Divacancy Binding Energy Research Articles

Bridging the gap between theory and experiment in vacancy concentration, C/N/O diffusivity, and divacancy interaction in tungsten: Role of vacancy-C/N/O interaction

Light element solutes such as carbon (C), nitrogen (N), and oxygen (O) are impurities commonly seen in tungsten alloys, often exerting great impact on materials properties due to their strong interaction with defects. In this work, we conducted a comprehensive investigation into the atomistic properties of small vacancy-impurity defect complexes in tungsten using rigorous first-principles calculations. Through the integration of a statistical approach, we meticulously examined the concentration distributions of these complexes across various temperatures and impurity concentrations. Our findings reveal that the introduction of C/N/O impurities significantly augments the concentration of vacancy-type defects to a level well above the thermal equilibrium vacancy concentration in pure tungsten. Moreover, we conducted a thorough assessment of the diffusivity of C/N/O atoms, conclusively demonstrating that the inclusion of the vacancy trapping effect is paramount in comprehending their unexpectedly low diffusivity observed in experiments. Finally, in conjunction with an object kinetics Monte Carlo method, we revisited the experimental determination of the divacancy binding energy in tungsten. We showed that the presence of C/N/O impurities may have conceivably impacted the experimentally ascertained divacancy binding energy. This work provides accurate data on evaluating vacancy-impurity interaction in tungsten, unveils the pivotal role played by C/N/O in affecting vacancy concentration and divacancy binding energy, and sheds new insights toward understanding the diffusion behavior of impurities in tungsten.

Investigation of vacancy formation energy and binding energy in fcc crystals by pseudopotential technique

The parameters of Heine and Abarenkov model potential have been computed for 21 face centered cubic (fcc) closed pack crystals and a comparison is made with their magnitudes to the Ashcroft model potential. The magnitudes of monovacancy and divacancy formation energy are calculated for different fcc crystals and compared with the literature. Huge values of divacancy formation energy and divacancy binding energy in some higher valence fcc metals are attributed to the probability of instability of these defects there. Lastly, the changes in monovacancy formation energy and divacancy formation energy in presence of an impurity in three simple metals Cu, Ag, Au are calculated.

Determining the diffusion behavior of point defects in zirconium by a multiscale modelling approach

Zr alloys are commonly used materials for in-core components of pressurized water reactors. The evolution of defects in Zr under irradiation conditions will directly contribute to the degradation of reactor components. For a better understanding of the evolution process of defects in Zr, a multiscale simulation was performed to study the diffusion behavior of radiation-induced point defects. Because the existing potentials cannot accurately describe the energy differences of stable states self-interstitial atoms (SIAs) in Zr, an embedded atom method potential that can exactly reproduce the energy difference between metastable states and ground states, and the binding energy of divacancy of Zr, was first developed by fitting a potential to point defect properties calculated by density functional theory and other basic crystal properties. Based on the constructed potential, molecular dynamics (MD) and kinetic Monte Carlo simulations were conducted to investigate the migration of point defects. We determined several frequent migration paths of SIAs by kinetic study, which is rarely reported in previous MD studies. The SIA exhibits obvious anisotropic diffusion characteristics at low temperatures (<600 K). The most frequent migration path for SIAs is the jump between the two nearest basal octahedral (BO) sites in the basal plane, which means that SIAs diffuse faster along the basal plane than along the c axis. It is found that diffusion within the basal plane tends to be two-dimensional (2D) diffusion rather than 1D diffusion, and there is a significant correlation effect for diffusion of SIAs along the basal plane. Additionally, the existence of trivacancy-SIA complex was found in the process of divacancy migration, which can inhibit divacancy migration. Monovacancies and divacancies exhibit anisotropic diffusion characteristics in the considered temperature range Divacancies have a much faster diffusion rate than monovacancies in the present MD simulation and can easily dissociate at high temperatures (>900 K). The rapid migration of the divacancies may contribute to the formation of vacancy dislocation loops. These results are meaningful to understand the evolution process of radiation-induced defects.

