Vacancy-wind Factors Research Articles

Composition dependence of tracer diffusion coefficients in Fe–Ga alloys: A case study by a tracer-diffusion couple method

The problem of estimation of the tracer diffusion coefficients is solved by utlizing a novel radiotracer-diffusion couple technique in the absence of a suitable radioisotope of one of the components and reliable thermodynamic parameters. This is demonstrated by generating reliable and reproducible mobility data in the alloys of the Fe–Ga system with a strong composition dependence of the diffusion coefficients. Tracer- (59Fe) and inter-diffusion are simultaneously measured in three couples Fe/Fe-16 Ga, Fe/Fe-24 Ga and Fe-16 Ga/Fe-24 Ga at 1143 K in at.%). The results obtained for the couples with different end-members are in an excellent agreement with each other for the overlapping composition intervals. The influence of the molar volume on the measured tracer- (59Fe) and inter-diffusion coefficients is evaluated. Using thermodynamic calculations, the Ga tracer diffusion coefficient and the vacancy wind factor are determined via the Darken-Manning relation for the composition range of 0–24 at.% Ga. The results in this study confirm the reliability of the radiotracer-diffusion couple technique for producing highly accurate diffusion data. As a result, this helped for optimizing the mobility description of the bcc phase of the Fe–Ga system. The Ga tracer diffusion coefficients are further estimated via experimental determination of the ratio of the Fe and Ga tracer diffusivities at the Kirkendall marker planes and utilizing the Fe tracer diffusion coefficients measured directly by the radiotracer method.

Comments on Experimental Assessment of the Thermodynamic Factor for Diffusion in Cocrfeni and Cocrfemnni High Entropy Alloys

In this contribution, as a response to the paper in Scripta Materialia, v.157, p.81, 2018, we show that the pseudo-binary approach for analysis of interdiffusion in multicomponent alloys has its reference value of the Manning factor (vacancy-wind factor) that is very different from true binary interdiffusion. In true binary interdiffusion, this factor is usually compared with unity. In the pseudo-binary analysis this reference value must be changed to the inverse of N1+N2, where Ni are the compositions (mole fractions), and components 1 and 2 are the atomic components that have different terminal compositions in the corresponding experiment.

The Vacancy-Wind Factor and the Manning Factor Occurring in Interdiffusion and Ionic Conductivity in Solids

In crystalline solids, during such processes as chemical interdiffusion in alloys, ionic conductivity and the annealing out of radiation damage there will inevitably be a net flux of vacancies. In most cases, when different species of atoms have different jump rates with vacancies within a net flux of vacancies, the phenomenon of the vacancy-wind effect will occur. This effect was first discovered in the 1960s by the late Dr John Manning. It is a subtle phenomenon that comes about because of the local redistribution of the equilibrium concentration of vacancies with respect to two or more species of drifting atoms in a driving force. The effect is captured in various ‘vacancy-wind factors’ (some of which are now sometimes called Manning factors) which formally arise from non-zero off-diagonal Onsager phenomenological transport coefficients and non-unity values of the tracer correlation factors. In this paper, the effect is reviewed and discussed.

Determining a Tracer Diffusivity by way of the Darken-Manning Equation for Interdiffusion in Binary Alloy Systems

The self-or tracer diffusivity of one component in a binary alloy is often required when there is knowledge of the other component’s self-or tracer diffusivity and the interdiffusivity (and the thermodynamic factor). In the present paper, this problem is addressed for the random alloy model by applying three possible approximations having different levels of accuracy: Darken (low level of accuracy), Manning (medium level of accuracy) and Moleko, Allnatt and Allnatt (MAA) (high level of accuracy). There are unexpectedly large differences between the results of these approximations that sometimes are reflected in the high sensitivity of the vacancy-wind factor to the level of approximation. Generally, for the application of Manning and the MAA approximations, it is found that there is a difference in the number of self-diffusivity roots depending on whether the tracer diffusivity is available for the faster diffuser or for the slower diffuser and depending on how close the composition is to the forbidden (according to Manning’s description) region. Provided that the interdiffusion coefficient (divided by the thermodynamic factor) is greater than the available self-diffusion coefficient multiplied by its complementary composition, the application of the Darken approximation always results in one self-diffusivity root.

