Thermodynamic Calculation of Phase Equilibria in the Nb-Ni-Ti-Zr Quaternary System

Transcription

1 Materials Transactions, Vol. 48, No. 2 (07) pp. 89 to 96 #07 The Japan Institute of Metals Thermodynamic Calculation of Phase Equilibria in the Nb--- Quaternary System Tatsuya Tokunaga 1; * 1, * 2, Satoshi Matsumoto 2; * 3, Hiroshi Ohtani 1;3 and Mitsuhiro Hasebe 1;3 1 Core Research for Evolutional Science and Technology (CREST), Japan Science and Technology Agency (JST), Kitakyushu , Japan 2 Graduate School of Engineering, Kyushu Institute of Technology, Kitakyushu , Japan 3 Department of Materials Science and Engineering, Kyushu Institute of Technology, Kitakyushu , Japan The phase equilibria in the Nb--- quaternary system have been studied using the CAPHAD method. Among the four ternary systems present in the quaternary phase diagram, the Nb-- ternary system was described using a simple ternary extrapolation of the constituent binary systems with no additional ternary parameters. The thermodynamic parameters of the -- ternary system were evaluated using data from first-principles calculations on the ternary,, and compound phases as well as available experimental data on the phase boundaries. The calculated isothermal and vertical section diagrams of both the Nb-- and -- ternary systems reproduced the experimental results satisfactorily. The thermodynamic parameters of the Nb-- and Nb-- ternary systems were adopted from previous studies. The liquidus surface in the Nb--- quaternary system was calculated based on the thermodynamic description of the ternary systems. According to the calculated liquidus surface of Nb x y x y alloys, in which a metallic glass was formed over a wide composition range, the liquidus temperature decreased with increasing content up to mol%. [doi:10.23/matertrans.48.89] (Received May 29, 06; Accepted October 18, 06; Published January 25, 07) Keywords: phase equilibria, niobium-nickel-titanium-zirconium, thermodynamic analysis, calculation of phase diagrams (CAPHAD) 1. Introduction Glassy or amorphous alloys form over a wide composition range in the Nb-- and Nb-- systems. 1,2) These amorphous alloys are expected to act as a membrane material for hydrogen purification due to their high hydrogen permeability. 3) In contrast, recent studies indicate that quaternary Nb--- glassy alloys exhibit high hydrogen impermeability, corrosion resistance, and formability for use in fuel cells as bipolar plates. 4,5) Apart from the formation of glassy or amorphous alloys, the hydrogen permeation characteristics of ternary Nb-- alloys have been investigated, and new hydrogen permeation alloys consisting of and bcc (Nb,) duplex phases have been developed. 6) Therefore, information on the phase equilibria in the Nb--- quaternary system is required for further development of these hydrogen energy-related materials. The Calculation of Phase Diagrams (CAPHAD) approach, 7) where phase diagrams are constructed using experimental data and thermodynamic analysis, provides a powerful tool for obtaining information about phase equilibria and the thermodynamic properties of alloy systems. In this study, a thermodynamic analysis of the constituent ternary systems making up the Nb--- quaternary system has been carried out, and the phase equilibria in this quaternary system have been calculated based on the thermodynamic descriptions of the constituent ternary systems using the CAPHAD method. * 1 Corresponding author, ttokunag@matsc.kyutech.ac.jp * 2 Present address: Department of Applied Science for Integrated System Engineering, Kyushu Institute of Technology, Kitakyushu , Japan. * 3 Graduate Student, Kyushu Institute of Technology, Present address: Kobe Steel, td., Shimonoseki , Japan. 