J. Cent. South Univ. (2021) 28: 39-47

DOI: https://doi.org/10.1007/s11771-021-4584-2

Stacking fault energy and electronic structure of molybdenum under solid solution softening/hardening

LIU Pan(刘攀), LIU Liu-cheng(刘柳成), GONG Hao-ran(龚浩然)

State Key Laboratory of Powder Metallurgy, Central South University, Changsha 410083, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2021

Abstract: Ab initio calculations are used to understand the fundamental mechanism of the solid solution softening/hardening of the Mo-binary system. The results reveal that the Mo-Ti, Mo-Ta, Mo-Nb, and Mo-W interactions are primarily attractive with negative heats of formation, while the interactions of Mo-Re, and Mo-Zr would be mainly repulsive with positive heats of formation. It is also shown that the addition of Re and Zr would cause the solid solution softening of Mo by the decrease of the unstable stacking fault energy and the increase of ductility. On the contrary, the elements of W, Ta, Ti, and Nb could bring about the solid-solution hardening of Mo through the impediment of the slip of the dislocation and the decrease of ductility. Electronic structures indicate that the weaker/stronger chemical bonding due to the alloying elements should fundamentally induce the solid solution softening/hardening of Mo. The results are discussed and compared with available evidence in literatures, which could deepen the fundamental understanding of the solid solution softening/hardening of the binary metallic system.

Key words: stacking fault energy; electronic structure; molybdenum; solid solution softening/hardening; ab initio calculation

Cite this article as: LIU Pan, LIU Liu-cheng, GONG Hao-ran. Stacking fault energy and electronic structure of molybdenum under solid solution softening/hardening [J]. Journal of Central South University, 2021, 28(1): 39-47. DOI: https://doi.org/10.1007/s11771-021-4584-2.

1 Introduction

Molybdenum (Mo) is a refractory transition metal and has been potentially used as turbine engines to replace nickel superalloys in the field of aerospace [1-7], structural components in nuclear fusion reactors [8, 9], heat sinks and heat elements in heating treating industry [10], etc. These important high-temperature applications are principally attributed to its high melting temperature of 2610 °C, good heat-conducting capability, excellent high-temperature strength, relatively low sputtering yield, high stability, and superior creep resistance [1, 5, 8, 11-20]. Nevertheless, the main disadvantage of Mo is its poor ductility at room temperature, and radiation may even aggravate the brittleness of Mo [2, 8, 21-25].

Generalized stacking fault energy (GSFE) has been well regarded as a suitable physical property to express the intrinsic ductility/brittleness of a given material, since the magnitude of GSFE is closely related to dislocation movement and subsequent plastic deformation [26-29]. In this respect, however, the theoretical GSFE values of Mo in literatures are not consistent with each other [30-33]. For instance, the unstable stacking fault energies of the (110)<111> slip system of Mo derived from the full-potential linear muffin-tin orbital method and ultrasoft pseudopotentials are 1.8 and 1.68 J/m² [30-32], respectively, while the corresponding value from the bond-order potential is 1.3 J/m² [33]. It is well known that the addition of alloying elements is an effective way to improve the ductility of Mo [10, 30, 34]. Specifically, the elements of Re, Ir, and Os could reduce the GSFE value of Mo and bring about the so-called solid solution softening [30, 34]. Experimental evidence also reveals that Re can considerably decrease the ductile-brittle transition temperature (DBTT) of Mo [10, 35]. On the other hand, the elements of Hf, Ta, and W increase the GSFE value of Mo and therefore induce the solid-solution hardening of Mo [30, 34]. It should be pointed out that further theoretical studies are needed to investigate the fundamental mechanism of the solid solution softening/hardening in Mo.

The present paper is therefore dedicated through ab initio calculations [36, 37] to find out the stacking fault energy and related ductility of Mo with the addition of several alloying elements. Particularly, two slip systems of <111> and <111> as well as six transition elements of W, Ta, Ti, Nb, Re, and Zr are intentionally selected as typical examples to find out solid solution hardening/softening of Mo [1-25, 29-32, 34, 38-43], and the elements of Ir, Os, and Hf are not included in the present study due to their clear effects in literatures [30, 34]. The heats of formation and electronic structure of these elements in Mo will be also derived and discussed, in order to deeply understand the fundamental mechanism of the solid solution softening/hardening of the Mo-binary system.

