Field-Based Simulations for the Design of Polymer Nanostructures

Glenn H. Fredrickson

Departments of Chemical Engineering & MaterialsMitsubishi Chemical Center for Advanced Materials (MC-CAM)

Complex Fluids Design Consortium (CFDC)University of California, Santa Barbara

The Equilibrium Theory of Inhomogeneous Polymers (Oxford, 2006)

The Mitsubishi Chemical Center for Advanced Materials (MC-CAM)

MC-CAM was created in 2001 to enable a research partnership between Mitsubishi Chemical and UCSBFocus is new organic, inorganic, and hybrid materials for applications in

Display technologiesSpecialty polymersSolid state lightingEnergy devices, e.g. photovoltaics

Funding has been ~$2.5M/yr ~50 patent disclosures to date

Polypropylene Block CopolymersA Mitsubishi Chemical—UCSB—Cornell collaboration

sPP-EPR-sPP

Mn=300K, wPP=.24

iPP-EPR-iPP

Mn=100K, wPP=.24

Complex Fluids Design Consortium

Academic partners:Fredrickson (Director), Banerjee, Ceniceros, Garcia-Cervera, Gusev (ETH), Cochran (Iowa St.)Industrial partners:Arkema, Mitsubishi Chemicals, Rhodia, General Electric, Dow Chemical, Kraton Polymers, Accelyrs, and NestléNational lab partners:Los Alamos (Lookman, Redondo)Sandia (Curro, Grest, Frischknecht)

The Complex Fluids Design Consortium (CFDC) is an academic-industrial-national lab partnership aimed at developing computational tools for designing soft materials and analyzing multiphase complex fluids

http://www.mrl.ucsb.edu/cfdc/

AcknowledgementsPostdocs:

Dr. Venkat Ganesan Dr. Scott SidesDr. Eric CochranDr. Jonghoon LeeDr. Yuri PopovDr. Kirill KatsovDr. Dominik Duechs

Students:A. Alexander-Katz, S. Hur E. Lennon, W. Lee, A. BosseT. Chantawansri, M. Villet

Collaborators:Prof. Edward KramerProf. Craig HawkerProf. Hector CenicerosProf. Carlos Garcia-Cervera

Funding:NSF DMR-CMMTNSF DMR-MRSECACS-PRF

Complex Fluids Design Consortium:

RhodiaMitsubishi ChemicalArkemaDow ChemicalGE CR&D NestléKraton PolymersAccelrys SNL, LANL

www.mrl.ucsb.edu/cfdc

The Problem—Design of Polymer FormulationsPolymer formulations are often inhomogeneous and multi-component

Multiphase plasticsSolution formulations

They exhibit complex phase behavior, including

Nanostructured mesophasesCoexistence of meso and macro phases (emulsions)

Relationship between formulation, self-assembled structure, and properties difficult to establish

Trial and error experimentation is norm

Can Theory/Simulation help?

• Microphase separation of block copolymers

Nanoscale Morphology Control: Block Copolymers

B

A

AAHolden & Legge(Shell – Kraton Polymers)

ABA Triblock Thermoplastic Elastomer

f

Enabling Chemistries to Create Nanostructured Polymers

The past decade has seen unprecedented advances in controlled (living or quasi-living) polymerization techniques:

Controlled free radical methodsSingle site metallocene catalystsImproved ring-opening techniques“Change of mechanism” strategiesPost-polymerization chemical modificationsLiving Ziegler-Natta methods

These synthetic techniques enable the creation of block and graft architectures from a broad range of commodity-priced monomers

Nanostructured Polymers via New Chemistry:sPP-b-EPR Block Copolymers

sPP minority HPL phase

P. Husted, J. Ruokolainen, R. Mezzenga, G. W. Coates, E. J. Kramer, GHF, Macrom. 38, 851 (2005)

Nanoparticles in Block CopolymersB.J. Kim et. al., Adv. Matl. 17, 2018 (2005)

-0.4 -0.2 0.0 0.2 0.40

50

100

150

200

Normalized Domain Size of PS-b-P2VP

Num

ber o

f Au

Parti

cles

PS PVPPVP

Central Interfacial

100nm 100nm

-0.4 -0.2 0.0 0.2 0.40

50

100

150

Num

ber o

f A

u Pa

rticl

es

Normalized Domain Size of PS-b-P2VP

PSPVP PVP

Au Au

Scales and Approaches to Fluids Simulation

Sub-atomic< 1Å

Fields(wavefunctions, density functionals)

Ab initio quantum chemistry, electronic structure

Atomic to mesoscopic1Å -- 1µm

Particles(positions, momenta)

Classical MD, MC, BD

Continuum> 1µm

Fields(densities, velocities, stresses)

PDEs of mass, momentum, energy flow, elasticity

Can we compute with fields in the atomic-mesoscopic regime?

