Entropy-Driven Mechanics of Crystalline and Biological Membranes

Abstract The relationship between continuum concepts and the microscopic behavior of materials has long intrigued researchers in both the mechanics and physics communities. While continuum mechanics typically assumes a well-defined reference (undeformed) configuration, materials at the atomic scale are never truly static. Even solid materials experience continuous random deformations–known as thermal fluctuations–driven by ambient thermal energy. When these fluctuations become comparable to at least one characteristic length scale of a nanostructure, they can significantly impact its mechanical and physical properties. Examples of such nanostructures include crystalline membranes (commonly referred to as two-dimensional materials), which appear in various morphologies such as nanotubes, nanoribbons, and form the foundational elements of nanoscale metamaterials, kirigami/origami structures, nanocomposites, among others. Flexible nanostructures also play crucial roles in biological systems, including biological membranes, microtubules, actin filaments, and DNA. In this paper, we aim to provide an overview of the fundamental concepts underlying entropy-driven mechanics in flexible nanostructures, focusing on biological and crystalline membranes–two classes of systems where thermal fluctuations are particularly significant. We will review the current state of continuum mechanics modeling of fluctuating surfaces, highlighting key technical challenges, open questions, and future research directions. Although this article is extensive, it is not meant to serve as a comprehensive literature review. Instead, its goal is to introduce a broad audience from mechanics, materials science and cell mechanics to the core ideas of entropy-driven mechanics and to lay the groundwork for incorporating statistical mechanics into continuum modeling of flexible nanostructures.