Equilibrium Vacancy Concentration Research Articles

Atom probe tomography-assisted kinetic assessment of spinodal decomposition in an Al-12.5 at.%Zn alloy

The rates of atomic clustering and precipitation hardening are closely related to the diffusivity of solutes and the concentration of vacancies during the natural aging of aluminum alloys. The measurement of the diffusivity of solutes at room temperature, especially in systems with an equilibrium vacancy concentration, is beneficial to the design of the aging process. However, this measurement has long been challenging because of the extremely low diffusion rates of solutes in aluminum at room temperature and the presence of supersaturated vacancies. In this work, we propose a method to quantify the diffusivity of solutes based on the kinetic evaluation of the spinodal decomposition process. This evaluation involves conducting atom probe tomography experiments, analyzing the radial distribution function, and modeling the phase separation process using the Cahn–Hilliard theory. The aging experiments were conducted on nanoscale samples, where excess vacancies can be eliminated at free surfaces due to a high surface-to-volume ratio. The results yielded a diffusivity of Zn in the Al-12.5 at.% Zn alloy of (1.32±0.46)×10−25 m2/s at 295 K. This work introduces a novel approach to assess the solute diffusivity under conditions of equilibrium vacancy concentration at room temperature and expands the temperature range for measuring the diffusivity in systems with spinodal decomposition, particularly in cases where kinetic data at low temperatures are scarce.

The effect of micromechanical stresses on vacancy formation and stress-driven mass-transport in polycrystalline Fe–Au alloy

In recent years, a new class of super saturated binary and ternary alloys have demonstrated the ability for the self-healing of creep-induced voids formed at the grain boundaries. However, a clear understanding of the parameters affecting the self-healing mechanism is still not yet complete. One of the main challenges is understanding the effect of microstructure and micromechanical stresses on the redistribution of the healing-solute and vacancies. To this end, we address this issue using a CALPHAD-informed diffusion model coupled with crystal plasticity. In principle, the approach is general and can be used for any binary Fe–X alloy, but in this work Fe–Au binary system is used since it experimentally showed the best healing efficiency. First, we present a multicomponent diffusion model considering cross and stress-driven diffusion. The effect of stress was also considered on the equilibrium vacancy concentration. To investigate the effect of the micromechanical stresses, a representative volume element (RVE) was obtained using the phase-field method. The results showed that the maximum vacancy concentration is at the grain boundaries (GBs) with the highest hydrostatic tensile stresses. These were also the regions of the highest Au enrichment. A crucial factor to achieve this is the high diffusivity of Au compared to the Fe matrix. Increasing the stresses, lead to an increase both in vacancy and Au concentration. The accompanying increased stress triaxiality is suggested to be the reason for the reduced self-healing efficiency observed in previous experimental studies.

Modeling the Effect of Excess Vacancies on Precipitation and Mechanical Properties of Al–Mg–Si Alloys

A model for vacancy annihilation during aging has been combined with a precipitation model for coupled nucleation, growth, and coarsening in AA 6xxx series aluminum alloys. The simulation results were compared with precipitation parameters from TEM measurements and hardness data obtained for various times during artificial aging. Both simulations and measurements indicated that a combination of an excess concentration of non-equilibrium vacancies at the start of aging and a fast vacancy annihilation rate significantly affected the resulting precipitation and strength evolution. Hence, the model reproduced the short aging time required to reach the maximum strength when direct artificial aging was applied (DAA). In contrast to the fast aging response of DAA, the hardness measurements showed a much slower aging response when artificial aging was performed after prolonged natural aging. This aging behavior was captured in the model simulations by assuming that an equilibrium vacancy concentration is present from the start of the aging.

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.

Defect properties of a body-centered cubic equiatomic TiVZrTa high-entropy alloy from atomistic simulations

