# The Impacts of Random Distributed Vacancy Defects in Steady-State Thermal Conduction of Graphene
## 3 Results and Discussion
### Random Distributed Vacancy Defects  
The center concentrated and periodic vacancy defects are two typical vacancy defects in graphene. Actually, the vacancy defects are stochastically distributed; thus, it is difficult to predict the placement in the entire graphene. By combing the finite element model with the Monte Carlo simulation [46], the vacancy defects are randomly dispersed in graphene with a specific amount. The statistical results of thermal energy for graphene with heat source in armchair and zigzag edges are listed in Table 2. In the procedure of the Monte Carlo based finite element method, the minimum and maximum values of the thermal energy are captured from many possible situations. The mean and variance values are computed from series of thermal energy from samples of the Monte Carlo based finite element method. Different than the center concentrated and periodic vacancy defects, the minimum, maximum, and mean values of thermal energy in graphene with random distributed vacancy defects have linear \begin{table}
\begin{tabular}{c c c c c c} \hline \hline _Type_ & _Per (\%)_ & **Minimum** & **Maximum** & **Mean** & **Variance** \\ \hline \multirow{4}{*}{A} & 0.2 & 1.1494 & 1.1577 & 1.1535 & 0.2181 \\  & 0.5 & 1.1335 & 1.1496 & 1.1428 & 0.5465 \\  & 0.8 & 1.1243 & 1.1419 & 1.1326 & 0.8798 \\  & 1 & 1.1163 & 1.1349 & 1.1252 & 1.0048 \\  & 3 & 1.0349 & 1.0752 & 1.0550 & 3.3332 \\  & 5 & 0.9571 & 1.0063 & 0.9831 & 6.0168 \\ \hline \multirow{4}{*}{Z} & 0.2 & 0.9232 & 0.9301 & 0.9271 & 0.1480 \\  & 0.5 & 0.9135 & 0.9234 & 0.9184 & 0.3278 \\ \cline{1-1}  & 0.8 & 0.9032 & 0.9161 & 0.9102 & 0.5540 \\ \cline{1-1}  & 1 & 0.8948 & 0.9130 & 0.9043 & 0.6667 \\ \cline{1-1}  & 3 & 0.8300 & 0.8589 & 0.8472 & 2.4754 \\ \cline{1-1}  & 5 & 0.7718 & 0.8050 & 0.7894 & 3.6630 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Statistical thermal energy results for random distributed vacancy defects (unit: \(\times 10^{5}\) J). Figure 11: Temperatures of graphene with its heat source in armchair edges (The four figures are for the graphene sheets with generally increasing periodic vacancy defects). Figure 12: Temperatures of graphene with its heat source in zigzag edges (The four figures are for the graphene sheets with generally increasing periodic vacancy defects). declined tendency with the increase of the amount of vacancy defects. The minimum, maximum, and mean values of graphene's thermal energy with its heat source in armchair edges are larger than when its heat source is in zigzag edges, but the linear results are parallel to each other in the two different boundary conditions, as presented in Figure 13. In addition, the divergences between the minimum and maximum values of the thermal energy with different amounts of vacancy defects are non-negligible (Table 2 and Figure 13). The unit of the thermal energy is \(10^{5}\) J. With the increase of the number of random distributed vacancy defects in graphene, the differences between the minimum and maximum values of the thermal energy are amplified. The exact computation results are provided and compared in the following discussion. Furthermore, the variance of thermal energy is amplified with the enlargement in the amount of stochastically distributed vacancy defects. However, the parallel characteristic between the results of the two boundary conditions disappears in the variance of thermal energy. The variances of graphene with its heat source in armchair edges have linear growth with an increase of the number of stochastic vacancy defects. However, the graphene with its heat source in zigzag edges is tougher to shield than the variation and fluctuation in thermal energy. The gradient of variance in Figure 13d declines when _Per_ equals 3%. The difference in variance of thermal energy in two boundary conditions becomes large. To be more exact, the probability density distribution of thermal energy in graphene under two boundary conditions are illustrated in Figures 14 and 15.