# Unjamming strongly compressed particle rafts

Gregor Plohl
&Mathieu Jannet
&Carole Planchette
<EMAIL_ADDRESS>

Institute of Fluid Mechanics and Heat Transfer, Graz University of Technology,
A-8010 Graz, Austria

###### Abstract

We experimentally study the unjamming dynamics of strongly compressed particle
rafts confined between two fixed walls and two movable barriers. The back
barrier is made of an elastic band, whose deflection indicates the local
stress. The front barrier is pierced by a gate, whose opening triggers local
unjamming. The rafts are compressed by moving only one of the two barriers in
the vicinity of which folds form. Using high speed imaging, we follow the
folded, jammed, and unjammed raft areas and measure the velocity fields inside
and outside of the initially confined domain. Two very different behaviors
develop. For rafts compressed by the back barrier, only partial unjamming
occurs. At the end of the process, many folds remain and the back stress does
not relax. The flow develops only along the compression axis and the particles
passing the gate form a dense raft whose width is the gate width. For rafts
compressed at the front, quasi-total unjamming is observed. No folds persist
and only minimal stress remains, if any. The particles flow along the
compression axis but also normally to it and form, after the gate, a rather
circular and not dense assembly. We attribute this difference to the opposite
orientation of the force chain network that builds up from the compressed side
and branchs. For rafts compressed at the gate side, keystone particles are
immediately removed which enhances local disentanglement and leads to large
scale unjamming. In contrast, for back compressed rafts, the force chain
network redirects the stress laterally forming arches around the gate and
resulting in a limited unjamming process.

##  1 Introduction

Capillary adsorbed particles at liquid interfaces have been since long used in
emulsions and foams Ramsden (1903); Pickering (1907); Pitois and Rouyer
(2019). Their potential for further applications, especially for producing
bijels Binks (2002a); Herzig et al. (2007); Cates and Clegg (2008), chemical-
free reversible encapsulation Aussillous and Quéré (2001); Abkarian et al.
(2013); Jambon-Puillet et al. (2018); Pike et al. (2002) and membranes
Kralchevsky et al. (2001) has renewed interest in them. In this context, one
key-attribute of the particles is their capacity to stabilize interfaces. This
stabilization partly results from the large reduction of surface energy caused
by the particle adsorption. Further stabilization originates from the physical
barrier the particles build that prevents direct contact with other surfaces.
The effects resulting from these two stabilization mechanisms remain
challenging to characterize at the macroscopic scale. The most common way
consists in characterizing these effects via the description of the
interfacial mechanical properties. Thanks to intensive research, these
properties are fairly well outlined for quasi-static regimes and moderate
compression. Yet, for general conditions, they remain poorly understood and
therefore poorly predictable. This lack of knowledge is particularly profound
for large strains and large strain rates or when strong gradients of particle
density exists. For these conditions, commonly encountered in natural
situations and industrial processes Garbin (2019), barely no data exist.
Consequently, key aspects such as self-healing capacity, self-healing dynamic
and processability of these interfaces remain widely unexplored. Our paper
aims to shed some light onto these topics.

Before detailing our method and findings, it is helpful to recall the existing
knowledge about the mechanical properties of particle-laden interfaces. At
moderate particle density, the interface is viscous. It becomes viscoelastic
upon the particle network percolation, and finally behaves as a solid for
larger particle density Reynaert et al. (2007); Cicuta et al. (2003); Lagubeau
(2010). The transition toward a solid-like behavior is associated to a
pressure collapse Aveyard et al. (2000); Monteux et al. (2007) and attributed
to a jamming process Liu and Nagel (1998), a phenomenon common to other
athermal assemblies, also called soft glassy materials Sollich et al. (1997);
Hébraud and Lequeux (1998). If further compressed, the interface buckles
giving rise to regular wrinkles whose amplitude regularly increases, until one
of them dramatically grows and eventually collapses into a large fold.

To date, most studies were motivated by practical interest such as bubble
dissolution arrest and foam stabilization Beltramo et al. (2017); Taccoen et
al. (2016); Abkarian et al. (2007); Timounay and Rouyer (2017) and therefore
focused on the mechanical properties of interfaces prepared between the
jamming and folding transitions. In this range, it has been shown that the
interface can be modelled as a continuous media. This approach is based on the
buckling of the interface subjected to quasi-static uni-axial compression.
Similar buckling is observed for monolayers of irreversibly adsorbed molecules
Milner et al. (1989), membranes Helfrich (1973) and solid sheets Cerda and
Mahadevan (2003); Pocivavsek et al. (2008). The energy minimization selects a
wavelength which results from the competition between bending and
gravitational ironing. The selected wavelength gives the elastic Young modulus
(for 3D approach) or the elastic bending modulus (for 2D approach) of the
particle-laden interface Vella et al. (2004). This elastic modeling proved to
be valid for dynamic regimes too Planchette et al. (2012) but fails when
mixtures of small and large particles were used. In this case, deviations from
the expected modulus are observed which were attributed to possible variations
of the stress transmission efficiency, potentially caused by the coexistence
of different types of particle-particle contacts Petit et al. (2016). This
observation with others, such as: finite size effects in uni-axial compression
Cicuta and Vella (2009), mechanical response under compression Pitois et al.
(2015); Taccoen et al. (2016), raft fractures Vella et al. (2006),
inhomogeneous stress propagation Basavaraj et al. (2006); Saavedra et al.
(2018), plasticity Jambon-Puillet et al. (2018)… evidence the limit of the
elastic model and points toward the granular character of these interfaces.
Indeed, these limits are probably related to the framework successfully
developed for 2D granular media. In dry granular matter, “force chains” emerge
which have been experimentally observed and numerically computed Tordesillas
et al. (2011); Peters et al. (2005). These force chains are suspected to build
up in particle rafts too, where they could significantly modify the dynamical
response Petit et al. (2016); Planchette et al. (2018). Yet, dedicated
investigations remain scarce and the present paper constitutes a first step in
this direction.

