First-principles study of mechanical and thermodynamic properties of intermetallic Pt3M (M = Al, Hf, Zr, Co, Y, Sc)

Computational Condensed Matter 23 (2020) e00462

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First-principles study of mechanical and thermodynamic properties of intermetallic Pt3M (M ¼ Al, Hf, Zr, Co, Y, Sc) Zongbo Li 1, Kai Xiong*, 1, Yingjie Sun, Chengchen Jin, Shunmeng Zhang, Junjie He, Yong Mao** Yunnan Provincial Engineering Laboratory of Copper-based and Advanced Conductive Materials, School of Materials Science and Engineering, Yunnan University, Kunming, 650091, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 3 December 2019 Received in revised form 21 January 2020 Accepted 28 January 2020

Pt-based alloys are a potential high-temperature structural material. The formation of second-phase intermetallic Pt3M (M ¼ Al, Hf, Zr, Co, Y, Sc) can further stabilize the high-temperature mechanical properties of Pt-based alloys. Here, we systematically studied the equilibrium geometry, elastic constants, electronic characteristics, Debye temperature and elastic wave velocity of intermetallic Pt3M using first-principles density function theory calculations. Elastic properties are estimated using the VoigtReuss-Hill approximation. Pt3Hf has the highest moduli and hardness, while Pt3Y has the lowest values. Pt3Co exhibits the strongest elastic anisotropy, while Pt3Zr is the weakest. Pt3Zr and Pt3Y have the highest and lowest Debye temperature. The metallicity and conductivity of these alloys mainly arise from d-state electronic contribution according to the electronic density of states. The charge density difference analysis shows that electrons flow from Pt to M atoms and accumulate into covalent-like bonds, which results in structural stability and mechanical anisotropy. © 2020 Elsevier B.V. All rights reserved.

Keywords: Platinum alloys First-principles Elastic constants Electronic structure Elastic anisotropy

1. Introduction As we all know it, intermetallic compounds with excellent mechanical properties at high temperatures [1]. At present, nickelbase superalloys are outstandingly successful high-temperature structural materials. Due to the limitation of nickels itself melting point, the high-temperature performance of nickel-based superalloys is approaching the limit. Therefore, materials that can maintain strength at high-temperatures that are ductile at room temperature are required. Corresponding to the high-temperature environments, solid solution strengthening and precipitation strengthening are considered as primary strengthening mechanisms in developing high-temperature alloys. Driven by the demand for intermetallic compounds with a much higher melting point have been considered because of their hightemperature strengths. We are attempting to design ultra-high temperature alloys based on platinum. Because of Platinum group

* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (K. Xiong), [email protected] (Y. Mao). 1 Z. Li and K. Xiong contributed equally to this work. https://doi.org/10.1016/j.cocom.2020.e00462 2352-2143/© 2020 Elsevier B.V. All rights reserved.

metals (platinum, iridium, and rhodium) are the best candidate because they have high melting points and good environmental resistance. Most importantly, they are more creep resistant associated with their face-centered cubic (fcc) structures at hightemperature. By alloying with additive elements, it is expected to develop a fine dispersion of small, preferably coherent, and hence stable, precipitates in alloys. The precipitation of second phases (intermetallic compounds, carbides, nitrides, oxides, etc.) at grain boundaries, which causes high stress/strain concentration due to the incompatibility of elastic and plastic deformation between particles and matrix [2e4]. In contrast to carbides and nitrides, intermetallic compoundeinduced embrittlement at room temperature can be relieved at high temperatures because plastic codeformation between these compounds and the matrix can reduce stress concentration and, thus, delay crack initiation. Among Pt-based alloys intermetallic compound, the L12-structure Pt3M compounds have received significant attentions due to their ability to refine grain, increase recrystallization temperature and improve the mechanical strength of alloys [5]. These advantageous properties appeal to come from the fine coherent dispersion of L12 precipitates in face-centered cubic (fcc) matrix to form fcc-L12 two-phase microstructure, similar to Ni-based alloys [6,7]. The fcc-L12 microstructure is expected to improve the mechanical

2

Z. Li et al. / Computational Condensed Matter 23 (2020) e00462 Table 1 Calculated equilibrium lattice constant (a in Å), mass density (r in g/cm3), equilibrium energy (E0 in eV/atom), formation enthalpy (DH in KJ/mol) elastic constants (Cij in GPa), bulk modulus (B in GPa), shear modulus (G in GPa).

a

Source

Pt3Al

Pt3Co

Pt3Hf

Pt3Sc

Pt3Y

Pt3Zr

This work Exp. Cal.

