# Evolution of Symmetrical Grain Boundaries under External Strain in Iron Investigated by Molecular Dynamics Method

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## Abstract

**:**

## 1. Introduction

## 2. Method

^{−10}and the specified force tolerance is 10

^{−10}eV/Å

^{3}. During the MD relaxation, the timestep is 1 fs and at least 20 ps relaxation is applied for each process to ensure the system is fully relaxed at the given temperature. It should also be noted that during relaxation, the constant number of atoms, pressure and temperature (NPT) ensemble is applied in order to relax both the atomic position and simulation volume. Furthermore, for volume relaxation, each direction of the box is allowed to relax independently to fully release the internal elastic stress and obtain a more stable state for further calculations. The GB formation energy, ${E}_{GB}$, is then calculated according to the following equation as applied in previous studies [13,29,30], which is defined as the difference between the potential energy ${E}_{total}$ of n atoms in the supercell containing GBs and the potential energy of a computational cell with the same number of atoms in a perfect crystal, divided by the cross-sectional area, S, of two GB planes.

_{p}is the cohesive energy of one atom in a perfect bcc Fe crystal, which is −4.12 eV according to Mendelev potential [27].

^{9}/s. The simulation temperature is set at 300 K. It should be noted that during the strain application along the normal direction of GB plane, the box length along the other two directions of the computational box is allowed to change automatically to ensure zero pressure along these two directions. The stress related to the increases in strain is also calculated and thus, the stress–strain curve is obtained. For comparison, this process is also performed for a perfect structure with strain direction along the same normal direction of GB plane. During the strain application along GB normal direction, the length of the box along the directions perpendicular to GB normal direction is also allowed to vary, which is similar to the real experimental condition and ensures the system is at a single stress–strain state [31].

_{max}) of each grid point to surrounding atoms. The free volume is then calculated as the sphere volume with radius of D

_{max}-r, where r is the atomic radius in perfect lattice. This method has been well used in previous studies [33]. The FV calculation under the application of external stress was performed for polymer and amorphous materials [34], which have an FV distribution through the whole system. In this work, this method is also applied firstly for GB investigation since the FV in GB is expected to change in the GB failure process. Furthermore, as stated in the Introduction, the results from the present work may be used for comparison with neutron diffraction measurements under the effect of external stress field in future. The stress of each atom is calculated according to the following equation and can be viewed via the Ovito software:

## 3. Results and Discussion

#### 3.1. Grain Boundary Energy and Structure after Different Relaxation Processes

#### 3.2. Evolution and Related Failure Mechanisms of GBs under the External Strain Effect

#### 3.2.1. Phase Transformation Induced Grain Boundary Failure

_{max}) is around 17.99 Å

^{3}and the average value of free volume is around 7.99 Å

^{3}. While with 15% strain, FV

_{max}reaches around 25.71 Å

^{3}with an average value around 11.98 Å

^{3}at the bcc-fcc interface induced by the phase transition. The stress field around this maximum free volume is around 21.48 GPa, resulting in the local stress concentration increasing and related activation of the slip system. Thus, the present results indicate that the increase in free volume induced by phase transition may be also one possible reason to induce the local stress to its critical value and the failure of GB system in bcc Fe.

_{max}up to 24.43 Å

^{3}. The local stress concentration is also observed above the maximum free volume region with a value around 21.65 GPa, resulting in the activation of slip system and related dislocation nucleation, as shown Figure 4d. Different to the activation of slip system initially in bcc phase for ∑3(112) GB, the activation of slip system under external stress along the <013> direction occurs initially in fcc phase in the {111} plane along <110> direction. Following the same method, the Schmid factor is also calculated for this case with external stress along <013> direction. The maximum Schmid factor for bcc and fcc phases are 0.4115 and 0.4899, respectively, in this case, which is the main reason for slip system activation near the interface from fcc phase side, as shown in Figure 4d.

#### 3.2.2. Mechanical Failure Induced by Activation of Slip System from GB Plane

_{max}is around 31.24 Å

^{3}, which is then increased to 33.36 Å

^{3}above the disordered region, as shown in Figure 5c Therefore, the maximum free volume change is also one possible factor relating to the failure of ∑5(012) GB from the activation of slip system directly in local GB plane region. Further analysis of ∑3(111) ∑9(221), ∑11(113) and ∑17(410) GB reaches a similar conclusion. The example of ∑3(111) GB has been shown in Figures S7 and S8 in the Supplementary Materials. Based on these results, the derivative of critical stress along the normal direction of GB plane of GB failure, Δσ, can be described as a function of GB formation energy, E

_{GB}, as shown in Figure 7a and the following equation:

## 4. Conclusions

## Supplementary Materials

**a**) and displacement distribution (

**b**) of ∑3(112) GB at state with peak stress, Figure S3: Example of FCC lattice structure in fcc phase after phase transition, Figure S4: Snapshots (y-z plane) shows ∑ = 5(013) undergoes phase transition and green atom is fcc struc-ture blue atom is bcc. Then the GB happens to crack at 31 ps, region a, b and c are most obvious, Figure S5: Atomic potential energy (