More precisely, this work aims to understand how the above mentioned granular
character of the interface, evidenced for moderate strains and mostly under
quasi-static regime, affects its mechanical properties when subjected to large
strains or fast strain rates.Said differently, is the granular character of
the interface relevant for the relaxation of strongly compressed rafts? and
what could be its effects? While answering these questions undoubtedly leads
to a gain of fundamental knowledge, it also addresses a crucial practical
aspect, known under the term of "self-healing". In contrast to commonly used
surfactants, the particles are irreversible absorbed to the interface and can
therefore not constitute reservoirs in the bulk from where to migrate in order
to feed freshly created interface areas Binks (2002b); Garbin (2013). While
this aspect is positive to arrest bubble dissolution, it remains challenging
for expending interfaces. A good strategy to efficiently overcome this
challenge might be to establish particle reservoirs outside the bulk, and more
particularly in interfacial folds. In this context, the questions can be
reformulated as such: can particles located within folds become available to
stabilize uncovered areas, providing self-healing capacity to these
interfaces? Is this capacity, i.e. the portion of stored particles that can be
efficiently released, influenced by the granular character of the interface?
What about the kinetic of this supply, i.e. the self-healing dynamic?

We answer these questions by studying the dynamical relaxation of strongly
compressed particle-laden interfaces and show that two typical behaviors are
obtained, which are solely selected by the direction of compression. First, we
describe the experimental methods. The results - obtained for two square rafts
of same compression performed either from the back or the front side - are
then detailed. We show that similar findings are also obtained for less
compressed rafts. We attribute the different behaviors to a purely granular
effect, namely the orientation of the chain force network, which either favors
unjamming via the removal of keystone particles, or prevents it by redirecting
the stress toward the sides. The paper ends with the conclusions which
underline the consequences of our findings in terms of interface
processability, self-healing capacity, and dynamic.

##  2 Experimental methods

###  2.1 Set-up

Figure 1: Experimental set-up sketched from (a) top and (b) side view. The
particle raft relaxed length, $L_{r}$, is equal to its width $W$. The raft is
compressed by moving either the back barrier (red) or the front one (blue) to
obtain a length $L_{c}<L_{r}$. The elastic deflection $\delta$ (exaggerated
here) enables to calculate the local stress.

As sketched in Fig. 1, the experimental set-up consists of a rectangular
trough with two fixed parallel walls separated by 6 cm and two movable
barriers that can be translated along the two walls to compress the confined
particle raft. The so-called "back" barrier is made of an elastic string
placed in the plane of the interface, perpendicularly to the side walls. The
string is produced in house by injecting a (1:1) mixture of Elite Double 8
basis and catalyst (Zhermack Spa) into a glass capillary, which is manually
removed after the elastomer reticulation has been completed. The elastic is
then fixed to a 6 cm broad structure and calibrated using known weights to
obtain its Young modulus, for details please see Appendix. During the
experiments, the elastic deflection, $\delta$, is measured and used to
calculate the stress developing at the back side of the raft, see Appendix.
The second barrier, refereed to as "front" barrier, is pierced in its center
by an orifice of width $w=1cm$. A vertically sliding gate made of a thin plate
enables to maintain the orifice closed. By suddenly lifting the gate, the
orifice gets unblocked and the applied stress is locally released.

###  2.2 Particles

The particles are sieved and silanized glass beads. The diameter distribution
measured over pictures of more than 1000 particles corresponds to a Gaussian
with a mean of 107 $\mu m$ and a standard deviation of 8.4%. After having
cleaned the particles with a piranha mixture, silanization is performed using
solution of trichloro-perfluorooctylsilane in anhydrous hexane. All chemicals
were purchased from Sigma-Aldrich and used as received. The resulting contact
angle measured on single particles placed at the apex of a pendant drop is
found to be $107^{\circ}\pm 10^{\circ}$.

###  2.3 Experimental procedure

The first step of the experimental procedure consists in obtaining square
rafts. To do so, the trough is filled with distilled water and particles are
sprinkled on the interface between the barriers. By gently blowing on them, we
insure that they distribute in a monolayer. This monolayer is then compressed
and decompressed by moving any of the two barriers and the quantity of
particles is adjusted in order to have a square relaxed raft. We define the
raft as relaxed as soon as the stress measured during decompression vanishes.
Noting $L_{r}$ the relaxed length and $W$ the trough width, we have
$L_{r}=W=6cm$. Once a square raft is formed, the distance between the barriers
is increased and the raft is annealed by stirring the particle assembly. The
raft is then compressed again, but by moving only one of the two barriers, the
other one remaining fixed until the end of the experiment. The state of
compression is then given by $K=(L_{r}-L_{c})/L_{r}$ with $L_{c}$ the distance
between the two barriers in their final position. Finally, the relaxation is
triggered by suddenly opening the gate. A high speed camera placed under the
raft records its evolution. Practically, 3000 frame per second are recorded.
The images have a resolution of 14.1 $\mu m/pixel$. The movies are then
analysed to provide different measurements.

Figure 2: Image sequences of strongly compressed rafts ($K=67\%$) after local
release of the stress achieved by lifting the gate of the front barrier. The
opening instant is taken as time origin. The red color indicates unjammed
particles area while the blue shows the "escaped" particles. Top: back
compression (subscript b) with from left to right: $t_{b,1}=70$ ms,
$t_{b,2}=120$ ms, $t_{b,3}=250$ ms, $t_{b,4}=360$ ms and $t_{b,5}=430$ ms.
Bottom: front compression (subscript f) with from left to right: $t_{f,1}=80$
ms, $t_{f,2}=410$ ms, $t_{f,3}=630$ ms, $t_{f,4}=1600$ ms, $t_{f,5}=2800$ ms.

##  3 Measurements

###  3.1 Unjammed, jammed, folded and escaped areas

During the raft relaxation, several quantities are measured, which first need
to be defined. The illustrative image sequences displayed in Fig. 2 are useful
to do so. Let us first consider what happens in the initially confined domain.
After the gate opens, some particles locally unjam. The corresponding surface
area, $A_{uj}^{*}$, colored in red in Fig. 2, is tracked using ImageJ and the
machine learning plug-gin called Trainable Weka Segmentation Arganda-Carreras
et al. (2017). After normalisation by the initial confined domain area
$A_{c}=WL_{c}$, we obtain $A_{uj}=A_{uj}^{*}/A_{c}$. Per definition, the
particles that remain jammed occupy the normalized area $A_{j}=1-A_{uj}$. They
can take the form of macroscopic folds or remain in the plane of the
interface. To characterize this distribution, we measure the so-called folded
surface area $A_{f}^{*}$. Practically, the folds correspond to interface
portions which make a significant angle to the horizontal plane and therefore
appear dark on the back-lighted pictures. Thus, $A_{f}^{*}$ is obtained by
using a threshold function and is then normalized by $A_{c}$ to provide
$A_{f}$. Note that this term refers to the projection of the folded surface
and not to the surface area contained in these folds, which is not accessible
with our images.

