J. Cent. South Univ. (2018) 25: 1573-1581

DOI: https://doi.org/10.1007/s11771-018-3849-x

Derivation and application of concentration formula of alloy genes of DO₃-type in BCC alloy systems

PENG Hong-jian(彭红建)^{1, 2}, LI Ling(李玲)¹, LI Xiao-bo(李小波)², XIE You-qing(谢佑卿)²

1. School of Chemistry and Chemical Engineering, Central South University, Changsha 410083, China;

2. School of Materials Science and Engineering, Central South University, Changsha 410083, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Abstract: Taking Ta–W alloy system as an example, the concentration formulae of alloy genes of DO₃-type alloys in BCC structures were derived on the basis of the theory of alloy genes and the number of coordination atoms. The concentrations of alloy genes of DO₃-TaW₃ and DO₃-Ta₃W ordered alloys were calculated as functions of composition x_W and ordering degree (s). When s=s_max, the concentrations of alloy genes of stoichiometric DO₃-TaW₃ compound are equal to those of alloys, that is,The concentrations of alloy genes of stoichiometric DO₃-Ta₃W compound are also equal to those of alloys, that is,As ordering degree decreases, alloy genes of DO₃-TaW₃ and DO₃-Ta₃W ordered alloys will split. And the degree of splitting of alloy genes increases with the ordering degree decreasing. The atomic states and properties of DO₃-TaW₃ and DO₃-Ta₃W ordered alloys were calculated as a function of composition x_W. The reason was pointed out that preparation of DO₃-TaW₃ and DO₃-Ta₃W intermetallic compounds is difficult due to small differences in their cohesive energies. It will provide theoretical guidance for the scientific designation to new candidate for ultra- high-temperature materials in areo-engine applications.

Key words: Ta–W alloy system; BCC structure; alloy gene

Cite this article as: PENG Hong-jian, LI Ling, LI Xiao-bo, XIE You-qing. Derivation and application of concentration formula of alloy genes of DO₃-type in BCC alloy systems [J]. Journal of Central South University, 2018, 25(7): 1573–1581. DOI: https://doi.org/10.1007/s11771-018-3849-x.

1 Introduction

The target of material science development is to continuously design and develop new materials to meet modernization and future needs. To develop material design by multi-structure levels and various properties, materials science should be systematic. After nearly 30 years of efforts by XIE et al, the framework of systematic science of alloys been had established on the basis of energy band theory, valence bond theory and statistical thermodynamics. The core is the theory of the characteristic crystals and points out direction of alloy design. The theory has been successfully applied to Ag–Cu alloys [1, 2] and Au–Cu alloys [3–6] in FCC structure, Ti–Al alloys with coexistence of FCC and HCP structures [7, 8].

Ta–W alloys show a series of desirable properties, such as high density, high melting point, good heat conductivity, good fracture toughness, excellent corrosion resistance and weldability. So, they are excellent candidates for ultra-high temperature structural materials in areo-engine applications [9–12]. The manufacturing technology for refractory metals is difficult. However, there is little research on alloys in BCC structure by the theory [13, 14]. The purpose of this paper is to choose Ta–W alloys in BCC structure as research subject. The concentration formulae of alloy genes of DO₃-type alloys in BCC structures were derived. The concentrations of alloy genes of DO₃-TaW₃ and DO₃-Ta₃W ordered alloys were calculated as functions of composition x_W and ordering degree (s). The atomic states and properties of DO₃-TaW₃ and DO₃-Ta₃W ordered alloys are calculated as a function of composition x_W. The reason that the preparation of DO₃-TaW₃ and DO₃-Ta₃W intermetallic compounds is difficult according to their cohesive energies analyzed. It will provide theoretical basis for the scientific designation to new candidate for refractory metals in areo-engine applications.

2 Crystal structure of DO₃-TaW₃ alloy

Figure 1 shows the crystal structure of DO₃-TaW₃ alloy. Its space group is Fm3m. Each Ta atom has 8 W2 atoms and each W1 atom also has 8 W2 atoms as the nearest neighbors. But each W2 atom has 4 Ta atoms and 4 W1 atoms.

Figure 1 Crystal structure of DO₃-TaW₃ alloy

The crystalline of DO₃-TaW₃ compound contains sublattices I and II. The numbers of sublattices I and II are represented by N¹ and N² , respectively, then N (=N¹+N²) denotes the total number of sublattices in the unit cell. v¹ and v² respectively represent the position fractions of sublattices I and II.andrepresent the numbers of sublattices I and II by A and B atoms. A and B atoms (A, B are Ta, W, respectively) have priority to occupy sublattices I and II. represents the probability with component j (j=A or B) in the sublattice point i (i= I, II). All the parameters should satisfy Eq. (1).

