Segregation in Materials - Kinetics of Segregation

Kinetics of Segregation

In some situations where segregation is important, the segregant atoms do not have sufficient time to reach their equilibrium level as defined by the above adsorption theories. The kinetics of segregation become a limiting factor and must be analyzed as well. Most existing models of segregation kinetics follow the McLean approach. In the model for equilibrium monolayer segregation, the solute atoms are assumed to segregate to a grain boundary from two infinite half-crystals or to a surface from one infinite half-crystal. The diffusion in the crystals is described by Fick’s laws. The ratio of the solute concentration in the grain boundary to that in the adjacent atomic layer of the bulk is given by an enrichment ratio, β. Most models assume β to be a constant, but in practice this is only true for dilute systems with low segregation levels. In this dilute limit, if is one monolayer, β is given as β = .

The kinetics of segregation can be described by the following equation:

Where F=4 four grain boundaries and 1 for the free surface, is the boundary content at time t, D is the solute bulk diffusivity, f is related to the atom sizes of the solute and the matrix, b and a, respectively, by . For short times, this equation is approximated by:

In practice, β is not a constant but generally falls as segregation proceeds due to saturation. If β starts high and falls rapidly as the segregation saturates, the above equation is valid until the point of saturation .