1 Introduction

Strengthening precipitates plays a crucial role in enhancing the strength of heat-treatment-strengthened aluminum alloys [1-3]. In practical applications, θʹ precipitate is the most common strengthening phase in Al-Cu alloys [4, 5]. Under non-stress aging treatment, the plate-like θʹ precipitates are distributed on three {100}_Al habit planes homogeneously [6]. However, θʹ precipitates prefer to grow on certain habit planes when an external stress is applied to the material during the aging treatment, such as creep age forming (CAF) [7, 8]. Aluminum alloys prepared using CAF are widely used in aerospace and transportation for weight reduction [9-11]. During the CAF process, the orientation effect of θʹ precipitates could lead to anisotropy in mechanical properties. Therefore, it is of importance to conduct in-depth research on the mechanism of the orientation effect of θʹ precipitates.

HOSFORD et al [12] first investigated the orientation effect of the θʹ precipitates in stress-aged Al-4wt.%Cu single crystal and their results indicate that the applied compressive stress promotes the formation of θʹ precipitates on the habit planes parallel to the loading direction, while the opposite result was observed when tensile stresses were applied. However, subsequent studies presented contradictory conclusions to HOSFORD’s results. ETO et al [13] found that applied compressive stress causes θʹ precipitates to grow preferentially on the habit planes perpendicular to the loading direction, while applied tensile stress favors the formation of θʹ precipitates that grow on the habit planes parallel to the loading direction. Moreover, they considered that orientation effect of θʹ precipitates is attributed to the influence of stress in the nucleation stage. SKROTZI et al [14] proposed the existence of threshold values of the distribution of θʹ precipitates with preferred orientation in stress aged Al-5wt.%Cu alloys, that is, the applied stress must exceed a certain threshold value to produce the orientation effect of θʹ precipitates. ZHU et al [15] investigated the influence of various factors on the orientation effect including stress, temperature and alloy composition. Their results supported Eto's findings. To explain the controversy about the orientation effect, CHEN et al [16] established a three-dimensional precipitation distribution of θʹ precipitates during stress aging, and they highlighted that the observation of two-dimensional images is not sufficient to determine the model of stress-orientating effect on θʹ precipitates. Additionally, the results obtained from the analysis of the three-dimensional model are consistent with Hosford’s work. SANKARAN [17] pointed out that the orientation effect is related to the dislocations induced by the applied stress. Furthermore, we showed in a previous work [18] that 1/2<110> perfect dislocations have an effect on the distribution of θʹ precipitates in Al-4wt.%Cu single crystal.

The TEM images can reveal the distribution of θʹ precipitates and provide evidence of the orientation effect phenomenon. However, the origin of the orientation effect cannot be determined. It is unclear whether the orientation effect is originated from the stress application process or by diffusion during post-loading aging treatment. In addition, a single TEM specimen cannot be observed from three different <100> axes simultaneously; in fact, only three different <100> axes can be observed separately from specimens at different positions. In order to improve the efficiency of microstructure research and investigate the accuracy of existing theories on the orientation effect, it is necessary to conduct molecular dynamics simulations.

Although CHEN et al [16] used a hypothetical 3D model to explain the previous conflicting conclusions, they lacked intuitive 3D images to prove their inference. Molecular dynamics (MD) simulations can effectively address this problem. After processing using visualization software, the results of MD simulations provide a comprehensive display of the 3D information of the model, which can provide direct and strong evidence for our conclusions. Therefore, the MD simulation is a powerful method for studying the orientation effect of θʹ precipitates. However, classic MD simulations that require solving Newton’s equations for each atom are computationally inefficient, requiring significant time to reach phase equilibrium and form precipitates [19]. Meanwhile, Monte Carlo (MC) simulations based on the metropolis algorithm can extend the timescale of simulation, but it ignores some critical kinetic information during precipitation, which leads to an inability to adequately capture the properties of the precipitates [20]. Combining MD and MC simulations is an effective method to solve the above problems and this approach is called the hybrid MC/MD simulation. In this paper, the variance-constrained semi-grand canonical ensemble Monte Carlo (VC-SGC-MC) method developed by SADIGH et al [21] is adopted to simulate the precipitation and its distribution in Al-Cu alloys. The VC-SGC-MC algorithm can be used to reach phase equilibrium in multiphase systems and can easily perform parallel calculations which improves the computational efficiency [22]. Therefore, the hybrid MC/MD algorithm combining VC-SGC-MC algorithm and MD simulations is ideal for simulating precipitation in large systems containing several million atoms.

