3.1 The geometry of Leidenfrost drops on curved surfaces

The shape of a Leidenfrost drop is determined by a competition between surface tension and gravity. It has been shown that a non-wetting drop with radius $R\ll l_{c}$ will exhibit a quasi-spherical profile except for the bottom region where the drop is slightly flattened by gravity, which still holds true for small Leidenfrost drops (Burton et al. Reference Burton, Sharpe, van der Veen, Franco and Nagel2012). In comparison, for large Leidenfrost drops, i.e. $R\gg l_{c}$ , the shape of the drops is dominated by gravity and resembles a circular puddle with a constant thickness of approximately $2l_{c}$ (Biance et al. Reference Biance, Clanet and Quéré2003). However, these large puddles are susceptible to the Rayleigh–Taylor instability, which is manifested by bubbles rising from the vapour layer beneath the drop (Biance et al. Reference Biance, Clanet and Quéré2003). In our experiment, we used curved surfaces (see § 2) in order to suppress the Rayleigh–Taylor instability and obtain large, stable Leidenfrost drops (Snoeijer et al. Reference Snoeijer, Brunet and Eggers2009; Quéré Reference Quéré2013; Trinh et al. Reference Trinh, Kim, Hammoud, Howell, Chapman and Stone2014).

Figure 2.

Schematic of the suppression of Rayleigh–Taylor instability at the liquid interface (solid blue curve) by a curved substrate (solid grey curve) with a curvature $\unicode[STIX]{x1D705}=1/R_{s}$ , where $R_{s}$ is the radius of curvature defined before, $\unicode[STIX]{x1D706}_{c}$ is the capillary wavelength.

The effects of surface curvature on the Rayleigh–Taylor instability can be seen through the following simple model, similar to the original model used by Biance et al. (Reference Biance, Clanet and Quéré2003). Assuming the liquid–vapour interface beneath a Leidenfrost drop is perturbed with axisymmetric, sinusoidal variations, as illustrated in figure 2, then the new drop interface can be expressed in cylindrical coordinates as

(3.1) $$\begin{eqnarray}S(r)=e+\frac{r^{2}}{2R_{s}}+\unicode[STIX]{x1D716}\cos \left(\frac{2\unicode[STIX]{x03C0}r}{\unicode[STIX]{x1D706}_{c}}\right),\end{eqnarray}$$

where $e$ is the mean vapour layer thickness, $\unicode[STIX]{x1D716}$ is the perturbation amplitude and $\unicode[STIX]{x1D706}_{c}$ is the capillary wavelength generated at the bottom of the drop (see figure 11). We also assume that $R_{s}\gg e\gg \unicode[STIX]{x1D716}$ , implying a small perturbation to the equilibrium shape of the drop (Duchemin et al. Reference Duchemin, Lister and Lange2005; Lister et al. Reference Lister, Thompson, Perriot and Duchemin2008). For a stable interface, the pressure at the drop centre should be smaller or equal to the pressure at $r\approx \unicode[STIX]{x1D706}_{c}/2$ in order to drive liquid back to the centre. If the pressure in the vapour layer is constant, then to leading order, the pressures at $r=0$ , and $r=\unicode[STIX]{x1D706}_{c}/2$ are

(3.2) $$\begin{eqnarray}\displaystyle & \displaystyle P_{0}\approx \unicode[STIX]{x1D6FE}\unicode[STIX]{x1D705}|_{r\rightarrow 0}=2\unicode[STIX]{x1D6FE}\left(\frac{1}{R_{s}}-\frac{4\unicode[STIX]{x1D716}\unicode[STIX]{x03C0}^{2}}{\unicode[STIX]{x1D706}_{c}^{2}}\right), & \displaystyle\end{eqnarray}$$

(3.3) $$\begin{eqnarray}\displaystyle & \displaystyle P_{1}\approx \unicode[STIX]{x1D6FE}\unicode[STIX]{x1D705}|_{r\rightarrow \unicode[STIX]{x1D706}_{c}/2}-\unicode[STIX]{x1D70C}_{l}g\unicode[STIX]{x0394}z=\frac{4\unicode[STIX]{x1D716}\unicode[STIX]{x03C0}^{2}\unicode[STIX]{x1D6FE}}{\unicode[STIX]{x1D706}_{c}^{2}}+\frac{2\unicode[STIX]{x1D6FE}+\unicode[STIX]{x1D70C}_{l}g\unicode[STIX]{x1D706}_{c}}{2R_{s}}-2\unicode[STIX]{x1D716}\unicode[STIX]{x1D70C}_{l}g, & \displaystyle\end{eqnarray}$$

where the height difference between the two points is $\unicode[STIX]{x0394}z=S(r)_{r\rightarrow 0}-S(r)_{r\rightarrow \unicode[STIX]{x1D706}_{c}/2}=2\unicode[STIX]{x1D716}-\unicode[STIX]{x1D706}_{c}^{2}/8R_{s}$ . The condition for stability can be found by equating (3.2) and (3.3), leading to an expression for the wavelength:

(3.4) $$\begin{eqnarray}\unicode[STIX]{x1D706}_{c}=2\left(2\unicode[STIX]{x1D716}R_{s}-\frac{\sqrt{2\unicode[STIX]{x1D70C}_{l}g\unicode[STIX]{x1D716}R_{s}(-3\unicode[STIX]{x03C0}^{2}\unicode[STIX]{x1D6FE}+2\unicode[STIX]{x1D70C}_{l}g\unicode[STIX]{x1D716}R_{s})}}{\unicode[STIX]{x1D70C}_{l}g}\right)^{1/2}.\end{eqnarray}$$

Since the perturbation amplitude and radius of curvature always appear as a product, (3.4) can also be written as

(3.5) $$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x1D706}_{c}}{l_{c}}}=2(2\unicode[STIX]{x1D712}-\sqrt{2\unicode[STIX]{x1D712}(2\unicode[STIX]{x1D712}-3\unicode[STIX]{x03C0}^{2})})^{1/2},\end{eqnarray}$$

where $\unicode[STIX]{x1D712}=\unicode[STIX]{x1D716}R_{s}/l_{c}^{2}$ . For finite $\unicode[STIX]{x1D716}$ , in the limit $R_{s}\rightarrow \infty$ , equation (3.4) simplifies to $\unicode[STIX]{x1D706}_{c}/2\approx 3.85l_{c}$ , in agreement with the prediction in Biance et al. (Reference Biance, Clanet and Quéré2003) for a flat surface. However, the addition of a curved surface couples the perturbation amplitude to the radius of curvature. The quantity $2\unicode[STIX]{x1D712}-3\unicode[STIX]{x03C0}^{2}$ must be positive in order for $\unicode[STIX]{x1D706}_{c}$ to be real, and thus represent the condition for instability. Subsequently, on a curved surface, the perturbation amplitude must satisfy $\unicode[STIX]{x1D716}\gtrsim 14.8l_{c}^{2}/R_{s}$ in order to lead to the Rayleigh–Taylor instability.

Figure 3.

The variations of drop radius $R$ with respect to volume $V$ for Leidenfrost water drops. The dashed blue line shows the theoretical prediction for a drop on a flat surface assuming that the contact angle of the drop is $180^{\circ }$ as indicated in the text.

