Self-Avoiding Walk of Plastic Wires Crumpled into Frictional Confinements
M.\ Reza Shaebani
Ali Farnudi
Javad Najafi
Daniel Bonn
Mehdi Habibi
The morphology of plastic wires packed into spherical confinements is investigated. The total length of the injected wire exhibits a power-law scaling with the system size (the ratio between container and wire radii $R{/r$), with an exponent dependent on the wire-wall friction. At high friction regime, coils rarely form and the number of folds follows a power-law dependence on $R {/ r$. During the injection process, the fold-size distribution gradually broadens and becomes more asymmetric, reflecting the increase of spatial exclusion effects. The cumulative wire length vs the fold size collapses onto a universal curve, allowing for the prediction of the total length of crumpled wire for different $R{/r$, material properties, and insertion forces. A self-avoiding random walk model for such a far-from-equilibrium process is introduced which remarkably reproduces the experimental data.
paper.tex
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\begin{document}

\title{Self-Avoiding Walk of Plastic Wires Crumpled into
Frictional Confinements}
\author{M.\ Reza Shaebani}
\email{shaebani@lusi.uni-sb.de}
\affiliation{Department of Theoretical Physics, Saarland
University, 66041 Saarbr\"ucken, Germany}
\author{Ali Farnudi}
\affiliation{Department of Physics, Institute for Advanced
Studies in Basic Sciences, Zanjan 45195, Iran}
\author{Javad Najafi}
\affiliation{Department of Physics, Institute for Advanced
Studies in Basic Sciences, Zanjan 45195, Iran}
\affiliation{Department of Experimental Physics, Saarland
University, 66041 Saarbr\"ucken, Germany}
\author{Daniel Bonn}
\affiliation{Van der Waals-Zeeman Institute, University of
Amsterdam, 1098 XH Amsterdam, The Netherlands}
\author{Mehdi Habibi}
\email{M.Habibi@uva.nl}
\affiliation{Department of Physics, Institute for Advanced
Studies in Basic Sciences, Zanjan 45195, Iran}
\affiliation{Van der Waals-Zeeman Institute, University of
Amsterdam, 1098 XH Amsterdam, The Netherlands}
\date{\today}

\begin{abstract}
The morphology of plastic wires packed into spherical confinements
is investigated. The total length of the injected wire exhibits a
power-law scaling  with the system size (the ratio between
container and wire radii $R{/}r$), with an exponent dependent on
the wire-wall friction. At high friction regime, coils rarely form
and the number of folds follows a power-law dependence on $R {/} r$.
During the injection process, the fold-size distribution gradually
broadens and becomes more asymmetric, reflecting the increase of
spatial exclusion effects. The cumulative wire length vs the fold
size collapses onto a universal curve, allowing for the prediction
of the total length of crumpled wire for different $R{/}r$, material
properties, and insertion forces. A self-avoiding random walk model
for such a far-from-equilibrium process is introduced which
remarkably reproduces the experimental data.
\end{abstract}

\pacs{05.40.-a, 62.20.F-, 89.75.Da, 46.65.+g}

%05.40.-a   Fluctuation phenomena, random processes, noise, and Brownian motion
%62.20.F-   Deformation and plasticity
%89.75.Da   Systems obeying scaling laws
%46.65.+g   Random phenomena and media

%46.25.-y   Static elasticity
%62.20.-x   Mechanical properties of solids
%46.32.+x   Static buckling and instability
%05.40.Fb   Random walks and Levy flights

\maketitle
Crumpling of thin objects in confined geometries is ubiquitously
observed in nature. Examples range from DNA packing in viral and
bacteriophage capsids \cite{DNA-viral,DNA-bacteriophage} to
folding of insect wings in cocoons \cite{Brackenbury94} and
plant leaves in buds \cite{Kobayashi98}. In
response to the applied forces and constraints, complex
morphologies may be adopted where friction, nonlinear response,
self-avoiding interactions, finite size effects, and boundary
conditions play crucial roles. There have been numerical efforts
to identify the separate contribution of self-avoidance on the
morphological properties of crumpled objects \cite{Vliegenthart06,
Tallinen09,Balankin,Sultan06}. It was found that the exponent of
the power-law governing the mass-size relation differs between
phantom elastic sheets and when self-avoidance and plasticity
are considered \cite{Vliegenthart06,Tallinen09}. Another
numerical study revealed that self-avoidance in highly compressed
elastic papers leads to self-correlations which modify the
hierarchical folding process \cite{Sultan06}. It is, however,
difficult to disentangle the effects of self-avoidance and
other key factors on the resulting structures of frictional
elasto-plastic materials in experiments.

