Machine-Learning-Driven Design of High-Elastocaloric NiTi-Based Shape Memory Alloys

Abstract

In recent years, the detrimental impact of traditional gas–liquid refrigerants on the environment has prompted a shift towards sustainable solid-state refrigeration technology. The elastocaloric effect, particularly in NiTi-based shape memory alloys (SMAs), presents a promising alternative due to its high coefficient of performance. However, conventional methods for alloy development are inefficient, often failing to meet the stringent requirements for practical applications. This study employed machine learning (ML) to accelerate the design of NiTi-based SMAs with an enhanced elastocaloric effect. Through active learning across four iterations, we identified nine novel NiTi-based SMAs exhibiting phase-transformation-induced entropy changes (ΔS) greater than 90 J/kg·K⁻¹, surpassing most existing alloys. Our ML model demonstrates robust interpretability, revealing key relationships between material features and performance. This work not only establishes a more efficient pathway for alloy discovery but also aims to contribute significantly to the advancement of sustainable refrigeration technologies.

1. Introduction

Traditional refrigeration systems predominantly rely on gas–liquid compression technology, utilizing refrigerants such as chlorofluorocarbon (CFC) and hydrochlorofluorocarbon (HCFC), which significantly harm the ozone layer. While some refrigerants like hydrofluorocarbon (HFC) do not deplete the ozone layer, they still contribute to the greenhouse effect, exacerbating climate change [1]. Moreover, these conventional refrigeration systems are plagued by issues such as excessive noise levels and high energy consumption. Refrigeration alone consumes over 15% of global electricity, a figure anticipated to escalate with economic growth [2]. In response, researchers are actively exploring alternative cooling materials and technologies.

Among the promising solutions are solid-state refrigeration technologies based on various mechanisms, including the magnetocaloric effect [3,4], electrocaloric effect [5], and elastocaloric effect [6], which have been proposed as alternatives due to their environmentally friendly nature. Particularly, the elastocaloric effect, specifically based on shape memory alloys (SMAs), has shown its unique promise [7] on account of a coefficient of performance (COP) exceeding 80% [8]. Notably, NiTi-based SMAs outperform Cu-based, Fe-based, and ferromagnetic SMAs in terms of the latent heat of phase change and elastomeric effects. As a result, prototype chillers utilizing NiTi-based SMAs have been successfully reported [9,10].

The composition [11,12,13,14,15,16,17,18] and preparative process [19,20] of NiTi-based SMAs wield a significant influence on the phase transition temperature, precipitation, and microstructure. These factors, consequently, play a crucial role in affecting the ΔS of the elastocaloric effect. In the case of multielement NiTi alloys, the variables concerning the type and content of the alloying elements span a wide range. Moreover, common heat treatment processes for NiTi-based SMAs encompass homogenization, solution, and aging treatments [21]. Each step in these processes has a broad range of selectable heating temperatures, times, and cooling methods. Wang et al. [22] and Mevada et al. [23] published review papers that summarized and commented on the latest research progress related to elastocaloric refrigeration and elastocaloric materials.

Given the extensive search space involved in combined composition and process parameters, machine learning (ML) has proven to be a highly effective tool in materials research and engineering practice [24,25,26,27,28,29,30,31,32,33]. For example, Yang et al. [34] proposed a machine learning workflow to design and optimize the elastocaloric properties of NiMn-based SMAs. An XGB Regressor model was trained and utilized to predict the adiabatic temperature change. Tang et al. [35] developed a non-isothermal phase-field model and machine learning to accelerate the prediction and optimization of magneto-elastocaloric materials. Raji et al. [36] employed various ML algorithms to predict the austenite finish temperature of NiTiHfPd SMAs and then validated them through experiments. Tian et al. [37] provided a potential selection of narrow thermal hysteresis TiNi-based SMAs screened using an effective strategy combining XGBoost and DFT calculation. Zhao et al. [38] showed an adaptive design strategy that tightly coupled ML with theoretical calculations to accelerate the process of the discovery of multi-component Cu-Al-based SMAs with high ΔS.

