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Ideal strengths and thermodynamic properties of W and W-Re alloys from first-principles calculation
Fusion Engineering and Design 155 (2020) 111579
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Ideal strengths and thermodynamic properties of W and W-Re alloys from first-principles calculation
T
Qian Wanga,b, Guoping Dua,*, Nan Chena, Changshuang Jianga, Liang Chenc,* a
School of Materials Science and Engineering, Nanchang University, Nanchang, Jiangxi, 330031, China School of Environmental and Chemical Engineering, Nanchang Hangkong University, Nanchang, Jiangxi, 330063, China c School of Materials Science and Engineering, Nanchang Hangkong University, Nanchang, Jiangxi, 330063, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: W W-Re alloys Ideal strengths Thermodynamic properties First-principles calculation
First-principles calculation, quasi-harmonic approximation and thermal electronic excitation have been combined to examine ideal strengths and thermodynamic properties of W and W-Re alloys. It is found that the ideal tensile and shear strengths of W are decreased due to Re alloying, with the easy cleavage plane of both W and WRe alloys found in (100) plane. Moreover, the derived temperature-dependent elastic properties explain the “Re softening effect”, and provide a deep understanding of the ductility of W-Re alloys as a function of the temperature. The results also reveal that Re alloying significantly increases the ductility of W, which is rare at low temperature in the temperature range 0−2000 K. The calculated results agree well with similar experimental observations in the literatures, and could explore their fundamental properties as well as various potential applications.
1. Introduction Tungsten (W) has been well regarded as one of the most promising plasma-facing component (PFC) under a fusion environment [1–4], mainly due to its advantageous properties such as high melting point, good creep resistance, low sputtering yield as well as deuterium/tritium retention, and excellent high-temperature mechanical properties [5–8]. Unfortunately, the low ductility at room temperature and poor radiation stability of tungsten severely limits its applications in fusion reactors [9–11]. Some researchers aim to solve the problems by means of the alloying and grain refinement of tungsten. Alloying elements (V, Ta, Mo, Re and Ti) [12–14] and ceramic phase (Y2O3, ZrC, TiC and La2O3) [15–17] are added in tungsten to improve the ductility and irradiation resistance. Rhenium (Re) is generated directly from the high fluxes of neutrons in fusion reactions, attracting attentions in the aspect of ductility of W [18,19]. There are already published work in the properties of W-Re alloys [14,20–23]. Experimental methods were performed to obtain the yield strength and the toughness of W-Re alloys, which predict that the ductility of W is enhanced by the alloying of Re at room temperature [22,23]. Theoretical calculations were used to explore the mechanism of the increased ductility of W in terms of stacking fault energy and dislocation movement [14,20]. Nevertheless, the influence of temperature in the results above is missing, which has a dominant role on ⁎
the properties of W-Re alloys in fusion reactors. In addition, the ideal strengths of W-Re alloys are important to understand the effects of Re alloying in W. There is no systematic predictions in the literatures to our best knowledge. Through a theoretical combination of first-principles calculation, quasi-harmonic approximation (QHA), and thermal electronic excitation (TEE), the pursuing aim is to systematically reveal the ideal strengths and thermodynamic properties of W and W-Re alloys. The derived results were compared with experimental observations in the literatures, giving a further research of ductility of W-Re alloys as a function of the temperature. The fundamental mechanism will be revealed by characterising electronic structures and elastic properties. 2. Theoretical methods The widely used VASP (Vienna Ab initio Simulation Package) code has been employed in the present calculations with a plane-wave basis set and the projected augmented wave (PAW) approach [24,25]. The exchange-correlation function was described using the Perdew–Burke–Ernzerhof (PBE) form of generalized gradient approximation (GGA) [26], and the cutoff energy was set at 400 eV. The first-order smearing method of Methfessel − Paxton [27] and the tetrahedron method of Blöchl − Jepsen − Andersen [28] were adopted for relaxation and static calculation, respectively.
Corresponding authors. E-mail addresses: [email protected] (G. Du), [email protected] (L. Chen).
https://doi.org/10.1016/j.fusengdes.2020.111579 Received 12 December 2019; Received in revised form 20 February 2020; Accepted 20 February 2020 Available online 28 February 2020 0920-3796/ © 2020 Elsevier B.V. All rights reserved.
