CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 41 (2013) 128–139
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CALPHAD: Computer Coupling of Phase Diagrams and
Thermochemistry
journal homepage: www.elsevier.com/locate/calphad
Thermodynamics of Ti–Ni shape memory alloys
E. Povoden-Karadeniz a,n, D.C. Cirstea b, P. Lang c, T. Wojcik d, E. Kozeschnik a,d
a
Christian Doppler Laboratory for Early Stages of Precipitation, Institute of Materials Science and Technology, Vienna University of Technology,
Favoritenstraße 9-11, 1040 Vienna, Austria
b
National Institute for Research and Development in Electrical Engineering INCDIE ICPE-CA, Advanced Materials Department, Splaiul Unirii, No. 313, Sector
3, Bucharest, Romania
c
Materials Center Leoben Forschung GmbH, Roseggerstraße 12, 8700 Leoben, Austria
d
Institute of Materials Science and Technology, Vienna University of Technology, Favoritenstraße 9-11, 1040 Vienna, Austria
art ic l e i nf o
a b s t r a c t
Article history:
Received 29 November 2012
Received in revised form
8 February 2013
Accepted 22 February 2013
Available online 23 March 2013
The thermodynamics of the Ti–Ni system are reviewed, and CALPHAD descriptions of metastable
intermetallic phases are presented. These phases play an important role as precipitates in shape memory
alloys. Metastable Ti3Ni4 and Ti2Ni3 are described as line compounds. Their thermodynamic model
parameters are optimized with experimental solvus data and molar enthalpies at 0 K from new firstprinciples analysis. Best results are obtained, when the thermodynamic description of the D024-ordered
TiNi3 phase is re-optimized with new thermodynamic data. This also requires adjustment of the other
phase descriptions, including B2 austenite and B19′ martensite. The modifications have important
consequences on the computed start temperature of the martensitic transformation, which is a crucial
property for the shape memory effect. R-phase, a metastable intermediate martensite, is considered in
the thermodynamic modeling. The following thermodynamic standard data for the metastable intermetallic phases are obtained at 298.15 K: ΔH1m(Ti3Ni4) ¼−34,714.5 J/mol, ΔH1m(Ti2Ni3) ¼−36,742 J/mol,
ΔH1m(R-phase) ¼−35,649 J/mol, S1m(Ti3Ni4) ¼31.91 J/mol K, S1m(Ti2Ni3)¼ 29.76 J/mol K, S1m(R-phase) ¼
27.87 J/mol K.
& 2013 Elsevier Ltd. All rights reserved.
Keywords:
Ti3Ni4
Ti2Ni3
TiNi3
Metastable phases
Shape memory
1. Introduction
Ti–50Ni to Ti–55Ni (at%) can be termed as the pioneer of shape
memory alloys (SMA) and a key system for studying phase
transformations and precipitate evolution in shape memory alloys.
Shape memory alloys are martensitic metals that “remember” the
original shape of their parent modification under specific conditions of temperature and mechanical loading/unloading. The
thermodynamics of the parent bcc-structured, ordered austenitic
B2 phase and the monoclinic martensitic B19′ phase are well
understood. In order to improve shape memory and mechanical
properties, SMA are usually aged at temperatures where precipitation of second phases from the supersaturated B2-ordered matrix
occurs. These phases are Ti3Ni4, Ti2Ni3 and the thermodynamically
stable TiNi3 (η–) phase. In particular, Ti3Ni4 plays an important role
for martensite formation. The martensite start temperature, Ms, is
strongly influenced by changes of plastic deformation limits
associated with precipitation hardening and the change of the
matrix composition due to precipitation [1–8]. For instance, an
n
Corresponding author. Tel.: þ 43 6763352362.
E-mail address: erwin.povoden-karadeniz@tuwien.ac.at
(E. Povoden-Karadeniz).
0364-5916/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.calphad.2013.02.004
increase of the martensite start temperature of approximately
30 K was reported due to aging of Ti–50.7 at% Ni [2]. Recently,
the fatigue failure of Ti–Ni SMA was related to the occurrence of
TiNi3 [9].
State-of-the-art precipitation simulation utilizes the thermodynamics of precipitates and matrix phase in the evolution
equations [10]. In order to evaluate the effects of precipitation,
thermodynamic descriptions of the precipitating phases are thus
required. Whereas the equilibrium thermodynamics of the Ti–Ni
system have been studied extensively [11–14], CALPHAD descriptions of the metastable intermetallic phases Ti3Ni4 and Ti2Ni3 are
missing. Gibbs energy polynomials of metastable Ti3Ni4 and Ti2Ni3
have not been published yet. The only Gibbs energy data are
available from Zhou et al. [15] and Guo et al. [16]. Both studies
derived the Gibbs energy of Ti3Ni4 simply from tangent construction and did not compare their results with other thermodynamic
and phase diagram data. Zhou et al. [15] did not consider
temperature-dependence, and Guo et al. [16] neglected heat
capacity in their formulation. In the present work, temperaturedependent Gibbs energy polynomials (in this paper, temperatures
are consistently given in Kelvin) of Ti3Ni4 and Ti2Ni3 are assessed
with first-principles thermodynamic data and metastable solvi
information. Metastability of Ti3Ni4 and Ti2Ni3 from the solvus to
room temperature is taken into account.
E. Povoden-Karadeniz et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 41 (2013) 128–139
2. Nature, structures and thermodynamics of Ti3Ni4 and Ti2Ni3
2.1. Nature and phase compositions of metastable phases in Ti–Ni
The discovery of metastable phases that occurred during aging
of Ti–Ni SMA dates back to the early seventies of the last century,
when Wasilewski et al. [17] identified Ti2Ni3 with X-ray diffraction
and X-ray probe micro-analyzing. Koskimaki et al. [18] detected
another intermetallic platelet phase in aged Ti–54.4 at% Ni, which
they called “x-phase”, by using electron microscopy. During that
time, the nature of phase relations among the new intermetallic
phases was not clear. This situation was changed by the accurate
experimental work of Nishida et al. [19]. These authors uncovered
the composition of “x-phase”, being Ti3Ni4 with energy dispersive
X-ray spectroscopy and found that Ti3Ni4 and Ti2Ni3 were essentially line compounds. Moreover, they understood that Ti3Ni4 and
Ti2Ni3 would be intermediate phases in the B2-structured Ti–Nimatrix evolving towards the B2 þTiNi3 equilibrium.
2.2. Structure
The structure of Ti3Ni4 was deciphered by Tadaki et al. [20]. The
phase is rhombohedral and has the space group R-3. The suggested atomic positions were later refined by Tirry et al. [21] using
least-squares optimization of diffracted beam intensities from
transmission electron microscopy. The preference of the refined
structure in terms of energy was confirmed by Density Functional
Theory (DFT) calculations by the same authors and later by
Wagner and Windl [22].
