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.
The motions of cars over networks of roads, blood cells in networks of blood vessels or data packets along networks of cabled routers can often show collective dynamics that may synchronize into system-wide oscillations1,2. While the emergence of such oscillations is still incompletely understood, it is known that they reflect not only system-specific factors (for example, driver behaviour, external regulation of arteries by muscles, route-selection computer algorithms) but also general features of the network, such as its topology, delays in flow related to channel length, common bottlenecks at merging junctions or competition between alternative paths (see refs. 3,4,5,6 for cars, refs. 7,8 for blood and refs. 9,10 for computer networks). Here, we study a related and important11,12,13 problem in the flow of droplets in ‘vascular’ microfluidic networks comprising long channels connected by diverging and converging junctions. We show experimentally, and confirm via modelling, that even with negligible interactions between neighbouring droplets, uniform droplet flows can spontaneously organize into long ‘trains’ that periodically exchange between a network’s various branches. Such oscillations are controlled by the network topology and dimensions, can be made to appear/disappear by reversing the flow direction and can synchronize into larger-scale flow patterns. Our models also rationalize (remarkably, for physiologically relevant parameters) the low amplitude of experimentally observed oscillations of blood flow in the microcirculation14,15,16,17,18 and ascribe it to the non-deterministic ‘turns’ red blood cells make at diverging capillary junctions. Overall, these results may have implications for the design of microfluidic systems directing droplet trains without the use of valves and also for the development of blood-replacement carriers19, determining physical parameters (particle size, concentration) that would ensure suppression of untoward oscillatory variations.
Motions of droplets in linear sequences, also known as single-lane flows20,21, have been studied by many authors—both experimentally21,22,23,24,25,26,27,28 and also via simulations21,22,23,24,25,26,27,28,29,30,31,32,33—but mostly in relatively simple arrangements of channels, such as two-channel loops21,22,23,24,25,26,27,28,29,30,33. Interestingly, even in such basic systems, the droplets were found to exhibit quite complex behaviours such as cyclic repetition of memorized sequences and transitions to chaotic dynamics for specific ranges of parameters21,22,23,24,25,26,29,30. Studies of more intricate networks have been limited to cases with low occupation of channels and included series of simple loops34, cascaded loops21,35 or large regular grids of short channels, mimicking porous materials36. There have been no systematic studies of less trivial networks with channels long enough (as compared to their cross-section) to contain a large number (hundreds and more) of droplets.
In the present work, we consider networks of branching channels connecting a single inlet and a single outlet, which may be perceived either as a standalone system or a subsystem of a more general network. Droplets were always confined between channel walls and we required that their numbers and loads of resistance be large enough to let the droplets explore all branches (in this way, we ignore trivial cases of the pure filtering regime37 in which all droplets follow the same path). The devices were fabricated in polycarbonate by milling and thermal bonding. In the experiments, channels accessible to droplets had a square cross-section ranging from 400 × 400 to 500 × 500 µm. To rule out any systematic dependence of the results on the method of droplet generation and/or small variations in droplet intervals and sizes, we generated droplet lane flows using three independent techniques: (1) a droplet-on-demand (DOD) system38 based on electronically controlled valves; (2) syringe pumps; (3) free flow from a capillary-stabilized hydrostatic pressure head. The results we describe in the following are independent of these technical details.
Figure 1 (Supplementary Video 1) illustrates a typical experiment with a vascular-like network. Dye-coloured droplets select one of four equal-length paths between the network’s inlet and outlet. Initially, the droplets distribute uniformly over the network (Fig. 1a,b) but gradually organize into longer sequences that enter the same branch at the first bifurcation. Ultimately, the lengths of these sequences stabilize and they periodically alternate between the two main branches of the network (Fig. 1c,d). These oscillations are quantified in Fig. 1f and are reproduced in simulations in Fig. 1g (see later in the text).
