Structure factor of fluctuating interfaces: From liquid surfaces to suspended graphene

We obtain the density-density correlation structure in molecular dynamics (MD) simulations of graphene, and analyze it within the capillary wave theory (CWT), developed for fluid surfaces, to describe the thermal corrugations of the graphene sheet with a wave-vector-dependent surface tension $\ensuremath{\gamma}({q}_{x})$. The density correlation function (from the atomic positions) is compared with the theoretical prediction by Bedeaux and Weeks (BW), within the CWT, in terms of $\ensuremath{\gamma}({q}_{x})$ and the density profile. The agreement is very good, even for relatively large ${q}_{x}\ensuremath{\approx}0.2{\AA{}}^{\ensuremath{-}1}$, and with very little role for the correlation background, which sets an important difficulty for liquid surfaces. We present and test a generic prediction for the structure factor $S({q}_{x},{q}_{z})$ from $\ensuremath{\gamma}({q}_{x})$, that contains and goes beyond the classical asymptotic expression, developed by Sinha, for the analysis of x-ray surface scattering. We compare our prediction with the formula used in the interpretation of experimental data, that assumes a direct relationship between $\ensuremath{\gamma}({q}_{x})$ and the correlation structure for the same wave vector ${q}_{x}$. That relation is exact only for the first (Wertheim's) term of the BW series, and we use our results to test the accuracy of the function $\ensuremath{\gamma}({q}_{x})$ estimated through that method.