Materials dimension extension
Reconstructing the 3D shape distribution of rock grains or other randomly packed objects with access to only a 2D slice through them.
Many geological processes can be inferred from the distribution of the size and shape of the grains in rocks. But figuring out this information involves costly and tedious imaging techniques. In one approach, a thin layer at the surface of the rock is repeatedly sliced off. In another, the structure is deduced by merging x-ray images taken at different angles.
In this project we develop a mathematical formalism for capturing the 3D structure of a rock’s grains from just one 2D slide through it. Our approach is based on the theory of packing spheres and other shapes. We derive exact relations between the distribution of the volume, surface area or shape characteristics of grains and simple measurements from a thin section through the rock. We also determine how many grains to measure to achieve a desired level of certainty. We test our method on real minerals and simulated materials made up of randomly oriented ellipsoids and blocks.
Our new approach will help geologists and material scientists determine the structure of a material’s grains at a fraction of the cost of current methods. It could shed light on fundamental processes of rock formation, help optimize oil extraction, and better connect geological percolation with percolation theory.
The distributions of size and shape of a material’s grains can be constructed from a 2D slice of the material and electron diffraction data.
Moment-based methods provide a simple way to describe a population of spherical particles and extract 3d information from 2d measurements.