Precipitation of Heusler Phase (Ni2TiAl) from B2-TiNi in
Ni-Ti-Al and Ni-Ti-Al-X (X=Hf, Zr) Alloys
J. Jung, G. Ghosh, D. Isheim and G.B. Olson
Department of Materials Science and Engineering
Robert R. McCormick School of Engineering and Applied Science
Northwestern University
2220 Campus Dr.
Evanston, IL 60208-3108, USA
E-mail: g-ghosh@northwestern.edu
Tel: (847)467-2595, Fax: (847)491-7820
Abstract: The precipitation of Heusler phase (L21: Ni2TiAl) from supersaturated B2 (TiNibased) matrix at 600 and 800˚C is studied using transmission and analytical electron microscopy
(AEM), and three-dimensional atom-probe (3DAP) microscopy in Ni-Ti-Al and Ni-Ti-Al-X
(X=Hf, Zr) alloys. The B2/L21 two-phase system, with ordered structures based on the bodycentered cubic lattice, is chosen for its microstructural analogy to the classical /’ system with a
face-centered cubic lattice. Knowledge of the temperature dependent partitioning of alloying
elements and their atomic volumes in the B2-TiNi and L21 phases is desired to support design of
high-performance shape memory alloys with controlled misfit strain and transformation
temperatures. After aging at 600˚C for up to 2000 hours, the L21 precipitates remain fully
coherent at a particle diameter of ~20 nm.
The observed effects of a misfit strain of –2% on the
microstructure of the B2/L21 system are similar to those theoretically predicted and
experimentally observed for the /’ system. The similarities are demonstrated in terms of the
precipitate shape, spatial distribution and a minimum distance of separation between L21
precipitates. However, all these effects disappear after aging the alloys at 800˚C for 1000 hours
when the L21 precipitates become semicoherent at particle diameters above ~400 nm.
A simple
analysis of the size evolution of L21 precipitates after an isochronal aging (1000 hours)
experiment suggests that they follow coarsening kinetics at 600˚C while growth kinetics at 800˚C,
1
consistent with the Langer-Schwartz theory of precipitation kinetics which predicts that a high
supersaturation suppresses the growth regime. Microanalysis using AEM and 3DAP define the
TiNi-Ni2TiAl phase boundaries at 800 and 600˚C. At 800°C, Hf and Zr partition to the B2-TiNi
while at 600°C they partition slightly to the L21 phase and strongly to the metastable phase
Ti2Ni3. To describe the composition dependence of the lattice parameter of multicomponent B2
and L21 phases the atomic volumes of Al, Hf, Ni, Ti and Zr in B2-TiNi and L21 phases are
determined. A simple model is proposed to predict the lattice parameters of these phases in
multicomponent systems.
2
1.
Introduction
The shape-memory effect (SME) is a consequence of a crystallographic reversible, thermoelastic
martensitic transformation.
Shape memory actuation occurs when a shape memory alloy is
deformed in its martensitic state, below its Ms temperature; the deformed shape is maintained
upon unloading. Once reheated beyond the reverse transformation temperature Af, a shape
memory alloy (SMA) will work against a resisting force to regain its original shape.
Recently,
the demand for powerful microactuators in MEMS devices has motivated significant SMA thin
film research
[1,2,3,4,5,6].
For engineering applications, it is essential that the shape-memory
behavior is repeatable and predictable after many cycles through the transformation.
Traditional
SMA microactuators used in MEMS devices suffer from limited cyclic life due to
accommodation slip.
To improve the output force and the cyclic lifetime of TiNi–based alloys, the strength of the
alloy must be improved.
Kajiwara et al.
[7, 8, 9]
Various types of precipitate strengthening may be considered. First,
found that subnanometric thin plate metastable bct precipitates formed
when sputter-deposited Ti-rich TiNi shape-memory films are annealed in the temperature range
of 377 to 827°C. With these fine precipitates in the parent phase, they could achieve recovery
strength as high as 670 MPa.
However, these precipitates have been observed only after
annealing of sputter deposited and amorphous TiNi thin films.
Second, the precipitation of
equilibrium Heusler phase (Ni2TiAl-type with L21 structure) in TiNi (B2) may be considered.
Since B2-TiNi (the nearest neighbor ordered structure based on the bcc lattice) and the Heusler
phase (the next-nearest neighbor ordered structure based on the bcc lattice) form an isomorphous
system, homogeneous precipitation of the Heusler phase is expected. In fact, Ohishi et al.
[10]
observed a very uniform distribution of the Heusler phase in a B2-TiNi matrix during aging of a
Ni-43Ti-7Al (in at%) alloy at 800˚C. Koizumi et al.
3
[11]
demonstrated that the precipitation of
Heusler phase increases the compressive yield strength of 50.71Ni-40.86Ti-8.43Al (in at%) by an
order of magnitude up to 2300 MPa. This strengthening method is applicable to both thin film
and bulk alloy processing. However, accurate knowledge of the TiNi-Ni2TiAl phase relations is
needed to design high-performance shape memory alloys, especially at lower temperatures (≤
800°C) where the processing of both bulk and thin film SMAs can be carried out.
According to known lattice constants
[12],
there is a lattice misfit between TiNi and Ni2TiAl,
as determined by the relation
a Ni2 TiAl  2aTiNi 
  0.0257

