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Diffusivities and atomic mobilities in the Ni-rich fcc Ni–Al–Cu alloys: experiment and modeling

  • Liang Zhong , Yuling Liu ORCID logo EMAIL logo , Huixin Liu , Shiyi Wen , Fei Wang , Changfu Du , Qianhui Min , Milena Premovic , Zhoushun Zheng , Jieqiong Hu and Yong Du ORCID logo EMAIL logo
Published/Copyright: April 28, 2022
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International Journal of Materials Research
From the journal International Journal of Materials Research Volume 113 Issue 5

Abstract

The diffusivities along the whole diffusion path in one diffusion couple of Ni-rich fcc Ni–Al–Cu alloys at 1273, 1333 and 1433 K are obtained by means of the numerical inverse method. In total 12 diffusion couples are used to obtain the diffusivities at the intersections in the diffusion paths by the Matano–Kirkaldy method. Two sets of atomic mobilities are assessed using the CALTPP program based on the diffusivities calculated by the numerical inverse method and Matano–Kirkaldy method, respectively. The validity of the obtained atomic mobilities is confirmed by comparing the model-predicted diffusivities, concentration profiles and diffusion paths with the measured ones. The whole calculation by the numerical inverse method in CALTPP with higher efficiency can achieve the same accuracy as the Matano–Kirkaldy method. The present study demonstrates that one diffusion couple combined with the numerical inverse method can reliably assess the atomic mobilities for fcc Ni–Al–Cu alloys.

Keywords: Atomic mobility; CALTPP program; Diffusivity; Experimental measurements; Ni–Al–Cu alloy

1 Introduction

As typical SMAs (shape memory alloys), Ni–Al–Cu alloys have an interesting characteristic, viz., “remembering” the shape they kept before pseudoplastic deformation. Like other SMAs, Ni–Al–Cu alloys present outstanding machinability [1], excellent stability of mechanical properties at elevated temperatures [2], and high thermal stability [3]. These properties at elevated temperatures applied in sensors, high damping material and actuators [4], [5], [6] can be remarkably improved by heat treatment, during which diffusion plays a key role. Therefore, a systematical study of the composition- and temperature-dependent interdiffusivities as well as the atomic mobilities for the Ni–Al–Cu alloys is of fundamental importance for these applications.

The interdiffusivities of the Cu-rich Cu–Ni–Al alloys have been investigated according to just two semi-infinite and one finite diffusion couples at 1273 K by Liu et al. [7, 8]. More experimental investigations on the temperature- and composition-dependent interdiffusivities and reliable diffusion kinetic parameters are required for the fcc Ni–Al–Cu alloys. Therefore, the Ni-rich fcc Ni–Al–Cu alloys are chosen as the target to study the interdiffusivities in the present work. It should be noted that after the present authors established the atomic mobilities of the Ni-rich fcc Ni–Al–Cu alloys, most recently the atomic mobilities of the Cu-rich Cu–Ni–Al alloys were reported by Zhang et al. [9]. They have performed eight diffusion couples of the Cu-rich Cu–Ni–Al alloys and established the kinetic parameters. However, the diffusion behavior in the Cu-rich alloys is different from that in the Ni-rich alloys, which will be demonstrated in Section 4. Therefore, it is necessary to study the diffusivities and atomic mobilities in the Ni-rich fcc Ni–Al–Cu alloys by experiment and modeling.

As is well-known, the traditional Matano–Kirkaldy method is the frequent way to calculate the diffusivities in a ternary system [10], [11], [12]. Although the Matano–Kirkaldy method is of great reliability and accuracy, its efficiency is low since two diffusion couples with an intersection in the diffusion paths are indispensable to calculate the diffusivities. To improve this situation, a novel numerical inverse method was proposed by our group [13] which can obtain the composition-dependent diffusion matrix along the whole diffusion path of a single diffusion couple with a high efficiency. The novel numerical inverse method and Matano–Kirkaldy method were implemented in the CALTPP (CALculation of ThermoPhysical Properties) [14]. CALTPP has been successfully utilized to calculate the interdiffusivities and optimize the atomic mobilities of various alloys [13, 15], [16], [17], [18], [19].

In the present work, just one diffusion couple at three temperatures is applied to obtain the interdiffusivities by the numerical inverse method in CALTPP. To confirm the results from the numerical inverse method, three new diffusion couples at each temperature are designed to create intersections. Then the interdiffusivities of five intersections in diffusion paths of all diffusion couples are calculated using the Matano–Kirkaldy method. Based on the obtained interdiffusivities, two sets of atomic mobilities are evaluated, separately.

The major purposes of this work are (i) to measure the interdiffusivities of Ni-rich fcc Ni–Al–Cu alloys along the whole diffusion path of one diffusion couple using the numerical inverse method and at intersections in diffusion paths of four diffusion couples using the Matano–Kirkaldy method, (ii) to evaluate two sets of atomic mobilities in CALTPP based on the interdiffusivities obtained from the numerical inverse method and Matano–Kirkaldy method, respectively, and (iii) to demonstrate the reliability of two sets of atomic mobilities by comprehensively comparing the calculated diffusivities, model-predicted concentration profiles, and diffusion paths with the experimental ones and validate the accuracy of the numerical inverse method.

2 Experimental procedure

Based on the Ni–Al–Cu isothermal sections at 1273, 1333 and 1433 K from Wang et al. [20], samples with the desired compositions listed in Table 1 were prepared by arc melting of appropriate mixtures of pure Ni (99.99 wt.%), Cu (99.99 wt.%) and Al (99.99 wt.%). The ingots were re-melted four times to ensure the homogeneity. Then the buttons were linearly cut into blocks of approximate dimensions of 5 × 5 × 3 mm3 by wire-electrode cutting. After mechanically removing the surface material, the buttons were sealed into evacuated quartz tubes, and homogenized at 1433 ± 5 K for 10 days to obtain large grains and minimize the effect of grain-boundary diffusion. Subsequently, all the samples were ground to move surface contamination and polished to get a smooth surface, and then bound together with Mo clamps to form diffusion couples, as listed in Table 1. After that, each diffusion couple was sealed into evacuated quartz tubes and annealed at 1273 K for 96 h, 1333 K for 36 h and 1433 K for 6 h, respectively, followed by quenching in cold water. After applying standard metallographic techniques, the concentration profiles were obtained by means of electron probe microanalysis with wavelength dispersive X-ray spectroscopy (EPMA-WDX, JXA-8530F, JEOL, Japan; accelerating voltage 20 kV, probe current 2.0 × 10−8 A) on a polished section, parallel to the diffusion direction. The software for EPMA measurements was Phi-Rho-Z Quantitative Analysis Program developed by JEOL with embedded ZAF method to perform corrections. Variations in alloy compositions were measured to be within ±0.5 at.% in the present work.

