Correlation Factor For Diffusion Research Articles

Vacancy diffusion barrier spectrum and diffusion correlation in multicomponent alloys

Vacancy diffusion serves a crucial role in many important kinetic behaviors and properties of multicomponent alloys. Essential questions, however, persist regarding how chemical complexity affects diffusion and what unique characteristics, if any, set these alloys apart from traditional metals. Using neural network kinetics model, we study vacancy diffusion in NbMoTa alloy across a broad temperature range (2600 to 800 K). Unlike pure metals, the two key diffusion parameters—diffusion correlation factor f and activation energy ΔGm—are not constant in alloys, but instead substantially decrease with decreasing temperature. This temperature dependence arises from a reduced number of active vacancy jump pathways at lower temperatures, leading to more correlated diffusion. Upon examining vacancy diffusion throughout the entire compositional space of the Nb-Mo-Ta system, we discover that the slowest vacancy diffusion surprisingly occurs in the non-equimolar region, rather than the equimolar concentration where the configurational entropy is highest. The diffusion barrier spectrum, characterizing the diffusion energy landscape, is an intrinsic material characteristic, which controls both f and ΔGm and, thereby, the diffusivity. Finally, we find that the vacancy diffusion rate drops noticeably in the presence of local chemical order in the NbMoTa system, particularly for MoTa alloys with long-range B2 order.

Migration processes in the glass forming melts of the Na2O–BaO–Ga2O3–SiO2 system

The specific conductivity and tracer of diffusion of sodium and barium ions have been determined and the values of the transport numbers and correlation factor for diffusion have been calculated in two melts of the Na2O–BaO–Ga2O3–SiO2 system.

Diffusion of sodium ions and electric conductivity in glassy and crystallized Na2O · Al2O3 · 2SiO2

The specific electric conductivity of glassy and crystallized samples of composition Na2O · Al2O3 · 2SiO2 has been determined. Diffusion coefficients of sodium ions have been determined by the sectioning method with the measurement of the residual activity 22Na isotope. The parameters of the temperature dependence and values of the diffusion correlation factor have been calculated. A significant difference of the diffusion mobility of sodium ions in the glassy and crystallized samples has been found.

ChemInform Abstract: Impurity Diffusion in Highly‐Ordered Intermetallic Compounds Studied by Nuclear Quadrupole Interactions

AbstractReview: 30 refs.

Correlation factor for diffusion in cubic crystals with solute–vacancy interactions of arbitrary range

Using a double Laplace and Fourier transform of the transport equation for the vacancy, we obtain the exact values of the return probabilities of the vacancy in the close vicinity of the tracer atom in the presence of a solute-vacancy interaction of arbitrary range. The study of model cases shows that taking into account the interaction up to the third neighbour shell is mandatory to obtain the solute diffusivity in BCC and FCC structures with a good precision. A thorough ab initio evaluation of all the migration barriers is rarely available in the literature; it is shown that the approximations often used to overcome this lack of data must be chosen with care in order to avoid puzzling conclusions. An examination of dilute systems studied in the recent literature is presented.

Impurity Diffusion in Highly-Ordered Intermetallic Compounds Studied by Nuclear Quadrupole Interactions

Diffusion of impurity atoms depends on the sublattices occupied, active diffusion mechanisms, and jump frequencies to neighboring sites. The method of perturbed angular correlation of gamma rays (PAC) has been applied over the past decade to study impurity diffusion through measurement of nuclear quadrupole interactions (NQI) at nuclei of 111In/Cd probe atoms. Extensive measurements have been made on highly-ordered compounds having the L12 crystal structure, including In3R, Sn3R, Ga3R, Al3R and Pd3R phases (R= rare-earth element). Measurements in thermal equilibrium at high temperature served to determine lattice locations of 111In parent probe-atoms, through characteristic NQIs, and to measure diffusional jump-frequencies of 111Cd daughter probe-atoms, through relaxation of the NQI. This paper summarizes results of the jump-frequency measurements and relates them to the conventional diffusivity that can be determined, for example, from penetration profiles of tracer species. In spite of chemical similarities of the series of rare-earth phases studied, remarkably large variations in jump frequencies have been observed especially along series of In3R and Pd3R phases. Most phases appear as “line compounds” in binary phase diagrams, but large differences in site-preferences and jump-frequencies were observed for samples prepared to have the opposing limiting phase boundary compositions. Comparisons of jump-frequencies measured at opposing boundary compositions can give insight into the predominant microscopic diffusional mechanisms of the impurity. A change in diffusion mechanism was proposed in 2009 to explain jump-frequency systematics for In3R phases. An alternative explanation is proposed in the present paper based on site-preferences of 111Cd daughter probes newly observed along the parallel Pd3R series. The diffusivity can be expressed as the product of a jump-frequency such as measured in these studies and a correlation factor for diffusion that depends on the diffusion mechanism. The correlation factor can be modeled for the L12 structure and diffusion sublattice of interest using a five-frequency model originally proposed for metals. Although the correlation factor is an essential parameter for the diffusion of impurities, it has never been measured. It is suggested that values of the correlation factor can be determined feasibly by combining results of jump-frequency measurements such as the present ones with diffusivity measurements made for the same host-impurity systems.

