Self-organized oscillations of Leidenfrost drops
In the Leidenfrost effect, a thin layer of evaporated vapor forms between a liquid and a hot solid. The complex interactions between the solid, liquid, and vapor phases can lead to rich dynamics even in a single Leidenfrost drop. Here we investigate the self-organized oscillations of Leidenfrost drops that are excited by a constant flow of evaporated vapor beneath the drop. We show that for small Leidenfrost drops, the frequency of a recently reported “breathing mode” (Caswell, Phys. Rev. E, vol. 90, 2014, 013014) can be explained by a simple balance of gravitational and surface tension forces. For large Leidenfrost drops, azimuthal star-shaped oscillations are are observed. Our previous work showed how the coupling between the rapid evaporated vapor flow and the vapor-liquid interface excites the star oscillations (Ma et al., Phys. Rev. Fluids, vol. 2, 2017, 031602). In our experiments, star-shaped oscillation modes of to 13 are observed in different liquids, and the number of observed modes depends sensitively on the viscosity of the liquid. Here we expand on this work by directly comparing the oscillations with theoretical predictions, as well as show how the oscillations are initiated by a parametric forcing mechanism through pressure oscillations in the vapor layer. The pressure oscillations are driven by the capillary waves of a characteristic wavelength beneath the drop. These capillary waves can be generated by a large shear stress at the liquid-vapor interface due to the rapid flow of evaporated vapor. We also explore potential effects of thermal convection in the liquid. Although the measured Rayleigh number is significantly larger than the critical Rayleigh number, the frequency (wavelength) of the oscillations depends only on the capillary length of the liquid, and is independent of the drop radius and substrate temperature. Thus convection seems to play a minor role in Leidenfrost drop oscillations, which are mostly hydrodynamic in origin.
When a volatile liquid drop is deposited on a sufficiently hot solid, it can survive for minutes due to the presence of a thermally-insulating layer of evaporated vapor beneath the drop. In this levitated state, commonly known as the Leidenfrost regime (Leidenfrost, 1756), the supporting vapor layer is maintained by the sustained evaporation of the liquid, and individual drops are free to undergo frictionless motion due to the absence of liquid-solid contact. The Leidenfrost effect can be easily observed by placing a water drop onto a hot pan over a cook stove in the kitchen, and has been the subject of numerous fundamental and applied studies due to the complex and rich interactions between the solid, liquid, and vapor phases (Quéré, 2013). Examples include the evaporation dynamics and geometry of the drop (Burton et al., 2012; Biance et al., 2003; Myers & Charpin, 2009; Pomeau et al., 2012; Xu & Qian, 2013; Sobac et al., 2014; Hidalgo-Caballero et al., 2016; Maquet et al., 2016; Wong et al., 2017), the stability of the vapor-liquid interface (Duchemin et al., 2005; Lister et al., 2008; Snoeijer et al., 2009; Bouwhuis et al., 2013; Trinh et al., 2014; Raux et al., 2015; Maquet et al., 2015), hydrodynamic drag-reduction (Vakarelski et al., 2011, 2012, 2014), self-propulsion of droplets (Linke et al., 2006; Dupeux et al., 2011b, a; Lagubeau et al., 2011; Cousins et al., 2012; Li et al., 2016; Soto et al., 2016; Sobac et al., 2017), impact dynamics (Biance et al., 2006; Tran et al., 2012; Castanet et al., 2015; Shirota et al., 2016), green nanofabrication (Abdelaziz et al., 2013), chemical reactions (Bain et al., 2016), fuel combustion (Kadota et al., 2007), quenching process in metallurgy (Bernardin & Mudawar, 1999), heat transfer (Talari et al., 2018; Shahriari et al., 2014), directional transport (Li et al., 2016), soft heat engines (Waitukaitis et al., 2017) and thermal control of nuclear reactors (Van Dam, 1992).
