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Composition-dependent diffusion coefficients are determined in B2-Ni(CoPt)Al system following the pseudo-binary and pseudo-ternary diffusion couple methods, which would not be possible otherwise in a quaternary inhomogeneous material fulfilling the conditions to solve the equations developed based on the Onsager formalism. The end-member compositions to produce ideal/near-ideal diffusion profiles are chosen based on thermodynamic details. The pseudo-binary interdiffusion coefficients of Ni and Al decrease in the presence of Co but increase in the presence of Pt. The pseudo-ternary interdiffusion coefficients indicate that the main interdiffusion coefficients increase significantly in the presence of Pt. Marginal changes of the cross interdiffusion coefficients substantiate a minor change of the diffusional interactions between the components. The thermodynamic driving forces show opposite trends with respect to composition as compared to the changes of the interdiffusion coefficients advocating a dominating role of the Pt(Co)-induced modifications of point defect concentrations. DFT-based calculations revealed that Pt alloying increases the Ni vacancy concentration and decreases the activation energy for the triple defect diffusion mechanism. These findings explain the increase in the thickness of the interdiffusion zone between the B2-Ni(Pt)Al bond coat and the single crystal superalloy Rene N5 because of Pt addition. Furthermore, the EPMA and TEM analyses reveal the growth of refractory elements-enriched precipitates.

Estimating the diffusion coefficients experimentally in a four-component inhomogeneous alloy following the conventional diffusion couple method by intersecting three couples at the same composition is difficult unless a small composition range of constant diffusivity is identified. Additionally, the intrinsic diffusion coefficients of the components cannot be estimated in a system with more than two components. To solve these issues, we have followed the pseudo-binary and pseudo-ternary diffusion couple methods for estimating the diffusion coefficients at the equiatomic composition of NiCoFeCr medium entropy alloy. Along with the pseudo-binary interdiffusion coefficients, we have estimated the intrinsic diffusion coefficients of all the components by designing the pseudo-binary couples such that Ni and Co develop the diffusion profiles keeping Fe and Cr constant in one couple and Fe and Cr develop the diffusion profiles keeping Ni and Co constant in another couple. Subsequently, we have proposed the relations for calculating the tracer diffusion coefficients utilizing the thermodynamic details. We have found a good match with the data estimated directly following the radiotracer method at the equiatomic composition. Following, we have produced three pseudo-ternary diffusion couples intersecting at the compositions close to the equiatomic composition. The main pseudo-ternary interdiffusion coefficients of Fe are found to be higher than Ni and Co. Therefore, we have estimated different types of diffusion coefficients highlighting the complex diffusion process in the four-component NiCoFeCr medium entropy alloy.

Diffusion couple technique is an efficient tool for the estimating the chemical diffusion coefficients. Typical experimental uncertainties of the composition profile measurements complicate a correct determination of the interdiffusion coefficients via the standard Boltzmann-Matano, Sauer-Freise or the den Broeder methods, especially for systems with a strong compositional dependence of the interdiffusion coefficient. A new approach for reliable fitting of the experimental profiles with an improved behavior at both ends of the diffusion couple is proposed and tested against the experimental data on chemical diffusion in the system Fe-Ga. An extension of the approach for reliable description of the up-hill diffusion phenomenon in multicomponent systems is presented.

The wrong estimation of the diffusion coefficients following the quasi-binary approach and subsequent analysis by Tsai et al. in Co-Cr-Fe-Mn-Ni high entropy alloy are explained. Since this approach, formulated by the present author before the publication by Tsai et al., has potential to emerge as an efficient technique in multicomponent systems, the correct steps of estimation of the diffusion coefficients are explained in detail. At present state, the diffusion data reported in high entropy alloy Tsai et al. has no significance unless estimated again following the correct steps.

