Distribution of Topological Types in Grain-Growth Microstructures

TL;DR

This paper proposes a thermodynamic-like model to predict the distribution of grain topologies in microstructures during normal grain growth, supported by numerical evidence.

Contribution

It introduces a novel energy-based probabilistic model for grain topology distribution in 2D and 3D microstructures, linking topology to physical energy considerations.

Findings

The distribution of grain topologies follows a specific probabilistic pattern.

Numerical simulations support the proposed energy-based model.

The model explains the asymptotic state of microstructure evolution.

Abstract

An open question in studying normal grain growth concerns the asymptotic state to which microstructures converge. In particular, the distribution of grain topologies is unknown. We introduce a thermodynamic-like theory to explain these distributions in two- and three-dimensional systems. In particular, a bending-like energy $E_i$ is associated to each grain topology $t_i$, and the probability of observing that particular topology is proportional to $\frac{1}{s(t_i)}e^{-\beta E_i}$, where $s(t_i)$ is the order of an associated symmetry group and $\beta$ is a thermodynamic-like constant. We explain the physical origins of this approach, and provide numerical evidence in support.