So far, very little is known about the structural evolution during
crumpling \cite{Sultan06,Lin09,Cambou11,Aharoni10}. Understanding
how the space-filling properties of crumpled objects develop is
of particular importance in technological applications such as
endovascular coiling treatment of cerebral aneurysms \cite{aneurysm}.
As expected from far-from-equilibrium processes, the imposed
constraints and initial conditions cannot uniquely determine the
crumpled state. There is indeed an ensemble of admissible
configurations, from which some structural properties of the
system can be derived.

\begin{figure}[b]
\centering
\includegraphics[trim=0cm 0.8cm 0cm 1.0cm,scale=0.4]{Figure1.eps}
\caption{(a) Morphologies of plastic wires packed into spherical
containers, at low (top) and high (bottom) wire-container friction
$\mu_{_\text{wall}}$. (b) The total length $L$ of the inserted
wire, scaled by the cross-section radius $r$ of the wire versus
the relative system size $R{/}r$. The dashed lines indicate the
best power-law fits. Inset: Schematic of the side view of the
setup.}
\label{Fig1}
\end{figure}

In this Letter, we study the injection of solder wires into
frictional rigid confinements, which enables us to isolate the
influence of self-avoiding interactions. By monitoring the
morphological evolution, a gradual crossover from random to
correlated folding events is observed due to increasing spatial
exclusion effects. The folding process of plastic wire can be
considered as a self-avoiding random walk (SAW) in a confined
geometry. While numerical sampling of the ensemble of such
SAW trajectories is unfeasible via stochastic Markovian dynamics
approaches, employed e.g.\ in the simple case of regular lattices,
we introduce an alternative approach which accounts for the
time evolution of the fold length. The results of Monte Carlo
simulations show satisfactory agreement with experiments. We
verify that the final length of the injected wire can be
estimated from the geometry, material properties, and imposed
constraints.

{\it Setup ---} The experimental setup consists of a rigid hollow
spherical container of inner radius $R$ with a small hole to
insert the plastic wire (see Fig.~\ref{Fig1}). Several polymeric
semi-transparent molds with radii $R$ ranging between $5$ and $20
\, \text{mm}$ are constructed. A small nozzle and two counterrotating
rollers are attached to the injecting hole, which facilitate the
control of the insertion force and speed. The friction coefficient
 between the wire and the container wall ($\mu_{_\text{wall}}$) is
$0.4$ ($0.2$) when the smooth inner wall of each mold is
roughened by sandpaper (lubricated with silicone oil). Traditional
solder wire $\text{Sn}_{60}\text{Pb}_{40}$ with Young's modulus
$Y$ of around $30 \, \text{GPa}$ and dry wire-wire friction
coefficient $\mu_{_\text{w-w}}{\simeq}0.2$ is used as working
material. We insert wires with cross-section radius
$r\,{=}\,0.4,\,0.5,\,0.6, \, \text{or}\,0.8\,\text{mm}$ into the
ball through the nozzle with feeding speed of about $1 \,
\text{mm/s}$. The insertion process continues until the applied
feeding force is unable to fold the wire anymore, and the wire
tends to buckle outside of the ball. In the absence of elastic
restoring energy, the final structure preserves its shape after
opening the mold, which allows for a detailed structural analysis.

{\it Friction-dependent structures ---} We observe distinct
morphological phases, depending on the wire-wall friction. At
low friction ($\mu_{_\text{wall}} {\simeq}0.2$), the wire can
slide on the container surface smoothly, which increases the
chance of coil formation during the early stages of injection
process [see Fig.~\ref{Fig1}(a)]. Even highly ordered coils form
at the outer layer of the crumpled structure in relatively small
spheres, which are reminiscent of the structures observed in
packings of elastic wires \cite{Vetter14,Stoop11,Najafi12,Habibi11}.
However, at higher friction ($\mu_{_\text{wall}}{\simeq}0.4$),
the wire randomly bends due to local constraints instead of
relaxing towards a global minimum energy. This effect, together
with excluded volume interactions, promote the formation of folds
rather than coils. By introducing the frequency of sharp changes of
curvature as the order parameter, it is recently shown that
a phase transition to disordered structures occurs as friction
is increased \cite{Vetter14}. We roughened the surface of ball
to ensure that $\mu_{_\text{wall}}$ is far above the transition
point, i.e. mixed fold-coil structures are avoided and folding
is the dominant process. The wire-wire friction  prevents coiling in
the core of the structure. This enables us to focus on the
growing role of self-avoidance by analyzing the folding statistics.

