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Mechanical properties of W–Ti alloys from first-principles calculations
Fusion Engineering and Design 106 (2016) 34–39
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Mechanical properties of W–Ti alloys from first-principles calculations D.Y. Jiang a,b,d , C.Y. Ouyang c , S.Q. Liu a,b,∗ a
Department of Materials Science and Engineering, Nanchang University, Nanchang 330047, China Department of Physics, Nanchang University, Nanchang 330047, China c Department of Physics, Jiangxi Normal University, Nanchang 330022, China d School of Basic Sciences, Jiangxi University of Technology, Nanchang 330098, China b
h i g h l i g h t s • The mechanical properties of the W1-x Tix alloys are calculated from DFT. • Ti alloying enhances the ductility of W metal substantially. • The mechanical strength of W-Ti alloys is slightly weaker than W while stronger than Ti.
a r t i c l e
i n f o
Article history: Received 8 October 2015 Received in revised form 29 January 2016 Accepted 5 March 2016 Available online 19 March 2016 Keywords: W-Ti alloy Mechanical properties Ductility The first wall First principles calculations
a b s t r a c t The effect of Ti concentration on the fundamental mechanical properties of W-Ti alloys has been studied from first principles calculations. The lattice constants, the cell volumes and the formation energies of the W1-x Tix (x = 0.0625, 0.125, 0.1875, 0.25, 0.5) alloys were calculated. It is shown that Ti alloying in bcc W lattice is thermodynamically favorable when the Ti concentration is lower than 25% and the W0.8125 Ti0.1875 have the lowest formation energy. With the optimized geometry and lattice, the elastic constants are calculated and then the elastic moduli and other mechanical parameters are derived. Results show that although the mechanical strength of the W-Ti alloys is lower than that of pure W metal, it is much higher than that of pure Ti metal. On the other hand, the B/G ratio and the Poisson’s ratio of the W-Ti alloys is much higher than that of pure W, and even higher than that of pure Ti, indicating that Ti alloying can improve the ductility of bcc W substantially. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Tungsten is receiving considerable attention as potential plasma facing material of the divertor and the first wall material in fusion power reactors, due to its high melting point, high strength at high temperatures, high sputtering threshold energy, good thermal conductivity and low thermal expansion coefficient [1–3]. In addition, tungsten is also thought as a shielding material in fusion reactors and other systems involving nuclear reactions [4]. Unfortunately, pure tungsten metal exhibits poor radiation stability and fracture toughness, low ductility associated with a high ductile-tobrittle transition temperature (DBTT). These properties are strongly dependent on the chemical composition and the micro-structural state [5]. Alloying is a good method to improve the physical properties of tungsten. For example, the thermal properties of tungsten are expected to be enhanced by alloying with other elements [6,7].
∗ Corresponding author at: Department of Materials Science and Engineering, Nanchang University, Nanchang 330047, China. E-mail address: [email protected] (L. S.Q.). http://dx.doi.org/10.1016/j.fusengdes.2016.03.028 0920-3796/© 2016 Elsevier B.V. All rights reserved.
Among various alloying strategies of W metal, W-Ti alloy is one typical one [8,9]. To our knowledge, most experimental efforts of the binary W-Ti alloys have been devoted to understanding the effect of Ti concentration on the grain growth behaviors [10–14]. However, little attention was paid to the effect of Ti-alloying and the Ti concentration on the fundamental mechanical properties, such as elastic properties and ductile/brittle behaviors. The density-functional theory (DFT) calculation can be used to predict the mechanical performance of materials with relatively low cost and high efficiency, comparing with the experimental evaluations. Since conventional experimental alloy design processes can be very expensive and time-consuming, DFT calculations can be used to study the structure-property relationships and help to design these materials. The computational data may serve as guidance for further optimizing the composition of W-Ti alloys. Computer simulation focus on improving specific alloy properties can narrow down the number of compositions that are needed to be prepared and characterized. In this paper, we studied the phase stability and mechanical properties of W1-x Tix alloys (x = 0.0625, 0.125, 0.1875, 0.25 and 0.5) from first principles calculations. The lattice constants, the cell vol-
J. D.Y. et al. / Fusion Engineering and Design 106 (2016) 34–39 Table 1 The elastic constant of bcc W.
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Table 3 The elastic constant for hcp Ti.
Method
C11 (GPa)
C12 (GPa)
C44 (GPa)
Method
C11 (GPa)
C12 (GPa)
C13 (GPa)
C33 (GPa)
C44 (GPa)
Present work Experiment [26] Theory [26,27]
529.94 533 553
211.19 205 207
139.44 163 163
Present work Experiment [28] Theory [29]
175.61 176.10 171.60
84.45 86.90 86.60
81.98 68.30 72.60
188.34 190.50 190.60
41.41 50.80 41.10
