Phenomenological coefficients in a concentrated alloy for the dumbbell mechanism

We present an adaptation of the self-consistent mean field (SCMF) theory to calculate the transport coefficients in a concentrated alloy for diffusion by the dumbbell mechanism. In this theory, kinetic correlations are accounted for through a set of effective interactions within a non-equilibrium distribution function of the system. Transport coefficients are calculated for the FCC and BCC multicomponent concentrated alloys for simple sets of jump frequencies, including different stabilities of the different defects. A first approximation leads to an analytical expression of the Onsager coefficients in a binary alloy, and a second approximation provides a more accurate prediction. The results of the SCMF theory are compared with existing models and available Monte Carlo simulations, and an interpretation of the set of effective interactions in terms of a competition between jump frequencies is proposed.     

A high accuracy diffusion kinetics formalism for random multicomponent alloys: application to high entropy alloys

In this paper, a new, lighter, version of the highly accurate Moleko, Allnatt and Allnatt formalism for describing both tracer (self) and collective diffusion kinetics in multicomponent random alloys is presented. Verification of the resulting expressions is performed by means of kinetic Monte Carlo simulation. The accuracy of the new formalism is much higher than that of the combined Manning and Holdsworth and Elliott formalism discussed recently. Using this formalism the possible range of the tracer diffusion ratio of the highest to the lowest atomic component is examined for equiatomic (or near equiatomic) binary, ternary, quaternary and quinary alloys. It is shown that in the random alloy model, the correlation effect is the highest with a reduction of the fastest tracer diffusion by 40–55%, when moving from two pure metals to their equiatomic binary alloy. By adding the third component (with an intermediate mobility) this effect can be further increased with a possible total reduction of the fastest tracer diffusion by up to 70% (depending on the combinations of mobilities), while adding the fourth component brings this reduction up to 80% and with a possible maximum of up to 85% reduction for the 5-component alloy (again depending on the combinations of mobilities). But the slowest diffusing components are not affected by this. This suggests that kinetics arguments alone are not enough for explaining the sluggish diffusion observed of all atomic components in (equiatomic) high-entropy alloys.     

Phenomenological coefficients in a dilute BCC alloy for the dumbbell mechanism

The self-consistent mean field (SCMF) method is applied to calculate the transport coefficients in a dilute BCC alloy with the dumbbell diffusion mechanism. The first degree of approximation (first shell) of the SCMF formalism coincides with the formerly derived pair association method, and the second degree of approximation (second shell) leads to a more accurate analytical formulation. The SCMF results are compared with other formalisms as well as existing and new Monte Carlo simulations, including the solute–dumbbell binding energy. This theory shows a good balance between accuracy and complexity in the investigated systems, and a simple criterion is proposed for the preferential use of the first and second shell approximations.     

A self-consistent mean field calculation of the phenomenological coefficients in a multicomponent alloy with high jump frequency ratios

We present an improvement of the self-consistent mean field (SCMF) approximation of the L _ij which extends its applications to alloys presenting high jump frequency ratios. The theory uses a vacancy–atom exchange model which depends on temperature and local composition through thermodynamic and kinetic parameters. Kinetic correlations due to the vacancy mechanism are represented by a time-dependent effective Hamiltonian. In the case of high jump frequency ratios it is shown that long return paths of the vacancy need to be considered, which is shown to be equivalent to introducing many-body long-range effective interactions. We compare this theory to existing formalisms and Monte Carlo simulations for systems both without and with atomic interactions.     

