Vacancy Formation Volume Research Articles

Strain effects on oxygen vacancy formation energy in perovskites

Oxygen vacancy formation energy is an important quantity for enabling fast oxygen diffusion and oxygen catalysis in technologies like solid oxide fuel cells. Both previous literature in various systems and our calculations in LaMnO3, La0.75Sr0.25MnO3, LaFeO3, and La0.75Sr0.25FeO3, show mixed results for the direction and magnitude of the change in vacancy formation energy with strain. This paper develops a model to make sense of the different trend shapes in vacancy formation energy versus strain. We model strain effects using a set of consistent ab initio calculations, and demonstrate that our calculated results may be simply explained in terms of vacancy formation volume and changes in elastic constants between the bulk and defected states. A positive vacancy formation volume contributes to decreased vacancy formation energy under tensile strain, and an increase in elastic constants contributes to increases in vacancy formation energy with compressive and tensile strains, and vice versa. The vacancy formation volume dominates the linear portion of the vacancy formation energy strain response, while its curvature is governed by the vacancy-induced change in elastic constants. We show results sensitive to B-site cation, A-site doping, tilt system, and vacancy placement, which contributions may be averaged under thermally averaged conditions. In general, vacancy formation energies for most systems calculated here decreased with tensile strain, with about a 30–100meV/% strain decrease with biaxial strain for those systems which showed a decrease in vacancy formation energy. Experimental verification is necessary to confirm the model outside of calculation.

Properties of vacancy formation in hcpH4ecrystals at zero temperature and fixed pressure

Equation of state of $^{4}\text{H}\text{e}$ hcp crystals with vacancies is determined at zero temperature using the diffusion Monte Carlo technique, an exact ground-state zero-temperature method. This allows us to extract the formation enthalpy and isobaric formation energy of a single vacancy in otherwise perfect helium solid. Results are obtained for pressures up to 160 bar. The isobaric formation energy is found to reach a minimum near 57 bar where it is equal to $10.5\ifmmode\pm\else\textpm\fi{}1.2\text{ }\text{K}$. At the same pressure, the vacancy formation volume exhibits a maximum and reaches the volume of the unit cell. This pressure coincides with the pressure interval over which a peak in the supersolid fraction of $^{4}\text{H}\text{e}$ was observed in a recent experiment.

First-principles study of vacancy-hydrogen interaction in Pd

Hydrogen absorption in face-centered-cubic palladium is studied from first principles, with particular focus on interaction between hydrogen atoms and vacancies, formation of hydrogen-vacancy complexes, and multiple hydrogen occupancy of a Pd vacancy. Vacancy formation energy in the presence of hydrogen, hydrogen trapping energy, and vacancy formation volume have been calculated and compared to existing experimental data. We show that a vacancy and hydrogen atoms form stable complexes. Further we have studied the process of hydrogen diffusion into the Pd vacancy. We find the energetically preferable position for hydrogen to reside in the palladium unit cell in the presence of a vacancy. The possibility of the multiple hydrogen occupancy (up to six hydrogen atoms) of a monovacancy is elucidated. This theoretical finding supports experimental indication of the appearance of superabundant vacancy complexes in palladium in the presence of hydrogen.

Semiempirical atomic potentials for the fcc metals Cu, Ag, Au, Ni, Pd, Pt, Al, and Pb based on first and second nearest-neighbor modified embedded atom method

Modified embedded atom method (MEAM) potentials for fcc elements Cu, Ag, Au, Ni, Pd, Pt, Al, and Pb have been newly developed using the original first nearest-neighbor MEAM and the recently developed second nearest-neighbor MEAM formalisms. It was found that the original MEAM potentials for fcc elements show some critical shortcomings such as structural instability and incorrect surface reconstructions on (100), (110), and/or (111) surfaces. The newly developed MEAM potentials solve most of the problems and describe the bulk properties (elastic constants, structural energy differences), point defect properties (vacancy and interstitial formation energy and formation volume, activation energy of vacancy diffusion), planar defect properties (stacking fault energy, surface energy, surface relaxation and reconstruction), and thermal properties (thermal expansion coefficients, specific heat, melting point, heat of melting) of the fcc elements considered, in good agreement with relevant experimental information. It has been shown that in the MEAM the degree of many-body screening ${(C}_{\mathrm{min}})$ is an important material property and that structural stability at finite temperatures should be included as a checkpoint during development of semiempirical potentials.

