A thermodynamic optimization of the Cu-In system

A Thermodynamic Optimization of the Cu-In SystemJean Hertz, Khadija El Aissaoui, and Lahcen Bouirden

(Submitted 5 March 2000; in revised form 1 October 2001)

The Cu-In liquid phase exhibits an important short range disordering between the liquidus lineand 1100°C. Kao et al. [1993Kao] have proposed a thermodynamic phase diagram optimizationusing a constant excess CP versus temperatures in the liquid phase, which seems unrealistic forlow and high temperatures far extrapolations. On the contrary, we suggest in this new model touse a description of the liquid phase without any excess CP: the considered liquid phase ismodelized only with high temperatures experimental data and the disordering enthalpy of theliquid is added to the melting enthalpy of the various solid phases. This artefact allows tocorrectly modelize the complete phase diagram using all the available thermodynamic data forformation of the solid phases.

1. Equilibrium Phase Diagram

A recent assessment of the (Cu-In) system by Subrama-nian and Laughlin [1989Sub] is based on 47 original refer-ences. The same phase diagram (see Fig. 1) was reproducedin the handbook by Massalski et al. [1990Mas]. This dia-

gram is very similar to the one published in Hansen andAnderko’s text [1958Han], which is based on the experi-mental works of Weibke and Egers [1934Wei], Hume-Rothery et al. [1940Hum], Reynolds et al. [1951Rey], andJones and Owen [1953Jon]. Okamoto has proposed twoupdates of this phase diagram. The first update [1991Oka]concerns the shape of the solidus in the primary (�Cu)domain taking into account thermodynamic considerationsthat are consistent with the experimental observations of[1987Mus]. The second update [1994Oka] simplifies thelow temperature transformations in the �� phase, accordingto the experimental work of Bolcavage et al. [1993Bol](Fig. 2), which also includes the changes proposed by Oka-moto [1991Oka].

Jean Hertz, Laboratoire de Thermodynamique Métallurgique, Chimiedu Solide Minéral, UMR 7555, Université Henri Poincaré Nancy I,BP 239 Vandoeuvre les Nancy Cedex 54506 France; and Khadija ElAissaoui and Lahcen Bouirden, Laboratoire de ThermodynamiqueMétallurgique et Rhéologie des Matériaux, Faculté des Sciences, Uni-versité Ibnou Zohr, BP 28/S, Agadir, 80000, Maroc. Contact e-mail:[email protected].

Fig. 1 The Cu-In experimental phase diagram according to Subramanian and Laughlin [1989Sub] and Massalski et al. [1990Mas]

Basic and Applied Research: Section I

Journal of Phase Equilibria Vol. 23 No. 6 2002 473

2. Crystal Structures

The phase diagram shows two primary solid solutions,the (�Cu) primary solution exhibits a large indium solubil-ity, up to 11 at.% In, while the (In) phase remains very closeto pure indium. Five intermediate phases: �, �, �, ��, andCu11In9 are well established. The �� phase exhibits one ormore variants at low temperature whose crystal structuresare not known. Table 1 summarizes various symbols andcrystal structures of the solid phases.

3. Thermodynamic Data for the Liquid Phase andModeling Strategy

The thermodynamic data concerning different Cu-Inphases have been collected in the work of [1989Sub]

along with the invariant reaction data and the experimentaltie lines.

According to Kang et al. [1976Kan] the enthalpy of mix-ing in the liquid phase varies strongly with temperatures.The literature gives different values for this quantity: forexample, at the composition xIn � 0.25 different publisheddata for the enthalpy of mixing vary from −2000 to −7000J/mol. It is clear that this binary liquid phase has short-rangeordering at lower temperatures, (just above the liquidus),and that this order gradually disappears at higher tempera-tures. Extrapolations of the liquid enthalpy data at high andlow temperatures need high precautions: Kao et al.[1993Kao] proposed a complete thermodynamic descriptionof the (Cu-In) system, optimized with the [1977Luk] pro-gram. The liquid phase was described with a constant excessCp vs temperature but varying with composition; this is

