# Design of (TiHfZr)(NiCoCu) High-Entropy Shape Memory Alloys: From Firstov's Experiments to Data-Driven Approach

Laurent Peltier, Léo Thiercelin, Fodil Meraghni

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Laurent Peltier, Léo Thiercelin, Fodil Meraghni. Design of (TiHfZr)(NiCoCu) High-Entropy Shape Memory Alloys: From Firstov's Experiments to Data-Driven Approach. Shape Memory and Supere-lasticity, 2025, 10.1007/s40830-025-00532-1. hal-04993403

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# Design of (TiHfZr)(NiCoCu) High-Entropy Shape Memory Alloys: From Firstov's Experiments to Data-Driven Approach

L. Peltier<fnref n="1" /> · L. Thiercelin<fnref n="1" /> · F. Meraghni<fnref n="1" />

Received: 19 September 2024 / Revised: 17 December 2024 / Accepted: 14 February 2025

© The Author(s) 2025

**Abstract** This paper deals with the design of (TiHfZr) (NiCoCu) high-entropy and high-temperature shape memory alloys (HE-HT-SMAs). It explains the chronology and the progress of this design starting from the experimental work of Georgi Firstov initiated in the 2015s until the advent of data-driven alloy approaches. A state-of-the-art (TiHfZr) (NiCoCu) HE-HT-SMA family is presented and enriched by a database used as input for a data-driven approach. The paper then focuses on the comparison of martensitic transformation temperatures provided by: (i) the experimental work of Firstov et al. started in 2015, (ii) other recent experimental studies and, (iii) those predicted by two numerical approaches. The first approach consists of a linear regression model proposed by Peltier et al., while the second one is proposed and enriched by Thiercelin et al. using a data-driven technique (random forest regression). The results from the data-driven approach yield accurate predictions that align with the experimental data from both the literature and previous studies. Thus demonstrating the importance of physics-informed, inspired techniques to optimize the design of future alloys, in particular HE-HT-SMAs.

**Keywords** High-entropy alloys · High-temperature shape memory alloys · Martensitic transformation · Shape memory effect · Physics-informed machine learning · Design of “NiTi-like” alloys

## Introduction

Quasi-equiatomic NiTi binary alloy discovered inadvertently by William Buehler in 1963 [1, 2] has become the most widely studied and used shape memory alloy (SMA) for actuation applications. Depending on the Ni content, the NiTi binary alloy can be either austenitic or martensitic at room temperature with a martensitic start temperature ($M_s$) ranging from – 100 °C for 51.1 at.% Ni to 70 °C for 49.5 at.% Ni [3, 4]. The NiTi binary system has been improved by adding alloying elements whose physical properties are quite close to Ni, such as Cu, Co, Fe, Pd, Pt, Au and close to Ti, such as Zr or Hf [5]. NiTi-like are alloys containing alloying elements in addition to Ni and Ti, but which retain the martensitic phase transformation. Based on the metallurgy of ternary and quaternary systems, the transformation temperatures are currently highly improved and led to the design of high-temperature SMAs (HT-SMAs) whose $M_s$ temperature is up to 300 °C [5]. However, these ternary or quaternary high-temperature shape memory alloys (HT-SMAs) have drawbacks, such as short service life [6–8] and require rare and expensive alloying elements. To overcome these limitations, some research works have been carried out using the strategy of high-entropy alloys (HEA). These alloys are generally containing at least five elements in equiatomic concentration. They are characterized by high mechanical strength due to the distortion of the crystallographic lattice, high thermal stability due to sluggish diffusion and the synergy of several properties of the alloying elements thanks to the cocktail effect. The first attempt to develop the concept of Complex Concentrated Alloys (CCAs) has been initiated by Achard in 1788 [9], who proposed more than 900 different combinations of alloys, including equiatomic alloys composed of 5 to 7 alloying elements. Several years after,

<figure><img src="image_2.png" /></figure>

<footnote>F. Meraghni
fodil.meraghni@ensam.eu</footnote>

<footnote n="1">¹ Arts Et Métiers Institute of Technology, CNRS, LEM3-UMR 7239, 57000 Metz, France</footnote>
in 2004, Brian Cantor [10, 11] promoted the HEA concept
as a particular case of equiatomic CCAs.

In 2015, Firstov et al. proposed an innovative study
devoted to high-temperature NiTi-like alloys containing at
least 5 elements, with the following compositions: (TiH-
fZr)₅₀ (NiCu)₅₀ and (TiHfZr)₅₀(NiCoCu)₅₀ [12]. These alloys
are called High-Temperature and High-Entropy Shape Mem-
ory Alloys (HE-HT-SMAs). Firstov has demonstrated that
HT-SMAs can be produced using the high-entropy strat-
egy, without expensive or strategic elements, such as gold,
platinum or palladium [13–20]. In Firstov et al. [12, 21–23],
the chemical composition of binary NiTi alloys has been
modified by partially replacing the titanium content with the
elements Zr and Hf, while Ni has been partially replaced by
Cu and Co. This equivalent element substitution strategy has
been explored and successfully implemented in recent stud-
ies by Peltier et al. [24–26]. The authors have demonstrated
that these new alloys exhibit similar reversible austenite/
martensite transformation features compared to binary NiTi
alloys. At high temperatures, the developed HEAs exhibit
stabilized superelastic and shape memory effects compared
to NiTi [24]. Indeed, the high-entropy strategy has been
demonstrated as a promising way to improve the high-tem-
perature thermal fatigue response of SMAs. It is worth notic-
ing that the alloys proposed by Firstov have been studied
in their as-cast metallurgical state. The as-cast state refers
to the metallurgical condition in which multiple phases are
present, including dendritic phases, precipitates, and inter-
metallic compounds, all of which contribute to delaying the
martensitic transformation. Indeed, the heterogeneity of the
metallurgical state brings about a progressive martensitic
transformation increasing hence the gap between the trans-
formation temperature M_s and M_f. For instance, the alloy
(TiHfZr)₅₀Ni₂₅Cu₁₅Co₁₀ proposed by Firstov [12] has been
characterized by a temperature M_s up to 157 °C. The same
alloy has been studied later by Chen [27], Yaacoub [28] and
Rehman in a homogenized metallurgical state. The homog-
enized state is a metallurgical state obtained after heat treat-
ments above 800 °C to achieve a predominant single-phase
state (austenite or martensite) by removing the residual
phases observed in the as-cast state (Peltier et al. [24], Pio-
runek et al. [29]).