ZHAO et al [23] conducted coupled MC/MD simulations using the VC-SGC approach and successfully simulated the precipitation of Al₃Mg and Al₅Mg in Al-Zn-Mg-Cu alloys. ERHART et al [24] simulated the precipitation of Cu in Fe-Cu system using hybrid VC-SGC-MC/MD algorithm. Their results were in very good agreement with experimental data. TURLO et al [25] performed hybrid VC-SGC-MC/MD simulations in body-centered cubic Fe-Ni alloys to simulate the formation of the metastable phase B2-FeNi and discovered the linear complexions mechanism. All of these studies above have implemented phase transition simulations using hybrid VC-SGC-MC/MD algorithms. In addition, the hybrid VC-SGC-MC/MD method has been widely used to explore the micro mechanism of materials, such as the short-range order structure in high-entropy alloys [26-28], precipitation of multiphase system [29, 30], dislocation mechanism [31, 32] and grain boundary behavior [33].

This paper aims to conduct a more in-depth and rigorous study of the stress-orientating effect in Al-Cu alloys by combining experiments and simulations. The distribution of θʹ precipitates in stress aged Al-4wt.%Cu samples was observed from three <100>_Al axes to determine the orientation effect. Corresponding simulations were carried out and results consistent with the experiments were obtained. Based on this consistency, the three-dimensional visualization characteristics of the simulation results were used to analyze the orientation effect and explain the experimental observation results. Furthermore, the interaction between dislocations and precipitates as well as the influence of dislocations on orientation effect was explored at the atomic level. This study aims to provide a better understanding of the orientation effect of precipitates based on the dislocation mechanism.

2 Experiments and simulation

2.1 Material preparation and characterization

Al-4wt.%Cu single crystal specimens were prepared by the cyclic strain annealing method [34]. The detailed preparation process for single-crystal samples was as described in our previous paper [18]. After 8-10 iterations of strain-annealing treatments, the number of dislocations inside the single-crystal specimens reaches a small value. Single crystal with 10 mm in length, 5 mm in width, and 5 mm in thickness were used for the experiments. The crystalline orientation of this single crystal was determined by electron backscattered diffraction in a FEI Nova 230 nano-lab scanning electron microscope. Before aging, the single crystal sample was solution-treated at 525 ℃ for 2 h and quenched in water. The single crystal sample was then subjected to stress-aging treatment at 180 ℃ for 66 h under a compressive stress of 40 MPa. The loading direction of the applied stress was parallel to the long axis of the sample, which was the [3, 2, 25] crystal orientation of the single crystal. To observe the morphology and distribution of precipitates in the stress-aged single crystals, several samples were observed by scanning transmission electron microscopy (STEM) using a Titan G2 60-300 with an image corrector operated at 300 kV. The STEM samples were prepared by twin-jet polishing at 20 V in a solution of HNO₃ and methanol (1:3 in volume) cooled to -25 ℃ using liquid nitrogen, which were collected from three faces of a rectangular cuboid single crystal sample, that is, [001]_Al axis (the [3, 2, 25] direction is approximated as the [001]_Al axis and the same treatment for the other two axes), [100]_Al axis, [010]_Al axis. Figure 1 shows the loading direction, crystal orientation, and STEM observation directions.

Figure 1全屏查看

Schematic of (a) stress aging treatment and STEM observation for Al-4wt.%Cu single crystal specimen; (b) Stress-aging treatment Al-1.8at.%Cu model for molecular dynamics simulation.

2.2 Modeling and hybrid MC/MD simulations

Atomic simulations in this study were performed using a large-scale atomic/molecular massive parallel simulator [35]. The interatomic interaction was described by the well-developed embedded-atom method potential for Al-Cu alloys and can be expressed as [36, 37]

(1)

where Ei is the total energy of atom i; α and β are the element types of the center atom i and neighbor atom j; ρβ is the charge density function of β; Fα is the embedding function of α; Φαβ is the pair potential function between α and β.

