Shape dependent phoretic propulsion of slender active particles

We theoretically study the self-propulsion of a thin (slender) colloid driven by asymmetric chemical reactions on its surface at vanishing Reynolds number. Using the method of matched asymptotic expansions, we obtain the colloid self-propulsion velocity as a function of its shape and surface physicochemical properties. The mechanics of self-phoresis for rod-like swimmers has a richer spectrum of behaviors than spherical swimmers due to the presence of two small length scales, the slenderness of the rod and the width of the slip layer. This leads to subtleties in taking the limit of vanishing slenderness. As a result, even for very thin rods, the distribution of curvature along the surface of the swimmer, namely, its shape, plays a surprising role in determining the efficiency of propulsion. We find that thin cylindrical self-phoretic swimmers with blunt ends move faster than thin prolate spheroid shaped swimmers with the same aspect ratio.

Figure 1

Comparison of swimmer propulsion speeds. The propulsion speed $U$ of a half-coated rod as a function of the slenderness ratio $\epsilon$. The speed $U$ is expressed in units of $|{\mu }^{*}{\alpha }^{*}/D|$. The blue solid line is the asymptotic result $\left(\epsilon \to 0\right)$ for the cylindrical rod [see Eq. (45)], while the black diamonds indicate the asymptotic result for the spheroid [see Eq. (40) and Ref. 22]. The dashed red line is the plot of the exact propulsion speed obtained in Ref. [24] of a half-coated spheroid. (inset) Plot of the ratio of the asymptotic speeds of the cylindrical rod to the spheroid asymptotic against the slenderness ratio [see Eqs. (40) and (45)].

Figure 2

(a) A slender catalytic microswimmer showing the variation of catalytic coverage and shape along the long axis oriented in the $z$ direction. The gray (shaded) half of the swimmer surface is made of a different catalyst than the unshaded half. The catalysts produce (or consume) molecules of the reactive species at different rates leading to an asymmetric distribution of molecules near the swimmer. ${\mathit{V}}^{\mathrm{slip}}$ is the phoretic slip velocity induced by the coupling of the asymmetric distribution of the molecules and their short-range interaction with the swimmer surface. $U{\widehat{\mathit{e}}}_z$ is the axial translational velocity of the swimmer in the laboratory frame. (b) The generic thin rod with rounded ends which interpolates between a thin spheroid and a thin cylinder. The limit $(\chi =0)$ is the spheroid, while $(\chi =1)$ is the cylindrical rod.

Figure 3

Slender swimmer showing the asymptotic regions. $\lambda =L^{*}/l$ and $\epsilon =a/l$ are dimensionless small parameters, where $2l$ is the swimmer length, $2a$ its diameter, and $L^{*}$ the range of the interaction between reactant (product) molecules and the swimmer surface (typically of the order of molecular size $\sim \AA )$.

Figure 4

Examples of slender self-phoretic swimmer shapes with different physicochemical properties on their surfaces. (a) $\chi =0$, (b) $0\le \chi \le 1$, and (c) $\chi =1$.

Figure 5

Swimmer blunt end, showing the matching regions. $\delta =1-\chi$ is the width of the near end region where the shape function decays to zero. Note that the intermediate layer (region I) vanishes near the ends.

Figure 6

Fore-aft asymmetric swimmer with uniform physicochemical properties over its surface.