The measured M_s temperatures provided by these authors
have been comparable but less than the ones given by Firstov
for the same alloy but in an as-cast metallurgical state. For
this reason, it is strongly recommended that all developed
HE-SMAs must be studied in a homogenized metallurgi-
cal state using appropriate heat treatments. In fact, this will
result in a narrow DSC thermogram peak corresponding to
the martensite transformation, providing hence a consistent
M_s temperature representative of the alloy being developed.

Artificial intelligence-based approaches are currently
being developed for the design of HE-SMAs and the

prediction of their transformation temperatures, and they
are particularly suitable for HT-SMAs, as presented in the
relevant review by Hart et al. [30]. It should be noted that
some of these physics-informed machine learning tools are
easy-to-use open-source online databases and models, such
as Thiercelin et al. [31] or Liu et al. [32], while others are
more difficult to use due to the lack of database access or
scripting [33–36].

In the present paper, the design process (TiHfZr)
(NiCoCu) HE-HT-SMAs family has been revisited using a
data-driven approach enriched by two databases. The first
one includes all the NiTi-like alloys combining the elements
Ni, Cu, Co, Ti, Zr, and Hf regardless of their metallurgical
state. This database (noted hereafter “global database”) con-
tains binary, ternary, quaternary, quinary, and senary (TiH-
fZr)(NiCoCu) HE-HT-SMAs. The second database noted
hereafter “reduced database” consists of a part of the global
one considering only the NiTi-like alloys having been sub-
mitted to homogenization heat treatment, where the differ-
ence between M_s and M_f is less than 50 °C [31]. To this end,
two prediction models have been deployed for estimating
the martensitic transformation temperature. The first model
is founded on linear regression proposed by Peltier et al.
[37]. It has been applied exclusively on homogenized HE-
HT-SMAs. The second prediction model utilizes a physics-
informed machine learning approach. It has been deployed
on the “global database” and then on the reduced one.

The paper aims to enhance the design process of the (TiH-
fZr)(NiCoCu) HE-HT-SMAs family, starting from Firstov’s
2015 experiments, by incorporating data-driven algorithms
and techniques to accelerate HE-SMA design. It traces the
evolution of the (TiHfZr)(NiCoCu) design strategy over
time, from one based entirely on empirical methods to a
more advanced approach that integrates predictive models.
The complexity of these models has progressively increased,
evolving from linear regression to physics-informed machine
learning algorithms.

The present paper is structured as follows: the second sec-
tion is devoted to the works performed by Firstov et al. [12,
21–23] that initiated the HE-HT-SMAs development and
design. The third section is devoted to experimental studies
inspired by the original works by Firstov emphasizing the
relevance of the metallurgical state on the martensitic trans-
formation in HE-HT-SMAs. The fourth part presents the
gathered databases of NiTi-like alloys that allow the jump
to a data-driven approach toward reliable representation of a
pseudo-ternary covering all the alloying composition space.
The comparison of the predicted M_s values, presented in
Sections, “Comparison of the Numerical Predictions Against
Rehman et al. Experimental Results” and “Comparison of
the Numerical Predictions Considering Other Composi-
tions,” highlights the impact of the metallurgical state on
the variability of M_s values found in the open literature and
those measured by the authors in previous studies. Some concluding remarks and prospects are drawn in the last part of the paper.

## Analysis of the Studies on HE-HT-SMAs Carried Out by Firstov

To study the compositional space of HE-SMAs (TiHfZr) (NiCoCu), Firstov started by setting the proportion of each Ti-like element (Ti, Hf, and Zr) to 16.67 at.%. The compositional space of HE-SMAs (TiHfZr)(NiCoCu) is thus reduced to 3 dimensions, i.e., the proportion of each Ni-like element (Ni, Cu, and Co) can vary from 0 to 25 at.% each. The system is shown in Fig. 1a as a pseudo-ternary

diagram (TiHfZr)₅₀(NiₓCoᵧCu_z)₅₀. Firstov et al. [12, 21–23] proposed and analyzed 8 NiTi-like alloys, whose chemical compositions are given in Table 1. For this purpose, X-ray diffraction (XRD) has been employed to identify the phases present at room temperature, for the developed 8 alloys. The results highlight a zone dominated by the martensite B19' phase, which is observed in alloys 1, 2, 3, 4, and 5, whose diffractograms are shown separately in Fig. 1b. In addition, those diffractograms highlight the presence of martensite peaks B19' and austenite peaks B2.

In the pseudo-ternary diagram, the widest zone is that of the austenitic B2 phase, which includes alloys 6, 7, and 8. The corresponding diffractograms of alloys 6, 7, and 8 are shown in Fig. 1c. It should be noted that Firstov did not provide the transformation temperatures of the 8 alloys shown

<figure><img src="image_3.png" /><figcaption>**Fig. 1** Pseudo-ternary diagram for (TiHfZr)₅₀(NiₓCoᵧCu_z)₅₀ alloys: a positioning of the compositions studied by Firstov on the NiₓCoᵧCu_z ternary diagram proposed at the ESOMAT 2018 international</figcaption></figure>

conference in Metz, b diffractograms of alloys with martensitic and austenitic phases, c diffractograms of austenitic alloys

