Also at ]Global Station for Soft Matter, GI-CoRE, Hokkaido University,
Sapporo, Japan

# Swelling induced debonding of thin hydrogel films grafted on silicon
substrates

Anusree Augustine  Yvette Tran  Emilie Verneuil  Antoine Chateauminois
<EMAIL_ADDRESS>Soft Matter Science and Engineering Laboratory
(SIMM), CNRS UMR 7615, ESPCI Paris, PSL University, Sorbonne Université,
F-75005 Paris, France \\\  Nicolas Gauthier  Marc Veillerot  University
Grenoble Alpes, CEA, LETI, F-38000 Grenoble, France
Bangguo Zhu  Chung-Yuen Hui  [  Department of Mechanical and Aerospace
Engineering, Cornell University, Ithaca, NY 14853, USA \\\

(August 28, 2024)

###### Abstract

We report on the delamination of thin ($\approx\mu$m) hydrogel films grafted
to silicon substrates under the action of swelling stresses.
Poly(dimetylacrylamide) (PDMA) films are synthesized by simultaneously cross-
linking and grafting preformed polymer chains onto the silicon substrate using
a thiol-ene reaction. The grafting density at the film/substrate interface is
tuned by varying the surface density of reactive thiol-silane groups on the
silicon substrate. Delamination of the films from well controlled line defects
with low adhesion is monitored under a humid water vapor flow ensuring full
saturation of the polymer network. A propagating delamination of the film is
observed under the action of differential swelling stresses at the debonding
front. A threshold thickness for the onset of this delamination is evidenced
which is increasing with grafting density while the debonding velocity is also
observed to decrease with an increase in grafting density. These observations
are discussed within the framework of a nonlinear fracture mechanics model
which assumes that the driving force for crack propagation is the difference
between the swelling state of the bonded and delaminated parts of the film.
Using this model, the threshold energy for crack initiation was determined
from the measured threshold thickness and discussed in relation to the surface
density of reactive thiol groups on the substrate.

hydrogel films, delamination, grafting

††preprint: APS/123-QED

## Introduction

Hydrogel coatings are suitable candidates for many applications in the
biomedical and engineering fields where properties such as bio-compatibility,
lubrication, hydrophilicity, and transparency are desired. Indeed these
hydrophilic polymer networks may absorb water by swelling to several times
their dry thickness. As such, hydrogel coatings are, for example, possibly
good candidates for anti-fog applications, as proposed by several teams [1, 2,
3, 4] who discussed their efficiency in terms of light transmission while
submitted to warm humid air. Indeed, the coating should act as a reservoir for
humidity and observation is made [3, 4, 5] that the higher the film thickness,
the more water is absorbed and the later mist appears. However, the viability
of such anti-fog coatings is conditioned by their ability to sustain stresses
without damage.
Inherent to the coating geometry is the very large aspect ratio which results
in the development of internal stresses upon swelling of a coating bound to a
rigid substrate. Indeed, swelling occurs along the thickness direction while
compressive stresses build up in the constrained in-plane direction. For
hydrogel films strongly anchored to the substrate, these swelling stresses
were observed to induce instabilities on the free surface in the form of
creases. [6, 7, 8, 9] In situations where the film is not strongly bound to
the substrate, swelling stresses can alternately result in delamination
followed by large scale folding and blistering of the hydrogel films. Such
mechanisms have been reported by Velankar et al. [10] for
poly(dimethylsiloxane) films swollen with toluene causing growth of sharp
folds of high aspect ratio. Other morphologies in the shape of craters were
also reported by Sharp et al. [11] for water immersed ultrathin films of
poly(d,l-lactide) as a result of a blistering process which occurs when the
films delaminate.
In a morphogenetic perspective, controlled debonding and blistering of swollen
hydrogel films on patterned substrates were also envisaged as a tool to
generate switchable actuators or micro/nanofluidic devices. As an example, Xu
and coworkers [12] elaborated poly(N-isopropylacrylamide)-co-sodium acrylate)
(pNIPAM) gel layers on micro-patterned electrode surfaces which allowed to
trigger electrochemically reversible delamination and blistering of the films.
Similarly, Takahashi et al. [13] explored in a systematic way the 3D
architectures which are generated from the swelling-induced buckling
instabilities within hydrogel films on patterned silanized substrates. In
these studies, the control of the architectures relies on the absence of any
debonding at the film/substrate interface outside the patterned zones.
In a different perspective, here we take advantage of patterning techniques to
promote swelling-induced delamination of thin
($\approx$\mathrm{\SIUnitSymbolMicro m}$$) hydrogel films grafted onto silicon
substrate from well controlled nucleation sites with low adhesion. Using this
methodology, we offer to quantitatively relate the delamination of swollen
hydrogel films to their chemical bonding to a rigid substrate that we vary
using well-controlled grafting densities.
We first focus on the analysis of the debonding and blistering mechanisms
which result from the swelling of the films under a water vapor flow ensuring
full saturation of the hydrogel network. In rigid film/substrate systems
experiencing stress mismatch, blistering and delamination are often considered
to be driven by the bending stresses induced within the buckled film (see e.g.
references [16, 17] for a review). For soft hydrogel films bonded to a rigid
substrate, we anticipate that the driving mechanisms can be different owning
to the comparatively higher compressive stresses and lower elastic modulus.
Indeed, we show here that for soft hydrogel films bonded onto rigid
substrates, delamination is instead triggered by localized differential
stresses which arise at the debonding front from the difference in the
swelling state in the bonded and delaminated parts of the film. We also show
that debonding only occurs above a thickness threshold which depends on the
grafting density.
As theoretically investigated by Gent et al. [19] for rubber layers, the
energy released when a swollen layer separates from a substrate and swells
further can indeed be quite high and justify the occurrence of spontaneous
debonding. Here, we develop a non linear fracture mechanics analysis of the
threshold energy required for crack extension at the subtrate/film interface
under the action of swelling stresses. Using this theory, we determine how the
threshold energy release rate for debonding is varying as a function of the
grafting density from the experimental threshold thickness.

## Experimental section

### Synthesis of grafted PDMA films

Poly(dimethylacrylamide) hydrogel films are synthesized by simultaneously
crosslinking and grafting preformed polymer chains onto silicon wafer
substrates with a thiol-ene click reaction. As fully described elsewhere [20,
21], ene-reactive PDMA is first synthesized by free radical polymerization of
dimethylacrylamide and acrylic acid which is then modified by amidification
using allylamine. The ene-functionalized PDMA is subsequently purified by
dialysis against water and recovered by freeze-drying.
The adhesion between the hydrogel film and the wafer is achieved through
thiol-modification of the silicon surface (Fig. 1a). In order to vary the
interface strength, we tuned the grafting density of the gel onto the silicon
wafer by playing with the density of active thiol-silane groups at the silicon
wafer surface. This goal is achieved using a mixture of
n-propyltrimethoxysilane and 3-mercaptopropyl trimethoxysilane for the
silanization of the silicon wafer (fig. 1a,b).

