Over the last 12 years, the possible existence of a tetrahedratic mesophase, involving a third-rank orientational order parameter and no positional order, has been addressed theoretically and predicted in some cases; no experimental realizations of a purely tetrahedratic phase are known at the time being, but various pieces of evidence suggest that interactions of tetrahedral symmetry do play a significant role in the macroscopic properties of mesophases resulting from banana-shaped (bent-core) mesogens. We address a very simple tetrahedratic mesogenic lattice model, involving continuous interactions; we consider particles possessing
Td symmetry, whose centers of mass are associated with a three-dimensional simple-cubic lattice; the pair potential is taken to be isotropic in orientation space and restricted to nearest-neighboring sites; we let the two orthonormal triads {
u
,
=1,2,3} and {
v
,
=1,2,3} define the orientations of a pair of interacting particles; we let the unit vectors
u be combined to yield four unit vectors {
ej,
j=1,2,3,4}, arranged in a tetrahedral fashion; we let the unit vectors
v be similarly combined to yield the four unit vectors {
fk,
k=1,2,3,4}; and finally we let
hjk=(
ej·
fk). The interaction model studied here is defined by the simplest nontrivial (cubic) polynomial in the scalar products
hjk, consistent with the assumed symmetry and favoring orientational order; it is, so to speak, the tetrahedratic counterpart of the Lebwohl-Lasher model for uniaxial nematics. The model was investigated by molecular field (MF) theory and Monte Carlo simulations; MF theory predicts a low-temperature, tetrahedrically ordered phase, undergoing a second-order transition to the isotropic phase at higher temperature; on the other hand, available theoretical treatments point to the transition being driven first order by thermal fluctuations. Simulations showed evidence of a first-order transition.
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