A thermodynamic analysis of the PbSn system and the calculation of the PbSn phase diagram

CALPHAD Printed

Vo1.5,

No.4,

pp.267-276,

0364-5916/81/040267-10$02.00/O (c) 1981 Pergamon Press Ltd.

1981.

in the USA.

A THERMODYNAMIC ANALYSIS OF THE Pb-Sn SYSTEM AND THE CALCULATION OF THE Pb-Sn PHASE DIAGRAM

Department

ABSTRACT:

T. Leo Ngai and Y. Austin Chang of Metallurgical and Mineral Engineering University of Wisconsin-Madison 1509 University Avenue Madison, Wisconsin 53706

Thermodynamic data of the liquid phase, the (Pb) phase and the (Sn) phases are analyzed in terms of the Marqules type For the liquid of equations for the excess Gibbs energies. phase, a quasi-sub-regular solution model is used. For the fee (Pb) phase and the bet (Sn) phase, a quasi-regular model The calculated Dhase diaqram of the Pb-Sn binarv is used. using these data agrees well with the experimental diagram. The present analysis yields a set of internally consistent thermodynamic data and the lattice stabilities of Pb and Sn, i.e. the Gibbs energy differences between the fee and bet forms of Pb and Sn.

Introduction Lead-acid storage batteries play an important role in the life of our society. These batteries are used universally for starting, lighting and ignition in automobiles and frequently as power sources in recreation, medicine and scientific research or as standby power sources in an increasingly more electrified economy. Potentially, leadacid storage batteries with innovative designs may be used to power electric vehicles in urban areas in order to minimize pollution and to maximize the efficiency of conventional power plants by acting as electric utility load leveling devices. To improve the performance and economics of these batteries, it is necessary both to develop new materials and to improve our understanding of the performance of the mateirals currently used in these batteries. Traditionally, lead-antimony alloys have been used in lead-acid storage batteries. The many years of engineering effort invested in the coimnercialization of these batteries have been largely spent in developing a manufacturing technology based on the casting and handling characteristics of antimonial leads. In those applications where water electrolysis is undesirable, both lead-calcium and lead-tin-calcium alloys are employed, but with the shortcomings such as the problems associated with their poor castability and low strength. And in spite of their widespread use in the battery industry, there is a lack of basic understanding of the metallurgy of lead-tin-calcium alloys as compared to that of ferrous alloys or titanium alloys. In order to fill

Received 26 September 1981

267

T.L. Ngai and Y.A. Chang

268

these gaps a researcn program was initiated at the University of Wisconsin with the collaboration of Johnson Controls, Inc., Milwaukee, Wisconsin. The objectives of the research program are fi) to determine experimentally the phase relationships of lead-tin-calciumalloys, (ii) to develope a thermodynamic framework for calculating and predicting the phase equilibria of lead-tin-metalalloys with the metal being calcium or another potentially useful alloying element (iii) to study the kinetics of precipitation of lead-rich alloys and (iv) to develop ultimately a process for making the best possible alloys for commercial use. This paper presents a thermodynamicdescription of the lead-tin system, a basic binary of the ternary lead-tin-metal systems. Thermodynamic Models To simplify the thermodynamic expressions, the following subscripts and superscripts are used: Subscript 1: the the Subscript 2: Superscript II:the Superscript ~1:the Superscript y: the 1.

Pb component Sn component liquid phase fee (Al) phase bet (A5) phase

The Liquid Phase

The following expressions for the Gibbs energy are used for the liquid phase: *xsGL = xsG" _ xOG& - x;G; 11 q

AXSg II=

RT{ ; x,x~Hw;~+w;,) + (w~2-w;l)(x2-xl)11

xsg _ 11

1

OGR

= RT lnyf

12 a. = RT (2 x21(w12+w&) + ~xs$ 2

q

Cl1

ca

xs$ _ OGL = RT In yi 2 2

= RT t; x;[(wf2+w;,) + (wf2-w&)(4x2-')]I where w!. =++B 1J

[31 c41

The subscript ij stands for 12 or 21 and A, B are constants. R is the gas constant and T is the temperature. 2.

