Solute-vacancy Binding Energy Research Articles

Initial structural transformation of Mg-Sm binary alloys during early isothermal aging

ABSTRACT The initial structural transformation of Mg-Sm binary alloys on early isothermal aging is investigated by first-principles calculations. There are three typical precipitate structures in this period, namely hexagonal structure, continuous zigzag structure, and discontinuous zigzag structure (˅˄ structure), which are the main transition state structures formed. The formation of clusters is a dynamic process needing to combine vacancy formation and migrations. Among these structures, the lower vacancy formation energy and dilute mixing energy (byproducts of the solute-vacancy binding energy calculation) make hexagonal clusters more possible to form and explain experimental observation of hexagonal structure abundant existing in the matrix, although its formation energy is slightly higher. Furthermore, vacancy-assisted diffusion in doped Mg lattices is calculated using the nudge elastic band method to investigate the kinetics of the cluster formation.

Investigation of vacancy trapping by solutes during quenching in aluminum alloys

In aluminum alloys, excess vacancies often form pairs or clusters with solute atoms. This is commonly referred to as vacancy trapping by solutes. Such trapping results in a change in the number of mobile or free vacancies, which in turn affects solute diffusion and precipitate formation during heat treatment. In the current work, such trapping is investigated indirectly in the Al-Sn binary model alloy via studying the dependence of Frank loop formation on the availability of free vacancies. Current experimental results show that the addition of 100 at. ppm. Sn to aluminum leads to a significant reduction in the number density of quenched-in Frank loops. Modeling of mono- and divacancy trapping by solutes during quenching demonstrates the influences of the vacancy-solute binding energy, solute concentration, and temperature on the trapping process. The results reveal that the binding between vacancies and solutes significantly reduces the concentration of free vacancies after quenching in Al-Sn.

Xenon mobility in γ-uranium and uranium–molybdenum alloys

Diffusion in bcc uranium and U–Mo alloys is of great interest because fission gas and other fission products impact the performance of nuclear fuels. We investigate the mobility of xenon and molybdenum in bcc uranium (γ-U) and metallic U–Mo alloys by calculating the migration energies of xenon and molybdenum for various local compositions using density functional theory. We also calculate the solute–vacancy binding energies of different solutes to vacancies in bcc uranium. We find that the solute–vacancy binding energy in bcc uranium is significantly higher than it is in other bcc metals (e.g., Fe and W). We also find that the migration energy of molybdenum is substantially higher than the migration energy of xenon, indicating that xenon is much more mobile than molybdenum in bcc uranium. The presence of molybdenum in the nearest-neighbor shell around a xenon atom typically increases the migration energy of xenon, which indicates a reduction of xenon mobility in U–Mo alloys compared to pure bcc uranium.

Hot deformation behavior and dynamic precipitation characteristics of Mg-Gd-Nd(-Sm)-Zr alloys

We deeply analyze the deformation behavior of Mg-Gd-Nd(-Sm)-Zr alloys under high temperature uniaxial compression, and explore the influence mechanism of Sm on the microstructure evolution of Mg-Gd-Nd-Zr alloy. We can draw the following conclusions: the addition of Sm delays the dynamic recrystallization of Mg-Gd-Nd-Zr alloy, resulting in the increase of stress and the decrease of machinability. There is no dynamic precipitation in Mg-Gd-Nd-Zr alloy at 400 °C/0.01 s−1/0.7, which is related to the high solute-vacancy binding energy of Nd. Under the same deformation condition, two kinds of dynamic precipitates with different size and shape appear in Mg-Gd-Nd-Sm-Zr alloy: fine particle phase and coarse lamellar phase. The two precipitates are Mg(Gd, Nd, Sm) with body-centered cubic structure, and are incoherent with the matrix. The coarse lamellar phase effectively hinders dislocation rearrangement, which explains the delay of dynamic recrystallization.

First-principles calculations of solute–vacancy interactions in aluminum**Project supported by the National Natural Science Foundation of China (Grant Nos. 51701095 and 51771185) and the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20170798).

