morphology of the current structure by simply clickingeither the Calculate button in the Growth Morphology subpanel(for attachment energy calculations) or the Equilibrium Morphol-ogy subpanel (for surface energy calculations).

Correct molecule It is important that you build the crystal with correct bonding,allowing the calculation to deduce the nature of each isolated mol-ecule present in the crystal.

Listing the growth faces The attachment or surface energy is calculated for all crystal faceslisted in the Edit/Add Faces control panel. This list could comefrom the BFDH calculation, which is frequently used as a screen-ing method to identify possible growth planes. The Do BFDHFirst option ensures that the BFDH method automatically gener-ates the list before running the calculation.

Alternatively, you can either enter your own list of growth planesor carry out the BFDH calculation separately, then edit the list. Forentering and removing faces and for compressing the list, optionsare provided on the Edit/Add Faces control panel (see Editing,adding and removing crystal faces). The attachment or surface energycalculations take much more computer time than the BFDHmethod, making it useful to limit the number of planes being con-sidered.

Calculating attachmentenergies

The morphology prediction using the Attachment Energy methodis controlled within the Growth Morphology subpanel. For a givenslice the program evaluates the attachment energy using Eq. 11.Two different algorithms for the attachment energy calculation areavailable.

1. The default mode of calculation (Full OFF Support) applies amethod which supports all features of the OFF module, includ-ing accurate Ewald summation techniques for the evaluation oflong range Coulomb interactions. The algorithm builds a tem-porary two-dimensional model for each growth slice and per-

General Methodology


forms a full force-field energy calculation on each of theseslices.

2. The second mode of calculation (Bond-Energy-List) computesthe lattice energy and the energy of the growth slice by apply-ing a force field to directly sum the interactions of pairs of mol-ecules having centers that lie within the specified interactionradius. Since these interactions are finite in range, they are pre-calculated and tabulated for each pair of molecules, making thecalculation of slice energies faster. However, this method is lessaccurate than a calculation using Ewald summation (see thefirst Note below).

Slice positioning The positioning of growth slices affects the calculated energies. Ifyou select Full OFF Support for the evaluation of the attachmentenergies (and thus use the default method above for attachmentenergy calculations) or if you calculate surface energies, all theo-retically possible slice positions leading to non-identical growthslices are found automatically. If you select the Bond-Energy-Listmode for the attachment energy calculation, an additional Prefer-ences ... button appears which allows you to set variables affectingthe Slice positioning.

Calculating surface ener-gies

The morphology prediction using the Surface Energy method iscontrolled within the Equilibrium Morphology subpanel. The sur-face energy of a crystal is calculated by simulating the semi-infi-nite crystal by a slab of finite thickness. The substrate thickness isa user-defined parameter. By increasing the substrate thickness,the accuracy of surface energies is systematically improved, at theexpense of larger computation times. The calculation is based onthe algorithm used for the attachment energy calculation and isalways performed by creating a temporary two-dimensionalmodel for each surface. (please see The Equilibrium Morphology formore information on surface energy calculation and see Calculat-ing attachment energies for more information about the attachmentenergy algorithm)

Deducing the morphol-ogy

Each growth face in the list is assigned a center-to-face distancethat is proportional to the attachment or surface energy calculated.The Edit/Add Faces list is updated with the new center-to-face dis-tances, and the calculated morphology is displayed in the modelwindow.


8. Morphology

Energy setup To calculate energies, you must load a force field and assign theforce field atom types to the model. Although the defaults do thisautomatically, it is often valuable to do this yourself. If you selectFull OFF Support for the evaluation of the attachment energies, orif you calculate surface energies, the energy setup, including thesettings of all variables affecting the energy expressions, must bedone using the OFF SETUP card deck. In particular, you specifythe method and parameters for evaluating the Coulomb and vander Waals interactions using the Energy Terms menu of the OPENFORCE FIELD card. On the other hand, if you use the Bond-Energy-List option for the calculation of attachment energies, anadditional Preferences... button appears which brings up a menuallowing you to specify the variables affecting the energy evalua-tion.

Note

Note

To calculate morphology with the Attachment or SurfaceEnergy methods

1. Place the crystal structure in the currently active model space.You can either load the crystal structure from a file or build itusing the Crystal Builder.

It is essential that you use EWALD summations (method 2) ifindividual molecules have a net charge, such as in ionic sys-tems, or for a morphology consisting of planes having a dipolemoment. In such cases, without Ewald summations, the calcu-lation does not converge properly, because the interactionradius will often contain a net charge different from zero, lead-ing to arbitrary energy oscillations as a function of interactionradius. For non-ionic systems, you must decide whether thegain in accuracy from Ewald summations justifies the increasedcomputation time.

The Morphology module currently does not support Ewaldsummation for evaluating van der Waals interactions. You maywish to increase the real space cutoff for the van der Waals inter-action from its default value on the OPEN FORCE FIELD/Energy Terms/van der Waals card. A large real space cut-off(15-30 A) will usually give converged results for the van derWaals interaction.

General Methodology


2. Load a force field.

3. Set up all variables affecting the generation of the energyexpression. The procedure depends on the calculation methodthat you want to apply. For surface energy calculations orattachment energy calculations with Full OFF Support you usethe OFF SETUP menu card (see the Cerius2 Simulation Toolsonline documentation). If you apply the Bond-Energy-Listmode for evaluating the attachment energy you edit the Bond-Energy-List Preferences menu accessible from the CalculateMorphology panel via the Preferences... button. This buttonappears as soon as you select the Bond-Energy-List mode onthe Growth Morphology subpanel.

4. If you want to automatically generate the list of growth planeswith the BFDH method, check the Do BFDH First box (defaultsetting). Otherwise, either enter your own list or do the BFDHcalculation separately, then edit the list (see Calculating Mor-phology with the BFDH method and Editing, adding and removingcrystal faces).

5. If you wish to calculate the growth morphology, select a calcu-lation mode: either Full OFF Support or Bond-Energy-List.

6. If you wish to calculate the growth morphology using theBond-Energy-List mode, specify how the growth slices areconsidered when calculating their energies (see Slice position-ing).

7. If you wish to calculate the equilibrium morphology, specifythe slab thickness for the surface energy calculation.

8. Depending on whether you wish to use the attachment or sur-face energy methods, click the Calculate button either on theGrowth Morphology or Equilibrium Morphology subpanel.

Note If you get nonsensical results, check that the lattice energy isnegative and that it has converged within the specified cutoffradius (especially if direct summations are used for the Cou-lomb interactions). It is possible that the chosen force field is notsuitable for the crystal under consideration or that the modeledstructure does not represent a stable crystal.


8. Morphology

Please see the online help for more detailed information on theCalculate Morphology control panel.

Setting up the energy calculations

The procedure for setting up the energy calculations depends onthe method used for morphology prediction. Except for the casewhere the Bond-Energy-List option for the calculation of attach-ment energies has been selected, variables affecting the attachmentenergy calculations are specified using the OPEN FORCE FIELDcard on the OFF SETUP card deck (see the Cerius2 SimulationTools manual). There, you can specify the non bond interactionterms included in the energy calculation, the method (EWALD,DIRECT, or SPLINE) to evaluate Coulomb interactions, the cutoffradius for direct non bond interactions, and a large number ofmore advanced parameters affecting your energy calculations.

Note

On the other hand, if you use the Bond-Energy-List option to cal-culate attachment energies, you specify variables affecting theenergy calculation using the options on the Bond-Energy-List Pref-erences control panel. This panel is accessible via the additionalPreferences... button which appears as soon as you specify Bond-Energy-List mode on the Growth Morphology subpanel. Theseoptions specify the nonbond interaction terms included in theenergy calculation and the interaction radius. A button for check-ing the lattice energy is also on this control panel.

In addition, the Attachment Energy Preferences panel (accessiblevia Preferences beside the calculate button) allows you to specifywhether an interaction energy file is created and whether the forcefield is automatically loaded and atom types assigned.

Choosing the force field To calculate attachment or surface energies, you must load a forcefield and assign atom types to the model. The choice of force field

To reduce the memory requirements when working with largesystems, you may wish to choose the NO LIST option for thesearch for neighbors on the Coulomb and van der Waals prefer-ences panels. This only slightly increases computation times forMorphology calculations and drastically reduces memoryrequirements.

General Methodology


is an essential variable in the morphology prediction. Most forcefields are parametrized for particular types of systems, thus twoforce fields can give widely diverging results for the same model.On the other hand, you can use suitably chosen force fields to val-idate each other. Roberts and Docherty (1988) have shown theeffects of using different force fields for predicting the morpholo-gies of a number of systems.

Auto force field switch An Auto Force Field switch is provided that, by default, automat-ically loads a default force field if you have not loaded one. If a youhave specified a force field, the Auto Force Field switch guaranteesthat the atoms are automatically assigned atom types appropriatefor this force field.

Note

You can use the OFF to load a different force field and to assignatom types and charges; you can use the Force Field Editor tochange parameters (see the chapters titled "Open Force Field" and"Force Field Editor" in the Cerius2 Simulation Tools manual).

Non bonded energyterms

The attachment and surface energy calculations use only the nonbonded terms from the chosen force field. They use all these termsby default (van der Waals, Coulomb, and hydrogen bonding), butyou can exclude some of these if you wish using either the EnergyTerms/Selection option on the OPEN FORCE FIELD card (allmethods except Bond-Energy-List mode) or the Bond-Energy-ListPreferences control panel (Bond-Energy-List mode).

Lattice energy and inter-action radius

For neutral molecules, you do not always require Ewald summa-tions to perform the Coulomb summations—direct treatment ofthe interactions may be sufficiently accurate and significantlyreduce the time required to perform the calculation. However, youthen need to ensure that the cutoff distance is large enough for thelattice energy to be converged. If you want to calculate the growthmorphology and you do not wish to use Ewald summation tech-niques, we recommend that you switch to Bond-Energy-Listmode. In this mode, the cutoff distance is set using the Bond-Energy-List Preferences control panel which is accessible via thePreferences... button. The Check Lattice Energy button allowsyou to plot the lattice energy and determine the convergence dis-

By default, Cerius2 uses the Universal force field. Note that youstill need to assign QEq charges (using the CHARGES card onthe OFF SETUP card deck) before using this force field.


8. Morphology

tance. This option is also useful in verifying that the value of thelattice energy is as expected for a stable crystal structure (it shouldbe negative).

If you use the Full OFF Support mode or if you want to predictequilibrium morphologies, the cutoff distance is set using the Pref-erences... of the Direct or Spline method for the Energy Terms/Coulomb or Energy Terms/van der Waals panels of the OPENFORCE FIELD card.

Saving the energy data A switch is provided that allows you to save essential informationto an output file named interactions.dat. For the format of this file,see "Morphology File Formats" in Appendix B of the Cerius2 Com-putational Instruments online documentation.

To set up the energy calculations

1. Go to the OPEN FORCE FIELD card, load a force field andselect options for the various non-bonded terms on the EnergyTerms ... menu. In particular, decide whether to use Ewaldsummations for the Coulomb term, and set the cutoff distancefor van der Waals and Coulomb interactions to a convergedvalue.

2. Depending on the force field you use, you may need to specifyappropriate charges for the crystal, for example by using theCHARGES card on the OFF SETUP card deck.

3. If you want to use the Full OFF Support mode for attachmentenergy calculations or if you want to calculate surface energies,specify the options affecting the energy calculation using theOFF SETUP deck (see the chapters titled "Open Force Field"and "Force Field Editor" in Cerius2 Simulation Tools).

4. If you wish to use the Bond-Energy-List mode for attachmentenergy calculations, proceed as follows:

Go to the MORPHOLOGY card and select the Calculate itemto bring up the Calculate Morphology control panel.

Select the Bond-Energy-List mode on the Growth Morphologysubpanel.

Click the Preferences... button to bring up the Bond-Energy-List Preferences control panel.

General Methodology


5. Click Preferences... next to the Calculate button to bring up theAttachment Energy Preferences panel. Check the Auto ForceField box to generate automatic atom typing for the structure.To do this manually, uncheck the box and use the options in theOFF; to edit force field parameters, use the Force Field Editor.To save the calculated energy data, check the Save InteractionsFile box.

To check the lattice energy

1. Go to the MORPHOLOGY card and select the Calculate itemto display the Calculate Morphology control panel (see theonline help for more control panel info).

2. Select the Bond-Energy-List mode on the Growth Morphologysubpanel.

3. Click the Preferences... button to bring up the Bond-Energy-List Preferences control panel.

4. Enter the number of data points to use in the plot.

5. Click the Check Lattice Energy button.

Slice positioning

The way that growth slices are positioned affects the calculatedenergies. If Full OFF Support has been selected for the evaluationof the attachment energies or if surface energies are calculated, alltheoretically possible slice positions leading to non-identicalgrowth slices are found automatically, and you do not need to setany parameters (you can skip this section). For the Bond-Energy-List mode for attachment energy calculations, you can set vari-ables affecting the slice positioning using the options on the SlicePositioning control panel (see the online help).

Finding the most stableslice

In calculating slice energies with the Attachment Energy method,it is important to find the most stable slice (that is, the one with themost negative energy), since this is the slice most likely to beinvolved in the growth process.

If you use the Bond-Energy-List mode for attachment energy cal-culations, Morphology finds the most stable slice by stepping theslice through the crystal in a direction normal to the slice, and


8. Morphology

repeatedly calculates the slice energy. You can specify the numberof steps used.

The size of each step is calculated so that the specified number ofsteps takes the slice through the crystal until the next slice wouldbe exactly equivalent to the first, due to the translational symmetrypresent:

Step size = d(hkl) / (Number of slice steps + 1) Eq. 9

Center slice on allmolecules

When working with mixed crystals, you can obtain more accurateresults by repeating the slice energy calculation with each of themolecules positioned at the center of the slice. The Center Slice onAll Molecules option is provided for this purpose.

Slice offset Positioning a slice so that the molecule is exactly at its center islikely to place other molecules on the edge of the slice. This canlead to difficulties in determining which interactions contribute tothe stability of the slice. You should therefore position the slice sothat the molecule is slightly offset from the center. You can use theSlice Offset option to specify the offset value (the default value is0.01 Angstroms).

To specify the slice positioning variables

1. Go to the MORPHOLOGY card and select the Calculate itemto display the Calculate Morphology control panel.

2. Select the Bond-Energy-List mode on the Growth Morphologysubpanel.

3. Click the Preferences... button to bring up the Bond-Energy-List Preferences control panel.

4. To calculate the slice energies with each of the molecules posi-tioned in the center of the slice in turn, check the Center SliceOn All Molecules box.

5. Specify the Slice Offset (in Angstroms).

6. Enter a value for the number of slice steps to be used.


General Methodology


Editing, adding and removing crystal faces

Face list Whenever you carry out a BFDH, Attachment or Surface Energycalculation or a morphology prediction using the Hartman-Per-dok method, the growth faces obtained from the calculationappear in the face list on the Edit/Add Faces control panel. Infor-mation listed for each face includes:

♦ Miller indices (h, k, and l) and multiplicity of the form

♦ The center-to-face distance and color of the plane

♦ Whether the form is visible given the calculated shape.

This list also provides the growth faces for the Attachment andSurface Energy calculations (unless the Do BFDH First box ischecked; see the online help). The center-to-face distances areupdated when the calculation is complete.

You can choose whether or not invisible faces will appear in the listby checking or unchecking the Include Invisible Faces in List but-ton.

Selecting faces To edit or remove faces in the list, you need to select these faces.You can select a face in the list by clicking the corresponding rowwith the left mouse button. Multiple faces can be selected by hold-ing down the < Ctrl > key while clicking additional rows. Holdingdown the < Shift > key during a second selection selects the wholerange of faces between the first and second selection. Double-click-ing the list selects all faces in the list.

Editing faces You can edit faces in the list using the three input fields immedi-ately below the face list. Selecting a single face in the list will fillthese input fields with the Miller indices, center-to-face distanceand color of the selected face. You can then modify the center-to-face distance (second) and color (third) fields for this selected face.You can also select a face by typing its Miller indices into the firstof the input fields. Changes are reflected in the displayed mor-phology. If multiple faces are selected you can set the center-to-face distance or the color of all selected faces to a common value.This provides an efficient way, for example, to change the displaycolor of the habit in the model window.

Adding faces If you type into the first input field below the list of faces the Millerindices of a face which is not symmetry related to any of the faces


8. Morphology

currently in the list, this form is added to the list of faces. Once theface is added, you need to set its center-to-face distance and colorattributes using the second and third input fields. If you do notenter a center-to-face distance, a default of 10 Angstroms isassigned. You can also compute the true value by running aGrowth or Equilibrium morphology calculation. If visible, theadded face is shown in the displayed morphology.

Removing faces You can remove one or more faces permanently from the list byfirst selecting them and then hitting the Remove Selected Facesbutton. You can also permanently remove all faces from the listthat are not present in the current morphology (that is, all thosethat are listed as not visible), using the Remove Invisible Facesbutton. This is often useful in decreasing the number of faces con-sidered in an Attachment or Surface Energy calculation. However,you should make sure that significant growth faces are notremoved accidentally. Note that in order to regenerate faces whichhave been removed, you either have to reenter their Miller indicesby hand or redo your morphology calculation.

Listing faces You can use the List All Faces to Text Window option to print amore detailed listing of all current faces in the text window. Thelisting is grouped into faces that are in the same form (that is, sym-metry related), and is ordered according to center-to-face distance.In addition to the Miller indices, the center-to-face distance andthe color of each face, the d-spacing, surface area and number ofcorners are also given.

To edit, add, and remove crystal faces

1. Go to the MORPHOLOGY card and select the Edit/Add Facesitem to display the Edit/ Add Faces control panel.

2. Add faces to the list entering the Miller indices of the new planeusing the input fields below the list. Adding a plane to the listautomatically adds all symmetry related planes, too.

3. Edit faces by selecting a single face and entering new values forcenter-to-face distance or color.

4. Remove faces by selecting them and executing RemoveSelected Faces.

5. Remove all invisible faces using the Remove Invisible Facesoption.

General Methodology


Please see the online help for more detailed information about thecontrol panels.

Calculating morphology using the Hartman-PerdokMethod

Morphology calculations applying the Hartman-Perdok methoduse the options on the Crystal Graph and Hartman-Perdok panels,both accessible via the Calculate Morphology control panel. Theprogram implements the Hartman-Perdok method for buildingstable growth planes from an analysis of strong intermolecularinteractions ("bonds") in the system. The Attachment Energymethod is then used to simulate the relative growth rate of thesegrowth planes.

Generating the CrystalGraph

The Crystal Graph menu is used to generate a set (graph) of strongcrystal bonds for the current structure. The Crystal Graph panel isaccessible via the Crystal Graph ... button on the Calculate Mor-phology menu. After you specify a spatial range, the program gen-erates a list of bonds between molecules in the crystal andcalculates bond energies. This bond list may be edited, and anenergy window can be specified determining those bonds withinthe bond list which are to be included in the subsequent identifica-tion of strong chains of bonds (periodic bond chains) and net-works of bonds (connected nets) (see Generating and editing theCrystal Graph).

Generating ConnectedNets

After a crystal graph has been defined, you may generate all pos-sible connected nets associated with that crystal graph by simplyclicking Calculate Morphology on the Hartman Perdok panel(accessible by clicking the Hartman Perdok ... button on the Cal-culate Morphology menu). The program selects only connectednets which have the correct stoichiometry, i.e., which include thesame ratio of different types of molecules as the full crystal. Inaddition, for a set of forms nh nk nl, n=1, 2, ... only those con-nected nets are shown which correspond to the smallest n forwhich the given form is not excluded due to symmetry con-straints. The connected nets generated are listed in the HartmanPerdok panel for further analysis (see Generating and editing Peri-odic Bond Chains and Connected Nets).


8. Morphology

Calculating the Morphol-ogy

After the connected nets resulting from a given crystal graph havebeen identified, the attachment energy of each net is calculated,and the energy of the most stable net for each generated form isused to assign a center-to-face distance for the corresponding face.You may select from two different methods to calculate the attach-ment energy of a connected net. In the default mode (E(Att)) theattachment energy is calculated using total energies for the evalu-ation of lattice and slice energies. Alternatively, the E(Bond) modeevaluates sums over bond energies as defined in the crystal graph.The list of planes and the predicted center-to-plane distances usedto calculate morphology is accessible via the Edit/Add Facesmenu on the MORPHOLOGY card (see Editing, adding and remov-ing crystal faces).

Creating a model of aconnected net

The program allows you to generate and visualize any net in thelist of connected nets within a new model space, which can then beused for further analysis. In addition to the molecules forming thenet, the model contains dummy atoms at the center of geometry ofeach molecule, connected by bonds, illustrating the two one-dimensional periodic bond chains defining the connected net.

Saving the crystal graphand the connected nets

Both the crystal graph as well as the structure of the connected netsare saved to the data model. You may permanently save this infor-mation by saving the Model to disk.

To calculate morphology with the Hartman-Perdok Method

1. Place the crystal structure in the current active model space.The crystal structure can either be loaded from a file or builtusing the Crystal Builder.

2. Load a force field and setup all variables affecting the genera-tion of the energy expression using the OFF SETUP menu card(see the Cerius2 Simulation Tools manual for more information).

3. Select Calculate on the MORPHOLOGY menu card and clickthe Crystal Graph ... button to bring up the Crystal Graphpanel.

4. Select a spatial region for which bonds between molecules inthe crystal are generated. You may specify the length alongeach of the crystal axis separately. Click the Generate Bondsbutton and the program automatically generates a list of bonds.

General Methodology


5. Select a set of bonds that are included in the Crystal Graph. Youmay specify a window of bond energies, change the energy ofindividual bonds and you may include/exclude individualbonds into/from the crystal graph.

6. Return to the Calculate Morphology panel and click the Hart-man Perdok... button to bring up the Hartman Perdok panel.

7. Click the Calculate Morphology button and the program gen-erates flat stoichiometric connected nets from the crystal graph,calculates the attachment energies and displays the morphol-ogy.

8. You can now analyze the generated connected nets, displayinformation about the periodic bonds chains defining a selectedconnected net and create models of connected nets.

Generating and editing the Crystal Graph

The first step of a Hartman-Perdok morphology prediction is theidentification of strong interactions ("bonds") between moleculesforming the crystal. The program generates a crystal graphdescribing the interactions between the molecules in the crystalstructure in terms of those bonds. Generating crystal bonds is doneusing the Crystal Graph panel accessible via the Calculate Mor-phology control panel.

Defining the spatial range Before generating the crystal graph, you need to specify the spatialrange of the bonds which you wish to be included in the crystalgraph. You define the spatial range by specifying a maximum dis-tance along each lattice vector using the entry fields below theGenerate Bonds button.

Generating crystal bonds If you click the Generate Bonds button, the program automati-cally generates all bonds that fall into the specified spatial range,calculates their energies and displays a list of representatives foreach family of symmetry related bonds in the Crystal Graph panel.The program also prints a list of all bonds into the text window.

Visualizing crystal bonds The visualization of the crystal graph is controlled by the DisplayCrystal Graph button. If checked bonds included in the crystalgraph are displayed in the model window as green lines connect-ing the centers of geometry of the molecules forming that bond.Selecting a bond in the bond list (left-clicking with the mouse on


8. Morphology

the corresponding line in the list box) highlights the correspond-ing bond (and all its symmetry related copies) in the model win-dow.

Editing crystal bonds From the list of bonds, you can select a set of "strong" bonds thatwill be used in subsequently determining periodic bond chainsand connected nets. By default, the program includes all bondswith energies larger than the thermal energy at room temperature(-0.596 kcal/mol). You can modify the energy window by settingthe energy of the strongest (weakest) included bonds using theentry fields beside the Apply Energy Window control and clickingthe Apply Energy Window button. You can also include (exclude)a single bond by selecting the bond from the list and select YES(NO) from the pull-down menu in the Include? column below thebond-list box. In addition you can modify the bond-energy usingthe entry field in the Energy column. In connection with theE(Bond) mode of calculating the attachment energy the latteroption is very useful to explore the effect of intermolecular inter-action strength on the crystal morphology. Editing the set of crys-tal bonds included in the crystal graph will change the bondsdisplayed in the model window accordingly.

Note

To Generate and edit the Crystal Graph

1. Place the crystal structure in the current active model space.The crystal structure can either be loaded from a file or builtusing the Crystal Builder.

2. Load a force field and setup all variables affecting the genera-tion of the energy expression using the OFF SETUP menu card(see the Cerius2 Simulation Tools manual for more informationabout using this card).

The crystal graph is associated with a model of a crystal struc-ture. Modifying the atomic arrangement of the crystal structure,for example by moving or deleting atoms, will invalidate thecurrent crystal graph. Before starting the Hartman-Perdok anal-ysis, the program checks the crystal graph for consistency. If aninconsistency is found, the program issues a warning messageand deletes the crystal graph as well as all connected nets gen-erated from that crystal graph.

General Methodology


3. Select Calculate on the MORPHOLOGY menu card and clickthe Crystal Graph ... button to bring up the Crystal Graphpanel.

4. Select a spatial region for which bonds between molecules inthe crystal are generated. You may specify the length alongeach of the crystal axes separately.

5. Click the Generate Bonds button and the program automati-cally generates a list of bonds.

6. Modify the set of bonds included in the crystal graph by settingan appropriate energy window, entering the strongest andweakest included bond beside the Apply Energy Windowcommand, and hitting the Apply Energy Window button.

7. Include single bonds in the crystal graph or exclude singlebonds from the crystal graph by selecting a bond and selectingYES or NO from the Include? pull down menu.

8. Modify the bond energy of a selected bond using the Energyentry field.

Generating and editing Periodic Bond Chains andConnected Nets

After a set of strong crystal bonds has been identified, you can gen-erate all two-dimensional connected nets of strong bonds. Thesenets define the stable growth planes of a crystal. Generating andediting connected nets is done using the Hartman-Perdok panelaccessible via the Calculate Morphology control panel.

Generating connectednets

If you click the Calculate Morphology button, the program auto-matically constructs all one-dimensional periodic bond chains(PBCs), then examines all combinations of pairs of PBCs to find allunique two-dimensional connected nets that are flat and stoichio-metric. For a set of forms nh nk nl, n=1, 2, ... only those connectednets are selected which correspond to the smallest n for which thegiven form is not excluded due to symmetry constraints. In addi-tion, the program calculates the attachment energy of the con-nected nets using the force field settings specified in the OFF. Inparticular, Ewald summations can be used to perform theseattachment energy calculations.


8. Morphology

Generating the crystalmorphology

From the attachment energies of the energetically most stable con-nected nets, the program determines the morphology. Two differ-ent methods may be used to calculate the attachment energy of aconnected net. In the default mode (E(Att)) the attachment energyis calculated using total energies for the evaluation of lattice andslice energies. This method is equivalent to the one applied if FullOFF Support is selected for the calculation of the growth morphol-ogy (see Calculating morphology with the Attachment or SurfaceEnergy methods). Alternatively, the E(Bond) mode evaluates sumsover bond energies as defined in the crystal graph. From the gen-erated list of connected nets the most stable one for each form isselected and used to predict the crystal morphology. These con-nected nets considered in the morphology calculation appear inthe face list on the Edit/Add Faces control panel (see Editing, add-ing and removing crystal faces).