Ti2Ni3 is orthorhombic at low temperatures (Ti2Ni3_L) and
tetragonal at high temperatures [23]. Hara et al. [24] analyzed
X-ray diffractograms of aged Ti–52.0 at% Ni during heating from
298 K and 373 K and determined space group, lattice parameters
and atomic coordinates. From the change of lattice parameters, a
transition temperature near room temperature was proposed. In
Table 1, structural properties of Ti3Ni4 and Ti2Ni3 are summarized.
2.3. Phase diagram data
The thermodynamic stabilities of Ti3Ni4 and Ti2Ni3 are reflected
by their metastable solvi. In a solution-treated and quenched Ti–Ni
SMA, a metastable precipitate does not form above its metastable
solvus temperature at any ageing time. In computational thermodynamics, the thermodynamic solvus of a metastable phase (here
denoted as “metastable solvus”) is simply obtained by suppressing
the stable phase. Then, the phase boundaries of the “next-stable”
phase can be calculated, providing that its Gibbs energy is defined.
In the real case of a continuous heating experiment after
solution treatment and quenching of an alloy, phases will precipitate, grow and, eventually, disappear at the kinetic solvus. The
kinetic solvus temperature of small particles in an alloy matrix is
always lower than the metastable solvus temperature. This is due
129
to the increase of the Gibbs energy of precipitates by contributions
of the interfacial energy, and, in the case of coherency, strain field
around the particles. Keeping in mind this constraint, experimental solvus data help to define the thermodynamic stability of
metastable phases. Nishida et al. [19] presented important experimental solvus data in terms of a time–temperature-transformation
diagram of a Ti–52 at% Ni alloy. Concluding from his work, the
kinetic solvi of Ti3Ni4 and Ti2Ni3 are situated at approximately
1023 K and 930 K, respectively. Pelton et al. [26] reported dissolution of Ti3Ni4 between 773 K and 873 K in Ti–50.8 at% Ni. Koskimaki's “x-phase”, representing Ti3Ni4, formed in Ti–54.4 at% Ni
below 898 K.
2.4. Thermodynamic data
Recently, Stott et al. [25] determined the enthalpy of formation of Ti3Ni4 at 0 K by first-principles analysis using the
generalized gradient approximation (GGA) with projected augmented wave pseudo-potentials from Perdew, Burke and Ernzerhof (PAW-PBE) [27]. These authors obtain ΔH(Ti3Ni4), 0 K ¼
−37,629 J/mol (in the present study, energy values are consistently given for 1 mol of atoms). Hatcher [28] calculated the
enthalpies of formation with full potential linearized augmented plane wave code (FLAPW) based on DFT and obtained ΔH
(Ti3Ni4), 0 K ¼−45,900 J/mol. In the light of exactness of modern
density functional theory-based calculations, this difference is
considerably large. For Ti2Ni3, the FLAPW-derived value was the
same as for Ti3Ni4. In the present study, a new dataset of molar
enthalpies is produced with first-principles analysis using PAWPBE exchange correlation functional and GGA. The molar enthalpies at 0 K are compared with the results being available up to
now. The analysis is also performed for the intermetallic
equilibrium phases≥x(Ni) ¼0.5, in order to obtain an internally
consistent comparison of the molar enthalpies of the important
phases in Ti–Ni SMA. The first-principles analysis is documented
in detail in the following section.
First-principles analysis of Ti3Ni4 and Ti2Ni3 enthalpies. Firstprinciples calculations, based on DFT, require knowledge of the
atomic species and crystal structure and yield quantities related to
the electronic structure and total energy of a given structure. When
the ground state electron density is known, all properties of a system
are completely determined. By solving the Schrödinger equation for
stoichiometric compounds and their respective constituents with
given atomic structures, enthalpies of formation of compounds with
respect to the constituting metals at 0 K are obtained.
In the present study, the plane wave pseudo-potential method
of the Vienna Ab initio Simulation Package (VASP, version 5.2)
[29–32] with GGA PAW-PBE is used for the calculations. The
crystal structures with their phase-specific atomic sites, as given
in the Appendix of this paper, are used in the model. For all
the structures investigated, energy and equivalent k-point
meshes 20 20 20 are utilized to obtain the required energy
Table 1
Crystal structures of Ti3Ni4 and Ti2Ni3.
Phase
Structure
Space group
Ti3Ni4
Rhombohedral
Rhombohedral
Rhombohedral
Rhombohedral
R-3
R-3
R-3
R-3
Ti2Ni3_L
Orthorhombic
Bbmm
Ti2Ni3
Tetragonal
I4/mmm
Lattice parameters, nm
Unit cell angles
Method
Reference
a ¼0.6684
a ¼0.66697
a ¼0.6718
a ¼0.671
α ¼ 113.93
α ¼ 113.84
α ¼ 113.97
α ¼ 113.883
Exp.
DFT
DFT
DFT
Exp.
Tadaki et al. [20]
Stott et al. [25]
Tirry et al. [21]
Wagner andWindl [22]
Khalil-Allafi et al. [3]
a ¼0.43985, b ¼ 0.43705, c¼ 1.35442
Exp.
Hara et al. [24]
a ¼0.30954, c ¼ 1.35852
Exp.
Hara et al. [24]
130
E. Povoden-Karadeniz et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 41 (2013) 128–139
convergences. This limits the uncertainty to 0.01 eV. The default
energy cutoff (ENMAX) values for Ni and for Ti, as stored in the
VASP-file POTCAR, are ENMAX(Ti) ¼222.338 eV and ENMAX(Ni) ¼
367.945 eV. These are the recommended cutoff energies for d-elements, where semi-core p states are treated as valence states. In
the VASP calculations, the maximum cutoff energy, ENMAX(Ni) ¼
367.945 eV is used. Gaussian smearing with a smearing width of
0.1 eV was used. All calculations are fully relaxed with respect to
volume and cell coordinates.
For the evaluations of ΔHf of Ti3Ni4 and Ti2Ni3, the structural
data listed in Table 1 are used. Positions of atoms in the Ti3Ni4
unit cell are taken from Tadaki et al. [20]. For Ti2Ni3, these
are evaluated with the program VESTA [33], based on the structural data from Hara et al. [24]. Atomic positions are listed in
the Appendix of this paper. The resulting unit cell geometry of
Ti3Ni4 is visualized in Fig. 1. Fig. 2 shows the crystallographic unit
cell for the high-temperature tetragonal modification of Ti2Ni3
and the low-temperature orthorhombic modification of Ti2Ni3_L,
respectively.
