Force ﬁelds in computer simulation of soft nanomaterialserjank/Lectures/557/557-Lecture2... · 2006. 12. 18. · Computational Nanoscience of Soft Materials ChE/MSE 557 Lecture

ChE/MSE 557 Lecture 2b Fall 2006 1Computational Nanoscience of Soft [email protected]

Force fieldsin computer simulation of soft

nanomaterials

Lecture 2 Part B

Recommended reading:

Leach Chapter 4


Force Field Methods

• Force field methods are simulation methods that useclassical force fields to describe the interactions betweenatoms or molecules or particles. (E.g. molecular mechanicsand molecular dynamics)

• Force field methods rely on Born-Oppenheimer, ignoreelectronic motions and calculate the energy of a system asa function of nuclear positions only.

• Thus simulations of much larger systems can be performed.– Up to billions of atoms possible on the largest supercomputers (nearly

a cubic micrometer - beyond that should not need to resolveeverything at the atomic or molecular level)!


Enter Force Field Methods

• In many cases, just as accurate as QM calculations, butmuch faster.

• Also allows statistical phenomena to be explored, andclassical and statistical mechanical theories to be explicitlytested.

• Thus force field methods are not just to save time or ifsystem too large for QM; also to probe qualitativelydifferent processes from what is probed with QM methods.

• If some phenomena not understood on a fundamental levelcan be mimicked accurately or even just qualitatively bymore “coarse-grained” methods, no reason to bring out “thebig guns”.


Force Fields

• What is a force field? An expression for the interactions between atoms (or

molecules, or particles).

• Force fields rely on:– Validity of Born-Oppenheimer approximation.– Relatively “simple” expressions that capture the stretching of

bonds, the opening and closing of angles, rotations aboutbonds…

– Transferability: the ability to apply a given form for a force fieldto many materials by tweaking parameters (e.g. polystyrenevs. polyethylene)


Particle-basedmesoscale simulation

Force Field MethodsTi

me

Scal

e (lo

g 10

seco

nds)

Length Scale (log10 meters)

-12

-9

-6

-3Simulationmethods that use force fields/interaction potentials

Field-basedmesoscale and continuummechanics

-10 -8 -6

Explicit atomistic models

Coarse-grained models

-3

Ab Initio/ElectronicStructure

Time and length scalesoverlap with neutronscattering, NMR, etc.

MD and MC BDDPD


Force Fields

• The force can be written in terms of the gradient of theinteraction energy (interaction potential):

• The term “force field” refers to the interaction potential U(i.e. the potential field from which forces are derived)!

• Recall when a force is derivable from the gradient of a scalarpotential, the force is conservative.

!

F = "#U({ri})


Force Fields

• Many molecular modeling force fields in use today(especially for soft materials) can be written interms of five simple types of interactions:

δ+

δ+

δ-

Bond stretching

Anglebending

Bond rotation(torsion)

Non-bonded(van der Waals)

Non-bonded(electrostatic)


A typical force field

!

U({ri}) =k jl

2(l j " l j

0)2

+k j#

2(

j

$j

$ # j "# j0)2

+Vn

2(1+ cos(n% " &))

torsions

$

+qiq j

rij+ 4'ij

( ijrij

)

* + +

,

- . .

12

"( ijrij

)

* + +

,

- . .

6/

0

1 1

2

3

4 4

i, j=1

N

$i, j=1

N

$

van der Waals(Lennard-Jones)

Electrostatic(Coulomb)

Intra- andInter-molecular

torsional

Valence angle bendBond stretch

Intra-Molec-ular


Example: Propane

• C3H8 has ten bonds: 2 C-C bonds and 8 C-H bonds.

The C-H bonds fall into two classes– The H’s bonded to the central (methylene) carbon– The H’s bonded to the end (methyl) carbons– Some fancier force fields treat these two differently.

• 18 valence angles & 18 torsional terms

• 27 non-bonded interactions (if 1,4’s neglected)

• 73 total terms to calculate for a single molecule: stillmany fewer than the number of integrals that would beinvolved in an equivalent ab initio calculation.

H-C-C-C-HHHH

H H H


General Features of Force Fields

• Defining a force field means defining both the functionalform and the parameters.

