Sauer Freise Method Research Articles

Interdiffusion behaviors and mechanical properties of Zr–Hf system

In this work, Zr–Hf diffusion couples were firstly prepared by Spark Plasma Sintering (SPS) method and the diffusion behavior and mechanical properties of the Zr–Hf system in temperatures range from 1300 to 1600 °C were investigated in detail. The microstructure characterization by SEM and TEM reveals that Zr–Hf diffusion zone form some amorphous and nanocrystalline grains in local area. Then the hardness and Young's modulus of the Zr–Hf system in the diffusion zone were tested and the results show that the hardness and Young's modulus of the alloy system was enhanced by solid solution Hf element, and the maximum hardness and Young's modulus in the diffusion zone reached to 10.74GPa and 203.5GPa, respectively. Finally, based on the profiles of Electron probe microanalysis (EPMA), the interdiffusion coefficient is composition-dependent. The value calculated by Sauer-Freise method is between 10−13 m2/s and 10−11 m2/s, which indicating that SPS diffusion annealing significantly enhances the diffusion rate of the Zr–Hf system. Meanwhile, the current direction has negligible effect on the diffusion of Zr–Hf system and the interdiffusion coefficient decreases slowly with increasing Hf concentration. The diffusion activation energy Q and the frequency factor D0 of the Zr–Hf binary system in this experiment ranges from 196 to 215 kJ/mol and 3.9 × 10−6 to 6.8 × 10−5 m2/s, respectively.

Study on diffusion and Kirkendall effect in diffusion triples for fcc Ni–Al–Ta alloys

In the present work based on two diffusion triples, the composition-dependent interdiffusivity matrices in the fcc Ni–Al–Ta alloys at 1373K and 1473K were efficiently deduced by using our newly developed two-dimensional (2D) inverse scheme. This scheme determines interdiffusivities from point and one-dimensional (1D) diffusion path to 2D composition region, yet requires much less experimental efforts, with the measured 2D composition profiles being well reproduced. Further, the interdiffusivities deduced from the inverse scheme were directly compared with those extracted by the traditional Sauer-Freise method and Whittle-Green method based on nine diffusion couples designed for verification. The interdiffusivities inferred from two distinct ways are fairly consistent, either at the binary boundaries or at the intersection points within the ternary composition range. Besides, the interdiffusion flux and the shift of the Kirkendall plane over the whole diffusion zone were simulated by applying the present 2D inverse scheme. The resulting 2D mapping presents a non-uniform but curved Kirkendall plane, which is in contrast to the flat shape generated in 1D diffusion couples and well explained by the calculated diffusion variables.

Interdiffusion behaviors and mechanical properties of Hf-X (X=Nb, Ta) binary systems

The diffusion behaviours and mechanical properties of the Hf-X (X = Nb, Ta) systems have been investigated using diffusion couples in the temperature range from 1150 °C to 1500 °C. The interdiffusion coefficients of Hf-X (X = Nb, Ta) systems were extracted using forward simulation analysis (FSA) based on the Sauer-Freise method. According to the results, the interdiffusion coefficients of the Hf-X (X = Nb, Ta) systems decreased as the Nb or Ta content increased. In the meantime, the diffusion activation energies of Hf-X (X = Nb, Ta) systems increased as the Nb or Ta content increased. Finally, the hardness and Young’s moduli of the Hf–Nb and Hf–Ta systems were determined by nanoindentation tests. The largest hardness values of the Hf–Nb and Hf–Ta were 5.2 GPa and 5.3 GPa, respectively. Young’s largest moduli of the Hf–Nb and Hf–Ta were 221 GPa and 238 GPa, respectively. The present results provide helpful diffusional data and mechanical properties for the materials design, research and development of hafnium-based alloys.

High-throughput measurements of interdiffusivities in fcc Co–Ni–Ta alloys at 1373K and 1473K

The composition-dependent interdiffusivities in the fcc Co–Ni–Ta alloys at 1373 K and 1473 K were deduced by using our newly developed numerical inverse scheme that combines two-dimensional (2D) simulations with diffusion triple experiments. This approach largely reduced the experimental efforts by analyzing only one single-phase diffusion triple at each temperature yet covering a much wider composition range than diffusion couples. The reliability of the high-throughput results was firstly verified by reproducing the experimental 2D composition profiles in each triple. In order to further validate the deduced interdiffusivities, several traditional one-dimensional (1D) diffusion couples were also devised, and the widely recognized Sauer-Freise method and Whittle-Green method were then employed to calculate the interdiffusivities for the binaries and at the intersection points within the ternary composition range, respectively. The interdiffusivities extracted from the inverse scheme agree well with those from the traditional approaches, which strongly proves the applicability of the present new scheme. Besides, the constructed main interdiffusivity planes at 1373 K and 1473 K provide an overview of the diffusion behavior in the fcc Co–Ni–Ta system and promote further kinetic studies.

