Curvature-induced smectic-C order of tangentially anchored hard spherocylinders on a sphere with a rigidly locked director field

Abstract

We study the strict locked-orientation limit of hard spherocylinders on a sphere, in which the rod axes are rigidly locked to a prescribed tangential director field and cannot reorient. Because the bulk hard-rod phase diagram contains no smectic-C phase, any coherent tilt isolates a geometric curvature mechanism rather than a finite-stiffness equilibrium effect. A ratio-symmetric recognition cost fixes the layer spacing at the bulk close-contact value and yields a hierarchy of geometric statements: the lower edge of the smectic-area window at $45^\circ$ follows from reciprocal symmetry; the upper edge at $58.3^\circ$ is a falsifiable channel-saturation hypothesis; the smectic-A to smectic-C boundary is a closed-form prediction; and the rod tilt angle is set by the rod-to-radius ratio, modulated by a chirality envelope peaking near $24^\circ$. Locked-orientation Monte Carlo across fifteen geometries confirms these predictions with no fitted elastic constants: the smectic area peaks at $55^\circ$, and a coherent smectic-C window is detected.