## Modeling CA Hysteresis

The first term in the right-hand part of the equation, which corresponds to the inherent contact angle hysteresis of a smooth surface, is proportional to the fraction of the solid-liquid contact area, 1 — fLA. The second term Hr is the effect of surface roughness, which is proportional to the length of the triple line. Thus Eq. 2.5 involves both the term proportional to the solid-liquid interface area and to the triple line length. It is observed from Eqs. 2.4 and 2.5 that increasingfLA ? 1 results in increasing the contact angle (cos h ? —1, h ? p) and decreasing the contact angle hysteresis (cos hadv - cos hrec ? 0). In the limiting case of a very small solid-liquid fractional contact area under the droplet, when the contact angle is large (cos h & — 1 ? (p — h)2/2, sin h & h — p) and where the contact angle hysteresis is small (hadv ~ h ~ hrec), based on Eq. 2.5 , p - h = yj2(1 -/la)(R/ COS h0 + 1) (2.6)

a a n , M, cos ha0 - cos hr0 n-— cOS hr0 - cOS ha0

For the homogeneous interface, fLA = 0, whereas for the composite interface fLA is a non-zero number. It is observed from Eqs. 2.6-2.7 that for a homogeneous interface, increasing roughness (high Rf) leads to increasing the contact angle hysteresis (high values of hadv — hrec), while for a composite interface, an approach to unity of fLA provides both high contact angle and small contact angle hysteresis [16, 26-28]. Therefore, the composite interface is desirable for self-cleaning.

A different semi-phenomenological model of the contact angle hysteresis has been proposed recently by Whyman et al. . According to their model, the i \ !=2

contact angle hysteresis is given by the equation hadv - hrec = iCj^J h(h*), where U is the height of the potential barrier connected with the motion of the triple line along a substrate, R0 is the initial radius of the spherical drop before deposition on the substrate, and h(h*) is the dimensionless function of the apparent contact angle h*.

Vedantam and Panchagunula  suggested a semi-empirical phase field method to calculate the CA hysteresis. In this method, the order-parameter g(x, y) is selected in such a manner that g = 0 for the non-wetted regions of the surface and g = 1 for wetted regions, whereas 0 < g < 1 for partially wetted regions. After that, the energy function f(g) is constructed, and its minima correspond to the equilibrium states of the system (e.g., the Wenzel and Cassie states). After that, the energy functional is written as

where k is the gradient coefficient. The functional that should be minimized involves the free energy and the gradient of the free energy. The latter term is needed to account for the fact that creating an interface between two phases is energetically unprofitable. The kinetic equation is given in the form bg = -dL = kV2g - 0f (2.9)

dg og where b > 0 is the kinetic coefficient. Vedantam and Panchagunula  showed that in the case of b = const for an axisymmetric drop flowing with the velocity V, Eq. 2.9 leads to cos hadc — cos hrec = 2abV (2.10)

In other words, assuming that the kinetic coefficient is constant, the contact angle hysteresis is expected to be proportional to the flow velocity. A more complicated form of the kinetic coefficient may lead to a more realistic dependence of the contact angle hysteresis on the velocity.

There is an asymmetry between the wetting and dewetting processes, since less energy is released during wetting than the amount required for dewetting due to adhesion hysteresis. Adhesion hysteresis is one of the reasons that leads to contact angle hysteresis, and it also results in the hysteresis of the Wenzel-Cassie state transition. The Cassie-Wenzel transition and CA hysteresis both may be considered as different manifestations of the same wetting-dewetting cycle behavior. Both the CA hysteresis and Cassie-Wenzel transition cannot be determined from the macroscale equations and are governed by micro- and nanoscale phenomena.

Note that the size of the surface roughness details is an important factor. It is generally assumed that the roughness factor Rf as well the fractional area of contact fSL can be determined by averaging the surface roughness over some area, which is itself small relative to the size of the liquid droplet. For Rf and fSL fractional areas changing with a spatial coordinate, special generalized Wenzel and Cassie equations, proposed by Nosonovsky , should be used. The size of the surface roughness also affects the ability of the interface to pin the triple line and thus affects the CA hysteresis. It could be claimed that CA hysteresis is a ''second order'' effect which is expected to vanish with the decreasing ratio of the size of

Fig. 2.2 a Optical images, b Scanning Microscope micrographs, and c Atomic Force Microscope roughness maps of petals of two roses [Rosa Hybrid Tea, cv. Bairage (Rosa, cv. Bairage), and Rosa Hybrid Tea, cv. Showtime (Rosa, cv. Showtime)] (adapted from )

Fig. 2.2 a Optical images, b Scanning Microscope micrographs, and c Atomic Force Microscope roughness maps of petals of two roses [Rosa Hybrid Tea, cv. Bairage (Rosa, cv. Bairage), and Rosa Hybrid Tea, cv. Showtime (Rosa, cv. Showtime)] (adapted from ) the surface roughness and heterogeneity details to the droplet radius. This, however, does not happen since surface roughness and heterogeneity is an inherent property of any surface. There is a deep similarity between the dry friction and the wetting of a solid surface . In the ideal situation of absolutely homogeneous and smooth surfaces there would be no friction and no CA hysteresis due to the absence of energy dissipation. However, in the real situations, surfaces are not ideal, and this leads to both dry friction and CA hysteresis. The development of quantitative relationships between the degrees of surface non-ideality (e.g., Shannon entropy of a rough surface) and CA hysteresis, remains an interesting task similar to the same task for friction . 