Diffusion-Based Height Analysis Reveals Robust Microswimmer-Wall Separation

Microswimmers typically move near walls, which can strongly influence their motion. However, direct experimental measurements of swimmer-wall separation remain elusive to date. Here, we determine this separation for model catalytic microswimmers from the height dependence of the passive component of their mean-squared displacement. We find that swimmers exhibit “ypsotaxis,” a tendency to assume a fixed height above the wall for a range of salt concentrations, swimmer surface charges, and swimmer sizes. Our findings indicate that ypsotaxis is activity induced, posing restrictions on future modeling of their still-debated propulsion mechanism.

Figure 1

Salt-dependent motion above a planar wall. (a) Schematic of the experiment. We obtain the swimmer-wall separation $h$ from the measured translational diffusion coefficient $D$ of the swimmer and its theoretically predicted dependence on wall separation. (b)–(d) Effect of salt concentration $c_{\text{NaCl}}$ on the motion of $2.77\pm 0.08\mu \mathrm{m}$ colloids with $4.4\pm 0.2\mathrm{nm}$ Pt. All reported values are medians, and error bars denote first quartiles. (b) Diffusion coefficient (orange diamonds) and separation (purple hexagons) in the Brownian state in water with $c_{\text{NaCl}}$. Lines show theoretical predictions based on balancing electrostatics and gravity. (c) Diffusion coefficient (circles) and separation (squares) in the active state in aqueous 10% ${\mathrm{H}}_2{\mathrm{O}}_2$ with $c_{\text{NaCl}}$. Dotted lines indicate mean values. (d) Speed decrease in 10% ${\mathrm{H}}_2{\mathrm{O}}_2$ with $c_{\text{NaCl}}$. The solid line is a least-squares fit with $V=A+[B/(C+c_{\text{NaCl}})]$, where $A$ is $0.35\pm 0.09\mu \mathrm{m}/\mathrm{s}$, the remaining speed in high salt, $B$ is a prefactor, and $C$ is $0.09\pm 0.07\mathrm{mM}$, the ion concentration already present in solution, following from ionic diffusio-osmosis along the wall.

Figure 2

Swimmer base zeta potential and size dependence of propulsion above a planar wall. All reported values are medians, error bars denote first quartiles, and dotted lines represent mean values. All experiments were performed in aqueous 10% ${\mathrm{H}}_2{\mathrm{O}}_2$. (a),(b) The base zeta potential ${\zeta }_b$ of TPM colloids with diameters between $2.70\pm 0.06$ and $2.77\pm 0.08\mu \mathrm{m}$ and Pt coating thicknesses $\approx 4.4\pm 0.2\mathrm{nm}$ was varied through surface functionalization. (a) Diffusion coefficient (circles) and separation (squares) with ${\zeta }_b$. (b) Speed for the same ${\zeta }_b$ range as in (a). The inset shows speed with separation (purple circles), with the white circle marking the intersection of mean values and the rectangle around it denoting standard deviations. (c),(d) Variation of the radius $R$ of active TPM spheres with similar ${\zeta }_b$ and Pt (coating thickness $\approx 4.5\pm 0.2\mathrm{nm}$) affects (c) the diffusion coefficient and (d) swimmer-wall separation. The inset in (c) shows swim speed with $R$. Solid lines in (c) are fits with $a/R^b$ with $a=0.120\pm 0.004\mu {\mathrm{m}}^3/\mathrm{s}$ and $b=1.3\pm 0.2$ (main) and the expected $a/R$ [29] with $a=2.2\pm 0.4\mu {\mathrm{m}}^2/\mathrm{s}$ (inset). The dotted line in (d) shows mean separation ($0.32\pm 0.08\mu \mathrm{m}$).

Figure 3

Activity-induced ypsotaxis. (a) Diffusion coefficient $D$ with increasing ${\mathrm{H}}_2{\mathrm{O}}_2$ (fuel) concentration. (b) Equivalent swimmer-wall separation $h$ for the same fuel concentration range. (c) Swim speed $V$ for the same fuel concentration range as in (a),(b). All experiments were performed using $2.23\pm 0.11\mu \mathrm{m}$ diameter TPM spheres with $4.5\pm 0.2\mathrm{nm}$ Pt. All reported values are medians and error bars denote first quartiles.