![Microstructural evolutions of austenite/martensite phases in [100]-oriented single crystalline NiTi alloy with 51.0% Ni under uniaxial compression. ① -⑦ correspond to the transient configurations in Fig. 11b. Blue, red, and green atoms represent BCC, HCP, and FCC lattices, respectively. From Fig. 11b, for the case with 51.0% Ni under 600 K, the deformation process can be categorized into six distinct stages as marked in the curve: (I) linear elastic region, (II) nonlinear elastic region, (III) second linear elastic region, (IV) unloading nonlinear elastic region, (V) stress plateau and (VI) the last linear elastic region. Seven critical points of ①-⑦ were selected to analyze the underlying deformation mechanisms, further shown in Fig. 13. One can note that the initiation of martensite phase transformation leads to a nonlinear stress-strain response, corresponding to stage II. The elastic deformation of formed martensite phase leads to a second linear stress-strain relation, corresponding to stage III. Then along with the release of the compressive load, martensite phase transfers back to austenite phase.](https://www.researchgate.net/profile/Tianyu-Yu-4/publication/363453363/figure/fig5/AS:11431281084122180@1663027572742/Microstructural-evolutions-of-austenite-martensite-phases-in-100-oriented-single_Q320.jpg)
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1
The effect of Ni content on phase transformation behavior of NiTi alloys: an
atomistic modeling study
Guotai Li1, 2, Tianyu Yu1, 2, *, Ning Zhang3, 4, Mingjun Chen1, 2
1State Key Laboratory of Robotics and System, Harbin Institute of Technology, 150001, P. R. China
2School of Mechatronics Engineering, Harbin Institute of Technology, Harbin, 150001, P. R. China
3Department of Aerospace Engineering and Mechanics, The University of Alabama, Tuscaloosa, AL 35487, USA
4Department of Mechanical Engineering, Baylor University, Waco, TX 76706, USA
* Corresponding author: T. Yu, tianyuyu@hit.edu.cn
In this study, the thermodynamic behavior of single- and poly-crystalline NiTi
shape memory alloys (SMAs) with different Ni contents was investigated by molecular
dynamics (MD) simulations. By employing the 2NN-MEAM potential, the influence
of Ni content on phase transformation temperature is analyzed for single- and poly-
crystalline NiTi. Our simulation results show that the characteristic phase
transformation temperature increases and then decreases as the binary NiTi alloy
changes from Ti-rich to Ni-rich state. In addition, the change of Ni content also
dramatically affects the formation of martensite variants and twins, whereas it shows
slight effect on the formed types of variants. According to the characteristic phase
transformation temperature obtained by thermally induced phase transformation,
temperatures of 500 K, 600 K, 700 K and 300 K, 400 K, 500 K, three for each type of
alloys, are selected to discuss the superelastic effect of single- and poly-crystalline NiTi.
By conducting uniaxial loading and unloading under different constant temperature
conditions, corresponding stress-strain responses are obtained. The microstructure and
grain orientation of each model are studied. The herringbone structure exists in
nanocrystals with different Ni contents, but is not identified in single crystals, whether
under thermally induced or stress-induced phase transformation conditions.
Keywords: NiTi shape memory alloy; Molecular dynamics simulation; Ni content;
Phase transformation; Superelasticity
1. Introduction
Owing to superior shape recovery strain of up to 8% and the excellent functional
properties, NiTi alloy has become one of the most popular and widely studied shape
memory alloys (SMAs). It also has wide applications in the areas of aerospace, medical,
mechanical, electrical, civil and many other fields [1-3]. At present, the NiTi SMAs
fabricated by conventional methods usually exhibit the shapes of wires, plates and bars,
considering the challenges in fabricating complicated structures such as period lattice
structures [4, 5], which has significantly limited the application of NiTi alloys. In order
to further expand the application of NiTi alloys, additive manufacturing (AM)
technique has been developed [4-7]. Compared with traditional manufacturing methods,
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AM methods have many advantages, such as high geometry freedom, decent
dimensional accuracy, and high raw material utilization. However, AM methods are
also sensitive to many factors, including the homogeneity of alloy powder mixer, laser
power, scanning rate, powder thickness, scanning spacing, etc., which can potentially
affect the material fusion and the final performance of the products. For example,
during laser melting and consolidation process, deviations of Ni content from the raw
powder mixture were reported due to the formation of Ni vapor. In addition, most AM
process builds a part in the vertical direction, each location goes through a different
thermal history, resulting in a possible content gradient in the building parts, especially
in a part with complex geometry. In order to solve these problems, many experimental
studies have been carried out on the preparation of NiTi SMA raw materials for AM [6-
9]. The ratio of Ni and Ti powder, as the basis for AM NiTi alloys, plays a crucial role
in determining the built part composition and performance, literatures show that a slight
increase of the content of Ni can lead to a decrease of the range of the martensitic phase
transformation temperature by 10 K [10]. Haberland et al. [11] studied the effect of
imported energy on phase transformation temperature of NiTi alloys with different
composition of Ni. The obtained experimental results indicated that Ni content can
significantly affect the phase transformation temperature of AMed-NiTi alloys. The
phase change temperature of near-isotonic composition of AMed-NiTi parts were less
than those of Ti-rich and Ni-rich AMed-NiTi parts. Lee et al. [12] analyzed the effects
of composition and post-heat treatment on the shape memory behavior and mechanical
properties of NiTi alloys. Their results demonstrated that the characteristic phase
transformation temperature of NiTi alloys with a composition of Ni between 49.5% and
51.0% increased first and then decreased subsequently, and a maximum value was
observed when the composition of Ni reaches about 50.5%. Frenzel et al. [13]
investigated Ni content on martensitic phase transformations in arc-melting produced
NiTi shape memory alloys. They found that it is important to account for the purity
levels of the starting materials as well as the oxygen pick-up during melting, which is
also important for AM since the oxygen level in the start metal powders can vary from
producer to producer. It was found that the martensite start temperature (Ms) decreases
with the increase of Ni content, and Ni contents affect the width of thermal hysteresis
and the transformation heats.
