Ageing and Ostwald ripening

Let us assume that precipitation of the A-rich a-phase has occurred to the extent that the matrix is no longer supersaturated, and that we now have a two-phase alloy which is nearly at equilibrium. In section 7.3.2, it was assumed that the precipitate particles were all of the same size. In reality, however, there will be a range of particle sizes, since not all nuclei were formed at the same time, and also since the supply of A to a precipitate particle will be influenced by the site of the particle in the matrix. According to the Gibbs-Thomson relation, smaller particles of precipitate have a higher chemical potential for component A than do the larger particles. For spherical particles of radius r and the difference in chemical potential is  

In the quantitative treatment of Ostwald ripening, two limiting cases may be distinguished. In the first case, the rate is controlled by diffusion between the particles. In the second case, the phase boundary reaction is rate-determining. If it is assumed that the A particles obey the laws of ideal dilute solutions in the matrix, then eq. (7-47) may be rewritten in the form of a concentration difference  

The relationship Ap = ARTItiCa RTA cJca (eq) has been used in this derivation. Eq. (7-48) says that there is a relative concentration gradient between two precipitating particles of radius r and r . In the case of a diffusion-controlled equilibration process, this concentration gradient leads to an increase in the average radius r of the particles of precipitate. Eq. (7-49) gives r as a function of time [33]  

This rate equation presupposes that a quasi-steady state distribution function /(r) exists for the particle radii r. As usual, the average particle radius r is defined as  

It seems worthwhile to comment upon the quasi-steady state distribution function f(r, t) in this Ostwald ripening process. The mathematical verification of the fact that this distribution function/(r, t) can be written as a product/i(0 fii lr) (and f2(rlf) is a universal function even in cases where the initial distribution / (r, 0) is of Gaussian shape and of moderate width) is rather cumbersome and must be studied from the original work [33]. However, one may conceive the shape of the quasi-steady state distribution (which has a maximum between 0 r/F 3/2 at r/F = 1.135 and is essentially zero at r/F 3/2) by realizing that it is the interplay between the activity difference of the average activity of A in the solution matrix 

---