Thermodynamic Analysis of Wetting Transitions on Micro/Nanopillared Superhydrophobic Surfaces

Abstract

The low adhesion of water drops on superhydrophobic surfaces is a prerequisite for their widespread potential industrial applications. The wetting transition between different wetting states significantly influences the dynamic behavior of water drops on solid surfaces. Although some theoretical studies have addressed wetting transitions, the underlying mechanisms by which local micro- and nanostructure parameters on superhydrophobic surfaces affect the wetting transition have not been fully elucidated. This study investigates three-dimensional micropillared and micro/nanopillared superhydrophobic surfaces, deriving thermodynamically the equation for the free energy barrier of wetting transition, which is influenced by the overall roughness of the entire superhydrophobic surface and its local micro/nanostructures. Theoretical calculations are performed to investigate the effects of various micro- and nanostructure parameters on the free energy barrier and wetting transition. Based on the principle of energy minimization and the calculated free energy barrier, the possible wetting states on superhydrophobic surfaces are analyzed and compared with experimental results. This study contributes to the theoretical understanding of wetting transitions and may guide the design of superhydrophobic surfaces for diverse applications.

1. Introduction

Superhydrophobic surfaces have garnered significant interest in materials science due to their excellent water-repellent properties, which arise from the synergistic effects of surface roughness and low surface energy materials [1,2,3,4]. When a water drop is in the Cassie state, characterized by an air layer trapped between the drop and the superhydrophobic surface, it demonstrates low adhesion, with its apparent contact angle θ described by the equation cos θ = fcos θ_Y + f − 1, where f represents the Cassie area fraction and θ_Y is the intrinsic contact angle [5,6]. This phenomenon enables numerous promising applications, including self-cleaning materials [6,7,8], anti-icing technologies [9,10,11], advanced microfluidic devices [12,13], and anti-flashover coatings [14,15]. Conversely, in the Wenzel state, the superhydrophobic surface becomes fully wetted by the water drop, resulting in significantly increased adhesion [6,16]. In this state, the apparent contact angle θ is expressed as cos θ = r_Wcos θ_Y, where r_W denotes the surface roughness [6,16]. While the Wenzel state may be desirable in certain drop manipulation applications to enhance surface adhesion, the Cassie state is generally preferred across most applications due to its facilitation of drop mobility [17,18]. Therefore, from a practical perspective, preventing the wetting transition from the Cassie to Wenzel state (also called C–W transition) on these surfaces, which fundamentally influences the movement behavior of water drops, is a prerequisite for the widespread application of superhydrophobic surfaces [19,20]. In order to develop strategies for maintaining the stability of the Cassie state and realizing the full potential of these surfaces, understanding the relationship between micro/nanocomposite structures and the C–W transitions has emerged as a critical area of study within surface science.

Currently, thermodynamic analysis based on free energy (FE) is a crucial approach for theoretically investigating the wetting properties of superhydrophobic surfaces [18,21]. Tie et al. conducted a comparative study on the wetting behavior of semicircular microstructured surfaces and micro/nanostructured surfaces, providing in-depth insights into the impact of nanostructures on wetting states and contact angle hysteresis [22]. Li et al. explored the driving forces behind contact line motion from the perspective of energy gradients, revealing the C–W transitions induced by water drop evaporation through force analysis [23]. Sui et al. focused on irregular micro-textured surfaces composed of six different pillar-shaped structures, identifying the critical parameters corresponding to C–W transitions [24]. Wu et al. examined the quantitative relationship between structural parameters and wetting equilibrium states on microstructured surfaces composed of fibers and nanohemispheres [25]. Zhang et al. investigated the FE changes associated with contact line motion on parallel-structured surfaces, examining the influence of sharp-edge and round structures on C–W transitions [26].

