Influence of micropillar array structures on evaporation and wetting transition of fakir drops: Recipe for superhydrophobicity

doi:10.21203/rs.3.rs-2097189/v1

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Influence of micropillar array structures on evaporation and wetting transition of fakir drops: Recipe for superhydrophobicity

https://doi.org/10.21203/rs.3.rs-2097189/v1

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published 09 Feb, 2023

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This study experimentally investigated the evaporation and wetting transition behavior of fakir drops on five different microstructured surfaces. Diamond-like carbon was introduced as the substrate, and the influence of varying the width, height, and pitch of the micropillars was assessed. The results showed that different evaporation modes emerged during the transition, which were influenced by the interfacial properties of the surfaces. In addition, the resistance of superhydrophobic surfaces to the Cassie–Baxter to Wenzel transition was strongly dependent on the depinning ability of the three-phase contact line of the liquid drop. The accuracy of the theoretical models for predicting the critical transition parameters was discussed, and a numerical model was developed in the surface evolver to compute the penetration of the drop bottom meniscus within the micropillars. Finally, a robust superhydrophobic surface capable of suppressing the Cassie–Baxter to Wenzel transition without a hierarchical nanostructure for microdroplets less than 0.37 mm in diameter was demonstrated as the key outcome of this study.

When a small millimeter-sized water drop is placed on a superhydrophobic surface (SHS) comprising micron-sized pillars, it initially maintains a quasi-spherical shape and exhibits a large apparent contact angle (CA). This drop is also known as the “fakir drop” [1] and is said to be in Cassie–Baxter (CB) state with high mobility. Under the influence of external perturbations (e.g., evaporation [2], vibration [3], acoustic wave [4], and electrical voltage [5]), the liquid invades the micropillars, and a fakir drop undergoes a wetting transition from the CB to the Wenzel state, as shown in Fig. 1a. As a result, the drop loses its mobility and the surface becomes wettable. Surface wettability and drop evaporation are key to many engineering applications, including spray cooling [6], self-assembly [7], DNA detection [8], biosensing [9], inkjet printing [10], nanoparticle coating [11], and microfluidic chips [12] design, as shown in Fig. 1b. Owing to this widespread implementation, understanding the physical mechanism of wetting phenomena has become a topic of fundamental interest [13, 14].

To date, two fundamental mechanisms, namely, “depinning impalement” and “sag or touch down impalement” were proposed to elucidate the wetting transition of evaporating drop on micropillared surfaces, as shown in Fig. 2 [15]. When the interpillar spacing is very small, the penetration of the drop bottom meniscus within the micropillars is negligible. During evaporation, when the CA between the liquid-air interface and micropillars ( ) reaches the advancing CA ( ), the three-phase contact line (TPCL) depins and vertically slides along the micropillar side wall and thus, the transition occurs. In contrast, for a sufficiently large interpillar spacing, the TPCL remains pinned with the micropillar side wall, and the deflection of the drop bottom meniscus becomes significant with increasing Laplace pressure during evaporation. The transition occurs when the meniscus touches the bottom of the substrate.

Several studies have investigated the wetting and evaporation phenomena of fakir drops on various micropatterned solid surfaces [2, 16-21]. Bormashenko [22] reviewed the advances in understanding the wetting transition phenomena on micropatterned surfaces. There are also numerous studies on the modeling and simulation of wetting transition [23-27]. Direct imaging [15, 19, 28-30] and analytical techniques [31-33] have been developed for the wetting transition of liquid drops on textured substrates.

In addition, many other studies have highlighted the essential properties of solid surfaces to prevent the collapse of the CB state. Zhang et al. [34] revealed that reducing the solid–liquid interaction or coating microstructure with nanostructures favored the Wenzel–CB transition for microdroplets. Zheng and Zhao [35] discussed the influence of hydraulic pressure on the stability and transition of wetting modes on textured surfaces. Zhang et al. [24] derived the theory of critical height to obtain a stable CB state. However, if the drop size is reduced significantly during evaporation, the Laplace pressure at the air–liquid interface rises above the critical limit, and the CB–Wenzel transition becomes inevitable.