Table 1:

Terminal compositions of the diffusion couples in the Ni-rich Ni–Al–Cu system.

Couple Composition (at.%)
C1 Ni/Ni-10.2Al-7.0Cu
C2 Ni/Ni-6.0Al-19.8Cu
C3 Ni-5.2Al/Ni-21.1Cu
C4 Ni-9.4Al/Ni-10.1Cu

3 Evaluation of ternary indiffusivities and atomic mobilities

3.1 Matano–Kirkaldy method for obtaining ternary diffusivities

The Boltzmann–Matano method has been applied to calculate diffusion coefficients in ternary and even high-order systems [10, 11]. Alternatively, with assumption of constant molar volume over the diffusion zone, interdiffusion fluxes of component i can be represented in the view of Kirkaldy [11] and Fick’s first law as follows:

(1) J ̃ i = 1 2 t ∫ c i − o r c i + c i x − x M d c i , i = A l , C u

where c i − and c i + denote terminal concentrations at the left and right side of diffusion couples, respectively. c i the concentration of element i at the certain position, x the distance and t the diffusion time. x M represents the Matano plane for the diffusion couple.

Theoretically, the positions of Matano planes of solutes Cu and Al should be the same. However, Matano planes of different solutes deviate from each other to a certain extent generally. Therefore, Whittle and Green [21] introduced the normalized concentration parameter Y i = c i − c i − / c i + − c i − to exclude the parameter x M from calculation. After that, the following formula could be derived from Equation (1):

(2) D ̃ C u C u N i + D ̃ C u A l N i d c A l d c C u = − 1 2 t d x d Y C u 1 − Y C u ∫ − ∞ x Y C u d x   + Y C u ∫ x + ∞ 1 − Y C u d x D ̃ A l A l N i + D ̃ C u A l N i d c A l d c C u = − 1 2 t d x d Y A l 1 − Y A l ∫ − ∞ x Y A l d x   + Y A l ∫ x + ∞ 1 − Y A l d x

where D ̃ i C u N i and D ̃ i A l N i i , j = A l , C u are the interdiffusion coefficients. The coefficients D ̃ C u C u N i and D ̃ A l A l N i are the main coefficients, which represent the influences of the concentration gradients of elements Cu and Al on their own fluxes, respectively, while the cross interdiffusion coefficients D ̃ A l C u N i and D ̃ C u A l N i reflect the influences of the concentration gradients of elements Cu and Al on the fluxes of elements Al and Cu, respectively.

The interdiffusion flux of component i (i = Al, Cu) in the Ni–Al–Cu ternary alloy can be expressed as follows [22]:

(3) J ̃ A l z * = c A l − − c A l + 2 t = 1 − Y A l * ∫ − ∞ z * Y A l d z + Y A l * ∫ z * + ∞ 1 − Y A l d z

(4) J ̃ C u z * = c C u − − c C u + 2 t = 1 − Y C u * ∫ − ∞ z * Y C u d z + Y C u * ∫ z * + ∞ 1 − Y C u d z

where t is the diffusion annealing time, the normalized composition variable Y i * of element i (i = Al, Cu) at section z* is the value of c k z * − c k − c k + − c k − .

As regards the standard deviation of interdiffusivities, Lechelle et al. [23] introduced a scientific method to calculate it, which is expressed as:

(5) u f A , B , … = ∑ α = A , B , … ∂ f ∂ α 2 u α 2

where A and B are the correlation quantities of function f, while u α α = A , B , … is the uncertainty in the measurements of variable α including concentration, interdiffusion flux and the gradient of the composition.

The thermodynamic stability of the diffusivities is examined by the following constraints [24]:

(6) D ̃ C u C u N i + D ̃ A l A l N i > 0 D ̃ C u C u N i D ̃ A l A l N i − D ̃ C u A l N i D ̃ A l C u N i ≥ 0 D ̃ C u C u N i − D ̃ A l A l N i 2 + 4 D ̃ C u A l N i D ̃ A l C u N i ≥ 0

3.2 Numerical inverse method for assessing ternary diffusivities

As mentioned before, the Boltzmann–Matano method can only obtain the interdiffusivity at intersection points. Consequently, the numerical inverse method was developed [13, 14, 25] to calculate interdiffusivity more efficiently.

In the Ni–Al–Cu ternary system, the interdiffusivity can be described by Equation (7) on the basis of the extended Fick’s law [26].

(7) ∂ c i ∂ t = ∂ ∂ x D ̃ i C u N i ∂ c C u ∂ x + ∂ ∂ x D ̃ i C u N i ∂ c A l ∂ x , i = A l , C u

The interdiffusivity D ̃ i j N i (i, j = Al, Cu) can be obtained by the following formula [13]:

(8) D ̃ i j N i = ∑ k = 1 3 δ k i − c i exp Φ k R T ϕ k j N i , i , j = A 1 , C u

where Φ k and ϕ k j N i denote the atomic mobility parameter and thermodynamic factor, respectively. R is the gas constant, T temperature and δ ki the Kronecker delta (δ ki = 1 if k = i, otherwise δ ki = 0). The numerical procedure is carried out by the integration of the finite element method (FEM) [27] and simple genetic algorithm (SGA) [28]. FEM and SGA are utilized to solve forward problems and optimize parameters, respectively.

3.3 Modeling of atomic mobility and diffusivity

Based on the absolute-reaction rate theory [29], the atomic mobility of element k expressed as M k could be described with frequency factor M k 0 and activation enthalpy Q k . Andersson and Ågren [30] reported that atomic mobility can be shown as Equation (9).

(9) M k = exp R T l n M k 0 R T exp − Q k R T 1 R T

Generally, Q k and R T l n M k 0 can be merged into one composition, temperature and pressure-dependent parameter, i.e., Φ k . In consideration of the CALPHAD (CALculation of PHAse Diagram) method, the composition dependency can be represented by the Redlich–Kister–Muggianu polynomial [31, 32] as follows:

(10) Φ k = ∑ i x i Φ k i + ∑ i ∑ j > i x i x j ∑ r Φ k i , j r x i − x j r + ∑ i ∑ j > i ∑ l > j x i x j x l ∑ i v i j l s r Φ k i , j , l , s = i , j , l

where Φ k i is the value of Φ k for pure i. Φ k i , j r and Φ k i , j , l r are interaction parameters for the mobility of element k, which could be represented by a polynomial of temperature and pressure if necessary. The parameter v i j l s is expressed as follows:

(11) v i j l s = x s + 1 − x i − x j − x l 3 , s = i , j , l

where x s , x i , x j and x l are mole fractions of elements s, i, j and l, respectively.

