On the validity of 2D reduction of the Fournier-Galatola planar anchoring model for polymer-dispersed nematic liquid crystals

The validity of directly reducing the three-dimensional Fournier-Galatola model of planar anchor- ing to two-dimensional nematic systems is investigated analytically and numerically. An alternative two- dimensional formulation based on a direct target surface Q-tensor is introduced for comparison. Analytical calculations reveal that the reduced Fournier-Galatola functional yields a cubic restoring torque at the poles of an elliptical droplet, whereas the direct tensor formulation preserves a linear response along the entire boundary. The influence of this difference on orientational relaxation is studied using the Landau-de Gennes simulations of mesoscale elliptical 5CB droplets. The reduced Fournier–Galatola model is shown to enhance boojum mobility, alter the final bipolar orientation at moderate aspect ratios, and systematically predict lower equilibrium energies. The results indicate that direct projection-based reduction can lead to qualitatively differ- ent relaxation scenarios in two-dimensional nematic models.

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