3.1 Typical time series

Figure 2 shows the time series of three drops impacting at similar Weber number, $We=\unicode[STIX]{x1D70C}D_{0}U^{2}/\unicode[STIX]{x1D6FE}$, onto a sapphire substrate with temperatures of (a) $T_{s}=249\,^{\circ }\text{C}$ (isothermal impact), (b) $150\,^{\circ }\text{C}$ and (c) $39\,^{\circ }\text{C}$. The top row of images shows the drop impact event in side view, and the bottom row shows images in bottom view. Before impact (panel (i) in each of figure 2(a–c)) the drop is not visible in bottom view since there is no contact with the substrate.

Figure 2. Time series of drops of similar Weber number impacting onto sapphire substrates of different temperatures: (a) $We=281$, $T_{s}=249\,^{\circ }\text{C}$; (b) $We=267$, $T_{s}=150\,^{\circ }\text{C}$; and (c) $We=278$, $T_{s}=39\,^{\circ }\text{C}$. The top row in each panel shows side-view images, and the bottom row shows bottom-view images. The scale bar represents 2 mm. The drop is shown just before impact (i) and spreads over the surface in time (ii) and (iii) until it reaches its maximum spreading (iv) and retracts or remains solidified at the substrate (v). The corresponding movies can be found in the supplementary material available at https://doi.org/10.1017/jfm.2019.886.

Figure 3. Microscope images of the solidified splat of a tin drop after impact onto a sapphire substrate. (a) For $We=135$ and $T_{s}\approx 50\,^{\circ }\text{C}$. The central bubble (black spot) is surrounded by a lighter defect-free zone. The border of this zone is marked by the dashed white arc. Further outwards, a pattern of circular ridges is observed. (b) Enlargement of the area marked by the red square in panel (a) to illustrate the border of the defect-free zone $D_{d}/D_{0}\approx 0.6$ (dashed white arc). (c) For $We=448$ and $T_{s}=150\,^{\circ }\text{C}$. Close-up of the splat showing that solidified radial ligaments consist of a sequence of air ridges and can be traced back towards the centre of the splat.

During isothermal impact (figure 2a, supplementary movie 1) the drop spreads smoothly over the substrate at early times (panel a.ii). Over time, ligaments become visible (panel a.iii), merge and grow until the drop reaches its maximum spreading (panel a.iv). As ligaments continue to appear, split and merge, their total number changes over time (Thoroddsen & Sakakibara Reference Thoroddsen and Sakakibara1998). After reaching its maximum spreading, the drop retracts and partially bounces off the substrate while small droplets detach (panel a.v). This bounce is caused by the repelling nature of the sapphire substrate, which has a contact angle with tin of approximately $130^{\circ }$ (as measured by side-view imaging of a deposited drop).

In figure 2(b) and supplementary movie 2 the substrate temperature is decreased to $T_{s}=150\,^{\circ }\text{C}$, which is below the melting temperature of tin. While the side view initially looks identical to the isothermal case (cf. panels a.ii–iv and b.ii–iv), the bottom view reveals striking differences: a grey-scale pattern appears that remains unchanged as the drop spreads and forms a clear mark of the solidification taking place at the interface with the substrate. Directly after impact a grey spot is visible at the impact centre of the drop (red arrow in panel b.ii; see also the close-up presented in figure 3a) surrounded by a lighter zone and grey radially outward-pointing stripes. This central grey spot is caused by the entrapment of an air bubble during first contact between the drop and the substrate (Chandra & Avedisian Reference Chandra and Avedisian1991; van Dam & Le Clerc Reference van Dam and Le Clerc2004). In the isothermal impact experiments, this air bubble could not be observed long enough in the bottom-view images, presumably due to pinch-off and detachment (Lee et al. Reference Lee, Weon, Je and Fezzaa2012). The light grey zone surrounding the bubble (red circle in panel b.iii; see also figure 3b) has no texture. A similar defect-free zone $D_{d}$ has been observed by Ruiter et al. (Reference Ruiter, Soto and Varanasi2017a). These authors attributed the existence of this zone to a delay in the solidification caused by the vertical flow of hot liquid in combination with the diverging contact line velocity.