Despite many theoretical models developed to investigate the C–W transitions of superhydrophobic surfaces, the influence of surface structural parameters on this phenomenon remains a topic of ongoing debate. For instance, some studies have suggested that increasing the pillar height can enhance the free energy barrier (FEB), thereby suppressing the occurrence of C–W transitions [27,28]. In contrast, other theoretical models propose that the critical pressure required to induce C–W transitions in a system is determined by the local microstructural parameters of the superhydrophobic surface rather than the pillar height [19,29]. This discrepancy highlights the need for a more comprehensive understanding of the underlying mechanisms governing C–W transitions on superhydrophobic surfaces. A deeper analysis of how different surface structural parameters affect the wetting state is essential to resolve this debate.

This study presents a thermodynamic framework for understanding the wetting transition on superhydrophobic surfaces, with a focus on the role of the overall roughness of the superhydrophobic surface and its local microstructures. A comparative analysis is performed to investigate the wetting transition of water drops on three-dimensional (3D) superhydrophobic surfaces featuring micropillar structures and hierarchical micro-nanopillar structures. Theoretical analysis is employed to elucidate the influence of various surface structural parameters, including microscale and nanoscale features, on the wetting transition. By exploring the relationship between surface topography and wetting behavior, this study contributes to a deeper understanding of the mechanisms by which surface structural parameters affect the stability of the Cassie state, providing valuable insights for subsequent optimization of surface structures.

2. Materials and Methods

The investigation of the C–W transitions on superhydrophobic surfaces was conducted using 3D textured surfaces, as illustrated in Figure 1. The micropillared surfaces were characterized by three primary geometric parameters: width (a₁), spacing (b₁), and height (h₁). When a water drop is deposited on the surface, its volume can be expressed as

$$V={\displaystyle \frac{\pi r^3}{3{\sin }^3\theta }}\left({\cos }^3\theta -3\cos \theta +2\right)$$

(1)

where r denotes the drop base radius and θ represents the apparent contact angle.

The wetting states of water drops, along with C–W transitions on micropillared surfaces, are illustrated in Figure 2. When the drop is in the Cassie state, as shown in Figure 2a, the air layer beneath the drop facilitates its easy rolling off from the superhydrophobic surface. Following the C–W transition, as depicted in Figure 2b, the liquid–vapor interface progressively descends a distance, denoted as h_x. Once h_x equals the micropillar height, the surfaces become fully wetted by the drop, leading to the emergence of the Wenzel state, as illustrated in Figure 2c. Generally, the system’s FE can be described by the following equation [30,31,32]:

$$F={\gamma }_{\mathrm{SV}}{S}_{\mathrm{SV}}+{\gamma }_{\mathrm{SL}}{S}_{\mathrm{SL}}+{\gamma }_{\mathrm{LV}}{S}_{\mathrm{LV}}$$

(2)

in which the interfacial tensions are denoted by γ_SV, γ_SL, and γ_LV, corresponding to the solid–vapor, solid–liquid, and liquid–vapor interfaces, respectively. The surface areas associated with these interfaces are given by S_SV, S_SL, and S_LV. The FEB associated with the C–W transition is conventionally defined as the difference in the system’s FE [25,27]. The volume of Wenzel drops consists of two parts: the liquid volume above the pillar structure and the liquid volume trapped between micro/nanostructures [16]. Notably, the millimeter-scale drops (several microliters in volume) are at least three orders of magnitude larger than the surface’s micro/nanoscale roughness features. As a result, the liquid volume trapped between the microstructures after the C–W transition becomes negligible, resulting in an approximately constant surface area of the water drop above the solid surface during the downward movement of the liquid–vapor interface. Using Young’s equation [33,34], for micropillared surfaces, the FEB for the C–W transition during the liquid–vapor interface descent of distance, specifically the drop penetration depth (h_x), can be expressed as:

$$F_{\mathrm{barr}}^{\mathrm{micro}}=-{\gamma }_{\mathrm{LV}}{\displaystyle \frac{4\pi r^2{f}_1{h}_x}{a_1}}\cos {\theta }_{\mathrm{Y}}$$

(3)

where f₁ represents the area fraction of micropillars, defined as

$$f_1={\displaystyle \frac{a_1^2}{{\left(a_1+b_1\right)}^2}}$$

(4)

