Biomedical Engineering Reference
In-Depth Information
Figure 3.76
Plot of the relation between cos(
q
real
) and cos(
q
Young
) for patterned surfaces. Reprinted
with permission from [47]. Copyright 1996 American Chemical Society.
Laplace relation has been derived. Next, the Young law for the contact angle of an
interface on a solid has been presented. From these two relations, an expression for
the capillary force on a triple line has been deduced. Such an expression has a key
role in determining the behavior of droplets on different substrates and geometry
of microsystems.
This chapter has shown the essential role of surface tension and capillarity at
the microscale. These forces often screen out forces such as gravity or inertia, which
are predominant at the macroscopic scale. Although we have taken the stance of
presenting capillarity and surface tension from an engineering point of view by con-
sidering global effects, one has to keep in mind that interactions at the nanoscopic
scale are the real underlying causes of these global effects.
Finally, it is stressed here that liquid-liquid or liquid-gas interfaces adopt a
shape that minimizes the interfacial area, taking into account the constraints at
the contact with the solid parts. Such surfaces encompass the concept of minimal
surfaces—surfaces with mean zero curvature (Figure 3.77) [48]—and extend it to
minimal energy surfaces, given the constraints acting on them.
The prediction of the shape of an interface results from the minimization of the
energy of the system (surface, gravitational, and so forth) under some constraints
imposed by external conditions, such as walls, wires, fixed volume, or fixed pres-
sure. When gravity is negligible, these surfaces have a constant mean curvature [49].
Before a droplet is deposited on a solid surface, the surface energy of the system is
E
=
γ
S
(3.97)
SG
,0
SG SG
,0
After deposition of the droplet, the surface energy is the sum of the three surface
energies
E E
=
+
E
+
E
(3.98)
LG
SL
SG
,1
