the phase equilibria in the mg–ni–ca...

Computer Coupling of Phase Diagrams and Thermochemistry 29 (2005) 289–302www.elsevier.com/locate/calphad

g–Ni–Caded to thethe liquid

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The phase equilibria in the Mg–Ni–Ca system

F. Islam, M. Medraj∗

Department of Mechanical and Industrial Engineering, Concordia University, 1455 de Maisonneuve Blvd. West, H-549, Montreal, QC, H3G 1M8, Canada

Received 10 April 2005; received in revised form 30 August 2005; accepted 31 August 2005Available online 21 September 2005

Abstract

The three binary systems Mg–Ni, Ca–Ni and Mg–Ca have been re-optimized. A self-consistent thermodynamic database of the Msystem is constructed by combining the optimized parameters of these three constituent binaries. Lattice stability values are not adpure elements Mg-hcp, Ni-fcc, Ca-fcc and Ca-bcc to construct this database. The Redlich–Kister polynomial model is used to describeand the terminal solid solution phases, and the sublattice model is used to describe the non-stoichiometric phase, in this system. Thedatabase is used to calculate the three binary and the ternary systems. The calculated binary phase diagrams along with their therproperties such as Gibbs energy, enthalpy, entropy and activities are found to be in good agreement with experimental data from theThis is the first attempt to construct the ternary phase diagram of the Mg–Ni–Ca system. The established database for this system preternary eutectic, five ternary quasi-peritectic, two ternary peritectic and two saddle points.c© 2005 Elsevier Ltd. All rights reserved.

Keywords: Mg–Ni–Ca system; Hydrogen storage; Thermodynamic modeling; Phase diagram

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1. Introduction

A metal hydride tank built from Mg–Ni–Ca alloys can methe demand for a cost effective method for storing hydrogenbecause of their higher storage capacity. This method offersdelivery of hydrogen at a constant pressure. In this processalloy absorbs and holds a large amount of hydrogen in hydform and releases it at room temperature or higher temperathrough heating the tank without changing the alloys’ structuMg and its alloys are considered to be the most promisbecause of their high capacity, light weight and low cost [1–4].However, Mg is prone to oxidation, and the hydriding adehydriding kinetics of Mg are slow due to low catalytactivity for dissociation chemisorption of hydrogen, whiclimit its potential for hydrogen storage [5]. Adding Ca and Nito Mg can solve these problems. Ni, for instance, improthe hydriding and dehydriding rates due to the improvemof catalytic activity [5,6]. Ca also has high affinities fohydrogen but forms very stable hydrides that bar its practapplication. Various binaries such as CaNi2, Ca2Ni7, MgNi2,

∗ Corresponding author. Tel.: +1 514 848 2424x3146; fax: +1 514 848 317E-mail address: [email protected] (M. Medraj).

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0364-5916/$ - see front matterc© 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.calphad.2005.08.006

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CaMg2 etc., form very stable hydrides and are thus unsuitafor practical application as hydrogen storage. However, soof the binary alloys of Mg and Ca with Ni such as Mg2Niand CaNi5 are less stable and hence can be better alternatas hydrogen storage [4]. Liang and Schulz [6] concluded that,under certain manufacturing conditions, the Mg–Ni–Ca basalloys can increase hydrogen capacity. However, exceptsome pseudobinaries, the Mg–Ni–Ca system has not binvestigated much [4] and no phase diagram of the ternarsystem is available [7].

Generally, it is necessary to have a good knowledge ofphase equilibria and thermodynamic properties of the materin order to optimize the production process and final propertof the materials. The objective of this work is to develoa thermodynamic description of the Mg–Ca–Ni systemcomplete as possible. For this purpose, this study first optimithe three constituent binaries, Mg–Ni, Ni–Ca and Mg–Cand then develops a self-consistent database. This is theattempt to construct the ternary phase diagram of this hpotential system and will provide a better understandingits alloys, which is necessary for their future technologicapplication.

http://www.elsevier.com/locate/calphad

290 F. Islam, M. Medraj / Computer Coupling of Phase Diagrams and Thermochemistry 29 (2005) 289–302

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2. Thermodynamic modeling

An equilibrium state of a system can be obtained whentotal Gibbs energy of all phases is minimized.

G =p∑

φ=1

nφGφ (1)

wherep is the number of phases,nφ is the number of moles andGφ is the Gibbs energy of phaseφ.

2.1. Unary phases

The Gibbs energy of the pure element,A, with a certainphaseφ, is described as a function of temperature by tfollowing equations:

0GφA(T ) = Gφ

A(T ) − H SERA

0GφA(T ) = a + bT + cT ln T + dT 2 + eT 3 + f T −1

+ gT 7 + hT −9 (2)

whereH SERA (the molar enthalpy of the stable element referen

(SER)) is at 298.15 K and 1 bar, andT is the absolutetemperature. The value of the coefficientsa to h are taken fromthe SGTE compilation by Dinsdale [8].

