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.

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.

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.

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.

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.

Improved Elastic Recovery from ABC Triblock Terpolymers.

The promise of ABC triblock terpolymers for improving the mechanical properties of thermoplastic elastomers is demonstrated by comparison with symmetric ABA/CBC analogs having similar molecular weights and volume fraction of B and A/C domains. The ABC architecture enhances elasticity (up to 98% recovery over 10 cycles) in part through essentially full chain bridging between discrete hard domains leading to the minimization of mechanically unproductive loops. In addition, the unique phase space of ABC triblocks also enables the fraction of hard-block domains to be higher (fhard ≈ 0.4) while maintaining elasticity, which is traditionally only possible with non-linear architectures or highly asymmetric ABA triblock copolymers. These advantages of ABC triblock terpolymers provide a tunable platform to create materials with practical applications while improving our fundamental understanding of chain conformation and structure-property relationships in block copolymers.

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.

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.

Predicting surfactant phase behavior with a molecularly informed field theory.

HYPOTHESIS: The computational study of surfactants and self-assembly is challenging because 1) models need to reflect chemistry-specific interactions, and 2) self-assembled structures are difficult to equilibrate with conventional molecular dynamics. We propose to overcome these challenges with a multiscale simulation approach where relative entropy minimization transfers chemically-detailed information from all-atom (AA) simulations to coarse-grained (CG) models that can be simulated using field-theoretic methods. Field-theoretic simulations are not limited by intrinsic physical time scales like diffusion and allow for rigorous equilibration via free energy minimization. This approach should enable the study of properties that are difficult to obtain by particle-based simulations. SIMULATION WORK: We apply this workflow to sodium dodecylsulfate. To ensure chemical fidelity we present an AA force field calibrated against interfacial tension experiments. We generate CG models from AA simulation trajectories and show that particle-based and field-theoretic simulations of the CG model reproduce AA simulations and experimental measurements. FINDINGS: The workflow captures the complex balance of interactions in a multicomponent system ultimately described by an atomistic model. The resulting CG models can study complex 3D phases like double or alternating gyroids, and reproduce salt effects on properties like aggregation number and shape transitions.

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.

Modeling Microstructure Formation in Block Copolymer Membranes Using Dynamical Self-Consistent Field Theory.

Block copolymers have attracted recent interest as candidate materials for ultrafiltration membranes, due to their ability to form isoporous integral-asymmetric membranes by the combined processes of self-assembly and nonsolvent-induced phase separation (SNIPS). However, the dependence of surface layer and substructure morphologies on the processing variables associated with SNIPS is not well understood nor is the interplay between microphase and macrophase separation in block copolymers undergoing such coagulation. Here, we use dynamical self-consistent field theory to simulate the microstructure evolution of block copolymer films during SNIPS and find that such films form the desired sponge-like asymmetric porous substructure only if the solvent and nonsolvent have opposite block selectivities and that otherwise they form a dense nonporous microphase-separated film. Our results could have important implications for the choices of solvent and nonsolvent in the processing of block copolymer membranes.

Field-Theoretic Simulation Method to Study the Liquid-Liquid Phase Separation of Polymers.

Liquid-liquid phase separation (LLPS) is a process that results in the formation of a polymer-rich liquid phase coexisting with a polymer-depleted liquid phase. LLPS plays a critical role in the cell through the formation of membrane-less organelles, but it also has a number of biotechnical and biomedical applications such as drug confinement and its targeted delivery. In this chapter, we present a computational efficient methodology that uses field-theoretic simulations (FTS) with complex Langevin (CL) sampling to characterize polymer phase behavior and delineate the LLPS phase boundaries. This approach is a powerful complement to analytical and explicit-particle simulations, and it can serve to inform experimental LLPS studies. The strength of the method lies in its ability to properly sample a large ensemble of polymers in a saturated solution while including the effect of composition fluctuations on LLPS. We describe the approaches that can be used to accurately construct phase diagrams of a variety of molecularly designed polymers and illustrate the method by generating an approximation-free phase diagram for a classical symmetric diblock polyampholyte.

Phase Separation, Reaction Equilibrium, and Self-Assembly in Binary Telechelic Homopolymer Blends.

We study a binary blend of telechelic homopolymers that can form reversible AB-type bonds at the chain ends. Reversibly bonding polymers display novel material properties, including thermal tunability and self-healing, that are not found in conventional covalently bonded polymers. Previous studies of reversibly bonding polymer systems have been limited by the computational demand of accounting for an infinite number of possible reaction products in a spatially inhomogeneous, self-assembled structure. We demonstrate that newly developed theoretical models and numerical methods enable the simultaneous computation of phase equilibrium, reaction equilibrium, and self-assembly via self-consistent field theory. Phase diagrams are computed at a variety of physically relevant conditions and are compared with nonreactive analogues as well as previous experimental studies of telechelic polymer blends.

Ionic Compatibilization of Polymers.

The small specific entropy of mixing of high molecular weight polymers implies that most blends of dissimilar polymers are immiscible with poor physical properties. Historically, a wide range of compatibilization strategies have been pursued, including the addition of copolymers or emulsifiers or installing complementary reactive groups that can promote the in situ formation of block or graft copolymers during blending operations. Typically, such reactive blending exploits reversible or irreversible covalent or hydrogen bonds to produce the desired copolymer, but there are other options. Here, we argue that ionic bonds and electrostatic correlations represent an underutilized tool for polymer compatibilization and in tailoring materials for applications ranging from sustainable polymer alloys to organic electronics and solid polymer electrolytes. The theoretical basis for ionic compatibilization is surveyed and placed in the context of existing experimental literature and emerging classes of functional polymer materials. We conclude with a perspective on how electrostatic interactions might be exploited in plastic waste upcycling.

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.

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.

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.

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.

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.

Liquid-liquid phase separation of Tau by self and complex coacervation.

The liquid-liquid phase separation (LLPS) of Tau has been postulated to play a role in modulating the aggregation property of Tau, a process known to be critically associated with the pathology of a broad range of neurodegenerative diseases including Alzheimer's Disease. Tau can undergo LLPS by homotypic interaction through self-coacervation (SC) or by heterotypic association through complex-coacervation (CC) between Tau and binding partners such as RNA. What is unclear is in what way the formation mechanisms for self and complex coacervation of Tau are similar or different, and the addition of a binding partner to Tau alters the properties of LLPS and Tau. A combination of in vitro experimental and computational study reveals that the primary driving force for both Tau CC and SC is electrostatic interactions between Tau-RNA or Tau-Tau macromolecules. The liquid condensates formed by the complex coacervation of Tau and RNA have distinctly higher micro-viscosity and greater thermal stability than that formed by the SC of Tau. Our study shows that subtle changes in solution conditions, including molecular crowding and the presence of binding partners, can lead to the formation of different types of Tau condensates with distinct micro-viscosity that can coexist as persistent and immiscible entities in solution. We speculate that the formation, rheological properties and stability of Tau droplets can be readily tuned by cellular factors, and that liquid condensation of Tau can alter the conformational equilibrium of Tau.