Matano Plane Research Articles

Novel General Solution for the Analysis of a Multicomponent Interdiffusion Couple

In this paper, we draw attention to the investigation of the novel exact solution [1.Scripta Mat. 210:114430; M.A. Dayananda, in JPED, this issue, (2022);] that is applicable to a multicomponent (n-component) interdiffusion couple where the interdiffusion matrix may change with alloy composition. In the derivation of this solution the interdiffusion flux J_{j} of a component j is related to (n-1) independent composition gradients for an isothermal, diffusion couple using the well-known continuity equation. Novel exact expressions are then derived for all of the interdiffusion coefficients, tilde{D}_{ij}^{n} (i, j = 1, 2, …..n − 1), where the partial derivatives of the product J_{j} left( {y - y_{0} } right) with respect to composition C_{i} (y_{0} is the Matano plane) are used. In this paper, it is shown that the novel solution leads to a computational procedure similar to the Boltzmann-Matano analysis. Note that the derivatives partial (J_{j} left( {y - y_{0} } right))/partial C_{i} , i,j = 1, ldots ,n - 1 (that are required for the solution) can only be calculated along the diffusion path and therefore, for n > 2, a single couple will not be enough to calculate all of them correctly.

Model for non-equilibrium vacancy diffusion applied to study the Kirkendall effect in high-entropy alloys

The effect of the non-equilibrium vacancy on the Kirkendall porosity formation was studied by means of a developed model of the multi-component diffusion with vacancies (MDV) which includes the intrinsic fluxes with vacancy gradient and non-ideal sources and sinks for vacancies. For this study, the diffusion couple experiments in multi-component alloys were chosen. To handle the case of concentration-dependent equilibrium vacancy concentration, we introduced the interaction parameters between the components and vacancies, which can have strong effects on the equilibrium vacancy concentration in alloys and on thermodynamic factors. The diffusion profiles of components and vacancies were simulated by using thermodynamic and kinetic data. The different intensity of the vacancy annihilation/generation and different initial vacancy distributions were considered. Furthermore, we show that the conventional model of diffusion in multi-component systems is a particular case of the MDV with a specific sink/source term. The conventional model was extended by the vacancy diffusion term, similar to the MDV, which significantly reduces the vacancy gradient and the pore formation near the Matano plane. The numerical results demonstrate that the diffusion profiles of substitutional components slightly depend on the sink/source intensity if the none-zero net flux of substitutional components is not significant and the sources and sinks of vacancies are not sparse, whereas the porosity depends very strongly and correlates with the vacancy distribution. For the simulation of variable equilibrium vacancy concentrations using the MDV, the corresponding interaction parameters related to vacancies are necessary to be included in the thermodynamic assessment.

A Direct Derivation of Fick's Law from Continuity Equation for Interdiffusion in Multicomponent Systems

A new analysis is presented where the interdiffusion flux J˜i of a component i is related to (n-1) independent concentration gradients for an isothermal, n-component diffusion couple with the aid of the continuity equation. This analysis may be considered as a derived statement of Fick's law applicable to the analysis of solid-solid and vapor-solid diffusion couples. New expressions are provided for both main and cross interdiffusion coefficients, D˜ijn(i,j=1,2,.....n−1), at section x in the diffusion zone involving the partial derivatives of [(J˜i)·(x−xo)] with respect to concentration Cj, where xo is the Matano plane. Such expressions for both binary and ternary interdiffusion coefficients are presented for various choices of the dependent concentration variable, Cn. The validity of the analysis is illustrated with a binary diffusion couple characterized by a constant interdiffusion coefficient. Diffusional interactions among components are reflected in the variations of [(J˜i)·(x−xo)] with Cj.