Fusion Engineering and Design 155 (2020) 111579
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According to the previous experimental and theoretical reports, the concentration of Re in W reaches above ∼25 % in the bcc structure [18,29]. However, the formation of Re precipitates should bring out in fusion reactors [30], and the small amount of Re alloying in W would decrease the solubility and increase the diffusivity of hydrogen isotopes [31]. Accordingly, a 2 × 2×2 (16 atoms) unit cell was chosen for pure W with the bcc structure, and one Re atom was introduced to substitute one W atom. The derived model of W15Re is close to the experimental composition of W-7%Re [22]. In addition, the ideal strengths (tensile and shear) of W and W15Re were calculated by incrementally deforming the unit cell in the direction of the performed strain and simultaneously relaxing both the atomic basis vectors orthogonal to the strain, as well as all internal atomic coordinates. To keep the strain path continuous, the starting position at each strain step was taken from the relaxed coordinates of the previous step [32]. The Monkhorst-Pack k meshes of 9 × 9×9 and 11 × 11 × 11 as well as the energy criteria of 0.01 and 0.001 meV were used for relaxation and static calculations, respectively. All internal atomic positions were fully optimized until the force on each atom is less than 0.025 eV/Å [33]. The thermodynamic properties of both W and W15Re are computed in terms of the quasi-harmonic approximation (QHA) and thermal electronic excitation (TEE). More specifically, the Helmholtz free energy F(V, T) is obtained through the summation of static total energy E0(V), the vibrational Fph(V, T) and electronic Fel(V, T) free energy, which could be further calculated according to the method used by Wang, et al. [34] and Liang, et al. [35]. After F(V, T) was derived at various volume of continuous intervals, the Vinet’s equation of states (EOS) were adopted to fit the temperature-dependent volume V(T)0 at ambient pressure,
F (V , T ) = F (V , T )0 +
Fig. 1. Tensile stress-strain curves of W and W15Re in the < 100 > , < 110 > and < 111 > directions.
4B (V , T )0 V (T )0 2B (V , T )0 V (T )0 − [B′ (V , T )0 − 1]2 [B′ (V , T )0 − 1]2
⎧3[B′ (V , T )0 − 1][( V (T ) )1/3 − 1] + 2⎫ exp ⎧ ⎨ ⎬ ⎨ V (T )0 ⎩ ⎭ ⎩ V (T ) 1/3 3 − [B′ (V , T )0 − 1][( ) − 1]⎫ ⎬ V (T )0 2 ⎭
(1)
Fig. 2. Shear stress-strain curves of W and W15Re along the [001] and [011] directions of (100) plane.
here B(V, T)0 and B’(V, T)0 is equal to bulk modulus and pressure derivative of bulk modulus, respectively. The coefficient of thermal expansion (CTE), α(T), was estimated by the following equation:
1 ∂V(T)0 α(T) = V(T)0 ∂T
deformation. The ideal shear strengths of W and W15Re in the (100) plane are thus investigated, and Fig. 2 plots the stress-strain curves of both W and W15Re along different directions under shear deformation. It can be seen clearly from Fig. 1 that the peak stresses of W15Re along (100) [001] and (100)[011] shear directions are 27.31 and 31.47 GPa, respectively, which are also lower than those of W (27.68 and 32.07 GPa) in these directions. Such a coincidence confirms that Re alloying reduces the shear resistance of W, which agrees well with other reported experimental results [37]. Specifically, geometric effects have a prominent role on the ideal strengths of W and W15Re with bcc structure, which may be the possible reason why the lowest shear strength displays along (100)[001] and the lowest tensile strength shows in < 100 > . In addition, a careful observation reveals that there is a different phenomenon between W and W15Re along the (100)[011] shear direction. In other words, the derived shear strength of W15Re gradually increases when the strain exceeds the ideal critical strain. The peak stresses of W have an opposite trend, probably due to the modified geometry of Re alloying in W during the process of shear deformation. It is of interest to have a deep understanding of ideal strengths at an electronic scale, and the electronic structures of both W and W15Re under the tensile and shear deformations are thus derived. Fig. 3 summarizes the comparison of total density of states (TDOS) of W15Re in the ideal critical strain under tensile and shear deformations. In order to investigate the intrinsic mechanism of the softness effects, the present charge density difference of W15Re is also derived and shown in
(2)
3. Results and discussions 3.1. Ideal strengths Prior to investigating the ideal strength, the lattice constants of W and W15Re with bcc structures were calculated as 3.172 and 3.167 Å, which are in accordance with the corresponding experimental results in the literature [31]. And the ideal tensile stress-strain curves of W and W15Re were calculated in three main crystal directions and shown in Fig. 1. Some characteristics are observed from this figure. Firstly, the tensile strengths of W are 28.97, 50.16, and 39.35 GPa along the < 100 > , < 110 > and < 111 > directions, respectively, which agree well with other computed values of 29.1, 49.2 and 37.6 GPa [36]. The excellent agreements above suggest that the present PAW-PBE approach reveals the structural and mechanical properties of W and W15Re. Secondly, the peak stresses of W15Re (27.51, 48.81 and 38.27 GPa) are lower than the corresponding values of W in the three directions, respectively, which may be due to the softness effects of the Re composition, and is consistent with the experimental observations of Ref. [22] and Ref. [23]. Thirdly, it is found that the weakest tensile strengths of W and W15Re are presented in the < 100 > direction, and thus an (100) plane is the easy cleavage plane under tensile 2
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Fig. 5. Temperature-dependent coefficient of linear thermal expansion (αL) of W and W15Re in different pressures. Triangles up and down denote experimental data from Refs. [39,40], respectively.