Figure 1: Silicon wafer grafted with (a) (3-mercaptopropyl)trimethoxysilane
and (b) propylsilane. (c) Allyl butyl ether used to passivate the grafted
thiol groups.

While mercaptosilane has a thiol group at one end which can react with the
ene-functionnalized PDMA chains to form covalent bonds at the interface,
propylsilane is not reactive. Here, 50 vol% and 100 vol% mercaptosilane in
volume is used in the silane mixture in order to vary the density of grafted
thiol groups. The silicon substrate is first cleaned in a UV ozone chamber and
quickly transferred into a sealed reactor filled with nitrogen where a
solution of 30%vol of silane in dry toluene is introduced. After 3 hours
soaking in the silane solution under nitrogen, the silicon wafer is rinsed and
sonicated in toluene before drying. Using a combinaison of X-ray Photoelectron
Spectroscopy (XPS) and Time of Flight Secondary Ion Mass Spectrometry (ToF
SIMS) analysis detailed elsewhere [22], the areal density of grafted thiol
groups was found to vary non linearly from 0.2 to 0.6 per ${\mathrm{nm}}^{2}$
when the percentage of mercaptosilane in the reactive silane mixture is
increased from 50 vol% to 100 vol%.
Prior to spin-coating on the thiol-modified silicon wafers, the ene-
functionalized PDMA and dithioerythritol crosslinker are dissolved in
dimethylformamide (DMF). In order to vary film thickness, spin coating was
performed at 3000 rpm using solutions with different polymer concentrations.
After spin-coating, polymer films were annealed at 150 ${}^{\circ}C$ for 24
hours under vacuum to activate the thiol-ene reaction. This procedure resulted
in PDMA films with thicknesses $h$ ranging from 100 $\mathrm{nm}$ to 1.2
$\mathrm{\SIUnitSymbolMicro m}$ in the dry state, as measured by ellipsometry.
All the film thicknesses $h$ reported in this paper refer to the dry state.
Unless otherwise stated, all the results reported in this study were obtained
with films with a swelling ratio of $2.7\pm 0.3$, as measured by additional
ellipsometry measurements with the films fully swollen in water. This value is
found to be independent on the grafting density and on film thickness within
the investigated range [22]. Such a constant swelling ratio ensures a constant
swelling stress within the films at all thicknesses.

### Patterning of the interface with strip defects

Debonding mechanisms were promoted from well controlled line defects on the
silicon substrate where the strength of the interface is reduced as a result
of localized passivation of the reactive thiol groups grafted on the silicon
substrate. Such a passivation is achieved by covalent bonding of the thiol-
silane goups on the silicon wafer with allyl butyl ether as a passivator (fig.
1c). As detailed in reference [22], selected strip areas with prescribed
widths in the range 2-100 $\mathrm{\SIUnitSymbolMicro m}$ were passivated on
the silicon substrate using UV photolithography techniques by means of a
thiol-ene click reaction. The actual width of the passivated lines ranged from
$9\pm 1$ to $101\pm 7\leavevmode\nobreak\ $ $\mathrm{\SIUnitSymbolMicro m}$ as
measured by ellipsometry and ToF SIMS measurements [22].

### Debonding set-up

Swelling-induced debonding of the hydrogel film is achieved by advecting a
humid air flow at room temperature on the cooled sample. Under such
conditions, the transfer of vapor to the coating was previously shown [5] to
result from a well-controlled advecto-diffusive mechanism which allows for
reproducible and fast (within less than 100 s) saturation of the gel layer.
More precisely, we set relative humidity at 60% and flow rate at 0.6
$\mathrm{L}\,\mathrm{m}\mathrm{i}\mathrm{n}^{-1}$. The water flux to hydrogel
coatings was achieved using a homemade setup consisting in a closed Plexiglas
chamber equipped with a diverging inlet and converging outlet on opposite
sides to obtain a laminar flow of humid gas with controlled rate. In the
bottom wall of the chamber, a Peltier module was inserted on which the samples
were set. The Peltier module allows to create an homogeneous temperature field
at the sample surface and to regulate the sample temperature to a value 10
$\mathrm{\SIUnitSymbolCelsius}$ lower than the ambient temperature. The upper
wall of the chamber is a glass window allowing observation of the film by
means of a zoom lens (Apo Z16, Leica) and CCD camera (1600 x 1200, 8 bits).
Water saturation within the films is first indicated by water condensation
droplets on the surface of the PDMA film followed by the formation of a water
film at the surface of the gel. In all the experiments to be reported,
debonding rates were measured with the films in a fully saturated state from
an average of at least three different locations along the patterned lines.

## Delamination mechanisms

Fig. 2a shows successive images taken during the initial stages of debonding
of a film 700 $\mathrm{nm}$ in thickness from a patterned line defect. The
width of the patterned line defect ($L_{0}=9\pm 1\>$\mathrm{\SIUnitSymbolMicro
m}$$) and its orientation are indicated in the figure by a blue box and dotted
line, respectively. Delamination propagates both along the passivated line
defect and perpendicular to it, i.e. on the grafted part of the film/substrate
interface. The debonding rate along the line defect ($y$-axis in Fig. 2a,b) is
typically of the order of 1 $\mathrm{\SIUnitSymbolMicro
m}\text{\,}{\mathrm{s}}^{-1}$ while in the transverse direction ($x$-axis in
Fig. 2a,b) it is only in the $10^{-3}\;$\mathrm{\SIUnitSymbolMicro
m}\text{\,}{\mathrm{s}}^{-1}$$ range or less for the considered grafting
density (0.6 ${\mathrm{nm}}^{-2}$). Indeed, this difference is consistent with
the expectation of a reduced interface strength along the passivated line
defect.

Figure 2: Phone cord debonding mechanisms from a defect line
$L_{0}=9\leavevmode\nobreak\ $\mathrm{\SIUnitSymbolMicro m}$$ in width
($h=700\leavevmode\nobreak\ $\mathrm{nm}$$, grafting density of 0.6
${\mathrm{nm}}^{-2}$, swelling ratio $S=2.7$). (a) images of the swollen film
taken (i) 360 , (ii) 420 and (iii) 620 $\mathrm{s}$ after exposure to humid
air. The blue areas and the dotted blue line indicates the line defect width
$L_{0}$ and orientation, respectively. (b) microscopic image of the dried
debonded hydrogel with the tip of the debonded area.

