Thermodynamic modelling of the La-Pb Binary system

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1 , 1 (29) DOI:1.151/jeep/291 Owned by the authors, published by EDP Sciences, 29 Thermodynamic modelling of the a-pb Binary system M.Idbenali 1, C.Servant 2, N.Selhaoui 1 and.bouirden 1 1 aboratoire de Thermodynamique Métallurgique et Rhéologie des Matériaux, Université Ibn Zohr, B.P. 496, Dcheira, Agadir, Morocco. 2 aboratoire de Physicochimie de l Etat Solide, ICMMO, Université de Paris-Sud, 9145 Orsay Cedex France. idbenalimohamed@yahoo.fr Abstract The thermodynamic modelling of the a-pb binary system was carried out with the help of CAPHAD (CAculation of PHAse Diagram) method. a 5 Pb 3, a 4 Pb 3, a 5 Pb 4, αa 3 Pb 4, βa3pb4, apb 2, apb 3 have been treated as stoichiometric compounds while a solution model has been used for the description of the liquid, BCC and FCC phases. The calculations based on the thermodynamic modeling are in good agreement with the phase diagram data and experimental thermodynamic values. Keywords: a-pb system, phase diagram, Thermodynamic modelling, Calphad method. 1 Introduction This study is part of a thermodynamic investigation of M-Pb systems (M=Ba [1], Ca [2], Yb [3]) which is intended to give a better understanding of the constitutional properties and potential technological applications of these alloys. The present work deals with an assessment of the thermodynamic description of the a-pb system using the CAPHAD technique [4]. Early investigations of the a-pb system included that of Canneri [5] who drew a diagram with only three intermetallic compounds a 2 Pb, apb and apb 2. No structural details were given and terminal solubilities were not studied. ater McMasters et al [6] determined the entire a- Pb system by differential thermal analysis (DTA), metallography and X-ray diffraction (XRD). The phase diagram of McMasters et al [6] contained six intermediate phases: a 5 Pb 3, a 4 Pb 3, a 11 Pb 1, αa 3 Pb 4, βa 3 Pb 4, apb 2, apb 3. a 11 Pb 1 existed only in the 1433 to 1623 K temperature range. The maximum solubility of Pb in a was about 2.5 at.% at the catatectic equilibrium temperature (T=1113K), less than 1% at the eutectic temperature of 148K and less than.5% at at ~973K. The solubility of Pb in a is about 5 % at the eutectic temperature of 598K. The maximum solubility of a in Pb was estimated to be less than.1 at.% according to McMasters et al [6]. McMasters et al [7] has proposed update of their diagram. The modifications involved deletion of a 11 Pb 1, addition of a 5 Pb 4, with incongruent melting at 1623K. Table 1. Symbols and Crystal Structures of the Stable Solid Phases in the (a-pb) Alloys from [8] Diagram Symbol Composition at Pb Symbol Used in Thermo-Calc Data File Pearson Symbol Space roup Strukturberic ht Symbol Prototype γa at 2.5 BCC_A2 ci2 Im3m A2 W βa from to 1 FCC_A1 cf4 Fm3m A1 Cu αa Dhcp hp4 P6 3 / mmc A3 a a 5 Pb a 5 Pb 3 hp16 P6 3 / mcm D8 8 Mn 5 Si 3 a 4 Pb a 4 Pb 3 ci28 I43d D7 3 Anti-Th 3 e 4 a 5 Pb a 5 Pb 4 op36 Pnma. Sm 5 e 4 αa 3 Pb αa 3 Pb 4 β a 3 Pb β a 3 Pb 4 apb apb 2 apb 3 75 apb 3 cp4 Pm3m (Pb) 99.9<x(Pb)<1 FCC_A1 cf4 Fm3m 1 2 AuCu 3 A1 Cu The assessment of the a-pb phase diagram (Fig.1) by Palenzona and Cirafici [8] is based on the investigation of [6] with modification of [7]. The equilibrium phases are: (1) the liquid, (2) the seven intermediate phases: a 5 Pb 3, a 4 Pb 3, a 5 Pb 4, αa 3 Pb 4, βa 3 Pb 4, apb 2, apb 3. and (3) the four terminal solid solutions : α(a), β (a), γ (a) and (Pb). The crystal structures of various phases are reported in Table 1. All data mentioned above are collected from studies [8]. In this paper, the thermodynamic modelling of the a-pb system was carried out using the CAPHAD [4] method. In this work, the ibbs energy of each phase is modelled using Parrot [9] module in the ThermoCalc software [1]. This is an Open Access article distributed under the terms of the Creative Commons Attribution-Noncommercial icense ( which permits unrestricted use, distribution, and reproduction in any noncommercial medium, provided the original work is properly cited. Article available at or