Although the curved surface suppresses the Rayleigh–Taylor instability, it still has an effect on the overall drop shape. We investigated the influence of surface curvature on the drop shape by depositing water drops of given volumes $V$ on a curved surface with $T_{s}=623~\text{K}$ . We measured the radius $R$ of the drops from recorded images taken immediately after deposition prior to the onset of oscillations. As can be seen in figure 3, stable Leidenfrost drops with $R\approx 11l_{c}$ are obtained on the curved surface, while on a flat substrate, the maximum radius of a stable Leidenfrost drop is $R\approx 3.95l_{c}$ (Biance et al. Reference Biance, Clanet and Quéré2003; Snoeijer et al. Reference Snoeijer, Brunet and Eggers2009; Burton et al. Reference Burton, Sharpe, van der Veen, Franco and Nagel2012), suggesting that surface curvature plays a crucial role in suppressing the Rayleigh–Taylor instability. The dashed blue line in figure 3 shows the theoretical prediction for a drop on a flat surface by solving the Young–Laplace differential equation numerically and assuming that the contact angle of the drop is $180^{\circ }$ (Burton et al. Reference Burton, Huisman, Alison, Rogerson and Taborek2010, Reference Burton, Sharpe, van der Veen, Franco and Nagel2012). The experimental results show excellent agreement with the theoretical prediction, indicating that the shape of the Leidenfrost drops is not strongly affected by the surface curvature. More specifically, when $R<l_{c}$ (2.5 mm), the drops are quasi-spherical and thus $R\propto V^{1/3}$ , whereas $R\propto V^{1/2}$ for $R>l_{c}$ , which is expected for puddle-like drops with constant thickness.

However, one can also notice that most of the experimental data are slightly less than the corresponding theoretical prediction, which we attribute to two possible reasons. First, for small drops, the evaporation begins from the moment of deposition on the hot surface, which removes a small amount of water prior to imaging. Second, large drops are thicker in the centre due to the underlying curved substrate, leading to a smaller apparent radius for a given volume. Thus, although surface curvature and evaporation may somewhat reduce the drop size, the effects seem to be a minor influence on the overall drop shape.

3.2 ‘Breathing mode’ of small Leidenfrost drops

Caswell (Reference Caswell2014) experimentally characterized axisymmetric oscillations in the radius of small Leidenfrost drops using interference imaging. The drops displayed small-amplitude changes in the radius of the flat region near the surface. Due to volume conservation, an increase (decrease) in the drop radius leads to a decrease (increase) in the thickness of the drop. Caswell (Reference Caswell2014) found that the oscillation frequency of the breathing mode, $f_{b}$ , obeyed a distinct power law, $f_{b}\propto R_{0}^{-0.68\pm 0.01}$ , where $R_{0}$ is the average drop radius during the oscillation. This dependence is distinctly different than the expected three-dimensional dispersion relation for inviscid spherical drops, $f\propto R_{0}^{-3/2}$ (Rayleigh Reference Rayleigh1879). Here we provide an analytical model which explains this contrast and fits the experimental data with no adjustable parameters.

To leading order, and due to the axisymmetry of the breathing mode, we model a Leidenfrost drop as an incompressible liquid cylinder of volume  $=V$ and a time-dependent radius, $R(t)$ . We will assume that the bottom of the cylinder is fixed at $z=0$ , which is reasonable if the thickness of the vapour layer varies much less than the radius. In cylindrical coordinates, $(r,\unicode[STIX]{x1D719},z)$ , the simplest form for the velocity which satisfies $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{v}=0$ and the boundary conditions $\boldsymbol{v}\boldsymbol{\cdot }\hat{\boldsymbol{r}}|_{r=R(t)}=R^{\prime }(t)$ and $\boldsymbol{v}\boldsymbol{\cdot }\hat{\boldsymbol{z}}|_{z=0}=0$ is:

(3.6) $$\begin{eqnarray}\boldsymbol{v}={\displaystyle \frac{R^{\prime }(t)}{R(t)}}(r,0,-2z).\end{eqnarray}$$

For simplicity, we will write $R(t)=R$ , and $R^{\prime }(t)=R^{\prime }$ . The total kinetic energy of the drop is then:

(3.7) $$\begin{eqnarray}\displaystyle T=\int \frac{1}{2}\unicode[STIX]{x1D70C}_{l}|\boldsymbol{v}|^{2}\,\text{d}V={\displaystyle \frac{\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}_{l}R^{2}}{R^{\prime 2}}}\int _{0}^{R}\int _{0}^{h}r(r^{2}+4z^{2})\,\text{d}z\,\text{d}r. & & \displaystyle\end{eqnarray}$$

Evaluating the integrals, and using the fact that $V=\unicode[STIX]{x03C0}R^{2}h$ , where $h$ is the time-dependent height of the cylinder, we obtain

(3.8) $$\begin{eqnarray}\displaystyle T={\displaystyle \frac{1}{12}}\unicode[STIX]{x1D70C}_{l}V\left(3+\frac{8V^{2}}{\unicode[STIX]{x03C0}^{2}R^{6}}\right)R^{\prime 2}. & & \displaystyle\end{eqnarray}$$

The total potential energy is the sum of gravitational potential energy and surface energy:

(3.9) $$\begin{eqnarray}U={\displaystyle \frac{1}{2}}\unicode[STIX]{x1D70C}_{l}Vgh+2\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FE}R(h+R)={\displaystyle \frac{gV^{2}\unicode[STIX]{x1D70C}_{l}+4\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FE}R(V+\unicode[STIX]{x03C0}R^{3})}{2\unicode[STIX]{x03C0}R^{2}}}.\end{eqnarray}$$

The equilibrium drop radius, $R_{0}$ , can be found by minimizing the potential energy with respect to $R$ ,

(3.10) $$\begin{eqnarray}{\displaystyle \frac{\text{d}U}{\text{d}R}}=0=4\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FE}R_{0}-{\displaystyle \frac{2V\unicode[STIX]{x1D6FE}}{R_{0}^{2}}}-{\displaystyle \frac{g\unicode[STIX]{x1D70C}_{l}V^{2}}{\unicode[STIX]{x03C0}R_{0}^{3}}}.\end{eqnarray}$$

For large drops where $R\gg h$ , the first and third terms must balance. Equating these two terms and using the fact that $V=\unicode[STIX]{x03C0}R^{2}h$ , we see that

(3.11) $$\begin{eqnarray}h\approx 2\sqrt{{\displaystyle \frac{\unicode[STIX]{x1D6FE}}{\unicode[STIX]{x1D70C}_{l}g}}},\quad R_{0}\rightarrow \infty ,\end{eqnarray}$$

which agrees with the expected asymptotic thickness of large Leidenfrost drops (Biance et al. Reference Biance, Clanet and Quéré2003). We may now define the Lagrangian of the system as $L=T-U$ , and apply the Euler–Lagrange equation to obtain a differential equation for $R$ :

(3.12) $$\begin{eqnarray}R^{\prime \prime }=\frac{6\unicode[STIX]{x1D70C}_{l}V^{2}g\unicode[STIX]{x03C0}R^{4}+12\unicode[STIX]{x1D6FE}V\unicode[STIX]{x03C0}^{2}R^{5}-24\unicode[STIX]{x1D6FE}\unicode[STIX]{x03C0}^{3}R^{8}+24\unicode[STIX]{x1D70C}_{l}V^{3}R^{\prime 2}}{8\unicode[STIX]{x1D70C}_{l}V^{3}R+3\unicode[STIX]{x1D70C}_{l}V\unicode[STIX]{x03C0}^{2}R^{7}}.\end{eqnarray}$$

To proceed further, we will linearize the equation by considering only small oscillations of the radius, $R=R_{0}(1+\unicode[STIX]{x1D716}\text{e}^{\text{i}\unicode[STIX]{x1D714}t})$ , where $R_{0}$ is equilibrium drop radius found by solving (3.10), $\unicode[STIX]{x1D716}$ is the perturbation amplitude and $\unicode[STIX]{x1D714}$ is the angular frequency of the oscillation. Assuming that $\unicode[STIX]{x1D716}\ll 1$ , to leading order (3.12) reduces to an expression for the angular frequency:

(3.13) $$\begin{eqnarray}\unicode[STIX]{x1D714}^{2}={\displaystyle \frac{576\unicode[STIX]{x1D6FE}^{2}(\unicode[STIX]{x1D6FE}+\unicode[STIX]{x1D6E4})+36\unicode[STIX]{x1D70C}_{l}^{2}g^{2}R_{0}^{4}(\unicode[STIX]{x1D6FE}+2\unicode[STIX]{x1D6E4})+384\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D70C}_{l}gR_{0}^{2}(5\unicode[STIX]{x1D6FE}+2\unicode[STIX]{x1D6E4})}{\unicode[STIX]{x1D70C}_{l}R_{0}^{3}(1120\unicode[STIX]{x1D6FE}^{2}+9\unicode[STIX]{x1D70C}_{l}^{2}g^{2}R_{0}^{4}+192\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D70C}_{l}gR_{0}^{2})}},\end{eqnarray}$$

where $\unicode[STIX]{x1D6E4}\equiv \sqrt{\unicode[STIX]{x1D6FE}^{2}+4\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D70C}_{l}gR_{0}^{2}}$ and we have substituted $V$ for the equilibrium radius $R_{0}$ using (3.10).