Fig. 2b shows post mortem observations of the tip of the delaminated strip for
which the crack is frozen by drying the film and imaging is made with a
microscope with enhanced resolution. The images show that the debonded part of
the film exhibits a regular folding pattern which looks similar to the so-
called telephone-cord architectures reported by Takahashi et al. [13] and Xu
et al. [12] for various kinds of thin, water swollen, hydrogel films on
patterned substrates. Briefly, these features were attributed to buckling
instabilities resulting from the swelling induced compressive stresses acting
over selected areas of the substrate where adhesion was purposely reduced. As
detailed in a theoretical stability analysis of straight-sided blisters by
Audoli et al. (the situation depicted Fig. 3a), the formation of such
telephone cord instabilities in bi-axially compressed films can be accounted
for by the residual compressive stress which remains along the longitudinal
direction after Euler buckling [14, 15]. This secondary buckling is driven by
the ratio of the compressive stress $\sigma$ to the buckling stress
$\sigma_{c}$ of a clamped-clamped wide plate. Here, $\sigma_{c}\propto
E(h/L_{0})^{2}$ where $L_{0}$ and $h$ are respectively the width of the
blister and the film thickness. As in our experiments $\sigma\approx E$ (see
below), the ratio $\sigma_{c}/\sigma$ thus scales as $(h/L_{0})^{2}$. Since
$h/L_{0}$ values ranges from $10^{-1}$ to $10^{-2}$, it turns out that
$\sigma/\sigma_{c}>>1$, a situation which was shown to be highly favorable to
the formation of telephone cord instabilities. [14, 15] In addition to the
action of compressive swelling stresses, the buckling of the film is also
promoted by the in-plane expansion of the film when it debonds from the
substrate as a consequence of the release of the lateral constraints.
Accordingly, calculations given below indicate that the stretch ratio along
the x-axis is increasing from 1 to 1.9 when the film delaminates.
As compared to the above mentioned experimental studies, the difference in our
experiments is that debonding and buckling over the passivated line defects
are followed by the progressive extension of delaminated areas over the
grafted film/substrate interface, as shown by the lateral extension of the
delaminated defect. The driving mechanisms for delamination have been widely
addressed in the context of rigid film/substrate systems where the film is in
a state of biaxial compression in the unbuckled state as a result, for
example, of thermal expansion mismatches. Interface failure in these systems
often results in the formation of regions, such as the phone cord blisters,
where the film has been buckled away from the substrate in order to minimize
its total energy since bending energy is much lower than compressive one. [17,
16, 18] However, when the film buckles away from the substrate, non zero
stress intensity factors are induced at the edges of the buckle as a result of
the bending moment which promotes further interfacial crack propagation, as
schematically depicted in Fig. 3a.
In our experiments, we noted that delamination also propagates from the free
edges of the film specimens with a velocity similar to that achieved along the
transverse direction of strip defects. As an example, the debonding velocity
from the edge of a film 900 $\mathrm{nm}$ in thickness and a swelling ratio
$S=2.9$ was found to be $0.008\pm 0.004$ $\mathrm{\SIUnitSymbolMicro
m}\text{\,}{\mathrm{s}}^{-1}$ while the transverse velocity from a strip
defect on the same film was $0.014\pm 0.004$ $\mathrm{\SIUnitSymbolMicro
m}\text{\,}{\mathrm{s}}^{-1}$. Such an observation clearly precludes the
hypothesis of buckling stresses as the driving force for delamination in our
film system. In addition, we also observed that the initial debonding velocity
$V_{0}$ in the transverse direction ($x$) is independent on the width $L_{0}$
of the passivated strip defect111In these experiments, the swelling ratio of
the film is $S=3.8$, which results in values of $V_{0}$ about two orders of
magnitude higher than for a swelling ratio of 2.7, probably as a result of
increased swelling stresses. (Fig. 4a) while the morphology of the debonded
film is strongly altered when $L_{0}$ increases: as shown in Fig. 4b, the
shape of the delaminated film is evolving from the single telephone-cord
pattern to much more complex morphologies characterized by extensive folding
when $L_{0}$ is increased from 9 to 100 $\mathrm{\SIUnitSymbolMicro m}$. As
schematically depicted in Fig. 3b, extensive folding of the delaminated film
is likely to relax crack-tip loading by buckling-induced stresses. Here again,
the hypothesis of delamination induced by buckling stresses can be discarded
from the independence of the debonding velocity on the morphology of the
delaminated blisters.

Figure 3: Schematic of potential delamination mechanisms from a defect line of
width $L_{O}$. (a) debonding induced by residual bending stresses in the
buckled blister which loads the tip of the interface cracks in a tensile mode;
(b) debonding induced by localized differential stress $\Delta\sigma_{sw}$
close to the delamination front. Extensive folding of the film is promoted by
the in-plane expansion of the film when it debonds. (b) is the relevant
mechanism for swelling induced debonding of soft hydrogel networks since
extensive folding of the delaminated film is likely to release crack tip
loading by buckling-induced bending stresses. Figure 4: Debonding from defect
strips with widths $L_{0}$ ranging from 9 to 100 $\mathrm{\SIUnitSymbolMicro
m}$ ($h=680$ $\mathrm{nm}$, grafting density of 0.6 ${\mathrm{nm}}^{-2}$,
swelling ratio $S=3.8$). (a) initial debonding velocity $V_{0}$ as a function
of $L_{0}$. (b) image of the debonded line for $L_{0}=100$
$\mathrm{\SIUnitSymbolMicro m}$ showing extensive folding of the debonded
film.

In what follows, we will rather make the hypothesis that debonding results
from localized differential swelling stresses which arise in the crack tip
region as a result of differences in the swelling state between the grafted
and delaminated parts of the film. This hypothesis will be discussed in the
light of experiments carried out for a defect width $L_{0}=9$
$\mathrm{\SIUnitSymbolMicro m}$, where debonding is initiated from well
controlled telephone-cord patterns.

### Time-dependence of the debonded width

We now focus on the time-dependence of the lateral extension $L(t)$ of the
delaminated strip for silicon substrates with grafting density of 0.2 and 0.6
${\mathrm{nm}}^{-2}$ with film thicknesses $h$ in the range 700 to 1130
$\mathrm{nm}$. In fig. 5, the width of the selected delaminated part is
measured over time. It turns out that delamination progressively stops at some
debonding width $L$ which increases with film thickness, except for the
thickest film with the lower grafting density ($h>480$ $\mathrm{nm}$, grafting
density 0.2 ${\mathrm{nm}}^{-2}$) where no delamination arrest is evidenced
(diamonds in fig. 5c).

Figure 5: Width $L$ of the debonded strip as a function of time ($L_{0}=9$
$\mathrm{\SIUnitSymbolMicro m}$). (a) schematic of delamination width $L(t)$
along $x$ direction from an interface defect width $L_{0}$ with $L(t=0)\approx
L_{0}$; (b) film thickness $h=$ ($\circ$) 770 and ($\bullet$) 1130
$\mathrm{nm}$ with a grafting density of 0.6 ${\mathrm{nm}}^{-2}$; (c) film
thickness $h=$ ($\square$) 480, ($\blacksquare$) 700 and ($\lozenge$) 1060
$\mathrm{nm}$ with a grafting density of 0.2 ${\mathrm{nm}}^{-2}$. Solid lines
are guides for the eye.

