Diffusion coefficients of multi-principal element alloys from first principles

Abstract

Vacancy-mediated diffusion in multi-principal element alloys (MPEAs) remains poorly understood. Existing computational methods face challenges in connecting electronic structure to macroscopic transport coefficients due to the large number of chemical elements. To address this, we introduce the embedded local cluster expansion (eLCE), which bridges first-principles calculations with kinetic Monte Carlo simulations to compute the matrix of multicomponent diffusion coefficients. Applying this approach to refractory MPEAs in the V-Cr-Nb-Mo-Ta-W system, we evaluate the complete mobility and diffusion tensors of a six-component alloy at finite temperatures. We find that local kinetic barriers, rather than thermodynamics or vacancy correlation factors, primarily control diffusion in these materials. Whether diffusion is sluggish or anti-sluggish depends on the mean vacancy migration barrier relative to the rule-of-mixtures estimate and on the availability of percolating pathways of fast-diffusing species. We use this insight to screen the senary composition space and identify compositions with anti-sluggish diffusion. This study presents a predictive, first-principles approach for computing non-dilute transport coefficients and designing MPEAs with targeted transport properties.

Figures (6)

• Figure 1: a) Energy landscape for a vacancy hop between initial and final states. The kinetically resolved activation (KRA) barrier, $\Delta E_{KRA}$, isolates the local contribution to the migration energy by subtracting the average end-state energy from the transition-state energy. The red circle denotes the hopping atom and the dashed circle denotes the vacancy. b) Local cluster expansion on a square lattice. The hop cluster $H$ (red) defines the migrating atom--vacancy pair. Symmetrically equivalent clusters in the surrounding environment are colored identically. c) Computational workflow. DFT calculations provide KRA training data for the eLCE parametrization, which enables rapid evaluation of migration barriers across arbitrary local environments. The parameterized eLCE model is then used in kMC simulations to compute transport coefficients ($\mathbf{\tilde{L}}$, $f_i$, $D^*_i$).
• Figure 2: a) Test mean absolute error (MAE) of the eLCE models as a function of the number of clusters, grouped by cluster type (points, pairs, triplets). Dashed vertical lines separate the cluster types. b) Test MAE as a function of the number of training data points. In both panels, each curve corresponds to a different embedding dimension for the local environment ($m = 2$--6), with the hop cluster embedding fixed at $n = 2$. Error bars indicate the standard deviation over five independent random initializations of the training set and neural network weights.
• Figure 3: a) Mean squared displacement of each element in equiatomic VCrNbMoTaW as a function of kMC simulation time, showing that group 5 elements (V, Nb, Ta) diffuse faster than group 6 elements (Cr, Mo, W). b) Onsager transport coefficient matrix $\mathbf{\tilde{L}}$ in the equiatomic VCrNbMoTaW alloy. Both panels are computed at 2000 K.
• Figure 4: a) Eigenvalues of the diffusion coefficient matrix $\mathbf{D}$ in equiatomic VCrNbMoTaW as a function of inverse temperature. The largest eigenvalue $\lambda_1$ corresponds to the vacancy tracer diffusion coefficient and is separated from the remaining five eigenvalues ($\lambda_{2}$--$\lambda_{6}$) by several orders of magnitude. b) Tracer diffusion coefficients $D^*_i$ of each element and the vacancy as a function of inverse temperature. The vacancy tracer diffusivity matches $\lambda_1$, while the elemental tracer diffusivities are of the same order as $\lambda_{2}$--$\lambda_{6}$.
• Figure 5: a) Schematic of four prototypical alloys used to disentangle thermodynamic and kinetic contributions to diffusion. Each panel shows the energy landscape for two hopping species, with shaded bands indicating environmental variability. Below each KRA energy landscape is the resulting distribution of KRA migration barriers. In the KRA-simple models, barriers are fixed at the pure-element values, while in the KRA-complex models they vary with local chemical environment. b) Enhancement factor $\eta_{Va}$ and c) vacancy correlation factor $f_{Va}$ as a function of Nb composition in Nb$_x$Mo$_{1-x}$ at 1500 K for all four model alloys. The dashed line in b) marks $\eta_{Va} = 1$, separating sluggish ($\eta_{Va} < 1$) from anti-sluggish ($\eta_{Va} > 1$) behavior. d) Distribution of energy barriers encountered (unfilled) and accepted (filled) by the vacancy during kMC simulations at the equiatomic composition. Vertical solid lines mark the pure-element KRA values for Nb and Mo, and the dashed line indicates the rule-of-mixtures (ROM) estimate.