The calculation of equilibrium phase diagrams from theoretical thermodynamic data involves the solution of a constrained minimization problem over a linear combination of functions for the free enthalpy, each of which are non-linear in both temperature and composition. The common iterative solution employed by CALPHAD codes requires these functions to be differentiable to obtain the Jacobian matrix. When the free energy data is obtained by computationally expensive methods such as ab-initio calculations, there is often not enough data to generate these differentiable functions with high resolution. Instead, data may only be available at discrete points in temperature and composition space and include a variable degree of model inaccuracies.

First, sources of this kind of data (such as ab-initio calculations) are discussed and the quality of the data is assessed.

Then this talk will present a solution approach to equilibrium calculation that is more suited for finding the equilibrium state over these sparse data points and in spite of model inaccuracies. This method is based on analytical geometry of higher-dimensional spaces and computationally simple and extendable. The resulting phase equilibria are found as states instead of transformation lines. Therefore, an additional step is required to correctly identify the types of phase boundaries and the phases involved in these reactions. For consistency between the discrete calculations, automatic matching of the results is carried out.

The method is demonstrated on a practical example and the software implementation is discussed briefly. The results are then discussed in relation to the initial source of such data and possible further application to kinetics of reactions.