2 3.2 Solution phases The solution phases ((γa), (βa), (Pb), and liquid) are modelled as substitutional solutions according to the polynomial Redlich-Kister model [17]. The ibbs energy of one mole of formula unit of phase φ is expressed as the sum of the reference part ref φ, the ideal part id φ and the excess part xs φ : ref φ id φ xs φ m φ = + + (3) Fig.1. The a-pb phase diagram as assessed by Palenzona et al [8] 2 Thermodynamic Data The thermodynamic data available for this system are: (i) the enthalpies of formation of the intermetallic compounds determined by many authors [11-14] using various techniques (see Table 4); (ii) heat capacities of apb 3 at constant pressure in the range 323 to 773K measured by ambino et al [15] using differential scanning calorimeter, which could be well represented by : Cp= T (J/g.atm.K) (1) 3 Thermodynamic Models 3.1 Pure elements The ibbs energy function ( ) SER φ i T = φ i H i (298.15K) for the element i ( i =a,pb) in the phase φ (φ = iquid, BCC_A2, DHCP or FCC_A1 ) is described by an equation of the following form: φ i() T = a+ bt+ ctlnt+ dt + et + ft + gt (2) SER Where H i (298.15K) is the molar enthalpy of the element i at K in its standard element reference (SER) state, FCC_A1 for Pb and DHCP for a. In this paper, the ibbs energy functions are taken from the STE compilation of Dinsdale [16]. As used in the Thermo-calc software [1]: ref φ φ φ ( T ) = x a a + x Pb Pb (4) id φ = RT ( x alnxa+ xpbln xpb) (5) Where: R is the gas constant; T is the temperature, in Kelvin; x a and x Pb are the mole fraction of elements a and Pb, respectively. φ i is the molar ibbs energy of elements (i=a, Pb) with the structure ø. The excess terms of all the phases were modelled by the Redlich-Kister [17] formula. [ φ ( T ) + 1 φ ( T )( x ) 1] id φ ( T ) = x a x Pb x a,pb a,pb a Pb i φ With: ( T ) = a a, Pb i + b T i (7) (6) i φ Where: ( T ) is the i th interaction parameter a, Pb between the elements a and Pb, which is evaluated in the presented work. a i, and b i are the coefficients to be optimized. 3.3 Stoichiometric compounds: The ibbs energy of the stoichiometric compound ApBq is expressed as follows: A p B q = p A + q B + a + bt (8) Where A and B are the ibbs energy of the pure elements a and Pb in the reference state DHCP and FCC respectively. a and b are parameters to be determined. 1-p.2