Despite the potential of NiTi-based SMAs, the traditional trial-and-error approach in alloy development is inefficient and often leads to suboptimal materials. The development of rapid and effective design methods is essential to address the growing environmental concerns associated with conventional refrigeration technologies. Therefore, this study aimed to leverage machine learning techniques to accelerate the discovery and optimization of NiTi-based SMAs with enhanced elastocaloric effects. By integrating machine learning into the alloy design process, we seek to not only improve the efficiency of material discovery but also to provide a deeper understanding of the relationships between compositional and processing parameters and elastocaloric performance. In this study, we developed machine learning models to predict the entropy change in phase transition (ΔS) in NiTi-based SMAs using both physical characteristics and heat treatment process parameters as shown in Figure 1. For comparison, three ML models were trained. The first model (M1) was constructed using only physicochemical characteristics. The second model (M2) was constructed using combined features including physicochemical characteristics and heat treatment process parameters. Based on M2, a segmentation fitting method was introduced and yielded the third model (M3), which showed superior results. Subsequently, adopting an active learning strategy [39,40,41], we iteratively optimized new materials, that is, nine new NiTi-based SMAs with ΔS values greater than 90 J/Kg·K⁻¹ after four iterations. We further optimized the corresponding heat treatment process parameters for these new alloys and conducted experimental verification. Finally, we delved into the physical interpretability of our ML models.

2. Methods

2.1. Datasets

The data used in present work were taken from published papers. M1 used 137 data samples including compositions and ΔS values. M2 and M3 used 200 additional data samples including heat treatment process parameters besides compositions and ΔS values.

For ΔS values, one part could be directly extracted from papers using text mining. The other part was obtained by measurement based on latent heat of phase transformation (ΔH) and martensitic transformation temperatures according to Equation (1)

$$\Delta S={\displaystyle \frac{\Delta H}{T_0}},$$

(1)

Here, T₀ = ${\displaystyle \frac{(M_s+A_f)}{2}}$ and $M_s$ and $A_f$ are the starting temperature of martensite phase transformation and the end temperature of reverse transformation, respectively.

2.2. Features

The original feature set for M1 included 60 physicochemical characteristics. M2 was constructed using 20 physical features and 9 heat treatment process parameters as shown in Table S1 in Supplementary Materials. These physicochemical characteristics encompassed information at the electron, atomic, and lattice levels. For NiTi-based SMAs, generally, there are three heat treatment stages: homogenization treatment (I), solution treatment (II) and aging treatment (III). Common cooling methods include water quenching (WQ), oil quenching (OQ), air cooling (AC), and cooling in the furnace (FC). To enable the ML models to handle string-type data [42], we used label encoding to encode WQ, OQ, AC, and FC as 4, 3, 2, and 1, respectively. Some of the data samples had only one or two of the three ways of heat treatment for the SMAs. To make more efficient use of the limited dataset, for some samples that had not undergone solution or/and age heat treatment, to align the data, a temperature of 293 K and a time of 0 min with air cooling were used as the temperature and time of heat treatment, as shown in

Table 1. Filling data for missing values of heat treatment process parameters.

Heat Treatment Process Parameter		Filling Value
${HT}_{\mathrm{I}}$ (K)	Temperature of homogenization treatment	1123/1223/1273/1323
${Ht}_{\mathrm{I}}$ (mins)	Time of homogenization treatment	4320
$W_{\mathrm{I}}$	Ways of quenching of homogenization treatment	WQ/OQ/AC/FC
${HT}_{\mathrm{II}}$ (K)	Temperature of solution treatment	293
${Ht}_{\mathrm{II}}$ (mins)	Time of solution treatment	0
$W_{\mathrm{II}}$	Ways of quenching of solution treatment	AC
${HT}_{\mathrm{III}}$ (K)	Temperature of aging treatment	293
${Ht}_{\mathrm{III}}$ (mins)	Time of aging treatment	0
$W_{\mathrm{III}}$	Ways of quenching of aging treatment	AC

.

