14 Plasmonics and Super-Hydrophobicity
509
towards the center of the drop, thus recovering the initial equilibrium contact angle
ε c (Fig. 14.2e). The total force along x that is generated is
F p = 2r
c
Δ lv (cos ε ne − cos ε c )
(14.10)
and accordingly the condition for depinning is F p > F c
a . If F p ≤ F c
a , the drop is stuck
on the surface. (This condition may be valid, strictly speaking, in close proximity of
ε c ) The ratio γ = F p /F c
a indexes the propensity of a drop to slide upon a surface.
When γ > 1 depinning dominate over friction, and thus the drop is recalled toward
its centre during evaporation. In Fig. 14.2f, γ is plotted against ε ne , that is the non
equilibrium contact angle as in Fig. 14.2e, at different φ ∼ s. At the initial time, γ < 1,
and this means that the drop is sticky. As evaporation keeps on, ε ne decreases. The
depinning force augments with time, and so does γ. When γ is sufficiently large,
that is γ >> 1, the contact line of the drop would recede, and the contact angle
would again coincide with the expected equilibrium contact angle ε c . Under these
conditions, γ = 0. The process would then starts over again with the repetition of
the same mechanism. Notice that the drop would experience a different story line on
changing φ. When φ is small, γ varies more rapidly with ε ne , and the condition for
radius downturn is met at earlier stages of evaporation. In any case, at each cycle,
the contact radius is reduced by a quantity that is the distance between two lines of
pillars. When r goes down below the critical radius of impalement, an irreversible
transition to Wenzel occurs, whereby the drop is stuck. For sufficiently high pillars
the minimum contact radius achievable is limited by energetic considerations being
[40–42]
r min √
ϕ
cos ε
(14.11)
where ϕ is the distance between a spot and another.
14.2.4 Consideration for an Optimal Design
The models above predict that the contact angle is large, and thus the contact area is
limited, provided that the pillars are diluted or, equivalently, that φ is small (Eq. 14.2).
Unfortunately, the collateral effect of small φ is that a drop would collapse at the early
stage of evaporation (Eq. 14.11). In this case, the drop would stick on the surface and,
upon the continuance of evaporation, the solute would eventually redistribute as to
form a ring like structure along the water-solid line of contact, therefore being spread
over a large area, with no or few concentration effects. The coffee ring effect is very
well understood and it is attributed to a fixed contact line and to an outward flow
within the drop [43, 44]. Here, a criterion is discussed that would guarantee the best
trade off between sufficiently dense forests of pillars, that prevent the early collapse
of the drop and avoid the dispersion, and diluted structures, that instead assure large
contact angle.
509
towards the center of the drop, thus recovering the initial equilibrium contact angle
ε c (Fig. 14.2e). The total force along x that is generated is
F p = 2r
c
Δ lv (cos ε ne − cos ε c )
(14.10)
and accordingly the condition for depinning is F p > F c
a . If F p ≤ F c
a , the drop is stuck
on the surface. (This condition may be valid, strictly speaking, in close proximity of
ε c ) The ratio γ = F p /F c
a indexes the propensity of a drop to slide upon a surface.
When γ > 1 depinning dominate over friction, and thus the drop is recalled toward
its centre during evaporation. In Fig. 14.2f, γ is plotted against ε ne , that is the non
equilibrium contact angle as in Fig. 14.2e, at different φ ∼ s. At the initial time, γ < 1,
and this means that the drop is sticky. As evaporation keeps on, ε ne decreases. The
depinning force augments with time, and so does γ. When γ is sufficiently large,
that is γ >> 1, the contact line of the drop would recede, and the contact angle
would again coincide with the expected equilibrium contact angle ε c . Under these
conditions, γ = 0. The process would then starts over again with the repetition of
the same mechanism. Notice that the drop would experience a different story line on
changing φ. When φ is small, γ varies more rapidly with ε ne , and the condition for
radius downturn is met at earlier stages of evaporation. In any case, at each cycle,
the contact radius is reduced by a quantity that is the distance between two lines of
pillars. When r goes down below the critical radius of impalement, an irreversible
transition to Wenzel occurs, whereby the drop is stuck. For sufficiently high pillars
the minimum contact radius achievable is limited by energetic considerations being
[40–42]
r min √
ϕ
cos ε
(14.11)
where ϕ is the distance between a spot and another.
14.2.4 Consideration for an Optimal Design
The models above predict that the contact angle is large, and thus the contact area is
limited, provided that the pillars are diluted or, equivalently, that φ is small (Eq. 14.2).
Unfortunately, the collateral effect of small φ is that a drop would collapse at the early
stage of evaporation (Eq. 14.11). In this case, the drop would stick on the surface and,
upon the continuance of evaporation, the solute would eventually redistribute as to
form a ring like structure along the water-solid line of contact, therefore being spread
over a large area, with no or few concentration effects. The coffee ring effect is very
well understood and it is attributed to a fixed contact line and to an outward flow
within the drop [43, 44]. Here, a criterion is discussed that would guarantee the best
trade off between sufficiently dense forests of pillars, that prevent the early collapse
of the drop and avoid the dispersion, and diluted structures, that instead assure large
contact angle.