Biomedical Engineering Reference
In-Depth Information
(a)
(b)
Figure 1.
Schematic close up showing surfaces decorated by (a) flat topped square pillars for which
f
1
=
f
and
f
2
=
(
1
−
f)
, (b) the general case of an arbitrary surface with rough pillar tops, for which
f
1
=
f
and
f
2
=
(
1
−
f)
.
⎛
⎝
=
⎞
⎠
f
2
cos 180
◦
cos
θ
c
=
f
1
cos
θ
y
+
f(
cos
θ
y
+
1
)
−
1
,
(3b)
See text
where
θ
c
is the Cassie contact angle. The weighting factors,
f
m
,arethe
total
areas
of each of the
m
materials under the drop, normalized by the total projected area.
As such,
∀
m
f
m
1. If a surface is rough and air is trapped under the drop, the
trapped air can be treated as a surface with
θ
y
=
180
◦
, as illustrated in Eq. (3b). This
is often employed to explain the superhydrophobicity of certain rough surfaces.
Naive applications of Eq. (3b) express
f
1
as
f
and
f
2
as
(
1
f)
, with
f
defined
as the
projected
solid-liquid area under the drop normalized by the total projected
area. Obviously then,
f
−
1. This simplification neglects the roughness of
the solid-liquid interface and the curvature of the liquid-vapor interface under the
drop, and is only valid for the case of flat topped pillars as seen in Fig. 1a. A more
general case of rough pillar tops is shown in Fig. 1b, for which
f
1
=
+
(
1
−
f)
=
f
and
f
2
=
(
1
f)
. To consider the difference in another way, consider what Cassie and Baxter
wrote [30]: “When [
f
2
is zero] equation (3b)
*
reduces to Wenzel's equation for the
apparent contact angle of a rough surface with the roughness factor
f
1
”. Further,
if
f
2
=
−
1, the Cassie equation reduces to the Young equation. This
idea will be considered later in the analysis of superhydrophobic surface wetting by
surfactant solutions, and in discussed in reference [31].
A given surface topography will have
f
1
,
f
2
and
r
values. The question of which
mode a drop will take is an interesting one. Following a similar strategy to Bico
et
al
. [32], one can combine Eqs (2) and (3), to yield:
0and
f
1
=
f
2
f
1
−
.
cos
θ
cr
=
(4)
r
Equation (4) suggests that the wetting mode will be Cassie or Wenzel depending
whether Young's (intrinsic) contact angle (controlled by liquid chemistry for a given
*
In reference [30], Eq. (3) of this chapter (denoted as Eq. (5) in reference [30]) is what has become known
as the Cassie equation. Further, Cassie and Baxter only considered a two material surface in their model.
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