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 Lrod2/(RsphDrod), 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∘ to 55∘. 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.
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BibTeX
@misc{washburn2026tilt,
title = {{Tilt control of coverage heterogeneity for hard spherocylinders locked on a sphere}},
author = {Jonathan Washburn and Hartmut Löwen and Elshad Allahyarov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05486},
eprint = {2609.05486},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05486},
}
RIS
TY - UNPB
TI - Tilt control of coverage heterogeneity for hard spherocylinders locked on a sphere
AU - Jonathan Washburn
AU - Hartmut Löwen
AU - Elshad Allahyarov
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.05486
UR - https://arxiv.org/abs/2609.05486
AB - 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/(RsphDrod), 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^∘ to 55^∘. 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.
ER -