<table><caption>**Table 1** Composition of HE-SMAs studied by Firstov [12, 21–23]</caption><thead><tr><th rowspan="2">Alloys</th><th colspan="6">Alloying composition (at.%)</th></tr><tr><th>Ti</th><th>Hf</th><th>Zr</th><th>Ni</th><th>Co</th><th>Cu</th></tr></thead><tbody><tr><td>(TiHfZr)₅₀Ni₂₅Cu₂₅</td><td>16.667</td><td>16.667</td><td>16.667</td><td>25</td><td>0</td><td>25</td></tr><tr><td>(TiHfZr)₅₀Ni₂₅Co₅Cu₂₀</td><td>16.667</td><td>16.667</td><td>16.667</td><td>25</td><td>5</td><td>20</td></tr><tr><td>(TiHfZr)₅₀Ni₂₅Co₁₀Cu₁₅</td><td>16.667</td><td>16.667</td><td>16.667</td><td>25</td><td>10</td><td>15</td></tr><tr><td>(TiHfZr)₅₀Ni₂₀Co₅Cu₂₅</td><td>16.667</td><td>16.667</td><td>16.667</td><td>20</td><td>5</td><td>25</td></tr><tr><td>(TiHfZr)₅₀Ni₂₀Co₁₀Cu₂₀</td><td>16.667</td><td>16.667</td><td>16.667</td><td>20</td><td>10</td><td>20</td></tr><tr><td>(TiHfZr)₅₀Ni₁₆.₆₆₇Co₁₆.₆₆₇Cu₁₆.₆₆₇</td><td>16.667</td><td>16.667</td><td>16.667</td><td>16.667</td><td>16.667</td><td>16.667</td></tr><tr><td>(TiHfZr)₅₀Ni₂₅Co₂₅</td><td>16.667</td><td>16.667</td><td>16.667</td><td>25</td><td>25</td><td>0</td></tr><tr><td>(TiHfZr)₅₀Co₂₅Cu₂₅</td><td>16.667</td><td>16.667</td><td>16.667</td><td>0</td><td>25</td><td>25</td></tr></tbody></table>
in Fig. 1. Only the X-ray diffraction technique is used to determine the present phases and to discriminate martensite B19' from austenite B2.

Analysis of Other Works on HT-SMAs Inspired
by Firstov Studies

Firstov’s work in 2015 initiated a promising new class of
high-entropy NiTi-like shape memory alloys, (TiHfZr)
(NiCoCu). Subsequently, several authors, such as Lee et al.
[38], Peltier et al. [24, 37], Thiercelin et al. [31], Yaacoub
et al. [28], Chen et al. [27], Rehman et al. [39], Chang
et al. [40, 41], Piorunek et al. [29, 42], designed new alloys
inspired from Firstov’s approach. Figure 2 shows the chemi-
cal composition of these alloys and their austenite-to-mar-
tensite transformation start temperatures (martensite start or
*M*s) obtained from differential scanning calorimetry (DSC)
measurements. The temperatures ranging from – 34 to
591 °C illustrate a transition zone between superelastic alloys
(*M*s below 25 °C) and shape memory alloys (*M*s above 25
°C) that correspond to a fully austenitic structure or a com-
bination of austenitic and martensitic phases, respectively,
as shown in Fig. 1. Nevertheless, by analyzing the *M*s tem-
perature given in the different works cited above, it must be
pointed out that the *M*s have not been measured necessarily

in the same metallurgical state of the alloy. In fact, this state
depends on the processing procedure, which can vary from
one study to another. That explains the noticed differences
between some alloys for an identical chemical composition.
A criterion of metallurgical state, based on the DSC meas-
urements, is given in the following section. Table 2 presents
all the alloys of the (TiHfZr)50(NiXCoYCuZ)50 system listed
in the literature. This table shows that certain alloys, such as
(TiHfZr)50Ni25Cu25 and (TiHfZr)50Ni25Co10Cu15 are studied
in several works. Table 2 shows that the results of the Ms
transformation temperatures of the same alloy vary from
one study to another, showing the importance of the metal-
lurgical state of the alloys considered to construct the SMAs
databases [31]. In that sense, one can mention the variation
of 191 °C between the Firstov study [12] and that carried
out by Yaacoub et al. [28] on the (TiHfZr)50Ni25Co10Cu15
alloy used for the training of the artificial intelligence-based
approach proposed by Thiercelin et al. [31]. It is important to
stress that the experimental results of Firstov et al. [12] and
Chang et al. [40] are based on the study of HE-SMAs in a
metallurgical state after casting or ‘as-cast’ state. These non-
homogenized alloys show wide hysteresis inducing hence a
large gap between the temperatures Ms and Mf as shown in
Table 2. Indeed, the mean value of |Ms-Mf| is up to 80 °C for
the result presented by Firstov et al. [12] and about 93 °C for
those presented by Chang et al. [40].

<figure><img src="image_4.png" /><figcaption>**Fig. 2** *M*<sub>s</sub> transformation temperatures of different alloys in studies based on the pseudo-ternary diagram proposed by Firstov. It is worth noticing that the temperatures ranging from – 34 to 591 °C illustrate a transition zone between austenitic state (*M*<sub>s</sub> below 25 °C) and a combination of austenitic and martensitic states (*M*<sub>s</sub> above 25 °C)</figcaption></figure>

<figure><img src="image_5.png" /></figure>
<figure><img src="image_6.png" /><figcaption>**Table 2** HE-SMAs of the (TiHfZr)₅₀NiCoCu system and the related *M*<sub>s</sub> temperatures found in the literature</figcaption></figure>

From Experimental Data to Numerical Predictive
Models

Considering the multi-component alloys, CCAs and HEAs,
metallurgy should take advantage of “machine learning”
concepts to explore the potential combination of the large
compositional space of chemical elements. This will lead
to optimized alloys to meet the industrial needs in terms
of application requirements or to assess other sustainability
criteria. For the NiTi-like alloy design, Peltier et al. [37]
proposed a linear regression model aimed at predicting mar-
tensitic transformation. Nevertheless, this model, called as
“Peltier model,” has been calibrated considering a small
database of only 16 alloys and it has been limited to only
some high-entropy alloys. In addition, the linear approach
was found to be unsuitable for exploring the full compo-
sitional space of multi-component alloys containing more
than 4 chemical elements. Thiercelin et al. overcome this
limitation [31], by improving the NiTi-like HE-SMAs data-
base proposed by Peltier et al. [43] (https://data.mendeley.
com/datasets/z6j6g3b9yf/1; https://doi.org/10.17632/z6j6g
3b9yf.1). This database has enriched the one given by Peltier
et al. [37] containing alloys of simple and complex chemical
compositions: 59% binary and ternary alloys, 28% quater-
nary alloys, and 13% alloys containing 5 and more elements,
such as Cu, Co, Fe, Pd, Pt, Au, Zr, Hf, Nb, and Ta. The
main advantage of this database resides in the addition of a
metallurgical criterion designed to retain only homogenized
alloys. This criterion measures the homogeneity of an alloy
by computing the difference between the start and the end
of the martensitic transformation. A suitable value has been
set to be below 50 °C, i.e., |*M*<sub>s</sub>−*M*<sub>f</sub>|<50 °C. This criterion
includes the metallurgical approach to the data, ensuring
more robustness of experimental data from different research
projects. Indeed, when the gap *M*<sub>s</sub>−*M*<sub>f</sub> does not exceed 50 °C,

