3.1. Phenomenology

3.1.1. Sequence of events

In figure 2(b,c), we present synchronized TIR (figure 2b) and RI (figure 2c) images recorded during the first instants of the impact presented in figure 2(a). The TIR snapshots combine the original grey scale image and the calculated height field colour map. The liquid first enters within the evanescent length scale as a faint ring ($t = 0.018$,ms). This ring expands as its height decreases, indicating that the liquid comes closer to the substrate as it starts spreading ($t = 0.035$ ms). The full film profile can be inferred by combining the information extracted from TIR with the axisymmetric interference patterns observed in RI images. The brightest fringes are localized at the same radial position as the TIR ring evidencing the region closest to the substrate which is called the neck. The liquid–air interface displays a dimple shape, sketched in the inset, as already observed in both isothermal and superheated conditions (Shirota et al. Reference Shirota, van Limbeek, Sun, Prosperetti and Lohse2016). Eventually, the whole bottom interface of the drop moves away from the substrate and escapes from the TIR measurement range ($t = 0.28$ ms) while we continue to observe the spreading liquid as the RI fringe pattern remains visible.

Combining TIR and RI, we are able to visualize the dynamics of the whole gas layer and to compare them with their isothermal counterpart. We azimuthally average each RI and TIR snapshot and stack the one-dimensional information obtained at each instant in space–time two-dimensional graphs (figures 3(a) and 3(b)). From this representation, we are able to distinguish three phases in the early evolution of the squeezed film:

1. (i) The initial approach during which the liquid–gas interface first deforms, creating the central dimple bordered by a region of high curvature (see the first profile of the liquid–gas interface, $\Delta t = 0$ ms, in figure 3c). This region, which is called the neck region, moves down and outwards until the minimum thickness is reached at the neck, here at $t = 0.031$ ms, marking the end of this first phase. During this initial phase, the motion of the liquid–gas interface is qualitatively similar to that reported in the absence of substrate heating (Mandre et al. Reference Mandre, Mani and Brenner2009; Hicks & Purvis Reference Hicks and Purvis2010; Mani et al. Reference Mani, Mandre and Brenner2010; Bouwhuis et al. Reference Bouwhuis, van der Veen, Tran, Keij, Winkels, Peters, van der Meer, Sun, Snoeijer and Lohse2012).

2. (ii) A second phase, for $0.031 < t<0.4$ ms, in which the centre of the dimple and the neck have markedly different vertical motion. The thickness at the centre ($r=0$) is constant, as evidenced by the absence of crossings between iso-height lines (i.e. fringes) and the centreline in figure 3(a). In the same time interval, the neck thickness increases, as shown by the TIR data (figure 3b) and by the multiple crossings between the dashed red line, materializing the neck location, and iso-height lines in figure 3(a). Here, the radial and upwards motion of the neck allows us to distinguish the superheated from the isothermal case where the neck has a fixed radial position and height (Kolinski et al. Reference Kolinski, Rubinstein, Mandre, Brenner, Weitz and Mahadevan2012,  Reference Kolinski, Mahadevan and Rubinstein2014b).

3. (iii) In the final phase ($t>0.4$ ms), both the thickness at the centre and at the neck increase as the liquid keeps spreading. This global film thickness increase, which occurs at different instants at the neck and at the dimple, further differentiates the hot and the cold case and is characteristic of the influence of vapour generation.

Figure 3. (a,b) Space–time plots of the short-time film dynamics obtained by azimuthally averaging the RI and TIR snapshots for the impact shown in figure 2(a) ($R = 1.1$ mm, $U = 0.5$ m s$^{-1}$, ${\textit {We}} = 9.9$ and $T_s = 164\,^\circ$C). The dashed lines in (a,b) and (d,e) are a guide to the eye indicating the position of the neck extracted from the RI space–time plots. (c) Successive liquid–gas profiles extracted from (b). The colour code stands for the time difference from the instant at which the minimum thickness is reached. (d,e) Space–time plots obtained for similar impact parameters as in (a,b) ($R = 1.0$ mm, $U = 0.48$ m s$^{-1}$ and ${\textit {We}} = 8.3$) but with $T_s = 105\,^\circ$C. ( f) Successive height profiles extracted from (e). The colour code is the same as in (c). Video (S3) is available in the supplementary material.

3.2. Neck dynamics

We track the azimuthally averaged neck radius $r_n(t)$ (figure 4a) and distance to the substrate (i.e. neck height) $h_n(t) = h(r_n(t),t)$ (figure 4b) for varying impact velocities and substrate temperatures at short times. As the neck spreads, its height decreases, with a velocity $\textrm {d} h_n/\textrm {d} t$ of the order of the impact velocity, until it reaches a minimum value $h_m$ that corresponds to the global minimum of the azimuthally averaged gas layer thickness. Later, the neck height grows at a slower rate. The evolution of $r_n(t)$ seems to be only affected by the impact velocity. In contrast, the neck height $h_n(t)$ is strongly influenced also by the substrate temperature $T_s$. This behaviour suggests that the spreading dynamics evidenced for isothermal impacts (Rioboo, Marengo & Tropea Reference Rioboo, Marengo and Tropea2002; Mongruel et al. Reference Mongruel, Daru, Feuillebois and Tabakova2009; Thoroddsen, Takehara & Etoh Reference Thoroddsen, Takehara and Etoh2012) could describe the neck motion. Notably, the Wagner prediction, $r_n(t) = \sqrt {3URt}$, shown to be accurate for the radius of the liquid–solid contact by Riboux & Gordillo (Reference Riboux and Gordillo2014) and Gordillo et al. (Reference Gordillo, Riboux and Quintero2019) on cold surfaces and by Shirota et al. (Reference Shirota, van Limbeek, Sun, Prosperetti and Lohse2016) on superheated substrates, could also be relevant in the levitated regime. We indeed observe a good agreement of that prediction with the data by plotting the normalized neck radius $r_n/R$ as a function of the dimensionless time $tU/R$ for three different impact velocities and superheats (figure 4c). The data collapse, in quantitative agreement with Wagner's prediction represented by the solid line. We conclude that vapour generation strongly affects the vertical motion of the liquid–gas interface at the neck, but has negligible influence on its horizontal dynamics.

