Development of a Neuroevolution Machine Learning Potential of Al-Cu-Li Alloys

Abstract

Al-Li alloys are widely used in aerospace applications due to their high strength, high fracture toughness, and strong resistance to stress corrosion. However, the lack of interatomic potentials has hindered systematic investigations of the relationship between structures and properties. To address this issue, we apply a neural network-based neuroevolutionary machine learning potential (NEP) and use evolutionary strategies to train it for large-scale molecular dynamics (MD) simulations. The results obtained from this potential function are compared with those from Density Functional Theory (DFT) calculations, with training errors of 2.1 meV/atom for energy, 47.4 meV/Å for force, and 14.8 meV/atom for virial, demonstrating high training accuracy. Using this potential, we simulate cluster formation and the high-temperature stability of the T1 phase, with results consistent with previous experimental findings, confirming the accurate predictive capability of this potential. This approach provides a simple and efficient method for predicting atomic motion, offering a promising tool for the thermal treatment of Al-Li alloys.

1. Introduction

As new lightweight and high-strength aluminum alloys, Al-Li alloys have attracted much attention for their low density and high strength, making them widely used in aerospace applications [1,2,3,4]. However, due to the lack of interatomic potential function, its corresponding aging characteristics and related properties are seldom investigated from the molecular dynamics (MD) point of view, and physical images at the atomic level are not available. To address these issues and facilitate the prediction of microstructural changes during the aging process of Al-Li alloys, this study employs machine learning to develop a ternary potential function for Al-Li alloys and validate it by comparing simulation results with experimental microscopic characterization data.

For the Al-Cu-Li ternary system, developing a tailored interatomic potential function is a prerequisite for conducting relevant molecular dynamics simulations. Density Functional Theory (DFT) based on quantum mechanics can accurately describe the motions of atoms of different elements [5], but it is limited by computational resources and cannot be computed on a large scale. DFT calculations are also expensive and time-consuming, whereas empirical potentials based on classical Newtonian mechanics have low prediction accuracy, which fails to satisfy the current simulation accuracy requirements [6,7]. In recent years, the development of machine learning has provided new ideas for solving the aforementioned problems, and machine learning potentials (MLPs) [6,8,9,10] have been widely adopted. The accuracy of most MLPs now rivals the computational results of density-functional theory (DFT) [11,12,13], with efficiency improvements on the order of several magnitudes [10]. Nonetheless, researchers continue to seek further improvements in computational efficiency. To this end, a neuroevolutionary machine learning potential (NEP) has been developed to achieve higher computational speeds, leveraging Image Processing Units (GPUs) and resulting in at least an order of magnitude improvement in efficiency [14]. Currently, the NEP model has been applied in various scenarios, including the superconductivities and superionic behaviors of LiAl compounds under high pressure [15], the mechanical properties and thermal conductivities of two-dimensional network structures of C₆₀ molecules [16,17], and the local structure and thermophysical properties of CaCl₂-KCl molten salt [18]. Overall, the NEP can leverage the powerful learning capability of neural networks combined with the automatic search characteristics of evolutionary algorithms to train more flexible and efficient machine learning potential functions. This approach significantly reduces the cost of alloy heat treatment studies while providing a faster and more accurate method for investigating the evolution of relevant thermodynamic state variables in materials. Therefore, this section performs DFT calculations on the obtained structures based on ab initio molecular dynamics (AIMD) sampling [19] and compiles the resulting dataset to train the NEP, ultimately obtaining the potential function for Al-Cu-Li alloys. Finally, MD simulations are carried out using GPUMD (or LAMMPS) [20,21,22], and the dynamic process is visualized using OVITO [23], etc.

2. Material and Methods

The experimental material used in this study is an Al-Cu-Li alloy plate with a thickness of 10 mm, and its chemical composition is shown in

Table 1. Chemical composition of experimental Al-Cu-Li alloy (wt.%).

Composition (%)	Cu	Li	Mn	Zr	Mg	Fe	Si	Al
	2.8	1.4	0.3	0.1	0.03	0.04	0.02	Bal.

. Dog-bone-shaped specimens with a specification length of 35 mm and a diameter of 5 mm were machined along the rolling direction of the received plate.

