Drop deformation during diffusiophoresis

Diffusiophoresis refers to the motion of a colloidal particle in a solute concentration gradient, animated by particle–solute interactions. We present a theoretical analysis of the diffusiophoretic motion of a viscous drop in a gradient of neutral solute at zero Reynolds number. In a spatially uniform gradient, the translational velocity of a spherical drop was found by Anderson et al. ( J. Fluid Mech. , vol. 117, 1982, pp. 107–121). Here, we show additionally that the drop experiences no tendency to deform, regardless of the magnitude of the interfacial tension at the interface of the drop and suspending fluid. Next, we consider a non-uniform gradient, where the ambient solute concentration takes the form of a quadrupole around the drop centroid. This gradient does not induce drop translation, due to symmetry, but does induce a deformation in the drop shape, which is spheroidal to first order in the capillary number $Ca=\beta k_B T R^2 K/\gamma$ , where $\beta$ is the magnitude of the quadrupolar variation in solute concentration, $k_B T$ is the thermal energy, $R$ is the drop radius, $K$ is the Gibbs adsorption length, and $\gamma$ is the interfacial tension. Whether the drop becomes prolate or oblate depends on whether the solute–drop interaction is attractive or repulsive. Therefore, our work shows that in principle, a drop could undergo deformation during diffusiophoresis in a non-uniform solute gradient.