TiVZrTa high-entropy alloys (HEAs) have been experimentally proven to exhibit excellent irradiation tolerance. In this work, defect energies and evolution were studied to reveal the underlying mechanisms of the excellent irradiation tolerance in TiVZrTa HEA via molecular statics calculations and molecular dynamics simulations. The atomic size mismatch of TiVZrTa is ∼6%, suggesting a larger lattice distortion compared to most face-centered cubic and body-centered cubic M/HEAs. Compared to pure Ta and V, smaller vacancy formation and migration energies with large energy spreads lead to higher equilibrium vacancy concentration and faster vacancy diffusion via low-energy migration paths. Vacancies in TiVZrTa have weaker abilities to form large vacancy clusters and prefer to form small clusters, indicating excellent resistance to radiation swelling. The formation energies of different types of dumbbells in TiVZrTa show significant differences and have large energy spreads. The binding abilities of interstitials in TiVZrTa are weaker compared to that in pure Ta and V. In TiVZrTa, fast vacancy diffusion and slow interstitial diffusion result in closer mobilities of vacancies and interstitials, significantly promoting point defect recombination. We further studied the effects of short-range ordered structures (SROs) on defect diffusion and evolution. SROs in TiVZrTa can effectively lead to higher fractions of defect recombination and fewer surviving defects. Our findings provide a comprehensive understanding of the underlying mechanisms of the high irradiation tolerance in body-centered cubic HEAs with large lattice distortion and suggest SROs are beneficial microstructures for enhancing irradiation tolerance.

Assessing the high concentration of vacancies in refractory high entropy alloys

Accurate determination of monovacancy formation enthalpy is vital for work on diffusion, melting point determination of high temperature materials, radiation damage, and thermophysical stability of alloys. These enthalpies take on a single value in pure metals but is made more complex in concentrated solid solutions due to the local chemical environments possible around each vacancy. Herein, using first-principles density functional theory, we report the distributions of vacancy formation enthalpies in 21 equiatomic 5-component solid-solution alloys from the Hf-Mo-Nb-Ta-Ti-W-Zr system. Chemical disorder is treated using the special quasi-random structure method, and the chemical potential of vacant element is treated by approximating it to its total energy in its standard state. We find that the highest vacancy formation enthalpies belong to the MoNbTaTiW alloy (which incidentally is the most reported in literature) and the lowest belongs to HfNbTaTiZr, with other alloys in-between. We use the whole distribution of formation enthalpies to estimate the equilibrium concentration of vacancies as a function of temperature.

Atomic diffusion in bcc Fe–Mn alloys: Theoretical analysis and experimental measurements across the Curie temperature

The influence of a magnetic transition on Fe and Mn diffusion in bcc Fe–Mn alloys as a function of Mn concentration is studied combining experimental measurements and DFT-informed modeling. The radiotracer technique in combination with precise mechanical sectioning or ion-beam sputtering is used to determine the diffusion rates of 59Fe and 54Mn in a series of Fe–Mn alloys with up to 1.9 at.%Mn. The solvent and solute enhancement factors are determined. While the diffusion rates of Fe atoms are found to be almost independent on the Mn content in the Fe–Mn alloys in both para- as well ferro-magnetic states, Mn diffusion is strongly enhanced by an increase of Mn concentration in these alloys, especially in the paramagnetic state. The experimental findings are supported by theoretical results, which are analyzed in terms of an equilibrium vacancy concentration, atomic jump frequencies, Mn–vacancy binding energy, short-range ordering tendency and kinetic correlation factors.

Microstructure and defect engineering of graphite anodes by pulsed laser annealing for enhanced performance of lithium-ion batteries

Nanosecond pulsed laser annealing significantly improves cyclability and current carrying capacity of lithium-ion batteries (LIBs). This improvement is achieved by engineering of microstructure and defect contents present in graphite in a controlled way by using pulsed laser annealing (PLA) to increase the number density of Li+ ion trapping sites. The PLA treatment causes the following changes: (1) creates surface steps and grooves between the grains to improve Li+ ion charging and intercalation rates; (2) removes inactive polyvinylidene difluoride (PVDF) binder from the top of graphite grains and between the grains which otherwise tends to block the Li+ migration; and (3) produces carbon vacancies in (0001) planes which can provide Li+ charging sites. From X-ray diffraction data, we find upshift in diffraction peak or reduction in planar spacing, from which vacancy concentration was estimated to be about 1.0%, which is higher than the thermodynamic equilibrium concentration of vacancies. The laser treatment creates single and multiple C vacancies which provide sites for Li+ ions, and it also produces steps and grooves for Li+ ions to enter the intercalating sites. It is envisaged that the formation of these sites enhances Li+ ion absorption during charge and discharge cycles. The current capacity increases from an average 360 mAh/g to 430 mAh/g, and C–V shows significant reduction in SEI layer formation after the laser treatment. If the vacancy concentration is too high and charge-discharge cycles are long, then trapping of electrons by Li+ may occur, which can lead to Li0 formation and Li plating causing reduction in current capacity.

