# A Thermal Transport Study of Branched Carbon Nanotubes with Cross and T-Junctions
## 2 Methodology  
All NEMD simulations were conducted using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) package [32]. The adaptive intermolecular reactive bond order (AIREBO) potential [33] was employed in this study to describe the interatomic interaction between carbon atoms. T-junctions and cross-junctions were created using the thermal welding method proposed in [17]. Some atoms were removed intentionally from one pristine 4 nm (6, 6) CNT to create a crossing region. The atoms were removed such that the topological requirement described by the Euler polyhedral formula was satisfied [28]. This formula regulates the numbers of vertices, faces, and edges in a convex three-dimensional polyhedron. The number of pentagons (\(n_{5}\)), heptagons (\(n_{7}\)), and octagons (\(n_{8}\)) at a junction must satisfy the following equality: \(2n_{8}+n_{7}-n_{5}=6\). A 1.5 nm (6, 6) CNT was placed vertically with respect to the crossing region of the defect-containing 4 nm (6, 6) CNT. Subsequently, the two aforementioned CNTs were moved closer and simultaneously heated up to 4000 K to perform thermal welding. A high temperature was adopted to accelerate the creation of interlinks and surface reconstruction. Finally, the T-junction model was cooled to 300 K.  
Through the sequential removal of carbon atoms, topological defects in the junction area could be quantitatively controlled to satisfy the Euler polyhedral formula. Thus, T-junctions with two atomic configurations containing different topological defects were constructed, as illustrated in Figure 1a,b. One atomic configuration, which is hereafter referred to as T-junction 1, contained two heptagon rings (one in the back) and two octagons in the corner region, as depicted in Figure 1a. The second atomic configuration, which is hereafter referred to as T-junction 2, contained four heptagon rings (two in the back), two pentagon rings (one in the back), and two octagons in the corner region, as depicted in Figure 1b. Cross-junctions were constructed by further removing atoms from the bottom region of the T-junction model, which is hereafter referred to as the defect-containing T-junction, as illustrated in Figure 2. For symmetry, the atoms removed from the bottom of the aforementioned model were the same as those in the top crossing region of this model. A 1.5 nm (6, 6) CNT was assembled with the defect-containing T-junction model by using the thermal welding process to form cross-junctions, as depicted in Figure 3a,b. The cross-junction models depicted in Figure 3a,b were called cross-junctions 1 and 2, respectively. The cross-junction models had twice the number of topological defects that their T-junction counterparts did. Figure 1: Atomic configurations of the two T-junctions: (**a**) T-junction 1 and (**b**) T-junction 2 (P, H, and O represent pentagons, heptagons, and octagons, respectively). Figure 2: Defect-containing T-junction model. Equal-length (6, 6) CNTs were connected to T-junctions and cross-junctions to form branched CNTs. To investigate the effect of branch length on the heat flow inside the branched CNTs, various branch lengths between 25 and 200 nm were considered for the simulations. The entire system was relaxed using a canonical (NVT) ensemble at the junction region under a temperature of 300 K. The time step used for all simulations was 1 fs. To release the stress inside branched CNTs, the atoms on each individual branch were allowed to move only along the axial direction of the CTNs under the microcanonical (NVE) ensemble and the equilibrating process was run for 1 ns. Through the aforementioned relaxation procedure, both longitudinal and transverse stresses inside the branches were eliminated. Relaxing the structure was crucial because the residual stress from the initial model can influence the material properties and can thus affect the heat flow [30]. The temperature setups of the cross and T-junctions are displayed in Figure 4a,b, respectively.