Diffusion coefficients and atomic mobilities in fcc Ag–Ge and Cu–Ge alloys: Experiment and modeling

The comprehensive investigations regarding the diffusion behaviors for Ag–Ge and Cu–Ge solders play key roles in their design and application. However, the diffusion behaviors in fcc Ag–Ge and Cu–Ge binary systems have not been carried out. To fill this gap, their diffusion characteristics were studied in this work based on the measured composition profiles of diffusion couples prepared for fcc Ag/Ag–Ge diffusion couples annealed at 873, 923 and 973 K as well as fcc Cu/Cu–Ge diffusion couples annealed at 1023, 1073 and 1123 K. On the basis of measured composition profiles, the composition-dependent interdiffusivities were calculated by the numerical inverse method incorporated in CALTPP (CALculation of ThermoPhysical Properties) program, which agree well with the ones calculated by the classic Sauer-Freise method. Besides, the atomic mobilities in fcc Ag–Ge and Cu–Ge alloys were assessed based on the diffusivities obtained from the literature and this work. Comparisons between model-predicted diffusion characteristics (viz. composition profiles and interdiffusivities) and experimental ones further validate the reliability of the presently obtained atomic mobilities. Insights gained in this work are of scientific significance and practical benefit for designing new Ge-bearing Ag and Cu solers.

Interdiffusion behaviors and mechanical properties of Zr-X (X Nb, Ta, Hf) binary systems

The diffusion behaviors and the mechanical properties of Zr-X (X Nb, Ta, Hf) systems have been systematically investigated at the annealing temperature range of 950–1200 °C. Based on the Sauer-Freise method and the Hall method, the interdiffusion coefficients of the Zr-rich alloys were evaluated, and the impurity diffusion coefficients on the Zr-rich side have been extracted by using the Darken equation. The presently obtained interdiffusion coefficients and impurity diffusion coefficients are all in good agreement with experimental data in the literature. According to the Arrhenius regression, the diffusion activation energies of the three systems have been calculated. It can be concluded that the diffusion activation energies of the Zr-Ta system are the largest value. Finally, the hardness and Young’s modulus of the Zr-X have been obtained by using the nanoindentation test. The present results can help the materials design, research and development for the Zr-based alloys. • Interdiffusion coefficients and activation energies are obtained from the measured composition profiles. • Hardness and Young’s modulus of Zr-X (X Nb, Ta, Hf) systems have been obtained. • The solid-solid diffusion couple has been used.

Interdiffusion and atomic mobilities in bcc V–X (X = Mn, Sn and Ni) alloys: Measurement and modeling

Totally 8 diffusion couples were prepared and the composition-dependent interdiffusion coefficients in bcc V–Mn, V–Sn and V–Ni alloys were measured between 1323 and 1503 K by means of Sauer-Freise method coupled with electronic probe microanalysis (EPMA) technique. The errors of the interdiffusion coefficients were computed by error propagation method. Based on the measured interdiffusivities and thermodynamic parameters from the literature, the atomic mobilities of bcc V–X (X = Mn, Sn and Ni) alloys were obtained by using the CALTPP (CALculation of ThermoPhysical Properties) program with the features of high efficiency and accuracy in the framework of CALPHAD (CALculation of PHAse Diagram) approach. For each system, the presently obtained atomic mobilities were validated by good agreement between the calculated interdiffusivities and the measured ones. The model-predicted concentration profiles in this work are consistent with the experimental ones. This work contributes to the establishment of the atomic mobility database containing V element and the key input for microstructure simulations in V-based alloys.

