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.
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The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon request.
The computer codes used for simulations are available from the corresponding authors upon request.
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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).
The authors declare no competing interests.
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.
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Supplementary Information, Supplementary Figs. 1–10 and Supplementary Refs. 1–9.
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.
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.
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.
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).
Stationary state (oscillations) of the system shown in Fig. 3a (top) alongside with corroborating simulation (bottom panel) from Fig. 3c.
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.
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.
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.
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.
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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
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