The benefits of using the pseudo-binary and pseudo-ternary diffusion couple methods in multicomponent inhomogeneous systems are demonstrated by estimating different types of composition-dependent diffusion coefficients. These are important for understanding the basic atomic mechanism of diffusion and complex compositional evolutions. These were otherwise considered impossible during the last many decades. Without any options previously, sometimes the average values over a composition range of random choice were estimated, which are not the material constants but depend on the composition range and also the end member compositions. The steps and analyses for utilising the pseudo-binary and pseudo-ternary methods are first described in the Ni-Co-Fe-Mo system by producing the ideal diffusion profiles fulfilling the concepts behind these methods. Following, the discussion is extended to the systems related to medium (Ni-Co-Cr) and high (Ni-Co-Fe-Mn-Al) entropy alloys. In fact, this is the first report showing a correct experimental method that should be followed for the estimation of the interdiffusion and intrinsic diffusion coefficients in inhomogeneous high entropy alloys. In the end, the limitations of following these methods because of the generation of non-ideal diffusion profiles are discussed based on experimental results. The steps are also suggested to avoid such complications. These methods are easy to adopt for research engineers. Most importantly, these give an opportunity to validate the data estimated following newly proposed numerical methods by different groups with experimentally estimated diffusion coefficients, which were not possible earlier.

The interdiffusion coefficients are estimated either following the Wagner's method expressed with respect to the composition (mol or atomic fraction) normalized variable after considering the molar volume variation or the den Broeder's method expressed with respect to the concentration (composition divided by the molar volume) normalized variable. On the other hand, the relations for estimation of the intrinsic diffusion coefficients of components as established by van Loo and integrated diffusion coefficients in a phase with narrow homogeneity range as established by Wagner are currently available with respect to the composition normalized variable only. In this study, we have first derived the relation proposed by den Broeder following the line of treatment proposed by Wagner. Further, the relations for estimation of the intrinsic diffusion coefficients of the components and integrated interdiffusion coefficient are established with respect to the concentration normalized variable, which were not available earlier. The veracity of these methods is examined based on the estimation of data in Ni-Pd, Ni-Al and Cu-Sn systems. Our analysis indicates that both the approaches are logically correct and there is small difference in the estimated data in these systems although a higher difference could be found in other systems. The integrated interdiffusion coefficients with respect to the concentration (or concentration normalized variable) can only be estimated considering the ideal molar volume variation. This might be drawback in certain practical systems. (C) 2018 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Major drawback of studying diffusion in multi-component systems is the lack of suitable techniques to estimate the diffusion parameters. In this study, a generalized treatment to determine the intrinsic diffusion coefficients in multi-component systems is developed utilizing the concept of a pseudo-binary approach. This is explained with the help of experimentally developed diffusion profiles in the Cu(Sn, Ga) and Cu(Sn, Si) solid solutions. (C) 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

When one starts to analyze the evolution of the interfacial reaction product layers between dissimilar materials it is often found out that as the number of interacting species grows, the complexity of the analysis increases extremely rapidly. It may even appear that the task is just too difficult to be completed. In this article we present the thermodynamic-kinetic method, which can be used to rationalize the evolution of interfacial reaction layers and bring back the physics to the analyses. The method is conceptually very simple. It combines energetics—what can happen—with kinetics—how fast things take place. Yet the method is flexible enough that it can utilize quantitative and qualitative data starting from the atomistic simulations up to the experiments carried out with bulk materials. Several examples about how to utilize this method in material scientific problems are given.

Thermodynamics and diffusion kinetics determine the microstructure of a given reaction couple between dissimilar materials under specific conditions. The microstructure, in turn, affects strongly the performance of multimaterial assemblies under different stress and environmental states. Hence it is shown here how the combination of thermodynamics and kinetics (namely the thermodynamic-kinetic method) can be used to rationalize the evolution of microstructure between dissimilar materials. The relevant thermodynamic considerations are given first, before moving to the foundations of diffusion kinetic considerations most relevant for the thermodynamic-kinetic method. The method itself is then briefly discussed, before finalizing this chapter with a couple of case studies from the field of electronics.

Diffusion couple method is one of the well-established techniques for estimation of the diffusion coefficients. In this chapter, the concept and definition of different types of diffusion coefficients are described along with experimental methods, tricks and the analysis in binary and pseudo-binary (multicomponent) systems. Experimental results in various systems are introduced. In the end, physical significance of the data estimated is described.

A study on reactive diffusion is conducted in the Re-Si system. According to the study, ReSi1.8 phase grows with much higher thickness than the Re2Si phase, in the interdiffusion zone of bulk diffusion couples. The activation energy for integrated diffusion of ReSi1.8 is estimated to be 605 +/- 23 kJ/mol. The growth of the Re2Si phase is studied by considering an incremental diffusion couple of Re/ReSi1.8. Analysis based on the calculation of integrated diffusion coefficients indicates the reason underlying the observed high difference between the growth rates of the ReSi1.8 and Re2Si phases.

