Suppression of Leidenfrost effect on superhydrophobic surfaces Open Access

Known for over two centuries, the Leidenfrost phenomenon entails the levitation of an evaporating liquid droplet on a hot surface such that the droplet's weight is counterbalanced by the pressure of the vapor film formed underneath.¹ This phenomenon has attracted interest across many disciplines,^2–12 and it has been applied to engineering applications, such as directional droplet propulsion,^7,9,13 electricity generation,¹⁴ and green chemistry,¹⁵ among others.^3,16,17 For liquids with strong intermolecular forces¹⁸ and a high boiling point such as water, the Leidenfrost phenomenon occurs at temperatures—the Leidenfrost point (T_L)—significantly higher than at normal temperature and pressure (NTP: 293 K and 1 atm). For example, for a smooth and flat silica surface, the T_L of water is in the range 270 °C–300 °C.^19,20 As a corollary, the Leidenfrost phenomenon can be realized at NTP with liquids that have ultralow boiling points, such as liquid nitrogen and 1,1,1,2-tetrafluoroethane with boiling points of −196 °C^21,22 and −26.1 °C,⁹ respectively. In either scenario, a continuous vapor film of ∼100 μm-thickness levitates the droplet above the surface, after which the droplet slowly evaporates, akin to film boiling.^3,9,21 Roughening of surfaces can have profound consequences on T_L: for example, hierarchical micro/nano-texturing furthers wettability of a silica surface with water, increasing the temperature needed for the water droplet to achieve the Leidenfrost state (e.g., T_L > 400 °C).^{19,20,23–25} Conversely, when hydrophobic coatings are applied to micro/nano-textured surfaces, they can give rise to superhydrophobicity, characterized by water contact angles, θ_r $≥$ 150°, and an empirical constraint on contact angle hysteresis, e.g., Δθ $≤$ 20°.^26–28 Superhydrophobicity lowers the T_L of water by facilitating (i) low adhesion energy per unit area at the solid–liquid–vapor (SLV) interface given by the Young–Dupré equation, $Wad=γLV×(1+cos θo)$⁠, where γ_LV is the surface tension, and $θo$ > 90° is the intrinsic contact angle of the liquid (water),²⁹ and (ii) reduced liquid–solid contact area (hence adhesion) due to the interfacial entrapment of air.^4,30–34 For instance, after a flat metallic surface was rendered superhydrophobic by coating it with the Glaco Mirror Coat Zero^TM coating (hereafter referred to as Glaco), the T_L of water reduced from 210 °C to 130 °C;³⁰ similarly, Glaco-coated superhydrophobic steel balls⁴ and rotors³¹ exhibited T_L approaching 100 °C and also afforded a significant reduction in frictional drag.⁴ Taken together, these findings give credence to the conventional wisdom that superhydrophobicity lowers the T_L of water.^30,32,33,35

Superhydrophobic surfaces have been exploited to enhance pool boiling;³⁶ however, their applications in thermal machinery operating at high temperatures remain limited as superhydrophobicity generally lowers the T_L of water. Specifically, the vapor film prevents the interfacial heat transfer because of its lower thermal conductivity than that of the liquid,³⁷ for example, in boiler pipes, heat exchangers, and turbines. This situation, known as boiling crisis or burnout,³⁸ can severely impact energy efficiency, damage machinery, and cause accidents.³⁹ Hence, it is of practical interest to realize superhydrophobicity without lowering the T_L of water, which could also facilitate frictional drag reduction.⁴⁰ However, this has remained unachievable thus far. We therefore wondered if the lowering of the T_L of water was a universal attribute of superhydrophobicity, similar to the water contact angles in the range θ_r > 150° and contact angle hysteresis^28,41 Δθ < 20°.

