Vacancies in metals: from first-principles calculations to experimental data

High-temperature effects for transition state calculations in solids

Transition state calculation is a critical technique to understand and predict versatile dynamical phenomena in solids. However, the transition state results obtained at 0 K are often utilized for the prediction or interpretation of dynamical processes at high temperatures, and the error bars of such an approximation are largely unknown. In this benchmark study, all the major temperature effects, including lattice expansion, lattice vibration, electron excitation, and band-edge shift, are evaluated with first-principles calculations for defect diffusion in solids. With the inclusion of these temperature effects, the notable discrepancies between theoretical predictions at 0 K and the experimental diffusivities at high temperatures are dramatically reduced. In particular, we find that lattice expansion and lattice vibration are the dominant factors lowering the defect formation energies and hopping barriers at high temperatures, but the electron excitation exhibits minor effects. In sharp contrast to typical assumptions, the attempt frequency with lattice expansion and vibration varies significantly with materials: several THz for aluminum bulk but surprisingly over 500 THz for 4H-SiC. For defects in semiconductors, the band-edge shift is also significant at high temperatures and plays a vital role in defect diffusion. We expect that this study would help accurately predict the dynamical processes at high temperatures.

Effect of doping Ti on the vacancy trapping mechanism for helium in ZrCo from first principles

The interactions of dopants with point defects such as that between vacancies and helium can affect helium evolution and ultimately the macroscopic properties of materials. Herein, the microscopic vacancy trapping mechanism for He defects and the formation of small He_mVac_n (consisting of m He atoms and n vacancies) clusters in pure and Ti-doped ZrCo systems are investigated by carrying out an extensive set of first-principles calculations based on density functional theory. Our results uncover the following: the helium atom can segregate from the adjacent interstitial (tetrahedral and octahedral) sites towards the vacancy center spontaneously, and therefore, a single He atom is energetically favorable to occupy a vacancy whether in the pure or in the doped system. The dopant Ti can act as a trapping center for He impurities similar to a vacancy. Moreover, it can improve the trapping ability and increase the trapping radius of the vacancies for helium. As for the effect of the Ti atom on the trapping of multiple helium atoms by the vacancy, the higher barrier in the doped systems than in the pure one implies that doping inhibits the formation of large He_mVac clusters. Furthermore, in order to evaluate the effect of dopant Ti on the stability of He atoms in multiple vacancies, the binding energies of a helium atom, a vacancy (Vac), and a self-interstitial atom (SIA) to a helium-vacancy cluster (He_mVac_n) were obtained and compared with that of the pure system. The results suggest that the cluster growth can be inhibited by the dopant Ti, and therefore, the formation of large helium bubbles is also hindered. All the binding energies do not depend much on the cluster size but primarily on the helium-to-vacancy ratio (m/n) of the clusters. The stability of the clusters is decided by the competitive processes among the emission of He atoms, vacancies, and SIAs, and also depends on the helium-to-vacancy ratio. The present results provide an in-depth explanation for the effect of the dopant on helium behavior and could aid future tritium storage material design.

A simple local expression for the prefactor in transition state theory

We present a simple and accurate computational technique to determine the frequency prefactor in harmonic transition state theory without necessitating full phonon density of states (DOS) calculations. The atoms in the system are partitioned into an "active region," where the kinetic process takes place, and an "environment" surrounding the active region. It is shown that the prefactor can be obtained by a partial phonon DOS calculation of the active region with a simple correction term accounting for the environment, under reasonable assumptions regarding atomic interactions. Convergence with respect to the size of the active region is investigated for different systems, as well as the reduction in computational costs when compared to full phonon DOS calculation. Additionally, we provide an open source implementation of the algorithm that can be added as an extension to Large-scale Atomic/Molecular Massively Parallel Simulator software.

Solid, liquid, and interfacial properties of TiAl alloys: parameterization of a new modified embedded atom method model

New interatomic potentials for pure Ti and Al, and binary TiAl were developed utilizing the second nearest neighbour modified embedded-atom method (MEAM) formalism. The potentials were parameterized to reproduce multiple properties spanning bulk solids, solid surfaces, solid/liquid phase changes, and liquid interfacial properties. This was carried out using a newly developed optimization procedure that combined the simple minimization of a fitness function with a genetic algorithm to efficiently span the parameter space. The resulting MEAM potentials gave good agreement with experimental and DFT solid and liquid properties, and reproduced the melting points for Ti, Al, and TiAl. However, the surface tensions from the model consistently underestimated experimental values. Liquid TiAl's surface was found to be mostly covered with Al atoms, showing that Al has a significant propensity for the liquid/air interface.