• Two or more FF’s with the same form but differentparameters, or with different forms, may give results ofcomparable accuracy for the same material and property.

• Typically, FF’s are developed to reproduce structures orspectra or thermodynamic quantities.– Just because it does one or two things well, doesn’t mean it

gets everything right! This is not a failure of the FF.– A single FF can’t do everything, even just for one material.


General Features of Force Fields

• Force fields are empirical; there is no one ‘correct’ form.

• Although most of the FF’s in use today have similar overallforms, that doesn’t mean better forms can’t be found.

• FF’s are a compromise between accuracy andcomputational efficiency.– We’ll see this when we look at the van der Waals’ term.– Forms should be “nice” since first and second derivatives of

the energy wrt atomic coordinates are required in molecularmechanics/molecular dynamics.


Force Fields

• Let’s delve more deeply into the individual terms thatmake up a force field (read Ch. 4 in Leach).

• In the many force fields in use today (MM2, MM3, Amber,Compass, Dreiding, UFF, …), some of these terms arewritten slightly differently, and treated with differentemphasis, and some force fields contain additional terms.

• When using a force field, it is important to know what isbeing included and how, and what isn’t.


Bond Stretching

The potential energy curve measured for a typical bond looks likethe red curve at theright.

U(l)

l


Bond Stretching

• This can be modeled by a Morse potential:

De = depth of potential energy minimuma = ω√(µ/2De)µ = reduced massω = frequency of bond vibration, related to stretching

constant by ω√(k/µ)l0 = reference bond length

Rarely used except for when bond-breaking needed*, sinceexponential is computationally slow.

!

u(l) = De1" exp "a l " l0( )[ ]{ }

2


Bond Stretching: Approximating Morse

!

u(l) =k

2l " l

0( )2

!

u(l) =k

2l " l

0( )2#

1" k' l " l0( ) + ...[ ]

Harmonic:

Harmonic + higher orders:

cubic - usedin MM2 MM3 uses quartic to fix large

l catastrophe


Bond Stretching

7191.345C - N

3671.438Csp3 - Nsp3

7771.208Csp3 = 0

6901.337Csp2 - Csp2

3171.497Csp3 - Csp2

3171.523Csp3 - Csp3

Bond l0 (Å) k (kcal mol-1 Å-2)

Allinger 1977 (from Leach)


Bond Angle Bending Interactions

• The deviation of the angle between two bonds is alsooften modeled with a harmonic (Hooke’s law) potential:

• Angles between different types of bonds have differentforce constants k and different reference angles θ0.

• As with bond stretching terms, higher accuracy isachieved by including higher order terms.!

u(") =k

2" #"

0( )2


Bond Angle Bending Interactions

0.0101122.5Csp3 - Csp2 = 0

0.0121121.4Csp3 - Csp2 = Csp2

0.0099117.2Csp3 - Csp2 - Csp3

0.0070109.47H - Csp3 - H

0.0079109.47Csp3 - Csp3 - H

0.0099109.47Csp3 - Csp3 - Csp3

Angle θ0 (deg) k (kcal mol-1 deg-1)

Allinger 1977 (from Leach) Smaller force constants comparedto bonds = much less energy req’dto bend than to stretch or compress


Bond Rotation (Torsional) Interactions

• Barriers to rotation about chemical bonds arise fromantibonding interactions between the H atoms on the(1,4) atoms.– Antibonding interactions are minimized

when conformation is staggered andmaximized when it is eclipsed.

• In organic macromolecules, torsional potentials are nearlyalways used for each A-B-C-D since major conformationalchanges are due to rotations about bonds.

• In inorganic materials, torsional potentials are oftenneglected and non-bonded interactions between every pairof (1,4) atoms are tweaked to achieve the desired energyprofile.


0

1

2

3

4

5

6

0 50 100 150

Rotation Angle

En

erg

y (

kcal/

mo

l)


Torsional energy profile along dihedral angle Si-C-C-C in propyl-POSS

From Cummings,et al.



• Torsional potentials modeled by a cosine series expansion:

• n is “multiplicity” = number of minima as bond rotated through360˚

• γ is phase factor: determines where ϕ passes through minimum• Vn is called “barrier height”, but not exactly, since non-bonded

contributions from (1,4) atoms affect rotation.– Vn for double bond > Vn for single bond

!

u(") =Vn

2(1+ cos(n" # $))

n= 0

%

!

u(") = Cncos(")n

n= 0

#or



• Again, different force fields may use different numbers of termsin cosine series (e.g. Amber contains just one, MM2 uses more).