The interactions of solute atoms with vacancies play a key role in diffusion and precipitation of alloying elements, ultimately influencing the mechanical properties of aluminum alloys. In this study, first-principles calculations are systematically performed to quantify the solute–vacancy interactions for the 3d–4p series and the 4d–5p series. The solute–vacancy interaction gradually transforms from repulsion to attraction from left to right. The solute–vacancy binding energy is sensitive to the supercell size for elements at the beginning. These behaviors of the solute–vacancy binding energy can be understood in terms of the combination and competition between the elastic and electronic interactions. Overall, the electronic binding energy follows a similar trend to the total binding energy and plays a major role in the solute–vacancy interactions.

First-principles study of interaction energies of atomic defects in bcc ferromagnetic iron

A comprehensive calculation of the solute formation energies, solute-solute and vacancy-solute binding energies in bcc iron is reported. An extended set of solutes with atomic numbers 1 to 54 has been considered. We used the projector augmented wave method of density functional theory with the generalized gradient approximation for the exchange-correlation energy functional. The prominent results are the following: (1) formation energies of solutes from fourth and fifth periods vary with their atomic numbers such that they reach maxima near the ends of the periods and a minimum in between, with a local increase near Cu and Ag (like a quasiparabolic valley). Solutes from second and third periods show similar trend like the elements near the ends of the fourth and fifth periods. (2) The size factors of the solutes also show similar variation with their atomic numbers like their formation energies. These trends corroborate the relatively smaller formation energies and size factors of the common alloying additions to Fe (such as $3d,4d$, and $sp$ elements) as compared to solutes that lack solubility (such as Li, Na, K, Rb, He, Ne, Ar, Kr, Xe, F, Cl, Br, I, Mg, Ca, Sr, Ag, Cd, In, Y). (3) The solubilities estimated from our formation energies are found to be in reasonable agreement with those from the phase diagram database. (4) Solute-solute and vacancy-solute binding energies are found to vary with the atomic number of the solutes in a manner inverse to solute formation energies and solute size factors, reaching strong binding energies near the ends of the periods which generally include the insoluble elements. (5) Another trend revealed by our work is that the size factors of isoelectronic sets of solutes increase down the groups with associated increase of formation energies, and strength of solute-solute and vacancy-solute binding energies. (6) A significant correlation is found between our vacancy-solute binding energies of $3d$ and $4d$ elements and the corresponding diffusion coefficients from literature whereby solutes with strong binding energies have higher diffusion coefficients and vice versa.

Enhancement of High Temperature Strength of 2219 Alloys Through Small Additions of Nb and Zr and a Novel Heat Treatment

This paper presents a detailed investigation on the effect of small amount of Nb and Zr additions to 2219 Al alloy coupled with a novel three-stage heat treatment process. The main aim of the work is to increase the high temperature strength of 2219 alloy by introducing thermally stable L12 type ordered precipitates in the matrix as well as by reducing the coarsening of metastable strengthening θ″ and θ′ precipitates. To achieve this, small amounts of Nb and Zr are added to 2219 alloy melt and retained in solid solution by suction casting in a water-cooled copper mould having a cooling rate of 102 to 103 K/s. The suction cast alloy is directly aged at 673 K (400 °C) to form L12 type ordered coherent Al3Zr precipitates. Subsequently, the alloy is solution treated at 808 K (535 °C) for 30 minutes to get supersaturation of Cu in the matrix without significantly affecting the Al3Zr precipitates. Finally, the alloy is aged at 473 K (200 °C), which results in the precipitation of θ″ and θ′. Microstructural characterization reveals that θ″ and θ′ are heterogeneously precipitated on pre-existing uniformly distributed Al3Zr precipitates, which leads to a higher number density of these precipitates. This results in a significant increase in strength at room temperature as well as at 473 K (200 °C) as compared to the 2219 alloy. Furthermore, the alloy remains thermally stable after prolonged exposure at 473 K (200 °C), which is attributed to the elastic strain energy minimization by the conjoint Al3Zr/θ′ or Al3Zr/θ″ precipitates, and the high Zr and Nb solute-vacancy binding energy, retarding the growth and coarsening of θ″ and θ′ precipitates.