Let us now look to what happens outside of the initially confined domain. Some
of the unjammed particles flow trough the orifice and migrate further to form
a more or less dense assembly, colored in blue in Fig. 2. The surface area
occupied by this assembly $A_{e}^{*}$, is tracked using a threshold function
applied after subtraction of the background taken in the absence of particles.
The normalized escaped area $A_{e}$ is obtained by dividing $A_{e}^{*}$ with
$(L_{r}-L_{c})W=A_{r}-A_{c}$, i.e. with the difference between the relaxed
raft area and the one of the confined domain. This difference represents the
surface that the excessive particles would occupy if forming a dense relaxed
raft.

###  3.2 Back stress

Additionally, the deflection of the elastic barrier $\delta$, is recorded and
used to compute $\Pi$, the lineic pressure (or stress) developing at the back
of the raft. For details about the conversion of $\delta$ into $\Pi$, please
read the Appendix.

###  3.3 Velocity fields

Finally, we use the PIVlab routine of Matlab Thielicke and Stamhuis (2014);
Thielicke and Sonntag (2021) to gain information about the velocity fields
developing in these systems. In practise, we perform three types of Particle
Image Velocimetry (PIV).

The first one focuses on the flow inside the confined area, and more precisely
within the unjammed area. To avoid wrong interpolations with movements taking
place in the jammed raft, such as fold translation, masks corresponding to the
identified unjammed areas (red regions in Fig. 2) are applied, limiting the
analysis to the region of interest. The second analysis is similar and
consists in performing PIV outside of the confined domain using masks that
correspond to the escaped assembly (blue regions in Fig. 2. Note that we
further reduce the masks of about 5 mm on the orifice side, since the
nonuniform background of this zone causes errors in the PIV.

These two PIVs, typically performed with a time resolution of $300Hz$, provide
- for every instant - the velocity field inside and outside the confined
domain. Due to the large number of frames, the instantaneous velocity fields
are difficult to visualize and interpret. To facilitate their analysis, we
divide each relaxation in four sub-phases, and compute the distribution of the
velocity components parallel and normal to the compression axis
($v_{\parallel}$ and $v_{\perp}$) for each sub-phase. The sub-phases
correspond to the time lapse between two consecutive images of Fig. 2, i.e to
$t_{b,i}\leq T_{b,i}\leq t_{b,i+1}$ and $t_{f,i}\leq T_{f,i}\leq t_{f,i+1}$,
with $1\leq i\leq 4$, for the back and front compression, respectively.

The third analysis deals with the velocity profile at the orifice. To obtain
this information, we limit the analysis to a small zone covering the orifice
and increase the time resolution to at least $600Hz$. The width of the
analysed area is the orifice width and its length is fixed to $10mm$. For each
picture, the obtained velocity field is then projected along the compression
direction to provide the velocity profile $u(x)$, with $-w/2\leq x\leq w/2$.
The flow-rate equivalent mean velocity $\overline{u}$ is then calculated as
$\overline{u}=\frac{1}{w}\int_{-w/2}^{w/2}{u(x)dx}$.

##  4 Results and discussion

Here and in the rest of this article, results obtained with front compression
are presented as solid lines and full symbols while those corresponding to
back compression are represented by dashed lines and empty symbols. The red
color systematically indicates quantities related to the confined domain while
blue is used for quantities defined outside.

###  4.1 Detailed analysis of rafts compressed at 67%

In this section, we present the results obtained for two rafts compressed at
67% by moving solely the front barrier or solely the back barrier.
Illustrative image sequences are reproduced in Fig. 2. To facilitate the
observation, the unjammed area found in the confined domain, $A_{uj}^{*}$, is
systematically colored in red and the surface occupied by the "escaped"
particle assembly, $A_{e}^{*}$, in blue. These colored masks, automatically
obtained by the image treatment (section 3.1), are in very good agreement with
the areas detected by eyes and can therefore be used for further analysis. The
two rafts behave differently. For the back compression (top sequence), only a
small zone found immediately behind the orifice umjams. The unjammed particles
flow through the orifice, along the compression axis, to form a dense assembly
whose width is the orifice width. The relaxation process stops while many
folds are still visible. In contrast, the relaxation of the raft compressed
from the front (bottom sequence) leads to important unjamming, which extends
to almost all the confined domain. The escaped particles form a large assembly
of reduced density and broad width. No folds remain indicating a full
relaxation of the raft. As the caption timestamps indicate, the relaxation of
the back compressed raft is much shorter. It almost stops after $0.5s$, i.e.
long before the one of the front compressed raft finishes, after more than
$3s$.

Figure 3:  Temporal evolution of $A_{uj}$ (red), $A_{f}$ (black) and $A_{e}$
(blue) for (a) back- and (b) front- compressed raft.

To quantify this observation, we plot the temporal evolution of the normalized
unjammed area $A_{uj}$ (red), folded area $A_{f}$ (black) and escaped assembly
area $A_{e}$ (blue). The curves, shown in Fig. 3, confirm the qualitative
findings. For the raft compressed from the back, slightly less than 15% of the
initially confined area unjams and approximately 35% of the confined area
appears folded at the end of the process. Interestingly, the unjammed area
does not significantly grow but rather fluctuates around its mean value. These
fluctuations originate the folds dynamics. The folds, initially found at the
back (Fig. 2, $t_{b,1}=70ms$) migrate towards the orifice and transiently
reduce the unjammed area ($t_{b,2}=120ms$) before disintegrating
($t_{b,3}=250ms$), explaining likewise the correlation between the local
minimums of $A_{uj}$, at $\approx 0.2s$ and $\approx 0.45s$, and the local
maximums of $A_{f}$. Finally, the particles that escape covers between 15% and
20% of the surface excessive particles would cover if forming a dense relaxed
raft. This means that the large majority of the particles stored in the folds
(at least 80%) cannot be made available to supply the neighboring surface
initially free of particles. From a practical point of view, the self-healing
capacity of such interfaces appear very limited under the present conditions.

