Journal of Nuclear Materials

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1 Journal of Nuclear Materials xxx (28) xxx xxx Contents lists available at ScienceDirect Journal of Nuclear Materials journal homepage: Thermodynamic modeling of the and systems X.J. iu, Z.S. i, J. Wang, C.P. Wang * Department of Materials Science and Engineering, College of Materials and Research Center of Materials Design and Applications, Xiamen niversity, Xiamen 3615, PR China article info abstract Article history: Received 1 March 28 Accepted 21 July 28 Available online xxxx PACS: s 81.3.Bx The thermodynamic assessments of the and binary systems were carried out by using the CAPHAD (Calculation of Phase Diagrams) method incorporating experimental thermodynamic properties and phase equilibria. The Gibbs free energies of the liquid, bcc, fcc, a and b phases were described by the subregular solution model with the Redlich Kister equation, and those of the intermetallic compounds ( 2 and 6 ) in the binary system were described by the two-sublattice model. The thermodynamic parameters of the and binary systems were optimized to reproduce the experimental data, and provide agreement with the experimentally determined phase diagram for each binary system. Ó 28 Elsevier B.V. All rights reserved. 1. Introduction The use of uranium in the production of energy from nuclear fission has given considerable impetus to the investigation of the alloys in recent decades. The investigations of -based alloys focus not only on the compounds used in the fuels, but also on the alloys of and common elements of the structural materials and fission products [1 3]. and are very important alloying elements for the -based alloys [4 9]. In order to develop new nuclear materials, it is necessary to understand the phase equilibria in -based alloy systems. The CAPHAD method is a powerful tool to reduce cost and time during development of materials [1]. As a result, it is of great importance to establish the thermodynamic database for the - based alloys system. In this paper, as a part of thermodynamic database of -based alloy systems, the thermodynamic descriptions for the phase equilibria in the and systems were carried out by means of the CAPHAD method. 2. Thermodynamic model The information of stable solid phases and the used models in the and the systems [11] is listed in Table Solution phases The Gibbs free energies of the solution phases in Me (Me:, ) system were described by the subregular solution model * Corresponding author. Tel.: ; fax: address: wangcp@xmu.edu.cn (C.P. Wang). [12]. The molar Gibbs free energy of each solution phase in the Me system is given as follows: G / ¼ x Me G/ Me þ x G/ þ RTðx Me ln x Me þ x ln x Þ X n þ x Me x m / Me; ðx Me x Þ m ; m¼ where G / Me and G / are the molar Gibbs free energy of pure element Me and with the structure / in a nonmagnetic state, which is taken from the compilation by Dinsdale [13] and shown in Table 2. The x Me and x are the mole fractions of Me and components, and m / Me; is the interaction energy between Me and atoms, and expressed as: m / Me; ð1þ ¼ a þ bt þ ct lnðtþ; ð2þ the parameters of a, b and c are evaluated based on the experimental data in the present work Stoichiometric intermetallic compounds The 2 and 6 compounds in the system are treated as stoichiometric phases. The Gibbs free energy of formation per mole of formula unit ðþ m ðþ n can be expressed by the two-sublattice model [14], as the following equation referring to the pure elements in their nonmagnetic state: DG mn f ¼ G mn f m G ref n G ref ¼ a þ b T; ð3þ where the DG mn f denotes the standard Gibbs free energy of formation of the stoichiometric compound from the pure elements /$ - see front matter Ó 28 Elsevier B.V. All rights reserved. doi:1.116/j.jnucmat