2. Calculation Procedures 2.1 Thermodynamic descriptions The Gibbs energy of the liquid and the terminal solid solution phases was described using the conventional regular solution model as follows: Gm ¼ x Nb G Nb þ x G þ x G þ x G þ RTðx Nb ln x Nb þ x ln x þ x ln x þ x ln x Þ þ x Nb x Nb, þ x Nbx Nb, þ x Nbx Nb, þ x x, þ x x, þ x x, þ x Nb x x Nb,, þ x Nbx x Nb,, þ x Nb x x Nb,, þ x x x,, ð1þ where G i denotes the Gibbs energy of element i in the phase, and is called the lattice stability. The descriptions of the lattice stability parameters were taken from the Scientific Group Thermodata Europe (SGTE) data file. 8) The term R is the universal gas constant, and x Nb, x, x and x are the mole fractions of Nb,,, and, respectively. The parameter i;j denotes the interaction energy between i and j in the phase. i;j;k is ternary interaction parameter between elements i, j, and k, respectively. The compositional dependency of the interaction parameters i;j and i;j;k is expressed using an nth degree Redlich-Kister 9) and Redlich-Kister- Muggianu 10) polynomials, respectively. i;j ¼ 0 i;j þ 1 i;j ðx i x j Þ 1 þ 2 i;j ðx i x j Þ 2 þþ n i;j ðx i x j Þ n ð2þ i;j;k ¼ x i 0 i;j;k þ x j 1 i;j;k þ x k 2 i;j;k ð3þ The interaction parameters for the quaternary system were not taken into account in our modelling. The Gibbs energy contributions due to ferromagnetic ordering of the -rich fcc solid solution phase were described using the model of Hillert and Jarl: 11)

2 90 T. Tokunaga, S. Matsumoto, H. Ohtani and M. Hasebe Table 1 The sublattice model formulae of the phases in the Nb--- quaternary system. Phase Formula Nb 3 (Nb,) 0:25 (Nb,) 0:75 Nb 6 7 (Nb,) 0:4615 (Nb,) 0: (,) 0:75 (,) 0:25 (Nb,,Va,) 0:5 (Nb,,,) 0:5 2 () 0:333 (,) 0:667 5 (,) 0:833 (Va,) 0:167 3 (,) 0:75 (Va,) 0: (,) 0:575 (Va,) 0:425 () 0:5 (,) 0:5 2 () 0:333 (,) 0:667 () (,) 0:333 (,) 0:667 mag G m ¼ RT lnð þ 1Þf ðþ; where mag Gm is the Gibbs energy of the magnetic ordering, is the average magnetic moment per atom in the phase (expressed in Bohr magnetons), and f ðþ is a polynomial function of. The term is defined as T=T C, where T C is the critical temperature of the magnetic transition in the phase. Both and T C can be concentration-dependent, and are represented by an equation similar to eq. (2). The Gibbs energy of compound phases with some homogeneity range was described using the sublattice model. 12) For the simple case of a phase with the formula (A,B) m (C,D) n, the Gibbs energy per mole of atoms of the compound is given by: ð4þ G m ¼ y 1 A y2 C G A:C þ y 1 B y2 C G B:C þ y 1 A y2 D G A:D þ yb 1 y2 D G B:D þ mrtðya 1 ln y1 A þ y1 B ln y1 B Þ þ nrtðyc 2 ln y2 C þ y2 D ln y2 D Þþex G where G A:C denotes the Gibbs energy of a hypothetical compound A m C n, in which all the sites in Sublattice 1 are occupied by element A, and all the sites in Sublattice 2 are occupied by element C, respectively. The colon separates the constituent elements in the sublattice. The site fraction of the elements on the sth sublattice is denoted by y s i. The term ex G is the excess Gibbs energy term containing the interaction energy between unlike atoms, and is expressed by the following equation: ex G ¼ ya 1 y1 B y2 C A,B:C þ ya 1 y1 B y2 D A,B:D þ yc 2 y2 D y1 A A:C,D þ yc 2 y2 D y1 B B:C,D; ð6þ where i;j:k (or i:j;k ) is the interaction parameter between unlike atoms on the same sublattice, and is described by an equation similar to