The stability of the CB state is key to the functionality of superhydrophobic surfaces in real-world applications [36]. To date, several efforts [37-40] have been made to demonstrate the stability of the CB state. However, in the above studies, liquid repellant surfaces were designed with complex microstructures, such as microposts with doubly re-entrant nano-overhangs [38], mushroom-like double re-entrant fibril arrays [39], or salvinia-like slippery surfaces [40], which enhance the fabrication difficulty. Thus, identifying a robust superhydrophobic surface with simple micropillars on which very small micron-sized droplets can stably maintain the CB state remains challenging. Existing literature [41-43] also reiterates that a hierarchical nanostructure is crucial for maintaining a stable CB state. However, such dual-scale nanostructures are fragile and can be easily degraded under real operating conditions. Hence, a complex armored design [44] was suggested to protect the nanostructure, thus increasing the fabrication difficulty.

The present study demonstrates that a stable CB state can be obtained without any hierarchical nanostructures or complex microstructures if the micropillar dimensions are properly selected. Here, the authors revealed a simple but robust SHS with micropillars that exhibited resistance to liquid penetration during evaporation, even for a minute microdroplet (drop diameter: < 0.37 mm). Furthermore, previous studies [18, 45-47] also reported that distinctive evaporation modes appeared during the wetting transition of water drops on SHS due to the interfaces of the solid surfaces, which is not thoroughly understood yet. Therefore, the current work also investigates the influence of the micropillar dimensions on the evaporation dynamics.

In addition, diamond-like carbon (DLC) was introduced as a novel substrate material to fabricate SHS owing to its excellent wear resistance, low friction coefficient, high hardness, and chemical inertness [48]. DLC has the potential for designing mechanically robust SHS for practical applications [49]. The accuracy of the existing theoretical models to predict the critical transition parameters on new surfaces has also been discussed. Finally, a numerical model was proposed to predict the penetration of the meniscus of the drop suspended on a unit cell comprising four micropillars based on Young’s CA [50].

Fabrication of DLC-based SHS. Schematics of the surface fabrication steps, and the drop experiment facility are shown in Fig. 3. The surfaces were designated as w3-p9-h3, w3-p9-h6, w3-p15-h3, w3-p15-h6, and w25-p75-h8, according to the design width (w), pitch (p) (center-to-center distance), and height (h) of the micropillars. Further details of the surface fabrication are provided in Supplementary Note 1. The conditions for reactive ion etching (RIE) using O₂ plasma in the DLC layer are summarized in Table S1.

However, after fabrication, the actual micropillar dimensions varied slightly from the design dimensions, as measured by scanning electron microscopy (SEM), as shown in Fig. S1. Detailed SEM images of the micropillar arrays on different surfaces are shown in Figs. S2 and S3. The actual micropillar dimensions of the different surfaces are summarized in Table S2. Furthermore, the surfaces were characterized in terms of the static CA, roll-off angle, advancing and receding angles, and CA hysteresis. The measured values are listed in Table S3. All details regarding the characterization of the fabricated surfaces are described in Supplementary Note 2.

Experimentation of wetting transition. A state-of-the-art experimental facility (Fig. 4) was assembled at the University of Tokyo to perform wetting transition experiments on DLC-based surfaces. The facility employed the latest sCMOS camera (Zyla 5.5, Andor-Oxford Instruments) to obtain the lateral view of the drop, and a high-resolution microscope (Olympus DSX 1000) to obtain the top view of the drop during evaporation. The lateral and top views of the evaporating drops were recorded at 10 and 1 fps, respectively. A real view of the experimental facility is shown in Fig. S4 in Supplementary Note 3.

First, a small droplet was generated using a syringe pump (Harvard Apparatus) and needle (Hamilton REF. 90031) connected via a Teflon tube, then gently placed on the surface. Then, the drop was left to evaporate under room condition (20–40% relative humidity and 20–25 ºC, measured by a humidity sensor (Rotronic HC2A-S)), and images were recorded from the side and top. An in-house image-processing code was developed in MATLAB to obtain the CA, contact length (CL) of the drop bottom with the surface, and drop diameter from the recorded data.

Wetting transition process. First, we elucidated the process of wetting transition on the surface w3-p9-h3 (baseline) through the recorded snapshots of the evaporating drop from the top and side, as shown in Fig. 5.

