2.1. Materials and methods

The experiment consists of releasing a drop at room temperature, $T_{ {d}}$ , onto a liquid film at a temperature $T_{\! {f}}$ lower than the melting temperature $T_{ {m}}$ of the drop. We choose two non-miscible liquids. The drop is hexadecane with a molar mass $M_{ {d}}=$ 226.45 g mol^–1, a melting temperature $T_{ {m}}=18.1\,^\circ {\textrm{C}}$ , a liquid density $ho _{ {d}}$ = 743 kg m $^{-3}$ , a viscosity ${\mu} _{ {d}}$ = 3.8 mPa s, a surface tension $\gamma _{ {d}}$ = 27 mN m^–1, a specific heat capacity $C_{p, {d}} = 2.20$ kJ (kg K^–1), a thermal conductivity $\lambda _{ {d}} $ = 0.14 W mK^–1 and an enthalpy of solidification ${\mathcal{L}}=$ 236 kJ kg^–1. We also introduce the surface tension between the liquid phase and solid phase of hexadecane, $\sigma _{ls} = 0.0068$ J m $^{-2}$ (Oliver & Calvert Reference Oliver and Calvert1975). The liquid and thermal properties of hexadecane are taken at room temperature, $T_{ {d}} = 20\,^\circ {\textrm{C}}$ (Berry et al. Reference Berry, Josserand, Salonen and Boulogne2024). The liquid film is a brine constituted of 23.3 wt % NaCl in pure water. The film temperature $T_{\! {f}}$ is measured with a K-type thermocouple (Radiospare) and ranges from room temperature, $20\,^\circ {\textrm{C}}$ , to $-21\,^\circ {\textrm{C}}$ . The liquid and thermal properties of the brine are taken at a mean temperature of $0\,^\circ {\textrm{C}}$ , giving a density $ho _{\! {f}}=$ 1184 kg m^– $^3$ , a viscosity ${\mu} _{\! {f}}=$ 2.6 mPa s, a specific heat capacity $C_{p, {f}} = 3.3$ kJ (kg K^–1) and a thermal conductivity $\lambda _{\! {f}} =$ 0.55 W mK^–1. All the liquid properties are summarised in the Appendix and the variation of the properties with temperature is discussed in the supplementary materials of Berry et al. (Reference Berry, Josserand, Salonen and Boulogne2024).

The drops are produced with a syringe pump, out of an 18G needle, which provides a drop diameter of $2r_0 = 2.70 \pm 0.04$ mm, measured by image analysis. A drop is released at an initial height $h_0$ ranging from 5 cm to 50 cm, and the impact velocity is calculated as $v_0 = \sqrt {2g h_0}$ where $g$ is standard gravity.

The solid surface below the liquid film is a 4 mm thick glass slide on top of a silicon wafer. The glass slide has similar thermal properties as the salt solution and is therefore used to ensure some thermal continuity. The optical reflectivity of the wafer enhances the visualisation. The wafer and the glass slide are placed in an aluminium Petri dish and covered with the salt solution. We measured the film thickness with an optical device (Chromapoint, STIL). We choose to work with a film thickness of $h_{\! {f}} = 0.27 \pm 0.02$ mm, which gives a dimensionless liquid thickness $H = h_{\! {f}} /( 2 r_0) = 0.1$ . For such dimensionless thickness, the bath is defined as a liquid film in the literature (Cossali, Coghe & Marengo Reference Cossali, Coghe and Marengo1997; Tropea & Marengo Reference Tropea and Marengo1999; Vander Wal et al. Reference Vander Wal, Berger and Mozes2006a ; Motzkus, Gensdarmes & Géhin Reference Motzkus, Gensdarmes and Géhin2009).

The cooling system is composed of a Peltier module (Adaptive, RS) rated at 340 W. The heat exchange between the Peltier module and the Petri dish is ensured by a homemade receptacle of copper covering the bottom and the perimeter of the Petri dish. On the other side of the Peltier module, heat is extracted by a heat sink (Laird Technologies) in which a refrigerant liquid circulates at a temperature of $1\,^\circ {\textrm{C}}$ by a refrigerated bath (VWR, AP15R-40).

