Mass-transfer equilibrium in multi-component closed systems at thermal and mechanical equilibrium requires that the chemical potential of each component be uniform over all phases in the system, yielding the classical Gibbs phase rule relating the number of independent components (i.e. chemical species), the number of externally unconstrained state variables (i.e. 'degrees of freedom'), and the number of coexisting phases at equilibrium. More general closed systems, however, may be subject to thermal or mechanical disequilibrium (e.g. non-zero temperature gradients), giving rise to non-zero chemical potential gradients. Such systems require 'intra-phase' equilibrium conditions upon the chemical potential gradients in addition to the usual 'interphase' conditions upon the chemical potentials themselves, yielding a modified phase rule restricting the stability of regions of multi-phase coexistence. The attainment of mass-transfer equilibrium under such conditions requires that isochemical multi-phase systems break down into chemically distinct juxtaposed single-phase regions, thermodynamically coupling multivariant phase transitions to chemical discontinuities. Hence, for example, phase change and chemical change hypotheses for seismic velocity discontinuities in the Earth's interior need not be mutually exclusive. The kinetic hindrance by slow diffusion of phase-transition-coupled chemical differentiation in the Earth may be alleviated by high temperatures, the presence of fluids or melts, convective mass flow, negative Clapeyron slopes, or stratification due to earlier episodes of differentiation. Radial mass-transfer equilibrium across the spinel-perovskite transition for an upper mantle with X(Mg) almost-equal-to 0.89, for example, yields a stable contrast in Fe/Mg composition with X(Mg) almost-equal-to 0.86 in the lower mantle.
