Spatial Distribution of Markers

The above-described experiment on the markers smearing in the diffusion couple is incorporated into the model describing the spatiotemporal evolution of the distribution of markers, p (t, x). Here p (t, x) dx is the number of markers in a thin layer (x, x- - dx). Obviously, from the condition of markers conservation, it follows that the continuity equation must be fulfilled 

Further, we treat (for simplicity) the case of constant and equal partial volumes for all constituents. Then, the time evolution of markers distribution is described by the system of two equations 

The second of these relations is not very convenient for numerical calculations, since the initial distribution is Gaussian with very small dispersion, rather resembling the Dirac i5-function. So, we rewrite it taking In p as the unknown function  

As proven, in order to find K-planes as well as the concentration dependencies for partial coefficients, one should have the velocity curve. The usual experimental method, providing this, is the multilayer method. The diffusion couple consists of several thin layers (generally, 10-20 gm thick) with inert markers placed between them. After annealing, the displacement of each marker plane is measured, and the velocity of each plane is determined by the equation [34, 35] 

Here the displacement is y (t, Xo) = x (t, Xo) — Xo, x being the marker s coordinate (at time t), and at the starting time the marker was at Xq point. AU coordinates are written in Matano s reference frame (x = 0 corresponding to the Matano plane). 

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