Folding pathways to crumpling in thermalized elastic frames.

The mechanical properties of thermally excited two-dimensional crystalline membranes can depend dramatically on their geometry and topology. A particularly relevant example is the effect on the crumpling transition of holes in the membrane. Here we use molecular dynamics simulations to study the case of elastic frames (sheets with a single large hole in the center) and find that the system approaches the crumpled phase through a sequence of origami-like folds at decreasing length scales when temperature is increased. We use normal-normal correlation functions to quantify the temperature-dependent number of folds.

Shape-Shifting Droplet Networks.

We consider a three-dimensional network of aqueous droplets joined by single lipid bilayers to form a cohesive, tissuelike material. The droplets in these networks can be programed to have distinct osmolarities so that osmotic gradients generate internal stresses via local fluid flows to cause the network to change shape. We discover, using molecular dynamics simulations, a reversible folding-unfolding process by adding an osmotic interaction with the surrounding environment which necessarily evolves dynamically as the shape of the network changes. This discovery is the next important step towards osmotic robotics in this system. We also explore analytically and numerically how the networks become faceted via buckling and how quasi-one-dimensional networks become three dimensional.

Viral nematics in confined geometries.

Motivated by recent experiments on the rod-like virus bacteriophage fd, confined to circular and annular domains, we present a theoretical study of structural transitions in these geometries. Using the continuum theory of nematic liquid crystals, we examine the competition between bulk elasticity and surface anchoring, mediated by the formation of topological defects. We show analytically that bulk defects are unstable with respect to defects sitting at the boundary. In the case of an annulus, whose topology does not require the presence of topological defects, we find that nematic textures with boundary defects are stable compared to defect-free configurations when the anchoring is weak. Our simple approach, with no fitting parameters, suggests a possible symmetry breaking mechanism responsible for the formation of one-, two- and three-fold textures under annular confinement.

Forming a cube from a sphere with tetratic order.

Composed of square particles, the tetratic phase is characterized by a fourfold symmetry with quasi-long-range orientational order but no translational order. We construct the elastic free energy for tetratics and find a closed form solution for ±1/4 disclinations in planar geometry. Applying the same covariant formalism to a sphere, we show analytically that within the one elastic constant approximation eight +1/4 disclinations favor positions defining the vertices of a cube. The interplay between defect-defect interactions and bending energy results in a flattening of the sphere towards superspheroids with the symmetry of a cube.

On the modeling of endocytosis in yeast.

The cell membrane deforms during endocytosis to surround extracellular material and draw it into the cell. Results of experiments on endocytosis in yeast show general agreement that 1) actin polymerizes into a network of filaments exerting active forces on the membrane to deform it, and 2) the large-scale membrane deformation is tubular in shape. In contrast, there are three competing proposals for precisely how the actin filament network organizes itself to drive the deformation. We use variational approaches and numerical simulations to address this competition by analyzing a meso-scale model of actin-mediated endocytosis in yeast. The meso-scale model breaks up the invagination process into three stages: 1) initiation, where clathrin interacts with the membrane via adaptor proteins; 2) elongation, where the membrane is then further deformed by polymerizing actin filaments; and 3) pinch-off. Our results suggest that the pinch-off mechanism may be assisted by a pearling-like instability. We rule out two of the three competing proposals for the organization of the actin filament network during the elongation stage. These two proposals could be important in the pinch-off stage, however, where additional actin polymerization helps break off the vesicle. Implications and comparisons with earlier modeling of endocytosis in yeast are discussed.

Vacancy localization in the square dimer model.

We study the classical dimer model on a square lattice with a single vacancy by developing a graph-theoretic classification of the set of all configurations which extends the spanning tree formulation of close-packed dimers. With this formalism, we can address the question of the possible motion of the vacancy induced by dimer slidings. We find a probability 57/4-10[sqrt2] for the vacancy to be strictly jammed in an infinite system. More generally, the size distribution of the domain accessible to the vacancy is characterized by a power law decay with exponent 9/8. On a finite system, the probability that a vacancy in the bulk can reach the boundary falls off as a power law of the system size with exponent 1/4. The resultant weak localization of vacancies still allows for unbounded diffusion, characterized by a diffusion exponent that we relate to that of diffusion on spanning trees. We also implement numerical simulations of the model with both free and periodic boundary conditions.

Grain boundary scars and spherical crystallography.

We describe experimental investigations of the structure of two-dimensional spherical crystals. The crystals, formed by beads self-assembled on water droplets in oil, serve as model systems for exploring very general theories about the minimum-energy configurations of particles with arbitrary repulsive interactions on curved surfaces. Above a critical system size we find that crystals develop distinctive high-angle grain boundaries, or scars, not found in planar crystals. The number of excess defects in a scar is shown to grow linearly with the dimensionless system size. The observed slope is expected to be universal, independent of the microscopic potential.

Crystalline order on a sphere and the generalized Thomson problem.

We attack the generalized Thomson problem, i.e., determining the ground state energy and configuration of many particles interacting via an arbitrary repulsive pairwise potential on a sphere via a continuum mapping onto a universal long range interaction between angular disclination defects parametrized by the elastic (Young) modulus Y of the underlying lattice and the core energy E(core) of an isolated disclination. Predictions from the continuum theory for the ground state energy agree with numerical simulations of long range power law interactions of the form 1/r(gamma) (0

Universal negative poisson ratio of self-avoiding fixed-connectivity membranes.

We determine the Poisson ratio of self-avoiding fixed-connectivity membranes, modeled as impenetrable plaquettes, to be sigma = -0.37(6), in statistical agreement with the Poisson ratio of phantom fixed-connectivity membranes sigma = -0.32(4). Together with the equality of critical exponents, this result implies a unique universality class for fixed-connectivity membranes. Our findings thus establish that physical fixed-connectivity membranes provide a wide class of auxetic (negative Poisson ratio) materials with significant potential applications in materials science.

Tubular phase of self-avoiding anisotropic crystalline membranes.

We analyze the tubular phase of self-avoiding anisotropic crystalline membranes. A careful analysis using renormalization group arguments together with symmetry requirements motivates the simplest form of the large-distance free energy describing fluctuations of tubular configurations. The non-self-avoiding limit of the model is shown to be exactly solvable. For the full self-avoiding model we compute the critical exponents using an epsilon expansion about the upper critical embedding dimension for general internal dimension D and embedding dimension d. We then exhibit various methods for reliably extrapolating to the physical point (D=2,d=3). Our most accurate estimates are nu=0.62 for the Flory exponent and zeta=0.80 for the roughness exponent.

The cosmological kibble mechanism in the laboratory: string formation in liquid crystals.

The production of strings (disclination lines and loops) has been observed by means of the Kibble mechanism of domain (bubble) formation in the isotropic-nematic phase transition of the uniaxial nematic liquid crystal 4-cyano-4'-n-pentylbiphenyl. The number of strings formed per bubble is about 0.6. This value is in reasonable agreement with a numerical simulation of the experiment in which the Kibble mechanism is used for the order parameter space of a uniaxial nematic liquid crystal.