3. Thermodynamic modeling and optimization
3.1. Modifications of previous thermodynamic descriptions
Experimental phase equilibria [17,34–47], calorimetric studies
[48–53] and activity measurements [54–56] delivered a sufficient
number of data for the CALPHAD assessment of thermodynamic
model parameters of the Gibbs energy polynomials of stable
phases. Phase diagram data of Ti2Ni, TiNi-B2 and TiNi3 are well
reproduced by the thermodynamic assessments [11–14]. Nevertheless, in the present study, the parameters of the thermodynamic description of D024-ordered TiNi3 are re-optimized. In
particular, it is found that with previous assessments the agreement between CALPHAD-optimized and DFT (density functional
theory)-based molar enthalpies is not sufficiently close in the light
of modern first-principles data accuracy, the difference being as
high as 10 kJ/mol. All optimizations of thermodynamic parameters
in this study were performed using the PARROT module of the
ThermoCalc [57] software, version S. PARROT minimizes the sum
of squared errors between thermodynamic input data and
Fig. 2. Unit cell geometry normal to crystallographic b of tetragonal Ti2Ni3 (a) and
orthorhombic Ti2Ni3_L (b) based on evaluated positions of atoms.
-30
Hf, kJ/mol of atoms
-35
De Keyzer09 [14]
Bellen96 [11]
Matsumoto05 [12]
-40
this w
-45
1st-princ.
-50
[25] GGA
[78] LMTO-ASA
[78] FP-LMTO
this work, GGA
-55
-60
0
Fig. 1. Unit cell
[111]-direction.
geometry
of
Ti3Ni4,
shown
along
the
crystallographic
ork
500
[58] direct synth. cal.
[58] heat content
[59] solution cal.
[49] cal.
[51] direct synth. cal.
1000
1500
Temperature, K
2000
Fig. 3. Re-assessed enthalpy of formation of TiNi3 compared with experimental
data, first-principles results and previous assessments. LMTO-ASA¼ linear-muffintin-orbital-atomic sphere approximation. FP ¼full-potential.
E. Povoden-Karadeniz et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 41 (2013) 128–139
assessment results. The assessment procedure includes the possibility of weighing input data for the optimization. If one tries a
better fit of the first-principles results, i.e. strong weight of these
data in the optimization, the metastable phases will become stable
at low temperatures. This, however, is in disagreement with the
experimental observations. As we prefer the newest experimental
enthalpies of formation (Fig. 3) from direct synthesis calorimetry
[58] and solution calorimetry [59] for the parameter optimization
of TiNi3 and experimental heat capacity data (Fig. 4) from Onderka
et al. [60], TiNi3 is stabilized at low temperatures. As a consequence, the optimized thermodynamics of Ti3Ni4 and Ti2Ni3 are
closer to our first-principles data, while they remain metastable.
Moreover, after the re-optimization, the molar enthalpy of TiNi3 is
in better agreement with first-principles results [25].
The modification of a single phase description, in fact, necessitates manipulations of all other model parameters as well, while
keeping the same sublattice descriptions as used in the assessment by De Keyzer et al. [14], and by Tang et al. [13] for the
martensite phase. These manipulations lead to slight deviations of
transition temperatures from earlier descriptions, which are listed
in Table 2.
Further differences concern the liquidus and solidus and the
martensite phase: In the present study, the course of liquidus and
solidus is adjusted to recent experimental results [63], which are
close to the early experimental work by Poole and Hume-Rothary
[35], as shown in Fig. 5.
This adjustment is obtained by reassessment of interaction
parameters of the liquid phase and the fcc-phase. The new
parameters are compared with previous assessments in Table 3.
With the re-assessed descriptions of the fcc phase and the
liquid phase the activities and the enthalpy of mixing remain
inside the experimental scatter [50,53,54,56].
For the re-optimization of the martensite phase, recent
enthalpy of formation at 0 K [64–67], the experimental decomposition temperature during slow cooling of TiNi-B2 [17] and
vibrational entropy data derived from low-temperature calorimetry and inelastic neutron-scattering [68,69] are taken into account.
As shown in the insert to the re-assessed equilibrium phase
diagram of the Ti–Ni system, Fig. 6, the composition of the B19′
131
phase does not depart from the composition of TiNi-B2 inside the
experimental detection limits.
In addition to B19′, the existence of intermediate martensite
phase modifications, orthorhombic B19 [1,64] and rhombohedral
R-phase [1,70], have been reported. These intermediate transformation stages are normally induced by additional alloying of third
elements [71,72]. Interestingly, R-phase has also been observed in
pure precipitation-hardened Ti–Ni SMA [73–77]. The intermediate
martensite modifications are included in the thermodynamic Ti–
Ni SMA database presented. Among the first-principles-results on
their relative stabilities [65–68,78] listed in Table 4, we prefer the
data from Hatcher et al. [66], who included R-Phase in the analysis
of molar enthalpies.
Further, reported formation temperature around 315 K during
slow cooling of aged Ti–Ni SMA [77] is accounted for in the
parameterization of R-phase. We obtain the following thermodynamic standard data of R-phase at 298.15 K: ΔH1m(R-phase) ¼
−35,722 J/mol, S1m(R-phase)¼ 27.6 J/mol K. Khalil-Allafi et al. [2]
clearly demonstrated that R-phase formation is controlled by
kinetics, involving delicate interplay between supersaturation of
the Ti–Ni B2 matrix, precipitate sizes and distribution and stress
fields. B19 is not found in pure Ti–Ni SMA at any temperatures. In
Table 2
Assessed transition temperatures in the Ti–Ni equilibrium phase diagram.
Phase transition
liq o –4bcc þ Ti2Ni
bcc o –4hcp þ Ti2Ni
liq þTiNi-B2 o –4Ti2Ni
liq o –4TiNi-B2
liq o –4TiNi-B2 þ TiNi3
liq o –4TiNi3
liq o –4TiNi3 þfcc
TiNi-B2 o –4 B19'
Transition temperatures (K)
This
work
Bellen
et al. [11]
Matsumoto
et al. [12]
Tang
et al. [13]
1221
1045
1250
1583
1419
1648
1559
353
1215
1040
1258
1584
1393
1653
1573
1215
1040
1258
1585
1393
1650
1576
1205
1040
1275
1580
1450
1660
1560
366
1750
45
35
u
Ne
30
ma
this
-K
nn
op
p’s
1700
work
Temperature, K
Heat capacity, J/molK atoms
40
25
20
1650
1600
15
10
5
1550
0
500
1000
1500
2000
Temperature, K
Fig. 4. Calculated heat capacity of TiNi3 compared with experimental data
[60-62,80] (symbols) and Neumann–Kopp's rule.
0
5
10
15
Mass percent ofTi
Fig. 5. Calculated liquidus and solidus in the Ni-corner. TL and Ts denote experimental liquidus and solidus data, respectively.
132
E. Povoden-Karadeniz et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 41 (2013) 128–139
the kinetic solvi are taken into account. The resulting values of A
and B are the result of a balanced weight of these input data.
