# Sliding of liquid droplets on thin viscoelastic soft layers

Menghua Zhao, Julien Dervaux, Laurent Limat, Matthieu Roché, Tetsuharu Narita, François Lequeux

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Menghua Zhao, Julien Dervaux, Laurent Limat, Matthieu Roché, Tetsuharu Narita, et al. Sliding of liquid droplets on thin viscoelastic soft layers. Physical Review Fluids, 2025, 10.1103/31j2-x7wy. hal-05019092v2

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# Sliding of liquid droplets on thin viscoelastic soft layers

Menghua Zhao

School of Information Mechanics and Sensing Engineering, Xidian University, Xi'an 710071, China
Université Paris Cité, CNRS, Matière et systèmes complexes, F-75013 Paris, France and
Sciences et Ingénierie de la Matière Molle, CNRS UMR 7615,
École supérieure de physique et de chimie industrielles de la Ville de Paris,
Sorbonne Université, Paris Sciences et Lettres Université, Paris 75005, France

Julien Dervaux, Laurent Limat, and Matthieu Roché*

Université Paris Cité, CNRS, Matière et systèmes complexes, F-75013 Paris, France

Tetsuharu Narita and François Lequeux

Sciences et Ingénierie de la Matière Molle, CNRS UMR 7615,
École supérieure de physique et de chimie industrielles de la Ville de Paris,
Sorbonne Université, Paris Sciences et Lettres Université, Paris 75005, France

Soft substrates are deformed by liquid-vapor surface tension upon contact with liquid droplets, forming the well-known wetting ridge. This ridge dynamically propagates with the moving contact line and critically influences liquid spreading. Here, we experimentally investigate gravity-driven sliding dynamics of water droplets on vertically tilted silicone layers whose viscoelasticity is characterized by the Chasset-Thirion model with the exponent *m*. At low Bond numbers, the sliding velocity scales with droplet size as $V_S \sim D_m^{2/m}$. While in the thin-film limit, velocity exhibits a pronounced power-law dependence on nominal substrate thickness, $V_S \sim \Pi(h)^{-1/m}$. We rationalize these observations by quantifying viscoelastic dissipation within the soft layer and balancing it against the gravitational driving force using an energy-conservation framework. Our findings offer novel avenues for designing advanced soft coatings, anti-fouling and self-cleaning surfaces, and biomedical devices.

## I. INTRODUCTION

Soft gels find wide applications in drug delivery [1], medical adhesives [2], soft robotics [3] due to their high programmability. When in contact with a droplet, they can be deformed by the liquid-vapor surface tension in the vicinity of the contact line, forming a wetting ridge [4–6] of a characterized height $\gamma_{LV}/E$, where $\gamma_{LV}$ and $E$ denote the liquid-vapor surface tension and Young’s modulus, respectively. While such deformation is negligible for rigid materials (less than one atom for materials of $E \sim \text{GPa}$), it becomes significant for soft substrates ($\mathcal{O}(10)\mu\text{m}$ for $E \sim \mathcal{O}(10) \text{ kPa}$). During contact line motion, the propagation of this deformation leads to energy dissipation in the substrate that can be greater than viscous dissipation in the liquid phase and profoundly influences spreading dynamics through the viscoelastic braking effect by slowing down the propagation velocity [4, 5, 7–9], challenging classical hydrodynamic models that only describe the flow in the liquid phase [6, 10–12]. The interplay between viscoelastic dissipation in the gel phase and viscous dissipation in the liquid phase introduces a complex spectrum of behaviors, while simultaneously enabling broader applications, such as directional transport [9, 13, 14], biomedical engineering [15, 16], and adaptive surface design [17].

