# The Impacts of Random Distributed Vacancy Defects in Steady-State Thermal Conduction of Graphene
## 3 Results and Discussion
### Center Concentrated Vacancy Defects  
The concentrated vacancy defects in the center of graphene are gradually amplified, as shown in Figure 5. According to the size amplification of the center concentrated vacancy defects, the thermal energy of entire graphene gradually decreases in both situations, no matter what its heat source is in the armchair or zigzag edges. The thermal energy of graphene with its heat source in armchair edges is larger than that of graphene with its heat source in zigzag edges, as shown in Figure 6a. The location of its heat source, as one of the boundary conditions, contributes to the difference of thermal energy in  
Figure 4: Graphene with vacancy defects. (**a–c**) are for center concentrated, periodic, and random distributed vacancy defects, respectively. Figure 3: Finite element model of pristine graphene for the thermal conduction. (**a,b**) is its heat source in the armchair and zigzag edge, respectively. the steady-state thermal conduction of graphene. Furthermore, the definition of equivalent coefficient of thermal conductivity for vacancy defected graphene can be expressed as:  
\[C=\frac{\sum\limits_{i=1}^{m}E_{i}^{T}/m}{\sum\limits_{i=1}^{n}E_{pi}^{T}/n} \tag{18}\]  
In addition, the percentage of vacancy defects in the entire graphene can be counted as:  
\[Per=\frac{n-m}{n}\times 100\% \tag{19}\]  
where \(n\) and \(m\) are the number of elements in pristine graphene and vacancy defected graphene. The total thermal energy is computed by the sum of the thermal energy of each element and \(E_{pi}^{T}\) in pristine graphene and vacancy defected graphene, independently. Based on Equation (18), the amount of vacancy defects is introduced in an equivalent coefficient of the thermal conduction. In Figure 6b, the reduction of an equivalent coefficient of thermal conductivity becomes sharper with the size amplification of a concentrated center. Under both boundary conditions, the impact of vacancy defects in a steady-state thermal conduction is non-linear. Further, the gradient of curve in the equivalent coefficient is augmented with the enlargement of center concentrated vacancy defects. Moreover, the vacancy defected graphene sheets with its heat source in armchair and zigzag edges have consistent results in the thermal condition's equivalent coefficient, especially when the size of the center concentrated vacancy defects is not large. With the size enlargement of the center concentrated vacancy defects, the difference between two boundary conditions becomes evident. Figure 5: Center concentrated vacancy defects. The size of center concentrated vacancy defects is gradually amplified from (**a–d**). Figure 6: The results of graphene with center concentrated vacancy defects (A—heat source in armchair edges, Z—heat source in zigzag edges, and (**a**) is for the thermal energy, (**b**) is for the equivalent coefficient, unit for thermal energy is J). The graphene with its heat source in armchair edges has higher equivalent coefficient of thermal conductivity than that with its heat source in zigzag edges. In addition, the contour of temperature in graphene is revealed in Figures 7 and 8. In order to track the exact impacts of center concentrated vacancy defects in the steady-state heat conduction of graphene, the local region around the center concentrated vacancy defects is compared. When the size of the vacancy defects is small, the effects in temperature are not obvious. Along with the size amplification of center concentrated vacancy, the influenced region in graphene become larger and the regularity and band characteristic of temperature results in pristine graphene is destroyed under both boundary conditions. The regions close to the center concentrated vacancy defects tend to have a higher temperature than the pristine graphene. Therefore, the center concentrated vacancy defects in graphene is a crucial factor to change the temperature field in the steady-state thermal conduction.