Random packing of model polymers: local structure, topological hindrance and universal scaling

The random packing of rigid objects has not only engrossed mathematicians since biblical times but is receiving attention for numerous applications and processes involving microgels, granular media, colloids, glasses, liquids, synthetic polymers and biomolecules. While dense random assemblies of single hard spheres have been extensively investigated both experimentally and theoretically over the past 50 years, it was only recently that analogous problems for chains of hard spheres have been addressed. We highlight the relevance of these recent advances, and describe the most salient characteristics of the "maximally random jammed'' state for hard sphere chains. Particular emphasis is placed on the scaling behavior of chain dimensions and topology with packing density. We also discuss the potentially far-reaching implications of an unexpected connection that has been found between entanglements (intermolecular constraints) and knots (of intramolecular origin) regarding their dependence on volume fraction.