On the abnormal fast diffusion of solute atoms in α-Ti: A first-principles investigation

Solute atoms such as Fe, Co, and Ni diffuse abnormally fast in α-Ti, which influences significantly the mechanical properties of the titanium alloys. Various mechanisms (e.g., the interstitial diffusion mechanism and solute-vacancy complex mechanism) have been proposed to account for the fast diffusion of these solutes in α-Ti. To elucidate such diffusion mechanism, a first-principles method is employed to calculate the formation energies, migration energy barriers, and solute-vacancy binding energies of the substitutional and interstitial solute atoms including Al, Si, Sn, V, Mn, Fe, Co, Ni, and Cu in α-Ti. Based on the calculated parameters, the diffusion mechanisms are discussed. Comparing the formation energies of the substitutional and interstitial solutes, we find that all the solute atoms prefer the substitutional configuration to the interstitial one. The interstitial migration energy barriers are quite low for all the solutes. Al and Sn diffuse mainly through normal vacancy mediated mechanism due to the high substitutional to interstitial preferential energy that leads to very low fraction of interstitial solutes (about 10−15∼10−16 at 1000 K) at thermal equilibrium state and high interstitial diffusion activation energy. The 3d metal solute atoms, especially Mn, Fe, and Co, are fast diffusers and the diffusion coefficients are dominated by the interstitial mechanism because of their sizable thermal equilibrium interstitial fractions (several percent at 1000 K). The solute-vacancy complex mechanism is not likely to account for the fast diffusions in α-Ti. We show that the direct chemical interaction between the solute and matrix atoms determines the site-occupancy of the solute atoms in α-Ti besides the atomic size effect that was commonly believed to be responsible for the fast diffusions.

Energetics of cobalt alloys and compounds and solute–vacancy binding in fcc cobalt: A first-principles database

Using extensive first-principles density functional calculations, we calculate thermodynamic properties of binary fcc cobalt-based alloys with 25 different solute elements. For each solute element X, we calculate its (a) nearest- and next-nearest-neighbor solute–vacancy binding energy, (b) dilute impurity mixing energy with respect to the equilibrium and hypothetical fcc-based reference states of the solute, (c) enthalpy of formation of cobalt-rich binary ordered compounds in the Co–X system, (d) solubility enthalpy, and other derived quantities. We find that the solute–vacancy binding energies of all the studied solutes in fcc cobalt are positive (indicating favorable binding), in contrast to nickel, aluminum, magnesium, and copper-based alloys, where mid 3d transition metal solutes have unfavorable solute–vacancy binding. We study the physical and chemical effects influencing solute–vacancy binding energy, and find that (a) it correlates broadly with solute size—larger solute atoms possess stronger binding with vacancies—and this is understood in terms of strain relief and secondary next-nearest-neighbor interactions, and (b) it follows a parabolic trend as a function of d-occupancy across the transition metal series—falling to a minimum at the middle of the series—and this is understood in terms of a d-band filling effect. We also find this d-band filling effect in other calculated quantities such as the dilute impurity volumes and dilute impurity mixing energies. To aid in building thermodynamic databases, we tabulate all the calculated thermodynamic quantities, and compare them with experimental phase diagrams data and previous literature, and find good agreement where such data is available.

Microalloying for the controllable delay of precipitate formation in metal alloys

A model is presented to predict the effectiveness of dilute solutes in delaying precipitate formation, with application to natural and artificial aging in metal alloys. Control of aging is achieved via the binding, at natural aging temperatures, and release, at artificial aging temperatures, of excess quenched vacancies by the solute atoms. The binding of vacancies to the solute atoms reduces the vacancy concentration in the bulk lattice, and thus reduces the rate of transport processes that control aging. To be useful, this strategy requires sufficiently strong, but not too strong, vacancy-solute binding, and sufficiently slow vacancy-solute diffusion, both quantified through application of the model. First-principles methods are used to compute the controlling materials properties (vacancy-solute binding energies and migration barrier) for a range of solutes in Al and Mg, and Pauli Electronegativity of the solute relative to the host metal is found useful for correlating properties. With the computed inputs, the model is applied to representative solutes (Ga, Sn, Pb) in Al that span weak to strong vacancy binding. Predictions of the model to Sn in Al are then shown to qualitatively agree with experiments, with quantitative agreement possible using a slightly stronger Sn-vacancy binding energy. The general model should be useful for identifying promising dilute solutes for control of precipitation/aging across many alloys in which quenched vacancies are used to control precipitation.

