# The macroscopic contact angle of water on ice

Wladimir Sarlin, Daniel Vito Papa, Rodolphe Grivet, Alexander Sascha Rosenbaum, Axel Huerre, Thomas Séon, Christophe Josserand

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Wladimir Sarlin, Daniel Vito Papa, Rodolphe Grivet, Alexander Sascha Rosenbaum, Axel Huerre, et al.. The macroscopic contact angle of water on ice. Journal of Fluid Mechanics, 2025, 1019, pp.A31. 10.1017/jfm.2025.10609. hal-05055151v2

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Distributed under a Creative Commons Attribution 4.0 International License
1 The macroscopic contact angle of water on ice

2 W. Sarlin¹, D. V. Papa¹, R. Grivet¹, A. Rosenbaum¹, A. Huerre², T. Séon³ and C.

3 Josserand¹

4 ¹Laboratoire d'Hydrodynamique, CNRS, École polytechnique, Institut Polytechnique de Paris, Palaiseau,

5 91120, France

6 ²Laboratoire Matière et Systèmes Complexes (MSC), Université Paris Cité, CNRS, UMR 7057, Paris,

7 75013, France

8 ³Institut Franco-Argentin de Dynamique des Fluides pour l'Environnement (IFADyFE), CNRS (IRL

9 2027), Universidad de Buenos Aires, CONICET, Buenos Aires, 1428, Argentina

10 Corresponding author: Wladimir Sarlin, wladimirsarlin@outlook.com

11 (Received xx; revised xx; accepted xx)

12 Wettability quantifies the affinity of a liquid over a substrate, and determines whether the
13 surface is repellent or not. When both the liquid and the solid phases are made of the
14 same chemical substance and are at thermal equilibrium, complete wetting is expected in
15 principle, as observed for instance with drops of molten metals spreading on their solid
16 counterparts. However, this is not the case for water on ice. Although there is a growing
17 consensus on the partial wetting of water on ice and several estimates available for the
18 value of the associated macroscopic contact angle, the question of whether these values
19 correspond to the contact angle at mechanical and thermal equilibrium is still open. In the
20 present paper, we address this issue experimentally and demonstrate the existence of such a
21 macroscopic contact angle of water on ice, from measurements and theoretical arguments.

22 Indeed, when depositing water droplets on smooth polycrystalline ice layers with accurately
23 controlled surface temperatures, we observe that spreading is unaffected by thermal effects
24 and phase change close enough to the melting point (namely, for undercoolings below 1 K)
25 so that conditions of thermal equilibrium are closely approached. Whereas the short time
26 motion of the contact line is driven by an inertial-capillary balance, the evolution towards
27 mechanical equilibrium is described by a viscous-capillary dynamics and is therefore
28 capillary - and not thermally - related. Moreover, we show that the resulting contact angle
29 remains constant for undercoolings below 1 K. By this way, we show the existence of
30 a non-zero macroscopic contact angle of water on ice under conditions of mechanical
31 and thermal equilibrium, that it is very close to 12°. We anticipate this key finding to
32 significantly improve the understanding of capillary flows in the presence of phase change,
33 which is of special interest in the realm of ice morphogenesis and glaciology, but also in
34 the aim of developing numerical methods for resolving triple-line dynamics.
## 1. Introduction

Wetting refers to the science of how a liquid deposited on a solid (or liquid) substrate spreads out. It plays an important role in many aspects of our everyday life such as the first breath of a newborn, underground flows, painting, chemistry, automobile, food industry, 3D printing, sap flow in plants or even eyes lubrication (de Gennes 1985; Bonn et al. 2009). A case of particular interest is that of the wetting of water on its own solid phase, ice. Indeed, a wide variety of ice structures are formed by the freezing of capillary flows (such as drops, rivulets and liquid films), including ice accretion on airplanes (Lynch & Khodadoust 2001), power lines (Laforte et al. 1998), bridge cables (Liu et al. 2019) or wind turbines (Wang 2017); ice falls (Montagnat et al. 2010); icicles (Chen & Morris 2011); cloud physics (Dash et al. 2006); frost heave (Wettlaufer & Worster 2006); frozen rivers (Beltaos 2013); and aufeis (Schohl & Ettema 1990). However, despite this ubiquitous coexistence of ice and capillary flows, the wetting behavior of water on ice remains largely unknown.

From a fundamental point of view, a crystal-clear investigation of the wetting properties of water on ice would require to be at thermodynamical equilibrium. In that purpose, an important piece of experimental developments has been devoted to the characterization of the shape of a water-like layer at the ice surface at small scales, particularly in the last decades (Sazaki et al. 2012; Asakawa et al. 2016; Slater & Michaelides 2019). Notably, these authors reported that as ice premelts a wetting quasi-liquid layer forms on the ice and thickens up to a nanometric thickness that depends on temperature (Murata et al. 2016), before the film free-surface gets attracted by the ice. This leads to the so-called pseudopartial wetting situation (Brochard-Wyart et al. 1991), where the quasi-liquid dewets and form micrometric water droplets that stand on the film. At such a small scale, the contact angle of micrometric quasi-liquid water droplets has been measured to range between 0.6° and 2.3° (Murata et al. 2016), while a recent theoretical analysis predicted a value of 3.4° (Luengo-Márquez et al. 2022).

