Equilibrium Vacancy Concentration Research Articles

Study on the Melting Temperature, the Jumps of Volume, Enthalpy and Entropy at Melting Point, and the Debye Temperature for the BCC Defective and Perfect Interstitial Alloy WSi under Pressure

The objective of this study is to determine the analytic expressions of the Helmholtz free energy, the equilibrium vacancy concentration, the melting temperature, the jumps of volume, enthalpy the mean nearest neighbor distance and entropy at melting point, the Debye temperature for the BCC defective, the limiting temperature of absolute stability for the crystalline state, and for the perfect binary interstitial alloy. The results obtained from the expressions are combined with the statistical moment method, the limiting condition of the absolute stability at the crystalline state, the Clausius–Clapeyron equation, the Debye model and the Gruneisen equation. Our numerical calculations of obtained theoretical results were carried out for alloy WSi under high temperature and pressure. Our calculated melting curve and relation between the melting temperature and the silicon concentration for WSi are in good agreement with other calculations. Our calculations for the jumps of volume, enthalpy and entropy, and the Debye temperature for WSi predict and orient experimental results in the future.

Self-diffusion of Fe and Pt in L10-Ordered FePt: Molecular Dynamics simulation

Vacancy-mediated lattice diffusion coefficients of Fe and Pt atoms in the chemically ordered L10-FePt phase at temperatures between 1300 and 1600 K were evaluated by means of Molecular Dynamics (MD) simulations. Due to the anisotropic structure of the L10-ordered FePt phase, Fe and Pt diffusion fluxes and the resulting self-diffusion coefficients were considered along and perpendicular to the [001] crystallographic direction. In view of a very low vacancy concentration in real FePt single crystals, steady state conditions of the simulated process were approximated by specifically scaling the self-diffusion coefficients estimated for higher vacancy concentrations to the equilibrium vacancy concentration. This procedure involved the calculation of vacancy formation energies which appeared temperature dependent. The validity of this approach was thoroughly tested and the final results were analyzed and compared to the relevant literature data. The evaluated temperature dependent Fe and Pt self-diffusion coefficients showed Arrhenius behavior, however, their values were much lower than the reported experimental ones. Apart from the inevitable effect of the applied quasi-empirical potentials, the discrepancies might originate from the fact that while the MD simulations addressed a single crystal of FePt defected exclusively with vacancies and antisites, the existence of fast diffusion paths along linear and planar defects cannot be excluded in real materials.

Hydrogen induced vacancy clustering and void formation mechanisms at grain boundaries in palladium

Hydrogen has a significant impact on the formation of vacancies, clusters and voids in palladium and other metals. The formation of vacancy-hydrogen complexes in bulk Pd and at Σ3 and Σ5 grain boundaries was investigated using first-principles calculations and thermodynamic models. Equilibrium vacancy and cluster concentrations were evaluated as a function of temperature and hydrogen partial pressure based on the Gibbs energy of formation including vibrational and configurational entropies. Vacancies were found to be significantly stabilized by association with interstitial hydrogen, leading to enhanced concentrations by several orders of magnitude. Vacancy clusters were further stabilized at grain boundaries, with equilibrium concentrations reaching site saturation for clusters comprising up to three vacancies. Nanovoids were investigated based on Wulff constructions from calculated surface energies of the (0 0 1) and (1 1 1) terminations as a function of temperature and coverage of hydrogen adsorbates. The most stable termination changed from (1 1 1) in vacuum to (0 0 1) in H2, and the surface energies were lowered due to hydrogen adsorbates. Consequently, voids were also stabilized in the presence of hydrogen. Coalescing of vacancies into nanovoids was found to be thermodynamically unfavorable due to the loss of configurational entropy. It was therefore concluded that enhanced concentrations of vacancies and clusters does not directly cause the formation of voids. The formation of voids in Pd-based membranes was discussed in terms of microstructural characteristics, and strain due to chemical expansion and plastic deformation.

