Insensitivity of active nematic liquid crystal dynamics to topological constraints

Confining a liquid crystal imposes topological constraints on the orientational order, allowing global control of equilibrium systems by manipulation of anchoring boundary conditions. In this article, we investigate whether a similar strategy allows control of active liquid crystals. We study a hydrodynamic model of an extensile active nematic confined in containers, with different anchoring conditions that impose different net topological charges on the nematic director. We show that the dynamics are controlled by a complex interplay between topological defects in the director and their induced vortical flows. We find three distinct states by varying confinement and the strength of the active stress: A topologically minimal state, a circulating defect state, and a turbulent state. In contrast to equilibrium systems, we find that anchoring conditions are screened by the active flow, preserving system behavior across different topological constraints. This observation identifies a fundamental difference between active and equilibrium materials.

Figure 1

(a),(b) Fluorescence microscopy images of the kinesin-microtubule system described in the text that motivate our theoretical investigation, with microtubules fluorescently labeled and confined to SU8 holes with radii of (a) $50\mu \mathrm{m}$ (see movie S1 [31] in the Supplemental Material) and (b) $250\mu \mathrm{m}$ (movie S2 [31]). Defects are labeled with magenta arrows ($+\frac{1}{2}$) and blue points with three spokes ($-\frac{1}{2}$).

Figure 2

(a) Director and order (top), and vorticity and streamlines (bottom), corresponding to the three dynamical steady states observed in the FEM simulations: (i) dipolar state (DS, movie S3), (ii) circulating state (CS, movie S4 [31]), and (iii) turbulent state (TS, movie S5 [31]). Results are shown for parallel anchoring. (b) Time-averaged, excess defect density ${\rho }_{\mathrm{\Sigma }}^{*}=\left(N_{\mathrm{\Sigma }}-2\right)/\pi R^2$ (where $N_{\mathrm{\Sigma }}$ is the total number of $\pm \frac{1}{2}$ defects) as a function of disk radius $R$ and active length scale ${\alpha }^{-1/2}$ [15, 25, 26]. We subtract 2 to offset by the number of topologically required $+\frac{1}{2}$ defects. The three cases shown in (a) are indicated with white circles. The inset of (b) plots the threshold activities for the transitions from DS to CS (${\alpha }^{†}$) and CS to TS (${\alpha }^{*}$) as a function of $R^{-2}$.

Figure 3

(a) Simulation results showing the director and degree of order (top) and flow field and vorticity (bottom) during the transition from the DS to the CS; $\alpha =5$ and $R=6.5$, the same as Fig. 2 ii (movie S4 [31]). The stills show the process in six steps: (1) The dipole configuration similar to that shown in Fig. 2 i. (2) Two $+\frac{1}{2}$ defects with a nascent $\pm \frac{1}{2}$ pair (purple region of disorder). (3) The ejection of a $+\frac{1}{2}$ defect (circled in red to distinguish it from the other $+\frac{1}{2}$ defects). (4),(5) The annihilation of one of the original DS $+\frac{1}{2}$ defects. (6) The beginning of the CS. (b) Total system viscous dissipation (blue), ${\mathcal{F}}_{\text{LDG}}$ (magenta), and circulation (black) as a function of time during the transition from the DS to the CS shown in panel (a) ($R=6.5$ and $\alpha =5$). The dashed lines show results when we impose left-right symmetry, suppressing the CS and thus forcing the system to remain in the DS.

Figure 4

Behaviors in three different container topologies (from left to right): saddle ($-1$), neutral (0), aster (+1). (a) Top: Equilibrium configuration for $\alpha =0$. Bottom: circulating states observed for $\alpha =5$. Animations of the circulating states are provided in movies S6–S9 [31]. (b) Excess defect number $N_{\mathrm{\Sigma }}^{*}$ as a function of activity for each topology with ${\alpha }^{†,*}$ labeled. $R=6.5$ for all cases.

Figure 5

(a) Spatial distributions of $+\frac{1}{2}$ (magenta), $-\frac{1}{2}$ (blue), and all (gray) defects normalized by the total, average defect concentration for the same conditions in the TS ($\alpha =12,R=15$). (b) Spatial distribution of defect nucleation rates (black) and annihilation rates (red) normalized by the system average. To illustrate the extent to which annihilation rates follow the law of mass action, the product of the spatial distributions of $\pm \frac{1}{2}$ defects (scaled by a factor of 4 to make the trend the same order of magnitude as the creation and destruction rates) is also plotted (blue). Panels (c) and (d) plot the normalized orientation distributions for each defect type at the radial points corresponding to their respective maxima (i) and (ii).

Figure 6

Top: Director field and defects in the TS ($\alpha =12,R=15$) for aster (+1) boundary condition (distributions for additional topologies are shown in Fig. 7). Labeled annuli have width $\mathrm{\Delta }r=0.83$ with inner radii $R-3\mathrm{\Delta }r$ and $R-\mathrm{\Delta }r$, and are used to calculate orientation probability distributions of $+\frac{1}{2}$ (bottom left) and $-\frac{1}{2}$ (bottom right) at regions dominated by hydrodynamic wall interactions ($i$) and anchoring energy ($\mathit{ii}$), respectively.

Figure 7

Orientation distributions for $+\frac{1}{2}$ (magenta) and $-\frac{1}{2}$ (blue) defects in the turbulent regime ($\alpha =12,R=15$) for three different topological boundary conditions; from left to right: natural boundary condition ${\mathbf{n}·\nabla \mathbf{Q}|}_{\partial \mathrm{\Omega }}=0$ (no topological preference), saddle ($-1$), neutral (0). Rows are different radial bins (see Fig. 6). In bin ($i$), defect orientations are dominated by hydrodynamic wall effects, and are therefore similar for all anchoring conditions. In bin ($\mathit{ii}$), the importance of the anchoring condition near the wall is revealed by differences in orientation distributions for different topological boundary conditions.

Figure 8

Left: Flow field and vorticity for $+(-)\frac{1}{2}$ defects near a wall at angle $\psi$. The area over which the force is imposed is circumscribed by a black circle in each plot. Right: The total vorticity integrated over the defect area as a function of orientation $\psi$, revealing both fixed points (roots) and stability (slope). Filled (open) circles denote stable (unstable) stationary defect orientations. The red circle in each plot corresponds to the configuration in the left column.