# Atomic-Scale Insights into the Deformation Mechanism of the Microstructures in Precipitation-Strengthening Alloys

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{−4}. The cut regimen prevails under the interaction between coherent precipitates and dislocations. In the case of a large lattice misfit of 19.3%, the dislocations tend to move toward the incoherent phase interface and are absorbed. The deformation behavior of the precipitate-matrix phase interface was also investigated. Collaborative deformation is observed in coherent and semi-coherent interfaces, while incoherent precipitate deforms independently of the matrix grains. The faster deformations (strain rate is 10

^{−2}) with different lattice misfits all are characterized by the generation of a large number of dislocations and vacancies. The results contribute to important insights into the fundamental issue about how the microstructures of precipitation-strengthening alloys deform collaboratively or independently under different lattice misfits and deformation rates.

## 1. Introduction

^{8}/s. In recent years, the phase-field crystal (PFC) model has naturally coupled all the physical properties generated by periodic structures and simulated the diffusive evolution of microstructures on the atomic spatial scale [37,38] during slow plastic deformation. The deformation of pure material systems has been successfully reproduced by PFC simulations [39,40].

^{−4}to 10

^{−2}. Then the deformation mechanisms of precipitate-matrix phase interface with different lattice misfits were also characterized. The results provide new insights at the atomic scale into the deformation mechanisms of alloys with precipitate-matrix structures.

## 2. Model and Simulation Details

#### 2.1. Phase-Field Crystal Model

#### 2.2. Simulation Details

_{p}are 0.92 (lattice misfit f = 4.3%), 0.88 (f = 9.2%), and 0.8 (f = 19.3%), which, respectively, gives rise to coherent, semi-coherent, and incoherent interfaces. Figure 1 presents the precipitation process of nano-precipitates. The precipitates size and volume fraction are representative of real alloys [53,54]. The initial grid spacing $d{x}_{0}$ = $d{y}_{0}$ = 0.125 is to ensure that each atom is resolved by eight mesh spacing. The dynamical Equations (9)–(11) are solved semi-implicitly in Fourier space, and the time step Δt is set as 0.05 [55].

^{5}Δt in Figure 1. Here, we set a tension force along the x direction with strain rate $\dot{{\epsilon}_{x}}$, and simultaneously compression force along the y direction with strain rate $\dot{{\epsilon}_{y}}$. At the $kd$ time steps, the changed grid sizes in the x direction and the y direction are calculated as,

^{−4}, 10

^{−3}and 10

^{−2}. Therefore, the strain is $\epsilon ={\epsilon}_{x}-{\epsilon}_{y}=\dot{\epsilon}\xb7kd\xb7\Delta t.$

## 3. Results and Discussion

#### 3.1. Deformation Mechanisms of Microstructure at Different Lattice Misfits

#### 3.1.1. Precipitate-GB Interactions

^{−4}. As we know, the second phases usually prefer to precipitate at GBs in alloys, which may profoundly affect the deformation behavior of GBs. In the case of a small lattice misfit of 4.3% (Figure 2a,b), the coherent precipitate-matrix interface structure exerts a weak force to pin the GB. As the deformation proceeds, the GB finally escapes from the precipitated phase and moves toward the matrix phase. In the case of the lattice misfit of 9.2% (Figure 2c,d), the obtained semi-coherent precipitate-matrix interface contains a bunch of misfit dislocations. The pinned GB penetrates through the precipitate-matrix interface, which releases the misfit strain of the semi-coherent interface. As shown in Figure 2c, the GB has been divided into two parts by the intersection point O, and bends sharply at this point due to deformation. Even though the pinned GB could migrate and slip during deformation, it is firmly pinned by the precipitate, and the two parts of the GB always connect. The situation is different from the case of a large lattice misfit of 19.2%. As shown in Figure 2e,f, the incoherent precipitate-matrix interface divides the GB into two parts, one segment in the precipitate and the other one in the matrix phase, but the two parts are totally separated by the precipitate/matrix interface and evolve independently. Moreover, the results above are compared with other deformation process of the GBs without precipitates simultaneously. Without the pinning of precipitates, under tensile deformation, these GBs move faster by rotation, straightening and merging, accompanied substantial changes of GB structure [57,58].

