Effects of Two-Level Surface Roughness on Superhydrophobicity

Abstract

Biomimetic superhydrophobic surfaces have become a focal point of recent research, driven by their promise in diverse applications. Among these, the lotus and rose effects are of particular interest due to their contrasting adhesion characteristics. Given that superhydrophobicity is closely related to the hierarchical structures of these surfaces, investigating the effects of two-level roughness on superhydrophobicity is crucial. In our previous work, we introduced a wetting parameter (W_Roughness), strongly correlated with the geometric characteristics of surface roughness, to elucidate the superhydrophobic behavior of solid surfaces. This parameter predicts the existence of a critical wetting parameter (W_Roughness,c) during the Wenzel–Cassie transition. For two-level surface roughness composed of primary and secondary roughness, the W_Roughness of the two-level surface is influenced by the geometric characteristics of both primary and secondary roughness. Furthermore, when secondary roughness is added to a primary roughness surface in the Wenzel state, the resulting two-level roughness can exhibit various superhydrophobic states, such as the Wenzel state, Wenzel–Cassie transition, or Cassie state, depending on the characteristics of the secondary roughness. To further investigate the influence of two-level roughness on superhydrophobicity, molecular dynamics (MD) simulations were also conducted.

1. Introduction

Wettability, a key characteristic of solid surfaces, is measured by the contact angle (CA, θ) [1], which is related to interfacial energies through Young’s Equation (1):

$$\cos \theta ={\displaystyle \frac{{\gamma }_{\mathrm{S}\mathrm{G}}-{\gamma }_{\mathrm{S}\mathrm{L}}}{{\gamma }_{\mathrm{L}\mathrm{G}}}}$$

(1)

In this equation, γ_SL, γ_SG and γ_LG denote the interfacial energies associated with the solid–liquid, solid–gas (or solid–air), and liquid–gas (or liquid–air) interfaces, respectively. Superhydrophobic surfaces, characterized by CA exceeding 150° and contact angle hysteresis (CAH) less than 10°, have become a focus of intense investigation. This heightened attention stems from their promising utility across a diverse range of applications. These include, but are not limited to, self-cleaning coatings [2,3], corrosion resistance [4,5], ice prevention [6,7], anti-flashover treatments [8,9], controlled micro-droplet movement [10], enhanced vapor condensation and collection [11], and selective oil/water separation using membranes and meshes [12,13,14,15].

The hydrophobicity of a solid surface is understood to be influenced by both its chemical makeup and the geometrical attributes of its surface texture. The effects of surface roughness on the apparent CA (θ*) of liquid droplets are commonly explained using the Wenzel model [16,17] and the Cassie–Baxter model [18]. Additionally, experimental evidence suggests that liquid droplets in the Wenzel state exhibit stronger adhesion to structured surfaces compared to those in the Cassie state [19,20]. Consequently, Cassie state is often favored for various practical applications.

Biomimetics utilizes designs, adaptations, or derivations inspired by biological systems [21]. Superhydrophobic surfaces are observed in various natural examples, including lotus leaves, rose petals, the legs of water striders, and butterfly wings [22,23,24,25,26]. Of these, the surfaces of lotus leaves and rose petals have been particularly well-studied [27,28,29,30]. While both possess high water CAs, they exhibit significantly different water adhesion characteristics. Lotus leaves display superhydrophobicity with low adhesion, allowing nearly spherical water droplets to form with contact angles exceeding 150° and to readily roll off [31]. This behavior, termed the Lotus Effect, is attributed to the dual-scale surface structure in conjunction with the waxy epicuticular layer. Rose petals also exhibit superhydrophobicity, achieving contact angles as high as 152.4°; however, water droplets remain pinned to the surface and do not roll off, a phenomenon known as the Petal Effect [32,33]. Therefore, understanding how hierarchical surface roughness affects superhydrophobicity is a key area of investigation.

The influence of hierarchical structures on superhydrophobicity has been extensively investigated through both experimental and theoretical approaches [34,35,36,37,38,39,40,41]. These investigations generally indicate that surfaces with hierarchical structuring can substantially reduce surface wettability, resulting in significantly increased CAs [37]. Compared to surfaces lacking hierarchical features, these structured surfaces exhibit enhanced hydrophobicity, thereby promoting the formation of droplets in the Cassie state [38,39]. Consequently, the presence of dual-scale roughness in hierarchical surfaces aids in preventing the transition from the Cassie state to the Wenzel state [40,41]. Furthermore, hierarchical structures can lead to the observation of superhydrophobic states beyond the traditional Wenzel and Cassie regimes. For instance, Wang and Jiang [42] proposed a classification of five superhydrophobic states, encompassing the Wenzel state, the Cassie state, the “Lotus” and “Gecko” states, as well as a transitional state between the Wenzel and Cassie configurations. Within this framework, the “Lotus effect” is equivalent to the Cassie–Baxter state, representing a droplet suspended on air pockets within the rough surface, while the “Petal effect” corresponds to the Cassie impregnating state, an intermediate condition between the Cassie–Baxter and Wenzel states. Given that varying surface morphologies result in differing wetting behaviors, the transition of liquid adhesion from the “Rose Petal Effect” to the “Lotus Effect” on a given surface is likely closely linked to the impact of hierarchical structures on superhydrophobicity. Therefore, for surfaces exhibiting two-level hierarchical roughness, it is essential to examine the individual contributions of the primary and secondary roughness features on the overall superhydrophobic characteristics.