Model parameters C–F are governed by the temperaturedependence of excess heat capacity. The heat capacities of these
metastable phases are not known. In such cases, normally Neumann–Kopp's rule is applied, and the heat capacity is equal to the
weighted sum of these parameters of the metals that constitute
the modeling, this is reflected when the same formation entropy
as in B19′ is assumed.
The re-optimized model parameters of the intermetallic equilibrium phases of the Ti–Ni system and are listed in Table 5.
Table 6 summarizes experimental [49,58–60,62,80], firstprinciples [32,33,64–67,78,81–83], previously assessed [11,12,14]
and re-assessed thermodynamic standard data of the intermetallic
equilibrium phases of the Ti–Ni system.
The experimental enthalpies of formation of TiNi3 presented in
Table 6 show a considerable spread among different studies [49,58]
using the same technique. The Guo and Kleppa´s value [58] was
reproduced recently by solution calorimetry [59] and lies closer to first
principles analysis. It is thus given a high weight in the thermodynamic parameter optimization. Accuracy of first-principles analysis
increased over the last years with the improvement of pseudopotentials by Perdew, Burke and Ernzerhof (PBE) [27]. FLAPW results seem
to have the same quality as first principle data with PBE. This was also
observed by Herper in a thorough study in Al–Cu–Zn [84].In contrast,
calculations with muffin-tin orbitals (MTO) and atomic sphere approximations (ASA) are a compromise between accuracy and computation
efficiency. In the present case, LDA-results for TiNi-B2 are closer to the
clalorimetric data than GGA-results. However, since TiNi-B2 is the only
phase with both LDA and GGA data available, general judgment of the
accuracy of GGA versus LDA data cannot be made.
Table 4
First-principles results of enthalpies of formation of martensite modifications at 0 K
from the literature and assessed values at 298.15 K referred to B2.
298.15 K
Method
Reference
−1700
−1800
−7815
−4728
−4052
−4438
−5347
LMTO-ASAa
FP-LMTOb
DFT-GGA
USPP-LDAc
USPP-GGA
LDA
FLAPW
Pasturel et al. [78]
Pasturel et al. [78]
Vishnu and Strachan [67]
Huang et al. [64]
Huang et al. [64]
Ye et al. [65]
Hatcher et al. [66]
CALPHAD assessed
This work
LMTO-ASA
FP-LMTO
DFT-GGA
USPP-LDA
USPP-GGA
LDA
FLAPW
Pasturel et al. [78]
Pasturel et al. [78]
Vishnu and Strachan [67]
Huang et al. [64]
Huang et al. [64]
Ye et al. [65]
Hatcher et al. [66]
CALPHAD assessed
This work
TiNi B19'
−3001
TiNi B19
þ 3600
þ 6800
−5114
−3859
−2895
−3281
−3911
3.2. Modeling of metastable phases
As there are no indications for off-stoichiometry of Ti3Ni4 and
Ti2Ni3, we model them as line compounds. Their Gibbs energy
polynomials read
−1564.3
R-Phase
GTixNiy
−x1H hcpðTiÞ
−y1H fmccðNiÞ ¼ A þ BT þ CTlnðTÞ þ DT 2 þET 3 þ FT ð−1Þ
m
m
f ccðNiÞ
þ x1GhcpðTiÞ
þ y1Gm
m
0K
−3275
−928.6
ð1Þ
a
The adjustable parameter A is optimized with molar enthalpies
of formation. Simultaneously, the constraint of metastability and
b
c
FLAPW
Hatcher et al. [66]
CALPHAD assessed
This work
LMTO-ASA¼ linear-muffin-tin-orbital-atomic sphere approximation.
FP ¼ full-potential.
USPP-LDA ¼ ultrasoft pseudopotentials-local-density approximation.
Table 3
Assessed interaction parameters (J/mol) of the fcc-phase and the liquid phase.
Order of interaction
This work
Bellen et al. [11]
Matsumoto et al. [12]
De Keyzer et al. [14]
0
−111400þ 5.66T
−52046
−160000 þ38T
−80000þ 18T
−5000−8T
þ3000
−99290.4 þ 6.21142T
−59449.5
Ref. [12]
Ref. [12]
Ref. [12]
Ref. [12]
−130333.64 þ20.22423T
−46714.31
−153707þ34.8594T
−81824.8þ 25.8099T
−10.0779T
0
−97427þ 12.112T
−32315
Ref. [12]
Ref. [12]
Ref. [12]
Ref. [12]
L(fcc)
L(fcc)
0
L(liquid)
1
L(liquid)
2
L(liquid)
3
L(liquid)
1
2000
1800
1648
1583
liquid
1559
1400
fcc
1200
354.0
B2
1419
TiNi3
1250
1221
bcc
1045
TiNi3+B2
B2+Ti2Ni
TiNi3+B19'
B19'+Ti2Ni
1000
353.0
800
Ti2Ni+hcp
600
400
353
200
0
0.2
0.4
B19'
Temperature, K
1600
0.6
0.8
1.0
352.0
0.4995
0.5000
Mole fractionTi
Fig. 6. Calculated equilibrium phase diagram of the Ti–Ni system.
0.5005
Table 5
Assessed model parameters of the intermetallic equilibrium phases.
Matsumoto et al. [12]
De Keyzer et al. [14]
This work
Tang et al.
[13]
Sublattice
description
Model type
(Ni,Ti)0.5(Ni,Ti)0.5
(Ni,Va)0.5 (Ni,Ti)0.5
Ref. [11]
(Ni,Ti)0.5(Ni,Ti)0.5
bcc-based split
Solid solution
Ref. [11]
bcc-based split
Ni:Ni
Ni:Ti
0
−33193.7 þ10.284T
Ref. [11]
Ref. [11]
0
−31000þ 11T
Ti:Ni
Ti:Ti
Va:Ni
−33193.7 þ10.284T
0
–
Ref. [11]
Ref. [11]
–
−31000þ 11T
0
–
−50100þ30T
0
–
Va:Ti
Va:Va
Ni:Va
Ti:Va
0
L Ni,Ti:Ni
0
L Ni:Ni,Ti
1
L Ni,Ti:Ni
1
L Ni:Ni,Ti
2
L Ni,Ti:Ni
2
L Ni:Ni,Ti
0
L Ti:Ni,Ti
0
L Ni,Ti:Ti
1
L Ti:Ni,Ti
1
L Ni,Ti:Ti
0
L Va:Ni,Ti
1
L Va:Ni,Ti
0
L Ni,Va:Ni
0
L Ni,Va:Ti
0
L Ti:Ni,Va
0
L Va:Ni,Ti
1
L Va:Ni,Ti
–
–
–
–
þ 55288.8 þ 25.4416T
þ 55288.8 þ 25.4416T
–
–
þ 6010.11þ 3.95974T
þ 6010.11þ 3.95974T
þ 60723.7−15.4024T
þ 60723.7−15.4024T
–
–
–
–
–
–
–
–
–
GNIBCCb
0.5GTIBCC þ 0.5GNIBCC þ 81198.65
−13.702875T
–
–
þ0.5GNIBCC þ 81198.65
−13.702875T
þ0.5GTIBCC þ 39351.22−8.296145T
–
–
–
–
−72295.24 þ23.47071T
–
−24442.75
–
–
–
–
–
–
−72295.24 þ23.47071T
−24442.75
−32012.19 þ13.247095T
−32012.19 þ13.247095T
–
−72295.24 þ23.47071T
−24442.75
(Ni,Ti,Va)0.5
(Ni,Ti,Va)0.5
bcc-based
split
0
−50100þ30T
–
–
–
–
Ref.
