THERMODYNAMICS OF FLUID DISPERSIONS IN EQUILIBRIUM

For a fluid dispersion, D, with fluid phases, i, of internal pressures p(i) and volumes V(i), and interfaces 8 of surface tensions sigma(s) and areas A(s), in contact with an ambient phase, i = 0, of pressure p0 and a rigid container wall w, the equilibrium equation is SIGMA(i)(p(i) - p0)delta V(i) = SIGMA(s)sigma(s)deltaA(s) - sigma(w0)deltaA(wD) From Gibbs' general criterion for equilibrium, we show that this equation is valid for an arbitrary continuous infinitesimal geometric displacement of the interfacial network, as long as it is consistent with the rigid wall constraints. In particular it is not necessary that the dispacement preserves equilibrium. This geometrical interpretation of the equilibrium equation allows a simple proof of Princen's virial equation for freely suspended dispersion, and its extension to a dispersion on a flat rigid substrate p0V(D) = SIGMA(i)p(i)V(i) - 2/3(SIGMA(s)sigma(s)A(s) - sigma(w0)A(wD)) When applied to ''dry'' foams this becomes Derjaguin's equation of state for foams. The proof is obtained by applying the equilibrium equation to an to an infinitesimal scaling transformation. Equivalent expressions for a fluid dispersion suspended at a flat fluid interface are also obtained.