  Colloidal particles dispersed in a liquid crystal lead to distortions of the
director field. The distortions are responsible for long-range effective
colloidal interactions whose asymptotic behaviour is well understood. The short
distance behaviour of the interaction, however, is sensitive to the structure
and dynamics of the topological defects nucleated near the colloidal particles
in the strong anchoring regime. The full non-linear theory is required in order
to determine the interaction at short separations. Spherical colloidal
particles with sufficiently strong planar degenerate anchoring nucleate a pair
of antipodal surface topological defects, known as boojums. We use the
Landau-de Gennes formalism in order to resolve the mesoscopic structure of the
boojum cores and to determine the pairwise colloidal interaction. We compare
the results in three (3D) and two (2D) spatial dimensions. The corresponding
free energy functionals are minimized numerically using finite elements with
adaptive meshes. Boojums are always point-like in 2D, but acquire a rather
complex structure in 3D which depends on the combination of the anchoring
potential, the radius of the colloid, the temperature and the LC elastic
anisotropy. We identify three types of defect cores in 3D which we call single,
double and split core boojums, and investigate the associated structural
transitions. In the presence of two colloidal particles there are substantial
re-arrangements of the defects at short distances, both in 3D and 2D. These
re-arrangements lead to qualitative changes in the force-distance profile when
compared to the asymptotic quadrupole-quadrupole interaction. In line with the
experimental results, the presence of the defects prevents coalescence of the
colloidal particles in 2D, but not in 3D systems.