The scene is illuminated with an LED panel (HSC Backlight, Phlox) positioned at $45^\circ$ pointing toward the liquid film in front of a high-speed camera. The high-speed camera (FASTCAM Nova S9 1024 $\times$ 1024 px, Photron) records at 6000 f.p.s. from a 3/4 point of view to measure the projected radius over time $r(t)$ , to count the number of crystals $N_{ {c}}$ and to estimate the time $t_{ \textit{layer}}$ to form a frozen layer. We obtain a resolution of 48 pixels mm^–1, which is the limiting factor in our experiments. Nevertheless, as we will show in the next section, this resolution is sufficient to identify the crystals, typically 1 or 2 ms after the impact.

The temperature at the interface $T_{ {c}}$ is constant in the case of two semi-infinite media and is given by Boeker & Grondelle (Reference Boeker and Grondelle2011):

(2.1) \begin{equation} T_{ {c}} = \frac {T_{ {d}} + T_{\! {f}} (e_{\! {f}}/e_{ {d}})}{1+(e_{\! {f}}/e_{ {d}})}, \end{equation}

where we used the effusivity $e_i = \sqrt {\lambda _i ho _i C_{p,i}} = \lambda _i/\sqrt {D_i}$ with $i = {d}, {f}$ . The hypothesis of a semi-infinite phase for the drop is correct for time scales smaller than the heat diffusion time scale, which is of the order of magnitude of a second for our typical length scale, whereas the time scale of impact is tens of milliseconds. Therefore, we will characterise the thermal shock by $\Delta T_{ \textit{mc}} = T_{ {m}} - T_{ {c}}$ (Berry et al. Reference Berry, Josserand, Salonen and Boulogne2024).

2.2. Observations

Figure 1.

(a) Time series showing a hexadecane drop impinging a liquid bath at a velocity $v_0$ = 2.8 m s^–1 for two different thermal shocks $\Delta T_{ \textit{mc}} =\{8, 12.5\}\,^\circ {\textrm{C}}$ . Time evolution of the experimental measurements of (b) the area covered by crystals and (c) the number of crystals, associated with the time series presented in (a) with light blue for $\Delta T_{ \textit{mc}} =8\,^\circ {\textrm{C}}$ and dark blue for $\Delta T_{ \textit{mc}} =12.5\,^\circ {\textrm{C}}$ . The associated movies are provided in the supplementary material available at https://doi.org/10.1017/jfm.2025.10508, with an overlay of the crystals’ outlines used for measurements. The inset of (c) shows the final number of crystals for different thermal shocks at $v_0$ = 2.8 m s^–1.

Figure 1(a) shows typical experiments at a fixed impact velocity $v_0$ = 2.8 m s^–1 for two different thermal shocks. Upon impact, the drop forms a liquid crown analogous to Edgerton’s observations (Edgerton Reference Edgerton1954). Additionally, we notice grey clusters that we attribute to crystals appearing at the drop–film interface (Berry et al. Reference Berry, Josserand, Salonen and Boulogne2024). In time, these crystals grow and additional crystals appear as the drop expands radially. The time series also show that for a cooler film, i.e. a larger thermal shock $\Delta T_{ \textit{mc}}$ , fewer crystals are observed, which is counterintuitive since lower temperatures should enhance crystal nucleation!

To verify this observation, we repeated these experiments at different thermal shocks $\Delta T_{ \textit{mc}} = \{8, 11.2, 12.5, 14.5\}\,^\circ {\textrm{C}}$ , for which we measured the surface area of the crystals and counted the number of crystals. The number of crystals and their area are measured by manually drawing their outline on the images with the software ImageJ. We cannot obtain measurements before 0.5 ms due to the limited resolution of the camera, and curvature of the drop that causes shadows and optically enlarges the surfaces. Additionally, near the end of the coverage, it becomes difficult to identify individual crystals because they are very close to one another. Therefore, as the boundary between crystals is not clear enough, we stop measuring the covered surface before the number of crystals.

