Diffusion couples

Darken assumed that the accumulated vacancies were annilrilated within the diffusion couple, and that during tlris process, tire markers moved as described by Smigelskas and Kirkendall (1947). His analysis proceeds with the assumption tlrat the sum of tire two concenuations of the diffusing species (cq - - cq) remained constant at any given section of tire couple, and tlrat the markers, which indicated the position of tire true interface moved with a velocity v. 

The constant of integration must be zero because at points on die diffusion couple far from the interface,... 

Now let us consider the stability of the two systems around fixed points of /(cr), and therefore around homogeneous solutions of the CML. From chapter 4 we recall that (7 = 0 is a stable fixed point for a < 1 and cr = 1 - 1/a is a stable fixed point for 1 < a < 3. Let us see whether our diffusive coupling leads to any instability. 

We see from both equations 8.32 and 8.33 that the most unstable mode is the mode and that ai t) = 1 - 1/a is stable for 1 < a < 3 and ai t) = 0 is stable for 0 < a < 1. In other words, the diffusive coupling does not introduce any instability into the homogeneous system. The only instabilities present are those already present in the uncoupled local dynamics. A similar conclusion would be reached if we were to carry out the same analysis for period p solutions. The conclusion is that if the uncoupled sites are stable, so are the homogeneous states of the CML. Now what about inhomogeneous states ... 

Table 8.5 summarizes some of the behaviors of the measures introduced in the sections above for each of the four regimes for the case where the diffusive coupling is of medium size I... 

We conclude this section with a few general observations about the zigzag regime (see [crutch87]) (i) For small diffusive-couplings, zigzag structures typically start... 

Localized Kink Regime when the diffusive coupling is too small for kinks to move, the initial kinks separating domains remain locked in position. The behavior is analogous to that of class c2 elementary CA. 

Multiparticle collision dynamics can be combined with full molecular dynamics in order to describe the behavior of solute molecules in solution. Such hybrid MPC-MD schemes are especially useful for treating polymer and colloid dynamics since they incorporate hydrodynamic interactions. They are also useful for describing reactive systems where diffusive coupling among solute species is important. 

In this section we demonstrate the utility of the Caco-2 cell monolayer to provide a systematic and quantitative assessment of passive diffusion coupled with intra-... 

The evolution of the isotope ratio at various depths is shown in Figure 8-11 for the first 200 years of the calculation. The shallowest depths depart little from the seawater value because they are diffusively coupled to open water. Respiration at somewhat greater depths drives the isotope ratio... 

Figure 5.2 Diffusion couple formed by two crystals separated by radioactive material (a) initially and (b) after heating. Short-circuit diffusion occurs down extended defects, and diffusion into the bulk occurs both from the surface and laterally from extended defects.

A plot of In cx versus x2 will have a gradient of [ —1/(4D t) (See also Supplementary Material S5.) A measurement of the gradient gives a value for the tracer diffusion coefficient at the temperature at which the diffusion couple was heated. 

Radioactive 22Na was coated onto a glass sample that was made into a diffusion couple (Fig. 5.2) and heated for 4 h at 411°C. The radioactivity perpendicular to the surface is given in the following table. Calculate the tracer diffusion coefficient, D, of 22Na in the glass. 

ZnO in a planar sandwich type of diffusion couple using the data Ar(0) =... 

Plot the concentration profile over the range 0 to 2 x 10 3 m for Fe diffusion into Ti02 (rutile) in a planar sandwich type of diffusion couple using the data ... 

For the diffusion couple experimental arrangement (Fig. 5.2), (ignoring all short-circuit diffusion), the solution is... 

The nonlinearity of the system of partial differential equations (51) and (52) poses a serious obstacle to finding an analytical solution. A reported analytical solution for the nonlinear problem of diffusion coupled with complexation kinetics was erroneous [12]. Thus, techniques such as the finite element method [53-55] or appropriate change of variables (applicable in some cases of planar diffusion) [56] should be used to find the numerical solution. One particular case of the nonlinear problem where an analytical solution can be given is the steady-state for fully labile complexes (see Section 3.3). However, there is a reasonable assumption for many relevant cases (e.g. for trace elements such as... 

A schematic illustration of the method, and of the correlation between binary phase diagram and the one-phase layers formed in a diffusion couple, is shown in Fig. 2.42 adapted from Rhines (1956). The one-phase layers are separated by parallel straight interfaces, with fixed composition gaps, in a sequence dictated by the phase diagram. The absence, in a binary diffusion couple, of two-phase layers follows directly from the phase rule. In a ternary system, on the other hand (preparing for instance a diffusion couple between a block of a binary alloy and a piece of a third... 

Figure 2.42. The Cu-Zn system phase diagram and microstructure scheme of the diffusion couple obtainable by maintaining Cu and Zn blocks in contact for several days at 400°C. Shading indicates subsequent layers, each one corresponding to a one-phase region. The two-phase regions are represented by the interfaces between the one-phase layers (adapted from Rhines 1956).

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