Stable Frank-Kasper phases of self-assembled, soft matter spheres.

Single molecular species can self-assemble into Frank-Kasper (FK) phases, finite approximants of dodecagonal quasicrystals, defying intuitive notions that thermodynamic ground states are maximally symmetric. FK phases are speculated to emerge as the minimal-distortional packings of space-filling spherical domains, but a precise measure of this distortion and how it affects assembly thermodynamics remains ambiguous. We use two complementary approaches to demonstrate that the principles driving FK lattice formation in diblock copolymers emerge directly from the strong-stretching theory of spherical domains, in which a minimal interblock area competes with a minimal stretching of space-filling chains. The relative stability of FK lattices is studied first using a diblock foam model with unconstrained particle volumes and shapes, which correctly predicts not only the equilibrium σ lattice but also the unequal volumes of the equilibrium domains. We then provide a molecular interpretation for these results via self-consistent field theory, illuminating how molecular stiffness increases the sensitivity of the intradomain chain configurations and the asymmetry of local domain packing. These findings shed light on the role of volume exchange on the formation of distinct FK phases in copolymers and suggest a paradigm for formation of FK phases in soft matter systems in which unequal domain volumes are selected by the thermodynamic competition between distinct measures of shape asymmetry.

Origins of low-symmetry phases in asymmetric diblock copolymer melts.

Cooling disordered compositionally asymmetric diblock copolymers leads to the formation of nearly spherical particles, each containing hundreds of molecules, which crystallize upon cooling below the order-disorder transition temperature (TODT). Self-consistent field theory (SCFT) reveals that dispersity in the block degrees of polymerization stabilizes various Frank-Kasper phases, including the C14 and C15 Laves phases, which have been accessed experimentally in low-molar-mass poly(isoprene)-b-poly(lactide) (PI-PLA) diblock copolymers using thermal processing strategies. Heating and cooling a specimen containing 15% PLA above and below the TODT from the body-centered cubic (BCC) or C14 states regenerates the same crystalline order established at lower temperatures. This memory effect is also demonstrated with a specimen containing 20% PLA, which recrystallizes to either C15 or hexagonally ordered cylinders (HEXC) upon heating and cooling. The process-path-dependent formation of crystalline order shapes the number of particles per unit volume, n/V, which is retained in the highly structured disordered liquid as revealed by small-angle X-ray scattering (SAXS) experiments. We hypothesize that symmetry breaking during crystallization is governed by the particle number density imprinted in the liquid during ordering at lower temperature, and this metastable liquid is kinetically constrained from equilibrating due to prohibitively large free energy barriers for micelle fusion and fission. Ordering at fixed n/V is enabled by facile chain exchange, which redistributes mass as required to meet the multiple particle sizes and packing associated with specific low-symmetry Frank-Kasper phases. This discovery exposes universal concepts related to order and disorder in self-assembled soft materials.

Role of Chain Length in the Formation of Frank-Kasper Phases in Diblock Copolymers.

The phase behavior of poly(styrene)-b-poly(1,4-butadiene) diblock copolymers with a polymer block invariant degree of polymerization N[over ¯]_{b}≈800 shows no evidence of Frank-Kasper phases, in contrast to low molar mass diblock copolymers (N[over ¯]_{b}<100) with the same conformational asymmetry. A universal self-concentration crossover parameter N[over ¯]_{x}≈400 is identified, directly related to the crossover to entanglement dynamics in polymer melts. Mean-field behavior is recovered when N[over ¯]_{b}>N[over ¯]_{x}, while complex low symmetry phase formation is attributed to fluctuations and space-filling constraints, which dominate when N[over ¯]_{b}

Accelerating self-consistent field theory of block polymers in a variable unit cell.

Self-consistent field theory (SCFT) is one of the most widely used tools to study the equilibrium phase behavior of block polymers. We have extended an existing version of the Anderson-mixing iteration scheme to solve the highly nonlinear SCFT equations while simultaneously optimizing the unit-cell dimensions. This improved scheme substantially increases the computational efficiency compared to existing schemes.

Commensurability and finite size effects in lattice simulations of diblock copolymers.

Lattice Monte Carlo (MC) simulations provide an efficient method for exploring the structure and phase behavior of block polymer melts. However, the results of such simulations may differ from the equilibrium behavior of a hypothetical infinite system as a consequence of the finite size of the simulation box. Standard finite-size scaling techniques cannot be employed to remove the effects of a small system size due to incommensurability between the ordered structure domain spacing and the periodicity of the simulation box. This work describes a systematic approach to estimating the equilibrium domain spacing in lattice MC simulations of symmetric diblock copolymers, and thereby minimize the effects of incommensurability. Results for simulations with commensurate simulation boxes, which are designed to be commensurate with the preferred lattice periodicity but contain different numbers of unit cells, show that once the effects of incommensurability are removed, the effects of finite size alone are relatively small.

