Biomedical Engineering Reference
In-Depth Information
Using (4.22), we derive
1
1
2
γ
æ
ö
æ
2(
γ γ
-
cos
θ
)
ö
2
2
SL
,0
SG
LG
0
V
=
=
ç
(4.25)
ç
÷
÷
sat
è
ø
è
C
ø
C
Equations (4.20) and (4.21) require the knowledge of the solid surface tension
g
SG
.
In the case of apolar surfaces, the Zisman's criterion produces a good approxima-
tion [16, 17]. In the particular case of EWOD, the coating of the substrate is real-
ized with apolar materials like Teflon, parylene, PET, PTFE, and so fourth, and we
can reasonably use the value of the wetting surface tension (Zisman's criterion) for
g
SG
. The PQRS model predicts values of the saturation contact angle in reasonably
good agreement with the experimental observations; however these are sometimes
somewhat overestimated.
The other plausible explanation for the saturation of the contact angle stems
from (4.17). When the vertical
F
y
force on the triple line becomes sufficiently larger
than the horizontal
F
x
force (i.e.,
F
y
/
F
x
= 1/tan
q
>>1) the droplet cannot spread
further on the substrate.
Modified BLY Law
To take into account the saturation limit, the BLY law can be modified to [18]
æ
2
ö
cos
θ
-
cos
θ
CV
0
0
=
L
(4.26)
ç
÷
(
)
cos
θ
-
cos
θ
2
γ
cos
θ
-
cos
θ
è
ø
S
S
0
where
L
is the Langevin function
L
(
X
) = coth (3
X
) - 1/3
X
[19], and
q
s
is the satura-
tion angle. Equation (4.22) reduces to the BLY law for small and moderate values of
the potential
V
. At large potentials, it satisfies the saturation asymptote. Equation
(4.22) is called the “modified” or “extended” Lippmann-Young law. It has been
verified that this function fits the experimental results [18]. Figure 4.10 shows the
fit between the experimental points and the modified Lippmann law.
Figure 4.10
Fit of the experimental results for (cos
q
- cos
q
0
) versus
V
2
obtained by Langevin's
functions.
