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Reactive blending is an efficient method for synthesizing polymer blends. Industrially, this process is carried out in extruders, where the reacting polymers and the generated copolymer are subjected to high shear stresses. The dynamics of the process, and the resulting morphology is dictated by a coupling of the hydrodynamic forces in the extruder, the thermodynamic interactions between species, and the reaction kinetics on a complex interfacial manifold. We use phase-field simulations to quantify the evolution of the reactive blending process under an external shear flow. Specifically, we consider a model system of two homopolymers of equal length, which react via an end-coupling reaction to form a diblock copolymer of double the length. We compare the morphology development in two different initial geometries of the homopolymers—a cylindrical thread and a drop of one homopolymer in a matrix of the second. We investigate the effect of flow strength, measured by the shear rate, and reaction kinetics, quantified by a Damkohler number, on the progress of the reaction and morphology development. Cylindrical threads are susceptible to breakup via the Rayleigh capillary instability. We demonstrate that this instability can be suppressed by imposing shear along the direction of the thread and increasing the extent of the reaction. The reaction rate in this geometry is unaffected by shear imposed along the cylinder axis. Drops deform significantly under an imposed flow, eventually stretching to long cylindrical threads for sufficient shear rates. In the case of drops, shear stresses enhance the reaction rate by deforming the drop, enabling more homopolymers to come in contact at the expanded interface. We show that shear stresses significantly impact the morphology development and reaction dynamics in reactive polymer blending.

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 (Da

This monograph provides an introduction to field-theoretic simulations in classical soft matter, including polymers, and Bose quantum fluids. The method represents a new class of molecular simulation in which continuous fields are sampled and evolved, rather than particle coordinates. Field-theoretic simulations are computationally advantaged over particle methods for dense phases of high molecular weight polymers and can reach length scales of microns or more. The monograph details analytical methods for converting classical and quantum many-body problems to equilibrium field theory models with a molecular basis. Numerical methods are described that enable efficient, accurate, and scalable simulations of such models on modern computer hardware, including graphics processing units (GPUs). Extensions to non-equilibrium systems are discussed, along with an introduction to advanced field-theoretic simulation techniques including free energy estimation, alternative ensembles, coarse-graining, and variable cell methods.

Starting with the simplest classical monatomic fluid, Chapter 2 discusses an “auxiliary field” technique that can be used to exactly convert equilibrium models of fluids framed in particle coordinates to statistical field theory models. The technique is applied to a variety of polymer and soft material models, including bonded, non-bonded, and electrostatic interactions among polymer segments. Field operators are identified that yield important structural and thermodynamic properties. A second “coherent states” field theory representation is introduced and shown to be useful in modeling polymers that can reversibly react

Chapter 5 provides an introduction to the numerical methods used to obtain mean-field solutions and conduct field-theoretic simulations of the field theory models developed in Chapters 2 and 3. The chapter discusses algorithms that exploit efficient FFT-based field operations and balance stability and accuracy. Mean-field solutions are obtained by deterministic relaxation, while the sign problem inherent to classical and quantum field theories mandates the use of complex Langevin sampling methods in stochastic field-theoretic simulations. Practical guidelines for conducting simulations, assessing errors, and troubleshooting are provided.

Chapter 1 contains a summary of the basic mathematical constructs necessary to derive and manipulate field theory models, including functional representations, and derivatives and integrals of functionals. A distinction is made between phenomenological and molecularly-informed field theories; the latter the primary focus of the book. Examples are provided of field theory models for classical polymers and quantum bosons at thermal equilibrium. The computational advantage of the field-theoretic simulation approach is illustrated for a polymer model in which both particle and field representations are available.

Chapter 4 provides an introduction to numerical methods for resolving fields and conducting field-based operations such as differentiation and convolution. Spectral collocation methods using Fourier and Chebyshev bases are emphasized, which permit efficient field operations based on highly-optimized fast Fourier transform (FFT) algorithms. Parallel implementations on multi-core central processing units (CPUs) and graphics processing units (GPUs) are described.