When the drop spreads further (panel b.iii), ligaments evolve from its edge. These ligaments immediately solidify on the substrate and elongate as time progresses while their number remains unchanged, in contrast to the isothermal case. The total number of ligaments present at maximum spread is smaller than for the isothermal impact (cf. panels a.iv and b.iv), as was also observed by Aziz & Chandra (Reference Aziz and Chandra2000). In fact, a close inspection shows that the ligaments in the solidified splat can be traced all the way back to the grey stripe pattern that was already visible at an early stage (panel b.ii and figure 3c). The imprint of the spreading drop also shows dark grey spots. At these locations, the spreading of liquid tin is hindered by imperfections at the substrate, which results in the entrapment of air pockets. Owing to solidification of the surrounding tin, most of these air pockets remain visible at the final splat (panel b.v). Solidification reduces the overall spreading velocity of the drop and the maximum spread as compared to isothermal impact. After the maximum expansion is reached, the bulk of the drop solidifies (panel b.v). A retraction phase is therefore not observed and the bottom-view image no longer changes over time.

When the substrate temperature is decreased further ($T_{s}=39\,^{\circ }\text{C}$, figure 2c and supplementary movie 3), we again observe a central air bubble (red arrow in panel c.ii), a defect-free zone (red circle in panel c.iii) and ligaments that form from a radial stripe pattern (panel c.iv). The number of ligaments is again smaller than for isothermal impact (figure 2a), but comparable to the experiment shown in figure 2(b). The maximum spreading diameter (panel c.iv) of the drop is smaller than in figure 2(b). Moreover, maximum spreading is followed by an inward-moving front (red arrow in panel c.v; see also supplementary movie). This self-peeling of the splat results from a build-up of bending stress due to shrinkage of the tin as it cools down (Ruiter et al. Reference Ruiter, Soto and Varanasi2017a). Indeed, the splats of the drops that show this self-peeling are much easier to remove from the substrate after the experiment than those that do not peel off. For impacted water drops, such build-up of bending stress due to solidification has been shown to result in dramatic cracking and fragmentation events (Ghabache et al. Reference Ghabache, Josserand and Séon2016).

3.2 Microscopy of the splat

Splats that self-peel could be removed from the substrate for evaluation under the microscope. An example of a microscope image for an experiment with $We=135$ is shown in figure 3(a). The black spot in the centre of the splat again corresponds to the central air bubble that is trapped in the solidified liquid. The dashed white arc marks the border of the defect-free zone. In the magnified view in figure 3(b), the defect-free zone and its border can be seen in more detail. Outside this zone a pattern of circular rings is observed, similar to what has been reported by Ruiter et al. (Reference Ruiter, Soto and Varanasi2017a). This pattern results from the entrapment of air during pinning events at the contact line: when solidification arrests the contact line, bulk liquid tin continues to spread over the substrate and forms a new contact. Each time a contact is formed, air gets entrapped, which leads to the formation of circular ridges in the splat. In this low-Weber-number experiment, radial stripes and ligaments are not observed. The drop edge, however, does show undulations with a clear wavelength from which ligaments can evolve once the Weber number is increased.

At larger $We$, we observe both circular ridge structures, radial stripes and ligaments (see figure 3c). One clearly observes that the solidified ligaments consist of a sequence of ridges. This observation suggests that ligaments formed by bulk liquid rapidly flow outwards over previously solidified undulations of tin. Thereby, a pattern of radial stripes is formed that can be traced all the way back to the defect-free zone in the centre of the drop. Hence, the early-time solidification of undulations of the spreading drop determines the fingering pattern observed in the final solidified splat. Note that such structures were not present in the experiments by Ruiter et al. (Reference Ruiter, Soto and Varanasi2017a), who considered low-Weber-number impacts only.

4.1 Experiments

We now analyse the spreading dynamics and determine the maximum spreading as a function of $U$ and $T_{s}$. At each point in time, the equivalent diameter $D$ of the area occupied by the spreading drop is determined from the bottom-view images. Figure 4(a) shows the typical dimensionless spreading diameter $\unicode[STIX]{x1D709}=D/D_{0}$ over time $t/(D_{0}/U)$ for three substrate temperatures at a similar impact velocity: $T_{s}=249\,^{\circ }\text{C}$ (which is above $T_{m}$, red dots), $150\,^{\circ }\text{C}$ (below $T_{m}$, yellow dots) and $39\,^{\circ }\text{C}$ (blue dots). For each curve, the moment in time when the drop diameter reaches $\unicode[STIX]{x1D709}=0.9\unicode[STIX]{x1D709}_{max}$ is marked by an arrow and labelled as $t_{90}$. As shown in figure 4(b), $t_{90}/(D_{0}/U)$ is of order unity, though it increases slightly with increasing substrate temperature and increasing Péclet number.