In our recent work [43], a wetting parameter (W_Roughness) is proposed, which is directly related to the geometric attributes of surface texture, and utilized it to elucidate the Wenzel–Cassie wetting transition. During the transition from the Wenzel state to the Cassie state, the existence of a critical wetting parameter (W_Roughness,c) is expected (Figure 1). Specifically, the Cassie state is predicted to occur when W_Roughness is below this critical value (W_Roughness < W_Roughness,c).

This work aimed to investigate the influence of two-level roughness on superhydrophobicity. For surfaces exhibiting two-level roughness, composed of primary and secondary roughness features, the overall W_Roughness value is determined by the geometric properties of both the primary and secondary structures. Moreover, we propose that introducing secondary roughness to a primary roughness already in the Wenzel state can induce various superhydrophobic states, including the Wenzel–Cassie transition or a fully established Cassie state, depending on the specific characteristics of the secondary roughness. To further investigate the impact of two-level structures on superhydrophobicity, molecular dynamics (MD) simulations were also conducted.

2. Methods

2.1. MD Simulations

MD simulations were performed using GROMACS 2019 [44,45] to investigate the effect of two-level surface roughness on superhydrophobicity. The interactions between carbon atoms in the graphite material were modeled using the Optimized Potential for Liquid Simulations-All Atom (OPLS-AA) force field. Several water models are available for simulating water molecules; in these simulations, we chose to represent water molecules using the 4-site transferable intermolecular potential (TIP4P) model.

The Lennard-Jones potential was used to calculate van der Waals forces, which is mathematically defined as:

$$U_{LJ\left(r_{ij}\right)}=4{\epsilon }_{ij}\left[{\left({\displaystyle \frac{{\sigma }_{ij}}{r_{ij}}}\right)}^{12}-{\left({\displaystyle \frac{{\sigma }_{ij}}{r_{ij}}}\right)}^6\right]$$

(2)

In this expression, ε_ij represents the characteristic energy, σ_ij the characteristic length, and r_ij the interatomic distance between atoms i and j. Within the framework of the OPLS force field, these parameters are derived from atomic values using the following geometric averaging rules:

$${\sigma }_{ij}=\sqrt{{\sigma }_{ii}{\sigma }_{jj}}$$

(3a)

$${\epsilon }_{ij}=\sqrt{{\epsilon }_{ii}{\epsilon }_{jj}}$$

(3b)

During the simulations, both primary and secondary square pillars are used to represent the two-level surface roughness. This roughness is characterized by several parameters, including the pillar side lengths along the x and y axes (a_x, a_y), the groove widths separating the pillars along the x and y axes (w_x, w_y), and the pillar height (h) (as depicted in Figure 2). In the present study, we investigated the influence of two-level roughness on superhydrophobicity by systematically varying the geometric properties of either the primary or secondary surface roughness features, such as their side lengths, separations, or pillar heights.

In the MD simulations, we initially positioned a cubic water box, measuring 7.0 nm × 7.0 nm × 7.0 nm, above the graphite substrate, which was configured with varying degrees of surface roughness. Upon reaching thermodynamic equilibrium, the system was expected to transition to either the Wenzel state, characterized by wetting of the surface texture, or the Cassie state, where the water remained perched on the surface asperities. Furthermore, recognizing that superhydrophobicity may be influenced by the dimensions of the water droplet, we performed additional simulations employing cubic droplets with dimensions of 4.0 nm × 4.0 nm × 4.0 nm, 5.0 nm × 5.0 nm × 5.0 nm, and 6.0 nm × 6.0 nm × 6.0 nm. These simulations enabled us to explore the dependence of superhydrophobicity on droplet size.

The simulations were conducted within a simulation box measuring 21.0 nm × 21.0 nm × 20.0 nm, with the graphite substrate constrained to a fixed position. The simulations were performed using a canonical ensemble (NVT), and a Nosé–Hoover thermostat was used to carefully maintain a constant temperature of 300 K. Periodic boundary conditions were imposed in all three spatial dimensions. Lennard-Jones interactions were truncated at a cutoff distance of 1.0 nm, and the particle mesh Ewald (PME) method was employed for calculating long-range electrostatic interactions. Each simulation was run for a duration of 2 ns, with a time step of 2 fs.

