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Solution formulations involving polymers are the basis for a wide range of products spanning consumer care, therapeutics, lubricants, adhesives, and coatings. These multicomponent systems typically show rich self-assembly and phase behavior that are sensitive to even small changes in chemistry and composition. Longstanding computational efforts have sought techniques for predictive modeling of formulation structure and thermodynamics without experimental guidance, but the challenges of addressing the long time scales and large length scales of self-assembly while maintaining chemical specificity have thwarted the emergence of general approaches. As a consequence, current formulation design remains largely Edisonian. Here, we present a multiscale modeling approach that accurately predicts, without any experimental input, the complete temperature–concentration phase diagram of model diblock polymers in solution, as established postprediction through small-angle X-ray scattering. The methodology employs a strategy whereby atomistic molecular dynamics simulations is used to parametrize coarse-grained field-theoretic models; simulations of the latter then easily surmount long equilibration time scales and enable rigorous determination of solution structures and phase behavior. This systematic and predictive approach, accelerated by access to well-defined block copolymers, has the potential to expedite in silicoscreening of novel formulations to significantly reduce trial-and-error experimental design and to guide selection of components and compositions across a vast range of applications.

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.

Chemically dissimilar polymers are rarely miscible due to a specific entropy of mixing that scales as 1/N, where Nis the degree of polymerization. As a consequence, the compatibilization of immiscible polymer blends presents a major challenge to plastics recycling efforts. We demonstrate that when the chain ends of two highly immiscible polymers are functionalized with reacting acid and base groups, ionic junctions form in situby proton transfer, leading to the stabilization of interfaces and the suppression of macroscopic phase separation. Due to the dynamic nature of proton transfer, the microstructure can be tuned via ionic bond energy (h) and segregation strength (χN). In this work, a library of ionically end-functionalized polystyrene (PS) and polydimethylsiloxane (PDMS) is used to demonstrate the importance of end-group acidity/basicity and molecular weight on compatibilization. Well-ordered lamellae result from blending polymers with strong acid and base units, while less ordered, more swollen microphases occur in systems with weak acid and base end groups or larger molar mass. Heating the ionic blends results in reverse proton transfer and destruction of the close-contact ion pairs. As a result, domains are swollen with the resulting unbonded homopolymers, which in sufficient amounts, leads to macrophase separation. Moreover, external salt ions screen the electrostatic attractions and disturb ionic compatibilization. The self-consistent field theory (SCFT) of this system reveals that the ionic interactions can be tuned to create a broad range of morphologies, from microphases to macrophases.

Bottlebrush copolymers, a class of high-density side-chain-grafted copolymers with high molecular weights, have recently attracted extensive research attention due to the dramatic reduction of chain entanglement in these copolymers, unique self-assembly behaviors, and distinctive physical properties. In this study, we employ self-consistent field theory simulations to systemically investigate the phase behavior of statistical bottlebrush copolymers A-stat-B, an intriguing subclass of bottlebrush copolymers with random or alternating sequences of A and B side chains. We find that a broader variety of ordered mesophases can be stabilized in the melts of these copolymers relative to the more extensively studied bottlebrush block copolymers. In particular, we observe that these statistical copolymers can stabilize sphere phases over a larger range of species volume fractions than their diblock and tetrablock counterparts but require much higher segregation strength to achieve this. The sphere and hexagonally packed cylinder phases formed by conformationally symmetric statistical bottlebrush copolymers possess interesting discrete core–shell structures with the backbone and lower volume fraction side-chain species forming shells and cores, respectively. Moreover, we uncover a new strategy of stabilizing the Frank–Kasper phase A15 by introducing conformational asymmetry through different side-chain lengths and also observe deflection of order–order phase boundaries and significant mixing between the backbones and short side chains as conformational asymmetry is introduced.

