Sphere-based representation of image-resolved porous media

Materials like the foam inside a coffee cup, the spongy rock beneath our feet, or the ceramic filters in a car engine share a hidden complexity: they are full of holes. Scientists call these porous media, and understanding how fluids move through them, how heat travels, or how chemicals react within their tiny tunnels is crucial for everything from cleaning oil spills to designing better batteries. To study these processes, researchers often turn to powerful 3D images taken by medical scanners or specialized microscopes, which reveal the intricate maze of solid walls and empty spaces inside a material. However, these images are massive digital files made of billions of tiny blocks, and trying to run physics simulations directly on such a massive grid is often too slow and computationally expensive. For decades, scientists have tried to simplify these images into smaller, easier-to-use models, but many of these simplifications lose the actual shape of the material, turning a complex, winding tunnel system into a simple list of disconnected dots and lines.

A team of researchers at Friedrich-Alexander-Universität Erlangen-Nürnberg has found a way to bridge this gap, creating a new method that keeps the true shape of the material while making it simple enough for computers to handle quickly. Instead of breaking the image down into a network of abstract connections, they used an algorithm to fill the empty spaces with a collection of overlapping spheres. Imagine taking a complex, irregular cave system and filling it with thousands of soft, rubber balls that press against one another; the balls don't need to fit perfectly into every nook, but together they trace the outline of the cave with surprising accuracy. This approach, which the researchers call the Multi-Sphere Shape algorithm, allows them to represent the entire internal structure of a material using a manageable number of these overlapping spheres. The result is a model that is light enough to run in simulations but detailed enough to tell scientists exactly how big the holes are, how wide the narrow passages between them are, and how many paths connect one area to another.

The researchers tested this method on a piece of ceramic foam, a material often used in high-temperature filters and catalysts. They started with a 3D image of the foam taken by a computed tomography scanner, which captured a tiny cube of the material measuring 3.14 millimeters on each side. This original image was incredibly detailed, containing 64 million tiny picture elements, or voxels, that defined every solid wall and empty space. Feeding this massive dataset into their new algorithm, the team generated a representation of the foam's internal pore space using just 22,468 overlapping spheres. Despite using fewer than 23,000 spheres to describe a structure made of 64 million blocks, the new model captured 95 percent of the actual empty volume. This means that for nearly every part of the original foam's internal tunnels, there was a corresponding sphere in the new model, preserving the overall shape and the principal connections that allow air or liquid to flow through.

What makes this discovery particularly useful is that the model does more than just look like the original material; it provides immediate answers to questions that usually require complex calculations. Because the spheres overlap, the points where they touch or intersect reveal the size of the narrowest passages, known as throats, and the size of the larger open areas, or pores. By counting how many spheres touch a single sphere, the researchers could determine the "coordination number," which tells them how many different paths branch out from any given point in the foam. In materials where the structure is dominated by these round, bubble-like shapes, these measurements translate directly into physical properties: the size of a sphere becomes the size of a pore, and the area where two spheres overlap becomes the size of the opening connecting them. This allows scientists to instantly see the distribution of hole sizes and connection widths without having to manually trace the entire 3D image.

The researchers demonstrated that this sphere-based representation is not just a static picture but a functional tool for future simulations. Because the model is built from spheres, it can be used directly in particle-based simulations, which are a common way to study how fluids or particles move through complex environments. Unlike older methods that reduced the material to a simplified map of nodes and links, losing the actual spatial geometry, this new approach retains the physical boundaries of the material. This means researchers can simulate how a fluid flows around the actual shapes of the pores rather than just moving between abstract points. Furthermore, the data extracted from these spheres—such as the sizes of the pores and the number of connections—can be used to generate new, parameterized models of porous materials. This creates a powerful loop where a real-world image is converted into a simple, data-rich model, which can then be used to design new materials with specific properties, all while keeping the connection to the original, real-world structure intact.