Predictive Modeling and Analysis of Cu–Be Alloys: Insights into Material Properties and Performance

Abstract

Cu–Be alloys are renowned for their exceptional mechanical and electrical properties, making them highly sought after for various industrial applications. This study presents a comprehensive approach to predicting the compositions of various types of Cu–Be alloys, integrating a Random Forest Regressor within a machine learning (ML) framework to analyze an extensive dataset of chemical and thermo-mechanical parameters. The research process incorporated data preprocessing, model training and validation, and robust analysis to discern feature significance. Cluster analysis was also conducted to illuminate the data’s intrinsic groupings and to identify underlying metallurgical patterns. The model’s predictive power was confirmed by high R² values, indicative of its capability to capture and explain the variance in the dataset for both testing (R² = 0.99375) and training (R² = 0.99858). Distinct groupings within the alloy data were uncovered, revealing significant correlations between composition, processing conditions, and alloy properties. The findings underscore the potential of ML techniques in advancing the material design and optimization of Cu–Be alloys, providing valuable insights for the field of material science.

1. Introduction

Copper-based alloys, integral to various high-performance applications, offer exceptional mechanical properties and electrical performance. These alloys are widely used in demanding fields, such as aerospace components [1], electrical connectors [2,3], and heat transfer systems [4], due to their mechanical robustness. Among these, copper–beryllium (Cu–Be) alloys are particularly valued for their unique precipitation behavior. This behavior underlies the development of extraordinary mechanical properties through aging processes, significantly enhancing characteristics such as hardness and tensile strength. The sophisticated age-hardening process, characterized by a detailed phase transformation sequence and the formation of nano-sized precipitates, is integral to the extensive industrial application of Cu–Be alloys, particularly in sectors requiring substantial mechanical strength [5]. Specifically, Cu–2 wt.% Be alloys have been shown to achieve a substantial increase in strength, exceeding 1000 MPa, after a brief aging at 320 °C, peaking at 4 h with the predominance of γ″ and γ′ phases before diminishing with extended aging [6]. This remarkable augmentation in strength, along with improved electrical conductivity, underscores the alloy’s adaptability, making it suitable for a variety of sectors, notably aerospace and electronics. Additionally, the precipitation process significantly affects Young’s modulus of Cu–1.95 wt.% Be alloys, resulting in a notable increase in stiffness when aged under optimal conditions. This transformation of the alloy’s microstructure during aging not only enhances stiffness but also sheds light on the strengthening mechanisms at play, including dislocation pinning phenomena such as by-shearing and by-passing, critical for applications within precision engineering [7].

In recent years, researchers have been paying a lot of attention to how machine learning (ML) can help create new materials. Through the synergy of ML and metallurgical expertise, novel approaches in material design are being pioneered to optimize the properties of copper alloys. For example, Zhao et al. [8] combined Gaussian process regression with phase diagram assessments to search for Cu–Co–Si alloys with enhanced strength and conductivity, demonstrating the potential of ML in identifying compositions for superior target properties. Deng and colleagues [9] extended ML’s predictive prowess to the mechanical properties of Cu–Al alloys, facilitating the rapid characterization of new materials. Marchand’s team developed neural network potentials that offer near-first-principles accuracy for Al–Cu systems, showcasing ML’s quantitative benefits in understanding alloy behavior [10]. Similarly, Wang and colleagues leveraged ML for property-oriented design, yielding high-performance copper alloys with targeted strength and conductivity [11]. Kolev’s study is particularly notable for presenting a hybrid deep learning and ensemble learning model that assesses the effects of composition and processing on Cu–Ni–Si alloys, incorporating feature importance analysis to pinpoint the most influential factors [12]. Blaschke et al. employed a combined model experimental approach to predict electrical conductivity in Cu/Nb composites, with a focus on understanding the impact of microstructural features such as layer thickness and grain sizes on conductivity, emphasizing the significant effects of bimetal interfaces and grain boundaries [13]. Lastly, Pan et al. employed a ML-driven design method to rapidly discover high-performance Cu–Ni–Co–Si alloys, proving the advantage of integrating Co into the Ni2Si phase to enhance the alloy’s performance [14]. These innovative methodologies highlight the transformative impact of ML on material science, charting a course toward the accelerated discovery and refinement of high-performance alloys.

