Elasticity of two-dimensional crystalline monolayers of fatty acid salts at an air-water surface

The elastic properties of organic-inorganic two-dimensional crystals floating at the water surface have been fully characterized by grazing incidence X-ray diffuse scattering and high-resolution diffraction. We show that the strong interaction between the organic molecules and the inorganic divalent cations is enough for these nm thick crystals to behave like true solids, with a residual tension of 1 x 10(-4)-10(-3) N m(-1). Their bending rigidity is renormalized as kappa(q) proportional to q(-eta k) with eta(k) = 0.25 +/- 0.07 and a microscopic value approximate to 100 k(B)T at q = 1 x 10(9) m(-1). The in-plane elastic constants behave like q(eta u) with eta(u) = 1.41 +/- 0.2, obeying the scaling relation eta(u) = 2 - 2 eta(k). These results are consistent with a long-range phonon-mediated interaction between out-of-plane fluctuations but the values of the exponents differ from those generally obtained in numerical simulations.