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Packing of elastic wires in spherical cavities

N. Stoop1, J. Najafi2, F. K. Wittel1, M. Habibi2, and H. J. Herrmann1,3

1Computational Physics for Engineering Materials, ETH Zurich,

Schafmattstr.6, HIF, CH-8093 Zurich, Switzerland

2Department of Physics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran

3Departamento de F´ ısica, Universidade Federal do Cear` a,

Campus do Pici, 60451-970 Fortaleza, Cear´ a, Brazil

(Dated: February 4, 2011)

We investigate the morphologies and maximum packing density of thin wires packed into spherical

cavities. Using simulations and experiments, we find that ordered as well as disordered structures

emerge, depending on the amount of internal torsion. We find that the highest packing densities are

achieved in low torsion packings for large systems, but in high torsion packings for small systems.

An analysis of both situations is given in terms of energetics and comparison is made to analytical

models of DNA packing in viral capsids.

PACS numbers: 87.10.Pq, 46.70.Hg, 89.75.Da

Thin objects are ubiquitous in nature and technology.

Driven by forces and external constraints, they can un-

dergo surprisingly complex spatial rearrangements, ob-

served, for example, in the folding of insect wings in co-

coons, crumpled wires and paper, or growing tissue [1–8].

The packing of long, slender objects in cavities emerges

in many situations in biology and mechanics. It occurs

in paper jams, when chromatin is stored in the cell nu-

cleus, DNA is injected into viral capsids [9–11], or when

endovascular coils are formed in aneurysma surgery [12].

Often, such systems are geared towards high packing den-

sities. For instance, the amount of DNA packed in a cap-

sid limits the genetic information the virus can spread,

and in endovascular surgery, high densities improve the

long-term stability of the treatment [13].

In this Letter, we consider the packing of a thin wire

in a rigid spherical container. First, the emergence of or-

dered as well as disordered packings depending on the

amount of internal torsion is shown.

the maximum packing density depends on morphology

and system size. We find that ordered packings provide

higher maximum packing densities at large system sizes,

while disordered ones offer higher maximum densities at

small system sizes.

We consider two setups in simulations and experi-

ments: In the first, straight wires are used that can axi-

ally rotate at the injection point, thereby having minimal

torsion during the entire packing process. We call this the

low torsion setup. In the second setup, we prohibit axial

rotation at the insertion point and inject precurved wires.

This setup is not only a good model for real-world ap-

plications, where intrinsic curvature is typically present,

but it also suppresses the release of internal torsion via

the free end [14]. Hence, we expect packings with large

internal torsion and thus term it the high torsion setup.

The experiment consists of a transparent, hollow rigid

sphere of inner radius R with a small hole to which a

nozzle is attached. We insert a Nylon or Silicon wire of

We show how

constant cross section radius a by two counter-rotating

rollers, which allow for a controlled insertion speed and

large forces. The Young’s modulus Y of the wires was de-

termined from axial extension tests and deflection mea-

surements [21]. For the high torsion setup, we use wires

of constant intrinsic radius of curvature Ri ≈ 2R. In

the torsion-free setup, we straightened the wires through

axial loading and allowed axial rotation between the noz-

zle and the sphere. The packing process is recorded by

a camera, and the insertion force is measured continu-

ously by a load cell. The packing process stops when the

force becomes so high that no further insertion is possi-

ble. Tomography images are taken from the final packing

to reconstruct the internal structure.

For the numerical model, we describe the wire by

its centreline γ, and an orthonormal director field di,

i = 1,2,3 that specifies the orientation of the cross sec-

tions along γ [15]. We parametrize γ by s, 0 ≤ s ≤ L,

with L the length of the wire. The position of a point on

the centreline is denoted as r(s). The director field is ori-

ented such that d3coincides with the centreline tangent,

r?(s), where?is short-hand for ∂s. It follows from the

theory of space-curves that a vector k exists such that

d?

denote its components in the director frame by ki. To

distinguish between the deformed and undeformed wire

configuration, we use the superscript0, i.e. the refer-

ence configuration is described by the smooth curve γ0,

directors d0

tion that the wire is thin compared to its curvature, the

internal elastic energy is given by

i= k × di. k is known as the Darboux vector and we

iand Darboux vector k0. Under the assump-

Eel=1

+ Y I?(k1− k0

Thefirst

ing/compression, with the cross section area A = πa2.