Vacancy formation enthalpy at high pressures in tantalum

Using a mixed basis pseudopotential method, total energy calculationswere performed to obtain the enthalpy of vacancy formation in Ta as afunction of pressure, which is important for understanding the effectsof pressure on mechanical properties. The vacancy formation enthalpyis found to increase from 2.95 eV at ambient pressures to 12.86 eV at300 GPa, and the vacancy formation volume decreases from being 53 ± 5%of the bulk volume per atom at ambient pressure to 20 ± 2% at300 GPa, for a 54-atom supercell. We also show that there is a strongcorrespondence between the vacancy formation enthalpy and the meltingtemperature in Ta.

Point-defect concentrations in B2 NiAl alloys under pressure

The pressure dependence of point-defect concentrations is calculated on the basis of two recently published models. It is demonstrated that the effective formation volume of vacancies shows asymmetrical character about stoichiometry. For Ni-rich and stoichiometric compositions, the concentration of Ni vacancies versus pressure can be described by a simple exponential function. The concentration of Ni antisite defects is independent of pressure. For Al-rich compositions, the concentration of Ni vacancies shows a more complex behaviour. At low pressures the vacancy concentration is independent of pressure, the effective formation volume is zero, while above a critical pressure the vacancy concentration becomes pressure dependent. This can be explained by the annihilation reaction of Ni vacancies. The pressure of complete replacement depends on the stoichiometry and has proved to be very sensitive to the input parameters used in the different models. The changes expected in diffusion activation energy and activation volume with composition are discussed on the basis of calculated temperature and pressure dependence of Ni vacancy concentration, assuming vacancy-mediated diffusion. The pressure effect on the boundary that separates regions in which triple defects or interbranch defects dominate was also investigated. It was observed that pressure expands the region in which Al interbranch defects dominate.

Vacancy formation volume in BCC 3He along the melting curve

The value of the vacancy formation volume is obtained near the melting curve in BCC solid 3He in the temperature range 0.5–2.7 K for molar volumes 19.32– 24.60 cm 3/ mol . The data are obtained both by measuring the temperature dependence of the self-diffusion coefficient in 3He crystals at various densities and by use of the NMR data obtained by H.A. Reich. The value of the vacancy formation volume is in disagreement with the values calculated from the standard thermodynamical relation.

First-principles calculations of the vacancy formation energy in transition and noble metals

The vacancy formation energy and the vacancy formation volume of the $3d,$ $4d,$ and $5d$ transition and noble metals have been calculated within the local-density approximation. The calculations employ the order-$N$ locally self-consistent Green's-function method in conjunction with a supercell approach and include electrostatic multipole corrections to the atomic sphere approximation. The results are in excellent agreement with available full-potential calculations and with the vacancy formation energies obtained in positron annihilation measurements. The variation of the vacancy formation energy through a transition-metal series and the effects of crystal and magnetic structure are investigated and discussed.

Theembedded-atom model applied to vacancy formation in bulk aluminium andlithium

The embedded-atom model (EAM) is applied to the study of vacancy formation in bulk aluminium and lithium. A systematic study is undertaken into the sensitivity of the EAM potentials and embedding energy functionals as a function of the unrelaxed vacancy formation energy which is normally obtained via ab initio density functional calculations. The effect of this `empirical' input parameter on the vacancy relaxation energy, formation volume and structural relaxation is also investigated using super-cell sizes not normally accessible in orbital-based ab initio relaxation studies. We find that for aluminium, for which at most a fifth-nearest-neighbour model is required, the vacancy relaxation energy and formation volume are not sensitive functions of the unrelaxed vacancy formation energy. For lithium, for which at least a ninth-nearest-neighbour model is needed, the situation is somewhat different: both the vacancy relaxation energy and the formation volume are found to be a noticeably related to the unrelaxed vacancy formation energy. For both solids, the structural relaxation was found to be largely insensitive to the unrelaxed vacancy formation energy, agreeing well with previous ab initio calculations. In particular for aluminium, the EAM result agrees extremely well with recent orbital-free density functional calculations which use super-cell sizes approaching those used here. Finally, we find that for lithium, the embedding energy functional has negligible curvature for a wide range of local electronic densities, justifying the use of a simpler pair potential description for lithium in mildly inhomogeneous systems.

