The thermo-wetting instability driving Leidenfrost film collapse

Significance

The insulating vapor film formed during film boiling dramatically reduces the heat flux from the surface. This can be highly detrimental in the quenching of metal alloys, the cooling of nuclear fuel rods, and the thermal control of electronic and photonic devices. On the other hand, the vapor film can also be harnessed to reduce liquid–solid drag as well as design levitating, self-rotating rotors and self-propelled droplets for lab-on-a-chip applications. Control of vapor film collapse in different systems and operating conditions is therefore highly desirable. Surface wettability is known to influence this collapse. However, a comprehensive thermo-wetting instability theory that predicts observed data has been elusive. This work proposes such a theory.

Abstract

Above a critical temperature known as the Leidenfrost point (LFP), a heated surface can suspend a liquid droplet above a film of its own vapor. The insulating vapor film can be highly detrimental in metallurgical quenching and thermal control of electronic devices, but may also be harnessed to reduce drag and generate power. Manipulation of the LFP has occurred mostly through experiment, giving rise to a variety of semiempirical models that account for the Rayleigh–Taylor instability, nucleation rates, and superheat limits. However, formulating a truly comprehensive model has been difficult given that the LFP varies dramatically for different fluids and is affected by system pressure, surface roughness, and liquid wettability. Here, we investigate the vapor film instability for small length scales that ultimately sets the collapse condition at the Leidenfrost point. From a linear stability analysis, it is shown that the main film-stabilizing mechanisms are the liquid–vapor surface tension-driven transport of vapor mass and the evaporation at the liquid–vapor interface. Meanwhile, van der Waals interaction between the bulk liquid and the solid substrate across the vapor phase drives film collapse. This physical insight into vapor film dynamics allows us to derive an ab initio, mathematical expression for the Leidenfrost point of a fluid. The expression captures the experimental data on the LFP for different fluids under various surface wettabilities and ambient pressures. For fluids that wet the surface (small intrinsic contact angle), the expression can be simplified to a single, dimensionless number that encapsulates the wetting instability governing the LFP.

As a surface is superheated above the boiling point of an adjacent fluid, vapor bubbles nucleate and grow. The boiling behavior of the liquid phase undergoes a fundamental change at a critical temperature known as the Leidenfrost point. Beyond this point, a film of insulating vapor forms between the liquid and the surface that suppresses heat transfer from the solid material. This heat flux reduction can be highly detrimental in the quenching of metal alloys by extending cooling rates and precluding the desired increase in strength and hardness (1). Alternatively, film boiling may be used to promote drag reduction as well as enable power generation through self-propulsion (2–4). Thus, modulation of the Leidenfrost point (LFP) through fluid choice, surface texture, and surface chemistry for the specific application is crucial (5).

On a fundamental level, the physical mechanism responsible for the LFP is still uncertain. Many theoretical frameworks have been used to characterize the Leidenfrost effect and estimate the LFP, including hydrodynamic instability (6, 7), superheat spinodal limits (8, 9), and the change of liquid wettability on the heated surface with temperature (10, 11). A thermocapillary model has also been proposed that attributes the film instability to fluctuations at micrometer length scales; however, the analysis posits that the thermocapillary effect is the dominant destabilizing term, which does not explain the significant change in LFP on surfaces with different wettabilities (12). For example, the LFP of water can vary from $300\hspace{0.17em}°\text{C}$ for hydrophilic surfaces to $145\hspace{0.17em}°\text{C}$ on hydrophobic surfaces (13, 14).

In this work, we introduce a stability analysis of the vapor film at the nanoscale regime. The dominant destabilizing term arises from the van der Waals interaction between the bulk liquid and the substrate across a thin vapor layer. On the other hand, liquid–vapor surface tension-driven transport of vapor and evaporation at the two-phase interface stabilize the film. The competition between these mechanisms gives rise to a comprehensive description of the LFP as a function of both fluid and solid properties. For fluids that wet the surface, such that the intrinsic contact angle is small, a single dimensionless number (Eq. 48) can be derived that encapsulates the instability determining the Leidenfrost point.

It is noted that the literature has proposed different names for the critical temperature associated with a droplet levitating on a heated plate (Leidenfrost point) versus the critical temperature for vapor film formation in pool boiling (minimum film temperature). The Leidenfrost point has been shown to be equivalent to the minimum film boiling temperature for saturated liquids on isothermal surfaces (15). In this work, the term LFP is used for both cases as a matter of convenience, with the understanding that no undercooling is applied to the liquid phase for pool boiling scenarios unless explicitly stated.

Film Instability

There are many approaches to examine the stability of a vapor film adjacent to a superheated wall in two dimensions. Models have been developed with a base solution imposing static equilibrium, where the interface is at the saturation temperature corresponding to the imposed, far-field liquid pressure (12, 16). Here, we consider the thickness of the vapor film to be in dynamic equilibrium, as in a vertical plate configuration (17) or vapor under a droplet (18). This appears to be a more general analysis since under experimental settings, droplet levitation occurs over a film that is continuously replenished by evaporation and depleted through escape of the buoyant vapor phase. Similarly in a horizontal setup for pool boiling, bubbles pinch off the film, necessitating a nonzero rate of evaporation to sustain a constant mean film thickness (17).

Fig. 1 shows a chosen “model” problem of film boiling on a vertical plate (8, 17). The vapor forms a laminar layer at the wall, with evaporation at the liquid interface sustaining the buoyant transport of vapor mass away from the base of the plate. The surrounding liquid is saturated and motionless with its properties fixed at the saturation temperature. The properties of vapor are assumed to be constant at the superheated wall temperature, an assumption discussed in SI Appendix.

Fig. 1.

Film boiling on a vertical plate, with the coordinate system delineated. The film thickness is denoted by $\overline{\delta }\left(x\right)$.

Governing Equations.