the alloy exhibits a homogenized metallurgical state. In con-
trast, as observed in the work of Firstov [12], Piorunek [29],
and Chang et al. [40] for as-cast or aged alloys, the DSC
curves exhibit a significantly broadened exothermic peak,
indicating slower and delayed transformations within the
material. As a result, this leads to a large M<sub>s</sub>-M<sub>f</sub> temperature
gap. Considering the |M<sub>s</sub>-M<sub>f</sub>|<50 °C, a physics-informed
machine learning model, called as “Thiercelin model” has
been developed to predict the martensitic transformation
temperature of NiTi-like shape memory alloys containing
several chemical elements [31]. This model is based on
Extremely Randomized Tree regression algorithm to pre-
dict M<sub>s</sub> temperature using a selection of 6 thermodynamic
descriptors defined using the chemical composition of the
alloying elements and the related physical features, namely:
the mixing enthalpy, the valence electron concentration, the
atomic radius (averaged with its variation coefficient), and
the electronegativity (averaged with its variation coefficient).
The model was trained using 80% of the total database. As
the amount of binary, ternary, quaternary, and high-entropy
alloys is not equal, a proportion of 80% is extracted for each
alloy class to be representative of the database. In addi-
tion, due to the reduced number of alloys in the database,
a K-cross validation method using 100 random splits was
used to assure the robustness of the algorithm. It is observed
that accuracy decreases as the number of elements increases,
with an average error of over 65 °C for the HEA predictions.

Figure 3 shows the extrapolation of this approach on all
(TiHfZr)₅₀(NiₓCoᵧCu_z)₅₀ alloy combinations. It is impor-
tant to mention that the Thiercelin model has been deployed
in two ways using different databases as mentioned in the
introduction. The first way exploited the “global database,”
which contains binary, ternary, quaternary, quinary, and
senary TiHfZrNiCoCu HE-HT-SMAs including data from
papers by Firstov et al. without any metallurgical state
<figure><img src="image_7.png" /><figcaption>**Fig. 3** Graphical representation of the Thiercelin model applied to (TiHfZr)₅₀(NiₓCoᵧCu_z)₅₀ alloys: **a** the results of the model exploiting the “reduced database,” **b** the results of the model based on the “global database” regardless the metallurgical state of the alloy</figcaption></figure>

criterion. The second has been limited to the “reduced database” considering only the NiTi-like alloys submitted to homogenization heat treatment for which the difference between $M_s$ and $M_f$ is less than 50 °C. The results limited to the homogenized alloys (reduced database) are illustrated in Fig. 3a, whereas the predictions using the reduced database version are plotted in Fig. 3b. It should be noted that the two representations (Fig. 3a and b) are very distinct. In the reduced version, the results, providing $M_s$ transformation temperatures range from 0 to 300 °C, whereas in the global version, $M_s$ transformation temperatures range from 0 to 600 °C. These differences are due to the use of data from Chang et al. [40], which shows a quaternary (TiHfZr)₅₀Ni₅₀ alloy with $M_s$ transformation temperatures of 591 °C considering the metallurgical state criterion, $|M_s - M_f|$ equal to 79 °C, which is above of the suitable limit value of 50 °C.

In Fig. 1a, Firstov et al. [21–23] demonstrated the existence of 2 zones on the pseudo-ternary diagram: one where the alloys are martensitic with transformation temperatures $M_s$ above 25 °C (presence of the B19' phase at room temperature in Fig. 1b) and the second, larger zone where the alloys are austenitic (presence of the B2 phase at room temperature in Fig. 1c). Conversely to Firstov et al., the Thiercelin model results in Fig. 3a show only one zone since all alloys have $M_s$ transformation temperatures above 50 °C. However, it should be noted that Thiercelin et al. explained that their model has a mean absolute error of 35 °C and 70 °C for 4 quaternary alloys, and HEAs, respectively. Therefore, a $M_s$ prediction below 100 °C implies necessarily that the

considered alloys exhibit an unknown metallurgical state, which is in an unhomogenized metallurgical state.

In addition, to further extend the work by Firstov et al. on NiTi-like HE-SMAs, Rehman et al. [39] have studied homogenized non-equimolar NiTi-like compositions, i.e., alloys with the proportion of Ti-like (Ti, Hf, Zr) and Ni-like (Ni, Co, Cu) different from 50 to 50 at.%. Indeed, the authors proposed the study of 4 alloys $(Ti_{1/3}Hf_{1/3}Zr_{1/3})_X(Ni_{1/2}Co_{1/5}Cu_{3/10})_{100-X}$ where X is replaced by 50, 50.5, 51, and 52 at.% given in Table 3.

In what follows, the numerical predictions are firstly compared to the experimental findings by Rehman et al. [39] considering 4 alloys (TiHfZr)(NiCoCu) where the alloying elements content, given in Table 3, are close to those studied by Firstov et al. In a second time, the $M_s$ temperatures predicted by Thiercelin model are compared to the experimental results provided by Resnina et al. [44–47], Zadeh et al. [36], Chang et al. [48], Kuo et al. [49], and Pang et al. [50] considering more than 30 new chemical compositions of (TiHfZr)(NiCoCu) whose contents are non-equiatomic or totally different from those studied by Firstov et al., as detailed in Table 4.