3.918 3.876a 3.928b 60.17 16.90 6.20 65.86 316.6 343k 177.2 179k 110.8 116k 139.5 223.7 94.3 89.7 92.0 248.2 237.3 242.7

3.881 3.873c 3.885d 58.49 18.29 6.41 7.19 321.2 e 188.3 e 117.9 e 132.9 232.6 97.3 90.0 93.6 256.2 239.1 247.7

4.023 4.02e 4.029f 65.13 19.47 8.16 106.58 353.6 e 178.7 e 129.9 e 174.9 236.9 112.9 108.8 110.8 292.4 283.0 287.7

3.999 3.958g 4.006h 63.95 16.36 7.17 99.41 296.5 292.2h 155.8 155.3h 114.5 110.6h 140.7 202.7 96.8 91.5 94.2 250.6 238.6 244.6

4.121 4.075g 4.123 h 70.03 15.99 7.18 44.13 244.9 243.4h 131.5 135.2h 88.7 86.7h 113.4 169.2 75.9 72.4 74.1 198.1 190.0 194.1

4.047 3.990i 4.055j 66.32 16.94 7.74 99.54 331.6 399l 166.3 211l 121.6 144l 165.3 221.4 106.0 102.3 104.1 274.3 265.9 270.1

V0

r

E0

DH C11

This work Cal. This work Cal. This work Cal.

C12 C44

Fig. 1. The crystal structure of L12 Pt3M. The green atoms represent M, and the red atoms are Pt. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

behavior and service lifetime of alloys as the role of interfacial dislocation interactions. Several high melting points Pt3M alloys have been identified from the existing experimental data. Such as the melting point of Pt3Al (1550  C) [8], Pt3Hf (2250  C) [9], Pt3Sc (1850  C) [10], Pt3Y (2020  C) [11], Pt3Zr (2250  C) [12] is higher than Ni3Al (1390  C). Pt3Co exhibits an order-disorder transformation from L12-order phase to fcc solid solution when temperature above 640  C and melting above 1800  C [13]. To develop Pt-based alloys with secondary phase strengthening precipitates, it is essential to understand the fundamental structure and properties of secondary phase particles Pt3M intermetallic. Experimental measurements of mechanical properties for secondary phase particles Pt3M intermetallic are seldom reported on literatures. Here, we systematically studied the equilibrium geometry, elastic constants, electronic characteristics, Debye temperature and elastic wave velocity of intermetallic Pt3M using firstprinciples density function theory calculations. These fundamental data of intermetallic Pt3M can provide important theoretical guidance and understanding of the development of Pt-based alloys.

C11eC12 B GV GR G EV ER E a b c d e f g h i j k l

[29]. [25]. [28]. [13]. [26]. [45]. [24]. [23]. [46]. [47]. [43]. [40].

incremental calculation was carried out k-points sampling to 19  19  19. The above parameters ensure the convergence of total energy to 1 meV and the relaxation force of atoms less than 0.01 eV/ Å. The elastic constant calculated by the stress-strain [18] approach. The stress-strain approach is based on Hooke’s law for elastic solids, which describes the linear relationship between the applied external stress components si and the small strain εj cased in the relaxed crystals as:

