Defect structures and transitions in active nematic membranes

TL;DR

This work addresses how anisotropic curvature coupling influences active nematic membranes and defect dynamics on deformable surfaces. It develops a minimal continuum model that combines Landau–de Gennes nematic order, bending energy, and curvature coupling, and it uses numerical simulations under low-Reynolds-number flow to explore the interplay between activity and geometry. The key finding is a continuous transition from a curvature-dominated, defect-trapped regime to an activity-dominated turbulent regime, occurring at a critical activity that scales as $\zeta_c \sim \alpha^2 / \kappa$; in the turbulent regime, strong spatial correlations persist between nematic walls and curvature, with walls driving wave-like membrane deformations. The results provide a physical framework for defect-mediated deformation and morphogenesis in nonequilibrium biological membranes, highlighting how topological defects can organize geometry under active conditions.

Abstract

We investigate the dynamics of active nematic liquid crystals on deformable membranes, focusing on the interplay between active stress and anisotropic curvature coupling. Using a minimal model, we simulate the coupled evolution of the nematic order parameter and membrane height. We demonstrate a continuous transition from a curvature-dominated regime, where topological defects are trapped by local deformation, to an activity-dominated regime exhibiting active turbulence. A scaling analysis reveals that the critical activity threshold $ζ_c$ scales as $α^2/κ$, where $α$ and $κ$ are the coupling constant and bending stiffness, respectively; this relationship is confirmed by our numerical results. Furthermore, we find that significant correlations between the orientational pattern and membrane geometry persist even in the turbulent regime. Specifically, we identify that "walls" in the director field induce characteristic wave-like curvature profiles, providing a mechanism for dynamic coupling between order and shape. These results offer a physical framework for understanding defect-mediated deformation in nonequilibrium biological membranes.

Figures (5)

• Figure 1: Snapshots of the Schlieren texture $Q_{xy}^2(\bm{r})$ (left column) and the height field $h(\bm{r})$ (right column) at times $t = 50, 100,$ and $400$. The parameters are set to $(\alpha, \zeta) = (10, 1.0)$. (a,c,e) The color map indicates the magnitude of $Q_{xy}^2$, where red and blue dots denote $+1/2$ and $-1/2$ defects, respectively. (b,d,f) The height field, where red and blue regions represent positive and negative values, respectively. In these panels, the $+1/2$ and $-1/2$ defects are marked by green and yellow dots, respectively.
• Figure 2: Time evolution of (a) the orientational correlation length $\xi_Q(t)$ and (b) the height correlation length $\xi_H(t)$ (log-log plot). The parameter $\alpha$ is fixed at $10$, while $\zeta$ is varied from $0$ to $1.0$ in increments of $0.2$. The green, orange, and blue lines correspond to $\zeta = 1.0$, $0.2$, and $0$, respectively. The data are averaged over 10 samples, and the colored regions indicate the standard deviation. In (a), the black dotted lines represent fitting results: a constant fit for $\zeta = 1.0$ (value $5.0$) and a power-law fit for $\zeta = 0.2$ (exponent $0.14$). (b) The inset shows the ratio $\xi_Q / \xi_H$.
• Figure 3: Activity dependence of the bending energy and fluid velocity. Each data point is time-averaged over the interval $t = 1000$ to $10000$. (a) The spatially averaged bending energy $\langle f_{\text{bend}} \rangle$ as a function of activity $\zeta$ for various $\alpha$. (b) Mean squared velocity $\langle v^2 \rangle$ for the same parameters. The red dashed line represents the linear fit $\langle v^2 \rangle = a (\zeta - 0.4) + b$ for $\alpha=10$, where $a = 3.61$, and $b = 0.544$.
• Figure 4: Phase diagram showing the time-averaged bending energy $\langle f_{\text{bend}} \rangle$ as a color map in the $(\alpha, \zeta)$ plane. The energy is averaged over the interval $t = 1000$ to $10000$. Crosses ($\times$) indicate transition points, defined as the intermediate activity values where the slope of $\langle f_{\text{bend}} \rangle$ with respect $\zeta$ is steepest These points are consistent within $\Delta\zeta = 0.04$ with the transition thresholds determined from the slope of mean squared velocity $\langle v^2 \rangle$. The red dotted line represents a fit to these transition points, consistent with the scaling $\zeta \propto \alpha^2 / \kappa$.
• Figure 5: Snapshots of the director field and the mean curvature field for a fixed $\alpha = 10$. Panels (a)--(c) correspond to $\zeta = 0$, $0.3$, and $1.0$, respectively. The background color in (a)--(c) indicates the mean curvature and the green lines represent the director field, while (d) and (e) show the correlation $\nabla^2 h \cdot J(\bm{r})$ between the mean curvature and the source term $J(\mathbf{r}) = \nabla\nabla:\bm{Q}$. Circles ($\circ$) and crosses ($\times$) denote $+1/2$ and $-1/2$ defects, respectively. The region enclosed by the black box highlights an example of the correlation between the director field and the sign of the curvature at the wall.