The a-Phase (fee, Al) The following Gibbs energy expressions are used, "G"- x;G; - x;G; = -x2 AGT' + RT Cxlx2worI

(53

x9-a "G';= RT In y';= RT {wo(l-~,)~} Gl *

051

xsTp _ *Gz = RT In $ 2

r71

where AGF'

= -AGy' + RT 1wo(l-~,)~l

= 'Gi - "G;

with C, D being constants,

and

WC1= ;+0

CR1

ANALYSIS OF THE Pb-Sn SYSTEM AND PHASE DIAGRAM

3.

-y-Phase

(bet, A5)

The following "Gy_

269

expressions

for the Gibbs energy are used,

x"Ga _ x"GY = x AG y" 11 22 1

"'Gy - "Gy = RT ln yy = ACT"

+ RT {xlx2wy}

[91

+ RT rw'(l-x,)*1

Cl01

"'GT - "Gz = RT In ys = RT {w'(l-x2)21

Cl11

cX+ll = 'G: - "Gy where AG,

II121

wy.;+F

and

with E, F being constants.

Evaluation

of the Thermodynamic

Properties

of the Various

Phases

The values for the solution parameters of the liquid-, a- and y-phases as well as those for the lattice stabilities, i.e. the Gibbs energy differences between the fee and bet structures, of Pb and Sn are obtained using the available thermodynamic data for the liquid- and the o-phases (1) as well as the well-known cr/R, y/a, and u/y phase boundaries (2). A detailed description is given below. 1.

The Pure Components

,+RThe Gibb$%energies of melting for the stable forms of Pb and Sn, i.e. and AG$ , are taken from Hultgren, Desai, Hawkins, Gleiser, Kelley AGl and Wagman (3). These values expressed in J/g-atom are given in Table I. TABLE Gibbs Energy Differences Between Stabilitieslof Subscript Subscript

1 - Pb 2 - Sn

I the Various Forms (the Lattice Lead and Tin

Superscript Superscript Superscript

9. - the liquid phase c1 - the fee (Al) phase y - the bet (A5) phase Ref.

A'GF"

=

4,799 -

7.99 T

AOGY+R 1

=

4,310 - 11.51 T

AOGFY

=

489 +

3.52 T

this study

A'G$f'

=

1,519 -

5.47 T

this study

*OGY+i 2

=

7,029 - 13.93 T

~0Go-+y 2

= -5,510 +

8.46 T

All values are in J/g-atom.

3 this study

3 this study

270

T.L. Ngai and Y.A. Chang

The values for the $kbbs ene&qies of melting for the unstable forms of Pb and Sn, i.e. AGI and AG2 are obtained from an analysis of the known thermodynamic properties if the liquid phase and the o-phase as well as the known al&, v/k, and CC/Y phase boundaries. The derived values are also given in Table I. Since these values are derived from an analysis of only the Pb-Sn systems over a relatively narrow range of temperatures, they may be subjected to rather large uncertainties. Furthermore, since the solubility of Pb in the y-phase is small, i.e. less than 2 at %, the uncertainties for AGl*' would be even larger. Nevertheless, as to be shown later, these values together with the solution parameters reported in the pres@ study a&consistent with the phase The values of A"G1 and A"G2 , given in Table I, are obtained diagram. from the Gibbs energies of melting for the stable and unstable forms of It is noteworthy to point out that Heumann and the component elements. Wostmann (4) measured the enthalpies of melting of Pb-rich Pb-Sn solid solutions with the aid of DTA. From these+measurements and the phase dia$Iam data, they obtained values of A’H$ y = -5.515 kJ/g-atom and These values correspond closely to the A'S2 y = -9.50 J/k g-atom. values obtained in the present analysis as shown in Table I. 2.

The Liquid

Phase

Values of wt2 and w& in Eqs. [l-3] are obtained from the known thermodynamic and p$ase equilibria data.available in the literature (1,2). Values of wl2 and w21 so obtained are given ln Table II. TABLE Solution

Parameters

Subscript Subscript

II

for the Liquid,

1 - Pb 2 - Sn

Superscript Superscript Superscript

wt2

=

v

t

wp;,

=

v

t 0.176

w*

=

q@

t

0.188

WY

=

2,058.80

_

, 42

T

fee and bet Phases

R - the liquid phase CL - the fee (Al) phase y - the bet (A5) phase

0.178

.