Experiments are conducted to study interdiffusion in a complete solid solution as well as in ordered phases in the Cu–Pt system. The variation of the interdiffusion coefficient and the activation energy with composition of the solid solution is explained with the help of melting points, vacancy formation energies and thermodynamic factors indicating the driving force for diffusion. Additionally, interdiffusion coefficients are also estimated in the CuPt and Cu3Pt phases at two different temperatures.

Solid diffusion couple experiments are conducted to analyse the growth mechanism of the phases and the diffusion mechanism of the components in the Ti-Si system. The calculation of the parabolic growth constants and the integrated diffusion coefficients substantiates that the analysis is intrinsically prone to erroneous conclusions if it is based on just the parabolic growth constants determined for a multiphase interdiffusion zone. The location of the marker plane is detected based on the uniform grain morphology in the TiSi2 phase, which indicates that this phase grows mainly because of Si diffusion. The growth mechanism of the phases and morphological evolution in the interdiffusion zone are explained with the help of imaginary diffusion couples. The activation enthalpies for the integrated diffusion coefficient of TiSi2 and the Si tracer diffusion are calculated as 190 +/- 9 and 197 +/- 8 kJ/mol, respectively. The crystal structure, details on the nearest neighbours of the components, and their relative mobilities indicate that the vacancies are mainly present on the Si sublattice.

Experiments are conducted in the W-Si system to understand the diffusion mechanism of the species. The activation energies from integrated diffusion coefficients are calculated as 152 +/- 7 and 301 +/- 40 kJ/mol in the WSi2 and W5Si3 phases, respectively. In both the phases, Si has a much higher diffusion rate compared to W. This is not surprising to find in the WSi2 phase, if we consider the number of nearest neighbors for both the elements in the crystal. The diffusion of W in this phase indicates the presence of W antisites. The faster diffusion rate of Si in the W5Si3 phase indicates the presence of higher concentration of vacancies on the Si sublattice compared to W sublattice.

Interdiffusion studies were conducted to understand the atomic mechanism of the diffusing species and the growth mechanism of the phases. Integrated diffusion coefficients and the ratio of tracer diffusion coefficients were estimated for these analyses. The activation energies for the integrated diffusion coefficients were calculated as 550 ± 70 and 410 ± 39 kJ/mol in the TaSi2 and Ta5Si3 phases, respectively. In the TaSi2 phase, Ta has a slightly lower but comparable diffusion rate with respect to Si, although no Ta–Ta bonds are present in the crystal. In the Ta5Si3 phase, Si has a higher diffusion rate, which is rather unusual, if we consider the atoms in the nearest-neighbour positions for both the elements. The ratio of Si to Ta tracer diffusion coefficients is found to be lower in the Si-rich phase, TaSi2, compared to the Si-lean phase, Ta5Si3, which is also unusual. This indicates the type of structural defects present. An analysis on the growth mechanism of the phases indicates that duplex morpholo...

A study has been conducted on a Cu(Sn) solid solution to examine the role of the vacancy wind effect on interdiffusion. First, the interdiffusion and the intrinsic diffusion coefficients are calculated. The trend of the interdiffusion coefficients is explained with the help of the driving force. Following this, the tracer diffusion coefficients of the species are calculated with and without consideration of the vacancy wind effect. We found that the role of the vacancy wind is negligible on the minor element in a dilute solid solution, which is the faster diffusing species in this system and controls the interdiffusion process. However, consideration of this effect is important to understand the diffusion rate of the major element, which is the slower diffusing species in this system.

Diffusion couple technique is used to study interdiffusion in Nb-Mo, Nb-Ti and Nb-Zr systems. Interdiffusion coefficients at different temperatures and compositions are determined using the relation developed by Wagner. The change in activation energy for interdiffusion with composition is determined. Further, impurity diffusion coefficient of the species are determined and compared with the available data in literature.