When a Cassie drop is deposited on superhydrophobic surfaces, its apparent contact angle can be calculated using the Cassie equation [5,35,36,37]. Typically, in characterizing the wetting properties of superhydrophobic surfaces, the water drop volume is typically known. In such cases, the base radius r can be determined by employing Equation (1) in conjunction with the calculated apparent contact angle of the water drop. When h_x just equals the micropillar height, the FEB reaches its maximum. Upon contact between the liquid–vapor interface and the bottom of the micropillar, the original liquid–vapor interface disappears, leading to the emergence of the Wenzel state [16]. The FE difference between the Wenzel and Cassie states can be expressed as

$$F_{\mathrm{W}-\mathrm{C}}^{\mathrm{micro}}=-{\gamma }_{\mathrm{LV}}{\displaystyle \frac{4\pi r^2{f}_1{h}_1}{a_1}}\cos {\theta }_{\mathrm{Y}}-{\gamma }_{\mathrm{LV}}\pi r^2\left(1-f_1\right)\left(\cos {\theta }_{\mathrm{Y}}+1\right)$$

(5)

Considering that some previous studies have suggested that the C–W transition is primarily dependent on the local microstructural parameters of the superhydrophobic surface [19,29], this study introduces the local FEB to elucidate the underlying physical mechanisms. The local FEB is defined as the FE difference per unit distance of the liquid–vapor interface movement, which can be expressed as

$$F_{\mathrm{barr},\mathrm{Loc}}^{\mathrm{micro}}=\underset{\Delta h_x\to 0}{\lim }{\displaystyle \frac{\Delta F_{\mathrm{barr}}^{\mathrm{micro}}}{\Delta h_x}}=-{\gamma }_{\mathrm{LV}}{\displaystyle \frac{4\pi r^2{f}_1}{a_1}}\cos {\theta }_{\mathrm{Y}}$$

(6)

Equation (6) demonstrates that the local FEB can be interpreted as an average value of the FEB generated by the overall rough structure of the surface, projected along the micropillar height direction. This averaging effect provides insight into how the local surface structure contributes to the macroscopic C–W transition observed on superhydrophobic surfaces.

For micro/nanopillared surfaces, nanopillars are distributed on both the sidewalls and top surfaces of the micropillars (Figure 3a). The nanostructure parameters, including width (a₂), spacing (b₂), and height (h₂), are defined in Figure 3b. For micro/nanopillared surfaces, when the wetting transition occurs, the liquid–vapor interface needs to wet both the micropillars and nanopillars simultaneously. The distances of the liquid–vapor interface movement along the micropillar and nanopillar height direction are denoted by h_x,m and h_x,n, respectively, as shown in Figure 4. Therefore, the FEB for the C–W transition can be expressed as

$$\begin{array}{ll}{F}_{\mathrm{barr}}^{\mathrm{micro}/\mathrm{nano}}& =-{\gamma }_{\mathrm{LV}}{\displaystyle \frac{4\pi r^2{f}_1{f}_2{h}_{x,\mathrm{n}}}{a_2}}\cos {\theta }_{\mathrm{Y}}-{\gamma }_{\mathrm{LV}}{\displaystyle \frac{4\pi r^2{f}_1{f}_2{h}_{x,\mathrm{m}}}{a_1}}\left(1+{\displaystyle \frac{4h_{x,\mathrm{n}}}{a_2}}\right)\cos {\theta }_{\mathrm{Y}}\\ & +{\gamma }_{\mathrm{LV}}{\displaystyle \frac{4\pi r^2{f}_1\left(1-f_2\right)h_{x,\mathrm{m}}}{a_1}}\end{array}$$

(7)

where f₂ is the area fraction of nanopillars, defined as

$$f_2={\displaystyle \frac{a_2^2}{{\left(a_2+b_2\right)}^2}}$$

(8)