Ref.
–
–
Ref.
Ref.
Ref.
Ref.
–
–
–
–
–
–
–
–
–
–
–
–
–
−56500þ 23T
−56500þ 23T
þ 13500−8T
þ 13500−8T
þ 37300−14T
þ 37300−14T
þ 70000−13T
þ 70000−13T
−10000þ 8T
−10000þ 8T
–
–
–
–
–
–
–
0
0
0
0
−108297þ 60T
−108297þ 60T
−7258
−7258
þ15347
þ15347
þ56480
þ56480
–
–
–
–
–
þ22666−41T
þ22666−41T
–
–
Ti3Ni
Sublattice
description
Ni:Ni
Ni:Ti
Bellen et al. [11]
(Ni,Ti)(Ni,Ti)3c
Matsumoto et al. [12]
(Ni,Ti)(Ni,Ti)3
De Keyzer et al. [14]
(Ni,Ti)(Ni,Ti)3
This work
(Ni,Ti)(Ni,Ti)3
þ 4GNIHCP
þ 3GNIHCP þ GHSERTI157744þ 18.6544T
þ GNIHCP þ 3GHSERTI þ157744
−18.6544T
–
þ 224163.6−166.7884T
þ 224788.4−152.9008T
þ 240000
þ 80000
þ4GNIHCP
þ3GHSERNI þ GHSERTI-165687.84
þ29.43472T
þ3GHSERTI þGHSERNI−22753.08
þ4GHSERTI
þ73099−64.85152T
–
þ200000
þ200000
þ 2GNIHCP þ 2GHSERNI
þ 1.5GNIHCP þ 1.5GHSERNI þ 0.5GHSERTI þ0.5GTIFCC
−157744þ 18.6544T
þ 0.5GNIHCP þ 0.5GHSERNI þ 1.5GHSERTI
þ1.5GTIFCCþ 157,744−18.6544T
–
þ 143216−101.776T
þ 109156−66.448T
þ 240000
þ 80000
þ 2GNIHCP þ 2GHSERNI
þ 1.5GNIHCP þ 1.5GHSERNI þ 0.5GHSERTI þ0.5GTIFCC
−181000þ19T þ Gexd
þ 0.5GNIHCP þ 0.5GHSERNI þ 1.5GHSERTI
þ1.5GTIFCCþ 181,000−19Tþ Gexd)
þ 2GTIFCC þ2GHSERTI
þ 143216−101.776T
þ 109156−66.448T
þ 50000
þ 50000
Bellen et al. [11]
(Ni)(Ti)2
Matsumoto et al. [12]
Ref. [11]
De Keyzer et al. [14]
(Ni,Ti)(Ni,Ti)2
This work
(Ni,Ti)(Ni,Ti)2
þ 2GHSERTI þ GHSERNI
−82542 þ 8.56035T
–
–
–
Ref. [11]
Ref. [11]
þ 2GHSERTI þ GHSERNI−83342 þ 8.56035T þGexe
–
–
–
GHSERNI−GHSERTI þ 127542.7−8.560357T
þ 60000
þ 60000
Ref. [14]
Ref. [14]
Ref. [14]
Ti:Ni
Ti:Ti
0
L Ni:Ni,Ti
1
L Ni:Ni,Ti
0
L Ni,Ti:Ti
0
L Ti:Ni,Ti
Ti2Ni
Sublattice
description
Ni:Ti
Ti:Ni
0
L Ni,Ti:Ni
0
L Ni:Ni,Ti
a
[11]
[11]
[11]
[11]
[11]
[11]
133
Bellen et al. [11]
E. Povoden-Karadeniz et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 41 (2013) 128–139
TiNi-B2
E. Povoden-Karadeniz et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 41 (2013) 128–139
the intermetallic phases, thus C ¼D ¼E ¼F¼ 0. However, Ti3Ni4 and
Ti2Ni3 are intermediate compounds to the equilibrium phase,
TiNi3, and the same deviation from Neumann–Kopp' rule as in
TiNi3 is assumed for their heat capacities, parameterizing C–F,
accordingly. In Table 7, optimized model parameters A–F of Ti3Ni4
and Ti2Ni3 are summarized.
Table 8 represents a comparison of the CALPHAD values of
molar enthalpies referred to the molar enthalpy of TiNi3 with the
results from DFT-calculated data. It is seen that the results of
Ti2Ni3_L and Ti2Ni3 lie inside the accuracy limits of the first
principles analysis of 0.01 eV. Hence, Ti2Ni3_L is not described
separately in the modeling.
The metastable phase diagrams of the Ti–Ni system presented in Figs. 7 and 8 complete this section. For the validation
of the modeled metastable solvi, experimental metastable subsolvus data [17–19,26,85] are included as symbols in the figures.
For the evaluation of the thermodynamic metastability of Ti2Ni3
in Fig. 5, the equilibrium TiNi3 phase was suppressed. Accordingly, in Fig. 8, Ti2Ni3 was also suppressed and the nextstable phase, Ti3Ni4 appears in the metastable equilibrium with
TiNi-B2.
The optimized thermodynamic database for Ti–Ni SMA alloys,
mc_SMA_v1.000.tdb is attached as supplement to this paper.
Ref. [14]
Ref. [14]
This work
Tang et al.
[13]
134
Disordered parameters of Ti–Ni bcc_a2 were given in published assessments [11,12,14].