At the beginning of the wetting experiment, the drop appeared circular from the top, maintained a quasi-spherical shape, and exhibited a small contact length (CL) between the drop and substrate (Figs. 5a and f). As time progressed, the drop became smaller owing to evaporation, the TPCL alternately pinned and depinned with the micropillars giving rise to a moving TPCL (Figs. 5b-c and g-h). During pinning and depinning, the Laplace pressure at the air–water interface continued to increase, and a metastable state appeared (i.e., liquid partly penetrated into the micropillars, but no transition took place yet) (Figs. 5c and h). As evaporation continued, the drop became very small and finally reached to the critical state. At this state, the TPCL was permanently pinned to the micropillars and finger-like spikes formed at the drop periphery (Fig. 5d). A magnified view of the finger-like spikes is shown in Fig. S5. The drop also lost its quasi-spherical shape at this stage, the CA decreased significantly, and the Cassie–Baxter to Wenzel transition occurred (Fig. 5i). Figure 5e shows traces of dust particles (trapped under the drop) remaining on the surface after evaporation. Reyssat et al. [17] also reported a similar image of the stain resulted in from the evaporation-driven flow of dust in water. Detailed pictures of the wetted area after complete evaporation of the drop on the five surfaces are shown in Fig. S6. Figure 5j shows a lateral view of the drop a few seconds prior to complete evaporation. Supplementary Movie S1 shows a video of the aforementioned wetting transition process.

Droplet evaporation dynamics. Next, we discuss the evaporation dynamics and appearance of distinct evaporation modes during the transition process on the five different surfaces. As shown in Figs. 6a-e, the droplets were initially in the CB state with a high apparent CA, and the evaporation initiated in a constant contact radius (CCR) mode (i.e., CL remained constant, but CA decreased with time) on all five surfaces. The appearance of the CCR mode at the onset of evaporation was also reported in another study [47]. It was elucidated by the fact that at the beginning of evaporation, the TPCL was strongly pinned, and CL did not change, but CA decreased due to the loss of drop volume during evaporation [46].

Subsequently, a mixed evaporation mode (i.e., both CL and CA decreased with time) appeared on the w3-p9-h3, w3-p9-h6, and w3-p15-h3 surfaces. Following the mixed mode, the transition began with a rapid decrease in CA on the w3-p9-h3 surface. However, on the w3-p9-h6 and w3-p15-h3 surfaces, a constant contact angle (CCA) mode (i.e., CL decreased but CA remained constant with time) appeared for a short duration before the transition started. Note that, during the mixed mode the CA decreased very slowly on the w3-p9-h6 and w3-p15-h3 surfaces than that on the w3-p9-h3 surface. Consequently, the CA remained over 140° on the w3-p9-h6 and w3-p15-h3 surfaces, whereas on the w3-p9-h3 surface the CA fell below 130 °until the transition started.

On the w3-p15-h6 surface, evaporation switched from the initial CCR to the CCA mode (Fig. 6d). Unlike on previous surfaces, here the transition did not occur, and the CA remained close to 150° until the range of the drop diameter was measurable.

Finally, the evaporation on the w25-p75-h8 surface switched from CCR to a distinct stick-slip mode [51] (i.e., the CL decreased in a step-wise manner, and the CA repeatedly increased and decreased until the transition started) owing to the highest magnitude of the depinning force (Table S4) (Fig. 6e). The CA before the start of the transition was ~ 118°. The appearance of above distinct evaporation modes on five surfaces under this study might be related to the different magnitude of depinning force on these surfaces (Table S4), which is further elaborated in section 4.3.

Evaluation of existing wetting transition models. In this section, we discuss the applicability of existing wetting transition models from the literature to predict the critical parameters during transition for the five surfaces covered in this study. For the theoretical critical pressure at transition (${\varDelta p}_{\text{c},\text{t}\text{h}}$), the following equation was employed [35, 52]:

$${\varDelta p}_{\text{c}, \text{t}\text{h}}=-\frac{4 {\sigma }_{\text{l}\text{g} }w \text{cos}{\theta }_{\text{A}}}{{p}^{2}-{w}^{2}}$$

where ${\sigma }_{\text{l}\text{g} }$is the surface tension at the liquid–gas interface of the drop, $w$ is the actual width of the square micropillars measured by SEM (Table S2), ${\theta }_{\text{A}}$ is the advancing CA on a flat surface (~ 115° for Teflon [53]), and $p$ is the actual pitch between micropillars. For the theoretical critical angle (${\theta }_{\text{c},\text{t}\text{h}}$), the following equation was selected, which was derived based on the surface free energy [46]:

$${\theta }_{\text{c},\text{t}\text{h}}={\text{cos}}^{-1}\left(\frac{{f}_{\text{s}}-1}{r-{f}_{\text{s}}}\right)$$

where ${f}_{\text{s}}$ is the solid fraction, and $r$ is the roughness factor (Table S4).