By contrast, at a macroscopic scale there is a much larger discrepancy in the reported values for the water-ice contact angle (Huerre et al. 2025). These configurations correspond to thicker water layers (typically of the order of the capillary length, i.e. few millimeters) and involve air as the ambient gas. As a result, even if most of these studies attempted to measure a contact angle close to the melting temperature, they are far from the thermodynamic equilibrium. Nevertheless, it should be emphasized that these conditions describe most practical situations where water and ice interact; therefore, characterizing the wetting properties of water on ice at mechanical and thermal equilibrium, with air as the ambient gas, is crucial. The first attempt to measure this angle was proposed 60 years ago by Knight (1967), who deposited a puddle of hot water on a cold copper substrate and measured a receding contact angle of 12° at a temperature slightly below 0°C. A few years later, the same author reported other measurements in various configurations (Knight 1971) and found different values, before ultimately concluding that measuring this angle unequivocally may be impossible, although he thought that the complete wetting scenario is very unlikely (Knight 1996). Since this seminal work, other authors conducted similar experiments, depositing either a water drop or film on ice and measuring the advancing or receding contact angle, and found values spanning a large range from 6° up to 40° (Makkonen 1997; Thiévenaz et al. 2020; Demmenie et al. 2023; Grivet et al. 2024). In a very recent study, Demmenie et al. (2025) performed droplet deposition experiments involving water and ice with varying surface temperature. These authors showed how the apparent angle varies with temperature and also identify 12° as a possible value for the contact angle of water on ice. However, these contributions did not approach the melting
<figure><img src="image_2.png" /><figcaption>Figure 1. (a) Schematics of the experimental setup. (b) Image sequence of the spreading of a water droplet on ice at a low surface temperature ($T_i = -20.08$ °C). Four representative stages are displayed: (i) Just before the initial contact with the ice surface ($t = 0$ ms), (ii) at short time ($t = 18.72$ ms), (iii) when the drop detaches from the needle ($t = 29.76$ ms) and (iv) when motion stops but before complete freezing ($t > 64.00$ ms). (c)-(d) Spreading of a droplet on ice at (c) a moderately low surface temperature ($T_i = -10.32$ °C) and (d) close to the melting point of water ($T_i = -0.06$ °C). The same four stages (i)-(iv) are depicted. The initial radius of curvature $R_0$ of the pendant drop, the radius $r(t)$ of the wetted area, the drop volume $V$ and the arrest radius $r_m$ are highlighted in white.</figcaption></figure>

point accurately enough, since reaching such a limit remains an experimental challenge, and thereby could not demonstrate that the macroscopic angle they report is observed in the absence of thermal or phase change effects.

Finally, the contact angle $\theta_e$ that a water drop makes with the ice can be expressed as a function of the three surface energies $\gamma_{iw}$, $\gamma$ and $\gamma_{iv}$ associated with the solid-liquid, liquid-vapour and solid-vapour interfaces, respectively, using the Young-Dupré relation

$$ \gamma_{iv} = \gamma_{iw} + \gamma \cos \theta_e. \quad (1.1) $$

From then on, a possible path to determine the contact angle of water on ice at thermodynamic equilibrium would be to deduce this angle by considering equation (1.1). Indeed, given that the surface tension of water at 0 °C is $\gamma = 75.6 \text{ mJ.m}^{-2}$ (Lide 2004), $\theta_e$ could in theory be estimated using relation (1.1), if both $\gamma_{iv}$ and $\gamma_{iw}$ are available. Furthermore, as the Young-Dupré relation remains valid out of thermodynamical equilibrium, provided that effective surface energies are used, this approach could also be used when investigating water-ice-air systems at mechanical and thermal equilibrium. Unfortunately, these values are not known precisely enough today to discriminate between 0 and 50°, as $\gamma_{iw}$ is currently found to be in the range 27–35 mJ.m$^{-2}$ (Espinosa et al. 2016; Ambler et al. 2017) and $\gamma_{iv}$ in the range 70–120 mJ.m$^{-2}$ (de Reuck 1957; Ketcham & Hobbs 1969; Van Oss et al. 1992; Pruppacher & Klett 2004; Djikaev & Ruckenstein 2017) at 0°C.

The goal of the present study is thus to characterize the contact angle, if it exists, of water on ice in the thermal equilibrium limit for macroscopic liquid layers. In that purpose, we perform experiments in which a water droplet is deposited on a smooth polycrystalline ice layer whose surface temperature is meticulously varied, in order to approach thermal equilibrium (i.e., temperature equilibrium between solid and liquid phases) as closely as possible. For temperatures near the melting point, i.e., for undercoolings below 1 K, we demonstrate that the spreading and arrest of the contact line occur independently of thermal effects and phase change. This allows us to report a so-called macroscopic contact angle
of water on ice that is robust, non-zero and observed under conditions of mechanical
and thermal equilibrium. Furthermore, for larger undercoolings we also characterize the
evolution of an apparent contact angle of water on ice with temperatures far from thermal
equilibrium.

2. Experimental setup and qualitative results

A schematic of the experimental setup is illustrated in figure 1(a). Experiments consist in depositing a pendent drop of pure demineralized water of density ρ, viscosity η and surface tension γ at vanishing impact velocity (using a linear stage) on a flat layer of ice made of the same pure demineralized water, whose surface temperature is Tᵢ. The spreading of the droplet is recorded with a Photron FASTCAM SA-X2 high-speed camera operating between 8000 and 12500 frames per second. Both drop and air temperatures (T_d and T_a, respectively) are controlled and kept as close as possible to the melting temperature of water, T_f = 273.15 K. Experiments are carried out in a reduced relative humidity environment (RH_a ≈ 16 ± 2 %) in order to limit the effect of this parameter on the ice and frost formation (Sebilleau et al. 2021). Each test is performed at a constant Tᵢ and repeated five times to ensure reproducibility of the results. In the present study, Tᵢ is varied from close to the melting point up to far out of T_f, i.e., Tᵢ ∈ [-0.03 °C, -20.42 °C], while all other parameters (T_d, T_a, RH_a) are kept constant.

