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1.
FieldBased Simulations for the
Design of Polymer Nanostructures
Glenn H. Fredrickson
Departments of Chemical Engineering & Materials
Mitsubishi Chemical Center for Advanced Materials (MCCAM)
Complex Fluids Design Consortium (CFDC)
University of California, Santa Barbara
The Equilibrium Theory of Inhomogeneous Polymers (Oxford, 2006)
2.
The Mitsubishi Chemical Center for
Advanced Materials (MCCAM)
MCCAM was created in 2001 to
enable a research partnership
between Mitsubishi Chemical and
UCSB
Focus is new organic, inorganic,
and hybrid materials for
applications in
Display technologies
Specialty polymers
Solid state lighting
Energy devices, e.g.
photovoltaics
Funding has been ~$2.5M/yr
~50 patent disclosures to date
4.
Complex Fluids Design Consortium
The Complex Fluids Design Consortium (CFDC) is an academic
industrialnational lab partnership aimed at developing computational
tools for designing soft materials and analyzing multiphase complex
fluids
Academic partners:
Fredrickson (Director), Banerjee,
Ceniceros, GarciaCervera,
Gusev (ETH), Cochran (Iowa St.)
Industrial partners:
Arkema, Mitsubishi Chemicals,
Rhodia, General Electric, Dow
Chemical, Kraton Polymers,
Accelyrs, and Nestlé
National lab partners:
Los Alamos (Lookman,
Redondo)
Sandia (Curro, Grest,
Frischknecht)
http://www.mrl.ucsb.edu/cfdc/
5.
Postdocs:
Acknowledgements
Dr. Venkat Ganesan
Funding:
Dr. Scott Sides
NSF DMRCMMT
Dr. Eric Cochran
NSF DMRMRSEC
Dr. Jonghoon Lee
ACSPRF
Dr. Yuri Popov
Dr. Kirill Katsov Complex Fluids Design
Dr. Dominik Duechs
Consortium:
Students: Rhodia
A. AlexanderKatz, S. Hur Mitsubishi Chemical
E. Lennon, W. Lee, A. Bosse Arkema
T. Chantawansri, M. Villet Dow Chemical
Collaborators: GE CR&D
Prof. Edward Kramer Nestlé
Prof. Craig Hawker Kraton Polymers
Prof. Hector Ceniceros Accelrys
Prof. Carlos GarciaCervera SNL, LANL
www.mrl.ucsb.edu/cfdc
6.
The Problem—Design of Polymer
Formulations
Polymer formulations are
often inhomogeneous and
multicomponent
Multiphase plastics
Solution formulations
They exhibit complex phase
behavior, including
Nanostructured mesophases
Coexistence of meso and
macro phases (emulsions)
Relationship between
formulation, selfassembled
structure, and properties
difficult to establish
Trial and error
experimentation is norm
Can Theory/Simulation help?
7.
Nanoscale Morphology Control: Block
Copolymers
• Microphase separation of block copolymers
ABA Triblock
Thermoplastic Elastomer
A
B
Holden & Legge A A
(Shell – Kraton Polymers)
f
8.
Enabling Chemistries to Create
Nanostructured Polymers
The past decade has seen unprecedented advances
in controlled (living or quasiliving) polymerization
techniques:
Controlled free radical methods
Single site metallocene catalysts
Improved ringopening techniques
“Change of mechanism” strategies
Postpolymerization chemical modifications
Living ZieglerNatta methods
These synthetic techniques enable the creation of
block and graft architectures from a broad range of
commoditypriced monomers
9.
Nanostructured Polymers via New Chemistry:
sPPbEPR Block Copolymers
sPP
minority
HPL
phase
P. Husted, J. Ruokolainen, R. Mezzenga, G. W.
Coates, E. J. Kramer, GHF, Macrom. 38, 851 (2005)
10.
Nanoparticles in Block Copolymers
B.J. Kim et. al., Adv. Matl. 17, 2018 (2005)
Central Interfacial
100nm 100nm
Au Au
200
150 PVP PS PVP PS PVP
PVP
Number of Au Particles
Number of Au Particles
150
100
100
50
50
0 0
0.4 0.2 0.0 0.2 0.4 0.4 0.2 0.0 0.2 0.4
Normalized Domain Size of PSbP2VP Normalized Domain Size of PSbP2VP
11.
Scales and Approaches to Fluids Simulation
Scale DOF Method
Subatomic Fields Ab initio quantum
< 1Å (wavefunctions, chemistry, electronic
density functionals) structure
Atomic to Particles Classical MD, MC, BD
mesoscopic (positions, momenta)
1Å  1µm
Continuum Fields PDEs of mass,
> 1µm (densities, velocities, momentum, energy
stresses) flow, elasticity
Can we compute with fields in the atomicmesoscopic regime?
12.
From Particles to Fields
Any classical “particlebased” model of an equilibrium fluid
can be exactly converted to a statistical field theory
E.g., monatomic fluid with invertable repulsive pair potential
v(r)  HubbardStratonovich transformation
microscopic
density
Particles are decoupled and rn coordinates can be traced out
of the partition function
Field theory is complex when repulsive interactions are
present
13.