The mass, momentum, and energy conservation equations in the vapor domain are given by the boundary-layer equations

$$\frac{\partial {ū}_V}{\partial x}+\frac{\partial {\overline{v}}_V}{\partial y}=0$$ [1]

$${ū}_V\frac{\partial {ū}_V}{\partial x}+{\overline{v}}_V\frac{\partial {ū}_V}{\partial y}=-\frac{1}{{\rho }_V}\frac{\partial \overline{\mathrm{\Phi }}}{\partial x}+\frac{\mathrm{\Delta }\rho g}{{\rho }_V}+\frac{{\mu }_V}{{\rho }_V}\frac{{\partial }^2{ū}_V}{\partial y^2}$$ [2]

$$\frac{\partial \overline{\mathrm{\Theta }}}{\partial t}+{ū}_V\frac{\partial {\overline{\mathrm{\Theta }}}_V}{\partial x}+{\overline{v}}_V\frac{\partial {\overline{\mathrm{\Theta }}}_V}{\partial y}=\frac{k_V}{{\rho }_V{c}_{p,V}}\frac{{\partial }^2{\overline{\mathrm{\Theta }}}_V}{\partial y^2}$$ [3]

where the parameters $\mu$, $\rho$, $k$, g, and $c_p$ represent the dynamic viscosity, density, thermal conductivity, gravitational acceleration, and specific heat of the fluid, respectively. The term $\mathrm{\Delta }\rho ={\rho }_L-{\rho }_V$ represents the density difference, the subscripts $L$ and $V$ denote the liquid and vapor field, and the temperature has been normalized as $\overline{\mathrm{\Theta }}=\frac{\overline{T}-T_s}{T_w-T_s}$, the difference between the temperature field and the saturation temperature $T_s$ at the interface over the difference between the wall temperature $T_w$ and $T_s$. Note that we take the liquid phase to be motionless due to its much greater viscosity relative to the vapor ($u_L\left(x,t\right)=0$), as well as isothermal at saturation temperature in the long-time limit (equilibrium) due to its larger thermal conductivity (${\mathrm{\Theta }}_L=0$) (19). Due to high thermal conductivity of the liquid relative to the vapor, the temperature variations in the liquid are much less than those in the vapor. Consequently, most of the temperature gradient would be observed within the vapor. These simplifications allow for an analytical solution for the base state per Burmeister (17).

Note that if no external temperature boundary conditions are imposed on the liquid reservoir, the saturation condition for the liquid is well established in experiment (15). The adjacent liquid is also generally assumed to be at the saturation temperature in physical models of film boiling (8, 17, 29).

The generalized pressure term $\overline{\mathrm{\Phi }}$ takes into account the fluid pressure arising due to surface tension as well as due to van der Waals interactions. To first order in the base solution, these terms are negligible since the liquid–vapor interface is assumed to be locally parallel to the wall (20, 21); this implies $\overline{\mathrm{\Phi }}\approx 0+\mathrm{\Phi }\prime$, where the overbar variables represent the general solution, the unbarred variables denote the base solution, and the primed variables give the perturbed solution. Additionally, the momentum and temperature equations are modeled as steady in the base solution and exhibit a time-varying term only in the linearized equations for the perturbations.

The boundary conditions at the superheated wall and the liquid–vapor interface at $\overline{\delta }\left(x\right)$ are given by

$$\text{at}\hspace{0.17em}y\hspace{1em}=0,\hspace{1em}{ū}_V\hspace{1em}=\hspace{1em}{\overline{v}}_{V\hspace{1em}}=0,\hspace{1em}{\overline{\mathrm{\Theta }}}_V=1$$ [4]

$$\text{at}\hspace{0.17em}y\hspace{1em}=\hspace{1em}\overline{\delta },\hspace{1em}{ū}_V+{\overline{v}}_V\frac{\text{d}\overline{\delta }}{\text{d}x}=0,\hspace{1em}{\overline{\mathrm{\Theta }}}_V=0.$$ [5]

Eq. 4 enforces the temperature and no interfacial slip at the impermeable wall, while Eq. 5 ensures that the tangential component of velocity is continuous and that the temperature is at saturation along the liquid–vapor interface. As with Burmeister (17), we neglect the “blowing” of vapor toward the plate by assuming $v\left(x,t\right)\approx 0$, such that the preceding set of boundary conditions is sufficient to fully specify the problem. Otherwise, the normal component of velocity would also need to be fixed at the interface; this leads to a cubic rather than a parabolic estimate to the velocity field.

The vapor generated due to phase change at this interface is balanced by the streamwise rate of change of the vapor flow in the film and the growth of the film thickness in time,

$${\rho }_V\frac{\partial \overline{\delta }}{\partial t}+\frac{\partial }{\partial x}{\int }_0^{\overline{\delta }}\left({\rho }_V{ū}_{V}dy\right)=-\frac{k_V}{h_{LV}}{\left.\frac{\partial {\overline{\mathrm{\Theta }}}_V}{\partial y}\right|}_{y=\overline{\delta }},$$ [6]

where $h_{LV}$ is the latent heat of vaporization. In the base state, the time variation of the film thickness is taken to be negligible, $\frac{\mathrm{d}\overline{\delta }}{\mathrm{d}t}\approx 0+\frac{\mathrm{d}\delta \prime }{\mathrm{d}t}$. An in-depth description of this setup is covered in SI Appendix.

Base Flow.

