Diffusion coefficient Intrinsic

Various diffusion coefficients have appeared in the polymer literature. The diffusion coefficient D that appears in Eq. (3) is termed the mutual diffusion coefficient in the mixture. By its very nature, it is a measure of the ability of the system to dissipate a concentration gradient rather than a measure of the intrinsic mobility of the diffusing molecules. In fact, it has been demonstrated that there is a bulk flow of the more slowly diffusing component during the diffusion process [4], The mutual diffusion coefficient thus includes the effect of this bulk flow. An intrinsic diffusion coefficient, Df, also has been defined in terms of the rate of transport across a section where no bulk flow occurs. It can be shown that these quantities are related to the mutual diffusion coefficient by... 

The self-diffusion coefficient is determined by measuring the diffusion rate of the labeled molecules in systems of uniform chemical composition. This is a true measure of the diffusional mobility of the subject species and is not complicated by bulk flow. It should be pointed out that this quantity differs from the intrinsic diffusion coefficient in that a chemical potential gradient exists in systems where diffusion takes place. It can be shown that the self-diffusion coefficient, Di, is related to the intrinsic diffusion coefficient, Df, by... 

The most commonly defined diffusion coefficients are the chemical or interdiffusion D, intrinsic diffusion coefficient (D1), and tracer or selfdiffusion coefficient (D ). The relevant coefficient depends on the driving... 

A driving force for diffusion includes any influence that increases the jump frequency. Examples of driving forces include chemical potential, thermal, and stress gradients. If a chemical potential gradient is present, the flux of species i at the diffusion plane is given by = —D (dc-Jdx), where D is the intrinsic diffusion coefficient of species i. Assuming Henry s law for the tracer element in binary alloys, Darken30 has shown that... 

Figure 15. Average enhanced- and retarded-diffusion coefficients ofP9 As, and Sb following direct nitridation experiments. Data are from Fahey (43). Abbreviations , time-averaged diffusion coefficient ) a, intrinsic diffusion coefficient. (Reproduced with permission from reference 118. Copyright 1984...

The intrinsic diffusion coefficients, Dk and DB, of a binary alloy A-B express the diffusion of the components A and B relative to the lattice planes [7], Therefore, during interdiffusion, a net flux of atoms across any lattice plane is present, where, normally, the diffusion rates of the diffusing particles A and B are different. Subsequently, this interdiffusion process provokes the shift of lattice planes with respect to a fixed axis of the sample, result which is named the Kirkendall effect [9],... 

As we have commented previously for metals, the diffusion in concentration gradients is described with the chemical diffusion coefficient, or interdiffusion coefficient. In this case, it is possible to consider that the interdiffusion and the intrinsic diffusion coefficients are equivalent, since we have only the movement of one species, that is, oxygen, by the vacancy mechanism. Subsequently, applying the Einstein relation... 

If we apply the Barrer and Jost model for the calculation of the self-diffusion coefficient [72] that is valid for a Langmuir system where the probability of a successful jump is proportional to the number of vacant sites [12], we get the corrected or intrinsic diffusion coefficient... 

The equations above imply that in the case of multiple diffusing species, each species will have its own intrinsic diffusion coefficient. Interestingly, when... 

Lawrence Stamper Darken (1909-1978) subsequently showed (Darken, 1948) how, in such a marker experiment, values for the intrinsic diffusion coefficients (e.g., Dqu and >zn) could be obtained from a measurement of the marker velocity and a single diffusion coefficient, called the interdiffusion coefficient (e.g., D = A ciiD/n + NznDca, where N are the molar fractions of species z), representative of the interdiffusion of the two species into one another. This quantity, sometimes called the mutual or chemical diffusion coefficient, is a more useful quantity than the more fundamental intrinsic diffusion coefficients from the standpoint of obtaining analytical solutions to real engineering diffusion problems. Interdiffusion, for example, is of obvious importance to the study of the chemical reaction kinetics. Indeed, studies have shown that interdiffusion is the rate-controlling step in the reaction between two solids. 

Lawrence Stamper Darken (1909-1978) subsequently showed how, in such a marker experiment, values for the intrinsic diffusion coefficients (e.g. Dcu and Dz )... 

In the two-component system gas + zeolite, in parallel with a general two-component system, one may consider 5 diffusion coefficients Dab (interdiffusion coefficient of sorbate and zeolite) Da, Db (intrinsic diffusion coefficients, respectively, of sorbate and zeolite) and Da, Db (tracer diffusion coefficients, respectively, of sorbate and zeolite). However, because Db and Db are effectively zero, as for a nonswelling crystal. 

