## 4 Properties of GR
### Thermal Properties  
The thermal studies were analyzed with the help of various equations and theories such as the Green-Kubo formula with the Boltzmann transport equation, nonequilibrium molecular dynamics, NEGF, and equilibrium molecular dynamics [141; 144; 145; 146; 147], GR shows a much lower value of thermal conductivity than what? Previously studied work [141; 144; 145; 146; 147] shows that at room temperature, GR has a thermal conductivity of 18-82 W/m K while G shows \(\sim\)1 \(\times\) 10\({}^{3}\) W/m K [146; 147]. It has been noticed that as the value of \(n\) in GR-n increases, thermal conductivity decreases [145; 147]. For graphyne-10 the value of thermal conductivity reaches 8 W/m K [145; 147] at room temperature. The behavior of the lower thermal conductivity of GR was studied with phonon normal mode analysis [145; 147].  
From Figure 19a, it can be seen that due to the combination of linkage atoms to lower frequency phonons, this combination dominates the heat transfer against the atoms present in the hexagonal ring, and as a result, lower thermal conductivity values of GR are observed [144]. The other reason is that due to these combinations, there is a mismatch observed between the hexagonal ring and linkage vibrations and a low value of energy transferred from one part to another. By increasing the linkage length, an enhancement is observed in mismatch vibrations [144; 145; 146], and a decrement is observed in phonon vibrational density because this density overlaps with the linkage and ring [147]. A higher linkage length causes lower values of thermal conductivity. It has been observed that some low-frequency coherent phonons (f \(\sim\) 0 THz) produce the same effect as linkage and ring atoms produced [147]. At lower temperatures (\(<\)30 K), the coherent phonons due to excitation produced a greater effect than that of delocalized low-frequency modes in G [145]. The same effect of dependence on linkage is observed in GR nanotubes (GRNT) [148]. The GRNT shows weak dependence on orientation, but in the case of GR nanoribbons (GRNR), because of the edge effects, the orientation can affect the thermal conductivity of GRNR. GRNR demonstrated that the thermal conductivity of the zigzag NR is much smaller than that of the armchair NR even though both have the same width (Figure 19b,c). The anisotropy factor of thermal conductivity is insensitive to temperature. In narrow GRNRs 30% more thermal conductivity was observed as compared to NRs that have a 5-nm width. This difference between GRNRs and GRNT's exhibited that the edge effect has an important impact on thermal conductance.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Name** & **Direction** & **\(\mu_{\text{h}}\) (\(\times\)10\({}^{4}\) cm\({}^{2}\)/Vs)** & **\(\mu_{\text{e}}\) (\(\times\)10\({}^{4}\) cm\({}^{2}\)/Vs)** \\ \hline Graphdiyne & X & 1.91 & 17.22 \\ \hline  & Y & 1.97 & 20.81 \\ \hline
6,6,12-Graphyne & X & 12.29 & 24.48 \\ \hline  & Y & 42.92 & 54.10 \\ \hline \(\beta\)-Graphyne & X & 0.856 & 0.798 \\ \hline  & Y & 1.076 & 0.892 \\ \hline \(\alpha\)-Graphyne & X & 2.960 & 2.716 \\ \hline  & Y & 3.316 & 3.3277 \\ \hline \end{tabular}
\end{table}
Table 3: _Cont._