eq. (2). The phases described by the sublattice model in this study and their formulae are listed in Table 1. The binary compounds, 7 2, 21 8, and 11 9, and the five ternary compounds, i.e., Nb ,Nb 8 9 3, Nb 5 75,Nb , and Nb 15 5, were treated as being stoichiometric compounds. For example, the Gibbs energy of the 7 2 phase was described as follows: G 7 2 m ¼ 0:78 G þ 0:22 G þ G f 7 2 ; ð7þ where G f 7 2 is the Gibbs energy of formation per mole of ð5þ formula unit of the compound, and is expressed by the following equation: G f 7 2 ¼ A þ B T: ð8þ The terms A and B correspond the enthalpy and entropy terms, respectively, and were evaluated in this study. 2.2 First-principles calculations In this analysis, the solubility of the third element in the, 2,, and 2 binary compound phases in the -- ternary system was taken into account based on experimental data. In addition, the ternary compound phase, (), with some range in homogeneity, is formed in this ternary system. In treating these phases using the sublattice model, the evaluation of the Gibbs energy parameters for metastable and/or unstable phases, as well as the stable phase, which are called end member compounds, is required, as shown in eq. (5). However, there is little information available for the precise evaluation of the thermodynamic parameters of the metastable phases. Therefore, in this study, the energy of formation for some stable and metastable compound phases in the -- system was obtained using first-principles calculations. The calculations were carried out using the WIEN2k software programme, 13) based on the Full Potential inearized Augmented Plane Wave (FAPW) method with a General Gradient Approximation (GGA). 14) Muffin-tin radii of 2.0 au (0.106 nm) for,, and were assumed, and the value of RK max was fixed at RK max ¼ 9:0, which corresponds to the cut-off energy of almost Ry (270 ev). 3. Results and Discussion 3.1 Nb-- ternary system In this ternary system, a liquid phase (), a bcc phase ((Nb,)), an fcc phase (()), an hcp phase (()), five binary compounds, and five ternary compounds are known. The thermodynamic analysis of this ternary system has been carried out by the authors group, 15) based on previous studies of the Nb-, 16) Nb-, 17) and - 18) binary systems, where all the ternary compounds with some range of homogeneity were treated as being stoichiometric phases. In this study, the assessed parameters were adopted, and are listed in Table Nb-- ternary system This ternary system is composed of a liquid phase (), a bcc phase ((Nb,)), an fcc phase (()), an hcp phase ((,)), and ten binary compounds. No data on the ternary compound is available in the literature. The thermodynamic parameters of our previous assessment, 19) based on the thermodynamic descriptions of the Nb-, 16) Nb-, ) and - 21) binary systems, were used in this study. The thermodynamic parameters of the Nb-- system are listed in Table Nb-- ternary system Isothermal sections of the Nb-- ternary system have been reported by several authors 22 27) in the temperature range of 843 to 1373 K over the entire composition range. In