## Comparison of the Numerical Predictions Against Rehman et al. Experimental Results

The experimental results of Rehman et al. [39], previously used as training and validation data [43] for the Thiercelin model, are investigated and compared with the numerical
<table><caption>**Table 3** Experimental data from Rehman [39] Vs calculated data [31, 37]</caption><thead><tr><th rowspan="3">X</th><th rowspan="3">Alloying compositions (at. %)</th><th colspan="4">M<sub>s</sub> temperature (°C)</th></tr><tr><th colspan="2">Experimental</th><th colspan="2">Calculated</th></tr><tr><th>Rehman et al.</th><th>Peltier et al.</th><th>Thiercelin et al.</th><th></th></tr><tr><th></th><th></th><th></th><th></th><th>Global Database</th><th>Reduced Database</th></tr></thead><tbody><tr><td>50.0</td><td>Ti<sub>16.7</sub>Hf<sub>16.7</sub>Zr<sub>16.7</sub>Ni<sub>25.0</sub>Co<sub>10.0</sub>Cu<sub>15.0</sub></td><td>54</td><td>153 <span style="color:red;">99</span></td><td>58 <span style="color:red;">4</span></td><td>93 <span style="color:red;">39</span></td></tr><tr><td>50.5</td><td>Ti<sub>16.8</sub>Hf<sub>16.8</sub>Zr<sub>16.8</sub>Ni<sub>24.7</sub>Co<sub>9.8</sub>Cu<sub>14.8</sub></td><td>168</td><td>170 <span style="color:red;">2</span></td><td>149 <span style="color:red;">19</span></td><td>152 <span style="color:red;">16</span></td></tr><tr><td>51.0</td><td>Ti<sub>17.0</sub>Hf<sub>17.0</sub>Zr<sub>17.0</sub>Ni<sub>24.5</sub>Co<sub>9.7</sub>Cu<sub>14.7</sub></td><td>170</td><td>185 <span style="color:red;">15</span></td><td>150 <span style="color:red;">20</span></td><td>159 <span style="color:red;">11</span></td></tr><tr><td>52.0</td><td>Ti<sub>17.3</sub>Hf<sub>17.3</sub>Zr<sub>17.3</sub>Ni<sub>24.0</sub>Co<sub>9.3</sub>Cu<sub>14.3</sub></td><td>183</td><td>220 <span style="color:red;">37</span></td><td>203 <span style="color:red;">20</span></td><td>183 <span style="color:red;">0</span></td></tr></tbody><tfoot><tr><td colspan="6" style="text-align:center;">Mean value for each model</td></tr><tr><td colspan="6" style="text-align:center;">Difference between model and experimental</td></tr><tr><td colspan="6" style="text-align:center;">39 16 17</td></tr></tfoot></table>

results of two prediction approaches, the linear approximation-based model of Peltier et al. [37] as well as the Thiercelin model [31].

The results are summarized and compared in Table 3. The latter shows that both the “global” and “reduced” versions of the results are the predictions using the Thiercelin model [31]. In fact for HE-SMAs, the predicted temperatures show a maximum absolute error of 39 °C with respect to the experimental values. One can remind that the maximum absolute error predicted using the linear approach proposed by Peltier et al. [37] reaches an error level of 99 °C for (TiHfZr)<sub>50</sub>(Ni<sub>x</sub>Co<sub>y</sub>Cu<sub>z</sub>)<sub>50</sub> HE-SMAs. However, it can be noticed that the linear model is still accurate as the data-driven model prediction (in both versions) for X=50.5, X=51, and X=52 at.%. This can be explained by the fact that the Peltier model has been based on a specific database of alloys containing only (TiHfZr)(NiCoCu) HE-SMAs with chemical compositions close to those elaborated by Firstov et al., whereas the Thiercelin model is interested in database including low, medium, and high-entropy SMAs.

Figure 4 shows the chemical composition map of NiTi-like alloys given by the Thiercelin model using the reduced database with different values of Ti-like content (X), namely: X=50, 50.5, 51, and 52 at.%. In this figure, the alloy compositions of Rehman et al. [39] are highlighted by red crosses surrounded by black circles. Table 3 and Fig. 4 demonstrate that increasing the Ti-like content increases the M<sub>s</sub> transformation temperature regardless the proportion of Ni, Cu, and Co elements.

## Comparison of the Numerical Predictions Considering Other Compositions

The numerical predictions are now compared with experimental studies recently published by Resnina et al. [44–47], Zadeh et al. [36], Chang et al. [48], Kuo et al. [49], Pang

et al. [50], and Li et al. [51] which provide more than 30 new chemical compositions whose contents are non-equiatomic or completely different from those studied by Firstov et al. as detailed in Table 4. It should be pointed out that the compositions proposed by Resnina et al. [44–47], Li et al. [51], and Zadeh et al. [36] do not allow the use of the linear model proposed by Peltier et al. [37]. In fact, these chemical compositions exceed the validity range of the model (in shaded boxes in Table 4). In fact, unlike the alloy compositions of Rehman et al. in Fig. 3 and Table 3, these new alloys cannot be represented on a pseudo-ternary diagram because their composition of Ti-like elements is not equiatomic. The list of these new alloys includes very specific chemical compositions (at.%), such as Ti<sub>9</sub>Hf<sub>39</sub>Zr<sub>4</sub>Ni<sub>30</sub>Co<sub>9</sub>Cu<sub>9</sub>, Ti<sub>6</sub>Hf<sub>38</sub>Zr<sub>8</sub>Ni<sub>17</sub>Co<sub>16</sub>Cu<sub>15</sub>, Ti<sub>9</sub>Hf<sub>32</sub>Zr<sub>11</sub>Ni<sub>16</sub>Co<sub>14</sub>Cu<sub>18</sub> [44–47], in which the element Hafnium predominates, in order to increase the transformation temperatures. It is worth noticing that compared to using the data in Table 3, the average error between the experimental results and the predictions of the two models is higher when using the experimental results in Table 4: 2.6 times higher for the Peltier model and 3.6 times higher for the Thiercelin model. This wide scatter of the calculated results can be explained by the fact that the experimental data of Resnina et al. and, Zadeh et al. are data obtained on alloys in their "as-cast" state without homogenization heat treatment.

