Stochastic representation of microstructure via higher-order statistics: Theory and application

At the core of materials science is the description of the internal structure (i.e. microstructure) of the material, which spans a multitude of length scales, from the atomistic to the macroscale. Recent advances have now made it possible to capture the rich three-dimensional details of the material structure at various length scales (e.g. X-ray micro-tomography, automated serial sectioning, 3-D atom probe). Given the several hundred thousands of distinct engineered and natural materials of interest, the critical need for new computationally efficient approaches for archival, retrieval, and real-time exploration of the microstructure datasets by the broader scientific community is self-evident. Efforts in these activities are hindered by the lack of a rigorous mathematical definition of the internal structure or microstructure of a material. To this end, the author proposes a definition of microstructure grounded in the formalism of stochastic processes. In this framework the microstructure can be thought of as a set of statistical rules that govern the spatial placement of microstructure features, and observed micrographs are different realizations of the overriding process. This interpretation of microstructure allows for the quantitative comparison of different materials based on structure and more importantly allows for the quantification of the observed variance in samples with the same nominal processing history.;The n-point correlation functions have been shown to be capable of recovering the original micrograph to within a linear translation and/or an inversion, and as such will serve as a primary descriptor of the statistics underlying the stochastic nature of the microstructure. Decomposition of the n-point statistics via principal component analysis (PCA) offers a highly efficient mathematical procedure for cataloguing the microstructure datasets. Subsequently, when a micrograph is selected for analyses, the search algorithms described above are able to identify instantly (in real-time) all of the structures in the database that are closest to the selected structure, and rank them by their distance in the PCA space. Such a database can dramatically increase the speed and efficacy with which we can build datasets that can be shared by the broader scientific community, while minimizing duplication of effort.