2.2. CA Measurement

CAs were determined using the ImageJ software package (version 1.54g) [46] in conjunction with a contact angle plugin. Following the attainment of thermodynamic equilibrium in each simulated system, a near-spherical water droplet with a radius of approximately 4.0 nm was observed. To quantify the CA between the droplet and the substrate, a procedure involving manual selection was employed. This procedure required the identification of two points to define the baseline and three points along the droplet profile. These selected points were then utilized to define the droplet profile, which was subsequently used for CA calculation. In this study, the CA was measured three times for each droplet, resulting in an estimated uncertainty of approximately 3°.

2.3. Density Calculation

Upon deposition of a water droplet onto a solid surface, a solid–water interface is established. This interface can alter the characteristics of the liquid water, leading to deviations from those observed in bulk water. The wetting transition from the Wenzel state to the Cassie state is accompanied by the changes in the solid–water interface. To investigate the influence of surface roughness on the structure of water, water density distributions were calculated based on the MD simulation data. These calculations were performed utilizing the Visual Molecular Dynamics (VMD 1.9.1) software [47].

3. Results

According to this study, the CA between a water droplet and a smooth graphite surface was 66.0° (Figure 3). This value is lower than the CAs traditionally reported for graphite samples (75–95°) [48]. However, it is in agreement with Kozbail et al. [48] for highly ordered pyrolytic graphite (HOPG), who reported a water CA of 64.4 ± 2.9° measured within 10 s after exfoliation in air.

During the simulations, primary and secondary square pillars are used to construct the two-level surface roughness. The geometric characteristics of the squared roughness are characterized by the pillar side lengths in the x and y axes (a_x, a_y), the groove widths between pillars in the x and y axes (w_x, w_y), and the pillar height (h) (Figure 2). To understand the effects of two-level roughness on superhydrophobicity, these characteristics are varied by modifying the geometric characteristics of either the primary or secondary surface roughness, such as side length, separations, or pillar height.

For primary surface roughness, increasing the height of the surface features resulted in an increase in CA from 66.0° to 139.7°. Furthermore, a sharp increase in CAs was observed during the wetting transition from the Wenzel state to the Cassie state (as illustrated in Figure 4 and detailed in

Table 1. The wettability characteristics of primary and secondary surface roughness. The square pillars are designed to simulate the surface roughness of them. The wetting parameter (W_Roughness) and revised parameter (r-W_Roughness) are presented.

System	ax (Å)	ay (Å)	wx (Å)	wy (Å)	h (Å)	Wettability	WRoughness	r-WRoughness	CA
secondary	2.45	12.76	9.82	8.51	13.4	Cassie	1.59	0.67	130.2°
	2.45	12.76	9.82	8.51	10.05	Wenzel	1.80	0.73	114.1°
primary	14.74	12.76	9.84	8.51	13.4	Cassie	0.49	0.37	118.2°
	14.74	12.76	9.84	8.51	10.05	Wenzel	0.56	0.41	109.7°

).

Superhydrophobicity is strongly linked to the geometric properties of surface roughness. Moreover, distinct solid–liquid contact configurations are associated with different superhydrophobic states (Figure 4). In the Wenzel state, the liquid phase entirely fills the spaces between surface roughness features. Conversely, in the Cassie state, the water droplet resides on top of the surface roughness elements, leaving the spaces on the surface unfilled by the liquid. Furthermore, during the Wenzel–Cassie transition, the lower surface of the penetrating liquid undergoes a gradual transformation from a conformal contact (complete penetration) to a final planar configuration (no penetration), indicative of a wetting transition from the Wenzel state to the Cassie state (partial penetration). This intermediate state can serve as a means of characterizing the Wenzel–Cassie wetting transition.

Water is often characterized as an anomalous liquid due to the formation of hydrogen bond. Upon the placement of a water droplet onto a solid surface, a solid–water interface is established. Drawing upon our previous studies [49,50,51] concerning the relationship between OH vibrations and hydrogen bonding, we observed that surface roughness primarily influences the structure of interfacial water (Figure 5). This conclusion is further supported by a range of experimental measurements detailed in the literature [52,53,54,55,56].

Surface roughness exerts a primary influence on the structure of interfacial water. To elucidate the properties of this interfacial water, the density distribution is calculated for water molecules filled in grooves between surface roughness using MD simulations. The results show that the interfacial water layer exhibits a higher density compared to bulk water (Figure 5). This observation aligns with both experimental measurements [57] and theoretical simulations [58] on confined water. In combination with our recent studies [43,51], this phenomenon is attributed to the dominance of single donor-single acceptor (DA) hydrogen bonding in interfacial water.

Because hydrogen bonds are stronger than van der Waals interactions, the liquid water phase plays a significant role in the Wenzel–Cassie transition. To assess the impact of the roughness–water interface on water structure, a parameter, W_Roughness, which is related to the ratio between the surface area and volume of the surface roughness, can be derived. Employing MD simulations, it was found that increasing the height of the surface roughness promotes the transition from the Wenzel state to the Cassie state (Figure 4). During the Wenzel–Cassie transition, this is accompanied by a reduction in W_Roughness, and an increase in CA (Table 1).