Figure 4. (a) Time evolution of the azimuthally averaged neck radius $r_n(t)$ for varying impact velocities $U$ and substrate temperatures $T_s$. The data are extracted from TIR images such as figure 3(b). The solid lines represent predictions from the Wagner theory, $r_n(t) = \sqrt {3URt}$. (b) Azimuthally averaged height at the neck $h_n(t)$ at short time. We denote $h_m$ the azimuthally averaged minimum film thickness. (c) Normalized azimuthally averaged neck radius $r_n/R$ as a function of the dimensionless time $tU/R$. The solid line shows the Wagner prediction. (d) Central dimple height $h_d$ plotted as a function of $U$ for the impact of drops for various $T_s$. Note that $h_d$ has a non-monotonic dependence on $T_s$. The solid line corresponds to the scaling $h_d/R \sim {\textit {St}}^{-2/3}$, taking the viscosity of the gas, $\eta _{g}$, as the viscosity of air at 20 $^\circ$C. The dashed and dotted lines correspond to two limiting choices for the gas viscosity, that of air at 230 $^\circ$C, and that of ethanol vapour at $T_b$, respectively. The square symbols are obtained in cases in which the liquid touches the solid. (e) Minimum film thickness $h_m$ as a function of the impact speed $U$ for various $T_s$. The dashed lines are guides to the eye with a slope of $-2$. ( f) Normalized minimum film thickness $h_m/R$ as a function of the prediction of (3.21), the data are consistent with this prediction and a fit gives a prefactor of $4.0 \pm 0.2$ in (3.21).

3.3. Thickness at the dimple and the neck

We now focus on the effect of $T_s$ on the thickness of the gas film. In figure 4(d), we show the dimple height $h_d$ measured at the same instant as the minimum thickness $h_m$ is reached and, for the two lowest substrate temperatures, also at the instant when contact occurs at the neck (square markers), enabling us to probe velocities up to 2 m s$^{-1}$. We obtain $h_d$ as the sum of the thickness at the neck $h_m$, determined using TIR, and of the height difference between the dimple and the neck, which we extract from the RI monochromatic fringe pattern. Coupling these two techniques, we are not limited by the 800 nm cutoff height and observe dimple heights of the order of 1 to 5 $\mathrm {\mu }$m, decreasing with increasing impact velocity. We compare our measurements with the scaling relation for $h_d$ derived in the absence of heating, $h_d/R \sim {\textit {St}}^{-2/3}$, where ${\textit {St}} = \rho _lRU/\eta _g$ is the Stokes number (Mandre et al. Reference Mandre, Mani and Brenner2009; Mani et al. Reference Mani, Mandre and Brenner2010; Bouwhuis et al. Reference Bouwhuis, van der Veen, Tran, Keij, Winkels, Peters, van der Meer, Sun, Snoeijer and Lohse2012), and $\eta _g$ the viscosity of the gas layer. Note that we define the Stokes number as the inverse of that proposed by Mandre et al. (Reference Mandre, Mani and Brenner2009) but consistent with other publications on the subject. The data are compatible with the isothermal scaling: for a fixed superheat $\Delta T$ and drop radius $R$, the dimple height decrease with increasing impact velocity $U$ is consistent with a power-law behaviour with an exponent $-2/3$. Yet, we observe a weak influence of the substrate temperature on $h_d$ that has a non-monotonic dependence with increasing superheat.

In figure 4(e), we show the minimum distance $h_m$ separating the drop from the substrate as a function of the control parameters $U$ and $T_s$. This minimum distance is of the order of a few hundred nanometres, and strongly increases with the superheat, in contrast to the weak non-monotonic influence observed for $h_d$. For example, increasing $T_s$ from 178 to 230 $^\circ$C leads to a doubling of the thickness at the neck for $U = 0.7$ m s$^{-1}$. For a fixed substrate temperature and drop radius, the data suggest a power-law decrease of $h_m$ with the impact velocity $U$ with an exponent $-2 \pm 0.2$. This exponent is close to that predicted for the minimum thickness at the neck in the isothermal case $-20/9$ (Mandre et al. Reference Mandre, Mani and Brenner2009; Mani et al. Reference Mani, Mandre and Brenner2010), but the strong influence of $\Delta T$ evidences the role of evaporation and distinguishes this case from an impact on a substrate at ambient temperature .

3.4. Model for the initial approach

We now seek to model the initial approach, i.e. the evolution of the gas film until the minimum thickness is reached at the neck. Typically, this phase lasts until $tU/R < 0.01$, a time much shorter than that associated with the purely inertial spreading that is valid up to $tU/R < 0.1$. We build on the work of Mani et al. (Reference Mani, Mandre and Brenner2010), who treated the case of isothermal impacts, and extend it to take into account the effect of vapour production. Indeed, our measurements of the dimple thickness in superheated conditions show the same scaling relation with the Stokes number as for isothermal impacts, indicating that liquid inertia and viscous drainage are still the relevant mechanisms when the substrate is heated. Yet, the thickness at the neck strongly depends on the superheat, suggesting that, to describe the movement of the liquid–gas interface, one also needs to capture the influence of vapour generation.

For completeness, we reproduce the equations of motion for the liquid and for the gas film in the incompressible regime as already derived by Mani et al. (Reference Mani, Mandre and Brenner2010). Following them, we consider a two-dimensional geometry and make further simplifications as in the isothermal case: in the liquid, we neglect viscosity and nonlinear inertia owing to, respectively, the large Reynolds number (${\textit {Re}} = \rho _l R U/\eta _l = 330$ for the impact of a millimetre-sized drop at $U = 0.5$ m s$^{-1}$) and the absence of velocity gradients within the drop before it interacts with the substrate. We also neglect the influence of surface tension, as for a typical impact the Weber number, ${\textit {We}} \approx 10$, is considerably larger than unity. With these assumptions, we have potential flow in the liquid and the equation of motion reads

(3.1a,b)\begin{equation} \rho_l \frac{\partial\boldsymbol{u}}{\partial t} + \boldsymbol{\nabla} p_l = 0,\quad \boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{u} = 0, \end{equation}

where $\boldsymbol {u} = (u,v)$ are the liquid velocity components in the $x$ (replacing $r$ in this two-dimensional model) and $z$ directions, respectively, and $p_l$ is the liquid pressure. Projecting (3.1a,b) in the vertical direction and using the kinematic boundary condition at the liquid–gas interface $\partial h/\partial t = v - u \partial h/\partial x$, we obtain at $z=0$

