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
Postdocs: Nabil Laachi Xingkun Man Rob Riggleman Scott Sides, Eric Cochran Yuri Popov, Jay Lee Venkat Ganesan Students: Mike Villet Erin Lennon Su-Mi Hur Debbie Audus Acknowledgements Collaborators: Kris Delaney (UCSB) Henri Orland (Saclay) Hector Ceniceros (UCSB) Carlos Garcia-Cervera (UCSB) Funding: NSF DMR-CMMT NSF DMR-MRSEC US Army (ICB) Complex Fluids Design Consortium (CFDC): Rhodia Mitsubishi Chemical Arkema Dow Chemical Nestlé Kraton Polymers DSM Intel, JSR, Asahi Kasei, Samsung IBM, SK Hynix SNL, LANL, ARL
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 selfassembly 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 B 10 nm Holden & Legge (Shell Kraton Polymers) S S 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 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 3x3x3 unit cells of Fddd (O70) phase in ABC triblock, K. Delaney ABA + A alloy, S. W. Sides
Models Starting point is a coarsegrained particle model Continuous or discrete chain models Pairwise contact interactions Excluded volume v, Flory parameters Easily added: Electrostatic interactions Incompressibility (melt) Arbitrary branched architectures Sam s pseudopotential 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 0 q(r,s) r s q (r,n-s) N
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 > 10 7 basis functions Unit cell calculations for ordered phases Large cell calculations for exploring self-assembly in new systems: discovery mode Flexible code base (K. Delaney) NVIDIA GPUs, MPI, or OpenMP 2.5 µm Block copolymer-homopolymer blend Complex Architectures Confined BC films Triply-periodic gyroid phase of BCs
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/cm 2 feasible? Jeong et al., ACS Nano, 4 5181, 2010
Directed Self-Assembly (DSA) for the hole shrink problem PS PMMA Vertical Interconnect Access (VIA) lithography: Use DSA to produce highresolution 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 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 (f PMMA = 0.3) Hole depth is ~100 nm Hole CD is varied between 50 and 75 nm Segregation strength χn Prepattern CD (in units of Rg) 15 20 25 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 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 (panel 8) PS-b-PMMA + PMMA, χn = 25, 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 H I [w]) dramatically slows the convergence of Monte Carlo methods based on the positive weight exp(-h R [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): In the field theory representation: Smear particles by a Gaussian of width a
Polyelectrolyte Complexation: Complex Coacervates Aqueous mixtures of polyanions and polycations complex to form dense liquid aggregates complex coacervates Applications include: Food/drug encapsulation Drug/gene delivery vehicles Artificial membranes Bio-sensors Bio-inspired adhesives + - - - + + Cooper et al (2005) Curr Opin Coll. & Interf. Sci. 10, 52-78. Herb Waite (UCSB) Bioadhesives: Sand Castle Worms, Marine Mussels
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 + - - - + + Interactions: Coulomb and excluded volume Uniform dielectric medium:
Corresponding Field-Theory Model w: fluctuating chemical pot. l B =e 2 / k B T: 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, Ermoshkin, Shklovskii, ) B=1, C = 1e-3 Polyampholyte and blend show same +- correlations at large E dimerization! K. Delaney
Other Field Theory Representations? There are at least two alternatives to Sam s Auxiliary Field (AF) representation of polymer field theory that have the nonlocal/nonlinear character of the Hamiltonian more simply expressed Another framework due to Sam is the Coherent States (CS) representation, an approach for branched polymers adapted from Edwards and Freed and inspired by quantum field theory Our innovation: adapting the CS to linear polymers and the first numerical simulations! S. F. Edwards, K. F. Freed J. Phys. C: Solid St. Phys. 3, 739 (1970)
Sam s Second Integral: Coherent States (Re L > 0) For The kernel of L -1 is a free polymer propagator or Green s function r' r s
CS Representation: Ideal Polymers In the CS framework, the degrees of freedom are D+1 dimensional forward and backward propagator-like fields:, CS representation of grand partition function of an ideal gas of linear polymers: L Chain end sources z: polymer activity Q 0 : single polymer partition function L -1 K. Delaney, X. Man, H. Orland, GHF
CS Representation: Interacting Polymers CS representation of Edwards Solution Model in Grand Canonical Ensemble: Segment density operator The same Edwards model, more simply expressed!
Complex Langevin for CS Framework The standard diagonal CL scheme is numerically unstable for the CS representation. A stable scheme has an off-diagonal mobility matrix: The complex noise terms are not uniquely specified by CL theory, but a suitable choice is η i are real, uncorrelated, D+1 dimensional Gaussian, white noises
Current Limitation of CS Framework for Polymers Population standard deviation of the density operator Ratio of simulation steps needed to produce a specified error of the mean Current CL sampling of CS models is currently too noisy to be practically useful but these are early days!
4 He Phase Behavior 4 He at low temperature has normal and superfluid liquid phases; the latter a manifestation of collective quantum behavior of cold boson systems A fraction of the fluid in the superfluid state has zero viscosity! The superfluid is weird in other ways with angular momentum quantized and thermal fluctuations/agitation spawning vortices ( rotons ) http://ltl.tkk.fi/research/theory/helium.html Many cool images of superfluid He fountains and vortices can be found online!
Quantum Statistical Mechanics: Diffusion in Imaginary Time (One Particle, N=1) density matrix Built from 1-particle eigenstates, energies Free particle diffusion in imaginary time in a closed cycle with τ = 1: thermal wavelength R. Feynman
Many Particles: Quantum Indistinguishability The many body wave function for identical Bose particles must be symmetric under pair exchange. For N indistinguishable particles: Each permutation of labels P can be decomposed into cycles ( ring polymers ) of imaginary time propagation these are exchange interactions Since Λ is large for He 4 at low T, these cycles dominate the thermodynamics and are responsible for superfluidity! Λ = 10 Å at 1K for He 4
Coherent States Formulation of QFT For interacting bosons in the grand canonical ensemble and periodic BCs on τ to enforce closed ring polymers, chemical potential pair potential Sums all closed cycles with both exchange and pairwise interactions among bosons! Unlike classical polymers, the interactions are at equal τ
Ideal gas (non-interacting) 4 He atoms Complex Langevin simulations, K. Delaney First direct simulation of a CS Quantum Field Theory! Classical Quantum BEC
Interacting 4 He system Contact potential Normal fluid Perturbation theory for λ line (S. Sachdev, 1999) Superfluid Quantum critical point K. Delaney
Future Work: Optical Lattices Our CS-CL simulation framework seems well positioned to tackle current research on cold bosons in periodic potentials It is likely advantaged over path integral quantum Monte Carlo techniques at low T, high ρ
Discussion and Outlook Field-based computer simulations are powerful tools for exploring self-assembly in polymer formulations SCFT well established Field-Theoretic Simulations (FTS) maturing Non-equilibrium extensions still primitive Good numerical methods are essential! Complex Langevin sampling is our main tool for addressing the sign problem; stable semi-implicit or exponential time differencing schemes mandatory Variable cell shape and Gibb s ensemble methods now exist Free energy accessible by thermo integration; flat histogram? Coarse-graining/RG techniques improving Coherent states (CS) formalism looks promising Framework due to Sam! Semi-local character may offer computational advantages, e.g. in coarse-graining, while still allowing for treatment of a wide range of systems Currently too noisy, but much remains unexplored Potential for advancing current research in cold bosons? G. H. Fredrickson, The Equilibrium Theory of Inhomogeneous Polymers (Oxford 2006)