While ML has been extensively employed to explore and optimize various alloy systems, there remains a notable absence in the literature concerning ML models specifically tailored to Cu–Be alloys. The unique precipitation processes and phase transformations within these alloys, critical to their mechanical and electrical properties, have not been thoroughly investigated through the lens of ML. Previous studies have primarily focused on experimental techniques and first-principles calculations to understand the precipitation behavior, phase transformations, and mechanical properties of a certain type of Cu–Be alloy. These methods, while providing valuable insights, often require extensive experimental setups or complex theoretical models. This gap represents a significant opportunity to integrate ML techniques, specifically a Random Forest Regressor (RF), to predict the compositions of different types of Cu–Be alloys. By leveraging a comprehensive dataset that includes chemical compositions, thermo-mechanical processing parameters, and resulting properties, the use of RF not only accelerates the discovery and optimization of alloy compositions but also provides a robust analysis of feature importance that is both hardware- and time-efficient, identifying key factors that influence the properties of the Cu–Be alloys.

This manuscript endeavors to delve into the application of an RF to accurately predict the compositions of Cu–Be alloys by analyzing all these chemical and thermo-mechanical processing parameters alongside the resulting electrical conductivity and mechanical characteristics. By employing predictive analytics, the study aims to establish a sophisticated prediction model that not only predicts the elemental composition but also aids in anticipating the resultant thermo-mechanical behavior and electrical and mechanical performance of Cu–Be alloys. Through meticulous data preprocessing and subsequent analysis, the manuscript seeks to validate the accuracy and efficacy of the predictive model in distinguishing between different alloy compositions. Additionally, it aims to elucidate the underlying patterns within the data through cluster analysis, providing insights into the intrinsic clustering of alloy properties, which may reflect fundamental metallurgical phenomena. The study is further enriched by a feature importance analysis that discerns the relative impact of chemical composition, processing, and thermo-mechanical parameters on the electrical conductivity and mechanical properties of the alloys, offering a comprehensive understanding of the factors driving alloy performance.

2. Materials and Methods

2.1. Data Collection and Preprocessing

The research utilized a comprehensive dataset on Cu–Be alloys for analysis and modeling, encompassing variables such as chemical composition and thermo-mechanical processing details, as well as properties including mechanical characteristics and electrical conductivity. This dataset, comprising 768 data points, was selected from a larger collection of various copper-based alloys detailed by Gorsse et al. [15] and is accessible via the open access platform at the Figshare repository [16]. During the preprocessing phase, specific variables were carefully chosen, including the alloys’ chemical compositions, processing methods, and crucial properties such as hardness, yield strength, ultimate tensile strength, and electrical conductivity. Interaction features were also added to capture the complex interdependencies among these variables effectively. The beryllium content range of 1.66 to 2.00 wt. % was selected based on the available data at the time of completing the dataset. In the predictive model, the variables included the weight percentages of the elemental compositions—‘Cu’ for Copper, ‘Be’ for Beryllium, ‘Al’ for Aluminum, ‘Mg’ for Magnesium, ‘Ni’ for Nickel, ‘Si’ for Silicon, and ‘Ti’ for Titanium. Thermo-mechanical processing parameters such as ‘Tss (K)’ for the solution temperature in Kelvin, ‘tss (h)’ for the solution duration, ‘Tag (K)’ for the aging temperature, and ‘tag (h)’ for the aging duration were also integrated. Additionally, the model analyzed properties such as Hardness (HV), Ultimate Tensile Strength (MPa), Yield Strength (MPa), and Electrical Conductivity (%IACS), which are essential for evaluating the performance characteristics of Cu–Be alloys. This approach ensured a comprehensive assessment of how compositional and processing parameters influence the alloys’ mechanical and electrical properties. In the dataset, instances where data were missing have been marked as NaN (not a number) to clearly indicate the absence of this data, ensuring these values are not misinterpreted as physically meaningful. Figure 1 outlines the workflow of data collection and preprocessing, model development, and analysis performed in this study.

