Curvature controlled defect dynamics in topological active nematics

Abstract

We study the spatiotemporal patterns that emerge when an active nematic film is topologically constraint. These topological constraints allow to control the non-equilibrium dynamics of the active system. We consider ellipsoidal shapes for which the resulting defects are 1/2 disclinations and analyze the relation between their location and dynamics and local geometric properties of the ellipsoid. We highlight two dynamic modes: a tunable periodic state that oscillates between two defect configurations on a spherical shape and a tunable rotating state for oblate spheroids. We further demonstrate the relation between defects and high Gaussian curvature and umbilical points and point out limits for a coarse-grained description of defects as self-propelled particles.Active systems are characterized by constant input of energy, which is converted by autonomous constituents into directed motion, leading to spatiotemporal patterns. These phenomena range from the macro-scale, e.g. flocks of birds 1 or schools of fish 2 to the micro-scale, e.g. bacterial colonies 3 , migrating tissue cells 4 or active nematic films 5 . If such systems are confined on curved surfaces, topological constraints strongly influence the emerging spatiotemporal patterns. Using these topological constraints to guide collective cell behavior might be a key in morphogenesis 6 and active nematic films on surfaces have been proposed as a promising road to engineer synthetic materials that mimic living organisms 7 . However, the complex dynamics of such topological active systems remains wildly unexplored. As in passive systems the mathematical Poincaré-Hopf theorem forces topological defects to be present in the nematic film. On a sphere this leads to an equilibrium defect configuration with four +1/2 disclinations arranged as a tetrahedron 8-10 , see Fig. 1 The disclinations repel each other and this arrangement maximizes their distance. In active systems unbalanced stresses drive this configuration out of equilibrium. But in contrast to planar active nematics with continuous creation and annihilation of defects 11-14 the creation of additional defect pairs can be suppressed on curved surfaces, which is demonstrated in refs 7 and 15 for an active nematic film of microtubules and molecular motors, encapsulated within a spherical lipid vesicle. This provides an unique way to study the dynamics of the four defects in a controlled manner and led to the discovery of a tunable periodic state that oscillates between the tetrahedral and a planar defect configuration. We confirm this finding by computer simulations, see Fig. 1.Within a coarse-grained model +1/2 disclinations in planar active nematic films can be effectively described by self-propelled particles with a velocity proportional to the activity 5 . In ref. 7 this relation is extended to spherical nematics. Four self-propelled particles on a sphere also oscillate between the planar and tetrahedral configuration. Both descriptions can be quantitatively linked to each other, but also differences can b...

Dynamically generated patterns in dense suspensions of active filaments

“…Finally, in an experiment by Keber et al [21], the microtubule-kinesin mixture was suspended onto the surface of a nearly spherical lipid vesicle. The presence of nonzero curvature leads to an even richer set of collective motion patterns that are only partly understood [27,28].…”

Collective transport of polar active particles on the surface of a corrugated tube

“…The position r i and the orientation n i are constrained to the tangent plane at every point. g(r) can be interpreted as a potential and the constraint trajectories will then lie on the isopotential surface with potential value 0 [12,14,46]. For each constraint there exists a constraint force that penalizes any deviations from the isopotential surface.…”

“…The presence of intrinsic surface curvature frustrates local order giving rise to novel physics. On the one hand, when the manifold is getting curved, defects emerge due to topological constraints (curvatureinduced defects) [11][12][13][14][15][16][17][18][19][20][21]. Keber and coworkers [11] studied the spatiotemporal patterns in active nematic vesicles and found that defects are largely static structures and move spontaneously.…”

“…Keber and coworkers [11] studied the spatiotemporal patterns in active nematic vesicles and found that defects are largely static structures and move spontaneously. In the active nematic film, Alaimo et al [12] found a relation between Gaussian curvature and defects. Sknepnek and Henkes [14] studied active swarms on a sphere and frustration due to curvature leads to stable elastic distortions storing energy in the band.…”

Properties of surface Landau–de GennesQ-tensor models

“…(8) and (9) provide a general surface Landau-de Gennes Q-tensor model with a minimum of a priori assumptions. Due to this generality we expect, as in 3D, the solution space for the tensorial order parameter Q(q, β) not to be restricted to the uniaxial eigenvalue spectra σ(Q) as defined in (1). As the curvature terms can also locally influence the state potential, see discussion above, we expect a simple criterion such as b < 0, which enforces uniaxiality in 3D, not to hold on surfaces.…”