Liquid-crystalline blue phase III and structures of broken icosahedral symmetry

The structure of the liquid-crystalline blue phase III (BPIII) is still unknown and remains one of the mysteries of liquid-crystal physics. We take all icosahedral space-group symmetries of the reciprocal space for BPIII and study their thermodynamic stability within the frame of an extended de Gennes--Ginzburg--Landau free-energy expansion. The stability of the icosahedral structures is compared with that of the cholesteric phase and of the cubic blue phases. Strikingly, even though the extended model contains three extra parameters, we could not detect a region of parameter space where icosahedral structures are absolutely stable just below the isotropic phase.