In many of these examples, transient and self-sustained capillary oscillations play an important role in the dynamics. The interplay between gravity, the flow of vapor beneath the drop, and the liquid surface tension can lead to both small- and large-amplitude oscillations with very little damping. An understanding of these detailed interactions is crucial for the stability of the vapor layer, the failure of which can lead to explosive boiling upon contact with the hot surface. However, the excitation mechanism of these oscillations is complicated by the presence of both hydrodynamic and thermal effects, for example, rapidly-flowing vapor can cause a strong shear stress at the liquid-vapor interface, and temperature gradients in the liquid can lead to convective and Marangoni forces. Here we focus on capillary oscillations in individual Leidenfrost drops, where the shape is mostly determined by the competition between the gravity and surface tension, as measured by the relative size of the drop with respect to the capillary length, , where and denote the surface tension and density of the liquid, and is the acceleration due to gravity. For drops with radius , surface tension forces are dominant, and the drop shape is essentially spherical except for a vanishingly small flat region near the solid surface (Burton et al., 2012). Caswell (2014) identified a planar vibrational mode (“breathing” mode) in the neck region of the drop closest to the solid substrate. The oscillation frequencies were found to obey a power law that is not consistent with a general three-dimensional dispersion relation for capillary waves (Rayleigh, 1879). In this paper, we provide an analytical expression for the breathing mode using a simple model based on gravity and surface tension which shows excellent agreement with the experimental data.
Large Leidenfrost drops form liquid puddles whose thickness is approximately 2. These puddles are known to spontaneously form large-amplitude, star-shaped oscillations (Holter & Glasscock, 1952; Adachi & Takaki, 1984; Strier et al., 2000; Snezhko et al., 2008; Strier et al., 2000; Ma et al., 2015, 2017). Similar oscillations have been observed in large drops on periodically-shaken, hydrophobic surfaces (Noblin et al., 2005, 2009), drops levitated by an underlying airflow (Bouwhuis et al., 2013), and drops excited by an external acoustic or electric field (Shen et al., 2010a, b; Mampallil et al., 2013). In studies where the frequency of external forcing is prescribed, the oscillations are excited by a parametric mechanism (Brunet & Snoeijer, 2011). The external forcing leads to variations in the drop radius with time. Since the drop radius appears in the dispersion relation for azimuthal, star-shaped oscillations, the evolution of the oscillation amplitude obeys an equation similar to the Mathieu equation. For Leidenfrost drops, however, the mechanism is less clear since there is no prescribed frequency, and the star oscillations are excited and sustained through the heat input and resulting evaporation of the liquid. It has been suggested that the star oscillations may result from modulations of the surface tension of the liquid due to temperature variations (Adachi & Takaki, 1984; Takaki & Adachi, 1985; Tokugawa & Takaki, 1994), or perhaps due to convective patterns (Snezhko et al., 2008; Strier et al., 2000). However, (Bouwhuis et al., 2013) observed star-shaped oscillations in drops which are supported by an external, steady air flow, suggest that a hydrodynamic coupling between the gas flow and liquid interface initiates the oscillations.
Given the importance of the Leidenfrost effect in basic fluid and thermal transport, or the numerous practical applications, we know surprisingly little about the coupling between the evaporated vapor flow and vapor-liquid interface that lead to rich dynamical phenomena. Here we explore this coupling by investigating the self-organized, star-shaped oscillations of Leidenfrost drops using six different liquids: water, liquid N, ethanol, methanol, acetone and isopropanol. The liquid drops were levitated on curved surfaces in order to suppress the Rayleigh-Taylor instability, and star-shaped oscillation modes with = 2 to 13 lobes along the drop periphery were observed. The number of observed modes depended sensitively on the liquid viscosity, whereas the oscillation frequency (wavelength) depended only on the capillary length but not the mode number, substrate temperature, or drop size. Accompanying pressure measurements in the center of the vapor layer indicate that the pressure variation frequency was approximately twice that the drop oscillation frequency for all of the observed modes, consistent with a parametric forcing mechanism. We show that the pressure oscillations are driven by capillary waves of a characteristic wavelength beneath the drop traveling from the drop center to the edge, and such capillary waves can be generated by a strong shear stress at the liquid-vapor interface. Additionally, we find that although thermal convection is expected to be quite strong, the robust frequency (wavelength) of star oscillations is only weakly affected by varying either the substrate or environmental temperature, suggesting that star-shaped oscillations of Leidenfrost drops are hydrodynamic in origin.
In the experiment, blocks of engineering 6061 aluminum alloy with dimensions of 7.6 cm 7.6 cm 2.5 cm were used as substrates. Resistive heaters were embedded with high-temperature cement into the aluminum to control the substrate temperature, . Six different liquids were used as Leidenfrost drops: deionized water, liquid nitrogen (liquid N), ethanol, methanol, acetone, and isopropyl alcohol (isopropanol). The relevant liquid properties at the boiling point, , such as the surface tension , density , dynamic viscosity , and capillary length , are listed in table 1. The substrates were heated to different temperatures based on for each liquid. For water, the temperature of the substrate was set from 523 K to 773 K, for ethanol, methanol, acetone, and isopropanol the temperature of the substrate was set to 523 K, whereas the substrate for liquid N was not heated due to its extremely low .