Although many experiments have been conducted on the AM of NiTi alloys with
different Ni contents, the effect of Ni content on the thermodynamic properties of NiTi
alloys were rarely studied at the atomic scale. Molecular dynamics (MD) simulations
were mainly carried out for equiatomic single crystalline NiTi and nanocrystalline NiTi.
Chen et al. [14, 15] investigated the superelastic effect and shape memory effect of
equiatomic single crystalline NiTi using atomistic modeling. The martensitic
reorientation, shape memory effect and superelastic behavior of single crystalline NiTi
were simulated by varying the temperature range. Besides, they studied the asymmetric
tension-compression behavior of equiatomic single- and poly-crystalline NiTi alloys.
Simulation results implied that the tension-compression asymmetry of single crystalline
NiTi was attributed to different stress-induced martensitic variants and different
3
deformation modes, while the phenomenological tension-compression asymmetry in
polycrystalline NiTi was mainly attributed to stress-induced martensitic variants.
Utilizing atomistic modeling method, Zhang et al. [16] investigated the superelasticity
of NiTi shape memory alloys with different microstructures, including bi-crystalline
grains, nanocrystalline grains, mixed grains and polymorphic structure. Simulation
results showed that the obvious stress plateau was only observed in the case of
nanocrystalline grains with a tensile strain of 8%. Wang et al. [17] conducted MD
simulations to study the cyclic superelasticity properties of nanocrystalline NiTi alloys.
Degradation of superelasticity was observed during the cyclic loading-unloading
process, and the residual strain accumulates progressively with the increase of
deformation cycles. Such degradation phenomena become more significant with the
decrease of grain size. In addition, MD simulations have been conducted to study the
mechanical behavior of single crystalline NiTi under bending [18, 19] and nano-
indentation [20]. However, limited studies have been focused on the effect of Ni
contents. Although Lee et al. [21] used MD technique to study the effect of Ni content
on phase transformation temperature of polycrystalline NiTi alloys, only the change of
Ni-rich content alloys was studied, no study was carried out on Ti-rich NiTi alloys.
Besides, the effect of Ni content on stress-induced phase transformation behavior has
not been studied.
Herein, we focused on the effect of Ni content on phase transformation
temperature and stress-induced transformation behavior of [100]-oriented single
crystalline NiTi, as well as randomly oriented polycrystalline NiTi alloys. These will
help to better understand the effect of different nominal Ni content, as well as providing
guidance of phase transformation and superelasticity behavior for composition gradient
structure produced by AM technique. First, in order to determine the characteristic
phase transformation temperature, the thermally induced transformation of single- and
poly-crystalline NiTi with different Ni contents are studied. Then, the influences of Ni
content on microstructure evolution during phase transformation are investigated.
Finally, the superelastic behavior of single- and poly-crystalline NiTi alloys with
different Ni content are investigated.
2. Simulation Details
2.1 Atomistic models of NiTi alloys
A single crystalline [100]-oriented NiTi pillar is first constructed at 600 K, at
which temperature the lattice structure of NiTi is stable in B2 (austenite) phase. The
lattice constant of the B2 NiTi alloy is 3.016 Å [22]. Certain percentages of Ni atoms
are added to obtain single crystalline NiTi pillars with different Ni contents i.e., 49.5%,
50.0%, 50.5%, and 51.0%, respectively. A typical supercell of NiTi alloy with 49.5%
Ni is shown in Fig. 1a, where the x, y, and z axes correspond to the [100], [010] and
[001] crystallographic orientations, respectively. Periodic boundary conditions were
applied in all three dimensions. Each model consists of 224,000 atoms with green atoms
represent nickel and the blue atoms represent titanium. The single crystalline NiTi
nanopillar has a dimension of 21 × 12 × 12 nm3. Tensile and compressive loads are
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applied along the x-axis. In addition, to investigate the model size effect on the phase
transformation behavior of single crystalline NiTi, a larger model with a dimension of
21 × 21 × 21 nm3 was build, consisting of 686,000 atoms. The polycrystalline NiTi
cubes with 8 randomly distributed nanograins are generated by using the Voronoi
method [23] through Atomsk [24].
