Abstract
We develop a statistical-mechanical model of one-dimensional normal grain growth that does not require any drift-velocity parameterization for grain size, such as used in the continuity equation of traditional mean-field theories. The model tracks the population by considering grain sizes in neighbour pairs; the probability of a pair having neighbours of certain sizes is determined by the size-frequency distribution of all pairs. Accordingly, the evolution obeys a partial integro-differential equation (PIDE) over ‘grain size versus neighbour grain size’ space, so that the grain-size distribution is a projection of the PIDE's solution. This model, which is applicable before as well as after statistically self-similar grain growth has been reached, shows that the traditional continuity equation is invalid outside this state. During statistically self-similar growth, the PIDE correctly predicts the coarsening rate, invariant grain-size distribution and spatial grain size correlations observed in direct simulations. The PIDE is then reducible to the standard continuity equation, and we derive an explicit expression for the drift velocity. It should be possible to formulate similar parameterization-free models of normal grain growth in two and three dimensions.
Highlights
Normal grain growth (NGG) refers to the gradual increase of the mean grain or crystal size x of a polycrystalline material, as grainboundary motion causes larger grains to consume smaller grains and small grains to be eliminated
NGG has been studied as a fundamental process affecting texture evolution in metals and geological materials [1,2], and more broadly in connection with coarsening dynamics in various physical, social and biological systems; e.g. [3e6]
The purpose of this paper is to provide a mean-field model for NGG in 1D that is complete in the above sense
Summary
Normal grain growth (NGG) refers to the gradual increase of the mean grain or crystal size x of a polycrystalline material, as grainboundary motion causes larger grains to consume smaller grains and small grains to be eliminated. In a grain system where the rules of grain-boundary migration and associated topological reorganization are all known or prescribed, the evolution can be tracked by a ‘complete’ statistical-mechanical model based on nothing besides the rules, i.e. not involving extraneous assumptions or approximations informed by the actual outcomes of the NGG dynamics This means that, if Eq (2) is a valid model, a self-contained recipe for the velocity v ought to exist (and hopefully can be found). The modified Hillert theories have engendered a tradition of invoking parameterizations to “close” the mean-field description Such approach is useful because an ansatz posed for the resulting model often yields an analytical solution that can be evaluated straightforwardly for the invariant grain-size distribution. We consider this avenue briefly at the end of the paper
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