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Downloaded by guest on October 1, 2021 nsrae ihdfeetwtaiiis(2.Freape the example, from For vary LFP 145 can (12). in water wettabilities change of significant different LFP the with explain surfaces not on destabiliz- does dominant which the term, is ing effect thermocapillary analysis the the instability that however, film posits scales; on length the micrometer attributes wettability at that fluctuations proposed to thermocapillary been of A also 11). change has (10, model the superheat with and 7), surface heated 9), (6, the estimate (8, instability and limits hydrodynamic effect Leidenfrost including the LFP, characterize have the frameworks to theoretical Many used uncertain. been still is is LFP application the for specific the for chemistry (5). crucial surface surface choice, fluid and through (LFP) texture, point modula- Leidenfrost enable the Thus, as of (2–4). well tion as self-propulsion reduction through drag generation promote power to film used Alternatively, be (1). the may hardness precluding and and strength rates in cooling increase extending quench- desired by the alloys in metal detrimental of highly be ing material. can reduction the flux from heat transfer and This heat liquid the suppresses between that forms surface the insulating Beyond change of point. film fundamental Leidenfrost a the point, a as this known undergoes temperature critical a liquid at the of behavior A instability fluid ednrs point Leidenfrost number LFP. dimensionless the governing single, instability a wetting the the to angle), encapsulates contact simplified that intrinsic be (small can surface the expression pressures. wet ambient that and fluids wettabilities For surface different various for under The LFP the fluids fluid. initio, on a data ab experimental of an the point captures derive expression Leidenfrost physical to the This us for allows collapse. expression dynamics mathematical film film drives sub- vapor solid phase into the insight vapor and van the liquid across Meanwhile, bulk the strate interface. between liquid–vapor interaction Waals the der and at mass the vapor are of mechanisms the transport film-stabilizing analy- tension-driven main stability surface the linear that liquid–vapor a shown From is it point. collapse sis, Leidenfrost the the sets film ultimately at vapor that condition the scales investigate length small we for Here, instability wettability. rough- surface liquid pressure, and system for by ness, dramatically affected varies is comprehensive LFP and the fluids truly that different given a difficult been formulating has model and However, rates, limits. nucleation that instability, models superheat Rayleigh–Taylor semiempirical the gener- of for variety through and a account mostly drag to occurred reduce rise has giving LFP to experiment, the harnessed of electronic Manipulation be of power. also of ate control may film thermal detrimen- but highly a and be devices, quenching above can metallurgical droplet film in vapor liquid tal insulating a The vapor. suspend own can its 2019) surface 14, (LFP), October point heated review Leidenfrost the for a as (received known 2020 temperature 15, critical April a approved Above and CA, Stanford, University, Stanford Moin, Parviz by Edited a Zhao Y. Tom film collapse Leidenfrost driving instability thermo-wetting The www.pnas.org/cgi/doi/10.1073/pnas.1917868117 eateto ehnclEgneig otwsenUiest,Easo,I 60208 IL Evanston, University, Northwestern Engineering, Mechanical of Department nafnaetllvl h hsclmcaimresponsible mechanism physical the level, fundamental a On ◦ etfli,vprbblsncet n rw h boiling The grow. and nucleate bubbles adja- an vapor of point fluid, boiling the cent above superheated is surface a s nhdohbcsrae 1,14). (13, surfaces hydrophobic on C a | iesols number dimensionless n els .Patankar A. Neelesh and | iiu l oln temperature boiling film minimum 300 ◦ o yrpii ufcsto surfaces hydrophilic for C a,1 | yai qiiru,a navria lt ofiuain(17) configuration in plate be to vertical film a vapor in the of as thickness equilibrium, the (12, dynamic tempera- consider pressure we saturation static liquid Here, far-field the imposing imposed, 16). at solution the Mod- to is base corresponding dimensions. interface ture a two the with where in developed equilibrium, wall vapor been a superheated have of a stability els to the examine adjacent to film approaches many are There pool Instability for Film phase liquid the stated. to understand- explicitly unless the applied scenarios used with is boiling convenience, is undercooling of LFP no term that a the ing as work, cases this saturated both In for for (15). equivalent temperature surfaces be boiling isothermal to film on shown minimum been the temper- has film to point (minimum Leidenfrost boiling The pool in ature). temperature formation critical film the vapor a versus for on point) levitating (Leidenfrost droplet plate a heated with associated temperature critical the point. num- Leidenfrost that dimensionless the such determining single fluid surface, a (Eq. small, both the is wet ber of angle that com- function contact fluids intrinsic a a For the as to properties. LFP solid rise The the and gives of film. description mechanisms the other prehensive stabilize these the interface between On two-phase competition layer. the vapor at and thin vapor evaporation of a transport liq- tension-driven across bulk surface the liquid–vapor substrate hand, between the interaction term Waals and destabilizing der uid dominant van The the regime. from arises nanoscale the at film is ulse a 7 2020. 27, May published First doi:10.1073/pnas.1917868117/-/DCSupplemental at online information supporting contains article This Submission.y 1 Direct PNAS a is article This the under Published interest.y con- competing no research, paper.y declare the authors performed wrote The and research, data, analyzed designed tools, reagents/analytic N.A.P. new and tributed T.Y.Z. contributions: Author owo orsodnemyb drse.Eal [email protected] Email: addressed. be may correspondence whom To rpsssc theory. work a This such elusive. been proposes has data observed instability col- predicts thermo-wetting that this theory comprehensive influence a to However, known lapse. is wettability in highly Surface therefore collapse desirable. is film conditions operating vapor and design systems of as different Control well for applications. droplets as self-propelled lab-on-a-chip drag and also rotors liquid–solid can self-rotating reduce film levitating, and vapor to electronic the harnessed hand, of other be control the of thermal On cooling the devices. the photonic and alloys, rods, metal fuel of highly nuclear be quenching can the This surface. in the detrimental from dramati- flux boiling heat film the reduces during cally formed film vapor insulating The Significance ti oe htteltrtr a rpsddfeetnmsfor names different proposed has literature the that noted is It vapor the of analysis stability a introduce we work, this In a edrvdta nasltsteinstability the encapsulates that derived be can 48) PNAS NSlicense.y PNAS | ue1,2020 16, June | . y o.117 vol. https://www.pnas.org/lookup/suppl/ | o 24 no. | 13321–13328

