Graphene Helicoid: The Distinct Properties Promote Application of Graphene Related Materials in Thermal Management

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1 Supporting Information Graphene Helicoid: The Distinct Properties Promote Application of Graphene Related Materials in Thermal Management Haifei Zhan 1,2, Gang Zhang 3,*, Chunhui Yang 1, and Yuantong Gu 2,** *Corresponding author. (Gang Zhang) **Corresponding author. (Yuantong Gu) 1 School of Computing, Engineering and Mathematics, Western Sydney University, Locked Bag 1797, Penrith NSW 2751, Australia 2 School of Chemistry, Physics and Mechanical Engineering, Queensland University of Technology (QUT), Brisbane QLD 4001, Australia 3 Institute of High Performance Computing, Agency for Science, Technology and Research, 1 Fusionopolis Way, Singapore , Singapore S1. Thermal conductivity of GH As expected, GH exhibits a much better thermal conductivity than MLG. As illustrated in Figure S1a, the absolute difference of κ between GH and MLG shows an increasing tendency when the thickness increases, i.e., κ increases from 0.07 to 0.24 W/mK. By comparing with the κ of MLG, and the κ of GH exhibits more than 30% average enhancement than that of MLG. Figure S1b shows the thermal conductivity of GH as a function of its width, from which κ shows an almost independent relationship with the width. The GH has a uniform thickness or effective turn number of 6, and the width changing from ~0.28 to ~1.14 nm. Two groups have been studied, including group one with varying inner radius r but fixed outer radius R, and verse versa in group two. 1

2 Figure S1. Thermal conductivity of GH. (a) The absolute difference of κ, i.e., κ = κ GH -κ GNR (upper panel), and the relative difference of κ, i.e., η= κ/κ GNR (bottom panel) between GH and MLG.; (b) The κ of GH as a function of width. Here, R or r equals C represents the GH has a constant outer or inner radius with varying width. S2. Logarithm relationship between heat current and sample thickness In line with the thermal conductivity, more heat is transferred in the GH compared with its MLG counterpart (as shown in Figure S2). The heat current decreases when the thickness or layer number increases. Specifically, there is a good power-law relationship between the heat current and the sample thickness of the GH structure. In principle, this feature is identified as an evidence of diverged thermal conductivity in one-dimensional lattice model, as reported earlier by Li et al. 1 Figure S2. The logarithm relationship between heat current (Q) and sample thickness (L) of the GH and MLG. S3. Thermal conductivity of graphene nanoribbon The thermal conductivity of graphene nanoribbon (GNR) is calculated with varying length (the GNR has a width of around 3.09 nm). As shown in Figure S3, κ exhibits a clear powerlaw relationship (i.e., κ ~ L β ) with length L for GNR, similar to that of GH. The estimated exponent is around

3 Figure S3. The power-law relationship between the thermal conductivity of GNR and its length (L). S4. Tensile and compressive deformation of GH and MLG For compressive/tensile deformation, a low constant velocity was applied to one end of the structure with the other end being fixed. Specifically, for compressive simulation, only one unit was taken as the fixed end or loading end. For tensile simulation, four adjacent units were grouped as the fixed end or loading end. The deformation was performed under the temperature of 300 K. Figure S4a and b show the force and strain energy as a function of compressive strain. The force is found to experience a clear reduction after the strain exceeds ~ 5%, which is resulted from the significant interlayer shift induced by the compression. Figures S4c and d show the force and strain energy as a function of tensile strain. Evidently, the MLG structure fails easily as there is only vdw interactions in the cross-plane direction, and the failure strain is only about 10%. In comparsion, the GH structure has large elastic deformation with strain exceeding 1200%. 3

4 Figure S4. Cross-plane deformation of GH and MLG. (a) The force, and (b) the strain energy as a function of the compressive strain; (c) the force, and (d) the strain energy as a function of the tensile strain. The strained GH structure was relaxed with fixed boundary conditions for 200 ps before thermal conductivity calculation (i.e., the two ends were fixed). As illustrated in Figure S5, the strain energy keeps a constant during the relaxation process. Such observation not only suggests that the applied strain rate is low enough to allow sufficient tensile deformation in the GH structure, but also indicates that the obtained strained structure is at an energy minimum status. Figure S5. Strain energy of GH as a function of relaxation time. S5. VDOS of GH and MLG under cross-plane strain Two unit cells located at the middle of the structures are selected for the VDOS calculation. As compared in Figure S6, we can clearly see the blue and red shift of the low-frequency phonon modes under compressive and tensile strain, respectively. For either GH or MLG structure, the Raman G-peak (~ 49 THz) is split due to the non-periodic boundary conditions. Figure S6. VDOS of GH and MLG under different cross-plane strain values. S6. Temperature profiles of GH at different tensile strain Different temperature profiles are observed in the deliminated and un-deliminated regions of the GH, as illustrated in Figure S7. 4

5 Figure S7. Temperature profile of GH at different tensile strain. (a) Strain of %, and (b) strain of 242%. Inset show the corresponding atomic configurations. References 1 Li, B., Wang, L. & Hu, B. Finite thermal conductivity in 1D models having zero Lyapunov exponents. Phys. Rev. Lett. 88, , (2002). 5