Mechanical Properties of Warped Membranes

Abstract

We explore how a frozen background metric affects the mechanical properties of planar membranes with a shear modulus. We focus on a special class of “warped membranes” with a preferred random height profile characterized by random Gaussian variables h(q) in Fourier space with zero mean and variance \(⟨| h(q)|^2〉\sim q^{−d_h}\) and show that in the linear response regime the mechanical properties depend dramatically on the system size L for \(d_h\geq 2\). Membranes with \(d_h=4\) could be produced by flash polymerization of lyotropic smectic liquid crystals. Via a self-consistent screening approximation we find that the renormalized bending rigidity increases as \(\kappa R\sim L^{(d_h−2)/2}\) for membranes of size L, while the Young and shear moduli decrease according to \(Y_R,\mu R \sim L^{−(d_h−2)/2}\) resulting in a universal Poisson ratio. Numerical results show good agreement with analytically determined exponents.