Efficient and transferable machine learning potentials for the simulation of crystal defects in bcc Fe and W

Data-driven, or machine learning (ML), approaches have become viable alternatives to semiempirical methods to construct interatomic potentials, due to their capacity to accurately interpolate and extrapolate from first-principles simulations if the training database and descriptor representation of atomic structures are carefully chosen. Here, we present highly accurate interatomic potentials suitable for the study of dislocations, point defects, and their clusters in bcc iron and tungsten, constructed using a linear or quadratic input-output mapping from descriptor space. The proposed quadratic formulation, called quadratic noise ML, differs from previous approaches, being strongly preconditioned by the linear solution. The developed potentials are compared to a wide range of existing ML and semiempirical potentials, and are shown to have sufficient accuracy to distinguish changes in the exchange-correlation functional or pseudopotential in the underlying reference data, while retaining excellent transferability. The flexibility of the underlying approach is able to target properties almost unattainable by traditional methods, such as the negative divacancy binding energy in W or the shape and the magnitude of the Peierls barrier of the $\frac{1}{2}\ensuremath{\langle}111\ensuremath{\rangle}$ screw dislocation in both metals. We also show how the developed potentials can be used to target important observables that require large time-and-space scales unattainable with first-principles methods, though we emphasize the importance of thoughtful database design and degrees of nonlinearity of the descriptor space to achieve the appropriate passage of information to large-scale calculations. As a demonstration, we perform direct atomistic calculations of the relative stability of $\frac{1}{2}\ensuremath{\langle}111\ensuremath{\rangle}$ dislocations loops and three-dimensional C15 clusters in Fe and find the crossover between the formation energies of the two classes of interstitial defects occurs at around 40 self-interstitial atoms. We also compute the kink-pair formation energy of the $\frac{1}{2}\ensuremath{\langle}111\ensuremath{\rangle}$ screw dislocation in Fe and W, finding good agreement with density functional theory informed line tension models that indirectly measure those quantities. Finally, we exploit the excellent finite-temperature properties to compute vacancy formation free energies with full anharmonicity in thermal vibrations. The presented potentials thus open up many avenues for systematic investigation of free-energy landscape of defects with ab initio accuracy.

Strain Dependence of Energetics and Kinetics of Vacancy in Tungsten.

We investigate the influence of hydrostatic/biaxial strain on the formation, migration, and clustering of vacancy in tungsten (W) using a first-principles method, and show that the vacancy behaviors are strongly dependent on the strain. Both a monovacancy formation energy and a divacancy binding energy decrease with the increasing of compressive hydrostatic/biaxial strain, but increase with the increasing of tensile strain. Specifically, the binding energy of divacancy changes from negative to positive when the hydrostatic (biaxial) tensile strain is larger than 1.5% (2%). These results indicate that the compressive strain will facilitate the formation of monovacancy in W, while the tensile strain will enhance the attraction between vacancies. This can be attributed to the redistribution of electronic states of W atoms surrounding vacancy. Furthermore, although the migration energy of the monovacancy also exhibits a monotonic linear dependence on the hydrostatic strain, it shows a parabola with an opening down under the biaxial strain. Namely, the vacancy mobility will always be promoted by biaxial strain in W, almost independent of the sign of strain. Such unexpected anisotropic strain-enhanced vacancy mobility originates from the Poisson effect. On the basis of the first-principles results, the nucleation of vacancy clusters in strained W is further determined with the object kinetic Monte Carlo simulations. It is found that the formation time of tri-vacancy decrease significantly with the increasing of tensile strain, while the vacancy clusters are not observed in compressively strained W, indicating that the tensile strain can enhance the formation of voids. Our results provide a good reference for understanding the vacancy behaviors in W.

The Vacancy Energy in Metals: Cu, Ag, Ni, Pt, Au, Pd, Ir and Rh

The predictive calculations of vacancy formation energies in metals: Cu, Ag, Ni, Pt, Au, Pd, Ir and Rh are presented. The energy is given as a function of electron density. Density functional theory underestimates the vacancy formation energy when structural relaxation is included. The unrelaxed mono-vacancy formation, unrelaxed di-vacancy formation, unrelaxed di-vacancy binding and low index surface energies of the fcc transition metals Cu, Ag, Ni, Pt, Au, Pd, Ir and Rh has been calculated using embedded atom method. The values for the vacancy formation energies agree with the experimental value. We also calculate the elastic constants of the metals and the heat of solution for the binary alloys of the selected metals. The average surface energies calculated by including the crystal angle between planes (hkl) and (111) correspond to the experiment for Cu, Ag, Ni, Pt and Pd. The calculated mono-vacancy formation energies are in reasonable agreement with available experimental values for Cu, Ag, Au and Rh. The values are higher for Pt and Ir while smaller values were recorded for Ni and Pd. The unrelaxed di-vacancy binding energy calculated agrees with available experimental values in the case of Cu, Ni, Pt and Au.

