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, $\tau _c = \sqrt {ho {R_0}^3/\gamma }$ is the inertial-capillary time scale. 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, Clanet & Quéré Reference Biance, Clanet and Quéré2004; Bird, Mandre & Stone Reference Bird, Mandre and Stone2008; Winkels et al. Reference Winkels, Weijs, Eddi and Snoeijer2012). This regime ends at a time $t_{\textit{ic}}$ , which depends on the ice temperature. For the dark blue curve in figure 2(a), corresponding to $T_i = -20.08\,^\circ {\textrm{C}}$ () and to sequence (b) in figure 1, a partial-pinning of the contact line is observed from $t/\tau _c \simeq 0.8$ up to $t/\tau _c \simeq 4.1$ , when the arrest radius is reached. Such a behaviour causes the asymmetry of the free-surface of the spreading drop in figure 1(b,ii) (compared with figures 1 c,ii or 1 d,ii). This observation suggests that, for a 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 $T_i = -2.92\,^\circ {\textrm{C}}$ () and $T_i = -10.30\,^\circ {\textrm{C}}$ () in figure 2(a). Indeed, for these experiments, the fast spreading dynamics abruptly stops around $t/\tau _c \simeq 3.8$ 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 $T_i = -0.06\,^\circ {\textrm{C}}$ () and to $T_i = -0.64\,^\circ {\textrm{C}}$ (), from $t/\tau _c \simeq 3.8$ onward. For all experiments, the horizontal dashed line whose colour corresponds to the symbols indicates the arrest radius $r_m$ reached at the end of the spreading process.

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 (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) $t_{\textit{ic}}$ as a function of $\Delta T$ . The solid line indicates $t_{\textit{ic}} = 25.84\ {\textrm{ms}}$ . The colours of the symbols denote the surface temperature $T_i$ of the ice, with darker shades corresponding to colder ice.

The short-time dynamics is well described by a balance between Laplace pressure, which drives the motion, and the inertial pressure that resists it (Biance et al. Reference Biance, Clanet and Quéré2004; Bird et al. Reference Bird, Mandre and Stone2008; Winkels et al. Reference Winkels, Weijs, Eddi and Snoeijer2012; Grivet et al. Reference Grivet, Monier, Huerre, Josserand and Séon2022). Indeed, equating the two effects gives the following inertial-capillary dynamics for the radius of the wetted area

(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 $\Delta T \lt 11\ {\textrm{K}}$ , $C$ plateaus at a constant value of 1.22, which is consistent with the measurements reported by Bird et al. (Reference Bird, Mandre and Stone2008) ( $C \in [0.75, 1.5]$ ) and Grivet et al. (Reference Grivet, Monier, Huerre, Josserand and Séon2022) ( $C = 1.15 \pm 0.06$ ) for droplet deposition experiments. For undercoolings beyond 11 K, a slight decrease of $C$ is noticed, which 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, $t_{\textit{ic}}$ , is then determined for all experiments. Its evolution with undercooling $\Delta T$ is illustrated in figure 2(c), where it can be seen that $t_{\textit{ic}}$ remains constant at approximately $25.84\ {\textrm{ms}}$ when $\Delta T \lt 11\ {\textrm{K}}$ , before dropping sharply for higher undercooling. This demonstrates that, for $\Delta T \gt 11\ {\textrm{K}}$ , 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 $\Delta T \lt 11\,{\textrm{K}}$ . In the intermediate range $1 \ {\textrm{K}} \lt \Delta T \lt 11\ {\textrm{K}}$ , the spreading of the droplet ceases right after the inertial-capillary regime, as highlighted by the experiments at $T_i = -2.92\,^\circ {\textrm{C}}$ () and $T_i = -10.3\,^\circ {\textrm{C}}$ () in figure 2(a), where no motion occurs after $t_{\textit{ic}}$ . However, a second and slower dynamics systematically takes place when $\Delta T \lt 1\,{\textrm{K}}$ , as illustrated by experiments where $T_i = -0.06\,^\circ {\textrm{C}}$ () and $T_i = -0.64\,^\circ {\textrm{C}}$ 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. Reference Biance, Clanet and Quéré2004; Bonn et al. Reference Bonn, Eggers, Indekeu, Meunier and Rolley2009). As discussed in particular by De Gennes (Reference De Gennes1984), this behaviour is intimately related to wettability that governs the manner the contact line relaxes to equilibrium. In the 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 Reference Tanner1979; De Gennes Reference De Gennes1984). 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 Reference Voinov1976; de Gennes Reference de Gennes1985; Cox Reference Cox1986; Bonn et al. Reference Bonn, Eggers, Indekeu, Meunier and Rolley2009). 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:

(3.2) \begin{align} \frac {{\textrm{d}}\tilde {r}}{{\textrm{d}}\tilde {t}} &= \frac {1}{3 \zeta } \tan \theta (\tilde {t}) \big ( \cos \theta _e - \cos \theta (\tilde {t}) \big ), \end{align}

(3.3) \begin{align} \frac {\tilde {r}^3}{4 \sin ^3 \theta (\tilde {t})} & \big ( 2 + \cos ^3 \theta (\tilde {t}) - 3 \cos \theta (\tilde {t}) \big ) = 1, \end{align}

where $\tilde {t} = t/\tau _v$ , $\tilde {r} = r/R_v$ and $\tau _v = \eta R_v / \gamma$ the viscous-capillary time scale, 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 (3.3) is the conservation of volume for a spherical cap (Van de Velde et al. Reference Van de Velde, Fabre-Parras, Josserand, Duprat and Protière2023). 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 nanometre; it enables matching between the microscopic and macroscopic solutions describing the shape of the contact line (Bonn et al. Reference Bonn, Eggers, Indekeu, Meunier and Rolley2009). 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, (3.2)–(3.3) can straightforwardly be solved numerically, with $\theta _e$ being evaluated from (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).

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 $T_i=-0.06\,^\circ {\textrm{C}}$ () and $T_i=-0.64\,^\circ {\textrm{C}}$ () (in logarithmic representation). The solid lines are the corresponding predictions obtained by solving (3.2)–(3.3), which are extended as 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. (Reference Quetzeri-Santiago, Castrejón-Pita and Castrejón-Pita2020). The red circles are the apparent angles estimated using (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 $\Delta T \lt 1\,{\textrm{K}}$ .

The evolution of $\tilde {r}$ with $\tilde {t}$ is illustrated in figure 3(a) for two experiments where $T_i=-0.06\,^\circ {\textrm{C}}$ () and $T_i=-0.64\,^\circ {\textrm{C}}$ (). The two curves highlight the relaxation of the wetted radius up to its arrest value $r_m$ , which is reported as horizontal dashed lines. The corresponding predictions from (3.2)–(3.3), with $\zeta =30$ , are also reported as 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 $\Delta T \lt 1$ K. Indeed, although it is based on physical grounds and expected to be constant for the reasons indicated previously, 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 stabilisation of the spreading drop. This result reveals that the slower spreading phase observed when $\Delta T \lt 1\ {\textrm{K}}$ 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 (3.2)–(3.3)) but only to fluctuations in $V$ or in the thermophysical properties of water that are captured by the present modelling. 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 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 time scale as $\tau _{\textit{sol}}=h^2/D_{\textit{eff}}$ , where $h$ is the typical thickness of the water film at maximal spreading and $D_{\textit{eff}}$ is the effective diffusion coefficient of ice growth when water is in contact with it (Sarlin et al. Reference Sarlin, Grivet, Xu, Huerre, Séon and Josserand2024). Then, in the small $\Delta T$ limit, one has $D_{\textit{eff}} \propto \Delta T^2$ (see more detail in Appendix B). However, the inertial-capillary and viscous-capillary dynamics are respectively based on the time scales $\tau _c = \sqrt {ho {R_0}^3 / \gamma }$ and $\tau _v = \eta R_v/\gamma$ that are mostly independent of temperature. Therefore, when comparing these two typical times to the solidification time scale, it is clear that in the limit $\Delta T ightarrow 0$ , the spreading dynamics becomes unaffected by phase change as thermal equilibrium is approached. In 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 time scales).

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. First, 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. (Reference Quetzeri-Santiago, Castrejón-Pita and Castrejón-Pita2020) 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. Second, an independent estimate is obtained by using (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 $\Delta T \lt 11\ {\textrm{K}}$ , due to the reasons mentioned in § 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 $\Delta T \simeq 11\ {\textrm{K}}$ , 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 $\Delta T \lt 1\ {\textrm{K}}$ , for both measurement methods. The average value and standard deviation for this constant macroscopic contact angle are

(3.4) \begin{align} \theta _0 = 11.8^\circ \pm 1.2 ^\circ . \end{align}

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, which 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. Reference Bonn, Eggers, Indekeu, Meunier and Rolley2009); 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 Reference Knight1967; Thiévenaz et al. Reference Thiévenaz, Josserand and Séon2020; Grivet et al. Reference Grivet, Huerre, Séon, Duchemin and Josserand2024; Demmenie et al. Reference Demmenie, Reus, Kolpakov, Woutersen, Bonn and Shahidzadeh2023, Reference Demmenie, Gorin, Kolpakov, Smith, Kellay and Bonn2025), which were performed under different drop and ambient gas temperatures and relative humidity conditions. Owing to this observation, we infer that the the macroscopic contact angle we report is not significantly impacted by humidity and evaporative effects.