Ordered packing of elastic wires in a sphere

In this paper we study the ordered packing of wires in a sphere. We propose an analytical model and compare the model predictions with the results of our experiments and simulations for the maximum packing fraction, the number of formed coils, the fractal dimension, and bending energy. We show that the relative system size [i.e., the ratio of the wire radius to the sphere radius ($a/R$)] is the most important control parameter for the maximum packing fraction. We find that the number of coils obeys a power-law relation of the form $N\sim {(R/a)}^{1.5}$ and the fractal dimension of the structures is 2.5, independent of the system size. Our theoretical results are in good agreement with the experimental data and the predictions of the numerical simulations.

Figure 1

(a) Side view of the packed wire at the first stages of packing. System properties are $L=764\text{cm}$, $a=0.5$ mm, and $R=19.9$ mm. (b) Cross section of the packed structure, obtained from x-ray tomography data. $L$, $a$, and $R$ are 955 cm, 2 mm, and $39.9$ mm, respectively.

Figure 2

Schematic of the proposed model. $a$, $r_1$, and $r_1-2a$ are the radius of the wire and the radii of the first and the second layers, respectively. ${\theta }_1$ is the angle between the lines that connects the center of the sphere to the axis of two successive coils in the first layer. ${\theta }_1^{\text{max}}$ is the angle between the lines connecting the center of the sphere to the largest coil in the first layer and the line connecting the center of the sphere to the last coil in the first layer. The formed coils in each layer are symmetric with respect to the center of the sphere.

Figure 3

Maximum packing fraction versus relative system size $(a/R)$. Solid curve is plotted by evaluating the integral of packing fraction [Eq. (2)], for $r_{\text{min}}$ equals to $2a$. Solid and hollow symbols are, respectively, experimental and numerical results.

Figure 4

Dimensionless maximum wire length $(L/a)$ versus the radius of the sphere over the radius of wire in logarithmic scale. Solid symbols and hollow squares show, respectively, the experimental and numerical results for different wires. In each series $a$ was fixed while $R$ has been changed. Solid curve is the theoretical prediction of Eq. (1) for the wires used in the experiments and simulation. The slope of the theoretical curve at large $R/a$ converges to the fractal dimension of 2.5.

Figure 5

Dimensionless bending energy versus packing fraction. The solid curve is obtained by Eq. (3) for a system of $a/R=1/40$ and $r_{\text{min}}=2a$. Black circles are experimental data points from Ref. [16].

Figure 6

Total number of coils ($N$) versus $R/a$ in logarithmic scale. Solid and hollow circles show, respectively, the experimental and numerical results. The solid curve is the theoretical prediction of Eq. (4). Theoretical results indicate that the number of coils in the limit of large $R/a$ has a power-law relation $N\sim {(R/a)}^{1.5}$. Inset: Number of coils in the outermost layer ($N_1$) versus wire radius. The solid curve represents the theoretical prediction and solid circles denote the experiments. The radius of the sphere ($R$) is $19.3$ mm. The number of coils is an integer, but it has been plotted continually in both plots to facilitate comparison.