2.1. Experimental set-up

The experimental set-up is composed of a dropper, an impact area and a high-speed imaging system. A schematic is provided in appendix A. The dropper consists of a needle (Gauge 21), connected through flexible tubes of polytetrafluoroethylene (PTFE) to a liquid reservoir. The liquid is set in motion by a peristaltic pump with a flow rate kept low enough to let a single droplet fall by gravity. The height from the wall-film surface to the needle tip is varied to obtain different droplet impact velocities ${V_{d}}{}$. The impact area is a shallow pool in which the liquid is progressively added with a micro-pipette, bounded by a metallic ring of 0.6 mm height and 60 mm diameter fixed on a sapphire plate. Sapphire is used for its high refractive index compared with the liquids used. This allows a reliable use of a confocal chromatic sensor (Micro-Epsilon, IFS2405-3) to measure the film thickness ${h_{f}}{}$ with an accuracy of approximately 1 % for the values investigated in this work (principle described in Lel et al. Reference Lel, Kellermann, Dietze, Kneer and Pavlenko2008).

The imaging system is designed for high-speed shadowgraphy, recording simultaneously two orthogonal perspectives to track asymmetric features such as holes in the crown. Typical high-speed images taken with a high-speed digital video camera (Photron FASTCAM SA1.1) and LED lights operated in backlit mode can be seen in figure 1 (only one perspective is shown). The spatial and temporal resolutions of the high-speed imaging were ${80}\ \mathrm {\mu }\textrm {m}\ \textrm {px}^{-1}$ and 0.05 ms, respectively combined with an exposure time of 1/92 ms.

The high-speed images are post-processed with an in-house MATLAB routine that extracts the primary geometrical parameters of the crown, as detailed in appendix A. The distances between the detected four edges of the crown, like the four orange dots in the schematic of figure 2, enables the evaluation of the crown rim radius ${R_{R}}{}$ and crown base radius ${R_{B}}{}$ for the horizontal distances, and the crown rim height ${H_{R}}{}$ in the vertical.

Figure 2. Schematic of a meridional section of a crown with droplet (green) and wall-film (blue) liquids, illustrating the full coverage of the droplet liquid on the inner part of the crown. The interface between them is marked by a red line. Note that the crown thickness is not representative of the reality. The crown edges (orange dots) extracted from the post-processing are used to define the crown geometrical parameters: rim radius ${R_{R}}{}$, rim height ${H_{R}}{}$ and base radius ${R_{B}}{}$.

The values of the geometrical parameters are averaged between the two perspectives of the experimental set-up to reduce the measurement uncertainty. The typical errors associated with the measurement of the crown geometrical parameters are between 1.5 % and 4.3 % for the radii, and between 1.6 % and 5.2 % for the crown height (Geppert Reference Geppert2019). The droplet diameter ${D_{d}}{}$ and impact velocity ${V_{d}}{}$ are also measured from the high-speed images shortly before impact (${\sim }1.5\ \textrm {ms}$). By adding up the systematic and random uncertainties, which maximizes the error, the overall uncertainty for the droplet diameter remains smaller than 2.8 %, and below 3 % for the impact velocity (Geppert Reference Geppert2019). A detailed description of the set-up (which has been used with the same operating conditions), the post-processing routine and the associated measurement errors can be found in Geppert et al. (Reference Geppert, Chatzianagnostou, Mesiter, Gomaa, Lamanna and Weigand2016, Reference Geppert, Terzis, Lamanna, Marengo and Weigand2017) and Geppert (Reference Geppert2019).

2.2. Variation of droplet/wall-film liquids

The liquids of the droplet/wall-film pairs have been chosen to vary the wettability and miscibility behaviours, keeping other liquid properties almost unchanged. The liquid properties are summarized in table 1, together with the abbreviations used for each liquid.

The droplet liquid is systematically composed of a mixture of glycerol and water (abbreviation GW), which is coloured with Indigotin 85 (E 132, BASF, Germany) similar to the liquids used by Baumgartner et al. (Reference Baumgartner, Bernard, Weigand, Lamanna, Brenn and Planchette2020). The only parameter varying significantly is the interfacial tension with the wall-film liquids $\sigma _{d/f}$ (see Baumgartner et al. (Reference Baumgartner, Bernard, Weigand, Lamanna, Brenn and Planchette2020) for more details on their determination), the subscript $d$ standing for droplet and $f$ for wall film. The interfacial tension is associated with the miscibility behaviour ($\sigma _{d/f}=0$ for miscible liquids) and the wettability behaviour through the spreading parameter $S$. The spreading parameter is defined as $S{}= \sigma _d- \sigma _f- \sigma _{d/f}$, which takes negative values for partial wetting (P.W.) and positive values for full wetting (F.W.) (De Gennes, Brochard-Wyart & Quéré Reference De Gennes, Brochard-Wyart and Quéré2013). For the wettability, n-hexadecane (abbreviation Hexa, third column in table 1) provides partial wetting with aqueous solutions as is the case with many alkanes. This is confirmed by the estimation of S for GW/Hexa which is negative with approximately ${-9}\ \textrm {mN}\ \textrm {m}^{-1}$, and the observation of n-hexadecane lenses on top of GW (Baumgartner et al. Reference Baumgartner, Bernard, Weigand, Lamanna, Brenn and Planchette2020). In contrast, silicone oil (abbreviation S.O., fourth column in table 1) provides full wetting with aqueous solutions (Ross & Becher Reference Ross and Becher1992; De Gennes et al. Reference De Gennes, Brochard-Wyart and Quéré2013). The spreading parameter S is positive and equal to approximately 15. The full wetting behaviour is confirmed by the observation of an oil drop spreading on the surface of a liquid bath of GW (Baumgartner et al. Reference Baumgartner, Bernard, Weigand, Lamanna, Brenn and Planchette2020).