Let us now consider the raft compressed to the same level but from the front
side. The evolution of the unjammed, folded and escaped areas appear to be
totally different, see Fig. 3(b). About 60% of the initially confined area
unjams. The folded area, which represents at maximum 15% of the confined area
totally disappears. Here as well, the evolution of these two quantities seem
to be correlated as indicated by the coincidence of the local minimums of
$A_{uj}$ with the local maximums of $A_{f}$, found around 1.4s and 2.2s. Yet,
in contrast to back compressed raft, these events do not correspond to the
migration, expansion and disintegration of individual folds but to the
elimination of two larger folded blocks, found on both sides of the orifice.
The first elimination takes place between the third ($t_{f,3}=0.63s$) and
fourth ($t_{f,4}=1.6s$) pictures of Fig. 2 while the second one occurs between
the fourth ($t_{f,4}=1.6s$) and fifth ($t_{f,5}=2.8s$) pictures. Beside these
two bumps, $A_{uj}$ continuously increases while $A_{f}$ decreases. Almost all
particles initially stored in folds unjam and migrate trough the orifice to
cover a surface that is 1.3 times the excessive surface given by
$A_{r}-A_{c}$. While a value larger than 1 can first be surprising, it is well
explained by the fact that the assembly of escaped particles is less dense
than the relaxed raft. The relaxed raft density is close to the one of
jamming, defined as $\varphi_{j}=\pi/2\sqrt{3}\approx 0.91$. The one of the
escaped assembly can be only roughly estimated from our pictures, providing
$\varphi_{e}\approx 0.74$, in agreement with 0.76, the value required to
obtain "particle surface conservation" under total unjamming and given by
$\varphi_{e}=\varphi_{j}/A_{e}$. These results clearly show that if the
compression direction is favorable, the self-healing capacity of particle-
laden interfaces can become very important and approach its theoretical
maximum. This maximum is found when all particles initially stored into folds
are efficiently released and mobilized to cover initially "free" interface.

At this stage, it is difficult to draw conclusions about the self-healing
dynamics. For front compressed raft, only $0.12s$ are required to obtain
$A_{e}=15\%$ by comparison to $0.33s$ for back compressed raft. Yet, this rate
quickly slows down. If calculated over a longer period of time ($2s$), it is
found to be $\dot{A}_{e}=0.5s^{-1}$, comparable to $0.45s^{-1}$, the rather
constant rate obtained for back compression.

To better understand the relaxation dynamics, we perform PIV on the unjammed
areas found inside and outside the initially confined domain. Representative
snapshots of the results are shown in Fig. 4. For the back compressed raft,
both the velocity magnitude and direction are uniform. The vectors clearly
indicate a flow in the compression direction, from the back of the raft to the
orifice and beyond. In contrast, the front compression gives rise to local
acceleration, especially close to the orifice, and components perpendicular to
the compression direction develop. Interestingly, the velocity field of the
unjamming particles inside the confined domain is not symmetric. Indeed, the
instant chosen here ($1.28s$) corresponds to the beginning of the already
mentioned elimination of one of the two folded blocks found on each side of
the orifice (here, up on the picture).

Figure 4: Exemplary results of the PIV for (a) back compressed raft,
$t_{b}=0.28s$ and (b) front compressed raft, $t_{f}=1.28s$.

Figure 5: Normalized velocity distributions for left: back compressed raft,
and rigth: front compressed raft. Red: unjammed area inside the confined
domain; blue: escaped particles outside of it. From top to bottom, the sub-
phases are: $T_{b,1}$, $T_{b,2}$ ,$T_{b,3}$, $T_{b,4}$ for the back compressed
raft; and $T_{f,1}$, $T_{f,2}$ ,$T_{f,3}$, $T_{f,4}$ for the front compressed
raft. The contours correspond to 0.1, 0.5 and 0.9 of the maximum value.

To visualize the PIV results obtained over the whole process, it may be useful
to compare the velocity distribution obtained on the sub-phases defined in
section 3.3. They are shown in Fig. 5, left and right, for back and front
compressed raft, respectively. For the back compressed raft, the velocity
fields inside (red) and outside (blue) of the confined domain are always very
similar, confirming that the unjammed particles form a cohesive assembly,
which moves as a block. The velocity component perpendicular to the
compression axis, $v_{\perp}$, is centered in zero and shows very small
fluctuations, the 50% contour being comprised in $\pm 0.01m/s$. This is not
the case of the parallel component, $v_{\parallel}$, which has positive values
along the whole process. The flow from the back to the front shows some
variations with a maximum during $T_{b,2}$, i.e. when the first fold
disintegrates. Its mean value ($0.07m/s$) and the fluctuations around it
($-0.03m/s$ and $+0.07m/s$) then decrease until the end of the process.

The velocity fields that develop for the front compressed raft are very
different, see Fig. 5. The correspondence between inside and outside is lost,
which may indicate a certain independence of the two flows. Inside the
confined domain (red), significant movements develop perpendicular to the
compression axis leading to $v_{\perp}$ values as large as $v_{\parallel}$
values. The flow is not symmetric across the compression axis, see for example
$T_{f,3}$ or $T_{f,4}$ for which $-0.01<v_{\perp}<+0.11m/s$ and
$-0.06<v_{\perp}<0.01m/s$, respectively. This profound asymmetry corresponds
to the successive dislocation of two large folded blocks found on each side of
the orifice. While passing through the orifice, the particles recover a
symmetric velocity field (see blue maps). One can suppose that the shear they
experience at this point overcome the capillary attraction, forcing them to
rearrange and explaining the reduced density. Finally, the focus of the
velocities toward zero can rather be attributed to a normalizing effect than
to changes in the flow close to the orifice. The increasing investigated
surface area ($A_{e}$) contains an increasing number of points with low
velocity, typically located at the assembly periphery, and therefore focuses
the normalized distribution.

Despite the important disparities of the velocity fields of the front and back
compressed rafts, the velocity profiles at the orifice seem first rather
similar. The profiles displayed in Fig. 6 do not evidence obvious differences.
The shape of the profile, the value of the maximum velocity and the importance
of the fluctuations are comparable in both cases.

Figure 6: Velocity profiles at the gate, $u(x)$, as a function of time, $t$,
for (a) back and (b) front compression.

Figure 7: Temporal evolution of $\overline{u}$ (blue) and $\Pi$ (red) for (a)
back compression and (b) front compression.