First, a single-crystal model of Al was created with dimensions of 162 Å×162 Å×324 Å, containing 512000 Al atoms. The crystal structure of the Al single-crystal model was face centered cubic with a lattice constant of 4.04 Å. The x, y, and z axes were oriented along the [100], [010] and [001] directions, respectively. Subsequently, 1.8% of the Al atoms were randomly replaced with Cu atoms, which means the Al-1.8at.%Cu alloy model is constructed (it is equal to Al-4wt.%Cu alloy), as shown in Figure 1. Before applying compressive stress to the model, energy minimization was performed using the conjugate gradient algorithm, and relaxation was performed at 453 K under an isothermal-isobaric (NPT) ensemble. The uniaxial compressive loading of the model was oriented in the [001] direction at a constant strain rate of, 2×10¹⁰ s^-1, with the lateral boundaries controlled using the NPT equations of motion to zero pressure, and the temperature fixed at 453 K. Figure 2(a) shows the model before the hybrid MC/MD simulations, and Figure 2(b) displays the distribution of dislocations within the model.

Figure 2全屏查看

Atomic snapshots: (a) A model before hybrid MC/MD simulations, colored according to PTM approach; (b) Distribution of dislocations in the model obtained by DXA method

To simulate precipitation during the aging process, some models under different stresses during the deformation process were selected as data files for the hybrid MC/MD simulations conducted in a VC-SGC ensemble. These simulation groups were termed M0, M1, M2, M3 and M4, respectively. In M0-M4, the larger the numerical part of the code, the higher the compressive stress exerted on the model of this group. The stress of M0-M4 is 0, 1.12, 1.22, 1.31, 1.41 GPa, respectively.

The temperature of the model was set to 453 K, and the momentum was conserved. The MC swap was then started, with 100 MD steps between each MC cycle. The chemical potential difference between Al and Cu was defined as -0.663 eV, which causes the concentration of Cu atoms to converge to 1.8% during the iteration and significantly accelerates the calculation process. MD was conducted in an NPT ensemble at a fixed temperature of 453 K. After 1×10⁷ MC/MD steps, the model was cooled to 273 K to acquire the microstructure of stress-aging Al-1.8at.%Cu at room temperature.

A set of comparative simulations were conducted to ensure the rigor of the study. During these simulations, all parameters were the same as those in the above process, except that the z- and x-axes were symmetrically exchanged (see Figure S5 and S6). These simulation groups were termed M5 and M6. The stress of M5-M6 is 1.31 and 1.42 GPa, respectively.

The open-source tool Atomsk was used to construct atomic models [38]. Visualization and post-processing of the simulation results were performed using the open-source software OVITO [39]. The polyhedral template matching (PTM) approach guarantees higher reliability than that of other post-processing methods such as common neighbor analysis in the presence of strong thermal fluctuations and strains [40]. Therefore, the PTM approach was adopted to identify local crystalline structures. All dislocation line defects in the crystal were visualized and analyzed using a dislocation extraction algorithm (DXA) [41].

3 Results

3.1 Orientation effect observed under STEM

Based on previous studies [7, 14, 16, 18], we know that θʹ/θʹʹ precipitation in Al-Cu alloys is disc-shaped and distributed on the three {100}_Al planes, and thus there are three variants of θʹ precipitates. When viewed along a certain axis, two variants of θʹ precipitates distributed on the planes parallel to the observation axis can be observed, while another variant is barely visible. According to Ref. [16], STEM images observed from three <100>_Al axes are required to determine the orientation effect of θʹ precipitates as the 3D distribution information of θʹ precipitates cannot be directly obtained from the 2D images. Figure 3 shows the high-resolution TEM (HRTEM) images and corresponding selected area electron diffraction (SAED) patterns of Al-4wt.%Cu single crystal specimens. These STEM images were taken from the [100]_Al, [010]_Al and [001]_Al axes, respectively. By analyzing the SAED patterns, it was easily determined that the precipitation in the Al-4wt.%Cu single crystal is θʹ-phase.