3 Table 2 The evaluated thermodynamic parameters in the Nb--- quaternary system. System Phase Thermodynamic parameters (J/mol) Reference Nb- Nb, ¼ þ 10T 15) Nb, ¼ 70007:4 7:39665T þðx Nb x Þðþ :07497TÞ T C(Nb,) ¼ 10 þðx Nb x Þðþ7Þ Nb 3 G Nb3 Nb:Nb G Nb ¼þ5000 G Nb3 : G ¼þ ) G Nb3 Nb: 0:25 G Nb 0:75 G ¼ 35300:6 þ 4:83322T G Nb3 :Nb 0:25 G 0:75 G Nb ¼þ45300:575 4:83322T Nb3 Nb,:Nb ¼ Nb3 Nb,: ¼ 3079:625 Nb3 Nb:Nb, ¼ Nb3 :Nb, ¼þ13505:625 Nb 6 7 G Nb67 Nb:Nb G Nb ¼þ98 G Nb67 Nb: 0:4615 G Nb 0:5385 G ¼ þ 0:305T 15) Nb67 Nb:Nb, ¼þ6850 þðy1 Nb y1 Þðþ264Þ Nb, ¼ 037:3 6:31498T þðx Nb x Þðþ97884:9 19:01069TÞþðx Nb x Þ 2 ðþ10000þ 16) Nb- Nb, ¼þ13150 Nb, ¼þ ) Nb, ¼þ3000 Nb- Nb, ¼þ15911 þ 3:35T þðx Nb x Þðþ3919 1:091TÞ Nb, ¼þ24411 ) Nb, ¼þ10311 þðx Nb x Þðþ6709Þ -, ¼ 000, ¼ 97427:4 þ 12:112T þðx x Þð 32315:3Þ 3 2, ¼ :64 þ :22423T þðx x Þð 46714:31Þ T C(,) ¼ 4670 : G ¼ G G Va: 0:5 G ¼þ81198:65 13:702875T ¼þ39351:22 8:296145T Va: 0:5 G : 0:5 G 0:5 G,Va: ¼,Va: :, ¼ ¼ 41847:43 þ 5:673T ¼ 312:19 þ 13:247095T Va:, ¼ 72295:24 þ 23:47071T þðy2 y2 Þð 24442:75Þ 18) G 3 : G ¼ G G G 3 : G ¼ 0 G 3 : 0:75 G 0:25 G ¼ 41421:96 þ 7:35868T G 3 : 0:25 G 0:75 G ¼ 5688:27 3 :, ¼ 3 :, ¼þ18274:75 16:21288T 3,: ¼ 3,: ¼þ : 0:333 G 0:667 G ¼ 27514:2 þ 2:85345T, ¼ þ 34:8594T þðx x Þð 81824:8 þ 25:99TÞþðx x Þ 2 ð 10:0779TÞ -, ¼ þ 2:3T þðx x Þð :6TÞ, ¼ 00 þ 2T þðx x Þð þ 2:5TÞ, ¼ þ 3:5T þðx x Þðþ :6TÞ G : 0:5 G 0:5 G ¼ þ 0:6T 2 G 2 : 0:333 G 0:667 G ¼ 650 þ 0:15T 21) 3 G 3 :Va 0:75 G ¼þ10000 þ 1:25T : G 3 G ¼ 0 G 3 :Va 0:5 G ¼þ47425 þ 5:68T 5 G 3 Thermodynamic Calculation of Phase Equilibria in the Nb--- system 91 : 0:75 G 0:25 G ¼ þ 0:88T 19) ¼ þ 4:5T 3 :Va, 3,:Va ¼þ ,: ¼þ7555 G 5 :Va 0:833 G ¼þ :428T G 5 : G ¼þ70 0:9T G 5 :Va 0:833 G ¼þ :22T G 5 : 0:833 G 0:167 G ¼ 335 þ 0:95T 21) ¼ þ 1:25T 5 :Va, 5,:Va ¼þ ,: ¼þ7850 continued on the next page

4 92 T. Tokunaga, S. Matsumoto, H. Ohtani and M. Hasebe System Phase Thermodynamic parameters (J/mol) Reference 7 2 G 72 : 0:78 G 0:22 G ¼ 775 þ 0:3T 10 7 G 107 :Va 0:575 G ¼þ051 5:8167T 21) G 107 : 0:575 G 0:425 G ¼ þ 1:75T G 107 :Va 0:575 G ¼þ275 5:674T G 107 : G ¼þ8750 2:556T 19) ,:Va ¼þ ,: ¼þ8500 G 119 :Va, ¼ ) 11 9 : 0:55 G 0:45 G ¼ 500 þ 0:74T 19) 21 8 G 218 : 0:725 G 0:275 G ¼ þ 0:5T 21), ¼ 0450 þ 10:35T þðx x Þð þ 3:58TÞþðx x Þ 2 ð 300 þ 32:37TÞ -, ¼ 4346 þ 5:489T, ¼þ ), ¼ 968 Nb-- Nb,, ¼ x ðþ100000þ Nb,, ¼ x Nb ðþ000þþx ðþ000þþx ðþ000þ Nb: 0:5 G Nb 0:5 G ¼ :Nb 0:5 G 0:5 G Nb ¼ 0 Nb: 0:5 G Nb 0:5 G ¼þ10000 Nb:Nb G Nb ¼þ Va:Nb 0:5 G Nb ¼ 0 :Nb, ¼ þ 85T :Nb,, ¼ T 15) Nb G Nb Nb:: 0:15 G Nb 0:56 G 0:29 G ¼ 500 þ 3:6T Nb G Nb893 Nb:: 0:4 G Nb 0:45 G 0:15 G ¼ 300 þ 2T Nb 5 75 G Nb575 Nb:: 0:05 G Nb 0:75 G 0:2 G ¼ 430 þ 8:1T Nb G Nb Nb:: 0:13 G Nb 0:75 G 0:12 G ¼ 900 þ 5:4T Nb 15 5 G Nb155 Nb:: 0:15 G Nb 0: G 0:05 G ¼ þ 5T Nb,, ¼ x ð 45000Þþx ðþ1000þ Nb-- Nb,, ¼ x ðþ2000þ 19) -- : G ¼þ Va: 0:5 G ¼þ4 : 0:5 G 0:5 G ¼ : 0:5 G 0:5 G ¼ : 0:5 G 0:5 G ¼þ10430 :, ¼ þ 5T þðy2 y2 Þð 100 þ 5TÞ 2 2 : 0:333 G 0:667 G ¼ :, ¼ 200 G : 0:5 G 0:5 G :, ¼þ12500 ¼ þ 10T G 2 : 0:333 G 0:667 G ¼ 261 þ 5T Present 2 :, ¼ 00 Work G : 0:333 G 0:667 G ¼ G : 0:333 G 0:667 G ¼þ17000 G : 0:333 G 0:667 G ¼ G : G ¼þ400 :, ¼þ000,: ¼ 0,: ¼þ10000 :, ¼ þ 35T þðy2 y2 Þðþ600 30TÞ,:, ¼ 0000 þ T,, ¼ x ð 50000Þþx ðþ000þþx ðþ000þ addition, the liquidus surface 27) and the solidus surface 28) projections have been investigated over the entire composition range. Previous studies indicated that the bcc miscibility gap (#1 þ #2) in the Nb- binary system extended into the ternary system, and that no ternary compound formed. Hence, this ternary system is composed of solution phases only: the liquid phase, the bcc phase ((Nb,,)), and the hcp phase ((,)). Although the compositions of the constituent phases in both the (#1 þ #2) two phase region and the tie triangle between the #1, #2, and (,)

