Diffusion, Fickian/Case

From the values of A listed in Table 4.1, only the two extreme values 0.5 and 1.0 for thin films (or slabs) have a physical meaning. When A = 0.5, pure Fickian diffusion operates and results in diffusion-controlled drug release. It should be recalled here that the derivation of the relevant (4.3) relies on short-time approximations and therefore the Fickian release is not maintained throughout the release process. When A = 1.0, zero-order kinetics (Case II transport) are justified in accord with (4.4). Finally, the intermediate values of A (cf. the inequalities in Table 4.1) indicate a combination of Fickian diffusion and Case II transport, which is usually called anomalous transport. 

Also, the following equation was used to simulate concurrent release mechanisms of Fickian diffusion and Case II transport throughout the release process ... 

We can speculate, however, on the diffusion or Case II contributions to the character of the observed profiles as a function of temperature. At the lowest temperature-penetrant activity plane, one would expect Fickian transport. At low temperatures the relaxation time is much greater than the time for diffusion. In this situation there is no possibility that the struc-... 

The problem of gas diffusion in, and permeation through, inhomogeneous polymers is more complex, but has been considered by a number of investigators, e.g., refs. (1-3,5,7). When the polymer is highly plasticized by the penetrant, the diffusion coefficient may also become a function of time and of "history", but these non-Fickian cases will not be discussed here (1,3,5,6,8-11). 

A limiting case of non-Fickian response in which a = 1.0 is typically referred to as Case II diffusion to differentiate it from normal Fickian (Case I) diffusion. Case I (Fickian) diffusion shows a linear increase in sorption as a function of the square root of time. By contrast. Case II kinetics are characterized by linear mass uptake with time as shown in Figure 38, where uptake of n-pentane in polystyrene at high activity (penetrant partial pressure) shows a Fickian response in small spheres and a non-Fickian response in larger diameter spheres (143). The data strongly suggest that diffusion into the small spheres is so rapid that there is insufficient time to generate a Case II concentration profile. Apparently the diffusional equilibration in the small spheres is essentially complete before the complex step concentration profile associated with Case II sorption can be established. This behavior is similar to other observations (144). 

Both types of diffusion i.e. Fickian and non-Fickian (case II), can be superimposed. Some penetrants will show Fickian behaviour in a given polymer whilst others will show non-Fickian (case II) behaviour. By increasing the crosslink density of a given polymer, the diffusion tends to move from Fickian type to non-Fickian (case II) type. 

Here, d is the pathlength of the diffusion front (in the present investigations, the distance ofthe NH/ND exchange front to the polyamide 11 (PAll)/butanol(OD) interface), A is a proportionality factor, t is the time, and a is the diffusion exponent. This exponent a is characteristic of the diffusion process and can take on the values of 0.5 and 1.0 for the limiting cases of Fickian (case I) and case II diffusion, respectively [3,4, 71]. 

The above case of dual-sorption model can be considered a special case of what has been termed anomalous diffusion in polymers. Because organic vapors or liquids can interact strongly with a polymer and cause it to swell, an extreme case of anomalous diffusion occurs when the mass uptake (time-integrated flux) into the polymer is totally controlled by the stress gradient between the swollen and unswollen regions rather than by the concentration gradient. This was first characterized by Alfrey and coworkers [31] and referred to as case II diffusion. Fickian diffusion leads to an initial mass uptake of a polymer film or sheet exposed to a swelling solvent that follows the expression [32] ... 

Release of tetracycUne hydrochloride from PCL fibers was evaluated as a means of controlled administration to periodontal pockets (69). Only small amounts of the drug were released rapidly in vitro or in vivo, and poly(ethylene-co-vinyl acetate) gave superior results. Because Fickian diffusion of an ionic hydrochloride salt in a UpophiUc polymer is unlikely, and because PCL and EVA have essentially identical Fickian permeabilities, we attribute this result to leaching of the charged salt by a mechanism similar to release of proteins from EVA (73). Poly-e-caprolactone pellets have been found unsuitable for the release of methylene blue, another ionic species (74,75). In this case, blending PCL with polyvinyl alcohol (75% hydrolyzed) increased the release rate. 

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... 

This relative importance of relaxation and diffusion has been quantified with the Deborah number, De [119,130-132], De is defined as the ratio of a characteristic relaxation time A. to a characteristic diffusion time 0 (0 = L2/D, where D is the diffusion coefficient over the characteristic length L) De = X/Q. Thus rubbers will have values of De less than 1 and glasses will have values of De greater than 1. If the value of De is either much greater or much less than 1, swelling kinetics can usually be correlated by Fick s law with the appropriate initial and boundary conditions. Such transport is variously referred to as diffusion-controlled, Fickian, or case I sorption. In the case of rubbery polymers well above Tg (De < c 1), substantial swelling may occur and... 

Hydrodynamic dispersion is in many cases taken to be a Fickian process, one whose transport law takes the form of Fick s law of molecular diffusion. If flow is along x only, so that vx = v and vy = 0, the dispersive fluxes (mol cm-2 s-1) along x and y for a component i are given by,... 

PMMA) film is quenched by permeation of methyl ethyl ketone (MEK), a good solvent for PMMA. A steady-state MEK concentration profile has been estimated from quenching data with existing sorption and light scattering data. The profile contains all the features of Case II diffusion the Fickian precursor, the solvent front, and the plateau region. However, the solvent front is not so steep as those observed in systems where penetrant diffusion is much slower. 

Consequently, we will follow the example of Mills et al. (29) who recently presented the first measurements of local solvent concentration using the Rutherford back-scattering technique. They analyzed the case of 1,1,1-trichloroethane (TCE) diffusing into PMMA films in terms of a simpler model developed by Peterlin 130-311, in which the propagating solvent front is preceded by a Fickian precursor. The Peterlin model describes the front end of the steady state SCP as ... 

When the diffusion coefficient D is dependent on concentration, the diffusion process is said to be Fickian. In such cases, D is inversely related to solubility, S, and to permeability, P, as follows ... 

Rate equations There are two basic types of kinetic rate expressions. The first and simpler is the case of linear diffusion equations or linear driving forces (LDF) and the second and more rigorous is the case of classic Fickian differential equations. 

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