When the substrate temperature is above the melting point, no solidification takes place (red dots in figure 4a). The drop spreads to a maximum diameter $\unicode[STIX]{x1D709}_{max}=D_{max}/D_{0}$, retracts and bounces off the substrate. We refer to this case as ‘isothermal spreading’.

At early times ($t\ll D_{0}/U$) all drops follow the isothermal spreading curve. As time progresses, the spreading of drops on the colder ($150\,^{\circ }\text{C}$ and $39\,^{\circ }\text{C}$) substrates slows down and comes to rest before the retraction phase is reached. Hence, $\unicode[STIX]{x1D709}$ remains at its maximum value $\unicode[STIX]{x1D709}_{max}$. The colder the substrate, the sooner the spreading deviates from isothermal spreading, the smaller $t_{90}$ and the lower the value of $\unicode[STIX]{x1D709}_{max}$ achieved.

Figure 4. (a) Drop spreading factor $\unicode[STIX]{x1D709}=D/D_{0}$ as a function of time $t/(D_{0}/U)$ for different substrate temperatures at similar impact velocity and size: $T_{s}=249\,^{\circ }\text{C}$, $U=3.9~\text{m}~\text{s}^{-1}$, $D_{0}=1.6~\text{mm}$ (red dots); $T_{s}=150\,^{\circ }\text{C}$, $U=3.5~\text{m}~\text{s}^{-1}$, $D_{0}=1.7~\text{mm}$ (yellow dots); and $T_{s}=39\,^{\circ }\text{C}$, $U=3.5~\text{m}~\text{s}^{-1}$, $D_{0}=1.7~\text{mm}$ (blue dots). The arrows mark the time $t_{90}$ at which the spreading diameter reaches 90 % of its maximum value. (b) Time $t_{90}/(D_{0}/U)$ as a function of $T_{s}$ for different values of the Péclet number $Pe=D_{0}U/\unicode[STIX]{x1D6FC}$ (colour bar).

The maximum spreading diameter reached by the drops is extracted from the curves in figure 4(a). For isothermal spreading, Laan et al. (Reference Laan, de Bruin, Bartolo, Josserand and Bonn2014) proposed the scaling law

(4.1)$$\begin{eqnarray}\unicode[STIX]{x1D709}_{max}\sim \frac{P^{1/2}}{A+P^{1/2}}Re^{1/5},\end{eqnarray}$$

where $A=1.24$ is a fitting constant and impact parameter $P=We\,Re^{-2/5}$, with $Re=UD_{0}/\unicode[STIX]{x1D708}$ the Reynolds number. In a large proportion of our experiments, $P=O(1)$, which implies that we are at the cross-over between the capillary and viscous spreading regimes and that we should use the full Padé function to describe our data (Laan et al. Reference Laan, de Bruin, Bartolo, Josserand and Bonn2014). Clearly, given the limited range of data, one could also fit $\unicode[STIX]{x1D709}_{max}\sim Re^{1/5}$, but for consistency we retain (4.1). In figure 5 we show the maximum dimensionless spreading $\unicode[STIX]{x1D709}_{max}$ as a function of impact parameter $P$ for different substrate temperatures, where we interpret $P$ as the dimensionless impact velocity (control parameter of the experiment) and $\unicode[STIX]{x1D709}_{max}$ as the response of the system. For isothermal impact, our data agree well with (4.1) when a prefactor of 0.9 is used. This prefactor is somewhat smaller than the prefactor of 1.0 obtained by Laan et al. (Reference Laan, de Bruin, Bartolo, Josserand and Bonn2014). We attribute this small discrepancy to the fact that we determine the spreading diameter from the equivalent area covered by the drop in the bottom-view images, thereby correcting for ligament formation, whereas Laan et al. (Reference Laan, de Bruin, Bartolo, Josserand and Bonn2014) used side-view images.

For substrate temperatures below $T_{m}$, the maximum spreading is reduced, in particular for larger $P$ and lower $T_{s}$ (see figure 5). For $T_{s}\approx 200\,^{\circ }\text{C}$, which is just below $T_{m}$, the maximum spreading is close to its isothermal value and only deviates a little for impact parameters larger than $P\approx 6$. By contrast, for a drop with $T_{s}\approx 50\,^{\circ }\text{C}$ and $P=10$, the maximum spreading decreases from $\unicode[STIX]{x1D709}_{max}=5.2$ (isothermal) to $\unicode[STIX]{x1D709}_{max}=3.3$, which is a reduction of 37 %.

