Diffusion with Discontinuous Swelling: III. Type II Diffusion as a Particular Solution of the Conventional Diffusion Equation

Very often a non-solvent diffuses into a glassy polymer with a steep concentration profile proceeding at an almost constant rate v yielding a weight gain proportional to time. Such a diffusion is called type II diffusion in order to distinguish it from the more usual “Fickian” diffusion proceeding without such a constant concentration front and yielding, at least in the beginning, a weight gain proportional to the square root of time. It turns out that the conventional diffusion equation without any special new term but with a diffusion coefficient rapidly increasing with concentration has a series of solutions representing exactly such type II diffusion with v as a completely free parameter which determines the steepness of concentration front. With the usual boundary conditions and infinite medium the diffusion coefficient has to become infinite at the highest penetrant concentration. This case can be considered as an extreme limit which is approached to a high degree in an actual experiment. The finite sample thickness, however, requires only a very large but not an infinite diffusion coefficient. Hence type II diffusion is only a special case of possible diffusion processes compatible with the conventional diffusion equation without any need for new terms if only the diffusion coefficient increases sufficiently fast with penetrant concentration.