2. Calculation methods

si ¼ The Vienna Ab-initio Simulation Package (VASP) [14] based on density functional theory (DFT) was used for our calculations. The electron-core interaction was treated with projector augmentedwave (PAW) method [15]. The exchange-correlation functional Perdew-Burke-Ernzerhof (PBE) was used to calculate the electronic structure with generalized gradient approximation (GGA) [16]. The Pt3M (M ¼ Al, Co, Hf, Sc, Y, Zr) binary intermetallic compounds were isostructural with L12-structured Cu3Au in the Pm3m space group (No. 221 in the X-Ray Tables). The atoms were located at the following Wyckoff locations: Pt and M atoms were located at 3c (0, 1/2, 1/2) and 1a (0, 0, 0), respectively. The structure models of Pt3M are shown in Fig. 1. The green atoms represent M, and the red atoms are Pt. The atomic basis PAW was selected as Pt 6s15p65d9, Al 3s23p1, Co 4s23p63d7, Hf 6s25p5d2, Sc 4s23p63d1, Y 5s24p64d1, Zr 5s24p64d2. Spin polarization was considered for Co atoms. The plane waves energy cut-off was established at 520 eV for all studied systems. A 9  9  9 Monkhorst-Pack grid of k-points was used for integration in the irreducible part of the Brillouin zone in geometry optimization [17]. For the electronic density of states (DOS), the structure was optimized by a single point calculation method, and the

6 X

Cij εj

(1)

j¼1

Table 2 Calculated Cauchy’s pressure (pc in GPa), Pugh’s ratio (B/G), Vickers hardness (Hv in GPa), elastic anisotropic factors (AZ, AU, AE), and acoustic wave velocities (VL, VT1, VT2 in m/s).

n B/G pc¼C12 - C44 Hv AZ AU AE V[100] L V[100] T1,2 V[110] L V[110] T1 V[110] T2 V[111] L V[111] T1,2

Pt3Al

Pt3Co

Pt3Hf

Pt3Sc

Pt3Y

Pt3Zr

0.319 2.430 66.4 7.0 1.588 0.261 0.336 4328 2561 4601 2561 2031 4688 2221

0.323 2.485 70.4 6.8 1.775 0.406 0.399 4191 2539 4514 2539 1906 4617 2138

0.298 2.138 48.8 9.9 1.485 0.190 0.291 4262 2583 4510 2583 2119 4590 2284

0.299 2.152 41.3 8.7 1.627 0.290 0.345 4257 2646 4563 2646 2074 4661 2280

0.309 2.283 42.8 6.4 1.564 0.244 0.325 3914 2355 4161 2355 1883 4241 2053

0.297 2.127 44.7 9.5 1.470 0.181 0.284 4424 2679 4677 2679 2209 4758 2376

Z. Li et al. / Computational Condensed Matter 23 (2020) e00462

The elastic constants Cij were obtained by making six kinds of finite distortions on the equilibrium lattice configuration. The polycrystalline Young’s moduli E, bulk modulus B, shear modulus G were estimated from the single crystal elastic constants according to the Voigt-Reuss-Hill approximation [19e21].

3. Results and discussion 3.1. Structure and elastic properties of Pt3M The equilibrium structure parameters for Pt3M alloys were obtained by the full relaxations of both lattice constants and internal atom coordinates, as summarized in Table 1. The results are fully consistent with experimental results [22e29] with the maximum deviation of less than 1.5%. In order to estimate thermodynamic stability of Pt3M compounds, the formation enthalpy (DH) was calculated in this paper, which is given by Refs. [30,31]:

1 4

DH ¼ ½EðPt3 MÞ  3EðPtÞ  EðMÞ

(2)