A first-principles study of the diffusion coefficients of alloying elements in dilute α-Ti alloys

We present a comprehensive investigation of the diffusion of alloying elements X in dilute α-Ti alloys. Besides the effect of solute size, two other key factors governing solute diffusion in dilute α-Ti are clarified: the chemical bonding characteristics and vibrational features of X–Ti pairs.

Using the electron localization function to correct for confinement physics in semi-local density functional theory

We have previously proposed that further improved functionals for density functional theory can be constructed based on the Armiento-Mattsson subsystem functional scheme if, in addition to the uniform electron gas and surface models used in the Armiento-Mattsson 2005 functional, a model for the strongly confined electron gas is also added. However, of central importance for this scheme is an index that identifies regions in space where the correction provided by the confined electron gas should be applied. The electron localization function (ELF) is a well-known indicator of strongly localized electrons. We use a model of a confined electron gas based on the harmonic oscillator to show that regions with high ELF directly coincide with regions where common exchange energy functionals have large errors. This suggests that the harmonic oscillator model together with an index based on the ELF provides the crucial ingredients for future improved semi-local functionals. For a practical illustration of how the proposed scheme is intended to work for a physical system we discuss monoclinic cupric oxide, CuO. A thorough discussion of this system leads us to promote the cell geometry of CuO as a useful benchmark for future semi-local functionals. Very high ELF values are found in a shell around the O ions, and take its maximum value along the Cu-O directions. An estimate of the exchange functional error from the effect of electron confinement in these regions suggests a magnitude and sign that could account for the error in cell geometry.

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

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

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

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

Stability and mobility of vacancy-H complexes in Al

The effect of hydrogen loading on the stability and mobility of vacancy-H complexes in aluminum is determined by applying DFT and the minimum-mode-following method. The binding energy per H-atom within a complex is found to range from -0.36 eV/atom to -0.34 eV/atom for an occupancy of, respectively, a single and eight H-atoms. When eight H-atoms are neighboring the vacancy the total binding energy becomes -2.72 eV. However, already at a load level of two H-atoms the total binding energy reaches -0.70 eV, which fully compensates the vacancy creation energy. It is observed that for complexes with four or more H-atoms the vacancy gets pinned, as the diffusion barrier increases by a factor of two, reaching a value of 1.03 eV or more. The explanation for the increased energy barrier is that at the higher hydrogen load levels the system must traverse an energetically unfavorable configuration where two or more H-atoms are separated from the vacancy. As a possible consequence of the decreased mobility and increased stability, highly loaded vacancy-H complexes are likely to act as nucleation sites for extended defects.

Ab initio local energy and local stress: application to tilt and twist grain boundaries in Cu and Al

The energy-density and stress-density schemes (Shiihara et al 2010 Phys. Rev. B 81 075441) within the projector augmented wave (PAW) method based on the generalized gradient approximation (GGA) have been applied to tilt and twist grain boundaries (GBs) and single vacancies in Cu and Al. Local energy and local stress at GBs and defects are obtained by integrating the energy and stress densities in each local region by the Bader integration using a recent algorithm (Yu et al 2011 J. Chem. Phys. 134 064111) as well as by the layer-by-layer integration so as to settle the gauge-dependent problem in the kinetic terms. Results are compared with those by the fuzzy-Voronoi integration and by the embedded atom method (EAM). The features of local energy and local stress at GBs and vacancies depend on the bonding nature of each material. Valence electrons in Al mainly located in the interatomic regions show remarkable response to structural disorder as significant valence charge redistribution or bond reconstruction, often leading to long-range variations of charges, energies and stresses, quite differently from d electrons in Cu mainly located near nuclei. All these features can be well represented by our local energy and local stress. The EAM potential for Al does not reproduce correct local energy or local stress, while the EAM potential for Cu provides satisfactory results.

Mechanisms of Defect Generation and Clustering in CH3S Self-Assembled Monolayers on Au(111)

Periodic density functional calculations probe that step edges play a key role as source of defects during self-assembly. It is shown that the self-assembly process strongly reduces the energy required to strip an atom from the gold surface, locally increasing the concentration of surface defects. The thermodynamic driving force for the atom stripping is considerably more favorable along step-edge lines within the self-assembly than on the higher-coordinated terrace sites. Furthermore, the clustering of surface defects is considered, and we probe that the formation of aggregates of vacancies in the form of vacancy pits significantly stabilizes the self-assembly on the terraces of gold, where the role of the step edges is expected to be less significant. The high stability of pit-like structures arises from a balance between the corrugation and the enhanced bonding of defect-rich substrates. Our results demonstrate the important role that step edges play during assembly and could be very valuable for discovering defect-free assembled structures.

A first-principles approach to transition states of diffusion

We propose a first-principles approach for treating the unstable vibrational mode of transition states in solid-state diffusion. It allows one to determine a number of fundamental quantities associated with the transition state, in particular the enthalpy and entropy of migration and the characteristic vibrational frequency, along with their temperature dependences. Application to pure face centered cubic Al shows good agreement with available experimental measurements and previous theoretical calculations. The procedure is further applied in calculations of the migration properties of Mg, Si and Cu impurities in Al, and the differences among Mg, Si and Cu are discussed.

