# Morphology and stability of droplets sliding on soft viscoelastic
substrates.

Mathieu Oléron  Laurent Limat  Julien Dervaux  Matthieu Roché
<EMAIL_ADDRESS>Université Paris Cité, Matière et Systèmes
Complexes, CNRS UMR 7057, F-75013 Paris, France, EU

###### Abstract

We show that energy dissipation partition between a liquid and a solid
controls the shape and stability of droplets sliding on viscoelastic gels.
When both phases dissipate energy equally, droplet dynamics is similar to that
on rigid solids. When only the solid dissipate, we observe an apparent contact
angle hysteresis, of viscoelastic origin. We find excellent agreement between
our data and a non-linear model of the wetting of gels of our own that also
indicates the presence of significant slip. Our work opens general questions
on the dynamics of curved contact lines on compliant substrates.

An ever-increasing number of applications such as biofouling repellency
MacCallum et al. (2015); Lavielle et al. (2021), dew harvesting Sokuler et al.
(2010); Phadnis and Rykaczewski (2017); Sharma et al. (2022) and anti-icing
Mittal and Choi (2020) relies on the use of viscoelastic coatings. Not only do
these materials confer controlled interfacial properties to their substrate,
they may be compliant enough to deform and dissipate energy after the
deposition of a droplet Carré et al. (1996); Long et al. (1996); Style et al.
(2013a); Park et al. (2014), affecting both the equilibrium shape of the
latter and the dynamics of the contact line between the solid, the liquid and
the ambient fluid Carré and Shanahan (2001); Karpitschka et al. (2015);
Dervaux and Limat (2015). This coupling leads to unique substrate-droplet and
droplet-droplet interactions Style et al. (2013b); Karpitschka et al. (2016);
Zhao et al. (2018); Smith-Mannschott et al. (2021); Khattak et al. (2022).

Figure 1: Top view of droplets sliding on a silicone gel as a function of the
Bond number $Bo_{\alpha}$ and the relaxation ratio $\mathcal{R}$. Each image
displays the value of $Bo_{\alpha}$. The pinch behind the front of the droplet
at large $\mathcal{R}$ is an artefact: the equilibrium contact angle is
greater than $\pi/2$ for these systems and the liquid/air interface hangs over
the moving contact line. Orange arrow: direction of motion. Scale bar: $2$
$\mathrm{mm}$.

The influence of substrate compliance on the shape and stability of moving
droplets is hardly known. The morphology of droplets sliding down a rigid
plate inclined at an angle $\alpha$ with the horizontal is dictated by the
dependence of the dynamic contact angle $\theta_{d}$ between the liquid-gas
and solid-liquid interfaces on droplet velocity $U$ and the ability of the
trailing edge contact line to form corners Podgorski et al. (2001); Rio et al.
(2005); Snoeijer et al. (2005); Le Grand et al. (2005); Peters et al. (2009);
Winkels et al. (2011). Beyond a threshold velocity, corners destabilize into
rivulets that fragment into tinier droplets known as pearls Podgorski et al.
(2001); Rio et al. (2005); Le Grand et al. (2005); Puthenveettil et al.
(2013); Kim et al. (2015); Liang and Tietze (2017). These results hold for
systems where energy dissipation occurs entirely in the liquid. However, on a
gel with surface energy $\gamma_{s}$ and shear modulus $\mu_{0}$, deformations
having a magnitude comparable to the elastocapillary length
$\ell_{s}=\gamma_{s}/(2\mu_{0})$ propagate with the contact line, leading to
dissipation in the droplet and the substrate and a different $\theta_{d}(U)$
relationship. Thus, we expect a more complex shape selection process.

Figure 2: (a) Dependence of the liquid capillary number $Ca$ on the Bond
number $Bo_{\alpha}$. (b) Focus on the case $\mathcal{R}\simeq 2$. Blue dashed
line: Eq. 1. (c) Dependence of the solid capillary number $Ca_{s}$ on the Bond
number $Bo_{\alpha}$. Dashed line: Eq. 33. In all panels, filled symbols
correspond to points measured above the pearling threshold. (a) and (c) share
the same legend.

Here we tune energy dissipation partition between the liquid and the solid and
show that this balance controls the morphology and stability of droplets
sliding on soft solids. When both materials dissipate energy equally, droplet
dynamics resemble those reported on a rigid substrate, despite gel compliance.
When dissipation occurs only in the substrate, we observe an apparent contact
angle hysteresis and the appearance of corners only beyond the pearling
instability threshold. We rationalize our results with a non-linear model of
the wetting of soft solids of our own. The very good agreement we obtain
brings strong evidence that the apparent hysteresis originates from the
viscoelastic force exerted by the solid on the moving contact line. Our
analysis also highlights the existence of significant slip due to the presence
of free polymer chains in the gel. Finally we discuss the perspectives of our
work regarding the role of deformable substrates in setting the properties of
curved contact lines.

