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Atomic diffusion mediated by vacancy defects in L12-Zr3Al: A first-principles study
Journal Pre-proof Atomic diffusion mediated by vacancy defects in L12-Zr3Al: A first-principles study Zhixin Ren, Zhe Xue, Xinyu Zhang, Jiaqian Qin, Mingzhen Ma, Riping Liu PII:
S0925-8388(19)34469-X
DOI:
https://doi.org/10.1016/j.jallcom.2019.153223
Reference:
JALCOM 153223
To appear in:
Journal of Alloys and Compounds
Received Date: 1 September 2019 Revised Date:
28 November 2019
Accepted Date: 29 November 2019
Please cite this article as: Z. Ren, Z. Xue, X. Zhang, J. Qin, M. Ma, R. Liu, Atomic diffusion mediated by vacancy defects in L12-Zr3Al: A first-principles study, Journal of Alloys and Compounds (2019), doi: https://doi.org/10.1016/j.jallcom.2019.153223. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.
Atomic diffusion mediated by vacancy defects in L12-Zr3Al: A first-principles study Zhixin Rena, Zhe Xuea, Xinyu Zhanga,*, Jiaqian Qin b, Mingzhen Maa, and Riping Liua a
State Key Laboratory of Metastable Materials Science and Technology, Yanshan University, Qinhuangdao 066004, P. R. China
b
Metallurgy and Materials Science Research Institute, Chulalongkorn University, Bangkok 10330, Thailand
*Corresponding Author. E-mail: [email protected] (X. Zhang)
ABSTRACT Systematic first-principles calculations based on density functional theory have been performed to determine the self-diffusion mechanism of Zr3Al. The formation energies of four intrinsic point defects were calculated and the relationship between concentration of different point defects and temperature was determined. Then, the effect of defect types on the stability of the system was explored after analyzing the electronic density of different supercells. In addition, the minimum energy pathways (MEPs) and the saddle point configurations during Zr or Al atom migration were calculated using the climbing image nudged elastic band method (CI-NEB). The energetic results show that the VZr-mediated nearest neighbor (NN) jump mechanism is the feasible migration mechanism for Zr atom and dominant Al atom diffusion mechanism is the antisite assisted (AS) mechanism because of the lowest diffusion activation energy barrier. The present study lays the foundation for further investigation, including solute-diffusion in Zr3Al and quantitative calculation of diffusion coefficients, and provides theoretical guideline for further design and improvement of the crucial diffusion-mediated properties of Zr3Al.
Keywords: Self - diffusion; Zr3Al; Migration barrier; First-principles;
1. Introduction Studies on the intermetallic compound Zr3Al are both of scientific and technological significance as it has attractive properties which make it used as nuclear structural material. Zr3Al is the first intermetallic on the Zr-rich side of the Zr-Al binary phase diagram. The equilibrium Zr3Al phase has an ordered fcc-like structure (Cu3Au type; L12 structure), whereby it exhibits isotropic physical and mechanical properties and higher strength than traditional zirconium alloys [1]. Unfortunately, it is difficult to produce a single-phase Zr3Al material. There are always some residual α-Zr, Zr2Al or both phases in Zr3Al based alloy, which limit the application of Zr3Al based materials [2]. In recent years, the ternary addition method is considered to be one of the effective ways to solve the above problems, which can alter the initial microstructure, control the grain size and shapes, and introduce new phases [1, 3-5]. A relatively mature system is to add a Nb element to form a Zr3Al-Nb alloy. In low concentration alloys, such as Zr3Al-1.8Nb, as the diffusion coefficients for solutes are of the same order, the formation of the Zr3Al phase appears to be dependent only on the difference of the concentration of solute species, and the controlling steps for advancing the interface would be the long-range diffusion of Al. In the case of Zr3Al-3Nb, which exhibits two-phase microstructure, the diffusion of Al is lower and therefore would be the controlling step in the growth of the Zr3Al phase [1, 3]. It can be seen that the related research on phase transformation of Zr3Al-based alloys involves atomic diffusion. However, as problems encountered in other materials, due to the limitations of sample corrosion, surface oxidation and impurity contamination, the exact experimental data of atomic diffusion in Zr3Al-based alloys has not been obtained
experimentally [6, 7]. Therefore, the suitable methods to explore the diffusion properties of Zr3Al are significant for better developing of Zr3Al-based alloys. Fortunately, as the development of computational material science in the last few decades, the design of materials to achieve optimal functionality theoretically has become a rational complement of the traditional empirical approach. Moreover, first-principles calculations based on density functional theory (DFT) developed in recent years have been widely used to study the diffusion properties of materials [8-17]. Perez et al. [18] summarized the solute diffusion anisotropy in Hcp-Zr and the correlation between solubility and diffusivity. Lu et al. [19] employed the 8-frequency model, performing a more complete diffusion behavior of solute atoms in Zr atoms, thus determined the main diffusion mechanism of each solute atom. In addition, Mantina et al. [20-22] predicted the self- and the dilute diffusion coefficient in the fcc-Al systems within the framework of the transition state theory. Nils Sandberg et al. [23] revealed the solute diffusion activation energy in Al. Shi et al. [24] studied the self- and solute diffusion of Al3Sc, and determined the diffusion mechanisms of the main alloying element Al and the minor alloying element Sc in Al3Sc, and further explored the effects of alloying elements such as Zr, Hf, Ti and Y on atomic diffusion. Wang and Priya Gopal [25, 26] concluded the self- and solute diffusion mechanisms of Ni3Al, and theoretically verified the relevant experimental research. Obviously, the aforementioned studies are usually coupled with transition state theory (TST) under the harmonic or the quasi-harmonic approximations, using the method of nudged elastic band (NEB) or the climbing-image nudged elastic band (CI-NEB) to find the
