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. 

Criteria 1-3 are the cardinal characteristics of Fickian diffusion and disregard the functional form of D(ci). Violation of any of these is indicative of non-Fickian mechanisms. Criterion 4 can serve as a check if the D(ci) dependence is known. As mentioned, it is crucial that the sorption curve fully adhere to Fickian characteristics for a valid determination of D from the experimental data. At temperatures well above the glass transition temperature, 7 , Fickian behavior is normally observed. However, caution should be exercised when the experimental temperature is either below or slightly above 7 , where anomalous diffusion behavior often occurs. 

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. 

The computed mole fractions as a function of distance along the tube are shown in Fig. 12.14. The mole fractions exhibit a nearly linear drop between their equilibrium values at the surface and zero at the top of the tube. This behavior is not unexpected for this simple system, in which the very dilute species diffuse into one dominant species (air) that is present in great excess. In such a case we expect (and observe) Fickian behavior. That is, we could solve this problem using one of the mixture-averaged formulations discussed in Section 12.7.4 with very little error. 

Consider the gel collapse in more detail. As hydrogen ions diffuse into the gel, they will rapidly react to form neutral carboxylic acid groups. Thus, a nonionic shell of collapsing gel will develop around a still-swollen ionized core. The diffusion of ions occurs freely in this nonionic shell, so that the collapse is limited by the Fickian diffusion of water out of the gel. We have confirmed this Fickian behavior by measuring the collapse of cylindrical gel samples of differing radii and ionic compositions [5]. The data for the fractional approach to equilibrium fall on a single curve against [Dt/R2]1/2, where D is the diffusion coefficient and R is the initial radius of the gel cylinder. 

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. 

Laatikainen and Lindstrom [100] used TSM devices to investigate absorption in cellulose acetate and poly-(hexamethylene adipamide). In addition to measuring absorption isotherms and partition coefficients, they reported on transient responses to changes in methanol concentration for a cellulose-acetate-coated TSM device (Figure 4.10). At low concentrations, the linear response with Vr is consistent with Fickian behavior, and diffusion coefficients can be evaluated (Z> =... 

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). 

One criterion of Fickian behavior holds that Mt/Moo vs. ti /2 plot should be linear up to 60% reduced sorption. 

Note that the dispersion terms described in equation (6.18) are valid only in the limit of Fickian behavior. From the central limit theorem, this regime is reached when every particle has amply sampled each region (wakes, gaps, recirculation zones). The average time-scale to advect through a wake is (a(u)Yl, and the average time-scale to experience trapping within a recirculation zone is r/ (yad). Then, the Fickian limit is reached at time t r/ (yad) and (fl(M 1. 

The observed longitudinal dispersion will also be influence by stem-wake dispersion. However, for ad < 0.1 (as is the case here), the recirculation zone and gap dispersion will be negligible (White and Nepf 2003 [643]). Further, the stem-wake dispersion can reach Fickian behavior at t > (a(/7 ) 1, or x > a x (see discussion above). For the canopy studied this requires x > 8 cm. Because all measurements were taken more than 50 cm downstream, stem-wake dispersion may be represented by the term shown in (6.18). Additionally using equation (6.8) to replace the velocity ratio in equation (6.27), the total longitudinal dispersion coefficient is,... 

To obtain diffusion coefficients, the following equation can be applied to the linear portion of the normalized sorption/desorption data assuming Fickian behavior with constant diffusion coefficient (Crank, 1975) ... 

In some cases deviations from Fickian behavior in glassy epoxy polymers (19-25) have been adequately described using dual mode sorption theory (22-24). This theory is based upon the premise that the sorbed penetrant exists in two thermodynamically distinct populations. These populations consist of molecules adsorbed in "holes", and species simply dissolved in the polymer matrix. 

The appearance of the conjugate sorption data presented so far qualitatively indicates a shift from a more to a less Fickian character as the initial surface moisture content increases. That is, the appearance of the conjugate sorption isotherms obtained by totally immersing the samples violate two of the criteria by which Fickian behavior is defined. The same cannot be said for those samples exposed to less than 100% R.H., particularly at 25°C. This qualitative trend for PMC is further demonstrated by Figures 16 and 17. Here MjVWq is presented as a function of time. For both the thick and the thin sample, as either temperature or relative humidity is increased, the character of the curves progresses towards pure Case II description. That is, the moisture uptake becomes linear with time up to the point where a plateau is achieved in the behavior. 

As described earlier, samples immersed in water, regardless of curing history or temperature, exhibited two deviations from the criteria defining Fickian behavior. These are 1) the initial higher value of D... 

For a Fickian behavior, the plots of Q(t) versus t should increase linearly up to about 50% sorption. Deviations... 

Considering the complexity of the sorption behavior of the bulk resin, the almost classical Fickian behavior of the composites comes as a surprise and represents a welcome simplification. At the present time the only explanation that can be made is by analogy with the size dependence of relaxation controlled diffusion effects reported by Berens (13) for diffusion of vinyl chloride in poly(vinylchloride). It was noted that as the particle size and, therefore, the path length diminished, the contribution of the... 

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. 

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