**a**) and displacement distribution (

**b**) of ∑5(013) GB at state with peak stress, Figure S6: (

**a**) The structure of ∑5(012) when slip system is activated with strain around 8%. The potential energy(b), stress(c) and atomic displacement distribution(d) at this state are shown respectively. Figure S7: (

**a**) The structure of ∑3(111) when slip system is activated with strain around 14%. The potential energy, stress and atomic displacement distribution at this state are shown in (

**b**), (

**c**) and (

**d**) respectively, Figure S8: Distribution of free volume near ∑3(111) GB region at (

**a**) 0 ps and (

**b**) at 28 ps (strain around 14%), Table S1: The peak tensile stresses of all cases studied in this work are listed in table.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**GB atomic structures before and after MD relaxations shown in the top and bottom panels, respectively: (

**a**) ∑3(112) GB, (

**b**) ∑3(111), (

**c**) ∑5(013), (

**d**) ∑5(012), (

**e**) ∑9(221), (

**f**) ∑11(113) and (

**g**) ∑17(410). Structure types are analyzed by the common neighbor analysis method. Atoms near the GB region are colored white, while the atoms in the grain are colored blue.

**Figure 3.**(

**a**) Strain–stress curves of ∑3(111), ∑3(112), ∑5(012), ∑5(013), ∑9(221), ∑ 11(113), ∑17(410). (

**b**) Strain–stress curves of single crystals corresponding to each grain boundary.

**Figure 4.**Snapshots of ∑3(112) and ∑5(013) GB under external strain effect. (

**a**) Shows the state of ∑3(112) GB at which the phase transition starts with a strain value of 10% and (

**b**) is the state at which the slip system in bcc phase near bcc-fcc interface is activated for ∑3(112). (

**c**) Shows the state of ∑5(013) GB at which the phase transition starts with a strain value of 8.5% and (

**d**) is the state at which the slip system is activated for ∑5(013) in fcc phase near the bcc-fcc interface. In the figure, the green, blue and white points are atoms in fcc, bcc and other phase states, respectively.

**Figure 5.**The maximum free volume and related stress distribution near the bcc-fcc interface when the stress is up to the peak value (shown by stress-strain curve in Figure 2 for (

**a**) ∑3(112), (

**b**) ∑5(013) and (

**c**) ∑5(012) GB respectively. The blue and green balls are atoms in bcc and fcc state respectively. The larger red balls are free volume higher than 20 Å

^{3}in GB region.

**Figure 6.**Snapshots of ∑5(012) GB evolution under external stress at different simulation times (t): (

**a**) t = 16 ps, (

**b**) t = 17 ps, (

**c**) t = 19 ps and (

**d**) t = 21 ps, respectively.

**Figure 7.**The Gaussian Fit of the relationship between GB energy and $\Delta \sigma $ (

**a**). The relationship between free volume and fit stress. Fit stress equals the sum of the stress peak value and $\Delta \sigma $ (

**b**).

NO. | GB Plane (hkl) | Sigma (∑) | Simulation Box Length(Å) | Number of Unit Cells along x, y, z and Normal Direction of GB Plane | Number of Atoms |
---|---|---|---|---|---|

1. | (112) | ∑3 | 121.15 × 74.2 × 209.8 | 30 × 30 × 15, Z | 162,000 |

2. | (111) | ∑3 | 121.2 × 148.4 × 209.8 | 30 × 30 × 30, Y | 324,000 |

3. | (310) | ∑5 | 85.8 × 271.8 × 271.8 | 30 × 30 × 15, Z | 270,000 |

4. | (210) | ∑5 | 85.8 × 191.5 × 191.5 | 30 × 30 × 15, Z | 540,000 |

5. | (221) | ∑9 | 121.2 × 363.4 × 256.9 | 30 × 30 × 15, Z | 972,000 |

6. | (113) | ∑11 | 133.9 × 80.8 × 189.4 | 10×20×20, Z | 176,000 |

7. | (410) | ∑17 | 235.4 × 57.2 × 235.4 | 20 × 20 × 10, Z | 272,000 |

**Table 2.**Grain boundary energy calculated by CG and CG-MD-CG relaxation processes. For comparison, the results from DFT calculations are also listed.

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**MDPI and ACS Style**

Ma, W.; Dong, Y.; Yu, M.; Wang, Z.; Liu, Y.; Gao, N.; Dong, L.; Wang, X.
Evolution of Symmetrical Grain Boundaries under External Strain in Iron Investigated by Molecular Dynamics Method. *Metals* **2022**, *12*, 1448.
https://doi.org/10.3390/met12091448

**AMA Style**

Ma W, Dong Y, Yu M, Wang Z, Liu Y, Gao N, Dong L, Wang X.
Evolution of Symmetrical Grain Boundaries under External Strain in Iron Investigated by Molecular Dynamics Method. *Metals*. 2022; 12(9):1448.
https://doi.org/10.3390/met12091448

**Chicago/Turabian Style**

Ma, Wenxue, Yibin Dong, Miaosen Yu, Ziqiang Wang, Yong Liu, Ning Gao, Limin Dong, and Xuelin Wang.
2022. "Evolution of Symmetrical Grain Boundaries under External Strain in Iron Investigated by Molecular Dynamics Method" *Metals* 12, no. 9: 1448.
https://doi.org/10.3390/met12091448