ENGINEERING 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, , 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 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 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). saturation along the liquid–vapor interface. As with Burmeis- Fig. 1 shows a chosen “model” problem of film boiling on a ter (17), we neglect the “blowing” of vapor toward the plate by vertical plate (8, 17). The vapor forms a laminar layer at the wall, assuming v(x, t) ≈ 0, such that the preceding set of boundary with evaporation at the liquid interface sustaining the buoyant conditions is sufficient to fully specify the problem. Otherwise, transport of vapor mass away from the base of the plate. The the normal component of velocity would also need to be fixed surrounding liquid is saturated and motionless with its properties at the interface; this to a cubic rather than a parabolic fixed at the saturation temperature. The properties of vapor are estimate to the velocity field. assumed to be constant at the superheated wall temperature, an The vapor generated due to phase change at this interface is assumption discussed in SI Appendix. balanced by the streamwise rate of change of the vapor flow in the film and the growth of the film thickness in time, Governing Equations. The mass, momentum, and energy con- servation equations in the vapor domain are given by the ¯ Z δ¯ ¯ boundary-layer equations ∂δ ∂ kV ∂ΘV ρV + (ρV u¯V dy) = − , [6] ∂t ∂x 0 hLV ∂y ¯ ∂u¯ ∂v¯ y=δ V + V = 0 [1] ∂x ∂y where hLV is the of . In the base state,

2 the time variation of the film thickness is taken to be negligible, ∂u¯ ∂u¯ 1 ∂Φ¯ ∆ρg µ ∂ u¯ ¯ 0 V V V V dδ ≈ 0 + dδ . An in-depth description of this setup is covered in u¯V +v ¯V = − + + 2 [2] dt dt ∂x ∂y ρV ∂x ρV ρV ∂y SI Appendix.

13322 | www.pnas.org/cgi/doi/10.1073/pnas.1917868117 Zhao and Patankar Downloaded by guest on October 1, 2021 Downloaded by guest on October 1, 2021 Ja thickness and normal- the in introducing (Θ field described After temperature is (17). approximately which variable Burmeister ized method, found by expansion detail be integral full can an solution using by developed steady, The h onaycniin ttewl are wall the at conditions boundary The following the yielding perturbed, are equations: thickness linearized film and ature, as Equations. fields Linearized temperature and steady. are velocity thickness the film the order, as first well to that Note equation value The scale. length the larger the much of streamwise to a in on the due encapsulated occurs in and is parabolic dependence conservation variation locally mass velocity under is the direction profile while force, velocity buoyancy the that Note are equations flow base Flow. Base hoadPatankar and Zhao approximation parallel (Eq. conditions temperature and location ity interface perturbed the At u ∂ c u V = 0 ∂ Θ 0 a efudb ovn h udai xrsin(17) expression quadratic the solving by found be can t

