  Hard helices can be regarded as a paradigmatic elementary model for a number
of natural and synthetic soft matter systems, all featuring the helix as their
basic structural unit: from natural polynucleotides and polypeptides to
synthetic helical polymers; from bacterial flagella to colloidal helices. Here
we present an extensive investigation of the phase diagram of hard helices
using a variety of methods. Isobaric Monte Carlo numerical simulations are used
to trace the phase diagram: on going from the low-density isotropic to the
high-density compact phases, a rich polymorphism is observed exhibiting a
special chiral screw-like nematic phase and a number of chiral and/or polar
smectic phases. We present a full characterization of the latter, showing that
they have unconventional features, ascribable to the helical shape of the
constituent particles. Equal area construction is used to locate the
isotropic-to-nematic phase transition, and results are compared with those
stemming from an Onsager-like theory. Density functional theory is also used to
study the nematic-to-screw-nematic phase transition: within the simplifying
assumption of perfectly parallel helices, we compare different levels of
approximation, that is second- and third-virial expansions and Parsons-Lee
correction.