Here E (Pt3M), E (Pt) and E(M) are the total energy of Pt3M, Pt atom and M atom at the ground state, respectively. Elastic constants play an important role in solid-state physics and also related to mechanical properties, thermal conductivity [32,33]. Experimentally, the elastic constants of crystals are detected by resonant ultrasound spectroscopy or ultrasonic pulseecho elastic wave velocity measurements [34]. Some technique challenges exist in ultrasonic tests, such as the influence of external factors, e.g. the accuracy of the ultrasonic resonance mass spectrometry is dependent on the samples [35]. Using atomistic simulations for a certain crystal solid with known structure, the elastic constant can be calculated by the first principles [36e39]. As listed in Table 1, the calculated elastic constants of Pt3M are consistent with available theoretical data [23,40e43]. The elastic constants of Pt3Co and Pt3Hf alloys are lack of available data. The calculated elastic constants of Pt3M satisfy the Born elastic stability criteria for the cubic system [33,44], namely, C11 > 0, C44 > 0, C11 : C12 > 0, C11þ2C12 > 0, implying that mechanical properties of the considered Pt-based alloys are expected to be stable. Based on the elastic constants of single crystal, the engineering elastic moduli of polycrystalline, i.e. bulk modulus B, shear modulus G, Young’s modulus E and Poisson’s ration n, were evaluated by the Voigt-Reuss-Hill approximation [19e21]. The Voigt approximation (index V) is obtained based on an assumption of uniform strain throughout the polycrystalline; it is the upper limit of the actual effective modulus [21]. The Reuss approximation (index R) is obtained by assuming the uniform stress, the lower bound of the actual modulus [20]. The Hill average (index H) is arithmetic or a geometric average of the Voigt and Reuss bounds [19]. It should be noted that the Hill average has no theoretical and physical meaning, but it is simple and reliable, and it is well agreements with experimental data. The calculated B, G and E results are shown in Table 1. Clearly, among the Pt3M alloys, Pt3Hf has the highest B, G and E; however Pt3Y has the lowest moduli. This indicates that Pt3Hf and Pt3Y have the strongest and weakest resistance, volume and shape deformation and uniaxial stretching, respectively. To judge the brittleness and ductility of materials, several ductile-brittle criterions had been proposed by difference authors, such as Pugh’s ratio [48], Poisson’s ratio [49] and Cauchy’s pressure [50]. The Pugh’s ratio [38] is the ratio of bulk modulus to shear modulus (B/G), which is widely used to provide information about brittle (ductile) nature of materials. Ductile materials often exhibit large Pugh’s ratio, while brittle materials tend to possess smaller value. The critical value of Pugh’s ratio that distinguish the ductile

3

and brittle materials is approximately 1.75, i.e. ductile behavior is predicted when B/G is large than 1.75, otherwise, the materials behavior is a brittle manner. According to the B/G in Table 2, all the alloys behave in a ductile manner, so that they can resistant to thermal shocks. The ductility of Pt3Co (B/G ¼ 2.484) and Pt3Al (B/ G ¼ 2.430) compounds are slightly better than Pt3Sc (B/G ¼ 2.153) and Pt3Hf (B/G ¼ 2.137). The Poisson’s ratio n is defined as the ratio of the relative shrinkage strain to the normal force of the relative expansion strain in the applied force direction [49], which is intimately linked to the B/G ratio by n ¼ (3B/G-2)/(6B/Gþ2), for the brittle/ductile transition, the critical value of 0.26. As listed in Table 2, the n values locate between 0.297 (Pt3Zr) and 0.323 (Pt3Co), further evidence the metallic bonding and ductile nature of Pt3M alloys. Similar to Pugh’s ratio, the Cauchy’s pressure (pc ¼ C12 e C44) is an indicator for evaluating the ductility of materials and reveals bonding type in materials. A positive pc represents more metal properties and trends to ductility. A negative pc occurs in covalent materials with strong bonds of directional characteristics, indicating the materials are expected to be brittle. As tabulated in Table 2, these alloys listed pc values are positive, indicating these alloys behave as ductile. Based on pc values, Pt3Co (pc ¼ 70.4 GPa) is found to be the most ductile, these results are good agreements with the Pugh’s ratio analysis. Hardness is a basic mechanical property of materials, represents the resistance of a material to deformation when a force is applied, which is closely related to the permanent plastic deformation or brittle fracture of materials [51]. Recently, an empirical formula of Vickers hardness has been proposed based on B and G, i.e. Hv ¼ 2 (G3/B2)0.585e3 [52]. Formula reveals that, if the shear modulus increases, the hardness would increase just the bulk modulus remains unchanged, and vice versa. Among the mentioned alloys, although the difference of bulk modulus between Pt3Co (232.6 GP) and Pt3Hf (236.9 GPa) is very small, the shear modulus of Pt3Hf (110.8 GPa) is about 18% larger than that of Pt3Co (93.6 GPa), which causes Pt3Hf (Hv ¼ 9.9 GPa) harder than Pt3Co (Hv ¼ 6.8 GPa). The elastic wave propagation in these crystals are discussed based on the Christoffel equations [53]. In cubic structure, the acoustic wave velocities propagating in <100>, <110> and <111> directions are given by the following equations: For <100> directions, the longitudinal (VL) and transverse (VT) wave velocities are defined as