2aTiNi

  
[1]
where a Ni2 TiAl is the lattice parameter of Ni2TiAl (a=0.5865 nm) and aTiNi is the lattice
parameter of TiNi (a=0.3010 nm).
Lattice misfit arising from different lattice parameters
between two coherent phases causes coherency strains with an associated volume strain energy
that can affect the precipitate shape, the spatial distribution and the coarsening behavior [13].
To
promote homogeneous precipitation, to retain coherency at larger particle size and to reduce the
interfacial frictional work for martensite nucleation and variant growth, it is necessary to
minimize the lattice mismatch. To achieve the lowest possible misfit between B2 and L21
phases, we consider the possibility of increasing the lattice parameter of the latter phase by
adding Hf or Zr in the alloy.
These additions are attractive because both Hf and Zr are known
to be martensite stabilizers [14,15], which can be used to control the transformation temperatures.
However, in a two-phase system it is necessary to consider the relative partitioning behavior.
The partitioning behavior of Cu, Fe, Hf, Nb, Ru and Ta between B2 and L21 has been studied [16]
at 1100°C. It is found that the elements with large atomic size such as Hf, Nb and Ta prefer to
partition to the B2 phase while small atoms such as Cu and Ru prefer L21. Since 1100°C is a
4
very high temperature, further information is needed regarding the partitioning behavior of these
elements at lower temperatures (≤ 800°C) relevant to processing of TiNi-based SMAs.
In this study, we report for model Ni-Ti-Al, Ni-Ti-Al-Hf and Ni-Ti-Al-Zr alloys: (i) the
microstructure evolution of L21 precipitates in a B2 matrix during isothermal aging at 800 and
600°C; (ii) the phase equilibria at 800 and 600°C, (iii) the partitioning behavior of Al, Hf and Zr
between B2 and L21, and (iv) a critical analysis of atomic volumes of Al, Hf, Ni, Ti and Zr in B2
and L21. Based on this analysis, a simple model is proposed to calculate the lattice parameter of
multicomponent TiNi-based alloys, and thereby predict the lattice misfit of a two-phase alloy.
2.
Experimental Procedures
Ni-Ti-Al and Ni-Ti-Al-X (X=Hf, Zr) alloys were prepared by arc-melting in an argon atmosphere
using pure elements (99.999 wt% Ni, 99.99 wt% Ti, 99.999 wt% Al, 99.9 wt% Hf, and 99.999
wt% Zr). The nominal compositions of alloys are Ni-45Ti-5Al, Ni-40Ti-5Al-5Hf and Ni-40Ti5Al-5Zr (at%, from here on compositions are in at%). Each as-cast specimen was sealed in an
evacuated quartz capsule and solution treated at 1100°C for 100 hours. After quenching by
crushing the capsules in oil, different sets of specimens were annealed at 800°C for 1000 hours
or 600°C for 1000 or 2000 hours in evacuated quartz capsules, and then quenched into oil. Thin
foils for TEM observation were prepared by standard twinjet electropolishing using a solution of
20% perchloric acid in 80% methanol as electrolyte at –40 to –50°C.
Conventional transmission electron microscopy (CTEM) was performed in a Hitachi H8100
microscope operated at 200kV. The centered dark field TEM micrographs were scanned and
5
the projected area of the L21 precipitates was measured using NIH-Image software version 1.62
[17].
Based on these measurements, the average equivalent spherical radius of the precipitates
was derived.
For specimens aged at 800°C, the analytical characterization was performed in a Hitachi HF2000 analytical electron microscope (AEM) equipped with an ultrathin-window Link energy
dispersive X-ray (EDS) detector and data processor (QX2000). The AEM was also operated at
200kV. The take-off angle for the EDS detector was 68°. The X-ray collection time was 100 s
and the electron probe size was about 8 nm.
Care was taken to ensure that the particle being
analyzed was not in a two-beam condition in order to minimize electron-channeling effects
[18].
Background correction was done using the Desktop Spectrum Analyzer (DTSA 2.5.1) software
[19].
The compositions of B2 and L21 phases in equilibrium at 800°C were determined by
analyzing the EDS data using a standard calibration method.
The background-subtracted
integrated intensities of the X-ray spectra were converted to compositions by using the CliffLorimer [20] equations:
wj
I
1
 k j / Ni j  ACF 
 I j 
w Ni
INi
1  
 I j 
[2a]
wAl  wHf  wNi  w Ti  wZr  1
[2b]
where j = Al, Hf, Ti and Zr; wj is the weight fraction of element j, Ij is the X-ray intensity of
element j, [ACF] is the absorption correction factor, Ij/Ij is the ratio of fluorescence intensity to
primary intensity, and kj/Ni is the proportionality constant or the Cliff-Lorimer factor which was
determined using thin foils of solution treated alloys with known compositions.
6
Although there
are several ways to define Ij, we have taken Ij as the background subtracted integrated intensity of
the K peak of element j . The absorption correction factor is given by [18]




Ni
  j  1  exp    SPEC t  cosec  
SPEC
ACF  
Ni

j
 
  SPEC  1  exp    SPEC t  cosec  
[3a]
  SPEC   Al wAl    Hf w Hf   Ni w Ni   Ti wTi   Zr w Zr
j
j
j
j
j
j
[3b]
where  is the density, t is the thickness of the sample, and  is the take-off angle of the X-ray
detector. The mass-absorption coefficients for the pure elements   A l , etc., in Eq. [3b]
j
were taken from those listed in Reference 21. The fluorescence yield was neglected. X-ray
spectra were collected from foils the thickness of which were 100 nm or less.
In the
aforementioned thickness ranges, Hf, Ni, Ti, and Zr satisfied the criteria of a thin foil and the
Cliff-Lorimer factor for Al was determined using the extrapolation method [22] due to the strong
thickness dependence.
The compositions of B2 and L21 phases were determined by analyzing about 30 EDS spectra
for each. The statistical accuracy of the composition determination from Eq. [2a] is primarily
limited by the counting statistics of the X-ray collection process
[23].
When the X-ray spectra
are collected for a sufficiently long time to obtain several thousand counts in each peak, the
counting statistics can be assumed to follow a normal distribution.
However, in a
multicomponent system it may be difficult to satisfy this criterion for each element if
experiments are to be carried out within a reasonable period of time. Nevertheless, since the
composition of the phases was determined by analyzing sets of about 30 data, the confidence
interval is estimated by the statistics of the student t-distribution. The total relative error (  wj )
in the determination of composition is given by
7
 wj   kj /Ni   Ij / INi
[4]
where  kj /Ni and  Ij /INi are the relative errors associated with k factors and in the counting
statistics of X-rays in the specimen of unknown composition, respectively. They are given by
[23]
 kj / N i 
 Ij /IN i 
where
n 1
t99
k
100
n k j /Ni
[5]
n 1
t99
I
100
n I j / I Ni
[6]
n 1
are the student t values for n measurements at 99% confidence level;  k and I
t99
are the standard deviations for the k factor and intensity ratio measurements, respectively; k j / Ni
is the mean of the n values of the k factor, and
I j / I Ni is the mean of the n values of I j / I Ni .
The relationship between the atomic fraction and the weight fraction was used to calculate the
total relative error in the former from that in the latter using a standard mathematical procedure.
A 3-Dimensional Atom-Probe (3DAP) field-ion microscope was employed to determine the
composition of the phases in specimens aged at 600°C. The 3DAP is equipped with a reflectron
lens for energy compensated time-of-flight mass spectrometry. After grinding the specimens to
roughly a 1 mm 1 mm cross section, field-ion microscopy (FIM) tips were electropolished in a
solution of 2 vol.% perchloric acid in butoxyethanol. For field-ion imaging, 110-5 Pa Ne was
used, and the tips were cooled between –193 and –233°C. Atom-probe analyses were carried
out at –223 and –243°C at a pulse voltage-to-d.c. voltage ratio f = 0.19-0.20.
The statistical
error in this analysis is caused by the uncertainty due to counting statistics. Following the
binomial distribution, the standard error is given by
8

