Vacancy Formation Entropy Research Articles

Prediction of the diffusion characteristics of the V-Cr system by molecular dynamics based on N-body interatomic potentials

Molecular dynamics (MD) is a promising method for predicting the diffusion characteristics of materials. However, the reliability of MD simulation results is largely determined by the technique of their interpretation and the interatomic potentials used. In this work, interatomic potentials are constructed for calculating the diffusion characteristics in alloys of the V-Cr system. We employed an approach that accurately takes into account the angular dependence of the potential energy of the atomic system. For monoatomic systems V and Cr, we used interatomic potentials that predict thermal expansion and melting point in good agreement with experimental values, along with good reproduction of BCC lattice parameters, sublimation energies, equations of state, elastic constants, and several other properties. The cross potential functions of the V-Cr binary system are optimized with using DFT calculations and experimental formation enthalpy values. The resulting V-Cr potential predicts the dependence of the BCC lattice parameter, the elastic constants of V-Cr solid solutions on the concentration of components and the thermal expansion of the stoichiometric V50Cr50 alloy in good agreement with the known experimental data. The constructed potentials were utilized to calculate the diffusion coefficients in V, Cr, V90Cr10 and V95Cr05 by the molecular dynamics method. We identified that the diffusion coefficient of V exceeds the diffusion coefficient of Cr in solid solutions of Cr in V. The calculated diffusion coefficients are consistent with the known results of diffusion experiments in V, Cr and Cr diffusion in V within the error of measurements. Inclusion of the temperature dependencies of the vacancy formation enthalpy and vacancy formation entropy into calculations of the diffusion coefficient leads to its deviation from the Arrhenius equation and improves the agreement with the experimental data.The good agreement between the independent results of the MD simulation and diffusion experiments confirms both the reliability of the experiment and the MD simulation.

Operational Limits of Redox Metal Oxides Performing Thermochemical Water Splitting

Solar thermochemical hydrogen production is an attractive technology that stores intermittent solar energy in the form of chemical bonds. Efficient operation requires the identification of a redox‐active metal oxide (MOx) material that can achieve high conversion of water to hydrogen at minimal energy input. Water splitting occurs by consecutive reduction and reoxidation reactions of MOx. MOx is reduced to MOx−δ and, in the second step, is reoxidized by water recovering the initial MOx and generate H2. The material must reduce at temperatures achievable in concentrated solar receiver/reactors, while maintaining a thermodynamic driving force to split water. At equilibrium, extent of reduction depends on temperature and oxygen partial pressure, and in this analysis, a set of thermodynamic properties, namely, enthalpy and entropy of oxygen vacancy formation, is sufficient to represent MOx. Herein, a method to easily classify materials based on these thermodynamic properties under any condition of oxygen partial pressure and temperature is presented. This method is based on fundamental thermodynamic principles and is applicable for any redox material with known thermodynamic properties. Despite the simplicity of the method, it is believed that this analysis will support future research in targeting thermodynamic properties of redox‐active metal oxides.

Vacancy diffusion in palladium hydrides.

The self-diffusion coefficients of palladium in PdHx (x = 0, 0.25, 0.5, 0.75, 1) were studied using density functional theory to obtain the required thermodynamic and kinetic parameters. The enthalpy of migration decreased from 0.95 eV for Pd to 0.78 eV for PdH. The enthalpy of vacancy formation exhibited a substantial decrease from about 1.1 eV in Pd to 0.4 eV in PdH, which was ascribed to successive filling of antibonding states weakening the Pd-Pd bonds. Concurrently, the Arrhenius pre-exponential was significantly reduced from 4.75 × 10-3 cm2 s-1 for Pd to 5.67 × 10-9 cm2 s-1 for PdH due to softening of the vibrational modes that determine the entropy of vacancy formation and initial/transition state frequencies. A linear correlation between the logarithm of the pre-exponential and the activation energy was interpreted as enthalpy-entropy compensation (Meyer-Neldel rule). The Pd self-diffusion coefficients in the hydrides were within 1 order of magnitude of that in pure palladium above 200 °C for hydrogen pressures up to at least 107 Pa.