The material preparation and testing procedures are as follows:

Prior to the experiment, the samples underwent solution treatment at a solutionizing temperature of 460 °C, with a holding time of 40 min to obtain a supersaturated solid solution. The samples are then water-quenched to room temperature. To avoid natural aging, the creep test should be conducted within half an hour after the solution treatment.

The creep aging experiments were conducted according to the high-temperature tensile testing method for metallic materials using a specialized RMT-10 creep testing machine(manufactured by Sansi Technology, located in Zhuhai, China). The tests were performed at aging temperatures of 155 °C, 210 °C, and 260 °C, with a test stress of 220 MPa and a duration of 18 h. After the creep aging tests, the samples were stored in a freezer to avoid natural aging. The creep test procedure is shown in

Table 2. Setting of steps in creep tests.

Step	Temperature (°C)	Load/N	Time/Min
1	35	200	2
2	35	$L=\sigma \times A$	5
3	155/210/260	$L=\sigma \times A$	18 × 60
4	50	$L=\sigma \times A$	2
5	50	200	2

.

To observe the relationship between dislocations and precipitates in the early stages of aging in Al-Li alloys, the S/TEM samples were first mechanically polished to a thickness of approximately 100 µm. Then, the samples were electropolished in a 30% nitric acid methanol solution at −25 °C using a Struers TenuPol-5 twin-jet electropolisher(manufactured by Struers A/S, located in Ballerup, Denmark) with an operating voltage of 15 V. Microstructural observations were conducted using a High-Resolution Transmission Electron Microscope (HRTEM; Titan G260-300; manufactured by FEI Company, located in Hillsboro, OR, USA) operating at 300 kV, equipped with a High-Angle Annular Dark-Field (HAADF; manufactured by FEI Company, located in Hillsboro, OR, USA) detector to explore precipitation.

3. Construction of the NEP for Al-Cu-Li Alloys

3.1. The Main Process of Constructing the NEP

• Sampling structures. For multicomponent alloys, obtaining atomic configurations corresponding to various compositions and atomic ratios is essential. These configurations can be acquired by performing AIMD simulations with variations in composition, temperature, and pressure, followed by equidistant sampling. Alternatively, structural perturbations can also generate additional configurations. Another important sampling method involves retrieving possible structures of the target system from databases, such as the Materials Project. This step is critical for determining whether the NEP potential function can accurately predict the behavior of the target alloy system. The more the collected structures comprehensively cover the potential scenarios of interest, the more accurate the subsequent MD simulations using the NEP function will be. The specific methodology is as follows: (i) AIMD simulations were performed using VASP to sample structures. AIMD simulations for Al-Cu-Li alloys were conducted over a temperature range of 50 K to 2000 K. The initial structures were generally 3 × 3 × 3 face-centered cubic supercells, with compositions (atomic ratios) including pure Al, Cu, and Li, as well as alloys such as Al₂₅Cu₅₀Li₂₅, Al₅₀Cu₂₅Li₂₅, Al₅₀Cu₅₀, Al₅₀Li₅₀, Al₉₅Cu₄Li₁, and Cu₅₀Li₅₀. The temperature range extended from 50 K to 2000 K, and the pressure conditions included ambient pressure and 10 GPa. (ii) Structures of elemental Al, Cu, and Li, as well as any compounds of these elements, were downloaded from the Materials Project database.
• DFT static calculation. The static DFT calculations were performed with the following parameters: KSPACING = 0.2, ENCUT = 600 eV, EDIFF = 1 × 10⁻⁶ eV, ISMEAR = 0, and SIGMA = 0.02. Subsequently, the selected training set structures were uniformly subjected to DFT calculations, and the results were compiled into a training dataset file. This step is the most computationally intensive part of the potential function process, typically accounting for approximately 70% of the total computation.
• Training and validation of the potential function. The data obtained from all the structural calculations in the previous step are organized to create a training set. This set is then used to train the potential function, which is subsequently validated by comparing its results with those from DFT calculations. The accuracy of the potential function is assessed using the root-mean-square error (RMSE) of the energy, force, and potential force.