c1 c 
N
[7]
where c is the composition in atomic fraction and N is the total number of atoms detected.
A reduction in the d.c. voltage field for ion imaging by 15% (compared to the voltage in use
for evaporation) of the field ion microscope was tried to reveal precipitate particles as suggested
by Warren et al. [24]; however, it did not create enough phase contrast.
Preferential retention of
Ti at the top plane of the crystallographic pole was detected and therefore compositional analysis
was conducted on a region carefully selected away from the pole. Homogeneous regions were
obtained by cutting out a volume of the reconstruction, which has no crystallographic pole or
phase boundaries. The peaks were deconvoluted in reference to the natural isotope abundances
to obtain correct compositions.
The 3DAP technique reconstructs a specimen volume of
typically 101050 nm3, based on the spatial coordinates and the chemical identity of each
detected atom.
Hence without enough phase contrast numerous trial analyses had to be
conducted to come across the precipitates, limiting the number of analyzed precipitates.
X-ray diffraction was performed using a Scintag machine with a copper target, excited to
40kV and 20mA. A step size of 0.01 degrees and a counting time of 30 seconds per step were
used for X-ray diffraction experiments.
The X-ray diffraction peaks were deconvoluted by the
MacDiff program [25], using a pseudo-Voigt method [26], to obtain the lattice parameter of B2 and
L21 phases. High purity silicon powder was used as a standard for correcting the diffractometer
misalignment. Silicon has a well-defined (220) diffraction peak at 2=47.302°, which is near to
but does not overlap with the expected peaks of the specimens.
3. Results
9
3.1 Phases and Microstructure: Conventional Transmission Electron Microscopy
The presence/absence of Heusler phase in TiNi can be investigated by electron diffraction along
the [011] or [112] zone axis.
Figure 1 shows a bright-field TEM micrograph of a solution
treated specimen and the [011] diffraction pattern, indicating the absence of any Heusler phase.
Figure 2 shows dark-field images of Heusler precipitates in the specimens aged at 800°C for
1000 hours. The presence of Heusler phase can be confirmed due to the extra superlattice
reflections of the diffraction pattern (see inset in Fig. 2 (a)). Centered dark-field images are
obtained by using the (111)-type superlattice reflection specific to Heusler ordering, shown in the
diffraction pattern. As seen in Fig. 2 (see especially Fig. 2 (c)), misfit dislocations are present at
the precipitate/matrix interfaces of these particles which are greater than ~400 nm in diameter.
Also, the L21 particles are larger than the foil thickness so that analytical electron microscopy can
be conducted without matrix overlap. The shape of the L21 precipitates ranges from nearly
spherical for the ternary alloy to irregular for the quaternary alloys. An apparent consequence of
loss of coherency is the physical coalescence of the precipitates, as shown in Fig. 2 (b). This
has been attributed to rapid diffusion interaction in which two or more particles become one by
particle migration [27] and/or anisotropic mass flow.
Figure 3 shows TEM images of Heusler precipitates taken in the B2 matrix of specimens
aged at 600°C for 1000 and 2000 hours.
In contrast to the 800°C aging treatment, the
precipitates remain sufficiently small (<30 nm diameter) to be fully coherent even after aging for
2000 hours at 600°C. As seen in Fig. 3, the strong dominance of the coherency strains in this
two-phase aggregate is manifested by (i) a cuboidal shape of L21 precipitates and (ii) a highly
ordered spatial distribution with alignment of L21 precipitates along the elastically soft <100>
directions of the B2 matrix. The precipitates are distributed through the thickness of the foil.
Due to the projection geometry along the
011 zone
10
axis, the precipitates appear to be
overlapping along 011 while they appear to be separated along [200] in Fig. 3(d).
An
inherent problem is that the dark field imaging cannot be performed along [001] zone axis as
there is no superlattice spot unique to the Heusler phase. Diffraction spots from a Ti2Ni3 phase
can be found in the insets of Fig. 3 (c) and (d). This phase will be discussed further in section
3.2.2.
The mean value of the precipitate radius, r , and the width of the particle size distribution as
a function of aging time and temperature are given in Table I. It is important to note that for an
isochronal aging treatment, r at 800˚C is larger than at 600˚C by a factor of 50 or more. This
result will be analyzed further (in section 5.1) to identify possible mechanisms governing the
microstructural dynamics.
By comparing r of three alloys at 600˚C, it is seen that both Hf and Zr have a retarding
effect on the microstructural dynamics of L21 precipitates, though Hf appears to be more
effective than Zr. This is consistent with our observation of partitioning of these elements into
the Heusler phase (as discussed in section 3.2). As shown in the example of Fig. 2 (b), the
semi-coherent L21 precipitates aged at 800˚C undergo physical coalescence. In a coalesced
particle, if apparent high angle boundaries (between L21 precipitates) were visible (see Fig. 2 (b))
then it was considered to consist of two or more separate particles; otherwise, it was treated as
one particle.
Table I:
Non-coalesced precipitates may be seen in Fig. 2 (a) and (c).
Average radius r (nm) of Heusler precipitates, with the width of the particle size
distribution.
The number in parenthesis indicates the number of precipitates used in the
measurement.
Aging Treatment
600°C for 1000 h
Ni-45Ti-5Al
7.50±1.97 (176)
Ni-40Ti-5Al-5Hf
6.24±1.53 (50)
11
Ni-40Ti-5Al-5Zr
5.65±1.00 (50)
600°C for 2000 h
12.0±4.10 (170)
8.40±1.58 (50)
9.94±1.58 (50)
800°C for 1000 h
332±158 (50)
312±105 (50)
233±81.3 (50)
394±145 (19)*
301±95.7 (34)*
207±51.8 (34)*
* Based on the particles that do not appear to have coalesced.
3.2 Phase Equilibria and Partitioning Behavior
3.2.1
Analytical Electron Microscopy
The k factors determined in this study were kTi/Ni=0.8598±0.0074, kAl/Ni=0.7043±0.0143,
kHf/Ni=3.4084±0.1214, and kZr/Ni=2.0302±0.0649. Figure 4 shows the EDS X-ray spectra of B2
and L21 obtained from the Ni-45Ti-5Al specimen aged at 800°C for 1000 hours.
The
qualitative difference in composition is clearly visible in the aluminum peak. The compositions
of B2 and L21 phases are listed in Table II.
Table II:
Equilibrium compositions of B2 (TiNi) and Heusler (Ni2TiAl) phases at 800°C
determined by AEM.
The error (99% confidence level) is according to Eq. [4].
Alloy (at%)
Phase
Al (at%)
Ti (at%)
Ni (at%)
Hf or Zr (at%)
Ni-45Ti-5Al
B2
3.80±0.39
43.15±1.52
53.04
------------
Heusler
19.79±1.47
27.94±0.36
52.27
------------
4.13±0.95
38.65±3.47
51.75
5.48±0.04
Heusler
20.52±1.75
25.23±0.39
51.72
2.53±0.05
B2
3.91±0.62
39.77±2.78
51.14
5.18±0.14
Heusler
21.80±0.12
23.80±0.38
50.56
3.84±0.13
Ni-40Ti-5Al-5Hf B2
Ni-40Ti-5Al-5Zr
12
It is found that the solubility of Al in TiNi is increased by the addition of Hf or Zr. Since Al is
needed to form Ni2TiAl, this increase of solubility means Hf / Zr is stabilizing B2-TiNi with
respect to Ni2TiAl.
The partition coefficients ( xB2 / L 21  x B2 x L21 ) of Hf and Zr at 800°C can be determined
B2 / L 2
based on the AEM data which give us  HfB2 / L 2 1  2.17 and  Zr 1  1.35 showing a tendency
to partition more to the B2 phase at this temperature. This weakens their effectiveness in
reducing the lattice misfit.
However, the stabilization of martensite phase can be expected,
allowing a higher transformation temperature [14,15].
The Heusler precipitates in all specimens aged at 600˚C for up to 2000 hours are too small to
conduct AEM experiments using thin foil specimens without having to consider matrix overlap
in the quantitative analysis of data. To overcome these difficulties, we have employed the
higher resolution 3DAP technique to determine the compositions of B2 and L21 phases in these
microstructures.
3.2.2 3D Atom-Probe Microscopy
Figure 5 displays an atom-by-atom 3D reconstruction of Heusler precipitates in a B2 matrix
obtained with the data analysis software ADAM
[28].
Overlaid on the reconstruction is an
isoconcentration surface that delineates the surface of the Heusler precipitate.
The
isoconcentration surface is constructed such that all points outside the surface have a
concentration of Al less than 9 at%, whereas all points inside the surface have a concentration
level of Al greater than 9 at%.
The average composition, based on 3DAP analysis, of the phases observed during aging at
600˚C is listed in Table III. An important finding is the presence of Ti2Ni3-based particles
13
(58Ni-31Ti-2Al-8Hf and 64Ni-24Ti-11Zr) in the quaternary alloys aged at 600˚C, but not in the
ternary alloy.
The presence of Ti2Ni3-based particles accounts for the low Hf, Zr content in
both B2 and L21 phases. Nishida et al. [29] have discussed the precipitation processes in Ni-rich
TiNi as follows:
0  1 + Ti11Ni14 (also known as Ni4Ti330)  2 + Ti2Ni3  3 + TiNi3
at aging temperature below 680  10C, where 0 is the original supersaturated Ni-rich alloy, 1
is the composition of the matrix in equilibrium with Ti11Ni14, and so on. They found Ti11Ni14
disappears after 100 hours of aging at 600˚C.
Ti2Ni3 is observed between 100 and 5000 hours
of aging at 600˚C; however it is a metastable phase since it dissolves upon further aging.
Because 3DAP analyzes a limited volume of the specimen, the whole morphology of the Ti2Ni3based particles could not be revealed. From the partial view of the Ti2Ni3-based particles, a
needle shape is suggested.
It is difficult to determine the shape of the Ti2Ni3-based particles
from TEM micrographs, because of the strain contrast generated by the Heusler precipitates.
Hara et al. [31] identified the crystal structure of Ti2Ni3 to be orthorhombic (a=0.4398, b=0.4370,
c=1.3544 nm) at room temperature. This is consistent with the splitting of diffraction spots
observed (see insets of Fig. 3 (c) and (d)). Due to the orientation relationship [29] between
Ti2Ni3 and the matrix and the low volume fraction it is difficult to produce better images. From
the present work it can be concluded that Hf and Zr stabilize while Al destabilizes the Ti2Ni3
phase.
B2 / L 2
The partitioning coefficient of Hf and Zr at 600˚C is  Hf 1  0.87 , and that of Zr is
 ZrB2 / L 2  0.75 , showing inversion of partitioning compared to 800˚C.  HfB2 / Ti2 Ni3  0.27 , and
1
 ZrB2 / Ti
2
Ni3
 0.26 reflecting the strong partitioning behavior of Hf and Zr to the Ti2Ni3 phase. At
600˚C, Hf and Zr slightly reduce the solubility of Al in TiNi as shown in Table III.
14
This is
opposite to the behavior at 800˚C, and indicates the stabilization of Heusler phase over TiNi.
The metastable Ti2Ni3 phase composition shows strong partitioning of Hf and Zr toward this
phase. The equilibrium between B2 and L21 alone shows a positive contribution of Hf and Zr to
the reduction of lattice misfit, and the tie-triangles defined in Table III define alloy composition
limits to avoid the competing Ti2Ni3 phase.
The alloys in this study were slightly Ni rich, and
the Ti2Ni3 phase could be avoided and lattice misfit be reduced in a Ni lean alloy composition.
Table III:
Compositions of the phases present at 600°C as determined by 3DAP.
The error
is according to Eq. [7].
Alloy (at%)
Ni-45Ti-5Al
Phase
B2
Heusler
Al (at%)
2.67±0.55
23.54±0.92
Ti (at%)
44.29±0.42
26.02±0.90
Ni (at%)
53.05±0.38
50.44±0.74
Hf or Zr (at%)
-----------------------
Ni-40Ti-5Al-5Hf
B2
Heusler
Ti2Ni3
2.14±0.27
16.56±0.61
2.17±1.54
44.32±0.20
27.89±0.57
31.20±1.30
51.34±0.19
53.02±0.46
58.37±1.01
2.20±0.27
2.52±0.66
8.26±1.50
Ni-40Ti-5Al-5Zr
B2
Heusler
Ti2Ni3
2.28±1.03
21.70±1.16
0.00±0.00
43.68±0.78
26.96±1.12
24.41±2.26
51.10±0.73
47.45±0.95
64.37±1.55
2.94±1.03
3.90±1.29
11.22±2.45
3.3
Lattice Parameter measurements by X-ray Diffraction
The measured lattice parameters obtained from the X-ray diffraction experiments, corrected for
instrumental factors, are listed in Table IV as a function of heat treatment.
The B2 matrix of Ni-
45Ti-5Al has a significantly smaller lattice parameter than stoichiometric TiNi, i.e., Al has a
strong effect in reducing the lattice parameter of TiNi. On the other hand, the quaternary alloys
15
show an increase in lattice parameter as one would expect due to the presence of relatively larger
Hf and Zr atoms.
The measured lattice parameter of Heusler phase in the specimens aged at 600°C for up to
2000 hours do not correspond to the unconstrained state, since the precipitates are apparently
fully coherent.
Therefore, it is of interest to consider the stress-free lattice mismatch. Since
the stress field around cuboidal precipitates is rather complex, we consider the spherical
geometry for the sake of simplicity. The shear modulus of the precipitate is assumed to be
equal to that of the matrix, since the shear modulus of Ni2TiAl at room temperature is not known.
Then the constrained mismatch, , for fully coherent, spherical precipitates is related to the
unconstrained mismatch, , assuming elastic isotropy by [32]:
 4  2