lowest energy path, and then carry out related diffusion studies. Therefore, our work is aimed at studying the self-diffusion mechanism of L12-Zr3Al intermetallic, exploring the relationship between point defect concentration and temperature, and determining the feasible diffusion path for Zr atom and Al atom, thus getting the diffusion activation energy. This is the basis for determining the selfand solute diffusion coefficients of Zr3Al, which helps to deepen the study of L12-Zr3Al intermetallic, theoretically guide the experiments to optimize performance design, improve the materials properties, especially diffusion-mediated properties, such as creep, plasticity and so on. 2. Computational details All calculations were carried out using the plane-wave-based density functional theory (DFT) code VASP [27, 28] with the pseudopotentials generated by the projector augmented wave method (PAW). Electron exchange and correlation is described within the generalized gradient approximation (GGA) using the Perdew-Burke-Ernzerhof (PBE) parametrization [29]. The plane-wave cutoff energy of 400 eV was set for all calculations. The Brillouin zone sampling was performed with a Γ-centered k-point mesh 5×5×5 for a 3a0×3a0×3a0 supercell containing 108 atoms used in the present work (i.e., 11×11×11 with respect to the four-atom fcc cell). The convergence criterion for Hellmann Feynman force and total energy were set as 0.01 eV/Å and 10-5 eV per atom, respectively. Furthermore, the migration barrier and the minimum energy pathways for atom diffusion were calculated using the climbing-image nudged elastic band (CI-NEB)
method [30, 31], which was developed by Henkelman et al. [32]. Five images were used to calculate the transition state (TS) structures, i.e., the saddle point structure, with the saddle point was determined from the image with a maximum energy, and the two end-point structures, i.e. the initial and final structures, were fully relaxed first. The same structural convergence criteria and settings as above-mentioned structure relaxed were used in our CI-NEB calculations. Table 1 shows the lattice constant, moduli and formation energy [33] of Zr3Al calculated under the parameter setting of this study. These agreements of our calculation results with others provide a confirmation that the computational methodology utilized in this work is suitable and reliable. Table 1 Crystal structure parameter for Zr3Al compared to experimental data and others computational study. Lattice parameters (Å)
Bulk modulus (GPa)
EF (eV/atom)
Present work
4.388
100.393
-0.300
[33]
4.381
102.476
-0.307
[34]
-
108.670
-
[35]
4.380
102.028
-0.300
3. Results and discussion 3.1 Point defects in Zr3Al The equilibrium Zr3Al phase has an ordered fcc-like structure (Cu3Au type; L12 structure), in which, atoms of the major element Zr occupy the sites at the center of the cube faces, and first-nearest neighbors of each Zr atom contains 8 Zr atoms and 4 Al atoms, while Al atoms at corner sites of the fcc lattice are surrounded by 12 Zr atoms as first-nearest neighbors. There are four intrinsic point defects in Zr3Al:
antisite ZrAl (Zr atoms on Al sublattice sites) and AlZr (Al atoms on Zr sublattice sites), vacancies VZr and VAl. We first calculated the formation energies of four intrinsic point defects using the Wagner-Schottky model, which is defined as: = Where
and
+∑
−
(1)
are the total energy of defective supercells and perfect
supercells, respectively, Ei is the energy per atom in hcp-Zr/fcc-Al unit cell (if remove atom from the supercell, Ei>0, or Ei<0). According to the calculation results presented in Table 2, the formation energy tendency was: AlZr < ZrAl < VZr < VAl. From the perspective of thermodynamic, a negative value of EF implies defects are more easily formed. Therefore, the antisite defects are easier to form than the vacancy defects, and the formation energy of AlZr defects is the lowest, only -0.230eV, which means that AlZr defects maybe the dominant point defects in Zr3Al. Table 2 The calculated formation energies of four intrinsic point defects. ∆H (eV) VZr
VAl
ZrAl
AlZr
1.989
3.216
1.886
-0.230
In order to verify the relative concentration relationship about different point defects revealed by the formation energy, and provide a basis for the discussion in the next section, we further investigated the change tendency of four intrinsic point defect concentrations with temperature, the dependence equation is as follows [36-40]: (
)
= exp −
(
)
= exp −
!" # !" #
$
(2)
$
(3)
%&'
(
)*
%)(
x+ Where ,/ , ,/ , ,+
− ,-.
= exp −
+
!" #
$
= ,+
(4) (5)
and ,-. + are the concentrations of VZr, VAl, ZrAl, and AlZr
in Zr3Al, respectively, and the ,+ represents the mole fraction of Zr in Zr3Al. Fig. 1 shows the variation of each defect concentration with temperature. It can be seen that since the defect formation is a thermal activation process, concentration rise with the increasing temperature for all defects. At the same temperature, the concentration of antisite defects is higher than the vacancy, and the concentration of VAl is the lowest, which qualitatively confirms the trend of the defect formation energy (AlZr < ZrAl < VZr < VAl) , that is, the higher the formation energy, the lower the concentration of the defect, the result is similar to the other L12 compounds. Furthermore, the effect of different defects on the stability of the system were investigated by comparing the electronic density of perfect and defective supercells. The calculation result is shown in Fig. 2, where the red dash line indicates the position of the Fermi level. From the point of view of quantum mechanics, the lower DOS value at the Fermi level means that more electrons participate in bonding process to produce electronic localization [41], so the stability of the system will be higher. Therefore, we can find that the stability of Zr3Al decreases once there are the point defects. The Fermi levels of ZrAl and AlZr are close to the peak position on the left side of the pseudopotential gap, that is, their Fermi level are in the bonding state, which indicates that the valence electron of Zr3Al fails to fill full of the valence band due to the formation of the antisite defects. Therefore, the stability of the system decreased.