Activation volume for arsenic diffusion in germanium

We have measured the effect of pressure on As diffusion in Ge. Diffusion anneals on ion-implanted samples were carried out in a high-temperature diamond anvil cell using fluid argon as a clean, hydrostatic pressure medium. At 575 °C over the pressure range 0.1–4 GPa, pressure slightly enhances the diffusivity, characterized by an activation volume of −1.7±1.4 cm3/mole or −0.12±0.10 times the atomic volume. The results call into question the prevailing view that diffusion of groups III, IV, and V elements are mediated entirely by vacancies. If diffusion of As is mediated entirely by vacancies then either the vacancy formation volume must be unexpectedly low or the energy of vacancy migration must be unexpectedly high.

Finite temperature vacancy formation thermodynamics: local harmonic and quasiharmonic studies

The vacancy formation thermodynamics in six FCC metals Ag, Au, Cu, Ni, Pd and Pt are determined from atomistic simulations as a function of temperature. The investigation is performed using the embedded atom method interatomic potentials and the finite temperature properties are determined within the local harmonic and the quasiharmonic frameworks. The temperature dependence of the vacancy formation free energy, entropy, enthalpy and vacancy formation volume are determined. The authors find that the temperature dependence of the vacancy formation energy can make a significant contribution to the vacancy concentration at high temperatures. An additional goal of the study is to evaluate the accuracy of the local harmonic method under circumstances in which the excess entropy associated with the formation of a defect is very small. The data demonstrate that while the errors associated with determining the vacancy formation entropy in the local harmonic model are large, a simple extension to the local harmonic method yields thermodynamic properties comparable to that obtained in the quasiharmonic model, but with much higher computational efficiency.

Pressure-Dependent Formation Volume and Compressibility of Vacancies in Solids

Pressure-Dependent Formation Volume and Compressibility of Vacancies in Solids

Formation Volume of Vacancies in Elements and the Defect Formation Mechanism of Melting

AbstractThe formation volume Vv of vacancies is given by Vv = (hv/L) ΔVf with hv = 8L (formation enthalpy hv of vacancies and heat L of fusion given in same units; ΔVf = change of volume due to melting). If there are phase transitions within the solid, L and ΔVf must be replaced by (L + Δ Ht) and by (ΔVf + ΔVt), respectively (Δ Ht and Δ Vt refer to the heat (s) of transition (s) and to the volume change(s) due to transition(s), resp.). The pressure dependence of the melting point is dTm/dp = (TmVv)/hv. Independent of the sign of Vv any increase of the vacancy concentration above the maximum concentration possible within the solid decreases the melting point thus resulting in the observed surface melting. The melting point is fixed by the characteristics of vacancy formation (hv, Sv, Vv) and by the bulk modulus of the solid (Sv = formation entropy of vacancies).

Application of the embedded atom method to Ni3Al

The embedded atom method [M. S. Daw and M. I. Baskes, Phys. Rev.B 29, 6443 (1984) used to calculate phase stability, lattice vibrational frequencies, point defect properties, antiphase boundary energies, and surface energies and relaxations for Ni3Al. The empirical embedding functions and core-core repulsions used by this method are obtained. The equilibrium phases for the Ni-rich half of the composition range of Ni–Al are determined for 1000 K and compared with experiment. The elastic constants and vibrational modes of Ni3Al are calculated and the elastic constants are compared with experiment. The formation energy, formation volume, and migration energies of vacancies are computed, and it is found that the formation energy of vacancies on the Ni sublattice is less than that on the Al sublattice. The (100) antiphase boundary is shown to be significantly lower in energy than the (111) antiphase boundary. The surface energies and atomic relaxations of the low index faces are computed, and it is shown that for the (100) and (110) faces that the preferred surface geometry corresponds to the bulk lattice with the mixed composition plane exposed.