 1
31    
   
[8]
where  = Poisson’s ratio of the precipitate. Substituting in  = 0.33, the above formulation
simplifies to   0.66. Unconstrained lattice parameters of the Heusler phase in the alloys aged
at 600°C were calculated using this relation and shown in Table IV.
Table IV: Lattice parameter of B2 phase in solution treated (at 1100°C) alloys, and B2 and L21
phases in aged (at 800°C for 1000 hours, at 600°C for 2000 hours) alloys. The corrected
unconstrained lattice parameters of the Heusler phase in the alloys aged at 600°C are shown in
parentheses.
16
Solution treated
Aged at 800˚C
Phase
B2
Ni-45Ti-5Al
0.30018 nm
Ni-40Ti-5Al-5Hf
0.30331 nm
Ni-40Ti-5Al-5Zr
0.30543 nm
B2
Heusler
0.30022 nm
0.59068 nm
-0.0163
0.30298 nm
0.59410 nm
-0.0196
0.30468 nm
0.59851 nm
-0.0178
B2
0.30132 nm
0.59358 nm
0.30171 nm
0.59518 nm
0.30255 nm
0.60351 nm
(0.58891 nm)
-0.0150
(0.59094 nm)
-0.0137
(0.60269 nm)
-0.0026
(-0.0228)
(-0.0207)
(-0.0040)
 at 25˚C
Aged at 600˚C
Heusler
 at 25˚C
(0.5909109
The observed misfit dislocations spacing () indicated in the lower left corner of Fig. 2 (b) is
consistent with the measured semi-coherent lattice parameters of B2 and L21 phases.
example, the expected value of  ( 
For
d1d2
, where d1 and d2 are d-spacings of the reflecting
d1  d2
planes of B2 and L21) is 15.18 nm, which compares favorably with the measured value of
16.150.81 nm.
4. Modeling the Lattice Parameter of B2 and Heusler Phases
4.1
Atomic Volumes in the B2 Phase
One approach to describe the composition dependence of the lattice parameter of a
multicomponent B2 is in terms of atomic size of the relevant species and their sublattice
occupancy in TiNi.
To quantify the effect of a third element on the lattice parameter of a B2
phase, at first it is necessary to understand and quantify the composition dependence of lattice
parameter of the binary B2 phase.
17
TiNi
Therefore, we derive the atomic volumes of Ni ( TiNi
Ni ) and Ti ( Ti ) in TiNi using the available
lattice parameter data [33,34,35,36].
For deviations from stoichiometry, the major structural defects
in TiNi are considered to be the constitutional vacancies in the Ni-sublattice and Ni antisite
atoms on the Ti-sublattice.
Figure 6 shows the variation of the lattice parameter of TiNi with
Ni-content. Since the Ni atoms are smaller than the Ti atoms, the lattice parameter decreases as
the Ti atoms are replaced with Ni.
In the following, we will consider the lattice parameter on
the Ni-rich side only. Due to the very small homogeneity range, a comprehensive analysis of
the lattice parameter of the Ti-rich side cannot be undertaken.
For modeling the lattice parameter of the B2 phase in terms of sublattice occupancy, a
fundamental assumption is that the atomic volume of an atomic species is independent of the site
it occupies.
Then, the volume of the unit cell is the weighted sum of volume of the species.
By adopting an approach similar to that of Kitabjian et al. [37] the atomic volumes of Ni and Ti in
TiNi can be derived.
From Fig. 6, a0 = 0.30152 nm and da/dxNi = -0.02109 nm on the Ni-rich
side of stoichiometry, where a0 is the TiNi lattice parameter at xNi = 0.5.
least square fit of selected experimental data shown in Fig. 6.
These are based on the
Then, we obtain
TiNi
Ni 
a0 3 3a0 2 da

