Direct free energy evaluation of classical and quantum many-body systems via field-theoretic simulation.

Free energy evaluation in molecular simulations of both classical and quantum systems is computationally intensive and requires sophisticated algorithms. This is because free energy depends on the volume of accessible phase space, a quantity that is inextricably linked to the integration measure in a coordinate representation of a many-body problem. In contrast, the same problem expressed as a field theory (auxiliary field or coherent states) isolates the particle number as a simple parameter in the Hamiltonian or action functional and enables the identification of a chemical potential field operator. We show that this feature leads a "direct" method of free energy evaluation, in which a particle model is converted to a field theory and appropriate field operators are averaged using a field-theoretic simulation conducted with complex Langevin sampling. These averages provide an immediate estimate of the Helmholtz free energy in the canonical ensemble and the entropy in the microcanonical ensemble. The method is illustrated for a classical polymer solution, a block copolymer melt exhibiting liquid crystalline and solid mesophases, and a quantum fluid of interacting bosons.

Investigating microstructure evolution in block copolymer membranes.

Block copolymer self-assembly in conjunction with nonsolvent-induced phase separation (SNIPS) has been increasingly leveraged to fabricate integral-asymmetric membranes. The large number of formulation and processing parameters associated with SNIPS, however, has prevented the reliable construction of high performance membranes. In this study, we apply dynamical self-consistent field theory to model the SNIPS process and investigate the effect of various parameters on the membrane morphology: solvent selectivity, nonsolvent selectivity, initial film composition, and glass transition composition. We examine how solvent selectivity and concentration of polymers in the film impact the structure of micelles that connect to form the membrane matrix. In particular, we find that preserving the order in the surface layer and forming a connection between the supporting and surface layer are nontrivial and sensitive to each parameter studied. The effect of each parameter is discussed, and suggestions are made for successfully fabricating viable block copolymer membranes.

Emergence of a Spin Microemulsion in Spin-Orbit Coupled Bose-Einstein Condensates.

We report the first numerical prediction of a "spin microemulsion"-a phase with undulating spin domains resembling classical bicontinuous oil-water-surfactant emulsions-in two-dimensional systems of spinor Bose-Einstein condensates with isotropic Rashba spin-orbit coupling. Using field-theoretic numerical simulations, we investigated the melting of a low-temperature stripe phase with supersolid character and find that the stripes lose their superfluidity at elevated temperature and undergo a Kosterlitz-Thouless-like transition into a spin microemulsion. Momentum distribution calculations highlight a thermally broadened occupation of the Rashba circle of low-energy states with macroscopic and isotropic occupation around the ring. We provide a finite-temperature phase diagram that positions the emulsion as an intermediate, structured isotropic phase with residual quantum character before transitioning at higher temperature into a structureless normal fluid.

A molecularly informed field-theoretic study of the complexation of polycation PDADMA with mixed micelles of sodium dodecyl sulfate and ethoxylated surfactants.

The self-assembly and phase separation of mixtures of polyelectrolytes and surfactants are important to a range of applications, from formulating personal care products to drug encapsulation. In contrast to systems of oppositely charged polyelectrolytes, in polyelectrolyte-surfactant systems the surfactants micellize into structures that are highly responsive to solution conditions. In this work, we examine how the morphology of micelles and degree of polyelectrolyte adsorption dynamically change upon varying the mixing ratio of charged and neutral surfactants. Specifically, we consider a solution of the cationic polyelectrolyte polydiallyldimethylammonium, anionic surfactant sodium dodecyl sulfate, neutral ethoxylated surfactants (C[Formula: see text]EO[Formula: see text]), sodium chloride salt, and water. To capture the chemical specificity of these species, we leverage recent developments in constructing molecularly informed field theories via coarse-graining from all-atom simulations. Our results show how changing the surfactant mixing ratios and the identity of the nonionic surfactant modulates micelle size and surface charge, and as a result dictates the degree of polyelectrolyte adsorption. These results are in semi-quantitative agreement with experimental observations on the same system.

Assessment of the partial saddle point approximation in field-theoretic polymer simulations.