Diffusion correlation effects of molybdenum and silicon in molybdenum disilicide

Diffusion data for both principal directions of silicon and molybdenum as well as germanium are briefly summarised. Analysis is performed of the defect formation energies (available from previous ab initio calculations and experimental measurements) for diffusion mechanisms via home and foreign sublattices. The home sublattice mechanism is shown to be the preferred one for both silicon and molybdenum. Tracer correlation factors for silicon and molybdenum diffusion via sublattice vacancies in the respective sublattices of the tetragonal C11b structure of molybdenum disilicide are calculated by a direct Monte Carlo simulation technique. Correlation factors for Si diffusion on its sublattice are compared with literature values that were calculated using a more complicated Monte Carlo method based on the matrix approach. It is shown that there is no need for this complicated approach and that the direct Monte Carlo simulation technique gives highly accurate correlation factors. Correlation factors and anisotropy ratios of vacancy-mediated diffusion in both sublattices are deduced and compared with experimental data. Tracer correlation in the tetragonal direction is shown to contribute 0.40 eV (i.e. over 55%) of the migration energy of the corresponding Si diffusivity. Two possible jump rates for Si diffusion are separately estimated. Mo diffusion correlation factors are calculated using the direct Monte Carlo technique. A comparison with experiment is made and the ratio of two possible jump rates is also estimated.

Correlation between self-diffusion in Si and the migration mechanisms of vacancies and self-interstitials: An atomistic study

The migration of point defects in silicon and the corresponding atomic mobility are investigated by comprehensive classical molecular-dynamics simulations using the Stillinger-Weber potential and the Tersoff potential. In contrast to most of the previous studies both the point defect diffusivity and the self-diffusion coefficient per defect are calculated separately so that the diffusion-correlation factor can be determined. Simulations with both the Stillinger-Weber and the Tersoff potential show that vacancy migration is characterized by the transformation of the tetrahedral vacancy to the split vacancy and vice versa and the diffusion-correlation factor ${f}_{V}$ is about 0.5. This value was also derived by the statistical diffusion theory under the assumption of the same migration mechanism. The mechanisms of self-interstitial migration are more complex. The detailed study, including a visual analysis and investigations with the nudged elastic band method, reveals a variety of transformations between different self-interstitial configurations. Molecular-dynamics simulations using the Stillinger-Weber potential show that the self-interstitial migration is dominated by a dumbbell mechanism, whereas in the case of the Tersoff potential the interstitialcy mechanism prevails. The corresponding values of the correlation factor ${f}_{I}$ are different, namely, 0.59 and 0.69 for the dumbbell and the interstitialcy mechanisms, respectively. The latter value is nearly equal to that obtained by the statistical theory which assumes the interstitialcy mechanism. Recent analysis of experimental results demonstrated that in the framework of state-of-the-art diffusion and reaction models the best interpretation of point defect data can be given by assuming ${f}_{I}\ensuremath{\approx}0.6$. The comparison with the present atomistic study leads to the conclusion that the self-interstitial migration in Si should be governed by a dumbbell mechanism.

