Oscillating droplet trains in microfluidic networks and their suppression in blood flow

Abstract

Droplets forming sequences in simple microfluidic networks are known to exhibit complex behaviours, but their dynamics are yet to be probed in channels long enough to accommodate many droplets simultaneously. Here we show that uniform sequences of liquid droplets flowing through microfluidic networks can spontaneously form ‘trains’ that periodically exchange between different branches of the network. Such system-wide oscillations do not rely on direct droplet–droplet interactions, are common to networks of various topologies, can be controlled or eliminated by adjusting network dimensions and can synchronize into larger flow patterns. The oscillations can also be suppressed via droplet collisions at diverging junctions. This mechanism may explain why red blood cells in microcapillaries exhibit only low-amplitude oscillations, preventing dangerous local hypertension or hypoxia that might otherwise ensue. Our findings are substantiated by a theoretical model that treats droplets as sets of moving points in one-dimensional ducts and captures the dynamics of large droplet ensembles without invoking the microscopic details of flows in or around the droplets. For blood flow, this simplified description offers more realistic estimates than continuous haemodynamic models, indicating the relevance of the discrete nature of blood to the excitation of oscillations.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

Fig. 1: Emergence and quantification of large-scale oscillations in a simple network.
Fig. 2: Regulation of flow patterns by side channels (bridges), non-accessible to droplets.
Fig. 3: Experiments and simulations of the simplest oscillating network.
Fig. 4: Suppressing and avoiding oscillations by collision-driven dynamics of droplets.
Fig. 5: Non-deterministic turns of droplets (or blood cells) at diverging junctions and oscillations in networks rescaled to mimic blood microcirculation.

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon request.

Code availability

The computer codes used for simulations are available from the corresponding authors upon request.

References

  1. 1.

    Wang, X., Lai, Y.-C. & Lai, C. H. Oscillations of complex networks. Phys. Rev. E 74, 66104 (2006).

    ADS  Article  Google Scholar 

  2. 2.

    Li, P., Yi, Z. & Zhang, L. Global synchronization of a class of delayed complex networks. Chaos Solitons Fractals 30, 903–908 (2006).

    ADS  MathSciNet  Article  Google Scholar 

  3. 3.

    Jin, W.-L. & Zhang, Y. Paramics simulation of periodic oscillations caused by network geometry. Transp. Res. Rec. J. Transp. Res. Board 1934, 188–196 (2005).

    Article  Google Scholar 

  4. 4.

    Nie, Y. & Zhang, H. M. Oscillatory traffic flow patterns induced by queue spillback in a simple road network. Transp. Sci. 42, 236–248 (2008).

    Article  Google Scholar 

  5. 5.

    Mauch, M. & Cassidy, M. J. in Transportation and Traffic Theory in the 21 st Century (ed. Taylor, M. A. P.) 653–673 (Emerald Group Publishing, 2002).

  6. 6.

    Ahn, S., Laval, J. & Cassidy, M. Effects of merging and diverging on freeway traffic oscillations. Transp. Res. Rec. J. Transp. Res. Board 2188, 1–8 (2010).

    Article  Google Scholar 

  7. 7.

    Davis, J. M. & Pozrikidis, C. Numerical simulation of unsteady blood flow through capillary networks. Bull. Math. Biol. 73, 1857–1880 (2011).

    MathSciNet  Article  Google Scholar 

  8. 8.

    Geddes, J. B., Carr, R. T., Karst, N. J. & Wu, F. The onset of oscillations in microvascular blood flow. SIAM J. Appl. Dyn. Syst. 6, 694–727 (2007).

    ADS  MathSciNet  Article  Google Scholar 

  9. 9.

    Zhang, L. & Clark, D. D. Oscillating behavior of network traffic: a case study simulation. J. Internetworking Res. Exp 1, 101–112 (1990).

    Google Scholar 

  10. 10.

    Wang, Z., Wang, Z. & Crowcroft, J. Analysis of shortest-path routing algorithms in a dynamic network environment. ACM Comput. Commun. Rev. 22, 63–71 (1992).

    Article  Google Scholar 

  11. 11.

    Kaminski, T. S., Churski, K. & Garstecki, P. Microdroplet Technology (Springer, 2012).

  12. 12.