All G-functions of unary metals are taken from Dinsdale [79].
c
Some original sublattice formulas are transformed to obtain unified descriptions for direct comparisons.
d
Excess terms optimized with experimental heat capacity data [60] read þ 0.0072T2−5 10−8T3 þ 450525T(−1).
e
Excess terms optimized with experimental heat capacity data [80] read þ5.5 10−3T2−2.78T·ln(T)−1.87 10-7T3 þ33285T(−1).
f
Excess terms optimized with experimental heat capacity data [62] read −0.002T2 þ50000T(−1).
b
a
þGNIBCC
þ0.5GNIBCC þ 0.5GTIBCC
−43394 þ12.069T
þ0.5GNIBCC þ50000
þ0.5GTIBCC þ22500
−6000þ 16T
−24255 þ8T
þ GNIBCC
þ 0.5GNIBCC þ0.5GTIBCC
−46750 þ 16.3T þ Gexf
[13]
[13]
þ 8000
−18000þ 8T
Va:Ni
Va:Ti
0
L Ni:Ni,Ti
0
L Ni,Va:Ti
Tang et al. [13]
(Ni,Va)0.5(Ni,Ti)0.5
This work
(Ni,Va)0.5(Ni,Ti)0.5
TiNi B19'
Sublattice
description
Ni:Ni
Ni:Ti
þ60000
þ60000
L Ti:Ni,Ti
L Ni,Ti:Ti
0
–
–
–
–
0
Table 5 (continued )
Matsumoto et al. [12]
Bellen et al. [11]
TiNi-B2
De Keyzer et al. [14]
4. Discussion
4.1. Thermodynamics of TiNi3
Previously in this paper, it was mentioned that the quality of
the descriptions of metastable phases are improved by the
re-optimization of the TiNi3 phase. The re-assessed thermodynamics of TiNi3 require some discussion. When assessing the set
of experimental and first-principles data for the TiNi3 phase, it
became clear that it would not be possible to obtain a description, which is consistent with the available set of data. The
discrepancies among different research groups [11,12,14,54–
56,86–89] are obvious when comparing the different Gibbs
energies of formation of TiNi3 as a function of temperature, as
shown in Fig. 9.
In fact, it is not easy to judge, which values should be given a
high weight in the optimization. In the early assessment work
by Bellen et al. [11], it was decided not to use any of the Gibbs
energies based on electro-magnetic force (EMF) experiments in
the optimization, but to focus on the approximation of modeled
and experimental enthalpies of formation. Agreement of both
sets of experimental data would lead to an unrealistic formation
entropy close to 0 J/mol K. This argumentation needs reconsideration in the light of new experimental heat capacities.
Indeed, it is important to take care of physically realistic entropy
values in the modeling, and, in the case of TiNi3, the strong
exothermic character of formation of TiNi3 should be coupled
with a considerably negative entropy of formation. Smith et al.
[62] derived a ΔSf(TiNi3) ¼ −14.9 J/mol K at 1653 K, close to the
melting point. Preferring experimental heat capacities over
Neumann–Kopp's rule, applied in all previous assessments,
now both groups of experimental thermodynamic data, part of
calorimetric experiments as well as experimental activities are
approximated by the modeling, while the entropy of formation
remains highly negative, ΔSf ¼ −10.3 J/mol K at 1653 K. The least
squares of errors between modeling and experimental data are
obtained, when the enthalpies of formation from Moser et al.
[59] and Guo and Kleppa [58], as well as EMF data from
Kubaschewski [86] and chrono-potentiometry results from
Barner et al. [87] is given a high weight in the optimization.
E. Povoden-Karadeniz et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 41 (2013) 128–139
135
The interfacial energy contributes to the nucleation barrier to
the third power. This means that precipitation is very sensitive
to changes of interfacial energy. Therefore, in the thermokinetic
simulation also lower (10 mJ/m2) and higher (100 mJ/m2)
4.2. Thermodynamics of metastable phases
Different types of data have been used for the thermodynamic
descriptions of Ti3Ni4 and Ti2Ni3. Among these, the first-principles
results are regarded as the most exact data, whereas the kinetic
solvi are to be taken with care due to typical experimental
insecurities such as the quantitative quenching rates of solutiontreated alloy. The difference between the kinetic solvus of a small
particle in Ti–Ni-Matrix and the metastable equilibrium solvus is
governed by properties, which are not directly determined by
experiments. The most important property in this context is the
interfacial energy. This means that by adopting the enthalpies of
formation from first-principles analysis in the modeling and
considering metastability from high to low temperature, still
considerable uncertainty remains for the entropic contribution to
the Gibbs energy. In the present study, we circumvent this
dilemma by iterative test runs of thermo-kinetic precipitation
simulations with MatCalc, version 5.51, release 0.0019, using
assessed mobility diffusion data [90]. A published interfacial
energy of 50 mJ/m2 for Ti3Ni4 [15] was used, and the same
interfacial energy was assumed for Ti2Ni3. For each simulation,
the T-dependent B parameters of the descriptions of Ti3Ni4 and
Ti2Ni3 were modified stepwise, until the best reproduction of
simulated and experimental precipitation [27] was obtained.
Table 7
Optimized model parameters of Ti3Ni4 and Ti2Ni3.
Phase
A
B
C
D
E
F
Ti3Ni4
Ti2Ni3
−265000
−200000
−6
þ11
0
0
0.0175
0.0086
−3.00E−07
−2.00E−07
þ 609713
þ 589836
Table 8
CALPHAD and DFT-calculated molar enthalpies of metastable phases referred to
Ti3Ni in the form ΔHm−ΔHm(Ti3Ni).
Phase
0 K,
FLAPW
[28]
þ6000
Ti3Ni4
Ti2Ni3_L
Ti2Ni3
þ6000
0 K, PAWPBE [25]
0 K, PAW-PBE,
this work
298.15 K, CALPHAD,
this work
þ8683.8
þ 8759
þ 5799
þ 5549
þ7418
þ5390
Table 6
Thermodynamic data of intermetallic equilibrium phases in Ti–Ni.