Scale MethodDOF

From Particles to Fields

Any classical “particle-based” model of an equilibrium fluid can be exactly converted to a statistical field theoryE.g., monatomic fluid with invertable repulsive pair potential v(r) -- Hubbard-Stratonovich transformation

Particles are decoupled and rn coordinates can be traced out of the partition functionField theory is complex when repulsive interactions are present

microscopicdensity

Why Field-Based Simulations of Polymer Fluids?

Relevant spatial and time scales cannot be accessed by atomistic “particle-based” simulationsUse of fluctuating fields, rather than particle coordinates, has potential computational advantages:

Simulations become easier at high density & high MWMore seamless connection to continuum mechanics Systematic coarse-graining by numerical RG appears feasible

Copolymer nanocomposite BJ Kim `06

Microemulsion, Bates ‘97

Coarse-Grained “Particle-Based” Model: Polymer SolutionTwo-parameter “Edwards” model of homopolymers in an implicit good solvent (v > 0):

N

0s

R(s)

2

vv

Edwards Field Theory (~1960)

Energy functional

Single-chain partition function

Fokker-Planck equation for chain propagator

GeneralizationsUsing similar methods, one can construct statistical field theories for a broad variety of polymer formulationsModels have been devised for:

Block and graft copolymers of arbitrary architectureMolten polymer alloysPolyelectrolytesLiquid crystalline polymers (worm-like chains)Polymer brushes, thin filmsSupramolecular polymers

Other ensembles, e.g. μVT, are straightforward

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

Mean-Field 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:

• “Mean-field” approximation (SCFT): F ≈ H[w*]

• Full stochastic sampling of the complex field theory: “Field-theoretic simulations” (FTS)

High-Resolution 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 symmetry

Large cell calculations for exploring self-assembly in new systems

Initial condition is randomComplex geometries can be addressed with a maskingtechnique

A. Bosse

S. Sides, K. Katsov

T. Chantawansri SPHEREPACK

Unit Cell Calculation, Ia3dSymmetry specified initial guess

(E. Cochran)

AB diblock melt, f = 0.39, χN = 20

-5

-4

-3

-2

-1

0

0 10 20 30 40Time

Log

Err

or

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0 10 20 30 40Time

Ene

rgy

9.8 Rg

Mean-Field A-B Diblock Copolymer Melt Phase Diagram Matsen-Bates (1995), Cochran (2006)

f

χ : strengthof A-B monomerRepulsion

N: degree of polymerization

+volume fraction of homopolymer

fraction of A monomers on each triblock fA χNh = 16.0

Nt /Nh = 2

ABA triblock + A homopolymerArkema (S. Sides)

light

dark

256 Rg∼2.5 μm

106 plane waves3000 field iterations

SW Sides and GHF,Polymer 44, 5859 (2003)

+ light

<φΒ > = 0.35

<φA > ~ 0.65 <φΒ > ~ 0.35

pdi =1.00

pdi=1.225

Polydispersity: Acrylic BCs

200 nm

<φA > ~ 0.65

PDI ~ 2.0-3.0

Simulation results (S. Sides)

Experimental data (Arkema/ESPCI)

PBA

PMMA

12 Rg

(light)

(dark)

<φPMMA > ~ 0.65<φPBA > ~ 0.35

~214 nmTEM data courtesy of A.-V. Ruzette

Chuanbing TangMaterials Research LaboratoryUniversity of California Santa Barbara

Basic steps

1. Coating polymers

2. Alignment of microdomains

3. Removal of one component

Expensive

Photolithography vs. Block Copolymer Lithography

Features5 -20 nm

LowCost

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)

Top-down lithography for creating μm scale “wells”, e.g. stripes, squares, or hexagonsBottom-up self-assembly to achieve perfect long-range registry of nm scale microdomains

Segalman et al. Macromolecules 36, 6831 (2003)

Here we examine f = 0.7, χ = 17, and χw = 17 (majority A-monomer 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

L = 14.75 Rg0 L = 16.25 Rg0 L = 18.00 Rg0

SCFT studies of hexagonal confinement: “A wetting”

Tetragonal Ordering by Square Confinement

AB block copolymers pack cylinders or monolayers of spheres in hexagonal latticesSCFT simulations show we can use graphoepitaxy with square wells to force tetragonal (square) packingLimitations:

Need to add majority block A homopolymer (φA=0.23, Nh/N =1.75)Surface/bulk competition restricts method to small lattices