Post-mortem microscope observation of the interface cracks frozen at various
stages of the debonding process by drying the film revealed that the decrease
in the delamination velocity $dL/dt$ is correlated to a progressive change in
the shape of the crack line. As an example, fig. 6 shows images of a film 1.2
$\mathrm{\SIUnitSymbolMicro m}$ in thickness with a grafting density of 0.6
${\mathrm{nm}}^{-2}$ together with the corresponding $L(t)$ relationship. It
turns out that the decrease in the delamination velocity is associated with
the development of a wavy crack line with an approximately constant wavelength
$\Lambda$ of about 40 $\mathrm{\SIUnitSymbolMicro m}$, i.e. about 10 times the
thickness of the swollen film. This ratio was preserved when the film
thickness was varied between 470 and 1100 $\mathrm{nm}$ and the grafting
density from 0.2 to 0.6 ${\mathrm{nm}}^{-2}$. As the delamination front moves
forward, fingers of adhering film with increased lengths are formed (shown by
red arrows in fig. 6). Noticeably, the width of these fingers, denoted $l$, is
close to the thickness of the swollen films. For such a situation, the
compressive swelling stress within the fingers is likely to be relaxed along
the longitudinal $y$-axis, thereby taking the driving mechanism for crack
propagation below the threshold stress.

Figure 6: Microscope observation of a dried debonded film at various stages of
the delamination process ($h=1.2$ $\mathrm{\SIUnitSymbolMicro m}$, grafting
density of 0.6 ${\mathrm{nm}}^{-2}$). Top: width of the debonded line as a
function of time. Bottom: images of the debonded lime taken at different times
denoted 1 to 5 in the top figure. Only half of the debonded line is shown in
(4) and (5). Adhering film fingers with a width about 5 times the swollen film
thickness are shown by arrows. The wavelength of the fingering instabilities
is about 10 times the swollen thickness.

The development of such a waviness in the delamination front could thus be
accounted for by differences in the stress state of the bounded film close to
a straight crack front and in the finger-like instabilities. As schematically
depicted in fig. 7, a wavy crack front exhibits two distinct areas. In area 1,
far from the fingers, the characteristic length in the $y$ direction
($\Lambda$) is much larger than the length scales in the other two directions,
i.e. $\Lambda>>h$. Conversely, at the finger tip (area 2), the width $l$ of
the finger is smaller or comparable with the characteristic lengths in the
other two directions i.e. $h\approx l$. This means that the material close to
the nearly straight crack front in between fingers (area 1) is rather in a
plane strain state while at the finger tip (area 2), it is under a plane
stress condition. Calculations detailed in appendix B indicate that the free
energy density of the swollen film under plane stress is lower than that under
plane strain which could account for the velocity difference between $V_{1}$
and $V_{2}$.
A prediction of the wavelength of the instability would require a more
quantitative analysis which is beyond the scope of this paper. However, one
could anticipate that this wavelength should be proportional to the film
thickness which is the only relevant length scale in the problem.
Interestingly, experimental [23] and theoretical [25, 24] investigations of
the peeling of thin films in confined geometries revealed the occurrence of
similar fingering instabilities with a characteristic wavelength of the order
of 3-4 times the film thickness, a value close to that measured in the present
study.

Figure 7: Schematic of debonding along a wavy crack front (blue and yellow
areas correspond to the delaminated and bounded part of the film,
respectively) corresponding to the situation depicted in fig. 6 (4). In the
fingers, the material is assumed to be in a plane stress state as a result of
the release of the swelling stress along the $y$ direction. Conversely, the
material close to a nearly straight crack front is under a plane strain state.
An estimate of the free energy density in both conditions indicates that crack
propagation is favored in plane strain (debonding rate $V_{1}>V_{2}$).

### Threshold thickness for delamination

Fig. 8 shows how the initial debonding velocity $V_{0}$ is varying as a
function of the film thickness and grafting density. Here, $V_{0}$ was
measured for a delaminated width of $L=20\leavevmode\nobreak\
$\mathrm{\SIUnitSymbolMicro m}$$ (the defect width is $L_{0}=9$
$\mathrm{\SIUnitSymbolMicro m}$), i.e. in a regime where the crack line is
still predominantly a straight line as in fig. 6 (1). It turns out that
delamination occurs only above a thickness threshold $h^{*}$ which is
decreasing with the grafting density ($h^{*}=390\pm 90\leavevmode\nobreak\
$\mathrm{nm}$$ and $h^{*}=650\pm 130\leavevmode\nobreak\ $\mathrm{nm}$$ for
0.2 and 0.6 ${\mathrm{nm}}^{-2}$, respectively). Above this threshold, orders
of magnitude changes in the debonding velocity are measured when the film
thickness is increased up to the $\mathrm{\SIUnitSymbolMicro m}$ range (fig.
8).
The existence of a thickness threshold for debonding and the dependence of
delamination rate on film thickness can be qualitatively accounted for by a
simple scaling argument where the elastic energy $U_{e}$ per unit length
released in the delamination of the constrained film is equated to the
fracture energy of the interface $\Gamma_{0}$. Taking $U_{e}=\sigma^{2}h/E$,
with $E$ the Young’s modulus, the threshold swelling stress for the occurrence
of debonding scales as $\sigma_{0}\approx\sqrt{\Gamma_{0}E/h}$, i.e. as
$1/\sqrt{h}$ (as detailed below, the swelling stress is independent on film
thickness $h$).
It should be noted that the concept that the strain energy release rate
increases in proportion of the film thickness and can only propagate an
interfacial crack when a critical thickness is exceeded is well established in
the literature dealing with hard coating. As an example, the scaling of
blister size with film thickness has been used as a tool to measure the
adhesion energy in Diamond-Like Carbon (DLC) films based on the estimate of
the elastic energy stored within the buckles [26, 27, 28].

Figure 8: Debonding velocity $V_{0}$ as a function of dry film thickness $h$
for films with a grafting density of ($\bullet$) 0.2 ${\mathrm{nm}}^{-2}$ and
($\bullet$) 0.6 ${\mathrm{nm}}^{-2}$.

In the next section, we develop a refined fracture mechanics analysis of
delamination which takes into account the non linearities of the swollen
polymer network and makes the assumption that debonding is induced by
localized differential swelling stresses instead of buckling stresses. From
this analysis, we will derive quantitative values of the interface fracture
energy $\Gamma_{0}$ from the measured thickness threshold $h^{*}$.

## Theoretical crack model

In what follows, a nonlinear fracture mechanics approach to the debonding
threshold is developed on the basis of a calculation of the amount of energy
which is released when the bonded, constrained layer, separates from the
substrate and then swells further. Here, this analysis is developed in the
limit of a straight crack line propagating along the direction denoted as $x$
in Fig. 9, i.e. for the initial stages of delamination. The main assumption of
the model is that debonding arises from localized stresses induced by the
differential swelling of the film in the grafted and delaminated states.
Specifically, we make the assumption that close to the crack tip, the
delaminated film is allowed to swell only in directions $x$ and $z$, i.e. the
film is in a plane strain deformation state, while the bonded part the film is
only allowed to swell in direction $z$.