The last wall-film liquid is a mixture of ethanol, water and glycerol (abbreviation EtOH, fifth column in table 1), which has been tuned to approach the two other wall-film liquid properties. The mixture of ethanol is miscible with the droplet liquid GW. In the case of miscible liquids, the interfacial tension is assumed to be zero because no interface, in a strict sense, exists between the liquids. Note that, for miscible liquids with composition gradients, $\sigma _{d/f}{}$ may transiently differ from zero before diffusion takes place (Truzzolillo & Cipelletti Reference Truzzolillo and Cipelletti2017), which may be the case for droplet impact processes of a few milliseconds. Nevertheless, the measurements of interfacial tensions for water and glycerol mixtures show values of $1\ \textrm {mN}\ \textrm {m}^{-1}$ or lower (Petitjeans & Maxworthy Reference Petitjeans and Maxworthy1996), and approximately $2\ \textrm {mN}\ \textrm {m}^{-1}$ for water and alcohols like butanol (Enders & Kahl Reference Enders and Kahl2008). These values remain within the experimental uncertainties, thus, the interfacial tension can be approximated to be zero for GW/EtOH.

In order to quantify the importance of the interfacial tension in the droplet/wall-film system, the non-dimensionalized interfacial tension $\sigma _{d/f}^{*}{}= \sigma _{d/f}{}/( \sigma _d{}+ \sigma _f{}+ \sigma _{d/f}{})$ is introduced. It corresponds to the ratio of the interfacial tension to the sum of the surface tensions of droplet and wall film. Thus, it is zero for miscible liquids such as GW/EtOH, and increases with increasing importance of the interfacial tension in the droplet/wall-film system, as for GW/SO and GW/Hexa in the last row of table 1. Note that, in some cases, it could be interesting to have a unified parameter combining the importance of the interfacial tension with the spreading parameter, e.g. as ${S^{*}}{}= \sigma _{d/f}{}/ S{}$ in the case where the spreading parameter is different from zero. This parameter could help to understand the kinetics of the receding where the wettability has a strong influence for example in the formation of a Worthington jet (Che & Matar Reference Che and Matar2018), whose magnitude could be related to ${S^{*}}{}$. In the present study, we rather focus on the crown extension phase where the value of the interfacial tension seems to play a significant role independently of the wetting behaviour of the liquid pair, probably because of the large Weber numbers involved (see § 2.3).

2.3. Experimental range

The single droplets of the glycerol/water mixture exhibit diameters of ${D_{d}}{} = 2.20\pm 0.07\ \textrm {mm}$ for the full database. The impact velocity is varied from ${V_{d}}{}=2.5$ to $3.7\ \textrm {m}\ \textrm {s}^{-1}$ with four different fall heights. For each impact velocity investigated, all three liquid pairs are used as a wall film. The dimensionless wall-film thickness $\delta {}={h_{f}}{}/{D_{d}}{}$ is kept quasi-constant at two investigated ranges, i.e. at $\delta =0.122\pm 0.011$, and at $\delta =0.259\pm 0.008$. In order to describe accurately the impact process, the initial state of the droplet/wall-film system needs to be quantified. Therefore, the range of Weber and Reynolds numbers based on droplet properties, $\mathit {We}_{d}{}= \rho _d {V_{d}}^{2} {D_{d}}{}/ \sigma _d$ and $\mathit {Re}_{d}=\rho_d {V_{d}}{}{D_{d}}{}/ \mu _d$ respectively, are given in the two first columns of table 2. Note that the range indicated includes the experimental conditions of all three liquids pairs of GW/Hexa, GW/SO and GW/EtOH. However, these numbers are not representative of the full binary droplet/wall-film systems since the liquid properties of droplet and wall film differ. A proper way of taking both droplet and wall-film liquid properties into account in these non-dimensional numbers is still under discussion in the literature (Geppert et al. Reference Geppert, Gomaa, Meister, Lamanna and Weigand2014; Kittel, Roisman & Tropea Reference Kittel, Roisman and Tropea2017). Recently, averaged liquid properties (subscript avg) such as $\rho _{avg}{}=( \rho _d{}+ \rho _f{})/2$, $\sigma _{avg}{}=( \sigma _d{}+ \sigma _f{})/2$ and $\nu _{avg}{}=\sqrt {{ \nu _d \nu _f}}$ have shown good ability to capture the crown rim dynamics for binary droplet/wall-film systems (Bernard et al. Reference Bernard, Vaikuntanathan, Weigand and Lamanna2020). Note that this approach was developed for miscible droplet/wall-film liquid pairs made of silicone oils. Hence, it does not consider wettability and miscibility effects which are considered separately in the present study (e.g. in terms of spreading parameter $S{}$ and dimensionless interfacial tension $\sigma _{d/f}^{*}{}$ summarized in table 1). These averaged properties can be used in the dimensionless numbers to form $\mathit {We}_{avg}{}$ and $\mathit {Re}_{avg}{}$ summarized in table 2 for each reference value of $\mathit {We}_{d}{}$ and all liquid pairs combined. Furthermore, they can be combined in a single impact parameter $K{}= \mathit {We}^{0.5} \mathit {Re}^{0.25}$ (also summarized in the last column of table 2), which, considered together with $\delta {}$, is fully representative of the droplet wall-film system for the onset of splashing (Cossali, Coghe & Marengo Reference Cossali, Coghe and Marengo1997). These non-dimensional impact parameters between the three liquid pairs indicate very similar droplet/wall-film systems for each set of experiments. The variation does not exceed ${\pm }8\,\%$ in general at a given $\mathit {We}_{d}{}$, while the non-dimensionalized interfacial tension $\sigma _{d/f}^{*}{}$ increases by more than 25 % between GW/SO and GW/Hexa, and is zero for GW/EtOH. Hence, the major varying parameter between the liquid pairs investigated is $\sigma _{d/f}^{*}{}$, which, combined with the surface tensions of droplet and wall films, defines the miscibility and wettability of the liquid pairs.

Table 2. The investigation range of typical non-dimensional parameters. The subscript avg refers to averaged liquid properties between droplet and wall film.