V C100D ¼ L

pffiffiffiffiffiffiffiffiffiffiffiffi C11 =r

(3)

V C100D T1;2 ¼

pffiffiffiffiffiffiffiffiffiffiffiffi C44 =r

(4)

The elastic waves velocities, which propagate in <110> directions are given as

V C110D ¼ L

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðC11 þ C12 þ 2C44 Þ=ð2rÞ

(5)

¼ V C110D T1

pffiffiffiffiffiffiffiffiffiffiffiffi C44 =r

(6)

V C110D ¼ T2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðC11  C12 Þ=ð2rÞ

(7)

The elastic waves velocities along <111> directions are written as

V C111D ¼ L

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðC11 þ 2C12 þ 4C44 Þ=ð3rÞ

(8)

V C111D T1;2 ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðC11  C12 þ C44 Þ=ð3rÞ

(9)

4

Z. Li et al. / Computational Condensed Matter 23 (2020) e00462

Fig. 2. Three-dimensional (3D) orientation of the surface related to Young’s modulus for Pt3M (M ¼ Al, Co, Hf, Sc, Y, and Zr) alloys.

3.2. Elastic anisotropy Elastic anisotropy reflects different bonding properties in different crystal directions, which is related to phase transformation and phonon modes in a perfect crystal. Zener [54] proposed an anisotropy factor for cubic single crystals AZ ¼ 2C44/(C11  C12). Physically, C44 represents the resistance of the shear stress applied to the (100) plane to the deformation in the [010] direction. Similarly, (C11  C12)/2 represents the resistance to shear deformation by shear stress applied across the (110) plane in the ½110 direction. It should be pointed out that AZ is only related to the anisotropic shear deformation; the anisotropy of bulk deformation is not involved. To unify the elastic anisotropy evaluation, a universal anisotropy index AU is proposed [55], i.e.

AU ¼ 5

Fig. 3. The (001) plane projections of Young’s modulus for these alloys.

Note that the two shear wave velocities (V<100> and V<110> ) are T1,2 T1 equivalent in both <100> and <110> directions. The wave velocities deduced from the above equations are also listed in Table 2. Obviously, the results of longitudinal wave velocities are larger than transverse wave. The longitudinal wave travels fastest and slowest in <111> and <100> directions, respectively. The slowest transverse wave propagates along <110> directions. The orientationdependent wave velocity in these alloys directly reflects the intrinsic anisotropy of elastic behavior.

GV BV þ 60 GR BR

(10)

The index AU is based on the fraction difference in the bulk (BV, BR) and the shear moduli (GV, GR) upper (Voigt) and lower (Reuss) bounds, which introduces the contribution of both shear and bulk deformation. As a generalization of the Zener anisotropy index, AU applies to all crystal symmetries. The calculated AZ and AU for Pt3M alloys are presented in Table 2. The anisotropy factor decreases in the following order: Pt3Co > Pt3Sc > Pt3Al > Pt3Y > Pt3Hf > Pt3Zr, which shows that elastic anisotropy for Pt3Co is the strongest and the anisotropy of Pt3Zr is the weakest. To further illustrate the elastic anisotropy of Pt3M intermetallic compounds, the three-dimensional (3D) surface of Young’s modulus are plotted in Fig. 2. The curved surface of Young’s modulus is obtained by Ref. [56].