Extensive studies focused on the static surface deformation close to the contact line using interferometry [4, 5], confocal microscopy [18, 19], X ray microscopy [20] and microscopic Schlieren imaging [9], revealing the logarithmic hape of the wetting ridge. Building upon these advances, more attention was subsequently devoted to dynamic scenarios where energy dissipation in the soft film phase was accounted for [5, 8, 9] and pertinent factors including the geometrical confinement [9], physico-chemistry [21, 22], non-linear effects [23, 24], the Shuttleworth effect [25] were thoroughly addressed. However, the majority of preceding research focused on a more idealized situation than the actual sliding process, overlooking the fact that a droplet can deform and substrates may undergo residual stress [26–29]. For example, in the case of sliding on thick soft substrates driven by gravity, droplet morphology has been shown to exhibit a pronounced dependence on the Bond number and relaxation ratio [27], i.e., the ratio between the viscocapillary relaxation in the liquid and that in the solid: droplets are more elongated at a higher Bond number and tend to be obtuser for a larger relaxation ratio. Such a morphology dependence is similarly recovered in alternative research context [29], where a droplet is forced to slide horizontally on the soft silicone substrate and the relaxation of elastocapillary ridges is additionally reported. In addition, recent studies have demonstrated that droplets exhibit anisotropic descent from a soft stretched substrate, moving faster in the stretched direction than in the perpendicular direction [30]. While in a compres-

<footnote>* Contact author: matthieu.roche@univ-paris-diderot.fr</footnote>
sion case, droplets exhibit a highly non-monotonic depen-
dence on the applied strain [28]. Most of those studies
were conducted on soft films of large thickness compared
to the elastocapillary length, despite the established sig-
nificance of small thickness in determining contact line
dynamics [9]. Furthermore, the driving force is typically
difficult to control in most contexts, which complicates a
precise estimation of the dissipation power.

The present research focuses on the gravity-driven slid-
ing of water droplets across vertically tilted soft sub-
strates where the driving power can be precisely con-
trolled. Sliding dynamics will be addressed by tailoring
the droplet size and the thickness of the film over three
orders of magnitude. The findings of this study demon-
strate that increasing the thickness of the soft layer re-
sults in enhanced energy dissipation, consequently lead-
ing to a reduction in sliding velocity. We propose a model
by balancing the viscoelastic dissipation and the gravi-
tational driving power to account for the experimental
observations. The study reveals a scaling law that relates
the droplet size and the thickness of the viscoelastic layer
to the sliding velocity in the limit of small Bond numbers.

II. EXPERIMENTAL SETUP

Droplet sliding experiments were performed in a
humidity-controlled chamber with soft films being
bonded onto vertically tilted glass slides, establishing a
90° sliding angle (Fig. 1). The tilting angle was mon-
itored using an inclinometer with a precision of ±0.1°.
The assembly was housed in a vertically oriented, en-
closed Petri dish containing additional water drops to
maintain the saturation vapor pressure, thereby sup-
pressing evaporation during sliding. To validate evap-
oration suppression, the smallest droplets (~1 µL) were
monitored and shown to exhibit a diameter reduction
from 1.25 mm to 1.23 mm over ~1 hour sliding, corre-
sponding to a negligible relative change of 1.6%. These
results confirm that our experiments were conducted in
a vapor-saturated environment, allowing evaporation ef-
fects to be safely neglected. To visualize and track the
sliding dynamics, a LED array was employed to pro-
vide homogeneous illumination from the back side and a
front view camera (DFK 23UX174, IMAGING SOURCE,
Germany) was installed facing the soft film with a spa-
tial resolution of 19 µm/pixel at a frame rate of 20 Hz.
Droplet tracking was implemented by means of the cor-
relation method with the sub-pixel technique enabling
a localization uncertainty of ±2 µm [31]. Droplet depo-
sition was achieved by gently injecting liquid onto soft
films with a micro-pipette that defines the droplet vol-
ume with a precision of ±0.2 µL. The liquid employed in
this study was distilled water (Milli-Q Integral; Millipore,
USA), with surface tension γ<sub>LV</sub> 72.8 mN/m and viscos-
ity η 1.0 mPa·s. The rheological response of the silicone
layer, prepared following a protocol we published pre-
viously [9], is described by the Chasset-Thirion model,

