First-principles calculation of phase equilibrium of V-Nb, V-Ta, and Nb-Ta alloys

Abstract

In this paper, we report the calculated phase diagrams of V-Nb, V-Ta, and Nb-Ta alloys computed by combining the total energies of 40-50 configurations for each system (obtained using density functional theory) with the cluster expansion and Monte Carlo techniques. For V-Nb alloys, the phase diagram computed with conventional cluster expansion shows a miscibility gap with consolute temperature T c = 1250 K. Including the constituent strain to the cluster expansion Hamiltonian does not alter the consolute temperature significantly, although it appears to influence the solubility of V-and Nb-rich alloys. The phonon contribution to the free energy lowers T c to 950 K (about 25%). Our calculations thus predicts an appreciable miscibility gap for V-Nb alloys. For bcc V-Ta alloy, this calculation predicts a miscibility gap with T c = 1100 K. For this alloy, both the constituent strain and phonon contributions are found to be significant. The constituent strain increases the miscibility gap while the phonon entropy counteracts the effect of the constituent strain. In V-Ta alloys, an ordering transition occurs at 1583 K from bcc solid solution phase to the V 2 Ta Laves phase due to the dominant chemical interaction associated with the relatively large electronegativity difference. Since the current cluster expansion ignores the V 2 Ta phase, the associated chemical interaction appears to manifest in making the solid solution phase remain stable down to 1100 K. For the size-matched Nb-Ta alloys, our calculation predicts complete miscibility in agreement with experiment.

Non-empirical phase equilibria in the Cr–Mo system: A combination of first-principles calculations, cluster expansion and Monte Carlo simulations

“…about one-third suppression when compared to the non-vibrational entropy calculations. Similar results can be found in many studies where the contributions of the vibrational entropy to the free energies lead to a percentage reduction in the critical temperature ranging from 5% to 45% [1,16,18,25,26,43].…”

“…The determination of the energy for such a random alloy system requires an evaluation of the energies of all possible configurations that are created by randomly placing Cr atoms and Mo atoms on N lattice sites of the underlying lattice (bcc) [1,36]. To get the desired bulk compositions and to avoid statistical errors, the configuration is normally chosen to be large enough which, however, results in a large number of possible configurations.…”

“…The study of a phase diagram would result in a full understanding of the governing conditions for phase stability and microstructure evolution that would aid in the material design [1]. Of particular interest is the solid part of the phase diagram in which the miscibility gap could occur.…”

Relative entropy as model selection tool in cluster expansions

“…5,6 Given the ECI, the energy of any configuration can be computed and subsequently used for thermodynamic simulations. [7][8][9][10][11][12][13] Let us provide an intuitive argument as to why fitting the ECI to equally weighted observed energies does not necessarily lead to an optimal description of the thermodynamics of the system. Consider the case of a canonical ensemble.…”

Machine-learning the configurational energy of multicomponent crystalline solids

“…In this way, complex energy landscapes as a function of configurational degrees of freedom can be reproduced by fitting to a relatively small number of first-principles electronic structure calculations. The approach has enabled accurate first-principles predictions of temperature-composition phase diagrams, [23][24][25][26][27][28][29][30][31][32][33][34] order-disorder phenomena, [35][36][37][38][39][40][41][42][43][44][45] and composition-dependent diffusion coefficients in alloys and complex inorganic compounds. [46][47][48][49][50][51][52] A cluster expansion is formulated as a linear series of cluster basis functions multiplied by constant expansion coefficients that are determined by the underlying chemistry and crystal structure of the multicomponent solid.…”