Herringbone ordering and lattice distortions in a planar-molecule model for Langmuir monolayers

A model of planar molecules, made up of ``atoms'' interacting by Lennard-Jones potentials and arranged to mimic the cross section of alkyl chains, is used to study the problem of backbone plane ordering in Langmuir monolayers. It is shown that two minima of the interaction energy are reached if molecules lie on the sites of a centered rectangular lattice in a herringbone configuration with two different dihedral angles. These orientationally ordered phases can be related to the so-called herringbone and pseudoherringbone structures, whose lattice distortions qualitatively agree with those determined by means of grazing incidence x-ray diffraction experiments on Langmuir monolayers. A third energy minimum is obtained for a configuration of parallel molecules on an oblique lattice, which has also been observed in some experiments. The competition between the three phases is investigated, upon varying geometric parameters of the model molecules and surface pressure. The effect of temperature is analyzed in a mean field approximation, by taking into account the orientational entropy contribution on a lattice system with variable unit cell parameters. In this framework the transition to an orientationally disordered phase is also pointed out.