Fig. 3. Total density of states (TDOS) of W15Re in the ideal critical strain under tensile and shear deformations, and the present charge density difference of W15Re are also shown in this figure. The gray and red balls represent W and Re atoms, respectively, and the isosurface value is set to 0.002 e/Bohr3 (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).
3.2. Thermodynamic properties The thermodynamic properties of W and W15Re were investigated, and Fig. 4 shows the obtained temperature-dependent total isochoric heat capacity (CV ), isobaric heat capacity (CP ), electronic (CVel ), vibrational (CVph ) heat capacities of W, and the comparative CP of W and W15Re at the temperature of 0−2000 K. It can be deduced from Fig. 4(a) that the present CP value of W at 500 K is 25.01 J/K/mol, which is in excellent agreement with the corresponding data of 24.79 and 25.88 J/K/mol from experimental measurements [38,39]. When the temperature is higher than 1000 K, the differences between calculated and experimental values become slightly bigger, suggesting that the calculated values from the present quasi-harmonic Debye model are generally underestimated at high temperature. Moreover, the curve value of CVph within the temperature range is much bigger than that of CVel , predicting that the electrons have much less contribution in the heat capacity of W. In addition, we can discern from Fig. 4(b) that the
Fig. 3. It should be noted that the yellow and slight blue regions correspond to the charge accumulation and depletion, respectively. It can be deduced from Fig. 3 that the main TDOS peaks of W15Re along < 111 > tensile direction located at about -3.0 and -5.0 eV below the Fermi level (Ef) are much higher than those of shear deformation, which implies that the W-Re bond has formed a stronger bonding under tensile deformation, providing a reasonable interpretation for the higher ideal strength shown in Figs. 1 and 2. In addition, it is clear from Fig. 3 that the strong charge accumulates around the Re atom, and the electron distribution is spherical and unobvious direction, which shows that the W-Re interaction strongly enhances metallic properties. The strong metallic bonds might explain why the Re alloying in W decreases the strength and improves the ductility from experimental observations [22,23].
Fig. 4. (a) Temperature-dependent CV , CP , CVel and CVph of W, and (b) the comparative CP of W and W15Re with the temperature range. The experimental results in the literatures [38,39] are also include for the sake of comparison. 3
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Fig. 6. Comparison of the temperature-dependent elastic properties of W and W15Re. (a)-(c) present elastic constants (C11, C12, C44), and d)-(f) denote polycrystalline elastic moduli (B, G, E), respectively.
which is determined through the fitting of EOS, is substituted with the Cij(V) and B(V), respectively. Through a simple mathematical change, the temperature-dependent elastic constants, Cij (T ) and bulk modulus, B (T ) , can be derived. The derived elastic constants are present to calculate the temperature-dependent shear modulus (G) and Young’s modulus (E) of both polycrystalline W and W15Re through the Voigt–Reuss–Hill’s approximations [42]. After a series of calculations, the obtained temperature-dependent elastic properties of both W and W15Re were shown in Fig. 6. It can be seen obviously from Fig. 6(a)–(c) that the elastic constants (C11, C12 and C44) of both W and W15Re decrease when the temperature increases from 0 K to 2000 K, and an almost linear decrease is observed above ∼200 K. Noting from Fig. 6(d)–(f) that at this temperature range, the same phenomenon can be applicable for the elastic moduli (B, G, E). It is deduced from Fig. 4 that the elastic constants and moduli of both W and W15Re have a descending sequence of C11 →C12 →C44 and E →B →G within the entire temperature range, respectively. Consequently, it was observed from Fig. 4(a)–(c) that the C11 and C44 curves of W15Re are located below those of W within the entire temperature range, and that the C12 values have an opposite trend. The derived elastic moduli of W15Re have a similar situation, W15Re has a slightly bigger B values and smaller E and G values than those of W at the temperature of 0−2000 K, respectively, which may be the possible reason why the tensile and shear strengths of W-Re alloys slightly drops from the above calculation. To further investigate the ductility of both W and W15Re as a function of temperature, the temperature-dependent G/B ratios were calculated, and the derived results were shown in Fig. 7. Generally, the G/B value from Pugh et al. has been extensively adopted as a parameter to describe the brittleness/ductility of materials [43], which means, a smaller G/B value means more ductility, and vice versa. According to our current results, the G/B values of both W and W15Re increase almost linearly with the temperature rising, with a dramatical increase above ∼200 K. The result indicates that the ductility of both W and W15Re decreases as the temperature increases, which is in good agreement with experimental investigation in the literature [44]. More importantly, at a low temperature, there is a much smaller G/ B value of W15Re than that of W, predicting that the alloying element of Re can strongly increase the ductility of W. It was observed from
Fig. 7. Calculated the G/B ratios of W and W15Re as a function of temperature.
