The Cassie-Baxter Model

FIGURE 9.3. Edge of a drop placed on a chemically composite surface. 

Letting 0 denote, here again, the apparent angle, the energy variation associated with a small displacement dx is 

Therefore, the apparent angle (which is indeed restricted to the interval [01,02]) is given by an average involving the angles characteristic of each constituent, but the average is applied to the cosines of these angles. For a review of the behavior of drops on chemically textured surfaces, the reader should consult a recent paper.  

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Figure 2. AmpUflcation of hydrophobicity due to surface roughness. In the Wenzel model, the liquid confonnaUy follows the surface topography (a), while in the Cassie-Baxter model, the air remains in the lower regions of the topographic features (b). In (a), r represents the ratio of the actual surface area of the rough substrate to the nominal surface area. In (b), fg is the sohd fraction that is in contact with the liquid under the droplet.

The Cassie-Baxter-Wenzel theory [44, 47,48] defines the critical contact angle value on the smooth surface above this value, the Cassie-Baxter model is more stable wetting state and below this value the Wenzel model is the most stable wetting state. If the measured contact angle on a smooth surface is lower than this critical value and if the superhydrophobic behaviour is observed, the transition between the two models should be possible like, for example, with the LDPE surface treated in both plasmas. In this case, the roughness factor is 1.043, the contact angle on the dried surface reaches a value of 171° and the contact angle on the same surface partially wetted with water vapour or dipped in water is only 140°. 

Modeling of Superhydrophobic Behavior by the Cassie-Baxter Model... 

Theoretically, the Cassie-Baxter model assumes that the droplet partially contacts with the air trapped in the pores of the surface when it sits on the peaks of surface feature [22]. This relationship between the apparent contact angle observed on a rough surface and the equilibrium contact angle 0 obtained on a smooth surface is based on the same chemical composition [13,14] ... 

Wenzel s equation applies to what is called homogeneous wetting, and it can only be applied to homogeneous, rough surfaces. Surfaces having heterogeneous character can be modeled with the Cassie-Baxter model [43] ... 

The Young equation cannot be used directly to explain the effect of surface roughness on the wettability of a material because it is valid only for ideal smooth solid surfaces. There are two wetting models that are proposed when a water droplet sits on rough surfaces, these are the Wenzel model and the Cassie-Baxter model. 

The surface roughness, r, is defined as the ratio of the actual surface area to the horizontal projection of the surface area. In the Cassie-Baxter model, the interaction of the fluid with the air trapped in the gaps of rough (or porous) surfaces is considered alongside the liquid-solid interaction. The effect is strongest in the case of water, which forms a contact angle of 180° with air. The Cassie-Baxter model introduces a surface fraction parameter, /, which represents... 

Figure 4.8 Schematics of a motion of the contact hne of a hquid droplet sitting on a solid surface, leading to a corresponding free energy variation for a smooth surface (Young model, a), complete penetration of the liquid into the recessed features of a pattern (Wenzel model, b), and for heterogeneous wetting regimes with absence of penetration (Cassie-Baxter model, c) and for partial penetration (d). In the Wenzel model, rfp = AB + BC + CD + DE)I AB + CD) according to the letters in (b). In the Cassie-Baxter model, (j) = (AB)I(AB + BE) according to the letters in (c).

In the heterogeneous wetting regime, air can instead be trapped inside the features underneath the liquid drop, which ideally stands on the top of the nanostructures. This situation (schematized in Figure 4.8c) is described by the Cassie-Baxter model. The free energy variation following the contact line displacement is ... 

Here we have proposed a modified Cassie-Baxter model to investigate the influence of prirticle size on superhydrophobic behavior of sfiica nanosphere arrays. An assembly technique enabled to prepare well-ordered silica nanosphere arrays, and then the silica arrays were fluorinated by a spin-coating process. Compared with a F-coated flat surface, the contact angle on fluorinated silica nanoarrays reached a value of 152 1.4°. A Cassie-Baxter parameter, surface firaction (4>s), was used to simulate the hydrophobicity. It was found that the nanosphere size played an important role in affecting the hydrophobicity of the sphere arrays. The present work demonstrates that the superhydrophobicity of nanoarrays is well correlated with the modified Cassie-Baxter model. 

Several frequentiy used models for wetting have been tested with the acquired data. The models investigated were the Cassie— Baxter approximation, the Furmidge equation for roll-off angles, and a proposition from Patankar for receding contact angles. 

Wettability of a rough surface can be described by the Wenzel [7] model if the surfaces are completely wetted by the liquid into the protrusions on the surface. The Cassie-Baxter [8,9] model gives an idea about wettability of rough hydrophobic surfaces. In the case of such surfaces, the air trapped into hierarchical roughness prevents water penetration into the surface protrusions. 

Wenzel and Cassie—Baxter models since the incremental advancing area is assumed to characterize the surface overall. And, as a result, features on either side of the TPL do not influence the contact angle over the increment but must be of sufficiently small size to ensure that an average of the overall topology is sampled by the TPL at any one time, as shown by McHale (2007) and Shirtcliffe et al. (2010) for a random surface, illustrated in Figure 2. 

In the present work, we aimed to examine how the structure of a nanosphere array affected the contact angle between a liquid droplet and fluorinated surfaces. Four types of silica nanospheres with various diameters were stacked to form different organized arrays using a self-assembly technique. To clarify the superhydrophobicity, we proposed a modified parameter into the conventional Cassie-Baxter equation. This mathematical model presented in this study can probably shed some light on how the variation of particle size would induce the superhydrophobicity of nanosphere surfaces. 

There remains a controversy in the field that can be summarized by the phrase area vs line . The models of Wenzel and Cassie— Baxter are based on area considerations. If they are tested by free-energy calculations and free-energy barriers, they correctly predict... 

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