• More accurate potentials will use more terms, since the moreparameters you have in your FF, the better you can generally fitexperimental data.

• Drawback of using more terms: Many more parameters to fit ismore complicated and time consuming. Necessary for somesystems.


Cross Terms b/t Stretch, Bend, Torsion

• Many force fields also include cross terms, such as:

– Stretch - stretch

– Stretch - torsion

– Stretch - bend

– Bend - torsion

– Bend - bend

θ θ

θ

θ

Again, in general, the more terms you include, the more accurate the FF, but the harder it is to parameterize and use.


Coulomb Interactions

• The distribution of charge in a molecule may berepresented by an arrangement of fractional point charges(partial charges).

• The electrostatic interaction between two molecules orbetween different parts of the same molecule is calculatedas a sum of interactions between pairs of point chargesusing Coulomb’s law:

!

V =qiq j

4"#0rijj=1

NB

$i=1

NA

$


Coulomb Interactions: multipole expansion

• Electrostatic interactions can be calculated via a multipoleexpansion, in which one sums over the individual electricmoments or multipoles in an efficient way.

• Charged molecules have the charge (a monopole) as theirlowest non-zero term; uncharged molecules may have thedipole or quadrupole or octupole or … as their lowest non-zero term, depending on the asymmetry of the chargedistribution within the molecule.

!

V =qiq j

4"#0rijj=1

NB

$i=1

NA

$


Coulomb Interactions: multipole expansion

• But, the multipole expansion approach to summing theelectrostatic interactions is typically used only for smallmolecules whose conformations are kept fixed during thecalculation, and where the interactions between moleculesact at their centers of mass.

• Leach describes several other approaches to dealing withcharges; it is a complicated business, and can beimportant when studying, e.g., charged macromolecules inaqueous solutions.

• For many studies, especially of uncharged species, theelectrostatic interaction is ignored.


van der Waals Interactions

• Electrostatic forces don’t account for all the non-bondedinteractions in a system.

• Van der Waals’ forces describe deviations from ideal gasbehavior not accounted for by electrostatics.

Johannes DiderikVan der Waals1837-1923



• The interaction energy between two isolated Argon atomshas the following dependence on the distance betweenthe atoms:

U = 0 at infinite distance,negligible at long distances.As separation is reduced, Udecreases, passing thru aminimum at r = .38 nm forArgon. U then rapidlyincreases as separationdecreases further.

Fij=-dUij/drij



• The VdW curve arises from a balance between long-rangeattractive forces and short-range repulsive forces.

• The attractive contribution is due to dispersive forces(London forces), which arise from instantaneous dipolesthat arise from fluctuations in the electron densitydistribution (electron cloud), and which induce dipoles inneighboring atoms.

• The Drude model predicts that the dispersion interactionvaries as 1/r6, and is negative (attractive). Additionalnegative, higher order even powers result if higher orderinduced multipoles are considered.



• The repulsive contribution has a quantum mechanicalorigin and arises due to the Pauli exchange principle, whichprohibits any two electrons in a system from having thesame set of quantum numbers.

• The exchange force prohibits the electrons in the twoatoms from occupying the same region of space.

• This results in a reduced electron density between the twonuclei, which leads to repulsion between the incompletelyshielded nuclei.

• At very small separations, the interaction energy due tonuclear repulsion varies as 1/r.

• At larger (but still small) separations, the energy decaysexponentially as exp(-2r/a0), where a0 is the Bohr radius.



• The dispersive (attractive) and exchange (repulsive)interactions can be calculated using QM (not trivial,however).

• There are several empirical models for the van der Waalspotential.

• Required for molecular simulations: a simple expression,because there are MANY non-bonded interactions toinclude (at least N*(N-1)/2, not counting Coulomb).


The Lennard-Jones Interaction Potential

• The most popular model of the van der Waals non-bondedinteractions between atoms is the Lennard-Jones model.

Sir John EdwardLennard-Jones1894-1954

!

