Orientational order of motile defects in active nematics

Abstract

The study of liquid crystals at equilibrium has led to fundamental insights into the nature of ordered materials, as well as to practical applications such as display technologies. Active nematics are a fundamentally different class of liquid crystals, driven away from equilibrium by the autonomous motion of their constituent rod-like particles. This internally generated activity powers the continuous creation and annihilation of topological defects, which leads to complex streaming flows whose chaotic dynamics seem to destroy long-range order. Here, we study these dynamics in experimental and computational realizations of active nematics. By tracking thousands of defects over centimetre-scale distances in microtubule-based active nematics, we identify a non-equilibrium phase characterized by a system-spanning orientational order of defects. This emergent order persists over hours despite defect lifetimes of only seconds. Similar dynamical structures are observed in coarse-grained simulations, suggesting that defect-ordered phases are a generic feature of active nematics.

Figure 1

Overview of experimental and simulation systems. a, Microtubules (MTs) are bundled together by the depletion agent PEG. Kinesin clusters crosslink MTs and induce inter-filament sliding. Bundles are confined to a surfactant-stabilized oil-water interface, where they form a quasi-2D active nematic film. b, Fluorescence microscope image of a MT active nematic with defects of charge +1/2 (red) and −1/2 (blue). c, Image sequence illustrating the creation (top) and annihilation (bottom) of a defect pair. Scale bars 50μm. d, Simulation microdynamics, consisting of hard rods which grow and split, reminiscent of the extension of MT bundles. L_max is the length at which a rod is split. e, Snapshot of simulated active nematic with marked defects. Rod colors indicate their orientations, and black streamlines guide the eye over the coarse-grained nematic field. f, Creation (top) and annihilation (bottom) events occur analogously to those in experiments. Scale bars 2L_max.

Figure 2

Defect-ordered phase in experiments. a, Retardance map of a thick MT film in the regime of weak defect alignment. Red and blue markers indicate locations and orientations of +1/2 and −1/2 defects. Scale bar 200μm. b, Thin MT film showing strong alignment of +1/2 defects. Scale bar 200μm. c, Orientational order in a large active nematic sample. Each red bar’s orientation and length indicates the mean direction and strength of defect alignment in one field of view. Scale bar 2mm. d, Defect alignment spans the largest samples studied (6cm × 2cm), containing ~20,000 defects. Scale bar 10mm. e, Normalized histogram of +1/2 (red) and −1/2 (blue) defect orientations, P(ψ). f, MT orientation, P(θ), for the sample shown in panels b–d. Measurements of P(θ) from 0 to π are duplicated from π to 2π. Both P(ψ) and P(θ) show strong nematic ordering. g, The preferred defect orientation (green) and magnitude of the order parameter (red) averaged over a field of view persists over the entire sample lifetime.

Figure 3

Defect-ordered phase in simulations. a, Snapshot of a high-area-fraction simulation in which +1/2 defects are not aligned. Red and blue markers indicate locations and orientations of +1/2 and −1/2 defects. Scale bar 5L_max. b, A low-area-fraction system in which defects show flocking-like polar alignment. Scale bar 5L_max. c, A large simulation with defects aligned over long distances. Scale bar 5L_max. d, Normalized histogram of +1/2 (red) and −1/2 (blue) defect orientations, P(ψ). e, Rod orientations P(θ) measured in the same sample. The former shows polar ordering, while the latter exhibits nematic ordering, and the preferred axes are offset by 90 degrees. Measurements of P(θ) from 0 to π are duplicated from π to 2π.

Figure 4

Quantitative measurements of defect alignment. a, The defect nematic order parameter, S, decreases as a function of the MT film’s retardance. The blue shaded region represents the “noise floor” (see Supplementary Information). Inset: The nematic order parameter of the underlying MT filaments, S, also decreases with the MT film’s retardance. b, The polar defect order parameter, P, showing a transition between ordered and isotropic regimes as a function of particle density. c, The two-point nematic correlation of defect orientation C_S(r) = 〈cos2(ψ(r) − ψ(0))〉 in MT films, which shows that orientational order persists over long distances. l_d indicates the mean inter-defect spacing. The magnitude of ordering falls as retardance increases. d, The polar correlation of defect orientation C_P(r) = 〈cos(ψ(r) − ψ(0))〉 in simulations. σ indicates the width of a single rod (see Supplementary Information). At short ranges, defects tend to point in opposing directions, but beyond the first shell of neighbors, defects are likely to be aligned in the same direction.