The ice surface is formed by freezing a small layer of water of thickness ~ 5 mm with a
40 × 40 mm Peltier cooler mounted on a heat exchanger and connected to a recirculating
thermal bath that operates at T_bath = 1 ± 0.1 °C. The thermal bath ensures a proper heat
evacuation from the Peltier cooler and therefore a precise control of the thermal flux
applied when imposing a voltage. A smooth and mirror-like surface of polycrystalline ice
is obtained by enforcing a low thermal flux on the Peltier modulus, typically of the order of
0.1 K/min, that leads to a slow and controlled growth of the solidification front from below.
The surface temperature of the ice layer, T_i, is measured by placing a class A flat PT100
RTD sensor (precision of 0.01 °C) on top of the ice surface and very close to the deposition
site of the droplet (see figure 1(a)). The thermal flux is then gently adapted to set the desired
ice temperature T_i. The temperature of the drop, T_d, is controlled by passing the needle
(at the tip of which the pendent drop is formed) through a heat exchanger also connected
to the recirculating bath. T_d is monitored with a type K thermoelectrical sensor located
inside the tip of the needle. Throughout the study, its value was kept at T_d = 2.5 ± 0.3
°C. The ambient gas temperature T_a is controlled by injecting cold gaseous nitrogen in
the box and is measured close to the deposition site with a type K thermoelectric sensor.
This parameter is roughly kept constant at T_a = 7 ± 1.5 °C. The same cold nitrogen influx
allows us to set the value of RH_a, that is controlled using a domestic humidity sensor. With
such a small value for the reduced relative humidity, we obtained an ice surface which is
almost completely dry, in addition of being very smooth. The values of T_a and T_d have
been chosen so as to be as close as possible to T_f. Based on preliminary tests, lower droplet
temperatures could lead to the partial freezing of its bottom part before the initial contact
with the ice, while a lower nitrogen temperature would have required an additional cooling
system and would have also lowered T_d.

The video recordings are used to monitor the spreading dynamics and to determine
all the variables of interest. The initial radius of curvature R₀ of the pendent drop and
the water-air surface tension γ are obtained from the last image preceding the onset of
spreading (column (i) in figures 1(b)-(d)) using the pendent drop method introduced by
Daerr & Mogne (2016). The radius r(t) of the wetted area is extracted from the contour
of the spreading droplets (column (ii) in figures 1(b)-(d)). The volume V of the drop is
<figure><img src="image_3.png" /><figcaption>Figure 2. (a) Dimensionless spreading radius, $r/R_0$, as a function of the rescaled time, $t/\tau_c$, in a logarithmic scale. The horizontal dashed lines highlight the arrest radius reached for each experiment. (b) Evolution of the numerical prefactor $C$ from equation (3.1) with the undercooling $\Delta T = T_f - T_i$, with $T_f$ the melting point of water. The solid line represents $C = 1.22$. (c) $t_{ic}$ as a function of $\Delta T$. The solid line indicates $t_{ic} = 25.84$ ms. The colors of the symbols denote the surface temperature $T_i$ of the ice, with darker shades corresponding to colder ice.</figcaption></figure>

computed from the first image following the droplet’s detachment from the tip of the needle, when the contrast in the images is at its maximum (column (iii) in figures 1(c)-(d)). For cold ice surfaces whose undercooling $\Delta T = T_f - T_i > 11$ K, the drop does not detach from the needle axisymmetrically, hence $V$ is not computed for these tests. At the end of the spreading process, an extra image is taken (column (iv) in figures 1(b)-(d)) to compute the arrest radius $r_m$ of the drop and the apparent contact angle $\theta_a$ that the droplet forms with the ice. $\theta_a$ is determined using an adapted version of the algorithm introduced by Quetzeri-Santiago et al. (2020). Furthermore, the density and viscosity of water are taken as $\rho = 1000 \text{ kg.m}^{-3}$ and $\eta = 1.8 \times 10^{-3} \text{ Pa.s.}$, respectively.

The spreading of the droplet is illustrated in figure 1 for three different surface temperatures $T_i = -20.08 \text{ }^{\circ}\text{C}$, $T_i = -10.32 \text{ }^{\circ}\text{C}$ and $T_i = -0.06 \text{ }^{\circ}\text{C}$ (panels (b), (c) and (d), respectively) and at four characteristic moments (columns (i)-(iv)). At the beginning of an experiment, a pendant drop of initial radius of curvature $R_0$ is put into contact with the ice surface (i) on top of which it starts spreading. The spreading rate is quantified by considering the radius of the wetted area of the drop on the ice surface, $r(t)$ (ii). At some point in the dynamics, a volume $V$ of liquid detaches from the needle (iii). When spreading ceases, this volume eventually stabilizes to form a sessile droplet that exhibits a spherical cap shape of arrest radius $r_m$ (iv). For large undercoolings $\Delta T$, the drop shape is similar to a hemisphere and its apparent contact angle $\theta_a$ with the substrate is high (figure 1(b)(iv)), whereas for ice surface temperature approaching the melting point of water $T_f$ the sessile drop resembles a very thin lens with a small but non-zero contact angle (figure 1(d)(iv)).

