  Many nanoparticle-based chiral liquid crystals are composed of polydisperse
rod-shaped particles with considerable spread in size or shape, affecting the
mesoscale chiral properties in, as yet, unknown ways. Using an algebraic
interpretation of Onsager-Straley theory for twisted nematics, we investigate
the role of length polydispersity on the pitch of nanorod-based cholesterics
with a continuous length polydispersity, and find that polydispersity enhances
the twist elastic modulus, $K_{2}$, of the cholesteric material without
affecting the effective helical amplitude, $K_{t}$. In addition, for the
infinitely large average aspect ratios considered here, the dependence of the
pitch on the overall rod concentration is completely unaffected by
polydispersity. For a given concentration, the increase in twist elastic
modulus (and reduction of the helical twist) may be up to 50% for strong size
polydispersity, irrespective of the shape of the unimodal length distribution.
We also demonstrate that the twist reduction is reinforced in bimodal
distributions, by doping a polydisperse cholesteric with very long rods.
Finally, we identify a subtle, non-monotonic change of the pitch across the
isotropic-cholesteric biphasic region.