Breakdown of Arrhenius law of temperature-dependent vacancy concentration in fcc lanthanum

We measured the temperature dependent equilibrium vacancy concentration using in-situ positron annihilation spectroscopy in order to determine the enthalpy $H_\text{f}$ and entropy $S_\text{f}$ of vacancy formation in elementary fcc-La. The Arrhenius law applied for the data analysis, however, is shown to fail in explaining the unexpected high values for both $S_\text{f}$ and $H_\text{f}$: in particular $S_\text{f}=17(2)~k_\text{B}$ is one order of magnitude larger compared to other elemental metals, and the experimental value of $H_\text{f}$ is found to be more than three standard deviations off the theoretical one $H_\text{f}=1.46~\text{eV}$ (our \acs{dft} calculation for La at $T=0~\text{K}$). A consistent explanation is given beyond the classical Arrhenius approach in terms of a temperature dependence of the vacancy formation entropy with $S_\text{f}^\prime=-0.0120(14)~k_\text{B}/\text{K}$ accounting for the anharmonic potential introduced by vacancies.

Structural and thermodynamic study of Ca A- or Co B-site substituted SrFeO3-δ perovskites for low temperature chemical looping applications.

Perovskite-structured materials, owing to their chemical-physical properties and tuneable composition, have extended their range of applications to chemical looping processes, in which lattice oxygen provides the oxygen needed for chemical reactions omitting the use of co-fed gaseous oxidants. To optimise their oxygen donating behaviour to the specific application a fundamental understanding of the reduction/oxidation characteristics of perovskite structured oxides and their manipulation through the introduction of dopants is key. In this study, we investigate the structural and oxygen desorption/sorption properties of Sr1-xCaxFeO3-δ and SrFe1-xCoxO3-δ (0 ≤ x ≤ 1) to guide the design of more effective oxygen carriers for chemical looping applications at low temperatures (i.e. 400-600 °C). Ca A- or Co B-site substituted SrFeO3-δ show an increased reducibility, resulting in a higher oxygen capacity at T ≤ 600 °C when compared to the unsubstituted sample. The quantitative assessment of the thermodynamic properties (partial molar enthalpy and entropy of vacancy formation) confirms a reduced enthalpy of vacancy formation upon substitution in this temperature range (i.e. 400-600 °C). Among the examined samples, Sr0.8Ca0.2FeO3-δ exhibited the highest oxygen storage capacity (2.15 wt%) at 500 °C, complemented by excellent redox and structural stability over 100 cycles. The thermodynamic assessment, supported by in situ XRD measurements, revealed that the oxygen release occurs with a phase transition perovskite-brownmillerite below 770 °C, while the perovskite structure remains stable above 770 °C.

Anharmonic thermodynamics of vacancies using a neural network potential

Lattice anharmonicity is thought to strongly affect vacancy concentrations in metals at high temperatures. It is however non-trivial to account for this effect directly using density functional theory (DFT). Here we develop a deep neural network potential for aluminum that overcomes the limitations inherent to DFT, and we use it to obtain accurate anharmonic vacancy formation free energies as a function of temperature. While confirming the important role of anharmonicity at high temperatures, the calculation unveils a markedly nonlinear behavior of the vacancy formation entropy and shows that the vacancy formation free energy only violates Arrhenius law at temperatures above 600 K, in contrast with previous DFT calculations.

Diffusion coefficients of transition metals in fcc cobalt

Using first-principles density functional theory (DFT), we calculate the diffusivities of 32 different solute elements—all transition metals, together with Al and Si—in fcc cobalt within the formalism of the five-frequency model. For self-diffusion in fcc cobalt, we compare the accuracy of various approximations to the exchange-correlation energy functional of DFT in estimating the activation energy, and find that only the Perdew-Burke-Ernzerhof (PBE) approximation agrees well with experimental reports and all other functionals largely overestimate it. Our calculations also show that an accurate estimation of the self-diffusion coefficient requires explicit calculation of the effective jump frequency and vacancy formation entropy via phonons. Using accurate self-diffusion data and scaling all solute-related attempt frequencies with respect to the attempt frequency for self-diffusion using a simple relation involving the atomic mass and melting temperature of the solute yields solute diffusivities in excellent agreement with experiments, where such data is available. We find that large solutes spontaneously relax toward the nearest neighbor vacancy to relieve the misfit strain, and the extent of this relaxation correlates negatively with the migration energy. Thus, in general, larger solutes have lower migration energies and diffuse faster than smaller solutes in fcc cobalt. However, extremely large solutes, e.g., group III elements Sc, Y, Lu, tend to be trapped in an energy valley located halfway toward the vacancy, and monovacancy mediated diffusion may no longer be valid in such cases. Finally, for all the solutes considered, we systematically tabulate the diffusion-related quantities calculated—diffusion prefactors, migration and activation energies—constructing an extensive and accurate first-principles database for solute diffusion in fcc cobalt.

