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Braess’s paradox and programmable behaviour in microfluidic networks

Abstract

Microfluidic systems are now being designed with precision as miniaturized fluid manipulation devices that can execute increasingly complex tasks. However, their operation often requires numerous external control devices owing to the typically linear nature of microscale flows, which has hampered the development of integrated control mechanisms. Here we address this difficulty by designing microfluidic networks that exhibit a nonlinear relation between the applied pressure and the flow rate, which can be harnessed to switch the direction of internal flows solely by manipulating the input and/or output pressures. We show that these networks— implemented using rigid polymer channels carrying water—exhibit an experimentally supported fluid analogue of Braess’s paradox, in which closing an intermediate channel results in a higher, rather than lower, total flow rate. The harnessed behaviour is scalable and can be used to implement flow routing with multiple switches. These findings have the potential to advance the development of built-in control mechanisms in microfluidic networks, thereby facilitating the creation of portable systems and enabling novel applications in areas ranging from wearable healthcare technologies to deployable space systems.

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Fig. 1: System schematics.
Fig. 2: Development of nonlinear flow.
Fig. 3: Braess’s paradox and flow switching.
Fig. 4: Experimental observation of flow switch and Braess’s paradox.
Fig. 5: Flow patterns in a multiswitch network.

Data availability

The datasets generated and/or analysed during the current study are available from the corresponding author on reasonable request.

Code availability

Custom Python code is available from the corresponding author on request.

References

  1. 1.

    Pennathur, S. Flow control in microfluidics: are the workhorse flows adequate? Lab Chip 8, 383–387 (2008).

    CAS  Google Scholar 

  2. 2.

    Stone, H. A. Microfluidics: tuned-in flow control. Nat. Phys. 5, 178–179 (2009).

    CAS  Google Scholar 

  3. 3.

    Perdigones, F., Luque, A. & Quero, J. M. Correspondence between electronics and fluids in MEMS: designing microfluidic systems using electronics. IEEE Ind. Electron. Mag. 8, 6–17 (2014).

    Google Scholar 

  4. 4.

    Thorsen, T., Maerkl, S. J. & Quake, S. R. Microfluidic large-scale integration. Science 298, 580–584 (2002).

    ADS  CAS  Google Scholar 

  5. 5.

    Geertz, M., Shore, D. & Maerkl, S. J. Massively parallel measurements of molecular interaction kinetics on a microfluidic platform. Proc. Natl Acad. Sci. USA 109, 16540–16545 (2012).

    ADS  CAS  Google Scholar 

  6. 6.

    Seker, E. et al. Nonlinear pressure-flow relationships for passive microfluidic valves. Lab Chip 9, 2691–2697 (2009).

    CAS  Google Scholar 

  7. 7.

    Weaver, J. A., Melin, J., Stark, D., Quake, S. R. & Horowitz, M. A. Static control logic for microfluidic devices using pressure-gain valves. Nat. Phys. 6, 218–223 (2010).

    CAS  Google Scholar 

  8. 8.

    Tanyeri, M., Ranka, M., Sittipolkul, N. & Schroeder, C. M. Microfluidic Wheatstone bridge for rapid sample analysis. Lab Chip 11, 4181–4186 (2011).

    CAS  Google Scholar 

  9. 9.

    Kim, S.-J., Lai, D., Park, J. Y., Yokokawa, R. & Takayama, S. Microfluidic automation using elastomeric valves and droplets: reducing reliance on external controllers. Small 8, 2925–2934 (2012).

    CAS  PubMed  PubMed Central  Google Scholar 

  10. 10.

    Li, L., Mo, J. & Li, Z. Nanofluidic diode for simple fluids without moving parts. Phys. Rev. Lett. 115, 134503 (2015).

    ADS  Google Scholar 

  11. 11.

    Chin, C. D., Linder, V. & Sia, S. K. Commercialization of microfluidic point-of-care diagnostic devices. Lab Chip 12, 2118–2134 (2012).

    CAS  Google Scholar 

  12. 12.

    Araci, I. E., Su, B., Quake, S. R. & Mandel, Y. An implantable microfluidic device for self-monitoring of intraocular pressure. Nat. Med. 20, 1074–1078 (2014).

    CAS  Google Scholar 

  13. 13.