2 2 X.J. iu et al. / Journal of Nuclear Materials xxx (28) xxx xxx Table 1 The stable solid phases and the models used in the and systems System Phase Prototype Struckturbericht designation Modeling phase sed models d W A2 (,) Subregular solution model c Cu A1 (,) Subregular solution model b b A13 (,) Subregular solution model a a A12 () Subregular solution model 2 Cu 2 Mg C15 () 2 () Two-sublattice model 6 6 D2 c ()() 6 Two-sublattice model c W A2 (,) Subregular solution model b b A b (,) Subregular solution model a a A2 () Subregular solution model (c,) W A2 (,) Subregular solution model b b A b (,) Subregular solution model a a A2 (,) Subregular solution model Table 2 Gibbs energy parameters of condensed pure elements [13] Gibbs free energy (J/mol) (K) iquid phase T Tln(T) T T T < T < T Tln(T) 955 < T < T Tln(T) T T T < T < T 48 Tln(T) 1519 < T < T Tln(T) T T T T < T < T Tln(T) 275 < T < 6 a phase T Tln(T) T T < T < T 48 Tln(T) T < T < 2 b phase T Tln(T) T T < T < T 48 Tln(T) T < T < 2 a phase T Tln(T) T T T < T < T Tln(T) 955 < T < 3 b phase T Tln(T) T T T < T < T Tln(T) < T < 3 Bcc_A2 phase T Tln(T) T T T < T < T Tln(T) 149 < T < T 23.7 Tln(T) T T < T < T 48 Tln(T) T < T < T Tln(T) T T T < T < T Tln(T) T < T < 6 Fcc_A1 phase T Tln(T) T T T < T < T Tln(T) 955 < T < T Tln(T).6 T T < T < T 48 Tln(T) T < T < 2 The terms G ref and G ref are the molar Gibbs free energy of pure element and with its defined reference structure in a nonmagnetic state. The parameters of a and b are evaluated in the present optimization.

3 X.J. iu et al. / Journal of Nuclear Materials xxx (28) xxx xxx ºC ºC 626ºC 6 716ºC 2 / at. % 135 ºC / ºC ºC 95 ± 2ºC () 647ºC / at. % Fig. 1. The phase diagram of the system reviewed by Massalski et al. [17]. Fig. 2. The phase diagram of the system reviewed by Koike et al. [31]. 3. Experimental information 3.1. The system The phase diagram of the system consists of seven solution phases (a, b, c, d, a, b, c), and two intermetallic compounds ( 6 and 2 phases). The phase diagram in the system was originally proposed by Wilhelm and Carlson [15], and then reassessed by Hanse et al. [16] and Massalski [17]. The maximum solubility of in the c phase was estimated to be less than 1 wt% based on metallography. Although the solubilities of in all the allotropes of have not been investigated thoroughly, some solid solubilities were found by Whilhelm using X-ray studies [15]. The 2 phase has three polymorphs: a 2, b 2 and c 2. The polymorphic transformation temperatures of the c 2 M b 2 and b 2 M a 2 were respectively reported to be 61 C and 161 C [17]. Because these transformation temperatures of the 2 phase were too low to use in the present assessment, the 2 phase is treated as one stoichiometric phase. The phase diagram of the system reviewed by Massalski [17] is shown in Fig. 1. In addition, the enthalpies and entropies of formation of the compounds ( 6 and 2 ) in the temperature range from 66 C to 86 C were determined by ebedev et al. [18] on the basis of Electromagnetic Fields (EMF) measurements The system The system consists of three solution phases (a, b and (c, )), and a phase separation (c + ()) in the bcc phase at lower temperature. The phase diagram in the system has been investigated by many researchers [16,19 31]. There are two different conclusions for the monotectoid reaction of c in this system as follows: the reaction of c M () + a was reported ºC 1111ºC Ref. [15] Ref. [17] 135ºC / at. % ºC 725 ºC 626 ºC 6 716ºC 2 698ºC / at. % 4 / at. % Fig. 3. Calculated phase diagram of binary system with experimental data [15,17].