Along with the boundary conditions Eqs. 4 and 5, the base flow equations are

$$\frac{\partial u_V}{\partial x}+\frac{\partial v_V}{\partial y}=0$$ [7]

$$u_V\frac{\partial u_V}{\partial x}+v_V\frac{\partial u_V}{\partial y}=\frac{\mathrm{\Delta }\rho g}{{\rho }_V}+\frac{{\mu }_V}{{\rho }_V}\frac{{\partial }^2{u}_V}{\partial y^2}$$ [8]

$$u_V\frac{\partial {\mathrm{\Theta }}_V}{\partial x}+v_V\frac{\partial {\mathrm{\Theta }}_V}{\partial y}=\frac{k_V}{{\rho }_V{c}_{p,V}}\frac{{\partial }^2{\mathrm{\Theta }}_V}{\partial y^2}.$$ [9]

The steady, developed solution can be found approximately by using an integral expansion method, which is described in full detail by Burmeister (17). After introducing the normalized variable $\eta =y/\delta$, we can determine the velocity ($u_V$) and temperature field (${\mathrm{\Theta }}_V$) in the base solution:

$$u_V=\frac{\mathrm{\Delta }\rho g{\delta }^2}{2{\mu }_V}\left(\eta -{\eta }^2\right)$$ [10]

$${\mathrm{\Theta }}_V=\frac{T_V-T_s}{T_w-T_s}=1+\left(-1-\frac{1-c}{2}\right)\eta +\frac{1-c}{2}{\eta }^3.$$ [11]

Note that the velocity profile is locally parabolic due to the buoyancy force, while the velocity variation in the streamwise direction under mass conservation is encapsulated in the $\delta {\left(x\right)}^2$ dependence and occurs on a much larger length scale. The value of $c$ can be found by solving the quadratic expression (17)

$$\frac{1}{3}\frac{c_{p,V}\mathrm{\Delta }T}{h_{LV}}c\left(1-\frac{3}{10}\left(1-c\right)\right)=1-c$$ [12]

and $\mathrm{\Delta }T=T_w-T_s$. For typical Jakob numbers around $Ja=\frac{c_{p,V}\mathrm{\Delta }T}{h_{LV}}=\frac{1}{10}$, $c$ can be found from a simplified linear equation $c=1-\frac{1}{3}\frac{c_{p,V}\mathrm{\Delta }T}{h_{LV}}$. Using this approximation, the film thickness $\delta$ is described by

$$\delta =2{\left(1-\frac{1}{3}\frac{c_{p,V}\mathrm{\Delta }T}{h_{LV}}\right)}^{1/4}{\left(\frac{x\mathrm{\Delta }T{\mu }_V{k}_V}{{\rho }_V{h}_{LV}g\mathrm{\Delta }\rho }\right)}^{1/4}.$$ [13]

Note that to first order, the velocity and temperature fields as well as the film thickness are steady.

Linearized Equations.

The base solutions for the velocity, temperature, and film thickness are perturbed, yielding the following linearized equations:

$$\frac{\partial u^{\prime }}{\partial x}+\frac{\partial v^{\prime }}{\partial y}=0$$ [14]

$$u^{\prime }\frac{\partial u}{\partial x}+u\frac{\partial u^{\prime }}{\partial x}+v^{\prime }\frac{\partial u}{\partial y}+v\frac{\partial u^{\prime }}{\partial y}=-\frac{1}{{\rho }_V}\frac{\partial \mathrm{\Phi }\prime }{\partial x}+\frac{{\mu }_V}{{\rho }_V}\frac{\partial u^{\prime }}{\partial y^2}$$ [15]

$$\frac{\partial {\mathrm{\Theta }}^{\prime }}{\partial t}+u^{\prime }\frac{\partial \mathrm{\Theta }}{\partial x}+u\frac{\partial {\mathrm{\Theta }}^{\prime }}{\partial x}+v^{\prime }\frac{\partial \mathrm{\Theta }}{\partial y}+v\frac{\partial {\mathrm{\Theta }}^{\prime }}{\partial y}=\frac{k_V}{{\rho }_V{c}_{p,V}}\frac{\partial {\mathrm{\Theta }}^{\prime }}{\partial y^2}.$$ [16]

The boundary conditions at the wall are

$$\text{at}\hspace{0.17em}y=0\hspace{1em}{u}_V^{\prime }=v_V^{\prime }=0,\hspace{1em}{\mathrm{\Theta }}_V^{\prime }=0.$$ [17]

At the perturbed interface location $\delta +\delta \prime$, the tangential velocity and temperature conditions (Eq. 5) after applying the locally parallel approximation $\frac{\mathrm{d}\delta }{\mathrm{d}x}\approx 0$ give

$${\left.u_V^{\prime }\right|}_{\eta =1}+{\left.\frac{\delta \prime }{\delta }\frac{\partial u}{\partial \eta }\right|}_{\eta =1}=0,\hspace{1em}{\left.{\mathrm{\Theta }}_V^{\prime }\right|}_{\eta =1}+{\left.\frac{\delta \prime }{\delta }\frac{\partial {\mathrm{\Theta }}_V}{\partial \eta }\right|}_{\eta =1}=0.$$ [18]

The phase-change equation at the interface (Eq. 6) is linearized as

$$\begin{array}{ll}\hfill & {\rho }_V\frac{\partial \delta \prime }{\partial t}+\frac{\partial }{\partial x}{\int }_0^{\delta }{\rho }_V{u}_V^{\prime }dy+\frac{\partial }{\partial x}{\int }_0^{\delta +\delta \prime }{\rho }_V{u}_{V}dy\hfill \\ \hfill & \hspace{1em}=-\frac{k\mathrm{\Delta }T}{h_{LV}}\left(\delta \prime {\left.\frac{{\partial }^2\mathrm{\Theta }}{\partial y^2}\right|}_{\delta }+{\left.\frac{\partial {\mathrm{\Theta }}^{\prime }}{\partial y}\right|}_{\delta }\right).\hfill \end{array}$$ [19]

Analogous to the base solution, the perturbed velocity is expanded in powers of $\eta$:

$$u_V^{\prime }=a_0^{\prime }+a_1^{\prime }\eta +a_2^{\prime }{\eta }^2$$ [20]

$$\text{at}\hspace{0.17em}\eta =0\hspace{1em}{u}_V^{\prime }=0\to a_0^{\prime }=0.$$ [21]

The terms $a_1^{\prime }$ and $a_2^{\prime }$ can be found as functions of the fluid properties and the generalized pressure gradient $\frac{\partial \mathrm{\Phi }\prime }{\partial x}=\frac{\partial p^{\prime }}{\partial x}+\frac{\partial {\varphi }^{\prime }}{\partial x}$ from the momentum equation (Eq. 15) evaluated at the wall ($\eta =0$) and the tangential velocity condition (Eq. 18):

$$a_1^{\prime }=\left(\frac{\mathrm{\Delta }\rho g\delta \delta \prime }{2{\mu }_V}-\frac{\frac{\partial \mathrm{\Phi }\prime }{\partial x}{\delta }^2}{2{\mu }_V}\right)$$ [22]

$$a_2^{\prime }=\frac{\frac{\partial \mathrm{\Phi }\prime }{\partial x}{\delta }^2}{2{\mu }_V}.$$ [23]