On the stability and mobility of di-vacancies in tungsten

Properties of small vacancy clusters in tungsten were studied with first-principles calculations. The binding and formation energies of the vacancy clusters increase with the cluster size. Dynamic characteristics of a di-vacancy were specified between room temperature and 700 K with lattice kinetic Monte Carlo calculations, which were parametrised with the present first-principles results for the dissociation barriers. An Arrhenius fit for the di-vacancy diffusion yielded , and for the mean lifetime, ps. The di-vacancy system was found to be stable up to 500 K, due to the high energy needed for its dissociation. Having a carbon impurity was found to increase the tungsten di-vacancy binding energy.

Thermodynamics of impurity-enhanced vacancy formation in metals

Hydrogen induced vacancy formation in metals and metal alloys has been of great interest during the past couple of decades. The main reason for this phenomenon, often referred to as the superabundant vacancy formation, is the lowering of vacancy formation energy due to the trapping of hydrogen. By means of thermodynamics, we study the equilibrium vacancy formation in fcc metals (Pd, Ni, Co, and Fe) in correlation with the H amounts. The results of this study are compared and found to be in good agreement with experiments. For the accurate description of the total energy of the metal–hydrogen system, we take into account the binding energies of each trapped impurity, the vibrational entropy of defects, and the thermodynamics of divacancy formation. We demonstrate the effect of vacancy formation energy, the hydrogen binding, and the divacancy binding energy on the total equilibrium vacancy concentration. We show that the divacancy fraction gives the major contribution to the total vacancy fraction at high H fractions and cannot be neglected when studying superabundant vacancies. Our results lead to a novel conclusion that at high hydrogen fractions, superabundant vacancy formation takes place regardless of the binding energy between vacancies and hydrogen. We also propose the reason of superabundant vacancy formation mainly in the fcc phase. The equations obtained within this work can be used for any metal–impurity system, if the impurity occupies an interstitial site in the lattice.

Interaction of Hydrogen Atoms with Vacancies and Divacancies in bcc Iron

The paper presents the results of both ab initio and thermodynamic analysis of vacancy and divacancy formation and hydrogen interaction with them in alpha (bcc) iron. Ab initio calculations were performed by DFT method using LAPW in WIEN2k package. Monovacancy formation energy was found to be 2.15 eV and divacancy binding energy 0.22 ± 0.01 eV. Equlibrium fraction of vacancies bound into divacancies is of the order of 10–5 even at the highest temperatures close to bcc → fcc transformation point. Hydrogen has a strong interaction with monovacancies (vacancy-hydrogen binding energy decreasing from 0.60 to 0.31 eV for the first–fifth H atom inside a single vacancy) but has only a small effect on divacancy formation energy that is equal to 0.28, 0.19 and 0.17 for the case of joining of VH + V, VH + VH and VH2 + VH2, respectively. This means that the presence of hydrogen cannot significantly increase the equilibrium concentration of divacancies.

Formation and binding energies of vacancies in the Al(111) surface: Density functional theory calculations confirm simple bond model

Abstract In terms of the density functional calculations on an Al(111) surface, we obtained reasonable results for the convergence of the formation energy of mono-vacancies with depth or supercell size and binding energy for di-vacancies in the first layer with supercell size. The formation energy of a mono-vacancy increases with the increase in the number of layers and the supercell size, and converges to the third layer and to a 4 × 4 supercell respectively. The binding energy of a di-vacancy on the surface will decrease with the supercell size. However, the lowest formation energy of a mono-vacancy and highest binding energy is in a 3 × 3 supercell because of the dimer formed by its two nearest neighbour Al atoms between two adjacent vacancies. The results are compatible with simple bond model theory, and the binding energy in a 2 × 2 supercell is in accord with experimental results.