Influence of the composition of a molten salt on the field-assisted diffusion of potassium ions in the 20Na2O · 80SiO2 glass

The kinetics of interaction between the sodium silicate glass 20Na2O · 80SiO2 and molten salts in a dc electric field is investigated using the electron-probe X-ray microanalysis and radioactive-tracer technique. The influence of the electric field strength, the temperature, the interaction time, and the composition of a molten salt on the concentration distribution of alkali cations in the 20Na2O · 80SiO2 glass in the course of field-assisted diffusion is elucidated. The effective electric mobilities of potassium ions are calculated, and the parameters of the temperature dependence of the electric mobility are determined. It is revealed that the application of an electric field does not affect the ion-exchange equilibrium in the system. The correlation factor for diffusion of impurity ions is determined for the first time. The obtained value of the correlation factor indicates that the processes of self-diffusion and heterodiffusion of alkali cations in binary silicate glasses occur through the same (relay-race) mechanism. It is established that the Nernst-Einstein equation, which relates the electric mobility to the diffusion coefficient and provides the basis for the description of the field-assisted diffusion, holds only for the processes bearing no relation to variations in the composition of the glass.

Correlation factor in diffusion in the L1 2 structure for atomic migration via triple defects

Expressions for the diffusion coefficients and correlation factors for diffusion by means of divacancies in compounds A 3B with the L1 2 structure are calculated. The divacancy is assumed not to dissociate and migrate via a triple-defect configuration. By applying a matrix method proposed by Howard [Phys Rev 1996;144:650], values for the correlation factors are calculated as a function of the ratio of the defect-atom exchange frequencies. The ratio of the diffusion coefficients of A and B atoms, D A/ D B, will lie between 0.061 and 4.9.

Diffusion correlation effects and the isotope effect in intermetallic compounds

There has been considerable discussion in the literature about the various roles and meanings of the diffusion correlation factors in ordered alloys. In this paper diffusion in a two-dimensional stoichiometric ordered alloy model is considered. Using a combination of Monte Carlo simulation and analytical development we consider how the correlation factor for diffusion, as would be found from an isotope effect experiment, is related to the actual correlation factor (all jumps included) and the correlation factor for effective diffusion by the six-jump-cycle mechanism. It is shown that the apparent correlation factor from an isotope effect experiment can deviate very greatly from the actual correlation factor but extrapolates precisely to that for the six jump cycle when the long range order parameter approaches unity. This clarifies the relationships between the various correlation factors in ordered alloys where intersublattice jumps predominate.

THEORY OF CORRELATION EFFECTS FOR DIFFUSION IN A TWO SUBLATTICE STRUCTURE

The tracer diffusion correlation factor has been studied for a simple cubic lattice gas with two energetically inequivalent sublattices and sites restricted to accommodate a maximum of one atom. The model serves as a paradigm for solids with site inequivalence but no algebraic formulae have previously been derived and compared with available Monte Carlo simulation data. An expression based on a kinetic equation theory which neglects the highest order in concentration fluctuations is shown to lead to a good description of the simulation results. The time dependence of the associated time correlation function appears to be unusually complex for diffusion in a two sublattice system. © 1997 Elsevier Science Ltd. All rights reserved

Symmetry properties of the collective diffusion correlation factors in the random alloy

Abstract In the present study a unifying mathematical framework for collective correlation factors in the random alloy is presented using principles of functional analysis. In the formal part of the analysis collective correlation factors are introduced and represented in terms of cosines of angles between atomic jumps. Relations among the correlation factors are discussed using the Onsager relations. Next, additional relations are considered using a sum rule. Finally, further relations between the correlation factors are introduced on physical grounds. Results of the analysis are applied first to the consideration of a number of previously derived relations between tracer correlation factors and collective correlation factors, or equivalently, between tracer diffusion coefficients and phenomenological coefficients. The analysis provides a particularly convenient way of comparing the relations. Secondly, the analysis is applied to the dilute binary alloy and consideration of the asymptotic behaviour of th...

Two approaches for the evaluation of collective diffusion correlation factors

Abstract The diffusion of probability and matrix methods for calculating collective (and tracer) correlation factors are compared using as an example the random alloy. It is shown that the method of diffusion probability can provide the general behaviour of the structure of the cosines (the cosine of the angle between a given jump of a species and a later jump) and of the summations, while the matrix method can give accurate values of the cosines. Accordingly, in cases of complicated systems and/or lattice structures the procedure of analysis of correlation effects should be a combination of these two methods.