In this study, bulk and multifoil diffusion couple experiments were conducted to examine the interdiffusion process in Ni–Pt and Co–Pt binary alloy systems. Inter-, intrinsic-, and tracer-diffusion coefficients at different temperatures, and as a function of the composition, were estimated by using the experimental data. Results show that in both the alloy systems, Pt is the slower diffusing species, and hence the interdiffusion process is controlled by either Ni or Co. The thermodynamic driving force makes the intrinsic diffusion coefficients of Co and Ni higher in the range of 30–70 at.%. The low activation energy for Co and Ni impurity diffusion in Pt compared with Pt in Ni and Co indicates that the size of the atoms plays an important role. The vacancy wind effects on the diffusion process are examined in detail, and it was demonstrated that its contribution falls within the experimental scatter and hence can be neglected.

Incremental diffusion couple experiments are conducted to determine the average interdiffusion coefficient and the intrinsic diffusion coefficients of the species in the Ni6Nb7 (μ phase) in the Ni-Nb system. Further, the tracer diffusion coefficients are calculated from the knowledge of thermodynamic parameters. The diffusion rate of Ni is found to be higher than that of Nb, which indicates higher defect concentration in the Ni sublattice.

Interdiffusion study is conducted in the V-Si system to determine integrated diffusion coefficients of the phases. Activation energy values are calculated from the experiments conducted at different temperatures. The average values are found to be 208, 240 and 141 kJ/mol, respectively, for the V(3)Si, V(5)Si(3) and VSi(2) phases. The low activation energy for the VSi(2) phase indicates very high concentration of defects or the significant contribution from the grain boundary diffusion. The error in calculation of diffusion parameters from a very thin phase layer in a multiphase diffusion couple is discussed. Further the data available in the literature in this system is compared and the problems in the indirect methodology followed previously to calculate the diffusion parameters are discussed.

There is a constant effort to understand the defect structure and diffusion behavior in intermetallic compounds with the A15 structure. Diffusion of elements in intermetallic compounds depends mainly on antisites and vacancies on different sublattices. In this article, we shall discuss the diffusion of elements in A3B compounds with the A15 structure.

Study of interdiffusion in the Co-Mo system is important to understand the performance of turbine blades in jet engine applications. Mo is added to superalloys to increase the solid solution strengthening and the creep resistance. In this study, the interdiffusion coefficient in the Co(Mo) solid solution and impurity diffusion coefficient of Mo in Co are determined. Further, the activation energy and pre-exponential factors are calculated, which provide an idea about the atomic mechanism of diffusion.

The Kirkendall-effect induced migration of inert markers during a diffusion-controlled growth of intermetallic compounds in the Cu/Sn and Au/Sn couples at 215 and 180°C were studied. It was shown that the behaviour of markers in the multiphase reaction zones can be rationalized in terms of the Kirkendall velocity construction. Observations on the microstructural features of the product intermetallic layers and the role of the Kirkendall effect in the morphogenesis of the interdiffusion systems are discussed. It was demonstrated that the velocity of markers in a product layer, the appearance of the Kirkendall plane(s), their location(s) and the morphological evolution of the reaction products can also be explained from a purely chemical point of view considering the diffusion-controlled interactions at the interphase interfaces. A representation of the reaction scheme and kinetics of intermetallic growth in each of the two systems is given.

The study on the formation and growth of topological close packed (TCP) compounds is important to understand the performance of turbine blades in jet engine applications. These deleterious phases grow mainly by diffusion process in the superalloy substrate. Significant volume change was found because of growth of the μ phase in Co–Mo system. Growth kinetics of this phase and different diffusion parameters, like interdiffusion, intrinsic and tracer diffusion coefficients are calculated. Further the activation energy, which provides an idea about the mechanism, is determined. Moreover, the interdiffusion coefficient in Co(Mo) solid solution and impurity diffusion coefficient of Mo in Co are determined.

Diffusion parameters, such as the integrated diffusion coefficient of the phase, the tracer diffusion coefficient of species at different temperatures and the activation energy for diffusion, are determined in V3Si phase with A15 crystal structure. The tracer diffusion coefficient of Si was found to be negligible compared to the tracer diffusion coefficient of V. The calculated diffusion parameters will help to validate the theoretical analysis of defect structure of the phase, which plays an important role in the superconductivity.

The confusion over the growth rate of the Nb3Sn superconductor compound following the bronze technique is addressed. Furthermore, a possible explanation for the corrugated structure of the product phase in the multifilamentary structure is discussed. Kirkendall marker experiments are conducted to study the relative mobilities of the species, which also explains the reason for finding pores in the product phase layer. The movement of the markers after interdiffusion reflects that Sn is the faster diffusing species. Furthermore, different concentrations of Sn in the bronze alloy are considered to study the effect of Sn content on the growth rate. Based on the parabolic growth constant at different temperatures, the activation energy for the growth is determined.