Cp value of W15Re at a certain temperature is generally bigger than that of W, suggesting that Re alloying can increase the heat capacity of W. Fig. 5 displays the temperature-dependent coefficients of linear thermal expansion (αL) of W and W15Re under different pressures. It is seen from Fig. 5 that our simulations of W listed match well with similar experimental results in the literatures [39,40]. The excellent agreements suggest that the Debye model can reveal the thermodynamic properties of both W and W15Re. It is also observed that αL values of W and W15Re increase dramatically in the temperature range from 0 to 250 K, followed by a gradual and almost liner increasing above 250 K. At a certain temperature, the αL decreases with the rising of pressure. It should be noted that there is a different relationship of αL between W and W15Re as a function of pressure, i.e., the αL curve of W is located below that of W15Re in the entire temperature range under the pressure of 0 and 50 GPa, with the opposite under 100 GPa. In order to further study the elastic properties of both W and W15Re as a function of temperature. The volume, in the range from V/V0 = 0.8–1.2 with an interval of 0.025, is used to obtain elastic constants Cij (V) by means of the universal-linear-independent coupling-strain (ULICS) method [41]. The volume-dependent bulk modulus, B(V), is computed in terms of Eq. (1) in theoretical methods section. The V(T)0, 4
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Fig. 6(d)–(f), at a certain temperature, that a smaller G and bigger B value of W15Re ultimately induce the increase of the ductility. It should also be pointed out from Fig. 6(d)–(e), with the increase of the temperature, that the B value of W15Re generally decreases and gets closer to that of W, and that the difference of G value between W and W15Re is almost unchanged. Such a comparison suggests that Re alloying brings improved ductility at the low temperature. Further studies are welcome to investigate the effects of dislocation and defect on ductility of W and W-Re alloys in the future.
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4. Conclusions In summary, first-principles calculation, quasi-harmonic approximation, and thermal electronic excitation have been combined to systematically investigate ideal tensile and shear strengths of W and WRe alloys, as well as temperature-dependent thermodynamic properties. The results demonstrate that Re alloying can soften W in the tensile and shear deformations. The G/B ratios of W-Re alloys are lower than those of W in the temperature range 0−2000 K, suggesting that Re alloying is an effective approach to improve the ductility of W under a fusion environment. The improved ductility is mainly due to the smaller G and bigger B values of W-Re alloys compared with those of W. Additionally, at the low temperature, the ductility of W-Re alloys is enhanced significantly, which is less obvious with the increase of temperature. The derived results were deeply explained by means of various discussions, and are in good agreement with similar experimental and theoretical observations in the literatures. Credit author statement Qian Wang as first author, performed first-principles calculation and analyses under the direction of Prof. Guoping Du and Prof. Liang Chen. All authors participated in the discussion and writing of this manuscript. Funding This work was supported by National Natural Science Foundation of China Grant No. 21571095, 61366004 and 51362020. Declaration of Competing Interest The authors declare that they have no conflict of interest. Appendix A. Supplementary data Supplementary material related to this article can be found, in the online version, at doi:https://doi.org/10.1016/j.fusengdes.2020. 111579. References [1] M. Rieth, J. Boutard, S. Dudarev, T. Ahlgren, S. Antusch, N. Baluc, et al., Review on the EFDA programme on tungsten materials technology and science, J. Nucl. Mater. 417 (2011) 463–467. [2] C. Linsmeier, M. Rieth, J. Aktaa, T. Chikada, A. Hoffmann, J. Hoffmann, et al., Development of advanced high heat flux and plasma-facing materials, Nucl. Fusion 57 (2017) 092007. [3] H. Bolt, V. Barabash, W. Krauss, J. Linke, R. Neu, S. Suzuki, N. Yoshida, Materials for the plasma-facing components of fusion reactors, J. Nucl. Mater. 329-333 (2004) 66–73. [4] L. Chen, J.L. Fan, H.R. Gong, Atomistic simulation of mechanical properties of tungsten-hydrogen system and hydrogen diffusion in tungsten, Solid State Commun. 306 (2020) 113772. [5] S.W.H. Yih, C.T. Wang, Tungsten-sources, Metallurgy, Properties and Applications, Plenum Press, New York, 1979. [6] S. Wurster, N. Baluc, M. Battabyal, T. Crosby, J. Du, C. García-Rosales, et al., Recent progress in R&D on tungsten alloys for divertor structural and plasma facing materials, J. Nucl. Mater. 442 (2013) S181–S189. [7] R.A. Pitts, S. Carpentier, F. Escourbiac, T. Hirai, V. Komarov, S. Lisgo, et al., A full tungsten divertor for ITER: physics issues and design status, J. Nucl. Mater. 438
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Ideal strengths and thermodynamic properties of W and W-Re alloys from first-principles calculation
T
Qian Wanga,b, Guoping Dua,*, Nan Chena, Changshuang Jianga, Liang Chenc,* a
School of Materials Science and Engineering, Nanchang University, Nanchang, Jiangxi, 330031, China School of Environmental and Chemical Engineering, Nanchang Hangkong University, Nanchang, Jiangxi, 330063, China c School of Materials Science and Engineering, Nanchang Hangkong University, Nanchang, Jiangxi, 330063, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: W W-Re alloys Ideal strengths Thermodynamic properties First-principles calculation
First-principles calculation, quasi-harmonic approximation and thermal electronic excitation have been combined to examine ideal strengths and thermodynamic properties of W and W-Re alloys. It is found that the ideal tensile and shear strengths of W are decreased due to Re alloying, with the easy cleavage plane of both W and WRe alloys found in (100) plane. Moreover, the derived temperature-dependent elastic properties explain the “Re softening effect”, and provide a deep understanding of the ductility of W-Re alloys as a function of the temperature. The results also reveal that Re alloying significantly increases the ductility of W, which is rare at low temperature in the temperature range 0−2000 K. The calculated results agree well with similar experimental observations in the literatures, and could explore their fundamental properties as well as various potential applications.
