Original Article 原文
Diffusivities and atomic mobilities in fcc Co–Cu–Mn alloys
fcc Co-Cu-Mn 合金中的扩散率和原子迁移率

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Abstract 摘要

Accurate diffusivities and atomic mobilities of fcc Co–Cu–Mn alloys are essential for developing novel high-entropy alloys. However, no ternary interdiffusion information of Co–Cu–Mn alloys is available in the literature. Twelve diffusion couples are prepared to determine the diffusivities in fcc Co–Cu–Mn alloys at the temperature ranges from 1373 to 1473 K. Two versions of diffusivity matrices are determined by the Matano-Kirkaldy method (version 1) and numerical inverse approach (version 2), respectively. And their corresponding atomic mobilities are extracted in CALTPP. The good consistency of the two versions of diffusivities demonstrates the reliability of diffusivities extracted by the numerical inverse approach. Further comparisons show that the numerical inverse approach can determine the diffusivities more efficiently, provide more composition-dependent diffusivities and obtain more reliable atomic mobilities than the Matano-Kirkaldy method. The atomic mobility parameters of version 2 are further used to construct the three-dimensional interdiffusivity matrix of the fcc Co–Cu–Mn system. The presently obtained kinetic parameters are key inputs for the computational design of high-entropy alloys.
fcc Co-Cu-Mn 合金的精确扩散率和原子迁移率对于开发新型高熵合金至关重要。然而,文献中并没有 Co-Cu-Mn 合金的三元互扩散信息。本研究制备了 12 个扩散对偶,以测定在 1373 至 1473 K 温度范围内 fcc Co-Cu-Mn 合金中的扩散系数。并在 CALTPP 中提取了相应的原子迁移率。两个版本的扩散系数具有很好的一致性,这表明数值反演法提取的扩散系数是可靠的。进一步的比较表明,与 Matano-Kirkaldy 方法相比,数值反演方法能更有效地确定扩散系数,提供更多与组成相关的扩散系数,并获得更可靠的原子迁移率。版本 2 的原子迁移率参数进一步用于构建 fcc Co-Cu-Mn 体系的三维互扩散矩阵。目前获得的动力学参数是高熵合金计算设计的关键输入。

Keywords 关键词

High-entropy alloys
fcc Co–Cu–Mn alloys
Diffusivity matrices
Matano-kirkaldy method
Numerical inverse approach

高熵合金fcc Co-Cu-Mn 合金扩散系数矩阵马托-柯卡尔迪法数值逆方法

1. Introduction 1.导言

The Co–Cu–Mn system is the basis for many HEAs (High Entropy Alloys) such as Co–Cu–Mn–Ni [1], Co–Cu–Mn–Fe–Ni [2,3], Co–Cu–Mn–Cr–Fe–Ni [4] and Co–Cu–Mn–Cr–Fe–Ni–Al [5]. These fcc-structured HEAs exhibit excellent mechanical properties such as toughness, strength, creep plastic strain, ductility and wear resistance, which are highly correlated with their thermodynamic and kinetic characteristics [6,7]. According to the CALPHAD (CALculation of PHAse Diagram) method [8], the diffusivity in multi-component system, can be extrapolated by its sub-systems in the form of atomic mobility. The thermodynamic description for the Co–Cu–Mn ternary system has been evaluated by Wang et al. [9]. However, there is no ternary diffusion investigation for the Co–Cu–Mn ternary system in the literature.
Co-Cu-Mn 系统是许多 HEA(高熵合金)的基础,如 Co-Cu-Mn-Ni [1]、Co-Cu-Mn-Fe-Ni [2,3]、Co-Cu-Mn-Cr-Fe-Ni [4] 和 Co-Cu-Mn-Cr-Fe-Ni-Al [5]。这些 fcc 结构的 HEA 具有优异的机械性能,如韧性、强度、蠕变塑性应变、延展性和耐磨性,这些性能与其热力学和动力学特性高度相关[6,7]。根据 CALPHAD(CALculation of PHAse Diagram,PHAse 图计算)方法[8],多组分系统中的扩散性可以通过其子系统以原子迁移率的形式进行推断。Wang 等人对 Co-Cu-Mn 三元体系的热力学描述进行了评估[9]。然而,目前还没有文献对 Co-Cu-Mn 三元体系进行三元扩散研究。

Therefore, the interdiffusion in the fcc Co–Cu–Mn system is experimentally investigated in the present work. The atomic mobility parameters of the Co–Cu–Mn system based on the experimental data can be used to extrapolate to HEAs systems containing these three elements.
因此,本研究对 fcc Co-Cu-Mn 体系中的相互扩散进行了实验研究。基于实验数据的 Co-Cu-Mn 体系原子迁移率参数可用于推断含有这三种元素的 HEAs 体系。

In general, two approaches are used to determine the diffusivity matrix (four independent interdiffusion coefficients) from the measured compositional profiles in the ternary diffusion couples. One way is the Matano-Kirkaldy method [10,11], by which the accurate interdiffusivity matrix can be determined [12]. However, this method is low efficient, since it can only measure one diffusivity matrix at the intersection point in diffusion paths of two diffusion couples. The other way is the numerical inverse approach [13] which can be used to retrieve the composition-dependent interdiffusivity matrices over the whole diffusion path from a single diffusion couple. Therefore, the diffusivity matrices can be determined with high efficiency. Moreover, the determined interdiffusion coefficients are usually employed to evaluate the atomic mobilities with the aim to efficiently storage the diffusivities via CALPHAD method [8]. Combining the atomic mobilities and consistent thermodynamic descriptions, the interdiffusion coefficients at wider composition and temperature ranges can be calculated. Both the Matano-Kirkaldy method and numerical inverse approach have been incorporated in CALTPP (CALculation of ThermoPhysical Properties) [14] for diffusivities determination. In addition, CALTPP can extract the atomic mobilities based on the experimental compositional profiles and/or interdiffusion coefficients through an intelligent model. During the atomic mobilities evaluation, the initial parameters to be optimized are automatically selected and the iteration termination is automatically performed [14,15]. For a comparative study of the Matano-Kirkaldy method and numerical inverse approach, two methods are used to extract the corresponding versions of interdiffusivities and atomic mobilities of fcc Co–Cu–Mn alloys by CALTPP.
一般来说,有两种方法可用于根据测得的三元扩散偶中的成分剖面确定扩散率矩阵(四个独立的相互扩散系数)。一种方法是 Matano-Kirkaldy 方法 [10,11],通过该方法可以确定精确的互扩散矩阵 [12]。然而,这种方法效率较低,因为它只能测量两个扩散偶扩散路径交叉点上的一个扩散率矩阵。另一种方法是数值反演法[13],它可以用来检索单个扩散偶在整个扩散路径上与成分相关的相互扩散系数矩阵。因此,可以高效地确定扩散系数矩阵。此外,确定的相互扩散系数通常用于评估原子迁移率,目的是通过 CALPHAD 方法[8]有效地存储扩散系数。结合原子迁移率和一致的热力学描述,可以计算出更大成分和温度范围内的相互扩散系数。Matano-Kirkaldy 方法和数值反演方法都被纳入了 CALTPP(热物理特性计算)[14],用于确定扩散系数。此外,CALTPP 还可以通过智能模型,根据实验成分剖面和/或相互扩散系数提取原子迁移率。在原子迁移率评估过程中,会自动选择要优化的初始参数,并自动执行迭代终止[14,15]。 为了对 Matano-Kirkaldy 方法和数值反演方法进行比较研究,我们使用这两种方法,通过 CALTPP 提取 fcc Co-Cu-Mn 合金相应版本的间扩散率和原子迁移率。

