Entropy and free energy of a polymer chain from dynamic Monte Carlo simulations on the lattice

Entropy and free energy of a polymer chain from dynamic Monte Carlo

simulations on the lattice. The extension of the statistical

counting method.

W. Nowicki, G. Nowicka and A. Mańka

Faculty of Chemistry, A. Mickiewicz University, Umultowska 89b , PL-61714 , Poland E-mail: [email protected]

• INTRODUCTION: ASSUMPTIONS OF THE SC METHOD

• SOME APPLICATIONS OF THE SC METHOD AND ITS LIMITATIONS

• THE MICRO-MODIFICATION PROBABILITIES (MMP) METHOD

• EXEMPLARY APPLICATIONS OF THE MMP METHOD

INTRODUCTION: ASSUMPTIONS OF THE SC METHOD

ji,

j1,iji,

1

1ji,jN,

N

i

the total number of conformations of the chain

Effective coordination number of the lattice =the number of empty lattice nodes

Zhao, D.; Huang, Y.; He, Z.; Qian, R. J. Chem. Phys., 1996, 104, 1672.

(001)(112)



n

j

n

j

nW

W

1ji,

1ji,ji,

i lim

jm,m

j1,m 1WW

)ln(1ln eff

1

1i

B

N

N

k

S N

ithe conforrmational entropy of the chain of N segments

the Rosenbluths’ weighting factors

weighted average



0Eeff )G(1 P the effective coordination number =

f(probability of a chain micromodification)

(112)

VERIFICATION OF THE SC METHOD


SOME APPLICATIONS OF THE SC METHOD

W Nowicki, G Nowicka, J. Narkiewicz-Michałek "Monte Carlo study of the translocation of a polymer chain through a hole", Eur. Polym. J, 46, 112-122 (2010)

W Nowicki, G Nowicka, J. Narkiewicz-Michałek "Influence of confinement on conformational entropy of a polymer chain and structure of polymer-nanoparticles complexes", Polymer, 50, 2161- 2171 (2009)

(112)

Chain anchored to the convex surface

Chain anchored to the concave surface

Chain with the forced position of the terminal segmet

Chain near the interface

Chain anchored to the interface

Chain translocationt through the narrow hole

21 0 SSS

SOME APPLICATIONS OF THE SC METHOD


)()( 21 NSNSS

21

2NN

N

Translocation

coordinate21 NNN

TRANSLOCATION THROUGH THE HOLE



The whole landscape of the conformational entropy

• the approach of the terminal segment to the interface• the translocation of the chain through the hole


approach

translocation

walk away

The whole landscape of the conformational entropy (escape from cavity)





21

2NN

N

Translocation

coordinate

l

TkF

2~ B

att

5/3attN

BdFF

TT

d

ST

d

AF

MACROMOLECULE NEAR THE INTERACE


NNaVp ~34

3 NNaVp ~34

3

Conformational pressure of the chain vs. cavity volume

Equation of state:

Equation of state:

constpV constpV

constbVp constbVp

MACROMOLECULE IN THE CAVITY


CHAIN ANCHORED TO A ROUGH SURFACE

Samples of Bm (a – c) and fBm (d – f) surfaces generated by the RMD method

W. Nowicki, G. Nowicka, M. Dokowicz, A. Mańka "Conformational entropy of a polymer chain grafted to rough surfaces", J. Mol. Model., 18(9), 337-348 (2013)

The probability distributions of finding a segment in the unperturbed coil (F), in the chain terminally attached (A) and the cumulative distribution of free sites near the surface (S). The difference =F–A is also indicated. The vertical line denotes the mean elevation of the surface (zmean), N=100, /a=10.

3

3222

EGF

72

1exp

9exp

)()()(

r

aN

b

r

Nb

rBN

rrr

2erf1

2

1)( GRx

zS

)()()( SFA zrr

dr

dr

k

S

F

A

Bln

CHAIN ANCHORED TO A ROUGH SURFACE

W. Nowicki, G. Nowicka, M. Dokowicz, A. Mańka "Conformational entropy of a polymer chain grafted to rough surfaces", J. Mol. Model., 18(9), 337-348 (2013)

ADVANATAGES AND DISADVANTAGES OF THE SC METHOD

Although the parameter values obtained from MC simulation are correct because of the employment of the Rosenbluths’ weighting factors, the generated chain conformations remain biased from the true population, which makes them unsuitable for detailed and realistic analysis of the polymer coil structure

Metropolis MC method • not biased conformations• the intermolecular interactions can be incorporated

Metropolis MC method • not biased conformations• the intermolecular interactions can be incorporated

DISADVANTAGES OF THE SC METHOD

Coil deformation:

- effective coordination number increases to – 1

- conformational entropy decreases

BASIS OF MMP METHOD: THE METROPOLIS MC

D. P. Landau, K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics (Cambridge University Press, New York, 2000).

The Metopolis MC method assumes that the appriopriate equilibration has take place before commencement of the averaging process. The average is simple (not weighted).

