Packing of elastic wires in flexible shells

The packing problem of long thin filaments that are injected into confined spaces is of fundamental interest for physicists and biologists alike. How linear threads pack and coil is well known only for the ideal case of rigid containers, though. Here, we force long elastic rods into flexible spatial confinement borne by an elastic shell to examine under which conditions recently acquired knowledge on wire packing in rigid spheres breaks down. We find that unlike in rigid cavities, friction plays a key role by giving rise to the emergence of two distinct packing patterns. At low friction, the wire densely coils into an ordered toroidal bundle with semi-ellipsoidal crosssection, while at high friction, it packs into a highly disordered, hierarchic structure. These two morphologies are shown to be separated by a continuous phase transition. Our findings demonstrate the dramatic impact of friction and confinement elasticity on filamentous packing and might drive future research on such systems in physics, biology and even medical technology toward including these mutually interacting effects. Copyright c © EPLA, 2015 Introduction. – Dense filament packing can be found in various natural systems. A well-known instance is the injection and subsequent coiling of long DNA in globules and viral capsids (e.g., [1–3]). A similar technique has been harnessed by neurosurgeons for the minimally invasive treatment of saccular aneurysms, into which detachable platinum wires are fed to initiate occlusion [4]. Extremely dense fiber packing can also be observed for example in gland thread cells of hagfish [5]. Albeit their morphology has recently been unraveled [6], the understanding of morphogenesis —the process of shape transformation and development— remains vague. In particular, the role of macroscopic material parameters is little understood. Several experimental and numerical studies [7–12] revealed how long wires form loop patterns and alignment between contacting segments when injected into rigid two-dimensional containers, depending on friction, plastic yield point and the precise insertion setup. In rigid threedimensional cavities [13–15], on the other hand, friction has been reported to have negligible impact on the packing process. In a recent study on morphogenesis of elastic ring filaments growing in flexible shells [16], however, we discovered four morphological phases strongly dependent (a)E-mail: [email protected] on friction, which brought it back into play in 3D. This naturally begs the question to which degree the knowledge previously acquired on the thread injection and packing problem in rigid cavities is applicable to less idealized, deformable ones, such as cell walls, vesicles, or arterial walls. In this paper, we perform a change of topology from ring-like (as in ref. [16]) to linear threads, which are more relevant in Nature and biomedical applications, to answer this question quantitatively using numerical simulations and simple table-top experiments. We show how confinement elasticity completely alters the packing process of a thin wire that is fed in. In strong contrast to rigid cavities, friction becomes the key macroscopic material property beyond a critical point, determining the packing process by giving rise to the spontaneous emergence of two different morphologies. We characterize them by comparing numerical measurements of their energetic and structural properties, finding that frictional forces lead to a highly disordered, hierarchic, crumpled structure governed by power laws, whereas the absence of friction yields a dense toroidal bundle with semi-ellipsoidal cross-section. We further show that the transition from rigid to flexible cavities as well as the transition from weak to strong friction in flexible cavities is continuous and accompanied by spontaneous symmetry breaking, and we identify the