The macroscopic contact angle of water on ice

TL;DR

This study resolves the macroscopic wetting behavior of water on ice by combining controlled droplet deposition experiments with a capillary-dominated dynamic framework. It demonstrates a finite macroscopic advancing angle, $ heta_e \\approx 12^\\circ$, for near-equilibrium conditions ($\\Delta T \\lesssim 1$ K), and shows that the early spreading is inertial-capillary while the later relaxation is viscous-capillary and largely insensitive to phase change. By linking the measured angle to the ice–air interfacial tension via the Young-Dupré relation, it provides a robust estimate of $\\gamma_{iv} \\approx 105 \\pm 4$ mN/m and offers a benchmark for three-phase flow simulations in glaciology and related capillary problems with phase change. The findings clarify the mechanism of partial wetting at the macroscopic scale and highlight the separation of thermal and capillary effects near equilibrium, with implications for modeling ice morphogenesis and triple-line dynamics.

Abstract

Wettability quantifies the affinity of a liquid over a substrate, and determines whether the surface is repellent or not. When both the liquid and the solid phases are made of the same chemical substance and are at thermal equilibrium, complete wetting is expected in principle, as observed for instance with drops of molten metals spreading on their solid counterparts. However, this is not the case for water on ice. Although there is a growing consensus on the partial wetting of water on ice and several estimates available for the value of the associated contact angle, the question of whether these values correspond to the equilibrium angle without thermal effects is still open. In the present paper, we address this issue experimentally and demonstrate the existence of a macroscopic contact angle of water on ice using theoretical arguments. Indeed, when depositing water droplets on smooth ice layers with accurately controlled surface temperatures, we observe that spreading is unaffected by thermal effects and phase change close enough to the melting point. Whereas the short time \C{motion of the contact line} is driven by an inertial-capillary balance, the evolution towards equilibrium is described by a viscous-capillary dynamics and is therefore capillary - and not thermally - related. Moreover, we show that this contact angle remains constant for undercoolings below 1 K. This way, we show the existence of a non-zero equilibrium contact angle of water on ice, that it is very close to 12$^\circ$. We anticipate this key finding to significantly improve the understanding of capillary flows in the presence of phase change, which is especially useful in the context of ice morphogenesis and of glaciology, but also in the aim of developing numerical methods for resolving triple-line dynamics.

Figures (3)

• 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 ($T_i = -20.08$ ℃). 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 ($T_i = -10.32$ ℃) and (d) close to the melting point of water ($T_i = -0.06$ ℃). The same four stages (i)-(iv) are depicted. The initial radius of curvature $R_0$ of the pendent drop, the radius $r(t)$ of the wetted area, the drop volume $V$ and the arrest radius $r_m$ are highlighted in white.
• 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 \ref{['inertio_capillary_scaling']} 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) $t_{\rm{ic}}$ as a function of $\Delta T$. The solid line indicates $t_{\rm{ic}} = 25.84\ \rm{ms}$. The colors of the symbols denote the surface temperature $T_i$ of the ice, with darker shades corresponding to colder ice.
• 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 ($\bullet$) $T_i=-0.06$ ℃ and ($\bullet$) $T_i=-0.64$ ℃ (in logarithmic representation). The solid lines are the associated predictions obtained by solving equations \ref{['cox_voinov']}-\ref{['spherical_cap_volume']}, 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 2020_quetzeri-santiago. The red circles are the apparent angles estimated using equation \ref{['spherical_cap_volume']} with $r=r_m$. The solid line is the plateau $\theta_0=11.8^\circ$, which is the mean contact angle measured from all experiments where $\Delta T < 1\mathrm{K}$.