LCP droplet dispersions: a two-phase, diffuse-interface kinetic theory and global droplet defect predictions

Advanced, predictive numerical tools for liquid crystalline polymers (LCPs) are well established. These tools yield descriptive features (steady and unsteady behavior, gradient morphology) of LCPs in the bulk and in confined geometries; some are applicable to LCP hydrodynamics. Our goal here is to expand the hydrodynamic toolkit (a predictive theory, algorithms, and simulations) to two-phase fluids consisting of an immiscible LCP phase (initialized as droplets) dispersed in a viscous or polymeric phase. To do so, we bridge the modeling and simulation gap between molecular dynamics and Monte Carlo discrete lattice models for polymer-dispersed liquid crystal (LC) droplets and continuum scale, diffuse interface methods for LC droplets in viscous fluids. We give a brief overview of molecular-to-continuum scales of modeling and simulations for single LCP phases, and of their relative benefits and limitations. We likewise recall the sharp versus diffuse interface methods for coupling of two immiscible fluid phases in a hydrodynamic theory. From this assessment, we present a kinetic scale, diffuse interface strategy for hydrodynamic simulations of LCP droplet dispersions. The Fokker–Planck or Smoluchowski equation due to Doi and Hess for the orientational distribution function of the LCP phase is generalized to a two-phase, energy-based theory. Although feasible for simulations by adaptation of existing kinetic solvers, we first put the two-phase kinetic model through analogous preliminary benchmarks that were applied to the Doi–Hess theory for LCPs. We project onto the scalar zeroth moment and tensorial second-moment of the rod number density function to achieve a generalized Landau-deGennes two-phase model for equilibria and imposed shear of LCP droplets in a viscous fluid. A numerical algorithm in two space dimensions is developed and implemented to study global droplet defect structure versus interfacial anchoring energy. Simulations reveal topological defects (surface boojums for tangential anchoring and interior degree 1/2 or radial defects for normal anchoring) and details of their respective cores. These predictions support the hypothesis in [I. Lin et al., Science, 2011, 332, 1297–1300] that anchoring boundary conditions between the LCP and surrounding fluid phase can be satisfied by the formation of extremely localized defect cores which self-organize the orientational order external to the cores, without the need for bend-splay-twist mediated, continuous elastic strain.