Table 1 Symbols and Crystal Structures of the Stable Solid Phases in the (Cu-In) Alloys

DiagramSymbol

OtherSymbol

Symbol Used inThermo-Calc Data File

PearsonSymbol

SpaceGroup

StrukturberichtSymbol Type Observations

�Cu fcc FCC_A1 cF4 Fm3m A1 Cu� bcc BCC_A2 cI2 Im3m A2 W� CUIN_GAMMA cP52 P43m InMn3 or

Al4Cu9

Distorted �-brassstructure

� CU7IN3 aP40 P1 Cu7In3 Ordered �-brassstructure

�� ETA-HT hP4 P63/mmc B81 NiAs� Not optimized in the present

work one or more variantshP6 ? P63/mmc? or

orthorhombic?B82? InNi2? Structure uncertainty

Cu11In9 Cu11In9 mC20 C2/m AlCu(In) TET_A6 tI2 I4/mmm A6 In

Fig. 2 The updated version of experimental Cu-In phase diagram [1994Oka]

Section I: Basic and Applied Research

474 Journal of Phase Equilibria Vol. 23 No. 6 2002

unrealistic. Therefore, the Kao et al. [1993Kao] descriptionshould not be used as the binary border in a multicomponentsystem calculation. As an example, this model gives a varia-tion of the enthalpy of mixing of the Cu.30In.70 liquid phaseof more than 3500 J/mol between 955 and 1272 K while[2000Dic] measured only 2260 J/mol in the same interval.Such a discrepancy between the variations of the mixingenthalpy of the liquid phase will become more and morepronounced with increasing the temperature interval. On theother hand, the bcc-� phase of the [1993Kao] model is a linecompound, which is incompatible with other binaries (likeAg-Zn, Cu-Sn, Cu-Zn...) or multicomponent systems wherethe bcc-� phase exhibits a wide composition range of so-lution. For these reasons a new optimization of the Cu-Insystem is needed.

It is easier to describe the liquid phase without any ex-cess Cp, adding the total order-disorder transformation en-thalpy of the liquid to the melting enthalpy of the solidphases. This means that the model for the liquid phase willrepresent an extrapolation of the high temperature disor-dered liquid phase. Moreover in the calorimetric experimen-tal work of Kang et al. [1976Kan], used by Kao [1993Kao]for his optimization, the variation of the liquid enthalpy ofmixing with temperature appears to be overestimated, andby comparison with the other experimental results some-what unlikely. In the optimization of the parameters for theliquid phase we have used the suggested enthalpy values ofHultgren et al. [1973Hul]. These values are based on theexperimental calorimetric results of Kleppa [1956Kle] at1073 K and of Itagaki and Yazawa [1971Ita] at 1373 K,which are very close and are also very close to the results ofKang et al. [1976Kan] at 1229 K. We have also used theindium activity data measured by Kang et al. [1977Kan]

(EMF technique using a zirconia’s solid electrolyte from1000-1400 K), and the indium activities at 1073 K selectedby Hultgren et al. [1973Hul], which are based on the EMFexperimental work of Azakami and Yazawa [1969Aza]. The

Fig. 3 The enthalpy of mixing (in J/mol) of the liquid (Cu-In)phase optimized in the present work using no excess Cp in themodel. Comparison with various experimental data: the Hultgrenet al. [1973Hul] compilation (measurements of Kleppa [1956Kle]and Itagaki and Yazawa [1971Ita] at 1073 K) and measurement ofKang et al. [1976Kan] at 1229 K

Fig. 4 The indium activity in the liquid phase calculated at 1073K with the present description and compared with experimentaldata by Azakami and Yazawa [1969Aza] (selected in Hultgren etal. [1973Hul]) and by Kang et al. [1977Kan]