3.1. Shift in the splashing limit

The three droplet/wall-film pairs in figure 1 differ qualitatively in their impact outcomes. A movie in the supplementary material shows the temporal evolution of the impact process for each liquid pair side by side. For GW/Hexa, the crown rim stays relatively stable, undulations become significant only in the receding phase (the maximum crown height is reached at approximately $\tau _{in,H_{max}}{} = {t_{H_{max}}}{} {V_{d}}{}/{D_{d}}{} = 5.40$) and fingers can barely be observed at the end of the impact process. Thus, the outcome of GW/Hexa can be categorized as a transition case (i.e. fingers formed) close to deposition (i.e. no fingers). In the case of GW/SO, undulations on the rim already occur in the ascending phase (up to $\tau _{in,H_{max}}{} = 6.53$), and at the end of the extension process, long fingers with droplets about to detach are formed. The long fingers only disintegrate into secondary droplets at the end of the impact process, leaving the droplets in the vicinity of the impact location. Thus, the outcome of GW/SO can again be categorized as a transition case, but close to splashing (i.e. secondary droplets are ejected far off the crown).

Note that the tiny droplets seen already at $\tau _{in}{} = 2.25$ are associated with prompt splashing, i.e. they are not considered as part of the crown-type outcome. The prompt splashing consists in the fragmentation of a thin and fast liquid lamella called ejecta sheet (Thoroddsen Reference Thoroddsen2002) formed at the base of the impacting droplet. The distinction between prompt and crown-type splashing relies first on their different time scales, since prompt splashing occurs quasi-immediately after the droplet impact onto the liquid surface (within the first $100\ \mathrm {\mu } \textrm {s}$, Liang & Mudawar (Reference Liang and Mudawar2016)). Second, the ejecta sheet and the crown wall arise from a different dynamics and can be observed distinctly (Zhang et al. Reference Zhang, Toole, Fezzaa and Deegan2012). Third, prompt and crown-type splashing can be observed independently of each other in a splashing regime map (Deegan, Brunet & Eggers Reference Deegan, Brunet and Eggers2007). Last, prompt splashing forms much smaller secondary droplets than crown-type splashing (Cossali et al. Reference Cossali, Coghe and Marengo1997; Motzkus, Gensdarmes & Géhin Reference Motzkus, Gensdarmes and Géhin2009). Hence, there is most likely no or small interdependence between the prompt splashing observed and the crown-type splashing studied in the present experiments. The movie of the impact experiments, similar to those of figure 1 in the supplementary material shows prompt splashing in all three cases upon droplet penetration into the wall film, although the tiny droplets produced are ejected beyond the crown size and thus observable later only for GW/SO and GW/EtOH.

For the last liquid pair GW/EtOH, fingers are already formed in the ascending phase (e.g. at $\tau _{in}{} = 4.50$), and some droplets are ejected from the fingers. Towards the end of the crown extension, holes are formed on the crown wall, as known from droplet impacts with aqueous and ethanol solutions (Thoroddsen et al. Reference Thoroddsen, Etoh and Takehara2006; Aljedaani et al. Reference Aljedaani, Wang, Jetly and Thoroddsen2018). As explained in the introduction, mixture composition gradients in the crown wall and/or fine droplets which cannot be observed with the current set-up impacting on the crown wall could cause these holes.

Despite the formation of holes in the crown (the first occurrence could be back tracked at $\tau _{in}{} = 6.30$), it is seen that a few, bigger secondary droplets are ejected from the fingers. Thus, the outcome of GW/EtOH can be categorized as a splashing case close to transition, since only few droplets are ejected. Hence, a small shift in the splashing limit is observed between the three cases despite similar liquid properties and impact conditions: the miscible pair GW/EtOH starts to splash, the full wetting pair GW/SO is close to splash and the partially wetting pair GW/Hexa is closer to deposition. Note that the smallest viscosity (despite the small differences between the pairs) is that of n-hexadecane. Since less viscous losses are expected, splashing with less kinetic energy should be observed (see the review article of Liang & Mudawar (Reference Liang and Mudawar2016)). Since this is not the case, small viscosity variations are not expected to modify the trend observed between the liquid pairs. This shift in the fragmentation limit, classically derived based on empirical observation of the outcome, is very small. Thus, it is hard to evaluate it in terms of the critical Weber or Reynolds number. Furthermore, there is a smooth transition from deposition to splashing (Cossali et al. Reference Cossali, Coghe and Marengo1997), so that the transition outcome (Geppert et al. Reference Geppert, Chatzianagnostou, Mesiter, Gomaa, Lamanna and Weigand2016) is used to refine the splashing criteria. However, qualitatively, the averaged Weber number $\mathit {We}_{avg}{}$ defined in § 2 could be modified taking into account all surface and interfacial tensions with $( \sigma _d{}+ \sigma _f{}+ \sigma _{d/f}{})/3$ instead of $( \sigma _d{}+ \sigma _f{})/2$. This would lead to a characteristic Weber number of the impact multiplied by 0.98 for GW/Hexa, 1.08 for GW/SO and 1.5 for GW/EtOH, highlighting that GW/EtOH would cross the splashing limit earlier than GW/SO and finally than GW/Hexa. This difficulty to assert the splashing limit for such a narrow difference highlights the need to consider another parameter to quantify the differences observed between the liquid pairs.

3.2. Shift in crown extension

In addition to the small shift in the fragmentation observed in figure 1, the liquid pairs show differences in the size of the crown. To quantify this observation, the crown surface is calculated from the high-speed images by extracting the geometrical parameters of the crown as shown in the schematic of figure 2. The crown rim height ${H_{R}}{}$, the rim radius ${R_{R}}{}$ and the base radius ${R_{B}}{}$ are extracted from the crown edges marked with orange circles. The crown surface, expressed as

(3.1)\begin{equation} \varSigma_{c}{}={\rm \pi} {R_{B}}^{2} + 2 {\rm \pi}\left({R_{B}}{}+{R_{R}}{}\right) \sqrt{{H_{R}}^{2}+\left({{R_{R}}{}}-{R_{B}}{}\right)^{2}}, \end{equation}

is calculated by summing the surface of the disc at the crown base, and the surface of the crown wall counted twice for the inner and outer parts. The crown wall is approximated by a conical frustum formed between the base and rim radii. This assumes having a straight line between the base and rim radii, although sometimes the crown wall is slightly bent inward. This bending and its influence on the determination of the crown surface is discussed in appendix B. However, the difference in crown morphologies (between the crown edges) is taken indirectly into account via the calculation of $\varSigma _{c}{}$. In addition, the surface area at the rim due to the crown thickness is neglected.