\begin{figure}[b]
\centering
\includegraphics[scale=0.43]{Figure2.eps}
\caption{Scaling of (a) the number of folds $N$, (b) the
dimensionless mean fold size $\langle \ell \rangle{/}R$, and (c) the
standard deviation $\text{w}{/}R$ versus the effective system
size $R{/}r$. The full (open) symbols indicate experimental
(simulation) results. The dashed lines are power-law fits
to the experimental data. The experiment is repeated 3
times for each value of $R{/}r$.}
\label{Fig2}
\end{figure}

The influence of friction on crumpling morphology is also
visible when looking at the total length $L$ of the inserted
wire. Figure \ref{Fig1}(b) shows $L{/}r$ versus the effective
system size $R{/}r$. The data-collapse on a common curve for
different combinations of $R$ and $r$ verifies that $R{/}r$
is an appropriate control parameter to describe the behavior.
For a given value of $R{/}r$, $L$ is expectedly smaller for
higher $\mu_{_\text{wall}}$ since randomly oriented folds
effectively fill the space with a lower packing fraction
than a mixed structure of folds and coils at low $\mu_{
_\text{wall}}$. At both wall friction coefficients, the
system-size dependence of $L{/}r$ is overally captured well
by the power-law
\begin{eqnarray}
L{/}r \sim (R{/}r)^D \, ,
\label{Eq:L-scaling}
\end{eqnarray}
with a friction-dependent exponent $D$. We obtain $D=2.48$
($2.62$) at high (low) $\mu_{_\text{wall}}$ by fitting the
experimental data to Eq.~\ref{Eq:L-scaling}. Exponent of $D=2.5$ was previously obtained for ordered packing of elastic wires in spherical cavities and similar
power-law scaling (with $D{\simeq}2.75$) was reported for
packing of plastic wires \cite{Aguiar91}. Balancing the
confining energy and self-avoidance in similar systems gives
a mean-field exponent $D{=}2.5$ \cite{deGennes85,Kantor86,
Gomes08}, because the lowest (harmonic) approximation of the
confining energy yields an energy density of the order of
$(R{/}r)^{-1}$, while the self-avoidance energy density is
proportional to $(L{/}r)^2 (R{/}r)^{-6}$, as it originates
from binary contacts between wires whose density nearly grows
as $\phi^2$ where $\phi$ is the packing fraction. Analogously,
an exponent $2.5$ can be obtained between the area of an
uncrumpled sheet and its radius of gyration after crumpling
\cite{Kantor86}. This suggests the independence of the
fractal dimension from the way the plastic wire is packed
into the container. For example, one can first construct a
sheet by regularly folding the plastic wire in a plane and
then crumple it. Another point is that from the scaling of
$L$ it follows that the packing fraction $\phi{=}\pi r^2 L
{/} (\frac43 \pi R^3)$ scales as $\phi \, {\sim}\,(R{/}
r)^{D{-}3}$ with a negative exponent, indicating the decay
of density with increasing the effective system size.

{\it Fold-size statistics ---} We now turn to the analysis
of folds statistics at high friction regime. The points of
folding are distinguishable by sharp changes of wire orientation.
A minimum threshold of ninety degrees is imposed for the
turning angle of the wire to be identified as a folding point.
We cut the wire at the folding points, straighten the segments,
and measure their length using digital calipers. Straightening
of the curved segments rarely allows for fold lengths $\ell$
longer than the container diameter, but we checked that
$\ell_\text{max}$ remains smaller than $\pi R$ in the absence
of coils.

\begin{figure}[t]
\centering
\includegraphics[scale=0.4,angle=0]{Figure3.eps}
\caption{(color online) (a) Probability distribution of the
normalized fold size $\ell{/}R$. Symbols (lines) are obtained
from experiments (simulations). (b) Comparison between the tail
of the data shown in part (a) and the corresponding best overall
fits to Gaussian, log-normal, and gamma functions. The solid
lines show the simulation results.}
\label{Fig3}
\end{figure}