A series of experiments and simulations with different network types allowed us to identify some regularities in the emergence and possible synchronization of droplet oscillations. First, the longest trains of droplets (that is, the largest non-uniformities in the distribution of droplets) are observed in networks in which the channels diverging from the inlet are shorter than channels converging to the outlet. Interestingly, when the direction of flow in such networks is reversed, the oscillations are much less pronounced (see Fig. 3a–d later and Supplementary Videos 5 and 6). In extreme cases, the same network might exhibit oscillations when droplets flow from shorter to longer channels, but no oscillations in the opposite direction (Supplementary Fig. 1). Second, oscillations are observed over a wide range of fluid properties (for example, on varying the ratio of viscosity of both phases by three orders of magnitude or including systems of gas bubbles in a liquid phase; Supplementary Fig. 2). Third, the oscillations are accompanied by rhythmic variations of pressures and flow rates between the network’s branches. This is vividly illustrated by the system in Fig. 2a–c, where two branches of the network are connected by a transverse channel. When this channel is ‘open’ (that is, filled by the same liquid as the rest of the network), its low resistance allows the pressure between the two branches to equalize at all times, resulting in uniform droplet flows. When, however, the bridge is partially filled by a plug of a very viscous liquid, its resistance becomes high enough such that oscillations in the main channels emerge (as in the system without the bridge, Fig. 1). The back-and-forth motions of the plug are clearly visible in Supplementary Video 3 and reflect small (−100–100 Pa), periodic changes of pressure difference between the ends of the control channel. Fourth, in more complex networks, ‘communication’ between different sub-branches—via transverse channels transmitting pressure but not droplets—can synchronize the droplets into larger-scale spatiotemporal patterns. In this spirit, Fig. 2d–f and Supplementary Video 4 show a network in which droplets flow from a central node into five radially oriented loops, all connected to the same-pressure outlet. If the intermediate nodes are not bridged, droplet trains in each loop emerge but are not ‘globally’ synchronized. When, however, the intermediate nodes are connected by transverse bridges, droplet sequences gradually synchronize and ultimately develop rotary patterns over the radial channels (Fig. 2e and Supplementary Video 4) with the symmetry-breaking into clockwise or anticlockwise directions being, as expected, random. Fifth, we have established that the simplest network supporting oscillations contains two branches, with one splitting and joining back to form a loop (Fig. 3).
To explain these observations, we adapted a simplified model20 that treats droplets as sets of moving points in one-dimensional ducts and captures the dynamics of large droplet ensembles without dealing with the microscopic details of flows inside or in the neighbourhood of droplets (note that volume of fluid (VOF)39 and boundary element methods33 are capable of modelling interfaces of flowing, colliding, coalescing or splitting droplets, but require excessive computational power and are largely redundant in explaining the phenomena we describe). In our model, the flow rate Qi in the ith channel is calculated as Qi = Δpi/Ri, where Δpi is the difference in pressures between the nodes at the ends of this channel, and Ri represents the hydraulic resistance to flow. Importantly, both nodal pressures and flow rates in the channels may be easily calculated by solving a system of linear equations, similar to the Ohm and Kirchhoff laws of electric circuits. Droplets then move along channels with superficial flow velocity Vi = βQi/A, where β is the so-called mobility factor40 (assumed here, due to the square cross-section of the channels, to be β = 1) and A is the cross-sectional area (here assumed constant). At diverging junctions, droplets turn into channels with the highest flow rate. At this level of description there is nothing non-trivial in the dynamics of such droplets—they will simply follow the optimal path. However, the presence of droplets in itself increases the hydraulic resistance of the channel in which they flow, so for n identical droplets \(R_i = R_i^0 + nr\), where \(R_i^0\) represents the hydraulic resistance of the ith channel without droplets and r is the contribution of an individual droplet to this resistance (assumed to be additive and independent of a droplet’s position or distance to neighbouring droplets41). In particular, droplets that follow a given path may elevate its resistance so much that subsequent ones must choose another route. Even though the model does not account for direct interactions between neighbouring droplets (which is justified as long as the gaps between them are larger than the droplet diameters), the ‘point loads of resistance’ carried by the droplets establish system-wide droplet–droplet dependencies. Such global effects are evidenced by the resistances, flow rates and nodal pressures evolving with the motion of every single droplet from channel to channel. Finally, to account for the imperfectness in manufacturing and the inherent fluctuations of the parameters, we emulated artificial noise, as detailed in Supplementary Section 3. The role of noise in simulations of the dynamics of droplets in microfluidic networks has been studied in greater detail in ref. 42.