The propagation of uncertainty from the measurements of ${H_{R}}{}$, ${R_{R}}{}$ and ${R_{B}}{}$ (given in § 2.1) to $\varSigma _{c}{}$ can be calculated based on the formula of Gauss–Laplace. The corresponding relative measurement uncertainty is dependent upon the crown morphology, i.e. ${H_{R}}{}/{R_{R}}{}$ and ${R_{B}}{}/{R_{R}}{}$. Note that these ratios vary during the impact process, and between different impact conditions. The typical crown configuration in the present experimental conditions (${H_{R}}{}/{R_{R}}{} \approx {} 0.5$ and ${R_{B}}{}/{R_{R}}{} \approx {} 1$) leads to an error of 5.7 %, while the extreme crown aspect ratios (${H_{R}}{}/{R_{R}}{}$ from 0.05 to 0.65, and ${R_{B}}{}/{R_{R}}{}$ from 0.65 to 1.30) lead to errors of up to 8.27 %, tendentially for high ${R_{B}}{}$ and/or small ${H_{R}}{}$.

The temporal evolution of $\varSigma _{c}{}$ is shown in figure 3(a) for the same experiments as in figure 1 that have similar impact conditions. Except at the early stage of impact, where kinetic energy dominates, the curves separate progressively. As observed qualitatively in the high-speed images, the crown surface of GW/Hexa is smaller than that of GW/SO, which is smaller than that of GW/EtOH. Note that this ranking of crown surface extensions correlates with the values of $\sigma _{d/f}^{*}{}$ reported in table 1: The larger is $\sigma _{d/f}^{*}{}$ (tendentially for immiscible liquid pairs with partial wetting), the smaller becomes the crown surface. A similar ranking in the spreading between miscible and immiscible droplet/wall films during the crown extension phase has also been measured by Chen et al. (Reference Chen, Chen and Amirfazli2017), without being explained. The formation of holes in the crown of GW/EtOH (grey symbols in figure 3) does not influence this trend since the values of $\varSigma _{c}{}$ for GW/EtOH are systematically above, but adds noise which prevents the exact determination of the maximum crown surface $\varSigma _{c,max}{}$ (filled symbols in figure 3a,b); $\varSigma _{c,max}{}$ for GW/EtOH is lying between the value where the first hole appears (filled grey symbols and grey dashed line), and the maximum value calculated before the destruction of the crown (filled red symbols and red dashed line). Furthermore, the determination of the crown parameters during extension remains repeatable despite stochastic holes (Geppert et al. Reference Geppert, Terzis, Lamanna, Marengo and Weigand2017).

Figure 3. (a) Temporal evolution of the crown surface $\varSigma _{c}{}$ for the same experiments as in figure 1. The filled symbols with yellow edge correspond to $\varSigma _{c,max}{}$. The dashed lines correspond to ${t_{H_{max}}}{}$, the time at which ${H_{R,max}}{}$ is reached. (b) Maximum crown surface $\varSigma _{c,max}{}$ at different Weber numbers $\mathit {We}_{d}{}$. (c) Time duration ${t_{H_{max}}}{}$ of the crown ascending phase (marked with dashed lines in (a)) at different Weber numbers $\mathit {We}_{d}{}$. The dimensionless wall-film thickness is kept constant for the data of this figure at $\delta =0.122\pm 0.011$. The grey symbols correspond to the measurements where holes in the crown are observed (only for GW/EtOH).

The values of $\varSigma _{c,max}{}$ are reported for different Weber numbers in figure 3(b). It is clear that the trend observed between the liquid pairs in (a) remains the same: at a given $\mathit {We}_{d}{}$, the values of $\varSigma _{c,max}{}$ are systematically smaller for GW/Hexa than for GW/SO, which are smaller than GW/EtOH. For a given liquid pair, $\varSigma _{c,max}{}$ increases with growing $\mathit {We}_{d}{}$, i.e. with increasing droplet kinetic energy. This corroborates the larger rim expansions observed at higher droplet kinetic energy for droplet impacts onto wall films (Bernard et al. Reference Bernard, Vaikuntanathan, Weigand and Lamanna2020), or for other impact configurations, e.g. onto dry surfaces (Huang & Chen Reference Huang and Chen2018), with another droplet (Roisman et al. Reference Roisman, Planchette, Lorenceau and Brenn2012) or with a continuous jet (Baumgartner et al. Reference Baumgartner, Bernard, Weigand, Lamanna, Brenn and Planchette2020).

3.3. Shift in duration of crown ascending phase

It is interesting to note that the duration of the crown ascending phase (i.e. the time at which ${H_{R,max}}{}$ is reached, marked by dashed lines in figure 3a) also exhibits a systematic ranking between the liquid pairs, although of smaller amplitude than the differences in crown surfaces. The ascending duration of GW/Hexa is smaller than that of GW/SO, which may be smaller than that of GW/EtOH (the grey dashed line corresponds to the time at which the first hole could be back tracked). Here again, this ranking correlates with the values of $\sigma _{d/f}^{*}{}$ reported in table 1. The higher is $\sigma _{d/f}^{*}{}$, the shorter becomes the ascending duration.

Note that a similar observation could not be made with ${t_{\varSigma _{c,max}}}{}$ (the time at which the maximum crown surface is reached). In fact, the exact time of crown surface receding is unclear because it has, in some cases, a plateau at its maximum value. This effect is known as the stabilization phase of the crown at the end of the surface extension (Zhang et al. Reference Zhang, Liu, Qu and Hu2019). This happens when the radial extension of the crown is longer than the axial one (Bernard et al. Reference Bernard, Geppert, Vaikuntanathan, Lamanna and Weigand2018). In this case, the decrease of ${H_{R}}{}$ is compensated by a continuous increase of ${R_{R}}{}$ and/or ${R_{B}}{}$. This effect shifts ${t_{\varSigma _{c,max}}}{}$ compared with ${t_{H_{max}}}{}$ (as in figure 3a for GW/Hexa between the blue filled symbol and the blue dashed line) and leads to scattered data. Hence, ${t_{H_{max}}}{}$ is preferred to ${t_{\varSigma _{c,max}}}{}$. The durations of the crown ascending phase ${t_{H_{max}}}{}$ are given in figure 3(c) for different $\mathit {We}_{d}{}$. A trend between the liquid pairs is observed, similar to the maximum crown surface $\varSigma _{c,max}{}$: the ascending phase of GW/Hexa is slightly shorter than that of GW/SO, which is (not systematically) slightly smaller than GW/EtOH. These durations are slightly increasing with growing $\mathit {We}_{d}{}$, as already observed for example with droplet impacts onto dry surfaces (Huang & Chen Reference Huang and Chen2018), but not for droplet head-on collisions or impacts onto a continuous jet, where the kinetics is fixed by the encapsulated drop only (Baumgartner et al. Reference Baumgartner, Bernard, Weigand, Lamanna, Brenn and Planchette2020).

