1. Reversibility of droplet trains in microfluidic networks Piotr Garstecki 1, Michael J. Fuerstman 2, George M. Whitesides 2 1 Institute of Physical Chemistry, PAS, Warsaw, Poland 2 Department of Chemistry and Chemical Biology, Harvard University

2. Kenis, Science (1999)

4. the simplest network – a single loop amplification and feedback: drop flows into the arm characterized by lower resistance (higher pressure gradient) once the drop enters a channel it increases its resistance

5. period-1period-2irregularperiod-3 f feed / f flow the simplest network – a single loop Phys. Rev. E (2006)

6. nonlinear dynamics embedded in a linear flow invariant under: x  - x, (or, equivalently V  - V, and p  -p) period 1 period N ??? period 1 period N

7. The “operation” of the system is stable against small differences in the incoming signal Science (2007)

8. there is amplification and feedback, but: the nonlinear events are isolated (very short) the long-range interactions are instantaneous (information is transmitted much faster than the flow proceeds) it is all embedded in a linear, dissipative flow

9. formation of bubbles – a single nozzle gas water Appl. Phys. Lett. 85, 2649 (2004) 1 mm height = 30  m

10. formation of bubbles – a single nozzle gas water Appl. Phys. Lett. 85, 2649 (2004) 1 mm height = 30  m nitrogen (p=8 psi) / 2% Tween20 in water (Q=3 mL/h), orifice width/length/height: 60/150/30  m.

11. gas liquid equilibrium shape for a given volume enclosed by the gas-liquid interface surface evolver end of the orifice end of the gas Inlet channel 50  m rate of collapse linear in the of inflow of the continuous phase only the very last (and short) stage is driven by interfacial tension Phys. Rev. Lett. 94, 164501 (2005)

12. coupled flow-focusing oscillators information (fast) evolution (slow) + dissipative dynamics (low to mod Re) final break-up takes ‘no’ time

13. period-29 coupled flow-focusing oscillators Nature Phys. 1, 168 (2005)

14. coupled flow-focusing oscillators Nature Phys. 1, 168 (2005)

15. 160 kfps – – 6.25  s The observed dynamics is (again) stable. Nature Phys. 1, 168 (2005)

16. dynamics of flow through networks: complicated (complex)  it is possible to design complex, automated protocols stable  the protocols can be executed in practice