Consequently, the major purposes of the present work are: (i) to experimentally determine the diffusivity matrices of fcc Co–Cu–Mn alloys by the Matano-Kirkaldy method and numerical inverse approach; (ii) to extract the atomic mobilities using the two versions of diffusivity matrices in CALTPP and confirm the reliability of the atomic mobilities by comparing the calculated diffusivities, predicted compositional profiles, diffusion fluxes and diffusion paths with the corresponding experimental data; and (iii) to show the temperature- and composition-dependent diffusional properties by plotting the 3-D (three-dimensional) main interdiffusion coefficients, activation energies and frequency–factor planes for the fcc Co–Cu–Mn ternary system.
因此,本研究的主要目的是(i) 通过 Matano-Kirkaldy 方法和数值反演方法,实验确定 fcc Co-Cu-Mn 合金的扩散系数矩阵;(ii) 在 CALTPP 中使用两种版本的扩散系数矩阵提取原子迁移率,并通过比较计算的扩散系数、预测的成分剖面、扩散通量和扩散路径与相应的实验数据,确认原子迁移率的可靠性;(iii) 通过绘制 fcc Co-Cu-Mn 三元体系的三维(三维)主要相互扩散系数、活化能和频率因子平面图,显示与温度和成分有关的扩散特性。

2. Experimental procedure
2.实验程序

Twelve diffusion couples were designed, as displayed in Table 1. Pure Co rod (purity: 99.99 wt.%), Cu rod (purity: 99.99 wt.%), and Mn block (purity: 99.98 wt.%) were taken as raw materials. First, the designed samples with different compositions were prepared by arc melting under an argon atmosphere to a button shape. The buttons are repeatedly melted four times to obtain homogeneity compositions. Then, the buttons were cut into cuboids with approximate dimensions of 3 × 5 × 8 mm3 and were homogenized under a vacuum atmosphere at 1373 ± 5 K for 15 days to acquire large size grains. After quenching in water, the surface contamination on each sample was removed with SiC paper. Subsequently, each surface was polished by diamond slurry. The polished cuboids were bound together with Mo fixtures to form the diffusion couples. These diffusion couples were sealed into evacuated quartz tubes and annealed at 1373 K for 58 h, 1423 for 54 h or 1473 K for 48 h according to the schedule listed in Table 1. The annealed alloys were quenched in cold water to keep the microstructure at annealed temperatures. Next, the diffusion couples were subjected to standard metallographic processing. At last, the EPMA (Electron Probe Micro-Analysis) was utilized to measure the compositional profiles of the solute elements parallel to the diffusion direction. The composition error measured by the EPMA is approximately ±5%.
如表 1 所示,共设计了 12 个扩散耦合器。原材料包括纯 Co 棒(纯度:99.99 wt.%)、Cu 棒(纯度:99.99 wt.%)和 Mn 块(纯度:99.98 wt.%)。首先,在氩气环境下用电弧熔化法制备不同成分的设计样品,使其呈纽扣状。纽扣反复熔化四次,以获得均匀的成分。然后,将钮扣切割成尺寸约为 3 × 5 × 8 mm 3 的长方体,并在 1373 ± 5 K 的真空环境下均质 15 天,以获得大尺寸晶粒。在水中淬火后,用碳化硅纸去除每个样品的表面污染物。随后,用金刚石浆对每个表面进行抛光。抛光后的立方体用 Mo 固定装置结合在一起,形成扩散耦合体。这些扩散耦合体被密封在抽真空的石英管中,并按照表 1 中列出的时间表在 1373 K 下退火 58 小时、1423 K 下退火 54 小时或 1473 K 下退火 48 小时。退火后的合金在冷水中淬火,以保持退火温度下的微观结构。然后,对扩散偶进行标准金相处理。最后,利用 EPMA(电子探针显微分析)测量与扩散方向平行的溶质元素的成分剖面。EPMA 测得的成分误差约为±5%。

Table 1. Terminal compositions of the diffusion couples and annealing schedule in the fcc Co–Cu–Mn system.
表 1.fcc Co-Cu-Mn 体系中扩散耦合的终端成分和退火时间表。

No.Assembly of diffusion couples (comp. in at.%)
扩散耦合剂的组装(单位:%)
T (K)Time (h)
A1Co-3.9Cu/Co-3.7Mn137358
A2Co-7.5Cu/Co-3.8Mn137358
A3Co-3.9Cu/Co-8.4Mn137358
A4Co/Co-2.9Cu-12.5Mn137358
B1Co-3.9Cu/Co-3.8Mn142354
B2Co-7.5Cu/Co-3.8Mn142354
B3Co-3.9Cu/Co-8.4Mn142354
B4Co/Co-2.9Cu-12.5Mn142354
C1Co-3.9Cu/Co-3.8Mn147348
C2Co-7.6Cu/Co-3.8Mn147348
C3Co-3.9Cu/Co-8.5Mn147348
C4Co/Co-2.9Cu-12.5Mn147348

3. Model description 3.模型说明

3.1. Matano-Kirkaldy method
3.1.马塔诺-柯卡迪方法

The Boltzmann-Matano method has been extended by Kirkaldy et al. [9,10] to retrieve interdiffusivity matrices in a ternary system. Supposing that the volume variation can be neglected, a normalized concentration parameter Yi is proposed by Whittle and Green [16] to avoid the calculation of the Matano plane. The diffusivity matrices of Co–Cu–Mn alloys can be solved by the following functions,(1)Yi=cicici+ci(i=CuorMn)(2)D˜CuCuCo+D˜CuMnCo(dcMndcCu)=12tdzdYCu[(1YCu)zYCudz+YCuz+(1YCu)dz](3)D˜MnMnCo+D˜MnCuCo(dcCudcMn)=12tdzdYMn[(1YMn)zYMndz+YMnz+(1YMn)dz]where, ci and ci+ are the concentrations of interests, the initial and terminal concentrations for element i, respectively. z denotes distance and t is diffusion time. D˜CuCuCo and D˜MnMnCo are main interdiffusion coefficients, D˜CuMnCo and D˜MnCuCo cross interdiffusivities. In order to determine the four independent diffusivities, four equations are needed. However, there are only two equations (Eqs. (2), (3)) in a single diffusion couple. Therefore, two diffusion couples, which are designed to have a common composition in their diffusion paths, are required.
Kirkaldy 等人[9,10]对波尔兹曼-马塔诺方法进行了扩展,以获取三元系统中的相互扩散矩阵。假设体积变化可以忽略,Whittle 和 Green[16]提出了一个归一化浓度参数 ,以避免计算马塔诺平面。Co-Cu-Mn 合金的扩散系数矩阵可通过以下函数求解: (1) (2) (3) 其中, 分别为相关浓度、元素 i 的初始浓度和末端浓度。 为主要相互扩散系数, 为交叉相互扩散系数。为了确定四个独立的扩散系数,需要四个方程。然而,单个扩散耦合中只有两个方程(公式 (2)、(3))。因此,需要两个在扩散路径上具有共同成分的扩散耦合。

