Ordered packing of elastic wires in a sphere

Packing of elastic rings with friction

We study the deformations of elastic filaments confined within slowly shrinking circular boundaries, under contact forces with friction. We perform computations with a spring-lattice model that deforms like a thin inextensible filament of uniform bending stiffness. Early in the deformation, two lobes of the filament make contact. If the friction coefficient is small enough, one lobe slides inside the other; otherwise, the lobes move together or one lobe bifurcates the other. There follows a sequence of deformations that is a mixture of spiralling and bifurcations, primarily the former with small friction and the latter with large friction. With zero friction, a simple model predicts that the maximum curvature and the total elastic energy scale as the wall radius to the − 3 / 2 and − 2 powers, respectively. With non-zero friction, the elastic energy follows a similar scaling but with a prefactor up to eight times larger, due to delayering and bending with a range of small curvatures. For friction coefficients as large as 1, the deformations are qualitatively similar with and without friction at the outer wall. Above 1, the wall friction case becomes dominated by buckling near the wall.

Tailoring relaxation dynamics and mechanical memory of crumpled materials by friction and ductility

Crumpled sheets show slow mechanical relaxation and long lasting memory of previous mechanical states. By using uniaxial compression tests, the role of friction and ductility on the stress relaxation dynamics of crumpled systems is investigated. We find a material dependent relaxation constant that can be tuned by changing ductility and adhesive properties of the sheet. After a two-step compression protocol, nonmonotonic aging is reported for polymeric, elastomeric and metal sheets, with relaxation dynamics that are dependent on the material's properties. These findings can contribute to tailoring and programming of crumpled materials to get desirable mechanical properties.

Effect of the material properties on the crumpling of a thin sheet

While simple at first glance, the dense packing of sheets is a complex phenomenon that depends on material parameters and the packing protocol. We study the effect of plasticity on the crumpling of sheets of different materials by performing isotropic compaction experiments on sheets of different sizes and elasto-plastic properties. First, we quantify the material properties using a dimensionless foldability index. Then, the compaction force required to crumple a sheet into a ball as well as the average number of layers inside the ball are measured. For each material, both quantities exhibit a power-law dependence on the diameter of the crumpled ball. We experimentally establish the power-law exponents and find that both depend nonlinearly on the foldability index. However the exponents that characterize the mechanical response and morphology of the crumpled materials are related linearly. A simple scaling argument explains this in terms of the buckling of the sheets, and recovers the relation between the crumpling force and the morphology of the crumpled structure. Our results suggest a new approach to tailor the mechanical response of the crumpled objects by carefully selecting their material properties.

Compaction of quasi-one-dimensional elastoplastic materials

Insight into crumpling or compaction of one-dimensional objects is important for understanding biopolymer packaging and designing innovative technological devices. By compacting various types of wires in rigid confinements and characterizing the morphology of the resulting crumpled structures, here, we report how friction, plasticity and torsion enhance disorder, leading to a transition from coiled to folded morphologies. In the latter case, where folding dominates the crumpling process, we find that reducing the relative wire thickness counter-intuitively causes the maximum packing density to decrease. The segment size distribution gradually becomes more asymmetric during compaction, reflecting an increase of spatial correlations. We introduce a self-avoiding random walk model and verify that the cumulative injected wire length follows a universal dependence on segment size, allowing for the prediction of the efficiency of compaction as a function of material properties, container size and injection force.

Principles underlying crumpling of one-dimensional objects may be relevant to both biomolecular processes and to design of mechanical devices. By compacting various wires under rigid confinement and modelling observed geometric features, the authors show how friction, plasticity and torsion enhance disorder and lead to a transition from coiled to folded geometries.

Scaling, crumpled wires, and genome packing in virions

The packing of a genome in virions is a topic of intense current interest in biology and biological physics. The area is dominated by allometric scaling relations that connect, e.g., the length of the encapsulated genome and the size of the corresponding virion capsid. Here we report scaling laws obtained from extensive experiments of packing of a macroscopic wire within rigid three-dimensional spherical and nonspherical cavities that can shed light on the details of the genome packing in virions. We show that these results obtained with crumpled wires are comparable to those from a large compilation of biological data from several classes of virions.

Cylindrical confinement of semiflexible polymers

Equilibrium states of a closed semiflexible polymer binding to a cylinder are described. This may be either by confinement or by constriction. Closed completely bound states are labeled by two integers: the number of oscillations, n, and the number of times it winds the cylinder, p, the latter being a topological invariant. We examine the behavior of these states as the length of the loop is increased by evaluating the energy, the conserved axial torque, and the contact force. The ground state for a given p is the state with n=1; a short loop with p=1 is an elliptic deformation of a parallel circle; as its length increases it elongates along the cylinder axis with two hairpin ends. Excited states with n≥2 and p=1 possess n-fold axial symmetry. Short (long) loops possess energies ≈pE(0)(nE(0)), with E(0) the energy of a circular loop with same radius as the cylinder; in long loops the axial torque vanishes. Confined bound excited states are initially unstable; however, above a critical length each n-fold state becomes stable: The folded hairpin cannot be unfolded. The ground state for each p is also initially unstable with respect to deformations rotating the loop off the surface into the interior. A closed planar elastic curve aligned along the cylinder axis making contact with the cylinder on its two sides is identified as the ground state of a confined loop. Exterior bound states behave very differently, if free to unbind, as signaled by the reversal in the sign of the contact force. If p=1, all such states are unstable. If p≥2, however, a topological obstruction to complete unbinding exists. If the loop is short, the bound state with p=2 and n=1 provides a stable constriction of the cylinder, partially unbinding as the length is increased. This motif could be relevant to an understanding of the process of membrane fission mediated by dynamin rings.

Morphogenesis of filaments growing in flexible confinements

Space-saving design is a requirement that is encountered in biological systems and the development of modern technological devices alike. Many living organisms dynamically pack their polymer chains, filaments or membranes inside deformable vesicles or soft tissue-like cell walls, chorions and buds. Surprisingly little is known about morphogenesis due to growth in flexible confinements--perhaps owing to the daunting complexity lying in the nonlinear feedback between packed material and expandable cavity. Here we show by experiments and simulations how geometric and material properties lead to a plethora of morphologies when elastic filaments are growing far beyond the equilibrium size of a flexible thin sheet they are confined in. Depending on friction, sheet flexibility and thickness, we identify four distinct morphological phases emerging from bifurcation and present the corresponding phase diagram. Four order parameters quantifying the transitions between these phases are proposed.

Tuning the ordered states of folded rods by isotropic confinement

The packing of elastic objects is increasingly studied in the framework of out-of-equilibrium statistical mechanics and thus these appear to be similar to glassy systems. Here, we present a two-dimensional experiment whereby a rod is confined by a parabolic potential. The setup enables spanning a wide range of folded configurations of the rod. Measurements of the distributions of length and curvature in the system reveal the importance of a stacking process whereby many layers of the rod are grouped into branches. The geometrical order of patterns increases with the confinement strength. Measurements of the distributions of energies lead to the definition of an energy scale that is correlated with the elastic energy of the stacked parts of the rod. This scale imposes energy partition in the system and might be relevant to the framework of the thermodynamics of disordered systems. Following these observations, we describe the patterns as excited states of a ground state corresponding to the most ordered geometry. Eventually, we provide evidence that the disordered state of a folded rod becomes spontaneously closer to the ground state as confinement is increased.