  We apply a recently proposed density functional for mixtures of parallel hard
cylinders, based on Rosenfeld's fundamental measure theory, to study the effect
of length-polydispersity on the relative stability between the smectic and
columnar liquid crystal phases.To this purpose we derive from this functional
an expression for the direct correlation function and use it to perform a
bifurcation analysis. We compare the results with those obtained with a second
and a third virial approximation of this function. All three approximations
lead to the same conclusion: there is a terminal polydispersity beyond which
the smectic phase is less stable than the columnar phase. This result is in
agreement with previous Monte Carlo simulations conducted on a freely rotating
length-polydisperse hard spherocylinder fluid, although the theories always
overestimate the terminal polydispersity because the nematic-columnar phase
transition is first order and exhibits a wide coexistence gap. Both, the
fundamental-measure functional and the third virial approximation, predict a
metastable nematic-nematic demixing. Conversely, according to second virial
approximation this demixing might be stable at high values of the
polydispersity, something that is observed neither in simulations nor in
experiments. The results of the fundamental-measure functional are
quantitatively superior to those obtained from the other two approximations.
Thus this functional provides a promising route to map out the full phase
diagram of this system.
