Revealing Nanostructures in High-Entropy Alloys via Machine-Learning Accelerated Scalable Monte Carlo Simulation

The computational cost of traditional first-principles method quickly becomes prohibitively expensive as the number of atoms increases. This challenge is further amplified by the need to evaluate finite-temperature properties with Monte Carlo (MC) simulations, which is inherently challenging to parallelize due to sequential Markov chain updates. Here, we introduce Scalable Monte Carlo (SMC), an efficient MC simulation method that overcomes the parallelization bottlenecks in conventional MC simulation, reducing the computational complexity of a MC sweep from quadratic to linear. We present a GPU implementation of the SMC method, SMC-GPU, which simultaneously harnesses the thousands of processing cores on a GPU to accelerate the computation. By adopting a data-driven workflow that surrogates the computationally expensive density functional theory (DFT) with ML models, we demonstrate that SMC-GPU is capable of simulating systems of more than one-billion atoms, while maintaining the accuracy of first-principles methods. Using this unprecedented capability, we performed billion-atom MC simulations to investigate the nanostructure evolution of two important high-entropy alloys (HEAs), FeCoNiAlTi and MoNbTaW, in which the nanostructures are believed to be responsible for their superb mechanical properties. Our results reveal a rich diversity of nanostructures, including nanoparticles (NP), 3D-connected NP, and disorder protected nanophases. We quantitatively analyze the size, composition, and morphology of the nanostructures, as well as directly simulate the atom-probe-tomography (APT) needle. The results align well with available experimental observations. This work underscores the promising potential of leveraging large-scale MC simulation to explore the largely uncharted territory of nanostructure evolution in HEAs.