Atomistic Modeling of Point Defects and Diffusion in Copper Grain Boundaries

The atomic structure of several symmetrical tilt grain boundaries (GBs) in Cu and their interaction with vacancies and interstitials as well as self-diffusion are studied by molecular statics, molecular dynamics, kinetic Monte Carlo (KMC), and other atomistic simulation methods. Point defect formation energy in the GBs is on average lower than in the lattice but variations from site to site within the GB core are very significant. The formation energies of vacancies and interstitials are close to one another, which makes the defects equally important for GB diffusion. Vacancies show interesting effects such as delocalization and instability at certain GB sites. They move in GBs by simple vacancy-atom exchanges or by long jumps involving several atoms. Interstitial atoms can occupy relatively open positions between atoms, form split dumbbell configurations, or form highly delocalized displacement zones. They diffuse by direct jumps or by the indirect mechanism involving a collective displacement of several atoms. Diffusion coefficients in the GBs have been calculated by KMC simulations using defect jump rates determined within the transition state theory. GB diffusion can be dominated by vacancies or interstitials, depending on the GB structure. The diffusion anisotropy also depends on the GB structure, with diffusion along the tilt axis being either faster or slower than diffusion normal to the tilt axis. In agreement with Borisov's correlation, the activation energy of GB diffusion tends to decrease with the GB energy.     

Fe-Cr合金辐照空洞微结构演化的相场法模拟

Fe-Cr合金作为包壳材料在高温高辐照强度等极端环境下服役,产生空位和间隙原子等辐照缺陷,辐照缺陷簇聚诱发空洞、位错环等缺陷团簇,引起辐照肿胀、晶格畸变,导致辐照硬化或软化致使材料失效.理解辐照缺陷簇聚和长大过程的组织演化,能更有效调控组织获得稳定服役性能.本文采用相场法研究Fe-Cr合金中空洞的演化,模型考虑了温度效应对点缺陷的影响以及空位和间隙的产生和复合.选择400—800 K温度区间、0—16 dpa辐照剂量范围的Fe-Cr体系为对象,研究在不同服役温度和辐照剂量下的空位扩散、复合和簇聚形成空洞的过程.在400—800 K温度区间,随着温度的升高,Fe-Cr合金空洞团簇形核率呈现出先升高后下降的趋势.考虑空位与间隙的重新组合受温度的影响可以很好地解释空洞率随温度变化时出现先升高后降低的现象.由于温度的变化将影响Fe-Cr合金中原子离位阀能,从而影响产生空位和间隙原子.同一温度下,空洞半径和空洞的体积分数随辐照剂量的增大而增大.辐照剂量的增大,级联碰撞反应加强,空位与间隙原子大量产生,高温下空位迅速的扩散聚集在Fe-Cr合金中将形成更多数量以及更大尺寸的空洞.     

Analysis of the effect of interatomic interaction energy in V-Fe alloys with different iron concentrations on segregation and swelling processes after ion irradiation

Comparative analysis of iron segregation profiles near the free surfaces of V-(2, 5, 7) at % Fe alloys irradiated with 50-keV V⁺ ions at temperatures 30–40°C with doses from 1 × 10¹⁹ to 1 × 10²¹ ions/m² has been performed. Data on the change in the moduli of normal elasticity and the values of swelling of these alloys with a change in Fe concentration in them are reported. A correlation between the character of the concentration dependence of the degree of Fe segregation, partial diffusion coefficients of the alloy components, the Young moduli, and the preference factors for vacancies and interstitials between dislocations and voids in these alloys is established.     

Statistical calculations of tracer and intrinsic diffusion coefficients in concentrated alloys and estimates of microscopic parameters of diffusion from experimental data

The earlier-developed statistical theory of diffusion in concentrated alloys based on the master equation approach is generalized to treat tracer diffusion in binary alloys. The theory developed is used to describe concentration dependencies of both tracer and intrinsic diffusion coefficients and to estimate microscopic parameters of diffusion in alloys CuNi, CuZn and AgCd for which necessary experimental data are available. We show that all main features of strong and peculiar concentration dependencies of diffusion coefficients observed in these alloys are naturally explained by the theory. Signs and scales of interatomic interactions important for diffusion in these alloys are found to strongly depend on the ratio of atomic sizes of alloy components, and types of these dependencies agree with simple physical considerations. We also discuss physical reasons for sharp concentration dependencies of diffusion coefficients characteristic of real alloys.     