Fig. 5 Comparison between the EMF/mV values calculated us-ing the present description (without excess Cp) and the measure-ment of Kang et al. [1977Kan]. The agreement is good for xIn >0.3, the mean discrepancy remains < 1 mV. For xIn < 0.21 weobserve a systematic deviation of 10% but it is difficult to con-clude whether this deviation is linked to any experimental error(like a small systematic loss of indium) or due to the model



optimized coefficients of all the phases are listed in theAppendix. Figure 3 shows the calculated enthalpy of mixingfor the liquid phase (independent of the temperature) to-gether with the experimental values selected by Hultgren[1973Hul] at 1073 K and the experimental values at 1229 Kof Kang [1976Kan]. Figure 4 shows the calculated indiumactivity in the liquid phase at 1073 K together with theexperimental values selected by Hultgren [1973Hul] ac-cording to Azakami [1969Aza] at 1073 K and the experi-mental values at 1073 K of Kang [1977Kan]. In Fig. 5,calculated EMFs for various alloys are compared with ex-perimental data due to Kang et al. [1977Kan]. For 0.331 <xIn < 0.802 the average difference between the model andthe experimental values is less than 1 mV. For xIn < 0.210we observe a systematic error of 10%, but it is well knownthat the EMF uncertainty increases at low concentration ofthe ionizable element.

After the optimization of the liquid phase the parametersfor the liquid were fixed for the optimization of the coeffi-cients of the different solid-phase descriptions. This precau-tion was taken while the thermodynamic Gibbs function ofthe liquid phase was well established and suitable to scalethe complete thermodynamic description of the otherphases.

4. Thermodynamic Data for the Solid Phases

The enthalpy of formation of the �, �, �, and � solidsolutions have been measured calorimetrically by variousauthors. In the present optimization we have used the en-thalpies of formation of the �−Cu7In3 and � phases mea-sured at 723 K by Vinakurova and Geiderick [1976Vin] andfor the � phase their result [1976Vin] is very close to that ofKleppa [1956Kle] at the same temperature. We have alsoused the experimental enthalpy of formation at 903 K of the� phase measured by Kang et al. [1976Kan]. The enthalpyof formation of the �-Cu phase can be considered to beclose to zero according to Kleppa [1956Kle] and the enthal-py of formation of the � phase is also small [1976Kan](maybe negative). The �-Cu and � phases have been mod-eled as substitutional disordered solutions while the otherintermediate phases have been described as ordered phaseswith two sublattices. A crystallographically correct descrip-tion of the � phase involves four sublattices [2000Ans].Nevertheless, this � phase occurs only in a small area of the

Fig. 6 The calculated Cu-In phase diagram according to the pres-ent work

Fig. 7 Comparison of the experimental enthalpy of the Cu0.7In0.3

alloy (in kJ/mol) measured by Bahari et al. [1999Bah] and the onecalculated from the present work. There is a good agreement of thetwo results in the solid � and � domains and also at high tempera-ture for the liquid phase. The curvature of the experimental lineabove the liquidus is linked to the disordering of the liquid whilethe model exhibits a straight line in this range



phase diagram and consequently, only few experimentalstudies concern this phase. This is the reason why we havechosen to describe the � phase with only two sublattices.We have taken the �-Cu7In3, and the Cu11In9 phases asstoichiometric compounds, (see Appendix). The low tem-perature � phase has not been considered here. The entropycontribution of all the solid phases has been fitted using thephase diagram data.

5. Thermodynamic Model

The unary Cu and In are described using the SGTEfunctions0G�

Cu and 0G�In [1991Din]. The disordered phases,

liquid, FCC-A1, BCC-A2, are modeled as single-lattice ran-dom solutions, using Redlich Kister polynomials:

Gm� � xCu

0GCu� � xln

0G ln� � RT (xCuln xCu � xlnln xln)

� xCuxln�n�0

nn LCu,ln

� (xCu − xln)n

The TET_A6 primary indium solution is considered as puresolid indium.

All the other intermediate phases are modeled using ap-propriate sublattice models [1970Hil] and [1981Sun].