Figure 3全屏查看

HRTEM images and SAED patterns of Al-4wt.%Cu single crystal aged under 40 MPa compressive stress along the [3, 2, 25] direction: (a) HRTEM image taken from [100] axis; (b) The enlarged view of the square area in (a) and corresponding SAED pattern; HRTEM images and SAED patterns taken from (c) [001] axis and (d) [010] axis, respectively

Figures 3(b) and (c) were obtained from the [100] and [010] axes respectively, where the observation planes were parallel to the loading direction. From these two perspectives, two variants of θʹ precipitates that are perpendicular to each other can be observed. In these two STEM images, it is obvious that the number of one variant is significantly higher than the other, and this situation is consistent with previous work [18], which is considered the typical basis for determining the stress-orienting effect. In Figures 3(b) and (c), there are more θʹ precipitates parallel to the [001] axis than those parallel to the [010] and [100] axes. Specifically, in Figure 3(b), the number density of variants parallel to the [001] axis is 345 μm^-2, while the number density of variants parallel to the [010] axis is 62 μm^-2; in Figure 3(c), the number density of variants parallel to the [001] axis is 384 μm^-2, while the number density of variants parallel to the [100] axis is 98 μm^-2.

Figure 3(d) was obtained from the [001] axes, where the observation planes were perpendicular to the loading direction. The distribution of the θʹ-phase here is different from that in Figures 3(b) and (c). From this perspective, two typical “edge-on” variants of θʹ-plates could be observed clearly. According to statistics, the number density of variants parallel to the [100] axis is 422 μm^-2 and the number density of variants parallel to the [010] axis is 441 μm^-2, which are very close. This result is consistent with those of previous work [13], however, the orientation effect cannot be reflected based on this image alone. Therefore, it is necessary to combine images captured from the three different axes to obtain an accurate orientation effect analysis.

According to the three-dimensional reconstruction model established by CHEN et al [16], if the θʹ-phase preferentially precipitates on the (001)_Al habit planes, less precipitation will be observed from the [001] axis, and if the θʹ-phase preferentially precipitates on the (100)_Al and (010)_Al habit planes, then more precipitates will be observed from the [001] axis. Obviously, our results match the latter. Based on the above analysis, it can be determined that the there is more θʹ precipitate on (100)_Al and (010)_Al habit planes in Al-4wt.%Cu single crystal aged under 40 MPa compressive stress along the [3,2,25] direction. Specifically, θʹ precipitates grow preferentially on the {100}_Al habit planes parallel to the direction of the applied compressive stress.

3.2 Atomic simulation results

Hybrid MC/MD simulation results of Al-1.8at.%Cu alloy model under a compressive stress are shown in Figure 4. According to the HRTEM images, the longest dimension of θʹ precipitate is usually 10-100 nm. The size of the simulation system reached almost 20 nm in all three directions, which is in the same order of magnitude as the experimental nano-precipitated phase. The simulation results are comparable to the experiments in terms of size.

Figure 4全屏查看

Atomic snapshots, microstructure evolution of M3 simulation group for perspective images of models at (a) 0 ns, (b) 1 ns and (c) 10 ns (PTM approach is used to analyze and color different structures)

Here, only the simulation results of the M3 group are shown; the following analysis is based on the simulation results of M3. The simulation results of the other groups are shown in Figures S1-S4. The OVITO software was used to visualize the simulation data, and the PTM approach was selected to identify the crystal structure in M3 because of its reliability for the simulation of heat treatment and strain. By using this method, the phase transition, interface and deformation generated by MD simulations can be analyzed. However, the PTM approach and other post-processing methods for identifying structures configured in the OVITO software currently have limitations. Some researchers have pointed out this problem and proposed solutions [42, 43]. Conventional post-processing methods allow the identification of the local crystalline structure of simple condensed phases but the complex phase structure cannot be identified, such as the tetragonal structure which is the crystal structure of precipitates in Al-Cu alloy. Thus, this study uses the radial distribution function (RDF) to assist in determining the precipitate structure of Al-1.8at.%Cu model in hybrid MC/MD simulations. Many researchers have used RDF curves to determine the structure of molecular dynamics simulations [44, 45].