Polyelectrolyte complexation offers unique opportunities to compatibilize polymers with very different backbone chemistries and to control the morphology of the resulting blend via electrostatic manipulation. In this study, we demonstrate the ability to formulate homogeneous complexes of a conjugated polyelectrolyte with a polymeric ionic liquid, utilizing the electrostatic attraction among their oppositely charged side chains. Variation of electrostatic parameters, such as counterion concentration or polymer charge fraction, tunes the morphology of these polymer complexes from homogeneously disordered blend to weakly structured microemulsion where the local ordering arises from backbone-immiscibility-induced microphase segregation. Our experimental observations are in qualitative agreement with both field-theoretic simulation and random-phase approximation calculations. Simulated morphology snapshots suggest and experimental evidence also indicates that the microphase-segregated complex likely takes on a cocontinuous microemulsion structure. Our findings show that ionic interactions are an effective pathway to compatibilize polymers at macroscopic length scales while achieving controlled nanostructures in these ionic blends. Such systems have great potential for engineering the nanostructure of polymers to tailor applications such as nanofiltration, catalysis, and energy storage, where local ordering can enhance the physical properties of an otherwise macroscopically homogeneous structure.

We use self-consistent field theory (SCFT) to map phase boundaries between periodic microphases for linear, comb-like, and bottlebrush diblock copolymers with continuous Gaussian, discrete Gaussian, and freely jointed chain statistics. By using a properly defined asymmetry parameter, composed of a variety of architectural parameters including side-chain length and segment length, we obtain a universal phase diagram for sphere phases that include A15 and σ phases. We do not observe a transition from comb-like to bottlebrush scaling with architectural parameter variation, which we attribute to the mean field approximation of SCFT.

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 route to control microstructure via parameters such as polymer charge density, salt concentration, and dielectric constant. Here, we use an equilibrium self-consistent field theory to study the phase behavior of such a binary blend, with or without counterions and added salt, and show that it exhibits the canonical ordered phases of a diblock copolymer melt. We demonstrate how differences in the charge density and the dielectric constant of the two polymers affect phase behavior in this system. In particular, we find that the phase windows for sphere phases are dramatically affected and that the Frank–Kasper phases σ and A15 can be stabilized when the minority component has a higher dielectric constant than the surrounding matrix. Since the domain length scale in this system is determined electrostatically and is not subject to chain-stretching limitations imposed by block architectures, our results suggest a possible route to large-unit-cell complex-sphere phases. These predictions will be most easily tested in oppositely charged polymeric ionic liquids, where the bulky and de-localized charges should aid equilibration in a solvent-free environment.

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 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 Tgthrough backbone modulation, separately from controlling ion-polymer interaction dynamics through ligand choice.

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 either thermodynamic phases at equilibrium or defective nonequilibrium states that are kinetically trapped. It is an important engineering challenge to predict key material properties from an underlying morphology, particularly when the material may have inherent defects from processing. Here, we focus on the self-diffusion of a tracer through porous membranes assembled from model triblock copolymers. We generate a large library of both equilibrium and nonequilibrium membrane structures using self-consistent field theory, and we simulate solute diffusion in these structures as a simple random walk on a lattice. We show that the solute self-diffusion coefficient strongly correlates with two of the Minkowski functionals characterizing the morphology of the pores: their volume and their integrated mean curvature. We find that, relative to the corresponding equilibrium morphologies, the structure and transport properties associated with nonequilibrium morphologies of gyroid-forming polymers are more tolerant of defects than those of lamella-forming polymers. However, the nonequilibrium morphologies of lamella-forming polymers exhibit a rich diversity of pore structures and corresponding diffusivities, which may prove helpful for designing membranes with targeted properties.

