The Thermo-Wetting Instability Driving Leidenfrost Film Collapse
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The thermo-wetting instability driving Leidenfrost film collapse Tom Y. Zhaoa and Neelesh A. Patankara,1 aDepartment of Mechanical Engineering, Northwestern University, Evanston, IL 60208 Edited by Parviz Moin, Stanford University, Stanford, CA, and approved April 15, 2020 (received for review October 14, 2019) Above a critical temperature known as the Leidenfrost point (LFP), In this work, we introduce a stability analysis of the vapor a heated surface can suspend a liquid droplet above a film of film at the nanoscale regime. The dominant destabilizing term its own vapor. The insulating vapor film can be highly detrimen- arises from the van der Waals interaction between the bulk liq- tal in metallurgical quenching and thermal control of electronic uid and the substrate across a thin vapor layer. On the other devices, but may also be harnessed to reduce drag and gener- hand, liquid–vapor surface tension-driven transport of vapor and ate power. Manipulation of the LFP has occurred mostly through evaporation at the two-phase interface stabilize the film. The experiment, giving rise to a variety of semiempirical models that competition between these mechanisms gives rise to a com- account for the Rayleigh–Taylor instability, nucleation rates, and prehensive description of the LFP as a function of both fluid superheat limits. However, formulating a truly comprehensive and solid properties. For fluids that wet the surface, such that model has been difficult given that the LFP varies dramatically for the intrinsic contact angle is small, a single dimensionless num- different fluids and is affected by system pressure, surface rough- ber (Eq. 48) can be derived that encapsulates the instability ness, and liquid wettability. Here, we investigate the vapor film determining the Leidenfrost point. instability for small length scales that ultimately sets the collapse It is noted that the literature has proposed different names for condition at the Leidenfrost point. From a linear stability analy- the critical temperature associated with a droplet levitating on a sis, it is shown that the main film-stabilizing mechanisms are the heated plate (Leidenfrost point) versus the critical temperature liquid–vapor surface tension-driven transport of vapor mass and for vapor film formation in pool boiling (minimum film temper- the evaporation at the liquid–vapor interface. Meanwhile, van ature). The Leidenfrost point has been shown to be equivalent der Waals interaction between the bulk liquid and the solid sub- to the minimum film boiling temperature for saturated liquids strate across the vapor phase drives film collapse. This physical on isothermal surfaces (15). In this work, the term LFP is used ENGINEERING insight into vapor film dynamics allows us to derive an ab initio, for both cases as a matter of convenience, with the understand- mathematical expression for the Leidenfrost point of a fluid. The ing that no undercooling is applied to the liquid phase for pool expression captures the experimental data on the LFP for different boiling scenarios unless explicitly stated. fluids under various surface wettabilities and ambient pressures. For fluids that wet the surface (small intrinsic contact angle), the Film Instability expression can be simplified to a single, dimensionless number There are many approaches to examine the stability of a vapor that encapsulates the wetting instability governing the LFP. film adjacent to a superheated wall in two dimensions. Mod- els have been developed with a base solution imposing static Leidenfrost point j minimum film boiling temperature j equilibrium, where the interface is at the saturation tempera- fluid instability j dimensionless number ture 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) s a surface is superheated above the boiling point of an adja- Acent fluid, vapor bubbles nucleate and grow. The boiling behavior of the liquid phase undergoes a fundamental change Significance at a critical temperature known as the Leidenfrost point. Beyond this point, a film of insulating vapor forms between the liquid and The insulating vapor film formed during film boiling dramati- the surface that suppresses heat transfer from the solid material. cally reduces the heat flux from the surface. This can be highly This heat flux reduction can be highly detrimental in the quench- detrimental in the quenching of metal alloys, the