The average, maximum, and minimum grain sizes of polycrystalline grains are
approximately 10.5 nm, 12 nm, and 9 nm, respectively. Likewise, polycrystalline NiTi
cubes with four different Ni contents, i.e., 49.5%, 50.0%, 50.5%, and 51.0%, are
generated, respectively. Fig. 1b presents a polycrystalline NiTi model with 50.0% Ni,
which is composed of ordered B2 austenite unit cells. The nanograins, grain boundaries
(GBs) and variant orientation are differentiated by a polyhedral template matching
(PTM) method [25], where the B2 austenite phase is colored in blue, B19’ martensite
phase is in red, and the FCC structure is in green (different phases can be referred to
the results section). Atoms with other structures are shown in white, which were
dominantly located on GBs. The polycrystalline NiTi specimens contains
approximately 675,220 atoms in total and the fraction of GB atoms is about 12.7%
(85,441 atoms).
Figure 1. Atomic configurations of (a) single crystalline NiTi nanopillar, and (b)
polycrystalline NiTi cube.
2.2 Interaction potential
The second nearest neighbor modified embedded-atom method (2NN-MEAM)
[22] potential is adopted to describe the interactions between atoms in the NiTi alloys.
2NN-MEAM has been demonstrated to be capable of reproducing the phase
transformation behaviors of martensitic transformation (B2→ B19’) as well as its
inverse transformation (B19’→B2) [22]. Compared to the EAM-FS potential, the lattice
constant predicted by 2NN-MEAM is more accurate [26, 27]. Besides of EAM-FS and
2NN-MEAM potential, RF-MEAM and EAM potential were proposed by Srinivasan
et al. and Ren et al., respectively [28, 29]. Since the 2NN-MEAM potential could
provide extra angular information and reflect the bonding directional characteristics,
the lattice and elastic constants predicted are more accurate. Using this potential, Wang
5
et al. [17, 30] studied the superelasticity and shape memory effect of nanocrystalline
NiTi alloy and its cyclic degradation behavior. Chen et al. [14, 15] simulated the shape
memory effect and superelasticity in single crystalline NiTi and the tension-
compression asymmetry of single-crystalline NiTi SMA. More details regarding the
2NN-MEAM potential can be referred to the existing literatures [31, 32].
2.3 Simulation procedure
The simulations were performed using the LAMMPS package [33]. First, periodic
boundary conditions were applied to remove spurious surface effects. In order to
minimize the energy of simulation systems and eliminate unstable atomic positions near
generated grain boundaries, the conjugate gradient method [34] is adopted. Then,
single- and poly-crystalline models are relaxed at 600 K for 50 ps and 450 K for 150 ps
using the NPT ensemble, respectively. A Parrinello-Rahman barostat [35] is
implemented for pressure control and Nose-Hoover thermostat [36] is used to control
the temperature. Thereafter, to investigate the thermally induced martensitic
transformation, the temperature was firstly gradually decreased to 50 K from 600 K and
450 K for single- and poly-crystalline models, respectively. The models are then heated
back to 600 K and 450 K with cooling/heating rates of ±5 K/ps. The time step used in
the simulation is controlled at 1 fs. Since the transform temperature of NiTi SMAs
varies with Ni content, the study of thermodynamic behavior is mainly focused on
transformation temperature and superelastic behavior. For mechanical loading, similar
to the thermally induced phase transformation, each model is fully relaxed to reach the
near zero-stress transformation state temperature. For single crystals, the deformation
rates of 1010/s and 109/s produce negligible difference on simulation results. The
mechanical loading is applied through strain-controlled method. This strain rate has
been proved to be slow enough (107/s – 109/s) [37, 38]. Maximum strains of 8% and
20% were applied for poly- and single-crystalline NiTi, respectively. When the strain
reaches maximum, the models were unloaded to zero stress to recover their original
shape. During loading and unloading, the pressures in y and z directions are controlled
at zero for fully relaxing the models during deformation.
OVITO [39] software was used to analyze the evolution of microstructures during
loading and unloading. Although Ko et al. [40] confirmed that the common neighbor
analysis (CNA) is suitable for distinguishing the structure between B2 austenite and
B19ʹ martensite, Larsen et al. [25] proved that PTM is more reliable than CNA when
there exist strong thermal fluctuations and strain. A cutoff of 0.128 was choose for
analysis using the PTM method. The lattice orientation of simulation cell was identified
by the local lattice orientation with PTM method.