(3.2)\begin{equation} \rho_l \frac{\partial^2 h }{\partial t^2} + \frac{\partial p_l}{\partial z} ={-} \rho_l \frac{\partial}{\partial t}\left(u\frac{\partial h}{\partial x}\right), \end{equation}

where, in the spirit of the boundary layer approximation (Prandtl Reference Prandtl1904), the term on the right-hand side can be neglected. Then, the vertical pressure gradient at the interface can be expressed as the Hilbert transform of the horizontal pressure gradient by taking advantage of the two-dimensional nature of the problem and the harmonic pressure field (Smith, Li & Wu Reference Smith, Li and Wu2003)

(3.3)\begin{equation} \rho_l \frac{\partial^2 h }{\partial t^2} -\mathcal{H}\left[\frac{\partial p_l}{\partial x}\right] = 0. \end{equation}

We now come to the description of the gas flow between the drop and the substrate. Here, the viscous lubrication approximation is justified as the gas film is thin ($h \ll R$) and the typical value of the Reynolds number is low in the gas phase (${\textit {Re}} = \rho _g h_d U/\eta _g = 0.05$ for air at 20 $^\circ$C). This approximation is valid when considering heating as the value of the Reynolds number is not significantly altered by the presence of ethanol vapour (${\textit {Re}} = \rho _v h_d U/\eta _v = 0.15$ for ethanol vapor at $T_b$), nor by the dependence of the gas viscosity and density on temperature. With these assumptions, we obtain the lubrication equation

(3.4)\begin{equation} \frac{\partial h}{\partial t} - \frac{\xi}{12 \eta_g}\frac{\partial}{\partial x} \left(h^3 \frac{\partial p_g}{\partial x}\right) = 0, \end{equation}

where $\xi$ is a numerical coefficient which accounts for the choice of boundary condition at the liquid–gas interface. If the no-slip condition holds at the interface, $\xi = 1$. In contrast, if a zero tangential stress condition is chosen, $\xi =4$ (Guyon, Hulin & Petit Reference Guyon, Hulin and Petit2012).

We now introduce an additional term in the lubrication equation to take into account the effect of evaporation. We consider that heat is transferred through the gas layer by conduction, similarly as in the static Leidenfrost situation, as this mechanism acts on a relevant time scale for impact, $h^2/\kappa _v \approx 0.1\,\mathrm {\mu }$s, much shorter than the initial approach time. The energy input from the heated solid is used to heat liquid at the bottom interface from the ambient temperature $T_a$ to the boiling temperature $T_b$, creating a thermal boundary layer with thickness $\sqrt {\kappa _l t}$, and to vaporize it. The Jakob number $Ja = C_{p,l}(T_b - T_a)/\mathcal {L}$, which compares the sensible heat with the latent heat $\mathcal {L}$, is $0.16$ so that we consider the energetic cost coming from the latent heat to be dominant. With this assumption, which overestimates the vapour production, the evaporation rate per unit area, $e$, is given by balancing the heat flux $k_g\Delta T/h$ with the released latent heat $\mathcal {L}e$, yielding $e = k_g\Delta T/(\mathcal {L}h)$. We thus obtain a modified lubrication equation

(3.5)\begin{equation} \frac{\partial h}{\partial t} - \frac{\xi}{12 \eta_g}\frac{\partial}{\partial x} \left(h^3 \frac{\partial p_g}{\partial x}\right) = \frac{1}{\rho_g}\frac{k_g \Delta T}{\mathcal{L}h}, \end{equation}

where the term on the right-hand side takes into account evaporation (Biance, Clanet & Quéré Reference Biance, Clanet and Quéré2003; Sobac et al. Reference Sobac, Rednikov, Dorbolo and Colinet2014).

3.4.1. Dimple height

Having obtained the governing equations for the liquid and the gas, we recall the dimple height scaling derived for isothermal impacts (Mani et al. Reference Mani, Mandre and Brenner2010; Bouwhuis et al. Reference Bouwhuis, van der Veen, Tran, Keij, Winkels, Peters, van der Meer, Sun, Snoeijer and Lohse2012). It is obtained by balancing liquid inertia with viscous drainage in the squeezed gas film. The pressure in the liquid can be estimated from (3.2) or (3.3)

(3.6)\begin{equation} p_l/L \sim \rho_l U^2/h_d, \end{equation}

where $L \sim \sqrt {h_dR}$ is the radial extent of the dimple computed as the radius of a spherical cap with height $h_d$. The gas pressure is deduced from the two-dimensional incompressible lubrication equation (3.4)

(3.7)\begin{equation} p_gh^3/\eta_gL^2 \sim U. \end{equation}

The dimple height is set when the liquid–gas interface first deforms, that is when the pressure in the liquid and in the gas become comparable

(3.8)\begin{equation} h_d \sim R\left(\frac{\eta_g}{\rho_lRU}\right)^{2/3} \sim R {\textit{St}}^{{-}2/3}, \end{equation}

where ${\textit {St}} = \rho _lRU/\eta _g$ is the Stokes number. The solid line (figure 4d) represents (3.8) with $\eta _g$ taken as the viscosity of air at 20 $^\circ$C and a prefactor of 2.8 extracted from the results of Bouwhuis et al. (Reference Bouwhuis, van der Veen, Tran, Keij, Winkels, Peters, van der Meer, Sun, Snoeijer and Lohse2012) for the impact of ethanol drops on a substrate at room temperature. The isothermal scaling gives a correct order of magnitude of $h_d$ and recovers the velocity dependence in superheated conditions, confirming that the viscous drainage of the gas layer dominates vapour generation as suggested by the weak influence of substrate temperature on the dimple height in our experiments.

We propose to explain the observed influence of $T_s$ also through (3.8), although it does not explicitly involve temperature. Indeed, $\eta _g$ depends on temperature in two ways. (i) The viscosity of the air squeezed between the drop and the surface increases with $T_s$, and (ii) as ethanol vapour is generated, the squeezed layer becomes a – possibly non-homogeneous – mixture of warm air and ethanol vapour, whose viscosity is lower than that of air at the same temperature. The interplay between these two antagonistic effects could be the cause of the non-monotonic behaviour of $h_d$ with $T_s$. In figure 4(d), we plot the scalings associated with each effect by taking $\eta _g$ as the viscosity of air at 230 $^\circ$C (dashed line) and as the viscosity of ethanol vapour at the boiling point (dotted line): all measurements lie in between the two bounds.