2.2. Disordered solution phases

The Gibbs energy of a disordered solution phase is descrby the following equation:

G = x A0Gφ

A +xB0Gφ

B +RT [xA ln x A + xB ln xB] + exGφ (3)

whereφ denotes the phases in question andxA andxB denotethe mole fractions of componentsA and B, respectively. Thefirst two terms on the right hand side of Eq.(3) represent theGibbs energy of the mechanical mixture of the components,third term is the ideal Gibbs energy of mixing, and the fourterm is the excess Gibbs energy. For disordered phasesexcess Gibbs energy is represented using the Redlich–Kequation:

exGφ = x A.xB

n=m∑n=0

[n LφA,B (x A − xB)n] (4)

with n LφA,B = an + bnT (n = 0, . . . , m)

wherean andbn are model parameters to be optimized usiboth experimental phase diagram and thermodynamic data

2.3. Stoichiometric phases

The Gibbs energy for stoichiometric compounds is describby the following equation:

Gφ = x A0Gφ1

A +xB0Gφ2

B +�G f (5)

with �G f = a + b × T

wherex A andxB are the mole fractions of componentsA andBand0Gφ1

A , 0Gφ2B represent the Gibbs energy of a component

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its standard state. However, the Gibbs energy of the compophase may refer to a different crystal structure from thosethe pure elements,φ1 and φ2. �G f is the Gibbs energy offormation per mole of atoms of the stoichiometric compouthe parametersa andb are obtained by optimization using phaequilibria and thermodynamic data.

2.4. Ordered intermetallic phase

The Gibbs energy of an ordered binary solution phasedescribed by:

G = Gref + G ideal+ Gexcess (6)

Gref =∑

ylA ym

B . . . yqF

0G(A:B:...:F) (7)

G ideal = RT∑

l

fl

∑A

ylA ln yl

A (8)

Gexcess=∑

ylA yl

B . . . ymF

∑γ=0

γ L(A,B):F ×(ylA − yl

B)γ (9)

whereA, B, . . . , F represent components or vacancy;l, m andq represent sublattices;yl

A is the site fraction of componenA on sublatticel; fl is the fraction of sublatticel relativeto the total lattice sites.0G(A:B:...:F) represents a real orhypothetical compound energy.L(A,B):F is the interactionparameter describing the interaction within the sublattice.

In this study only one terminal solid solution phasemodeled, this is the Mg-hcp phase in the Mg–Ca system.other terminal phases are considered as stoichiometric phthese are: Ca-fcc, Ca-bcc, and Ni-fcc. For these stoichiomephases the Gibbs energy of the corresponding GHSER funcfrom Dinsdale is used.

In this work, the computer program WinPhaD [9] is used foroptimizing the binary sub systems Mg–Ni, Ni–Ca and Mg–C

3. Mg–Ni System

3.1. Phase diagram

Voss [10] was the first researcher who investigated the phequilibria of the Mg–Ni system in 1908 by thermal analysand reported the liquidus points in the composition ran0.04 < XNi < 0.98. But in his work, the purity of Mgwas not specified and the purity of Ni was low (97.7 wt%Moreover, the homogeneity of the mixture was not obtainLater, further investigations was performed by [11–15]. Amongthem, the phase equilibria were investigated by [10–12,14] andthe melting point of intermediate phase MgNi2 was determinedby [10,13,14]. Haugton and Pyne [12] determined the liquidustemperature more accurately on the Mg-rich end(0 ≤ XNi≤ 0.34) by thermal analysis (heating and cooling curves) wpurity of elements and homogeneity of mixture. Bagnoud aFeschotte [13] investigated the homogeneity range of MgN2by means of XRD, metallography, electron microprobe analyand DTA. They mentioned that the homogeneity range extefrom 66.2 at.% Ni at the termination of the peritectic linto 67.3 at.% Ni at the starting point of the eutectic linon the Ni-rich side. From the previous studies it has be

F. Islam, M. Medraj / Computer Coupling of Phase Diagrams and Thermochemistry 29 (2005) 289–302 291

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observed that the liquidus temperature in the compositrange 0.30 < XNi < 0.60 cannot be fixed with certainty duto discrepancies in the experimental data. Thus, Micke aIpser [15] determined the magnesium vapor pressure over liqMg–Ni alloys in the composition rangeXMg > 0.65 by theisopiestic method and they obtained the shape of the liquicurve between 0.30 < XNi < 0.40 from their measurementsDifferent researchers reported different solid solubility betwethe two end members. Among them, Haughton and Pyne [12]mentioned that the solubility of Ni-fcc in Mg-hcp is less tha0.04 mol% Ni at 773 K and Merica and Waltenberg [11]reported that the solubility of Mg-hcp in Ni-fcc is less tha0.2 mol% Mg at 1373 K. In the present work, terminal solsolutions are not considered during optimization, as exvalues are not given. Moreover, the ferromagnetic behavof Ni is not included in the optimization as Wollam anWallace [16] and Buschow [17] disputed the ferromagneticbehavior. They investigated the system by heat capacitymagnetic susceptibility measurements and did not find aanomaly in magnetic susceptibility of MgNi2 or heat capacityat any temperature.

Nayeb-Hashemi and Clark [18] calculated the phase diagramby thermodynamic modeling. Though their work adequatdescribed the phase equilibria and the thermodynamic datathermodynamic description for the MgNi2 Laves phase was nomentioned. In order to provide a more reliable thermodynamdescription, Jacob and Spencer [19] re-optimized the systemComparison between the work of Jacob and Spencer [19] andthe current investigation is given inSection 3.3.1.

The phases present in the Mg–Ni system are: the liqphase, the stoichiometric compound Mg2Ni-C16 type structure,the MgNi2-C36 hexagonal Laves structure with narrohomogeneity range and the terminal solid solution phases Mhcp, Ni-fcc with insignificant solubility.

