Drift of a semi-permeable vesicle through an osmotic gradient: Anomalous velocity amplification due to a proximate wall

A spherical vesicle is made up of a liquid core bounded by a semi-permeable membrane that is impermeable to solute molecules. When placed in an externally imposed gradient of solute concentration, the osmotic pressure jump across the membrane results in an inward trans-membrane solvent flux at the solute-depleted side of the vesicle, and and outward flux in its solute-enriched side. As a result, a freely suspended vesicle drifts down the concentration gradient, a phenomenon known as osmophoresis. An experimental study of lipid vesicles observed drift velocities that are more than three orders of magnitude larger than the linearised non-equilibrium prediction (Nardi et al., Phys. Rev. Lett., vol. 82, 1999, pp. 5168-5171). Inspired by this study, we analyse osmophoresis of a vesicle in close proximity to an impermeable wall, where the vesicle-wall separation is small compared with the vesicle radius. Due to intensification of the solute concentration gradient in the narrow gap between the membrane and the wall, the 'osmophoretic' force and torque on a stationary vesicle scale as an irrational power, of. Both the rectilinear velocity and the angular velocity of a freely suspended vesicle scale as the ratio of that power to. In contrast to the classical problem of sedimentation parallel to a wall, where the ratio approaches as, here the ratio approaches unity, as though the vesicle performs pure rigid-body rolling without slippage. Our approximations are in excellent agreement with hitherto unexplained numerical computations in the literature.