2.2. Model Training, Testing, and Validation

For the analysis of Cu–Be alloys, an ML approach was employed utilizing an RF—a robust method suitable for regression tasks involving complex datasets with interactions between variables. The RF is particularly adept at handling non-linear relationships, which are prevalent in material property prediction [17,18]. The initial step involved defining the features and labels from the dataset. The features selected for the model included several metal compositions and processing parameters as follows: ‘Cu’, ‘Be’, ‘Al’, ‘Mg’, ‘Ni’, ‘Si’, ‘Ti’, ‘Tss (K)’, ‘tss (h)’, ‘Tag (K)’, ‘tag (h)’, ‘Hardness (HV)’, ‘Yield strength (MPa)’, ‘Ultimate tensile strength (MPa)’, ‘Electrical conductivity (%IACS)’. The labels, intended for prediction, included the compositions of the metals (‘Cu’, ‘Be’, ‘Al’, ‘Mg’, ‘Ni’, ‘Si’, ‘Ti’). The dataset was partitioned into training and testing sets, with 40% of the data allocated for training and the remaining 60% for testing, ensuring a diverse distribution of data points. This split was determined by a random seed to ensure reproducibility of the model. An exhaustive search for the optimal hyperparameters was conducted through GridSearchCV, iterating over a predetermined range of values for the number of trees (n_estimators: [50, 100, 200]), tree depth (max_depth: [5, 10, 15]), minimum samples to split a node (min_samples_split: [2, 5, 10]), and minimum samples for a leaf node (min_samples_leaf: [1, 2, 5]). The best model was selected based on the lowest negative mean squared error score, a measure of model accuracy. GridSearchCV employed cross-validation during this tuning process, ensuring a robust evaluation of the hyperparameters. After the training phase, the model’s performance was evaluated using a set of established metrics to quantify the accuracy and predictive quality. The primary metrics used for evaluation were the mean absolute error (MAE), mean squared error (MSE), root mean squared error (RMSE), and the coefficient of determination, denoted as R². To ensure the model’s validity, both the training and testing datasets were subjected to these evaluations. This process helped ascertain the model’s capacity to learn from the training data and its generalizability to new and unseen data.

2.3. Analysis

Cluster analysis was performed to identify inherent groupings within the Cu–Be alloy dataset, employing the K-Means clustering algorithm for its efficacy in identifying clusters [19]. The data were first standardized using a StandardScaler to ensure that each feature contributed equally to the distance computations involved in the K-Means algorithm. The scaled dataset included alloy compositions as well as various physical properties such as yield strength, hardness, ultimate tensile strength, and electrical conductivity. To ascertain the optimal number of clusters, the elbow method was utilized. This method involved plotting the sum of squared errors (SSE) for a range of cluster counts and identifying the point where the decrease in SSE slowed significantly, indicating diminishing returns with additional clusters. The elbow point indicated an appropriate quantity of clusters for the algorithm to utilize. With the optimal number of clusters determined, K-Means was applied to the scaled data. The resulting cluster labels were then appended to the original dataset, enabling a detailed analysis of the common characteristics and differences among the identified clusters. Following clustering, Principal Component Analysis (PCA) was conducted to decrease the data’s dimensionality for visualization purposes. PCA reconfigured the data into a new coordinate system. In this system, the maximum variance by any data projection was situated on the first coordinate (known as the first principal component or PC1), the second maximum variance was on the second coordinate (known as the second principal component or PC2), and this pattern continued in the same manner. The loading matrix was calculated, encapsulating the influence of each initial variable on the principal components. The results were visualized in a 2D scatter plot, with the principal components on the axes and cluster labels denoting the grouping of each data point. This visualization aided in understanding the spread and separation of clusters within the reduced feature space.

To gain insights into the relative importance of different features in predicting the physical properties of the Cu–Be alloys, a feature importance analysis was conducted using a Random Forest Regressor. This analysis helped us to identify which features contribute most significantly to the model’s predictions [20]. For each target property of the alloys, such as hardness and electrical conductivity, a separate RF model was trained. The importance of each feature was then extracted, indicating its contribution to the model’s predictive ability. The importance of each feature was graphically represented through bar plots, which displayed the ranking of features for each physical property predicted by the model.

