Fickian diffusion process

The release of steroids such as progesterone from films of PCL and its copolymers with lactic acid has been shown to be rapid (Fig. 10) and to exhibit the expected (time)l/2 kinetics when corrected for the contribution of an aqueous boundary layer (68). The kinetics were consistent with phase separation of the steroid in the polymer and a Fickian diffusion process. The release rates, reflecting the permeability coefficient, depended on the method of film preparation and were greater with compression molded films than solution cast films. In vivo release rates from films implanted in rabbits was very rapid, being essentially identical to the rate of excretion of a bolus injection of progesterone, i. e., the rate of excretion rather than the rate of release from the polymer was rate determining. 

While the advection-dispersion equation has been used widely over the last half century, there is now widespread recognition that this equation has serious limitations. As noted previously, laboratory and field-scale application of the advection-dispersion equation is based on the assumption that dispersion behaves macroscopically as a Fickian diffusive process, with the dispersivity being assumed constant in space and time. However, it has been observed consistently through field, laboratory, and Monte Carlo analyses that the dispersivity is not constant but, rather, dependent on the time or length scale of measurement (Gelhar et al. 1992),... 

From the standpoint of using multicomponent diffusion in a numerical simulation, it can be beneficial to pose the multicomponent diffusion in terms of an equivalent Fickian diffusion process [72,422]. To do this, imagine that a new mixture diffusion coefficient can be defined such that the first term (summation) in Eq. 12.166 can be replaced with the right-hand side of Eq. 12.162. An advantage of the latter is that the diffusion of the fcth species depends on its own mole fraction gradient, rather than on the gradients of all the other species the Jacobian matrix is more diagonally dominant, which can sometimes facilitate numerical solution. 

In many cases of transport in solids, the atoms (ions) of one sublattice of the crystal are (almost) immobile. Here, we can identify the crystal lattice with the external (laboratory) frame and define the fluxes relative, to this immobile sublattice (to = 0). v° is bk-Xk (Eqn. (4.51)) where Xk is the sum of all local forces which can be applied externally (eg., an electric field), or which may stem from fields induced by the, (Fickian) diffusion process itself (eg., self-stresses). An example of such a diffusion process that leads to internal forces is the chemical interdiffusion of A-B. If the lattice parameter of the solid solution changes noticeably with concentration, an elastic stress field builds up and acts upon the diffusing particles, it depends not only on the concentration distribution, but on the geometry of the bounding crystal surfaces as well. 

To dearly distinguish between these two modes of solvent penetration of the gel, we immersed poly(acrylamide-co-sodium methacrylate) gels swollen with water and equilibrated with either pH 4.0 HQ or pH 9.2 NaOH solution into limited volumes of solutions of 10 wt % deuterium oxide (DzO) in water at the same pHs. By measuring the decline in density of the solution with time using a densitometer, we extracted the diffusion coefficient of D20 into the gel using a least squares curve fit of the exact solution for this diffusion problem to the data [121,149]. The curve fit in each case was excellent, and the diffusion coefficients obtained were 2.3 x 10 5cm2/s into the ionized pH 9.2 gel and 2.4 x 10 5 cm2/s into the nonionized pH 4.0 gel. These compare favorably with the self diffusion coefficient of D20, which is 2.6 x 10 5 cm2/s, since the presence of the polymer can be expected to reduce the diffusion coefficient about 10% in these cases [150], In short, these experiments show that individual solvent molecules can rapidly redistribute between the solution and the gel by a Fickian diffusion process with diffusion coefficients slightly less than in the free solution. 

Various types of coupled non-linear Fickian diffusion processes were numerically simulated using the free-volume approach given by equation [12.8], as well as non-Fickian transport. The non-Fickian transport was modeled as a stress-induced mass flux that typically occurs in the presence of non-uniform stress fields normally present in complex structures. The coupled diffusion and viscoelasticity boundary value problems were solved numerically using the finite element code NOVA-3D. Details of the non-hnear and non-Fickian diffusion model have been described elsewhere [14]. A benchmark verification of the linear Fickian diffusion model defined by equations [12.3]-[12.5] under a complex hygrothermal loading is presented in Section 12.6. 

For the adsorption process governed by a single Fickian diffusion process, the time constant is defined as the ratio t = L /D, where L is the characteristic half-dimension and D the diffusion coefficient Accordingly, the time constant for the micropore diffusion model would be... 

For the sake of simplicity, we only consider here the Fickian diffusion process which may occnr in the glassy state far from T. Bnt there are exceptions to this process and we invite the reader to refer to extensive literature reviews for the treatment of other transport modes of small molecnles into polymers (Miiller-Plathe, 1994 Masaro and Zhn, 1999 George and Thomas, 2001). 

Copolymers of tetrafluoroethylene and sulfonic acid functional per-fluorinated monomers (e.g., Nafion, Dow s perfluorosulfonic acid (PFSA)) have high water permeability. Water transport through these ionomer membranes has been investigated. The non-Fickian diffusion process is analyzed by a thermodynamic approach. The results provide some useful insights into the behavior of these materials as dehydration membranes. 

Fig. 2.13. Non-Fickian diffusion processes illustrated by the interaction of polycarbonate with methyl methacrylate vapour at 25 C. Sequential experiments are (A) sorption, (B) desorption in vacuum and (C) resorption.

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