Traditionally, research in phoresis has diverged into two distinct avenues: (a) active systems where self-generated gradients drive microswimmers via surface chemistry or asymmetry in polarizability, and (b) passive systems where externally imposed gradients propel colloids. Despite being governed by the same underlying equations, these two domains are often treated independently.
In our work, we seek to theoretically reconcile both of these directions of research and understand micro-swimmer propulsion in a combination of active and passive systems. By employing Lorentz’s reciprocal theorem, we derive integral expressions for translation and rotation velocities in terms of phoretic body forces, eliminating the need to explicitly solve flow fields—especially advantageous at finite interaction length scales, where solving from first principles quickly becomes cumbersome.
Firstly, for passive spherical particles, we can use these integral expressions to recover well-known mobility relationships in external electrophoresis and external diffusiophoresis for arbitrary double-layer thickness. Secondly, at the thin interaction length limit, these equations reduce to the well-known slip expressions used to study micro-swimmer motion in active particle literature. Thirdly, for self-diffusiophoretic systems, our calculations reveal that the translation velocity of the particle depends directly on the size of the catalytic patch and interaction length scales. Additionally, at moderate interaction length scales, there is a significant over-prediction of translation velocities from slip velocity calculations on self-diffusiophoretic Janus particles.
Finally, we explore the dynamics of phoretically driven spheroidal particles both in external and self-generated field gradients. We investigate how particle eccentricity, surface heterogeneities, external gradients, and interaction length scales impact their swimming speeds. Interestingly, particle motion in external fields depends not just on the external gradient direction but also on particle eccentricity—unlike spheres, which always align with the imposed gradient. In summary, our approach offers a robust, scalable method to analyze phoretic motion across a spectrum of active and passive systems, bridging a key theoretical gap in the field.