Theoretical Discussion—Diffusion Couples

Now suppose that intermetallic compounds are suppressed in Fig. 2.16. Consider the metastable equilibrium between an amorphous phase and the terminal solutions of a-Zr (bcc Zr) and Ni. Under these conditions, the common tangent construction yields a metastable equilibrium state which consists of a single-phase a-solution for xZr x4, a two-phase region of liquid (glass)/a- 

The solutions of these equations have the following asymptotic forms [2.77-80]  

Such studies as those described above using RBS have convincingly shown that amorphous interlayers grow by diffusion limited growth. Further, such 

In addition to allowing measurements of 2), calorimetry studies also allow direct determination of the total enthalpy evolved when Ni and Zr layers are completely converted to amorphous phase. This is the enthalpy of mixing of the amorphous phase (referred to the starting pure metals). Cotts et al. [2.68, 69] showed that, for example, the enthalpy of mixing of an amorphous alloy of composition Ni68Zr32 is 35 5 kj/mol. Since the entropy of mixing of this alloy at a temperature of 300°C (the typical growth temperature) is much smaller, one can approximate 

Nucleation and the phase sequence problem. We showed above that many possible reactions can occur in a diffusion couple. Any new phase for which AG 0 is thermodynamically allowed. In fact, the thermodynamic driving force for intermetallic compounds (e.g., crystalline NiZr) is greater than that for the formation of an amorphous phase 

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The transformation of the crystalline into the glassy state by solid-state reactions is extensively reviewed in its theoretical and experimental aspects. First, we give some historical background and describe the thermodynamics of metastable phase formations, adding as well the kinetic requirements for the amorphization process. Then we discuss the different experimental routes into the amorphous state hydriding, thin diffusion couples, and other driven systems. In the discussion and the summary, we close the gap between the melting phenomena and the amorphization and provide a tentative outlook. 

The discussion of SSAR in thin-film diffusion couples will be divided into two parts. In the first subsection, a review of experimental results will be presented for bilayers and multilayers. The discussion will include cases for which the reactant layers are both metals and cases in which one of the reactants is a non-metal. The second subsection consists of a theoretical discussion of the... 

In the first case (a), a miscibility gap M occurs in the phase diagram, and there will thus be a phase boundary in the diffusion couple with a discontinuity in concentration In the second case (b), there are several phases in the phase diagram, and several phase boundaries with discontinuities in concentration occur in the diffusion couple [25]. In the limiting case, the phases in Fig. 7-5a and 7-5b may have only very narrow ranges of homogeneity. The theoretical expressions will thus be simplified, inasmuch as the end phases a and p can then be treated as ideal dilute solutions, and the rest of the phases can be treated as nearly stoichiometric compounds. All the cases just mentioned will now be discussed in detail. 

In Chapter 14, we discussed the case of a single-component band. In practice, there are almost always several components present simultaneously, and they have different mass transfer properties. As seen in Chapter 4, the equilibrium isotherms of the different components of a mixture depend on the concentrations of all the components. Thus, as seen in Chapters 11 to 13, the mass balances of the different components are coupled, which makes more complex the solution of the multicomponent kinetic models. Because of the complexity of these models, approximate analytical solutions can be obtained only under the assumption of constant pattern conditions. In all other cases, only numerical solutions are possible. The problem is further complicated because the diffusion coefficients and the rate constants depend on the concentrations of the corresponding components and of all the other feed components. However, there are still relatively few papers that discuss this second form of coupling between component band profiles in great detail. In most cases, the investigations of mass transfer kinetics and the use of the kinetic models of chromatography in the literature assume that the rate constants and the diffusion coefficients are concentration independent. This seems to be an acceptable first-order approximation in many cases, albeit separation problems in which more sophisticated theoretical approaches are needed begin to appear as the accuracy of measru ments improve and more interest is paid to complex... 

Fig. J Theoretical concentration profiles obtained by plotting Eq. (2) at various values oft, as indicated on the figure. Panel A shows the result for the pure diffusion case. Panel B includes the kinetics of a coupled chemical reaction. Values of h, k, and D are discussed in the text.

Theoretically we can use the same treatment as for the more general case of energy transfer however, experimentally it may be more difficult to distinguish between sensitizers and activators. So any of the microscopic processes discussed above may happen with a maximum overlap when an identical couple of levels are involved. From the macroscopic point of view we have for the diffusion-limited case, from eq. (152) ... 

The theoretical approaches aimed at a description of nonlinear adsorption regimes are discussed next—in particular the elassieal, random sequential adsorption (RSA) model. The role of polydispersity, particle shape, orientation, and electrostatic interactions is elucidated. Then, the generalized RSA model is presented, which reflects the coupling of the surface transport with the bulk transfer step governed by external force or diffusion. 

Theoretical studies of defect-mediated turbulence in two-dimensional systems have generally followed two approaches. One starts with local oscillators that couple through a diffusion process [36] another starts from a stationary patterned state [49] (see also the discussion of [10]). Both approaches lead to a turbulent state. However, our data do not distinguish in an obvious way between these two approaches. We do not observe any hint of global oscillations in the neighborhood of the transition from hexagons or stripes to turbulence, as would be expected in the first hypothesis. Nor do we observe any indication of a spatially biperiodic pattern, as would be expected in the second hypothesis. More experiments and analyses are needed to characterize and understand the mechanism of chemical turbulence. 

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