Influence of cooling rate during cryogenic treatment on the hierarchical microstructure and mechanical properties of M54 secondary hardening steel

Cryogenic treatment (CT) is an essential heat treatment process in improving the mechanical properties of secondary hardening ultra-high strength steel (SH-UHSS). The present study investigated the influence of cooling rate during CT on the hierarchical microstructure and strength-toughness of M54 SH-UHSS. Increasing cooling rate can synchronously refine martensite matrix and M2C-type (M = Mo, Cr, W, V) carbides, while there are the most refined martensite blocks and highest-density precipitation of the carbides at cooling rate of 3 °C·min-1. Further increased cooling rate weakens the effect of CT on the refinement and precipitation of martensite matrix. The refinement is related to the high-level segregation of carbon atoms and favorable equilibrium concentration of vacancies during CT and pinning effect of carbides on mobile dislocation during tempering. Besides, uniform carbon clusters and high-nucleation rate by the refined martensite matrix and high-density of dislocations mostly contribute to the above beneficial precipitation of carbides. Precipitation and martensitic matrix strengthening mostly contribute to the ultra-high strength (yield strength of 1730 MPa, ultimate tensile strength of 2018 MPa) at the cooling rate of 3 °C·min-1; meanwhile, the refined blocks as "effective" controlling unit is identified to be the major toughening mechanism contributing to a desirable impact toughness (V-notched Charpy impact energy of 30 J). This study would be instructive for processing the engineering components with large cross-section.

Equilibrium vacancy concentration and thermodynamic quantities of BCC defective alloys FeCrSi and VWSi under pressure

In this paper, we present the analytic expressions of the cohesive energy, the alloy parameters, the equation of state, the mean nearest neighbor distance, the Helmholtz free energy, equilibrium vacancy concentration and thermodynamic quantities such as the isothermal compressibility, the thermal expansion coefficient, the heat capacities at constant volume and at constant pressure for BCC defective ternary substitutional and interstitial alloy ABC derived by the statistical moment method. The obtained thermodynamic quantities depend on temperature, pressure, concentration of substitutional atoms, concentration of interstitial atoms and equilibrium vacancy concentration. Thermodynamic quantities of BCC defective metal A, BCC defective substitutional alloy AB, BCC defective interstitial alloy AC and BCC defective metal A are specific cases for thermodynamic quantities of BCC defective ternary substitutional and interstitial alloy ABC. The theoretical results are calculated numerically to alloys FeCrSi and VWSi. Our calculated results of thermal expansion coefficient and heat capacities at constant pressure for main metals Fe, V are in good agreement with experimental data. Our other calculated results for thermodynamic quantities of alloys FeCrSi and VWSi at different temperature, pressure, concentration of substitutional atoms and concentration of interstitial atoms orient and predict new experimental data in the future

Mapping defects during phase transformation in high Cr content NiCr solid-solution through positron trapping

Understanding the role of point defects towards hardening –and influence on secondary phase evolution demands a systematic investigation of defects during aging. In the present work, we investigate defects present/evolved during thermal aging in high Cr content NiCr solid-solution using positron annihilation lifetime and coincidence Doppler broadening spectroscopy. Thermal aging at the studied temperature results in phase transformation of the matrix primarily by discontinuous precipitation leading to Cr precipitation as revealed from field-effect scanning electron microscopy. Positron annihilation results reveal that non-equilibrium thermal (excess) vacancies are predominant defects in the studied alloy. These vacancies are retained in the alloy due to the formation of vacancy-solute complex. During solutionization treatment, the excess vacancies are largely recovered attaining an equilibrium vacancy concentration. On the other hand, excess vacancies are not recovered during thermal aging and stable up to 24 h aging period. In addition to vacancy-solute complex, dislocations are evolved at higher aging times and suggested to be present at the interface of Cr precipitate and matrix. Under the consideration of positron trapping model, the estimated atomic concentration of vacancies (vacancy-solute complex) in the phase transformed NiCr is ~ 10−5 and the dislocation density is ~ 4 × 109 cm−2.