Modification of the statistical moment method for the high-pressure melting curve by the inclusion of thermal vacancies

Melting behaviors of defective crystals under extreme conditions are theoretically investigated using the statistical moment method. In our theoretical model, heating processes cause missing atoms or vacancies in crystal structures via dislocating them from their equilibrium positions. The coordination number of some atoms is assumed to be removed by one unit and the defect depends on temperature and external pressure. We formulate analytical expressions to directly connect the equilibrium vacancy concentration, the elastic modulus, and the melting temperature. Numerical calculations are carried out for six transition metals including Cu, Ag, Au, Mo, Ta, and W up to 400 GPa. The obtained results show that vacancies strongly drive the melting transition. Ignoring the vacancy formation leads to incorrect predictions of the melting point. The good agreement between our numerical data and prior experimental works validates the accuracy of our approach.

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.

Hydrogen Induced Vacancy Clustering and Void Formation Mechanisms at Grain Boundaries in Palladium

Hydrogen has a significant impact on the formation of vacancies, clusters and voids in palladium and other metals. The formation of vacancy-hydrogen complexes in bulk Pd and at Σ3 and Σ5 grain boundaries was investigated using first-principles calculations and thermodynamic models. Equilibrium vacancy and cluster concentrations were evaluated as a function of temperature and hydrogen partial pressure based on the Gibbs energy of formation including vibrational and configurational entropies. Vacancies were found to be significantly stabilized by association with interstitial hydrogen, leading to enhanced concentrations by several orders of magnitude. Vacancy clusters were further stabilized at grain boundaries, with equilibrium concentrations reaching site saturation for clusters comprising up to three vacancies. Nanovoids were investigated based on Wulff constructions from calculated surface energies of the (0 0 1) and (1 1 1) terminations as a function of temperature and coverage of hydrogen adsorbates. The most stable termination changed from (1 1 1) in vacuum to (0 0 1) in H2, and the surface energies were lowered due to hydrogen adsorbates. Consequently, voids were also stabilized in the presence of hydrogen. Coalescing of vacancies into nanovoids was found to be thermodynamically unfavorable due to the loss of configurational entropy. It was therefore concluded that enhanced concentrations of vacancies and clusters does not directly cause the formation of voids. The formation of voids in Pd-based membranes was discussed in terms of microstructural characteristics, and strain due to chemical expansion and plastic deformation.

On the Melting of Defective FCC Interstitial Alloy γ-FeC under Pressure up to 100 GPa

A statistical moment method has been applied to study the melting of defective face-centered cubic (FCC) interstitial alloy γ-FeC under pressure up to 100 GPa. The theory of melting is based on the absolute stability limiting condition of the crystalline state. The relationship between the pressure, temperature, and concentration of equilibrium vacancies at the melting point of alloy γ-FeC is discussed and evaluated. The numerical results in limit cases are in good agreement with experiments and calculations by other authors, allowing reliable prediction of the melting features of alloy γ-FeC at high pressures.

Self-diffusion in a triple-defect A-B binary system: Monte Carlo simulation

In this comprehensive and detailed study, vacancy-mediated self-diffusion of A- and B-elements in triple-defect B2-ordered ASB1-S binaries is simulated by means of a kinetic Monte Carlo (KMC) algorithm involving atomic jumps to nearest-neighbour (nn) and next-nearest-neighbour (nnn) vacancies. The systems are modelled with an Ising-type Hamiltonian with nn and nnn pair interactions completed with migration barriers dependent on local configurations. Self-diffusion is simulated at equilibrium and temperature-dependent vacancy concentrations are generated by means of a Semi Grand Canonical MC (SGCMC) code. The KMC simulations reproduced the phenomena observed experimentally in Ni-Al intermetallics being typical representatives of the 'triple-defect' binaries. In particular, they yielded the characteristic ‘V’-shapes of the isothermal concentration dependencies of A- and B-atom diffusivities, as well as the strong enhancement of the B-atom diffusivity in B-rich systems. The atomistic origins of the phenomenon, as well as other features of the simulated self-diffusion such as temperature and composition dependences of tracer correlation factors and activation energies are analyzed in depth in terms of a number of nanoscopic parameters that are able to be tuned and monitored exclusively with atomistic simulations. The roles of equilibrium and kinetic factors in the generation of the observed features are clearly distinguished and elucidated.

