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High order elastic terms, boojums and general paradigm of the elastic interaction between colloidal particles in the nematic liquid crystals

Theoretical description of the elastic interaction between colloidal particles in NLC with incorporation of the higher order elastic terms beyond the limit of dipole and qudrupole interactions is proposed. The expression for the elastic interaction potential between axially symmetric colloidal particles, taking into account of the high order elastic terms, is obtained. The general paradigm of the elastic interaction between colloidal particles in NLC is proposed so that every particle with strong anchoring and radius $a$ has three zones surrounding itself. The first zone for $a<r\lessapprox 1.3a$ is the zone of topological defects; the second zone at the approximate distance range $1.3a \lessapprox r \lessapprox 4a$ is the zone where crossover from topological defects to the main multipole moment takes place. The higher order elastic terms are essential nere (from 10% to 60% of the total deformation). The third zone is the zone of the main multipole moment, where higher order terms make a contribution of less than 10%. This zone extends to distances $r\gtrapprox 4a=2D$. The case of spherical particles with planar anchoring conditions and boojums at the poles is considered as an example. It is found that boojums can be described analitically via multipole expansion with accuracy up to $1/r^{7}$ and the whole spherical particle can be effectively considered as the multipole of the order 6 with multipolarity equal $2^{6}=64$. The correspondent elastic interaction with higher order elastic terms gives the angle $θ_{min}=34.5^{\circ}$ of minimum energy between two contact beads which is close to the experimental value of $θ_{min}=30^{\circ}$.