Fickian

A molecule colliding with the pore wall is reflected in a specular manner so that the direction of the molecule leaving the surface has no correlation with that of the incident molecule. This leads to a Fickian mechanism, known as Knudsen diffusion, in which the flux is proportional to the gradient of concentration of partial pressure. The Knudsen diffusivity is independent of pressure and varies only weaMy with temperature ... 

Sorption Rates in Batch Systems. Direct measurement of the uptake rate by gravimetric, volumetric, or pie2ometric methods is widely used as a means of measuring intraparticle diffusivities. Diffusive transport within a particle may be represented by the Fickian diffusion equation, which, in spherical coordinates, takes the form... 

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

The objective of the immersion test is to determine the moisture content (percent weight gain) of a material as a function of its immersion time. To interpret immersion test data, moisture diffusion through the thickness of a test specimen can be described using a one-dimensional Fickian equation... 

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. 

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. 

Malek K, Coppens MO. 2003. Knudsen self- and Fickian diffusion in rough nanoporous media. J Chem Phys 119 2801-2811. 

The development of the theory of solute diffusion in soils was largely due to the work of Nye and his coworkers in the late sixties and early seventies, culminating in their essential reference work (5). They adapted the Fickian diffusion equations to describe diffusion in a heterogeneous porous medium. Pick s law describes the relationship between the flux of a solute (mass per unit surface area per unit time, Ji) and the concentration gradient driving the flux. In vector terms. 

Fig. 2.3.9 A time series of profiles showing the is shown by the lowest trace. The inset shows ingress from right (stratum corneum) to left the advance of the glycerine front against the (viable epidermis) of glycerine into human skin square root of time from which Fickian in vitro. The skin before application of glycerine diffusion is inferred.

Transport in the polymeric system is assumed to be by Fickian diffusion, although the diffusivity of the various species depends on the extent of hydrolysis of the polymeric linkages. 

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... 

Permeability A rate of mass transfer, usually expressed per unit surface area. For Fickian diffusion in a membrane, the permeability is proportional to the diffusion coefficient and inversely proportional to the membrane thickness. 

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... 

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. 

Diffusion of small molecular penetrants in polymers often assumes Fickian characteristics at temperatures above Tg of the system. As such, classical diffusion theory is sufficient for describing the mass transport, and a mutual diffusion coefficient can be determined unambiguously by sorption and permeation methods. For a penetrant molecule of a size comparable to that of the monomeric unit of a polymer, diffusion requires cooperative movement of several monomeric units. The mobility of the polymer chains thus controls the rate of diffusion, and factors affecting the chain mobility will also influence the diffusion coefficient. The key factors here are temperature and concentration. Increasing temperature enhances the Brownian motion of the polymer segments the effect is to weaken the interaction between chains and thus increase the interchain distance. A similar effect can be expected upon the addition of a small molecular penetrant. 

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... 

Solutions to the Fickian sorption problem for a variety of systems are given by Crank [142], For a flat sheet with an aspect ratio greater than 10, initially free of solvent, the solvent uptake as a function of time is given as... 

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. ... 

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