Wenzel state

The Law of Reciprocal Proportioxs.—The ponderable quantities in which substances unite -with the saine substance express the relation, or a simple multiple thereof in which they unite with each other. Or, as Wenzel stated it, the weights b, b, b of several bases which neutralize the same weight a of an acid are the same which will neutralize a constant weight a of another acid and the weights a, a , a of different acids which... 

Before proceeding further we note that the problem of drops on top of rough surfaces with pillars or protrusions is different from the one in which rough surfaces have cavities. In case of rough substrates (made of hydrophilic materials) with pillars, a Cassie-Baxter state will typically transition to a Wenzel state by simply displacing the air which is part of the ambient, thus trapping of air is not possible. 

A few remarks regarding the bubble state are in order. When the air in the cavity is completely enclosed within a bubble, there is no net surface tension force acting on it. In this case, gravity does play a role of imposing a net buoyancy force that will cause the bubble to rise. Assuming that the bubble escapes the cavity, it may eventually be released into the ambient. This will result in a state in which the cavities are completely filled up by water and there is no trapped air. We will call this the Wenzel state in the cavities (equation (1)). cav as defined before, corresponding to the Wenzel state is given by... 

Note that is the change in energy corresponding to the transition from the Cassie-Baxter state to the Wenzel state. Since all air in both the Cassie-Baxter and Wenzel states is at the ambient conditions, there is no contribution to from the pressure terms in equation (14). The only contribution comes from the surface energy change. 

It is seen in Fig. 3 that, when fo > 0.67 (corresponding to 0o = 70°), there are five possible equilibrium states while for < 0.67 there are only three equilibrium states. The Cassie-Baxter state is not a possible equilibrium state for < 0.67 because, as discussed above, the liquid-air interface cannot be piimed at the edge of the spherical cavity. In this case, a Cassie-Baxter state is defined simply for the purpose of calculations. Note that cav is the energy relative to the Cassie-Baxter state, irrespective of whether the Cassie-Baxter state is an equilibrium state or not. The Wenzel state is always at the lowest energy. 

The equilibrium states in Fig. 3 can be listed in the likely order in which they are physically encountered as the air is trapped in the cavity (Fig. 4). This order gives the possible stable and barrier (unstable) states. For example, when > 0.67, the Cassie-Baxter state will be encountered first followed by the first equilibrium state, the second equilibrium state, the bubble state and lastly the Wenzel state (Fig. 4). Thus, the first equilibrium and the bubble states are expected to be the energy barrier states that separate the remaining stable equilibrium states. Similarly, for 0 < 0.67 the bubble state is expected to be the barrier state. These expectations will be checked below by exploring a part of the energy landscape and finding the minimum (or stable) energy states. 

Figure 5. Plots of cav vs On at P = 0.1739 and ffe = 70°. (a) fo = 0.88 (i.e. 0q = 40.5°) where there are five equilibrium states including the Wenzel state, and (b) fo = 0-5 (i-e. 0o = 90°) where there are three equilibrium states including the Wenzel state.

Wenzel case. This indicates a contact angle of 0° in the Wenzel state. Figure 6 also shows the experimental values of Abdelsalam et al. [15]. Their results in the context of our theory will be discussed in the following section. 

Figure 8a and 8b shows the plots of cav % for = 0.88 and 0.5, respectively. These plots can be directly compared to Fig. 5a and 5b where the value of Pa is smaller. Figure 8a shows that the Cassie-Baxter state is the border minimum followed by the bubble state which presents a significant energy barrier before the Wenzel state. Thus, the Cassie-Baxter state will be more robust for Pa = I compared to the case in Fig. 5a where Pa = 0.17. Figure 8b shows that the Cassie-Baxter state is a border maximum. The stable equilibrium state in this case has much higher energy, compared to the case in Fig. 5b. A comparison between Figs 5b and 8b also shows that the energy barrier represented by the bubble state is much larger when Pa is larger.

As expected, it has been measured that the large soUd-liquid friction achieved in the impregnated Wenzel state, results in a no-slip BC at the surface as illustrated by the values of fceff — 0 reported in Fig. 3 ( ). In the Fakir (Cassie) state, however, the trapped air is responsible for a finite slip length, the evolution of which is shown in Fig. 3 (o) as a function of the roughness lateral scale L. The measured values... 

Figure 6. This graph shows the results of the uncoated and coated tapered SU8 pillars. The figure is separated in two different sections. The first block on the left side shows the CA values on the uncoated SU8 pillars (probe). The second block shows the SU8 pillars coated with Au and ODT Additionally, because of the different wetting states achieved here, the uncoated SU8 structures are labeled W for the Wenzel-state and CB for the Cassie-Baxter-state.

Figure 1. Behavior of a droplet on a perfectly flat surface (a), on a rough surface according to the Wenzel state (h) and on a rough surface according to the Cassie-Baxter state (c).

Figure 2. Photographs of (a) Si nanoposts and (b) saline droplets in the Cassie and electrowetted Wenzel states. Images are reprinted with permission from [13], copyright 2004 American Chemical Society.

To reiterate, the above-described techniques provide extrinsic means of creating reversibility. Other techniques listed in Table 1 have shown that intrinsic reversibility is limited to 10-20°. As diagrammed and described for Fig. 1, the limit of intrinsic reversibility for the structures reviewed here occurs at the threshold for the transition from the Cassie to Wenzel state. 

Despite the strong hysteresis, all drops evaluated were still in the CB state. In Supporting Information 6, examples of light microscopy images taken through a water drop are presented and clearly show the air enclosure over the full gradient. Despite its lower hydrophobicity, the epoxy surface is clearly in the CB state, since it shows qualitatively the same behavior as observed on the more hydrophobic substrates, which would not be the case if it were in the Wenzel state. ... 

Problem Show that the fakir state is metastabh on a surface such as in Fig. 9.12. in most cascis. [For that, c-onipare the iiiterfacial eiu-rgy of th( Wenzel state and that of tlu Chtssie states]. ... 

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