6. Results and Discussion

Figure 6 shows the calculated phase diagram. Theadopted strategy (without excess Cp in the liquid phase)cannot reproduce exactly the shape of the liquidus at lowtemperature. The peritectic (� ↔ � + L) temperature iscalculated at 982 K (experimental 983 K) and the (� ↔ �+ �) eutectoid temperature at 852.5 K (experimental 847 K).All the other invariant temperatures in the composi-tion range xIn ∈ (0.25-0.35) exhibit a mean discrepancyof ±15 K.

Table 2 shows the calculated enthalpies for the solidphases in comparison with different experimental valuesfrom literature. The consistency between the model and theexperimental enthalpies of formation of the solid phases isgenerally good except for the � phase with a discrepancy of∼20%.

7. Discussion of the Liquid Model

It is clear that the disordered model of the liquid phasethat was used in this work is not satisfactory at lower tem-peratures (T < 900 K) and low copper content (xCu < 0.4).Nevertheless, it allows a complete calculation of the phasediagram and provides a correct thermodynamic representa-tion of the Gibbs function of the liquid above 1000 K, usefulfor various extensions to multicomponent calculations. Ac-cording to Kang et al. [1976Kan] the greatest excess Cp inthe liquid phase is observed by calorimetry in the concen-tration range 0.2 < xIn < 0.3. Recently [1999Bah], measuredthe heat content of the Cu7In3 alloy with high accuracy fromroom temperature up to 1272 K. It is interesting to comparethe experimental heat content of this alloy to the model (Fig.7). In the solid � or � domains there is no significant dis-crepancy between the model and the experimental data. Forthe liquid phase there is an important difference: the calcu-lated curve is a straight line, while the experimental oneexhibits a curvature between the liquidus and ∼1000 °C. Athigh temperature the two curves become very close. Thismeans that the disordering enthalpy of the liquid phase,which occurs between the liquidus and ∼1000 °C is directlyadded by the model to the melting enthalpy of the � phase.According to Bahari et al. [1999Bah] the measured enthalpyof melting of the � phase at 955.5 K is �Hmelt(�, xIn �0.300) � 8340 ± 168 J/mol; while the model gives�Hmelt(�, xIn � 0.300) � 11 325 J/mol. It also means thatthe total disordering enthalpy of the liquid phase for xIn�0.3 is less than 3000 J/mol, which is lower than the value4472 J/mol measured by Kang et al. [1976Kan] between955 and 1348 K for the same composition.

Table 2 Calculated Enthalpies for the Solid Phases in Comparison With the Literature Values (Reference StatesCu*FCC and In*TET_A6)

Literature (J/mol) Reference Calculation From the Model (J/mol)

�Hf[�, 903 K, x(In) � 0.2086] � −431 [1976Kan] �Hf[�, 903 K, x(In) � 0.2086] � −48�Hf[�, 941 K, x(In) � 0.3087] � −3874 [1976Kan] �Hf[�, 935 K, x(In) � 0.3087] � −4218�Hf[�, 723 K, x(In) � 0.306] � −8745 [1976Vin] �Hf[�, 723 K, x(In) � 0.300] � −8520�Hf[�, 298 K, x(In) � 0.300] � −8620 [2000Dic] �Hf[�, 298 K, x(In) � 0.300] � −8520�Hf[�, 723 K, x(In) � 0.300] � −8535 [1956Kle] �Hf[�, 723 K, x(In) � 0.300] � −8520�Hf[�, 903 K, x(In) � 0.343] � −6636 [1976Kan] �Hf[�, 903 K, x(In) � 0.343] � −5837�Hf[�, 903 K, x(In) � 0.348] � −6402 [1976Kan] �Hf[�, 903 K, x(In) � 0.348] � −5916�Hf[�, 773 K, x(In) � 0.352] � −7548 [1976Vin] �Hf[� + �, 773 K, x(In) � 0.352] � −7055�Hf[�, 298 K, x(In) � 0.360] � −8211 [2000Dic] �Hf[� + �, 298 K, x(In) � 0.360] � −7519�Hf[�, 723 K, x(In) � 0.390] � −7382 [1976Vin] �Hf[�, 723 K, x(In) � 0.390] � −6540�Hf[�, 723 K, x(In) � 0.390] � −7531 [1956Kle] �Hf[�, 723 K, x(In) � 0.390] � −6540