The Metopolis MC method assumes that the appriopriate equilibration has take place before commencement of the averaging process. The average is simple (not weighted).

otherwise,

/,

I

I 1I 1II Γ

ΓpΓprΓΓ

otherwise,

/,

I

I 1I 1II Γ

ΓpΓprΓΓ

Transition acceptance based on Boltzmanndistribution

Transition acceptance based on Boltzmanndistribution

n

A

A

n

j 1

ji,

i n

A

A

n

j 1

ji,

i

Metropolis importance sampling Monte Carlo scheme

0. Create a certain conformatin 1. Modify the conformation and calculate its energy U2. If U<0 then accept the new conformation and go to 13. If U>0 then generate a random number4. If numer X<p then accept the new conformation 5. If numer X>p then reject the new conformation 6. Go to 1

Metropolis importance sampling Monte Carlo scheme

0. Create a certain conformatin 1. Modify the conformation and calculate its energy U2. If U<0 then accept the new conformation and go to 13. If U>0 then generate a random number4. If numer X<p then accept the new conformation 5. If numer X>p then reject the new conformation 6. Go to 1

Tk

Up

Bexp

Tk

Up

Bexp

C R E

K

IF

CHAIN MODIFICATIONS IN THE METROPOLIS MC METHOD

Verdier-Stockmayer local/bilocal micromodifications (non-ergodic)

Cut-and-paste non-local modifications: inversion and reflection (ergodic)

P. H. Verdier, W. H. Stockmayer, J. Chem. Phys. 36, 227 (1962).P.H. Verdier, J. Comput. Phys. 4, 204 (1969).

The kink-jump motion trials for two different locations of a monomer in the chain a) the forbidden move, b) the permitted move

PROBABILITIES OF MICROMODIFICATIONS (KINK-JUMP)

PS(K)

PE(K)

The skeleton effect

The excluded volume effect

A. Mańka, W. Nowicki, G. Nowicka, J. Mol. Model. 19, 3659 (2013).

Kink-jump

The dependencies of PE(K) and PE(C) on LH.

The horizontal lines indicate the corresponding probabilities determined for the free chain – PE(K)0 and PE(C)0,

respectively (SAW model, N=100).

The influence of LH value on probabilities

PS(K) and PS(C). The horizontal lines

indicate corresponding probabilities determined for the free chain (SAW, N=100).

PROBABILITIES OF MICROMODIFICATIONS (DEFORMED COIL)


Plots of: a) PS(K)/PS(K)0 against L (inset: enlarged part of the dependence around L=0, coordinates are not marked) and b) (1–PS(K)/PS(K)0) against L in log-log scale for positive values of variables; N=100.

1

0S

S

0S

S

)K(

)K(1

)K(

)K(1

P

P

P

P

bN

L

5

1

Tested for 2D RW, 3D RW, 2D SAW and 3D SAW

1

2HH

1

vM

RLL

The values of the expression ((–1)–eff)1/2 plotted against 1–PE(K), obtained for a chain of N=100 in theta and poor solvents. The regression coefficient of the straight line indicated in the figure is equal to 2.

2Eeff (K)141 P

3Eeff (C)151 P

PS(K) – > LH PE(K) –> eff

1122

21

PPeff

N

U

2

1 SSPP

1i 1jji,

B

NN

Tk

U


standardizationstandardization

CONFORMATIONAL ENTROPY VERSUS L

AND eff ZYXZYX MMMMMMM

!!!!!!

!

ZYXZYX MMMMMM

M

!!

!

