Building on Sir Sam’s Formalism: Molecularly-Informed Field-Theoretic Simulations of Soft Matter Glenn H. Fredrickson Departments of Chemical Engineering & Materials Materials Research Laboratory (MRL) University of California, Santa Barbara R&D Strategy Office Mitsubishi Chemical Holdings Corporation Tokyo, Japan TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA Acknowledgements Postdocs: Nabil Laachi Funding: Xingkun Man Rob Riggleman NSF DMR-CMMT Scott Sides, Eric Cochran NSF DMR-MRSEC Yuri Popov, Jay Lee US Army (ICB) Venkat Ganesan Complex Fluids Design Consortium (CFDC): Students: Rhodia Mike Villet Mitsubishi Chemical Erin Lennon Arkema Su-Mi Hur Dow Chemical Debbie Audus Nestlé Kraton Polymers DSM Collaborators: Intel, JSR, Asahi Kasei, Samsung Kris Delaney (UCSB) IBM, SK Hynix Henri Orland (Saclay) SNL, LANL, ARL Hector Ceniceros (UCSB) Carlos Garcia-Cervera (UCSB) Sam’s Favorite Complex Gaussian Integrals Representing: Pair interaction (Re v > 0) Inverse operator (Re L > 0) True for scalars, vectors, and functions w and ! Outline Field-theoretic simulations Why? Methodology Applications Advanced lithography – directed self- assembly Polyelectrolyte complexation Fun Superfluid He We aim to develop simulation tools that can guide the design of nano/meso-structured polymer formulations and soft materials Why nano-structured polymers? Nano-structuring is a way to achieve functionality that differentiates and adds value to existing and new families of polymers and derivative materials Nanoscale Morphology Control: Block Copolymers • Microphase separation of block copolymers SBS Triblock Thermoplastic Elastomer S 10 nm B Holden & Legge S S (Shell – Kraton Polymers) Elastic, clear f Rigid, tough, clear Why Field-Based Simulations? Nano/meso: 1 nm to 1 μm Relevant spatial and time scales challenging for fully atomistic, “particle-based” simulations 3x3x3 unit cells of Fddd (O70) phase in ABC triblock, K. Delaney Use of fluctuating fields, rather than particle coordinates, has computational advantages: Simulations become easier at high density & high MW – access to a mean-field (SCFT) solution Systematic coarse-graining more straightforward 2.5 µm ABA + A alloy, S. W. Sides Models Sam’s pseudo- Starting point is a coarse- potential grained particle model Continuous or discrete chain models Pairwise contact interactions Excluded volume v, Flory parameters Easily added: Electrostatic interactions Incompressibility (melt) Arbitrary branched architectures A branched “multiblock” polymer Sam’s First Integral: Auxiliary Field Formalism Representing: Pair interaction (Re v > 0) : an “auxiliary field” True for scalars, vectors, and functions w! From Particles to Fields A “Hubbard-Stratonovich-Edwards” transformation is used to convert the many-body problem into a statistical field theory Polymers decoupled! microscopic particle density Boltzmann weight is a complex number! Edwards Auxiliary Field (AF) Model Sam’s classic model of flexible homopolymers dissolved in good, implicit solvent (S. F. Edwards, 1965) Field-theoretic form “Effective Hamiltonian” Q[iw] is the single-chain partition function for a polymer in an imaginary potential field iw Single-Chain Conformations Q[iw] calculated from propagator q(r,s) for chain end probability distribution s Propagator obtained by integrating a complex diffusion (Fokker-Planck) equation along chain contour s Numerically limiting “inner loop” in field-based simulations! Observables and Operators •Observables can be expressed as averages of operators O[w] with complex weight exp(-H[w]) •Density and stress operators (complex) can be composed from solutions of the Fokker-Planck equation s N 0 q(r,s) r q (r,N-s) Types of Field-Based Simulations • The theory can be simplified to a “mean-field (SCFT)” description by a saddle point approximation: • SCFT is accurate for dense, high MW melts • 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/FTS Simulations By spectral collocation methods and FFTs we can resolve fields with > 107 basis functions Unit cell calculations for ordered 2.5 µm phases Large cell calculations for exploring self-assembly in new systems: “discovery mode” Flexible code base (K. Delaney) NVIDIA GPUs, MPI, or OpenMP Block copolymer-homopolymer blend Triply-periodic gyroid Confined BC films phase of BCs Complex Architectures Directed Self-Assembly An emerging sub 20 nm, low cost patterning technique for electronic device manufacturing Grapho-epitaxy Chemo-epitaxy Unlike bulk BCP assembly, must manage: • Surface/substrate interactions • Commensurability 1. Can we understand defect energetics and kinetics at a fundamental level? 