Anisotropic microstructure-induced reduction of the Rayleigh instability for liquid crystalline polymers

Abstract

We analyze a macroscopic 3D model for flows of liquid crystalline polymers (LCPs), deduced from Doi-type [3,4] kinetic equations. The Doi model accounts for rigid-rod microstructure, which introduces elastic relaxation and polymer-induced viscosity in addition to a Newtonian solvent viscosity, thus capturing all effects contained in standard isotropic viscoelastic models for Maxwell and Oldroyd B fluids. The rod-like microstructure further introduces anisotropic effects in the form of drag on the rods, together with a short-range, Maier-Saupe intermolecular potential, whose critical points vary with LCP concentration and yield stable isotropic (at low density) and nematic (at high density) equilibrium phases. From this single model, we compare various physical mechanisms for reducing the capillary instability of inviscid cylindrical jets: solvent viscosity as studied by Rayleigh and Chandrasekhar; isotropic viscoelasticity, both with and without Newtonian solvent viscosity; anisotropic polymer friction; and finally, the nematic, highly aligned prolate phase at high LCP density. Realistic parameter values for LCPs correspond to a regime in which the LCP capillary number (polymer bulk free energy relative to surface tension) is above an identified critical value; in such regimes, the unstable growth rates of the isotropic and nematic phases are lowered arbitrarily close to zero if the molecular drag is sufficiently anisotropic even in the absence of solvent viscosity. In low capillary number regimes, where surface tension dominates LCP bulk free energy, the LCP growth rates are sandwiched below the inviscid Rayleigh curve and above an explicit positive lower bound.