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 $h(\mathbf{q})$ in Fourier space with zero mean and variance $<|h(\mathbf{q}){|}^2>\sim q^{-d_h}$ and show that in the linear response regime the mechanical properties depend dramatically on the system size $L$ for $d_h\ge 2$. 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 modulii 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.

27 pages, 10 figures, submitted to Physical Review E

References in corpus (2)

• The Structure of Physical Crystalline Membranes within the Self-Consistent Screening Approximation
• Self-Consistent Screening Approximation for Flexible Membranes: Application to Graphene

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