cooling of ing of metal alloys by extending cooling rates and precluding the nuclear fuel rods, and the thermal control of electronic and desired increase in strength and hardness (1). Alternatively, film photonic devices. On the other hand, the vapor film can also boiling may be used to promote drag reduction as well as enable be harnessed to reduce liquid–solid drag as well as design power generation through self-propulsion (2–4). Thus, modula- levitating, self-rotating rotors and self-propelled droplets for tion of the Leidenfrost point (LFP) through fluid choice, surface lab-on-a-chip applications. Control of vapor film collapse in texture, and surface chemistry for the specific application is different systems and operating conditions is therefore highly crucial (5). desirable. Surface wettability is known to influence this col- On a fundamental level, the physical mechanism responsible lapse. However, a comprehensive thermo-wetting instability for the LFP is still uncertain. Many theoretical frameworks have theory that predicts observed data has been elusive. This work been used to characterize the Leidenfrost effect and estimate proposes such a theory. the LFP, including hydrodynamic instability (6, 7), superheat spinodal limits (8, 9), and the change of liquid wettability on Author contributions: T.Y.Z. and N.A.P. designed research, performed research, con- the heated surface with temperature (10, 11). A thermocapillary tributed new reagents/analytic tools, analyzed data, and wrote the paper.y model has also been proposed that attributes the film instability The authors declare no competing interest.y to fluctuations at micrometer length scales; however, the analysis Published under the PNAS license.y posits that the thermocapillary effect is the dominant destabiliz- This article is a PNAS Direct Submission.y ing term, which does not explain the significant change in LFP 1 To whom correspondence may be addressed. Email: [email protected] on surfaces with different wettabilities (12). For example, the This article contains supporting information online at https://www.pnas.org/lookup/suppl/ ◦ LFP of water can vary from 300 C for hydrophilic surfaces to doi:10.1073/pnas.1917868117/-/DCSupplemental.y ◦ 145 C on hydrophobic surfaces (13, 14). First published May 27, 2020. www.pnas.org/cgi/doi/10.1073/pnas.1917868117 PNAS j June 16, 2020 j vol. 117 j no. 24 j 13321–13328 Downloaded by guest on October 1, 2021 2 @Θ¯ @Θ¯ V @Θ¯ V kV @ Θ¯ V +u ¯V +v ¯V = 2 [3] @t @x @y ρV cp,V @y where the parameters µ, ρ, k, g, and cp represent the dynamic vis- cosity, density, thermal conductivity, gravitational acceleration, and specific heat of the fluid, respectively. The term ∆ρ = ρL − ρV represents the density difference, the subscripts L and V denote the liquid and vapor field, and the temperature has been ¯ normalized as Θ¯ = T−Ts , the difference between the tempera- Tw −Ts ture field and the saturation temperature Ts at the interface over the difference between the wall temperature Tw and Ts . Note that we take the liquid phase to be motionless due to its much greater viscosity relative to the vapor (uL(x, t) = 0), as well as isothermal at saturation temperature in the long-time limit (equi- librium) due to its larger thermal conductivity (ΘL = 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 simplifi- cations 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 Φ¯ 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 Φ¯ ≈ 0 + Φ0, where the overbar variables represent the general solution, the unbarred variables denote the base solution, and the primed vari- ables 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 Fig. 1. Film boiling on a vertical plate, with the coordinate system liquid–vapor interface at δ¯(x) are given by delineated. The film thickness is denoted by δ¯(x). at y = 0,u ¯V =v ¯V = 0, Θ¯ V = 1 [4] or vapor under a droplet (18). This appears to be a more gen- ¯ eral analysis since under experimental settings, droplet levitation dδ at y = δ¯,u ¯V +v ¯V = 0, Θ¯ V = 0: [5] occurs over a film that is continuously replenished by evapora- dx tion and depleted through escape of the buoyant vapor phase. Eq. 4 enforces the temperature and no interfacial slip at the Similarly in a horizontal setup for pool boiling, bubbles pinch off impermeable wall, while Eq. 5 ensures that the tangential com- the film, necessitating a nonzero rate of evaporation to sustain a ponent of velocity is continuous and that the temperature is at constant mean film thickness (17).