The relationship between atomic mobility M i and interdiffusion coefficients D ̃ k j n is described with the following equation [33]:

(12) D ̃ k j n = ∑ i = 1 n δ i k − x k x i M i ∂ μ i ∂ x j − ∂ μ i ∂ x n

where x i and μ i denote the mole fraction and chemical potential of element i, respectively,

4 Results and discussion

4.1 Interdiffusion coefficients at 1273, 1333 and 1433 K

Based on the isothermal sections from Reference [20], the terminal compositions of all the diffusion couples were designed to locate in a single fcc Ni-rich Ni–Al–Cu phase region. The concentration profiles perpendicular to the interface direction for all diffusion couples were acquired by EPMA-WDX. Only the experimental concentration profiles of C1 diffusion couple at three temperatures were taken directly to calculate the interdiffusivities along the diffusion path using the numerical inverse method in CALTPP.

For calculating the interdiffusivities by the Matano–Kirkaldy method, the Boltzmann or additive Boltzmann function is utilized to fit all the measured concentration profiles of C1, C2, C3 and C4 diffusion couples which is expressed as:

(13) c i = p 1 + p 2 1 + exp x − p 3 / p 4 + p 5 1 + exp x − p 6 / p 7

where p 1–p 7 are the parameters to be fitted. While for relatively symmetrical curves, only p 1–p 4 are to be fitted. Note that the composition profiles for the solvent Ni are obtained by 1–c(Al)–c(Cu). The fitting results are listed in Table 2. Here, the estimation for the error of the main diffusion coefficients D ̃ A 1 A 1 N i at the intersection of diffusion couples C1 and C3 is given in detail as an example. According to the Matano–Kirkaldy method, D ̃ A 1 A 1 N i can be calculated using the following function:

(14) D ̃ A l A l N i = J ̃ A l C 3 ∂ c C u ∂ z C 1 − J ̃ A l C 1 ∂ c C u ∂ z C 3 ∂ c A l ∂ z C 1 ∂ c C u ∂ z C 3 − J ̃ A l C 1 ∂ c A l ∂ z C 3 ∂ c C u ∂ z C 1

where J ̃ A l C 3 and J ̃ A l C 1 are the fluxes of Al in C3 diffusion couple and C1 diffusion couple at the intersection of diffusion couples A1 and A3, respectively; ∂ c A 1 ∂ z C 1 and ∂ c C u ∂ z C 1 are the concentration gradients of Al and Cu in C1 diffusion couple at the intersection of diffusion couples C1 and C3, respectively; ∂ c A 1 ∂ z C 3 and ∂ c C u ∂ z C 3 are the concentration gradient of Al and Cu element of C3 diffusion couple at the intersection of diffusion couples C1 and C3, respectively. Therefore, the error of the main diffusion coefficient, u D ̃ A 1 A 1 N i contains the contributions of individual variables and could be expressed as follows:

(15) u D ̃ A l A l N i 2 = ∂ D ̃ ∂ J ̃ A l C 3 2 u J ̃ A l C 3 2 + ∂ D ̃ ∂ J ̃ A l C 1 2 u J ̃ A l C 1 2 + ∂ D ̃ ∂ c A l ∂ z C 1 2 u ∂ c A l ∂ z C 1 2 + ∂ D ̃ ∂ c C u ∂ z C 1 2 u ∂ c C u ∂ z C 1 2 + ∂ D ̃ ∂ c A l ∂ z C 3 2 u ∂ c A l ∂ z C 3 2 + ∂ D ̃ ∂ c C u ∂ z C 3 2 u ∂ c C u ∂ z C 3 2

where u J ̃ is the uncertainty of flux at the intersection point. According to Equations (3) (5), J ̃ z * and u J ̃ 2 can be calculated as follows:

(16) J ̃ z * = c − − c + 2 t 1 − Y k * ∫ − ∞ z * Y k d z + Y k * ∫ z * + ∞ 1 − Y k d z

(17) u J ̃ 2 = ∂ J ̃ ∂ c 2 u c 2 + ∂ J ̃ ∂ t 2 u t 2 + ∂ J ̃ ∂ z 2 u z 2

Table 2:

The fitting results for the experimentally measured composition profiles by the Boltzmann function.

Couple p 1 p 2 p 3 p 4 p 5 p 6 p 7
A1-Al 0 4.265 329.1 27.96 5.986 382.7 16.09
A1-Cu 0.0389 6.961 360.8 16.69 / / /
A2-Al 0.0140 2.259 254.8 25.94 3.569 316.4 11.53
A2-Cu 0 19.80 294.1 13.39 / / /
A3-Al 5.095 −3.187 371.2 18.26 −1.905 323.3 27.00
A3-Cu 0 21.25 353.8 16.60 / / /
A4-Al 9.370 −9.370 354.1 18.22 / / /
A4-Cu 0.0323 10.22 352.8 12.24 / / /
B1-Al 10.40 −5.526 388.5 26.82 −4.864 337.9 13.38
B1-Cu 6.960 −6.921 362.1 16.52 / / /
B2-Al 0 4.209 363.1 14.36 1.930 278.8 27.20
B2-Cu 0.0037 19.81 337.6 15.948 / / /
B3-Al 5.121 5.080 313.2 28.96 / / /
B3-Cu 0 21.13 307.5 19.36 / / /
B4-Al 0 9.426 350.9 29.51 / / /
B4-Cu 10.16 −10.09 350.2 18.26 / / /
C1-Al 10.16 −4.848 363.0 29.01 −5.306 309.4 15.30
C1-Cu 7.011 −6.976 332.9 17.01 / / /
C2-Al 6.204 −2.504 389.6 29.97 −3.685 318.2 13.55
C2-Cu 19.38 −19.38 346.4 16.38 / / /
C3-Al 0 2.118 375.4 22.75 3.216 330.2 16.20
C3-Cu 20.12 −20.12 350.1 17.03 / / /
C4-Al 0 7.080 360.1 21.32 2.475 313.1 22.33
C4-Cu 10.02 −9.980 352.2 17.19 / / /

Since the diffusion annealing time t and diffusion distance z can be measured very accurately, the errors for two terms are assumed to be zero (i.e., u t = 0 , u z = 0 ). Consequently, u J ̃ is estimated as follows:

(18) u J ̃ 2 = ∂ J ̃ ∂ c 2 u c 2

where u c is the uncertainty of concentration, which is associated with the error in EPMA measurement and fitting function. In this work, the double iterations of Boltzmann function Equation (13) were used to smooth the experimental data, so the error calculation for fitting concentration is given by the following equation:

(19) c = c z , p 1 , p 2 , … , p n u c 2 = λ ⋅ ∑ k = 1 n ∂ c ∂ p k 2 ⋅ u p k 2

where the parameter λ is estimated as 1.02 in view of composition measurement error (about 0.02) from EPMA. According to present calculations, the uncertainty of calculating the ternary interdiffusion coefficients are estimated as: 60% uncertainty from interdiffusion flux and 40% uncertainty from concentration.

