Chemistry Reference
In-Depth Information
Fig. 7.16
The ensemble average (of 7 droplets) of the mean square displacement as a function of
time. The
solid blue line
is a linear fit used to calculate the effective diffusivity of the droplets. The
dotted blue
like has a slope of 2. The
black cross-lines
are a guide to identify the cross-over from
the ballistic to diffusive regimes. The
red
trace with the axis on the right is the autocorrelation of
the velocity
average of the initial velocity of the droplets when the time variable is rescaled
according to
7.2
as discussed above.
However, it must be noted that the present experiments are limited by a finite
system size, while the theory is for the thermodynamic limit with
N
.More
importantly, there are strong hydrodynamic effects which can change the interactions
between the swimmers, that are not present in the theory. These could be the reason
for the discrepancy between the theoretical and experimental values found above and
the statistical inferences drawn might be improved when such effects are taken into
account or if the experiments are modified.
Finally, we show that as we change the one-dimensional confinement of the
droplets to what may be termed as a quasi 1.5-D confinement, the dynamics of the
droplet motion (as measured in one-dimension) become diffusive. This can be seen in
Fig.
7.17
where we progressively reduce the droplet size, while still maintaining the
condition that neighbouring droplets do not pass each other, such that the motion of
the droplets is not strictly one-dimensional. In addition to the reduction of the droplet
size, the number density also increases. In such a setting, droplets change direction
not only in response to a collision, but also due to changes in swimming direction and
therefore their dynamics appears more diffusive. As we can see from the MSD calcu-
lated for these experiments, the short time dynamics change from being ballistic for
the strictly one-dimensional motion, through enhanced diffusive dynamics, finally
to a close to diffusive dynamics when the confinement is significantly reduced. The
MSD is calculated only for one dimension in all these experiments. When the short
time dynamics tend to be diffusive, as we see for the data in red, we see a crossover
to the anomalous diffusive regime as described in Eq. (
7.5
). Further, since the density
increases, the mean time between collisions reduces, thus shifting the crossover to
earlier times.
→∞
