  The mobility of externally-driven phoretic propulsion of particles is
evaluated by simultaneously solving the solute conservation equation,
interaction potential equation, and the modified Stokes equation. While
accurate, this approach is cumbersome, especially when the interaction
potential decays slowly compared to the particle size. In contrast to external
phoresis, the motion of self-phoretic particles is typically estimated by
relating the translation and rotation velocities with the local slip velocity.
While this approach is convenient and thus widely used, it is only valid when
the interaction decay length is significantly smaller than the particle size.
Here, by employing the Lorentz reciprocal theorem, we combine the benefits of
two approaches and derive unified mobility expressions with arbitrary
interaction potentials such that the expressions predict the translation and
rotation velocities for both externally driven and self-propelling particles.
We show that these expressions can conveniently recover the well-known mobility
relationships of external electrophoresis and diffusiophoresis for arbitrary
double-layer thickness. Additionally, we show that for a spherical
microswimmer, our derived expressions relax to the slip velocity calculations
in the limit of thin interaction lengthscales. We also employ the derived
mobility expressions to calculate the velocities of an autophoretic Janus
particle. We find that there is significant dampening in the translation
velocity even when the interaction length is an order of magnitude larger than
the particle size. Finally, we study the motion of a catalytically
self-propelled particle, while it also propels due to external concentration
gradients, and demonstrate how the two propulsion modes compete with each
other.