To better analyse the particle flux and its fluctuations, we now focus on the
mean velocity $\overline{u}$. The results are plotted in Fig. 7 together with
$\Pi$, the stress measured at the raft back. Two points are worth being
discussed. First, the different evolution of $\overline{u}$ for the front and
back compressed rafts. For the back compression, after a short transitory
phase and before the relaxation stops, $\overline{u}$ seems rather constant.
The two peaks, attributed to the disintegration of two successive folds (see
Fig. 2-3), can be identified but no clear increase or decrease of
$\overline{u}$ over time can be seen. It is not the case for the front
compressed raft for which $\overline{u}$ clearly decreases over time.
Fluctuations are perturbing this evolution but they can be well explained by
the already mentioned elimination of the two folded blocks found on each side
of the orifice.This confirms that the compression side does not only influence
the self-healing capacity but also its kinetics. The second important point
evidenced by Fig. 7 concerns the stress measured at the back of the raft. The
latter remains almost unchanged for the back compressed raft, indicating that
an important fraction of the initially applied stress remains. This remaining
stress most likely originates the still compressed portions found on each side
of the central unjammed corridor. A closer look at the curve indicates a
slight and probably step-wise decrease of $\Pi$ after each fold
disintegration. If confirmed, such variations are typical of a solid-like
behavior, for which the stress and strain are linearly connected via the Young
modulus. For the front compressed raft, the stress totally relaxes.
Interestingly, fluctuations are observed that correspond to the ones of
$\overline{u}$, indicating a direct and almost instantaneous transmission of
the stress trough the entire raft. This could be interpreted as a liquid-like
behavior, for which the stress and strain rate are proportional, the
proportionality coefficient being the viscosity.

###  4.2 Generalisation and interpretation

The detailed results presented above were obtained on two similar rafts and
generalizing these findings requires more data. Thus, these experiments were
repeated with two other pairs of raft, compressed either from the front or
from the back, with a compression level of 33% and 50%, respectively.

Figure 8:  Plots (central raw): final values of $A_{e}$, $A_{uj}$ and $A_{f}$
as a function of $K$ with empty (full) symbols for back (front) compression.
Pictures: final relaxation states obtained for back (left raw) and front
(rigth raw) compressed rafts. From top to bottom, the raft compression is :
K=33%, 50% and 67%.

The final relaxation states are characterized in Fig. 8 and appear to be
mostly fixed by the compression side. For rafts compressed on the back side
(left picture raw), the relaxation is partial, folds remain and the escaped
particles form a rather dense assembly, whose width is the orifice width.
Similar rafts compressed from the front side give rise to a quite different
outcome, see right picture raw. The relaxation is almost total, only a few
folds remain if any, and the escaped particles form less dense assemblies
whose width is much greater than the orifice width. A more quantitative
description is shown in the central plots where $A_{e}$ (squares), $A_{uj}$
(diamonds) and $A_{f}$ (triangles) are reported as a function of $K$. Whatever
the level of compression, the relaxation of the front compressed rafts (full
symbols) is always much more complete than the one of the back compressed
rafts (empty symbols). In other words, $A_{e}$ and $A_{uj}$ are systematically
greater for front compression (factor 4 to 8 and 3 to 8, respectively) while
the opposite is observed for $A_{f}$. Focusing on the back compressed rafts,
$K$ has a limited influence on $A_{e}$ and $A_{uj}$ but strong effects on
$A_{f}$. For front compression, the influence of $K$ on $A_{e}$ and $A_{uj}$
is significant. The greater the compression, the greater these two quantities.
The effect is stronger for $A_{e}$ than $A_{uj}$, which could be explained by
variations of the assembly density. Indeed, the surface occupied by a given
number of escaped particles is modulated by the packing density of the
assembly, which possibly decreases with increasing compression (or flow
velocity). Folds are very limited, which is expected given the large scale
unjamming.

From a practical point of view, these results indicate that for the studied
geometry, the self-healing capacity of particle laden interfaces is
principally a function of the compression direction. For a given direction,
the greater the compression, the greater the fraction of stored particles that
can flow and cover initially particle free regions. This released fraction is
always much larger for front than for back compression.

To go further and better assess the potential of folds as particle reservoirs,
the dynamics of this release should be characterized. We therefore plot in
Fig. 9 the temporal evolution of $A_{e}$ for all studied rafts.

Figure 9:  $A_{e}(t)$ for (a) back and (b) front compressed rafts. Symbols are
experimental data with triangles for $K=33\%$, diamonds for $K=50\%$ and
circles for $K=67\%$. The dashed and continuous lines are linear and
exponential fits given by Eq. (3) and (4), respectively. Insets:
$\dot{A}_{e}(t=0)$ (squares) and $\tau$ (stars, if applicable) as a function
of $K$.

The back compression does not only limit the magnitude of the particle release
but also its duration. The latter is always less than $0.5s$, to be compared
with $2s$ to $4s$ for similar compression achieved from the front side. Said
differently, the incomplete character of the relaxation could, at first
glance, be explained by a premature arrest of the process. Yet, a more
detailed analysis of our data points toward more complex phenomenon.

Let us start considering the front compressed rafts, see Fig. 9(b). For
interpreting the release dynamics, it may be useful to make an analogy with an
electric system. In this frame, the instantaneous rate of release, or flow,
$-\varphi_{e}\dot{A}_{e}$, corresponds to the current $i$. The flow is
subjected to some resistance, $R$, which we assume to be constant. The
confined area can be modeled as a capacitor of capacitance $C$, whose charge
$q$ corresponds, at first order, to the excess of particles. Thus, for any
instant $t$, $q=\varphi_{j}(A_{r}-A_{c})-\varphi_{e}A_{e}$, where
$\varphi_{j}(A_{r}-A_{c})$ represents the initial excess of particles and
$\varphi_{e}A_{e}$ the escaped particles. Noting $U$ the potential difference,
the capacitor discharge trough a resistor $R$ is given by

$C\frac{dU}{dt}+\frac{U}{R}=0\\\ $ (1)

This equation can be solved for given initial conditions and provide $U(t)$,
and by extension $i(t)=U(t)/R$.

In the present system, the first term of Eq. (1), $C{dU}/{dt}={dq}/{dt}=i$,
corresponds to $-\varphi_{e}\dot{A}_{e}$ and the second one,
${U}/{R}={q}/{RC}$, to $[\varphi_{j}(A_{r}-A_{c})-\varphi_{e}A_{e}]/RC$. The
discharge equation therefore becomes

$\dot{A}_{e}+\frac{A_{e}-(A_{r}-A_{c})\varphi_{j}/\varphi_{e}}{RC}=0$ (2)

and the solution verifying the initial condition $A_{e}(t=0)=0$ is then:

$A_{e}=A_{0}(1-e^{-t/\tau})$ (3)

with $A_{0}={(A_{r}-A_{c})}{\varphi_{j}}/{\tau}{\varphi_{e}}$, the final
escaped surface and $\tau=RC$, the typical release time constant.