The upper surfaces of the substrates were machined into a concave, spherical shape in order to suppress the buoyancy-driven Rayleigh-Taylor instability at the vapor-liquid interface and keep the drops stationary (Snoeijer et al., 2009; Quéré, 2013; Trinh et al., 2014). The curved surfaces for different liquids were designed to satisfy = 0.03, where is the radius of curvature of the surface whose top and cross-sectional views are schematically shown in figures 1(a) and 1(b), respectively. Following this principle, we fabricated three types of curved substrates, i.e. one for water, one for ethanol, methanol, acetone, and isopropanol, and one for liquid N, considering their respective capillary lengths () listed in table 1. For some experiments, a plano-concave, fused silica lens (focal length = 250 mm) was used as the heated substrate in order to allow for optical imaging of the capillary waves underneath the Leidenfrost drop. A T-type thermocouple (maximum measurable temperature: 473 K, Omega Engineering) was used to measure the internal temperature profile of Leidenfrost drops.
For most of the substrates, a pressure sensor (GEMS Sensors, response time: 5 ms, sensitivity: 2 mV/Pa) was connected to a hole (diameter = 1 mm) at the center of the curved substrate in order to measure the pressure variations in the vapor layer at sample rates of 500-1000 Hz during quiescent and oscillatory phases of drops as illustrated in figure 1(b). We used a high-speed digital camera (Phantom V7.11, Vision Research) with a resolution of about 132 pixels/cm to image the motions of drops from above at frame rates of 1000 frames per second. Recorded videos were then analyzed with NIH ImageJ software to obtain the frequency and wavelength of the star-shaped oscillations (A typical star-shaped oscillation mode = 8 is schematically shown in figure 1c).
3 Results and discussions
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 nonwetting drop with radius 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., 2012). In comparison, for large Leidenfrost drops, i.e. , the shape of the drops is dominated by gravity and resembles a circular puddle with a constant thickness of approximately 2 (Biance et al., 2003). However, these large puddles are susceptible to the Rayleigh-Taylor instability, which is manifested by bubbles rising from the vapor layer beneath the drop (Biance et al., 2003). 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., 2009; Quéré, 2013; Trinh et al., 2014).
The effects of surface curvature on the Rayleigh-Taylor instability can seen through the following simple model, similar to the original model used by Biance et al. (2003). Assuming the liquid-vapor interface beneath a Leidenfrost drop is perturbed with axisymmetric, sinusoidal variations, as illustrated in figure 3, then the new drop interface can be expressed in cylindrical coordinate as
where is the mean vapor layer thickness, is the perturbation amplitude, and is the capillary wavelength generated at the bottom of the drop (see figure 11). We also assume that , implying a nearly-flat interface with an infinitesimal perturbation. For a stable interface, the pressure at the drop center should be smaller or equal to the pressure at in order to drive liquid back to the center. If the pressure in the vapor layer is constant, then to leading order, the pressures at = 0, and = are
Since the perturbation amplitude and radius of curvature always appear as a product, (4) can also be written as
where . For finite , in the limit , (4) simplifies to , in agreement with the prediction in Biance et al. (2003) for a flat surface. However, the addition of a curved surface couples the perturbation amplitude to the radius of curvature. The quantity must be positive in order for to be real, and thus represent the condition for instability. Subsequently, on a curved surface, the perturbation amplitude must satisfy in order to lead to the Rayleigh-Taylor instability.
Although the curved surface suppresses the Rayleigh-Taylor instability, it still has an affect on the overall drop shape. We investigated the influence of surface curvature on the drop shape by depositing water drops of given volumes on a curved surface with = 623 K. We measured the radius of the drops from recorded images taken immediately after deposition prior to the onset of oscillations. As can be seen in figure 2, stable Leidenfrost drops with are obtained on the curved surface, while on a flat substrate, the maximum radius of a stable Leidenfrost drop is (Snoeijer et al., 2009; Biance et al., 2003; Burton et al., 2012), suggesting that surface curvature plays a crucial role in suppressing the Rayleigh-Taylor instability. The dashed blue line in figure 2 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 (Burton et al., 2010, 2012). 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 (2.5 mm), the drops are quasi-spherical and thus , whereas for , which is expected for puddle-like drops with constant thickness.
