Dynamical arrest in active nematic turbulence

TL;DR

This paper investigates defect-free active nematic turbulence, revealing dynamical arrest driven by emergent domain-wall networks. Using a defect-free director-based model and its unconstrained Q-tensor extension, it shows two regimes governed by the product $S\nu$: strong large-scale turbulence for $S\nu>1$ and grid-locked wall networks leading to arrested turbulence for $S\nu<-1$, with walls channeling flow and walls forming tree-like patterns. The authors introduce pseudo-defects—startpoints, branchpoints, and endpoints—with a ν-dependent pseudo-charge $Q(\nu)$ that constrains wall connectivity, enabling a topological description of arrest in the defect-free limit. They further demonstrate robustness by showing arrested-wall networks persist in the full Q-tensor model at small defect-core sizes and identify a defect-core threshold near $\epsilon/\ell_{a} \approx 0.25$ above which true defects nucleate and disrupt arrest. The work also uncovers a mechanism by which defects can nucleate near branchpoints, linking wall geometry to defect dynamics and highlighting the transitional regime between defect-free and defect-laden turbulence with potential experimental realization at low defect densities.

Abstract

Active fluids display spontaneous turbulent-like flows known as active turbulence. Recent work revealed that these flows have universal features, independent of the material properties and of the presence of topological defects. However, the differences between defect-laden and defect-free active turbulence remain largely unexplored. Here, by means of large-scale numerical simulations, we show that defect-free active nematic turbulence can undergo dynamical arrest. This state is characterized by an emergent network of nematic domain walls that channels coherent streams and suppresses chaotic flows. As the system evolves, the branched wall network produces a large-scale pattern with tree-like topological properties. We find that flow alignment -- the tendency of nematics to reorient under shear -- enhances large-scale chaotic jets in contractile rodlike systems while promoting dynamical arrest in extensile systems. We further show that dynamical arrest persists regardless of whether defects are prohibited by construction or simply fail to form due to a high energy cost of defect cores. Taken together, our findings reveal a striking pattern-formation mechanism, with labyrinths emerging from active turbulence, and illuminate the rich transitional regime between defect-free and defect-laden dynamics. These behaviors call for the experimental realization of active nematics at vanishing or low defect densities, and underscore that, in extensile rodlike nematics, topological defects enable turbulence by preventing dynamical arrest.

Figures (14)

