Microfluidic
Crystals

Expression
on a Chip

State of
a Living Cell

Protein
Nano-
Structures

Methods

 
 

 

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).