  The capillary energy landscape for particles on curved fluid interfaces is
strongly influenced by the particle wetting conditions. Contact line pinning
has now been widely reported for colloidal particles, but its implications in
capillary interactions have not been addressed. Here, we present experiment and
analysis for disks with pinned contact lines on curved fluid interfaces. In
experiment, we study microdisk migration on a host interface with zero mean
curvature; the microdisks have contact lines pinned at their sharp edges and
are sufficiently small that gravitational effects are negligible. The disks
migrate away from planar regions toward regions of steep curvature with
capillary energies inferred from the dissipation along particle trajectories
which are linear in the deviatoric curvature. We derive the curvature capillary
energy for an interface with arbitrary curvature, and discuss each contribution
to the expression. By adsorbing to a curved interface, a particle eliminates a
patch of fluid interface and perturbs the surrounding interface shape. Analysis
predicts that perfectly smooth, circular disks do not migrate, and that
nanometric deviations from a planar circular, contact line, like those around a
weakly roughened planar disk, will drive migration with linear dependence on
deviatoric curvature, in agreement with experiment.