Analytical Bond-order Potential for hcp-Y

The lattice parameters, elastic constants, cohesive energy, structural energy differences, as well as the properties of point defects and planar defects of hexagonal close-packed yttrium (hcp-Y) have been studied with ab initio density functional theory for constructing an extensive database. Based on an analytical bond-order potential scheme, empirical many-body interatomic potential for hcp-Y has been developed. The model is fitted to some properties of Y, e.g., the lattice parameters, elastic constants, bulk modulus, cohesive energy, vacancy formation energy, and the structural energy differences. The present potential has ability to reproduce defect properties including the self-interstitial atoms formation energies, vacancy formation energy, divacancy binding energy, as well as the bulk properties and the thermal dynamic properties.

Analytical Interatomic Potential for HCP-Scandium

A reactive interatomic potential based on an analytical bond-order scheme is developed for hexagonal close-packed (hcp) scandium, and the model is fitted to the lattice parameters, elastic constants, cohesive energy and vacancy formation energy of scandium. The potential was used to calculate the structural energy differences of bcc-hcp, fcc-hcp, sc-hcp and diamond-hcp, as well as self-interstitial atom (SIA) formation energy, vacancy migration energy, divacancy binding energy, surface energy and stacking fault energy. The developed potential is shown to be able to reproduce energetics and structural properties of hcp-scandium.

Do the atoms at second layer block the path of vacancies in the bulk? - The DFT study of vacancies below the Mg (0001) surface

The formation energy of monovacancy and binding energy of divacancy at and below Mg (0001) surface are examined by ab initio calculations based on the density-functional theory. Appropriate dense of k-point mesh and large supercell is used to obtain reasonable values. During calculation, we find that higher formation energy is needed to form monovacancy at the second layer. The electron density contour plots shows that electron tends to accumulate in the vicinity of vacancy at the second layer and accumulation charges are also impact on the stableness of the divacancies located at different layers. The migration barrier energy of transition state for the vacancy at the second layer is counted to be very low because of the accumulation electrons apply the power to make the vacancy to migrate. Perhaps, our computational results will be useful to account for the diffusion of hydrogen atoms at bulk of Mg and the mechanism of catalysis of impurities like carbon for the diffusion of hydrogen atoms.

Role of Macroscopic Deformations in Energetics of Vacancies in Aluminum

Electronic structure calculations on million-atom samples are employed to investigate the effect of macroscopic deformations on energetics of vacancies in aluminum. We find that volumetric strain associated with a deformation largely governs the formation energies of monovacancies and divacancies. The calculations suggest that nucleation of these defects is increasingly favorable under volumetric expansion, so much to the point that they become thermodynamically favorable under large positive volumetric strains. On the contrary, on an average, existing vacancies are found to bind preferentially under compressive volumetric strains. Shear deformations did not affect the formation energies of vacancies, but strongly influenced the 110 divacancy binding energies, causing them to orient under energetically preferential directions.

Thermodynamics of mono- and di-vacancies in barium titanate

The thermodynamic and kinetic properties of mono- and di-vacancy defects in cubic (para-electric) barium titanate BaTiO3 are studied by means of density-functional theory calculations. It is determined which vacancy types prevail for given thermodynamic boundary conditions. The calculations confirm the established picture that vacancies occur in their nominal charge states almost over the entire band gap. For the dominating range of the band gap the di-vacancy binding energies are constant and negative. The system, therefore, strives to achieve a state in which, under metal-rich (oxygen-rich) conditions, all metal (oxygen) vacancies are bound in di-vacancy clusters. The migration barriers are calculated for mono-vacancies in different charge states. As oxygen vacancies are found to readily migrate at typical growth temperatures, di-vacancies can be formed at ease. The key results of the present study with respect to the thermodynamic behavior of mono- and di-vacancies influence the initial defect distribution in the ferroelectric phases and therefore the conditions for aging.

A General Embedded Atom Method and Application to Prediction for Thermodynamic Properties of Fe-Eu System

An analytic embedded-atom potentials was developed. It was applied to calculating mono-vacancy formation energy, divacancy binding energy, elastic constants, energy difference of different structures, the surface energy, and the phonon spectra of iron and europium. The formation enthalpies of Fe-Eu binary alloy were also calculated. The calculated physical properties are in agreement with the experiments available or other theoretical results. The formation enthalpies are in good agreement with the results obtained by Miedema’s theory.