### Swelling equilibrium

As a starting point, we first focus on the equilibrium swelling state in the
delaminated and bonded parts of the film denoted as $A$ and $B$ in fig. 9. As
detailed by Treloar [29], the swelling equilibrium of the gel network in these
two swelling states can be derived from a balance of the free energy of
mixing, $W_{m}$, and the free energy associated to the stretching of network,
$W_{e}$. Following Flory and Rehner [30], we take for the free energy of
mixing

Figure 9: Reference configuration of the delaminated gel film of thickness $h$
on a rigid substrate (before swelling). In zone $A$, the material undergoes
free swelling in plane strain, whereas in zone $B$ material is constrained to
swell only in direction $z$. By virtue of the plane strain assumption, the gel
film at $B$ is under in-plane bi-axial stress. $\mathcal{C}$ is the contour
(red line) over which the J-integral in Eq. (6) is evaluated.

$W_{m}=-\frac{kT}{v}\left[vC\ln\left(1+\frac{1}{vC}\right)+\frac{\chi}{1+vC}\right],$
(1)

where $\nu C$ is the volume of the solvent molecules with respect to the
volume of the dry polymer network (i.e., $\nu C=\phi/(1-\phi)$ with $\phi$ the
water volume fraction within the swollen gel system) and $\chi$ is a
dimensionless measure of the enthalpy of mixing known as the Flory parameter.
For the free energy due to the stretching of the polymer network, we used the
following expression [31, 32]

$W_{e}=\frac{G}{2}\left(\lambda_{x}^{2}+\lambda_{z}^{2}+\lambda_{y}^{2}-3-2\ln\lambda_{x}\lambda_{z}\lambda_{y}\right),$
(2)

where $\lambda_{i=x,y,z}$ are the stretch ratios and $G$ is the shear modulus
of the drained network. In the above equation, the logarithmic term accounts
for swelling (this term would be vanishing for an incompressible rubber, i.e.
when $\lambda_{x}\lambda_{z}\lambda_{y}=1)$.
For one dimensional swelling of the bonded film, $\lambda_{z}=\lambda_{B}$,
$\lambda_{x}=\lambda_{y}=1$ and molecular incompressibility yields $1+\nu
C=\lambda_{B}$. Using equations (1) and (2), the swelling equilibrium is given
by

$\frac{kT}{v}\left[\ln\left(1-\frac{1}{\lambda_{B}}\right)+\frac{1}{\lambda_{B}}+\frac{\chi}{\lambda_{B}^{2}}\right]=-G\left(\lambda_{B}-\frac{1}{\lambda_{B}}\right).$
(3)

For plane strain swelling of the debonded film,
$\lambda_{x}=\lambda_{z}=\lambda_{A}$, $\lambda_{y}=1$ and $1+\nu
C=\lambda_{A}^{2}$ which yields for equilibrium

$\frac{kT}{v}\left[\ln\left(1-\frac{1}{\lambda_{A}^{2}}\right)+\frac{1}{\lambda_{A}^{2}}+\frac{\chi}{\lambda_{A}^{4}}\right]=-G\left(1-\frac{1}{\lambda_{A}^{2}}\right).$
(4)

In Fig. 10, the theoretical values of the swelling ratio in the bonded and
delaminated parts of the films are plotted as a function of $G$ for a value of
the interaction parameter, $\chi=0.57$ which corresponds to the experimental
value of the synthesized PDMA network [5]. Although the volume fraction of
water $\phi$ within the swollen network (see inset) is, as expected, greater
in the delaminated zone ($\phi_{A}>\phi_{B}$), it turns out that the release
of the lateral constraints results in $\lambda_{A}<\lambda_{B}$: the thickness
of the debonded film ($A$) is smaller than that of the bounded film ($B$). As
mentioned above, the swelling ratio is found to be $\lambda_{B}=2.7$ from
ellipsometry measurements carried out in water. According to eqn (3), this
swelling ratio corresponds to a shear modulus of the drained network
$G=800\leavevmode\nobreak\ $\mathrm{kPa}$$ 222It should be noted that the dry
PDMA network is glassy at room temperature. The obtained value of $G$ is thus
to be related to that which would be measured above $T_{g}$ (i.e. about
110$\mathrm{\SIUnitSymbolCelsius}$).. The calculated thickness of the debonded
film is decreased by a factor 1.4 while it is expanding in the direction $x$
by a factor 1.9 with respect to the grafted film (which corresponds to a net
increase in the water volume fraction from 0.63 to 0.71). In the bonded part
of the film, the true compressive stress (i.e. the stress applied to the
current, swollen, material configuration) can be written as

$\sigma_{xx}=\sigma_{yy}=G\left(\lambda_{B}-\frac{1}{\lambda_{B}}\right).$ (5)

With $\lambda_{B}=2.7$, $\sigma_{xx}=1.9\leavevmode\nobreak\
$\mathrm{MPa}$\approx 2.4G$.

Figure 10: Theoretical swelling ratios $\lambda$ calculated for the bonded
(red, $\lambda_{B}$, Eq. (3)) and the plane strain delaminated parts of the
film (blue, $\lambda_{A}$, Eq. (4)) as a function of the shear modulus $G$ of
the drained network ($\chi=0.57$, $kT/\nu=1.38\>10^{8}\>$\mathrm{Pa}$$).
Inset: corresponding volume fractions $\Phi_{A}$ and $\Phi_{B}$ of water
within the swollen network.

### Threshold energy release rate $\Gamma$ for crack extension

In what follows, we derive the energy $\Gamma$ available for unit crack
extension in the reference (i.e. dry) configuration using a non linear
fracture mechanics approach. Following the work by Bouklas et al. [33] for
fracture in hydrogels, we use for that purpose a modified form of the path-
independent $J$-integral approach developed by Rice [34]. Accordingly,
$\Gamma$ is evaluated under equilibrium conditions (i.e. the chemical
potential of the solvent is constant) using the following contour integral $J$
where the usual elastic strain energy density is replaced by the Grand
potential $\widehat{W}$

$\Gamma\equiv
J=\int_{\mathcal{C}}\left(\widehat{W}_{B}(\mathbf{F}_{B},\mu)N_{1}-s_{iJ}N_{J}\frac{\partial
x_{i}}{\partial X_{1}}\right)dS$ (6)

where according to Eqn 2.24 in [33] $X$ and $x$ denote the coordinates in the
reference and deformed configuration, respectively, $\mathbf{F}$ is the
deformation gradient tensor, $N_{J}$ is the $J$ component of outward unit
vector normal on the boundary of the reference configuration and $s_{iJ}$ is
the nominal stress. $\widehat{W}=W-\mu C$ (with $W=W_{m}+W_{e}$) is the Grand
free energy density in which the independent variables are the deformation
gradient tensor $\mathbf{F}$ and the chemical potential $\mu$ of the solvent
molecules [35].
Evaluation of the $J$-integral along the contour $\mathcal{C}$ (Fig. 9) yields

$\Gamma=h\left[\left(\widehat{W}_{B}(\mathbf{F}_{B},\mu)-s_{xx}(B)\right)-\left(\widehat{W}_{A}(\mathbf{F}_{A},\mu)\right)\right]$
(7)

where the subscripts $A$ and $B$ denote that the quantity $\widehat{W}$ is
evaluated at $A$ and $B$ (see fig. 9). Here we have considered that the gel is
in a state of equilibrium and that the chemical potential of the solvent
molecules within the gel is homogeneous and equal to that of the solvent
outside the network. In the above expression, we also made the assumption that
the only relevant stress in the bonded part of the gel is the compressive
swelling stress $s_{xx}$ and that the delaminated part of the gel film is
traction free. The latter is justified by the extensive folding of the
debonded film we observed experimentally. After some calculations detailed in
Appendix A, the strain energy release rate writes

$\frac{\Gamma}{Gh}=\left[\frac{\lambda_{B}^{2}-1-2\ln\lambda_{B}}{2}+2\ln\lambda_{A}+\frac{1}{\lambda^{2}_{A}}-\frac{1}{\lambda_{B}}+\lambda_{B}-1\right]\\\
+\frac{kT}{\nu
G}\left[\left(2\chi-1\right)\left(\frac{1}{\lambda^{2}_{A}}-\frac{1}{\lambda_{B}}\right)+\frac{\chi}{\lambda^{2}_{B}}-\frac{\chi}{\lambda^{4}_{A}}\right].$
(8)

We used the results in fig.10 to obtain $\lambda_{A}$ and $\lambda_{B}$ to
calculate the normalized strain energy release rate $\Gamma/Gh$ which is
reported in fig.11 as a function of the shear modulus $G$ of the network for a
value of the Flory interaction parameter $\chi=0.57$.

Figure 11: Non dimensional strain energy release rate $\Gamma/Gh$ for the
delamination of a film with dry thickness $h$ as a function of the shear
modulus $G$ of the drained hydrogel network ($\chi=0.57$,
$kT/\nu=1.37\>10^{8}\leavevmode\nobreak\ $\mathrm{Pa}$$).

## Discussion and conclusion

Swelling stresses within the investigated PDMA hydrogel films were observed to
induce delamination from pre-existing strip defects with low adhesion. The
velocity and the extent of debonding depend on both film thickness and the
amount of reactive thiol groups grafted on the silicon substrate prior to film
deposition. A threshold thickness $h^{*}$ for the initiation of debonding was
especially evidenced. This threshold thickness corresponds to a situation
where the strain energy release rate reaches the fracture energy of the
interface, $\Gamma=\Gamma_{0}$. Taking $G=800$ $\mathrm{kPa}$, we can
determine $\lambda_{A}$ and $\lambda_{B}$ from fig. 10 and use this value to
calculate the non dimensional strain energy release rate $\Gamma/Gh$ from Eqn
8. We find $\Gamma/Gh=8.4$ which means that at the onset of debonding
$\Gamma_{0}=8.4Gh^{*}$. For the measured values of $h^{*}$ (from $390$ to
$650$ $\mathrm{nm}$), this value corresponds to a threshold fracture energy
which ranges from $\Gamma_{0}$=2.6 to 4.3
$\mathrm{J}\text{\,}{\mathrm{m}}^{-2}$ when the thiol density is increased
from $\Sigma$=0.2 to 0.6 ${\mathrm{nm}}^{-2}$.
For a comparison, an independent estimate of the fracture energy can be
derived from simple molecular arguments taking into account the energy which
is dissipated in the breaking of a stretched network chain grafted onto the
rigid silicon substrate. For that purpose, we follow an argument initially
proposed by Lake and Thomas [36] for rubber fracture which was recently
extended by Akagi et al. [37] to the molecular description of fracture energy
within water swollen polymer gels. Basically, this theory postulates that when
any of the chain bonds breaks, the total bond energy of each bond in the chain
is irreversibly lost. Assuming that only chains grafted onto the silicon
substrate break, the threshold energy for crack initiation is

$\Gamma_{0}=\eta U_{b}\Sigma$ (9)

where $\Sigma$ is the areal density of chains grafted on the silicon wafer,
$U_{b}$ is the dissociation energy of a carbon-sulfur bond ($\approx 4\times
10^{-19}\>$\mathrm{J}$$) and $\eta$ is the average number of monomers of the
chains. From the value of the shear modulus, $G=800$ $\mathrm{kPa}$, we can
estimate $\eta$ as the average monomer number between crosslinks: $\eta=\rho
RT/Gm_{m}\approx 30$ with $m_{m}$=99 $\mathrm{g}\text{\,}{\mathrm{mol}}^{-1}$
the monomer mass and $\rho$ the density. We assume here that the surface
density of the grafted chains and of the thiol groups are equal. According to
eqn. 9, for $\Sigma$=0.1 to 1 ${\mathrm{nm}}^{-2}$, the threshold fracture
energy is found to be of the order of $\Gamma_{0}\approx$1 to 10
$\mathrm{J}\text{\,}{\mathrm{m}}^{-2}$ which is consistent with the values
derived from eqn (8).
The order of magnitude agreement between the threshold fracture energy derived
from our model and a Lake and Thomas estimate of the energy required to break
covalent bonds at the interface tends to support our assumption of a vanishing
poroelastic dissipation close to the crack tip at the onset of crack
propagation. The validation of our fracture model from experiments would,
however, deserve further investigations, especially regarding the effects of
changes in the swelling ratio as a result of varying network crosslink
densities. From Eq. 8, we expect the threshold thickness $h^{*}$ to depend non
linearly on the associated changes in the shear modulus and swelling ratio.
Experimental results show that above the crack threshold, order of magnitude
changes in the initial debonding rate $V_{0}$ are also induced when the
thickness or the grafting density are varied (Fig. 8). A theoretical analysis
of $V_{0}$ would require much more involved calculations taking into account
the dissipative processes at skate when the crack front is advancing. As
detailed by Bouklas et al., [33] the determination of the energy release rate
in such situations requires to take into account solvent diffusion at the
crack tip, i.e. to separate the energy lost in diffusion from the enregy
available to drive the crack growth. Such an approach would require extensive
numerical simulations which are beyond the scope of this study.

## Acknowledgments

This project has received funding from the European Union’s Horizon 2020
research and innovation program under the Marie Skłodowska-Curie grant
agreement No 754387. It was also granted by Saint Gobain Research Paris. C.-Y.
Hui and B. Zhu are supported by NSF CMMI grant no.-1903308. Part of this work
was carried out on the Platform for Nano-Characterization (PFNC) and was
supported by the “Recherche Technologique de Base” program of the French
National Research Agency (ANR). One of us (A. Augustine) is indebted to
Ekkachai Martwong for his kind support during the synthesis of the hydrogels
systems. C.-Y. Hui also thank ESPCI Paris for hosting his visit.

## Appendix

###  A Derivation of the strain energy release rate $\Gamma$

In the reference (i.e. dry) configuration the energy $\Gamma$ available for
unit crack extension is derived using the following J-integral expression

$\Gamma=h\left[\left(\widehat{W}_{B}(\mathbf{F}_{B},\mu)-s_{xx}(B)\right)-\left(\widehat{W}_{A}(\mathbf{F}_{A},\mu)\right)\right]$
(A.1)

According to Hong et al. [38], the grand free energy can be written as

$\displaystyle\widehat{W}(\mathbf{F},\mu)=\frac{G}{2}\left(I-3-2\ln J\right)$
(A.2)
$\displaystyle-\frac{kT}{\nu}\left[\left(J-1\right)\ln\frac{J}{J-1}+\frac{\chi}{J}\right]$
$\displaystyle-\frac{\mu}{\nu}\left(J-1\right)$

where $J=\det\mathbf{F}$ and $I=\operatorname{Tr}(\mathbf{F}^{T}\mathbf{F})$.
Using the Flory-Rehner model [30] one can shown that, in plane strain

$\displaystyle\widehat{W}_{A}(\mathbf{F}_{A},\mu)=G\left(\lambda_{A}^{2}-1-\ln\lambda_{A}^{2}\right)$
(A.3)
$\displaystyle-\frac{kT}{\nu}\left[\left(\lambda_{A}^{2}-1\right)\ln\frac{\lambda_{A}^{2}}{\lambda_{A}^{2}-1}+\frac{\chi}{\lambda_{A}^{2}}\right]$
$\displaystyle-\frac{\mu}{\nu}\left(\lambda_{A}^{2}-1\right)$

with

$\mathbf{F}_{A}=\begin{vmatrix}\lambda_{A}&0&0\\\ 0&1&0\\\
0&0&\lambda_{A}\end{vmatrix}$ (A.4)

and

$\displaystyle\widehat{W}_{B}(\mathbf{F}_{B},\mu)=\frac{G}{2}\left(\lambda_{B}^{2}-1-\ln\lambda_{B}^{2}\right)$
(A.5)
$\displaystyle-\frac{kT}{\nu}\left[\left(\lambda_{B}-1\right)\ln\frac{\lambda_{B}}{\lambda_{B}-1}+\frac{\chi}{\lambda_{B}}\right]$
$\displaystyle-\frac{\mu}{\nu}\left(\lambda_{B}-1\right)$

with

$\mathbf{F}_{B}=\begin{vmatrix}1&0&0\\\ 0&1&0\\\
0&0&\lambda_{B}\end{vmatrix},$ (A.6)

where the relationships between the equilibrium swelling ratios
$\lambda_{A}$,$\lambda_{B}$ and the chemical potential of the swollen gel
$\mu$ is given by

$\frac{\mu_{A}}{\nu}=G\left(1-\frac{1}{\lambda_{A}^{2}}\right)-\frac{kT}{\nu}\left[\ln\frac{\lambda_{A}^{2}}{\lambda_{A}^{2}-1}-\frac{1}{\lambda_{A}^{2}}-\frac{\chi}{\lambda_{A}^{4}}\right]$
(A.7)
$\frac{\mu_{B}}{\nu}=G\left(\lambda_{B}-\frac{1}{\lambda_{B}}\right)-\frac{kT}{\nu}\left[\ln\frac{\lambda_{B}}{\lambda_{B}-1}-\frac{1}{\lambda_{B}}-\frac{\chi}{\lambda_{B}^{2}}\right]$
(A.8)

with $\mu_{A}=\mu_{B}$. The free energy at $A$ can be expressed in term of
$\lambda_{A}$ only by substituting (A.7) into (A) to eliminate the chemical
potential,i.e.

$\displaystyle\widehat{W}_{A}(\mathbf{F}_{A},\mu)=G\left(\frac{\lambda_{A}^{2}-1}{\lambda_{A}^{2}}-\ln\lambda_{A}^{2}\right)$
(A.9)
$\displaystyle-\frac{kT}{\nu}\left[1+\frac{2\chi-1}{\lambda_{A}^{2}}-\frac{\chi}{\lambda_{A}^{4}}\right]$

In the same way, substituting (A.8) into (A) and taking $s_{xx}(B)=\partial
W_{e}/\partial\lambda_{B}$, it comes

$\displaystyle\widehat{W}_{B}(\mathbf{F}_{B},\mu)-s_{xx}(B)=\frac{G}{2}\left(\lambda_{B}^{2}-1-2\ln\lambda_{B}\right)$
(A.10)
$\displaystyle+G\left(\frac{\lambda_{B}^{2}-1}{\lambda_{B}}\right)-\frac{kT}{\nu}\left[1+\frac{2\chi-1}{\lambda_{B}}-\frac{\chi}{\lambda_{B}^{2}}\right]$

From Eq. A.1, we obtain Eq. 8 that gives $\Gamma$ as a function of
$\lambda_{A}$ and $\lambda_{B}$.

###  B Estimate of the free energy density in the finger-like instabilities

In order to estimate the free energy density of the adhering film in the
finger-like instabilities, we assume that the swelling stresses are largely
relaxed due to the release of lateral constraints when $\ell\approx h$ (area 2
in fig. 6). Such a situation is approximated by considering that the film
within the finger is under a plane stress condition equivalent to isotropic
swelling. The corresponding free energy $W_{i}so$ is calculated from Eq. A
with

$\mathbf{F}_{iso}=\begin{vmatrix}\lambda_{iso}&0&0\\\ 0&\lambda_{iso}&0\\\
0&0&\lambda_{iso}\end{vmatrix},$ (B.1)

where the stretch ratio $\lambda_{iso}$ is obtained from the Flory Rehner
theory [30] as

$\frac{kT}{v}\left[\ln\left(1-\frac{1}{\lambda_{iso}^{3}}\right)+\frac{1}{\lambda_{iso}^{3}}+\frac{\chi}{\lambda_{iso}^{6}}\right]=-G\left(\frac{1}{\lambda_{iso}}-\frac{1}{\lambda_{iso}^{3}}\right).$
(B.2)

Conversely, when the adhering film is close to a straight crack front (area 1
in fig. 6), we consider that the film is under a plane strain condition with
the strain deformation tensor given by Eq. A.4. The corresponding free energy
density is obtained from Eq. A.
Taking $G=800\>$\mathrm{kPa}$$, $kT/\nu=1.37\>10^{8}\leavevmode\nobreak\
$\mathrm{Pa}$$, $\lambda_{A}=1.87$ and $\lambda_{iso}=1.58$, the difference in
the free energy between plane strain $W_{PE}$ and plane stress $W_{PS}$ is
$\Delta W=W_{PS}-W_{PE}=6.37\>10^{5}\leavevmode\nobreak\
$\mathrm{J}\text{\,}{\mathrm{m}}^{-3}$=637\leavevmode\nobreak\
$\mathrm{kPa}$\approx G$ which means that the energy density within the
adhering fingers is less than that close to a straight crack front and that
the difference is the order of magnitude of the shear modulus.