2. Is the ITRS target of < 0.01 defects/cm2 feasible? Jeong et al., ACS Nano, 4 5181, 2010 Directed Self-Assembly (DSA) for the “hole shrink” problem Vertical Interconnect Access (VIA) lithography: Use DSA to produce high- resolution cylindrical holes with reduced critical dimensions relative to a cylindrical pre-pattern created with conventional lithography Current metrology is top-down and cannot probe 3D structures! ~20nm ~50nm ~100nm PS PMMA SiN / SiON SEM images courtesy of J. Cheng of IBM Almaden Research Center Search for basic morphologies: PMMA-attractive pre-patterns SCFT simulations of PS-b-PMMA diblock copolymer in cylindrical prepattern (fPMMA = 0.3) Hole depth is ~100 nm Hole CD is varied between 50 and 75 nm Prepattern CD (in units of Rg) Segregation 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 strength χN 15 20 25 Top view Side view N. Laachi, S. Hur High defect energies for PMMA selective walls in pure DB and blends PS-b-PMMA, χN = 25, CD ~ 68nm PS-b-PMMA + PMMA, χN = 25, (panel 8) CD ~ 75nm (panel 11) 20 kT defect formation energy = parts per billion defect levels Nabil Laachi “String method” for VIAs: Transition pathways PMMA-selective prepattern, CD ≈ 85 nm Metastable defects Perfect state 48 kT 35 kT 15 kT 0 kT - Multi-barrier pathway - Barriers for defect melting are < 5kT Weinan, E., Ren, W. and Vanden-Eijnden, E. J. Chem. Phys., 126, 164103 (2007) N. Laachi et. al., J. Polym. Sci: Part B, Polym. Phys. 53, 142 (2015) Beyond Mean-Field Theory: the “Sign Problem” When sampling a complex field theory, the statistical weight exp( – H[w]) is not positive semi-definite Phase oscillations associated with exp(- i HI[w]) dramatically slows the convergence of Monte Carlo methods based on the positive weight exp(-HR[w]) This sign problem is encountered in other branches of chemistry and physics: QCD, lattice gauge theory, correlated electrons, quantum rate processes Complex Langevin Dynamics (G. Parisi, J. Klauder 1983) A Langevin dynamics in the complex plane for sampling complex field theories and avoiding the sign problem Thermal noise is asymmetrically placed and is Gaussian and white satisfying a fluctuation-dissipation relation: The stochastic field equations are stiff, nonlocal, and nonlinear E. M. Lennon et. al., SIAM Multiscale Modeling and Simulation 6, 1347 (2008) M. Villet and GHF, J. Chem. Phys. 132, 034109 (2010) Curing UV Divergences FT models (beyond mean-field) with infinite interactions at contact have no well-defined continuum limit It is critical to “regularize”, i.e. remove, these singularities for results independent of the computational grid A simple and universal procedure is due to Z.G. Wang (2010): Smear particles by a Gaussian of width a In the field theory representation: Polyelectrolyte Complexation: Complex Coacervates • Aqueous mixtures of - polyanions and - polycations complex to - form dense liquid aggregates – complex coacervates + + + • Applications include: Cooper et al (2005) Curr • Food/drug Opin Coll. & Interf. Sci. encapsulation 10, 52-78. • Drug/gene delivery vehicles Herb Waite (UCSB) • Artificial membranes Bioadhesives: Sand Castle Worms, • Bio-sensors Marine Mussels • Bio-inspired adhesives A Symmetric Coacervate Model • In the simplest case, assume symmetric polyacids & polybases mixed in equal proportions - - - • Polymers are flexible and carry total charge + + + • Implicit good solvent Uniform dielectric • Interactions: Coulomb medium: and excluded volume Corresponding Field-Theory Model 2 w: fluctuating chemical pot. lB =e / kBT: Bjerrum length : fluctuating electrostatic pot. v: excluded volume parameter σ: charge density polymer partition function Harmonic Analysis • Three dimensionless parameters appear in the model (and a/Rg ): • Model has a trivial homogeneous mean-field solution, with no coacervation predicted • Expanding H to quadratic order in w and we recover the “RPA” result of Castelnovo, Joanny, Erukimovich, Olvera de la Cruz, … • These attractive electrostatic correlations provide the driving force for complexation However, we can numerically simulate the exact model! FTS-CL Simulations of Complex Coacervation C = 2, B = 1, E = 64000 Y. O. Popov, J. Lee, and G. H. Fredrickson, J. Polym. Sci. B: Polym. Phys. 45, 3223 (2007) Electrostatic potential fluctuations necessary to obtain coacervation! Complex Coacervation vs “Self-Coacervation” Using FTS-CL, we have generated the first “exact” phase diagrams for complexation of blends and polyampholytes (B=1, a/Rg = 0.2) vs. RPA RPA fails qualitatively on the dilute branches! K. Delaney Evidence for Dimerization Many authors have speculated on dimerization in the dilute branch prior to complexation (Rubinstein, Dobrynin,