Wetting States

where y, ySL, ySA are the surface tensions at the liquid/air (vapor), solid/liquid, and solid/air interfaces, respectively. The contact angle of a droplet deposited on a solid depends on external parameters, such as temperature. The temperature change may stimulate the transition from the partial to complete wetting of a solid substrate. In this case we observe a wetting transition on a smooth solid surface

The Young equation supplies the sole value of contact angle for a certain combination of solid, liquid, and gaseous phases. Regrettably, the experimental situation is much more complicated; even on the atomically flat surfaces the diversity of contact angles is observed. This is due to the long-range interaction between molecules forming the triple (three-phase) line of the droplet and molecules forming the solid substrate [100]. It was shown that the droplet-surface attraction is time-dependent due to re-orientation of the molecules constituting the solid substrate; this fact hinders experimental tribology investigations of a solid/ liquid contact and calls for novel experimental techniques such as the recently reported centrifugal adhesion balance [95].

The maximal contact angle observed on the surface is called the advancing angle, hadv; the minimal one is called the receding angle, hrec [32, 38, 44, 47]. The advancing and receding contact angles are equilibrium (though metastable) angles [69]. The difference between advancing and receding contact angles hadv — hrec is called the contact angle hysteresis [93, 94]. The experimental establishment of advancing and receding angles is a challenging task, and it should be mentioned that reported contact angles are sensitive to the experimental technique used for their measurement [19, 35, 60].

Chemical heterogeneities and roughness strengthen the contact angle hysteresis [4, 36]. Various models explaining the phenomenon of hysteresis were proposed [54, 74, 96, 99]. The effect was related to the pinning of the triple line by defects, which produces the potential barrier U to be surpassed by the droplet under its displacement [99]. The general expression for the contact angle hysteresis is supplied by Eq. 6.2

where R0 is the initial radius of the spherical drop before deposition on the substrate; the function h(h) of the equilibrium contact angle is given in [99]. However, a general theory of the contact angle hysteresis is still not built. For experimental study of hysteresis, the manufacture of well-defined microscopically scaled defects is necessary, which is a complicated technological task [85].

The wetting of flat, chemically heterogeneous surfaces is characterized by APCA hC predicted by the Cassie-Baxter wetting model [29, 30]. Consider the wetting of a composite flat surface comprising several materials. Each material is characterized by its own surface tension coefficients y;,SL and yj,SA, and by the fraction f in the substrate surface, f1 + f2 + ••• + fn = 1. The APCA in this case is supplied by the Cassie-Baxter equation:

Fig. 6.2 Various wetting states occurring on rough surfaces. a Cassie air trapping state. b Wenzel state. c Cassie impregnating wetting state. d mixed wetting state

Cassie (air trapping)


Cassie (air trapping)



Liquid mixed



1 A A

where 0, are equilibrium (Young) contact angles for the ith material. The Cassie-Baxter equation can also be applied to the solid surface comprising pores (the contact angle for pores equals p and cos 0 = — 1, see Fig. 6.2a). In this case the Cassie-Baxter equation yields:

where fS and 1—fS are the relative fractions of the solid and air fractions underneath the droplet. In the case of pillar reliefs, the Cassie air trapping wetting state is also called the ''fakir state''. The Cassie-like air trapping wetting results in unusual tribology of the surface providing an easy sliding of water droplets. The slip lengths as high as 200-400 im were reported recently [63, 64].

It is noteworthy that the derivation of Eq. 6.4 from Eq. 6.3 is not straightforward, because the triple (three phase) line could not be in rest on pores [20]. Taking into account the fine structure of the triple line justifies the success of the Cassie-Baxter formula for predicting APCAs on porous surfaces [15, 20, 34, 92]. When the relief is hierarchical the Cassie-Baxter equation obtains more complicated forms taking into account the interrelation between scales constituting the topography of the relief [12, 21, 22, 48]. It should be mentioned that the dependence of the APCA on equilibrium contact angles is weak on the hierarchical surfaces when compared to those on single-scale surfaces [21, 22]. Modifications of the Cassie-Baxter equation considering the peculiarities of complicated topography of biomimetic surfaces were reported recently [34, 62, 76].

The wetting of rough chemically homogeneous surfaces is governed by the Wenzel model [98]. According to the Wenzel model, the surface roughness r defined as the ratio of the real surface in contact with liquid to its projection onto the horizontal plane, always magnifies the underlying wetting properties (see Fig. 6.2b). Both hydrophilic and hydrophobic properties are strengthened by surfaces textures. The Wenzel apparent contact angle is given by Eq. 6.5:

Actually, pure Cassie and Wenzel wetting situations are rare in occurrence [37], and Marmur introduced a mixed wetting state [67]. In the mixed wetting state, a droplet partially wets the side surface of pores and partially sits on air pockets as described in Fig. 6.2d. The APCA is supplied in this case by Eq. 6.6:

In this equation f is the fraction of the projected area of the solid surface that is wetted by the liquid. When f = 1, Eq. 6.6 turns into the Wenzel Eq. 6.5.

One more wetting state has been introduced [7, 50, 52]. This is the Cassie impregnating state depicted in Fig. 6.2c. In this case liquid penetrates into grooves of the solid and the drop finds itself on a substrate viewed as a patchwork of solid and liquid (solid ''islands'' ahead of the drop are dry, as shown in Fig. 6.2c). The APCA of Cassie impregnating state is established as (the contact angle at pores is zero and cos h = 1; [7]):

The Cassie impregnating wetting is possible when the Young angle satisfies Eq. 6.8 [7]:

It should be stressed that Eqs. 6.3-6.7 could be applied when the radius of the droplet is much larger than the characteristic scale of surface heterogeneities [75].

Fig. 6.3 Multiple minima of the Gibbs energy of a droplet deposited on a rough surface

Fig. 6.3 Multiple minima of the Gibbs energy of a droplet deposited on a rough surface

The rigorous thermodynamic derivation of Eqs. 6.3-6.7 was obtained in the series of theoretical works [7, 24, 25, 46, 66, 71, 99].

Various wetting states featured by very different APCA can co-exist on the same heterogeneous surface. The diversity of APCA could be easily understood if one takes into account that the Gibbs energy curve for a droplet on a real surface is characterized by multiple minima points [22, 68]. It could be shown that the Wenzel state is energetically favorable compared to the Cassie state when [6]:

The lowest minimum of the Gibbs energy usually corresponds to the Cassie impregnating APCA given by Eq. 6.7 (see Fig. 6.3). When APCA changes spontaneously or under external stimuli we observe the wetting transition (WT). It should be emphasized that physical mechanisms of WT on flat and rough surfaces are quite different.

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