Biomedical Engineering Reference
In-Depth Information
Using sine and cosine functions expansions, the total capillary force is then
F
=
e
γ θ
[cos
-
cos
θ
]
-
e
γα θ θ
[sin
+
sin
]
(4.34)
0
0
The first term on the right hand side of (4.34) is the usual “Lippmann” force. The
second term is a resistance force depending on the value of the hysteresis contact
angle. It can be shown that this second term is always negative, because sin
q
and
sin
q
0
are positive. A consequence is that hysteresis reduces the capillary force, as
expected. Because the minimum potential corresponds to the linear part of the BLY
relation, the “Lipmmann” force can be expressed by
eC
2
F
=
V
(4.35)
EWOD
2
A criteria for drop displacement is then
eC
V
2
-
e
γα θ θ
[sin
+
sin
] 0
>
(4.36)
0
2
Without hysteresis (
a
= 0) the drop would move even with an infinitely small elec-
tric actuation. Taking into account the hysteresis (
a
¹ 0), (4.36) shows that the
minimum electric potential is given by
2
γ
α θ
2
V
=
[sin (
V
)
+
sin
θ
]
(4.37)
min
0
min
C
Using the Lippmann-Young law, (4.37) can be cast under the form
C
V
2
=
α θ
[sin (
V
)
+
sin
θ
]
(4.38)
min
0
min
2
γ
Equation (4.38) is somewhat cumbersome because it is an implicit equation due to
the fact that
q
depends on
V
. In the case of a sufficiently small V
min
(4.38) can be
simplified
γα θ
sin
0
V
=
2
(4.39)
min
C
A large capacitance, a low liquid surface tension, and a small value of the hysteresis
angle minimize the value of the voltage required to move droplets.
4.2.2.6 Working Range
Equation (4.39) gives an expression of the minimum actuation that is required to
move droplets. On the other hand, there exists also a maximum actuation volt-
age—noted
V
max
—above which the electrocapillary force on a drop does not in-
crease anymore due to the saturation phenomenon. For the moment we do not take
into account the dielectric breakdown voltage, which is closely related to saturation
and will be treated later on. Hence
V
max
=
V
sat
. Thus, for a given type of EWOD
microdevice characterized by its capacitance
C
and its surface properties, and for a
given electrically conductive liquid immersed in a surrounding nonconductive gas
