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Thermochimica Acta 578 (2014) 35– 42

Contents lists available at ScienceDirect

Thermochimica Acta

jo ur nal ho me page: www.elsev ier .com/ locate / tca

hermodynamic modeling of the Cr–Ni–Ti system using aour-sublattice model for ordered/disordered bcc phases

iao Hua,b, Yong Dua,∗, J.C. Schusterc, Weihua Suna, Shuhong Liua, Chengying Tangd

State Key Laboratory of Powder Metallurgy, Central South University, Changsha, Hunan 410083, PR ChinaSchool of Materials Science and Engineering, Anhui University of Science and Technology, Huainan, Anhui 232001, PR ChinaInnovative Materials Group, Universität Wien, Währinger Str. 42, A-1090 Wien, AustriaGuangxi Key Laboratory of Information Materials, Guilin University of Electronic Technology, Guilin, Guangxi 541004, PR China

r t i c l e i n f o

rticle history:eceived 24 July 2013eceived in revised form 4 January 2014ccepted 6 January 2014

a b s t r a c t

A thermodynamic assessment of the ternary Cr–Ni–Ti system together with a refinement for the ther-modynamic description of the binary Ni–Ti sub-system is performed following the CALPHAD methodand using the compound energy formalism (CEF). The ordered/disordered transition between disorderedBcc A2 and ordered Bcc B2 phases is described using a four-sublattice (4SL) model. An optimal set of ther-

vailable online 16 January 2014

eywords:r–Ni–Ti systemhermodynamic calculationour-sublattice modelrdered/disordered transition

modynamic parameters for the Cr–Ni–Ti system is obtained by considering the experimental data fromthe literature. Comparisons between the calculated and measured phase diagrams indicate that almostall of the reliable experimental information is satisfactorily accounted for by the present modeling. Thecomplete liquidus projection and reaction scheme of the Cr–Ni–Ti system are also presented.

© 2014 Elsevier B.V. All rights reserved.

. Introduction

The Cr–Ni–Ti ternary system is of great interest because Ni–Crased alloys are important corrosion-resistant high-temperatureaterials, and the addition of Ti into Ni–Cr alloys enhances their

igh-temperature corrosion resistance and the respective phys-cal and mechanical properties [1]. The application of the Ni–Crlloys in combination with the Ti-based alloys might lead to aignificant reduction in the specific weight of various structuralomponents [2]. Technologically important multi-component Ni-ased superalloy, which contains Cr and Ti, is currently usedo manufacture parts of aircraft engines and land-based tur-ines [3]. Knowledge of the phase equilibria and thermodynamicroperties of the Cr–Ni–Ti system is essential in developing theew Ni-based and Ti-based high-temperature structural materi-ls.

Therefore, a thorough assessment of the Cr–Ni–Ti system isecessary in order to provide a reliable set of thermodynamicarameters for thermodynamic extrapolations to higher order sys-

ems. The purposes of the present work are (i) to critically evaluatehe measured phase diagram data available for the Cr–Ni–Ti system,ii) to describe the ordered/disordered transition between Bcc A2

∗ Corresponding author. Tel.: +86 731 88836213; fax: +86 731 88710855.E-mail address: yongducalphad@gmail.com (Y. Du).

040-6031/$ – see front matter © 2014 Elsevier B.V. All rights reserved.ttp://dx.doi.org/10.1016/j.tca.2014.01.002

and Bcc B2 phases using a four-sublattice model, and (iii) to obtainan optimal set of thermodynamic parameters for the ternary sys-tem by means of CALPHAD approach.

2. Evaluation of experimental phase diagram data

In the Cr–Ni–Ti system, there are 10 stable phases, i.e.liquid, (Ni), (�Ti), (Cr, �Ti), Ni3Ti, NiTi, NiTi2, �Cr2Ti, �Cr2Tiand �Cr2Ti. The disordered phase (Cr, �Ti) with Bcc A2 struc-ture and the ordered phase NiTi with Bcc B2 structure aretreated as the same phase in the present thermodynamicmodeling.

There have been several experimental determinations of thephase equilibria in the Cr–Ni–Ti system since the early work byTaylor and Floyd [4]. Using X-ray diffraction (XRD) and opticalmicroscopy, Taylor and Floyd [4] established the Ni-rich isother-mal sections at 1150, 1000 and 750 ◦C. The 44 alloys were preparedfrom pure elements Cr (99.5 wt.%), Ni (99.9 wt.%) and Ti (99.7 wt.%)in an induction furnace or in an argon-arc furnace. The alloys werehomogenized in vacuum for up to 2 h at 1150 ◦C, 1 day at 1000 ◦Cand 3 weeks at 750 ◦C, followed by water quenching. Since theexperimental procedure [4] was well described and the measured

phase relationships along the Cr–Ni and Ni–Ti binary sides agreewith the generally accepted ones [5], the ternary phase diagramdata published by Taylor and Floyd [4] are used in the presentthermodynamic modeling.