   1 1 ¼ S11  2 S11  S12  S44 l21 l22 þ l22 l23 þ l23 l21 E 2

(11)

Z. Li et al. / Computational Condensed Matter 23 (2020) e00462

5

Fig. 4. The total density of states and partial density of states of the Pt3M (M ¼ Al, Co, Hf, Sc, Y, and Zr) alloys. The vertical dash line represents Fermi level Ef.

where Sij is calculated from the elastic compliance matrix. l1, l2, and l3 are the directional cosines to the [100], [010], [001] axes, respectively. For isotropic crystals, the 3D surface of E is spherical. Obviously in Fig. 2, all Pt3M alloys exhibit relatively anisotropic as the 3D surface deviating from the spherical shape. It is noted that the maximum of Young’s modulus (Emax) for these compounds locates in <111> directions, with a descending order of E111> E110 > E100. The elastic anisotropy of crystals also is able to evaluate by the anisotropic factor AE ¼ (EmaxeEmin)/Emax. As is shown in Table 2, Pt3Co has the largest value of AE, while the elastic anisotropy of Pt3Zr is the weakest. This value is in good accordance with the analysis of AZ and AU. Fig. 3 plots the (001) plane projections of Young’s modulus for Pt3M alloys. Obviously in Fig. 3, the orientation-dependent Young’s modulus of these alloys exhibits similar orientation distribution as their same crystal structure symmetry, have minimum value along the <100> direction, and increase gradually from the <100> direction to the diagonal <110> direction. The outermost and inmost curves in Fig. 3 illustrate that Pt3Hf and Pt3Y has the highest and lowest stiffness, respectively. 3.3. Electronic structure Electronic structures are the important parameters in predicting and controlling mechanical properties, especially for transition metals with the orientation-dependent bonding characteristics [57,58]. The total density of states (t-DOS) and partial density of states (p-DOS) diagrams of these alloys are plotted in Fig. 4. The vertical dotted line indicates the Fermi level (Ef), which is considered as the boundary between bonding and anti-bonding states. In Fig. 4, the t-DOS diagrams show no energy gap at Ef, indicating their metallic character. For Pt3Al, Pt3Co, Pt3Sc and Pt3Y alloys, there is a peak near the Fermi level of the t-DOS curves, indicating their good

conductivity. The partial density of states (p-DOS) diagrams plotted in Fig. 4 elucidate the contribution of each atom to the t-DOS. It is clear that the metallic nature of Pt3M alloys is primarily due to the Ptd states and M-d states (except Pt3Al), indicating significant hybridization between Pt and M atoms, forming the Pt-M metallic bonds along the d-d direction, which are in a fine consistent with previous theoretical investigations [23,59]. In the case of Pt3Al, the main bonding peaks of Pt3Al appear from 8 eV to 5 eV, originating mainly from the hybridized Pt-5d and Al-3p states. The s and p states of Pt and Al finitely contribute to metallic nature. For Pt3Co, both spin-up and spin-down band structures were plotted in Fig. 4b, and the t-DOS is mainly dominated by Pt-5d and Co-3d states. The t-DOS curves in Fig. 4 show that a deep valley named pseudogap appears in the vicinity of Ef due to the transfer of electrons to lower energy range. The position of Ef on the pseudogap is related to the structural stability of the crystals. If Ef locates in pseudogap, the structural of an intermetallic compound is considered more stable [60]. As shown in Fig. 4c and f, the Fermi level is located at the bottom of pseudogap, while Ef lies in the region of high DOS in other four alloys. A part of d-states electrons transfers from Pt atoms to the empty d orbits of Hf and Zr atoms, which leads to high d-states overlap and structure stability for Pt3Hf and Pt3Zr. Furthermore, the pseudogap at Ef can directly reflect the covalent bonding strength of materials. A wilder pseudogap implies a stronger bonding strength and higher resistance to deformation [61]. By comparing Fig. 4c and d, it is found that the pseudogap of Pt3Hf is a little wider than Pt3Zr, implying the covalent bond strength of PteHf is stronger than PteZr, which can help to explain the highest B, G, E and Hv of Pt3Hf. A more intuitive understanding of the bonding properties and charge transfer of these compounds can be obtained by calculating

