Soft, coarse-grained models for multi-component polymer ... · multi-component polymer melts: free energy and single chain ... see also Grochola, JCP 120, ... • soft generic coarse-grained

Soft, coarse-grained models for

multi-component polymer melts:

free energy and single chain dynamics

Marcus Müller and Kostas Ch. Daoulas

Development and Analysis of Multiscale Methods

Minneapolis, Nov 3, 2008

outline:

• soft, coarse-grained models

• free energy of self-assembled structures

• kinetics (single-chain and collective)

conformational

rearrangements ~ 10-12 - 10-10 s

diffusion

~ 10-9 -10-6 s

bond

vibrations

~ 10-15 s

ordering kinetics

~ hours/days

coarse-grained models: time and length scales

Edwards, Stokovich, Müller, Solak, de Pablo, Nealey,

J. Polym. Sci B 43, 3444 (2005)

minimal coarse-grained model that captures only relevant interactions: connectivity, excluded volume,

repulsion of unlike segments

• incorporate essential interactions through a

small number of effective parameters,

chain extension, Re, compressibility kN and

Flory-Huggins parameter cN

• soft potentials, elimination of degrees of freedom

efficient techniques for large systems

(106 segments)

conformational

rearrangements ~ 10-12 - 10-10 s

diffusion

~ 10-9 -10-6 s

bond

vibrations

~ 10-15 s

coarse-grained models: time and length scales

a small number of atoms is

lumped into an effective

segment (interaction center)

MC,MD, DPD, LB,SCFT

relevant interactions: connectivity, excluded volume, repulsion of unlike segments

systematic coarse-graining procedures (RG) demonstrates that effective

interactions become weaker as one increases the degree of coarse-graining

no (strict) excluded volume, effective segments can overlap,

rather: enforce low compressibility on length scale of interest, ##

terms of order generate pair-wise interactions, more general density functionals

can be employed for polymer solutions or solvent-free models of bilayer membranes

soft, coarse-grained models

with

molecular architecture:

Gaussian chain

starting point for field-theoretic

and particle-based description

express density through particle coordinates particle-based descriptionamenable to MC, BD, DPDor SCMF-simulations

• regularize d-function by lattice with discretization DL MC or SCMF-simulation

• regularize d-function by a weighting function (WDA) DPD-like models

soft, coarse-grained models

computationally fast

but not translationally invariant

slower in dense systems (factor 10-100)

but translationally invariant

“ ´’’

Daoulas, Müller, JCP 125, 184904 (2006),

PM-methods (electrostatics), PIC (plasma physics)

Laradji, Guo, Zuckermann PRE 49, 3199 (1994) Groot,Warren, JCP 107, 4423 (1997)

Single-Chain-in-Mean-Field (SCMF) simulations

algorithm:

1. simulate an ensemble of many independent

molecules in real, fluctuating, external fields,

wA and wB, for a predefined number of

MC steps

2. calculate (coarse-grained) densities, fA and fB,

on a grid, and calculate new, external fields

3. goto 1.

if it converges and the ensemble is very large the average densities

will relax towards a solution of the (equilibrium) SCF equations

Do SCMF simulations describe fluctuations for a finite ensemble of molecules?

SCMF simulations provide a controlled approximation

quasi-instantaneous field approximation depends on discretization

Daoulas, Müller, JCP 125, 184904 (2006)

correlation hole and long-ranged correlations

• 1/r-behavior of total correlation at short scales and gtot(r )=1 for r>x

• ginter exhibits correlation hole on length scale Re and depth

• long-ranged correlations between bonds

Wittmer, Meyer, Baschnagel, Johner, Obukhov, Mattoni, Müller, Semenov, PRL 93, 147801 (2004)

r

definition of length without referring to a definition of a segment: interdigition

• Ginzburg-parameter that controls regime of critical, Ising-like fluctuations

in a binary blend, or shift of ODT from first-order transition in block copolymer

• broadening of interfaces by capillary waves

• bending rigidity of interfaces, formation of micro-emulsions near Lifshitz-points

• depth of correlation hole and amplitude of long-range bond-bond correlations

• tube diameter, packing length for Gaussian coils

invariant degree of polymerization

or (tricritical)