Total A segment concentration

A homopolymer segment concentrationSupport: FENA-

MARCO, UCLA

Polymer – substrate interaction

Multi-layer Films of Spherical AB Diblocks

Gila Stein and Ed Kramer

BCC spheres – 110 planeHCP spheres – 111 plane

(p6m 2D symmetry)

1 layermany layers

Polymer – airinteraction

HCP spheres (p6m 2D symmetry)

a2

a1

a1 / a2 = 1

a2

a1

BCC spheres – 110 plane

a1 / a2 = 2 /√3 = 1.155

a2

a1

Fm3m spheres – 100 planeFace-centered orthorhombic

1 < a1 / a2 < 2 /√3

Stein-Kramer experiments reveal 3 structures:

HCP

BCC

Fmmm

a1 a2

1.00

1.04

1.08

1.12

1.16

0 5 10 15 20# Layers

a 1 /

a 2

Experiment

hcp Fmmm bcc (bulk behavior)

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 1-layer system (d1,f1) and a unit cell calculation of a bulk system (d1

b, fb)

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)

X Ia3dX Lam

X DIS

Physicalpath

w plane

Beyond Mean-Field TheoryIn many situations, mean-field theory is inaccurate

Polymer solutionsMelts near a critical point or ODT

In such cases, the field theory is dominated by w configurations far from any saddle point w*

How do we statistically sample the full field theory?

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

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 planeCompute averages by:

The CL method is a stochastic dynamics that serves toVerify the existence of the real, positive weight P[wR,wI] To importance sample the distribution

Complex Langevin DynamicsA 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 fluctuation-dissipation relation:

Order-disorder 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 stress-free gyroid from SCFT

Polyelectrolyte Complexation: Complex Coacervates

Aqueous mixtures of polyanions and polycations complex to form dense liquid aggregatesFluctuation-dominated: SCFT fails!Applications include:

Food/drug encapsulationDrug/gene delivery vehiclesPurification/separationsBio-inspired adhesives

Cooper et al (2005) Curr Opin Coll. & Interf. Sci.10, 52-78.

H. Waite (UCSB) “Sandcastle worms”

+ + +

--

-

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§ =§ σNImplicit solventInteractions: Coulomb and excluded volume

+ +

+

+

- -

--

Uniform dielectric medium: ε

Corresponding Field-Theory Model

lB =e2 /ε kBT: Bjerrum length

v: Excluded volume parameter

w: fluctuating chemical pot.

φ: fluctuating electrostatic pot.

Complex CoacervationCL Simulations of the Field Theory Model

• 2D, 32x32 Rg02

• C=2.0 B=1.0 E=64000

Future workWe believe that our CL simulation method can be used to explore the phase behavior of a broad range of PE complexation phenomena:

Block copolyampholytesBlock copolymers with charged blocks and uncharged blocks (hydrophobic or hydrophilic)Charged graft, star, and branched polymersPolymer-surfactant complexes Delivery vehicles:

EnzymesDrugsGenes

Complexation of oppositely charged diblocks

2D, 16x16 Rg02

C=11.0 B=0.1 E=64000

A Hybrid Particle-Field Simulation ApproachS. W. Sides et. al. Phys. Rev. Lett. 96, 250601 (2006)

Combine a field-based description of a polymeric fluid with a particle-baseddescription of the nanoparticlesThe particles are described as cavitiesin the fluid. They can:

Be of arbitrary size, shape, and aspect ratioHave a surface treatment to attract or repel any fluid componentHave grafted polymers of any architecture on their surfaces

The fluid field equations are solved inside the cavities for computational efficiencyThe forces on the particles can computed in a single sweep of the fluid fieldA variety of MC and BD update schemes can be applied

Block Copolymer Morphology Change Induced by Nanoparticles (BJ Kim, et. al. PRL 96, 250601 (2006))

By adding PS coated nanoparticles

from S. W. Sides, G. H. Fredrickson

Hybrid FTS PS-b-P2VP 58k-57k/ Au-PSLow particle conc. Low particle conc.

High particle conc. High particle conc.

Lamellar

Hexagonal

Summary“Field-based” computer simulations are powerful tools for exploring equilibrium self-assembly in complex polymer formulationsGood numerical methods are essential!

Free energy evaluation, multiscale methods, and numerical RG remain to be explored

Emerging areas areHybrid simulations with nanoparticles and colloidsPolyelectrolyte complexesSupramolecular polymersNonequilibrium extensions to coupled flow and structure

This is an exciting frontier research area that brings together topics from

Theoretical physics and applied mathNumerical and computational sciencesMaterials scienceReal-world applications!

The Equilibrium Theory of Inhomogeneous Polymers (Oxford, 2006)G. H. Fredrickson et. al., Macromolecules 35, 16 (2002)