As given in the evaluation by Hultgren et al. (l), many investigators had'studied the enthalpy an Gibbs energies of mixing of the liquid alloys. The values of AH!! selected by Hultgren et al. are primarily based on the calorimetric values of Kleppa (5). These values are temperature-independent and an uncertainty of +125 J/g-atom was estimated by Hultgren et al. As shown in Fig. 1, values of AHi calculated from Eqs. [ll and [43 using values of A are in agreement with the experimental values within the estimated uncertainty of the data.

ANALYSIS OF THE Pb-Sn SYSTEM AND PHASE DIAGRAM

Pb 0.1 0.2 0.3 0.4 Xsn 0.6 0.7 0.8 0.9 Sn

Fig. 1.

Enthalpy of mixing of liquid Pb-Sn alloys; comparison between experimental and calculated values.

Since the compilation of Hultgren et al. (l), Bros, Castanet and Laffitte (6) and Yazawa, Kawashima and Itagaki (7) measured the enthalpy of mixing of liquid Pb-Sn alloys. As shown in Fig. 1, the values calculated from the model are in good agreement with the data of Bros et al. and slightly more positive than the data of Yazawa et al. There are large discrepancies between the activity data reported in the literature by various groups of investigators (8-19). The activity data were obtained by one of three methods, vapor pressure, emf or gas-equilibrationwith mixlures of H O/Hz. As shown in Fig. ZA, the calculated values of AX%Pb/(l-xpb) E agree with the vapor pressure data of Predel (9). The vapor pressure data of Hawkins and Hultgren (8) at high Sn concentrations are in good agreement with the data of Predel (9) and the calculated values but deviate significantly as the concentration of Pb increases. The data of Sooner et al. (16) show similar compositional dependence. Hultgren et al.'s selected values were based on the data of Hawkins and Hultgren. Attempts were made to use their data to describe the thermodynamic properties of liquid Pb-Sn alloys. However, these data are not consistent with the

271

Fig. 2A.

Pb

30

0.2 0.3

0.4

IOSOK

X,, 0.6

0.7

0.8

0.9

1050

I

MODEL, 1050K

'I

HULTGREN.

2. PREDEL , 1007 K 3. SOMMER ET AL. 0 HAWKINS

I

FROM THE

I

K

Sn

The excess Gibbs energy of Pb in liquid Pb-Sn alloys in terms of the function: AXSiipQ;/(1-Xpb)2; comparison between experimental and calculated values.

0.1

_

I I I 1.CALCULATED

c

Fig. 2B.

a

Z!? ii

I

Pb 0.1

A

0.2

DAS 8

0.3

0.4

X,,

IOSOK

0.7

I

MODEL,

0.6

FROM THE

GHOSH,

1. CALCULATED

0.8

I

0.9

I

I IO50 K -

The excess Gibbs Energy of Sn in liquid Pb-Sn alloys in terms of the function: nx%S&/(l-xS#; comparison between experimental and calculated values.

0

IO

20-

c,

N-

=

I

Sn

ANALYSIS OF THE Pb-Sn SYSTEM AND PHASE DIAGRAM

273

phase equilibria data. Fig. 26 shows comparisons between the calculated and experimental values of the excess Gibbs energy of Sn. Within the scatter of the data, the calculated values agree with all the experimental data. All of the'investigators used an emf method except Atarashiya, Uta, Shimoji and Niwa (11) who used a gas-equilibration method. The data of Voronin and Evseev (10) are suspected to be in error. Their Pb activity data show negative deviations from ideal behavior at high Pb concentrations and then positive deviations at high Sn concentrations. This is unlikely in view of the positive entha~py of mixjng over the entire compositional range, 3.

0 0 Fig. 3.