Figure 5. Maximum spreading $\unicode[STIX]{x1D709}_{max}$ as a function of impact parameter $P=We\,Re^{-2/5}$ for different substrate temperatures: $T_{s}\approx 250\,^{\circ }\text{C}$ (red dots), $T_{s}\approx 200\,^{\circ }\text{C}$ (orange dots), $T_{s}\approx 150\,^{\circ }\text{C}$ (yellow dots), $T_{s}\approx 100\,^{\circ }\text{C}$ (light blue dots) and $T_{s}\approx 50\,^{\circ }\text{C}$ (dark blue dots). The dashed red line corresponds to the model (4.1) for isothermal spreading by Laan et al. (Reference Laan, de Bruin, Bartolo, Josserand and Bonn2014) with a prefactor of 0.9.

4.2 Model for solidification-limited spreading

To explain the behaviour observed in figure 5, we derive a scaling relation for drop spreading in a regime where solidification is the limiting factor. Isothermal spreading dynamics upon impact has been modelled successfully by Eggers et al. (Reference Eggers, Fontelos, Josserand and Zaleski2010). Inspired by this model, Laan et al. (Reference Laan, de Bruin, Bartolo, Josserand and Bonn2014) derived the expression (4.1) for the maximum spreading. Eggers et al. (Reference Eggers, Fontelos, Josserand and Zaleski2010) assumed that spreading arrests when the thickness of the viscous boundary layer that grows from the substrate equals the height of the drop. Similarly, here we assume that, during solidification-limited spreading, the drop arrests when the solidified layer that grows from the cold substrate reaches the drop height, as shown schematically in figure 6. Note that until solidification takes place, the increase in kinematic viscosity of liquid tin due to its cooling is less than $5\,\%$ (Assael, Kalyva & Antoniadis Reference Assael, Kalyva and Antoniadis2010) and therefore too small to account for the observed change in the drop spreading.

Figure 6. Schematic view of the growth of the solidified layer inside the spreading drop. The drop is assumed to have a pancake shape (in red) with height $h(t)$ and diameter $D(t)$. A solidified layer with thickness $s(t)$ starts growing from the moment the drop contacts the substrate. Arrest occurs when the solidified layer reaches the pancake thickness and bulk liquid can no longer overflow the contact line: $s(t)=h(t)$. Assuming heat transfer in the radial direction $r$ is negligible, one obtains a square-root dependence of the solidified layer on $r$ (see text).

To find the moment of arrest, we first describe the spreading dynamics of the drop. In our experiments, arrest occurs at early times $t_{a}\sim D_{0}/U$ during the expansion phase (see figure 4b). In this phase, the rim still has a negligible mass and the drop shape can be approximated well by a pancake of uniform thickness $h(t)$ (Eggers et al. Reference Eggers, Fontelos, Josserand and Zaleski2010). From mass conservation, it then follows that the pancake thickness scales as

(4.2)$$\begin{eqnarray}\frac{h}{D_{0}}\sim \left(\frac{D_{0}}{D}\right)^{2}.\end{eqnarray}$$

The next step is to find an expression for $D(t)$. Explicit scaling relations that cover the entire expansion phase do not exist, as the spreading diameter is the result of a complex interplay between inertial, capillary and viscous forces and eventually also wettability of the substrate (Rioboo, Marengo & Tropea Reference Rioboo, Marengo and Tropea2002; Roisman, Berberovic & Tropea Reference Roisman, Berberovic and Tropea2009; Eggers et al. Reference Eggers, Fontelos, Josserand and Zaleski2010). We therefore look for a scaling relation that applies solely to the regime of interest $t\sim D_{0}/U$. At this time, viscous effects are confined to a tiny boundary layer of thickness $\unicode[STIX]{x1D6FF}/D_{0}\sim Re^{-1/2}\sim 10^{-2}$ and can be neglected (Riboux & Gordillo Reference Riboux and Gordillo2014). Moreover, as $t_{a}/t_{c}\sim We^{-1/2}\sim 10^{-1}$, with $t_{c}=\sqrt{\unicode[STIX]{x1D70C}D_{0}^{3}/\unicode[STIX]{x1D6FE}}$ the capillary time scale, we also neglect the influence of surface tension on the spreading dynamics. Since at $t_{a}$ we have $h\ll D$, we describe the pancake dynamics in the thin-film approximation. The mass and momentum conservation equations then read

(4.3a)$$\begin{eqnarray}\displaystyle & \displaystyle r\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}t}+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}(ruh)=0, & \displaystyle\end{eqnarray}$$

