Chemistry Reference
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straightforward explanation why they found velocity correlations ranging up to more
than five droplet diameters, in marked contrast to our results.
Correlations become even more pronounced as the density of the droplets is
increased to an areal density of 0.78. As the black curve in Fig.
7.2
shows, we observe
two significant changes. First, the range of the correlation comes significantly larger,
extending clearly beyond four droplet diameters. Second, we observe the appearance
of distinct peaks in the correlation function. The black arrows are at multiples of 1.08
droplet diameters, which is close to what one would expect in case of lateral 'layer-
ing' effects. Also this marked texture, which may be described as ordered rafts, has
not been reported before from simulations of similar systems. An obvious possible
reason is the particularly high areal density in our experiment (it was up to 0.5 in
the study of Ishikawa and Pedley [
7
]). These results suggest the presence of a phase
transition occurring at a density somewhere between 0.46 and 0.78. Experiments are
under way to search for this transition.
Next, we discuss the effect of the collective behaviour on the effect diffusivities
of the monolayers of swimmers in the two dimensional setting as we just discussed.
We calculate the mean square displacement
x
2
to quantify the motion of
the swimmers. We note here that since the velocity of the droplet is time dependant
as we discussed in Chap.
6
i.e.
v
(
t
−
t
)
where
v
is the velocity of the droplets. In
particular we note that the velocity of the droplet swimmers decreases gradually over
time. Therefore, in order to calculate the mean squared displacement, we introduced
a rescaled time variable
=
v
(
t
)
τ
which is related to the velocity
v
(
t
)
as
t
τ
=
(
)
v
t
dt
(7.2)
0
such that in the frame of the rescaled time, the velocity
v
of the droplets will
remain constant at all times. The thus calculated ensemble averaged MSD for exper-
iments with populations of swimmers at three different areal densities,
(τ )
, are plotted
in Fig.
7.3
. From the MSD plots, it can be seen that the motion of the swimmers tran-
sitions from being ballistic (MSD
φ
t
2
) at the short time scales to being diffusive
∼
(MSD
t
) at long times. The crossover to the diffusive regime clearly happens at
different times for the different concentrations. As the concentration of the swimmers
increases, the crossover shifts to shorter times. At low concentration, the crossover is
much later because the swimmers travel in directionally persistant paths before the
weak hydrodynamic fluctuations can significantly alter their trajectories. As as result,
at the lower concentration the transport is ballistic at short times, and at longer times
a crossover to diffusive behavior occurs. However, as the concentration is increased
the diffusivity of the swimmers decreases, as their naturally ballistic trajectories are
increasingly perturbed by hydrodynamic interactions with other swimmers.
For the diffusive regime, an effective diffusivity can be calculated using a linear
fit to the data at the long times (
∼
80) and this is plotted as a function of the areal
density in the right panel of Fig.
7.4
. As we expect from the argument above and
also evident from the MSD traces in Fig.
7.3
, the effective diffusivity of the swimmer
populations reduces with increasing swimmer density. However, we see a levelling
τ >