Figure 4 shows perspective view of the model for different simulation steps. In Figure 4, the matrix atoms are colored green, precipitate atoms are colored red, and atoms of other structures are shown in white. The starting configuration for hybrid MC/MD simulation is shown in Figure 4(a). This initial configuration is comprised primarily of “other” structure, suggesting that the material is in a highly defected configuration. The atomic configuration consisting mainly of amorphous structure in molecular dynamics simulations corresponds to the supersaturated solid solution state of the material in the experiment [46, 47].

Figure 5 shows the curve of potential energy and the element percentage as a function of the simulation time in M3 simulation group. The total energy of the system decreases continuously as the simulation progresses, eventually reaching a stable value. This suggests that the model has ultimately reached a thermodynamic equilibrium state. The atomic concentration of the system remained almost constant throughout the entire simulation process. This ensures the consistency of the model and experimental results in terms of composition.

Figure 5全屏查看

Potential energy variation curve and the element percentage variation curve of M3

Figure 6(a) shows the RDF curves of the precipitates obtained from M3 simulation group. Figure 6(b) shows the RDF peak maps of precipitates in simulation and three types of precipitates in Al-Cu alloys with standard structure, i.e., θʹʹ-phase, θʹ-phase and θ-phase. The crystallographic information files for the three types of precipitates mentioned above were download from the Materials Project Database [48]. By comparing the four peak maps in Figure 6(b), it can be found that the peaks of the θʹ-phase have the best matching degree with the peaks of precipitates generated in the simulation. Through statistical measurements, the lattice parameter a of the precipitates in the simulation is 4.025 Å, which is between 4.075 Å for θʹ-phase and 3.925 Å for θ″-phase. Obviously, this lattice parameter is closer to that of θʹ-phase. Therefore, the structure of precipitates in the simulation can be considered between the precipitation sequences of θʹʹ-phase and θʹ-phase, and closer to the structure of θʹ-phase.

Figure 6全屏查看

(a) Radial distribution function diagram of three pair of atoms of precipitates in M3 simulation group; (b) Peak maps of precipitates in simulation and three precipitates in Al-Cu alloys with standard structure (The color scale in (b) is the same as (a))

Furthermore, Figure 4 shows that the precipitates were disc-shaped and grew on three {100}_Al habit planes (see Figures S1-S6 for details), which is in perfect agreement with the morphology and distribution of the θʹ-phase observed in the experiments. At this point, it can be concluded that the precipitates in the simulation results are indeed the θʹ-phase, which is the most common enhanced precipitate in the aged Al-Cu alloys. It must be noted that the peaks of the precipitates in the simulation do not exactly coincide with that of the θʹ-phase with standard structure, as the precipitates and matrix form a semi-coherent interface, resulting in lattice distortion.

Figure 7 shows the simulation results observed from the three <100> axes for the different simulation steps. This observation enables the simulation results to correspond better with the HRTEM images in Figure 3, and the evolution process of the structure can also be seen. In Figure 7, each column contains three views of the same simulation step. By comparing the images of different columns, it can be found that the proportion of red atoms in the images increases as the simulation progresses, indicating an increase in the number of θʹ precipitates. Several precipitates can be seen in Figures 6(b), (e) and (h), and distribution of the θʹ precipitates in Figures 6(c), (f) and (i) is approximately consistent. Therefore, it can be assumed that the system containing precipitates reaches a stable state when the number of simulation steps reaches 10⁷ (i.e., t=10 ns). In Figures 6(c) and (f), almost all precipitates are parallel to the [001] axis, while no precipitates are parallel to the [100] and [010] axes. As shown in Figure 7(i), several precipitates are perpendicular to each other and, parallel to the [100] and [010] axes. In Figure 7(c), the number density of variants parallel to the [001] axis is 27500 μm^-2, while the number density of variants parallel to the [010] axis is 2500 μm^-2; in Figure 7(f), the number density of variants parallel to the [001] axis is 32500 μm^-2, while the number density of variants parallel to the [100] axis is 2500 μm^-2; in Figure 7(i), the number density of variants parallel to the [100] axis is 42500 μm^-2 and the number density of variants parallel to the [010] axis is 30000 μm^-2. This situation is similar to that shown in Figure 3, which indicates that the simulation and experimental results are in good agreement. According to the theory proposed by CHEN et al [16], the distribution of precipitates in Figures 6(c), (f) and (i) indicates that almost all θʹ precipitates grow on the (100) and (010) habit planes. Through the three-dimensional images in Figure 4 and Figures S1-S4, it can be seen that the majority of the precipitates are distributed in the (100) and (010) planes, which proves the correctness of CHEN et al’s theory and indicates that external compressive stress can cause precipitates to prefer to grow on planes parallel to the loading direction.