In this study, we use self-consistent field theory to demonstrate that the A(BA′)nmiktoarm architecture can strongly deflect order–order phase boundaries to large volume fractions fA. The A(BA′)narchitecture achieves this strong deflection by combining the effects of miktoarm frustration and block bidispersity and is shown to stabilize discrete spheres and cylinders of the A block up to values of fA= 0.58 and fA= 0.78, respectively. We next analyze the prevalence of chain bridging νBin both neat miktoarm melts and in homopolymer blends that form the fluctuation-stabilized “bricks and mortar” phase. These calculations demonstrate that high νBand fAcan both be simultaneously achieved with highly asymmetric miktoarm stars, a property especially useful for the design of tough thermoplastic elastomers. Finally, we show that these miktoarms exhibit large windows of phase space where the σ and A15 Frank–Kasper phases are stable.

A new class of thermoplastic elastomers possessing unusual mechanical properties has recently been discovered in binary blends of A-b-(B-b-A′)nmiktoarm star block copolymers and A homopolymers that spontaneously form an unusual, thermodynamically stable, aperiodic “bricks-and-mortar” (B&M) mesophase morphology. The B&M mesophase is believed to be stabilized by thermal fluctuations as in the well-known case of the bicontinuous microemulsion phase. Here, two-dimensional field-theoretic simulations are used to study the equilibrium self-assembly of such miktoarm polymer binary blends. As expected, the B&M mesophase is not present in the mean-field phase diagram obtained with self-consistent field theory, but complex Langevin (CL) simulations, which fully incorporate thermal fluctuation effects, reveal dramatic changes to the phase diagram. A region of strong fluctuations results in the emergent stabilization of the B&M mesophase in a broad composition channel positioned between microphase separation and macrophase separation envelopes, consistent with experimental observations. Our simulations clarify the topology of the blend phase diagram and suggest that the B&M mesophase, at least as observed near the order–disorder transition, has no long-range or quasi-long-range positional or orientational order.

Control of interfacial interactions leads to a dramatic change in shape and morphology for particles based on poly(styrene-b-2-vinylpyridine) diblock copolymers. Key to these changes is the addition of Au-based surfactant nanoparticles (SNPs) which are adsorbed at the interface between block copolymer-containing emulsion droplets and the surrounding amphiphilic surfactant to afford asymmetric, ellipsoid particles. The mechanism of formation for these novel nanostructures was investigated by systematically varying the volume fraction of SNPs, with the results showing the critical nature that the segregation of SNPs to specific interfaces plays in controlling structure. A theoretical description of the system allows the size distribution and aspect ratio of the asymmetric block copolymer colloidal particles to be correlated with the experimental results.

Self-consistent field theory (SCFT) is a versatile framework for investigating self-assembly and thermodynamic properties in broad classes of polymer systems. Nonetheless, it is expensive to implement in higher space dimensions das propagator fields of d+ 1 dimensions must be resolved in a numerical SCFT simulation. In principle, SCFT can be exactly reduced to a phase field (or density functional) theory involving only d-dimensional species density fields, but the density functional of the reduced theory is not known analytically. Although asymptotic approximations to the functional are available in the literature, e.g., the Ohta–Kawasaki (OK) model for diblock copolymers, these functionals tend to produce inaccurate and unreliable predictions. Here, we present a phase field mapping technique adapted from force-matching concepts in the particle coarse-graining literature to best-fit coefficients in a trial density functional using data obtained from low-dimensional SCFT simulations. Such a mapping scheme results in an optimized phase field (OPF) model with improved accuracy over ad hoc choices of the fit parameters. We demonstrate the method by generating an OPF model for diblock copolymer melts, either symmetric or asymmetric, and we study its accuracy and transferability. We show that compared to the OK model, the OPF model makes significantly more accurate predictions of microphase domain periods. The OPF model also semi-quantitatively captures densities and energies, making it useful for computationally intensive applications, either on its own or in a multiscale approach alongside SCFT.

