# Friction through molecular adsorption at the sliding interface of hydrogels:
Theory and experiments

###### Abstract

Lola Ciapa, Ludovic Olanier, Yvette Tran, Christian Frétigny, Antoine
Chateauminois, Emilie Verneuil
Soft Matter Science and Engineering (SIMM), CNRS UMR 7615, ESPCI Paris, PSL
University, Sorbonne Université, F-75005 Paris, France
<EMAIL_ADDRESS><EMAIL_ADDRESS>

We report on the frictional properties of thin ($\approx\mu m$)
poly(dimethylacrylamide) hydrogel films within contacts with spherical silica
probes. In order to focus on the contribution to friction of interfacial
dissipation, a dedicated rotational setup is designed which allows to suppress
poroelastic flows while ensuring an uniform velocity field at the sliding
interface. The physical-chemistry of the interface is varied from the grafting
of various silanes on the silica probes. Remarkably, we identify a velocity
range in which the average frictional stress systematically varies with the
logarithm of the sliding velocity. This dependency is found to be sensitive to
the physical-chemistry of the silica surfaces. Experimental observations are
discussed in the light of a molecular model where friction arises from
thermally activated adsorption of polymer chains at the sliding interface,
their elastic stretching and subsequent desorption. From this theoretical
description, our experimental data provide us with adhesion energies and
characteristic times for molecular adsorption that are found consistent with
the physico-chemistry of the chemically-modified silica surfaces.

00footnotetext: † Electronic Supplementary Information (ESI) available: SI1:
Scaling of friction force with contact radius; SI2: Frictional shear stress in
the low velocity regime.].

##  I Introduction

Hydrogels consist of hydrophilic polymer networks that can absorb large
quantities of water. As such, they have recently been developed as promising
materials for varied applications such as optical engineering, bioengineering,
coatings, microfluidics,… When used as substitute for cartilages,[1, 2, 3] as
contact lenses [4, 5] or as anti-fog coatings for windows and windshields,[6,
7] a good control of their frictional properties is needed. Recently, attempts
have been made to tune the frictional properties of hydrogels by changing the
physico-chemistry of their surface or of the counter-surface. Hence, among
other examples, changing the counter surface hydrophobicity [8] or the
electrostatic charges at the two interfaces [9] was found to change both the
shape and the magnitude of the friction-velocity curves. However, the friction
force was found to result from at least two intricate mechanisms: interfacial
friction and viscous dissipation. Interfacial friction is reminiscent of the
Schallamach model for rubber friction [10]: it results from the molecular
adsorption, stretching and desorption of the gel polymers at the interface.
Viscous dissipation occurs within a thin lubrication film of solvent possibly
squeezed at the sliding interface.[11, 12] While the former is expected to
dominate at low velocity, the latter arises at larger velocity where
lubrication is forced, [13] but their relative weight still lacks a
quantitative description. This is due to two especially challenging
experimental characterizations: the molecular parameters controlling the
adsorption mechanisms across the confined interface, [14] and the
thickness[13, 15, 16] and rheological behavior[17] of a lubrication film when
it exists.[18, 19, 20] In addition, a third effect comes into play when
contacts are displaced at the hydrogel surface: poroelasticity. Indeed, the
normal load translates into a porous flow within the gel network until the
load balances with the elastic response of the polymer network. These flows
result in a velocity- and time-dependent contact area [21, 22] and possibly
extra frictional dissipation. Hence, in friction experiments where a
macroscopic solid or an AFM probe slides on a hydrogel [23], the three
mechanisms should a priori operate simultaneously. Hence, understanding
hydrogel friction still lacks a precise description of each one of the
dissipative phenomena, and of their possible couplings.

In the present paper, we offer to gain insights into the physical mechanisms
at play in hydrogel friction by designing experiments aiming at isolating the
molecular adsorption/desorption mechanisms and measuring their molecular
parameters (adsorption energy and time, surface density). To do so, we will
show that our experimental set-up and system allows to minimize lubrication
and poroelastic transport so that the friction-velocity data we measure can be
interpreted solely in terms of interfacial friction as defined above. In the
past, a large variety of experiments have been developed in order to measure
polymer/substrate interactions, at both macroscopic and single molecule
scales. Macroscopic tests include static and dynamic adhesion experiments[24],
flows of polymer melts[25], or friction on elastomers[26] or glassy
polymers[27]. Their interpretation is not always straightforward due to
complex couplings between interfacial and bulk dissipations, non-uniform
interfacial velocities, or possibly collective effects. In an alternate and
interesting way to investigate the role of pinning/depinning molecular
processes on friction, Bennewitz and collaborators designed functionalised
silica surfaces whose frictional and adhesive properties were tuned through
switchable guest/host interactions by applying either an electric potential
[28] or light [29]. Single-molecule experiments have been helpful at assessing
the molecular parameters governing the break of single bonds in pull-out
experiments[30] or friction of single molecules using AFM tips[31]. The force-
extension spectra measured in these works - sometimes combining both in- and
off-plane displacements- were successfully interpreted with models based on
thermally activated force-induced desorption to obtain molecular adhesion
energies or molecular mobility. The dependency of these adsorption energies
with extension rate[30] or orientation of the imposed displacement[31] remains
however unclear. Finally, friction between two monolayers of C12-alkyl chains
obtained by the strong adsorption of cationic surfactants on mica was measured
using a Surface Force Apparatus nanotribometer over decades in sliding
velocities [32] and convincingly interpreted with a thermally activated model
derived from Schallamach.[10] The authors however point out that (i) for such
dense monolayers, correlation between neighboring bonds cannot be avoided and
(ii) for these short alkyl chains for which the in-plane and out-of-plane
characteristic lengths are close, the bond stiffness and the density of bonds
cannot be separated so that the dependence with the surfactant concentration
could not be elucidated.

Here, by working on friction of hydrogels, we expect that the relatively low
density of active polymer chains at the interface will support the hypothesis
of independent bonds, while the stiffness of the bond will be set by the
length of the interfacial chains: this enhanced compliance will allow for
relatively larger displacements before break that can be modelled. We will
show that the molecular adhesion, the typical adsorption times, and the
surface density involved in an irreversible adsorption-stretching-desorption
dissipative mechanism can be measured between a variety of solid silica
surfaces functionalised with different molecules and a swollen hydrogel.