3 Table 2. The thermodynamic parameters of the a-pb system Phase Thermodynamic Models Parameters (units in J.mol -1. at and J/mol.at.K) iquid (a,pb) 1 iq a, Pb = T 1 iq a, Pb = T BCC_A2 (a,pb) 1 (Va) * 3 FCC_A1 (a,pb) 1 (Va) 1 a 5 Pb 3 (a).625 : (Pb).375 a 4 Pb 3 (a).571 : (Pb).429 a 5 Pb 4 (a).555 : (Pb).445 αa 3 Pb 4 (a).429 : (Pb).571 β a 3 Pb 4 (a).429 : (Pb).571 apb 2 (a).333 : (Pb).667 apb 3 (a).25 : (Pb).75 * (Va) for vacancy BCC _ A2 a, Pb = T 1 BCC _ A2 a, Pb = T FCC _ A1 a, Pb = T 1 FCC _ A1 a, Pb = T a5pb3 a: Pb =.375 FCC _ A1 DHCP Pb a T a4pb3 a: Pb =.429 FCC _ A1 DHCP Pb a T a5pb4 a: Pb =.445 FCC _ A1 DHCP Pb a T a3pb4 α a: Pb =.571 FCC _ A1 DHCP Pb a T a3pb4 β a: Pb =.571 FCC _ A1 DHCP Pb a T apb2 a: Pb =.667 FCC _ A1 DHCP Pb a T apb3 a: Pb =.75 FCC _ A1 DHCP Pb +.25 a T 4. Results and discussions The parameters of a-pb binary system were optimized using the Parrot module [9] in the ThermoCalc [1] software. This program is able to take various kinds of experimental data in one operation and minimizes the sum of squared error. Each of the selected data values is given a certain weight, which is chosen and adjusted based on the data uncertainties given experimentally. No enthalpy of mixing of the liquid was available in the literature. In the present parameter optimization procedure we first imposed the conditions d 2 /dx 2 > for modeling the liquid phase to avoid the appearance of an unwanted inverted miscibility gap in that phase during the phase diagram calculation as recommended in [18-21]. Therefore we grid the space in x( every.5 at.) and T (every 25K) (6<T<5K). The phase boundary data reported by [8] and thermodynamic data were used. Then the thermodynamic parameters for the intermediate phases were optimized based on the enthalpies of formation of the intermetallic compounds measured by [14] and the invariant equilibrium from [8]. All the parameters were evaluated and listed in Table 2. The calculated phase diagram is shown in Fig. 2. The experimental and calculated temperatures of the invariant reactions are compared in Table 3. The enthalpy of formation of the intermetallic compounds calculated with the experimental values is listed in Table 4 and presented in Fig. 4. A satisfactory agreement is noted. Fig. 3 shows the calculated molar heat capacity of apb3 at different temperatures with the experimental data. It can be seen that the calculated results agree well with the experimental values [15]. The singular point at 6.61K corresponds to the joining temperature of two thermodynamic descriptions of pure element Pb for different temperature ranges in [16]. As first stated by Chen et al [2], we verified that when the liquid phase is suspended, the 1-p.3

4 intermetallic compounds are no more stable at very high temperatures, because the enthalpy of Fig.2.Calculated a-pb phase diagram with data from [8] Fig.3. Calculated molar heat capacity of apb 3 at different temperatures, together with experimental data Table 3. Invariant reactions in the a-pb System. from [16] Reaction [8] This work T (K) x(pb). T (K) x(pb) αa βa * γa βa+liq γa.25 γa βa.1 liq βa.8 liq.46 iq βa+ a 5 Pb 3 βa.1 βa iq iq.8 iq a 5 Pb a 4 Pb 3 iq+ a 5 Pb iq iq.438 a 5 Pb 4 iq+ a 4 Pb iq iq.514 βa 3 Pb 4 iq+ a 5 Pb iq iq.587 βa 3 Pb 4 αa 3 Pb iq βa 3 Pb 4 + apb iq iq.656 iq+ apb3 apb iq iq.674 iq apb iq (Pb) + apb 3 (Pb).999 (Pb) iq.995 iq.988 formation of these compounds is approximately the same but their entropy is negative (i.e. b> in Eq. 8) but less negative than -12J/mol.K. In Fig. 9, therefore the computed phase diagram gives the BCC_A2 phase stable. At high temperatures, only this solid solution is calculated on the whole Pb composition range and not the FCC_A1 one in the Pb-rich region. This is due to the power series in terms of temperature for the Pb element in the BCC_A2 state which becomes more stable compared to the FCC_A1 state around 2K and for higher temperatures [16]. 1-p.4

