Phase coexistence

Phase coexistence means that distinct equilibrium phases can persist together under common external conditions. For exchanging material, the phases have equal temperature and equal chemical potentials of each exchanging species. A flat unstressed interface also requires equal pressure; curvature changes the mechanical balance through capillary pressure.

A sealed pure fluid below its liquid-gas critical point can contain both liquid and vapor. Changing the total amount or volume can change their proportions without changing their equilibrium densities at fixed temperature. In a binary mixture, the two phases can instead have different compositions. The common-tangent construction determines their coexistence compositions by enforcing the appropriate chemical and mechanical equalities. Coexistence concerns competing phases with matching intensive conditions, rather than equality of their total extensive free energies for arbitrary sample amounts.

If two equilibrium phases have continuous entropy and volume but jumps in second derivatives of Gibbs free energy, differentiate their entropy equality along the transition line. Since $dS=(C_P/T)dT-V{\alpha }_{T}dP$, one obtains the title slope, for nonzero $\Delta C_P$. Before imposing equal volume, the numerator is $T\Delta (V{\alpha }_T)$. Equality of Gibbs free energies with equal entropy implies equal volume along a differentiable coexistence curve on which pressure varies. A constant-pressure curve or zero numerator and denominator needs separate analysis. This entropy-continuity condition is one Ehrenfest relation; volume continuity gives another independent relation.

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.

Formation of finite-scale composition domains when complete macroscopic separation is frustrated, for example by the covalent connection of incompatible blocks in a diblock copolymer. Interfacial cost and chain entropy jointly select the domain scale.

Alternating sheet-like domains of the components of a microphase-separated material. In diblock systems the two layer thicknesses can be unequal and reflect unequal block fractions. Confinement, surface wetting and kinetic trapping can produce layered films outside the thermodynamic equilibrium bulk lamellar region.

A microdomain pattern of approximately parallel cylinders arranged on a two-dimensional hexagonal lattice in the other block's matrix. It is conventionally labelled H in many diblock phase diagrams. Cylinder orientation and defects are additional properties beyond the local morphology.

An ordered morphology in which a minority block forms approximately spherical domains in a continuous majority-block matrix. The lattice packing and phase stability depend on segregation strength, composition and chain statistics. The label S in a phase diagram denotes a morphology, not a spherical shape of the entire specimen.

The transition between a compositionally disordered state and an ordered microdomain state as effective segregation strength increases. The parameter $\chi N$ combines unlike-segment interaction preference and chain length; its transition value depends on composition and model assumptions. A bulk thermodynamic equilibrium boundary is not by itself an evaporation or vitrification kinetic model.

A domain structure resembling lamellae pierced by an ordered array of perforations. It is distinct from an unperforated lamellar phase and from a double gyroid.

A bicontinuous domain morphology containing two interpenetrating, nonintersecting networks of one component within the other. In diblock copolymers it can appear between lamellar and cylindrical morphologies in part of the phase diagram.

Separation of a biomolecular mixture into compositionally distinct phases. Concentration dependence, partitioning and reversibility should be tested; a fluorescent punctum or recovery after bleaching alone does not establish the mechanism.

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.

Vapor-liquid equilibrium is coexistence of a liquid and its vapor with no net macroscopic transfer of matter between them. The two phases have equal temperature, equal pressure for a flat interface without external stress, and equal chemical potential for each exchanging species.

For a pure substance, fixing the temperature below its liquid-gas critical point selects the saturation vapor pressure. Molecules still cross the interface in both directions; equilibrium means equal average opposing rates, rather than halted molecular motion. In a mixture, the vapor and liquid generally have different compositions, while each component separately satisfies the chemical-potential equality. This distinction underlies separation by evaporation and condensation and explains why equal total pressure alone is insufficient to determine phase compositions.