Diffusive mass transfer from a Janus sphere

We study the mixed boundary-value problem of finding the concentration distribution surrounding a two-sided (Janus) sphere, with a uniform concentration imposed on one segment of it and the no-mass-flux condition enforced on the other one. This mass transfer problem arises in the analysis of the Marangoni propulsion phenomenon, where a chemically active particle propels itself along a liquid-gas interface via nonuniform release of a chemical species that locally alters the surface tension distribution. Assuming that the mass transfer is purely driven by diffusion (i.e., neglecting advection), we present an approximate form for the concentration field derived from the integral representation of the solution to the corresponding Laplace equation. We demonstrate the high fidelity of the proposed approach by comparing its results with those obtained from a collocation method based on the series solution of the problem. Beyond the motivating problem, our findings are expected to be applicable to the self-diffusiophoresis of catalytic colloids, as well as to the conduction heat transfer and electrostatics problems involving partially insulated spheres. Moreover, our approach can lead to the derivation of similarly accurate approximate solutions to analogous problems in mathematical physics.

Figure 1

Mass release from a Janus sphere into an unbounded domain. The sphere has active and inert sides that are modeled by uniform concentration and no-flux boundary conditions, respectively.

Figure 2

(a) The plot of $\tau$ versus $\alpha$ overlaid by a dashed line of slope $3/14$ passing through the point $(\pi ,2)$. (b) The plots of approximation error, as defined by Eq. (20), versus $\theta$ for $\alpha =\pi /6,\pi /3,\pi /2,2\pi /3,5\pi /6$.

Figure 3

The plots of the dimensionless (a) concentration distribution ${\varphi }_s$ and (b) local mass flux $j$ versus $\theta$ on, respectively, the impermeable and active segments of the Janus sphere. The subfigure numbers $1,2,...,5$ correspond to $\alpha =\pi /6,\pi /3,\pi /2,2\pi /3,5\pi /6$, respectively.

Figure 4

The plots of the dimensionless (a) total mass flux $J$ [see Eq. (25)] and (b) representative of the net surface tension force $F_{\gamma }$ [see Eq. (26)] versus $\alpha$. The insets show the percentage difference (denoted by $\delta$) between the results of our approximation and those obtained from the series solution described by Eq. (4), with $N=1000$.