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Multiscale Materials Modeling

Lecture 07

Development of Coarse Grained PotentialsDevelopment of Coarse-Grained Potentials

Fundamentals of Sustainable Technology These notes created by David Keffer, University of Tennessee, Knoxville, 2012.

Outline

Development of Coarse-Grained Potentials

I. IntroductionII. Techniquesq

II.A. Direct Potential Fitting II.B. Iterative Boltzmann InversionII C Integral Equation TheoryII.C. Integral Equation Theory

III. ApplicationsIII.A. Small MoleculesIII B N ti lIII.B. NanoparticlesIII.C. Polymers

IV. Conclusions

Fundamentals of Sustainable Technology

Introduction

The purpose of coarse-grained simulations is to provide computationally tractable simulations of materials with a broad spectrum of dynamic modes p ythat eliminate degrees of freedom, which are not crucial to understanding the particular physics of interest.

These coarse grained (CG) simulations require potentials to describe theThese coarse-grained (CG) simulations require potentials to describe the interaction between the CG particles.

The topic of this module is the procedures by which these CG interaction potentials are generated.


Techniques: Define Beads

polystyrene PET

Fundamentals of Sustainable Technology Wang, Q., et al. J.Phys.Chem.B, 2010.Harmandaris et al., Macromolecular Chem. Phys., 2007.

Techniques: Direct Potential Generation

To generate the interaction potential between a CG bead of type A and a CG bead of type B:

1.Take an isolated fragment of the molecule corresponding to A and an isolated molecule corresponding to B and separate their center-of-masses by a distance r. 2. Compute the energy for each possible orientation and configuration, where the orientation m st be a eraged o er all possible rotations and the config ration o erorientation must be averaged over all possible rotations and the configuration over all internal degrees of freedom (bending, stretching, torsion).3. Compute the average A-A interaction energy as the potential of mean force.

4. Repeat for the next separation r.5. This must be done again for each temperature.

Fundamentals of Sustainable Technology Harmandaris et al., Macromolecules, 2006.

Techniques: CG Potential Functional Forms

Sometimes the CG potentials are left in tabular form as a function of separation, r.Sometimes, they are fit to a given functional form.

Fundamentals of Sustainable Technology Harmandaris et al., Macromolecular Chem. Phys., 2007.

Techniques: Iterative Boltzmann Inversion

To generate the interaction potential between a CG bead of type A and a CG bead of type B:

1.Perform an atomistic simulation of a small system or short chain system and generate the pair correlation function (PCF) for the center-of-mass of molecules or molecule fragments corresponding to CG beads.2 Estimate the CG potential as2. Estimate the CG potential as

rrU αβBαβ,0 glnTk

3. Perform CG simulations and generate PCFs.

4 Compare PCFs from atomistic and CG simulations If they are the same then4. Compare PCFs from atomistic and CG simulations. If they are the same, then you are done. If not, improve the potential estimate iteratively by

r

UU iαβglTk


rrUrU

αβ

iαβ,Biαβ,1iαβ, g

glnTk

Techniques: Iterative Boltzmann Inversion

The IBI method cannotreproduce the LJ potential ina simple test of self-consistency, due to cut-offand statistical noise

Fundamentals of Sustainable Technology Reith et al., J. Comput. Chemistry, 2003.

problems and poor sensitivityof the PCF to the potential.

Techniques: Integral Equation Theory

3γβγαγαβα d1gncc1'g r''r'',r'r''r,r''r,r'r,r

The Ornstein-Zernike (OZ) Integral Equation is an exact relation between energy, u(r),and structure, g(r), because it introduces a new unkonwn, c(r).

The Percus‐Yevick approximation expresses the direct correlation function as

r,r'r r'r r' αβuexp1gc

γ

γβγαγαβα gg

TBkr,rr,r β

αβαβ exp1gc

r,r''' βu

Cavity correlation function defined as

TBk

r,rr,r'r,r'y αβuαβαβ expg

Total correlation function defined as

1gαβαβ r,r'r,r'h

OZPY


r''r'',r'hr''r,r''r,ry 3γβγγ αγαβ dnc1'

Techniques: Integral Equation Theory

The Ornstein-Zernike (OZ) Integral Equation

exact relationship between

Vthsnrcr dc)(1g the pair correlation function (PCF) and the interaction potential. splits PCF into direct and indirect parts

2.0

Wang et al Physical Review E 2010

PCF: probability of finding an neighboring particle at distance r, as shown right typical PCF of Argon from MD simulation (r

)

1.0

1.5

Wang et al.. Physical Review E 2010

typical PCF of Argon from MD simulation at T* = 2.0, * = 0.005.

