  We address the question: How does capillarity propel microspheres along
curvature gradients? For a particle on a fluid interface, there are two
conditions that can apply at the three phase contact line: Either the contact
line adopts an equilibrium contact angle, or it can be pinned by kinetic
trapping, e.g. at chemical heterogeneities, asperities or other pinning sites
on the particle surface. We formulate the curvature capillary energy for both
scenarios for particles smaller than the capillary length and far from any
pinning boundaries. The scale and range of the distortion made by the particle
are set by the particle radius; we use singular perturbation methods to find
the distortions and to rigorously evaluate the associated capillary energies.
For particles with equilibrium contact angles, contrary to the literature, we
find that the capillary energy is negligible, with the first contribution
bounded to fourth order in the product of the particle radius and the
deviatoric curvature. For pinned contact lines, we find curvature capillary
energies that are finite, with a functional form investigated previously by us
for disks and microcylinders on curved interfaces. In experiments, we show
microsphere migrate along deterministic trajectories toward regions of maximum
deviatoric curvature with curvature capillary energies ranging from $6
\times10^3 - 5 \times 10^4~k_BT$. These data agree with the curvature capillary
energy for the case of pinned contact lines. The underlying physics of this
migration is a coupling of the interface deviatoric curvature with the
quadrupolar mode of nanometric disturbances in the interface owing to the
particle's contact line undulations. This work is an example of the major
implications of nanometric roughness and contact line pinning for colloidal
dynamics.

Soft Condensed Matter