First-principles calculations of rare earth (Y, La and Ce) diffusivities in bcc Fe

The impurity diffusivities of rare earth elements, Y, La and Ce, in bcc Fe have been investigated by the first-principles calculations within nine-frequency model. The microscopic parameters in the pre-factor and activation energies have been calculated. For the three elements, the first nearest-neighbor solute-vacancy interactions are all attractive, in which Y and La solute atoms more favorably bond to the vacancy. The solute-vacancy binding energy can be explained in terms of the combination of the distortion binding energy and the electronic binding energy, and the decomposition results of the total solute-vacancy binding energy suggest that the strain-relief effect accounts for larger portion of the binding energy for Y and La than that for Ce. The diffusion coefficients of Y are one order of magnitude larger than that of La, and predicted to be comparable to that of Fe self-diffusion. Compared with Y and La, Ce shows large migration energy and small solute-vacancy attractive interaction, which accounts for the lowest diffusivity of this element.

Mobility of U Solutes in fcc Al: A Theoretical Model

We propose a theoretical model for characterizing the diffusion of solute atoms, driven by vacancy migration. Our model requires only a few static calculations, in the same line of ideas as the five-frequency model. We obtain microscopic parameters, namely: i) the free energy of vacancy formation and the vacancy-solute binding energy, ii) the tracer correlation factor, (ii) the solvent-enhancement factor, and iii) the involved jump frequency. In this paper, we specifically perform calculations in (Al,U) diluted fcc alloy. Our results show excellent agreement with available experimental data for pure Al solvent, and also with result from molecular dynamic calculations for the U mobility diluted in Al. Qualitatively, we conclude that the U solutes diffuse through a drag mechanism.

Diffusion behavior in Nickel–Aluminum and Aluminum–Uranium diluted alloys

Impurity diffusion coefficients are entirely obtained from a low cost classical molecular statics technique (CMST). In particular, we show how CMST is appropriate in order to describe the impurity diffusion behavior mediated by a vacancy mechanism. In the context of the five-frequency model, CMST allows to calculate all the microscopic parameters, namely: the free energy of vacancy formation, the vacancy-solute binding energy and the involved jump frequencies, from them, we obtain the macroscopic transport magnitudes such as: correlation factor, solvent-enhancement factor, Onsager and diffusion coefficients. Specifically, we perform our calculations in f.c.c. diluted Ni–Al and Al–U alloys. Results for the tracer diffusion coefficients of solvent and solute species are in agreement with available experimental data for both systems. We conclude that in Ni–Al and Al–U systems solute atoms migrate by direct interchange with vacancies in all the temperature range where there are available experimental data. In the Al–U case, a vacancy drag mechanism could occur at temperatures below 550K.

Ageing Characteristics of Al-Mg-(Ge,Si)-Cu Alloys

In order to elucidate some of the differences between Al-Mg-Si and Al-Mg-Ge alloys and the role of Cu, a series of Al-Mg-Ge, Al-Mg-Si and Al-Mg-Ge-Si alloys, some of them containing Cu, are investigated by positron annihilation lifetime spectroscopy during natural ageing. Al-Mg-Ge alloys show qualitatively the same evolution of positron lifetime τ1C with time as Al-Mg-Si alloys, namely an initial decrease, followed by a re-increase, after which τ1C drops to an equilibrium value. However, for alloys with equal Mg contents, Ge gives rise to a notably slower ageing kinetics than Si, pointing at effects of atomic size or solute-vacancy binding energies. Adding Cu to both Al-Mg-Ge and Al-Mg-Si alloys slows down the initial formation of clusters but promotes their further growth.

Thermo-Kinetic Simulation of the Yield Strength Evolution of AA7075 during Natural Aging

The yield strength evolution in aluminum alloy 7075 is investigated during natural aging. The thermo-kinetic simulation, capable of predicting nucleation, growth, coarsening and dissolution of metastable and stable hardening precipitates in Al-Zn-Mg-Cu during natural aging, is outlined briefly. A recent strengthening model for shearing and bypassing of precipitates by dislocations is utilized to calculate the evolution of the macroscopic yield strength at room temperature. The simulation accounts for vacancy-solute binding energies calculated with the help of first principles simulations that influence the diffusivity of the system due to the presence of excess quenched-in vacancies. These results provide predictions about the amount of excess vacancies trapped by solid solution alloying elements and how the lifetime of vacancies changes due to attractive or repelling binding forces between vacancies and different solid atoms in the aluminum matrix. In our approach, we calculate the strength evolution after quenching due to interaction between dislocations and changes in the microstructure by precipitation of different kinds of secondary phases. The predicted evolution of yield strength is finally verified on experimental measurements.