Figure 7全屏查看

Atomic snapshots, structure evolution of M3 simulation group: (a, b, c) Observed from [100] axes; (d, e, f) Observed from [010] axes; (g, h, i) Observed from [100] axes

4 Discussion

Previous studies have proposed that dislocations have an effect on the distribution of θʹ precipitates [17, 18]. This study uses the results of simulations to explore how dislocations interact with precipitates. Figure 8 shows the curve of the volume fraction of θʹ precipitates and density of all types of dislocations as a function of the simulation time, with data obtained from the M3 simulation group. The volume fraction of precipitates and density of dislocations increased sharply within the simulation time of 0-1 ns, after which the volume fraction of precipitates remained stable, while the values of the density of dislocations fluctuated within a certain range. According to Figure 8, it is not possible to obtain a definite relationship between the precipitates and number of dislocations, but it can be speculated that there exists a pair of mutually constraining dislocation mechanisms that cause fluctuations in the number of dislocations. In order to better investigate the mechanism of interaction between dislocations and precipitates, the research scope must be narrowed. The close relationship between the 1/2<110> perfect dislocations and orientation effect of θʹ precipitates has been noted in several papers [17, 18]. GUO et al [18] found that the 1/2<110> dislocations will provide the preferential nucleation sites for the θʹʹ phase and θʹ phase. The θʹ phase reacts with 1/2<110> dislocations during nucleation and growth. A theory of the interaction between the θʹ precipitate and 1/2<110> dislocations has been proposed. The HRTEM images presented in this work demonstrate that 1/2<110> dislocations invariably accompany θʹ precipitates, providing compelling evidence for the interaction between the θʹ precipitates and 1/2<110> dislocations. Therefore, it is necessary to focus on 1/2<110> perfect dislocations, especially the relationship of various precipitation growth and dislocation evolution, which would be discussed later.

Figure 8全屏查看

The volume fraction of θʹ precipitates and density of dislocation as a function of simulation time in M3 simulation group

Figure 9(a) shows the HRTEM image of the θʹ precipitate. Performing inverse fast Fourier transform (IFFT) on the matrix region near the θʹ precipitate in Figure 9(a) yields the result shown in Figure 9(b). In Figure 9(b), the lattice fringes near the dislocations are marked with yellow lines, and the Burgers circuit of the dislocations is drawn with white lines to determine the Burgers vector.

Figure 9全屏查看

(a) HRTEM image of θʹ precipitate and its adjacent area in Al-4wt.%Cu single crystal; (b) IFFT of the red border region in (a); (c) Enlarged image of the 1/2<110> dislocations in (b) (The yellow line represents the crystal plane, while the white border represents the Burgers circuit of the dislocation)

The two white borders in Figure 9(c) represent the Burgers circuits of the two dislocations, respectively. It can be observed that both of the Burgers circuits contain segments that exceed the closed path, indicating that they are not closed loops. In Figure 9(c), the long red arrow represents the direction of the Burgers vector, while the short red arrow represents the magnitude and direction of the Burgers vector. It can be determined that the two Burgers vectors are 1/2[110] and , respectively. The experimental observations presented in this paper indicate a correlation between θʹ precipitates and 1/2<110> perfect dislocations.