              (1)

The arrangement of atoms in strict stoichiometric alloys will be destroyed due to elevated temperature or deviation of compositions from stoichiometric ratio. Alloys that deviate from stoichiometric ratio maybe form ordered alloys again. Therefore, we must consider influence of ordering degree and establish the formula between the concentration of alloy genes and composition and ordering degree (s). Only considering the influence of the nearest neighbor atoms, the ordering degree is defined as

                       (2)

According to Eq. (2), we can know as follows: ①When x_A=v¹ andthe alloys are completely stoichiometric and sublattice I is occupied by A atom and ordering degree (s) is equal to 1.② When the compositions deviate from stoichiometric, the maximal ordering degree is

a) s_max=x_A/v¹ for the alloy x_A1;

b) s_max=(1–x_A)/(1–v¹) for the alloy x_A>v¹, that is, x_A1 or x_A>v², the ordering degree is unequal to 1.

The function can be derived as follows according to ordering degree and the definition of atomic occupancy probability:

                     (3)

According to the sublattice point model, various of sublattice occupancy fractions of several types of intermetallic compounds are determined. For AB type, v¹=1/2; for A₃B type, v¹=3/4; for AB₃ type, v¹=1/4.

3 Concentrations and of DO₃- type ordered alloys

3.1 Derivation of concentrations formulae and

According to the sublattice model, the concentration of the ith alloy gene A is equal to the summation of concentrations in sublattices I and II, that is,

                            (4)

whereandrespectively represent concentrations of alloy genes A in sublattices I and II which has i B atoms in its neighbor configuration. andrepresent the occupancy probabilities of alloy genes A in sublattices I and II. and represent the occupancy probabilities of alloy genes B in sublattices I and II. The concentrationof alloy genes A is

           (5)

According to the same method of concentration of alloy genes A, the concentration of alloy genes B is

           (6)

According to Eqs. (5) and (6), the concentrationsandof Ta and W alloy genes of DO₃-TaW₃ ordered alloys as a function of composition x_W are shown in Figure 2. When s=s_max, the distribution characteristics are as follows.

1) When x_W=0.75, alloy is stoichiometric DO₃-TaW₃ compound. There are only three types of alloy genes, that is, The concentrations of alloy genes are equal to the compositions of alloys, that is,

2) When x_W>0.75, andin DO₃-TaW₃ ordered alloys will split into and the concentration of W alloy genes will be while Ta alloy gene will be still the concentration of Ta alloy genes will be

3) When x_W<0.75,in DO₃-TaW₃ ordered alloys will split into the concentration of Ta alloy genes will be and will also split into the concentration of W alloy genes will be

From the above, the alloy genes of splitting are different in rich Ta and rich W DO₃-TaW₃ ordered alloys, which leads to great influence on atomic states, crystal structure parameters and physical properties.

Figure 2 Concentrations (a) and (b) of alloy genes of DO₃-TaW₃ ordered alloys as a function of composition x_W with maximal ordering degree

3.2 Concentrations and of alloy genes of DO₃-TaW₃ ordered alloys as functions of composition x_W and ordering degree (s)

The concentrationsandof alloy genes as functions of composition x_W and ordering degree (s=0.8s_max, s=0.6s_max and s=0) of DO₃-TaW₃ ordered alloy are shown in Figure 3. From Figure 3, when ordering degree decreases, alloy genes and in DO₃-TaW₃ ordered alloys will split intoThe splitting degree of alloy genes will be larger with the ordering degree decreasing. As ordering degree decreases, the concentrations of various alloy genes of ordered alloys are approaching to those of disordered alloys.

4 Concentrationsandof alloy genes of DO₃-Ta₃W ordered alloys as functions of composition x_W and ordering degree (s)

On the basis of Eqs. (5) and (6), the concentrations and of Ta and W alloy genes of DO₃-Ta₃W ordered alloys as a function of composition x_W are shown in Figure 4. When s=s_max, the distribution characteristics are as follows.

Figure 3 Concentrations(a, c, e) and(b, d, f)of alloy genes of DO₃-TaW₃ type ordered alloys as functions of composition x_W and ordering degree (s):

Figure 4 Concentrations (a) and (b) of alloy genes of DO₃-Ta₃W ordered alloys as a function of composition x_W with maximal ordering degree

1) When x_W=0.25, alloy is stoichiometric DO₃-Ta₃W compound. There are only three types of alloy genes, that isandThe concentrations of alloy genes are equal to the compositions of alloy, that is,

2) When x_W<0.25,andof DO₃-TaW₃ ordered alloys with a maximal ordering degree split intoand the concentration of Ta alloy genes is whiledoes not split and the concentration of W alloy genes is

3）When x_W>0.25,atoms of DO₃- TaW₃ ordered alloys with a maximal ordering degree split intoand the concentration of W alloy genes is andare also split into and the concentration of Ta alloy genes is

From the above, the alloy genes of constitute are different between rich Ta and rich W DO₃-TaW₃ ordered alloys, which leads to great influence on atomic state, crystal structure parameters and physical properties.

