An algorithm for determining cloud and shadow curves using continuous thermodynamics

A computer algorithm has been developed to determine the cloud and shadow point curves of polydisperse polymer/solvent systems using continuous thermodynamics to represent the polymer. The algorithm is based upon the work of Michelsen [M.L. Michelsen, Fluid Phase Equilibria, 4 (1980), pp. 1–10] and Koak [N. Koak, Polymer Solution Phase Behaviour, PhD Thesis, University of Calgary, Calgary, 1997] where a Newton–Raphson procedure is used to solve the equations defining phase equilibrium for fixed phase segment fractions between 0 and 1. The equations have been developed with the Sanchez–Lacombe equation of state which allows the continuous system to be defined by a set of five equations in five unknowns. Numerical integration is used to determine the values of the indefinite integrals involved. The cloud and shadow curves for a polydisperse polyethylene/ n -hexane system were found at 6 bar using both a log-normal distribution and a Schulz–Flory distribution to represent the polyethylene. Compositions of the cloud point curve covered the entire composition range showing the model's behavior in high temperature regions, a hypothetical liquid–vapour critical point and a three-phase point when the log-normal distribution was used. The cloud and shadow point curves of a polyethylene/ethylene system were also found and indicated a high pressure critical point.