Determination of interdiffusion coefficients of Si and Al in Ti3SiC2–Ti3AlC2 diffusion couple at 1373‐1673 K

AbstractIn the diffusion couple of Ti3SiC2 and Ti3AlC2, only interdiffusion of Si and Al occurred during diffusion treatment process. Based on the concentration profiles of Si and Al measured by electron probe microanalysis (EPMA), the interdiffusion coefficients of Si and Al at 1373‐1673 K in Ti3SiC2–Ti3AlC2 diffusion couple were determined by both the Boltzmann‐Matano (B‐M) method and the Saucer‐Freise (S‐F) method. At the position of Matano plane with the composition of Ti3Al0.5Si0.5C2, the interdiffusion coefficient could be expressed as Dint (m2/s) = 5.6 × 10−4⋅exp [−246 ± 14 (kJ/mol)/RT]. Based on the two methods, the calculated interdiffusion coefficients increased with increasing temperature, and the magnitudes of their absolute values were on the order of 10–13‐10–11 m2/s at 1373‐1673 K. At 1373‐1573 K, the calculated interdiffusion coefficients decreased monotonously with the increase of Si concentration, that is, xSi/(xAl + xSi). But at 1673 K, the variation trend of interdiffusion coefficients with xSi/(xAl + xSi) was no longer monotonous, probably due to the presence of Ti5Si3 phase and voids on Ti3AlC2 side.

On the Matano Plane Position in Multicomponent Diffusion Couples

Even though several methods of diffusion analysis avoid a necessity for the Matano plane determination, the Matano plane locations are of interest in the multicomponent couples and when tracer experiments are performed. The positions of the Matano plane calculated from the concentration profiles should be exactly the same. However, due to experimental errors, the results can differ significantly. In the paper we consider Matano plane for multi-component couples of conserved and non-conserved overall volumes. We use Darken method and Sauer-Freise scheme, respectively and show that looking for the Matano plane as an inverse problem leads to a system of linear algebraic equations which are over-determined. The relations for the Matano plane position are derived by virtue of the least-squares methodology. The exemplary computations shows that the method is particularly useful in analysis of experimental data and allows decreasing computing errors.

Seconds after impact: Insights into the thermal history of impact ejecta from diffusion between lechatelierite and host glass in tektites and experiments

Tektites contain inclusions of lechatelierite, nearly pure SiO2 glass formed by quenching of quartz grains melted during hypervelocity impacts. We report the discovery in a tektite of chemically zoned boundary layers (ca 20 μm) between lechatelierite and host felsic glass. These boundary layers in tektites formed by chemical diffusion between molten silica inclusions (quenched to lechatelierite on cooling) and surrounding felsic melt. We reproduced the details of these boundary layers via experiments on mixtures of powdered natural tektite plus quartz grains heated to 1800–2400 °C for 1–120 s using an aerodynamic levitation laser heating furnace. The results of these experiments were used to provide quantitative constraints on possible thermal histories of the natural sample.The experiments successfully reproduced all major aspects of the concentration profiles from the natural sample including diffusion length scale, strong asymmetry of the concentration profiles with respect to the Matano plane (due to the strong concentration dependence of the diffusivities of all oxides on SiO2 content), similarities in lengths of the diffusive profiles (due to control by the diffusion of SiO2 on the diffusivity of the other oxides), and differences in the shapes of the profiles among the oxides (including a maximum in the diffusion profile of K2O due to uphill diffusion). The characteristic lengths of all non-alkali oxide profiles are proportional to t from which diffusivities and activation energies can be derived; these results are consistent with measurements in melts with lower SiO2 contents and at lower temperatures reported in the literature. We also fit the experimental profiles of SiO2 and Al2O3 using simple formulations of the dependence of their diffusivities on SiO2 content and temperature, yielding results similar to those obtained from the t dependence of the characteristic profile lengths.The quantitative characterization of diffusion in boundary layers based on our experiments allow us to set limits on the thermal history of the natural tektite in which the boundary layers were discovered. If the interdiffusion between the silica and felsic melts occurred at constant temperature, the duration of heating experienced by the natural tektite we studied depends on temperature; possible solutions include heating at ∼2000 °C for ∼70 s, −2400 °C for ∼3 s. We also explored non-isothermal, asymptotic cooling histories; for a maximum temperature of 2400 °C, a characteristic cooling time scale of ∼50 s is implied, whereas, for 2000 °C, the time scale is ∼1400 s. Further, a maximum temperature of ∼2360 °C yields an effective diffusive time scale of ∼5 s, a cooling time scale of ∼90 s, and a cooling rate at the glass transition temperature of ∼5 °C/s; results that are consistent with independent estimates of cooling time scales for ∼1 cm clasts (Xu and Zhang, 2002), as well as cooling rates at the glass transition temperature (Wilding et al., 1996) – thus satisfying all currently available relevant data. More complex T-t paths are possible and can also be modeled using our experimental results and compared with and used as tests of the accuracy of physical models of tektite-forming impact events.