The concentrationsandof alloy genes as functions of composition x_W and ordering degree (s=0.8s_max, s=0.6s_max and s=0) of DO₃-Ta₃W ordered alloys are shown in Figures 5. From Figure 5, when ordering degree decreases, alloy genes and in DO₃-Ta₃W ordered alloys will split intoThe splitting degree of alloy genes will be larger with the ordering degree decreasing. As ordering degree decreases, the concentrations of various alloy genes of ordered alloys are approaching to those of disordered alloys.

5 Atomic states and properties of DO₃- TaW₃ and DO₃-Ta₃W ordered alloys as a function of composition x_W

The properties of disordered and ordered alloys are described by average properties of components. The properties of disordered alloys are obtained by the additive law of alloy genes. Therefore, the atomic states and properties of DO₃-TaW₃ and Ta₃W ordered alloys as a function of composition x_W are listed in Tables 1 and 2. The variation trend of properties of DO₃-TaW₃ and Ta₃W ordered alloys is almost the same as that of atomic states. Cohesive energies of disordered Ta_1–xW_x alloys DO₃-Ta₃W and DO₃-TaW₃ ordered alloys as functions of composition x_W are shown in Figure 6. From Figure 6, we can know that the differences of cohesive energies among disordered Ta_1–xW_x alloys, DO₃-Ta₃W and DO₃-TaW₃ ordered alloys are very small. Therefore, it is difficult to prepare DO₃-Ta₃W and DO₃-TaW₃ intermetallic compounds. The results are excellent agreement with experimental results. That will provide theoretical guidance for preparing Ta–W alloys materials.

Figure 5 Concentrations(a, c, e) and (b, d, f) of alloy genes of DO₃-Ta₃W ordered alloys as functions of composition x_W and ordering degree (s):

6 Conclusions

1) Taking Ta–W alloy system as an example, the concentration formulae of alloy genes of DO₃-type alloys in BCC structures were derived on the basis of the theory of alloy genes and the number of coordination atoms. The concentrations of alloy genes of DO₃-TaW₃ and DO₃-Ta₃W ordered alloys were calculated as functions of composition x_W and ordering degree (s).

2) When s=s_max, the concentrations of alloy genes of stoichiometric DO₃-TaW₃ compound are equal to those of alloys, that is,The concentrations of alloy genes of stoichiometric DO₃-Ta₃W compound are also equal to those of alloys, that is, 0.25,As ordering degree decreases, alloy genes of DO₃-TaW₃ and DO₃-Ta₃W ordered alloys will split. And the degree of splitting of alloy genes increases with the ordering degree decreasing.

Table 1 Atomic states and properties of DO₃-TaW₃ ordered alloys as a function of composition x_W

Table 2 Atomic states and properties of DO₃-Ta₃W ordered alloys as a function of composition x_W

Figure 6 Cohesive energies of disordered Ta_1–xW_x alloys, DO₃-Ta₃W and DO₃-TaW₃ ordered alloys as functions of composition x_W

3) The atomic states and properties of DO₃-TaW₃ and DO₃-Ta₃W ordered alloys were calculated as a function of composition x_W. The main reason is that it is difficult to prepare DO₃-Ta₃W and DO₃-TaW₃ intermetallic compounds due to their small differences in cohesive energies. It will provide theoretical guidance for the scientific designation to new candidate for ultra-high- temperature materials in areo-engine applications.

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(Edited by YANG Hua)

中文导读

BCC 合金系中DO₃-型合金基因浓度公式的推导和应用

摘要：本文以Ta–W合金系为例, 根据合金基因理论和配位原子数，推导出了BCC结构中DO₃-型合金基因浓度的公式。计算了DO₃-TaW₃和DO₃-Ta₃W有序合金的基因浓度随成分x_W和有序度(s)的变化。当s=s_max时，化学计量比的DO₃-TaW₃化合物的合金基因浓度等于合金的浓度，即 化学计量比的DO₃-Ta₃W化合物的合金基因浓度也等于合金的浓度，即随着有序度的降低，DO₃-TaW₃和DO₃-Ta₃W有序合金将会发生分裂，且随着有序度的降低，合金基因分裂程度增加。计算了DO₃-TaW₃和DO₃-Ta₃W有序合金的原子状态和性质随成分x_W的变化，指出了DO₃-TaW₃和DO₃-Ta₃W 金属间化合物的制备非常困难的原因主要是由于它们的结合能差异小，为新一代超高温材料在航空发动机上的应用设计提供科学理论指导。

关键词：Ta–W合金系；BCC结构；合金基因

Foundation item: Project(50954006) supported by the National Natural Science Foundation of China; Project (2014-12) supported by the Environmental Protection Department of Hunan, China; Project(2016TP1007) supported by the Hunan Provincial Science and Technology Plan Project, China; Project(2016-01) supported by the Development and Reform Commission of Hunan Province, China

Received date: 2016-11-02; Accepted date: 2017-02-23

Corresponding author: PENG Hong-jian, PhD, Associate Professor; Tel: +86–731–88879287; E-mail: hongjianpeng@126.com