## 3. Results and discussion

### 3.1. Spreading dynamics

The evolution of the dimensionless radius $r/R_0$ with the rescaled time $t/\tau_c$ is presented in figure 2(a) for five representative ice surface temperatures, $T_i$. Here, $\tau_c = \sqrt{\rho R_0^3}/\gamma$ is the inertial-capillary timescale. A first phase of fast spreading is observed for all experiments (before the black arrows), which is reminiscent of the short-time evolution observed for
isothermal droplet deposition experiments (Biance et al. 2004; Bird et al. 2008; Winkels et al. 2012). This regime ends at a time $t_{ic}$ which depends on the ice temperature. For the dark blue curve in figure 2(a), corresponding to $T_i = -20.08$ °C (●) and to sequence (b) in figure 1, a partial-pinning of the contact line is observed from $t/\tau_c \approx 0.8$ up to $t/\tau_c \approx 4.1$, when the arrest radius is reached. Such a behavior causes the asymmetry of the free-surface of the spreading drop in figure 1(b)(ii) (compared to figures 1(c)(ii) or 1(d)(ii)). This observation suggests that, for cold enough ice surface, the contact line can undergo pinning and unpinning due to interactions between the advancing contact line and the ice layer that grows beneath the spreading drop. For moderate ice surface temperatures, a different evolution is noticed, as illustrated by the two curves corresponding to $T_i = -2.92$ °C (●) and $T_i = -10.30$ °C (●) in figure 2(a). Indeed, for these experiments the fast spreading dynamics abruptly stops around $t/\tau_c \approx 3.8$ without any significant partial-pinning of the contact line afterwards. Finally, for ice surface temperatures closer to the melting point, a second and slower dynamics takes place after the sudden stop of the initial fast spreading regime. This is illustrated by the light blue curves in figure 2(a), corresponding to $T_i = -0.06$ °C (●) and to $T_i = -0.64$ °C (●), from $t/\tau_c \approx 3.8$ onward. For all experiments, the horizontal dashed line whose color corresponds to the symbols indicates the arrest radius $r_m$ reached at the end of the spreading process.

The short-time dynamics is well described by a balance between Laplace pressure, that drives the motion, and the inertial pressure that resists it (Biance et al. 2004; Bird et al. 2008; Winkels et al. 2012; Grivet et al. 2022). Indeed, equating the two effects gives the following inertial-capillary dynamics for the radius of the wetted area

$$ \frac{r(t)}{R_0} = C \left( \frac{t}{\tau_c} \right)^{1/2}, \qquad (3.1) $$

with C a numerical prefactor. It can be observed in figure 2(a) that the short-time evolution of the dimensionless radius, before each of the black arrows, effectively displays a square root of time evolution, as evidenced by the 1/2 slope followed by all curves in this first dynamical regime. It has been verified that this 1/2 power law is insensitive to errors in determining the time origin, given the large temporal resolution at which the experiments were recorded. The prefactor C is systematically extracted and presented in figure 2(b) as a function of the undercooling $\Delta T = T_f - T_i$. For $\Delta T < 11$ K, C plateaus at a constant value of 1.22, that is consistent with the measurements reported by Bird et al. (2008) ($C \in [0.75, 1.5]$) and Grivet et al. (2022) ($C = 1.15 \pm 0.06$) for droplet deposition experiments. For undercoolings beyond 11 K, a slight decrease of C is noticed, that can be attributed to the increasing rate of solidification. Strikingly, the constant value of C over two decades of $\Delta T$ between 0.06 and 11 K reveals that this inertial-capillary regime is unaffected by the ice surface temperature. In other words, the contact line dynamics of the spreading droplet in this regime is insensitive to both temperature gradients and phase change at short times. The arrest time of this inertial-capillary dynamics, $t_{ic}$, is then determined for all experiments. Its evolution with undercooling $\Delta T$ is illustrated in figure 2(c), where it can be seen that $t_{ic}$ remains constant at approximately 25.84 ms when $\Delta T < 11$ K, before dropping sharply for higher undercooling. This demonstrates that, for $\Delta T > 11$ K, solidification does induce an early pinning of the advancing contact line during the inertial-capillary regime, whereas for smaller undercooling the arrest of this dynamics is not thermally-related.

Then, two different situations are observed when $\Delta T < 11$ K. In the intermediate range $1$ K $< \Delta T < 11$ K, the spreading of the droplet ceases right after the inertial-capillary regime, as highlighted by the experiments at $T_i = -2.92$ °C (●) and $T_i = -10.3$ °C (●)
in figure 2(a), where no motion occurs after $t_{ic}$. However, a second and slower dynamics systematically takes place when $\Delta T < 1$K, as illustrated by experiments where $T_i = -0.06$ °C (•) and $T_i = -0.64$ °C in figure 2(a). We suspect this second regime to correspond to a viscous-capillary spreading. Indeed, in the absence of thermal effects, such dynamics usually develops once inertia becomes negligible (Biance et al. 2004; Bonn et al. 2009). As discussed in particular by De Gennes (1984), this behavior is intimately related to wettability that governs the manner the contact line relaxes to equilibrium. In case of total wetting, one would expect Tanner's law to be valid, so that $r/R_v \propto (\gamma t/(\eta R_v))^{1/10}$, with $R_v = (3V/(4\pi))^{1/3}$ the equivalent radius for a droplet of initial volume V (Tanner 1979; De Gennes 1984). However, if partial wetting is at play, a viscous-capillary spreading (similar to a Cox-Voinov dynamics) is expected for the moving contact line (Voinov 1976; Cox 1986; Bonn et al. 2009; de Gennes 1985). To describe the motion of the contact line for a spreading droplet of initial volume V and equivalent radius $R_v = (3V/(4\pi))^{1/3}$, we consider the following set of equations:

