Clarifying the Ti-V Phase Diagram Using First-Principles Calculations and Bayesian Learning

TL;DR

This work resolves a long-standing controversy over the Ti–V binary phase diagram by delivering an ab initio–level phase diagram with quantified uncertainties. It combines an actively trained Moment Tensor Potential with a Bayesian framework that fuses MD, phonon, and melting-point data to reconstruct free-energy surfaces and phase boundaries across the full composition range. The resulting Ti–V diagram exhibits a BCC miscibility gap terminating near $T\approx980$ K and $c\approx0.673$, reproduced without including oxygen impurities, thereby challenging impurity-based explanations. The study demonstrates a robust, transferable workflow for uncertainty-aware first-principles phase-diagram construction that can be applied to other alloys and ternaries.

Abstract

Conflicting experiments disagree on whether the titanium-vanadium (Ti-V) binary alloy exhibits a body-centred cubic (BCC) miscibility gap or remains completely soluble. A leading hypothesis attributes the miscibility gap to oxygen contamination during alloy preparation. To resolve this disagreement, we use an ab initio + machine-learning workflow that couples an actively-trained Moment Tensor Potential with Bayesian inference of free energy surface. This workflow enables construction of the Ti-V phase diagram across the full composition range with systematically reduced statistical and finite-size errors. The resulting diagram reproduces all experimental features, demonstrating the robustness of our approach, and clearly favors the variant with a BCC miscibility gap terminating at T = 980 K and c = 0.67. Because our simulations model a perfectly oxygen-free Ti-V system, the observed gap cannot originate from impurity effects, in contrast to recent CALPHAD reassessments.

Figures (10)

• Figure 1: Comparison of two conflicting interpretations of the Ti–V binary phase diagram. Version A predicts a BCC-phase miscibility gap, whereas Version B shows complete solid solubility across the entire composition range. Version A is adapted from Murray (1989) murray1989phase, and Version B is adapted from Lindwall et al. (2018) Lindwall2018.
• Figure 2: Schematic illustration of our algorithm. Thermodynamic data of different natures (MD simulations, melting points, phonon computations) along with the asymptotics of the free energies is fed as input data to the Gaussian process regression. We next predict the free energies with their uncertainties to construct the phase diagram.
• Figure 3: Evaluation of the MTP against DFT on the training dataset. A dashed black line represents a perfect linear dependence.
• Figure 4: Dependence of $\frac{1}{N} \log\det \hat{H}$ (left panel), obtained from phonon computations, and the melting point (right panel) for BCC vanadium on the number of atoms in the simulation cell. Gaussian process regression is used for extrapolation, yielding the following estimates in the thermodynamic limit ($N \rightarrow \infty$): $\frac{1}{N} \log\det \hat{H} = 5.94765 \pm 6\times10^{-5}$ and $T_{\mathrm{m}}(\mathrm{V}) = 2180 \pm 2 K$.
• Figure 5: Dataset used to train the Bayesian algorithm, plotted atop the Ti–V phase diagram to illustrate the phase-stability domains. The dataset comprises grand-canonical MD simulations, phonon calculations at 0 K for the solid phases, and melting points obtained from coexistence simulations. Green markers denote the data used to reconstruct the HCP free-energy surface (MD simulations + phonon calculations), blue markers do the same for the BCC phase, and red markers (MD simulations + melting points) pertain to the liquid phase.