In the last few years, there has been considerable theoretical and experimental interest in inverted nematic emulsions, i.e. colloidal dispersions in anisotropic host fluids .
Since the temperature changes the anisotropic properties of the host fluid, the dispersed particles exhibit a rich variety of collective behaviors, leading to unexpected phase diagrams. Depending on the details of the system, chain-like, crystal or cellular structures have been observed.
In the case of nematic hosts, the colloidal interactions result from the competition between the anchoring properties of the spherical colloidal surfaces favoring e.g., radial (homeotropic) nematic orientation (anchoring) and the bulk elasticity that favors a uniform nematic.
For sufficiently strong homeotropic anchoring W, the spherical colloids may induce topological defects or singularities in the nematic director field. If the size of the colloidal particle a is large when compared to the nematic correlation length zeta, a hyperbolic point defect appears at a certain distance from the particle (satellite dipolar configuration).
For smaller particles, however, a ring disclination around the equatorial plane of the sphere may have a lower energy (saturn-ring quadrupolar configuration).
Finally, if Wa is smaller than the typical nematic elastic constant K, the director field varies smoothly and there are no defects (surface-ring quadrupolar configuration).
For 2D nematic hosts and homeotropic anchoring conditions, only quadrupolar configurations are stable. Dipolar configurations may be found in 2D smectic C films, a system that is similar to a 2D nematic, but where the director does not exhibit mirror symmetry, thus excluding configurations with half integer topological charges.
Figure 2. satellite dipolar, saturn-ring quadrupolar and surface-ring quadrupolar configurations.
Depending on the symmetry of the distortion, the dominant long range interactions are dipolar or quadrupolar.
Pettey et al.  obtained analytic expressions for the free energy of a 2D smectic C film with multiple disk-defect pairs, valid in the linear regime where the particles are sufficiently far apart.
The short range interactions are more difficult to predict, as nonlinear terms in the elastic free energy come into play.
We investigated the short range interactions in this system by using a numerical method capable of describing the nematic orientation profiles induced by a small number of colloids with arbitrary shape in 2D. Numerical investigations of this type are faced with a problem due to the very different length scales characterizing the colloids and the topological defects.
We have found a strong repulsion at short range followed by the expected long range dipolar interaction with a pronounced minimum at an intermediate distance. The equilibrium director field configuration exhibits a topological defect at the mid-point between the disks.
Figure 3. Interaction between two aligned satellite dipolar configurations. Exact numerical results.
For strong homeotropic anchoring, each disk generates a pair of defects with one-half topological charge responsible for the 2D quadrupolar interaction between the disks at large distances. At short distance, the position of the defects may change, leading to unexpected complex interactions with the quadrupolar repulsive interactions becoming attractive. This short range attraction in all directions is still anisotropic. As the distance between the disks decreases their preferred relative orientation with respect to the far-field nematic director changes from oblique to perpendicular.
Figure 4. Interaction between two saturn-ring quadrupolar configurations, separated by a distance (R cosα, R sinα).
Figure 5. Interaction between two initially saturn-ring configurations. The coloured regions represent the liquid crystal defects. As the particles approach, the defects move, giving rise to a complex non linear attraction.