    Mashaghi, S., Abbaspourrad, A., Weitz, D. A. & Van Oijen, A. M. Droplet microfluidics: a tool for biology, chemistry and nanotechnology. TrAC Trends Anal. Chem. 82, 118–125 (2016).

    Article  Google Scholar 

  13. 13.

    Chou, W.-L., Lee, P.-Y., Yang, C.-L., Huang, W.-Y. & Lin, Y.-S. Recent advances in applications of droplet microfluidics. Micromachines 6, 1249–1271 (2015).

    Article  Google Scholar 

  14. 14.

    Popel, A. S. & Johnson, P. C. Microcirculation and hemorheology. Annu. Rev. Fluid Mech. 37, 43–69 (2005).

    ADS  MathSciNet  Article  Google Scholar 

  15. 15.

    Pries, A. R., Secomb, T. W., Gaehtgens, P. & Gross, J. F. Blood flow in microvascular networks. Experiments and simulation. Circ. Res. 67, 826–834 (1990).

    Article  Google Scholar 

  16. 16.

    Kiani, M. F., Pries, A. R., Hsu, L. L., Sarelius, I. H. & Cokelet, G. R. Fluctuations in microvascular blood flow parameters caused by hemodynamic mechanisms. Am. J. Physiol. 266, H1822–H1828 (1994).

    Google Scholar 

  17. 17.

    Forouzan, O., Yang, X., Sosa, J. M., Burns, J. M. & Shevkoplyas, S. S. Spontaneous oscillations of capillary blood flow in artificial microvascular networks. Microvasc. Res. 84, 123–132 (2012).

    Article  Google Scholar 

  18. 18.

    Podgoreanu, M. V., Stout, R. G., El-Moalem, H. E. & Silverman, D. G. Synchronous rhythmical vasomotion in the human cutaneous microvasculature during nonpulsatile cardiopulmonary bypass. Anesthesiology 97, 1110–1117 (2002).

    Article  Google Scholar 

  19. 19.

    Palmer, A. F. & Intaglietta, M. Blood substitutes. Annu. Rev. Biomed. Eng. 16, 77–101 (2014).

    Article  Google Scholar 

  20. 20.

    Schindler, M. & Ajdari, A. Droplet traffic in microfluidic networks: a simple model for understanding and designing. Phys. Rev. Lett. 100, 44501 (2008).

    ADS  Article  Google Scholar 

  21. 21.

    Amon, A., Schmit, A., Salkin, L., Courbin, L. & Panizza, P. Path selection rules for droplet trains in single-lane microfluidic networks. Phys. Rev. E 88, 13012 (2013).

    ADS  Article  Google Scholar 

  22. 22.

    Cybulski, O. & Garstecki, P. Dynamic memory in a microfluidic system of droplets traveling through a simple network of microchannels. Lab Chip 10, 484–493 (2010).

    Article  Google Scholar 

  23. 23.

    Glawdel, T., Elbuken, C. & Ren, C. Passive droplet trafficking at microfluidic junctions under geometric and flow asymmetries. Lab Chip 11, 3774–3784 (2011).

    Article  Google Scholar 

  24. 24.

    Sessoms, D. A., Amon, A., Courbin, L. & Panizza, P. Complex dynamics of droplet traffic in a bifurcating microfluidic channel: periodicity, multistability and selection rules. Phys. Rev. Lett. 105, 154501 (2010).

    ADS  Article  Google Scholar 

  25. 25.

    Parthiban, P. & Khan, S. A. Filtering microfluidic bubble trains at a symmetric junction. Lab Chip. 12, 582–588 (2012).

    Article  Google Scholar 

  26. 26.

    Maddala, J., Vanapalli, S. A. & Rengaswamy, R. Origin of periodic and chaotic dynamics due to drops moving in a microfluidic loop device. Phys. Rev. E 89, 23015 (2014).

    ADS  Article  Google Scholar 

  27. 27.

    Wang, W. S. & Vanapalli, S. A. Millifluidics as a simple tool to optimize droplet networks: case study on drop traffic in a bifurcated loop. Biomicrofluidics 8, 64111 (2014).

    Article  Google Scholar 

  28. 28.

    Glawdel, T. & Ren, C. Global network design for robust operation of microfluidic droplet generators with pressure-driven flow. Microfluid. Nanofluid. 13, 469–480 (2012).

