Fast anharmonic free energy method with an application to vacancies in ZrC

Abstract

We propose an approach to calculate the anharmonic part of the volumetric-strain and temperature dependent free energy of a crystal. The method strikes an effective balance between accuracy and computational efficiency, showing a ×10 speed-up on comparable free energy approaches at the level of density functional theory, with average errors less than 1 meV/atom. As a demonstration we make new predictions on the thermodynamics of substoichiometric ZrCx, including vacancy concentration and heat capacity.

On the stoichiometry of zirconium carbide

“…The very long heat treatments in graphite or vacuum at high temperatures conducted by Sara 13 and Storms & Wagner 14 may have acted in the same way as the inert gas fusion technique: temperature and time allowed conversion of most of the oxygen in contact with free carbon or graphite to escape the system as CO/CO 2 leading to oxygen free ZrC x samples. These authors 13,14 both found a relationship between the lattice parameter and ratio C/Zr that exhibited a maximum at ~ 0.83 and this was recently confirmed by Mellan et al 41 . Mellan et al 41 reported the lattice parameter of ZrC x to decrease from x = 0.97 to x = 1 using computational calculations, as the volume of vacant carbon site (in sub-stoichiometric ZrC x compounds) is larger than the corresponding volume of the perfect crystal.…”

“…These authors 13,14 both found a relationship between the lattice parameter and ratio C/Zr that exhibited a maximum at ~ 0.83 and this was recently confirmed by Mellan et al 41 . Mellan et al 41 reported the lattice parameter of ZrC x to decrease from x = 0.97 to x = 1 using computational calculations, as the volume of vacant carbon site (in sub-stoichiometric ZrC x compounds) is larger than the corresponding volume of the perfect crystal. This means that the lattice parameter should expand, or increase, for x < 1.…”

Stability and structural properties of vacancy-ordered and -disordered ZrCx

“…where x Zr and x C are the atomic fractions of Zr and C respectively, and E hcp Zr and E graphite C are the enthalpies of pure Zr and C in their ground state structures. As calculations of graphite C lack accuracy using conventional pseudopotentials due to the interlayer van der Waals forces, 35 the energy was obtained by calculating the energy of diamond C and adjusting by the experimentally determined energy difference of 27 meV per atom, 36 as done by Mellan et al 37 The mixing enthalpy (energy) with respect to the fully occupied and empty carbon sublattice (fcc Zr and stoichiometric ZrC) at 0 K is dened as…”

Erosion of refractory carbides in high‐temperature hydrogen from ab initio computations

“…Of note, the simulation values are obtained from structures that are ideal single crystals, whereas experimental values will include effects of polycrystal grain boundaries and other defects, which may account for some overprediction in the simulations, especially at high temperatures. Additionally, some overprediction may result from the generalized gradient approximation (GGA) used in the AIMD simulations here, which differ from previous DFT studies considering ZrC thermal expansion using the local‐density approximation (LDA) 59,60 . Although LDA optimizes for structure, it was preferable to use GGA here, which optimizes for energy, as accurate relative energies were needed to estimate erosion, the main focus here.…”

“…For the materials of interest here, there is precedent for the computation of thermodynamic properties 53–58 . In particular, a number of density functional theory (DFT) studies accurately computed high‐temperature free energies of ZrC and ZrC 1‐ x , with special regard being given to the anharmonic components of free energy that are difficult to measure experimentally 59 . Such studies have provided high quality, experimentally comparable values of heat capacity and thermal expansion 60 .…”