4 4 X.J. iu et al. / Journal of Nuclear Materials xxx (28) xxx xxx by Roger et al. [16,22,25,28], and the c M () + b was reported by Terekhov [29]. Massalski [17] reviewed the phase diagram according to Rogers [22] and Terekhov [29], where the monotectoid reaction of the c M () + b is accepted [29]. However, in the Koike review [31], the monotectoid reaction of the c M N- b+a is adopted according to the previous work [16,17,19 28,3]. The liquidus line was only estimated by Rogers [22], because of the difficulty in the experiment. In this work, the assessment was carried out on the basis of the experiment data by [17,21 23,3], and the monotectoid reaction of the c M () + a was adopted. The phase diagram reviewed by Koike, et al. [31] is shown in Fig. 2. In addition, the physical properties such as the Debye temperature and Young s modulus of the solid solution alloys were determined based on the fused salt EMF measurements by Fedorov and Smirnov [32] and by Vamberskij et al. [33]. Vamberskij et al. [33] also calculated Gibbs free energies of formation of the system at 775 C and 9 C by the Gibbs Duhem equations according to their experimental data. 4. Optimized results and discussion Optimization of thermodynamic parameters describing the Gibbs free energies of each phase is carried out using PARROT [34] module in the Thermo-Calc software [35], a computer program that can accept different types of data, such as any specific thermodynamic quantities and phase equilibria, in the same operation. Each piece of the selected data is given a certain weight and Table 3 Thermodynamic parameters for the system optimized in this work Parameters in each phase (J/mol) Enthalpy of formation, kj / mol / at. % Ref. [18] Fig. 4. Calculated enthalpies of formation of intermetallic compounds at 677 C in the system compared with the experimental data [18]: the reference states are c phase and b phase. iquid phase, format (,) 1 iq ; ¼ 234 þ 11:7T 1 iq ; ¼ 32 1:77T 2 iq ; ¼ þ 1:4T A2 (c,d) phase, format (,) 1 (Va) 3 bcc ; ¼ 1 þ 9:83T 1 bcc ; ¼ 28 þ 4:7T A b b phase, format (,) 1 G b ¼ Ga þ 25 b ; ¼ 29614:8 12:5T 1 b ; ¼ :7T 2 b ; ¼ 49519:7 þ 47:7T A1 c phase, format (,) 1 (Va) 1 fcc ; ¼ 25 A13 b phase, format (,) 1 (Va) 1 G bðþ ¼ G a þ 5 b ; ¼ 7971 þ 6:97T 2 phase, format ().6667 ().3333 G 2 : ¼ :6667Ga þ :3333Ga 91 :32T 6 phase, format ().1429 ().8571 G 6 : ¼ :1429Ga þ :8571Ga 59 þ 2:71T Entropy of formation, J / mol K Ref. [18] Temperatuer Phase Boundary Ref. [22] Ref. [23] Ref. [17] Single Phase Ref. [3] Two Phase Ref. [3] 958ºC 647ºC () / at. % Fig. 5. Calculated entropies of formation intermetallic compounds at 677 C in the system compared with the experimental data [18]: the reference states are c phase and b phase. / at. % Fig. 6. Calculated phase diagram of binary system with experimental data [17,21 23,3].

