Effective spring constant of bubbles and droplets

It is shown that gas bubbles and liquid droplets respond as hookean springs to applied loads in the experimental weak force regime. An analytic expression is obtained for the spring constant that is identical for both, linear in the surface tension gamma, and logarithmically dependent on the lengths (the decay length of the interaction, the radius of the bubble or droplet, and the radius of the particle or probe). For acute interior contact angles greater than about 20 degrees, it is typically in the range 0.8-1.2 gamma, in agreement with published atomic force microscopy data, and it increases logarithmically for smaller contact angles. Analytic expressions are also obtained for the deformed profile, the extent of the dimple and of the interaction region, the wrapping radius, and the rupture force.