Hydrodynamics Defines the Stable Swimming Direction of Spherical Squirmers in a Nematic Liquid Crystal

We present a study of the hydrodynamics of an active particle—a model squirmer—in an environment with a broken rotational symmetry: a nematic liquid crystal. By combining simulations with analytic calculations, we show that the hydrodynamic coupling between the squirmer flow field and liquid crystalline director can lead to reorientation of the swimmers. The preferred orientation depends on the exact details of the squirmer flow field. In a steady state, pushers are shown to swim parallel with the nematic director while pullers swim perpendicular to the nematic director. This behavior arises solely from hydrodynamic coupling between the squirmer flow field and anisotropic viscosities of the host fluid. Our results suggest that an anisotropic swimming medium can be used to characterize and guide spherical microswimmers in the bulk.

Figure 1

(a) A cartoon showing the squirmer in a nematic liquid crystal, defining the angle $\varphi$ used in the text, between the particle swimming direction ${\mathbf{u}}_0$ and the nematic director $\widehat{\mathbf{n}}$. Examples of (b) the trajectory in the $x\text{−}z$ plane and (c) $\varphi (t)$ observed in simulations of a puller ($\beta =+0.2$) and of a pusher ($\beta =-0.2$), with an initial orientation ${\varphi }_0=45°$.

Figure 2

(a) $\varphi (t)$ exhibits an $S$-shaped evolution, with a stable configuration $\varphi \approx 90°$ ($\varphi \approx 0°$) for a puller (pusher). (b) The rotational velocity $\mathrm{\Omega }(\varphi )$ is symmetric around $\varphi =45°$ and vanishes for $\varphi \to 0$ and 90°, in agreement with theoretical arguments (see text for details).

Figure 3

The rotational velocity $\mathrm{\Omega }(\beta )$ shows a linear dependence of the squirming parameter $\beta$ for a fixed $\varphi$. For $\beta =0$, $\mathrm{\Omega }$ takes a value $a\approx 8\times {10}^{-6}[\mathrm{rad}/(t)]$ (see inset and text for details).

Figure 4

The angle $\varphi (t)$ for particles with (a) homeotropic and (b) planar anchoring of the nematic director at the particle surface ($WR/K\approx 4$) for both a puller ($\beta =+5$) and a pusher ($\beta =-5$). The insets show the defect structure around a passive particle: (a) Saturn ring defect for a homeotropic anchoring and (b) two boojums for a planar anchoring at the particle surface. (The arrow denotes the orientation of the far-field nematic director).