These measurements are reported in figures 1(b) and 1(c), where we plot, for two different thermal shocks, the time evolution of the surface area covered by the crystals $S_c$ and the number of crystals $N_c$ , respectively. For the sake of clarity, error bars are not plotted on these curves but we note that uncertainty for $S_{ {c}}$ and $N_{ {c}}$ would be of 10 % on each data point. We observe that increasing the thermal shock (dark blue points) leads to faster dynamics. In the inset of figure 1(c), more measurements of the final number of crystals are given, highlighting a decrease with thermal shock. For larger thermal shocks, crystals would appear earlier and grow faster, leading to the observation of fewer crystals. The purpose of the next section is to establish a model able to render these experimental findings and particularly to explain more quantitatively why the cooler the liquid, the fewer the number of crystals nucleated.

3.1. Dynamics of contact opening

In this section, we describe the impact of a drop on the liquid film in terms of contact surface area and time scale to reach maximum opening. Classically, drop impact on thin films can be characterised by a combination of Reynolds and Weber numbers (Cossali et al. Reference Cossali, Coghe and Marengo1997; Weiss & Yarin Reference Weiss and Yarin1999; Vander Wal et al. Reference Vander Wal, Berger and Mozes2006b ; Roisman, van Hinsberg & Tropea Reference Roisman, van Hinsberg and Tropea2008; Chen, Chen & Amirfazli Reference Chen, Chen and Amirfazli2017). The Reynolds number is defined as ${\textit{Re}} = 2{ho _{ {d}} v_0 r_0}/{{\mu} _{ {d}}}$ , with ${\mu} _{ {d}}$ the viscosity of the drop, and varies between 936 and 1872 in this study. The Weber number is defined as ${\textit{We}} = 2 ho _{ {d}} v_0^2 r_0 / \gamma _{ {d}}$ , with $\gamma _{ {d}}$ the surface tension of the drop, and varies between 145 and 578 in this study.

Figure 2.

(a) Time evolution of the drop contact area $S_{ {d}}(t) = \unicode{x03C0} r(t)^2$ upon impact on a liquid film, for different substrate temperatures and impact velocities. Each scatter curve represents one experiment and the black line represents (3.1). (b) Maximum contact area between the drop and the film and (c) time to reach the maximum surface as a function of the impact velocity, with each point representing an experiment. The different markers correspond to impact velocity. The descriptions of Roisman et al. (Reference Roisman, van Hinsberg and Tropea2008) adapted for thin films are plotted by solid black lines.

We measured through image analysis the dynamics of the surface area $S_{ {d}}(t)$ that corresponds to the contact between the drop and the liquid film as illustrated in figure 2(a). During the opening phase of contact, $S_{ {d}}(t)$ grows linearly at short times, which agrees with the literature (Weiss & Yarin Reference Weiss and Yarin1999; Roisman et al. Reference Roisman, van Hinsberg and Tropea2008) and quadratically to saturation, as shown in figure 2(a). For a more accurate description of the whole dynamic until the extension reaches the maximum surface area $ S_{ {m}}$ at a corresponding impact time of $\tau _{ {m}}$ , we choose to write the surface area of contact as

(3.1)

which is valid for $t \lt \tau _{ {m}}$ .