Field-theoretic simulations are numerical treatments of polymer field theory models that go beyond the mean-field self-consistent field theory level and have successfully captured a range of mesoscopic phenomena. Inherent in molecularly-based field theories is a "sign problem" associated with complex-valued Hamiltonian functionals. One route to field-theoretic simulations utilizes the complex Langevin (CL) method to importance sample complex-valued field configurations to bypass the sign problem. Although CL is exact in principle, it can be difficult to stabilize in strongly fluctuating systems. An alternate approach for blends or block copolymers with two segment species is to make a "partial saddle point approximation" (PSPA) in which the stiff pressure-like field is constrained to its mean-field value, eliminating the sign problem in the remaining field theory, allowing for traditional (real) sampling methods. The consequences of the PSPA are relatively unknown, and direct comparisons between the two methods are limited. Here, we quantitatively compare thermodynamic observables, order-disorder transitions, and periodic domain sizes predicted by the two approaches for a weakly compressible model of AB diblock copolymers. Using Gaussian fluctuation analysis, we validate our simulation observations, finding that the PSPA incorrectly captures trends in fluctuation corrections to certain thermodynamic observables, microdomain spacing, and location of order-disorder transitions. For incompressible models with contact interactions, we find similar discrepancies between the predictions of CL and PSPA, but these can be minimized by regularization procedures such as Morse calibration. These findings mandate caution in applying the PSPA to broader classes of soft-matter models and systems.

Machine learning and polymer self-consistent field theory in two spatial dimensions.

A computational framework that leverages data from self-consistent field theory simulations with deep learning to accelerate the exploration of parameter space for block copolymers is presented. This is a substantial two-dimensional extension of the framework introduced in the work of Xuan et al. [J. Comput. Phys. 443, 110519 (2021)]. Several innovations and improvements are proposed. (1) A Sobolev space-trained, convolutional neural network is employed to handle the exponential dimension increase of the discretized, local average monomer density fields and to strongly enforce both spatial translation and rotation invariance of the predicted, field-theoretic intensive Hamiltonian. (2) A generative adversarial network (GAN) is introduced to efficiently and accurately predict saddle point, local average monomer density fields without resorting to gradient descent methods that employ the training set. This GAN approach yields important savings of both memory and computational cost. (3) The proposed machine learning framework is successfully applied to 2D cell size optimization as a clear illustration of its broad potential to accelerate the exploration of parameter space for discovering polymer nanostructures. Extensions to three-dimensional phase discovery appear to be feasible.

Coarsening dynamics of ternary polymer solutions with mobility and viscosity contrasts.

Using phase-field simulations, we investigate the bulk coarsening dynamics of ternary polymer solutions undergoing a glass transition for two models of phase separation: diffusion only and with hydrodynamics. The glass transition is incorporated in both models by imposing mobility and viscosity contrasts between the polymer-rich and polymer-poor phases of the evolving microstructure. For microstructures composed of polymer-poor clusters in a polymer-rich matrix, the mobility and viscosity contrasts significantly hinder coarsening, effectively leading to structural arrest. For microstructures composed of polymer-rich clusters in a polymer-poor matrix, the mobility and viscosity contrasts do not impede domain growth; rather, they change the transient concentration of the polymer-rich phase, altering the shape of the discrete domains. This effect introduces several complexities to the coarsening process, including percolation inversion of the polymer-rich and polymer-poor phases-a phenomenon normally attributed to viscoelastic phase separation.

Multiscale modeling of solute diffusion in triblock copolymer membranes.

We develop a multiscale simulation model for diffusion of solutes through porous triblock copolymer membranes. The approach combines two techniques: self-consistent field theory (SCFT) to predict the structure of the self-assembled, solvated membrane and on-lattice kinetic Monte Carlo (kMC) simulations to model diffusion of solutes. Solvation is simulated in SCFT by constraining the glassy membrane matrix while relaxing the brush-like membrane pore coating against the solvent. The kMC simulations capture the resulting solute spatial distribution and concentration-dependent local diffusivity in the polymer-coated pores; we parameterize the latter using particle-based simulations. We apply our approach to simulate solute diffusion through nonequilibrium morphologies of a model triblock copolymer, and we correlate diffusivity with structural descriptors of the morphologies. We also compare the model's predictions to alternative approaches based on simple lattice random walks and find our multiscale model to be more robust and systematic to parameterize. Our multiscale modeling approach is general and can be readily extended in the future to other chemistries, morphologies, and models for the local solute diffusivity and interactions with the membrane.

Molecularly informed field theory for estimating critical micelle concentrations of intrinsically disordered protein surfactants.

