Active Extensile Stress Promotes 3D Director Orientations and Flows

We use numerical simulations and linear stability analysis to study an active nematic layer where the director is allowed to point out of the plane. Our results highlight the difference between extensile and contractile systems. Contractile stress suppresses the flows perpendicular to the layer and favors in-plane orientations of the director. By contrast extensile stress promotes instabilities that can turn the director out of the plane, leaving behind a population of distinct, in-plane regions that continually elongate and divide. This supports extensile forces as a mechanism for the initial stages of layer formation in living systems, and we show that a planar drop with extensile (contractile) activity grows into three dimensions (remains in two dimensions). The results also explain the propensity of disclination lines in three dimensional active nematics to be of twist type in extensile or wedge type in contractile materials.

Figure 1

Snapshots from simulations of 2D active nematics layers. Color denotes the magnitude of the in-plane order ($S_{\mathrm{in}}=[1-n_z^2{]}^{1/2}$) from in-plane (black) to out-of-plane (white): (a) extensile stress, defects in cyan. 3D twist-type defects are represented by circles. Inset: contractile stress, $2\mathrm{D}\pm 1/2$ topological defects are shown in white. (b) Area fraction of regions with out of plane director as a function of activity. (c) In-plane $u_{\mathrm{in}}$ and out-of-plane $u_z$ speed for contractile (yellow, $u_z=0$) and extensile (pink, orange) driving. (d) Variation of the number of snakes ${\mathcal{N}}_s$ with time. (e) Number of snakes as a function of activity at steady state. (f) Distribution of the angle between the director and the tangent to the boundary of snakes. The histogram peaks at $\cos (\delta )=1$, indicating parallel alignment of the director with the boundary. (g) Distribution of the angle $\alpha$ in extensile systems. (h) Director in a snake. Cyan outline: moving along the blue dashed arrow, crossing the width of a snake, the director twists by $\pi$. Yellow outline: top view of the director across the width of a snake. Magenta outline: 3D twist type defects are commonly found at the ends of the snakes. The blue circle shows the core of the defect, $\widehat{t}$ represents the normal to the layer, and the blue dashed line connects the center of the twist type defect to the position with an in-plane director. $\alpha$ is the angle between this line and the local director. The director rotates out of the plane around the rotation vector $\mathrm{\Omega }$ to form the twist defect.

Figure 2

(a) Average flow along the direction of the elongation of snakes (measured at the ends, i.e., within the yellow box). The origin corresponds to the blue dot and $x>0$ always corresponds to outside the snake. $⟨{\mathbf{u}}_{\mathrm{in}}·\widehat{r}⟩>0$, indicating that active flows elongate the snakes. (b) Flows around a snake. (c)–(e) Evolution of a snake. Elongated snakes undergo a bend instability and form defects. The director then moves to the third direction dividing the snake into two with twist-type defects at their ends.

Figure 3

Twist angle $\beta$ in (a) 2D, (b) 3D. In the extensile system most defects are twist type ($\beta =\pi /2$) while in contractile systems most defects are wedge type ($\beta =0$, $\pi$). Right and left axis shows $p(\cos [\beta ])$ in the contractile and extensile system.

Figure 4

3D views of a growing drop with (a) contractile, (b) extensile activities. Defects are represented in cyan, green shows the out-of-plane concentration, and the red shading indicates out-of-plane speed. [Note (b) is a close-up.] (c) The difference in the number of out-of-plane particles in an active growing system ($\zeta \ne 0$) and in a passive growing system ($\zeta =0$) for extensile (solid lines) and contractile (dash-dotted lines) activity. The semitransparent colors show error bars calculated using the standard deviation from five simulations.