  Using scaled-particle theory for binary mixtures of two-dimensional hard
particles with rotational freedom, we analyse the stability of nematic phases
and the demixing phase behaviour of a variety of mixtures, focussing on cases
where at least one of the components consists of hard rectangles or hard
squares. A pure fluid of hard rectangles may exhibit, aside from the usual
uniaxial nematic phase, an additional (tetratic) oriented phase, possessing two
directors, which is the analogue of the biaxial or cubatic phases in three-
dimensional fluids. There is computer simulation evidence that the tetratic
phase might be stable with respect to phases with spatial order for rectangles
with low aspect ratios. As hard rectangles are mixed with other particles not
possessing stable tetratic order by themselves, the tetratic phase is
destabilised, via a first- or second-order phase transition, to uniaxial
nematic or isotropic phases; for hard rectangles of low aspect ratio tetratic
order persists in a relatively large range of volume fractions. The order of
these transitions depends on the particle geometry, dimensions and
thermodynamic conditions of the mixture. The second component of the mixture
has been chosen to be hard discs or disco-rectangles, the geometry of which is
different from that of rectangles, leading to packing frustration and demixing
behaviour, or simply rectangles of different aspect ratio. These mixtures may
be good candidates for observing thermodynamically stable tetratic phases in
monolayers of hard particles. Finally, demixing between fluid
(isotropic--tetratic or tetratic--tetratic) phases is seen to occur in mixtures
of hard squares of different sizes when the size ratio is sufficiently large.