3.2. Thermodynamic properties

Sommer et al. [20] determined the enthalpy of mixing byhigh-temperature calorimetry at 980 and 1120 K. Feufel aSommer [21] measured the integral enthalpy of mixing bycalorimetric method at 1002 and 1008 K. Both of them [20,21] reported consistent experimental data only up to ab74 at.% Mg because of the experimental limitation causedthe relatively high pressure of Mg. From the measured integenthalpy of mixing by Feufel and Sommer [21], the partialenthalpy of mixing of Ni between 2 and 17 at.% Ni was derivby the intercept method. Micke and Ipser [15] obtained thepartial enthalpy of mixing of Mg in the liquid by the isopiestimethod. Reasonable agreement was found between their reand those of Tkhai and Serebryakov [22] in the compositionrange XNi ≤ 0.30. Hultgren et al. [23] also measured thepartial enthalpy of Mg and their results are considerably mnegative than those of Micke and Ipser [15]. Hultgren et al. [23]and Sieben and Schmahl [24] measured the thermodynamiactivity of Mg and Ni from Mg vapor pressure. Micke anIpser [15] determined the activity of Mg at several temperaturusing the isopiestic method. The results of Hultgren et al. [23]

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and Micke and Ipser [15] were found to be consistentThe heat capacity of MgNi2 was measured by Wollam anWallace [16] in the low temperature range 10.6–558 K. Feufand Sommer [21] also determined the heat capacity of thtwo intermetallic phases. Several researchers used diffemethods to find out the enthalpy and entropy of formationMgNi2 and Mg2Ni compounds. Among them, Schmahl anSieben [25] obtained data from vapor pressure measuremenMg over solid Mg–Ni alloys using the transpiration methoSmith and Christian [26] measured vapor pressure using thKnudsen effusion method, King and Kleppa [27] used tinsolution calorimetry, Lukashenko and Eremenko [28] used emfmeasurements. King and Kleppa [27] mentioned that, fromthe appearance of the phase diagram it is expected thatformation of MgNi2 intermediate phase is more exothermthan that of Mg2Ni. The thermodynamic formation propertieof the two intermediate phases measured by LukashenkoEremenko [28] can be considered the most reliable as they usthe emf method and their results are comparatively most recand consistent with other experimental results.

3.3. Optimized phase diagram and thermodynamic properties

3.3.1. Phase diagramMost recently, Jacob and Spencer [19] re-optimized the sys-

tem and reported the optimized parameters. But their paraters cannot be used in this study, as they considered latticebility values for pure elements and this study does not consany lattice stability value for pure components in Mg–Ca aNi–Ca binaries in order to maintain consistency. AccordingHari Kumar and Wollants [29], it is quite necessary to considethe common set of Gibbs energy descriptions for unary stems when different binaries have to be combined to form ladatabase. Similarly, the thermodynamic description of substems, which is a part of the higher order systems, must be sif the higher order systems are to be combined. Also, Jaand Spencer [19] used too many parameters to optimize thesystem. As physically sound models are more informative wfewer adjustable parameters to fit the experimental data [29],this system is re-optimized with a simpler model and smanumber of model parameters to reproduce the phase diagand thermodynamic properties.

In the present work, for the optimization of the Mg–Ni phadiagram, the experimental data of Haughton and Payne [12],Bagnoud and Feschotte [13], Micke and Ipser [15] and afew data from Voss [10] in the composition range(XNi >

60 at.%), have been considered. This is because the liquipoints reported by Voss in the liquid-Mg and liquid-MgNi2regions do not enable the reproduction of the compositionthe eutectic point precisely. Moreover, between 30 and 60 aNi the reported liquidus values are much higher than the valfrom other researches. The reference state of the Gibbsenergy of formation for Mg2Ni and MgNi2 compounds areconsidered as Mg-hcp and Ni-fcc and lattice stability valuesnot added to Mg-hcp and Ni-fcc phases. The optimized phdiagram (Fig. 1) shows very good agreement with experimenand calculated works mentioned in the literature. To mo


Fig. 1. Calculated Mg–Ni phase diagram with data reported in the literature.

Table 1Crystal structure and lattice parameters of MgNi2 [31]

Structure type MgNi2Pearson symbol hP24Space group P63/mmc, No 194Lattice parameter (nm) a = 0.4824, c = 1.5826Angles α = 90, β = 90, γ = 120

Atoms WPa CNb PScAtomic position

Mole fraction Combiningmole fraction

X Y Z

Mg1 4e 16 3m 0 0 0.09400 0.1660.33

Mg2 4f 16 3m 1/3 2/3 0.84417 0.166

Ni1 4f 12 3m 1/3 2/3 0.12524 0.1660.67Ni2 6g 12 2/m 1/2 0 0 0.250

Ni3 6h 12 mm2 0.16429 0.32858 1/4 0.250

a WP= Wyckoff position.b CN = Coordination number.c PS= Point symmetry.

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non-stoichiometric phase MgNi2 in this system, the followingprocess has been followed during optimization.

3.3.2. Thermodynamic modeling of the MgNi2 as non-stoichiometric phase

In the Mg–Ni system, the intermediate phase MgNi2 isfirst modeled as a stoichiometric compound, since, fromCALPHAD point of view, it is a good strategy to start thoptimization using a simple model with fewer parameters [29].Once the satisfactory thermodynamic description is obtainthe model for the MgNi2 phase is changed to a more appropriasublattice model. Hari Kumar et al. [30] mentioned that whileimplementing the sublattice model, two important aspemust be considered: the crystallographic information and

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solubility range of the phase. The mixing characteristicscommensurate with the data on the homogeneity range andcrystallographic information is mainly required for deciding thnumber of sublattices to be used and for assigning constituspecies to each of them [30]. The crystallographic and the siteoccupancy data for MgNi2 phase have been collected from [31]and shown inTable 1and the homogeneity range is considerfrom the work of Bagnoud and Feschotte [13]. In the unitcell of MgNi2 phase, atoms are distributed among the ficrystallographically distinct lattice sites in the ratio 2:2:2:3with 16, 16, 12, 12, 12 coordination number, respectively,shown inFig. 2.

