Thermal Equilibrium Vacancy Concentration in an Alloy with Chemical Short-Range Order

TL;DR

The paper addresses the challenge of predicting the equilibrium vacancy concentration in multicomponent alloys where chemical short-range order (CSRO) alters local vacancy energetics. It derives an exact, atomistically implementable expression for $X_{ m V}^{\rm eq}(T)$ based on a semiclassical partition function and a grand-canonical Monte Carlo thought experiment, yielding $X_{ m V}^{\rm eq}(T) \approx \left\langle \sum_{i=1}^{\tilde{n}} e^{-\Delta f({\rm site}\; i)/k_{\rm B}T}/\tilde{n} \right\rangle$ with site-dependent $\Delta f$. Applied to equiatomic CrCoNi in the FCC structure, it combines Monte Carlo sampling of CSRO-embedded configurations with DFT-based evaluation of vacancy formation energies, producing vacancy concentrations across $T=300{-}900$ K and a polynomial fit for $\log X_{ m V}^{\rm eq}(T)$. The results validate the approach and enable prediction of diffusion-related relaxation and CSRO formation timescales in compositionally complex alloys, with open-source code provided for broader use.

Abstract

The equilibrium vacancy concentration in multi-principal element alloys remains a controversial and nontrivial subject, primarily because of chemical complexity and chemical short-range order (CSRO). Here we derive an exact expression that is amenable to atomistic calculations, using multiple perspectives. We applied this expression to equiatomic CrCoNi alloys in the face-centered cubic structure. The derived equilibrium vacancy concentration is used in our recent work, which predicts the chemical short-range order formation timescale consistent with experimental observation. The results demonstrate the practical utility of the approach for predicting equilibrium vacancy concentrations in compositionally complex alloys.

Figures (7)

• Figure 1: Monte Carlo sampling of low-energy configurations of CrCoNi. Energy minimization of CrCoNi at various temperature with a simulated annealing schedule, defined as $T = 1200 - \frac{1200 - T_{\rm anneal}}{4500} \times {\rm step}$ over 4500 simulation steps, followed by equilibration at a constant annealing temperature $T_{\rm anneal}$ for an additional 3500 steps. The dashed lines denote the energies obtained by averaging the last 1000 frames of each simulation.
• Figure 2: Distribution of vacancy formation energy ($E_{\rm V}$) with different temperatures. The solid lines represent Gaussian fits to the distributions, while the dashed lines indicate the corresponding mean values.
• Figure 3: Change in equilibrium vacancy concentration $X_{\rm V}^{\rm eq}(T)$ with temperature.
• Figure S1: Distribution of WC parameters $\alpha_{ij}^1$ for $\rm Cr_{85}Co_{85}Ni_{86}$ at equilibrium across different temperatures.
• Figure S2: Distribution of WC parameters $\alpha_{ij}^1$ for $\rm Cr_{85}Co_{86}Ni_{85}$ at equilibrium across different temperatures.