For primary surface roughness (a_x = 14.74 Å, a_y = 12.76 Å, w_x = 9.84 Å, w_y = 8.51 Å), the wettability transition from the Wenzel to Cassie states is observed as the roughness height increases from 10.05 Å to 13.4 Å (Figure 4). This is due to the structural transition between interfacial and bulk water. This also suggests the existence of a critical height h_c for the specific roughness. In combination with the above discussion, this can be represented by the critical wetting parameter of roughness, W_Roughness,c (Table 1).

Based on the MD simulations, the W_Roughness,c of primary roughness is determined to fall within the range of 0.49–0.56. This value is slightly larger than the theoretical W_Roughness,c (0.47) at 0.1 MPa and 300 K [43]. From the definition of W_Roughness, the molecular (or atomic) size of surface roughness is neglected. In combination with Figure 4, it is found that, with decreasing the separations between grooves, it is necessary to take into account the effects of atom size of surface roughness on W_Roughness. Following the correction for the van der Waals radius of the substrate, the revised W_Roughness, denoted as r-W_Roughness, falls within the range of 0.37–0.41 for primary roughness. In the MD simulations conducted by Ren et al. [59], employing the TIP4P model for water molecules, the W_Roughness,c parameter was determined to be in the range of 0.65–1.05, and the corresponding r-W_Roughness,c lies in the range of 0.45–0.68 during the wetting transition from the Wenzel to Cassie states. Additionally, in our recent study [43], utilizing the SPC/E water model for water molecules, the W_Roughness,c parameter was calculated to be in the range of 0.76–1.26, and the corresponding r-W_Roughness,c lies in the range of 0.49–0.71 during the Wenzel–Cassie transition.

Based on the preceding discussion, we anticipate that the Cassie state will be favored when W_Roughness is less than W_Roughness,c. This relationship provides a framework for understanding the influence of surface roughness on hydrophobic states. In other words, the Cassie state is expected when the pillar height exceeds the critical height (=a_c, =w_c, >h_c), the separation is less than the critical width (=a_c, <w_c, =h_c), or the size of the pillar exceeds the critical size (>a_c, =w_c, =h_c) [43]. This prediction aligns with the results obtained from our MD simulations. Furthermore, this understanding can also be applied to analyze other MD simulations [59,60] concerning the dependence of superhydrophobicity on the geometric characteristics of surface roughness.

In the simulations, to mimic two-level surface roughness, secondary square pillars were incorporated onto the primary roughness, thus creating a two-level surface roughness. The geometric characteristics of secondary roughness are characterized by the pillar side lengths in the x and y axes (a_x, a_y), the groove widths between pillars in the x and y axes (w_x, w_y), and the pillar height (h) (Figure 4 and Table 1). For the secondary roughness, increasing the height leads to an increase in the CA. Furthermore, the Wenzel–Cassie transition is observed as h increases from 10.05 Å to 13.4 Å. Consequently, during the Wenzel–Cassie transition, the corresponding W_Roughness,c values were determined to fall within the range of 1.59–1.80, while r-W_Roughness,c lies in the range of 0.67–0.73. It can be found that, with decreasing the roughness size, it may be necessary to take into account the effects of the molecular (or atomic) size of surface roughness on W_Roughness,c.

To investigate the effects of two-level surface roughness on superhydrophobicity, secondary roughness is added to various primary roughness surfaces exhibiting either Wenzel or Cassie states. For two-level roughness built on a primary roughness surface in the Wenzel state (h = 6.7 Å), increasing the height of the secondary roughness leads to an increase in the CA and a decrease in the value of W_Roughness (Figure 6 and

Table 2. Two-level surface roughness. During the simulations, primary and secondary square pillars were used to model the two-level surface roughness. The wetting parameter (W_Roughness) and revised W_Roughness (r-W_Roughness) are shown for primary, secondary, and two-level roughness.

