Thermodynamics of multilayer fluctuating graphene metamaterials

The rational design of graphene-based metamaterials with programmable thermal and mechanical responses hinges on a fundamental understanding of how collective fluctuations govern their macroscopic behavior. This work develops a unified statistical mechanics framework to decode the anomalous thermomechanical behavior of multilayer graphene membranes and metamaterials. Combining variational perturbation theory with molecular dynamics simulations, we demonstrate how the competition governs thermal expansion characteristics across different temperature regimes and interlayer spacings. The theoretical approach reveals distinct power-law scaling regimes characterized by critical exponents. Results show that in large separation regimes, thermal fluctuations induce significant nonlinear strain energy leading to negative thermal expansion, while in small separation regimes, the tension term becomes dominant. Molecular dynamics simulations provide atomic-scale verification, confirming that both 2D and 3D graphene structures exhibit negative thermal expansion originating primarily from out-of-plane displacements. The established framework bridges atomic-scale fluctuations with macroscopic thermomechanical response, offering powerful design principles for tailoring thermal expansion in graphene-based metamaterials and advancing the fundamental understanding of fluctuation-induced thermodynamics in such metamaterials.