5 Table 4. Calculated and measured enthalpies formation of the intermetallic compounds. Phase apb 3 apb 2 Enthalpy of formation kj/mol Technique used Ref Dynamic differential calorimetric [11] Differential direct isoperibol calorimeter [12] Emf [13] -49 Semi empirical model of Miedema [14] optimization This work -6.3 Differential direct isoperibol calorimeter [12] -63 Semi empirical model of Miedema [14] optimization This work αa 3 Pb optimization This work -64 Differential direct isoperibol calorimeter [12] βa 3 Pb 4-72 Semi empirical model of Miedema [14] optimization Differential direct isoperibol calorimeter [12] a 5 Pb 4-72 Semi empirical model of Miedema [14] optimization This work Differential direct isoperibol calorimeter [12] a 4 Pb 3 a 5 Pb 3-71 Semi empirical model of Miedema [14] optimization This work Differential direct isoperibol calorimeter [12] -65 Semi empirical model of Miedema [14] optimization This work. Fig. 4. Calculated and measured enthalpies of formation of the intermetallic compounds Fig.5. Calculated a-pb phase diagram when the liquid phase is suspended Conclusion A consistent set of thermodynamic parameters of the different phases of the a-pb binary system has been optimized. The computed values are in good agreement with the experimental data. Further thermodynamic determinations, in particular the 1-p.5

6 enthalpy of mixing of the liquid phase, will be necessary to improve the assessment. References [1] M. Idbenali, C. Servant, N. Selhaoui,. Bouirden, CAPHAD, 31 (27) 479. [2] M. Idbenali, C. Servant, N. Selhaoui,. Bouirden, CAPHAD, 32 (28) 64. [3] M. Idbenali, C. Servant, N. Selhaoui,. Bouirden, CAPHAD, 33 (29) 57. [4]. Kaufman and H. Bernstein, Computer Calculations of Phase Diagrams, Academic Press, New-York, NY, (197). [5]. Canneri, Metall. Ital, 23, (1931) in Italian. (Equi. diagram; Experimental) [6] O.D. McMasters, S.D. Soderquist, and K.A. schneider, Jr., ASM Trans. Quart., 61, (1968). [7] O.D. McMasters and K.A. schneidner, jr., J.ess-Common Met., 45, (1976).(Equi. Diagram, Crys Structure; Experimental) [8] A. Palenzona and S. Cirafici ; J. Phase Equilibria Vol.13, (1992) [9] B. Jansson Thesis, Royal Institute of Technology, Stockholm; 1984 [1] B. Sundman, J-O. Andersson, CAPHAD 9 (1985) [11] A. Palenzona and S. Cirafici, Thermochim, Acta, 6, (1973) [12] R. Ferro, A. Orsese, R. Capelli, and S. Delfino, Z.Anorg.Allg.Chem, 413, (1975). [13] A. Morisson, C. Petot, and A. Percheron-uegan, Thermochim.Acta, 11, (1986) [14] C. Colinet, A. Pasturel, A. Percheron-uegan, and J.C. Achard, J.ess-Common Met, 12, (1984) [15] M. ambino, P. Rebouillon, J.P. Bros,. Borozone,. Caccicamani, and R. Ferro, J. ess-common Met, 154, (1989) in French. (Thermo;Experimental) [16] A.T. Dinsdale, Calphad 15 (1991) 317. [17] O. Redlich and A. Kister, Ind. Eng. Chem, 4, (1948). [18] R. Arroyave and Z.K. iu, Calphad 28 (26) [19] C.H.P.UPIS, Chemical Thermodynamics of Materials,(1983) Elsevier Science Publishers B.V,P.O. Box 211,1 AE Amsterdam, the Netherlands. [2] S.. Chen, S. Daniel, F. Zhang, Y.A. Chang, W.A. Oates and R. Schmid-Fetzer, J. Phase Equilibria, 22 (21) [21] K.C. Kumar, P. Wollants, J. Alloys Compds, 32 (21), p.6

Thermodynamic description of the Gallium-Lanthanum binary system

JEEP, 8 () DOI:.5/jeep/ 8 Owned by the authors, published by EDP Sciences, Thermodynamic description of the llium-anthanum binary system M. Idbenali, C.Servant ycée oummane Elftwaki, Dcheira, Agadir, Maroc.