Determination of eigenvalues, eigenvectors, and interdiffusion coefficients in ternary diffusion from diffusional constraints at the Matano plane

New diffusional constraints interlinking interdiffusion fluxes, concentration gradients and concentrations of the components at the Matano plane are derived for an isothermal, solid-solid ternary diffusion couple. The derivations are based on the diagonalization of the interdiffusion coefficient matrices employed for the description of selected regions of the diffusion zone and on the integration of the decoupled concentration gradients and interdiffusion fluxes over those regions. By selecting two diffusion regions, one on either side of the Matano plane, the diffusional constraints are employed for the direct determination of eigenvalues, eigenvector parameters and a matched pair of interdiffusion coefficient matrices for the generation of concentration profiles over the two regions. For a diffusion couple assembly, the input data needed for the calculations are the concentration gradients, interdiffusion fluxes and composition developed at the Matano plane at a given time. The diffusion parameters determined by the new analysis help ensure that the calculated concentration profiles, interdiffusion fluxes and concentration gradients of the components are continuous everywhere including the Matano plane. The application of the analysis is illustrated and discussed with selected experimental diffusion couples investigated in the Cu-Ni-Zn ternary system.

Ternary diffusion path in terms of eigenvalues and eigenvectors

Based on the transfer matrix methodology, a new analysis is presented for the description of slopes of the ternary diffusion path for a solid–solid diffusion couple. Concentration profiles and diffusion paths for isothermal, ternary diffusion couples are examined in the context of eigenvalues and eigenvectors obtained from the diagonalisation of the ternary interdiffusion coefficients employed for their representation. New relations are derived relating the decoupled interdiffusion fluxes to combinations of concentration gradients through the major and minor eigenvalues, and the diffusion path becomes parallel to the major eigenvector at each path end. General expressions for the slope of the ternary diffusion path at any section of the couple are also derived in terms of eigenvalue and eigenvector parameters. Expressions for the path slope at the Matano plane involve only concentrations, major and minor eigenvalues and eigenvector parameters. New constraints relating the eigenvalues and the concentration gradients of the individual components are also presented at selected sections, where the diffusion path is parallel to the straight line joining the terminal composition points on an isotherm. Applications of the various relations are illustrated with the aid of a hypothetical couple and an experimental Cu–Ni–Zn diffusion couple.

Effect of alloying elements on interdiffusion phenomena in explosive clads of 304LSS/Ti–5Ta–2Nb alloy

Dissimilar joints of 304L austenitic stainless steel (SS) and Ti–5Ta–2Nb alloy were fabricated using explosive cladding process with an aim to avoid the formation of brittle intermetallic phases at the interface. Subsequently, diffusion annealing heat treatments were carried out at temperatures in the range of 550–800 °C for various durations. In the present study, concentration and temperature dependence of the distinct diffusion zones, formed at the clad interface, due to interdiffusion of the alloying elements, have been established using Electron Probe Micro Analysis and imaging of the interface. Molar volume showed a close match with the ideal Vegard’s law at low temperatures while a non-linear negative deviation has been observed at high temperatures (800 °C) due to the formation of secondary phases such as FeTi, Fe2Ti and βTi(Fe) phases. Interdiffusion parameters evaluated by Wagner’s method showed sluggish diffusion kinetics of Fe and Ti at concentration corresponding to a two phase mixture of FeTi and Fe2Ti. Further, an attempt has been made to understand multicomponent diffusion using Dayananda’s approach and estimation of effective interdiffusion coefficients revealed ~22 µm shift of the interface from Matano plane at 800 °C.