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6 B. Hu et al. / Thermoch

By means of XRD and differential thermal analysis (DTA) meth-ds, Nartova et al. [6] measured the vertical section at 8 wt.%i below 10 wt.% Cr. Within the investigated composition range,

eutectoid reaction (Cr, �Ti) = (�Ti) + NiTi2 + �Cr2Ti at 650 ◦C isbserved. The thermal effect peaks corresponding to the phaseoundary between (�Ti) and (�Ti) + NiTi2 were excluded from theodeling since the boundary along the Ni–Ti binary side derived

rom these peaks is higher by about 60 ◦C than the generallyccepted Ni–Ti phase diagram [5]. The other experimental datarom Nartova et al. [6] are utilized in the present modeling butssigned a relatively low weight factor.

Using a diffusion couple technique, Xu and Jin [7] constructed partial isothermal section at 927 ◦C. Bars of Cr (99.4 wt.%) and Ti99.7 wt.%), and plates of Ni (99.98 wt.%) were used to prepare dif-usion triples. Electron probe microanalysis (EPMA) method waspplied to characterize the diffusion zones existing in the diffu-ion couple annealed at 927 ◦C for 580 h. No ternary compoundas found at 927 ◦C. Since the phase boundary data reported by

hem are very scattering, they are not used for the thermodynamicptimization.

In order to provide more accurate phase diagram data at27 ◦C, Tan et al. [3] prepared four diffusion couples (Cr15Ni85/Ti,r36Ni64/Ti, Cr18Ti82/Ni and Cr18Ti82/Cr36Ni64, at.%) supplementedith eight key ternary alloys. The prepared diffusion couples were

nnealed at 927 ◦C for 8 days and then analyzed with EPMA method.ased on the experimental results from diffusion couples, eight keylloys were prepared and annealed at 927 ◦C for 16 days and thenubjected to EPMA measure. The phase equilibrium data deter-ined in [3] are more accurate than these previously reported [7],

nd these data [3] which are employed in the present optimiza-ion.

A complete isothermal section at 850 ◦C was provided by Beekt al. [2] using diffusion couple and equilibrated alloys, whichere analyzed via optical microscopy (OM), scanning electronicroscopy (SEM), XRD, EPMA and polarized light microscopy. No

ernary phase was formed at this temperature. Since high-puritytarting materials (99.98 wt.% Ni, 99.95 wt.% Cr and 99.97 wt.%i) were utilized and the experimental results from diffusionouple and equilibrated alloys were consistent with each other,heir experimental data were employed in the present optimiza-ion.

Haour et al. [8] reported a ternary eutectic at the compositionf Cr3.7Ni61Ti35.3 (at.%) with a melting temperature of 1220 ◦C.ince the phases present in the obtained microstructures wereot identified, this information was of limited value. Recently,rendelsberger et al. [9] prepared more than 40 ternary alloysnd systematically investigated the nature of solid–liquid phasequilibira in the ternary Cr–Ni–Ti system using SEM/EDS, XRD andTA, and constructed the reaction scheme and liquidus projection

or the entire Cr–Ni–Ti system. The C14-type Laves phase �Cr2Tias found to coexist with the liquid phase in the ternary system.

n addition, the types of Cr2Ti Laves phase coexisting with Ni–Tiinary phases were confirmed to be hexagonal C14-type �Cr2Tind cubic C15-type �Cr2Ti. No C36-type Laves phase �Cr2Ti wasbserved. The experimental data published by Krendelsbergert al. [9] are used in the present thermodynamic modeling.

Based on the limited experimental data [4], Kaufman andesor [10] carried out a preliminary thermodynamic calculation

or this ternary system. Recently, Isomäki et al. [1] performed complete thermodynamic assessment of the Cr–Ni–Ti systemased on the experimental phase diagram data available in the

iterature [2–4]. However, the parameters obtained by Isomäki

t al. [1] cannot describe satisfactorily the new experimental dataeasured by Krendelsberger et al. [9]. Hence, it is necessary to

eassess the Cr–Ni–Ti system in order to obtain a more reasonablehermodynamic description.

