About the Alkemade Theorem and the Limits of its Applicability for the Construction of Ternary Liquidus Surfaces

The Alkemade theorem goes back to a very fundamental paper on the graphical description of thermodynamic equilibrium problems from 1893 (van Rijn van Alkemade in Z Phys Chem 11: 289-327, 1893). It is one of the most helpful implements for the construction of the liquidus surface of ternary phase diagrams. In its original form, it allows to find the direction of falling or increasing temperature along the monovariant reaction lines forming the boundaries of the primary crystallization fields. The theorem is valid for systems with any number of phases; however, its geometrical construction rule is only defined for the case of stoichiometric phases and it is not clear how to apply the theorem in the case of phases with extended homogeneity ranges. Some examples from a ternary, transition-metal-based system containing phases with large homogeneity ranges are presented, and the usefulness and limits of applicability of the theorem are discussed.