Hm (J/mol), 298.15 K
Hm (J/mol), 0 K
TiNi3
−42,132
−36,871
−40,043
−36,978
−43,800±1600
−42,200±1200
−34,700
Sm (J/mol K), 298.15 K
Cp (J/mol K), 298.15 K
Method
Reference
27.44
27.01
25.09
27.01
21.42
25.02
25.02
25.02
Calphad assessment
Calphad assessment
Calphad assessment
Calphad assessment
Solution calorimetry
Direct synthesis calorimetry
Direct synthesis calorimetry
calorimetry
Calorimetry
1st-princ., GGA
1st-princ. (LMTO-ASA)
1st-princ. (FP-LMTO)
1st-princ. (PBE GGA)
This work
Bellen et al. [11]
Matsumoto et al. [12]
De Keyzer et al. [14]
Moser et al. [59]
Guo and Kleppa [58]
Kubaschewski [49]
Onderka et al. [60]
Smith et al. [62]
This work
Pasturel et al. [78]
Pasturel et al. [78]
Stott et al. [25]
Calphad assessment
Calphad assessment
Direct synthesis calorimetry
Calorimetry
Calorimetry
1st-princ. (LMTO-ASA)
1st-princ. (FP-LMTO)
This work
Bellen et al. [11]
Kubaschewski [49]
Zabdyr et al. [80]
Smith et al. [62]
Pasturel et al. [78]
Pasturel et al. [78]
21.79
23.4
−45,928
−47,100
−47,600
−46,313
Ti2Ni
−26,977
−26,901
−26,800
29.22
28.63
24.71
25.09
23.4
24.6
−33,700
−28,200
TiNi-B2
−34,721
−34,052
−33,339
−33,900
30.82
31.19
31.47
25.52
25.52
25.58
Calphad assessment
Calphad assessment
Calphad assessment
direct synthesis calorimetry
1st-princ., GGA
1st-princ. (LMTO-ASA)
1st-princ. (FP-LMTO)
1st-princ. (FP-LMTO)
1st-princ. (FLAPW)
1st-princ. (PW/PBE)
1st-princ. (EMTO/GGA)
1st-princ. (PW/LDA)
This work
Bellen et al. [11]
Matsumoto et al. [12]
Kubaschewski [49]
This work
Pasturel et al. [78]
Pasturel et al. [78]
Sanati et al. [81]
Hatcher et al. [66]
Stott et al. [25]
Lu et al. [82]
Huang et al. [83]
22.31
25.54
24.2
Calphad assessment
Calorimetry
1st-princ. (LMTO-ASA)
1st-princ. (FP-LMTO)
1st-princ. (FLAPW)
This work
Smith et al. [62]
Pasturel et al. [78]
Pasturel et al. [78]
Hatcher et al. [66]
−32,784
−38,100
−36,000
−38,000
−34,100
−31,840
−36,665
−33,770
TiNi B19'
−37,721
−39,800
−37,800
−39,500
136
E. Povoden-Karadeniz et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 41 (2013) 128–139
1600
Sf, J/mol K
Wasilewski1971 [17]
Nishida1986 [19]
Pelton2000 [26]
Zheng2008 [5]
Paryab2010 [6]
1500
1400
3
Ti3Ni4
1
liquid
1200
1100
1000
Ti2Ni+
TiNi-B2
800
700
0.45
TiNi-R
TiNi3
-7
TiNi-B19'
0.50
0.55
0.60
0.65
0.70
0.75
-9
Fig. 7. Metastable phase diagram of the Ti–Ni system. The TiNi3 phase is suspended from
the calculation. Symbols denote conditions of experimentally observed TiNi-B2þTi2Ni3.
1600
400
1400
1300
350
300
Temperature,K
1500
liquid
1200
fcc
250
200
150
700
0.45
Ti3Ni4
0.50
0.55
50
0.60
0.65
0.70
0.75
10-3
Fig. 8. Metastable phase diagram of the Ti–Ni system. TiNi3 and Ti2Ni3 are
suspended from the calculation. Symbols denote conditions of experimentally
observed TiNi-B2 þ Ti3Ni4.
-20
Gf,kJ/mol
-25
-30
-35
-40
-45
-50
1300
1500
(th
is
wo
rk
wo
)
rk
)
T0, exp.:
Tang99 [13]
1700
Mole fraction of Ni
Fig. 11. Calculated T0 and Ms of B19′ martensite (lines) compared with experiments
(symbols).
-15
1100
s
0
480 485 490 495 500 505 510 515 520
Mole fraction of Ni
900
(th
is
9]
800
M
Ms[
Ti2Ni+
TiNi-B2
0
Otsuka2005 [1]
Khalil-Allafi2006 [3]
Wasilewski71 [17]
Nishida86 [19]
Bogdanoff2001 [69]
Miyazaki86 [93]
Hanlon71 [94]
Melton81 [95]
Tang2000 [96]
Kornilov71 [97]
Smith91 [98]
Saburi82 [99]
]
[9
100
900
T
M s, exp.:
T0
TiNi-B2
Temperature, K
Fig. 10. ΔHf versus ΔSf of Ti–Ni phases at 298.15 K.
Nishida [19] Zheng2008 [5]
Pelton [26]
Cao2012 [7]
Khalil-Allafi2002 [4]
Koskimaki1969 [18]
Otsuka2002 [84]
Zheng2008 [5]
1000
-45
-40
-5
Mole fraction of Ni
1100
-35 Ti2Ni3
Ti2Ni
-3
Ti2Ni3
900
-30
-1
fcc
Hf, kJ/mol
TiNi-B2
0
TiNi-B2
Temperature, K
1300
1900
Temperature, K
Fig. 9. Gibbs energy of formation of TiNi3 from various sources.
E. Povoden-Karadeniz et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 41 (2013) 128–139
-43
Gm, kJ/mol
-44
Rph
B1
as
e
9'
-45
T0
-46
Tc
hand, the re-assessed data agree with experimental data from
Refs. [1,3,17,19,94–99]. As seen in Fig. 11, the kink of T0-course and
derived Ms-temperatures above x(Ni) ¼ 0.5, using the same
assumptions as Tang et al. [13] for the calculation of Ms, is less
pronounced than proposed by these authors. The slope of T0 is
resulting from optimized 1st and 2nd order interaction parameters
0
for the ordering of TiNi-B2, as well as0 LB19
Ni:Ni,Ti .
The thermodynamic relations between B19′ and R-phase are
shown in Fig. 12. The critical temperature of equal Gm is calculated
at 388 K, which is above the thermodynamic equilibrium transformation to TiNi-B2. This result is quite reasonable, as slow
cooling of stress- and precipitate-free Ti–Ni SMA results in B19′
martensite only, indicating the metastable nature of R-phase. This
means that the cross-over between Gm(B19′) and Gm(R-phase)
must be kept above the B2-formation temperature.
B2
-47
137
260
300
340
380
Temperature, K
Fig. 12. Molar Gibbs energies of B19′ and R-phase as a function of temperature.
T0 is calculated at 353 K. Gm(B19′) ¼ Gm(R-phase) at Tc ¼388 K.
interfacial energy values were tested. Those lead to worse overall
reproduction of available data, and for the time being, the
suggested interfacial energy [15] is adopted.
In Fig. 10, the thermodynamic standard data of equilibrium and
metastable phases resulting from the present assessment are
shown. A ΔHf versus ΔSf plot chosen reveals a typical dependence
for the phases, which form precipitates in Ti–Ni SMA (black
diamonds in Fig. 10). ΔSf decreases with increasing phase stabilities. Ti3Ni4, the least-stable precipitate, has high positive entropy
of formation, whereas the equilibrium phase TiNi3 has a strongly
negative entropy of formation. Similar trends were shown in other
systems with metastable precipitates, e.g. Refs. [91,92]. An even
stronger tendency of decreasing ΔSf is calculated from TiNi-B2
austenite via intermediate martensitic TiNi-R-phase, to TiNi-B19′
martensite. This is a result of stability considerations explained in
the next section.