Here, contrary to conventional wisdom, we demonstrate that superhydrophobic surfaces do not always lower the T_L of water. We utilize silica surfaces adorned with a specialized microtexture comprising doubly reentrant pillars (DRPs) that exhibit superhydrophobicity despite their hydrophilic surface chemistry.^27,28,42,43 The DRP microtexture is inspired by cuticles of springtails (Collembola)⁴⁴ and microtrichia of sea-skaters (Halobates),⁴⁵ and its super liquid-repellent properties have been studied.^27,43 Counterintuitively, water's T_L on these superhydrophobic silica surfaces is significantly higher than that on flat hydrophilic silica surfaces and even higher than that on superhydrophilic microtextured silica surfaces (see Table I for the apparent contact angles of water droplets). We combine experiment and theory to pinpoint the contributions of the microtexture, the interfacial heat transfer, and the surface chemistry to the onset of the Leidenfrost phenomenon on these surfaces. These findings advance the current understanding of the Leidenfrost phenomenon in the context of superhydrophobic surfaces and may broaden the scope of superhydrophobicity for applications in thermal machinery/engineering.

In this work, silicon wafers (p-doped, 4-in. diameter, and 500 μm thickness) with a 2.4-μm-thick thermally grown silica layer (hereafter referred to as SiO₂/Si wafers) were used as substrates [Fig. 1(a)]. The SiO₂/Si wafers [Fig. 1(a)] and Glaco-coated SiO₂/Si wafers [Fig. 1(b)] served as hydrophilic and superhydrophobic surfaces⁴⁶ (hereafter referred to as flat silica and Glaco-coated silica, respectively). Glaco-coated silica surface is rougher than flat silica [Figs. 1(a) and 1(b)] because the Glaco coating is comprised of perfluorinated silica nanoparticles of average diameter ∼50 $n$m.⁴ Additionally, we microfabricated arrays of cylindrical pillars [Fig. 1(c)] and DRPs [Figs. 1(d)–1(f)] with diameter, D = 10 μm, height, H = 50 μm, and pitch, L = 50 μm. To fabricate DRPs [Fig. 1(d)], we applied photolithography, dry etching, and thermal oxide growth, followed by the protocols reported in the previous studies.^27,28,47,48 The surface adorned with arrays of cylindrical pillars exhibits superhydrophilicity, but the surface decorated with arrays of DRPs exhibits superhydrophobicity, even though they have the compositions of the same materials. After micro fabrication, all the samples were cleaned with Piranha solution (H₂SO₄: H₂O₂ = 3:1 by volume) for 15 min at 110 °C and stored in a vacuum oven for 24 h at 50 °C and ∼0.03 bar. Consequently, the apparent contact angles (⁠$θr$⁠) of water on the flat and smooth silica surface were ∼40°.

The effects of surface chemistry were studied by coating perfluorodecyltrichlorosilane (FDTS) on DRPs using a molecular vapor deposition system (ASMT 100E), as described previously.⁴⁹ All the surfaces were examined via a scanning electron microscope (Helios Nanolab field-emission SEM operated at an acceleration voltage of 5 kV, beam current of 86 pA, and a working distance of 4 mm). Microtextured substrates were cut into square samples (dimensions: ∼10 × 10× 0.5 mm) using a diamond-tip scriber for further experiments, and the debris generated was blown away using a nitrogen gun.

We characterized the wettability of these surfaces [Figs. 1(a)–1(d)] with water by measuring apparent contact angles (CAs) of water droplets (6 μl) and advancing and receding angles^50,51 at a rate of 0.2 μl s⁻¹. The water CAs on the surfaces of different samples were measured using a drop shape analyzer (Kruss: DSA100) interfaced with the Advance software. The sliding angles of water droplets (10 μl) on these surfaces were measured by an automatic tilting stage, where a digital camera was used to record the movement of droplets. All experiments were repeated at least three times.

To generate water droplets, we used a Harvard Apparatus syringe pump (PHD ULTRA: 703007INT) and a 60-ml plastic syringe connected by a plastic syringe extension tube. A stainless-steel capillary (diameter: ∼0.51 mm) was used to dispense water perpendicular to heated test surfaces. The flow speed was set as 100 μl min⁻¹, and the syringe system continuously generated ∼15 μl droplets (radius: ∼1.5 mm, which is smaller than the capillary length of water ∼2.7 mm at NTP).

A high-speed camera (model number: Phantom V1212) with a zoomed lens was applied to record images at a high frame rate. To ensure that all the images have the same configuration, the imaging parameters of the camera, such as frames per second (FPS), image pixels, and exposure time in all experiments, were set to 1000 FPS, 512 × 512 pixels, and 200 μs, respectively. A triaxial stage was used to adjust the position of the samples with respect to the high-speed camera. The hot plate was leveled using a bubble leveler, and the images were calibrated by the stainless-steel capillary. For image processing, Phantom Camera Control 3.3 software was used.