The critical micelle concentration (CMC) is a crucial parameter in understanding the self-assembly behavior of surfactants. In this study, we combine simulation and experiment to demonstrate the predictive capability of molecularly informed field theories in estimating the CMC of biologically based protein surfactants. Our simulation approach combines the relative entropy coarse-graining of small-scale atomistic simulations with large-scale field-theoretic simulations, allowing us to efficiently compute the free energy of micelle formation necessary for the CMC calculation while preserving chemistry-specific information about the underlying surfactant building blocks. We apply this methodology to a unique intrinsically disordered protein platform capable of a wide variety of tailored sequences that enable tunable micelle self-assembly. The computational predictions of the CMC closely match experimental measurements, demonstrating the potential of molecularly informed field theories as a valuable tool to investigate self-assembly in bio-based macromolecules systematically.

A phase field model for dynamic simulations of reactive blending of polymers.

A facile way to generate compatibilized blends of immiscible polymers is through reactive blending of end-functionalized homopolymers. The reaction may be reversible or irreversible depending on the end-groups and is affected by the immiscibility and transport of the reactant homopolymers and the compatibilizing copolymer product. Here we describe a phase-field framework to model the combined dynamics of reaction kinetics, diffusion, and multi-component thermodynamics on the evolution of the microstructure and reaction rate in reactive blending. A density functional with no fitting parameters, which is obtained by adapting a framework of Uneyama and Doi and qualitatively agrees with self-consistent field theory, is used in a diffusive dynamics model. For a symmetric mixture of equal-length reactive polymers mixed in equal proportions, we find that depending on the Flory χ parameter, the microstructure of an irreversibly reacting blend progresses through a rich evolution of morphologies, including from two-phase coexistence to a homogeneous mixture, or a two-phase to three-phase coexistence transitioning to a homogeneous blend or a lamellar copolymer. The emergence of a three-phase region at high χ leads to a previously unreported reaction rate scaling. For a reversible reaction, we find that the equilibrium composition is a function of both the equilibrium constant for the reaction and the χ parameter. We demonstrate that phase-field models are an effective way to understand the complex interplay of thermodynamic and kinetic effects in a reacting polymer blend.

Interfacial reaction-induced roughening in polymer thin films.

Reactive blending of immiscible polymers is an important process for synthesizing polymer blends with superior properties. We use a phase-field model to understand reaction dynamics and morphology evolution by diffusive transport in layered films of incompatible, end-reactive polymers. We thoroughly investigate this phenomenon over a large parameter space of interface shapes, layer thicknesses, reaction rates specified by a Damkohler number (Daf), and Flory-Huggins interaction parameter (χ), under static conditions with no external fields. For films of the same thickness, the dynamics of the system is not significantly influenced by the length of the film or the initial shape of the interface. The interface between the polymers is observed to roughen, leading to the formation of a spontaneous emulsion. The reaction progresses slower and the interface roughens later for thicker films, and systems with higher χ. Increasing Daf increases the reaction rate and hastens the onset of roughening. The quasi-static interfacial tension decreases with the extent of reaction, but does not become vanishingly small or negative at the onset of roughening. Simulations with reversible reactions and systems where only a fraction of the homopolymers have reactive end groups show that a critical diblock (reaction product) concentration exists, below which interfacial roughening and spontaneous emulsification is not observed. We also demonstrate that thermal fluctuations accelerate the onset of interfacial roughening, and help sustain the system in an emulsified state.

Molecular design of self-coacervation phenomena in block polyampholytes.

Coacervation is a common phenomenon in natural polymers and has been applied to synthetic materials systems for coatings, adhesives, and encapsulants. Single-component coacervates are formed when block polyampholytes exhibit self-coacervation, phase separating into a dense liquid coacervate phase rich in the polyampholyte coexisting with a dilute supernatant phase, a process implicated in the liquid-liquid phase separation of intrinsically disordered proteins. Using fully fluctuating field-theoretic simulations using complex Langevin sampling and complementary molecular-dynamics simulations, we develop molecular design principles to connect the sequenced charge pattern of a polyampholyte with its self-coacervation behavior in solution. In particular, the lengthscale of charged blocks and number of connections between oppositely charged blocks are shown to have a dramatic effect on the tendency to phase separate and on the accessible chain conformations. The field and particle-based simulation results are compared with analytical predictions from the random phase approximation (RPA) and postulated scaling relationships. The qualitative trends are mostly captured by the RPA, but the approximation fails catastrophically at low concentration.

