Title: Distribution of ripples in graphene membrane

Abstract: Intrinsic ripples with various configurations and sizes were reported to affect the physical and chemical properties of 2D materials. By performing molecular dynamics simulations and theoretical analysis, we use two geometric models of the ripple shape to explore numerically the distribution of ripples in graphene membrane. We focus on the ratio of ripple height to its diameter (t/D) which was recently shown to be the most relevant for chemical activity of graphene membranes. Our result demonstrates that the ripple density decreases as the coefficient t/D increases, in a qualitative agreement with the Boltzmann distribution derived analytically from the bending energy of the membrane. Our theoretical study provides also specific quantitative information on the ripple distribution in graphene and gives new insights applicable to other 2D materials.