ume, the formation energies and the elastic constants of the W1-x Tix alloys are calculated. Based on the mechanical characteristic of B/G ratio, Poisson’s ratio () and Cauchy pressure (C’), the ductile/brittle properties of the W-Ti alloys are evaluated. 2. Model and computational details Computations in the present work were done within the density-functional theory (DFT) and the plane-wave pseudopotential method, which were implemented in the Vienna ab initio simulation package (VASP) [15,16]. The core ion and valence electron interaction was described by the projector augmented wave method (PAW) [17], and the exchange-correlation part was described with the generalized gradient approximation (GGA) by Perdew and Wang (PW91) [18]. All models were described by a 2 × 2 × 2 supercell containing 16 atoms in a body centered cubic (bcc) structures. Part of W atoms were substituted by the Ti atoms to obtain various alloys with different Ti concentrations, that is pure W, W0.9375 Ti0.0625 , W0.875 Ti0.125 , W0.8125 Ti0.1875 , W0.75 Ti0.25 , W0.5 Ti0.5 and pure Ti. Except for the case of pure Ti, which was modeled with a hexagonal supercell, all the other systems were modeled with bcc supercells. For the Brillouin-zone sampling, we used the 11 × 11 × 11 Monkhorst-Pack mesh [19] for all W1-x Tix alloys with bcc structures and a 6 × 6 × 4 Monkhorst-Pack mesh for hexagonal close packing (hcp) phase titanium metal. A cutoff energy of 350 eV was used for all systems. Gaussian smearing method with a smearing width of 0.05 eV is applied in all calculations in this study. Within the framework of the continuum elasticity theory [20,21], for the cubic structure, there are three independent elastic constants, i.e. C11 , C12 and C44 . In order to calculate these elastic constants, three sets of specific strains (␦) along different directions are applied to the cubic supercell, and the total energy changes are calculated as a function of the applied strains. Then, the elastic constants can be obtained through fitting the energy changes (E) vs the applied strains (␦). For the hexagonal structure, there are five independent elastic constants, i.e. C11 , C12 , C13 , C33 and C44. Similarly, five sets of specific strains (␦) along different directions are applied to the hexagonal supercell, and the elastic constants can be obtained through fitting the energy changes (E) vs the applied strains (␦). The mechanical properties can be calculated from these single crystal elastic constants, according to the Voigt-Reuss-Hill scheme [22–25].The bulk modulus (B), shear modulus (G), Young’s modulus (E), Poisson’s ratio () and Cauchy pressure (C’) of the cubic W-Ti alloys are calculated with C11 , C12 and C44 and given by the following formula: B=
C11 + 2C12 3C 44 + C11 − C12 ,G = , 3 5
Table 2 Bulk modulus (B), shear modulus (G), Young’s modulus (E), B/G ratio, Poisson’s ratio (), Cauchy pressure C’ of bcc W. Method
B (GPa)
G (GPa)
E (GPa)
B/G
C’ (GPa)
Present work Experiment [26] Theory [26,27]
317.44 314.33 322.33
147.42 163.40 173.00
382.96 417.80 440.24
2.15 1.92 1.86
0.30 0.28 0.27
35.87 21.00 22.00
Table 4 Bulk modulus(B), shear modulus (G), Young’s modulus (E), B/G ratio,Poisson’s ratio (), Cauchy pressure C’ of hcp Ti. Method
B (GPa)
G (GPa)
E (GPa)
B/G
Present work Experiment [28] Theory [29]
115.12 109.96 110.81
44.94 50.16 44.68
119.30 130.62 118.15
2.56 2.19 2.48
0.327 0.302 0.322
E=
9BG E C12 − C44 , V = −1 , C = 3B + G 2G 2
Similarly, the mechanic parameters B, G, E and of the hexagonal Ti are calculated from elastic constants C11 , C12 , C13 , C33 and C44. The specific methods can be found in the literature [22–25]. 3. Results and discussion 3.1. The mechanical properties of bcc tungsten (W) and hcp titanium (Ti) The calculated single crystal elastic constants and mechanical property parameters of W and Ti are listed in Tables 1–4. For comparison purpose, we also list the data from experiments, as well as other theoretical results in the tables. Notice that the elastic constants as well as the mechanical parameters of the bcc W and hcp Ti metal from our calculations are basically in agreement with the experimental and other theoretical results. This confirms that the model and computational parameters are reasonable in the present work and therefore the following results on the W-Ti alloys are reliable. Furthermore, the elastic constants C44 for the bcc W is underestimated while the C13 for the hcp Ti is overestimated comparing with those experimental data. This is due to the GGA pseudopotentials we used in the current study. As GGA always underestimate the lattice constants of metal systems, it transfers this error to the calculation of the elastic constants. 3.2. The energetics and stability of the binary W-Ti alloys First of all, the atomic positions and W-Ti configuration in the supercell are considered and optimized, and the configuration with the lowest total energy is chosen for further study. The schematic diagrams of the atomic configurations with the lowest total energy of the W1-x Tix alloys at different x value are shown in Fig. 1. As it is shown, the W-Ti alloys keep the bcc lattice, and the Ti atoms prefer to locate at positions to ensure that the structure is in a highest symmetry. This shows that the solid solution of the W-Ti alloy can be formed at the atomic level. In addition, to obtain the equilibrium lattice constant, we calculated the total energies as a function of the volume and then fit the results using Birch-Murnaghan equation of state. Fig. 2 shows the variation of the equilibrium lattice constants of W1-x Tix alloys as a function of the Ti concentration. When Ti concentration is less than 25%, the lattice constant of the W1-x Tix alloy decreases almost monotonically with x increases, due to the smaller radius of the Ti atom comparing with that of the W atom. However, when titanium concentration reaches 50%, the lattice starts to expand. The expanded lattice of the W1-x Tix alloys when x is high is related with the weakened bonding interaction between the W and Ti atoms, which will be discussed in the
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J. D.Y. et al. / Fusion Engineering and Design 106 (2016) 34–39 Table 5 The cell volumes, elastic constants, and formation energies of bcc W1-x Tix alloys. Composition
C11 (GPa)
C12 (GPa)
C44 (GPa)
V (Å3 )
Ef (eV/Cell)
Pure W W0.9375 Ti0.0625 W0.875 Ti0.125 W0.8125 Ti0.1875 W0.75 Ti0.25 W0.5 Ti0.5
529.94 509.01 480.94 436.59 385.92 248.78
211.19 199.28 193.75 192.92 195.69 174.94
139.44 127.95 114.32 100.50 82.43 39.46
31.94 31.88 31.81 31.74 31.71 31.85
0.000 −0.798 −1.141 −1.310 −1.149 0.166
Fig. 1. The energetically most favorable atomic arrangements of the bcc W1-x Tix (x = 0.0625, 0.125, 0.25 and 0.5) alloys in a 2 × 2 × 2 supercell. Small blue and large grey balls are W and Ti atoms, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. Formation energy of bcc W-Ti alloys as function of Ti concentration.
Fig. 2. Equilibrium lattice constants of bcc W-Ti alloys as a function of Ti concentration.