Here the larger Mg atoms prefer sites with higher CN (1whereas the smaller Ni atoms prefer lattice site with lower C


Fig. 2. The sub-structures in the MgNi2 crystal: (a) Mg1, (b) Mg2, (c) Ni1, (d) Ni2 and (e) Ni3.

ee

totnaeorti

museimeh

,

the

antNi

y

iel

(12). An intermediate phase usually has ideal stoichiomewhere each sublattice is occupied with only one constituspecies [29]. Hence, at first, the sublattice model for the Lavphase MgNi2 can be represented as five sublattices:

(Mg1)2 : (Mg2)2 : (Ni1)2 : (Ni2)3 : (Ni3)3

According to Hari Kumar and Wollants [29] the numberof sublattices can be reduced by grouping sublattices togewith similar crystallographic characteristics such as same psymmetry criteria and/or same coordination number. Assimplified version is more practical from a modeling poiof view, the sublattices of Mg1 and Mg2 are combined,they have the same coordination number and point symmcriteria, and considered in sublattice I with mole fracti0.3333. Similarly, the sublattices of Ni1, Ni2 and Ni3 acombined in sublattice II, as they have same coordinanumber, with mole fraction 0.6667.

(Mg)4 : (Ni)8

(Mg)1 : (Ni)2

(Mg%, Ni)0.3333 : (Mg, Ni%)0.6667

In reality, in order to model the homogeneity range, sosublattices should be allowed to mix. The presence of structdefects in certain lattice sites may cause the mixing of specieone or more sublattices for the phase with narrow homogenrange [30]. To model the narrow homogeneity range of MgN2phase, the substitution of Mg atoms on Ni sites and Ni atoon Mg sites is considered as the only defect in this study. Hthe ‘%’ denotes the major constituent of the sublattice. Tmodel of two sublattices covers the whole composition ranand therefore the homogeneity range of 0.665≤ XNi ≤ 0.676could be obtained for the MgNi2 phase from this model. Hencethe Gibbs energy per mole of formula unit can be written as

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in

Table 2The optimized model parameters for the liquid and intermetallic phases inMg–Ni system

Phase Term a (J/g atom) b (J/g atom K)

Liquid L0 −50 899.73 21.586L1 −15 084.82 10.784

Mg2Ni �G f −16 208.80 5.403MgNi2 G(Mg:Mg) 8333.333 12.667

G(Mg:Ni) −21 665.00 7.160G(Ni:Mg) 0 0G(Ni:Ni ) 5466.666 6.733

Eq.(10).

Gm = yIMgyII

Mg0G

MgNi2Mg:Mg +yI

MgyIINi

0GMgNi2Mg:Ni

+ yINi y

IIMg

0GMgNi2Ni:Mg +yI

Ni yIINi

0GMgNi2Ni:Ni

+ RT

(0.333

Ni∑i=Mg

yIi ln yI

i + 0.666Ni∑

i=Mg

yIIi ln yII

i

)

+ yIMgyI

Ni(yIIMg

0LMgNi2Mg,Ni:Mg +yII

Ni0L

MgNi2Mg,Ni:Ni)

+ yIIMgyII

Ni(yIMg

0LMgNi2Mg:Mg,Ni +yI

Ni0L

MgNi2Ni:Mg,Ni)

(10)

The optimized parameters and comparison of the invaripoints between current results and other works on the Mg–system are shown inTables 2and3, respectively.

3.3.3. Thermodynamic propertiesDuring optimization, the enthalpy of mixing reported b

Sommer et al. [20] and Feufel and Sommer [21], partial en-thalpies of mixing obtained by [15,21,23], the activity valuesof [15,23] and partial Gibbs energies reported by [24,32] areconsidered. The calculated enthalpy of mixing of liquid Mg–Nat 1008 K inFig. 3(a) shows very good agreement with Feuf

mmedraj

Cross-Out

mmedraj

Inserted Text

mole

mmedraj

Cross-Out

mmedraj

Inserted Text

mole


k

rk

k

Fig. 3. Calculated (a) enthalpy of mixing, (b) activities, (c) partial Gibbs free energy of Mg and Ni, and (d) partial enthalpy of mixing of Mg and Ni alloy in relationwith experimental results available in literature.

Table 3Comparison between calculated and different works of the Mg–Ni system reported in the literature

Reaction Temp. (K) (XNi -liquid) (XNi -Mg2Ni) (XNi -MgNi2) Reference

Eutectic 781.53 0.1048 0.333 – This worL → Mg-hcp+ Mg2Ni 781.0 0.113 0.333 – [12]

779.0 – – [13]785.0 0.176 0.333 – [10]

Peritectic 1031.0 0.291 0.665 0.333 This woL + MgNi2 → Mg2Ni 1033.0 0.290 0.667 0.333 [12]

1033.0 – 0.662 0.333 [13]1041.0 0.260 0.667 0.333 [10]

Eutectic 1377.7 0.797 0.676 This worL → Ni-fcc + MgNi2 1370.0 – 0.673 [13]

1355.0 0.772 – 0.667 [10]

Melting point 1422.84 0.666 – This workMgNi2 → L 1420± 3 – – [13]

ithon

and Sommer [21]. It also shows reasonable agreement wSommer et al. [20]. The calculated activities of Mg and Ni in

the liquid phase at 1073 K are shown inFig. 3(b). The activityof Mg and Ni in the liquid phase has shown negative deviati


