Computational studies of the phase behavior and dynamics of rodlike liquid crystals

Abstract

In this thesis, the unusual physics of rodlike liquid crystals are explored through simulations of rigid-rod kinetic theory discretized by the finite element method. As their name suggests. these substances retain crystalline. anisotropic microstructure but deform as a liquid: the microstructure results from the anisotropic shape and interactions of the constituent molecules. The molecules are modeled as Brownian. interacting, rigid rods in a kinetic theory formulation. A solution of such rods undergoes a phase transition from a disordered (isotropic) state to an ordered (nematic) state as the density of rods is increased: such a solution also exhibits interesting, complex. non-Newtonian rheological behavior. Liquid-crystalline substances are used in a number of high-performance industrial applications because the aligned, liquid-crystalline structure yields superior mechanical properties in the final product. The properties can be compromised by any interfaces or defects present in the system. The processing of these systems is poorly understood, and further progress requires accurate modeling of the coupled evolution of the microstructure and non-Newtonian flow field. The main goal of this thesis is the simulation of liquid-crystalline phenomena featuring sharp nonhomogeneities in structure on the length scale of a single rod. Such critical phenomena include the interfaces and defects that are prevalent in isotropic-nematic coexistence, phase transitions. aligning boundaries such as walls. and nonhomogeneous flows. In order to capture this level of detail, simulations evolve the rod distribution function, which describes the distribution of rod positions and orientations, coupled with the velocity field. The evolution equation for the distribution function is known as the Doi diffusion equation.

Prior studies of rigid rod behavior have typically avoided this level of detail: instead. a series of approximations and assumptions have been used to simplify the diffusion equation and avoid evolving the distribution function since it is a quantity that varies in both spatial dimensions and orientation dimensions. These prior studies evolve the order tensor (the second moment of the distribution function) and invoke closure approximations to obtain a closed form for an evolution equation for this tensor. However, these techniques suffer from inaccuracy, often of unknown magnitude, because of the closure approximations, and they lose the ability to describe phenomena where gradients in structure on the length scale of a single rod are important or where variations in density are important. This thesis avoids these approximations and their attending limitations. A parallel, finite-element method is used to discretize the rod distribution function and the rod interaction potential. However, these simulations differ from traditional finite-element-based simulations of complex fluids because of the coupling between the physical space for rod location and orientation space for rod alignment and because of the difficulty of computing extended rod-rod interactions in a tractable manner. Instead of approximating the rod-rod interaction potential via Taylor expansions as previous studies have done, novel numerical methods are developed to meet these challenges and allow rapid, parallel computation of the full, nonhomogeneous Doi diffusion equation. These methods are applied to the equilibrium phase behavior of solutions of rigid rods in both periodic systems and system bounded by hard walls. Newton's method is used to solve the nonlinear set of equations for extrema of the system free energy, and the properties of the Jacobian matrix describe a local free energy surface.

Simulation results show that periodic, isotropic-nematic coexistence states are stable in a given concentration range; the bulk phase properties and interfacial properties are independent of the average concentration or size of the system if the solution is stable. Results for confined systems show a number of surprising results when hard wall boundary conditions are used; in particular, the phase behavior of an extremely large system bounded by walls does not converge to the phase behavior of a homogeneous system without bounds, because the walls fundamentally alter the free energy surface of the system. Spinodal decomposition, the phase transition from an unstable isotropic state to a stable nematic state, is also studied. Prior studies of this process have been limited to linear stability analyses and have yielded sharp conflicts about the dominant mechanism of the process. The linear stability analysis is re-examined and prior conflicts are resolved by numerically showing that the process mechanism is a strong function of the system concentration and diffusivities for rotational and translational rod motion. A semi-implicit, finite-element-based time stepper is then used for full dynamic simulations of the spinodal decomposition process. These simulations mark the first results for the nonhomogeneous Doi diffusion equation in the spinodal decomposition process, and the results show how restricted rod motion can cause the process to become kinetically trapped in nonhomogeneous intermediate states. The method is also applied to the related problem of coarsening between large nematic domains of aligned rods, and results show the effects of domain misalignment on the final structure. This dynamic method is then extended to simulations of rigid rods in wall-driven, rectilinear shear flow and pressure-driven flow between parallel plates. These simulations mark the first results for the full nonhomogeneous Doi diffusion equation in nonhomogeneous shear flow.

Silmulation results show that the accurate treatment of wall-rod interactions is critically important: aphysical anchoring conditions at walls can suppress out-of-plane instabilities in planar shear flow. Also. simulations of pressure-driven flow show that misaligned domains marked by "'twist" interfaces can spontaneously form as a result in gradients in the shear rate: this concept lends important insight into the formation of domains in macroscopic rheological experiments. Results also show that miisaligned. tumnbling domains are not stable. Phase diagrarns for wall-driven. rectilinear shear flow and pressure-driven. rectilinear shear flow are given for varying shear rate and system size.