following analysis. Here we mention that we also tested the alloys with hcp lattice, and our results show that the calculated energies of the W-Ti alloys with hcp lattice is much higher than that of with bcc lattice. For example, the energy of W0.5 Ti0.5 in the bcc lattice (−164.22 eV) is much smaller than that of W0.5 Ti0.5 in the hcp lattice (−161.31 eV). For other concentrations, the energy difference is even larger. To evaluate the stability of the alloys, we calculated the formation energy of the W1-x Tix alloys, which is defined as: Ef = EW −Ti − (1 − x) EW − xETi where x is the concentration of the alloying element, EW , ETi and EW-Ti are the total energies of bcc W, hcp Ti and W-Ti alloy in the bcc lattice, respectively. From this definition, negative value of the formation energy means that the alloy is thermodynamically more stable. The calculated results are summarized in Table 5 and the variation of the formation energy as a function of the Ti concentration are given in Fig. 3, from which we can see that at least 25% atomic ratio of Ti can be alloyed in the bcc lattice of the W. However, the formation energy becomes positive when the atomic ratio reaches 1:1 (W0.5 Ti0.5 ), indication of thermodynamically unstable at this concentration. When the Ti concentration is lower than 18.75%, the formation energy decreases when the concentration
is increased. In addition, we also tested the influence of the supercell size to the formation energy, and our results show that the 2 × 2 × 2 supercell is large enough in this study. For example, the formation energy difference is less than several meV for one Ti atom in 2 × 2 × 2 and 3 × 3 × 3 supercells. This is reasonable because Ti only interacts with its nearest neighboring W atoms, which will be shown in the following of this paper. The formation energy discussed above is based on the calculated ground state total energy. In real systems, rather than the total energy, the Gibbs free energy accounts for the phase formation and stability. In order to clarify that the above mentioned phases can be stable and dominated in real systems, we compared it with other disordered configurations and evaluated the entropy contribution of a disordered system. We take the 25% Ti concentration as our example and created a supercell that contains 128 atoms. The ordered configuration in the supercell is the same as the W0.75 Ti0.25 , while several other disordered configurations are tested. Our results shows that the total energy of the system in the ordered configuration (−1498.12 eV) is much lower than that of the disordered configurations (for example, we obtained total energies of −1491.64 eV, −1495.00 eV, −1495.06 eV, −1495.65 eV and −1496.43 eV for different configurations). On the other hand, the entropy of the system can be evaluated with S = kB ln, where is the total number of the configurations. Therefore, we can estimate that the contribution of the entropy to the Gibbs free energy is around 1 eV under room temperature, which is clearly lower than that of the energy difference between the ordered and disordered configurations. Therefore, the disordered systems are not considered in this paper. The formation energy of an alloy is dependent on the bonding interactions among the alloying atoms with the host atoms, which can be studied from analysis of the bond lengths in the alloy. The average bond lengths of the alloying Ti atom to its nearest neighboring W atoms are 2.747 Å, 2.744 Å, 2.742 Å, 2.741 Å and 2.745 Å for the Ti concentration x of 6.25%, 12.5%, 18.75%, 25% and 50%, respectively. As it is shown, when the concentration is lower than 25%, the W Ti bond length decreases with increasing Ti concentration, indicating that the bonding interaction is strengthened with higher Ti concentration. When the Ti concentration becomes 50%, the TiW bond length becomes larger and the Ti-W bonding interaction
J. D.Y. et al. / Fusion Engineering and Design 106 (2016) 34–39
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Fig. 4. The atomic projected density of states (PDOS) of W and Ti atoms in bccW1-x Tix. (a) W0.9375 Ti0.0625 ; (b) W0.875 Ti0.125 ; (c) W0.8125 Ti0.1875 ; (d) W0.75 Ti0.25 . The energy is with respect to the Fermi level. “W-n”, “W-f” and “W-m” are the PDOS of W nearest neighboring to Ti, far away from Ti, and pure W metal, respectively. “Ti” and “Ti-m” are the PDOS of Ti atom in the W-Ti alloy and pure Ti metal, respectively.
is weakened. On the other hand, as all W atoms are surrounded by Ti atoms at the nearest neighboring sites, W-W bonding are not existed in the case of W0.5 Ti0.5 and therefore the formation energy becomes positive. The bonding interaction can be illustrated with the density of states. Fig. 4 presents the atomic projected density of states (PDOS) to W and Ti atoms with different atomic surrounding environments. From Fig. 4 we can see that the alloying effect lowers the energy level of the Ti states comparing with that of in pure Ti metal in the case of W0.9375 Ti0.0625 , while the energy level of the W atoms are not changed much comparing with that of W metal. As a result, the energy of the W0.9375 Ti0.0625 alloy is lower than that of the pure metal and gives rise to the negative formation energy. When the concentration of the Ti becomes higher, more Ti-states (as more Ti atoms are involved) become lower in energy, which further decreases the energy of the alloy. On the other hand, the energy levels of W-states getting higher in energy, which increases the energy of the alloy. Therefore, the decreased energy from Ti atoms is compensated by the increased energy from W atoms, and finally the formation energy states to increase when the Ti concentration reaches certain value (25%). Furthermore, Fig. 4 also demonstrates that the degree of overlapping of the W-states and Ti-states in the case of W0.75 Ti0.25 is higher than that of in W0.9375 Ti0.0625 , indicating that the W Ti bonding strength is higher in W0.75 Ti0.25 comparing with that of in W0.9375 Ti0.0625 . Charge transfer between W and Ti is essential for the lowered energy of the alloy. As the electronegativity of W (1.7) is higher than that of Ti (1.5), it is expected that small charge transfer from Ti to W in the alloy. Fig. 5 gives the differential charge density that illustrates the charge transfer clearly. As it is shown, the 8 nearest neighboring W atom to the Ti atom obtains most of the charge from the Ti atom. Charge transfer can also be analyzed with the Bader charge analysis. Our results show that the Bader charge of Ti is 8.93 e, indicating that Ti losses of about 1.1 electrons (Ti atom has 10 valence electrons). On the other hand, the Bader charge of the W atoms nearest to the Ti atom is 6.20 e, indicating that each W atom obtains about 0.2 electrons (W atom has 6 valence electrons). The charge transfer gives rise to slightly positively charged Ti and negatively charged W, which in turn leads to the slightly shortened
Fig. 5. Charge transfer upon Ti alloying in the case of W0.9375 Ti0.0625 . The color and size of W and Ti atoms are the same as in Fig. 1. The green and red color charge density contours indicate gain and loss charge, respectively. The value of the isosurface is 0.075 e/Å3 . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
W Ti bond length, as indicated above. However, as the amount of the charge transfer is very small (the isosurface value is 0.075 e/Å3 ), the major interaction among the W and Ti atoms is metallic bonding. Therefore, the shortened W Ti bond length strengthened the W Ti metallic bonding. 3.3. The mechanical properties of the binary W Ti alloys For each Ti concentration x, we calculated the single crystal elastic constants and the mechanical property parameters of the W1-x Tix alloys in a bcc structure. The calculated elastic constants are shown in Table 5 and plotted in Fig. 6 as a function of x. As it can be seen, the elastic constants C11 and C44 decrease obviously and
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J. D.Y. et al. / Fusion Engineering and Design 106 (2016) 34–39
Fig. 6. Elastic constants of bcc W-Ti alloys as a function of Ti concentration.
Fig. 8. B/G ratio and Poisson’s ratio of bcc W-Ti alloys as function of Ti concentration.
Table 6 Bulk modulus(B), shear modulus (G), Young’s modulus (E), B/G ratio,Poisson’s ratio (), Cauchy pressure (C’) for bcc W1-x Tix alloys.