There are essentially two different phenomenological models available to describe the interdiffusion process in binary systems in the olid state. The first of these, which is used more frequently, is based on the theory of flux partitioning. The second model, developed much more recently, uses the theory of dissociation and reaction. Although the theory of flux partitioning has been widely used, we found that this theory does not account for the mobility of both species and therefore is not suitable for use in most interdiffusion systems. We have first modified this theory to take into account the mobility of both species and then further extended it to develop relations or the integrated diffusion coefficient and the ratio of diffusivities of the species. The versatility of these two different models is examined in the Co-Si system with respect to different end-member compositions. From our analysis, we found that the applicability of the theory of flux partitioning is rather limited but the theory of dissociation and reaction can be used in any binary system.

The role of the molar volume on the estimated diffusion parameters has been speculated for decades. The Matano–Boltzmann method was the first to be developed for the estimation of the variation of the interdiffusion coefficients with composition. However, this could be used only when the molar volume varies ideally or remains constant. Although there are no such systems, this method is still being used to consider the ideal variation. More efficient methods were developed by Sauer–Freise, Den Broeder, and Wagner to tackle this problem. However, there is a lack of research indicating the most efficient method. We have shown that Wagner’s method is the most suitable one when the molar volume deviates from the ideal value. Similarly, there are two methods for the estimation of the ratio of intrinsic diffusion coefficients at the Kirkendall marker plane proposed by Heumann and van Loo. The Heumann method, like the Matano–Boltzmann method, is suitable to use only when the molar volume varies more or less ideally or remains constant. In most of the real systems, where molar volume deviates from the ideality, it is safe to use the van Loo method. We have shown that the Heumann method introduces large errors even for a very small deviation of the molar volume from the ideal value. On the other hand, the van Loo method is relatively less sensitive to it. Overall, the estimation of the intrinsic diffusion coefficient is more sensitive than the interdiffusion coefficient.

Diffusion couple experiments are conducted to study phase evolutions in the Co-rich part of the Co-Ni-Ta phase diagram. This helps to examine the available phase diagram and propose a correction on the stability of the Co2Ta phase based on the compositional measurements and X-ray analysis. The growth rate of this phase decreases with an increase in Ni content. The same is reflected on the estimated integrated interdiffusion coefficients of the components in this phase. The possible reasons for this change are discussed based on the discussions of defects, crystal structure and the driving forces for diffusion. Diffusion rate of Co in the Co2Ta phase at the Co-rich composition is higher because of more number of Co-Co bonds present compared to that of Ta-Ta bonds and the presence of Co antisites for the deviation from the stoichiometry. The decrease in the diffusion coefficients because of Ni addition indicates that Ni preferably replaces Co antisites to decrease the diffusion rate. (C) 2014 Elsevier B.V. All rights reserved.

A diffusion-controlled growth of intermetallic phases and the role of the Kirkendall effect in morphological evolution of the product phase layers can be described in terms of an alternative theory considering chemical reactions at the interphase interfaces. Application of this “physicochemical” treatment to diffusional growth of intermediate phases with fairly wide homogeneity ranges is illustrated by the example of interaction in the Ag-Zn system. The model is purely phenomenological, and its use is convenient, since no explicit assumption of the underlying diffusion mechanism is required.

The partial and integral enthalpies of mixing of liquid Bi–Sn–Zn alloys were determined at 500°C by a drop calorimetric technique using a Calvet-type microcalorimeter. The ternary interaction parameters in the Bi–Sn–Zn system were fitted using the Redlich-Kister-Muggianu model for substitutional solutions, and isoenthalpy curves of the integral molar enthalpy of mixing at 500°C were constructed. Furthermore, a DSC technique was used to determine the liquidus temperatures in three sections (3, 5, and 7 at.% Zn) as well as the invariant reaction temperature of the ternary eutectic L ⇄ (Bi) + (Sn) + (Zn). The ternary eutectic reaction was found at 135°C.

