A Metric on the Polycrystalline Microstructure State Space

TL;DR

This work introduces a general framework to compare polycrystalline microstructures by modeling each microstructure as a probability distribution of local windows sampled from micrographs, forming a microstructure state space. A Wasserstein-based metric, specifically using an unbalanced formulation for grain-boundary mass, quantifies the distance between window distributions, enabling statistically meaningful comparisons of grain geometry below a chosen length scale. The authors implement a practical pipeline—defining a grain boundary mass function on windows, discussing computational strategies (OT, entropic regularization, and assignment-based methods), and validating with DREAM.3D-generated microstructures, including a proof-of-concept microstructure database to demonstrate querying capabilities. This framework aims to support ICME by enabling reproducibility checks, pathway optimization, and property interpolation across broadly defined microstructures, with future work to incorporate phase and orientation information.

Abstract

Material microstructures are traditionally compared using sets of statistical measures that are incomplete, e.g., two visually distinct microstructures can have identical grain size distributions and phase fractions. While this is not a severe concern for materials fabricated by traditional means, the microstructures produced by advanced manufacturing methods can depend sensitively and unpredictably on the processing conditions. Moreover, the advent of computational materials design has increased the frequency of synthetic microstructure generation, and there is not yet a standard approach in the literature to validate the generated microstructures with experimental ones. This work proposes an idealized distance on the space of single-phase polycrystalline microstructures such that two microstructures that are close with respect to the distance exhibit statistically similar grain geometries in all respects below a user-specified length scale. Given a pair of micrographs, the distance is approximated by sampling windows from the micrographs, defining a distance between pairs of windows, and finding a window matching that minimizes the sum of pairwise window distances. The approach is used to compare a variety of synthetic microstructures and to develop a procedure to query a proof-of-concept database suitable for general single-phase polycrystalline microstructures.

Figures (10)

• Figure 1: Two probability distributions on the window space for two distinct materials can be approximated by points clouds of sampled windows (left). Even without a coordinate system, the shape of a point cloud can be inferred from the pairwise distances between the points (right, blue and yellow lines). Moreover, the similarity of the underlying probability distributions for two materials can be inferred from the pairwise distances between points in the two clouds (right, red lines).
• Figure 2: A schematic showing the various concepts involved in the construction of the microstructure state space. Windows of a consistent size are sampled uniformly at random from the interiors of the micrographs on the bottom row. The windows are represented as points in the window spaces on the middle row, with the points distributed according to underlying probability distributions $\mu$ and $\nu$ and distances on the window space given by $c$. Window distributions are represented as points in the microstructure state space on the top row, with the distances on the state space given by $W(\mu, \nu | c)$ (the notation is introduced in Sec. \ref{['sec:wasserstein']}).
• Figure 3: A schematic showing the use of the Wasserstein metric to compare microstructures $X$ and $Y$. The window distributions $\mu(x)$ and $\nu(y)$ (top) are actually two functions on a single window space (bottom left). The coupling $\gamma(x, y)$ can be viewed as a transport plan to convert $\mu(x)$ into $\nu(y)$ by transporting the probability mass from a position given by the vertical coordinate to a position given by the horizontal coordinate (bottom right).
• Figure 4: Distances induced by the Wasserstein metric using the $L^1$ ground distance between subsets of $4 \times 4$ windows with a total mass of four. Examples of optimal transport plans (orange vector fields) are shown on the left. A subset of the window space on the right shows that the Wasserstein metric induces a self-consistent notion of window proximity (i.e., a window space topology). Each two-sided arrow indicates a distance of one between adjacent windows, and the window at position $(0, 0)$ is a distance of one from $(0, 1)$, two from $(1, 1)$, and four from $(2, 2)$, both in terms of the Wasserstein metric and locations on the grid.
• Figure 5: A visual representation of the optimal transport plan between the blue (left) and red (right) grain boundary configurations. Cyan arrows indicate transport in the window interior, yellow arrows indicate transport to the window boundary, and purple pixels (darkest, center) are not transported.