A hot plate (temperature: 20 °C–540 °C) was placed on a triaxial stage and balanced by a bubble level. The temperatures of the hot plate surface were calibrated by a K-type thermocouple and recorded by a temperature data logger (Omega: RDXL6SD). The samples were placed at the center of the hot plate, and their temperatures were monitored via a K-type thermocouple. After the samples reached the hot-plate temperature (±3 °C) and became stable, droplet experiments were started.

Water droplets (15 μl) were generated by the syringe pump and released from the capillary tip to the heated sample surfaces. The capillary tip was placed as close to the surface as possible to gently release the droplet to minimize the droplet impact. Thus, the droplets had a low Weber number (We ≈ 1.7, defined as the ratio of the droplet's inertia to capillarity, $We=ρV2Rd/γLV$⁠, where ρ, R_d, γ_LV, and V is the density, radius, water surface tension, and impact velocity, respectively) to minimize its effect on the Leidenfrost point. A high-speed camera was used to record the impact, bounce, and vibration of droplets and observe vapor film at the interface between droplets and surfaces. All the experiments were repeated at least three times.

The classical lifetime method¹⁰ was used to measure T_L on these surfaces. The highest peak in the curve formed by the lifetimes of droplets at various surface temperatures yields T_L.^10,30 In these measurements, the temperature of the hot plate was increased stepwise by 10 °C, and the droplet lifetime was recorded by a timer. All experiments were repeated at least five times. Commonly, the lifetime method does not yield a sharp peak for the Leidenfrost transition on superhydrophobic surfaces. In response, we used a high-speed camera to detect T_L on Glaco-coated flat silica and silica with DRPs by noting the temperature at which a continuous vapor film formed and the droplet bounced.

A Zeiss upright laser confocal microscope (Model number: LSM710) was used to detect the meniscus at the interface between a droplet and pillars. A ∼100 μl water droplet with Rhodamine B (Acros) was gently placed on the sample using a manual syringe, and a 20× micro-objective lens was immersed into the droplet before image acquisition. A 561-nm laser in the Z-stack mode was used to scan the droplet–surface interface. Subsequently, using the Imaris v.8.1 software by Bitplane, we obtained cross-sectional images to visualize the meniscus shape.

We utilized the heat transfer module in the COMSOL^® (COMSOL Multiphysics,⁵² version 5.4) software to simulate temperature distributions in our experiments. Boundary conditions were set based on high-speed imaging experiments and reasonable assumptions ( Appendix, Fig. A1 and the modeling section). The meshes' independence was tested⁵³ by varying the number of mesh elements (for DRPs: 33 406, 96 178, and 379 527 elements; for cylindrical pillars: 20 748, 43 158, and 120 045 elements), and the relative errors were found to be < 1%. Next, we built an analytical model based on Eqs. (1)–(13) to predict the Leidenfrost points in our experiments; these equations were solved using the Wolfram Mathematica software (version 11.0). In the calculation, we utilized the thermophysical properties of water at 99 °C and those of solid and gas phases at 225 °C, which is an average temperature between the water and the hot substrate.^3,54

First, we characterized the wettability of these surfaces [Figs. 1(a)–1(d)] with water by measuring the apparent (⁠$θr$⁠), advancing (⁠$θA$⁠), and receding (⁠$θR$⁠) contact angles of droplets. On flat silica, we observed $θA$ = 45° ± 2°, $θR$ = 0°, and $θr$ = 40° ± 2° [Fig. 1(g)]; the advancing contact angle (⁠$θA$= 45°) was considered as the intrinsic contact angle (⁠$θo$⁠) in our theoretical analysis (presented later).^26,50,51 Glaco-coated silica exhibited superhydrophobicity with θ_r = 156° ± 1° and contact angle hysteresis Δθ = $θA−θR$ = 19° ± 2° [Fig. 1(h)]. Conversely, silica surfaces with cylindrical pillars were superhydrophilic—they imbibed water, thus reaching the fully filled (or the hemiwicking) state⁵⁵ [Fig. 1(i)]; apparent CAs of water droplets were ultralow, θ_r < 10°. Silica surfaces with DRPs exhibited superhydrophobicity with θ_r = 150° ± 1° and Δθ = 19° ± 2° [Fig. 1(j)], similarly to Glaco-coated silica. The sliding angles of water droplets (10 μl) on Glaco-coated silica and silica with DRPs were 7° ± 3° and 13° ± 2°, respectively (Table I).

