soft coarse-grained models for multi-component polymer systemsnsasm10/mueller.pdf · soft...

Soft coarse-grained models for multi-component polymer systems

Marcus Müller and Kostas Ch. Daoulas

outline:• minimal, soft, coarse-grained models• free-energy of self-assembled structures• pattern replication by quasi-block copolymers

Dresden, September 23, 2010

thanks to A. Cavallo, R. Shenhar, J.J. de Pablo, P.F. Nealey, M.P. Stoykovich

structure formation of amphiphilic molecules

1-100 nanometer(s)

amphiphilic molecules: two, incompatible portions covalently linked into one molecule, e.g., block copolymers or biological lipids

no macroscopic phase separationbut self-assembly into spatiallystructured, 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

conformational rearrangements ~ 10-12 - 10-10 s

diffusion ~ 10-9 -10-4 s

bond vibrations~ 10-15 s

ordering kinetics~ hours/days

minimal, soft, coarse-grained models

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 κN andFlory-Huggins parameter χN universality

• elimination of degrees of freedom soft interactions

conformational rearrangements ~ 10-12 - 10-10 s

diffusion ~ 10-9 -10-4 s

bond vibrations~ 10-15 s

minimal, soft, coarse-grained models a small number of atoms islumped into an effectivesegment (interaction center)MC,MD, DPD, LB, SCFT

Daoulas, Müller, JCP 125, 184904 (2006)

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 κN andFlory-Huggins parameter χN universality

• elimination of degrees of freedom soft interactions

conformational rearrangements ~ 10-12 - 10-10 s

diffusion ~ 10-9 -10-4 s

bond vibrations~ 10-15 s

minimal, soft, coarse-grained models a small number of atoms islumped into an effectivesegment (interaction center)MC,MD, DPD, LB, SCFT

Daoulas, Müller, JCP 125, 184904 (2006)

10-15s minimal, soft, coarse-grained model 10-5s simulation 102s

effective interactions become weaker for large degree of coarse-graining no (strict) excluded volume, soft, effective segments can overlap, rather enforce low compressibility on length scale of interest,

`` ´´ -terms generate pairwise interactionsparticle-based description for MC, BD, DPD, or SCMF simulations

minimal, soft, coarse-grained models

with

molecular architecture: Gaussian chain

Müller, Smith, J. Polym. Sci. B 43, 934 (2005); Daoulas, Müller, JCP 125, 184904 (2006); Detcheverry, Kang, Daoulas, Müller, Nealey, de Pablo, Macromolecules 41, 4989 (2008); Pike, Detcheverry, Müller, de Pablo, JCP 131, 084903 (2009); Detcheverry, Pike, Nealey, Müller, de Pablo, PRL 102, 197801 (2009)

bead-spring model with soft, pairwise interactions

definition of chain length without referring to a definition of a segment

coarse-grained model invariant under refinement of contourdiscretizationnumber of interaction centers, N, irrelevant but important

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

• broadening of interfaces by capillary waves • bending rigidity of interfaces, formation of

micro-emulsions near Lifshitz-points• Ginzburg-parameter that controls critical fluctuations in binary blends

or shift of first-order ODT in block copolymer• tube diameter, packing length for Gaussian coils

invariant degree of polymerization

not an additional coarse-grained parameter

corresponds to SCFTsmall exaggerates fluctuations

invariant degree of polymerizationtypical length scale

typical value for a coarse-grained model with excluded volume:necessary condition: or less (otherwise crystallization, glass)

typical values for a coarse-grained model with soft cores:

large requires soft potentials

chain discretization

dense melt of long chains reptation

chains are crossable Rouse-dynamics

choose

1020 elementary moves

4 108 elementary moves

crystallization vs self-assembly

order parameter:Fourier mode of density fluctuation Fourier mode of composition fluctuationideal ordered state: ideal crystal (T=0) SCFT solutiondisordered state: ideal gas homogeneous fluid/melt

ordered state: particles vibrate ordered phase: composition fluctuatesaround ideal lattice positions around reference state (SCFT solution),

but molecules diffuse (liquid)