 0.0123 nm3
2
4 dxNi
[9]
TiNi
Ti 
a0 3 3a0 2 da

 0.0151 nm3
2
4 dxNi
[10]
Taking the lattice parameter of stoichiometric TiNi as reference, the atomic volume of Al in
TiNi TiNi
can be derived from the knowledge of the lattice parameter of ternary B2 with known
Al
composition.
Using our measured lattice parameter of the solution treated ternary alloy Ni-
45Ti-5Al, we obtain
18
 Al  Ti 
TiNi
TiNi
3a0 2 da
3
 0.0114 nm
2 dx Al
[11]
The atomic volume of Hf or Zr in TiNi can be obtained in a similar way to Al in TiNi.
As
there is no literature lattice parameter data of B2 TiNi(Hf) and TiNi(Zr) alloys, we use the B2
lattice parameter of our solution treated Ni-45Ti-5Al specimen as a reference for an analysis of
the B2 lattice parameters of our solution treated quaternary alloys in order to derive the atomic
volumes of Hf and Zr. The results are summarized in Table V.
For comparison, the atomic
volumes are also calculated for the pure elements based on their lattice parameter in the
respective stable structure and also in B2-NiAl.
4.2
Atomic Volumes in the Heusler Phase
The homogeneity range of Ni2TiAl along the pseudobinary section TiNi-NiAl can be accounted
for by Ti atoms occupying sites of the Al sublattice or vice versa. Since we are interested in the
TiNi-Ni2TiAl system, the lattice parameter of Ti-rich Ni2TiAl will be modeled. Our approach is
an extension of the lattice parameter model of B2 phase proposed by Kitabjian et al.
L21 phase.
Ohishi et al.
[10]
[37]
to the
found that the lattice parameter of Ni2TiAl increases with
increasing Ti content, and these results are used to derive atomic volumes of the species in the
L21 phase.
As in the B2 system, the atomic volume of a solute atom is assumed to be independent of the
site it occupies.
Then, the volume (V) of N unit cells is the weighted sum of the volume of the
species.
2 TiAl
2 TiAl
V  8N Ni
 4N  nTi* TiNi2 TiAl  4(N  n*Ti ) Ni
 Na3
Ni
Al
19
[12]
where n*Ti is the number of antisite Ti defects.
It is assumed that the concentration of
constitutional vacancies in the L21 phase is either negligibly small or they are absent.
For the
L21 structure with 16 atoms per unit cell, the atomic fraction of Ti and Al can be expressed by
4N  nTi*  1 nTi*
xTi 
 
16N
4 4N
[13a]
4N  nTi*  1 n*Ti
x Al 
 
16N
4 4N
[13b]
Differentiating Eqs. [12] and [13a] with respect to n*Ti yields
3Na2
da
Ni 2 TiAl
2 TiAl
  Ni

*  4Ti
Al
dnTi
[14a]
dx Ti
1
* 
dnTi 4N
[14b]
Combining Eqs. [14a] and [14b], we obtain
2 TiAl
2 TiAl
Ni
 Ni

Ti
Al
3a 2 da 3a0 2 da

16 dx Ti
16 dxTi
[15]
For the stoichiometric compound the volume of the unit cell can be expressed by
2 TiAl
2 TiAl
2 TiAl
a0 3  8Ni
 4Ni
 4Ni
Ni
Ti
Al
[16]
where a0 is the Ni2TiAl lattice parameter at xTi=0.25. From Eqs. [15] and [16]

Ni2 TiAl
Ti
 Ni2
Ni TiAl


Ni2 TiAl
Al
3a0 2 da

16 dxTi
[17a]
a0 3
3a 2 da
Ni TiAl
  Al2  0
8
32 dxTi
[17b]
20
2 TiAl
2 TiAl
2 TiAl
Since there are two equations with three unknowns (  Ni
,  Ni
, and  Ni
), further
Ni
Ti
Al
2 TiAl
simplification is needed. We approximate  Ni
as
Al
2 TiAl
 Ni

Al
Al
 Ni
 TiNi
Al
Al
 0.0124 nm3
2
[18]
Because Al occupies the least amount of sublattice in the given specimen, this assumption is
reasonable. Then, the quantities a0 and da/dxTi are determined from a linear best fit to the data
of Ohishi et al. [10] to be 0.58934 nm and 0.05081 nm, respectively.
The atomic volumes of the
species obtained through this model are summarized in Table V.
In the absence of any experimental lattice parameter data of single phase Ni2TiAl(Hf or Zr)
alloys, we use the following approximations based on the substitution behavior to derive the
atomic volume of Hf ( Hf2
Ni Ti Al
2 TiAl
Ni
Hf
2 TiAl
 Ni

Ti
2 TiAl
2 TiAl
Ni
 Ni

Zr
Ti
2 TiAl
) and Zr ( Ni
) in the L21 phase
Zr
TiNi
Hf
[19a]
TiNi
Ti
TiNi
Zr
TiNi
Ti
[19b]
As seen in Table V, Al has a smaller atomic volume in TiNi than in its stable fcc state,
reflecting a strong bonding interaction. For the atomic volume of Ni it is interesting to note that
 NiNi 2 TiAl 
Ni Al
 TiNi
Ni   Ni
although no such assumption has been imposed in our anlysis. The
2
atomic volume of Ti in TiNi or Ni2TiAl is smaller than in its stable hcp state, while Ti in NiAl
has a similar atomic volume to the hcp state. Hf and Zr belong to the same atomic group and
show similar trends in the atomic volume as expected. Both Hf and Zr show an increase of
atomic volume in TiNi and Ni2TiAl over their atomic volume in the stable hcp state, while in
21
NiAl they exhibit a reduction of atomic volume. The atomic volume data will be discussed in
more detail in section 5.4.
Table V: The atomic volumes (in nm3) of Ni, Al, Ti, Hf and Zr at 25˚C derived from the lattice
parameter data of TiNi, Ni2TiAl and NiAl compared with the atomic volume derived from the
lattice parameter of the respective pure element.
Species
TiNi
Ni2TiAl
NiAl
Pure Element
Ni
0.0123
0.0115
0.0106 [37]
0.0109 (fcc)
Al
0.0114
0.0124
0.0134 [37]
0.0166 (fcc)
Ti
0.0151
0.0157
0.0180 [37]
0.0177 [38] (hcp)
Hf
0.0236
0.0245
0.0210 [37]
0.0224 [39] (hcp)
Zr
0.0293
0.0305
0.0221 [37]
0.0233 [40] (hcp)
4.3
The Lattice Parameter of Multicomponent B2 Phase
Since the thermodynamic model predicts the site occupancies for various species in the B2
sublattices fairly accurately, the composition dependence of lattice parameter can be described in
22
terms of site fractions of the species.
In other words, the volume of the unit cell is the weighted
sum of the species (including constitutional vacancy)
a3 
n
I
 (y j
j1
with
y
I
j
 y IIj )TiNi
j