Interdiffusion coefficients in the ß-NiAl phase over the homogeneity range are determined by the diffusion couple technique in the temperature range of 1000–1200 °C. Intrinsic diffusivities of the species at 1000 °C at different compositions are measured by Kirkendall marker experiments. The variations of the molar volume with composition and partial molar volumes of the species, required for the determination of the diffusion parameters, are calculated using lattice parameter and vacancy concentration. Tracer diffusion coefficients of the species were calculated from the knowledge on intrinsic diffusivities and compared with the data available in the literature, which were measured directly by tracer methods. The influence of the vacancy wind effect is determined on the calculated results.

The partial and integral enthalpies of mixing of binary liquid Ag–Pd alloys (with compositions up to about 55 at.% Pd) were determined at 1400 °C. A Calvet type microcalorimeter was used for the measurements employing a drop calorimetric technique. Additionally, eight sections in the Sn-rich corner of the ternary Ag–Pd–Sn system, which is relevant for lead-free soldering, were investigated at 900 °C. The ternary interaction parameters were fitted using the Redlich–Kister–Muggiano model for substitutional solutions. The isoenthalpy curves for Ag–Pd–Sn at 900 °C were constructed for the integral molar enthalpy of mixing.

Interdiffusion experiments were performed between different Ni–Al two-phase alloys. A method is given to estimate the tracer diffusion coefficient of one of the constituents in a binary alloy knowing the tracer diffusion coefficient of the other species. The procedure is based on the so-called Darken-Manning analysis. The tracer diffusion coefficient of Al was determined in this way in Ni3Al: D Al ∗ =5.05×10 −7 −1.117×10 −7 +2.28×10 −6 exp −243±16 KJ / mol RT m 2 s Our results are compared with others available in the literature. The values for the self diffusivity of Al calculated from interdiffusion experiments give consistent result. All these measurements support the theory that diffusion of the minor element occurs through a vacancy mechanism in Ni3Al.

In this chapter, the growth of the line compounds and the phases with narrow homogeneity range by reactive diffusion is discussed. The concept of the integrated diffusion coefficient is introduced. In the end, few case studies are given to explain the estimation procedure of the diffusion coefficients.

A detailed diffusion study was carried out on Cu(Ga) and Cu(Si) solid solutions in order to assess the role of different factors in the behaviour of the diffusing components. The faster diffusing species in the two systems, interdiffusion, intrinsic and impurity diffusion coefficients, are determined to facilitate the discussion. It was found that Cu was more mobile in the Cu–Si system, whereas Ga was the faster diffusing species in the Cu–Ga system. In both systems, the interdiffusion coefficients increased with increasing amount of solute (e.g. Si or Ga) in the matrix (Cu). Impurity diffusion coefficients for Si and Ga in Cu, found out by extrapolating interdiffusion coefficient data to zero composition of the solute, were both higher than the Cu tracer diffusion coefficient. These observed trends in diffusion behaviour could be rationalized by considering: (i) formation energies and concentration of vacancies, (ii) elastic moduli (indicating bond strengths) of the elements and (iii) the interaction parameters and the related thermodynamic factors. In summary, we have shown here that all the factors introduced in this paper should be considered simultaneously to understand interdiffusion in solid solutions. Otherwise, some of the aspects may look unusual or even impossible to explain.

This chapter deals with diffusion in the phases with wide homogeneity range. Different approaches, which could be used to estimate the variation of interdiffusion coefficients with composition, are described. Following the Kirkendall effect is introduced along with the estimation of the intrinsic diffusion coefficients. The estimation of the tracer diffusion coefficients from a diffusion couple is also explained.

Estimation of the diffusion parameters in the multicomponent system is explained. A pseudobinary approach simplifying the mathematical conditions is described. The use of diffusion couple establishing the phase diagram is presented.

In this chapter, the microstructural evolution in the interdiffusion zone in a diffusion couple is explained based on thermodynamics. A two-phase mixture cannot develop in a binary system, whereas this is possible to find in a ternary system. Condition for finding an uphill diffusion is explained.