(Fredrickson-Helfand)

corresponds to SCFT

typical experimental values:

• invariant degree of polymerization

• typical length scale

typical value for a coarse-grained model with excluded volume (BFM, bead-spring):

necessary condition: or less (otherwise crystallization, glass)

typical values for a coarse-grained model with soft cores

large requires soft potentials

eff. interaction centers (segments)/chain

chain discretization

dense melt of long chains reptation

chains are crossable Rouse-dynamics

otherwise

choosere-entrant melting of underlying

soft-spheres fluid + lattice effects



outline:

• soft generic coarse-grained models for multi-component polymer melts

o coarse-grained model that incorporates the relevant interactions:

chain connectivity/molecular architecture

low compressibility of melt / excluded volume

repulsion between unlike species

o invariant degree of polymerization controls fluctuation effects

o experimentally large values of are conveniently described

with soft interactions

large time and length scales: structure formation,

phase separation kinetics, self-assembly

• free energy of self-assembled structure


summary I

free energy of self-assembled structures

relevance:

properties of coarse- macroscopic behavior

grained model

examples:

• free energy of morphologies accurate location of 1st order

phase transitions

phase diagram

input for coarser models (phase field)

• interface and surface wetting behavior (Young’s equation)

free energies nucleation

• defect free energies free energies of intermediates

kinetics of structure formation

self-assembly vs. crystallization

order parameter:

Fourier mode of composition fluctuation Fourier mode of density fluctuation

ideal ordered state: SCFT solution ideal crystal (T=0)

ideal disordered state: homogeneous melt ideal gas

in ordered phase, composition fluctuates in ordered state, particles vibrate around

around reference state (SCFT solution), ideal lattice positions

but molecules diffuse (liquid)

no simple reference state for self- Einstein crystal is reference state

assembled morphology use thermodynamic integration wrt

to uniform, harmonic coupling of particles

to ideal position

(Frenkel & Ladd, Wilding & Bruce)


order parameter:










free energy per molecule (ex vol) N kBT to ideal position

relevant free energy differences 10-3 kBT (Frenkel & Ladd, Wilding & Bruce)

absolute free energy must be measured with a relative accuracy of 10-5


order parameter:










free energy per molecule (ex vol) N kBT to ideal position

relevant free energy differences 10-3 kBT (Frenkel & Ladd, Wilding & Bruce)

absolute free energy must be measured with a relative accuracy of 10-5

measure free energy differences between disordered and ordered phase

(10-3 relative accuracy needed)


order parameter:



Ideal disordered state: homogeneous melt ideal gas







to ideal position

(Frenkel & Ladd, Wilding & Bruce)

PRE 51, R3795 (1995)

see also Grochola, JCP 120, 2122 (2004)

calculating free energy differences

Müller, Daoulas, JCP 128, 024903 (2008)

optimal choice of external field (Sheu et al):

structure does not change along 2nd branch

SCFT:

TDI vs expanded ensemble/replica exchange

• only replica exchange is

impractical because one

would need several 100

configurations

• at initial stage, where weights h

are unknown (DF~104kBT),

replica exchange guarantees

more uniform sampling

• expanded ensemble technique

is useful because it provides

an error estimate

TDI vs expanded ensemble/replica exchange

no kinetic barrier, ie no phase transition

roughly equal probability

first-order fluctuation-induced ODT

cNODT<14 at fixed spacing

cNODT=13.65(10) hysteresis

soft, off-lattice model:

measure chemical potential m

via inserting method in NpT-

ensemble

two structures will coexist,

if they have same p and mLennon, Katsov, Fredrickson,

PRL 101, 138302 (2008)

Einstein-integration for

fluctuations of lattice-based

density fields around SCFT

Pike, Detcheverry, Müller,

de Pablo, submitted (2008)

further applications: T-junctions

Duque, Katsov, Schick, JCP 117, 10315 (2002)SCF theory:

0.19(2)