The thermodynamical constraint equations [17] are used to validate the thermodynamic stability of interdiffusivity matrices, and the error propagation method [18] is applied to evaluate the SDs (standard deviations) of the determined interdiffusivities. The detailed description can be referred to our previous work of ternary diffusivities measurements [[13], [14], [15]].
热力学约束方程 [17] 用于验证互扩散系数矩阵的热力学稳定性,误差传播方法 [18] 用于评估所确定的互扩散系数的 SDs(标准偏差)。详细描述可参考我们以前的三元扩散系数测量工作[[13], [14], [15]]。

3.2. Numerical inverse approach
3.2.数值反演法

Inspired by the work of Bouchet and Mevrel [19], the composition-dependent diffusion coefficients can be defined by the concept of basic functions [13]. The interdiffusion coefficients in fcc Co–Cu–Mn alloys are given by the following formula,(4)D˜ijCo=k(δikci)exp(ΦkRT)φkjCo,(i=CuorMn)where Φk is the mobility parameters of element k to be optimized, φkjCo is the thermodynamic factor, T denotes the absolute temperature, R is the gas constant and δik is the Kronecker delta (δik=1 if i=k, otherwise δik=0).
受 Bouchet 和 Mevrel [19] 工作的启发,与成分相关的扩散系数可以用基本函数的概念来定义 [13]。fcc Co-Cu-Mn 合金中的相互扩散系数由以下公式给出: (4) 其中, 是待优化元素 k 的迁移率参数, 是热力学系数,T 表示绝对温度,R 是气体常数, 是克朗内克δ( 如果是 ,否则是 )。

The numerical inverse approach deals with the forward problem by minimizing the difference between calculated concentrations and experimental ones. The cost function can be described by the RMSE (Root Mean Square Errors),(5)RMSE=i=121gm=1g(ci(zm)cical(zm))2where ci(zm) and cical(zm) are the experimental and calculated concentration data at the position of zm. g is the totally number of experimental data. During each step of calculation, the thermodynamical reliability is validated by thermodynamical constraint equations [17]. A detailed description of the numerical inverse approach is available in Ref. [13].
数值反演法通过最小化计算浓度与实验浓度之间的差值来处理正演问题。成本函数可以用 RMSE(均方根误差)来描述, (5) 其中 位置处的实验浓度数据和计算浓度数据。4# 是实验数据的总数。在每一步计算中,热力学可靠性都要通过热力学约束方程来验证 [17]。关于数值反演方法的详细描述,请参阅参考文献[13]。[13].

3.3. Atomic mobility model
3.3.原子迁移率模型

Based on the theory of Jönsson [20] and Andersson and Ågren [8], the atomic mobility Mk of element k can be evaluated by the following function,(6)Mk=exp(RTlnMk0RT)exp(QkRT)1RTwhere Mk0 is frequency factor and Qk is activation energy. Generally, Mk0 and Qk can be combined into the mobility parameters Φk, thus can be given as,(7)Φk=Qk+RTlnMk0
根据 Jönsson [20] 和 Andersson 与 Ågren [8] 的理论,元素 的原子迁移率 可用以下函数 (6) 评估,其中 是频率因子, 是活化能。一般来说, 可以合并成迁移率参数 ,因此可以表示为: (7)

On the basis of the CALPHAD (CALculation of PHAse Diagrams) method, Φk can be descripted by the Redlich-Kister polynomial [21],(8)Φk=iciΦki+ij>icicj[rmΦki,jr(cicj)r]+ij>il>jcicjcl[svijlsΦki,j,ls];(s=i,jorl)vijls=cs+(1cicjcl)/3(s=i,jorl)where ci, cj, cl and cs denote the component of elements i, j, l and s, respectively. Φki is the value of Φk for pure i, Φki,jr and Φki,j,kr are the interaction parameters to be evaluated. The relationship between interdiffusion coefficient D˜ijn and the atomic mobility are connected through Eq. (4).
在 CALPHAD(PHAse Diagrams 的计算)方法的基础上, 可以用 Redlich-Kister 多项式[21]来描述, (8) 其中 , , 分别表示元素 , , 的分量。 是纯 i 的 值, 是要评估的相互作用参数。相互扩散系数 与原子迁移率之间的关系通过公式 (4) 来表示。

4. Results and discussion
4.结果和讨论

4.1. Diffusivity matrices determination
4.1.确定扩散系数矩阵

4.1.1. Diffusivity matrices determination by the Matano-Kirkaldy method
4.1.1.用 Matano-Kirkaldy 方法确定扩散系数矩阵

It is believed that the Matano-Kirkaldy method can be used to accurately determine the ternary interdiffusivities at the intersection composition of the diffusion paths of two diffusion couples [12]. In the present work, the determination is carried out through the following steps. First, the compositional profiles of diffusion couples are measured by EPMA technique. Next, the Boltzmann functions are employed to fit the compositional profiles of solute elements Cu and Mn. Finally, the diffusivity matrix is determined by the Matano-Kirkaldy method (version 1) using fitted curves. Totally 12 intersection points are obtained from the diffusion paths of the present 12 diffusion couples at three temperatures. The interdiffusivity matrices determined by the Matano-Kirkaldy method and their corresponding SDs calculated by the error propagation method [18] are listed in Table 2. Noted that all the interdiffusivity matrices in Table 2 can satisfy the thermodynamic constraint functions [17], which indicates that the diffusivity matrices of version 1 are thermodynamically stable. As shown in Table 2, the SD of the main interdiffusivities ranges from 10% to 49%, while the SD of the cross interdiffusivities varies from 13 to 98%.
一般认为,Matano-Kirkaldy 方法可用于精确测定两个扩散偶的扩散路径交叉组成处的三元互扩散系数[12]。本研究通过以下步骤进行测定。首先,利用 EPMA 技术测量扩散偶的成分剖面。然后,利用玻尔兹曼函数拟合溶质元素 Cu 和 Mn 的成分曲线。最后,利用拟合曲线通过 Matano-Kirkaldy 方法(版本 1)确定扩散系数矩阵。在三个温度下,从目前 12 个扩散偶的扩散路径中总共得到了 12 个交叉点。表 2 列出了用 Matano-Kirkaldy 方法确定的互扩散系数矩阵以及用误差传播法[18]计算的相应标度。注意到表 2 中的所有间扩散系数矩阵都能满足热力学约束函数[17],这表明版本 1 的扩散系数矩阵在热力学上是稳定的。如表 2 所示,主扩散系数的 SD 值在 10% 到 49% 之间,而交叉扩散系数的 SD 值在 13% 到 98% 之间。

