Reinterpretation of the Mean Field Hypothesis in Analytical Models of Ostwald Ripening and Grain Growth
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ic and comprehensive exposition some recent results in the description of the Ostwald ripening process[1,2] in conjunction with a previous treatment of grain growth.[3–5] The two subjects are strictly related since, in both cases, the newly developed analytical models based on a pairwise interaction framework have been proven to be a valid alternative to the respective classical theories based on the mean field approach. The aim is to show how the hypothesis of ‘‘mean field,’’ in the way it is classically intended, is actually unsuitable since it has been
PAOLO EMILIO DI NUNZIO is with RINA Consulting - Centro Sviluppo Materiali S.p.A., Via di Castel Romano, 100, 00128 Rome, Italy. Contact e-mail: [email protected] Manuscript submitted October 17, 2018.
METALLURGICAL AND MATERIALS TRANSACTIONS A
responsible for the failure in predicting the asymptotic shape of size distributions of particles and grains. Such a feature is instead very well predicted by the new approaches that can be classified as mean field theories as well. Generally speaking, analytical models provide a representation of the evolution of the system from a statistical viewpoint in that they imply the homogeneity of the system topology. Thus, the growth rate of a particle or grain depends only on its size irrespective of its position. In other words, the local environment obtained from averaging operations is representative of the system as a whole. On the contrary, numerical models operate on a system whose size and initial configuration are defined. The microstructural evolution results from a precise description of any single local environment. Therefore, additional phenomena such as gradients of physical properties (e.g., temperature, solute concentration) on a scale longer than the characteristic length of the microstructure can be taken into
account. A discussion on merits and disadvantages of the two approaches can be found in Reference 6. However, one unquestionable advantage of analytical models relies in their ability of synthesizing the underlying physics of the processes and to put in evidence in an explicit fashion the main factors affecting the phenomena without resorting to extensive simulation plans and to a large computational power. Especially for grain growth, analytical models of coarsening, starting from the basic approach by Hillert,[7] are often compared with numerical models such as Potts and Monte Carlo,[8–15] cellular automata,[16,17] phase field,[18] molecular dynamics,[19–22] or the Surface Evolver.[23,24] The literature on this subject is huge and many theoretical and computational works are continuously published. Some recent results are reported in References 25 through 37. The majority of approaches to Ostwald ripening are instead based on the original formulation by Lifshitz, Slyozov, and Wagner (LSW)[38,39] modified through more or less complex analytical formulations, apart from the model by Voorhees and Glicksman[40,41] which is numerical in nature. Also in this case, the literature offers a great deal of theoretic
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