Antipolar ordering of topological defects in active liquid crystals

Abstract

ATP-driven microtubule-kinesin bundles can self-assemble into two-dimensional active liquid crystals (ALCs) that exhibit a rich creation and annihilation dynamics of topological defects, reminiscent of particle-pair production processes in quantum systems. This recent discovery has sparked considerable interest but a quantitative theoretical description is still lacking. We present and validate a minimal continuum theory for this new class of active matter systems by generalizing the classical Landau-de Gennes free-energy to account for the experimentally observed spontaneous buckling of motor-driven extensile microtubule bundles. The resulting model agrees with recently published data and predicts a regime of antipolar order. Our analysis implies that ALCs are governed by the same generic ordering principles that determine the non-equilibrium dynamics of dense bacterial suspensions and elastic bilayer materials. Moreover, the theory manifests an energetic analogy with strongly interacting quantum gases. Generally, our results suggest that complex nonequilibrium pattern-formation phenomena might be predictable from a few fundamental symmetry-breaking and scale-selection principles.Active materials [1] assembled from intracellular components, such as DNA [2], microtubules and motor proteins [3-5] promise innovative biotechnological applications, from microscale transport and medical devices [2] to artificial tissues [1] and programmable soft materials [6][7][8]. Beyond their practical value, these systems challenge theorists to generalize equilibrium statistical mechanics to far-from-equilibrium regimes [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]. Recent experimental advances in the self-assembly and manipulation of colloids with DNA-mediated interactions [27][28][29] have stimulated theoretical analysis that may eventually help clarify the physical principles underlying self-replication [30-32] and evolution in viruses [33][34][35] and other basic biological systems. Yet, despite some partial progress [10,18,19,[36][37][38][39], our conceptual understanding of active materials, and living matter in general, remains far from complete. We do not know whether, or under which conditions, 'universality' ideas [40] that have proved powerful in the description of equilibrium systems can be generalized to describe collective dynamics of active matter not just qualitatively but also quantitatively. This deficit may be ascribed to the fact that mathematical models have been successfully tested against experiments in only a few instances [3,4,15,[41][42][43].Recently discovered 2D active liquid crystal (ALC) analogs [5,[44][45][46][47] comprise an important class of nonequilibrium systems that allows further tests of general theoretical concepts [40] and specific models. ALCs are assemblies of rod-like particles that exhibit non-thermal collective excitations due to steady external [44,45] or internal [5, 47] energy input. At high concentrations, ALCs form an active nematic phase characterized by...

Negative stiffness and modulated states in active nematics

“…A negative value of the order parameter stiffness was assumed in Refs. 23,28,42 as the mechanism responsible for driving pattern formation in continuum models of dry polar and nematic active fluids inspired by the Toner-Tu equations. Mechanisms that can lead to a negative stiffness have recently been identified for polar fluids 29,30 .…”

“…Numerical studies of dry systems have reported both nematic 23 and polar 21,27 order of the defect orientations and a remarkable defect-ordered state accompanied by a flow-vortex lattice at the crossover between wet and dry regimes 26 . This range of results indicates that more work is needed to understand what controls the nature of defect ordering in these systems.Finally, the dynamics of dry polar 28 and nematic 23 …”

“…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 examined both wet 5,7,12,20 and dry [21][22][23][24][25] systems, as well as the crossover between the two 26 , revealing a rich dynamics driven by orientational instabilities and by the unbinding and proliferation of topological defects. A detailed summary of recent work most closely related to ours is given in Section 4.…”

“…In experiments in suspensions of microtubule bundles the +1/2 defects were observed to organize in nematically ordered states 27 . Numerical studies of dry systems have reported both nematic 23 and polar 21,27 order of the defect orientations and a remarkable defect-ordered state accompanied by a flow-vortex lattice at the crossover between wet and dry regimes 26 . This range of results indicates that more work is needed to understand what controls the nature of defect ordering in these systems.…”

“…Finally, the dynamics of dry polar 28 and nematic 23 has also been studied via minimal models where pattern formation is driven by diffusive currents in the order parameter equations, with the assumption of a negative order parameter stiffness, which provides an appealing and simple explanation for the instability mechanisms. Recently, it has been shown that in systems with polar symmetry such a negative stiffness can arise from competing aligning and antialigning interactions 29 or from the interplay of polar alignment and intermediate range hydrodynamic interactions 30 .…”