In summary, slight differences between the liquid pairs were observed for the onset of splashing, but more prominently, differences in maximum crown surface and in the duration of the crown ascending phase despite similar impact conditions and liquid properties were measured. Thus, these differences can be attributed to their wettability and miscibility behaviours.

4.1. Energy storage

In order to estimate if the differences in the crown extension observed in § 3.2 can be explained by the differences in interfacial tension $\sigma _{d/f}$, the temporal evolution of the crown surface energy ${E_{\sigma ,c}}{}$ is calculated. Each portion of the crown (i.e. base disc and conical frustum) is multiplied by the corresponding surface/interfacial tension (i.e. $\sigma _d$, $\sigma _f$ and/or $\sigma _{d/f}$). The droplet liquid is assumed to completely cover the wall-film liquid on the inside of the crown, as represented in figure 2. This full coverage by the droplet liquid on the inner crown wall has been observed numerically for aqueous droplet/wall-film systems (Zhang et al. Reference Zhang, Liu, Qu and Hu2019), and experimentally for dyed water droplet impacting onto silicone oil films (Shaikh et al. Reference Shaikh, Toyofuku, Hoang and Marston2018), for $Re_{f}=\rho _{f}{V_{d}}{}{h_{f}}/\mu _{f}$ below 400 (Kittel et al. Reference Kittel, Roisman and Tropea2018a). In the current work, the maximum value of $\mathit {Re}_{f}{}$ is 196. Hence, it can reasonably be assumed that the droplet liquid completely covers the inner part of the crown, which leads to the following expression for the crown surface energy:

(4.1)\begin{equation} {E_{\sigma,c}}{}= \left( \sigma_d+\sigma_{d/f} \right) {\rm \pi}{R_{B}}{}^{2} + \left( \sigma_d+\sigma_{d/f}+\sigma_f \right){\rm \pi} \left({R_{B}}{}+{R_{R}}{}\right) \sqrt{{H_{R}}{}^{2}+\left({R_{R}}{}-{R_{B}}{} \right)^{2}} . \end{equation}

The propagation of uncertainty for ${E_{\sigma ,c}}{}$ can be calculated similar to that of the crown surface $\varSigma _{c}{}$, by considering additionally the measurement uncertainty associated with the surface and interfacial tensions given in table 1. This provides a propagation of error for the reference geometrical configuration (${H_{R}}{}/{R_{R}}{} \approx {} 0.5$ and ${R_{B}}{}/{R_{R}}{}\approx {} 1$) of 6.1 % for GW/Hexa, 7.2 % for GW/SO and 7.6 % for GW/EtOH. By considering the extreme values of the crown geometrical configuration, the maximum propagation of uncertainty to ${E_{\sigma ,c}}{}$ becomes 9.6 % (obtained for GW/EtOH).

The temporal evolution of ${E_{\sigma ,c}}{}$ is shown in figure 4(a) for the same cases as in figure 3(a). In contrast to $\varSigma _{c}{}$, the maxima are now comparable (note that at a constant Weber number, ${E_{k,d,0}}{}$ is constant for the three liquid pairs). Indeed, the overall surface energy of GW/Hexa, which has the smallest crown surface, is relatively increased due to the high value of $\sigma _{d/f}$ for this liquid pair. Likewise, the overall surface energy of GW/EtOH, which has the biggest crown surface, is relatively decreased since the interfacial tension is zero (miscible liquid pair). In between lies the case of GW/SO. For all $\mathit {We}_{d}{}$ investigated in our study, all ${E_{\sigma ,c,max}}{}$ normalized with the droplet kinetic energy ${E_{k,d,0}}{}=\rho _d {\rm \pi}{D_{d}}^{3} {V_{d}}^{2}/12$ are systematically coming together. Hence, the interfacial tension appears as a good candidate to explain the differences in maximum crown extensions. This means that, for immiscible liquid pairs, the interface between droplet and wall film stores a non-negligible amount of energy during the extension phase, proportionally to $\sigma _{d/f}^{*}{}$. This stored energy neither participates in the crown extension, nor in the splashing process. Hence, liquid pairs with smaller interfacial tension (the limit being zero for miscible cases) expand more, and splash tendentially at lower droplet kinetic energy, as observed in figure 1. For impact processes with smaller extensions (e.g. at very small $\mathit {We}_{d}{}$, or onto another impacted substrates), the interfacial energy could be less significant and its effect may not be noticeable.

Figure 4. (a) Temporal evolution of the crown surface energy ${E_{\sigma ,c}}{}$ normalized with the initial droplet kinetic energy ${E_{k,d,0}}{}$ as a function of $\tau _{cap}{}$ (see (4.2)) for the same experiments as in figures 1 and 3(a). The filled symbols with yellow edge correspond to ${E_{\sigma ,c,max}}{}/{E_{k,d,0}}{}$. The dashed lines correspond to $\tau _{cap,H_{max}}{}$, the time at which ${H_{R,max}}{}$ is reached normalized with ${t_{cap}}{}$. (b) Maximum crown surface energy ${E_{\sigma ,c,max}}{}$ normalized with the initial droplet kinetic energy ${E_{k,d,0}}{}$ at different Weber numbers $\mathit {We}_{d}{}$. (c) Capillary time duration $\tau _{cap,H_{max}}{}$ of the crown ascending phase (highlighted with dashed lines in (a)) at different Weber numbers $\mathit {We}_{d}{}$. The capillary time scale ${t_{cap}}{}$ used has a typical spring constant of $\sigma _d{}+ \sigma _{d/f}{}+ \sigma _f{}$ and a characteristic mass ${M_d}{}+{M_f}{}=( \rho _d+ \rho _f ) {\rm \pi}{D_{d}}^{3}/6$. The dimensionless wall-film thickness is kept constant for the data of this figure at $\delta =0.122\pm 0.011$. The grey symbols correspond to the measurements where holes in the crown are observed (only for GW/EtOH).