    Article  Google Scholar 

  29. 29.

    Djalali Behzad, M. et al. Simulation of droplet trains in microfluidic networks. Phys. Rev. E 82, 37303 (2010).

    ADS  Article  Google Scholar 

  30. 30.

    Smith, B. J. & Gaver, D. P. III Agent-based simulations of complex droplet pattern formation in a two-branch microfluidic network. Lab Chip 10, 303–312 (2010).

    Article  Google Scholar 

  31. 31.

    Maddala, J. & Rengaswamy, R. Design of multi-functional microfluidic ladder networks to passively control droplet spacing using genetic algorithms. Comput. Chem. Eng. 60, 413–425 (2014).

    Article  Google Scholar 

  32. 32.

    Kasule, J. S., Maddala, J., Mobed, P. & Rengaswamy, R. Very large scale droplet microfluidic integration (VLDMI) using genetic algorithm. Comput. Chem. Eng. 85, 94–104 (2016).

    Article  Google Scholar 

  33. 33.

    Kadivar, E., Herminghaus, S. & Brinkmann, M. Droplet sorting in a loop of flat microfluidic channels. J. Phys. Condens. Matter 25, 285102 (2013).

    Article  Google Scholar 

  34. 34.

    Jeanneret, R., Vest, J.-P. & Bartolo, D. Hamiltonian traffic dynamics in microfluidic loop networks. Phys. Rev. Lett. 108, 34501 (2012).

    ADS  Article  Google Scholar 

  35. 35.

    Choi, W. et al. Bubbles navigating through networks of microchannels. Lab Chip 11, 3970–3978 (2011).

    Google Scholar 

  36. 36.

    Champagne, N., Vasseur, R., Montourcy, A. & Bartolo, D. Traffic jams and intermittent flows in microfluidic networks. Phys. Rev. Lett. 105, 44502 (2010).

    ADS  Article  Google Scholar 

  37. 37.

    Parthiban, P. & Khan, S. A. Bistability in droplet traffic at asymmetric microfluidic junctions. Biomicrofluidics 7, 44123 (2013).

    Article  Google Scholar 

  38. 38.

    Churski, K., Nowacki, M., Korczyk, P. M. & Garstecki, P. Simple modular systems for generation of droplets on demand. Lab Chip 13, 3689–3697 (2013).

    Article  Google Scholar 

  39. 39.

    Hoang, D. A., van Steijn, V., Portela, L. M., Kreutzer, M. T. & Kleijn, C. R. Benchmark numerical simulations of segmented two-phase flows in microchannels using the volume of fluid method. Comput. Fluids 86, 28–36 (2013).

    Article  Google Scholar 

  40. 40.

    Jakiela, S., Makulska, S., Korczyk, P. & Garstecki, P. Speed of flow of individual droplets in microfluidic channels as a function of the capillary number, volume of droplets and contrast of viscosities. Lab Chip 11, 3603–3608 (2011).

    Article  Google Scholar 

  41. 41.

    Labrot, V., Schindler, M., Guillot, P., Colin, A. & Joanicot, M. Extracting the hydrodynamic resistance of droplets from their behavior in microchannel networks. Biomicrofluidics 3, 12804 (2009).

    Article  Google Scholar 

  42. 42.

    Cybulski, O., Jakiela, S. & Garstecki, P. Between giant oscillations and uniform distribution of droplets: the role of varying lumen of channels in microfluidic networks. Phys. Rev. E 92, 063008 (2015).

    ADS  Article  Google Scholar 

  43. 43.

    Pries, A. R. & Secomb, T. W. Microvascular blood viscosity in vivo and the endothelial surface layer. Am. J. Physiol. Heart Circ. Physiol. 289, H2657–H2664 (2005).

    Article  Google Scholar 

  44. 44.

    Balogh, P. & Bagchi, P. Direct numerical simulation of cellular-scale blood flow in 3D microvascular networks. Biophys. J. 113, 2815–2826 (2017).

    Article  Google Scholar 

  45. 45.

    Balogh, P. & Bagchi, P. Analysis of red blood cell partitioning at bifurcations in simulated microvascular networks. Phys. Fluids 30, 051902 (2018).

    ADS  Article  Google Scholar 

  46. 46.