When the liquid–vapor interface touches the bottom of the micropillar and nanopillar structure, forming the Wenzel state, the difference in the system’s FE can be expressed as

$$\begin{array}{ll}{F}_{\mathrm{W}\text{-}\mathrm{C}}^{\mathrm{micro}/\mathrm{nano}}& =-{\gamma }_{\mathrm{LV}}{\displaystyle \frac{4\pi r^2{f}_1{f}_2{h}_2}{a_2}}\cos {\theta }_{\mathrm{Y}}-{\gamma }_{\mathrm{LV}}\pi r^2{f}_1\left(1-f_2\right)\left(\cos {\theta }_{\mathrm{Y}}+1\right)\\ & -{\gamma }_{\mathrm{LV}}{\displaystyle \frac{4\pi r^2{f}_1{h}_1}{a_1}}\left(1+{\displaystyle \frac{4a_2{h}_2}{{\left(a_2+b_2\right)}^2}}\right)\cos {\theta }_{\mathrm{Y}}-{\gamma }_{\mathrm{LV}}\pi r^2\left(1-f_1\right)\left(\cos {\theta }_{\mathrm{Y}}+1\right)\end{array}$$

(9)

Clearly, when the liquid–vapor interface moves downward along the micropillars, the nanostructure can produce the maximum FEB when h_x,n = h₂. In this case, the maximum FEB and the maximum local FEB produced by the surface’s local micro/nanostructure can be derived as

$$\begin{array}{ll}{F}_{\mathrm{barr},\max }^{\mathrm{micro}/\mathrm{nano}}& =-{\gamma }_{\mathrm{LV}}{\displaystyle \frac{4\pi r^2{f}_1{f}_2{h}_2}{a_2}}\cos {\theta }_{\mathrm{Y}}-{\gamma }_{\mathrm{LV}}{\displaystyle \frac{4\pi r^2{f}_1{f}_2{h}_{x,\mathrm{m}}}{a_1}}\left(1+{\displaystyle \frac{4h_2}{a_2}}\right)\cos {\theta }_{\mathrm{Y}}\\ & +{\gamma }_{\mathrm{LV}}{\displaystyle \frac{4\pi r^2{f}_1\left(1-f_2\right)h_{x,\mathrm{m}}}{a_1}}\end{array}$$

(10)

and

$$F_{\mathrm{barr},\mathrm{Loc},\max }^{\mathrm{micro}/\mathrm{nano}}=\underset{\Delta h_{x,\mathrm{m}}\to 0}{\lim }{\displaystyle \frac{\Delta F_{\mathrm{barr}}^{\mathrm{micro}/\mathrm{nano}}}{\Delta h_{x,\mathrm{m}}}}=-{\gamma }_{\mathrm{LV}}{\displaystyle \frac{4\pi r^2{f}_1{f}_2}{a_1}}\left(1+{\displaystyle \frac{4h_2}{a_2}}\right)\cos {\theta }_{\mathrm{Y}}+{\gamma }_{\mathrm{LV}}{\displaystyle \frac{4\pi r^2{f}_1\left(1-f_2\right)}{a_1}}$$

(11)

3. Results and Discussion

3.1. Three Typical Cases of Wetting Transition

The wettability of superhydrophobic surfaces is significantly influenced by variations in their microstructural parameters. For micropillared surfaces, the analysis of contact angle can be conducted using the classical Cassie and Wenzel equations [5,16]. The Cassie area fraction is expressed by Equation (4), while the surface roughness can be represented as 1 + 4a₁h₁/(a₁ + b₁)². The surface hydrophobicity can be comprehensively analyzed by incorporating these parameters into the Cassie and Wenzel equations [5,16]. Furthermore, the wetting transition can be theoretically investigated through FEB analysis, and three typical cases of wetting transition on superhydrophobic surfaces are identified, as illustrated in Figure 5. Taking a micropillared superhydrophobic surface as an example, a detailed explanation is provided below. Consider a scenario where a water drop with a volume of 10 μL is carefully deposited on the material surface. As previously discussed, the FEB arises from the difference in the system’s FE [25,27]. For the first case, as shown in Figure 5a, the FEB increases significantly as the liquid–vapor interface moves downward. When h_x just equals the micropillar height h₁, the FEB reaches its maximum. After the liquid–vapor interface reaches the bottom of the micropillars, forming a Wenzel state, the contact between the water drop and the solid surface leads to a negative difference in FE (−41.8 nJ), indicating that the Wenzel state is more favorable for minimizing the system’s FE. According to the principle of minimum energy, the surface should be completely wetted by the liquid drop. However, considering that the positive FEB can separate the Cassie and Wenzel states during the downward movement of the liquid–vapor interface, the Cassie state could also appear. If the external environment provides sufficient energy input, such as mechanical vibration or drop impact, to overcome the FEB, the Wenzel state can emerge [19,29]. Conversely, the positive FEB can lead to the formation of a metastable Cassie state. Thus, the final wetting state depends on the influence of external factors [19,29,38]. The second case, as shown in Figure 5b, also exhibits an increasing FEB as the liquid–vapor interface moves downward. However, unlike the first case, the difference in FE between the Wenzel and Cassie states is 38.1 nJ when the liquid–vapor interface reaches the bottom of the micropillars. This indicates that the positive FEB separates the Cassie and Wenzel states, and the Cassie state is stable. The third case, as shown in Figure 5c, features a negative FEB regardless of the position of the liquid–vapor interface on the micropillars, indicating that the Cassie state is unstable and the system can eventually transition to the Wenzel state.