Chapter 3 shows how an equilibrium collection of quantum particles with Bose statistics can be converted into an imaginary-time, coherent states field theory. Field operators are identified that yield structural and thermodynamic properties. Extensions are provided to continuum models with external potentials and artificial gauge fields. Quantum lattice and spin models are shown to be described by the same framework.

Chapter 6 discusses the extension of field-theoretic representations, models, and numerical methods to treat the dynamical evolution of systems that are out of equilibrium. For Bose fluids and magnets, this entails the use of coherent state fields on complex contours with real and imaginary time segments. Complex Langevin sampling of such non-equilibrium field theories provides access to finite temperature dynamics with full quantum fluctuations, both near and far from equilibrium. For classical polymers, approximations are required to address chain entanglement and time-dependent couplings between chain conformations and collective field variables. Current approaches to mesoscopic field-based polymer dynamics are described, along with their numerical implementation.

Advanced methods for conducting field-theoretic simulations are surveyed. This includes a discussion of alternative ensembles, variable cell shape methods, free energy estimation, and coarse-graining methodologies. A strategy for linking atomistic particle simulations to field-theoretic simulations is also provided.

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.

A wide range of field-update algorithms for polymer self-consistent field theory (SCFT) and field theoretic simulations (FTSs) are analyzed. We provide the first direct comparison between Anderson ...

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.

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.

Microphase separation in a binary blend of oppositely charged polymers can in principle be stabilized electrostatically without the need for connected block polymer architectures. This provides a r...

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.

We present a new methodology for polymer self-consistent field theory (SCFT) that has spectral accuracy in the contour dimension while retaining linear scaling of computational effort with system size. In contrast, traditional linear-scaling algorithms only have polynomial order accuracy. The improved accuracy allows for faster simulations and lower memory costs compared to traditional algorithms. The new spectral methods are enabled by converting from an auxiliary field representation to a recently developed “polymer coherent states” framework.

Molecular architecture plays a key role in the self-assembly of block copolymers, but few studies have systematically examined the influence of chain connectivity on tetrahedrally close-packed (TCP) sphere phases. Here, we report a versatile material platform comprising two blocks with substantial conformational asymmetry, A = poly(trifluoroethyl acrylate) and B = poly(dodecyl acrylate), and use it to compare the phase behavior of AB diblocks, ABA triblocks, and (AB)

The effect of dispersity on block polymer self-assembly was studied in the monodisperse limit using a combination of synthetic chemistry, matrix-assisted laser desorption ionization spectroscopy, and small-angle X-ray scattering. Oligo(methyl methacrylate) (oligoMMA) and oligo(dimethylsiloxane) (oligoDMS) homopolymers were synthesized by conventional polymerization techniques and purified to generate an array of discrete, semidiscrete, and disperse building blocks. Coupling reactions afforded oligo(DMS–MMA) block polymers with precisely tailored molar mass distributions spanning single molecular systems (Đ = 1.0) to low-dispersity mixtures (Đ ≈ 1.05). Discrete materials exhibit a pronounced decrease in domain spacing and sharper scattering reflections relative to disperse analogues. The order–disorder transition temperature (TODT) also decreases with increasing dispersity, suggesting stabilization of the disordered phase, presumably due to the strengthening of composition fluctuations at the low molar mas...

Molecular architecture plays a key role in the self-assembly of block copolymers, but few studies have systematically examined the influence of chain connectivity on tetrahedrally close-packed (TCP) sphere phases. Here, we report a versatile material platform comprising two blocks with substantial conformational asymmetry, A = poly(trifluoroethyl acrylate) and B = poly(dodecyl acrylate), and use it to compare the phase behavior of AB diblocks, ABA triblocks, and (AB)n radial star copolymers with n = 3 or 4. Each architecture forms TCP sphere phases at minority A block compositions (fA < 0.5), namely, σ and A15, but with differences in the location of order-order phase boundaries that are not anticipated by mean-field self-consistent field theory simulations. These results expand the palette of polymer architectures that readily self-assemble into complex TCP structures and suggest important design considerations when targeting specific phases of interest.

