Locomotion and transport in a hexatic liquid crystal

The swimming behavior of bacteria and other microorganisms is sensitive to the physical properties of the fluid in which they swim. Mucus, biofilms, and artificial liquid-crystalline solutions are all examples of fluids with some degree of anisotropy that are also commonly encountered by bacteria. In this article, we study how liquid-crystalline order affects the swimming behavior of a model swimmer. The swimmer is a one-dimensional version of G. I. Taylor's swimming sheet: an infinite line undulating with small-amplitude transverse or longitudinal traveling waves. The fluid is a two-dimensional hexatic liquid-crystalline film. We calculate the power dissipated, swimming speed, and flux of fluid entrained as a function of the swimmer's wave form as well as properties of the hexatic film, such as the rotational and shear viscosity, the Frank elastic constant, and the anchoring strength. The departure from isotropic behavior is greatest for large rotational viscosity and weak anchoring boundary conditions on the orientational order at the swimmer surface. We even find that if the rotational viscosity is large enough, the transverse-wave swimmer moves in the opposite direction relative to a swimmer in an isotropic fluid.

Figure 1

Illustration of an idealized one-dimensional swimmer immersed in a two-dimensional hexatic liquid crystal (not to scale). The propagating transverse wave has wave vector $q$, frequency $\omega$, and amplitude $b\ll 2\pi /q$. The angle field $\theta$ is defined up to rotations by $2\pi /6$ by the angle between the $x$ axis and any of the lines connecting a particle at $(x,y)$ with its six nearest neighbors.

Figure 2

Log-log plot of dimensionless power $P_T/\left(\mu q{\omega }^2{b}^2\right)$ vs Ericksen number Er for $\gamma =\mu$ and various dimensionless anchoring strengths $w=W/\left(Kq\right)$ for a transverse wave.

Figure 3

Log-log plot of dimensionless power $P_{\mathrm{L}}/\left(\mu q{\omega }^2{b}^2\right)$ vs Ericksen number Er for $\gamma =\mu$ and various dimensionless anchoring strengths $w=W/\left(Kq\right)$ for a longitudinal wave.

Figure 4

Semilog plot of dimensionless swimming speed $U/\left(\omega q{b}^2\right)$ vs $\gamma /\mu$ for $\mathrm{Er}\ll 1$ (left panel), $\mathrm{Er}=1$ (middle panel), and $\mathrm{Er}\gg 1$ (right panel), for a transverse wave. Note that the strong anchoring case $W/\left(Kq\right)=\infty$ is independent of Er and that anchoring effects are small once Er is of order unity.

Figure 5

Log-log plot of dimensionless swimming speed $U/\left(\omega q{b}^2\right)$ for a transverse wave with $\gamma =\mu$ and various dimensionless anchoring strengths $W/\left(Kq\right)$.

Figure 6

Log-log plot of dimensionless flux $Q/\left(\omega b^2\right)$ for a transverse wave with $\gamma =\mu$ and various dimensionless anchoring strengths $W/\left(Kq\right)$.

Figure 7

Log-log plot of dimensionless flux $Q/\left(\omega b^2\right)$ vs $\gamma /\mu$ for a transverse wave for $\mathrm{Er}\ll 1$ (left panel), $\mathrm{Er}=1$ (middle panel), and $\mathrm{Er}\gg 1$ (right panel). Note that the strong anchoring case $W/\left(Kq\right)=\infty$ is independent of Er, and the flux is very small for all cases when $\gamma /\mu <0.1$.

Figure 8

Dimensionless flux for a longitudinal swimmer vs Ericksen number Er for $\gamma =\mu$ and various dimensionless anchoring strengths $W/\left(Kq\right)$.

Figure 9

Left panel: Plot of dimensionless first-order velocity field $\omega {\mathbf{v}}^{\left(1\right)}/q$ for ${\epsilon }_b=0.3$ for a transverse traveling wave in an isotropic viscous fluid. The reference frame is the swimmer's frame, and the wave is moving to the right. The color denotes the magnitude of the velocity field. Right panel: Plot of dimensionless first-order velocity field $\omega {\mathbf{v}}^{\left(1\right)}/q$ for ${\epsilon }_a=0.3$ for a longitudinal traveling wave in an isotropic viscous fluid. The wave is traveling to the right.

Figure 10

Plots of the dimensionless velocity field $\omega \mathbf{v}/q$ for ${\epsilon }_b=0.3$ for a transverse traveling wave in a hexatic film with (left panel) $\gamma \ll \mu$ and (right panel) $\gamma \gg \mu$. In both panels, the reference frame is the swimmer's frame, the wave is traveling to the right, and the color denotes the magnitude of the velocity field. Note the direction of the flow for large values of $q\left|y\right|$. The swimmer swims to the left for $\gamma \ll \mu$ and to the right for $\gamma \gg \mu$.