The model thus formulated reproduces faithfully the emergence of oscillations in networks of different dimensions and connectivities (Figs. 1f,g and 3a–d and Supplementary Videos 5 and 6) and enables a systematic study of the parameters involved—here, the results are illustrated for the simplest network topology for which oscillations are observed (Fig. 3a–e). The requirement that all three paths traversing this network must be accessible even to the first droplet (we call it pre-balancing; Supplementary Section 3.2) reduces the number of independent parameters to two. Choosing these parameters as a and c, key flow characteristics can be captured in triangular maps (Fig. 3e) plotting calculated amplitudes/lengths and periods of the droplet trains (nsqr and tfft, respectively; for formal definitions see Supplementary Section 3.3). The key observation from all maps shown in Fig. 3f–j is that oscillations are quite ubiquitous and not limited to small regions of parameters. The maps also confirm the experimental result that oscillations are more pronounced when channel a between two diverging junctions is shorter than channel c between two converging junctions. Even in this regime, however, there are combinations of parameters for which oscillations are weak or non-existent. For example, in the maps in Fig. 3f,g there are no oscillations along the lines defined by equations a + c = 0 or 3a + 4c − 3 = 0. For these parameters, the choices made by the droplets at the junctions are akin to coin flipping, although we note that they have different statistics. Furthermore, as long as the number of droplets residing in the network at a given time is large enough, the maps do not change strongly when the resistive loads of the droplets, the amplitude of the noise or the frequencies of dripping (that is, linear densities of droplets, fdrop) are varied (Fig. 3h–j and Supplementary Section 5). Finally, we confirmed that oscillations are present in many (we tested hundreds) other, more complex networks. In particular, the simulations detailed in Supplementary Section 6 and Supplementary Video 7 evidence that all combinatorial rewirings of the innermost channels between diverging and converging ‘trees’ (rendering the network non-planar) also exhibit system-wide oscillations, despite the fact that equations describing them become much more complicated. These results indicate that our model can be applicable to the study of non-planar networks such as those often found in natural systems.
In this context, one particularly interesting problem comprises the systematic (0.07–0.13 Hz) but low-amplitude (up to ±20% of average flux) oscillations of red blood cells (RBCs)16,18, which have been demonstrated experimentally in systems mimicking the microcirculation in vitro17 in animals with functioning heart and lungs16 and in humans during non-pulsatile cardiopulmonary bypass18—a condition that eliminates a potential relation between the observed regular oscillations of the haematocrit and rhythmic operation of these organs. The mechanisms of these self-sustaining oscillations are not completely understood and—even excluding the abovementioned rhythms of large organs—several physiological explanations may apply14,18. Although some haemodynamic models of blood flow predict the emergence of oscillations in certain regimes7,8, the values of key parameters required for these oscillations are not realistic for real blood (see the discussion below).
