A Novel Differential Geometry Methodology for Curvature and Shape Characterization of Ellipsoids: Shape Transition Symmetry Breaking and Mechanistic Insights into Self-Assembly Curvature Driving Force

This paper develops, implements, and uses a novel methodology that integrates global and local geometric modeling of ellipsoids and transforms curvatures and shape descriptors into energy metrics that provide curvature-driven mechanistic insight into self-assembly driving force pathways associated with fiber and film formation. Global parameterization, based on eccentricities, and local parameterization, based on mean and Gaussian curvatures, are mapped into shape and curvedness parameterizations that clearly distinguish critical shape effects from curvedness effects in contrast to classical mean and Gaussian curvature methodologies. Finally, the mechanistic insights are illustrated using a liquid membranology model, transforming the geometric descriptors into bending energy densities that point the way to fiber and film assembly pathways.