Next, we investigated the response of water droplets on the following substrates placed on a leveled hot plate to control the surface temperature precisely (±1 °C): (i) flat silica, (ii) Glaco-coated silica, and silica surfaces with arrays of (iii) cylindrical pillars and (iv) DRPs [Figs. 1(a)–1(d)]. We arbitrary chose an initial test temperature of 347 °C, which is beyond the T_L of water on common surfaces.⁵⁶ Water droplets (15 μl) were gently placed onto these surfaces, and the subsequent behaviors were recorded via high-speed imaging. The placement of the droplets ensured that the Weber numbers were as low as possible (∼1.7).³ Contrary to our expectation, all the samples except superhydrophobic DRPs exhibited the Leidenfrost phenomenon at 347 °C [Figs. 2(a)–2(d) (Multimedia view)] Specifically, after the initial impact and followed by 3–6 bounces due to inertia, water droplets reestablished contact with the DRPs with slight vibrations [Fig. 2(d)].

To confirm this result, we ascertained T_L on these surfaces by tracking the lifetime of water droplets on them as a function of surface temperature to determine how long it took for the water droplets to evaporate on these surfaces. As the surface temperature [T_s (°C)] increases, the droplet lifetime decreases due to the faster evaporation. In the vicinity of T_L, the droplet lifetime starts increasing due to the formation of an insulating vapor layer underneath the droplet that diminishes the interfacial heat transfer. When the surface temperature goes beyond T_L, the droplet lifetime decreases gradually. By gradually increasing the temperature in the range 70 °C–440 °C, the T_L values for hydrophilic silica surfaces—flat and those adorned with cylindrical pillars—were pinpointed to be ∼286 °C and ∼314 °C, respectively, [Fig. 2(e)]. These results are consistent with previous studies.^19,20

For superhydrophobic surfaces—DRPs and Glaco-coated silica—there were no distinct peaks in the droplet lifetime curves [Fig. 2(f)]. Therefore, we utilized high-speed imaging to monitor the droplet bounce and the vapor film formation at the droplet-solid interface to pinpoint the onset of the Leidenfrost phenomena ( Appendix, Figs. A2 and A3). For water droplets, superhydrophobic Glaco-coated silica exhibited the Leidenfrost phenomenon at ∼130 °C. In stark contrast, however, superhydrophobic silica surfaces with DRPs did not exhibit the Leidenfrost phenomenon until the surface temperature was raised to ∼363 °C. This dramatic variation in the T_L values of superhydrophobic surfaces challenges our current understanding of the factors and mechanisms underlying the Leidenfrost phenomenon.

Next, we observed the behavior of water droplets (15 μL) on silica with DRPs at T = 347 °C via high-speed imaging complemented with high-magnification optics (Fig. 3). We found that a typical droplet exhibited an up-and-down vibratory motion under the influence of its vapor force built underneath, weight, and adhesion at the SLV interface [Fig. 3(a) (Multimedia view)]. After the initial contact, when the center of mass of the droplet was at the highest point, it was still pinned to ∼3–10 DRPs, which preempted the onset of the Leidenfrost phenomenon [Figs. 3(b) and 3(c)]. As the droplet moved upward/downward, the number of DRPs in contact with the droplet decreased/increased, and the air–water interface assumed concave/convex curvatures [Figs. 3(d) and 3(e)]. We considered that as a liquid droplet is placed on a hot substrate, its vapor flow is obstructed by the surface microtexture, which builds up an upward vapor force, F_v (N). If F_v is greater than the droplet's weight, G (N), it tends to lift it, whereas the liquid–solid adhesive force, F_ad (N), impedes this motion [Fig. 3(d)]. Conversely, if F_v < G, the droplet tends to sink into the microtexture, which the adhesive force, F_ad, also counters [Fig. 3(e)]. In the concave configuration, the vapor has adequate space to escape; thus, F_v drops [Figs. 3(f) and 3(h)]. Subsequently, the droplet advances downward into the microtexture, assuming a near-flat convex shape [Figs. 3(g) and 3(i)] with F_ad acting upward, which indicates F_v < G.