Viscosity and diffusivity in melts: from unary to multicomponent systems

Viscosity and diffusivity, two important transport coefficients, are systematically investigated from unary melt to binary to multicomponent melts in the present work. By coupling with Kaptay’s viscosity equation of pure liquid metals and effective radii of diffusion species, the Sutherland equation is modified by taking the size effect into account, and further derived into an Arrhenius formula for the convenient usage. Its reliability for predicting self-diffusivity and impurity diffusivity in unary liquids is then validated by comparing the calculated self-diffusivities and impurity diffusivities in liquid Al- and Fe-based alloys with the experimental and the assessed data. Moreover, the Kozlov model was chosen among various viscosity models as the most reliable one to reproduce the experimental viscosities in binary and multicomponent melts. Based on the reliable viscosities calculated from the Kozlov model, the modified Sutherland equation is utilized to predict the tracer diffusivities in binary and multicomponent melts, and validated in Al–Cu, Al–Ni and Al–Ce–Ni melts. Comprehensive comparisons between the calculated results and the literature data indicate that the experimental tracer diffusivities and the theoretical ones can be well reproduced by the present calculations. In addition, the vacancy-wind factor in binary liquid Al–Ni alloys with the increasing temperature is also discussed. What’s more, the calculated inter-diffusivities in liquid Al–Cu, Al–Ni and Al–Ag–Cu alloys are also in excellent agreement with the measured and theoretical data. Comparisons between the simulated concentration profiles and the measured ones in Al–Cu, Al–Ce–Ni and Al–Ag–Cu melts are further used to validate the present calculation method.

The Nature of the Vacancy-Wind Effect Occurring in Diffusion via Six-Jump-Cycles in B2 Intermetallics

First discovered by the late Dr John Manning, the vacancy-wind effect is a subtle phenomenon that occurs when two or more atomic species compete for vacancies in a net vacancy flux. The vacancy-wind effect is incorporated in (for example) the vacancy-wind or Manning factor that appears in the Darken-Manning Equation relating the interdiffusivity, the tracer diffusivities and the thermodynamic factor. The mechanism of the vacancy-wind phenomenon has long been very poorly understood. Recently, a moving reference frame Monte Carlo method was used to illustrate graphically how the vacancy-wind effect operates in both ionic conductivity in an ionic solid with a dilute solute and chemical interdiffusion in concentrated alloys and ionic compounds. That strategy is extended in this paper to show graphically how the vacancy-wind effect operates in interdiffusion in a stoichiometric intermetallic taking the B2 structure. A simple 4-frequency vacancy diffusion model is used. In previous work, it was shown that depending on composition and temperature, this model can exhibit the six-jump-cycle mechanism. It is shown that in the limit of perfect order that there is no vacancy-wind effect associated with this mechanism when both types of cycle operate equally (zero net vacancy flux). The non-unity value of the vacancy-wind factor found for this mechanism under zero vacancy flux conditions is purely a consequence of a particular geometric mix of tracer and collective atom displacements. The concept that a non-zero off-diagonal phenomenological coefficient provides the vacancy-wind effect is verified.

Non-Random Interaction of Vacancies with Atoms during Interdiffusion and Ionic Conductivity in Materials

The non-random interaction of vacancies with atoms during interdiffusion and ionic conductivity is referred to as the vacancy-wind effect. This effect, first discovered by the late Dr John Manning, is a subtle phenomenon arising from the non-random distribution of vacancies with respect to a given moving atom within a net flux of vacancies. Recently, a good deal of progress has been made in determining accurate expressions for vacancy-wind factors in binary and ternary alloys, and in mixed cation ionic systems. The present paper provides an overview of these recent findings and puts them into a broader and historical context.

Interdiffusion in strongly ionic insulating compounds: The Nernst–Planck equation

In strongly ionic insulating materials, the Nernst–Planck Equation relates the interdiffusion coefficient of the cations (having the same charge) with the corresponding tracer diffusivities and the thermodynamic factor. In this paper, we explore the Nernst–Planck Equation for ionic ternary (quasi-binary) and ionic quaternary (quasi-ternary) systems using the diffusion kinetics formalisms of Darken [Trans. Am. Inst. Min. (Metall.) Eng. 175 184 (1948)], Manning [Phys. Rev. B 4 1111 (1971)] and Moleko, Allnatt and Allnatt [Phil. Mag. A 59 141 (1989)]. It is shown that for the binary system, the Darken and Manning formalisms both give the usual form of the Nernst–Planck Equation. However, the almost exact Moleko, Allnatt and Allnatt formalism (for randomly mixed systems) provides an additional correction factor analogous to the vacancy-wind factor in the well-known Darken–Manning Equation used in binary alloy systems. Nernst–Planck-type equations are also derived for strongly ionic insulating ternary systems and are found to behave similar to the binary case.

Analysis of interdiffusion data in multicomponent alloys to extract fundamental diffusion information

This paper reviews new procedures based around the random alloy model that have been established recently for analyzing chemical diffusion data in binary and ternary alloy systems. The authors show how atom-vacancy exchange frequency ratios, individual tracer correlation factors, and vacancy-wind factors can be extracted from the chemical diffusion data. Examples are taken from intrinsic diffusion data in the Ag-Cd and Ag-Cd-Zn alloy systems and from interdiffusion data in the Fe-Ni-Cr and Cu-Fe-Ni alloy systems.

