Statistical Mechanics of Two-Dimensional Materials; From Biological Membranes to Graphene
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2D materials are fascinating for numerous reasons. Their geometrical and mechanical characteristics along with other associated physical properties have opened up fascinating new application avenues ranging from electronics, energy harvesting, biological systems among others. Due to the 2D nature of these materials, they are known for their unusual flexibility and the ability to sustain large curvature deformations. Further, they undergo noticeable thermal fluctuations at room temperature. In the following, we highlight both the characteristics and implications of thermal fluctuations in 2D materials and discuss the following problems in biological physics and material science: (i) The minimum electric field that can be detected by a biological membrane: Using a nonlinear continuum electromechanical model, and methods of statistical mechanics, we developed a variational approximation to analytically obtain the benchmark results for model fluid membranes as well as physically reasonable estimates of the minimum electric field that can be detected by a biological membrane. (ii) Thermal fluctuations of vesicles and nonlinear curvature elasticity---Implications for sized-dependent renormalized bending rigidity and vesicle size distribution: In this work, we discuss the statistical mechanics of closed membranes (vesicles) incorporating both constitutive and geometrical nonlinearities. Our closed-form results may also be used to determine nonlinear curvature elasticity properties from either experimentally measured fluctuation spectra or microscopic calculations such as molecular dynamics. (iii) Fluctuations and effective bending stiffness of solid membranes within nonlinear elasticity: The study of thermal fluctuations of graphene is rendered rather complicated due to the necessity of accounting for geometric deformation nonlinearity in its deformation. Coupling of stretching and flexural modes leads to a highly anharmonic elastic Hamiltonian. In this study, using a variational perturbation method, we present a ”mechanics-oriented” novel treatment of the thermal fluctuations of graphene, fully accounting for deformation nonlinearities, and evaluate their effect on the effective bending stiffness. (iv) The quest for the determination of the Gaussian modulus—exploiting membrane edge fluctuations: In this work, recognizing that the Gaussian modulus plays a non-trivial role in the fluctuations of a membrane edge, we derive closed-form expressions for edge fluctuations. Combined with atomistic simulations, we use the developed approach to extract Gaussian modulus of graphene.