Transcript Soft, coarse-grained models for multi-component polymer melts: free energy and single chain dynamics Marcus Müller and Kostas Ch.

Soft, coarse-grained models for
multi-component polymer melts:
free energy and single chain dynamics
Marcus Müller and Kostas Ch. Daoulas
outline:
• soft, coarse-grained models
• free energy of self-assembled structures
• kinetics (single-chain and collective)
Development and Analysis of Multiscale Methods
Minneapolis, Nov 3, 2008
coarse-grained models: time and length scales
diffusion
~ 10-9 -10-6 s
bond
vibrations
~ 10-15 s
conformational
rearrangements ~ 10-12 - 10-10 s
Edwards, Stokovich, Müller, Solak, de Pablo, Nealey,
J. Polym. Sci B 43, 3444 (2005)
ordering kinetics
~ hours/days
coarse-grained models: time and length scales
diffusion
~ 10-9 -10-6 s
bond
vibrations
~ 10-15 s
a small number of atoms is
lumped into an effective
segment (interaction center)
MC,MD, DPD, LB,SCFT
conformational
rearrangements ~ 10-12 - 10-10 s
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)
soft, coarse-grained models
relevant interactions: connectivity, excluded volume, repulsion of unlike segments
starting point for field-theoretic
and particle-based description
molecular architecture:
Gaussian chain
with
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
express density through particle coordinates
“
•
´’’
particle-based description
amenable to MC, BD, DPD
or SCMF-simulations
regularize d-function by lattice with discretization DL
MC or SCMF-simulation
Daoulas, Müller, JCP 125, 184904 (2006),
PM-methods (electrostatics), PIC (plasma physics)
computationally fast
but not translationally invariant
• regularize d-function by a weighting function (WDA)
DPD-like models
Laradji, Guo, Zuckermann PRE 49, 3199 (1994) Groot,Warren, JCP 107, 4423 (1997)
slower in dense systems (factor 10-100)
but translationally invariant
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
r
•
•
•
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)
invariant degree of polymerization
definition of length without referring to a definition of a segment: interdigition
corresponds to SCFT
•
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
(Fredrickson-Helfand)
•
broadening of interfaces by capillary waves
•
bending rigidity of interfaces, formation of micro-emulsions near Lifshitz-points
or
(tricritical)
•
depth of correlation hole and amplitude of long-range bond-bond correlations
•
tube diameter, packing length for Gaussian coils
large
requires soft potentials
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)
eff. interaction centers (segments)/chain
chain discretization
dense melt of long chains
reptation
typical values for a coarse-grained model with soft cores
re-entrant melting of underlying
soft-spheres fluid + lattice effects
choose
chains are crossable
otherwise
Rouse-dynamics
summary I
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
o experimentally large values of
controls fluctuation effects
are conveniently described
with soft interactions
large time and length scales: structure formation,
phase separation kinetics, self-assembly
• free energy of self-assembled structure
• kinetics (single-chain and collective)
Development and Analysis of Multiscale Methods
Minneapolis, Nov 3, 2008
free energy of self-assembled structures
relevance:
properties of coarsegrained model
macroscopic behavior
examples:
•
free energy of morphologies
accurate location of 1st order
phase transitions
phase diagram
input for coarser models (phase field)
•
interface and surface
free energies
wetting behavior (Young’s equation)
nucleation
•
defect free energies
free energies of intermediates
kinetics of structure formation
self-assembly vs. crystallization
order parameter:
Fourier mode of composition fluctuation
ideal ordered state:
SCFT solution
ideal disordered state: homogeneous melt
Fourier mode of density fluctuation
ideal crystal (T=0)
ideal gas
in ordered phase, composition fluctuates
around reference state (SCFT solution),
but molecules diffuse (liquid)
in ordered state, particles vibrate around
ideal lattice positions
no simple reference state for selfassembled morphology
Einstein crystal is reference state
use thermodynamic integration wrt
to uniform, harmonic coupling of particles
to ideal position
(Frenkel & Ladd, Wilding & Bruce)
self-assembly vs. crystallization
order parameter:
Fourier mode of composition fluctuation
ideal ordered state:
SCFT solution
ideal disordered state: homogeneous melt
Fourier mode of density fluctuation
ideal crystal (T=0)
ideal gas
in ordered phase, composition fluctuates
around reference state (SCFT solution),
but molecules diffuse (liquid)
in ordered state, particles vibrate around
ideal lattice positions
no simple reference state for selfassembled morphology
Einstein crystal is reference state
use thermodynamic integration wrt
to uniform, harmonic coupling of particles
to ideal position
(Frenkel & Ladd, Wilding & Bruce)
free energy per molecule (ex vol) N kBT
relevant free energy differences 10-3 kBT
absolute free energy must be measured with a relative accuracy of 10-5
self-assembly vs. crystallization
order parameter:
Fourier mode of composition fluctuation
ideal ordered state:
SCFT solution
ideal disordered state: homogeneous melt
Fourier mode of density fluctuation
ideal crystal (T=0)
ideal gas
