Effective diffusion coefficient penetrant concentration

Figure 7. The effective diffusion coefficient for small molecules in glassy polymers as a function of penetrant concentration. Dp is the mutual diffusion coefficient for the penetrant population dissolved by the Henry s-law mode. Other symbols are defined in the text.

The time for diffusion (without reaction) to increase the average concentration within a fiber wall to one-half of its ultimate value, with the fiber treated as a fully collapsed slab of thickness equal to two fiber wall widths, gives a 0.19. For a fiber wall thickness of a = 4.0 x 10 m and an effective diffusion coefficient of D = 1.3 X 10 m /s, Xd = 0.0023 s. The ozonation of pulp fibers follows a shrinking core mechanism, limited by the rate of ozone diffusion through the fiber wall. Exposed fibers are completely penetrated and reacted in seconds under typical operating conditions (Bennington et al., 1999). 

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. 

The diffusion theory states that interpenetration and entanglement of polymer chains are additionally responsible for bioadhesion. The intimate contact of the two substrates is essential for diffusion to occur, that is, the driving force for the interdiffusion is the concentration gradient across the interface. The penetration of polymer chains into the mucus network, and vice versa, is dependent on concentration gradients and diffusion coefficients. It is believed that for an effective adhesion bond the interpenetration of the polymer chain should be in the range of 0.2-0.5 pm. It is possible to estimate the penetration depth (/) by Eq. (5),... 

In the absense of any dependence of S and DT on penetrant concentration, the sorption and steady-state permeation properties of the membrane are described by effective solubility and permeability or diffusion coefficients given by... 

Polymers Mixed by Milling. The effect of EVA concentration in the blends on gas permeation and light transmission through the film was studied. The permeability and the diffusion coefficients at 50 °C for the penetrants helium, argon, and carbon dioxide are shown in Figures 1, 2,... 

In an attempt to justify the assumption of plasticization put forth in their interpretation of 3 in Eq (A-2), Raucher and Sefcik compare transport data and NMR data for the C02/pvC system This comparison has several questionable aspects To relate local molecular chain motions to the diffusion coefficient of a penetrant, one should use the so-called local effective coefficient, Deff O such as shown in Figure 5 rather than an average or "apparent" diffusion coefficient as was employed by these authors Deff(C) describes the effects of the local sorbed concentration on the ability of the average penetrant to respond to a concentration or chemical potential gradient in that region ... 

To improve topical therapy, it is advantageous to use formulation additives (penetration enhancers) that will reversibly and safely modulate the barrier properties of the skin. Fick s first law of diffusion shows that two potential mechanisms are possible. The two constants that could be altered significantly are the diffusion coefficient in the stratum corneum and the concentration in the outer regions of the stratum corneum. Thus, one of mechanisms of action of an enhancer is for it to insert itself into the bilayer structures and disrupt the packing of the adjacent lipids, thereby, reducing the microviscosity. The diffusion coefficient of the permeant will increase This effect has been observed using ESR and fluorescence spectroscopy [16,17]. 

According to the thermodynamics of irreversible processes, the mutual diffusion coefficient D may be a function of penetrant concentration ct, position x, and time t. In the present chapter we shall discuss sorption behavior of systems in which D varies with cx only, and shall use the notation D (cx) to indicate this condition. It is assumed that the sample film is so thin that diffusion takes place effectively in the direction of its thickness. At the beginning of an absorption or a desorption experiment the film is conditioned so that Cj is uniform everywhere in it. This initial concentration is denoted by cf. Then we have... 

It has long been a mystery why diffusion coefficients of polymer-diluent systems, especially when the diluent is a good solvent for a given polymer, exhibit so pronounced a concentration dependence that it looks extraordinary. Several proposals have been made for the interpretation of this dependence. Thus Park (1950) attempted to explain it in terms of the thermodynamic non-ideality of polymer-diluent mixtures, but it was found that such an effect was too small to account for the actual data. Fujita (1953) suggested immobilization of penetrant molecules in the polymer network, which, however, was not accepted by subsequent workers. Recently, Barrer and Fergusson (1958) reported that their diffusion coefficient data for benzene in rubber could be analyzed in terms of the zone theory of diffusion due to Barrer (1957). Examination shows, however, that their conclusion is never definitive, since it resorted to a less plausible choice of the value for a certain basic parameter. 

For simple gases the interactions with polymers are weak, with the result that the diffusion coefficient is independent of the concentration of the penetrant. In this case the penetrant molecules act effectively as "probes of variable size" which can be used to investigate the polymer structure. 

Recently Koros and Hopfenberg have considered explicitely the effect of dual mode sorption on the local effective concentration dependent diffusion coefficient for low activity penetrant migration in glassy polymers. They showed that... 

The models most frequently used to describe the concentration dependence of diffusion and permeability coefficients of gases and vapors, including hydrocarbons, are transport model of dual-mode sorption (which is usually used to describe diffusion and permeation in polymer glasses) as well as its various modifications molecular models analyzing the relation of diffusion coefficients to the movement of penetrant molecules and the effect of intermolecular forces on these processes and free volume models describing the relation of diffusion coefficients and fractional free volume of the system. Molecular models and free volume models are commonly used to describe diffusion in rubbery polymers. However, some versions of these models that fall into both classification groups have been used for both mbbery and glassy polymers. These are the models by Pace-Datyner and Duda-Vrentas [7,29,30]. 

SBR filled with intercalated montmorillonite had substantially lower toluene uptake compared with the same rubber filled with carbon black (see Figure 15.42). Figure 5.28 shows that the diffusion coefficient of kerosene, which defines penetration rate, decreases when the concentration of carbon black in SBR vulcanizates is increased. Figure 15.33 compares the uptake rate of benzene by unfilled rubber and by silica and carbon black filled rubber. Both fillers reduce the solvent uptake but carbon black is more effective. 

Vrentas and Duda s theory formulates a method of predicting the mutual diffusion coefficient D of a penetrant/polymer system. The revised version ( 8) of this theory describes the temperature and concentration dependence of D but requires values for a number of parameters for a binary system. The data needed for evaluation of these parameters include the Tg of both the polymer and the penetrant, the density and viscosity as a function of temperature for the pure polymer and penetrant, at least three values of the diffusivity for the penetrant/polymer system at two or more temperatures, and the solubility of the penetrant in the polymer or other thermodynamic data from which the Flory interaction parameter % (assumed to be independent of concentration and temperature) can be determined. An extension of this model has been made to describe the effect of the glass transition on the free volume and on the diffusion process (23.) ... 

A number of assumptions were made and then verified to develop a model (i) The rate-limiting step for transport is drug diffusion through pores (other steps such as water penetration into the matrix and drug dissolution occur in less than 40 hours), (ii) The effect of concentration dependence on the drug diffusion coefficient is not significant. This was verified by an analysis of diffusion effects at the concentrations in the... 

Fig. 5.4 Oxygen microgradient (data points) at the sediment-water interface compared to the ratio, E/D (logarithmic scale), between the vertical eddy diffusion coefficient, E, and the molecular diffusion coefficient, D. Oxygen concentration was constant in the overflowing seawater. It decreased linearly within the diffusive boundary layer (DEL), and penetrated only 0.7 mm into the sediment. The DEL had a diickness of 0.45 mm. Its effective thickness, 8 is defined by the intersection between the linear DEL gradient and die constant bulk water concentration. The diffusive boundary layer occurs where E becomes smaller than D, i.e. where E/D = 1 (arrow). Data from Aarhus Eay, Denmark, at 15 m water depth during fall 1990 (Gundersen et al. 1995).

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