## 4 Self - Folding of Graphene  
The previous section was devoted to graphene in free space--a situation that is not of direct interest. However, most of what has been predicted remains qualitatively correct for graphene deposited on a substrate. Of course, the shape of the folded sheet changes from that of Figure 1b to that of Figure 1a due to the underlying material [38]. The case of SiO\({}_{2}\) is very frequent: the driving force for folding is the difference \(\sigma_{\text{gr-gr}}-\sigma_{\text{gr-SiO}_{2}}\). This difference can be small and even negative according to recent accurate measurements: \(\sigma_{\text{gr-SiO}_{2}}=0.45\) J/m\({}^{2}\) for monolayer graphene and 0.31 J/m\({}^{2}\) for few-layer (2-5) graphene [40]. Accordingly, folding should be favorable for few-layer graphene, but not for monolayer graphene (see Table 2). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(D\) **(eV)** & \(\sigma\) **(J/m\({}^{2}\))** & \(h\) **(nm)** & \(m\) & \(\alpha\) & \(R\) **(nm)** & \(w_{b}\)/L **(eV/nm)** & \\ \hline
1.6 & 0.28 & 0.335 & 8.16 & 0.855 & 0.400 & 6.37 & a \\
1.4 & 0.23 & 0.335 & 8.60 & 0.861 & 0.413 & 5.44 & b \\
35.5 & 0.35 & 0.67 & 36.2 & 0.944 & 1.95 & 37.72 & c \\ \hline \hline \end{tabular} [a] this work; [b] nominal parameters of ref. [32]; [c] bilayer (see text).  
\end{table} 
Table 1: Variation of the dimensionless parameter \(m=D/\sigma h^{2}\), folding parameter \(\alpha\) (Equation 3), curvature radius \(R\) of the folded edge at its apex, and the bending energy per unit length \(w_{b}\) (first term in Equation 2) versus graphene parameters (flexural rigidity \(D\), interlayer binding energy \(\sigma\), inter-sheet distance \(h\)). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(n\) & **1** & **2** & **3** & \(\infty\) \\ \hline \(\sigma_{n,n}\) (J/m\({}^{2}\)) & 0.28 & 0.35 & 0.36 & 0.37 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Binding energy per unit area of an \(n\)-layer graphite sheet deposited on another \(n\)-layer graphite sheet for different values of \(n\). Nanoinendation of few-layer graphene on SiO\({}_{2}\) by a triangular AFM tip gives rise to interesting observations. When the tip has small-amplitude horizontal movements during the punch, three graphene ribbons are peeling and growing away from the edges of the indentation [41]. The driving force is the difference between the graphene-graphene and graphene-silica adhesion energies. While the fold moves away from its origin, the ribbon gets narrower so as to reduce the bending energy of the fold. The tapering angle \(\theta\) is given by the equation [41]  
\[\sin\frac{\theta}{2}=\frac{w_{b}}{G} \tag{5}\]  
where \(w_{b}\) is the bending energy per unit width of the folded region and \(G\) is the bond breaking energy introduced in Section 2. The value of \(\theta\) measured experimentally for bi- to tetra-layer graphene samples was \(12^{\circ}\)[41]. Assuming that the bond breaking energy scales linearly with the number of layers, one obtains \(G\approx 42\) eV/nm for a bilayer. One therefore needs \(w_{b}\approx 4.5\) eV/nm to obtain a tapering angle of \(12^{\circ}\) for a self-folded bilayer. This value is a factor of 8.3 smaller than that given in the last row of Table 1. However, the bending constant \(D\) of the self-folded bilayer could be around 3 eV--not 35 eV--if the atomic layers were bending without coupling, as predicted by molecular dynamics [38]. This would bring \(w_{b}\) close to the value estimated from the tapering angle. Instead of making a single fold, a monolayer graphene can scroll, but this transformation works only in solution with alcohol, which helps the graphene layer to detach from the substrate [42]. The gr-gr interlayer interaction is about a factor of two larger in a scroll than in a single fold, because there is an interaction on both faces of graphene in most part of the scroll. Then, Van der Waals interaction favors the scroll over the fold, the more so that the bending energy in a scroll is not a strong penalty compared to a folded sheet [5; 43].