Acta 578 (2014) 35– 42

3. Thermodynamic modeling

In the thermodynamic assessments of the fcc and bcc orderedphases, the two-sublattice (2SL) model [11] in the frameworkof the compound energy formalism (CEF) [12] has been gener-ally accepted in the CALPHAD community due to its simplicityand computational efficiency. However, the 2SL model is notenough to simultaneously describe the fcc ordered phases L12and L10, or bcc ordered phases B2, B32 and D03 observed inmulticomponent system. Recently, the developed four-sublattice(4SL) model is becoming more preferable [13–17] in view ofthe following two aspects. On the one hand, the 4SL model,(A, B)0.25(A, B)0.25(A, B)0.25(A, B)0.25, which reflects the crystal struc-tures of the fcc and bcc lattices, is more physically sound thanthe 2SL model. On the other hand, 4SL model can describe all theordered fcc states that only depend on the nearest neighbors, suchas L12 and L10, or the ordered bcc states that only depend on thenearest and next nearest neighbors, such as B2, B32 and D03. Forthe Cr–Ni–Ti system, the 2SL model is enough to describe both thestable disordered bcc and B2 phases. However, when it is extrap-olated into high-order systems, such as the Al–Fe–Cr–Ni–Ti andAl–Li–Cr–Ni–Ti quinary systems, which contain the stable orderedD03 and B32 phases, the 4SL is necessary. Hence, it is appropri-ate to apply the 4SL model to describe the bcc ordered/disorderedtransitions in the present work.

In the present modeling, the Gibbs energy functions for the ele-ments Cr, Ni and Ti are taken from the SGTE database compiledby Dinsdale [18]. The thermodynamic parameters in the Cr–Ni andCr–Ti systems are taken from Lee [19] and Ghosh [20], respectively.The thermodynamic parameters in the Ni–Ti system are taken fromKeyzer et al. [21], except for the ordered NiTi phase. The model ofthe NiTi phase is modified from 2SL to 4SL in the present work. Inthe following, the analytical expressions for Gibbs energies of theternary phases are briefly presented.

3.1. Liquid and (Ni) phases

The solution phases, i.e. liquid, (Ni) and (�Ti) are describedby the substitutional solution model. Taking the liquid phaseas example, its molar Gibbs energy is described by theRedlich–Kister–Muggianu polynomial [22]:

0GLm = xCr · 0GL

Cr + xNi · 0GLNi + xTi · 0GL

Ti

+ R · T · (xCr · ln xCr + xNi · ln xNi + xTi · ln xTi)

+ R · T · (xCr · ln xCr + xNi · ln xNi + xTi · ln xTi)

+ xCr · xNi · LLCr,Ni + xCr · xTi · LL

Cr,Ti + xNi · xTi · LLNi,Ti

+ exGLCr,Ni,Ti (1)

where R is the gas constant, xCr, xNi and xTi are the molar fractionsof the elements Cr, Ni and Ti, respectively. The standard elementreference (SER) state [18], i.e. the stable structure of the element at25 ◦C and 1 bar, is used as the reference state of Gibbs energy. Theparameters LL

i,j(i, j = Cr, Ni, Ti) are the interaction parameters from

binary systems. The excess Gibbs energy exGLCr,Ni,Ti

is expressed asfollows:

exGLCr,Ni,Ti = xCr · xNi · xTi

·(xCr · 0LLCr,Ni,Ti + xNi · 1LL

Cr,Ni,Ti + xTi · 2LLCr,Ni,Ti) (2)

in which the interaction parameters 0LLCr,Ni,Ti

, 1LLCr,Ni,Ti

and 2LLCr,Ni,Ti

are linearly temperature-dependent. These parameters will beoptimized in the present work.

An equation similar to Eq. (1) can be written for the Gibbs energyof (�Ti). For (Ni), its Gibbs energy is described by splitting it into

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B. Hu et al. / Thermoch

nonmagnetic contribution (0Gnmg) and a magnetic one (�Gmg).Gnmg is given by an equation similar to Eq. (1). �Gmg is describedy Hillert–Jarl–Inden model [23].

.2. Binary phases extending into the ternary system

In view of the large solubility for Ni in �Cr2Ti [2,3], this phase isescribed with the sublattice model (Cr, Ni, Ti)2(Cr, Ti)1, where theoldfaces Cr and Ti mean the normal atoms (i.e., major species)

n the sublattices. In order to reduce the number of adjustablearameters, it is assumed that Ni only substitutes for Cr in therst sublattice. In accordance with the formula for sublatticeodel [24,25], the Gibbs energy of �Cr2Ti per mol-formula can be

xpressed as:

�Cr2Tim = y′

Cry′′CrG�Cr2Ti

Cr:Cr + y′Cry′′

TiG�Cr2TiCr:Ti + y′

Niy′′CrG�Cr2Ti

Ni:Cr

+ y′Niy

′′TiG

�Cr2TiNi:Ti + y′

Tiy′′CrG�Cr2Ti

Ti:Cr + y′Tiy

′′TiG

�Cr2TiTi:Ti

+ y′Cry′

Niy′′Cr[0LCr,Ni:Cr + (y′

Cr − y′Ni)