ulus decreases from 383.0 GPa for pure W down to 235.9 GPa for the case of W0.75 Ti0.25 . The shear modulus decreases from 147.4 GPa for pure W down to87.5 GPa for W0.75 Ti0.25 . Although, these data suggest that the mechanical strength of the W1-x Tix alloys decreased comparing with that of pure W, it is still much better than that of pure Ti metal, whose elastic modulus are 115.12 GPa (B), 44.94 GPa (G), and 119.30 GPa (E), as given in Table 4. On the other hand, the ductility of the W-Ti alloys is improved substantially compared with the pure W metal. According to Pugh theory, the bulk modulus to shear modulus ratio (B/G ratio) is strongly related with the ductility of metal materials. The larger the B/G ratio is, the better ductility the material exhibits [31]. Fig. 8 presents the B/G ratio and the Poisson’s ratio () as functions of the Ti concentration x. With increasing Ti concentrations, the B/G value increases gradually and reaches a value of 2.96 at x = 25%. This value is already higher than that of pure Ti metal (2.56), indicating that the ductility of the W0.75 Ti0.25 alloy is better than that of pure Ti metal. Poisson’s ratio can also characterize the ductility of bulk materials. As it is shown in Fig. 8, the variation of the Poisson’s ratio to the Ti concentration exhibits similar trend to that of the B/G ratio. When the Ti concentration x reaches 0.25, the Poisson’s ratio is 0.348 and larger than that of pure Ti (0.327). Qualitatively, the metallic bonding and covalent bonding behaviors of the W-Ti alloy can be characterized by the calculated Cauchy pressure, which is shown in Fig. 9. When the Cauchy pressure (C’) has a positive value, metallic bonding interaction dominates the materials and therefore the bulk material exhibits ductility. The larger the C’ value is, the stronger of the metallic bond and thus better ductility properties [32]. On the contrary, as for brittle materials, the Cauchy pressure is negative. The smaller negative value, the stronger covalent bonding and the more serious degree of brittleness of materials[32]. From Fig. 9 we can see that the Cauchy
Composition
B (GPa)
G (GPa)
E (GPa)
B/G
C’ (GPa)
Pure W W0.9375 Ti0.0625 W0.875 Ti0.125 W0.8125 Ti0.1875 W0.75 Ti0.25 W0.5 Ti0.5
317.44 302.52 289.48 274.14 259.10 199.56
147.42 138.72 126.03 109.03 87.50 38.44
382.96 360.97 330.18 288.81 235.95 108.37
2.15 2.18 2.30 2.51 2.96 5.19
0.299 0.301 0.310 0.324 0.348 0.410
35.87 35.67 39.71 46.21 56.63 67.74
Fig. 7. Bulk modulus(B), shear modulus (G) and Young’s modulus (E) of bcc W-Ti alloys as function of Ti concentration.
almost monotonically with the increase of the Ti concentration.The C11 is an important elastic coefficient related with the stiffness of cubic crystals. The C11 value changes from 529.9 GPa for pure W to 248.8 GPa for W0.5 Ti0.5 alloy, indicating that the stiffness of the W1-x Tix alloys decreases with the increase of the Ti concentration. The C12 represents the ability of resistance to lateral deformation of the cubic lattice. As it is shown, the C12 value of the bcc W does not change much upon Ti alloying. With the above elastic constants, we calculated the elastic moduli as shown in Table 6 and Fig. 7. The elastic modulus include bulk modulus (B), shear modulus (G) and Young’s modulus (E), which represent the ability of resistance to deformation of the bulk material [30]. For the bulk modulus of pure W, the calculated value (317.4 GPa) matches very well with experimental data (314.3 GPa) [26], as it is indicated above. Obviously, we can see that the bulk modulus decreases monotonically with the increase of Ti concentration. When Ti concentration reaches 25%, bulk modulus of W-Ti alloy decreased to 259.1 GPa. The changes of Young’s modulus and shear modulus are similar, and the slope of curve for the Young’s modulus is steeper than that of the bulk modulus. The Young’s mod-
Fig. 9. Cauchy pressure(C ) of bcc W-Ti alloys as function of Ti concentration.
J. D.Y. et al. / Fusion Engineering and Design 106 (2016) 34–39 Table 7 Mechanical parameters of the W0.75 Ti0.25 different Ti distributions in the unitcell. The elastic constants, B, G, E, and c’ are in GPa. W0.75 Ti0.25 C11
C12
C44
B
G
E
C’
B/G
Ordered Disorder-1 Disorder-2 Disorder-3 Disorder-4
195.69 196.14 195.84 195.70 192.39
82.43 78.38 80.28 82.30 86.25
259.10 258.51 258.68 258.95 257.87
87.50 84.45 85.87 87.33 91.04
235.95 228.47 231.95 235.51 244.35
0.348 0.353 0.351 0.348 0.342
56.63 58.88 57.78 56.70 53.07
2.96 3.06 3.01 2.97 2.83
385.92 383.24 384.35 385.43 388.83
pressures of the binary W1-x Tix alloys are positive, because metallic bonding dominates the binary W1-x Tix alloys.Furthermore, the Cauchy pressure increases with the increased Ti concentration, showing that the metallic bonding is strengthened upon alloying higher concentration of Ti in the bcc W. This is consistent with the above analysis of the bond length and PDOS in Section 3.2. Finally, it is worthwhile to mention that the distribution of the Ti atoms does not affect much to the mechanical properties of the W-Ti alloys. Taken as an example, Table 7 gives the mechanical parameters of the W0.75 Ti0.25 concentration with four different disordered random distributions of the Ti atoms in the lattice, comparing with that of the ordered distribution. As it can be seen, the mechanical parameters of those disordered system is very close to that of the ordered W0.75 Ti0.25 configuration, indicating that the distribution of the Ti atoms do not affect much to the mechanical properties. Therefore, in real application, although some kind of disorder of the Ti distribution may occur due to various preparing conditions, the mechanical properties can be retained. 4. Summaries and conclusions In summary, the effect of titanium alloying on the fundamental mechanical properties of bcc W has been studied by first principles calculations. Within a 16 atoms solid solution model, the lattice constants, the cell volumes, the formation energies and the elastic constants of W1-x Tix (x = 0.0625, 0.125, 0.1875, 0.25, 0.5) alloys were calculated, and the Ti concentration to the performance of the W-Ti alloys are specifically addressed. When the concentration is lower than 25%, the W Ti bond length decreases with higher Ti concentration, indicating that the bonding interaction is strengthened with higher Ti concentration. When the Ti concentration becomes 50%, the Ti W bond length becomes larger and the Ti W bonding interaction is weakened, which gives rise to the positive formation energy of the W0.5 Ti0.5 . From the obtained elastic constants, the elastic moduli and the mechanical parameters are calculated. The elastic constants and the elastic moduli of W1-x Tix alloys decrease with the increase of Ti concentration. Although these data suggest that the mechanical strength of the W1-x Tix alloys decreased comparing with that of pure W, it is still much better than that of pure Ti metal. On the other hand, based on the calculated B/G and Poisson’s ratio, it is found that the ductility of bcc W can be improved through Ti alloying. The ductility of the W1-x Tix alloys increases with increasing, and it is even better than that of pure Ti metal when the Ti concentration reaches 25%.Through analysis of the Cauchy pressure(C’), the metallic bonding is strengthened upon alloying higher concentration of Ti in the bcc W. Acknowledgements This study was supported by the National Natural Science Foundation of China (No.11178002), the International S&T Cooperation Program of China (No.2015DFA61800), the program for Innovative Research Team in Jiangxi Province (20133BCB2403), the Scientific & Technological Innovation Research Fund of Jiangxi Province
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Fusion Engineering and Design journal homepage: www.elsevier.com/locate/fusengdes
Mechanical properties of W–Ti alloys from first-principles calculations D.Y. Jiang a,b,d , C.Y. Ouyang c , S.Q. Liu a,b,∗ a
Department of Materials Science and Engineering, Nanchang University, Nanchang 330047, China Department of Physics, Nanchang University, Nanchang 330047, China c Department of Physics, Jiangxi Normal University, Nanchang 330022, China d School of Basic Sciences, Jiangxi University of Technology, Nanchang 330098, China b
h i g h l i g h t s • The mechanical properties of the W1-x Tix alloys are calculated from DFT. • Ti alloying enhances the ductility of W metal substantially. • The mechanical strength of W-Ti alloys is slightly weaker than W while stronger than Ti.