The pressure gradient arises from the liquid–vapor surface tension ${\sigma }_{LV}$ at the two-phase interface due to capillary pressure induced by local nonzero curvature:

$$\frac{\partial p^{\prime }}{\partial x}=-{\sigma }_{LV}\frac{{\text{d}}^3\delta \prime }{\text{d}{x}^3}.$$ [24]

This implies that positive curvature corresponds to the center of curvature lying in the vapor domain, such that the vapor bulges into the liquid. Here, we also introduce the disjoining pressure term $\overline{\varphi }$, which describes the van der Waals interaction between the fluid and the substrate:

$$\overline{\varphi }=\frac{A}{6\pi {\overline{\delta }}^3}.$$ [25]

The streamwise derivative of this term is negligible in the base state under the locally parallel interface approximation. The Hamaker constant $A$ is typically positive, denoting attractive interactions between dipoles (22). The perturbed component is

$$\frac{\partial {\varphi }^{\prime }}{\partial x}=-\frac{A}{2\pi {\delta }^4}\frac{\text{d}\delta \prime }{\text{d}x}.$$ [26]

This gives an expression for the perturbed, generalized pressure term evaluated at the liquid–vapor interface:

$$\frac{\partial \mathrm{\Phi }\prime }{\partial x}=\frac{\partial p^{\prime }}{\partial x}+\frac{\partial {\varphi }^{\prime }}{\partial x}=-{\sigma }_{LV}\frac{{\text{d}}^3\delta \prime }{\text{d}{x}^3}-\frac{A}{2\pi {\delta }^4}\frac{\text{d}\delta \prime }{\text{d}x}.$$ [27]

Next, the expanded perturbed temperature is

$${\mathrm{\Theta }}_V^{\prime }=b_0^{\prime }+b_1^{\prime }\eta +b_2^{\prime }{\eta }^2+b_3^{\prime }{\eta }^3.$$ [28]

From the wall boundary condition (Eq. 17) and energy conservation equation (Eq. 16) at $\eta =0$, we find that $b_0^{\prime }=b_2^{\prime }=0$. Similarly, the temperature condition (Eq. 18) and energy conservation (Eq. 16) at the interface $\eta =1$ lead to an expression for $b_1^{\prime }$ in terms of $\delta \prime$:

$$\begin{array}{ll}\hfill & \frac{\delta }{4}\frac{\partial b_1^{\prime }}{\partial t}+\frac{1}{4}\left(1-2b_3\right)\frac{\partial \delta \prime }{\partial t}+\left(\frac{1}{6}-\frac{2b_3}{15}\right)\frac{\mathrm{\Delta }\rho g{\delta }^2}{2{\mu }_V}\frac{\text{d}\delta \prime }{\text{d}x}\hfill \\ \hfill & \hspace{1em}+\left(\frac{1}{12}-\frac{b_3}{20}\right)\left(\frac{{\delta }^3}{2{\mu }_V}\right)\left({\sigma }_{LV}\frac{{\text{d}}^4\delta \prime }{\text{d}{x}^4}+\frac{A}{2\pi {\delta }^4}\frac{{\text{d}}^2\delta \prime }{\text{d}{x}^2}\right)\hfill \\ \hfill & \hspace{1em}+\frac{a_1}{30}\left(1-2b_3\right)\frac{\text{d}\delta \prime }{\text{d}x}+\frac{a_1\delta }{20}\frac{\text{d}{b}_1^{\prime }}{\text{d}x}=\frac{3k_V}{{\rho }_V{c}_{p,V}{\delta }^2}\delta \prime -\frac{3k_V}{{\rho }_V{c}_{p,V}\delta }{b}_1^{\prime }.\hfill \end{array}$$ [29]

The time evolution for the perturbed $\delta \prime$ as a function of $b_1^{\prime }$ follows from the linearized phase-change expression (Eq. 19) and the expressions for $u_V^{\prime }$ and ${\mathrm{\Theta }}_V^{\prime }$ (Eqs. 20 and 28):

$$\begin{array}{ll}\hfill & {\rho }_V\frac{\partial \delta \prime }{\partial t}+\frac{{\rho }_V\mathrm{\Delta }\rho g{\delta }^2}{4{\mu }_V}\frac{\text{d}\delta \prime }{\text{d}x}+\frac{{\rho }_V{\delta }^3{\sigma }_{LV}}{12{\mu }_V}\frac{\text{d}{\delta }^{\prime 4}}{\text{d}{x}^4}\hfill \\ \hfill & \hspace{1em}+\frac{{\rho }_{V}A}{24\pi {\mu }_V\delta }\frac{\text{d}{\delta }^{\prime 2}}{\text{d}{x}^2}+\frac{3k_V\mathrm{\Delta }T}{h_{LV}{\delta }^2}\delta \prime -\frac{2k_V\mathrm{\Delta }T}{h_{LV}\delta }{b}_1^{\prime }=0.\hfill \end{array}$$ [30]

The perturbation Eqs. 29 and 30 give two homogeneous conditions for $\delta \prime$ and $b_1^{\prime }$. The perturbations can now be expressed in terms of normal modes:

$$\delta \prime ={\delta }_a^{\prime }\text{exp}\left(i\left(kx+\omega t\right)\right)$$ [31]

$$b_1^{\prime }=b_{1a}^{\prime }\text{exp}\left(i\left(kx+\omega t\right)\right).$$ [32]

To avoid introducing new notation, we will represent the amplitudes without subscripts ${\delta }_a^{\prime }\to \delta \prime$ and $b_{1a}^{\prime }\to b_1^{\prime }$. Here, $k$ is the wave number and $\omega$ the time rate of growth of the perturbation. We combine Eqs. 29 and 30 to obtain a single equation with the coefficient $\delta \prime$. To simplify the representation, we introduce the following dimensionless parameters:

$${\pi }_{LP}=\frac{3A^2{h}_{LV}{\rho }_V}{{\left(24\pi \right)}^2{\delta }^3{k}_V{\mu }_V\mathrm{\Delta }T{\sigma }_{LV}}$$ [33]

$${\pi }_{LB{\sigma }_{LV}}=\sqrt{\frac{A}{\pi {\sigma }_{LV}}}\left(\frac{\mathrm{\Delta }\rho g{\delta }^2{h}_{LV}{\rho }_V}{2k_V{\mu }_V\mathrm{\Delta }T}\right)$$ [34]

$$k^{″}=k{\delta }^2\sqrt{\frac{4\pi {\sigma }_{LV}}{A}}$$ [35]

$$\omega \prime =\omega \left(\frac{h_{LV}{\rho }_V{\delta }^2}{k_V\mathrm{\Delta }T}\right).$$ [36]

This leads to the general, characteristic equation for the temporal growth rate $i\omega$ of the perturbation after eliminating $b_1^{\prime }$ using Eqs. 29 and 30:

$$\begin{array}{ll}\hfill & \left(\frac{i\omega \prime }{8}+\frac{i{k}^{″}{\pi }_{LB{\sigma }_{LV}}}{80}+\frac{3}{2Ja}\right)\hfill \\ \hfill & \hspace{1em}\left(i\omega \prime +i{k}^{″}\frac{{\pi }_{LB{\sigma }_{LV}}}{4}+{k″}^4{\pi }_{LP}-2{\pi }_{LP}{k″}^2+3\right)\hfill \\ \hfill & \hspace{1em}+i\omega \prime \frac{c}{4}+i{k}^{″}\frac{\left(1+c\right){\pi }_{LB{\sigma }_{LV}}}{20}+{k″}^4\frac{3{\pi }_{LP}}{20}\left(\frac{7}{3}+c\right)\hfill \\ \hfill & \hspace{1em}-{k″}^2\left(\frac{7}{3}+c\right)\frac{3{\pi }_{LP}}{10}-\frac{3}{Ja}=0.\hfill \end{array}$$ [37]

The marginal state occurs when the real part $\text{Re}\left(i\omega \right)=0$, separating zones of stability ($\text{Re}\left(i\omega \right)<0$), where the perturbation amplitude decays in time, from regions of instability ($\text{Re}\left(i\omega \right)>0$), where the base state becomes unstable (Fig. 2A). Note that only three dimensionless numbers $Ja$, ${\pi }_{LP}$, and ${\pi }_{LB{\sigma }_{LV}}$ govern the stability of the perturbed solution. With the inclusion of van der Waals interactions, the buoyancy terms described by ${\pi }_{LB{\sigma }_{LV}}$ become negligible at nanoscale, as will be discussed shortly.

Fig. 2.

The stability of the perturbed solution. (A) The variation of Re($i\omega$) with ${\pi }_{LP}$ and the dimensionless wavenumber $k^{″}$ for ${\pi }_{LB{\sigma }_{LV}}=2\times 10^{-9}$ and $Ja=0.1$. A critical ${\pi }_{LP,crit}$ can be defined, for which lower values lead to unconditional stability, and greater values allow the coexistence of stable and unstable zones. (B) The Jakob number vs. the critical ${\pi }_{LP,crit}$, as predicted by the numerical solution to the full stability solution (Eq. 44) and by taking ${\pi }_{LP,crit}$ to occur at $k^{″}=1$ (Eq. 47). The agreement is excellent. (C) The marginal stability curves are calculated by taking the locus of points where the real value of $i\omega$ changes sign. The diffusive expression (Eq. 45) is a good approximation for small Jakob numbers.

This analysis incorporated time variation and convective transport in the energy equation. We can obtain a simpler expression for the stability problem by neglecting these two terms,

$$i\omega \prime =-\left({k″}^4{\pi }_{LP}-2{k″}^2{\pi }_{LP}+1\right)-i{k}^{″}\frac{{\pi }_{LB{\sigma }_{LV}}}{4},$$ [38]

where the buoyancy term ${\pi }_{LB{\sigma }_{LV}}$ is explicitly shown to give the dimensionless traveling-wave velocity, signifying that the base flow acts only to convect the perturbation and does not affect its growth. The full derivation of Eq. 38 is provided in SI Appendix.

The diffusive expression (Eq. 38) is a good estimate to the full stability equation (Eq. 37) for small Jakob numbers, as demonstrated in Fig. 2C. Since the dimensionless parameters are calculated from the vapor properties at the superheated wall temperature, the Jakob number is small ($Ja⪅\frac{1}{10}$) for the vapor phase of most fluids, implying that the thermal energy imparted by the heated solid is predominantly consumed through the latent heat of phase change rather than as sensible heat in raising the temperature of the vapor. Physically, dropping the time-varying term in the energy equation implies that the heat conduction timescale is much longer (quasi-steady) than that for the perturbation growth in the phase-change equation.

The Leidenfrost point corresponds to the lowest, critical ${\pi }_{LP,\text{crit}}$ on the marginal stability curve, below which the flow becomes unconditionally stable for all values of $k^{″}$. For Eq. 37, a good approximation for the critical ${\pi }_{LP}$ can be derived by noting that due to how we scaled the dimensionless parameters, ${\pi }_{LP,\text{crit}}$ occurs at $k^{″}=1$. This leads to an algebraic equation for ${\pi }_{LP,\text{crit}}$:

$$\begin{array}{ll}\hfill & \hspace{0.17em}\hspace{0.17em}\left.\left(\frac{Ja}{10}\left(\frac{7}{3}+c\right)+1\right){\pi }_{LP,\text{crit}}-1\right)\hspace{0.17em}{\left(1+\frac{Ja}{4}+\frac{Ja}{6}c-\frac{Ja}{12}{\pi }_{LP,\text{crit}}\right)}^2\hfill \\ \hfill & ={\left(\frac{3{\pi }_{LB{\sigma }_{LV}}}{20}\right)}^2{\left(\frac{Ja}{12}\right)}^2\left(1+\frac{Ja}{9}\left(2+c\right)\right)\left(\frac{1}{3}+\frac{2c}{3}-{\pi }_{LP,\text{crit}}\right).\hfill \end{array}$$ [39]

