Stability of the columnar and smectic phases of length-bidisperse parallel hard cylinders

Abstract

The effect of length-bidispersity on the stability of nematic, smectic and columnar phases of rod-like particles is studied in the perfect alignment limit using Onsager's second virial theory. The rod-like particles are modelled as hard cylinders of equal diameters (D) but different lengths (). Three different smectic structures are observed: (i) the conventional smectic (S1) phase, where both components accommodate in the same layer; (ii) the microsegregated smectic (S2) phase, which can be considered as an alternation of fluid layers rich in short and long rods, respectively; and (iii) two layers of short rods accommodate inside one layer of long rods, which gives the third smectic (S3) structure. Due to the inefficient packing of the short and long rods into a layered structure along the symmetry axes of the rods, the smectic phase is destabilised with respect to nematic and columnar phases upon mixing the short and long components. With decreasing length ratio () the smectic phase is destabilised with respect to the nematic phase at compositions rich in short rods and two forms of smectic phases, namely S1 and S2, take place in alternation. The alternation of the structure is the consequence of the minimisation of the number of overlapping layers of the short rods with one long rod. In mixtures rich in long rods, the short and long rods are in the same layer up to l = 0.39, while the short rods can accommodate into the interstitial region of long rods for l < 0.39 and the system forms a S2 phase. The S3 phase is observed in the range 0.57 < l < 0.39 and is due to the efficient packing of two layers of short rods inside one smectic layer of long rods. Our theoretical predictions for the three smectic structures are in close agreement with the smectic phase behaviour of a binary mixture of short and long helical polysilanes (Okoshi et al., Macromolecules 42, 3443 (2009)). It seems reasonable that the helical polysilanes can be considered as an ideal system for testing the hard-body theories. Finally, it is interesting that the stabilisation process of the columnar phase with respect to the smectic ordering with decreasing length ratio turns over at l = 0.3.