Researchers solve model that can improve sustainable design, groundwater management, nuclear waste storage, and more

In an approach reminiscent of the classic board game Battleship, Stanford researchers have discovered a way to characterize the microscopic structure of everyday materials such as sand and concrete with high precision.

Heterogeneous, or mixed, materials have components in random locations. For example, concrete – the most abundant human-made material – is composed of cement, water, sand, and coarse stone. Predicting where a particular component appears in a jumbled mosaic of concrete or in Earth’s subsurface can help researchers understand how to design stronger materials, evaluate the long-term viability of potential sites for underground storage of carbon dioxide or nuclear waste, and answer other critical questions about the behavior of complex systems. But previous modeling efforts have fallen short.

In an Oct. 9 study in Physical Review Letters, researchers show a new mathematical approach to unlocking information about the composition of a material based on knowledge of any other random point – like taking a shot in Battleship. The approach is based on a common statistical method known as a Poisson model.

“With this study, we’ve solved the famous Poisson model for heterogenous materials,” said lead study author Alec Shelley, a PhD student in applied physics in Daniel Tartakovsky’s lab at the Stanford Doerr School of Sustainability. “Our result could have a broad impact on several areas of science, because heterogenous materials are common and their models almost never have exact solutions.”

Because a vast range of useful properties stem from microstructural arrangements like those in concrete, the new findings could enable the design of better, stronger, cheaper materials.

“What Alec has succeeded in doing in this study is quite remarkable,” said Tartakovsky, a professor of energy science and engineering. “Using his approach, you could design a composite material to your specifications and obtain certain properties based on the proper mixture of components.”

Abundant applications

Looking ahead, Shelley and Tartakovsky are interested in applying the mathematical solution to predict the compositions of several materials. The model reveals “a huge list” of properties tied to microstructure, Shelley said, including hardness, elasticity, tensile strength, electrical and heat conductivity, how quickly a substance moves through another substance, magnetic susceptibility, light transmittance, and more.

With concrete, the approach could guide engineers toward optimizing microstructure. Concrete is full of little air-pocket voids that if well-modeled could be filled with supplementary materials, such as fly ash, slag, or biochar, thereby reducing the overall cement content. That, in turn, would lower carbon dioxide emissions related to cement manufacturing and overall boost the concrete’s strength while lowering costs.

Additional applications include modeling fractured and porous media, a central challenge in groundwater management, as well as in nuclear waste disposal, geothermal energy, and carbon sequestration. “These systems are complex and difficult to model,” said Tartakovsky. “However, the Poisson model’s multipoint functions that we solve in this study offer a new tool for understanding and predicting their behavior.”

Predictions via Poisson

The Poisson model is named after Siméon-Denis Poisson, a French mathematician and physicist from the 1800s. He developed what became known as Poisson statistics, which describe independent events, such as snowflakes landing on one’s tongue or radioactive clicks from a Geiger counter. The Poisson model follows these statistics in describing a space that is broken up into a pattern of shapes with perfectly straight borders, where the borders are rendered independently of each other.

In this way, as a microstructural model, the Poisson model can accurately simulate a wide range of heterogenous materials, including everything from the appearance and distribution of ice fragments on a frozen lake to the marbling in a juicy steak.

Shelley described a simple way to create a realization of a Poisson model from scratch, which he did often as part of his work for the new study: Take a piece of paper and draw random lines across it to create disjointed regions separated by the lines as borders, then color those regions arbitrarily to get a mosaic.

The new research proceeds from that setup by then metaphorically placing a piece of paper over the colorful mosaic. Poking a single hole in that top paper reveals a certain color of the mosaic beneath. That information, in turn, can be mathematically leveraged through multipoint correlations to predict the mosaic pattern with increasing accuracy, based on knowing some context of the mosaic and poking more holes, what colors subsequent holes would likely reveal – as one would for a heterogenous material. “It’s like we’ve created the perfect Battleship player for guessing colors in this model,” Shelley said.

In real life, predicting where certain colors will appear equates to credibly knowing where components are in a heterogenous material’s microstructure. “If you can predict that microstructure and know where stuff is located microscopically, you can intentionally control macroscopic properties related to it,” said Shelley. “That’s what this paper contributes.”

To arrive at the mathematical solution for the Poisson model’s multipoint correlations, Shelley drew upon tools in the field of stochastic geometry, which concerns random point patterns. Initially, Shelley relied on just pen and paper, sketching points, lines, and formulas in a notebook with a four-color pen. To evaluate his solution for two points that have known colors, he added eight different numbers and variables by hand. For three points, though, the number-crunching extended to 128 different terms, and for four points, he turned to computer simulations, lest he spend weeks or months on end doing manual calculations.

According to Shelley, the seemingly painstaking work was anything but. “I love math, and I was a math double major in undergrad, so I had the knowledge to go in and try this problem out,” he said.

Shelley is a doctoral student in the School of Humanities and Sciences. The research was supported by an Oak Ridge Institute for Science and Education Fellowship and Sandia National Laboratories.