  The phase behavior of a model suspension of colloidal polydisperse platelets
is studied using density-functional theory. Platelets are modelled as parallel
rectangular prisms of square section $l$ and height $h$, with length and height
distributions given by different polydispersities $\delta_l$ and $\delta_h$. We
obtain the phase behavior of the model, including nematic, smectic and columnar
phases and its dependence with the two polydispersities $\delta_l$ and
$\delta_h$. When $\delta_l>\delta_h$ we observe that the smectic phase
stabilises first with respect to the columnar. If $\delta_h>\delta_l$ we
observe the opposite behavior. Other more complicated cases occur, e.g. the
smectic stabilises from the nematic first but then exists a first-order
transition to the columnar phase. Our model assumes plate-rod symmetry, but the
regions of stability of smectic and columnar phases are non-symmetric in the
$\delta_l-\delta_h$ plane due to the different dimensionality of ordering in
the two phases. Microsegregation effects, i.e. different spatial distribution
for different sizes within the periodic cell, take place in both phases.