Model for non-equilibrium vacancy diffusion applied to study the Kirkendall effect in high-entropy alloys

The effect of the non-equilibrium vacancy on the Kirkendall porosity formation was studied by means of a developed model of the multi-component diffusion with vacancies (MDV) which includes the intrinsic fluxes with vacancy gradient and non-ideal sources and sinks for vacancies. For this study, the diffusion couple experiments in multi-component alloys were chosen. To handle the case of concentration-dependent equilibrium vacancy concentration, we introduced the interaction parameters between the components and vacancies, which can have strong effects on the equilibrium vacancy concentration in alloys and on thermodynamic factors. The diffusion profiles of components and vacancies were simulated by using thermodynamic and kinetic data. The different intensity of the vacancy annihilation/generation and different initial vacancy distributions were considered. Furthermore, we show that the conventional model of diffusion in multi-component systems is a particular case of the MDV with a specific sink/source term. The conventional model was extended by the vacancy diffusion term, similar to the MDV, which significantly reduces the vacancy gradient and the pore formation near the Matano plane. The numerical results demonstrate that the diffusion profiles of substitutional components slightly depend on the sink/source intensity if the none-zero net flux of substitutional components is not significant and the sources and sinks of vacancies are not sparse, whereas the porosity depends very strongly and correlates with the vacancy distribution. For the simulation of variable equilibrium vacancy concentrations using the MDV, the corresponding interaction parameters related to vacancies are necessary to be included in the thermodynamic assessment.

Distinct point defect behaviours in body-centered cubic medium-entropy alloy NbZrTi induced by severe lattice distortion

The point defect properties of body-centered cubic medium-entropy alloy NbZrTi were studied by first-principles calculations. Due to severe lattice distortion, a significant portion of conventional vacancy and interstitial structures are unstable and require large structural relaxation, indicating an irregular energy landscape with large site-to-site variations. The average vacancy and interstitial formation energy are 0.95 eV± 0.34 eV and 1.92 eV ± 0.39 eV, respectively, much lower than that of Nb (2.77 eV and 4.38 eV). The vacancy migration energy exhibits a wide distribution extending to 0 eV, resulting in preferential vacancy migration through low barrier sites. The interstitial diffusion is slower than that of pure Nb due to the reduction of long <111> diffusion induced by the site-to-site variations in stable interstitial orientations. Ti atoms diffuse much faster than Nb and Zr atoms due to the preferential interstitial binding with Ti. The effect of atomic composition and short-range order on elemental and total interstitial diffusion was also investigated. The obtained first-principles results are important for the development of interatomic potentials for radiation damage studies. When irradiated with 3-MeV Fe ions at 675 ∘C to a peak dose of ∼100 dpa, NbZrTi reduced the void formation at high temperature compared to Nb owing to its higher equilibrium vacancy concentration and closer mobility between vacancies and interstitial atoms.

МОДЕЛЮВАННЯ ПОРОУТВОРЕННЯ ПРИ РЕАКЦІЙНІЙ ДИФУЗІЇ У БІНАРНІЙ СИСТЕМІ

The proposed model of void formation takes into account existence two types of sinks/sources of non-equilibrium vacancies, depending on their location: in the phase volume and at the interfacial boundaries. At the interdiffusion and reaction diffusion which happens on the vacancy atomic diffusion mechanism, the inequality of atoms fluxes is caused by their differential mobility, give rise to a directional flux of vacancies. This flux of vacancies cause an appearance of areas in a diffusion zone with supersaturation and deficiency in vacancies, where sinks/sources of non-equilibrium vacancies act. It is believed that the voids arise with a certain periodicity near the interfacial boundary, where there is a vacancy supersaturation due to the different mobility of the components. The voids move in volume of growing phase, their sizes change. The void radius increases as long as void is in the region of the diffusion zone where there is a vacancy supersaturation. The void radius begins to decrease if the void is in the area of the diffusion zone, where there is a negative vacancy supersaturation (the concentration of vacancies is less than the equilibrium) until it disappears. The study of the influence on the kinetics of void formation during reaction diffusion in the binary system of sinks/sources of vacancies was carried out by computer simulation. In the case of efficient operation of vacancies sinks/sources only at the borders (vacancies sinks/sources do not work in the phase volume) we get a significant saturation of vacancies in the diffusion zone, which leads to rapid voids growth and the existence of significant pore size differences. If the vacancies sinks/sources in the volume work well (the efficiency of their work at the interfacial boundaries does not affect the result in this case), then almost everywhere in the system an equilibrium concentration of vacancies is established, as a result of which the vast majority of pores have the same (maximum) size. The pore sizes are much smaller than with the previous case.