$$ \frac{d\tilde{r}}{d\tilde{t}} = \frac{1}{3\zeta} \tan \theta(\tilde{t}) (\cos \theta_e - \cos \theta(\tilde{t})), \quad (3.2) $$

$$ \frac{\tilde{r}^3}{4 \sin^3 \theta(\tilde{t})} (2 + \cos^3 \theta(\tilde{t}) - 3 \cos \theta(\tilde{t})) = 1, \quad (3.3) $$

where $\tilde{t} = t/\tau_v$, $\tilde{r} = r/R_v$ and $\tau_v = \eta R_v/\gamma$ the viscous-capillary timescale, and where $\theta(\tilde{t})$ and $\theta_e$ are the instantaneous and equilibrium macroscopic contact angles, respectively. Equation (3.2) is obtained by balancing viscous dissipation at the contact line with the action of capillary forces that drive the motion (see Appendix A for details on the derivation), whereas relation (3.3) is the conservation of volume for a spherical cap (Van de Velde et al. 2023). In the former relation, $\zeta$ is a dimensionless parameter related to the ratio between the typical length scale of the system (e.g., the droplet radius $R_v$) and a microscopic length of the order of a nanometer; it enables matching between the microscopic and macroscopic solutions describing the shape of the contact line (Bonn et al. 2009). Since all the present experiments involve water droplets with a rather constant initial radius that are deposited on the same substrate (ice), $\zeta$ can reasonably be considered constant. From then on, equations (3.2)-(3.3) can straightforwardly be solved numerically, with $\theta_e$ being evaluated from equation (3.3) when $r = r_m$ and $\theta = \theta_e$ (thus, $\theta_e$ is the macroscopic contact angle that the drop reaches at the end of spreading).

The evolution of $\tilde{r}$ with $\tilde{t}$ is illustrated in figure 3(a) for two experiments where (•) $T_i = -0.06$ °C and (●) $T_i = -0.64$ °C. The two curves highlight the relaxation of the wetted radius up to its arrest value $r_m$, that is reported in horizontal dashed lines. The corresponding predictions from equations (3.2)-(3.3), with $\zeta = 30$, are also reported in solid black lines in figure 3(a). This value for $\zeta$ has been obtained by adjusting a theoretical curve to one experiment, and has then been used to describe all tests where $\Delta T < 1$ K. Indeed, although it is based on physical grounds and expected to be constant for the reasons indicated above, there is no analytical estimate available to determine its value. The theoretical solutions closely match the experimental measurements for the two ice surface temperatures, from the onset of the second regime to the stabilization of the spreading drop. This result reveals that the slower spreading phase observed when $\Delta T < 1$ K is also insensitive to phase change or to any thermal effect. The differences between the two curves, for instance in terms of the final radius, are not due to the ice temperature (as this parameter is not involved in equations (3.2)-(3.3)) but only to fluctuations in V or in the thermophysical properties of water that are captured by the present modelling.
<figure><img src="image_4.png" /></figure>

Figure 3. (a) Rescaled radius $\tilde{r} = r/R_v$ as a function of $\tilde{t} = t/\tau_v$, in the viscous-capillary regime, for two experiments where (•) $T_i = -0.06$ °C and (•) $T_i = -0.64$ °C (in logarithmic representation). The solid lines are the corresponding predictions obtained by solving equations (3.2)-(3.3), that are extended in dotted lines prior to the observed viscous-capillary spreading. The 1/10 slope that would correspond to complete wetting (Tanner's law) is also indicated. The horizontal lines highlight the dimensionless arrest radius $\tilde{r}_m = r_m/R_v$ reached for each experiment. (b) Evolution of the apparent contact angle $\theta_a$ as a function of the undercooling $\Delta T$. The black and white triangles respectively correspond to measurements of the right and left angles between the water droplet and the ice once the final state is reached, using the method described by Quetzeri-Santiago et al. (2020). The red circles are the apparent angles estimated using equation (3.3) with $r = r_m$, i.e., assuming the final shape is a spherical cap. The solid line is the plateau $\theta_0 = 11.8^\circ$, which is the mean contact angle measured from all experiments where $\Delta T < 1$K.

Furthermore, the dynamics is well described by a viscous-capillary evolution and the two experiments significantly depart from a 1/10 power law during the relaxation of the contact line. This key observation demonstrates that the arrest of the drop is solely due to capillary effects, with no influence from temperature, and therefore that water partially wets ice. In fact, one can easily argue that thermal effects related to solidification vanish towards the capillary ones as the melting point is approached. Indeed, we can estimate the solidification timescale as $\tau_{sol} = h^2/D_{eff}$, where h is the typical thickness of the water film at maximal spreading and $D_{eff}$ is the effective diffusion coefficient of ice growth when water is in contact with it (Sarlin et al. 2024). Then, in the small $\Delta T$ limit, one has $D_{eff} \propto \Delta T^2$ (see more detail in Appendix B). On the other hand, the inertial-capillary and viscous-capillary dynamics are respectively based on the timescales $\tau_c = \sqrt{\rho R_0^3/\gamma}$ and $\tau_v = \eta R_v/\gamma$ that are mostly independent of temperature. Therefore, when comparing these two typical times to the solidification timescale, it is clear that in the limit $\Delta T \to 0$ the spreading dynamics becomes unaffected by phase change as thermal equilibrium is approached. By contrast, for large $\Delta T$ solidification will become dominant, which agrees with the observation of the partial-pinning effect observed near the triple line at the lowest ice surface temperatures investigated (see Appendix B for details about the comparison of the timescales).