Table 4Calculated enthalpy and entropy of formation of Mg2Ni and MgNi2 compounds with the reported data in the literature

Mg2Ni MgNi2 Methoda

(�H f ) (�S f ) (�G f ) (�H f ) (�S f ) (�G f ) [Reference](kJ/g atom) (J/g atom K) (kJ/g atom) (kJ/g atom) (J/g atom K) (J/g atom)

−16.20 −5.40 −14.73 −21.55 −7.11 −19.61 Cal. [This work]−16.13 −4.96 −14.78 −21.47 −6.65 −19.65 emf [28]−20.06 −9.20 −17.54 −25.92 −10.89 −22.95 Vap. press. [25]−13.59 – – −19.23 – – Soln. calo. [27]−19.23 −4.32 −18.05 −20.62 −2.79 −19.86 Vap. press. [26]−14.20 – – −21.073 – – Soln. calo. [21]

a Cal.: Calculated, Vap. press.: Vapor pressure and Soln. calo.: Solution calorimetry.

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from ideal behavior in the whole composition range. The callated activity of Mg shows a good agreement with experimetal data of Hultgren et al. [23] and Micke and Ipser [15]. Forthe activity of Ni, however, not many experimental results areported in the literature. The calculated activity of Ni showvery good agreement with the data of Hultgren et al. [23]. Thecalculated partial Gibbs energies of Mg and Ni are shownFig. 3(c). There is very good agreement between the calculavalues and experimental data of Sieben and Schmahl [24] andSryvalin et al. [32] for the partial Gibbs energy of Mg. Thepartial enthalpy of mixing of the two components is shownFig. 3(d). The partial enthalpy of Ni shows very good agrement with Feufel and Sommer [21]. There is also good agreement of enthalpy of mixing of Mg with Hultgren et al. [23] andMicke and Ipser [15].

The calculated enthalpy and entropy of formation of the twMgNi2 and Mg2Ni intermetallic compounds show are listein Table 4 and very good agreement with Lukashenko aEremenko [28]. They obtained the emf values using a moltesalt galvanic cell. There is a reasonable agreement betwthe current work and Schmahl and Sieben [25] and Kingand Kleppa [27] who determined the values by transportatiovapor pressure technique and solution calorimetry methrespectively. On the other hand, the enthalpy and entropyformation of the MgNi2 compound show excellent agreemenwhereas the results for the Mg2Ni compound show reasonablagreement with the works of Feufel and Sommer [21] andSmith and Christian [26] who used solution calorimetry andthe Knudsen effusion vapor method, respectively. Howevif the Gibbs energy of formation of the MgNi2 compoundis considered, the deviation from the experimental resultsSmith and Christian [26] is less significant as can be seenTable 4.

4. Ca–Ni system

4.1. Phase diagram

Takeuchi et al. [33] proposed a phase diagram witcongruent CaNi5 and incongruent Ca2Ni5 linear compoundsin the Ca–Ni system first. Later, Buschow [34] determinedthe existence of the four intermetallic compounds wcrystal structure CaNi2-cubic, CaNi3-rhombohedral, Ca2Ni7-rhombohedral and CaNi5-hexagonal by X-ray diffraction.The Ca2Ni5 compound was not found in the investigatio

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d-en

d,of,

r,

of

of Buschow [34]. Later, Saindrenan et al. [35] and otherresearchers indicated that CaNi5 melts incongruently throughperitectic transformation. Based on this suggestion aexperimental points related to the other three compouproposed by Notin and Hertz [36], a new version of the Ca–Nphase diagram was published by Okomoto [37]. However,Notin et al. [38] investigated the possibility of congruent oincongruent melting nature of CaNi5 by series of experimentsand suggested the congruent melting nature of this compoand a eutectic point between CaNi5 and Ni-fcc. But intheir work, the linear compound CaNi5 terminated at around850 K, which contradicts many of the previous publishphase diagrams where all the intermetallic compoundsstable down to room temperature. Besides, they [38] didnot consider the phase transformation of Ca-fcc to Ca-at 716 K. Hence, reassessment of this system is requwhich will reproduce the most acceptable phase diagramthermodynamic properties.


For the Ni–Ca system, the experimental data in the literatis scarce. The experimental results on enthalpy of mixing wreported by Sommer et al. [39] at 0 ≤ XNi ≤ 37 at.% and havebeen used in the present work. Notin et al. [38] calculated theenthalpy of mixing. But, Predel [40] mentioned a discrepancin the enthalpy of mixing between data of Notin et al. [38]and Sommer et al. [39]. Meysson et al. [41] reported partialactivity values of calcium, which did not match with the typicform of a partial activity curve. But the data points extractedHultgren et al. [42] for partial activity of Ca produced a feasiblcurve and were used to determine the partial activity of Niusing the Gibbs–Duhem equation — these activity data are uby this study for optimization. The partial Gibbs energies wealso calculated by Hultgren et al. [42] from the partial activitydata of Meysson and Rist [41]. Notin and Hertz [36] reportedthe enthalpy, entropy and Gibbs energy of formation ofcompounds in the Ca–Ni system.


4.3.1. Phase diagramThe calculated Ca–Ni phase diagram in comparison w

experimental data from the literature is shown inFig. 4.