3.2. Influence of Micropillar Structural Parameters on C–W Transitions

The physical structural parameters can significantly impact the wetting state of superhydrophobic surfaces. The effect of micropillar height on the wetting state is shown in Figure 6. The drop volume is set to 10 μL. It can be observed that as the micropillar height increases from 10 μm to 40 μm, although the FEB increases continuously with the increasing drop penetration depth, the FEB curves overlap, and the local FEB remains constant at 0.0033 J/m. Theoretically, increasing the micropillar height should lead to an increase in the FEB [24,27], which in turn should help suppress the appearance of the Wenzel state. However, previous studies on the de-pinning mechanism of the C–W transition have shown that external factors, such as vibrations, can exert external pressure on the water drop, causing the C–W transition to occur when the pressure on the liquid–vapor interface exceeds the critical pressure [19,29,39,40]. The critical pressure depends only on the local microstructure of the material surface surrounding the liquid–vapor interface, not on the micropillar height [19,29]. Consequently, utilizing the FEB generated by the overall rough structure of the micropillared surface to elucidate the de-pinning mechanism of the C–W transition presents inherent limitations. In contrast, the local FEB generated by the surface’s local microstructure remains a constant value, and its value is independent of micropillar height. Thus, the constant nature of the local FEB renders it more pertinent to providing a thermodynamic explanation for the previously proposed de-pinning mechanism of the C–W transition [19,29,39,40].

Although the micropillar height does not affect the local FEB, increasing the micropillar height can enhance the stability of the Cassie state. As shown in Figure 6, when the micropillar height is 10 μm and 20 μm, the difference in FE between the Wenzel and Cassie states is −61.8 nJ and −28.4 nJ, respectively. However, when the micropillar height increases to 30 μm and 40 μm, the difference in FE becomes positive. This observation indicates that increasing the micropillar height leads to an increase in the system’s FE associated with the Wenzel state, rendering it higher than that of the Cassie state. As a result, the Cassie state can transition from a metastable to a stable regime with the increasing micropillar height. In this case, even if the wetting transition occurs, the system can easily revert from the Wenzel state to the Cassie state under external forces, such as vibrations [41,42]. Although the micropillar height may not help suppress the wetting transition caused by the de-pinning mechanism, increasing the micropillar height can help maintain the energy minimization of the Cassie state. Consequently, when analyzing the wetting state of superhydrophobic surfaces, it is necessary to consider both the difference in the system’s FE, FEB, and the local FEB.

