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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
0.01 0.1
a/R
0.1
0.2
0.3
0.4
max
Φ
low torsion
setup
low torsion setuphigh torsion setup
high torsion
setup
5 10 20 50
0.002
0.01
0.05
0.2
R
M/aα
0.005
0.01
0.015
E /G
tors
00.10.20.3
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.
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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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