Dynamics of a self-diffusiophoretic particle in shear flow

Colloidal particles can achieve autonomous motion by a number of physicochemical mechanisms. For instance, if a spherical particle acts as a catalyst with an asymmetric surface reactivity, a molecular solute concentration gradient will develop in the surrounding fluid that can propel the particle via self-diffusiophoresis. Theoretical analyses of self-diffusiophoresis have mostly been considered in quiescent fluid, where the solute concentration is usually assumed to evolve solely via diffusion. In practical applications, however, self-propelled colloidal particles can be expected to reside in flowing fluids. Here, we examine the role of ambient flow on self-diffusiophoresis by quantifying the dynamics of a model Janus particle in a simple shear flow. The imposed flow can distort the self-generated solute concentration gradient. The extent of this distortion is quantified by a Peclet number, Pe, associated with the shear flow. Utilizing matched asymptotic analysis, we determine the concentration gradient surrounding a Janus particle in shear flow at a small, but finite, Peclet number and the resulting particle motion. For example, when the symmetry axis of the particle is aligned with the imposed flow, the Janus particle experiences an $\text{O}\left(\text{Pe}\right)$ cross-streamline drift and an $\text{O}\left({\text{Pe}}^{3/2}\right)$ reduction in translational velocity along the flow direction. We then analyze the in-plane trajectory of the Janus particle in shear. We find that the particle performs elliptical orbits around its initial position in the flow, which decrease in size with increasing Pe.

Figure 1

A spherical Janus particle of radius $a$ immersed in an imposed shear flow, $\mathbf{u}=\dot{\gamma }y{\mathbf{e}}_x$. One side of the particle generates hard-sphere solute particles while the other side consumes them. The symmetry axis of the particle $\widehat{\mathbf{d}}$ lies in the shear plane and is oriented at an angle $\psi$ from the direction of the flow. The particle coordinate system is described as $(r^{\prime },{\theta }^{\prime },{\varphi }^{\prime })$ (spherical polar) and $(x^{\prime },y^{\prime },z^{\prime })$ (Cartesian), while the laboratory fixed coordinate system is $(r,\theta ,\varphi )$ and $(x,y,z)$.

Figure 2

Solute concentration around a self-diffusiophoretic particle aligned along the flow direction of the imposed shear, $\psi =0$. (a) The solute concentration at $\text{Pe}=0$ is dependent solely on solute diffusion and is axially symmetric. This is sketched in the top image of (a), and the bottom image of (a) plots the solute concentration in the shear plane for $\text{Pe}=0$ from (13). The color scheme indicates increasing concentration in green (light gray) and decreasing concentration in blue (dark gray). As a result of the axial symmetry, the particle moves solely along its symmetry axis. (b) In a shear flow ($\text{Pe}\ne 0$), the axial symmetry is broken as the shear flow aids and restricts solute flux on the bottom and top halves of the particle, respectively. The resulting top-bottom asymmetry results in particle migration across fluid streamlines, as depicted in the top image of (b). The bottom image of (b) plots the solute concentration at $\text{Pe}=0.5$ from (25): the broken symmetry is clearly evident.

Figure 3

Solute concentration around a self-diffusiophoretic particle aligned along the gradient direction of the imposed shear, $\psi =\pi /2$. (a) The solute concentration at $\text{Pe}=0$ is dependent solely on diffusion and is therefore axially symmetric, leading to particle motion solely in the velocity gradient $y$ direction. This is sketched in the top image of (a), and the bottom image of (a) plots the solute concentration in the shear plane for $\text{Pe}=0$ (color scheme is the same as Fig. 2). (b) In a shear flow ($\text{Pe}\ne 0$), the left-right symmetry of the concentration field is broken, inducing an $\text{O(Pe)}$ phoretic velocity contribution along the flow $x$ direction, as depicted in the top figure of (b). The asymmetric concentration is plotted in the bottom figure of (b) for $\text{Pe}=0.5$ from (25).

Figure 4

Planar trajectory of a self-diffusiophoretic particle in a shear flow: (a) the particle is initially aligned with the shear flow (${\psi }_0=0$); and (b) the particle is aligned with the flow gradient (${\psi }_0=\pi /2$). Trajectories are plotted for $\text{Pe}=0$ and $0.5$; in both cases, the trajectories are elliptical; however, the area of the orbit decreases at finite Pe. The arrows depict the initial translation direction of the particle.