A key observation is the scaling of the total number of folds
$N$ with the effective system size $R{/}r$. As shown in
Fig.~\ref{Fig2}, a power-law relation of the form
\begin{eqnarray}
N\,{\sim}\,(R{/}r)^\beta
\label{Eq:N-scaling}
\end{eqnarray}
holds with $\beta{\simeq}1.86$. Notably, the number of steps in
a self-avoiding random walk on a cubic lattice scales in a similar
way with the size of the  visited region, even though with a
somewhat smaller exponent $\beta{\simeq}5{/}3$ \cite{Havlin82}
(for an ordinary random walk $\beta$ equals 2). Similar scaling
laws also hold for the number of coils in 3D ordered elastic packing ($\beta{\simeq}1.5$) \cite{Najafi12},
or the number of loops in 2D packing of crumpled
wire at high plasticity regime \cite{Donato02,Gomes08,Stoop08}.
From the scaling of $L$ and $N$ with $R{/}r$ one finds that the
normalized mean fold size $\langle {\ell} \rangle {/} R$ follows
$(R{/}r)^{D{-}\beta{-}1{\simeq}-0.4}$. The best fit to the
standard deviation of $\ell$ versus the system size also follows
a power-law: $\,\text{w}{/} R\,{\sim}\,(R{/}r)^{0.9}$. The same
exponent is reported for the roughness of crumpled surfaces
\cite{Plouraboue96} which might indicate a universality class
for roughness exponent in crumpled systems independent of
dimensionality.

The probability distribution $P(\ell {/}R)$ of the normalized
fold size is plotted for different values of $R{/}r$ in
Fig.~\ref{Fig3}. $P(\ell{/}R)$ becomes more asymmetric with
the peak shifting towards smaller fold sizes by increasing
$R{/}r$. At earlier stages of the injection process in a given
sphere size, the injected wire proceeds in the container
without interacting with the accumulated wire. Assuming that
the wire bends at a random point between the injecting hole
and a contact point at the container surface, the resulting
fold length is a random variable, symmetrically distributed
between $0$ and $\ell_\text{max}$. By increasing the length
of accumulated wire, spatial exclusion effects grow and the
injected wire cannot easily proceed through the sphere without
touching the crumpled structure. Hence, long folds gradually
become less probable and $P(\ell{/}R)$ becomes more asymmetric
due to relatively large populations of smaller folds. The
evolution of $P(\ell{/}R)$ with increasing $R{/}r$ in
Fig.~\ref{Fig3}(a) should similarly indicate the growth of
spatial exclusion effects, which is counterintuitive since
the packing fraction $\phi$ decreases with increasing the
system size. The circular cross section of wires allows them
to slide over each other rather easily (which results in
slightly curved folds). Indeed, the strength of self-avoidance
effects is better captured by the length of the crumpled wire,
rather than the volume excluded by it. Thus, we can ignore the
role of wire thickness as a spatial exclusion constraint, and
assign the entire contribution to the length of wire, reflected
in the dimensionless quantity $\lambda{=}\frac{L}{R}{\sim}(R{/}
r)^{1.5}$ (while $\phi{=} \frac{\pi r^2 L}{\frac43 \pi R^3}
{\sim}(R{/}r)^{{-}0.5}$).

A closer look at the tail behavior of $P(\ell{/}R)$ in
Fig.~\ref{Fig3}(b) verifies that the correlations grow
with $R{/}r$. The experimental data is fitted to Gaussian,
log-normal, and gamma distributions over the entire range
of $\ell{/}R$. While the Gaussian function represents the
distribution of random (uncorrelated) data, the gamma and
log-normal distributions are associated with random events
in the presence of self-correlations or those that occur
hierarchically \cite{Vliegenthart06,Sultan06,Wood02,Blair05,
Adda-Bedia10,Deboeuf13}. The tail behavior is better captured
by the Gaussian for small $R{/}r$, while there is a gradual
crossover towards the two other functions as $R{/}r$ increases.