Hence, for undercoolings below 1 K, the inertial-capillary dynamics is followed by a viscous-capillary regime that is well described by a model independent of thermal effects and phase change, which strongly suggests that we are closely approaching thermal equilibrium in this range of ice surface temperature.

## 3.2. The macroscopic contact angle of water on ice

To further discuss the value of the macroscopic contact angle of water on ice under the present experimental conditions, the apparent angle $\theta_a$ between the droplet and the ice layer
at the end of the spreading process is systematically extracted in two different manners. Firstly, the right and left angles are evaluated from the contours of the final photograph taken for each experiment, using the algorithm developed by Quetzeri-Santiago et al. (2020) to ensure reliable measurements. By doing so, the contact line is approximated by first or second order polynomials, and the most robust tangent at the triple line is retained. Secondly, an independent estimate is obtained by using equation (3.3) with the initial volume V and the final measured radius $r_m$, assuming the final liquid film has a spherical cap shape. This second method is only applied to experiments where $\Delta T < 11$ K, due to the reasons mentioned in section 2. The results are illustrated in figure 3(b), where $\theta_a$ is presented as a function of the undercooling $\Delta T$. The apparent angle is large when $\Delta T$ is greater than 11 K, as the contact line gets pinned during the short-time dynamics where its curvature is important. It then decays when the undercooling is reduced. A first inflection of the curve is observed around $\Delta T \approx 11$ K, i.e., when contact line pinning occurs at the end of the inertial-capillary regime. Finally, $\theta_a$ reaches a constant value $\theta_0$ that is seen over more than a decade when $\Delta T < 1$ K, for both measurement methods. The average value and standard deviation for this constant macroscopic contact angle are

$$ \theta_0 = 11.8^{\circ} \pm 1.2^{\circ}. \qquad (3.4) $$

Interestingly, this estimate is significantly above zero, even including measurement errors from both methods.

Therefore, $\theta_a$ is independent of temperature for undercoolings up to approximately 1 K. Beyond this limit, however, temperature begins to play a dominant role and the apparent contact angle increases with $\Delta T$. Based on the observations in Figure 3(b) and the aforementioned discussion on the spreading dynamics, that is insensitive to thermal effects close enough to the melting point, we conclude that $\theta_0$ is the advancing contact angle of water on ice under conditions of mechanical and thermal equilibrium. Due to the nano-smooth surface state of ice at small undercoolings, contact line hysteresis must be very small in this case (Bonn et al. 2009), hence $\theta_0$ is expected to be very close to the macroscopic equilibrium contact angle. Finally, it is striking to note that the present estimate for the macroscopic angle of water on ice is strongly supported by the results from numerous similar studies (Knight 1967; Thiévenaz et al. 2020; Grivet et al. 2024; Demmenie et al. 2023, 2025), that were performed under different drop and ambient gas temperatures and relative humidity conditions. Owing to this observation, we infer that the macroscopic contact angle we report is not significantly impacted by humidity and evaporative effects.

## 4. Conclusion

In the present investigation, droplet deposition experiments are reported that demonstrate the partial wetting of water on ice. Indeed, close enough to the melting point of water, i.e., when $\Delta T < 1$ K, the spreading dynamics is found to be independent of thermal effects, causing the liquid film to relax into a sessile droplet shape. In this temperature range, a short-time inertial-capillary regime takes place first and is followed by a viscous-capillary dynamics. At the end of the spreading process, a macroscopic advancing contact angle of water on ice very close to 12° is identified. If this value agrees with previous estimates, the key findings here are to show that the slow evolution of the contact line in the second regime corresponds to a viscous-capillary dynamics, which implies that the observed angle is only due to wetting without any thermal effects, and that there is a smooth asymptotic convergence of the contact angle with decreasing undercooling. These
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two elements put together allow us to assert that for $\Delta T < 1$ K we reach mechanical and thermal equilibrium. Furthermore, because the ice layer is dry and nano-smooth due to the reduced humidity and slow formation rate, the contact angle hysteresis is expected to be small in this case. Although reducing the relative humidity implies that the present experiments are not performed at vapor pressure equilibrium (i.e., far from thermodynamic equilibrium), our measurements remarkably agree with numerous similar studies that operated under different relative humidity conditions (Knight 1967; Thiévenaz et al. 2020; Demmenie et al. 2023; Grivet et al. 2024; Demmenie et al. 2025), thereby suggesting that this parameter does not have a significant influence. Therefore, we infer that the value of the advancing contact angle $\theta_0$ reported here is robust and close to the macroscopic equilibrium contact angle of water on its solid phase.

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As mentioned in the introduction of the present work, two main groups of results exist. In the first, a microscopic contact angle around or less than one degree has been reported by thermodynamic studies, whereas in the second a larger macroscopic contact angle is obtained from fluid mechanics studies (Huerre et al. 2025). Although the present results suggest that the macroscopic contact angle of water on ice is very close to 12°, it already appears that there is a clear need to understand the origin of the discrepancy between small- and large-scale observations. Solving this issue is of great interest, and definitely deserve further investigations.