List of connected nets All generated connected nets are listed in the Hartman Perdokpanel. Since the Hartman-Perdok analysis may generate severalconnected nets for a form (hkl) all connected nets of a given formare grouped together and ordered according to their stability. Inthe list of connected nets the forms are ordered according to thestability of their most stable connected net. Both the attachmentenergy based on the total energy of the connected net E(Att) andthe attachment energy calculated using the bond energies of thecrystal graph E(Bond) is given for each connected net.

Deleting less stable con-nected nets

The Hartman-Perdok analysis may generate a large number of dif-ferent connected nets for a given form (hkl) some of which are con-siderably more unstable than the most stable connected net of thisform (appearing as Type 1). Those relatively unstable nets mayusually be ignored for the analysis of crystal morphology. In addi-tion, since the connected nets are saved to the data model a largenumber of connected nets may affect the performance of Cerius2.You may delete all nets within each form which are less stable thanthe most stable connected net of this form by a user-specifiedenergy value by clicking the Delete Types less stable by button.The energy value is specified using the entry field beside theDelete Types less stable by control. Note, that the energy criterionis applied to each form separately and that no forms are deletedfrom the list.

General Methodology


Analyzing connectednets

Selecting a connected net from the list on the Hartman Perdokpanel (left-clicking with the mouse on the corresponding line inthe list box) gives you access to two modes of analysis:

1. You can create a model of the connected net.

2. Information about the two PBCs forming the selected con-nected net is displayed in the lower part of the Hartman-Per-dok panel. This information includes the overall direction of aPBC and a list of the bonds forming this PBC. In case the currentmodel is a model of a connected net, you can select each bondin the list of bonds forming a PBC, which highlights the corre-sponding bond in the model window by adding it to the list ofselected objects.

Create a model of a con-nected net

If you have selected a connected net from the list on the Hartman-Perdok panel (left-clicking with the mouse on the correspondingline in the list box), you may create an atomic model of this con-nected net using the Create Selected Face option. In addition tothe molecular arrangement, the model illustrates the PBCs form-ing the connected nets by dummy atoms at the centers of geometryof the molecules forming the PBC and bonds between them.

Display style of a con-nected net

The Display Style pull-down menu allows you to change the dis-play style for a model of a connected net. You can show the atomsonly (Atoms), the PBCs only (PBCs) or both the atoms and thePBCs together (Atoms + PBCs).

Removing dummy atoms You can permanently remove the dummy atoms from a model ofa connected net by clicking the Remove PBCs button. This isimportant if you want to further manipulate the model of the con-nected net. For example, it is helpful if you want to perform energyevaluations, since the dummy atoms may affect the result of theoperations performed.

Note The list of connected nets on the Hartman-Perdok panel is inde-pendent of the face list on the Edit/Add Faces control panel. Alltools for analyzing the morphology (see Analyzing the mor-phology) are available for the morphologies generated usingthe Hartman-Perdok method. The only exception is the CleaveSelected Face option, which is replaced by Create SelectedFace on the Hartman-Perdok panel. Editing the face list on theEdit/Add Faces control panel will modify the morphology, butwill not change the list on the Hartman-Perdok panel.


8. Morphology

Note

To Generate and edit Periodic Bond Chains and ConnectedNets

1. Generate a crystal graph (see Generating and editing the CrystalGraph).

2. Go back to the Calculate Morphology panel and click the Hart-man-Perdok... button to activate the Hartman-Perdok panel.

3. Click the Calculate Morphology button on the Hartman-Per-dok panel. This generates all connected nets corresponding tothe current crystal graph and calculate the morphology on thebasis of attachment energies of these connected nets.

4. .Select the attachment energy method used to simulate the crys-tal morphology using the Display Morphology pulldownmenu.

5. Get more information about a particular connected net byselecting the corresponding line in the list of connected nets(left-clicking with the mouse on the corresponding line).

6. Generate a model of the selected connected net by clicking theCreate Selected Face button.

Displaying the morphology

The calculated morphology is displayed in the model window.You can specify the way the morphology is displayed using theoptions on the Morphology Display control panel.

Visualization With visualization options you can choose to display the externalmorphology either with or without the internal molecular struc-ture, or display the molecular structure alone. By default, both aredisplayed.

The model of a connected net created using the Create Face but-ton on the Hartman-Perdok panel contains dummy atoms toillustrate the PBCs. Before manipulating this model with otherCerius2 tools, it might be necessary to remove the dummyatoms from the model using the Remove PBCs button (seeRemoving dummy atoms).

General Methodology


Scale factor The morphology is displayed with the center-to-face distance foreach plane represented in Angstrom units multiplied by a scalefactor. The units are used to enable easy comparison of molecularstructure and morphology and should not be regarded as repre-senting the true size of the crystal. The larger the scale factor, thelarger the size of the morphology relative to the molecular model.A scale factor of 1.000 is used by default.

Transparency You can change the transparency of the crystal faces, allowing themolecular structure to be viewed inside the morphology model.An entry box or slider bar may be used to specify the value (0 isopaque, 1 is transparent). The default setting is 0.400.

Face label The face label options allow you to label the faces of the modelshowing their Miller indices (Indices) or their surface area as apercentage of the total area (Area). Use the None option if you donot want labels.

Redisplay morphology Usually, any change to the list of faces on the Edit/Add Facespanel results in an automatic update of the morphology display inthe model window. However, in some cases, such as a manualchange of lattice parameters, you may wish to update the mor-phology display manually. You can do this using the RedisplayMorphology button.

To specify the display controls

1. Go to the MORPHOLOGY card and select the Display item tobring up the Morphology Display control panel.

2. Specify what to display by selecting one of the options from theVisualization popup (Morphology, Molecular, or Both).

3. Enter the scale factor to apply.

4. Specify the transparency of the crystal faces by either movingthe slider bar or entering a value in the entry box (0 to 1.0).

5. Label the faces by selecting one of the options from the FaceLabel popup (None, Indices, or Area).

6. If you have made changes to the model outside the morphol-ogy module, use the Redisplay Morphology option.



8. Morphology

Analyzing the morphology

Several properties of the crystal morphology can be calculatedusing the options on the Morphology Analysis control panel (seethe online help). These include inter planar angles, aspect ratioand the percentage area accounted for by certain forms.

Interplanar angles andsurface areas

Buttons are provided that allow you to calculate the inter planarangle between two faces or the surface area of a face. Faces areidentified by entering their Miller indices. Results are reported inthe text window.

List Areas by Form option You can use the List Areas by Form option to list all forms presentin the current morphology and to display the percentage of thetotal surface area accounted for by certain forms.

Aspect ratio You can also calculate the aspect ratio. The ratio is defined as D/d,where D is the maximum center-to-corner distance and d is theminimum center-to-face distance. Knowledge of shape and aspectratio is essential to understanding packing, flow problems, clog-ging of filters, and related questions.

Cleave selected face If the list of faces has been generated from an attachment energycalculation, it is possible to cleave a growth face from the crystal,thereby creating a new model which can subsequently be manip-ulated using the Surface Builder control panel (see the onlinehelp). This feature, which is also useful for visualizing growthfaces, is accessed by clicking the Cleave Selected Face button.

Note

To analyze the crystal morphology

1. Go to the MORPHOLOGY card and select the Analysis item tobring up the Morphology Analysis control panel.

2. To calculate the angle between two faces, enter the Miller indi-ces of the two faces in the entry boxes, then click the CalculateAngle between __ and __ button.

If you perform a Hartman-Perdok analysis the Cleave SelectedFace control is not active and is replaced by the Create SelectedFace command on the Hartman-Perdok panel.

General Methodology


3. To calculate the surface area of a face, enter the Miller indices ofthe face in the entry box, then click the Calculate Area of Facebutton.

4. To list the forms and their percentage areas, click the List Areasby Form button.

5. To calculate the aspect ratio of the crystal, click the AspectRatio button.

6. To cleave a face out from the last attachment energy calculation,enter its Miller indices, then click the Cleave Selected Face but-ton.

Please see the online help for more detailed information about thecontrol panels.

Storing morphologies

Saving You can save the current morphology in two different ways. Theusual way is to use the File/Save Model option of the main Cerius2

menu. If you choose the .msi format, the morphology is savedtogether with the underlying molecular structure, the crystalgraph and any connected nets which may have been generated.

It is possible to save the morphology in a habit (.hab) file that usesa subset of the Crystallographic Information File (CIF) format. Todo this, open the Save Morphology control panel (see the onlinehelp) and specify a file name in the browser box. The .hab exten-sion is automatically added.

Only the morphology, cell parameters, and symmetry are stored inthe .hab file. The CIF format is described in Appendix B, "File For-mats", of the Cerius2 3.8 Computational Instruments manual.

Other structural parameters, such as atomic coordinates, are notsaved in the .hab file.

Loading If a morphology has been saved as part of an .msi file, it can beloaded using the File/Load Model option of the main Cerius2

menu. You can also read morphologies saved in CIF-formattedfiles using options on the Load Morphology control panel (see theonline help). You can add a morphology to the current modelspace or put it into a new one. However, you can add a morphol-


8. Morphology

ogy to the current structure only if both have the same cell andspace group symmetry.

To save the current morphology

1. Go to the File/Save Model item on the main menu to bring upthe Save Model control panel if you wish to save a morphologyas part of an .msi file, or go to the MORPHOLOGY card andselect the File/Save item to bring up the Save Morphology con-trol panel.

2. Use the browser box to specify the file name and save the file(the extension is automatically added).

To load a morphology

1. Go to the File/Load Model item on the main menu to bring upthe Load Model control panel if you wish to load a morphologysaved as part of an .msi file, or go to the MORPHOLOGY cardand select the File/Load item to bring up the Load Morphologycontrol panel.

2. Select the model store to be used (click either the Current orNew button).

3. Use the browser to specify the file name and load the file.

Note

Theory

This section provides detailed theoretical information for the fourmethods used by Morphology: the Bravais Friedel Donnay Harkermethod, the Attachment Energy method, the Surface Energymethod, and the Hartman-Perdok method.

If you are loading a .hab file into the current model store, andthe cell parameters or space-group symmetry are not the sameas the current crystal, you can either replace the current crystalor ignore the load command (that is, choose Replace or Ignorefrom the dialog box).

Theory


Bravais Friedel Donnay Harker method

The Bravais Friedel Donnay Harker (BFDH) method is a geomet-rical calculation that uses the crystal lattice and symmetry to gen-erate a list of possible growth faces and their relative growth rates.From this, crystal morphology can be deduced.

Growth rates - BravaisFriedel rules

Generally, observed crystal morphology is found to be dominatedby lower-order faces. Bravais (1913) and Friedel (1907) observedthat the center-to-face distance for a given plane tended to berelated to the inverse-plane spacing,

D ~ 1/d Eq. 10

where:

D = Center-to-face distance.

d = Lattice-plane spacing.

This is easily rationalized. Assume that growth involves consecu-tively adding growth planes of atoms and molecules. If energeticeffects are discounted, the ease of adding a plane is proportional toits thickness. Thus, a thinner growth plane grows faster and has alarger center-to-face distance.

Growth planes - DonnayHarker rules

Donnay and Harker (1937) refined this approach by developingrules that related the crystal symmetry to the possible growthplanes. These rules account for the effect of translational symme-try operators, meaning that higher-order planes can grow in pref-erence to lower-order ones. For example, in a body-centered cell, amolecule at the origin repeats in the center of the cell. This impliesthat growth in the [1 0 0] direction could occur by adding the (2 00) slice, rather than the thicker (1 0 0) slice.

The BFDH method combines these observations, using the Don-nay Harker rules to isolate the likely growth planes, then the Bra-vais Friedel rules to deduce their relative growth rates. Themethod is an approximation, and does not account for the energet-ics of the system. The stronger the bonding effects in the crystal,the less accurate the method becomes. In many cases, however,you can get good approximations, and the method is always use-ful for identifying important faces in the growth process.


8. Morphology

The Attachment Energy method

The Attachment Energy method can predict the shape of the crys-tal more accurately because it takes the energetics of the systeminto account.

Calculating Eatt The attachment energy, Eatt, is defined as the energy release on theattachment of a growth slice to a growing crystal surface(Docherty et al. 1991). Eatt is computed as (Berkovitch-Yellin 1985):

Eatt = Elatt - Eslice Eq. 11

where:

Elatt = Lattice energy of the crystal

Eslice = Energy of a growth slice of thickness dhkl.

Growth rate ~ Eatt

The growth rate of the crystal face is proportional to its attachmentenergy. That is, faces with the lowest attachment energies are theslowest growing and, therefore, have the most morphologicalimportance.

Deducing morphology The attachment energy is calculated for a series of suitable slices(hkl) that are chosen either by performing a Donnay-Harker pre-diction, or by entering your own data. From the energy calculationand, hence, the growth rate, a center-to-face distance is assigned toeach face. This information is used to deduce the morphologyusing a Wulff plot (Wulff 1901).

Assumptions The attachment energy model assumes that the surface is a perfecttermination of the bulk and that no surface relaxation takes place.This has been shown to be significant in the case of inorganic sys-tems such as a-Al2O3 and a-Fe2O3. The attachment energy modelworks well in the case of many organic molecular systems.

The Equilibrium Morphology

The equilibrium morphology of a crystal is determined by theminimum of the surface free energy for a given volume and tem-perature (Gibbs 1928). If the surface free energies, or, at zero tem-perature, the surface energies are known for all relevant crystal

Theory


faces the morphology of a crystal in equilibrium with its surround-ing can be visualized using a Wulff plot (Wulff 1901).

Calculating Esurf In the program, the surface energy is calculated from the energy ofa slab of finite thickness:

Eq. 12

where:

Elatt(M) = energy of a M-layer thick slab inside the infinite crystal.

Eslice(M) = energy of a M-layer thick slab in vacuum.

Ahkl = surface area of a plane with Miller indices hkl.

The factor 1/2 in front of Eq. 12 accounts for the fact that the slabhas two surfaces.

It is readily seen that Eq. 12and Eq. 11 are closely related. In thelimit of a slab thickness of one growth layer, a very rough approx-imation of the surface energy may be obtained using the attach-ment energy:

Esurf(1) = 1/2 Eatt / Ahkl = 1/2 dhkl Eatt/V Eq. 13

where

dhkl = interplanar spacing of a surface with Miller indices hkl

V = unit cell volume.

In the program, the surface energy is calculated using a finite andfixed slab thickness, M. The number of growth layers, M, formingthe slab is a user defined parameter. According to Eq. 12, the accu-racy of the surface energy can be systematically improved byincreasing the slab thickness M.

Assumptions The equilibrium morphology is calculated at zero temperature. Itis assumed that the surface is a perfect termination of the bulk andthat no surface relaxation takes place. In the slab method, the cal-culated surface energy is an average between the surfaces withMiller indices h k l and -h -k -l. The latter restriction is importantfor crystal structures not having a center of inversion.

Esurf12---limM infinity→ Elatt M( ) Eslice M( )–[ ] Ahkl⁄=


8. Morphology

The Hartman-Perdok method

The Hartman-Perdok method (Hartman-Perdok 1955, Bennema1996, Grimbergen et al. 1998) provides a systematic way to gener-ate the stable growth planes of a crystal. It removes one of theassumptions of the common Attachment Energy Method, namelythat the growth planes are always ideal, flat crystal planes. Oncethe stable growth planes of a crystal have been identified, theattachment energy is used to simulate the growth morphology ofthe crystal (see The Attachment Energy method). In addition, theHartman-Perdok method gives you critical insight into why sur-faces are stable and thus provides you with a powerful tool (forexample) for designing growth additives.

The implementation of the Hartman-Perdok theory is based on theprogram FACELIFT (Grimbergen and Meekes 1997) developed atthe University of Nijmegen.

Crystal bonds and thecrystal graph

The Hartman-Perdok theory starts from the concept that crystalsare stable because of the ability of the growth units to form strongattractive bonds between each other. The network of these bondsin the crystal defines the crystal graph.

The program determines the bond energy between two growthunits by placing them into a temporary model with the same dis-tance and orientation as in the crystal and calculating the differ-ence between the total energy of the interacting growth units andthe two total energies of the separated growth units. In molecularcrystals, the sum of the bond energies over all bonds in a crystaladds up to the lattice energy Elatt, since the Coulomb, van derWaals and H-bond interactions are pair interactions.

From the list of bonds, a set of "strong bonds" has to be selectedwhich are included in the crystal graph. In the context of crystalgrowth, bonds are strong if they are unlikely to be broken at agiven temperature, that is, if the bond strength larger then aboutkT.

Periodic bond chains(PBCs) and connectednets

According to the Hartman-Perdok theory, the stable growthplanes are those planes which consist of two-dimensional con-nected nets of strong bonds in the surface plane. Generally, thoseplanes will have a non-zero step energy hindering nucleation anda low attachment energy (Grimbergen et al. 1998).

References


Once the crystal graph is defined, it is in principle possible to gen-erate all two-dimensional connected nets of strong crystal bonds.This is done in two steps. First, all one-dimensional periodic bondchains (PBCs) are constructed. A PBC represents an uninterruptedchain of strongly bonded growth units of which only the end-points are identical. In the second step, connected nets are gener-ated from all combinations of two non-parallel intersecting PBCs.Only stoichiometric connected nets are kept as growth planes.

Deducing morphology The attachment energy is calculated for all connected nets (see TheAttachment Energy method). From the attachment energy calcula-tion and, hence, the growth rate, a center-to-face distance isassigned to each face. This information is used to deduce the mor-phology using a Wulff plot (Wulff 1901). If more than one con-nected net exists for given Miller indices hkl, the most stable (thelowest absolute value of the attachment energy) is defined as thegrowth plane.

Assumptions The program assumes that the growth units are single molecules(or single atoms if not part of a molecule).

References

Bravais, A., Etudes Crystallographiques, Paris (1913).

Bennema, P., J. Cryst. Growth, 166, 17 (1996).

Berkovitch-Yellin, Z., J. m. Chem. Soc., 107, 8239 (1985).

Docherty, R.; Roberts, K. J., J.Cryst. Growth, 88, 159 (1988).

Docherty, R.; Clydesdale, G.; Roberts, K.J.; Bennema, P., J. Phys. D:Appl. Phys., 24, 89 (1991).

Donnay, J. D. H.; Harker, D., Am. Mineral., 22, 463 (1937).

Friedel, G., Bull. Soc. Fr. Mineral., 30, 326 (1907).

Gibbs, J. W., Collected Works, (Longman, New York, 1928).

Grimbergen, R. F. P., Acta Cryst, A54, (1998)

Grimbergen, R. F. P.; Bennema, P.; Meekes H., Acta Cryst A54(1998).


8. Morphology

Grimbergen, R. F. P. and Meekes and Boerrigter, S, C-programFACELIFT for PBC analysis, University of Nijmegen (1997).

Hall, S. R.; Allen, F. H.; Brown, I. D., Acta Cryst., A47, 655 (1991).

Hartman, P. and Perdok, W., Acta Cryst., 8, 49, 521, 525 (1955).

Strom, C. S. and Vogels, L. J. P., Acta Cryst A54 (1998)

Wulff, G. Z., Krystallogr., 34, 449 (1901).


9 Flexisorb

Introduction

The Flexisorb module allows you to study the sorptive propertiesof long chain saturated hydrocarbons, both linear and branched,within orthorhombic microporous host lattices, such as zeolites.The ranges of properties that can be calculated include the isostericheat of adsorption, Henry's constant, and the uptake isotherm as afunction of temperature and pressure. Flexisorb implements aConfigurationally Biased Grand Canonical Monte Carlo proce-dure (Maginn, 1995). This method was developed to overcomeproblems with steric hindrance and subsequent low acceptanceprobabilities which plague conventional Monte Carlo methodswhen dealing with larger molecules.

Using Flexisorb

Running a Flexisorb calculation generally involves a sequence ofsteps outlined below. The basic operations involved are:

♦ Preparing the model for calculation.

♦ Selecting a forcefield and assigning atom types.

♦ Generating energy maps.

♦ Calculating gas phase chemical potentials for the sorbates.

♦ Predicting properties: Isosteric Heat, Henry constant, andUptake Isotherm

♦ Analyzing results.


9. Flexisorb

To gain a better understanding of the process involved in settingup and running calculations, you may wish to work through thetutorial below.

Tutorial

Physi-sorption II - simulating large hydrocarbons

Calculating the sorptive properties of large molecules usingMonte Carlo methods can be difficult, due to the large probabilityof overlap between the sorbate and framework. One approachwhich has been developed to overcome this problem is the Config-urationally Biased Grand Canonical Monte Carlo (CBGCMC)method, featured in the FLEXISORB module.

In this tutorial you will:

♦ Sketch a pentane molecule.

♦ Run a Sorption calculation of this molecule in silicate.

♦ Generate potential energy maps to be used in a Flexisorb calcu-lation.

♦ Calculate gas phase properties of the sorbates.

♦ Calculate a single point on an uptake isotherm.

♦ Analyze the energy grids and sorbate distributions.

1. Loading the model

First you must load a model of the zeolite.

Next you will want to visualize the structure of the zeolite. This zeo-lite has a narrow c-dimension so you need to make a larger supercell

Select the File/Load Model... command from the menu barin the Visualizer window. Go to the Cerius2-Models/zeo-lites directory, select MFI.msi, and click the LOAD button.

Tutorial


to allow a larger van der Waals cutoff.

You will next make this into a superlattice.

This generates a 1x1x2 supercell.

With the current supercell, it becomes easier to visualize the porestructure of MFI. You will use the mouse to rotate the model in thewindow.

2. Sketching a pentane molecule

Go to the CRYSTAL BUILDER card located in the BUILD-ERS 1 card deck and select the Visualization item. In thepanel that opens, extend the display range in the c directionto 2 cells by typing a 2 in the third box to the right underCrystal Cell Display Range. Click ENTER.

Close the Crystal Visualization panel.

Select the Crystal Building item, and click the CrystallineSuperlattice button on the resulting panel.

Close the Crystal Building panel.

Select View/Display Attributes from the Visualizer menu.In the Display Attributes panel, change the visualizationStyle from STICK to POLYHEDRA and use the mouse torotate the model space to view the zeolite structure in thisdifferent style. Change back to STICK style when you arefinished.


9. Flexisorb

Next you sketch a pentane molecule in a new model window.

Now you clean the structure.

Click the + button in the Visualizer window to create a new,empty model space.

Open the Sketcher panel by selecting the Build/3D-Sketcher command from the Visualizer menu.

Select the Draw with Hydrogens option, then select theSketch with option. Click the left mouse button in the modelwindow five times in a zig-zag fashion to construct the pen-tane molecule.

Click the arrow button at the top left of the Sketcher panel.

Click and hold down the CLEAN button until the atoms areno longer moving.

Rename the model by going to the Visualizer panel, high-lighting the name of the model with the mouse, and typingpentane.

Close the Sketcher panel.

Tutorial


Next you will load in the appropriate forcefield for pentane sorption.

With the forcefield loaded, you can input the parameters for your sorp-tion study.

Now you will want to reduce the number of MC steps so that your jobwill run in a reasonable amount of time.

3. Running the calculation

Go to the OFF SETUP card deck and click the OPENFORCE FIELD card forward. Select the Load item to openthe Load Force Field panel. Select the sor_yashonath1.01force field and click the LOAD button.

Close the Load Force Field panel.

Go to the SORPTION TOOLS deck and select the Run itemfrom the SORPTION card.

Click the arrow to the right of the first Sorbate, and selectpentane.

Set the Pressure (kPa) to 0.1 for the pentane sorbate selec-tion.

In the Run Sorption panel, change the Length of run from100000 to 25000.

Make the MFI model active by clicking the diamond next toit in the Visualizer panel.

Now go back to the Run Sorption panel and click the RUNSIMULATION button.


9. Flexisorb

Note that the number of accepted creation attempts is very low, andthe calculation is far from convergence. This problem progressivelyworsens at higher pressures, for larger molecules, and for mixed sor-bate systems.

You will now perform a similar calculation using the FLEXISORBmodule.

4. Loading a model of the zeolite

Now you will reduce the symmetry to P1. There is no need to create asupercell, since the Flexisorb code extends the structure internally.

5. Assigning atom types

Next you assign atom types to the crystal lattice, using the defaultcbmc.frc forcefield file.

Select File/Load Model... from the menu in the Visualizerpanel. Go to the Cerius2-Models/zeolites directory andselect MFI.msi.

Click LOAD.

Close the Load Model panel.

Go to the BUILDERS 1 deck and select the CRYSTALBUILDER card. Select the Crystal Building item and clickthe Crystalline Superlattice button in the resulting panel.

Close the Crystal Building panel.

Go to the SORPTION TOOLS deck and select the FLEX-ISORB card. Select the Forcefield/Typing item to open thepanel and click the Calculate Atom Types button.

Tutorial


Now you must check that atom types have been assigned.

The Si atoms should be assigned sx and the O assigned oz. Now youneed to set the atom types for the pentane sorbate.

6. Sketching the sorbate

Now you will add the pentane sorbate to the sorbate list.

7. Creating the potential energy maps

Now you will set the current model to be MFI, since the Flexisorb cal-culation always runs using the current model. Then you will run thepotential energy map calculation. This will generate energy mapswhich for use in subsequent calculations. This should take about 10

Use the Atom Labels popup in the Visualizer panel tochange the atom labeling from NO LABEL to FFTYPE.

Change the Model window to the pentane molecule. Selectthe Forcefield/Typing item from the FLEXISORB card andclick the Calculate Atom Types button on the resultingpanel.

Close the Atom Typing panel.

Select the Run item from the FLEXISORB card and use thepulldown selection tool in the Sorbate List to select the pen-tane sorbate molecule.

Make sure that the Task is set to Create Energy Maps, andSet the File Prefix to MFI_C5.


9. Flexisorb

minutes to complete.

8. Analyze the output

Once the calculation has completed, you can view the output.

Now you will load in one of the potential energy grids using the Anal-ysis isosurface commands.

After visualizing the surface you can either analyze other surfaces bychanging the Isosurface Value or deleting the surface from the modelwindow.

Make the MFI_1 model the current model. Change theFFTYPE atom labeling back to NO LABEL using the popupin the Visualizer panel.

Now go to the Run Flexisorb panel and click the RUN but-ton.

Select the Analysis/View Output item from the FLEX-ISORB card and select the MFI_C5.log file.

Select the Analysis/Surfaces item from the FLEXISORBcard and set the Isosurface Value to 10. Now click the Cre-ate New Surface button.

Delete the isosurface by clicking the Delete Surface radiobutton in the Isosurfaces panel.

Tutorial


9. Map properties onto surfaces

The Free Volume surface is now color-coded by the values of theenergy map. Once you are finished visualizing the surface you maydelete it from the model window.

10.Perform the ideal gas calculation

Having generated energy maps, the next step is to calculate the gasphase properties of the sorbate, for comparison with the bulk phase

Select the Geometry/Free Volumes... item from the Visual-izer menu bar, and calculate a surface for the zeolite byclicking the CALCULATE button in the Free Volume panel.

