Orientational dependence of the surface free energy of the L(alpha)-L(3) interface

Geometrical, topological, and energetical properties on an interface between the lamellar L(alpha) and the isotropic sponge phase L(3) are considered. The sponge phase shows a characteristic interbilayer separation d(3) that is larger than the interbilayer separation d(alpha) in the L(alpha) phase. This feature controls the experimentally observed phenomenon of orientational epitaxy (Quilliet, C.; Blanc, C.; Kleman, M. Phys. Rev. Lett. 1996, 77, 522) that keeps the lamellar L(alpha) bilayers tilted with respect to the interface. We discuss the concentration profile near the interface, the geometry of smooth matching between the bilayers of both phases and estimate the surface tension of the L(alpha)-L(3) interface at equilibrium orientation of layers. The energy needed to deviate the layers from the equilibrium orientation (''anchoring'' energy) is calculated using the Frenkel-Kontorova dislocation model. The model also predicts faceting, which is observed experimentally. Both the surface tension and anchoring energies are of the order of K/d(alpha), where K is the bend elastic constant. The role of layer undulation in tangential anchoring is discussed and illustrated experimentally. Finally, an outcome of the theory is that the orientation is related to the topological parameter of the sponge phase (coordination number of the passages introduced by Forte et al., J. Phys. (Paris) 1988, 49, 511).