We perform experiments on a silicone gel with shear modulus $\mu_{0}\simeq 1$
kPa, relaxation time $\tau=18.2$ ms and a 62-wt% free chain content. We use
Newtonian liquids with surface tensions $\gamma$, densities $\rho$ and
viscosities $\eta$. Details about the materials and protocols can be found in
the Supplemental Materials suppMat (30). Sliding experiments are characterized
by two dimensionless numbers, the Bond number $Bo_{\alpha}=\rho
gR_{0}^{3}\sin{\alpha}/(\gamma R_{c})$ and the liquid capillary number
$Ca=\eta U/\gamma$, that compare capillary stresses to gravitational and
viscous ones respectively. Here, $g$ is the acceleration of gravity, $R_{0}$
is the radius of the spherical droplet before deposition, and $R_{c}$ is the
contact radius between the droplet and the substrate. Energy dissipation
partition between the two media is characterized by the relaxation ratio
$\mathcal{R}$ Karpitschka et al. (2015); Dervaux et al. (2020) that compares
the viscocapillary relaxation velocity in the liquid, $U_{l}=\gamma/\eta$, to
that in the solid, $U_{s}=\ell_{s}/\tau$, giving
$\mathcal{R}=\gamma\tau/(\eta\ell_{s})$. Contact line motion has been studied
in the limit $\mathcal{R}\rightarrow\infty$ in the literature Style et al.
(2017); Andreotti and Snoeijer (2020), i.e. energy dissipation in the liquid
is neglected. The liquids we use allow us to vary $\mathcal{R}$ over four
orders of magnitude.

Droplet shapes carry obvious signatures of changes in the magnitude of
$\mathcal{R}$ (Fig. 1). While droplets remain nearly axisymmetric at small
Bond numbers, symmetry is lost as $Bo_{\alpha}$ increases. When
$\mathcal{R}\simeq 2$, a corner appears at the trailing edge. In contrast, the
aft and fore radii of curvature of the droplets are comparable when
$10^{2}\leq\mathcal{R}\leq 10^{4}$. Besides, the droplet contour contains
portions parallel to the direction of motion, leading to shapes similar to
those observed in the case of droplets sliding on hysteretic surfaces Dussan
V. and Chow (1983); Podgorski (2000); Puthenveettil et al. (2013); Ahmed et
al. (2014). A further increase of $Bo_{\alpha}$ at all values of $\mathcal{R}$
leads to the observation of the pearling instability.

Figure 2a shows that, for equivalent Bond numbers, liquid capillary numbers
vary over four orders of magnitude as the relaxation ratio changes by the same
amount. The data for $\mathcal{R}\simeq 2$ suggest an affine relation between
$Ca$ and $Bo_{\alpha}$ (Fig.2b), with a non-zero y-intercept, similar to the
rigid case Podgorski et al. (2001); Le Grand et al. (2005). The functional
form for the other datasets is more complex. We multiply $\mathcal{R}$ with
$Ca$ to obtain a capillary number for the solid, $Ca_{s}=U\tau/\ell_{s}$ and
plot the data in Figs. 2a-b as a function of this quantity. Accounting for
variations of the equilibrium contact angle $\theta_{eq}$ from one system to
another suppMat (30), we observe a collapse of the large-$\mathcal{R}$ data on
a single master curve (Fig. 2c). We can discriminate the curve obtained for
$\mathcal{R}\simeq 2$, in line with the assumption that the power balance
relevant to these experiments differs from the one tested in this
representation.

Another way to characterize the dynamics of sliding droplets is to measure the
dependence of the dynamic contact angle on droplet velocity. Figure 3 shows
the deviation from the equilibrium contact angle $\theta_{eq}$ suppMat (30) as
a function of the capillary number $Ca$, for each system. We observe that the
apparent dynamic contact angle $\theta_{d}$ increases smoothly as the
capillary number goes from negative to positive values when $\mathcal{R}\simeq
2$. Corners appear when $Ca\geq 7.5\times 10^{-3}$, a value of the same order
of magnitude as those reported for fluoropolymer-coated silicon wafers
Podgorski et al. (2001); Rio et al. (2005); Le Grand et al. (2005). The other
curves display a steep jump of several tens of degrees around $Ca=0$ that
brings to mind results obtained in the case of significant wetting hysteresis
Eral et al. (2013) and when a contact line moves on low-modulus natural rubber
and cis-butadiene Extrand and Kumagai (1997). In a vein similar to what we
observed in figure 2, the datasets collapse on a master curve when plotted
against the solid capillary number $Ca_{s}$ suppMat (30). The curve obtained
at $\mathcal{R}\sim 10^{4}$ displays plateaus in the advancing and receding
branches, similar to those reported in earlier studies Karpitschka et al.
(2015).

Figure 3: (a) Dependence of the deviation from the equilibrium contact angle
$\theta_{d}-\theta_{eq}$ on the liquid capillary number. Positive (resp.
negative) $Ca$ values correspond to leading-edge (resp. trailing-edge) angles,
and advancing (resp. receding) contact lines. Filled symbols: points measured
above the pearling threshold. Red dashed line : fit of the Cox-Voinov law (Eq.
3) to the $\mathcal{R}\simeq 2$ data. (b)-(f) Comparison between experimental
dynamic contact angles $\theta_{d}$ and Eq. 4. The fitting parameter is the
microscopic cutoff length scale, $\lambda=100$ nm for all systems. Colored
areas: fit spread reflecting experimental uncertainties.

