Spatiotemporal Control of Charge +1 Topological Defects in Polar Active Matter

Abstract

Topological defects are a conspicuous feature of active liquid crystals that have been associated with important morphogenetic transitions in organismal development. Robust development thus requires a tight control of the motion and placement of topological defects. In this manuscript, we study a mechanism to control +1 topological defects in an active polar fluid confined to a disk. If activity is localized in an annulus within the disk, the defect moves on a circular trajectory around the center of the disk. Using an ansatz for the polar field, we determine the dependence of the angular speed and the circle radius on the boundary orientation of the polar field and the active annulus. Using a proportional integral controller, we guide the defect along complex trajectories by changing the active annulus size and the boundary orientation.

Figures (4)

• Figure 1: Effect of heterogeneous activity patterning. (a) Illustration of the system. At the boundary $|\mathbf{p}|=1$ and $\psi_b=\pi/4$. Light blue lines: polarization field lines. Dark purple curved arrow: circular segment with radius $r_d$. Gray shaded: region of non-zero activity $\Delta\mu\neq0$. (b) Initial configuration of the numerical solver. White arrows: polarization field. Color code: orange refers to $\Delta\mu=0$, black to $\Delta\mu\neq0$. (c) Trajectory of the defect center. Color code: time. (d) Average distance $r_d$ and angular speed $\omega$ of the defect center as a function of activity. Unless specified differently $(\zeta + \nu\gamma\lambda)\Delta\mu=-100 K_S/r_o^2$, $r_i/r_o=8/25$ and $K_B=K_S$.
• Figure 2: Mechanism underlying circular defect movement. (a) Stable polarization field from simulation with heterogeneous activity and $\psi_b=\pi/4$. White arrows show the orientation and color shows the magnitude of $\mathbf{p}$. Insets: localized splay (top) and bend (bottom) distortions in the orientation field. (b) Stable flow field corresponding to simulation shown in (a). Black lines: streamlines, color: normalized speed. (c) Polarization field given by our ansatz for $\psi_b=\pi/4$ shows good agreement with (a). Insets: same as (a). $\Delta P=0.5$, $r_p=r_o/20$(d) Stokes flow field calculated from the active force around polar field shown in (c). This shows good agreement with (b).
• Figure 3: Controlling defect motion with boundary angle. (a) Profile of the elastic potential $F$ associated with placing the defect at radius $r_d$ for different values of $r_i\in \{0., 0.65\}$ using the ansatz described in the text. (b) Location of the minimum in the elastic potential as a function of $r_i$. (c) Angular speed of the defect as a function of boundary angle. (d) Radial position of the defect as a function of boundary angle. (e) Oscillating defect by switching boundary angle between $\psi_b=\pm\pi/4$ and using $\psi_b=0$, Tab. S II SI, as an intermittent value to stop the defect before return. (f) Defect tracing a figure eight pattern by switching between boundary angle $\psi_b=(0.45\pm0.25)\pi$, Tab. S III SI.
• Figure 4: Dynamic feedback control of the defect location. (a) Schematic of the PI-Controller. The controller maintains the radial distance of the defect at a desired setpoint $r_d^*$ by dynamically adjusting the radius of the passive core, $r_i$. For each $r_i$, the system is simulated for $n=50$ time steps to determine the resulting effective $r_d$, which yields the error that is fed back into the controller. The new $r_i$ is calculated as $r_i=r_{i, old}+P+I$, where the $P$ denotes the proportional part and $I$ the integral part of the controller. $k_P=10^{-3}$ and $k_I=9\cdot 10^{-3}$. (b) Independently controlling $r_d$ and azimuthal motion of a defect. The setpoint changes from $r_d^*/r_o=7/25$ to $r_d^*/r_o=11/25$ at time step $1200$ (indicated by black horizontal lines). Meanwhile, the boundary angle increases each $500\gamma/\chi$ by $\pi/5$ starting from $\psi_b=\pi/4$ (indicated by gray background). (c) Path of a defect tracing a closed loop with right angle corners, Tab. S IV SI. (d) Path of a defect tracing a hanagata shape with internal angles over $180^\circ$ and curved edges, Tab. S V SI. Circles in c) and d) correspond to the extreme values for $r_d^*$.