##  II Material and methods

###  II.1 Hydrogel films

All the experiments to be reported were carried out using
poly(dimethylacrylamide) (PDMA) hydrogel films covalently grafted onto glass
substrates. These films were synthesized by simultaneously crosslinking and
grafting preformed ene-functionalised polymer chains onto glass substrates
using a thiol-ene click reaction which is fully described elsewhere [33, 21]
and allows to separately control the film thickness and swelling ratio. Strong
adhesion between the hydrogel film and its substrate was achieved by
functionalising borosilicate glass slides with thiol groups. Through a thiol-
ene reaction, the gel film was covalently bound to the glass slide, thereby
preventing interfacial debonding during swelling and friction. In order to
assess the reproducibility of our results, two different PDMA films with
slightly different thicknesses and swelling ratio were synthesized and their
dry and swollen thicknesses were measured by spectroscopic ellipsometry. The
dry thickness $e_{0}$ and swelling ratio $Sw$ are $e_{0}=2.8$
$\mathrm{\SIUnitSymbolMicro m}$ and $Sw=2.3$ for one film, and $e_{0}=2.3$
$\mathrm{\SIUnitSymbolMicro m}$ and $Sw=1.8$ for the other one. The elastic
response of these two films was measured by indentation experiments using a
procedure published elsewhere [33]. For normal loads up to 200 mN, the
indentation curve was successfully described using a linear stress/strain
relationship characterized by an elastic oedometric modulus $\tilde{E}=21$ MPa
and $30$ MPa respectively. Finally, the shear modulus of the drained network
was deducted[34] from the swelling ratio $Sw$ to be $G_{0}=1.9$ MPa and $4.0$
MPa respectively. From the shear modulus, we estimate the number of Kuhn
segments between crosslinks $\nu_{c}=6$ and $\nu_{c}=13$ respectively, so that
$\nu_{c}$ will be approximated as $\nu_{c}\sim 10$ in what follows. The large
value of these moduli, together with the high crosslink density, were chosen
to avoid film damage upon swelling [34] and sliding [21, 22].

###  II.2 Functionalised silica surfaces

As solid probe, we used fused silica lenses (Newport, UV fused silica SPX114)
with curvature radius $R=23$ mm. In order to vary their molecular interactions
with PDMA hydrogel, the silica surface was grafted with three different
silanes, namely aminopropyltriethoxysilane (APTES, ABCR GmbH),
propyltriethoxysilane (PTES, ABCR GmbH) and octadecyltrichlorosilane (OTS,
ABCR GmbH).
The silica surface is first activated with a freshly prepared piranha
(H2SO4/H2O2) solution at 150 $\mathrm{\SIUnitSymbolCelsius}$, rinsed and
sonicated in Milli-Q water prior to drying under nitrogen flow and subsequent
UV/ozone treatment. Then, the freshly cleaned silica lenses are quickly
transferred into a sealed reactor filled with nitrogen where a solution of dry
toluene with the silane is introduced. For APTES and PTES silanes, the lenses
are soaked for three hours in solutions containing 5%vol and 3%vol of silane,
respectively. For OTS, the lens is immersed for 15 minutes only in a 0.25%vol
solution. These conditions are selected in order to promote the formation of
dense monolayers without extensive polycondensation of the silanes. The silica
lens is then rinsed and sonicated in toluene before drying.
In what follows, the PTES, APTES and OTS treated surfaces will be denoted as
”propyl”, ”aminopropyl” and ”octadecyl”, respectively.
The contact angle of deionized water droplets on these surfaces were found to
be 56, 62 and 104 o for the propyl, aminopropyl and octadecyl surfaces,
respectively. The grafting of the silica lenses was assessed from
ellipsometric microscopy performed on silanized silicon wafers which were
grafted simultaneously with the silica lenses. The ellipsometric images of the
silanized surfaces allow to measure an average thickness which combines the
molecular size of the grafted molecules and their surface density, and the
homegeneity of the silane layer. Both propyl and octadecyl surfaces are
homogeneous over areas of the order of 1 ${\mathrm{mm}}^{2}$ and a pixel size
of 2 $\mu$m2 with variations below 0.1 $\mathrm{nm}$. The average thickness is
0.4 $\mathrm{nm}$ and 2.3 $\mathrm{nm}$ for propyl and octadecyl
respectivement, in agreement with the literature[27, 35]. Accounting for the
molecular size of the silanes, this shows that, compared to propyl, octadecyl
is grafted more densily and exhibits longer chains extending outwards,
consistently with the literature [36]. For aminopropyl, the average thickness
is 1.6 $\mathrm{nm}$, greater than the monolayer thickness measured in the
literature at 0.7 $\mathrm{nm}$ [37], and the layer is more heterogeneous with
sparse localized defects about 5 $\mathrm{nm}$ in thickness and 10
$\mathrm{\SIUnitSymbolMicro m}$ wide which can be interpreted as multilayered
islands. Since propyl and aminopropyl silanes have about the same molecular
size, we conclude that aminopropyl layers are denser than propyl layers.
Hence, the surface densities rank as follows: propyl ¡ aminopropyl ¡
octadecyl, consistently with the contact angle measurements which show
hydrophobicity increases likewise.

As a reference, a non-functionalised silica lens cleaned with piranha solution
and UV/ozone was also considered. The frictional properties of this freshly
cleaned silica lenses were observed to change as a function of time as a
result of progressive surface contamination (timescale $\sim$ hour). The
friction results reported below corresponds to stabilized values of the
lateral force. In comparison, the silanated glass lenses provided friction
forces that were stable over time.

###  II.3 Friction set-up

A dedicated setup was designed in order to perform sliding experiments within
a contact between a spherical silica lens and a gel film without any
contribution of poroelastic transport to friction. Such a constraint requires
that the contact remains fixed within the framework of the gel film. As
schematically depicted in Fig. 1, this goal was achieved by rotating the lens
about an axis which is slightly tilted by an angle $\alpha=5$ deg. with
respect to the normal to the film. For the considered radius of curvature of
the lens ($R=23$ $\mathrm{mm}$) and typical values of the contact radius
$a\approx 100$ $\mathrm{\SIUnitSymbolMicro m}$, geometrical calculations
indicate that the radius of curvature of the sliding trajectories within the
contact is about ten times the contact radius, i.e. sliding trajectories can
be considered as rectilinear within the contact. As compared to more
conventional rotational experiments using plate rheometers in which sliding
velocities vary from zero at the center to the set value at the edge, the
advantage here is that the sliding velocity field within the contact is
uniform to a very good approximation.
All the experiments are carried out under imposed normal load and sliding
velocity with the contact immersed in a droplet of deionized water. As shown
in Fig. 1, an accurate alignment of the apex of the lens (b) with respect to
the axis of the rotating stage (M060-DG, PI) is obtained by means of a home-
made, two-axis, micro-metric translation stage (c) with high stiffness. During
experiments, the normal and lateral forces are continuously monitored from the
measurement (using capacitive displacement sensors CS02 and CS05,
Microepsilon) of the deflection of two crossed double cantilever beams (d)
with stiffnesses $k_{\perp}=1.01\>10^{5}$ N.m-1 and
$k_{\parallel}=1.13\>10^{5}$ N.m-1 in the normal and lateral directions,
respectively. A double cantilever beam actuated in servo-loop control by a
piezoelectric actuator (e) allows to maintain a constant normal force during
the rotation of the lens and to compensate the effects of small residual
misalignments of the lens with respect to the rotation axis.

Figure 1: Schematic of the friction rotation set-up. The coated glass
substrate (a) is contacting a silica lens (b) fixed on centering mount (c)
which is actuated by a rotating stage. The rotation axis is inclined by an
angle $\alpha=$5 deg. with respect to the $z$ axis by means of a goniometric
stage. The normal and lateral force components ($F_{n}$ and $F_{t}$) are
measured from the deflection of two crossed double-cantilever arrangements
using two capacitive displacements sensors (d). A constant normal load is
applied by means of a double cantilever beam system and a piezoelectric
actuator (e) operated in feedback loop control. The contact immersed in a
deionized water droplet is lightened by a white light source and images are
recorded using a zoom lens (f), a CMOS camera (g) and an optical setup
consisting in a semi-reflecting plate, two crossed polarizers (h) and a
quarte-wave plate.

Reflection Interference Contrast Microscopy (RICM) [38] images of the immersed
contact were continuously recorded through the glass substrate using a CMOS
camera (SVS Vistek Exo250 MU3, 2048x2048 pixels with 12 bits resolution), a
combination of crossed-polarizers and a quarter-wave plate, a macro zoom (APO
Z16, Leica), and white light illumination (HXP, Leica). A typical image of the
steady-state contact is shown in Fig. 1(g) where the contact appears as a dark
disk whose radius is denoted $a$. The optical contrast measured at the
gel/probe interface showed that no lubrication water film is trapped at this
interface.
Friction experiments were carried out with sliding velocities ranging from 0.4
$\mathrm{\SIUnitSymbolMicro m}\text{\,}{\mathrm{s}}^{-1}$ to 450
$\mathrm{\SIUnitSymbolMicro m}\text{\,}{\mathrm{s}}^{-1}$ and under imposed
normal load $F_{n}$ from 20 $\mathrm{mN}$ to 200 $\mathrm{mN}$. The values of
the normal load were chosen so that the hydrogel remains sufficiently
hydrated[33]. More precisely, the water volume fraction $\phi$ in the film is
always larger than the threshold corresponding to the glass transition of the
PDMA network ($\phi\approx 0.2$). In the range of loads tested, the contact
radius $a$ varies between 100 and 150 $\mu$m so that it is always far larger
than the film thickness $e_{0}$. Under such a mechanical confinement, the
polymer network compresses along the normal direction solely with no
significant lateral strain. As a consequence, the number density of polymer
chains in the plane of the interface can be assumed to remain constant,
independently of the level of film compression.

In all experiments, sliding motion is initiated after a static contact time
greater that the poroelastic time $\tau$ (of the order of a few tens of
seconds for the contact conditions under consideration [22]) in order to
ensure that indentation equilibrium is achieved. However, when sliding starts,
we systematically observe that the lens further sinks into the film: a small
increase of the indentation depth $\delta$ (or equivalently of the contact
radius $a$) occurs over a time close to the poroelastic time (results not
shown). Typically, the increase in indentation depth is about 30 $\mathrm{nm}$
for an initial indentation depth $\delta_{0}\approx 410$ $\mathrm{nm}$ under
an imposed normal load $F_{n}=100$ $\mathrm{mN}$ (the initial static contact
radius $a_{0}=132$ $\mathrm{\SIUnitSymbolMicro m}$ increases by 5
$\mathrm{\SIUnitSymbolMicro m}$). It was carefully checked that this
phenomenon is not an artifact due to a cross-talk between the normal and
lateral directions. Instead, it reflects an intrinsic coupling between the
normal and lateral stress components at the sliding interface which will be
discussed at the end of this paper in Section V. In what follows, the average
sliding stress $\sigma_{t}$ was systematically determined from the ratio of
the steady-state friction force $F_{t}$ to the actual contact area $\pi a^{2}$
(see Fig. 1g) measured under steady-sliding.

##  III Friction model

We aim at discussing the variations of the friction stress with both
interfacial physical-chemistry and velocity. For that purpose, we choose as a
framework the friction model initially developed by Schallamach[10] and
revisited by Chernyak and Leonov[39, 40], and Singh and Juvekar [41] in which
friction of elastomers is ascribed to a molecular mechanism, namely the
elastic stretching and desorption of polymer chains at the sliding interface.
In Section III.1, we recall the main assumptions and results of the model.
Then, in Section III.2, we identify various velocity regimes and we
conveniently derive approximate analytical expressions for the frictional
shear stress versus velocity, with parameters related to the molecular
interactions and to the stretching behavior of polymer chains. We especially
identify an intermediate velocity range in which $\sigma_{t}$ depends
logarithmically on sliding velocity.

###  III.1 Steady-state frictional stress

In the following, an infinite extended gel/solid interface is considered. At
rest, the adsorption of the polymer chains in the gel is thermally favored and
reversible: the adsorption and desorption rates are described by probability
distributions of Arrhenius’ type where the energy barriers are denoted $E$ and
$E+W$ respectively, as depicted in Fig. 2: Here, $W$ is a molecular adhesion
energy and we define $u=e^{-W/kT}$. Hence, at rest, the lifetimes of an
unbound chain $\tau$ and of a bound chain $\tau_{ads}$ can be written as:

$\displaystyle\tau$ $\displaystyle=\tau_{0}e^{E/kT}$ (1)
$\displaystyle\tau_{ads}$ $\displaystyle=\frac{\tau}{u}=\tau_{0}e^{(E+W)/kT}$
(2)

Figure 2: Schematics of the Schallamach’s model: (a) The silica surface slides
at a velocity $V$ on the gel surface; if a chain from the polymer network
adsorbs on silica, it stretches and experiences a force $f$ assumed to be
oriented along the interface plane; (b) Energy profiles for adsorption and
desorption of the polymer chains at $V=0$ and $V>O$. The energy difference
between bound and unbound states at rest is $W$. Upon sliding, this energy is
biased by an amount $\lambda f$ where $\lambda$ is an activation length.

where $\tau_{0}$ is a molecular characteristic time of exploration often taken
as $h/kT$, with $h$ the Planck’s constant. Upon sliding at a velocity $V$, the
adsorbed chains are stretched and the energy landscape is biased accordingly.
If a chain remains attached for a time $t_{a}$, it experiences a force $f$
that increases with $t_{a}$ and favors desorption. Then, following Eyring’s
model, the bias can be written as the stretch energy stored in the bond
$\lambda f$ where $\lambda$ is an activation length. For polymeric gels, this
length is expected to be of the order of the gel mesh size. Here, we assume
the chains are stretched within the plane of the interface so that the force
$f$ is directed along the interface plane as depicted in Fig. 2a). Hence,
denoting the chain length at rest $L$, and the stretch, $\Delta L$, the
stretch varies linearly with the bond time and the velocity as $\Delta
L=Vt_{a}$. In the present derivation of the model, assumption is made that the
unbound to bound transition is not biased by stretching111The effect of
stretching on the unbound to bound transition is cautiously accounted for in
models derived for the detachment force of adhesive patches and is found to
affect the very low velocity regime. See for example [42]. while the unbinding
rate depends on the lifetime of the bond $t_{a}$ through $f(t_{a})$. In
addition, $t_{a}$ is the age of the bond, but we assume no chemical ageing :
the energies $E$ and $W$ do not depend on $t_{a}$. Then, during sliding, the
binding and unbinding rates per unit time follow Eyring’s model[43] and are
written as:

$\displaystyle r_{b}$
$\displaystyle=\frac{1}{\tau_{0}}e^{-E/kT}=\frac{1}{\tau}$ (3) $\displaystyle
r_{u}$ $\displaystyle=\frac{1}{\tau_{0}}e^{-\left(E+W-\lambda
f\left(t_{a}\right)\right)/kT}=\frac{u}{\tau}e^{-\lambda
f\left(t_{a}\right)/kT}$ (4)

We now derive the equations governing the distribution $n(t,t_{a})$ that
describes the number of bonds that have been attached for a time $t_{a}$ at a
time $t$. Indeed, this distribution allows to express the friction stress
$\sigma_{t}$, which is the sum of the forces exerted by the stretched chains
at a given time $t$, according to:

$\sigma_{t}=\int_{0}^{\infty}n\left(t_{a},t\right)f\left(t_{a}\right)dt_{a}$
(5)

We first define the total number of bonds $N(t)$ existing at time $t$ and
given by:

$N(t)=\int_{0}^{\infty}n\left(t_{a},t\right)dt_{a}$ (6)

Between $t$ and $t+dt$, the number of bonds aged $t_{a}$ increases, first,
because of the ageing of the bonds that were aged $t_{a}-dt$ at time $t$.
Second, old enough bonds unbind with a rate $r_{u}$ and in proportion to the
number of bonds. Third, new bonds are created with a rate $r_{b}$, in
proportion to the total number of available sites $(N_{0}-N(t))$, and are aged
$t_{a}=0$ so that we use the Dirac function $\delta(t_{a})$. Using Eqs. 4, the
time evolution of the number of bonds obeys:

$\frac{\partial n\left(t_{a},t\right)}{\partial t}=-\frac{\partial
n\left(t_{a},t\right)}{\partial
t_{a}}+\frac{1}{\tau}\left[N_{0}-N\left(t\right)\right]\delta\left(t_{a}\right)-\frac{u}{\tau}e^{\lambda
f\left(t_{a}\right)/kT}n\left(t_{a},t\right)$

In steady state, the time derivative vanishes so that:

$\frac{dn\left(t_{a}\right)}{dt_{a}}=\frac{1}{\tau}\left(N_{0}-N\right)\delta\left(t_{a}\right)-\frac{u}{\tau}e^{\lambda
f\left(t_{a}\right)/kT}n\left(t_{a}\right)$ (7)

This differential equation is solved by setting: $n(t_{a})=C(t_{a})g(t_{a})$
where $g(t_{a})$ is defined by:

$g(t_{a})=\exp\left(-\frac{u}{\tau}\int_{0}^{t_{a}}e^{\lambda
f(\xi)/kT}d\xi\right)$ (8)

Equation 7 becomes
$C^{\prime}(t_{a})=\frac{1}{\tau}\left(N_{0}-N\right)\frac{\delta(t_{a})}{g(t_{a})}$
which yields:

$C(t_{a})=\frac{1}{\tau}\left(N_{0}-N\right)\leavevmode\nobreak\
\text{for}\leavevmode\nobreak\ t_{a}>0$ (9)

Then, the number distribution of bonds aged $t_{a}$ is given by:

$n(t_{a})=\frac{1}{\tau}\left(N_{0}-N\right)g(t_{a})$ (10)

We define $G$ as $G=\frac{1}{\tau}\int_{0}^{\infty}g(t_{a})dt_{a}$. Equation 6
gives the total number of active bonds $N$ and, from this, the distribution
$n(t_{a})$:

$\displaystyle N=\frac{G}{1+G}N_{0}$ (11) $\displaystyle
n(t_{a})=\frac{\frac{1}{\tau}g(t_{a})}{1+G}N_{0}$ (12)

The friction stress can then be expressed from Eq. 5 as:

$\sigma_{t}=\frac{\frac{1}{\tau}\int_{0}^{\infty}g(t_{a})f(t_{a})dt_{a}}{1+G}N_{0}=\frac{N}{G}\frac{1}{\tau}\int_{0}^{\infty}g(t_{a})f(t_{a})\,dt_{a}$
(13)

which corresponds to Eq. 17 in reference [41]. In the particular case of a
Hookean force $f$ depending linearly of the stretch $\Delta L=t_{a}V$ though a
stiffness denoted $M$, the force $f(t_{a})=MVt_{a}$ and $g(t_{a})$, $G$ and
$N$ can be further derived. The hookean model well describes the case of
polymer chains in a poor solvent[44] as PDMA in water for which the Flory
parameter is $\chi=0.57$ [6]. A characteristic velocity $V^{*}$ comes up, for
which the typical elastic energy stored in the stretched polymer chain
$MV^{*}\tau\lambda$ is equal to the thermal energy $kT$. It corresponds to a
characteristic stretch denoted $l=\frac{kT}{\lambda M}$. This allows to define
two non-dimensional variables in time $s$ and space $z$ so that:

$\displaystyle V^{*}=\frac{l}{\tau}=\frac{kT}{\tau\lambda M}$ (14)
$\displaystyle t_{a}=\tau\frac{V^{*}}{V}s$ (15)
$\displaystyle\xi=\tau\frac{V^{*}}{V}z$ (16)

and Eqs. 8 and $G(t_{a})$ become:

$\displaystyle
g(t_{a})=\exp{\left(-\frac{uV^{*}}{V}\int_{0}^{s}e^{z}\,dz\right)}=\exp{-\frac{uV^{*}}{V}\left(e^{s}-1\right)}$
(17) $\displaystyle
G=\frac{V^{*}}{V}\int_{0}^{\infty}\exp{\left(-\frac{uV^{*}}{V}(e^{s}-1)\right)}\,ds$
(18)

The velocity dependence of the number of bonds $N=N_{0}\frac{G}{1+G}$ and of
the friction stress

$\sigma_{t}=\frac{kT}{\lambda}\frac{V^{*}}{V}\frac{N}{G}\int_{0}^{\infty}\exp{\left(-\frac{uV^{*}}{V}(e^{s}-1)\right)}s\,ds$
(19)

are set by two parameters, namely $V^{*}/V$ and $uV^{*}/V$. The former is
actually the ratio between the advective time $\tau_{V}=l/V$ built upon the
characteristic stretch $l=\frac{kT}{\lambda M}$ and the lifetime of a detached
bond $\tau$. The latter is a ratio between this advective time $\tau_{V}=l/V$
and the lifetime of an active bond $\tau_{ads}$ defined in Eq. 2. Eq. 19 is
evaluated in Appendix, and the result is plotted in Fig. 3. A bell shaped
curve is obtained for $\sigma_{t}(V/V^{*})$ with a maximum at $V=V^{*}$.

Figure 3: Friction stress $\sigma_{t}$ normalized by $N_{0}kT/2\lambda$ as a
function of the normalized velocity $\hat{V}=V/V^{*}$ in logarithmic scale
(Eq. 19). Here, $u=10^{-7}$ which corresponds to an adhesion energy $W\sim
15kT$. Intermediate velocities correspond to the logarithmic regime.

###  III.2 Approximate expressions for $\sigma_{t}(V)$

In the following, we offer to derive approximate analytical expressions of
$\sigma_{t}$ at low, intermediate, and large velocities.

####  III.2.1 Low velocity regime $V\ll uV^{*}$

At low velocities, the integrand in Eq. 18 almost always vanishes except for
vanishing $s$ values. Hence, Watson’s lemma yields
$G\sim\frac{V^{*}}{V}\int_{0}^{\infty}\exp\left(-s\frac{uV^{*}}{V}\right)\,ds\sim\frac{1}{u}$.
Remember that $u$ is defined as $u=e^{-W/kT}$ where $W$ is a molecular
adsorption energy which is typically equal to a few $kT$ [44]. Hence, $u\ll 1$
and, from this, $G>>1$. It follows that the total number of bonds $N\sim
N_{0}$: almost all possible sites are bound. Here, the sliding is slow enough
so that the advective time is large compared to both the adsorption and
desorption times. It follows that the friction stress increases linearly with
the velocity as:
$\sigma_{t}=\frac{N_{0}kT}{\lambda}\frac{V}{uV^{*}}=N_{0}MV\tau_{ads}$.
Experimentally, this low velocity regime is not observed for all the silica
lenses and will not be further examined.

####  III.2.2 Intermediate velocities $uV^{*}\ll V\ll V^{*}$

At larger velocities for which $uV^{*}\ll V$, the integrand in Eq. 18 is a
step-down function: its value is zero for large enough $s$ and 1 below a
threshold value $S$ given by:
$\exp\left(-\frac{uV^{*}}{V}(e^{S}-1)\right)=1/2$ so that $S=\ln\frac{V\ln
2}{uV^{*}}$. It follows that the integral in $G$ and $\sigma_{t}$ is simply
approximated as:

$G=\frac{V^{*}}{V}\ln\frac{V\ln 2}{uV^{*}}$ (20)
$\sigma_{t}=\frac{N_{0}kT}{2\lambda}\frac{S^{2}}{\frac{V}{V^{*}}+S}$ (21)

This allows to define two velocity regimes, above and below $V^{*}$.
Below $V^{*}$, this intermediate regime is clearly logarithmic in velocity. We
find $G>>1$ and again, the total number of bonds is almost equal to the number
of possible sites $N\sim N_{0}$: the thermal energy controls the binding and
unbinding transitions and the bias on the energetic landscape due to the
elastic force $f$ is negligible. Hence, the mean lifetime of a bond is still
equal to $\tau_{ads}$ but this time, the advective time is smaller than the
lifetime of a bond at rest: $\tau\ll\tau_{V}\ll\tau_{ads}$ so that the sliding
effectively stretches the chains and, during the adsorption time, the elastic
stretch increases with velocity. The friction stress increases faster with
velocity, and is now written as:

$\sigma_{t}=\frac{N_{0}kT}{2\lambda}\ln\frac{V\ln 2}{uV^{*}}$ (22)

####  III.2.3 High velocities $uV^{*}\ll V^{*}\ll V$

Finally, above $V^{*}$, Eqs. 20,21 still hold but, now, the large velocity
regime corresponds to a decrease in the number of bonds due to the detrimental
effect of sliding on the binding transition: here,
$\tau_{V}\ll\tau\ll\tau_{ads}$. It follows from Eq. 20 and $V\gg>V^{*}$ that
$G\ll 1$ and $N\sim N_{0}G\ll N_{0}$: the number of active bonds decreases. In
the meantime, the stretching also diminishes the mean bond time which
decreases with $V$. Altogether, the friction stress decreases with velocity:
$\sigma_{t}\sim\frac{N_{0}kT}{2\lambda}\frac{V^{*}}{V}\ln^{2}\frac{V\ln
2}{uV^{*}}$. In this case, this equation only partially approximates the exact
solution since the active bonds number decreases slowly. In the literature,
this decreasing and large velocity regime was positively compared [41] to data
collected in friction experiments of silicone elastomers on alkyl-silanized
glass slides [45]. Qualitative comparisons were also made with friction
experiments on hydrogels of varied chemistry where bell-shaped curves were
observed [13].

##  IV Experimental results

For each of the functionalised lenses, the steady state friction force $F_{t}$
and contact radius $a$ were measured as a function of the sliding velocity
$V$. From this raw data (detailed in Supplementary Information), the average
frictional shear stress $\sigma_{t}=F_{t}/\pi a^{2}$ was calculated and
reported in Fig. 4 as a function of the sliding velocity $V$ plotted on a
logarithmic scale, for the film with thickness $e_{0}$=2.3 $\mu$m and $Sw=1.8$
for varied normal loads $F_{n}$. The four data sets clearly exhibit
logarithmically increasing branches. Both their slope and intercept with the
$x$-axis strongly depend on the physical-chemistry of the silica surface. The
slope also depends on the normal load. Accordingly, we offer to analyse our
data in the light of the model derived for intermediate velocities (Eq. 22)
for which the thermal energy controls the transitions between free and bound
states and stretching energy shortens the unbinding.

Figure 4: Friction stress $\sigma_{t}$ as a function of the sliding velocity
$v$. Film thickness $e_{0}$=2.3 $\mu$m, swelling ratio $Sw=1.8$. Surface
treatment of the silica lens: (a) bare silica; (b) propyl; (c) aminopropyl;
(d) octadecyl. Normal load $F_{n}$: ($\bullet$) 20 $\mathrm{mN}$;
($\blacksquare$) 50 $\mathrm{mN}$; ($\blacklozenge$) 100 $\mathrm{mN}$;
($\blacktriangle$) 150 $\mathrm{mN}$; (✦) 200 $\mathrm{mN}$. Solid and dotted
lines are fits to Eq. 23.

Since the two gel films under consideration have similar swelling ratios, the
order of magnitude of all the parameters related to the polymeric network will
be kept as a constant. Noticeably, the stiffness $M$ of the bonds can here be
precisely modelled from the stiffness of the network chains, at variance with
studies on patches of self-assembled monolayers (SAM) where the bond stiffness
could not be separated from the bonds surface density.[32] The chain stiffness
$M$ is estimated as $M=kT/\nu_{c}b^{2}$ with $b\approx 1$ nm the Kuhn length
for PDMA and $\nu_{c}=10$ the number of segments of a chain at the interface,
which we consider equal to the segment number between crosslinks measured in
Section II.1. Hence, the characteristic stretching length $l=\frac{kT}{\lambda
M}$ can be rewritten as $l\sim\frac{\nu_{c}b^{2}}{\lambda}$, with $\lambda\sim
1$ nm (the mesh size): we find $l\sim 10$ nm. This value is 20 times larger
than the stretch obtained on SAMs of short dodecyl chains, of 0.45 nm.[32]
This relatively large value reflects the compliance of the coiled polymer
chains in the present hydrogel system, which conveniently shifts the velocity
range for molecular friction towards larger values since $V^{*}=l/\tau$.
On the other hand, the lifetime of a bond will depend on the physical
chemistry of the gel/silica interface though $\tau_{ads}=\tau/u$, the lifetime
of a bond at zero velocity, defined by Eq. 2. Hence, omitting the $\ln{2}\sim
1$ factor, Eq. 22 is conveniently rewritten as:

$\sigma_{t}=N_{0}\frac{kT}{2\lambda}\ln\frac{V}{l/\tau_{ads}}$ (23)

in which $l$ depends on the polymeric network only, while $N_{0}$ and
$\tau_{ads}$ characterize the interfacial adsorption. Equation 23 was fitted
to the logarithmic branch of the friction stress data from Fig. 4 with the
adsorption time $\tau_{ads}$ and the surface density of bonds $N_{0}$ as
fitting parameters. For propyl and bare silica (Fig. 4a,b), the logarithmic
branch extends over more than three decades in velocity, and over one decade
for aminopropyl and octadecyl (Fig. 4c,d). The fitting parameters $\tau_{ads}$
and $N_{0}$ are plotted in Fig. 5 and 6a as a function of the normal force
$F_{n}$ for the four different silica chemistries (colors) and the two PDMA
films (empty or full markers).
Remarkably, for each type of grafted silica, the stress-velocity curves
intercept the $x$-axis at the same value of the critical velocity $uV^{*}$
independently of the normal force or of the hydrogel films. It follows that
each silica treated lens is characterized by a single value of $\tau_{ads}$.
For propyl silica, the mean value is $\tau_{ads}=0.2$ s. Noting for example
that the propyl curve exhibits a logarithmic branch over 4 decades and no
maximum in the experimental window allows to estimate that $u$ exceed
$10^{-4}$ so that the molecular adhesion energy $W$ exceeds $4kT$.

Figure 5: Lifetime in the bound state $\tau_{ads}$ as a function of normal
load $F_{N}$. $\bullet$ bare silica, • propyl, • aminopropyl, • octadecyl.
Open and filled symbols refer to films with $e_{0}=2.8$
$\mathrm{\SIUnitSymbolMicro m}$, $\lambda=2.3$ and $e_{0}=2.3$
$\mathrm{\SIUnitSymbolMicro m}$, $\lambda=1.8$, respectively. Dotted lines are
guides for the eye.

On the other hand, the number of available sites $N_{0}$ is found in Fig. 6a
to increase with $F_{n}$, a surprising result which suggests a coupling
between the frictional and normal stress responses. It nevertheless reduces to
a linear dependence on the elastic deformation of the film which is squeezed
by the lens: when $N_{0}$ is normalized by the ratio between the actual
indentation depth of the lens into the film $\delta$ and the swollen thickness
of the hydrogel film $e_{0}$, the data for the two different gel films no
longer varies with the normal load as plotted in Fig. 6b. This result allows
to compare the different types of surface in terms of number density of
available sites: bare silica, propyl, and aminopropyl silica engage about the
same number of bonds with the hydrogel, while for octadecyl silica, this
number is 3 to 4 times larger: $N_{0,propyl}\sim N_{0,silica}\lesssim
N_{0,amino}\ll N_{0,octadecyl}$. We will see that this ranking is reversed
when adhesion energy is considered.

Figure 6:  Density of available binding sites on the silica surface as a
function of the normal load $F_{N}$. (a) Number of binding sites per surface
area $N_{0}$; (b) Reduced number of binding sites
$\tilde{N_{0}}=N_{0}e_{0}/\delta$. $\bullet$ bare silica, • propyl, •
aminopropyl, • octadecyl. Open and filled symbols refer to films with
$e_{0}=2.8$ $\mathrm{\SIUnitSymbolMicro m}$, $\lambda=2.3$ and $e_{0}=2.3$
$\mathrm{\SIUnitSymbolMicro m}$, $\lambda=1.8$, respectively.

In the following, we offer to compare the adsorption times depending on the
silica chemistry. The largest value of $\tau_{ads}$ being obtained for bare
silica and propyl, these interfaces exhibit the largest adhesion energy and a
ranking in terms of adhesion is established: $W_{propyl}\sim
W_{silica}>W_{amino}>W_{octadecyl}$ with an estimate of the differences:
$W_{propyl}-W_{silica}\sim\pm 0.5kT$, $W_{propyl}-W_{amino}\sim 4.6kT$ and
$W_{propyl}-W_{octadecyl}\sim 6.5kT$.

##  V Discussion

The interface of PDMA with bare silica and propyl silica having close adhesion
energy are likely to engage molecular interactions of the same type, namely
H-bonds between amides groups of PDMA and silanols SiOH on the silica side
(see Fig. 7). Indeed, the silanization of silica by short alkyl chains is only
partial, disordered and unreacted silanol groups are still available as
compared to longer alkyl chains as detailed in Section II.2. Here, the grafted
propyl chains appear to be short enough not to disturb the H-bonds with
neighboring silanols. This is at variance with what is observed with the long
octadecyl C18 tails: H-bonds with the buried silanol groups are probably
unfavored. Instead, an entropic attraction is likely to develop between the
densely grafted layer of C18 tails and the PDMA chains which excludes water.
It is also thought of as an ”hydrophobic interaction” and expected to
correspond to much lower adhesion energies than H-bonds.[46] Recalling from
Section II.2 that the silanization by OTS provides both larger and more
uniform grafting densities compared to propylsilane,[35, 36] the surface
density $N_{0,octadecyl}$ we measure is consistently larger than
$N_{0,propyl}$. Finally, the aminopropyl-grafted surface has an intermediate
behavior. The tails of the aminopropyl and propyl silanes have the same
length, 3 carbons. In water at pH=6, part of the amine groups of $pKa\simeq
9.5$ [47] are protonized into NH3+ while part of the silanol are deprotonized
into SiO-. Electrostatic interactions between neighboring charged aminopropyl
silanes and silanols lead to the bending of the former on the latter,
preventing the formation of H-bonds with part of the silanols which are then
capped by hydrophobic C3 tails. Hence, on the aminopropyl-silica surface, the
average adhesion energy with PDMA in water results from a combination of
H-bonds with the remaining non-capped silanols and low hydrophobic
interactions elsewhere: this could explain the low value of $W_{0,amino}$ as
compared to bare silica or propyl silica.

Figure 7: Schematics of the available groups for binding at the surface of the
PDMA network (top) and of the functionalised silica surfaces (bottom). The
proton acceptors and donors involved in hydrogen bonds are indicated together
with the groups possibly involved in entropic (or hydrophobic) interactions.

We shall now briefly comment on the two observations evidencing a coupling
between the normal and friction forces: (i) Upon sliding, the silica lens
further sinks into the hydrogel film, whatever its surface chemistry. The
contact radius increases. (Section II.3) (ii) The prefactor $N_{0}$ of the
logarithmic $V$-dependence of the friction stress (Eq. 23) increases with the
normal force and linearly depends on the normal deformation of the gel (Fig.
6). A complete description of these coupling mechanisms would deserve a
comprehensive study, but we suggest here to link these observations with the
theoretical descriptions derived by Leonov [40, 39] of the adhesive friction
of elastomers. Based on Schallamach’s model, Leonov accounts for the finite
non-zero thickness of the interface whereas both Singh, Schallamach and, from
them, the present model, assume it to be zero. Then, if the polymer chains are
attached to the gel network at a certain non-zero depth from the silica
interface, upon sliding and stretching, they are likely to exert an elastic
force $f$ with a component both along the interface and normal to it: Leonov
shows that this mechanism results in not only a friction stress but also a
normal stress: the gel pulls the interface in with a normal stress that
increases by up to 10$\%$ when the velocity is ten-fold. The derivation of a
model combining molecular adsorption, stretching, and desorption à la Singh to
an interface with finite non-zero thickness à la Leonov, and comparison with
our experimental data on normal/friction coupling will be the subject of a
forthcoming paper.
On the experimental side in the literature, friction was found to depend on
the load when measured between two monolayers of alkyl chains[32], but the
load-dependence was ascribed to an increase of the adsorption energy or,
equivalently, of the bond lifetime, at variance with the present findings,
with no explanation of the underlying mechanisms. We offer next to summarize
our results by expressing the friction force as a function of the contact
radius, and extend their validity to another geometry where the silica lens
slides on the gel film following a linear trajectory. The rectilinear set-up
we used was described in earlier works [21, 22] that evidenced a poroelastic
flow within the gel film in response to the lens displacement. First, using
the geometrical relationship between indentation depth $\delta$ and contact
size $a$, $\delta=a^{2}/R$, our experimental data show that $\sigma_{t}$ vary
with the contact radius as $a^{2}$ or equivalently, $F_{t}$ varies as :

$F_{t}=a^{4}\tilde{N_{0}}\frac{kT}{2e_{0}\lambda}\ln\frac{V}{l/\tau_{ads}}$
(24)

with $\tilde{N_{o}}$ the reduced number of bonding sites per surface area that
only depends on physical-chemistry. As detailed in Supplementary Material, for
both geometries, namely rotational and rectilinear friction, we find that (i)
rectilinear friction depends on surface chemistry, (ii) a master curve in
$F_{t}/a^{4}$ allows to collapse the data as expected from Eq. 24. These
findings show that (i) friction by molecular adsorption at the interface
occurs even when poroelastic flows come into play, and (ii) even dominates
over poroelastic dissipation. Poroelasticity, however, does set the contact
size in transient sliding and steady state rectilinear sliding [21, 22].

Finally, we briefly discuss the friction data in the very low velocity regime
which is not accounted for by the present model (Fig.4c,d and SI2), in which
the effect of stretching on the unbound to bound energy barrier is neglected.
From the literature on pull-off experiments on small adhesive patches[30],
this hypothesis, however, is expected to fail at very low velocity where the
pinning energy barrier should be also be biased. This remark calls for the
development of a model combining biased interaction potentials as in Friddle
_et al._ [30] and variable number of available sites as in here[39, 32, 41].
We also note that ageing effects are likely to exist in this low velocity
regime where sites on the functionalised silica surface spend longer times
within the contact[48].

