Composition-dependent Interdiffusion Coefficients Research Articles

Method for extracting the intrinsic diffusion coefficient from grain boundary diffusion depth profile

Numerical research is done on grain boundary diffusion of Cu-Ni bilayer thin films. Together, the composition-dependent interdiffusion coefficient and the position of the grain boundary are considered. Fick’s second law provides a quantitative assessment of the effects of grain size, volume, grain boundary diffusion coefficient, diffusion temperature, and time on the depth profile. Regarding the situation where the plateau values of the Ni concentration in the Cu layer and the Cu concentration in the Ni layer are almost the same for the case where the Ni intrinsic diffusion coefficient is significantly bigger than that for Cu, a detailed explanation is provided. We may produce a profile that is like the direct volume diffusion profile with a relative inaccuracy of 7.53% by using a method that is provided to eliminate the impacts of grain boundaries and grain size on the grain boundary diffusion profile. For most of the circumstances covered in this study, the technique is effective. The transformation of the grain boundary diffusion profile yields a composition-dependent interdiffusion coefficient, and the calculation’s outcome is consistent with the theoretical conclusion drawn from the Darken equation. Thin films of Cu (150 nm)/Ni (150 nm)/SiOx/Si (100 nm) were produced as examples, and Auger electron spectroscopy was used to determine the respective depth profiles. The calculated diffusion parameters, namely, the pre-exponential factor and the activation energy, are well within the bounds of the available data.

Influence of solute-source composition on composition-dependent interdiffusion coefficient

The validity of a long-held implicit assumption that the composition-dependent interdiffusion coefficient is a material parameter that is isothermally constant, irrespective of solute-source composition, is studied theoretically and experimentally. Solute concentration distributions under the influence of diffusion-induced stress generation and relaxation are numerically simulated, and the results are used to compute and compare composition-dependent interdiffusion coefficients under different solute-source compositions. The results of both the theoretical and experimental analyses show that solute-source concentration can significantly change the composition dependency of interdiffusion coefficient, in contrast to what is often implicitly assumed. Accordingly, the use of concentration-dependent interdiffusion functions, obtained from pure-metal/pure-metal diffusion couples, to predict or analyze diffusion effects when the solute-source is an alloy instead of a pure-metal can be grossly inaccurate.

Experimental investigations on the quinary interdiffusivities in diffusion couples of NiAlCoCr alloy/CoCrFeNi high-entropy alloys

The diffusion couples composed of Ni–Al–Co–Cr alloys and low-entropy (pure Ni, and NiCo, NiCr, and NiCoCr)/medium-entropy (CoCrNi)/high-entropy (CoCrFeNi) alloys were fabricated, and subjected to annealing at 1473 K for 72 h. Their composition profiles were measured by using the electron microprobe analysis technique. By combining the measured concentration profiles and the free-accessible High-throughput Determination of Interdiffusion Coefficients (HitDIC) software, the composition-dependent interdiffusion coefficients in the diffusion couples along the entire diffusion paths were determined in a high-throughput manner. The general relation, D˜CrCrNi>D˜FeFeNi>D˜CoCoNi, exists in most diffusion couples. A comprehensive comparison of the presently interdiffusion coefficients of elements in presently low-, medium- and high-entropy alloys and the available data in the literatures were conducted.

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.

Experimental investigation of diffusion behaviors in γ and γ’ Ni–Al–Co alloys

On the basis of 13 groups of single-phase diffusion couples together with electron probe microanalysis technique, the composition profiles in γ and γ′ Ni–Al–Co systems at 1373–1523 K were measured. Atomic mobilities in γ Ni–Al–Co alloys were directly assessed from 9 groups of concentration profiles by means of HitDIC software. The established atomic mobilities contain only two interaction parameters, and their reliability was verified by comprehensively comparing the model-predicted concentration profiles with the experimental data in the literature and the present work. After that, the composition-dependent interdiffusion coefficients along the entire diffusion path in γ and γ′ Ni–Al–Co alloys were also evaluated at 1373–1523 K by using HitDIC. D˜AlAlNi and D˜CoCoNi in γ Ni-Al-Co alloys were validated by the results obtained by Matano-Kirkaldy method and the corresponding experimental values in Ni-Al and Ni-Co binary alloys. Comprehensive comparisons between the interdiffusion coefficients in the present work and those reported the literature validated the reliability of the presently evaluated interdiffusivities in γ’ Ni-Al-Cr alloys. The influences of Al and X contents on the D˜XXNi in γ Ni-Al-2 at% X (X= Co, Rh, Ta, W, Re, Os, and Ir) alloys and D˜AlAlNi in γ Ni-2 at% Al-X alloys were further discussed.