2 Method of calculation

The present calculation is performed via the well-known software of Vienna ab initio Simulation Package [44] with the projector- augmented wave (PAW) method under the framework of density functional theory [45, 46]. The generalized gradient approximation (GGA) is used to express the exchange and correlation function [47], and the plane-wave basis is chosen with a cutoff energy of 400 eV. Moreover, the temperature-smearing method [48] is selected for the relaxation calculations, and the modified tetrahedron method [49] is conducted for the static calculations.

A supercell of 48 atoms is first constructed for pure BCC Mo bulk, and a Mo atom is replaced by an alloying X atom (X=W, Ta, Ti, Nb, Re and Zr) to form the binary Mo₄₇X₁ (X=W, Ta, Ti, Nb, Re, and Zr) bulk. The lattice constants of pure Mo and Mo₄₇X₁ bulks are obtained through fully optimization, and the heats of formation of Mo₄₇X₁ bulks are also calculated for comparison. It should be pointed out that the Mo₄₇X₁ (X=W, Ta, Ti, Nb, Re, and Zr) bulks can maintain the BCC structure during optimization, and the derived lattice parameters of BCC Mo and Mo₄₇X₁ bulks will be utilized in the subsequent calculations.

A surface unit cell of 2×2 with 12 surface layers and 12 vacuum layers is then chosen for the slip systems of<111> and <111> after a series of test calculations. Consequently, Figure 1 displays theandsurface models of Mo₄₇X₁ (X=W, Ta, Ti, Nb, Re, and Zr) with 12 surface layers. One can observe from this figure that the alloying element is located at the seventh layer of the surface model, and that the x, y, and z axes of thesurface model are [111], , and , respectively, whereas the x, y and z axes of thesurface are [111], and respectively.

In order to create the stacking fault, the upper half (from the seventh to twelfth layers) of each surface in Figure 1 is set to move along the [111] direction (x axis) with a certain displacement ranged from 0.0 to 1.0 with a spacing of 0.1, whereas the lower half (from the first to sixth layers) remains unchanged. It should be noted that the above displacement is a normalized value, and 1.0 corresponds to the length of the Burgers vector of a/2<111>. After the displacement, the atoms in the whole surface model are only permitted to relax along theordirection (z axis) [50]. The total energy of the surface model after relaxation is calculated before and after each displacement, for the purpose of deriving the stacking fault energy. During each relaxation and static calculation, the k meshes of Monkhorst-Pack [45] are 7×7×1 and 9×9×1, respectively, while the energy criteria are 0.01 and 0.001 meV, respectively.

3 Results and discussions

To begin with, the lattice parameters of Mo and Mo₄₇X₁ (X=W, Ta, Ti, Nb, Re, and Zr) bulks with the BCC structure are optimized and the derived values are summarized in Table 1. One can see that the obtained lattice constant of Mo from the present PAW-GGA method is 3.150 , which is in good agreement with the corresponding value of 3.147 from experimental measurements [51]. Interestingly, the lattice constants of Mo₄₇X₁ bulks are similar to that of Mo. This similarity would be probably attributed to the low composition of the alloying atom as well as its close atomic radius to that of Mo.

Figure 1 (a) and(b) surfaces of Mo₄₇X (X=W, Ta, Ti, Nb, Re, and Zr) with 12 surface layers, seventh layers of(c) and(d) surfaces with one Mo atom replaced by the X atom (The green and yellow spheres refer to Mo and X atoms, respectively)

Table 1 Lattice constants (a), unstable stacking fault energy (E_us),and ductility (D) of BCC Mo and Mo₄₇X₁ (X=W, Ta, Ti,Nb, Re, and Zr) bulks. The heats of formation (△H_f) of Mo₄₇X₁ bulks from the present study and the thermodynamic model [52] are also listed for comparison. (<111> and <111>are two slip systems)

For each BCC Mo₄₇X₁ (X=W, Ta, Ti, Nb, Re, and Zr) bulk, the heat of formation (△H_f) could be obtained as follows:

(1)

where E_Mo and E_X are total energies of BCC Mo₄₇X₁, BCC Mo, and alloying atom of W, Ta, Ti, Nb, Re, or Zr at the equilibrium state, respectively. Consequently, Table 1 lists the derived ΔH_f values of Mo₄₇X₁ bulks as well as other available data in Ref. [52]. One can discern from this table that the present heats of formation (-1.917 kJ/mol) of Mo₄₇Ti₁ are in nice agreement with the corresponding value (-2 kJ/mol) of Mo₅Ti₁, while ΔH_f of Mo₄₇Ta₁, Mo₄₇Nb₁ and Mo₄₇W₁ bulks from the present calculation are different, to some extent, from the thermodynamic values of Mo₅Ta₁, Mo₅Nb₁, and Mo₅W₁ bulks [52]. On the other hand, Mo₄₇Re₁ and Mo₄₇Zr₁ possess the positive heats of formation of 0.257 and 1.191 kJ/mol, respectively, which are quite distinct from the negative values (-4 kJ/mol) of Mo₅Re₁ and Mo₅Zr₁ by the Miedema model [52]. These divergences would be probably due to the computational errors as well as the compositional differences. The above negative or positive values of ΔH_f from the present study also indicate that the Mo-Ti, Mo-Ta, Mo-Nb and Mo-W interactions are primarily attractive, while the interactions of Mo-Re and Mo-Zr would be mainly repulsive.

The generalized stacking fault energies (GSFE) [41] of both BCC Mo and Mo₄₇X₁ (X=W, Ta, Ti, Nb, Re, and Zr) surfaces are calculated according to the following form:

(2)

where E_f and E_b are total energies of the surface after and before a certain displacement along the Burgers vector (a/2<111>), respectively; and A is the area of the surface model. After a series of calculations, the derived GSFE values of <111> and <111> slip systems are shown in Figure 2 for BCC Mo and Mo₄₇X₁ (X=W, Ta, Ti, Nb, Re, and Zr) under several displacements along the Burgers vector (a/2<111>). The summit of each curve in Figure 2 has been regarded as the energy barrier of the dislocation emission, and named as the unstable stacking fault energy (E_us) [36]. The calculated E_us values of BCC Mo and Mo₄₇X₁ are thus included in Table 1.

Based on the derived E_us value, the ductility (D) of BCC Mo and Mo₄₇X₁ (X=W, Ta, Ti, Nb, Re, and Zr) can be estimated by [28, 50]:

(3)

where E_s is the surface energy of the or surfaces of BCC Mo and Mo₄₇X₁. In the present calculation, the surface energies of the orsurfaces of BCC Mo are 2.829 and 2.893 J/m², respectively, which agree well with the corresponding value (2.807 J/m²) of the surface of BCC Mo from experimental measurements [53]. It should be noted that the E_s value of BCC Mo₄₇X₁ is very close to that of pure Mo (results not shown). The derived D values of the slip systems of <111> and <111> of both Mo and Mo₄₇X₁ (X=W, Ta, Ti, Nb, Re, and Zr) are therefore summarized in Table 1.

Figure 2 Generalized stacking fault energies (GSFE) of <111> (a) and <111> (b) slip systems of BCC Mo and Mo₄₇X₁ (X=W, Ta, Ti, Nb, Re, and Zr) under several displacements along the Burgers vector (a/2<111>)

Several characteristics can be discerned from Figure 2 and Table 1. Firstly, the calculated E_us value (1.373 J/m²) of the <111> slip system of BCC Mo from the present PAW-GGA method is in good agreement with the corresponding value of 1.3 J/m² from the bond-order potential [33], while much smaller than the values of 1.8 and 1.68 J/m² from the full-potential linear muffin-tin orbital method and ultrasoft pseudopotentials,respectively [30-32]. Such a difference of E_us values of BCC Mo should be due to the different theoretical methods, and further studies are welcome to clarify this issue.

Secondly, theandsurfaces of both Mo₄₇Re₁ and Mo₄₇Zr₁ possess smaller unstable stacking fault energy (E_us) than those of pure Mo, whereas their D values are bigger. That is to say, the addition of Re and Zr in Mo is beneficial to the movement of the dislocations and therefore increases the ductility of Mo. This kind of solid solution softening of Re or Zr in Mo from the present calculation matches well with similar experimental observations of Re in Mo from experiments [10, 35] and theoretical calculations of Re [30, 34].

Moreover, for both and <111> slip systems, the E_us and D values of Mo₄₇X₁ (X=W, Ta, Ti, and Nb) are higher and lower than those of BCC Mo, respectively. This comparison implies that the addition of W, Ta, Ti, and Nb could impede the slip of the dislocation and thus decrease the ductility of Mo. Such a solid-solution hardening of Mo due to the alloying elements of W, Ta, Ti, and Nb is consistent with similar predictions of Ta and W [30, 34]. Note that the theoretical methods in Refs. [30] and [34] are full-potential linear muffin-tin orbital and ultra-soft pseudopotentials, respectively, which are quite different from the present PAW-GGA method, the above agreements regarding Re, Ta, and W in Mo [30, 34] provide further support of the relevance of the present results.