a r t i c l e
i n f o
Article history: Received 8 October 2015 Received in revised form 29 January 2016 Accepted 5 March 2016 Available online 19 March 2016 Keywords: W-Ti alloy Mechanical properties Ductility The first wall First principles calculations
a b s t r a c t The effect of Ti concentration on the fundamental mechanical properties of W-Ti alloys has been studied from first principles calculations. The lattice constants, the cell volumes and the formation energies of the W1-x Tix (x = 0.0625, 0.125, 0.1875, 0.25, 0.5) alloys were calculated. It is shown that Ti alloying in bcc W lattice is thermodynamically favorable when the Ti concentration is lower than 25% and the W0.8125 Ti0.1875 have the lowest formation energy. With the optimized geometry and lattice, the elastic constants are calculated and then the elastic moduli and other mechanical parameters are derived. Results show that although the mechanical strength of the W-Ti alloys is lower than that of pure W metal, it is much higher than that of pure Ti metal. On the other hand, the B/G ratio and the Poisson’s ratio of the W-Ti alloys is much higher than that of pure W, and even higher than that of pure Ti, indicating that Ti alloying can improve the ductility of bcc W substantially. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Tungsten is receiving considerable attention as potential plasma facing material of the divertor and the first wall material in fusion power reactors, due to its high melting point, high strength at high temperatures, high sputtering threshold energy, good thermal conductivity and low thermal expansion coefficient [1–3]. In addition, tungsten is also thought as a shielding material in fusion reactors and other systems involving nuclear reactions [4]. Unfortunately, pure tungsten metal exhibits poor radiation stability and fracture toughness, low ductility associated with a high ductile-tobrittle transition temperature (DBTT). These properties are strongly dependent on the chemical composition and the micro-structural state [5]. Alloying is a good method to improve the physical properties of tungsten. For example, the thermal properties of tungsten are expected to be enhanced by alloying with other elements [6,7].
∗ Corresponding author at: Department of Materials Science and Engineering, Nanchang University, Nanchang 330047, China. E-mail address: [email protected] (L. S.Q.). http://dx.doi.org/10.1016/j.fusengdes.2016.03.028 0920-3796/© 2016 Elsevier B.V. All rights reserved.
Among various alloying strategies of W metal, W-Ti alloy is one typical one [8,9]. To our knowledge, most experimental efforts of the binary W-Ti alloys have been devoted to understanding the effect of Ti concentration on the grain growth behaviors [10–14]. However, little attention was paid to the effect of Ti-alloying and the Ti concentration on the fundamental mechanical properties, such as elastic properties and ductile/brittle behaviors. The density-functional theory (DFT) calculation can be used to predict the mechanical performance of materials with relatively low cost and high efficiency, comparing with the experimental evaluations. Since conventional experimental alloy design processes can be very expensive and time-consuming, DFT calculations can be used to study the structure-property relationships and help to design these materials. The computational data may serve as guidance for further optimizing the composition of W-Ti alloys. Computer simulation focus on improving specific alloy properties can narrow down the number of compositions that are needed to be prepared and characterized. In this paper, we studied the phase stability and mechanical properties of W1-x Tix alloys (x = 0.0625, 0.125, 0.1875, 0.25 and 0.5) from first principles calculations. The lattice constants, the cell vol-
J. D.Y. et al. / Fusion Engineering and Design 106 (2016) 34–39 Table 1 The elastic constant of bcc W.
35
Table 3 The elastic constant for hcp Ti.
Method
C11 (GPa)
C12 (GPa)
C44 (GPa)
Method
C11 (GPa)
C12 (GPa)
C13 (GPa)
C33 (GPa)
C44 (GPa)
Present work Experiment [26] Theory [26,27]
529.94 533 553
211.19 205 207
139.44 163 163
Present work Experiment [28] Theory [29]
175.61 176.10 171.60
84.45 86.90 86.60
81.98 68.30 72.60
188.34 190.50 190.60
41.41 50.80 41.10
ume, the formation energies and the elastic constants of the W1-x Tix alloys are calculated. Based on the mechanical characteristic of B/G ratio, Poisson’s ratio () and Cauchy pressure (C’), the ductile/brittle properties of the W-Ti alloys are evaluated. 2. Model and computational details Computations in the present work were done within the density-functional theory (DFT) and the plane-wave pseudopotential method, which were implemented in the Vienna ab initio simulation package (VASP) [15,16]. The core ion and valence electron interaction was described by the projector augmented wave method (PAW) [17], and the exchange-correlation part was described with the generalized gradient approximation (GGA) by Perdew and Wang (PW91) [18]. All models were described by a 2 × 2 × 2 supercell containing 16 atoms in a body centered cubic (bcc) structures. Part of W atoms were substituted by the Ti atoms to obtain various alloys with different Ti concentrations, that is pure W, W0.9375 Ti0.0625 , W0.875 Ti0.125 , W0.8125 Ti0.1875 , W0.75 Ti0.25 , W0.5 Ti0.5 and pure Ti. Except for the case of pure Ti, which was modeled with a hexagonal supercell, all the other systems were modeled with bcc supercells. For the Brillouin-zone sampling, we used the 11 × 11 × 11 Monkhorst-Pack mesh [19] for all W1-x Tix alloys with bcc structures and a 6 × 6 × 4 Monkhorst-Pack mesh for hexagonal close