In order to explore the specific mechanism of the interaction between dislocations and precipitates, the results of the M0–M6 simulation group were analyzed and listed in Table 1. In Table 1, N is the number of 1/2<110> dislocations; ΔN represents the difference in the number of dislocations before and after simulation; and ΔV represents the difference in the volume fraction of θʹ precipitates before and after simulation. The volume fraction of θʹ precipitates is calculated and counted using the PTM algorithm. For quantitative characterization of the degree of the orientation effect, the orientation factor is defined and its expression is as

(2)

where is the volume fraction of all three θʹ precipitate variants; is the volume fraction of θʹ precipitates parallel to the loading direction; and is the volume fraction of θʹ precipitates perpendicular to the loading direction. When the number of two variants is equal, i.e., , then , which means that here is no orientation effect. When there is a difference in the number of two variants, i.e., , then , and the smaller the , the higher the degree of the orientation effect. It should be noted that has a minimum value of 0. The value of in Table 1 is between 0 and 1, which represents a complete orientation effect and the non-orientation effect, respectively. The of M0 is 1, which means that there is no orientation effect, and the number of dislocations in M0 is consistently 0. This indicates that if there are no 1/2<110> dislocations, the θʹ precipitates do not exhibit the orientation effect. On the other hand, of M4 reaches the minimum value of 0, which means there is a complete orientation effect. At this time, the value of ΔN in M4 is the minimum value among all simulation groups. This may indicate a positive correlation between the degree of the orientation effect and reduction of the 1/2<110> dislocations. However, there are also results that contradict this speculation, such as those for M2 and M5. The value of ΔN in M2 is higher than that in M5, and according to previous assumptions, the degree of the orientation effect in M2 should be higher than that in M5, i.e., of M2 is smaller than that of M5. However, the statistical results in Table 1 contradict this inference. In addition, another value ΔV is worth noting. M2 has the maximum ΔN-value, but its ΔV-value is also the highest and M5 has the minimum ΔV-value. Therefore, it can be inferred that an increase in the volume fraction of θʹ precipitates may offset the effect of the 1/2<110> dislocations on the orientation effect.

Figures 10 and 11 show two cases of interaction between the θʹ precipitates and 1/2<110> dislocations. From Figures 10(a₁), (a₂) and (a₃), it can be seen that the θʹ precipitate plate grows on the (010)_Al crystal plane, and as the simulation time increases, the precipitate plate grows along the [010] axis, i.e., the thickness expands. Apparently, this growth pattern does not increase the degree of the orientation effect. During the process of θʹ precipitate plate growing thicker, a 1/2<110> dislocation is generated in the vicinity of the θʹ precipitate, and this interaction between the dislocation and precipitate can be clearly observed from the Figures 10(b₁) and (b₂). Two 1/6<112> Shockley dislocations were produced by the matrix near the θʹ precipitate.

Figure 10全屏查看

Snapshots showing (a₁-a₃) Perspective view of the θʹ precipitate that grow thick with increasing simulation time; (b₁, b₂) θʹ precipitate before and after simulation, observed from the [001] axes; (c₁, c₂) View from the [010] axes of the θʹ precipitate in (b₁, b₂)

Figure 11全屏查看

Snapshots showing (a₁-a₃) Perspective view of the θʹ precipitate that grow radially with increasing simulation time; (b₁, b₂) θʹ precipitate before and after simulation, observed from the [001] axes; (c₁, c₂) Enlarged view of dislocations in (b₁) and (b₂) observed from the [100] axes, the yellow arrows on the dislocations are their Burgers vectors

In Figures 11(a₁), (a₂) and (a₃), it can be seen that the θʹ precipitate plate grows on the (100)_Al crystal plane, and as the simulation time increases, the precipitate plate grows along the [010] axis, i.e., the length increases. In contrast to the result in Figure 10, this growth pattern will lead to an enhanced degree of the orientation effect. During the process of θʹ precipitate plate growing longer, the 1/2<110> perfect dislocation near the θʹ precipitate decomposes into two 1/6<112> Shockley dislocations, and this reaction of dislocations can be clearly observed in Figures 11(b₁) and (b₂). Figures 11(c₁) and (c₂) are the observation of dislocations, where the Burgers vector of each dislocation is identified by yellow arrows. The 1/2<110> perfect dislocation dissociates to produce two 1/6 Shockley dislocations, according to the reaction in Eq. (3):

(3)

Through the above analysis, two mechanisms of interaction between the 1/2<110> dislocations and θʹ precipitates have been revealed. These two dislocation mechanisms can be summarized as follows:

Growth in the thickness direction of θʹ precipitates induces the generation of 1/2<110> perfect dislocations, while the reaction of 1/2<110> perfect dislocations decomposing into 1/6<112> Shockley dislocations leads to the growth of θʹ precipitates along the length direction.