Given that—akin to our droplets—RBCs flow in single file43, are strictly confined/deformed by the walls of the surrounding microvessels, and increase these vessels’ resistance to flow, the small amplitude of their oscillations might be surprising (although certainly beneficial from a physiological point of view by, for example, preventing local hypoxia or hypertension). We suggest that the reason for such small amplitudes might lie in the fact that the RBCs at the junction of arterioles do not necessarily follow the paths of least fluidic resistance. In particular, collisions between two or more RBCs can cause them to move into a channel of lower flow rate, an effect that we actually reproduced in experiments with closely spaced droplets colliding at junctions. As illustrated in Fig. 4 (and with further discussion in Supplementary Section 7), this phenomenon is fully deterministic; as long as intervals between incoming droplets are controlled, it is possible to enforce the transition from fully developed oscillations (as in Fig. 4b) to an entirely uniform distribution of droplets (Fig. 4c and Supplementary Video 9) and vice versa (Supplementary Video 9), even without changing the average density of droplets or average intervals between them. Beyond these extremes, all intermediate distributions of droplets can be arranged by purposefully placing closely separated droplets among longer intervals in the incoming stream (Supplementary Section 7). Even for constant intervals, if they are short enough, collisions may prevent development of the oscillations, as in Fig. 4d, although this mode of flow is metastable (applying the same intervals to the system already in oscillatory dynamics would not be enough to quench it).
Collisions of RBCs in blood are much less deterministic. First, due to the flattened shape of an RBC, the process of passing a junction depends on the initial orientation of the RBC, which introduces some randomness, even at the level of single cells. Second, due to the high concentration and elevated stiffness, mutual collisions at areas of bifurcation may take longer and involve more than two RBCs, leading to flow obstruction by temporary structures44,45. This intrinsic (or effective—as it results from the lack of complete knowledge) indeterminacy can be accounted for in our model by introducing the so-called ‘plasma skimming function’7,8 (Fig. 5a). This function, widely used in haemodynamics, quantifies ‘how deterministic’ is the division of the haematocrit between two branching arterioles with volumetric flows Q1 and Q2. The probabilities of RBCs entering these two channels are, respectively, \(Q_1^S/(Q_1^S + Q_2^S)\) and \(Q_2^S/(Q_1^S + Q_2^S)\), where parameter s controls the level of determinism. For s → ∞ the choice is fully deterministic, and droplets always enter the channel with the largest flow rate, as in our experiments described above (provided that there are no collisions). For s = 1, the probabilities of choosing a channel are proportional to the flow rates, and for s = 0, they are equal regardless of the flow rates (which does not make physical sense). Mathematical models of blood microcirculation based on in vivo data43 estimate that for bifurcations at the smallest capillaries s = 2–2.5. At the same time, studies based on continuous haemodynamic models report that oscillations are observed only at s > 5 (ref. 7) or s > 4 (ref. 8; where the threshold s could be lowered at the expense of an unrealistically large exponent in the nonlinear dependence of apparent blood viscosity on the haematocrit). Simulations for our networks quantified in Supplementary Fig. 11 show a general trend (holding for different parameter values) that as s decreases—that is, when choices at junctions become more random—the oscillations indeed fade away. Importantly, with the network from Fig. 1 rescaled to the size typical for the blood microcirculation (Fig. 5b) and with parameters fdrop and r chosen to mimic the rheological properties of blood (Supplementary Section 9), clearly visible oscillations are observed for s as small as 2.2 (Fig. 5c,d)—that is, for the value of s matching the in vivo data (cf. above). The frequency of these oscillations is 0.1 Hz, which also matches the frequencies observed in vivo16,18.
The fact that our simplified, discrete model gives more realistic estimates than sophisticated continuous haemodynamic models points to an intriguing conclusion that the discrete nature of RBCs may play a more important role in haematocrit oscillations than many subtle interactions and nonlinearities underlying continuous descriptions (for further discussion, including the consequences of the discreteness of RBCs, see Supplementary Section 9).
We envision three possible developments from this work: (1) in further studies of the mechanism of oscillations and their suppression in blood, incorporating alternative or coexisting mechanisms (for example, the role of leukocytes17, varying lumen of vessels42 and their active control46); (2) the design of microfluidic circuits in which the network topology alone would direct the droplet cargos to desired locations without the use of any valves (cf. the example in Fig. 2d–f); (3) the development of blood substitutes19, especially in estimating the concentrations and sizes of oxygen-carrying particles ensuring that they undergo collisions at capillary junctions, thus preventing large-scale oscillations.
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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