Expressions for vacancy-wind factors occurring in interdiffusion in ternary and higher-order alloys

In this paper, an analysis of the vacancy-wind factors occurring in the expression relating interdiffusion and tracer diffusion coefficients in ternary (and higher-order) alloys is provided for the first time. It is shown that for the ternary alloy the vacancy-wind factors occur as a matrix of four elements. The diffusion kinetics formalism developed by Manning for a random alloy is used to find closed-form expressions for the vacancy-wind factors in terms of the tracer diffusion coefficients. These reduce to the well-known expression for the vacancy-wind factor occurring in the expression relating interdiffusion and tracer diffusion coefficients in the binary random alloy. The more recent self-consistent diffusion kinetics formalism of Moleko, Allnatt and Allnatt (which is near exact) is also used to calculate the vacancy-wind factors from the tracer diffusion coefficients. It is found that the off-diagonal vacancy-wind factors can actually be negative with high absolute values. Accordingly, the use of the Darken approach (which ignores the off-diagonal phenomenological coefficients) to modelling of the interdiffusion process would then give an incorrect direction of the cross-terms in the corresponding interdiffusion flux expressions. Tracer diffusion data in the ternary Cu–Fe–Ni and Cu–Ni–Zn systems are used to calculate the vacancy-wind factors in these systems.

Vacancy-Wind Factors and Collective Correlation Factors in Nonstoichiometric B2 Intermetallic Compounds

The six-jump-cycle (6JC) diffusion mechanism is used to analyze the behavior of vacancy-wind factors and collective correlation factors in partially ordered B2 intermetallic compounds at stoichiometric and near- stoichiometric compositions. Expressions for the vacancywind factors are obtained in the framework of the four-frequency model where the two sublattices exist a priori. The phenomenological coefficients on the two sublattices that remain hitherto independent in 6JC mechanism are connected through a microscopic detailed balance condition. The present results for collective correlation factors when compared with our earlier calculation based on taking the harmonic mean of the sublattice correlation factors show much better agreement with Monte Carlo simulation results. The collective correlation factors and tracer correlation factors are used to calculate the vacancy-wind factors. Our results for vacancy-wind factors agree qualitatively with the simulation data when the frequency ratio ( α ) of structural and antistructural atoms jumps decreases up to the order of unity.

Theoretical analysis of experimental tracer and interdiffusion data in Cu–Ni–Fe alloys

In this paper, we present strategies to extract fundamental atomistic information from measured diffusion coefficients in a ternary alloy system. The strategies are exemplified with Cu–Ni–Fe alloys at 1271K where recent extensive interdiffusion coefficients and tracer diffusion coefficients for all three components have become available. We develop new defining phenomenological expressions for the vacancy-wind factors in terms of the diffusion coefficients. We show that the measured tracer diffusion coefficients can be processed using the Manning and Moleko, Allnatt and Allnatt random alloy diffusion kinetics formalisms (with and without the assumption of the Gibbs–Duhem relation between the thermodynamic activities) to give jump frequencies, tracer correlation factors, vacancy-wind factors and phenomenological coefficients. It is shown for example that Cu is generally the most correlated component in its diffusion behavior and that the off-diagonal phenomenological coefficients can be as high as 64% of the smallest of the diagonal phenomenological coefficients. It is also shown that the Darken formalism (which ignores off-diagonal phenomenological coefficients) is in fact a reasonable approximation for expressing the diagonal phenomenological coefficients in terms of the tracer diffusion coefficients. It is then shown how the measured interdiffusivities can be processed with these formalisms to give tracer diffusivities, vacancy-wind factors and phenomenological coefficients. Finally, we show how a straightforward strategy starting with the Darken analysis that is then followed by the Manning or Moleko, Allnatt and Allnatt analysis can be used to gain access to the vacancy-wind factors and the off-diagonal phenomenological coefficients.

Interdiffusion and Al Self-Diffusion in Iron-Aluminides

We have investigated interdiffusion in iron-aluminium alloys using single-phase interdiffusion couples of FexAl1−x–FeyAl1−y for three combinations of x and y for Al contents between 18 and 49.5 at. % Al. Experimental diffusion profiles were obtained from electron-microprobe analysis of the diffusion zone. Interdiffusion coefficients were deduced via the Sauer-Freise method taking into account volume changes. A temperature interval between 997 and 1447 K was covered in our experiments. Thermodynamic factors were obtained from two theoretical models and judged by an analysis of the Kirkendall effect in the diffusion couples. The Darken-Manning equation was used to deduce self-diffusion coefficients of aluminium from the present interdiffusion coefficients, the thermodynamic factors, the vacancy-wind factors, and the iron tracer diffusivities obtained recently at the M¨unster laboratory. The results show that Al diffusion is always slightly faster than Fe diffusion. The difference never exceeds a factor of three. This small difference indicates that Fe and Al diffusion in B2 ordered iron-aluminides are closely coupled.