Select the surface named CAVITY from the Edit Surface listin the Isosurfaces panel. Click the Property Map... button tobring up the Property Map panel.

On the Property Map panel, select the potential energy mapcg-MFI.grd and click the LOAD button.

Add this property map to the currently selected isosurfaceby clicking the Add Property button in the Property Mappanel.

Go back to the Isosurfaces panel and click the Delete Sur-face button.


9. Flexisorb

properties.

Now you will set up the simulation parameters.

In more realistic simulations much larger numbers of steps should berun to ensure convergence.

You are now ready to run the calculation. The program will first tryto generate a lookup file for the generation of branch points. If this filehas already been created, the calculation will proceed much morequickly.

While the calculation is running, you can monitor its progress usingthe Job Control item on the FLEXISORB card. Click the UPDATEbutton periodically to check whether a calculation has completed.

Once the calculation completes, the program creates file called <run_name>.IGdat, which contains information about the density of thesorbate in the gas phase. This file must exist for subsequent calcula-tions.

Now you will set up the parameters for the simulation.

In the Run Flexisorb panel, toggle the Task from CreateEnergy Maps to Ideal Gas.

Press the Input... button next to the Task on the Run Flex-isorb panel. On the resulting Ideal Gas panel, set the Num-ber of Steps to 25000, so the calculation will be fairly short,and the Block Average Frequency to 1000.

Click the RUN button in the Run Flexisorb panel.

Tutorial


11. Compute the uptake isotherm

Statistics will not begin to be gathered until this number of steps hasbeen reached.

You can now run the calculation. Note that this will take some time.While the calculation is running you can monitor the output fileusing the Job Control command as previously described.

Once the calculation has finished, you can load in the density distri-bution grid. One such grid is produced for each sorbate in the system.Here only one grid exists as only one sorbate is being studied in thistutorial.

12.Analyze the results

During the calculation, snapshots of conformations of the sorbates

Change the Task in the Run Flexisorb panel to Uptake Iso-therm and make sure the Mol Fraction of the sorbate is setto 1.00.

Click the Input... button and set the Total Pressure (KPa) to0.01. Set the Ideal Gas File to MFI_C5.IGdat clicking theBrowse... button and selecting it from the list.

Set the Equilibration Steps to 10000.

Select the Analysis/Cloud Points item from the FLEX-ISORB card, and load the MFI_C5_1.grd grid file. Click theCreate Cloud Points button to display the data cloud show-ing the sorbate distributions.

When you are finished, click the Delete Could Points but-ton.


9. Flexisorb

were saved to .car files. You will now load in one of these files.

The file only contains the sorbates. To view the sorbates overlaid on theframework, you must change the view to OVERLAY mode.

The sorbate snapshots encompass a greater range than a single unitcell, since the program internally generates a supercell. You canincrease the display range of the crystal so that it encompasses the sor-bates.

This completes the tutorial lesson.

Select the File/Load Model... command from the Visualizermenu bar.

Change the File Format in the resulting panel from MSI toCAR, select the MFI_C5_4.car file, and click LOAD.

Click the Select OVERLAY Display icon to the left of theVisualizer panel.

Select MFI as the current model by clicking the diamondnext to it in the Visualizer window. Turn on the visibility ofthe sorbate snapshots by checking the Visible box to theright of the MFI_C4_4 model.

Go to the BUILDERS 1 card deck and select the CRYSTALBUILDER card. Select the Visualization item. On theresulting panel, change the Crystal Cell Display Range to 22 3 and click the ENTER button.

General Methodology


General Methodology

Preparing a model

The first step in preparing a model is to sketch, or load from a file,one or more sorbate molecules. If you desire a united-atom modelfor the sorbate, then you may need to remove the hydrogen atomsfrom the sorbate (see below). Once you have done this, you shouldload the framework structure. Some commonly used sorbates andframework structures can be found in the Cerius2-Models directo-ries: /sorbates, /catalysts, and /zeolites. Before you can run a cal-culation, you must reduce the host lattice in symmetry to triclinic,P1, symmetry. You can accomplish this using the Crystal Superlat-tice command in the CRYSTAL BUILDER card (found in theBUILDERS 1 card deck).

Note

Selecting a forcefield

You must assign forcefield parameters to the host lattice and thesorbates. First you must select a forcefield, using the Forcefield/Select item on the FLEXISORB card (found in the SORPTIONTOOLS card deck). The default forcefield selection is cbmc.frc.This forcefield is based on the familiar cvff.frc forcefield, but con-tains additional parameters for simulating alkanes in zeolites. Ifyou wish, you may select other CVFF format forcefields using thebrowser (click the SELECT button), though their ability to simu-late sorption data may be limited.

Note

The sorbate molecules must not contain rings, and the hostframework must be orthorhombic.

This calculation does not use OFF. Selecting forcefields in theOFF SETUP card deck has no bearing on Flexisorb forcefieldcalculations.


9. Flexisorb

Atom typing

Once you select an appropriate forcefield, you must assign atomtypes to the molecules. Selecting the Forcefield/Typing item fromthe FLEXISORB card opens the Atom Typing panel, which allowsyou to calculate atom types, or to have them be automaticallydetermined when you launch calculation. If you select the Per-form Automatic Typing option when you have manually set anyatom types, these manual settings will be overridden (if they donot coincide).

Note

United-atom versus all-atom

The cbmc.frc forcefield contains some united atom parameters(ch1, ch2, ch3, ch4) for the interaction of hydrocarbons with zeo-lites. Using these potentials versus all atom potentials will yieldsignificantly shorter calculation and convergence times. Thesepotentials are automatically assigned if you have constructed aunited-atom sorbate molecule (that is, a sorbate molecule with thehydrogens stripped).

Note

Calculating energy maps

The Flexisorb program uses energy maps in conjunction withlookup and interpolation for computational expediency, ratherthan calculating the energies explicitly. Prior to running any calcu-lation, you must first generate these energy maps.

Selecting the Run item from the FLEXISORB card opens the RunFlexisorb panel. Within this panel, you should set the Task popupmenu to Create Energy Maps. You can access and set additionalcontrol parameters by clicking the Input... button to open theEnergy Map panel. Though the defaults should be suitable formost purposes, you can change the Temperature (K) and Boltz-

No charges are used in the simulations.

It is best not to mix united-atom and all-atom models within thesame simulation.

General Methodology


mann Threshold parameters. These are used for data compres-sion, to filter high energy values from the resultant energy maps,since these will contribute little to the Boltzmann factor.

The generated map files are named according to the host name andaccording to the unique atom type name (that is, the equivalencetype in the forcefield file). One map file is generated per uniqueinteraction. In addition to the map files, the program also producesa .grd file for each interaction, which you can graphically displayusing the FLEXISORB/Analysis item. If you do not wish to gener-ate these files, you can click the Output... button on the Run Flex-isorb panel and unselect the option: Create Grid File for EnergyMaps.

Once you add the sorbate molecules to the Sorbate List on the RunFlexisorb panel, you are ready to run the calculation.

Job control

You can monitor the progress of the calculations by selecting theFLEXISORB/Job Control item and using the options in the result-ing Flexisorb Job Control panel. The default behavior is to run allcalculations in the background on the local machine. Since manyfiles required for the calculation are binary, it is not currently pos-sible to launch calculations on remote hosts.

Calculating gas phase chemical potential

The Grand Canonical Monte Carlo method relies upon determina-tion of the equilibrium between the chemical potential for a sor-bate molecule in the gas phase, and its chemical potential within ahost lattice. You must determine the chemical potential in the gasphase before running any further calculations.

You should select the FLEXISORB/Run item to open the Run Flex-isorb panel. Within this panel, the Task parameter should now beset to Ideal Gas. You can set additional parameters by clicking theInput... button and changing the parameters found within theresulting panel. The Temperature (K) should be set to the temper-ature at which you wish to perform the subsequent sorption calcu-lation. If you wish to use a united atom model, select the UseUnited Atom Model option. If you then add the sorbates to the


9. Flexisorb

Sorbate List, you may run the calculation. The run creates a file,<File Prefix>.Igdat, which contains density and chemical potentialdata for each sorbate listed.

Note

Using existing map files

This calculation uses the energy information stored in the previ-ously computed map files. The program will try to determine themap files to use based upon the names of the host lattice and theatom types. However, if the host name is not the same as that youused in the generation of the map files, you can explicitly specifythe maps using the Energy maps panel (accessed by selecting theFlexisorb/Run item and clicking Input... on the panel, then click-ing Map Files... on the second panel). You should be careful whenexplicitly specifying the maps, that they were generated for thesame system. Information stored within the maps, such as theatom types and forcefield, can be queried as an additional check,using the INFO buttons on the Selected Maps list within theEnergy maps panel.

Calculating the isosteric heat

The setup for the calculation of the isosteric heat of adsorption issimilar to that for the Ideal Gas calculation. You must first ensurethat the Task is set to Isosteric Heat, and that the sorbates are listedin the exact same order used for the Ideal Gas computation. In thepanel accessed by clicking the Input... button (Isoteric Heat), youshould verify that the appropriate Ideal Gas File is selected, orselect it using the browser. Set the Temperature (K) to the samevalue as that used for the Ideal Gas simulation.

The order of the sorbates in subsequent calculations which usethis data file must be the same as specified here, else theincorrect chemical potentials will be associated with thesorbates. To avoid possible conflicts, it is best to perform thiscalculation (which is very quick) prior to any furthercalculations.

General Methodology


Running the calculation generates an output file called <file_pre-fix>.log, which contains details of the statistics, as well as the isos-teric heat and Henry constant for each sorbate.

Note

Predicting the uptake isotherm

To begin this calculation, you must once again select the Flexisorb/Run item from the FLEXISORB card, then set the Task (on the RunFlexisorb panel) to Uptake Isotherm. The calculation of the uptakeisotherm requires that you know the temperature and pressure ofthe system. You can set these parameters, together with other con-trol parameters, using the commands in the panel accessed byclicking the Input... button. As with the calculation of the isostericheat (see Calculating the isosteric heat), the Ideal Gas File must bespecified on the Uptake Isotherm panel, and be consistent with thecurrent simulation. If you plan to select multiple sorbate mole-cules, you must also specify the mole fraction of each, from whichthe partial pressures of each sorbate can be calculated. The totalmole fraction must sum to unity.

The GCMC calculation attempts four different moves: translation,insertion, deletion, and re-growth. You can modify the relativeratios of these moves to try to improve convergence of the system.However, the total probabilities must always sum to unity. In cer-tain cases, it may be a good idea to bias the insertion attempts tolow energy regions of the system, since there may be a greaterprobability of acceptance. However, at higher pressures this maybe a disadvantage, since the low energy sites may already be occu-pied. It may also be undesirable in certain frameworks, where alow energy sites may be physically inaccessible to a larger mole-cule. Therefore, you should exercise care in using this feature.

If you specify more than a single sorbate for an Isosteric Heatcalculation, the isosteric heat and Henry's constant arecalculated for each sorbate in turn. Competitive sorptioncalculations are not done here, since the sorbates are assumedto be at infinite dilution. This contrasts with the UptakeIsotherm calculation, where the sorbates are simultaneouslyintroduced into the system.


9. Flexisorb

Non-Ideal gases

Many gases exhibit non-ideal behavior, particularly at higher pres-sures. At low pressures, the observed behavior is often close toideal. This deviation from ideality is described using the equationof state of the gas, via the BWR formalism. You can load a data filefor each sorbate in question by clicking the BWR Files... button onthe Uptake Isotherm panel. Several files are provided, and youmay select others using the file browser. If this data is unavailable,ideal gas behavior is assumed.

Output files

In additional to an output file, <file_prefix>.log, the uptake iso-therm calculation may produce several other output files, for usein analysis. Files containing conformational snapshots of the sys-tem may be written. Number density grids giving information onthe preferential siting of the sorbates may also be generated. Youcan prevent the creation of these files using the commands in thepanel opened by clicking the Output... button on the Run Flex-isorb panel. Table files containing details of statistics and energiesare also be produced by default. You may plot these using thepanel accessed by going to the GRAPHS card in the TABLES &GRAPHS card deck, and selecting the File/Import Insight TBLitem.

Analysis

Depending on the type of calculation you perform, several outputfiles may be produced. The Analysis pullright on the FLEXISORBcard contains items that allow you to view the output log files andvisualize the generated grid data.

The Analysis/View Output item provides a convenient means ofloading and displaying the information in the log file. A new win-dow is created containing the output file.

For .grd files, several different display options are available:

♦ Surfaces

General Methodology


♦ Slices

♦ Cloud Points

Creating an isosurface

The Analysis menu of the FLEXISORB card allows you to accessseveral methods for displaying gridded data. You can also accessthese methods from the ISOSURFACES card in the TABLES &GRAPHS card deck. To display an isosurface, you must first loada grid file. To do this, you must select the Analysis/Surfaces itemon the FLEXISORB card. The Grid Files... button opens a panelthat allows you to select .grd and .mbk files. Once you select thefile, click the LOAD button. The name of the selected grid file isdisplayed at the top of the Isosurfaces panel, with the Data Rangeand the mid-point of the data selected as the Isosurface Value. Youmay change this to any desired value within the data range. Togenerate a new surface, click the Create New Surface button. Thename of this new surface appears in the Edit Surface list. If isosur-faces already exist and you wish to create a new surface, you mustensure that no surfaces are selected in the Edit Surface box, other-wise the selected surface (highlighted in black) is replaced. Tomodify the properties of an existing surface, such as its IsosurfaceValue or Color, first click the surface in the Edit Surface list tohighlight it, then change the desired values either by typing themin, or by clicking the arrows.

You can delete an isosurface by selecting it from the Edit Surfacelist and clicking the Delete Surface button.

Mapping a property onto a surface

You can color an isosurface using the data stored within a secondgrid. This can be useful, for example, if you want to display areasof high occupancy of a sorbate on a van der Waal’s surface. Youmust first create the isosurface onto which such a property is to bemapped (as described above), and select its name from the EditSurface list on the panel accessed by selecting the Analysis/Sur-faces item. Clicking the Property Map... button on this same panelopens another panel from which you can select the property gridfrom the list and open it by clicking the LOAD button. You should


9. Flexisorb

then make sure the correct surface is selected, and click the AddProperty button. The selected isosurface is colored according tothe value of the data in the property grid at the surface coordinate,using a spectral range. You can modify this spectral range usingthe commands accessed by clicking the Preferences... button onthe Property Map panel. These commands dictate the correspon-dence between the color spectrum and the data range.

Note

Creating a slice plane

A slice plane is a two-dimensional property map. The data is col-ored by value, using a spectral range, at points on a user-definedplane. You can create multiple slice planes, using the panelaccessed by clicking the Analysis/Slices item. You must first loadan appropriate grid file by clicking the Files... button, selecting thedesired file from the Grid Files panel, and clicking LOAD. Youshould designate a Position and Direction for the slice (in Carte-sian coordinates). The Position specifies a point in space throughwhich the plane will pass, while the Direction specifies a planenormal. To calculate the average position of the selected atoms,select a group of atoms, then click the radio button next to the Posi-tion parameter. Similarly, you can calculate the best fit plane for aselected group of atoms by clicking the radio button next to theDirection parameter. You can then create the slice plane by click-ing the Create New Slice button on the Slices panel.

As with isosurfaces, you can modify the properties of existingslices by selecting the appropriate slice from the Edit Slice list,then changing the value. You can change the Position of the sliceby entering a new value, or by clicking the arrows.

You can display an alternative representation of the slice plane inthe graph window, either as a contour plot or a continuous plot, byclicking the Create Slice Plot in Graph Window button.

Once an isosurface is colored by property, you cannotsubsequently recolor it unless you recreate the isosurface.

Theory


Displaying a cloud point diagram

The cloud point diagram displays grid data points colored by theirdata value. If a grid is very dense, the data points are automaticallysparsed. To view a cloud point diagram, you should first select theAnalysis/Cloud Points item on the FLEXISORB card. You mustthen load a grid file, using the panel accessed when you click theGrid Files... button. You the should specify the Property range toplot and the Color mapping, then click the Create Cloud Pointsbutton.

Note

Troubleshooting Note that each of the grid display options has its own grid brows-ing and loading capability, so different display options may besimultaneously used on different grids. You must be careful toselect the appropriate grid for the display style of interest. Forexample, you could load a grid for Isosurfaces, but neglect to loada grid for Cloud Points.

Theory

Monte Carlo methods are widely used to compute phase equilib-ria. Common to all these methods is the insertion and deletion ofmolecules to relate the chemical potential, a "statistical" property,to "mechanical" properties such as density and pressure. In stan-dard canonical ensemble Monte Carlo, this is accomplishedthrough the use of Widom insertions (Widom, 1963). In grandcanonical Monte Carlo (GCMC) simulations, molecules are cre-ated and destroyed to maintain a fixed chemical potential, µ. InGibbs ensemble Monte Carlo (GEMC) (Panangiotopoulos, 1987),molecules are transferred between two systems representinghomogenous regions of the different phases.

For dense or confined phases and for situations where multi-atommolecules are involved, standard particle insertion and transfermoves become impractical due to steric overlap. To overcomethese problems, configuration-biased Monte Carlo (CBMC) meth-

Data points lying outside the specified range are not plotted.Also, you may only display a single cloud point per model.


9. Flexisorb

ods have been developed (Siepmann, 1992; Frenkel, 1992; dePablo, 1992), based upon the Rosenbluth sampling scheme (Rosen-bluth, 1956). These biasing techniques have been successfullyapplied to GEMC in order to calculate phase behavior of mole-cules with varying complexity, including a series of n-alkanes(Siepmann, 1993). Configurational-biased techniques have alsobeen adapted to the study of adsorption in microporous solidssuch as zeolites. A configurational-biased grand canonical MonteCarlo (CB-GCMC) method was used to compute adsorption iso-therms of hydrocarbons in zeolites (Smit, 1995; Frenkel and Smit,1996), while configurational-biased Monte Carlo integration (CB-MCI) has been used to examine the infinite dilution behavior oradsorbate molecules in zeolites (Maginn et al., 1995).

GCMC is the most natural technique for examining sorption inmicroporous materials such as zeolites. In this approach, theadsorbed phase (solid) is assumed to be rigid and in contact witha vapor phase at constant chemical potential. Given an accurateequation of state for the vapor phase, one can directly relate thechemical potential to the pressure, thus eliminating the need tosimulate the fluid phase altogether.

Here we describe the methodology used in implementing CB-GCMC in the Flexisorb module. Importantly, this implementationallows one to properly simulate all-atom or united-atom models oflinear and branched alkanes. Previous CB-GCMC formulationswere limited to linear alkanes. The Flexisorb method and codewere developed by Michael Macedonia and Edward Maginn(Macedonia and Maginn, 1998).

General CB-GCMC Acceptance Rules

Here we derive the form of the Monte Carlo acceptance rules thatremove the bias introduced by preferentially generating low-energy conformations. Let Πmn be the one-step transition probabil-ity of going from state m to state n in a Monte Carlo simulation. Letαmn be the probability of attempting such a move, and let ρm andρn be the grand canonical ensemble probability distribution func-tions of states m and n having Nm and Nn molecules, respectively.Microscopic reversibility requires that:

Theory


Eq. 14

For the grand canonical ensemble,

Eq. 15

where µ is the chemical potential of the system and ν is the poten-tial energy of the system which is a function of r, the multi-dimen-sional vector containing the Cartesian coordinates of all the atomsin the Nm molecules. The acceptance rule for moves which guaran-tees that Eq. 14 is met is:

Eq. 16

If conformations are generated randomly, then αmn = αnm andstandard Monte Carlo acceptance rules are recovered. In the Flex-isorb code, the form of the attempt probabilities αmn and αnm arealtered such that the total compute time required to take properthermodynamic averages is minimized. To see how this methodworks, consider the case of moving between states m and n, whereNn = Nm + 1. Through substitution of Eq. 15, into Eq. 16, the accep-tance rule for insertions is:

Eq. 17

and for deletions:

Eq. 18

where ∆ν is the difference in energy between states m and n.Acceptance rules for thermal equilibrations can also be obtainedby considering the case when states m and n contain the same

Πmnαmnρm Πnmαnmρn=

ρmαβµNm[ ] V

Nmexp

Λ3Nm

Nm!

------------------------------------------ βν r( )–[ ]exp

Πmn min 1αnmρn

αmnρm-----------------,

=

Π mnins

min 1αnm

αmn---------- βµ( ) Vexp

Λ3Nm 1+( )

------------------------------- β∆νmn–

exp,

=

Π nmdel

min 1αmn

αnm----------

Λ3Nm

βµ( ) Vexp--------------------------- β∆νnm

–

exp,

=


9. Flexisorb

number of molecules. The general acceptance rule for these movesis:

Eq. 19

Pressure-explicit forms of the insertion and deletion acceptancerules can be obtained as follows. The chemical potential of a spe-cies in the gas phase is the sum of an ideal gas and residual term:

Eq. 20

Note that for mixtures at low to moderate pressures, it is reason-able to assume that the gas phase forms an ideal solution, and thusthe chemical potential of each component can be computed fromits partial pressure by Eq. 20. Calculation of µRES can be accom-plished through integration of an equation of state at the simula-tion pressure and temperature via the following relation:

Eq. 21

where φ is the fugacity coefficient, Zc is the compressibility factor,and ρ is the molar density of the gas phase. In the Flexisorb mod-ule, the empirical (but accurate) Benedict-Webb-Rubin (BWR)equation of state is used to evaluate Zc. If BWR coefficients do notexist for the species under consideration, then the pressure set atthe start of a simulation will actually be the fugacity. For suffi-ciently low pressures, φ ≈ 1 and µRES can be safely neglected. Theideal gas chemical potential is given by:

Eq. 22

where the first term is a kinetic energy term comprised of knownquantities. The second term in Eq. 22 is zero for simple spheres,but is nonzero for molecules having internal degrees of freedom.ZIG is the ideal gas configurational integral of the molecule:

Π mnNVT

min 1αnm

αmn---------- β∆νnm

–

exp,

=

µ µIG µRES+=

µRES 1β--- φln

1β--- Zc 1– Zc Zc 1–[ ] ρd

ρ------

0

P

∫+ln– = =

µIG 1β--- PΛ3

kBT----------

1

β--- Z

IG

Ω---------

ln–ln=

Theory


Eq. 23

and Ω is the volume associated with the generalized coordinates,q, given by:

Eq. 24

The cbmc.frc forcefield employs a classical flexible model in thelimit of infinitely stiff bond lengths, which results in 2M + 1 gener-alized coordinates per molecule of M atoms. It is shown in Maginnet al. (1995) that for a known sampling distribution, ρ(q):

Eq. 25

While ρ(q) can be any probability distribution, here it representsthe configurational-biased distributions described in later sec-tions. Evaluation of ZIG requires the simulation of a single mole-cule in the ideal gas phase. Notice that ZIG is only a function oftemperature and sorbate identity. Since it is possible to obtain ZIG

for a range of temperatures during a single integration, this calcu-lation needs to be performed only once for each sorbate examined.Since the calculation is fast, however, in the present implementa-tion of Flexisorb, ZIG is evaluated for each sorbate and tempera-ture with an independent simulation.

Using biasing strategies similar to those used to compute the idealgas configurational integral, the configurational integral of the sor-bate molecules in the zeolite phase may also be computed with theFlexisorb module. Such a calculation is very rapid, and allows oneto compute the Henry’s constant and the infinite dilution isostericheat and partial molar entropy of sorption. This calculation cantypically be performed in a fraction of the time required to com-pute a full isoterm, and can provide a good check on the conver-gence of a full GCMC simulation. The theoretical underpinningsof the CB-MCI method may be found in Maginn et al. (1995).

ZIG βνIG

q( )–exp2M 1+

qd∫=

Ω 2M 1+qd∫=

ZIG

Ω---------

βνIGq( )–exp

ρ q( )------------------------------------------⟨ ⟩

ρ q( )1

ρ q( )-------------⟨ ⟩

ρ q( )

-----------------------------------------------------------=


9. Flexisorb

For the insertion acceptance rule, substitution of Eq. 20 into Eq. 17yields:

Eq. 26

while for deletions, substitution of Eq. 20 into Eq. 18 gives:

Eq. 27

where f is the fugacity of the gas phase. Eq. 26 and Eq. 27, alongwith Eq. 19, are the acceptance rules used in the Flexisorb module.One is at liberty to use any form of the attempt probabilities αmnand αnm that optimize the efficiency of the algorithm. Care must betaken, however, to ensure that the form of the attempt probabilitiesthat appear in these equations actually corresponds to the attemptprobabilities used in the simulations, so that the biasing is prop-erly removed. A combination of energy and conformational bias-ing moves are used in the Flexisorb module. These moves exploitthe fact that the zeolite lattice is rigid, so certain locations withinthe pores are more favorable than others. The code also uses thefact that the internal degrees of freedom are much "stiffer" thanexternal degrees of freedom, therefor biasing with these degrees offreedom is more aggressive. Complete details of the form of theattempt probabilities can be found in Macedonia and Maginn(1998).

Model details

Forcefield Following Kiselev and coworkers (1985), the zeolite is modeled asa rigid lattice of oxygen atoms, with lattice atoms taken from theMSI databank. Adsorbate molecules are modeled using either anall-atom (AA) forcefield, or a united-atom (UA) forcefield withLennard-Jones parameters optimized for zeolite adsorption (Vlugtet al., 1998). The UA forcefield parameters are summarized inTable 2.

The non-bonded interactions are calculated for interaction sites indifferent molecules or those separated by more than three bonds

Π mnins

min 1αnm

αmn---------- Ω

ZIG

--------- fβVNm 1+( )

----------------------- β∆νmn–

exp,

=

Π nmdel

min 1αmn

αnm---------- Z

IG

Ω---------

Nm

fβV--------- β∆νnm

–

exp,

=

Theory


in the same molecule by the 12-6 Lennard-Jones potential function(Allen and Tildesley, 1987), truncated at a radius of 13.8 Å. Due tothe periodicity of the system, tail corrections are not applied. Bondlengths are fixed at a value of 1.54 Å. Bond angle bending interac-tions are calculated by a harmonic bending potential given by:

Eq. 28

Dihedral angle potentials are treated with the form (Jorgenson etal., 1984) given by:

Eq. 29

Parameters of the all-atom forcefield are given in Table 3.

As with the UA forcefield, non-bonded interactions are treatedwith the 12-6 Lennard Jones function, truncated at 13.8 Å with notail corrections. Bond lengths are fixed at 1.105 Å for C-H and 1.526Å for C-C. Bond angle bending interactions are described byEq. 28, while the dihedral angle potential function is given by:

Table 2. United-atom forcefield parameters

Non-Bond ε/kB[K] α[Å]

CH3 98.1 3.77

CH2 47.0 3.93

CH 12.0 4.10

CH3-O 80.0 3.60

CH2-O 58.0 3.60

CH-O 58.0 3.60

Bond Angle kθ [kcal rad-2] θo [degrees]

CHχ-CH2-CHχ 62.1 114

CHχ-CH-CHχ 62.1 112

Torsion ao [kcal] a1 [kcal] a2 [kcal] a3 [kcal]

CHχ-CH2-CH2-CHχ 0 0.70544461 -0.13549350 1.57235284

ν θ( ) kθ θ θo–( ) 2=

νUA φ( ) ao a1 1 φcos+( ) a2 1 2φcos–( ) a3 1 3φcos+( )+ + +=


9. Flexisorb

Eq. 30

Unlike Eq. 29, this function does not favor the trans state over thegauche state. Yet, through combination of Eq. 30 and intramolecu-lar non-bond interactions, the appropriate barrier of rotation isultimately realized.