To evaluate the relevance of our analogy, the experimental evolution of
$A_{e}$ is fitted by Eq. (3) letting $A_{0}$ and $\tau$ be adjusted to
minimize the sum of square residuals. The agreement is excellent, see Fig.
9(b). The biggest deviations are indeed observed either at the very beginning
or at the end. The former is attributed to initial perturbations caused for
example by capillary wave, by a possible short transient regime or by the
difficulty to precisely identify the time origin. The discrepancy found in the
last instants may be caused by a premature arrest of the relaxation, which is
discussed later. The values of $\tau$ and $A_{0}/\tau$ produced by the fitting
procedure are plotted in the inset as stars and squares, respectively. Both
quantities seem to increase roughly linearly with $K$. Given the limited
number of points, the interpretation remains tentative but one could postulate
that $R$, the resistance to the flow, is similar for all (front compressed)
rafts, while $C$, the system capacitance, is directly proportional to the
amount of particles stored in the folds, and thus to $K$. The initial
discharge rate given by $A_{0}/\tau$ also increases with $K$, but the linear
character is less pronounced. If confirmed, it would indicate that
$\varphi_{j}\varphi_{e}$ varies as $K^{2}$. Future experiments could be used
to probe this scaling, the variations of $\varphi_{e}$ being important in view
of using folds as particles reservoir.

Let us now focus on back compressed rafts. The experimental points of Fig.
9(a) cannot be immediately identified with an exponential decay, which
questions the previous interpretation. The latter considers the emptying of a
reservoir subjected to a pressure, which decreases linearly with $A_{e}$, and
to losses, which are proportional to the flow, i.e. scaling as $\dot{A}_{e}$.
Given the very limited decrease of $\Pi$ observed during the relaxation of
back compressed rafts, it is legitimate to consider a constant pressure.
Keeping the rest unchanged leads to a simple linear increase of $A_{e}$ with
$t$, which reads:

$A_{e}=\frac{\Pi_{0}}{\varphi_{e}R}t$ (4)

Here $\Pi_{0}$ is the constant pressure applied at the back of the raft,
$\varphi_{e}$ and $R$ are unchanged and represent the escaped particle density
and the resistance to the flow, respectively. To test this model, the
experimental results are fitted by linear functions, see dashed lines in Fig.
9(a). Note that not zero intercepts are enabled since the experimental curves
show an initial step. Beside this, the agreement is reasonable with the
largest deviations observed at the end of the process. Interestingly, assuming
that $\Pi_{0}\propto A_{r}-A_{c}$, we expect that the slope
${\Pi_{0}}/(\varphi_{e}R)$ of fixed size rafts (constant $A_{r}$) is
proportional to $K$. This is indeed in good agreement with our data, see inset
of Fig. 9(a), which shows - despite the limited number of points - a linear
variation of the fitted slopes with $K$. These curves can therefore be seen as
classical emptying of granular silos. Indeed, the well known Beverloo law
predicts for given particle and orifice sizes, a constant flow rate Beverloo
et al. (1961). The origin of this law remains controversial and neither the
Janssen effect Janssen (1895) nor the free falling arch approach seem to
provide the correct view Rubio-Largo et al. (2015). Furthermore, the
dependency of the flow rate ($\dot{A}_{e}$) with the pressure remains unclear.
Some experiments evidence a total independence Aguirre et al. (2010), while
others show - under certain circumstances - a proportional relation Peng et
al. (2021), in agreement with our findings. Yet, it is worth noting that these
results, i.e. $A_{e}(t)\propto t$ and $\dot{A}_{e}\propto K$, are also
compatible with the premature arrest of exponential release as found for front
compressed rafts. One must here keep in mind that the limited character of the
current data does allow to choose one of the two models.

Whatever the chosen function, the question of why and when the relaxation
process get arrested remains open. We attribute the ending to the existence of
a yield stress, at least for back compressed rafts. Thus as long as the
pressure $\Pi$ is above certain critical value, flow can occur. Small decrease
of $\Pi$, even limited and at first order negligible (hypothesis of constant
$\Pi_{0}$), can then stop the flow. Coming back to the electrical analogy,
everything happens as if compressed rafts were granular diodes. When the diode
is mounted in the appropriate direction, here corresponding to the front
compression, the capacitor constituted by the particles stored in the folds
can (quasi)-totally discharge, the current passing through a constant resistor
$R$. Yet, if the compression occurs from the back, the diode, mounted in the
opposite direction, stops - after some leaking current has passed - the
capacitor discharge.

In the context of compressed particle-laden interfaces, this diode effect can
be understood in the light of granular framework, which describes the
development of force chain network during the compression. By moving one
barrier while keeping the other fixed, chains build up starting from the
moving barrier and propagate via particle-particle contacts. For some of these
contacts, chains divide forming two or more branches. Considering the
monodisperse character of the particles, and by extension their hexagonal
close packing, the orientation of these chains and branches are expected to be
found in a cone, whose axis is the compression axis and angle is $60^{\circ}$.
Consequently, for rafts compressed from the back side, the network transmits
the stress toward the front barrier but an important portion of it is
redirected on each side of the orifice. The keystones, i.e. particles for
which the chains branch, are in the back of the raft and therefore not easily
removed. Arches may form that block the unjamming, which only occurs in a
narrow corridor whose width is more or less the orifice width. In contrast,
when rafts are compressed from the front, the gate opening causes the removal
of keystone particles found right behind it, leading the force chain network
to collapse, and therefore triggering a quasi-total unjamming of the confined
area. A careful inspection of the unjammed zone observed at the beginning of
the process (see for example Fig. 2, $t_{f,2}$) let see a conical shape whose
angle is very close to $60^{\circ}$, in perfect agreement with the expected
network structure. Finally, this interpretation also explains why under
certain circumstances, such as very strong compressions, also front compressed
rafts may give rise to a force chain structure that can, at least partially,
arrest the unjamming, see Fig. $K=67\%$ (circles) in Fig. 9(b).

##  5 Conclusions

The different behaviors observed during the relaxation of strongly compressed
rafts evidence the importance of the compression direction on these systems.
For back compressed rafts, the relaxation is incomplete while it is quasi-
total for front compressed rafts. Practically, these results demonstrate that
the processability of these interfaces highly depends on their history. In the
present configuration, the flowability is primarily set by the compression
direction, the level of compression playing a secondary role. This important
finding should be accounted for in industrial processes. It is also of
importance while considering the self-healing properties of these interfaces.
Indeed, in this simple uni-axial geometry, the capacity of particles stored
into large folds to migrate and stabilize uncovered areas is almost total for
favorable compression direction, while it is strongly hindered in the opposite
case. The dynamics of the particle release itself is strongly affected by the
compression direction. While the current data remain limited, we observe
similar trends in on-going tests performed using different raft size, orifice
width and particle size.

