  We perform extensive Monte Carlo simulations to investigate the phase
behaviour of colloidal suspensions of hard board-like particles (HBPs). While
theories restricting particle orientation or ignoring higher ordered phases
suggest the existence of a stable biaxial nematic phase, our recent simulation
results on monodisperse systems indicate that this is not necessarily the case,
even for particle shapes exactly in between prolate and oblate geometries,
usually referred to as self-dual shape. Motivated by the potentially striking
impact of incorporating biaxial ordering into display applications, we extend
our investigation to bidisperse mixtures of short and long HBPs and analyse
whether size dispersity can further enrich the phase behaviour of HBPs,
eventually destabilise positionally ordered phases and thus favour the
formation of the biaxial nematic phase. Not only do our results indicate that
bidisperse mixtures of self-dual shaped HBPs cannot self-assemble into biaxial
nematic phases, but they also show that these particles are not able to form
uniaxial nematic phases either. This surprising behaviour is also observed in
monodisperse systems. Additionally, bidisperse HBPs tend to phase separate in
coexisting isotropic and smectic phases or, at relatively large pressures, in a
smectic phase of mostly short HBPs and a smectic phase of mostly long HBPs. We
conclude that limiting the particle orientational degrees of freedom or
neglecting the presence of positionally ordered (smectic, columnar and crystal)
phases can dramatically alter the phase behaviour of HBPs and unrealistically
enlarge the region of stability of the biaxial nematic phase.
