Biomedical Engineering Reference
In-Depth Information
Figure 3.39
Sketch of an interface pinned by pillars. (a) Microfabricated pillars; (b) interface be-
tween two slowly flowing liquids stabilized by pillars; and (c) detail of an interface between two
pillars. Depending on the pressure difference
P
1
-
P
2
the interface bulges more or less. If the pressure
difference is too large, the interface breaks down.
Let us examine the case of triangular (or diamond shaped) pillars with hydro-
phobic surface. According to the Laplace theorem, the curvature radius of the inter-
face is related to the pressure difference across the interface. When the pressure on
one side of the interface is increased, the curvature increases until the pinning limit
is reached. Then the interface is disrupted and the high pressure liquid penetrates
into the low-pressure channel (Figure 3.40).
Take an interface pinned between the two facing edges of two similar micro-
pillars, and suppose that the pressure P
1
in one liquid is progressively increased.
Two conditions govern the pinning: the first condition is related to capillarity and
to the phenomenon of canthotaxis [8], for example the pinning is effective if the
condition
θ θ
£
(3.56)
C
is met, and the interface does not slide on the pillar walls. In (3.56)
q
C
is the (static)
contact angle. Above this value, the interface slides irreversibly along the two fac-
ing walls of the pillars (Figure 3.41). The second condition is geometrical and cor-
responds to the minimum possible curvature of the interface. This curvature is
obtained when the interface has the shape of a half-circle with a radius
d
/2. In such
a case,
q
=
a
+
π
/2. The second condition is then
θ α π
£
+
2
(3.57)
Figure 3.40
When the water pressure is increased, the interface is disrupted and water invades the
solvent channel.