Table 2. Diffusivity matrices at the intersection composition and corresponding SD in fcc Co–Cu–Mn alloys determined by the traditional Matano-Kirkaldy method (version 1) and numerical inverse approach (version 2).
表 2.用传统的 Matano-Kirkaldy 方法(版本 1)和数值反演方法(版本 2)确定的 fcc Co-Cu-Mn 合金中交点成分的扩散系数矩阵和相应的 SD。

Diffusion couples 扩散耦合Intersection comp. (at.%)
交叉路口复合值 (at.%)
Interdiffusion coefficient (×10-15m2/s)
相互扩散系数
Methods
CuMnD˜CuCuCo (SD)D˜CuMnCo (SD)D˜MnCuCo (SD)D˜MnMnCo (SD)
A1/A40.692.162.90 (0.48)
2.85
−0.56 (0.31) -0.56 (0.31)
−0.05 -0.05
0.84 (0.13)
−0.12 -0.12
4.55 (0.69)
4.46
Version 1 版本 1
Version 2 第 2 版
A2/A33.500.992.03 (0.47)
2.28
−0.81 (0.77) -0.81 (0.77)
−0.05 -0.05
−0.40 (0.21) -0.40 (0.21)
−0.16 -0.16
5.80 (1.32)
4.48
Version 1 版本 1
Version 2 第 2 版
A2/A40.762.282.59 (0.45)
2.68
−0.44 (0.41) -0.44 (0.41)
−0.10 -0.10
−0.30 (0.29) -0.30 (0.29)
−0.10 -0.10
4.64 (0.91)
3.82
Version 1 版本 1
Version 2 第 2 版
A3/A41.794.562.46 (0.24)
2.33
−0.32 (0.21) -0.32 (0.21)
−0.08 -0.08
−0.04 (0.03) -0.04 (0.03)
−0.17 -0.17
4.33 (0.62)
3.84
Version 1 版本 1
Version 2 第 2 版
B1/B40.612.656.36 (3.13)
6.51
−0.09 (0.06) -0.09 (0.06)
−0.09 -0.09
0.42 (0.41)
−0.21 -0.21
9.01 (1.73)
9.52
Version 1 版本 1
Version 2 第 2 版
B2/B33.391.995.63 (1.13)
5.36
−0.25 (0.13) -0.25 (0.13)
−0.28 -0.28
−0.12 (0.11) -0.12 (0.11)
−0.01 -0.01
6.94 (1.25)
5.90
Version 1 版本 1
Version 2 第 2 版
B2/B40.753.257.10 (0.87)
6.28
−0.30 (0.23) -0.30 (0.23)
−0.01 -0.01
−0.04 (0.01) -0.04 (0.01)
−0.28 -0.28
8.43 (1.88)
9.48
Version 1 版本 1
Version 2 第 2 版
B3/B41.435.774.63 (0.69)
5.28
−0.32 (0.04) -0.32 (0.04)
−0.03 -0.03
1.52 (1.31)
−0.28 -0.28
7.71 (1.99)
9.10
Version 1 版本 1
Version 2 第 2 版
C1/C40.762.7113.88 (1.56)
14.23
−0.72 (0.61) -0.72 (0.61)
−0.15 -0.15
4.24 (0.73)
−0.32 -0.32
16.82 (2.47)
18.75
Version 1 版本 1
Version 2 第 2 版
C2/C32.821.7810.05 (2.11)
12.26
−2.52 (1.56) -2.52 (1.56)
−0.40 -0.40
2.89 (2.28)
−0.23 -0.23
19.70 (7.01)
12.92
Version 1 版本 1
Version 2 第 2 版
C2/C40.852.9513.02 (4.68)
13.82
−0.70 (0.42) -0.70 (0.42)
−0.15 -0.15
−0.65 (0.62) -0.65 (0.62)
−0.32 -0.32
17.52 (2.72)
18.34
Version 1 版本 1
Version 2 第 2 版
C3/C41.374.6412.52 (2.16)
12.05
−0.68 (0.45) -0.68 (0.45)
−0.15 -0.15
−0.74 (0.69) -0.74 (0.69)
−0.48 -0.48
16.18 (2.52)
18.01
Version 1 版本 1
Version 2 第 2 版

SD = standard deviation, which was evaluated using the method considering the error propagation method [18].
SD = 标准偏差,采用误差传播法进行评估 [18]。

4.1.2. Diffusivity matrices determination by the numerical inverse approach
4.1.2.用数值反演法确定扩散矩阵

The thermodynamic parameters from Wang et al. [9] are used to provide thermodynamic factors in the fcc Co–Cu–Mn system. The numerical inverse approach uses the compositional profiles measured by EPMA as input. During the diffusion coefficients determination, the corresponding atomic mobility parameters are simultaneously extracted. The interdiffusivity matrices at 1373, 1423 and 1473 K along the whole diffusion path extracted by the numerical inverse approach (version 2, denoted in lines) are displayed in Fig. 1, Fig. 2, Fig. 3, the interdiffusivity matrices of version 1 denoted in symbols are superimposed.
Wang 等人[9]的热力学参数用于提供 fcc Co-Cu-Mn 系统的热力学系数。数值反演方法使用 EPMA 测量的成分剖面作为输入。在确定扩散系数的过程中,同时提取相应的原子迁移率参数。图 1、图 2 和图 3 显示了数值反演法(版本 2,用线条表示)提取的 1373、1423 和 1473 K 时沿整个扩散路径的扩散系数矩阵,图中叠加了用符号表示的版本 1 的扩散系数矩阵。

Fig. 1
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Fig. 1. The interdiffusivity vs diffusion distance determined by the Matano-Kirkaldy method (version 1, denoted in symbols) and numerical inverse method (version 2, denoted in lines) for diffusion couples A1-A4 annealed at 1373 K for 58 h.
图 1.在 1373 K 下退火 58 小时的扩散偶 A1-A4 通过 Matano-Kirkaldy 方法(版本 1,用符号表示)和数值反演方法(版本 2,用线条表示)测定的互扩散率与扩散距离的关系。

Fig. 2
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Fig. 2. The interdiffusivity vs diffusion distance determined by the Matano-Kirkaldy method (version 1, denoted in symbols) and numerical inverse method (version 2, denoted in lines) for diffusion couples B1–B4 annealed at 1423 K for 54 h.
图 2.在 1423 K 下退火 54 小时的扩散偶 B1-B4 用 Matano-Kirkaldy 方法(版本 1,用符号表示)和数值逆方法(版本 2,用线条表示)测定的相互扩散率与扩散距离的关系。