The data presented in figure 2 can be discussed in terms of scaling laws. We
focus first on the case $\mathcal{R}\simeq 2$, and we follow a rationale
proposed in studies of droplets sliding on rigid substrates Podgorski et al.
(2001); Le Grand et al. (2005). We assume that dissipation occurs entirely in
the liquid and balances the gravitational force experienced by the droplet.
Contact-angle hysteresis may also be present. The resulting force balance
leads to suppMat (30):

$Ca\sim\frac{1-\cos\theta_{eq}}{\sin\theta_{eq}}[Bo_{\alpha}-Bo_{c}].$ (1)

where $Bo_{c}$ is a threshold Bond number capturing hysteresis effects.
Scaling 1 captures well the trend of the data obtained for $\mathcal{R}\simeq
2$ (Fig. 2c). We can estimate the magnitude of the contact angle hysteresis of
silicone gels from $Bo_{c}$ Le Grand et al. (2005); Dussan V. (1985) and we
find $\Delta\theta=\theta_{a}-\theta_{r}\simeq 3.5$
$\mathrm{\SIUnitSymbolDegree}$, a value compatible with the data displayed in
Fig. 3, the rounded shapes of droplets at $\mathcal{R}\simeq 2$ (Fig. 1) and
other reports in the literature regarding silicone gels MacCallum et al.
(2015); Zhao et al. (2018).

In the limit $\mathcal{R}\rightarrow\infty$, energy dissipates in the
substrate. A reasoning similar to the one used in the previous paragraph leads
to a scaling that balances viscous dissipation in the solid and droplet weight
suppMat (30):

${Bo_{\alpha}}\sim\left(\frac{\gamma}{\gamma_{s}}\right)^{2}(\sin{\theta_{eq}})^{3}{Ca_{s}}^{m},$
(2)

where $m$ is the exponent of the power law describing the loss modulus of the
substrate as a function of strain frequency suppMat (30). For systems where
$\mathcal{R}\geq 10^{2}$, Eq. 33 captures our data up to $Bo_{\alpha}\simeq
8\times 10^{-2}$ (Fig. 2c).

Given the similarities between the $\mathcal{R}\simeq 2$ data and the rigid
case, we compare the contact angle dependence on the capillary number to the
Cox-Voinov law Voinov (1976); Cox (1986); Le Grand et al. (2005):

$\theta_{d}^{3}-\theta_{eq}^{3}=9Ca\ln\left(\frac{h}{\lambda}\right),$ (3)

where $h$ is the height on the liquid/vapor interface at which the angle is
measured and $\lambda$ is a nanoscopic length scale introduced to circumvent
stress divergence at the contact line. The agreement is qualitatively
excellent (dashed line in Fig. 3), provided the logarithmic term has an
amplitude around 15. As we measure the contact angle at $h\sim 100$
$\mathrm{\SIUnitSymbolMicro m}$, we obtain an unreasonable cutoff length scale
$\lambda\simeq 30$ $\mathrm{pm}$, smaller than an interatomic bond. The large
value of the logarithmic term likely results from the fact that
$\mathcal{R}\sim 1$: dissipation in the solid is of the same order of
magnitude as in the liquid. Failure of the Cox-Voinov law is then expected, as
it does not account for all dissipation sources.

We compare the data in Fig. 3 to a model that we proposed recently Dervaux et
al. (2020) that describes the wetting of soft solids in a regime of finite
strains under the assumption that $\gamma_{s}$ is independent of strain: there
is no Shuttleworth effect Shuttleworth (1950). The model provides a prediction
for the dependence of $\theta_{d}$ on $Ca$ and $\mathcal{R}$ :

$g(\theta_{d})=g\left(\frac{\pi}{2}+\arctan{\sqrt{\frac{\sqrt{1+\mathcal{A}^{2}(\mathcal{R},{Ca},\Lambda)}-1}{2}}}\right)\\\
+{Ca}\ln\left(\frac{h}{\lambda}\right),$ (4)

with $g(x)=\int_{0}^{x}\frac{z-\sin{(z)}\cos{(z)}}{2\cos{(z)}}dz$,
$\mathcal{A}$ a function that accounts for the capillary and viscoelastic
forces at the contact line, and $\Lambda$ the ratio between the thickness of
the substrate and the elastocapillary length $\ell_{s}$; here
$\Lambda\rightarrow\infty$. This equation is formally similar to the general
form of the Cox-Voinov relation, where the first r.h.s. term is related to the
microscopic contact angle and the second results from dissipation in the
liquid Bonn et al. (2009). Here, the microscopic angle term is a dynamic
quantity set by the response of the solid to the propagation of the ridge. The
correction to this term increases as $\mathcal{R}$ increases.

Figure 3b-f show that the agreement between the experimental data of Fig. 3a
and Eq. 4, setting $\lambda=$ 100 nm, is good to excellent for all datasets.
In the case $\mathcal{R}=119$, the model fails to capture the receding branch
$Ca<0$. The trailing edge of these droplets oscillate, and our model does not
predict this response. The steepness of the $\mathcal{R}>>1$ curves close to
$Ca=0$ is characteristic of soft hysteresis Dervaux et al. (2020). This
hysteresis is apparent as contact angles remain defined at all velocities.
However, its signature on droplet shape is akin to that of real hysteresis
(Fig. 1). These results and their interpretation clarify the nature of the
hysteresis reported for soft materials in the past literature Extrand and
Kumagai (1997). They highlight the necessity to reach the smallest possible
velocities to characterize wetting dynamics on soft solids accurately. Sliding
droplets are a good system in this respect, as the sign of velocity near the
contact line switches from positive to negative continuously along their
contour Rio et al. (2005); Snoeijer et al. (2005).

