Tight packing of a flexible rod in two-dimensional cavities

Abstract

The present work deals with the injection and packing of a flexible polymeric rod of length $L$ into a simply connected rectangular domain of area $XY$. As the injection proceeds, the rod bends over itself and it stores elastic energy in closed loops. In a typical experiment $N$ of these loops can be identified inside the cavity in the jammed state. We have performed an extensive experimental analysis of the total length $L(N, X, Y)$ in the tight packing limit, and have obtained robust power laws relating these variables. Additionally, we have examined a version of this packing problem when the simply connected domain is partially occupied with free discs of fixed size. The experimental results were obtained with 27 types of cavities and obey a single equation of state valid for the tight packing of rods in domains of different topologies. Besides its intrinsic theoretical interest and generality, the problem examined here could be of interest in a number of studies including packing models of DNA and polymers in several complex environments.