A single-volume approach for vacancy formation thermodynamics calculations

The vacancy formation Gibbs free energy, enthalpy and entropy in fcc Al, Ag, Pd, Cu, and bcc Mo are determined by first-principles calculations using the quasi-harmonic approximation to account for vibrational contributions. We show that the Gibbs free energy can be determined with sufficient accuracy in a single-volume approach using the fixed equilibrium volume of the defect-free supercell. Although the partial contributions to the Gibbs free energy, namely, the formation enthalpy and entropy exhibit substantial errors when obtained directly in this approach, they can be computed from the Gibbs free energy using the proper thermodynamic relations. Compared to experimental data, the temperature dependence of the vacancy formation Gibbs free energy is accounted for at low temperatures, while it overestimates the measurements at high temperature, which is attributed to the neglect of anharmonic effects.

Енергія формування вакансій у квазігармонійному наближенні з перших принципів

Background. This work is related to the theoretical calculation of the vacancy formation energy in the context of finding an answer to two unsolved questions. Firstly, there is no single answer to the question of calculations accuracy of the vacancy formation energy. Secondly, the existing phenomenological theories tacitly assume that the vacancy formation energy and entropy of vacancy formation do not depend on temperature.Objective. The aim of the study was to investigate the effect of temperature factor on the free energy of vacancy formation and its components in a number of metals within the quasi-harmonic approximation.Methods. The vacancy formation energy and entropy in FCC Al, Ag, and Pd are determined as a function of temperature using ab initio methods within a quasi-harmonic approximation.Results. It was shown, that the vacancy formation energy substantially increases with temperature in all cases, which is related to the thermal lattice expansion. Such increase of the vacancy formation energy is compensated by the vibrational vacancy formation entropy contribution. The latter appears to be strongly increasing with temperature in the case of Mo.Conclusions. However, a thermodynamic analysis shows that such increase of vacancy formation energy in the process of calculation is largely underestimated. The latter is compensated by contribution from vibrating entropy of vacancy formation and strongly increases with temperature. Such underestimation is related to certain problems in quasi-harmonic approximation.

Contribution of the entropy on the thermodynamic equilibrium of vacancies in nickel

The equilibrium vacancy concentration in nickel was determined from ab initio calculations performed with both generalized gradient approximation and local density approximation up to the melting point. We focus the study on the vacancy formation entropy expressed as a sum of a vibration and an electronic contribution, which were determined from the vibration modes and the electronic densities of states. Applying a method based on the quasi-harmonic approximation, the temperature dependence of the defect formation energy and entropy were calculated. We show that the vibrations of the first shell of atoms around the defect are predominant to the vibration formation entropy. On the other hand, the electronic formation entropy is very sensitive to the exchange-correlation potential used for the calculations. Finally, the vacancy concentration is computed at finite temperature with the calculated values for the defect formation energy and entropy. In order to reconcile point-defects concentration obtained with our calculations and experimental data, we conducted complementary calorimetric measurements of the vacancy concentration in the 1073-1273 K temperature range. Close agreement between theory and experiments at high temperature is achieved if the calculations are performed with the generalized gradient approximation and both vibration and electronic contributions to the formation entropy are taken into account.

Vibrational Properties of Vacancy in Na and K Using MEAM Potential

AbstractThe modified embedded atom method (MEAM) with the universal form of embedding function and a modified energy term along with the pair potential has been employed to determine the potentials for alkali metals: Na, K, by fitting to the Cauchy pressure (C12 − C44)/2, shear constants Gν = (C11 − C12 + 3C44)/5 and C44, the cohesive energy and the vacancy formation energy. The obtained potentials are used to calculate the phonon dispersions of these metals. Using these calculated phonons we evaluate the local density of states of neighbours of vacancy using Green’s function method. The local density of states of neighbours of vacancy has been used to calculate mean square displacements of these atoms and formation entropy of vacancy. The calculated mean square displacements of both 1st and 2nd neighbours of vacancy are found to be lower than that of host atom. The calculated phonon dispersions agree well with the experimental phonon dispersion curves and the calculated results of vacancy formation entropy compare well with the other available results.