Interdiffusion studies become increasingly difficult to perform with the increasing number of elements in a system. It is rather easy to calculate the interdiffusion coefficients for all the compositions in the interdiffusion zone in a binary system. The intrinsic diffusion coefficients can be calculated for the composition of Kirkendall marker plane in a binary system. In a ternary system, however, the interdiffusion coefficients can only be calculated for the composition where composition profiles from two different diffusion couples intersect. Intrinsic diffusion coefficients are possible to calculate when the Kirkendall markers are also present at that composition, which is a condition that is generally difficult to satisfy. In a quaternary system, the composition profiles for three different diffusion couples must intersect at one particular composition to calculate the diffusion parameters, which is a condition that is almost impossible to satisfy. To avoid these complications in a multicomponent system, the average interdiffusion coefficients are calculated. I propose a method of calculating the intrinsic diffusion coefficients and the variation in the interdiffusion coefficients for multicomponent systems. This method can be used for a single diffusion couple in a multicomponent pseudobinary system. The compositions of the end members of a diffusion couple should be selected such that only two elements diffuse into the interdiffusion zone. A few hypothetical diffusion couples are considered in order to validate and explain our method. Various sources of error in the calculations are also discussed.

Diffusion parameters such as the interdiffusion coefficients and the ratio of the tracer diffusion coefficients are calculated in the Co2Ta Laves phase. The activation energy for the interdiffusion coefficients is calculated as 186 +/- 29 kJ/mol. The ratio of tracer diffusion coefficients indicates that Co has higher diffusion rate than that of Ta. This is explained with the help of possible point defects and the crystal structure of the phase: The phase boundary compositions measured in this study is different from the compositions published previously. (C) 2012 Elsevier Ltd. All rights reserved.

The integrated diffusion coefficient of the phases and the tracer diffusion coefficients of the species are determined in the Nb-Si system by the diffusion couple technique. The diffusion rate of Si is found to be faster than that of Nb in both the NbSi2 and Nb5Si3 phases. The possible atomic mechanism of diffusion is discussed based on the crystal structure and on available details of the defect concentration data. The faster diffusion rate of Si in the Nb5Si3 phase is found to be unusual. The growth mechanism of the phases is also discussed on the basis of the data calculated in this study. (C) 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

The knowledge of diffusion parameters, such as integrated diffusion coefficient and the activation energy for diffusion is important to understand the growth rate of the product phase and the atomic mechanism of diffusion. These parameters are determined in Ti3Au phase with A15 crystal structure. The calculated diffusion parameters will help in validating the theoretical analysis on defect structure of the phase.

The variation of the interdiffusion coefficient with the change in composition in the Nb-Mo system is determined in the temperature range of 1800 °C to 1900 °C. It was found that the activation energy has a minimum at around 45 at. pct Nb. The values of the pre-exponential factor and the activation energy for diffusion are compared with the data available in the literature. Further, the impurity diffusion coefficients of Nb in Mo and Mo in Nb are calculated.

The complete understanding of the phenomenological process related to the bifurcation of the Kirkendall marker plane is still lacking. In this study a physico-chemical approach is used to elucidate this behavior in a single phase of \beta-NiAl and to calculate the diffusion parameter at the Kirkendall marker planes in the interdiffusion zone. To apply the physico-chemical approach, Ni- and Al-rich part of the phase is treated as two different phases and the plane corresponding to equiatomic composition is considered as virtual interface between them. Possible dissociation and reaction equations are considered to combine with the flux equations to derive the relation for diffusion coefficient.

A physico-chemical approach is developed, which can be used in binary diffusion couples to determine diffusion parameters of product phases with a wide homogeneity range, as well as phases with a narrow homogeneity range. It is demonstrated that this approach is basically equivalent to the diffusion-based treatment. However, the physico-chemical approach pedagogically sheds light on the chemical reactions occurring during interdiffusion at the interphase interfaces. This theory can be used in any binary system for any end-member condition to explain single-phase or multiphase diffusion-controlled growth. Ni–Al and Ag–Zn systems are considered here to calculate diffusion parameters following the physico-chemical approach. It is evident from our theoretical analysis and experimental evidence that in the case of presence of a stable Kirkendall marker plane one should expect the presence of duplex grain morphology in a particular phase layer.

A general treatment of a diffusion-controlled growth of a stoichiometric intermetallic in reaction between two two-phase alloys is introduced. A reaction couple, in which a layer of $Co_2Si$ is formed during interdiffusion from its adjacent saturated phases is used as a model system. On the basis of chemical reaction equations occurring at the interphase interfaces, data on relative mobilities of diffusing species and the integrated diffusion coefficient in the product phase are deduced. The analysis yields numerical results identical to those calculated from classical Wagner’s theory for the case in which the terminal phases of the diffusion couple are initially saturated.