The surface free energy of a nematic system in presence of microstructurated susbstrates, the Frank-Oseen model and the boundary elements method
By: Oscar Rojas
From: Universidad de Sevilla
At: Instituto de Investigação Interdisciplinar, Anfiteatro
[2013-10-16]
The behavior of nematic liquid crystals in the presence of microstructured substrates has been the subject of intensive research in recent times. We present a generalization of Berreman’s model for the elastic contribution to the surface free-energy density of a nematic liquid crystal in presence of sawtoothed and crenellated substrates which favors homeotropic anchoring as a function of the wave number of the surface structure q, certain geometrical characteristics and the surface anchoring strength w. In addition to the nonanalytic contribution proportional to −q ln q, due to the nucleation of disclination lines at the substrate, the next-to-leading contribution is proportional to q for a given substrate roughness, in agreement with Berreman’s predictions.We characterize this term, finding that it has two contributions: the deviations of the nematic director field with respect to a reference field corresponding to the isolated disclination lines and their associated core free energies. Comparison with the results obtained from the Landau-de Gennes model shows that our model is quite accurate in the limit wL > 1, when strong anchoring conditions are effectively achieved.
For the other hand, if the variations of the nematic order parameter, S, are restricted to the neighborhood of the substrate of a width typically of order to a few correlation lengths and inside the defect cores, S takes the bulk value, Sbulk, everywhere. Thus, the surface free-energy functional to minimize is reduced to a Frank-Oseen functional, so we can solve the Laplace equation associated with appropriate boundary conditions.
Finally, a very suitable method to use in order to characterize the first said contribution is the Boundary Elements Method. With this we can discretize the space and reduce the Laplace equation to a linear system equations for the border of the liquid cristal system and reaching the solution in the whole system after to solve the Laplace equation.