    Bhatia, S. N. & Ingber, D. E. Microfluidic organs-on-chips. Nat. Biotechnol. 32, 760–772 (2014).

    CAS  PubMed  Google Scholar 

  14. 14.

    Sackmann, E. K., Fulton, A. L. & Beebe, D. J. The present and future role of microfluidics in biomedical research. Nature 507, 181–189 (2014).

    ADS  CAS  Google Scholar 

  15. 15.

    Leslie, D. C. et al. Frequency-specific flow control in microfluidic circuits with passive elastomeric features. Nat. Phys. 5, 231–235 (2009).

    CAS  Google Scholar 

  16. 16.

    Mosadegh, B. et al. Integrated elastomeric components for autonomous regulation of sequential and oscillatory flow switching in microfluidic devices. Nat. Phys. 6, 433–437 (2010).

    CAS  PubMed  PubMed Central  Google Scholar 

  17. 17.

    Duncan, P. N., Nguyen, T. V. & Hui, E. E. Pneumatic oscillator circuits for timing and control of integrated microfluidics. Proc. Natl Acad. Sci. USA 110, 18104–18109 (2013).

    ADS  CAS  Google Scholar 

  18. 18.

    Duncan, P. N., Ahrar, S. & Hui, E. E. Scaling of pneumatic digital logic circuits. Lab Chip 15, 1360–1365 (2015).

    CAS  Google Scholar 

  19. 19.

    Doh, I. & Cho, Y.-H. Passive flow-rate regulators using pressure-dependent autonomous deflection of parallel membrane valves. Lab Chip 9, 2070–2075 (2009).

    CAS  Google Scholar 

  20. 20.

    Collino, R. R. et al. Flow switching in microfluidic networks using passive features and frequency tuning. Lab Chip 13, 3668–3674 (2013).

    CAS  Google Scholar 

  21. 21.

    Stroock, A. D. et al. Chaotic mixer for microchannels. Science 295, 647–651 (2002).

    ADS  CAS  Google Scholar 

  22. 22.

    Squires, T. M. & Quake, S. R. Microfluidics: fluid physics at the nanoliter scale. Rev. Mod. Phys. 77, 977–1026 (2005).

    ADS  CAS  Google Scholar 

  23. 23.

    Amini, H., Lee, W. & Di Carlo, D. Inertial microfluidic physics. Lab Chip 14, 2739–2761 (2014).

    CAS  Google Scholar 

  24. 24.

    Zhang, J. et al. Fundamentals and applications of inertial microfluidics: a review. Lab Chip 16, 10–34 (2016).

    CAS  Google Scholar 

  25. 25.

    Tesař, V. & Bandalusena, H. C. H. Bistable diverter valve in microfluidics. Exp. Fluids 50, 1225–1233 (2011).

    Google Scholar 

  26. 26.

    Amini, H. et al. Engineering fluid flow using sequenced microstructures. Nat. Commun. 4, 1826 (2013).

    ADS  Google Scholar 

  27. 27.

    Sudarsan, A. P. & Ugaz, V. M. Multivortex micromixing. Proc. Natl Acad. Sci. USA 103, 7228–7233 (2006).

    ADS  CAS  Google Scholar 

  28. 28.

    Di Carlo, D., Edd, J. F., Humphry, K. J., Stone, H. A. & Toner, M. Particle segregation and dynamics in confined flows. Phys. Rev. Lett. 102, 094503 (2009).

    ADS  PubMed  PubMed Central  Google Scholar 

  29. 29.

    Wang, X. & Papautsky, I. Size-based microfluidic multimodal microparticle sorter. Lab Chip 15, 1350–1359 (2015).

    CAS  Google Scholar 

  30. 30.

    Xia, H. M. et al. Analyzing the transition pressure and viscosity limit of a hydroelastic microfluidic oscillator. Appl. Phys. Lett. 104, 024101 (2014).

    ADS  Google Scholar 

  31. 31.

    Braess, D. Über ein Paradoxon aus der Verkehrsplanung. Unternehmensforschung 12, 258–268 (1968).

    MathSciNet  MATH  Google Scholar 

  32. 32.

    Braess, D., Nagurney, A. & Wakolbinger, T. On a paradox of traffic planning. Transport. Sci. 39, 446–450 (2005).

