Non-Fickian behavior

If De crr 1, non-Fickian behavior may appear, ie, a dependence other than linearity with the square root of time may be observed. 

Another important point which may be inferred from the limited data of Tables IV, V, and VI is that for compounds which are chemically similar, e.e., congeners or isomers as was the case for the halogenated ethanes, there may be appreciable differences in the rate at which they permeate given materials, and in many cases will show non-Fickian behavior as shown in the graphs of Fig. 9-18. 

To obtain quantitative information about sorption processes controlled by time-dependent diffusion Crank (1953) solved numerically Eq. (1), coupled with Eqs. (4), (5) and (14), for some assumed forms of >i(Cj), De(Cl) and a(cj. His results agreed reasonably with many typical non-Fickian features known at that time (Park (1953)]. However, when a new type of non-Fickian behavior, now generally called the "two-stage type, was discovered in 1953 by Long and his coworkers, it soon became evident that the concept of time-dependent diffusion was too simple to explain every non-Fickian behavior. This situation remains unaltered at present, and so we shall not go further into this subject. 

This non-Fickian behavior arises from the fact that the range of fluid motions responsible for dispersion of a tracer depends on the size of the tracer patch in relation to the spectrum of fluid motions occurring, and the distinction between stirring and mixing (e.g., see Csanady, 1972 Rhines and Young, 1983 Young et al., 1982). 

It can be further extended to describe simultaneous diffusion of multiple species in a multicomponent solid. However, Pick s law may not effectively handle individual atomic jumps at short spatial scales or on very short time scales (as determined by some spectroscopic methods or computer simulations), nor can it address cases where diffusion occurs in a medium whose structure is changing, like the glass transition region. These may lead to what is termed non-Fickian behavior (e g. Crank 1975). 

Diffusion of small molecules, usually solvents, into glassy polymers exhibits anomalous or non-Fickian behavior 34). As the solvent penetrates, the diffusion coefficient increases because the glass transition temperature is lowered. The solvent acts as a plasticizer, increasing the free volume and the mobility of the solvent. Thus we have an autocatalytic diffusion process. This can be relevant in Isothermal Frontal Polymerization, which we discuss below. 

Improved Water Resistance, and the Non-Fickian Behavior of Epoxy Systems During Water Ageing," Brit. Polym. J., 14, 1983. 

Using sorption-desorption kinetics, the extent of non-Fickian behavior decreases as the thickness of the film increases. Presumably, as the ratio of the time scale for diffusion to the time scale for stress-induced relaxation I ID) becomes large, the time-dependent relaxation occurs so rapidly (relatively) that it has a small effect on the diffusion process. In other words, in the limit where molecular-scale responses to swelling stresses are rapid compared to the rate of diffusion, the process appears to be Fickian. This is the case for totally amorphous rubbery polymers that behave as high molecular weight liquids. 

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