3.2. Construction of Al-Cu-Li Ternary Alloy Dataset

The atomic configurations for different element types and compositions are defined, including the structures of pure Al, Cu, and Li alloys, as well as Al-Cu, Cu-Li, and Al-Li binary alloys and Al-Cu-Li ternary alloys under varying temperatures and pressures. For instance, within a pressure range of 0–5 GPa and a temperature range of 300–800 K, the Al-Cu, Cu-Li, and Al-Li ratios are set to 4:1, 3:2, and 1:4, respectively, covering the full range of component ratios. Additionally, face-centered cubic (FCC), body-centered cubic (BCC), and hexagonal close-packed (HCP) structures are considered based on different elemental contents. A total of 1637 structures are sampled, comprising approximately 180,000 atoms. The sampled structures are primarily sourced from the Materials Project database, the ISCD database, MD simulation iterative generation, and AIMD sampling, covering alloy compositions from unary to ternary and including atomic configurations under a range of pressure and temperature conditions.

3.3. Principle of the NEP Model

The potential energy surface function $U_i$ for atom $i$ in the NEP model is derived from modeling the descriptor vector $q_{\nu }^i$, defined as [24]

$$U_i={\sum }_{\mu =1}^{N_{\mathrm{n}\mathrm{e}\mathrm{u}}}{w}_{\mu }^{\left(1\right)}\mathit{tanh}\left({\sum }_{\nu =1}^{N_{\mathrm{d}\mathrm{e}\mathrm{s}}}{w}_{\mu \nu }^{\left(0\right)}{q}_{\nu }^i-b_{\mu }^{\left(0\right)}\right)-b^{\left(1\right)}$$

(1)

where $\mathit{tanh}\left(x\right)$ is the activation function of the hidden layer; $N_{\mathrm{d}\mathrm{e}\mathrm{s}}$ is the number of components of the descriptor vector; $N_{\mathrm{n}\mathrm{e}\mathrm{u}}$ is the number of neurons; $w^{\left(0\right)}$ is the matrix of connection weights from the input layer to the hidden layer; $w^{\left(1\right)}$ is the vector of connection weights from the hidden layer to the output layer node $U_i$; $b^{\left(0\right)}$ is the bias vector of the hidden layer, and $b^{\left(1\right)}$ is the bias of the output layer node $U_i$.

In the training of the NEP model, the loss function is optimized with a separable natural evolution strategy (SNES), and the total loss function L is defined as the sum of the weights of multiple loss functions [25],

$$L={\lambda }_1{L}_1+{\lambda }_2{L}_2+{\lambda }_e\Delta U+{\lambda }_f\Delta F+{\lambda }_v\Delta W$$

(2)

Here $\Delta U$, $\Delta F$, and $\Delta W$ are root mean square errors (RMSEs) between predictions and reference values of energy, force, and virial, respectively. $L_1$ and $L_2$ are proportional to the 1 norm and 2 norm of the parameters in the model, respectively, and ${\lambda }_1$, ${\lambda }_2$, ${\lambda }_e$, ${\lambda }_f$, and ${\lambda }_v$ are weighting terms.

4. Validation of NEP for Al-Cu-Li Alloys

The evolution of various loss functions during the training of the Al-Cu-Li alloy NEP function, as a function of iteration number, is shown in Figure 1a. These include the total loss of the training dataset, L1 and L2 regularization, as well as the energy, force, and virial loss functions, all of which converge after approximately 800,000 iterations. The MD simulations used for validation were performed under zero pressure, periodic boundary conditions, and an NPT ensemble with a time step of 1 fs. The system contained between 5000 and 200,000 atoms. In Figure 1b–d, the converged RMSE values for energy, force, and virial in the training dataset are 2.1 meV/atom, 47.7 meV/Å, and 14.8 meV/atom, respectively. These results indicate that the trained NEP potential function is highly consistent with DFT calculations, demonstrating excellent predictive accuracy. Furthermore, the energy, force, and virial distributions in the training dataset span a broad range, ensuring the stability of NEP-based MD simulations across a variety of scenarios.

5. MD Simulation Results and Experimental Validation of Al-Cu-Li Alloys Based on NEP Function

The accuracy of the NEP function was verified in the previous subsection, and MD simulations of the aging process using this NEP function are conducted in this section. Due to limitations in computational resources and time constraints, the molecular dynamics simulation operates at a nanosecond (ns) time scale. Although this restricts direct comparison with the macro-scale molding process, it does not hinder the study of underlying laws and mechanisms. The molecular dynamics simulation conditions and parameter settings are shown in

Table 3. Boundary condition settings for MD simulations.