Meantime, the Fermi levels of VZr and VAl are close to the valley position, and the DOS value is significantly lower. In other words, although from the perspective of formation energy that antisite defects are easily formed, the stability of the vacancy defects is higher, which explains the reason why the vacancy-mediated diffusion is the dominant diffusion mechanism. 3.2 Atomic diffusion mechanism in Zr3Al Point defect is the simplest crystal defect, which forms for the reason that atoms deviates from the normal alignment of the crystal structure at or near the microscopic region. Particularly, if the vibration of an atom is large sufficiently, it may overcome the restriction of the surrounding atoms and jump away from its original site, and then forms a vacancy. Once vacancies formed in the lattice, the surrounding atoms will lose their balance, then they will relax toward the vacancy direction. Therefore, the atom migration will be easier with the existence of vacancies. In general, the dominant diffusion mechanism in alloys and compounds is based on vacancies. There are two steps in the process of vacancy-mediated diffusion: first, a vacancy is formed in the crystal lattice; then, the site of diffusion atom and its first nearest neighbor vacancy exchange, i.e., a thermal activation jump occurs. Therefore, the energy required for vacancy-mediated diffusion consists of two parts, one is the defect formation energy, and the other is the diffusion migration energy barrier. Based on the defect formation energy and the diffusion migration barrier, the diffusion activation energy can be determined by the Eq. (6): 0
=∑
1
+
2
(6)
Where
0,
and
2
are the diffusion activation energy, the sum of the formation
energy of all point defects and migration energy in certain path. The point defect formation energies have been calculated in 3.1. In the following work, we use the climbing-image nudged elastic band (CI-NEB) method to calculate the migration barriers of different diffusion mechanism. Finally, the most feasible diffusion mechanism of Zr and Al atoms in Zr3Al will be determined by comparing the diffusion activation energy of different mechanism. 3.2.1 Zr atom diffusion The trend of the faster diffusion of the major element than that of the minor element in ordered L12 alloys has been termed “the ordered Cu3Au effect” by Heurle and Gas [42]. For example, in Ni3Ge [43-45], Pt3Mn [45] and Co3Ti [46], Ni, Pt and Co occupying the face center, the diffusion rate of them are faster than the minor element by an order of magnitude. In addition, the diffusion coefficient of Ni is 103 times that of the Al diffusion coefficient in Ni3Al [47, 48]. Therefore, we first studied the possible diffusion paths of Zr atom in Zr3Al, which are VZr-mediated diffusion mechanism, VAl-mediated diffusion mechanism and antisite bridge mechanism, the schematic diagram of different diffusion paths is given in the Fig. 3. The diffusion of the major element Zr can be regarded as the self-diffusion in a hypothetical pure crystal possessing the structure of the a sublattice, or ‘NbO lattice’ [49]. The exchange between the Zr atom and the vacancy in its sublattice is the simplest diffusion mechanism. As shown in Fig. 3, there are 8 equivalent Zr atoms in the nearest neighbor of each Zr atom. When VZr formed in these sites, the Zr atom
could diffuse by the VZr-mediated nearest neighbor (VZr-NN) [50, 51] jump mechanism (ie, the Zr atom at Z1 site makes a nearest neighbor jump to VZr at the Z2 site), and the degree of lattice order does not change. The energy profile of minimum energy paths (MEP) is shown in Fig. 3(b), the migration energy barrier of this mechanism is only 1.146eV. In addition to the NN jump, there are 4 equivalent next nearest neighbor diffusion (VZr-NNN) [52] paths (the Zr atom at Z1 site jumps to the VZr at Z3 site) due to the existence of 4 next nearest neighbor Zr atoms around the VZr, and the atomic migration barrier of this path is 6.40 eV, which is significantly higher than the migration barrier of the NN jump. This is because when the Zr atom make the nearest neighbor jump, it can jump directly from the initial state to the final state, but in the NNN mechanism, it needs to pass through a quadrilateral made by 4 Zr atoms at the nearest neighbor of it, during this process, the migration resistance increases with the larger lattice distortion, then the migration barrier became higher. As shown in the Fig .3(a), nearest neighbors of Zr atom has 4 equivalent Al atoms besides of 8 Zr atoms. Therefore, when VAl is formed, Zr atoms can also be diffused by VAl (VAl-NNN). After calculation, the migration barrier is 1.056 eV. In addition to the above three migration paths, Zr atom may also diffuse through the antisite bridge mechanism (ASB) [53-57] induced by VAl. The initial condition of this mechanism is that ZrAl antisite and VAl vacancy exist in the crystal lattice, and ZrAl is located in the NNN site of VAl. During the diffusion process, the Zr atom is firstly exchanged with the nearest neighbor VAl to produce the intermediate state VZr + ZrAl, and then the VZr and ZrAl sites are interchanged to form the final structure of VAl +
ZrZr. After the two steps, the degree of lattice order has not changed. By calculation, its diffusion energy barrier is 0.739 eV. Fig 3. (d) shows the diffusion activation energy calculated by the equal (5), the activation energies of the four diffusion paths of Zr atom are 3.135 eV, 8.393 eV, 4.272 eV and 5.841 eV, respectively. Therefore, from the point of energy, the NNN diffusion mechanism induced by VZr is the most feasible diffusion mechanism for Zr atom. This conclusion is consistent with others’ reports that the self-diffusion of the major element A is most possible via the ordinary vacancy mechanism on its own sublattice (a sublattice) in A3B alloys of the L12 type structure. 3.2.2 Al atom diffusion After determining the optimal diffusion path for Zr atom, we next analyzed and determined the feasible diffusion path for Al atom. Through the analysis of the electronic structure we can find that VZr has the highest stability in Zr3Al, so the VZr-mediated nearest neighbor (VZr-NN) jump for Al atom is studied firstly. As we can see from the Fig .4 (a), because of nearest neighbor sites of each Al atom are 12 Zr atoms, the Al atom can make the NN jump with the help of the existence of VZr. The migration energy of this path is 1.787 eV. However, it is worth noting that the migration of Al atoms appears not so easy as that of Zr atoms, a jump of an Al atom to a nearest-neighbor site will inevitably destroy the ordered arrangement. Although the concentration of VAl is higher than VZr for the stochiometric Zr3Al, the proportion of VAl in all point defects may rise while the composition of the alloy