Eq. 39 was verified against a numerical solution to the full stability equation (Eq. 37) with $\text{Re}\left(i\omega \right)=0$ and was found to give the same solution for ${\pi }_{LP,\text{crit}}$ up to machine precision for all parameter sets tested (Fig. 2B). For small Jakob numbers, Fig. 2C shows that the critical ${\pi }_{LP}$ can also be estimated from the diffusive expression (Eq. 38):

$${\pi }_{LP,\text{crit}}=1.$$ [40]

The implication of Eq. 48 as a good estimate is threefold. First, it means that the boundary layer approximation is not required if film boiling is assumed to be diffusion dominated in both energy and momentum via lubrication theory. Second, the use of the locally parallel assumption (20, 21) is self-consistent since the film thickness $\delta$ in the base solution changes over a much longer length scale compared to the critical perturbation wavelength. Note that the wave number $k$ for the critical stability criterion $k^{″}=1$ is on the order of 100 (1/$\mathrm{\mu }$m), corresponding to a wavelength of around 10 nm. Over this wavelength, the relative change in film thickness $\frac{\mathrm{\Delta }\delta }{\delta }$ is around $10^{-9}\ll 1$, showing that the slow growth of the film thickness is negligible in the perturbation analysis.

Finally, perturbation growth is independent of the base flow, which carries vapor generated at the liquid–vapor interface out of the local control volume. This explains why the LFP is not found to be strongly dependent on the configuration of the experimental setup. Any configuration eventually takes the vapor out of the film by some buoyancy-driven force (even if it is horizontal or upside down—vapor eventually finds its way up). The strength of that driving force would indeed depend on configuration, but it does not matter for the perturbation solution.

Stabilizing Terms.

The diffusive approximation to the critical ${\pi }_{LP}$ (Eq. 40) reveals that the main stabilizing terms are the liquid–vapor surface tension ${\sigma }_{LV}$, the evaporative phase change that replenishes local vapor mass $\frac{k_V\mathrm{\Delta }T}{h_{LV}{\rho }_V}$, and the viscous shear $\mu$ that reduces mass transport away from any given point in the vapor field.

The liquid–vapor surface tension acts as a restoring force against oscillatory modes imposed on the basic, locally parallel solution. Positive curvature of the liquid–vapor interface with its center in the vapor region (curving into the liquid) induces high pressure locally with an adjacent low-pressure zone due to the negative curvature of the continuous two-phase interface. This creates a pressure gradient that attempts to restore the basic state by dampening all possible oscillatory frequencies.

Similarly, a perturbed interface that bulges into the vapor steepens the thermal gradient in the vapor film, triggering an increase in the rate of evaporation locally that restores the base state and vice versa. Viscous shear is larger for smaller film thicknesses, therefore reducing mass transport away from a local bulge into the vapor domain and enhancing transport away from a bulge into the liquid field; this also acts to dampen perturbed modes.

van der Waals Interaction.

The heterogeneous Hamaker constant $A_{SVL}$ is used to characterize the van der Waals dispersion forces between an uncharged surface and an adjacent liquid separated by vacuum. It is incorporated into this analysis via the generalized pressure gradient (Eq. 27). From the diffusive expression of the perturbation stability (Eq. 38), it is shown that these dispersion forces between the liquid and solid substrate across the vapor film are the only destabilizing term for attractive interactions $A>0$. The film is unconditionally stable if the interaction is purely repulsive $A<0$. The former case holds in general for a liquid separated from a solid by a vacuum or an intermediate gas phase (23, 24).

The relationship between the heterogeneous Hamaker constant and the contact angle $\theta$ of the substrate has been derived using Lifshitz theory (25, 26),

$$1+\text{cos}\left(\theta \right)=\frac{A_{SVL}}{12\pi {\sigma }_{LV}{H}_{SVL}^2},$$ [41]

where $H_{SVL}$ is the equilibrium contact separation between the solid substrate (S) and the liquid (L) across vacuum (V) and takes on values in the order of magnitude of 1 nm for most materials. Eq. 41 can thus be used to account for the effect of surface wettability on the stability of the perturbed solution. As the van der Waals interaction plays a significant role only for film thicknesses that have reached the same order of magnitude as $H_{SVL}$, we approximate the ratio $\frac{H_{SVL}}{\delta }\approx 1$.

In this nanoscale regime, the neutral curve described by the full perturbation solution (Eq. 37) is insensitive to ${\pi }_{LB{\sigma }_{LV}}$, which encapsulates the buoyancy force on the vapor film and is on the order of $10^{-14}$. The diffusive expression of the perturbation equation (Eq. 38) has an explicit dependence on ${\pi }_{LB{\sigma }_{LV}}$ only in the imaginary part of the temporal growth rate, such that the marginal state predicted is completely agnostic to changes in ${\pi }_{LB{\sigma }_{LV}}$. This implies that the stability criterion (Eq. 39 or Eq. 40) can be applied to capture the Leidenfrost point on plates of arbitrary orientation, as the direction and magnitude of the gravitational field do not play a significant role in the instability mechanism examined. The further implications of the nanoscale regime on this analysis are presented in SI Appendix.

