# Phase Transition of Single-Layer Molybdenum Disulfide Nanosheets under Mechanical Loading Based on Molecular Dynamics Simulations

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Models and Methods

^{−5}ps

^{−1}, respectively, referring to our previous work [21,23].

^{−6}eV.

## 3. Results and Discussions

#### 3.1. Nanoindentation of SLMoS2 Nanosheets

_{max}). Different from the monotonic increase of loading force in simulations of nanoindentation for single-layer graphene films [28], there is another sudden decrease in loading force at the deflection of 29.72 Å, in which the deformation is defined as the first critical deflection (δ

_{a}). The sudden decrease also appears in the curve between the total system energy and the deflection, which is not shown here [21].

_{a}, as shown in Figure 2a. Obviously, the unloading curve is coincident with the loading curve, and the deformed SLMoS2 nanosheets can recover to their initial states (see the inset of Figure 2a). Perfect elastic deformation is appeared, as the deflection is smaller than the first critical deflection under nanoindentation. Another unloading process was conducted by moving the indenter back when the initial deflection was greater than δ

_{a}but less than δ

_{max}, as seen in Figure 2b. A residual pit can be observed in the inset of Figure 2b, which inferred that the plastic deformation occurred.

_{a}, and then suddenly changes upward or downward until it returns to a stable terminal condition, as seen in Figure 4a,b. The order of their contribution to the residual deformation can be evaluated by their deviations from their initial bond lengths. The maximum contribution of the bond length is found to be the bond of Mo(5)–S(6) for its deviations of 7.69%, whereas the maximum contribution of bond angle is granted to the Mo(3)–S(4)–Mo(5) for its deviations of 20.07%. It can be seen that the maximum contributions of the bond lengths and bond angles result from the 3th, 4th, 5th, and 6th atoms, which are around the waist of the bell. Here, we also display the evolution of the layer thickness, as shown in Figure 4c. The layer thickness is defined as the distance between the top and the bottom S layers, as shown in Figure 5. Herein, the vertical distance between the top S atom (labeled 2 in Figure 3b) and its relative bottom S atom right underneath the indenter are measured and plotted to show the evolution of the layer thickness during the indentation. The layer thickness is found to suddenly decrease during the loading process and then gradually increases to a stable value, with the deviations of 10.49% after the unloading process. Though significant residual deformation of the layer thickness was observed in this paper and our previous work [21], the bond length and bond angle of the labeled atoms around the waist of the bell also have considerable residual deformation. The maximum deviation is attributed to the bond angle (20.07%), which is much greater than that of the layer thickness (10.49%). Combining the atomic structure and the evolutions of the bond lengths and bond angles, it can be inferred that the lattice distortion around the waist of the bell is substantially worse than that around the bottom of the bell.

#### 3.2. Effect of Indenter Size on the Plastic Deformation

_{r}, i.e., the distance between the fixed Mo atoms and the lowest Mo atom, (2) the width of the residual bell, w

_{r}, i.e., the distance between the two Mo atoms around the half depth of the bell, and (3) the ratio of r

_{d}= w

_{r}/h

_{r}. The greater the values of h

_{r}, w

_{r}, and r

_{d}are, the much more substantial the residual deformation will be. Increasing the indenter size from 10 Å to 15 Å, to 20 Å, and to 40 Å, the deformations become more substantial as the depth of the residual bell h

_{r}increases from 10.29 Å to 12.18 Å, to 15.93 Å, and to 17.19 Å, the width of the residual bell w

_{r}increases from 20.5 Å, to 27.0 Å, to 33.6 Å, and to 44.9 Å, and the ratios of r

_{d}become 2.0, 2.2, 2.1, and 2.6. It is worth noting that a saddle-shape bulge occurs at the edge of the 15 Å indenter, which has the possibility of reducing its h

_{r}and w

_{r}. Moreover, when the radius of the indenter increases to more than 20 Å, the ratio between the radius of the SLMoS2 sheet and the indenter radius will become smaller than 5, which indicates that the indenter cannot be regarded as a point loading and will decrease the precision and accuracy of the MD simulation. When we looked into the evolutions of bond lengths and bond angles around the distorted lattices (given in the supplementary materials), it was found that the variation trends of layer thickness, every bond length, and every bond angle are the same as those shown in Figure 4. The above results again demonstrate that the phase transitions result not only from the sudden reduction in layer thickness but also from the distorted lattice around the waist of the bell.