The Percus-Yevick (PY) approximationg

0.5

1.0

TBkrurr exp1gc estimation of direct correlation function

so that the OZ equation can be solved.

The Percus Yevick (PY) approximationr/

0 1 2 30.0


Outline

Development of Coarse-Grained Potentials

I. IntroductionII. Techniquesq

II.A. Direct Potential Fitting II.B. Iterative Boltzmann InversionII C Integral Equation TheoryII.C. Integral Equation Theory

III. ApplicationsIII.A. Small MoleculesIII B N ti lIII.B. NanoparticlesIII.C. Polymers

IV. Conclusions


OZPY: Consistency Test

An alternative method that we develop and investigate here is to use the Ornstein-Zernike (OZ) equation with the Percus-Yevick approximation (OZPY) to extract the non-bonded potential from the PCFs.

inputfinteraction

potentials

performsimulation

generatePCFs

foutput performinverseOZPY

outputinteractionpotentials


compareConsistency test before applying to CG.

OZPY: Typical Application

Given U, find PCF.Comparison of PCFs from OZPY with MD simulation for monatomic fluid.

3.0 OZPY (= 0.55)MD (= 0.55) monatomic

)

2.0

2.5OZPY (= 0.90)MD (= 0.90)

monatomicLennard‐Jonesfluid

g(r)

1.0

1.5

0 1 2 3 40.0

0.5


r/0 1 2 3 4

Good agreement in PCFs from OZPY and MD at low density, deviation at high density

OZPY: Consistency Test

rrU αβBαβ glnTkSimple inversion : only works at very low density of simple fluids

4

= 0.55

)

2

3

= 0.35

monatomic Lennard-Jones

u(r)

1= 0.15

0 005

only able to reproduce thecorrect potentialat very low

1 0 1 5 2 0 2 5 3 0 3 5-1

0 = 0.005 at very low

density.


r/

1.0 1.5 2.0 2.5 3.0 3.5

OZPY: Consistency Test with Simple Fluid

OZPY-1: test of consistency on monatomic Lennard-Jones fluid

3.5

OZPY: Given U, find PCF OZPY‐1: Given PCF, find U1.5

= 0.55

g(r)

2.0

2.5

3.0

= 0.55

= 0.45

= 0.35 (r)

0.5

1.0

= 0.15

= 0.25

= 0.35

= 0.45

g

0.5

1.0

1.5 = 0.25

= 0.15

= 0.005

u(

-0.5

0.0= 0.005

r/0 1 2 3

0.0

Pair Correlation Functions (PCFs) obtained by Comparison of non‐bonded interaction r/

1 2 3 4-1.0

solving OZPY equation directly, under T* =2.0, * from 0.005 to 0.55, here T* = T/; * = r 3, the data has been shifted in vertical direction for clarity.

potential obtained by solving the OZPY equation inversely (symbol) and potential predicted by Lennard‐Jones model (line), under the same conditions


The OZPY‐1 is capable of reproducing the potentials in the density range studied here.

OZPY: Consistency Test: Diatomic Fluids

The OZPY equation for multi‐component is

γ 0 γβαγαγγ

αβ dtt1dss2

1sr

srthsysh

rn

ry

are the particles’ relative possible positions, assume we have three particles,

r

TBk

rrry αβuαβαβ expg 1gαβαβ rrh

we fix particle 1 and 2 at positions and , particle 1 and 2 interact via non‐bonded. The summation over counts all the possible particles that could sit at .

To get 1‐2 non‐bondedInteraction, we have toIntegrate over all the possible particle 3.



3

OZPY-1 (= 0 07341)

MD simulation OZPY‐10.20

non-bonded (simulation, = 0.07341)

u(r)

1

2OZPY ( 0.07341)Lennard-Jonesfit OZPY-1 (= 0.07341)

g N'(r

)

0.10

0.15

)

30

40 stretching (simulation)stretching (Gaussian Fit)

2

-1

0

0 00

0.05

0.20 0.25 0.30 0.35 0.40

g S'(r

)

0

10

20

r

r/

0.8 1.0 1.2 1.4 1.6 1.8 2.0-2

Bonded and non‐bonded PCFs obtained Raw data has some deviation

r0 1 2 3 4 5

0.00

from MD simulation of N2 under T* = 8.3333, * = 0.07341.

