First-principles calculations of the thermodynamic factor of multicomponent alloys

Abstract

Computing non-dilute diffusion coefficients from Onsager transport coefficients of a multicomponent alloy requires the thermodynamic factor, which measures the curvature of the free energy with respect to composition. As diffusion in metals is mediated by vacancies, the curvature that matters is not that of the vacancy-free alloy, which is straightforward to compute, but that of an alloy carrying a dilute concentration of vacancies. The resulting matrix is nearly singular, and composition fluctuations from which it is obtained converge slowly in Monte Carlo simulations. Here we develop two routes that avoid sampling vacancy composition fluctuations. The first constructs the thermodynamic factor from two quantities, the free-energy curvature of the vacancy-free alloy and the vacancy concentration. The second truncates the semi-grand canonical partition function at a single vacancy and recovers the full thermodynamic factor of the vacancy-containing alloy. The same separation shows that the largest eigenvalue of the diffusion matrix is the vacancy tracer diffusion coefficient for any number of components and any degree of non-ideality, while the remaining eigenvalues scale with the vacancy concentration. Across binary through quinary alloys of the Hf--Mo--Nb--Ti--Zr system, both routes reproduce eigenvalues and eigenvectors of diffusion matrices obtained from fully converged Monte Carlo simulations. The single-vacancy expansion reaches this accuracy with roughly one tenth of the sampling effort. A scaling relation extends both routes to local vacancy chemical potentials away from equilibrium, without additional simulations. These results allow thermodynamic descriptions from atomistic models or CALPHAD assessments to be used directly in mesoscale simulations of mass transport.