First principles investigation of the vacancy-mediated impurity diffusion in dilute Zr-X (X = Sc, Y, Ce) alloys

The vacancy-mediated diffusion in dilute Zr-X (X = Sc, Y, Ce) alloys were investigated by adopting the eight-frequency model based on density functional theory calculations within the general gradient approximation. The correlation between the diffusion related properties and the intrinsic properties of the solute was revealed in terms of size and chemical effects. It is found that Sc is beneficial to restrain the diffusion in the Zr alloy, and the parameters governing diffusion in these alloys correlate well with the atomic radius difference (ΔR) and the absolute electronegativity difference (|ΔEl|) between X and Zr. With the increasing of ΔR and |ΔEl| from Sc to Ce, the equilibrium vacancy concentration increases while the X-vacancy migration energy decreases, originating from the local lattice distortion near X and the chemical interaction between Zr and X. As a result, the diffusion coefficients (D) show an order of DZr-Sc < DZr-Y < DZr-Ce. Moreover, the diffusion within the basal plane is larger than that between the adjacent basal planes except for Ce, and the difference between them becomes smaller from Sc to Ce, resulting from the anisotropy of the HCP-Zr structure as well as the combination of size and chemical effects.

Thermal expansion coefficient of BCC defective substitutional alloy AB with interstitial atom C

The analytic expressions for the free energy, the concentration of equilibrium vacancies, the isothermal compressibility, the thermal expansion coefficient and the heat capacities at constant volume and at constant pressure of BCC substitutional alloy AB with interstitial atom C and with vacancy under pressure are derived by the statistical moment method. The theoretical results are applied to the thermal expansion coefficient of alloy FeCrSi in the interval of temperature from 700 K to 1100 K, in the interval of substitutional atom concentration from 0% to 10%, in the interval of interstitial atom concentration from 0% to 5% and in the interval of pressure from 0 GPa to 10 GPa. Our calculated results for main metal Fe are compared with the experimental data.

The Vacancy-Wind Factor and the Manning Factor Occurring in Interdiffusion and Ionic Conductivity in Solids

In crystalline solids, during such processes as chemical interdiffusion in alloys, ionic conductivity and the annealing out of radiation damage there will inevitably be a net flux of vacancies. In most cases, when different species of atoms have different jump rates with vacancies within a net flux of vacancies, the phenomenon of the vacancy-wind effect will occur. This effect was first discovered in the 1960s by the late Dr John Manning. It is a subtle phenomenon that comes about because of the local redistribution of the equilibrium concentration of vacancies with respect to two or more species of drifting atoms in a driving force. The effect is captured in various ‘vacancy-wind factors’ (some of which are now sometimes called Manning factors) which formally arise from non-zero off-diagonal Onsager phenomenological transport coefficients and non-unity values of the tracer correlation factors. In this paper, the effect is reviewed and discussed.

Study on the Melting of the Defective Interstitial Alloys TaSi and WSi with BCC Structure

The statistical moment method is used to study the melting of defective interstitial AB alloys, where A is the main element and B is an interstitial atom, with a body-centered-cubic (BCC) structure. The melting temperature of the AB alloy with defects is obtained from the temperature of absolute stability for the crystalline state and the equilibrium vacancy concentration. Numerical calculations are performed for the interstitial alloys TaSi and WSi. Our calculated results are in good agreement with other calculations.

Electrical Resistance of the Most Refractory Carbide Ta0.8Hf0.2C in the Solid and Liquid States (2000–5000 K)

Carbide Ta0.8Hf0.2C in the form of a thin magnetron sputtering layer (~1 μm) was studied under rapid heating (5 μs) by an electric current pulse. The resistivity of this carbide (referred to initial dimensions) and temperature coefficient of resistance were obtained up to 5000 K for the first time. The temperature was measured by surface radiation with the help of high-speed pyrometer calibrated by a temperature tungsten lamp. The sharp rise of temperature coefficient of resistance before melting indicates an increase in the concentration of defects before melting. This confirms the assumption of the appearance of non-stationary paired Frenkel defects in the lattice of rapidly heated solids, which does not have time to establish the equilibrium concentration of vacancies in the lattice.