0.21

further applications: rupture of lamellar ordering

0.01(3)

further applications: stalks in solvent-free

membrane models

with Yuki Norizoe



outline:



o general computational scheme to calculate free-energies

of self-assembled structures relies on reversibly converting

one structure into another via an external ordering field

o does not rely on soft interactions

(alternative: measure p and m simulateneously)

o does not require a field-theoretic formulation

o accurate and suitable for parallel computers

metastablity: observed structures may depend on kinetics of structure formation

What is the coarse-grained parameter that parameterizes dynamic properties?


summary II

length set by bulk lamellar spacing, L0

time set by time it takes to diffuse L0

experiment: PS-PMMA diblock

L0=48nm, D0=42nm

DPS =6.8 10-12cm2/s tPS=0.56s

DPMMA=9.5 10-15cm2/s tPMMA=404s

Mw=100 000, Mwe(PS)= 35 000

observation: stripe period matches L0 hexagonal order @ 3h, registration @ 6h

SCMF simulation:

L0=1.786Re DPMMA=DPS/10=3.3 10-5Re2/MCS tloc=L0

2/6D=16 100 MCS

match time scale via PS: 1s = 287.5 MCS 1h = 1 035 000 MCS

match time scale via PMMA: 1s = 40 MCS 1h = 143 000 MCS

cN=36.7, kN=50, N=32=15+17, LN=-3, npoly=44 000

10 stripes with period 1.7Re=0.95L0,

system size: 1.2Re * (17Re)2=32nm * (0.457mm)2

ordering kinetics in thin films:

lamellar-forming copolymer on stripe pattern

Edwards, Stokovich, Müller, Solak, de Pablo, Nealey, J. Polym. Sci B 43, 3444 (2005)

ordering kinetics: SCMF simulations

10000 MCS

PS-rich regions (red)

PS-PMMA interface

(green)

ordering kinetics: SCMF simulations

ordering kinetics: SCMF simulationsmatch time scale via PMMA: 1s = 40 MCS, 1h = 143 000 MC

(slow component dictates the ordering kinetics)

• time scale to fast (no entanglement effects)

• defects anneal out from substrate to surface hexagonal surface morph. morphology no lateral diffusion of defects

1000500 1500

5000 10000

100

20000=8min

=0.66 t

towards a more realistic dynamics: slip-links

Likhtman, Macro 38, 6128 (2005)

single chain theory (ensemble of independent chains)

natural dynamics is Rouse-like, entanglements are

not predicted but have to be introduced ``by hand’’

also: softer interactions in coarse-grained models do not

guarantee non-crossability

idea: restrict lateral motion by tethering to chain contour

anchor points aj are fixed in space; attachment points r(sj)

hop from one segment to another


single chain theory (ensemble of independent chains)

natural dynamics is Rouse-like, entanglements are

not predicted but have to be introduced ``by hand’’

also: softer interactions in coarse-grained models do not

guarantee non-crossability

idea: restrict lateral motion by tethering to chain contour

anchor points aj are fixed in space; attachment points r(sj)

hop from one segment to another


comparison of the simulation results with predictions of the tube model

yields entanglement lengths, depends on quantity because of approximations

invoked in the tube model (e.g., constraint release). For N=128 and Nsl=32

one obtains: self-diffusion coefficient: Ne ≈ 3.5

early mean-square displacements: Ne ≈ 7

dynamic structure factor: Ne ≈ 12

single-chain theory entanglements cannot be predicted but are input parameter

1) concept of packing length:

purely dynamic relation between and

2) primitive path analysis using the

multi-chain configurations

(SL)

(SS)

towards a more realistic dynamics: slip-linkssingle-chain dynamics in the lamellar phase without and with slip-links

entangled dynamics in spatially inhomogeneous systems

lamellar phase: cN=80, N=128

Rouse: dynamics parallel and perpendicular decouple, parallel dynamics unaltered

Reptation: coupling of directions and significant slowing

down of parallel and perpendicular dynamics


towards a more realistic dynamics: slip-linksstructure formation with Rouse-dynamics, slip-links and slithering snake-dynamics

idea: describe over-damped motion in a polymer melt by Brownian dynamics

Langevin-thermostat with respect to local velocity field

determine self-consistently from particle displacements

in a short time interval Dt , average over all particles and time interval T

(self-consistent Brownian dynamics)

replace non-bonded interactions by self-consistent external fluctuating field, W,

which is frequently updated (quasi-instantaneous field approximation)

hydrodynamic velocity field must not fluctuate (dense systems and large T)

limited to quasi-stationary flows

Miao, Guo, Zuckermann (1996), Doyle, Shaqfeh, Gast (1997)

