Lyogel equilibria with Helmholtz equations of state. Part I: A robust second-order framework

The thermodynamic calculation of gel phase equilibria is numerically challenging because of the strong asymmetry of polymer–solvent systems and the coupling of mixing and elastic contributions. In this work, a framework for gel phase equilibria is developed using the PC-SAFT equation of state. Building on strategies for polymer–solution liquid–liquid equilibria, the method is extended to gel systems by incorporating the elastic contribution of the polymer network through a Poynting-like correction. To improve numerical stability, the framework combines a second-order Newton–Raphson scheme with homotopy-based phase-envelope tracing and alternating-tangent initialization. The method is applicable to multicomponent systems and allows continuation in composition and temperature space. Its performance is demonstrated for pNIPAM–water and pNIPAM–water–ethanol systems. The results show that continuation strategies established for conventional polymer–solution phase equilibria can be transferred consistently to gel equilibria without repeated reinitialization.