## References

  * Grube _et al._ [2015] S. Grube, K. Siegmann and M. Hirayama, _Journal of Coatings Technology and Research_ , 2015, 12, 669–680
  * Chevallier _et al._ [2011] P. Chevallier, S. Turgeon, C. Sarra-Bournet, R. Turcotte and G. Laroche, _ACS Appl Mater Interfaces_ , 2011, 3, 750–758
  * Shibraen _et al._ [2016] M. H. M. A. Shibraen, H. Yagoub, X. Zhang, J. Xu and S. Yang, _Applied Surface Science_ , 2016, 370, 1–5
  * Park _et al._ [2016] S. Park, S. Park, D. H. Jang, H. S. Lee and C. H. Park, _Materials Letters_ , 2016, 180, 81–84
  * Delavoipière _et al._ [2018] J. Delavoipière, B. Heurtefeu, J. Teisseire, A. Chateauminois, Y. Tran, M. Fermigier and E. Verneuil, _Langmuir_ , 2018, 34, 15238–15244
  * Trujillo _et al._ [2008] V. Trujillo, J. Kim and R. C. Hayward, _Soft Matter_ , 2008, 4, 564–569
  * Tanaka _et al._ [1987] T. Tanaka, S.-T. Sun, Y. Hirokawa, S. Katayama, J. Kucera, Y. Hirose and T. Amiya, _Nature_ , 1987, 325, 796–798
  * Yoon _et al._ [2010] J. Yoon, J. Kim and R.C. Hayward, _Soft Matter_ , 2010, 6, 5807–5816
  * Xu _et al._ [2015] X. Xu, X. Yao, F. Chen and Q. Fu, _RSC Advances_ , 2015, 5, 3733–3742
  * Velankar _et al._ [2012] S. S. Velankar, V. Lai and R. A. Vaia, _ACS Applied Materials & Interfaces_, 2012, 4, 24–29
  * Sharp and Jones [2002] J. S. Sharp and R. A. L. Jones, _Physical Review E_ , 2002, 66, 011801
  * Xu _et al._ [2016] B. B. Xu, Q. Liu, Z. Suo and R. C. Hayward, _Advanced Functional Materials_ , 2016, 26, 3218–3225
  * Takahashi _et al._ [2019] R. Takahashi, H. Miyazako, A. Tanaka and Y. Ueno, _ACS Applied Materials & Interfaces_, 2019, 11, 28267–28277
  * Audoly [2019] B. Audoly, _Physical Review Letters_ , 1999, 83, 4124–4127
  * Audoly _et al._ [2002] B. Audoly, B. Roman, and A. Pocheau, _The European Physical Journal B_ , 2002, 27,7–10
  * Gioia and Ortiz [1997] G. Gioia and M. Ortiz, _Advances in Applied Mechanics_ , 1997, 33, 119–192
  * Hutchinson and Suo [1992] J. W. Hutchinson and Z. Suo, _Advances in applied mechanics_ , 1991, 29, 63–191
  * Moon _et al._ [2002] M.W. Moon, H.M. Jansen, J.W. Hutchinson, K.H. Oh and A.G. Evans _Journal of The Mechanics and Physics of Solids_ , 2002, 50, 2355-2377
  * Gent _et al._ [2001] A. N. Gent, W. J. Hung and M. F. Tse, _Rubber Chemistry and Technology_ , 2001, 74, 89–99
  * Chollet _et al._ [2016] B. Chollet, M. Li, E. Martwong, B. Bresson, C. Fretigny, P. Tabeling and Y. Tran, _ACS Appl. Mater. Interfaces_ , 2016, 8, 11729–11738
  * Li _et al._ [2015] M. Li, B. Bresson, F. Cousin, C. Fretigny and Y. Tran, _Langmuir_ , 2015, 31, 11516–11524
  * Augustine _et al._ [2023] A. Augustine, M. Veillerot, N. Gauthier, E. Verneuil, A. Chateauminois and Y. Tran, _to be submitted_ , 2023
  * Ghatak _et al._ [2000] A. Ghatak, M. K. Chaudhury, V. Shenoy and A. Sharma, _Physics Review Letters_ , 2000, 85, 4329–4332
  * Mönch and Herminghaus [2001] W. Mönch and S. Herminghaus, _Europhysics Letters_ , 2001, 53, 525 – 531
  * Vilmin _et al._ [2010] T. Vilmin, F. Ziebert and E. Raphaël, _Langmuir_ , 2010, 26, 3257–3260
  * Matuda _et al._ [1981] N. Matuda and S. Baba and A. Kinbara, _Thin Solid Films_ , 1981, 81, 301–305
  * Iyer _et al._ [1995] S.B. Iyer and K.S. Harshavardhan and V. Kumar, _Thin Solid Films_ , 1995, 256, 94–100
  * Nir [1984] D. Nir, _Thin Solid Films_ , 1984, 112, 41-49
  * Treloar [2005] L. Treloar, _The physics of rubber elasticity_ , Oxford University Press, 3rd edn, 2005
  * Flory and Rehner [1943] P. J. Flory and J. Rehner, _journal of Chemical Physics_ , 1943, 11, 521–526
  * Flory [1961] P. J. Flory, _Transactions of the Faraday Society_ , 1961, 57, 829
  * Wall and Flory [1951] F. Wall and P. Flory, _Journal of Chemical Physics_ , 1951, 19, 1435
  * Bouklas _et al._ [2015] N. Bouklas, C. M. Landis and R. Huang, _Journal of Applied Mechanics_ , 2015, 82, 081007
  * Rice [1968] J. R. Rice, _Journal of Applied Mechanics_ , 1968, 35, 379–386
  * Hong _et al._ [2009] W. Hong, Z. Liu and Z. Suo, _International Journal of Solids and Structures_ , 2009, 46, 3282–3289
  * J. and G. [1967] L. G. J. and T. A. G., _Proceedings of the Royal Society London Seies A_ , 1967, 300, 108–119
  * Akagi _et al._ [2013] Y. Akagi, H. Sakurai, J. P. Gong, U. Chung and T. Sakai, _Journal of Chemical Physics_ , 2013, 139, 144905
  * Hong _et al._ [2008] W. Hong, X. Zhao, J. Zhou and Z. Suo, _Journal of the Mechanics and Physics of Solids_ , 2008, 56, 1779–1793