Machine Learning and Kinetic Monte Carlo Models of Transport in Concentrated Alloys

With the increasing interest in so-called high entropy, multi-principal component, or compositionally complex alloys, there is a greater need to understand how transport is affected by the complex chemistry of these materials. As compared to a simple elemental solid, in which every lattice site is identical and transport can be described by a handful of saddle points, these materials exhibit a very rugged potential energy surface in which every site has a unique chemical environment. This means that every site exhibits a different defect formation energy and corresponding migration barriers for motion to neighboring sites. As the number of elements in these systems increases, it becomes increasingly challenging, and soon impossible, to enumerate the energetics of every site in the system. Alternative approaches are necessary.<br/><br/>Machine learning has become a popular choice to describe the chemistry-dependent properties of defects in these alloys. By training on atomistic data, machine learning models can predict the energetics of defects in unseen environments and be subsequently used in kinetic Monte Carlo simulations of defect transport. However, for the transport simulations to be thermodynamically valid, the underlying machine learning model must obey detailed balance.<br/><br/>Here, we describe various ways that detailed balance can be enforced in a machine learning model of defect formation and migration energies. Specifically, we consider a brute force model, a “soft constraint” model in which penalty associated with detailed balance is added to the loss function, and two “hard constraint” models in which detailed balance is explicitly included in the architecture of the model. We find that these physically-constrained models exhibit superior performance in maintaining detailed balance. This is demonstrated by determining the error in the energy of the system as defects traverse closed loops. We conclude that detailed balance must be considered to obtain valid trajectories.<br/><br/>We then use the machine learning model to determine the kinetic properties of defects. Using a recently-developed approach in which the complex correlated+uncorrelated kinetic Monte Carlo problem is mapped rigorously onto an uncorrelated surrogate, we then use this to identify which atomic scale events are most critical in describing the trajectory of the defect. This provides a route to close the loop, so to speak: to build an autonomous workflow in which the events that the trajectory is most sensitive to are further refined to improve the description of the material. We demonstrate the elements of this workflow for a vacancy in the simple Cu-Ni alloy, highlighting how we can then quickly determine the diffusion tensor for defects in complex alloys when given appropriate computational resources.