Multiscale Mechanism of Fatigue Fracture of Ti—6A1-4V Titanium Alloy within the Mesomechanical Space-Time-Energy Approach

Ultrasonic impact treatment (UIT) of alloy Ti-6Al-4V (VT6) causes a high lattice curvature, nanostructuring of thin surface layers, and the formation of complex band structures of T Al pre-precipitates in the a phase of the underlying sublayer, as well as the formation of the martensitic a ' phase. In so doing, the fatigue life of the alloy increases only by a factor of 1.3 due to the negative influence of complex band structures. Positron annihilation spectroscopy revealed a nonequilibrium vacancy concentration in the treated surface layer equal to 10-5, which is by five orders of magnitude greater than the equilibrium vacancy concentration. This makes possible reversible structural transformations through plastic distortion under cyclic loading of VT6 and underlies the increase in fatigue life. There is a convergence of the electron energy distribution curves for VT6 + UIT and Al obtained from the Doppler broadening spectra of annihilation radiation. This result suggests the formation of Ti-Ti-Al clusters and Ti3Al pre-precipitates in etch-resistant banded structures. Hydrogen charging of the ultrasonically treated VT6 surface layers leads to a 4-fold decrease in the fatigue life of the material. This effect is due to the formation of α"-phase martensite laths in the a phase which rearranges the hcp lattice into an orthorhombic structure under the functional influence of hydrogen, with the segregation of vanadium atoms in the α"-phase bands. The segregation causes a convergence of the electron energy distribution curves of VT6 + UIT + HN and V, as evidenced by the Doppler broadening spectra of annihilation radiation. Bundles of α"-phase bands reinforce the nanostructured surface layer, which drastically reduces the fatigue life of the alloy. Its microhardness in the zone of fatigue fracture greatly increases. The multiscale structural analysis of fatigue fracture is carried out on the basis of the mesomechanical space-time-energy approach.

Diffusion and Point Defects in Elemental Semiconductors

Elemental semiconductors play an important role in high-technology equipment used in industry and everyday life. The first transistors were made in the 1950ies of germanium. Later silicon took over because its electronic band-gap is larger. Nowadays, germanium is the base material mainly for γ-radiation detectors. Silicon is the most important semiconductor for the fabrication of solid-state electronic devices (memory chips, processors chips, ...) in computers, cellphones, smartphones. Silicon is also important for photovoltaic devices of energy production.Diffusion is a key process in the fabrication of semiconductor devices. This chapter deals with diffusion and point defects in silicon and germanium. It aims at making the reader familiar with the present understanding rather than painstakingly presenting all diffusion data available a good deal of which may be found in a data collection by Stolwijk and Bracht [1], in the author’s textbook [2], and in recent review papers by Bracht [3, 4]. We mainly review self-diffusion, diffusion of doping elements, oxygen diffusion, and diffusion modes of hybrid foreign elements in elemental semiconductors.Self-diffusion in elemental semiconductors is a very slow process compared to metals. One of the reasons is that the equilibrium concentrations of vacancies and self-interstitials are low. In contrast to metals, point defects in semiconductors exist in neutral and in charged states. The concentrations of charged point defects are therefore affected by doping [2 - 4].

Free energy and concentration of crystalline vacancies by molecular simulation

ABSTRACTWe present an approach for evaluating the concentration of vacancy defects in crystalline materials by molecular simulation. The proposed method circumvents the problem of equilibration of the number of ‘lattice sites’ M, which characterises the trade-off between more, smaller lattice cells (with some vacant), versus fewer, larger cells. Working in a grand-canonical framework, we instead fix M and solve for the chemical potential μ that ensures thermodynamic consistency of the ensemble-averaged pressure and the grand potential. Having determined μ this way for the given M, the equilibrium vacancy concentration and free energy are easily determined. Methods are demonstrated for the classical Lennard–Jones fcc crystal, examining all states where the crystal is stable. We find for this system that the effect of equilibrating M is negligible at all conditions. Also, although the vacancy fraction varies by many orders of magnitude with temperature and density, we find that the value at melting is nearly independent of density, equal to about . Results further show that a lattice-energy approximation (ignoring entropic effects) underestimates the correct concentration by four orders of magnitude at almost all conditions; ignoring only anharmonic contributions underestimates the vacancy concentration at melting by nearly one order of magnitude.