Figure 4.

(a) Breathing mode oscillation frequency $f_{b}$ of small Leidenfrost drops as a function of the average drop radius $R_{0}$ . The dashed red curve represents the power law $f_{b}\propto R_{0}^{-0.68}$ , and the solid blue curve represents the dispersion relation from (3.13). (b) A zoomed-in view of the data in (a). With permission, the data shown here are taken from Caswell (Reference Caswell2014).

Taking typical values of the parameters in (3.13) for water, we can plot the oscillation frequency of the breathing mode, $f_{b}=\unicode[STIX]{x1D714}/2\unicode[STIX]{x03C0}$ , as a function of $R_{0}$ , which is shown by the solid blue curve in figure 4(a). The theory shows excellent agreement with the data. For comparison, the power law $f_{b}\propto R_{0}^{-0.68}$ found by Caswell (Reference Caswell2014) is plotted as the dashed red line in figure 4(a). A closer view of the comparison between the prediction by (3.13) and $f_{b}\propto R_{0}^{-0.68}$ can be seen in figure 4(b). The data lie in the transition regime between two asymptotic limits, $R_{0}\ll l_{c}$ and $R_{0}\gg l_{c}$ , which may explain the anomalous power law reported by Caswell (Reference Caswell2014). In these two limits, equation (3.13) reduces to

(3.14) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\unicode[STIX]{x1D714}\approx \left({\displaystyle \frac{36\unicode[STIX]{x1D6FE}}{35\unicode[STIX]{x1D70C}_{l}R_{0}^{3}}}\right)^{1/2},\quad R_{0}\rightarrow 0,\\ \unicode[STIX]{x1D714}\approx {\displaystyle \frac{4}{R_{0}}}\left({\displaystyle \frac{g\unicode[STIX]{x1D6FE}}{\unicode[STIX]{x1D70C}_{l}}}\right)^{1/4},\quad R_{0}\rightarrow \infty .\end{array}\right\} & & \displaystyle\end{eqnarray}$$

The scaling for small drops is independent of $g$ , which is expected based on their nearly spherical shape. For large drops where $h\approx 2l_{c}$ (3.11), the scaling is determined essentially by gravity: $\unicode[STIX]{x1D714}\sim \sqrt{2gh}/R_{0}$ .

Figure 5.

Star-shaped oscillations of Leidenfrost water drops at $T_{s}=623~\text{K}$ . (a) A snapshot of the oscillation mode with $n=11$ . (b,c) The star-shaped oscillation wavelength $\unicode[STIX]{x1D706}_{d}$ , and frequency $f_{d}$ , with respect to drop radius $R$ . The data points from left to right in (b,c) represent the increasing mode number $n$ as indicated by the red arrows, and the error bars in (b,c) are the standard deviations of multiple drops. The data for $\unicode[STIX]{x1D706}_{d}$ in (b) are indirectly measured using the relation $2\unicode[STIX]{x03C0}R=n\unicode[STIX]{x1D706}_{d}$ , and the red circles in (c) represent the theoretical prediction by (3.16) for drops with different radii.

3.3 Star-shaped oscillations of Leidenfrost water drops

In addition to the breathing mode, large Leidenfrost drops may also develop azimuthal, star-shaped oscillations. Similar oscillations have been observed in a variety of systems involving liquid drops (Brunet & Snoeijer Reference Brunet and Snoeijer2011). In our experiments with Leidenfrost water drops using a curved substrate, we observe star-shaped oscillation modes with $n=2{-}13$ lobes along the drop periphery at $T_{s}=623~\text{K}$ . A typical star-shaped oscillation mode ( $n=11$ ) is shown in figure 5(a). Figures 5(b) and 5(c) show the drop oscillation wavelength $\unicode[STIX]{x1D706}_{d}$ and frequency $f_{d}$ for different $R$ , respectively, which are measured by analysing the high-speed videos of the oscillating drops. Surprisingly, both $\unicode[STIX]{x1D706}_{d}$ and $f_{d}$ remained nearly constant as we varied $R$ . Increasing $R$ thus led to an increase in the allowable number of lobes $n$ , as indicated by the red arrows. This similar trend also applies to other liquids used in our experiments, which will be discussed in § 3.5.

For the free oscillations of an incompressible, axisymmetric spherical drop with infinitesimal deformations, the natural resonance frequency, $f_{n}$ , of the $n$ th mode is given by (Rayleigh Reference Rayleigh1879):

(3.15) $$\begin{eqnarray}f_{n}=\frac{1}{2\unicode[STIX]{x03C0}}\sqrt{\frac{n(n-1)(n+2)}{\unicode[STIX]{x1D70C}_{l}R^{3}}}.\end{eqnarray}$$

For a liquid puddle with $R\gg h\approx 2l_{c}$ , where $h$ is the thickness of the liquid puddle (Biance et al. Reference Biance, Clanet and Quéré2003), the resonance frequency $f_{n}$ takes the form (Yoshiyasu, Matsuda & Takaki Reference Yoshiyasu, Matsuda and Takaki1996):

(3.16) $$\begin{eqnarray}f_{n}=\frac{1}{2\unicode[STIX]{x03C0}}\sqrt{\frac{n(n^{2}-1)}{\unicode[STIX]{x1D70C}_{l}R^{3}}}\sqrt{\frac{1}{1+\left(2-{\displaystyle \frac{\unicode[STIX]{x03C0}}{2}}+{\displaystyle \frac{n-3}{4}}\right){\displaystyle \frac{l_{c}}{R}}}}.\end{eqnarray}$$

The first term under the square root is for a strictly two-dimensional drop, whereas the correction factor, $1/\sqrt{1+(2-(\unicode[STIX]{x03C0}/2)+(n-3/4))l_{c}/R}$ , is due to the quasi-two-dimensional nature of the puddle.

As shown in figure 5, for water drops where we observed $n=2$ modes, the average radius was $R\approx 4~\text{mm}$ , which is greater than $l_{c}$ . Hence, it is reasonable to use (3.16) to predict the oscillation frequency for all star-shaped modes in our experiments. The comparison between theory and experiment is shown in figure 5(c), which indicates an excellent agreement. For smaller values of $n$ , both $\unicode[STIX]{x1D706}_{d}$ and $f_{d}$ vary non-monotonically with $R$ . There are a few potential reasons for this. One possibility is that nonlinear effects play a more significant role for smaller values of $n$ because the ratio of the mode amplitude to the drop radius is larger (Becker, Hiller & Kowalewski Reference Becker, Hiller and Kowalewski1991; Smith Reference Smith2010). However, the excellent agreement with the linear theory, i.e. (3.16), suggests that nonlinear effects may not be important.