3. Result and discussion
3.1 Thermally induced phase transformation
3.1.1 Transformation temperature of single crystalline NiTi SMAs
Temperature plays an important role in the thermomechanical behavior of NiTi
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shape memory alloys [41, 42]. In general, the characteristic phase transformation
temperatures consist of four types: martensite start temperature (Ms), martensite finish
temperature (Mf), austenite start temperature (As), and austenite finish temperature (Af).
B2 austenite phase is stable when the temperature is greater than Af , or between Af and
As. Superelastic behavior displays when an external load is applied. The superelastic
behavior can be described as: stress-induced phase transformation can happen when
stress is applied, forming sheared derivative structure/stress-induced martensite phase.
Martensite phase is stable only in the presence of external stress, martensite becomes
unstable when removing the stress. NiTi SMAs will transform to austenite, recovering
the structure to its original shape. No thermal cycling is needed for phase transformation,
and large applied strains can be recovered by unloading [43]. Shape memory effect is
triggered when the temperature is between Mf and Af. For temperature lower than Mf,
the B19ʹ martensitic state is stable and martensitic reorientation can occur when
subjected to external stress. In order to study the effect of Ni content on superelastic
behavior of single crystalline NiTi SMAs, four characteristic phase transformation
temperatures will need to be determined first. Fig. 2a displays the specimen volume
evolution of NiTi SMAs with different Ni contents, i.e., 49.5%, 50.0%, 50.5%, and
51.0%, during cooling and heating processes. The phase transformation temperatures
of Ms, Mf, As, and Af are identified and labeled in Fig. 2a. It can be seen that the phase
transformation temperatures vary with Ni content, particularly for As and Af as shown
in Fig. 2b. Ms increases from 198 K to 208K when Ni content increases from 49.5%, to
51.0%, and then drop to 196 K. The Mf first increases from 158 K to 168 K, then
gradually decreases to 163 K and 150 K, respectively. Comparing to Ms and Mf, the Ni
content exhibits stronger effect on the austenite phase transformation temperatures, As
and Af, increasing from 395 K to 494 K, and decreases to 375 K, increasing from 447
K to 512 K, and decreases to 478 K, respectively.
7
Figure 2. (a) The specimen volume evolution for single crystalline NiTi (21× 12 × 12
nm3) with different Ni contents during cooling and heating processes. (b) Phase
transformation temperatures of single crystalline NiTi with different Ni contents. (c)
Microstructural evolution of single crystalline NiTi alloys showing phase boundaries
during phase transformation. The models are processed with PTM method, where blue
and red atoms represent B2 and B19’ structure, respectively.
One can note from Fig. 2b that the transformation temperature increases first, then
decreases with the increase of Ni content. As and Af reach the highest values for model
with 50% Ni content, which was consistent with experimental findings [12]. To further
investigate the effect of Ni content on the microstructural evolution, Fig. 2c displays
the microstructure evolution of NiTi alloys with different Ni contents. In equiatomic
model with 50% Ni, phase boundaries only appear in the cooling process (Fig. 3). As
shown in Fig. 3a, phase transformation starts at around 200 K, and the dominant phase
is austenite. During cooling, phase transformation initiates from the center and then
propagates toward the marginal area, leaving the reamined austenite phase boundaries.
These boundaries disappear when temperature reaches 165 K (around Mf), and pure
B19’ martensitic state is observed, indicating the completion of the phase
transformation. However, for non-equiatomic models, the austenite phase boundaries
8
stay until heating to As during thermal cycles. The Remaining phase boundaries could
inhibit the martensitic transformation [44], thus, decreasing the phase transformation
temperature As and Af. Moreover, austenite phase boundaries of rich-Ni phase are
observed during martensite phase formation. This phenonmenon is similar to Ni4Ti3
precipitates formation during martensitic transformation. Experimental results show
that Ni4Ti3 precipitates can impede the occurrence of martensitic phase transformation
[45]. The Ni content of austensite phase boundaries decreases with the martensite phase
transformation procedes, reaching about 50.0% when martensite phase transformation
finishes. This again implies that austensite phase boundaries reamined in NiTi lead to
the variation of phase transformation temperature. Larger proportion of the left
austenite boundaries results in a lower As and Af. Furthermore, Fig. 4 shows the
formation of austenite phase boundaries in the single crystalline NiTi alloy with
different Ni contents at a temperature near Mf. The volume fractions of the austenite
phase boundaries for NiTi alloys with 49.5%, 50.0%, 50.5% and 51% Ni contents are
8.3%, 0.4%, 4.9% and 5.3%, respectively. The Ni content in each austenite phase
boundaries is 50.9%, 52.4%, 51.8% and 53.4%, respectively. Therefore, higher volume
fraction of austenite phase boundaries and Ni content could inhibit the martensitic phase
transformation, consequently decreasing the phase transformation temperature As and
Af.