Then, the fitted results are then applied to obtain D ̃ C u C u N i , D ̃ A 1 A 1 N i , D ̃ C u A 1 N i and D ̃ A 1 C u N i at five intersections in diffusion paths of all the diffusion couples at each temperature by the Matano–Kirkaldy method. The obtained diffusivities in fcc Ni–Al–Cu alloys together with their standard error calculated by Equation (5) are listed in Table 3. To make a quantitative comparison, the diffusivities of the same compositions obtained by the numerical inverse method are also listed. As shown in the table, both negative and positive values for cross interdiffusivities can be found, which indicates an attractive and a repelling interaction between Cu and Al atoms, respectively. Furthermore, all these diffusivities meet the thermodynamically stable constraints i.e., Equation (6), and thus they are thermodynamically stable. Moreover, the results calculated by the Matano–Kirkaldy method listed in Table 3 show that the value of the main interdiffusion coefficients D ̃ A 1 A 1 N i is larger than D ̃ C u C u N i at each temperature, indicating that Al diffuses faster than Cu in Ni-rich Ni–Al–Cu alloys.

Table 3:

Diffusion coefficients in fcc Ni–Al–Cu alloys obtained by the Matano–Kirkaldy method (denoted by “M-K”) and numerical inverse method (denoted by “N-I”).

Temperature (K) Diffusion couple Composition (at.%) Interdiffusion coefficient (×10−15 m2 s−1)a
Cu Al D ̃ C u C u N i SD D ̃ C u A 1 N i SD D ̃ A 1 C u N i SD D ̃ A 1 A 1 N i SD method
1273 C1/C3 1.92 4.20 1.40 ± 0.04 0.16 ± 0.37 0.035 ± 1.14 2.26 ± 0.02 M-K
1.10 0.07 0.30 2.23 N-I
C1/C4 3.32 5.63 0.83 ± 0.01 0.32 ± 0.03 0.61 ± 0.04 2.27 ± 0.01 M-K
1.03 0.10 0.42 2.63 N-I
C2/C3 6.57 3.01 0.96 ± 0.01 1.04 ± 0.08 0.022 ± 1.40 2.22 ± 0.07 M-K
1.31 0.29 0.22 2.36 N-I
C2/C4 7.40 3.16 0.71 ± 0.01 0.12 ± 0.14 0.12 ± 0.16 1.65 ± 0.02 M-K
1.32 0.32 0.22 2.45 N-I
C3/C4 8.05 2.74 1.26 ± 0.02 0.84 ± 0.07 0.23 ± 0.05 1.14 ± 0.03 M-K
1.38 0.37 0.20 2.43 N-I
1333 C1/C3 1.91 4.07 4.84 ± 0.05 −1.31 ± 0.18 −1.05 ± 0.12 6.68 ± 0.02 M-K
3.43 −0.21 −0.92 6.84 N-I
C1/C4 3.52 5.71 3.34 ± 0.01 0.52 ± 0.09 0.26 ± 0.54 9.00 ± 0.02 M-K
3.20 0.32 1.33 8.30 N-I
C2/C3 6.07 3.09 3.60 ± 0.01 −3.23 ± 0.07 0.20 ± 0.32 7.98 ± 0.04 M-K
3.94 −0.80 0.71 7.21 N-I
C2/C4 7.40 3.38 3.00 ± 0.01 1.16 ± 0.08 0.18 ± 0.80 9.24 ± 0.02 M-K
3.96 0.96 0.79 7.66 N-I
C3/C4 8.36 2.69 4.05 ± 0.01 0.029 ± 7.96 0.18 ± 0.30 10.03 ± 0.03 M-K
4.22 1.14 0.62 7.47 N-I
1433 C1/C3 2.17 4.32 22.67 ± 0.03 −2.83 ± 0.25 3.88 ± 0.12 37.68 ± 0.02 M-K
17.97 −1.22 5.54 38.06 N-I
C1/C4 3.45 5.58 19.05 ± 0.01 −0.58 ± 0.29 3.93 ± 0.16 48.64 ± 0.02 M-K
17.08 −1.62 7.02 43.44 N-I
C2/C3 6.06 3.13 18.83 ± 0.01 −3.38 ± 0.27 1.96 ± 0.26 39.27 ± 0.05 M-K
20.52 −4.09 3.93 38.08 N-I
C2/C4 6.87 3.31 18.00 ± 0.01 −2.29 ± 0.16 2.40 ± 0.23 41.59 ± 0.02 M-K
20.50 4.54 4.18 39.37 N-I
C3/C4 7.54 2.80 18.51 ± 0.02 −2.40 ± 0.28 1.31 ± 0.26 39.70 ± 0.02 M-K
21.41 5.25 3.51 38.64 N-I

  1. aSD = standard deviation, which was evaluated using the method considering the error propagation [23].

4.2 Assessment of atomic mobilities

The thermodynamic descriptions of the Ni–Cu, Cu–Al and Ni–Al binary systems were obtained from Miettinen [34], Chen et al. [35] and Ansara et al. [36], respectively, and the Ni–Al–Cu ternary system was reported by Wang et al. [20]. These thermodynamic parameters are applied in this work. The adopted atomic mobility parameters of fcc Cu–Ni, Cu–Al and Ni–Al binary systems are taken from Zhang et al. [37], Liu et al. [38] and Zhang et al. [39], respectively.

The optimization of the ternary atomic mobilities for fcc Ni–Al–Cu alloys was carried out using CALTPP [13, 14] using the presently obtained concentration profiles or interdiffusivities combined with binary kinetic parameters from References [37], [38], [39]. The atomic mobilities of version 1 were obtained during the calculation of interdiffusivities using the measured concentration profiles of C1 diffusion couple at three temperatures, as expressed in Equation (8). It should be noted that the calculation speed for CALTPP is very fast. For the three diffusion couples in the present work, it only takes 40 s for the calculation with a basic desktop computer (CPU: Intel Core i7-8700HQ). The algorithms for the optimization of atomic mobilities based on the interdiffusivities in CALTPP contain two types: deterministic algorithm and heuristic algorithm, such as the Levenberg–Marquardt method and Simulated Annealing algorithm. Principally, the Levenberg–Marquardt method is advisable to adopt first [15] and was utilized in the present work to optimize the atomic mobilities of version 2. In addition, the initial parameters to be optimized and the iteration stop criteria are automatically selected in CALTPP. Using the above algorithms, CALTPP can obtain the optimal atomic mobility values automatically. The summary of the two versions of atomic mobility parameters for fcc Ni–Al–Cu alloys is shown in Table 4.

Table 4:

Summary of the atomic mobility parameters for fcc Ni–Al–Cu alloys assessed in the present work as well as those from the literatures (all in SI units).

