## Energy Barrier and Cassie Wenzel and Young Contact Angles

It is well established experimentally that wetting transitions are usually irreversible. This conveys the suggestion that some general reasons for such irreversibility exist. It turns out that a variety of wetting states, as well as transitions between them, may be described on the same mathematical basis that gives the possibility to elucidate their features which are independent of peculiarities of a particular substrate. Starting with the spherical model for the droplet shape, it can be shown that the surface-energy dependence E(h) on the (non-equilibrium) apparent contact angle h looks like [99].

9pV 2

and its minimum E0i is expressed as

where V is the droplet volume and 0t is the equilibrium APCA in a given wetting state. In particular, Eq. 6.12 supplies equilibrium energies in the Cassie (i = C) and Wenzel (i = W) states or in the wetting state on flat surfaces (i = Y, Young's angle 0i = 0Y) with the corresponding angles in Eqs. 6.4, 6.5, and 6.1, respectively. Moreover, for a definite mechanism of transition, Eqs. 6.11 and 6.12 give the energies of the transition state (i = trans). For hydrophobic materials and orthogonal reliefs it can be shown that the energy in the transition state is also expressed by (6.11), (6.12) with

The mentioned mechanism of the Cassie-Wenzel transition is described as wetting the side surfaces of hydrophobic relief [3] accompanied with the energy increase. A transition (composite) state corresponds to the almost complete filling of relief asperities. A transition barrier is overcome when liquid touches their bottoms, and the high-energy liquid-air interface under the droplet disappears.

Equations 6.11-6.13 enable one to calculate the energy barrier of transition by using the measured or calculated values of contact angles in the wetting states without entering geometrical details of a substrate relief. In this way, e.g., the results [3] of the barrier calculation can be reproduced.

As mentioned in Sect. 6.4, wetting transitions may proceed quickly or slowly. Accordingly, two types of wetting transitions may proceed in principle: adiabatic transitions with a fixed value of the contact angle, and slow non-adiabatic transitions when a droplet has time to relax and the contact angle changes in the course of liquid penetration into depressions (or going out from them). Both these types of energy barriers can be calculated on the basis of the presented model, e.g., for the transition from the Cassie state to the Wenzel one, as

The irreversibility of wetting transitions is seen from peculiarities of the dependence (6.12) of the equilibrium surface energies on the equilibrium APCAs (Fig. 6.6). The function E0i(cos 0,) is a monotonically decreasing one, with a weak dependence for low values of cos 0; (*-1) and a strong one for higher values. Furthermore, as it can be proven, cos 0trans \ cos 0c, cos 0trans \ cos 0w, i.e. cos 0trans is located out of the interval between cos 0C and cos 0W, closer to the lower limit. Consequently, the energy barrier is very asymmetric, low from the side of cos 0trans = cos 0W + cos 0C — cos 0Y

Wadia = Etrans(cos 0c)— Eoc, Wionadia = Eotrans — Eoc (6.14)

:ans

Fig. 6.6 Dependence (Eq. 6.12) of the equilibrium surface energy (in units of y(9ftV 2)1/3) on the equilibrium APCA and barriers of wetting transitions. Numerical values of APCAs are 107.4°(Wenzel), 134.4° (Cassie), and 101.5° (Young). The transition state angle calculated according (6.13) is 143.1°. The heights of the highly asymmetric energy barrier for a water droplet of a volume of 3 il are: from the side of the (metastable) Cassie state WC = 8 nJ and from the side of the (stable) Wenzel state WW = 70 nJ

Fig. 6.6 Dependence (Eq. 6.12) of the equilibrium surface energy (in units of y(9ftV 2)1/3) on the equilibrium APCA and barriers of wetting transitions. Numerical values of APCAs are 107.4°(Wenzel), 134.4° (Cassie), and 101.5° (Young). The transition state angle calculated according (6.13) is 143.1°. The heights of the highly asymmetric energy barrier for a water droplet of a volume of 3 il are: from the side of the (metastable) Cassie state WC = 8 nJ and from the side of the (stable) Wenzel state WW = 70 nJ

the metastable (higher energy) state and high from the side of the stable state. Calculations of real transitions based on (6.12-6.14) gave the difference of almost one order of magnitude. Taking into account exponential (Arrhenius-type) dependence of the transition probability on the barrier height shows that the reverse transition is impossible.

The results of this section remain true for inherently hydrophilic substrates, where the transition mechanism is different compared to hydrophobic substrates. In this case the existence of a barrier may be due to the pinning of liquid at discontinuities, such as pore or pillar borders, that leads to the formation of a new liquid-vapor interface at additional energy expense, as shown in Fig. 6.7. The wetting transition takes place when the menisci touch the bottoms of the relief [12, 13, 15, 34]. The connection between APCAs for hydrophilic substrates is given by cos 0tans = 1 + cos 0C - ^. (6.15)

6.5.3 Critical Pressure Necessary for Wetting Transition

Consider a single-scale pillar-based biomimetic surface, similar to that studied by Yoshimitsu with pillar width a, and groove width b [101]. Analysis of the balance

Fig. 6.7 Pressure induced displacement of the water front leading to the collapse of the Cassie air trapping wetting state pillars pillars

of forces at the air-liquid interface, at which the equilibrium is still possible, yields for the critical pressure pc [102]:

where k = (A/L), A and L are pillar cross-sectional area and perimeter respectively. As an application of Eq. 6.16 with h = 114 (Teflon), a = 50 im, b = 100 im we obtain pc = 296 Pa, in excellent agreement with experimental results [101, 102]. Recalling that the dynamic pressure of rain droplets may be as high as 104-105 Pa, which is much larger than pc « 300 Pa, we conclude that creating biomimetic reliefs with very high critical pressure is of practical importance [102]. The concept of critical pressure leads to the conclusion that reducing the micro-structural scales (e.g., the pillars diameters and spacing) is the most efficient measure needed to enlarge the critical pressure [72, 102]. The energy barrier separating the Cassie and Wenzel states is given by an expression similar to (6.10) and scales as R2 [ ]. It is noteworthy that neither Eq. 6.10 nor Eq. 6.16 explains the existence of Cassie wetting on inherently hydrophilic surfaces [12, 13, 22, 58, 87]. Indeed, Wtrans and pc calculated according to Eqs. 6.10 and 6.16 are negative for h<(p/2), this makes the traditional Cassie wetting inapplicable to hydrophilic surfaces, and the alternative physical reasoning explaining experimentally observed high APCAs should be involved, as discussed below.

An alternative mechanism of WT based on the concept of critical pressure was proposed [65, 81]. It was supposed that as the pressure applied to the droplet increased, the meniscus will move towards the flat substrate as shown in Fig. 6.7. The meniscus will eventually touch the substrate, this will cause the collapse of the water water

trapped air

Fig. 6.8 Geometrical air trapping on hydrophilic reliefs trapped air

Fig. 6.8 Geometrical air trapping on hydrophilic reliefs

Cassie wetting, and consequently lead to the Cassie-Wenzel transition ([65, 81]; see also the preceding section).

## Renewable Energy 101

Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable. The usage of renewable energy sources is very important when considering the sustainability of the existing energy usage of the world. While there is currently an abundance of non-renewable energy sources, such as nuclear fuels, these energy sources are depleting. In addition to being a non-renewable supply, the non-renewable energy sources release emissions into the air, which has an adverse effect on the environment.

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