Other simulation details Prior to the start of a simulation, the simulation box size is auto-matically computed to ensure that the minimum image conditionis not violated. A potential energy map is tabulated on a three-dimensional grid for fast potential interpolation (June et al., 1990).An ensemble of molecular fragments are also generated for use inthe construction of molecules during insertion and thermal equil-ibration steps. In addition, ideal gas configuration integrals (andthus chemical potentials) for each sorbate are evaluated via MCintegration. If these pre-calculation items have already been com-puted, the code does not execute these calculations. Each MC cycleconsists of an attempted molecule insertion, deletion, translation,

Table 3. All-atom forcefield parameters

Non-Bond ε/kB[K] σ[Å]

CHχ 19.6276 3.88

H 19.1243 2.45

O 114.7422 2.86

Bond-Angle kθ [kcal rad-2] θo [degrees]

H-C3-H 39.5 107.67

H-C2-H 39.5 106.4

Cχ-C3-H 44.4 111.26

Cχ-C2-H 44.4 109.91

Cχ-C1-H 44.4 109.22

Cχ-C2-Cχ 46.6 110.55

Cχ-C2-Cχ 46.6 109.8

Torsion kφ [kcal] n-φo [degrees]

[*] - Cχ - Cχ - [*] 1.4225 3-0

νAA φ( ) kφ 1 nφ φo–( )cos+[ ]=

References


or cut and regrow thermal equilibration move. For good statistics,it is recommended that the system be allowed to equilibrate foranywhere from two million to ten million moves, prior to takingaverages over the last two million moves. Based on past experi-ence, however, the biasing is so effective that within 500,000moves, all but the bulkiest molecules at the highest loadings willbe essentially equilibrated. The magnitude of molecule transla-tions is adjusted automatically throughout the simulation suchthat 50% of translations are accepted.

References

Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids, Claren-don Press: Oxford (1987).

dePablo, J. J.; Laso, M.; Suter, U. W. J.Chem.Phys., 96, 2395 (1992).

Frenkel, D.; Mooij, G. C. A. M.; Smit, B. J.Phys.: Condens. Matter, 4,3053 (1992).

Frenkel, D.; Smit, B. Understanding Molecular Simulation: From Algo-rithms to Applications, Academic Press: San Diego (1996).

Jorgenson, W. L.; Madura, J. D.; Swenson, C. J. J. Am. Chem. Soc.,106, 6638 (1984).

June, R. L.; Bell, A. T.; Theodorou, D. N. J. Phys. Chem., 94, 8232(1990).

Kiselev, A.V.; Lopatkin, A. A.; Shulga, A. A. Zeolites, 5, 251 (1985).

Macedonia, M. D.; Maginn, E. J. Mol. Phys. In Press.

Maginn, E. J.; Bell, A. T.; Theodorou, D. N. J.Phys.Chem., 99, 2057(1995).

Panangiotopoulos, A.Z. Mol. Phys., 63, 527 (1987).

Rosenbluth, M. N.; Rosenbluth, A. W. J.Chem.Phys., 23, 356 (1956).

Siepmann, J. I.; Frenkel, D. Mol. Phys., 75, 59 (1992).

Siepmann, J. I.; Karaborni, S.; Smit, B. Nature, 365, 330 (1993).

Smit, B. Mol. Phys., 85, 153 (1995).


9. Flexisorb

Vlugt, T. J. H.; Zhu, W.; Kapteijn, F.; Moulijn, J. A.; Smit, B.;Krishna, R. J.Am.Chem.Soc., 120, 5599 (1998).

Widom, B. J. Chem. Phys., 39, 2808 (1963).


10 Sorption

The Sorption module simulates adsorption of molecules intoporous crystal structures such as zeolites. More than one type ofsorbate molecule can be used in the simulation, and pressure andtemperature conditions can be varied.

Introduction

Information obtained Information obtained from sorption simulations includes adsorp-tion isotherms, binding sites, binding energies, loading capacity asa function of pressure, and Henry’s constant.

Output forms Output can be viewed in four forms:

♦ The positions of the sorbate molecules adsorbed in the frame-work can be displayed in the model window and files contain-ing snapshots of these conformations saved.

♦ Various plots pertaining to the simulation can be created anddisplayed in the graph window.

♦ A sorption trajectory file can be written.

♦ Information can be sent to the text window both during andupon completion of the simulation.

Sections in this chapter Using Sorption

Running a sorption simulation

Settings for the energy calculation

Output during the simulation

Setting up and running the simulation

Analysis of sorption trajectory files

Theory


10. Sorption

References

Using Sorption

Sorption is used to simulate sorption of small molecules, calledsorbates, into porous 3D frameworks. The frameworks are typicallymicroporous inorganic structures such as zeolites and alumino-phosphates. Characterizing the behavior of these materials hasimportant application in fields of catalysis and separations tech-nology.

The Sorption module has been designed to be used by experimen-talists, as well as computational chemists. Output-analysis fea-tures, such as automatic calculation and display of isotherms,make it easy to directly compare Sorption data with bench results.

Sorption implements a rapid Monte Carlo statistical mechanicsmethod (Metropolis et al. 1953, Allen and Tildesley 1987); threedifferent types of simulation can be performed:

♦ Fixed loading (canonical ensemble) — The total number ofsorbates in the system remains constant throughout, thoughthe positions and orientations of the sorbates are changed ateach step.

♦ Fixed pressure (grand canonical ensemble) — Again, sorbatepositions and orientations are varied but, additionally, sorbatesare created and destroyed during the course of the simulation.

♦ Henry’s constant — This simulation is used to calculateHenry’s constant. Henry’s constant is defined as the simula-


Building crystals Cerius2 Builders bookSketching (sorbate) molecules Cerius2 Modeling Environment bookEnergy calculations The “Open Force Field” chapter in

the Cerius2 Simulation Tools bookAssigning charges The Cerius2 Simulation Tools bookSaving and modifying graphs Cerius2 Modeling Environment book

Using Sorption


tion-cell loading divided by the sorbate pressure in the limit ofvanishing pressure.

Running a sorption simulation

The general steps involved in a sorption simulation are outlinedbelow. Detailed procedural directions on how to use the modulefollow in subsequent sections.

1. Prepare models.

a. Create or load from a file one or more sorbate molecules. Thesorbates must be both nonperiodic and of neutral charge. Ifthe sorbate is flexible, a trajectory file containing representa-tive conformations of the sorbate may be generated, usingmolecular dynamics or conformational analysis.

b. Create or load from a file the framework model. The frame-work model must be a 3D periodic structure of both primi-tive (P1) symmetry and neutral charge. If a surfacecalculation is desired, a vacuum-slab model must be gener-ated.

Some commonly used sorbates and framework structurescan be found in the Cerius2-Models /sorbates, /catalysts,and /zeolites directories.

2. Specify the simulation type and set up various simulationoptions (that is, temperature, length of run, move sizes, moveprobabilities, and output).

3. Choose a forcefield and other energy calculation options to beused during the simulation. A number of forcefields developedspecifically for Sorption energy calculations are provided in theCerius2-Resources/FORCE-FIELD directory. For more infor-mation on the sor_yashonath1.01, sor_demontis1.01, sor_pickett1.01, and watanabe-austin1.01 forcefields, see the Force-field-Based Simulations book.

4. Run the simulation.

First, Sorption generates random configurations by translatingand rotating, and, depending on the ensemble type, creatingand destroying sorbate molecules in the framework. The maxi-mum size of translation and rotation moves is set through the


10. Sorption

Sorption Step Sizes control panel. (See the online help for moredetailed information on control panels.)

Next, Sorption either accepts or rejects the configurations onthe basis of energy and, in the grand canonical ensemble, pres-sure. Configurations with lower energy are more likely to beaccepted.

As the simulation proceeds, a large set of configurations is gen-erated. Typically, 105–107 configurations are generated in a sim-ulation.

5. Upon completion of the simulation, configurations saved to asorption trajectory file can be analyzed: energy distribution,average interaction energies, positional distributions, and so oncan be extracted from these trajectories.

General methodology

Settings for the energy calculation

Energy calculation in Sorption is confined to intermolecular ornonbond energies; that is, sorbate-framework energies and sor-bate-sorbate energies. Because both the framework and the sor-bates are held internally rigid during the simulation, no valenceenergy terms are required in the sorption energy calculation.

Sorption relies on the Open Force Field (OFF) for loading theforcefield, calculating and assigning forcefield atom types, and, ifthe Universal forcefield is used, assigning bond order. However,unlike modules such as Minimizer and Dynamics, Sorptiondoesn’t use the energy terms selection or any of the calculationpreferences from OFF.

You probably want to be somewhat familiar with OFF before run-ning sorption simulations. For information about OFF, see theCerius2 Simulation Tools book.

Bump checks Before energy is calculated, Cerius2 looks for bad contacts betweensorbates and the framework. Configurations in which sorbate andframework atoms are very close together have high energies, andconformations with high energies are very likely to be rejected.

General methodology


The bad contact check (sometimes called a bump check) simplysaves calculation time by rejecting the high-energy configurationswithout having to do the energy calculation.

The default value for the bad contact rejection factor is 0.5; that is,the configuration is rejected if any sorbate and framework atomsare closer to each other than half of their van der Waals radii. Ingeneral, the rejection factor should be between 0.5 and 0.8. The aimis to find the right balance between needlessly rejecting acceptableconfigurations and wasting time calculating high-energy configu-rations.

Excluded volumes Certain areas within framework structures may be physically inac-cessible to a sorbate molecule diffusing through the lattice. Due tothe random nature of the insertion method, these areas might stillbe sampled during a Sorption calculation, leading to an over-esti-mation of loading. To counter this effect, you can calculate anaccessibility grid using the Free Volumes panel, accessed byselecting the Simulation Controls/Excluded Volumes item fromthe SORPTION card, and clicking the Free Volumes... button. Youmay then use this grid to prohibit sampling of inaccessible regions.

van der Waals radiusreduction

Before running a sorption simulation, you may want to reduce thevan der Waals radii of some elements, particularly atoms withlarge positive charges where the actual radii are significantlysmaller than the van der Waals radii of the neutral atoms (see theCerius2 Modeling Environment book).

Sorption calculates the van der Waals and Coulomb energy of thesorbate/framework system. Hydrogen bonds are not explicitlycalculated, but hydrogen-bonding effects can be included throughcareful choice of van der Waals forcefield parameters and charges.

van der Waals energy VDW energy is calculated by summing all pair interactions withina specified sphere, the radius of the sphere being determined bythe interaction cutoff distance.

Minimum image conven-tion

The minimum-image convention is used to approximate van derWaals energy in the periodic framework. In the minimum imageconvention, an atom is considered to interact only with its closestneighbor atoms in a periodic box around it. For more about theminimum image convention, see the Forcefield-Based Simulationsbook and Allen and Tildesley (1987).


10. Sorption

When the minimum-image convention is applied to the evaluationof inter-sorbate energy, the sorbate molecules are treated as wholeunit, and not split across minimum-image borders. That is, if thecenter of a molecule lies within the border, the energy interactionsof the whole sorbate molecule are considered, not just those ofatoms within the border. This differs from the treatment of frame-work/sorbate interactions, because sorbates are not considered torepeat periodically in the way that the framework does.

Note

Coulomb energy Including Coulomb interactions in the sorption-energy expressionis optional. The Coulomb energy calculation takes time and, thus,you should check if the Coulomb energy contribution is significantcompared to the VDW energy before beginning a long simulation.For weakly charged systems, omitting the Coulomb energy cangreatly reduce simulation time with minimal loss of accuracy.

Ewald summation method Sorbate/framework electrostatics are evaluated using the Ewaldsummation method. This method accelerates the long-range Cou-lomb calculation by splitting the slowly converging real-spacesum into a quickly converging real-space sum and a reciprocal-space sum. If a long calculation, or series of calculations, is to beperformed using Ewald summation, it may be expeditious to pre-calculate the reciprocal space component over a grid. Thereafter,lookup and interpolation will be used rather than explicit calcula-tion. The duration of the calculation must be sufficiently long tojustify the time required to calculate the grid. For more informa-tion on the Ewald sum, see Forcefield-Based Simulations book.

For the interaction cutoff to be consistent with the minimumimage convention, the cutoff sphere must be smaller than theperiodic cell. If the framework cell is small (dimensions lessthan roughly 20 Å) then, rather than reduce the interactioncutoff distance, it is preferable to enlarge the periodic cell of theframework by making a superlattice of a larger number of cells.

For information about making a superlattice, see the CrystalBuilder chapter in the Cerius2 Builders book. In addition togiving more accurate van der Waals energy values, using alarger cell improves the simulation statistics.

General methodology


Note

Setting energy calculation options for a sorption simulation

1. Open the Sorption Energy control panel (see the online help formore information) by selecting the Energy Calculation itemfrom the SORPTION card.

Forcefield 2. If the default forcefield and atom typing rules are adequate forthe simulation, check the Use Automatic Force-Field Optionsbox and skip to step 5, below.

The default forcefield is usually the Universal 1.02 forcefield. Tosee which forcefield is set as the default, open the Load Prefer-ences control panel in the Open Force Field module. (Select theEnergy Expression/Automate Setup item on the OPENFORCE FIELD card to open the Automate Energy Setup con-trol panel, then select the Preferences… button to open theLoad Preferences control panel.)

3. To use a forcefield other than the default, but have atom typesfor that forcefield automatically calculated:

a. Check the Use Automatic Force-Field Options box.

b. Load your choice of forcefield. For more information onloading forcefields, see the Cerius2 Simulation Tools book.

Atom types 4. To override or alter calculated atom types:

a. Uncheck the Use Automatic Force-Field Options check box.

b. Load your choice of forcefield.

c. Assign and alter forcefield atom types (see the Cerius2 Simu-lation Tools book).

Bad contacts 5. If you want, alter the bad contact rejection factor from itsdefault value of 0.5.

If there are cations in the framework or strongly positive atomsin the sorbate molecules, you may want to reduce the VDWradii of these elements to values approaching the ionic radii.

Because the nonperiodicity of the sorbates precludes the use ofthe Ewald summation method, sorbate/sorbate electrostaticsare evaluated directly.


10. Sorption

VDW radii can be edited through the Edit Elements controlpanel accessed by choosing Element Defaults… from theBuild pulldown.

van der Waals energy 6. If you want, alter the interaction cutoff from its default value.Because the interaction cutoff must be consistent with the min-imum image convention, its size is limited by the cell dimen-sions.

7. If charges have been assigned to the framework and sorbatemodels, then you may want to check the Include CoulombEnergy check box.

Coulomb energy 8. If Coulomb energy is included, set the Ewald parameters forthe electrostatic energy summation:

a. Open the Sorption Ewald control panel (see the online help)by selecting the Ewald Parameters… button on the SorptionEnergy control panel.

b. The real-space cutoff is the same as the interaction cutoff onthe Sorption Energy control panel.

c. Under the Automatic Parameter Estimation heading, enterthe required accuracy for the Coulomb energy calculation inkcal/mol.

d. Click the Estimate Ewald Parameters action button toupdate both the reciprocal-space cutoff and the Ewald sumconstant to suitable values.

Using Ewald Grids 9. If long calculations involving electrostatic interactions are to beperformed, it may be desirable to pre-compute part of theEwald summation and use interpolation. The steps outlinedabove should first be followed for specifying the Ewald accu-racy.

a. Open the Sorption Ewald Grid control panel by selecting theEwald Grid... button on the Sorption Energy control panel.

b. Ensure that the current model is set to the host lattice. Spec-ify a name for the grid file, then click the Calculate Ewalddata grid action button.

c. To ensure that the grid is used in later calculations, check theUse Ewald data grid checkbox, and select the grid from

General methodology


which to read the Ewald data. If the grid already exists froma previous calculation, step b above can be omitted.

Output during the simulation

There are several ways to monitor and record the progress of thesimulation. Before beginning the simulation, use the Sorption Out-put control panel to set these forms of output.

♦ Model updates — The position of sorbates in the frameworkcan be displayed in the model window at regular intervals. Thisallows you to see if the sorbate is occupying a range of positionsor if it is confined to a single location in the host.

♦ Plots — Plots that allow you to monitor the progress of the sim-ulation can be sent to the graph window during the simulation.Plots tell you how many configurations have been completedand are useful for determining if the simulation has reachedequilibrium, or whether a simulation should be carried to com-pletion or terminated early. The values plotted vary accordingto the type of simulation, as shown in the following table.

♦ Sorption trajectory files — A trajectory file must be written ifyou want to use the analysis facilities of Sorption (see “Analysisof sorption trajectory files” on page 245). Usually, only a small(but representative) fraction of the configurations is written tothe trajectory file; typically, about one percent.

Trajectory files can get very large. For example, in a simulationcell of 624 atoms with a maximum of 250 water sorbate mole-cules run for 10 million configurations with output every 10,000steps, the trajectory file would be about 11 MB.

Simulation Plot type

Fixed pressure Average total energy versus number of configurationsAverage loading versus number of configurationsAverage energy per sorbate molecule vs. number of configurations.

Fixed loading Average total energy versus number of configurations

Henry’s constant Henry’s constant versus number of trials


10. Sorption

Sorption trajectory files are non-ASCII and as such cannot beviewed with a text editor.

♦ Text — Output to the text window varies according to the typeof simulation performed. Whether or not you choose text out-put during the simulation, a summary of results is reportedboth during the course of the simulation, and upon completionof the simulation. These summaries allow you to determinewhether or not the calculation has converged. This text will alsobe written to a log file, which can be reviewed following the cal-culation.

♦ Sorption snapshot files — A file containing a snapshot of theconformation at predetermined intervals can be generated.

Note

Although control panels cannot be opened or closed during a sim-ulation, the text, model, and graph windows can be resized, lay-ered, shrunk to icons, and expanded out from icons withoutinterrupting the simulation. This allows you to inspect output inall three windows, even if they are overlapping at the start of thesimulation.

Setting output options for the simulation

1. Open the Sorption Output control panel by selecting Simula-tion Controls/Output from the SORPTION card.

Model window output 2. If you want to see the position of sorbates in the frameworkupdated in the model window during the course of the simula-tion, check the Update Model During Simulation box andspecify a model update frequency. This option is not recom-mended for long sorption runs.

3. If you want to see the position of sorbates in the model windowat the conclusion of the simulation, check the Update Model atEnd of Simulation box.

If the model is updated in this way, the sorbates become part ofthe framework model, and, if another sorption simulation is

With any of the above four options, it is important to rememberthat processing and writing output takes time. To minimize thelength of a simulation, select only the types and frequency ofoutput that you absolutely need.

General methodology


run on the same framework, you must first delete the old sor-bate molecules.

Plotted output 4. If you want progress of the simulation to be sent to the graphwindow during the simulation, choose the Use Graphical Sim-ulation Monitor option and specify an update frequency. Plotssent to the graph window depend upon the type of simulationchosen.

Trajectory file output 5. If you want to save the simulation to a trajectory file so that itcan be analyzed later, choose the Write Trajectory File Duringthe Simulation box and specify the trajectory write frequency.Remember that trajectory files can get quite large. Also, specifythe root name of the trajectory file name in the Filename Seedentry box.

Snapshot file output 6. If you want to save sample conformations to a file so that it canbe analyzed later, choose the Write .msi snapshots box andspecify the snapshot frequency. Remember that these files canget quite large. Also, specify the root name of the snapshot filename in the Filename Seed entry box.

Note

Text output 7. If you want output sent to the text window during the courseof the simulation, choose the Output Text During the Simula-tion box and specify the write frequency for the text. This textwill also be echoed to a log file.

Setting up and running the simulation

Basic run parameters for the simulation (that is, run length, tem-perature, and pressure) can be found on the Run Sorption controlpanel. The more advanced options, which rarely need altering, canbe found on smaller control panels accessed by selecting the Sim-ulations Controls item.

Framework The sorbing framework for the simulation must be in the currentmodel (in the model window) at the start of the simulation. It mustbe a 3D periodic model with primitive symmetry, overall neutral

Trajectory files cannot be saved for Henry’s constantsimulations.


10. Sorption

in charge, and of appropriate size; that is, not too small (see Noteon page 234).

Surface If a surface constrained calculation is to be performed, the 2D sur-face model must first have been converted to a vacuum-slabmodel. This can be achieved easily using the CRYSTAL BUILDERmodule.

a. Having generated a 2D surface, via the SURFACEBUILDER module (see the Builder manual on line), openthe Preferences... subpanel on the Crystal Building panel.

b. Specify the Vacuum Thickness. This should be at least twiceas big as the direct cutoff which will be subsequently used.

c. Specify the surface Orientation.

d. Convert the surface to a 3D vacuum-slab model by clickingthe BUILD CRYSTAL button.

Surface Setup In a surface constrained calculation, the sorption placements arerestricted to lie within a specified distance above a defined surface.The surface height plus short-range cutoff distance should notexceed the width of the vacuum region. Prior to calculation, acheck is made to determine if the specified surface orientationappears to be correct, and a warning will be issued if the surfaceorientation seems inconsistent with the model.

Sorbate Sorbate molecules are typically small (neutral) molecules. Thoughthey can reside in any model space, to participate in the simula-tion, they must appear on the Run Sorption control panel.

Move probabilities Move probabilities control the choice of move and, during a simu-lation, are used to generate a new trial configuration from the pre-vious configuration.

In the fixed-loading simulation, the only choice of move isbetween a translation or a rotation.

In the fixed-pressure or Henry’s constant simulations, the movecan be the creation of a new sorbate, the destruction of an existingone, a rotation, or a translation. Creation and destruction probabil-ities are always set equal, ensuring microscopic reversibility.

By default, each type of move is given equal probability. Nonethe-less, you can bias the choice by editing the probabilities for eachmove type in the Move Probabilities control panel. For example, if

General methodology


the sorbate is a single atom or a spherically shaped molecule,reducing the rotation probability shortens the simulation with noloss of information.

Maximum step sizes The size of a random translation or rotation move for a given sor-bate is limited by the maximum step size. Typical default startingvalues are 1 Å for the maximum translation step and 50˚ for themaximum rotation.

This maximum step size can be a significant parameter in the sim-ulation. If it is high, the risk of the new sorbate making bad con-tacts with the framework is large, thus generation of valid newconfigurations may be slow. However, if the maximum step size islow, the simulation takes a long time to generate configurations farfrom the starting point; in effect, too many similar configurationsare generated. Both these cases cause the simulation to take a longtime to equilibrate; thus, it is desirable to strike a balance betweenthem.

The maximum step-size values can be held constant throughout asimulation, if you are satisfied that they give an acceptable equili-bration period. By default, however, they are used as starting val-ues for the simulation and are automatically rescaled during therun to try to minimize the equilibration period.

Rescaling step sizes How to determine the correct bias for the step size, such that equi-librium is achieved most quickly? The equilibration rate is conven-tionally held to be highest when the success rate for the sorbateand move type in question is approximately 30%–50%. By default,the target rate is set for acceptance of half the steps sampled in thesimulation.

Three kinds of control parameters can be specified on the SorptionStep Scaling control panel:

♦ Target success rate — The actual success rate is defined as thenumber of accepted configurations for the sorbate divided bythe number of trial steps. (Only steps using the relevant movetype are considered.) When the actual success rate meets thedefined target success rate, no step-size rescale is required.

♦ Rescale factor — If the actual success rate is greater than thetarget rate, the step size is increased by multiplying the stepsize by the rescale factor. Alternatively, if the actual success rateis low, the step size is divided by the rescale factor.


10. Sorption

♦ Rescale frequency — The rescale frequency is the number ofMonte Carlo steps (rejected and accepted) between step-sizerescales.

A single rescale frequency is set per simulation, but a uniquerescale factor and target success rate can be set for each move typefor each sorbate.

To set up and run the simulation

Once the sorbate and framework models have been loaded (see“Running a sorption simulation” on page 231), energy options set(see “Setting energy calculation options for a sorption simulation”on page 235), and output options set (see “Setting output optionsfor the simulation” on page 238), you are almost ready to begin thesimulation.

1. Make the model that contains the framework use the currentmodel.

2. Open the Run Sorption control panel (see the online help) fromthe Run item on the SORPTION card.

3. Choose the type of simulation from the popup.

4. Enter the simulation temperature in the entry box.

5. Enter the length of run. This is the total number of steps, includ-ing both accepted and rejected configurations.

6. Enter sorbate molecules into the list box by entering the modelspace number of each sorbate into the Toggle Model in SorbateList entry box.

To remove a sorbate from the list, simply re-enter the modelspace number in the entry box.

If you use multiple sorbates for the fixed loading calculation,the program first attempts to add the requested number of spe-cies. If it cannot add these due to energetic considerations, theprogram issues a warning message and the calculation contin-ues.

If a fixed pressure simulation, enter the partial pressure (or, fora non-ideal gas, the fugacity) for the sorbate into the list box.

General methodology


7. If flexible sorbate molecules are to be used, a trajectory file con-taining representative conformations must be associated withthe sorbate. This trajectory file may have been generated byusing molecular dynamics in OFF, or by conformation analysis.To associate the trajectory file with the sorbate, open the Simu-lation Controls/Sorbate Conformers panel. Select the desiredsorbate from the Sorbate List, which will be highlighted. Selecta trajectory file from the Conformer Files browser. The selectedsorbate and conformer file will be added to the list. To specifythat the calculation should use the structures from the con-former file, rather than the sorbate structure, as loaded, clickthe Use checkbox adjacent to the conformer file of interest.

The program will then extract a conformer with random prob-ability for each specified conformer file/sorbate combination. Ifa conformer file is not provided, the single conformation of thesorbate, as loaded, will be used.

8. Open the Move Probabilities control panel (see the online help)by selecting the Simulation Controls/Move Probabilities itemfrom the SORPTION card. By default, all moves are set withequal probability; if you want to alter the probability, enter newvalues in the entry boxes.

9. To set the maximum step size per move, open the Sorption StepSizes control panel (see the online help) by choosing Simula-tion Controls/Step Sizes from the SORPTION card. TheHenry’s constant simulation does not use step-size control; forthis simulation type, skip to Step 11.

Default step sizes are 1 Å for translations and 50˚ for rotations.If you want to specify other step sizes, edit the entry boxes.

10.To modify the procedure for step-size rescaling, open the Sorp-tion Step Scaling control panel (see the online help) by selectingthe Rescale Controls… button on the Sorption Step Sizes con-trol panel:

a. By default, step sizes are automatically rescaled to help thesimulation reach equilibrium most efficiently. If you wantthe maximum step size to remain constant throughout thesimulation, deselect the Automatically Rescale Step Sizescheck box.


