  Self-phoretic Janus particles move by inducing -- via non-equilibrium
chemical reactions occurring on their surfaces -- changes in the chemical
composition of the solution in which they are immersed. This process leads to
gradients in chemical composition along the surface of the particle, as well as
along any nearby boundaries, including solid walls. Chemical gradients along a
wall can give rise to chemi-osmosis, i.e., the gradients drive surface flows
which, in turn, drive flow in the volume of the solution. This bulk flow
couples back to the particle, and thus contributes to its self-motility. Since
chemi-osmosis strongly depends on the molecular interactions between the
diffusing molecular species and the wall, the response flow induced and
experienced by a particle encodes information about any chemical patterning of
the wall. Here, we extend previous studies on self-phoresis of a sphere near a
chemically patterned wall to the case of particles with elongated shape. We
focus our analysis on the new phenomenology potentially emerging from the
coupling -- which is inoperative for a spherical shape -- of the elongated
particle to the strain rate tensor of the chemi-osmotic flow. Via numerical
calculations, we show that the dynamics of a rod-like particle exhibits a novel
"edge-following" steady state: the particle translates along the edge of a
chemical step at a steady distance from the step and with a steady orientation.
Within a certain range of system parameters, the edge-following state co-exists
with a "docking" state (the particle stops at the step, oriented perpendicular
to the step edge), i.e., a bistable dynamics occurs. These findings are
rationalized as a consequence of the competition between fluid vorticity and
the rate of strain by using analytical theory based on the point-particle
approximation which captures quasi-quantitatively the dynamics of the system.