The fitting procedure also suggests that all systems, even at small values of
$\mathcal{R}$, exhibit a steep asymptote at ${Ca}=0$. This is because
viscoelastic dissipation in the solid, $\propto U^{m}$ with $m<1$, always
exceeds viscous dissipation in the liquid, $\propto U$, at vanishing $Ca$. We
note that the value of $\Delta\theta$ obtained from Eq. 1 is compatible with
the jump magnitude seen in the fit of the model to the data for
$\mathcal{R}\simeq 2$, indicating again that signaturess of soft hysteresis
are akin to those of real, defect-induced, hysteresis. The model can also
predict the full range of the data displayed in Fig. 2 suppMat (30).

Finally, fits to the datasets with Eq. 4 are obtained while keeping $\lambda$
constant. We can also set experimental parameters to their nominal value an
leave $\lambda$ free to adjust. We find best-fit values $50\leq\lambda\leq
1000$ nm, larger than the molecular sizes of the liquids we use, of the order
of 10 nm at most, and smaller than the elastocapillary length $\ell_{s}\simeq
20$ $\mathrm{\SIUnitSymbolMicro m}$. The large magnitude of $\lambda$ is
likely the result of the presence of free chains that are known to lubricate
the contact between the droplet and the gel and induce significant slip
Lavielle et al. (2021); Cai and Pham (2022); Hauer et al. (2023).

Figure 4: (a) Receding contact angle $\theta_{c}$ and (b) capillary number
$Ca_{c}$ at the pearling threshold as a function of the relaxation ratio
$\mathcal{R}$. $\theta_{c}$: receding contact angle of the last stable point;
${Ca_{c}}$: mean of the capillary numbers of the last stable and the first
unstable points. Error on the latter is estimated with its standard deviation.
Dashed lines: prediction of the model Dervaux et al. (2020) for the vanishing-
contact-angle capillary number.

The characterization of the properties of the system at the threshold of the
pearling instability shows how different the transition to liquid deposition
on soft substrates is from that on rigid ones. First, the contact angle just
before pearling, $\theta_{c}$, is around $32$$\mathrm{\SIUnitSymbolDegree}$
when the relaxation ratio $\mathcal{R}\lesssim 10^{2}$ and rises up to
$50$$\mathrm{\SIUnitSymbolDegree}$ when $\mathcal{R}\simeq 10^{4}$ (Fig. 4a).
These values are much larger than those, around
$10\text{\,}\mathrm{\SIUnitSymbolDegree}$, reported for rigid substrates Le
Grand et al. (2005). Second, the threshold capillary number $|Ca_{c}|$ depends
on both the equilibrium contact angle $\theta_{eq}$ and $\mathcal{R}$ (Fig.
4b). While the former is expected Snoeijer et al. (2007) and may at least
partly explain the jump of around an order of magnitude in $|Ca_{c}|$ that we
see for $\mathcal{R}\sim 100$, the latter remains to be investigated. For
$\mathcal{R}\simeq 2$, $|Ca_{c}|\simeq 1.1\times 10^{-2}$ and it is around
twice as large as in the rigid case for a comparable equilibrium contact angle
$\theta_{eq}$ Le Grand et al. (2005). Keeping the latter constant, a one-
hundredfold increase of $\mathcal{R}$ decreases $|Ca_{c}|$ tenfold. This
result stems from the fact that receding branches in the $\theta_{d}(Ca)$
curves tend to be steep, and their steepness increases with an increase of
$\mathcal{R}$ Dervaux et al. (2020). As a consequence, for a constant
equilibrium contact angle, the capillary number at which a zero receding
contact angle is attained decreases as $\mathcal{R}$ increases. Thus, we
expect the rivulet transition to occur at smaller values of $Ca$ for larger
values of $\mathcal{R}$. This rationale explains why the prediction of our
model for the zero-contact-angle receding capillary number captures quite well
the data (Fig. 4b). It also suggests that dissipation in the substrate
increases the sensitivity of droplets to the pearling instability.

In conclusion, we document how droplets slide on soft viscoelastic gels as a
function of energy dissipation partition between the liquid and the substrate.
While the substrate is always deformable, sliding droplets display shapes and
dynamics akin to those observed on a rigid solid when dissipation occurs
equally in both materials. When the substrate is the main dissipative element,
straight lines parallel to the direction of motion appear in the droplet
contour as the signature of an apparent hysteresis that appears when the
dependence of dynamic contact angles on velocity is measured. A non-linear
model that we proposed recently is able to describe the data very well. Thus,
accounting for geometrical non-linearities is enough to describe droplet
dynamics on soft substrates, without the need to assume a dependence of the
surface energy of the solid on strain, in line with recent experimental
results Bain et al. (2021). The model accounts at least qualitatively for the
lubricating effect of free chains present in our system.

This study raises questions regarding the physics of curved contact lines. On
rigid substrates, the trailing edge contact angle decreases almost to zero at
the pearling transition threshold. The path to fragmentation is different on a
soft gel. Our observations suggest that the curvature of the trailing edge is
constrained by the substrate, an issue that calls for future work. In
particular, contributions of the dry cross-linked network and the free chains
should be controlled separately to investigate how elasticity via the
elastocapillary length and viscous dissipation via the slip length set the
shape of curved receding contact lines. The tenfold increase of the capillary
number when passing the pearling threshold at $\mathcal{R}\simeq 10^{4}$ also
deserves investigation, as we were not able to find a smooth transition.