3.4.2. Neck thickness

We next focus on the subsequent formation of the neck. Here, we extend the calculation of Mani et al. (Reference Mani, Mandre and Brenner2010) to incorporate the effect of evaporation, as our experimental observations show a strong influence of the substrate temperature on the thickness at the neck. We non-dimensionalize the governing equations (3.3) and (3.5) with the scales involved in the dimple formation, namely with the transformations

(3.9a–e)\begin{align} h = R {\textit{St}}^{{-}2/3}\tilde{h},\quad x = R {\textit{St}}^{{-}1/3}\tilde{x},\quad t = \frac{R {\textit{St}}^{{-}2/3}}{U}\tilde{t}, \quad p_l = \frac{\eta_g U}{R {\textit{St}}^{{-}4/3}}\tilde{p}_l, \quad p_g = P_0 \tilde{p}_g, \end{align}

where $P_0$ is the atmospheric pressure. Then, the equations of motion of the liquid and gas respectively become

(3.10)$$\begin{gather} \frac{\partial^2 \tilde{h}}{\partial \tilde{t}^2} - \varepsilon \mathcal{H}\left[ \frac{\partial \tilde{p}_l}{\partial \tilde{x}}\right] = 0, \end{gather}$$

(3.11)$$\begin{gather}\frac{\partial \tilde{h}}{\partial \tilde{t}} - \delta \frac{\xi}{12}\frac{\partial}{\partial \tilde{x}}\left(\tilde{h}^3\frac{\partial \tilde{p}_g}{\partial \tilde{x}}\right) = \frac{\mathcal{E}{\textit{St}}^{5/3}}{{\textit{We}} \tilde{h}}. \end{gather}$$

Here, we have introduced two extra dimensionless quantities:

1. (i) the ratio of the atmospheric pressure to the pressure build-up below the drop,

   (3.12)\begin{equation} \delta = \frac{P_0 RSt^{{-}4/3}}{\eta_gU}; \end{equation}

2. (ii) and the evaporation number $\mathcal {E}$ defined by Sobac et al. (Reference Sobac, Rednikov, Dorbolo and Colinet2014) in the study of static Leidenfrost drops,

   (3.13)\begin{equation} \mathcal{E} = \frac{\eta_g k_g \Delta T}{\gamma \rho_g R \mathcal{L}}. \end{equation}

The pressure buildup in the gas film below the drop is a small correction to the atmospheric pressure. We thus consider the limit $\delta \gg 1$, that is the incompressible limit, and assume the following pressure expansion $\tilde {p}_g = 1 + \tilde {p}/\delta$ which we introduce in the governing equation:

(3.14a,b)\begin{equation} \frac{\partial^2 \tilde{h}}{\partial \tilde{t}^2} - \mathcal{H}\left[ \frac{\partial \tilde{p}}{\partial \tilde{x}}\right] = 0,\quad \frac{\partial \tilde{h}}{\partial \tilde{t}} - \frac{\xi}{12}\frac{\partial}{\partial \tilde{x}} \left(\tilde{h}^3\frac{\partial \tilde{p}}{\partial \tilde{x}}\right) = \frac{\mathcal{E}{\textit{St}}^{5/3}}{{\textit{We}} \tilde{h}}, \end{equation}

where we used the pressure continuity at the interface (i.e. $p_g = p_l$). As the motion of the liquid–gas interface is similar in the isothermal and the superheated cases, we adopt the same self-similar ansatz as in Mani et al. (Reference Mani, Mandre and Brenner2010) and construct a self-similar solution for the behaviour in the neck region. The height and the pressure are given by

(3.15a,b)\begin{equation} \tilde{h}(\tilde{x},\tilde{t}) = \tilde{h}_n(\tilde{t})H(\varTheta),\quad \tilde{p}(\tilde{x},\tilde{t}) = \tilde{p}_n(\tilde{t})\varPi(\varTheta), \end{equation}

where $\varTheta (\tilde {x},\tilde {t}) = (\tilde {x}-\tilde {x}_n(\tilde {t}))/\widetilde {\ell }(\tilde {t})$ is the self-similarity variable, $\tilde {x}_n(\tilde {t})$ the neck's radial coordinate, $\tilde {p}_n(\tilde {t})$ the pressure at $\tilde {x} = \tilde {x}_n(\tilde {t})$ and $\widetilde {\ell }(\tilde {t})$ the horizontal length scale associated with the high-curvature region, i.e. the neck region. The time derivatives of the height or pressure field have three contributions coming respectively from the height's temporal variation, the change of horizontal extent, and the radial motion of the neck

(3.16)\begin{equation} \frac{\partial \tilde{h}}{\partial \tilde{t}} = \frac{\textrm{d} \tilde{h}_n}{\textrm{d}\tilde{t}}H - \frac{\tilde{h}_n \varTheta}{\widetilde{\ell}} \frac{\textrm{d} \widetilde{\ell}}{\textrm{d}\tilde{t}} \frac{\textrm{d} H}{\textrm{d} \varTheta} - \frac{\tilde{h}_n}{\widetilde{\ell}} \frac{\textrm{d} \tilde{x}_n}{\textrm{d}\tilde{t}}\frac{\textrm{d} H}{\textrm{d} \varTheta}. \end{equation}

Mani et al. (Reference Mani, Mandre and Brenner2010) hypothesized that the advection term is dominant and proposed that the neck region has a wave-like behaviour: $\textrm {d}/\textrm {d}\tilde {t} \approx \tilde {c} \textrm {d}/\textrm {d}\tilde {x}$ where $\tilde {c} = \textrm {d} \tilde {x}_n/\textrm {d}\tilde {t}$ is the neck radial velocity that is considered to be constant during the initial approach. This hypothesis seems to be at odds with our description of the neck dynamics using Wagner theory where $x_n$ evolves as $\sqrt {t}$. This apparent contradiction can be resolved by comparing the time scales associated with the initial approach phase and the motion captured by Wagner's theory. Typically, the minimum thickness is reached for $tU/R < 0.01$ while Wagner's theory is valid up to dimensionless times of the order of $0.1$. This separation of time scales validates the use of a linear approximation for the neck position during the initial approach. Introducing the self-similar fields in (3.14a,b) and using the wave like nature of the solution we obtain

