Solid solution mechanism and thermodynamic properties of Ti–Re alloy system: Experiment and theory

Intermetallics 15 (2007) 1116e1121 www.elsevier.com/locate/intermet

Solid solution mechanism and thermodynamic properties of TieRe alloy system: Experiment and theory Yurong Wu a, Wangyu Hu a,*, Shanglei Yang b, Songnian Lou b a b

Department of Applied Physics, Hunan University, Changsha 410082, China Institute of Welding, Shanghai Jiao Tong University, Shangshai 20030, China Received 11 April 2006; accepted 25 January 2007 Available online 26 March 2007

Abstract Button ingots of Ti100xRex (x ¼ 4.26, 13.87, 27.78, 47.48 and 73.88 at.%) alloy samples were prepared by an arc melting method. The analyses of X-ray diffraction (XRD) patterns were used to estimate the lattice parameter of b-Ti, the results indicated that the lattice constant of b-Ti decreased with increasing Re content. The solid solution mechanism for Re atoms in b-Ti lattice is studied by molecular dynamics (MD) and the first-principles, the simulation results showed that Re atoms in b-Ti lattice should occupy the substitute sites, not the interstice sites. Moreover, the thermodynamic data, such as the formation enthalpies of disordered solid solutions and intermetallic compounds and the lattice parameters for TieRe alloy were also calculated with a modified analytic embedded atom method (EAM) and the first-principles. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: A. Intermetallics, miscellaneous; B. Thermodynamic and thermochemical properties; E. Simulations, atomistic; F. Diffraction; G. Aero-engine components

1. Introduction Recently, rhenium has received increasing attention due to their exceptional properties, including a high melting point of 3180  C and the highest elastic modulus of all the refractories, which makes rhenium good candidate for using as reinforcement in some alloys. Substantial advance has been made, such as, low rhenium content can improve Ni base superalloys’s creep and fatigue behaviors [1e4], and enhances the strength at high temperatures in tungsten and molybdenum alloys [5,6], as well as increases the superconducting transition temperature Tc in some alloys [7]. Rhenium and titanium alloys are utilized extensively in aerial engine materials, due to high melting point and strength. Welding of Re and Ti alloys are usually obtained using electron beam melting, which is a preferred way in welding Ti

* Corresponding author. Fax: þ86 731 882 3971. E-mail address: [email protected] (W. Hu). 0966-9795/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.intermet.2007.01.007

alloy and other different metals, because the physical and crystallization properties of Ti and Re are quite different, the disadvantage, such as residual stress and brittle phase Ti5Re24 will occur in welding Re and Ti alloys. In order to understand the physical properties of this alloy, the thermodynamic data and solid solution mechanism are absolutely necessary. In this paper, the relation between lattice parameter of b-Ti and Re content has been estimated from the X-ray diffraction (XRD) patterns of Ti100xRex alloys. And this relation is studied with the modified analytical embedded-atom method (EAM) by Hu et al. [8,9]. Using the constructed TieRe alloy potential, the solid solution characteristic of TieRe and thermodynamic properties have been studied by molecular dynamics (MD) and the first-principles VASP method [10]. 2. Experiment and theory techniques Button ingots of Ti100xRex alloy samples with nominal compositions of x ¼ 4.26, 13.87, 27.78, 47.48 and 73.88 at.% were prepared by an arc melting method. The purities of initial

Y. Wu et al. / Intermetallics 15 (2007) 1116e1121 Table 1 Parameters of the many-body potentials for Ti and Re

1117

The superscripts a and b indicate the a- and b-type atoms in binary alloy. faa ðrÞ and fbb ðrÞ are the monatomic potentials which could be given by the monatomic models. ra and rb are a- and b-type atoms’ parameters, respectively. m and rc are alloy adjustable parameters, which is obtained from fitting ab-initio enthalpies of formation data for TiRe type intermetal˚ , respectively. By lic phase, their values are 4.15 and 2.8133 A using the fitting alloying parameters, some physical properties, heats of formation, lattice parameters, and the solid solution mechanism of the TieRe binary alloy can be studied. The model parameters are reported in Tables 1 and 2. Calculations employed the local density approximation (LDA) to density functional theory embodied in the Vienna ab initio simulation package (VASP) [10]. The projectoraugmented wave (PAW) pseudopotentials were used.