<figure><img src="image_3.png" /><figcaption>FIG. 1. Schematic of the sliding experiments. (a) 3D dia-
gram of the water droplet sliding on the silicone gel layer along
the gravity direction. The layer thickness *h* is controlled by
spin coating, ranging from ~ *µm* to ~ *mm*. (b) 2D sketch of
the surface deformation induced by the liquid-vapor surface
tension close to the contact line.</figcaption></figure>

$G(\omega) = G_s(1 + (i\omega\tau)^m)$. We find a shear modulus $G_s = 1.2$ kPa, a relaxation time $\tau = 8.6$ ms, and an exponent $m = 0.62$. The soft gel film was prepared by the spin coating technique and its thickness was tuned by spinning speed, allowing a continuous control of the thickness from ~ $\mu$m to ~ $\mm$. That was quantitatively measured by a 3D Profiler (FOCAL 3D Pilot, FOCAL Nanotech) in white light scanning mode.

III. EXPERIMENTAL OBSERVATIONS

We first report the collective sliding behaviors on a
soft film of fixed thickness in Fig. 2 by varying droplet
diameter, *D*. A transient acceleration follows the initial
droplet deposition, subsequently giving way to a steady
state characterized by a constant sliding velocity, *V*<sub>S</sub>.
This transient regime happens during the very first tens
of seconds and is not the primary subject of interest in
this study. Fig. 2(a) presents the sliding trajectories of
droplets with varying sizes (0.9 µL to 6.0 µL) from 60 s
to 400 s after their initial deposition on soft film with the
thickness of 18.5 µm. We observe that all those droplets
slide down along the gravitational direction and maintain
a spherical shape. Red arrows denote sliding displace-
ments over a fixed 340 s observation period, indicating
that *V*<sub>S</sub> increases with droplet size. To analyze droplet
morphology, two dimensionless parameters are critical:
the Capillary number (Ca=η*V*<sub>S</sub>/γ<sub>LV</sub>) characterizing the
predominance of the viscous force and the surface tension
at the interface, and the Bond number (Bo=(*D*/*l*<sub>c</sub>)<sup>2</sup>),
with *l*<sub>c</sub>=√γ<sub>LV</sub>/ρg the capillary length. The calculated
Ca value ranges from ~1.4×10<sup>-8</sup> to ~1.4×10<sup>-5</sup>, sug-
gesting negligible viscous effects during sliding. Mean-
while, Bo values near unity highlight the importance
of droplet size in shaping morphology. As shown in
Fig. 2(b), droplet shape evolves with volume: droplets re-
<figure><img src="image_4.png" /><figcaption>FIG. 2. Droplets sliding on silicone gel layers. (a) Superimposed images of droplets recorded with a time lag of 340 s. The initial droplet position (gray) and its position after 340 s (black) illustrate the sliding motion. The red arrow indicates the sliding trajectory. Film thickness is 18.5 µm. (b) Morphology of steady sliding droplets as a function of droplet size. (c) Sliding velocity as a function of sliding displacement. Position 0 corresponds to the initial deposition position of the first droplet. (d) Sliding velocity as a function of the soft layer thickness.</figcaption></figure>

main spherical for $D < l_c$, while elongating in the gravita-
tional direction at $D > l_c$. However, a precise prediction
of the exact shape as a function of large Bo still remains
challenging and will not be the main focus of this study.
Inspired by Oléron et al.'s work on droplet morphology
during sliding [27], we further estimate the relaxation ra-
tio $R$ ($R = \gamma_{LV}\tau/\eta l_s$ with $l_s = \gamma_s/(2G_s)$ and $\gamma_s$ the
surface energy of the soft gel.). In the present study,
$R \approx 2 \cdot 10^4$, representing the ratio between viscoelastic
dissipation in the soft film and viscous dissipation within
the droplet, indicates that viscoelastic dissipation in the
gel phase overwhelmingly dominates the overall energy
dissipation [9, 23, 27]. Interestingly, the trailing edge
of running droplets exhibits different morphology than
the previous report [27]: the rear thins down collectively
rather than forming pinched-off pearlings at a high Bo
number [26, 27], possibly pointing to a different instabil-
ity mechanism which deserves further study.

