Chaos-generating periodic orbits of topological defects in confined active nematics

Abstract

Active nematics in two dimensions stir themselves efficiently through internally generated chaotic flows, largely driven by motile $+1/2$ disclinations. We investigate how this tendency toward chaotic fluid stirring can, counterintuitively, produce certain ordered, periodic flows in confinement, characterized by stable periodic orbits of $+1/2$ disclinations. We computationally study two-dimensional active nematics in systems with boundary conditions requiring a prescribed number $n$ of excess $+1/2$ disclinations, using Beris-Edwards nematohydrodynamics simulations alongside an agent-based simulation approach. We find that when confinement is sufficiently strong to prevent defect pair-nucleation, but not strong enough to arrest all flow, then $n=3$ defects generically follow a "golden braid" orbit as observed recently in experiments, and we predict a "silver braid" orbit of $n=4$ defects. For these results and for greater numbers of defects, we show that the periodic or chaotic nature of the dynamics is determined by a balance between the number of defects and the number of vortices in the flow field, suggesting a new design criterion for ordered flows in active nematics.

Figures (9)

• Figure 1: (a) Exponential stretching of a passively advected line, as a measure of topological entropy production, in simulated bulk active nematic dynamics beginning along an arbitrary contour of the nematic director field. Left inset shows illustrations of the local director field around $\pm 1/2$ topological defects. Right inset shows the semi-log plot of the advected contour length over time. Green dotted lines indicate the divergence of the contour length between the two dotted green points. Purple arrows show the recent swapping of two $+1/2$ defects. (b) Schematic illustration of the divergence of two passive tracers whose separation distance $D$ grows exponentially in time $t$, giving a positive Lyapunov exponent $\lambda$. (c) Schematic mixing dynamics for "stirring rod" motion described by the braidword $\{\sigma_{2}^{-1} \sigma_{1}\}$. The exponential stretching is deducible from the growth in length of the black line. (d) Worldlines of topological defects for the braidword shown in (c).
• Figure 2: Braiding terminology demonstrated with two $+1/2$ defects in an active nematic confined in a circle with tangential anchoring, from a $100 \times 100$ Beris-Edwards nematohydrodynamic simulation. (a) Snapshot of the nematic director showing the two defects in a periodic orbit. (b) Trajectory top view of defects (blue and green) over simulation time (335,000 time-steps). (c) Worldlines of the trajectories in (b). Spatial coordinates and time are labeled on the $x, y$, and $z$ axes respectively. (d) Defect worldlines spatially projected onto the $x$ axis. Crossings (all clockwise) are labeled by a red "x"; the structure is represented by the one-element braidword $\{\sigma_1\}$.
• Figure 3: Periodic braiding orbits of $+1/2$ defects in a simulated active nematic confined in a disk with an anchoring direction that winds through angle $2\pi q$, for $q=3/2$ (top row) and $q=4/2$ (bottom row). (a) Snapshot of the nematic director showing the three ($q=3/2$) and four ($q=4/2$) defects in a periodic orbit. (b) The anchoring direction along the circular boundary and defect trajectories traced over the simulation time of $7.5\times10^5$ ($q=3/2$) and $5.0\times10^5$ ($q=4/2$) time-steps. Each simulation was performed on a $100\times100$ lattice with a dimensionless active length of $0.045$, and a dimensionless nematic coherence length of $0.011$. Arrows indicate defect direction of motion. (c) (left) The projection of the trajectories onto the $x$ axis, where swaps between defects are labeled with a red marker if clockwise and a black marker if counter-clockwise; (right) schematic diagram summarizing the braid exhibited by the defects $\{\sigma_2^{-1}\sigma_1\}$ for $q=3/2$ and $\{\sigma_1 \sigma_3 \sigma_2 \sigma_1^{-1} \sigma_3^{-1} \sigma_2^{-1}\}$ for $q=4/2$. (d) The numerically calculated topological entropy using the E-tec and Line Stretching (LS) algorithms (shown in Movies S2 and S3), as well as the calculated Lyapunov exponent, in units reciprocal to the time between defect swaps. Each braiding pattern consists of two effective swaps corresponding to a co-linear arrangement of defects. The numerical values of average topological entropy per swap using E-tec are $0.5277 \pm 0.0005$ ($q=3/2$) and $0.8610 \pm 0.0004$ ($q=4/2$) for samples of 3000 randomly initialized advected trajectories. The numerical values of average topological entropy per swap using the LS algorithm are $0.52 \pm 0.03$ ($q=3/2$) and $0.90 \pm 0.02$ ($q=4/2$). Errors are standard error of the mean taken over five advected curves. The numerical values of average Lyapunov exponent per swap are $0.3873 \pm .0003$ ($q=3/2$) and $.8006 \pm .0004$ ($q=4/2$),both of which are, as required, below their respective analytic values of topological entropy for ideal stirring rods, shown in the dashed blue line. Each $\lambda$ calculation uses 350 pairs of randomly initialized passive tracers. Standard deviation is shown in red and are smaller than the marker size.
• Figure 4: Defect dynamics in circular confinement with winding number $q\geq 5/2$ in the anchoring direction. Columns show the different studied values of $q$. (a) Examples of the aperiodic trajectories. For the $q = 5/2$ system, trajectories over 40 time-steps are shown. For the $q = 6/2,7/2,8/2,9/2$ systems, trajectories are shown over 100 time-steps and for $q = 10/2$ 300 time-steps are shown. (b) The passive ground state configurations of the defects. In all studied geometries, the ground states show a symmetric placement of defect cores about the boundary. However, orientations of the defects are not symmetric, with orientations varying locally to match the fixed anchoring against the circular boundaries. (c) The time averaged vorticity shown throughout the simulation time of $10^6$ time-steps. Each simulation was performed on a $100\times100$ lattice, at a dimensionless active length of .0003, and a dimensionless nematic coherence length of 0.011.
• Figure 5: (a) Schematic illustration of a $+1/2$ defect in three orientations and positions consistent with a line of effective tangential anchoring; arrows mark defect self-propulsion direction. The only defect orientation which does not break an anchoring line is oriented parallel to that line and away from the fixed anchoring point on the boundary. (b) The set of lines parallel to the anchoring direction (anchoring lines) at the associated point on the surface, showing the cardioid, nephroid, and trefoiloid as emergent effective boundaries. Anchoring lines are colored by angle with respect to the radial direction $\hat{r}$ and have opacity decaying with distance from the anchoring point. (c) Envelope curves (dashed) extracted from the lines of (b), together with simulated defect trajectories. Defect trajectories tend to stay approximately within the envelopes and to intersect with the cusps in the envelope. (d) The active force (Eq. \ref{['eq:f_active_epicycloid']}) imposed by the winding anchoring conditions, with splay-dominated regions in pink and bend-dominated regions in cyan. In (b)-(d), each row corresponds to the anchoring winding $q$ labeled at left.