example, according to TieRe binary phase diagram [12], a complex phase can be found when the Re content increase and a lot of Ti5Re24 phase were observed in the present experiment, as shown in Fig. 1(e), in which there are three peaks of the Ti5Re24 phase, corresponding to (330), (332), and (431), respectively. Furthermore, the lattice constant of b-Ti decreased with increasing Re content, which implied that elements Ti and Re can form solid solutions, and Re atoms should locate at substitutable sites in b-Ti lattice. It may be ascribed to different atomic size: the atomic radius of Re ˚ ) is smaller than that of Ti (r ¼ 1.448 A ˚ ). By com(r ¼ 1.370 A paring the EAM calculated lattice parameters with the experimental values, obviously, the EAM calculated values are lower than the experimental data. It fact, it can be found that the EAM calculated lattice parameter of pure b-Ti (a ¼ 0.3258 nm) is lower than the experimental data (a ¼ 0.3306 nm), which is mainly due to the element Ti potential that is constructed by a-Ti in the present model. In spite of discrepancy, the tendency in the compositional dependence of lattice parameter is similar. Two intermetallics can be obtained according to the TieRe phase diagram. TiRe intermediate phase is cubic, cP2-CsCl type. Ti5Re24 is cubic, cI58-a-Mn type. The lattice parameter of these two intermetallics are calculated and listed in Table 3. For the TiRe intermediate phase, the EAM and VASP calculations are slightly larger than the experiment [13], with the error ranging from 1.0% in EAM to 0.5% in VASP. For the Ti5Re24 phase, the EAM and VASP results are somewhat larger than the experimental values, with the error ranging from 0.56% in EAM to 0.8% in VASP. In general, the lattice constants of intermetallic for both the EAM and VASP are closer to the experimental data.

3. Results and discussions

3.2. Heats of formation for the TieRe binary systems

3.1. X-ray diffraction and lattice parameter

In order to calculate the heats of formation for disordered solid solutions, their lattice structures can be assumed as the structures of their constituent elements: bcc (b-Ti) or ideal hcp (dhcp). Thus, the lattice structure for TieRe alloy systems may take a bcc or dhcp structure in the whole composition range. The calculated heats of formation for the disordered solid solutions are demonstrated in Fig. 3 as the solid and dot curves for bcc and dhcp structures, respectively. For comparison, the Miedema’s data (B) [12] are also indicated in Fig. 3. The heats of formation for the disordered solid solutions are negative values, and the calculated heats of formation of disordered bcc are larger than that of dhcp. The results also showed elements Ti and Re can form solid solutions. This is in good agreement with the above discussion. As shown in Fig. 3, it

Ti Re

n

F0

a  106

b  106

kc

rp

fe

0.51 0.57

3.0303 5.7039

9.1514 3.6111

4.2872 1.6661

0.9 0.1

2.9881 3.7698

3.5118 6.3316

˚. n, fe and kc are dimensionless, F0, a, and b are in eV, rp( p ¼ a, b) is in A

materials were 99.5% Ti and 99.99% Re. XRD measurements were carried out with a Bruker D8 Advance system. The modified analytic embedded atom potentials for Re, Ti elements, and ReeTi alloys were employed, which were described in detail in Refs. [8,9,11]. In the present EAM model, the alloy potential is taken as   a  b  1 r r aa bb f ðrÞ ¼ m f r c þ f r c 2 r r ab

The obtained XRD patterns for alloys Ti100xRex with x ¼ 4.26, 13.87, 27.78, 47.48 and 73.88 at.% are shown in Fig. 1(aee). As shown in Fig. 1, there are three peaks for b-Ti, corresponding to (110), (211) and (200), respectively, and (110) is the main peak. According to these XRD patterns, the lattice parameters of b-Ti are estimated and shown in Fig. 2 as solid circles (C). For comparison, the EAM calculations are given in Fig. 2 as solid squares (-). As shown in Fig. 2, it can be clearly seen that the measured lattice constants of b-Ti decreased with increasing Re content, and the lattice constant of b-Ti versus Re content curve is nonlinear. It can be considered that this nonlinear behavior is due to structure transition with increasing Re content. For Table 2 Parameters of the many-body potentials for Ti and Re. ki (i ¼ 1, 0e6) in eV

Ti Re

k1

k0

k1

k2

k3

k4

k5

k6

169.417 756.654

776.198 3523.036

1522.923 6942.170

1665.388 7525.282

1096.771 4855.499

434.364 1866.96

95.571 396.360

8.991 35.854

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Y. Wu et al. / Intermetallics 15 (2007) 1116e1121

Fig. 1. X-ray diffraction patterns of b-Ti for Ti100xRex sample: (a) x ¼ 4.26; (b) x ¼ 13.87; (c) x ¼ 27.78; (d) x ¼ 47.78; and (e) x ¼ 73.88 at.%.

can be seen that the values obtained by EAM are lower than those from Miedema’s theory, but the tendency in the compositional dependence of heats of formation for disordered solid solutions is similar. The triangle (:) and square (-) in Fig. 3 represent the calculated results for a series of ordered intermetallic compounds with EAM and VASP, respectively. The heat of formation of Ti5Re24 intermediate phase from EAM calculation is very small, which may be originated from that the elastic

relaxation and temperature effect are not considered in the present calculations. 3.3. Solid solution mechanism Elements Ti and Re can form solid solutions according to the above discussion, and Re atoms in b-Ti lattice occupied the substitute site. In this section, the solid solution mechanism is studied with MD and the first-principles VASP.

Y. Wu et al. / Intermetallics 15 (2007) 1116e1121

Fig. 2. Lattice constants of b-Ti as a function of Re content.

Fig. 3. Calculated formation enthalpies of the TieRe alloy.