However, one can also notice that most of the experimental data is 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 center 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 (2014) experimentally characterized 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. He found that the oscillation frequency of the breathing mode, , obeyed a distinct power law, , where is the average drop radius during the oscillation. This dependence is distinctly different than the expected three-dimensional dispersion relation for inviscid spherical drops, (Rayleigh, 1879). Here we provide an analytical model which explains this contrast and fits the experimental data with no adjustable parameters.
To leading order, we model a Leidenfrost drop as an incompressible liquid cylinder of volume = and a time dependent radius, . We will assume that the bottom of the cylinder is fixed at , which is reasonable if the thickness of the vapor layer varies much less than the radius. In cylindrical coordinates, , the simplest form for the velocity which satisfies and the boundary conditions and is:
For simplicity, we will write , and . The total kinetic energy of the drop is then:
Evaluating the integrals, and using the fact that , where is the time-dependent height of the cylinder, we obtain
The total potential energy is the sum of gravitational potential energy and surface energy:
The equilibrium drop radius, , can be found by minimizing the potential energy with respect to ,
For large drops where , the first and third terms must balance. Equating these two terms and using the fact that , we see that
which agrees with the expected asymptotic thickness of large Leidenfrost drops (Biance et al., 2003). We may now define the Lagrangian of the system as , and apply the Euler-Lagrange equation to obtain a differential equation for :
To proceed further, we will linearize the equation by considering only small oscillations of the radius, , where is equilibrium drop radius found by solving (10), is the perturbation amplitude, and is the angular frequency of the oscillation. Assuming that , to leading order (12) reduces to an expression for the angular frequency:
where and we have substituted for the equilibrium radius using (10).
Taking typical values of the parameters in (13) for water, we can plot the oscillation frequency of the breathing mode, , as a function of , which is shown by the solid blue curve in figure 4. The theory shows excellent agreement with the data. For comparison, the power law found by Caswell (2014) is plotted as the dashed red line in figure 4. The data lies in the transition regime between two asymptotic limits, and , which may explain the anomalous power law reported by Caswell (2014). In these two limits, (13) reduces to
The scaling for small drops is independent of , which is expected based on their nearly-spherical shape. For large drops where (11), the scaling is determined essentially by gravity: .
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, 2011). In our experiments with Leidenfrost water drops using a curved substrate, we observe star-shaped oscillation modes with with = 2 to 13 lobes along the drop periphery at K. A typical star-shaped oscillation mode () is shown in figure 5(a). Figures 5(b) and 5(c) show the drop oscillation wavelength and frequency for different , respectively, which are measured by analyzing the high-speed videos of the oscillating drops. Surprisingly, both and remained nearly constant as we varied . Increasing thus led to an increase in the allowable number of lobes , 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, , of the th-mode is given by (Rayleigh, 1879):
The first term under the square root is for a strictly two-dimensional drop, whereas the correction factor, , is due to the quasi-two-dimensional nature of the puddle.
As shown in figure 5, an = 2 water drop has an average radius of 4 mm , hence it is reasonable to use (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 , both and vary non-monotonically with . There are a few potential reasons for this. One possibility is that nonlinear effects play a more significant role for smaller values of because the ratio of the mode amplitude to the drop radius is larger (Becker et al., 1991; Smith, 2010). However, the excellent agreement with the linear theory, i.e. (16), suggests that nonlinear effects may not be important.
Instead, we suggest that the behavior 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 or is selected, (16) will determine the other. As will be shown in §3.6, there are strong capillary waves excited at the liquid-vapor interface beneath the drop. These waves produce pressure oscillations which parametrically couple to the star-shaped modes. A determination of the expected behavior of on is thus complicated by the physics of the vapor flow beneath the drop, nevertheless, we would expect the effects to be less variable for , which agrees with the data shown in figure 5.
3.4 Pressure oscillations in the vapor layer
It has been well-documented that star oscillations of large liquid drops can be initiated by a parametric forcing mechanism (Brunet & Snoeijer, 2011). 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 oscillations will be excited when the drop oscillation frequency is approximately half that the excitation frequency (Yoshiyasu et al., 1996; Brunet & Snoeijer, 2011). In the experiment, we observed star oscillations of Leidenfrost water drops (e.g. see figure 5a) in the absence of an external excitation. Therefore, we hypothesize that the star oscillations are driven by fluctuations in the vapor layer pressure that supports the drop.
