First-order phase transition

At a first-order phase transition, the Gibbs free energy is continuous while at least one first derivative, such as entropy or volume, jumps. A nonzero entropy jump produces latent heat $L=T\Delta S$.

A smooth scalar Landau free energy can end a two-phase phase coexistence curve by merging two minima and their intervening maximum. At a generic stable endpoint, its first three derivatives vanish and its fourth derivative is positive. A field shift and two mixed control parameters reduce the local form to a quartic Landau free energy; the order-parameter jump then vanishes and the magnetic susceptibility diverges. This conclusion assumes there is no competing third phase and that the endpoint lies inside the control-parameter domain. It is not a general theorem about every first-order line: a sextic potential with fixed negative quartic coefficient has a symmetry-related ordered coexistence line that ends at a triple point with a finite jump.

Classical nucleation theory balances the positive surface free energy of a droplet against its negative bulk free-energy gain. In three dimensions,has critical radius $R^{\ast }=2\sigma /\Delta$ and barrier $F^{\ast }=16\pi {\sigma }^3/(3{\Delta }^2)$.

A critical nucleus is the unstable droplet at the maximum of the nucleation free energy. Smaller droplets shrink, while larger droplets lower their free energy by growing.

Thermal crossing of a nucleation barrier $F^{\ast }$ has rate proportional to $e^{-F^{\ast }/(k_{B}T)}$ and mean waiting time proportional to $e^{F^{\ast }/(k_{B}T)}$, up to a kinetic prefactor.