Active nematic materials with substrate friction

Active turbulence in dense active systems is characterized by high vorticity on a length scale that is large compared to that of individual entities. We describe the properties of active turbulence as momentum propagation is screened by frictional damping. As friction is increased, the spacing between the walls in the nematic director field decreases as a consequence of the more rapid velocity decays. This leads to, first, a regime with more walls and an increased number of topological defects, and then to a jammed state in which the walls deliminate bands of opposing flow, analogous to the shear bands observed in passive complex fluids.

Figure 1

(a) Steady-state director field configurations for different values of activity $\zeta$ and friction coefficient $\gamma$ for a 1D system with periodic boundary conditions. The increase in thresholds of ${\zeta }_{C_{\mp }}$ is calculated analytically as a function of $\gamma$ and is plotted as continuous lines. The symbols on these lines represent the thresholds obtained numerically by simulating the full Eqs. (1) and (2). (b) Director angle measured with respect to the flow direction (top) and magnitude of the velocity field associated with a single bend deformation (bottom). The vertical continuous lines, drawn at $l_{\text{wall}}+2l_{\text{diss}}$, show the screening effects on the velocity field.

Figure 2

Left panel: director field and topological defects (with $+1/2$ and $-1/2$ defects denoted by red and blue dots); right panel: vorticity, denoted by the color scale from +ve (red) to $-\mathrm{ve}$ (blue) normalized to its maximum value, and velocity field (arrows). (a), (b) No solid friction $\left(\gamma =0\right)$; (c), (d) $\gamma =0.01$; and (e), (f) $\gamma =0.5$. Panels (a), (c), and (e) correspond to the left bottom quarter of panels (b), (d), and (f), respectively. The formation of a pair of walls is identified in (a).

Figure 3

(a) The RMS velocity decreases and (b) the defect density increases as the substrate friction, $\gamma$ increases. At very large $\gamma$, parallel walls become more stable with decreasing velocity and fewer defect creation events. (c) The distance between the walls, $l_n$ decreases when it becomes comparable to the dissipation screening length $l_{\text{diss}}=\sqrt{\mu /\gamma }$. The shaded area between $l_n^{\text{wet}}$ and $2l_n^{\text{wet}}$ shows the transition region where the screening effects start to dominate.

Figure 4

Effect of friction on the director field (and topological defects) and flow field (both vorticity and streamlines) at two different temperatures are illustrated here. On the left-hand side [panels (a)–(d)], $A=-0.025$ or, equivalently, $\beta =AC/B^2=-0.083$, corresponding to a temperature $T<T^{*}$, where $T^{*}$ is the temperature close to isotropic nematic transition temperature $T_{NI}$, about which the free energy is expanded. Below $T^{*}$ the isotropic phase is completely unstable. On the right-hand side [panels (e)–(h)], $A=0.0124$ or $\beta =AC/B^2=0.0413$, corresponding to a temperature $T$ very close to $T^{**}$, where $T^{**}$ is the highest temperature at which the nematic phase exists. Above $T^{**}$ the nematic phase is completely unstable.