These results are interpreted in the light of the framework developed for
granular matter, which demonstrates the importance of chain forces to transmit
stress. Thus, the direction of these chains, and more precisely their
branching, is found to be essential as evidenced by triggering a local
relaxation along the compression axis. For front compressed rafts, the local
unjamming causes keystone particles to be removed leading to the network
collapse and large scale unjamming. In contrast, for back compressed rafts the
network redirects the stress laterally. As a result, only a limited number of
particles found directly behind the orifice can flow, limiting the relaxation
to a narrow corridor. To visualize these effects, the analogy with an
electrical circuit can be made. The folds are like a capacitor and the
particle flow like a current through a resistor. The force chain network can
then be seen as a diode which let, or not, the current flow depending on its
orientation.

Finally, it is worth noting that our results and interpretation give rise to
many open questions which need to be treated in future research. From our
point of view, future investigations should aim in understanding what
influences the properties of the chain force network. In this frame, we can of
course think of the raft and trough geometry, and especially of the raft
length, its width and the size of the orifice. A second important aspect is
the influence of the individual particles, and more particularly, the role of
friction, shape, but also possible contact lubrication, which could for
example cause network aging.

## Acknowledgments

We would like to thank the Austrian Science Fund (FWF) for the financial
support under Grant No. P33514-N. We also acknowledge Graz University of
Technology for its technical support, especially the Institute of Hydraulic
Engineering and Water Resources Management for having borrowed us the high-
speed camera and the Institute of Process and Particle Engineering for having
helped us sieving the particles. C.P acknowledges Elise Lorenceau and Anne-
Laure Biance for fruitful discussions.

## Appendix

This Appendix presents 1) the preparation of the elastic barrier, 2) the
method used to calibrate it and 3) the deformation equations, which are then
used to deduce the lineic pressure acting on it.

### 1) Elastic rubber preparation

The elastic rubber strings are produced in house by injecting a freshly
prepared (1:1) mixture of Zhermack Elite Double 8 basis and catalyst (Zhermac
Spa) into glass capillaries. After the elastomer reticulation has been
completed, the glass capillaries are manually removed. Using capillaries of
various diameter, we produce strings whose diameter ranges between $0.5$ and
$1$ $\mathrm{mm}$. Their typical length is in the range of $10$ $\mathrm{cm}$,
which is sufficient to be installed in $6$ $\mathrm{cm}$ wide trough.

### 2) Rubber calibration

The calibration of the rubber mostly consists in determining its Young’s
modulus. The principle, detailed below, is simple, and consists in measuring
the deflections produced by hanging known weights to the rubber, see Fig. 10.
It was first presented in the PhD thesis of Petit (2014).

Figure 10: Principle of the elastic calibration: deformation is measured as a
function of the applied force.

Consider an elastic string of relaxed length $s_{0}$ (not precisely known),
attached between two holders separated by $s_{1}$ (black rectangles in Fig.
10). The elastic is further loaded using a known mass $m$ ($F_{g}=mg$) and
reaches a stretched length $s_{f}$, which can easily be measured. The tensile
stress, $\sigma$, is linearly proportional to the strain, $\epsilon$, and the
proportionality coefficient, $E$, is the Young’s modulus. This reads:

$\sigma=E\epsilon$ (5)

The strain is equal to the relative extension and the tensile stress derives
from the tension $T$, providing:

$\epsilon=\frac{s_{f}-s_{0}}{s_{0}}\qquad\mathrm{and}\qquad\sigma=\frac{T}{\Sigma}$
(6)

with $\Sigma$, the cross-section area of the elastic string under tension. The
Hook’s law may now be rewritten as

$T=E~{}\Sigma~{}\frac{s_{f}-s_{0}}{s_{0}}=E~{}\Sigma_{0}~{}\frac{s_{f}-s_{0}}{s_{f}}$
(7)

where the variation of the cross-section area due to the stretching was
introduced as

$\Sigma=\Sigma_{0}\frac{s_{0}}{s_{f}}$ (8)

$\Sigma_{0}$ is the cross-section area of the non-deformed elastic string.
Under a small load, the force equilibrium may be written as:

$F_{g}=2T\sin{\beta}\qquad\mathrm{with}\qquad\sin{\beta}=\sqrt{1-\left(\frac{s_{1}}{s_{f}}\right)^{2}}$
(9)

Here, $\beta$ is the angle between the rubber string and the horizontal.
Combining the above equations, one obtains the following relationship:

$mg=2\lambda_{0}\cdot\frac{s_{f}-s_{0}}{s_{f}}\sqrt{1-\left(\frac{s_{1}}{s_{f}}\right)^{2}}$
(10)

with the newly introduced parameter $\lambda_{0}$, defined by
$\lambda_{0}=E\Sigma_{0}$.

Practically, the length of the deformed rubber $s_{f}$, is measured from
pictures taken for different known masses $m$. The undeformed rubber length,
$s_{0}$, which is not precisely known and the parameter $\lambda_{0}$ are then
obtained by finding the best fit to the measured data. This is done with the
help of the fminsearch function of Matlab. An illustrative curve is displayed
in Fig. 11. The coefficient of regression is always very close to one, greater
than 0.99. The 95% confident intervals corresponds to an uncertainty of $\pm
1\%$ on $\lambda_{0}$ and $s_{0}$.

Figure 11: Typical calibration curve and its linear fit providing $s_{0}$
(here $44.46mm$) and $\lambda_{0}$ (here $0.3152N$). Function $f$ is defined
as
$f(s_{f})=2\frac{s_{f}-s_{0}}{s_{0}}\sqrt{\left(1-\frac{s_{1}^{2}}{s_{f}^{2}}\right)}$.
The coefficient of regression is $R^{2}=0.9995$.

Knowing $\Sigma_{0}$, the cross section of the undeformed string, the modulus
of elasticity is finally deduced as:

$E=\frac{\lambda_{0}}{\Sigma_{0}}$ (11)

Using the Elite Double 8, the measured $E$ is found to be in the expected
range of $0.1MPa$. Note, that the two parameters $s_{0}$ and $\lambda_{0}$ are
also necessary to describe the deformation of the elastic string subjected to
a constant lineic force, as shown in the next section.

