Fickian plots

Figure 2. Fickian plots for 3-methyl-pentane adsorption at room temperature (letters as in figure 1 and Table I).

Uptake plots of the type shown in Fig. 2.12 are typical of diffusion of substances in leathery polymers (i.e. T>Tg) and are termed Fickian. This is in contrast to non-Fickian diffusion which is generally expected for diffusion of organic solvents in glassy polymers. Plot C in Fig. 2.13 shows a typical non-Fickian plot. Here the solvent begins to plasticise the polymer and lower its Tg, so that once Tg is lowered to the experimental temperature there is a sharp increase in the rate of uptake. 

The rate and type of release can be analyzed by the expression Mt/Moo=ktn (76). In the case of pure Fickian diffusion n = 0.5, whereas n > 0.5 indicates anomalous transport, i.e., in addition to diffusion another process (or processes) also occurs. If n = 1 (zero order release), transport is controlled by polymer relaxation ("Case II transport") (76). The ln(Mt/Mco) versus In t plots, shown in Figure 4, give n = 0.47 and 0.67 for samples A-9.5-49 and A-4-56, respectively. Evidently theophylline release is controlled by Fickian diffusion in the former network whereas the release is... 

For a classical diffusion process, Fickian is often the term used to describe the kinetics of transport. In polymer-penetrant systems where the diffusion is concentration-dependent, the term Fickian warrants clarification. The result of a sorption experiment is usually presented on a normalized time scale, i.e., by plotting M,/M versus tll2/L. This is called the reduced sorption curve. The features of the Fickian sorption process, based on Crank s extensive mathematical analysis of Eq. (3) with various functional dependencies of D(c0, are discussed in detail by Crank [5], The major characteristics are... 

This solution is valid for the initially linear portion of the sorption (or desorption) curve when MtIM is plotted against the square root of time. These equations also demonstrate that for Fickian processes the sorption time scales with the square of the dimension. Thus, to confirm Fickian diffusion rigorously, a plot of MJM vs. Vt/T should be made for samples of different thicknesses a single master curve should be obtained. If the data for samples of different thicknesses do not overlap despite transport exponents of 0.5, the transport is designated pseudo-Fickian. ... 

Figure 14 Master curve generated from mean-square displacements at different temperatures, plotting them against the diffusion coefficient at that temperature times time. Shown are only the envelopes of this procedure for the monomer displacement in the bead-spring model and for the atom displacement in a binary Lennard-Jones mixture. Also indicated are the long-time Fickian diffusion limit, the Rouse-like subdiffusive regime for the bead-spring model ( ° 63), the MCT von Schweidler description of the plateau regime, and typical length scales R2 and R2e of the bead-spring model.

An industrial oil-extended synthetic cis-polyisoprene was investigated by von Meerwall and Ferguson30). Following Boss, et al. 29), they substracted the unattenu-atable spin echo arising from the rubber, obtaining the diffusivity of the extender oil from the remainder. They demonstrated that no departures from Fickian diffusion occur, and measured the diffusion of the oil, both in the rubber and in the pure liquid, between —10 °C and 130 °C. Since the plot of log D vs. 1/T was not a straight line it was necessary to invoke the Williams-Landel-Ferry temperature dependence,... 

When the fractional drug release from an initially dehydrated hydrogel sheet is plotted as a function of square root of time as shown in Figure 1 for thiamine HC1 release from a poly(2-hydroxyethyl methacrylate) sheet, linearity in the plot is observed only at large times. This illustrates the non-Fickian and time-dependent nature... 

The magnitude of D. in relation to D also has a profound effect on the overall release behavior. As shown in Figures 5 and 6, where the time dependence of fractional release is plotted as a function of D /D for both the dissolved and dispersed systems at D /kJt2 = 10, as D./D ->-1 the release behavior becomes more Fickian... 

If D only depends on temperature (and thus not on concentration or time), the diffusion process is called Fickian. Simple gases show Fickian diffusion and so do many dilute solutions (even in polymers).The diffusivity can be determined directly either from sorption or from permeation experiments. In the first case the reduced sorption, c(t) / (cfX, cG), is plotted versus the square root of the sorption time and D is calculated from the equation ... 

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

Mixing in a stream or river can be quantitatively illustrated by instantaneously releasing a mass of chemical uniformly throughout the cross section of a channel this pulse injection is best expressed as mass per cross section of river, [M/L2]. Ideally, one chooses a conservative tracer (i.e., a chemical that does not undergo degradation in the river and is not absorbed to the river channel or suspended particles). The lower panel of Fig. 2-4 shows a chemical concentration in a stream at several different times after a pulse injection. At any instant, the plot of concentration versus distance is bell-shaped ideally, if the mixing is truly Fickian, the curve has the shape of a Gaussian, or normal, curve,... 

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

From a least-squares analysis, the values of n and k have been estimated and these are included in Table I, and Figure 8 represents a typical plot for benzene and toluene. The average uncertainty in the estimation of n is around 0.007. The value of n do not indicate any systematic variation with temperature. However, a general variation of n from a minimum value of 0.53 to a maximum of 0.74 indicates that the anomalous type transport mechanism is operative and the diffusion is slightly deviated from the Fickian trend. This fact can be further substantiated... 

Figure 12.12 An illustration of a typical Fickian diffusion behavior of water absorption by two WPG deck boards, a solid Trex board (lower plot), and a hollow GeoDeck board filled with only rice hulls (see Table 12.8). The horizontal axis—a square root from underwater exposure time.

In Fickian diffusion, the absorption behaviour is as shown in Fig. 7.6. The absorption (weight gain) and desorption (weight loss) curves can be plotted against (square root of time) and the resulting graphs are linear until 60% of the maximum absorption, has been reached, but later it is concave towards the axis and asymptotically approaches the final equilibrium value. is a constant when the material is fully submerged in a... 

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