4.3. Martensitic transformation
The present re-assessment of the Ti–Ni system has also strong
influence on the martensitic transformation as follows. With the
present model parameters, the two-phase region of TiNiB2 þmartensite becomes smaller compared to the work of Tang
et al. [13]. This is more realistic, since the physical meaning of two
transformations involving TiNi-B2 and martensite is in question in
the light of first-order martensitic transformation. For the model
parameterization of B19′, enthalpies from first principles were
given a high weight due to their consistency among different
groups (see Tables 4 and 6). This requires a high negative entropy
of formation of B19′ relative to TiNi-B2 (see Fig. 10), ΔSf ¼−8.5J/
mol K for the reproduction of T0-temperatures. The entropy data
available [68,69] from low-T calorimetry and inelastic neutron
scattering is significantly lower, ΔSf ¼ −3.9±0.75 J/mol K. However,
adopting of this value would lead to a large difference between
assessed and first principles enthalpies or T0-temperatures. The
re-assessed T0-temperatures for the martensitic transformation,
where the parent phase TiNi-B2 and the B19′ phase have the same
Gibbs free energy, lead to higher Ms-temperatures compared to
Tang et al. [13] and experiments from Refs. [93–95]. On the other
5. Conclusions
The thermodynamic assessment of metastable Ti3Ni4 and
Ti2Ni3 phases, based on new first-principles results and metastable
solvi, is coupled with necessary re-considerations of the thermodynamic equilibrium descriptions.
New experimental heat capacities were given high weight in
the re-optimization of TiNi3. These data deviate significantly from
Neumann–Kopp's rule, and former constraints for the modeling
were loosened. New calorimetric enthalpies of formation and
enthalpy increments, ΔTHf−Δ298.15 KHf, are preferred, and activity
measurements are included in the parameter optimization of
TiNi3. The re-calculated enthalpy of formation shows a pronounced temperature dependence.
A systematic dependence of ΔSf of the thermodynamic stability
of the intermetallic phases precipitating in SMA is identified.
Ti3Ni4 has the highest entropy and enthalpy of formation, followed
by Ti2Ni3 and TiNi3. The presented thermodynamic descriptions of
metastable phases in Ti–Ni are essential prerequisites for the
evaluation of the changing martensite start temperature in Ti–Ni
SMA associated with precipitation during ageing.
Acknowledgment
The contribution of the Vienna Computational Materials Laboratory
(VICOM) is acknowledged in allocating computational capacities,
maintenance and technical support. Financial support by the Christian
Doppler Forschungsgesellschaft (CDG), and by the Austrian Federal
Government (in particular from the Bundesministerium für Verkehr,
Innovation und Technologie and the Bundesministerium für
Wirtschaft, Familie und Jugend) and the Styrian Provincial Government, represented by Österreichische Forschungsförderungsgesellschaft mbH and by Steirische Wirtschafts-förderungsgesellschaft
mbH, within the research activities of the K2 Competence Centre on
“Integrated Research in Materials, Processing and Product Engineering”, operated by the Materials Center Leoben Forschung GmbH in the
framework of the Austrian COMET Competence Centre Programme, is
gratefully acknowledged.
Appendix A
See appendix Table A1.
138
E. Povoden-Karadeniz et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 41 (2013) 128–139
Table A1
Positions of atoms, as used in the first-principles analysis.
Ni3Ti
DFT structure
x
y
z
Occupancy
Ni1
Ni2
Ni3
Ni4
Ni5
Ni6
Ni7
Ni8
Ni9
Ni10
Ni11
Ni12
Ti1
Ti2
Ti3
Ti4
0.5000
0.5000
0.0000
0.5000
0.5000
0.0000
0.1645
0.8355
0.3289
0.6711
0.1645
0.8355
0.0000
0.0000
0.6667
0.3333
0.5000
0.5000
0.5000
0.0000
0.0000
0.5000
0.8355
0.1645
0.1645
0.8355
0.3289
0.6711
0.0000
0.0000
0.3333
0.6667
0
0.5
0
0
0.5
0.5
0.25
0.75
0.75
0.25
0.25
0.75
0
0.5
0.25
0.75
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
b
0.5101
c
0.8307
α
901
β
901
γ
1201
Ni1
Ni2
Ni3
Ni4
Ni5
Ni6
Ni7
Ni8
Ti1
Ti2
Ti3
Ti4
Ti5
Ti6
Lattice parameters: a¼ b¼ c ¼0.672 nm; α ¼β ¼ γ¼ 113.971
0
0.5
0.0605
0.7574
0.5931
0.5931
0.0605
0.7574
0.4989
0.2513
0.1125
0.1125
0.4989
0.2513
0
0.5
0.5931
0.0605
0.7574
0.0605
0.7574
0.5931
0.1125
0.4989
0.2513
0.4989
0.2513
0.1125
0
0.5
0.7574
0.5931
0.0605
0.7574
0.5931
0.0605
0.2513
0.1125
0.4989
0.2513
0.1125
0.4989
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Ti2Ni3
Ni1
Ni2
Ni3
Ni4
Ni5
Ni6
Ti1
Ti2
Ti3
Ti4
0
0
0.5
0.5
0
0.5
0
0
0.5
0.5
0
0
0.5
0.5
0
0.5
0
0
0.5
0.5
0.196
0.804
0.696
0.304
0
0.5
0.391
0.609
0.891
0.109
1
1
1
1
1
1
1
1
1
1
b
0.3095
c
1.3585
α
901
β
901
γ
901
Ni1
Ni2
Ni3
Ni4
Ni5
Ni6
Ni7
Ni8
Ni9
Ni10
Ni11
Ni12
Ti1
Ti2
Ti3
Ti4
Ti5
Ti6
Ti7
Ti8
0.725
0.275
0.225
0.775
0.232
0.768
0.232
0.768
0.732
0.268
0.732
0.268
0.227
0.773
0.227
0.773
0.727
0.273
0.727
0.273
0.25
0.75
0.25
0.75
0.25
0.75
0.25
0.75
0.25
0.75
0.25
0.75
0.25
0.75
0.25
0.75
0.25
0.75
0.25
0.75
0
0
0.5
0.5
0.3
0.7
0.7
0.3
0.8
0.2
0.2
0.8
0.106
0.894
0.894
0.106
0.606
0.394
0.394
0.606
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
b
0.437
c
1.3544
α
901
β
901
γ
901
Lattice parameters (nm)
a
0.5101
Ti3Ni4
Lattice parameters (nm)
a
0.3095
Ti2Ni3_L
Lattice parameters (nm)
a
0.4398
E. Povoden-Karadeniz et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 41 (2013) 128–139
Appendix A. Supporting information
Supplementary data associated with this article can be found in
the online version at http://dx.doi.org/10.1016/j.calphad.2013.02.
004.
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