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, 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” s r' r 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

Q0: 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! 0 0 0 0 0 h´ i ( r ; s; t) ´ j ( r ; s ; t ) i = ±i j ±( r ¡ r ) ±( s ¡ s )

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

ηi are real, uncorrelated, D+1 dimensional Gaussian, white noises Current Limitation of CS Framework for Polymers

Population standard deviation Ratio of simulation steps needed to of the density operator 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! 4He 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”)

Many cool images of superfluid He fountains and vortices can be found online! http://ltl.tkk.fi/research/theory/helium.html 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 He4 at low T, these cycles dominate the thermodynamics and are responsible for superfluidity! Λ = 10 Å at 1K for He4 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) 4He atoms

Complex Langevin simulations, K. Delaney

First direct simulation of a CS Quantum Field Theory!

Classical Quantum BEC Interacting 4He 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 Framework due  Field-Theoretic Simulations (FTS) maturing

 Non-equilibrium extensions still primitive to Sam!

 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

 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)