(3.17)$$\begin{gather} \tilde{h}_n\frac{\tilde{c}^2}{\widetilde{\ell}^2}\frac{\textrm{d} H}{\textrm{d}\varTheta} = \frac{\tilde{p}_n}{\widetilde{\ell}}\mathcal{H}\left[\frac{\textrm{d}\varPi}{\textrm{d}\varTheta}\right], \end{gather}$$

(3.18)$$\begin{gather}\tilde{h}_n\frac{\tilde{c}}{\widetilde{\ell}}\frac{\textrm{d} H}{\textrm{d}\varTheta} - \frac{\xi}{12}\frac{\tilde{h}_n^3\tilde{p}_n}{\widetilde{\ell}^2} \frac{\textrm{d}}{\textrm{d}\varTheta}\left(H^3\frac{\textrm{d}\varPi}{\textrm{d}\varTheta}\right) = \frac{\mathcal{E}{\textit{St}}^{5/3}}{{\textit{We}}\tilde{h}_n H}. \end{gather}$$

This set of equations is identical to that obtained by Mani et al. (Reference Mani, Mandre and Brenner2010) in the isothermal case except for the additional term on the right hand side of (3.18) that incorporates the influence of evaporation in the viscous lubrication flow.

If the liquid's latent heat tends towards infinity or no superheat is applied, the evaporation number $\mathcal {E}$ tends towards zero and the terms on the left-hand side balance as in the isothermal case (see Mani et al. (Reference Mani, Mandre and Brenner2010) and Appendix C). On the contrary, if evaporation plays a dominant role as suggested by the strong influence of superheat on the minimum thickness at the neck, the evaporation term balances the downwards motion of the interface (the first term in (3.18)). Equation (3.18) then gives a relationship between the horizontal and the vertical length scales

(3.19)\begin{equation} \widetilde{\ell} \sim \tilde{c} \tilde{h}_n^2 \mathcal{E}^{{-}1} {\textit{St}}^{{-}5/3} {\textit{We}}, \end{equation}

that we combine with the scaling extracted from (3.17), $\tilde {p}_n \sim \tilde {h}_n \tilde {c}^2/\widetilde {\ell }$, to obtain a relation between the pressure and height at the neck

(3.20)\begin{equation} \tilde{p}_n \sim \tilde{c}\tilde{h}_n^{{-}1}\mathcal{E}{\textit{St}}^{5/3}{\textit{We}}^{{-}1}. \end{equation}

As $h_n$ decreases, the initially neglected Laplace pressure at the neck $\gamma h_n/\ell ^2$, that scales as $h_n^{-3}$, diverges quicker than the gas pressure at the neck $p_n$, that evolves as $h_n^{-1}$. The hypothesis to neglect surface tension is no longer valid as the drop approaches the solid, similarly as for isothermal impacts. The balance between $p_n$ and the Laplace pressure sets the minimum film thickness $h_m$

(3.21)\begin{equation} \frac{h_m}{R} \sim {\textit{We}}^{{-}1}\mathcal{E}^{1/2}. \end{equation}

Equation (3.21) predicts a power-law decrease of the minimum thickness as a function of impact velocity $U$ with an exponent $-2$ when fixing the drop radius and superheat, consistent with the experimental findings displayed in figure 4(e), where the dashed lines are guides to the eye with a slope of $-2$. This power-law decrease, $h_m \propto U^{-2}$, is close to that numerically predicted in the isothermal case where the minimum thickness follows a power law with exponent $-20/9$ (Mani et al. Reference Mani, Mandre and Brenner2010). In Appendix C, we derive this isothermal scaling and show that it does not allow us to capture the effect of substrate temperature, contrary to (3.21) which explicitly involves superheat. Testing the effect of the substrate temperature $T_s$ requires us to take into account the temperature-dependent values of the gas viscosity, density and thermal conductivity as well as the value of the liquid surface tension. The viscous effects are introduced by the gas drainage associated with dimple formation. We thus use the temperature-dependent gas viscosity extracted from our measurements of the dimple height. The gas density and thermal conductivity, on the contrary, are related to vapour generation at the neck. Given the conduction time scale $h^2/\kappa _v\approx 0.1\,\mathrm {\mu }$s, we consider steady heat transfer in the lubrication layer. The temperature profile is linear between the substrate temperature $T_s$ and the temperature at the liquid–gas interface where evaporation occurs, that is $T_b$. We then evaluate the gas density $\rho _g$ and thermal conductivity $k_g$ at $(T_s+T_b)/2$, assuming that the gas phase in the neck region is constituted of ethanol vapour only, and taking the surface tension $\gamma$ at temperature $T_b$. In figure 4( f), we compare the normalized minimum thickness $h_m/R$ with the prediction of (3.21). The data for different superheats and impact velocities collapse onto a line with prefactor $4.0\pm 0.2$. This prefactor can be rationalized by coming back on the hypothesis to neglect the subdominant gas drainage contribution to levitation at the neck. This assumption leads us to underestimate the minimum thickness. We thus expect a prefactor larger than one in (3.21), which is consistent with what we see in experiments. The influence of drainage might also explain the deviation from the scaling, observed in figure 4( f) for the largest values of $We^{-1}\mathcal {E}^{1/2}$, as these discrepancies correspond to the largest thicknesses, for which we expect the relative importance of drainage compared with evaporation to increase.

