Self-diffusiophoresis with bulk reaction

We consider phoretic self-propulsion of a chemically active colloid where solute is produced on the colloid surface (with a spatially varying rate) and consumed in the bulk solution (or vice versa). Assuming first-order kinetics, the dimensionless transport problem is governed by the surface Damköhler number $\mathcal{S}$ and the bulk Damköhler number $\mathcal{B}$. The dimensionless colloid velocity $U$, normalized by a self-phoretic scale, is a nonlinear function of these two parameters. In the limit of small $\mathcal{S}$, the solute flux is effectively prescribed by the surface activity distribution, resulting in an explicit expression for $U$ that is proportional to $\mathcal{S}$. In the limit of large $\mathcal{B}$, the deviations of solute concentration from the equilibrium value are restricted to a narrow layer about the active portion of the colloid boundary. The associated boundary-layer analysis yields another explicit expression for $U$. Both asymptotic predictions are corroborated by an eigenfunction expansion solution of the exact problem for the cases when all physical parameters are held fixed except for a varying colloid size (resulting in $\mathcal{S}\propto {\mathcal{B}}^{1/2}$) or a varying solute diffusivity (resulting in $\mathcal{S}\propto \mathcal{B}$). The boundary-layer structure breaks down near the transition between the active and inactive portions of the boundary. The local solution in the transition region partially resembles the classical Sommerfeld solution of wave diffraction from an edge.

Figure 1

Schematic showing the particle geometry and coordinates. The zoomed region (rotated) describes the transition-region coordinates.

Figure 2

$U$ versus $\mathcal{S}$ using linkage-by-size (5.1) for the Janus profile (7.1). Solid: exact result (4.4) for the indicated values of $\alpha$. Dashed: small-$\mathcal{S}$ approximation (5.6) and large-$\mathcal{B}$ approximation (7.3).

Figure 3

$U$ versus $\mathcal{S}$ using linkage-by-diffusivity (5.2) with $\beta =1$ for both the Janus (7.1) and the generalized Janus (7.4) activity profiles. Solid: exact result (4.4). Dashed: small Damköhler-number approximation (5.6).