In order to explore the dynamics of pressure fluctuations in the vapor layer, we utilized a pressure sensor attached to the center of the curved substrate, schematically shown in figure 1(b). We measured the pressure variations in the vapor layer of water drops at = 623 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 about 18.5 s.
The snapshots on the top panel of figure 6(a) show images of an = 4 drop at different stages during the oscillation. More specifically, the drop shape was nearly circular immediately upon placement on the substrate ( = 0 s), then the pressure fluctuations in the vapor layer increased as the drop shape developed well-defined lobes. The amplitude continued to grow with time ( =12 s) until the formation of a steady-state, star-shaped oscillation with large amplitude ( = 16.1 s).
Figure 6(a) shows the pressure variations in the vapor layer during the whole process. The mean pressure required to support the drop is Pa. However, the mean pressure we measure is slightly larger than this value because the pressure is measured at the center of the substrate where the the pressure is larger in order to drive the viscous vapor 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 (about = 14 s).
In order to gain insight into the underlying relationship between the oscillations of the drop and pressure in the vapor 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 figures 6(b)-6(e). Initially, the pressure remains nearly constant, and there are no sharp peaks in the power spectrum (figure 6b). Between = 5-6 s, the pressure fluctuations become stronger and more periodic, and several small peaks are visible (figure 6c). Then, as the star oscillation is further developed, a sharp peak which is located at 28 Hz shows up in the power spectrum (figure 6d). Finally, when the star oscillation is fully developed, the pressure fluctuations stop growing, and a sharp peak located at 28 Hz dominates the spectrum (figure 6e).
It is interesting to note that the location of the sharp peak is approximately twice the oscillation frequency of a fully-developed, = 4 mode Leidenfrost water drop (see figure 5c). This is consistent with a parametric forcing mechanism for the excitation of the star 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 shows the pressure oscillation frequency in the vapor layer of Leidenfrost water drops for all of the observed modes. The data points from left to right correspond the oscillation modes from = 2 to 13, as indicated by the red arrows. By comparing this with the drop oscillation frequency (figure 5c), we find . This robust relationship suggests that the star-shaped oscillations are parametrically driven by the pressure (Miles & Henderson, 1990; Kumar & Tuckerman, 1994; Yoshiyasu et al., 1996; Brunet & Snoeijer, 2011; Terwagne & Bush, 2011). Consider the radial position of a point on the perimeter of the drop during a star-shaped oscillation, . This point will oscillate in time due to the azimuthal standing wave, and obey the following equation:
To leading order, is the resonant frequency of the mode, (16):
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:
When , 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 vapor layer (figure 7), which are approximately twice that the drop oscillation frequencies (figure 5c) that we observed in our experiment. We note that it is possible that the pressure oscillations are a consequence of the star 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 vapor layer. Thus we expect that the pressure oscillations are the source of the star oscillations. §3.6 will explore the potential source of the pressure oscillations.
3.5 Star-shaped oscillations of different liquids
In addition to water, we performed similar experiments with five other liquids: liquid N, acetone, methanol, ethanol, and isopropanol. For these liquids, the curved surfaces were also machined to satisfy (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 analyzing the images of the star-shaped oscillations, we find that star-shaped oscillations of acetone, methanol, ethanol, and isopropanol share a similar wavelength cm and frequency Hz, whereas liquid N shows a wavelength of cm and frequency Hz. This suggests that the star oscillations may depend on the capillary length of the liquid.
Figure 9 shows the dependence of star-shaped oscillation mode , rescaled wavelength , and rescaled frequency on the rescaled drop radius of the six liquids used in the experiment. The mode number increases linearly with 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 for the six liquids collapses onto a straight line as shown in figure 9(b), which indicates that the wavelength of the star oscillations only depends on of the liquid. Figure 9(c) shows the oscillation frequency of both the azimuthal star oscillations, , and the corresponding pressure, , in the vapor layer. The solid symbols denote the drop oscillations, whereas open symbols with the same color represent the corresponding pressure oscillations in the vapor layer for a specific liquid. Both and collapse fairly well, and the relationship is consistent with the data, as indicated by the dashed black and red lines. Additionally, by comparing different values of for water, figure 9 shows that , , and are mostly independent of substrate temperature.