5 Thermodynamic Calculation of Phase Equilibria in the Nb--- system 93 Nb β#1 + β#2 (α, α) (Nb, β) content (mol%) Ref. 22) β#1 + β#2 Ref. 25) (α, α)+β#1+β#2 Fig. 1 (a) Calculated and (b) experimental isothermal section diagrams of the Nb-- system at 843 K K Nb Fig (β) K content (mol%) phases were not in agreement in the reported data, the phase relationships were well determined. Thermodynamic modelling of this ternary system has been carried out using a simple ternary extrapolation, 29) based on the Nb-, 17) Nb-, ) and - 29) binary systems. In this modelling, the same parameters as those used in previous modelling were used. The thermodynamic parameters are listed in Table 2. The calculated isothermal section diagram of the Nb-- ternary system at 843 K is shown in Fig. 1, together with the experimental data. 22,25) Figure 2 shows the calculated liquidus surface projection of the Nb-- system ternary system An isothermal section of this ternary system has been Calculated liquidus surface projection in the Nb-- system. Fig. 3 2a ( and ) 4f ( and ) 6h ( and ) Crystal structure for () with a MgZn 2 -type structure. investigated in the composition region below 50 mol% at 973 K. 30,31) The phase equilibria in the ,32) and - 33) sections have been studied using X-ray diffraction (XRD), electron probe microanalysis (EPMA), and differential thermal analysis (DTA). Investigations into the 3-3 section have also been carried out. 34,35) These experimental data on the -- ternary system has been critically reviewed by Gupta. 36) According to the experimental results, the solubility of in both the 2 and phases reaches about 5 mol% at 973 K. The 2 and phases dissolve about 9 mol% and 30 mol% at 973 K, respectively. The extensions of the 3 and 3 phases into the ternary system were found to be about 3 mol% and 2 mol% at 1173 K, respectively. With regard to the ternary compound phase, a MgZn 2 -type aves phase, (), and 3 ( 0:67 0:33 ) with a BaPb 3 -type structure, were found to form. The reported homogeneity ranges for () were found to be in good agreement in the experimental investigations, whereas whether the () melts congruently was not clarified, and a discrepancy in temperature range of stability of 3 ( 0:67 0:33 ) was found. In our analysis, the solid solubility of the third element in,, 2, and 2 was taken into account, and the Gibbs energy was described using sublattices with the formulae (,Va,) 0:5 (,,) 0:5, () 0:5 (,) 0:5, () 0:333 (,) 0:667, and () 0:333 (,) 0:667, respectively. The other binary compounds were treated as being pure binary phases due to either negligible solubility of the third element or due to a lack of experimental information. In regard to the ternary () phase, a detailed structural investigation using neutron diffraction has revealed that the 2a and 6h sites were occupied by both and atoms, whereas 4f site was occupied by and atoms in the unit cell shown in Fig. 3, and hence, the formula (,) 0:333 (,) 0:167 (,) 0:5 for the sublattice model was proposed. 37) However, in our modelling, the formula (,) 0:333 (,) 0:667 was applied to reduce the thermodynamic parameters to be evaluated and to maintain consistency with other thermodynamic modelling of the aves phases. The another ternary compound, 3 ( 0:67 0:33 ), was not considered in our work, because both the temperature range of stability and the phase equilibria with the other phases have not been clarified. The thermodynamic analysis was carried out based on the experimental data of Eremenko et al. 30,31,33) and our first-principles calculations. The calculated energy of formation for the various compounds in the

