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