Mobility Parameters (J mol−1) Reference
Mobility of Ni Φ N i N i = − 2 71 377.6 − 81.79 * T [39]
Φ N i C u = − 2 29 936.8 − 78.83 * T [37]
Φ N i A 1 = 1 44 600.0 − 64.85 * T [39]
Φ N i N i , C u = 39 620.8 − 24.19 * T [37]
Φ N i N i , A 1 = − 29 571.8 [39]
Mobility of Al Φ A 1 A 1 = − 1 23 111.6 − 97.34 * T [39]
Φ A 1 C u = − 1 81 583.4 − 99.8 * T [38]
Φ A 1 N i = − 2 68 381.0 − 71.04 * T [39]
Φ A 1 A 1 , C u = − 1 83 094.3 − 159.04 * T [38]
Φ A 1 A 1 , N i = − 3 08 067.5 − 111.52 * T [39]
Φ A 1 N i , C u = − 32 688.2 This work (version 1)
Φ A 1 N i , C u = − 45 329.1 This work (version 2)
Mobility of Cu Φ C u C u = − 2 05 872.0 − 82.52 * T [38]
Φ C u N i = − 2 63 689.7 − 77.04 * T [37]
Φ C u A 1 = − 1 33 184.4 − 83.65 * T [38]
Φ C u C u , N i = 14 204.2 − 4.98 * T [37]
Φ C u C u , A 1 = − 31 461.4 + 78.91 * T [38]
Φ C u N i , A 1 = − 2 07 745.5 This work (version 1)
Φ C u N i , A 1 = − 2 26 583.1 This work (version 2)

With the aim of confirming the reliability of the obtained atomic mobilities, the application to predict the diffusion characteristics is further performed by the CALTPP program. Figure 1 shows the interdiffusivity planes at 1273, 1333, 1433 K calculated by the numerical inverse method. The interdiffusivities at the intersections in diffusion paths obtained through the Matano–Kirkaldy method are overlaid by symbols in Figure 1, which manifests a consistency between the diffusivities calculated by the numerical inverse method and the experimental ones. From the figure, it can be inferred easily that the values of main interdiffusivities D ̃ C u C u N i and D ̃ A 1 A 1 N i are affected by temperature and concentration. With temperature rising, both D ̃ C u C u N i and D ̃ A 1 A 1 N i increase. Furthermore, the increases of Cu and Al concentrations contribute to the slight decreases of D ̃ C u C u N i and the increases of D ̃ A 1 A 1 N i in general. The comparisons between the calculated main diffusivities based on the two versions of atomic mobilities and the ones measured by the Matano–Kirkaldy method in fcc Ni–Al–Cu alloys at 1273, 1333 and 1433 K are shown in Figure 2. Dashed lines refer to the general acceptable deviation of model-predicted diffusivities, i.e., with a factor of 2 or 0.5. As illustrated in the figure, the calculated main diffusivities D ̃ C u C u N i and D ̃ A 1 A 1 N i from two versions can fit well with the experimental ones.

Figure 1: Interdiffusivity planes predicted by the numerical inverse method (denoted as surface) at (a) 1273 K, (b) 1333 K and (c) 1433 K compared with the results obtained by the Matano–Kirkaldy method (denoted in symbols).
Figure 1:

Interdiffusivity planes predicted by the numerical inverse method (denoted as surface) at (a) 1273 K, (b) 1333 K and (c) 1433 K compared with the results obtained by the Matano–Kirkaldy method (denoted in symbols).

Figure 2: Comparisons between the calculated main interdiffusivities using the kinetic parameters of the present (a) version 1 (based on the numerical inverse method), and (b) version 2 (based on the Matano–Kirkaldy method) and the measured ones by the Matano–Kirkaldy method in fcc Ni–Al–Cu alloys at 1273, 1333 and 1433 K. Dashed lines refer to the diffusivities with a factor of 2 or 0.5 from the model-predicted diffusivities.
Figure 2:

Comparisons between the calculated main interdiffusivities using the kinetic parameters of the present (a) version 1 (based on the numerical inverse method), and (b) version 2 (based on the Matano–Kirkaldy method) and the measured ones by the Matano–Kirkaldy method in fcc Ni–Al–Cu alloys at 1273, 1333 and 1433 K. Dashed lines refer to the diffusivities with a factor of 2 or 0.5 from the model-predicted diffusivities.

The comparisons between the model-predicted concentration profiles using the present two versions of kinetic parameters and the measured ones are illustrated. Figures 3 –5 exhibit the model-predicted concentration profiles of Cu and Al for fcc Ni–Al–Cu alloys, compared with the measured data for the diffusion couples annealed at 1273 K for 96 h, 1333 K for 36 h, 1433 K for 6 h, respectively. The solid and dashed lines represent the calculated results utilizing the kinetic parameters from the version 1 and version 2, respectively. As revealed by these figures, the model-predicted concentration profiles from version 1 and version 2 present nearly the same profiles and agree well with the corresponding measured ones. Also, all of the concentration profiles of Al and Cu have an “S” shape and symmetry is obvious to be seen in Cu and Al diffusion. The diffusion distances of Al are about 100 μm longer than the ones of Cu.

Figure 3: Comparisons between the model-predicted concentration profiles using the kinetic parameters of the present version 1 (based on the numerical inverse method) and version 2 (based on the Matano–Kirkaldy method) in diffusion couples of (a) Ni/Ni-10.2Al-7.0Cu, (b) Ni/Ni-6.0Al-19.8Cu, (c) Ni-5.2Al/Ni-21.1Cu, and (d) Ni-9.4Al/Ni-10.1Cu (at.%) annealed at 1273 K for 96 h, compared with the measured data in this work.
Figure 3:

Comparisons between the model-predicted concentration profiles using the kinetic parameters of the present version 1 (based on the numerical inverse method) and version 2 (based on the Matano–Kirkaldy method) in diffusion couples of (a) Ni/Ni-10.2Al-7.0Cu, (b) Ni/Ni-6.0Al-19.8Cu, (c) Ni-5.2Al/Ni-21.1Cu, and (d) Ni-9.4Al/Ni-10.1Cu (at.%) annealed at 1273 K for 96 h, compared with the measured data in this work.

Figure 4: Comparisons between the model-predicted concentration profiles using the kinetic parameters of the present version 1 (based on the numerical inverse method) and version 2 (based on the Matano–Kirkaldy method) in diffusion couples of (a) Ni/Ni-10.2Al-7.0Cu, (b) Ni/Ni-6.0Al-19.8Cu, (c) Ni-5.2Al/Ni-21.1Cu, and (d) Ni-9.4Al/Ni-10.1Cu (at.%) annealed at 1333 K for 36 h, compared with the measured data in this work.
Figure 4:

Comparisons between the model-predicted concentration profiles using the kinetic parameters of the present version 1 (based on the numerical inverse method) and version 2 (based on the Matano–Kirkaldy method) in diffusion couples of (a) Ni/Ni-10.2Al-7.0Cu, (b) Ni/Ni-6.0Al-19.8Cu, (c) Ni-5.2Al/Ni-21.1Cu, and (d) Ni-9.4Al/Ni-10.1Cu (at.%) annealed at 1333 K for 36 h, compared with the measured data in this work.