###### Acknowledgements.

We thank L. Bergougnoux and E. Guazzelli for providing UCON U90. We are
grateful to H. Montès for performing the time-temperature rheology of Sylgard
527. ANR (Agence Nationale de la Recherche) and CGI (Commissariat à
l’Investissement d’Avenir) are gratefully acknowledged for their financial
support through the GelWet grant (ANR-17CE30-0016).

##  I Supplemental materials

###  I.1 Materials and Methods

#### Liquids

We use pure glycerol (G100, Sigma Aldrich, G5516), a 60-%-glycerol-in-water
mixture (G60), polyethylene glycol-ran-propylene glycol (P25, PEG-ran-PPG,
average molecular weight $M_{w}\sim 2500$
$\mathrm{g}\text{\,}{\mathrm{mol}}^{-1}$, Sigma Aldrich), a 70%-polyethylene
glycol-ran-propylene glycol monobutyl ether-in-water mixture (P7, PEG-ran-PPG
ME, Sigma Aldrich), and the UCON lubricant 75-H-90,000 (U90, Dow corning).
Table 1 summarizes their properties. All the liquids are insoluble in silicone
gels. We measure the liquid-vapor surface tension $\gamma$ with the pendant
drop technique. We estimate density $\rho$ by weighing a volume $V=10\pm 0.5$
mL of liquid with a 0.01-$\mathrm{g}$-accurate scale. We measure the dynamic
viscosity $\eta$ with a capillary viscosimeter sitting next to the set-up
twice a day to account for hygroscopic and thermal effects.

Table 1: Properties of liquids used in our experiments. | Surface | Viscosity | Density | Equilibrium | Relaxation
---|---|---|---|---|---
| tension |  |  | contact angle | ratio
| $\gamma$ | $\eta$ | $\rho$ | $\theta_{eq}$ | $\mathcal{R}$
|  mN m-1 | mPa s |  $10^{3}$ kg m-3 | $\mathrm{\SIUnitSymbolDegree}$ |
U90 | 40.7 | 36900 | 1.08 | 61.6 | 2.24
|  $\pm$ 0.9 |  $\pm$ 690 |  $\pm$ 0.05 |  $\pm$ 0.7 |  $\pm$ 0.42
P7 | 37.3 | 842 | 1.05 | 56.6 | 92.7
|  $\pm$ 0.2 |  $\pm$ 99 |  $\pm$ 0.05 |  $\pm$ 0.7 |  $\pm$ 11.0
G100 | 63.1 | 631 | 1.27 | 97.2 | 119
|  $\pm$ 0.5 |  $\pm$ 104 |  $\pm$ 0.06 |  $\pm$ 1.1 |  $\pm$ 20
P25 | 37.1 | 491 | 1.05 | 64.3 | 165
|  $\pm$ 0.4 |  $\pm$ 34 |  $\pm$ 0.01 |  $\pm$ 0.2 |  $\pm$ 11
G60 | 62.9 | 6.8 | 1.16 | 98.1 | 10700
|  $\pm$ 0.1 |  $\pm$ 0.2 |  $\pm$ 0.06 |  $\pm$ 0.9 |  $\pm$ 400

#### Equilibrium contact angle measurements

Wetting equilibrium is difficult to identify on these systems using classical
techniques such as droplet deposition. We see the contact line moving even a
few hours after deposition. We circumvent this issue by deducing equilibrium
contact angles $\theta_{eq}$ from experimental dynamic contact angles (Fig.
3a): we fit the points around $U=0$ with a linear law, and take $\theta_{eq}$
equal to the value of the fitted contact angle at $U=0$.

#### Gel preparation

Gel slabs are prepared with a two-part commercial silicone kit (Dow Corning
Sylgard 527). We mix equal volumes of each part of the kit together, as
recommended by the manufacturer, in a weighing boat previously cleaned with
ethanol and water, and dried. The gel mixture is degassed under vacuum for 2
$\mathrm{h}$ to remove bubbles. It is then poured in a $60\times 40$-mm2
plastic vessel (Caubère), also cleaned with ethanol and distilled water and
let to dry in a vacuum before use. Then, we leave the sample in an oven at
$65\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for 15 to 18 $\mathrm{h}$. We
perform experiments exclusively on dust-free unmarked gels.

#### Free-chain extraction

We extract free chains from silicone gels using the process described in ref.
Hourlier-Fargette et al. (2017). After weighing pieces of gels, we dip them
into toluene (VWR, AnalaR NORMAPUR), a good solvent for PDMS. Free chains
migrate to the solvent. We renew toluene everyday for five days to accelerate
the extraction process. Then the sample is immersed in a mixture of toluene
and ethanol (VWR, AnalaR NORMAPUR) to the bath to remove toluene from the gel.
Ethanol is added progressively to avoid damaging the sample. We start with a
solution of 20wt% ethanol in toluene, and we increase the ethanol proportion
by steps of 20wt% every day, until the sample sits in 100% ethanol. After
three baths in pure ethanol, the gel stops shrinking. We dry the gel under
vacuum to remove remaining solvent and weigh it again. From this procedure, we
find that our materials contain $62$ wt% free chains. This large amount of
free chains explains likely the absence of transition between two sliding
regimes Hourlier-Fargette et al. (2017); hourlier-fargette2018 in our
experiments.