Fig. 3
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Fig. 3. The interdiffusivity vs diffusion distance determined by the Matano-Kirkaldy method (version 1, denoted in symbols) and numerical inverse method (version 2, denoted in lines) for diffusion couples C1–C4 annealed at 1473 K for 48 h.
图 3.在 1473 K 下退火 48 小时的扩散偶 C1-C4 用 Matano-Kirkaldy 方法(版本 1,用符号表示)和数值反演方法(版本 2,用线条表示)测定的相互扩散率与扩散距离的关系。

4.1.3. Comparison between the Matano-Kirkaldy method and numerical inverse approach
4.1.3.Matano-Kirkaldy 方法与数值反演法的比较

The diffusivities at the intersection compositions extracted by the numerical inverse approach are listed in Table 2 for a direct comparison. As displayed in Table 2, all the main interdiffusivities extracted by the numerical inverse approach are within the experimental uncertainties of accurate ones determined by the Matano-Kirkaldy method. Besides, most of the signs (19/24) of cross-diffusion coefficients determined from the numerical inverse approach are consistent with that from the Matano-Kirkaldy method. According to the work of Du et al. [13], the diffusivity matrices determined by the numerical inverse approach methods are reliable.
表 2 列出了数值反演法提取的成分间扩散系数,以便进行直接比较。如表 2 所示,数值反演法提取的所有主要交叉扩散系数都在 Matano-Kirkaldy 方法确定的精确交叉扩散系数的实验不确定性范围之内。此外,数值反演法确定的大部分交叉扩散系数的符号(19/24)与 Matano-Kirkaldy 方法确定的交叉扩散系数的符号一致。根据 Du 等人的研究[13],数值反演方法确定的扩散系数矩阵是可靠的。

As listed in Table 2, the diffusivity at each intersection points of diffusion paths determined by the Matano-Kirkaldy method in version 1 exhibit that the value of D˜MnMnCo is larger than D˜CuCuCo. The diffusivities extracted by the numerical inverse approach in version 2 show that most of D˜MnMnCo are larger than D˜CuCuCo at the same composition. However, D˜MnMnCo are smaller than D˜CuCuCo at the right side of the diffusion couples A2, B1–B3 and D1-D3 displayed in Fig. 1, Fig. 2, Fig. 3. The numerical inverse approach can effectively capture the diffusion coefficients along the whole diffusion path and provide more information on the composition-dependent diffusivities than the Matano-Kirkaldy method.
如表 2 所示,第 1 版中用 Matano-Kirkaldy 方法确定的扩散路径各交点的扩散率表明, 的值大于 。版本 2 中通过数值反演法提取的扩散率显示,在相同成分下,大部分 的值都大于 。然而,在图 1、图 2 和图 3 所示的扩散耦合 A2、B1-B3 和 D1-D3 的右侧, 小于 。与 Matano-Kirkaldy 方法相比,数值反演方法可以有效地捕捉整个扩散路径上的扩散系数,并提供更多有关成分依赖性扩散系数的信息。

4.2. Atomic mobility assessment
4.2.原子迁移率评估

The kinetic parameters of fcc Co–Cu [22], Co–Mn [23] and Cu–Mn alloys [24,25] have been assessed in the literature. The parameters of fcc Co–Cu alloys from Liu et al. [22] are directly adopted in this work. Recently, the atomic mobility parameters of fcc Cu–Mn alloys were updated by the present authors [26], considering that the more reliable self-diffusivity of fcc-Mn calculated by first-principle calculations [27] than the result calculated by empirical analysis [28]. For the same reason, the atomic mobilities for the fcc Co–Mn system are re-assessed in this work. The impurity diffusivity of Co in the metastable fcc-Mn is assumed to be equal to that of self-diffusivity fcc-Mn calculated by first-principles calculations [27], according to our previous experiences [13,15]. The impurity diffusivity of Mn in fcc-Co assessed in Ref. [23] is adopted. The binary interaction parameters are re-assessed based on the experimental data from Refs. [23,29,30]. The re-assessed atomic mobilities for fcc Co–Mn alloys are listed in Table 3. Fig. 4 exhibits the calculated interdiffusivities in fcc Co–Mn alloys using the mobility parameters obtained in the present work and our previous work [23], compared with the determined ones [23,29,30]. As displayed in Fig. 4, the calculated interdiffusivities show good consistency with the experimental ones, indicating that the presently obtained atomic mobilities for fcc Co–Mn alloys are reliable. Noted that the calculated interdiffusivities of fcc Co–Mn alloys at the temperature ranges from 1073 to 1323 K have some kinks due to the effects of magnetic-ordering across the curie points. The same phenomenon can also be found in the fcc Co–Al [31] and Co–W [32] systems.
文献中评估了 fcc Co-Cu[22]、Co-Mn[23] 和 Cu-Mn 合金[24,25]的动力学参数。本研究直接采用了 Liu 等人[22]的 fcc Co-Cu 合金参数。最近,本文作者更新了 fcc 铜锰合金的原子迁移率参数[26],认为第一原理计算[27]计算出的 fcc 锰的自扩散性比经验分析[28]计算出的结果更可靠。出于同样的原因,本文对 fcc Co-Mn 体系的原子迁移率进行了重新评估。根据我们以前的经验[13,15],假定钴在可蜕变的 fcc-Mn 中的杂质扩散率等于第一性原理计算得出的 fcc-Mn 的自扩散率[27]。采用了参考文献[23]中评估的 Mn 在 fcc-Co 中的杂质扩散率。[23] 中评估的 Mn 在 fcc-Co 中的杂质扩散率。二元相互作用参数是根据参考文献 [23,29,30] 中的实验数据重新评估的。表 3 列出了重新评估的 fcc Co-Mn 合金的原子迁移率。图 4 展示了使用本研究和我们之前的研究 [23] 所获得的迁移率参数计算出的 fcc Co-Mn 合金中的相互扩散率,并与文献 [23,29,30] 中测定的相互扩散率进行了比较。如图 4 所示,计算的间扩散系数与实验的间扩散系数显示出良好的一致性,表明目前获得的 fcc Co-Mn 合金原子迁移率是可靠的。注意到在 1073 至 1323 K 温度范围内,ccc Co-Mn 合金的计算间扩散率由于跨居里点的磁排序效应而出现了一些扭结。同样的现象也出现在 fcc Co-Al [31] 和 Co-W [32] 系统中。

Table 3. Atomic mobility parameters for fcc Co–Cu–Mn alloys obtained in the present work as well as those from the literature [[22], [23], [24],26,27] (all in SI units).
表 3.本研究以及文献[[22], [23], [24],26,27] 中获得的 fcc Co-Cu-Mn 合金的原子迁移率参数(单位均为国际单位)。(均为国际单位制)。