    Buchin, V. A. & Shadrina, N. K. Regulation of the lumen of a resistance blood vessel by mechanical stimuli. Fluid Dyn. 45, 211–222 (2010).

    Article  Google Scholar 

Download references

Acknowledgements

O.C. and B.A.G. acknowledge support from the Institute for Basic Science Korea, project code IBS-R020-D1. P.G. was supported by the European Research Council starting grant 279647 and by the National Science Centre Symfonia grant (DEC2014/12/W/NZ6/00454).

Author information

Affiliations

Authors

Contributions

O.C. conceived the idea and performed experiments and simulations. P.G. and B.A.G. supervised and guided the research. All authors participated in the writing of the manuscript.

Corresponding authors

Correspondence to O. Cybulski or P. Garstecki or B. A. Grzybowski.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Journal peer review information: Nature Physics thanks Carolyn Ren and the other anonymous reviewer(s) for their contribution to the peer review of this work.

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Information, Supplementary Figs. 1–10 and Supplementary Refs. 1–9.

Supplementary Video 1

Onset of oscillations in the system from Fig. 1. Droplets of water (dyed blue) suspended in hexane (dyed yellow) flow into the network, which is initially filled solely with hexane. At the beginning, the droplets distribute uniformly all over the network. Subsequently, longer and longer droplet sequences develop with clear oscillations between the two branches visible from approximately 1 min 5 s.

Supplementary Video 2

Oscillations in networks of the same topology as in Fig. 1 and Supplementary Video 1, with three different cross-sections of channels. Original frames from experiments are shown side by side with false-colour images for improved visibility of channels. See Supplementary Section 2 for a detailed description.

Supplementary Video 3

Accelerated (5×) record of the system from Fig. 2a–c, before, during and after blocking the bridge using a plug of very viscous liquid (red dyed glycerol). Oscillations occur when the bridge is blocked—the variation of pressure accompanying them makes the plug move back and forth. After removing the plug, the oscillations disappear.

Supplementary Video 4

System from Fig. 2d–f. The first part of the video shows version without transverse bridges (see Fig. 2d). The second part shows the stationary state of the system with bridges (see Fig. 2e).

Supplementary Video 5

Stationary state (oscillations) of the system shown in Fig. 3a (top) alongside with corroborating simulation (bottom panel) from Fig. 3c.

Supplementary Video 6

Stationary oscillations of the system shown in Fig. 3b (experiment, top panel) and Fig. 3d (simulation, bottom panel), that is, of the same microfluidic chip as in Supplementary Video 5 and Figs. 3a,c, but traversed in the opposite direction.

Supplementary Video 7

Visualization of short fragments of simulations of variously routed versions of the network from Supplementary Fig. 9a, each encompassing a single cycle of oscillation. Sequence of digits in the lower left corner shows the currently presented connectivity and direction of flow. Selected sequences are 01234567, 01452367, 02134567, 02143567, 02361457 and 04263517, each traversed in both directions.

Supplementary Video 8

Impact of mutual collisions between two droplets at the area of a diverging T-junction on the selection of a branch the trailing droplet enters. Initially, flow rate in the left branch is larger than that in the right, so that the majority of incoming droplets turn left. However, every eighth interval between subsequent droplets is shorter, causing these two droplets to collide at the area of the junction, and redirecting the trailing droplet into the opposite branch. Flow difference between branches gradually decreases, becoming negative in the middle of the movie. From this moment on, most droplets turn to the right, whereas the droplets ‘scattered’ off their predecessors turn left.

Supplementary Video 9

Mutual collisions between droplets as a tool for controlling oscillations. As long as droplets are generated pairwise (that is, with alternating short and long intervals), droplets from each pair are directed to two different branches of the network. Consequently, droplets are distributed uniformly all over the network. Upon switching to generation of droplets with constant intervals (the average distance between droplets remains the same), the oscillations slowly arise, reaching the stationary state within a few minutes. Then, pairwise droplet generation is turned on again, leading to systematic (but not immediate) suppression of oscillations. The system comes back to the uniform distribution of droplets.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Cybulski, O., Garstecki, P. & Grzybowski, B.A. Oscillating droplet trains in microfluidic networks and their suppression in blood flow. Nat. Phys. 15, 706–713 (2019). https://doi.org/10.1038/s41567-019-0486-8

Download citation

Further reading

Search

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing