Classical Mean-Field Theory of Coarsening

In 1961, the classical theory of particle coarsening was developed at about the same time, but independently, by Lifshitz and Slyozov [1] and Wagner [2], Most of the 

Kinetics of Materials. By Robert W. Balluffi, Samuel M. Allen, and W. Craig Carter. 

Consider a binary system at an elevated temperature composed of A and B atoms containing a distribution of spherical /0-phase particles of pure B embedded in an A-rich matrix phase, a. The concentration of B atoms in the vicinity of each /0-phase particle has an equilibrium value that increases with decreasing particle radius, as demonstrated in Fig. 15.1. Because of concentration differences, a flux of B atoms from smaller to larger particles develops in the matrix. This flux causes the smaller particles to shrink and the larger particles to grow. 

In the following, this model is used to analyze the kinetics for the two cases where the particle growth is either diffusion- or source-limited. Each of the two cases yields a different growth law for the particles in the distribution. 

At any time t, a distribution of particle sizes will exist which can be quantified by defining a particle-size distribution function, f(R,t) [units, (length)-4], such that the number of particles per unit volume with radii between R and R + dR, n(R, R + dR t), is given by 

---

Beyond the Classical Mean-Field Theory of Coarsening... 

---