Figure 3. The evolutions of atomic structure with 50.0% Ni content at the temperatures
of (a) 200 K, (b) 195 K, (c) 190 K, (d) 175 K and (e) 165 K. The solid red wireframe is
shown as a sectional plane, parallel to the xz plane, moving a distance of 6 nm in the y
direction, the black borders showing model boundaries. Orange dotted line is the face
boundary line at the195 K.
9
Figure 4. The left austenite phase boundaries in single crystalline NiTi alloy at a
temperature near Mf with Ni contents of (a) 49.5%, (b) 50.0%, (c) 50.5%, and (d) 51.0%,
respectively. The red and blue atoms represent Ni and Ti, respectively.
To investigate the influence of grain size on the phase transition temperature of
single crystalline NiTi, models with a 21 × 21 × 21 nm3 size (S2 models) have been
simulated. Fig. 5a shows the volume evolution for models with two different sizes (S1
and S2) of different Ni contents during cooling and heating processes. Results show
that the effect of grain size impacts more on the austenite phase transition temperature
rather than the martensitic phase transition temperature due to changing the volume
fractions of the austenite phase boundary (brown colored region in Fig. 5b). Different
from the smaller S1 models, the maximum As temperature of S2 models reached at a Ni
content of 50.5%. The volume fractions of the austenite phase boundary for different
Ni contents of 49.5%, 50.0%, 50.5% and 51% have been analyzed. As shown in Fig.
5b, the austenite phase boundary content in S1 model decreased from 5.3% to zero, then
increased to 4.5% at the temperature of 50K. The volume fractions of the austenite
phase boundary for the S2 model with the Ni content increase are 3.2%, 3.0%, 2.1%
and 2.6%, respectively. As a result, As increased from 367K to 445K, to 470K and
decreases to 450K with the Ni content increases. The grain size of the single crystalline
model with rich-Ti increases will lead an increase of As. The results of the temperature
evolution of As further validate that the presence of austenite phase boundary inhibits
the martensitic phase transition process.
Figure 5. (a) The volume evolution for single crystalline NiTi model with different Ni
contents during cooling and heating processes. (b) The volume fractions and the front
view of austenite phase boundary for S1 and S2 model with 49.5%, 50.0%, 50.5% and
51% Ni content. (S2 represents the larger model, and S1 represents the smaller one).
Furthermore, Fig. 6 displays the atomic structure of S2 model with different Ni
contents. The results of S1 model are shown in Fig. 7, respectively. The change of Ni
content results in the formation of different variants. In the S2 model with 50.0% Ni
content, the variants were labeled as V2 and V3 according to the atomic structure and
variant orientation as shown in Fig. 6. It could be observed that there is a certain angle
between variant 1 and variant 4, about 4°. The variant boundary, dislocation loop (green
10
color), and the dislocation structure were also marked in Fig. 6 and Fig. 7. Norfleet et
al. experimentally observed the presence of dislocation loop, suggesting that loop
formation may be driven by local stress field at a scale of individual variants [46]. Our
study shows that the increase of the grain size of single crystalline NiTi promotes the
formation of dislocation loops. The change of Ni content results in the formation of
different variants and loops. The dislocation loop region presents a low Von Mises stress,
which validates the suggestion by Norfleet et al. [46].
Figure 6. The atomic structure of S2 model (21 × 21 × 21 nm3) with 49.5%, 50.0%,
50.5% and 51% Ni contents at the temperature of 50 K.
Figure 7. The atomic structure of S1 model (21 × 12 × 12 nm3) with 49.5%, 50.0%,
50.5% and 51% Ni contents at the temperature of 50 K.
Fig. 8 displays the phase structure and Von Mises stress contour of the single
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crystalline NiTi model with 51.0% Ni content during cooling process. At the beginning,
the model is in austenite state and the Mises stress decreases with the temperature
decreases from 600 K to 400 K. Then the Mises stress increases from the martensitic
nucleus region and spreads around as the martensitic phase transition proceeds (Fig.
8d2). The austenite phase boundary (dislocation loop) region always presents in a low
Mises stress state due to a higher Young's modulus of martensite than the austenite. The
untransformed austenite is continuously transformed into martensite, and the residual
austenite forms the austenite phase boundary during martensitic transformation. The
austenite phase boundaries formed in different variants are different, resulting in
different phase transformation temperatures.
Figure 8. (a1-f1) phase structure of single crystalline NiTi model with 51.0% Ni content
during cooling process, where blue and red atoms represent austenite and martensite
structure, respectively. (a2-f2) are the corresponding Von Mises stress contour plot of
(a1-f1).
Waitz et al. experimentally investigated the effect of grain size on the phase
transformation behavior of nanocrystalline NiTi. Results show that grain size less than
about 50nm did not transform to martensite even upon large undercooling [47, 48].
