Interface Source-Limited Growth

Four basic situations can occur both 0i s small, both large, and two cases of one large and the other small. When 0X and 02 are both small, c = cf and c/3t a c 7) and the /3-phase layer should grow under diffusion-limited conditions. Putting these conditions into Eqs. 20.26 and 20.28 and summing them, 

When 0i and 02 are both large, c 7 = c a = c and the growth should be interface-limited. Putting these conditions into Eqs. 20.25 and 20.26 and equating them, doing the same with Eqs. 20.27 and 20.28, and then subtracting the results, 

Integration of Eq. 20.34 then yields linear growth at a rate that is controlled by the rate constants at the two interfaces. Further analysis (see Exercise 20.3) shows that interface-limited growth also prevails when one 0j is small and the other large. In this case, the /3-phase growth rate is controlled by the interface that possesses the larger value of 0j (and the smaller rate constant), as might be expected. 

The parameter 0j therefore emerges as the critical parameter determining the mode of growth. The magnitude of 0j, in turn, depends directly upon the magnitude of the rate constant Ki. Determining the magnitude of Ki requires the construction of a detailed model for the source action of the interface based on one (or more) of the mechanisms described in Section 13.4. 

Experimental results for the growth of layers has been reviewed [6, 7]. It is recognized that all layer growth should be interface-limited in the early stages when 

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Source-Limited Coarsening. During source-limited coarsening, the interfaces surrounding the particles behave as poor sources and sinks, and the coarsening rate then depends upon the rate at which the diffusion fluxes between the particles can be created or destroyed (accommodated) at the particle interfaces. In a simple model, the same assumptions can be made about the source action at the particles as those that led to Eq. 13.24. The rate of particle growth can then be written... 

The growth of spherical precipitates under diffusion-limited conditions has been observed in a number of systems, such as Co-rich particles growing in Cu supersaturated with Co (see Chapter 23). In these systems, the particles are coherent with the matrix crystal and the interfaces possess high densities of coherency dislocations, which are essentially steps with small Burgers vectors, The interfaces therefore possess a high density of sites where atoms can be exchanged and the particles operate as highly efficient sources and sinks. 

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