#### Rheology

Figure 5: Rheology of Sylgard 527. Storage modulus $G^{\prime}$ and loss
modulus $G"$ as function of pulsation $\omega$. Strain amplitude: 1%.
Continuous black lines: Chasset-Thirion law, Eqs. 6 and 7.

We perform small amplitude oscillatory shear rheology on our gels with an
Anton Paar MCR 501 rheometer mounted with a plate-plate geometry (diameter
$d=25$ mm). Strain amplitude is set at 1%. We extend the frequency range using
a time-temperature superposition procedure Rubinstein and Colby (2003). We fit
the data with the Chasset-Thirion model Curro and Pincus (1983):

$G(\omega)=\mu_{0}(1+(i\omega\tau)^{m}),$ (5)

with $G$ the complex modulus, $\omega$ the strain angular frequency and $\tau$
a characteristic relaxation time. The exponent $m$ depends on the degree of
polymerization of the polymer network and on the coordination number of
monomers Curro and Pincus (1983); Curro et al. (1985).

If we decompose the complex modulus $G(\omega)$ in a real elastic part
$G^{\prime}(\omega)$ and an imaginary dissipative part $G"(\omega)$, we obtain

$\displaystyle G^{\prime}(\omega)$
$\displaystyle=\mu_{0}(1+\cos\left(m\pi/2\right)(\omega\tau)^{m}),$ (6)
$\displaystyle G"(\omega)$
$\displaystyle=\mu_{0}\sin\left(m\pi/2\right)(\omega\tau)^{m}.$ (7)

We determine $\tau$ and $m$ by fitting the loss factor deduced from the model
to its experimental value:

$\frac{G"}{G^{\prime}}(\omega)=\tan{(\delta)}=\frac{\sin\left(m\pi/2\right)(\omega\tau)^{m}\tan\left(m\pi/2\right)}{\sin{\left(m\pi/2\right)}(\omega\tau)^{m}+\tan{\left(m\pi/2\right)}}.$
(8)

We inject the values of $\tau$ and $m$ to fit equations 6 and 7 to the
rheological data (Fig. 5). We find $\mu_{0}=1.077$ kPa, $\tau=18.2$ ms, and
$m=0.626$.

#### Sliding experiments

We deposit a liquid droplet with a micropipette on the gel. The spherical
radius $R_{0}$ of the droplets before deposition is of the order of the
capillary length $\ell_{c}=(\gamma/(\rho g))^{1/2}\sim\mathcal{O}(1.5)$ mm of
the liquids in all cases.The experiment starts when we tilt the gel at an
angle $\alpha$ with the horizontal. A LED panel (Effilux) shines light on the
sample from below, and a camera (Imaging Source, DMK 33UX174) records top
views of the droplet with a spatial resolution of 32
$\mathrm{\SIUnitSymbolMicro m}\text{\,}{\mathrm{px}}^{-1}$. We take side views
on some experiments (Imaging Source, DMK 33UX174, spatial resolution 4
$\mathrm{\SIUnitSymbolMicro m}\text{\,}{\mathrm{px}}^{-1}$). The thickness of
all the samples, $h_{s}\sim$ 4 mm, is much larger that the elastocapillary
length of our material, $\ell_{s}\sim 10$ $\mathrm{\SIUnitSymbolMicro m}$, to
avoid small-thickness effects Zhao et al. (2018). Samples are covered with a
polystyrene lid that we find able to prevent surface ageing and dust
deposition. We unmold gel layers and cut their edges so the meniscus is not in
the way of side views. We obtain identical results when the silicone gel is in
the box or unmolded.

We check the volume of droplets by weighing samples before droplet deposition
and after. We tracked the motion of droplets with the software package FiJi
Schindelin et al. (2012). Droplet velocities range from $10^{-3}$ to 1 mm s-1.
In most cases, the trajectories that we observe are linear functions of time:
droplets move at constant speed. For the longest experiments, drops may lose
or gain water from surrounding air. In that case, we fit only the part of the
trajectory that is unaffected. Thus we extract a single value $U$ of the
droplet velocity from each experiment. Each set of experimental conditions is
tested three times to ensure reproducibility.

###  I.2 Derivation of scaling laws describing the dynamics of sliding
droplets.

####  I.2.1 Vanishing values of the relaxation ratio $\mathcal{R}$

Figure 6: Different configurations for a liquid drop.  (a) A spherical droplet
of volume $V=4R_{0}^{3}/3\pi$. (b) The same droplet as a spherical cap after
spreading on a substrate, with radius of curvature $R_{\rm cal}$, height at
the center $h$, contact radius $R_{c}$ and contact angle $\theta_{eq}$.

In experiments, a droplet of volume $V$ is deposited on the substrate. This
volume corresponds to a sphere of radius $R_{0}$ (Fig. 6a). Once it has spread
on the surface of the substrate, the droplet reaches its equilibrium shape, a
spherical cap with radius $R_{\rm cal}$, height $h$, contact radius $R_{c}$
and contact angle $\theta_{eq}$ (Fig. 6b).

