Chemistry Reference
In-Depth Information
The lines at which three membranes meet, at an angle of 120
◦
at equilibrium, is
called a Plateau border [
16
]. These objects are clearly visible in Fig.
2.5
a, b. In the
central image, they show up as dark regions surrounded by the lines of glow from
the surfactant layers and the oil. It is clear that as long as there is no membrane,
the surfactant layers delimiting adjacent droplets will meet tangentially, i.e., at zero
contact angle. This is illustrated in Fig.
2.5
c, where the grey shaded area corresponds
to the dark Plateau border in the fluorescence micrograph. The latter reveals that
upon membrane formation, the size of this region shrinks significantly. We can use
this effect to determine the contact angle at which the surfactant layers meet the
membranes, as illustrated in the bottom panel of Fig.
2.5
. For the shaded area,
A
, one
readily finds
R
2
√
3 cos
2
+
3
2
(θ
−
θ)
−
2
A
=
sin
(2.3)
The radius of curvature,
R
, is directly linked to the pressure,
p
, in the droplet via
p
is the free energy per unit area of the surfactant-laden water/oil
interface.
R
can thus be considered to remain unchanged as the membrane forms. It
is then easy to obtain the contact angle,
=
γ/
R
, where
γ
θ
, from the size of the Plateau borders. We
3
◦
.
At the three-phase contact line, where the membrane meets the adjacent water/oil
interfaces, force balance yields
obtain
θ
=
48
±
cos
2
=
2
γ
(2.4)
where
is the free energy per unit area of the membrane. We can thus directly infer
the membrane free energy from inspection of the Plateau borders if
γ
is known.
Using the standard pendant drop method, we obtained
γ
=
1
.
77
±
0
.
11mN/m.
Our result for the membrane free energy is thus
20mN/m. The
formation of the membrane is therefore accompanied by a gain in free energy of
=
3
.
23
±
0
.
02mN/m.
The interfacial free energies involved are of interest for the potential performance
of devices designed in the way proposed here. In order to take full advantage of the
concept, droplets have to be formed and moved relative to the geometry provided by
the micro-fluidic channel structure. The excess free energy of the droplet interfaces
sets the scale for the pressures to be applied in these processes. The Laplace pressure
is given by
p
L
F
=
2
γ
−
=
0
.
31
±
0
.
≈
γ/
l
, where
l
is the size of the smallest orifice to be passed. For
γ
=
1 bar for a 30nm orifice, which appears well feasible.
As the sharp rise in the capacitance trace shown in the inset of Fig.
2.6
suggests,
the formation of the membrane is a rapid process. Figure
2.7
a displays a series of
images captured with a high speed camera (1000 frames per second, Photron SA 3).
The zipper-like transition is clearly discernible, although details on the scale of the
membrane thickness are of course not accessible to optical imaging. The total time
of formation in this run was about 150ms. Figure
2.7
b shows the membrane diameter
as a function of time, as observed during a single formation event. The approximate
3mN/m, we obtain
p
L
=