Reactivity-Guided Depercolation Processes Determine Fracture Behavior in End-Linked Polymer Networks.

The fracture of polymer networks is tied to the molecular behavior of strands within the network, yet the specific molecular-level processes that determine the mechanical limits of a network remain elusive. Here, the question of reactivity-guided fracture is explored in otherwise indistinguishable end-linked networks by tuning the relative composition of strands with two different mechanochemical reactivities. Increasing the substitution of less mechanochemically reactive ("strong") strands into a network comprising more reactive ("weak") strands has a negligible impact on the fracture energy until the strong strand content reaches approximately 45%, at which point the fracture energy sharply increases with strong strand content. This aligns with the measured strong strand percolation threshold of 48 ± 3%, revealing that depercolation, or the loss of a percolated network structure, is a necessary criterion for crack propagation in a polymer network. Coarse-grained fracture simulations agree closely with the tearing energy trend observed experimentally, confirming that weak strand scissions dominate the failure until the strong strands approach percolation. The simulations further show that twice as many strands break in a mixture than in a pure network.

Random Forest Predictor for Diblock Copolymer Phase Behavior.

Physics-based models are the primary approach for modeling the phase behavior of block copolymers. However, the successful use of self-consistent field theory (SCFT) for designing new materials relies on the correct chemistry- and temperature-dependent Flory-Huggins interaction parameter χAB that quantifies the incompatibility between the two blocks A and B as well as accurate estimation of the ratio of Kuhn lengths (bA/bB) and block densities. This work uses machine learning to model the phase behavior of AB diblock copolymers by using the chemical identities of blocks directly, obviating the need for measurement of χAB and bA/bB. The random forest approach employed predicts the phase behavior with almost 90% accuracy after training on a data set of 4768 data points, almost twice the accuracy obtained using SCFT employing χAB from group contribution theory. The machine-learning model is notably sensitive toward the uncertainty in measuring molecular parameters; however, its accuracy still remains at least 60% even for highly uncertain experimental measurements. Accuracy is substantially reduced when extrapolating to chemistries outside the training set. This work demonstrates that a random forest phase predictor performs remarkably well in many scenarios, providing an opportunity to predict self-assembly without measurement of molecular parameters.

Order and Disorder in ABCA' Tetrablock Terpolymers.

Self-assembly of poly(styrene)-block-poly(isoprene)-block-poly(lactide)-block-poly(styrene) (PS-PI-PLA-PS' or SILS') tetrablock terpolymers, where the volume fractions of the first three blocks are nearly equivalent, was studied both experimentally and using the self-consistent field theory (SCFT). SCFT indicates that addition of the terminal PS' chain to a low-molecular-mass, hexagonally packed cylinders forming, SIL precursor can produce a disordered state due to preferential mixing of the polystyrene end-blocks with the PI and PLA midblocks in the SILS' tetrablock, alleviating the unfavorable contact between the highly incompatible PI and PLA segments. In contrast, SCFT predicts that higher-molar-mass triblock precursors will maintain an ordered morphology upon addition of the terminal PS' block due to stronger overall segregation strengths. These predictions were tested using three sets of SILS' polymers that were synthesized based on three precursor SIL triblock polymers differing in total molar mass (14, 30, and 47 kg mol-1) and varying the length of the terminal PS' chain. In the lowest-molar-mass set of tetrablock polymers, the shift from order to disorder was observed in the materials at ambient temperature as the molar mass of the terminal PS' block was increased, consistent with SCFT calculations. Disorder with longer S' chain lengths was not found in the two higher-molar-mass polymer sets; the medium-molar-mass set showed both microphase separation and long-range order based on transmission electron microscopy (TEM) and small-angle X-ray scattering (SAXS), while the largest of these block polymers microphase separated but showed limited long-range order. The combination of the experimental and theoretical results presented in this work provides insights into the self-assembly of ABCA'-type polymers and highlights potential complications that arise from frustration in accessing well-ordered materials.

Thermal processing of diblock copolymer melts mimics metallurgy.

Small-angle x-ray scattering experiments conducted with compositionally asymmetric low molar mass poly(isoprene)-b-poly(lactide) diblock copolymers reveal an extraordinary thermal history dependence. The development of distinct periodic crystalline or aperiodic quasicrystalline states depends on how specimens are cooled from the disordered state to temperatures below the order-disorder transition temperature. Whereas direct cooling leads to the formation of documented morphologies, rapidly quenched samples that are then heated from low temperature form the hexagonal C14 and cubic C15 Laves phases commonly found in metal alloys. Self-consistent mean-field theory calculations show that these, and other associated Frank-Kasper phases, have nearly degenerate free energies, suggesting that processing history drives the material into long-lived metastable states defined by self-assembled particles with discrete populations of volumes and polyhedral shapes.

Cornucopia of Nanoscale Ordered Phases in Sphere-Forming Tetrablock Terpolymers.

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 N is 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.