The experimental wetting pressure (${\varDelta p}_{\text{w}, \text{e}\text{x}\text{p}}$) was approximated as the Laplace pressure at the liquid-air interface, according to the following equation [54]:

$${\varDelta p}_{\text{w}, \text{e}\text{x}\text{p}}= \frac{2{\sigma }_{\text{l}\text{g} }}{{R}_{\text{c},\text{e}\text{x}\text{p}}}$$

where ${R}_{\text{c},\text{e}\text{x}\text{p}}$ is the critical drop radius at transition from experiment.

The experimental critical angle (${\theta }_{\text{c},\text{e}\text{x}\text{p}}$) was estimated from the image analysis at the onset of the transition. Table 1 shows a comparison between the theoretical predictions and experimental values of above critical parameters. The experimental wetting pressure and critical angle for the w3-p15-h6 surface were not reported in Table 1 because no transition was observed up to a drop diameter of 0.37 mm. From Table 1 it is evident that, Eq. (1) highly overpredicted the critical pressure for the first two surfaces with smaller pitch (9 µm). The deviation became smaller for the w3-p15-h3 surface as the pitch increased from 9 µm to 15 µm, and minimal for the w25-p75-h8 surface having the largest pitch (75 µm). On the other hand, Eq. (2) underestimated the ${\theta }_{\text{c},\text{e}\text{x}\text{p}}$for the w3-p9-h6 surface and overestimated for all other surfaces. Nevertheless, the deviations were reasonable for the first three surfaces but was largest for the w25-p75-h8 surface. Thus, neither theoretical model could precisely capture the critical transition parameters of this work.

The key reason behind the high discrepancy between ${\varDelta p}_{\text{w}, \text{e}\text{x}\text{p}}$ and ${\varDelta p}_{\text{c}, \text{t}\text{h}}$ for surfaces with small pitch might be as follows: Eq. (1) was originally derived from the vertical force balance between hydraulic pressure and surface tension along the micropillar walls acting on the interpillar floating water column of a drop in equilibrium with pinned TPCL [35]. However, for an evaporating drop on surfaces with small pitch under this study, a moving TPCL appeared with alternate pinning and depinning. During the movement of TPCL the Laplace pressure continued to increase due to evaporation and the liquid partly penetrated into the micropillars which led to a metastable state. Therefore, in experiment transition occurred at a much lower pressure than ${\varDelta p}_{\text{c}, \text{t}\text{h}}$. This situation was not accounted in Eq. (1).

Table 1

Comparison of critical parameters during the transition
Surface	Theoretical		Experimental
Surface	${\varDelta p}_{\text{c},\text{t}\text{h}}$ (Pa)	${\theta }_{\text{c},\text{t}\text{h}}$ (degree)	${\varDelta p}_{\text{w},\text{e}\text{x}\text{p}}$ (Pa)	${\theta }_{\text{c},\text{e}\text{x}\text{p}}$ (degree)
w3-p9-h3	5207.09	131.35	560.31	125.56
w3-p9-h6	3580.72	125.88	417.39	139.96
w3-p15-h3	1777.92	148.30	738.46	139.06
w3-p15-h6	992.08	146.76	-	-
w25-p75-h8	570.57	150.40	782.60	118.03

Evaluation of suspended meniscus penetration. In this section, a finite-element-based open source code Surface Evolver (SE) [55] was used to numerically approximate the penetration of the meniscus of the drop suspended on a unit cell under different Laplace pressures. The governing equations of the SE model are described in Supplementary Note 5. First, the bottom of the drop was defined as a planar liquid surface whose edges were pinned to a unit cell comprising four micropillars (Fig. 7a). The final equilibrium planar shape (Fig. 7b) was achieved by successive refinement of the triangular meshes and iterative energy-minimizing steps. The iterations were repeated until the energy change of the evolved surface was < 10^− 8 J. The numerical penetration (${\varDelta }_{\text{S}\text{E}}$) was defined as the vertical distance from the pillar top to the minimum point of the evolved equilibrium surface owing to the energy minimization (Fig. S8).

