A Novel AI Based Method for Calculating Maximum Solid State Solubility: A Case Study of the Al–Zr System

In the 1920s, Hume Rothery advanced metallurgy toward a predictive science by proposing a set of rules for estimating solubility in alloy systems. Because of their simplicity and broad applicability, these rules became fundamental principles in materials science. In the years following their introduction, many studies sought to refine the Hume Rothery rules in order to extend their validity to a wider variety of alloy systems. In this study, different amounts of zirconium were added to aluminum, and X‑ray diffraction patterns were recorded from the produced samples at various milling times. Variations in the diffraction peak positions were used to calculate the lattice parameter as a function of zirconium concentration and milling time. Artificial intelligence techniques were then applied to obtain a mathematical relationship describing the dependence of the lattice parameter on zirconium percentage and milling duration. This relationship was further optimized using a bee colony algorithm to identify the zirconium contents and milling times corresponding to the maximum lattice parameter. These optimal values were documented, and a best‑fit linear equation was derived. The resulting equation establishes a distinctive relationship that can predict the milling time required to reach the maximum lattice parameter for any given zirconium concentration.