Our research is of theoretical and computational nature, and focuses on mesoscale theories of nonequilibrium phenomena in extended systems, with applications to Statistical Mechanics, Soft Matter Physics, and Materials Science. Generically, systems outside of thermodynamic equilibrium involve unstable interfaces and moving topological defects. We develop theories at the mesoscale of the relevant phases, analyze their macroscopic scale asymptotics, and obtain their non equilibrium properties through large scale computation. Our research is currently motivated by topologically driven flows in liquid crystals, unstable interfaces in complex fluids, and a mesoscale description of plasticity in defected materials.

### Smectic film Instabilities

A smectic is a complex fluid phase of a liquid crystal that exhibits broken translational symmetry in only one direction (a one dimensional solid or a layered phase). Interfaces separating smectic and isotropic phases can be quite complex, such as the array of focal conics shown left. Equilibrium interfacial shape depends on the complex interplay between mean and Gaussian curvatures, and bending and torsion energies. These are effects largely absent in normal fluid or solid phases. We are investigating an order parameter model of the smectic-isotropic system that can describe the equilibrium thermodynamics of its complex interfacial shapes as a function of intrinsic metric properties of the interface. The same model can then be extended to describe surface instabilities and the ensuing pattern formation. The model is especially useful in describing instabilities in which the topology of the domains change as a function of time. The figure at left shows an evolving surface pattern in three dimensions from an initial toroidal focal domain. New smectic layers are being formed at the center of the domain by layer growth from the isotropic phase.

### Defect motion and plasticity at the mesoscale

A current major objective in Materials Science, “reversing the arrow: materials by design”, calls for the development of theoretical and computational inverse methods to design materials that have specified properties and function. Such an inverse optimization problem requires robust solutions to the forward problem involved in the determination of macroscopic properties as a function of the nano and mesoscopic properties and composition of any given material. Although great progress has been achieved in the inverse design of materials close to thermodynamic equilibrium, most materials, are rarely uniform, but rather exhibit complex and often evolving microstructure. Therefore the standard tools of Solid State Physics and equilibrium Statistical Mechanics do not generally apply. In fact, the strong response of non equilibrium phases to external stresses and perturbations means that their properties strongly depend on interactions with boundaries, preparation methods, and prior history. The aim of our work is to establish the theoretical framework within which to predict structure evolution in non-equilibrium solids, and thus to develop the foundation of reliable forward models of processing-structure-property relationships.

For a defected solid, we adopt a coarse-grained or continuum description of the defect density that encodes the topological content of a given element of volume [PE16]. Phenomenological equations of motion that respect topological constraints are developed to describe the temporal evolution of an ensemble of interacting defects. It is possible to reformulate these equations at a more microscopic level by introducing Hamiltonians that allow for defected ground states, and that can lead to macroscopic equations of defect motion while still resolving mesoscopic defect cores and their evolution [AS18]. The figure shows amplitudes and phases of a dislocation dipole in a hexagonal lattice. Two of the three wave amplitudes decay to zero in the defect core region, whereas the phase exhibit a discontinuity along a line joining the core. These solutions are not imposed as in classical treatments; rather they appear as a direct solution of the mesoscopic theory.

### Topology driven flows in nematic fluids

The study of of liquid crystal based suspensions is motivated by applications in materials science as well as in biological systems. At a fundamental level, and in contrast with normal suspending fluids, nematic order in a liquid crystalline matrix leads to long range elastic interactions either among colloidal particles or with bounding walls, resulting in a variety of unexpected phenomena. Furthermore, long range order in the matrix is distorted by the suspended particles, resulting in anisotropic response, and unavoidable topological defects that must move with the particles. On the one hand, the existence of structure in the liquid matrix affords new opportunities for flow control, processing, and suspension stability. At the same time, and for the same reasons, efficient engineering of these systems requires major advances to our current understanding of nematic fluid colloids.

The figure above (left) shows the spatial charge distribution induced by a (+1, -1) disclination pair under a uniform but oscillatory electric field normal to the line joining the defects. A (+1) disclination represents a suspended particle with homeotropic anchoring, and the (-1) disclination is the associated hyperbolic hedgehog. Both defects lead to spatial charge separation which, in turn, results in *streaming* flows (figure on the right), resulting in transport normal to the direction of the applied field [CO17]. Thus a nematic matrix allows nonlinear anisotropic hydrodynamic mobilities. Effects like this one are being developed to engineer designer flows for use in microfluidic devices, including particle sorting and aggregation (including cell sorting), and stirring and the micro scale [PE15]. The same effects are being investigated to control, dynamically and on demand, the motion of self-propelled (or active) particles such as swimming bacteria in a nematic matrix [CO18].