InfPolyn, a Nonparametric Bayesian Characterization for Composition-Dependent Interdiffusion Coefficients.

Composition-dependent interdiffusion coefficients are key parameters in many physical processes. However, finding such coefficients for a system with few components is challenging due to the underdetermination of the governing diffusion equations, the lack of data in practice, and the unknown parametric form of the interdiffusion coefficients. In this work, we propose InfPolyn, Infinite Polynomial, a novel statistical framework to characterize the component-dependent interdiffusion coefficients. Our model is a generalization of the commonly used polynomial fitting method with extended model capacity and flexibility and it is combined with the numerical inversion-based Boltzmann–Matano method for the interdiffusion coefficient estimations. We assess InfPolyn on ternary and quaternary systems with predefined polynomial, exponential, and sinusoidal interdiffusion coefficients. The experiments show that InfPolyn outperforms the competitors, the SOTA numerical inversion-based Boltzmann–Matano methods, with a large margin in terms of relative error (10× more accurate). Its performance is also consistent and stable, whereas the number of samples required remains small.

An Effective Approach to Acquire the Impurity Diffusion Coefficients in Binary Alloys with Quantified Uncertainties

In this paper, we started from the composition-dependent interdiffusion coefficients with quantified uncertainties in binary alloys by integrating the Matano-based method, distribution functions, and uncertainty propagation approach. After carefully defining the numerically stable region for the interdiffusion coefficients, the suitable pre-set functions were screened to achieve the reasonable fit to the D-c and μ-c data according to the Akaike information criterion. With the fitted D-c and μ-c curves, the impurity diffusion coefficients with uncertainties can be directly determined. Benchmark tests in five hypothetical binary systems with different preset D-c relations were then utilized to validate the presently effective approach, followed by practical applications in five real cases, i.e., fcc Ni-Co, fcc Cu-Al, fcc Pt-Ni, hcp Mg-Zn, and bcc Ti-V alloys. The impurity diffusion coefficients with uncertainties derived by the presently effective approach were found to be in excellent agreement with the data by tracer experiments, indicating that this effective approach can serve as a standard one for acquiring the high-quality impurity diffusion coefficients in binary alloys with quantified uncertainties, especially for the noble metals and the cases without suitable radioactive tracer isotopes.

Diffusion behavior of fcc and L12 Ni–Al–Cr alloys

In this paper, 12 groups of single-phase (γ and γ′) Ni–Al–Cr diffusion couples were carefully fabricated, and their concentration profiles at 1423, 1473, and 1523 K were measured by means of electron probe microanalysis technique. Atomic mobilities of γ phase in ternary Ni–Al–Cr system were then established directly from the experimental concentration profiles by using the HitDIC (High-throughput Determination of Interdiffusion Coefficients) software. The reliability of the atomic mobilities was confirmed by the coincidence between the calculated and experimental properties including the composition profiles and diffusion paths. Moreover, based on the atomic mobilities together with the available thermodynamic descriptions, the three-dimensional interdiffusivities planes at 1423–1523 K were constructed, from which the activation energies in γ Ni–Al–Cr alloys were extracted. Lower diffusion activation energy corresponds to higher interdiffusion coefficient. After that, the composition-dependent interdiffusion coefficients in γ′ Ni–Al–Cr alloys were determined and the magnitude relation of D˜AlAlNi and D˜CrCrNi in γ’ Ni–Al–Cr alloys was verified.

An efficient method of calculating composition-dependent inter-diffusion coefficients based on compressed sensing method

Composition-dependent inter-diffusion coefficients are key parameters in many physical processes. Due to the under-determinedness of the governing diffusion equations, numerical methods either impose strict physical conditions on the samples or require a computationally onerous amount of data. To address such problems, we propose a novel inverse framework to recover the diffusion coefficients using a compressed sensing method, which in principle can be extended to alloy systems with arbitrary number of species. Comparing to conventional methods, the new approach does not impose any priori assumptions on the functional relationship between diffusion coefficients and concentrations, nor any preference on the locations of the samples, as long as it is in the diffused zone. It also requires much less data compared to least-squares approaches. Through a few numerical examples of ternary and quandary systems, we demonstrate the accuracy and robustness of the new method.