Θ ∂ ∂ η c 0 ∆ p =1 V u x + ,V h T LV = + u ∆T δ + δ c = 0 ln ihtebudr odtosEqs. conditions boundary the with Along 2 = u T 1 = sdsrbdby described is T ∂ ∂ u δ T δ ∂ u V = w ∂ Θ x 0 3 1 V η w V u  x ∂ at ∂η − − − c = ∂ 0 10 + − 1 1 ∂ u ∂ p u + y T ∂ T ,V Θ h y − 3 1 x ,

u V T x η LV 0 = /δ s s v c V ∆T c ∂ =1 s p 1 3 ∂ V 0 h aesltosfrtevlct,temper- velocity, the for solutions base The + + 1 = o yia ao ubr around numbers Jakob typical For . ,V h u Θ ecndtrietevlct (u velocity the determine can we , ∂ ∂ x + LV c a efudfo ipie linear simplified a from found be can ntebs solution: base the in ) V ,Θ 0, = v 0 u ∆T y p dx dδ ∂ c V v ,V h + = ∂ ∂ u ∂ + V u LV  V x u ∂ 0 ≈ sn hsapoiain h film the approximation, this Using . x V ∆ v  ∂ ∆ρg ∂ 1 v u 0 0 = 2µ y ∂ 0 T −1 Θ V + + − ∂ ∂ ∂ ∂ y give v Θ u V y  y V = V ∂ ∂ 10 δ V 0 0 ∂ 0 − 3 1/4 ∂ 2 v v + = y =

,Θ 0, = y ∆ V (η 0 (1 η ρ 1  v =1 ρ − 0 = ρg V δ fe pligtelocally the applying after 5) − 0 = 2 − − V ∂ ρ x ∂ + ρ k + Θ V c c ∆ 1 y V V c η + p δ  0 ) h 2 T ,V 0 δ  ∂ ) δ LV = µ h agnilveloc- tangential the , ρ η ∂ 0 V 0 µ Φ V 1 = V x + ∂ ∂ ρ V g 0 = 0 ∂ ∂η Θ 2 V ∂ ∆ρ k + 1 Θ y ∂ k − 2 V V c V 2 − y u V . 2 p µ ρ

2  c V ,V η V c V . 1/4 =1 η 4 ∂ ∂ ∂ ∂ 3 . Θ y and 0 = y . u 2 2 0 0 V . . the 5, δ and ) [12] [10] [18] [16] [15] [14] [13] [11] [17] (x [8] [7] [9] ) 2 risadtegnrlzdpesr gradient pressure generalized the and erties terms The nlgu otebs ouin h etre eoiyis velocity perturbed the solution, base of powers in the expanded to Analogous et h xaddprubdtmeaueis temperature perturbed expanded the Next, interface: pressure liquid–vapor generalized the perturbed, at the evaluated term for expression an gives This (Eq. interface as the at equation phase-change The neatosbtendpls(2.Teprubdcmoetis component The perturbed The approximation. (22). dipoles interface between base interactions parallel the in constant locally negligible Hamaker the is term under this state of derivative streamwise The substrate: pressure the and disjoining fluid the the introduce bulges also vapor we the term that Here, such liquid. domain, the of vapor into center the the in to lying corresponds curvature curvature positive that implies This curvature: nonzero local by ten- induced surface liquid–vapor the from sion arises gradient pressure The (Eq. equation (η momentum the from b (Eq. (Eq. condition equation boundary servation wall the From iial,tetmeauecniin(Eq. (Eq. condition servation temperature the Similarly, 1 0 0 = ntrsof terms in σ hc ecie h a e al neato between interaction Waals der van the describes which φ, ¯ n h agnilvlct odto (Eq. condition velocity tangential the and ) LV ρ V ttetopaeitraedet ailr pressure capillary to due interface two-phase the at ∂ = ∂ a ∂δ Φ ∂ 1 x 0 − t 0 and 0 δ tteinterface the at 16) = k + 0 h : ∆ LV Θ ∂ ∂ ∂ a T ∂ at p V 0 a x PNAS x 2 0 0 A 1 0 η a efuda ucin ftefli prop- fluid the of functions as found be can = u  Z + = V 0 = 0 0 ∂φ δ η stpclypstv,dntn attractive denoting positive, typically is ∂ ∂ b ∂ δ

∂φ 0 : ∂ 0 p = 0 x | x ρ at 16) 0 a x ∂ 0 + ∂ ∆ V ue1,2020 16, June φ a 0 2 ¯ = 0 2 = y 2µ 0 u ρg b 0 = Θ u = = 2 1 − V 0 + −σ V 0 0 V η

−σ δδ 6π dy δ ∂ 2πδ 0 = a 2µ ∂ + η Φ A + LV x 0 1 A 0 LV 0 δ η 0 = ¯ η + b − V δ 3 4 2 1 = 0 → ∂ + d 2 dx ∂ . η ∂ d Θ dx dδ 3 ∂ dx efidthat find we , . ∂ y vlae ttewall the at evaluated 15) 2 2µ ∂ a 3 x δ Φ a 3 | 0 x δ 2 + 0 0 0 edt nepeso for expression an to 0 3 0 0

Z 0 η . V . δ δ o.117 vol. 0 = b 0 2 −  2 n nrycon- energy and 17) 3 δ 0 n nrycon- energy and 18) ! . +δ η 2πδ . 3 A 0 . ρ ∂ 4 ∂ V | Φ slinearized is 6) 18): x dδ dx u 0 o 24 no. V = 0 b . dy 0 0 ∂ ∂ = p x 0 | b + 2 0 13323 [21] [20] [19] [22] [26] [27] [25] [24] [23] [28] 0 = ∂φ ∂ x 0 .