deviates from the stochiometric dueto external conditions such as high temperature or radiation, so the VAl-mediated Al atom diffusion (VAl-NNN) cannot be ignored (the Al atom at the A6 site jump to the Z3 site). After calculation, the migration barrier is 4.442 eV, which is higher than the NN jump of Al atom induced by VZr. The reason may result from two aspects, on the one hand, the distance traveled by the diffusion Al atom is farther than the one of the VZr-mediated diffusion as the VAl is located at the NNN site of the Al atom, and on the other hand, the radius of Zr atom is larger than Al atom, so the distortion caused by VZr in the crystal lattice is more than the one induced by VAl. Therefore, the lattice of the VZr-containing lattice has less resistance to the Al atom. Although the jump of an Al atom from its regular position to a nearest neighbor vacancy disturbs the ordered arrangement, an antisite AlZr atom can exchange its position with a vacancy on a neighboring a sublattice site without causing further disordering. Therefore, Al atom may also diffuse through the antisite bridge (ASB) mechanism and the antisite assisted (AS) [43, 58] mechanism. In the ASB mechanism, AlZr and VZr are located at the NN site of each other, firstly, VZr exchange position with Al atom to form a composite defect AlZr+VAl, and then the newly formed VAl jump to the site of another AlZr, so that when the diffusion is finally completed, even if the two Al atoms are exchanged, the degree of lattice order has not changed, and the migration energy barrier is 1.91 eV. Moreover, the AS mechanism is another Al atom diffusion mechanism induced by VZr with the help of AlZr. When an AlZr exists in the crystal lattice, the diffusion Al atom jumps to the nearest neighbor VZr to accomplish
the migration (the Al atom at the A2 site jumps to the VAl at the A3 site), obviously, the jump mode does not affect the order of the crystal lattice. The calculated migration barrier is extremely low, only 0.649 eV. This may be due to the degree of distortion inward and outward is superimposed because of there are both antisite defects and vacancy defects in the crystal lattice, the lattice arrangement became more lose, therefore, the resistance caused by atom migration is small. Since the NN diffusion of Al atom mediated by VZr destroys the order of the structure, it is further considered that Al atom diffuse through a six-jump cycle (6JC) [55, 56] mechanism, which involves six atomic jumps: VZr + Al → VAl + AlZr; VAl + Zr → VZr + ZrAl; VZr + Al → VAl + AlZr; VAl + AlZr → VZr + Al; VZr + ZrAl → VAl + Zr; VAl + AlZr → VZr + Al, the schematic can be seen from the Fig .5 (a). Although the first three jumps reduce the order of the structure, the last three jumps can be regarded as the inverse of the first three jumps. Therefore, the order of the structure does not change when 6JC mechanism is completed. The 6JC mechanism can be subdivided into a straight 6JC and a bent 6JC. The former occurs on the (100) plane, while atomic diffusion occurs outside the (100) plane in the bent 6JC mechanism, which caused three-dimensional atom diffusion. After calculation, the energy barriers of them are 2.91 eV and 3.53 eV, respectively. By calculation, among all possible diffusion mechanisms of Al atom, the diffusion activation energy of the anti-assisted mechanism is the lowest, only 2.408eV, which indicates the difference between the most feasible diffusion mechanism is that the NN jump mechanism induced by VZr for Zr atom, but the anti-assisted mechanism for Al
atom, which is consistent with the diffusion mechanism of the secondary elements in other L12 phase structures. 4. Conclusions It should be remarked that a normal first-principles study based on density functional theory have been used to study the self-diffusion of the L12-Zr3Al in this work. Firstly, the formation energies of four intrinsic point defects in Zr3Al were calculated, and their relative stability was explained from the perspective of electronic structure. Next, the relationship between point defect concentration and temperature at stochiometric was considered. Finally, the atom diffusion barrier of different diffusion paths for Zr and Al atom in Zr3Al were calculated by CI-NEB method, the most feasible diffusion mechanism of Zr atom and Al atom were determined by comparing the diffusion activation energy. The specific conclusions are as follows: 1.
Although the results of point defect formation energy indicate that the AlZr defect
is the highest concentration defect type in Zr3Al, the stability of VZr and VAl is higher. Therefore, the vacancy-mediated diffusion mechanism is the main diffusion mechanism in Zr3Al. 2.
The concentration of the four intrinsic point defects in Zr3Al increases with the
increasing temperature, and at the same temperature, the concentration of the antisite defects is higher than the vacancy defects, and the concentration of VAl is the lowest. 3.
From the perspective of energy, the NN jump mechanism induced by the VZr is
the most feasible diffusion mechanism for Zr atom, and the antisite assisted mechanism for Al atom has the lowest diffusion activation energy barrier, so it is the
most dominant diffusion mechanism.
Acknowledgments This work was supported by the National Key R&D Program of China (grant 2018YFA0703602), NSFC (grant 51571174), National Science Foundation for Distinguished Young Scholars for Hebei Province of China (grant E2016203376), and Hundred Excellent Innovative Talents Support Program in Hebei Province (grant SLRC2017056).
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Figure captions Fig. 1. The calculated equilibrium point defect concentrations in stoichiometric L12-Zr3Al as a function of temperature. Fig. 2. The total density of states (DOS) of perfect and defective Zr3Al, in which the red dash line indicates the position of Fermi level. Fig. 3. The schematic diagram, minimum energy paths (MEPs) and activation energy of different diffusion mechanism for Zr atom in Zr3Al. (a) A schematic of different diffusion mechanism for Zr atom; (b) The MEPs of VZr (NN) and VZr (NNN); (c) The MEP of VAl (NN); (d) The MEP of ASB. Fig. 4. The schematic diagram, minimum energy paths (MEPs) and activation energy of different diffusion mechanism for Al atom in Zr3Al. (a) A schematic of different diffusion mechanism for Al atom; (b) The MEP of VZr (NN); (c) The MEP of VAl (NNN); (d) The MEPs of ASB and AS. Fig. 5. The schematic diagram, minimum energy paths (MEPs) and activation energy of six-jump cycle (6JC) mechanisms in Zr3Al. (a) A schematic of straight and bent 6JC mechanisms; (b) The MEP of straight 6JC; (c) The MEP of bent 6JC.
Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Highlights 1. The point defects formation energies are determined in Zr3Al. 2. The concentration of the four intrinsic point defects increases with the increasing of temperature. 3. The nearest-neighbor jump mechanism induced by VZr is the most feasible diffusion mechanism for Zr atom. 4. Al mainly diffuses by the antisite assisted mechanism.
Zhixin Ren: Methodology, Investigation, Writing-Original Draft, Writing-Review & Editing. Zhe Xue: Investigation, Writing-Review & Editing. Xinyu Zhang: Methodology, Writing-Review & Editing, Project administration; Funding acquisition. Jiaqian Qin: Writing-Review & Editing. Mingzhen Ma: Writing-Review & Editing. Riping Liu: Writing-Review & Editing, Funding acquisition.
