Tilt control of coverage heterogeneity for hard spherocylinders locked on a sphere

Abstract

We study hard spherocylinders on a sphere with axes rigidly locked to a tangential director field at fixed angle $\tilt$ to the meridian, necessarily singular at the poles. Three lengths set the problem: the rod length $\Lrod$, the diameter $\Drod$, and the host radius $\Rsph$. Monte Carlo simulations across fifteen geometries and four coverages give two main results for the polar marginal, the azimuthal average of rod-center density. First, the tilt is a continuous handle on the width of the depleted region that packing induces around each singularity. Under meridian locking it is set principally by the rod length; turning the director toward the latitude contracts it substantially, the contraction being spent well before latitude locking. Second, how uniform the polar marginal can be made is limited by geometry, not tilt. The smallest variance on the sampled grid follows a power law in $\Lrod^{2}/(\Rsph\Drod)$, which measures how far a straight rod's ends stand off the curved surface in rod diameters, with an effective exponent between $1.1$ and $1.3$. Long rods on small hosts cannot be made uniform at any sampled tilt; the remedy is geometric, not orientational. The polar marginal is equator-heavy almost everywhere, inverting only at high coverage and tilt; meridian locking is the least uniform choice, and the variance-minimizing tilt usually lies in a sampled band from $31.7^{\circ}$ to $55^{\circ}$. The golden-ratio tilt $\arctan(1/\phigold)$ is one of those angles: a benchmark, not one the model selects. Both results describe the infinitely locked athermal ensemble.