4.1. Phenomenology

4.2. A comment on the dynamic Leidenfrost temperature

Having described the phenomenology of the three collapse modes present in the phase diagram, we now discuss their influence on the dynamic Leidenfrost transition. Figure 5(a) contains two key pieces of information that give us new insight on the dynamic Leidenfrost temperature. (i) In the low-velocity limit, the static Leidenfrost temperature, that is approximately 150 $^\circ$C for ethanol, does not act as a lower asymptote for the dynamic Leidenfrost temperature. The later can fall below the static value as a consequence of the transient stability of the gas cushion promoted by drainage. Thus, remarkably, dynamic levitation can be less demanding than static levitation. This observation reveals the existence of metastable Leidenfrost drops at impact (Baumeister, Hamill & Hendricks Reference Baumeister, Hamill and Hendricks1966; Harvey, Harper & Burton Reference Harvey, Harper and Burton2021) and evidences that the static Leidenfrost temperature does not express the hydrodynamic ability of the vapour layer to support the liquid, in agreement with the recently proposed understanding of the Leidenfrost effect as a directed percolation phase transition (Chantelot & Lohse Reference Chantelot and Lohse2021). (ii) The gas film collapse can be driven by multiple mechanisms: namely short-time contact, late-time contact and contact induced by vertical oscillations of the gas film. Below 150 $^\circ$C, the transition to the Leidenfrost state is controlled by short and late-time contact, similarly as on isothermal substrates. For larger substrate temperatures, gas film oscillations occur for lower impact velocities than short-time contact, thus becoming the relevant mechanism in the transition towards the dynamic Leidenfrost effect. More generally, we expect the interplay between the different types of gas film collapse to be sensitive to liquid and substrate properties, such as roughness. Predicting the dynamic Leidenfrost temperature thus requires a better understanding of each collapse mode, especially the oscillation induced contact mode, which we further investigate.

4.3. Oscillations: a minimal model

We now focus on understanding the appearance of oscillations in the squeezed gas layer, a feature specific to impacts on superheated surfaces unlike short or late-time contact. Our observations are reminiscent of the rapid vibrations that appear at the base of soft sublimable solids and liquid drops, as contact occurs with a hot substrate during an impact (Khavari & Tran Reference Khavari and Tran2017; Waitukaitis et al. Reference Waitukaitis, Zuiderwij, Souslov, Coulais and Van Hecke2017; Waitukaitis, Harth & Van Hecke Reference Waitukaitis, Harth and Van Hecke2018). Oscillations can also spontaneously develop, without contact, at the bottom interface of drops in the static Leidenfrost state (Liu & Tran Reference Liu and Tran2020; Bouillant et al. Reference Bouillant, Cohen, Clanet and Quéré2021; Graeber et al. Reference Graeber, Regulagadda, Hodel, Küttel, Landolf, Schutzius and Poulikakos2021) or levitated by a steady airflow (Bouwhuis et al. Reference Bouwhuis, Winkels, Peters, Brunet, Van Der Meer and Snoeijer2013). Such vibrations are responsible for the vertical rebounds of Leidenfrost drops (Liu & Tran Reference Liu and Tran2020; Graeber et al. Reference Graeber, Regulagadda, Hodel, Küttel, Landolf, Schutzius and Poulikakos2021), and for the subsequent emergence of star shapes (Holter & Glasscock Reference Holter and Glasscock1952; Brunet & Snoeijer Reference Brunet and Snoeijer2011; Bouillant et al. Reference Bouillant, Cohen, Clanet and Quéré2021). For both levitation mechanisms, the appearance of vertical oscillations has been linked to the coupling of drop motion and flow in the thin gas film. Although the frequencies reported in these systems are of the order of $100$ Hz, two orders of magnitude smaller than those we measure at impact, our observations can be related to this hydrodynamic mechanism. (i) Oscillations grow as the dimple height starts to increase, that is when the vapour flow influences the whole liquid–gas interface. (ii) There is a temperature threshold below which vibrations do not appear, i.e. a sufficient gas flow is needed to make the interface unstable.

We thus build a hydrodynamic model that accounts for the motion of the bottom interface after the minimum thickness is reached at the neck by adapting the model of Bouillant et al. (Reference Bouillant, Cohen, Clanet and Quéré2021) to the impact situation. We reproduce here for completeness the derivation of the equation coupling the drop motion and the lubrication flow created by vapour generation. We apply Newton's second law to the vaporizing drop in the vertical direction. The momentum variation has two sources: the motion of the drop's centre of mass, $m\mathrm {d}^2 z_{cm}/\mathrm {d} t^2$, and the ejection of vapour, $v_{e}\mathrm {d} m/\mathrm {d} t$ where $m(t)$ is the drop mass, $z_{cm}(t)$ the vertical position of its centre of mass and $v_{e}(t)$ the vertical ejection velocity of vapour in the reference frame of the drop. We consider a simplified geometry, that of a gas layer with uniform thickness $h(t)$ and constant radial extent $R$, allowing us to derive the evaporation rate and the lubrication force analytically (Biance et al. Reference Biance, Clanet and Quéré2003). Indeed, we obtain the evaporation rate $\mathrm {d} m/\mathrm {d} t$ by integrating the evaporation rate per unit area $e = - k_v\Delta T/(\mathcal {L} h)$, derived in § 3.4, over the bottom interface of the drop assuming that evaporation predominantly occurs in the gas layer. The lubrication force is obtained by integrating the pressure $p_g(r,t)$, derived from the lubrication equation (3.5), over the gas layer: $F_L(t) = \int _0^R p_g(r,t) 2{\rm \pi} r \,\mathrm {d}r$. We then write the momentum balance

(4.1)\begin{equation} m\frac{\mathrm{d}^2 z_{cm}}{\mathrm{d} t^2} + \frac{k_v \Delta T{\rm \pi} R^2}{\mathcal{L} h} v_{e} ={-}\frac{6\eta_v{\rm \pi} R^4}{\xi h^3} \left(\frac{\mathrm{d}h}{\mathrm{d}t} - \frac{1}{\rho_v}\frac{k_v\Delta T}{\mathcal{L} h}\right) -mg, \end{equation}

and express the ejection velocity $v_e$ as the sum of the absolute ejection velocity $-k_v\Delta T/(\rho _v\mathcal {L}h)$, derived from a mass balance, and of the interface velocity $-\mathrm {d}h/\mathrm {d}t$

(4.2)\begin{align} m\frac{\mathrm{d}^2 z_{cm}}{\mathrm{d} t^2} + \left(\frac{6\eta_v{\rm \pi} R^4}{\xi h^3}-\frac{k_v \Delta T {\rm \pi}R^2}{ \mathcal{L} h} \right) \frac{\mathrm{d}h}{\mathrm{d}t} - \frac{k_v^2 \Delta T^2 {\rm \pi}R^2}{\rho_v \mathcal{L}^2 h^2} - \frac{6\eta_v k_v \Delta T {\rm \pi}R^4}{\xi\rho_v \mathcal{L} h^4} - mg = 0. \end{align}