Declaration of interests ☒ 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. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
S0925-8388(19)34469-X
DOI:
https://doi.org/10.1016/j.jallcom.2019.153223
Reference:
JALCOM 153223
To appear in:
Journal of Alloys and Compounds
Received Date: 1 September 2019 Revised Date:
28 November 2019
Accepted Date: 29 November 2019
Please cite this article as: Z. Ren, Z. Xue, X. Zhang, J. Qin, M. Ma, R. Liu, Atomic diffusion mediated by vacancy defects in L12-Zr3Al: A first-principles study, Journal of Alloys and Compounds (2019), doi: https://doi.org/10.1016/j.jallcom.2019.153223. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.
Atomic diffusion mediated by vacancy defects in L12-Zr3Al: A first-principles study Zhixin Rena, Zhe Xuea, Xinyu Zhanga,*, Jiaqian Qin b, Mingzhen Maa, and Riping Liua a
State Key Laboratory of Metastable Materials Science and Technology, Yanshan University, Qinhuangdao 066004, P. R. China
b
Metallurgy and Materials Science Research Institute, Chulalongkorn University, Bangkok 10330, Thailand
*Corresponding Author. E-mail: [email protected] (X. Zhang)
ABSTRACT Systematic first-principles calculations based on density functional theory have been performed to determine the self-diffusion mechanism of Zr3Al. The formation energies of four intrinsic point defects were calculated and the relationship between concentration of different point defects and temperature was determined. Then, the effect of defect types on the stability of the system was explored after analyzing the electronic density of different supercells. In addition, the minimum energy pathways (MEPs) and the saddle point configurations during Zr or Al atom migration were calculated using the climbing image nudged elastic band method (CI-NEB). The energetic results show that the VZr-mediated nearest neighbor (NN) jump mechanism is the feasible migration mechanism for Zr atom and dominant Al atom diffusion mechanism is the antisite assisted (AS) mechanism because of the lowest diffusion activation energy barrier. The present study lays the foundation for further investigation, including solute-diffusion in Zr3Al and quantitative calculation of diffusion coefficients, and provides theoretical guideline for further design and improvement of the crucial diffusion-mediated properties of Zr3Al.
Keywords: Self - diffusion; Zr3Al; Migration barrier; First-principles;
1. Introduction Studies on the intermetallic compound Zr3Al are both of scientific and technological significance as it has attractive properties which make it used as nuclear structural material. Zr3Al is the first intermetallic on the Zr-rich side of the Zr-Al binary phase diagram. The equilibrium Zr3Al phase has an ordered fcc-like structure (Cu3Au type; L12 structure), whereby it exhibits isotropic physical and mechanical properties and higher strength than traditional zirconium alloys [1]. Unfortunately, it is difficult to produce a single-phase Zr3Al material. There are always some residual α-Zr, Zr2Al or both phases in Zr3Al based alloy, which limit the application of Zr3Al based materials [2]. In recent years, the ternary addition method is considered to be one of the effective ways to solve the above problems, which can alter the initial microstructure, control the grain size and shapes, and introduce new phases [1, 3-5]. A relatively mature system is to add a Nb element to form a Zr3Al-Nb alloy. In low concentration alloys, such as Zr3Al-1.8Nb, as the diffusion coefficients for solutes are of the same order, the formation of the Zr3Al phase appears to be dependent only on the difference of the concentration of solute species, and the controlling steps for advancing the interface would be the long-range diffusion of Al. In the case of Zr3Al-3Nb, which exhibits two-phase microstructure, the diffusion of Al is lower and therefore would be the controlling step in the growth of the Zr3Al phase [1, 3]. It can be seen that the related research on phase transformation of Zr3Al-based alloys involves atomic diffusion. However, as problems encountered in other materials, due to the limitations of sample corrosion, surface oxidation and impurity contamination, the exact experimental data of atomic diffusion in Zr3Al-based alloys has not been obtained
experimentally [6, 7]. Therefore, the suitable methods to explore the diffusion properties of Zr3Al are significant for better developing of Zr3Al-based alloys. Fortunately, as the development of computational material science in the last few decades, the design of materials to achieve optimal functionality theoretically has become a rational complement of the traditional empirical approach. Moreover, first-principles calculations based on density functional theory (DFT) developed in recent years have been widely used to study the diffusion properties of materials [8-17]. Perez et al. [18] summarized the solute diffusion anisotropy in Hcp-Zr and the correlation between solubility and diffusivity. Lu et al. [19] employed the 8-frequency model, performing a more complete diffusion behavior of solute atoms in Zr atoms, thus determined the main diffusion mechanism of each solute atom. In addition, Mantina et al. [20-22] predicted the self- and the dilute diffusion coefficient in the fcc-Al systems within the framework of the transition state theory. Nils Sandberg et al. [23] revealed the solute diffusion activation energy in Al. Shi et al. [24] studied the self- and solute diffusion of Al3Sc, and determined the diffusion mechanisms of the main alloying element Al and the minor alloying element Sc in Al3Sc, and further explored the effects of alloying elements such as Zr, Hf, Ti and Y on atomic diffusion. Wang and Priya Gopal [25, 26] concluded the self- and solute diffusion mechanisms of Ni3Al, and theoretically verified the relevant experimental research. Obviously, the aforementioned studies are usually coupled with transition state theory (TST) under the harmonic or the quasi-harmonic approximations, using the method of nudged elastic band (NEB) or the climbing-image nudged elastic band (CI-NEB) to find the