Instead, we suggest that the behaviour may be related to the mode selection mechanism. For a given radius, either the frequency or the wavelength is preferentially selected by the excitation mechanism. Once $f_{d}$ or $\unicode[STIX]{x1D706}_{d}$ is selected, equation (3.16) will determine the other. As will be shown in § 3.6, there are strong capillary waves excited at the liquid–vapour interface beneath the drop. These waves produce pressure oscillations which parametrically couple to the star-shaped modes. A determination of the expected behaviour of $f_{d}$ on $R$ is thus complicated by the physics of the vapour flow beneath the drop, nevertheless, we would expect the effects to be less variable for $R\gg \unicode[STIX]{x1D706}_{d}$ , which agrees with the data shown in figure 5.

3.4 Pressure oscillations in the vapour layer

It has been well documented that star-shaped oscillations of large liquid drops can be initiated by a parametric forcing mechanism (Brunet & Snoeijer Reference Brunet and Snoeijer2011). For instance, when a liquid puddle is deposited onto a superhydrophobic, vertically vibrating substrate with a prescribed frequency, then the equation of motion of the drop can be described by an equation similar to Mathieu’s equation, and star-shaped oscillations will be excited when the drop oscillation frequency is approximately half that of the excitation frequency (Yoshiyasu et al. Reference Yoshiyasu, Matsuda and Takaki1996; Brunet & Snoeijer Reference Brunet and Snoeijer2011). In the experiment, we observed star-shaped oscillations of Leidenfrost water drops (e.g. see figure 5 a) in the absence of an external excitation. Therefore, we hypothesize that the star-shaped oscillations are driven by fluctuations in the vapour layer pressure that supports the drop.

In order to explore the dynamics of pressure fluctuations in the vapour layer, we utilized a pressure sensor attached to the centre of the curved substrate, schematically shown in figure 1(b). We measured the pressure variations in the vapour layer of water drops at $T_{s}=623~\text{K}$ during the initiation of the star-shaped oscillations. We started recording the pressure immediately after the drops were placed on the substrate, and stopped after the well-defined star-shaped oscillations had been initiated for several seconds. The whole process lasted approximately 18.5 s.

The snapshots on the top part of figure 6(a) show images of an $n=4$ drop at different stages during the oscillation. More specifically, the drop shape was nearly circular immediately upon placement on the substrate ( $t=0~\text{s}$ ), then the pressure fluctuations in the vapour layer increased as the drop shape developed well-defined lobes. The amplitude continued to grow with time ( $t=12~\text{s}$ ) until the formation of a steady-state, star-shaped oscillation with large amplitude ( $t=16.1~\text{s}$ ).

Figure 6.

(a) The pressure variations in the vapour layer during the initiation of a 4-mode star-shaped oscillation of a Leidenfrost water drop within ${\approx}18.5~\text{s}$ . Panels (b), (c), (d) and (e) represent the Fourier power spectra of the pressure in the vapour layer during the time intervals of 0–1 s, 5–6 s, 12–13 s and 16–17 s, respectively. The snapshots of the top panel represent the drop profile at 0 s, 5.1 s, 12 s and 16.1 s, respectively, during the initiation process.

Figure 6(a) shows the pressure variations in the vapour layer during the whole process. The mean pressure required to support the drop is $\unicode[STIX]{x1D70C}_{l}gh\approx 2\unicode[STIX]{x1D70C}_{l}gl_{c}=47~\text{Pa}$ . However, the mean pressure we measure is slightly larger than this value because the pressure is measured at the centre of the substrate where the pressure is larger in order to drive the viscous vapour to flow out to the drop edge. The pressure fluctuations around the mean increase with time until the formation of a steady star-shaped oscillation with a large amplitude (approximately $t=14~\text{s}$ ).

In order to gain insight into the underlying relationship between the oscillations of the drop and pressure in the vapour layer, we performed a fast Fourier transform on the pressure data in four different time intervals: 0–1 s, 5–6 s, 12–13 s and 16–17 s. The results are shown in figure 6(b–e). Initially, the pressure remains nearly constant, and there are no sharp peaks in the power spectrum (figure 6 b). For $t=5{-}6~\text{s}$ , the pressure fluctuations become stronger and more periodic, and several small peaks are visible (figure 6 c). Then, as the star-shaped oscillation is further developed, a sharp peak which is located at ${\approx}28~\text{Hz}$ shows up in the power spectrum (figure 6 d). Finally, when the star-shaped oscillation is fully developed, the pressure fluctuations stop growing, and a sharp peak located at ${\approx}28~\text{Hz}$ dominates the spectrum (figure 6 e).

It is interesting to note that the location of the sharp peak is approximately twice the oscillation frequency of a fully developed, $n=4$ mode Leidenfrost water drop (see figure 5 c). This is consistent with a parametric forcing mechanism for the excitation of the star-shaped oscillations. Moreover, harmonics at higher frequencies are also visible as indicated by the secondary peaks in figure 6(e). This is likely due to nonlinear effects involved in the star-shaped oscillations. In addition, the location of the peak is robust. As will be shown in § 3.7, the location of the dominant peak in the power spectrum is mostly independent of the substrate temperature and the environmental temperature.

Figure 7.

(a) Summary of the pressure oscillation frequencies $f_{p}$ in the vapour layer of star-shaped Leidenfrost water drops with different radii at $T_{s}=623~\text{K}$ . The data points from left to right represent the oscillation modes from $n=2$ to 13, respectively, as indicated by the red arrows. (b) The drop oscillation frequency $f_{d}$ with respect to drop radius $R$ (figure 5 c). The error bars of $f_{p}$ are taken from the full width at half-max of the highest peak in the Fourier power spectra computed over a time interval of 10 s, and the error bars of $R$ are defined as the standard deviation of multiple measurements of different drops after the star-shaped oscillations are fully developed for each mode. The dashed lines in (a) and (b) are visual guides to indicate that $f_{p}\approx 2f_{d}$ .

Figure 7(a) shows the pressure oscillation frequency $f_{p}$ in the vapour layer of Leidenfrost water drops for all of the observed modes. The data points from left to right correspond the oscillation modes from $n=2$ to 13, as indicated by the red arrows. By comparing this with the drop oscillation frequency $f_{d}$ (figure 7 b), we find $f_{p}\approx 2f_{d}$ as indicated by the dashed blue and red lines in 7(a) and 7(b), respectively. This robust relationship suggests that the star-shaped oscillations are parametrically driven by the pressure (Miles & Henderson Reference Miles and Henderson1990; Kumar & Tuckerman Reference Kumar and Tuckerman1994; Yoshiyasu et al. Reference Yoshiyasu, Matsuda and Takaki1996; Brunet & Snoeijer Reference Brunet and Snoeijer2011; Terwagne & Bush Reference Terwagne and Bush2011). Consider the radial position of a point on the perimeter of the drop during a star-shaped oscillation, $r(t)$ . This point will oscillate in time due to the azimuthal standing wave, and obey the following equation:

(3.17) $$\begin{eqnarray}\frac{\text{d}^{2}r}{\text{d}t^{2}}+\unicode[STIX]{x1D714}^{2}r=0.\end{eqnarray}$$

To leading order, $\unicode[STIX]{x1D714}$ is the resonant frequency of the mode, equation (3.16):

(3.18) $$\begin{eqnarray}\unicode[STIX]{x1D714}_{0}^{2}={\displaystyle \frac{n(n^{2}-1)\unicode[STIX]{x1D6FE}}{\unicode[STIX]{x1D70C}_{l}R^{3}}}.\end{eqnarray}$$

For simplicity, we have ignored the correction due to the quasi-two-dimensional nature of the drop. The pressure variations will induce vertical oscillations of the drop, and in turn oscillations of the drop radius:

(3.19) $$\begin{eqnarray}R=R_{0}(1+\unicode[STIX]{x1D716}\cos \unicode[STIX]{x1D714}_{p}t),\end{eqnarray}$$

where $R_{0}$ is the average radius of the drop, and $\unicode[STIX]{x1D716}$ is the amplitude of the small perturbation. Plugging (3.19) into (3.18), to leading order in $\unicode[STIX]{x1D716}$ we obtain $\unicode[STIX]{x1D714}^{2}=\unicode[STIX]{x1D714}_{0}^{2}(1-3\unicode[STIX]{x1D716}\cos \unicode[STIX]{x1D714}_{p}t)$ . Therefore, equation (3.17) now becomes:

(3.20) $$\begin{eqnarray}\frac{\text{d}^{2}r}{\text{d}t^{2}}+\unicode[STIX]{x1D714}_{0}^{2}(1-3\unicode[STIX]{x1D716}\cos \unicode[STIX]{x1D714}_{p}t)r=0.\end{eqnarray}$$

When $\unicode[STIX]{x1D714}_{p}\approx 2\unicode[STIX]{x1D714}_{0}$ , the parametric resonance is the strongest, i.e. the oscillation amplitude will exponentially increase with time until the star-shaped oscillations are fully developed. This is consistent with our measurements of the pressure oscillation frequencies in the vapour layer (figure 7), which are approximately twice those of the drop oscillation frequencies (figure 5 c) that we observed in our experiment. We note that it is possible that the pressure oscillations are a consequence of the star-shaped oscillations. If the star oscillations were excited by some other mechanism, then the symmetry of the star shape would necessitate a pressure extrema when the star shape reaches its maximum amplitude. However, the only energy which is available to drive the oscillation comes from the evaporation and gas flow in the vapour layer. Thus we expect that the pressure oscillations are the source of the star-shaped oscillations. Section 3.6 will explore the potential source of the pressure oscillations.

Figure 8.

Snapshots of star-shaped oscillation modes with $n=2{-}13$ of six different liquids when the lobes are at their maximum displacement. Panels (a), (b), (c), (d), (e) and (f) represent water, acetone, methanol, ethanol, liquid $\text{N}_{2}$ and isopropanol, respectively. For water: $T_{s}=623~\text{K}$ , for acetone, methanol, ethanol and isopropanol: $T_{s}=523~\text{K}$ . The substrate for liquid $\text{N}_{2}$ was not heated.

3.5 Star-shaped oscillations of different liquids

In addition to water, we performed similar experiments with five other liquids: liquid $\text{N}_{2}$ , acetone, methanol, ethanol and isopropanol. For these liquids, the curved surfaces were also machined to satisfy $l_{c}/R_{s}\approx 0.03$ (see § 2), and the relevant physical properties and substrate temperatures are listed in table 1. Figure 8 shows snapshots of star-shaped oscillations for the six different liquids we used in the experiments, where the other five liquids show similar star-shaped patterns to water drops but with different observable oscillation modes. By analysing the images of the star-shaped oscillations, we find that star-shaped oscillations of acetone, methanol, ethanol and isopropanol share a similar wavelength $\unicode[STIX]{x1D706}_{d}\approx 0.9~\text{cm}$ and frequency $f_{d}\approx 14~\text{Hz}$ , whereas liquid $\text{N}_{2}$ shows a wavelength of $\unicode[STIX]{x1D706}_{d}\approx 0.6~\text{cm}$ and frequency $f_{d}\approx 17~\text{Hz}$ . This suggests that the star-shaped oscillations may depend on the capillary length of the liquid.

Figure 9.

Dependence of star-shaped oscillation mode $n$ (a), and dimensionless oscillation wavelength $\unicode[STIX]{x1D706}_{d}/l_{c}$ (b) and frequency $f/\sqrt{g/l_{c}}$ (c) with respect to the dimensionless drop radius $R/l_{c}$ for six different liquids. In (c), the solid symbols represent the drop oscillation frequency $f_{d}$ , whereas the open symbols denote the corresponding pressure oscillation frequency $f_{p}$ . The error bars of $R$ , $\unicode[STIX]{x1D706}_{d}$ and $f_{d}$ come from the standard deviations of multiple measurements of different drops, and the error bars of $f_{p}$ are defined as the full width of the highest peak in the power spectrum at half the maximum value. The dashed lines in all panels are visual guides.

Figure 9 shows the dependence of star-shaped oscillation mode $n$ , rescaled wavelength $\unicode[STIX]{x1D706}_{d}/l_{c}$ and rescaled frequency $f/\sqrt{g/l_{c}}$ on the rescaled drop radius $R/l_{c}$ of the six liquids used in the experiment. The mode number $n$ increases linearly with $R/l_{c}$ as indicated by the dashed black line in figure 9(a), showing that large modes can only be observed in larger drops, as supported by figure 8. The rescaled wavelength $\unicode[STIX]{x1D706}_{d}/l_{c}$ for the six liquids collapses onto a straight line as shown in figure 9(b), which indicates that the wavelength of the star-shaped oscillations only depends on $l_{c}$ of the liquid. Figure 9(c) shows the oscillation frequency of both the azimuthal star-shaped oscillations, $f_{d}$ , and the corresponding pressure, $f_{p}$ , in the vapour layer. The solid symbols denote the drop oscillations, whereas open symbols with the same colour represent the corresponding pressure oscillations in the vapour layer for a specific liquid. Both $f_{d}$ and $f_{p}$ collapse fairly well, and the relationship $f_{p}\approx 2f_{d}$ is consistent with the data, as indicated by the dashed black and red lines. Additionally, by comparing different values of $T_{s}$ for water, figure 9 shows that $\unicode[STIX]{x1D706}_{d}$ , $f_{d}$ , and $f_{p}$ are mostly independent of substrate temperature.

The main influence of $T_{s}$ is the through the number of observable modes. For instance, at $T_{s}=673~\text{K}$ , we did not observe oscillation modes $n=12$ and $n=13$ in water, which we attribute to the extremely fast evaporation rate of the liquid, as will be discussed in § 3.6. Although the robust relationship between $f_{d}$ and $f_{p}$ suggests that the star-shaped oscillations of the drops are parametrically driven by the pressure in the vapour layer, there is an interesting variation in both $f_{d}$ and $f_{p}$ at small mode numbers. The data for smaller modes (e.g. $n=2,3,4$ ) deviate further from the average compared to larger modes. This behaviour may be nonlinear in nature since the amplitude of the mode relative to the drop radius is much larger for small  $n$ (Becker et al. Reference Becker, Hiller and Kowalewski1991; Smith Reference Smith2010). In addition, the quasi-two-dimensional nature of the drop is more important at small $n$ , which may also contribute to the variations in this regime. A more quantitative explanation for this dependence is left for future studies.

Figure 10.

Scaling behaviour for the minimum observed star mode $n_{min}$ with respect to $Re_{l}$ of different liquids (table 1). The dashed line represents the fit to the data, $2600Re_{l}^{-1}$ .

As shown in table 1, different oscillation modes are observed in different liquids. In particular, some liquids failed to display smaller mode numbers. This may be due to inherent viscous damping that prevents the sustained excitation of large-amplitude oscillations. Following the analysis in Ma et al. (Reference Ma, Liétor-Santos and Burton2017), the role of damping can be characterized by the Reynolds number associated with the liquid oscillation. For a star-shaped oscillation, the characteristic length scale and time scale are $l_{c}$ and $\sqrt{l_{c}^{3}\unicode[STIX]{x1D70C}_{l}/\unicode[STIX]{x1D6FE}}$ , respectively. Thus, the inertial term in the Navier–Stokes equation is estimated as $|\unicode[STIX]{x1D70C}_{l}(\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{v}|=\unicode[STIX]{x1D6FE}/l_{c}^{2}$ , and the viscous term is $|\unicode[STIX]{x1D702}_{l}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{v}|=\unicode[STIX]{x1D702}_{l}/\sqrt{l_{c}^{5}\unicode[STIX]{x1D70C}_{l}/\unicode[STIX]{x1D6FE}}$ . The Reynolds number of the oscillating liquid is defined as the ratio of inertial to viscous terms: $Re_{l}=\sqrt{l_{c}\unicode[STIX]{x1D70C}_{l}\unicode[STIX]{x1D6FE}}/\unicode[STIX]{x1D702}_{l}$ , and the values of $Re_{l}$ for different liquids are listed in table 1.