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The outcomes of the present study also allow us to provide an estimate of the difference between the ice-air and ice-water surface energies (respectively, $\gamma_{ia}$ and $\gamma_{iw}$). For that purpose, one can use the Young-Dupré relation (equation (1.1)) to write $\gamma_{ia} - \gamma_{iw} = \gamma \cos \theta_0 \approx 73.9 \text{ mN.m}^{-1}$, using $\theta_0 = 12^\circ$ and $\gamma = 75.6 \text{ mN.m}^{-1}$. From then on, it can be mentioned that estimates for $\gamma_{iw}$ available in the literature lie in the small range [27–35] $\text{mN.m}^{-1}$ (Huerre et al. 2025). Using those two extreme values to compute the ice-air surface energy, this suggests that $\gamma_{ia}$ is between $101 \text{ mN.m}^{-1}$ and $109 \text{ mN.m}^{-1}$. So far, several studies gave different estimates for this parameter, spanning a large range from $70 \text{ mN.m}^{-1}$ to $120 \text{ mN.m}^{-1}$ notably due to the lack of consensus on the contact angle of water on ice (Huerre et al. 2025). The findings reported here, in particular the robust measurement of $\theta_0$, thereby allow us to infer a narrower range for $\gamma_{ia}$, that goes in the sense of Ketcham & Hobbs (1969) and Pruppacher & Klett (2004).

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We anticipate these results to be of significant importance to better understand the formation of ice structures resulting from the interaction between capillary flows and phase change. The knowledge of the contact angle of water on ice in environmental conditions as well as the ice-air surface energy is also expected to be valuable in order to improve current numerical simulations dealing with three-phase problems at the macroscopic scale. Given the detailed description of the spreading dynamics of a water droplet on ice provided here, this model experiment could constitute a benchmark test for the development of numerical methods. More broadly, the outcomes of the present study bring new insights on the manner water covers ice, and could thereby be applied to the study of ice melt in glaciology.

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**Acknowledgments.** The authors warmly thank C. Frot and A. Garcia for their help in the elaboration of the experimental set-up.

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**Author contributions.** W. Sarlin and D. V. Papa contributed equally to the present study.

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**Funding.** This work was supported by Agence de l'Innovation de Défense (AID) - via Centre Interdisciplinaire d'Etudes pour la Défense et la Sécurité (CIEDS) - (project 2021 - ICING).

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**Competing Interests.** The authors report no conflict of interest.

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**Data availability statement.** The data that support the findings of this study are available from the corresponding author, upon reasonable request.

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390 Author ORCIDs. W. Sarlin, https://orcid.org/0000-0002-2668-2279; D. V. Papa, https://orcid.org/0000-0002-7852-3157; R. Grivet, https://orcid.org/0000-0002-1489-7336; A. Huerre, https://orcid.org/0000-0003-4702-5128; T. Séon, https://orcid.org/0000-0001-6728-6072; C. Josserand, https://orcid.org/0000-0003-1429-4209

391 7852-3157; R. Grivet, https://orcid.org/0000-0002-1489-7336; A. Huerre, https://orcid.org/0000-0003-4702-5128; T. Séon, https://orcid.org/0000-0001-6728-6072; C. Josserand, https://orcid.org/0000-0003-1429-4209

392 5128; T. Séon, https://orcid.org/0000-0001-6728-6072; C. Josserand, https://orcid.org/0000-0003-1429-4209

393 **Appendix A. The viscous-capillary spreading**

394 Once the inertial-capillary dynamics end, a second regime takes place that involves this time a competition

395 between surface tension, that drives the motion, and viscous dissipation. In what follows, we denote by $U$ the

396 velocity of the contact line, $r$ the radius of the wetted area, $\theta$ the macroscopic contact angle and $u$ the velocity

397 of the fluid (all these quantities depend on time). We assume that $u$ has a parabolic profile with respect to the

398 vertical direction, so that

$$u(z) = \frac{3}{2} U \left( \frac{2z}{\xi} - \frac{z^2}{\xi^2} \right), \qquad (A 1)$$

399 with $z$ the vertical distance from the substrate and $\xi = \alpha \tan \theta$ the position of the free-surface, that depends on

400 the horizontal distance $\alpha$ from the triple point. In that situation, the power density is $\eta(\partial u/\partial z)^2$ (Guyon et al.

401 2012), so that the dissipation per unit length reads

$$D = \int_{\alpha_m}^{\alpha_M} \int_{0}^{\xi} \eta \left( \frac{\partial u}{\partial z} \right)^2 dz d\alpha = \frac{3\eta U^2}{\tan \theta} \zeta, \qquad (A 2)$$

402 where $\zeta = \ln(\alpha_M/\alpha_m)$ depends on two quantities $\alpha_m$ and $\alpha_M$ (of the order of a nanometer and of the typical

403 size of the system, respectively) and allows one to match the microscopic and macroscopic shapes of the contact

404 line (Bonn et al. 2009).