The numerical penetration results for the three different surfaces with increasing Laplace pressure during evaporation are shown in Fig. 8a. The w3-p9-h6, w3-p15-h6, and w25-p75-h8 surfaces were selected as representative surfaces with varying interpillar distances and micropillar widths. When the interpillar spacing was the largest (w25-p75-h8), sagging became dominant, and the penetration increased significantly with increasing ${\varDelta P}_{\text{L}\text{a}\text{p}\text{l}\text{a}\text{c}\text{e}}$. On the w25-p75-h8 surface, ${\varDelta }_{\text{S}\text{E}}$ was 8.36 µm at ${\varDelta P}_{\text{L}\text{a}\text{p}\text{l}\text{a}\text{c}\text{e}}$ = 871 Pa, which was slightly larger than the micropillar height (8.23 µm). Hence, it can be inferred that the suspended meniscus touched the substrate bottom at this pressure, and the transition occurred.

An experimental plot of CA vs. ${\varDelta P}_{\text{L}\text{a}\text{p}\text{l}\text{a}\text{c}\text{e}}$ obtained for this surface is shown in Fig. 8b. At ${\varDelta P}_{\text{L}\text{a}\text{p}\text{l}\text{a}\text{c}\text{e}}$ = 871 Pa, the corresponding CA was ~ 104° which was much smaller than the ${\theta }_{\text{c}, \text{e}\text{x}\text{p}}$ of this surface (118.03°). Thus, the developed SE model could successfully predict the penetration of the suspended meniscus in the Wenzel state for a large interpillar distance. Conversely, in case of small interpillar distance (i.e., w3-p9-h6 and w3-p15-h6 surfaces), the increase in ${\varDelta }_{\text{S}\text{E}}$ was insignificant, even at a large ${\varDelta P}_{\text{L}\text{a}\text{p}\text{l}\text{a}\text{c}\text{e}}$. Therefore, it can be said that the transition on these surfaces did not occur by the sagging of the bottom meniscus. Rather, the TPCL depinned and vertically slid along the pillars when ${\theta }_{\mu }$ reached ${\theta }_{\text{A}}$ during evaporation.

Furthermore, the theoretical penetration (${\varDelta }_{\text{t}\text{h}}$) was approximated by the following equation for comparison with the numerical penetration [31]:

$${\varDelta }_{\text{t}\text{h}}=\frac{L}{2}\frac{1-\text{sin}(\pi -{\theta }_{\mu })}{\text{cos}(\pi -{\theta }_{\mu })} \left(=-\frac{\left(1-\text{sin}{\theta }_{\mu }\right)L}{2\text{cos}{\theta }_{\mu }}\right)$$

where $L$ is the interpillar spacing and ${\theta }_{\mu }$ is the CA with the micropillar, which can be approximated as follows for square pillars [31]:

$${\theta }_{\mu }={\text{cos}}^{-1}\left(-{\varDelta P}_{\text{L}\text{a}\text{p}\text{l}\text{a}\text{c}\text{e}}\frac{{p}^{2}-{w}^{2}}{4{w\sigma }_{\text{l}\text{g}}}\right)$$

where $p$ is the pitch between pillars, $w$ is the width of the square pillar, ${\sigma }_{\text{l}\text{g}}$ is the surface tension at the liquid–gas interface, and ${\varDelta P}_{\text{L}\text{a}\text{p}\text{l}\text{a}\text{c}\text{e}}$ is the Laplace pressure at the liquid–air interface. A comparison between the numerical and theoretical penetrations of the above three surfaces at the critical transition pressure is presented in Table 2. The theoretical model overestimated the penetration of the bottom meniscus for all three surfaces.

