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

1. Introduction

Traditional refrigeration relies on gas-liquid compression technology, utilizing common refrigerants like chlorofluorocarbon (CFC) and hydrochlorofluorocarbon (HCFC), which pose a serious threat to the ozone layer. While some refrigerants such as hydrofluorocarbon (HFC) don’t harm the ozone layer, they contribute significantly to the greenhouse effect [1]. Moreover, 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 escalated with economic growth [2]. In response, researchers are actively exploring alternative cooling materials and technologies.

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

The composition [11,12,13,14,15,16,17,18] and preparative process [19,20] of NiTi-based SMAs wield a significant influence on phase transition temperature, precipitation, and microstructure. These factors, consequently, play a crucial role in affecting the ΔS of 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 have a broad range of selectable heating temperature, time and cooling method. 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 [22,23,24,25,26,27,28,29,30,31].

In this study, we developed machine learning models to predict the entropy change of phase transition (ΔS) in NiTi-based SMAs using both physical characteristics and heat treatment process parameters. Adopting an active learning strategy [32,33], we iteratively optimized new materials that is 9 new NiTi-based SMA with ΔS 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 and Features

The first ML model for prediction of ΔS constructed using physical features (denoted as M1) contain 137 data points taken from published literatures, and including the ΔS values and 60 physicochemical characteristics. The second ML model (denoted as M2) constructed using physical features combined with heat treatment process parameters has 200 data points, including the ΔS values, 20 physical features and 9 heat treatment process parameters, as shown in Table S1 in Supplementary Materials. These physicochemical characteristics encompass 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 [34], we used label encoding to encode WQ, OQ, AC and FC as 4, 3, 2, and 1 respectively. Some of the data have only one or two of the three ways of heat treatment for the SMAs. To make more efficient use of the limited dataset, we complemented the missing solution and aging heat treatment parameters as heating at room temperature of 293 K for 0 min with air cooling, and the homogenization was also added, as shown in Table 1.

2.2. Experiments

In this work, our experiments focused on the exploration and extrapolation of the composition and process of NiTi-based binary and ternary SMAs, which guided by machine learning models and are used for model validation and iterative learning. The ingots of SMAs were prepared using 99.99% pure nickel, titanium and other alloying metals by arc melting under argon atmosphere. The specimens corresponding to the predicted results from M1 weren’t heat treated, and the specimens corresponding to the predicted results of M2 were heat-treated according to the predicted process parameters. Samples with a size of 5 mm×5 mm×3 mm and Ф3 mm×1.5 mm cut from the ingots were ground and polished. And then their elemental contents were verified using an energy dispersive spectrometer (EDS) and phase transformation temperatures and latent heat ΔH were measured using a differential scanning calorimetry (DSC).

3. Results and Discussion

3.1. ML Model Selection and Evaluation

The Figure 1(a) and Figure 2(a) illustrate the Pearson correction between features for M1 and M2, respectively. After removing the strongly correlated features with a Pearson correlation coefficients threshold of 0.8 for M1, we retained 13 features (Table S2) for details). Similarly, 10 physicochemical features and 9 heat treatment process features were preserved for M2 (pink-marked in Table S1).

Notably, the temperature, time and quenching methods of 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, from Support Vector Machine with radial kernel (SVR. rbf), Random Forest (RF), Gradient Boosting (GBR), Ridge Regression (Ridge), Least absolute shrinkage and selection operator (LASSO), Elastic Network (EN), and Linear regression model (Lin). We finetuned the hyperparameters of these models using leave-out cross-validation [35] and Grid Search or Random Search.

As shown in Figure 1(b) and 1(c), for M1, the top performer was SVR. rbf. After a more exhaustive features screening, we identified 4 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, which were obtained according to Eq. (1) and (2).

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

(1)

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

(2)

where $\widehat{y_i}$ is the predicted value of the output of each cluster, y_i is the experimental data, and y̅ 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 model is R2 over 0.8. Therefore, to enhance the R2 of M2, we explored ensemble learning [36,37,38] (Boosting, Bagging and Stacking), cluster learning (K-Means [39] and DBSCAN [40]) and Neural Networks [41]. The segmentation fitting method showed superior results, and donated as M3. Using M1 as a reference, we clustered points with similar heat treatment process parameters into four clusters with 45,50,49,56 data points, respectively. The R2 and RMSE of the optimal models for these clusters were showcased in Figure 3 and Table 2. 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 Eq. (3) [37]

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

(3)

where $\Delta S_{vib}$ represents the entropy change caused by atomic vibration, $\Delta S_{ele}$ represents the entropy change caused by electron, $\Delta S_{mag}$ represents the entropy change caused by magnetic properties of material, and $\Delta S_{conf}$ represents the entropy change caused by structural alteration.