$R_{c}$ is difficult to measure in experiments, especially when the
equilibrium contact angle is larger than $\pi/2$. In contrast, we can obtain
$R_{0}$ by weighing the drop. If we know the equilibrium contact angle
$\theta_{eq}$, we can write, assuming that droplets form spherical caps:

$\left\\{\begin{aligned}
R_{c}&=R_{\mathrm{cal}}\,\cos\left(\frac{\pi}{2}-\theta_{eq}\right)\\\
R_{\mathrm{cal}}-h&=R_{\mathrm{cal}}\,\sin\left(\frac{\pi}{2}-\theta_{eq}\right).\end{aligned}\right.$
(9)

Rearranging, we can link the contact radius and the height of the droplet to
the spherical cap radius:

$\left\\{\begin{aligned} R_{c}&=R_{\rm cal}\sin\theta_{eq}\\\ h&=R_{\rm
cal}(1-\cos\theta_{eq}).\end{aligned}\right.$ (10)

Now, we can express the volume for both a sphere and a spherical cap:

$\left\\{\begin{aligned} V&=\frac{4\pi}{3}R_{0}^{3}\\\
V&=\frac{\pi}{3}h^{2}(3R_{\rm cal}-h)\end{aligned}\right.$ (11)

in terms of $R_{c}$ and $\theta_{eq}$:

$\left\\{\begin{aligned} V&=\frac{4\pi}{3}R_{0}^{3}\\\ V&=\frac{\pi
R_{c}^{3}}{3}\frac{(2+\cos\theta_{eq})(1-\cos\theta_{eq})^{2}}{\sin^{3}\theta_{eq}}\end{aligned}\right.$
(12)

Volume conservation then leads to:

$\frac{R_{c}}{R_{0}}=\frac{1}{f(\theta_{eq})}=\sin\theta_{eq}\left(\frac{(2+\cos\theta_{eq})(1-\cos\theta_{eq})^{2}}{4}\right)^{-\frac{1}{3}}.$
(13)

For a droplet sliding down an inclined plane, the ratio between its weight and
the capillary force along its perimeter, called the Bond number, writes:

$Bo_{\alpha}=\frac{\rho gR_{0}^{3}}{\gamma R_{c}}\sin\alpha.$ (14)

Injecting Eq. 13 into Eq. 14, we obtain an expression for the Bond number that
accounts for changes in the equilibrium contact angle:

$\boxed{Bo_{\alpha}=f(\theta_{eq})\frac{\rho gR_{0}^{2}}{\gamma}\sin\alpha.}$
(15)

Now, let’s assume that the gravitational force experienced by the droplet is
balanced dissipation in the liquid and contact angle hysteresis:

$\rho
gR_{0}^{3}\sin\alpha\sim\eta\frac{U}{h}R_{c}^{2}+\gamma(\cos{\theta_{a}}-\cos{\theta_{r}})R_{c}.$
(16)

Here $U/h$ estimates the velocity gradient in the droplet, and $\theta_{a}$
and $\theta_{r}$ are the advancing and receding dynamic contact angles, i.e.
the threshold values of the contact angle above and below which contact line
motion occurs.

Figure 7: Comparison between our data for $\mathcal{R}\simeq 2$ and the data
of Podgorski et al.  Podgorski et al. (2001) once corrections related to a
different equilibrium contact angle are accounted for.

In what follows, we assume that the values of dynamic contact angles are close
to that of the equilibrium contact angle, a hypothesis that is valid at low
droplet velocities, $Ca<<1$. Thus, we can use the description of the
equilibrium shape of the droplet to relate the contact angle and the contact
radius. Using Eq. 10, we obtain:

$\frac{R_{c}^{2}}{h}=R_{c}\frac{\sin{\theta_{eq}}}{1-\cos{\theta_{eq}}}$ (17)

Dividing both sides by the liquid-vapor surface tension $\gamma$, replacing
$R_{c}^{2}/h$, and using Eq. 13, it follows that:

$Ca\sim\frac{1-\cos{\theta_{eq}}}{\sin{\theta_{eq}}}[Bo_{\alpha}-Bo_{c}],$
(18)

where $Bo_{c}$ is a threshold Bond number below which contact hysteresis pins
the droplet to the surface. This scaling was first suggested in ref. Podgorski
et al. (2001). We observe that this correction leads to the overlap of our
data for $\mathcal{R}\simeq 2$ and those of ref. Podgorski et al. (2001) (Fig.
7). Following Dussan Dussan V. (1985) and Le Grand et al. Le Grand et al.
(2005), we can evaluate contact angle hysteresis from the experimental value
of $Bo_{c}$ using:

$Bo_{c}=\left(\frac{24}{\pi}\right)^{1/3}\frac{(\cos{\theta_{r}}-\cos{\theta_{a}})(1+\cos{\theta_{a}})^{1/2}}{(2+\cos{\theta_{a}})^{1/3}(1-\cos{\theta_{a}})^{1/6}}$
(19)

Le Grand et al. performed this estimation accounting for the prefactor
appearing in front of $Bo_{\alpha}$ when fitting their data with Eq. 18. We
use the same procedure to obtain our estimate.