Saphiannikova, Prymitsyn, Crosgrove (1998), Narayanan, Prymitsyn, Ganesan (2004)

towards a more realistic dynamics: flow

flow of a melt over brush of identical chains

• flow through channels coated with a

brush/network

• reduction of friction

• motion of block-copolymer saturated

interfaces in AB homopolymer-diblock

copolymer blends (mechanical strength

of interfaces)

hydrodynamic boundary condition

at brush-melt interface

how does a polymer brush

respond to shear flow?

Couette and Poiseuille flows

and consistency of Navier

slip boundary condition

velocity profiles

Poiseuille and Couette flow

velocity profiles

inversion of flow direction inside the brush

top of the brush is dragged along by the flow, vb>0 for large x

average velocity of brush vanishes (grafted chains)

vb<0 close to grafting surface

confirmed by MD simulation of bead-spring model

tumbling motion of isolated, grafted chains in shear flow

Doyle, Ladoux, Viovy, 2000, Gerashchenko, Steinberg,

2006, Delgado-Buscaliono 2006, Winkler, 2006

hydrodynamic interactions not important, non-Gaussian distribution of orientations

inversion of flow direction inside the brush

tumbling motion of grafted chains

v



outline:




o explicit dynamics of molecules

(field-theoretic descriptions require Onsager coefficients)

o soft potentials do not enforce non-crossability and result in Rouse dynamics

o slip-link model can be utilized to describe the entangled dynamics in melt

entanglement length/number of slip-links is an input parameter

o basic characteristics of hydrodynamic flow in dense systems can be

captured by self-consistent Brownian dynamics

(only quasi-stationary flows and no hydrodynamic interactions of solvent)

summary III

SCMF simulations vs SCF theory

ensemble of independent molecules in independent molecules in static, real

fluctuating, real fields representing quasi- fields, self-consistently determined from

instantaneous interactions with surrounding average densities

result: explicit many body configuration spatial density distribution

(including intermolecular correlations)

statics: QIF-approximation is controlled by MF-approximation controlled by

a small parameter Ginzburg-parameter

that depends on discretization DL and N that is an invariant of the system

correlations and fluctuations no correlations or fluctuations

dynamics: evolution of explicit molecular time evolution of collective densities

conformations (Rouse-like dynamics) non-local Onsager-coeffizient required

dynamic asymmetries and ``freezing’’ molecular conformations are assumed

of one component feasible to be in equilibrium with external fields

structure formation of amphiphilic moleculesamphiphilic molecules:

two, incompatible portions covalently

linked into one molecule, e.g.,

block copolymers or biological lipids

no macroscopic phase separation

but self-assembly into spatially

structured, periodic microphases

universality:

systems with very different molecular interactions

exhibit common behavior (e.g., biological lipids in

aqueous solution,high molecular weight amphiphilic

polymers in water, diblock copolymer in a melt)

use coarse-grained models that only incorporate the relevant interactions:

connectivity along the molecule and repulsion between the two blocks

1-100 nanometer(s)

free energy difference via TDI

Rouse-like dynamics via SMC simulations

SMC: Brownian dynamics as

smart MC simulation

Rossky, Doll, Friedman, 1978

idea: uses forces to construct trial

displacements Dr

SMC or force bias MC allow for a much larger

time step (factor 100) than Brownian dynamics


orientation distributions of tumbling chains

Gerashchenko, Steinberg, 2006, Delgado-Buscaliono 2006, Winkler, 2006

?

v

static tilt vs cyclic motion

brush does not act

like a static, porous

medium

Soft, coarse-grained models for multi-component polymer ... · multi-component polymer melts: free energy and single chain ... see also Grochola, JCP 120, ... • soft generic coarse-grained

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