Ab initiostudy on divacancy binding energies in aluminum and magnesium

The divacancy binding energies in fcc Al and hcp Mg have been examined by ab initio calculations based on the density-functional theory. Dense k-point meshes and large supercells are used enough to obtained converged values. For Al, the first-nearest-neighbor (1NN) divacancy is indeed unstable with a negative binding energy similar to the recent calculations by Carling et al. [Phys. Rev. Lett. 85, 3862 (2000)]. However, the second-nearest-neighbor (2NN) divacancy is stable with a positive binding energy larger than their value. For Mg, both the 1NN and 2NN divacancies corresponding to the 1NN divacancy in fcc structures are stable with positive binding energies in accordance with the conventional view. The difference in the stability of divacancies in Al and Mg has been analyzed clearly through the calculations of divacancies in hcp Al and fcc Mg. The tendency to form local directional bonds with covalency at defects is the electronic origin of the peculiar nature of divacancies in Al.

Determination of solid-state sulfidation mechanisms in ion-implanted copper

Ion-beam irradiation and ion implantation were used to evaluate the influence of point defects and alloying elements on the sulfidation rate of copper films in atmospheric environments containing H 2S. Low-energy ions from an oxygen plasma were used to grow thin metal oxide passivation layers on Cu films that were subsequently irradiated and exposed to sulfidizing environments (50–600 ppb H 2S in air with 0.5–85% relative humidity). The type of oxide proved to be important in that a CuO layer essentially prevented sulfidation whereas a Cu 2O layer permitted sulfidation. For the native copper oxide (Cu 2O), density-functional theory modeling of Cu divacancy binding energies suggested that alloying with In or Al would cause vacancy trapping and possibly slow the rate of sulfidation. This finding was then experimentally verified for an In-implanted Cu film. A series of marker experiments using unalloyed Cu showed that sulfidation proceeds by solid-state transport of Cu from the substrate through the Cu 2S product layer.

Force fields in close-packed crystals and their melts in relation to defects, surface energies and mechanical properties

Abstract Pair potentials afford a quantitative starting point for studying the vacancy formation energy E v in hot close-packed crystals such as Ar. A summary will be given of the relation of E v to the melting temperature T m, via liquid structure, and a brief comment made on the role of three-body-forces. This leads into a discussion of ‘criteria’ characterizing the solid—liquid phase transition, one of these being the assertion that a close-packed crystal melts when the internal energy required to create a localized hole (the unrelaxed vacancy in the hot crystal) is equal to the change in internal energy at T m required to expand the liquid by one atomic volume. When N-body forces become important and are treated by so-called glue models, exemplified in the work of R. A. Johnson (1988, Physical Review B, 37, 3924), on a model of Cu metal, the important role of the shear modulus in determining both E v (now at T = 0) and the divacancy binding energy is stressed. Finally, E v in, for example, Al is argue...

Force fields in close-packed crystals and their melts in relation to defects, surface energies and mechanical properties

Abstract Pair potentials afford a quantitative starting point for studying the vacancy formation energy E v in hot close-packed crystals such as Ar. A summary will be given of the relation of E v to the melting temperature T m, via liquid structure, and a brief comment made on the role of three-body-forces. This leads into a discussion of ‘criteria’ characterizing the solid—liquid phase transition, one of these being the assertion that a close-packed crystal melts when the internal energy required to create a localized hole (the unrelaxed vacancy in the hot crystal) is equal to the change in internal energy at T m required to expand the liquid by one atomic volume. When N-body forces become important and are treated by so-called glue models, exemplified in the work of R. A. Johnson (1988, Physical Review B, 37, 3924), on a model of Cu metal, the important role of the shear modulus in determining both E v (now at T = 0) and the divacancy binding energy is stressed. Finally, E v in, for example, Al is argued to be closely connected with surface energy, through the similarity of conduction-electron density profiles around the vacant site and through the planar surface. This leads to a brief account of unpublished work on mechanical properties in the liquid at T m, and in particular shear viscosity, which is related to surface tension via the velocity of sound and the thickness of the liquid vapour interface.