In the present study, we adopt a bcc solid solution model, which is set to consist of 10  10  10  2 ¼ 2000 atoms. In this model, the [100], [010], and [001] atomic crystal directions are parallel to the x, y, and z axes, respectively, and periodic boundary conditions are adopted in three dimensions. Employing the constructed TieRe potential, MD simulation is carried out with a ParrinelloeRahman constant temperature and constant pressure (NPT) ensemble, and the equations of motion are solved using a fourth-order predictorecorrector algorithm of gear with a time step of t ¼ 2  1015 s [14]. The bcc b-Ti model is formatted the original lattice. To simulate the solid solution mechanism for TieRe binary alloy, the formation energies of three kinds of sites are evaluated, substitutable site and two possible interstitial sites for Re in b-Ti, the octahedral site and the tetrahedral site, respectively. To calculate the formation energies of the two interstitial sites, the interstitial atom was fixed, in order to avoid movement of them. In addition, the two possible formation energies with the first-principles VASP were employed. The formation energy of substitutable site was evaluated by means of a supercell approach based on a cell size of 2a  2a  2a, with a being the crystal lattice spacing. The supercell contain 16 atoms. For octahedral site 2a  2a  5a supercell with 40 atoms is adopted. The calculations of the formation energies for both EAM and VASP are listed in Table 4. At 300 K, for substitutable site the formation energy calculation of EAM is negative, while the formation energies are positive for the two interstitial sites, and the calculation of octahedral site is lower than that of Table 3 Lattice constants for TieRe intermetallics Phase

tetrahedral, showing octahedral site may be easier to form than the tetrahedral site. The VASP calculations are similar to the EAM results, although there are small discrepancies between the EAM calculations and VASP. And the results of the calculations for both EAM and VASP indicated that Re atoms in b-Ti lattice should occupy the substitutable site. The results are in good agreement with the above discussion. To describe the snapshot of atomic positions, we make the defects move freely. The initial states of tetrahedral and octahedral sites are in shown in Fig. 4(a and b), respectively. Fig. 5 displays the snapshot of atomic positions for the octahedral site at 300 K for 50 000 time steps. The snapshot of atomic positions for the substitute sites at 300 K for 50 000 time steps is similar to Fig. 5 (not shown). It is of interest to note that the lattice distortion next to defects takes place when a Re atom replaces a Ti atom. And after 50 000 time steps, Re atom locating at the octahedral site will move to the substitute site, which implies that substitute site is a stable configuration. For the tetrahedral site, the mode of movement is different from the octahedral site. Fig. 6 displays the snapshot of atomic positions for tetrahedral site at 300 K for 20 000 time steps, which can provide a visualized atomic picture of the process. The snapshot of atomic positions for tetrahedral site at 300 K for 50 000 time steps is similar to Fig. 5 (not shown). It can be seen that Re atom occupied the tetrahedral site moves to the octahedral site after 20 000 time steps firstly as shown in Fig. 6, and then it moves to the substitute site after 50 000 time steps as shown in Fig. 5. Accordingly, the interstice sites are the unstable states, the substitute site is a stable configuration.

˚ ) Ref. Pearson symbol Space group Prototype Lattice constants (A a

TiRe

1119

cP2

Ti5Re24 cI58

Pm3m

CsCl

I43m

a-Mn

3.104 3.135 3.121 9.606 9.66 9.68

b

c [13] EAM VASP [13] EAM VASP

Table 4 The calculated formation energies of Re in Ti lattice with three site for MD and first-principle simulation Approaches

Substitute

Octahedral

Tetrahedral

EAM VASP

1.733(300 K) 1.578(0 K)

0.833(300 K) 0.295 (0 K)

1.02 (300 K)

Energies are in units of eV.

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Y. Wu et al. / Intermetallics 15 (2007) 1116e1121

Fig. 6. The snapshot of atomic positions of tetrahedral site at 300 K for 20 000 time steps: Ti (green circles), Re (grey circle). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

4. Conclusions

Fig. 4. The snapshot of atomic positions of the starting configuration for (a) the tetrahedral site, (b) the octahedral site, respectively: Ti (green circles), Re (grey circle). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

In the present paper, the relation of the lattice constant of b-Ti with Re content was estimated by the experiments and EAM model. The results indicated that the lattice constant of b-Ti decreased with increasing Re content. The heats of formation for disordered solid solution and ordered intermetallic compounds of TieRe are calculated with a modified analytical EAM model and first-principles VASP. The results agreed well with the experiment and other theory’s values, and showed that elements Ti and Re can form solid solutions. Moreover, the solid solution mechanism of TieRe binary alloy system is studied by MD and first-principles method, Re atoms in b-Ti lattice should occupy the substitutable site, and which is demonstrated also by the atomic positions at 300 K. In addition, the lattice constants of intermediated phase from EAM calculations are in good agreement with the available experimental and the first-principles VASP values. Those consistencies between the EAM calculations and the experiment data demonstrate that the modified analytic EAM is an effective and successful method for predicting the physical properties of TieRe binary alloy system. Acknowledgements This work is financially supported by the National Natural Science Foundation under contract Nos. 50371026 and 50571036. References

Fig. 5. The snapshot of atomic positions for Re occupied the octahedral site at 300 k for 50 000 time steps: Ti (green circles), Re (grey circle). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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