Biomedical Engineering Reference
In-Depth Information
which all lead to the derivation of the BLY law. Basically three approaches exist:
thermodynamical [4], energy minimization [5], and electromechanical approaches
[6, 7]. We present here the thermodynamical and electromechanical approaches
because they shed an interesting light on the physics of the phenomenon. Energy
approach is more mathematical and the reader can refer to the work of Shapiro et
al. [5].
Thermodynamical Approach
The thermodynamical approach is based on the existence of an electric double
layer in the conductive liquid along the substrate surface. This layering of charges
stretches the droplet. First, we assume a perfectly smooth solid surface at the con-
tact of the conductive liquid. The solid is a metal directly at the contact of the liquid
and the potential difference is small enough so that no electric current is flowing
through the liquid (no hydrolysis if the liquid is aqueous). Upon applying an el-
ementary electric field, an elementary potential difference builds up at the interface
and an electric double layer forms in the liquid at the contact of the surface. Gibbs'
interfacial thermodynamics yields
eff
d
γ
= -
ρ
dV
(4.2)
SL
SL
where
g
eff
denotes the effective surface tension at the liquid-solid interface,
r
SL
the
surface charge density in counter-ions, and
V
the electric potential. If we make the
Helmholtz simplifying assumption that the counter-ions are all located at a fixed
distance
d
H
from the surface (
d
H
is of the order of a few nanometers), the double
layer has a fixed specific capacitance (capacitance par unit area)
ε ε
0
l
C
=
(4.3)
H
d
H
where
e
l
is the relative permittivity of the liquid and
e
0
is the permittivity of vacuum:
e
0
= 8.8541878176 × 10
−12
F/m. Integration of (4.2) yields
V
V
C
eff
( )
H
2
ò
ò
γ
V
=
γ
-
ρ
dV
=
γ
-
C V dV
=
γ
-
(
V V
-
)
(4.4)
SL
SL
SL
H
SL
pzc
SL
2
V
V
pzc
pzc
Figure 4.2
(a) in absence of electric charges, a droplet of water shows a contact angle larger than
90° on a hydrophobic solid substrate. (b) the contact angle of the water with the substrate notably
decreases when the electrode is actuated.