(4.3b)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}t}+u\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}r}=0, & \displaystyle\end{eqnarray}$$

$r$

$u$

$u=r{\dot{D}}/D$

$\ddot{D}=0$

(4.4)$$\begin{eqnarray}D\sim Ut.\end{eqnarray}$$

From (4.2) we then find the spreading dynamics of the pancake for $t\sim D_{0}/U$ to be given by

(4.5)$$\begin{eqnarray}\frac{h}{D_{0}}\sim \left(\frac{D_{0}}{Ut}\right)^{2}.\end{eqnarray}$$

Second, we model the growth of the solidified layer from the cold substrate. Here we employ a one-dimensional heat transfer model similar to that of Pasandideh-Fard et al. (Reference Pasandideh-Fard, Bhola, Chandra and Mostaghimi1998) and Aziz & Chandra (Reference Aziz and Chandra2000) based on equilibrium solidification, which is justified in the regime under study (Fedorchenko & Wang Reference Fedorchenko and Wang2007). The model neglects heat transfer in the radial direction and therefore strictly applies to the dynamics of the solidification front $s(t)$ in the centre of the drop, as illustrated in figure 6. As the drop and its surroundings are kept at approximately $T_{m}$, the heat transfer from the solidification front occurs solely through the solidified layer. In our experiments, the Stefan number of the solid

(4.6)$$\begin{eqnarray}Ste_{s}=c_{p,s}\frac{T_{m}-T_{s}}{L},\end{eqnarray}$$

is typically small ($Ste_{s}<1$) and therefore the heat transfer is assumed to be quasi-steady. The temperature dependence of the thermal properties of solidified tin is neglected and as reference the values at room temperature are used (specific heat $c_{p,s}=228~\text{J}~\text{kg}^{-1}~\text{K}^{-1}$, density $\unicode[STIX]{x1D70C}_{s}=7310~\text{kg}~\text{m}^{-3}$ and thermal conductivity $k_{s}=67~\text{W}~\text{m}^{-1}~\text{K}^{-1}$). Furthermore, the contact resistance between the sapphire and the tin is neglected (Aziz & Chandra Reference Aziz and Chandra2000), which can be justified thanks to the smooth sapphire substrates used in our experiment. The sapphire substrate will, in general, not be isothermal. The interface temperature at the contact is given by the transient convective heat transfer problem between the spreading solidifying tin drop and the conductive sapphire, and lies in between the temperature of the liquid drop and that of the substrate. To avoid this complexity, we use the substrate temperature $T_{s}$, which is a known experimental input parameter, as a reference temperature in our scaling analysis. The dynamics of the solidification front is then given by (Bejan Reference Bejan1993, pp. 185–188)

(4.7)$$\begin{eqnarray}\unicode[STIX]{x1D70C}_{s}L\frac{\text{d}s}{\text{d}t}=k_{s}\frac{T_{m}-T_{s}}{s},\end{eqnarray}$$

with $L$ the latent heat of solidification. From (4.7) we find the thickness of the solidified layer to scale as

(4.8)$$\begin{eqnarray}\frac{s}{D_{0}}\sim \left(\unicode[STIX]{x1D6FD}\tilde{t}\frac{Ste_{s}}{Pe}\right)^{1/2},\end{eqnarray}$$

with $\tilde{t}=t/(D_{0}/U)$, $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D6FC}_{s}/\unicode[STIX]{x1D6FC}=2.1$ the ratio of solid and liquid thermal diffusivities and $Pe=UD_{0}/\unicode[STIX]{x1D6FC}$ the Péclet number. Solidification at each radial distance $r$ from the impact centre starts as soon as the liquid contacts the substrate. Neglecting heat transfer in the radial direction and using (4.4) and (4.8), we find the radial profile of the solidified layer to scale as $s(r,t)\sim [D_{0}(\frac{1}{2}D(t)-r)]^{1/2}$, as illustrated in figure 6.

We assume that the spreading arrests as soon as there is no bulk liquid left to overflow the solidified liquid at the contact line of the drop. In terms of our scaling model, this condition translates to $h(t_{a})=s(t_{a})$. Combining (4.5) and (4.8) one finds for the time of arrest $t_{a}/(D_{0}/U)\sim (Pe/\unicode[STIX]{x1D6FD}Ste_{s})^{1/5}$. This scaling of $t_{a}$ motivated us to show the dependence of $t_{90}$ on the impact parameters in figure 4(b) using $Pe$. Combining the scaling for $t_{a}$ with $D_{max}\sim Ut_{a}$ one obtains

(4.9)$$\begin{eqnarray}\unicode[STIX]{x1D709}_{max}\sim \left(\frac{Pe}{\unicode[STIX]{x1D6FD}Ste_{s}}\right)^{1/5}.\end{eqnarray}$$

In the regime where $P\gg 1$ and hence the isothermal data follow the scaling $\unicode[STIX]{x1D709}_{max}\sim Re^{1/5}$, relation (4.9) can be obtained straightforwardly: replacement of the kinematic viscosity by the diffusion coefficient of the solidification front propagation $\unicode[STIX]{x1D6FC}_{s}Ste_{s}$ directly results in (4.9).