The effect of the micropillar width/spacing ratio (a₁/b₁) on the FEB, local FEB, and apparent contact angle is shown in Figure 7. The micropillar spacing is fixed at 20 μm, while the micropillar width varies between 40 μm and 1 μm. The drop volume is set to 10 μL. As the ratio a₁/b₁ decreases from 2 to 0.05, the maximum FEB (corresponding to the FEB at a penetration depth h_x of 20 μm) decreases from 40.0 nJ to 0.05 nJ, as shown in Figure 7a. Concurrently, the local FEB decreases from 0.002 J/m to 2.4 × 10⁻⁶ J/m, accompanied by an increase in the apparent contact angle from 129.3° to 176.5°, as depicted in Figure 7b. From the perspective of superhydrophobic surface fabrication, it is generally desirable to reduce the Cassie area fraction and enhance the apparent contact angle. Although reducing a₁/b₁ can help reduce the Cassie area fraction, it simultaneously reduces the FEB of the wetting transition, as shown in Figure 7. Generally, external environmental factors can also impact the wetting state of superhydrophobic surfaces [19]. If the environment provides energy input to the system, such as through drop impact, wetting transitions may occur even when the FEB is positive [19]. Thus, when designing superhydrophobic surfaces, one should not rely solely on the FEB but rather adjust structural parameters according to the specific application requirements. For instance, if an extremely high contact angle is not necessary, increasing the ratio a₁/b₁ can enhance the FEB, thus stabilizing the Cassie state. Conversely, if an extremely high contact angle is required, reducing this ratio while increasing pillar height can help maintain Cassie state stability. Therefore, in the design and fabrication of superhydrophobic surfaces, it is crucial to optimize the ratio a₁/b₁ within an appropriate range. This optimization aims to achieve a delicate balance between superhydrophobicity and the stability of the Cassie state, ensuring both the desired wetting properties and resistance to C–W transitions.

3.3. Analysis of FEB on Micro/Nanopillared Superhydrophobic Surfaces

Experimental results of Surmeneva et al. demonstrated that surface wettability undergoes significant modifications when nanoscale roughness is hierarchically distributed over microscale structures [43]. Thus, when nanopillar structures are distributed on the surface of micropillar structures, the FEB of the C–W transition undergoes significant changes, as shown in Figure 8. The drop volume is set to 10 μL. For both micropillared and micro/nanopillared surfaces, a positive FEB exists to separate the Cassie and Wenzel states as the penetration depth increases. The local FEB for the micro/nanopillared surface is 0.0047 J/m, which is significantly higher than the 7.1 × 10⁻⁴ J/m for the micropillared surface. However, after the solid surface is completely wetted by the liquid, the FE difference between the Wenzel and Cassie states is −41.8 nJ for the micropillared surface, whereas it is 14.5 nJ for the micro/nanopillared surface. This indicates that the presence of nanostructures, without changing the microstructure parameters, can not only increase the FEB to suppress the wetting transition but also help maintain the Cassie state at the minimum FE state, thereby enhancing its stability. This may be one reason why researchers strive to fabricate micro/nanohierarchical superhydrophobic surfaces during material preparation [1,2,3,4].

The effect of nanopillar height on the maximum FEB and maximum local FEB (calculated via Equations (10) and (11)) is shown in Figure 9. The drop volume is set to 10 μL. As the liquid–vapor interface moves downward along the micropillar height (i.e., as h_x,m gradually increases), a positive FEB can appear. Compared to the micropillar height, nanopillar height has a significant impact on the FEB. On the one hand, as shown in Figure 9a,b, increasing the nanopillar height can increase both the maximum FEB and the maximum local FEB. On the other hand, increasing the nanopillar height can also help maintain the Cassie state at the minimum energy state (Figure 9a). When the nanopillar height is 5 nm and 10 nm, the FE difference between the Wenzel and Cassie states is −16.6 nJ and −11.0 nJ, respectively. However, when the nanopillar height increases to 20 nm and 30 nm, the FE difference between the Wenzel and Cassie states increases to 0.3 nJ and 11.5 nJ, respectively. This indicates that increasing the nanopillar height helps maintain the stability of the Cassie state. Therefore, during material preparation, it is desirable to increase the nanopillar height as much as possible.

The effect of the nanopillar width/spacing ratio (a₂/b₂) on the maximum FEB, local maximum FEB, and apparent contact angle is shown in Figure 10. The nanopillar spacing is fixed at 30 nm, while the nanopillar width varies between 30 μm and 1 μm to control the ratio a₂/b₂. The drop volume is set to 10 μL. Similar to the micropillared surface, as the ratio a₂/b₂ decreases, the FEB (Figure 10a) and local FEB (Figure 10b) decrease while the apparent contact angle increases. This indicates that for nanostructured surfaces, it is also necessary to set the ratio a₂/b₂ within an appropriate range to ensure that the superhydrophobic surface has a high contact angle while maintaining the stability of the Cassie state.