The q-Phase Values of Wa in Eqs. [5-i’]

are obtained

0.1

X

0.2

0.3

sn

The enthalpy of mixing of the (Pb) phase; comparison between experimental and calculated values.

from the known ._

the~odynamic and phase equilibria data vailabl in the literature (1,2) as is done for obtaining the values of w% 2 and wk 1. Fig. 3 shows the values of nHa selected by Hultgren et al. and those calculated from Eq. [S] using the values of Wa given in Table II and the lattice stability of Sn given in Ta$le I. The values of AHa are calculated in a similar manner as those of AH . Hultgren et al. selected the AHa values based on the experimental data of Murphy and Oriani (20), Kendall and Hultgren (21), SchDnann and Gilhaus (22) and Gilhaus (23). An uncertainty of +I70 J/g-atom was estimated by Hultgren et al. for these data. As shown in Fig. 3, values of AH" calculated from the model are in reasonable agreement with the experimental data at low and high Sn concentration, but somewhat less positive in the mid-concentration range. The more recent calorimetric data of Heumann and W~stmann (4) are between the values selected by Hu~tgren et al. and the calculated values. However, they are closer to the va'luesof Hultgren et al. 4.

The y-Phase

Values of wY in Eqs. [8-91 are obtained in a similar manner as for those of @ and given in Table If. Phase Diagram Calculations The Pb-Sn Phase diagram is calculated using the condition that the chemical Potentials of the component elements in a two-phase field are equal, i.e. q or

=?$

"Gs f RT In a; = OGk + RT In a? 1

[12Al

I12Bl

T.L. Ngai and Y.A. Chang

274

where the subscript i stands for either Pb or Sn, the superscript a, B indicates any of the three phases and ai is the activity of either component. Using the Gibbs energies for any of the two phases, Eq. [12] may be used to solve for the compositions of the co-existing phases as a function of temperature. Numerical evaluation is carried out using a standard nonlinear regressional analysis technique (24). As shown in Fig. 4, the calculated solidus and liquidus curves are in good agreement with the experimental data of Jeffery (25), Fisher and Phillips (26), Hultgren and Lever (27) Honda and Abe (28), Stockdale (29) and Borelius, Larris and Ohlsson (30). The calculated eutectic temperature is one degree lower than that selected by Hans&n and Anderko (2) as given in Table III. The calculated values of x;, x2 and xj at TABLE III Comparison Between the Calculated and Experimental Values (according to Hansen and Anderko) at the Eutectic Temperature

T

eut.

K

Calculated

455.2

0.289

0.746

0.976

Hansen & Anderko

456.2

0.29

0.739

0.986

600

*nn x

““‘I

I

I f I Wbj+

L

I

PII

- I

.

E 450

z h 2 4or

P

Pb

Fig. 4.

0.1

0.2

0.3

0.4

X,_

0.6

0.7

0.8

0.9

Sn

The Pb-Sn phase diagram; comparison between experimental and calculated values.

ANALYSIS OF THE Pb-Sn SYSTEM AND PHASE DIAGRAM

the eutectic temperature are also compared with those of Han en and Anderko given in the same table. The calculated values of xh is 0,007 higher while that of x3 is 0.01 lower. However, when one exami es all the liquidus data, it is difficult to sl;aqethat the value of xh = 0.739 is more preferable than that of x2 - 0.746. The literature data on the solvus curve of (Pb) in equilibrium with (Sn) are in disagreement (25,28-32). The descrepancies are due to the difficulty in reaching equilibrium during the solubility experiments. Cahn and Treaftis (31) discussed the problems associated with this type of experiment. As shown in Fig. 4, the calculated solvus curve above 400 K is in reasonable agreement with the data of Stockdale, except cooling curve (29), Cahn and Treaftis (31), Borelius et al. (30). Below 400 K, the calculated curve is in reasonable agreement with the data of Stockdale, heating curve (29), and Parravano and Scortecci (32). The data of Cahn and Treatis are somewhat higher than the calculated values. The calculated solvus curve for (Sn) in equilibrium with (Pb) is in reasonable agreement with the data of Jeffery (25) but slightly greater in solubility than the data of Borelius et al. (30). Acknowledgement The authors wish to thank Or. Larry Kaufman for many useful discussions during the course of this study and the National Science Foundation for financial support through an Industry-UniversityCooperative Research Activity Grant (NSF-DMR-79-08333). One of the authors, Y.A. Chang, wishes to thank Dr. W.O. Gentry of Johnson Controls, Milwaukee, Wise. who first introduced him the subject of lead metallurgy. References 1.

