  Colloidal rods immersed in a thermotropic liquid-crystalline solvent are at
the basis of so-called hybrid liquid crystals which are characterized by
tunable nematic fluidity with symmetries ranging from conventional uniaxial
nematic or anti-nematic to orthorhombic [Mundoor et al., Science 360, 768
(2018)]. We provide a theoretical analysis of the elastic moduli of such
systems by considering interactions between the individual rods with the
embedding solvent through surface-anchoring forces, as well as steric and
electrostatic interactions between the rods themselves. For uniaxial systems
the presence of colloidal rods generates a marked increase of the splay
elasticity which we found to be in quantitative agreement with experimental
measurements. For orthorhombic hybrid liquid crystals we provide estimates of
all twelve elastic moduli and show that only a small subset of those elastic
constants play a relevant role in describing the nemato-elastic properties. The
complexity and possibilities related to identifying the elastic moduli in
experiments are briefly discussed. The results are expected to be helpful in
stimulating future studies on defect-dressed "topological" colloids and the
analysis of boundary-driven phenomena where the non-trivial director topology
generated by biaxial nematics plays a key role.