ENGINEERING 0 0   2 0 δ ∂b1 1 ∂δ 1 2b3 ∆ρgδ dδ This leads to the general, characteristic equation for the tempo- + (1 − 2b3) + − 0 4 ∂t 4 ∂t 6 15 2µV dx ral growth rate iω of the perturbation after eliminating b1 using   3  4 0 2 0  Eqs. 29 and 30: 1 b3 δ d δ A d δ + − σLV 4 + 4 2 12 20 2µV dx 2πδ dx  0 00  iω ik πLBσ 3 0 0 + LV + a1 dδ a1δ db1 3kV 0 3kV 0 8 80 2Ja + (1 − 2b3) + = 2 δ − b1. 30 dx 20 dx ρV cp,V δ ρV cp,V δ  π  iω0 + ik 00 LBσLV + k 004π − 2π k 002 + 3 [29] 4 LP LP c (1 + c)π 3π  7  0 0 + iω0 + ik 00 LBσLV + k 004 LP + c The time evolution for the perturbed δ as a function of b1 follows 4 20 20 3 from the linearized phase-change expression (Eq. 19) and the 0 0  7  3π 3 expressions for uV and ΘV (Eqs. 20 and 28): − k 002 + c LP − = 0. [37] 3 10 Ja 0 2 0 3 04 ∂δ ρV ∆ρgδ dδ ρV δ σLV dδ ρV + + 4 The marginal state occurs when the real part Re(iω) = 0, sep- ∂t 4µV dx 12µV dx (iω) < 0 02 arating zones of stability (Re ), where the perturbation ρV A dδ 3kV ∆T 0 2kV ∆T 0 amplitude decays in time, from regions of instability (Re(iω) > + 2 + 2 δ − b1 = 0. [30] 24πµV δ dx hLV δ hLV δ 0), where the base state becomes unstable (Fig. 2A). Note that

only three dimensionless numbers Ja, πLP , and πLBσLV govern The perturbation Eqs. 29 and 30 give two homogeneous condi- the stability of the perturbed solution. With the inclusion of van 0 0 tions for δ and b1. The perturbations can now be expressed in der Waals interactions, the buoyancy terms described by πLBσLV terms of normal modes: become negligible at nanoscale, as will be discussed shortly. This analysis incorporated time variation and convective trans- 0 0 δ = δa exp(i(kx + ωt)) [31] port in the energy equation. We can obtain a simpler expression for the stability problem by neglecting these two terms, b0 = b0 exp(i(kx + ωt)). [32] 1 1a 0 004 002  00 πLBσLV iω = − k πLP − 2k πLP + 1 − ik , [38] To avoid introducing new notation, we will represent the ampli- 4 0 0 0 0 tudes without subscripts δa → δ and b1a → b1. Here, k is the where the buoyancy term πLBσ is explicitly shown to give the wave number and ω the time rate of growth of the perturbation. LV dimensionless traveling-wave velocity, signifying that the base We combine Eqs. 29 and 30 to obtain a single equation with the flow acts only to convect the perturbation and does not affect its coefficient δ0. To simplify the representation, we introduce the growth. The full derivation of Eq. 38 is provided in SI Appendix. following dimensionless parameters: The diffusive expression (Eq. 38) is a good estimate to the

2 full stability equation (Eq. 37) for small Jakob numbers, as 3A hLV ρV demonstrated in Fig. 2C. Since the dimensionless parameters πLP = 2 3 [33] (24π) δ kV µV ∆T σLV are calculated from the vapor properties at the superheated 1 wall temperature, the Jakob number is small (Ja / 10 ) for the vapor phase of most fluids, implying that the thermal energy r A  ∆ρgδ2h ρ  LV V imparted by the heated solid is predominantly consumed through πLBσLV = [34] πσLV 2kV µV ∆T the latent heat of phase change rather than as sensible heat in r raising the temperature of the vapor. Physically, dropping the 00 2 4πσLV time-varying term in the energy equation implies that the heat k = kδ [35] A conduction timescale is much longer (quasi-steady) than that for the perturbation growth in the phase-change equation.  h ρ δ2  The Leidenfrost point corresponds to the lowest, critical ω0 = ω LV V . [36] kV ∆T πLP, crit on the marginal stability curve, below which the flow

A B C 2.5 = 2e-9 and Ja = 0.1 1 LB LV Numerical solution = 2e-9 and Ja = 10.7 Analytical solution (eqn. 39) 2 LB LV 0.8 Diffusive eqn.

