Phase diagram

A phase diagram partitions a space of thermodynamic variables into regions occupied by different equilibrium thermodynamic phases and records the phase coexistence curves between them. For a pure, one-component substance, a pressure–temperature phase diagram gives one point for each pair $(T,p)$. In a single-thermodynamic phase region both controls can vary independently. On a two-thermodynamic phase phase coexistence curve, they obey one relation; at a triple point, both are fixed. These statements assume bulk thermodynamic equilibrium, fixed composition and no additional applied fields.^[ref]

At fixed temperature and pressure, the stable bulk thermodynamic phase minimizes Gibbs free energy for a fixed amount of substance. For a pure substance this is a comparison of chemical potentials on a common normalization. Two candidate thermodynamic phases coexist where ${\mu }_{\alpha }(T,p)={\mu }_{\beta }(T,p)$ and neither is exceeded in stability by a third thermodynamic phase. Differentiating their equality gives the Clapeyron equation. Thus a phase coexistence curve is a locus of equal stability, rather than a trajectory that every heated sample must follow.^[ref]

The following original cartoon uses dimensionless positive coordinates $\theta =T/T_{\ast }$ and $\pi =p/p_{\ast }$, with arbitrary reference scales $T_{\ast },p_{\ast }$. It represents no particular substance and contains no measured data. Stipulate a triple point at $(1,1)$, a liquid-gas critical point at $(3,5)$, and these drawing rules:They specify sublimation, liquid–gas phase coexistence and melting boundaries for the picture. They are not an equation of state, do not predict latent heat or mass density, and do not supply a quantitative critical theory. In particular, the melting slope is merely an illustrative choice.

For example, at fixed $\pi =3$ the melting boundary is at $\theta =1.2$ and the liquid–gas boundary is at $\theta =2$. Heating from $(\theta ,\pi )=(1.6,3)$ to $(3.6,3)$ therefore starts in liquid, crosses phase coexistence at $(2,3)$, and ends in gas. In an equilibrium conversion at that crossing, added heat can change the phase fraction at fixed temperature through latent heat. The pressure–temperature point alone does not specify how much of each thermodynamic phase is present: for that one needs an additional amount or volume constraint.

Now connect exactly the same endpoints by raising $\pi$ from $3$ to $6$ at $\theta =1.6$, moving to $\theta =3.6$ at $\pi =6$, then lowering $\pi$ to $3$. The first leg stays on the liquid side because ${\theta }_{\mathrm{melt}}$ ranges only from $1.2$ to $1.5$ and ${\pi }_{\mathrm{vap}}(1.6)=2.2<3$. The second leg lies above the entire liquid–gas boundary, whose largest drawn pressure coordinate is $5$; the third lies beyond its largest temperature coordinate, $3$. No leg crosses a first-order phase transition boundary. The middle leg passes through a supercritical fluid region, with $\theta >3$ and $\pi >5$, and the final leg returns to a gaslike state continuously.

Avoiding a first-order phase transition does not mean that no heat or work is exchanged. Along the detour, mass density, enthalpy and heat capacity may change substantially but continuously. The liquid-gas critical point itself has singular behavior, so the detour goes around it rather than through it. This connected fluid region is the striking lesson of a phase diagram: a familiar change from liquid to gas need not involve boiling across a sharp boundary.

An equilibrium phase diagram also does not predict how rapidly a conversion occurs. Metastability and a barrier described by classical nucleation theory can delay the observed phase transition. Nor should a binary-composition phase diagram be read as the same picture: its coexistence regions and lever rule encode composition and phase fractions, and can occupy an area even though pure-substance two-thermodynamic phase coexistence occupies a curve in the pressure–temperature plane.

A diagram of stable mineral assemblages as functions of specified aqueous thermodynamic activities, at fixed temperature and pressure. The coordinates often use logarithmic activity ratios. Mass-action equations give pairwise coexistence lines; a line segment is stable only when no omitted mineral has a lower relevant thermodynamic potential under the same imposed chemical potentials.

Suppose a reference mineral $C$ can dissolve to either $G$ or $K$, with the same conserved solid component, and normalized Gibbs energies of reaction ${\Delta }_r{G}_G/A=y-{\ell }_G(x)$ and ${\Delta }_r{G}_K/A=y-{\ell }_K(x)$ for $A>0$. Choose relative solid potentials ${\Phi }_C=0$, ${\Phi }_G=y-{\ell }_G$ and ${\Phi }_K=y-{\ell }_K$. The stable state minimizes $\Phi$. Thus $C$ is stable above $y=\max ({\ell }_G,{\ell }_K)$; $G$ is stable below ${\ell }_G$ where ${\ell }_G>{\ell }_K$; and $K$ is stable below ${\ell }_K$ where ${\ell }_K>{\ell }_G$. The $G$–$K$ coexistence line is stable below their triple intersection and metastable above it. These inequalities follow directly from the sign of each forward dissolution reaction. Reversing reaction directions reverses the corresponding sign convention, so one must establish the reaction direction before choosing an upper or lower envelope.

A pairwise equilibrium line restricted to conditions where both minerals are stable against the other allowed phases. Compute the Gibbs energy of reaction for independent conversions to compare them. A pairwise line can continue into a region where a third mineral is favoured; this continuation is a metastable equilibrium relation, rather than a boundary of stable fields.