First‐Principles Characterization of Equilibrium Vacancy Concentration in Metamagnetic Shape Memory Alloys: An Example of Ni2MnGa

Despite the fact that there is evidence for the important role that vacancies play in the martensitic transformation (MT) behavior of metamagnetic shape memory alloys (MMSMAs), little theoretical – and even experimental – work on the thermodynamics and kinetics of point defects in these systems has been carried out. Since the MT behavior of MMSMAs has a great influence on their magneto‐caloric response, investigating the vacancy evolution in MMSMAs has potentially a significant technological impact. Scarcity of studies may be due to the limited characterization capability available for studying vacancy properties as well as their impacts on the materials performance. The current work seeks to introduce the application of the grand‐canonical dilute‐solution model to the investigation of equilibrium (thermal) populations of point defects in Ni2MnGa, used as a prototypical MMSMA. The thermodynamic model is coupled to first‐principles calculations of the energetics of defect‐containing supercell structures. Such characterization capability allows for more realistic investigations of MMSMAs with the vacancy degree of freedom taken into account and subsequently opens up many interesting research topics. Here we demonstrate the capability of the first principles based characterization method by investigating the role of vacancy concentration in the kinetics of order–disorder (ODO) process and the MT temperature of Ni2MnGa.

Phase and vacancy behaviour of hard "slanted" cubes.

We use computer simulations to study the phase behaviour for hard, right rhombic prisms as a function of the angle of their rhombic face (the "slant" angle). More specifically, using a combination of event-driven molecular dynamics simulations, Monte Carlo simulations, and free-energy calculations, we determine and characterize the equilibrium phases formed by these particles for various slant angles and densities. Surprisingly, we find that the equilibrium crystal structure for a large range of slant angles and densities is the simple cubic crystal-despite the fact that the particles do not have cubic symmetry. Moreover, we find that the equilibrium vacancy concentration in this simple cubic phase is extremely high and depends only on the packing fraction and not the particle shape. At higher densities, a rhombic crystal appears as the equilibrium phase. We summarize the phase behaviour of this system by drawing a phase diagram in the slant angle-packing fraction plane.

Thermodynamics of vacancies and clusters in high-entropy alloys

By means of thermodynamic analyses, we provide the intriguing evidence that the equilibrium vacancy concentrations and their clusters in high entropy alloys (HEAs) are substantially larger than those in pure metals and simple binary alloys. The increased defect concentrations strongly change the thermodynamic and kinetic properties of the HEAs. With these findings, we predict a strong recombination tendency of vacancies and interstitials after excitation from radiation bombardments, and the results are favorably compatible with very recent experiments on the superior radiation resistance of HEAs. The thermodynamics of vacancies and clusters in HEAs will be of great interest to the design of advanced materials involving many thermodynamic and kinetic properties such as thermal conductivity, diffusivity, precipitation, creep, and irradiation damage.

Effect of nitrogen incorporation on the ordering transformation of CoPt in CoPt/TiN bilayer films

The effect of nitrogen incorporation on the L10 ordering transformation of CoPt thin films deposited by dc magnetron sputtering on MgO substrate with/without TiN seed layers has been investigated. Without TiN seed layer, CoPt film is subjected to a tensile stress, and the perpendicular magnetic anisotropy (PMA) decreases with N2 gas flow ratio. This is due to the degradation of tensile stress, which is proven to be beneficial to the ordering transformation of L10 CoPt. On the contrary, when sputtered on TiN seed layer, CoPt film is subjected to a compressive stress and the PMA increases with N2 gas flow ratio. The increase of PMA is due to the increase in equilibrium vacancy concentration, which also promotes the diffusion of Co and Pt atoms during annealing. On quartz substrate, the maximum perpendicular coercivity of 11kOe was obtained at the CoPt/TiN bilayer films with 50% partial N2 gas flow ratio after post annealing at 700°C. By partially flowing nitrogen during the deposition, a 200°C reduction of ordering temperature was also obtained.