## Concluding Remarks

This paper provides an overview of the design strategy for (TiHfZr)(NiCoCu) HE-HT-SMAs, starting with the experimental work presented by Firstov at the ESOMAT 2018 international conference in Metz. At the conference, he introduced eight quinary and senary alloys in a pseudo-ternary diagram, (TiHfZr)<sub>50</sub>(Ni<sub>x</sub>Co<sub>y</sub>Cu<sub>z</sub>)<sub>50</sub>, which exhibited
<table><caption>**Table 4** Experimental data from the recent literature Vs calculated data [31, 37]</caption><thead><tr><th colspan="2">(TiHfZr)<sub>x</sub>(NiCoCu)<sub>100-x</sub></th><th rowspan="3">X</th><th rowspan="3">Alloying compositions (at.%)</th><th rowspan="3"></th><th colspan="4">M<sub>s</sub> temperature (°C)</th></tr><tr><th colspan="2">Experimental</th><th colspan="2">Calculated</th></tr><tr><th></th><th></th><th>Peltier et al. [37]</th><th>Thiercelin et al. [31]</th><th>Global Database</th><th>Reduced Database</th></tr></thead><tbody><tr><td>52.0</td><td>Ti<sub>9</sub>Hf<sub>39</sub>Zr<sub>4</sub>Ni<sub>30</sub>Co<sub>9</sub>Cu<sub>9</sub></td><td>Resnina et al. [44-47]</td><td></td><td></td><td>419</td><td></td><td>308 <span style="color:red;">111</span></td><td>418 <span style="color:red;">1</span></td></tr><tr><td>50.0</td><td>Ti<sub>12</sub>Hf<sub>26</sub>Ni<sub>41</sub>Cu<sub>9</sub></td><td>Zadeh et al. [36]</td><td></td><td></td><td>300</td><td></td><td>167 <span style="color:red;">133</span></td><td>130 <span style="color:red;">170</span></td></tr><tr><td>50.3</td><td>Ti<sub>25.3</sub>Hf<sub>25</sub>Ni<sub>42.7</sub>Cu<sub>7</sub></td><td>Zadeh et al. [36]</td><td></td><td></td><td>299</td><td></td><td>195 <span style="color:red;">104</span></td><td>150 <span style="color:red;">149</span></td></tr><tr><td>51.5</td><td>Ti<sub>31.5</sub>Hf<sub>25</sub>Zr<sub>5</sub>Ni<sub>48.5</sub></td><td>Pang et al. [50]</td><td></td><td></td><td>291</td><td></td><td>274 <span style="color:red;">17</span></td><td>277 <span style="color:red;">14</span></td></tr><tr><td>50.0</td><td>Ti<sub>23.9</sub>Hf<sub>26</sub>Ni<sub>41.3</sub>Cu<sub>7</sub></td><td>Zadeh et al. [36]</td><td></td><td></td><td>286</td><td></td><td>217 <span style="color:red;">69</span></td><td>132 <span style="color:red;">154</span></td></tr><tr><td>51.5</td><td>Ti<sub>31.5</sub>Hf<sub>25</sub>Zr<sub>5</sub>Ni<sub>43.5</sub>Cu<sub>5</sub></td><td>Pang et al. [50]</td><td></td><td></td><td>221</td><td></td><td>139 <span style="color:red;">82</span></td><td>204 <span style="color:red;">17</span></td></tr><tr><td>50.0</td><td>Ti<sub>20</sub>Hf<sub>15</sub>Zr<sub>15</sub>Ni<sub>35</sub>Cu<sub>15</sub></td><td>Kuo et al. [49]</td><td></td><td></td><td>206</td><td>150 <span style="color:red;">56</span></td><td>180 <span style="color:red;">26</span></td><td>187 <span style="color:red;">19</span></td></tr><tr><td>50.0</td><td>Ti<sub>23.1</sub>Hf<sub>27</sub>Ni<sub>21.9</sub>Cu<sub>8</sub></td><td>Zadeh et al. [36]</td><td></td><td></td><td>192</td><td></td><td>226 <span style="color:red;">34</span></td><td>182 <span style="color:red;">10</span></td></tr><tr><td>52.0</td><td>Ti<sub>17.3</sub>Hf<sub>17.7</sub>Zr<sub>17.3</sub>Ni<sub>24.0</sub>Co<sub>14.4</sub>Cu<sub>9.6</sub></td><td>Rehman et al. [39]</td><td></td><td></td><td>183</td><td>212 <span style="color:red;">29</span></td><td>183 <span style="color:red;">0</span></td><td>203 <span style="color:red;">20</span></td></tr><tr><td>52.0</td><td>Ti<sub>6</sub>Hf<sub>38</sub>Zr<sub>8</sub>Ni<sub>27</sub>Co<sub>16</sub>Cu<sub>15</sub></td><td>Resnina et al. [44-47]</td><td></td><td></td><td>172</td><td></td><td>285 <span style="color:red;">113</span></td><td>426 <span style="color:red;">254</span></td></tr><tr><td>52.0</td><td>Ti<sub>9</sub>Hf<sub>32</sub>Zr<sub>11</sub>Ni<sub>16</sub>Co<sub>14</sub>Cu<sub>18</sub></td><td>Resnina et al. [44-47]</td><td></td><td></td><td>172</td><td>450 <span style="color:red;">278</span></td><td>258 <span style="color:red;">86</span></td><td>298 <span style="color:red;">126</span></td></tr><tr><td>51.0</td><td>Ti<sub>17.0</sub>Hf<sub>17.0</sub>Zr<sub>17.0</sub>Ni<sub>24.5</sub>Co<sub>14.7</sub>Cu<sub>9.8</sub></td><td>Rehman et al. [39]</td><td></td><td></td><td>170</td><td>180 <span style="color:red;">10</span></td><td>159 <span style="color:red;">11</span></td><td>150 <span style="color:red;">20</span></td></tr><tr><td>50.5</td><td>Ti<sub>16.8</sub>Hf<sub>16.8</sub>Zr<sub>16.8</sub>Ni<sub>24.7</sub>Co<sub>14.8</sub>Cu<sub>9.9</sub></td><td>Rehman et al. [39]</td><td></td><td></td><td>168</td><td>166 <span style="color:red;">2</span></td><td>152 <span style="color:red;">16</span></td><td>126 <span style="color:red;">42</span></td></tr><tr><td>51.5</td><td>Ti<sub>31.5</sub>Hf<sub>15</sub>Zr<sub>5</sub>Ni<sub>38.5</sub>Cu<sub>10</sub></td><td>Pang et al. [50]</td><td></td><td></td><td>161</td><td></td><td>99 <span