6 94 T. Tokunaga, S. Matsumoto, H. Ohtani and M. Hasebe Table 3 The calculated energy of formation for various compounds in the -- ternary system. Phase Compound Energy of formation (kj/mol) (Va) 0:5 0:5 þ82:8 0:5 0:5 þ10:4 0:5 0:5 33:8 0:5 0:5 38:7 2 0:333 0:667 26:1 0:333 0:667 þ31:8 0:333 0:667 27:2 () 0:333 0:667 38:1 0:333 0:667 þ17:0 0:333 0:667 þ44:0 -- ternary system is listed in Table 3. Each value is referred to as fcc, hcp, and hcp, and thus the term A in eq. (8) corresponds to the calculated energy of formation listed in Table 3. The thermodynamic parameters evaluated in this study are listed in Table 2. The calculated isothermal section diagram at 973 K is shown in Fig. 4, together with the experimental diagram. 31) A comparison of the calculated and experimental results shows that the calculated diagram reproduces the experimental phase relationships. However, the composition ranges for and () are narrower than those determined experimentally. Although a larger solid solubility of in could be attained by introducing a more negative interaction parameter, the () phase would disappear from the phase diagram due to the increased phase stability of. However, the phase stability of () was well determined based on the energy of formation of 2, ( 2=8 6=8 ) 2, and 2, listed in Table 3. Thus, further experimental study is required to determine the solid solubility of in. With regard to the composition range of (), an extension of () towards 2 has been observed experimentally. On the other hand, although the formula for the aves phase used in our modelling could not be used to describe its behaviour, the energy of formation of the hypothetical aves phase 2 was calculated to be 31.8 kj/mol, and thus this parameter is unlikely to allow the () phase to extend towards the composition range of 2. Figure 5 shows the calculated isopleths at the 33.3 mol% and 50 mol% sections, along with the experimental data obtained from DTA experiments. Such a comparison shows that the calculated liquidus temperatures at the 33.3 mol% section agree well with the experimental data, 31,32) whereas the calculated results at the 50 mol% section are lower than the experimental values. 33) The reason for this discrepancy is not clear, and the thermodynamic parameters used could not reproduce both the experimental isothermal section and the isopleth at 50 mol%. The calculated liquidus surface projection of the -- system is shown in Fig. 6. The liquidus surface was composed of 14 ternary invariant reactions, and these points are summarized in the data shown in Table 4. Except for Points E1, U1, and U7, the invariant reactions occurred at relatively low temperature when compared with the melting points of the constituent elements. (a) (γ) (b) Calculated 2 (α,α) 3 2 content (mol%) Experimental 2 (β,β) (α,α) 3.5 Nb--- quaternary system There is no experimental information on the phase equilibria in the Nb--- quaternary system. However, the prediction of this quaternary phase diagram is of interest for the development of hydrogen energy-related alloys, as described in the introduction to this paper. In general, the glass-forming ability of alloys is related to the liquidus temperature, and amorphous or glassy alloys are formed in the composition ranges that have relatively low liquidus temperature. Thus, information on the phase equilibria involving the liquid phase is useful. Figure 7 shows the liquidus surface projection in the alloy composition range Nb x y x y, in which a metallic glass is formed over a wide composition range. 38) According λ λ content (mol%) (α,α) 2 Ref. 31) single-phase two-phase three-phase Ref. 31) single-phase two-phase three-phase (β,β) (α,α) Fig. 4 (a) Calculated and (b) experimental isothermal section diagrams of the -- system at 973 K.

