Uniaxial and biaxial nematic phases of banana-shaped molecules and the effects of an external field

A mean field theory is introduced to describe uniaxial and biaxial nematic phases of banana-shaped liquid crystalline molecules. From a geometric point of view, we derive the tensor order parameter of the banana-shaped molecules as a function of a bend angle and four orientational order parameters. Using the nematic free energy of the Maier-Saupe type, we calculate the phase diagram on the temperature $-$bend angle ( $0<{\theta }_B<\pi$) plane and find three uniaxial nematic ( $N_U^i$) and three biaxial nematic ( $N_B^i$) phases, where $i(=a,b,c)$ is the primary axis of the orthogonal molecular axis. We also derive the approximate formulas of a nematic-isotropic phase transition ( $NIT$) and a biaxial-uniaxial phase transition ( $BUT$) temperature as a function of the bend angle. We also discuss the effects of the external field on the $NIT$ temperature and find the negative dependence of the $NIT$ temperature on the external field.