Defect Line Coarsening and Refinement in Active Nematics

Active matter is naturally out of equilibrium which results in the emergence of diverse dynamic steady states, including the omnipresent chaotic state known as the active turbulence. However, much less is known how active systems dynamically depart out of these configurations, such as get excited or damped to a different dynamic steady state. In this Letter, we demonstrate the coarsening and refinement dynamics of topological defect lines in three-dimensional active nematic turbulence. Specifically, using theory and numerical modeling, we are able to predict the evolution of the active defect density away from the steady state due to time-dependent activity or viscoelastic material properties, establishing a single length scale phenomenological description of defect line coarsening and refinement in a three-dimensional active nematic. The approach is first applied to growth dynamics of a single active defect loop, and then to a full three-dimensional active defect network. More generally, this Letter provides insight into the general coarsening phenomena between dynamical regimes in 3D active matter, with a possible analogy in other physical systems.

Figure 1

Coarsening dynamics of active nematic turbulence. (a) A snapshot of a defect line network (gray) during coarsening process. Additionally, selected defect line segments are drawn in different colors at earlier time intervals $50\mathrm{\Delta }{x}^2/(\mathrm{\Gamma }L)$ apart. (b) The director field rotates for an angle of $\pi$ around the defect lines: from $+1/2$ profiles (red), to twist (green) and $-1/2$ profiles (blue).

Figure 2

Shrinking and expanding dynamics of isolated active nematic defect loop. (a) Depending on activity $\alpha$, the defect loop either shrinks or expands in time; the panels show the defect loop as isosurface of degree of order (red), director field (yellow rods), and velocity field (green arrows). The defect loop has $+1/2$ profile at the left side, and $-1/2$ profile on the right side, which generates an overall self propulsion velocity and a local active stress on the loop toward the left. (b) Single loop radius as function of time for different activities. Radius is determined as the loop left-to-right dimension in panel (a). The dashed line is a fit [Eq. (2)] to the numerical data with fit parameters ${\tau }_{\mathrm{loop}}$ and activity-independent parameter $r_c^2/{\tau }_{\mathrm{loop}}$, obtaining $r_c^2/{\tau }_{\mathrm{loop}}=2.78\mathrm{\Gamma }L$. (c) Critical radius $r_c$ and initial radius $r_0=35.3\mathrm{\Delta }x$ determine shrinkage ($r_0<r_c$) and expansion ($r_0>r_c$) regimes. A linear dependence of $1/{\tau }_{\mathrm{loop}}$ (or equally $1/r_c^2$) on activity is obtained with the slope of $1/{\tau }_{\mathrm{loop}}=0.186\mathrm{\Gamma }\alpha$.

Figure 3

Coarsening dynamics of 3D active nematic turbulence. (a) Defect network at short time in the coarsening dynamics. A selected connected defect segment is colored in red. (b) Defect network at later time in the coarsening dynamics. Defect line segments are further apart and show smaller curvature compared to (a). (c) Inverse defect density over time for different activities. Dashed lines are fits of Eq. (5) to the simulation data. Black dashed line represents the activity-independent initial density dynamics. The fluctuations of the defect density at later times is the effect of the finite simulation volume but with a well determined average. (d) Steady-state defect density as dependent on activity; linear fit (dashed line) has the slope of $0.0065/L$. (e) Linear dependence of the inverse characteristic time on activity; the linear fit has coefficient of $0.036\mathrm{\Gamma }$. Points in (d) and (e) are obtained from fits in (c).

Figure 4

Active refinement induced by time-varying activity. Defect density (top graph) is shown for the three different time-varying activities (bottom graph). Note the matching colors. The dynamics is initiated at $t=0$ from a random director field, leading first to coarsening which upon an activity increase changes to refinement. Dashed lines are fits with Eq. (4).