The ratio of ${E_{\sigma ,c,max}}{}$ with ${E_{k,d,0}}{}$ in figure 4(b) indicates how much the incoming kinetic energy has been transferred into surface energy. On average, this ratio is approximately 50 % of the initial kinetic energy in our experimental range. This amount corroborates numerical simulations at similar $\mathit {We}_{d}{}$ and wall-film thickness of aqueous droplet/wall-film systems (Zhang et al. Reference Zhang, Liu, Qu and Hu2019). Similar values have also been found in numerical simulations of droplet impacts onto dry surfaces with slip condition (Wildeman et al. Reference Wildeman, Visser, Sun and Lohse2016), i.e. with negligible surface friction which can be likened to a liquid interface.

At the end of crown extension, the quantity $1-{E_{\sigma ,c,max}}{}/{E_{k,d,0}}{}$ is representative of the remaining energy and losses before the onset of retraction. Some kinetic energy might remain due to internal flow motions and the complex shape of the crown. Furthermore, the motion of a liquid mass in the vertical direction leads to a conversion of kinetic energy into gravitational energy. Numerical studies of water droplet impacts onto wall films at $\delta {}=0.1$ and $\mathit {We}_{d}{}=250$ (Zhang et al. Reference Zhang, Liu, Qu and Hu2019) show that, at the end of the expansion, the remaining kinetic energy is approximately 8 % of ${E_{k,d,0}}{}$, and the gravitational energy 2 %. Combined with a surface energy of approximately 50 %, this corresponds to an energy loss of approximately 40 % of the droplet kinetic energy during the extension phase. This corroborates the study on dry surfaces with slip condition at the wall (Wildeman et al. Reference Wildeman, Visser, Sun and Lohse2016), where they found 10 % remaining energy (no gravitational energy since the spreading is horizontal), and 40 % of ${E_{k,d,0}}{}$ converted into heat. Hence, in the present droplet/wall-film experiments, the remaining kinetic and gravitational energies at the end of crown extension are expected to be approximately 10 % of ${E_{k,d,0}}{}$. Note that the gravitational energy is thus minor for $\delta {}=0.1$, but for a higher wall-film thickness, such as $\delta {}=0.3$, it can represent up to 15 % of the crown surface energy (Cossali et al. Reference Cossali, Marengo, Coghe and Zhdanov2004).

A full energy balance of the impact process (always assuming a zero kinetic energy at maximum crown surface) should also consider the initial surface energies of droplet and wall film. The initial surface energy of the droplet can be calculated with ${E_{\sigma ,d,0}}={\rm \pi} {D_{d}}^{2} \sigma _{d}$. For the wall film, the initial surface energy can be approximated with the free surface of the wall film covered by the crown base at maximum extension, i.e. ${E_{\sigma ,f,0}}\approx {\rm \pi}R^{2}_{{B,max}} \sigma _{f}$ (see e.g. similar approaches for droplet impact onto dry surfaces by Vaikuntanathan & Sivakumar (Reference Vaikuntanathan and Sivakumar2016)). This surface corresponds to the free surface of the wall film covered by the crown base at maximum extension and is therefore an impact-dependent variable that can be estimated a posteriori in the present database. The values of ${E_{\sigma ,c,max}}{}/({E_{k,d,0}}+{E_{\sigma ,d,0}}+{E_{\sigma ,f,0}})$ taking all initial energies into account would be diminished from 5 % to 10 % compared with ${E_{\sigma ,c,max}}{}/{E_{k,d,0}}{}$ alone given in figure 4(b). Since this shift roughly corresponds to the value of the remaining kinetic and gravitational energies of approximately 10 % of ${E_{k,d,0}}{}$ at the end of the crown extension discussed previously, this means that ${E_{\sigma ,c,max}}{}/{E_{k,d,0}}{}$ gives an estimation of the losses at the end of the crown extension, of approximately 50 % of all initial energies for these impact conditions.

Looking at the influence of the droplet kinetic energy, the maximum crown surface energy normalized with the droplet kinetic energy slightly decreases with increasing $\mathit {We}_{d}{}$. Hence, the percentage of total losses is increasing with growing $\mathit {We}_{d}{}$, which can be explained by higher deformation losses of the impacting droplet (Gao & Li Reference Gao and Li2015) and a more pronounced influence of the boundary layer because of a higher shear due to the higher velocity gradient, and a deeper penetration of the droplet bringing the flow closer to the wall (Lamanna, Geppert & Weigand Reference Lamanna, Geppert and Weigand2019).

The extension of the experimental database to $\delta {} \approx {} 0.26$ given in appendix B shows a similar data reduction of the normalized crown surface energies between the different liquid pairs at a given Weber number. However, the values are now higher compared with $\delta {} \approx {} 0.12$, of approximately 0.6 and above. This corroborates the influence of shear during the extension process since it is expected to be reduced for higher wall-film thickness because of a decreased influence of the solid wall. However, a more refined quantitative analysis is not possible here since the potential energies and remaining kinetic energy at the end of crown extension might be larger for a wall-film thickness above 0.2 (Zhang et al. Reference Zhang, Liu, Qu and Hu2019).

To conclude, the interface (if any) between the droplet and wall film stores a non-negligible amount of energy during the extension. Taking into account the interfacial energy generalizes the estimation of viscous losses made for miscible liquid pairs to immiscible ones with different interfacial tensions. With approximately 50 % loss of the initial surface and kinetic energies for $\delta {} \approx {} 0.1$, the viscous effects prevent the crown extension as much as the capillary forces for thin wall films. Hence, the droplet/wall-film system should be considered as an inertio-capillary-viscous system.