{\it Self-avoiding random walk model ---} To better understand
the mechanism of increasing spatial exclusion effects,
we model the folding process as a self-avoiding random walk
(SAW) of the wire inside the confinement. While the existing
SAW algorithms mainly follow stochastic Markovian dynamics
to sample the ensemble of trajectories on regular lattices,
here we propose an alternative approach which accounts for
the time evolution of the step size $\ell$. We first suppose
that the strength of self-avoidance effects after the $n$-th
fold is mainly controlled by the length $L_n$ of the inserted
wire, i.e.\ the larger is the parameter $\lambda_n{=}
\frac{L_n}{R}$, the smaller is the success probability for
the fold $n{+}1$ to be a long fold. The size $\ell$ of the
next fold is obtained via the following algorithm: The
distance $\ell$ between two randomly chosen points inside
the sphere is taken as the trial fold size, and the proposal
is accepted according to a Metropolis-like criterion with
probability
\begin{figure}[t]
\centering
\includegraphics[scale=0.40,angle=0]{Figure4.eps}
\caption{(color online). The cumulative length $L_n$ of
the inserted wire after the $n$-th fold. The full (open)
symbols indicate experimental (simulation) results. The
simulation results are multiplied by $10$ for clarity.
Inset: $L_n{/}R$ averaged over all experiments (solid line)
and its standard deviation (shaded area). The dotted
line is obtained from Eq.~(\ref{Eq:Sum}).}
\label{Fig4}
\end{figure}
\begin{eqnarray}
P\!\!_{n{+}1}(\ell) = \mathcal{N}^{-1}
\exp\big[{-}\beta \lambda_n \, \ell{/}R \big],
\label{Eq:AcceptanceProb}
\end{eqnarray}
where $\mathcal{N}{=}\frac{R}{\beta \lambda_n} (1{-}\exp
[-\beta \lambda_n \ell_\text{max}/R])$ is the normalization
factor, and $\beta$ is a coefficient which depends on wire
properties. In the case of rejection, a new  $\ell$
is tried. Finally, the cumulative length is updated as
$L_{n{+}1}{=}L_n{+}\ell$ before starting the next step.
Equation (\ref{Eq:AcceptanceProb}) assumes an exponentially
lower acceptance chance for larger trial lengths $\ell$.
Moreover, the acceptance probability decreases with
increasing $\lambda_n$, as the self-avoidance effects become
more pronounced. The method indeed samples the fold-length
landscape according to a Boltzmann-like distribution.
Initially, the wire walks in free space ($\lambda_n{=}0$),
thus, $P\!\!_{n{+}1}(\ell){=}\frac{1}{\ell_\text{max}}$
independently of the trial length $\ell$. However, larger
values of $\ell$ become gradually less probable with
increasing $\lambda_n$. We perform extensive Monte Carlo
simulations by adjusting $\ell_\text{max}$ and the
threshold value of $\phi$ to the experimental data. The
results shown in Figs.~\ref{Fig2}-\ref{Fig4} are in
remarkable agreement with experiments (the straight folds
in simulations lead to slightly lower values of $\langle
{\ell} \rangle {/} R$ in Fig.~\ref{Fig2}(b) compared to
curved folds in experiments). The exponents are not
affected by the choice of $\beta$, however, we treat it
as a free parameter to quantitatively reproduce the
experimental data. We also checked that the power-law
scalings can not be reproduced when replacing $\lambda_n$
with $\phi_n$ in Eq.~(\ref{Eq:AcceptanceProb}). Importantly,
from the simulation results in Fig.~\ref{Fig3}(b), we can
specify the growing of self-correlations as dominant
mechanism rather than hierarchical events. A numerical
study of crumpled elastic papers also revealed that
self-avoidance at high compression leads to
self-correlations which modify the hierarchical folding
process and the shape of the segment length distribution
\cite{Sultan06}.