10. Sorption

b. Specify the frequency of the rescale and, for each sorbateand move type, both a rescale factor and a target successrate. The maximum step size is multiplied or divided by thisrescale factor if the requested acceptance rate is not met. Theacceptance rate is based on the number of accepted to totalmoves during the previous rescale.

11. If a surface constrained calculation is being run, the surfaceregion must first be defined.

a. Open the Simulation Controls/Surface Setup panel. Checkthe Surface Constrained Calculation.

b. Specify the Surface Height and Surface Orientation.Ensure that the orientation is consistent with that used togenerate the vacuum-slab representation of the model.

Begin the simulation 12.Return to the Run Sorption control panel and select the RunSimulation button.

13.During the simulation, you can move and rescale windows,enabling you to look at the output in the model, graph, and textwindows.

Stopping the simulation 14.The simulation can be stopped before completion by selectingthe Interrupt button in the dialog box that appears during therun. Choose the Stop Current Process ASAP option. This endsthe simulation at the end of the current step, writes the normalinformation summary to the text window, and updates themodel window and trajectory file (if appropriate).

Restarting the simulation 15.If the calculation has not converged over the specified numberof steps, or if it has ended prematurely (perhaps due to a prob-lem with disk space availability), you can restart or resume thecalculation using data stored in the restart data file. To restart acalculation, first load the following saved coordinates and sta-tus files back into Cerius2, for use in the Sorption calculation:

♦ <run_name>_host.msi

♦ <run_name>_mol_<#>.msi (one file per sorbate)

♦ <run_name>.rst

These files contain the coordinates of the framework and sor-bates molecules, as well as the status of the calculation. Loadingthe files before restarting ensures that the framework structure

General methodology


and sorbate models exactly match the models stored in the .msifiles.

16.Be sure to add the sorbate models to the sorbate list in the exactsame order as originally, that is, in their correct numericalsequence, as specified by the file name.

17.When you restart a calculation using the Restart panel, the sta-tistics from the terminated calculation are read from the .rst file,and the calculation proceeds from the last stored configuration.The calculation statistics are written to the restart file at thesame frequency as the textport output.

Analysis of sorption trajectory files

Sorption analysis tools help you analyze the simulation by pre-senting the trajectory file data in several ways. Five sets of plotscan be created:

♦ Trajectory file

♦ Mass distribution

♦ Energy distribution

♦ Loading curve

♦ Mass cloud

Each is described on the following pages.

Trajectory file plots Trajectory file data are plotted as energy vs. step number, and, forthe fixed-pressure simulation, as cell loading vs. step number.

These plots present the raw data of the trajectory file; that is, theenergy and loading are instantaneous values rather than thecumulative average values that are displayed in other plots.

Plotting the trajectory file in this way is often the first step in sim-ulation analysis. These plots give an overall view of the trajectoryfile and are particularly helpful in estimating where the simulationequilibrated.


10. Sorption

Mass distribution plots The mass distribution of all sorbate molecules in the analysisrange is plotted. The distribution is projected down a particularzone axis onto a periodic cell. You specify the zone axis and thetwo basis vectors for the plot cell.

Each cell axis is divided evenly into bins, and the number of mol-ecules having centers that fall within each bin square is counted.This value is plotted on the graph.

Energy distribution plots A histogram of energy distribution for each sorbate is plotted. Youspecify the maximum energy, minimum energy, and resolution.

General methodology


Loading-curve plots Loading-curve plots combine average cell-loading values from aset of fixed sorption simulations. The framework and sorbatemodels must be identical in each set. In the example below, threeloading curves from sets of simulations are presented on the samegraph. Six fixed-pressure simulations are made for each tempera-ture.

Because the same analysis range is chosen for each simulation, besure to use the range based on the simulation that was slowest toequilibrate


10. Sorption

.

Mass cloud plots In mass-cloud analysis, the center of mass of each sorbate mole-cule in each configuration is displayed as a dot in the model space.The mass cloud can be displayed concurrently with the frameworkmodel or alone in the model window. The mass cloud results in a3D picture that shows the preferred positions of the sorbates in thelattice. The dots can be colored according to sorbate type or config-uration energy.

General methodology


Plotting sorption trajectory files

All graphs, with the exception of mass cloud plots, created in theprocedures outlined below can be saved, reloaded, and editedusing the Graphs module.

Note

Plotting trajectory files

Plotting trajectory files creates a raw plot of instantaneous energyas a function of configuration and, for the fixed pressure simula-tion, a plot of instantaneous loading versus configuration.

1. Open the Sorption Analysis control panel (see the online help)by selecting Analysis from the SORPTION card.

2. Load the trajectory file of the simulation using the browser boxin the Sorption Analysis control panel.

3. Specify the range of steps you want to include in the analysis.

4. Click the Plot Trajectory File button.

Because sorption trajectory files cannot be saved for a Henry’sconstant simulation, none of the following analysis plots can beapplied to Henry’s constant simulations.


10. Sorption

Plotting mass distribution of sorbates in framework

This procedure produces a 2D plot in which sorbate positions areprojected onto a specified crystal plane.

1. Open the Sorption Analysis control panel by selecting Analysisfrom the SORPTION card.


3. Specify the range of steps you want to include in the analysis(probably excluding the initial pre-equilibrium steps of the sim-ulation).

4. Open the Sorption Mass Plot control panel by choosing the firstPreferences… button of the Sorption Analysis control panel.

5. Choose the axis of projection for the mass distribution.

6. Cerius2 suggests suitable first and second axes vectors based onthe projection vector, but you can override these suggestions byediting the entry boxes.

7. The default of 20 bins on each axis gives a mass distributiongraph of 21 by 21 points. Increase these bin numbers for a graphof higher resolution.

8. Select the Plot Mass Distribution button on the Sorption Anal-ysis control panel to create the plot.

If the simulation contains more than one sorbate type, a sepa-rate plot for each sorbate is sent to the graph window. Pleasesee the online help for more information about the control pan-els.

Plotting energy distribution of sorbates

This procedure plots a graph showing the number of configura-tions as a function of energy.




General methodology


4. Click the Plot Energy Distribution button.

This produces a default energy distribution plot with an energyrange of -50 to 50 kcal/mol divided into 100 bins (10 kcal/molper bin).

5. To rescale the energy graph:

a. Open the Sorption Energy Plot control panel (see the onlinehelp) by choosing the Preferences… button next to the PlotEnergy Distribution button on the Sorption Analysis con-trol panel.

b. Edit the entry boxes to set more appropriate minimum andmaximum energy limits and bin groupings.

c. Display the rescaled graph by clicking Plot Energy distribu-tion on the Sorption Analysis control panel.

Plotting a loading curve

Plots of this type are available only if you have a series of fixed-pressure trajectory files. The framework and sorbates must beidentical in all of the files; file names must all have the same seedand be numbered in consecutive order starting from one, that is:

<run_name>_1.sor, <run_name>_2.sor, <run_name>_3.sor…

If you have the correct files, but incorrect names, simply renamethe files to form an appropriately numbered set (use the UNIX mvcommand).


2. Load any sorption trajectory file in the set using the browserbox on the Sorption Analysis control panel.

3. Specify the range of steps you want to include in the analysis(probably excluding the initial pre-equilibrium steps of the sim-ulation). This range applies to all simulations in the set.

4. Select the Plot Loading Curve button.

The loading curve plot of sorbate molecules per cell as a func-tion of pressure is sent to the text window. A separate data setfor each sorbate type is generated, but all data sets are pre-sented on the same graph.


10. Sorption

Plotting mass clouds (dot density maps) of sorbate positions

Mass clouds are displayed in the Model window. They allow youto view preferred sorbate locations in 3D. (Mass clouds are mostoften shown in the model space that contains the framework.)




4. Open the Mass Cloud control panel (see the online help) bychoosing the second Preferences… button on the SorptionAnalysis control panel.

5. Choose to color the cloud by sorbate type or by energy. Cerius2

automatically sets the property range to that of the trajectoryfile; edit this range if you want.

The default color is red. In the Color Cloud by Energy selec-tion, dots representing the highest and lowest energy sorbatesare red, and dots representing intermediate energies are a rain-bow of colors in between. On some monitors, red dots canappear very faint; if the red dots are hard to see, try a differentcolor setting (for example, white (0) or a yellow shade (45through 85)).

6. With the framework model in the current model space or in anempty model space, select the Plot Mass Cloud button on theSorption Analysis control panel.

Mass clouds can take some time to calculate and, once calcu-lated, can take up a lot of memory.

Theory

Simulation methods

As mentioned above, three simulation methods are available in theSorption module:

Theory


♦ Fixed loading (canonical ensemble).

♦ Fixed pressure (grand canonical ensemble).

♦ Henry’s constant simulation.

Details of each method are discussed below.

Fixed loading (canonical ensemble) simulation

In the fixed loading (or canonical ensemble) simulation, the num-ber of sorbates in the framework is held constant throughout thesimulation.

Note

The initial configuration is generated by placing the sorbate at anarbitrary position in the framework cell. The initial coordinates ofthe sorbate in the framework are the same as the coordinates of thesorbate in its model window. In this way, you can choose to begina simulation by placing the sorbate in a pore of the framework.

Each subsequent configuration is generated by either a randomtranslation or rotation of the sorbate molecule. The choice of moveis governed by move probabilities and limited by the maximumtranslation and rotation step size. The maximum step size values canbe adjusted automatically during the course of the simulation inorder to achieve the target acceptance rate (equilibrium). For moreinformation about move probabilities and acceptance rates, see“Setting up and running the simulation” on page 239 and onlinehelp for the Move Probabilities control panel.

Each generated configuration is accepted or rejected using aMetropolis (Metropolis 1953) algorithm based on the configura-tional energy change:

Eq. 1

Where:

P = Probability of the move being accepted.

You may use multiple molecules and types. The programattempts to load the desired number of molecules prior to thecalculation starting, but may not load the specified number ofmolecules if there is too much steric hindrance.

P min 1 exp∆EkT-------–

;=


10. Sorption

E = Energy change between the new configuration and the previ-ous configuration.

k = Boltzmann’s constant.

T = Temperature of the simulation.

If the change in energy (E) resulting from the move is negative, thenew configuration is accepted. If the move results in an energyincrease (E>0), the Boltzmann factor (exp(-E/kT)) is computed andcompared to a randomly generated number between zero and one.If the random number is less than the Boltzmann factor, the moveis selected; otherwise it is rejected.

The simulation takes several steps to equilibrate from its initialposition; this equilibration can be recognized by the convergenceof the configurational energy. Simulation statistics are valid onlyfor results taken from a sufficiently large number of configurationsafter this equilibration period.

Fixed pressure (grand canonical ensemble) simulation

The second simulation method available in Sorption uses a grandcanonical ensemble; that is, the number of particles in the systemis not fixed, but the chemical potential of each species is fixed. Sorp-tion translates the chemical potential into the partial pressure (orfugacity) of each component. The basis for the simulation is as fol-lows:

♦ Equilibrium is achieved when the temperature (Tframe) and thechemical potential (µframe) of the gas inside the framework areequal to the temperature (Tgas) and chemical potential (µgas) ofthe free gas outside the framework.

♦ For an ideal gas, the chemical potential depends upon the pres-sure of the gas:

pgas ⇒ µ gas = µframe

♦ For a non-ideal gas, the chemical potential depends uponfugacity (f), which is a function of both temperature and pres-sure:

f(Tgas, pgas) ⇒ µ gas = µframe

Theory


Note

The initial configuration is generated by one of four moves, forwhich the acceptance criteria are different. Each type of move isdescribed below:

♦ Create a molecule — A random molecule is picked from the listof sorbates and is placed in a random position and orientationin the framework. The new configuration is accepted withprobability P:

Eq. 2

Where:

P = Probability of the new configuration being accepted.

E = Energy change between the new configuration and the previ-ous configuration.

k = Boltzmann’s constant.

T = Temperature of the simulation.

Ni = Current number of molecules of component i in framework.

fi = Fugacity of component i in the gas phase.

V = Cell volume. If a surface constrained calculation is performed,the volume used corresponds only to the framework and surfaceregions — i.e., the vacuum layer above the specified surface regionis not included.

♦ Destroy a molecule — A molecule is removed from the frame-work. The simulation first randomly chooses which sorbatetype to remove, then randomly chooses a molecule of that typein the framework. The new configuration is accepted withprobability P:

Eq. 3

When the sorbate is a non-ideal gas, fugacity values shouldalways be used in preference to pressure values.

P min 1 exp∆EkT-------– ln

Ni 1+( ) kT

fiV----------------------------–

;=

P min 1 exp ∆EkT-------–

lnNikT

fiV-------------

+;=


10. Sorption

♦ Translate — A sorbate molecule in the framework is chosen atrandom and translated by a random amount within a cube ofsize 2 δ (where δ is the maximum step size). The new configu-ration is accepted with the probability P. The acceptance crite-rion is the same as for the fixed loading simulation (see Eq. 1 onpage 253).

♦ Rotate — A random sorbate molecule is chosen in the frame-work. The rotation axis is chosen at random, and the moleculeis rotated by a random amount within the range -δ to +δ (whereδ is the maximum step size). The new configuration, based onthe energy change, is accepted with the same probabilityapplied to the translation move above (see Eq. 1 on page 253).

For fixed loading, the move is chosen according to given moveprobabilities. For translations and rotations, a maximum value isplaced on the allowed movement. This value can be adjusted auto-matically during the simulation to achieve a target acceptance rate(see “Setting up and running the simulation” on page 239 andonline help for the Sorption Step Scaling control panel).

The result of this simulation is a set of configurations that convergetowards the specified fugacity and temperature. The simulationtakes several steps to equilibrate from its original random posi-tion. For accurate statistical results, the steps made prior to equili-bration should be excluded in the analysis, and a large number ofconfigurations should be generated after the equilibration period.

Henry’s constant simulation

A third type of sorption simulation calculates Henry’s constant,which is defined as the simulation-cell loading divided by the sor-bate pressure in the limit of vanishing pressure:

Eq. 4

Where:

K = Henry’s constant with units of molecules/cell/kPa.

P = Pressure in units of kPa.

Loading = Number of sorbate molecules in the cell.

K Loading at pressure PP

----------------------------------------------------P 0→lim

≡

Theory


Henry’s constant can be calculated by performing a series of fixedpressure simulations, and extrapolating the ratio to zero pressure.More conveniently, it can be found from the expression (Bezus etal. 1978):

Eq. 5

Where:

K = Henry’s constant.

r = Positional degrees of freedom for the sorbate molecule.

Θ = Rotational degrees of freedom for the sorbate molecule.

U(r,Θ) = Energy of a sorbate molecule in the cell at position rorientated by Θ.

The integral of Eq. 5 can be approximated by a finite sum:

Eq. 6

Where:

N = Number of steps in the simulation.

KN = Henry’s constant approximated after N steps.

Vcell = Volume of the cell

In the simulation, the positions ri and the orientations Θi are cho-sen at random. No step size control is used, and the Metropolisalgorithm is not applied. A check for bad contacts is made to iden-tify high energy positions that give negligible contribution to theintegrand (see “Settings for the energy calculation” on page 232).

K1

kT------ dr

dΘ8π2---------e

U r Θ,( )–kT

------------------------

∫cell∫≈

KN

Vcell

kT----------- 1

N---- e

U ri Θi

,( )–

kT----------------------------

i 1=

N

∑≈


10. Sorption

If the graphical-simulation monitor is selected, KN is displayed asN increases. This can be used to assess the number of steps, N,needed to obtain a well-converged estimate of K.

References

Metropolis, N.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A. H.;Teller, E. J. Chem. Phys., 21 1087 (1953).

Allen, M.P.; Tildesley, D.J. Computer Simulation of Liquids, Claren-don Press, Oxford, (1987).

Karavias, F.; Myers, A.L. Molecular Simulation, 8 23 (1991).

Stapleton, M. R.; Tildesley, D. J.; Quirke, N. J. Chem. Phys., 92 4456(1990).

Bezus, A. G.; Kiselev, A. V.; Lopatkin, A. A.; Pham Quang Du J.Chem.Soc., Faraday Trans.2, 74 367 (1978).

Ewald, P.P. Ann. d. Physik, 64 253 (1921).


11 MesoDyn

MesoDyn is a new tool for the prediction of mesoscale structuresof soft-condensed matter. These are the patterns of size 10 to 100nm which can be found for example in polymer blends, block-copolymer systems, surfactant aggregates in detergent materials(e.g. shampoo), latex particles, or drug delivery systems.

Sections in this chapter Introduction

Using MesoDyn

General methodology

Overview

Building a molecular ensemble

Specifying the Run parameters for a MesoDyn simulation

Setting the host machine and parallel nodes, job monitoring and out-put handling

Analyzing the results

Theory

Introduction

Dynamics

Thermodynamics

Parameterization: Mapping of the atomistic level to the mesoscale

Numerics

References


11. MesoDyn

Introduction

In contrast to previous approaches aiming at classifying morphol-ogies by means of equilibrium theories, the present approach rec-ognizes the fact that by their very nature these patterns areirregular, and hence can only be characterized via the dynamicproperties of the systems. From an industrial perspective thisapproach is much more realistic, since typical processing times areorders of magnitude shorter than the thermodynamic relaxationtime, and thus such non-perfect states contribute substantially tothe behavior of the final material.

A typical scenario is that of a quenched block-copolymer meltwhich rapidly undergoes an initial phase separation (`spinodaldecomposition'), but subsequently gets stuck in a defect-rich mor-phology, i.e. not reaching equilibrium on a realistic time-scale.MesoDyn aims to bridge the gap between the fast molecular kinet-ics on the one hand, and slow thermodynamic relaxations of mac-roscale properties on the other. This is done by means of a well-defined coarse grained representation of the molecular modelwhich forms the basis of a simulation of the phase separation pro-cess leading to mesoscale morphologies. These can then be linkedto macroscale properties.

Of special interest in this version of the MesoDyn method is theability to study the effects of externally applied fields on the kinet-ics of phase separation. The principal field is a simple shear, butalready this can lead to an increase in the defect annihilation rate,provided the shear rate is less than the characteristic polymerrelaxation rate, or an increase in the overall disorder. It has beenlong known that material properties are often strongly determinedby processing; MesoDyn now allows the polymer scientist andchemical engineer to quantify this interdependence.

The molecules are defined on a coarse-grained level as `chains ofbeads'. Each bead is of a certain component type representingcovalently bonded groups of atoms such as given by one or a fewmonomers of a polymer chain. Chemically specific informationabout the molecular ensemble enters into MesoDyn via materialparameters such as the self-diffusion coefficients of the bead-com-ponents, the Flory-Huggins interaction parameters, the bead sizes,and the molecular architecture (chain length, branching etc.).

Using MesoDyn


The dynamics of the system is described by a set of so-called func-tional Langevin equations. In simple terms these are diffusionequations in the component densities which take account of thenoise in the system. By means of numerical inversions, the evolu-tion of the component densities is simulated, starting from an ini-tially homogeneous mixture in a cube of typical size 100-1000 nmand with periodic boundary conditions.

The ensuing mesoscale phase separated morphologies can be anal-yses in Cerius2 by means of slices through the cubic box, or displayof isodensity surfaces. These can be compared directly with exper-imental observations e.g. from electron microscopy. The methodallows the chemical engineer to assess the effects of changes to themolecular composition of a formulation on the microstructure andhence the expected macroscopic properties.

The implementation of the theoretical tools used in MesoDyn iscomputationally very intensive. Hence, in order to bring downsimulation times as well as to satisfy the huge memory require-ments the MesoDyn code has been design to run on HPCN-machines (parallel computers with distributed memory). TheCerius2 interface has been designed in such a way that even theuser new to such techniques can utilize the full power of HPCNwith the same ease as using any other remote host application.

Using MesoDyn

This reveals a menu consisting of Run, Build, Analysis and Job

Start with a new session of Cerius2.

From the Cerius2•Visualizer card deck menu go to theMESOSCALE card deck and choose the MESODYN card.


11. MesoDyn

Control items.

Next you conclude the definition of the molecular ensemble.

First, make sure that you have a molecular ensemble definedfor the MesoDyn simulation. From the Build submenuchoose Beads.

Check that three beads, named A, B and W with diffusioncoefficients are defined here.

Next, from the Build submenu, choose Molecules.

Check that one “triblock” molecule consisting of the twobeads A and B is set (e.g. A 3 B 9 A 3).

Press the right mouse button on the Architecture field to lookat the help available for the chain architecture input.

Note the way in which branched architectures can be defined.

Select the Build item from the MESODYN card and thenselect the Interactions item to open the panel.

Check that the AA, BB and WW interaction energies are set tozero and that the AB, AW and BW interaction energies areeach set to some positive value (e.g. 7.3 kcal/mol for AB).

Using MesoDyn


The repulsive interaction between the components leads to microphaseseparation of the block copolymer molecule defined.

There you find the main parameters for a particular run.

This is all you need to do for now. The subpanels accessible from theRun panel provide further more detailed control which does not con-cern us for now.

Before you press the RUN button, however, you need to make sure youhave a suitable host selected for this job.

At the top of the Job Control panel the currently-selected host isshown. This host must be a parallel machine, with the Message Pass-ing Interface (MPI) software installed. The pulldown on the rightreveals other machines that are available. If your local machine(localhost) is a properly configured, parallel machine, choose that. Ifnot, and you have a parallel remote machine available, select that. No

Close all open panels.

Go to the MESODYN card and select the Run item.

First type the name FirstTest in the File prefix text entrybox. (You may also enter a Title such as À short test run ofMesoDyn'.)

Next check that the Lattice Dimensions are set to reason-ably small values, such as 16 16 16. Finally, set the Numberof Steps to 20 (or some other small value).

Go to the MESODYN card and select the Job Control item.


11. MesoDyn

further action need be taken for this first test.

You should get the following message in the textport:

MESODYN job FirstTest has been started.

On the Job Control panel the job is listed in the Job Status box.

You can monitor the progress of the job by clicking the Monitor sta-tus file button. This opens a window displaying the bottom of the sta-tus file.

The Status indication in the Job Control Box changes to complete, ifthe job has finished.

The next step is to prepare the output files for analysis. This is neces-sary since MesoDyn writes out binary files on the remote host.

In the default setting this converts all density output files. A short

Finally, go to the MesoDyn Run panel and press the RUNbutton.

When the number of steps you had selected has beenreached, press the UPDATE Job status button.

Click the Collect & Transfer button and then click the COL-LECT button on the subpanel.

Using MesoDyn


interactive job is started to perform this task.

An appropriate message tells you what happened.

You are now ready to analyses the results of the run.

This opens a file browser from which you select the MesoDyn_stat filefrom your run (that is, FirstTest.MesoDyn_stat). This shows thename, title, and date. This is from now on the selected system for anyof the other analysis functionalities.

For this short example we just consider the Morphology.

The Morphology item on the Analysis submenu has two pullrightoptions: Isodensities and Profiles.

This displays the name of the selected system, the time step and bead

If the file systems of the remote host on which the job wasexecuted, and that of the local host are different, then youneed to click TRANSFER to move back the scalar outputfiles.

If you are not sure, press the button anyway.

Close all panels.

Go to the MESODYN card and select the Analysis item,then select the System item from the submenu.

Go to the MesoDyn card and Select the Analysis/Morphol-ogy/Profiles item.


11. MesoDyn

name.

This produces a slice through the simulation box in the Model win-dow. Different colors represent different concentrations of the bead.

The density boxes show the lowest and highest concentration of A inthe slice.

You can do further analysis by looking at the isodensity surfaces.

The isodensity value must fall between the minimum and maximumvalues you found in the Profiles analysis of that bead.

You will see some closed surfaces which indicate the regions where theA bead preferably resides. You may edit the display style of the surface,and create a similar surface for the bead B, in the same or in a differentmodel space.

Click the Create New Profile button.

You may not see much evidence of phase separation for thisshort run, but you can enhance the contrast by clicking theMore Editing Options... button and clicking Reset Map toFull Range.

Move the slice by using the sliders.

Select the Analysis/ Morphology/Isodensities item.

Set the isodensity of A to 0.4.

Make sure that you are in an empty model space, and thenclick the Create Isodensity Display button.

General methodology


General methodology

Overview

A graphical user interface module to the MesoDyn simulation hasbeen provided within the Cerius2 molecular modelling environ-ment. This interface handles the input generation, (i.e. the defini-tion of the molecular ensemble, setting of the simulation runparameters, control of the numerical algorithm parameters), thejob control (choice of machine for run, parallel nodes and outputpaths, job monitoring), the output file handling from the parallelprocessors, and the analysis of the simulation output (3D densityfields and scalar thermodynamic functions).

In addition, Cerius2 provides access to a range of molecular mod-elling tools which can furnish most of the input parametersrequired for MesoDyn.

In common with the Cerius2 standard, MesoDyn can be accessedby selecting the MESOSCALE card deck from the deck of cardsmenu. Choosing MESOSCALE reveals the MESODYN card,


11. MesoDyn

which contains the following items: Run, Build, Analysis, JobControl, and Reset (see Figure 5).

The first action in many cases will be to build a molecular ensem-ble. This is done from the Build sub-menu. The Beads, Moleculesand Interactions are defined here, as are Geometrical Constraints,i.e. regions representing an organic pigment or inorganic filler par-ticle (see Figure 6 and Figure 7).

The MesoDyn Run control panel (Figure 8) allows you to editparameters governing the simulation run, such as run time andoutput frequency.

The Job Control control panel (Figure 9 and Figure 10) allows youto set the run host, associated parameters, the job to be monitored,and the output to be handled and prepared for further analysis.

In the Analysis section, a particular system for which output isavailable can be selected, its output files inspected (Figure 11), out-put functions plotted (Figure 12), and the three-dimensional den-sity profiles of the beads analyzed graphically both by slicing(Figure 13) and isodensity surfaces (Figure 14).

Figure 5. The Cerius2 Visualizer window showing the MESOSCALE cardand the main MESODYN menu.

General methodology



MesoDyn works on an ensemble of Gaussian chains, which aredefined in terms of the constituent beads (covalently bondedgroups of atoms, e.g. monomers), their respective diffusion coeffi-cients, the way the beads are joined up in molecules (the moleculararchitecture), the bond length (Gaussian chain parameter), theconcentrations of the molecules and the Flory-Huggins interactionenergies between the beads.

The Build submenu of the MESODYN card provides access to theBeads, Molecules, and Interactions control panels which allow allof the above parameters to be set (Figure 6).


11. MesoDyn

.

Beads, as stated before, represent the individual chemical speciesin the system, and for a homopolymer or block copolymer mole-cule are identical with a statistical (or Kuhn) segment. For a sol-vent, a bead represents the collective degrees of freedom of a

Figure 6. The MesoDyn Build panels for defining beads, molecules andinteractions.

General methodology


group of solvent molecules. At the moment, all beads are restrictedto having the same volume and diffusion coefficient.

The molecule architecture is entered in the form of a tree string bywhich even complex branched structures may be defined as inputto MesoDyn. Comprehensive on-line help is available for this task.

It should be noted that the interaction parameters are essentiallythe Flory-Huggins parameters for the system (times kBT), and, assuch, these parameters are known experimentally for a large num-ber of systems. (The chapter by Gundert and Wolf in the PolymerHandbook, VII 173 is a useful first reference here.)