System *	Roughness	ax (Å)	ay (Å)	wx (Å)	wy (Å)	h (Å)	Wettability	CA	Wettability	WRoughness	r-WRoughness
5p-4s	Secondary	2.45	12.76	9.82	8.51	13.4	Cassie	129.3°	Cassie	0.58	0.38
	Primary	14.74	12.76	9.84	8.51	16.75	Cassie
5p-3s	Secondary	2.45	12.76	9.82	8.51	10.05	Wenzel	130.5°	Cassie- Wenzel	0.59	0.39
	Primary	14.74	12.76	9.84	8.51	16.75	Cassie
5p-2s	Secondary	2.45	12.76	9.82	8.51	6.7	Wenzel	123.1°	Cassie- Wenzel	0.62	0.41
	Primary	14.74	12.76	9.84	8.51	16.75	Cassie
5p-1s	Secondary	2.45	12.76	9.82	8.51	3.35	Wenzel	121.2°	Cassie- Wenzel	0.66	0.45
	Primary	14.74	12.76	9.84	8.51	16.75	Cassie
2p-4s	Secondary	2.45	12.76	9.82	8.51	13.4	Cassie	131.8°	Cassie	0.69	0.42
	Primary	14.74	12.76	9.84	8.51	6.7	Wenzel
2p-3s	Secondary	2.45	12.76	9.82	8.51	10.05	Wenzel	124.8°	Cassie- Wenzel	0.71	0.44
	Primary	14.74	12.76	9.84	8.51	6.7	Wenzel
2p-2s	Secondary	2.45	12.76	9.82	8.51	6.7	Wenzel	121°	Cassie- Wenzel	0.76	0.47
	Primary	14.74	12.76	9.84	8.51	6.7	Wenzel
2p-1s	Secondary	2.45	12.76	9.82	8.51	3.35	Wenzel	113.8°	Wenzel	0.83	0.53
	Primary	14.74	12.76	9.84	8.51	6.7	Wenzel

* In this work, an np-ms system is used to represent two-level roughness, where primary and secondary roughness are composed of n and m layers of graphene, respectively.

). Furthermore, the addition of secondary roughness can influence the superhydrophobicity of the two-level surface, causing a change from the Wenzel state to the Wenzel–Cassie transition or Cassie state (Figure 6).

Based on the MD simulations of primary and secondary surface roughness, these findings can be applied to investigate the dependence of superhydrophobicity in two-level roughness on both primary and secondary roughness characteristics. When the primary surface roughness is in the Wenzel state, the superhydrophobicity of the two-level surface can be related to either the Wenzel or Cassie state, depending on the characteristics of the secondary roughness. It is observed that the superhydrophobicity of the two-level roughness is influenced by both primary and secondary roughness features (Figure 6 and Table 2).

For two-level roughness built on a primary roughness surface exhibiting the Cassie state (h = 16.75 Å), increasing the height of the secondary roughness can also affect the CA of the two-level structure, which is reflected in the changes of W_Roughness (Figure 7 and Table 2). Furthermore, compared to the primary roughness alone, the addition of secondary roughness can alter the superhydrophobicity of the two-level surface, causing a transition from the Wenzel state to the Wenzel–Cassie transition, or resulting in a Cassie state. The influence of secondary roughness on the superhydrophobicity of the two-level surface is closely related to the characteristics of the primary surface roughness.

Based on MD simulations, when the primary surface roughness exhibits the Cassie state, the superhydrophobicity of the two-level surface is influenced by the superhydrophobicity of the secondary roughness. The final state of the hierarchical surface is expected to be either Wenzel or Cassie, depending on whether the secondary roughness exhibits the Wenzel or Cassie state (Figure 7 and Table 2). However, when both primary and secondary roughness exhibit the Cassie state, the resulting superhydrophobicity of the two-level surface is definitively attributed to the Cassie state (Table 2).

In comparison with the Wenzel state, the Cassie state is often preferred in many practical applications. Therefore, it is important to study the superhydrophobicity of two-level roughness built upon a primary roughness surface in the Wenzel state. The superhydrophobicity of two-level roughness is related to the combined effects of primary and secondary roughness on superhydrophobicity. Based on the above discussion on two-level roughness, adding secondary roughness to primary roughness can decrease W_Roughness. Unlike the Wenzel state for primary roughness alone, other superhydrophobic states, such as the Wenzel–Cassie transition and the Cassie states, can be achieved for two-level roughness (Figure 6). These principles can be utilized to rationally improve the superhydrophobicity of a solid surface.

In our recent study [43], the superhydrophobicity of solid roughness was found to be dependent on the size of the water droplet. To investigate this dependence, water droplets of varying cubic sizes were, respectively, placed on primary and secondary roughness, specifically 40 Å × 40 Å × 40 Å, 50 Å × 50 Å × 50 Å, 60 Å × 60 Å × 60 Å, and 70 Å × 70 Å × 70 Å (Figure 8). It was observed that increasing the size of the water droplet resulted in a wetting transition from the Wenzel to the Cassie state (Figure 8). The Wenzel–Cassie wetting transition was observed when the droplet diameter reached approximately 75 Å for primary roughness or 45 Å for secondary roughness, corresponding to an initial water box of 70 Å × 70 Å × 70 Å or 50 Å × 50 Å × 50 Å. Therefore, the Cassie state is expected where the droplet size significantly exceeds the geometric features of the surface roughness. To maintain the Cassie state, a lower size limit for the water droplet is expected. Given that the size of the secondary roughness is smaller than that of the primary roughness, it follows that the addition of secondary roughness may decrease this lower size limit for water droplet size. In other words, compared to primary roughness alone, the addition of secondary roughness reduces the dependency of superhydrophobicity on droplet size.