MobilityParametersRef.
Mobility of Cu 铜的流动性ΦCuCu=205872.082.52×T[22]
ΦCuCo=27501376.57×T[22]
ΦCuMn=25012585.3×T[24]
ΦCuCu,Co=89735.64[22]
ΦCuCu,Mn=302249.1113.05×T[26]
ΦCuCo,Mn=113099.2This work (version 1) 本作品(第 1 版)
ΦCuCo,Mn=84276.4This work (version 2) 本作品(第 2 版)
Mobility of Co Co 的流动性ΦCoCu=215759.2980.35×T[22]
ΦCoCo=296542.974.48×T[22]
ΦCoMn=31260053.62×TThis work
ΦCoCo,Mn=229192.743.69×TThis work
ΦCoCo,Mn1=151896.67This work
ΦCoCu,Co=117927.63[22]
Mobility of Mn 锰的迁移率ΦMnCu=193085.3281.86×T[24]
ΦMnCo=234008.6103.4×T[23]
ΦMnMn=31260053.62×T[27]
ΦMnCu,Mn=249619.8459.18×T[26]
ΦMnCo,Mn=381817.95+253.42×TThis work
ΦMnCo,Cu=135679.35This work (version 1) 本作品(第 1 版)
ΦMnCo,Cu=302641.97This work (version 2) 本作品(第 2 版)
Fig. 4
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Fig. 4. Comparison between the calculated composition-dependent interdiffusion coefficients of fcc Co–Mn alloys using the kinetic parameters in this work (denoted in solid line) and experimental data [[23], [29], [30]]. The calculated results (denoted in dash line) based on our previous assessment [23] are also superimposed.
图 4.使用本研究中的动力学参数计算得出的 fcc Co-Mn 合金随成分变化的相互扩散系数(实线表示)与实验数据[[23], [29], [30]]的比较。根据我们之前的评估[23]得出的计算结果(用虚线表示)也叠加在一起。

Coupled with the atomic mobilities of the three binary subsystems mentioned above, and thermodynamic description of Ref. [9], two versions of atomic mobility parameters for the fcc Co–Cu–Mn system are assessed based on the interdiffusivity matrices of version 1 and version 2, respectively. The kinetic parameters for the fcc Co–Cu–Mn system are summarized in Table 3.
结合上述三个二元子系统的原子迁移率和参考文献[9]的热力学描述,根据版本 1 和版本 2 的相互扩散矩阵,分别评估了 fcc Co-Cu-Mn 系统的两个版本的原子迁移率参数。[9],根据版本 1 和版本 2 的相互扩散矩阵,分别评估了 fcc Co-Cu-Mn 体系的两个版本的原子迁移率参数。表 3 总结了 fcc Co-Cu-Mn 体系的动力学参数。

In order to validate the quality of the two versions of atomic mobility parameters, the diffusion behaviors are model-predicted in CALTPP by the two versions of atomic mobility parameters, and compared with the corresponding experimental data. A comparison between the main interdiffusion coefficients calculated by atomic mobilities of version 1 and version 2 together with the ones determined by the Matano-Kirkaldy method at three temperatures is illustrated in Fig. 5. Open symbols represent the diffusion coefficients calculated by the atomic mobilities of version 1, and those calculated by version 2 are denoted in closed symbols. The diagonal line represents the calculated interdiffusion coefficients show perfect agreement with the experimental ones. The dash lines represent deviations between the calculated and experimental diffusion coefficients with factors of 2 or 0.5, which is a generally accepted experimental error for the interdiffusivities in the ternary system. As exhibited in Fig. 5, all of the calculated results are within the experimental error, which indicates the presently assessed two versions of atomic mobilities are reliable. Besides, compared with the open symbols, the closed symbols are closer to the diagonal line, indicating that the interdiffusivities calculated by the atomic mobilities of version 2 show better agreement with the experimentally measured ones than version 1.
为了验证两个版本的原子迁移率参数的质量,在 CALTPP 中用两个版本的原子迁移率参数对扩散行为进行了模型预测,并与相应的实验数据进行了比较。图 5 展示了在三个温度下,由版本 1 和版本 2 的原子迁移率计算出的主要相互扩散系数与由 Matano-Kirkaldy 方法确定的系数之间的比较。开口符号表示根据版本 1 原子迁移率计算的扩散系数,闭口符号表示根据版本 2 计算的扩散系数。对角线代表计算出的相互扩散系数与实验数据完全一致。虚线表示计算扩散系数与实验扩散系数之间存在 2 或 0.5 倍的偏差,这是三元体系中公认的相互扩散系数实验误差。如图 5 所示,所有的计算结果都在实验误差范围之内,这表明目前评估的两个版本的原子迁移率是可靠的。此外,与开放式符号相比,封闭式符号更接近对角线,这表明与版本 1 相比,版本 2 的原子迁移率计算出的互电阻与实验测量出的互电阻更吻合。

Fig. 5
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Fig. 5. Comparison between the main diffusion coefficients calculated by the atomic mobilities of version 1 and the ones determined by the Matano-Kirkaldy method in the fcc Co–Cu–Mn system at 1373, 1423 and 1473 K. The diagonal line represents the calculated interdiffusion coefficients equal to the experimental ones. Dashed lines refer to a factor of generally accepted experimental error for the measurement of interdiffusivities.
图 5.在 1373、1423 和 1473 K 下的 fcc Co-Cu-Mn 体系中,用版本 1 原子迁移率计算的主要扩散系数与用 Matano-Kirkaldy 方法测定的扩散系数的比较。虚线表示测量间扩散系数时公认的实验误差系数。