Interdiffusion in Cr–Fe–Co–Ni medium-entropy alloys

Diffusion in multi-component alloys is attracting renewed attention because of the worldwide interest in high- and medium-entropy alloys (HEAs/MEAs). In the present work, we used diffusion multiples made from MEAs of the quaternary Cr–Fe–Co–Ni system arranged as six distinct pseudo-binary diffusion couples (Cr29Fe13Co29Ni29–Cr29Fe29Co29Ni13, Cr29Fe29Co13Ni29–Cr29Fe29Co29Ni13, and so on, where the interdiffusing elements are italicized for clarity). In the two halves of each couple, the starting concentrations of the interdiffusing elements (Fe,Ni and Co,Ni in the above examples) were different while those of the background elements (Cr,Co and Cr,Fe in the above examples) were the same. The diffusion multiples were annealed at temperatures between 1153 and 1355 K at times from 100 to 900 h, after which the concentrations of the different elements were measured as a function of distance across each couple. Interdiffusion coefficients were derived from such concentration profiles using the standard Sauer-Freise method and compared with literature data as well as with published tracer diffusion coefficients. Although the background elements were homogeneously distributed initially, some of them developed distinct sine-wave shaped concentration gradients near the interfaces after annealing, implying that uphill diffusion of these elements had occurred. We show using a kinetic model for substitutional diffusion via vacancy hopping that such uphill diffusion can occur even in the absence of thermodynamic interactions, i.e. in ideal solid solutions in which the thermodynamic factor Φ of each element is equal to one (Φi=1+∂lnfi/∂lnci where fi and ci are the activity coefficient and mole fraction of element i, respectively). The model accounts for all elemental fluxes and also rationalizes the diffusion profiles of the major interdiffusing elements.

An experimentally-validated numerical model of diffusion and speciation of water in rhyolitic silicate melt

The diffusion of water through silicate melts is a key process in volcanic systems. Diffusion controls the growth of the bubbles that drive volcanic eruptions and determines the evolution of the spatial distribution of dissolved water during and after magma mingling, crystal growth, fracturing and fragmentation, and welding of pyroclasts. Accurate models for water diffusion are therefore essential for forward modelling of eruptive behaviour, and for inverse modelling to reconstruct eruptive and post-eruptive history from the spatial distribution of water in eruptive products. Existing models do not include the kinetics of the homogeneous species reaction that interconverts molecular (H2Om) and hydroxyl (OH) water; reaction kinetics are important because final species distribution depends on cooling history. Here we develop a flexible 1D numerical model for diffusion and speciation of water in silicate melts. We validate the model against FTIR transects of the spatial distribution of molecular, hydroxyl, and total water across diffusion-couple experiments of haplogranite composition, run at 800–1200 °C and 5 kbar. We adopt a stepwise approach to analysing and modelling the data. First, we use the analytical Sauer-Freise method to determine the effective diffusivity of total water DH2Ot as a function of dissolved water concentration CH2Ot and temperature T for each experiment and find that the dependence of DH2Ot on CH2Ot is linear for CH2Ot≲1.8 wt.% and exponential for CH2Ot≳1.8 wt.%. Second, we develop a 1D numerical forward model, using the method of lines, to determine a piece-wise function for DH2OtCH2Ot,T that is globally-minimized against the entire experimental dataset. Third, we extend this numerical model to account for speciation of water and determine globally-minimized functions for diffusivity of molecular water DH2OmCH2Ot,T and the equilibrium constant K for the speciation reaction. Our approach includes three key novelties: (1) functions for diffusivities of H2Ot and H2Om, and the speciation reaction, are minimized simultaneously against a large experimental dataset, covering a wide range of water concentration (0.25≤CH2Ot≤7 wt.%) and temperature (800°C≤T≤1200°C), such that the resulting functions are both mutually-consistent and broadly applicable; (2) the minimization allows rigorous and robust analysis of uncertainties such that the accuracy of the functions is quantified; (3) the model can be straightforwardly used to determine functions for diffusivity and speciation for other melt compositions pending suitable diffusion-couple experiments. The modelling approach is suitable for both forward and inverse modelling of diffusion processes in silicate melts; the model is available as a MATLAB script from the electronic supplementary material.

In-situ transmission electron microscopy determination of solid-state diffusion in the aluminum-nickel system

Using in-situ techniques, which couple energy dispersive spectroscopy mapping with a heated transmission electron microscope stage, solid-state diffusion of Ni-Al is studied in the temperature range 623–723 K with characteristic diffusion lengths of 1–100 nm. When the concentration profiles are analyzed using the Sauer-Freise method, and evaluated for 50 atomic percent Ni, the diffusion coefficients follow an Arrhenius temperature dependence: DNi→Al=8.2x10−9exp(−111kJ/RT)m2/s. Additionally, the formation of Al-rich intermetallic phases (Al3Ni & Al3Ni2) is shown to occur within heating durations of a second at the studied temperatures.

Interdiffusion and atomic mobilities in fcc Ag–Mg and Ag–Mn alloys

On the basis of Ag/Ag–Mg and Ag/Ag–Mn semi-infinite diffusion couples together with electron probe microanalysis technique, the interdiffusion coefficients in fcc phases of Ag–Mg and Ag–Mn alloys were measured between 873 and 1173 K via the Sauer-Freise method. A statistic method was applied to evaluate the errors of the identified interdiffusivities in terms of the propagation of errors. Based on available thermodynamic information and the experimental interdiffusion coefficients, the atomic mobilities for the fcc Ag–Mg and Ag–Mn systems are obtained by using the DICTRA software package. The quality of the assessed atomic mobilities was confirmed by the comprehensive comparisons between calculated concentration profiles and the experimental data.