Fit of short‐range data to LJ potential yields original parameters within 1%


The OZPY‐1 is generally able to reproduce the non‐bonded potential in the presence of a bonded potential at low density.


3

OZPY-1 (= 0.5057)1.4

1.6

non bonded ( = 0 5057)

MD simulation OZPY‐1

u(r)

1

2( )

Lennard-Jonesfit OZPY-1 (= 0.5057)

g N'(r

)

0.8

1.0

1.2

non-bonded ( = 0.5057)

100

120

stretching( = 0.5057)

2

-1

0

0 0

0.2

0.4

0.6

r/0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40

g S'(r

)

0

20

40

60

80

r/

0.8 1.0 1.2 1.4 1.6 1.8 2.0-2

Bonded and non‐bonded PCFs obtained Raw data has some deviation

r/0 1 2 3 4 5

0.0

from MD simulation of N2 under T* = 5.5556, * = 0.5057.

Fit of short‐range data to LJ potential yields original parameters within 1%


The OZPY‐1 is generally able to reproduce the non‐bonded potential in the presence of a bonded potential at high density.

Applications: Nanoparticles

Nanoparticles: a uniform distribution of particles interacting with a Lennard‐Jones interaction

Total interaction between two nanoparticlesincludes attractive and repulsive parts.

Total:

Attractive:

Repulsive:



Each nanoparticle is now a bead. The solvent is now implicit. It only exists in the interaction potential.


Comparison of the atomistic and CG PCFs for nanoparticles.


The CG interaction potential between nanoparticles can capture effects of the strength of the solvent for the nanoparticle.


Applications: Polymers

Direct Potential Fitting: Polystyrene

Fundamentals of Sustainable Technology Harmandaris et al., Macromolecular Chem. Phys., 2007.


Inverse Boltzmann Inversion: Polyisoprene

Fundamentals of Sustainable Technology Reith et al., J. Comput. Chemistry, 2003.


Ornstein-Zernike Percus-Yevick Integral Equation Theory: PETn

2.0 2.0

dist

ance

dis

yrib

utio

n

1.0

1.5

BB-targetBB-CGMD

BB

d di

stan

ce d

istri

butio

n

1.0

1.5

AA-targetAA-CGMD

AA

0 2 4 6 8 10 12 14 16

nonb

onde

d

0.0

0.5

0 2 4 6 8 10 12 14 16

nonb

onde

d

0.0

0.5

distance (Angstroms)

brib

utio

n

1.5

2.0

BA-targetBA-CGMD

distance (Angstroms)

nbon

ded

dist

ance

dis

b

0.5

1.0BA

Fundamentals of Sustainable Technology distance (Angstroms)

0 2 4 6 8 10 12 14 16

non

0.0

Wang et al., Macromolecules, 2010.


OZPY Integral Equation Theory: PEG

Fundamentals of Sustainable Technology Wang et al., J. Chem. Phys., 2011.


IBI initialized by OZPY Integral Equation Theory: PEG




Subsequent conversions do not get first peak height correct.There are limitations in the convergence of the iterative method due to sensitivity to

i d l i t ti


noise and long range interactions.



In comparing the different methods, the structures look very similar but the potentials are very different.


The PCF has poor sensitivity to the shape of the potential.


IBI: PCHD

Molecular Model CG Model Cross-linked CG PCHD chainsMolecular Model CG Model Cross-linked CG PCHD chains

A

(C6H9SO3H)

(1)

DA

BC

D

DC C C C

CCC C

A A A AA AA

BCC C C C C C C C CB

A

A

(C6H9SO3H)

(1)

DA

BC

D

DC C C C

CCC C

A A A AA AA


AD

A

BC

D

DC C C C

CCC C

A A A AA AA


A

B

(C6H10OHSO3H)

(2)

D DC C C CCC

C C

B A A A A A A A B

AD

C C C C

B

A A A A A

B

(C6H10OHSO3H)

(2)

D DC C C CCC

C C

B A A A A A A A B

AD

C C C C

B

A A A A A

AD

C C C C

B

A A A A A

C(C6H10)