Table 2

Comparison of numerical and theoretical penetration of the bottom meniscus of the drop
Surface	${\varDelta }_{\text{t}\text{h}}$ (µm)	${\varDelta }_{\text{S}\text{E}}$(µm)
w3-p9-h6	0.082	0.067
w3-p15-h6	1.11	0.53
w25-p75-h8	8.09	7.39

Proposed mechanism for the stability of the Cassie–Baxter state. To elucidate the stability of the CB state, a plot of CA vs. drop diameter ($D$) during evaporation on the five surfaces is shown in Fig. 9. The superiority of the five surfaces in terms of maintaining CA ≥ 140° were as follows: w3-p15-h6 > w3-p15-h3 > w3-p9-h6 > w3-p9-h3 > w25-p75-h8. Here, we used CA = 140° as the reference value for the following reasons: (i) at 140° one can assume the surface close to superhydrophobic, and (ii) CA decreased rapidly after 140° on the w3-p15-h3 and w3-p9-h6 surfaces. The CAs on the w3-p9-h3 (CA < 140° at $D$ = 1.25 mm) and w25-p75-h8 (CA < 140° at $D$ = 1.30 mm) surfaces decreased much faster than those on the other three surfaces. On the other hand, the CAs on the w3-p9-h6 and w3-p15-h3 surfaces remained close to 140° until $D$ reached 0.69 mm and 0.44 mm, respectively. Thus, the latter two surfaces demonstrated a better ability to maintain high CA until the drop size became considerably small. However, on the w3-p15-h6 surface, the CA remained close to 150°, even when $D$ reached 0.37 mm, yet no decreasing trend in CA was observed; thus, this surface outperformed all other surfaces. The CA could not be measured for smaller drops with $D$ < 0.37 mm using the developed image processing code because the image quality deteriorated significantly. Supplementary Movie S2 (lateral view) and Movie S3 (top view) shows a comparison of the evaporation and transition behavior of the fakir drop on the w3-p9-h3 and w3-p15-h6 surfaces.

Another comparison of the diameter of the wetted areas on the five surfaces after the evaporation completed is shown in Fig. S6. From this figure one can see that the diameter of the wetted area on the best performed w3-p15-h6 surface was only 75 µm, which was approximately five times smaller than that of the baseline w3-p9-h3 surface. Thus, for simplicity we assumed that the CA on the w3-p15-h6 surface was maintained above 140° until $D$ reached 75 µm.

The above observations can be explained as follows: the stability of the CB state primarily depends on the depinning ability of the TPCL [36]. The force required for depinning of TPCL becomes significantly smaller with a decrease in the solid–liquid contact area. A small pillar width and larger pitch will result in a small solid fraction (${f}_{\text{s}}$) and shorter normalized TPCL ($\delta$) (i.e., the ratio of perimeter to pitch) [56]. Consequently, the depinning force, ${F}_{\text{D}}$ would be significantly reduced (Fig. S7) [56], and a rapidly moving TPCL would appear which in turn would inhibit the impalement of liquid within micropillars (further explanation in Supplementary Note 6).

Table S4 shows that the ${F}_{\text{D}}$ of the five surfaces under this study decreases in the following order: w25-p75-h8 > w3-p9-h3 > w3-p9-h6 > w3-p15-h3 > w3-p15-h6. This contrasts with the previously mentioned order for maintaining CA ≥ 140°. Thus, it can be inferred that reducing the depinning force was the key to maintaining high CA. Among the five surfaces, w3-p15-h6 had the lowest actual solid fraction (${f}_{\text{s},\text{a}}$ = 1.51%, Table S4), resulting in a very low ${F}_{\text{D}}$ (0.80 mN/m) on this surface. Table S2 shows that the actual micropillar width was the smallest (w = 1.87 µm) for this surface. During fabrication, RIE on the w3-p15-h6 surface was continued for a longer duration to achieve 6 µm pillar height. The top of the micropillars was etched faster than the bottom, which led to the formation of conical micropillars (Fig. S3(d)), resulted in the smallest micropillar width. In addition, the interpillar spacing of the w3-p15-h6 surface was sufficiently large to significantly reduce $\delta$ ($\delta$ = 0.49, Table S4). Owing to this combined effect, a nearly super-slippery surface having a very small ${F}_{\text{D}}$ was obtained. As a result, a high CA of ≥ 150 °could be maintained on the w3-p15-h6 surface even for small micron-sized droplets.