Roisman et al. (Reference Roisman, van Hinsberg and Tropea2008) proposed a description for the crown radius on miscible liquid films $H \in$ [0.5, 2], which is also used on immiscible liquid films $H \in$ [0.1, 0.5] by Bernard et al. (Reference Bernard, Baumgartner, Brenn, Planchette, Weigand and Lamanna2021). For simplicity, we choose to adapt their expressions for $H ightarrow$ 0, which gives dimensionless expressions as

(3.2a) \begin{align} {S_{ {m}}} / \big(\unicode{x03C0} r_0^2\big) &\sim \, \big(\beta \sqrt {H}/2\big)^2 {\textit{We}}, \end{align}

(3.2b) \begin{align} \tau _{ {m}} \frac {v_0}{2 r_0} &\sim \, (\beta {H}/6) {\textit{We}}, \end{align}

where $\beta \sim H^{-1/3}$ . Since we only vary the impact velocity, we have ${S_{ {m}}} \propto v_0^2$ and ${\tau _{ {m}}}\propto v_0$ , which is compared with our data in figure 2(b,c). This description is acceptable although the slope against velocity is slightly overestimated, which is not caused by the approximation $Hightarrow 0$ .

3.2. Crystal growth velocity

Crystal growth can be assumed to be limited by the reaction at the crystal–melt interface (Kirkpatrick Reference Kirkpatrick1975). To pass from the melt to the crystal, a molecule must go through an activated state before releasing energy. The free-energy difference between the melt and the activated state is denoted $\Delta G'$ and between the melt and the crystal $\Delta G_{ {c}}$ . We define a constant rate $\alpha$ , equivalent to a diffusion coefficient for transport across the melt–crystal interface, which is defined with the Stokes–Einstein relation as $\alpha = RT_{ {c}} / (3\unicode{x03C0} \mathcal{N}_a a_0 {\mu} _{ {d}})$ with $a_0$ the intermolecular distance, $R$ the gas constant and $ \mathcal{N}_a$ the Avogadro number (Kirkpatrick Reference Kirkpatrick1975). Based on the reaction-rate theory, the crystal growth velocity writes

(3.3)

By assuming that entropy and enthalpy differences are independent of temperature, $\Delta G_{ {c}}$ can be approximated as $M_{ {d}} {\mathcal{L}} \Delta T_{ \textit{mc}} /T_{ {m}}$ (Wagstaff Reference Wagstaff1968) with $M_{ {d}}$ the molar mass of the liquid. The small undercooling approximation ( $\Delta G_{ {c}} \ll RT_{ {c}}$ ) is often used to linearise the crystal growth velocity as

(3.4) \begin{equation} v_{ {c}} \simeq \kappa \Delta T_{ \textit{mc}}, \end{equation}

in which we introduce the kinetic undercooling coefficient $\kappa = M_{ {d}} {\mathcal{L}} / (3\unicode{x03C0} a_0^2 {\mu} _{ {d}} \mathcal{N}_a T_{ {m}})$ . We note that both expressions of crystal growth do not depend on the problem geometry and are valid for two- and three-dimensional growth (Kant et al. Reference Kant, Koldeweij, Harth, van Limbeek and Lohse2020). The mean free path in a liquid is assumed to be of the same order of magnitude as the intermolecular distance. Therefore, $a_0 \sim (M_{ {d}} /( ho _{ {d}} \mathcal{N}_a))^{1/3} \simeq 8 \times 10^{-10}$ m. The theoretical value of $\kappa$ from (3.4) gives $ 1.3 \times 10^{-2}$ (m s^–1) K^–1.

Additionally, de Ruiter et al. (Reference de Ruiter, Colinet, Brunet, Snoeijer and Gelderblom2017) and Koldeweij et al. (Reference Koldeweij, Kant, Harth, de Ruiter, Gelderblom, Snoeijer, Lohse and van Limbeek2021) studied crystal growth in hexadecane on solid substrates and they measured experimental values for $\kappa$ of $1.1 \times 10^{-2}$ and $ 0.45 \times 10^{-2}$ (m s^–1) K^–1, respectively. In a previous study on alkanes solidifying on a liquid substrate (Berry et al. Reference Berry, Josserand, Salonen and Boulogne2024), the value was measured at $0.6 \times 10^{-2}$ (m s^–1) K^–1. In the present study, the coefficient can be determined experimentally from the images as shown in figure 3(a) by measuring directly the growth velocity of individual crystals. In figure 3(b), we plot these measurements, each point representing a mean value of multiple crystals growing in one experiment, and a linear fit provides $\kappa = (0.65 \pm 0.20) \times 10^{-2}$ (m s^–1) K^–1.