Boundary Condition	Parameters
periodic boundary condition (X, Y, Z)	P, P, P (P: periodic boundary condition)
simulated pressure	220 MPa
simulated temperature	90, 155, 210, 260 °C

. The simulated tensile fracture process uses the NVT ensemble, while the remaining simulations use the NPT ensemble, with a time step of 1 fs.

5.1. Dislocation Defect-Induced Inhomogeneous Precipitation

The random solid solution model of the Al-Cu-Li alloy consists of 181,440 atoms, with approximate dimensions of 181.61 Å, 105.92 Å, and 159.78 Å along the x, y, and z axes, respectively. To simulate the effect of dislocation defects on precipitates, dislocations were introduced into the model (see Figure 2a; the left image shows the Al-Cu-Li alloy random solid solution model with an atomic ratio of Al:Cu:Li = 95:4:1, while the right image illustrates the positions of the introduced dislocations within the model). In this section, the stress state of a central, localized region of the sample is used as the basis for applying boundary conditions in the MD simulation. Under realistic processing conditions, the specimen is subjected to uniaxial tensile stress, resulting in contraction in the direction perpendicular to the applied tensile force due to Poisson’s effect. This induces tensile forces in the surrounding material, creating a triaxial tensile stress state. Consequently, tensile stresses are applied to the model in all three directions.

Figure 2b₁,b₂ present the initial distributions of Cu and Li atoms in the random solid solution model, respectively. Both figures show that Cu and Li atoms are randomly and uniformly distributed in the initial structure, with no evidence of segregation. Figure 2c₁,c₂ depict the distributions of Cu and Li elements after 5 ns of MD simulations performed on the random solid solutions. These figures clearly show Cu segregation. Due to the positive mixing enthalpy and high reactivity of Cu with other metallic elements [26], Cu atoms are more prone to segregation and diffusion, and dislocations provide diffusion pathways, making the segregation of Cu atoms more prominent. By comparing the regions with dislocations in Figure 2a and the regions with elemental enrichment in Figure 2c₁,c₂, it is evident that the regions of elemental enrichment correspond to the locations of dislocations, highlighting the role of dislocations in promoting elemental enrichment and precipitation. This observation is consistent with findings from previous studies [27,28,29,30,31]. The T1 phase, the primary precipitate in the Al-Cu-Li alloy, has a chemical composition of Al₂CuLi [32,33,34], and the enrichment of Cu and Li elements provides the necessary conditions for the formation of the T1 phase. Figure 2d₁,d₂ show TEM images before and after the formation of the T1 phase. It is evident from the images that the preferential precipitation of the T1 phase always occurs at dislocations, further demonstrating the significant role of dislocations in precipitation. This is in agreement with the phenomena observed in the MD simulations and provides additional validation for the accuracy of the NEP function.

5.2. Dissolution and Secondary Precipitation of the T1 Precipitation Phase

The atomic structure of the T1 phase is shown in Figure 3a, with structural data sourced from the Inorganic Crystal Structure Database (ICSD). It exhibits a hexagonal crystal structure, with partial occupancies of Cu and Al at specific local sites. To better reflect realistic conditions, Al atoms in these positions are substituted with Cu atoms. Figure 3b presents the structure obtained by eliminating the partial occupancies in the T1 phase. In a typical T1 unit cell, six positions are partially occupied by both Al and Cu. When one of the sites is occupied by Cu and the other by Al, this structure is referred to as Cu1. If two sites are occupied by Cu and the others by Al, it is termed T1-Cu2. If all six sites are occupied by Cu with only one Al site, the structure is called Cu6.

Based on the partial occupancy of the T1 structure, it is evident that at least half of the Al atoms will be replaced by Cu atoms. Therefore, the Cu4, Cu5, and Cu6 structures (as shown in Figure 3c₁–c₃, respectively) are used as the unit cell structures for a 10 × 10 × 6 cell expansion in this section. Taking the Cu6 unit cell structure as an example, the expansion details are illustrated in Figure 4, with Figure 4b showing the Cu6 supercell structure after the expansion. Finally, MD simulations of dissolution and precipitation during multistage aging were performed using the supercell T1 phase structures of Cu4, Cu5, and Cu6.