The intrinsic diffusivities in multi component systems

The complete understanding of the phenomenological process related to the position of the Matano and Kirkendall plane in multicomponent system is still lacking. In this paper some aspects of Matano and lattice plane migration are studied by means of numerical simulations. New approach to calculate intrinsic diffusion coefficients in binary Ni–Au system and ternary Cu–Fe–Ni and Co–Fe–Ni is presented. Diffusivities were calculated using modified Boltzmann–Matano method. Self diffusion coefficients of the elements in diffusion couple must be known as well as position of the Kirkendall plane after the experiment to use this method. Diffusion profile was calculated using presented equations. Calculations compared with experimental data prove validity of the presented approach. This work proves that intrinsic diffusion coefficients should be given by the concentration function, not the exact value.

Erratum to: Anisotropic Diffusion Behavior of Al in Mg: Diffusion Couple Study Using Mg Single Crystal

IN the calculation of the interdiffusion coefficients of two intermediate phases and the impurity diffusion coefficients ofAl in hcpMg, a small numerical errorwas found. The error does not change the main conclusions of the work but can lead to small errors in the diffusion calculations. The following is a revision of the interdiffusion coefficients and impurity diffusion coefficient. The numbering of figures, tables and references used in this erratum are the same as the original paper for convenience. Interdiffusion coefficients for two intermediate phases (Mg17Al12 and Mg2Al3) were calculated using the Heuman-Matano method by identifying the Matano plane from Al concentration profile throughout the whole diffusion zone quantified by Electron Probe Micro Analyzer (EPMA). For example, the position of the Matano plane for the sample annealed after 72 hours at 693 K (420 C) is marked in Figure 1. The revised interdiffusion coefficients for the Mg17Al12 and Mg2Al3 intermediate phases are presented in Table III and in Figure 4 with comparison to previous studies. Our results are in good agreement with Brennan et al. The revised values for the Al impurity diffusion coefficient in hcp Mg from the multiphase diffusion calculation used in the original paper are plotted in Figure 7(a). In order to confirm the impurity diffusion, a Gaussian solution of Fick’s second law is also applied to analytically calculate impurity diffusion coefficient in this erratum. In the analytical method, the impurity diffusion coefficient for each sample was calculated from the slope of ln (Al concentration) vs square of penetration distance in dilute Al concentration region of hcp Mg solid solution. As can be seen in Figure 7(a), the analytical results are in good agreement with the results obtained from multiphase diffusion simulation. The revised Al impurity diffusion coefficients, depending on the orientation of hcp Mg, were plotted in Figure 7(b) and compared with previous studies. The revised optimized mobilities of Al in hcp Mg along the aand c-axis of Mg crystal are

Study on Diffusion of Copper of SOFC Cu-Based Anodes in Co-Sintering with Electrolyte

Cu-based anode has no or little catalytic activity to reaction of precipitating carbon during using carbon-hydrogen fuel, which can avoid failure or disintegration of SOFC caused by catalytically precipitating carbon resulting from Ni-based anode. However, the melting point of CuO, that is 1148°C, is far below the sintering temperature of electrolyte which limits the use of Cu-based anode. So the property of sintering of Cu-based anode between 1050°C and 1150°C was studied by co-pressing and co-sintering Cu-based anode with electrolyte. The result shows that the diffusion of copper in the electrolyte is not apparent when the co-sintering temperature is below 1130°C. At 1140°C, the Matano plane moved to anode side and the diffusion coefficient of copper in the electrolyte is relevant to concentration of copper which is 0.543×10-8-1.74×10-8(cm2•s-1) when the content of copper is 1.51%~18.7%.