Suppose two dissolution reactions to a reference mineral $C$ have been normalized to the same amount of a conserved reference component. After division by the same positive energy factor, let their Gibbs energies of reaction be $y-{\ell }_1(x)$ and $y-{\ell }_2(x)$. Their relative thermodynamic potentials can then be written as displayed. The stable mineral has the smallest potential.

It follows that $C$ is stable below $y=\min ({\ell }_1,{\ell }_2)$, mineral 1 is stable where ${\ell }_1<{\ell }_2$ and $y>{\ell }_1$, and mineral 2 is stable where ${\ell }_2<{\ell }_1$ and $y>{\ell }_2$. The two nonreference minerals coexist stably where ${\ell }_1={\ell }_2<y$. Their equality below that height is a metastable continuation. This comparison proves the stable lower envelope and selects which portions of the pairwise mass-action lines belong on the mineral activity diagram.

A boundary giving the equilibrium limit of one solid solution against coexistence with another solid phase. Moving along a solvus during cooling changes equilibrium solid compositions even after all liquid has disappeared.

The lower endpoint of a two-liquid coexistence region. In the lower-critical case, demixing can be induced by heating above the endpoint.

The upper endpoint of a two-liquid coexistence region in a temperature-composition diagram. In the ordinary upper-critical case, heating above this temperature removes liquid-liquid demixing.

The phase rule counts independent intensive controls compatible with coexistence of $P$ distinct bulk thermodynamic phases made from $C$ independent components:Here $P$ counts thermodynamic phases, not pressure, and $F$ does not count phase fractions. For the nonreacting one-component case, $C=1$ and $F=3-P$: a single thermodynamic phase occupies a region with $F=2$, two-thermodynamic phase coexistence a curve with $F=1$, and a triple point an isolated point with $F=0$. Fixing pressure externally removes one adjustable control wherever that constraint is independent.^[ref]

For an elementary count without chemical reactions, take $C$ chemical species. Each thermodynamic phase has $C-1$ independent composition fractions, since they sum to one. Shared temperature and pressure add two controls, giving $2+P(C-1)$. Equality of each species' chemical potential between one reference thermodynamic phase and the other $P-1$ thermodynamic phases supplies $C(P-1)$ constraints. Subtraction gives $2+P(C-1)-C(P-1)=C-P+2$. This argument assumes the constraints are independent and treats interfaces as negligible; chemical reactions, external fields or critical mergers require appropriate qualifications.

The solidus bounds the entirely solid equilibrium region of a phase diagram. For a linear liquidus $T_L(C_l)=-m{C}_l$ and a constant segregation coefficient $C_s=k_D{C}_l$, the corresponding solidus is $T_S(C_s)=-m{C}_s/k_D$. The concentration argument is the solid concentration, not the neighboring liquid concentration.

The lever rule finds phase fractions from a horizontal tie line on a phase diagram. For solid and liquid concentrations $C_s,C_l$ and bulk concentration $\overline{C}$, conservation gives $\overline{C}=(1-\varphi )C_s+\varphi C_l$. Thus $\varphi =(\overline{C}-C_s)/(C_l-C_s)$ is the liquid fraction, provided $C_s\le \overline{C}\le C_l$. For pure solid, $\varphi =\overline{C}/C_l$.

A bicritical point is where two continuous transition lines and a first-order line meet, typically separating a disordered phase from two competing ordered phases.

The liquidus is the locus in a phase diagram above which the equilibrium state is entirely liquid. In a dilute binary solution it is often approximated locally by a linear relation such as $T_L=T_0-mC$.

Freezing-point depression is the lowering of a solvent's equilibrium freezing temperature by dissolved solute. In the linear dilute approximation, $T_L=T_0-mC$ with $m>0$.

An eutectic system has a composition and temperature at which a liquid is in equilibrium with two distinct solid phases. Cooling liquid of that composition through the eutectic temperature produces both solid phases together.

The transformation $L\to \alpha +\beta$ in which one liquid produces two solid phases at a binary eutectic composition and temperature under fixed pressure. The resulting two-phase constituent must be distinguished from either individual thermodynamic phase.

The invariant liquid-plus-two-solids equilibrium of a binary eutectic system at fixed pressure. It specifies both the eutectic temperature and the eutectic composition.

A two-solid microstructure formed by the coupled transformation of liquid at a eutectic point. Lamellae and rods are common morphologies. A hypoeutectic alloy can contain both primary crystals and a eutectic microstructure; those microconstituents are not the same accounting categories as its total phase fractions.

The fraction of an alloy that solidifies through the eutectic reaction, equal to the remaining liquid fraction immediately before that reaction in a simple equilibrium solidification path. It differs from the total fraction of either solid phase, and later precipitation can change the phase inventory.

The eutectic composition is the liquid concentration at the eutectic temperature, where liquid and two solid phases coexist. It is a composition coordinate on a phase diagram, not the overall concentration of every sample in an eutectic system.

The eutectic temperature is the temperature of simultaneous liquid–two-solid phase coexistence in an eutectic system. In directional freezing of a solution away from the eutectic composition, a mushy layer can end at this temperature, where its remaining liquid solidifies.