Long-time anomalous swimmer diffusion in smectic liquid crystals

The dynamics of self-locomotion of active particles in aligned or liquid crystalline fluids strongly deviates from that in simple isotropic media. We explore the long-time dynamics of a swimmer moving in a three-dimensional smectic liquid crystal and find that the mean-square displacement transverse to the director exhibits a distinct logarithmic tail at long times. The scaling is distinctly different from that in an isotropic or nematic fluid and hints at the subtle but important role of the director fluctuation spectrum in governing the long-time motility of active particles. Our findings are based on a generic hydrodynamic theory and Brownian dynamics computer simulation of a three-dimensional soft mesogen model.

Figure 1

MSDs of a swimmer moving through a nematic and smectic medium. Shown is the contribution (a) perpendicular and (b) parallel to the layer normal at various temperatures $T^{*}$. The rod densities corresponding to the various phases are ${\rho }^{*}=0.22$ (Iso), ${\rho }^{*}=0.3$ (N), and ${\rho }^{*}=0.5$ (SmA). The long-time dynamics transverse to the director is anomalous and exhibits a distinct long-time logarithmic behavior with scaling exponent $\alpha \approx 1$ for the nematic and $\alpha \approx 0.5$ for the smectic phase. For the isotropic phase, standard long-time swimmer diffusion $\langle \mathrm{\Delta }{\mathbf{r}}^2\rangle \propto \tau$ is observed, as it should.

Figure 2

Van Hove correlation function $G(z_{\parallel },\tau )$ for the host rods forming a (a) nematic phase ($T^{*}=1.8$, ${\rho }^{*}=0.3$) and (b) smectic phase ($T^{*}=1.8$, ${\rho }^{*}=0.5$). Curves are depicted at various time $\tau$. The distinct shoulders at discrete rod lengths $\ell$ indicate the diffusive barriers imparted by the lamellar microstructure of the smectic phase.