packing (hcp) phase titanium metal. A cutoff energy of 350 eV was used for all systems. Gaussian smearing method with a smearing width of 0.05 eV is applied in all calculations in this study. Within the framework of the continuum elasticity theory [20,21], for the cubic structure, there are three independent elastic constants, i.e. C11 , C12 and C44 . In order to calculate these elastic constants, three sets of specific strains (␦) along different directions are applied to the cubic supercell, and the total energy changes are calculated as a function of the applied strains. Then, the elastic constants can be obtained through fitting the energy changes (E) vs the applied strains (␦). For the hexagonal structure, there are five independent elastic constants, i.e. C11 , C12 , C13 , C33 and C44. Similarly, five sets of specific strains (␦) along different directions are applied to the hexagonal supercell, and the elastic constants can be obtained through fitting the energy changes (E) vs the applied strains (␦). The mechanical properties can be calculated from these single crystal elastic constants, according to the Voigt-Reuss-Hill scheme [22–25].The bulk modulus (B), shear modulus (G), Young’s modulus (E), Poisson’s ratio () and Cauchy pressure (C’) of the cubic W-Ti alloys are calculated with C11 , C12 and C44 and given by the following formula: B=
C11 + 2C12 3C 44 + C11 − C12 ,G = , 3 5
Table 2 Bulk modulus (B), shear modulus (G), Young’s modulus (E), B/G ratio, Poisson’s ratio (), Cauchy pressure C’ of bcc W. Method
B (GPa)
G (GPa)
E (GPa)
B/G
C’ (GPa)
Present work Experiment [26] Theory [26,27]
317.44 314.33 322.33
147.42 163.40 173.00
382.96 417.80 440.24
2.15 1.92 1.86
0.30 0.28 0.27
35.87 21.00 22.00
Table 4 Bulk modulus(B), shear modulus (G), Young’s modulus (E), B/G ratio,Poisson’s ratio (), Cauchy pressure C’ of hcp Ti. Method
B (GPa)
G (GPa)
E (GPa)
B/G
Present work Experiment [28] Theory [29]
115.12 109.96 110.81
44.94 50.16 44.68
119.30 130.62 118.15
2.56 2.19 2.48
0.327 0.302 0.322
E=
9BG E C12 − C44 , V = −1 , C = 3B + G 2G 2
Similarly, the mechanic parameters B, G, E and of the hexagonal Ti are calculated from elastic constants C11 , C12 , C13 , C33 and C44. The specific methods can be found in the literature [22–25]. 3. Results and discussion 3.1. The mechanical properties of bcc tungsten (W) and hcp titanium (Ti) The calculated single crystal elastic constants and mechanical property parameters of W and Ti are listed in Tables 1–4. For comparison purpose, we also list the data from experiments, as well as other theoretical results in the tables. Notice that the elastic constants as well as the mechanical parameters of the bcc W and hcp Ti metal from our calculations are basically in agreement with the experimental and other theoretical results. This confirms that the model and computational parameters are reasonable in the present work and therefore the following results on the W-Ti alloys are reliable. Furthermore, the elastic constants C44 for the bcc W is underestimated while the C13 for the hcp Ti is overestimated comparing with those experimental data. This is due to the GGA pseudopotentials we used in the current study. As GGA always underestimate the lattice constants of metal systems, it transfers this error to the calculation of the elastic constants. 3.2. The energetics and stability of the binary W-Ti alloys First of all, the atomic positions and W-Ti configuration in the supercell are considered and optimized, and the configuration with the lowest total energy is chosen for further study. The schematic diagrams of the atomic configurations with the lowest total energy of the W1-x Tix alloys at different x value are shown in Fig. 1. As it is shown, the W-Ti alloys keep the bcc lattice, and the Ti atoms prefer to locate at positions to ensure that the structure is in a highest symmetry. This shows that the solid solution of the W-Ti alloy can be formed at the atomic level. In addition, to obtain the equilibrium lattice constant, we calculated the total energies as a function of the volume and then fit the results using Birch-Murnaghan equation of state. Fig. 2 shows the variation of the equilibrium lattice constants of W1-x Tix alloys as a function of the Ti concentration. When Ti concentration is less than 25%, the lattice constant of the W1-x Tix alloy decreases almost monotonically with x increases, due to the smaller radius of the Ti atom comparing with that of the W atom. However, when titanium concentration reaches 50%, the lattice starts to expand. The expanded lattice of the W1-x Tix alloys when x is high is related with the weakened bonding interaction between the W and Ti atoms, which will be discussed in the
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J. D.Y. et al. / Fusion Engineering and Design 106 (2016) 34–39 Table 5 The cell volumes, elastic constants, and formation energies of bcc W1-x Tix alloys. Composition
C11 (GPa)
C12 (GPa)
C44 (GPa)
V (Å3 )
Ef (eV/Cell)
Pure W W0.9375 Ti0.0625 W0.875 Ti0.125 W0.8125 Ti0.1875 W0.75 Ti0.25 W0.5 Ti0.5
529.94 509.01 480.94 436.59 385.92 248.78
211.19 199.28 193.75 192.92 195.69 174.94
139.44 127.95 114.32 100.50 82.43 39.46
31.94 31.88 31.81 31.74 31.71 31.85
0.000 −0.798 −1.141 −1.310 −1.149 0.166
Fig. 1. The energetically most favorable atomic arrangements of the bcc W1-x Tix (x = 0.0625, 0.125, 0.25 and 0.5) alloys in a 2 × 2 × 2 supercell. Small blue and large grey balls are W and Ti atoms, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. Formation energy of bcc W-Ti alloys as function of Ti concentration.
Fig. 2. Equilibrium lattice constants of bcc W-Ti alloys as a function of Ti concentration.
