Limit conditions for abnormal grain growth in the presence of Zener drag

Most of the grain growth processes in technical polycrystalline materials occur in the presence of restraining forces, opposing the grain boundary movement, which can be generated by the presence of a distribution of second phase particles and/or by a certain amount of solute elements segregating at the grain boundary. The mechanisms by which the restraining forces operate on the grain boundary are quite well established. However the effects of such forces on grain growth kinetics are often discussed on heuristic basis and mostly considering a simplified microstructure described by average parameters (as the mean grain size) which typically do not allow to consider the various microstructural features playing a role on the kinetics of grain growth and eventually on the conditions for the onset of abnormal growth. Here the approach of the statistical theory of grain growth is used where the effect of the grain growth inhibition is introduced at the very basic level of the single grain boundary (i, j) shared between the grain i and the grain j. The calculation of the growth kinetics of a given grain i, shows a very relevant feature of this approach: for each grain class i a well defined critical radius appears as a consequence of the grain growth modulation, induced by the specific restraining force, on the different size classes of the grain distribution and consequently the concept that only a single critical radius for the whole size distribution exists, as it is obtained for the normal grain growth case and as it has been introduced by Hillert, is no longer valid. The latter result has significant consequences on the predicted grain growth kinetics which, although for some aspects qualitatively similar to those obtained by the simplified approaches, show significant quantitative differences, particularly in respect to the kinetics of approaching grain growth stagnation and to the conditions for the onset of abnormal grain growth. In particular it is worth to mention the formation of a peculiar grain size distribution with a very high peak centered on the smallest grain class still in equilibrium with the largest grains present in the system (about 2 times the average grain size). In this case of "critical" equilibrium condition quantitative prescription on the level of grain growth inhibition, in relationship with the shape and relevant parameters of the grain size distribution and a derivation of the analytical expression of the corresponding grain size distribution are presented and compared with simulations.