Einstein crystal is reference state no simple reference state foruse thermodynamic integration wrt self-assembled morphologyto uniform, harmonic coupling ofparticles to ideal position (Frenkel & Ladd)

order parameter:Fourier mode of density fluctuation Fourier mode of composition fluctuationideal ordered state: ideal crystal (T=0) SCFT solutiondisordered state: ideal gas homogeneous fluid/melt

ordered state: particles vibrate ordered phase: composition fluctuatesaround ideal lattice positions around reference state (SCFT solution),

but molecules diffuse (liquid)

free energy per molecule N kBTrelevant free-energy differences 10-3 kBT

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

crystallization vs self-assembly

order parameter:Fourier mode of density fluctuation Fourier mode of composition fluctuationideal ordered state: ideal crystal (T=0) SCFT solutiondisordered state: ideal gas homogeneous fluid/melt

ordered state: particles vibrate ordered phase: composition fluctuatesaround ideal lattice positions around reference state (SCFT solution),

but molecules diffuse (liquid)

free energy per molecule N kBTrelevant free-energy differences 10-3 kBT

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

measure free energy differences by reversibly transforming one structure into another (10-3 relative accuracy needed)

crystallization vs self-assembly

order parameter:Fourier mode of density fluctuation Fourier mode of composition fluctuationideal ordered state: ideal crystal (T=0) SCFT solutiondisordered state: ideal gas homogeneous fluid/melt

ordered state: particles vibrate ordered phase: composition fluctuatesaround ideal lattice positions around reference state (SCFT solution),

but molecules diffuse (liquid)

crystallization vs self-assembly

see also Grochola, JCP 120, 2122 (2004)

PRE 51, R3795 (1995)

calculating free energy differences

Müller, Daoulas, JCP 128, 024903 (2008)

1st ordertransition

intermediate state:independent chains in static, external field (SCFT)

branch 1:“non-interacting

= no collective phenomena”

branch 2:ideally: no structural change

condition for ordering field

free energy difference via TDIwith

SCFT:

use SCFT to predict optimal field and path

optimal choice of external field (Sheu et al):structure does not change along 2nd branch

TDI vs expanded ensemble/replica exchange

• only replica exchange isimpractical because one would need several 100configurations

• at initial stage, where weightsare unknown (ΔF~104kBT), replica exchange guarantees more uniform sampling

• expanded ensemble techniqueis useful because it providesan error estimate

Müller, Daoulas, Norizoe, PCCP 11, 2087 (2009)

accuracy of the methodno kinetic barrier, ie no phase transitionroughly equal probability

• reweighting technique removes large free energy change along the path• probability distribution of reweighted simulation estimates accuracy• kinetics demonstrates the absence of first-order transition

alternative methods • Einstein crystal of grid-discretized fields conjugated to composition::1) HS transformation of particle-based description to field-theoretic model2) discretize the field theory on a lattice3) at each lattice site, complex fields fluctuate around the mean-field solution like atoms

in a crystal fluctuate around the ideal lattice position use Einstein integration

• Einstein crystal around a single, representative liquid configuration:1) tether particles to a frozen, liquid snapshot 2) tether update: swap association between liquid particle and reference particle3) thermodynamic integration to calculate free energy difference to reference system

• absolute free energies:1) calculate

with in nVT-esemble

2) calculate in npT-ensemble

Lennon, Katsov, Fredrickson, PRL 101, 138302 (2008)

Detcheverry, Pike, Nealey, Müller, de Pablo, PRL 102, 197801(2009)

Wilding, Bruce, PRL 85, 5138 (2000), Schilling, Schmid, JCP 131, 231102 (2009)

Martínez-Veracoechea, Escobedo, JCP 125, 104907 (2006)

Einstein-integration for fluctuations of lattice-based density fields

Detcheverry, Pike, Nealey, Müller,de Pablo, PRL 102, 197801(2009)

first-order fluctuation-induced ODTχNODT<14 at fixed spacingχNODT=13.65(10) hysteresis

Lennon, Katsov, Fredrickson, PRL 101, 138302 (2008)

Müller, Daoulas, JCP 128, 024903 (2008)

soft, off-lattice model:measure chemical potential μvia inserting method in NpT-ensemble

grain boundaries

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

0.19(2)

0.21Müller, Daoulas, Norizoe, PCCP 11, 2087 (2009)

reconstruction of soft morphology at patterned surface

0.01(3)

Müller, Daoulas, Norizoe, PCCP 11, 2087 (2009)

rupture of lamellar ordering at 19.5% stretch

particle simulation and Ginzburg-Landau descriptionsystem: symmetric, binary AB homopolymer blenddegrees of freedom:

particle coordinates, composition field (and density),

model definition:intra- and intermolecular potentials free-energy functional,(here: soft, coarse-grained model, SCMF) (here: Ginzburg-Landau-de Gennes functional)single-chain dynamics time-dependent GL theory(here: Rouse dynamics) (model B according to Hohenberg & Halperin) segmental friction, Onsager coefficient,

projection:

time

free e

nerg

y

short simulations of constrained particle model, δt:relax configuration & measure GL parameters