Advancing density functional theory to finite temperatures: methods and applications in steel design

The performance of materials such as steels, their high strength and formability, is based on an impressive variety of competing mechanisms on the microscopic/atomic scale (e.g. dislocation gliding, solid solution hardening, mechanical twinning or structural phase transformations). Whereas many of the currently available concepts to describe these mechanisms are based on empirical and experimental data, it becomes more and more apparent that further improvement of materials needs to be based on a more fundamental level. Recent progress for methods based on density functional theory (DFT) now makes the exploration of chemical trends, the determination of parameters for phenomenological models and the identification of new routes for the optimization of steel properties feasible. A major challenge in applying these methods to a true materials design is, however, the inclusion of temperature-driven effects on the desired properties. Therefore, a large range of computational tools has been developed in order to improve the capability and accuracy of first-principles methods in determining free energies. These combine electronic, vibrational and magnetic effects as well as structural defects in an integrated approach. Based on these simulation tools, one is now able to successfully predict mechanical and thermodynamic properties of metals with a hitherto not achievable accuracy.

Density functional study of vacancies and surfaces in metals

We compare the performances of three common gradient-level exchange-correlation functionals for metallic bulk, surface and vacancy systems. We find that approximations which, by construction, give similar results for the jellium surface, show large deviations for realistic systems. The particular charge density and density gradient dependence of the exchange-correlation energy densities are shown to be the reason behind the obtained differences. Our findings confirm that both the global (total energy) and the local (energy density) behavior of the exchange-correlation functional should be monitored for a consistent functional design.

Efficacy of surface error corrections to density functional theory calculations of vacancy formation energy in transition metals

We calculate properties like equilibrium lattice parameter, bulk modulus and monovacancy formation energy for nickel (Ni), iron (Fe) and chromium (Cr) using Kohn-Sham density functional theory (DFT). We compare the relative performance of local density approximation (LDA) and generalized gradient approximation (GGA) for predicting such physical properties for these metals. We also make a relative study between two different flavors of GGA exchange correlation functional, namely PW91 and PBE. These calculations show that there is a discrepancy between DFT calculations and experimental data. In order to understand this discrepancy in the calculation of vacancy formation energy, we introduce a correction for the surface intrinsic error corresponding to an exchange correlation functional using the scheme implemented by Mattsson et al (2006 Phys. Rev. B 73 195123) and compare the effectiveness of the correction scheme for Al and the 3d transition metals.

Predicting Diffusion Coefficients from First Principles via Eyring’s Reaction Rate Theory

A simplified approach to predicting diffusion coefficients directly from first-principles is proposed. In this approach, the atomic jump frequencies are calculated through the Eyring’s reaction rate theory while the temperature dependence of diffusion coefficients are accounted using phonon theory within the quasi-harmonic approximation. The procedure can be applied to both self-diffusion and impurity diffusion coefficients and different crystal systems. Applications to self-diffusion coefficients in fcc Cu, bcc Mo, hcp Mg and impurity diffusion coefficients of Li in fcc Al, W in bcc Mo and Cd in hcp Mg show agreement with experimental measurements.

Role of the defect core in energetics of vacancies

Electronic structure calculations at macroscopic scales are employed to investigate the crucial role of a defect core in the energetics of vacancies in aluminium. We find that vacancy core energy is significantly influenced by the state of deformation at the vacancy core, especially volumetric strains. Insights from the core electronic structure and computed displacement fields show that this dependence on volumetric strains is closely related to the changing nature of the core structure under volumetric deformations. These results are in sharp contrast to mechanics descriptions based on elastic interactions that often consider defect core energies as an inconsequential constant. Calculations suggest that the variation in core energies with changing macroscopic deformations is quantitatively more significant than the corresponding variation in relaxation energies associated with elastic fields. Upon studying the influence of various macroscopic deformations, which include volumetric, uniaxial, biaxial and shear deformations, on the formation energies of vacancies, we show that volumetric deformations play a dominant role in governing the energetics of these defects. Further, by plotting formation energies of vacancies and di-vacancies against the volumetric strain corresponding to any macroscopic deformation, we find that all variations in the formation energies collapse on to a universal curve. This suggests a universal role of volumetric strains in the energetics of vacancies. Implications of these results in the context of dynamic failure in metals through shock-induced spalling are analysed.

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.