The second term is the bending energy Eb, with

2

?L

0

ds

?

1)2+ (k2− k0

term

Y A?r?

3− r?0

3

2)2?+G(k3− k0

accounts

?2

3)2?

.

(1)

foraxialstretch-

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2

a) c) d)

low torsion setup

high torsion setup

b)

e)g)h)f)

Eb/(103YI)

12.7

0

4.3

8.7

FIG. 1: Morphologies of wires packed into spherical cavities. Top row: The low torsion setup results in an ordered morphology,

characterized by ring-like coiling (a). A cut through the packing (c) reveils the shell-like inner structure. X-ray tomography

scans from experimental realizations are shown in (b, d). Bottom row: The high torsion setup produces disordered structures

(e, g), with corresponding experiments in (f,h). Color represents the bending energy Eb. System parameters are: φ = 0.23,

a/R = 0.02 (simulations); φ = 0.24, a/R = 0.025 (low torsion, experiment); φ = 0.18, a/R = 0.017 (high torsion, experiment)

I = πa4/4 the 2nd moment of inertia. The components

k0

configuration. The last term accounts for torsion, with

G =

ν the Poisson ratio. For naturally straight rods, the

bending energy simplifies to the well-known expression

1

2

centreline.

For an efficient simulation including self-contact, we

discretize the wire into N mass-points ri, connected by

straight edges ei. Tensile energy is modeled by linear

springs connecting mass-points. For bending and torsion,

we use the quaternion group to represent the rotation of

the director frame of each edge, with resulting moments

and forces given by the gradient of the discretized form

of Eq. 1 [16]. Contact of the wire with itself or with the

cavity is modelled by a linear repelling force law. We

use the same stick-slip friction model as in Ref. [2] with

friction coefficients µs= 0.2 for static and µd= 0.18 for

dynamic Coulomb friction - a choice which yields best

agreement with experiments. Forces and moments are

integrated in time using a standard predictor-corrector

method of 6th order, with small viscous damping added

for equilibration and numerical stability. We fixed the

wire radius to a = 1 and chose 7 different sphere radii

R = (4, 5, 10, 13, 20, 40, 50). The Young’s modulus

was set to Y = 5 and ky = 0.001 for the high torsion

setup, corresponding to an intrinsic radius of curvature

of Ri= 100. The insertion speed was sufficiently small

to ensure being in the quasi-static regime, and the sim-

ulation was stopped when the measured injection force

1and k0

2describe the intrinsic curvature of the reference

Y

2(1+ν)being the torsional shear modulus, and

?L

0Y Iκ2ds, where κ is the geometric curvature of the

reached a fixed threshold.

The packing process starts with one end of the wire

being inserted into the cavity. To break symmetry of

the straight wires in simulations, small random displace-

ments are initially imposed on all nodes. When the wire

contacts the cavity walls for the first time, a loop forms

along a cavity wall. The orientation of this loop is ran-

dom for naturally straight wires depending on initial con-

ditions, whereas for curved wires it is in the prefered,

curved direction. As more wire is inserted, distinct mor-

phologies emerge for the two setups:

Low torsion setup: Here, the wire continues to form

loops which align in parallel with each other, similar to

the coiling of DNA in spherical capsids [11]. This pro-

cess leads to ring-like structures with decreasing coiling

radius. Whenever the coiling radius becomes too small, a

new coil is started at a different orientation (Fig. 1, a,b).

The result is an ordered packing in layers from outside

inwards, similar to the shells of an onion (Fig. 1, c,d).