1. Introduction Tungsten (W) has been well regarded as one of the most promising plasma-facing component (PFC) under a fusion environment [1–4], mainly due to its advantageous properties such as high melting point, good creep resistance, low sputtering yield as well as deuterium/tritium retention, and excellent high-temperature mechanical properties [5–8]. Unfortunately, the low ductility at room temperature and poor radiation stability of tungsten severely limits its applications in fusion reactors [9–11]. Some researchers aim to solve the problems by means of the alloying and grain refinement of tungsten. Alloying elements (V, Ta, Mo, Re and Ti) [12–14] and ceramic phase (Y2O3, ZrC, TiC and La2O3) [15–17] are added in tungsten to improve the ductility and irradiation resistance. Rhenium (Re) is generated directly from the high fluxes of neutrons in fusion reactions, attracting attentions in the aspect of ductility of W [18,19]. There are already published work in the properties of W-Re alloys [14,20–23]. Experimental methods were performed to obtain the yield strength and the toughness of W-Re alloys, which predict that the ductility of W is enhanced by the alloying of Re at room temperature [22,23]. Theoretical calculations were used to explore the mechanism of the increased ductility of W in terms of stacking fault energy and dislocation movement [14,20]. Nevertheless, the influence of temperature in the results above is missing, which has a dominant role on ⁎
the properties of W-Re alloys in fusion reactors. In addition, the ideal strengths of W-Re alloys are important to understand the effects of Re alloying in W. There is no systematic predictions in the literatures to our best knowledge. Through a theoretical combination of first-principles calculation, quasi-harmonic approximation (QHA), and thermal electronic excitation (TEE), the pursuing aim is to systematically reveal the ideal strengths and thermodynamic properties of W and W-Re alloys. The derived results were compared with experimental observations in the literatures, giving a further research of ductility of W-Re alloys as a function of the temperature. The fundamental mechanism will be revealed by characterising electronic structures and elastic properties. 2. Theoretical methods The widely used VASP (Vienna Ab initio Simulation Package) code has been employed in the present calculations with a plane-wave basis set and the projected augmented wave (PAW) approach [24,25]. The exchange-correlation function was described using the Perdew–Burke–Ernzerhof (PBE) form of generalized gradient approximation (GGA) [26], and the cutoff energy was set at 400 eV. The first-order smearing method of Methfessel − Paxton [27] and the tetrahedron method of Blöchl − Jepsen − Andersen [28] were adopted for relaxation and static calculation, respectively.
Corresponding authors. E-mail addresses: [email protected] (G. Du), [email protected] (L. Chen).
https://doi.org/10.1016/j.fusengdes.2020.111579 Received 12 December 2019; Received in revised form 20 February 2020; Accepted 20 February 2020 Available online 28 February 2020 0920-3796/ © 2020 Elsevier B.V. All rights reserved.