    Google Scholar 

  33. 33.

    Rojas, S. & Koplik, J. Nonlinear flow in porous media. Phys. Rev. E 58, 4776–4782 (1998).

    ADS  CAS  Google Scholar 

  34. 34.

    Andrade, J. S. Jr, Costa, U. M. S., Almeida, M. P., Makse, H. A. & Stanley, H. E. Inertial effects on fluid flow through disordered porous media. Phys. Rev. Lett. 82, 5249–5252 (1999).

    ADS  CAS  Google Scholar 

  35. 35.

    Fourar, M., Radilla, G., Lenormand, R. & Moyne, C. On the non-linear behavior of a laminar single-phase flow through two and three-dimensional porous media. Adv. Water Resour. 27, 669–677 (2004).

    ADS  Google Scholar 

  36. 36.

    Adams, M. L., Johnston, M. L., Scherer, A. & Quake, S. R. Polydimethylsiloxane based microfluidic diode. J. Micromech. Microeng. 15, 1517–1521 (2005).

    ADS  CAS  Google Scholar 

  37. 37.

    Calvert, B. & Keady, G. Braess’s paradox and power-law nonlinearities in networks. J. Aust. Math. Soc. Ser. B 35, 1–22 (1993).

    MathSciNet  MATH  Google Scholar 

  38. 38.

    Penchina, C. M. Braess’s paradox and power-law nonlinearities in five-arc and six-arc two-terminal networks. Open Transplant. J. 3, 8–14 (2009).

    Google Scholar 

  39. 39.

    Ayala, L. F. & Blumsack, S. The Braess paradox and its impact on natural-gas-network performance. Oil Gas Facilities 2, 52–64 (2013).

    Google Scholar 

  40. 40.

    Cohen, J. E. & Horowitz, P. Paradoxical behavior of mechanical and electrical networks. Nature 352, 699–701 (1991).

    ADS  Google Scholar 

  41. 41.

    Youn, H., Gastner, M. T. & Jeong, H. Price of anarchy in transportation networks: efficiency and optimality control. Phys. Rev. Lett. 101, 128701 (2008).

    ADS  Google Scholar 

  42. 42.

    Nicolaou, Z. G. & Motter, A. E. Mechanical metamaterials with negative compressibility transitions. Nat. Mater. 11, 608–613 (2012).

    ADS  CAS  Google Scholar 

  43. 43.

    Pala, M. G. et al. Transport inefficiency in branched-out mesoscopic networks: an analog of the Braess paradox. Phys. Rev. Lett. 108, 076802 (2012).

    ADS  CAS  Google Scholar 

  44. 44.

    Motter, A. E. & Timme, M. Antagonistic phenomena in network dynamics. Annu. Rev. Condens. Matter Phys. 9, 463–484 (2018).

    ADS  PubMed  PubMed Central  Google Scholar 

  45. 45.

    Crane Co.  Engineering Division. Flow of Fluids through Valves, Fittings, and Pipe. Technical paper no. 410 (Crane Co., 2010).

  46. 46.

    Khodaparast, S., Borhani, N. & Thome, J. R. Sudden expansions in circular microchannels: flow dynamics and pressure drop. Microfluid. Nanofluidics 17, 561–572 (2014).

    CAS  Google Scholar 

  47. 47.

    Bhargava, K. C., Thompson, B. & Malmstadt, N. Discrete elements for 3D microfluidics. Proc. Natl Acad. Sci. USA 111, 15013–15018 (2014).

    ADS  CAS  Google Scholar 

  48. 48.

    OpenFOAM v4.1 (OpenFOAM Foundation, 2016).

  49. 49.

    Geuzaine, C. & Remacle, J.-F. Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Methods Eng. 79, 1309–1331 (2009).

    MATH  Google Scholar 

  50. 50.

    Oh, K. W., Lee, K., Ahn, B. & Furlani, E. P. Design of pressure-driven microfluidic networks using electric circuit analogy. Lab Chip 12, 515–545 (2012).

    CAS  Google Scholar 

  51. 51.

    Zeitoun, R. I., Langelier, S. M. & Gill, R. T. Implications of variable fluid resistance caused by start-up flow in microfluidic networks. Microfluid. Nanofluidics 16, 473–482 (2014).