1LCr,Ni:Cr]

+y′Cry′

Niy′′Ti[

0LCr,Ni:Ti + (y′Cr − y′

Ni)1LCr,Ni:Ti]

+ y′Cry′

Tiy′′Cr[0LCr,Ti:Cr + (y′

Cr − y′Ti)

1LCr,Ti:Cr]

+y′Cry′

Tiy′′Ti[

0LCr,Ti:Ti + (y′Cr − y′

Ti)1LCr,Ti:Ti]

+ y′Niy

′Tiy

′′Cr[0LNi,Ti:Cr + (y′

Ni − y′Ti)

1LNi,Ti:Cr]

+y′Niy

′Tiy

′′Ti[

0LNi,Ti:Ti + (y′Ni − y′

Ti)1LNi,Ti:Ti]

+ y′Cry′′

Cry′′Ti[

0LCr:Cr,Ti + (y′′Cr − y′′

Ti)1LCr:Cr,Ti]

+y′Niy

′′Cry′′

Ti[0LNi:Cr,Ti + (y′′

Cr − y′′Ti)

1LNi:Cr,Ti]

+ y′Tiy

′′Cry′′

Ti[0LTi:Cr,Ti + (y′′

Cr − y′′Ti)

1LTi:Cr,Ti] (3)

here y′Cr , y′

Niand y′

Tiare the site fractions of Cr, Ni and Ti in the

rst sublattice, and y′′Cr and y′′

Tiin the second one.

The binary phases NiTi2, Ni3Ti and �Cr2Ti, which werexperimentally observed [2,3] to show noticeable solubilitiesor the third element, are described with the sublattice mod-ls (Ni, Ti)2(Cr, Ni, Ti)1Va0.5, (Cr, Ni, Ti)0.75(Cr, Ni, Ti)0.25Va0.5, andCr, Ni, Ti)2(Cr, Ti)1, respectively. The Gibbs energies of these phasesre defined in an analogous equation to Eq. (3).

.3. Disordered (Cr, ˇTi) and ordered NiTi phases

In the present work, the ordered NiTi phase with Bcc B2tructure is modeled as (Cr, Ni, Ti, Va)0.25(Cr, Ni, Ti, Va)0.25Cr, Ni, Ti, Va)0.25(Cr, Ni, Ti, Va)0.25Va3, where Va means vacancy.n order to represent the Gibbs energies of both orderediTi phase and disordered (Cr, �Ti) with Bcc A2 structuresing a single function, (Cr, �Ti) is described by a modelCr, Ni, Ti, Va)1Va3.

The Gibbs energy of both the ordered NiTi and disorderedCr, �Ti) phases is given by a single function of the following form:

bccm = Gdis

m (xi) + �Gordm (ys

i ) = Gdism (xi) + G4sl

m (ysi ) − G4sl

m (ysi = xi) (4)

n which Gdism (xi) is the Gibbs energy of the disordered (Cr, �Ti)

hase. �Gordm (ys

i) = G4sl

m (ysi) − G4sl

m (ysi= xi) denotes the contribution

f the ordered phase to the Gibbs energy. This difference is identicalo zero when the disordered state is stable. Hence, the param-

ters of the ordered and disordered phases can be evaluatedndependently.

It is assumed that atoms on the first sublattice have nearesteighbors on the third and fourth sublattice and next nearest

Acta 578 (2014) 35– 42 37

neighbors on the second sublattice. This leads to the followingGibbs energies for the ordered compounds:

G4slNi:Ni:Ti:Ti:Va = G4sl

Ti:Ti:Ni:Ni:Va = GB2Ni2Ti2

= 4 · uNiTi (5)

G4slNi:Ti:Ni:Ti:Va = G4sl

Ni:Ti:Ti:Ni:Va = G4slTi:Ni:Ni:Ti:Va = G4sl

Ti:Ni:Ti:Ni:Va

= GB32Ni2Ti2

= 2 · uNiTi + 3 · vNiTi (6)

G4slNi:Ni:Ni:Ti:Va = G4sl

Ni:Ni:Ti:Ni:Va = G4slNi:Ti:Ni:Ni:Va = G4sl

Ti:Ni:Ni:Ni:Va

= G4slNi3Ti = 2 · uNiTi

+ 1.5 · vNiTi + �GNi3Ti (7)

G4slNi:Ti:Ti:Ti:Va = G4sl

Ti:Ni:Ti:Ti:Va = G4slTi:Ti:Ni:Ti:Va = G4sl

Ti:Ti:Ti:Ni:Va

= G4slNiTi3

= 2 · uNiTi + 1.5 · vNiTi + �GNiTi3 (8)