4.2. Recoiling force

Droplet impact processes can be considered as oscillating mass–spring systems (Okumura et al. Reference Okumura, Chevy, Richard, Quéré and Clanet2003; Planchette et al. Reference Planchette, Hinterbichler, Liu, Bothe and Brenn2017), where surface tension is the recoiling force opposing the deformation induced by the impacting droplet. Hence, the droplet and the wall film could be seen as springs in parallel whose spring constants are given by the surface and interfacial tensions. Following this reasoning, the system during impact could be represented by an equivalent spring constant equal to the sum of each, i.e. $\sigma _d+\sigma _f+\sigma _{d/f}$. Considering the crown as an inertio-capillary system, the capillary time scales of the crown ascending phase (i.e. increase of ${H_{R}}{}$) can be compared for the three liquid pairs to see whether the trend between the liquid pairs observed in figure 3(c) is related to the differences in interfacial tensions. Thus, we propose to evaluate the capillary time based on the equivalent spring constant as

(4.2)\begin{equation} \tau_{cap}{}=\frac{t}{{t_{cap}}{}}=\frac{t}{\sqrt{\dfrac{{M_d}{}+{M_f}{}}{\sigma_d+\sigma_{d/f}+\sigma_f}}}. \end{equation}

The characteristic mass of the droplet/wall-film system entering ${t_{cap}}{}$ is most likely a combination of droplet and wall-film masses as (${M_d}{}+{M_f}{}$). While the characteristic mass of the droplet can be easily calculated before impact as ${{M_d}{}}={ \rho _d} {\rm \pi}{D_{d}}^{3}/6$, the determination of wall-film mass ${M_f}{}$ is more difficult. By assuming an equal volume of droplet and wall-film liquid participating in the oscillation process, we get ${{M_d}{}}+{{M_f}{}}=({ \rho _d}+{ \rho _f} ) {\rm \pi}{D_{d}}^{3}/6$. This rough estimation has the advantage of being determined with pre-impact parameters. In figure 4(a), the time is non-dimensionalized with ${t_{cap}}{}$ to form $\tau _{cap}{}$. Compared with ${t_{H_{max}}}{}$ in figure 3(a), we observe that $\tau _{cap,H_{max}}{}$ (the non-dimensional time at which ${H_{R,max}}{}$ is reached) are coming together. Indeed, the differences of $\sigma _{d/f}{}$ in ${t_{cap}}{}$ compensate the differences in extension duration for each liquid pair, similar to the compensating effect for ${E_{\sigma ,c,max}}{}$ in figure 4(b). Hence, the interfacial tension is responsible for the different durations of ascending phases.

This means that, for immiscible liquid pairs, the interface between droplet and wall film acts as a non-negligible force preventing the extension, proportionally to $\sigma _{d/f}^{*}{}$. This scaling leads to a unified temporal evolution for the three liquid pairs of the normalized crown surface energy with the capillary time during the ascending phase at a given impact condition. The role of interfacial tension is observed for all impact conditions investigated in our study as shown in figure 4(c), all data points coming together at a given $\mathit {We}_{d}{}$; $\tau _{cap,H_{max}}{}$ is approximately 0.4, similar to what has been also observed for much lower Weber numbers (below 10) for the droplet oscillation alone impacting a shallow and deep pool (Tang et al. Reference Tang, Saha, Sun and Law2019).

The values of $\tau _{cap,H_{max}}{}$ are slightly increasing with increasing $\mathit {We}_{d}{}$. This indicates that $\tau _{cap}{}$ cannot properly capture the process for all impact conditions, but only highlights the influence of the interfacial tension during the extension. The consideration of the inertial time $\tau _{in,H_{max}}{}={t_{H_{max}}}{}{V_{d}}{}/{D_{d}}{}$ would lead to an even stronger dependency on the Weber number since $\mathit {We}_{d}{}$ increases with ${V_{d}}{}$. Furthermore, it would not bring the experiments together at a given $\mathit {We}_{d}{}$ since no surface tensions are considered. A viscous time as ${D_{d}}^{2}/\nu$ would also let the ranking of ${t_{H_{max}}}{}$ and the trend with $\mathit {We}_{d}{}$ unchanged given the similar kinematic viscosities involved for all impact conditions (see table 1).

The extension of the experimental database to ${\delta {}} \approx {} 0.26$ represented in figure 5 shows a similar data reduction of the ascending phase durations between the different liquid pairs at a given Weber number. However, the values are now higher compared with ${\delta {}} \approx {} 0.12$, lying around 0.6. Furthermore, the trend with increasing Weber number becomes weaker compared with ${\delta {}} \approx {} 0.12$, exhibiting almost constant values.

Figure 5. (a) Time duration ${t_{H_{max}}}{}$ of the crown ascending phase at different Weber numbers $\mathit {We}_{d}{}$. (b) Capillary time duration $\tau _{cap,H_{max}}{}$ of the crown ascending phase at different Weber numbers $\mathit {We}_{d}{}$. The dimensionless wall-film thickness is kept constant at ${\delta }=0.259\pm 0.008$ for the data of this figure.

The discrepancies in oscillation periods between different Weber numbers and wall-film thicknesses suggest that the scaling coefficient ${t_{cap}}{}$ defined above does not capture the full complexity of the impact process, especially regarding the role of inertia and damping effects. In the following, reasons for these discrepancies are proposed, focusing on the role of the interacting mass during the crown extension, of the viscous damping, and of an altered isochronism of oscillations.

