Kinetics of grain-boundary nucleated transformations in rectangular geometries and one paradox relating to Cahn's model

TL;DR

This paper derives volume-fraction expressions for systems of parallel planes, including a non-Poissonian process, and reveals limitations of Cahn's model in predicting transformation kinetics in various grain structures.

Contribution

It introduces a new volume-fraction expression for a non-Poissonian plane process and demonstrates the limitations of Cahn's model in different grain structures.

Findings

Cahn's equation underestimates transformation kinetics.

Non-Poissonian plane arrangements require different modeling approaches.

Packing structure affects transformation rates despite identical elements.

Abstract

Volume-fraction expressions are obtained for the systems of an infinite number of parallel planes arranged both regularly and randomly. As a special case of random arrangement, a non-Poissonian point process (the second-order Erlang process) of arrangement of planes is considered for the first time. The exact volume-fraction expression obtained for this case shows that it cannot be derived by the Cahn method, i. e. the extended-volume approach is applicable only to Poisson processes. The volume fraction equations for regular planes are used to study cubic grain structures, both regular and random. It is shown that the Cahn equation underestimates the transformation kinetics in both regular and random structures with four different size distributions of cubes; the degree of underestimation depends on the size distribution, being the largest in the regular structure. The paradox of packing is described: the structures composed of the same elements (random parallelepipeds) transforming in the same way, but packed differently, give different transformation rates.