New calculation theory established for concentration limits for repulsive elements — Expectations for new guidelines in alloy material development

When creating an alloy, the placement of additive elements determines properties such as strength and corrosion resistance. However, when repulsive forces act between additive elements, they form a complex pattern where they distribute uniformly while avoiding adjacency. If the concentration is increased to the point where additive elements are forced to be adjacent, mechanical properties such as strength and toughness can change significantly.

Research Scientist Atsushi Kubo and Manager Yosuke Abe of the Nuclear Science and Engineering Center at the Nuclear Science Research Institute of the Japan Atomic Energy Agency (JAEA) have established a new theory to calculate the upper concentration limit (saturation concentration) of repulsive additive elements by numerical simulation and mathematical modeling. This can be applied to various alloys as a general theory. Representing the extent to which randomly placed atoms can avoid adjacency through a rigorous, yet simple theoretical formula is an achievement of great academic significance.

Kubo says, "We believe that it is useful because it provides a theoretical approach to items that are quite difficult to analyze, such as high-entropy alloys (alloys that are mixtures of five or more metallic elements in approximately equal proportions)." The results were published in Scientific Reports.

Accurately evaluating saturation concentration using conventional experiments or material simulations is extremely difficult and previously required individual examination for each material. Consequently, comprehensive research on saturation concentration had not been conducted, and the question of "what determines saturation concentration?" remained unresolved.

The research group focused on the mathematical characteristics of crystal lattices, such as how atoms "connect" and their "shape," and constructed a theory to estimate saturation concentration based on the geometry of the crystals.

First, they established a methodology for probability simulation to place additive elements randomly under the condition that they do not sit adjacent to each other, and they conducted simulations for various crystal structures. In addition to typical crystal structures such as body-centered cubic, face-centered cubic, and diamond structures in each crystal structure, virtual crystal structures such as one-dimensional lattices and higher-dimensional hypercubic lattices were also targeted, and saturation concentrations were evaluated numerically.

As a result, they clarified a correlation between the coordination number of the crystal structure and the saturation concentration. Specifically, an inverse correlation where the larger the coordination number, the smaller the saturation concentration.

In order to understand the nature of the inverse correlation between saturation concentration and coordination number obtained from simulations and to establish it as a general-purpose theory, a mathematical study was conducted using a graphical model.

First, they replaced the crystal lattice with a random regular graph, formulated it as a differential equation, and solved this equation to derive a rigorous theoretical solution for the saturation concentration. As a result, the correlation between saturation concentration and coordination number in the graphical model was successfully obtained as an equation.

"Since it is very difficult to solve equations for every single crystal lattice structure, we used graph theory to understand the properties that various crystal structures would approximately satisfy through their average behavior," said Kubo.

Next, the theoretical solution of the obtained saturation concentration was numerically verified to be correct by performing a non-adjacent random placement simulation on the graph model.

The obtained theoretical solution explains very well the inverse correlation between saturation concentration and coordination number. However, there was a discrepancy between the theoretical solution for the random regular graph and the saturation concentration obtained from simulations for the crystal lattice model.

When the crystal lattice contains three-membered rings, the saturation concentration deviates downward from the theoretical formula. When the lattice contains four-membered rings and no three-membered rings, it deviates upward. These analyses reveal that the saturation concentration of the crystal lattice is governed by two types of crystal geometrical parameters: the coordination number of the lattice and the ring size.

When they actually estimated how much aluminum (Al) could be added to an Fe-Cr-Al alloy to prevent it from becoming brittle, they obtained a calculation result of 13%, which was found to be in good agreement with experimental results from previous research.

A theoretical understanding of the saturation concentrations of additive elements that repel each other in alloys has been obtained. Some of the factors that determine the short-range ordering of additive elements have been clarified. The research results are expected to be used as a new guideline for the development of alloy materials.

Journal Information
Publication: Scientific Reports
Title: Graph-theoretic analyses of saturation fraction of repulsive dopants in solid solutions
DOI: 10.1038/s41598-025-30829-1