Topological structure and dynamics of three-dimensional active nematics

Abstract

Topological structures are effective descriptors of the nonequilibrium dynamics of diverse many-body systems. For example, motile, point-like topological defects capture the salient features of two-dimensional active liquid crystals composed of energy-consuming anisotropic units. We dispersed force-generating microtubule bundles in a passive colloidal liquid crystal to form a three-dimensional active nematic. Light-sheet microscopy revealed the temporal evolution of the millimeter-scale structure of these active nematics with single-bundle resolution. The primary topological excitations are extended, charge-neutral disclination loops that undergo complex dynamics and recombination events. Our work suggests a framework for analyzing the nonequilibrium dynamics of bulk anisotropic systems as diverse as driven complex fluids, active metamaterials, biological tissues, and collections of robots or organisms.

Fig. 1:

Assembling 3D active nematics and imaging their director field. (A) Schematic of the 3D active nematic system: active stress generating extensile microtubule bundles are dispersed in a passive colloidal liquid crystal. (B) Active 3D nematic imaged with widefield fluorescent microscopy (left) and polarized microscopy (right). Birefringence indicates local nematic order. (C) Multi-view light sheet microscopy allows for 3D imaging of millimeter sized samples with single-bundle resolution. (D) (left) A 2D slice of fluorescent microtubule bundles, with highlighted elastic distortions. (right) Corresponding elastic distortion energy map, with an overlaid nematic director field (red). (E) Three-dimensional elastic distortion map reveals presence of curvilinear rather than point-like singularities. An entangled network of lines coexists with isolated loops. (F) Hybrid lattice Boltzmann simulations yield a similar structure of 3D active nematics. All experimental samples consist of passive fd viruses at 25mg/mL and microtubules at 1.33mg/mL.

Fig. 2:

Dynamics of experimentally observed disclination loops. (A) Loop nucleation from a defect-free region. (B) Loop self-annihilation leaves behind a defect-free nematic. (C) A disclination line self-intersects, reconnects, and emits a loop. (D) A disclination loop intersects, reconnects and merges with a disclination line. Each bounding box is 30 μm *30 μm *38 μm. The time interval between two pictures is 12 seconds.

Fig. 3:

Structure of disclinations lines, wedge-twist and pure-twist loops. (A) Disclination line where a local +1/2 wedge winding continuously transforms into −1/2 wedge through an intermediate twist winding. The director field winds by π about the rotation vector Ω (black arrows), which makes angle β with the tangent t (orange arrow), and which is orthogonal to director field everywhere in each slice. For ±1/2 wedge windings, β=0 and π. β=π/2 indicates local twist winding. Reference director n₀ (brown) is held fixed. Color map indicates angle β. (B) The wedge-twist loop where local winding as reflected by angle β varies along the loop. Ω is spatially uniform and forms an angle γ=π/2 with the loop’s normal, N. The winding in the four illustrated planes corresponds to the profiles of the same colors shown in (A), with dashed edges of squares aligned to show the local director field. Double-headed brown arrows indicate n_out, the director just outside the loop. (C) Pure-twist loop, with Ω both uniformly parallel to loop normal N (γ=0) and perpendicular to the tangent vector.

Fig. 4:

Structure of disclination loops in experiments and theory. (A) Two orthogonal views of an experimental wedge-twist loop overlaid onto a fluorescent image of the microtubules. The nematic director is in red. (B and E) Structure of wedge-twist disclination loops in experiments and simulation. (C and F) Structure of pure-twist disclination loops from experiment and simulation Panels show the director field’s winding in the corresponding cross-sections on the experimental loops. (D) Distribution of loop types extracted from experiment (N=268) and hybrid lattice Boltzmann simulations (N=94). ∣cos(γ)∣=0 for wedge-twist loops and 1 for pure-twist loops. Distributions of standard deviations of ∣cos(γ)∣ are shown in Fig. S3. The count of simulated loops includes analysis of some loops at multiple time points, as we do not track loop identity in the complex flow dynamics. Coloring of loops indicates the angle β. Scales and bounding boxes for the loops are shown in Fig. S4.

Fig. 5:

Nucleation mechanism of wedge-twist and pure-twist loops. (A) Nucleation and growth of a wedge-twist disclination loop through a self-amplifying bend distortion. Purple rods represent the 2D director field through the local ±1/2 wedge profiles. (B) Schematic of a wedge-twist loop and the director field in the plane that intersects ±1/2 wedge profiles. (C) A pure-twist disclination loop nucleates and grows from a local twist distortion (Movie S10). Black arrows indicate the local buildup of the twist distortion. Insert shows the top view of a growing twist disclination loop. (D) Schematic of a pure-twist loop and the director field in the loop’s plane.