Contributions to Diffusion in Complex Materials Quantified with Machine Learning

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Contributions to Diffusion in Complex Materials Quantified with Machine Learning

Soham Chattopadhyay and Dallas R. Trinkle

Phys. Rev. Lett. 132, 186301 – Published 30 April 2024

Abstract

Using machine learning with a variational formula for diffusivity, we recast diffusion as a sum of individual contributions to diffusion—called “kinosons”—and compute their statistical distribution to model a complex multicomponent alloy. Calculating kinosons is orders of magnitude more efficient than computing whole trajectories, and it elucidates kinetic mechanisms for diffusion. The density of kinosons with temperature leads to new accurate analytic models for macroscale diffusivity. This combination of machine learning with diffusion theory promises insight into other complex materials.

Received 11 January 2024
Revised 17 March 2024
Accepted 5 April 2024

DOI:https://doi.org/10.1103/PhysRevLett.132.186301

Physics Subject Headings (PhySH)

Diffusion

Machine learning Materials modeling Monte Carlo methods

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Soham Chattopadhyay and Dallas R. Trinkle ^*

Department of Materials Science and Engineering, University of Illinois, Urbana-Champaign, Illinois 61801, USA

^*dtrinkle@illinois.edu

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Issue

Vol. 132, Iss. 18 — 3 May 2024

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Images

Figure 1
Predicting diffusivity in an “ordered” (left) vs a “random” (right) five-component model system. Diffusivity is shown as a fraction of the one-step diffusivity prediction, with smaller values indicating increased trapping. The five atomic components have equal concentration, and they move via exchange with a single vacancy, either with a fixed rate (ordered) or a distributed rate (random) that depends on the chemistry. The fastest species dominates the exchanges with the vacancy, but it has difficulty moving over longer distances, as it needs to connect with other fast species to escape trapping. The kinetic Monte Carlo results converge to the true diffusivity as the number of steps increases (lower diffusivity is more accurate). In the ordered case, a neural network (NN) is most accurate, with only a single step; in the random case, the rate-informed scaled bias basis (SBB) method is the most accurate. A complex system lies in between ordered and random, and we expect to need a combination of methods to predict diffusivity.
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Figure 2
Predicting vacancy (left) and Mn (right) diffusivity in a five-component complex high-entropy alloy. Diffusivity is shown as a fraction of the one-step diffusivity prediction, with smaller values indicating increased trapping. The real alloy has atomic exchanges with a vacancy that depend on the local atomic environment, behaving as a combination of an ordered and a random system. The kinetic Monte Carlo results converge to the true diffusivity as the number of steps increases (lower diffusivity is more accurate). By combining a neural network with a rate-informed scaled residual bias correction ( $NN + SRBC$ , stars), it is possible to predict diffusivity very close to the true value, while only taking a single transition from one state to another. The neural network trained on higher-temperature states can work at lower temperatures in the same phase, with similar accuracy (orange stars). This accuracy allows the computation of the density of kinosons for this complex alloy, and an understanding of the diffusion process.
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Figure 3
Density of kinosons $κ$ (top) and optimized displacements $\tilde{δ x}$ (bottom) for the vacancy in a high-entropy alloy. Without the optimized displacements, the distribution of exchange rates provides an “uncorrelated” estimate of the density of kinosons that contribute to diffusivity; for this alloy, this is a nearly log-normal distribution. Once the displacements are known, the kinoson distribution skews to lower values and follows a log–exponentially modified Gaussian. The optimized displacements show a distinct distribution depending on whether they are with a fast Mn or one of the slower elements; the Mn displacements collapse to a peak at zero, while the other elements are distributed around the jump distance in the face-centered cubic lattice. This change is a signature of percolation-like behavior for Mn.
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Figure 4
The density of kinosons’ mean, standard deviation, and decay parameters with temperature for the complex high-entropy alloy, and extrapolation to lower temperatures. The log–exponentially modified Gaussian form in Fig. 3 is quantified with three parameters, and their temperature dependence is empirically fit. This reveals a new non-Arrhenius analytic form for the vacancy diffusivity [Eq. (5)] that can be extrapolated accurately to lower temperatures.
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Figure 5
Ratio of the diffusivity of Mn to those of Fe and Cr in the cantor alloy. Using the average rates (“uncorrelated”) disagrees with experimental tracer diffusion measurements [75], while the neural network calculations of the optimized kinosons are within experimental uncertainty.
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