Effect of Ti and Fe Alloying on Kirkendall Voids Feature in Ni/Ni<sub>3</sub>Al Welded Joints after the High Temperature Annealing

The study of influence of Fe and Ti alloying elements in the Ni3Al intermetallic alloy on diffusion feature in Ni/Ni3Al-Me (Me = Fe or Ti) joints was performed using the diffusion couple technique. Four sets of diffusion couples of diverse compositions prepared by means of resistance welding or electron beam welding in vacuum were used. The concentrations profiles and diffusivities of welded Ni/Ni3Al-Fe and Ni/Ni3Al-Ti joints were determined after annealing at 1050°C for 100 hours. The concentration profiles were smoothed using suitable types of polynomials. The Kirkendall voids in the area between the Matano plane and the new  interface created after annealing were observed. Metallographic study and qualitative evaluation of voids by means of AFM method revealed different feature of Kirkendall regions in Ni/Ni3Al-Fe couples unlike Ni/Ni3Al-Ti joints.

Role of Ti diffusion on the formation of phases in the Al2O3–Al2O3 brazed interface

Diffusivities of Ti, Cu, Al and Ag in the interface of Al2O3–Al2O3 braze joints using Ag–Cu–Ti active filler alloy, have been calculated by Matano–Boltzman method. The Matano plane has been identified for each elemental diffusion at various brazing temperatures. The diffusivities of Ag, Cu and Al are almost insignificant on formation of interface during brazing, whereas the diffusivity of Ti changes significantly with the brazing temperature and controls the formation of different reaction product in the interface. Presence of TiO and Ti3Cu3O phases in the interface has been confirmed by transmission electron microscopy (TEM).

Study of Kirkendall Effect in Ni/Ni<sub>3</sub>Al Welded Joint after the High Temperature Annealing

The study of composition and microstructure of welded joints was performed before and after the diffusion annealing at elevated temperatures for different annealing time. The Kirkendall effect in the Ni/Ni3Al diffusion couples was observed by means of different methods, e.g. using light, scanning electron and atomic force microscopies. The study suitably completes and specifies the morphology features of Kirkendall voids at different evolution stages, i.e. at nucleation, growing and coalescence. Kirkendall voids occurred in the region between the Matano and γ/γ´interface planes. The location of the γ/γ´ interface that moved in the direction of Ni3Al phase during the annealing resulted from the Al concentration profile measured by EDAX. The Matano plane location was determined by means of Boltzmann-Matano’s method using concentration profile data. It was observed that the void size was increasing in the direction from the Matano plane to the γ/γ´ interface. The obtained results were completed by surface topography of Kirkendall voids of slightly etched specimens by atomic force microscopy (AFM).

A transfer matrix analysis of quaternary diffusion

A transfer matrix method (TMM), developed recently by Ram-Mohan and Dayananda, has been employed for the first time to generate error function solutions for quaternary diffusion in solid–solid metallic diffusion couples. The method was validated with the aid of a hypothetical couple with known quaternary interdiffusion coefficients and applied to two experimental Cu–Ni–Zn–Mn quaternary couples annealed at 775°C. Quaternary interdiffusion coefficients were determined by the Dayananda analysis from the concentration profiles of the couples over selected composition ranges in the diffusion zone and employed for the subsequent TMM calculation of error function solutions. For the hypothetical test couple, identical sets of constant interdiffusion coefficients were determined on either side of the Matano plane and utilized for the calculation of concentration profiles by TMM. For the experimental Cu–Ni–Zn–Mn couples, interdiffusion coefficients determined over selected regions in the diffusion zone were employed for the TMM generation of the concentration profiles.

