Diffusion of heat or impurities

When the rate of advance of the phase boundary is limited by the removal of latent heat or impurities, then the diffusion field modifies the interfacial values of the relevant field variables and therefore the growth shape. The physical picture is as follows As a phase boundary advances, latent heat and/or impurities are emitted at the front. Their build up occurs over a characteristic dimension, or boimdary layer, that depends on their respective diffusivities and the rate of advance. The motion itself is limited by the speed with which this excess is depleted by diffusion. These diffusion-limited xt gimes are generally studied using non-geometric formulations. The reason is that the... 

Spin diffusion to paramagnetic impurity is an important mechanism, not only in solids but in some large molecules which exist in solution. In fact, there can be spin diffusion not only to impurities but to any center which has good coupling to the lattice and this could be a sub-molecular unit with its own motion conducive to rapid relaxation or a quad-rupolar nucleus. Because of the sheer size of the electron moment, spin diffusion to paramagnetic impurities is more common than to these other heat sinks. See the review by Noack (1971) for references to this mechanism. 

The diffusion of impurities or heat is governed by differential equations, that can be derived from the molecular models just described. We only... 

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