4. Discussion

In the Wenzel state, the water phase completely occupies the surface indentations, effectively submerging the surface roughness elements within the liquid medium. This immersion has a direct effect on the local structure of the surrounding water. Conversely, the Cassie state is characterized by a liquid droplet that does not penetrate the cavities of the surface. Instead, air becomes trapped within these cavities, forming a heterogeneous interface. As a transition occurs from the Wenzel configuration to the Cassie configuration, the area of direct contact between the surface roughness and the water is reduced, while the proportion of water exhibiting bulk-like characteristics is increased. This suggests a correlation between Wenzel–Cassie transition and structural rearrangements within the water phase. Thus, it is necessary to study the structure of water and the modifying influence of surface roughness upon that structure.

The sensitivity of OH vibrations to hydrogen bonding makes them a valuable tool for investigating the structure of water. In our Raman spectroscopic studies [49,50,51], we observed that OH vibrations are primarily influenced by the hydrogen bonds within a water molecule’s immediate surroundings, reflecting its local hydrogen bonding environment. As a result, distinct OH vibrations correspond to OH groups participating in different local hydrogen-bonded networks. In contrast to both mixture and continuum models [61,62], our findings indicate that a water molecule interacts with neighboring water molecules through various local hydrogen-bonded arrangements [50,51], such as DDAA, DDA, DAA, and DA hydrogen bondings.

Given the primary dependence of OH vibrations on the local hydrogen-bonded networks surrounding a water molecule, surface roughness primarily influences the structure of interfacial water (Figure 1 and Figure 5). This is also demonstrated by experimental measurements concerning the structure and dynamics of water surrounding ions [52,53,54,55,56]. Additionally, based on our previous studies [63,64] on air–water interfaces and confined water, the formation of a roughness–water interface may be linked to the reduction in DDAA hydrogen bonding in interfacial water. Once the ratio of interfacial water to bulk water (R_{Interfacial water/Bulk water}) is established, the Gibbs energy of interfacial water (ΔG_{Roughness-water}) is given as, ΔG_{Roughness-water} = R_{Interfacial water/Bulk water}∙ΔG_DDAA∙n_HB, in which ΔG_DDAA represents the Gibbs energy of DDAA hydrogen bonds, n_HB indicates the average number of hydrogen bonds per water molecule [43].

As a droplet is placed on a solid surface, the water is divided into interfacial and bulk water. During the Wenzel–Cassie wetting transition, the structural rearrangement between interfacial and bulk water is expected, which is thermodynamically expressed as, ΔG_{Roughness-water} = ΔG_Water-water. For squared roughness (Figure 1), the following equation may be derived,

$$\underset{W_{\mathrm{R}\mathrm{o}\mathrm{u}\mathrm{g}\mathrm{h}\mathrm{n}\mathrm{e}\mathrm{s}\mathrm{s}}}{\underbrace{{\displaystyle \frac{1}{h}}\cdot {\displaystyle \frac{\left(a_{\mathrm{x}}+w_{\mathrm{x}}\right)\left(a_{\mathrm{y}}+w_{\mathrm{y}}\right)}{a_{\mathrm{x}}{a}_{\mathrm{y}}}}+{\displaystyle \frac{2\left(a_{\mathrm{x}}+a_{\mathrm{y}}\right)}{a_{\mathrm{x}}{a}_{\mathrm{y}}}}}}=\underset{W_{\mathrm{W}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}}}{\underbrace{{\displaystyle \frac{3}{8r_{\mathrm{H}2\mathrm{O}}}}\cdot {\displaystyle \frac{\Delta G_{\mathrm{W}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}\text{-}\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}}}{\Delta G_{\mathrm{D}\mathrm{D}\mathrm{A}\mathrm{A}}}}}}$$

(4)

where W_Roughness and W_Water are defined as the wetting parameters related to surface roughness and water [43]. This relationship is predicated on the assumption that the size of the water droplet is significantly larger than the characteristic dimensions of the surface roughness, and that gravitational effects on wettability are negligible.

From Equation (4), a critical W_Roughness (W_Roughness,c) value is expected during the Wenzel–Cassie transition (Figure 1). With reference to W_Roughness,c, it may be classified as exhibiting either the Wenzel or Cassie state (Figure 1). Furthermore, the W_Roughness,c value is characterized by critical geometric parameters of the surface roughness, such as a_c, w_c, and h_c [43]. This can be demonstrated by MD simulations discussed as above (Figure 4 and Table 1). This framework can be leveraged for the rational design of superhydrophobic surfaces with tailored properties.

Upon the placement of a water droplet onto a solid surface, it separates into interfacial and bulk water regions. In our recent work [43], it is found that, during the wetting transition from Wenzel to Cassie states, this leads to the decrease in interfacial water, and the increase in bulk water. Therefore, a transition between interfacial and bulk water may be expected during the Wenzel–Cassie transition. Indeed, this phenomenon arises from the higher degree of hydrogen bonding present in bulk water as compared to interfacial water [43]. This indicates that the Wenzel–Cassie wetting transition may be driven by the structural competition between interfacial and bulk water. Drawing upon our recent studies [51,65], we propose that hydrophobic interactions play a significant role in driving the Wenzel–Cassie transition.

