Eppur si muove: Shape of topological defects -- and consequent motion -- in active nematics

Abstract

Topological defects in systems with liquid-crystalline order are crucial in determining their large-scale properties. In active systems, they are known to have properties impossible at equilibrium: for example, $+1/2$ defects in nematically-ordered systems self-propel. While some previous theoretical descriptions relied on assuming that the defect shape remains unperturbed by activity, we show that this assumption can lead to inconsistent predictions. We compute the shape of $-1/2$ defects and show that the one of $+1/2$ is intimately related to their self-propulsion speed. Our analytical predictions are corroborated via numerical simulations of a generic active nematic theory.

Figures (7)

• Figure 1: Nematic director ${\bf n}=(\cos\theta,\sin\theta)$ orientation for defects with charges $+1/2$ (left) and $-1/2$ (right) with $\phi_0=\pi/2$ obtained by numerical simulations of eq. \ref{['eq:minimal-nematics']} in a circular domain. Red lines denote the director orientation at the boundary that is kept fixed in time; this is chosen to correspond to the far field of the defect in a single Frank constant liquid crystal, which ensures the presence of a $\pm 1/2$ defect in the bulk.
• Figure 2: Panel (a): Motion of the $+1/2$ defects in three active systems and three different system-sizes $R$, showing that defects always self-propel with a well-defined speed (i.e., independent of $R$), irrespectively of whether the active non-linearity $L_2$ is present. At late times, defects reach the boundary of the domain and stop. Only the defect with $L_2\neq0$ self-propels along its polarity while the other two do so anti-parallel to it. Here, $K_i\equiv K_1, K_2$. Panel (b): Defect self-propulsion speed $v_d$ in systems with various levels of activity, obtained by varying $L_2$ (blue), $K_1=K_2=L_1=\tilde{K}$ (red), and $L_1$ (green) and setting other coefficients to $0$, showing that $v_d$ has a comparable magnitude in all cases. Dots are measurements from the simulations, lines analytical predictions at small activity obtained from eq. \ref{['eq:defect-speed-perturbative']}, showing very good agreement, while eq. \ref{['eq:v_d-wrong']} gives a wrong prediction at arbitrarily low activity.
• Figure 3: Shape of the $-1/2$ topological defect in the far field encoded in the functions $A(\phi)$ and $B(\phi)$ of eq. (\ref{['eq:generic_far']}), (\ref{['eq:generic_far-2']}) found by solving eq. \ref{['eq:defect-12']} and \ref{['eq:active-minus-half-B']}. Analytical predictions (lines) and numerical simulations (dots) are in excellent agreement. In blue: passive system with two Frank constants ($K=1/2$); in red and green, respectively, active systems with $L_{2}=1/2, L_{1}=K_{1}=K_{2}=0$ and $K_1=K_2=L_1=0.1, L_2=0$. The shape of the defects was measured at $r=15$ in domains with $R=100$ (blue and red) and $R=120$ (green).
• Figure 4: Angular shape of the core of $+1/2$ defect as predicted theoretically (lines) by eq. (\ref{['eq:vd-shape']}) and found in our numerical simulations (points), showing perfect agreement. Red symbols correspond to $K_1=K_2=L_1=0, L_2=0.1$, blue ones to $K_1=K_2=L_1=0.1, L_2=0$, the dashed line is the defect shape for a two Frank constants defect with $K=0.1$ and the dashed-dotted line to the shape of a single Frank constant defect ($K=0$). Bars correspond to the error in measuring $b(\phi)$ coming from the uncertainty in the defect location due to the spatial discretisation employed in simulations. Other parameters: $R=10$, spatial discretisation $\Delta x_0=2\times 10^{-3}$ for $r<3$ and $\Delta x_0=0.5$ for $r>3$. $b(\phi)$ measured at $r=0.5$.
• Figure 5: Far field shape of the angle field for passive $+1/2$ defect. The perturbative solution at order four discussed in eq. \ref{['app:eq-perturbative-sol']} is indistinguishable from the one obtained by numerical solution of the ODE for $B(\phi)$ up to $K\approx 1$.