G4slCr:Cr:Ni:Ti:Va = G4sl

Cr:Cr:Ti:Ni:Va = G4slNi:Ti:Cr:Cr:Va = G4sl

Ti:Ni:Cr:Cr:Va

= GL21Cr2NiTi = 2 · uCrNi + 2 · uCrTi + 1.5 · vNiTi (9)

G4slCr:Ni:Cr:Ti:Va = G4sl

Cr:Ni:Ti:Cr:Va = G4slNi:Cr:Cr:Ti:Va

= G4slNi:Cr:Ti:Cr:Va = G4sl

Cr:Ti:Cr:Ni:Va = G4slCr:Ti:Ni:Cr:Va

= G4slTi:Cr:Cr:Ni:Va = G4sl

Ti:Cr:Ni:Cr:Va = GF43̄mCr2NiTi

= uCrNi + uCrTi + uNiTi + 1.5 · vCrNi + 1.5 · vCrTi (10)

where uij and vij (i = Cr, Ni; j = Ni, Ti; i /= j) are the bond energiesbetween nearest neighbors and between next nearest neighbors,respectively. �GNi3Ti and �GNiTi3 are the extra parameters for theend members in order to give an asymmetrical description forreal systems. It is noteworthy that the end members involvingvacancies are set to be zero in the present work.

Similarly, the interaction parameters are subjected to the fol-lowing equations:

L4slNi,Ti:∗:∗:∗:Va = L4sl

∗:Ni,Ti:∗:∗:Va = L4sl∗:∗:Ni,Ti:∗:Va = L4sl

∗:∗:∗:Ni,Ti:Va = L4slNiTi

(11)

and for the reciprocal parameters:

L4slNi,Ti:∗:Ni,Ti:∗:Va = L4sl

Ni,Ti:∗:∗:Ni,Ti:Va = L4sl∗:Ni,Ti:Ni,Ti:∗:Va

= L4sl∗:Ni,Ti:∗:Ni,Ti (for nearest neighbors) (12)

L4slNi,Ti:Ni:Ti:∗:∗:Va = L4sl

∗:∗:Ni,Ti:Ni,Ti:Va (for next nearest neighbors) (13)

The character “*” in a sublattice means that the parameter isindependent of the occupation of that sublattice.

4. Results and discussion

The thermodynamic parameters were evaluated by the opti-mization module PARROT [26] of the program Thermo-Calc, whichworks by minimizing the square sum of the differences betweenmeasured and calculated values. The step-by-step optimizationprocedure carefully described by Du et al. [27] was utilized in the

present assessment.

For the Ni–Ti binary system, the thermodynamic parametersare taken from Keyzer et al. [21], except for the NiTi phase.Start values for the NiTi phase have to be found be trial and

38 B. Hu et al. / Thermochimica Acta 578 (2014) 35– 42

Fig. 1. Calculated phase diagrams of the Ni–Ti system (the presently calculatedresult along with the one from Keyzer et al. [21] in dashed lines and experimentald

eonresfi

o[dt

Pcootit[i(titarwsCtrit

emaeT

Fig. 2. Calculated isothermal section at 1000 ◦C, compared with the experimentaldata from Taylor and Floyd [4] and Krendelsberger et al. [9].

Fig. 3. Calculated isothermal section at 927 ◦C, compared with the experimentaldata from Tan et al. [3].

ata [28–33]).

rror. It turns out to be necessary to use reciprocal parametersf the NiTi phase, otherwise the NiTi phase field remains veryarrow. For the sake of simplification, the two different types ofeciprocal parameters are set to be equal. In addition, the twoxtra parameters �GNi3Ti and �GNiTi3 turn out to have a rathertrong influence on the shape and symmetry of the NiTi phaseeld.

Fig. 1 presents the calculated Ni–Ti phase diagram along with thene from Keyzer et al. [21] in dashed lines and experimental data28–33]. As can be seen from this figure, most of the experimentalata can be well accounted for by the present calculation withinhe estimated experimental errors.

In order to validate the high accuracy of the present CAL-HAD assessment for the thermodynamic properties of the binaryompounds, in the present work, the first-principles calculationsf the enthalpy of formation (�fH) at 0 K for possible bcc basedrdered phases NiTi(B2), NiTi(B32), Ni3Ti(D03) and NiTi3 (D03) inhe Ni–Ti system are performed by the highly efficient Vienna abnitio simulation package (VASP) [34–36]. The electron–ion interac-ions are described by the full potential frozen-core PAW method37], and the exchange-correlation is treated within the general-zed gradient approximation (GGA) of Perdew–Burke–ErnzerhofPBE) [38]. The plane wave cut-off energy is set to be 350 eV andhe Monkhorst–Pack k-point mesh scheme for the Brillouin-zonentegrations is used. The total energy differences are convergedo be within 0.1 kJ/mole-atoms. The calculated �fH at 298.15 Kfter optimization for possible bcc based ordered phases witheference to the stable states for the pure elements comparedith first-principles data [39] and experimental data [40–42] are

hown in Table 1. It can be seen that the calculated �fH byALPHAD is in good agreement with the first-principles calcula-ions results except for the ordered B32 phase. The comparisonesults show that the B32 phase predicted by CALPHAD methods much more unstable than that by first-principles calcula-ions.