Tracer diffusion coefficients for the dumb-bell transport mechanism in a concentrated f.c.c. alloy

Abstract A model of matter transport by dilute dumb-bell interstitial defects in a concentrated f.c.c. alloy of two components A and B where the defects and substitutional atoms mix randomly is studied within the framework used earlier to study the Manning random alloy model for vacancy migration. The tracer diffusion correlation factors are calculated by an approximation in the kinetic equations whereby the tracer atoms and dumb-bells encounter substitutional A and B atoms with the equilibrium probability at each jump. The final expressions contain Brillouin zone integrals which are functions of the alloy composition and of the defect jump frequencies; these integrals must be calculated numerically for each thermodynamic state point of interest. When the ratio of the largest to the smallest jump frequencies is 10:1, the results are in moderate agreement with earlier Monte Carlo simulations but a theory which gives results for larger jump frequency ratios remains an unsolved problem. The essential differe...

Correlation factors for diffusion in binary random alloys with fcc structure

Tracer correlation factors have been calculated by simulating the diffusion of atoms and defects on a computer using a Monte Carlo technique. The correlated motion of the atomic species by a vacancy mechanism have been investigated for a binary random alloy with fcc structure. The tracer correlation factors for the atoms, fA and fB, and for the vacancies, fV, are calculated as functions of the jump frequencies, and the mixing ratio of A and B atoms for vacancy concentrations of cV = 0.001 and 0.1. The results are compared with theories and Monte Carlo simulations from the literature. Substantial discrepancies with previous Monte Carlo studies are observed. It is concluded that the correlation factors given in the literature are systematically too high because the jump sequences in those calculations were too short.

Correlation factor for diffusion via sublattice vacancy mechanism in the L12-type ordered alloy

Abstract In A3B alloys of the L12 type, while the B atoms have only unlike nearest neighbours, the A atoms have a mixed neighbourhood. Therefore, the A atoms can diffuse by exchanging sites with vacancies on its own sublattice without concomitant disordering. The impurity correlation factor for such a mechanism (termed the sublattice vacancy mechanism) are calculated exactly by adopting the five-jump frequency model. The expression is compared with those for other cubic lattices derived previously.

Diffusion mechanisms in ordered alloys—a detailed analysis of six-jump vacancy cycle in the B2 type lattice

The diffusion coefficients of A and B atoms in the B2 type AB alloy are derived for the case that highly correlated vacancy jump cycle is operative. The effective frequencies of 100- and 110-type jumps of A and B atoms are expressed in terms of frequencies for individual vacancy jumps; the calculation is made by applying the concept of the mean first passage time known in the theory of stochastic processes. The correlation factors for diffusion of A and B atoms are calculated as a function of the wrong bond energy. The present analysis provides a sound and quantitative basis for the discussion of the ratio of diffusion coefficients for two types of atoms, D A D B , in ordered alloys.

Monte Carlo evaluation of the diffusion correlation factor in concentrated lattices and comparison with an improved theory.

A theoretical discussion of the difficulty inherent in obtaining an accurate estimate of the tracer diffusion correlation factor in concentrated, single-component systems from Monte Carlo data involving relatively short runs is given. The theory is tested on the available Monte Carlo data of Murch, who used runs lasting approximately 50 Monte Carlo steps per particle (MCS/p). To avoid these difficulties, new precision Monte Carlo data, collected from runs lasting 500 MCS/p, are supplied. The relevant estimates for the diffusion correlation factor are compared with those supplied by a properly renormalized version of the earlier theory, which also includes the small but noticeable effects of spatial constraints on the scattering of a coupled tracer-vacancy pair. The agreement between the improved theory and the numerical estimates is within the overall precision, 2 parts per thousand, of the new Monte Carlo experiments.

Isotope separation during diffusion along an edge dislocation in a simple cubic crystal

During diffusion in crystalline lattices, isotope separation is well known to be proportional to the correlation factor. In a recently published analysis of vacancy-based transport along an edge dislocation in a simple cubic system, the expression for the diffusion coefficient takes on a much more complicated form than is found in the lattice, and it is uncertain whether the isotope separation factor is indeed proportional to the correlation factor. This effect is studied analytically and numerically, and shown to equal the correlation factor for diffusion along the dislocation. In addition, the isotope separation factor for electromigration is also determined to be proportional to the correlation factor for diffusion.