The final element in understanding diffusion is the effect of the surface. If for any reason molecules cannot penetrate quickly through the surface then diffusion will be limited not by diffusion coefficients but by the surface resistance, or both simultaneously in series. This elementary truth is often ignored. Happily, it is relatively easy to model within the same system as before. An extra "h" term is added which is the mass transfer coefficient in 1/s. As few of us know what this is, the idea is to play with values to see if there is any effect. When h is large it has no impact, when it is very small it is dominant. A useful guide value for a constant diffusion coefficient is B, the ratio of diffusion resistance (L/D) to surface resistance (1/h). This is then given by B= hL/D for a constant diffusion coefficient. When B approaches unity then, typically, surface resistance becomes visible in the graphs. Because surface resistance often shows as being linear in time rather than square root, the Linear T option allows you to plot in Linear Time.
The point about modelling surface resistance is that although most times you don't need it, when you find "anomalous" diffusion curves (e.g. sigmoidal rather than square root dependency on time) you will most likely find it readily explained via a significant surface resistance perhaps also coupled with concentration dependence of the diffusion coefficient.
Surface resistance can be very complex. Two cases illustrate the issues.
- Diffusion is not concentration dependent. While the potential flux to the surface is largest at the start because of the highest concentration gradient, diffusion can sometimes remove the absorbed solute still faster into the bulk of the film than it arrives from the external phase. Surface resistance means that transport in the external phase limits the absorption rate. As time goes on the diffusion resistance can become more significant because of transport over larger distances within the film, and can eventually exceed the surface resistance, in which case the surface concentration increases to the equilibrium value and the surface resistance is no longer significant in the final stages of absorption.
- Diffusion is concentration dependent. When the diffusion coefficient depends very strongly on concentration, the surface resistance may not be particularly significant at the start, but becomes very much so at the last. At the start the diffusion coefficients are low and diffusion can control or be significant. As time goes on the concentrations in the film and at the surface increase and the diffusion resistance can become less significant or even disappear leaving the final stages of absorption entirely controlled by the surface resistance at a very low concentration difference across the surface. The concentration profiles will be very flat when surface resistance controls. The terminology for such cases in the literature is Case II or Super Case II, depending on whether the uptake becomes linear with time or increases more rapidly than with linear time, respectively.