For the Cr–Ni–Ti ternary system, by using the reliablexperimental data [2–4,6,9] selected from the literature, theodel parameters for liquid, (Cr, �Ti), NiTi2, NiTi, Ni3Ti, �Cr2Ti

nd �Cr Ti could be optimized. The thermodynamic param-

2ters finally obtained in the present work are listed inable 2.

Fig. 4. Calculated isothermal sections at 850 ◦C, compared with the experimentaldata from Beek et al. [2] and Krendelsberger et al. [9].

B. Hu et al. / Thermochimica Acta 578 (2014) 35– 42 39

Table 1Experimental and calculated enthalpies of formation for bcc based ordered phases in the Ni–Ti system. The reference states are Fcc Al Ni and Hcp A3 Ti.

Phase Temperature, K �fH (kJ/mole-atom) Method References

NiTi(B2) – −46.0 TBBa [39]– −40.9 LMTOb [39]

1460 −34.0 Calorimetry [40]298.15 −33.9 Calorimetry [41]999 −34.15 Calorimetry [42]

0 −33.24 First-principles This work298.15 −33.85 CALPHAD This work

NiTi(B32) – −36.5 TBB [39]– −38.5 LMTO [39]0 −29.29 First-principles This work

298.15 −9.80 CALPHAD This workNi3Ti(D03) – −44.4 TBB [39]

– −46.8 LMTO [39]0 −29.5 First-principles This work

298.15 −30.68 CALPHAD This workNiTi3(D03) 0 −12.11 First-principles This work

298.15 −13.68 CALPHAD This work

a

8ttbiammosaapN

ascb(bt

Fd

TBB: tight-binding-bond.b LMTO: linear muffin-tin orbitals.

Figs. 2–5 are the calculated isothermal sections at 1000, 927,50 and 750 ◦C along with the experimental data [2–4,9], respec-ively. Most of the experimental data can be well reproduced byhe calculation within the estimated experimental errors. It shoulde noted that the calculated types of Cr2Ti Laves phase coexist-

ng with Ni–Ti binary phases are to be hexagonal C14-type �Cr2Tind cubic C15-type �Cr2Ti, which are consistent with the experi-ental results of Krendelsberger et al. [9]. In addition, it is worthentioning that the Ni3Ti phase field extends both in the directions

f Cr3Ti and Ni3Cr in these figures. This is due to fitting the compo-itions of Ni3Ti for the two three-phase fields, i.e. Ni3Ti + NiTi + (Cr)nd Ni3Ti + (Ni) + (Cr), at 927 ◦C isothermal section in Fig. 3. Actu-lly, more experiments are necessary to determine whether thehase field of Ni3Ti extends both in the directions of Cr3Ti andi3Cr.

The calculated vertical section at 8 wt.% Ni is shown in Fig. 6long with thermal effect data from Nartova et al. [6]. As can beeen from this diagram, due to the differences between the cal-ulated and measured boundaries of (�Ti)/(�Ti) + NiTi2 along the

inary Ni–Ti side, except for the data points of the boundaries of�Ti)/(�Ti) + NiTi2, the other experimental data are well describedy the present thermodynamic modeling in view of the experimen-al errors.

ig. 5. Calculated isothermal section at 750 ◦C, compared with the experimentalata from Taylor and Floyd [4].

Fig. 6. Calculated vertical section at 8 wt.% Ni, compared with the experimental datafrom Nartova et al. [6].

Fig. 7. Calculated liquidus projection in the Cr–Ni–Ti system, compared with theexperimental data from Krendelsberger et al. [9].

4 imica Acta 578 (2014) 35– 42

wdtlptFAtptbettIpa

st

os

Fig. 9. Calculated sites occupancy in the 4SL model of the ordered B2 phase at 850 ◦C,50 at.% Ti and 10 at.% Cr.