Measurements have been made of tracer or intrinsic diffusion coefficients of water in a number of zeolites. Most refer to crystals nearly saturated with zeolitic water, while smoothed rather than adsorption areas were used in calculating the coefficients. From the temperature dependence of Da or Da, the activation energy, E, may be found using the Arrhenius equations Da = Do exp — E/RT or Da = Do exp — E/RT. Some results for tracer diffusion are summarized in Table VI. (iS). These shed considerable light upon certain aspects of intracrystalline diffusion for small polar molecules ... 

Once an appropriate frame of reference is chosen, a two components (A, B) system may be described in terms of the mutual diffusion coefficient (diffusivity of A in B and vice versa). Unfortunately, however, unless A and B molecules are identical in mass and size, mobility of A molecules is different with respect to that of B molecules. Accordingly, the hydrostatic pressure generated by this fact will be compensated by a bulk flow (convective contribution to species transport) of A and B together, i.e., of the whole solution. Consequently, the mutual diffusion coefficient is the combined result of the bulk flow and the molecules random motion. For this reason, an intrinsic diffusion coefficient (Da and Db), accounting only for molecules random motion has been defined. Finally, by using radioactively labeled molecules it is possible to observe the rate of diffusion of one component (let s say A) in a two component system, of uniform chemical composition, comprised of labeled and not labeled A molecules. In this manner, the self-diffusion coefficient (Da) can be defined [54]. Interestingly, it can be demonstrated that both Da and Da are concentration dependent. Indeed, the force/acting on A molecule at point X is [1]... 

Equation (9.43a) can easily be converted in terms of occupancies by dividing numerator and denominator of the In term by qgau- Note again that Dq is the intrinsic diffusion coefficient and that Dpick = Do/(1 - 0) and so Dp increases strongly when 0 assumes larger values. 

Here Do, is the pre-exponential coefficient of the intrinsic diffusion coefficient. 

To obtain Eqs. (9.43), (9.47) and (9.48) it was necessary to assume that the intrinsic diffusion coefficient should be independent of the concentration (occupancy). This is only correct when there are no intermolecular interactions, so for lower values of the occupancy 9. An extensive discussion has been given by Xiao and Wei [86]. Based on model calculations and analysis of experimental results, they showed that is approximately constant imtil 0 = 0.5-0.6 and then... 

Polymer interdiffusion above Tg was then monitored [93] by using technique (ii) at ambient probe temperature following various times during which the polymer system was held at 190°C. Initially, the sample consisted of stacks of discs cut from PMMA and PVDF sheet (average thicknesses 92.5 and 88.2 p,m, respectively) alternately. Evaluation of the diffusion equation gave intrinsic diffusion coefficients (7 5) x 10 and (15 5) x 10... 

Fredrickson and Helfand incorporated coupling terms in the transport equation and showed that "these terms provide mobility to molecules that are absorbed into microvoids, even if the molecules have an intrinsic diffusion coefficient in the hole phase that vanishes." These investigators proposed experiments that could provide information on the size and topological connectivity of microcavities (51). Related studies have also been made by Chern, Koros, et al. (52) and by Barrer (53). However, Petropoulos (54) has expressed the opinion that the treatment of Fredrickson and Helfand (51 ) as well as that of Barrer (53) "introduces more diffusion parameters than can reasonably be expected to be measurable on the basis of past experience". Petropoulos also showed that these treatments impose certain limitations on the physical meaning of the diffusion parameters. In the same study, Petropoulos has examined the modification of his dual-mode sorption model ( 5) which are necessary if the Langmuir domains are sufficiently extensive to constitute a macroscopically recognizable phase rather than scattered individual sites (or microcavities) (54). ... 

The tensiometric data are adjusted to the diffusion data at 100 mm where the diffusion coefficient is known from experiment to be 1.79 x 10 by calculating the intrinsic diffusion coefficient A),. It is assumed that = (/i -f constant) at any temperature... 

Diffusion coefficients involving only free diffusion are called true or intrinsic diffusion coefficients processes involving both free and forced diffusion are called mutual diffusion processes. Experimentally, one cannot usually evaluate free diffusion in kinetic studies on keratin fibers. Therefore, the usual practice is to apply equations derived from Fick s laws for free diffusion to data involving mutual diffusion. This provides apparent or approximate diffusion coefficients, instead of intrinsic diffusion coefficients, and compromises the fundamental significance or interpretations of these processes involving molecular motion (e.g., the activation energies or entropies of activation). 

In a porous medium the value of diffusion coefficient, beside this, is affected by porosity and dual electric layer at the separation surface of water with other media. They obstruct the diffusion in underground water and decrease the diffusion coefficient. The diffusion coefficient taking this effect into account is called intrinsic diffusion coefficient. Its value is equal to... 

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