Chaos in droplet based microfluidics Patrick TABELING, Herve WILLAIME, Valessa BARBIER, Laure MENETRIER, Alice Mc DONALD ESPCI, MMN, Paris

Complex dynamical behavior is not unfrequent in droplet based microfluidics

From a dynamical point of view, a droplet emitter is a a non linear oscillator Droplets are emitted periodically H. Willaime, V. Barbier, P. Tabeling, Phys. Rev. Lett., 96, (2006) Local intensity measurement

Perturbing the emitter with a mechanical actuator gives rise to complex dynamics PDMS Working channel Actuation channel Local intensity measurement H. Willaime, V. Barbier, P. Tabeling, Phys. Rev. Lett., 96, (2006) GLASS

Water + Fluorescein Oil + Span 80 actuator Movie slowed down three times Natural emission frequency = 5 Hz There exists frequency locking states

Periodic state f f = 1.5 Hz Quasi-periodic state f f =0.7 Hz There also exists quasi periodic states

Different regimes obtained at different forcing amplitudes and frequencies Quasi periodic Periodic 1/3 Periodic 1/1 Periodic 2/3

(a) (b) W Arnold tongues and devil staircases in microdroplet flows H. Willaime, V. Barbier, P. Tabeling, Phys. Rev. Lett., 96, (2006)

Why is it so ? The parametric excitation Oscillating plate

The mechanism leading to complex behavior is parametric excitation Step 1 : The emission frequency of an individual emitter depends of the water flow-rate QwQw QwQw QOQO Step 2 : The actuator modulates Q w and therefore modulates the frequency

Devil staircases also exist with electric fields

Tongues staircases…

water oil Feeding the droplet computer requires droplet emittors connected to each other through the device oil water Droplet-based microfluidic computer Output

The behaviour of an elementary parallel system - Section of the channels : 250 x 40 µm 2 water oil sensors 10 mm V. Barbier, H. Willaime, F. Jousse, P. Tabeling, Phys Rev E (2006)

SYNCHRONIZED REGIMES

Synchronized regimes are favorable for the production of monodisperse emulsions

Surprisingly, chaos frequently appears in this system V. Barbier, H. Willaime, F. Jousse, P. Tabeling, Phys Rev E (2006)

 P/(P w -P 0) ≈ 1% Synchronized regimes may be sensitive to small imperfections

RORO RORO R’ S RSRS QWQW QWQW QOQO qoqo q’ o q’ s qsqs P P’ PSPS P0P0 The mechanism is also parametric excitation Model g = coupling parameter

The main regimes (locked, QP, chaos) are qualitatively reproduced by the dynamical system we used Modelling an elementary parallel system g=0.7 g=0.35 g=0.05

Dynamical phenomena must be taken seriously in droplet-based microfluidic systems

water oil An ubiquitous presence of chaos may jeopardize the possibility of devising droplet based computers oil water Droplet-based microfluidic computer Output R R R R

Microfluidic computer M does not include just resistance-type terms, but something generally more complicated InputOutput May microfluidic computers have a rich dynamics ?

Microfluidic computer InputOutput Should we decouple each element from each other to avoid complex behavior ? X + R?

Complex dynamical phenomena must be taken seriously in droplet-based microfluidic systems Conclusion

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