## 3 Graphene Folding  
Folding of graphene has often been observed in high-resolution electron microscopy images [23; 24; 25]. It has been shown by electron nano-diffraction that graphene folds preferentially along either the armchair or the zigzag direction [26]. The folded sheet forms a bilayer with continuous curved connection. The elastic energy in the curved fold is a penalty that can be more then balanced by the Van der Waals adhesion of the overlapping flat layers. When this happens, the adhesion energy acts as a driving force to increase the area of the overlap. In the absence of pinning defect, the system will evolve spontaneously until the overlapping area reaches a maximum. With a simple dimensional reasoning, one may estimate the average curvature radius of the folded edge to be \(R\approx\sqrt{D/\sigma}\), where\(D\) is the flexural rigidity of graphene (1.6 eV) and \(\sigma\) is the interlayer binding energy. In the case of graphite, the Van der Waals cohesion can be measured accurately from the cleavage energy (the energy needed to separate two parts of graphite along a basal plane): \(\sigma_{\text{Gr}}=0.37\,\text{J}/\text{m}^{2}\)[27]. Experimentation is much more difficult to conduct for bilayer graphene. Electronic-structure calculations yield dispersed values that may depend strongly on the particular AB, AA, or Moire stacking [28]. To circumvent the problem, we have used a relationship demonstrated in the Appendix within the Girifalco model [29]: \(\sigma+\sigma_{\text{Gr}}=hc_{33}/20\), where \(C_{33}\) is the elastic constant of graphite along the \(c\) axis and \(h\) is the interlayer distance. Using the experimentally-measured value of \(\sigma_{\text{Gr}}\)[27] and \(C_{33}=38.7\,\text{GPa}\)[30], we obtain \(\sigma=0.28\,\text{J}/\text{m}^{2}\) for the graphene-graphene interaction (see also Table 2). The folding mechanism can be examined with a simple continuous-mechanical model [24; 26; 31; 32; 33]. Assuming a graphene ribbon of width \(L\) that is folded back along a line perpendicular to its length, neglecting the in-plane strain energy, we simply have to minimize the difference between the bending energy and the Van der Waals energy  
\[\frac{\Delta E}{L}=\frac{D}{2}\int_{\mathcal{C}}\kappa^{2}\,\text{d}s-\sigma \left[l_{0}-\int_{\mathcal{C}}\,\text{d}s\right] \tag{2}\]  
where \(D\) is the flexural rigidity, \(\sigma\) is the interlayer binding energy, \(\mathcal{C}\) is the curve drawn by the folded edge (see Figure 1), \(l_{0}\) is the initial length of this curve, and \(\kappa\) is the local curvature. During the evolution of the folding, the curve changes in such a way that its length \(l=\int_{\mathcal{C}}\,\text{d}s\) decreases, thereby increasing the overlapping area and then the Van der Waals adhesion. At the same time, the bending energy increases until an equilibrium is reached. We have a variational problem to solve [25]. This problem can be further simplified by assuming an a-priori mathematical form of the curve \(\mathcal{C}\), evaluating the right-hand side of Equation (2) and minimizing the result with respect to the parameter(s) on which the curve depends. We have chosen a one-parameter racket-like curve whose expression in polar coordinates \((\rho,\varphi)\) is  
\[\rho(\varphi)=\frac{h}{4\cos(\alpha\pi/2)}\,\frac{\cos(\alpha\varphi/2)}{\cos (\varphi/2)}\quad-\pi<\varphi<+\pi \tag{3}\]  
where \(h\) is the Van der Walls spacing between two graphene planes, \(\alpha\) is a free parameter in the interval \((0,1)\). The shape of the curve is shown in Figure 1b. \(\varphi=0\) corresponds to the apex of the folded edge where the local curvature radius is \(R=h/[\cos(\alpha\pi/2)(\alpha^{2}+3)]\). The initial length \(l_{0}\) plays no role in the minimization of Equation (2), because it is a constant term. For mathematical reasons, \(l_{0}\) was chosen as the length of the curve defined by Equation (3) for \(\alpha=0\).