Scientists are working to better understand how components in mixed materials like concrete or underground rock are distributed, which could lead to stronger materials and safer storage of substances like carbon dioxide or nuclear waste. A key tool in this effort is the Poisson model, which randomly divides space using flat surfaces to simulate how materials mix. Although useful in fields like radiation transport, the model lacked precise formulas to describe relationships between multiple points—limiting its accuracy and application in complex systems until recent breakthroughs addressed this gap.
In a new study, Stanford researchers introduced a clever math method that helps reveal what a material is made of, just by knowing details from one random spot. They used a well-known statistical model, called the Poisson model, which randomly divides space, to study materials such as sand and concrete. This new approach enables scientists to understand the tiny structure of these materials with very high accuracy, which could aid in designing stronger and more reliable materials.
Lead study author Alec Shelley said, “With this study, we’ve solved the famous Poisson model for heterogenous materials.”
“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.”
The tiny structure inside materials affects how strong, durable, and useful they are. Thanks to the new research, scientists can now understand these structures more precisely.
“What Alec has succeeded in doing in this study is quite remarkable,” said Daniel 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.”
Shelley and Tartakovsky plan to use their new math method to predict what different materials are made of. Their model can reveal a long list of important properties that depend on a material’s tiny inner structure, such as hardness and elasticity, tensile strength (how much it can stretch before breaking), electrical and heat conductivity, how fast one substance moves through another, magnetic behavior, and how much light passes through.
Concrete has tiny air pockets inside it. If engineers can accurately model these spaces, they could use materials like fly ash, slag, or biochar to fill them in. This would reduce the amount of cement needed, helping to lower carbon dioxide emissions from cement production, while also making the concrete stronger and more affordable.
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.”
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 shared a neat way to understand the Poisson model. Imagine taking a blank piece of paper and randomly drawing lines across it to create different sections. Then, color each section however you want, it’s like creating a colorful mosaic! The new research takes this idea a step further by picturing another piece of paper laid over that mosaic.
If you poke a hole in the top sheet, you see one color below. That small peek gives a clue about the whole pattern. By making more holes and using a math method called multipoint correlations, you can predict the full design more accurately each time. This approach mirrors how scientists study heterogeneous materials, such as concrete or rock, by using small samples to understand the bigger picture.
“It’s like we’ve created the perfect Battleship player for guessing colors in this model,” Shelley said.
To handle the intricate math involved in the Poisson model’s multipoint correlations, Shelley took a hands-on approach. He began by sketching ideas in a notebook to help picture the problem. Figuring out two points was simple, but things got complicated fast, by the time he worked on three points, he had to deal with 128 different terms.
By the time he reached the four-point scenarios, the complexity was overwhelming, pushing him to turn to computer simulations. It was a necessary shift that saved him from spending months buried in calculations. on manual work.
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.
For more information: Physical Review Letters
