**Department of Physics**

### University of Qom

I have numerically studied the behavior of colloidal systems with degenerate planar anchoring in a nematic fluid by numerically minimizing the Landau-de Gennes bulk free energy and Fournier

*et al.*surface energy [Europhys.Lett., 72, 403, (2005)] using a 3D or a 2D finite element methods.

### Interaction of spherical colloidal particles in nematic media with degenerate planar anchoring

As shown in Fig.1(a), the nematic-induced effective potential energy of two immersed colloidal particles in a nematic liquid crystal has been obtained for different inter-particle separations and orientations with respect to the far-field nematic director. By studying the free energy landscape of the system, we found that the system assumes its unique global minimum of energy when the particles are in close contact and are oriented at an angle 28° ± 2° with respect to the far-field nematic director. The results are in a very close agreement with the experimental results [Phys.Rev.Lett., 95, 157801, (2005)]. For large inter-particle separations, the quadrupolar structure of the defect-pair on each of the particles is essentially preserved, resulting in a monotonous attractive or repulsive inter-particle net force, depending on the orientation angle. As shown in Fig.1(b), for smaller orientation angles (≤15°) and at smaller inter-particle separations, the axial symmetry of the defect-pairs is continuously broken, resulting in the emergence of an attractive interaction due to the tendency of the system to reduce the volume of distorted fluid.
Fig.1: The free energy landscape of the system of two particles as a function of the inter-particles distance, d/R, and θ, the angle between the line joining the center of particles and the far-field director. (b) Director profile and topological defects on the surface of the colloidal particles when they approach each other along the nematic direction θ=0 at d/R=2.10. |

### Interaction of long parallel prism particles in nematic media with degenerate planar anchoring

Here, the interaction between two sections of long parallel prism particles with degenerate planar anchoring in a nematic host is studied. The triangular prisms prefer to align one of their faces with the director while square ones align their diagonal to the nematic director. As shown in Fig.2, we find effective pair interactions between the particles and show that numerically obtained minimum energy configurations are in agreement with the recent experiments[Science, 326, 1083, (2009)]. In large separations, the defects around the particles have dipolar and quadrupolar symmetry for triangular and square particle, respectively. In close contacts and due to present the degenerate surface anchoring, the director is aligned in long prisms axes.
Fig.2: (a) Effective potential energy(EPE) between two long triangle prism particles as a function of dimensionless inter-particle distance, d/R, for four given configurations. (b) EPE between two long square prism particles as a function of dimensionless inter-particle distance, d/R, at eleven fix orientations. |

### Spherical shells of nematic liquid crystals

We studied special arrangements of defect configuration in a confined nematic shell with different thickness. In a symmetric shell, as shown in Fig.3(a-b), the calculations reveal two main defect configurations: two pair boojums in thick shells and tetrahedral structure when the shells are thin. We have investigated the boojums and tetrahedral defect configurations by compare their dimensionless free energy densities (FED). The energy-crossing, h_c=0.66R, indicates how stability of the boojum and the tetrahedral regimes to shell thickness(see Fig.3(d)). Furthermore, for both boojums and tetrahedral configurations, the inner droplet has not energetically a steady state when the nematic shell is symmetric. In agreement with experimental observation, as shown Fig.3(e), the inner droplet in boojums regime moves in the director direction until reaches to the outer surface [Phys.Rev.Lett. 99, 15780, 2007]. By studying the FED in close of the energy-crossing, we found isosceles triangle defect arrangements(see Fig.3(c)) that show good agreement with Fernandez et al. observations[Nature phys. 7, 391, (2011)].
Fig.3: The regions in bulk and on surface where the scalar order parameter drops below 0.4S _{b} is highlighted, signaling the existence of a topological defect for: (a) boojums, (b) tetrahedral and (c) isosceles triangle configurations, (d) Calculated FED as a function of dimensionless thickness h/R for boojums and tetrahedral defect configurations, (e) Calculated FED as a function of dimensionless center-center displacement of surfaces, Δ/R, in thickness of boojums regimes. |