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

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

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

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

(4)

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

Features like 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 the larger the 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.

3.2. Validation

To evaluate the 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 5 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 ($Di{s}_{ij}=\sqrt{{(D_j-C_i)}^2}$) of each data point (D1, …, Dj, …, D30) to the four clustering centers separately. We considered a data $D_j$ to belong to cluster $C_i$ when the distance $Di{s}_{ij}$ was minimal.

The validation results for the new data of M1, M2, and M3 are shown in Figure 4 (a), (b) and (c), respectively, with average error of 13.84%,13.51%, 2.91%. Notably, the incorporation of heat treatment process features led to a significant reduction in the extrapolation error for the model. Moreover, the segmented fitting model proved to be even more effective in minimizing extrapolation error, providing clear evidence of the improved extrapolation ability of the model.

3.3. Materials Design toward High Elastocaloric SMA

In pursuit of NiTi-based SMAs with heightened elastocaloric effects, we apply machine learning models M1, M2, and M3 to an adaptive design approach. The exploration encompassed the search space for binary and ternary SMAs, comprising 51 NiTi-based SMA system. The third group of elemental predictions excluded gases, radioactive elements, toxic elements, alkali metal elements, halogens with lanthanum representing the lanthanide system. The elemental concentration ranges are detailed in Table 3, 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. Notably, we constrained the parameters for homogenization heat treatment to 1323 K, 4320 mins, and water quenching, optimizing for computational efficiency and cost-effectiveness. Additionally, for solution and aging heat treatment, we introduced values not only from the Table 4 but also used the heat treatment process parameters with temperature of 293 K, time of 0 mins, and cooling method of air cooling to complete the missing process parameters.

For M1, candidate alloys within the search space were selected based on the maximum expected improvement EI (μ, σ) [32] as defined by Eq.6. Active learning, as shown in Figure 1 (e), led to the discovery of 9 new NiTi-based SMAs with $\Delta$S>90 J/Kg·K-1 in four iterations, as detailed in Supplementary Table 3.

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

(6)

Here, $z=\left(\mu -{\mu }^{*}\right)/ \sigma$, where ${\mu }^{*}$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$\varphi \left(z\right)$, representing the standard normal density and distribution function, respectively. Similarly, active learning was also employed for M2 and M3, and the initial results closely mirrored those of M1. Due to timeliness consideration, the subsequent experimental results will be published separately.

4. Summary

In this work, we harnessed the power of machine learning to expedite the discovery of novel NiTi-based SMAs with high elastocaloric effect, as well as optimizing the corresponding heat treatment processes. Our approach involved constructing machine learning prediction models, specifically the SVR.rbf model (M1), which utilized four physicochemical features (aven, en, Nsvalence, Modulus compression). M1 exhibited superior performance with an R2=0.860 and an RMSE=8.472. Leveraging an active learning strategy through four iterations, we identified 9 new NiTi-based SMAs with phase transformation entropy change $\Delta$S>90 J/Kg·K-1, surpassing the majority of alloys in the original dataset.

For practical engineering consideration, we extended our efforts to establish the RF model (M2) for $\Delta$S, incorporating 4 physicochemical features (aw, dor, rcov, Valence). To enhance interpretability, we performed feature selection and introduced 9 heat treatment process features as inputs. Consequently, the model’s performance slightly diminished, yielding an R2= 0.746 and an RMSE=10.422. To address this, we implemented K-Means clustering to formulate the model M3, achieving improved prediction performance with an R2= 0.866 and an RMSE= 7.544. Active learning was similarly applied to guide alloys design, yielding results in line with M1 in the two iterations.

It’s noteworthy that the selected features in M1, M2 and M3 encompass information about NiTi-based SMA at the electronic, atomic and lattice levels. Furthermore, M2 and M3 incorporate the influence of the heat treatment process. Depending on specific requirements, practitioners can choose different models to assist their decision-making processes.