Chen et al. and Nie et al. found that the martensite transformation temperature decreases
with reducing grain size. The martensite transformation was constrained by the grain
boundary with a grain size of 4nm [44, 49].
3.1.2 Transformation temperature of polycrystalline NiTi
Fig. 9a displays the change of atomic volume with temperature in polycrystalline
NiTi with different Ni contents. Comparing to the single crystalline NiTi, the range of
phase transformation temperature in polycrystals is wider. In addition, grain sizes can
influence phase transformation temperature, i.e., each individual grains may exhibit
different phase transformation temperature [47]. The existence of grain boundaries may
inhibit the progress of phase transformation as well [44]. Similar phenomenon has also
been reported in shape memory ceramics [50-52]. As indicated in Fig. 9a, the
martensitic phase transformation can still happen under 50 K. Such behavior is similar
to the hypothesis phenomenon reported by Kulin et al. [53] that the martensitic phase
12
transformation was observed at temperatures approaching absolute zero. Since Mf and
As were hard to determine from Fig. 9, only Ms and Af were calculated. The modeling
results of Ms and Af for polycrystalline NiTi alloys with different Ni contents are
displayed in Fig. 9b. The Ms of polycrystalline NiTi models with Ni contents of 49.5%,
50.0%, 50.5%, and 51.0% are 190 K, 197 K, 205 K, and 188 K, respectively. The atomic
volumes for models with different Ni contents during cooling and heating processes
were analyzed, as shown in Fig. 9a. Atomic volumes do not change abruptly at Ms
during the cooling process. Compared with single crystalline samples, the process of
phase transformation becomes slower. In the early stage of cooling, atomic volume
changes are dominated by the effect of thermal expansion. The A → M phase
transformation at this stage is mainly a tangential lattice change, which insignificantly
affects the atomic volume. In the later stage of cooling process, the A→ M phase
transformation is dominated by the unit cell volume change induced by phase
transformation, resulting in the sudden turning of the atomic volume-temperature curve.
During the process of heating, the atomic volume change can be devided into four
stages: continuous A → M phase transformation, thermal expansion, M → A phase
transformation, and thermal expansion, respectively. Af for polycrystalline alloys are
274 K, 297 K, 305 K, and 300 K, respectively. Ms and Af increase first and then decrease
with the increase of Ni content, which is consistent with the experimental results [12].
Figure 9. Evolution of atomic volume during cooling-heating cycles in polycrystalline
NiTi with different Ni contents.
To study the effect of Ni content on martensitic phase transformation behavior,
martensitic variants for alloys with different Ni contents at 50 K are shown in Fig. 10.
It can be seen that Ni content has a significant effect on the formation of distinct
martensitic variants and twin structures. The morphology of martensitic variants in
grain Ⅰ (black dotted region) presents almost a pure variant. While in Fig. 10b, c and d,
three or more variants are observed in grain Ⅰ. Further analysis of grain Ⅰ and grain Ⅱ
(white dotted region) in Fig. 10a and Fig. 10b with Ni contents of 49.5% and 50.0% are
shown in Fig. 10 Ⅰ-Ⅳ. The structure similarities in grain Ⅰ between Fig.10 Ⅰ and Fig. 10
Ⅲ exist, showing compound twins and twin boundaries that are parallel to each other,
which is consistent with experimental observations [54]. The martensite variants type
and microstructure present differently. Only one martensite variant exists in grain Ⅰ with
13
49.5% Ni content and two variants exists with 50.0% Ni content. Moreover, the angle
of twin orientation in grain Ⅰ with 49.5% Ni content (Fig.10 Ⅰ) is about 86°, which
changes to 82° for Ni content of 50.0% (Fig.10 Ⅲ). It is also found that the martensitic
variants in grain Ⅱ are significantly different from those in grain Ⅰ. As shown in Fig. 10
Ⅱ and Fig. 10 Ⅳ, martensitic variants in grain Ⅱ present a fishbone-like structure and
herringbone structure, respectively. The change of Ni content makes the type of variants
formed during temperature-induced martensitic transformation different, resulting in
different variant boundaries and structures.
Figure 10. Cross-section of polycrystalline NiTi with Ni contents of (a) 49.5%, (b)
50.0%, (c) 50.5%, and (d) 51.0% at 50 K. Results are processed by PTM method and
colored with angles respect to the x-axis. Black dotted and white dotted regions
represent grain Ⅰ and grain Ⅱ, respectively. Twin boundaries are marked by yellow
dashed lines and twin orientation are marked by solid white line connected with arrows.
Black dashed lines in Ⅰ-IV represent fishbone or herringbone structures.