Stability of the A15 phase in diblock copolymer melts.

The self-assembly of block polymers into well-ordered nanostructures underpins their utility across fundamental and applied polymer science, yet only a handful of equilibrium morphologies are known with the simplest AB-type materials. Here, we report the discovery of the A15 sphere phase in single-component diblock copolymer melts comprising poly(dodecyl acrylate)-block-poly(lactide). A systematic exploration of phase space revealed that A15 forms across a substantial range of minority lactide block volume fractions (fL = 0.25 - 0.33) situated between the σ-sphere phase and hexagonally close-packed cylinders. Self-consistent field theory rationalizes the thermodynamic stability of A15 as a consequence of extreme conformational asymmetry. The experimentally observed A15-disorder phase transition is not captured using mean-field approximations but instead arises due to composition fluctuations as evidenced by fully fluctuating field-theoretic simulations. This combination of experiments and field-theoretic simulations provides rational design rules that can be used to generate unique, polymer-based mesophases through self-assembly.

Emergence of Hexagonally Close-Packed Spheres in Linear Block Copolymer Melts.

The hexagonally close-packed (HCP) sphere phase is predicted to be stable across a narrow region of linear block copolymer phase space, but the small free energy difference separating it from face-centered cubic spheres usually results in phase coexistence. Here, we report the discovery of pure HCP spheres in linear block copolymer melts with A = poly(2,2,2-trifluoroethyl acrylate) ("F") and B = poly(2-dodecyl acrylate) ("2D") or poly(4-dodecyl acrylate) ("4D"). In 4DF diblocks and F4DF triblocks, the HCP phase emerges across a substantial range of A-block volume fractions (circa fA = 0.25-0.30), and in F4DF, it forms reversibly when subjected to various processing conditions which suggests an equilibrium state. The time scale associated with forming pure HCP upon quenching from a disordered liquid is intermediate to the ordering kinetics of the Frank-Kasper σ and A15 phases. However, unlike σ and A15, HCP nucleates directly from a supercooled liquid or soft solid without proceeding through an intermediate quasicrystal. Self-consistent field theory calculations indicate the stability of HCP is intimately tied to small amounts of molar mass dispersity (D); for example, an HCP-forming F4DF sample with fA = 0.27 has an experimentally measured D = 1.04. These insights challenge the conventional wisdom that pure HCP is difficult to access in linear block copolymer melts without the use of blending or other complex processing techniques.

Open-source platform for block polymer formulation design using particle swarm optimization.

Facile exploration of large design spaces is critical to the development of new functional soft materials, including self-assembling block polymers, and computational inverse design methodologies are a promising route to initialize this task. We present here an open-source software package coupling particle swarm optimization (PSO) with an existing open-source self-consistent field theory (SCFT) software for the inverse design of self-assembling block polymers to target bulk morphologies. To lower the barrier to use of the software and facilitate exploration of novel design spaces, the underlying SCFT calculations are seeded with algorithmically generated initial fields for four typical morphologies: lamellae, network phases, cylindrical phases, and spherical phases. In addition to its utility within PSO, the initial guess tool also finds generic applicability for stand-alone SCFT calculations. The robustness of the software is demonstrated with two searches for classical phases in the conformationally symmetric diblock system, as well as one search for the Frank-Kasper [Formula: see text] phase in conformationally asymmetric diblocks. The source code for both the initial guess generation and the PSO wrapper is publicly available.

Self-consistent field theory study of polymer-mediated colloidal interactions in solution: Depletion effects and induced forces.

Polymer-mediated colloidal interactions control the stability and phase properties of colloid-polymer mixtures that are critical for a wide range of important applications. In this work, we develop a versatile self-consistent field theory (SCFT) approach to study this type of interaction based on a continuum confined polymer solution model with explicit solvent and confining walls. The model is formulated in the grand canonical ensemble, and the potential of mean force for the polymer-mediated interaction is computed from grand potentials. We focus on the case of non-adsorbing linear polymers and present a systematic investigation on depletion effects using SCFT. The properties of confined polymer solutions are probed, and mean-field profiles of induced interactions are shown across different physical regimes. We expose a detailed parametric dependence of the interaction, concerning both attractive and repulsive parts, on polymer concentration, chain length, and solvent quality and explore the effect of wall surface roughness, demonstrating the versatility of the proposed approach. Our findings show good agreement with previous numerical studies and experiments, yet extend prior work to new regimes. Moreover, the mechanisms of depletion attraction and repulsion, along with the influence of individual control factors, are further discussed. We anticipate that this study will provide useful insights into depletion forces and can be readily extended to examine more complex colloid-polymer mixtures.

