Biomedical Engineering Reference
In-Depth Information
Case 2: Microscopic Drops, Bo << 1
As mentioned earlier, a microdrop has the form of a spherical cap. A spherical cap is
a surface of minimum energy if only surface tension is taken into account. This can
be checked by using the Surface Evolver software. The result is shown in the Figure
3.48 for a contact angle of 110°.
Figure 3.49 shows a cross section of the droplet on nonwetting and wetting sur-
faces. The volume
V
of such a droplet is a function of two parameters in the set of
the four parameters {
q
,
a
,
R
,
h
}, where
q
is the contact angle,
a
is the contact radius
(i.e., the radius of the circular base),
R
is the curvature radius (i.e., the sphere radius),
and
h
is the height of the droplet. We shall not demonstrate again the derivation of
the liquid volume; it has already been done in [8]. The most used formula is
π
3
3
(
, )
θ
=
(
2 3cos
-
θ
+
cos
θ
)
(3.72)
V R
R
3
The other ones are
π
2
2
V a h
( , )
=
h a
(
3
+
h
)
6
æ
2
ö
π
è
è
2
2
2
2
2
V a R
( ,
)
=
R
±
R
-
a
3
a
+
R
±
R
-
a
ç
÷
6
è
ø
(3.73)
3
π
(
2 3cos
-
θ
+
cos
θ
)
3
V a
( , )
θ
=
a
3
3
sin
θ
3
2 3cos
cos
π
(
θ
θ
)
-
+
3
V h
( , )
θ
=
h
3
3
(
)
1 cos
-
θ
Note that in (3.73) the plus sign corresponds to a nonwetting case (lyophobic)
and the minus sign corresponds to the wetting case (lyophilic).
The surface area of the spherical cap is also of importance since the surface
energy is proportional to this surface.
(3.74)
E
=
γ
S
surf
Figure 3.48
Shape of a microdrop calculated with the software Surface Evolver (
q
= 110°).
