Glenn Fredrickson of UCSB

Field-Based Simulations for the Design of Polymer Nanostructures Glenn H. Fredrickson Departments of Chemical Engineering & Materials Mitsubishi 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 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

Polypropylene Block Copolymers A Mitsubishi Chemical—UCSB—Cornell collaboration sPP-EPR-sPP iPP-EPR-iPP Mn=300K, wPP=.24 Mn=100K, wPP=.24

Complex Fluids Design Consortium 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 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 Nestlé National lab partners: Los Alamos (Lookman, Redondo) Sandia (Curro, Grest, Frischknecht) http://www.mrl.ucsb.edu/cfdc/

Postdocs: Acknowledgements Dr. Venkat Ganesan Funding: Dr. Scott Sides NSF DMR-CMMT Dr. Eric Cochran NSF DMR-MRSEC Dr. Jonghoon Lee ACS-PRF Dr. Yuri Popov Dr. Kirill Katsov Complex Fluids Design Dr. Dominik Duechs Consortium: Students: Rhodia A. Alexander-Katz, 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 Garcia-Cervera SNL, LANL www.mrl.ucsb.edu/cfdc

The Problem—Design of Polymer Formulations Polymer formulations are often inhomogeneous and multi-component Multiphase plastics Solution formulations They exhibit complex phase behavior, including Nanostructured mesophases Coexistence 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?

Nanoscale Morphology Control: Block Copolymers • Microphase separation of block copolymers ABA Triblock Thermoplastic Elastomer A B Holden & Legge A A (Shell – Kraton Polymers) 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 methods Single site metallocene catalysts Improved ring-opening techniques “Change of mechanism” strategies Post-polymerization chemical modifications Living 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 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 PS-b-P2VP Normalized Domain Size of PS-b-P2VP

Scales and Approaches to Fluids Simulation Scale DOF Method Sub-atomic 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 atomic-mesoscopic regime?

From Particles to Fields Any classical “particle-based” 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) -- Hubbard-Stratonovich 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

Why Field-Based Simulations of Polymer Fluids? Relevant spatial and time scales cannot be accessed by atomistic “particle-based” 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 coarse-graining by numerical RG appears feasible Microemulsion, Bates ‘97

Coarse-Grained “Particle-Based” Model: Polymer Solution Two-parameter “Edwards” model of homopolymers in an implicit good solvent (v > 0): 2 s 0 v N R(s) v

Edwards Field Theory (~1960) Energy functional Single-chain partition function Fokker-Planck equation for chain propagator

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 (worm-like chains) Polymer brushes, thin films Supramolecular 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 S. Sides, K. Katsov symmetry Large cell calculations for exploring self-assembly in new systems Initial condition is random Complex geometries can be addressed with a masking technique T. Chantawansri A. Bosse SPHEREPACK

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

Mean-Field A-B Diblock Copolymer Melt Phase Diagram Matsen-Bates (1995), Cochran (2006) f χ : strength of A-B monomer Repulsion N: degree of polymerization

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

SW Sides and GHF, Polymer 44, 5859 (2003) + light 106 plane waves 3000 field iterations 256 Rg ∼2.5 μm

Simulation results (S. Sides) Polydispersity: Acrylic BCs <φA > ~ 0.65 <φΒ > = 0.35 (dark) PBA pdi = PMMA (light) 1.00 Experimental data (Arkema/ESPCI) <φPMMA > ~ 0.65<φPBA > ~ 0.35 <φA > ~ 0.65 <φΒ > ~ 0.35 PDI ~ pdi= 2.0- 1.225 3.0 TEM data courtesy of A.-V. Ruzette 200 nm 12 Rg ~214 nm

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

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 hexagons Bottom-up self-assembly to achieve perfect long-range registry of nm scale microdomains Segalman et al. Macromolecules 36, 6831 (2003)

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 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

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

Multi-layer 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

Stein-Kramer experiments reveal 3 structures: HCP spheres Fm3m spheres – 100 plane BCC spheres – 110 plane (p6m 2D symmetry) Face-centered orthorhombic a2 a2 a2 a1 a1 a1 a1 / a2 = 1 1 < a1 / a2 < 2 /√3 a1 / a2 = 2 /√3 = 1.155

1.16 BCC 1.12 a1 / a2 bcc 1.08 hcp Fmmm Fmmm (bulk behavior) 1.04 Experiment HCP 1.00 0 5 10 15 20 # Layers a1 a2

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 (d1b, 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)

Beyond Mean-Field Theory In many situations, mean-field 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?

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 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

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 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 aggregates + + + Fluctuation-dominated: SCFT fails! Applications include: Cooper et al (2005) Curr Opin Coll. & Interf. Sci. Food/drug encapsulation 10, 52-78. Drug/gene delivery vehicles Purification/separations Bio-inspired adhesives 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§ =§ σN - Implicit solvent - - Interactions: Coulomb and excluded volume Uniform dielectric medium: ε

Corresponding Field-Theory Model w: fluctuating chemical pot. lB =e2 /ε kBT: Bjerrum length φ: fluctuating electrostatic pot. v: Excluded volume parameter

Complex Coacervation CL Simulations of the Field Theory Model • 2D, 32x32 Rg02 • C=2.0 B=1.0 E=64000

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 Polymer-surfactant complexes Delivery vehicles: Enzymes Drugs Genes

Complexation of oppositely charged diblocks 2D, 16x16 Rg02 C=11.0 B=0.1 E=64000

A Hybrid Particle-Field Simulation Approach S. W. Sides et. al. Phys. Rev. Lett. 96, 250601 (2006) Combine a field-based description of a polymeric fluid with a particle-based 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

Block Copolymer Morphology Change Induced by Nanoparticles (BJ Kim, et. al. PRL 96, 250601 (2006)) Hybrid FTS PS-b-P2VP 58k-57k/ Au-PS 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

Summary “Field-based” computer simulations are powerful tools for exploring equilibrium self-assembly 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 Real-world applications! The Equilibrium Theory of Inhomogeneous Polymers (Oxford, 2006) G. H. Fredrickson et. al., Macromolecules 35, 16 (2002)

Glenn Fredrickson of UCSB

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