Diffusion growth of ϕ ternary intermetallic compound in the Mg-Al-Zn alloy system: In-situ observation and modeling

Study on the diffusion growth of ternary intermetallic compounds in Mg-Al-Zn based light-weight alloys is important due to its close interrelation with alloy property. However, there is a very lack of existing data due to difficulties in both experimental and computational aspects. The current work aims at presenting the experimental observation on the diffusion growth behavior of ϕ phase at 360 °C as well as calculating its composition-dependent interdiffusion coefficients. We designed and successfully fabricated four Mg-τ ternary diffusion couples annealed at 360 °C for different times, where the diffusion path goes across the ϕ phase region and the diffusion growth of ternary intermetallic compound can be solely detected. In-situ observation of the time-dependent growth of ϕ phase was performed to accurately determine the parabolic growth constant. The experimental data were then subjected to a numerical inverse method to generate a set of self-consistent interdiffusivities of the ternary intermetallic compounds, which can reproduce the presently observed diffusion growth behavior of ϕ ternary intermetallic compound in Mg-τ diffusion couples.

A novel approach to calculate diffusion matrix in ternary systems: Application to Ag–Mg–Mn and Cu–Ni–Sn systems

Diffusion behavior is of fundamental importance in material science and engineering. As a result, extensive attention has been paid to develop methodologies for obtaining the composition-dependent interdiffusivities. The present work proposed a novel approach with two steps to extract the interdiffusion coefficients from a single ternary diffusion couple. The concept of basis function is introduced, and this novel method is based on the utilization of both finite element method and optimization algorithm. Utilizing eight groups of diffusion couples together with electron probe microanalysis technique, the composition-dependent interdiffusion coefficients in fcc Ag–Mg–Mn alloys at 973 and 1073 K were determined via both the present method and Matano-Kirkaldy method. A home-made code is written to implement the novel numerical approach. The obtained interdiffusion coefficients based on the present method agree with the reliable ones from the Matano-Kirkaldy method better than previous methods for fcc Ag–Mg–Mn alloys in the present work as well as fcc Cu–Ni–Sn alloys from the literature, especially for the main interdiffusivities. In most cases, the presently computed main interdiffusivities by means of the composition-dependent diffusivities agree with the reliable ones from the Matano-Kirkaldy method within 50%. Besides, the sign of cross interdiffusivities using this novel method is usually the same as that from the Matano-Kirkaldy method. Such a high calculation precision and efficiency cannot be obtained using previous methods.

Experimental Investigations on the Quaternary Interdiffusion Coefficients, Young’s Modulus and Hardness in bcc Ti-Nb-Ta-Zr Quaternary Alloys

Accurate interdiffusion coefficient, Young’s modulus and hardness are essential for precisely controlling the production of a homogeneous bio-metallic alloys with excellent properties. In this work, we utilized a combinational approach by combining the advanced diffusion multiple technique and the pragmatic numerical inverse method to efficiently determine the composition–dependent interdiffusion coefficient matrices in bcc quaternary Ti-Nb-Ta-Zr alloys at 1373 K. Moreover, the composition-dependent Young’s modulus and hardness of bcc Ti-Nb-Ta-Zr alloys were obtained by means of the nanoindentation technique.

Interdiffusion and Atomic Mobilities of fcc Co-V-Mo Alloys: Measurement and Modeling

Based on 18 bulk diffusion couples, the composition-dependent interdiffusion coefficients in the fcc Co-V-Mo alloys at 1273, 1373 and 1473 K were obtained from the intersection points of the diffusion couples by means of EPMA technique coupled with the Whittle and Green method. With the experimentally determined interdiffusion coefficients and the critically reviewed experimental diffusivities available in the literature, the atomic mobilities of fcc Co-V-Mo alloys were assessed by means of Diffusion Controlled Transformation (DICTRA) software. The quality of the assessed atomic mobilities was confirmed by the comprehensive comparisons between various DICTRA-calculated diffusion behaviors and the experimental ones, including concentration profiles and diffusion paths.

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.

Interdiffusion Coefficients in FCC Co-Rich Co-Ti-V Alloys at 1273, 1373 and 1473 K

Based on 18 bulk diffusion couples, the composition-dependent interdiffusion coefficients in the FCC Co-rich Co-Ti-V alloys at 1273, 1373 and 1473 K were obtained from the intersection points of the diffusion couples by means of EPMA technique applied to Whittle and Green method. The reliability of the experimental interdiffusivities is validated via thermodynamic constraints. Taking Co as the solvent element, the present results show that when the temperature is from 1273 to 1473 K, the ternary interdiffusion coefficients increases from 10−16 to 10−14 m2/s and the diffusion of Ti is generally faster than V. The ternary main interdiffusion coefficients of $$\tilde{D}_{\text{TiTi}}^{\text{Co}}$$ and $$\tilde{D}_{\text{VV}}^{\text{Co}}$$ at different compositions of Ti and V at 1473 K were compared with the values obtained for boundary binary Co-Ti and Co-V systems in the literature. A composition-dependent decreasing-increasing tendency was found for $$\tilde{D}_{\text{VV}}^{\text{Co}}$$ .

