case [126]. Finally, infra-red spectroscopy has been used to show the

In order to calculate the force due to a bridging bubble, one must first calculate the bubble shape. This is done by optimizing the appropriate constrained thermodynamic potential [133, 134]. In this case, the external atmospheric pressure, po, the temperature, 7\ the liquid-vapour surface energy, y, and the difference in solid surface energies, Ay > 0 (the contact angle at equilibrium is 9 = cos - 1 [—Aj//y]), are fixed, as is the number of gas molecules, N. The last condition is important, as assuming diffusive equilibrium of the gas with the atmosphere leads to the prediction that all bubbles are unstable [106, 125, 127]. The constrained Gibbs free energy for an arbitrary bubble profile z(r) is

where &B is Boltzmann's constant, V[z] is the volume of the bubble, A\w[z] is the liquid-vapour surface area, Asv[z] is the solid-vapour surface area, X represents the fixed variables listed above, and h0 is the separation between the solid surfaces.