Chemistry Reference
In-Depth Information
trajectories shown on the right. This is in sharp contrast to other schemes, where
the propulsion mechanism itself changes the surrounding medium strongly enough
to prevent self-crossing of the path [
12
]. That this is clearly not the case here is
demonstrated in the quasi one-dimensional setup in the bottom panel, which shows a
time lapse representation of seven droplets moving in a micro-channel. As two drops
touch each other, they reverse their direction of motion and perambulate the channel
again, without significant reduction in velocity.
In order to gain some insight into the propulsion mechanism, let us consider a
spherical droplet with radius
R
. The total coverage,
c
, of the droplet surface with
the mono-olein, either brominated or not, is assumed to be roughly constant and in
equilibrium with the micellar phase in the oil. The brominated fractional coverage
shall be called
b
. If the droplet moves, there is (in the rest frame of the droplet) an
axisymmetric flow field,
u
(θ)
, along its surface. The equation of motion for
b
is
∂
b
∂
t
=
k
(
b
0
−
b
)
+
div
(
D
i
grad
b
−
ub
)
(6.1)
where
b
0
is the equilibrium coverage with brominated mono-olein (brMO). It is
determined by the bromine supply from inside the droplet and the rate constant,
k
,
of escape of brMO into the oil phase.
D
i
is the diffusivity of the surfactant within
the interface.
The droplet motion is accompanied by a flow pattern within the droplet and in
the neighboring oil, which can be determined from
u
[
27
,
28
]. The correspond-
ing viscous tangential stress exerted on the drop surface must be balanced by the
Marangoni stress, grad
(θ)
γ(θ)
=
M
grad
b
(θ)
, where
γ
is the surface tension of the
surfactant-laden oil/water interface, and
M
db
is the Marangoni coefficient
of the system. Expanding the bromination density in spherical harmonics,
=
d
γ/
∞
(θ)
=
b
m
P
m
(
θ)
b
cos
(6.2)
m
=
0
we can express the velocity field [
27
,
28
] at the interface as
∞
b
m
C
−
1
/
2
m
(
m
+
1
)
1
(
cos
θ)
M
m
+
u
(θ)
=
(6.3)
μ
sin
θ
2
m
+
1
m
=
1
where
C
n
is the sum of the liquid viscosi-
ties outside and inside the droplet. Inserting this into Eq. (
6.1
) and exploiting the
orthogonality relations of Gegenbauer and Legendre polynomials, we obtain
denote Gegenbauer polynomials, and
µ
m
k
b
m
db
m
dt
b
0
M
D
i
R
2
=
(
m
+
1
)
µ
−
−
(6.4)
(
2
m
+
1
)
R
