Phase transition

A phase transition is a nonanalytic change in equilibrium behavior as a thermodynamic control parameter varies. A continuous phase transition has a diverging correlation length and no latent heat.

Melting is a solid-to-liquid phase transition that absorbs latent heat. It can be driven by heating, changing phase pressures, or adding solute that causes freezing-point depression. The ice layer between freshwater and cold brine illustrates melting into a cold liquid through dilution and a composition-dependent liquidus.

Freezing is a liquid-to-solid phase transition. A moving phase boundary releases latent heat; its rate is constrained by heat transport and the Stefan condition. In a solution the liquidus depends on composition, so being below the pure solvent's melting temperature does not imply that the solution must freeze.

A morphological instability amplifies shape perturbations of a solidification interface. Protrusions may receive a larger diffusive supply or encounter larger undercooling, while curvature-induced melting-temperature depression suppresses fine corrugations. Solute rejection can produce constitutional supercooling, destabilizing a planar front and leading to cells or crystal dendrites.

A direct transition from a solid phase to a gaseous phase. Heated interplanetary dust or a comet can lose material through this process, changing its size and radiation-pressure coefficient or destroying it before gravitational or drag-driven evolution is complete.

For a locally linear ice temperature with positive gradient $G$ into the solid, a corrugated sublimation front receives a perturbation in heat flux proportional to its horizontal wavenumber $\alpha$. The Gibbs-Thomson effect suppresses short waves, giving the displayed quasi-static growth rate. It requires $V/(\kappa \alpha )\ll 1$ and $|\sigma |/(\kappa {\alpha }^2)\ll 1$, as well as a perturbation depth short enough for the linear background approximation. Unstable modes have $0<\alpha <\sqrt{G/\Gamma }$.

For a sublimating surface with exponentially absorbed volumetric heating, use a frame moving into the solid at speed $V$. Its steady heat equation is $k{T}^{\prime \prime }+\rho c_{p}V{T}^{\prime }+q_0{e}^{-\lambda z}=0$. The surface Stefan condition is $k{T}^{\prime }(0)=\rho LV$. Integrating over the half-space gives the displayed energy-limited speed. The surface gradient is positive, but the far field is colder, so the temperature has a subsurface maximum where melting can begin.

An irradiated grain loses solid material by sublimation. Its lifetime depends strongly on temperature and composition. In a planetary dust tail, a short lifetime can truncate the distribution well before a relative orbital wrap.

Phase coexistence occurs when two or more thermodynamic phases have equal free energy under the same external conditions. A first-order transition crosses a coexistence locus, where the competing phases are simultaneously stable.

A system can split into macroscopic regions with distinct local order parameters, densities or compositions. In a short-range system at phase coexistence, regions of two phases with means $M_1,M_2$ and volume fractions $q,1-q$ have spatial mean $M=q{M}_1+(1-q)M_2$. Their interface contribution is subextensive, so their thermodynamic free-energy density is the corresponding convex combination of the phase densities. This can lie below the homogeneous-branch Landau free energy at the same $M$. The equilibrium constrained order-parameter free energy is consequently convexified, with a straight coexistence segment described by the Maxwell construction. A nonconvex local potential remains useful for homogeneous branches, metastability and interfaces, but must not be identified with the full equilibrium constrained density throughout a coexistence interval.

A pure thermodynamic phase is an extremal equilibrium Gibbs state rather than a probabilistic mixture of distinct macroscopic phases. On an ordered Ising branch, taking the thermodynamic limit before removing a selecting conjugate field gives a nonzero magnetization. A symmetric mixture can instead have zero mean but a nondecaying two-point contribution. The connected correlation function and correlation length describing ordinary critical fluctuations should be evaluated in a chosen pure phase when this distinction matters.

For a bulk free-energy density $f(\varphi )$, two compositions coexist across a flat interface whenThe first condition equates the chemical potentials, and the second equates the pressure. Equivalently the same straight line of slope $\mu$ is tangent to the graph of $f$ at both compositions. A stable coexistence pair uses a supporting tangent, so the resulting mixture lowers the bulk free energy.

Adding $d\varphi$ to a free-energy density adds a constant at fixed total compositional order parameter. In the common-tangent construction for phase coexistence, it shifts the tangent slope by $d$ while preserving its points of contact. Thus coexistence compositions are unchanged, whereas their chemical potential shifts. This does not assert unchanged equilibrium composition at a fixed externally imposed chemical potential.

A continuous phase transition has no latent heat and develops fluctuations on a diverging correlation length. Its singular observables are characterized by critical exponents.