Chemistry Reference
In-Depth Information
Fig. 7.11
A time lapse series
of droplets confined to motion
in a single dimension in
a microchannel. Droplets
retrace their paths at each
collision, reminiscent of single
file diffusion
[
29
] that the expression for the mean squared displacement (MSD) of each particle
undergoing SFD can be written as
x
2
(
t
−
t
)
=|
x
(
t
)
|
/ρ
(7.3)
with
|
x
(
t
)
|
denoting the absolute displacement of a free particle. For ballistic
particles,
is the ensemble average taken over the
distribution of initial velocities. Therefore, the MSD for a ballistic single file (bSF)
particle can be written as
|
x
(
t
)
| = |
v
|
t
, where
···
x
2
(
t
−
t
)
=|
v
|
t
/ρ
(7.4)
such that a bSF particle diffuses apparently like a Brownian particle with normal
diffusion coefficient
D
.
For a stochastic single file (SSF) i.e. a single file of Brownia
n particl
es with a
damping constant
=|
v
|
/ (
2
ρ)
)
| =
√
4
D
0
t
η
at temperature
T
, the equality
|
x
(
t
/π
results in
the anomalous diffusion law
2
F
√
t
x
2
(
t
−
t
)
=
/ρ
(7.5)
where
F
is a mo
bility
factor related to the single particle diffusion constant,
D
0
=
κ
=
√
D
0
/π
/η
.
The moving droplets in a microchannel are reminiscent of a bSF, such that they
move ballistically between collisions. This can be seen in Fig.
7.12
where the velocity
of 4 representative droplets in a single file are plotted as a function of time. The
changes in droplet direction are marked by droplet collisions, corresponding with
the zero crossings of the velocity traces. Between the collisions, the velocity of
the droplet stays roughly constant, characteristic of the ballistic regime. From the
crossover times, we get an estimate of the most probable collision time between the
droplets as shown in Fig.
7.13
and we see that there are peaks in the distribution
corresponding to collision times of
T
,as
F
∼
48,
∼
100 and
∼
220 s. The various peaks are
