Gregory M. Grason

Geometric theory of diblock copolymer phases.

We analyze the energetics of spherelike micellar phases in diblock copolymers in terms of well-studied, geometric quantities for their lattices. We argue that the A15 lattice with Pm3;n symmetry should be favored as the blocks become more symmetric and corroborate this through a self-consistent field theory. Because phases with columnar or bicontinuous topologies intervene, the A15 phase, though metastable, is not an equilibrium phase of symmetric diblocks. We investigate the phase diagram of branched diblocks and find that the A15 phase is stable.

Soft spheres make more mesophases

We use both mean-field methods and numerical simulation to study the phase diagram of classical particles interacting with a hard-core and repulsive, soft shoulder. Despite the purely repulsive interaction, this system displays a remarkable array of aggregate phases arising from the competition between the hard-core and shoulder length scales. In the limit of large shoulder width to core size, we argue that this phase diagram has a number of universal features, and classify the set of repulsive shoulders that lead to aggregation at high density. Surprisingly, the phase sequence and aggregate size adjusts so as to keep almost constant inter-aggregate separation.

Chirality and Equilibrium Biopolymer Bundles

We use continuum theory to show that chirality is a key thermodynamic control parameter for the aggregation of biopolymers: chirality produces a stable disperse phase of hexagonal bundles under moderately poor solvent conditions, as has been observed in in vitro studies of F actin [O. Pelletier et al., Phys. Rev. Lett. 91, 148102 (2003)]. The large characteristic radius of these chiral bundles is not determined by a mysterious long-range molecular interaction but by in-plane shear elastic stresses generated by the interplay between a chiral torque and an unusual, but universal, nonlinear gauge term in the strain tensor of ordered chains that is imposed by rotational invariance.

ABC triblock copolymer vesicles with mesh-like morphology.

Polymer vesicles made from poly(isoprene-b-styrene-b-2-vinyl pyridine) (PI-b-PS-b-P2VP) triblock copolymer confined within the nanopores of an anodic aluminum oxide (AAO) membrane are studied. It was found that these vesicles have well-defined, nanoscopic size, and complex microphase-separated hydrophobic membranes, comprised of the PS and PI blocks, while the coronas are formed by the P2VP block. Vesicle formation was tracked using both transmission and scanning electron microscopy. A mesh-like morphology formed in the membrane at a well-defined composition of the three blocks that can be tuned by changing the copolymer composition. The nanoscale confinement, copolymer composition, and subtle molecular interactions contribute to the generation of these vesicles with such unusual morphologies.

Interfaces in Diblocks: A Study of Miktoarm Star Copolymers

We study ABn miktoarm star block copolymers in the strong segregation limit, focusing on the role that the AB interface plays in determining the phase behavior. We develop an extension of the kinked-path approach which allows us to explore the energetic dependence on interfacial shape. We consider a one-parameter family of interfaces to study the columnar to lamellar transition in asymmetric stars. We compare with recent experimental results. We discuss the stability of the A15 lattice of spherelike micelles in the context of interfacial energy minimization. We corroborate our theory by implementing a numerically exact self-consistent-field theory to probe the phase diagram and the shape of the AB interface.

Cooperativity and Frustration in Protein-Mediated Parallel Actin Bundles

We examine the mechanism of bundling of cytoskeletal actin filaments by two representative bundling proteins, fascin and espin. Small-angle x-ray studies show that increased binding from linkers drives a systematic overtwist of actin filaments from their native state, which occurs in a linker-dependent fashion. Fascin bundles actin into a continuous spectrum of intermediate twist states, while espin only allows for untwisted actin filaments and fully overtwisted bundles. Based on a coarse-grained, statistical model of protein binding, we show that the interplay between binding geometry and the intrinsic flexibility of linkers mediates cooperative binding in the bundle. We attribute the respective continuous (discontinuous) bundling mechanisms of fascin (espin) to difference in the stiffness of linker bonds themselves.

Self-consistent field theory of multiply branched block copolymer melts.

We present a numerical algorithm to evaluate the self-consistent field theory for melts composed of block copolymers with multiply branched architecture. We present results for the case of branched copolymers with doubly functional groups for multiple-branching generations. We discuss the stability of the cubic phase of spherical micelles, the A15 phase, as a consequence of the tendency of the AB interfaces to conform to the polyhedral environment of the Voronoi cell of the micelle lattice.

Measuring cohesion between macromolecular filaments one pair at a time: depletion-induced microtubule bundling.

In the presence of nonadsorbing polymers, colloidal particles experience ubiquitous attractive interactions induced by depletion forces. Here, we measure the depletion interaction between a pair of microtubule filaments using a method that combines single filament imaging with optical trapping. By quantifying the dependence of filament cohesion on both polymer concentration and solution ionic strength, we demonstrate that the minimal model of depletion, based on the Asakura-Oosawa theory, fails to quantitatively describe the experimental data. By measuring the cohesion strength in two- and three- filament bundles, we verify pairwise additivity of depletion interactions for the specific experimental conditions. The described experimental technique can be used to measure pairwise interactions between various biological or synthetic filaments and complements information extracted from bulk osmotic stress experiments.

Non-euclidean geometry of twisted filament bundle packing.

Densely packed and twisted assemblies of filaments are crucial structural motifs in macroscopic materials (cables, ropes, and textiles) as well as synthetic and biological nanomaterials (fibrous proteins). We study the unique and nontrivial packing geometry of this universal material design from two perspectives. First, we show that the problem of twisted bundle packing can be mapped exactly onto the problem of disc packing on a curved surface, the geometry of which has a positive, spherical curvature close to the center of rotation and approaches the intrinsically flat geometry of a cylinder far from the bundle center. From this mapping, we find the packing of any twisted bundle is geometrically frustrated, as it makes the sixfold geometry of filament close packing impossible at the core of the fiber. This geometrical equivalence leads to a spectrum of close-packed fiber geometries, whose low symmetry (five-, four-, three-, and twofold) reflect non-Euclidean packing constraints at the bundle core. Second, we explore the ground-state structure of twisted filament assemblies formed under the influence of adhesive interactions by a computational model. Here, we find that the underlying non-Euclidean geometry of twisted fiber packing disrupts the regular lattice packing of filaments above a critical radius, proportional to the helical pitch. Above this critical radius, the ground-state packing includes the presence of between one and six excess fivefold disclinations in the cross-sectional order.

Mesophases of soft-sphere aggregates

Soft spheres interacting via a hard core and a range of attractive and repulsive ‘soft-shoulder’ potentials self-assemble into clusters forming a variety of mesophases. We combine a mean-field theory developed from a lattice model with a level surface analysis of the periodic structures of soft-sphere aggregates to study stable morphologies for all clustering potentials. We develop a systematic approach to the thermodynamics of mesophase assembly in the low-temperature, strong-segregation and predict a generic sequence of phases including lamella, hexagonal-columnar and body-center cubic phases, as well as the associated inverse structures. We discuss the finite temperature corrections to strong segregation theory in terms of Sommerfeld-like expansion and how these corrections affect the thermodynamic stability of bicontinuous mesophase structures, such as gyroid. Finally, we explore the opposite limit of weakly-segregated particles, and predict the generic stability of a bicontinuous cluster morphology within the mean-field phase diagram.