Crystal growth Stefan problem

H. Kopetsch. A numerical method for the time-dependent Stefan-problem in Czochralski crystal growth. J Crystal Growth SS 71, 1988 H. Kopetsch. Phy-sico-Chem Hydrodyn 77 357, 1989 H. Kopetsch. J Cryst Growth 102 500, 1990. 

If the boundary motion is controlled by an independent process, then the boundary motion velocity is independent of diffusion. This can happen if the magma is gradually cooling and crystal growth rate is controlled both by temperature change and mass diffusion. This problem does not have a name. In this case, u depends on time or may be constant. If the dependence of u on time is known, the problem can also be solved. The Stefan problem and the constant-w problem are covered below. 

If crystal growth or dissolution or melting is controlled by diffusion or heat conduction, then the rate would be inversely proportional to square root of time (Stefan problem). It is necessary to solve the appropriate diffusion or heat conduction equation to obtain both the concentration profile and the crystal growth or dissolution or melting rate. Below is a summary of how to treat the problems more details can be found in Section 4.2. 

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. 

Equation (2.3-47), the pseudo steady-slate solution for the flux, could be used to predict the diflusion-conuojled growth or dissolution rate of the crystal in a manner analogous to the Stefan problem solution. The result would indicate that the sqnene of the particle sedius varies linearly with time. 

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