The calculation of the interfacial free energy between two thermodynamic phases is crucial across various fields, including materials science, chemistry, and condensed matter physics. In this study, we apply an existing thermodynamic approach, the Gibbs-Cahn integration method, to determine the interfacial free energy under different coexistence conditions, relying on data from a single-state calculation at specified pressure and temperature. This approach developed by Laird et al. [J. Chem. Phys. 131, 114110 (2009)] reduces the computational demand and enhances efficiency compared to methods that require separate measurements at each thermodynamic state. The integration scheme computes the excess interfacial free energy using unbiased constant volume, temperature, and number of particle simulations (NVT), where the two phases coexist, to provide input for the calculations. We apply this method to the Lennard-Jones and mW water models for liquid-solid interfaces, as well as the Lennard-Jones and TIP4P/2005 models for liquid-vapor interfaces. Our results demonstrate the accuracy and effectiveness of this integration route for estimating the interfacial free energy along a coexistence line.
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