Geometrical constraints are a new feature in this version of Meso-Dyn. They provide one of the first opportunities to study adsorp-tion and adhesion phenomena in realistic complex liquids. Assuch, the number of validation studies is still small, however, earlyresults are encouraging.

The Cerius2 interface provides a shortcut to creating two types ofconstraints: a flat wall at z = 0, and a random distribution of mon-odisperse spheres, representing particles in a polymer melt or sus-pension. In this latter case, the sphere radius and volume fractionof spheres are both under the control of the user.

Figure 7. The MesoDyn Constraints panels for defining regions fromwhich the beads are excluded.


11. MesoDyn

Note

In addition to the above two geometries, it is possible to use anarbitrary constraint. This is accomplished by editing the mask file(with the file extension MesoDyn_ascii) which describes the con-straint (as a grid of ones and zeros). The mask file is created in theworking directory when a job is submitted which uses a con-straint; this is a simple ASCII file which may be edited with a texteditor. The ones represent regions accessible to the beads and thezeros regions denied to the beads. The mask file may be saved andused in a later simulation. The Constraints Panel also allows for anarbitrary mask file to be visualized in the Cerius2 Model Window.

Specifying the Run parameters for a MesoDyn simulation

Having defined a molecular ensemble, a number of parametersneed to be set that govern the particular simulation to be done.These parameters are accessed from the MesoDyn Run panel(Figure 8). In particular, each run is identified by a certain nameand given as file prefix to all input and output files generated. Themodel system is further defined by specifying its temperature andthe size of the simulation box (with periodic boundary condi-tions), in terms of the side length of its cubic cells and the numberof cells in each cartesian dimension. Typically, the cell size is of thesame order as the bead size (i.e. the bond length). Furthermore, thestep size of the numerical integration and the length of the run,either in terms of its total time or in terms of the number of steps,must be set.

New in this release of MesoDyn is the facility to shear the system.This allows you to study the interplay between phase separationkinetics and processing timescales. The shear may be toggled onand the shear rate set. The units of the shear rate are inverse to thedimensionless units of time which the simulation uses. Please seethe on-line help for a definition. The default shear rate, 0.001, wasthat used to successfully reproduce the known ordering of the hex-agonal phase of pluronics under shear (Zvelindovsky, et al., sub-mitted). By default, shearing is assumed to be imposed

Important: In the current version of MesoDyn, the particles arefixed permanently in space. They may thus only modelsituations in which the particles don't move appreciably on thetimescale of the phase separation.

General methodology


throughout the simulation; however, this may be changed byusing the facility mentioned below which allows the user to editthe parameter file, and changing the shear_start and shear_endtimes.

Note

Finally, a title for the simulation may be added, to include com-ments about the specific run. The date and time are added to thisautomatically.

Further “expert” parameters are accessible via the Noise... andSolver... buttons. The output level of the run can be set from theOutput... control panel. In particular, the frequency in terms oftime steps, with which the status information, density files, poten-tials files and restart files are written to disk is set here. The amountof disk space required for frequent output of three-dimensionalfields, such as contained in density, potentials and restart files,may be very large.

The implementation of shear for parallel machines requires thesystem to be decomposed in the y-direction only. Please see thesection below on the Job Control Options Panel for a furtherexplanation.

Figure 8. The MesoDyn Run control panel.


11. MesoDyn

When a job is run, all system parameters are written to a parameterfile which is picked up by MesoDyn. You may access existingparameter files from the Files... control panel, in order to save andedit them, update the panels from a file, or run MesoDyn directlyfrom an existing parameter file, thereby circumventing the currentinterface settings.

Usually, a job is started from the initial homogeneous mixture.However, you may wish to continue a previous run. This can bedone by choosing the Restart option on the Restart... control panel.A file for restarting must be selected on this panel before this typeof MesoDyn computation is permitted.

Finally, the simulation is started by pressing the RUN button.

Setting the host machine and parallel nodes, jobmonitoring and output handling

The Job Control panel (Figure 9) (accessed by clicking Job Controlon the MESODYN card) shows the name of the host runningMesoDyn and the mode of execution (INTERACTIVE or BACK-

General methodology


GROUND). Access information to the remote host (user name andpassword) may also be set there, if required.

Also on the Job Control panel, you can set the Base directory onthe remote host. The Job Status box shows previous and currentjobs and related information such as process ID and status -- infor-mation which is required by Cerius2•MesoDyn to collect theresults of a MesoDyn run.

During a run, the execution can be monitored by inspecting thestatus file output by MesoDyn. This shows the input parameter setand the time steps completed so far along with a solver parameterwhich monitors the numerical stability of the run: the so-called“Crank-Nicholson norm” must remain close to zero.

To see the current status of a job, select it in the MesoDyn Job Sta-tus box and press the UPDATE Job status button.

Figure 9. The MesoDyn Job Control panel.


11. MesoDyn

During a run, each of the processors writes out files to its specifiedoutput path directory. These are binary files which contain thedensities of the corresponding sectors of the simulation box.

The Job Control Options control panel allows you to edit the pro-cessor configuration, the node names, and the output paths foreach node. On parallel machines with shared memory, such as anSGI Octane or Power Challenge, the UI will by default store thedata for each node in a subdirectory of the run directory. In mostcases, with a file server such as NFS, this will be the same as theworking directory even if the job is launched from a differentmachine.

On parallel machines with distributed memory, such as the IBMsp2, correct use of the Options control panel syntax is much moreimportant. Here the default run directory is typically the user'slogin directory, which is mounted on the control workstation ofthe sp2, and usually has little disk space. Instead, the output pathsshould correspond to the local disks on each processor, which aretypically very large. The Options control panel displayed inFigure 10 above is correctly set to run on two nodes of the sp2 at

Figure 10. The MesoDyn Job Control Options subpanel, with nodes andoutput paths specified for running on two nodes of a particular IBM sp2.

General methodology


MSI, where the directory /msidata has been mounted on each ofthe local disks.

For sheared systems, as stated before, the system must be decom-posed in the y-direction only. Thus, for running the above job ontwo nodes of the sp2 with shear, the number of processors wouldbe specified as 1 2 1. The node and path definitions would remainunchanged. For four nodes, the processors would be 1 4 1, and soon.

Before the density profiles can be analyzed, the binary output filesfrom the processors must be collected and converted to ASCII den-sity files. This is done from the Collect & Transfer control panel,accessible from the Job Control control panel. You may choose tocollect the density fields for ALL or for SELECTED time steps, aschosen from the Time Steps... control panel which lists the timesteps for which output has been produced. Pressing the COLLECTbutton then starts up a short interactive run of MesoDyn on the rel-evant processors. This generates a collected density file for eachtime step on the host base directory. If the host base directory dif-fers from the local run directory, files are transferred back to thecurrent local directory.

Before taking this step, you need to make sure that sufficient diskspace is available on your local disk for the collected density files.These can be quite large, e.g. a system consisting of 8 beads, outputat 100 different time steps, on a 50 x 50 x 50 grid requires 400Mbstorage space.

Finally, any scalar output files that reside on a remote host may betransferred to the current local directory by pressing the TRANS-FER button. These files include the status file (extensionMesoDyn_stat), the thermodynamics file (extension MesoDyn_ther), the restart file (extension MesoDyn_rst), the log file, contain-ing information about parallel communications (extensionMesoDyn_log) and the output file (extension MesoDyn_out). Thislast file will not be transferred if you are running on an sp2 but willbe placed in the output path specified for the first node in theOptions control panel. In practice the results in this file will berarely needed; they are not used by the Analysis section of theinterface.


11. MesoDyn

Analyzing the results

Having completed a simulation run, collected and transferred theoutput files, you are ready to analyze the results. This functionalityis accessed via the Analysis submenu on the MESODYN card.

Selecting a system The first step is always to open the System Analysis panel(Figure 12).

This provides a file browser with a filter for the MesoDyn statusoutput files. Selecting the status file from the desired run sets thename of the system to be analyzed further. The name and title ofthe system selected will be shown in the box below, and you mayexamine the log and status files by pressing the relevant pushbut-ton.

Plotting the thermody-namics functions

As a next step you can see the thermodynamics of the system as itevolved from its initial to its final state. Open the Thermodynam-

Figure 11. The MesoDyn Systems Analysis panel.

General methodology


ics control panel (Figure 12) and plot the free energy, potentialenergy and entropy over the whole or a selected time interval.

Exploring the morphology Finally, and often most importantly, the morphology of the molec-ular ensemble can be investigated and analyzed in two ways.

First, the MesoDyn Profiles panel (Figure 13) lets you create slicesthrough the simulation box for each time step and for each bead.The profiles are displayed in the model window and any of thetools provided by the Cerius2 Visualizer to alter the display of

Figure 12. The MesoDyn Thermodynamics panel for plotting the freeenergy, entropy, and other time-dependent averages.


11. MesoDyn

models (such as rotation, translation, magnification, colors, light-ing, clipping, depth cueing, etc.) may be used.

Furthermore, although the current MesoDyn interface does notprovide an animation facility directly, a sequence of models can besaved in a Log file via the Utilities/Record commands from themenu bar, and then replayed via the Utilities/Playback Scriptfacility.

The sliders on the MesoDyn Profiles control panel allow you tochange the position and direction of the selected slice. More Edit-ing Options... shows the way in which the density is mapped tocolor, and lets you edit this mapping. Adjust the Profile grid scal-ing to control the factor by which the actual grid is multiplied to

Figure 13. The MesoDyn Profiles panels for creating and analyzing slicesthrough the density fields.

General methodology


generate the triangular mesh. A higher number gives a smootherappearance but takes longer to process.

You can create isodensity surfaces of the bead concentrations fromthe Isodensities panel (Figure 14).

Open the Isodensities control panel from the Morphology pull-right of the Analysis submenu. For each bead type you can createan isodensity surface at a number of time steps and for up to fourdifferent values of the concentration at once.

Typical analyses involve displays of:

♦ the different beads, each at a concentration close to its maxi-mum (to be determined from the slice display),

Figure 14. The MesoDyn Isodensities control panel for creating andanalyzing isosurfaces of the density fields.


11. MesoDyn

♦ one bead type at a number of time steps, and

♦ one particular bead at different isodensities.

Each surface in the current model can be edited in terms of its vis-ibility, transparency and color.

Finally, the main MESODYN card provides a Reset button, whichreinitializes all the panels and sets all values back to their default.

Theory

Introduction

MesoDyn is based on a dynamic variant of mean-field densityfunctional theory. The latter is based on a theorem which basicallystates that there is a one-to-one mapping between the distributionfunctions of the system, the densities and an external potentialfield. Furthermore, a real system, i.e. a system with interactions,can be equated to an ideal system, i.e. without interactions, via aneffective external potential.

The reason for this is that with the above theorem an externalpotential can always be found such that the distribution of theideal system equals that of the real system at the same density. Thistheory can be used to great effect in the description of polymer flu-ids.

We take the polymer chain as the fundamental building blocks ofthe model. In this description, the intrachain correlations can inprinciple be treated by any suitable model. In practice, a Gaussianchain model is used

♦ because it allows a factorization of the interactions, hence iscomputationally very efficient, and

♦ because it can be shown that the Gaussian chain may be usedas a statistical model for a real chain, i.e. for each real, atomisticforce-field model, a Gaussian chain representation with thesame response function can be found.

The non-interacting Gaussian chains are hence the ideal system.Any interchain, i.e. non-bonded interactions are treated as non-

Theory


ideal, that is, they enter into the effective external potential. Hence,the molecular ensemble is represented by a number n of Gaussianchains, made up of a number of different beads of types I, with atotal number of N beads per chain. At an instant of time there willbe a certain distribution Ψ of bead positions in space, resulting inthree-dimensional concentration fields ρI(r). The evolution ofthese fields is the result of the dynamics outlined in the followingsection, in combination with the thermodynamic driving forcedescribed in the section thereafter. For simplicity of presentation,in the following we are going to limit the number of bead types totwo, named A and B, but the theory will equally apply to any num-ber of bead types.

The following description of the theory is based on a paper byFraaije et al (1996) to which the interested reader is referred also forfurther references.

It should be noted that some of the approximations imposed inthis paper (e.g. the assumption of perfect incompressibility) havenow been lifted. The interested reader should consult the refer-ences below.

Dynamics

The derivation of the diffusive dynamics of the molecular ensem-ble is based on the assumption that for each type of bead I the localflux is proportional to the local bead concentration and the localthermodynamic driving force:

Eq. 7

where JI is a stochastic flux (related to thermal noise). Togetherwith the continuity equation

Eq. 8

this leads to simple diagonal functional Langevin equations (sto-chastic diffusion equations) in the density fields:

JI MρI µI JI+∇–=

t∂∂ρI ∇ JI⋅+ 0=


11. MesoDyn

Eq. 9

with a Gaussian distribution of the noise.

However, the fluctuations in the total density of this simple systemare not realistic since finite compressibility is not enforced by themean-field potential chosen (see below). Therefore, total densityfluctuations are simply removed by introducing an incompress-ibility constraint:

Eq. 10

where νB is the average bead volume. This condition then leads to“exchange” Langevin equations:

Eq. 11

Eq. 12

Here M is a bead mobility parameter. The kinetic coefficientMνρAρB models a local exchange mechanism. Hence the model isstrictly valid only for Rouse dynamics. (Effects such as reptationlead to kinetic coefficients which extend over a range of roughlythe coil size. They lead to computationally expensive non-localoperators which in addition, are very complex in the non-linearregime.)

The distribution of the Gaussian noise satisfies the fluctuation-dis-sipation theorem:

Eq. 13

Eq. 14

and ensures that the time-integration of the Langevin equationsgenerates an ensemble of density fields with Boltzmann distribu-tions.

t∂∂ρI M∇ ρ I µI η I+∇⋅=

ρA r t,( ) ρB r t,( )+( ) 1νB------=

t∂∂ρA MvB∇ ρ AρBI∇ µ A µB–[ ] η+⋅=

t∂∂ρB MvB∇ ρ AρB∇ µ B µA–[ ] η–⋅=

η r t,( )⟨ ⟩ 0=

η r t,( ) η r ′ t ′,( )⟨ ⟩2MvB

β--------------δ t t ′–( ) ∇ r δ r r ′–( ) ρAρB∇ r ′⋅–=

Theory


Thermodynamics

The above Langevin equations contain the bead chemical potentialas the thermodynamic driving force of the diffusive dynamics.These chemical potentials can be derived from the thermodynam-ics of the molecular ensemble.

The first step is to derive an expression for the free energy of thesystem in terms of the bead distribution functions Ψ. Since thepositions of the beads are correlated to each other this amounts toa multi-dimensional many-body problem. To overcome this, anyinterchain correlations are neglected, and the system is approxi-mated by a set of independent Gaussian chains embedded in amean-field.

The distribution functions of the independent Gaussian chains fac-torize exactly, and hence the density functional can be simplifiedto a product of single-chain density functionals. In this approxima-tion, the free energy functional can be written as

Eq. 15

The first term is the average value of the Hamiltonian for the idealsystem, comprising the internal Gaussian chain interactions:

Eq. 16

where HγG is the Gaussian chain Hamiltonian of chain:

Eq. 17

here a is the Gaussian bond length parameter and the index s goesover all N segments of the chain. The second term in the freeenergy functional stems from the Gibbs entropy of the distribu-tion. The third term is the non-ideal contribution related to theinterchain interactions.

F ψ[ ] 1Q---- R ψH

id β 1– ψ ψln+

Fnid ψ[ ]+d∫≡

Hid

HγG

γ 1=

n

∑=

βHγG 3

2a2

-------- Rγs Rγ s 1–,–( ) 2

s 2=

N

∑=


11. MesoDyn

In the present mean-field approximation, the latter is independentof the particular distribution ψ. In the spirit of the particular appli-cation of density functional theory taken here, namely treating thechains as the ideal system, the correlations between the chains areneglected, and the density functional method applies to the corre-lations within the Gaussian chain only.

The key rudiment of dynamic density functional theory is nowthat on a coarse-grained time-scale the distribution function ψ issuch that the free energy functional F[ψ] is minimized. Hence ψ isindependent of the history of the system, and is fully characterizedby the constraints that it represents the density distribution andminimizes the free energy functional. This constraint on the den-sity fields is realized by means of an external potential UI .

The constraint minimization of the free energy functional leads toan optimal distribution, which in turn, and by the one-to-one rela-tion between densities, distributions and external potential can bewritten as:

Eq. 18

Finally, a Flory-Huggins type interaction is introduced for the non-ideal (inter-chain) interactions:

Eq. 19

where εIJ(|r-r’|) is a mean-field energetic interaction betweenbeads type I at r and J at r’, defined by the same Gaussian kernelas in the ideal chain Hamiltonian:

Eq. 20

βF ρ[ ] n Φ β 1–lnn! UI r( ) ρI r( ) r βF

nid ρ[ ]+d∫I

∑–+ln=

Fnid ρ[ ] 1

2--- εAA r r ′–( ) ρA r( ) ρA r ′( ) +∫∫=

εAB r r ′–( ) ρA r( ) ρB r ′( ) +

εBA r r ′–( ) ρB r( ) ρA r ′( ) +

εBB r r ′–( ) ρB r( ) ρB r ′( ) drdr ′

εIJ r r ′–( ) εIJo 3

2πa2

------------

32---

e

3

2a2

--------- r r ′–( ) 2–

≡

Theory


Parameterization: Mapping of the atomistic level to themesoscale

As a result of the model of the mesoscopic molecular ensembledynamics outlined above, the following parameters arise as char-acterizing the system:

♦ The molecular architecture of the Gaussian chain in terms of anumber of different beads. This includes the chain or blocklengths of each bead type, as well as the possibility of branch-ing.

♦ The Gaussian bond length parameter.

♦ The self-diffusion coefficient of each bead type.

♦ The Flory-Huggins interaction parameters between the differ-ent bead types.

Fortunately, the theory does not end here, but actually tells us howto interpret these parameters in terms of atomistic information.The crucial theorem which helps at this point was already men-tioned above. It says that for each atomistic, force field model, aGaussian chain representation can be found, such that this Gauss-ian chain has the same response function (or correlation functions)as the real system.

At this point it is important to emphasize, that this Gaussian chainmight differ considerably from the real chain. For example, theGaussian chain may be branched while the real chain is not. This,however, does not need to concern us. We only need to know whatthe mapping is, and in particular what each bead represents interms of bonded atoms.

On this basis the next steps are to derive in some way, either exper-imentally, by Molecular Dynamics or other methods, the diffusioncoefficients of the beads. Furthermore, by the Cerius2 AmorphousBuilder or Blends module or by group contribution methods theinteraction coefficients can be determined.

In conclusion, MesoDyn offers a complete path from the atomisticlevel through to the simulation of mesoscale phase morphologies.


11. MesoDyn

Numerics

The Gaussian chain density functional constitutes a one-to-onerelation between the external potential fields and the density fieldsfor each bead type. In addition, the intrinsic chemical potentials µare functionals of the external potentials and the density fields.The coupled Langevin equations constitute a relation between thetime derivatives and the intrinsic chemical potentials. Finally, thenoise source is related to the exchange kinetic coefficients.

Together these equations form a closed set, which can be inte-grated efficiently on a cubic mesh by a Crank-Nicholson scheme.

Since a very large number of equations ( about106 nested Fredholintegrals per time step) has to be solved and memory requirementsfor systems of up to 10 beads on meshes of the order of 1003 is alsovery high, a domain decomposition method has been used andimplemented with MPI (Message Passing Interface) for parallelplatforms with distributed memory.

Briefly, the cubic grid is divided into subgrids, each associatedwith a processor. Communications scale only linearly with thetotal mesh length, and thereby an efficiency of more than 75% isachieved on an 8-processor IBM SP2.

References

The best general introduction to MesoDyn is the paper by Profes-sor Hans Fraaije, et al. of the University of Groningen:

J.G.E.M. Fraaije, B.A.C. van Vlimmeren, N.M. Maurits, M. Postma,O.A. Evers, C. Hoffman, P. Altevogt, and G. Goldbeck-Wood,“The dynamic mean-field density functional method and itsapplication to the mesoscopic dynamics of quenched blockcopolymer melts,” Journal of Chemical Physics, 106, 4260 (1997).

Although most of the method description in this paper has beenadapted for this manual, the reader may wish to consult the origi-nal paper for some early application work.

In addition, five of the more recent applications from the Univer-sity of Groningen group are also available on MSI's Web site, in the

References


Mesoscale Products area. These include several which explain theimplementation of shear, now available in the present release.These papers are available in electronic form here.

A.V.Zvelindovsky, G.J.A.Sevink, B.A.C.van Vlimmeren,N.M.Maurits, J.G.E.M.Fraaije, “Lammelar phase of diblockcopolymer melt under shear: kinetics and conformationalanalysis,” Accepted by Progress in Colloid and Interface Science.Title: “Trends in Colloid and Interface Science XII.”

N.M.Maurits, A.V.Zvelindovsky, G.J.A.Sevink, B.A.C.van Vlim-meren and J.G.E.M.Fraaije, “Hydrodynamic effects in 3dmicrophase separation of block copolymers: dynamic mean-field density functional approach,” Journal of Chemical Physics,108, 91,500 (1998).

A.V.Zvelindovsky, B.A.C.van Vlimmeren, G.J.A. Sevink,N.M.Maurits and J.G.E.M.Fraaije, “3D simulation of hexago-nal phase of a specific polymer system under shear: thedynamic density functional approach,” Submitted to the Jour-nal of Chemical Physics (rapid communications).

B.A.C. van Vlimmeren, N.M.Maurits, A.V. Zvelindovsky, G.J.A.Sevink, J.G.E.M.Fraaije, “Micro-phase separation kinetics inconcentrated aqueous solution of the triblock polymer surfac-tant (EO)13(PO)30(EO)13: an application of dynamic mean-field density functional theory,” Submitted to Phys. Rev. Lett.

A.V. Zvelindovsky, G.J.A. Sevink, B.A.C. van Vlimmeren, N.M.Maurits and J.G.E.M.Fraaije, “3D mesoscale dynamics of blockcopolymers under shear: the dynamic density functionalapproach,” Phys. Rev. E 57 4699 (1998).

Also available in the above-mentioned MesoDyn Product area isthe following recent review of the MesoDyn ESPRIT project:

P. Altevogt, O.A. Evers, J.G.E.M.Fraaije, N.M.Maurits andB.A.C.van Vlimmeren, “The MesoDyn project: software formesoscale chemical engineering,” Published in Theochem.

A complete list of the papers of the University of Groningen groupis available at the website of the MesoDyn ESPRIT project, whichis found here. These include the following three papers, the firsttwo of which explain the numerical method behind MesoDyn andthe last of which explains the modeling of compressible systems,which is implemented in the current Cerius2 MesoDyn release.


11. MesoDyn

N.M. Maurits, P. Altevogt, O.A. Evers and J.G.E.M. Fraaije, “Sim-ple numerical quadrature rules for gaussian chain polymerdensity functional calculations in 3d and implementation ofparallel platforms” Comp. Polymer Sci. 6, 1 (1996).

B.A.C. Van Vlimmeren and Fraaije, “Calculation of Noise Distribu-tion in mesoscopic dynamics models for phase separation ofmulticomponent complex fluids” Comput. Phys. Commun. 99,21 (1996).

N.M. Maurits, B.A.C. van Vlimeren and J.G.E.M. Fraaije, “Mesos-copic phase separation dynamics of compressible copolymermelts” Accepted by Phys. Rev. E.


12 Dissipative Particle Dynamics(DPD)

Dissipative Particle Dynamics (DPD) is a technique for simulatingcomplex fluids such as surfactant solutions and copolymer melts.A development of Molecular Dynamics (MD) and lattice gasautomata, DPD represents long-range hydrodynamic forcesdirectly in its equations of motion allowing more realistic model-ing of the dynamics of phase separation and other processesdepending on large length-scale interactions.

Sections in this chapter Introduction

Using DPD

General methodology


Specifying the Run parameters for a DPD simulation

Setting the host machine, job monitoring and output handling

Analysing the result: Selecting a system, inspecting the files, plottingthe thermodynamic functions, and exploring the morphology

Theory

Introduction

Equations of motion for a DPD system

Integration scheme in DPD

Choosing the repulsion parameters

Choosing the dissipation and random noise magnitudes

Mapping the interactions onto Flory-Huggins theory

Calculation of Flory-Huggins c parameters as input to DPD simula-tions


12. Dissipative Particle Dynamics (DPD)

References

Introduction

In the DPD methodology, the fundamental particles are “beads”that represent small regions of fluid material rather than the atomsand molecules familiar from MD simulations. All degrees of free-dom smaller than a bead radius are assumed to have been inte-grated out leaving only coarse-grained interactions betweenbeads. There are three types of force between pairs of beads, eachof which conserves both bead number and linear momentum: anharmonic conservative interaction, a dissipative force represent-ing the viscous drag between moving beads (i.e., fluid elements),and a random force to maintain energy input into the system inopposition to the dissipation. All forces are short-ranged with afixed cut-off radius. By a suitable choice of the relative magnitudesof these forces, a system can be shown to evolve to a steady-statethat corresponds to the Gibbs Canonical ensemble. Integration ofthe equations of motion for the beads generates a trajectorythrough the system's phase space from which all thermodynamicobservables (e.g., density fields, order parameters, correlationfunctions, stress tensor, etc) may be constructed from suitableaverages. An immense advantage over conventional MolecularDynamics and Brownian Dynamics simulations is that all forcesare "soft" allowing the use of a much larger time-step and corre-spondingly shorter simulation times.

Polymers may be constructed out of the beads in a DPD simulationallowing the investigation of the morphologies of, for example,surfactants and branched polymers. Just as a bead represents asmall fluid element whose interactions with other beads includedissipative and random (thermal) terms, so a DPD polymer repre-sents a segment of a real polymer whose size is of the order of thepersistence length. The interactions of a

polymer with other polymers occur via the conservative, dissipa-tive and random forces between their component beads. Typically,the beads making up a DPD polymer are bound to each other withharmonic forces. Because the fundamental objects in DPD arecoarse-grained beads and polymers, mapping the physical andchemical properties of polymers onto the DPD simulation param-

Using DPD


eters is non-trivial: there is however a correspondence between thebeads in a DPD simulation and the atoms and molecules in a realpolymeric system. We deal with this aspect of the methodology atlength later in this guide.

The mesoscale morphologies resulting from a DPD simulation canbe analyzed in Cerius2 by means of a three-dimensional display ofthe beads and polymers making up the system; or by drawingdensity slices through the simulation box; or by display of isoden-sity surfaces. The latter can be compared directly with experimen-tal observations e.g. from electron microscopy. The method allowsthe chemical engineer to assess the effects of changes to the molec-ular composition of a formulation on the microstructure and hencethe expected macroscopic properties. In addition, the dynamicalpathway followed by the system in evolving to its equilibriumstate may also be visualized using the same tools.

Using DPD

Note At any time when running DPD, online help is available byclicking the right mouse button when the cursor is over a panelor a command.

Start with a new session of Cerius2.

From the Cerius2•Visualizer card deck menu go to theMESOSCALE card deck and choose the DPD card.