The boundary conditions for the MD simulation are set based on the temperature settings for multistage aging described in this chapter. The temperature is divided into several stages: 428 K, 428–483 K, 483 K, 483–513 K, 513 K, 513–533 K, 533 K, 533–513 K, 513 K, 513–483 K, 483 K, and 483–428 K. These stages encompass all processes, including 155 °C (428 K), 210 °C (483 K), 240 °C (513 K), 260 °C (533 K), and the entire heating and cooling cycle. The simulation fully models the complete aging process from 155 °C to 210/240/260 °C and back to 155 °C. The applied stress is 220 MPa, and the simulation time for each process is 1 ns.

Figure 5 shows the trend of the T1 phase in the Cu6 supercell structure as a function of temperature, with the horizontal axis representing simulation time and the vertical axis representing the number of atoms in the T1 structure. The decrease in the number of atoms in the T1 structure indicates that, due to enhanced thermal motion, more atoms have broken free from the constraints of the local atomic environment in the original hexagonal structure of the T1 phase. These atoms have migrated away from their original lattice sites, transitioning into an amorphous atomic state. This suggests that the T1 phase is gradually dissolving and becoming unstable. From the figure, it can be seen that there is no rapid re-dissolution of clusters at 210 °C; the number of atoms stabilizes at this temperature, indicating that the T1 phase neither continues to precipitate nor begins to dissolve at 210 °C. At 240 °C and 260 °C, dissolution becomes more pronounced, which is consistent with the known dissolution temperature of the T1 phase at 210 °C. [35]. Furthermore, the re-dissolution of the T1 phase inevitably leads to a reduction in strength, which is also in agreement with previous research findings [36,37,38,39].

6. Conclusions

In this work, a NEP for the Al-Cu-Li alloys has been developed based on the full permutation combination of the three elements—Al, Cu, and Li—and a training set of atomic configurations comprising various compositions. The results obtained from this potential function are compared with DFT calculations, showing training errors for energy, force, and potential force of 2.1 meV/atom, 47.4 meV/Å, and 14.8 meV/atom, respectively, indicating high training accuracy. Simulations of cluster formation and the high-temperature stability of the T1 phase, performed using this potential function, are in good agreement with experimental results. These findings demonstrate the high predictive accuracy of the potential function and provide a simple, efficient tool for predicting atomic motion in the heat treatment of aluminum–lithium alloys.

Based on the above results, NEP plays a crucial role in the potential application of alternative alloy systems. First, NEP offers a more efficient and accurate simulation method, particularly in high-temperature and high-pressure environments, serving as an effective tool for the design and optimization of high-performance alloy materials. For example, in predicting the thermomechanical properties of high-temperature and lightweight alloys, NEP can rapidly simulate phase transitions, thermal expansion coefficients, melting points, and other key properties under various conditions, significantly enhancing simulation efficiency and accuracy. Furthermore, NEP shows considerable potential in exploring new lightweight alloys by adjusting elemental ratios and simulating the interactions between different elements, which aids in predicting mechanical properties, fatigue strength, corrosion resistance, and other important characteristics.

In the future, to further enhance the application of NEP modeling in alloy systems, several potential improvement directions can be considered:

(1)

Enhance model generalizability and accuracy. By incorporating data with varied chemical compositions and structural morphologies, the model can be adapted to a wider range of alternative alloy systems. Additionally, exploring more advanced neural network architectures, such as graph convolutional networks (GCNs), could help capture complex inter-atomic interactions and further improve prediction accuracy.

(2)

Improve training efficiency and data quality. Evolutionary algorithms, such as genetic algorithms and Bayesian optimization, can be employed to intelligently select training data, thus improving model training efficiency. Furthermore, data augmentation techniques (e.g., perturbing atomic positions or simulating structures under different environments) can be introduced during the training process to enhance data diversity, thereby improving the model’s generalization ability.

(3)

Enable multi-scale simulation. Combining NEP with coarse-grained models or finite element methods can facilitate multi-scale coupling calculations, providing a more comprehensive simulation of alloy behavior in practical applications.