THERMODYNAMIC ASSESSMENT OF THE Pb–Sr SYSTEM

a State Key Laboratory of Powder Metallurgy, Central South University, Changsha, Hunan, PR China b Collaborative Innovation Center of Steel Technology, University of Science and Technology Beijing, Beijing, PR China c Company of Hunan Liugu Grand Pharmacy, Hunan, PR China d College of Mechanical and Electrical Engineering, Hunan University of Science and Technology, Xiangtan, PR China e School of Materials Science and Engineering, Anhui University of Science and Technology, Huainan, PR China


Introduction
Knowledge of the physical properties in Pb-based alloys is of great importance in industrial applications such as grid material and lead-acid batteries.One of important Pb-based binary alloy systems is the Ca-Pb system, which is widely used in the maintenance-free lead-acid batteries, as it can improve storage life and reduce self-discharge.The disadvantages are its poor strength, casting difficulties and low resistance to discharge cycle capacity [1,2].The Pb-Sr alloy can satisfy the maintenance of free performance and realize deep cycle discharge performance as well as corrosion resistance requirements [3,4].
There have been several experimental determinations on the phase equilibria of the Pb-Sr system [5][6][7][8].The comprehensive determinations by Bruzzone et al. [6] were redrawn by Massalski [9].However, there is no general agreement between the proposed phase diagram and thermodynamic data.The present work is thus devoted to the assessment of experimental phase diagram and thermodynamic data available for the Pb-Sr system via the CALPHAD approach, which is a common method in the thermodynamic field and has been used for many systems [10][11][12], and provide two optimal sets of thermodynamic parameters for this system by using both Redlich-Kister linear [13] and exponential polynomials [14].

Literature review 2.1 Phase diagram data
In the present assessment, the reported experimental phase diagram and thermodynamic data of the Pb-Sr system were critically reviewed.The experimental phase diagram data have been measured by four groups of authors [5][6][7][8].Seven intermetallic compounds, i.e.SrPb 3 , Sr 3 Pb 5 , Sr 2 Pb 3 , SrPb, Sr 5 Pb 4 , Sr 5 Pb 3 and Sr 2 Pb, were identified to be stable in this system.

Dedicated to the memory of Professor Dragana Živković
the composition range of 0 at.% to 25 at.%Sr, a mixture of primary SrPb 3 crystals and β (the solid solution of Sr in Pb) eutectic was formed.At 900 °C, there is a peritectic reaction, L + Sr 2 Pb ↔ SrPb.The congruent melting of Sr 2 Pb compound with a melting point of 970 °C was observed [5], while the melting point for Sr 2 Pb is 1155 °C according to Bruzzone et al. [6].The experimental results given by Vakhobov et al. [5] are different from those of Bruzzone et al. [6] and Marshall and Chang [7].Moreover, only the L + Sr 2 Pb ↔ SrPb peritectic reaction was detected by Vakhobov et al. [5], and the other four peritectic reactions were not determined.The temperature is 630 °C for the eutectic reaction in the Sr-rich part of the system [5].However, the temperature for the above eutectic reaction is 725 °C according to Bruzzone et al. [6].Since the method for the preparation of Pb-Sr alloys by Vakhobov et al. [5] could lead to the melting of Pb and Sr in corundum crucibles, in which the specimens were obtained by combining the processes of vacuum distillation and directional solidification, this processes cannot ensure the homogeneous specimens.Therefore, the data published by Vakhobov et al. [5] were not used in the present optimization.
Marshall and Chang [7] studied the phase equilibria of the Pb-Sr system in the composition range up to 36 at.%Sr by using DTA, OM, XRD and electron probe micro analysis (EPMA).An eutectic at 324.5 °C and 1.0 at.%Sr is formed between (Pb) and SrPb 3 .According to DTA results [7], the maximum Sr content in the lead solid solution is approximate 0.33 at.%. SrPb 3 was found to melt congruently at 677 °C, which is in good agreement with the values 676 °C [8] and 675 °C [6].The liquidus from 0 to 25 at.%Sr [7] is consistent with those given by Piwowarsky [8] and Bruzzone et al. [6].Those experimental data are used in the present thermodynamic modeling.