While this small undercooling approximation operates at the boundary of its conventional validity range, it still provides meaningful insights. The expression of $v_{ {c}}$ will not be replaced in the rest of the model and the approximation will mainly be used to study the relation between the thermal shock and the number of crystals during the discussion, keeping in mind that the order of magnitude of $\kappa$ is $10^{-2}$ (m s^–1) K^–1 and that the crystal growth rate $v_{ {c}}$ is proportional to thermal shock $\Delta T_{ \textit{mc}}$ .

Figure 3.

(a) Time series illustrating the evolution and measurements of the crystals for $\Delta T_{ \textit{mc}}=11\,^\circ {\textrm{C}}$ . The edges of the crystals are indicated by a thin white line and some crystals are coloured for clarity. (b) Crystal growth velocity as a function of $\Delta T_{ \textit{mc}}$ . Each point represents the mean number of individual crystals growing in one experiment, as highlighted on (a). The black dashed line is a fit, giving $\kappa = (0.65 \pm 0.20) \times 10^{-2}$ (m s^–1) K^–1. The grey area indicates the effect of the uncertainty on $\kappa$ .

3.3. Number and surface area covered by crystals

The increase of the number of crystals over a duration ${\textrm {d}}t$ is assumed to be proportional to a nucleation rate $\dot n$ , which has the dimension of number of molecules per unit area per time, and to the available surface area $S_{ {d}}(t) - S_{ {c}}(t)$ , such that

(3.5)

The nucleation rate $\dot {n}$ is related to the number of crystals that develop beyond a critical size to become stable and is written as

(3.6)

with $A$ the attempt frequency per unit area and the nucleation activation energy $E_{{n}} = (16\unicode{x03C0} /3) \sigma _{ls}^3f(\theta ) \mathcal{N}_a / \Delta G_{ {c}}^2$ (Mullin Reference Mullin2001; Koldeweij et al. Reference Koldeweij, Kant, Harth, de Ruiter, Gelderblom, Snoeijer, Lohse and van Limbeek2021). We note that $f(\theta )$ is a function of the contact angle between the solid crystal and the substrate, and is accounting for heterogeneous nucleation in homogeneous activation energy $E_{hom}$ as $E_{n} = f(\theta) E_{hom}$ , i.e. a value of $f(\theta )$ close to one in an undercooled liquid is equivalent to homogeneous nucleation whereas a small value of $f(\theta )$ is associated with heterogeneous nucleation near the substrate (Mullin Reference Mullin2001). In our approach, we consider heterogeneous nucleation in the drop because the crystals appear on the interface due to the temperature gradient growing as a diffusive process in the drop. The temperature gradient perpendicular to the interface, induced by the cold substrate, dominates over the much more localised in-plane gradient near growing crystals. Consequently, we model nucleation as spatially uniform over the available surface, treating crystals as independent entities. This simplification may lead to an overestimation of the nucleation rate, particularly during the final stages of surface coverage.

We introduce the surface area $\sigma (t,t_0)$ of a single crystal, at a time $t$ , that appeared at $t_0\leqslant t$ :

(3.7) \begin{equation} \sigma (t,t_0) = \unicode{x03C0} v_{ {c}}^2 (t-t_0)^2, \end{equation}

where the crystal growth velocity is given by (3.4). The surface area $S_{ {c}}(t)$ covered by crystals is the result of the time evolution of each germ, described by (3.7), which appeared at a rate ${\textrm {d}}N_{ {c}}(t')$ :

(3.8) \begin{equation} S_{ {c}}(t) = \int ^{N_{ {c}}(t)}_{N_{ {c}}(t'=0)} \sigma (t,t') \, {\textrm {d}}N_{ {c}}(t'). \end{equation}

This covered surface area grows until it reaches the surface area of the drop at a time $t_{ \textit{end}}$ , i.e. $S_{ {c}}(t_{ \textit{end}}) = S_{ {d}}(t_{ \textit{end}})$ .

