Periodic systems of frustrated fluid films and « bicontinuous » cubic structures in liquid crystals

Abstract

We consider periodic organizations of two fluid media separated by interfaces in which interactions between the two media, normal to the interfaces, maintain constant distances between interfaces and constraints within each medium, parallel to the interfaces, control interfacial curvatures. The structures must therefore conciliate the constant interfacial distances and curvatures imposed by the thermodynamical parameters of the systems. This is a purely geometrical problem whose solutions constitute the foundation of the structures of periodic systems of fluid films. When the interfacial curvature is null, the obvious solution is a periodical stacking of parallel layers. When the curvatures are not null, adjacent interfaces must have curvatures with the same concavities relative to the two media because of the symmetry of the layers and the constant distances between them can no longer be maintained if the lamellar geometry is kept. This is a typical case of frustration which implies a change of structure. We have recently proposed to look for the solutions to this frustration following a geometrical approach which provides solutions whose topologies are similar to those of liquid crystalline phases and which leads to consider the latter as structures of disclinations. We now develop this approach to study the particular case of solutions with bicontinuous topology, i.e. where a film without self-intersection built by one medium separates two labyrinthine nets built by the second medium. We demonstrate that they correspond to configurations in which the film is supported by ordered hyperbolic surfaces having topologies and symmetries similar to those of three infinite periodic minimal surfaces calculated by mathematicians. We discuss the relation between these ordered bicontinuous solutions and the structures determined for cubic liquid crystalline phases, either in amphiphilic systems (Qα) where the two media are paraffinic and polar, or in mesogenic ones (SmD) where the two media are aliphatic and aromatic.