U(r) = 4"#

r

$

% &

'

( )

12

*#

r

$

% &

'

( )

6$

% & &

'

( ) )

U(r)

r = rij

minimum at r=1.122σεσ

typical cutoffat 2.5σ

Collision diameter


Lennard-Jones Model

• The 1/r6 term in the LJ potential has the same form as thatfound from theoretical calculations of the Drude model forthe dispersive interactions.

• There are no strong theoretical arguments in support of the1/r12 term, which is not a particularly good approximation to1/r times an exponential.

– While the 1/r12 term works well for rare gases, it is usually too steepfor hydrocarbons.


Lennard-Jones Model

• Nevertheless, the 6-12 LJ potential is widely used,especially for large simulations, since the 1/r12 is rapidlycalculated by squaring the 1/r6 term, which in turn is foundfrom r2 without any costly square roots.

These are the types of considerations that go into developing force fields!


Lennard-Jones Model

• LJ original potential was written in the general form:

• Several force fields use n ≠ 12.• Several formulations in which 1/r12 term is replaced by a

more realistic potential have been proposed, such as theBuckingham potential:

!

U(r) = k"#

r

$

% &

'

( )

n

*#

r

$

% &

'

( )

m$

% & &

'

( ) ) k =

n

n *m

n

m

$

% &

'

( )

m n*m( )

!

U(r) = "6

# $ 6e

$#r

rm

$1%

& '

(

) *

$#

# $ 6

r

rm

%

& '

(

) *

6%

&

' '

(

)

* *

Careful, whenseparationvery small U issteeply attractive.


Lennard-Jones Model

• Between different types of atoms, the LJ interaction takesthe form:

• A system with N different types of atoms has N(N-1)/2different sets of parameters for interactions between unlikeatoms.

• Since vdW parameters are difficult to obtain, we assumeparameters for cross interactions between unlike atoms canbe obtained from those for like atoms through mixing rules.!

U"# (r) = 4$"#%"#r

&

' (

)

* +

12

,%"#r

&

' (

)

* +

6&

' ( (

)

* + +


Lorenz-Berthelot Mixing Rules

• A common mixing rule used to calculate LJ parametersbetween unlike atoms is the Lorenz-Berthelot mixing rule.

• Collision diameter σAB = (σAA+ σBB)/2 - (arithmetic mean)

• Well depth εAB = (εAAεBB)1/2 - (geometric mean)

• These mixing rules are most successful between similartypes of atoms.– Major drawback: well depth can be overestimated by geometric

mean. Some force fields also use geometric mean for collisiondiameter.

• VdW parameters are typically optimized to reproducethermodynamic properties such as densities and enthalpiesof vaporization, and structure.


Many Body Effects b/t Non-bonded Atoms

• In real materials, there are many body effectsbetween non-bonded atoms that cannot be simplydescribed by pairwise forces.

• The interaction between two atoms can be affectedby the presence of other atoms.


Many Body Effects b/t Non-bonded Atoms

• Some force fields include three-body terms at theexpense of greatly increased computational effort.

– # of intramolecular terms increases linearly with N

– # of non-bonded forces increases as N2• N(N-1)/2 interactions to evaluate in a pair-wise potential.

– N(N-1)(N-2)/6 unique three-body interactions.

– For 1000 atom system:• 499,500 pairwise terms• 166,167,000 three-body terms• In gen’l, N/3 times more three-body than two-body terms.


Comparison of Force Fields and QM

• UFF (Universal Force Field)– A.K. Rappe, et al, “UFF, a Full Periodic-Table Force-Field for

Molecular Mechanics and Molecular Dynamics Simulations,” J.Am. Chem. Society, 114, 10024-10035.

– Used for many nanoscale carbon structures (Buckyballs,nanotubes)

• COMPASS (Condensed-phase Optimized MolecularPotentials for Atomistic Simulation Studies)– H. Sun and D. Rigby, “Polysilsesquioxanes: Ab Initio Force

Field and Structural Conformational and ThermophysicalProperties,” Spectrochimica Acta (A) 1779, 53, 130, andrefs therein.

– http://www.accelrys.com/cerius2/compass.html


Comparison of Force Fields and QM

Structure of POSS(H)8 cubedetermined from various QM andMM energy minimizations.