2.3. Experiments

In this work, our experiments focused on the exploration and extrapolation of the compositions and processes of NiTi-based binary and ternary SMAs, which are guided by machine learning models and are used for model validation and iterative learning. Ingots of SMAs, each with a weight of about 100 g, were prepared using 99.99% pure nickel, titanium, and other alloying metals by arc melting at 1723 K under argon atmosphere using a WK-II-type non-consumable vacuum arc melting furnace. The specimens corresponding to the predicted results from M1 were not heat-treated, and the specimens corresponding to the predicted results of M2 were heat-treated according to the predicted process parameters. Specimens with the size of 5 mm × 5 mm × 3 mm and Φ3 mm × 1.5 mm, cut from the ingots by electrospark cutting, were ground and polished. Elemental contents were verified using an energy-dispersive spectrometer (EDS) embedded in a field-emission scanning electron microscope (FE-SEM, ZEISS Sigma 300, Zeiss Greater China headquarters, Shanghai, China) using square specimens. Phase transformation temperatures and latent heat ΔH were measured using differential scanning calorimetry (DSC, Q2000 V24.11 Build 124, TA Instruments, New Castle, DE, USA) at heating and cooling rates of 10 K/min in the temperature range of 188 K~573 K using circular specimens.

3. Results and Discussion

3.1. ML Model Selection and Evaluation

Figure 2a and Figure 3a illustrate the Pearson correction between features for M1 and M2, respectively. After removing the strongly correlated features with a Pearson correlation coefficient threshold of 0.8 for M1, we retained 13 features (Table S2) for details. Similarly, ten physicochemical features and nine heat treatment process features were preserved for M2 (pink-marked in Table S1).

Notably, the temperatures, times, and quenching methods of the three heat treatment processes were independent and only related to the experimenter’s setup. Consequently, these factors were not considered in the feature screening process.

Subsequently, we evaluated the R² and RMSE of various regression models using the standardized and pre-screened dataset. The optimal model, selected based on the largest R² and smallest RMSE, was drawn from a Support Vector Machine with a radial kernel (SVR.rbf), a Random Forest (RF), Gradient Boosting (GBR), Ridge Regression (Ridge), a least absolute shrinkage and selection operator (LASSO), an Elastic Network (EN), and a linear regression model (Lin). We finetuned the hyperparameters of these models using leave-out cross-validation [43] and Grid Search or Random Search.

As shown in Figure 2b,c, for M1, the top performer was SVR.rbf. After a screening of more exhaustive features, we identified four key physical features for constructing M1: ΔS = f1(aven, en, Nsvalence, Modulus, compression). The resulting R² for M1 was 0.860, and the RMSE was 8.472, and these were obtained according to Equations (1) and (2).

$$R^2=1-{\displaystyle \frac{{\sum }_{i=1}^n(y_i-\widehat{y_i})}{{\sum }_{i=1}^n(y_i-\overline{y})}}$$

(2)

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n{(y_i-\widehat{y_i})}^2}$$

(3)

Here, $\widehat{y_i}$ is the predicted value of the output of each cluster, y_i represents the experimental data, and $\stackrel{-}{y}$ is the mean value of y.

Similarly, for M2, the best-performing model was RF, with ΔS = f₂(aw, dor, rcov, Valence, HT_I,II,III, Ht_I,II,III, W_I,II,III). The R² of M2 was 0.746 and the RMSE was 10.422. However, the generally accepted threshold for better interpretability in regression models is an R² value over 0.8. Therefore, to enhance the R² of M2, we explored ensemble learning [44,45,46] (Boosting, Bagging, and Stacking), cluster learning (K-Means [47] and DBSCAN [48]), and Neural Networks [49]. The segmentation fitting method showed superior results and was designated as M3. Using M1 as a reference, we clustered points with similar heat treatment process parameters into four clusters with 45, 50, 49, and 56 data points. The R² and RMSE values of the optimal models for these clusters are showcased in Figure 4 and

Table 2. The best models for four clustering subsets, their R² and RMSE values, and comprehensive results for K-Means clustering model M3.

