The Lifshitz–slyozov–wagner Equation for Reaction-controlled Kinetics

The late-stage behavior of a material undergoing a first-order phase transition (due to changes in temperature and/or pressure for example) is characterized by thermodynamic instability resolved through phase separation and consequent coarsening of the emerging phase. In the case of the new phase occupying much smaller volume fraction, and thus appearing as well-separated particles, this coarsening process (known as Ostwald ripening) is driven by the minimization of surface energy at the interface via diffusional mass exchange between particles while the total mass or volume of each phase is conserved. The result of this kind of mass diffusion from regions of high to regions of low interfacial curvature is the growth of large particles and the shrinkage and final extinction of smaller ones. For a review of some aspects of Ostwald ripening, mainly from the physical and modeling viewpoint, see the survey by Voorhees [21] or the book by Ratke and Voorhees [18]. In this coarsening scenario the mass-diffusion process can be controlled by two different mechanisms: either by the diffusion of atoms away from the particles and into the bulk, or by the reaction-rate of attachment of atoms at the phase interface. In the former case (diffusion control), the random exchange of atoms between the particles and the bulk is sufficiently rapid and the surrounding of each particle is in thermal equilibrium with the atoms in it; in the latter (interface-reaction control), detachment and attachment are slow compared to diffusion and the surrounding bulk can be out of equilibrium with the particle interface. We refer to the physics literature for more details, for example, Slezov and Sagalovich [19], Bartelt, Theis, and Tromp [3]; for a related mathematical treatment see Dai and Pego [5]. The classical theory for Ostwald ripening was developed by Lifshitz and Slyozov [9] and Wagner [22] in the case of supersaturated solid solutions in three dimensions. The Lifshitz–Slyozov–Wagner theory statistically characterizes the evolution by the particle-radius density n(t, R), where n(t, R) dR is defined to be the number of particles with radii between R and dR at time t per unit volume. In the late stages of the phase transition nucleation and coalesence of particles can be neglected since new nuclei dissolve immediately and since particles cannot merge because of the large distances between them. Thus, the particle-radius density satisfies the