lowest energy path, and then carry out related diffusion studies. Therefore, our work is aimed at studying the self-diffusion mechanism of L12-Zr3Al intermetallic, exploring the relationship between point defect concentration and temperature, and determining the feasible diffusion path for Zr atom and Al atom, thus getting the diffusion activation energy. This is the basis for determining the selfand solute diffusion coefficients of Zr3Al, which helps to deepen the study of L12-Zr3Al intermetallic, theoretically guide the experiments to optimize performance design, improve the materials properties, especially diffusion-mediated properties, such as creep, plasticity and so on. 2. Computational details All calculations were carried out using the plane-wave-based density functional theory (DFT) code VASP [27, 28] with the pseudopotentials generated by the projector augmented wave method (PAW). Electron exchange and correlation is described within the generalized gradient approximation (GGA) using the Perdew-Burke-Ernzerhof (PBE) parametrization [29]. The plane-wave cutoff energy of 400 eV was set for all calculations. The Brillouin zone sampling was performed with a Γ-centered k-point mesh 5×5×5 for a 3a0×3a0×3a0 supercell containing 108 atoms used in the present work (i.e., 11×11×11 with respect to the four-atom fcc cell). The convergence criterion for Hellmann Feynman force and total energy were set as 0.01 eV/Å and 10-5 eV per atom, respectively. Furthermore, the migration barrier and the minimum energy pathways for atom diffusion were calculated using the climbing-image nudged elastic band (CI-NEB)
method [30, 31], which was developed by Henkelman et al. [32]. Five images were used to calculate the transition state (TS) structures, i.e., the saddle point structure, with the saddle point was determined from the image with a maximum energy, and the two end-point structures, i.e. the initial and final structures, were fully relaxed first. The same structural convergence criteria and settings as above-mentioned structure relaxed were used in our CI-NEB calculations. Table 1 shows the lattice constant, moduli and formation energy [33] of Zr3Al calculated under the parameter setting of this study. These agreements of our calculation results with others provide a confirmation that the computational methodology utilized in this work is suitable and reliable. Table 1 Crystal structure parameter for Zr3Al compared to experimental data and others computational study. Lattice parameters (Å)
Bulk modulus (GPa)
EF (eV/atom)
Present work
4.388
100.393
-0.300
[33]
4.381
102.476
-0.307
[34]
-
108.670
-
[35]
4.380
102.028
-0.300
3. Results and discussion 3.1 Point defects in Zr3Al The equilibrium Zr3Al phase has an ordered fcc-like structure (Cu3Au type; L12 structure), in which, atoms of the major element Zr occupy the sites at the center of the cube faces, and first-nearest neighbors of each Zr atom contains 8 Zr atoms and 4 Al atoms, while Al atoms at corner sites of the fcc lattice are surrounded by 12 Zr atoms as first-nearest neighbors. There are four intrinsic point defects in Zr3Al:
antisite ZrAl (Zr atoms on Al sublattice sites) and AlZr (Al atoms on Zr sublattice sites), vacancies VZr and VAl. We first calculated the formation energies of four intrinsic point defects using the Wagner-Schottky model, which is defined as: = Where
and
+∑
−
(1)
are the total energy of defective supercells and perfect
supercells, respectively, Ei is the energy per atom in hcp-Zr/fcc-Al unit cell (if remove atom from the supercell, Ei>0, or Ei<0). According to the calculation results presented in Table 2, the formation energy tendency was: AlZr < ZrAl < VZr < VAl. From the perspective of thermodynamic, a negative value of EF implies defects are more easily formed. Therefore, the antisite defects are easier to form than the vacancy defects, and the formation energy of AlZr defects is the lowest, only -0.230eV, which means that AlZr defects maybe the dominant point defects in Zr3Al. Table 2 The calculated formation energies of four intrinsic point defects. ∆H (eV) VZr
VAl
ZrAl
AlZr
1.989
3.216
1.886
-0.230
In order to verify the relative concentration relationship about different point defects revealed by the formation energy, and provide a basis for the discussion in the next section, we further investigated the change tendency of four intrinsic point defect concentrations with temperature, the dependence equation is as follows [36-40]: (
)
= exp −
(
)
= exp −
!" # !" #
$
(2)
$
(3)
%&'
(
)*
%)(
x+ Where ,/ , ,/ , ,+
− ,-.
= exp −
+
!" #
$
= ,+
(4) (5)
and ,-. + are the concentrations of VZr, VAl, ZrAl, and AlZr
in Zr3Al, respectively, and the ,+ represents the mole fraction of Zr in Zr3Al. Fig. 1 shows the variation of each defect concentration with temperature. It can be seen that since the defect formation is a thermal activation process, concentration rise with the increasing temperature for all defects. At the same temperature, the concentration of antisite defects is higher than the vacancy, and the concentration of VAl is the lowest, which qualitatively confirms the trend of the defect formation energy (AlZr < ZrAl < VZr < VAl) , that is, the higher the formation energy, the lower the concentration of the defect, the result is similar to the other L12 compounds. Furthermore, the effect of different defects on the stability of the system were investigated by comparing the electronic density of perfect and defective supercells. The calculation result is shown in Fig. 2, where the red dash line indicates the position of the Fermi level. From the point of view of quantum mechanics, the lower DOS value at the Fermi level means that more electrons participate in bonding process to produce electronic localization [41], so the stability of the system will be higher. Therefore, we can find that the stability of Zr3Al decreases once there are the point defects. The Fermi levels of ZrAl and AlZr are close to the peak position on the left side of the pseudopotential gap, that is, their Fermi level are in the bonding state, which indicates that the valence electron of Zr3Al fails to fill full of the valence band due to the formation of the antisite defects. Therefore, the stability of the system decreased.
