Phoresis kernel theory for passive and active spheres with nonuniform phoretic mobility†

By introducing geometry-based phoresis kernels, we establish a direct connection between the translational and rotational velocities of a phoretic sphere and the distributions of the driving fields or fluxes. The kernels quantify the local contribution of the field or flux to the particle dynamics. The field kernels for both passive and active particles share the same functional form, depending on the position-dependent surface phoretic mobility. For uniform phoretic mobility, the translational field kernel is proportional to the surface normal vector, while the rotational field kernel is zero; thus, a phoretic sphere with uniform phoretic mobility does not rotate. As case studies, we discuss examples of a self-phoretic axisymmetric particle influenced by a globally-driven field gradient, a general scenario for axisymmetric self-phoretic particle and two of its special cases, and a non-axisymmetric active particle.