3.4. Experimental Validation

To validate the theoretical analysis,

Table 1. Comparison of experimental results from Yeh et al. [44] with theoretical predictions in this study.

Different Cases	Micropillar Spacing (μm)	Micropillar Height (μm)	Micropillar Width (μm)	Observed Wetting State	Theoretical Results
					Local FEB (J/m)	FE Difference Between the Wenzel and Cassie States (nJ)	Calculated Wetting State
Case 1	3	1	6	Wenzel	0.0243	−21.3	Wenzel
	3	1.41	6	Cassie	0.0243	−11.3	Wenzel
	3	3.05	6	Cassie	0.0243	28.6	Cassie
Case 2	6	1.5	6	Wenzel	0.008	−24.0	Wenzel
	6	2.76	6	Cassie	0.008	−13.9	Wenzel
	6	5.02	6	Cassie	0.008	4.2	Cassie

compares the experimental results of C–W transition on micropillared surfaces reported by Yeh et al. [44] with the theoretical analysis results of this study. The theoretical wetting states in Table 1 are determined based on the minimum energy principle. As illustrated in Table 1, for Case 1, when the micropillar height is 1 μm and 3.05 μm, both the experimentally observed and theoretically predicted wetting states are in complete agreement. However, at a micropillar height of 1.41 μm, there is a discrepancy between the theoretical and experimental results. Similarly, for Case 2, the experimental and theoretical results align well, except when the micropillar height is 2.76 μm. The observed deviation can be attributed to the combined effects of local FEBs and external environmental factors. As previously discussed, in Case 1, with a micropillar height of 1.41 μm, although the minimum energy principle indicates that the wetting system should be in the Wenzel state, a positive local FEB (0.0243 J/m) can cause the Cassie state to be metastable when the external environmental influence is minimal, such as when the water drop is placed gently on the material surface. This may lead to the possibility of observing the Cassie state in experiments. Similarly, for Case 2 at a micropillar height of 2.76 μm, the presence of a positive local FEB can also result in the observation of the metastable Cassie state. Therefore, as discussed in Figure 5, accurately determining the wetting state of a solid–liquid–vapor system requires considering both the FEB, the FE difference between the Wenzel and Cassie states, and the influence of external environmental factors.

In this study, FEB analysis was employed to investigate wetting transitions on micro/nanopillared superhydrophobic surfaces. While the current theoretical analysis focused on flat superhydrophobic surfaces, many practical applications could involve curved surfaces in 3D space. Consequently, investigating wetting transitions on curved superhydrophobic surfaces is a crucial direction for future research. Additionally, considering that water drops can be in the Cassie state on porous surfaces [18,19], studies of wetting transitions on such surfaces are essential for advancing a comprehensive understanding of wetting phenomena.

4. Conclusions

In this study, the C–W transition on superhydrophobic surfaces with micro/nanopillar structures was theoretically analyzed. The FEBs generated by the overall rough structure and local surface micro/nanostructure were thermodynamically modeled and calculated. The theoretical results demonstrate that for micropillared surfaces, although increasing the micropillar height does not enhance the local FEB, it helps to minimize the system FE corresponding to the Cassie state, thereby maintaining the stability of the Cassie state. For the micropillar width and spacing, it is necessary to set the micropillar width/spacing ratio to an appropriate value, which not only maintains a high contact angle on the superhydrophobic surface but also ensures the stability of the Cassie state. For micro/nanopillared surfaces, it is also necessary to set the nanopillar width/spacing ratio to an appropriate value. Increasing the nanopillar height can not only enhance the FEB of the C–W transition but also help transform the Cassie state from a metastable state to a stable state. When predicting the wetting state of superhydrophobic surfaces, it is necessary to apply the minimum energy principle and combine the FEB analysis with the influence of external environmental factors. This study can provide helpful insights for designing superhydrophobic surface structures to achieve stable Cassie states.