0 B. Hu et al. / Thermoch

The calculated liquidus projection of the Cr–Ni–Ti system alongith the experimental data [9] according to the present thermo-ynamic parameters is presented in Fig. 7. A comparison betweenhe calculated and experimentally measured invariant equilibria isisted in Table 3. An excellent agreement is obtained between theresent calculations and the experimental data [6,9], except theemperature of the pseudobinary eutectic e1,max: L = �Cr2Ti + NiTi.or this discrepancy, it can be explained as the following reasons.ccording to the experimental results of Krendelsberger et al. [9],

he points between e1,max and U1 are very close, but these twooints show a large temperature difference. It is difficult to measurehe temperature of e1,max with the steep boundary of �Cr2Ti + NiTiy DTA technique. Hence, to some extent, the temperature for1,max measured by Krendelsberger et al. [9] is doubtful. In addition,he CALPHAD method cannot well reproduce all of experimen-al data unless each piece of experimental data is very accurate.t is concluded that the presently calculated temperature of theseudobinary eutectic e1,max is within experimental errors andcceptable.

Finally the reaction scheme for the whole Cr–Ni–Ti system ishown in Fig. 8, which has proved to be a useful tool in describing

ernary and higher component systems.

Fig. 9 presents the calculated sites occupancy in the 4SL modelf the ordered B2 phase at 850 ◦C, 50 at.% Ti and 10 at.% Cr. As can beeen from this figure, Ni is the major species in the first and second

Fig. 8. Reaction scheme for the entire Cr–Ni–T

i system according to the present work.

B. Hu et al. / Thermochimica Acta 578 (2014) 35– 42 41

Table 2Summary of the optimized thermodynamic parameters in the Cr–Ni–Ti system.a

Liquid: Model (Cr, Ni, Ti)10LLiquid

Cr,Ni,Ti= −61, 425.282 + 11.998 · T

1LLiquidCr,Ni,Ti

= 20, 436.948 − 0.492 · T2LLiquid

Cr,Ni,Ti= −21, 343.364

(Cr, ˇTi): Model (Cr, Ni, Ti, Va)1Va30G(Cr,ˇTi)

Va:Va= 80 · T

0L(Cr,ˇTi)Cr,Va:Va

= 150, 0000L(Cr,ˇTi)

Cr,Ni,Ti:Va= −41, 956.674

1L(Cr,ˇTi)Cr,Ni,Ti:Va

= 137, 331.9782L(Cr,ˇTi)

Cr,Ni,Ti:Va= −8270.623

NiTi: Model (Cr, Ni, Ti, Va)0.25(Cr, Ni, Ti, Va)0.25(Cr, Ni, Ti, Va)0.25(Cr, Ni, Ti, Va)0.25Va3

uCrNi = 45, 674.201vCrNi = 45, 393.541uCrTi = 17, 601.914vCrTi = 10, 365.184uNiTi = −8298.42 + 2.571 · TvNiTi = −2148.766 + 9.819 · T�GNi3Ti = −20, 596.7895�GNiTi3

