Bioinspired Self-cleaning Materials - Biotechnologies and Biomimetics for Civil Engineering - page 212

where h W is the contact angle observed on a rough surface following Wenzel's
model, h 0 is the contact angle on an ideally smooth surface of same chemistry, and
r / is the average roughness ratio, i.e., the ratio between actual surface in contact
with the liquid and projected area of the wetted region (Wenzel 1936 ). Wenzel's
theory has been extended during years to take into account the different apparent
contact angles that may be observed on surfaces with nonuniform roughness
(Johnson and Dettre 1964 ; Wolansky et al. 1999 ; Gao et al. 2007 ; Liu et al. 2012 ).
Wenzel's model refers to a complete wetting of the surface, and the hysteresis in
contact angle measurements can reach 100.
The second model, proposed by Cassie and Baxter ( 1944 ), considers the for-
mation of air pockets in roughness valleys under the drop, which leads to higher
contact angle values and—theoretically—almost null hysteresis, as shown in
Fig. 9.5 d (Johnson and Dettre 1964 ; Callies et al. 2005 ; Verplanck et al. 2007 ).
Similarly to Wenzel's model, a change in contact angle is computed owing to the
presence of air-liquid interfaces at the droplet-roughness interface. Again, a
parameter is identified to quantify the excess interfacial surface free energy per
unit area, substituting the term c sg - c sl of Young's equation with terms that
consider / s —the fraction of solid surface area available for direct contact—and
r / —as in the previous model, the average roughness ratio. The Cassie-Baxter
equation then results to be (Cassie et al. 1944 ; Marmur 2003; Choi et al. 2009 ):

cos h CB ¼ r u u s cos h 1 þ 1 u s ð Þ cos h 2 ð 3 Þ

where h 1 and h 2 represent the equilibrium contact angle of the liquid on solid (h 0 )
and air (180), respectively. r / / s and (1-/ s ) come to represent the fractions of
drop surface in contact with the solid and with air, respectively.
Indeed, the Cassie-Baxter model also required corrections and extensions in
order to justify the hysteresis that is observed on rough, nonwetting surfaces (Choi
et al. 2009 , and references therein).
Another important consideration to understand surface wetting phenomena is
the metastability of most Cassie-Baxter states, with consequent transitions to the
Wenzel state: water tends to penetrate the air pockets to minimize surface energy,
gradually filling all roughness cavities starting from the center of the drop until it
reaches its outer limits. This phenomenon is subject of several research studies
aimed at achieving a controlled and reversible Cassie-Baxter to Wenzel state
transition (Peters et al. 2009 ; Giacomello et al. 2012 ; Bormashenko et al. 2013 ).
In spite of all theories, validations and confutations of Cassie-Baxter and
Wenzel models that may be found on the topic(Extrand 2002 ; Gao and McCarthy
2007 , 2009 ; Panchagnula et al. 2007 ; Nosonovsky 2007 ; McHale 2007 ; Whyman
et al. 2008 ), what really causes self-cleaning on superhydrophobic surfaces—in
practical terms—is the achievement of the spontaneous bouncing and rolling of
water drops on a horizontal surface, as will be described in next paragraph (Morra
et al. 1989 ; Solga et al. 2007 ; Balu et al. 2008 ; Myint et al. 2014 ).