# Effect of Defects on the Mechanical and Thermal Properties of Graphene

^{1}

^{2}

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## Abstract

**:**

^{0.28}, and decreased monotonously with the increase of defect concentration. Compared with the pristine graphene, the thermal conductivity of defective graphene showed a low temperature-dependent behavior since the phonon scattering caused by defect dominated the thermal properties. In addition, the corresponding underlying mechanisms were analyzed by the stress distribution, fracture structure during the deformation and phonon vibration power spectrum.

## 1. Introduction

^{2}bonded carbon atoms.

^{3}-type defects; however, it decreased significantly with the existence of vacancy defect. Using the molecular dynamic (MD) simulations, Mortazavi et al. [11] indicated that the TC of Gr decreased ~50% at a defect concentration of ~0.25%, and the Young’s modulus, fracture strength and fracture strain decreased with the increase of defect concentration. Zhao et al. [12] found that the oxygen plasma treatment could reduce the TC of Gr significantly at a low defect concentration (~83% reduction for ~0.1% defect concentration) through MD simulations and non-contact optothermal Raman measurement. Jing et al. [13] indicated that the vacancy defect could decrease the Young’s modulus while the reconstruction of vacancy could stabilize the modulus. Although there are some pioneering reports on the mechanical and thermal properties of Gr, as described above, there is still a lack of comprehensive study about some important factors, e.g., how the type and concentration of defects are related to the mechanical and thermal properties, especially at different temperatures. The reduction or enhancement mechanisms for the TC of defective Gr at different temperature were also not well understood. Meanwhile, the Gr is commonly used as nanofiller to enhance the mechanical and thermal properties of polymer materials, e.g., Gr/epoxy nanocomposites [14,15], etc., thus it is of great significance to fully understand the mechanical and thermal properties of Gr and effects of various defects.

## 2. Computational Methods

#### 2.1. Molecular Model of Gr

#### 2.2. Calculation of Mechanical and Thermal Properties

^{6}timesteps followed by a microcanonical NPT ensemble along x/y directions (i.e., constant number of atoms, pressure and energy) for another 10

^{6}timesteps. Followed by the equilibration, a constant uniaxial strain was applied along the x- or y-direction with a strain rate of 5 × 10

^{−3}ps

^{−1}. The atomic stress of the Gr during the uniaxial tension were calculated by the viral theorem using Equation (1) [19]:

_{i}, m

_{i}and v

_{i}represent the volume, mass and velocity of atom i, respectively; F

_{ij}and r

_{ij}are the force and distance between atom i and j, respectively; and indices α and β denote the Cartesian coordinate components. The thickness of Gr was determined by van der Waals interaction between the single layers, i.e., 3.35 Å. Then, the mechanical properties including Young’s modulus, fracture strength and fracture strain could be obtained from the stress–strain curves.

_{H}= T

_{0}(1 + Δ) and T

_{C}= T

_{0}(1 − Δ) by Langevin thermostat, respectively (as shown in Figure 1b). T

_{0}is the average temperature and Δ is the normalized temperature difference. To evaluate the effect of temperature, T

_{0}varied from 300 K to 900 K, while Δ was fixed at 0.03. During the NEMD simulations, the energies removed from the cold bath and added to the hot bath as a function of time were calculated. The sum of added/removed energy is equal to zero, thus the total energy is conserved. The heat flux along the x-direction J

_{x}can be expressed by:

^{6}timesteps), the error bars were determined by the standard deviation.

## 3. Results and Discussion

#### 3.1. Validation of Models

_{A}was greater than that with an angle of 60° to the loading direction F

_{Z}, thus Mode I ruptured first, leading a greater fracture strength (106 GPa) and strain (0.20) along the zigzag direction than those (93 GPa and 0.14, respectively) along the armchair direction.

^{−1}K

^{−1}at the temperature of 300 K. The measured value of TC with respect to time during the steady-state simulations is provided in the Supplementary Materials. The TC and relevant values obtained by simulations or experiments are listed in Table 2. The results show that the TC of Gr was related with the calculation method, system size, potentials, etc. The present results were consistent with the previous simulation results [24,25] (with same potential and size). Therefore, the above results confirmed the validity of the AIREBO potential, molecular models and calculation method.