Bottlebrush block polymers are a promising platform for self-assembled photonic materials, yet most work has been limited to one-dimensional photonic crystals based on the lamellar phase. Here we demonstrate with simulation that nonfrustrated ABC bottlebrush block polymers can be used to self-assemble three-dimensional photonic crystals with complete photonic band gaps. To show this, we have developed a computational approach that couples self-consistent field theory (SCFT) simulations to Maxwell's equations, thereby permitting a direct link between molecular design, self-assembly, and photonic band structures. Using this approach, we calculate the phase diagram of nonfrustrated ABC bottlebrush block polymers and identify regions where the alternating gyroid and alternating diamond phases are stable. By computing the photonic band structures of these phases, we demonstrate that complete band gaps can be found in regions of thermodynamic stability, thereby suggesting a route to realize these photonic materials experimentally. Furthermore, we demonstrate that gap size depends on volume fraction, segregation strength, and polymer architecture, and we identify a design strategy based on symmetry breaking that can achieve band gaps for lower values of refractive index contrast. Taken together, the approach presented here provides a powerful and flexible tool for predicting both the self-assembly and photonic band structures of polymeric materials.

Block copolymers self-assemble into a variety of morphologies that can serve as templates for preparing conductive or porous materials such as batteries and membranes. These morphologies can be eit...

Motivated by the much discussed, yet unexplained, presence of macrovoids in polymer membranes, we explore the impact of Marangoni flows in the process of nonsolvent induced phase separation. Such flows have been hypothesized to be important to the formation of macrovoids, but little quantitative evidence has been produced to date. Using a recently developed multifluid phase field model, we find that roll cells indicative of a solutal Marangoni instability are manifest during solvent/nonsolvent exchange across a stable interface. However, these flows are weak and subsequently do not produce morphological features that might lead to macrovoid formation. By contrast, initial conditions that lead to an immediate precipitation of the polymer film coincide with large Marangoni flows that disturb the interface. The presence of such flows suggests a new experimental and theoretical direction in the search for a macrovoid formation mechanism.

We report the first simulations of nonsolvent-induced phase separation (NIPS) that predict membrane microstructures with graded asymmetric pore size distribution. In NIPS, a polymer solution film is immersed in a nonsolvent bath, enriching the film in nonsolvent, and leading to phase separation that forms a solid polymer-rich membrane matrix and polymer-poor membrane pores. We demonstrate how mass-transfer-induced spinodal decomposition, thermal fluctuations, and glass-transition dynamics-implemented with mobility contrast between the polymer-rich and polymer-poor phases-are essential to the formation of asymmetric membrane microstructures. Specifically, we show that the competition between the propagation of the phase-separation and glass-transition fronts determines the degree of pore-size asymmetry. We also explore the sensitivity of these microstructures to the initial film composition, and compare their formation in 2D and 3D.

Polymer formulations possessing mesostructures or phase coexistence are challenging to simulate using atomistic particle-explicit approaches due to the disparate time and length scales, while the predictive capability of field-based simulations is hampered by the need to specify interactions at a coarser scale (e.g., χ-parameters). To overcome the weaknesses of both, we introduce a bottom-up coarse-graining methodology that leverages all-atom molecular dynamics to molecularly inform coarser field-theoretic models. Specifically, we use relative-entropy coarse-graining to parametrize particle models that are directly and analytically transformable into statistical field theories. We demonstrate the predictive capability of this approach by reproducing experimental aqueous poly(ethylene oxide) (PEO) cloud-point curves with no parameters fit to experimental data. This synergistic approach to multiscale polymer simulations opens the door to de novo exploration of phase behavior across a wide variety of polymer solutions and melt formulations.

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.

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.

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

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.

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.