1.5 0.6 k''

LP,crit 1 0.4

0.2 0.5

0 0 0204060801000 0.5 1 1.5 2 2.5 Ja LP

The stability of the perturbed solution. ( ) The variation of Re( ω) with π and the dimensionless wavenumber 00 for π = 2 × 10−9 and Fig. 2. A i LP k LBσLV Ja = 0.1. A critical πLP,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 πLP,crit , as predicted by the numerical solution to the full stability solution (Eq. 44) and by taking πLP,crit to occur at k00 = 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 iω changes sign. The diffusive expression (Eq. 45) is a good approximation for small Jakob numbers.

13324 | www.pnas.org/cgi/doi/10.1073/pnas.1917868117 Zhao and Patankar Downloaded by guest on October 1, 2021 Downloaded by guest on October 1, 2021  aeegho rud1 m vrti aeegh h relative the wavelength, thickness this film Over in change nm. 10 around of wavelength a number wave wave- the perturbation criterion that critical the Note to length. compared scale both length since longer self-consistent in thickness is use dominated 21) film the (20, diffusion the Second, assumption parallel theory. be locally lubrication the to of via assumed momentum required and is not energy is boiling approximation film layer boundary if the that means it Eq. of implication The (Eq. 2C expression Fig. critical numbers, Jakob the small For that 2B). (Fig. tested sets ter Eq. hoadPatankar and Zhao perturbed dampen to acts also this field; film liquid modes. from smaller the away into transport for enhancing bulge and larger a domain local vapor is a the from into shear away bulge transport Viscous mass base reducing an the versa. therefore restores thicknesses, triggering vice that film, locally and evaporation vapor of state the rate the in in gradient increase thermal basic the the steepens restore frequencies. to oscillatory attempts possible all that dampening This by gradient state interface. the pressure two-phase to a continuous due creates the zone of low-pressure high curvature adjacent induces liquid) negative an the with into locally (curving region pressure its vapor with the parallel interface in locally liquid–vapor center the basic, of the curvature on Positive solution. imposed modes oscillatory against vapor the in point given any from field. away transport mass reduces mass vapor local replenishes tension surface vapor (Eq. perturbation the Terms. Stabilizing for depend matter indeed not would does force it driving solution. way but if its that configuration, (even finds of force on eventually strength buoyancy-driven down—vapor The upside some up). or by horizontal film LFP of is the configuration the the it takes the of why eventually on out configuration explains dependent vapor Any strongly This setup. interface be experimental volume. liquid–vapor to the control found the not local at is the generated of vapor out carries which perturbation the in negligible is thickness analysis. film the of growth slow for solution same (Eq. equation ity parameters, dimensionless the at scaled occurs we how to due that critical the for of approximation values good all for stable unconditionally becomes = Ja 10 iial,aprubditraeta ugsit h vapor the into bulges that interface perturbed a Similarly, force restoring a as acts tension surface liquid–vapor The flow, base the of independent is growth perturbation Finally,  39 eel httemi tblzn em r h liquid– the are terms stabilizing main the that reveals 40)  3π a eie gis ueia ouint h ulstabil- full the to solution numerical a against verified was 3 7 LB 20 k + k 00 σ 00 c LV 1 = 1 =   1 + 2 hslast nagbaceuto for equation algebraic an to leads This . 38): ) orsodn to corresponding (1/µm), 100 of order the on is π  h ifsv prxmto otecritical the to approximation diffusive The π ihRe with 37)  LP δ LP Ja 12 π ntebs ouincagsoe much a over changes solution base the in LP , a lob siae rmtediffusive the from estimated be also can  crit σ 2 48 , ∆δ LV crit  δ pt ahn rcso o l parame- all for precision machine to up π sago siaei hefl.First, threefold. is estimate good a as + 1 h vprtv hs hnethat change phase evaporative the , − LP saround is h k (i 1 , LV V Ja  crit ω 9 ∆T ρ 0 = ) V 1 = 2+ (2 + 1 n h icu shear viscous the and , π LP . 10 k Ja n a on ogv the give to found was and c 4 −9 a edrvdb noting by derived be can ) o h rtclstability critical the for  +  Ja 3 1 6 hwn htthe that showing 1, + c k 00 − 2c 3 o Eq. For . Ja 12 − π π LP LP π π LP µ shows LP , , crit a 37, crit [40] [39] π , that , crit LP  crit  2 : . sn isizter 2,26), (25, theory Lifshitz using essta aerahdtesm re fmgiueas magnitude of ratio order the thick- approximate same film we the for reached only van have role the that significant As a nesses solution. plays perturbed interaction the Waals of der and stability mate- the (V) most on for vacuum nm wettability 1 across of magnitude (L) Eq. of order rials. liquid the in the values on and takes (S) substrate solid where angle contact the and stant intermediate an or vacuum 24). a (23, by phase solid a from separated liquid interac- repulsive purely attractive is for term the destabilizing across only tions substrate the solid are and film liquid vapor the between forces (Eq. persion general- stability the perturbation via the analysis (Eq. of this gradient into pressure incorporated separated ized is liquid It adjacent an vacuum. and by surface uncharged an between A Interaction. Waals der van hr ehv endtodmninesprmtr htwe that parameters significant: dimensionless be to two show defined will have we where the on angle contact intrinsic the of terms in substrate, LFP the for sion Eq. in Substituting a erwitni em fmtra properties: material of terms in rewritten be can Eq. data, experimental against compare To in Simulation and presented Experiment by are Verification analysis this in on role implications regime significant magni- further Appendix. nanoscale a The and play the examined. not direction of mechanism do the field instability as gravitational the the orientation, of arbitrary tude of plates changes to agnostic Eq. completely that is such predicted rate, in growth state temporal marginal the of on the part imaginary is the and in only film (Eq. vapor equation the tion of on order force the buoyancy the encapsulates (Eq. solution perturbation full SVL h eainhpbtentehtrgnosHmkrcon- Hamaker heterogeneous the between relationship The nti aocl eie h eta uv ecie ythe by described curve neutral the regime, nanoscale this In π 4 3 LB a eapidt atr h ednrs on on point Leidenfrost the capture to applied be can 40) sue ocaatrz h a e al iprinforces dispersion Waals der van the characterize to used is A |  σ H > H LV SVL ≈1 41 {z SVL δ h l sucniinlysal fteinteraction the if stable unconditionally is film The 0. hsipista h tblt rtro (Eq. criterion stability the that implies This . (24π a hsb sdt con o h feto surface of effect the for account to used be thus can steeulbimcnatsprto ewe the between separation contact equilibrium the is  10 4 } 3  | −14 ) 2 h + 1 A PNAS LV  h ifsv xrsino h perturba- the of expression diffusive The . σ a nepii eedneon dependence explicit an has 38) π {z < eoti h orsodn expres- corresponding the obtain we 41, LV 1 δ 2 ρ cos(θ  V h omrcs od ngnrlfra for general in holds case former The 0. π π 4 | δ 2 1 h eeoeeu aae constant Hamaker heterogeneous The h H  } LV ue1,2020 16, June = = SVL δ σ 1+ (1 = ) θ .Fo h ifsv expression diffusive the From 27). LV h k ρ ftesbtaehsbe derived been has substrate the of sisniieto insensitive is 37) ≈ V LV V σ 12πσ µ σ cos(θ δ 1. LV ρ ,i ssonta hs dis- these that shown is it 38), V LV 2 A V ∆ A 2 δ LV k SVL T )) . V | H 2 ∆T o.117 vol. | SVL k 1 2 V 40 µ ∆T σ , π {z V o h critical the for LV 2 1 1 = µ | π V o 24 no. } LB . =1, σ LV π which , | LB 39 H 13325 [41] [44] [42] [43] [45] π SVL σ LP LV or SI ,