The grafting-through copolymerization of two distinct macromonomers via ring-opening metathesis polymerization is typically used to form statistical or diblock bottlebrush polymers with large total backbone degrees of polymerization (NBB) relative to that of the side-chains (NSC). Here, we demonstrate that Grubbs-type chemistry in the opposite limit, namely NBB≪ NSC, produces well-defined materials with excellent control over ensemble-averaged properties, including molar mass, dispersity, composition, and number of branch points. The dependence of self-assembly on these molecular design parameters was systematically probed using small-angle X-ray scattering and self-consistent field theoretic simulations. Our analysis supports the notion that two-component bottlebrush copolymers with small NBBbehave like miktoarm star polymers. The star-to-bottlebrush transition is quantifiable for both statistical and diblock sequences by unique signatures in the experimental scaling of domain spacing and simulated distribution of backbone/side-chain density within lamellar unit cells. These findings represent a conceptual framework that simplifies the synthesis of miktoarm star polymers when dispersity in the number of arms and composition can be tolerated. The analytical approach introduced to distinguish chain conformations in complex macromolecules also complements previous methods, for example, form factor scattering and rheology.

Abstract

It is curious that first-principles quantum simulations for establishing the electronic structure and bonding patterns of molecules and materials are conducted using fields, yet the standard theoretical approach to understanding their thermal behavior, phase transitions, and self-assembly on larger length- and time scales relies on classical force fields acting on particle degrees of freedom. This article discusses how equilibrium models of classical particle assemblies can be exactly reframed as statistical field theories, and how these theories can be numerically simulated. Today, such field-theoretic simulations have emerged as a highly efficient way to study phase transitions and self-assembly behavior in broad classes of soft materials, including block polymers, polyelectrolyte complexes, and polymeric emulsions.

We report the phase behavior of a series of poly(styrene)-b-poly(isoprene)-b-poly(styrene)′-b-poly(ethylene oxide) (SIS′O) tetrablock terpolymers. This study was motivated by self-consistent field theory (SCFT) calculations that anticipate a rich array of sphere-forming morphologies with variations in the molecular symmetry parameter τ = NS/(NS+ NS′), where Nis the block degree of polymerization and the volume fraction of O is less than about 0.22. Eight SIS′O samples, with τ ranging from 0.21 to 0.73, were synthesized and investigated using small-angle X-ray scattering and transmission electron microscopy, yielding evidence of nine different spherical phases: hexagonal, FCC, HCP, BCC, rhombohedral (tentative), liquid-like packing, dodecagonal quasicrystal, and Frank–Kasper σ and A15 phases. At temperatures close to the order–disorder transition, these tetrablocks behave as pseudo-[SIS′]-O diblocks and form equilibrium morphologies mediated by facile chain exchange between micelles. Transition from equilibrium to nonequilibrium behavior occurs at a temperature (Terg) several tens of degrees below the order–disorder transition temperature, speculated to be coincident with the loss of ergodicity, as chain exchange is arrested due to increased segregation strength between the core (O) and corona (SIS′) blocks. Nonequilibrium ordered structures form when T< Terg; these are interpreted using SCFT calculations to elucidate the free energy landscape driving ordering in the S and I block matrix. These experiments demonstrate a profound dependence on phase stability with variations in τ and temperature, providing insights into the formation of ordered phase symmetry in this class of asymmetric multiblock polymers.

Ternary blends and energy cascades are gaining popularity as ways to engineer absorption as well as exciton and charge collection in organic solar cells. Here, we use kinetic Monte Carlo simulations to investigate energy cascade designs for improving exciton collection in bilayer solar cells via a Förster energy transfer mechanism. We determine that an interfacial monolayer (C) between the donor and acceptor with a D → A → C energy cascade will lead to good exciton collection, allowing for >90% collection, even for energy donor layers up to 75 nm thick. We further examine how roughening the interface, increasing the exciton diffusion length, and using other energy cascade designs affect the enhancement from the energy transfer. We also propose using the inherent charge transfer states at the interfaces as energy acceptors and estimate that the Förster radius could be as large as 3.4 nm, leading to nearly 70% improvement in exciton collection, without the need for a third material.