Hydrodynamic Mechanism of Colloidal Propulsion through Momentum Exchange

Abstract

Propulsion of colloidal particles due to momentum transfer from localized surface reactions is investigated by solving the exact unsteady Stokes equation. We model the effect of surface reactions as either a {\it force dipole} acting on the fluid or a {\it pair force} acting on both the colloid and the fluid. Our analysis reveals that after a single reaction event the colloid's velocity initially decays as $\sim t^{-1/2}$, followed by a long-time tail decay $\sim t^{-5/2}$. This behavior is distinct from the $\sim t^{-3/2}$ decay seen for simple impulsively forced particles, a result of the force-free nature of the reaction mechanism. The velocity and transient dynamics are strongly controlled by the distance of the reaction from the colloid surface. For a colloid subject to periodic reactions, the theory predicts a steady-state velocity that is comparable to experimental results and previous simulations, suggesting that direct momentum transfer is a relevant mechanism for self-propulsion in systems like Janus particles. Finally, our study shows that fluid compressibility is not required for momentum transfer to produce colloidal propulsion.

Figures (4)

• Figure 1: Schematic view of the two scenarios considered in this work for a colloid with radius $a$: (a) a dipole of point force densities act on the fluid at distances $d_1$ and $d_2$ from the colloidal surface; (b) a pair of forces with $\mathbf{K} _{ext}$ acting directly on the colloid and an opposite force $\mathbf{F} _{ext}$ on the fluid at a distance $d$ from the surface.
• Figure 2: (a) Colloidal velocity following an impulsive pair force with $d^{\prime}=0.1$ and dipole scenario with $d_1^{\prime}=0.1$ and different values of $d_2^{\prime}$. (b) Colloidal velocity on a log-log scale for different reaction displacement $d^{\prime}$, scaled with $\chi(d^{\prime})$ (see Eq. \ref{['eq:chid']}), along with the mid-time scaling $\sim t^{-1/2}$ and long-time tail prediction $\beta^{-1} t^{-5/2}$. Vertical lines indicate the respective values of $t=t_d/10$ for each value of $d^{\prime}$.
• Figure 3: Colloidal velocity $u(t)$ under a periodic forcing with three values of the period $T_0$. Black dashed lines for each $T_0$ are the period-averaged velocity $u_n$. The inset is a zoom on the area of the black box within the $T_0=10^{-1}$ curve.
• Figure 4: Orientationally averaged colloidal velocity $\langle u\rangle_{T_0, \varphi}$ as a function of the dipole displacement $d$ from the colloidal surface.