y
II
j
[20]
 1 where the superscripts I and II refer to two sublattices.
Thus, in a Ni-rich
TiNi(Al) alloy where there are no constitutional vacancies, and the Ni-sublattice is fully occupied
by Ni atoms and Al atoms replace Ti atoms on the Ti-sublattice, Eq. [20] becomes
II
II
TiNi
II
TiNi
a 3  1 y IINi TiNi
Ni  1  yNi  y Al Ti  y Al Al
[21]
Knowing the atomic volumes of the species and their site occupancies in the sublattices in the
B2 structure, it is possible to model lattice parameters in the multicomponent system. For
example, in the Ni-45Ti-5Al alloy aged at 800˚C for 1000 hours, the composition of the B2 phase
is 53.0Ni-43.2Ti-3.9Al. Applying Eq. [21], its lattice parameter is calculated to be 0.29980 nm,
in good agreement with the experimental value 0.30022 nm.
Similarly, knowing the
composition of the B2 phase and atomic volume of the species, the lattice parameter of the B2
phase in the quaternary alloys were also calculated using Eq. [20].
All calculated lattice
parameters are compared with the experimental values in Fig. 7 (a). It is seen that they agree
within ±0.4%.
4.4
Modeling Composition Dependence of Lattice Parameter of Heusler Phase
As in the B2 system, the composition dependence of the lattice parameter can be described in
terms of site fractions of the species. Knowing the atomic volume of all species and their site
occupancies on the sublattices of the Heusler structure, the lattice parameter in the
multicomponent system is modeled by an extension of Eq. [16]. For example, the composition
23
of the Heusler phase in Ni-45Ti-5Al aged at 800˚C for 1000 hours is 52.3Ni–27.9Ti–19.8Al and
the lattice parameter is 0.59024 nm.
This model can be easily extended to quaternary systems.
The unconstrained lattice parameter values calculated based on atomic volumes of species are
presented together with experimental lattice parameter values in Fig. 7 (b). The experimental
values and calculated values are in good agreement, with deviations less than 0.7%.
4.5
Modeling Temperature Dependence of Lattice Parameter of B2 Phase
The temperature dependence of the lattice parameter of TiNi has been reported by Klopotov et al.
[41].
They found a nonlinear dependence of the lattice parameter on the temperature, and the
deviation from linearity might be related to transformation of the alloy at lower temperatures.
Hence, in the present work, their high temperature data has been used to find the following linear
best fit:
a (nm) = 0.30206+1.834610-6 (T-298)
[22]
where T is the absolute temperature in K. Assuming that the composition dependence and the
temperature dependence of the lattice parameter are independent from each other, both
dependencies can be superposed linearly to estimate the lattice parameter at any given
composition and temperature.
4.6
Modeling Temperature Dependence of Lattice Parameter of Heusler Phase
The temperature dependence of the lattice parameter of Ni2TiAl has been reported by Boettinger
et al. [42]. A least square fit of their data yields the following relation:
a (nm) = 0.58758+7.740310-6 (T-298)
[23]
24
Knowing both the equilibrium composition and the temperature dependence of the lattice
parameter, the unconstrained lattice misfit between B2 and Heusler phase of the ternary Ni-Ti-Al
specimens at the aging temperature 800 and 600˚C are estimated to be -1.10 and -1.88%,
respectively. The unconstrained lattice misfit between B2 and Heusler phase of the quaternary
Ni-Ti-Al-(Hf, Zr) specimens at the aging temperature 600˚C are estimated to be -1.67 and –
0.01%, for Hf and Zr containing specimens respectively.
5. Discussion
5.1
Microstructure of the TiNi – Ni2TiAl system
5.1.1
Morphology and Spatial Distribution of L21 Precipitates
The elastic interactions between misfitting particles not only affect the shape and the coarsening
behavior, but also the spatial arrangement as shown in Fig. 3. The effect of coherency strain on
the shape and spatial arrangement of precipitates is well established for the /’ system in Nibased superalloys. Voorhees et al. [43] have derived a dimensionless parameter L defining the
relative dominance of strain and interfacial energies on the shape of coherent precipitates in cubic
systems.
L
The parameter L is given by
= 2 C44 l / 
[24]
where  is the dilatational misfit strain, C44 is an element of the cubic symmetric elastic constant
tensor, l is a characteristic length of the precipitate (equivalent radius), and  is the isotropic
interfacial energy. For small L, i.e. L << 1, the interfacial energy dominates the equilibrium
shape giving spherical morphology. For large L, i.e. L >> 1, the elastic energy dominates the
25
equilibrium, and the resulting morphology may vary from cuboidal to plate shaped. Using the
estimated =-0.0188 at 600˚C as described in section 4.6, the estimated C44 for TiNi 70.77109
N/m2 at 600˚C
[44],
l=7.50 nm (see Table I) and assuming a reasonable value for the coherent
interfacial energy =2010-3 J/m2, we obtain L=9.38. Therefore, the observation of cuboidal
shape of the L21 precipitates in Fig. 3 is consistent with theoretical prediction.
Besides the cuboidal shape of the L21 precipitates, another important observation in Fig. 3 is
their spatial arrangement where the precipitates not only align along elastically soft directions of
the matrix but also maintain a minimum distance of separation. Theoretical analysis by Su and
Voorhees
[45]
shows that aligned microstructures arise from (i) elastic stress induced particle
migration and (ii) preferential coarsening. They also examined the effect of configurational
forces on the morphology and spatial distribution of coherent precipitates. In their theoretical
analysis, the elastic self-energy, total interfacial energy and elastic interaction energies are all
functions of particle shape and interparticle separation. They found that when the coherent
precipitates are aligned along elastically soft directions of the matrix, elastic stress introduces a
long-ranged attractive and a short-ranged repulsive force between the particles. The latter is
responsible for a minimum distance of separation and prevents particle coalescence. These
theoretical predictions, made for the classical /’ systems, are also consistent with our
experimental observations in the B2/L21 system. Of course, when the L21 precipitates become
semicoherent (as in Fig. 2) the influence of elastic stress becomes minimal, and then the shape
and spatial distribution of the precipitates are dominated by the total interfacial energy alone.
The irregular shaped particles due to coalescence, shown in Fig. 2 (b), most likely represent a
transient state. If the alloys are aged further at 800˚C, the precipitates should eventually achieve
the expected equiaxed shape.
26
5.1.2
Kinetics of L21 Precipitation
It is well established that the kinetics of precipitation from a supersaturated solid solution is
governed by the interplay of three processes: (i) nucleation of precipitates, (ii) their growth
kinetics, and (iii) coarsening (or Ostwald ripening) of the precipitates.
Our transmission
electron diffraction confirms that the nucleation of L21 precipitates was suppressed during
quenching.
Therefore, all three processes may take place during isothermal aging.
As
mentioned before, a striking observation is that r of L21 precipitate after isochronal aging at 600
and 800˚C differ by more than an order of magnitude (see Table I).
It is important to point out
that the coalescence process at 800˚C is not responsible for such a large difference.
For
example, even the non-coalesced particles at 800˚C, shown in Fig. 2 (a) and (c), are more than an
order of magnitude larger than the coherent particles at 600˚C shown in Fig. 3. Therefore, the
question arises if this result is consistent with only one process, either growth or coarsening,
operating at both temperatures. Even though the distinction between the growth and coarsening
stages are somewhat arbitrary, we will provide simple arguments to identify the governing
processes at 600 and 800˚C.
Since both growth and coarsening processes are diffusion controlled, it is necessary to
consider the diffusion data in B2-TiNi.
Unfortunately, the diffusion data for all relevant
elements in B2-TiNi are not available. Even in the binary B2-TiNi, the diffusivities of both
elements are not known. Available diffusion data for other B2 intermetallics show that the
diffusivities of the components do not differ appreciably.
For example in B2-FeAl the
Arrhenius parameters for 59Fe are D0=5.310–3 m2/s and Q=265 kJ/mol, while for 114mIn which is
isoelectronic with Al, they are 6.410–3 and 258 respectively [46]. Nonetheless, Erdelyi et al. [47]
measured the tracer diffusivity of
63Ni
in TiNi and observed the Arrhenius behavior with
27
D0=2.110-9 m2/s, Q=155.6 kJ/mol.
This gives the diffusivity (D) at 600 and 800˚C to be
1.0310-18 and 5.5910-17 m2/s, respectively. In other words, D800/D600 is about 54.3. These D
values are also used in the following analysis.
For an isochronal heat treatment, if we assume that the diffusion controlled growth process is
operating at both temperature, then we have
r80 0
D80 0