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According to the previous experimental and theoretical reports, the concentration of Re in W reaches above ∼25 % in the bcc structure [18,29]. However, the formation of Re precipitates should bring out in fusion reactors [30], and the small amount of Re alloying in W would decrease the solubility and increase the diffusivity of hydrogen isotopes [31]. Accordingly, a 2 × 2×2 (16 atoms) unit cell was chosen for pure W with the bcc structure, and one Re atom was introduced to substitute one W atom. The derived model of W15Re is close to the experimental composition of W-7%Re [22]. In addition, the ideal strengths (tensile and shear) of W and W15Re were calculated by incrementally deforming the unit cell in the direction of the performed strain and simultaneously relaxing both the atomic basis vectors orthogonal to the strain, as well as all internal atomic coordinates. To keep the strain path continuous, the starting position at each strain step was taken from the relaxed coordinates of the previous step [32]. The Monkhorst-Pack k meshes of 9 × 9×9 and 11 × 11 × 11 as well as the energy criteria of 0.01 and 0.001 meV were used for relaxation and static calculations, respectively. All internal atomic positions were fully optimized until the force on each atom is less than 0.025 eV/Å [33]. The thermodynamic properties of both W and W15Re are computed in terms of the quasi-harmonic approximation (QHA) and thermal electronic excitation (TEE). More specifically, the Helmholtz free energy F(V, T) is obtained through the summation of static total energy E0(V), the vibrational Fph(V, T) and electronic Fel(V, T) free energy, which could be further calculated according to the method used by Wang, et al. [34] and Liang, et al. [35]. After F(V, T) was derived at various volume of continuous intervals, the Vinet’s equation of states (EOS) were adopted to fit the temperature-dependent volume V(T)0 at ambient pressure,
F (V , T ) = F (V , T )0 +
Fig. 1. Tensile stress-strain curves of W and W15Re in the < 100 > , < 110 > and < 111 > directions.
4B (V , T )0 V (T )0 2B (V , T )0 V (T )0 − [B′ (V , T )0 − 1]2 [B′ (V , T )0 − 1]2
⎧3[B′ (V , T )0 − 1][( V (T ) )1/3 − 1] + 2⎫ exp ⎧ ⎨ ⎬ ⎨ V (T )0 ⎩ ⎭ ⎩ V (T ) 1/3 3 − [B′ (V , T )0 − 1][( ) − 1]⎫ ⎬ V (T )0 2 ⎭
(1)
Fig. 2. Shear stress-strain curves of W and W15Re along the [001] and [011] directions of (100) plane.
here B(V, T)0 and B’(V, T)0 is equal to bulk modulus and pressure derivative of bulk modulus, respectively. The coefficient of thermal expansion (CTE), α(T), was estimated by the following equation:
1 ∂V(T)0 α(T) = V(T)0 ∂T
deformation. The ideal shear strengths of W and W15Re in the (100) plane are thus investigated, and Fig. 2 plots the stress-strain curves of both W and W15Re along different directions under shear deformation. It can be seen clearly from Fig. 1 that the peak stresses of W15Re along (100) [001] and (100)[011] shear directions are 27.31 and 31.47 GPa, respectively, which are also lower than those of W (27.68 and 32.07 GPa) in these directions. Such a coincidence confirms that Re alloying reduces the shear resistance of W, which agrees well with other reported experimental results [37]. Specifically, geometric effects have a prominent role on the ideal strengths of W and W15Re with bcc structure, which may be the possible reason why the lowest shear strength displays along (100)[001] and the lowest tensile strength shows in < 100 > . In addition, a careful observation reveals that there is a different phenomenon between W and W15Re along the (100)[011] shear direction. In other words, the derived shear strength of W15Re gradually increases when the strain exceeds the ideal critical strain. The peak stresses of W have an opposite trend, probably due to the modified geometry of Re alloying in W during the process of shear deformation. It is of interest to have a deep understanding of ideal strengths at an electronic scale, and the electronic structures of both W and W15Re under the tensile and shear deformations are thus derived. Fig. 3 summarizes the comparison of total density of states (TDOS) of W15Re in the ideal critical strain under tensile and shear deformations. In order to investigate the intrinsic mechanism of the softness effects, the present charge density difference of W15Re is also derived and shown in
(2)
3. Results and discussions 3.1. Ideal strengths Prior to investigating the ideal strength, the lattice constants of W and W15Re with bcc structures were calculated as 3.172 and 3.167 Å, which are in accordance with the corresponding experimental results in the literature [31]. And the ideal tensile stress-strain curves of W and W15Re were calculated in three main crystal directions and shown in Fig. 1. Some characteristics are observed from this figure. Firstly, the tensile strengths of W are 28.97, 50.16, and 39.35 GPa along the < 100 > , < 110 > and < 111 > directions, respectively, which agree well with other computed values of 29.1, 49.2 and 37.6 GPa [36]. The excellent agreements above suggest that the present PAW-PBE approach reveals the structural and mechanical properties of W and W15Re. Secondly, the peak stresses of W15Re (27.51, 48.81 and 38.27 GPa) are lower than the corresponding values of W in the three directions, respectively, which may be due to the softness effects of the Re composition, and is consistent with the experimental observations of Ref. [22] and Ref. [23]. Thirdly, it is found that the weakest tensile strengths of W and W15Re are presented in the < 100 > direction, and thus an (100) plane is the easy cleavage plane under tensile 2
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Fig. 5. Temperature-dependent coefficient of linear thermal expansion (αL) of W and W15Re in different pressures. Triangles up and down denote experimental data from Refs. [39,40], respectively.
