Elasticity and excitations of minimal crystals

(Received 2 July 1991, revised 21 October 1991, accepted 22 November 1991)

Abstract
The elastic properties of the unusual crystals encountered in surfactant-rich solutions are investigated. Triply-periodic minimal surfaces provide a convenient frame-work for the understanding of such materials but, as is shown, degeneracy leads to vanishing elastic coefficients in the framework of the classical Helfrich energy. This degeneracy is lifted by higher-order corrections and by finite temperature effects. We show that, as a result, thermodynamic stability can be achieved at low levels of dilution but that with increasing dilution the P surface inevitably melts. The degeneracy also leads to an unusual collective excitation spectrum which has a smectic-like undulation dispersion, except at very long wavelengths where it becomes sound-like. The elastic moduli are found to have the same dependence on temperature and concentration as those of tethered stacked membranes and the shear moduli are shown to have a temperature and material independent ratio.