Active topological strings in renewing nematopolar fluids

TL;DR

The study shows that in nematopolar fluids with continuous material renewal, stable topological strings—lines of polar order where $p=0$ connecting nematic defects—emerge and persist. Renewal sustains hydrodynamic flows that generate an effective pressure $\Pi^{\rm e}$, and charge-dependent forces (via the antisymmetric part of the Ericksen stress) set the steady-state string length, differentiating strings with charges $\pm 1$ from those with charge $0$. The interplay between renewal, Coulomb-like defect interactions, and charge-dependent hydrodynamic stresses stabilizes strings and can yield lattices of alternating charges, while active stress can reorient defects and promote patterns such as vortices or asters; at high activity, chaotic string networks arise. Overall, renewal acts as a generic mechanism stabilizing topological defect structures in active matter with mixed nematic-polar order, with potential implications for cytoskeletal organization and developmental processes, and suggests broader relevance for systems with renewing components and multiple order parameters.

Abstract

Active matter often simultaneously exhibits different kinds of orientational order and, in many cases of biological interest, undergoes continuous material renewal. In renewing nematopolar fluids we find stable topological strings, structures consisting of two nematic point defects connected by a defect line in the polar field. We identify the mechanism underlying string stabilization and unveil how string length is determined. In the presence of active stress, we observe active-string chaos. Our work identifies continuous material renewal as a generic mechanism underlying the stabilization of topological defect structures in systems with mixed order parameters. It could be used for orchestrating living matter during development and other biological processes.

Figures (6)

• Figure 1: Renewal stabilizes topological strings. (a-c) Heatmaps of the polar order parameter $p$ for topological strings of charge $-1$ (a), $+1$ (b), and $0$ (c) in the presence of material renewal. White arrows: polar director, black dashes: nematic director, scale bar: unit length. (d) Steady-state length $\ell$ of $\pm1$ strings ($+$/$-$) as a function of the nematopolar coupling strength $\chi$. Parameters: $\rho_0=0.6$, $\tau^{-1}=5$, and $\chi=0.1$ (a-c).
• Figure 2: Differences in effective pressure $\Pi^{\rm e}$ drive flows stabilizing topological strings. (a) Effective pressure $\Pi^{\rm e}$ around a topological string. Top panel: Heatmap of $\Pi^{\rm e}$ for a $-1$ string in the presence of renewal. Black arrows: velocity field $\mathbf{v}$, scale bar: unit length. Bottom panel: Profile of $\Pi^{\rm e}$ as a function of the coordinate $s$ along the string. Pressure differences $\Delta \Pi^{\rm e}_\mathrm{in}$ and $\Delta\Pi^{\rm e}_\mathrm{out}$ are, respectively, taken between the nematic point defect and the string interior or the defect-free bulk. (b) Pressure differences as a function of the nematopolar coupling parameter $\chi$. Dashed lines: analytical estimates, see text. Parameters as in Fig. \ref{['fig:single-defect']}.
• Figure 3: Charge-dependent hydrodynamic forces control the length and stability of topological strings. (a) Heatmap of the antisymmetric part of the Ericksen stress $\sigma^{\rm e,a}_{xy}$ for $\pm1$ strings ($+$/$-$). Arrows: polar field $\mathbf{p}$. (b) Longitudinal forces resulting from $\mathsf{\sigma}^\mathrm{e,a}$ as a function of the coordinate $s$ along $\pm1$ strings (red/blue). (c) Heatmap of the polar order parameter $p$ for an unstable $-1$ topological string. White arrows: $\mathbf{p}$. (d) Effective pressure $\Pi^{\rm e}$ adjacent to an unstable $-1$ topological string. Inset: zoom on unstable wiggling string. White lines indicate where profiles of $\Pi^{\rm e}$ were measured (line styles match). Arrows indicate net forces resulting from pressure imbalance. Parameter values: As in Fig. \ref{['fig:single-defect']} and $\chi=0.1$ (a,b) and $\chi=0.15$ (c,d). Scale bars: unit length.
• Figure 4: States of coexisting strings are stabilized by hydrodynamic repulsion. Heatmap of the polar order parameter $p$ for steady-state configurations. ( a) String lattice for $\tau^{-1}=0.5$. ( b) Coexistence between integer-charge stings and loops for $\tau^{-1}=2$. Parameter values: $\rho_0=0.8$ and $\chi=0.15$, scale bar: unit length.
• Figure 5: Active stress generates spontaneous flows and align nematic point defects with topological strings in nematopolar fluids. (a) Stability boundary of the homogeneous, uniformly orientationally ordered state as a function of the nematopolar coupling parameter $\chi$. Snapshots of numerical solutions for $\zeta=5 \times 10^{-4}$ (pentagon) and $\zeta = 0.02$ (star). Parameters: $\rho_0=0.6$ and $\tau^{-1}=0$. (b) In the presence of renewal, defect lattices with either vortices or lattices are stabilized. Same parameters as in Fig. \ref{['fig:multiple-defect']}a except for $\tau^{-1}=5$ and $\zeta=5 \times 10^{-4}$ (top), $\zeta=-5 \times 10^{-4}$ (bottom). Scale bars: unit length.