style="color:red;">62</span></td><td>136 <span style="color:red;">25</span></td></tr><tr><td>50.0</td><td>Ti<sub>20</sub>Hf<sub>15</sub>Zr<sub>15</sub>Ni<sub>35</sub>Cu<sub>15</sub></td><td>Chang et al. [48]</td><td></td><td></td><td>160</td><td>150 <span style="color:red;">10</span></td><td>180 <span style="color:red;">20</span></td><td>187 <span style="color:red;">27</span></td></tr><tr><td>50.0</td><td>Ti<sub>25</sub>Hf<sub>12.5</sub>Zr<sub>12.5</sub>Ni<sub>35</sub>Cu<sub>15</sub></td><td>Kuo et al. [49]</td><td></td><td></td><td>111</td><td>-6 <span style="color:red;">117</span></td><td>101 <span style="color:red;">10</span></td><td>111 <span style="color:red;">0</span></td></tr><tr><td>51.5</td><td>Ti<sub>31.5</sub>Hf<sub>15</sub>Zr<sub>5</sub>Ni<sub>33.5</sub>Cu<sub>15</sub></td><td>Pang et al. [50]</td><td></td><td></td><td>72</td><td>-81 <span style="color:red;">153</span></td><td>24 <span style="color:red;">48</span></td><td>90 <span style="color:red;">18</span></td></tr><tr><td>49.5</td><td>Ti<sub>22.6</sub>Hf<sub>27</sub>Ni<sub>35.4</sub>Cu<sub>15</sub></td><td>Zadeh et al. [36]</td><td></td><td></td><td>71</td><td></td><td>170 <span style="color:red;">99</span></td><td>108 <span style="color:red;">37</span></td></tr><tr><td>51.0</td><td>Ti<sub>28</sub>Hf<sub>27</sub>Ni<sub>27</sub>Co<sub>1</sub>Cu<sub>1</sub></td><td>Resnina et al. [44-47]</td><td></td><td></td><td>56</td><td></td><td>44 <span style="color:red;">12</span></td><td>32 <span style="color:red;">24</span></td></tr><tr><td>50.0</td><td>Ti<sub>16.7</sub>Hf<sub>16.7</sub>Zr<sub>16.7</sub>Ni<sub>25.0</sub>Co<sub>15.0</sub>Cu<sub>10.0</sub></td><td>Rehman et al. [39]</td><td></td><td></td><td>54</td><td>145 <span style="color:red;">91</span></td><td>104 <span style="color:red;">50</span></td><td>58 <span style="color:red;">4</span></td></tr><tr><td>50.5</td><td>Ti<sub>32.5</sub>Hf<sub>18</sub>Ni<sub>32.5</sub>Cu<sub>17</sub></td><td>Zadeh et al. [36]</td><td></td><td></td><td>43</td><td></td><td>-2 <span style="color:red;">45</span></td><td>79 <span style="color:red;">36</span></td></tr><tr><td>50.5</td><td>Ti<sub>38.4</sub>Hf<sub>12</sub>Ni<sub>34.6</sub>Cu<sub>15</sub></td><td>Zadeh et al. [36]</td><td></td><td></td><td>39</td><td></td><td>42 <span style="color:red;">3</span></td><td>58 <span style="color:red;">19</span></td></tr><tr><td>52.0</td><td>Ti<sub>14</sub>Hf<sub>23</sub>Zr<sub>15</sub>Ni<sub>18</sub>Co<sub>15</sub>Cu<sub>15</sub></td><td>Resnina et al. [44-47]</td><td></td><td></td><td>35</td><td>292 <span style="color:red;">257</span></td><td>314 <span style="color:red;">279</span></td><td>291 <span style="color:red;">256</span></td></tr><tr><td>50.0</td><td>Ti<sub>48</sub>Hf<sub>2</sub>Zr<sub>1</sub>Ni<sub>48</sub>Co<sub>1</sub>Cu<sub>1</sub></td><td>Resnina et al. [44-47]</td><td></td><td></td><td>34</td><td></td><td>44 <span style="color:red;">10</span></td><td>27 <span style="color:red;">7</span></td></tr><tr><td>50.0</td><td>Ti<sub>25</sub>Hf<sub>25</sub>Ni<sub>25</sub>Cu<sub>25</sub></td><td>Li et al. [51]</td><td></td><td></td><td>27</td><td></td><td>49 <span style="color:red;">22</span></td><td>42 <span style="color:red;">15</span></td></tr><tr><td>50.5</td><td>Ti<sub>36.5</sub>Hf<sub>24</sub>Ni<sub>32.5</sub>Cu<sub>17</sub></td><td>Zadeh et al. [36]</td><td></td><td></td><td>13</td><td></td><td>5 <span style="color:red;">8</span></td><td>3 <span style="color:red;">10</span></td></tr><tr><td>49.0</td><td>Ti<sub>47</sub>Hf<sub>2</sub>Zr<sub>1</sub>Ni<sub>29</sub>Co<sub>1</sub>Cu<sub>1</sub></td><td>Resnina et al. [44-47]</td><td></td><td></td><td>-1</td><td></td><td>21 <span style="color:red;">22</span></td><td>22 <span style="color:red;">23</span></td></tr><tr><td>51.5</td><td>Ti<sub>31.5</sub>Hf<sub>15</sub>Zr<sub>5</sub>Ni<sub>28.5</sub>Cu<sub>20</sub></td><td>Pang et al. [50]</td><td></td><td></td><td>-2</td><td>-130 <span style="color:red;">128</span></td><td>-3 <span style="color:red;">1</span></td><td>30 <span style="color:red;">32</span></td></tr><tr><td>50.0</td><td>Ti<sub>40</sub>Hf<sub>5</sub>Zr<sub>5</sub>Ni<sub>20</sub>Co<sub>5</sub>Cu<sub>5</sub></td><td>Resnina et al. [44-47]</td><td></td><td></td><td>-42</td><td></td><td>-31 <span style="color:red;">11</span></td><td>35 <span style="color:red;">77</span></td></tr><tr><td>51.0</td><td>Ti<sub>42</sub>Hf<sub>5</sub>Zr<sub>5</sub>Ni<sub>39</sub>Co<sub>5</sub>Cu<sub>5</sub></td><td>Resnina et al. [44-47]</td><td></td><td></td><td>-46</td><td></td><td>-42 <span style="color:red;">4</span></td><td>47 <span style="color:red;">93</span></td></tr><tr><td>49.0</td><td>Ti<sub>39</sub>Hf<sub>5</sub>Zr<sub>5</sub>Ni<sub>41</sub>Co<sub>5</sub>Cu<sub>5</sub></td><td>Resnina et al. [44-47]</td><td></td><td></td><td>-99</td><td></td><td>-28 <span style="color:red;">71</span></td><td>33 <span style="color:red;">132</span></td></tr></tbody></table>