= 42, 787.269 − 30.713 · T0G4sl

Cr:Cr:Cr:Cr:Va= 0G4sl

Ni:Ni:Ni:Ni:Va= 0G4sl

Ti:Ti:Ti:Ti:Va= 0

0G4slVa:Va:Va:Va:Va

= 0G4slCr:Va:Va:Va:Va

= 0G4slCr:Cr:Va:Va:Va

= 0G4slCr:Cr:Cr:Va:Va

= · · · = 0L4sl

Ni,Ti:∗:Ni,Ti:∗:Va= L4sl

Ni,Ti:∗:∗:Ni,Ti:Va= L4sl

∗:Ni,Ti:Ni,Ti:∗:Va= L4sl

∗:Ni,Ti:∗:Ni,Ti:Va=

−2447.1371 − 12.274 · TL4sl

Ni,Ti:Ni:Ti:∗:∗:Va= L4sl

∗:∗:Ni,Ti:Ni,Ti:Va= −2447.1371 − 12.274 · T

NiTi2: Model (Ni, Ti)2(Cr, Ni, Ti)10GNiTi2

Ni:Cr= −4000 + 2 · 0GFcc A1

Ni+ 0GBcc A2

Cr0GNiTi2

Ti:Cr= 500 + 2 · 0GHcp A3

Ti+ 0GBcc A2

Cr

Ni3Ti: Model (Cr, Ni, Ti)0.75(Cr, Ni, Ti)0.250GNi3Ti

Cr:Ni= −500 + 0.75 · 0GHcp A3

Cr+ 0.25 · 0GHcp A3

Ni0GNi3Ti

Ni:Cr= −3000 + 0.75 · 0GHcp A3

Ni+ 0.25 · 0GHcp A3

Cr0GNi3Ti

Cr:Ti= 24, 000 + 0.75 · 0GHcp A3

Cr+ 0.25 · 0GHcp A3

Ti0GNi3Ti

Ti:Cr= 10, 000 + 0.75 · 0GHcp A3

Ti+ 0.25 · 0GHcp A3

Cr0GNi3Ti

Cr:Cr= +0GHcp A3

Cr0LNi3Ti

Cr,Ni:Ti= −13, 955.820

0LNi3TiNi:Cr,Ti

= −3538.353

˛Cr2Ti (C15): Model (Cr, Ni, Ti)2/3(Cr, Ti)1/30G˛Cr2Ti

Ni:Cr= +2/3 · 0GFcc A1

Ni+ 1/3 · 0GBcc A2

Cr0G˛Cr2Ti

Ni:Ti= −18, 243.123 + 2/3 · 0GFcc A1

Ni+ 1/3 · 0GHcp A3

Ti0L˛Cr2Ti

Cr,Ni:Ti= −10, 077.725

ˇCr2Ti (C36): Model (Cr, Ni, Ti)2/3(Cr, Ti)1/30G˛Cr2Ti

Ni:Cr= +2/3 · 0GFcc A1

Ni+ 1/3 · 0GBcc A2

Cr0G˛Cr2Ti

Ni:Ti= +2/3 · 0GFcc A1

Ni+ 1/3 · 0GHcp A3

Ti

�Cr2Ti (C14): Model (Cr, Ni, Ti)2/3(Cr, Ti)1/30G�Cr2Ti

Ni:Cr= +2/3 · 0GFcc A1

Ni+ 1/3 · 0GBcc A2

Cr0G�Cr2Ti

Ni:Ti= −16, 938.266 − 6.6667 · T + 2/3 · 0GFcc A1

Ni+ 1/3 · 0GHcp A3

Ti0L�Cr2Ti

Cr,Ni:Ti= −6136.709

a In J/(mole of atoms); temperature (T) in K. The Gibbs energies for the pureelements are taken from the compilation of Dinsdale [18]. The thermodynamicparameters in the Cr–Ni, Cr–Ti, and Ni–Ti systems are taken from Lee [19], Ghosh[20], and Keyzer et al. [21] (the NiTi phase was modified in the present work),respectively.

ssets

docott

Table 3Comparisons between the calculated and experimentally measured invariant reac-tion temperatures in the Cr–Ni–Ti system.

Type Invariant equilibrium Temperature(◦C)

Source

p1,max L + (Cr, �Ti) = �Cr2Ti 1389 Measured [9]1425.3 Calculated (this work)

e1,max L = NiTi + �Cr2Ti 1202 Measured [9]1348.4 Calculated (this work)

e4,max L = (Cr, �Ti) + Ni3Ti 1223 Measured [9]1216.2 Calculated (this work)

E1 L = (Ni) + (Cr, �Ti) + Ni3Ti 1216 Measured [9]1216.0 Calculated (this work)

U1 L + �Cr2Ti = NiTi + (Cr, �Ti) 1109 Measured [9]1109.1 Calculated (this work)

E2 L = Ni3Ti + NiTi + (Cr, �Ti) 1100 Measured [9]1099.9 Calculated (this work)

U2 L + �Cr2Ti = NiTi + (Cr, �Ti) 1043 Measured [9]1043.1 Calculated (this work)

U3 L + NiTi = NiTi2 + (Cr, �Ti) 976 Measured [9]976.1 Calculated (this work)

ublattices, and Ti in the third and fourth sublattices. In addition, theite fractions of the Cr, Ni and Ti in first and second sublattices arequal, respectively, which means the first and second sublattices ofhe ordered B2 phase are equivalent. Similarly, the third and fourthublattices are also equivalent.

These phases diagrams calculated in the present workemonstrate the successful application of 4SL model forrdered/disordered bcc phases. It is expected that the 4SL model

an be applied to model the ordered/disordered transformations inther systems in order to develop reliable thermodynamic descrip-ion of multicomponent alloys with various bcc ordered/disorderedransitions.

[

[

E (Cr, �Ti) = (�Ti) + NiTi2 + �Cr2Ti 650 ± 2 Measured [6]643.1 Calculated (this work)

5. Summary

• Based on all the reliable data available in the literature, athermodynamic assessment of the Cr–Ni–Ti system is per-formed and an optimal set of thermodynamic parameters isobtained.

• The complicated ordered/disordered transition between Bcc A2and Bcc B2 has been molded satisfactorily using a four-sublatticemodel. Comprehensive comparisons show that most of theexperimental data are well accounted for by the present ther-modynamic description.

• The liquidus projection and the reaction scheme for the wholeCr–Ni–Ti system have been constructed, which are of inter-est for engineering applications as well as basic materialsresearch.

Acknowledgements

The financial support from the National Basic Research Programof China (Grant No. 2011CB610401), the National Natural ScienceFoundation of China (Grant Nos. 51021063 and 50971135), and Sci-entific Research Foundation for the Introduced Talents of AnhuiUniversity of Science and Technology are greatly acknowledged.

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