Prediction of solute diffusivity in Al assisted by first-principles molecular dynamics

Ab initio calculations of the solid-state diffusivity of solute atoms in bulk aluminium have previously been based on transition state theory (TST), employing transition state searches and systematic assessments of single jumps together with appropriate models of jump frequencies and correlation factors like the five-frequency model. This work compared TST benchmark predictions of diffusivities with first-principles molecular dynamics (FPMD). The TST calculations were performed at unprecedented high precision, including the temperature dependent entropy of vacancy formation which has not been included in previous studies of diffusion in Al; this led to improved agreement with experimental data. It was furthermore demonstrated that FPMD can yield sufficient statistics to predict the frequency of single jumps, and FPMD was used to successfully predict the macroscopic diffusivity of Si in Al. The latter is not possible in systems with higher activation energies, but it was demonstrated that FPMD in such cases can identify which jumps are prevalent for a given defect configuration. Thus, information from FPMD can be used to simplify the calculation of correlation terms, prefactors and effective transition barriers with TST significantly. This can be particularly important for the study of more complicated defect configurations, where the number of distinct jumps rapidly increases to be intractable by systematic methods.

Vibrational properties of vacancy in bcc Nb using embedded atom method

Embedded atom method (EAM) with universal form of embedding function along with Morse form of pair potential have been employed to determine the potential for Nb by fitting to Cauchy pressure, shear constants, cohesive energy and vacancy formation energy. Obtained potential is used to calculate the phonon dispersion spectra of this metal. On comparing with experimental phonons, the results using EAM show large discrepancies. In order to overcome this inadequacy of EAM model, we have further employed a modified embedded atom method (MEAM) in which a modified term along with pair potential and embedding function are added in the total energy. Calculated phonon dispersions using the potential obtained from MEAM are found to be closer to experimental results compared to those obtained from EAM. With application of MEAM potential, we have computed local vibrational density of states of atoms in ideal lattice and neighbouring atoms for vacancy. Vacancy formation entropy and mean square thermal displacements of the neighbouring atoms of vacancy are obtained using Green’s function method.

Equilibrium vacancy parameters and limit temperature of nonequilibrium melting in osmium

For osmium, based on experimental data on enthalpy and the averaged heat capacity obtained by the method of mixing within the temperature range of 1150–2960 K, for the first time the parameters of the equilibrium vacancies for this metal have been determined: the vacancy formation energy E = 1.8 eV, the vacancy concentration at melting c = 3.3%, and the vacancy formation entropy S = 25.6 J/(mol K). The limit temperature of the onset of nonequilibrium melting of osmium T m = 4256 K and its relative value T lim/T m = 1.30 has been calculated.

Oxidation–reduction behavior of La0.8Sr0.2ScyMn1 − yO3 ± δ (y = 0.2, 0.3, 0.4): Defect structure, thermodynamic and electrical properties

Oxygen non-stoichiometry of La0.8Sr0.2ScyMn1−yO3±δ (y=0.2, 0.3, 0.4) oxide was studied by coulometric titration as a function of oxygen partial pressure, p(O2), and temperature in a range of 923–1023K. Depending on the Sc doping amount, p(O2), and temperature, oxygen non-stoichiometry varies significantly. Under a reducing condition, La0.8Sr0.2ScyMn1−yO3±δ shows both oxygen excess and oxygen deficient compositions. At the higher p(O2) region, the oxygen excess composition is due to metal ion vacancies, whereas in the lower p(O2) region, the oxygen deficient composition is due to the formation of oxygen vacancies. The experimental data were analyzed by a random defect model. Partial molar enthalpy and partial molar entropy of oxygen vacancy formation are calculated using the Gibbs–Helmholtz equation from the non-stoichiometric data. The electrical conductivity was measured as the function of the oxygen partial pressure and temperature. In the lower p(O2) region, electrical conductivity strongly depends on the oxygen non-stoichiometry.

