Medial packing and elastic asymmetry stabilize the double-gyroid in block copolymers

Triply-periodic networks are among the most complex and functionally valuable self-assembled morphologies, yet they form in nearly every class of biological and synthetic soft matter building blocks. In contrast to simpler assembly motifs – spheres, cylinders, layers – networks require molecules to occupy variable local environments, confounding attempts to understand their formation. Here, we examine the double-gyroid network phase by using a geometric formulation of the strong stretching theory of block copolymer melts, a prototypical soft self-assembly system. The theory establishes the direct link between molecular packing, assembly thermodynamics and the medial map, a generic measure of the geometric center of complex shapes. We show that “medial packing” is essential for stability of double-gyroid in strongly-segregated melts, reconciling a long-standing contradiction between infinite- and finite-segregation theories. Additionally, we find a previously unrecognized non-monotonic dependence of network stability on the relative entropic elastic stiffness of matrix-forming to tubular-network forming blocks. The composition window of stable double-gyroid widens for both large and small elastic asymmetry, contradicting intuitive notions that packing frustration is localized to the tubular domains. This study demonstrates the utility of optimized medial tessellations for understanding soft-molecular assembly and packing frustration via an approach that is readily generalizable far beyond gyroids in neat block copolymers

Non-affinity of liquid networks and bicontinuous mesophases

Amphiphiles self-assemble into a variety of bicontinuous mesophases whose equilibrium structures take the form of high-symmetry cubic networks. Here, we show that the symmetry-breaking distortions in these systems give rise to anomalously large, non-affine collective deformations, which we argue to be a generic consequence of mass equilibration within deformed networks. We propose and study a minimal liquid network model of bicontinuous networks, in which acubic distortions are modeled by the relaxation of residually-stressed mechanical networks with constant-tension bonds. We show that non-affinity is strongly dependent on the valency of the network as well as the degree of strain-softening/stiffening force in the bonds. Taking diblock copolymer melts as a model system, liquid network theory captures quantitative features of two bicontinuous phases based on comparison with self-consistent field theory predictions and direct experimental characterization of acubic distortions, which are likely to be pronounced in soft amphiphilic systems more generally.Comment: 23 pages, 9 figure

Extreme thermodynamics with polymer gel tori:Harnessing thermodynamic instabilities to induce large-scale deformations

When a swollen, thermoresponsive polymer gel is heated in a solvent bath, it expels solvent and deswells. When this heating is slow, deswelling proceeds homogeneously, as observed in a toroid-shaped gel that changes volume whilst maintaining its toroidal shape. By contrast, if the gel is heated quickly, an impermeable layer of collapsed polymer forms and traps solvent within the gel, arresting the volume change. The ensuing evolution of the gel then happens at fixed volume, leading to phase-separation and the development of inhomogeneous stress that deforms the toroidal shape. We observe that this stress can cause the torus to buckle out of the plane, via a mechanism analogous to the bending of bimetallic strips upon heating. Our results demonstrate that thermodynamic instabilities, i.e., phase transitions, can be used to actuate mechanical deformation in an extreme thermodynamics of materials.Comment: 5 pages, 4 figures. To appear in Physical Review E (2018

Source Data for End exclusion zones in strongly stretched, molten polymer brushes of arbitrary shape

Supplementary code for solving the constraint equations that describe curved polymer brushes. Also contains software for analyzing the resulting solutions.https://scholarworks.umass.edu/data/1148/thumbnail.jp

Source Data for Xueyan Feng, Michael S. Dimitriyev & Edwin L. Thomas, Soft, malleable double diamond twin

Source data and code for Xueyan Feng, Michael S. Dimitriyev & Edwin L. Thomas, Soft, malleable double diamond twinhttps://scholarworks.umass.edu/data/1172/thumbnail.jp

Swelling thermodynamics and phase transitions of polymer gels

We present a pedagogical review of the swelling thermodynamics and phase transitions of polymer gels. In particular, we discuss how features of the volume phase transition of the gel's osmotic equilibrium are analogous to other transitions described by mean-field models of binary mixtures, and the failure of this analogy at the critical point due to shear rigidity. We then consider the phase transition at fixed volume, a relatively unexplored paradigm for polymer gels that results in a phase-separated equilibrium consisting of coexisting solvent-rich and solvent-poor regions of gel. Again, the gel's shear rigidity is found to have a profound effect on the phase transition, here resulting in macroscopic shape change at constant volume of the sample, exemplified by the tunable buckling of toroidal samples of polymer gel. By drawing analogies with extreme mechanics, where large shape changes are achieved via mechanical instabilities, we formulate the notion of extreme thermodynamics, where large shape changes are achieved via thermodynamic instabilities, i.e.phase transition

Supplementary Code for Chain trajectories, domain shapes and terminal boundaries in block copolymers

Supplementary code used for the publication Chain trajectories, domain shapes and terminal boundaries in block copolymers by Benjamin R. Greenvall, Michael S. Dimitriyev, and Gregory M. Grason. Includes code for extracting and analyzing polar order and chain trajectories from self-consistent field calculations of block copolymers.https://scholarworks.umass.edu/data/1184/thumbnail.jp

Medial Packing, Frustration, and Competing Network Phases in Strongly Segregated Block Copolymers

Self-consistent field theory (SCFT) has established that for cubic network phases in diblock copolymer melts, the double-gyroid (DG) is thermodynamically stable, relative to the competitor double-diamond (DD) and double-primitive (DP) phases, and exhibits a window of stability intermediate to the classical lamellar and columnar phases. This competition is widely thought to be controlled by “packing frustration”the incompatibility of uniformly filling melts with a locally preferred chain packing motif. Here, we reassess the thermodynamics of cubic network formation in strongly segregated diblock melts based on a recently developed medial strong segregation theory (mSST) approach that directly connects the shape and thermodynamics of chain packing environments to the medial geometry of tubular network surfaces. We first show that medial packing significantly relaxes prior SST upper bounds on the free energy of network phases, which we attribute to the spreading of terminal chain ends within network nodal regions. By exploring geometric and thermodynamic metrics of chain packing in network phases, we show that mSST reproduces effects dependent on the elastic asymmetry of the blocks that are consistent with SCFT at large χN. We then characterize geometric frustration in terms of the spatially variant distributions of local entropic and enthalpic costs throughout the morphologies, extracted from mSST predictions. By analyzing these distributions, we found that the DG morphology, due to its unique medial geometry in the nodal regions, is stabilized by the incorporation of favorable, quasi-lamellar packing over much of its morphology, motifs which are inaccessible to DD and DP morphologies due to “interior corners” in their medial geometries. Finally, we use our results to analyze “hot spots” of chain stretching and discuss implications for network susceptibility to the uptake of guest molecules