The scaling relation (4.9) is valid as long as the spreading is solidification-limited, which requires that the solidified layer grows faster than the viscous boundary layer and hence $\unicode[STIX]{x1D6FC}_{s}Ste_{s}>\unicode[STIX]{x1D708}$. In addition, surface tension forces should have a negligible influence on the maximum spreading, which requires $t_{a}<t_{c}$ or $Pe/\unicode[STIX]{x1D6FD}Ste_{s}<We^{5/2}$. When these conditions are not met and longer spreading times are considered, scaling (4.5) is no longer valid and the full spreading dynamics of the drop needs to be taken into account.

Figure 7. (a) Maximum spreading $\unicode[STIX]{x1D709}_{max}$ as a function of $Pe$ for different $Ste_{s}$: $Ste_{s}\approx -0.070$ (corresponding to $T_{s}\approx 250\,^{\circ }\text{C}$, red dots), $Ste_{s}\approx 0.12$ ($T_{s}\approx 200\,^{\circ }\text{C}$, orange dots), $Ste_{s}\approx 0.32$ ($T_{s}\approx 150\,^{\circ }\text{C}$, yellow dots), $Ste_{s}\approx 0.51$ ($T_{s}\approx 100\,^{\circ }\text{C}$, light blue dots) and $Ste_{s}\approx 0.70$ ($T_{s}\approx 50\,^{\circ }\text{C}$, dark blue dots). The solid lines correspond to scaling (4.9) for solidification-limited spreading with a prefactor of unity. The dashed line is the isothermal spreading (4.1) rewritten in terms of $Pe$ with prefactor 0.9. (b) When rescaled in terms of $(Pe/\unicode[STIX]{x1D6FD}Ste_{s})^{1/5}$ for $Ste_{s}>0$, all data collapse onto a single curve given by (4.9) with a prefactor of unity (solid line). The data points marked by crosses are not solidification-limited and therefore do not follow the master curve (see text).

In figure 7, scaling model (4.9) is compared to the experimental data. Figure 7(a) shows the same data as in figure 5 but now plotted in terms of $Pe$ for different values of $Ste_{s}$. For $Ste_{s}>0$ scaling relation (4.9) is shown with a prefactor of unity (solid lines). In addition, we also show the isothermal scaling (4.1) with prefactor 0.9 (red dashed line). Note that by rescaling (4.1) in terms of $Pe$, we introduce an explicit $D_{0}$ dependence, which results in a somewhat poorer collapse of the isothermal experimental data. Scaling (4.9) is in good agreement with the experimental data. As expected for smaller $Pe$, a deviation is observed, in particular, at smaller $Ste_{s}$ (higher substrate temperatures). For these experiments, equation (4.1) predicts a smaller spreading than (4.9) and hence solidification is not the rate-limiting factor, but viscous and/or surface tension effects dominate such that the data actually follow (4.1).

To further validate (4.9) we plot all data with $Ste_{s}>0$ rescaled in terms of $(Pe/\unicode[STIX]{x1D6FD}Ste_{s})^{1/5}$ as shown in figure 7(b). The experiments where solidification is not rate-limiting are marked by crosses. For each $Ste_{s}$, the data corresponding to solidification-limited spreading now collapse onto a single master curve given by (4.9) with a prefactor of unity.

5.1 Experiments

In figure 2 we observed that solidification has a strong influence on both the development of the ligaments and their final number. To quantify these observations, we manually count the number of ligaments $N$ as a function of the Weber number for different substrate temperatures, as shown in figure 8. For isothermal impact, ligaments appear, split and merge as time progresses (see also § 3). To find $N$, we therefore count the number of ligaments at the moment the drop reaches its maximum expansion, before retraction of the rim causes merger. This ambiguity in the definition of $N$ does not appear when the drop solidifies during impact and the number of ligaments is constant.