    CAS  Google Scholar 

  52. 52.

    Zovatto, L. & Pedrizzetti, G. Flow about a circular cylinder between parallel walls. J. Fluid Mech. 440, 1–25 (2001).

    ADS  CAS  MATH  Google Scholar 

  53. 53.

    Gervais, T., El-ali, J., Gunther, A. & Jensen, K. F. Flow-induced deformation of shallow microfluidic channels. Lab Chip 6, 500–507 (2006).

    CAS  Google Scholar 

  54. 54.

    Christov, I. C., Cognet, V., Shidhore, T. C. & Stone, H. A. Flow rate–pressure drop relation for deformable shallow microfluidic channels. J. Fluid Mech. 841, 267–286 (2018).

    ADS  MathSciNet  CAS  MATH  Google Scholar 

  55. 55.

    Amstad, E., Datta, S. S. & Weitz, D. A. The microfluidic post-array device: high throughput production of single emulsion drops. Lab Chip 14, 705–709 (2014).

    CAS  Google Scholar 

  56. 56.

    Haudin, F., Callewaert, M., De Malsche, W. & De Wit, A. Influence of nonideal mixing properties on viscous fingering in micropillar array columns. Phys. Rev. Fluids 1, 074001 (2016).

    ADS  Google Scholar 

  57. 57.

    Zhao, H., Liu, Z., Zhang, C., Guan, N. & Zhao, H. Pressure drop and friction factor of a rectangular channel with staggered mini pin fins of different shapes. Exp. Therm. Fluid Sci. 71, 57–69 (2016).

    Google Scholar 

  58. 58.

    Kim, M., Huang, Y., Choi, K. & Hidrovo, C. H. The improved resistance of PDMS to pressure-induced deformation and chemical solvent swelling for microfluidic devices. Microelectron. Eng. 124, 66–75 (2014).

    CAS  Google Scholar 

  59. 59.

    Johnston, I. D., McCluskey, D. K., Tan, C. K. L. & Tracey, M. C. Mechanical characterization of bulk sylgard 184 for microfluidics and microengineering. J. Micromech. Microeng. 24, 035017 (2014).

    ADS  Google Scholar 

  60. 60.

    Martin, R. S., Gawron, A. J., Lunte, S. M. & Henry, C. S. Dual-electrode electrochemical detection for poly(dimethylsiloxane)-fabricated capillary electrophoresis microchips. Anal. Chem. 72, 3196–3202 (2000).

    CAS  Google Scholar 

  61. 61.

    Duffy, D. C., McDonald, J. C., Schueller, O. J. A. & Whitesides, G. M. Rapid prototyping of microfluidic systems in poly(dimethylsiloxane). Anal. Chem. 70, 4974–4984 (1998).

    CAS  Google Scholar 

  62. 62.

    Lachaux, J. et al. Thermoplastic elastomer with advanced hydrophilization and bonding performances for rapid (30 s) and easy molding of microfluidic devices. Lab Chip 17, 2581–2594 (2017).

    CAS  Google Scholar 

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Acknowledgements

This research was supported by the US National Science Foundation (grants PHY-1001198 and CHE-1900011), the Simons Foundation (award number 342906) and a Northwestern University Presidential Fellowship.

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D.J.C., J.-R.A. and A.E.M. designed the overall study and formulated the theory. Y.L. and I.Z.K. designed and performed the experiments. D.J.C. implemented the numerical simulations and analyses. All authors contributed to the writing of the manuscript, which was led by D.J.C. and A.E.M. All authors reviewed and approved the final manuscript.

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Correspondence to Adilson E. Motter.

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Peer review information Nature thanks Sujit Datta and Dino Di Carlo for their contribution to the peer review of this work.

Supplementary information

Supplementary Information

Additional theoretical, simulation, and experimental results across six sections, fifteen figures, and one table are included. It contains two sections with details of the theoretical network model and four sections on further simulations and experiments of nonlinear flow behaviour, switching, Braess’s paradox, and multiswitch networks.

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Case, D.J., Liu, Y., Kiss, I.Z. et al. Braess’s paradox and programmable behaviour in microfluidic networks. Nature 574, 647–652 (2019). https://doi.org/10.1038/s41586-019-1701-6

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