Figure 10 shows the dependence of the minimum mode number $n_{min}$ on $Re_{l}$ for each liquid. The dashed line represents a suggested scaling $Re^{-1}$ , indicating that the liquid viscosity damps the oscillations of smaller drops and thus sets the minimum mode number of stable star-shaped oscillations. However, the substrate temperature and thus evaporation rate will likely affect the number of observed modes as well. A higher evaporation rate will induce a stronger driving of the oscillation modes, which would reduce $n_{min}$ . This can be seen in figure 10 for liquid $\text{N}_{2}$ , which undergoes much more rapid evaporation due to the large difference between the boiling and substrate temperature, $T_{s}-T_{b}$ . In addition, as mentioned previously, the stronger evaporation may also inhibit larger modes if the pressure oscillations are less coherent. In the next section we show how the flow in the vapour layer is linked to the pressure oscillations.

Figure 11.

Sketch of the cross-sectional view of a large Leidenfrost drop levitating on a flat hot surface. The red dashed line and black solid line represent the drop profiles before and after the initiation of capillary waves at the bottom surface of the drop. The symbols are defined in the main text.

3.6 Origin of pressure oscillations in the vapour layer

As evidenced by figure 9(c), the star-shaped oscillations of Leidenfrost drops are parametrically driven by the pressure variations in the vapour layer. Star-shaped oscillations induced by a parametric coupling have been observed in a variety of systems where variations of the drop radius are induced by external, periodically modulated fields (Brunet & Snoeijer Reference Brunet and Snoeijer2011). However, in our experiment, there are no obvious external fields. It is then crucial to understand the source of the pressure oscillations. In this section we show how capillary waves of a characteristic wavelength at the liquid–vapour interface induce the pressure oscillations in the vapour layer.

Figure 11 shows a sketch of a large, axisymmetric Leidenfrost drop with a maximum radius  $=R$ and thickness ${\approx}2l_{c}$ . The mean vertical velocity of the gas at the liquid surface is $v$ , $e$ is the mean thickness of the vapour layer, and $u$ is the outward radial velocity of the gas near $r=R$ . For such a large Leidenfrost drop, both the bottom surface of the drop and the substrate surface are assumed to be approximately flat to simplify the analysis. Following the model of Biance et al. (Reference Biance, Clanet and Quéré2003), the mass loss rate of the drop due to evaporation can be expressed as:

(3.21) $$\begin{eqnarray}\frac{\text{d}m}{\text{d}t}=\frac{\unicode[STIX]{x1D705}_{v}}{L}\frac{\unicode[STIX]{x0394}T}{e}\unicode[STIX]{x03C0}R^{2}=\unicode[STIX]{x1D70C}_{v}\unicode[STIX]{x03C0}R^{2}v,\end{eqnarray}$$

where $\unicode[STIX]{x1D705}_{v}$ is the thermal conductivity of the vapour, $L$ is the latent heat of the evaporation, $\unicode[STIX]{x0394}T=T_{s}-T_{b}$ and $\unicode[STIX]{x1D70C}_{v}$ is the density of the vapour at the liquid interface. Since $e\approx 100~\unicode[STIX]{x03BC}\text{m}$ , and thus $R\gg e$ (Burton et al. Reference Burton, Sharpe, van der Veen, Franco and Nagel2012), we employ lubrication theory, leading to an expression relating the flow rate to the mean pressure, $P$ , in the vapour layer:

(3.22) $$\begin{eqnarray}\frac{\text{d}m}{\text{d}t}=\unicode[STIX]{x1D70C}_{v}\frac{2\unicode[STIX]{x03C0}e^{3}}{3\unicode[STIX]{x1D702}_{v}}P=\unicode[STIX]{x1D70C}_{v}\frac{2\unicode[STIX]{x03C0}e^{3}}{3\unicode[STIX]{x1D702}_{v}}2l_{c}\unicode[STIX]{x1D70C}_{l}g=\unicode[STIX]{x1D70C}_{v}\unicode[STIX]{x03C0}R^{2}v,\end{eqnarray}$$

where $\unicode[STIX]{x1D702}_{v}$ is the dynamic viscosity of the vapour. We note here that parameters such as $\unicode[STIX]{x1D705}_{v}$ , $\unicode[STIX]{x1D70C}_{v}$ and $\unicode[STIX]{x1D702}_{v}$ will vary in the vapour layer due to the temperature gradient between the liquid and solid surfaces. For simplicity, we will assume that these represent average values of the vapour layer properties. A more detailed treatment, possibly including full numerical simulations, would be necessary to extend this simplified model.

Figure 12.

Dependence of vapour film thickness (a), radial velocity (b), vertical velocity (c) and the Reynolds number in the vapour (d) on temperature difference $\unicode[STIX]{x0394}T$ , respectively.

In addition, due to mass conservation, the volumetric flow rate of gas from the bottom of the drop must be equal to the flow rate exiting the perimeter of drop:

(3.23) $$\begin{eqnarray}\unicode[STIX]{x03C0}R^{2}v=2\unicode[STIX]{x03C0}eRu.\end{eqnarray}$$

Solving (3.21)–(3.23) for $u$ , $v$ and $e$ yields:

(3.24) $$\begin{eqnarray}\displaystyle & \displaystyle u=\left(\frac{\unicode[STIX]{x1D70C}_{l}g\unicode[STIX]{x1D705}_{v}l_{c}}{3\unicode[STIX]{x1D70C}_{v}\unicode[STIX]{x1D702}_{v}L}\right)^{1/2}\unicode[STIX]{x0394}T^{1/2}, & \displaystyle\end{eqnarray}$$

(3.25) $$\begin{eqnarray}\displaystyle & \displaystyle v=\left[\frac{4l_{c}\unicode[STIX]{x1D70C}_{l}g}{3\unicode[STIX]{x1D702}_{v}}\left(\frac{\unicode[STIX]{x1D705}_{v}\unicode[STIX]{x0394}T}{L\unicode[STIX]{x1D70C}_{v}}\right)^{3}\right]^{1/4}R^{-1/2}, & \displaystyle\end{eqnarray}$$

(3.26) $$\begin{eqnarray}\displaystyle & \displaystyle e=\left(\frac{3\unicode[STIX]{x1D705}_{v}\unicode[STIX]{x0394}T\unicode[STIX]{x1D702}_{v}}{4Ll_{c}\unicode[STIX]{x1D70C}_{l}\unicode[STIX]{x1D70C}_{v}g}\right)^{1/4}R^{1/2}. & \displaystyle\end{eqnarray}$$

In this model, the steady-state, linear temperature profile in the vapour layer is valid since the characteristic time scale associated with thermal diffusion, $e^{2}/D_{v}\approx 1~\text{ms}$ , is smaller than the typical residence time of the gas in the vapour layer, $R/u\approx 10~\text{ms}$ , where $D_{v}$ is the thermal diffusivity of the vapour. Let us assume a water Leidenfrost drop with $R=0.01~\text{m}$ , and for the vapour, we use the properties of steam at 373 K and 1 atm of pressure: $\unicode[STIX]{x1D70C}_{l}=959~\text{kg}~\text{m}^{-3}$ , $\unicode[STIX]{x1D70C}_{v}=0.45~\text{kg}~\text{m}^{-3}$ , $\unicode[STIX]{x1D6FE}=0.059~\text{N}~\text{m}^{-1}$ , $g=9.8~\text{m}~\text{s}^{-2}$ , $L=2.26\times 10^{6}~\text{J}~\text{kg}^{-1}$ , $\unicode[STIX]{x1D702}_{v}=1.82\times 10^{-5}~\text{Pa}~\text{s}$ and $\unicode[STIX]{x1D705}_{v}=0.04~\text{W}~\text{m}^{-1}~\text{K}^{-1}$ . Although the properties of the vapour may vary somewhat in the vapour layer due to the temperature gradient, we have verified that this does not significantly affect our analysis.

