  The effective interaction between spherical colloids in nematic liquid
crystals is investigated in the framework of the Landau-de Gennes theory. The
colloids differ through their interaction with the nematic. While both
particles induce quadrupolar far-field distortions in the nematic matrix, with
unlike quadrupole moments, one favours homeotropic and the other degenerate
planar anchoring of the nematic director. In the strong anchoring regime the
colloids with homeotropic anchoring are accompanied by an equatorial
disclination line defect, known as "Saturn-ring", while the colloids with
degenerate planar anchoring nucleate a pair of antipodal surface defects,
called "Boojums". In the linear (large-distance) regime the colloidal
interactions are of the quadrupolar type, where the quadrupoles have opposite
signs. These are attractive when the colloids are aligned either parallel or
perpendicular to the far-field director. At short distance, non-linear effects
including "direct" interactions between defects give rise to a repulsion
between the particles, which prevents them from touching. This finding supports
the stability of nematic colloidal square crystallites the assembly of which
has been reported recently.