Departures from Joule's Law in Dense Liquids near Freezing Related to Vacancy Properties in Hot Crystals

Abstract The departures from Joule's Law that the internal energy U is independent of volume V are first expressed in dense liquids in terms of specific heats and the isothermal compressibility. For condensed rare gases, described by density independent pair potentials, (∂U/∂V)T is shown to be closely related to the vacancy formation energy Ev , a correction term dependent on the density dependence of the pair function being displayed explicitly. For metals with density-dependent pair potentials, (∂U/∂V)T is shown to involve the vacancy formation volume.

Point-defect and stacking-fault properties in body-centred-cubic metals withn-body interatomic potentials

Abstract The properties of vacancies, self-interstitial atoms and translation stacking faults have been studied by computer simulation using the n-body empirical potentials recently developed for the body-centred-cubic transition metals by Finnis and Sinclair. Vacancy formation energy, formation volume and migration energy have been computed and the results are compared with other theoretical and known experimental values. Formation energy and volume have been calculated for six possible interstitial configurations and, although the expected 〈110〉 dumbbell is most stable in Mo and α-Fe, the 〈111〉 crowdion is favoured in Ta, Cr and W. The energetics of interstitial migration have been studied in detail for Mo, and the migration steps which lead to two-dimensional diffusion do not have the lowest migration energy. The n-body potentials predict the absence of stable translation stacking faults on the {110} and {112} planes.

Embedded-atom-method functions for the fcc metals Cu, Ag, Au, Ni, Pd, Pt, and their alloys.

A consistent set of embedding functions and pair interactions for use with the embedded-atom method [M.S. Daw and M. I. Baskes, Phys. Rev. B 29, 6443 (1984)] have been determined empirically to describe the fcc metals Cu, Ag, Au, Ni, Pd, and Pt as well as alloys containing these metals. The functions are determined empirically by fitting to the sublimation energy, equilibrium lattice constant, elastic constants, and vacancy-formation energies of the pure metals and the heats of solution of the binary alloys. The validity of the functions is tested by computing a wide range of properties: the formation volume and migration energy of vacancies, the formation energy, formation volume, and migration energy of divacancies and self-interstitials, the surface energy and geometries of the low-index surfaces of the pure metals, and the segregation energy of substitutional impurities to (100) surfaces.

Erratum: Connection between the formation volume and formation Gibbs energy in noble-gas solids

In the past the authors advanced a macroscopic model that connects defect parameters of solids with the bulk modulus. Recently, the formation volume of vacancies was measured by Macrander in solid krypton. The value agrees with that calculated from the macroscopic model in conjunction with the experimental enthalpy measured by Losee and Simmons and thus provides a rigorous test of the model.

Vacancy formation volume of sodium, aluminium and argon

The whole theory of vacancy formation volume, as developed in parts I, II, III, IV and V of this series, is applied at Al, Na and Ar. In Na the anharmonic terms are found to be responsible for about 35% of the formation volume at 300 K and bring the dominant contribution to its temperature derivative at constant pressure. In fact, we find in this case a negative expansion coefficient for the vacancy, in conflict with the results of Audit and Gilder (1978), who found a positive value, having neglected anharmonicity. The important implications of this result in the interpretation of self-diffusion data are discussed. The analysis of Al confirms the relevant weight of anharmonicity, but also the inadequacy of second-order pseudopotential theory for trivalent metals. Finally, for Ar our conclusions are in full agreement with those reached by Zubov (1980).

Role of high-order anharmonic terms in the theory of vacancy formation volume

The analysis of the anharmonic contributions to the vacancy formation volume in metals, partially developed in part III of this series, is now accomplished with particular emphasis on the high-temperature limit of a second-order anharmonic term, not yet considered before. This term requires the knowledge of structure-dependent dimensionless coefficients, whose values for b.c.c. and f.c.c. lattices are explicitly given.