There has been a great deal of discussion in the literature about “anomalies” of diffusion in polymers. I have not found any anomalies. This is true for absorption as well as for desorption (film drying). Appropriate solutions to the diffusion equation, Fick’s second law, describe all of these. Evidence for this is found in the second edition CRC handbook (back to home page) in Chapter 16 and earlier publications. The diffusion equation is a second order partial differential equation that requires two boundary conditions and an initial condition for solutions. The diffusion equation is derived in a very general way. There is no problem involved with using concentration dependent diffusion coefficients or a surface condition. The fact that diffusion coefficients can vary by a factor of a million or more within a limited concentration range is therefore no problem. Boundary conditions can be chosen to account for important physical phenomena at the film surface. A significant surface resistance can be caused by factors exterior to the film surface itself such as rate of heat transfer, air velocity, diffusion in a stagnant air film, and the like. The mass transfer coefficients are relatively high, but can still be significant if diffusion within the film is rapid. A significant entry resistance may be found within the film surface itself and is caused by the morphology of the polymer in the surface. The mass transfer coefficient in such a case is very low. This surface morphology can be a result of rapid cooling after an injection molding processing or just be natural to given polymer types where larger and bulkier molecules have difficulty finding an adsorption site from which to subsequently absorb into the bulk of the film. Both surface resistance and entry resistance are described by the same kind of boundary condition for solutions to the diffusion equation. Some definitions follow: 

Fickian Diffusion usually means that there is a straight line describing the uptake of a liquid or gas into a polymer film when the amount taken up at a given time is plotted against the square root of time. The straight line passes through the origin. This is common and expected when the diffusion coefficient is constant or nearly so, and there is no surface or entry resistance.

 

The S-Curve Uptake  “anomaly” involves delayed absorption. A plot of amount uptake versus the square root of time does not produce a linear relation that passes through the origin. The reason for this delay is a surface or entry resistance that has become significant relative to the diffusion resistance. The surface or entry resistance is expressed by the inverse of the mass transport coefficient (1/h), and the diffusion resistance is given by the film thickness divided by the diffusion coefficient (L/D). Their ratio has been called B = hL/D and can be used in generalized solutions to the diffusion equation. S-curves are readily generated by solutions to the diffusion equation with a boundary condition that accounts for significant surface or entry resistance.

 

Case II diffusion is the term generally applied to a linear uptake of solvent using a plot of amount absorbed versus linear time (not square root of time). This is obviously a deviation from the terminology discussed above for Fickian diffusion. This behavior is described by appropriate solutions to the diffusion equation, however, and is found when the diffusion coefficient is very dependent on solvent concentration. The diffusion coefficient in rigid polymers increases by a factor of about 10 for each additional 3 percent by volume solvent that is locally present. In elastomers the diffusion coefficient increases by a factor of approximately 10 for each additional 15 percent by volume of solvent. There is a limit to this rule of thumb when the solvent concentration approaches 100% and the diffusion coefficient approaches the self-diffusion value.

As the name indicates, Super Case II is a modification of the Case II type behavior that has just been discussed. In this case the uptake is initially linear on a plot using linear time, but at some time well into the absorption process, the rate of uptake suddenly increases. An explanation for this astounding behavior has challenged many and there are many explanations. Since straightforward solutions to the diffusion equation can readily describe this behavior as well as all of the above, I see no reason to be surprised. An example from a diffusion modeler written by Prof. Steven Abbott (MacDermid /Autotype) is given in the figure below. This figure confirms that one can reproduce this type of behavior when a significant entry resistance is combined with a concentration dependent diffusion coefficient. This figure will appear in a forthcoming book written by Prof. Abbott and myself. The diffusion coefficient varies with concentration as shown in the upper right figure. These values are essentially those measured for the system chlorobenzene in polyvinylacetate as described in numerous of my publications including the second edition of the handbook. Combinations of absorption and desorption were used to arrive at these results. The surface condition in this case involves an entry resistance and not an effect external to the film surface. This is described by the given B and h values in the figure. The concentration gradients that develop as a function of time can be seen at the lower left. The uptake curve is plotted as a function of linear time showing the very dramatic increase in the rate of uptake well into the absorption process. It would not be surprising if the surface mass transfer coefficient increased with increasing surface concentration. This would only reinforce the astounding effect.