Molecular Mechanics

109.6˚

146.1˚

1.47

1.61

KiefferRFF

110.2˚

146.9˚

1.473

1.624

Cerius2Compass

109.6˚

147.5˚

1.48

1.619

Exp.

110.0˚

146.8˚

1.470

1.592

Cerius2UFF

109.1˚

148.7˚

1.462

1.650

RHF(cc-pVdz)

(GAUSSIAN 98)

109.6˚109.6˚O-Si-O

145.9˚146.7˚Si-O-Si

1.4811.463Si-H

1.6541.630Si-O

AtomicOrbital(DMOL)

PlaneWave(VASP)


Biological and Biochemical Force Fields

• CHARMM, GROMOS, AMBER, OPLS, …– Optimized to work in the temperature and density range

relevant to living systems• Little concern for accuracy beyond this regime


United Atom Force Fields

• There are essentially three “classes” of force field models,each of which corresponds to a different level of detail:

– Explicit atom (all atoms represented explicitly)• Used to model a specific system

– United atom (coarse-grained)• Used to model a specific system

– Mesoscopic (coarse-grained)• Used to model phenomena characteristic

of a class of systems


United Atom Force Fields

• Since the number of non-bonded interactions scales as N2,for large systems explicitly including every atom isprohibitive.

• In united atom force fields, some atoms (usually just thehydrogens) are subsumed into the atoms to which they arebonded.– E.g. CH3 is treated as a single “united atom”.

• The interaction parameters for each term in the force fieldare modified to account for the “subsumed” atoms.

• Many hydrocarbons are modeled with united atom forcefields. In some UA models of proteins, only some of thehydrogens are eliminated.

• Good alkane UA models: Toxvaerd, Siepmann, Smith & Paul


Explicit Atom & United Atom Force Fields

• Force field development is something of an art.• Experimental data and QM calculations are used to help

parameterize force fields.• Incredibly important to have good force fields, since your

simulation is only as good as the model you use.• Few simulators develop force fields; many use them.• Force fields developed for metals, ceramics, polymers,

liquids, solids, gases have different requirements and thereis a different culture.

• Challenge for nanoscience: Force fields for dissimilarmaterials are not easy to obtain. Future studies rely on newforce fields. Major area of research for nanoscienceresearchers.


Explicit Atom & United Atom Force Fields

• How well can these force fields do?– Structure: within a few percent considered good.– Thermodynamics: within a few percent considered good.– Dynamics: within 20% considered good.

• Limitations– With MD, 1016 atom-steps on large supercomputer possible,

(1012 atom-steps usual) depending on terms in force fields,approximations, tricks used, etc.

– For many problems, less computationally intensive models areneeded in order to study larger systems.


Mesoscopic Molecular Models

• Explicit atom and united atom force fields are used to modelparticular systems, and to predict properties withquantitative accuracy.

• Sometimes we do not seek to model specific systems, butrather we seek to model phenomena characteristic of aclass of systems.

• For such studies, further coarse-graining is done andmesoscopic “molecular” models are used. These modelscontain some of the same terms used in force fields andthus look like force fields; they’re used like force fieldmodels, but not generally called force fields.


A Mesoscopic Polymer Model

•“Bead-spring” polymer

V(r)

V(r) = 4ε[(σ/r)12 – (σ/r)6] - (k/2)Ro2ln[1-(r/Ro)2]all monomer pairs bonded pairs only

Kremer-Grest/Rudisill-Cummings

Coarse-grained“mesoscopic” molecular model of a polymer melt.

Eg. k=30, Ro=1.5, ε=1, σ=1


A mesoscopic model for colloids

• The Yukawa potential is commonly used tomodel the interaction between charged colloidsin solvent where the interactions are screeneddue to the solvent:

Here σ = particle diameter, κ = inverse screening length, andε = surface potential, r = distance between two colloid centers.


Mapping Mesoscopic Models toEA or UA Models

• Important area of current research:– Coarse-graining detailed EA or UA models to

mesoscopic models– Mapping mesoscopic models to EA or UA models

• We’ll learn more about this later in the class.

Documents

Force ﬁelds in computer simulation of soft nanomaterialserjank/Lectures/557/557-Lecture2... · 2006. 12. 18. · Computational Nanoscience of Soft Materials ChE/MSE 557 Lecture