The cumulative length $L_n$ of the inserted wire after the
$n$-th fold qualitatively collapses onto a master curve for
different values of $R{/}r$, as shown in Fig.~\ref{Fig4}.
The folds are initially independent of each other and $L_n
{/}R$ grows linearly with $n$. The steps however become
more correlated with increasing $n$, leading to a slower
growth of $L_n{/}R$. From the scaling of $L$ and $N$ with
$R{/}r$ [Eqs.~(\ref{Eq:L-scaling}) and (\ref{Eq:N-scaling})]
one obtains the asymptotic scaling $L{/}R \,{\sim}\,
N^{(D{-}1){/}\beta{\simeq}0.8}$, which is consistent with
experiments (see inset). Starting from $\langle {\ell_1}
\rangle{=}\frac{\ell_\text{max}}{2}$, the mean fold size
at next steps can be estimated in terms of the accumulated
wire length $L_n$ as
\begin{eqnarray}
\langle {\ell_{n{+}1}} \rangle = \int_0^{\ell_\text{max}}
\!\!\!\!\!\!\!\!\!\!\!\! P\!\!_{n{+}1}(\ell) \, \ell \,
\text{d}\ell = \frac{R}{\beta \lambda_n} {+} \frac{
\ell_\text{max}}{1{-}e^{\beta \lambda_n \ell_\text{max}/R}}.
\label{Eq:ln-Average}
\end{eqnarray}
Hence, we obtain the following recursive analytical
expression for the injected length of wire after the
$n$-th fold
\begin{eqnarray}
\displaystyle \frac{L_n}{R} {=}
\frac{\ell_\text{max}}{2R} {+} \sum_{i{=}1}^{n{-}1}
\bigg[ \frac{R}{\beta L_i} {+} \frac{\ell_\text{max}{/}R}
{1{-}\exp[\beta\ell_\text{max}L_i{/}R^2]}\bigg],\,\,\,\,\,
\label{Eq:Sum}
\end{eqnarray}
in excellent agreement with the data [see Fig.~\ref{Fig4}(inset)].

\begin{figure}[b]
\centering
\includegraphics[scale=0.42,angle=0]{Figure5.eps}
\caption{(color online). Collapse of $L_n{/}R$ vs.\
the scaled fold length $\ell_n{/}R$ for different values
of $R{/}r$ in experiments (background gray lines) and 
simulations (thick colored lines). Same colors as in 
Fig.~\ref{Fig4}. The symbols show the experimental cutoff 
values $\ell_c{/}R$ for $r{=} 0.5\,\text{mm}$. Inset: 
$\ell_c$ versus the sphere radius $R$. The horizontal line 
shows the average value $\bar\ell_c{=}5.8\,\text{mm}$.}
\label{Fig5}
\end{figure}

{\it Buckling threshold ---} Finally, we clarify how
the injection of plastic wire eventually stops. According
to the experimental results shown in Fig.~\ref{Fig5}, the
cutoff fold length $\ell_c$ (averaged over the last five
folds) is independent of the system size $R$ for a given
wire radius. We suggest that the process stops when the
fold size becomes so small that the applied feeding force
$F$ is unable to fold the wire anymore. The minimum threshold
length can be obtained from the Euler buckling theory as
$\ell_c{=}\sqrt{\frac{\pi^3Y}{16F}}r^2$ for fixed-free end
conditions, thus, $\ell_c$ is determined by the applied
force and wire properties independent of $R$. For $r{=}
0.5\,\text{mm}$ and a typical force of $F{=}100N$, we obtain
$\ell_c{\simeq}6.0\,\text{mm}$, comparable to the
experimental values.

Figure~\ref{Fig4} interestingly evidences a universal filling
mechanism independent of the relative system size. The
normalized cumulative wire length $L_n{/}R$ versus the
normalized length of the $n$-th fold $\ell_n{/}R$ collapses
onto a universal curve for different values of $R{/}r$
(Fig.~\ref{Fig5}). As the wire injection continues, $\ell_n
{/}R$ gradually decreases until it reaches the minimum
threshold value $\ell_c{/}R$, where the process eventually
stops. For a given value of $r$, $\ell_c$ is
independent of $R$, thus, $\ell_c{/}R$ decreases (i.e.
shifts to the left in Fig.~\ref{Fig5}) with increasing $R$.
Consequently, the cutoff number of the steps at which
the process stops increases with the system size. By
calculating $\ell_c$ from the insertion force and wire
properties, one can predict the total length of the crumpled
wire for different system sizes.

In conclusion, we studied the growing self-avoidance
effects during the injection of plastic wires in frictional
containers. The direct measurement of the evolution of
fold-length distributions verified that the process goes on
in a correlated way, such that the longer the injected wire,
the stronger the spatial exclusion effects. The results provide
a new insight into underlying mechanisms of crumpling, which
goes beyond a simple hierarchical description of the process.
The proposed sampling method of SAW trajectories would open
the door to feasible Monte Carlo simulations of self-avoiding
random walks in continuous space inside arbitrary confinements
to obtain macroscopic quantities of interest, as an alternative
to computationally expensive DEM methods.

The experiments were done in the Complex Fluid Laboratory
of IASBS. The authors gratefully acknowledge support from the
IASBS Research Council and DFG within SFB 1027 (A7).

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