### 3) Deformation equations

The calibrated rubber string can then be used as a pressure sensor for the
particle rafts in our experiments. To be quantitative, the relationship
between the elastic deflection, $\delta$, and the applied lineic pressure,
$\Pi$, is needed. These quantities are defined in Fig. 12. The length of the
stretched elastic is $s_{f}$, as previously defined. The position along the
elastic can be described by the curvilinear ordinate $s$ or by the cartesian
coordinates $(x,y)$, the $x-$ and $y-$axes have unit vectors $\hat{x}$ and
$\hat{y}$, respectively.

Figure 12: Deformation of an elastic string due to the action of constant
lineic force $\Pi$.

The following derivation is based on the well-known mathematical problem of a
classical catenary Borggräfe et al. (2015). The tension $\vec{T}$ has a norm
($T$) and is collinear to the string at any point $s\in[-s_{f}/2;s_{f}/2]$.
Thus, naming $T_{x}$ and $T_{y}$ its horizontal and vertical components, we
get:

$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{T_{y}}{T_{x}}\qquad\mathrm{and}\qquad
T=\sqrt{{T_{x}}^{2}+{T_{y}}^{2}}$ (12)

Using the definition of the lineic pressure, $\Pi$, and applying the force
balance to a string element of infinitesimal length $ds$ (see Fig. 12)
provides:

$T_{y}=\int^{s_{f}/2}_{0}\Pi\mathrm{d}s=\Pi\int^{s_{f}/2}_{0}\mathrm{d}s$ (13)

which leads with Eq. (12) to:

$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\Pi}{T_{x}}\int^{s_{f}/2}_{0}\mathrm{d}s=\frac{\Pi}{T_{x}}\int^{s_{1}/2}_{0}\frac{\mathrm{d}s}{\mathrm{d}x}\mathrm{d}x$
(14)

Eq. (14) can be differentiated with respect to $x$, leading to:

$\frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}=\frac{\Pi}{T_{x}}\frac{\mathrm{d}s}{\mathrm{d}x}$
(15)

To go further, the curvilinear ordinate is eliminated using:

${\mathrm{d}s}^{2}={\mathrm{d}x}^{2}+{\mathrm{d}y}^{2}={\mathrm{d}x}^{2}\left(1+\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^{2}\right)$
(16)

which also writes:

$\frac{\mathrm{d}s}{\mathrm{d}x}=\sqrt{1+\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^{2}}$
(17)

Eq. (15) becomes:

$\frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}=\frac{\Pi}{T_{x}}\sqrt{1+\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^{2}}$
(18)

It is integrated with boundary conditions $\mathrm{d}y/\mathrm{d}x(x=0)=0$ and
${y(x=0)=0}$, providing:

$y(x)=\frac{T_{x}}{\Pi}\left(\cosh{\left(\frac{\Pi}{T_{x}}x\right)}-1\right)$
(19)

Evaluating this expression for $x=s_{1}/2$ gives the central string
deflection:

$\delta=\frac{T_{x}}{\Pi}\left(\cosh{\left(\frac{\Pi
s_{1}}{2T_{x}}\right)}-1\right)$ (20)

In practise, to use this expression, it is necessary to explicit $T_{x}$ as a
function of the known parameters, i.e. $s_{1}$, $s_{0}$ and $\lambda_{0}$.
Combining Eq. (12) and (13), we get:

$T=\sqrt{{T_{x}}^{2}+\left(\Pi\int^{s_{f}/2}_{0}\mathrm{d}s\right)^{2}}$ (21)

Using the stress-strain relation from the calibration section (Eq. (7)) and
the definition of the parameter $\lambda_{0}$ given by Eq. (11), we identify
$T$ with:

$T=\lambda_{0}\frac{s_{f}-s_{0}}{s_{f}}$ (22)

and obtain:

$\lambda_{0}\frac{s_{f}-s_{0}}{s_{f}}=\sqrt{{T_{x}}^{2}+\left(\Pi\int^{s_{f}/2}_{0}\mathrm{d}s\right)^{2}}$
(23)

We then eliminate $s_{f}$ using:

$\begin{split}\frac{s_{f}}{2}=\int^{s_{f}/2}_{0}\mathrm{d}s=\int^{s_{1}/2}_{0}{\sqrt{1+\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^{2}}\mathrm{d}x}\\\
=\frac{T_{x}}{\Pi}\sinh{\left(\frac{\Pi s_{1}}{2T_{x}}\right)}\end{split}$
(24)

Eq. (23) and(24) provide the condition

$\lambda_{0}\frac{\frac{T}{\Pi}\sinh{\left(\frac{\Pi
s_{1}}{2T_{x}}\right)}-\frac{s_{0}}{2}}{\frac{T_{x}}{\Pi}\sinh{\left(\frac{\Pi
s_{1}}{2T_{x}}\right)}}=\sqrt{T^{2}_{x}\left(1+\sinh^{2}\left(\frac{\Pi
s_{1}}{2T}\right)\right)}$ (25)

or equivalently

$T_{x}\cosh\left(\frac{\Pi
s_{1}}{2T_{x}}\right)=\lambda_{0}\left(1-\frac{s_{0}\Pi}{2T_{x}\sinh{\left(\frac{\Pi
s_{1}}{2T_{x}}\right)}}\right)$ (26)

The relation between $\delta$, the central deflection of the elastic string
and $\Pi$, the applied lineic pressure is therefore given by the two equations
(20) and (26). Note that, using Eq. (19) and (26), the entire shape of the
elastic is indeed described as a function of known and measurable parameters,
namely $s_{1}$, $s_{0}$ and $\lambda_{0}$.

For small deformations, Eq. (20) and (26) can be simplified by a third-order
series expansion with the approximation

$\frac{\mathrm{d}y}{\mathrm{d}x}\approx\frac{s_{1}\Pi}{T_{x}}$ (27)

which provides the following relation between the lineic pressure $\Pi$ and
the (small) central deflection $\delta$:

$\Pi=\frac{8\lambda_{0}}{s_{1}}\left[\left(1-\frac{s_{0}}{s_{1}}\right)\frac{\delta}{s_{1}}+\left(\frac{4s_{0}}{3s_{1}}-1\right)\left(\frac{2\delta}{s_{1}}\right)^{3}\right]$
(28)

When applying the lineic pressure measurement method described above, a non-
negligible uncertainty must be taken into account. The accuracy depends
strongly on the difference $s_{1}-s_{0}$ as shown below:

$\frac{\Delta\Pi}{\Pi}=2\frac{\Delta s_{1}}{s_{1}}+\frac{s_{1}\Delta
s_{0}+s_{0}\Delta s_{1}}{s_{1}(s_{1}-s_{0})}$ (29)

In the present work, we evaluate this uncertainty to approximately 15%.

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