- Nano Express
- Open Access
Modeling the Effects of Nanopatterned Surfaces on Wetting States of Droplets
© The Author(s). 2017
Received: 31 March 2017
Accepted: 16 April 2017
Published: 26 April 2017
An analytic thermodynamic model has been established to quantitatively investigate the wetting states of droplets on nanopatterned surfaces. Based on the calculations for the free energies of droplets with the Wenzel state and the Cassie-Baxter state, it is found that the size and shape of nanostructured surfaces play crucial roles in wetting states. In detail, for nanohole-patterned surfaces, the deep and thin nanoholes lead to the Cassie-Baxter state, and contrarily, the shallow and thick nanoholes result in the Wenzel state. However, the droplets have the Wenzel state on the patterned surfaces with small height and radii nanopillars and have the Cassie-Baxter state when the height and radii of nanopillars are large. Furthermore, the intuitive phase diagrams of the wetting states of the droplet in the space of surface geometrical parameters are obtained. The theoretical results are in good agreement with the experimental observations and reveal physical mechanisms involved in the effects of nanopatterned surfaces on wetting states, which implies that these studies may provide useful guidance to the conscious design of patterned surfaces to control the wetting states of droplets.
Hydrophobic materials can be turned into superhydrophobic ones if their surfaces are decorated with micro- or nanocorrugations. Superhydrophobic surfaces have attracted much interest in both fundamental research and industry. This is mainly due to their unique wetting properties and promising technologic application, e.g., in coating, slef-cleaning, and ultra liquid repellent [1–8]. In reality, the superhydrophobicity of lotus leaves is owing to their own hierarchical pillared structure; more specifically, a solid surface can be significantly enhanced by the presence of nano- or micro-scale pillars . On the other hand, the superhydrophobicity also can be influenced by the coulomb repulsion at nanometer-sized contact . A water droplet sitting on a micro-structure surface generally fully wets the substrate topography or is suspended atop the surface topography without penetration, namely, either is the Wenzel (W)  state or Cassie-Baxter (CB)  state.
Understanding the mechanism of the wetting transition between the W and CB states on nanodecorated surfaces is of essential importance for the design and fabrication of nanostructured surfaces. Much effort has been devoted to this issue in recent years in experiments [13–15] and theories [16–20]. Experimentally, Martines et al.  investigated the hydrophilicity, hydrophobicity, and sliding behavior of water droplets on nanoasperities of controlled dimensions and found that the droplet is in the CB state on the pillared surface with large radii and height, but the droplet is in the W state when the pillar radii and height are small. He et al.  fabricated a series of micro-rod surfaces with different geometric parameters to study the relationship between micro-rod geometry and wetting state transition and found the similar phenomenon with that observed by Martines et al. Theoretically, based on energy balance, Patankar  studied how the surface roughness impacts the wetting state transition of a droplet from a higher energy of the CB state to a lower energy of the W state. Bhushan and Jung  demonstrated that hierarchical roughness can result in different superhydrophobic states due to the effects of multiscale roughness on the droplet wetting. In addition, for a pillar-patterned surface, Extrand  and Wang and Chen  respectively proposed some theoretical criterions to predict wetting states of droplets according the surface geometrical morphology. In spite of much progress in experiments and theories, the effects of the nanopatterned surfaces on wetting states of droplets have not been well understood and there are some fundamental issues in the wetting phenomena on nanopatterned surfaces. For example, the specific relations between the wetting states and geometrical morphology of the patterned surface are unclear, especially for the nano- and micro-holed surfaces. An intuitive phase diagram of wetting states of a droplet in the space of surface geometrical parameters is eagerly expected.
