Mark J. Bowick

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.

Topology and dynamics of active nematic vesicles

Liquid crystals on a deformable substrate The orientation of the molecules in a liquid crystalline material will change in response to either changes in the substrate or an external field. This is the basis for liquid crystalline devices. Vesicles, which are fluid pockets surrounded by lipid bilayers, will change size or shape in response to solvent conditions or pressure. Keber et al. report on the rich interactions between nematic liquid crystals placed on the surface of a vesicle. Changes to the vesicle size, for example, can “tune” the liquid crystal molecules. But conversely, the shape of the vesicles can also change in response to the activity of the nematic molecules. Science, this issue p. 1135 Dynamical shape-changing materials result from merging active liquid crystals with soft deformable vesicles. Engineering synthetic materials that mimic the remarkable complexity of living organisms is a fundamental challenge in science and technology. We studied the spatiotemporal patterns that emerge when an active nematic film of microtubules and molecular motors is encapsulated within a shape-changing lipid vesicle. Unlike in equilibrium systems, where defects are largely static structures, in active nematics defects move spontaneously and can be described as self-propelled particles. The combination of activity, topological constraints, and vesicle deformability produces a myriad of dynamical states. We highlight two dynamical modes: a tunable periodic state that oscillates between two defect configurations, and shape-changing vesicles with streaming filopodia-like protrusions. These results demonstrate how biomimetic materials can be obtained when topological constraints are used to control the non-equilibrium dynamics of active matter.

Two-dimensional matter: order, curvature and defects

Many systems in nature and the synthetic world involve ordered arrangements of units on two-dimensional surfaces. We review here the fundamental role payed by both the topology of the underlying surface and its Gaussian curvature. Topology dictates certain broad features of the defect structure of the ground state but curvature-driven energetics control the detailed structure of the ordered phases. Among the surprises are the appearance in the ground state of structures that would normally be thermal excitations and thus prohibited at zero temperature. Examples include excess dislocations in the form of grain boundary scars for spherical crystals above a minimal system size, dislocation unbinding for toroidal hexatics, interstitial fractionalization in spherical crystals and the appearance of well-separated disclinations for toroidal crystals. Much of the analysis leads to universal predictions that do not depend on the details of the microscopic interactions that lead to order in the first place. These predictions are subject to test by the many experimental soft- and hard-matter systems that lead to curved ordered structures such as colloidal particles self-assembling on droplets of one liquid in a second liquid. The defects themselves may be functionalized to create ligands with directional bonding. Thus, nano- to meso-scale superatoms may be designed with specific valency for use in building supermolecules and novel bulk materials. Parameters such as particle number, geometrical aspect ratios and anisotropy of elastic moduli permit the tuning of the precise architecture of the superatoms and associated supermolecules. Thus, the field has tremendous potential from both a fundamental and materials science/supramolecular chemistry viewpoint.

The statistical mechanics of membranes

Abstract The fluctuations of two-dimensional extended objects (membranes) is a rich and exciting field with many solid results and a wide range of open issues. We review the distinct universality classes of membranes, determined by the local order, and the associated phase diagrams. After a discussion of several physical examples of membranes we turn to the physics of crystalline (or polymerized) membranes in which the individual monomers are rigidly bound. We discuss the phase diagram with particular attention to the dependence on the degree of self-avoidance and anisotropy. In each case we review and discuss analytic, numerical and experimental predictions of critical exponents and other key observables. Particular emphasis is given to the results obtained from the renormalization group e-expansion. The resulting renormalization group flows and fixed points are illustrated graphically. The full technical details necessary to perform actual calculations are presented in the Appendices. We then turn to a discussion of the role of topological defects whose liberation leads to the hexatic and fluid universality classes. We finish with conclusions and a discussion of promising open directions for the future.

The Holomorphic Geometry of Closed Bosonic String Theory and Diff S1 / S1

Abstract We present a proposal for a classical non-perturbative bosonic closed string field theory based on Kahler geometry. Motivated by the observation that the loop space of Minkowski space-time is a Kahler manifold, we conjecture that infinite-dimensional complex (Kahler) geometry is the right setting for closed string field theory and that the correct dynamical variable (closed string field) is the Kahler potential. To incorporate reparametrization invariance, one must consider the space of complex structures Diff S1/S1. Geometrical considerations then lead us to a (non-linear) equation of motion for the Kahler potential which is that the curvature of a certain vector bundle over Diff S1/S1 vanish. This is basically the requirement of conformal invariance. Loops on flat Minkowski space are shown to be a solution only if the space-time dimension is 26. We also discuss geometric quantization since our approach can be viewed as an application of geometric quantization to string theory. Previously announced mathematical results that Diff S1/S1 is a homogeneous Kahler manifold are established in more detail and its curvature is computed explicitly. We also give an axiomatic formulation of the minimal geometric setting we require — this is an attempt to avoid basing the theory on loops of a given riemannian manifold. Einsteins field equations are derived in an adiabatic approximation. The relation of our work to some other approaches to string theory is briefly discussed.

Interacting topological defects on frozen topographies

We propose and analyze an effective free energy describing the physics of disclination defects in particle arrays constrained to move on an arbitrary two-dimensional surface. At finite temperature the physics of interacting disclinations is mapped to a Laplacian sine-Gordon Hamiltonian suitable for numerical simulations. We discuss general features of the ground state and thereafter specialize to the spherical case. The ground state is analyzed as a function of the ratio of the defect core energy to the Youngs modulus. We argue that the core energy contribution becomes less and less important in the limit

TOPOLOGICAL MASS GENERATION IN 3+1 DIMENSIONS

R\ensuremath{\gg}a,

High-temperature strings

where R is the radius of the sphere and a is the particle spacing. For large core energies there are 12 disclinations forming an icosahedron. For intermediate core energies unusual finite-length grain boundaries are preferred. The complicated regime of small core energies, appropriate to the limit

Defect annihilation and proliferation in active nematics.

R/\stackrel{\ensuremath{\rightarrow}}{a}\ensuremath{\infty},

Crystalline order on a sphere and the generalized Thomson problem

is also addressed. Finally we discuss the application of our results to the classic Thomson problem of finding the ground state of electrons distributed on a two sphere.