Modeling And Numerical Simulations Of Active Liquid Crystals

We systematically derive a continuum model for active liquid crystals that is consistent with thermodynamics and is capable of describing polar as well as apolar active liquid crystals. We then focus on one of its limits, a model for polar active liquid crystals. With this model, we explore its linear stability properties near constant steady states, flow-driven, and flow-activity coupled dynamics in1-D and2-D spatial domain as well as in a free surface jet geometry.Firstly, we carry out a linear stability analysis about constant steady states to study near hydrodynamic equilibrium dynamics, revealing long-wave instabil-ity inherent in this model system and how the active parameters in the model affect the instability. We then study model predictions for1-D spatial-temporal structures of active liquid crystals in a channel subject to physical boundary con-ditions. We discuss the model prediction in two selected regimes, one is the viscous stress dominated regime, also known as the flow-driven regime, while the other is the full regime, in which all active mechanisms are included. In the viscous stress dominated regime, the polarity vector is driven by the prescribed flow field. Dynamics depend sensitively on the physical boundary condition and the type of the driven flow field. Bulk-dominated temporal periodic states and spatially homogeneous states are possible under weak anchoring conditions while spatially inhomogeneous states exist under strong anchoring conditions. In the full model regime, flow-orientation interaction generates a host of planar as well as out-of-plane spatial-temporal structures related to the spontaneous flows due to the molecular self-propelled motion; these results provide contact with the recent literature on active nematic suspensions.In addition, symmetry breaking patterns emerge as an additional viscosity due to the polarity vector is included in the force balance. A rich set of mechanisms for generating and limiting the flow structures as well as the spatial-temporal structures predicted by the model are displayed. An asymptotic analysis is carried out to investigate the onset of the asymmetric pattern in addition to the numerical investigation.Next, we devise a numerical scheme to calculate various patterns and their dependence on model parameters in2spatial dimensions. We use two types of boundary conditions, the half periodic and half physical BC as well as the full physical BC. For the system with half periodic and half physical BC, we obtain solutions which are homogeneous in the periodic direction, consistent with those in the1-D simulations, and in addition identify some2-D spatial structures different from the results in the1-D simulations. For the system with the full physical BC, the stronger activity is needed to induce the spontaneous flow caused by instability. We give a linear stability analysis for three defect patterns, known as aster, vortex and spiral, and devise appropriate initial states to obtain their numerical simulations. We then study the three defect patterns:aster, vortex and spiral and their stability in2-D and3-D analytically and numerically.In the end, we study the linear stability of an infinitely long, axisymmetric, cylindrical active liquid crystal (ALC) jet in a passive isotropic fluid matrix using the polar active liquid crystal model. We identify three possible unstable modes as the result of flow and active motion interaction. The first unstable mode is related to the polarity vector instability when coupled to the flow field at the presence of molecular activity. It can be traced back to the single mode, inherent polarity vector instability in a bulk active liquid crystal flow. However, it can be amplified in the ALC jet to demonstrate up to infinitely many unstable modes when the long range elastic interaction is weak in certain parameter regimes; it can also be suppressed in other parameter regimes completely. The second unstable mode is related to the classical capillary or Rayleigh instability, which exists in a finite wave interval [0,kcutoff]. The new feature for this instability lies in the curoff wave number (kcutoff) dependence on the activity of the active matter system. For ALC jets with sufficiently strong contractile activity, the instability can be completely suppressed. The third unstable mode is due to the active viscous stress. This unstable mode can emerge in the intermediate wave number regime at sufficiently strong active viscosity and even extend all the way to the zero wave number limit when the other unstable modes are absent. It can also be suppressed in the regime of weak active viscous stress. At any given values of the model parameters, the three types of instabilities can show up either individually or in certain combinatoric pattern, or be completely suppressed. We discuss the positive growth rates associated with each type of instabilities, windows of instability and their dependence on model parameters through extensive numerical computations aided by asymptotic analysis.In the last chapter, we summary the comprehensive study and discuss the next phase of modeling and simulations for active liquid crystals in the near future.