#### 3.1.2. The Dislocation-GB/Precipitate Interactions

^{−4}. Dislocation dynamics plays a crucial role in the deformation of precipitates and grains. As the deformation proceeds, a great number of dislocations are generated continuously from the deformed precipitates and grains, and subsequently are absorbed by GBs and precipitate-matrix phase interface. The coherent precipitate-matrix interface shows a very limited capacity to accommodate the generated dislocation during deformation. The moving dislocations have to penetrate through the precipitate/matrix interface quickly without a duration of stay, implying that the cut regimen prevails under the interaction between coherent precipitate and dislocations. This is in line with previous theories and studies that the coherent precipitate tends to be cut by moving dislocations [59,60]. They move toward the GBs and finally are absorbed as shown in Figure 3a–c. The dislocation pairs A1 are absorbed by the GB first, as shown in Figure 3a,b. As shown in Figure 3b,c, the dislocation pairs A2 are also absorbed by the GB. It can be observed that the two dislocations with opposite signs react and merge into a new dislocation, which then move towards the GB driven by the external forces during the evolution of the dislocation. In reverse, the GB can also emit dislocations continuously. Some of these dislocations move to other GBs and some are annihilated by the dislocations with the opposite burgers vector. This has also been reported in previous studies [61].

#### 3.2. Deformation Mechanisms at Higher Strain Rate

^{−4}, the original GBs are undermined and broken into small segments, with dislocations and vacancies at a strain rate of 10

^{−2}. The pinned GBs are decomposed soon, leaving abundant dislocations and vacancies around the phase interface. Even for the case of a small lattice misfit of 4.3%, large lattice distortions, abundant dislocations and vacancies, and even micro-cracks could be observed in the severely deformed precipitate-matrix interface. Because of the decomposition of GBs and high dislocation density in the deformed precipitate-matrix phase interface, there is a limited space to accommodate such newly generated dislocations. Consequently, as quantified in Figure 6, the number density of dislocations of fast deformation increases to a much higher peak value than the slow deformation. The steep increases in dislocation number density contribute to the fast increase of storage energy of the system that increases with deformation rates, as shown in Figure 6c. This is in line with the classical deformation theory of alloys, which regards that the yield strength of alloys increases with deformation rates [64,65,66].

^{15}m

^{−2}after tensile deformation in aluminum alloys [66,67]. The dynamic recrystallization is not observed in the slow deformation, exemplified by the observation of dynamics of dislocations and GBs in Figure 5, and the evolution of dislocation density, free energy-strain curves, and free energy-dislocation density curves in Figure 6. The free energy of the system is calculated as the current free energy minus the free energy of the reference state, with negative values indicating a decrease in energy relative to the reference state [44,45]. The energy of the regularly arranged crystal is the lowest, and the propagation of new dislocations produces additional free energy. As shown in Figure 6d, dislocation density is proportional to free energy in the initial deformation stage.

#### 3.3. Deformation Mechanisms of Precipitate-Matrix Phase Interface

## 4. Conclusions

^{−4}, the pinning effect of precipitate can actually hinder the motion of deforming GBs, and becomes increasingly strong with the increase of lattice misfit. In the faster deformation with a strain rate of 10

^{−2}, the original GBs are undermined and broken into small segments, dislocations, and vacancies. Therefore, the dislocation density of fast deformation is higher.

## Supplementary Materials

^{−4}with lattice misfit of f = 4.3%, f = 9.2%, f = 19.3% respectively.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The evolution of precipitation process with different lattice misfits: (

**a**) f = 4.3%; (

**b**) f = 9.2%; (

**c**) f = 19.3%. The deformation simulations are conducted after 100,000 Δt, as shown in the Videos S1–S3 in the Supplementary Materials. Note that the “con” on the right side of the color bar is the abbreviation for concentration.