To test the crucial role of F_ad, we reduced the liquid–solid adhesion on silica with DRPs by functionalizing it with perfluorodecyltrichlorosilane (FDTS), which yielded apparent CA for water droplets $θr$ = 162 ± 2°. When we placed water droplets onto FDTS-coated DRPs, T_L dropped to ∼172 °C [ Appendix, Fig. A4(a) (Multimedia view)]. The local adhesion force between a micropillar and water in our experimental configuration is given by $Fad=γLV×πD×(1+cos θo)$⁠.^41,57,58 As the surface chemistry changed from silica to FDTS, the intrinsic CA (⁠$θo$⁠) of water (on smooth and flat surfaces) increased from 45° for silica to 118° for FDTS, which caused a 69% reduction in F_ad and thus lowered T_L significantly. Also, after FDTS coating, the liquid–solid contact area was reduced slightly as water cannot penetrate the doubly reentrant edges, i.e., the vertical overhangs of the cap region [ Appendix, Figs. A4(b) and A4(c), images obtained from the confocal microscope experiments]. We also quantified the effects of the Glaco coating on the DRPs' liquid–solid adhesion and the Leidenfrost point. The surfaces were maintained at ∼347 °C (similarly to the results presented in Fig. 2). Due to the lower adhesion, droplets detached from the surfaces and attained the Leidenfrost state [ Appendix Fig. A5(a)]; Glaco-coated cylindrical pillars also exhibited the same behavior [ Appendix Fig. A5(b)]. These trends were quite similar to those on FDTS-coated microtextured surfaces.

These points demonstrate the sensitivity of the Leidenfrost phenomenon on the adhesion force rather than empirical cutoffs set for the macroscopic apparent CAs of water droplets on them.^28,41 This implies that not all superhydrophobic surfaces reduce the T_L of water. But why did the onset of the Leidenfrost phenomenon on silica with DRPs [superhydrophobic, Fig. 2(d)] get delayed in comparison to that on silica with cylindrical pillars [superhydrophilic, Fig. 2(c)] despite their identical surface chemistry and similar dimensions?

To address this question, first, we pinpoint some crucial differences in the ways DRPs interact with liquids in comparison to cylindrical pillars. The bioinspired DRP geometry is peculiar in the sense that despite the hydrophilicity of the substrate, it exhibits superhydrophobicity when liquids are placed on top because (i) the mushroom shape of the pillars prevents the liquid imbibition such that the droplet rests on the pillars and air [ Appendix, Fig. A4(b)], and (ii) as the droplet recedes, it has to detach from individual pillars, which requires small energy barriers to surpass, affording low contact angle hysteresis.^27,42,59,60 In the DRPs studied in this work, the pillar cap and the base were comprised of silica (∼2 $μ$m thickness) and silicon (∼50 $μ$m), respectively. When a water droplet lands onto DRPs, it only touches the pillars for a depth of ∼2 $μ$m [Fig. 4(a)], after which the air–water interface assumes a convex shape and any further liquid movement does not increase the liquid–solid contact [Fig. 4(b)]. Particularly, the liquid–solid contact on the DRPs is sensitive only to the dimensions of the pillars' caps and not to the penetration depth (L_p); this peculiarity would cease to exist at ultrahigh Weber numbers, where the drops would penetrate the microtexture, and those cases are beyond the scope of this study. In stark contrast to DRPs, when a water droplet lands on superhydrophilic cylindrical pillars [ Appendix, Fig. A6 (Multimedia view)], it tends to penetrate them [Fig. 4(c)], thus increasing the liquid–solid contact area and deepening the liquid penetration depth [Fig. 4(d)]. The enhanced penetration depth remarkably impacts the heat transfer, which we next investigate computationally by simulating the temperature distributions in DRPs and cylindrical pillars during droplet penetration. This is followed by presenting an analytical model for deriving F_v to gain mechanistic insights into the onset of the Leidenfrost phenomenon.

