Vortex Pattern Stabilization in Thin Films Resulting from Shear Thickening of Active Suspensions

The need for structuring on micrometer scales is abundant, for example, in view of phononic applications. We here outline a novel approach based on the phenomenon of active turbulence on the mesoscale. As we demonstrate, a shear-thickening carrier fluid of active microswimmers intrinsically stabilizes regular vortex patterns of otherwise turbulent active suspensions. The fluid self-organizes into a periodically structured nonequilibrium state. Introducing additional passive particles of intermediate size leads to regular spatial organization of these objects. Our approach opens a new path toward functionalization through patterning of thin films and membranes.

Figure 1

Snapshots of the vorticity field $\omega (\mathbf{x},t)$ at $a=1.1$ and $n=3$ at arbitrarily chosen times $t$ for different values of the reference shear rate: (a) ${\dot{\gamma }}_2=0.1$, (b) ${\dot{\gamma }}_2=0.9$, and (c) ${\dot{\gamma }}_2=2.5$. Arrows denote the velocity field $\mathbf{v}$. Both quantities are rescaled for visualization purposes. (a) For very small ${\dot{\gamma }}_2$, the flow field settles into a stationary vortex lattice, where vortices are elongated along the director $\mathbf{n}$ shown as the dashed line in the inset. The structure can be represented by two wave vectors forming an angle $\phi =\pi /4$ (inset). For larger ${\dot{\gamma }}_2$, the flow field becomes increasingly irregular. In (b), the vortices are still visibly aligned along a common director $\mathbf{n}$ (inset), whereas the rotational order is lost in (c). The size of the snapshots is $16\pi \times 10\pi$.

Figure 2

Characteristic quantities as a function of $a$ and ${\dot{\gamma }}_2$ for different values of $n$. (a) Correlation time $\tau$ on a log scale. (b) Nematic order parameter $|q|$ for the elongated vortices. High nematic order for small values of ${\dot{\gamma }}_2$ signifies a globally ordered vortex structure with broken rotational symmetry. The gray circles in the plots for $n=3$ indicate the parameter values where the snapshots shown in Fig. 1 are taken. The column on the right-hand side includes the Newtonian case ($\nu =1$) for comparison.

Figure 3

Results obtained from the amplitude equations. (a) Maximum growth rate of perturbations with respect to the stationary solution for $n=3$ [Eq. (8)] and $n=5$ [67] as a function of the angle $\phi$ between the two modes representing the vortex pattern. (b) Amplitude $|A_{\mathrm{s}}|$ of the stationary solution at $\phi =\pi /4$ as a function of $a$ and ${\dot{\gamma }}_2$ for $n=3$ [Eq. (7)] and for $n=5$ [67].

Figure 4

Spatial organization of passive particles (dark dots) of diameter $d=0.06{\mathrm{\Lambda }}_{\mathrm{c}}$ at $\mathrm{St}=0.033$ in an active shear-thickening carrier fluid at time $t=16000$ after starting from random initial distributions. (a) Lighter particles (here half the fluid density, $R=1$) cluster in vortex centers, whereas (b) heavier particles (here twice the fluid density, $R=2/5$) accumulate between vortices. Snapshot size is $6\pi \times 6\pi$.