Theory of bent-core liquid-crystal phases and phase transitions

We study phases and phase transitions that can take place in the recently discovered bow-shaped or bent-core liquid-crystal molecules. We show that to completely characterize phases exhibited by such bent-core molecules a third-rank tensor $T^{\mathrm{ijk}}$ order parameter is necessary in addition to the vector and the nematic (second-rank) tensor order parameters. We present an exhaustive list of possible liquid phases, characterizing them by their space-symmetry group and order parameters, and catalog the universality classes of the corresponding phase transitions that we expect to take place in such bent-core molecular liquid crystals. In addition to the conventional liquid-crystal phases such as the nematic phase, we predict the existence of other liquid phases, including the spontaneously chiral nematic ${(N}_T{+2)}^{*}$ and chiral polar ${(V}_T{+2)}^{*}$ phases, the orientationally ordered but optically isotropic tetrahedratic T phase, and a nematic $N_T$ phase with $D_{2d}$ symmetry that is neither uniaxial nor biaxial. Interestingly, the isotropic-tetrahedratic transition is continuous in mean-field theory, but is likely driven first order by thermal fluctuations. We conclude with a discussion of smectic analogs of these phases and their experimental signatures.