**Figure 2.**The interactions between GB and precipitate during slow deformation with strain rate $\dot{\epsilon}$ = 10

^{−4}with lattice misfit (

**a**,

**b**) f = 4.3%, (

**c**,

**d**) f = 9.2%, (

**e**,

**f**) f = 19.3%. Note that the time evolution of deformation process is characterized by the increase of applied deformation strain (ε). The black arrows in (

**a**,

**b**) are the direction of GB movement. For more details of the three deforming precipitates, please refer to Videos S4–S6 in the Supplementary Materials.

**Figure 3.**The dislocation dynamics around the deformed precipitate at different lattice misfits: (

**a**–

**c**) f = 4.3% (

**d**–

**f**) f = 19.3%. Note that the black arrows mark the moving directions of dislocation, the white circles are signs, the red arrows are the indicators.

**Figure 4.**Dislocation density ρ-strain ε curves of the slow deformation process with $\dot{\epsilon}$ = 10

^{−4}at different lattice misfits of 4.3%, 9.2% and 19.3%.

**Figure 5.**The deformation process of precipitates at a higher strain rate of $\dot{\epsilon}$ = 10

^{−2}. (

**a**–

**c**) The evolution of the precipitate with f = 4.3%, the evolution of the precipitate with f = 19.3% containing a GB (

**d**–

**f**), and the evolution of the precipitate without intergranular GB with f = 19.3% (

**g**–

**i**).

**Figure 6.**Dislocation density ρ and free energy evolution as a function of deformation strain ε for deformation processes under different degree of lattice misfit f and strain rates $\dot{\epsilon}$. (

**a**,

**b**) the ρ-ε curves for lattice misfit f = 4.3% and f = 19.3%, respectively, (

**c**) free energy-ε curves for deformation process, (

**d**) free energy-dislocation density curves for strain rate $\dot{\epsilon}=0.01$.

**Figure 7.**Emitting and separation of dislocation pairs at the semi-coherent interface of different deformation strains during deformation with a strain rate $\dot{\epsilon}$ = 10

^{−4}, (

**a**) ε = 0.086, (

**b**) ε = 0.118, (

**c**) ε = 0.119, (

**d**) ε = 0.196. The white arrows mark the moving directions of dislocation, the red arrows are signs.

Parameters | Symbols | Values/Expressions |
---|---|---|

Reference density | ${\rho}^{0}$ | 0.01 |

Reference composition | c_{0} | 0.5 |

Polynomial fitting parameters | $\mathsf{\eta},\mathsf{\chi}$ | $\mathsf{\eta}=$ 1.4, $\mathsf{\chi}=$ 1 |

Entropy of mixing coefficient | $\mathsf{\omega}$ | 0.005 |

Parameters for correlation function | ${\sigma}_{{M}_{j}},{\alpha}_{i}$ | ${\sigma}_{{M}_{j}}=$0.8, ${\alpha}_{i}=2.0$ |

Gradient energy coefficient | $\alpha $ | 1 |

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**MDPI and ACS Style**

Wei, C.; Tang, S.; Kong, Y.; Shuai, X.; Mao, H.; Du, Y.
Atomic-Scale Insights into the Deformation Mechanism of the Microstructures in Precipitation-Strengthening Alloys. *Materials* **2023**, *16*, 1841.
https://doi.org/10.3390/ma16051841

**AMA Style**

Wei C, Tang S, Kong Y, Shuai X, Mao H, Du Y.
Atomic-Scale Insights into the Deformation Mechanism of the Microstructures in Precipitation-Strengthening Alloys. *Materials*. 2023; 16(5):1841.
https://doi.org/10.3390/ma16051841

**Chicago/Turabian Style**

Wei, Chenshuang, Sai Tang, Yi Kong, Xiong Shuai, Hong Mao, and Yong Du.
2023. "Atomic-Scale Insights into the Deformation Mechanism of the Microstructures in Precipitation-Strengthening Alloys" *Materials* 16, no. 5: 1841.
https://doi.org/10.3390/ma16051841