r60 0
D60 0
[25]
Based on the measured r data for the ternary alloy (see Table I), r800 / r600 is about 52.6
(considering non-coalesced particle data in Table I) which is much bigger than the expected
value of 7.4 from Eq. [25]. This is also true for other alloys.
For an isochronal heat treatment, if we assume that the coarsening process is operating at
both temperatures, then we have
r8 00
f  D
 3 8 00 S 8 00
r6 00
f 6 00  C D6 00
[26]
where the ratio f800/f600 accounts for the effect of volume fraction on the coarsening rate constant,
and S and C are the semicoherent and coherent interfacial energies. At this point, we neglect
the difference in molar volume between 600 and 800˚C. Since the volume fraction of L21
decreases with increasing temperature, the ratio f800/f600 will be always less than unity.
Assuming S =5C, an upper bound value for the right hand side of Eq. [26] is evaluated to be
6.47, which is again much smaller than experimental r800 / r600 .
If we assume that the growth process is operating at 800˚C while coarsening at 600˚C, then
we have
28
800
X1100
B 2  X B2
800
X 800
H  XB 2
r800

r600
3
D800 t
a3
, Vm  N A
600
16
8D600 CtVm X 600
B2 1  XB 2 
[27]
9RTXH600  X 600
B2 
2
1100
800
600
where X B2 is the concentration Al in the solution treated specimen, X B2 and X B2 the
800
concentration of Al in B2 of the specimens aged at 800 and 600˚C respectively, X H
600
and X H
the concentration of Al in Heusler phase of the specimens aged at 800 and 600˚C respectively.
These are listed in Table II and III.
R is the gas constant, T the temperature in K, Vm the average
molar volume of the Heusler phase, NA Avogadro’s number and a the lattice parameter of
Heusler phase at 600˚C. The concentration of Al is chosen because Al is the solute which
undergoes the largest partitioning between the phases. This gives us r800 / r600 =30.5, which is
closer to the experimental value of 52.6 than the two possible cases discussed. Considering the
assumptions made for D800, D600 and C, (2010-3 J/m2) the calculated value compares favorably
with the experimental value and thereby the mechanisms for the kinetics of precipitation can be
reasonably identified as diffusion controlled growth at 800˚C while Ostwald ripening at 600˚C.
It has been argued that the splitting of precipitation kinetics into three distinct regimes is
somewhat artificial, and in reality all three processes may overlap. To address this issue, Langer
and Schwartz
[48]
developed a general theory of precipitation kinetics.
In this theory an
important parameter governing the precipitation kinetics is the degree of supersaturation. At
high supersaturation, the nucleation rate is very high causing the supersaturation to drop rapidly.
The decrease in supersaturation causes the particles smaller than the critical size to dissolve.
The average particle size is initially governed by the nucleation process and smoothly changes to
the regime governed by coarsening.
The growth stage is bypassed because all the
supersaturation is consumed during nucleation and coarsening. This seems to be the operating
29
process at 600˚C. On the other hand at low supersaturation, the precipitation kinetics proceeds
through distinct stages of growth and coarsening. This may be the operating mechanism at
800˚C, where the Al supersaturation ratio is decreased by 60% based on our measured tie lines.
It is also worth considering a fourth case where it may be assumed that the coarsening
process is operating at both temperatures after 1000 hours of aging, but only after a distinct
growth period. In that case Eq. [26] can be rewritten by taking into account an initial size
defining the transition from growth to coarsening, i.e.,
r803 0  r0,380 0 f 80 0  S D80 0 (1000 - t' )

