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rectification under non-equilibrium conditions. The influences of the length and taper
of the walls and the ratio of droplet to gap size are currently under investigation from
the data already at hand. Further, it must be noted that due to the limited 'lifetime' of
the swimmers in this experiment, due to the consumption of fuel, higher distribution
ratios could not be achieved.
The droplet squirmers are not just a simple physical model to study the hydrody-
namic effects of collective swimming, but also model SPPs well suited for the study
of non-equilibrium phenomena. The rectified motion in this system has similarities
to Brownian ratchets where particles undergoing Brownian motion in the presence
of an asymmetric potential can exhibit a net drift in the presence of an additional
ac drive or flashing potential substrate. Whereas the laws of thermodynamics pro-
hibit extraction of useful work from the Brownian motion of particles in equilibrium,
these motions can be 'rectified' under nonequilibrium conditions as demonstrated
here. Our system is not a Brownian ratchet in this sense; instead, it can be considered
to be a realization of a correlation ratchet. The rectification described above is an
example for a ratchet created by a broken spatial symmetry (due to the asymmet-
ric barriers) in combination with a broken temporal symmetry (due to the ballistic
motion of the swimmers along walls). In correlation ratchets, an overdamped particle
can exhibit dc drift on an asymmetric substrate in the absence of an ac flashing or
rocking provided that the fluctuations of the particle motion have certain proper-
ties that break detailed balance. In our self propelled droplets, the force due to the
Marangoni stresses that drives the droplet motion causes the fluctuations to break
detailed balance and thus resulting in the broken temporal symmetry.
It has been suggested that such effects could also be at work in biological lipid
membraneswhere asymmetric ion poresmay allowfor unidirectional diffusion across
the membranes [
28
]. However, in such settings, the pore size is comparable to the size
of the molecule traversing it, such that together with a high molecular concentration
additional effects arise due to 'stacking up' of the molecules along the length of the
ion pore. In the next section, we study the dynamics of themoving droplets when their
motion is confined to a single dimension, which is reminiscent of such a situation.
7.3.4 Swimmers in One-Dimension
We can confine the droplet squirmers to a microchannel, such that their motion is
restricted to one dimension as shown in Fig.
7.11
. In such a geometry, the droplets
cannot pass each other, but whenever they collide, they retrace their paths till they
meet with a collision again. Since the initial positions, directions and speeds are
random, interesting statistics emerge from such motion. In fact, the diffusion of
particles in a single dimension where the mutual passage is forbidden and thus the
sequence of particles remains the same over time is known as single file diffusion
(SFD) and has been studied for a long time [
29
,
30
]. In the thermodynamic limit
(when the density of particles
L
is constant where
N
is the number of
particles moving on a segment of length
L
, and
N
ρ
=
N
/
,
→∞
L
), it has been shown