Diffusion multiple study of the Co-Fe-Ni system at 800 °C

The Co-Fe-Ni system is a key system in a wide range of industrial applications. Knowledge of the thermodynamic and kinetic properties of the system is crucial for the alloy and process design. Although the system has been studied extensively, there remain several unexplained discrepancies between different literature data and, for mediate low temperatures, the information is scarce. In this work, a high throughput diffusion multiple approach was applied. The isothermal phase diagram section at 800 °C was determined using the Co-Fe-Ni multiple. The interdiffusion coefficients of the binary Co-Fe, Co-Ni, Fe-Ni systems and their composition dependence were calculated using the Sauer-Freise method based on the compositional profiles obtained from the diffusion multiple. Different from previous experimental results for mediate low temperature, our results coincide with the extrapolated Arrhenius temperature dependence from diffusion coefficient data at high temperature range. These observations are important for a better understanding and modelling of the interdiffusion behavior in this key alloy system.

Interdiffusion behaviors and mechanical properties of Cu-Zr system

In this work, solid-to-solid diffusion couples were assembled and annealed to investigate diffusion behavior and mechanical properties of the Cu-Zr system in temperatures range from 1043 K to 1113 K. Six intermetallic compounds (IMCs) Cu9Zr2, Cu51Zr14, Cu8Zr3, Cu10Zr7, CuZr, and CuZr2 were observed in the diffusion zone. Composition-dependent interdiffusion coefficients of IMCs have been calculated based on the measured composition Cu profiles of the diffusion zones by using Sauer–Freise method. And the average effective interdiffusion coefficients for each phase were also calculated by using Wagner method. The activation energies of diffusion are evaluated according to the average effective interdiffusion coefficient. Finally, the load-displacement curves measured by nano-indentation are obtained to characterize mechanical properties of Cu9Zr2 and Cu51Zr14, which have similar hardness and elastic moduli.

Experimental measurements of the interdiffusivities in fcc Co–rich Co–Ti, Co–W and Co–Ti–W systems

Utilizing 8 bulk single–phase diffusion couples together with the electron probe microanalysis technique, the composition–distance profiles of fcc binary Co–Ti and Co–W systems as well as ternary Co–Ti–W system at 1373K were first measured. The measured composition profiles for Co–rich Co–Ti, Co–W and Co–Ti–W systems at 1373K were then adopted to evaluate the corresponding binary and ternary interdiffusivities by using the newly proposed pragmatic numerical inverse method. The comprehensive comparisons between various model–predicted diffusion properties based on the evaluated interdiffusivities due to the pragmatic numerical inverse method, including composition profiles, interdiffusion fluxes and diffusion paths, and the experimental data were conducted to validate their reliability. Moreover, the presently obtained binary and ternary interdiffusivities by the numerical inverse method were further validated by the literature data and the calculated results by using the traditional Sauer–Freise method and Matano–Kirkaldy method.

Interdiffusion in the Ni-Re System: Evaluation of Uncertainties

Diffusion couple experiments between Ni and Re at 1200 and 1350 °C were performed. These experiments established the limits of the two-phase FCC + HCP region. No intermediate phase was observed at these temperatures. Composition-dependent interdiffusion coefficients and associated uncertainties were estimated by three methods. The first employed fitting of the penetration curves in conjunction with the Sauer-Freise (SF) method. The second method employed a numerical solution of the Boltzmann-Matano ordinary differential equation for composition-dependent interdiffusion coefficient functions whose parameters were optimized by a least squares fitting to the data. Discrepancies between the results of these methods indicate typical uncertainties in experimental determination of diffusion coefficients. To further assess such discrepancies, a third method was employed to perform an uncertainty quantification of the diffusion coefficients via a statistical analysis based on the SF method.

