Equilibrium solid-state diffusion

One possibility for increasing the minimum porosity needed to generate disequilibria involves control of element extraction by solid-state diffusion (diffusion control models). If solid diffusion slows the rate that an incompatible element is transported to the melt-mineral interface, then the element will behave as if it has a higher partition coefficient than its equilibrium partition coefficient. This in turn would allow higher melt porosities to achieve the same amount of disequilibria as in pure equilibrium models. Iwamori (1992, 1993) presented a model of this process applicable to all elements that suggested that diffusion control would be important for all elements having diffusivities less than... 

The immobilization of dissolved chemical species by adsorption and ion exchange onto mineral surfaces is an important process affecting both natural and environmentally perturbed geochemical systems. However, sorption of even chemically simple alkali elements such as Cs and Sr onto common rocks often does not achieve equilibrium nor is experimentally reversible (l). Penetration or diffusion of sorbed species into the underlying matrix has been proposed as a concurrent non-equilibration process (2). However, matrix or solid state diffusion is most often considered extremely slow at ambient temperature based on extrapolated data from high tem-... 

When the solid-liquid interface moves too fast to maintain equilibrium, it results in a chemical composition gradient within each grain, a condition known as coring (Figure 4.6b). Without solid-state diffusion of the solute atoms in the material... 

One possibility that would lead to larger inferred porosities for the U-series was introduced by Qin (1992, 1993), who proposed that the retained melt was only in complete equilibrium with the surface of minerals and that solid-state diffusion limited the re-equilibration of the retained melt with the solid. In other respects, this model is identical to the ACM model of Williams and Gill (1989). Qin introduced a specific microscopic melting/diffusion model for spherical grains and coupled it to the larger-scale dynamic melting models. The net affect of this... 

The phenomenon of compensation is not unique to heterogeneous catalysis it is also seen in homogeneous catalysts, in organic reactions where the solvent is varied and in numerous physical processes such as solid-state diffusion, semiconduction (where it is known as the Meyer-Neldel Rule), and thermionic emission (governed by Richardson s equation ). Indeed it appears that kinetic parameters of any activated process, physical or chemical, are quite liable to exhibit compensation it even applies to the mortality rates of bacteria, as these also obey the Arrhenius equation. It connects with parallel effects in thermodynamics, where entropy and enthalpy terms describing the temperature dependence of equilibrium constants also show compensation. This brings us the area of linear free-energy relationships (LFER), discussion of which is fully covered in the literature, but which need not detain us now. 

From this equation, it becomes obvious that the ratio of the equilibrium ion activities in the solution is linked with the alloy composition as expressed by the bulk atom fractions of the components, Xa and Xb = 1 — Xa- In general, therefore, the establishment of complete equilibrium for an alloy electrode requires a change of composition both of the alloy phase and of the electrolyte solution [1]. For solid alloys at ambient temperature, compositional changes (due to the preferential dissolution of one alloy component) will be restricted to the uppermost atomic layers. Further equilibration between the surface and the bulk of the alloy is prevented by solid-state diffusion limitations. Complete thermodynamic equilibrium for both components is therefore expected to evolve only with liquid alloys in which the diffu-sivity at ambient temperature is extremely high (for dilute Zn-amalgams, e.g., inter-diffusion coefficients t>zn of the order of 10 cm s have been reported under these conditions [2]). 

For most engineering alloys, the ambient temperature only corresponds to a small fraction of the melting temperature, Tm-As outlined above, this implies a very low solid-state diffusivity under these conditions that impedes the establishment of complete equilibrium of the alloy electrode according to Eq. (3). At anodic... 

In this section, we develop an atomistic picture to understand solid-state diffusion in more detail. At the most fundamental level, a solid-state diffusion coefficient D is a measure of the intrinsic rate of the hopping process by which atoms/molecules can move from one site to another in a solid medium. Even in the absence of any driving force, hopping of atoms from site to site within the lattice still occurs at a rate that is characterized by the diffusivity. Of course, without a driving force, the net movement of atoms is zero, but they are still exchanging lattice sites with one another. This is another example of a dynamic equilibrium, compare it to the dynamic reaction equilibrium processes that we discussed in Chapter 3. 

The nucleation rate, growth rate, and transformation rate equations that we developed in the preceding sections are sufficient to provide a general, semiquantitative understanding of nucleation- and growth-based phase transformations. However, it is important to understand that the kinetic models developed in this introductory text are generally not sufficient to provide a microstructurally predictive description of phase transformation for a specific materials system. It is also important to understand that real phase transformation processes often do not reach completion or do not attain complete equilibrium. In fact, extended defects such as grain boundaries or pores should not exist in a true equilibrium solid, so nearly all materials exist in some sort of metastable condition. Many phase transformation processes produce microstructures that depart wildly from our equilibrium expectation. The limited atomic mobilities associated with solid-state diffusion can frequently cause (and preserve) such nonequilibrium structures. In this section, we will focus more deeply on solidification (a liquid-solid phase transformation) as a way to discuss some of these issues. In particular, we will examine a few kinetic concepts/models... 

In sintering, matter transport by evaporation and condensation is normally treated alongside the solid-state diffusion mechanisms. The rate of transport is taken as proportional to the equilibrium vapor pressure over the surface, which can be related to the value of /Xa - / v beneath the surface. Suppose a number dNa of atoms is removed from the vapor and added to the surface with an accompanying decrease in the number of vacancies beneath the surface. The free energy change for this virtual operation must be zero, so that... 

It is assumed here that solidification is being carried out as a series of equilibrium states in which the solid continually adjusts through solid-state diffusion to the value corresponding to the intersection of the tie-line with the solidus. Eventually as the temperature falls to T3, the final solid will have a uniform composition equal to the starting value. 

The first-to-freeze will have composition Cq/K where Q is the initial composition of the melt. Equilibrium solidification in which the solid has the same composition as the melt is only possible if the solidification is carried out very slowly so that solid-state diffusion can take place. Otherwise, the grains are cored, meaning that centers of the first-to-freeze component are surroimded by the last-to-freeze composition. 

Unfortunately, an equilibrium microstructure is often not realized in Pb-Sn solders because of the conditions that prevail in manufacturing processes and applications that utilize these materials. First, manufacturing processes result in cooling rates that are relatively fast. Therefore, the microstructure is far from equilibrium. Second, applications, even those at room temperature (25 °C) are close to the melting temperatures of Pb-Sn alloys. As a result, there is sufficient thermal energy to support solid-state diffusion processes, which will, in turn, alter the microstructure of the solder and thus the performance of interconnections. 

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