A similar phenomenon was observed on the w3-p9-h6 surface (Fig. S3(b)), where the actual micropillar width also contracted to 2.29 µm from the design width of 3 µm, owing to the prolonged time for RIE (Table S2). However, the pitch of this surface was smaller than the w3-p9-h6 surface, resulted in the third-best performance. Although the actual and design pillar widths on the w3-p15-h3 surface were identical, the solid fraction was smaller than that on the w3-p9-h6 surface because of the larger pitch. As a result, it exhibited the second-best performance in maintaining a high CA. Thus, the current experiments revealed that reducing the solid-liquid contact by selecting an appropriate pitch and designing conical micropillars with small widths would significantly enhance the stability of the CB state. Figure 10 shows the relationship between $\delta$, ${F}_{\text{D}}$, and the minimal drop diameter that could maintain CA ≥ 140° (${D}_{\text{m}\text{i}\text{n}, \text{C}\text{A}\ge {140}^{\text{o}}}$) for the five surfaces. Evidently, as $\delta$ and ${F}_{\text{D}}$ decreased, ${D}_{\text{m}\text{i}\text{n}, \text{C}\text{A}\ge {140}^{\text{o}}}$ also reduced remarkably.

The wetting behavior of a fakir drop on micropillared surfaces with five different geometrical arrangements was investigated. DLC was introduced as a new substrate material and suitable recipes were developed for cleanroom fabrication. The evaporation dynamics and wetting transition behavior of the developed surfaces were experimentally investigated. The effect of micropillar dimensions on the evaporation and transition behavior was studied. The key finding of this study is that a stable CB state can be obtained for micron-sized minute droplets on a superhydrophobic surface with simple micropillars by designing a nearly super-slippery surface with a very low depinning force.

Critical parameters for the Cassie–Baxter to Wenzel transition for five surfaces were estimated theoretically and experimentally, and the accuracy of the theoretical models was discussed. The results showed that the theoretical models could not precisely predict the experimental transition parameters, particularly for surfaces with short interpillar spacings. Furthermore, a numerical model was proposed and validated to predict the penetration of the suspended meniscus of the drop on a unit cell of micropillars. The authors anticipate that the outcomes of this study will be helpful to advance various engineering applications of drop evaporation on SHS.

Acknowledgment

This work was supported by the Mitsubishi Heavy Industries Thermal Systems, Japan. In addition, the authors would like to express their gratitude to the following persons for their support: Dr. Eric Lebrasseur (Project Researcher) and Md. Shamim Sarkar (Doctoral Student) from the Department of Electrical Engineering and Information Systems, and Soumyadeep Paul (Doctoral Student) from the Department of Mechanical Engineering at the University of Tokyo.

Author contributions

All the authors contributed equally.

Competing interests

The authors declare no competing interest.

Data Availability

The image processing code in MATLAB, as well as the script in the surface evolver for the numerical calculation of drop meniscus penetration, is available from the authors for academic purposes upon reasonable request.

Supplementary Information

Details of the fabrication of DLC-based SHS, details of the characterization of DLC-based SHS, details of the wetting transition experiment, parameters to evaluate the drop dynamics on micropillars, governing equations to predict the equilibrium shape of the drop bottom meniscus within the micropillars, and explanation of transition delay due to moving TPCL are available in the supplementary material.

Supplementary movies, top and side views of water drop evaporation and wetting transition process on w3-p9-h3 surface (Movie S1), comparison of side view of water drop evaporation process on w3-p9-h3 and w3-p15-h6 surfaces (Movie S2), and comparison of top view of water drop evaporation process on w3-p9-h3 and w3-p15-h6 surfaces (Movie S3) are also available.

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Influence of micropillar array structures on evaporation and wetting transition of fakir drops: Recipe for superhydrophobicity

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Surface	Theoretical		Experimental
Surface	\({\varDelta p}_{\text{c},\text{t}\text{h}}\) (Pa)	\({\theta }_{\text{c},\text{t}\text{h}}\) (degree)	\({\varDelta p}_{\text{w},\text{e}\text{x}\text{p}}\) (Pa)	\({\theta }_{\text{c},\text{e}\text{x}\text{p}}\) (degree)
w3-p9-h3	5207.09	131.35	560.31	125.56
w3-p9-h6	3580.72	125.88	417.39	139.96
w3-p15-h3	1777.92	148.30	738.46	139.06
w3-p15-h6	992.08	146.76	-	-
w25-p75-h8	570.57	150.40	782.60	118.03