It is of importance to find out the fundamental mechanism of the above solid solution softening/ hardening. The electronic structures of both Mo and Mo₄₇X₁ (X=W, Ta, Ti, Nb, Re and Zr) are thus calculated and compared with each other. As a typical example, Figure 3 shows the density of states (DOS) of the Mo atom in the seventh layer of thesurface of BCC Mo and Mo₄₇Zr₁. One can see clearly from Figure 3 that compared with BCC Mo, the DOS peaks of Mo in Mo₄₇Zr₁ experience right shifts and have slightly smaller binding energies. These features suggest that the chemical bonding of Mo in the seventh layer of Mo₄₇Zr₁ should become weaker due to the addition of Zr atom. Such a weaker chemical bonding would therefore bring about the decrease of the E_us and solid solution softening of Mo₄₇Zr₁ or Mo₄₇Re₁, and is also consistent with the positive heats of formation of Mo₄₇Zr₁ or Mo₄₇Re₁ as related before.

Figure 3 Density of states of Mo atom in seventh layer ofsurface of BCC Mo and Mo₄₇Zr₁

In addition, Figure 4 displays another example, the charge density plots of Mo and Mo₄₇W₁ under a displacement of a/4<111> through the<111> slip system. It can be discerned clearly from this figure that the W-Mo bonding between the seventh and sixth layers of Mo₄₇W₁ seems much stronger than the original Mo-Mo bonding in pure Mo. This stronger chemical bonding of W-Mo would be originated from the attractive interaction between W and Mo with negative heat of formation as shown in Table 1. In other words, it is the stronger bonding due to the attractive Mo-X (X=W, Ti, Ta,and Nb) interaction which intrinsically impedes the slip of the dislocation, increases the stacking fault energy, decreases the ductility, and fundamentally causes the solid solution hardening of Mo₄₇X₁ (X=W, Ta, Ti, and Nb).

Figure 4 Charge density plots of Mo (a) and Mo₄₇W₁ (b) under a displacement of a/4<111> through <111> slip system

4 Conclusions

The present Ab initio calculation has been performed to find out the stacking fault energy and related ductility of Mo with the addition of several alloying elements such as W, Ta, Ti, Nb, Re, and Zr. It is found that the addition of Re and Zr in Mo is beneficial to the movement of the dislocations and therefore increases the ductility of Mo. On the other hand, the addition of W, Ta, Ti, and Nb could impede the slip of the dislocation and thus decrease the ductility of Mo. The heats of formation and electronic structure of these elements in Mo have been discussed and compared with each other, which could therefore bring about a deep understanding of the fundamental mechanism of the solid solution softening/hardening of the Mo-binary system.

Contributors

GONG Hao-ran provided the concept and edited the draft of manuscript. LIU Pan conducted the literature review and wrote the first draft of the manuscript. LIU Liu-cheng edited the draft of manuscript.

Conflict of interest

LIU Pan, LIU Liu-cheng and GONG Hao-ran declare that they have no conflict of interest.

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(Edited by FANG Jing-hua)

中文导读

固溶软化/硬化下钼的层错能与电子结构

摘要：用第一性原理计算方法研究了钼二元系固溶体软化/硬化的基本机理。结果表明，Mo-Ti、Mo-Ta、Mo-Nb和Mo-W之间的作用主要是相互吸引，具有负的生成热，而Mo-Re和Mo-Zr之间的作用主要是相互排斥，具有正的生成热。Re和Zr的加入会降低钼的堆垛层错能，提高合金的塑性，从而导致钼的固溶软化；而W、Ta、Ti、Nb等元素通过阻碍位错滑移和塑性下降，使Mo固溶硬化。电子结构表明，合金元素导致的弱/强化学键合从根本上诱导了钼的固溶体软化/硬化。本文对计算结果进行讨论，并与文献中已有的证据进行了分析比较，加深了对二元金属体系固溶体软化/硬化的基本认识。

关键词：层错能；电子结构；钼；固溶体软化/硬化；第一性原理计算方法

Foundation item: Project(51801129) supported by the National Natural Science Foundation of China; Project supported by the State Key Laboratory of Powder Metallurgy, China

Received date: 2020-05-17; Accepted date: 2020-07-31

Corresponding author: GONG Hao-ran, PhD, Professor; Fax: +86-731-88710855; E-mail: gonghr@csu.edu.cn; ORCID: https://orcid. org/0000-0001-9767-6776