3-19. Ostwald Ripening In a supersaturated solid solution, precipitation may lead to the formation of grains or particles of a new phase. It is found that, after an initial stage of this process that involves nucleation, there is a second stage in which the particles grow to much larger sizes. In the second stage the degree of supersaturation is slight, and the rate of particle growth is controlled by diffusion. This coarsening of particles, called Ostwald ripening, has important effects on the physical properties of various alloys. An extensive analysis of the kinetics of the second stage of this process was given by Lifshitz and Slyozov (1961). The objective here is to derive one of their simpler results. Consider a dilute solid solution consisting of components A and B, with B the abundant species. The particles being formed by precipitation are composed of pure B. Assume that at any given time the precipitate consists of widely dispersed spherical particles of radius R(t). Accordingly, it is sufficient to model one representative particle, for which CA=CA(r,t) in the surrounding solution. Far from the particle the concentration of A is constant at CA. A key aspect of this problem is that the concentration of A at the particle surface is influenced by the interfacial energy and therefore depends on particle size. The relationship is CA[R(t),t]=CA+R(t) where the constant is proportional to the interfacial energy. It is the interfacial energy term that provides the driving force for diffusion. (a) Derive an expression for the growth rate of the particle, dR/dt, valid when growth is slow enough that the diffusion process is pseudosteady. (Assume that the densities of the particle and solution are the same.) (b) Show that, eventually, R(t)t1/3, which is the result of Lifshitz and Slyozov. (c) Use order-of-magnitude reasoning to determine when the pseudosteady assumption is valid. "This problem was suggested by R. A. Brown. Problems 109 x=0x=(t) Figure P3-20. Bleaching of wood pulp. The ClO2 diffuses into a mat of fibers, where it reacts with lignin to cause the bleaching. 3-19. Ostwald Ripening In a supersaturated solid solution, precipitation may lead to the formation of grains or particles of a new phase. It is found that, after an initial stage of this process that involves nucleation, there is a second stage in which the particles grow to much larger sizes. In the second stage the degree of supersaturation is slight, and the rate of particle growth is controlled by diffusion. This coarsening of particles, called Ostwald ripening, has important effects on the physical properties of various alloys. An extensive analysis of the kinetics of the second stage of this process was given by Lifshitz and Slyozov (1961). The objective here is to derive one of their simpler results. Consider a dilute solid solution consisting of components A and B, with B the abundant species. The particles being formed by precipitation are composed of pure B. Assume that at any given time the precipitate consists of widely dispersed spherical particles of radius R(t). Accordingly, it is sufficient to model one representative particle, for which CA=CA(r,t) in the surrounding solution. Far from the particle the concentration of A is constant at CA. A key aspect of this problem is that the concentration of A at the particle surface is influenced by the interfacial energy and therefore depends on particle size. The relationship is CA[R(t),t]=CA+R(t) where the constant is proportional to the interfacial energy. It is the interfacial energy term that provides the driving force for diffusion. (a) Derive an expression for the growth rate of the particle, dR/dt, valid when growth is slow enough that the diffusion process is pseudosteady. (Assume that the densities of the particle and solution are the same.) (b) Show that, eventually, R(t)t1/3, which is the result of Lifshitz and Slyozov. (c) Use order-of-magnitude reasoning to determine when the pseudosteady assumption is valid. "This problem was suggested by R. A. Brown. Problems 109 x=0x=(t) Figure P3-20. Bleaching of wood pulp. The ClO2 diffuses into a mat of fibers, where it reacts with lignin to cause the bleaching