Collective
Dynamics of Microfluidic Droplets
Far from Thermal Equilibrium

Non-equilibrium systems with many interacting particles
often exhibit highly complex patterns and dynamics. Many such patterns
are known from our everyday life: weather and turbulence, global
economics, zebra stripes and, ultimately, life itself, which is probably
the most striking non-equilibrium phenomenon. The understanding of
non-equilibrium dynamics lags well behind the advanced theory of systems
at thermal equilibrium. A system at equilibrium can be described by a
free-energy functional, whose minimization predicts the system’s
behavior. Beyond equilibrium, however, this approach breaks down –
energy functionals are unavailable, which renders the formulation of a
general non-equilibrium theory a difficult problem.

Movie 1: Formation of water-in-oil droplets in a microfluidic
T-junction.
The ordered array is, in fact, a one-dimensional crystal of interacting
droplets.

Microfluidic two-phase flow and, specifically, the flow
of water-in-oil droplets in quasi-2D channels, offer a convenient
experimental tool to investigate non-equilibrium dynamics. The flow of
oil drives the system far from equilibrium, breaks its translational
symmetry and induces a long-range hydrodynamic dipolar interactions
between droplets that decay as r^{-2}
. The flow is heavily dissipative with Reynolds number 10^{-4}
, which implies tractable, linear flow equations. Finally, its planar
geometry with lengths of tens of microns and time scales of tenths of a
second, make it experimentally accessible using standard optical
microscopy.

A microfluidic 1D crystal is probably the simplest
many-body system made of droplets. The disc-like droplets are formed at
a T-junction between water and oil channels, emanating at constant size
and frequency due to the force balance on the tip of the water at the
junction (Movie 1). The Reynolds number of this system is very low
, hence vibrations are expected to be over-damped. Against such
expectations, we found that the crystal carries phonon-like
vibrational modes that propagate at sound velocity of
~100microns/s and frequencies of ~1Hz
(Fig. 1). These phonons exhibit unusual dispersion markedly different
than those of harmonic crystals and give rise to a variety of non-linear
crystal instabilities. We developed a first-principles theory
showing how these phonons arise from the symmetry-breaking flow field,
which induces long-range inter-droplet hydrodynamic interactions.

Movie 2: Following the crystal as it flows downstream reveals
phonons that propagate along the crystal. Both longitudinal and
transversal phonons can be seen. When the transversal amplitude
increases the crystal breaks due to the non-linear interactions between
its modes.

a.

c.

b.

Figure 1: (a)-(b) snapshots
of phonons in a 1D microfluidic crystal. Both transversal and
longitudinal modes are easily seen. Channel width was 250µm and
its height 10µm. Scale-bars are 100µm. (c) the power-
spectrum of longitudinal phonons. The sine-like curve corresponds
to the phonons dispersion relation ω(k).

Movie 3: Zoon-in of a longitudinal phonons. Flow is from left to
right and the camera follows the crystal in its mean velocity. Long
longitudinal modes propagate against the flow, as seen also from their
dispersion relation (Figure 1c).

Movie 4: Zoom-in of a transversal phonon. Long transversal modes
propagate in the direction of flow.

Later, we studied the boundary effect of the channel
sidewalls on the phonon (Figure 2). We measured the phonon spectra of 1D
microfluidic crystals under different degrees of confinement,
ranging from unconfined flow in 2D to 1D flow, where the channel is
nearly blocked by the droplets ('plug-flow'). Long range forces, such as
the hydrodynamic dipolar force, are known to be radically affected by
boundaries and dimensionality. Hence, the inter-droplet forces that fall
off as r^{-2} without boundaries, cross over under
confinement to decay as exp(–2πr/W), where W
is the channel width. However, close to plug-flow, and despite the
weakening of interactions due to screening, the magnitude of
inter-droplet forces increases as tan(πR/W)
due to the crystal’s incompressibility, R being the droplet
radius. This interplay between hydrodynamic screening and
incompressibility is reflected in a non-monotonous behavior of
the phonon spectra. Confinement breaks the translational invariance,
which is manifested in the breaking of the x–y anti-symmetry of
unconfined spectra. Additionally, the approach to incompressibility in
the 1D limit implies a divergence of sound velocity and, indeed, we
observed its marked increase.

d.

Figure 2: (a)-(c) Microfluidic
crystals in different degrees of confinement. The three images
correspond to a confinement parameter (γ=2R/W) of γ=0.62,
0.58 and 0.46 respectively. Phonons are readily observed in (b)
and (c). Scale-bars are 50µm.
(d) the transversal power
spectrum at γ=0.63. The faint sine-like curve is phonon
dispersion. In black is our theoretical prediction. In white –
the prediction for an unconfined crystal of the same parameters.
Notice the marked increase in sound velocity and frequency due to
confinement.

Finally, we measured the complex dynamics of a 2D disordered
ensemble of droplets, where we observed ultra slow shock waves and sound
propagating at ~100µm/s. These modes obey the renounced Burgers
equation in one dimension.

This part is still under construction, on the meantime,
see our PRL paper "Burgers Shock Waves
and Sound in a 2D Microfluidic Droplets Ensemble", Phys.
Rev. Lett 103, 114502 (2009) (including movies).