#### 3.3. Uniaxial Compression of SLMoS2 Nanosheets

^{2}is shown in Figure 7, where a new quadrilateral lattice occurs at a compressive strain of 27.7% (Figure 7b). Before the quadrilateral lattice, the initial hexagonal lattice gradually becomes prolate, as shown in Figure 7a. New bonds are formed by the localized Mo–Mo atoms when the compressive strain is about 32.7%, as seen in Figure 7c. Note that there are several folds in the stress–strain curve, such as the points of 8.2%, 15%, 18.5%, and so on, which means that a slightly distorted bond length and bond angle are present. The quadrilateral lattice is then found to be stable when subjected to the energy minimization and then the tension. It is worth noting that, unlike [24], the ambient temperature in our simulations was set to 0.1 K. In [24], they focused their studies on the effect of temperature on the mechanical properties of SLMoS2 and found that a phase transition under the uniaxial compression occurred at below 40 K, e.g., 4.2 K. Herein, we focused our simulations on the phase transition of SLMoS2 under compression with a temperature of 0.1 K, which was expected to maximally eliminate the interference of thermal fluctuation and reveal the intrinsic phase transition of SLMoS2. It is also worth noting that the new quadrilateral phase occurs only when the compressive strain is greater than 27.7% and that the ideal limit of compressive strain of SLMoS2 is higher than 30%, and this is basically owed to the small size of the model, i.e., a size of less than 40 Å. Currently, such simulation results are useless for practical applications. However, to the best of our knowledge, by carefully preparing a sample of one-dimensional Si nanowires and precisely controlling the experimental procedure, one can obtain a higher elastic strain that approaches the theoretical elastic limit of silicon nanowires [30].

## 4. Conclusions

## Supplementary Materials

**a**,

**d**and

**h**), S-Mo-S bond angles and Mo-S-Mo angles (

**b**,

**e**and

**i**), and layer thickness of S-S (

**c**,

**f**and

**j**) versus deflection during the loading process and unloading process with indenter radii of 15 Å, 20 Å and 40 Å (The labeled atoms are shown in Figure 3).

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 2.**The force–deflection curves for the loading–unloading–reloading process of SLMoS2 nanosheets: (

**a**) The deflection is smaller than δ

_{a}; (

**b**) The deflection is greater than the δ

_{a}but smaller than δ

_{max}.

**Figure 3.**The atomic configuration at the cross section of the deformed SLMoS2 (

**a**); the labeled atoms in the distorted area (

**b**); the zoom in view of the residual hollow (

**c**).

**Figure 4.**The S–Mo bond lengths (

**a**), S–Mo–S bond angles and Mo–S–Mo angles (

**b**), and the layer thickness of S–S (

**c**) versus deflection during the loading process and unloading process (the labeled atoms are shown in Figure 3).

**Figure 5.**The atomic structures of SLMoS2 nanosheets in the process of nanoindentation: (

**a**) before loading; (

**b**) after unloading.

**Figure 6.**The sectional view of atoms in residual indentation with different indenter radii of 10 Å, 15 Å, 20 Å, and 40 Å at the deflection exceeding δ

_{a}after the unloading process. The rectangular, prismatic, triangular, and pentagonal points are the atomic positions of the Mo atoms close to the plane (yz plane) crossing the center of the indenter.

**Figure 7.**The stress–strain curves of the SLMoS2 nanosheets under uniaxial compression along the armchair direction. The insets show (

**a**) The distorted hexagonal lattice; (

**b**) the quadrilateral lattice; (

**c**) the buckling.

**Figure 8.**The band structures of (

**a**) the initial hexagonal SLMoS2 nanosheets (semiconducting) and (

**b**) the new phase of the quadrilateral structure (metallic).

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**MDPI and ACS Style**

Pang, H.; Li, M.; Gao, C.; Huang, H.; Zhuo, W.; Hu, J.; Wan, Y.; Luo, J.; Wang, W.
Phase Transition of Single-Layer Molybdenum Disulfide Nanosheets under Mechanical Loading Based on Molecular Dynamics Simulations. *Materials* **2018**, *11*, 502.
https://doi.org/10.3390/ma11040502

**AMA Style**

Pang H, Li M, Gao C, Huang H, Zhuo W, Hu J, Wan Y, Luo J, Wang W.
Phase Transition of Single-Layer Molybdenum Disulfide Nanosheets under Mechanical Loading Based on Molecular Dynamics Simulations. *Materials*. 2018; 11(4):502.
https://doi.org/10.3390/ma11040502

**Chicago/Turabian Style**

Pang, Haosheng, Minglin Li, Chenghui Gao, Haili Huang, Weirong Zhuo, Jianyue Hu, Yaling Wan, Jing Luo, and Weidong Wang.
2018. "Phase Transition of Single-Layer Molybdenum Disulfide Nanosheets under Mechanical Loading Based on Molecular Dynamics Simulations" *Materials* 11, no. 4: 502.
https://doi.org/10.3390/ma11040502