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k

mmedraj

Sticky Note

units are wrong. Please see corrections.

mmedraj

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Inserted Text

mole

mmedraj

Cross-Out

mmedraj

Inserted Text

mole

mmedraj

Cross-Out

mmedraj

Inserted Text

mole

mmedraj

Cross-Out

mmedraj

Inserted Text

mole

mmedraj

Cross-Out

mmedraj

Inserted Text

mole

mmedraj

Cross-Out

mmedraj

Inserted Text

mole


Fig. 4. Calculated Ca–Ni phase diagram with experimental results from the literature.

Table 5Comparison of experimental [38] and calculated invariant points of the Ca–Ni system

Type Reaction XNi -liquid Temp. (K) Reference

Eutectic Liquid→ Ca-bcc+ CaNi2 0.160 896.0 [38]0.170 900.0 This work

Eutectic Liquid→ Ni-fcc + CaNi5 0.880 1459.0 [38]0.877 1460.1 This work

Peritectic Liquid+ CaNi3 → CaNi2 0.390 1138.6 [38]0.378 1145.0 This work

Peritectic Liquid+ Ca2Ni7 → CaNi3 0.541 1287.2 [38]0.561 1280.1 This work

Peritectic Liquid+ CaNi5 → Ca2Ni7 0.715 1400.1 [38]0.715 1400.2 This work

Congruent Liquid→ CaNi5 0.832 1477.3 [38]0.834 1478.9 This work

e[hidys,,rk

thC-bNi-fs

mic

nta

es

eryareymse

The uncertainty is considered±0.02-mole fraction for allpoints in the calculated phase diagram. A good agreembetween calculated and experimental values of Notin et al.38]is found in Fig. 4. But the values reported by Takeucet al. [33] contradict with the calculated values in this stubecause they [33] reported only two intermetallic compoundCaNi5 and Ca2Ni5, instead of four. For all critical pointsthe comparison between calculated and experimental woshown inTable 5.

The reference states of the Gibbs energy of formation forfour intermetallic compounds are considered as Ni-fcc andbcc and lattice stability values are not added to Ca-fcc, Caand Ni-fcc phases. In this calculation, the stability of the Ca5

compound at room temperature and transformation of Cato Ca-bcc are evident. The optimized parameters for all pha

nt

is

ea-cc

cces,

which have reproduced the phase diagram and thermodynaproperties, are shown inTable 6.

4.3.2. Thermodynamic propertiesThe calculated enthalpy of mixing shows negative deviatio

towards the Ni rich corner and agrees well with the dareported by Sommer et al. [39] as shown inFig. 5(a). Thecalculated activities of Ca and Ni are shown inFig. 5(b),which also shows comparison with the experimental valuof Hultgren et al. [42]. Although a deviation exists betweencalculated and experimental data, an exact match is often vdifficult to obtain. Moreover, only one set of experimental datis available for comparison with the calculated curve. Theis a possibility that the deviation is within the uncertaintlimits of the measured values. It is unusual that a systewith such negative enthalpy has activity plots, which are clo


perimal data
Fig. 5. Calculated (a) enthalpy of mixing, (b) activities, (c) entropy of mixing, and (d) partial Gibbs free energy of Ni and Ca in relation with exentavailable in literature.
sf

aminanlto

es

ea

he

ed

to

to ideal. This means that the excess entropy just happennearly balance out the excess enthalpy at the temperature oactivity measurements (1750 K). It is evident fromFig. 5(c),that the calculated entropy of mixing of the liquid phase1750 K shows a low value in the Ni-rich part of the diagraNo experimental data for the entropy of mixing were foundthe literature. The calculated partial Gibbs energies of CaNi (Fig. 5(d)) show good agreement with experimental resuof Hultgren et al. [42]. These values are predicted and are nincluded in the optimization.

The calculated Gibbs energy of formation(�G f ) for thefour intermetallic compounds is shown inTable 7. These valuesare in close agreement with Notin and Hertz [36]. A graphicalcomparison between the calculated and experimental valushown inFig. 6.

tothe

t.

dst

is

5. The Mg–Ca system

5.1. Phase diagram

Baar [43] determined the complete liquidus data of thMg–Ca system using low purity of the components andsignificant loss of Ca was noticed. Paris [44] also determinedthe complete liquidus temperature but without mentioning tpurity of the elements. Haughton [45] determined the liquidustemperature for the Mg rich region(0 ≤ XCa ≤ 0.17) withoutloss of Ca. Klemm and Dinkelacker [46] accomplished the lastexperimental measurement of liquidus points, which agrewell with Haughton [45]. Nayeb-Hashemi and Clark [47]critically evaluated the system. Later, Agarwal et al. [48] re-optimized the system considering their experimental data


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htitae

se

ng.

l

er

at

oth

ueere

o

re

ofnceAstersereof

ated

asetes

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ser

ng.

ingre

Fig. 6. Comparison of the calculated Gibbs energy of formation ofstoichiometric compounds in the Ni–Ca system with the results fromliterature.

Table 6Optimized parameters of the Ca–Ni liquid

Phase Parameter a (J/g atom) b (J/g atom K)

Liquid L0 −25 023.00 15.13L1 13 652.60 −1.91L2 −6362.16 −4.05

CaNi2 �G f −7347.86 0.717CaNi3 �G f −7197.58 0.358Ca2Ni7 �G f −7047.45 0.2856CaNi5 �G f −5998.24 ∼0Ca-fcca �G f 0 0Ca-bcca �G f 0 0Ni-fcca �G f 0 0

a Ca-fcc, Ca-bcc and Ni-fcc are considered as stoichiometric compouwhere for Ca-fcc:�G f = 0+0T +G0

Ca-fcc, Ca-bcc:�G f = 0+0T +G0Ca-bcc

and Ni-fcc:�G f = 0 + 0T + G0Ni-fcc.

verify the compatibility of the results. In the Mg–Ca system, tstable phases are: the liquid phase, the Mg-hcp solid soluphase, the Ca-fcc and Ca-bcc phases and the linear intermecompound Mg2Ca with C14 hexagonal type Laves structurSeveral researchers measured the solubility of Ca in Mg [45,49,50]. Among them Vossk¨uhler [49] and Burke [50] reportedlimited solubility and their results agreed fairly well, whereaothers reported larger solubility. Hence, in this study the limitsolubility is adopted during optimization.


