14 Plasmonics and Super-Hydrophobicity
505
faces may have, via chemical modifications, a contact angle that is 120 ∪ at most
[27, 28]. Artificial super hydrophobic states are possible solely for surfaces that also
incorporate a regular pattern or texture [29]. Super hydrophobic states are very well
described by the models of Wenzel [30] and Cassie Baxter [31]. In 1936, Wenzel
explained the increased contact angle in terms of geometrical effects only and especially as a function of the solid roughness r (r is the real interfacial area over projected
area, and it is always larger than one),
cos ε w = r cos ε
(14.1)
and thus the effect of surface roughness is to amplify the wetting. Differently from
Wenzel, Cassie described super hydrophobicity on account of the pockets of air that
get trapped between the drop and the substrate
cos ε c = −1 + λ (cos ε + 1)
(14.2)
where φ is the fraction of solid in contact with the drop (and notice that φ is always
smaller than one). The less φ the larger the apparent contact angle. At the limit of
φ ≈ 0, the drop would paradoxically float in air. The Cassie model is intuitive
in that predicts that a drop upon a patterned surface develops a contact angle that is
proportional to the fraction of air in contact with the drop. In spite of the fact that they
do both induce hydrophobicity, these situations are very different from their adhesive
properties, in that Wenzel drops are strongly pinned. On the contrary, in the Cassie
state the drop sits mainly upon air, and this dramatically reduces the contact angle
hysteresis thus favouring the self cleaning properties of the surface. On comparing
Eq. 14.1 and 14.2, one can notice that a transition from Cassie to Wenzel occurs when
cos (ε ) =
λ − 1
r − λ
(14.3)
Equation 14.3 claims that the Cassie state is assured provided that the contact angle
of the flat, unmodified material ε is sufficiently large; that is, larger than a certain,
non linear combination of the geometrical parameters of the substrate. In particular,
large φ ∼ s prompt a non wetting behaviour of the substrate [29–32]. In other terms,
if the forest of posts or pillars is sufficiently dense on the surface, a Cassie state
is assured. Notice though that large φ would induce small contact angles, and thus
the choice for the best parameters in Eq. 14.3 is not trivial, and it is indeed a matter
of optimization. Equation 14.3 holds on average. Situations are widely reported in
literature where Cassie holds albeit φ is small, meaning that it is a metastable state.
Practically this means that, if positioned with sufficient care upon a surface, a drop
would not collapse even if it is supposed to, in that an energy barrier inhibits the
spontaneous transition of the droplet to the level of lower energy content [33, 34].
505
faces may have, via chemical modifications, a contact angle that is 120 ∪ at most
[27, 28]. Artificial super hydrophobic states are possible solely for surfaces that also
incorporate a regular pattern or texture [29]. Super hydrophobic states are very well
described by the models of Wenzel [30] and Cassie Baxter [31]. In 1936, Wenzel
explained the increased contact angle in terms of geometrical effects only and especially as a function of the solid roughness r (r is the real interfacial area over projected
area, and it is always larger than one),
cos ε w = r cos ε
(14.1)
and thus the effect of surface roughness is to amplify the wetting. Differently from
Wenzel, Cassie described super hydrophobicity on account of the pockets of air that
get trapped between the drop and the substrate
cos ε c = −1 + λ (cos ε + 1)
(14.2)
where φ is the fraction of solid in contact with the drop (and notice that φ is always
smaller than one). The less φ the larger the apparent contact angle. At the limit of
φ ≈ 0, the drop would paradoxically float in air. The Cassie model is intuitive
in that predicts that a drop upon a patterned surface develops a contact angle that is
proportional to the fraction of air in contact with the drop. In spite of the fact that they
do both induce hydrophobicity, these situations are very different from their adhesive
properties, in that Wenzel drops are strongly pinned. On the contrary, in the Cassie
state the drop sits mainly upon air, and this dramatically reduces the contact angle
hysteresis thus favouring the self cleaning properties of the surface. On comparing
Eq. 14.1 and 14.2, one can notice that a transition from Cassie to Wenzel occurs when
cos (ε ) =
λ − 1
r − λ
(14.3)
Equation 14.3 claims that the Cassie state is assured provided that the contact angle
of the flat, unmodified material ε is sufficiently large; that is, larger than a certain,
non linear combination of the geometrical parameters of the substrate. In particular,
large φ ∼ s prompt a non wetting behaviour of the substrate [29–32]. In other terms,
if the forest of posts or pillars is sufficiently dense on the surface, a Cassie state
is assured. Notice though that large φ would induce small contact angles, and thus
the choice for the best parameters in Eq. 14.3 is not trivial, and it is indeed a matter
of optimization. Equation 14.3 holds on average. Situations are widely reported in
literature where Cassie holds albeit φ is small, meaning that it is a metastable state.
Practically this means that, if positioned with sufficient care upon a surface, a drop
would not collapse even if it is supposed to, in that an energy barrier inhibits the
spontaneous transition of the droplet to the level of lower energy content [33, 34].