Purely viscous acoustic propulsion of bimetallic rods

Synthetic microswimmers offer models for cell motility and their tunability makes them promising candidates for biomedical applications. Here, we measure the acoustic propulsion of bimetallic microrods that, when trapped at the nodal plane of a MHz acoustic resonator, swim with speeds of up to $300\mu \mathrm{m}{\mathrm{s}}^{-1}$. While past acoustic streaming models predict speeds that are more than one order of magnitude smaller than our measurements, we demonstrate that the acoustic locomotion of the rods is driven by a viscous, nonreciprocal mechanism relying on shape anisotropy akin to that used by swimming cells and that reproduces our data with no adjustable parameters.

Figure 1

(a) Diagram of the acoustic resonators. The amplitude of the acoustic field displacement at the pressure nodal plane is denoted by $\xi$ so the corresponding transverse velocity is ${\mathit{U}}_e=\xi \omega exp\left(i\omega t\right)\widehat{\mathit{z}}$. (b) Geometry of the rods of types R1 and R2. The values of $L$, $L_1$, $L_2$, and $L_G$ and corresponding standard deviations for type R2 are provided in the Supplemental Material [43]. (c) Scanning electron microscopy pictures of bimetallic rods: (i) type R2-A; (ii) type R2-B; (iii) type R2-C; (iv) type R1, $L=587\mathrm{nm}$; (v) type R1, $L=2458\mathrm{nm}$; (vi) type R1, $L=990\mathrm{nm}$. The contrast between the gold stripe (yellow) and the ruthenium sections (gray) has been enhanced. In (c)(i)–(vi) the white scale bars are $300\mu \mathrm{m}$ long.

Figure 2

(a)–(d) Propulsion speed of type R1 rods, $V_{\mathrm{prop}}$, as a function of the acoustic displacement amplitude at the pressure nodal plane, $\xi$ (nm), for four different rod lengths. The experimental data are plotted in solid symbols. The viscous model depicted in the text [Eq. (4)], for which $V_{\mathrm{prop}}\sim {\xi }^2$, is shown in solid lines. Blue, red, and green symbols/lines correspond to the frequencies $f=2.17$, 3.17, and 5.0 MHz, respectively.

Figure 3

Propulsion speed of type R2 rods, $V_{\mathrm{prop}}$ ($\mu \mathrm{m}{\mathrm{s}}^{-1}$), as a function of the asymmetry distance $\delta$ (nm), for $f=3.7\mathrm{MHz}$. Due to fluctuations in the manufacturing process the length of type R2 rods lies in the range 539–1284 nm. The theory (viscous model) is plotted with a solid line for the mean value of the length $L=892$ nm. Inset: Dimensionless form of the plot.

Figure 4

(a) Equivalent ellipsoid model of the cylindrical rods used to derive an expression of the propulsion speed $V_{\mathrm{prop}}$. (b) Physical principle of the rectified viscous mechanism based on the combined effects of shape anisotropy and the flapping motion [47]. Regardless of the direction of the vertical velocity within a period of oscillation, the shape anisotropy yields a rectified force oriented towards the less dense end of the solid (green arrow).

Figure 5

Dimensionless propulsion velocity $V_{\mathrm{prop}}/\left(\xi \omega \right)$ for type R1 rods rescaled by the dimensionless field amplitude $\varepsilon$. The experimental data are plotted in solid symbols; the theory is plotted in solid lines. The color code is the same as in Fig. 2.