Clusters	Models	R2	RMSE_Train	RMSE_Test
cluster_0	GBR_1	0.834	3.178	7.598
cluster_1	RF	0.812	2.899	7.998
cluster_2	Lasso	0.893	5.678	6.861
cluster_3	GBR_2	0.895	1.859	6.393
all		0.866		7.544

. The final R² for M3 was 0.866, with an RMSE of 7.544. The divide-and-conquer approach demonstrated better fitness and model performance compared to the traditional RF model M2.

In physics, entropy is commonly understood as the degree of disorder within a system. The entropy ΔS of an alloy is influenced by various factors, as expressed in Equation (3) [45].

$$\Delta S={\Delta S}_{vib.}+{\Delta S}_{el.}+{\Delta S}_{mag.}+{\Delta S}_{dis.}+{\Delta S}_{conf.}$$

(4)

Here, ${\Delta S}_{vib}$ represents the entropy change caused by atomic vibration, ${\Delta S}_{ele}$ represents the entropy change caused by electrons, ${\Delta S}_{mag}$ represents the entropy change caused by the magnetic properties of the material, and ${\Delta S}_{conf}$ represents the entropy change caused by structural alteration.

Given that the NiTi-based SMAs in this study were non-magnetic, we have ${\Delta S}_{mag}=0$. Additionally, since the martensitic phase change was a non-diffusive phase transformation, ${\Delta S}_{conf}=0$.

While the highest-scoring four features for M1 differed from those for M2 and M3, all three models encapsulated information about the alloys at the electronic, atomic, and lattice levels. Electronegativity (en), for instance, denotes an atom’s ability to attract electrons. Electronegativity, the valence electron number (ven), and the covalent radius (rcov) exhibited periodic trends, with main group elements of the same period witnessing an increase in en and a decrease in rcov as ven increased. In the same main group, as periodicity increased, en decreased while rcov increased.

The average electron concentration (aven) and the s orbital electron filling number (NsValence) are related to an element’s position in the periodic table. In addition, the compression modulus has a relationship, as per Equation (4), with the elastic modulus.

$$K={\displaystyle \frac{E}{3\left(1-2v\right)}}$$

(5)

Here, $E$ represents the modulus of elasticity, $K$ represents the modulus of compression, and $v$ is the Poisson’s ratio of the alloys, signifying the lateral shrinkage capacity. The elastic modulus indicates the ease of atoms leaving their equilibrium positions, characterizing the strength of interatomic bonding. Crystals with strong interatomic bonding, such as covalently bonded crystals like diamonds, have a high elastic modulus.

Features like the aven, en, Nsvalence, compression modulus, rcov, and ven all directly or indirectly reflect information about ${\Delta S}_{ele}$ at the electronic level. Additionally, the effect of atomic mass (am) on vibrational entropy shows that a larger atomic mass in a compound leads to less violent vibration under the same conditions, resulting in a smaller ${\Delta S}_{vib}$. Despite the non-identical feature sets, both provide a reasonable explanation for the phase transformation entropy change $\Delta$S in terms of electron, atomic, and lattice aspects.

In terms of the generalization ability of the model, the machine learning framework and methodology that we have developed can be adapted to other elastocaloric materials directly. The core principles of feature selection, model training, and active learning can be transferred to other caloric materials, e.g., magnetocaloric and electrocaloric materials. However, each type of caloric material may require specific features relevant to its mechanisms. For instance, magnetocaloric materials would involve magnetic properties while electrocaloric materials would focus on dielectric properties.

