Detachment force of particles with pinning of contact line from fluid bubbles/droplets

Deformation of a spherical droplet or bubble, containing a pair of particles on its surface is considered when equal but opposite forces are applied to the particles. The particles are placed opposite each other thus providing a symmetrical problem which is more amenable to analytical treatment. We extend our previous calculations, concerning such arrangements with constant contact angles, to situations where now it is the contact line that is pinned on the surface of the particles. The force-displacement curves are calculated as the particles are pulled apart and was found to be linear for small displacements. However, it is also found that the "Hookean constant" for the pinned contact line problem is different to one derived for systems with a constant contact angle, being larger if the pinned line is at the equator of the particles.