6

Z. Li et al. / Computational Condensed Matter 23 (2020) e00462

Fig. 5. Three-dimensional isosurfaces and two-dimensional plane view of charge density difference distribution for Pt3M (M ¼ Al, Co, Hf, Sc, Y, and Zr) alloys. The color scheme is red for positive charge density and blue for negative charge density. The blue portion represents the charge density decreases, and yellow represents the charge density increases. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

the charge density difference, Dr ¼ r(Pt3M) e r(Pt3) e r(M). In the equation, r(Pt3M), r(Pt3) and r(M) are the charge densities of the corresponding substances with the same atoms coordinated. Physically Dr shows the difference in electron density between the entire system and some isolated atoms that are undisturbed. The charge density difference distribution is visualized by VESTA [62]. Fig. 5 shows the three-dimensional (3D) isosurfaces and twodimensional (2D) contour maps of charge density difference distribution for Pt3M alloys. From the 3D isosurfaces, these alloys exhibit similar features of charge density distribution as their identical crystal structural. The yellow and cyan isosurfaces represent gain and depletion charge density, respectively. The electrons are collecting (yellow region) the octahedral interstice of lattices, the electron depletion (cyan region) occurs around Pt and M atoms. This suggests that electrons flow from both Pt and M atoms and accumulate into covalent-like Pt-M bonds between Pt and M atoms. The 2D contour lines in Fig. 5 are plotted from lower density (blue) region to higher density (red) region. The 2D contour plots show that a significant amount of charges is transferred from Pt and M atoms and accumulated into interstitial regions, while there is no significant charge aggregation between M atoms. These findings confirmed that the bonding between Pt and M atoms is covalentlike. As described in DOS, the covalent-like bonds in these alloys lead to their structural stability. Fig. 5 also shows that charge accumulation in Pt3M alloys exhibits robust directionality, mainly appearing along the <111> and <110> direction with 4-fold rotational symmetry about the <100> direction. The directional bonding characteristics in these alloys result in their anisotropic mechanical properties. At first sight, the charge density distribution

Table 3 The calculated longitudinal sound velocity (vl in m/s), transversal sound velocity (vt in m/s), average sound velocity (vm in m/s), and Debye temperature (qD in K) of Pt3M (M ¼ Al, Co, Hf, Y, Zr) alloys.

vl vt vm

qD

a b c d

Pt3Al

Pt3Co

Pt3Hf

Pt3Sc

Pt3Y

Pt3Zr

4527 2334 2613 315.1 349.0a

4421 2263 2535 308.6

4445 2386 2664 312.9

4479 2399 2679 316.6 318.0b

4095 2154 2408 276.1 271.9c

4612 2410 2769 323.2 346.0d

[64]. [23]. [23]. [36].

of these alloys is almost identical in shape, but the extent of charge accumulation is slightly different by carefully scrutinizing. The charge accumulation differences in these alloys result in the different resistance to deformation. For example, the charge accumulation in Pt3Hf and Pt3Zr is significantly higher than that in Pt3Y. This may account for the moduli (E, B and G) of Pt3Hf and Pt3Zr are higher than those of Pt3Y. 3.4. Debye temperature Thermodynamic properties of materials provide valuable information for understanding the phase stability and hightemperature mechanical properties of Pt3M alloys. Especially for the Debye temperature (qD), it is associated with the elasticity and

Z. Li et al. / Computational Condensed Matter 23 (2020) e00462

melting temperature of materials. The Debye temperature can be estimated from the average sound velocity vm by Refs. [63,64].

qD ¼

   h 3n NA r 1=3 vm kB 4p M

(12)