King and Kleppa [51] determined the enthalpy of formationfor Mg2Ca compound by tin solution calorimetry as−40.50±1.25 kJ/mol at 298 K. Davison and Smith [52] also measured it

ee

ds

eonllic.

d

by the acid solution method at room temperature as−39.38±2.63 kJ/mol and mentioned that the value measured by Kiand Kleppa [51] for enthalpy of formation is the most reliableBy high temperature calorimetry, Sommer et al. [53] measuredthe heat of mixing of liquid Mg–Ca at 1150 K. Agarwaet al. [48] measured the enthalpy of mixing at 1023 Kcalorimetrically but their results contradicted with Sommet al. [53]. Hultgren et al. [54] and Sommer [55] measuredthe activity of Mg and Ca by vapor pressure calorimetry1010 K and 1200 K, respectively. Sommer [55] used themodified Ruff method to measure the vapor pressure of bcomponents. Mishra et al. [56] calculated the activity of Mg andCa at 1200 K. Mashovets and Puchkov [57] also determinedthe activity of Mg and Ca by using a gas carrier techniqto measure the vapor pressure but their reported values wslightly lower than those of Sommer [55]. Moreover, theactivity of Ca measured by Mashovets and Puchkov [57]showed positive deviation from Raoult’s law which alscontradicted with Sommer [55]. For our optimization, theenthalpy of mixing and activity values of Sommer et al. [53]and Sommer [55], respectively, are used, as these values amore reliable and consistent with other works.


5.3.1. Phase diagramThough Agarwal et al. [48] reported the optimized

parameters for the Mg–Ca system, the total numberparameters is higher and they did not mention which referestate they used during optimization of different phases.physically sound models need fewer adjustable parameto fit the experimental data, an attempt has been made hto re-optimize the system to reduce the total numberparameters. Good agreement between the model calculphase diagram and experimental data is shown inFig. 7. Theoptimized parameters are shown inTable 8. This indicates thatthe thermodynamic model is capable of reproducing the phdiagram within experimental error limits. The reference staof the Gibbs energy of formation of Mg2Ca are considered asMg-hcp and Ca-fcc. Lattice stability values are not addedthe pure elements in all the binary subsystems. This enabcombining these systems with other binaries modeled in ogroup to build a multi-component database for Mg alloys.

Comparison between calculated and experimental valuethe critical points of the Mg–Ca phase diagram is shownTable 9.

A narrow solid solubility is observed in the calculated phadiagram shown inFig. 7. The solidus and solvus lines appeaalmost linear, where the solidus line starts from the meltipoint of Mg and terminates at 99.44 mol% Mg at 789.80 KThis agrees with the results of Vossk¨uhler [49] as 99.50 mol%Mg at 789.80 K and of Haughton [45] as 98.70 mol% Mg at790 K.

5.3.2. Thermodynamic propertiesGood agreement between the calculated enthalpy of mix

of Mg–Ca liquid and the experimental values from the literatuis shown inFig. 8(a). The activities of Mg and Ca at 1010 K

mmedraj

Sticky Note

correct units should be kJ/mole atom

mmedraj

Cross-Out

mmedraj

Inserted Text

mole

mmedraj

Cross-Out

mmedraj

Inserted Text

mole


Fig. 7. Calculated Mg–Ca phase diagram in relation to data from the literature.

Table 7Optimized Gibbs energy of formation parameters for the linear compounds

Gibbs energy of formation Temp. (K) CaNi2 CaNi3 Ca2Ni7 CaNi5

�G f (J/g atom) 1050 −6595.01 −6821.68 −6747.57 −5998.241200 −6487.46 −6767.98 −6704.73 −5998.241350 −6379.91 −6714.28 −6661.89 −5998.24

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Table 8Parameters for the liquid phase, Mg2Ca stoichiometric phase of the Mg–Csystem

Phase Terms a (J/g atom) b (J/g atom K)

Liquid L0 −24 018.60 1.9432L1 1785.73 4.4724L2 14 387.50 −22.9827

Mg-hcp L0 7150.90 −9.4012Mg2Ca �G f −12 704.40 1.8093Ca-fcca �G f 0 0Ca-bcca �G f 0 0

a Ca-fcc and Ca-bcc are considered as stoichiometric compounds wheCa-fcc:�G f = 0 + 0T + G0

Ca-fcc, Ca-bcc:�G f = 0 + 0T + G0Ca-bcc.

are shown inFig. 8(b). For the activity of Mg, there is gooagreement between the calculated values of this study anexperimental values of Hultgren et al. [54] and Mashovets andPuchkov [57] and reasonable agreement with Sommer [55].On the other hand, the activity of Ca agrees well withwork of Sommer [55] where he measured the activity of Conly for Ca rich alloys. The experimental data inFig. 8(b)are scattered. This is probably due to the fact that they ua different experimental method to measure the activity.the partial Gibbs energy of Mg and Ca, a good agreembetween calculated and experimental values of MashovetsPuchkov [57] is shown in Fig. 8(c). This figure also show

for

the

e

edornt

and

good agreement for the partial Gibbs energy of Ca andin the composition range(0.5 ≤ XMg ≤ 1) with the valuesof Sommer [55]. There is also good agreement for the partenthalpy of mixing of Ca with the work of Agarwal et al. [48]as shown inFig. 8(d). The partial enthalpy data is not usefor the optimization and is predicted by the model. The goagreement is evidence of the quality of the model.

