Defect Spirograph: Dynamical Behavior of Defects in Spatially Patterned Active Nematics

Topological defects in active liquid crystals can be confined by introducing gradients of activity. Here, we examine the dynamical behavior of two defects confined by a sharp gradient of activity that separates an active circular region and a surrounding passive nematic material. Continuum simulations are used to explain how the interplay among energy injection into the system, hydrodynamic interactions, and frictional forces governs the dynamics of topologically required self-propelling $+1/2$ defects. Our findings are rationalized in terms of a phase diagram for the dynamical response of defects in terms of activity and frictional damping strength. Different regions of the underlying phase diagram correspond to distinct dynamical modes, namely immobile defects, steady rotation of defects, bouncing defects, bouncing-cruising defects, dancing defects, and multiple defects with irregular dynamics. These dynamic states raise the prospect of generating synchronized defect arrays for microfluidic applications.

Figure 1

(a) An snapshot of defect trajectories within the activated region of nematic confined in a solid disk. Purple arrows show the orientation of $+1/2$ defects ($\widehat{u}$). The color coding corresponds to the nematic ordering ($S$). (b) A phase diagram of defect dynamics as a function of activity and friction coefficients.

Figure 2

The effect of activity variation on (a) normalized defect separation, (b) average angle of the defect orientation with the radial vector, and (c) average elastic free energy of the system, for ID and SR states. Points are sampled from the red line in Figs. 1, 1, Velocity field in the domain corresponding to the points marked with I–III. The background color represents the nematic scalar order parameter ($S$); for the color scale refer to Fig. 1. Sample trajectory of two steadily orbiting defects are shown in the inset of Fig. 2 ($\zeta =0.0008$, $f=0.05$).

Figure 3

Effect of friction on ID and SR states. (a) Defect separation normalized by $R$, (b) average angle between the defect orientation and the radial vector, (c) average of the angles that the active, elastic, and friction forces exerted on the defects make with the radial vector; $⟨{\theta }_a⟩$, $⟨{\theta }_e⟩$, $⟨{\theta }_f⟩$, respectively. Points are sampled from the blue line in Fig. 1. (d)–(f) Director field and the trajectories of the defects corresponding to the points marked with I–III. Active, elastic, and frictional forces are denoted by black, pink, and green arrows, respectively. Panels (d),(e) correspond to the SR, where defects orbit in circles and, after the completion of each circle, retrace their steps. Only a part of the circular trajectories is plotted. Elastic and frictional forces in (e),(f) have been magnified by a factor of 2.

Figure 4

(a) Defect trajectory, and (b) snapshots of the director field in the dancing state ($\zeta =0.0016$, $f=0.04$). (c)–(f) Long-time defect trajectories for TB state for different values of activity ($\zeta =0.0005$, 0.0006, 0.0006, 0.0008) and friction ($f=0.0$, 0.001, 0.002, 0.002), respectively. Elastic free energy of the system (g) and snapshots of the director with forces exerted on the defect and the defect trajectory (blue) (h); (g),(h) correspond to (f). Active, elastic, and frictional forces are shown with black, pink, and green arrows, respectively (frictional forces are relatively small). (i) Defect trajectories, and (j) snapshots of the director in the BC state ($\zeta =0.001$, $f=0.01$). The orientations of positive (negative) charge defects are identified by purple arrows (a triad of brown arrows). Full blown animations of all of these states are provided in Supplemental Material [61], Movies 5–9.