  We discuss theoretically, using density-functional theory, the phase
stability of nematic and smectic ordering in a suspension of platelets of the
same thickness but with a high polydispersity in diameter, and study the
influence of polydispersity on this stability. The platelets are assumed to
interact like hard objects, but additional soft attractive and repulsive
interactions, meant to represent the effect of depletion interactions are also
considered. The aspect (diameter to thickness) ratio is taken to be very high,
in order to model solutions of mineral platelets recently explored
experimentally. In this regime a high degree of orientational ordering occurs;
therefore the model platelets can be taken as completely parallel and are
amenable to analysis via a fundamental-measure theory. Our focus is on the
nematic vs. smectic phase interplay, since a high degree of polydispersity in
diameter suppresses the formation of the columnar phase. When interactions are
purely hard, the theory predicts a continuous nematic-to-smectic transition,
regardless of the degree of diameter polydispersity. However, polydispersity
enhances the stability of the smectic phase against the nematic phase.
Predictions for the case where an additional soft interaction is added are
obtained using mean-field perturbation theory. In the case of the one-component
fluid, the transition remains continuous for repulsive forces, and the smectic
phase becomes more stable as the range of the interaction is decreased. The
opposite behaviour with respect to the range is observed for attractive forces,
and in fact the transition becomes of first order below a tricritical point.
Also, for attractive interactions, nematic demixing appears, with an associated
critical point. When platelet polydispersity is introduced the tricritical
temperature shifts to very high values.
