  Using density-functional theory, we have analyzed the phase behavior of
binary mixtures of hard rods of different lengths and diameters. Previous
studies have shown a strong tendency of smectic phases of these mixtures to
segregate and, in some circumstances, to form microsegregated phases. Our focus
in the present work is on the formation of columnar phases which some studies,
under some approximations, have shown to become thermodynamically stable prior
to crystallization. Specifically we focus on the relative stability between
smectic and columnar phases, a question not fully addressed in previous work.
Our analysis is based on two complementary perspectives: on the one hand, an
extended Onsager theory, which includes the full orientational degrees of
freedom but with spatial and orientational correlations being treated in an
approximate manner; on the other hand, we formulate a Zwanzig approximation of
fundamental-measure theory on hard parallelepipeds, whereby orientations are
restricted to be only along three mutually orthogonal axes, but correlations
are faithfully represented. In the latter case novel, complete phase diagrams
containing regions of stability of liquid-crystalline phases are calculated.
Our findings indicate that the restricted-orientation approximation enhances
the stability of columnar phases so as to preempt smectic order completely
while, in the framework of the extended Onsager model, with full orientational
degrees of freedom taken into account, columnar phases may preempt a large
region of smectic stability in some mixtures, but some smectic order still
persists.
