Alignment of a topological defect by an activity gradient

As a method for controlling active materials, researchers have suggested designing patterns of activity on a substrate, which should guide the motion of topological defects. To investigate this concept, we model the behavior of a single defect of topological charge $+1/2$, moving in an activity gradient. This modeling uses three methods: (1) approximate analytic solution of hydrodynamic equations, (2) macroscopic, symmetry-based theory of the defect as an effective oriented particle, and (3) numerical simulation. All three methods show that an activity gradient aligns the defect orientation, and hence should be useful to control defect motion.

Figure 1

(a) Steady-state defect orientation $\mathrm{\Psi }$ as a function of activity gradient $Z^{\prime }$, at $r_{\text{max}}=2$, compared with a linear fit. (b) Steady-state defect orientation $\mathrm{\Psi }$ as a function of system size $r_{\text{max}}$, at fixed $Z^{\prime }=1$, compared with a cubic fit.

Figure 2

Examples of defect trajectories, as a defect moves outward from the origin and exits the system with free boundary conditions. The arrows indicate the defect position and orientation at times $t=0,2.0,2.5,3.0,3.5,4.0,...$ .

Figure 3

Time evolution of the defect orientation angle $\mathrm{\Psi }$, in a system with free boundary conditions, for several positive and negative values of the activity gradient $Z^{\prime }$.

Figure 4

Snapshots of the dynamic evolution of the circular motion of $+1/2$ defects. (a)–(d) show the process with negative activity, and (e) and (f) show the process with positive activity. The pink (central) circle is the area with activity, and the red (larger) arrows show the direction of activity gradient. The blue (smaller) arrows show the location and orientation of defects.