Polymer Membrane Formation through the Thermal-Inversion Process. 2. Mathematical Modeling of Membrane Structure Formation
Formation of anisotropic membranes by thermal inversion is simulated by a model developed here. Structured membranes are formed when a polymer/solvent film is thermally coagulated from one face while the opposite face is thermally insulated. Both kinetics and thermodynamics of liquid-liquid phase separation are included in the analysis. Only noncrystalline polymers are considered, and the composition of the starting polymer solution is close to the critical or θ composition of the polymer/solvent system. Hence, spinodal decomposition is taken as the mechanism of phase separation. For a given point in the membrane cross section, the degree of phase separation is characterized by the deviation of the fluctuating concentration profile from the concentration of the starting polymer solution. The use of the Flory-Huggins model for solution thermodynamics and Cahn-Hilliard theory for spinodal decomposition leads to a one-dimensional fourth-order nonlinear partial differential equation that describes the phase-separation behavior induced by thermal inversion. Periodic boundary conditions are employed as well as infinitesimal sinusoidal perturbations for the initial condition. The solution scheme includes local linearization (from a previous time step) through Fourier transformation, followed by the application of the Euler method to solve the resulting set of ordinary differential equations for the frequency components of the concentration. With this mathematical model, a reasonable membrane pore size profile consistent with experimental observations is obtained. © 1985, American Chemical Society. All rights reserved.
Polymer Membrane Formation through the Thermal-Inversion Process. 2. Mathematical Modeling of Membrane Structure Formation.
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