We propose a fully ab initio based integrated approach to determine the volume and temperature dependent free-energy surface of nonmagnetic crystalline solids up to the melting point The approach is based on density-functional theory calculations with a controlled numerical accuracy of better than 1 meV/atom It accounts for all relevant excitation mechanisms entering the free energy including electronic, quasiharmonic, anharmonic, and structural excitations such as vacancies To achieve the desired accuracy of $l1\text{ }\text{meV}/\text{atom}$ for the anharmonic free-energy contribution without losing the ability to perform these calculations on standard present-day computer platforms, we develop a numerically highly efficient technique: we propose a hierarchical scheme---called upsampled thermodynamic integration using Langevin dynamics---which allows for a significant reduction in the number of computationally expensive ab initio configurations compared to a standard molecular dynamics scheme As for the vacancy contribution, concentration-dependent pressure effects had to be included to achieve the desired accuracy Applying the integrated approach gives us direct access to the free-energy surface $F(V,T)$ for aluminum and derived quantities such as the thermal expansion coefficient or the isobaric heat capacity and allows a direct comparison with experiment A detailed analysis enables us to tackle the long-standing debate over which excitation mechanism (anharmonicity vs vacancies) is dominant close to the melting point

https://physics.aps.org/featured-article-pdf/10.1103/PhysRevB.79.134106

Ab initio up to the melting point: Anharmonicity and vacancies in aluminum

Vacancy concentrations in metals

The molar volume and thermal expansion of selected metallic elements with the bcc, fcc and hcp structures were studied by means of the Calphad approach. Experimental data were critically assessed, and model parameters were obtained yielding reasonable descriptions of all experimental data on molar volume and thermal expansion. The descriptions are valid between room temperature and the melting points at atmospheric pressure.

Assessments of molar volume and thermal expansion for selected bcc, fcc and hcp metallic elements

Two models have been proposed for the interconnection of the defect Gibbs energy gi with bulk properties almost 60 and 30years ago, respectively. The one, proposed by Zener [J. Appl. Phys. 22, 372 (1951)], assumes that gi can account for the work that goes into straining the lattice and hence that it is proportional to the shear modulus of the solid. The other considers that since gi corresponds to an isobaric and isothermal process, it should be proportional to the isothermal bulk modulus and the mean volume per atom. The results of these two models are compared for different processes (defect formation, self-diffusion activation, and heterodiffusion) in a variety of solids including metals (fcc, bcc, and tetragonal) as well as solids that exhibit a superionic behavior. We find that the latter model does better than the former.

Comparison of models that interconnect point defect parameters in solids with bulk properties

Point defects can significantly alter the behavior of solid-state materials, but a theoretically and experimentally consistent understanding of their formation energy has been lacking so far. Taking into account anharmonic lattice vibrations, a new state-of-the-art theoretical effort makes a very significant advance toward filling that gap and demonstrates a critical need to revise the official international point-defect database.

https://link.aps.org/pdf/10.1103/PhysRevX.4.011018

Breakdown of the Arrhenius law in describing vacancy formation energies: The importance of local anharmonicity revealed by Ab initio thermodynamics

Thermal expansion on high purity aluminum has been measured to study defects in thermal equilibrium by the (ΔL/L
 0−Δa/a
 0) method. Measurements of the changes in macroscopic length ΔL and lattice parameter Δa were made from room temperature to 637° C. Length changes were measured by a laser interferometer and lattice parameter changes by a X-ray diffractometer technique on the same crystal at identical temperatures. At temperatures above 400° C ΔL/L
 0 becomes greater than Δa/a
 0, indicating the generation of a noticeable amount of new lattice sites due to vacancy formation. Extrapolation gives a vacancy concentration ΔN/N
 0=9.8·10−4 at the melting point (660° C). The experimental findings can be explained by assuming formation of monovacancies and divacancies. Values for the enthalpy and entropy of formation for mono- and divacancies are given.

R. Zitzmann

Papers

Equilibrium vacancy concentration measurements on aluminum