Thermodynamic data
Using the first-principles method, Duan et al. [24] and Peng et al. [25] calculated the enthalpies of formation for the intermetallic compounds.The calculated enthalpies of formation for Sr 2 Pb are -63.56kJ/(mol•atom) at 0 K and 0 Gpa and -42.98 kJ/(mol•atom) at 0 K and 10 Gpa by Duan et al. [24].The data by Duan et al. [24] are used in this work.The enthalpies of formation for the seven intermetallic compounds have been calculated by Peng et al. [25].The enthalpy of formation for Sr 2 Pb at 0 K is -61.951kJ/(mol•atom) according to Peng et al. [25], which is more positive than the results by Duan et al. [24].In this work, the enthalpies of formation for SrPb 3 , Sr 2 Pb 3 , SrPb, Sr 5 Pb 4 , Sr 5 Pb 3 and Sr 2 Pb were also calculated and the detail will be discussed in Section 4.
The phase diagram and thermodynamic data of the Pb-Sr binary system evaluated above are summarized (Sr)rt Cu / Fm-3m 6.082 6.082 6.082 [22] (Sr)ht W / Im-3m 4.85 4.85 4.85 [23] in Table 2, together with the indications whether the experimental data were used or not in this optimization.

Thermodynamic modeling 3.1. Unary phases
The Gibbs energy function (298.15K) for the element i (i = Pb, Sr) in the phase φ (φ = liquid, bcc_A2 and fcc_A1) is described by an equation of the form: (1) where (298.15K) is the molar enthalpy of the element i at 298.15 K in its standard element reference (SER) states.In this work, the Gibbs energy functions are taken from the SGTE compilation by Dinsdale [26].

Solution phases
The liquid, fcc and bcc phases are described by a substitution solution model, and their molar Gibbs energies are given by the following formula: (2) in which is the contribution to the Gibbs energy from pure components, is the ideal mixing contribution to Gibbs energy , and is the excess Gibbs energy corresponding to the non-ideal interactions between the components.
For the Pb-Sr system, the following equations hold: ( (4) where R is the gas constant, T is temperature in K, and x Pb and x Sr are the mole fractions of Pb and Sr, respectively.
The excess terms of the solution phases were modeled by the Redlich-Kister linear polynomial [13].To avoid the artificial miscibility gap at high temperatures, the exponential equation [14] was also used for the excess Gibbs energy of the liquid phase.(5) in which the interaction parameters are described as follows: (6) or, (7) where is the ith interaction parameter between the elements Pb and Sr.The coefficients a i , h i are the enthalpy part of the interaction energy, b i is the entropy part of the interaction energy and τ i (τ i >0) is a special temperature.These parameters are to be optimized in the present work.

Stoichiometric compounds
In view of the negligible solubilities, all the intermetallic phases in the Pb-Sr system were modeled as stoichiometric compounds.The Gibbs energy of the stoichiometric compound is expressed as follows: (8) in which and are the Gibbs energies of the Sr and Pb pure elements, respectively.And A and B are the parameters to be evaluated.

Results and discussion
The evaluation of model parameters in the Pb-Sr binary system has been carried out by recurrent runs of the PARROT module in Thermo-Calc software [27], which works by minimizing the square sum of the differences between measured and calculated values.The step-by-step optimization procedure described by Du et al. [28] was utilized in the present assessment.Each piece of selected information was given a certain weight based on the uncertainties of the experimental data, and changed by trial and error during the assessment, until most of the selected experimental information was reproduced within the expected uncertainty limits.
In the present work, two methods were used to avoid the formation of an undesired miscibility gap in the liquid phase for the Pb-Sr phase diagram.Using the linear equation for the excess Gibbs energy of the liquid, constraints must be imposed during the optimization procedure [29,30].A positive curvature of the liquidus by restricting d 2 G/dx 2 > 0 in the atomic composition range was applied in the thermodynamic optimization.Another method is to apply the exponential equation [14] directly to describe the excess Gibbs energy of the liquid.The advantage of the exponential equation is that constraint is needed during the thermodynamic optimization for the sake of avoiding the undesired miscibility gap in the liquid phase.
The optimization started with the liquid phase, and a thermodynamic parameter was adjusted based on the liquidus data [6][7][8].Then the intermetallic compounds were taken into consideration.The enthalpies of formation for all intermetallic phases via first-principles calculations were used as initial values for the parameters A in Eq. ( 8) and the initial values of B obtained at random.After that, all the coefficients in Eq. ( 8) were optimized according to the phase diagram data [6][7][8].At the same time, the parameters and were adjusted in order to have a good agreement between calculation and experiment.The optimized parameters for the liquid and compounds were then fixed during the next optimization procedure.Both (Pb) and (Sr)rt have fcc_A1 structure, in order to make the thermodynamic property of (Pb) independent from that of (Sr)rt, the parameters and for the fcc phase were optimized subsequently.Finally, all parameters for the individual phases were optimized simultaneously to achieve a global self-consistent thermodynamic description.
The presently obtained thermodynamic parameters for the Pb-Sr system are given in Table 3.Using these two sets of parameters, the phase equilibria, enthalpies of mixing for the liquid as well as the enthalpies of formation for the intermetallic phases in the Pb-Sr system are calculated to show the rationality of the present modeling.30.94 at.%Sr and 725.6 °C at 95.53 at.%Sr, respectively.And the calculated temperatures using exponential model are 325.9°C at 0.65 at.%Sr, 631.0 °C at 31.13 at.%Sr and 724.0 °C at 94.95 at.%Sr, respectively.It is noted that small discrepancies exist between the calculated results using linear and exponential models.The optimized results agree well with the experimental data [6][7][8] except for the composition of the eutectic point (L ↔ Sr 2 Pb + (Sr)ht).