Figure 5: Comparisons between the model-predicted concentration profiles using the kinetic parameters of the present version 1 (based on the numerical inverse method) and version 2 (based on the Matano–Kirkaldy method) in diffusion couples of (a) Ni/Ni-10.2Al-7.0Cu, (b) Ni/Ni-6.0Al-19.8Cu, (c) Ni-5.2Al/Ni-21.1Cu, and (d) Ni-9.4Al/Ni-10.1Cu (at.%) annealed at 1433 K for 6 h, compared with the measured data in this work.
Figure 5:

Comparisons between the model-predicted concentration profiles using the kinetic parameters of the present version 1 (based on the numerical inverse method) and version 2 (based on the Matano–Kirkaldy method) in diffusion couples of (a) Ni/Ni-10.2Al-7.0Cu, (b) Ni/Ni-6.0Al-19.8Cu, (c) Ni-5.2Al/Ni-21.1Cu, and (d) Ni-9.4Al/Ni-10.1Cu (at.%) annealed at 1433 K for 6 h, compared with the measured data in this work.

Figure 6 illustrates the model-predicted diffusion paths utilizing presently obtained parameters at 1273, 1333 and 1433 K, compared with the experimental ones. Almost all the diffusion paths are observed as the serpentine shape, which results from the difference in diffusion rates of Al and Cu atoms. From the figure, it can be concluded that two sets of the assessed atomic mobilities of Ni–Al–Cu alloys are quite reliable since the model-predicted diffusion paths can attach a general agreement with the measured ones.

Figure 6: Comparisons between the model-predicted diffusion paths using the kinetic parameters of the present version 1 (based on the numerical inverse method) and version 2 (based on the Matano–Kirkaldy method) in fcc Ni–Al–Cu alloys annealed at (a) 1273 K for 96 h, (b) 1333 K for 36 h, and (c) 1433 K for 6 h, compared with the experimental data in this work.
Figure 6:

Comparisons between the model-predicted diffusion paths using the kinetic parameters of the present version 1 (based on the numerical inverse method) and version 2 (based on the Matano–Kirkaldy method) in fcc Ni–Al–Cu alloys annealed at (a) 1273 K for 96 h, (b) 1333 K for 36 h, and (c) 1433 K for 6 h, compared with the experimental data in this work.

As mentioned in Section 1, the diffusion behaviors in Cu-rich Cu–Ni–Al alloys at 1073, 1173, 1273 K were studied and the corresponding atomic mobilities were reported by Zhang et al. [9]. The individual experimental investigations at the common temperature of 1273 K are performed in the present work focusing on Ni-rich alloy and in Reference [9] on Cu-rich alloy. To further analyze the differences in the diffusion behaviors and atomic mobilities of Cu-rich and Ni-rich alloys, the model-predicted concentration profiles at 1273 K together with the measured data are displayed in Figures 7 and 8. Figure 7 presents the concentration profiles predicted by the atomic mobilities from the present work and Reference [9], together with the presently measured data in Ni-rich alloys annealed at 1273 K for 96 h. It is obvious that the concentration profiles simulated by the atomic mobilities from the present work can fit the experimental data well for Ni-rich alloys. The concentration profiles calculated by the atomic mobilities from Reference [9] lead to noticeable discrepancies from the experimental ones. Figure 8 presents the concentration profiles predicted by the atomic mobilities from the present work and Reference [9], together with the measured data in the Cu-rich alloys from Reference [9] annealed at 1273 K for 48 h. It is obvious that the concentration profiles simulated by the atomic mobilities from Reference [9] can fit the experimental data well for Cu-rich alloys since the experimental data are used in the optimization of the atomic mobilities. It is noted that the presently obtained atomic mobility parameters in Ni-rich alloys can predict the concentration profiles in Cu-rich alloys reasonably although the experimental data in Cu-rich side from Reference [9] are not used in the present optimization.

Figure 7: Comparisons between the model-predicted concentration profiles using the kinetic parameters of the present version 1 (denoted in solid line) and in Reference [9] (denoted in dashed line) in diffusion couples of (a) Ni/Ni-10.2Al-7.0Cu, (b) Ni/Ni-6.0Al-19.8Cu, (c) Ni-5.2Al/Ni-21.1Cu, and (d) Ni-9.4Al/Ni-10.1Cu (at.%) annealed at 1273 K for 96 h, compared with the measured data in this work.
Figure 7:

Comparisons between the model-predicted concentration profiles using the kinetic parameters of the present version 1 (denoted in solid line) and in Reference [9] (denoted in dashed line) in diffusion couples of (a) Ni/Ni-10.2Al-7.0Cu, (b) Ni/Ni-6.0Al-19.8Cu, (c) Ni-5.2Al/Ni-21.1Cu, and (d) Ni-9.4Al/Ni-10.1Cu (at.%) annealed at 1273 K for 96 h, compared with the measured data in this work.

Figure 8: Comparisons between the model-predicted concentration profiles using the kinetic parameters of the present version 1 (denoted in solid line) and in Reference [9] (denoted in dashed line) in diffusion couples of (a) Cu-5.07Al/Cu-16.53Ni, (b) Cu-5.56Al/Cu-7.62Al-8.92Ni, (c) Cu/Cu-3.83Al-21.57Ni (at.%) annealed at 1273 K for 48 h, compared with the measured data in Reference [9].
Figure 8:

Comparisons between the model-predicted concentration profiles using the kinetic parameters of the present version 1 (denoted in solid line) and in Reference [9] (denoted in dashed line) in diffusion couples of (a) Cu-5.07Al/Cu-16.53Ni, (b) Cu-5.56Al/Cu-7.62Al-8.92Ni, (c) Cu/Cu-3.83Al-21.57Ni (at.%) annealed at 1273 K for 48 h, compared with the measured data in Reference [9].

As shown in Figures 7 and 8, the concentration profiles in the Ni-rich alloys show symmetrical characteristics, while the concentration profiles in the Cu-rich alloys display the complex feature that obvious uphill diffusion appears in the concentration profile of Al. The different diffusion characteristics in the Ni-rich and Cu-rich alloys lead to the incompatibility of the established atomic mobilities of different side alloys. Therefore, it is necessary to study the diffusivities and atomic mobilities in the Ni-rich fcc Ni–Al–Cu alloys. It is worth mentioning that the diffusion experiment of four diffusion couples annealed at three temperatures for Cu-rich Ni–Al–Cu alloys has been finished by the present authors and further work regarding the establishment of the universal atomic mobility parameters in the Cu-rich and Ni-rich alloys will be proposed.