Why FieldBased Simulations of Polymer
Fluids?
Relevant spatial and time scales
cannot be accessed by atomistic
“particlebased” simulations
Use of fluctuating fields, rather
than particle coordinates, has
potential computational
advantages: Copolymer nanocomposite
BJ Kim `06
Simulations become easier at high
density & high MW
More seamless connection to
continuum mechanics
Systematic coarsegraining by
numerical RG appears feasible
Microemulsion,
Bates ‘97
14.
CoarseGrained “ParticleBased” Model:
Polymer Solution
Twoparameter “Edwards” model of homopolymers in an
implicit good solvent (v > 0):
2
s
0
v
N
R(s) v
15.
Edwards Field Theory (~1960)
Energy functional
Singlechain partition function
FokkerPlanck equation for chain propagator
16.
Generalizations
Using similar methods, one can construct statistical
field theories for a broad variety of polymer
formulations
Models have been devised for:
Block and graft copolymers of arbitrary architecture
Molten polymer alloys
Polyelectrolytes
Liquid crystalline polymers (wormlike chains)
Polymer brushes, thin films
Supramolecular polymers
Other ensembles, e.g. μVT, are straightforward
17.
Structure of the Field Theories
The field theories have “saddle point”
configurations w*(r) corresponding to stable and
metastable phases of the system
Saddle points can be homogeneous (disordered
phase) or inhomogeneous (ordered phase)
Saddle points lie in the complex plane such that
H[w*] is real
18.
MeanField Approximation: SCFT
• SCFT is derived by a saddle point approximation to the
field theory:
• The approximation is asymptotic for
• We can simulate a field theory at two levels:
• “Meanfield” approximation (SCFT): F ≈ H[w*]
• Full stochastic sampling of the complex field theory:
“Fieldtheoretic simulations” (FTS)
19.
HighResolution SCFT
By the above methods we can
compute saddle points using
~107 or more plane waves
Unit cell calculations for high
accuracy with variable cell shape
to relax stress
Initial condition has desired S. Sides, K. Katsov
symmetry
Large cell calculations for
exploring selfassembly in new
systems
Initial condition is random
Complex geometries can be
addressed with a masking
technique
T. Chantawansri
A. Bosse SPHEREPACK
20.
Unit Cell Calculation, Ia3d 0.00
Symmetry specified initial guess 0.05
(E. Cochran)
0.10
Energy
AB diblock melt,
f = 0.39, χN = 20
0.15
9.8 Rg
0.20
0.25
0 10 20 30 40
Time
0
1
Log Error
2
3
4
5
0 10 20 30 40
Time
21.
MeanField AB Diblock Copolymer Melt Phase
Diagram MatsenBates (1995), Cochran (2006)
f
χ : strength
of AB monomer
Repulsion
N: degree of
polymerization
22.
dark
ABA triblock + A homopolymer
Arkema (S. Sides) +
light
volume fraction of homopolymer
Nt /Nh = 2
fraction of A monomers on each triblock fA
χNh = 16.0
25.
Photolithography vs. Block Copolymer Lithography
Basic steps
1. Coating polymers
2. Alignment of
Expensive
microdomains
3. Removal of
one component
Low Features
Cost 5 20 nm
Chuanbing Tang
Materials Research Laboratory
University of California Santa Barbara
26.
Defects in Laterally Confined Block
Copolymer Thin Films
Large 2D arrays of spheres or cylinders
will exhibit defect populations, even at
equilibrium
However, lateral confinement can be
used to induce order in smaller 2D
systems—”graphoepitaxy” (Kramer,
Segalman, Stein)
Topdown lithography for creating μm
scale “wells”, e.g. stripes, squares, or
hexagons
Bottomup selfassembly to achieve
perfect longrange registry of nm scale
microdomains
Segalman et al. Macromolecules 36, 6831 (2003)
27.
SCFT studies of hexagonal confinement: “A wetting”
L = 14.75 Rg0 L = 16.25 Rg0 L = 18.00 Rg0
Here we examine f = 0.7, χ = 17, and χw = 17
(majority Amonomer is attracted to the wall)
We have identified “commensurability windows” of
side length L, for which various annealing conditions
always produced a defect free configuration
28.
Tetragonal Ordering by Square Confinement
AB block copolymers pack cylinders or monolayers of
spheres in hexagonal lattices
SCFT simulations show we can use graphoepitaxy with
square wells to force tetragonal (square) packing
Limitations:
Need to add majority block A homopolymer (φA=0.23, Nh/N
=1.75)
Surface/bulk competition restricts method to small lattices
Total A A homopolymer
segment segment
Support: FENA concentration concentration
MARCO, UCLA
29.
Multilayer Films of Spherical AB Diblocks
Gila Stein and Ed Kramer Polymer – air
interaction
Polymer – substrate
interaction
1 layer
many layers
HCP spheres – 111 plane
(p6m 2D symmetry) BCC spheres – 110 plane
32.