3.2 Stress induced phase transformation and superelastic behavior
3.2.1 Superelasticity of single crystalline NiTi alloys
Superelasticity is a key property of shape memory alloys and ceramics, which are
mediated by the reversible martensitic phase transformation [55]. As shown in section
3.1.1, for single crystalline NiTi specimens, the maximum Af is less than 520 K. Thus,
three temperatures of 500 K, 600 K and 700 K are used here to study the superelastic
effect, maintaining the austenite phase with the absence of external stress. Fig. 11
displays the compressive stress-strain responses of [100]-oriented NiTi alloys with
different Ni contents and temperatures. Different hysteresis behaviors were observed
under different operating temperatures (Fig. 11a-c).
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Figure 11. Stress-strain responses of the single crystalline NiTi alloys with different Ni
contents of 49.5%, 50.0%, 50.5%, 51.0% under uniaxial compression tests at 500 K,
600 K, 700 K.
Figure 12. (a) Phase evolution in the single crystal NiTi model with different Ni content
at 500 K. (b) Ni content in different phases of the single crystal NiTi model with
different Ni content at 500 K.
It is observed that stress induced phase transformation has similarity with
thermally induced phase transformation, the last formed martensite variant will undergo
the first reverse phase transformation. Furthermore, the change of Ni content results in
the change of elastic modulus of B2 structure. Young’s modulus changes from 41.9 GPa
to 41.6 GPa, and then to 36.9 GPa with Ni content increasing from 49.5% to 51.0%.
Young’s modulus of single crystalline NiTi with different Ni content falls into a range
of 35-55 GPa, which is consistent with published results [21]. For further discuss this
phenomenon, the proportion of each phase in the single crystal NiTi model with
different Ni content and the Ni content in each atomic structure at the temperature of
500 K were analyzed. As shown in Fig. 12a, the amount of martensite phase structure
increases and austenite phase structure decreases with the increase of Ni content. The
Ni content in the structure (HCP and BCC are the majority parts) also increases with
the increase of Ni content (Fig. 12b). From the rich-Ti to rich-Ni model, the increase of
Ni content reduces the NiTi2 structure in the model, and the Ni3Ti and NiTi structures
increase, which leads to the decrease of Young’s modulus.
15
Figure 13. Microstructural evolutions of austenite/martensite phases in [100]-oriented
single crystalline NiTi alloy with 51.0% Ni under uniaxial compression. ①-⑦
correspond to the transient configurations in Fig. 11b. Blue, red, and green atoms
represent BCC, HCP, and FCC lattices, respectively.
From Fig. 11b, for the case with 51.0% Ni under 600 K, the deformation process
can be categorized into six distinct stages as marked in the curve: (I) linear elastic region,
(II) nonlinear elastic region, (III) second linear elastic region, (IV) unloading nonlinear
elastic region, (V) stress plateau and (VI) the last linear elastic region. Seven critical
points of ①-⑦ were selected to analyze the underlying deformation mechanisms,
further shown in Fig. 13. One can note that the initiation of martensite phase
transformation leads to a nonlinear stress-strain response, corresponding to stage II. The
elastic deformation of formed martensite phase leads to a second linear stress-strain
relation, corresponding to stage III. Then along with the release of the compressive load,
martensite phase transfers back to austenite phase.
Figure 14. (a1-d1) Stress-strain curves during compression and tension tests.
Microstructures of single-crystalline NiTi with different Ni content under uniaxial
compression at a strain of 8% (a2-d2) and tension at a strain of 20% (a3-d3) at 700 K.
Different colors correspond to grain and variant orientations as shown in the legend.
White arrows represent variant orientation in x-axis and white dotted line represent
variant boundaries.
16
Uniaxial tension simulations were carried out to further study the phase
transformation behavior of single crystalline NiTi alloy. Fig. 14 compares the stress-
strain curves and microstructures of single crystalline NiTi with different Ni content
under uniaxial compression and tension at 700 K. The critical compressive and tensile
phase transformation stresses for models with 49.5%, 50.0%, 50.5% and 51.0% Ni are
3.2 GPa and 1.0 GPa (Fig. 14a), 2.9 GPa and 0.9 GPa (Fig. 14b), 2.5 GPa and 0.8 GPa
(Fig. 14c), and 2.3 GPa and 0.7 GPa (Fig. 14d), respectively. By comparing different
microstructures (a2-d2, a3-d3), it can be seen that different martensite variants formed
due to different loading conditions. The variants formed during compression and
tension are marked by V1, V2 and V3, V4, respectively. The formation of different
types of variants is related to deformation modes and is found independent of Ni content
and temperature. To explain this phenomenon, Chowdhury et al. [57] proposed the
twinning unidirectionality where effect of orientation between a certain transformation
direction and 180° to the transformation direction is totally different. If one favors a
variant formation, the other is unfavorable. The sizes and positions of the variants
formed in the models with different Ni contents are found to be different. Moreover,
different from thermally induced phase transformation, B2 austenite structure
transformed into B19 martensite structure (detwinned structure) instead of B19’
martensite structure (twin structure) during compression (Fig. 11). The detwinned
structure has not been observed under tension, which is consistent with the experimental
results [58]. It is noted that the orientation of detwinned structure formed in single
crystalline NiTi models under compression is parallel to the loading direction, which
impedes further deformation, resulting in a higher stress required in compression than
in tension.