Another finding is that the sliding velocity increases
with the droplet volume, consistent with prior stud-
ies [26, 27]. A simplified scaling argument supports
this trend. The dissipation power during the sliding
should scale with the total length of the contact line,
$P \sim DV_S^{m+1}$, based on the coarse assumption that the
dissipation in the gel phase is uniform along the con-
tact line [9, 27]. Concurrently, the gravitational driv-
ing power scales with the droplet volume, $P_d \sim D^3V_S$,
provided that the pinning force during the sliding is ne-
glected. Balancing the two yields a power-law relation-
ship between the sliding velocity and the droplet size,
$V_S \sim D^{2/m}$. This analysis, however, offers only a qualita-
tive heuristic framework. Quantitative predictions, espe-
cially when considering the additional effect of substrate
thickness, would require a more rigorous treatment of
dissipation mechanisms.

Focusing on velocity along the sliding trajectory shown
in Fig. 2(c), we observe no significant physicochemical variations during steady-state sliding. This result contrasts with previous studies suggesting that uncrossed free chains within the silicone elastomer may migrate and coat the sliding droplets once they are in contact [21, 22]. A plausible resolution to this discrepancy lies in the high concentration of free chains within the silicone gel used here [27], which may have contaminated the droplet surface during the initial acceleration phase. Consequently, the steady sliding regime likely corresponds to a droplet in a "contaminated" state. Critically, this surface modification does not compromise our analysis of droplet size and thickness effects, as the droplet surface state stabilizes after the transient regime. Furthermore, as discussed later, the impact of this contamination is minor and can be attributed to a slight adjustment of surface tension rather than a fundamental alteration of interfacial dynamics.

Furthermore, the present study demonstrates a pronounced effect of the gel layer thickness on the sliding velocity across all droplet sizes, for both $D > l_c$ and $D < l_c$, as illustrated in Fig. 2(d). This observation is in qualitative accordance with our previous report [9]: thinner films lead to faster sliding velocities. Overall, increasing the film thickness slows down droplet motion, and this effect manifests in two distinct regimes. In the thin-film limit ($h < l_s$), thickness exerts a strong influence on dissipation, consistent with the scaling laws reported in our earlier work [9]. In contrast, in the thick-film regime ($h < l_s$), increasing the thickness still reduces the sliding velocity, but the effect becomes more gradual. These observations confirm a monotonic control of sliding dynamics by substrate thickness, highlighting its governing role and suggesting the presence of a consistent underlying dissipation mechanism—an aspect we address in the following analysis.

While conventional analysis estimates driving energy from deviations in the liquid-vapor interface relative to static conditions, our system enables direct manipulation of the driving power thanks to the absence of contact line pinning on silicone gels [27, 32, 33]. This departure from conventional approaches highlights the unique control of interfacial dynamics in our experiments. However, due to variations in the contact angle and the misalignment between the sliding direction and the axis of deformation symmetry along the contact line, dissipation along the contact line is expected to be position-dependent. This raises questions about the universality of existing dissipation models where the contact line moving direction is the same as the deformation symmetric axis. Systematically adapting existing dissipation models to this asymmetric regime presents an intriguing avenue for future investigation.

