A constant of motion for ideal grain growth in three dimensions

TL;DR

This paper establishes a new rigorous relationship in 3D grain structures linking Gaussian curvature integral to the number of grains and junctions, verified through numerical simulation.

Contribution

It introduces a novel, mathematically rigorous constant of motion for three-dimensional grain growth, connecting geometric and topological properties.

Findings

Derived a new relation between Gaussian curvature and grain topology.

Numerically verified the relation using periodic truncated octahedra.

Provides a theoretical foundation for understanding grain boundary evolution.

Abstract

Most metallic and ceramic materials are comprised of a space-filling collection of crystalline grains separated by grain boundaries. While this grain structure has been studied for more than a century, there few rigorous results regarding its global properties available in the literature. We present a new, rigorous result for three-dimensional grain structures that relates the integral of the Gaussian curvature over the grain boundaries to the numbers of grains and quadruple junctions. The result is numerically verified for a grain structure consisting of periodic truncated octahedra.