Phase diagrams are powerful tools to understand the multi-scale behaviour of complex systems. Yet, their determination requires in practice both experiments and computations, which quickly becomes a daunting task. Here, we propose a geometrical approach to simplify the numerical computation of liquid-liquid ternary phase diagrams. We show that using the intrinsic geometry of the binodal curve, it is possible to formulate the problem as a simple set of ordinary differential equations in an extended 4D space. Consequently, if the thermodynamic potential, such as Gibbs free energy, is known from an experimental data set, the whole phase diagram, including the spinodal curve, can be easily computed. We showcase this approach on four ternary liquid-liquid diagrams, with different topological properties, using a modified Flory-Huggins model. We demonstrate that our method leads to similar or better results comparing those obtained with other methods, but with a much simpler procedure. Acknowledging and using the intrinsic geometry of phase diagrams thus appears as a promising way to further develop the computation of multiphase diagrams.
Keywords: LLE ternary diagrams; binodal curves; differential path-following method; ordinary differential equations; spinodal curves.