Abstract
Active nematics are a class of nonequilibrium systems which have received much attention\r in the form of continuum models in recent years. For the dense, highly ordered case which is\r of particular interest, these models focus almost exclusively on suspensions of active particles\r in which the flow of the medium plays a key role in the dynamical equations. Many active\r nematics, however, reside at an interface or on a surface where friction excludes the effects\r of long-range flow. In the following pages we shall construct a general model which describes\r these systems with overdamped dynamical equations. Through numerical and analytical\r investigation we detail how many of the striking nonequilibrium behaviors of active nematics\r arise in such systems.\r \r We shall first discuss how the activity in these systems gives rise to an instability in\r the nematic ordered state. This instability leads to phase-separation in which bands of\r ordered active nematic are interspersed with bands of the disordered phase. We expose the\r factors which control the density contrast and the stability of these bands through numerical\r investigation.\r \r We then turn to the highly ordered phase of active nematic materials, in which striking\r nonequilibrium behaviors such as the spontaneous formation, self-propulsion, and ordering\r of charge-half defects occurs. We extend the overdamped model of an active nematic to\r describe these behaviors by including the advection of the director by the active forces in\r the dynamical equations. We find a new instability in the ordered state which gives rise to\r defect formation, as well as an analog of the instability which is seen in models of active\r nematic suspensions. Through numerical investigations we expose a rich phenomenology in\r the neighborhood of this new instability. The phenomenology includes a state in which the\r orientations of motile, transient defects form long-range order. This is the first continuum\r model to contain such a state, and we compare the behavior seen here with similar states\r seen in the experiments and simulations of Stephen DeCamp and Gabriel Redner et. al. [1]\r \r Finally, we propose the measurement of defect shape as a mechanism for the comparison\r between continuum theories of active nematics and the experimental and simulated realiza-\r tions of these systems. We present a method for making these measurements which allows\r for averaging and statistical analysis, and use this method to determine how the shapes of\r defects depend on the parameters of our continuum theory. We then compare these with the\r shapes of defects which we measure in the experiments and simulations mentioned above in\r order to place these systems in the parameter space of our model. It is our hope that this\r mechanism for comparison between models and realizations of active nematics will provide\r a key to pairing the two more closely.