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Curvature Capillary Migration of Microspheres

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.