ENGINEERING 350 scale vapor film thicknesses where the bulk liquid appears to wet Diffusive expression the substrate. Although instabilities on the micrometer scale and Kim et al. (2013) above perturb the liquid–vapor interface and induce frequent Liu et al. (2010) 300 Nagai et. al (1996) local contact between the liquid and the solid, only when the Kruse et al. (2013) vapor film becomes unstable on the smallest length scales where Kim et al. (2011) van der Waals interactions dominate will the film completely col- Nagai et. al (1996) lapse. Further discussion on the timescales associated with the 250 Liu et al. (2010) instability theory as well as the residence time and frequency of Vakarelski et al. (2012) liquid–solid contact observed in experiment is presented in SI Kruse et al. (2013) Appendix. 200 Vakarelski et al. (2012) Finally, we note that our theoretical analysis predicts that the MD simulations (this work) main effects governing the LFP operate in the nanoscale regime, which is accessible by molecular dynamics (MD). Fig. 4 shows 150 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 correspond- ing intrinsic contact angle of the substrate. Details of the MD 100 implementations are provided in SI Appendix. Note that a vapor 0 20 40 60 80 100 120 140 160 180 film forms when the liquid water adjacent to the bottom plate is heated above the LFP, whereas liquid contact with the solid Fig. 3. The Leidenfrost temperature vs. the contact angle for water, from surface is preserved below the LFP due to the attractive hetero- experiments (13, 14, 30–32), the diffusive prediction of the LFP (Eq. 43), geneous van der Waals interactions. The relationship between and molecular dynamics simulations from this work. The equilibrium sep- the contact angle and the Leidenfrost point of the extended sim- aration HSVL and its variation associated with changes in the contact angle ple point charge (SPC/E) water model is in good agreement with dHSVL dθ can be estimated from experimental data (25, 33). Typical errors in the diffusive prediction of the LFP (Eq. 43). These simulations the LFP and the contact angle measured from experiment are around 5 ◦C show that vapor film stability is ultimately determined at the and 2◦, respectively, although many sources do not explicitly report an proposed nanometric length scale where fluid–surface van der error value for either quantity. The data point corresponding to a con- ◦ Waals interactions cannot be discounted and where the effect of tact angle of 160 from Vakarelski et al. (31) corresponds to the only gravity-driven instabilities is nonexistent. surface which was textured with nanoparticles to achieve superhydropho- bicity; the other data points correspond to flat surfaces with random roughness. 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 θ ≈ 20◦. Nonetheless, the Hamaker constant must be found to determine the equilibrium Eq. 43 provides an explicit relationship between the intrinsic con- HSVL separation HSVL. Although the assumption δ ≈ 1 is made, the tact angle of a fluid on a substrate and the Leidenfrost point for base film thickness δ still needs to be incorporated into our insta- the system. Since each fluid property (kV (T ), µV (T ), σLV (T ), bility expression via π2 (Eq. 45). We can find the homogeneous etc.) is calculated at the superheated wall temperature, the Hamaker constant of the fluid (acetone, ethanol, benzene, etc.) left-hand side of Eq. 43 is in general a nonlinear function of temperature. The temperature at which Eq. 43 is satisfied cor- responds 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 Tc and pressure pc , 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 . Without considering van der Waals interaction between the liquid and substrate surfaces, such a relationship cannot be explained or predicted from first principles. 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 lifes- pan between formation and collapse showed that film collapse Fig. 4. Molecular dynamics simulation of vapor film formation adjacent to on the macroscopic level is preceded by submicrometer length- a heated surface. The system is pressurized at 1 atm.