#### 3.2. Effects of Temperature and Strain Rate on the Mechanical Properties

^{−5}−3 × 10

^{−3}ps

^{−1}), the Young’s modulus was insensitive to the strain rate, while the fracture strength and strain slightly increased at higher strain rate. The lower strain rate means a longer response time for Gr, which would increase the number of atoms that could overcome the energy barrier required for fracture, leading a lower fracture strength and strain. However, the effect of strain rate was less significant than that of temperature. The relation between fracture strength and strain rate can be described by Arrhenius equation [29]:

#### 3.3. Effect of Defects on the Mechanical Properties

^{3}-type defect, while decreases significantly with the increase of vacancy defect. Meanwhile, further increase of defect concentration has little effect on the fracture strength and strain.

^{2}and sp

^{2}-sp

^{3}network structures were formed, and fracture behavior for Gr changed from brittle to ductile.

#### 3.4. Effects of Temperature and System Size on the Thermal Properties

^{0.28}, which was similar to the previous results of λ~L

^{0.35}obtained by Guo et al. [30]. Such phenomenon was also consistent with the previous studies about the carbon nanotube [31], λ~L

^{β}and β~0.3–0.4. The size-dependent behavior was due to the long mean free path of phonons (MFP) of Gr, ~775 nm [32], which was much longer than that used in MD. Therefore, apart from the phonon–phonon scattering, the phonon scattering existed at the boundary of Gr. With the increase of length, more phonons will be excited and contribute to the increase of TC. We also compared the TCs calculated at different boundary conditions along the width direction (y), as shown in Figure 11a. The fixed boundary condition means that the particle could not interact across the boundary and move from one side of the box to the other, i.e., the length of Gr was finite in the width direction. This clearly indicated that the TCs obtained at the fixed boundary condition were smaller than those obtained at the periodic boundary condition. For instance, when the length was 80 nm, the TCs at the periodic and fixed boundary condition were 273.1 and 168.6 W/mK, respectively. At the fixed boundary condition, the power law between the TCs and length, λ~L

^{0.18}, the index of power law was smaller than that at the periodic boundary condition. This was because, at the periodic boundary condition, the phonons can across the regions perpendicular to the direction of the heat flow without boundary scattering, thus the width of Gr had negligible effect on the calculation results. In addition, the vibration density of states (VDOS) of Gr at the two different boundary conditions were calculated. The VDOS can be obtained by calculating the Fourier transformation of atomic velocities autocorrelation function at the equilibrium state:

#### 3.5. Effect of Defects on the Thermal Properties

_{phonon-phonon}and l

_{defect-phonon}are the length of phonon–phonon scattering and scattering caused by defects, respectively. According to Equations (10) and (11), the TC of defective Gr can be obtained:

^{2}structure of the local lattice, while a DV was formed by removing two adjusted atoms as the local structure can rearrange to restore the three-coordinated sp

^{2}bonding by creating an octagon and two pentagon structures [35]. The two-coordinated atoms were less stable, leading to higher level of defect scattering. Previously, Haskin et al. [35] found that the Gr containing SV decreased ~80% at a concentration of 0.1%, while the Gr containing DV and SW decreased ~70%. Zhang et al. [24] also found that different types of defects had different effects on TC at a low concentration (≤0.2%). When the concentration increased (>0.2%), they had similar effect on TC, which was mainly because the heat transport mechanism changed from propagating to diffusive, and the TC was insensitive to the defect type in diffusive form.

^{−α}; the corresponding index is shown in Figure 14b. Compared with the pristine Gr, the power law index of defective Gr decreased significantly, i.e., showing a weak temperature-dependent behavior. The main reason was that, compared with the phonon scattering caused by temperature, the scattering caused by defects was the dominant factor for the TC of defective Gr, which was also consistent with previous results obtained by Zhang [24] and Hu et al. [33].

## 4. Conclusions

^{0.28}: more phonons were excited with the increase of length, contributing to the TC. The results also indicate that the TC of Gr containing SV, DV and SW decreased ~57.6%, ~43.4% and ~31.9% even at the low concentration of 0.23%, respectively, which was mainly due to the reduced MFP caused by defect scattering. Besides, the TCs of Gr with/without defect at different temperature were calculated, showing that, compared with the phonon scattering caused by temperature, the phonon scattering caused by defect dominated the thermal properties. The TCs of defective Gr showed a low temperature-dependent behavior. The above findings can provide a comprehensive understanding in the mechanical and thermal properties of Gr.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Molecular model of: (