Fig. 3. Total density of states (TDOS) of W15Re in the ideal critical strain under tensile and shear deformations, and the present charge density difference of W15Re are also shown in this figure. The gray and red balls represent W and Re atoms, respectively, and the isosurface value is set to 0.002 e/Bohr3 (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).
3.2. Thermodynamic properties The thermodynamic properties of W and W15Re were investigated, and Fig. 4 shows the obtained temperature-dependent total isochoric heat capacity (CV ), isobaric heat capacity (CP ), electronic (CVel ), vibrational (CVph ) heat capacities of W, and the comparative CP of W and W15Re at the temperature of 0−2000 K. It can be deduced from Fig. 4(a) that the present CP value of W at 500 K is 25.01 J/K/mol, which is in excellent agreement with the corresponding data of 24.79 and 25.88 J/K/mol from experimental measurements [38,39]. When the temperature is higher than 1000 K, the differences between calculated and experimental values become slightly bigger, suggesting that the calculated values from the present quasi-harmonic Debye model are generally underestimated at high temperature. Moreover, the curve value of CVph within the temperature range is much bigger than that of CVel , predicting that the electrons have much less contribution in the heat capacity of W. In addition, we can discern from Fig. 4(b) that the
Fig. 3. It should be noted that the yellow and slight blue regions correspond to the charge accumulation and depletion, respectively. It can be deduced from Fig. 3 that the main TDOS peaks of W15Re along < 111 > tensile direction located at about -3.0 and -5.0 eV below the Fermi level (Ef) are much higher than those of shear deformation, which implies that the W-Re bond has formed a stronger bonding under tensile deformation, providing a reasonable interpretation for the higher ideal strength shown in Figs. 1 and 2. In addition, it is clear from Fig. 3 that the strong charge accumulates around the Re atom, and the electron distribution is spherical and unobvious direction, which shows that the W-Re interaction strongly enhances metallic properties. The strong metallic bonds might explain why the Re alloying in W decreases the strength and improves the ductility from experimental observations [22,23].
Fig. 4. (a) Temperature-dependent CV , CP , CVel and CVph of W, and (b) the comparative CP of W and W15Re with the temperature range. The experimental results in the literatures [38,39] are also include for the sake of comparison. 3
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Fig. 6. Comparison of the temperature-dependent elastic properties of W and W15Re. (a)-(c) present elastic constants (C11, C12, C44), and d)-(f) denote polycrystalline elastic moduli (B, G, E), respectively.
which is determined through the fitting of EOS, is substituted with the Cij(V) and B(V), respectively. Through a simple mathematical change, the temperature-dependent elastic constants, Cij (T ) and bulk modulus, B (T ) , can be derived. The derived elastic constants are present to calculate the temperature-dependent shear modulus (G) and Young’s modulus (E) of both polycrystalline W and W15Re through the Voigt–Reuss–Hill’s approximations [42]. After a series of calculations, the obtained temperature-dependent elastic properties of both W and W15Re were shown in Fig. 6. It can be seen obviously from Fig. 6(a)–(c) that the elastic constants (C11, C12 and C44) of both W and W15Re decrease when the temperature increases from 0 K to 2000 K, and an almost linear decrease is observed above ∼200 K. Noting from Fig. 6(d)–(f) that at this temperature range, the same phenomenon can be applicable for the elastic moduli (B, G, E). It is deduced from Fig. 4 that the elastic constants and moduli of both W and W15Re have a descending sequence of C11 →C12 →C44 and E →B →G within the entire temperature range, respectively. Consequently, it was observed from Fig. 4(a)–(c) that the C11 and C44 curves of W15Re are located below those of W within the entire temperature range, and that the C12 values have an opposite trend. The derived elastic moduli of W15Re have a similar situation, W15Re has a slightly bigger B values and smaller E and G values than those of W at the temperature of 0−2000 K, respectively, which may be the possible reason why the tensile and shear strengths of W-Re alloys slightly drops from the above calculation. To further investigate the ductility of both W and W15Re as a function of temperature, the temperature-dependent G/B ratios were calculated, and the derived results were shown in Fig. 7. Generally, the G/B value from Pugh et al. has been extensively adopted as a parameter to describe the brittleness/ductility of materials [43], which means, a smaller G/B value means more ductility, and vice versa. According to our current results, the G/B values of both W and W15Re increase almost linearly with the temperature rising, with a dramatical increase above ∼200 K. The result indicates that the ductility of both W and W15Re decreases as the temperature increases, which is in good agreement with experimental investigation in the literature [44]. More importantly, at a low temperature, there is a much smaller G/ B value of W15Re than that of W, predicting that the alloying element of Re can strongly increase the ductility of W. It was observed from
Fig. 7. Calculated the G/B ratios of W and W15Re as a function of temperature.