It should be pointed out that the alloys developed by Resnina et al. [44–47], Zadeh et al. [36], Chang et al. [48], Kuo et al. [49], Pang et al. [50] and Li et al. [51] have non-equiatomic contents or completely different from those studied by Firstov et al. nor those proposed by Rehman et al. [39]

remarkable properties, particularly in terms of the $M_s$ temperature. The database for (TiHfZr)(NiCoCu) system has since been expanded and enriched through additional research studies, incorporating variations in chemical composition and homogenization heat treatments. Since 2021, numerical models predicting $M_s$ temperature have emerged, paving the way for a data-driven strategy for $(TiHfZr)_{50}(Ni_xCo_yCu_z)_{50}$ alloys. One such model, proposed by Thiercelin et al., utilizes a database of NiTi-like alloys and physical material descriptors to predict the $M_s$ temperature. In this paper, two predictive models have

been applied to the investigated (TiHfZr)(NiCoCu) family which have led to the following concluding remarks:

(1) The Thiercelin model exploits the experimental data of numerous researchers through a detailed database on the metallurgical system of (TiHfZr)(NiCoCu) shape memory alloys. The Thiercelin model is suitable for the design of HE-HT-SMAs with complex (TiHfZr)(NiCoCu) type compositions. This model represents a breakthrough in HE-SMA design since it is now pos-
<figure><img src="image_8.png" /><figcaption>**Fig. 4** Graphical representation of the $M_s$ transformation temperatures given by the Thiercelin model (using the reduced database) applied to $(TiHfZr)_x(Ni_xCo_yCu_z)_{100-x}$ alloys. The representation</figcaption></figure>

shows the model response for 4 values of X: X=50, X=50.5, X=51, and X=52. The blue circles show the 4 alloys designed by Rehman et al. [39]

sible to consistently optimize alloys without having to produce a large number of alloys experimentally.

(2) The results from the Thiercelin model are far from the experimental results proposed by Firstov in

2015, Resnina and Zadeh in 2024. This difference is explained by the fact that these authors used as-cast alloys without homogenization treatment.
(3) The results given by the Thiercelin model on (TiHfZr) (NiCoCu) alloys are very close to the experimental results of Rehman et al., Pang et al., Kuo et al., and Chang et al. who use alloys in their homogenized state.

(4) The database used in this work has been built in such a way that it can be easily enriched by input from other researchers (collaborative works). These further developments and data inputs will be beneficial for the HEAs community to expand the compositional space studied in the (TiHfZr)(NiCoCu) alloys. It will be carried out by exploring further alloys whose contents of Hf, Co, Zr, and Cu exceed those of the current (TiHfZr) (NiCoCu) alloys. This will enrich the training and testing stages of the Thiercelin model.

The design strategy for the (TiHfZr)(NiCoCu) HE-HT-SMA family is increasingly being generalized to the development of Multi-Principal Element Alloys (MPEAs). The developed approaches should combine numerical methods and experimental data to create data-driven tools that can accelerate alloy design optimization. Additionally, the critical need for sourcing materials sustainably must be integrated into alloy design strategies, potentially guided by computational approaches. While metallurgy has long been a phenomenological and experimental science, it now requires efficiency and acceleration through artificial intelligence, as reported in a review paper by Dewangan et al. [52].

## Supplementary data

Supplementary data related to this article can be found at: https://data.mendeley.com/datasets/z6j6g3b9yf/1; https://doi.org/10.17632/z6j6g3b9yf.1 [43]. The authors offer their help in using the model, encouraging readers to produce their HE-SMAs.

**Acknowledgements** This research sounds like a tribute to the work of Firstov et al.. It was supported by the M²ARTS research group of the LEM3 laboratory. The authors thank the facilities of the LEM3 laboratory ("Procédés", "MicroMat", and "MécaRhéo") and the "Cassopiée" intensive computing platform at the Ecole Nationale Supérieure d'Arts et Métiers. The authors would like to thank our KIT colleagues Eva Guilbert and Nicolas Ruchty for their help, which greatly facilitated this study. The authors will enrich the database proposed by Thiercelin et al. [43] by appending the recent state of the art on the new HE-SMAs compositions presented in the paper. The updated database will be achieved to improve the existing Thiercelin model to make it more efficient and predictive for the benefits of the scientific community. Although this article deals in part with artificial intelligence, the authors vouch for the authenticity of its contents. Artificial intelligence was used in the context of metallurgy and alloy design and in no way in the writing of this article.

**Funding** Open access funding provided by Arts et Metiers Institute of Technology.

## Declarations

**Conflict of interest** The authors have no conflicts to disclose.

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