In summary, comparing the diffusion behaviors, viz., main interdiffusivities, concentration profiles and diffusion paths adopting two versions of atomic mobility parameters with the experimental ones, it can be concluded that these atomic mobility parameters obtained through both methods can predict the diffusion behaviors of fcc Ni–Al–Cu alloys well. What is more, the diffusion characteristics predicted through the numerical inverse method are nearly the same as those predicted by the Matano–Kirkaldy method. However, the numerical inverse method possesses two advantages in comparison with the Matano–Kirkaldy method. One advantage is that the numerical inverse method does not need to fit the concentration profiles with functions. Regarding the Matano–Kirkaldy method, the adoptions of different fitting functions of concentration profiles may result in different diffusion coefficients and thereby influence the establishment of atomic mobilities. On the other hand, the numerical inverse method is devoted to applying the experimental concentration profiles directly to the assessment of interdiffusivities and atomic mobilities, without fitting periods. Another advantage associated with the numerical inverse method is that the intersection in diffusion paths is not a necessity in its application and the atomic mobilities thus obtained utilizing the experimental concentration profile of just one diffusion couple can reproduce the accurate diffusion behaviors of the other three diffusion couples in the present work. It is noted that the choice of C1 diffusion couple is arbitrary and the optimized results of the other three diffusion couples are presented in the Appendix. Thus, fewer diffusion couples combined with the numerical inverse method can reliably determine the atomic mobilities for fcc Ni–Al–Cu alloys in an effective way.

5 Conclusions

In the present work, a total of 12 diffusion couples of fcc Ni–Al–Cu alloys were prepared and the concentration profiles after annealing at 1273, 1333 and 1433 K were measured by EPMA-WDX. The main conclusions are as follows:

  1. The numerical inverse method and the Matano–Kirkaldy method were utilized to obtain the diffusion coefficients along the C1 diffusion path and at the intersections of all the diffusion couples at each temperature, respectively.

  2. Two sets of atomic mobilities were obtained through an intelligent model using the diffusivities from the numerical inverse method and the Matano–Kirkaldy method implemented in CALTPP.

  3. The validity of the present two versions of atomic mobilities was confirmed by comparing the assessed diffusion characteristics including diffusivities, concentration profiles and diffusion paths of four diffusion couples with the measured ones, which showed good agreement. These diffusion behaviors predicted through the numerical inverse method and the Matano–Kirkaldy method achieved good consistency. The numerical inverse method combined with fewer diffusion couples can obtain accurate diffusion coefficients and thus reliably assess the atomic mobility parameters for the Ni-rich fcc Ni–Al–Cu alloys.

Corresponding authors: Yuling Liu and Yong Du, State Key Laboratory of Powder Metallurgy, Central South University, Changsha, Hunan, 410083, P. R. China, E-mail: (Y. Liu) (Y. Liu) and (Y. Du) (Y. Du)

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: The financial support by grant from Genetic Engineering Project of Materials in Yunnan Province 2019ZE001-1 and the Research Fund for foreign young scholars of National Natural Science Foundation of China (No. 51950410600) are gratefully acknowledged.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix

Figures A1–A8

Figure A1: Comparisons between the model-predicted concentration profiles using the kinetic parameters of the Model 1, Model 2, Model 3 and Model 4 in diffusion couples of (a) Ni/Ni-10.2Al-7.0Cu, (b) Ni/Ni-6.0Al-19.8Cu, (c) Ni-5.2Al/Ni-21.1Cu, and (d) Ni-9.4Al/Ni-10.1Cu (at.%) annealed at 1273 K for 96 h, compared with the measured data in this work.
Figure A1:

Comparisons between the model-predicted concentration profiles using the kinetic parameters of the Model 1, Model 2, Model 3 and Model 4 in diffusion couples of (a) Ni/Ni-10.2Al-7.0Cu, (b) Ni/Ni-6.0Al-19.8Cu, (c) Ni-5.2Al/Ni-21.1Cu, and (d) Ni-9.4Al/Ni-10.1Cu (at.%) annealed at 1273 K for 96 h, compared with the measured data in this work.

Figure A2: Comparisons between the model-predicted concentration profiles using the kinetic parameters of the Model 1, Model 2, Model 3 and Model 4 in diffusion couples of (a) Ni/Ni–10.2Al–7.0Cu, (b) Ni/Ni–6.0Al–19.8Cu, (c) Ni–5.2Al/Ni–21.1Cu, and (d) Ni–9.4Al/Ni–10.1Cu (at.%) annealed at 1333 K for 36 h, compared with the measured data in this work.
Figure A2:

Comparisons between the model-predicted concentration profiles using the kinetic parameters of the Model 1, Model 2, Model 3 and Model 4 in diffusion couples of (a) Ni/Ni–10.2Al–7.0Cu, (b) Ni/Ni–6.0Al–19.8Cu, (c) Ni–5.2Al/Ni–21.1Cu, and (d) Ni–9.4Al/Ni–10.1Cu (at.%) annealed at 1333 K for 36 h, compared with the measured data in this work.

Figure A3: Comparisons between the model-predicted concentration profiles using the kinetic parameters of the Model 1, Model 2, Model 3 and Model 4 in diffusion couples of (a) Ni/Ni–10.2Al–7.0Cu, (b) Ni/Ni–6.0Al–19.8Cu, (c) Ni–5.2Al/Ni–21.1Cu, and (d) Ni–9.4Al/Ni–10.1Cu (at.%) annealed at 1433 K for 6 h, compared with the measured data in this work.
Figure A3:

Comparisons between the model-predicted concentration profiles using the kinetic parameters of the Model 1, Model 2, Model 3 and Model 4 in diffusion couples of (a) Ni/Ni–10.2Al–7.0Cu, (b) Ni/Ni–6.0Al–19.8Cu, (c) Ni–5.2Al/Ni–21.1Cu, and (d) Ni–9.4Al/Ni–10.1Cu (at.%) annealed at 1433 K for 6 h, compared with the measured data in this work.

Figure A4: Comparisons between the model-predicted diffusion paths using the kinetic parameters of the Model 1, Model 2, Model 3 and Model 4 in fcc Ni–Al–Cu alloys annealed at (a) 1273 K for 96 h, (b) 1333 K for 36 h, and (c) 1433 K for 6 h, compared with the experimental data in this work.
Figure A4:

Comparisons between the model-predicted diffusion paths using the kinetic parameters of the Model 1, Model 2, Model 3 and Model 4 in fcc Ni–Al–Cu alloys annealed at (a) 1273 K for 96 h, (b) 1333 K for 36 h, and (c) 1433 K for 6 h, compared with the experimental data in this work.

