Fickian diffusion kinetics

Drug release from soluble polymers is accompanied by the gradual erosion-type dissolution of the polymer. Therefore, polymer dissolution and drug diffusion may be the overall hybrid mechanism of release. Drug release from nonsoluble hydrogels generally follows Fickian or non-Fickian diffusion kinetics [51]. The mechanism of... 

After all drug is dissolved, the osmotic pressure decays and the beads shrink to their equilibrium swelling value, as observed for beads with high water swelling (11,12) during this second release phase normal Fickian diffusion kinetics becomes more important, characterized by a very low rate which depends on the hydro-philicity of the polymer. 

We can predict the distribution of moismre in a laminate using Fickian diffusion kinetics. Figme 12.3 shows the calculated distribution of moisture in a carbon fibre laminate of thickness 0.94 mm after 3 days at 96% RH and 50 °C. As result of Fickian analysis, we can observe that it would take 13 years to reach equilibrium if the material had a thickness of 12 mm. However, this does not reflect the acmal environment since the temperature and RH can change. 

Of particular importance is the timescale over which diffusion occurs under various conditions of relative humidity (RH) and temperature. The RH determines the equilibrium moisture concentration, whereas higher temperatures will accelerate the moisture sorption process. In order to predict the moisture profile in a particular structure, it is assumed that Fickian diffusion kinetics operate. It will be seen later that many matrix resins exhibit non-Fickian effects, and other diffusion models have been examined. However, most resin systems in current use in the aerospace industry appear to exhibit Fickian behaviour over much of their service temperatures and times. Since the rate of moisture diffusion is low, it is usually necessary to use elevated temperatures to accelerate test programmes and studies intended to characterize the phenomenon. Elevated temperatures must be used with care though, because many resins only exhibit Fickian diffusion within certain temperature limits. If these temperatures are exceeded, the steady state equilibrium position may not be achieved and the Fickian predictions can then be inaccurate. This can lead to an overestimate of the moisture absorbed under real service conditions. 

First, Fig. 15.2 shows the diffusion profile that does match Fickian diffusion kinetic model. What expected is that the slope should become smaller and smaller with after initial linearity. Fig. 15.2 is just opposite. Fig. 15.3 demonstrates the diffusion index n is far higher than the value n = 0.5. All these facts indicate that polymer degradation is companying with the in vitro water diffusion progressing, which... 

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. 

The addition of a filler changes the kinetics of the water absorption by an epoxy binder, water absorption becoming a multistage process (Fig. 12). Crank and Park150) have given the equation for the kinetics of water sorption by a thin plate, as well as a solution of the Fickian diffusion differential Equation as ... 

The importance of adsorbent non-isothermality during the measurement of sorption kinetics has been recognized in recent years. Several mathematical models to describe the non-isothermal sorption kinetics have been formulated [1-9]. Of particular interest are the models describing the uptake during a differential sorption test because they provide relatively simple analytical solutions for data analysis [6-9]. These models assume that mass transfer can be described by the Fickian diffusion model and heat transfer from the solid is controlled by a film resistance outside the adsorbent particle. Diffusion of adsorbed molecules inside the adsorbent and gas diffusion in the interparticle voids have been considered as the controlling mechanism for mass transfer. 

In principle the mobility B and therefore the corrected diffusivity D0 are also concentration-dependent, so Eq. (12) does not necessarily predict quantitatively the concentration dependence of D even for a system where the isotherm obeys the Langmuir equation. Nevertheless, the concentration dependence of B is generally modest compared with that of the thermodynamic factor, so a monatonic increase in diffusivity with adsorbed-phase concentration is commonly observed (Fig. 5). Clearly in any attempt to relate transport properties to the physical properties of the system it is important to examine the corrected, diffusivity D0 (or the mobility B) rather than the Fickian diffusivity, which is in fact a product of kinetic and thermodynamic factors. 

Equation (6.94) illustrates that zero-order release kinetics are obtained if drug dissolution controls the release kinetics. However, as soon as the last particle in the matrix dissolves, the controlling mechanism of drug release shifts to Fickian diffusion. Figure 6.19 shows the dissolution-controlled release of KC1 at the early stage of release and the diffusion-controlled release at the later stage of release from an ethyl cellulose tablet. 

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. 

However, the multicomponent Fickian diffusivities, Dgy, do not correspond to the approximately concentration independent binary diffusivities, Dsr, which are available from binary diffusion experiments or kinetic theory determined by the inter-molecular forces between s —r pair of gases. Instead, these multicomponent Fickian diffusion coefficients are strongly composition dependent. 

A multicomponent Fickian diffusion flux on this form was first suggested in irreversible thermodynamics and has no origin in kinetic theory of dilute gases. Hence, basically, these multicomponent flux equations represent a purely empirical generalization of Pick s first law and define a set of empirical multi-component diffusion coefficients. 

The kinetic reactions occurring in the sorption of Ni, Cd, and Zn on goethite during a period of 2 hours to 42 days at pH 6 were hypothesized to occur via a three-step mechanism using a Fickian diffusion model (1) sorption of trace elements on external surfaces (2) solid-state diffusion of trace elements from external to internal sites and (3) trace element binding and fixation at positions inside the goethite particle (Bruemmer et al., 1988). 

Farnworth [14] reported a numerical model describing the combined heat and water-vapor transport through clothing. The assumptions in the model did not allow for the complexity of the moisture-sorption isotherm and the sorption kinetics of fibers. Wehner et al [30] presented two mechanical models to simulate the interaction between moisture sorption by fibers and moisture flux through the void spaces of a fabric. In the first model, diffusion within the fiber was considered to be so rapid that the fiber moisture content was always in equilibrium with the adjacent air. In the second model, the sorption kinetics of the fiber were assumed to follow Fickian diffusion. In these models, the effect of heat of sorption and the complicated sorption behavior of the fibers were neglected. 

For 3-methyl-pentane, the initial ranges of the adsorption kinetics are shown in figure 1. It is observed that except for sample C, the kinetics strongly depend on the crystallite size of the samples. The curves may be analyzed in terms of the Fickian diffusion law (16) at small time values (17) ... 

Chapter 3 dealt with the problem of the reaction kinetics for different gas-solid reactions, while chapter 5 dealt with the mass and heat transfer problems for porous as well as non-porous catalyst pellets. In chapter 5 different degrees of complexities and rigor were used. In chapter 5, the analysis started with the simplest case of non-porous catalyst pellets where the only mass and heat transfer Coefficients are those at the external surface which depend mainly on the flow conditions around the catalyst pellet and the properties of the reaction mixture. It was shown clearly that j-factor correlations are adequate for the estimation of the external mass and heat transfer coefficients (k, h) associated with these resistances. For the porous catalyst pellets different models with different degrees of rigor have been used, starting from the simplest case of Fickian diffusion with constant diffusivity, to the rigorous dusty gas model based on the Stefan-Maxwell equations for multicomp>onent diffusion. 

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