**a**) the armchair and zigzag Gr for uniaxial tensile test; and (

**b**) Gr for the calculating TC.

**Figure 2.**Types of defect studied in this work: (

**a**) Single vacancy (SV); (

**b**) Double vacancy (DV); and (

**c**) Stones–Wales (SW).

**Figure 3.**(

**a**) Stress–strain curves for Gr along the armchair and zigzag directions; and (

**b**) the total energy variations during the loading process.

**Figure 4.**The fracture process and distributions of von Mises along: (

**a**) the armchair direction and; (

**b**) the zigzag direction.

**Figure 5.**(

**a**) Steady-state temperature profile of Gr without defect obtained using the NEMD simulations at 300 K. The color bars highlight the hot/cold bath in simulation. (

**b**) Energies added to the hot bath and removed from the cold bath with respect to the time.

**Figure 6.**The stress–strain curves of Gr at different temperature along: (

**a**) armchair; and (

**b**) zigzag.

**Figure 7.**(

**a**) The Young’s modulus; (

**b**) the fracture strength; and (

**c**) the fracture strain of Gr along the armchair and zigzag directions at different temperatures.

**Figure 8.**(

**a**) The Young’s modulus; (

**b**) the fracture strength; and (

**c**) the fracture strain of Gr along the armchair and zigzag directions at different strain rates.

**Figure 9.**Under different defect types and concentrations, the variations of: (

**a**) Young’ modulus; (

**b**) fracture strength; and (

**c**) fracture strain of Gr along the armchair direction. The variations of: (

**d**) Young’ modulus; (

**e**) fracture strength; and (

**f**) fracture strain of Gr along the zigzag direction with respect to defect concentration at different defect types.

**Figure 10.**The fracture process of Gr containing: (

**a**) SV; (

**b**) DV; and (

**c**) SW. The red ring indicates the location of defect.

**Figure 11.**(

**a**) The variation of TCs of pristine Gr with respect to length; and (

**b**) the VDOS of Gr under periodic/fixed boundary conditions.

**Figure 12.**(

**a**) The TCs of Gr; and (

**b**) the energies added to the hot bath and removed from the cold bath during the NEMD at different temperature.

**Figure 14.**(

**a**) The TCs of Gr with/without defect at different temperature; and (

**b**) the corresponding power law index α.

References | Method | Direction | Young’s Modulus (GPa) | Fracture Strength (GPa) | Fracture Strain |
---|---|---|---|---|---|

Lee (2008) [4] | Experiment | / | 1000 | 130 ± 10 | 0.25 |

Liu (2007) [23] | DFT | Armchair | 1050 | 110 | 0.19 |

Zigzag | 1050 | 121 | 0.26 | ||

Q.X. Pei (2010) [16] | MD (AIREBO) | Armchair | 890 | 105 | 0.17 |

Zigzag | 830 | 137 | 0.27 | ||

Ansari (2012) [21] | MD (Tersoff) | Armchair | 790 | 123 | 0.23 |

Zigzag | 807 | 127 | 0.22 | ||

This paper | MD (AIREBO) | Armchair | 961 | 93 | 0.14 |

Zigzag | 911 | 106 | 0.20 |

References | Potentials | Method | Size | TC at 300 K (Wm^{−1} K^{−1}) |
---|---|---|---|---|

Balandin (2008) [2] | / | Experiment | ~0.5–1 um | ~4840–5300 |

Wei (2011) [25] | AIREBO | RNEMD | 102 × 102Å^{2} | 77.3 |

Yang (2012) [26] | AIREBO | EMD | (90~270) × (40~180) Å^{2} | ~3200–5200 |

Xu (2014) [27] | Tersoff | NEMD | 50 × (2 × 150) Å^{2} | ~400–1800 |

Zhang (2012) [24] | AIREBO | RNEMD | 61 × 200 Å^{2} | ~170 |

This paper | AIREBO | NEMD | 60 × 200 Å^{2} | 182 |

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**MDPI and ACS Style**

Li, M.; Deng, T.; Zheng, B.; Zhang, Y.; Liao, Y.; Zhou, H.
Effect of Defects on the Mechanical and Thermal Properties of Graphene. *Nanomaterials* **2019**, *9*, 347.
https://doi.org/10.3390/nano9030347

**AMA Style**

Li M, Deng T, Zheng B, Zhang Y, Liao Y, Zhou H.
Effect of Defects on the Mechanical and Thermal Properties of Graphene. *Nanomaterials*. 2019; 9(3):347.
https://doi.org/10.3390/nano9030347

**Chicago/Turabian Style**

Li, Maoyuan, Tianzhengxiong Deng, Bing Zheng, Yun Zhang, Yonggui Liao, and Huamin Zhou.
2019. "Effect of Defects on the Mechanical and Thermal Properties of Graphene" *Nanomaterials* 9, no. 3: 347.
https://doi.org/10.3390/nano9030347