in ordered phase, composition fluctuates
around reference state (SCFT solution),
but molecules diffuse (liquid)
in ordered state, particles vibrate around
ideal lattice positions
no simple reference state for selfassembled morphology
Einstein crystal is reference state
use thermodynamic integration wrt
to uniform, harmonic coupling of particles
to ideal position
(Frenkel & Ladd, Wilding & Bruce)
free energy per molecule (ex vol) N kBT
relevant free energy differences 10-3 kBT
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)
self-assembly vs. crystallization
order parameter:
Fourier mode of composition fluctuation
ideal ordered state:
SCFT solution
Ideal disordered state: homogeneous melt
Fourier mode of density fluctuation
ideal crystal (T=0)
ideal gas
in ordered phase, composition fluctuates
around reference state (SCFT solution),
but molecules diffuse (liquid)
in ordered state, particles vibrate around
ideal lattice positions
no simple reference state for selfassembled morphology
Einstein crystal is reference state
use thermodynamic integration wrt
to uniform, harmonic coupling of particles
to ideal position
(Frenkel & Ladd, Wilding & Bruce)
PRE 51, R3795 (1995)
see also Grochola, JCP 120, 2122 (2004)
calculating free energy differences
optimal choice of external field (Sheu et al):
structure does not change along 2nd branch
SCFT:
Müller, Daoulas, JCP 128, 024903 (2008)
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
Pike, Detcheverry, Müller,
de Pablo, submitted (2008)
soft, off-lattice model:
measure chemical potential m
via inserting method in NpTensemble
two structures will coexist,
Lennon,
Katsov,
if they
have Fredrickson,
same p and m
PRL 101, 138302 (2008)
Einstein-integration for
fluctuations of lattice-based
density fields around SCFT
further applications: T-junctions
0.19(2)
SCF theory:
0.21
Duque, Katsov, Schick, JCP 117, 10315 (2002)
further applications: rupture of lamellar ordering
0.01(3)
further applications: stalks in solvent-free
membrane models
with Yuki Norizoe
summary II
outline:
• soft generic coarse-grained models for multi-component polymer melts
• free energy of self-assembled structures
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?
• kinetics (single-chain and collective)
Development and Analysis of Multiscale Methods
Minneapolis, Nov 3, 2008
ordering kinetics in thin films:
lamellar-forming copolymer on stripe pattern
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=L02/6D=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
Edwards, Stokovich, Müller, Solak, de Pablo, Nealey, J. Polym. Sci B 43, 3444 (2005)
ordering kinetics: SCMF simulations
10000 MCS
PS-PMMA interface
(green)
PS-rich regions (red)
ordering kinetics: SCMF simulations
ordering kinetics: SCMF simulations
match time scale via PMMA:
1s = 40 MCS, 1h =
(slow component dictates the ordering kinetics)
•
•
time scale to fast (no entanglement effects)
defects anneal out from substrate to surface
morphology
100
5000
500
10000
143 000 MC
hexagonal surface morph.
no lateral diffusion of defects
1000
20000=8min
=0.66 t
1500
towards a more realistic dynamics: slip-links
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
Likhtman, Macro 38, 6128 (2005)
towards a more realistic dynamics: slip-links
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
towards a more realistic dynamics: slip-links
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:
(SL)
(SS)
purely dynamic relation between
2) primitive path analysis using the
multi-chain configurations
and
towards a more realistic dynamics: slip-links
single-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
Müller, Daoulas, JCP 129, 164906 (2008)
towards a more realistic dynamics: slip-links
structure formation with Rouse-dynamics, slip-links and slithering snake-dynamics
towards a more realistic dynamics: flow
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)
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
inversion of flow direction inside the brush
tumbling motion of grafted chains
v
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
summary III
outline:
• soft generic coarse-grained models for multi-component polymer melts
• free energy of self-assembled structures
• kinetics (single-chain and collective)
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)
Development and Analysis of Multiscale Methods
Minneapolis, Nov 3, 2008
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
(including intermolecular correlations)
spatial density distribution
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
conformations (Rouse-like dynamics)
time evolution of collective densities
non-local Onsager-coeffizient required
dynamic asymmetries and ``freezing’’
of one component feasible
molecular conformations are assumed
to be in equilibrium with external fields
structure formation of amphiphilic molecules
amphiphilic molecules:
two, incompatible portions covalently
linked into one molecule, e.g.,
block copolymers or biological lipids
1-100 nanometer(s)
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
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
Müller, Daoulas, JCP 129, 164906 (2008)
orientation distributions of tumbling chains
v
?
static tilt vs cyclic motion
brush does not act
like a static, porous
medium
Gerashchenko, Steinberg, 2006, Delgado-Buscaliono 2006, Winkler, 2006