7 Thermodynamic Calculation of Phase Equilibria in the Nb--- system 95 Temperature, T/K Temperature, T/K (a) (b) Ref. 31) Ref. 32) λ λ + λ + 2+ λ λ Ref. 33) Fig. 5 Calculated isopleths of the -- system at the: (a) 33.3 mol% and (b) 50 mol% sections. Table 4 The invariant reaction points of the calculated liquidus surface projection in the -- system. Composition of Type Reaction liquid phase Temperature (mol%) (K) E 1, () E 2, E 3, E 4, E 5, E 6, () E 7, U , U , U , U 4 + 3, U , U 6 +(), U , Fig (γ) E U 7 U E 5 E 4 3 U 3 E U 2 U U E 6 E 5 λ E U 6 (β) content (mol%) Calculated liquidus surface projection in the -- system. to our calculations, the liquidus temperature decreases with increasing content up to mol%. Although there is no comparative information, it is interesting that the compositional range corresponding to a low liquidus temperature is consistent with the experimental region showing a high glassforming ability. 38) The glass-forming ability of various alloys has been evaluated by coupling the Davies-Uhlmann kinetic formulations 39,) with the CAPHAD approach. 41) In the computations, time-temperature-transformation (TTT) curves were obtained utilizing the driving force for crystallization derived from the thermodynamic calculations, and the critical cooling rates could be calculated from the TTT curves. Therefore, the Gibbs energy functions formulated in this study enable us to evaluate the glass-forming ability of Nb--- alloys quantitatively. 4. Conclusions A thermodynamic analysis of the constituent ternary systems constructing the Nb--- quaternary system has been carried out, and the phase equilibria in the Nb-- - quaternary system were studied using the thermodynamic descriptions of four ternary systems. The parameters used in our previous evaluations were adopted for the thermodynamic descriptions of the Nb-- and Nb-- ternary systems. The results are summarized as follows: (1) The phase equilibria in the Nb-- ternary system could be calculated using a simple extrapolation of the constituent binary systems with no additional ternary parameters. (2) The thermodynamic parameters for, 2,,

8 96 T. Tokunaga, S. Matsumoto, H. Ohtani and M. Hasebe Nb Nb 2, and () were evaluated using firstprinciples calculations as well as the experimental data on the phase boundaries. The calculated phase diagrams of the -- ternary system agree well with the available experimental data. (3) The calculated liquidus surface of the Nb--- quaternary system in the composition range Nb x y x y showed that the liquidus temperature decreases with content up to about mol%. Acknowledgement This work is partially supported by the Ministry of Education, Culture, Sports, Science and Technology, Grantin-Aid for Scientific Research on Priority Areas, Materials Science of Bulk Metallic Glasses. REFERENCES Nb Nb content (mol%) ) W. Zhang and A. Inoue: Mater. Trans. 43 (02) ) H. Kimura, A. Inoue, S. Yamaura, K. Sasamori, M. shida, Y. Shinpo and H. Okouchi: Mater. Trans. 