## IV. MODEL AND DISCUSSION

To rationalize the aforementioned results, we move towards an analytical solution by balancing the gravitational energy with the dissipation at the contact line. Within this framework, the dissipation power per unit length at the contact line is expressed, at small spreading velocity for the silicone gel characterized by the Chasset-Thirion model [9, 27, 33], as,

$$P_{\text{film}} \sim \left(\frac{\gamma_{LV}}{\gamma_S}\right)^2 G_S V_S^{m+1} \tau^m \Pi(h) \quad (1)$$

with

$$\Pi(h) = \int_{0}^{\infty} \frac{|k|^{m+2}}{\left(k^2 + \frac{G_S}{\gamma_S K(k)}\right)^2} dk \quad (2)$$ $$K(k) = \frac{\sinh(2hk) - 2hk}{2k\left[\frac{2h^2k^2}{2hk^2 + \cosh(2hk)} + 1\right]}$$

in which $k$ is a wavenumber. Now we estimate the overall dissipation by considering multiple dissipation channels and contact line morphology. The overall viscous dissipation in the droplet, estimated using lubrication theory, is given by $P_{\text{viscous}} \approx \eta DV_S^2 \ln(\frac{D}{L_m})$ with $L_m$ a microscopic cutoff length (on the order of nanometers) [6, 34]. The total viscoelastic dissipation at the wetting ridge in the gel phase can be estimated as $P_{\text{viscoelastic}} \approx \pi D P_{\text{film}}$. For the range of sliding velocities and film thicknesses studied here, the ratio $P_{\text{viscous}}/P_{\text{viscoelastic}}$ is significantly less than one, confirming that viscoelastic dissipation in the gel phase dominates over viscous dissipation in the liquid. This conclusion is consistent with the large relaxation ratio $R \gg 1$ for the silicone gel used in this work and our preceding research [9, 27]. Consequently, we will exclusively analyze dissipation in the soft gel phase. In the case of small droplets at $Bo<1$ where surface tension dominates, the contact line adopts a round shape, in which case the total length of the contact line scales with $D$. For $Bo>1$, the sliding droplet elongates along the sliding direction, making the aforementioned claim questionable [27, 29]. A precise description of the droplet shape on the soft substrate at $Bo>1$ remains elusive, preventing the formulation of an exact relation between the droplet size and the contact line length. In light of these difficulties, we simply assume that the total contact line length scales with $D$ across all Bond numbers. In the following, we examine the second assumption. It is important to note that Eq. (1) is only valid for a dynamic contact angle of $\pi/2$, which is not applicable to a sliding droplet. The lack of a description of the three-dimensional shape of the droplet shape on soft substrates further hinders precise characterization of surface deformation and local dissipation along the contact line during the sliding. We thus approximate the dissipation power per unit contact line length as uniform, yielding a total dissipation power scaling as $P_{\text{film}}D$. This assumption is reasonable on the whole by potential mutual compensatory effects between
deformation-induced dissipation at the advancing and receding contact line. The driving force for sliding arises solely from gravity, as pinning forces are negligible on silicone gels [27, 33]. Balancing the dissipation energy and gravitational work over a sliding duration $\Delta t$, we derive,

$$DP_{\text{film}}\Delta t = mgV_S\Delta t. \quad (3)$$

with $m, g$ denoting the droplet mass and gravitational acceleration. By injecting Eq. (1) and Eq. (2) into Eq. (3), we obtain the following relationship,

$$V_S^m \sim \left(\frac{\gamma_S}{\gamma_{LV}}\right)^2 \left(\frac{\rho D^2 g}{G_S \Pi(h) \tau^m}\right). \quad (4)$$

where $\rho$ is the density of the distilled water, and $\Pi(h)$ is a nominal thickness that quantifies the thickness effect. $\Pi(h)$ is evaluated numerically. To reach a dimensionless relation, we rearrange terms on the left side and right side and finally obtain the following scaling,

$$\frac{V_S \tau}{l_c} \sim \left(\frac{\gamma_S}{\gamma_{LV}}\right)^{\frac{2}{m}} \left(\frac{\rho g}{G_S l_c^{m-2} \Pi(h)}\right)^{\frac{1}{m}} \left(\frac{D}{l_c}\right)^{\frac{2}{m}}. \quad (5)$$