5 X.J. iu et al. / Journal of Nuclear Materials xxx (28) xxx xxx 5 Table 4 A comparison of calculated invariant reactions and special points in the system with experimental results Reaction type Reaction (at.%) T ( C) References Catatectic c? b [17] [This work] Peritectic b+? [17] [This work] Eutectoid b? a [17] Eutectic? [17] [This work] Eutectic? b [17] [This work] Melting? [17] [This work] Peritectic d +? c [This work] Peritectic c +? b [This work] Eutectoid b? a [This work] is given in Table 3. And all invariant reactions and special points in the system are summarized in Table 4, in which the experimental data are also listed for comparison [17]. The calculated congruent melting temperature of the 2 phase and the eutectic temperatures are in agreement with the corresponding experimental data [18] The system The calculated phase diagram with the experimental data is shown in Figs. 6 and 7. The calculated phase diagram, especially the phase boundary of miscibility gap region (c + ), is in agreement with the experimental data [17,21 23,3]. The calculated liquidus line agrees with Rogers [22] and the Massalski review [17]. Figs. 8 and 9, respectively indicate the calculated Gibbs free energies at 775 C and 9 C compared with the data obtained from the Gibbs Duhem equations [33]. It is seen from Figs. 8 and 9 that there are some differences between the present calculated results and data reported by Vamberskij [33]. In particular, in the the weight can be changed until a satisfactory description for most of the selected data is achieved The system The calculated phase diagram of the system compared to all the experimental data [15,17] used in the present optimization is shown in Fig. 3. It is seen from Fig. 3 that the calculated results are in agreement with Massalski [17], but there are some differences with Wilhelm [15] in the -rich region. The calculated largest solid solubilities of in the b, c and d phases are respectively.7 at.%,.4 at.% and.3 at.%. In the -rich region, a little solubility of in the b and c was given in this work, although there are no experimental data about this point. The calculated enthalpies and entropies of formation of the compounds with the experimental data at 677 C are shown in Figs. 4 and 5. The calculated results are in good agreement with the experimental data [18]. A complete set of the thermodynamic parameters describing the Gibbs free energy of each phase, including the elements as well, Gibbs free energy of formation, kj / mol Ref. [33] () / at. % Temperatuer ºC Phase Boundary Ref. [22] Single Phase Ref. [3] Two Phase Ref. [3] 647ºC Fig. 8. Calculated Gibbs free energy at 775 C in the system compared with the calculation data by Vamberskij et al. [33]: the reference states are bcc c phase and bcc () phase. Table 5 Thermodynamic parameters for the system optimized in this work Parameters in each phase (J/mol) iquid phase, format (,) 1 iq ¼ 39836:8 41:2T ; 1 iq ¼ 14923:3 þ 47:2T þ 4:66T lnðtþ ; 2 iq ¼ 3891:3 4:33T ; A2 (c,) phase, format (,) 1 (Va) 3 bcc ; ¼ 1776:2 23:99T 1 bcc ; ¼ 54699:5 þ 3:2T 2 bcc ; ¼ 42938:27 þ 9:6T 3 bcc ; ¼ þ 11:6T / at. % Fig. 7. Calculated phase diagram of binary system in the -rich portion with experimental data [21,22,3]. A b b phase, format (,) 1 G b ¼ Gbcc þ 15 b ; ¼ 418:4 A2 a phase, format (,) 1 G a ¼ Gbcc þ 25 a ; ¼ 5

6 6 X.J. iu et al. / Journal of Nuclear Materials xxx (28) xxx xxx Table 6 A comparison of calculated invariant reactions and special points in the system with experimental results Reaction type Reaction (at.%) T ( C) References Eutectoid b? a+c [31] [This work] Monotectoid c? a + () [31] [This work] Critical (c, )? c + () [31] [This work] Gibbs free energy of formation, kj / mol c + () miscibility gap region, the calculated Gibbs free energies are higher than ones by Vamberskij. However, the present calculations are reasonable by considering the error range of Vamberskij s data. A complete set of the thermodynamic parameters describing the Gibbs free energy of each phase in this system is given in Table 5, and all invariant reactions in the system are summarized in Table 6, in which the experimental data are also listed for comparison [31]. 5. Conclusions The phase diagrams and thermodynamic properties in the and binary systems were evaluated by combining the thermodynamic models with the available experimental information in literature. A consistent set of optimized thermodynamic parameters has been derived for describing the Gibbs free energy of each solution phase and intermetallic compounds in the and binary systems. Good agreement between the calculated results and most of the experimental data is obtained. Acknowledgements () / at. % Ref. [33] Fig. 9. Calculated Gibbs free energy at 9 C in the system compared with the calculation data by Vamberskij et al. [33]: the reference states are bcc c phase and bcc () phase. This work was supported by the National Natural Science Foundation of China (Nos and ) and Fujian Provincial Department of Science and Technology (Grant No. 22I18), and Xiamen Bureau of Science and Technology (Grant No. 2Z25516). 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