Curvature-controlled defect dynamics in active systems

Abstract

We have studied the collective motion of polar active particles confined to ellipsoidal surfaces. The geometric constraints lead to the formation of vortices that encircle surface points of constant curvature (umbilics). We have found that collective motion patterns are particularly rich on ellipsoids with four umbilics where vortices tend to be located near pairs of umbilical points to minimize their interaction energy. Our results provide a new perspective on the migration of living cells, which most likely use the information provided from the curved substrate geometry to guide their collective motion.Introduction -Active particles are known to spontaneously form complex dynamic patterns at length scales ranging from the molecular [1], to the cellular [2,3] up to macroscopic patterns seen in flocking birds [4], schooling fish [5] or humans in crowded environments [6,7]. The key feature of these active systems is the constant energy input on each individual unit, which renders the system completely out of equilibrium. Collective phenomena in such active systems have been successfully described using so-called self-propelled particle models [8] that are limited to close neighbour interactions only [9]. In unconstrained 2D and 3D systems these models display selforganised pattern formation resembling experimental observations [9]. The behaviour of active particles confined to a surface has been mainly studied on planar surfaces of zero Gaussian curvature. It is known however, that the presence of intrinsic surface curvature frustrates local order giving rise to novel physics [10], as has been shown for 2D fluids confined to curved surfaces [11]. As a consequence of the Poincaré-Hopf theorem, for instance, it is not possible to have continuous fluid flow on the entire surface of a sphere, which requires the presence of two +1 defects (vortices) [12]. The effect of non-zero Gaussian curvature on self-propelled particles remains poorly understood, with only a few recent examples studying the effect of spherical constraints [13,14]. In living systems, cells are influenced by surface curvature as demonstrated by cell movements in the developing corneal epithelium leading to vortex patterns [15] or by the coordinated collective migration of cells during embryonic development [16]. The emergent behaviour of moving cells is not only the result of intercellular interactions, but is crucially influenced by geometrical constraints [2,17,18]. The aim of the current work is to investigate the impact of non-constant Gaussian curvature constraints on the collective behaviour of self-propelled particles. Our restriction of the geometry of the surfaces to ellipsoids allows an analysis of how geometrical cues (represented by the umbilical points of the surface, Fig. 1c) effectively interact with defects in the director-field (e.g., vortices). The strong coupling between vortex position and umbilical points demonstrates the importance of surface geometry on the emergence of patterns in active systems.

Curvature controlled defect dynamics in topological active nematics

“…We consider an extension of these models which includes excluded volume [15][16][17][18] and classify systems by the head-tail symmetry of their particles in polar or nematic. For active polar particles these models have been formulated on a sphere 19 and on ellipsoidal surfaces 20 .…”

Aging and rejuvenation of active matter under topological constraints

“…These latter processes are generally understood in terms of an energy-landscape picture, whereby aging and rejuvenation correspond to relaxation toward deeper and shallower energy minima, respectively [32]. However, owing to the non-Hamiltonian nature of particle activity, the potential (or free) energy is generally not a useful metric for active matter, and hence it remains unclear if and how aging and rejuvenation might be manifested in an active glass.A different avenue of research concerns the effects of geometric [33][34][35][36][37][38] and topological [39][40][41][42][43][44][45] constraints on active matter. For passive soft matter systems, it is well established that confining a system to a curved surface can both frustrate and promote long-range orientational order [46][47][48], induce complex topological-defect structures [49][50][51][52], and affect a system's glass-forming properties [53].…”

“…For active softmatter systems, however, only a limited number of experimental and theoretical studies has addressed the role of curvature and topology. Explicitly, recent experimental work has focused on active nematic microtubuli confined to a deformable droplet interface [39], and subsequent theoretical [40][41][42] and simulation [43][44][45] studies have explored the dynamics of nematic and polar active particles under a spherical or ellipsoidal constraint. These developments point toward a rich array of topological-defect patterns and curvature-driven dynamics in the presence of strong aligning interactions between the particles.…”

Liquid Crystals on Deformable Surfaces

“…On a sphere they are known as global minimizers, subject to rotation 6 . On an ellipsoid the two +1 defects are know to be located at umbilical points 44 , which here correspond with the maxima of curvature at the long axis. The equilibrium configuration is thus uniquely determined.…”