We can now plot $u$ , $v$ and $e$ with respect to $\unicode[STIX]{x0394}T$ based on (3.24)–(3.26). The results are shown in figures 12(a), 12(b) and 12(c), respectively. The inertial term in the Navier–Stokes equation is estimated as $|\unicode[STIX]{x1D70C}_{v}(\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{v}|=\unicode[STIX]{x1D70C}_{v}v^{2}/e$ , and the viscous term as $|\unicode[STIX]{x1D702}_{v}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{v}|=\unicode[STIX]{x1D702}_{v}v/e^{2}$ . Thus the Reynolds number in the vapour layer is $Re_{v}=\unicode[STIX]{x1D70C}_{v}ve/\unicode[STIX]{x1D702}_{v}$ . Plugging in (3.25) and (3.26), we arrive at a surprisingly simple expression for the Reynolds number in the vapour: $Re_{v}=\unicode[STIX]{x0394}T\unicode[STIX]{x1D705}_{v}/L\unicode[STIX]{x1D702}_{v}$ , which is independent of the drop size. This expression is plotted in figure 12(d). Under typical experimental conditions, $Re_{v}\approx 0.2$ , suggesting that the original lubrication flow assumption is valid, although inertial forces are non-negligible.

Based on figure 12, we can estimate the Bernoulli pressure in the vapour layer of Leidenfrost water drops as $\unicode[STIX]{x1D70C}_{v}u^{2}/2\approx 1~\text{Pa}$ , which is much smaller than the pressure variations ${\approx}10~\text{Pa}$ (see figure 6 a), suggesting that the inertial force does not account for the pressure variations. The viscous pressure is $P_{vis}\sim \unicode[STIX]{x1D702}_{v}vR^{2}/e^{3}$ , thus the pressure variation due to the local variations of the film thickness is $\unicode[STIX]{x0394}P_{vis}=\text{d}P_{vis}/\text{d}e\unicode[STIX]{x0394}e$ . Then $\unicode[STIX]{x0394}P_{vis}=10~\text{Pa}$ corresponds to $\unicode[STIX]{x0394}e\approx 15~\unicode[STIX]{x03BC}\text{m}$ considering the typical values of $v$ (figure 12 c) and $R$ . Although this local film thickness could also be possibly induced by the vertical motion of the centre of mass of the drops, the fact that small and large drops (see figure 9 c) share a nearly constant oscillation frequency suggests this is not the case. Therefore, the pressure variations in the vapour layer are likely to be induced by the local variations of the vapour film thickness.

We propose that capillary waves with a characteristic wavelength, $\unicode[STIX]{x1D706}_{c}$ , travelling from the centre to the edge of the drop lead to local variations in vapour film thickness, and thus pressure variations in the vapour layer, as schematically shown in figure 11. To confirm this possibility, we used a heated, plano-concave, fused silica lens as a substrate to image the capillary waves from below. The results for acetone and ethanol Leidenfrost drops are shown in figure 13. The images were produced by averaging all frames in a given video sequence, then subtracting this background image in order to enhance contrast in the centre of the drop. The white ‘halo’ surrounding the drop is a consequence of this subtraction process. The Fourier spectra for the capillary waves in the central region of the drops were computed in both time and space using the pixel intensity as the signal. Using the Nyquist sampling theorem, the maximum frequency was limited by half the video frame rate (500–1000  Hz), and the maximum spatial frequency was set by half the camera magnification ( $132~\text{pixel}~\text{cm}^{-1}$ ).

Figure 13.

Panels (a), (c) and (e) show the capillary waves imaged beneath 8-mode acetone drop, 11-mode ethanol drop and 12-mode ethanol drop, respectively. Panels (b), (d) and (f) represent the corresponding Fourier power spectra of the capillary waves shown in (a), (c) and (e), respectively. The images of capillary waves were enhanced for visibility, as described in the text.

We observed a sharp peak in the measured Fourier spectrum for all the analysed video sequences. In figure 13(a,b) (8-mode, acetone), this peak is located at a capillary wave frequency $f_{c}\approx 24~\text{Hz}$ and within a range of wavenumbers, $k_{c}=2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D706}_{c}$ , from $7~\text{cm}^{-1}$ to $12~\text{cm}^{-1}$ , where $\unicode[STIX]{x1D706}_{c}$ is the wavelength of the capillary waves. Figure 13(c,d) shows the capillary waves beneath a 11-mode Leidenfrost ethanol drop and a sharp peak at a capillary wave frequency $f_{c}\approx 25~\text{Hz}$ with $k_{c}\approx 7{-}10~\text{cm}^{-1}$ . Similarly, for the capillary waves beneath a 12-mode Leidenfrost ethanol drop (figure 13 e), a sharp peak exists at $f_{c}\approx 27~\text{Hz}$ and $k_{c}\approx 10{-}11~\text{cm}^{-1}$ as shown in figure 13(f). These frequencies show excellent agreement with the corresponding typical pressure oscillation frequencies measured in the vapour layer of star-shaped Leidenfrost ethanol and acetone drops (see figure 9 c).

It is well known that capillary waves can be generated at a liquid–vapour interface due to a strong shear stress in the vapour (Miles Reference Miles1957; Zhang Reference Zhang1995; Chang & Demekhin Reference Chang and Demekhin2002; Zeisel, Stiassnie & Agnon Reference Zeisel, Stiassnie and Agnon2008; Paquier, Moisy & Rabaud Reference Paquier, Moisy and Rabaud2015). A similar mechanism underlies the Kelvin–Helmholtz instability, however, the intermediate Reynolds number in the Leidenfrost vapour layer complicates the analysis. Nevertheless, we can estimate the strength of this shear stress. Typically, the ‘friction velocity’ is generally used to measure the strength of shear, which is defined as $u_{\ast }=\sqrt{\unicode[STIX]{x1D70F}/\unicode[STIX]{x1D70C}_{v}}$ , where $\unicode[STIX]{x1D70F}$ is the shear stress at the liquid–vapour interface. The maximum shear stress at the interface is:

(3.27) $$\begin{eqnarray}\unicode[STIX]{x1D70F}=\frac{6\unicode[STIX]{x1D702}_{v}u}{e},\end{eqnarray}$$

assuming a parabolic flow profile in the vapour layer with mean velocity $u$ near the edge of the drop. Using water as an example, and plugging (3.24) and (3.26) and the value of $\unicode[STIX]{x1D702}_{v}$ of water vapour into (3.27) yields $u_{\ast }\approx 2~\text{m}~\text{s}^{-1}$ . This friction velocity is quite strong and sufficient to lead to the growth of unstable modes with wavelengths of millimetre scale (Zhang Reference Zhang1995; Zeisel et al. Reference Zeisel, Stiassnie and Agnon2008).

The general dispersion relation of gravity–capillary waves with a dense upper layer is:

(3.28) $$\begin{eqnarray}f_{c}=\frac{1}{2\unicode[STIX]{x03C0}}\sqrt{\left(-gk_{c}+\frac{\unicode[STIX]{x1D6FE}k_{c}^{3}}{\unicode[STIX]{x1D70C}}\right)\tanh (k_{c}h)},\end{eqnarray}$$

where $k_{c}=2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D706}_{c}$ , and $h\approx 2l_{c}$ is the thickness of the drop. For simplicity, we have assumed that the normal velocity is zero at the upper surface of the drop. For a large Leidenfrost water drop whose characteristic pressure oscillation frequency in the vapour layer is $f_{p}\approx 26~\text{Hz}$ (see figure 7), the corresponding capillary wavelength is calculated to be $\unicode[STIX]{x1D706}_{c}\approx 3.03l_{c}$ , so that $k_{c}\approx 8.3~\text{cm}^{-1}$ (Ma et al. Reference Ma, Liétor-Santos and Burton2017). Similarly, for the Leidenfrost acetone and ethanol drops shown in figure 13(a,c,e), the corresponding $k_{c}\approx 13~\text{cm}^{-1}$ , which is slightly larger than the positions of the peaks indicated in figure 13(b,d,f), where $\unicode[STIX]{x1D706}_{c}\approx 4l_{c}$ . The agreement between the estimate for $k_{c}$ and the measurements shown in figure 13 is good considering the simplicity of (3.28), which is derived using an inviscid, semi-infinite flow in both phases.