3.4. Dimensionless equations

To obtain dimensionless equations, we introduce the dimensionless time $\overline {t} = t/\tau _{ {m}}$ , the dimensionless surface $\overline {S_i} = S_i / S_{ {m}}$ , where $i=\{{ {c}}, {{d}}\}$ . Thus, (3.1) becomes $\overline {S_{ {d}}}(\overline {t}) = 1 - ( \overline {t} - 1 )^2$ for $\overline {t}\lt 1$ . We then obtain the dimensionless form of (3.5) on the number of crystals:

(3.9) \begin{equation} {\textrm {d}}\overline {N_{ {c}}}(\overline {t}) = \big(1 - \big( \overline {t} - 1 \big)^2 - \overline {S_{ {c}}}(\overline {t})\big) {\textrm {d}}\overline {t}, \end{equation}

with the dimensionless number of crystals written as $\overline {N_{ {c}}} = N_{ {c}} / N_{ {m}}$ , with $N_{ {m}} = \dot {n} \tau _{ {m}} S_{ {m}}$ .

With (3.8) on the surface area covered by crystals, we have

(3.10) \begin{equation} \overline {S_{ {c}}}(\overline {t}) = \chi ^3 \int ^{\overline {t}}_{0} (\overline {t}-\overline {t}')^2 \big( 1 - \big( \overline {t}' - 1 \big)^2 - \overline {S_{ {c}}}(\overline {t}')\big) {\textrm {d}}\overline {t}', \end{equation}

with $\chi ^3 = \dot {n} \tau _{ {m}} \unicode{x03C0} v_c^2 \tau _{ {m}}^2$ . The parameter $\chi ^3$ is the nucleation rate multiplied by impact time and by $\unicode{x03C0} v_c^2 \tau _{ {m}}^2$ , which is the surface covered by one crystal that grew between $t=0$ and $\tau _{ {m}}$ .

To obtain (3.9) and (3.10) in a form more appropriate for a numerical resolution, we use the Leibniz integral rule, which gives

(3.11a) \begin{align} \frac {{\textrm {d}}\overline {S_{ {c}}}}{{\textrm {d}}\overline {t}} &= 2 \chi ^3 \int ^{\overline {t}}_{0} (\overline {t}-\overline {t}') \big( 1 - \big( \overline {t}' - 1 \big)^2 - \overline {S_{ {c}}}(\overline {t}')\big) {\textrm {d}}\overline {t}', \end{align}

(3.11b) \begin{align} \frac {{\textrm {d}}\overline {N_{ {c}}}}{{\textrm {d}}\overline {t}} &= 1 - \big( \overline {t} - 1 \big)^2 - \overline {S_{ {c}}}(\overline {t}) . \end{align}

These equations are complemented by the initial condition $\overline {S_{ {c}}}(0) = \overline {N_{ {c}}}(0) = 0$ . With (3.1), the end condition $S_{ {c}}(t_{ \textit{end}}) = S_{ {d}}(t_{ \textit{end}})$ reads

(3.12) \begin{equation} \overline {S_{ {c}}}(\overline {t}_{ \textit{end}}) = 1 - \big(\overline {t}_{ \textit{end}} - 1\big)^2 \qquad {\textrm {with}}\quad \overline {t}_{ \textit{end}}\lt 1. \end{equation}

If $\overline {t}_{ \textit{end}} \gt 1$ , crystals have not covered the drop during the opening phase of the impact and the model ceased to be valid due to the description adopted for the hydrodynamics.

In the next section, we solve these equations, discuss the results of the model, and compare the predictions with the experimental measurements.

Figure 4.

Variation of the solutions of (3.11) with $\chi$ . (a) Dynamics of the surface area covered by the crystals. The black line represents the dimensionless evolution of the contact area. (b) Time evolution of the number of crystals.