The thermodynamic calculation of gel phase equilibria is numerically challenging, because polymer and solvent systems are strongly asymmetric and because mixing and elastic contributions are coupled. In this work, the authors develop a framework for gel phase equilibria using the PC-SAFT equation of state. Building on strategies established for polymer solution liquid and liquid equilibria, the method is extended to gel systems by including 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 initialisation. The method is applicable to multicomponent systems and allows continuation in composition and temperature space. Its performance is demonstrated for pNIPAM and water as well as pNIPAM, water and ethanol systems. The results show that continuation strategies established for conventional polymer solution phase equilibria can be transferred consistently to gel equilibria without repeated reinitialisation.
The study is a collaboration between the Institute of Thermal Separation Processes at Hamburg University of Technology and Mercator Fellow Walter G. Chapman from Rice University, carried out within project B01 of the Collaborative Research Centre CRC 1615 (SMART Reactors). It is the first of two connected publications on lyogel equilibria.
Nanning Jaeschke, Irina Smirnova, Walter G. Chapman (2026). Lyogel Equilibria with Helmholtz Equations of State. Part I: A Robust Second-Order Framework. Ind. Eng. Chem. Res. 65 (31), 16843-16851.
