Negative stiffness and modulated states in active nematics

Abstract

We examine the dynamics of an active nematic liquid crystal on a frictional substrate. When frictional damping dominates over viscous dissipation, we eliminate flow in favor of active stresses to obtain a minimal dynamical model for the nematic order parameter, with elastic constants renormalized by activity. The renormalized elastic constants can become negative at large activity, leading to the selection of spatially inhomogeneous patterns via a mechanism analogous to that responsible for modulated phases arising at an equilibrium Lifshitz point. Tuning activity and the degree of nematic order in the passive system, we obtain a linear stability phase diagram that exhibits a nonequilibrium tricritical point where ordered, modulated and disordered phases meet. Numerical solution of the nonlinear equations yields a succession of spatial structures of increasing complexity with increasing activity, including kink walls and active turbulence, as observed in experiments on microtubule bundles confined at an oil-water interface. Our work provides a minimal model for an overdamped active nematic that reproduces all the nonequilibrium structures seen in simulations of the full active nematic hydrodynamics and provides a framework for understanding some of the mechanisms for selection of the nonequilibrium patterns in the language of equilibrium critical phenomena.Active systems are continuously driven out of equilibrium by energy injected at the local scale and generate collective motion at the large scale. Examples include bacterial colonies, in vitro extracts of cytoskeletal filaments and motor proteins, and living cells 1 . Elongated active particles can order in states with liquid crystalline order. The emergent behavior of such 'living' liquid crystals has been described using liquid crystal hydrodynamics, modified to include the local energy input from active processes 2,3 . This work has highlighted a rich variety of collective phenomena, including spontaneous laminar flow, pattern formation, spontaneous unbinding of topological defects, and active turbulence 4-13 . Previous studies have established that bulk active nematics are generically unstable 1,3,14,15 . Confinement, on the other hand, has profound effects on active fluids. It damps the flow 16,17 and suppresses the generic instability of unbound systems, resulting in a finite activity threshold for the onset of spontaneously flowing states 8,18 . Frictional damping due to confinement plays a key role in experimental realizations of active systems. In bacterial suspensions channel confinement was shown to stabilize vortex In confined nematics the energy input from active stresses is dissipated both via viscous flows that mediate hydrodynamic interactions between the active units and via friction with a substrate. Systems where viscous dissipation dominates so that frictional damping is negligible and momentum is conserved are referred to as 'wet', while those where dissipation is mainly controlled by friction are referred to as 'dry'. Previous work has ex...

Onset of meso-scale turbulence in active nematics

“…From the lattice Boltzmann simulations, we find that the rate of puff formation is (0.54±0.10) × 10 −7 ∼10 −7 , which is the value used in the main text. Although P c moves the system slightly away from the critical point, the directed percolation scaling exponents are not observed to change and are known to be independent of P 0 in this weak field limit32. Measuring and N || , as for the lattice Boltzmann simulations, supplies the critical exponents reported in the main text.…”

“…The emergence of the intermediate vortex lattice in active matter has been observed experimentally in motility assays of microtubles28, in bacterial suspension in a channel confinement6, and also numerically by short-range attraction of self-propelled particles29 and hydrodynamic screening of activity-induced flows due to frictional damping30. To focus on the effect of the confining channel on the transition to meso-scale turbulence, we consider the ideal ‘wet' limit and neglect additional frictional damping113132. This is because in the experimental systems studied so far, there is no obvious qualitative effect on the active turbulence, friction appears to be a small effect, and the meso-scale turbulent state we consider here is unaffected261011.…”

Hydrodynamics of Active Defects: From Order to Chaos to Defect Ordering

“…This state, which we refer to as "polar defect order", has no giant fluctuations in either the defect charge or number density and provides an intriguing realization of a "Malthusian defect flock". While polar defect order has been reported before in numerical models of active nematics [32][33][34][35], the mechanism driving it has remained unexplained. Our work identifies a mechanism for polar order as arising from both active self-aligning torques (derived perturbatively in activity in Ref.…”

Low-noise phase of a two-dimensional active nematic system

“…This would signal a modulational instability, possibly giving rise to a smectic array of bend-splay distortions, about which one would have to reorganize the low noise fluctuation expansion, far beyond the scope of this paper. Note that unlike the linear Lifshitz instability prediction for an overdamped 2d active nematic without a conserved density at the mean field level [47], here the theory is linearly stable to begin with and only destabilized nonlinearly in the presence of noise.…”

“…These stability lines correspond to splay-bend instabilities that have a finite threshold due to the presence of a frictional substrate and have been extensively studied (see for instance Refs. [46,47] and reference therein), so we shall not discuss them any further. Note that, as we are deep in the ordered phase, we do not concern ourselves with the density banding instability which only occurs near the mean-field transition.…”