End-to-End Distance Probability Distributions of Dilute Poly(ethylene oxide) in Aqueous Solution.

We introduce a powerful, widely applicable approach to characterizing polymer conformational distributions, specifically the end-to-end distance distributions, P(Ree), accessed through double electron-electron resonance (DEER) spectroscopy in conjunction with molecular dynamics (MD) simulations. The technique is demonstrated on one of the most widely used synthetic, disordered, water-soluble polymers: poly(ethylene oxide) (PEO). Despite its widespread importance, no systematic experimental characterization of PEO's Ree conformational landscape exists. The evaluation of P(Ree) is particularly important for short polymers or (bio)polymers with sequence complexities that deviate from simple polymer physics scaling laws valid for long chains. In this study, we characterize the Ree landscape by measuring P(Ree) for low molecular weight (MW: 0.22-2.6 kDa) dilute PEO chains. We use DEER with end-conjugated spin probes to resolve Ree populations from ∼2-9 nm and compare them with full distributions from MD. The P( Ree)'s from DEER and MD show remarkably good agreement, particularly at longer chain lengths where populations in the DEER-unresolvable range (<1.5 nm) are low. Both the P(Ree) and the root-mean-square R̃ee indicate that aqueous PEO is a semiflexible polymer in a good solvent, with the latter scaling linearly with molecular weight up to its persistence length (lp ∼ 0.48 nm), and rapidly transitioning to excluded volume scaling above lp. The R̃ee scaling is quantitatively consistent with that from experimental scattering data on high MW (>10 kDa) PEO and the P(Ree)'s crossover to the theoretical distribution for an excluded volume chain.

Rapid Generation of Block Copolymer Libraries Using Automated Chromatographic Separation.

A versatile and scalable strategy is reported for the rapid generation of block copolymer libraries spanning a wide range of compositions starting from a single parent copolymer. This strategy employs automated and operationally simple chromatographic separation that is demonstrated to be applicable to a variety of block copolymer chemistries on multigram scales with excellent mass recovery. The corresponding phase diagrams exhibit increased compositional resolution compared to those traditionally constructed via multiple, individual block copolymer syntheses. Increased uniformity and lower dispersity of the chromatographic libraries lead to differences in the location of order-order transitions and observable morphologies, highlighting the influence of dispersity on the self-assembly of block copolymers. Significantly, this separation technique greatly simplifies the exploration of block copolymer phase space across a range of compositions, monomer pairs, and molecular weights (up to 50000 amu), producing materials with increased control and homogeneity when compared to conventional strategies.

The Role of Backbone Polarity on Aggregation and Conduction of Ions in Polymer Electrolytes.

The usual understanding in polymer electrolyte design is that an increase in the polymer dielectric constant results in reduced ion aggregation and therefore increased ionic conductivity. We demonstrate here that in a class of polymers with extensive metal-ligand coordination and tunable dielectric properties, the extent of ionic aggregation is delinked from the ionic conductivity. The polymer systems considered here comprise ether, butadiene, and siloxane backbones with grafted imidazole side-chains, with dissolved Li+, Cu2+, or Zn2+ salts. The nature of ion aggregation is probed using a combination of X-ray scattering, electron paramagnetic resonance (in the case where the metal cation is Cu2+), and polymer field theory-based simulations. Polymers with less polar backbones (butadiene and siloxane) show stronger ion aggregation in X-ray scattering compared to those with the more polar ether backbone. The Tg-normalized ionic conductivities were however unaffected by the extent of aggregation. The results are explained on the basis of simulations which indicate that polymer backbone polarity does impact the microstructure and the extent of ion aggregation but does not impact percolation, leading to similar ionic conductivity regardless of the extent of ion aggregation. The results emphasize the ability to design for low polymer Tg through backbone modulation, separately from controlling ion-polymer interaction dynamics through ligand choice.