Diffusion behavior of Cu in Cu/electroless Ni and Cu/electroless Ni/Sn-37Pb solder joints in flip chip technology

For Cu pads used as under bump metallization (UBM) in flip chip technology, the diffusion behavior of Cu in the metallization layer is an important issue. In this study, isothermal interdiffusion experiments were performed at 240°C for different times with solid-solid and liquid-solid diffusion couples assembled in Cu/electroless-Ni (Ni-10 wt.% P) and Cu/electroless Ni (Ni-10 wt.% P)/ Sn-37Pb joints. The diffusion structure and concentration profiles were examined by scanning electron microscopy and electron microprobe analysis. The interdiffusion fluxes of Cu, Ni and P were calculated from the concentration profiles with the aid of Matano plane evaluation. The values of JCu, JNi, and JP decreased with increasing annealing time. The average effective interdiffusion coefficients on the order of 10−14 cm2/s were also evaluated within the diffusion zone. The amounts of Cu dissolved in the intermetallic compounds (IMCs) Ni3Sn4 and Ni3P that precipitate after annealing the Cu/electroless Ni/Sn-37Pb joints were about 0.25 at.% and 0.5 at.%, respectively. For the short period of annealing, it appears that the presence of electroless Ni (EN) with the Sn-Pb soldering reaction assisted the diffusion of Cu through the EN layer.

An examination of a multicomponent diffusion couple

New expressions relating the interdiffusion flux of a component to its own concentration gradient in a multicomponent diffusion couple have been derived and applied to a diffusion couple investigated in the Cu-Ni-Zn system. From these relations, effective interdiffusion coefficients were determined at selected sections in the diffusion zone directly from the locations of the sections relative to the Matano plane. The Cu-Ni-Zn couple was analyzed for interdiffusion fluxes and interdiffusion coefficients with the aid of “MultiDiFlux” program developed for the analysis of interdiffusion in multicomponent systems. The couple was examined for zero-flux plane development, interdiffusion against activity gradients, and diffusion path representation. Diffusion path slopes at selected sections in the diffusion zone were related to the interdiffusion coefficients; slopes at path ends were determined from eigenvectors evaluated from limiting ratios of interdiffusion fluxes. Expressions for internal consistency among the concentration profiles or flux profiles of the individual components were also developed in terms of the terminal alloy compositions and applied to the Cu-Ni-Zn couple in the diffusion zone.

Zero-Flux Plane Kinetics at Multicomponent Interfaces

Zero flux planes (ZFP's) are the locations in a diffusion couple where fluxes of individual components vanish. The conditions required for developing ZFP's between interdiffusing multicomponent alloys are now well understood through the works of Dayananda and Morral. In this paper we analyze the kinetics of multicomponent diffusion near ZFP's. In contrast to the usual mixing that occurs between end-member alloys in a diffusion couple, where the average component compositions are approached through “global” exchange of atoms, the presence of a stationary ZFP requires instead that mixing of the blocked component occurs through the release of a pair of coupled diffusion waves. In classical (thick) couples these waves spread symmetrically away from the Matano plane. The wave pair consists of a “depletion” wave that reduces the blocked component concentration in the component-rich alloy, and a conjugate “repletion” wave that increases the concentration in the adjacent component-poor alloy. Curiously, each of these waves establishes the average (equilibrium) concentration unilaterally on either side of the Matano plane. A stationary ZFP precludes net transport of one component across the Matano plane—a circumstance that has practical and theoretically interesting implications for the design of stable multicomponent films and coatings that must resist diffusive loss of a component.

Dynamics of multicomponent diffusion with zero flux planes

Multicomponent diffusion induces zero flux planes (ZFPs) that are locations in a diffusion couple where fluxes of individual components vanish. We analyze the kinetics of multicomponent diffusion near ZFPs. In contrast to the usual mixing that occurs between alloys in a diffusion couple, where the average component compositions are approached through ‘global’ exchange of atoms, the presence of a ZFP requires that mixing of the blocked component occurs through the release of coupled diffusion waves consisting of a ‘depletion’ wave that reduces the blocked component concentration in the component-rich alloy, and a ‘repletion’ wave that increases the concentration in the component-poor alloy. Each wave establishes the average (equilibrium) concentration unilaterally on either side of the Matano plane. Multicomponent diffusion behaviors, such as slowed diffusive spreading and reduced atomic transport, are associated with the near collocation of multiple ZFPs, and the proximity of a zero flux plane for the major component.