First, the characteristic mass used in ${t_{cap}}{}$ is very likely dependent upon the impact conditions. Indeed, the kinetic energy of the droplet influences the droplet penetration inside the film. This could lead to an increased interacting wall-film mass, which is not captured with the current capillary scaling assuming an equal amount of droplet and wall-film liquids. Similarly, larger wall-film thicknesses increase the available amount of wall-film liquid that could enter the crown. Hence, the estimation of the interacting mass of wall film needs to be refined to get a deeper insight into the oscillating behaviour. Therefore, the maximal interacting wall-film mass ${M_f}{}$ can be estimated by considering the mass contained in a cylinder of radius ${R_{B}}{}$ at ${t_{H_{max}}}{}$ and of height equal to the difference of the wall-film height ${h_{f}}{}$ with the residual film thickness ${h_{res}}{}$, as ${{M_f}{}}= { \rho _f{}} {\rm \pi}{R_{B}}^{2}({{t_{H_{max}}}{}}) ({{h_{f}}{}} - {{h_{res}}{}})$. The residual thickness corresponds to the thickness of the wall film which is not set in motion by the droplet impact. It can be approximated with ${{h_{res}}{}} \approx (0.098{\delta {}}^{4.0413}+0.79){ \mathit {Re}}^{-2/5} {{D_{d}}{}}$ (van Hinsberg et al. Reference van Hinsberg, Budakli, Göhler, Berberović, Roisman, Gambaryan-Roisman, Tropea and Stephan2010). Hence, the residual thickness increases with increasing initial wall-film thickness, and decreases with increasing Reynolds number due to a deeper penetration of the droplet in the wall film.

In the present database, the Reynolds number can be approximated with averaged liquid properties as introduced in § 2.3 on the experimental range, corresponding to $\mathit {Re}_{avg}$. For the value of ${{R_{B}}{}}({{t_{H_{max}}}{}})$, it has now the drawback that it should be extracted post-impact. For predicting purposes, however, one could use available theoretical models predicting ${R_{B}}{}$ in the literature, e.g. that of Roisman, van Hinsberg & Tropea (Reference Roisman, van Hinsberg and Tropea2008) predicting the maximum value of ${R_{B}}{}$ during the impact process in function of $\mathit {We}_{d}{}$, $\delta {}$ and the Froude number $\mathit {Fr}_{d}{}$.

The experimental data scaled with the modified capillary time scale $\tau _{cap,H_{max},m}{}$ based on the maximum mass of the wall film are given in figure 6 for ${\delta {}} \approx {} 0.12$ (a) and ${\delta {}} \approx {} 0.26$ (b). It can be seen that the values are now comparable for both wall-film thicknesses. Hence, the modified scaling captures the oscillating dynamics for different wall-film thicknesses, due to the increase of the characteristic mass. This correction also enables us to reduce the dependency with $\mathit {We}_{d}{}$ for ${\delta {}} \approx {} 0.1$. Yet, a slight slow down of the crown dynamics with increasing $\mathit {We}_{d}{}$ can still be observed. Hence, additional effects might be responsible for this remaining trend.

Figure 6. (a) Time duration ${t_{H_{max}}}{}$ of the crown ascending phase at different Weber numbers $\mathit {We}_{d}{}$. (b) Capillary time duration $\tau _{cap,H_{max}}{}$ of the crown ascending phase at different Weber numbers $\mathit {We}_{d}{}$. The dimensionless wall-film thickness is kept constant at $\delta =0.259\pm 0.008$ for the entire database.

Considering a classical oscillating mass–spring system, the damping can increase the inviscid oscillating period ${t_{cap}}{}$ as ${{t_{cap,\nu }}{}} = 2 {\rm \pi}/ ({\omega _{cap}}\sqrt {1-{\eta }^{2}})$, where $\omega _{cap}{}$ is the inviscid pulsation and $\eta {}$ the damping ratio. Applied to oscillating droplets during an impact, the damping ratio can be expressed as ${\eta {}} \approx {} 32\nu {{t_{cap}}{}}/(2{\rm \pi} {D_{d}}^{2})$ by approximating a damping coefficient of $32\nu /{D_{d}}^{2}$ (Tang et al. Reference Tang, Saha, Sun and Law2019). By replacing ${t_{cap}}{}$ in $\eta {}$, the damping ratio can be re-expressed with the impact parameters of the present study in terms of a modified Ohnesorge number as $\eta {}=16{\rm \pi} { \textit {Oh}}/9$, with $\textit {Oh}=\nu {}\sqrt {({ \rho _d{}}+{ \rho _f{}})/(2{D_{d}}{}( \sigma _d{}+ \sigma _f{}+ \sigma _{d/f}{}))}$. The maximum value of Oh in the present database (considering averaged kinematic viscosity as $\sqrt { \nu _d \nu _f}$ suitable for binary droplet/wall-film system (Bernard et al. Reference Bernard, Vaikuntanathan, Weigand and Lamanna2020)) is less than 0.015, leading to $\eta {}=0.084$, and an increase of ${t_{cap}}{}$ of less than 0.5 %. Hence, the damping due to viscosity alone is negligible. Note, however, that this damping does not take shear into account during the extension, which has been shown to play a significant role in the crown extension dynamics at the base (Geppert et al. Reference Geppert, Bernard, Weigand and Lamanna2020) and at the rim (Bernard et al. Reference Bernard, Vaikuntanathan, Weigand and Lamanna2020). This viscous losses due to shear might increase the damping, and with it the oscillation period when shear increases. Since shear losses are expected to increase with increasing $\mathit {We}_{d}{}$, as highlighted by the increasing trend of $1-{{E_{\sigma ,c,max}}{}}/{{E_{k,d,0}}{}}$ in figure 4(b), this could explain the increasing trend of $\tau _{cap,H_{max}}{}$ with growing $\mathit {We}_{d}{}$. This also corroborates that this trend is weaker for $\delta {}=0.2$ where shear is smaller because the influence of the solid wall is reduced.

Another mechanism that could lead to the increase of $\tau _{cap,H_{max}}{}$ with growing $\mathit {We}_{d}{}$ is the amplitude of the oscillation. Since droplet/wall-film systems are characterized by large extensions, the isochronism of small oscillations might be altered. In this case, the oscillation period increases with increasing amplitude (Fulcher & Davis Reference Fulcher and Davis1976) which corresponds in our case to the Weber number.

To conclude, the interface (if any) between the droplet and wall film brings a supplementary recoiling force during the extension. Similar to a mass–spring oscillating system, the interfacial tension influences the oscillation behaviour of immiscible droplet/wall-film systems. Furthermore, the impact configuration (especially the wall-film thickness and the droplet kinetic energy) changes the characteristic mass of the system by modifying the mass of wall film interacting in the crown, and the influence of shear, which might influence the damping significantly.