The main influence of is the through the number of observable modes. For instance, at = 673 K, we did not observe oscillation modes = 12 and 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 and suggests that the star oscillations of the drops are parametrically driven by the pressure in the vapor layer, there is an interesting variation in both and at small mode numbers. The data for smaller modes (e.g. ) deviates further from the average compared to larger modes. This behavior may be nonlinear in nature since the amplitude of the mode relative to the drop radius is much larger for small (Becker et al., 1991; Smith, 2010). In addition, the quasi-two-dimensional nature of the drop is more important at small , which may also contribute to the variations in this regime. A more quantitative explanation for this dependence is left for future studies.
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. (2017), the role of damping can be characterized by the Reynolds number associated with the liquid oscillation. For a star oscillation, the characteristic length scale and time scale are and , respectively. Thus, the inertial force density in the Navier-Stokes equation is estimated as , and the viscous force density is . The Reynolds number of the oscillating liquid is defined as the ratio of inertial to viscous forces: , and the values of for different liquids are listed in table 1.
Figure 10 shows the dependence of the minimum mode number on for each liquid. The dashed line represents a suggested scaling , indicating that the liquid viscosity damps the oscillations of smaller drops and thus sets the minimum mode number of stable star 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 . This can be seen in figure 10 for liquid N, which undergoes much more rapid evaporation due to the large difference between the boiling and substrate temperature, . 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 vapor layer is linked to the pressure oscillations.
3.6 Origin of pressure oscillations in the vapor layer
As evidenced by figure 9(c), the star-shaped oscillations of Leidenfrost drops are parametrically driven by the pressure variations in the vapor layer. Star oscillations induced by a parametric coupling have been observed in a variety a systems where variations of the drop radius are induced by external, periodically-modulated fields (Brunet & Snoeijer, 2011). 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-vapor interface induce the pressure oscillations in the vapor layer.
Figure 11 shows s sketch of a large, axisymmetric Leidenfrost drop with a maximum radius = and thickness . The mean velocity of the gas at the liquid surface is , is the mean thickness of the vapor layer, and is the radial velocity of the gas near . 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. (2003), the mass loss rate of the drop due to evaporation can be expressed as:
where is the thermal conductivity of the vapor, is the latent heat of the evaporation, , and is the density of the vapor. Since 100 m, and thus (Burton et al., 2012), we employ lubrication theory, leading to an expression relating the flow rate to the pressure in the vapor layer:
where is the dynamic viscosity of the vapor.
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 the perimeter of drop:
In this model, the steady-state, linear temperature profile in the vapor layer is valid since the characteristic time scale associated with thermal diffusion, 1 ms, is smaller than the typical residence time of the gas in the vapor layer, 10 ms, where is the thermal diffusivity of the vapor. Let us assume a water Leidenfrost drop with = 0.01 m, and for the vapor, we use the properties of steam at 373 K and 1 atm of pressure: = 959 kg/m, = 0.45 kg/m, = 0.059 N/m, = 9.8 m/s, = 2.26 J/kg, = 1.82 10 Pa s, and = 0.04 W/m/K. Although the properties of the vapor may vary somewhat in the vapor layer due to the temperature gradient, we have verified that this does not significantly affect our analysis.
We can now plot , , and with respect to based on (24), (25) and (26). The results are shown in figures 12(a), 12(b), and 12(c), respectively. The inertial force in the Navier-Stokes equation is estimated as , and the viscous force as . Thus the Reynolds number in the vapor layer is . Plugging in (25) and (26), we arrive at a surprisingly simple expression for the Reynolds number in the vapor: , which is independent of the drop size. This expression is plotted in figure 12(d). Under typical experimental conditions, , 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 vapor layer of Leidenfrost water drops as Pa, which is much smaller than the pressure variations 10 Pa (see figure 6a), suggesting that the inertial force does not account for the pressure variations. The viscous pressure is , thus the pressure variation due to the local variations of the film thickness is . Then 10 Pa corresponds to 15 m considering the typical values of (figure 12c) and . Although this local film thickness could also be possibly induced by the vertical motion of the center of mass of the drops, the fact that small and large drops (see figure 9c) share a nearly constant oscillation frequency suggests this is not the case. Therefore, the pressure variations in the vapor layer are likely to be induced by the local variations of the vapor film thickness.
We propose that capillary waves with a characteristic wavelength, , traveling from the center to the edge of the drop lead to local variations in vapor film thickness, and thus pressure variations in the vapor 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 center 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 px/cm).
