Activity-assisted barrier crossing of self-propelled colloids over parallel microgrooves

We report a systematic study of the dynamics of self-propelled particles (SPPs) over a one-dimensional periodic potential landscape $U_0\left(x\right)$, which is fabricated on a microgroove-patterned polydimethylsiloxane (PDMS) substrate. From the measured nonequilibrium probability density function $P(x;F_0)$ of the SPPs, we find that the escape dynamics of the slow rotating SPPs across the potential landscape can be described by an effective potential $U_{\text{eff}}(x;F_0)$, once the self-propulsion force $F_0$ is included into the potential under the fixed angle approximation. This work demonstrates that the parallel microgrooves provide a versatile platform for a quantitative understanding of the interplay among the self-propulsion force $F_0$, spatial confinement by $U_0\left(x\right)$, and thermal noise, as well as its effects on activity-assisted escape dynamics and transport of the SPPs.

Figure 1

(a) A schematic side view of a Pt-${\mathrm{SiO}}_2$ Janus particle of diameter $d$ moving over a microgroove-patterned substrate with groove spacing $L$ and period $\lambda$. The gray hemisphere indicates the Pt coating and the black arrow indicates the direction of the self-propulsion velocity $v_0$. (b,c) Two representative trajectories of the Janus particle in the aqueous solution of 0.2% ${\mathrm{H}}_2{\mathrm{O}}_2$ [(b) duration $\sim 200$ s] and in water [(c) duration $\sim 0.5$ h], respectively. The data are taken from Sample S1. The parallel stripes in the background show the microgroove pattern with alternating ridges (darker stripes) and grooves (brighter stripes). The dark circle shows the Janus particle image at the beginning of the trajectory. The red dots indicate the end of the trajectory. Also shown in (a) and (b) are the coordinate systems used in the experiment. The scale bars are 2 $\mu \mathrm{m}$.

Figure 2

(a) Comparison of the effective potentials $U_{\text{eff}}(x;v_0)/k_{B}T$ that are measured in experiments (open symbols) and calculated in simulation (dashed lines) across one microgroove period. Here $x=0$ is set at the bottom center of the microgroove. The data are taken from Sample S1. The color-coded symbols show the obtained $U_{\text{eff}}(x;v_0)/k_{B}T$ for increasing values of $v_0$. (b) Measured barrier height difference $\mathrm{\Delta }{E}_b/k_{B}T=[E_b\left(0\right)-E_b\left(v_0\right)]/k_{B}T$ as a function of the normalized self-propulsion work $W/k_{B}T=v_0\lambda /\left(2D_0\right)$ for Samples S1 (black circles) and S2 (red triangles). The solid line shows a linear fit $\mathrm{\Delta }{E}_b=aW$ with $a=0.42$ to the data points with small values of $W/k_{B}T$. The error bars show the experimental uncertainties of the measurements. (c) Reconstructed potential $U_0\left(x\right)/k_{B}T$ from the measured $U_{\text{eff}}(x;v_0)/k_{B}T$ using Eq. (3) for Sample S1 with different values of $v_0$. The black solid line shows the directly measured $U_0\left(x\right)/k_{B}T$ for passive particles with $v_0=0$.

Figure 3

(a) Measured PDF $P\left(t_d\right)$ of the dwell time $t_d$ for Sample S1. In the plot, $t_d$ is normalized by its mean value $\langle t_d\rangle$ obtained for different values of $v_0$. The solid line shows an exponential fit $P\left(t_d\right)\propto exp(-t_d/\langle t_d\rangle )$ to the tail part of the measured $P\left(t_d\right)$. (b) Experimentally measured (black and red symbols) and numerically simulated (blue symbols) mean dwell time, $\langle t_d\rangle /t_0$, as a function of the effective barrier height $E_b\left(v_0\right)/k_{B}T$ obtained for different values of $v_0$. Here $\langle t_d\rangle$ is normalized by the relaxation time $t_0$. The two rightmost data points (black circle and blue triangle) are obtained for passive particles with $v_0=0$. The solid line shows an exponential fit $\langle t_d\rangle /t_0\propto exp(E_b/k_{B}T)$ to the black circles. The error bars show the experimental uncertainties of the black circles.