First-principles calculation of self-diffusion coefficients

We demonstrate a first-principles method to compute all factors entering the vacancy-mediated self-diffusion coefficient. Using density functional theory calculations of fcc Al as an illustrative case, we determine the energetic and entropic contributions to vacancy formation and atomic migration. These results yield a quantitative description of the migration energy and vibrational prefactor via transition state theory. The calculated diffusion parameters and coefficients show remarkably good agreement with experiments. We provide a simple physical picture for the positive entropic contributions.

Many-body potential for point defect clusters in Fe-C alloys

Modeling the consequences of crystalline defects requires efficient interaction sampling. Empirical potentials can identify relevant pathways if the energetics and configurations of competing defects are captured. Here, we develop such a potential for an alloy of arbitrary point defect concentration, body-centered cubic alpha-Fe supersaturated in C. This potential successfully calculates energetically favored defects, and predicts formation energies and configurations of multicarbon-multivacancy clusters that were not attainable with existing potentials or identified previously via ab initio methods.

Energetics and kinetics of vacancy diffusion and aggregation in shocked aluminium via orbital-free density functional theory

A possible mechanism for shock-induced failure in aluminium involves atomic vacancies diffusing through the crystal lattice and agglomerating to form voids, which continue to grow, ultimately resulting in ductile fracture. We employ orbital-free density functional theory, a linear-scaling first-principles quantum mechanics method, to study vacancy formation, diffusion, and aggregation in aluminium under shock loading conditions of compression and tension. We calculate vacancy formation and migration energies, and find that while nearest-neighbor vacancy pairs are unstable, next-nearest-neighbor vacancy pairs are stable. As the number of nearby vacancies increases, we predict that vacancy clusters preferentially grow through next-nearest-neighbor vacancies. The energetics are found to be greatly affected by expansion and compression, leading to insight as to how vacancies behave under shock conditions.

Hydrogen embrittlement of aluminum: the crucial role of vacancies

We report first-principles calculations which demonstrate that vacancies can combine with hydrogen impurities in bulk aluminum and play a crucial role in the embrittlement of this prototypical ductile solid. Our studies of hydrogen-induced vacancy superabundant formation and vacancy clusterization in aluminum lead to the conclusion that a large number of H atoms (up to 12) can be trapped at a single vacancy, which overcompensates the energy cost to form the defect. In the presence of trapped H atoms, three nearest-neighbor single vacancies which normally would repel each other, aggregate to form a trivacancy on the slip plane of Al, acting as embryos for microvoids and cracks and resulting in ductile rupture along these planes.

Structural changes and viscoplastic behavior of a generic embedded-atom model metal in steady shear flow

We study equilibrium and nonequilibrium properties of a simple "generic embedded-atom model" (GEAM) for metals. The model allows to derive simple analytical expressions for several zero-temperature constitutive properties--in overall agreement with real metals. The model metal is then subjected to shear deformation and strong flow via nonequilibrium molecular dynamics simulation in order to discuss the origins of some qualitative properties observed using more specific embedded-atom potentials. The "common neighbor analysis," based on planar graphs is used to obtain information about the transient structures accompanying viscoplastic behavior on an atomic level. In particular, pressure tensor components and plastic yield are investigated and correlated with underlying structural changes. A simple analytical expression for the isotropic pressure at finite temperatures is proposed. A nonequilibrium phase diagram is obtained by semianalytic calculation.

Derivation of the density functional theory from the cluster expansion

The density functional theory is derived from a cluster expansion by truncating the higher-order correlations in one and only one term in the kinetic energy. The formulation allows self-consistent calculation of the exchange correlation effect without imposing additional assumptions to generalize the local density approximation. The pair correlation is described as a two-body collision of bound-state electrons, and modifies the electron- electron interaction energy as well as the kinetic energy. The theory admits excited states, and has no self-interaction energy.

Comprehensive ab initio thermodynamic treatment of impurities in ordered alloys: application to boron in B2 Fe-Al

A rigorous description of impurities in nearly stoichiometric ordered systems, relying on independent-point-defect thermodynamics and density-functional-theory ab initio calculations, is presented and applied to the study of the equilibrium point defect structure of Fe-Al-B alloys in the B2 composition domain (x(Al) approximately 50 at. %, x(B) approximately 0), including the influences of (i) magnetism and (ii) the local density approximation (LDA) or generalized gradient approximation (GGA) exchange-correlation functionals. Whatever the alloy composition, B substitutes for Al, except in the nonmagnetic GGA scheme predicting a switch towards interstitial occupancy for sufficient Al contents.

Self-diffusion rates in Al from combined first-principles and model-potential calculations

Monovacancy diffusion alone dominates over diffusion due to divacancies and interstitials in Al for all temperatures up to the melting point. Deviations from a single Arrhenius dependence are due to anharmonicity. The conclusion is based on a combination of theoretical methods, from density functional theory to thermodynamic integration, without fitting to experimental data. The calculated diffusion rate agrees with experimental data over 11 orders of magnitude.