R. Hultgren, P.D. Desai, D.T. Hawkins, M. Gleiser and K.K. Kelley, Selected Values of The Thermodynamic Properties of Binary Alloys, American Society for Metals, Metals Park, Ohio, 1973.

2.

M. Hansen and K. Anderko, Constitution of Binary Alloys, 2nd Edition, McGraw-Hill, New York, 1958.

3.

R. Hultgren, P.D. Desai, D.T. Hawkins, M. Gleiser, K.K. Kelley and D.D. Wagman, Selected Values of the Thermodynamic Properties of the Elements, American Society for Metals, Metals Park, Ohio 1973.

4.

T. Heumann and H. WBstmann, Z. Metallk., 1972, 63, 332.

5.

O.J. Kleppa, J. Phys. Chem., 1955, 59, 175.

6.

J.P. Bros, R. Castanet and M. Laffitte, C.R. Acad. Sci., Paris, Ser. C., 1967, 264, 1804.

7.

A. Yazawa, T. Kawashima and K. Itagaki, J. Japan Inst. Metals, 1968, 32, 1281.

8.

D.T. Hawkins and R. Hultgren, Trans. TMS-AIME, 1967, 239, 1046.

9.

B. Predel, Z. Metallk., 1960, 51, 381.

10. :;g. Voronin and A.M. Evseev, Russ. J. Inorg. Chem., 1959, 33_,

. 11.

K. Atarashiya, M. Uta, M. Shimoji and K. Niwa, Bull..Chem. Sot. Japan, 1960, 33, 706.

12.

R.A. Schaefer and F. Hovorka, Electrochem Sot. Reprint NO. 87, 1945, 23, 267.

13.

J.F. Elliott and J. Chipman, J. Am. Chem. Sot., 1951, 73, 2684.

275

276

T.L. Ngai and Y.A. Chang

14.

S.K. Das and A. Ghosh, Met. Trans., 1972, 3, 803.

15. G.W. Padgaokar, P.S. Kinnerkar and D.L. Roy, Trans. Indian Inst. Metals, 1968, 2l_, 36. 16.

F. Sommer, Y.H. Suh and B. Predel, Z. Metallk., 1978, 69, 470.

17.

K. Okajima and H. Sakao, Trans. Japan Inst. Metals, 1968, 9, 325.

18. D.F. Tkhai, Trudy Leningrad. Politekhn. Inst., 1967, 272, 61. 19.

N.I. Shurov, N.G. Ilyushchenko,A.I. Anfinogenov and T.L. Mitrofanova, Deposited Doc., 1974, VINITI 1039-74, 10 pp.

20. W.K. Murphy and R.A. Oriani, Acta Met., 1958, 6, 556. 21. W.B. Kendall and R. Hultgren, J. Phys. Chem., 1959, 63, 1158. 22.

E. Schiirmannand F. Gilhaus, Arch. Eisenhtittenw., 1961, 32, 867.

23.

F. Gilhaus, Z. Naturforsch., 1959, a,

1001.

24. Non-linear Regression Routines, Mathematical Routines Series, Academic Computer Center, The University of Wisconsin-Madison,1972. 25.

F.H. Jeffery, Trans. Faraday Sot., 1928, 24, 209.

26.

H.J. Fisher and A. Phillips, Trans. AIME, 1954, 3,

27.

R. Hultgren and S.A. Lever, Trans. AIME, 1949, 185, 67.

28.

K. Honda and H. Abe, Science Repts. Tohoku Imp. Univ., 1930, 19, 315.

29.

D. Stockdale, J. Inst. Metals, 1932, 49, 267; J. Inst. Metals, 1930, 43, 193.

30.

G. Borelius, F. Larris and E. Ohlsson, Arkiv. Mat., Astron. Fysik, 1944, 3& 1.

1062.

31. J.W. Cahn and H.N. Treaftis, Trans. Met. Sot. AIME, 1960, 218, 376. 32.

N. Parravano and A. Scortecci, Gazz. Chim. ital., 1920, a,

83.