This reveals a menu consisting of Run, Build, Analysis and Job Con-trol and Reset.

First, make sure that you have a molecular ensemble definedfor the DPD simulation. From the Build submenu chooseBeads.

Check that three beads, named A, B and W are defined there.

Next, from the Build submenu, choose Molecules.

Check that one “triblock” molecule consisting of the twobeads A and B is set (i.e., A 3 B 9 A 3).

Press the right mouse button on the Architecture field to lookat the help available for the chain architecture input.

Note the way in which branched architectures can be defined.Also, notice the concentrations of the molecules and thedefault spring constant defining the strength of the bondinginteraction between beads in a molecule.

Using DPD


Next you conclude the definition of the molecular ensemble.

The repulsive interaction between the components leads to microphaseseparation of the block copolymer molecule defined.For the moment,ignore the interactions of the beads with the wall: these have no effectunless the Constraints control panel is used to select the wall.

There you find the main parameters for a particular run.

This is all you need to do for now. The control panels accessible fromthe Run panel provide further more detailed control which does notconcern us for now.

Before you press the RUN button, however, you need to make sure you

Select the Build item from the DPD card and then select theInteractions and Dissipations control panels.

Check that the AA, BB and WW interaction energies are set to25 kBT and that the AB, AW and BW interaction energies areeach set to larger positive values.

Close all open panels.

Go to the DPD card and select the Run item.

First type the name FirstTest in the File prefix text entrybox. (You may also enter a Title such as À short test run ofDPD'.)

Next check that the Cell Size is set to a reasonable value,such as 10 10 10. Finally, set the Number of Steps to 20 (orsome other small value).



have a suitable host selected for this job.

At the top of the Job Control panel the currently-selected host isshown. This should be localhost. The pulldown on the right will revealother machines available. If you have a more powerful remote machineavailable, select that. No further action need be taken for this first test.

You should get the following message in the textport:

DPD job FirstTest has been started.

On the Job Control panel the job is listed in the Job Status box.

You can monitor the progress of the job by clicking the Monitor sta-tus file button. This opens a window displaying the bottom of the sta-tus file.

The Status indication in the Job Control Box changes to complete, ifthe job has finished.

The next step is to analyze the results of the run.

This will open a file browser from which you select the DPD “stat” file

Go to the DPD card and select the Job Control item.

Finally, go to the DPD Run panel and press the RUN button.

When the number of steps you had selected has beenreached, press the UPDATE Job status button.

For that purpose, choose System from the Analysis sub-menu on the DPD card.

Using DPD


from your run, i.e. FirstTest.Dpd_stat. The name and title, as well asdate will be shown. This is, from now on, the selected system for anyof the other analysis functionalities. In this short example we considerjust the morphology.

You will see the name of the selected system, the time step and beadname.

This produces a slice through the simulation box in the Model win-dow. Different colors represent different concentrations of the bead.

The density boxes show the lowest and highest concentration of A inthe slice.

The Morphology entry on the Analysis submenu shows twopullright options: Isodensities and Profiles. Choose Profilesfirst.

Press Create New Profile.

You may not see much evidence of phase separation for thisshort run, but you can enhance the contrast by opening theMore Editing Options... control panel and pressing ResetMap to Full Range.

Move the slice by using the sliders.



You can do further analysis by looking at the isodensity surfaces.

The isodensity value must fall between the minimum and maximumvalues you found in the Profiles analysis of that bead.

You will see some closed surfaces which indicate the regions where theA bead preferably resides. You may edit the display style of the surface,and create a similar surface for the bead B, in the same or in a differentmodel space.

Notice that the isodensity surface is superimposed on the snapshotdisplayed in the model window unless you select an empty model win-dow first.

General methodology

A graphical user interface module to the DPD simulation has beenprovided within the Cerius2 molecular modeling environment.This interface handles the input specification (i.e., the definition ofthe beads and polymers and their interactions, setting of the sim-ulation run parameters, control of the numerical algorithm param-eters), the job control (choice of machine for run, and output paths,job monitoring), the output file handling from the (possiblyremote) hosts, and the analysis of the simulation output (3-D den-sity fields and scalar thermodynamic functions). In addition,Cerius2 provides access to a range of molecular modeling toolsthat can furnish the Flory-Huggins parameters required for calcu-lating the repulsion parameters between beads in DPD. The rela-tionship between these two sets of parameters is described indetail in the preceding Theory section.

In accordance with the Cerius2 standard, DPD can be accessed viaa card, called MESOSCALE, from the Deck of Cards menu. Choos-

Select the Analysis/ Morphology/Isodensities item.

Set the isodensity of A to 0.4.

Make sure that you are in an empty model space, and thenclick the Create Isodensity Display button.

General methodology


ing this cards reveals the main DPD menu card which contains thefollowing five main sections: Run, Build, Analysis, Job Control,and Reset (see Figure 15).

The first action in many cases will be to build a molecular ensem-ble. This is done from the Build sub-menu. The Beads, Mole-cules, Interactions, Dissipations, and Constraints menu itemsare defined here (see Figure 16).

The Run panel (Figure 17) then provides for editing of the param-eters governing the actual simulation, such as the length of run,time step-size, density, shear rate (if required), output frequencyand the name of the run.

In the Analysis panel a particular system for which output is avail-able can be selected and its output files inspected (Figure 19), out-put functions plotted (Figure 20), and the three-dimensionaldensity profiles of the beads be analysed graphically by slicing(Figure 18) and isodensity surfacing (Figure 23).

The Job Control section (Figure 19) allows the host for the run,and its associated parameters to be set, the job to be monitored,

Figure 15. The Cerius2 Visualizer window showing the MESOSCALE cardand the main DPD menu.



and output to be retrieved from remote hosts for analysis. It alsoprovides for two modes of simulation: Interactive mode, in whichthe DPD simulation continuously updates the Cerius2 model win-dow with snapshots of the system, and Background mode, inwhich the simulation runs independently of Cerius2. Advancedoptions here allow the user to modify the paths to the backgroundexecutable and the spawner program required for licensing pur-poses, as well as the username and password for runs on remotehosts.


A DPD simulation consists of a system of beads which may be con-nected together to form polymers. The number and type of beadsand their connectivity into polymers together with the forcesbetween them are specified in the sub-panels of the Build menuitem.

The Build menu item on the DPD card provides access to theBeads, Molecules, Interactions, Dissipations and Constraintspanels which allow all of the above parameters to be set(Figure 16). In particular, the chain architecture is entered in theform of a `tree string' by which even complex branched structurescan be defined as input to DPD. Comprehensive on-line help isavailable for this task. The spring constant for the Hookean forcelaw between beads connected into a polymer is set in the Mole-cules panel, as are the concentrations of each molecule type(including single-bead species such as water). The interactionparameters between all pairs of beads, and between beads and thewall if it is selected in the Constraints panel, are entered here aswell as the dissipative forces for all bead pairs.

Specifying the Run parameters for a DPD simulation

Having defined a molecular ensemble, a number of parametersneed to be set to define the particular simulation to be carried out.These parameters are specified in the DPD Run panel (Figure 17).Each run is identified by a seed name, given as the file prefix to allinput and output files generated. The model system is furtherdefined by specifying its temperature, and the size of the simula-tion box (with periodic boundary conditions), in terms of the side

General methodology


length of its cubic cells and the number of cells in each cartesiandimension. Furthermore, the step size for the numerical integra-tion and the length of the run, either in terms of its total time, or interms of the number of steps, must be set. A title for the simulationmay be added, to include comments about the specific run. Thedate and time are added to this automatically.

The output level of the run can be set from the Output... controlpanel. In particular, the period, in units of time steps, with whichthe output is sampled and the status information, density files,and restart files are written to disk is set here. The amount of disk

Figure 16. The DPD Build panels for defining beads, molecules,interactions and dissipations.



space required for frequent output of three-dimensional fields,such as contained in density, and restart files, may be very large.

The colors of the beads and polymers displayed in the model win-dow during an interactive run (and also during analysis) may beset using the Display submenu item of the Analysis menu(Figure 20). There are two modes of display: Bead and Molecule.In Bead mode, the beads and their connecting bonds are colouredby bead type whereas in Molecule mode, all the beads in the spec-ified molecules take on the colour of their parent molecule. The“Show beads as dots” and “Show bonds as lines” functions oper-ate in both modes, and can be used to render specific bead typesand bonds invisible.

A simulation is started by pressing the RUN button.

All the system parameters are written to a parameter file (*.Dpd_par) that is picked up by DPD at the start of the simulation. Youmay access existing parameter files from the Files... control panel,in order to save and edit them, update the panels from a file, or run

Figure 17. The DPD Run panel.

General methodology


DPD directly from an existing parameter file, thereby circumvent-ing the current interface settings.

Usually, a job is started from the default initial homogeneous mix-ture. However, you may wish to continue a previous run. This canbe done by choosing the Restart option on the Restart... controlpanel. A file for restarting (*.Dpd_rst) should be selected on thispanel and the Restart button selected. If any changes are requiredto the parameter file the Edit button may be used to call up an edi-tor window before selecting restart. Note that a run can only berestarted from this panel and not by using the Run button on theRun panel.

An èxpert' parameter accessible solely via the command line(DPD/RANDOM_SEED nnn) allows the user to set the seed (tonnn) for the random number generator. This enables repeated runsof a simulation with the same sequence of pseudo-random num-bers. Notice that the seed that appears in the parameter file as aresult of executing this command is not the number entered by theuser: however, the same number entered via this command alwaysproduces the same seed in the parameter file.

Figure 18. The DPD bead and molecule Display panels.



Setting the host machine, job monitoring and outputhandling

Opening the Job Control panel (Figure 19) from the DPD menushows the current host used for running DPD and the mode of exe-cution (interactive or background). The Options... control panelallows you to set the paths to the background executable code andthe spawner program required to run the licensed DPD code.Access information to the remote host (user name and password)can also be set here.

Further on the Job Control panel, you can set the base directory onthe remote host. The Monitor box shows previous and current jobsand related information such as process id and status.

The extent of a currently-running simulation can be monitored byinspecting the status file output by DPD. This shows the tempera-

Figure 19. The DPD Job Control panel.

General methodology


ture and pressure of the selected system and the number of timesteps so far completed. In addition, the output file produced byDPD may be monitored. This contains more information on thepolymer endpoint distributions and stress tensor calculation.

The current status of the job selected in the DPD Job Status box canbe obtained by pressing the UPDATE Job status button. A com-pleted job may be removed from the status window by selectingthe job and using the Remove button.

Finally, any output files that reside on a remote host may be trans-ferred to the current local directory by pressing the TRANSFERbutton.

Analysing the result: Selecting a system, inspecting thefiles, plotting the thermodynamic functions, andexploring the morphology

Having completed a simulation and transferred the output files tothe local host, you are ready for analysis of the results. This func-tionality is accessed via the Analysis submenu on the main DPDcard.

The first step is always to open the System Analysis panel(Figure 20). This provides a file browser with a filter for the DPDstatus files. Selecting the status file from the desired run sets thename of the system to be analysed further. The name and title ofthe system selected will be highlighted, and you may examine thestatus file by pressing the pushbutton.

As a next step, you can inspect the time evolution of the diffusioncoefficients for all beads in the system by selecting the PLOT/DIF-FUSION menu item (Figure 21). Alternatively, you can display thepolymer endpoint distribution or the bond-length distribution foreach polymer type.

Finally, and usually most importantly, the morphology of themolecular ensemble can be investigated and analysed. Two meth-ods are provided for this.

First, the DPD Profiles panel (Figure 22) lets you create slicesthrough the simulation box for each time step and for each bead.



Figure 20. The DPD System Analysis panel.

Figure 21. The DPD Plot panel for plotting the bead diffusion coefficients,and polymer endpoint and bond length distributions.

General methodology


The profiles are displayed in the model window, and any of thetools provided by the Cerius2 Visualizer to alter the display ofmodels (such as rotation, translation, magnification, colors, light-ing, clipping, depth cueing etc), to save and print the models, isavailable to you. Furthermore, although the current DPD interfacedoes not provide an animation facility directly, a sequence of mod-els can be saved in a Log file via Utilities/Record Commands fromthe Menu bar, and then replayed via the Utilities/Playback Scriptfacility.

On the DPD Profiles panel, you may change the position anddirection of the selected slice by means of the sliders. More EditingOptions... shows the way in which the density is mapped to color,and lets you edit this mapping. The Profile grid scaling is a factor

Figure 22. The DPD Profiles panel for creating and analyzing slicesthrough the density fields.



by which the actual grid is multiplied to generate the triangularmesh for the graphics. A higher number gives a smoother appear-ance but takes longer to process.

Second, you can create isodensity surfaces of the bead concentra-tions from the Isodensities panel (Figure 23).

You can open this panel from the Morphology menu item of theAnalysis menu. For each bead type you can create an isodensitysurface at a number of time steps and for up to four different val-ues of the concentration at once. Typical analyses involve displaysof (a) the different beads, each at a concentration close to its maxi-mum (to be determined from the slice display), (b) one bead typeat a number of time steps, and (c) one particular bead at different

Figure 23. The DPD Isodensities panel for creating and analyzingisosurfaces of the density fields.

Theory


isodensities. Each surface in the current model can be edited interms of its visibility, transparency and color.

Finally, the main DPD card provides a Reset button, which reini-tializes all the panels, and sets all values back to their defaults.

Theory

Introduction

The most accurate method of calculating the dynamical behaviorof an atomistic system is to integrate the equations of motion of allthe atoms in the system. This is the basis of the Molecular Dynam-ics simulation technique. Its major drawback is that it often pro-vides far more detail of small-scale fluctuational motion of atomsthan is necessary for an understanding of many physical pro-cesses. It also requires computer processor speeds and memorycapacities that currently limit its applicability to a few nanosec-onds of molecular motion. This is inadequate for many chemicalprocesses that occur on the microsecond (or longer) time-scales.Some years ago, Hoogerbrugge and Koelman (1992) introduced anew simulation technique, derived from Molecular Dynamicssimulations and Lattice Gas Automata, that effectively opens upthe mesoscopic length and time regimes in complex fluids to sim-ulation. It achieves this goal by keeping the principle of integrat-ing the equations of motion for a system but integrating out thesmallest spatial degrees of freedom first. The fast motion of theatoms in a system is averaged over and the remaining structure isrepresented by a set of “beads”, of given mass and size, that inter-act via soft potentials with other beads. A bead represents a smallregion of fluid matter and its motion is assumed to be governed byNewton’s laws in which the total force on a bead is a sum of itsdirect interactions and dissipative and random forces between itand other beads. The dynamical behavior of the system is followedalong a trajectory through its phase space by integrating its equa-tions of motion. The equilibrium properties are calculated by per-forming suitable averages along this trajectory. In this section wedescribe in detail the simulation technique and how it can berelated to the Flory-Huggins theory of polymeric fluids. The fol-



lowing description of the theory is based on a paper by Groot andWarren (1997).

Equations of motion for a DPD system

We consider a set of beads interacting by specified forces andwhose dynamical evolution is governed by Newton’s laws

Eq. 21

where ri, vi and fi and are the position vector, velocity and totalforce on the ith bead. For simplicity, we set all bead masses miequal to unity. Each bead is subject to three forces from its neigh-bors: a conservative interaction which is linear in the bead-beadseparation; a dissipative force proportional to the relative velocityof two beads and a random force between a bead and each of itsneighbors. The total force on a bead is

Eq. 22

where the sum is over all beads within a distance rc of the ith bead.This short-range cut-off makes the interactions local. From now onwe set rc equal to unity so that all lengths are measured relative tothe bead radius. The conservative force is a repulsive central forcewith a maximum magnitude aij

Eq. 23

where rij is the magnitude of the bead-bead vector rij , and r-hatij isthe unit vector joining beads i and j.

The dissipative force is proportional to the relative velocity of twobeads and acts so as to reduce their relative momentum

Eq. 24

ri∂t∂

------- vi= , mi

vi∂t∂

------- fi,=

fi FijC

FijD

FijR

+ +

j 1≠∑=

FijC

aij 1 rij–( ) rij rij 1<

0 rij 1,

>

=

FijD γωD

– rij( ) rij vij⋅( ) rij rij 1<

0 rij 1,

>

=

Theory


where ωD(rij) is a short-range weight function. Because of the formchosen for the dissipative force it conserves the total momentumof each pair of particles, and hence also of the system. The randomforce also acts between all pairs of beads subject to a similar short-range cut-off, with a possibly different function ωR(rij), and acts soas to pump energy into the system

Eq. 25

where ζij(t) is a delta-correlated stochastic variable with zero mean

Eq. 26

Note the unusual property of the random force that it is pairwisecentral (in contrast to Brownian Dynamics in which white noise isadded to the overdamped equation of motion for each particleindependently) and hence conserves total linear momentum evenwhile adding energy to the system. At this stage, there are twounknown functions, ωD(rij) and ωR(rij), and two unknown con-stants γ and σ. Espagnol and Warren (1995) have shown that inorder for the steady-state solution to the equations of motion to bethe Gibbs ensemble, and for the fluctuation-dissipation theorem tobe satisfied, only one of the two weight functions ωD(rij), ωR(rij)can be chosen arbitrarily while this choice fixes the other; and thetwo multiplicative constants γ, σ are related by the temperature.That is, we must impose the relations

Eq. 27

where T is the absolute temperature and kB is Boltzmann’s con-stant. For simplicity we choose

Eq. 28

At this point a word about the role of units in DPD simulations isin order.

Unlike Molecular Dynamics simulations in which the particlesrepresent atoms with a known size subject to experimentally-mea-

FijR σωR

rij( ) ζ ijrij rij 1<

0 rij 1,

>

=

ζ ij t( )⟨ ⟩ 0 ζ ij t( ) ζkl t ′( )⋅⟨ ⟩ δ ikδjl δilσjk+( ) δ t t ′–( )= =

ωDr( ) ωR

r( )2

= , σ22γkBT,=

ωDr( ) ωR

r( )2

= 1 r–( ) 2r 1.≤=



surable forces, the beads in DPD do not correspond to real atomsor molecules. They represent small regions of fluid material inter-acting via phenomonological forces. Starting from Newton’s lawsabove, we have chosen the bead mass and radius (actually therange of the bead-bead interactions) to be unity. Quantities withunits of mass and length are rendered dimensionless by divisionby the bead mass and radius in all that follows unless stated oth-erwise. There remains still the time unit to specify. Instead of set-ting this to unity, we can use the theorem of equipartition of energyto normalize the velocities of all the beads by the temperature. Thisis equivalent to measuring time in units of SQRT[mrc

2/kBT]. Allthe quantities in the DPD simulation are then dimensionless. Thetemperature parameter fixes the mean of the initial velocity distri-bution which is, up to the numerical accuracy of the integrationscheme, constant. Notice that increasing the temperature reducesthe above dimensionless time interval and this requires a smallertime-step in the integration scheme to maintain accuracy. To relatethe results of a simulation to a real system we have to put back theunits of mass, length and time by choosing the mass and radius ofa bead and the temperature. The bead positions, velocities and dis-tributions can then be converted into physical units by scalingwith the appropriate combinations of these three fundamental val-ues. If (r,v,t) are a length, velocity and time in physical units, thecorresponding quantities in the DPD simulation are given by

Eq. 29

The results of a single DPD simulation may thus correspond tomany physical systems depending on the values chosen for thebead mass and radius, the temperature and the interaction anddissipation parameters.

Integration scheme in DPD

A modified velocity-Verlet algorithm has been used to integratethe equations of motion. In this scheme, the current values of posi-tion, velocity and the force on a particle are used to calculate theposition and velocity of the particle at the next time-step; the new

rrrc----= , v

v

kBTm----

--------------, tt

mrc2

kBT---------

--------------= = .

Theory


position and velocity are then used to calculate the new force, andthis then corrects the velocity.

Eq. 30

The factor λ is set to 1/2 in the DPD simulation code, althoughGroot and Warren (1997) discuss the effect of alternative values forthis factor in taking the stochastic nature of the force into accountwhen calculating the order of the integration scheme.

Choosing the dissipation and random noise magnitudes

It was stated above that the magnitudes of the dissipative and ran-dom forces are connected by the fluctuation-dissipation theorem.This still leaves one of them as a free parameter. We take this to bethe magnitude of the dissipative forces in the system. Groot andWarren (1997) found that there was no observable differencebetween the effects of Gaussian noise and uniform noise and so thesimpler uniform noise has been used. When the noise magnitudewas larger than approximately σ = 8, they found that the integra-tion scheme was unstable. It was found that a noise amplitude ofσ = 3 gave reasonably fast relaxation for temperatures betweenKBT = 1 and KBT = 10. When choosing values for the dissipationparameters, care must be taken that the chosen temperature anddissipation parameters do not result in noise that is much largerthan σ = 3 or the simulation will be unreliable.

Choosing the repulsion parameters

Once the mass and radius of the beads, the architecture of the poly-mers in the system, and the physical constraints of temperature,dissipation magnitude and system size have been fixed, thereremains only one parameter to characterize the interactions

ri t ∆t+( ) ri t( ) ∆tvi t( ) 12--- ∆t( )2

fi t( ) ,+ +=

vi t ∆t+( ) vi t( ) λ∆ tfi t( ) ,+=

fi t ∆t+( ) fi r t ∆t+( ) ,vi t ∆t+( ) ,=

vi t ∆t+( ) vi t( ) 12---∆t fi t( ) fi t ∆t+( )+( ) .+=



between the beads: the repulsion parameters aij. These are all thatappear in the DPD methodology to represent the complex forcesthat act between the atoms and molecules in a physical system.

In order that a DPD fluid correspond to a typical liquid, such aswater, its density fluctuations should correspond to those of thereal liquid. We fix the repulsion parameters by using the equationof state of the DPD fluid. Groot and Warren (1997) found from sim-ulations that for moderate densities (approximately 3-10 beads perunit volume of the simulation box) and repulsion parameters (a =15 - 30) the equation of state of a simple DPD fluid is

Eq. 31

where p is the pressure, ρ is the density and α = 0.101 ± 0.001. Inorder to fix the magnitude of the repulsion parameters we com-pare the dimensionless compressibility of the DPD fluid with thatof water using the equation

Eq. 32

For water at room temperature (300 K) the dimensionless com-pressibility has the value κ-1 = 15.9835. By differentiating the equa-tion of state and comparing with the compressibility of water wefind aρ/kBt ≈ 75 . In principle, we could choose the density freely,but to minimise the number of interactions between beads in asimulation we take the smallest value at which this relationshipholds reasonably well: Groot and Warren have shown that ρ ≈ 3 issufficient. In order that a DPD fluid has the compressibility ofwater, we need a repulsion parameter of a ≈ 25kBT at ρ ≈ 3; for otherdensities we use a ≈ 75kBT/ρ.

Mapping the interactions onto Flory-Huggins theory

Having established that a DPD simulation can represent the vol-ume fluctuations of a simple fluid, we would like next to be able tomodel the interactions between unlike fluids such as copolymermelts or surfactant-oil-water mixtures. We need to find a relationbetween the interactions between DPD beads and the theory ofpolymer mixtures so that the behaviour observed in a simulation

p ρkBT αaρ2,+=

κ 1– 1kBT--------- p∂

ρ∂------

T

= .

Theory


can be mapped onto the physical phase diagram of real fluids. Oneway of doing this is to compare the free energy of a DPD fluid withthe mean-field theory of polymer mixtures, Flory-Huggins theory.We give a brief summary of this theory.

The Flory-Huggins theory of polymers is a lattice theory in whicheach lattice site is occupied either by an A or B type monomer. LetNA and NB be the number of monomers per polymer of each type,and φA and φB the volume fractions of each polymer. The freeenergy of the mixture in excess of the ideal gas contribution is

Eq. 33

where the filled lattice condition means φA + φB = 1. The χ param-eter controls the interaction between the polymers: if positive, thetwo species prefer to phase separate, whereas if it is negative theyprefer to mix. The equilibrium state of the mixture is found byminimising the free energy which leads to an implicit relation forthe density

Eq. 34

As χ increases from a negative value the system starts to phaseseparate. The critical value of χ at which phase separation firstoccurs may be found by minimising the free energy and notingwhere the spinodal points coincide. As in van der Waal’s theory ofgases, this occurs when the first and second derivatives of the freeenergy are zero. We find

Eq. 35

The free energy density of a single-component DPD fluid whoseequation of state, as given above, is quadratic in the density is

Eq. 36

and for a two-component fluid we expect

FkBT---------

φA

NA-------lnφA

φB

NB-------lnφB χφAφB,+ +=

χNA

ln 1 φA–( ) φA⁄[ ]1 2φA–( )

--------------------------------------------= .

χcrit 12--- 1

NA

----------- 1

NB

-----------+

.=

fV

kBT--------- ρlnρ ρ–

αaρ2

kBT-------------,+=



Eq. 37

Setting aAA = aBB and assuming that the total density is approxi-mately constant, we find (up to constant terms)

Eq. 38

where we have written x = ρA/(ρA + ρB), and identified

Eq. 39

Comparing the DPD free energy with the Flory-Huggins equiva-lent suggests that the two theories correspond if the χ parameter isproportional to the repulsion parameters as specified by this equa-tion. Groot and Warren (1997) find that the excess pressure of abinary mixture of DPD monomers and polymers is proportional tothe term x(1-x) for repulsion parameters aAA = aBB = 15, 25, , butthat the constant of proportionality is not linear in the repulsionparameters (aAB = aAA). However, for large enough values of the χparameter we expect the mean field theory to be reasonably accu-rate, and we can use the above relation between the density of theDPD fluid and the equivalent χ parameter to relate the two theo-ries. The average density is measured in two homogenous regions,distant from any interfaces, in a DPD simulation of a binary mix-ture. The quantity χNA is obtained from the density via the loga-rithmic expression above (noting that NA = 1 for monomericsystems), and is plotted against the repulsion parameter differ-ence, (aAB = aAA) to test the correspondence between the Flory-Huggins theory and the simulation. Groot and Warren (1997) find

Eq. 40

fV

kBT---------

ρA

NA------- ρA

ρA

NA-------

ρB

NB-------lnρB

ρB

NB-------–+–ln=

αkBT--------- aAAρA

22aABρAρB aBBρB

2+ +

.+

fV

ρA ρB+( ) kBT------------------------------------ x

NA-------lnx

1 x–( )NB

------------------ln 1 x–( ) χx 1 x–( ) ... ,+ ++≈

χ2α aAB aAA–( ) ρA ρB+( )

kBT-----------------------------------------------------------------=

χ 0.286 0.002±( ) aAB aAA–( )=

χ 0.689 0.002±( ) aAB aAA–( ) ,=

Theory


for densities of ρ = 3.5 respectively. Notice that although the coef-ficient is not linear in the density as expected, such a relation canbe measured from a set of simulations at a fixed density and usedto relate the repulsion parameters in the DPD fluid to the χ param-eter in the Flory-Huggins theory.

