Theory of the evaporation/condensation transition of equilibrium droplets in finite volumes

A phenomenological theory of phase coexistence of finite systems near the coexistence curve that occurs in the thermodynamic limit is formulated for the generic case of d-dimensional ferromagnetic Ising lattices of linear dimension L with magnetization m slightly less than m_coex. It is argued that in the limit L→∞ an unconventional first-order transition occurs at a characteristic value m_t< m_coex, where a large equilibrium droplet ceases to exist, and the thermodynamically conjugate variable to m, the magnetic field H, exhibits a jump from H_t⁽¹⁾ to H_t⁽²⁾. It is found that H_t^(1,2) scale like L^{- d/( d+1)} their ratio being simply H_t⁽¹⁾/ H_t⁽²⁾=( d+1)/( d-1), and m_coex- m_t∝ L^{- d/( d+1)} as well, while the excess thermodynamic potential (relative to its value according to the double-tangent construction) varies as g_t∝ L^{-2 d/( d+1)} . The prefactors in all these relations are derived and it is shown that near the bulk critical point this transition shows a standard scaling behavior and the prefactors can be expressed in terms of known universal constants. Also the rounding of this transition at very large but finite L is considered and it is found that the jump in H at H_t is rounded over an interval ∆ m∝ L^{- d2/( d+1)} . Various simulations are interpreted in the light of these predictions, and the possibility to extract the surface free energy of liquid droplets coexisting in a finite volume with supersaturated gas is critically discussed.