All parameters are determined experimentally, with the exception of surface tension $\gamma_{LV}$, due to uncertainties in the surface state caused by the substantial presence of free polymer chains in the silicone gel that can migrate and coat the droplet interface [21, 22, 27]. Fig. 3(a) shows the experimental scaling relation between the droplet size and sliding velocity on thin films of different thickness. Their extensive dimensionless comparison is further demonstrated in Fig. 3(c) where the dimensionless sliding velocity is associated with the normalized droplet size via the same simple scaling law, $V_S \sim D^{3.3}$. That aligns closely with our analytical solution presented by Eq. (5) for $D < l_c$ where $V_S \sim D^{2/m}$. Using $m=0.62$ derived from rheological measurement shown in appendix Fig. A1, the theoretical exponent $2/m=3.2$, matches the experimental value of 3.3. This agreement supports the assumptions we made in writing our model. However, for large droplets of $D > l_c$, systematic deviations from the theoretical curve emerge, likely linked to droplet elongation and morphological changes in the contact line in Fig. 2(b). A simplified explanation for this deviation is illustrated in the inset of Fig. 3(c). For elongated droplets, dissipation perpendicular to the sliding direction (x-axis) differs fundamentally from that along the sliding direction (y-axis). In particular, surface deformation of the gel phase does not propagate uniformly with the droplet motion: it remains stationary along the x direction (no dissipation) and cycles along the y direction as the droplet moves downward under gravity (strong dissipation as assumed in our model). This results into asymmetric dissipation along the contact line. Consequently, the actual overall energy dissipation is effectively lower than that predicted by the symmetric theoretical model, leading

to a higher sliding velocity observed in experiments at high Bond numbers. Notably, while advancing and receding contact angles deviate from the $\pi/2$ reference value, their combined effects seem to compensate spatially, preserving the proportionality between total dissipation and contact line length. We further comment on the surface tension that is left as a fitting parameter. The fitting of the experimental data to the theoretical prediction yields $\gamma_{LV}=55$ mN/s, a value that closely approximates the measured value from the prior contamination test [21], reinforcing the consistency of the proposed model. This concordance underscores the robustness of the simplified framework in explaining the observed scaling dynamics.

For the thickness effect, we first report the experimental scaling relation between the sliding velocity and the nominal thickness, $\Pi(h)$ for three typical droplet sizes in Fig. 3(b): $V_S$ scales with $\Pi(h)$ via a power law with the exponent of -1.6. To generalize a consistent way to report our observations, we reformulate Eq. (5) and further show the dimensionless velocity as a function of the dimensionless thickness $\Pi(h)l_c^{2-m}$ in Fig. 3(d). We observe good agreement between experiments and the model outlined in Eq. (5) for small droplet volume (<10 µL). Direct comparison with $h$ is avoided, as the model focuses on $\Pi(h)$ rather than $h$ itself, though prior work [9] links $\Pi(h)$ to $h$ at small $h$. Our analysis hereby extends to broader thickness ranges, revealing an experimental scaling exponent of -1.6, matching the theoretical prediction, $-1/m=-1.6$. This confirms thickness-dependent control of dissipation. Two systematic deviations merit discussion. First, nondimensionalized sliding velocity still exhibits the droplet size dependence, and such dependence seems to persist in Fig. 3(d) in line with the thickness effect, in sharp contrast to the report in Fig. 3(c) where the effects from $\Pi(h)$ and $D$ can be separately demonstrated. The entanglement of the two effects in Fig. 3(d) suggests other second-order mechanisms in operation other than the simple model proposed here and the droplet size effect must overshadow the $\Pi(h)$ effect and other possible effects. Given the circular contact morphology, this other possible effect likely comes from the dynamic contact angle, a considerable deviation from the model assumption. Second, significant departures from theoretical predictions occur for large droplets (>8 µL), exceeding explanations above. Given the fact of the contact line morphology dependence on the droplet volume shown in Fig. 2(b), contact line morphology likely dominates here. What is interesting though is that both the dynamic contact angle and the contact line morphology trigger less the total energy dissipation compared to the case of the $\pi/2$ sliding droplet. In Fig. 3(c), we provide a preliminary interpretation of this deviation. However, the current theoretical understanding remains insufficient to fully incorporate of dynamic contact angles and droplet deformation effects, necessitating future refinements.
<figure><img src="image_5.png" /><figcaption>FIG. 3. Scaling law of droplet sliding. (a) Sliding velocity as a function of the droplet size on soft films with different thickness: 8.3 µm, 62 µm, 596 µm. The dotted line corresponds to the power law fitting with the exponent of 3.3. (b) Sliding velocity as a function of nominal thickness Π(h) for droplets of different volume: 4 µL, 6 µL, 8 µL. The dotted line corresponds to the power law fitting with the exponent of -1.6. (c) Power-law relationship between the nondimensionalized sliding velocity and the droplet size normalized by the capillary length, $Dl_c^{-1}$. The dotted gray line represents the theoretical scaling. Insert: Schematic of the asymmetric dissipation mechanism. (d) Power-law relationship between the nondimensionalized sliding velocity and the nominal thickness of the thin film Π(h)$l_c^{2-m}$. The dotted gray line represents the model scaling.</figcaption></figure>