Verification by Experiment and Simulation

To compare against experimental data, Eq. 40 for the critical ${\pi }_{LP}$ can be rewritten in terms of material properties:

$$\frac{3}{{\left(24\pi \right)}^2}{\left(\frac{1}{\delta }\right)}^4\frac{h_{LV}{\rho }_V\delta }{{\sigma }_{LV}}{A}^2\frac{1}{k_V\mathrm{\Delta }T{\mu }_V}=1.$$ [42]

Substituting in Eq. 41, we obtain the corresponding expression for the LFP in terms of the intrinsic contact angle on the substrate,

$$\frac{3}{4}\underset{\approx 1}{\underbrace{{\left(\frac{H_{SVL}}{\delta }\right)}^4}}\underset{{\pi }_2}{\underbrace{\left(\frac{h_{LV}{\rho }_V\delta }{{\sigma }_{LV}}\right)}}{\left(1+\text{cos}\left(\theta \right)\right)}^2\underset{{\pi }_1}{\underbrace{\frac{{\sigma }_{LV}^2}{k_V\mathrm{\Delta }T{\mu }_V}}}=1,$$ [43]

where we have defined two dimensionless parameters that we will show to be significant:

$${\pi }_1=\frac{{\sigma }_{LV}^2}{k_V{\mu }_V\mathrm{\Delta }T}$$ [44]

$${\pi }_2=\frac{h_{LV}{\rho }_V\delta }{{\sigma }_{LV}}.$$ [45]

Eq. 43 provides an explicit relationship between the intrinsic contact angle of a fluid on a substrate and the Leidenfrost point for the system. Since each fluid property ($k_V\left(T\right),{\mu }_V\left(T\right),{\sigma }_{LV}\left(T\right)$, etc.) is calculated at the superheated wall temperature, the left-hand side of Eq. 43 is in general a nonlinear function of temperature. The temperature at which Eq. 43 is satisfied corresponds to the predicted LFP; this can be found numerically with the temperature- and pressure-dependent fluid properties available from databases like National Institute of Standards and Technology and tabulations from literature (27–29).

For ease of use, Eq. 43 can also be written using corresponding states correlations (SI Appendix, Eq. S18), which express fluid properties in terms of the critical temperature $T_c$ and pressure $p_c$, the applied saturation pressure $p$, and the molar mass of the fluid.

Surface Dependence of the LFP.

The LFP for water has been demonstrated to vary dramatically with changes in the liquid wettability of the solid surface (13, 14). Fig. 3 shows that the diffusive prediction of the LFP (Eq. 43) accurately captures the relationship between the LFP and the contact angle as delineated by experiments (13, 14, 30–32). Physically, larger contact angles indicate a hydrophobic substrate, which exhibits less attractive van der Waals interaction with the bulk liquid and presents a smaller destabilizing effect to the vapor film; the LFP thus decreases to near the boiling point. Without considering van der Waals interaction between the liquid and substrate surfaces, such a relationship cannot be explained or predicted from first principles.

Fig. 3.

The Leidenfrost temperature vs. the contact angle for water, from experiments (13, 14, 30–32), the diffusive prediction of the LFP (Eq. 43), and molecular dynamics simulations from this work. The equilibrium separation $H_{SVL}$ and its variation associated with changes in the contact angle $\frac{\text{d}{H}_{SVL}}{\text{d}\theta }$ can be estimated from experimental data (25, 33). Typical errors in the LFP and the contact angle measured from experiment are around $5\hspace{0.17em}°\text{C}$ and $2^{○}$, respectively, although many sources do not explicitly report an error value for either quantity. The data point corresponding to a contact angle of $160^{○}$ from Vakarelski et al. (31) corresponds to the only surface which was textured with nanoparticles to achieve superhydrophobicity; the other data points correspond to flat surfaces with random roughness.

Further evidence of the significant role played by van der Waals forces in governing the LFP arises from X-ray imaging of the vapor film collapse (34). Images spanning the film lifespan between formation and collapse showed that film collapse on the macroscopic level is preceded by submicrometer length-scale vapor film thicknesses where the bulk liquid appears to wet the substrate. Although instabilities on the micrometer scale and above perturb the liquid–vapor interface and induce frequent local contact between the liquid and the solid, only when the vapor film becomes unstable on the smallest length scales where van der Waals interactions dominate will the film completely collapse. Further discussion on the timescales associated with the instability theory as well as the residence time and frequency of liquid–solid contact observed in experiment is presented in SI Appendix.

Finally, we note that our theoretical analysis predicts that the main effects governing the LFP operate in the nanoscale regime, which is accessible by molecular dynamics (MD). Fig. 4 shows one of the boiling heat-transfer simulations performed using Large-scale Atomic/Molecular Massively Parallel Simulator (35) software to numerically determine the LFP and the corresponding intrinsic contact angle of the substrate. Details of the MD implementations are provided in SI Appendix. Note that a vapor film forms when the liquid water adjacent to the bottom plate is heated above the LFP, whereas liquid contact with the solid surface is preserved below the LFP due to the attractive heterogeneous van der Waals interactions. The relationship between the contact angle and the Leidenfrost point of the extended simple point charge (SPC/E) water model is in good agreement with the diffusive prediction of the LFP (Eq. 43). These simulations show that vapor film stability is ultimately determined at the proposed nanometric length scale where fluid–surface van der Waals interactions cannot be discounted and where the effect of gravity-driven instabilities is nonexistent.

Fig. 4.

Molecular dynamics simulation of vapor film formation adjacent to a heated surface. The system is pressurized at 1 atm.

Fluid Dependence of the LFP.

For most experimentally available data on the Leidenfrost point, the contact angle of the fluid on the substrate material is low, around $\theta \approx 20^{○}$. Nonetheless, the Hamaker constant must be found to determine the equilibrium separation $H_{SVL}$. Although the assumption $\frac{H_{SVL}}{\delta }\approx 1$ is made, the base film thickness $\delta$ still needs to be incorporated into our instability expression via ${\pi }_2$ (Eq. 45). We can find the homogeneous Hamaker constant of the fluid (acetone, ethanol, benzene, etc.) and the substrate (gold, aluminum, copper) and take the geometric mean to obtain the heterogeneous value (24, 36). From the relationship between the surface energy and the homogeneous Hamaker constant, we can obtain the homogeneous contact separations via

$${\sigma }_{LV}=\frac{A_{LVL}}{24\pi H_{LVL}^2}$$ [46]

$${\sigma }_{SV}=\frac{A_{SVS}}{24\pi H_{SVS}^2}.$$ [47]

SI Appendix, Fig. S1 shows that it is possible to determine either the heterogeneous Hamaker constant given the Leidenfrost point for a fluid on a solid substrate or vice versa with knowledge of the homogeneous Hamaker constants of both species.