Diffusion in substitutionally disordered B2 intermetallics

In this paper, a comprehensive diffusion kinetics theory is formulated to describeseamlessly tracer and chemical diffusion in antistructurally disordered B2intermetallics showing positive and negative deviations from stoichiometry. Thetheory is based around unit processes consisting of six-jump cycles that can beassisted by intrinsic and extrinsic antistructural atoms of either atomic species.The Ising alloy model is used to illustrate the formalism, but the formalism can beadapted to other models. Expressions are developed for the tracer diffusioncoefficients, the phenomenological coefficients, the intrinsic diffusion coefficients,the interdiffusion coefficient and the various correlation factor components.Results for the tracer and collective correlation factors and the vacancy windfactor (in interdiffusion) are in excellent agreement with results fromMonte Carlo computer simulations based around single vacancy jumps.

Behaviour of the diffusion vacancy-wind factors in the concentrated random alloy

Abstract In this paper, the vacancy-wind factors which appear in the well known expressions relating the interdiffusion coefficient and intrinsic diffusion coefficients to the tracer diffusion coefficients are examined in the context of the random alloy model. It is shown by Monte Carlo simulation that the formalism of Moleko et al. provides a much better description of these factors than the well known and commonly used formalism of Manning. Even when the ratio of the tracer diffusion coefficients is relatively close to unity, it is preferable to use the Moleko et al. formalism. The latter formalism is re-expressed in this paper to be suitable for direct experimental use in determining vacancy-wind factors. The limiting behaviour of the ratios of the tracer diffusivities and intrinsic diffusivities is also discussed. In the latter case the ratio is given simply (and generally) by the ratio of the exchange frequencies.

Behaviour of the diffusion vacancy-wind factors in the concentrated random alloy

Behaviour of the diffusion vacancy-wind factors in the concentrated random alloy

The vacancy wind effect for chemical diffusion in intermetallic compounds

In this paper we examine the behaviour of the vacancy wind factors in the intrinsic diffusivities and the chemical or interdiffusivity in highly ordered stoichiometric AB intermetallic compounds. We show that the six-jump cycle gives quite different limits for these factors from those given by the well-known Manning expressions. Remarkably, there is non-trivial behaviour of these factors even when the vacancy flux is zero. The findings are all verified by computer simulations of a four-frequency model in the square planar lattice and an Ising model in the B2 structure. The consequences of the findings for the interpretation of experimental data are commented on.

The vacancy wind effect for chemical diffusion in intermetallic compounds

Abstract In this paper we examine the behaviour of the vacancy wind factors in the intrinsic diffusivities and the chemical or interdiffusivity in highly ordered stoichiometric AB intermetallic compounds. We show that the six-jump cycle gives quite different limits for these factors from those given by the well-known Manning expressions. Remarkably, there is non-trivial behaviour of these factors even when the vacancy flux is zero. The findings are all verified by computer simulations of a four-frequency model in the square planar lattice and an Ising model in the B2 structure. The consequences of the findings for the interpretation of experimental data are commented on.

Tracer Correlation, Collective Correlation and Vacancy Wind Factors in Intermetallics Taking the B2 Structures

Tracer Correlation, Collective Correlation and Vacancy Wind Factors in Intermetallics Taking the B2 Structures

Relating tracer and chemical diffusion coefficients in intermetallic compounds taking the DO 3 and A15 structures

In this article we employ computer simulation to explore the validity of the Darken/Manning relation between the chemical diffusion coefficient and the tracer diffusion coefficients of the components in stoichiometric intermetallic compounds A 3B taking the DO 3 and A15 structures at vanishingly small vacancy contents. The analysis centres on the validity of Manning’s random alloy expression for the vacancy wind factor. The models for both the DO 3 and A15 structures use eight atom-vacancy exchange frequencies. For the DO 3 structure it is found that the actual vacancy wind factor is usually somewhat larger than that predicted by Manning but overall the agreement is good. At worst the use of Manning’s expression would underestimate this factor by about 30 or 40%. For the A15 structure a similar result is found except when diffusion along the chains in the structure is rapid. Then Manning’s expression fails badly when a constant geometrical tracer correlation factor is employed. In both the structures if the geometrical correlation factor is varied to reflect the structure actually explored by the atoms (mainly the majority atoms A) the agreement is improved very dramatically.