In order to address these problems, in this paper, we took two typical nanodecorated surfaces as an example, i.e., periodic nanohole-patterned surface and nanopillar-patterned surface, respectively. By calculating the free energy difference between the W and CB states, the dependence of wetting behaviors on the periodic nanopatterned geometrical parameters is studied. In detail, we found that for nanohole-patterned surfaces, they tend to be in Cassie-Baxter state as the hole is higher and thinner and tend to be in the Wenzel state as the hole is shorter and thicker. However, for nanopillar-patterned surfaces, the short and thin nanoholes lead to the Wenzel state, and contrarily, the long and thick nanoholes result in the Cassie-Baxter state. In addition, we discussed in detail the intrinsic mechanism of the wetting states of droplets on different patterned surfaces. Furthermore, the intuitive phase diagrams of wetting states of the droplet on different patterned surfaces are obtained. In this way, the phase diagrams offer a simple method to evaluate the wetting states of droplets on nanopatterned surfaces.
Analytical Model of Nanohole-Patterned Surface
With the aim of investigating the influence of geometrical morphology parameters of nanohole-patterned surfaces on wetting states, the free energy of the W and CB states are calculated according to Eqs. (5) and (10). Furthermore, in order to obtain a phase diagram of wetting states of a water droplet on a nanohole-patterned surface, the free energy difference between the W and CB states are calculated. Here, we define the free energy difference between the W and CB states as ΔE = E W − E CB. Thus, by combining Eqs. (5) and (10), the free energy difference can be obtained as ΔE h = E W-h − E CB-h for a nanohole-patterned analytical model. When ΔE h > 0, i.e., E CB-h < E W-h, it indicates that the CB state will be the final equilibrium state. When ΔE h < 0, it implies that the W state is more stable. Accordingly, when ΔE h = 0 the boundary of the phase diagram of wetting states is attained to separate the Wenzel state and the Cassie-Baxter state.
Analytical Model of Nanopillar-Patterned Surface
to better understand the effects of nanopillar-patterned surface factors on droplet wetting behaviors and obtain a certain phase diagram of wetting states of a water droplet on a nanopillar-patterned surface. A similar strategy as the previous section is applied to investigate the intrinsic mechanism of wetting states on a nanopillar-patterned surface in this part. By combining Eqs. (17) and (18), the free energy difference is given by ∆E p = E W-p − E CB-p. When ∆E p > 0, it represents CB state. When ∆E p < 0, it represents the W state. According to ∆E p = 0, the boundary of the phase diagram of wetting states is obtained to separate the Wenzel state and the Cassie-Baxter state as well.
Here, we consider that the droplets’ volumes have the same value and are constant. The contact angles θ W and θ CB are expressed by cosθ W = rcosθ 0 and cosθ CB = fcosθ 0 − (1 − f), respectively, where r and f are the respective Wenzel roughness factor and Cassie-Baxter roughness factor. It is notable that there might be energy barriers when the transition between the CB and W states happens and the lower energy state will occur as long as the energy barriers are overcome. However, the energy barriers are usually rather low; in some instances, the energy barriers can be overcome with the help of sufficient conditions such as vibration, enough pressure, and extra field. Accordingly, we consider that the lower energy state is more stable in our analytical models.
Results and Discussion
Wetting States on the Nanohole-Patterned Surface
In summation, the theoretical results are in good agreement with the experimental observations. Figure 4b displays a phase diagram of the W and CB states of a water droplet on a nanohole-patterned surface, which can clarify the interrelated effects of the radius and the height of the nanohole on wetting states. The phase diagram clearly shows the relationship between the geometrical morphology parameters of a nanopatterned surface and the wetting states. Therefore, Fig. 4b is an explicit wetting state phase diagram for the nanohole-patterned analytical model and is applicable for predicting the initial wetting state under the circumstance of a water droplet sitting on different sizes of nanohole-patterned surfaces.