As is clear from figure 8, no ligaments are observed for $We\lesssim 150$. For such small $We$, the impact is not violent enough and therefore the rim retracts before ligaments become apparent. For larger Weber numbers, $N$ increases with increasing $We$ and is spread in two clouds: isothermal impacts ($T_{s}>T_{m}$, red data points) and solidification-limited impacts ($T_{s}<T_{m}$, yellow and blue data points). The orange data points correspond to experiments on substrates close to $T_{m}$ and appear in both clouds.

Even though each cloud has a considerable spread in $N$, we systematically observe that solidification leads to fewer ligaments. For example, $N$ (almost) halves when we compare the impact onto the coldest substrate ($T_{s}=50\,^{\circ }\text{C}$) to isothermal impact ($T_{s}=250\,^{\circ }\text{C}$).

Figure 8. Number of ligaments $N$ as a function of $We$ for different $Ste_{s}$: $Ste_{s}\approx -0.070$ (corresponding to $T_{s}\approx 250\,^{\circ }\text{C}$, red dots), $Ste_{s}\approx 0.12$ ($T_{s}\approx 200\,^{\circ }\text{C}$, orange dots), $Ste_{s}\approx 0.32$ ($T_{s}\approx 150\,^{\circ }\text{C}$, yellow dots), $Ste_{s}\approx 0.51$ ($T_{s}\approx 100\,^{\circ }\text{C}$, light blue dots) and $Ste_{s}\approx 0.70$ ($T_{s}\approx 50\,^{\circ }\text{C}$, dark blue dots). The red dashed line shows the number of ligaments for isothermal impact as predicted by (5.1) with a prefactor of 4.3. For $We\lesssim 150$ no ligaments are observed.

5.2 Qualitative explanation

To predict the number of ligaments formed during isothermal impact, several models based on a Rayleigh–Taylor instability of the decelerating rim have been proposed (see e.g. Allen Reference Allen1975; Marmanis & Thoroddsen Reference Marmanis and Thoroddsen1996; Aziz & Chandra Reference Aziz and Chandra2000; Kim, Feng & Chun Reference Kim, Feng and Chun2000; Villermaux & Bossa Reference Villermaux and Bossa2011). In these models, complicated effects such as the splitting and merging of fingers (Thoroddsen & Sakakibara Reference Thoroddsen and Sakakibara1998) and the influence of ambient pressure (Xu, Zhang & Nagel Reference Xu, Zhang and Nagel2005; Xu Reference Xu2007) or surface roughness (Range & Feuillebois Reference Range and Feuillebois1998) have been neglected. Despite these shortcomings, we now derive a modified version of the model by Aziz & Chandra (Reference Aziz and Chandra2000) as a ‘guide to the eye’ to describe the isothermal data shown in figure 8.

A Rayleigh–Taylor instability of the rim gives rise to a preferred wavenumber that scales as $k\sim \sqrt{\unicode[STIX]{x1D70C}(-\ddot{D})/\unicode[STIX]{x1D6FE}}$, where $\ddot{D}$ is the acceleration of the rim. The number of ligaments is then given by $N\sim kD_{\ell }$, where $D_{\ell }$ is the spreading diameter at the moment of ligament expulsion. For isothermal impact, ligaments are expelled from the rim close to the maximum drop expansion, such that $D_{\ell }\sim D_{max}$ (Aziz & Chandra Reference Aziz and Chandra2000). At that moment, we estimate the rim deceleration as $\ddot{D}\sim D_{max}/t_{c}^{2}$ (Villermaux & Bossa Reference Villermaux and Bossa2011) and hence $k\sim \sqrt{D_{max}/D_{0}^{3}}$. Using the maximum expansion given by (4.1) we find

(5.1)$$\begin{eqnarray}N\sim \left(\frac{P^{1/2}}{A+P^{1/2}}Re^{1/5}\right)^{3/2}.\end{eqnarray}$$

In the limit of small $P$ (large $Re$) one recovers that $N\sim We^{3/4}$, as was derived by Villermaux & Bossa (Reference Villermaux and Bossa2011) for drops impacting on a pillar. Scaling (5.1) differs somewhat from the one proposed by Aziz & Chandra (Reference Aziz and Chandra2000), who used $U^{2}/D_{0}$ as an estimate for the rim deceleration as well as a different expression for $D_{max}$. In figure 8 we show that (5.1) with a prefactor of 4.3 describes the isothermal experimental data.