3.2. Validation

To evaluate each machine learning model, we not only assessed its performance on the training data but also prioritized examining its extrapolation ability. For this purpose, we collected five new data points that were not present in the training dataset to assess the extrapolation ability of M1. Similarly, 30 new data points were gathered for validating M2 and M3. For the M3 segmented-fit model, a preliminary step involved assigning each data point to the relevant cluster before entering it into the prediction model. Here, we computed the clustering center of each cluster model Ci (i = 1, 2, 3, 4), where Ci represented a sequence of 13 × 1. The next step involved calculating the Euclidean distance (${Dis}_{ij}=\sqrt{{{(D}_j-C_i)}^2}$) of each data point (D1, …, Dj, …, D30) from the four clustering centers separately. We considered a data point $D_j$ to belong to cluster $C_i$ when the distance ${Dis}_{ij}$ was minimal.

The validation results for the new data of M1, M2, and M3 are shown in Figure 5a–c, respectively, with average errors of 13.84%, 13.51%, and 2.91%. Notably, the incorporation of heat treatment process features led to a significant reduction in the extrapolation error for the models. Moreover, the segmented fitting model proved to be even more effective in minimizing the extrapolation error, providing clear evidence of the improved extrapolation ability of the model.

3.3. Material Design Toward High-Elastocaloric SMA

In pursuit of NiTi-based SMAs with heightened elastocaloric effects, we applied the machine learning models M1, M2, and M3 to an adaptive design approach. The exploration encompassed the search space for binary and ternary SMAs, comprised of 51 NiTi-based SMAs. The third group of elemental predictions excluded gases, radioactive elements, toxic elements, alkali metal elements, and halogens, with lanthanum representing the lanthanide system. The elemental concentration ranges are detailed in

Table 3. Types and concentration ranges of third group elements in NiTi-based SMA for M1, M2, and M3.

Element	Range
Ni	balance
Ti	45 at. %~60 at. %
Cu, V, Mn, Fe, Hf, Pd	0~30 at. %
Other	0~10 at. %

, with a step of 0.5 at.% for each element. In addition, we defined the search space for heat treatment process parameters for M2 and M3 as outlined in

Table 4. Range and steps of 9 heat treatment process parameters for M2 and M3. HT1, Ht1, and CQ1 indicate the temperature, time, and way of cooling of homogenization; HT2, Ht2, and CQ2 indicate the parameter of solution; and HT3, Ht3, and CQ3 indicate the parameter of aging.

Heat Treatment	Range	Step
${HT}_{\mathrm{I}}$	1323 K	/
${Ht}_{\mathrm{I}}$ $W_{\mathrm{I}}$	4320 min WQ	/ /
${HT}_{\mathrm{II}}$	323 K~1223 K	300 K
${Ht}_{\mathrm{II}}$ $W_{\mathrm{II}}$	60 min~720 min WQ/AC	60 min /
${HT}_{\mathrm{III}}$	323 K~1223 K	300 K
${Ht}_{\mathrm{III}}$	30 min~420 min	60 min
$W_{\mathrm{III}}$	WQ/AC	/

. Notably, we constrained the parameters for homogenization heat treatment to 1323 K, 4320 min, and water quenching, optimizing for computational efficiency and cost-effectiveness. Additionally, for solution and aging heat treatment, we not only introduced values from Table 4 but also used the heat treatment process parameters with a temperature of 293 K, time of 0 min, and cooling method of air cooling to complete the missing process parameters.

For M3, candidate alloys within the search space were selected based on the maximum expected improvement EI (μ, σ) as defined by Equation (6). Active learning was used as shown in Figure 2e.

$$EI\left(\mu ,\sigma \right)=\sigma [\varphi \left(z\right)+z\mathrm{\Phi }(z\left)\right]$$

(6)

Here, $z=\left(\mu -{\mu }^{\mathrm{*}}\right)/\sigma$, where ${\mu }^{\mathrm{*}}$ is the maximum value in the training set, $\mu$ is the mean value of the predicted data, and $\sigma$ is the standard deviation of $\varphi \left(z\right)$ and $\mathrm{\Phi }\left(z\right)$, representing the standard normal density and distribution function, respectively.

