Diffusion interdiffusion problem

Unfortunately, the interpretation of Giletti et al. (1978) does not solve the problem of differential diffusivities of 0 and To do this, their experimental results should be interpreted in terms of interdiffusion of 0 and O. Application of Pick s first law to interdiffusion of the two species would in fact lead to the definition of an interdiffusion coefficient/), so that... 

Matano developed a graphical method which, for certain classes of boundary value problems, relates the form of the diffusion profile with the concentration dependence of the interdiffusivity, D(c), introduced in Section 3.1.3 [5]. This method can determine D(c) from the diffusion profile in chemical concentration-gradient diffusion experiments where atomic volumes are sufficiently constant so that changes in overall specimen volume are insignificant and diffusion can be formulated in a F-frame. The method uses scaling, as discussed in Section 4.2.2. 

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. 

MOCVD process as well as the oxidation problem of the diffusion barrier, the low temperature MOCVD process is required which usually results in amorphous or weekly crystallized as-deposited thin films. Therefore, high temperature post-annealing is an absolute necessity. The upper limit temperature of the post-annealing is about 800°C considering the interdiffusion between the BST and electrodes at higher temperature and process integration issues such as degradation of the metal contact resistance. 

Most of the known IE kinetic problems have been solved by the use of a single mass-balanced diffusion equation [1-3,7-11,14-24,34-43]. They are, on this basis, identified as one component systems and the diffusion rate for the invading B ion is controlled by the concentration gradient of this ion alone. In these cases the effective interdiffusion coefficient depends on the ion concentrations and the equilibrium constants of the chemical reaction between both ions in the ion exchanger [2-3,7-12,16-22,23,23,30,32,34,42,32-34]. 

The description in terms of a substrate that self-diffuses plus components that interdiffuse permits a further distinction it is the substrate that has continuum properties, to which the reasoning in Chapter 11 applies, and for which we use eqn. (12.7) specifically along one direction or another. The interdiffusive effects, here mimicked by the motion of the additive a, do not resemble continuum behavior in isotropic materials, the additive a affects only the volume of a sample-element and cannot affect its shape the additive responds directly to the mean stress and produces only an isotropic change in mean strain. In the cylinder problem treated above, the symmetry and uniformity assumed are such that this distinction leaves the mathematical solution unchanged in form. But if a less regular physical situation were to be treated, the distinction between the behavior of BX and the behavior of the additive would have more noticeable consequences. 

The final complication in the problem of polymer interface formation is to consider the very early stages of interdiffusion, when the interface is broadened on a length scale comparable to or less than the radius of gyration of the polymers involved. At such small distances diffusion ideas, which deal solely... 

Transient Interdiffusion in Two Semi-Infinite Bodies The transient diffusion problem illustrated in Figure 4.8, which involves the interdiffusion of two semi-infinite bodies in contact with one another, is closely related to the previous semi-infinite transient diffusion problem. In fact, if you consider just one-half of the problem domain (e.g., consider the evolution of the diffusion profiles for species A for X > 0), diffusion proceeds exactly like the previous semi-infinite diffusion problem. The only difference is that in this case the interfacial concentration of species A is assumed to be pinned at half of its bulk (i.e., pure material A) value. 

Solution To answer this question, it is helpful to first establish a sketch of the problem, as shown in Figure 4.9. Considering first the diffusion of A into B from left to right, we wish to determine the right-side boundary of the interdiffusion region. As detailed in the problem statement, this corresponds to the location (x value) where the concentration of species A falls to 1/10 of its initial bulk value. As shown on the schematic illustration, we will call this location x = 8. Applying this criteria to the solution for the transient interdiffusion of two semi-infinite bodies gives... 

In the form of eq. (5-30), Pick s second law applies only to one-dimensional problems in an isotropic medium. The index i on the diffusion coefficient has been removed in order to make it clear that this is no longer the component diffusion coefficient D,-, but rather, it is the chemical interdiffusion coefficient. Normally, the chemical interdiffusion coefficient will be a function of the individual component diffusion coefficients Di because of the coupling of the fluxes in the lattice system. When local thermodynamic equilibrium prevails, the coefficients Di are, in turn, unique functions of the composition. From the thermodynamics of irreversible processes it can be shown [6] that in binary systems there is only one independent transport coefficient, and in general, in n-component systems there can only be (n - 1) /2 independent transport coefficients. 

We have been concerned hitherto with so much solvent that its amount for practical computations of D can be reckoned as infinite. Such a supposition limits the applicability of the equations given, for it will be much more usual to work with small amounts of diffusion media. The solutions now to be given will concern themselves wdth problems such as the following. A solute diffuses from a solution bounded between the planes x =0 x = h into a solvent bounded between the planes x and x = 1, The concentration-distance-time curves may be measured readily enough and have now to be interpreted so that the diffusion constants, 2), may be evaluated. The new solutions of Fick s law will apply to numerous cases of the interdiffusion of metals, and salts, so long as D does not depend on the concentration, and whenever the amount of metal or salt is limited. 

A mathematical explanation for the severity of the room temperature diffusion problem, exemplified by silver-gold, can be summarized as follows Data [76,77] demonstrate that interdiffusion at room temperature is too slow to account for deallo5dng. Diffusion coefficients previously vised were extrapolated from values taken above 600°C. Data taken nearer room temperature cast doubt on earlier extrapolations and allow much more reliable values for diffusion coefficients to be obtained. These values could not account for diffusion of the silver to the solution interface, which would require diffusion of many orders of magnitude faster for removal of all of the silver from the alloy. Since, in some alloy compositions at room temperature, all of the silver can be ionized and removed from the alloy within a few minutes, bulk diffusion cannot be the means for moving the silver to the surface for ionization. There must be a much faster method by which the less noble metal can reach the solution metal interface, because volume diffusion can only account for a fraction of the movement required to ionize all of the less noble metal at the alloy-solution interface. 

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