Meantime, the Fermi levels of VZr and VAl are close to the valley position, and the DOS value is significantly lower. In other words, although from the perspective of formation energy that antisite defects are easily formed, the stability of the vacancy defects is higher, which explains the reason why the vacancy-mediated diffusion is the dominant diffusion mechanism. 3.2 Atomic diffusion mechanism in Zr3Al Point defect is the simplest crystal defect, which forms for the reason that atoms deviates from the normal alignment of the crystal structure at or near the microscopic region. Particularly, if the vibration of an atom is large sufficiently, it may overcome the restriction of the surrounding atoms and jump away from its original site, and then forms a vacancy. Once vacancies formed in the lattice, the surrounding atoms will lose their balance, then they will relax toward the vacancy direction. Therefore, the atom migration will be easier with the existence of vacancies. In general, the dominant diffusion mechanism in alloys and compounds is based on vacancies. There are two steps in the process of vacancy-mediated diffusion: first, a vacancy is formed in the crystal lattice; then, the site of diffusion atom and its first nearest neighbor vacancy exchange, i.e., a thermal activation jump occurs. Therefore, the energy required for vacancy-mediated diffusion consists of two parts, one is the defect formation energy, and the other is the diffusion migration energy barrier. Based on the defect formation energy and the diffusion migration barrier, the diffusion activation energy can be determined by the Eq. (6): 0
=∑
1
+
2
(6)
Where
0,
and
2
are the diffusion activation energy, the sum of the formation
energy of all point defects and migration energy in certain path. The point defect formation energies have been calculated in 3.1. In the following work, we use the climbing-image nudged elastic band (CI-NEB) method to calculate the migration barriers of different diffusion mechanism. Finally, the most feasible diffusion mechanism of Zr and Al atoms in Zr3Al will be determined by comparing the diffusion activation energy of different mechanism. 3.2.1 Zr atom diffusion The trend of the faster diffusion of the major element than that of the minor element in ordered L12 alloys has been termed “the ordered Cu3Au effect” by Heurle and Gas [42]. For example, in Ni3Ge [43-45], Pt3Mn [45] and Co3Ti [46], Ni, Pt and Co occupying the face center, the diffusion rate of them are faster than the minor element by an order of magnitude. In addition, the diffusion coefficient of Ni is 103 times that of the Al diffusion coefficient in Ni3Al [47, 48]. Therefore, we first studied the possible diffusion paths of Zr atom in Zr3Al, which are VZr-mediated diffusion mechanism, VAl-mediated diffusion mechanism and antisite bridge mechanism, the schematic diagram of different diffusion paths is given in the Fig. 3. The diffusion of the major element Zr can be regarded as the self-diffusion in a hypothetical pure crystal possessing the structure of the a sublattice, or ‘NbO lattice’ [49]. The exchange between the Zr atom and the vacancy in its sublattice is the simplest diffusion mechanism. As shown in Fig. 3, there are 8 equivalent Zr atoms in the nearest neighbor of each Zr atom. When VZr formed in these sites, the Zr atom
could diffuse by the VZr-mediated nearest neighbor (VZr-NN) [50, 51] jump mechanism (ie, the Zr atom at Z1 site makes a nearest neighbor jump to VZr at the Z2 site), and the degree of lattice order does not change. The energy profile of minimum energy paths (MEP) is shown in Fig. 3(b), the migration energy barrier of this mechanism is only 1.146eV. In addition to the NN jump, there are 4 equivalent next nearest neighbor diffusion (VZr-NNN) [52] paths (the Zr atom at Z1 site jumps to the VZr at Z3 site) due to the existence of 4 next nearest neighbor Zr atoms around the VZr, and the atomic migration barrier of this path is 6.40 eV, which is significantly higher than the migration barrier of the NN jump. This is because when the Zr atom make the nearest neighbor jump, it can jump directly from the initial state to the final state, but in the NNN mechanism, it needs to pass through a quadrilateral made by 4 Zr atoms at the nearest neighbor of it, during this process, the migration resistance increases with the larger lattice distortion, then the migration barrier became higher. As shown in the Fig .3(a), nearest neighbors of Zr atom has 4 equivalent Al atoms besides of 8 Zr atoms. Therefore, when VAl is formed, Zr atoms can also be diffused by VAl (VAl-NNN). After calculation, the migration barrier is 1.056 eV. In addition to the above three migration paths, Zr atom may also diffuse through the antisite bridge mechanism (ASB) [53-57] induced by VAl. The initial condition of this mechanism is that ZrAl antisite and VAl vacancy exist in the crystal lattice, and ZrAl is located in the NNN site of VAl. During the diffusion process, the Zr atom is firstly exchanged with the nearest neighbor VAl to produce the intermediate state VZr + ZrAl, and then the VZr and ZrAl sites are interchanged to form the final structure of VAl +
ZrZr. After the two steps, the degree of lattice order has not changed. By calculation, its diffusion energy barrier is 0.739 eV. Fig 3. (d) shows the diffusion activation energy calculated by the equal (5), the activation energies of the four diffusion paths of Zr atom are 3.135 eV, 8.393 eV, 4.272 eV and 5.841 eV, respectively. Therefore, from the point of energy, the NNN diffusion mechanism induced by VZr is the most feasible diffusion mechanism for Zr atom. This conclusion is consistent with others’ reports that the self-diffusion of the major element A is most possible via the ordinary vacancy mechanism on its own sublattice (a sublattice) in A3B alloys of the L12 type structure. 3.2.2 Al atom diffusion After determining the optimal diffusion path for Zr atom, we next analyzed and determined the feasible diffusion path for Al atom. Through the analysis of the electronic structure we can find that VZr has the highest stability in Zr3Al, so the VZr-mediated nearest neighbor (VZr-NN) jump for Al atom is studied firstly. As we can see from the Fig .4 (a), because of nearest neighbor sites of each Al atom are 12 Zr atoms, the Al atom can make the NN jump with the help of the existence of VZr. The migration energy of this path is 1.787 eV. However, it is worth noting that the migration of Al atoms appears not so easy as that of Zr atoms, a jump of an Al atom to a nearest-neighbor site will inevitably destroy the ordered arrangement. Although the concentration of VAl is higher than VZr for the stochiometric Zr3Al, the proportion of VAl in all point defects may rise while the composition of the alloy