Figure A5: The main interdiffusivity versus the diffusion distance for C1 at (a) 1273 K, (b) 1333K, (c) 1433 K, which are calculated by the Matano–Kirkaldy method (denoted by symbols) and Model 1 (denoted by lines).
Figure A5:

The main interdiffusivity versus the diffusion distance for C1 at (a) 1273 K, (b) 1333K, (c) 1433 K, which are calculated by the Matano–Kirkaldy method (denoted by symbols) and Model 1 (denoted by lines).

Figure A6: The main interdiffusivity versus the diffusion distance for C2 at (a) 1273 K, (b) 1333 K, (c) 1433 K, which are calculated by the Matano–Kirkaldy method (denoted by symbols) and Model 2 (denoted by lines).
Figure A6:

The main interdiffusivity versus the diffusion distance for C2 at (a) 1273 K, (b) 1333 K, (c) 1433 K, which are calculated by the Matano–Kirkaldy method (denoted by symbols) and Model 2 (denoted by lines).

Figure A7: The main interdiffusivity versus the diffusion distance for C3 at (a) 1273 K, (b) 1333 K, (c) 1433 K, which are calculated by the Matano–Kirkaldy method (denoted by symbols) and Model 3 (denoted by lines).
Figure A7:

The main interdiffusivity versus the diffusion distance for C3 at (a) 1273 K, (b) 1333 K, (c) 1433 K, which are calculated by the Matano–Kirkaldy method (denoted by symbols) and Model 3 (denoted by lines).

Figure A8: The main interdiffusivity versus the diffusion distance for C4 at (a) 1273 K, (b) 1333 K, (c) 1433 K, which are calculated by the Matano–Kirkaldy method (denoted by symbols) and Model 4 (denoted by lines).
Figure A8:

The main interdiffusivity versus the diffusion distance for C4 at (a) 1273 K, (b) 1333 K, (c) 1433 K, which are calculated by the Matano–Kirkaldy method (denoted by symbols) and Model 4 (denoted by lines).

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Received: 2021-06-20
Revised: 2022-03-04
Accepted: 2021-11-25
Published Online: 2022-04-28
Published in Print: 2022-05-26

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. Preface of the issue of the 19th national symposium on phase diagram and materials design
  4. Review
  5. Improvement of the thermoelectric properties of GeTe- and SnTe-based semiconductors aided by the engineering based on phase diagram
  6. Original Papers
  7. Diffusivities and atomic mobilities in the Ni-rich fcc Ni–Al–Cu alloys: experiment and modeling
  8. Composition-dependent interdiffusivity matrices of ordered bcc_B2 phase in ternary Ni–Al–Ru system at 1273∼1473 K
  9. Investigation of interdiffusion behavior in the Ti–Zr–Cu ternary system
  10. Measurement of the diffusion coefficient in Mg–Sn and Mg–Sc binary alloys
  11. Thermodynamic calculation of phase equilibria of rare earth metals with boron binary systems
  12. Thermodynamic modeling of the Bi–Ca and Bi–Zr systems
  13. Redetermination of the Fe–Pt phase diagram by using diffusion couple technique combined with key alloys
  14. Experimental determination of the isothermal sections and liquidus surface projection of the Mo–Si–V ternary system
  15. Experimental determination of isothermal sections of the Hf–Nb–Ni system at 950 and 1100 °C
  16. Experimental investigation and thermodynamic assessment of the Al–Ca–Y ternary system
  17. Phase equilibria of the Ni–Cr–Y ternary system at 900 °C
  18. Phase constituents near the center of the Co–Cr–Fe–Ni–Ti system at 1000 °C
  19. Metastable phase diagram of the Gd2O3–SrO–CoO x ternary system
  20. Crystallization kinetic and dielectric properties of CaO–MgO–Al2O3–SiO2 glass/Al2O3 composites
  21. Investigation of the phase relation of the Bi2O3–Fe2O3–Nd2O3 system at 973 K and the microwave absorption performance of NdFeO3/Bi25FeO40 with different mass ratios
  22. The influence of SrCl2 on the corrosion behavior of magnesium
  23. Retraction
  24. Retraction of: Electrolytic synthesis of ZrSi/ZrC nanocomposites from ZrSiO4 and carbon black powder in molten salt
  25. News
  26. DGM – Deutsche Gesellschaft für Materialkunde
https://doi.org/10.1515/ijmr-2021-8426
Keywords for this article
Atomic mobility; CALTPP program; Diffusivity; Experimental measurements; Ni–Al–Cu alloy

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. Preface of the issue of the 19th national symposium on phase diagram and materials design
  4. Review
  5. Improvement of the thermoelectric properties of GeTe- and SnTe-based semiconductors aided by the engineering based on phase diagram
  6. Original Papers
  7. Diffusivities and atomic mobilities in the Ni-rich fcc Ni–Al–Cu alloys: experiment and modeling
  8. Composition-dependent interdiffusivity matrices of ordered bcc_B2 phase in ternary Ni–Al–Ru system at 1273∼1473 K
  9. Investigation of interdiffusion behavior in the Ti–Zr–Cu ternary system
  10. Measurement of the diffusion coefficient in Mg–Sn and Mg–Sc binary alloys
  11. Thermodynamic calculation of phase equilibria of rare earth metals with boron binary systems
  12. Thermodynamic modeling of the Bi–Ca and Bi–Zr systems
  13. Redetermination of the Fe–Pt phase diagram by using diffusion couple technique combined with key alloys
  14. Experimental determination of the isothermal sections and liquidus surface projection of the Mo–Si–V ternary system
  15. Experimental determination of isothermal sections of the Hf–Nb–Ni system at 950 and 1100 °C
  16. Experimental investigation and thermodynamic assessment of the Al–Ca–Y ternary system
  17. Phase equilibria of the Ni–Cr–Y ternary system at 900 °C
  18. Phase constituents near the center of the Co–Cr–Fe–Ni–Ti system at 1000 °C
  19. Metastable phase diagram of the Gd2O3–SrO–CoO x ternary system
  20. Crystallization kinetic and dielectric properties of CaO–MgO–Al2O3–SiO2 glass/Al2O3 composites
  21. Investigation of the phase relation of the Bi2O3–Fe2O3–Nd2O3 system at 973 K and the microwave absorption performance of NdFeO3/Bi25FeO40 with different mass ratios
  22. The influence of SrCl2 on the corrosion behavior of magnesium
  23. Retraction
  24. Retraction of: Electrolytic synthesis of ZrSi/ZrC nanocomposites from ZrSiO4 and carbon black powder in molten salt
  25. News
  26. DGM – Deutsche Gesellschaft für Materialkunde
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