V. CONCLUSION

In this study, we experimentally investigate the vertical sliding dynamics of water droplets on silicone gel substrates by systematically varying droplet size and gel layer thickness. Quantitative characterization of droplet morphology and sliding velocity reveals distinct controls by both parameters. For Bond numbers Bo<1, the contact line of sliding droplets exhibits a circle shape, while it is elongated with the increase of the Bond number at Bo>1. The sliding velocity increases with the droplet size by a power law with the exponent 3.3, and scales with the nominal thickness Π(h) by a power law with exponent -1.6 at the limit of Bo<1. Using the Chasset-Thirion model within a linear viscoelastic framework, we formulate a dissipation model for the gel phase and balance it against gravitational driving energy. Key assumptions includes: (1) a circular contact line, (2) a fixed dynamic contact angle of π/2, (3) constant liquid-vapor surface tension. Despite these simplifications, the model successfully reproduces the observed scaling laws between the sliding velocity and the droplet size, thickness. This research sketches a framework for the sliding dynamics by con-

sidering the energy dissipation from the gel phase, with implications for applications such as dew harvesting, self-cleaning surfaces. The following challenges include the incorporation of three-dimensional droplet morphology during sliding and dissipation mechanisms under multi-directional substrate deformations.
ACKNOWLEDGMENTS

This work was supported by the National Natural
Science Foundation of China grant No. 12388101,
Agence Nationale de la Recherche and Commissariat à
l'Investissement d'Avenir Multiscale Modeling of Mate-
rial Interfaces (MMEMI) Grant (LabEx Science and En-
gineering for Advanced Materials and Devices) ANR-11-
LABX-086, ANR-11-IDEX05-02 and Poroelasticity and
Non-linearity in Gel Wetting (GELWET) Grant ANR-
17-CE30-0016.

DATA AVAILABILITY STATEMENT

All data are included in the manuscript and can be
accessed from the Supplementary Dataset.

Appendix A: Rheology characterization

<figure><img src="image_6.png" /><figcaption>FIG. A1. Rheology of the silicone gel. Rheological mea-
surements: Blue dots correspond to the storage modulus and
the red dots correspond to loss modulus; The solid lines rep-
resent fits to the Chasset-Thirion model with the best fit of
m=0.62. Silicone gel rheology was characterized by small am-
plitude oscillatory shear measurements by a strain-controlled
rheometer (Physica MCR 500; Anton Paar, Austria) with a
parallel plate geometry (PP20-MRD) at controlled tempera-
ture, 25±0.2°C. The gap was set at 0.6 mm and the strain
was fixed as 1%. The gel sample was directly crosslinked onto
the rheometer plate.</figcaption></figure>

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