Based on these two mechanisms, the fluctuations in the density of dislocations in Figure 8 as well as the relationship between and the values of ΔN (Table 1) can be explained. Specifically, is negatively correlated with the values of ΔN, while ΔN is positively correlated with the values of ΔV. When ΔV is large, it means that the θʹ precipitates have a large growth and generate several 1/2<110> perfect dislocations. Thus, even if the θʹ precipitates have a strong orientation effect and consume a lot of 1/2<110> perfect dislocations, the value of ΔN may not be very small. Hence, when considering the impact of ΔN values on , it is necessary to also consider the ΔV values simultaneously.

The quantitative relationship between dislocations and the size of θʹ precipitates is shown in Figure 12. In Figure 12(a), the θʹ precipitate selected primarily growing along the thickness direction and the value of thickness is close to the diameter. A large number of 1/2<110> perfect dislocations are present in the early stages of the simulation, indicating that increasing thickness leads to the generation of 1/2<110> perfect dislocations. As the simulation progresses, the number of 1/2<110> perfect dislocations decreases, while the number of 1/6<112> Shockley dislocations gradually increases. This is due to the fact that the θʹ precipitate grows along the thickness as well as the length, causing 1/2<110> perfect dislocations to decompose into 1/6<112> Shockley dislocations. In Figure 12(b), the θʹ precipitate selected primarily growing along the length direction and the thickness value is significantly smaller than the diameter. Throughout the simulation, the number of 1/6<112> Shockley dislocations is high while the number of 1/2<110> perfect dislocations stays at a very low level. The phenomenon occurs because the precipitates mainly grow in the length direction, leading to significant decomposition of 1/2<110> perfect dislocations into 1/6<112> Shockley dislocations.

Figure 12全屏查看

The size the of θʹ precipitate and number of dislocations as a function of simulation time. Two precipitates with different growth patterns are selected: (a) The θʹ precipitate primarily growing along the thickness direction; (b) The θʹ precipitate primarily growing along the length direction. Here we use the maximum length of θʹ precipitates as the value of the diameter

5 Conclusions

In this paper, we use the atomistic simulation approach to investigate the microstructure and dislocation mechanism of the orientation effect of θʹ precipitates in Al-4wt.%Cu alloy. Our simulation results were identical to the experimental results. Based on the consistency between the simulation and experiments, we studied the impact patterns of stress aging treatment on θʹ precipitates of Al-Cu alloys and the mechanism of interaction between θʹ precipitates and dislocations. The results of this work support the three-dimensional precipitation distribution of θʹ precipitates during stress aging would distribute along the habit planes perpendicular to the loading direction. The findings are as follows.

1) θʹ precipitates prefer to grow on the {100}Al habit planes parallel to the direction of the applied compressive stress, rather than the habit planes perpendicular to the direction of the compressive stress.

2) Growth in the thickness direction of θʹ precipitates plates in stress-aged Al-4wt.%Cu alloys induces the generation of 1/2<110> perfect dislocations, while the reaction of 1/2<110> perfect dislocations decomposing into 1/6<112> Shockley dislocations leads to the growth of θʹ precipitates along the length direction.

3) The orientation effect of θʹ precipitates in stress-aged Al-4wt.%Cu alloys is dominated by the decomposition of 1/2<110> dislocations within or near the θʹ precipitates and the degree of orientation effect is determined by the reduction of 1/2<110> dislocations, i.e., the higher the number of 1/2<110> dislocations decomposing, the higher the degree of the orientation effect of θʹ precipitates.