1

2

efficiency of the scheme is quantified by ratio Δt / δt

δt δt

propagate GL model by large time step, Δt

orde

r par

amet

er

GL

free

ener

gy

heterogeneous multiscale modeling

Δt

E, Ren, van den Eijnden, J. Comp. Phys. 228, 5437 (2009), E et al, Com. Comp. Phys 2, 367 (2007)

estimate GL free energy by

restraint particle

simulations

propagate the order

parameter of the GL model by a large Δt

relax the particle model

to the GL configuration in a small δt

to show: steps that involve particle simulation require a time of the order

heterogeneous multiscale modeling

estimate GL free energy by

restraint particle

simulations

propagate the order

parameter of the GL model by a large Δt

relax the particle model

to the GL configuration in a small δt

heterogeneous multiscale modeling

free-energy functional from restraint simulationsidea: restrain the composition, , of particle model to fluctuate

around the order-parameter field, , of the continuum description(umbrella sampling for order-parameter field, , yields )

strong coupling between particle model and continuum description

inspired by Maragliano, van den Eijnden, Chem. Phys. Lett. 426, 168 (2006)

bead-spring model

soft, non-bonded

restrain composition

free-energy functional from restraint simulationsidea: restrain the composition, , of particle model to fluctuate

around the order-parameter field, , of the continuum description(umbrella sampling for order-parameter field, , yields )

comparison yields parameters of GL model

inspired by Maragliano, van den Eijnden, Chem. Phys. Lett. 426, 168 (2006)

impose a periodic order-parameter field,(RPA: coll. structure factor, )

free-energy functional from restraint simulations

Müller, EPJ SpecialTopics 177, 149 (2009)

impose an arbitrary order-parameter field, , and calculate

free-energy functional from restraint simulations

copolymer

φc=0.7

blend

φc=1/3

impose an arbitrary order-parameter field, , and calculate

free-energy functional from restraint simulations

copolymer

φc=0.7

blend

φc=1/3

concurrent coupling• estimate GL free energy by restraint particle simulations

increasing one decreases the decorrelation time and magnitude of fluctuations of the restraint system, but also has to be sampled with accuracy

to measure accurately one needs a time of order independent of (no “equation-free” description à la Kevrekidis, but GL model is required)

use average over many q-vectors to determine the few parameters of the Ginzburg-Landau model

spatial average instead of time average (fast)

Ginzburg-Landau model for a particle-based model at a specific TD state(more complex architectures feasible) Müller, EPJ Special Topics 177, 149 (2009)

(cf. later)

concurrent coupling

estimate GL free energy by

restraint particle

simulations

propagate the order

parameter of the GL model by a large Δt

relax the particle model

to the GL configuration in a small δt

propagate order parameter: GL simulationsidea: relaxation of free energy with conserved order parameter

model B à la Hohenberg & Halperin

early stages of spinodal decomposition:

in Fourier space:

integrate with Heun-algorithmOnsager coefficient

concurrent coupling

estimate GL free energy by

restraint particle

simulations

propagate the order

parameter of the GL model by a large Δt

relax the particle model

to the GL configuration in a small δt

Onsager coefficient from relaxation of simulationsidea: study relaxation of restraint system towards equilibrium,

relaxation time of the constraint system is speeded-up by a factor

constraint system exponentially relaxes towardswith a fast relaxation time scale (fraction of Rouse time)

concurrent coupling

estimate GL free energy by

restraint particle

simulations

propagate the order

parameter of the GL model by a large Δt

relax the particle model

to the GL configuration in a small δt

fast because of spatial average instead of time averageparameters of GL free energy

to show: steps that involve particle simulation require a time of the order

particle model relaxes towards GL configuration on time scale

speed-up and scale separationquestion: What limits the increase of ?• accurate measurement of the chemical potential• forces due to the restraint must be smaller than the original forces

that dictate the intrinsic kinetics of the particle modelbonded force per segmentnon-bonded, thermodynamic forcerestraint force

speed-up and scale separationquestion: What limits the increase of ?• accurate measurement of the chemical potential• forces due to the restraint must be smaller than the original forces

that dictate the intrinsic kinetics of the particle modelbonded force per segmentnon-bonded, thermodynamic forcerestraint force

question: where is the scale separation?• there is no scale separation between GL

order parameter, ,and microscopic• there is a scale separation between the

segmental motion, , and the order-parameter field,

numerical results: fluctuations in disordered stateconcurrent coupling with fixed parameters of GL modelquestion: how much faster can the concurrent scheme run

compared to the original particle model?

time step of thecoupled model

relaxation time ofparticle simulation

speed-up

Müller, EPJ SpecialTopics 177, 149 (2009)

numerical results: early, spinodal decompositionconcurrent coupling with fitting the parameters of the Ginzburg-Landau model

from fitting of restraint relaxation

results: • speed-up 5.8•

from fits• rough estimate of

Onsager coefficient