  The present study investigates the arrangement of hollow pyramidal cone
shells and their interactions with degenerate planar anchoring on the inner and
outer surfaces of particles within the nematic host. The shell thickness is in
order of the nematic coherence length. The numerical behavior of colloids is
determined by minimizing the Landau-de Gennes free energy in the presence of
the Fournier surface energy and using the finite element method. Colloidal
pyramidal cones can orient parallel and perpendicular with the far director
orientation. In the parallel alignment, we found the splay director distortion
into the pyramid with two boojum defects at the inner and outer tips. The
director shows bending distortion without defect patterns when the pyramid is
aligned perpendicularly. They induce long-range dipolar interaction and can
form nested structures in close contact.