Appendix: The Thermo-Calc DescriptionElement Stable Reference Mass H298-H0 S298

VA VACUUM 0.0E+00 0.00E+00 0.00E+00CU FCC_A1 6.3546E + 01 5.0041E + 03 3.3150E + 01IN TETRAGONAL_A6 1.1482E + 02 6.6100E + 03 5.7650E + 01

Liquid1 Sublattice, Sites 1 Constituents: Cu,In

0GCuLiquid − HCu

SER � G(LIQUID,CU;0) − H298(FCC_A1,CU;0)298.13 < T < 1358.00: +12 964.84 − 9.510243 · T − 5.83932E − 21 · T7 + GHSERCU1358.00 < T < 3200.00: +13 495.4 − 9.920463 · T − 3.64643E + 29 · T−9 + GHSERCU0GIn

Liquid − HInSER � G(LIQUID,IN;0) − H298(TETRAGONAL_A6,IN;0) �

298.12 < T < 429.75: +3282.152 − 7.63649 · T-5.21918E − 20 · T7 + GHSERIN429.75 < T < 3800.00: +3283.66 − 7.640174·T − 3.30026E + 22 · T−9 + GHSERIN298.13 < T < 3000.00: 0LCu,In

Liquid � L(LIQUID,CU,IN;0) � − 5323 − 2.319 � T298.13 < T < 3000.00: 1LCu,In

Liquid � L(LIQUID,CU,IN;1) � − 19 188 + 3.534 · T298.13 < T < 3000.00: 2LCu,In

Liquid � L(LIQUID,CU,IN;2) � −9975

BCC_A22 Sublattices, Sites 1: 3 Constituents: Cu,In: Va

0GCuBCC − HCu

SER � G(BCC_A2,CU:VA;0) − H298(FCC_A1,CU;0) �

298.13V < T < 3200.00: + 4017− 1.255 · T + GHSERCU0GIn

BCC − HInSER � G(BCC_A2,IN:VA;0) − H298(TETRAGONAL_A6,IN;0) �

298.13 <T < 3000.00: + 800 − 0.8·T + GHSERIN298.13 < T < 3000.00: 0LCu,In

BCC � L(BCC_A2,CU,IN:VA;0) � 0298.13 < T < 3000.00: 1LCu,In

BCC � L(BCC_A2,CU,IN:VA;1) � − 43 423 − 5.2367 · T298.13 < T < 3000.00: 2LCu,In

BCC � L(BCC_A2,CU,IN:VA;2) � +13 980 + 6.97 · T

CU11IN92 Sublattices, Sites 11: 9 Constituents: Cu: In

0GCu : InCu11In9 − 11 HCu

SER − 9 HInSER � G(CU11IN9,CU:IN;0) − 11 H298(FCC_A1,CU;0) − 9 H298(TETRAGONAL_A6,IN;0) �

298.13 < T < 3000.00: − 121 000 + 0.3985 · T + 11 · GHSERCU + 9 · GHSERIN

CU7IN32 Sublattices, Sites 7: 3 Constituents: Cu: In

0GCu : InCu7In3 − 7 HCu

SER − 3 HInSER � G(CU7IN3,CU:IN;0) − 7 H298(FCC_A1,CU;0) − 3 H298(TETRAGONAL_A6,IN;0) �

298.13 < T < 3000.00: − 85217 + 23.9 · T + 7 · GHSERCU + 3 · GHSERIN

CUIN_GAMMA2 Sublattices, Sites 1: 3 Constituents: In: Cu,In

0GIn : CuGamma − 3 HCu

SER − HInSER �

G(CUIN_GAMMA,IN:CU;0) − 3 H298(FCC_A1,CU;0) − H298(TETRAGONAL_A6,IN;0) �

298.13 < T < 3000.00: −19 000 − 2.9 · T + GHSERIN + 3 · GHSERCU0GIn : In

Gamma − 4 HInSER � G(CUIN_GAMMA,IN:IN;0) − 4 H298(TETRAGONAL_A6,IN;0) �

298.13 < T < 3000.00: +40 450 + 4 · GHSERIN298.13 < T < 3000.00: 0LIn : Cu,In

Gamma � L(CUIN_GAMMA,IN:CU,IN;0) � −35 000

ETA_HT2 Sublattices, Sites 6: 4 Constituents: Cu,In: Cu,In

0GCu : CuEta − 10 HCu

SER � G(ETA_HT,CU:CU;0) − 10 H298(FCC_A1,CU;0) �

298.13 < T < 3000.00: +32 554 + 10 · GHSERCU0GCu : In

Eta − 6 HCuSER − 4 HNi

SER � G(ETA_HT,CU:IN;0) − 6 H298(FCC_A1,CU;0) − 4 H298(TETRAGONAL_A6,IN;0) �

298.13 < T < 3000.00: −68 524 + 7 · T + 6 · GHSERCU + 4 · GHSERIN0GNi : Ni

Eta − 10 HNiSER � G(ETA_HT,IN:IN;0) − 10 H298(TETRAGONAL_A6,IN;0) �

298.13 < T < 3000.00: +122 565 + 10 · GHSERIN298.13 < T < 3000.00: 0LCu : Cu,In

Eta � L(ETA_HT,CU:CU,IN;0) � −38 926298.13 < T < 3000.00: 0LCu,In : In

Eta � L(ETA_HT,CU,IN:IN;0) � 30 356



FCC_A12 Sublattices, Sites 1: 1 Constituents: Cu,In: Va

0GCuFCC − HCu

SER � G(FCC_A1,CU:VA;0) − H298(FCC_A1,CU;0) �

298.13 < T < 3000.00: +GHSERCU0GIn

FCC − HInSER � G(FCC_A1,IN:VA;0) − H298(TETRAGONAL_A6,IN;0) �

298.12 < T < 3800.00: +123 − 0.1988 · T + GHSERIN298.13 < T < 3000.00: 0LCu,In

FCC � L(FCC_A1,CU,IN:VA;0) � 0298.13 < T < 3000.00: 1LCu,In

FCC � L(FCC_A1,CU,IN:VA;1) � −1737 − 9.025 · T298.13 < T < 3000.00: 2LCu,In

FCC � L(FCC_A1,CU,IN:VA;2) � +4.8745 · T

TET_A61 Sublattice, Sites 1 Constituents: In

0GInTet_A6 − HIn

SER � G(TET_A6,IN;0) − H298(TETRAGONAL_A6,IN;0) �

298.13 < T< 3800.00: +GHSERIN

SYMBOLS:GHSERCU298.13 < T < 1358.00: − 7770.458 + 130.485403 · T − 24.112392 · T · LN(T)−0.00265684 · T2 + 1.29223E − 07 · T3 + 52478 · T−1

1358.00 < T < 3200.00: −13542.33 + 183.804197 · T−31.38 · T · LN(T) + 3.64643E + 29 · T−9

GCULIQ298.13 < T < 1358.00: +12 964.84 − 9.510243 · T − 5.83932E-21 · T7 + GHSERCU1358.00 < T < 3200.00: +13 495.4 − 9.920463 · T − 3.64643E + 29 · T−9 + GHSERCUGCUBCC298.13 < T < 3000 + 4017 − 1.255 · T + GHSERCUGHSERIN298.13 < T < 429.75: −6978.89 + 92.338115 · T − 21.8386 · T · LN(T)−0.00572566 · T2 − 2.120321E − 06 · T3 − 22906 · T−1

429.75 < T < 3800.00: −7033.47 + 124.476492 · T − 27.4562 · T · LN(T)+5.4607E − 04 · T2 − 8.367E − 08 · T3 − 211 708 · T−1 + 3.30026E + 22 · T−9

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A thermodynamic optimization of the Cu-In system