Phase Equilibria and Solidification

Gibbs energy–composition curves and phase diagram

In a binary system at constant temperature and pressure, each phase has its own molar Gibbs energy curve as a function of composition X_B. The equilibrium state of an alloy of a given overall composition is the one that minimizes the total Gibbs energy, and this is found graphically with a common tangent. A single common tangent touching two curves gives two-phase equilibrium, and the tangent points give the compositions of the two coexisting phases. A common tangent that touches three curves at once, for example the liquid, \alpha and \beta curves, represents three-phase equilibrium. In a binary system this can occur at only one temperature, which is the invariant isotherm such as the eutectic temperature, where the phase diagram shows a horizontal line.

The following patterns in a G–X plot help to identify the state of the alloy at a given temperature.

• Single minimum — one phase is stable over the whole composition range, for example a fully liquid alloy.
• Two tangent points — a two-phase field such as L + \alpha, L + \beta or \alpha + \beta. A curve lying above the common tangent at every composition is not stable.
• One straight tangent touching three curves — a three-phase invariant condition.

For example, if the \alpha and \beta curves have their minima on opposite sides of the composition axis and the liquid curve lies above both, the equilibrium at that temperature is a mixture of \alpha and \beta. Its overall composition is fixed by the points where the common tangent touches the two curves.

Chemical potential equilibrium between phases

The chemical potential of component i is the partial molar Gibbs energy, \mu_i = (\partial G / \partial n_i)_{T,P,n_j}. At constant temperature and pressure, two phases \alpha and \beta are in equilibrium when the total Gibbs energy of the system is a minimum. This condition requires that the chemical potential of each component be the same in every coexisting phase, so that \mu_A^{\alpha} = \mu_A^{\beta} and \mu_B^{\alpha} = \mu_B^{\beta}. Each component must be paired with itself across the phases. Pairing A in one phase with B in the other gives no meaningful equilibrium condition.

The other criteria in common use apply to different constraints, and they are often confused with the Gibbs energy condition.

• Gibbs energy minimum — the equilibrium criterion at constant T and P.
• Helmholtz energy minimum — the criterion at constant T and V, not at constant P.
• Entropy maximum — the criterion for an isolated system at constant energy and volume.

For a one-component system, the free energy of each phase can be plotted against temperature at a fixed pressure. Since (\partial G / \partial T)_P = -S, the phase with the largest entropy has the steepest negative slope. For water, the vapour has the largest entropy, followed by the liquid and then the solid, so S_V > S_L > S_S. The three lines meet at the triple point T_I. Below T_I the solid has the lowest free energy, and above T_I the vapour is lowest at the triple-point pressure. The liquid line touches the lowest envelope only at the triple point itself, so it is not the stable phase on either side of that point at that pressure.

Lever rule weight fraction

In a two-phase region of a binary diagram, the lever rule gives the amount of each phase from the overall composition C_0 and the compositions of the two phases C_\alpha and C_\beta. The mass fractions are W_\alpha = \dfrac{C_\beta - C_0}{C_\beta - C_\alpha} and W_\beta = \dfrac{C_0 - C_\alpha}{C_\beta - C_\alpha}, where the composition unit is weight percent or atomic percent. The two fractions always sum to 1.

When the ratio of the amounts of two phases is given, convert it into fractions first. For example, if the weight ratio of \alpha to \beta is 4, then W_\alpha = 0.8 and W_\beta = 0.2. The overall weight percent of A is then C_0 = 0.8\,C_\alpha + 0.2\,C_\beta. Because the alloy is binary, the weight percent of B is 100 - C_0.