A Simple Theory
• Assume that the surface excess free energy
contributions are negligible beyond a single layer film,
n=1
• The free energy per chain as a function of the order
parameter η =a1/a2 is:
• The model can be parameterized by SCFT simulations
of a 1layer system (d1,f1) and a unit cell calculation of a
bulk system (d1b, fb)
33.
Theory vs. Experiment
• The theory + SCFT explains the observation of a 1st
order transition!
• The transition is predicted at n=7 (χN=60) vs. n=4 (expt.)
G. E. Stein et. al. Phys. Rev. Lett. 98, 158302 (2007)
34.
Beyond MeanField Theory
In many situations, meanfield theory is inaccurate
Polymer solutions
Melts near a critical point or ODT
In such cases, the field theory is dominated by w
configurations far from any saddle point w*
w plane X Ia3d Physical
X Lam path
X DIS
How do we statistically sample the full field theory?
35.
The “Sign Problem”
When sampling a complex field theory, the
statistical weight exp( – H[w]) is not positive
definite
Phase oscillations associated with the factor
exp(i HI[w]) dramatically slows the convergence
of stochastic sampling methods, e.g. MC
techniques
This sign problem is encountered in other
branches of chemistry and physics: QCD, lattice
gauge theory, correlated electrons, quantum rate
processes
36.
Complex Langevin Sampling
(Parisi, Klauder 1983)
A method to circumvent the sign problem in polymer
simulations (V. Ganesan)
Extend the field w(r) to the complex plane
Compute averages by:
The CL method is a stochastic dynamics that serves to
Verify the existence of the real, positive weight P[wR,wI]
To importance sample the distribution
37.
Complex Langevin Dynamics
A Langevin dynamics in the complex plane for generating
Markov chains with stationary distribution P[wR, wI]
Thermal noise is asymmetrically placed and is Gaussian and
white satisfying usual fluctuationdissipation relation:
38.
Orderdisorder transition of diblock copolymers
(E. Lennon)
f=0.396
χN = 14 ! 11
C=nRgd/V =60.0
L=17.8 Rg
483 lattice
IC: 23 unit cells of
stressfree gyroid
from SCFT
39.
Polyelectrolyte Complexation: Complex
Coacervates
Aqueous mixtures of 
polyanions and 

polycations complex to
form dense liquid
aggregates + + +
Fluctuationdominated:
SCFT fails!
Applications include: Cooper et al (2005) Curr
Opin Coll. & Interf. Sci.
Food/drug encapsulation 10, 5278.
Drug/gene delivery
vehicles
Purification/separations
Bioinspired adhesives H. Waite (UCSB)
“Sandcastle worms”
40.
A Symmetric Model of Coacervation
In the simplest case,
assume symmetric
polyacids & polybases +
mixed in equal + +
proportions (no
counterions) +
Polymers are flexible
and carry total charge 
Z§ =§ σN 
Implicit solvent  
Interactions: Coulomb
and excluded volume Uniform dielectric
medium: ε
42.
Complex Coacervation
CL Simulations of the Field Theory Model
• 2D, 32x32 Rg02
• C=2.0 B=1.0 E=64000
43.
Future work
We believe that our CL simulation method can be used to
explore the phase behavior of a broad range of PE
complexation phenomena:
Block copolyampholytes
Block copolymers with charged blocks and uncharged blocks
(hydrophobic or hydrophilic)
Charged graft, star, and branched polymers
Polymersurfactant complexes Delivery vehicles:
Enzymes
Drugs
Genes
45.
A Hybrid ParticleField Simulation Approach
S. W. Sides et. al. Phys. Rev. Lett. 96, 250601 (2006)
Combine a fieldbased description of a
polymeric fluid with a particlebased
description of the nanoparticles
The particles are described as cavities
in the fluid. They can:
Be of arbitrary size, shape, and aspect
ratio
Have a surface treatment to attract or
repel any fluid component
Have grafted polymers of any
architecture on their surfaces
The fluid field equations are solved
inside the cavities for computational
efficiency
The forces on the particles can
computed in a single sweep of the
fluid field
A variety of MC and BD update
schemes can be applied
46.
Block Copolymer Morphology Change Induced
by Nanoparticles (BJ Kim, et. al. PRL 96, 250601 (2006))
Hybrid FTS PSbP2VP 58k57k/ AuPS
Low particle conc. Low particle conc.
Lamellar
By adding PS
coated nanoparticles
High particle conc. High particle conc.
Hexagonal from S. W. Sides, G. H. Fredrickson
47.
Summary
“Fieldbased” computer simulations are powerful tools for
exploring equilibrium selfassembly in complex polymer
formulations
Good numerical methods are essential!
Free energy evaluation, multiscale methods, and numerical RG
remain to be explored
Emerging areas are
Hybrid simulations with nanoparticles and colloids
Polyelectrolyte complexes
Supramolecular polymers
Nonequilibrium extensions to coupled flow and structure
This is an exciting frontier research area that brings together
topics from
Theoretical physics and applied math
Numerical and computational sciences
Materials science
Realworld applications!
The Equilibrium Theory of Inhomogeneous Polymers (Oxford, 2006)
G. H. Fredrickson et. al., Macromolecules 35, 16 (2002)
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