Interactions of solute (3p, 4p, 5p and 6p) with solute, vacancy and divacancy in bcc Fe

Solute–vacancy binding energy is a key quantity in understanding solute diffusion kinetics and phase segregation, and may help choice of alloy compositions for future material design. However, the binding energy of solute with vacancy is notoriously difficult to measure and largely unknown in bcc Fe. With first-principles method, we systemically calculate the binding energies of solute (3p, 4p, 5p and 6p alloying solutes are included) with vacancy, divacancy and solute in bcc Fe. The binding energy of Si with vacancy in the present work is in good consistent with experimental value available. All the solutes considered are able to form stable solute–vacancy, solute–divacancy complexes, and the binding strength of solute–divacancy is about two times larger than that of solute–vacancy. Most solutes could not form stable solute–solute complexes except S, Se, In and Tl. The factors controlling the binding energies are analyzed at last.

First-principles study of solute–vacancy binding in Cu

Solute–vacancy binding is a key quantity in understanding diffusion kinetics, and may also have a considerable impact on the hardening response in Cu alloys. However, the binding energies between solute atoms and vacancies in Cu are largely unknown and difficult to measure accurately. A large database of solute–vacancy binding energies in Cu from first-principles calculations based on density functional theory was presented in the paper. The trends in the binding energies in terms of super cell size, solutes size and magnetic moments are analyzed. The calculated binding energies agree well with experimental measurements available.

First-principles study of 4d solute diffusion in nickel

Diffusion of the 4d transition elements in Ni has been investigated within the five-frequency model framework using migration energy barriers calculated from the first principles. Agreement with counterintuitive experimental/calculated data is observed; atoms in the middle of 4d row have the smallest atomic radii while exhibiting the lowest diffusivity as compared to larger atoms at the beginning and the end of 4d row. We show that 4d solute diffusion is controlled mainly by the size misfit. The larger atoms have higher solute–vacancy binding energies and lower migration barriers. Both were shown to correlate with a displacement of the equilibrium solute position toward the adjacent vacancy. The difference in mechanisms controlling sp- and transition elements diffusion rates in Ni is discussed.

Solute-Vacancy Interactions in Nickel

ABSTRACTSolute trapping of vacancies in a Ni host lattice is a first step to understanding how solute additions may effect ordering phase transformation in Ni-based alloys. Minor additions of solutes (<3 wt.%) may not have an appreciable effect on the thermodynamics of phase stability but strong binding between the vacancy and solute may significantly alter the kinetics of phase transformations. In this work vacancy-solute binding energies are calculated for multiple elements (Cr, Fe, Mn, Si, Ti, Nb, Ta, Cu, Al, Mo) in a pure Ni face-centered cubic lattice. These elements are common in corrosion-resistant Ni-based alloys. Binding enthalpies for the first four nearest-neighbor vacancy-solute pairs are reported for all elements. The results of this study show that Si, Ti, Nb and Ta bind to vacancies in the first nearest-neighbor position. Alloys containing these solute additions may experience delayed phase transformation due to vacancy trapping that slows diffusion.

Diffusion of Co, Ru and Re in Ni-based superalloys: A first-principles study

Self and impurity diffusivities in Ni-based superalloys have been investigated by first-principles calculations within five-frequency model. The vacancy formation energy, migration energy, correlation factor, solute–vacancy binding enthalpies and entropies in vacancy-mediated diffusions have been considered. The positive solute–vacancy binding energies for Co, Ru and Re indicated that the solute–vacancy pair is not a favorable binding. From the electronic point of view, the origination of the repulsive interaction between solute and vacancy has been comprehensibly expatiated on, which possibly accounts for the low diffusivity of Re element. The calculated self and impurity diffusion coefficients show a good consistent with the reported data from experiments and it is found that the trend in diffusivity is DCo>DNi-self>DRu>DRe.