following analysis. Here we mention that we also tested the alloys with hcp lattice, and our results show that the calculated energies of the W-Ti alloys with hcp lattice is much higher than that of with bcc lattice. For example, the energy of W0.5 Ti0.5 in the bcc lattice (−164.22 eV) is much smaller than that of W0.5 Ti0.5 in the hcp lattice (−161.31 eV). For other concentrations, the energy difference is even larger. To evaluate the stability of the alloys, we calculated the formation energy of the W1-x Tix alloys, which is defined as: Ef = EW −Ti − (1 − x) EW − xETi where x is the concentration of the alloying element, EW , ETi and EW-Ti are the total energies of bcc W, hcp Ti and W-Ti alloy in the bcc lattice, respectively. From this definition, negative value of the formation energy means that the alloy is thermodynamically more stable. The calculated results are summarized in Table 5 and the variation of the formation energy as a function of the Ti concentration are given in Fig. 3, from which we can see that at least 25% atomic ratio of Ti can be alloyed in the bcc lattice of the W. However, the formation energy becomes positive when the atomic ratio reaches 1:1 (W0.5 Ti0.5 ), indication of thermodynamically unstable at this concentration. When the Ti concentration is lower than 18.75%, the formation energy decreases when the concentration
is increased. In addition, we also tested the influence of the supercell size to the formation energy, and our results show that the 2 × 2 × 2 supercell is large enough in this study. For example, the formation energy difference is less than several meV for one Ti atom in 2 × 2 × 2 and 3 × 3 × 3 supercells. This is reasonable because Ti only interacts with its nearest neighboring W atoms, which will be shown in the following of this paper. The formation energy discussed above is based on the calculated ground state total energy. In real systems, rather than the total energy, the Gibbs free energy accounts for the phase formation and stability. In order to clarify that the above mentioned phases can be stable and dominated in real systems, we compared it with other disordered configurations and evaluated the entropy contribution of a disordered system. We take the 25% Ti concentration as our example and created a supercell that contains 128 atoms. The ordered configuration in the supercell is the same as the W0.75 Ti0.25 , while several other disordered configurations are tested. Our results shows that the total energy of the system in the ordered configuration (−1498.12 eV) is much lower than that of the disordered configurations (for example, we obtained total energies of −1491.64 eV, −1495.00 eV, −1495.06 eV, −1495.65 eV and −1496.43 eV for different configurations). On the other hand, the entropy of the system can be evaluated with S = kB ln, where is the total number of the configurations. Therefore, we can estimate that the contribution of the entropy to the Gibbs free energy is around 1 eV under room temperature, which is clearly lower than that of the energy difference between the ordered and disordered configurations. Therefore, the disordered systems are not considered in this paper. The formation energy of an alloy is dependent on the bonding interactions among the alloying atoms with the host atoms, which can be studied from analysis of the bond lengths in the alloy. The average bond lengths of the alloying Ti atom to its nearest neighboring W atoms are 2.747 Å, 2.744 Å, 2.742 Å, 2.741 Å and 2.745 Å for the Ti concentration x of 6.25%, 12.5%, 18.75%, 25% and 50%, respectively. As it is shown, when the concentration is lower than 25%, the W Ti bond length decreases with increasing Ti concentration, indicating that the bonding interaction is strengthened with higher Ti concentration. When the Ti concentration becomes 50%, the TiW bond length becomes larger and the Ti-W bonding interaction
J. D.Y. et al. / Fusion Engineering and Design 106 (2016) 34–39
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Fig. 4. The atomic projected density of states (PDOS) of W and Ti atoms in bccW1-x Tix. (a) W0.9375 Ti0.0625 ; (b) W0.875 Ti0.125 ; (c) W0.8125 Ti0.1875 ; (d) W0.75 Ti0.25 . The energy is with respect to the Fermi level. “W-n”, “W-f” and “W-m” are the PDOS of W nearest neighboring to Ti, far away from Ti, and pure W metal, respectively. “Ti” and “Ti-m” are the PDOS of Ti atom in the W-Ti alloy and pure Ti metal, respectively.
is weakened. On the other hand, as all W atoms are surrounded by Ti atoms at the nearest neighboring sites, W-W bonding are not existed in the case of W0.5 Ti0.5 and therefore the formation energy becomes positive. The bonding interaction can be illustrated with the density of states. Fig. 4 presents the atomic projected density of states (PDOS) to W and Ti atoms with different atomic surrounding environments. From Fig. 4 we can see that the alloying effect lowers the energy level of the Ti states comparing with that of in pure Ti metal in the case of W0.9375 Ti0.0625 , while the energy level of the W atoms are not changed much comparing with that of W metal. As a result, the energy of the W0.9375 Ti0.0625 alloy is lower than that of the pure metal and gives rise to the negative formation energy. When the concentration of the Ti becomes higher, more Ti-states (as more Ti atoms are involved) become lower in energy, which further decreases the energy of the alloy. On the other hand, the energy levels of W-states getting higher in energy, which increases the energy of the alloy. Therefore, the decreased energy from Ti atoms is compensated by the increased energy from W atoms, and finally the formation energy states to increase when the Ti concentration reaches certain value (25%). Furthermore, Fig. 4 also demonstrates that the degree of overlapping of the W-states and Ti-states in the case of W0.75 Ti0.25 is higher than that of in W0.9375 Ti0.0625 , indicating that the W Ti bonding strength is higher in W0.75 Ti0.25 comparing with that of in W0.9375 Ti0.0625 . Charge transfer between W and Ti is essential for the lowered energy of the alloy. As the electronegativity of W (1.7) is higher than that of Ti (1.5), it is expected that small charge transfer from Ti to W in the alloy. Fig. 5 gives the differential charge density that illustrates the charge transfer clearly. As it is shown, the 8 nearest neighboring W atom to the Ti atom obtains most of the charge from the Ti atom. Charge transfer can also be analyzed with the Bader charge analysis. Our results show that the Bader charge of Ti is 8.93 e, indicating that Ti losses of about 1.1 electrons (Ti atom has 10 valence electrons). On the other hand, the Bader charge of the W atoms nearest to the Ti atom is 6.20 e, indicating that each W atom obtains about 0.2 electrons (W atom has 6 valence electrons). The charge transfer gives rise to slightly positively charged Ti and negatively charged W, which in turn leads to the slightly shortened
Fig. 5. Charge transfer upon Ti alloying in the case of W0.9375 Ti0.0625 . The color and size of W and Ti atoms are the same as in Fig. 1. The green and red color charge density contours indicate gain and loss charge, respectively. The value of the isosurface is 0.075 e/Å3 . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
W Ti bond length, as indicated above. However, as the amount of the charge transfer is very small (the isosurface value is 0.075 e/Å3 ), the major interaction among the W and Ti atoms is metallic bonding. Therefore, the shortened W Ti bond length strengthened the W Ti metallic bonding. 3.3. The mechanical properties of the binary W Ti alloys For each Ti concentration x, we calculated the single crystal elastic constants and the mechanical property parameters of the W1-x Tix alloys in a bcc structure. The calculated elastic constants are shown in Table 5 and plotted in Fig. 6 as a function of x. As it can be seen, the elastic constants C11 and C44 decrease obviously and
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J. D.Y. et al. / Fusion Engineering and Design 106 (2016) 34–39
Fig. 6. Elastic constants of bcc W-Ti alloys as a function of Ti concentration.
Fig. 8. B/G ratio and Poisson’s ratio of bcc W-Ti alloys as function of Ti concentration.
Table 6 Bulk modulus(B), shear modulus (G), Young’s modulus (E), B/G ratio,Poisson’s ratio (), Cauchy pressure (C’) for bcc W1-x Tix alloys.
