Entropic pressure on fluctuating solid membranes

Biological and crystalline membranes exhibit noticeable fluctuations at room temperature due to their low bending stiffness. These fluctuations have a significant impact on their overall mechanical behavior and interactions with external objects. When two membranes come into proximity, they mutually suppress each other's fluctuations, leading to a repulsive force that plays a pivotal role in the mechanical behavior of these membranes. From the mechanics point of view, crystalline membranes are modeled as solid membranes with inherent shear resistance, whereas biological membranes are commonly described as fluidic entities without shear resistance. Under this premise, the entropic force between two fluctuating biological membranes is proposed to scale as p∝1/d³, where d is the intermembrane distance. Yet, there are numerous instances where these membranes display shear resistance and behave akin to solid membranes. In this paper, we develop a statistical mechanics model within nonlinear elasticity to study the entropic force acting on a confined, fluctuating solid membrane. We demonstrate that, due to the nonlinear elasticity of solid membranes, the entropic force scales differently compared to that of fluid membranes. Our predictions align well with the results obtained from molecular dynamics simulations involving graphene, a representative of a solid membrane, confined between two rigid walls.