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Sluggish diffusion is postulated as an underlying mechanism for many unique properties in high-entropy alloys (HEAs). However, its existence remains a subject of debate. Due to the challenges of exploring the vast composition space, to date most experimental and computational diffusion studies have been limited to equiatomic HEA compositions. To develop a high-throughput approach to study sluggish diffusion in a wide range of non-equiatomic compositions, this work presents an innovative artificial neural network (ANN) based machine learning model that can predict the vacancy migration barriers for arbitrary local atomic configurations in a model FeNiCrCoCu HEA system. Remarkably, the model utilizes the training data exclusively from the equiatomic HEA while it can accurately predict barriers in non-equiatomic HEAs as well as in the quaternary, ternary, and binary sub-systems. The ANN model is implemented as an on-the-fly barrier calculator for kinetic Monte Carlo (KMC) simulations, achieving diffusivities nearly identical to the independent molecular dynamics (MD) simulations but with far higher efficiency. The high-throughput ANN-KMC method is then used to study the diffusion behavior in 1,500 non-equiatomic HEA compositions. It is found that although the sluggish diffusion is not evident in the equiatomic HEA, it does exist in many non-equiatomic compositions. The compositions, complex potential energy landscapes (PEL), and percolation effect of the fastest diffuser (Cu) in these sluggish compositions are analyzed, which could provide valuable insights for the experimental HEA designs.