Toward a theoretical framework for self-diffusiophoretic propulsion near a wedge

TL;DR

This work addresses self-diffusiophoretic propulsion of a catalytically active sphere confined in a wedge, aiming to understand how corner geometry governs autonomous motion. It develops a systematic theoretical framework using the Fourier–Kontorovich–Lebedev transform and the method of images to solve the diffusion equation for the concentration field, keeping monopole and dipole surface activities and deriving wall-induced phoretic velocities. The authors obtain leading-order expressions for the self-induced velocity as a function of wedge opening angle $\alpha$ and particle position, and recover the planar-wall limit ($\alpha=\tfrac{\pi}{2}$) as a consistent special case. The results reveal that wedge geometry can qualitatively alter both magnitude and direction of motion, offering insights for microfluidic design and control of autophoretic particles near corners, while highlighting the need to incorporate hydrodynamic contributions for a complete dynamics description.

Abstract

We investigate the self-diffusiophoretic motion of a catalytically active spherical particle confined within a wedge-shaped domain. Using the Fourier-Kontorovich-Lebedev transform, we solve the Laplace equation for the concentration field in the diffusion-dominated regime. The method of images is employed to obtain the first and second reflections of the concentration field, accounting for both monopole and dipole contributions of the particle's surface activity. Based on these results, we derive leading-order expressions for the self-induced phoretic velocity in the far-field limit and examine how it varies with the wedge opening angle and the particle's position within the domain. Our findings reveal that the wedge geometry significantly affects both the magnitude and direction of particle motion. Our study provides a systematic framework for understanding active particle dynamics near corners, with implications for microfluidic design and control of autophoretic particles in confined geometries.

Figures (8)

• Figure 1: A catalytically active particle of radius $R$ is located near a wedge formed by two walls intersecting along the $z$-axis, with the wedge having a semi-opening angle $\alpha$. The evaluation point is denoted by $(r,\theta,z)$ in cylindrical coordinates, while the position of the active colloid is specified by the polar distance $\rho$ and polar angle $\beta$. No-flux boundary conditions are assumed at the surfaces of the walls.
• Figure 2: Contour plots of the scaled monopole concentration field around an active particle in the radial-azimuthal plane for (a) an obtuse wedge with $\alpha = \pi/6$ and the particle located at $\beta = \pi/8$, and (b) a salient wedge with $\alpha = 2\pi/3$ and the particle located at $\beta = \pi/4$. In both cases, the results are shown in the axial plane $z/\rho = 0.1$.
• Figure 3: Contour plots of the scaled monopole concentration field around an active particle in the radial-axial plane for $\alpha = \pi/6$ and $\beta = \pi/12$. Results are shown in the azimuthal plane $\theta = 0$.
• Figure 4: Contour plots of the scaled dipole concentration field around an active particle in the radial-azimuthal plane for (a) an obtuse wedge with $\alpha = \pi/6$, where the particle is located at $\beta = 0$ and has angular orientation $(\lambda,\delta) = (\pi/6, \pi/2)$, and (b) a salient wedge with $\alpha = 2\pi/3$, where the particle is located at $\beta = \pi/4$ and has angular orientation $(\lambda,\delta) = (0, \pi/2)$. In both cases, the results are shown in the axial plane $z/\rho = 0.1$.
• Figure 5: Contour plots of the scaled dipole concentration field around an active particle in the radial-axial plane for $\alpha = \pi/6$ and $\beta = \pi/12$, shown for the orientations (a) $\delta = 0$ and (b) $(\lambda,\delta) = (\pi/12, \pi/2)$. Results are displayed in the azimuthal plane $\theta = 0$.