Interdiffusion in fcc Ni–X (X = Rh, Ta, W, Re and Ir) alloys

The interdiffusion coefficients in binary face-centered cubic (fcc) Ni–X (X = Rh, Ta, W, Re and Ir) alloys at 1473, 1523, 1573 and 1623 K were measured by using semi-infinite diffusion couples together with the Sauer–Freise method. A scientific method was employed to evaluate the errors of the determined interdiffusivities by considering the error propagation. The measured interdiffusion coefficients agree in general with the literature data. In order to find out the possible substitutional element for Re in Ni-based superalloys, a comprehensive comparison among the interdiffusion coefficients in fcc Ni–X (X = Nb, Mo, Ru, Rh, Pd, Ta, W, Re, Os, Ir and Pt) alloys with 6 wt% X at 1473–1623 K was conducted. The comparison results indicate that fcc Ni-6 wt% Os alloys exhibit the lowest diffusion coefficient, followed by Ni-6 wt% Re and Ni-6 wt% Ir alloys. The further analysis on the variation of interdiffusion coefficient with its influence factors including alloying concentration, temperature, atomic number, activation energy, atomic radius and compressibility was also performed.

Measurement of tracer diffusion coefficients in an interdiffusion context for multicomponent alloys

A recently developed novel approach of simultaneous analysis of isotope and interdiffusion profiles in binary alloy systems is significantly advanced in order to apply to the case of multicomponent alloy systems. The resulting relations for the tracer or self-diffusion coefficients allow for the avoiding of the explicit solution of the interdiffusion equations. This remarkable result means that in the experimental implementation of this new technique there is no need for multiple diffusion couples. Only two profiles of the same component are necessary for the complete analysis. These can be two different isotopes or just two spatially different parts of the same atomic species. Descriptions of three possible experimental implementations of the novel technique combined with the Sauer–Freise method are discussed. Therefore, the new development is ready to be applied experimentally and can provide valuable insight into otherwise very difficult diffusion investigations into multicomponent alloys including high-entropy alloys.

Thermodynamic description, diffusivities and atomic mobilities in binary Ni–Os system

A systematical study of thermodynamics and diffusion kinetics in the binary Ni–Os system was performed in the present work by means of CALculation of PHAse Diagram (CALPHAD) method supported by key experiments. Based on all the available experimental phase equilibria, a full thermodynamic assessment of binary Ni−Os system was conducted. A reasonable set of thermodynamic description for the binary Ni−Os system was obtained. Comparisons between the calculated and the measured phase equilibria showed that most of the experimental information can be satisfactorily accounted for by the present thermodynamic description. Moreover, the interdiffusion coefficients in face-centered cubic (fcc) Ni−Os alloys at 1373, 1473, 1523, 1573 and 1623K were measured by using five groups of Ni−Os semi-infinite diffusion couples together with the Sauer–Freise method. On the basis of the interdiffusion coefficients from the present work and the literature as well as the presently obtained thermodynamic description, atomic mobilities in fcc Ni−Os alloys were then assessed by means of DIffusion-Controlled TRAnsformation (DICTRA) software package. Comprehensive comparisons between the calculated and the measured diffusivities show that most of the experimental data can be well reproduced by the presently obtained atomic mobilities. In addition, the reliability of the atomic mobilities obtained in the present work were further validated by comparing the model-predicted concentration profiles of fcc Ni−Os diffusion couples with the experimental data.

Extracting interdiffusion coefficients from binary diffusion couples using traditional methods and a forward-simulation method

A MatLab program is developed to allow effective extraction of both interdiffusion coefficients and impurity diffusivities from concentration profiles of binary diffusion couples. In addition to implementing the Boltzmann–Matano method, the Sauer–Freise method, the Hall method, and the Wagner method, a forward-simulation method was developed and implemented in the program. Eight binary systems (Fe–Ni, Co–Ni, Nb–W, Ni–W, Ni–Mo, Co–Mo, Fe–Nb and Ni–Nb) were used to test the program and to evaluate the advantages and disadvantages of the various methods. The forward-simulation method shows advantages including the ability to extract the impurity diffusion coefficients. Valuable impurity diffusivities and interdiffusion coefficients data at 1100 °C for the eight binary systems are obtained and they are in good agreements with available literature data. The successful diffusivity extraction from several complicated composition profiles with several intermetallic compounds demonstrates the reliability and robustness of the program.

Interdiffusion and Atomic Mobility for Face-Centered-Cubic Co-Al Alloys

Interdiffusion in the face-centered-cubic (fcc) Co-Al binary alloys was studied by the diffusion-couple approach in the temperature range of 1173 K to 1573 K (900 °C to 1300 °C). Interdiffusion coefficients of the fcc Co-Al alloys were then evaluated by using the Sauer–Freise method. The effect of magnetic ordering on the Co-Al interdiffusion was observed at 1273 K (1000 °C) by examining the Arrhenius plots. The interdiffusion data were assessed to develop the atomic mobility for the fcc Co-Al alloys, and their validity was tested by simulating the diffusion-couple experiments.