Fickian diffusion characteristics

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. 

As seen from Fig. 19.3, a sharp increase of optical density of richlocaine coincides with volume transition of NlPA-AA (see also Fig. 19.1). The percentage of released richlocaine was 44.5% and 7.0% at 40°C and 35°C, respectively. The n value equal to 0.52 at 40 °C reflects the Fickian diffusion while the n=0.26 at 35°C is characteristic of anomalous diffusion of richlocaine [7]. The rate of drug release from NIPA-AA was minimal at pH 5. It gradually increased with increasing pH and leveled off at pH 8 (Fig. 19.5). 

Figure 14.4 Shape of a Fickian diffusion curve. Definition of the characteristic time of diffusion.

Very thorough investigations of Eq. (1) subject to conditions (4) and (5) have been made by Crank and others for various assumed forms of D (cj), of which an excellent summary has been given by Crank (1956). The information we need here is not the detailed mathematical expressions for such solutions of Eq. (1) but the characteristic features of sorption processes predicted from this set of equations. Customarily, the sorption processes in which D is a function of ct only and the initial and boundary conditions are given by Eqs. (4) and (5) are referred to as of the Fickian type. Moreover, it is often said that such processes are controlled by the Fickian diffusion mechanism1. 

Next to it is the concentration-dependent Fickian diffusion zone, which is characteristic of many small organic molecules of moderate to high activity at temperatures above or sufficiently below the effective Tg of the system. 

The characteristic of a facilitated or carrier-mediated transport is the occurrence of a reversible chemical reaction or complexation process in combination with a diffusion process. This implies that either the diffusion or the reaction is rate limiting The total flux of a permeant A will thus be the sum of both the Fickian diffusion and the carrier-mediated diffusion as illustrated in Equation 4.19 [46] ... 

Ash composition and size affect the deposition characteristics, along with reactor geometry, gas and wall temperature, and velocity. Fickian diffusion is important for the deposition of vapors and small particles (< pm), thermophoresis is important for the deposition of intermediate-sized particles (1-lO an), and inertial... 

Fickian diffusion and first order relaxation rate process. These two processes are linked through a coupling constant, 2 - D/0R The quantity 0D = L2/4D is the characteristic time for diffusion. The quantity 0 = l/kR, where kR is the first order relaxation rate constant, is the characteristic time for relaxation. The fractional weight uptake is give by the following equation ... 

Fickian diffusion with constant effective diffusivities, which is the simplest and most widely used approach. This is quite useful in studying the general characteristics of the problem, but is not valid for realistic simulation of industrial systems except for cases which can be approximated by the assumption of components diffusion in a stationary bulk phase. 

Fick s Laws may not apply at cry short times and diffusion distances. Certainly, at times much less than 10 s, w here a typietd characteristic Fickian diffusion distance (Dr) - is of the order ol 1 A or less, the extents of oscillatory motions equal or exceed those of net displacements. Also, since molecular clusters lend to persist on this timescale and since motions within and b clusters of real molecules may be suhjeel to gear cflccls. there m iy be non-random contributions to rcl.itivc displacements on very short timescales, fhcsc effects arc neglected here. Most of the phenomena in which we are interested occur on longer timescales. 

Ueberreiter found that in the polymer systems he studied, the diffusion of the solvent across the gel layer is the rate-determining step (i.e., it is the slowest of all the involved steps),and in those cases solvent uptake and the inward movement of the glass-gel interface depend on the square root of time, as is characteristic of Fickian diffusion. In other systems, the processes in the glass-gel interface are rate determining, and solvent uptake and interface movement are linear functions of time. Alfrey et al. termed it case II diffusion or polymer relaxation-controlled mass transfer.Figure 11.31 shows a schematic of time-concentration profiles for the two dissolution modes. 

SBS Interphase. Since 20°C is below the 0-temperature for the polystyrene-cyclohexane systems, it was expected that the PBD phase would be permeable to cyclohexane, but the PS domains would be relatively impermeable. (It is known that PS swells almost fourfold in liquid cyclohexane and that SBS may be dissolved even in cyclohexane. However, the maximum uptake of cyclohexane vapor by SBS was approximately 40% of its original weight. Furthermore, a sample of pure PS did not absorb any vapor within the time scale of these experiments. It was concluded then that the pure PS domain was not penetrated by cyclohexane vapor in these experiments and that, except for the interface, the PS domains may be considered an impermeable phase dispersed within a permeable continuum.) Thus the diffusion coefficient would be expected to reflect the structure of the PBD phase and to be characteristic of diffusion in elastomers (i.e., Fickian diffusion). 

Figure 3.1 shows an example of a Fickian diffusion curve [4]. In this case, the resin had been previously conditioned and subjected to moisture absorption at 50°C and 96% RH, followed by desorption in vacuum at 50°C, prior to resorption [4]. One-dimensional diffusion can therefore be described by the following characteristics of the M, versus curve ... 

Above the the rate of polymer chain relaxation is faster than the diffusion of CO2, and hence Fickian diffusion is to be expected. The diffusion of CO2 is believed to occur within the amorphous domains of the polymer matrix, and for this reason the diffusion in semi-crystalline polymers may be more complex than it in the case for glassy polymers. In the case of semi-crystalline polymers, CO2 is not soluble in the crystalline domains, and therefore the degree of crys-taUinity and hence the amorphous fraction available for CO2 molecules may influence the diffusion characteristics. Furthermore, C02-induced crystallization is likely to lead to an increase in the tortuosity factor, and thus the diffusion path length may increase as a function of time. Syndiotactic polystyrene and poly(4-methyl-l-pentene) [45] are semi-crystalline polymers which have crystalline phases (helical in the case of sPS) with lower densities than that of the amorphous phase and are exceptions, as CO2 access is not restricted to the amorphous domains, in fact CO2 diffuses faster in the helical sPS than in the amorphous polymer [46]. 

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... 

Our group also developed rapamycin (RM)-eluting stents using electrospun PU vascular grafts that could effectively suppress local smooth muscle cell proliferation. We observed that the release kinetics was characteristic of a Fickian diffusion for at least 77 days in vitro. RM-PU fibers generafed via powder blending showed the highest encapsulation efficiency. 

Polymers whose diffusion characteristics can be described by Pick s first and second laws are said to follow Fickian diffusion behaviour. 

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