Penetration of solvents into polymers is sometimes characterized by steep concentration gradients that move into the polymer and last for long times. The behavior of these fronts cannot be explained by standard diffusion equations, even with concentration dependent diffusion coefficients. The addition of stress terms to the diffusive flux can produce such progressive fronts. Model equations are proposed that include solvent flux due to stress gradients in addition to the Fickian flux. The stress in turn obeys an concentration dependent evolution equation.
The model equations are analyzed in the limit of small diffusivity for the problem of penetration into a semi-infinite medium. Provided that the coefficient functions obey certain monotonicity conditions, the solvent concentration profile is shown to have a steep front that progresses into the medium. A formula governing the progression of the front is developed. After the front decays away, the long time behavior of the solution is shown to be a similarity solution. Two techniques for approximating the solvent concentration and the front position are presented. The first approximation method is a series expansion; formulas are given for the initial speed and deceleration of the front. The second approximation method uses a portion of the long time similarity solution to represent the short time solution behind the front.
The addition of a convective term to the solvent flux is shown to raise the possibility of a traveling wave solution. The existence of the traveling wave solution is shown for certain types of coefficient functions. The way the initial front speed evolves onto the traveling wave speed is sketched out.