Finally, to examine the relationships between all features, a correlation heatmap was generated. The heatmap displayed Pearson’s correlation coefficients between pairs of features, providing insights into the linear relationships within the data. Positive values represented a direct relationship, whereas negative values expressed an inverse relationship. The heatmap was created using Seaborn and Matplotlib libraries to provide a clear, color-coded representation of the feature correlations.

3. Results

3.1. Model Performance and Validation

The assessment of the predictive model is two-fold, involving both quantitative metrics and visual analysis. The validation process is crucial to establish the model’s reliability in accurately predicting the compositions of Cu–Be alloys. The scatter plot, designated as Figure 2, methodically contrasts the predicted values with their actual counterparts. Its depiction of data points conforming closely to the diagonal line is emblematic of the model’s high precision. Quantitatively, the model’s prowess is validated by impressive metrics outlined in Table 1. It exhibits strong accuracy, as indicated by low MAE values of 0.01997 for the testing set and 0.01591 for the training set, and RMSE values of 0.04542 and 0.04275 for the testing and training sets, respectively. Additionally, the MSE values of 0.00206 for the testing set and 0.00183 for the training set further validate the model’s precision. These low error values, coupled with high R² scores of 0.99375 for the testing set and 0.99858 for the training set, demonstrate not only the model’s precision but also its ability to explain a significant proportion of the variance in the dataset. The consistent R² values across testing and training datasets confirm that the model is generalizable and not merely tailored to the noise within the data. This indicates reliable performance across various data samples, highlighting the model’s robustness. The predictive modeling results indicate that the optimal composition and heat treatment parameters for Cu–Be alloys are as follows: Cu 92.95 wt. %, Be 1.95 wt. %, Mg 2.00 wt. %, Ni 0.10 wt. %, Ti 3.00 wt. %. The optimal heat treatment involves a solution heat treatment at 1193 K, followed by an aging temperature at 593 K. These conditions provide the best combination of mechanical properties and electrical conductivity.

Moreover, the comparison of predicted versus actual composition values in Table 2 underscores the model’s precision. The model’s capacity to accurately predict the compositions of Cu–Be alloys is particularly evident in the minimal percentage differences between the predicted and actual values for all elements. The model’s precision is particularly notable for copper, the primary component of the alloy, with an average discrepancy of only 0.015%, underscoring its potential utility in critical applications. Beryllium, another essential element, had a slightly larger variance at 0.27%. The model incurred a deviation of 2.56% for both aluminum and silicon, while nickel had a deviation of 1.45%. For magnesium and titanium, the differences were 0.46%.

3.2. Cluster Analysis and Principal Component Insights

The application of cluster analysis reveals distinct groupings within the Cu–Be alloy dataset, which can be interpreted as representing the underlying patterns related to alloy composition and processing conditions. The elbow method, as illustrated in Figure 3a, indicates that a model with three clusters represents the data most effectively. This determination is based on the sum of squared errors, which displays a significant bend at the point of three clusters, suggesting that additional clusters would not significantly increase the model’s explanatory power. The cluster analysis results, presented in Table 3 and Figure 3c, decomposes the Cu–Be alloys into three clusters based on their chemical and physical properties. Specifically, on PC1, the strong positive loadings of Cu, Be, yield strength, and ultimate tensile strength, combined with the negative loadings of Al, Mg, Ni, Si, and Ti indicate that variations in these elements and mechanical properties play a significant role in differentiating the alloys. In contrast, PC2 is characterized by high positive loadings for Cu, Al, Ni, and Si, while electrical conductivity and mechanical properties exhibit negative loadings. The PCA loading matrix, detailed in Table 4, categorizes the Cu–Be alloys into three clusters, each with distinct chemical compositions and physical properties. Cluster 0 is distinguished by its composition, exhibiting a balance of Cu, Be, Al, Mg, Ni, Si, and Ti, and by its moderate electrical conductivity and mechanical properties. Cluster 1, which lacks Al, Mg, Ni, Si, and Ti, displays higher mechanical strength than Cluster 0 but is marked by the absence of hardness values. Cluster 2 is characterized by higher ultimate tensile strength and yield strength compared with the other two clusters, as well as increased electrical conductivity, suggesting an advanced stage of processing where the mechanical properties have been optimized. The PCA cluster visualization, as depicted in Figure 3b, confirms the clear demarcation between these clusters, corroborating the cluster analysis findings.

