Beyond the Classical Mean-Field Theory of Coarsening

2 Beyond the Classical Mean-Field Theory of Coarsening 

The mean-field theory has a number of shortcomings, including the approximations of a mean concentration around all particles and the establishment of spherically symmetric diffusion fields around every particle, similar to those that would exist around a single particle in a large medium. The larger the particles total volume fraction and the more closely they are crowded, the less realistic these approximations are. No account is taken in the classical model of such volume-fraction effects. Ratke and Voorhees provide a review of this topic and discuss extensions to the classical coarsening theory [8]. 

Work by Voorhees and Glicksman concludes that the classical theory is correct in the limit of zero volume fraction of the coarsening phase and that both the kinetics and the size distributions are significantly dependent on the precipitate volume fraction, p [9—12]. The temporal law for diffusion-limited coarsening, given by Eq. 15.18, remains valid for all volume fractions, but the rate constant Kp is a monotonically increasing function of j , as in Fig. 15.8. 

Volume-fraction effects on particle coarsening rates have been observed experimentally. For comparisons between theory and experiment, data from liquid+solid systems are far superior to those from solid+solid systems, as the latter are potentially strongly influenced by coherency stresses. Hardy and Voorhees studied Sn-rich and Pb-rich solid phases in Pb-Sn eutectic liquid over the range p = 0.6-0.9 and presented data in support of the volume-fraction effect, as shown in Fig. 15.9 [7], 

Voorhees s experimental study of low-volume-fraction-solid liquid+solid Pb-Sn mixtures carried out under microgravity conditions during a space shuttle flight enabled a wider range of solid-phase volume fractions to be studied without significant influence of buoyancy (flotation and sedimentation) effects [13]. The rate of approach to the steady-state particle-size distribution in 0.1-0.2 volume-fraction 

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