MOVING BOUNDARY DIFFUSION

It is known that diffusion in liquids is usually faster than in solids of the same solute at the same temperature. However, the liquid geometry is not weU understood because of the imperfect atomic arrangement. Despite of this problem, Einstein derived the diffusivity based on Brownian motion of fine particles in liquid water by assuming random direction and jump length of particles [43]. The main objective of a diffusion problem is to understand the problem and subsequently, determine the proper boundary conditions needed for developing a suit le mathematical solution of Pick s second law equation. The boundary conditions, however, depend on the electrochemical system and the type of diffusion taking place. 

In this section, surface boundary motion is considered as an essential feature for solving such an equation. Thus, diffusion accompanied by an electrode thickness increment due to diffusion of metal cations (solute) and immobile species is considered to be a particular diffusion problem that resembles mass transfer diffusion toward the faces of cathode sheets in an electrowinning EW) cell. 

Metal diffusion can be simulated by letting a stationary electrode sheet be the cathode and be immersed in an electrolyte at temperature T and pressure 

the diffusion molar flux (Tick s first law of diffusion) is 

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Figure 7.16 Model for moving boundary diffusion in infinite media.

These two processes provide examples of the moving boundary problem in diffusing systems in which a solid solution precedes the formation of a compound. The diickness of the separate phase of the product, carbide or... 

Figure 19.2 shows, at a microscopic level, what is going on. Atoms diffuse from the grain boundary which must form at each neck (since the particles which meet there have different orientations), and deposit in the pore, tending to fill it up. The atoms move by grain boundary diffusion (helped a little by lattice diffusion, which tends to be slower). The reduction in surface area drives the process, and the rate of diffusion controls its rate. This immediately tells us the two most important things we need to know about solid state sintering ... 

The mathematical solution to moving boundary problem involves setting up a pseudo-steady-state model. The pseudo-steady-state assumption is valid as long as the boundary moves ponderously slowly compared with the time required to reach steady state. Thus, we are assuming that the boundary between the salt solution and the solid salt moves slowly in the tablet compared to the diffusion... 

The most complete model to date for describing Case II diffusion is that of Thomas and Windle (13-16). They envision the process as a coupled swelling-diffusion problem in which the swelling rate is treated as a linear viscoelastic deformation driven by osmotic pressure. This model leads to the idea of a precursor phase propagating ahead of a moving boundary, as we have depicted in Figure 4. While Thomas and Windle have used numerical methods to examine in detail the predictions of their model, this model is difficult to test with the data obtained here. 

Mathematically, diffusive crystal dissolution is a moving boundary problem, or specifically a Stefan problem. It was treated briefly in Section 3.5.5.1. During crystal dissolution, the melt grows. Hence, there are melt growth distance and also crystal dissolution distance. The two distances differ because the density of the melt differs from that of the crystal. For example, if crystal density is 1.2 times melt density, dissolution of 1 fim of the crystal would lead to growth of 1.2 fim of the melt. Hence, AXc = (pmeit/pcryst) where Ax is the dissolution distance of the crystal and Ax is the growth distance of the melt. 

The reaction scheme at and near the phase boundary during the phase transformation is depicted in Figure 10-14. The width of the defect relaxation zone around the moving boundary is AifR, it designates the region in which the relaxation processes take place. The boundary moves with velocity ub(f) and establishes the boundary conditions for diffusion in the adjacent phases a and p. The conservation of mass couples the various processes. This is shown schematically in Figure 10-14b where the thermodynamic conditions illustrated in Figure 10-12 are also taken into account. The transport equations (Fick s second laws) have to be solved in both the a and p... 

Proceeding systematically, diffusion controlled a-fi transformations of binary A-B systems should be discussed next when a and / are phases with extended ranges of homogeneity. Again, defect relaxations at the moving boundary and in the adjacent bulk phases are essential for their understanding (see, for example, [F. J. J. van Loo (1990)]). The morphological aspects of this reaction type are dealt within the next chapter. 

Atoms are diffusing into the boundary laterally from its edges and can diffuse out through its front face into the forward grain. At the same time, atoms will be deposited in the backward grain in the wake of the boundary. In the quasi-steady state in a coordinate system fixed to the moving boundary, the diffusion flux in the forward grain is J = — DXL(dc/dx) — vc and the diffusion equation is... 

As in surface diffusion (Eq. 14.6), flux accumulation during grain-boundary diffusion leads to atom deposition adjacent to the grain boundary. The resulting accumulation causes the adjacent crystals to move apart at the rate2... 

Here rmax is the radial distance to the moving boundary and texp is an experimental time relative to some time t0. The diffusion constant D was determined from knowledge of the highest (dc/dr) coordinate measured at different times and by applying the following relationship ... 

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