3.2.2 Superelasticity of polycrystalline NiTi alloys
According to the Af of polycrystalline NiTi alloys with different Ni contents (Fig.
9), three temperature of 300 K, 400 K, and 500 K are selected to investigate the
superelastic behavior. Fig. 15 displays the compressive and tensile stress-strain curves
of polycrystalline NiTi with different Ni contents at different temperatures. It can be
seen that the critical phase transformation stress decreased with the increase of Ni
content. Ni content does not show significant effect on changing the tensile or
compressive elastic modulus of B2 structure. The influence of Ni content on phase
transformation stress in polycrystals is less than that of single crystals due to the
interaction of multiple grains and the effect of grain boundaries. At higher temperature,
the effect of Ni content on the superelastic effect becomes trivial. Notably, not all
polycrystals exhibit superelasticity at 300 K (in Af range), some samples showing shape
memory effect. The higher Ni content, the more obvious shape memory effect is shown
(Fig. 15a). When temperature is controlled at 400 K or 500 K, the specimen displayed
pseudoelasticity. Different from single crystalline model, polycrystalline NiTi only
exhibits tension-compression asymmetry in stress, existing residual strain after
unloading, resulting from the presence of grain boundary. The models with different Ni
content almost show the same residual strains under compression or tension, 1.0% and
17
0.5%, respectively. The change of Ni content had almost no effect on tension-
compression asymmetry at a strain of 8%, and the increase in temperature will narrow
the gap between the maximum tensile stress and the maximum compressive stress. As
indicated in Fig. 15, the maximum compressive stress decreased when temperature
increases.
Figure 15. Stress-strain curves of polycrystalline NiTi with different Ni content under
tension and compression at (a) 300 K, (b) 400 K and (c) 500 K. Stress decreases with
the increase of temperature when strain reaches 8% under compression.
Figure 16. Structural evolution of polycrystalline NiTi with different Ni content under
compression and tension at 500 K. Different colors correspond to different grain and
variant orientations.
The structural evolution during tension is similar to that of compression, except
the type of variants generated and different lattice orientation. The microstructure of
polycrystalline samples with different Ni content under compression and tension at
500K was shown in Fig. 16. It can be observed that martensite varients formed are
different for compression and tension cases. The change of Ni content leads to the
formation of different twin boundaries and variant sizes in the same grain, as indicated
in Fig. 16. Martensite variants formed under compression and tension are different, i.e.,
three types under compression and four new types under tension (marked by V1-V7 in
Fig. 16). The change of Ni content promotes the formation of different detwinned
boundaries and variant sizes in the same grain, but nearly no effect on variant types, as
shown in Fig. 16. Different from single crystalline alloys, the angle between variants
18
orientation and loading direction in polycrystalline is about 20°, resulting in easier
deformation in tension than compression at the same stress level.
4. Conclusions
The thermally induced and stress induced phase transformation behaviors of
[100]-oriented single crystalline NiTi and randomly oriented polycrystalline NiTi with
different Ni content were investigated by atomistic modeling. The phase transformation
temperature and superelasticity behavior were investigated against the Ni content,
which is a critical issue in laser additive manufacturing where Ni element can evaporate
during melting, influencing the actual Ni content locally. Martensite variants formation
phenomenon in twining and detwinned structure were discussed. Major findings in this
work are summarized as follows:
• Ni content variation in single crystalline NiTi alloys leads to the formation of
austinite phase boundaries. The larger volume of austenite phase boundaries, the
higher Ni content in this region. Remained phase boundaries could inhibit the
martensitic transformation, resulting a lower phase transformation temperature.
• The Mises stress increases from the martensitic nucleus region and spreads around
as the thermally-induced martensitic phase transition proceeds. During phase
transformation, the austenite phase boundary (dislocation loop) region is always in
a low Mises stress state.
• The change of Ni content has no effect on the types of variants formed by external
load in both single- and poly-crystalline models. It almost has no influence on the
tensile asymmetry but the formation of twin and variant structures in single crystals
and nanocrystalline grains is affected. Thus, Ni content affects the phase
transformation temperature and critical phase transformation stresses.
Acknowledgements
This work is supported by State Key Laboratory of Robotics and Systems (HIT)
under Grant No. SKLRS-2022-KF-10, State Key Laboratory of Mechanics and Control
of Mechanical Structures (Nanjing University of Aeronautics and astronautics) under
Grant No. MCMS-E-0522Y01, and by the Oak Ridge Associated Universities (ORAU)
under award number of A21-0435.
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