See Self-diffusiophoresis.

At steady-state, in the reference frame of the object, and neglecting distortions induced by the flow (small Peclet number), the solute concentration in the liquid is given by

$D\nabla^2 c = 0$ (steady state diffusion)

$-D\mathbf{n}\cdot\nabla c(\mathbf{r}_s) = \alpha(\mathbf{r}_s)$

i.e. the flux of solute is given by some space-dependent function that measures the 'surface activity' at the surface of the colloid, i.e. the generation or consumption of solute by a chemical reaction. In general, describing this process involves additional coupled transport problems for other species involved in the surface reactions.) Some variations are needed for the cases of Self-electrophoresis and Self-thermophoresis. Approximately, these equations give, $V\sim \alpha \mu /D$. In particular, once the surface properties, $\mu$ and $\alpha$ and the shape are given, the velocity turns out to be independent of the size $R$ of the object, showing that this method of propulsion is robust under downscaling.