Solid atomic hydrogen: Point defect formation and elastic stability

Within the framework of the density functional theory, the resistance of solid atomic hydrogen with an I41/amd crystal structure to the formation of point defects, such as a vacancy, an interstitial site, and a double vacancy, was studied. At the pressures below 265 GPa, the equilibrium vacancy concentration per atom reaches a critical value, even near the temperature of liquid helium. Thus, the destruction of the metastable atomic state occurs at higher pressures than predicted by the calculations based on the phonon spectra. For atomic hydrogen at room temperature, pressures exceeding 280 GPa are required. The elastic stability of the I41/amd atomic phase in the pressure range from 250–600 GPa was investigated. It is shown that there are regions of instability of the I41/amd atomic hydrogen phase, where a transition to the structure with FDDD symmetry occurs.

A Review of Platinum Diffusion in Silicon and Its Application for Lifetime Engineering in Power Devices

In silicon power semiconductors, platinum is used to improve the switching characteristics of the devices by adjusting the lifetime of the minority charge carriers. Platinum diffusion in silicon has been a research topic for many scientists over the past decades. Herein, the current state of knowledge is presented together with the concepts developed for its use in silicon power devices. The relevant processes include diffusion from a platinum silicide, postimplantation annealing, and phosphorus‐diffusion gettering. The evaluation of corresponding experiments and the calibration of the models have not only resulted in more narrow limits for the equilibrium concentrations of vacancies and self‐interstitials in silicon and the transport capacity of interstitial platinum, but also in improved devices.

EQUILIBRIUM VACANCY CONCENTRATION AND THERMODYNAMIC QUANTITIES OF FCC DEFECTIVE ALLOYS AuCuSi AND PtCuSi UNDER PRESSURE AND TEMPERATURE

We present the analytic expressions of the cohesive energy, the alloy parameters, the equation of state, the mean nearest neighbor distance, the Helmholtz free energy, equilibrium vacancy concentration, and thermodynamic quantities such as the isothermal compressibility, the thermal expansion coefficient, the heat capacities at constant volume and constant pressure for facecentered cubic (FCC) defective ternary substitutional and interstitial alloy ABC derived by the statistical moment method (SMM). The obtained thermodynamic quantities depend on temperature, pressure, the concentration of substitutional atoms, the concentration of interstitial atoms, and equilibrium vacancy concentration. Thermodynamic quantities of FCC defective metal A, FCC defective substitutional alloy AB, and FCC defective interstitial alloy AC are specific cases for thermodynamic quantities of FCC defective ternary substitutional and interstitial alloy ABC. The theoretical results are calculated numerically to alloys AuCuSi and PtCuSi. Our calculated results of thermal expansion coefficient and heat capacities at constant pressure for main metals Au, Pt are in good agreement with experimental data. Our other calculated results for thermodynamic quantities of alloys AuCuSi and PtCuSi at different temperatures, pressure, the concentration of substitutional atoms, and concentrations of interstitial atoms orient and predict new experimental data in the future.

Effects of magnetic excitations and transitions on vacancy formation: Cases of fcc Fe and Ni compared to bcc Fe

Vacancy is one of the most frequent defects in metals. We study the impacts of magnetism on vacancy formation properties in fcc Ni, and in bcc and fcc Fe, via density functional theory (DFT) and effective interaction models combined with Monte Carlo simulations. Overall, the predicted vacancy formation energies and equilibrium vacancy concentrations are in good agreement with experimental data, available only at the high-temperature paramagnetic regime. Effects of magnetic transitions on vacancy formation energies are found to be more important in bcc Fe than in fcc Fe and Ni. The distinct behaviour is correlated to the relative roles of longitudinal and transversal spin excitations. At variance with the bcc-Fe case, we note a clear effect of longitudinal spin excitations on the magnetic free energy of vacancy formation in fcc Fe and Ni, leading to its steady variation above the respective magnetic transition temperature. Below the N\'{e}el point, such effect in fcc Fe is comparable but opposite to the one of the transversal excitations. Regarding fcc Ni, although neglecting the longitudinal spin excitations induces an overestimation of the Curie temperature by 220 K, no additional effect is visible below the Curie point. The distinct effects on the three systems are closely linked to DFT predictions of the dependence of vacancy formation energy on the variation of local magnetic-moment magnitudes and orientations.

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.