Fig. 6, Fig. 7, Fig. 8 present the model-predicted compositional profiles in fcc Co–Cu–Mn alloys, compared with the data measured in the present work at 1373, 1423 and 1473 K, respectively. The solid and dash lines denote the results model-predicted by the atomic mobility parameters of version 1 and version 2, respectively. As be seen in Fig. 6, Fig. 7, Fig. 8, the model-predicted concentration profiles using the present two versions of atomic mobilities agree with the measured ones well. The further analysis of the compositional profiles of Mn near the interface of diffusion couples in Fig. 7 (c) and (d), Fig. 8 (a), (c) and (d) demonstrate that the compositional profiles of diffusion couples B3, B4, C1, C3 and C4 predicted by the atomic mobility parameters of version 2 show better consistent with the measured ones than version 1. Moreover, the RMSEs calculated by Eq. (5) of version 1 and version 2 are 0.001789 and 0.001443, respectively, which further indicates that the mobilities of version 2 can better reproduce the measured data. The comparisons of the model-predicted diffusion fluxes at 1373, 1423 and 1473 K with the experimental data are exhibited in supplementary Fig. S1-S3, respectively. It can be more clearly reflected in the better agreement between the results of the version 2 atomic mobility predictions and the experimental data. It should be noted that the experimental diffusion fluxes are calculated from the fitted compositional profiles. As exhibited in supplementary Figs. S1-S3, almost all of the results predicted by atomic mobilities of version 2 show a higher degree of fitting with experimental data (except for the diffusion couple A4) than the diffusion fluxes predicted by the parameters of version 1.
图 6、图 7 和图 8 分别显示了在 1373、1423 和 1473 K 下,ccc Co-Cu-Mn 合金中模型预测的成分分布,并与本研究中测量到的数据进行了对比。实线和虚线分别表示根据版本 1 和版本 2 的原子迁移率参数模型预测的结果。从图 6、图 7 和图 8 中可以看出,使用这两个版本的原子迁移率参数进行模型预测的浓度曲线与测量结果非常吻合。进一步分析图 7 (c) 和 (d)、图 8 (a)、(c) 和 (d)中扩散耦合界面附近的锰组成剖面可以发现,用版本 2 的原子迁移率参数预测的扩散耦合界面 B3、B4、C1、C3 和 C4 的锰组成剖面与实测值的一致性要好于版本 1。此外,根据公式(5)计算出的版本 1 和版本 2 的均方根误差分别为 0.001789 和 0.001443,这进一步表明版本 2 的迁移率能更好地再现测量数据。模型预测的 1373、1423 和 1473 K 时的扩散通量与实验数据的比较分别见补充图 S1-S3。第 2 版原子迁移率的预测结果与实验数据的一致性更好,这一点可以更清楚地反映出来。需要注意的是,实验中的扩散通量是根据拟合的成分曲线计算得出的。如补充图 S1-S3 所示,与版本 1 的参数预测的扩散通量相比,版本 2 的原子迁移率预测的几乎所有结果(扩散偶 A4 除外)与实验数据的拟合程度都更高。

Fig. 6
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Fig. 6. Model-predicted compositional profiles using the mobility parameters of version 1 (denoted in solid line) and version 2 (denoted in dash line) in diffusion couples A1-A4 annealed at 1373 K for 58 h, compared with the measured data in the present work.
图 6.在 1373 K 下退火 58 小时的扩散偶 A1-A4 中,使用版本 1(实线表示)和版本 2(虚线表示)的迁移率参数建立的模型预测成分曲线,与本研究中的测量数据进行比较。

Fig. 7
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Fig. 7. Model-predicted compositional profiles using the mobility parameters of version 1 (denoted in solid line) and version 2 (denoted in dash line) in diffusion couples B1–B4 annealed at 1423 K for 54 h, compared with the measured data in the present work.
图 7.在 1423 K 下退火 54 h 的扩散偶 B1-B4 中,使用版本 1(实线表示)和版本 2(虚线表示)的迁移率参数建立的模型预测成分曲线,与本研究中的测量数据进行比较。

Fig. 8
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Fig. 8. Model-predicted compositional profiles using the mobility parameters of version 1 (denoted in solid line) and version 2 (denoted in dash line) in diffusion couples C1–C4 annealed at 1473 K for 54 h, compared with the measured data in the present work.
图 8.在 1473 K 下退火 54 h 的扩散偶 C1-C4 中,使用版本 1(实线表示)和版本 2(虚线表示)的迁移率参数建立的模型预测成分曲线,与本研究中的测量数据进行比较。

Fig. 9 presents the comparisons between the diffusion paths simulated using the two versions of atomic mobilities and the measured ones at 1373, 1423 and 1473 K in fcc Co–Cu–Mn alloys. A generally good agreement is found in all these comparisons. Besides, the diffusion paths of diffusion couples A1, B2, C2 and C3 model-predicted by version 2 can better reproduce the measured ones than the results predicted by mobility parameters of version 1. It can be concluded that all the two versions of atomic mobility parameters are reliable and the mobility parameters of version 2 can better describe the diffusion behaviors in the fcc Co–Cu–Mn system.
图 9 显示了在 1373、1423 和 1473 K 下,在ccc Co-Cu-Mn 合金中使用两种原子迁移率模拟的扩散路径与测量值之间的比较。在所有这些比较中发现,两者的一致性总体良好。此外,第 2 版模型预测的扩散偶 A1、B2、C2 和 C3 的扩散路径比第 1 版迁移率参数预测的结果更好地再现了测量结果。可以得出结论:所有两个版本的原子迁移率参数都是可靠的,版本 2 的迁移率参数能更好地描述 fcc Co-Cu-Mn 体系中的扩散行为。

Fig. 9
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Fig. 9. Comparison between the model-predicted diffusion paths of diffusion couples by the mobility parameters of version 1 (denoted in solid line) and version 2 (denoted in dash line) with the measured ones in diffusion couples annealed at (a) 1373 K for 58 h, (b) 1423 K for 54 h and (c) 1473 K for 48 h.
图 9.用第 1 版(实线表示)和第 2 版(虚线表示)的迁移率参数模型预测的扩散偶的扩散路径与在 (a) 1373 K 退火 58 小时、(b) 1423 K 退火 54 小时和 (c) 1473 K 退火 48 小时条件下测量的扩散偶的扩散路径的比较。

4.3. Three-dimensional diffusion coefficients
4.3.三维扩散系数

Moreover, based on the atomic mobility parameters of version 2 and the thermodynamic parameters [9] for fcc Co–Cu–Mn alloys, the 3-D surfaces for the main interdiffusion coefficients D˜CuCuCo and D˜MnMnCo at 1373, 1423 and 1473 K are presented in Fig. 10 (a) and (b), respectively. As illustrated in Fig. 10, both the values of D˜CuCuCo and D˜MnMnCo decrease with the increasing composition of Cu. The value of D˜CuCuCo varies slightly with the concentration of Mn while D˜MnMnCo increases with the increasing composition of Mn. In addition, the main interdiffusion coefficients D˜CuCuCo in the fcc Co–Cu–Ni and Co–Cu–Fe systems are calculated based the assessed atomic mobilities in the literature [33,34]. As shown in Fig. 11 (a) and (b), the main interdiffusion coefficients of D˜CuCuCo in the fcc Co–Cu–Fe [33] and Co–Cu–Ni [34] systems decrease with the increasing composition of Cu, while increase with the increasing of third element (Ni or Fe). In comparison with Fig. 10 (a), we can see that the variation tendency of D˜CuCuCo along composition Cu is similar in Co–Cu–Fe, Co–Cu–Ni and Co–Cu–Mn systems. While due to the chemical difference of the third elements, the variation tendency of D˜CuCuCo along composition of the third element (Mn, Ni or Fe) has some deviations in the three systems. Therefore, we suppose that there is no universal variation tendency of interdiffusivities along the composition.
此外,根据版本 2 的原子迁移率参数和 fcc Co-Cu-Mn 合金的热力学参数[9],图 10(a)和(b)分别给出了 1373、1423 和 1473 K 时主要相互扩散系数 的三维表面。如图 10 所示,随着铜成分的增加, 的值都会降低。 的值随锰的浓度变化而略有不同,而 则随锰成分的增加而增大。此外,根据文献[33,34]中评估的原子迁移率,计算出了 fcc Co-Cu-Ni 和 Co-Cu-Fe 体系中的主要相互扩散系数 。如图 11 (a) 和 (b) 所示,ffc Co-Cu-Fe [33] 和 Co-Cu-Ni [34] 体系中的主要相互扩散系数 随着铜成分的增加而减小,而随着第三元素(镍或铁)的增加而增大。与图 10(a)相比,我们可以发现在 Co-Cu-Fe、Co-Cu-Ni 和 Co-Cu-Mn 体系中, 随 Cu 成分的变化趋势相似。而由于第三元素的化学性质不同, 随第三元素(锰、镍或铁)组成的变化趋势在三个体系中存在一定的偏差。因此,我们认为相互电阻率并不存在普遍的成分变化趋势。