It remains to show how we can simulate polymers using the DPDbead-bead interactions. Typically, beads are strung together intopolymers using harmonic forces, and the spring constant is chosenso that the average monomer-monomer distance has a reasonablevalue, such as corresponding with the first peak in the bead-beadcorrelation function. The optimum value must be extracted fromthe simulations in the same manner as the equation of state for agiven system. The default value for this parameter is 4. The equi-librium length for the springs is chosen to be zero for simplicity,although this is not required. The repulsion parameters betweenbeads at a given density ρ are chosen to be

Eq. 41

where the second term is absent for interactions between likebeads. The χ parameter for polymers is found from simulations byvarying the repulsion parameters and the polymer length andplotting χNkBT/(aAB - aAA) against N. Groot and Warren (1997)find

Eq. 42

This relation enables the properties of real polymers, with lengthsof the order of ten thousand monomers, to be simulated usingpolymers with only 10 or so DPD beads per polymer, if the χparameter from the Flory-Huggins theory of the polymers isincreased such that the product χN remains constant. The input tothe DPD simulations may be obtained from MD simulations ofrealistic models of the polymers that generate only the χ parame-ters. We next indicate how these may be obtained from micro-scopic simulations and other techniques within Cerius2.

aij

75kBT

ρ--------------- aAB aAA–( )+= ,

χNkBT

aAB aAA–( )------------------------------- 0.306 0.003±( ) N.=



Calculation of Flory-Huggins χ parameters as input toDPD simulations

There are several methods of calculating the χ parameters for poly-mers of known structure. Three of these are available within theCerius2 software suite. In order of increasing complexity they are:

♦ QSPR using C2•Synthia

♦ Monte Carlo simulations of pair contact energies usingC2•Blends

♦ Molecular Mechanics using C2•Polymer Builder andC2•Amorphous Builder plus the Open Force Field or C2•Dis-cover

The first method involves correlating the structure of known poly-mers with their measured properties and then using this knowl-edge to predict the behavior of other polymers whose structure isknown (Bicerano, 1996). This is essentially an empirical structure-property correlation method.

The second method relies on calculating the free energy of mixingof one species of polymer in another from their pair contact ener-gies. Flory-Huggins theory assumes that the χ parameter for amixture of A and B polymers represents the repulsive energy of anAB pair averaged over all possible configurations: the pair is con-sidered to feel only the mean field of all the other polymers and toadopt that configuration that minimises the mean field free energy.Hence, it is natural to try and calculate the pair contact energy bysampling over the most probable conformations of a pair of poly-mers in contact using Monte Carlo simulations. By suitably choos-ing the set of trial conformations, the average interaction energycan be calculated and the χ parameter extracted.

The final method involves using Molecular Mechanics simulationsto find the cohesive energy of the two species of polymer in theirpure phase and when mixed together. Comparing these with thegas phase energy of each species allows us to calculate the energyof mixing of the polymers and extract the χ parameter. Thismethod is more accurate than the other two, including as it doeshydrogen bonding effects and chain structure in atomistic detail,but takes considerably longer to calculate.

References


Further details on these methods can be found in the Cerius2 doc-umentation for the relevant products.

References

Bicerano. J. Prediction of polymer properties, 2nd edition. Marcel Dek-ker. New York (1996).

Case, F. H. and D. J. Honeycutt. TRIPS. 2 259 (1994).

Espagnol, P. and P Warren. Statistical mechanics of Dissipative Parti-cle Dynamics (1995).

Europhys. Lett. 30 191-196.

Fan. C. F., B. D. Olafson, M. Blanco, and S. L. Hsu. Macromolecules.25 366 (1987)..

Groot R. D. and P. B. Warren. “Dissipative Particle Dynamics:Bridging the gap between atomistic and mesoscopic simula-tion” J. Chem. Phys. 107 4423-4435 (1997).

Hoogerbrugge, P. J. and J. M. V. A. Koelman. “Simulating micro-scopic hydrodynamic phenomena with Dissipative ParticleDynamics” Europhys. Lett. 19 155-160 (1992).




A File Formats

Mechanical Properties files

Output files

In addition to the files used to save the results (.txt and .dat), anumber of other files are generated each time a mechanical prop-erties calculation is performed. These include a model file (.msi), afile containing the run parameters (.mp), and trajectory files (.trj).In default mode, six trajectory files are output (one for eachsweep). In custom mode, only one trajectory file is output.

File names The files are all named with the file name prefix (seed) specified onthe Mech Props Run control panel (see the online help for moreinfo). The method used (CStress, CStrain, or SecDer) is appendedto this prefix and the appropriate extension is included at the end.If more than one sweep is done, the varying component is insertedinto the trajectory file names.

For example, a calculation performed using the Const Stress Minmethod in default mode without accumulating averages wouldgenerate the following files:

♦ Seed_CStress.mp — File containing the run parameters

♦ Seed_CStress.msi — Model file used by the calculation

♦ Seed_CStress.xx.trj — Trajectory file from sweep 1

♦ Seed_CStress.yy.trj — Trajectory file from sweep 2

♦ Seed_CStress.zz.trj — Trajectory file from sweep 3

♦ Seed_CStress.yz.trj — Trajectory file from sweep 4

♦ Seed_CStress.xz.trj — Trajectory file from sweep 5

♦ Seed_CStress.xy.trj — Trajectory file from sweep 6


A. File Formats

♦ Seed_CStress.txt — Text file containing the calculation results(output by default)

♦ Seed_CStress.dat — Tables file containing the calculation results(only output when the Output Results To File box is checked)

When the Accumulate Averages box is checked, a generation num-ber is also inserted into the .mp, .msi, and .trj file names. The initialpart of the file names becomes:

seed_CStress_n

Where seed is the files prefix and n is an integer indicating that ncalculations have been averaged.

For example, a file called MechProps_CStress_4.mp would containthe run parameters for the fourth calculation in a series of accumu-lated calculations where the files prefix is MechProps.

Morphology files

CIF format

Cerius2 uses a subset of the Crystallographic Information File(CIF) format (Hall et al. 1991) for storing habit (.hab) data. The fol-lowing data records are used:

Record Keyword Format Description

1 _audit_creation_method 1X,1A,1A Records origin of file. Currently, this isMORPHOLOGY FROM Cerius2

2 _symmetry_space_group_name_H-M

1X,1A,1A Hermann-Maugin name for spacegroup (for example, P 1 21/a 1), fol-lowed by option number

3 _symmetry_Int_Tables_number

1X,1A,1I12 Space group number in internationaltables

4 _cell_length_a 1X,1A,F12.6 Size of crystal cell a dimension5 _cell_length_b 1X,1A,F12.6 Size of crystal cell b dimension6 _cell_length_c 1X,1A,F12.6 Size of crystal cell c dimension7 _cell_angle_alpha 1X,1A,F12.6 Crystal cell alpha angle8 _cell_angle_beta 1X,1A,F12.6 Crystal cell beta angle

Morphology files


All the initial records (1 through 15) begin with a blank space fol-lowed by a keyword. The remaining records (16 and above) list theMiller plane indices and center-to-face distance for each crystalface. The file is terminated with a blank line.

A sample .hab file is shown below.

_audit_creation_method MORPHOLOGY FROMCerius2 _symmetry_space_group_name_H-M P 2/b 21/c 21/mOption 1 _symmetry_Int_Tables_number 57 _cell_length_a 4.076000 _cell_length_b 13.380000 _cell_length_c 6.721000 _cell_angle_alpha 90.000000 _cell_angle_beta 90.000000 _cell_angle_gamma 90.000000 _exptl_crystal_colour PIN loop_ _exptl_crystal_face_index_h _exptl_crystal_face_index_k _exptl_crystal_face_index_l _exptl_crystal_face_perp_dist 2 1 0 49.633644 1 0 0 24.533857 1 5 1 47.114166 1 5 0 44.703106 1 4 2 48.797031 1 4 1 41.436939 1 4 0 38.673546

9 _cell_angle_gamma 1X,1A,F12.6 Crystal cell gamma angle10 _exptl_crystal_colour 1X,1A,1A Color used in morphology display11 loop_ 1X,1A Indicates beginning of loop12 _exptl_crystal_face_index_h 1X,1A Label for crystal face data13 _exptl_crystal_face_index_k 1X,1A Label for crystal face data14 _exptl_crystal_face_index_l 1X,1A Label for crystal face data15 _exptl_crystal_face_index_

perp_dist1X,1A Label for crystal face data

16 …(one/face)

3I8,1F12.6 Miller plane indices (h,k,l), center-to-face distance (Å)

Record Keyword Format Description


A. File Formats

1 3 2 44.611012 1 3 1 36.414444 1 3 0 33.236050 1 2 2 41.362431 1 2 1 32.353054 1 2 0 28.728790 1 1 2 39.284550 1 1 1 29.650385 1 1 0 25.646997 1 0 2 38.567051 2 0 2 57.385956 0 6 1 47.246967 0 2 0 14.947683 0 10 4 95.539833 0 10 2 80.444626

Interactions.dat

Interactions.dat files are used to store intermolecular interactionsenergy information. These files can be saved after performing anattachment energy calculation or checking the lattice energy. Thefollowing data records are used:

Record* Format Description

1 1A,1I4,1A Header stating number of molecules incrystal cell

2 … N+1(1/molecule)

1A,1I4,1A,3F8.3,1A

Molecule number and position withincell

N+2 to N+4 Blank linesN+5 1A Title for lattice energy dataN+6 1A Line of dashesN+7 1A,1F14.5 Total lattice energyN+8 1A,1F14.5 Attractive energy contributionN+9 1A,1F14.5 Repulsive energy contributionN+10 1A,1F14.5 Coulombic energy contributionN+11 1A,1F14.5 Hydrogen bond contributionN+12 1A,1I14 Number of intermolecular bondsN+13 to N+15 Blank linesN+16 2A Labels for intermolecular energy data

Polymorph files


Polymorph files

Trajectory file usage and naming

The Monte Carlo simulation, cluster analysis, and minimizationprocedures all produce output in the form of trajectory files. Thecluster analysis and minimization phases also take trajectory filesas input.

N+17 Blank lineN+18 …

(1/slice)2I4,2X,3I4,2X,5F10.

5Number of molecule I, number of mole-

cule J, their UVW translation, and theirattractive, repulsive, Coulombic, H-bond, and total intermolecular ener-gies

(then for eachslice)

1A,3I4 Slice indices (h k l)

1A,F12.4 Lattice energy1A,F12.4 Slice energy1A,F12.4 Attachment energy1A,1I4 Central molecule in slice1A,F12.4 Offset1A,1I4 Number of contributing bonds

Blank line2A Labels for intermolecular energy data

Blank line2I4,2X,3I4,2X,5F10.

5Number of molecule I, number of mole-

cule J, their UVW translation, and theirattractive, repulsive, Coulombic, H-bond, and total intermolecular ener-gies

*N is the number of molecules in the crystal cell

(Continued)

Record* Format Description


A. File Formats

Note

The Monte Carlo simulation produces one trajectory file for eachspace group searched using the following naming convention:

FileNameSeed.SpaceGroup.pmp

where FileNameSeed is a user-definable text string; the default isPMorph (see the online help for the Polymorph MC Output con-trol panel for more detailed information). SpaceGroup is a modifiedrepresentation of the name of the space group that was searched inthe simulation run (in which spaces and forward slashes (/) arereplaced by underscores).

For example (default FileNameSeed assumed):

♦ A search of space group P 21/c produces a trajectory file namedPMorph.P_21_c.pmp

♦ A search of space group I -4 c 2 produces a trajectory file namedPMorph.I_-4_c_2.pmp

Subsequent clustering and minimization operations should beperformed for each space group searched using the appropriatetrajectory files as input. Unless file names are specifically definedfor cluster analysis and energy minimization output, the name ofthe input file is used as the template for naming the output file.Specifically, a period and a two-letter identifier is added immedi-ately before the .pmp extension to identify the operation that cre-ated the trajectory file:

InputFileName.SourceIdentifier.pmp

where InputFileName is the name of the input trajectory file onwhich the procedure was performed, and SourceIdentifier is:

♦ cl — For files output by cluster analysis

♦ mi — For files output by energy minimization

For example:

The trajectory file format currently implemented is proprietaryand specific to Polymorph Predictor. These trajectory files (with.pmp extension) therefore cannot be read into or written byother Cerius2 modules.

Polymorph files


♦ A trajectory file output by cluster analysis of PMorph.P_21_c.pmp (output from Monte Carlo simulation) is namedPMorph.P_21_c.cl.pmp

♦ A trajectory file output from an energy minimization procedureperformed on PMorph.P_21_c.cl.pmp (output from clusteranalysis) is named PMorph.P_21_c.cl.mi.pmp

♦ A trajectory file output by a final cluster analysis procedureperformed on PMorph.P_21_c.cl.mi.pmp (output from energyminimization) is named PMorph.P_21_c.cl.mi.cl.pmp

Tip By default, the last trajectory file generated by any prediction pro-cedure is selected as the default input file for subsequent clusteranalysis and energy minimization operations. For operational con-venience, when manually running a prediction that searches poly-morphs in several space groups (each involving its own trajectoryfiles), you may want to follow the complete sequence (MonteCarlo, cluster analysis, minimization, and final clustering) for eachspace group in turn.


A. File Formats


B Using MSI Online Documentation

To use MSI online documentation, you must first have a webbrowser, such as NCSA Mosaic or Netscape Navigator.

MSI Hypertext Locations

You can find additional documentation for MSI products over theinternet at the MSI corporate web site:

HTTP://www.msi.com/doc/index.html


B. Using MSI Online Documentation


Aaccumulate averages of mechanical properties

6alignment of polymers

measuring 39alloys, polymer, see Blendsall-trans conformation 79analysing trajectory files 149analyzing

stress/strain data in Mechanical Properties14

applied strain table 10, 12applied stress table 10, 11asymmetric unit 128, 130

defining molecules in the 134atomic correction terms 89atomic indices 86atomistic interactions 85averaging

of mechanical properties 6axial packing for Blends 63

Bbackbone

flags 94, 96torsion 101

bad contacts check in Crystal Packer, see closecontacts

bad contacts check in Sorption 232, 235Biased Metropolis averaging in Pairs Method

for Blends 60Bicerano, Dr. Jozef 69binodal curve, see phase diagramsBlends interaction energy (.enr) files 44

analysis of 49, 53extracting molecular pairs from 53loading 48, 54saving 45

Blends module 41–67

analysis functions 43, 65applications for 42–43calculation functions 43, 58interaction parameter 55, 63, 64phase diagrams 57, 65theory 56–57

Boltzmann averaging in Pairs Method forBlends 59

bond indices 87bond selection rules for polymer orientation

function 32branched polymer chain 98bulk modulus

definition 17to calculate, see mechanical properties

bump check, see bad contacts check

Ccalculating

chain properties of polymers 23coordination numbers 45crystal energy 116dihedral angle distributions in polymers 26enthalpy of mixing 51entropy of mixing 51free energy of mixing 51, 66interaction parameter 65mechanical properties 5phase diagrams, two component 65physical properties of polymers 23polymer orientation functions 38Voronoi volume of polymers 25

canonical ensemble sorption simulation, seefixed loading

catalysts database 231cell parameters

constraining during Crystal Packer mini-mization 112

chain connectivity 88chain continuation 86charges, ESP 130


D

Chi, see interaction parameterCIF file format 322cluster analysis 128, 144

procedure 144cluster analysis preferences

clustering parameters 146input file 145output options 147

cluster, lowest energy representative 128clustering energy minimization output 144clustering Monte Carlo output 144cohesive energy 80, 89combination rules, see VDW energy termcomparing powder spectra 151compliance matrix 2

definition 17to calculate, see mechanical properties

compressibility, see volume compressibilityconnectivity indices 69, 86, 87

backbone 89first-order (bond) 88side group 89zeroth-order (atomic) 87

constant strain minimization (for propertyprediction) 4setting up 9–12

constant stress minimization (for propertyprediction) 3setting up 9–11

constraints for Crystal Packer minimization112

contact table, see nonbond listcontributions 80convergence accelerating smearing 119coordination numbers in blends 55, 61

procedure for calculating 45copolymer, statistical 69correlations, general forms of 89Coulomb energy calculation in sorption 234,

236Coulomb energy term, Crystal Packer 120crystal constraints, effect on mechanical prop-

erties 7determining 9

crystal modifications 157Crystal Packer module 105–126curve fitting, see data fitting

Ddat files

output from Mechanical Properties 13data fitting (Mechanical Properties) 15defining space groups 143degree of polymerization 55density of polymers 34deriving mechanical properties 17Designer Correlations 81dihedral angle distributions in polymers 26

selecting torsions 26torsion selection rules 28

dihedrals, see also torsionsdipole moment 100dipole moment in polymers 33displaying

mechanical properties 12Dow Units 81Dreiding force field 130

Eelectronic environment 86electrostatic interactions 130electrostatic, see Coulombempirical and semi-empirical methods 69end-to-end distance

probability distribution 99end-to-end distance of polymers 36energy

minimization, see minimizingprocedure for setting up Crystal Packer cal-

culation 108setting calculation for Crystal Packer 117setting calculation for Mechanical Proper-

ties 19setting calculation for Sorption 232

energy distribution histogram, sorption 247,250

energy expression in Crystal Packer, settingup 117

energy minimization 129, 147optimizing the current model 147

energy of mixingfitting 64

enthalpy of mixing

F


calculating and plotting 51entropy of mixing

calculating and plotting 51ESP charges 130Ewald long-range summation method 130Ewald summation method

in Crystal Packer 109, 121in Sorption 234, 236

excluded atom constraints in Blends packing64

excluded volume constraint, Blends 59extensive property 89

additive constant term 89extracting models from trajectory files 149

FFedors-like, cohesive energy 80file formats

crystal morphology 322interactions.dat 324

files.arc 99.rislog 99.rismnt 99.tab 99_dih.tbl 99_pr.tbl 99_rsq.tbl 99

finding probable crystal structures 159fixed loading sorption simulation 230, 253–254fixed pressure sorption simulation 230, 254–

256flexible molecules 132flexible molecules, handling 131Flory-Huggins model for polymer blends 54–

55force field

Crystal Packer's own 105, 117for Blends calculation 57for mechanical properties 19for sorption simulation 231, 235

force fields 132free energy of mixing 55

calculating 51, 66plotting 51

fugacity in fixed pressure sorption simula-tions 254

GGaussian ab initio quantum mechanics pack-

age 130geometrical parameters 89grand canonical ensemble sorption simula-

tion, see fixed pressuregraph

hydrogen-suppressed 86graphs

applied strain profile (Mechanical Proper-ties) 12

applied stress profile (Mechanical Proper-ties) 11

Blends pair energy analysis 49Crystal Packer energy 116energy of mixing 65interaction parameter χ(T) 65phase diagrams, two component 65polymer dihedral angle plots 26polymer orientation angle histogram 30polymer orientation vector vs. time 30, 32polymer properties vs. time 24sorption analysis 245sorption output 237, 239thermodynamics of mixing 51Voronoi volume vs. time 37

group additive methods 75, 85group contributions 69group correction terms 89

HH-bond energy term, Crystal Packer 121–123Henry constant sorption calculation 230, 256–

258Hermans orientation function, see orientation

functionhomopolymer, bulk amorphous 69hydrogen-suppressed graph 86hydrostatic energy term, see pressure term


I

Iindices

atomic 86bond 87connectivity 86valence 87

initialization of model in Crystal Packer 107intensive properties 89interaction energies

file format 324interaction energies, see pair interaction ener-

giesinteraction parameter

calculating and plotting 64definition of 55expression for 63

interactions.dat file format 324interrupting

a sorption simulation 244intramolecular symmetry 133isotherms, sorption see loading curve 247isotherms, thermodynamic for Blends 52

to plot 53isotropic packing for Blends 63

LLamé constants

equation for 18to calculate, see mechanical properties

lattice cluster theory for polymer blends 55Lennard-Jones functional form 119liquid crystal polymers

orientation function of 38packing variables for Blends 60, 64

loadingBlends interaction energy (.enr) files 48, 54Mechanical Properties .mp files 16off-diagonal VDW pair potentials file 120on-diagonal VDW pair potentials file 120trajectory file for Polymer Properties analy-

sis 22loading curve, sorption 247, 251lowest energy cluster representative 128

Mmacromolecules, see polymersmass clouds, sorption 248, 252mass distribution plot, sorption 246, 250Max Bonds 95mechanical properties 1, 69, 79

analyzing stress/strain data 14calculating 5deriving from stiffness matrix 17models for calculating 19output 6, 12prediction methods described 2–18saving data to file 12

Mechanical Properties module 1merging trajectory files 155Metropolis algorithm 138Metropolis averaging in Pairs Method for

Blends 60Metropolis selection in sorption simulation

253Min Bonds 95minimization algorithms 113

Modified Newton 113Steepest Descent 113

minimization preferencesinput file 147output options 148termination criteria 147

minimizing, crystal energy 106, 112constraints setup 112setup for 113

minimizing, in Mechanical Propertiesmodel first 6under constant strain, see constant strain

minimizationunder constant stress, see constant stress

minimizationminimum image convention 233mixing energy, see energy of mixingmixtures

polymer-polymer, see Blendspolymer-solvent, see Blendssolvent-solvent, see Blends

model initialization, Crystal Packer 107model window

sorption updates 237, 238modifications, crystal 157

N


Modified Newton, minimization algorithm113

modulussee bulk modulussee shear modulussee Young's modulus

molarheat capacity 89volume 89

molecular pair configurations 44energy distribution 49selection by energy 53

molecular properties 86molecular silverware algorithm 59molecular weight of polymers 34Monte Carlo packing simulation 128, 138

trial steps 139Monte Carlo preferences

output options 144search parameters 141space groups to searcb 142

morphology of crystalsfile format 322

move probabilities, sorption 240mp files for Mechanical Properties 321msi model files output by Mechanical Proper-

ties 321

Nnearest neighbors packing in Blends 62nonbond list

updating during minimization

Ooff-diagonal VDW parameters 120on-diagonal VDW parameters 119order parameter of polymers, see orientation

functionorientation angle, polymer 40orientation function, polymer 38

bond selection rules 32calculating and plotting 29definition 39–40

oriented polymerspacking variables for Blends 60, 64

output for sorption simulation 237output from Mechanical Properties module 6,

12analyzing 14msi model files 321reloading for analysis 16run parameter (.mp) files 321specifying 13table (.dat) files 321text (.txt) files 321trajectory (.trj) files 321

Ppacking calculations for Blends

axial packing 63excluded atom constraints 64isotropic packing 63procedure to specify variables 46specifying variables 63

packing calculations for crystals, see CrystalPacker module

pair interaction energies 55calculating 58calculation output 60plotting distribution of 49procedure to calculate 44–45procedure to plot 51

Pairs Method, Monte Carlo sampling forBlends 58–61Biased Metropolis averaging 60Boltzmann averaging 59Metropolis averaging 60

permeability units 81persistence length 100phase diagrams, two component 57, 65.pmp files, see Polymorph Predictor trajectory

filesPoisson's ratio


polymer blends 41–67polymer properties

density 34dihedral angle distributions 26dipole moment 33end-to-end distance 36estimation 69


Q

molecular weight 34quadrupole tensor 34radius of gyration 35

Polymer Properties module 21polymer solutions, see Blendspolymers, crystalline

dimensionality in Crystal Packer 118polymers, liquid crystal

Blends packing variables 60, 64polymers, oriented

Blends packing variables 60, 64polymorph prediction 127–163polymorph prediction steps

cluster analysis 144cluster minimized structures 129, 144energy minimization 128, 147Monte Carlo packing simulation 128, 138setup 128, 129

polymorph prediction theory 158Polymorph Predictor limitations

force fields 132intramolecular symmetry 133molecule flexibility 132processing speed 133

Polymorph Predictor module 127–163Polymorph Predictor trajectory files 128

analysis 149extracting models from 149merging 129, 155properties of structures in 149

polymorphism 157potential of mean force 97powder spectra 151predicting polymorphs manually 136predictive correlations 85preparing models for polymorph prediction

130pressure term, Crystal Packer 125properties of polymers, see polymer proper-

tiesproperty prediction, see mechanical proper-

ties

Qquadrupole tensor in polymers 34

Rradius of gyration of polymers 35reliability checks 129, 156, 159repeat unit

length computation 79rescaling step sizes, sorption 241restart files 133restarting interrupted polymorph prediction

procedures 133rigid body minimization 148rigid units, Crystal Packer 106RIS Metropolis Monte Carlo

computing dihedral distribution functions98

energy calculation 95output files 99parameters 96rotatable bonds 94

RIS Metropolis Monte Carlo (RMMC) simula-tion 99

RIS Monte Carloproperties calculated from 100

RIS_Metrop_MC_Run command, RIS_Monte_Carlo pulldown 91

RISM theory for polymer blends 55.rst files, see restart filesrun parameter (.mp) files (Mechanical Proper-

ties) 321running a complete polymorph prediction se-

quence 134

Ssaving

Blends interaction energy (.enr) files 45Mechanical Property data to file 12

search level 141, 146searching space groups 128, 326second derivative method for property predic-

tion 2selecting

frames from trajectory files for PolymerProperties analysis 23

torsions for dihedral angle distributions inpolymers 26

shear modulus

T


not from second derivative 5simulated annealing 138

solutions provided by 160theory 160

solubility parameter 80sorbates database 231sorption

energy calculations for 232fixed loading simulation 230, 253–254fixed pressure simulation 230, 254–256Henry constant simulation 230, 256–258models for 231, 239output 237running a simulation 239trajectory file analysis 245trajectory files 237, 239

Sorption module 229–258sound, velocity of, see velocity of soundspace group 128, 326

defining 143speed of sound, see velocity of soundspinodal curve, see phase diagramsSteepest Descent, minimization algorithm 113stiffness matrix, methods for calculating 2–4strain

analyzing data in Mechanical Properties 14applying in Mechanical Properties 9see also stress/strain

stressanalyzing data in Mechanical Properties 14applying in Mechanical Properties 9

subrotations, Crystal Packer 106, 123–125constraining 112defining 124

sweepparameters for mechanical properties cal-

culation 6, 9Synthia

properties calculable 78

T12-10 functional form 122table (.dat) files

output from Mechanical Properties 13text

Mechanical Properties output 12sorption output 238, 239

thermodynamic functionscalculating for blends 51

thermodynamic properties 69topological information of polymers 85torsional energy term, Crystal Packer 123–125torsional subrotations, see subrotationstorsions, see also dihedralstrajectory (.trj) files

for Mechanical Properties 321trajectory files, dynamics

loading for Polymer Properties analysis 22selecting frames for Polymer Properties

analysis 23trajectory files, sorption 237, 239

analysis of 245plotting 245, 249

transport properties 69

Uusing the Polymorph Predictor 128

Vvalence indices 87van der Waals interactions 130van Krevelen-like, solubility parameter 80VDW energy calculation in sorption 233VDW energy term, Crystal Packer 118–120VDW radius reduction for bad contacts check

236velocity of sound


Voigt notation for stress/strain tensors 10volume compressibility


Voronoi volume analysis of polymers 36

YYoung's modulus



Z

Zzeolites database 231

Documents

Cerius2 - ESIvi Cerius2 Property Prediction/December 1998 Applied stress/strain tables 10 Standard Voigt notation 10 Editing the defaults 10 Plot stress/strain profile 10 Choosing