The CALPHAD modeling of thermodynamics was pioneered by Kaufman and Bernstein [47] and has been reviewed in detail by Saunders and Miodownik [48] and Lukas, Fries, and Sundman [49]. Information on features of software tools for CALPHAD modeling can be found in two series of publications in the CALPHAD journal [50, 51]. The key feature of the CALPHAD method is the modeling of the Gibbs energy of individual phases using both thermodynamic and phase equilibrium data. The main significance of the CALPHAD method is the following.

1. i. It enabled the development of the concept of lattice stability, i.e. the energy difference between the stable and non-stable crystal structures of a pure element.

2. ii.The Gibbs energy expression of each phase covers the full temperature, pressure, and composition spaces including both stable and non-stable regions of the phase. This enables the evaluation of the Gibbs energy of a system as a function of non-equilibrium state, i.e. with ξ as an independent variable.

3. iii.Thermodynamic data are usually obtained by measurements of heat such as the enthalpy of transition and heat capacity, as discussed in Section 4.2, which have large uncertainties typically in the range of kilojoules per mole-of-atoms. On the other hand, phase equilibrium data as discussed in Section 4.1, though more accurate, only contain information on compositions of phases at equilibria, i.e., the relative Gibbs energy of phases at equilibrium. The combination of these two sets of data is foundational in CALPHAD modeling and allows for the accurate modeling of thermodynamic properties of individual phases and reliable calculations of phase stability and driving forces.

4. iv. CALPHAD provides a framework to model thermodynamic properties of multi-component systems of industrial importance, enabling computational materials design. It has also been extended to model a range of properties of individual phases in multi-component systems such as diffusion coefficients, elastic coefficients, and thermal expansion, supplying input data for computational simulations of phase transformations during materials processing.

In this chapter, the basics of CALPHAD modeling of the Gibbs energy of individual phases are presented. For detailed implementations in various software packages and modeling procedures, readers are referred to the references given above.

Importance of lattice stability

For modeling of the Gibbs energy of individual phases, it is necessary to define the values of °Gi in Eq. 2.48.