Temperature distributions in silica surfaces adorned with DRPs and cylindrical pillars as a function of water penetration depth were simulated using COMSOL^®. In this approach, we assumed that (i) a quasi-steady state, (ii) the water temperature was 99 °C,³ (iii) the convective heat transfer coefficient for the vapor inside microtextures was typically set as 20 W m⁻² K⁻¹)³⁷ ( Appendix, Fig. A1), and (iv) the evaporation at the air–water interface was considered as an outflow of heat flux. The experimentally observed penetration depths of water droplets into the microtextures comprising DRPs and cylindrical pillars were about 2 and 20 μm, respectively [Figs. 4(a) and 4(c)]. Consequently, the top regions of the DRPs [Fig. 4(f)] were significantly hotter than those of the cylindrical pillars [Fig. 4(h)]. In other words, the cylindrical pillars were effectively cooled by water due to the high penetration depth, but the DRPs were not. This explains why the heat transfer in the case of the DRPs was significantly lower than that in the case of cylindrical pillars [Figs. 4(e) and 4(g)]. The higher heat transfer in the case of cylindrical pillars generated higher vapor flux, which induced higher upward vapor force, thereby lifting the droplet at lower temperatures than that in the case of DRPs.

Next, we developed an analytical model to estimate the vapor force (F_v) and the Leidenfrost point (T_L). The assumptions underlying this quasi-steady state analytical model include (i) the temperature of the pillars in Region 1, wherein water penetrates, is the same as that of the water [based on simulation results in Figs. 4(e)–4(h)], (ii) the heat transfer is dominated by conduction and the contribution of convection is negligible,¹² (iii) the vapor flow across the microtexture is laminar; i.e., the characteristic Reynolds number is low, which is typical for the Leidenfrost phenomenon (at 100 °C, the density of the water vapor is ρ_v = 0.4 kg m⁻³, the viscosity of the water vapor is μ = 2 × 10⁻⁵Pa s, and the speed of the vapor in the microtexture is typically U $≈$ 1 m s⁻¹;^3,54 thus, Re = ρ_vUH/μ is ∼1). The energy for water evaporation (Q_eva (W)) is received through the heat influx at the water interface, Q₁ (W) [Region 1 in Figs. 4(b) and 4(d)]. This heat transfer rate, Q₁, is also equal to the heat transfer rate absorbed from the substrate underneath Q₂ (W) [Region 2 in Figs. 4(b) and 4(d)]. Therefore,

where $Q2=qHAo$⁠, and $qH$(W m⁻²) is the heat flux from the hot surface. A₀ = L² (m²) is the projected area of a periodic unit of the microtexture [Fig. 5(a)], m_e (kg) is the mass of water in a periodic cell, t (s) is time, and h_fg (J kg⁻¹) is the latent heat of the evaporation of water. Accounting for the heat conduction across the vertical direction, Eq. (1) yields

where ρ_v (kg m⁻³) is the density of the water vapor, u_z (m s⁻¹) is the vapor velocity along the z-axis, A_v = $L2−πD/22$(m²) is the vapor escape area inside the periodic cell, and T (°C) is the temperature. The thermal gradient, dT/dz (°C m⁻¹), depends on the penetration depth of water into the microtexture. Considering the droplet keeps moving up and down during heating [Fig. 5(a)], the average thermal gradient, dT/dz was taken as (T_s − T_sat)/(H − L_p/2) in the calculation. Here, T_s (°C) and T_sat (°C) are the surface temperature of substrates and the boiling point of water, respectively; H (m) is the height of DRPs, and L_p (m) is the penetration depth when the droplet moves down, ranging within, 0 < L_p< H. Note that k_eff (W m⁻¹ K⁻¹) is the effective heat transfer coefficient determined by the area average of air and the solid fractions,²⁰ 

where k_v (W m⁻¹ K⁻¹) and k_s (W m⁻¹ K⁻¹) are the thermal conductivities of the vapor and the solid, respectively, and ε is the ratio of the vapor area to the projected area, $ε=L2−πD/22/L2$⁠. Since the radius of the droplet's base (R_b (m), ∼0.7 mm from experimental observation) is dramatically larger than the height of the pillars, i.e., R_b ≫ H, the lubrication analysis can be applied.⁶¹ As the vapor is produced from the bottom of the droplet and released from the spaces between pillars, the continuity equation can be written as

where u_r (m s⁻¹) is the radial vapor velocity. Considering an approximately linear temperature distribution along the pillar height,¹² we can obtain the vapor velocity along the z-axis