The element space to be predicted involved all elements in the periodic table except gases, halogen elements, alkali metal elements, radioactive elements, and toxic elements. This element space was too large. Therefore, based on domain knowledge and the time cost of model running, the alloys of the third component such as Cu, V, Mn, Fe, Hf, Pd, and Zr were limited to the range of 0~5 at.%, and the other alloys were limited to the range of 0~2 at.% near the NiTi-based SMAs. The selected prediction material included about 300,000 compositions.

3.4. Experimental Validation

We conducted experiments to validate the prediction results and provide feedback to each next iteration. At least four compositions were experimentally validated for each iteration. In four iterations, we found nine new NiTi-based SMAs with ΔS > 90 J/Kg·K⁻¹, as detailed in

Table 5. SMAs designed with ΔS greater than 90 J/Kg·K⁻¹ in 4 iterations.

Alloys (at. %)	ΔS_pred (J/Kg·K−1)	ΔS_exp (J/Kg·K−1)
Ni49.5Ti49.5Ga1	89.03	97.91
Ni49Ti50Ge1	89.80	93.8
Ni49Ti50.5Si0.5	84.63	93.78
Ni49.5Ti50Te0.5	88.27	93.77
Ni49Ti50.5In0.5	84.33	93.41
Ni48.5Ti51Sb0.5	87.56	92.77
Ni49Ti50.5Sn0.5	84.60	92.45
Ni49Ti49Be2	91.35	91.48
Ni49Ti50.5Sc0.5	84.62	90.72

. As shown in the third column in Table 5, a small amount of Ge, Ga, In, Te, Sb, Sc, Si, Sn, and Be can improve the elastocaloric effect of NiTi-based SMAs. These third group elements are mainly meta-metals and poor metals, which generally have higher electronegativity, greater atomic masses, and larger covalent radii.

The above validation results showcase the efficacy of machine learning in accelerating material discovery, significantly reducing the time and resources typically required in traditional experimental methods.

However, the datasets for the tree models were relatively small, which may have limited the generalizability of the models. Although the introduction of active learning and cluster learning improved the performance of the models and assisted us in finding new materials, the dependence of the prediction reliability on the dataset volume and data quality cannot be ignored. In addition, the performance of M2 did not significantly improve despite incorporating heat treatment parameters; the main reason could be attributed to the dimensional disaster problem. That is, the introduction of process parameters led to an increase in the feature dimension, but the incrementation of process data was not enough to match it, indicating that the further exploration of additional features or methods may be necessary for enhanced prediction.

4. Conclusions

In this work, we focused on the application of machine learning to accelerate the discovery of novel NiTi-based shape memory alloys (SMAs) with enhanced elastocaloric effect. Three models were compared and validated. The following conclusions were drawn:

(1)

The SVR.rbf model (M1) utilized four physicochemical features (aven, en, Nsvalence, modulus compression) exhibited superior performance with an R² = 0.860 and an RMSE = 8.472.

(2)

The RF model (M2), incorporating four physicochemical features (aw, dor, rcov, Valence) along with nine heat treatment parameters, achieved R² = 0.746 and RMSE = 10.422.

(3)

Combining the M2 model and K-Means clustering, we formulated the model M3, which achieved improved prediction performance with an R² = 0.866 and an RMSE = 7.544.

(4)

Based on the M3 model, leveraging an active learning strategy through four iterations, we identified nine new NiTi-based SMAs with phase transformation entropy changes ΔS > 90 J/Kg·K⁻¹, surpassing the majority of alloys in the original dataset.

This research demonstrates the efficacy of machine learning in material discovery, offering improved model performance and the potential for practical applications in engineering. Practitioners can select from different models based on their specific requirements, allowing for informed decision making in alloy design.