Diffusive behaviors of circle-swimming motors

Spherical catalytic micromotors fabricated as described in Wheat et al. [Langmuir 26, 13052 (2010)] show fuel concentration dependent translational and rotational velocity. The motors possess short-time and long-time diffusivities that scale with the translational and rotational velocity with respect to fuel concentration. The short-time diffusivities are two to three orders of magnitude larger than the diffusivity of a Brownian sphere of the same size, increase linearly with concentration, and scale as $v^2/2\omega$. The measured long-time diffusivities are five times lower than the short-time diffusivities, scale as $v^2/\left\{2D_r\left[1+{(\omega /D_r)}^2\right]\right\}$, and exhibit a maximum as a function of concentration. Maximums of effective diffusivity can be achieved when the rotational velocity has a higher order of dependence on the controlling parameter(s), for example fuel concentration, than the translational velocity. A maximum in diffusivity suggests that motors can be separated or concentrated using gradients in fuel concentration. The decrease of diffusivity with time suggests that motors will have a high collision probability in confined spaces and over short times; but will not disperse over relatively long distances and times. The combination of concentration dependent diffusive time scales and nonmonotonic diffusivity of circle-swimming motors suggests that we can expect complex particle responses in confined geometries and in spatially dependent fuel concentration gradients.

Figure 1

Representative traces of the 3-μm spherical bimetallic motor's path over 75 s at each concentration. The hydrogen peroxide concentration increases from (a) to (h) (0.063, 0.135, 0.253, 0.391, 0.5, 0.75, 1.0, 1.25 vol $\%$).

Figure 2

(a) Average bimetallic spherical micromotor velocity versus hydrogen peroxide concentration. The error bars represent one standard deviation of the ensemble of time averaged velocities. (b) Average bimetallic spherical micromotor rotational velocity versus hydrogen peroxide concentration. Each individual motor velocity (•) is plotted along with the mean value (□), and a linear fit of the velocity. The error bars represent one standard deviation of the ensemble of time averaged velocities.

Figure 3

Squared displacement of individual bimetallic spherical micromotors versus time for hydrogen peroxide concentrations of 0.135$\%$ (Δ), 0.253$\%$ (○), and 0.5$\%$ (□). For each concentration there is a short-time (open symbols) and a long-time (filled symbols) diffusivity region. The short-time region is marked by the sharp increase of the SD and corresponds to the motor completing half of a rotation. The second region is marked by dampened oscillations that correspond to displacement of the circular motor trajectories.

Figure 4

(a) MSD of bimetallic spherical micromotors versus time for all concentrations of hydrogen peroxide at short times ($t$ < π/ω). (b) MSD versus time at all times. The hydrogen peroxide concentrations shown are 0.063$\%$ (◊), 0.135$\%$ (Δ), 0.253$\%$ (○), 0.5$\%$ (□), 0.756$\%$ (×), 1.0$\%$ (*), and 1.25$\%$ (•). The slope of the MSD gives the effective diffusivity. The slope of the MSD at short times ($t$ < π/ω) is the short-time effective diffusivity. The slope of the MSD at long times ($t$ > π/ω) is the long-time effective diffusivity.

Figure 5

(a) Short-time effective diffusivity of bimetallic spherical micromotors scaled by Brownian diffusivity versus the controlling parameter, $v^2$/2ω, scaled by the Brownian diffusivity. (b) Short-time effective diffusivity of bimetallic spherical micromotors scaled by the Brownian diffusivity versus hydrogen peroxide concentration. The experimental data (○) are plotted along with steady Brownian dynamics simulations with exact experimental velocities (□), steady Brownian dynamics simulations with velocities determined from fits of experimental values (•), unsteady Brownian dynamics simulations with velocities determined from fits of experimental values (Δ), and the fit of Eq. (7) (solid line). The short-time effective diffusivity is the slope of the MSD at times less than π/ω shown in Fig. 4a.

Figure 6

(a) Long-time effective diffusivity of bimetallic spherical micromotors scaled by the Brownian diffusivity versus the controlling parameter, $v^2/\left\{2D_r\left[1+{(\omega /D_r)}^2\right]\right\}$, scaled by the Brownian diffusivity. (b) Long-time effective diffusivity of bimetallic spherical micromotors scaled by the Brownian diffusivity versus hydrogen peroxide concentration. The experimental data (○) are plotted along with steady Brownian dynamics simulations with exact experimental velocities (□), steady Brownian dynamics simulations with velocities determined from fits of experimental values (•), unsteady Brownian dynamics simulations with velocities determined from fits of experimental values (Δ), and the scaling shown in Eq. (6) (solid line). The long-time effective diffusivity is the slope of the MSD at times longer than π/ω shown in Fig. 4b.

Figure 7

(a) Space-field map of the normalized long-time effective diffusivity calculated from Eq. (6) as a function of hydrogen peroxide concentration, and order of power dependence of the rotational velocity on concentration (ω $=$ $K_1$$C^b$). The long-time diffusivity is normalized by the maximum diffusivity within the sample space. The translational velocity is linearly dependent on concentration for all values of $b$. (b) The long-time diffusivity calculated from Eq. (6) vs rotational diffusivity. The diffusivity is scaled by the maximum diffusivity within the sample space and the rotational diffusivity is scaled by the rotational velocity.