3.3. Feature Importance and Inter-Feature Relationships

Understanding the intricate web of inter-feature relationships within the Cu–Be alloy dataset is crucial for unraveling the complex dependencies that govern the material’s properties. Figure 4a presents a detailed correlation heatmap, offering a panoramic view of how the compositional and processing parameters interact. Notable among these is the strong positive correlation between Cu and Be, indicative of their synergistic effect on the alloy’s characteristics. In particular, Be shows a substantial correlation (0.74) with the solution duration (tss h), underscoring the significance of solution time in the development of Be’s properties within the alloy. Furthermore, Be exhibits a modest positive correlation (0.18) with the aging duration (tag h), suggesting that while less pronounced, the duration of aging is nonetheless a meaningful factor in the final properties of the alloy. Figure 4b shifts the focus to the realm of electrical properties, revealing that aging time (0.6) and aging temperature (0.31) are the dominant factors affecting electrical conductivity (%IACS). This finding accentuates the pivotal role of thermo-mechanical processing, likely through its impact on the microstructure during aging and on the electrical attributes of the Cu–Be alloys. Hardness (HV) of the alloys, as demonstrated in Figure 4c, appears to be most strongly influenced by aging time (0.34), followed closely by the elemental composition, with Cu (0.17), Mg (0.14), Ti (0.12), Ni (0.11), and Be (0.10). The preeminence of aging time corresponds with metallurgical principles, where the duration of aging is a key factor in the microstructural evolution that governs hardness. In terms of ultimate tensile strength (MPa), highlighted in Figure 4d, the aging temperature (Tag K) emerges as the feature with the highest importance score (0.34), suggesting a critical parameter that can be fine-tuned to optimize tensile performance. Solution temperature (Tss K) and the presence of Al also have significant importance scores, indicating their influence alongside compositional factors such as Cu and Be. Finally, Figure 4e examines yield strength (MPa), revealing that the aging temperature (Tag K) with a score of 0.19 is the most impactful. The aging time (tag h) and solution temperature (Tss K), with importance scores of 0.15 and 0.14, respectively, further underscore the critical influence of thermal treatments on the yield strength. Compositional elements such as Al, Cu, and Be also contribute to this property, demonstrating a nuanced interplay with thermal processing.

4. Conclusions

This study successfully deployed a Random Forest Regressor to predict the compositions of Cu–Be alloys with high accuracy, marked by low MAE values of 0.01997 for the testing set and 0.01591 for the training set, and low RMSE values of 0.04542 for the testing set and 0.04275 for the training set. The model effectively elucidated the variance within the dataset, as reflected in the R² values of 0.99375 for testing and 0.99858 for training. The precision of compositional prediction, especially for the primary component, copper, with a minimal average discrepancy of only 0.015%, underscores the model’s robustness and reliability. The variance results for other elements, such as beryllium with a variance of 0.27%, aluminum and silicon both at 2.56%, nickel at 1.45%, and notably low variances for magnesium and titanium at 0.46%, further corroborate the model’s precision. Cluster analysis delineated three distinct groups within the Cu–Be alloys, each with characteristic mechanical and electrical property profiles, revealing the maturity levels of processing stages. The strong positive correlations between elements and processing conditions, such as the substantial correlation between beryllium and solution duration, highlighted the critical impact of thermo-mechanical processing on the alloy’s attributes. Feature importance analysis provided insights into the significant drivers of alloy properties. Aging time and temperature emerged as dominant factors influencing both electrical conductivity and mechanical strength, reflecting their paramount role in microstructural development and consequently in the alloy’s performance. The analysis of feature importance also sheds light on the nuanced contributions of alloying elements to the strength characteristics. Future investigations could expand the model’s domain to encompass a wider spectrum of copper-based alloys, assessing its predictive adaptability and refining its precision to foster advancements in alloy design and innovation across the broader metallurgical field.