High torsion setup: In this setup, torsion can only be

minimized on the expense of bending deformations, lead-

ing to frequent reorientations of loops. In certain cases,

torsion becomes so large that figure-eight patterns ap-

pear, or loops form with smaller radius than imposed by

geometric conditions. As a consequence, the packing is

disordered (Fig. 1, e,f) and fills the cavity homogeneously

(Fig. 1, g,h) with a large amount of accumulated torsion

(Fig. 2, d).

We turn to the study of the maximum packing den-

sity φmax and its dependence on the morphology and

effective system size a/R.

φmax follows from the

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3

0.01 0.1

a/R

0.1

0.2

0.3

0.4

max

Φ

0.010.1

a/R

0.1

0.2

0.3

0.4

max

Φ

low torsion

setup

low torsion setup high torsion setup

high torsion

setup

5102050

0.002

0.01

0.05

0.2

R

M/aα

0.005

0.01

0.015

E /G

tors

0 0.10.20.3

0

Φ

a)b)

d)

c)

0.38

2.62

?

?

?

?

?

?

?

FIG. 2: Top: Maximum packing density φmax as a function

of the effective system size a/R for the low torsion (a) and

the high torsion setup (b). Experimental data are shown with

empty, numerical data with filled symbols. DNA packing den-

sities (?) are added in the low torsion setup for reference [17].

Bottom: Length-radius scaling in the high torsion setup (c),

and a comparison between the total accumulated internal tor-

sional energy Etors as function of φ (d).

inserted wire length L as φmax

and is shown in Fig. 2 (a,b) as function of a/R for

both setups.The system sizes in experiments were

in the range 0.0069 < a/R

of R[mm] = (6.8,14.0,19.9,22.9,28.5,30.3,38.5,48.5) and

a[mm] = (0.16,0.25,0.35,0.4,0.5,1.5,1.93,2.0,3.25).

simulations a/R was within 0.02 and 0.25. Since data

points in each setup collapse on a common curve, we

conclude that φmaxin either setup only depends on a/R

and not on a and R individually. Furthermore, no de-

pendence on Young’s modulus Y and on the amount

of intrinsic curvature was found within the tested range

R ≤ Ri≤ 5R.

In comparison, the low torsion setup (Fig. 2, a) yields

higher maximum packing densities for large systems

(small a/R), that are comparable with experimental data

from ordered DNA coiling (triangles).

creased, φmax increases slower than φmax for the high

torsion setup. Consequently, around a/R = 0.2, the high

torsion setup starts to provide larger φmax, with packing

fractions up to 58%. This could be due to the fact that

torsion supports the bending of the wire and thus reduces

the required buckling force, similar to the formation of a

DNA plectoneme [18].

The strong dependence of φmax on a/R in the high

torsion setup is captured well by the power-law φmax∼

(a/R)α, with α = 0.38±0.04 (Fig. 2, b). We can further

= Lπa2/(4/3πR3)

< 0.23, with values

In

As a/R is in-

Φ

???????????

?

?

0.00.1

?

? ?

?

?????

0.20.30.4

0

500

1000

1500

2000

EbR

Y I

???

???

0.1

???

???????????

?

Φ

1

5

10

50

0.00.20.30.4

FR2/Y I

FIG. 3: Rescaled bending energy and injection force (inset)

of the wire for the low torsion setup (simulations: continuous

line, experiments: ?) and the high torsion setup (simulations:

dotted line, experiments: ?), with a/R = 1/40. The dashed

line is the prediction of the DNA spool model [11]. Simulation

data are averaged over 9 runs.

conclude that if φmaxscales as a power-law in a/R, then

the packed wire mass scales as M ∼ a2L ∼ R3(a/R)α,

i.e. M/aα∼ R3−α(Fig. 2, c). The exponent 3 − α =

2.62±0.04 is close to the experimental value of 2.75 found

by Gomes et al [19] for the forced crumpling of wires in

three dimensions.

