Cassie-Baxter State

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. 

When a drop is deposited on a surface with cavities, it will be assumed that the air is trapped in the cavities under the drop by a liquid-air interface at top of the cavities. This state will be termed the Cassie-Baxter state as shown in Fig. 1. The interface at the top of the cavities will have a curvature equal to that of the drop itself. However, compared to the length scale of the cavities, the radius of curvature of this interface is large — it will be assumed to be flat. As such the pressure in the drop will be approximated to be equal to Fo as far as computing the equilibrium inside the cavities is concerned. This assumption is no different from the flat interface assumption in case of surfaces with pillars or bumps [14,29, 30]. In the Cassie-Baxter state the air in the cavity will be assumed at pressure Pq and temperature To — same as the ambient. 

When the interface at the top of the cavity, in the Cassie-Baxter state, moves into the cavity, the availability Acav of the new state can be written as Acav = Acb -I- A A where... 

Note that the summation in (8) is only for the cavities below the drop. The remaining cavities give zero contribution. In equation (8), R T = Pq Vcav has been used by assuming that the amount of air trapped in the cavities under the drop is determined by the Cassie-Baxter state. In the Cassie-Baxter state, the air is at pressure Pq and volume Vcav- An ideal gas law is used. AUy. denotes the change in the surface energy in cavity i as the liquid-air interface moves into the cavity. 

Table 1 lists the equilibrium values of 0h at different values of Co (i e. at different values of 9q) for = 0.1739 and 0e = 70°. Once % is known, equation (16) or (17) can be used to obtain the equilibrium value of. The first two columns of Table 1 specify the size of the spherical cavity. An interface will remain pinned at the top edge of the spherical cavity if 180° 0e 0o (see [31] for details). Thus, whenever this condition is satisfied the Cassie-Baxter state is possible in the cavities because the local contact angle condition is satisfied at the top edge of the spherical cavities. The corresponding value of Ou is equal to Oq. These values for 0 = 0o are listed in Table 1. For 0q > (= 70°), the local contact angle condition cannot...

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. 

Figure 3. Plots of cav vs fo for different equilibrium states Pa = 0.1739 and 6t = 70°). cav is the energy relative to the Cassie-Baxter state.

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. 

To explore the entire energy landscape, Ecav could be plotted as a function of and f. The extrema on this landscape will be the equilibrium solutions, discussed above, of which some will be the stable solutions. In this work, we will select a probable path in this landscape. The values of Ecav will be plotted for this path. To this end, assume that the liquid-air interface is first in the Cassie-Baxter state. As the liquid-air interface moves toward the other equilibrium states, the value of 9u will increase until it reaches the next available equilibrium state given in Table 1. At each intermediate state, between the equilibrium states, it will be assumed that the interface is spherical, the gas pressure is such that it is in accordance with the ideal gas law (equation (16)) and that the interface is in mechanical equilibrium (equation (17)). Thus, equations (16) and (17) are satisfied, however, equation (18) is not. This implies that in the intermediate states the local contact angle condition... 

There are two types of situations in Table 1 or Fig. 3 — one with five equilibrium states and the other with three. Fcav vs % is plotted for one case of each type in Fig. 5a and 5b. Figure 5a shows the five equilibrium states case at Co = 0.88 (i.e. 00 = 40.5°). It is seen that the Cassie-Baxter state at 0h = 0o = 40.5° is a border minimum. If the liquid-air interface moves in, at mechanical equilibrium, as assumed here, then the energy Ecav increases until it reaches a maximum value at 0H = Thus, represents the energy barrier state where the equi-... 

In this work we have identified the various possible states within the cavities and compared their energies. In case of the experiments of Abdelsalam et al. [15], the Cassie-Baxter state in the cavities (see Fig. 1) is possible only for (fo > 0.67. However, Abdelsalam et al. [15] report an apparent contact angle corresponding to the... 

Cassie-Baxter formula for all cases. The following conclusion is apparent from the comparison between theory and experiments in Fig. 6 even if the experimentally observed contact angle matches the Cassie-Baxter formula, it does not necessarily imply that the cavities below the drop are also in the Cassie-Baxter state (see Fig. 1). A possible scenario is discussed below. 

In light of the above discussion, the experimental data of Abdelsalam et al [15] at smaller values of i iay be considered. Initially, the deposited drop may trap air in the Cassie-Baxter state in the cavities. The advancing front of the drop attains the apparent contact angle corresponding to the Cassie-Baxter state. Since the... 

Comparing the results in Tables 1 and 2, it is clear that by increasing the value of Pa, the second equilibrium state, which is the stable state, occurs at a smaller value of %. Thus, as expected, the intrusion of water into the cavities is reduced. For 0 > 0.69, the Cassie-Baxter state is the only stable equilibrium state with trapped air. In Table 2, there is a very small range in which there are five equilibrium stales as in Table 1. 

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.

For three different structure distances of 60 pm, 75 pm and 90 pm the Cassie-Baxter state is already achieved with the pure SU8 structures on glass (hrst section in Fig. 6 CB-30/60, CB-30/75, CB-30/90). This is confirmed by the reduced hysteresis values of 29°, 24° and 20° and the observed possible and easy drop removal... 

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.

For the first two structure distances of 30 and 45 pm, the final structure forms are close and interconnected enough to produce a kind of capillary effect which forces the water to fill the cavities, leading to very high hysteresis values (Fig. 4). This interconnection between the pillars leads, especially for the 30 pm distance, to an area more fikely made of holes than free-standing pillars. For the next distances of 60, 75 and 90 pm, both Wenzel and Cassie-Baxter states are observable depending on how carefully the drop is placed on the surface (Fig. 6). For the last dimension of 30/120 pm we could not achieve a Cassie-Baxter state because of the large distance between the pillars. 

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).

Surfaces of certain plants — such as the leaf of the Lotus plant — have a surface topography with two scales of roughness in the form of a base profile with peak-to-peak distances of the order of several micrometers and a superposed fine structure with peak-to-peak distances significantly below one micrometer [7-11]. Given this, the Lotus leaf follows the Cassie-Baxter state as sketched in Fig. Ic. Surfaces with... 

Figure 9 a) A drop of water on a lotus-leaf surface, b) Cassie Baxter state of a droplet on a hydrophobic, rough surface. Note the air enclosed under the drop. Pictures courtesy of Doris Spori, ETH Zurich. 

In addition to its influence on surface reactivity, surface structure is also seen to affect wettability on the micrometer scale, as is best illustrated by the lotus effect (see Chapter 3b). The lotus leaf is superhydrophobic, i.e. has a water contact angle of about 160°, thanks to the combination of the waxes on the surface with a characteristic dual micrometer- and nanometer-scale surface topography. Without the structure, the wax chemistry would only impart mild hydrophobicity to the surface. Superhydrophobicity comes about only when a water droplet is in contact with a rough surface with a substantial enclosure of air beneath the drop (Figure 9). This is the so-called Cassie-Baxter state, named after the authors of the work that described the contact angle of water droplets in this state by means of the equation ... 

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