Defect engineering on the electronic and transport properties of one-dimensional armchair phosphorene nanoribbons

We investigate the electronic and transport properties of one-dimensional armchair phosphorene nanoribbons(APNRs) containing atomic vacancies with different distributions and concentrations using ab initio density functional calculations. It is found that the atomic vacancies are easier to form and detain at the edge region rather than a random distribution through analyzing formation energy and diffusion barrier. The highly local defect states are generated at the vicinity of the Fermi level, and emerge a deep-to-shallow transformation as the width increases after introducing vacancies in APNRs.Moreover, the electrical transport of APNRs with vacancies is enhanced compared to that of the perfect counterparts. Our results provide a theoretical guidance for the further research and applications of PNRs through defect engineering.     

Spinodal decomposition in Monte Carlo simulations of a binary alloy

Recently Cahn's generalized diffusion equation theory of spinodal decomposition in binary alloys has been modified to include the effects of thermal fluctuations. This paper reports studies of a one-dimensional binary alloy system in which fluctuations can be observed on an atomic time scale. The system, a computer-simulated linear chain binary alloy which evolves from an initially random atomic arrangement through interchange of unlike nearest neighbors via the Monte Carlo technique, rapidly develops grains of two different concentrations and then slowly experiences coarsening. A numerical solution of the diffusion equation successfully predicts the development of grain structure, but only predicts coarsening to the extent present as fluctuations in the initial atomic arrangement. The simulated alloy coarsens further than the prediction of the diffusion equation because of thermal fluctuations which develop naturally during its evolution. This suggests that thermal fluctuations may play an important role in coarsening in real alloys.     

Resonant modes due to anion interstitials and anion vacancies in fluorites

The resonant modes due to anion interstitials and anion vacancies in fluorites are computed on the basis of a Green's function formalism. The Green's functions are calculated on the basis of shell model fitted to the phonon dispersion curves of the respective crystals. The force constant changes are calculated by finding the relaxations of the different host atoms and then using the short range potential due to Catlow and Norgett. The calculated frequencies of the resonant mode due to anion interstitials in BaF₂; shows good agreement with the experimental results. The results discussed in a comparative fashion for the three crystals CaF₂, SrF₂; and BaF₂; along with the possibility of their experimental detection.     

Statistical theory of diffusion in concentrated bcc and fcc alloys and concentration dependencies of diffusion coefficients in bcc alloys FeCu,FeMn, FeNi,and FeCr

The statistical theory of diffusion in concentrated bcc and fcc alloys with arbitrary pairwise interatomic interactions based on the master equation approach is developed. Vacancy–atom correlations are described using both the second-shell-jump and the nearest-neighbor-jump approximations which are shown to be usually sufficiently accurate. General expressions for Onsager coefficients in terms of microscopic interatomic interactions and some statistical averages are given. Both the analytical kinetic mean-field and the Monte Carlo methods for finding these averages are described. The theory developed is used to describe sharp concentration dependencies of diffusion coefficients in several iron-based alloy systems. For the bcc alloys FeCu, FeMn, and FeNi, we predict the notable increase of the iron self-diffusion coefficient with solute concentration c, up to several times, even though values of c possible for these alloys do not exceed some percent. For the bcc alloys FeCr at high temperatures T ? 1400 K, we show that the very strong and peculiar concentration dependencies of both tracer and chemical diffusion coefficients observed in these alloys can be naturally explained by the theory, without invoking exotic models discussed earlier.     

Analysis of interdiffusion via vacancy-pairs in strongly ionic solids

In this paper, we analyse chemical interdiffusion via Schottky vacancy-pairs in strongly ionic crystals for diffusion couples (AY–BY), where A and B take the same valence. We derive a sum-rule relating the phenomenological coefficients that is based on an earlier sum-rule derived by Moleko and Allnatt (Phil. Mag. A, 58 () 677) for diffusion in the multicomponent random alloy via the agency of isolated vacancies. With this sum-rule and other relationships derived by us, we show that the ratio of the intrinsic diffusivities can be expressed in exactly the same simple form as the case for diffusion via isolated vacancies. The functional form for the interdiffusion coefficient when expressed in terms of atom-vacancy exchange frequencies for diffusion is found to be essentially the same as that for isolated vacancies. The difference centres only on the relative differences of the local jump coordination of the vacancies in each case.     