• Figure 1: Strong and arrested regimes of active nematic turbulence.\ref{['Fig velocity-non-arrested']}--\ref{['Fig splay-bend-arrested']}, Snapshots from simulations of defect-free active nematic turbulence in contractile (\ref{['Fig velocity-non-arrested']}--\ref{['Fig splay-bend-non-arrested']}) and extensile (\ref{['Fig velocity-arrested']}--\ref{['Fig splay-bend-arrested']}) flow-aligning systems. Parameter values were set to $R=1$, $\nu=-1.1$, and $A=3.2\times10^5$. Top panels (\ref{['Fig velocity-non-arrested']},\ref{['Fig velocity-arrested']}) show the flow field; black curves are streamlines, and the color indicates the speed (see \ref{['movies']} and \ref{['movies']}). Middle panels (\ref{['Fig Frank-non-arrested']},\ref{['Fig Frank-arrested']}) show the Frank free energy density $\sim|\nabla\theta|^2$, with high-intensity lines corresponding to nematic domain walls (see \ref{['movies']} and \ref{['movies']}). Bottom panels (\ref{['Fig splay-bend-non-arrested']},\ref{['Fig splay-bend-arrested']}) are zooms highlighting the type of nematic distortion as well as the interplay between nematic walls and flows. The gray-scale background is the line integral convolution representation of the director field $\bm{n}$. Magenta and cyan intensities respectively represent splay $(\nabla\cdot\bm{n})^2$ and bend $|\nabla\times\bm{n}|^2$ contributions to the Frank energy density. The black arrows represent the flow field $\bm{v}$, which localizes along the nematic walls in the arrested regime. Black circles indicate stagnation points of the flow. White scale bar represents the selected wavelength $\lambda_i$. \ref{['Fig velocity-spectra']}--\ref{['Fig correlation-times']}, Spectra characterizing fully-developed active nematic turbulence (see details in \ref{['spectra']}). The lines in (\ref{['Fig Frank-spectra']},\ref{['Fig correlation-times']}) represent a smoothed (Gaussian) interpolation of the computed data points. We compare the flow-aligning contractile (red, as in \ref{['Fig velocity-non-arrested']}--\ref{['Fig splay-bend-non-arrested']} and \ref{['movies']}) and extensile (blue, as in \ref{['Fig velocity-arrested']}--\ref{['Fig splay-bend-arrested']} and \ref{['movies']}) cases with the $\nu=0$ case (black, as in Fig. S1SM and \ref{['movies']}), for which contractile and extensile stresses are equivalent up to a rotationEdwards2009Giomi2014aLavi2024 . \ref{['Fig velocity-spectra']}, Velocity power spectrum on a log-log scale, showing (i) the universal low-$q$ scaling law and (ii) the distinct organization of flows across scales in the different cases. The wider scaling regime in the contractile case captures the strong large-scale jets (see \ref{['Fig velocity-non-arrested']}). The peak in the extensile case is representative of wall streams (see \ref{['Fig velocity-arrested']}). \ref{['Fig Frank-spectra']}, Frank energy spectrum, showing that (i) the selected wavelength (peak position) depends on $\nu$ but not on the sign of active stress, and (ii) the peak width depends on the sign of active stress when $\nu\neq0$. \ref{['Fig correlation-times']}, Spectrum of correlation times associated with the flow $\bm{v}$ (light colored points and lines) and the nematic tensor $\hat{q}_{\alpha\beta}$ (darker points and lines). This log-log plot reveals strong differences in decay times between the regimes, as well as the differences between the flow and nematic tensor within a regime. Correlation times are extracted from exponential fits to the corresponding space-time autocorrelation functions in Fourier space (see \ref{['spectra']}).
• Figure 2: Wall coarsening, branching, and dynamical arrest. Sequential snapshots from a single simulation in the extensile flow-aligning regime, showing the coarsening of bending walls (\ref{['Fig wall-coarsening']}, top row), the zigzag instability of the coarsened configuration, followed by wall branching and rapid tip growth (\ref{['Fig wall-branching']}, second row), and the evolution towards the arrested tree pattern (\ref{['Fig wall-arrest']}, bottom row). The initial condition was set to a striped pattern of wavelength that matches the typical length selected in the chaotic regime (see first and final panels). A perturbation along the $y$-axis triggers the coarsening of one-dimensional stripes, confirming that straight anti-parallel walls do not have a preferred separation. The length selected in the tree-like pattern is determined by two-dimensional interactions, with growing branches avoiding preexisting walls. Parameter values are: $R=1$, $S=1$, $\nu=-1.1$, and $A=19692$ (chosen so that the system size roughly equals six times the selected wavelength).
