Thermodynamic equilibrium calculation is one of the nonlinear minimization problems, for which robust and stable calculation is difficult in general. Calculations for metal systems composed of elements are relatively easy and stable, but calculations for compound systems such as ceramics and intermetallic compounds, and even for metal systems, such as pseudo-binary phase diagrams calculations, are difficult problems.

　The reason for the problem is because chemical potentials of elements are not fully determined by the usual Gibbs energy minimization (GEM). An example is shown in Fig.1a. When the solid SiO2 is the only stable phase in the Si-O system, then the tangent plane cannot be fixed because the overall Gibbs energy function bends steeply at the composition of SiO2, and the chemical potentials of Si and O are not definitely determined. However, chemical potentials of system elements are actually needed in equilibrium calculation to determine the correct set of active stable phases. If they are not available, stable calculation is very difficult.

  CaTCalc determines all element chemical potentials by assuming gas-phase that are in equilibrium with the SiO2 phase in such cases, as shown in Fig. 1b. Note that this gives physically correct chemical potentials when the gas phase becomes active when the pressure is sufficiently lowered, or when an inert gas is introduced to the system. The algorithm makes calculation very stable and robust in general.

The calculated element chemical potentials are, of course, given in the result.