Superhydrophobicity is dependent on the geometric characteristics of the roughness of a solid surface. This can be evaluated using the wetting parameter described above. This approach can also be extended to investigate the effects of two-level surface roughness on the superhydrophobicity of a solid surface. In this work, primary and secondary square pillars were used to model the two-level surface roughness. The geometric characteristics of the surface roughness are characterized by the pillar side lengths in the x and y axes (a_x, a_y), the groove widths between pillars in the x and y axes (w_x, w_y), and the pillar height (h) (Figure 2). To understand the dependence of superhydrophobicity on two-level roughness, the geometric characteristics of either the primary or secondary surface roughness, such as side length, separations, or pillar height, were varied.

For two-level roughness, the solid–liquid interface is influenced by both primary and secondary roughness. Unlike interfaces with single-level roughness, the Gibbs energy of the interfacial water of two-level roughness may be related to both primary and secondary roughness. Of course, this may be reflected on the wetting behavior of the water droplet. Additionally, various superhydrophobic states can exist for two-level surfaces, such as Wenzel, Cassie, and Wenzel–Cassie transition states. These states arise from the combined superhydrophobic characteristics related to primary and secondary roughness, such as the Wenzel–Wenzel (W-W) state, Wenzel–Cassie (W-C) state, Cassie–Wenzel (C-W) state, and Cassie–Cassie (C-C) state (Figure 9). Therefore, compared to the superhydrophobicity of single-level roughness, different Wenzel–Cassie transition states can be expected for two-level roughness.

Based on the above discussion, a corresponding wetting parameter (W_Two-level) can be defined for two-level surface roughness (Figure 2), which can be expressed as follows,

$$\begin{array}{l}{W}_{\mathrm{Two}\text{-}\mathrm{level}}={\displaystyle \frac{W_{\mathrm{Primary}}+\raisebox{1ex}{$\left(W_{\mathrm{Secondary}}\cdot m\cdot V_{\mathrm{Secondary}}\right)$}\!\left/ \!\raisebox{-1ex}{$V_{\mathrm{Primary}}$}\right.-\raisebox{1ex}{$S_{\mathrm{Overlap}}$}\!\left/ \!\raisebox{-1ex}{$V_{\mathrm{Primary} }$}\right.}{1+\raisebox{1ex}{$\left(m\cdot V_{\mathrm{Secondary}}\right)$}\!\left/ \!\raisebox{-1ex}{$V_{\mathrm{Primary}}$}\right.}}\\ ={\displaystyle \frac{W_{\mathrm{Secondary}}+\raisebox{1ex}{$\left(W_{\mathrm{Primary}}\cdot V_{\mathrm{Primary}}\right)$}\!\left/ \!\raisebox{-1ex}{$\left(m\cdot V_{\mathrm{Secondary}}\right)$}\right.-\raisebox{1ex}{$S_{\mathrm{Overlap}}$}\!\left/ \!\raisebox{-1ex}{$\left(m\cdot V_{\mathrm{Secondary}}\right)$}\right.}{1+\raisebox{1ex}{$V_{\mathrm{Primary}}$}\!\left/ \!\raisebox{-1ex}{$\left(m\cdot V_{\mathrm{Secondary}}\right)$}\right.}}\end{array}$$

(5)

where W_Primary and W_Secondary represent the wetting parameters of the primary and secondary surface roughness, V_Primary and V_Secondary mean the volume of primary and secondary roughness, respectively, S_Overlap is the surface area overlapped by the primary and secondary roughness, and m is number of secondary roughness built upon per primary roughness (Figure 2). From the calculated W_Two-level, this may be applied to investigate the effects of two-level surface roughness on superhydrophobicity of solid surface.

As discussed above, the W_Roughness parameter can be used to investigate the dependence of superhydrophobicity on the geometric characteristics of surface roughness. To understand the superhydrophobicity of two-level roughness, it is essential to study the influential factors of W_Two-level.

(I) Based on Equation (5), the W_Two-level parameter is related to the wetting parameters of primary and secondary roughness (W_Primary and W_Secondary). Additionally, it is also influenced by the correlation between primary and secondary roughness (V_Secondary/V_Primary, S_Overlap/V_Primary). Therefore, it may be derived that the wetting parameter of two-level roughness, W_Two-level, mainly lies in the range from W_Primary to W_Secondary. Of course, the value of W_Two-level may also be influenced by the relation between primary and secondary roughness, especially as W_Primary (or W_Secondary) near to the critical wetting parameter.