Fig. 10
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Fig. 10. Composition-dependent main interdiffusivities at temperature range from 1373 to 1473 K, symbols are the interdiffusivities determined by the Matano-Kirkaldy method, surfaces are the interdiffusivities determined by the numerical inverse approach: (a) D˜CuCuCo and (b) D˜MnMnCo.
图 10.在 1373 至 1473 K 温度范围内与成分有关的主要间离度,符号为通过 Matano-Kirkaldy 方法确定的间离度,面为通过数值反演方法确定的间离度:(a) 和(b)

Fig. 11
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Fig. 11. Composition-dependent main interdiffusion planes for D˜CuCuCo in the fcc (a) Co–Cu–Fe [33] and Co–Cu–Ni [34] systems at 1373, 1423 and 1473 K.
图 11.在 1373、1423 和 1473 K 下,fcc (a) Co-Cu-Fe [33] 和 Co-Cu-Ni [34] 体系中 的主要相互扩散平面与成分有关。

The obtained 3-D main interdiffusivities are used to derive the pre-exponential factors D0i and activation energies QiiCo for elements i (i = Cu or Mn) in terms of the Arrhenius equation [35]:(9)D˜iiCo=D0iexp(QiiCoRT)
根据阿伦尼乌斯方程 [35],得到的三维主要间扩散率可用于推导元素 i(i = 铜或锰)的前指数 和活化能 (9)

Fig. 12, Fig. 13 display the composition-dependent QiiCo and D0i for elements i, respectively. As illustrated in Fig. 12, both the values of QCuCuCo and QMnMnCo increase with the increasing composition of Cu and Mn. The value of D0Cu decrease with the increasing composition of Mn while D0Mn increases with the increasing composition of Mn, as shown in Fig. 13.
图 12 和图 13 分别显示了元素 i 随成分变化的 。如图 12 所示,随着铜和锰成分的增加, 的值都会增加。如图 13 所示, 的值随 Mn 成分的增加而减小,而 则随 Mn 成分的增加而增大。

Fig. 12
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Fig. 12. Composition-dependent activation energies in the fcc Co–Cu–Mn system (a) QCuCuCo and (b) QMnMnCo.The lines are isopleths for activation energy.
图 12.fcc Co-Cu-Mn 体系(a) 和(b) 中随成分变化的活化能。

Fig. 13
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Fig. 13. Composition-dependent frequency factors in the fcc Co–Cu–Mn system (a) D0Cu and (b) D0Mn.The lines are isopleths for frequency factor.
图 13.fcc Co-Cu-Mn 体系(a) 和(b) 中与成分有关的频率因数。

5. Conclusions 5.结论

Totally twelve singe-phase diffusion couples in fcc Co–Cu–Mn alloys are prepared at 1373, 1423 and 1473 K, and their compositional profiles are measured by EPMA technique.
在 1373、1423 和 1473 K 下制备了共钴锰合金中的十二个单相扩散偶,并利用 EPMA 技术测量了它们的成分分布。

Version 1 and version 2 of the interdiffusivity matrices are determined by the Matano-Kirkaldy method and numerical inverse approach, respectively, and the corresponding versions of atomic mobilities are extracted in CALTPP. The good agreement between the diffusivities of version 2 and the accurate ones in version 1 demonstrates the reliability of diffusivities extracted by the numerical inverse approach.
相互扩散系数矩阵的版本 1 和版本 2 分别由 Matano-Kirkaldy 方法和数值反演方法确定,并在 CALTPP 中提取相应版本的原子迁移率。版本 2 的扩散系数与版本 1 的精确扩散系数之间的良好一致性证明了数值反演法提取的扩散系数的可靠性。

The numerical inverse can determine the diffusivities more efficiently and give more composition-dependent diffusivities. Besides, the diffusion behaviors predicted by the atomic mobilities of version 2 show a better agreement with experimental results than version 1. The atomic mobilities of version 2 are recommended in the present work.
数值反演可以更有效地确定扩散系数,并给出更多依赖于成分的扩散系数。此外,与版本 1 相比,版本 2 的原子迁移率预测的扩散行为与实验结果的一致性更好。本研究推荐使用版本 2 的原子迁移率。

The 3-D surfaces of the main interdiffusivities, activation energies and frequency-factors indicated that the values of D˜CuCuCo and D˜MnMnCo decrease with the increasing concentration of Cu. Both the values of QCuCuCo and QMnMnCo increase with the increasing composition of Cu and Mn. The value of D0Cu decreases with the increasing composition of Mn, while D0Mn increases with the increasing composition of Mn.
主要间离散度、活化能和频率因子的三维表面表明,随着铜浓度的增加, 的值减小。 的值随着铜和锰成分的增加而增加。 的值随 Mn 成分的增加而减小,而 则随 Mn 成分的增加而增大。

Declaration of Competing Interest
竞争利益声明

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
作者声明,他们没有任何可能会影响本文所报告工作的已知经济利益或个人关系。

Acknowledgments 致谢

This work was supported by the National Natural Science Foundation of China (Grant No. 5210011609), the young scholars of National Natural Science Foundation of Hunan Province (Grant No. 2021JJ40749), the Major Science, Technology Project of Precious Metal Materials Genome Engineering Project in Yunnan Province (Grant No. 2019ZE001-1), the Postgraduate Scientific Research Innovation Project of Hunan Province (Grant No. CX20210147) and the Fundamental Research Funds for the Central Universities of Central South University (Grant No. 1053320211417).
本研究得到了国家自然科学基金项目(批准号:5210011609)、湖南省青年学者国家自然科学基金项目(批准号:2021JJ40749)、云南省重大科技专项贵金属材料基因组工程项目(批准号:2019ZE001-1)、湖南省研究生科研创新项目(批准号:CX20210147)和中南大学中央高校基本科研业务费项目(批准号:105105)的资助。2019ZE001-1)、湖南省研究生科研创新项目(批准号:CX20210147)和中南大学中央高校基本科研业务费(批准号:1053320211417)。

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附录 A.补充数据

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References 参考资料