In this study, the calculated enthalpy of formation of Mg2Caintermetallic compound at 298 K is−12.70 kJ/g atom, whichis in close agreement with King and Kleppa [51] and Davisonand Smith [52] who reported this value as−13.5 kJ/g atom and−13.17 kJ/g atom, respectively.

6. The Mg–Ni–Ca ternary system

The Mg–Ca–Ni system has been modeled in this studythe first time. A self-consistent database has been construfor this system by combining thermodynamic propertiesthe three constituent binaries.Fig. 9 shows the calculatedMg–Ni–Ca ternary phase diagram by projecting liquidus linfrom isothermal sections on Gibbs triangle. In this systethree ternary eutectic, five ternary quasi-peritectic, two ternperitectic and two saddle points are calculated and shownFig. 9. The estimated invariant points and resultant reactioare listed inTable 10. Unfortunately, there are no experimentresults for this system in the published literature. Thereforecomparison has been possible.

mmedraj

Cross-Out

mmedraj

Inserted Text

mole

mmedraj

Cross-Out

mmedraj

Inserted Text

mole

mmedraj

Cross-Out

mmedraj

Inserted Text

mole


Fig. 8. Calculated (a) enthalpy of mixing, (b) activities, (c) partial Gibbs free energy of Mg and Ca alloys, and (d) partial enthalpy of mixing of Ca alloy in relationwith the results available in literature.

Table 9Comparison between calculated and experimental values of invariant reactions in the Mg–Ca system

Reaction Reaction type Temp. (K) XMg-liquid Reference

L ↔ Mg2Ca Congruent

987.0 0.667 [43]998.0 . . . [44]987.0 . . . [49]984.6 0.667 This work

L ↔ Mg2Ca+ Ca-fcc Eutectic

719.0 0.309 [43]733.0 0.266 [44]718.0 0.270 [46]718.0 . . . [49]719.0 0.288 This work

L ↔ Mg2Ca+ Mg-hcp Eutectic

787.0 0.878 [43]798.0 0.886 [44]790.0 0.895 [45]789.0 0.894 [49]789.8 0.896 This work


n).
Fig. 9. The calculated Mg–Ni–Ca ternary phase diagram with the invariant points (based on mole fractio
5

7

0884

Table 10Calculated invariant reactions in the Mg–Ni–Ca system

Reaction type Reactions Temp. (K) XCa XMg XNi

E1 L → Mg-hcp+ Mg2Ni + Mg2Ca 737.2 0.0597 0.8628 0.077E2 L → Mg2Ni + Mg2Ca+ MgNi2 859.1 0.1913 0.6275 0.1812E3 L → Mg2Ca+ MgNi2 + Ca-fcc 682.0 0.6276 0.2887 0.083U1 L + Ni-fcc → CaNi5 + MgNi2 1275.9 0.0939 0.1469 0.7591U2 L + CaNi5 → MgNi2 + Ca2Ni7 1171.2 0.2369 0.1897 0.5734U3 L + Ca2Ni7 → CaNi3 + MgNi2 1051.5 0.3714 0.1971 0.4315U4 L + CaNi3 → CaNi2 + MgNi2 962.6 0.4817 0.1843 0.3340U5 L + CaNi2 → Ca-bcc+ MgNi2 780.6 0.6881 0.1428 0.1691P1 L + MgNi2 + Ca-bcc→ Ca-fcc 715.9 0.6499 0.2491 0.101P2 L + Mg2Ca+ Ca-bcc→ Ca-fcc 715.9 0.7038 0.2882 0.007Max1 L → Mg2Ni + Mg2Ca 863.2 0.1823 0.6469 0.170Max2 L → MgNi2 + Mg2Ca 867.9 0.2518 0.5718 0.176

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From the database, vertical sections and other diagramcalculated to verify the critical points in the system. It is clefrom the ternary phase diagram that the MgNi2 compoundhas the dominating phase field. This calculated phase diagprovides a basis for more theoretical and experimeinvestigations and can be improved by including ternexperimental results in the analysis after they become availa

7. Conclusions

A self-consistent thermodynamic database has beenstructed using the Redlich–Kister polynomial model for

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amalyle.

n-l

the disordered phases and the sublattice model for thestoichiometric compound and without adding lattice stabilvalues to the pure components. The model parameters areuated by incorporating all experimental data available in theerature. The phase diagrams and the thermodynamic propeof all three binaries show good agreement with the experimtal data. The ternary phase diagram of the Mg–Ni–Ca systecalculated by combining the databases of the three constitbinaries. This is the first attempt to construct the ternary phdiagram of the Mg–Ni–Ca system and lays down the fountion for more developed evaluation. There is no experimedata for the ternary phase diagram in the literature, these


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essential to verify the current model and thus to produce a mreliable thermodynamic description.

Acknowledgments

This study is carried out with the support of NSERC anNATEQ grants. The authors would like to thank D. Uremovicof INCO Ltd, Canada for his help in the assessment ofNi–Ca system.

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the phase equilibria in the mg–ni–ca...

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