##  VI Conclusions

Steady-state friction experiments were conducted on hydrogel thin films using
a combination of model interfaces with controlled and varied surface chemistry
and an original experimental set-up. The latter relies on the sliding of a
rotating solid sphere at the hydrogel surface in a geometrical arrangement
that allows to (i) obtain a uniform sliding velocity within the sliding
interface, (ii) reach the low velocity regime in which interfacial friction
dominates, (iii) provide direct and contrasted images of the gel/probe
interface, (iv) provide a fine measurement, over time, of the friction and
normal forces, and of the contact area, indentation depth, and possibly
lubrication layers. In this geometry, by working at the bare surface of model
hydrogel films sliding against functionalised silica surfaces, we show that
friction is dominated by adhesive molecular interactions resulting in
irreversible adsoprtion-stretching-desorption mechanisms and thus dissipation
at the sliding interface.
An analytical model built upon the work of Schallamach [10] reformulated by
Singh and Juvekar [41] allowed us to derive analytical expressions for the
friction stress as a function of velocity. A logarithmic branch was clearly
identified in both the model and the experimental data. By fitting the
friction stress versus velocity data to the model for the four surface
chemistries tested and under varied load, we successfully assess the molecular
parameters involved in an adsorption/stretching/desorption mechanism of the
polymer chains of the hydrogel network at the solid interface. We insist that
this was made possible notably because the stiffness of the bonds is set by
the elasticity of the polymeric chains independently of the counter-surface
physico-chemistry, and the relatively dilute surface of hydrogels allows to
hypothetise that bonds are independent, two conditions underlying the model
that are not easily met in experimental systems.[32, 27]

Both the adsorption energies and the surface densities of bonds we measure are
consistent with the physico-chemical characteristics of the functionalised
silica surfaces. Hence, for the first time, the dissipative effects of
molecular interactions at a sliding hydrogel interface were isolated from
other sources of dissipation involved in friction (no viscous dissipation, no
poroelastic flow).
We believe our work has strong implications from both applied and fundamental
points of view. In material science, when frictional properties of hydrogels
are to be controlled, our results provide a quantitative tool to assess the
most relevant parameters to be tuned. In soft matter physics, friction on
hydrogel films in a rotating sliding geometry appears as a model situation to
explore the interaction energy landscapes at a polymer/solid interface.

## Acknowledgements

The authors gratefully acknowledge the support of Jean-Claude Mancer in the
realization of the frictional set-up. We also thank R. Bennewitz and C.
Drummond for fruitful discussions and for kindly making us aware of the work
of Friddle and coworkers.

## Appendix

###  A Explicit evaluation of $\sigma_{t}(V)$

We derive here the explicit calculation of the friction stress, starting from
the equations for $g(t_{a})$, $G$, and $\sigma_{t}$, Eq. 18 and 19:

$\displaystyle g(t_{a})=-\frac{uV^{*}}{V}\left(e^{s}-1\right)$ (A.1)
$\displaystyle
G=\frac{V^{*}}{V}\int_{0}^{\infty}ds\exp{\left(-\frac{uV^{*}}{V}(e^{s}-1)\right)}$
(A.2)
$\displaystyle\sigma_{t}=\frac{kT}{\lambda}\frac{V^{*}}{V}\frac{N}{G}\int_{0}^{\infty}sds\leavevmode\nobreak\
\exp{\left(-\frac{uV^{*}}{V}(e^{s}-1)\right)}$ (A.3)

Here, we introduce the ratio $\hat{V}=V/V^{*}$ so that $s=t_{a}/\tau\hat{V}$ .
It follows:

$\displaystyle g(t_{a})$
$\displaystyle=\exp\left[-\frac{u}{\hat{V}}\left(\exp\left(\frac{\hat{V}t_{a}}{\tau}-1\right)\right)\right]$
(A.4) $\displaystyle G$
$\displaystyle=\frac{\tau}{\hat{V}}\exp\left(\frac{u}{\hat{V}}\right)E_{1}\left(\frac{u}{\hat{V}}\right)$
(A.5)

where we use the exponential integral

$E_{1}\left(x\right)=\int_{x}^{\infty}\frac{e^{-y}}{y}dy$ (A.6)

The calculation arrives at:

$\sigma_{t}=\frac{kTN_{0}}{\lambda}\frac{\frac{1}{\hat{V}}\exp\left(\frac{u}{\hat{V}}\right)}{1+\frac{1}{\hat{V}_{0}}\exp\left(\frac{u}{\hat{V}}\right)E_{1}\left(\frac{u}{\hat{V}}\right)}\left[G_{1}\left(\frac{u}{\hat{V}}\right)-\ln\left(\frac{u}{\hat{V}}\right)E_{1}\left(\frac{u}{\hat{V}}\right)\right]$
(A.7)

with, using Maple, 222 This result can also be derived using DLMF [49]
${{}_{p+1}F_{q+1}}\left({a_{0},\dots,a_{p}\atop b_{0},\dots,b_{q}};z\right)=$
$\frac{\Gamma\left(b_{0}\right)}{\Gamma\left(a_{0}\right)\Gamma\left(b_{0}-a_{0}\right)}\int_{0}^{1}t^{a_{0}-1}(1-t)^{b_{0}-a_{0}-1}{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop
b_{1},\dots,b_{q}};zt\right)\mathrm{d}t$ to express
${{}_{3}F_{3}}\left({1,1,1\atop 2,2,2};z\right)$ from Wolfram [50]
${}_{2}F_{2}(1,1;2,2;z)=\frac{1}{z}\left(\operatorname{Ei}(z)+\frac{1}{2}\left(\log\left(\frac{1}{z}\right)-\log(z)\right)-\gamma\right)$

$\displaystyle G_{1}\left(x\right)=$
$\displaystyle\int_{x}^{\infty}\frac{e^{-y}}{y}\ln ydy$ (A.8) $\displaystyle=$
$\displaystyle\ln\left(x\right)E_{1}\left(x\right)+x\Bigg{[}\frac{1}{2x}\left(2\gamma\ln\left(x\right)+\ln^{2}\left(x\right)+\gamma^{2}+\frac{\pi^{2}}{6}\right)$
(A.9) $\displaystyle-{}_{3}F_{3}\left(1,1,1;2,2,2;-x\right)\Bigg{]}$ (A.10)

Finally,

$\sigma_{t}=\frac{kTN_{0}}{2\lambda}\frac{\frac{1}{\hat{V}}\exp\left(\frac{u}{\hat{V}}\right)}{1+\frac{1}{\hat{V}_{0}}\exp\left(\frac{u}{\hat{V}}\right)E_{1}\left(\frac{u}{\hat{V}}\right)}\\\
\left[\left(\gamma+\ln\frac{u}{\hat{V}}\right)^{2}+\frac{\pi^{2}}{6}-\frac{2u}{\hat{V}}\,_{3}F_{3}\left(1,1,1;2,2,2;-\frac{u}{\hat{V}}\right)\right]$
(A.11)

The variations of $\sigma_{t}$ with $\hat{V}$ are plotted in Figure 3 as a
lin-log plot and show the well-known bell-shaped curve with an increasing
logarithmic branch between $uV^{*}$ and $V^{*}$. Similar equations were
derived in the context of friction of adhesive patches of SAMs.[32]

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