Equation (4.2) is identical to that obtained by Bouillant et al. (Reference Bouillant, Cohen, Clanet and Quéré2021) in the static Leidenfrost situation. Relating the motion of the bottom interface of the drop with that of its centre of mass is the critical step where the static and impact situations differ. At short times, i.e. for $t \ll \tau _i = R/U$, the drop is in the kinematic phase. The motion of the drop's centre of mass is approximately ballistic: $z_{cm}(t) \approx h(t) + R - Ut$, allowing us to recover an equation for the vertical motion of the liquid–gas interface from (4.2) which we non-dimensionalize using the following transformations:

(4.3a,b)\begin{equation} h = h_d \hat{h},\quad t = \tau \hat{t}, \end{equation}

where we chose the dimple height $h_d$ as length scale as it gives the correct order of magnitude of the film thickness after the initial approach and denote as $\tau$ the characteristic time of the oscillations of the gas film. Plugging these non-dimensional variables in (4.2) allows us to identify the physical phenomena and time scales involved in the gas film dynamics

(4.4)\begin{align} & \frac{\mathrm{d}^2 \hat{h}}{\mathrm{d} \hat{t}^2} + \tau \frac{\beta}{m} \left(\frac{6{\rm \pi}}{\xi}\left(\frac{R}{h_d}\right)^3\hat{h}^{{-}3} - {\rm \pi}\frac{\rho_v}{\rho_l}\frac{\mathcal{E}{\textit{St}}^2}{{\textit{We}}}\frac{R}{h_d} \hat{h}^{{-}1} \right) \frac{\mathrm{d}\hat{h}}{\mathrm{d}\hat{t}} \nonumber\\ &\quad -\tau^2 \frac{\gamma}{m}\left( {\rm \pi}\frac{\rho_v}{\rho_l}\frac{\mathcal{E}^2{\textit{St}}^2}{{\textit{We}}} \left(\frac{R}{h_d}\right)^3\hat{h}^{{-}2} + \frac{6{\rm \pi}}{\xi}\mathcal{E} \left(\frac{R}{h_d}\right)^5\hat{h}^{{-}4}\right) - \tau^2\frac{g}{h_d} = 0, \end{align}

where we introduced the damping coefficient $\beta = \eta _v R$, and the evaporation number $\mathcal {E}$ already defined in (3.13). We further simplify (4.4) by noticing that the change of momentum linked to the absolute ejection velocity, that is proportional to $1/h^2$, and the gravity term are orders of magnitude smaller than the contribution from the lubrication equation, that varies as $1/h^4$. We can therefore approximate (4.4) by

(4.5)\begin{align} \frac{\mathrm{d}^2 \hat{h}}{\mathrm{d} \hat{t}^2} + \tau \frac{\beta}{m} \left(\frac{6{\rm \pi}}{\xi}\left(\frac{R}{h_d}\right)^3\hat{h}^{{-}3} - {\rm \pi}\frac{\rho_v}{\rho_l}\frac{\mathcal{E}{\textit{St}}^2}{{\textit{We}}}\frac{R}{h_d} \hat{h}^{{-}1} \right) \frac{\mathrm{d}\hat{h}}{\mathrm{d}\hat{t}} - \tau^2 \frac{\gamma}{m}\frac{6{\rm \pi}}{\xi}\mathcal{E}\left(\frac{R}{h_d}\right)^5\hat{h}^{{-}4} = 0. \end{align}

Equation (4.5) describes a nonlinear oscillator and allows us to identify the three time scales involved in the evolution of the gas film. We start by discussing the characteristic time linked to the $\hat {h}^{-4}$ term, that we denote $\tau _o$. It is proportional to $\sqrt {m/\gamma }$, indicating that it can be understood as the characteristic time of a spring–mass system. This suggests that the lubrication force acts as a nonlinear spring with time scale $\tau _o$. Plugging in typical values, we obtain $\tau _o \approx 10\,\mathrm {\mu }$s, a value of the same order of magnitude as the period reported in figure 5(d). Equation (4.5) thus gives us insight into the mechanism that can lead to vertical oscillations. If the bottom interface is perturbed towards the substrate, we compress the spring and the lubrication force increases. The interface is then repelled and it can either come back to its initial position or overshoot it. In the latter situation, the lubrication pressure decreases and the interface moves down starting an oscillatory motion. The appearance of these vibrations is controlled by the damping term, whose time scale is proportional to $m/\beta$. It is composed of a viscous damping term, with characteristic time $\tau _d = \xi m h_d^3/(6{\rm \pi} \beta R^3)$, and an amplification term (i.e. negative damping), with time scale $\tau _a = m\rho _l{\textit {We}} h_d/({\rm \pi} \beta \rho _v\mathcal {E}{\textit {St}}^2 R)$, associated with vapour ejection. Using typical numerical values at impact, we find that the viscous damping time scale $\tau _d \approx 1\,\mathrm {\mu }$s is one order of magnitude smaller than the oscillation period $\tau _o$ and that the amplification characteristic time $\tau _a \approx 1$ s is larger than the inertio-capillary time $\tau _c$. These findings are in contradiction to the observations reported in figure 5(d) where we observe fast oscillations modulated by a slower envelope that acts on a characteristic time of the order of 1 ms (figure 5d). Yet, the qualitative interplay of damping and amplification reproduces some of our observations. For a fixed thickness $h_d$, viscous dissipation always dominates at low substrate temperatures, but, as the superheat increases (i.e. $\mathcal {E}$ increases), the damping coefficient decreases, leading to the appearance of damped oscillations and finally to unbounded oscillations as the total damping term becomes negative. This behaviour is in qualitative agreement with the existence of a temperature threshold, here $T_s = 150\,^\circ$C, below which we do not observe oscillations (figure 5a). When fixing the superheat $\Delta T$ and investigating the role of the film thickness, viscous damping becomes dominant as $h_d$ decreases. This evolution contradicts our observations which show that with increasing impact velocity, that is decreasing film thickness, damped oscillations appear first, followed by unbounded ones (figure 5a). Our overestimation of viscous dissipation likely results from the assumption of a simplified, time-independent geometry. Similarly as for static Leidenfrost drops, we expect the detailed interplay of viscous dissipation and evaporation to depend on the film geometry (Sobac et al. Reference Sobac, Rednikov, Dorbolo and Colinet2014).