13326 | www.pnas.org/cgi/doi/10.1073/pnas.1917868117 Zhao and Patankar Downloaded by guest on October 1, 2021 Downloaded by guest on October 1, 2021 hoadPatankar and Zhao for point angles: Leidenfrost sim- contact intrinsic a the low, with to to systems small leads fluids/substrate prediction approximations are dimensionless above angles surface the plified, contact the numbers, Jakob intrinsic wet low that their fluids that feature (θ such regime contact, boiling in film the in depen- functional a dence exists Leidenfrost respective there their that at suggests fluids This most . that for 0.06 observed around is is sion It simplification: quantity for dimensionless avenue interest. the of an substrate or note fluid we a for Here, available be not may constants species. knowledge both with of Leidenfrost versa constants vice Hamaker or the homogeneous substrate solid the given a of on fluid constant a for Hamaker point heterogeneous the S1 Fig. Appendix, SI contact homogeneous the obtain can homogeneous via the separations we the From and constant, 36). energy (24, Hamaker surface value the heterogeneous between the relationship obtain geomet- to the mean take and ric copper) aluminum, (gold, substrate the and 37–43). 27–29, (15, potassium liquid and benzene, are sodium, RC318, data liquid , R113, property R134a, R11, fluid pentane, , and ethanol, interaction LFP acetone, Waals Experimental for der fluids. available van angle the the contact which of low at role for value destabilizing critical the the describes while occurs LFP transport, viscous generation), and mass (vapor tension, change surface phase evaporative number of dimensionless effects stabilizing The the error. 5% within (Eq. to expression experiment diffusive the from imation 5. Fig. ietevprfim nldn ufc eso,paecag,and change, phase tension, surface including stabi- film, that vapor terms a metals. the the liquid for lize describes number and LFP dimensionless cryogens in the single including on The discussed fluids, data different experimental equal- as of the this variety which captures system, at satisfied the temperature is the the of ity that shows to application 5 Fig. by theorem Appendix. arises Pi also Buckingham quantity dimensionless This ngnrl xeietldt ntehmgnosHamaker homogeneous the on data experimental general, In ≈ 1000 1200 1400 -400 -200 200 400 600 800 20 H 0 h iesols criterion dimensionless The ◦ SVL 2) rmtedfuieepeso (Eq. expression diffusive the From (29). ) Acetone = Ethanol F Diffusive expression Experiment ( Pentane h LV σ LV ρ hw hti spsil odtrieeither determine to possible is it that shows V

π R134a diinly oteprmna setups experimental most Additionally, ). 1 σ σ Nitrogen = SV LV k π = V = 2 RC318 ∆ σ = 24π 24π π LV 2