Activated-process parameters for diamond, silicon, and germanium crystals

Vacancy-formation parameters are calculated for crystals belonging to group IV elements: C (diamond), Si, Ge, α-Sn, and Pb. A method is proposed for dealing with both the quantum phenomena at low temperatures and atomic delocalization at high temperatures. The latter feature is shown to result in larger predicted values of vacancy-formation enthalpy, entropy, and volume. Considerable temperature dependence of vacancy-formation parameters is predicted for temperatures below a certain point, including sign reversal for vacancy-formation entropy. Applied at high temperatures, the method is found to show close agreement with previous experimental and theoretical results.

Contribution of equilibrium vacancies to vanadium caloric properties

For the first time, data have been obtained on the parameters of the contribution of equilibrium vacancies to the caloric properties of vanadium: the vacancy formation energy, E = 1.22 eV, the vacancy formation entropy, S = 26.8 J/(mole K)−1, and the temperature dependence of the vacancy concentration (at the vanadium melting temperature, T m = 2220 K, the concentration equals to c = 4.2%). These values are determined on the basis of experimental measurements of the average heat capacity of vanadium. From analysis of the interconnection between the vacancy contribution and the limit temperature of superheating of the beginning of melting, T sh, we found that the most reliable vacancy-free straight line of the average heat capacity corresponds to (T sh/T m) ≈ 1.25 and may serve as a reliability criterion for calculation of the vacancy contribution in metals.

Self-Diffusion Coefficient of fcc Mg: First-Principles Calculations and Semi-Empirical Predictions

First-principles calculations and semi-empirical equations are employed to determine the self-diffusion mobility of fcc Mg. All factors entering the vacancy-mediated self-diffusion coefficient, which include the equilibrium lattice parameter, the enthalpy of vacancy formation and atom migration, and the vibrational entropy of vacancy formation as well as the effective frequency, are evaluated with the local density approximation (LDA) and generalized gradient approximation (GGA) in first-principles calculations. These computed quantities are then utilized to calculate the self-diffusion coefficient of fcc Mg. For comparison, four widely used semi-empirical equations are also used to estimate the self-diffusion coefficient of fcc Mg. The comparisons show that first-principles calculations and semi-empirical equations yield values close to the activation energy Q for self-diffusion of fcc Mg from the LDA calculation, but the diffusion prefactor D 0 predicted from the four semi-empirical equations are all about one order of magnitude larger than those from the first-principles calculations. Based on the comparison for the self-diffusion coefficients of fcc Al computed with the first-principles method and semi-empirical approaches, it is concluded that the self-diffusion coefficient of fcc Mg calculated with the LDA method is more accurate than the estimation from semi-empirical approaches.

Schottky Defects in Nanoparticles

Vacancies play a major role in the electrical and thermal transport as well as the mechanical behavior of materials. To understand the processes occurring in nanomaterials during heat treatment and mechanical deformation, the size effects on the vacancy formation energy and entropy have to be considered. As these material properties are hardly measurable experimentally, particularly in nanoparticles, a theoretical model calculating the size-dependent vacancy formation energy and entropy has been developed in this study. It has been found that size reduction makes the vacancy much easier to form; then the vacancy concentration increases when size reduces and temperature increases.

Formation energies of point defects at finite temperatures

AbstractThe dominant entropy contribution affecting defect concentrations is configurational entropy. Other contributions such as harmonic and anharmonic lattice vibrations are second order effects and computationally expensive to calculate. Therefore, such contributions have been rarely considered in defect investigations. However, to achieve the next accuracy level in defect calculations and thus significantly improve the agreement with experiment, the inclusion of these contributions is critical. In this paper, we present the methods needed to compute highly accurate free energies of point defects from first principles. We demonstrate how to include all relevant free energy contributions up to the melting point. The focus will be on nonmagnetic metals and point defects in the dilute limit. We consider all relevant excitation mechanisms: electronic excitations and ionic vibrations both in the quasiharmonic approximation and explicitly including anharmonicity (i.e., phonon–phonon interaction). Since computing such interactions requires to sample large parts of the phase space, straightforward ab initio based simulations (such as molecular dynamics) are in most cases out of reach even on supercomputers. To overcome this difficulty, a recently developed hierarchical scheme will be presented which allows to coarse grain the configuration space and thus to efficiently calculate anharmonic contributions to defect formation. We discuss the performance and accuracy of the developed methodology for the example of vacancies in aluminum. An important insight is that the entropy of vacancy formation is significantly affected by anharmonicity. We further show that the inclusion of all the aforementioned excitation mechanisms is critical to guarantee an accurate description of thermodynamic properties up to the melting point.