ulus decreases from 383.0 GPa for pure W down to 235.9 GPa for the case of W0.75 Ti0.25 . The shear modulus decreases from 147.4 GPa for pure W down to87.5 GPa for W0.75 Ti0.25 . Although, these data suggest that the mechanical strength of the W1-x Tix alloys decreased comparing with that of pure W, it is still much better than that of pure Ti metal, whose elastic modulus are 115.12 GPa (B), 44.94 GPa (G), and 119.30 GPa (E), as given in Table 4. On the other hand, the ductility of the W-Ti alloys is improved substantially compared with the pure W metal. According to Pugh theory, the bulk modulus to shear modulus ratio (B/G ratio) is strongly related with the ductility of metal materials. The larger the B/G ratio is, the better ductility the material exhibits [31]. Fig. 8 presents the B/G ratio and the Poisson’s ratio () as functions of the Ti concentration x. With increasing Ti concentrations, the B/G value increases gradually and reaches a value of 2.96 at x = 25%. This value is already higher than that of pure Ti metal (2.56), indicating that the ductility of the W0.75 Ti0.25 alloy is better than that of pure Ti metal. Poisson’s ratio can also characterize the ductility of bulk materials. As it is shown in Fig. 8, the variation of the Poisson’s ratio to the Ti concentration exhibits similar trend to that of the B/G ratio. When the Ti concentration x reaches 0.25, the Poisson’s ratio is 0.348 and larger than that of pure Ti (0.327). Qualitatively, the metallic bonding and covalent bonding behaviors of the W-Ti alloy can be characterized by the calculated Cauchy pressure, which is shown in Fig. 9. When the Cauchy pressure (C’) has a positive value, metallic bonding interaction dominates the materials and therefore the bulk material exhibits ductility. The larger the C’ value is, the stronger of the metallic bond and thus better ductility properties [32]. On the contrary, as for brittle materials, the Cauchy pressure is negative. The smaller negative value, the stronger covalent bonding and the more serious degree of brittleness of materials[32]. From Fig. 9 we can see that the Cauchy
Composition
B (GPa)
G (GPa)
E (GPa)
B/G
C’ (GPa)
Pure W W0.9375 Ti0.0625 W0.875 Ti0.125 W0.8125 Ti0.1875 W0.75 Ti0.25 W0.5 Ti0.5
317.44 302.52 289.48 274.14 259.10 199.56
147.42 138.72 126.03 109.03 87.50 38.44
382.96 360.97 330.18 288.81 235.95 108.37
2.15 2.18 2.30 2.51 2.96 5.19
0.299 0.301 0.310 0.324 0.348 0.410
35.87 35.67 39.71 46.21 56.63 67.74
Fig. 7. Bulk modulus(B), shear modulus (G) and Young’s modulus (E) of bcc W-Ti alloys as function of Ti concentration.
almost monotonically with the increase of the Ti concentration.The C11 is an important elastic coefficient related with the stiffness of cubic crystals. The C11 value changes from 529.9 GPa for pure W to 248.8 GPa for W0.5 Ti0.5 alloy, indicating that the stiffness of the W1-x Tix alloys decreases with the increase of the Ti concentration. The C12 represents the ability of resistance to lateral deformation of the cubic lattice. As it is shown, the C12 value of the bcc W does not change much upon Ti alloying. With the above elastic constants, we calculated the elastic moduli as shown in Table 6 and Fig. 7. The elastic modulus include bulk modulus (B), shear modulus (G) and Young’s modulus (E), which represent the ability of resistance to deformation of the bulk material [30]. For the bulk modulus of pure W, the calculated value (317.4 GPa) matches very well with experimental data (314.3 GPa) [26], as it is indicated above. Obviously, we can see that the bulk modulus decreases monotonically with the increase of Ti concentration. When Ti concentration reaches 25%, bulk modulus of W-Ti alloy decreased to 259.1 GPa. The changes of Young’s modulus and shear modulus are similar, and the slope of curve for the Young’s modulus is steeper than that of the bulk modulus. The Young’s mod-
Fig. 9. Cauchy pressure(C ) of bcc W-Ti alloys as function of Ti concentration.
J. D.Y. et al. / Fusion Engineering and Design 106 (2016) 34–39 Table 7 Mechanical parameters of the W0.75 Ti0.25 different Ti distributions in the unitcell. The elastic constants, B, G, E, and c’ are in GPa. W0.75 Ti0.25 C11
C12
C44
B
G
E
C’
B/G
Ordered Disorder-1 Disorder-2 Disorder-3 Disorder-4
195.69 196.14 195.84 195.70 192.39
82.43 78.38 80.28 82.30 86.25
259.10 258.51 258.68 258.95 257.87
87.50 84.45 85.87 87.33 91.04
235.95 228.47 231.95 235.51 244.35
0.348 0.353 0.351 0.348 0.342
56.63 58.88 57.78 56.70 53.07
2.96 3.06 3.01 2.97 2.83
385.92 383.24 384.35 385.43 388.83
pressures of the binary W1-x Tix alloys are positive, because metallic bonding dominates the binary W1-x Tix alloys.Furthermore, the Cauchy pressure increases with the increased Ti concentration, showing that the metallic bonding is strengthened upon alloying higher concentration of Ti in the bcc W. This is consistent with the above analysis of the bond length and PDOS in Section 3.2. Finally, it is worthwhile to mention that the distribution of the Ti atoms does not affect much to the mechanical properties of the W-Ti alloys. Taken as an example, Table 7 gives the mechanical parameters of the W0.75 Ti0.25 concentration with four different disordered random distributions of the Ti atoms in the lattice, comparing with that of the ordered distribution. As it can be seen, the mechanical parameters of those disordered system is very close to that of the ordered W0.75 Ti0.25 configuration, indicating that the distribution of the Ti atoms do not affect much to the mechanical properties. Therefore, in real application, although some kind of disorder of the Ti distribution may occur due to various preparing conditions, the mechanical properties can be retained. 4. Summaries and conclusions In summary, the effect of titanium alloying on the fundamental mechanical properties of bcc W has been studied by first principles calculations. Within a 16 atoms solid solution model, the lattice constants, the cell volumes, the formation energies and the elastic constants of W1-x Tix (x = 0.0625, 0.125, 0.1875, 0.25, 0.5) alloys were calculated, and the Ti concentration to the performance of the W-Ti alloys are specifically addressed. When the concentration is lower than 25%, the W Ti bond length decreases with higher Ti concentration, indicating that the bonding interaction is strengthened with higher Ti concentration. When the Ti concentration becomes 50%, the Ti W bond length becomes larger and the Ti W bonding interaction is weakened, which gives rise to the positive formation energy of the W0.5 Ti0.5 . From the obtained elastic constants, the elastic moduli and the mechanical parameters are calculated. The elastic constants and the elastic moduli of W1-x Tix alloys decrease with the increase of Ti concentration. Although these data suggest that the mechanical strength of the W1-x Tix alloys decreased comparing with that of pure W, it is still much better than that of pure Ti metal. On the other hand, based on the calculated B/G and Poisson’s ratio, it is found that the ductility of bcc W can be improved through Ti alloying. The ductility of the W1-x Tix alloys increases with increasing, and it is even better than that of pure Ti metal when the Ti concentration reaches 25%.Through analysis of the Cauchy pressure(C’), the metallic bonding is strengthened upon alloying higher concentration of Ti in the bcc W. Acknowledgements This study was supported by the National Natural Science Foundation of China (No.11178002), the International S&T Cooperation Program of China (No.2015DFA61800), the program for Innovative Research Team in Jiangxi Province (20133BCB2403), the Scientific & Technological Innovation Research Fund of Jiangxi Province
39
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