The dominant energy in both setups is the bending

energy Eb. It is directly obtainable from simulations,

whereas experimentally, an approximation of Ebfollows

from numerical integration of the measured injection

force over the wire length. To compare results of different

cavity radii, we show in Fig. 3 the dimensionless quantity

EbR/Y I, which yields good agreement between simula-

tions and experiments. It is natural to compare these

findings to the packing of DNA in viral capsids, where

spool-like, ordered structures are assumed to emerge due

to the attractive part of the electronic interaction of dif-

ferent DNA beads. A recent analytical model by Purohit

et al [11] predicts, for the bending energy,

?√

Eb∼ −RY I

d2

s

kφ1/3+ log

?

1 −√kφ1/3

?

The only remaining

1 − kφ2/3

??

,

(2)

in which k = [3d4

parameter is the average segment distance ds, which

we determined from cross sections of the simulations as

ds= 2.81a. The prediction agrees well with our data, c.f.

Fig. 3, dashed line. The dimensionless insertion force

FR2/Y I obtained analytically from Eq. 2 is, however,

somewhat smaller than the measurements (Fig. 3, inset),

due to the fact that we include contact friction.

The distribution of the bending energy measured at

φmaxgives further insights into the statistical properties

of the morphologies (see Fig. 4). The low torsion setup

(+) features a strong concentration of energies around

s/(4π2a4)]1/3.

Page 4

4

?????

?????????????????

0.001

?????????????

??

????????????????????????????????

0.11

?

?

?

?

?

?

???????????????????????

p(

)

Eb/?Eb?

???? ?? ???????? ?? ? ????

??????????????????????

0.1

???

10

???

?

?????????????????????????????????????????

0.010.001

0.01

0.1

1

1

0.01

0

1

2

3

4

Eb/?Eb?

FIG. 4: Bending energy distribution of the packed wire in

log-linear scale and log-log scale (inset) for a/R = 1/40. Data

for the high torsion setup (∗) can be approximated by a log-

normal distribution with parameters µ = −0.905 and σ =

1.275 (continuous line). For the low torsion setup (+, dotted

line is shown as a guide to the eye) small bending energies

dominate due to the ordered coiling.

Eb/?Eb? = 0.1 due to the presence of ordered coils with

large radii. Subsequent coiling rings are squeezed into

existing ones. They take on a shape reminding of sta-

dium racepaths, with almost straight parts that lead to

a high probability for small Eb. In contrast, the high

torsion setup (∗) and resulting disordered packings yield

a distribution fitted well by a log-normal function of the

type

p(x = Eb/?Eb?) =

1

σx√2πexp

?

−(ln(x) − µ)2

2σ2

?

,

(3)

with µ = −0.901 and σ = 1.275. Log-normal distribu-

tions are usually associated with hierarchical events and

are found in similar systems of densely packed objects,

for instance in the ridge-length distribution of crumpled

paper or two-dimensional crumpled wires [4, 5, 20].

To summarize, we investigated the packing of elastic

wires into spherical cavities using a high torsion and a

low torsion setup. Low torsion led to ordered packings,

comparable to analytic predictions, while high torsion led

to disordered structures. We found highest packing den-

sities in the first case for large cavities and in the second

case for small cavities, with a cross-over as the system

size is varied. Our work elucidates the importance of

torsion in the dense packing regime, and provides novel

insights into the role of the system size with relevance

for, e.g., the surgical treatment of aneurysms. The pre-

sented results are largely independent on Young’s mod-

ulus, the amount of intrinsic curvature within the tested

range, and friction. The role of material nonlinearities,

however, remains an important open question in view of

such applications.

This work was supported by Grant No. TH-0607-3 of

the Swiss Federal Institute of Technology Zurich, FUN-

CAP, and grant No. G2010IASBS103 of the Institute for

Advanced Studies in Basic Sciences (IASBS) Research

Council. The authors would like to thank R.L. Stoop, S.

Kusuma and EMPA for their help in the analysis part of

this work.

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Teschner,in ACMSIG-