Here, h, kB, and NA are the Plank’s constant, the Boltzmann’s constant and Avogadro constants, respectively; r is the density, M is the molecular weight, n is the number of atoms. The average wave velocity vm is estimated from elastic moduli by

"

1 2 1 vm ¼ þ 3 v3t v3 l

!#1=3

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B 4G þ vl ¼ r 3r

vt ¼

(13)

(14)

sffiffiffiffi G

wave velocity of intermetallic Pt3M (M ¼ Al, Co, Hf, Sc Y, Zr) using first-principles density function theory calculations. (1) These alloys satisfy elastic mechanical stability constraints and exhibit ductility due to their positive Cauchy pressure, high Poisson’s ratio. Among these alloys, Pt3Hf has the highest bulk modulus (B), shear modulus (G), Young’s modulus (E), while Pt3Y has the lowest values. (2) These alloys exhibit similar anisotropy features in the 3D surface of Young’s Modulus. The maximum (Emax) and minimum (Emin) Young’s modulus are along <111> and <100> directions, respectively. The anisotropy of Pt3Co is the highest, while Pt3Zr is the weakest. Pt3Zr and Pt3Y have the highest and lowest Debye temperature. (3) The metallic nature of Pt3M alloys mainly originates from Ptd states and M-d states (or p states) electrons. Electrons flow from Pt and M atoms and accumulate to form directionally bonds that lead to structural stability and mechanical anisotropy.

(15)

r

Declaration of competing interest

where vl and vt are the longitudinal and transverse elastic wave velocities of polycrystalline materials. At low temperatures the vibrational excitations arise mainly from acoustic vibrations. Therefore, the Debye temperature calculated from the elastic constant is equal to the Debye temperature determined by the specific measurement. The calculated Debye temperature, the longitudinal, transversal, average sound velocities are tabulated in Table 3. The longitudinal sound velocities of all considered alloys are much faster than transversal velocities. Among these compounds, Pt3Zr has the supreme average sound velocity (vm ¼ 2769 m/s). According the Debye theory, lattice vibration in crystals can be described as elastic wave propagation. The vibration frequencies of the elastic wave are distributed continuously within the range of 0 to vm. The Debye theory assumes that all lattice vibration modes have same vibration frequencies (vm) when elevated temperature above Debye temperature (T > qD). As a key physical parameter of metals, the magnetite of qD directly reflects the intensity of lattice vibrations, which is proportional to vm, see Eq. (11). According to Lindeman equation, qD is directly proportional to the melting temperature (Tm) [65] of crystals, i.e.

constant ffiffiffiffi qD ¼ p 3 V

7

rffiffiffiffiffiffi Tm M

(16)

Here, M and V are the atomic weight, atomic volume respectively. Accordingly, a higher qD indicates a larger Tm, which indirectly demonstrates a stronger atomic bonding. Debye temperature (qD ¼ 323 K) which implies a relative strength bonding in Pt3Zr. Knowing by the Debye theory, the higher Debye temperature means a greater associated thermal conductivity [66]. Lowtemperature phonons are the main conduction mechanism, and Debye temperature is a limit stability of all phonons. At this time, the thermal conductivity is approximately proportional to the third power of Debye temperature [67]. Therefore, Pt3Zr has the best thermal conductivity among all these alloys. Pt3Y possess the lowest qD (276 K) that show the Pt3Y has the worst metallic bonding than others. 4. Conclusions We systematically studied the equilibrium geometry, elastic constants, electronic characteristics, Debye temperature and elastic

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. Acknowledgements This work was financially supported by the National Natural Science Foundation of China (Nos. 51801179 and U1502272), the Yunnan Science and Technology Projects (Nos. 2018ZE001, 2018ZE006, 2018ZE016, 2018ZE021, 2018ZE023 and 2019ZE001), the Yunnan Applied Basic Research Projects (Nos. 2018FB083 and 2018FD011). References [1] R. Cahn, P. Haasen (Eds.), Physical Metallurgy, Elsevier Science, Amsterdam, 1996.
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