Chemistry Reference
In-Depth Information
Therefore we conclude that the droplet swimmers that are propelled byMarangoni
stresses, as described earlier, indeed behave as purely neutral squirmers. We note,
however, that by careful adjustment of conditions, such as adding suitable solvents to
the oil phase, one should be able to tune the exchange rate of the surfactant molecules
k
(see Sect.
6.3.1
), and thus control which modes become unstable. Thereby the flow
pattern in the droplet and its vicinity can be adjusted to include the second order term
in Eq.
6.5
and thus get a pusher or puller type of squirmer.
6.3.3 Swimmer Velocity
In this section, we present results concerning the velocity of the droplets—first we
discuss the speed, followed by a discussion about the directionality.
To discuss the droplet speed,
V
, we reconsider the total surfactant coverage,
c
.This
adjusts itself as a balance between the molecular adsorption energy at the water/oil
interface and the mutual repulsion of the adsorbed surfactant molecules. The equilib-
riumcoverage thus represents aminimum in the interfacial energy. As a consequence,
the interfacial tension will not change to first order if
c
is varied. Deviations of
c
from
its equilibrium value, which come about necessarily for any finite
V
, will thus be
replenished from the surrounding oil phase without noticeable Marangoni stresses.
For a flow profile as shown in Fig.
6.5
,wehavediv
u
(θ)
∝
cos
θ
. The density of
ρ
≈
ρ
0
+
δρ
(
)
θ
δρ
∝
(
)
surfactant thus takes the form
is some
function of the radius. If
D
m
is the diffusivity of the micelles, the diffusion current,
D
m
∂ρ/∂
f
r
cos
, where
V
.
f
r
3
Vc
2
R
θ
[
24
]. As long as this holds, the only source of appreciable Marangoni stresses is
the gradient in bromination density,
b
r
, must balance the depletion rate at the drop interface,
c
div
u
(θ)
=
cos
. The drop thus keeps taking up speed
according to Eq. (
6.4
). This comes to an end when
(θ)
δρ
≈
ρ
0
. The surfactant layer at
the leading end of the drop surface can then not be replenished anymore, and
c
comes
substantially below its equilibrium value. This leads to an increase of surface tension
accompanied by a 'backward' Marangoni stress, and thus finally to a saturation of
the velocity. According to the reasoning above, we expect that
V
3
c
.
There is no literature value for the diffusivity of MO micelles in squalane, but we
can estimate it on the basis of the Stokes-Einstein relation assuming the radius of
the micelles to be similar to the length of a MO molecule (2.3nm). Using 36 mPa.s
for the viscosity of the squalane, we obtain
D
m
=
≈
2
ρ
0
D
m
/
10
−
12
m
2
/s. We thus predict
2
.
6
×
27
µ
m
/
sec
mM
V
l
.
This can be measured experimentally. In order to measure
V
, we used a reac-
tion mixture within the droplets similar to the Belousov-Zhabotinski (B-Z) reaction,
with reactant concentrations adjusted such as to prevent chemical oscillations (with
concentrations as described in Sect.
6.2
). This results in a spatially and temporally
constant bromine release rate in the aqueous phase for an extended period of time.
However, since the system is closed, one expects a time dependant swimmer behav-
ior. Indeed, as shown in Fig.
6.8
, the speed of the swimmer gradually reduces over
time, eventually coming to a halt due to the consumption of fuel. The top panel of
/ρ
0
≈
0
.
/