Z5

Z

MM

M

11

1

Z5

-Z

MM

M

ZZ 5MMM

ZZ MMb

L

1/11

1/1

lnln

eff

eff1

eff

B ff

bLNbLN

N

k

Se

P. Atkins, Pchysical Chemistry, Oxfeord University Press,

2D RW eff =43D RW eff =62D NRRW eff =33D NRRW eff =52D SAW eff =2.6383D SAW eff =4.684

1/11

1/1

lnln

eff

eff1

eff

B ff

bLNbLN

N

k

Se

1

0S

S

0S

S)K(

)K(1

)K(

)K(1

P

P

P

P

bN

L

2Eeff (K)141 P

Bmax / kS B*max / kSBmax / kS B

*max / kS

System eff N=1000 N=10000

100

100

2D RW3D RW2D NRRW3D NRRW3D SAW

46466

46354.6839

13771776109215971533

13851790109816081544

0.580.780.540.680.71

1385017896109771607715426

1386317918109861609415441

0.090.120.080.110.10

lnB

*max Mk

S

1ln1lnB

*max Mk

S

eff,B

*max lnln1'ln MMCk

S

CONFORMATIONAL ENTROPY VS. PS(K) AND PE(K) = MMP METHOD


VERIFICATION OF THE MMP METHOD

The relationships of a) S vs LH and b) F vs LH; SAW chain of N=50. For comparison, the same dependencies calculated by means of expanded ensemble MC method (squares) are shown.

Vorontsov-Velyaminov, P.N.; Ivanom, D.A.; Ivanom, S.D.; Broukhno, A.V. Colloids Surfaces A: Physicochemical and Engineering Aspects 1999, 148, 171-177.

Dependencies of conformational entropies vs. fixed end-to-end distance calculated for 2D RW, 3D RW and 3D SAW (N=100). Perpendicular lines indicate rms end-to-end distances of RW and SAW chains equal to 10b and 15.8b, respectively. The rms end-to-end distance for RW model is independent on the dimensionality of the lattice.


rms end-to-end distance for free chain


The comparison of relationships of a) S vs LH, b) U vs LH and c) A vs LH, which were determined for three different solvent conditions: =0 (athermal), =1/2 (theta) and 1/2<2 (poor), N=50.

The entropy of SAW chain (N=100) with ends attached to the opposite parallel surfaces versus the surface separation (circles). For comparison, the corresponding results for the chain with fixed ends but in the open space are indicated (squares).

Different solvents

Geometical constraints(chain between two parallel plates)

VERIFICATION OF THE MMP METHOD




Free energy contributions:• non-electrostatic (London) energy• energy of electrostatic interactions

between ions of electrolyte, polyion and charged nanoparticles

• conformational entropy of polyion• entropy of the DLVO layer

APPLICATIONS OF THE MMP METHOD

+/–/+

The same dependencies as in previous slide for the case when nanoparticles have the electric charge of the opposite sign to that of the chain (Q/e=−20, qP/e=+1, NC=20, I=5.910-6 (squares) and I=5.910-5 (triangles)).

W. Nowicki, G. Nowicka, “Conformation and elasticity of a charged polymer chain bridging two nanoparticles”, J. Chem. Phys., 139, 214903 (2013)

Entropy contributions:- chain conformational entropy reduction caused by chain deformation by the approach of nanoparicles- chain cinformational entropy reduction caused by charged segments adsorption- configurational entropy of the electrical double layer

APPLICATIONS OF THE MMP METHOD

W. Nowicki, G. Nowicka, “Conformation and elasticity of a charged polymer chain bridging two nanoparticles”, J. Chem. Phys., 139, 214903 (2013)

+/–/+

APPLICATIONS OF THE MMP METHOD – NON-EQUILIBRIUM MC

Umbrella sampling

Metropolis algorithm

The standard Boltzmann weighting for Monte Carlo sampling is replaced by a potential chosen to cancel the influence of the energy barrier present.

Ph. Attard, Non-Equilibrium Computer Simulation Algorithms, Oxfrord University Press, 2012

otherwise,

/,

I,

I, 1,I 1,II

t

ttt

Γ

ΓpΓprΓΓ

otherwise,

/,

I,

I, 1,I 1,II

t

ttt

Γ

ΓpΓprΓΓ

Transition acceptance based on entropy of conformation

Transition acceptance based on entropy of conformation

BtI,NE /exp kSp BtI,NE /exp kSp

n

j

n

j

np

Ap

A

1ji,NE,

1ji,ji,NE,

i lim

n

j

n

j

np

Ap

A

1ji,NE,

1ji,ji,NE,

i lim

otherwise,

/,

I

I 1I 1II Γ

ΓpΓprΓΓ

otherwise,

/,

I

I 1I 1II Γ

ΓpΓprΓΓ

Translocation through the long channel of an assumed width

Thank you for your attention

Documents

Entropy and free energy of a polymer chain from dynamic Monte Carlo simulations on the lattice