For impacts on cold substrates, $N$ falls below the isothermal curve (5.1). This decrease in $N$ was already observed from top-view photography of tin splats by Aziz & Chandra (Reference Aziz and Chandra2000), who attributed it to the smaller maximum spreading $D_{max}$ for solidification-limited impacts. Our bottom-view images, however, allow us to follow the growth of the ligaments over time and reveal a remarkable feature: for the solidified drops, the number of ligaments is set by the early-time freezing of undulations and remains unchanged during the subsequent drop spreading, thereby forming a pattern of radially outward-pointing stripes (recall figures 2 and 3c). By contrast, for isothermal impact, these early-time undulations remain invisible and ligaments evolving from the rim split at later times (Thoroddsen & Sakakibara Reference Thoroddsen and Sakakibara1998), thereby increasing $N$ (compare figure 2a.iv with figure 2c.ii). These two mechanisms qualitatively explain the two clouds of data for isothermal and solidification-limited spreading in figure 8. For temperatures close to the melting temperature of tin (orange data in figure 8), both mechanisms are at play: in some drops solidification is fast enough to prevent splitting of the fingers, while in other drops new ligaments emerge at later times.

6.1 Experiments

An impact event may have several outcomes, depending on the impact conditions. We observed deposition of the drop, which means that the drop remains intact and adheres to the substrate after impact. Splashing occurs when secondary droplets are generated. Bouncing takes place when (part of) the drop lifts off from the substrate. In figure 9(a) we quantify these impact phenomena as a function of the Stefan and Weber numbers, i.e. the dimensionless substrate temperature and impact velocity.

For the isothermal impact events ($Ste_{s}\leqslant 0$, $T\geqslant T_{m}$) and $We<350$, the drop bounces. For $We>375$, drops always splash during isothermal impact. For intermediate Weber numbers, a transition regime exists where both bouncing and splashing can occur. Drop deposition is not observed in the isothermal regime. When the substrate temperature is decreased below the melting temperature ($Ste_{s}=0.12$, $T_{s}=200\,^{\circ }\text{C}$), only a few bounces occur at low $We$, while most of the drops deposit and stick to the substrate. For higher Weber numbers, we again observe a transition to splashing. On even colder substrates ($Ste_{s}>0.12$, $T_{s}<200\,^{\circ }\text{C}$), the bouncing regime completely disappears and we only observe freezing-induced sticking of the drop. The Weber number at which the transition from bouncing or deposition to splashing takes place varies with the Stefan number. Importantly, for solidification-limited impacts, there is no longer a sharp boundary between bouncing/deposition and splashing, but both can be observed even at the highest Weber numbers accessible by our set-up.

Figure 9. (a) Phase diagram showing the outcome of drop impact as a function of $We$ and $Ste_{s}$. The three impact behaviours observed are bouncing (orange triangles), deposition (green circles) and splashing (purple squares). (b) Plot of the first splashing event observed as a function of $We$ and $Ste$ as extracted from the data in (a).

The splashing threshold is analysed in more detail in figure 9(b), where the critical Weber numbers corresponding to the first splashing event (i.e. the lowest $We$ where splashing is observed) are plotted as a function of $Ste_{s}$. The splashing threshold shows a non-monotonic trend: it first increases up to $Ste_{s}\approx 0.3$ and then decreases again, to a value that lies below the isothermal splashing threshold.

6.2 Qualitative explanation

To explain the observations in figure 9(b), we hypothesise that the influence of solidification on the splashing threshold is twofold. First, solidification limits the spreading of the drop and the growth of ligaments as was observed in §§ 4 and 5. Consequently, one expects splashing due to ligament breakup to be suppressed, as was already suggested by Aziz & Chandra (Reference Aziz and Chandra2000). This effect becomes stronger with increasing $Ste_{s}$ (lower substrate temperatures), which leads to the observed increase in the critical Weber number for the first splash.

Second, rapid solidification of the ejecta sheet forms local irregularities on the substrate, thereby increasing the roughness as discussed in § 5. These irregularities interact with the moving contact line and may thereby provoke a freezing-induced splash (Dhiman & Chandra Reference Dhiman and Chandra2005; Dhiman et al. Reference Dhiman, McDonald and Chandra2007; Chandra & Fauchais Reference Chandra and Fauchais2009). This effect becomes more severe with increasing $Ste_{s}$ such that at some point the splashing threshold decreases below that for isothermal impact. As the freezing-induced splash depends on nucleation events at the substrate, it has a stochastic nature and is not observed in each experiment.

These two competing effects, the splash suppression by solidification and the freezing-induced splash, can qualitatively explain the non-monotonic trend in the splashing threshold observed in figure 9(b).