####  I.2.2 Large-$\mathcal{R}$ sliding

We expect that the relation between injected energy and dissipation be
modified when the effective viscosity of the solid exceeds that of the liquid.
The collapse of the $Ca-Bo_{\alpha}$ curves in the $Ca_{s}-Bo_{\alpha}$ space,
with $Ca_{s}=\mathcal{R}Ca$, suggests that the latter is a good metric.
Besides, we know from the previous section that the equilibrium contact angle
matters. Hence we should derive a scaling law for the solid-dominated case
that accounts for all of these modifications. Note that we use again the
approximation that the dynamic contact angles remain close to the equilibrium
contact angle.

Inspired by Ref. Zhao et al. (2018), we estimate the power dissipated per unit
of volume of the solid when the drop moves by a length $\ell$:

$d\mathcal{P}_{\rm diss}\sim\sigma\epsilon^{2}\omega$ (20)

with $\sigma$ the viscous stress, $\epsilon$ the strain and $\omega$ the
pulsation. Using the Chasset-Thirion model:

$G(\omega)=\mu_{0}(1+(i\omega\tau)^{m}),$ (21)

and taking the typical strain scale to be:

$\epsilon\sim\frac{\gamma}{\gamma_{s}}\sin{\theta_{eq}}$ (22)

and the characteristic pulsation of the experiment as:

$\omega=\frac{U}{\ell_{s}},$ (23)

we have the following estimate for viscous stresses in the solid:

$\sigma\sim\mu_{0}\left(\frac{U\tau}{\ell_{s}}\right)^{m}.$ (24)

Then we have:

$d\mathcal{P}_{\rm
diss}\sim\mu_{0}\left(\frac{U\tau}{\ell_{s}}\right)^{m}\left(\frac{\gamma}{\gamma_{s}}\sin{\theta_{eq}}\right)^{2}\frac{U}{\ell_{s}}\\\
$ (25)

Dissipation takes place in a half-torus having a radius $R_{c}$, width $\ell$
and height $\ell_{r}$. As elasticity balances the vertical component of the
resulting capillary force per unit length at the contact line
$\gamma\sin{(\theta_{eq})}$, the height of the ridge scales as:

$\ell_{r}\sim\gamma\sin{\theta_{eq}}/\mu_{0}.$ (26)

Then, we can estimate the power dissipated in the solid, neglecting numerical
prefactors:

$\displaystyle\mathcal{P}_{\rm diss}\sim\sigma\epsilon^{2}\omega
R_{c}\ell_{r}\ell$ (27) $\displaystyle\mathcal{P}_{\rm
diss}\sim\mu_{0}UR_{c}\ell\left(\frac{\gamma}{\gamma_{s}}\sin{\theta_{eq}}\right)^{3}\left(\frac{U\tau}{\ell_{s}}\right)^{m}$
(28)

Now, we can write the force balance that a droplet sliding on a viscoelastic
substrate should obey. Dividing $\mathcal{P}_{\rm diss}$ by the sliding
velocity $U$, we have:

$\rho
gR_{0}^{3}\sin{\alpha}\sim\mu_{0}R_{c}\ell\left(\frac{\gamma}{\gamma_{s}}\sin{\theta_{eq}}\right)^{3}\left(\frac{U\tau}{\ell_{s}}\right)^{m}.$
(29)

Dividing by the liquid-vapor surface tension $\gamma$ on both sides and
rearranging, we obtain:

$Bo_{\alpha}\sim\frac{\mu_{0}\ell}{\gamma}\left(\frac{\gamma}{\gamma_{s}}\sin{\theta_{eq}}\right)^{3}\left(\frac{U\tau}{\ell_{s}}\right)^{m}$
(30)

and using:

$\frac{U\tau}{\ell_{s}}=\mathcal{R}Ca=Ca_{s},$ (31)

we end up with the following prediction:

${Bo_{\alpha}\sim\frac{\mu_{0}\ell}{\gamma}\left(\frac{\gamma}{\gamma_{s}}\sin{\theta_{eq}}\right)^{3}{Ca_{s}}^{m}}.$
(32)

The prefactor $[(\gamma/\gamma_{s})\sin{(\theta_{eq})}]^{3}$ in Eq. 32 should
capture the dependence on equilibrium contact angles. In the limit of thick
samples, $\ell=\ell_{s}=\frac{\gamma_{s}}{2\mu_{0}}$ Zhao et al. (2018), and
we obtain Eq. 7 in the main text when neglecting numerical prefactors:

${Bo_{\alpha}}\sim\left(\frac{\gamma}{\gamma_{s}}\right)^{2}(\sin{\theta_{eq}})^{3}{Ca_{s}}^{m}.$
(33)

### Comparison of experimental $Ca-Bo_{\alpha}$ curves with the non-linear
model

Figure 8 shows a comparison between experimental data for the $Ca-Bo_{\alpha}$
displayed in Fig. 2 in the main text and theoretical predictions from our non-
linear model Dervaux et al. (2020). Theoretical fits cover a surface in the
$Ca-Bo_{\alpha}$ space because we account for uncertainties in experimental
parameters such as the viscosity and the surface tension of the liquids, etc.
Fits are seen to be in good agreement with experimental data.

Figure 8: Comparison between the data of Fig. 2 in the main text with the non-
linear model of Dervaux et al. Dervaux et al. (2020). Closed symbols represent
data points obtained beyond the pearling threshold. The fitting domains
represent the variation of the fitting results due to uncertainty on the
experimental parameters. Dashed lines: pearling threshold capillary number.

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