# A Thermal Transport Study of Branched Carbon Nanotubes with Cross and T-Junctions
## 3 Results  
The heat current ratios occurring after the cross and T-junctions were examined through NEMD simulations. The effects of branch length, atomic configurations at the junctions, and temperature on the heat flow inside the branched CNTs were examined. The length dependence of the heat current ratios for the two cross-junctions is illustrated in Figure 5a,b. Regardless of the atomic configuration at the junctions, the heat current ratios \(R_{2}\) and \(R_{4}\) (for branches 2 and 4, respectively) were approximately equal and larger than \(R_{3}\). This result indicates that the heat currents did not flow equally into the three branches after the junction even though all of the branches had identical length and chirality. The heat flow into the top (branch 2) and bottom (branch 4) branches was higher than that into the straight branch (branch 3), especially for short branch lengths. However, as the branch length increased, the heat current ratio tended to approach 1/3. This value was equal to the prediction result obtained using conventional thermal circuit theory under the assumptions that the thermal resistances along all the branch directions at the junction were identical and that the thermal properties of all of the branches were the same. Moreover, the thermal transport behavior was investigated by creating a hot region placed at the bottom branch (branch 4) and cold regions at the ends of the other branches for the branched CNT with cross-junctions. Different temperature settings produced different  
Figure 4: Illustration of the temperature-controlled setup for non-equilibrium molecular dynamics simulations. A hot region was created at branch 1, whereas cold regions were created in the remaining branches of the branched CNTs with (**a**) cross-junction and (**b**) T-junction. heat current directions, along which different topological defects were encountered. For instance, the heat current from branch 1 encountered two heptagons in parallel at cross-junction 1; however, the heat flow from branch 4 encountered two heptagons in series. The heat current results for two temperature settings were within the standard deviation error, as shown in Figures 5a and 6a. For both settings, the heat current preferentially flowed sideways rather than straight. We attempted to explain the aforementioned observations by using thermal circuit theory and by considering the anisotropic thermal conductance at the junction. As the topological defect locations were horizontally (along branches 1 and 3) and vertically (along branches 2 and 4) asymmetric, the thermal resistances at the junctions along the branches may have been different. A previous study [29] indicated that defects in series have lower thermal resistance than those in parallel. When the asymmetric junction resistance is considered (Figure 6b), the temperature difference between the hot and cold regions (\(\Delta T\)) (hot region placing at branch 1) can be expressed as follows:  
\[ \begin{split}\Delta T=T_{hot}-T_{cold}=I_{1}(\Omega_{1}+\Omega_ {CNT})+I_{2}(\Omega_{2}+\Omega_{CNT})\\ =I_{1}(\Omega_{1}+\Omega_{CNT})+I_{3}(\Omega_{3}+\Omega_{CNT})=I _{1}(\Omega_{1}+\Omega_{CNT})+I_{4}(\Omega_{4}+\Omega_{CNT}),\end{split}  \tag{2}\]  
where \(\Omega_{CNT}\) is the thermal resistance of the pristine CNT branch and \(I_{i}\) is the heat current at branch \(i\). The following expression is obtained:  
\[I_{2}(\Omega_{2}+\Omega_{CNT})=I_{3}(\Omega_{3}+\Omega_{CNT})=I_{4}(\Omega_ {4}+\Omega_{CNT}) \tag{3}\]  
As the heat currents of branches 2 and 4 were larger than that of branch 3 (\(I_{2,4}>I_{3}\)), we deduced that the thermal resistance at the junction satisfied the following expression: \(\Omega_{2,4}<\Omega_{3}\). However, a contradictory result (i.e., \(\Omega_{1,3}<\Omega_{2}\)) was obtained when we considered the heat current situation in which a hot region was created at branch 4. Our simulation results cannot be explained using an account of conventional thermal circuits with or without the consideration of thermally anisotropic junctions.