Cp value of W15Re at a certain temperature is generally bigger than that of W, suggesting that Re alloying can increase the heat capacity of W. Fig. 5 displays the temperature-dependent coefficients of linear thermal expansion (αL) of W and W15Re under different pressures. It is seen from Fig. 5 that our simulations of W listed match well with similar experimental results in the literatures [39,40]. The excellent agreements suggest that the Debye model can reveal the thermodynamic properties of both W and W15Re. It is also observed that αL values of W and W15Re increase dramatically in the temperature range from 0 to 250 K, followed by a gradual and almost liner increasing above 250 K. At a certain temperature, the αL decreases with the rising of pressure. It should be noted that there is a different relationship of αL between W and W15Re as a function of pressure, i.e., the αL curve of W is located below that of W15Re in the entire temperature range under the pressure of 0 and 50 GPa, with the opposite under 100 GPa. In order to further study the elastic properties of both W and W15Re as a function of temperature. The volume, in the range from V/V0 = 0.8–1.2 with an interval of 0.025, is used to obtain elastic constants Cij (V) by means of the universal-linear-independent coupling-strain (ULICS) method [41]. The volume-dependent bulk modulus, B(V), is computed in terms of Eq. (1) in theoretical methods section. The V(T)0, 4
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Fig. 6(d)–(f), at a certain temperature, that a smaller G and bigger B value of W15Re ultimately induce the increase of the ductility. It should also be pointed out from Fig. 6(d)–(e), with the increase of the temperature, that the B value of W15Re generally decreases and gets closer to that of W, and that the difference of G value between W and W15Re is almost unchanged. Such a comparison suggests that Re alloying brings improved ductility at the low temperature. Further studies are welcome to investigate the effects of dislocation and defect on ductility of W and W-Re alloys in the future.
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4. Conclusions In summary, first-principles calculation, quasi-harmonic approximation, and thermal electronic excitation have been combined to systematically investigate ideal tensile and shear strengths of W and WRe alloys, as well as temperature-dependent thermodynamic properties. The results demonstrate that Re alloying can soften W in the tensile and shear deformations. The G/B ratios of W-Re alloys are lower than those of W in the temperature range 0−2000 K, suggesting that Re alloying is an effective approach to improve the ductility of W under a fusion environment. The improved ductility is mainly due to the smaller G and bigger B values of W-Re alloys compared with those of W. Additionally, at the low temperature, the ductility of W-Re alloys is enhanced significantly, which is less obvious with the increase of temperature. The derived results were deeply explained by means of various discussions, and are in good agreement with similar experimental and theoretical observations in the literatures. Credit author statement Qian Wang as first author, performed first-principles calculation and analyses under the direction of Prof. Guoping Du and Prof. Liang Chen. All authors participated in the discussion and writing of this manuscript. Funding This work was supported by National Natural Science Foundation of China Grant No. 21571095, 61366004 and 51362020. Declaration of Competing Interest The authors declare that they have no conflict of interest. Appendix A. Supplementary data Supplementary material related to this article can be found, in the online version, at doi:https://doi.org/10.1016/j.fusengdes.2020. 111579. References [1] M. Rieth, J. Boutard, S. Dudarev, T. Ahlgren, S. Antusch, N. Baluc, et al., Review on the EFDA programme on tungsten materials technology and science, J. Nucl. Mater. 417 (2011) 463–467. [2] C. Linsmeier, M. Rieth, J. Aktaa, T. Chikada, A. Hoffmann, J. Hoffmann, et al., Development of advanced high heat flux and plasma-facing materials, Nucl. Fusion 57 (2017) 092007. [3] H. Bolt, V. Barabash, W. Krauss, J. Linke, R. Neu, S. Suzuki, N. Yoshida, Materials for the plasma-facing components of fusion reactors, J. Nucl. Mater. 329-333 (2004) 66–73. [4] L. Chen, J.L. Fan, H.R. Gong, Atomistic simulation of mechanical properties of tungsten-hydrogen system and hydrogen diffusion in tungsten, Solid State Commun. 306 (2020) 113772. [5] S.W.H. Yih, C.T. Wang, Tungsten-sources, Metallurgy, Properties and Applications, Plenum Press, New York, 1979. [6] S. Wurster, N. Baluc, M. Battabyal, T. Crosby, J. Du, C. García-Rosales, et al., Recent progress in R&D on tungsten alloys for divertor structural and plasma facing materials, J. Nucl. Mater. 442 (2013) S181–S189. [7] R.A. Pitts, S. Carpentier, F. Escourbiac, T. Hirai, V. Komarov, S. Lisgo, et al., A full tungsten divertor for ITER: physics issues and design status, J. Nucl. Mater. 438
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