We observed a sharp peak in the measured Fourier spectrum for all the analyzed video sequences. In figures 13(a) and 13(b) (8-mode, acetone), this peak is located at a capillary wave frequency 24 Hz and within a range of wave numbers, , from 7 cm to 12 cm, where is the wavelength of the capillary waves. Figures 13(c) and 13(d) show the capillary waves beneath a 11-mode Leidenfrost ethanol drop and a sharp peak at a capillary wave frequency 25 Hz with 7-10 cm. Similarly, for the capillary waves beneath a 12-mode Leidenfrost ethanol drop (figure 13e), a sharp peak exists at 27 Hz and 10-11 cm as shown in figure 13(f). These frequencies show excellent agreement with the corresponding typical pressure oscillation frequencies measured in the vapor layer of star-shaped Leidenfrost ethanol and acetone drops (see figure 9c).
It is well-known that capillary waves can be generated at a liquid-vapor interface due to a strong shear stress in the vapor (Miles, 1957; Zhang, 1995; Paquier et al., 2015; Zeisel et al., 2008; Chang & Demekhin, 2002). A similar mechanism underlies the Kelvin-Helmholtz instability, however, the intermediate Reynolds number in the Leidenfrost vapor 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 , where is the shear stress at the liquid-vapor interface. The maximum shear stress at the interface is:
assuming a parabolic-flow profile in the vapor layer with mean velocity near the edge of the drop. Using water as an example, and plugging (24) and (26) and the value of of water vapor into (27) yields m/s. This friction velocity is quite strong and sufficient to lead to the growth of unstable modes with wavelengths of millimeter scale (Zhang, 1995; Zeisel et al., 2008).
The general dispersion relation of gravity-capillary waves with a dense upper-layer is:
where = 2/, and 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 vapor layer is 26 Hz (see figure 7), the corresponding capillary wavelength is calculated to be , so that 8.3 cm (Ma et al., 2017). Similarly, for the Leidenfrost acetone and ethanol drops shown in figure 13(a), 13(c), and 13(e), the corresponding cm, which is slightly larger than the positions of the peaks indicated in figures 13(b), 13(d), and 13(f), where . The agreement between the estimate for and the measurements shown in figure 13 is good considering the simplicity of (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 = 2 mode. More specifically, the radius we measured our experiments is rather than as illustrated in figure 11. The relationship between these two lengths is (Snoeijer et al., 2009; Burton et al., 2012). Thus, the minimum drop size required to fit one capillary wavelength beneath the drop is . Using from above, we find that the radius of an = 2 mode drop should be . This is in good agreement with figure 9(a), which shows is slightly less than 2 for the smallest drops. Taken together, this analysis suggests a purely hydrodynamic origin for the star oscillations based on capillary waves generated by a strong shear stress in the rapidly-flowing vapor beneath the drop.
3.7 Thermal effects
In the Leidenfrost effect, thermal energy is transfered 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 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, 1901; Rayleigh, 1916). The convection pattern is strongly affected by the variation of surface tension with temperature (Marangoni, 1871; Schatz & Neitzel, 2001; Maroto et al., 2007). Given that our data collapses 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, , and Rayleigh number, , are used to characterize Bénard-Marangoni convection. These dimensionless numbers are defined as:
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-vapor interface is characterized by the Biot number:
The definitions of the symbols in , , and are listed in table 2, as well as their values for water at the boiling point.
|liquid kinematic viscosity||2.9410 m/s|
|liquid thermal diffusivity||1.74310 m/s|
|thermal expansion coefficient||7.5210 /K|
|liquid thermal conductivity||3.210 W/m/K|
|drop thickness||0.005 m|
|heat transfer coefficient||depends on|
|surface tension gradient||1.4610 N/m/K|
|surface temperature||367 K|
Figure 14 shows the temperature profile of a large Leidenfrost water drop levitating on its vapor layer with K, in which the substrate surface is defined as . 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 and , respectively, and the temperature of the position which is slightly above the top surface of the drop is denoted as the environment temperature, , as shown in figure 11. From figure 14 we can obtain = 3.8 K, the environment temperature measured to be 367 K, thus we can calculate , , and 1.1. Both the values of and are much larger than their critical values, which are typically of order 100 for the initiation of convective instability (Maroto et al., 2007; Leal, 2007). 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, , 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 aluminum foil around the substrate and Leidenfrost drop, which dramatically increased the environment temperature, , near the top of drop.