deviates from the stochiometric dueto external conditions such as high temperature or radiation, so the VAl-mediated Al atom diffusion (VAl-NNN) cannot be ignored (the Al atom at the A6 site jump to the Z3 site). After calculation, the migration barrier is 4.442 eV, which is higher than the NN jump of Al atom induced by VZr. The reason may result from two aspects, on the one hand, the distance traveled by the diffusion Al atom is farther than the one of the VZr-mediated diffusion as the VAl is located at the NNN site of the Al atom, and on the other hand, the radius of Zr atom is larger than Al atom, so the distortion caused by VZr in the crystal lattice is more than the one induced by VAl. Therefore, the lattice of the VZr-containing lattice has less resistance to the Al atom. Although the jump of an Al atom from its regular position to a nearest neighbor vacancy disturbs the ordered arrangement, an antisite AlZr atom can exchange its position with a vacancy on a neighboring a sublattice site without causing further disordering. Therefore, Al atom may also diffuse through the antisite bridge (ASB) mechanism and the antisite assisted (AS) [43, 58] mechanism. In the ASB mechanism, AlZr and VZr are located at the NN site of each other, firstly, VZr exchange position with Al atom to form a composite defect AlZr+VAl, and then the newly formed VAl jump to the site of another AlZr, so that when the diffusion is finally completed, even if the two Al atoms are exchanged, the degree of lattice order has not changed, and the migration energy barrier is 1.91 eV. Moreover, the AS mechanism is another Al atom diffusion mechanism induced by VZr with the help of AlZr. When an AlZr exists in the crystal lattice, the diffusion Al atom jumps to the nearest neighbor VZr to accomplish
the migration (the Al atom at the A2 site jumps to the VAl at the A3 site), obviously, the jump mode does not affect the order of the crystal lattice. The calculated migration barrier is extremely low, only 0.649 eV. This may be due to the degree of distortion inward and outward is superimposed because of there are both antisite defects and vacancy defects in the crystal lattice, the lattice arrangement became more lose, therefore, the resistance caused by atom migration is small. Since the NN diffusion of Al atom mediated by VZr destroys the order of the structure, it is further considered that Al atom diffuse through a six-jump cycle (6JC) [55, 56] mechanism, which involves six atomic jumps: VZr + Al → VAl + AlZr; VAl + Zr → VZr + ZrAl; VZr + Al → VAl + AlZr; VAl + AlZr → VZr + Al; VZr + ZrAl → VAl + Zr; VAl + AlZr → VZr + Al, the schematic can be seen from the Fig .5 (a). Although the first three jumps reduce the order of the structure, the last three jumps can be regarded as the inverse of the first three jumps. Therefore, the order of the structure does not change when 6JC mechanism is completed. The 6JC mechanism can be subdivided into a straight 6JC and a bent 6JC. The former occurs on the (100) plane, while atomic diffusion occurs outside the (100) plane in the bent 6JC mechanism, which caused three-dimensional atom diffusion. After calculation, the energy barriers of them are 2.91 eV and 3.53 eV, respectively. By calculation, among all possible diffusion mechanisms of Al atom, the diffusion activation energy of the anti-assisted mechanism is the lowest, only 2.408eV, which indicates the difference between the most feasible diffusion mechanism is that the NN jump mechanism induced by VZr for Zr atom, but the anti-assisted mechanism for Al
atom, which is consistent with the diffusion mechanism of the secondary elements in other L12 phase structures. 4. Conclusions It should be remarked that a normal first-principles study based on density functional theory have been used to study the self-diffusion of the L12-Zr3Al in this work. Firstly, the formation energies of four intrinsic point defects in Zr3Al were calculated, and their relative stability was explained from the perspective of electronic structure. Next, the relationship between point defect concentration and temperature at stochiometric was considered. Finally, the atom diffusion barrier of different diffusion paths for Zr and Al atom in Zr3Al were calculated by CI-NEB method, the most feasible diffusion mechanism of Zr atom and Al atom were determined by comparing the diffusion activation energy. The specific conclusions are as follows: 1.
Although the results of point defect formation energy indicate that the AlZr defect
is the highest concentration defect type in Zr3Al, the stability of VZr and VAl is higher. Therefore, the vacancy-mediated diffusion mechanism is the main diffusion mechanism in Zr3Al. 2.
The concentration of the four intrinsic point defects in Zr3Al increases with the
increasing temperature, and at the same temperature, the concentration of the antisite defects is higher than the vacancy defects, and the concentration of VAl is the lowest. 3.
From the perspective of energy, the NN jump mechanism induced by the VZr is
the most feasible diffusion mechanism for Zr atom, and the antisite assisted mechanism for Al atom has the lowest diffusion activation energy barrier, so it is the
most dominant diffusion mechanism.
Acknowledgments This work was supported by the National Key R&D Program of China (grant 2018YFA0703602), NSFC (grant 51571174), National Science Foundation for Distinguished Young Scholars for Hebei Province of China (grant E2016203376), and Hundred Excellent Innovative Talents Support Program in Hebei Province (grant SLRC2017056).
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Figure captions Fig. 1. The calculated equilibrium point defect concentrations in stoichiometric L12-Zr3Al as a function of temperature. Fig. 2. The total density of states (DOS) of perfect and defective Zr3Al, in which the red dash line indicates the position of Fermi level. Fig. 3. The schematic diagram, minimum energy paths (MEPs) and activation energy of different diffusion mechanism for Zr atom in Zr3Al. (a) A schematic of different diffusion mechanism for Zr atom; (b) The MEPs of VZr (NN) and VZr (NNN); (c) The MEP of VAl (NN); (d) The MEP of ASB. Fig. 4. The schematic diagram, minimum energy paths (MEPs) and activation energy of different diffusion mechanism for Al atom in Zr3Al. (a) A schematic of different diffusion mechanism for Al atom; (b) The MEP of VZr (NN); (c) The MEP of VAl (NNN); (d) The MEPs of ASB and AS. Fig. 5. The schematic diagram, minimum energy paths (MEPs) and activation energy of six-jump cycle (6JC) mechanisms in Zr3Al. (a) A schematic of straight and bent 6JC mechanisms; (b) The MEP of straight 6JC; (c) The MEP of bent 6JC.
Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Highlights 1. The point defects formation energies are determined in Zr3Al. 2. The concentration of the four intrinsic point defects increases with the increasing of temperature. 3. The nearest-neighbor jump mechanism induced by VZr is the most feasible diffusion mechanism for Zr atom. 4. Al mainly diffuses by the antisite assisted mechanism.
Zhixin Ren: Methodology, Investigation, Writing-Original Draft, Writing-Review & Editing. Zhe Xue: Investigation, Writing-Review & Editing. Xinyu Zhang: Methodology, Writing-Review & Editing, Project administration; Funding acquisition. Jiaqian Qin: Writing-Review & Editing. Mingzhen Ma: Writing-Review & Editing. Riping Liu: Writing-Review & Editing, Funding acquisition.
Declaration of interests ☒ 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. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: