Coupled Diffusion Processes

Because of the interwoven coupling between forces and fluxes stipulated by NET, the mathematical treatment of transport can quickly get complicated. For example. 

TABLE 4.4 Summary of the Various Transient Solutions Discussed in This Chapter 

Electrodiffusion Electrodiffusion, or electromigration, occurs when an apphed electrical field provides an additional driving force for diffusion. There are two major categories of electric-field-assisted diffusion  

Thermodiffusion Thermodiffiision reflects the fact that a temperature gradient can act as a driving force for diffusion. Incorporating both concentration gradient and temperature gradient driving forces into the ID diffusion equation yields 

Stress-Driven Diffusion Stress and diffusion can be coupled in a number of ways. In a uniform stress field, the dijfusivity of the diffusing species can become directionally dependent. This is because the stress field can affect the amount work required for the species to move in different directions (e.g., parallel vs. perpendicular to the stress field). Movements in directions that cause the greatest distortions to the stress field will be penalized, while movements in directions that minimize the distortion to the stress field will be favored. 

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Figure 4.19 summarizes these important coupled diffusion processes, each of which is briefly treated below. 

Here is another example. The diffusion of one salt, say NaCl, affects the diffu.sion of another, such as KCl. Table 18.2 shows careful measurements of such coupled diffusion processes, and shows that at least in this case the reciprocal relations hold to within experimental error. 

It now remains to calculate the diffusion currents, zrequired times. An apparently simple way would be to use a substance fairly similar to Ox (or having a similar diffusion coefficient) capable of being reduced simply by a diffusion process (or, without coupled chemical reactions) through a process involving n + 2 electrons. A solution of this substance could therefore be prepared with the same molarity as that containing Ox, such that one can measure the potentiostatic current at the required times. In practice, however, this method is quite laborious. 

Far less information is available on the mechanism of the growth and stabilisation of structured flows once the nucleation process of coupled diffusion-mediated density inversion occurs. At this stage, only speculative arguments can be offered. 

The entry-length region is characterized by a diffusive process wherein the flow must adjust to the zero-velocity no-slip condition on the wall. A momentum boundary layer grows out from the wall, with velocities near the wall being retarded relative to the uniform inlet velocity and velocities near the centerline being accelerated to maintain mass continuity. In steady state, this behavior is described by the coupled effects of the mass continuity and axial momentum equations. For a constant-viscosity fluid,... 

If the diffusion process is coupled with other influences (chemical reactions, adsorption at an interface, convection in solution, etc.), additional concentration dependences will be added to the right side of Equation 2.11, often making it analytically insoluble. In such cases it is profitable to retreat to the finite difference representation and model the experiment on a digital computer. Modeling of this type, when done properly, is not unlike carrying out the experiment itself (provided that the discretization error is equal to or smaller than the accessible experimental error). The method is known as digital simulation, and the result obtained is the finite difference solution. This approach is described in more detail in Chapter 20. 

Modeling diffusion-coupled vaporization processes associated with non-stoichiometric carbides requires the use of the chemical diffusion coefficient, D, for the calculation of temporal C concentrations. Clearly D will have a strong concentration dependence. In principle, the concentration dependence should be calculable from measured D (NC,T) and ac(Nc,T) values. However, our attempts and those of Wakelkamp31 to verify the correlation have been unsuccessful for TiC, ZrC, VC, and TaC. The disparity is probably based in the approximations used to derive these equations. For example, Howard and Lidiard32 have shown that the right hand sides of equations 3.10 and 3.11 are approximate and proposed that additional terms are needed. 

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