Liquid crystals are known for their anisotropic mechanical and optical properties which
originate from the long-range orientational molecular ordering. If a liquid crystal is used
as a host liquid in a colloidal suspension, this ordering gives rise to long-range effective
interactions between the colloidal particles. The type of the interaction is controlled by
the presence and symmetry of
in the alignment field of anisotropic liquid crystal molecules. Particles clustering, formation
of superstructures, and even new phases are immediate consequences of these anisotropic interactions.
For these liquid crystalline systems molecular structure of the host liquid is usually not important,
and often a continuum description of liquid crystals is used. Several continuum models,
each characterized by its own order parameter, such as the director field, tensorial order parameter,
or particle density exist and describe phenomena occurring on a particular length-scale.
We base our study on the Landau-de Gennes model with tensorial order parameter. This formalism is
applicable for intermediate distances between colloidal particles, where nonlinear effects become
important. Liquid crystal colloids are characterized by widely separated length scales (the
particle's size and the nematic correlation length). In order to resolve this problem we use adaptive
finite elements methods in order
to minimize the corresponding Landau-de Gennes free energy functional
and obtain the equilibrium
nematic configurations around particles.
With this technique at hand we analyze the structure of topological defects as a function of boundary
conditions at particles surfaces, particles size,
, temperature or anisotropy of elastic
constants. We are also studying nematic-mediated effective interaction between two colloidal particles
and the role played by the topological defects at the intermediate particle-particle separations. In
particular, topological defects which often accompany colloids bind colloidal particles into "colloidal
molecules" via defects-sharing mechanism an analogue of a covalent bond in atomic systems.