Wetting States on Nanopillar-Patterned Surface
Finally, to further validate our analytical model, we compare our model predictions with several relevant experimental studies of the wetting properties of water droplets on pillar-patterned silicon surfaces. Martines et al.  experimentally investigated the hydrophobicity behavior of water droplets on nanoasperities with fixed center-to-center pitch d p = 300 nm and various pillar diameters and heights. They observed that the droplet is in the CB state when the pillar radii and height are 78 and 286 nm, respectively (the red dot in Fig. 5a), and the droplet is in the W state when the pillar radii and height are 73.5 and 239 nm, respectively (the black dot in Fig. 5a), which are located in the corresponding regions that were theoretically predicted. Next, we use the experimental data in micro-scale to confirm our model. Fürstner and co-workers  studied the wetting and self-cleaning properties of water droplets on three types of artificial superhydrophobic surfaces with different micro-structure geometries. The experimental results reported that the wetting model is the W state when the geometric parameters are r p = 0.5 μm, h p = 1.0 μm, and d p = 2.0 μm (the black dot in Fig. 5b) and the wetting model is the CB state when the geometric parameters are r p = 0.5 μm, h p = 2.0 μm, and d p = 2.0 μm (the red dot in Fig. 5b). They also found that as r p = 0.5 μm, d p = 5.0 μm, and h p = 1.0, 2.0, and 4.0 μm (the black dots in Fig. 5c, respectively), the wetting models are the W state. In the end, we compare the theoretical results with the experimental results measured by Jopp et al.  for water droplets on micro-texture structured surfaces when the pillar height and spacing distance are 250 and 110 μm, respectively; the water droplets are in the W state when the pillar radius is 30, 40, 50, and 60 μm (the black dots in Fig. 5d, respectively). The theoretical results predicted by our model are in excellent agreement with those results observed by experiments. Moreover, the model results can qualitatively explain some experiment results. For instance, He et al. [14, 15] found that increasing the height and decreasing the space of micro-rods may result in the Cassie-Baxter wetting state, while decreasing the height and increasing the space may result in the Wenzel wetting state. The comparison between the related experiment data and the prediction of the theoretical model in this paper are well matched, and the theoretical results’ variation tendency agrees well with the experimental results, which validate our theoretical model.
The above discussions suggest that the droplet wetting behaviors on a nanopillar surface are determined by the surface geometric structure, including nanopillar radius, height, and spacing between the nanopillars. Figure 5a shows a phase diagram of the wetting states on the nanopillar-patterned surface, and it explicitly expresses the relationship between the geometry of the nanopillar-patterned surface and the corresponding wetting state. Therefore, with the help of the wetting state phase diagram, it would be very easy to obtain the initial equilibrium wetting state when a droplet is sitting on a nanopatterned surface by knowing the size of the surface nanostructure.
In conclusion, we applied the thermodynamic method to study the wetting states of a water droplet on the two typical nanopatterned surfaces. We calculated the free energy difference between the W and CB states. Analyzing the free energy difference, we found that the wetting states of water droplets on nanopatterned surfaces are sensitive to the geometrical morphology and the intrinsic mechanism of wetting states which depends on the periodic nanopatterned geometrical parameters is elucidated in detail. To systematically understand the dependence of wetting states on the geometrical parameters of nanopatterned surfaces, two phase diagrams of the W and CB states for water droplets on nanopatterned surfaces are given, which would be quite useful in predicting the initial wetting state. The theoretical results agree well with the reported experimental results; we hope that the results can provide some useful guidance to the design and fabrication of nanopatterned surfaces with certain wetting characteristics. In this work, we focused on the phase diagrams of water droplets on the nanohole and nanopillar-patterned surfaces with a periodic square lattice distribution. Nonetheless, the thermodynamic method is general, and it can be applied to other nanopatterned surfaces and obtain the corresponding phase diagrams as well.
This work was financially supported by National Natural Science Foundation of China (grant nos. 11104084 and 11574080) and Opening Project of Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control (Hunan Normal University).
All the authors participated in the discussion of the results and in the writing of the paper. All authors read and approved the final manuscript.
The authors declare that they have no competing interests.
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