Size effect of topological charge in disc-like nematic liquid crystal films

Algebraic topology, algebraic geometry, and category theory are new branches of mathematics that have developed in the last hundred years and have had profound collisions with modern physics in recent decades. A large number of topological phenomena are found in systems such as viruses, bacteria, fingerprints, fish school, typhoons, and the galaxies. Topological phenomena play a significant role in the spatial distribution of viral particles, the formation of nanovesicles of polymer, and Bose-Einstein condensates. In this paper, based on Landau-de Gennes theory, models have been constructed to simulate the topological charge distribution and other topological phenomena in liquid crystals. The research indicates that as the radius of the liquid crystal panel grows, the ratio of the optimal distance between the topological charge to the radius gradually increases and tends to stabilize. The size of the disc affects the equilibrium position of the topological load. The relative equilibrium position of topological load is between 0.542 and 0.558, in which the ratio of the distance between the two +1/2 topological loads in the 0-5 mm disc increases from 0.542 to 0.558, and then in the 5-12 mm section the ratio is almost stable at 0.558. As the size of the disc increases, the influence of the boundary anchoring energy decreases, and the equilibrium position, i.e. the distance between the two topological charges and the diameter of the disc, approaches a constant value. This equilibrium position is the result of the repulsive force of the disc boundary on the +1/2 topological load and the repulsive force between the two topological loads. The angle between two topological charges in a liquid crystal disc is between 140 degrees and 180 degrees. The trajectory of the topological charge is the process of finding the lowest free energy point, and the end of the trajectory is in the region of minimum free energy. The result is instructive significance in the design of classification containers by using topological charge condensate effect. And it is helpful to further understand the topological phenomena in soft materials including topological colloids, liquid crystals, and liquid crystal copolymers.

Automated summary

Algebraic topology, algebraic geometry, and category theory are new branches of mathematics that have had significant interactions with modern physics in recent decades. Topological phenomena, such as the distribution of viral particles in spatial environments and the formation of nanovesicles and condensates, play important roles in various systems. Research based on Landau-de Gennes theory constructs models to simulate topological charge distribution in liquid crystals, revealing how the size of the disc affects equilibrium positions and the angle between topological charges in the range of 140 to 180 degrees.