Curvature analysis of liquid crystal phase ordering energy landscapes: application to the direct isotropic-smectic-A transition

Liquid crystals are ordered mesophases that present polymorphisms characterised by creation and loss of orientational and positional order. Transitions are mainly driven by mesogen structure, shape, size, temperature, concentration, and external fields. Different mesophase sequences show a direct transition involving positional and orientational orders. The direct isotropic-to-smectic-A transition is characterised by a strong coupling between these two order parameters. This work studies this transition through a geometric-thermodynamic approach. We construct a thermody5namic surface energy landscape with orientational and positional order parameters. The mean and Gaussian curvatures are computed to locally describe the surface geometry and phase stability. Experimental data from a smectogenic thermotrope validates computations for different quenching regimes. The energy landscape provides a unique insight into interaction and transition between phases. The mean curvature shows a tendency of sequential phase stability order, while the Gaussian curvature assigns the stable phase to the highest curvature values as the transition is approached. The interplay between these curvatures allows phase classification and stability, identifying stable configurations where curvature is maximised for stable liquid crystal regimes. These results contribute to the evolving characterisation of multi-order transitions where computations may not capture the presence of all critical points on the highly complex energy landscape.