  Building a general theoretical framework to describe the microscopic origin
of macroscopic chirality in (colloidal) liquid crystals is a long-standing
challenge. Here, we combine classical density functional theory with Monte
Carlo calculations of virial-type coefficients, to obtain the equilibrium
cholesteric pitch as a function of thermodynamic state and microscopic details.
Applying the theory to hard helices, we observe both right- and left-handed
cholesteric phases that depend on a subtle combination of particle geometry and
system density. In particular, we find that entropy alone can even lead to a
(double) inversion in the cholesteric sense of twist upon changing the packing
fraction. We show how the competition between single-particle properties
(shape) and thermodynamics (local alignment) dictates the macroscopic chiral
behavior. Moreover, by expanding our free-energy functional we are able to
assess, quantitatively, Straley's theory of weak chirality, used in several
earlier studies. Furthermore, by extending our theory to different lyotropic
and thermotropic liquid-crystal models, we analyze the effect of an additional
soft interaction on the chiral behavior of the helices. Finally, we provide
some guidelines for the description of more complex chiral phases, like
twist-bend nematics. Our results provide new insights on the role of entropy in
the microscopic origin of this state of matter.