The capillary wave origin for the star-shaped oscillations also agrees with the minimum size of the $n=2$ mode. More specifically, the radius we measured in our experiments is $R$ rather than $R_{neck}$ as illustrated in figure 11. The relationship between these two lengths is $R=R_{neck}+0.53l_{c}$ (Snoeijer et al. Reference Snoeijer, Brunet and Eggers2009; Burton et al. Reference Burton, Sharpe, van der Veen, Franco and Nagel2012). Thus, the minimum drop size required to fit one capillary wavelength beneath the drop is $2R_{neck}\approx \unicode[STIX]{x1D706}_{c}$ . Using $\unicode[STIX]{x1D706}_{c}\approx 3.03l_{c}$ from above, we find that the radius of an $n=2$ mode drop should be $R\approx 2.05l_{c}$ . This is in good agreement with figure 9(a), which shows $R/l_{c}$ is slightly less than 2 for the smallest drops. Taken together, this analysis suggests a purely hydrodynamic origin for the star-shaped oscillations based on capillary waves generated by a strong shear stress in the rapidly flowing vapour beneath the drop.

3.7 Thermal effects

In the Leidenfrost effect, thermal energy is transferred from the substrate to the drop, inducing evaporation and enabling the sustained star-shaped oscillations. Here we consider the role of thermal effects, such as convection in the liquid, which may play a role in the star-shaped oscillations. At sufficiently large temperature gradients in liquid layers, convective structures can develop with well-defined length scales. A well-known example is Bénard–Marangoni convection, where hexagonal patterns are generated in a thin layer of liquid when heated from the bottom (Bénard Reference Bénard1901; Rayleigh Reference Rayleigh1916). The convection pattern is strongly affected by the variation of surface tension with temperature (Marangoni Reference Marangoni1871; Schatz & Neitzel Reference Schatz and Neitzel2001; Maroto, Pérez-Munuzuri & Romero-Cano Reference Maroto, Pérez-Munuzuri and Romero-Cano2007). Given that our data collapse when scaled by the capillary length of the liquid (figure 9), this particular type of convection may be important for Leidenfrost drops since the capillary length also appears in the length scale which characterizes the size of the convective patterns.

Generally, the Marangoni number, $Ma$ , and Rayleigh number, $Ra$ , are used to characterize Bénard–Marangoni convection. These dimensionless numbers are defined as:

(3.29) $$\begin{eqnarray}\displaystyle & \displaystyle Ma=\frac{\left(\unicode[STIX]{x2202}\unicode[STIX]{x1D6FE}/\unicode[STIX]{x2202}T\right)\unicode[STIX]{x1D6FF}Th}{\unicode[STIX]{x1D708}_{l}\unicode[STIX]{x1D70C}D_{l}}, & \displaystyle\end{eqnarray}$$

(3.30) $$\begin{eqnarray}\displaystyle & \displaystyle Ra=\frac{g\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FF}Th^{3}}{\unicode[STIX]{x1D708}_{l}D_{l}}. & \displaystyle\end{eqnarray}$$

These numbers characterize the strengths of thermocapillary and buoyancy effects, respectively. Additionally, the ratio of thermal conduction inside the liquid to the conduction at the liquid–vapour interface is characterized by the Biot number:

(3.31) $$\begin{eqnarray}Bi=\frac{\unicode[STIX]{x1D6FD}h}{\unicode[STIX]{x1D705}_{l}}=\frac{T_{hot}-T_{cold}}{T_{cold}-T_{1}}.\end{eqnarray}$$

The definitions of the symbols in $Ma$ , $Ra$ and $Bi$ are listed in table 2, as well as their values for water at the boiling point.

Figure 14.

The temperature profile of a large Leidenfrost water drop with the volume of $10^{-6}~\text{m}^{3}$ . The error bars come from the standard deviations of multiple measurements.

Figure 15.

Fourier power spectra of the pressure variations in the vapour layer at different environment temperature $T_{1}$ . Panels (a), (b) and (c) represent pressure variations of a 4-mode Leidenfrost water drop at $T_{1}\approx 367~\text{K}$ , 483 K and 623 K, respectively. Panels (d), (e) and (f) represent pressure variations of a 5-mode Leidenfrost water drop at $T_{1}\approx 367~\text{K}$ , 483 K and 623 K, respectively. All of the data shown here were obtained at the same scanning rate within the same time interval, 20 s.

Table 2.

Physical properties of water at the boiling points (Lemmon et al. Reference Lemmon, McLinden, Friend, Linstrom and Mallard2011).

Figure 14 shows the temperature profile of a large Leidenfrost water drop levitating on its vapour layer with $T_{s}=673~\text{K}$ , in which the substrate surface is defined as $z=0$ . The temperature was measured with a fine-point thermocouple, as described in § 2. We define the temperatures of the bottom and top surfaces of the drop as $T_{hot}$ and $T_{cold}$ , respectively, and the temperature of the position which is slightly above the top surface of the drop is denoted as the environment temperature, $T_{1}$ , as shown in figure 11. From figure 14 we can obtain $\unicode[STIX]{x1D6FF}T=T_{hot}-T_{cold}\approx 3.8~\text{K}$ , the environment temperature measured to be $T_{1}=367~\text{K}$ , thus we can calculate $Ma\sim 2\times 10^{5}$ , $Ra\sim 7\times 10^{4}$ and $Bi\approx 1.1$ . Both the values of $Ma$ and $Ra$ are much larger than their critical values, which are typically of order 100 for the initiation of convective instability (Leal Reference Leal2007; Maroto et al. Reference Maroto, Pérez-Munuzuri and Romero-Cano2007). Thus it is possible that thermal convection plays a role in initiating the star-shaped oscillations. In this case, one may expect the star-shaped wavelength, $\unicode[STIX]{x1D706}_{d}$ , to be related to the critical wavelength of the convective instability. This critical wavelength depends on the Biot number. We implemented a qualitative test for this dependence by wrapping aluminium foil around the substrate and Leidenfrost drop, which dramatically increased the environment temperature, $T_{1}$ , near the top of drop.

Figures 15(a) and 15(d) show Fourier power spectra of the pressure oscillations in the vapour layer of Leidenfrost water drops during $n=4$ and $n=5$ oscillations at $T_{1}\approx 367~\text{K}$ , respectively. Figures 15(b) and 15(e) are the spectra of pressure oscillations during $n=4$ and $n=5$ oscillations at $T_{1}\approx 483~\text{K}$ , respectively. Finally, figure 15(c,f) represents the same modes at $T_{1}=623~\text{K}$ . For $T_{1}=483~\text{K}$ and 623 K, the direction of heat transfer at the upper surface of the drop has been reversed; energy is added to the drop. Overall, dramatically increasing the surrounding temperature affects the appearance of higher harmonics in the spectra. The behaviour is non-monotonic, and is likely due to the highly non-equilibrium conditions (high evaporation rate) induced by the high temperatures of the substrate ( $T_{s}$ ) and environment ( $T_{1}$ ). A more detailed understanding of the pressure oscillation spectra is left for future studies. Nevertheless, the position of the main peak in the spectra is independent of $T_{1}$ , indicating that convection and the details of thermal transport in Leidenfrost drops play a secondary role in the star-shaped oscillations.