Symmetry, Thermodynamics, and Topology in Active Matter

The name active matter refers to any collection of entities that individually use free energy to generate their own motion and forces. Through interactions, active particles spontaneously organize in emergent large-scale structures with a rich range of materials properties. The active-matter paradigm is applied to living and nonliving systems over a vast dynamic range, from the organization of subnuclear structures in the cell to collective motion at the human scale. The diverse phenomena exhibited by these systems all stem from the defining property of active matter as an assembly of components that individually and dissipatively break time-reversal symmetry. This article outlines a selection of current and emerging directions in active matter research. It aims at providing a pedagogical and forward-looking introduction for researchers new to the field and a road map of open challenges and future directions that may appeal to those established in the area.

Figure 1

Experimental realizations of active turbulence, from top to bottom: a layer of Bacillus subtilis (adapted from Ref. [26]), a suspension of kinesin-powered microtubule bundles (courtesy of Linnea Lemma and Zvonimir Dogic), and a layer of MDCK cells (adapted from Ref. [36]). The left column shows images of the system, while the right column displays snapshots of the local vorticity.

Figure 2

Topological defects in active nematic liquid crystals. Top and middle rows: configuration of the director texture for a $+1/2$ (left) and a $-1/2$ (right) defect (top row) and associated flow fields (middle row). The topological charge $q$ is defined as the net rotation (in units of $2\pi$) of the order parameter as one encircles the defect. In the left frame, the director rotates by $\pi$ in the same direction as the path is traversed, yielding $q=+1/2$. In the right frame, the director rotates by $\pi$ in the opposite direction as the path is traversed; hence, $q=-1/2$. The magnitude of the flow field around an isolated defect is shown in blue, with the white lines representing the streamlines of the flow (adapted from Ref. [61]). The red lines are tangent to the director field. Bottom row: $+1/2$ defects as self-propelled polar particles. The red arrow shows the self-propulsion speed of a $+1/2$ defect powered by flows induced by extensile (left) and contractile (right) active stresses.

Figure 3

Aligning an active nematic with a magnetic field by interfacing it with a passive nematic layer. The figure, adapted from Ref. [92], shows, from top to bottom, the configuration and an optical micrograph of the underlying passive liquid crystal aligned in the smectic A (SmA) phase by application of a magnetic field (top); a snapshot of the active nematic obtained by fluorescent confocal microscopy showing the alignment in “kink walls” (also referred to “arches” in the literature) induced by the coupling to the SmA structure of the passive layer, and the time average of the dynamical pattern in the active nematic layer, with arrows indicating antiparallel flow directions (scale bar, $100\mu \mathrm{m}$).

Figure 4

The simplest example of an active colloid is a micron-size polystyrene bead half coated with platinum immersed in a hydrogen peroxide solution and powered by solvent concentration gradients that interact with the particle’s surface. The platinum catalyzes the breakdown of ${\mathrm{H}}_2{\mathrm{O}}_2$ in oxygen and water, generating anisotropic concentration and charge currents that turn the micron-size bead into a swimmer. The speed of these synthetic swimmers is controlled by the concentration of ${\mathrm{H}}_2{\mathrm{O}}_2$ and is typically of order of tens of microns per second, comparable to that of flagellated bacteria like Escherichia coli.

Figure 5

Nonequilibrium phase diagram of polar active colloids in the plane spanned by adimensionalized “chemotactic” reorienting response coefficient $A$ (horizontal axis) and phoretic motility $B$ (vertical axis). Reproduced from Ref. [133].

Figure 6

Examples of nonreciprocal interactions. Top: Active colloids can experience NR effective interactions due to different surface reactivities. Here, particle $A$ is attracted to $B$, but $B$ is repelled by $A$. Bottom: Birds interact with other birds that are within their vision cone. Here, bird $A$ interacts with $B$, which is within the vision cone of $A$, but $B$ does not interact with $A$.

Figure 7

Developing starfish embryos self-organize into living chiral crystals. Time sequence of still images showing crystal assembly and dissolution ($t=0$ h corresponds to the onset of clustering; scale bar, 1 mm). Embryo morphology and flow fields change with developmental time (shape scale bar, $100\mu \mathrm{m}$; flow field scale bar, $200\mu \mathrm{m}$). Embryos assembled in a crystal perform a global collective rotation (scale bar, 2 mm). Spinning embryos (yellow arrows) in the crystal form a hexagonal lattice, containing fivefold (red) and sevenfold (cyan) defects (scale bar, 0.5 mm). Spinning embryos exchange forces (brown arrows) and torques (red arrows) due to hydrodynamic interactions. Adapted from Ref. [164].

Figure 8

Morphogenesis of the Drosophila pupal wing. The cells in the tissue undergo shape changes, cell divisions, cell rearrangements, and cell extrusions during wing morphogenesis. Colors denote cell contact dynamics during tissue morphogenesis. Adapted from Ref. [252].

Figure 9

A pictorial depiction of the connections among the various aspects of active-matter physics discussed in this article.