In general, experimental data on the homogeneous Hamaker constants may not be available for a fluid or substrate of interest. Here, we note an avenue for simplification: It is observed that the dimensionless quantity ${\pi }_2=\frac{h_{LV}{\rho }_V\delta }{{\sigma }_{LV}}$ in the diffusive expression is around 0.06 for most fluids at their respective Leidenfrost temperatures. This suggests that there exists a functional dependence $H_{SVL}=F\left(\frac{h_{LV}{\rho }_V}{{\sigma }_{LV}}\right)$. Additionally, most experimental setups in the film boiling regime feature fluids that wet the surface in contact, such that their intrinsic contact angles are small ($\theta \approx 20^{○}$) (29). From the diffusive expression (Eq. 43) valid for low Jakob numbers, the above approximations leads to a simplified, dimensionless prediction to the Leidenfrost point for fluids/substrate systems with low, intrinsic contact angles:

$${\pi }_1=\frac{{\sigma }_{LV}^2}{k_V\mathrm{\Delta }T{\mu }_V}\approx 6.$$ [48]

This dimensionless quantity also arises by application of the Buckingham Pi theorem to the system, as discussed in SI Appendix. Fig. 5 shows that the temperature at which this equality is satisfied captures the experimental data on the LFP for a variety of different fluids, including cryogens and liquid metals. The single dimensionless number describes the terms that stabilize the vapor film, including surface tension, phase change, and viscous transport, while the critical value corresponding to the LFP denotes the destabilizing effect of attractive van der Waals interaction between the bulk liquid and solid substrate. Larger values of ${\pi }_1$ above the critical imply the system is the film boiling regime, since the stabilizing terms dominate.

Fig. 5.

The dimensionless criterion ${\pi }_1=6$ as a low contact angle approximation from the diffusive expression (Eq. 43) captures the LFP data from experiment to within 5% error. The dimensionless number ${\pi }_1$ encapsulates the stabilizing effects of evaporative phase change (vapor mass generation), surface tension, and viscous transport, while the critical value at which the LFP occurs describes the destabilizing role of the van der Waals interaction for low contact angle fluids. Experimental LFP and fluid property data are available for acetone, ethanol, pentane, R134a, nitrogen, RC318, benzene, helium, R11, R113, liquid sodium, and liquid potassium (15, 27–29, 37–43).

Pressure Dependence of the LFP.

Experimental work has shown that the LFP depends on the ambient pressure applied, such that the Leidenfrost temperature gradually increases from near the boiling point toward the critical point of the fluid (45). For low contact angle fluid/substrate systems, we find that ${\pi }_2{\left(1+\text{cos}\left(\theta \right)\right)}^2$ scales linearly with pressure such that the LFP corresponds to

$${\pi }_1=6\frac{p_{\text{ref}}}{p_{\text{applied}}},$$ [49]

where $p_{\text{ref}}$ and $p_{\text{applied}}$ are 1 atm and the applied, operating pressure, respectively. Essentially, Eq. 49 is a small contact angle estimate to the full diffusive expression (Eq. 43) obtained by empirical observation of the data and reported for ease of use, since fewer terms are computed in this approximation. Excel sheets that apply the full diffusive expression and the small contact angle approximations (Eqs. 48 and 49) are provided in Dataset S1. Fig. 6 demonstrates that Eq. 43, a simplified estimate to the diffusive expression (Eq. 43), captures the LFP of various fluids for both subatmospheric and superatmospheric pressures up to the critical point.

Fig. 6.

The LFP from experimental data of R113, R12, hexane, and nitrogen at different pressures normalized by the critical pressure $p_c$ of the fluid (44–46) vs. the predictive ability of the small contact angle approximation (Eq. 49) to the diffusive expression (Eq. 43).

Conclusion

The dynamic stability of a vapor film on a heated vertical wall under the effects of gravity was considered. The only possible instability at nanoscale was driven by attractive van der Waals interaction between the bulk liquid and the substrate, which could be stabilized by the liquid–vapor surface tension, evaporative phase change, and viscous transport. The marginal or neutral state can be found analytically (Eq. 39) for the most general case or simplified for small Jakob number flows to a diffusive approximation (Eq. 40). The resulting theoretical solution for the LFP captures the variation of experimental data with surface wettability, fluid properties, and pressure.

A single, dimensionless number ${\pi }_1$ is found to encapsulate the physical instability mechanism of the Leidenfrost phenomenon for wetting fluids. The value of ${\pi }_1$ with respect to the critical denotes regimes in which the vapor film is stable or unstable, providing a useful characterization of both the thermodynamic state and the physical means by which transition to the pool boiling regime occurs.

This insight into the nanoscale mechanisms inducing the transition from film to nucleate boiling enables control of the phase adjacent to the surface (47). It would be of interest to extend the instability mechanism toward surface roughness, which experiment has shown to effect dramatic changes in the LFP beyond what can be explained by variation in surface wettability (5, 14, 48). In addition, a theoretical treatment of the Nukiyama temperature corresponding to the critical heat flux may reveal the mechanism underlying transition boiling and provide a comprehensive understanding of the entire boiling curve under a unified, physical framework.

Materials and Methods

Data Availability.

The data and procedures are freely available in SI Appendix and Dataset S1.

Usage of the Excel Sheets.

The Excel sheets provided in Dataset S1 can be used for different systems by substituting the correct homogeneous Hamaker constants for the fluid and solid substrate, the surface energy of the solid, and the ambient pressure, as well as the temperature-dependent vapor thermal conductivity, vapor viscosity, saturation pressure, vapor density, liquid and vapor enthalpies, and surface tension for the fluid. The Excel sheet will report the corresponding predicted LFP and the percentage of error compared with the experimental data (which must also be provided by the user). Please check that the formulas in the top row cover the entire length of the column for the new datasets provided.