Cholesteric liquid crystal roughness models: from statistical characterization to inverse engineering

The surface geometry, particularly the curvature and roughness, play crucial roles in the functionalities of bio-compatible cholesteric liquid crystal (CLC) substrates. For example, experiments show increased alignment of hBMSCs (human bone-marrow-derived stromal cells) with larger curvature on a cylindrical manifold [Callens <i>et al.</i>, <i>Biomaterials</i>, 2020, <b>232</b>, 119739]. Previous studies on cholesteric liquid crystal surfaces have primarily focused on an elastic approach, which does not fully capture the anisotropic nature and multiscale wrinkling profiles. The objective of this research is to characterize the surface geometry of CLCs based on a generalized anisotropic anchoring model (the Rapini-Papoular model). In this paper, we propose both analytic approximations and direct numerical solutions for surface wrinkling, curvature profiles, and surface roughness characterization. We also explore the important limits of the Rapini-Papoular model, including lower bounds for the kurtosis and Willmore energy. The inverse problem offers an alternative approach to measuring the anchoring coefficients, which are difficult to determine experimentally. These findings suggest that surface anchoring is the key determinant of multiscale surface wrinkling patterns. This paper sheds light on the applications and functionalities of surface wrinkling patterns in liquid crystals and their solid analogues. Furthermore, this research incorporates a novel coordinate-free differential geometric approach and provides a general framework for studying dynamic properties and surface evolution.