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Interplay between columnar and smectic stability in suspensions of polydisperse colloidal platelets

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 $δ_l$ and $δ_h$. We obtain the phase behavior of the model, including nematic, smectic and columnar phases and its dependence with the two polydispersities $δ_l$ and $δ_h$. When $δ_l>δ_h$ we observe that the smectic phase stabilises first with respect to the columnar. If $δ_h>δ_l$ 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 $δ_l-δ_h$ 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.