On Sluggish Diffusion in Fcc Al–Co–Cr–Fe–Ni High-Entropy Alloys: An Experimental and Numerical Study

High-throughput measurement using the numerical inverse method combined with the diffusion multiple technique was employed to determine the composition-dependent interdiffusion coefficients in fcc Al–Co–Cr–Fe–Ni high-entropy alloys at 1273, 1323, and 1373 K. The reliability of the obtained interdiffusion coefficients was confirmed by comparing the model-predicted composition/interdiffusion flux profiles with the experimental data. The tracer diffusivities of the components were then predicted based on the obtained interdiffusion coefficients and the simplified thermodynamic description. The diagonal interdiffusion coefficients in the present high-entropy alloys were comprehensively compared with those in conventional fcc alloys. Therefore, the sluggish diffusion effect does not apply for all the elements in the present fcc Al–Co–Cr–Fe–Ni high-entropy alloys. A similar result was also observed for the evaluated tracer diffusivities.

Application of numerical inverse method in calculation of composition-dependent interdiffusion coefficients in finite diffusion couples

The previously developed numerical inverse method was applied to determine the composition-dependent interdiffusion coefficients in single-phase finite diffusion couples. The numerical inverse method was first validated in a fictitious binary finite diffusion couple by pre-assuming four standard sets of interdiffusion coefficients. After that, the numerical inverse method was then adopted in a ternary Al-Cu-Ni finite diffusion couple. Based on the measured composition profiles, the ternary interdiffusion coefficients along the entire diffusion path of the target ternary diffusion couple were obtained by using the numerical inverse approach. The comprehensive comparisons between the computations and the experiments indicate that the numerical inverse method is also applicable to high-throughput determination of the composition-dependent interdiffusion coefficients in finite diffusion couples.

Composition-dependent interdiffusion coefficient, reduced elastic modulus and hardness in γ-, γ′- and β-phases in the Ni-Al system

A combinatorial approach using diffusion couples and advanced characterization was carried out to investigate the composition-dependent interdiffusion and mechanical properties of γ-Ni(Al) solid solution, γ′-Ni3Al and β-NiAl. The diffusion couples between pure Ni and Ni52Al48 were annealed at temperatures of 1000 °C, 1100 °C and 1200 °C for 96 h, 48 h and 24 h, respectively. The γ-Ni(Al), γ′-Ni3Al and β-NiAl have developed in the diffusion zone. The β-NiAl transformed to martensite above room temperature when Al is less than 37 at.%. The composition-dependent interdiffusion coefficients for all phases corresponded closely to previous research. A systematic composition-dependent elastic modulus and hardness for γ-Ni(Al), γ′-Ni3Al and β-NiAl phases were obtained by nanoindentation. Variation of elastic modulus and hardness in the γ-Ni(Al) solid solution is influenced by solid solution strengthening and hardening due to precipitation of γ′-Ni3Al while cooling. The evolution of elastic modulus and hardness in the β-NiAl phase exhibited a minimum value of elastic modulus and hardness near the phase boundary between martensite and austenite. Lattice softening associated with martensitic transformation was recognized to lower the value of elastic modulus and hardness.

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.

Simultaneous tracer diffusion and interdiffusion in a sandwich-type configuration to provide the composition dependence of the tracer diffusion coefficients

In this paper, a new formalism of a combined tracer and interdiffusion experiment for a binary interdiffusion couple is developed. The analysis requires an interdiffusion couple that initially contains a thin layer of tracers of one or both of the constituent elements at the original interface of the couple (sandwich interdiffusion experiment). This type of interdiffusion experiment was first performed in 1958 by J.R. Manning. The theoretical analysis presented in this paper is based on a newly developed phenomenological theory of isotopic interdiffusion combined with the Boltzmann–Matano formalism. This new analysis now provides the means to obtain the composition dependent interdiffusion coefficient and tracer diffusion coefficients simultaneously from analysis of the interdiffusion and tracer profiles in a single sandwich interdiffusion experiment. The new analysis is successfully applied to the results of Manning’s original ‘sandwich interdiffusion’ experiment in the Ag–Cd system (six couples in total) and is validated with an independent determination of the Ag and Cd tracer diffusion coefficients by Schoen using the conventional thin film technique. Suggestions for further development of the sandwich interdiffusion experiment and analysis to the case of multicomponent alloys are provided.