(II) Furthermore, based on Equation (5), it is found that W_Two-level is also related to the ratio between S_Overlap and V_Primary (S_Overlap/V_Primary), or between S_Overlap and V_Secondary (S_Overlap/(m∙V_Secondary)). Increasing this ratio (S_Overlap/V_Primary) decreases the value of the W_Two-level parameter, which can promote the Cassie state, especially as the wetting parameter of surface roughness near to critical value. This can be achieved by either increasing the area overlapped between the primary and secondary roughness or decreasing the volume of the primary surface roughness. This understanding may be used to understand and design the superhydrophobicity of two-level structured roughness.

According to the preceding discussion regarding the relationship between hydrophobicity and W_Roughness, the Wenzel state is anticipated when W_Roughness exceeds the critical value (W_Roughness > W_Roughness,c), while the Cassie state is observed when W_Roughness is below the critical value (W_Roughness < W_Roughness,c). Based on Equation (5), this can be applied to investigate the relationship of superhydrophobicity between two-level, primary, and secondary surface roughness.

When the primary surface roughness is in the Wenzel state, the superhydrophobicity of the two-level surface is influenced by the characteristics of the secondary roughness. The resulting two-level surface is expected to exhibit Wenzel, Wenzel–Cassie transition, or Cassie states, depending on whether the secondary roughness is in the Wenzel or Cassie state (Figure 6 and Table 2).

Furthermore, if the primary surface roughness is in the Cassie state, the superhydrophobicity of the two-level surface roughness is influenced by the superhydrophobicity of the secondary roughness. If the secondary roughness is in the Wenzel state, the superhydrophobicity of the two-level surface can be in Wenzel, Wenzel–Cassie transition, or Cassie states. However, when both primary and secondary roughnesses are in the Cassie states, the resulting superhydrophobicity of the two-level surface will be the Cassie state (Figure 7 and Table 2).

Based on the discussion of superhydrophobicity in two-level surface roughness, these principles can also be extended to understand the difference between the Lotus and Petal effects. Although both Lotus and Petal effects are related to two-level hierarchically roughness, the Lotus effect is associated with the Cassie state, while the Petal effect is attributed to the Wenzel–Cassie transition state (Figure 9). This is due to the combined effects of primary and secondary roughness on superhydrophobicity, which are in turn related to structural differences in the primary and secondary roughness features. In fact, these differences can be explained by the ratio between the surface overlap (S_Ovelap) and the volume of the primary roughness (V_{Roughness,primary}), i.e., S_Overlap/V_{Roughness,primary}. Compared to hierarchical roughness exhibiting the Petal effect, a larger S_Overlap and a smaller V_{Roughness,primary} are typically observed for the surface roughness exhibiting the Lotus effect [66,67] (Figure 9).

To understand the mechanism of hierarchical roughness on superhydrophobicity, various views have been proposed [68,69,70,71]. In combination with our recent work [43], roughness primarily affects the structure of interfacial water. This suggests that super-hydrophobicity arises from the competition between interfacial and bulk water, a phenomenon effectively described by the W_Roughness. Consequently, the Wenzel–Cassie transition is significantly influenced by the ratio of surface area to volume of the roughness features. When comparing one-level and two-level roughness structures of the same volume, the improved superhydrophobicity observed in the two-level system is related to the increased surface area provided by the two-level structure.

In our recent work [43], a wetting parameter (W_Roughness), closely related to the geometric characteristics of roughness, was proposed and used to understand the superhydrophobicity of a solid surface. During the Wenzel–Cassie wetting transition, a critical parameter, W_Roughness,c, is expected. These parameters can also be applied to understand the effects of two-level roughness on superhydrophobicity. For two-level roughness composed of primary and secondary roughness, the W_Two-level of the two-level surface is related to the geometric characteristics of both primary and secondary roughness (Figure 10). In other words, the resulting superhydrophobicity of two-level structured roughness arises from the combined effects of primary and secondary roughness.

5. Conclusions

Building on our recent studies of superhydrophobicity on solid surfaces, this work investigates the effects of two-level surface roughness on superhydrophobicity. In our previous research, we proposed and applied a wetting parameter, W_Roughness, closely related to the geometric characteristics of roughness, to understand the superhydrophobicity of solid surfaces. We also identified a critical W_Roughness,c associated with the Wenzel–Cassie transition. For two-level roughness composed of primary and secondary roughness, the W_Two-level parameter is related to the wetting parameters of primary and secondary roughness (W_Primary and W_Secondary). Additionally, it is also influenced by the correlation between primary and secondary roughness (V_Secondary/V_Primary, S_Overlap/V_Primary). Therefore, it may be derived that the wetting parameter of two-level roughness, W_Two-level, mainly lies in the range from W_Primary to W_Secondary. Consequently, the W_Two-level of the two-level surface is related to the geometric characteristics of both primary and secondary roughness. Additionally, given that the size of the secondary roughness is less than that of the primary roughness, the addition of secondary roughness may lower the dependence of superhydrophobicity on droplet size compared to surfaces with only primary roughness. Additionally, regarding the mechanism of hierarchical roughness on superhydrophobicity, the enhanced superhydrophobicity observed in the two-level system results from the increased surface area provided by the two-level structure.