H. Zhang et al. / JMM 53 (3) B (2017) 179 -187
183 Table 3 continues from the previous page a All parameters are given in J/mole and temperature (T) in K.The Gibbs energies for the pure elements are from the SGTE compilation [26].Peng et al. [25] used the local density approximation (LDA) CA-PZ function [36,37].The calculated enthalpies of formation according to the present CALPHAD modeling are a good compromise among these first-principles calculation values, as shown in Fig. 3. Fig. 4 shows the calculated enthalpy of mixing in the liquid phase at 1200 °C by using the linear and exponential parameters, respectively.Although there are no experimental data, the interaction parameters for liquid are reasonable in view of its compatible magnitude with the computed enthalpies of formation for the solid solution phases, as shown in Fig. 3.
The present work demonstrates that both the R-K linear equation and exponential equation for the excess Gibbs energy of liquid can describe the properties of liquid phase satisfactory.In comparison with the linear equation, the unique feature of the exponential equation is that no constraint is imposed during the thermodynamic optimization in order to avoid the possible formation of undesired miscibility gap in the liquid phase.Thus the exponential equation is preferable for the future thermodynamic modeling.

Conclusions
The phase diagram and thermodynamic data available for the Pb-Sr system have been critically evaluated.The enthalpies of formation for the six intermetallic compounds have been calculated by means of the first-principles calculations.On the basis of reliable phase diagram data and the first-principles computed enthalpies of formation values, two optimal sets of thermodynamic functions for the system were obtained by using both Redlich-Kister linear and exponential formulations.The comprehensive comparisons show that the experimental phase diagram data and thermodynamic data are reasonably accounted for by the present description of the Pb-Sr system.

Table 3 .S
Summary of the optimized thermodynamic parameters in the Pb-Sr systemTable 3 is continued on the next page.The calculated Pb-Sr phase diagram using the present two sets of thermodynamic parameters are shown in Fig. 1 and Fig. 2 together with the invariant reaction temperatures and experimental data.Fig. 1 (a) presents the calculated phase diagram using the linear equation, while Fig. 1 (b) displays the calculated Pb-rich part of the phase diagram.Fig. 2 (a) is the computed phase diagram based on the exponential model and Fig. 2 (b) is the Pb-rich part of the diagram.The calculated temperatures of three eutectic points using linear model are 324.7 °C at 0.86 at.%Sr, 628.4 °C at

HFigure 1 .
Figure 1.(a) Calculated Pb-Sr phase diagram with linear parameters, compared with the experimental data [5-8] in the whole composition range; (b) The Pb-rich part of Pb-Sr phase diagram in comparison with the experimental data [5-8] from Pb to 8 at.%Sr.

Figure 3 .
Figure 3. Calculated enthalpies of formation forstoichiometric compounds at 298.15 K using two kinds of models in the Pb-Sr system together with previous first-principles calculations[24,25] and present calculations.

Figure 4 .
Figure 4. Calculated enthalpies of mixing for the liquid using two kinds of parameters at 1200 °C.

Table 1 .
Phase designation and crystal structure data for the solid phases in the Pb-Sr system.

Table 2 .
Summary of the reported experimental data for the Pb-Sr system.
a For the experimental techniques: DTA = Differential thermal analysis; XRD = X-ray diffraction; OM = Optical metallography; CA = Chemical analysis; MHM = Microhardness measurement; SA = Spectrophotometric analysis; TA = Thermal analysis and EPMA = Electron probe micro analysis.b Indicates whether the data were used or not in the optimization process: ■ used; □ not used.

Table 4 .
Calculated invariant equilibria in the Pb-Sr system along with the experimental data.

Table 4
is continued on the next page.