Although the minimal model does not correctly capture the detailed balance of viscous dissipation and evaporation, it provides a mechanism for oscillations that we can test against our observations. We assume in the following that damping and amplification act on a longer time scale than the fast oscillations, as observed experimentally. We introduce this hypothesis in (4.4) by choosing $\tau _o$ as characteristic time and taking the ratio $\varepsilon = \tau _o/\tau _d$ as the fundamental small parameter of the problem

(4.6)\begin{equation} \frac{\mathrm{d}^2 \hat{h}}{\mathrm{d} \hat{t}^2} + \varepsilon\left(\hat{h}^{{-}3} - \frac{\tau_d}{\tau_a}\hat{h}^{{-}1} \right) \frac{\mathrm{d}\hat{h}}{\mathrm{d}\hat{t}} - \hat{h}^{{-}4} = 0. \end{equation}

We separate the fast oscillations from the slow envelope using a multiscale approach (Hinch Reference Hinch1991). We define a fast time $\hat {t}_0 = \hat {t}$ and a slow time $\hat {t}_1 = \varepsilon \hat {t}$ and introduce the asymptotic expansion $\hat {h}(\hat {t},\varepsilon ) = \hat {h}_0(\hat {t},\hat {t}_1) + \varepsilon \hat {h}_1(\hat {t},\hat {t}_1) + O(\varepsilon \hat {t})$ into (4.6). At leading order, we obtain an equation that describes the dynamics of the gas film on the fast time scale

(4.7)\begin{equation} \frac{\partial^2 \hat{h}_0}{\partial \hat{t}^2} -\hat{h}_0^{{-}4} = 0. \end{equation}

We now look for oscillations as a perturbation to the film thickness. We define a perturbation of the form $\hat {h}_0(\hat {t},\hat {t}_1) = \hat {H}_0(\hat {t},\hat {t}_1) + \zeta \hat {H}_0'(\hat {t},\hat {t}_1)$ where $\zeta \ll 1$ and $\hat {H}_0(\hat {t},\hat {t}_1)$ is a solution of (4.7) that we assume to verify the initial conditions set at the end of the initial approach. At leading order, (4.7) becomes a harmonic oscillator

(4.8)\begin{equation} \frac{\partial^2 \hat{H}'_0}{\partial \hat{t}^2} + \frac{4}{\hat{H}_0^5}\hat{H}'_0 = 0, \end{equation}

which admits oscillatory solutions with pulsation $\hat {\omega } = \sqrt {4/\hat {H}_0^5}$. Physically, the film thickness after the initial approach varies between the constant dimple height $h_d$ and the increasing thickness at the neck $h_n(t)$. As the oscillations appear at the dimple, we choose $h_d$ as a typical scale for the initial film thickness. With this choice, we obtain the oscillation frequency

(4.9)\begin{equation} f_{th} = \frac{1}{2{\rm \pi}}\sqrt{\frac{24{\rm \pi}}{\xi}\frac{\gamma}{m}\mathcal{E} \left(\frac{R}{h_d}\right)^5} = \frac{1}{2{\rm \pi}\tau_c}\sqrt{ \frac{18}{\xi}\mathcal{E} \left( \frac{R}{h_d} \right)^5}. \end{equation}

We now compare the observed value of $f$ with the prediction of (4.9). In figure 6(a), we show the measured frequency $f$, extracted from the space–time representation of the RI data (figure 5d), as a function of the impact velocity $U$ for various substrate temperatures $T_s$. The frequency $f$ ranges from $22$ to $48$ kHz. For a fixed superheat, $f$ increases with the impact velocity. This variation is in qualitative agreement with the prediction of (4.9) as, combining it with the scaling relation linking the dimple height to the Stokes number (3.8), we obtain a power-law relationship with exponent $5/3$ between $f_{th}$ and the impact velocity $U$, or in dimensionless form, the Stokes number St

(4.10)\begin{equation} f_{th}\tau_c \sim {\textit{St}}^{5/3}\mathcal{E}^{1/2}.\end{equation}

This power-law behaviour is consistent with the experimental variation displayed in figure 6(a), where the dashed lines are drawn as guides to the eye with a slope $5/3$. Equation (4.10) also reveals the power law, with exponent $-1/3$, that links the frequency $f_{th}$ and the radius $R$. This weak dependence justifies, a posteriori, our hypothesis to disregard the time variation of the vapour film radius. Probing the effect of the substrate temperature $T_s$ is more subtle, as it affects both the evaporation number $\mathcal {E}$ and the dimple height $h_d$. To quantitatively test the frequency prediction, we evaluate the temperature-dependent gas properties. As the conduction time scale $h_d^2/\kappa _v$ is of the order of $0.1\,\mathrm {\mu }$s, we consider steady heat transfer in the vapour film. We evaluate the density, thermal conductivity and viscosity at $(T_s+T_b)/2$ and plot the measured frequency $f$ normalized by the prediction $f_{th}$ as a function of $U$ (figure 6b). In this plot, the dimple heights $h_d$ are measured. We choose $\xi = 4$, as the ratio of the liquid to the vapor viscosity $\eta _l(T_b)/\eta _v((T_s+T_b)/2) \approx 35$ is smaller than the viscosity contrast at the liquid/air interface under ambient conditions ($\eta _l/\eta _g \approx 55$), favouring slip. The data collapse on a constant $0.25 \pm 0.1$ when varying $T_s$. The hydrodynamic model captures both the effect of superheat and impact velocity, but consistently overpredicts by a factor four the oscillation frequency. This discrepancy can be explained by the simplicity of the model that disregards the spatial variation of the gas film thickness and the temporal variation of its radial extent. More elaborate models could also introduce a slip length at the liquid/gas interface, and discuss liquid entrainment driven by the vapour flow.

Figure 6. (a) Oscillation frequency $f$ of the gas film for impacts at velocity $U$ on substrates with different $T_s$ (same legend as in (b)). Dashed lines with slope $5/3$ are drawn as a guide to the eye. (b) Measured frequency $f$ divided by the frequency $f_{th}$ predicted from (4.9) as a function of $U$ for various substrate temperatures. The data collapse on a constant value $0.25 \pm 0.1$. Although the data are consistent with the scaling of (4.9), the prefactor in that expression is approximately four times too large.