r603 0  r0,360 0 f 60 0  C D60 0 (1000 - t'' )
[28]
where r0, 80 0 and r0, 80 0 are radii at the onset of coarsening which might have ensued at times t'
and t'' during at 800 and 600˚C, respectively. Based on the aforementioned argument, it is very
unlikely that t'<<t'' while it is likely that t'≥t''. In the limit t'≈t'', it turns out that r800 and r0, 80 0
have to be very close to account for the experimentally observed r800 / r600 . Once again, this
implies that the growth process had occurred at 800˚C for most (if not all) of the isochronal aging
period of 1000 hours.
Further experimental work is needed to clearly demonstrate the
fundamental mechanism of microstructural dynamics as a function of aging temperature.
5.2
Phase boundaries in the TiNi-NiAl pseudobinary system
The phase boundaries of TiNi-NiAl pseudobinary phase diagram have been determined
previously based upon calculations and estimates
[49,50,51,52].
However there is a significant
discrepancy among the previous works, particularly concerning the Heusler phase field.
The
data sets obtained in the present work are plotted in Fig. 8, where the previously reported phase
boundaries are included.
As for the Heusler phase, our experimented observation agrees well
30
with the boundaries suggested by the calculation of Ansara [52].
However the solubility limit of
Al in TiNi is suggested to be smaller than those predicted by the previous reports.
5.3
Partition of Hf or Zr between TiNi and Ni2TiAl phases
While the mean compositions of the phases in equilibrium at 800˚C were determined by AEM
using 30 positions for each phase, the partition coefficients of Hf and Zr at 600˚C were obtained
by 3DAP based on a limited number of precipitates observable in FIM samples.
In both B2 and L21 structures the stacking sequence of the atomic layers of the {001} and
{111} planes is the alternation of pure Ni planes and Ti-Al mixed planes (50%Ti, 50%Al).
Therefore, if the concentration depth profile is obtained with atomic layer resolution either along
the {001} or {111} plane, the site preference of a quaternary element can be determined.
If the
quaternary element exclusively substitutes for Ti, that element will be detected only from the TiAl mixed planes.
On the other hand, if a quaternary element exclusively substitutes for Ni, it
will be detected only from the Ni planes.
phase boundary.
Fig. 9 and 10 show the atomic layers across a B2-L21
Fig. 9 (c) exhibits a proxigram analysis [53] of the concentration of species as a
function of distance to the isoconcentration surface.
This proxigram composition profile is
generated from the entire volume displayed in Fig. 9 (d) to easily identify each phase.
In order
to obtain the site occupation behavior, the number of Hf or Zr atoms detected from the Ti planes
was divided by the total number of Hf or Zr atoms in the B2 or L21 phase from the quaternary
specimens aged at 600˚C for 2000 h.
and Zr in both B2 and L21 phases.
Table VI summarizes the site occupation behavior of Hf
The error expected in this analysis is defined by Eq. [7].
Both Hf and Zr have a very strong preference for the Ti sites in both B2 and L21.
Table VI: Site occupancy of Hf and Zr in B2 and L21. Error according to Eq. [7].
31
Site
Hferror
Zrerror
B2
Ti
824%
835%
L21
Ti
8910%
8315%
This has been predicted by Hosoda et al.
[54]
by analyzing the heat of formation. In B2-
type intermetallic compounds (AB) X occupies preferentially A sites only when
HBX<HAB+HAX (HAB stands for the heat of formation between A and B), and B sites only
when HAX<HAB+HBX. Heat of formation calculations were carried out by Hosoda et al. [53]
using the pseudo-ground state analysis based on the nearest-neighbor, pair-approximation. The
calculated heats of formation of Ti-Zr (-2 kJ/mol) and Ti-Hf (-3 kJ/mol) are one to two orders of
magnitude different from those of Ni-Zr (-101 kJ/mol) and Ni-Hf (-107 kJ/mol). Thus, both Hf
and Zr prefer to have Ni as nearest neighbor, which in the B2 lattice results in a substitution on
the Ti sublattice.
The change in the electronic structure due to the substitutional atoms could be
studied by a more sophisticated analysis of the site occupation behavior.
First-principles
calculations analogous to the work of Medvedeva et al. [55] on the NiAl and FeAl systems would
be desirable in the TiNi-based system.
5.4 Atomic volumes of Al, Hf, and Zr in TiNi
In the case of Hf or Zr substituting for Ti, the increase of lattice parameter can be explained by a
simple atomic size effect associated with atomic cores. The size factor of Hf and Zr in TiNi can
be determined as follows:
 ( Hf ,Zr ) 
*( Hf , Zr )  Ti
Ti
 +0.56 for Hf and +0.94 for Zr.
32
In other words, Hf atoms are
calculated to be 56% larger than the Ti atoms they replace, and Zr atoms are 94% larger.
This
large size effect accounts for the increase of lattice parameter. It is interesting to note that the
4d element Zr has a larger atomic volume than the 5d element Hf, in its pure state, as well as in
the compounds TiNi, NiAl, and Ni2TiAl.
The behaviors of Al, Ti and Ni reflect the differences in bonding interactions. For B2-TiNi the
bonds between nickel and titanium atoms [56] are Ti d-Ti d, Ni d-Ni d, Ti p-Ni d, and Ni p – Ti d
types. The lower energy occupied set of peaks of the density of states (DOS) comes mainly
from the Ni d states, and the higher energy peaks are mainly due to the d states of Ti. There is a
significant intensity for the Ti site at the lower energies and the Ni site at the higher energies,
indicative of hybridization and some covalent bonding between the orbitals on the two sites.
There are negligible contributions from the s states.
On the other hand, for B2-NiAl the bonds between nickel and aluminum atoms
[57]
are Ni d-
Ni d, Ni d-Al p, Al p-Al p, and Al s – Ni d types. The Ni d DOS accounts for a large portion of
the total DOS, especially in Ni-rich compounds. The lowest pronounced peak is due to the s-d
bonds between Al and Ni. The characteristic of the total DOS is mainly determined by the d-d
bonds (interplanar connection between Ni atoms), and by the d-p bonds (the interaction between
Ni and Al atoms).
The change of the total DOS affects the bonding between the atoms and results in
variations in the lattice parameters.
This is best demonstrated by the atomic volume of Ni that
is different in TiNi and NiAl phases. Also, the atomic volume of Ni in NiAl is smaller than that
in pure fcc Ni. The atomic volume of Ni is smallest in NiAl, reflecting a strong interatomic
interaction with Al. The electronic structure changes if a solute atom replaces a Ni or Ti atom.
In the case of Al atoms substituting on the Ti sublattice as in Ni-45Ti-5Al, the total DOS will be
altered because the DOS of Ni and Ti will be modified by the Al addition.
33
The reduction of
lattice parameter by the alloying of Al would suggest that d electrons take part in the d-p
hybridization, and the Ni-Al bond is stronger than that of the Ni-Ti bond.
6.
Conclusions
The precipitation of Ni2TiAl Heusler phase in a TiNi based matrix has been investigated for NiTi-Al and Ni-Ti-Al-X (X=Hf and Zr) alloys. The following conclusions are drawn:
1. Precipitation of Heusler phase Ni2TiAl with L21-structure in a supersaturated B2-TiNi matrix
forms a coherent two-phase aggregate at the early stages. The observed cuboidal precipitate
morphology and precipitate alignment along the soft [001]-type matrix directions are similar
to the ones reported for ’precipitates in Ni-based superalloys. The B2-L21 system, based on
a bcc lattice, is directly analogous to the A1-L12 (fcc) system of ’precipitates in a matrix in
Ni-based superalloys.
2. During isochronal aging for 1000 hours, the microstructural dynamics of L21 precipitates is
governed by the growth process at 800˚C and by coarsening at 600˚C.
This behavior is
qualitatively consistent with the general theory of precipitation by Langer and Schwartz.
3. The phase compositions after aging at 800 or 600˚C determined by AEM and 3DAP show
that the solubility of Al in TiNi is smaller than predicted by previous thermodynamic
modeling or extrapolation.
4. For partitioning of Hf and Zr between L21 precipitates and B2 matrix at 800˚C, the
B2 / L 2
B2 / L 2
partitioning ratio is determined by AEM to be  Hf 1  2.17 for Hf, and  Zr 1  1.35 for
Zr.
At 600˚C, however, Hf and Zr show the inverse partitioning behavior, with values of
0.87 for Hf and 0.75 for Zr, as measured by 3DAP.
34
5. The site occupancy of the quaternary additions Hf and Zr in the ordered phases TiNi and
Ni2TiAl determined by 3DAP show that both Hf and Zr exhibit a strong preference for the Ti
sublattice in both phases. This is consistent with a prediction based on an analysis of the
enthalpy of formation by Hosoda et al. [54].
6. The addition of Hf or Zr to the TiNi-Ni2TiAl system at 800˚C is undesirable for lattice misfit
reduction but useful for control of martensite phase stability.
At 600˚C the partition
behavior of Hf and Zr is more amenable for lattice misfit reduction.
7. The atomic volumes of Al, Hf, and Zr in TiNi and Ni2TiAl are obtained based on lattice
parameter measurements.
phases are proposed.
Simple models to predict the lattice parameters of B2 and L21
The calculated and experimental values are found to agree within
±0.7%. The temperature dependence of the lattice parameter is modeled also for both
phases to allow for an estimation of the lattice misfit at the actual aging temperature.
8. While the reduction of lattice parameter by alloying of B2-TiNi with Al can be explained
qualitatively by a Ni-Al bonding that is shorter and stronger than for a Ni-Ti pair, the increase
of lattice parameter after alloying with Hf or Zr is attributed to the atomic core size effect.
9. Ti2Ni3-type precipitates highly enriched in Hf or Zr with compositions 58Ni-31Ti-2Al-8Hf or
64Ni-24Ti-11Zr (at%) are observed in the quaternary alloys after aging at 600˚C, setting a
limit on the allowable Ni content of B2-L21 two-phase alloys.
Acknowledgements—Financial support from the National Science Foundation (Grant No.
DMR-9806749) is gratefully acknowledged.
35
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