Figures 15(a) and 15(d) show Fourier power spectra of the pressure oscillations in the vapor layer of Leidenfrost water drops during = 4 and = 5 oscillations at 367 K, respectively. Figures 15(b) and 15(e) are the spectra of pressure oscillations during = 4 and = 5 oscillations at 483 K, respectively. Finally, figures 15(c) and 15(f) represent the same modes at = 623 K. For = 483 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 behavior is non-monotonic, and is likely due to the highly nonequilibrium conditions (high evaporation rate) induced by the high temperatures of the substrate () and environment (). 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 , indicating that convection and the details of thermal transport in Leidenfrost drops play a secondary role in the star-shaped oscillations.
4 Conclusion and outlook
Both large and small Leidenfrost drops display self-organized oscillations due to the constant input of thermal energy and continuous evaporation and flow beneath the drop. Here we have focused on radial oscillations (i.e. “breathing” mode) of small Leidenfrost drops, and the large-amplitude, star-shaped oscillations that appear in large Leidenfrost drops. We have characterized the number of observed modes for various volatile liquids, the frequency and wavelength of the oscillations, and the pressure variations in the vapor layer beneath the drops. The number of observed modes is sensitive to the properties of the liquid (see table 1), i.e. the star-shaped oscillations of smaller Leidenfrost drops are dissipated by the liquid viscosity, which sets the minimum oscillation mode number that can be observed in experiment. The relationship between the frequency and wavelength agrees very well the quasi-two-dimensional theory proposed by Yoshiyasu et al. (1996). The dominant frequency associated with the pressure oscillations is approximately twice the drop oscillation frequency, consistent with a parametric forcing mechanism for the star oscillations.
One of the main findings of our work is identifying the underlying cause of the pressure oscillations. By imaging the liquid-vapor interface from below the drop, and using a simplified model for the flow in the thin vapor layer, we conclude that capillary waves of a characteristic wavelength, , lead to pressure oscillations at the experimentally measured frequency. The flow in the vapor layer is quite rapid. Near the edge of the drop, the mean radial velocity can reach 1-2 m/s or more. In the small gap ( 100 m) between the liquid and the solid surface, this flow applies a large shear stress to the liquid-vapor interface, and can easily excite capillary waves with millimeter-scale wavelengths. The dispersion relation for the capillary waves then leads to a characteristic frequency for the pressure oscillations, which in turn parametrically drive the star-shaped oscillations. Furthermore, although the vapor flow is inherently driven by evaporation and heat transfer, the substrate and surrounding temperature have little effect on the dominant frequency and wavelength of the oscillations, suggesting they are purely hydrodynamic in origin.
Although the work presented here has focused mostly on the the origin of star-shaped oscillations, the coupling of the flow in the vapor layer and the liquid-vapor interface underlies a rich spectrum of dynamical phenomena observed in both Leidenfrost liquid layers and drops. In particular, of key interest is understanding the failure of the Leidenfrost vapor layer which can lead to explosive boiling. If the observed capillary waves beneath the drop act as the precipitant to vapor-layer failure, then it is possible that geometrical tailoring of the surface to be commensurate with may inhibit the generation of capillary waves. In addition, patterned, ratchet-shaped substrates with with wavelengths 1-3 mm are known to induce propulsion of small Leidenfrost drops (Cousins et al., 2012; Linke et al., 2006; Lagubeau et al., 2011), however, less is known about transport of large quantities of liquid, and the dependence on the wavelength of the surface patterns. We leave these questions open to future experiments. More generally, our results may offer insight into the direct control of oscillations in levitated drops in many other systems (Haumesser et al., 2002; Duchemin et al., 2005; Paradis & Ishikawa, 2005; Ishikawa et al., 2006; Lister et al., 2008; Langstaff et al., 2013), for example, precise control is crucial when levitating high-temperature or harmful liquids using a gas film. We also expect our results to enhance the understanding of dynamics that couple a thin, supporting gas film, a liquid interface, and a solid surface, a scenario which occurs through forced wetting and gas entrainment in liquid coating (Xu et al., 2005; Driscoll & Nagel, 2011; Marchand et al., 2012; Kolinski et al., 2012; Liu et al., 2013, 2015).
We thank Tom Caswell for providing the experimental data related to the breathing mode of Leidenfrost drops, and Juan-José Liétor-Santos for helpful discussions. This work was supported by the National Science Foundation through grant NSF DMR-1455086.
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