Revisiting Derivatives at Equilibrium to Establish a Differentiable Metric for Phase Stability

While methods of calculating analytic derivatives of thermodynamic properties at equilibrium with respect to conditions of the equilibrium calculation have been implemented in various computational thermodynamics software packages for decades, the documentation of the mathematical underpinnings of such methods remains sparse. In this work, the mathematical formalism behind the “dot derivative” technique as implemented in Thermo-Calc, OpenCalphad, and PyCalphad is rigorously presented. This procedure is then leveraged to construct general forms of derivatives of the residual driving force, a popular metric for measuring phase stability, with respect to overall system and phase vertex compositions. Applied examples--- calculating heat capacity in the Al-Fe system, thermodynamic factors in the Nb-V-W system, and residual driving force derivatives in the Ni-Ti system---demonstrate the versatility, accuracy, and extensibility of this method.

As the reframing of residual driving force as a differentiable quantity shows, a thorough understanding of the mathematical theory behind this decades-old technique holds the key for its future extensibility. Looking ahead, derivatives with respect to model parameters would allow for gradient-based optimization approaches, and reapplication of the mathematical arguments presented in this work could lead to the derivation of methods for calculating higher-order derivatives at equilibrium.