• Figure 3: Domain wall nodes and their pseudo-topology. In all diagrams, black directed lines represent nematic bending walls, with gray lines tracing the director and cyan indicating the bending energy. \ref{['Fig startpoint']}--\ref{['Fig endpoint']}, Nodes are either startpoints (green, negative pseudo-defects), branchpoints (blue, negative pseudo-defects), or endpoints (red, positive pseudo-defects). The orange curves illustrate the paths excluding walls used to define the pseudo-charge (\ref{['pseudo-charge']}). \ref{['Fig inception']}--\ref{['Fig deletion']}, Two ways in which a startpoint-endpoint pair of opposite charge can be born. The inverse processes of (\ref{['Fig inception']}) (complete wall dissolution) and (\ref{['Fig deletion']}) (local wall completion) result in the annihilation of such pairs. \ref{['Fig branching']}, Wall branching gives birth to a branchpoint-endpoint pair, also of opposite charge. A connected pair may annihilate via branch retraction. \ref{['Fig dislocation']}, A branchpoint (\ref{['Fig branch point']}) transitions into a startpoint when one outgoing wall disjoins. The inverse process corresponds to a startpoint (\ref{['Fig startpoint']}) joining with a bare wall. \ref{['Fig T1']}, Connected branchpoints can shrink their connecting branch, transiently creating a $-2Q$ pseudo-charged structure, exchange outgoing walls and extend again in the perpendicular direction. The processes (\ref{['Fig inception']}--\ref{['Fig T1']}) are further illustrated in \ref{['animations']} to \ref{['animations']}.
• Figure 4: Motifs of arrested wall networks and experimental snapshots of microtubule-kinesin active nematics. In all diagrams, lines, nodes and colors are as defined in \ref{['Fig topology']}. \ref{['Fig anchoring']}--\ref{['Fig uncommon2']}, Basic network motifs. The anchoring motif (\ref{['Fig anchoring']}) is made of an endpoint and branchpoint that meet head-on, with the endpoint trapped between the two outgoing walls of the branchpoint. In motif (\ref{['Fig common']}) the endpoint meets the branchpoint from one of its sides, i.e. between the incoming wall and an outgoing wall. The dashed line traces a weak distortion, indicating that the wall associated with the endpoint tends to align its direction with the outgoing wall on the opposite side of the branchpoint. The motifs (\ref{['Fig uncommon1']},\ref{['Fig uncommon2']}) involve a single pseudo-defect interacting with a bare wall. These, along with (\ref{['Fig common']}), do not follow the tendency to have strictly antiparallel walls. \ref{['Fig Composite']}, Composite motif schematic (left) and one formed spontaneously in a simulation (right). The stream plot on the right represents the flow, with black indicating maximal $|\bm{v}|$ and full transparency indicating $|\bm{v}|=0$. The gray background is the line-integral-convolution representation of the nematic director $\bm{n}$. Parameter values and color legend for splay and bend distortions are as in \ref{['Fig splay-bend-arrested']}. \ref{['Fig Chandrakar']}--\ref{['Fig Guillamat']}, Raw fluorescence images from experiments (left panels) and overlaid schematic drawings (right panels) depicting domain walls, pseudo-defects and actual $\pm1/2$ defects in white. \ref{['Fig Chandrakar']}, Taken from a movie in Chandrakar2020 (courtesy of Guillaume Duclos), which shows the evolution of the microtubule-based nematic following the bending instability of the aligned state. \ref{['Fig Guillamat']}, Taken from a movie in Guillamat2016 (courtesy of Pau Guillamat), which shows a turbulent transient with all types of pseudo-defects and actual nematic defects. Note how walls may also originate from true $+1/2$ defects and be absorbed by true $-1/2$ defects.
• Figure 5: Ageing of the arrested wall network.\ref{['Fig initial']},\ref{['Fig final']}, Skeleton of the domain walls (black) with startpoints, branchpoints and endpoints (green, blue and red triangular nodes) at an early time (\ref{['Fig initial']}) and a late time (\ref{['Fig final']}). The detection of the network skeleton and its nodes is described in \ref{['image processing']}, \ref{['Fig node detection']}. \ref{['Fig branchpoint-evolution']}, Evolution of the number of startpoints (green), branchpoints (blue) and endpoints (red). In the initial transient, sequential 'zig-zag' instabilities result in the proliferation of both branchpoints and endpoints. Once the wall pattern establishes a wavelength, the system ages slowly as some endpoints retract and annihilate with their connected branchpoint, while others extend (\ref{['movies']}). Throughout the simulation, there are frequent transitions between branchpoints and startpoints, though the number of startpoints remains low. Additionally, the detection algorithm is not perfect, occasionally misidentifying endpoints or branchpoints as startpoints and vice versa. Parameter values were set to $R=1$, $S=1$, $\nu=-0.9$, and $A=3.2\times 10^5$.