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 $\theta_{min}=34.5^{\circ}$ of minimum energy between two contact beads which is close to the experimental value of $\theta_{min}=30^{\circ}$.Comment: 9 pages, 13 figure

Elastic interaction between colloidal particles in confined nematic liquid crystals

The theory of elastic interaction of micron size axially symmetric colloidal particles immersed into confined nematic liquid crystal has been proposed. General formulas are obtained for the self energy of one colloidal particle and interaction energy between two particles in arbitrary confined NLC with strong anchoring condition on the bounding surface. Particular cases of dipole-dipole interaction in the homeotropic and planar nematic cell with thickness $L$ are considered and found to be exponentially screened on far distances with decay length $\lambda_{dd}=\frac{L}{\pi}$. It is predicted that bounding surfaces in the planar cell crucially change the attraction and repulsion zones of usual dipole-dipole interaction. As well it is predicted that \textit{the decay length} in quadrupolar interaction is \textit{two times smaller} than for the dipolar case.Comment: 4 pages,2 figure

Theory of elastic interaction between colloidal particles in the nematic cell in the presence of the external electric or magnetic field

The Green function method developed in Ref.[S. B. Chernyshuk and B. I. Lev, Phys. Rev. E \textbf{81}, 041707 (2010)] is used to describe elastic interactions between axially symmetric colloidal particles in the nematic cell in the presence of the external electric or magnetic field. General formulas for dipole-dipole, dipole-quadrupole and quadrupole-quadrupole interactions in the homeotropic and planar nematic cells with parallel and perpendicular field orientations are obtained. A set of new results has been predicted: 1) \textit{Deconfinement effect} for dipole particles in the homeotropic nematic cell with negative dielectric anisotropy $\Delta\epsilon<0$ and perpendicular to the cell electric field, when electric field is approaching it's Frederiks threshold value $E\Rightarrow E_{c}$. This means cancellation of the confinement effect found in Ref. [M.Vilfan et al. Phys.Rev.Lett. {\bf 101}, 237801, (2008)] for dipole particles near the Frederiks transition while it remains for quadrupole particles. 2) New effect of \textit{attraction and stabilization} of the particles along the electric field parallel to the cell planes in the homeotropic nematic cell with $\Delta\epsilon<0$ . The minimun distance between two particles depends on the strength of the field and can be ordinary for . 3) Attraction and repulsion zones for all elastic interactions are changed dramatically under the action of the external field.Comment: 15 pages, 17 figure

Theory of Elastic Interaction of the Colloidal Particles in the Nematic Liquid Crystal Near One Wall and in the Nematic Cell

We apply the method developed in Ref. [S.B.Chernyshuk and B.I.Lev, Phys.Rev.E, \textbf{81}, 041701 (2010)] for theoretical investigation of colloidal elastic interactions between axially symmetric particles in the confined nematic liquid crystal (NLC) near one wall and in the nematic cell with thickness $L$. Both cases of homeotropic and planar director orientations are considered. Particularly dipole-dipole, dipole-quadrupole and quadrupole-quadrupole interactions of the \textit{one} particle with the wall and within the nematic cell are found as well as corresponding \textit{two particle} elastic interactions. A set of new results has been predicted: the effective power of repulsion between two dipole particles at height $h$ near the homeotropic wall is reduced gradually from inverse 3 to 5 with an increase of dimensionless distance $r/h$; near the planar wall - the effect of dipole-dipole \textit{isotropic attraction} is predicted for large distances $r>r_{dd}=4.76 h$; maps of attraction and repulsion zones are crucially changed for all interactions near the planar wall and in the planar cell; one dipole particle in the homeotropic nematic cell was found to be shifted by the distance $\delta_{eq}$ from the center of the cell \textit{independent} of the thickness $L$ of the cell. The proposed theory fits very well with experimental data for the confinement effect of elastic interaction between spheres in the homeotropic cell taken from [M.Vilfan et al. Phys.Rev.Lett. {\bf 101}, 237801, (2008)] in the range $1\div1000 kT$.Comment: 18 pages, 23 figure

Ordered droplet structures at the liquid crystal surface and elastic-capillary colloidal interactions

We demonstrate a variety of ordered patterns, including hexagonal structures and chains, formed by colloidal particles (droplets) at the free surface of a nematic liquid crystal (LC). The surface placement introduces a new type of particle interaction as compared to particles entirely in the LC bulk. Namely, director deformations caused by the particle lead to distortions of the interface and thus to capillary attraction. The elastic-capillary coupling is strong enough to remain relevant even at the micron scale when its buoyancy-capillary counterpart becomes irrelevant.Comment: 10 pages, 3 figures, to be published in Physical Review Letter

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