44 (03) ) S. Yamaura, Y. Shinpo, H. Okouchi, M. shida, O. Kajita, H. Kimura and A. Inoue: Mater. Trans. 44 (03) ) S. Yamaura, H. Kimura and A. Inoue: Met. Technol. 75 (05) (in Japanese). 5) C. Qin, W. Zhang, H. Nakata, H. Kimura, K. Asami and A. Inoue: Mater. Trans. 46 (05) Fig. 7 Calculated liquidus surface projection in the composition range of Nb x y x y alloys. 6) K. Hashi, K. Ishikawa, T. Matsuda and K. Aoki: J. Alloys Compd. 368 (04) ) N. Saunders and A. P. Miodownik, CAPHAD, A Comprehensive Guide, (Pergamon, Elsevier Science td., Oxford, UK, 1998). 8) A. T. Dinsdale: CAPHAD 15 (1991) ) O. Redlich and A. T. Kister: Ind. Eng. Chem. (1948) ) Y.-M. Muggianu, M. Gambino and J. P. Bros: J. Chim. Phys. 72 (1975) ) M. Hillert and M. Jarl: CAPHAD 2 (1978) ) M. Hillert and.-i. Staffansson: Acta Chem. Scand. 24 (1970) ) P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka and J. uitz: WIEN2k, An Augmented Plane Wave plus ocal Orbitals Program for Calculationg Crystal Properties (Tech. Universität Wien, Austria, 02), ISBN ) J. P. Perdew, K. Burke and Y. Wang: Phys. Rev. B 54 (1996) ) S. Matsumoto, T. Tokunaga, H. Ohtani and M. Hasebe: Mater. Trans. 46 (05) ) A. Bolcavage and U. R. Kattner: J. Phase Equilib. 17 (1996) ) K. C. H. Kumar, P. Wollants and. Delaey: CAPHAD 18 (1994) ) T. Tokunaga, K. Hashima, H. Ohtani and M. Hasebe: Mater. Trans. 45 (04) ) T. Tokunaga, S. Matsumoto, H. Ohtani and M. Hasebe: J. Japan Inst. Metals 70 (06) ) A. F. Guillermet: Z. Metallk. 82 (1991) ) G. Ghosh: J. Mater. Res. 9 (1994) ) T. Doi, H. Ishida and T. Umezawa: J. Japan Inst. Metals 30 (1966) (in Japanese). 23) N. Y. Alekseyevskiy, O. S. Ivanov, I. I. Rayevskiy and M. V. Stepanov: Fiz. Met. Metall. 23 (1967) ) G. N. Kadykova and G. N. Surovaya: Metall. Obr. Met. 8 (1967) ) T. Doi, M. Kitada and F. Ishida: J. Japan Inst. Metals 32 (1968) (in Japanese). 26) U. Zwicker, T. Meier and E. Röschel: J. ess-common Met. 14 (1968) ) I. K. Khegay and P. B. Budberg: Russ. Metall. (1971) ) V. S. Mikheev and O. K. Belousov: Russ. J. Inorg. Chem. 6 (1961) ) K. C. H. Kumar, P. Wollants and. Delaey: J. Alloys Compd. 6 (1994) ) V. N. Eremenko, E.. Semenova and. A. Tretyachenko: Dop. Akad. Nauk Ukr. RSR, Fiz. Met. Tekh. (No. 2) (1988) (in Russian). 31) V. N. Eremenko, E.. Semenova and. A. Tretyachenko: Russ. Akad. Nauk. Metall. (No. 6) (1992) (in Russian). 32) V. V. Moloknov, V. N. Chebotnikov and Yu. K. Kovneristyi: Izv. Akad. Nauk SSSR, Neorganic. Mater. 25 (1989) (in Russian). 33) V. N. Eremenko, E.. Semenova,. A. Tretyachenko and Z. G. Domatyrko: Izv. V.U.Z. Tsvetn. Metall. (No. 6) (1989) (in Russian). 34) J. H. N. van Vucht: J. ess-common Met. 11 (1966) ) J.. Glimois, P. Forey, R. Guillen and J.. Feron: J. ess-common Met. 134 (1987) ) K. P. Gupta: J. Phase Equilib. (1999) ) M. Bououdina, B. ambert-andron, B. Ouladdiaf, S. Pairis and D. Fruchart: J. Alloys Compd (03) ) A. Inoue, W. Zhang and T. Zhang: Mater. Trans 43 (02) ) D. R. Uhlmann: J. Non-Cryst. Solids 7 (1972) ) H. A. Davies: Phys. Chem. Glasses 17 (1976) ) e.g. T. Tokunaga, H. Ohtani and M. Hasebe: Mater. Trans. 46 (05)