(C6H10OHSO3H)D

A

C

C C C C C

CC

CC

C

B

C

A

AA

DB

CCC

CC

A

AA

C(C6H10)

(C6H10OHSO3H)D

A

C

C C C C C

CC

CC

C

B

C

A

AA

DB

CCC

CC

A

AA

D

( 6 10)

B D

CC

C

A AA

A

A

CC

CC

B

A

AAD

C

D

( 6 10)

B D

CC

C

A AA

A

A

CC

CC

B

A

AAD

C

Fundamentals of Sustainable Technology Wang et al., polymer, 2012.This complex polymer has different types of A and C beads.

(C6H10SCl)(C6H10SCl)


IBI:PCHD

rrel

atio

n fu

nctio

n

1.0

1.5

2.0

AA

rrel

atio

n fu

nctio

n

1.0

1.5

2.0

AB

Red line:Atomistic‐

separation (Angstroms)0 5 10 15 20 25

pair

co

0.0

0.5

2.0


pair

co

0.0

0.5

2.0

Atomistic‐two chains cross‐linked

Black line:

pair

corr

elat

ion

func

tion

0 5

1.0

1.5AC

pair

corr

elat

ion

func

tion

0 5

1.0

1.5AD

Black line: CG‐two chainscross‐linked


0.0

0.5

2.0

BB


0.0

0.5

2.0

BC

Blue line:CG‐three chains cross‐linked

pair

corr

elat

ion

func

tion

0.5

1.0

1.5

pair

corr

elat

ion

func

tion

0.5

1.0

1.5 cross linked

Fundamentals of Sustainable Technology Wang et al., polymer, 2012.

The OZPY method failed due to complex nature of polymer. Results for IBI for 6 of 10 modes.separation (Angstroms)

0 5 10 15 20 250.0

separation (Angstroms)

0 5 10 15 20 250.0

Transferability2.0 2.0

IBI:PCHD

air c

orre

latio

n fu

nctio

n1.0

1.5

AA

air c

orre

latio

n fu

nctio

n

1.0

1.5

AB

Red line:Atomistic‐


pa

0.0

0.5

2.0

AC


pa

0.0

0.5

2.0

AD

Atomistic‐two chains cross‐linked

Black line:

pair

corr

elat

ion

func

tion

0.5

1.0

1.5AC

pair

corr

elat

ion

func

tion

0.5

1.0

1.5AD Black line:

CG‐two chainscross‐linked


0.0

ctio

n 1.5

2.0

BB


0.0

ctio

n 1.5

2.0

BC

Blue line:CG‐three chains cross‐linked

pair

corr

elat

ion

func

0 0

0.5

1.0

pair

corr

elat

ion

func

0 0

0.5

1.0

cross linked


CG potentials are generally transferable aross chaing lengths, so parameterization to short chains is okay..


0 5 10 15 20 250.0


0 5 10 15 20 250.0

Transferability

IBI: hydrated PCHD


Here we look at a CG hydrated PCHD membrane at ( = 15 H2O/HSO3) using CG potentials parameterized at ( = 10 H2O/HSO3) and ( = 20 H2O/HSO3) . The structure is different so the results are not transferable across different water contents.

Wang et al., under review, 2012.

Dynamics

When one uses structure to validate CG potentials, other features like dynamics may not match.

In order to get a good comparison for relaxation times, time was scaled in the CG simulation by an empirically determined factor of 7.5. 1 CG fs = 7.5 atomistic fs. Why? CG molecules are smoother and thus generate less friction allowing them to move more quickly All dynamic


smoother and thus generate less friction, allowing them to move more quickly. All dynamic properties, relaxation times, diffusivities, viscosities, thermal conductivities are impacted by this.

Conclusions

Before one can perform coarse-grained simulations, one must generate course-grained interaction potentials.g p

There are several different methods for generating these potentials. No one method is perfect. Work still needs to be done.

The CG potentials can be solvent explicit (solvent molecules treated as CG beads) or solvent implicit (solvent incorporated in the CG potential).

The CG potentials are in general not transferable to other temperatures, densities and compositions. This must be validated.

The dynamics of CG simulations are skewed Currently empirical scalingThe dynamics of CG simulations are skewed. Currently, empirical scaling factors are used to correct the dynamics.


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