Interdiffusion and thermotransport in Ni–Al liquid alloys

In this paper, we present extensive self-consistent results of molecular dynamics (MD) simulations of diffusion and thermotransport properties of Ni–Al liquid alloys. We develop a new formalism that allows easy connection between results of the MD simulations and the real experiments. In addition, this formalism can be extended to the case of ternary and higher component liquid alloys. We focus on the temperature and composition dependence of the self-diffusion coefficients, interdiffusion coefficients, thermodynamic factor, Manning factor and the reduced heat of transport. The two latter quantities both represent measures of the off-diagonal Onsager phenomenological coefficients. The Manning factor and the reduced heat of transport can be related to experimentally obtainable quantities provided the thermodynamic factor is available. The simulation results for the reduced heat of transport show that for all compositions, in the presence of a temperature gradient, Ni tends to migrate to the cold end. This is in agreement with an available experimental study for a Ni_21.5Al_78.5 melt (only qualitative result is available so far).     

Theory of the Gorsky effect for low interstitial concentrations

The formalism of the preceding paper is applied to work out the theory of the Gorsky effect, or anelastic relaxation due to the long-range diffusion of interstitials in a host lattice, for non-interacting (low-concentration) interstitials (e.g., H in Nb). It is shown how linear response theory (LRT) provides a number of advantages that simplify the solution of the problem and permit the handling of complications due to specimen geometry and stress inhomogeneity. The multiple-relaxation time creep function of Alefeldet al is first re-derived. Next, the dynamic responseand the short-time behaviour of the creep function are deduced exactly, and theω ^−1/2 fall-off of the internal friction at high frequencies is exhibited. Finally, it is pointed out that the true asymptotic behaviour of the dynamic response must be found by going beyond the diffusion equation model. A two-state random walk analysis is used to predict a cross-over to a trueω ⁻¹ asymptotic behaviour, and the physical reasons for this phenomenon are elucidated.     

Tracer diffusion in the presence of segregating obstacles

Tracer diffusion in an alloy in which the trajectories of one of the species is biased is examined as a model of mass transport with attendant segregation to extended defects (e.g., dynamic strain ageing, grain-boundary segregation). More specifically, we employ Monte Carlo simulation to describe the nonequilibrium diffusive behavior of the components of a two-dimensional lattice gas comprising A and B atoms wherein one of the species (B) interacts with randomly distributed line defects to create equilibrium atmospheres at late times. Various kinetic assumptions and defect densities are explored to highlight the role of B-atom mobility and defect interaction strength on the transport behavior of the A atoms. From the calculated instantaneous diffusivity, several diffusive regimes are then identified and related to evolving segregation profiles and, in particular, to the free area available for diffusion.     

Gorsky relaxation in the presence of traps

It has been shown recently that a linear response theoretic formalism leads to a straightforward derivation of the theory of the Gorsky effect at low interstitial concentrations, for arbitrary specimen geometry and applied stress inhomogeneity. We now extend this formalism to the practically important situation in which traps (e.g., N interstitials) inhibit the diffusion of the H interstitials, in order to motivate the experimental application of the Gorsky relaxation technique to the determination of the trap parameters. An explicit calculation is done for a typical specimen geometry, using Schroeder's phenomenological model for diffusion in the presence of traps. Exact expressions are obtained for the anelastic creep function, the relaxation strength and the internal friction. The structure and physical implications of these expressions are discussed using numerical values for the parameters deduced by other methods (in particular, neutron scattering). Noteworthy features of the predicted dynamic response include shifts in the position and height of the Gorsky peak owing to the occurrence of an effective diffusion constant and the difference in the values of the trace of the elastic dipole tensor in the free and trapped states; and an additional Snoek-like contribution due to local free trapped transitions of the diffusing particles.