Thus, the average vapor velocity at the r-direction (⁠$ur¯$⁠) can be obtained:

After applying the lubrication analysis, the momentum equation for the flow in the porous media can be reduced to the classical Brinkman Equation that has been widely used to describe flows across microtextures,^62–64 

where P_r (Pa) is the pressure distribution at the r-direction, μ_v (Pa s) is the dynamic viscosity of vapor, and K (m²) is the permeability of the porous media.⁶⁵ We apply the no-slip boundary conditions and estimate the average radial velocity as

where $β=ε/K$⁠. Next, we substitute Eq. (6) into Eq. (8), apply the boundary conditions, and obtain the radial pressure profile as

Thus, the vapor force, F_v (N), exerted by the vapor film onto the droplet can be estimated by

Furthermore, the adhesive force on each DRP, F_adi (N), is estimated as

where D (m) is the diameter of the top surface of DRPs, S_h (m) is the horizontal length of edge DRPs, γ_LV (N m⁻¹) is the surface tension of water, and $θo$ is the intrinsic contact angle of water on the silica surface. The Leidenfrost phenomenon occurs when the vapor force (F_v) overcomes the adhesive force (F_ad) and gravitation force (G), which allows the droplet to be levitated by a continuous vapor layer,

where G = mg is the gravitation force, m (kg) is the mass of the droplet, g (m s⁻²) is the gravitation acceleration, and n is the number of DRPs in contact with the water droplet and can be estimated as $n=πRb2L2$⁠. At the onset of the Leidenfrost phenomenon, T_s is equal to T_L. Substituting Eqs. (10) and (11) into Eq. (12), the temperature corresponding to T_L is

Equation (13) captures the dependence of the T_L on the thermal, structural, chemical, and inertial characteristics of the system, especially the heat conduction at the water–substrate interface that scales with the penetration depth (L_p), the microtexture dimensions, the liquid–solid adhesion, and the vapor flow through the microtexture. As the penetration depth, L_p, increases, T_L drops for both microtextures [Fig. 5(b)]. To disentangle the effects of geometry and surface chemistry, we utilized the model to predict the variation in the T_L of water with pillar height and surface chemistry. We found that T_L in both cases increased with increased pillar height [Fig. 5(c)] because the taller the pillars are for a given pitch, the larger is the space to escape for the vapor, lowering F_v. The model also pinpointed that as the surface hydrophobicity increases, T_L drops [Fig. 5(d)]. Therefore, the onset of the Leidenfrost phenomenon depends on a delicate balance among F_v, G, and F_ad, which cannot be satisfied by all superhydrophobic surfaces. Thus, these forces could be tuned to engineer superhydrophobic surfaces that can prevent the onset of the Leidenfrost phenomenon even better than hydrophilic and superhydrophilic surfaces.

Here we consider various factors and mechanisms that are responsible for the suppression of Leidenfrost effect on superhydrophobic surfaces. First of all, we discuss the elemental differences between DRPs and cylindrical pillars in the context of wetting. Microtextures comprised of arrays of cylindrical pillars get spontaneously and irreversibly filled by wetting liquids. In contrast, microtextured surfaces comprised of doubly reentrant pillars (DRPs) can prevent the intrusion of wetting liquids by flattening the liquid–vapor interface curvature at their overhanging feature/lip.²⁷ If the liquid is pushed further, the liquid–vapor curvature is reversed, which presents a kinetic barrier against liquid intrusion.²⁷ Thus, DRPs designed with low liquid-solid area fraction (ϕ_LS), e.g., ϕ_LS ≈ 3% in the current study, can exhibit superomniphobicity, characterized via advancing and receding contact angles.^28,41 These DRPs therefore lie inside the fourth quadrant of the classical Kao's diagram with the cosine of the intrinsic angle on the x-axis and the cosine of the apparent contact angles on the y-axis.^27,66 It should also be realized that the air-filled (Cassie) states on DRPs are metastable and could transition irreversibly to the fully filled or the Wenzel state if the kinetic barrier is overcome or if the liquid approaches the stems of DRPs laterally.^27,66,67 Of course, if silica microtextures—cylindrical pillars or DRPs—are lined with a hydrophobic coating, such as FDTS or Glaco, then the combination of surface roughness and hydrophobic chemical makeup can further enhance water repellency, and this has been investigated extensively.