T Benzene 1 A A h = LV µ SVS LVL H σ H salwcnatageapprox- angle contact low a as 6 V LV ρ SVS LVL 2 2

atrsteLPdt from data LFP the captures 43) Helium V ≈ δ . 6. ntedfuieexpres- diffusive the in R11 Cyclohexane R113 π ai for valid 43) 1 Sodium encapsulates Potassium [47] [46] [48] SI hsclisaiiymcaimo h ednrs phenomenon Leidenfrost the of mechanism instability physical surface pressure. with and diffusive data properties, experimental fluid a of wettability, variation to the flows captures LFP number the Jakob small (Eq. for approximation (Eq. simplified analytically or found case be neutral which or can marginal The substrate, state transport. evapora- viscous the tension, and surface change, phase and liquid–vapor Waals tive the liquid der by stabilized van bulk be possible attractive could the only by wall The between driven vertical considered. was interaction heated was nanoscale a gravity at on of instability film effects vapor the a under of stability dynamic The Conclusion point. pressures critical the superatmospheric to and up subatmospheric both for small fluids (Eq. expression the diffusive the and Eq. to that demonstrates expression 6 Fig. diffusive Excel S1. Dataset (Eqs. full approximation. approximations this the angle in contact use, apply of computed that ease are for sheets terms reported and fewer (Eq. data since expression the of diffusive observation full empirical the to Eq. estimate Essentially, respectively. sure, where 6 s h rdcieaiiyo h ml otc nl prxmto (Eq. approximation angle contact (Eq. small expression diffusive the the of to ability 49) predictive the pressure vs. critical the 46) by normalized pressures different at 6. Fig. to corresponds LFP the low that such For pressure (45). with fluid linearly scales the that find of we systems, point fluid/substrate the angle critical contact near the from toward increases point that boiling gradually such temperature applied, pressure Leidenfrost ambient the the on depends LFP. LFP the the that of Dependence Pressure dominate. terms stabilizing Larger the since substrate. regime, solid and Waals liquid of der values bulk van the the attractive to between of effect corresponding interaction destabilizing value the critical denotes the LFP while transport, viscous ige iesols number dimensionless single, A -200 -150 -100 100 150 200 250 -50 50 0 p h F rmeprmna aao 13 1,hxn,adnitrogen and hexane, R12, R113, of data experimental from LFP The ref . . . . 1 0.8 0.6 0.4 0.2 0 π and 1 bv h rtclipytesse stefimboiling film the is system the imply critical the above p applied PNAS .Tersligtertclslto for solution theoretical resulting The 40). r t n h ple,oeaigpres- operating applied, the and atm 1 are | π 1 ue1,2020 16, June 6 = 43). p ,cpue h F fvarious of LFP the captures 43), applied xeietlwr a shown has work Experimental p p/p π ref 1 49 48 sfudt naslt the encapsulate to found is c o h otgeneral most the for 39) , | sasalcnatangle contact small a is and ipie estimate simplified a 43, o.117 vol. Experiment Diffusive expression r rvddin provided are 49) bandby obtained 43) p π | c 2 ftefli (44– fluid the of o 24 no. 1+ (1 cos(θ | 13327 [49] )) 2

ENGINEERING for wetting fluids. The value of π1 with respect to the critical Usage of the Excel Sheets. The Excel sheets provided in Dataset S1 can denotes regimes in which the vapor film is stable or unstable, be used for different systems by substituting the correct homogeneous providing a useful characterization of both the thermodynamic Hamaker constants for the fluid and solid substrate, the surface energy of state and the physical means by which transition to the pool the solid, and the ambient pressure, as well as the temperature-dependent boiling regime occurs. vapor thermal conductivity, vapor viscosity, saturation pressure, vapor den- sity, liquid and vapor enthalpies, and surface tension for the fluid. The Excel This insight into the nanoscale mechanisms inducing the tran- sheet will report the corresponding predicted LFP and the percentage of sition from film to enables control of the phase error compared with the experimental data (which must also be provided adjacent to the surface (47). It would be of interest to extend the by the user). Please check that the formulas in the top row cover the entire instability mechanism toward surface roughness, which experi- length of the column for the new datasets provided. ment has shown to effect dramatic changes in the LFP beyond what can be explained by variation in surface wettability (5, 14, ACKNOWLEDGMENTS. This research was supported in part through the 48). In addition, a theoretical treatment of the Nukiyama tem- computational resources and staff contributions provided for the Quest high performance computing facility at Northwestern University which is perature corresponding to the critical heat flux may reveal the jointly supported by the Office of the Provost, the Office for Research, and mechanism underlying transition boiling and provide a compre- Northwestern University Information Technology. Partial support from the hensive understanding of the entire boiling curve under a unified, US Department of Energy (DOE), Office of Energy Efficiency and Renewable physical framework. Energy, Advanced Manufacturing Office, under contract DE-LC-000L059 is gratefully acknowledged. The US government retains, and the publisher, by Materials and Methods accepting the article for publication, acknowledges that the US government retains, a nonexclusive, paid-up, irrevocable, worldwide license to publish or Data Availability. The data and procedures are freely available in SI reproduce the published form of this paper or allow others to do so, for US Appendix and Dataset S1. government purposes.

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