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Braess’s paradox and programmable behaviour in microfluidic networks


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.


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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.

Author information




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.

Corresponding author

Correspondence to Adilson E. Motter.

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The authors declare no competing interests.

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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).

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