Our experimental and theoretical findings might aid the rational design of superhydrophobic surfaces in thermal power engineering, for instance, for reducing frictional drag without compromising interfacial heat transfer. We illustrate this by comparing the time it took to evaporate a drop of water placed on silica with DRPs with that on Glaco-coated silica, both maintained at 200 °C. Although both surfaces were superhydrophobic, the water evaporated much faster on silica with DRPs than that on Glaco-coated silica (Fig. 6). Silica with DRPs facilitated a 300% higher heat transfer than Glaco-coated silica because the former suppressed the Leidenfrost phenomenon to well above 200 °C, whereas the latter experienced film boiling at T_L ≈ 130 °C.

Penetration depth was recognized as a crucial factor influencing the water–substrate heat transfer and, for a given microtexture, it depends on the Weber number.⁶⁸ At low Weber numbers, such as in the experiments reported here, DRPs pinned the liquid meniscus at the doubly reentrant edge such that any further penetration did not increase the liquid–solid contact area. Therefore, the DRP architecture helped us disentangle the effects of the penetration depth and the droplet–substrate contact area on the onset of the Leidenfrost phenomenon. However, at higher Weber numbers, the liquid meniscus might de-pin from doubly reentrant edges and land onto the DRPs' stems.⁶⁹ Subsequently, akin to the case of cylindrical pillars, the curvature of the liquid–vapor interface will reverse, the liquid–solid contact area will increase, and the conductive heat transfer will yield the Leidenfrost phenomenon. However, once the droplet's momentum dissipates and it lands on the surface, DRPs' original behavior, suppressing the Leidenfrost phenomenon, might get restored. These predictions necessitate a systematic investigation. Furthermore, the effects of the various geometrical aspects of the DRPs along with the surface chemistry on the Leidenfrost point warrant an investigation. In  Appendix, Fig. A7, we present predictions based on our theoretical model on the effects of the pitch (L), the height (H), and the cap diameter (D) on the Leidenfrost point of water droplets. We find that the Leidenfrost point (i) increases dramatically with H, (ii) increases gradually with L, and (iii) reduces with D. Here, it should be realized that the variations in D and L not only change the vapor force but also vary the adhesion force. This makes it hard to disentangle the contributions of the geometry and the surface chemistry on the Leidenfrost point. Therefore, in this contribution, we have focused on the effects of varying the pillar height, H, which only affects the vapor force but not the adhesion force [Fig. 5(c)]. In the future, we plan to do a comprehensive study toward creating a regime map for the Leidenfrost points on DRPs surfaces of varying D, L, and H.

In conclusion, this study demonstrates that it is possible to realize superhydrophobic surfaces that do not reduce the T_L of water via a judicious selection of surface microtexture and chemistry. Such a design can simultaneously reduce frictional drag and preempt boiling crisis in thermal machinery, such as heat exchangers, boiler pipes, and turbine blades. The coating-free aspect of the DRP architecture could also offer resilience against harsh operational and cleaning protocols and organic fouling⁷⁰ that perfluorinated coatings are vulnerable to.^71,72

The co-authors acknowledge research funding from King Abdullah University of Science and Technology (KAUST). M.S. thanks Professor Sigurdur Thoroddsen from KAUST and Professor Shangsheng Feng from Xi'an Jiaotong University for fruitful discussions.

Two of the co-authors, H.M. and M.S., have filed a provisional patent application (USPTO No. 63/141,101); the other co-authors declare no competing financial interest.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Figure A1 describes the geometrical models and boundary conditions of heat transfer simulations. Figures A2 and A3 exhibit the droplets' behaviors on Glaco-coated flat silica and DRPs surfaces maintained at different temperatures, respectively. Figure A4 demonstrates the effect of hydrophobic coating (FDTS) on the Leidenfrost phenomenon on silica with doubly reentrant pillars. Figure A5 shows the effect of Glaco coating on the Leidenfrost phenomenon on silica surfaces with cylindrical pillars and DRPs. Figure A6 demonstrates the droplet detachment from cylindrical pillars surface maintained at 347 °C. Figure A7 exhibits the theoretical predictions on the effects of the DRPs' pitch, height, and cap diameter on the Leidenfrost point of water droplets.