405 On the other hand, the capillary force per unit length which acts on the contact line can be expressed

406 as $F_c = \gamma_{sg} - \gamma \cos \theta - \gamma_{st}$, with $\gamma_{sg}$ and $\gamma_{st}$ the surface energies associated with the solid-gas and solid-

407 liquid interfaces, respectively. As $F_c$ vanishes when the equilibrium contact angle $\theta_e$ is reached, we also have

408 $\gamma_{sg} = \gamma_{st} + \gamma \cos \theta_e$. Thus, we have $F_c = \gamma (\cos \theta_e - \cos \theta)$ and the power (per unit length) this force induces

409 is $F_c U$. If it gets fully dissipated, we thereby have

$$\gamma (\cos \theta_e - \cos \theta) U = \frac{3\eta\zeta}{\tan \theta} U^2, \qquad (A 3)$$

410 which can be rewritten, as $U = dr/dt$, in the form

$$\frac{d\bar{r}}{d\bar{t}} = \frac{1}{3\zeta} \tan \theta (\cos \theta_e - \cos \theta), \qquad (A 4)$$

411 with $\bar{r} = r/R_v$ and $\bar{t} = t/\tau_v$ ($R_v = (3V/(4\pi))^{1/3}$ being the equivalent radius based on the initial volume $V$

412 of liquid deposited, and $\tau_v = \eta R_v/\gamma$ the typical viscous-capillary timescale). Equation (A 4) is the same as

413 equation (3.2) in the main text.

414 A case of particular interest is the situation of complete wetting, i.e., when $\theta_e = 0$. When this is the case,

415 assuming $\theta \ll 1$, equation (A 4) reduces to

$$\frac{dr}{dt} = \frac{\gamma}{6\eta\zeta} \theta^3 \propto \frac{\gamma}{\eta} \theta^3, \qquad (A 5)$$

416 as $\zeta$ can approximately be considered constant (De Gennes 1984). Then, by observing that, for small angles, we

417 have $\theta \approx 2e/r$ for a spherical cap of radius $r$ and uppermost height $e$, and that a rough conservation of volume

418 implies $r^2 e \propto R_v^3$, equation (A 5) leads to

$$r^9 \frac{dr}{dt} \propto \frac{\gamma}{\eta} R_v^9, \qquad (A 6)$$

419 which, after integration, gives

$$\frac{r}{R_v} \propto \left( \frac{t}{\tau_v} \right)^{1/10}. \qquad (A 7)$$

420 Equation (A 7), widely known as Tanner's law, describes the spreading of a viscous drop of a completely wetting

421 liquid (Tanner 1979).
## 422 Appendix B. Spreading and solidification timescales

The time needed to solidify a droplet of volume V and radius $R_v = (3V/(4\pi))^{1/3}$ can be estimated via the self-similar solution of the Stefan problem, that gives in particular the evolution of the solidification front $h = \sqrt{D_{\text{eff}}t}$. In this expression, $D_{\text{eff}}$ is an effective diffusion coefficient that is solution to a transcendental equation involving the thermal properties of the different phases and the undercooling $\Delta T$ (Thiévenaz et al. 2019; Sarlin et al. 2024). Considering that, at maximal spreading $r_m$, the droplet has a characteristic height $h \propto R_v^3/r_m^2$, the typical solidification timescale writes

$$ \tau_{\text{sol}} = \frac{R_v^6}{r_m^4 D_{\text{eff}}}. \qquad (B1) $$

Similarly, it is possible to estimate the timescale $\tau_{ic}$ needed to reach the maximal spreading diameter in the inertial-capillary regime from the scaling law $r/R_0 \propto (t/\tau_c)^{1/2}$, that yields (here, we take $R_0 \approx R_v$ for the sake of simplicity)

$$ \tau_{ic} = \sqrt{\frac{\rho R_v^3}{\gamma} \left(\frac{r_m}{R_v}\right)^2}. \qquad (B2) $$

Conversely, the viscous-capillary timescale $\tau_{vc}$ is obtained from the scaling $r/R_v \propto (\gamma t/(\eta R_v))^{1/10}$ (see Appendix A), that leads this time to

$$ \tau_{vc} = \frac{\eta r_v}{\gamma} \left(\frac{r_m}{R_v}\right)^{10}. \qquad (B3) $$

With these three times, it is possible to derive two dimensionless ratios comparing the solidification timescale to the inertial-capillary or viscous-capillary ones. They respectively write

$$ \frac{\tau_{\text{sol}}}{\tau_{\text{ic}}} = \frac{1}{D_{\text{eff}}} \sqrt{\frac{\gamma R_v}{\rho}} \left(\frac{R_v}{r_m}\right)^6 \quad \text{and} \quad \frac{\tau_{\text{sol}}}{\tau_{\text{vc}}} = \frac{1}{D_{\text{eff}}} \frac{\gamma R_v}{\eta} \left(\frac{R_v}{r_m}\right)^{14}. \qquad (B4) $$

In both cases, one can observe that they are inversely proportional to $D_{\text{eff}}$. Then, for undercoolings approaching 0, one has

$$ D_{\text{eff}} \sim \frac{4 D_i c_{p,i}^2}{\pi \mathcal{L}_f^2} \Delta T^2, \qquad (B5) $$

with $D_i$ and $c_{p,i}$ the ice thermal diffusivity and heat capacity, and $\mathcal{L}_f$ the latent heat of fusion for water (Thiévenaz et al. 2019). As a result, when $\Delta T \to 0$ the ratios of timescales will tend to infinity as fast as $\Delta T^2$.

This implies that, close to the melting point, the solidification rate rapidly becomes negligible compared to the

spreading dynamics, regardless of the drop size or the maximal spreading radius. On the other hand, for large

undercoolings the solidification rate becomes important and causes the partial-pinning effect observed near the

triple line.

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