Main

Fulfilment of the promise of microfluidics to operate as autonomous microscale networks in which fluids can be transported, mixed, reacted, separated and processed is no longer limited by experimental fabrication challenges, but rather by difficulties in creating built-in controls1,2,3. The importance of this limitation can be appreciated by noting that the development of the modern microelectronics that form the basis of computer microprocessors was ultimately determined by the creation of integrated circuits, with all components fabricated on the same substrate. Microfluidics have already reached a level of integration in which networks with thousands of components, including control devices, are built on a single compact chip. However, in contrast with electronic integrated circuits, existing on-chip fluid control devices still need to be actuated externally. For example, microfluidic circuits fabricated from flexible polydimethylsiloxane (PDMS) can now incorporate a large number of control valves, which nevertheless have to be operated using control fluids through a control layer that lies on top of the working fluid network4,5. As a result, microfluidics are still predominantly controlled by external hardware, despite substantial efforts over the past 20 years to develop systems with new control schemes6,7,8,9,10. The construction of systems that forgo the current reliance on external hardware is crucial to further the development of portable microfluidic systems for pressing applications, ranging from point-of-care diagnostics and health monitoring wearables to analysis kits for field research11,12,13,14. This requires developing next-generation integrated circuits in which not only the control devices but also the operation of those devices is integrated on-chip. The development of such a level of integration has been fundamentally limited by the fact that, at the microscale, fluid flows tend to respond linearly to pressure changes and thus cannot be easily amplified or switched.

In this Article, we explore new physics that emerges by combining network theory and fluid mechanics to induce nonlinear behaviour in microfluidics and effectively create a passive two-terminal flow-switch device that is entirely operated on-chip, directly by the working fluid. Previous work that has achieved built-in control capabilities (often externally actuated), including oscillatory flows15,16,17,18 and flow rate regulation19,20, generally relied on flexible membranes and surfaces. Microfluidics with such flexible components require flows with very low Reynolds numbers—a regime in which fluid inertia, and thus the only nonlinear term of the Navier–Stokes equations for incompressible fluids, becomes negligible. This has led researchers to often discount the potential effects of fluid inertia on the flows (as reviewed, for example, in refs. 21,22). Recent work has shown, however, that inertial forces can serve as a powerful on-chip tool to manipulate microfluidic dynamics locally23,24, including shaping streamlines25,26, mixing fluids27 and directing particles28,29. Here, we present networks designed to amplify inertial effects by incorporating properties of porous media that can be used for non-local fluid routing and manipulation of output patterns.

Figure 1a shows a schematic representation of a microfluidic system with the fundamental network structure we consider. It consists of five segments arranged as two parallel channels connected by a linking channel, where the inlets are kept at a common pressure Pin and the outlets are held at a common, lower, pressure, Pout. One of the outlet channels is modified to generate a nonlinear pressure–flow relationship, which is achieved by introducing an array of cylindrical obstacles. Our principal results are supported by theory, simulations and experiments, and they show that we can: (i) induce a flow direction switch through the linking channel solely by varying the pressure difference between the inlets and outlets; and (ii) identify a pressure difference above which the total flow rate between the inlets and outlets increases on closing the linking channel. We also predict negative conductance transitions when the linking channel is equipped with an offset fluidic diode, which are transitions associated with non-monotonic pressure–flow relations analogous to those previously realized using flexible diaphragm valves30. The counter-intuitive behaviour described in (ii) is formally equivalent to the so-called Braess’s paradox originally established for traffic networks31,32, where closing a shortcut road has the possible effect of increasing net traffic flow. We demonstrate integration of the flow switch described in (i) by considering larger microfluidic networks, as illustrated in Fig. 1b, which incorporate multiple linking channels and are thus capable of exhibiting multiple flow switches. Flows through these networks are driven by a single pressure difference and yet can be designed to exhibit a variety of flow states by programming the pressure at which each flow switch occurs.

Fig. 1: System schematics.
figure 1

a, Microfluidic network consisting of two parallel channels, joined by a linking channel, that connect high- and low-pressure fluid reservoirs. Grey filled circles represent stationary cylindrical obstacles. The labels denote pressures (P), channel lengths (L) and flow rates (Q), with arrows indicating the positive flow direction. b, Generic multiswitch microfluidic network consisting of an array of parallel channels interconnected by multiple linking channels. A subset of channel segments contain cylindrical obstacles. Flow is driven through the network by a single pressure difference (Pin − Pout).

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System design and nonlinearity

We consider conditions under which all channel segments have the same width w, the working fluid is water, and all surfaces (including obstacles) have no-slip boundaries. We assume, without loss of generality, that the pressure Pout at the outlets is zero, and consider scenarios in which either the static or the total pressure is controlled at the inlets (Methods). We examine two network configurations of the system in Fig. 1a: the connected configuration, in which the two parallel channels are allowed to exchange fluid through the linking channel; and the disconnected configuration, in which the linking channel is closed or removed. In our theoretical analysis and simulations, the flows are assumed to be two-dimensional, yet the main results carry over to three dimensions, as verified in our experiments.

For a straight microfluidic channel of length \(L\gg w\)  without obstacles, an approximate steady-state solution of the Navier–Stokes equations in two dimensions yields a linear relation between the total volumetric flow rate per unit depth Q and the pressure drop ΔP along the channel:

$$-\Delta P=\frac{12\mu L}{{w}^{3}}Q$$
(1)

where μ is the dynamic viscosity of the fluid. To induce deviations from this linear regime, we consider the effect of introducing multiple stationary obstacles in the channel. Figure 2a, b shows simulations of the Navier–Stokes equations for a channel with ten cylindrical obstacles of radius r = w/5 (Methods). We observe recirculation regions forming near the obstacles for sufficiently large Reynolds number Re ≡ 2ρQ/μ, where ρ is the fluid density. The recirculation regions first appear for Re of the order of 10, and their number and size depend on Re. These localized structures are hallmarks of fluid inertia effects (and thereby of nonlinearity). We investigate how fluid inertia effects compound to impact the total flow rate by performing simulations across moderate values of Re when different numbers of obstacles are present. We find that a nonlinear relation between the pressure drop ΔP and flow rate Q = μRe/2ρ emerges as soon as obstacles are introduced, and that the nonlinearity becomes more pronounced as the number of obstacles is increased (Supplementary Information section S3.1 and Supplementary Fig. 3).

Fig. 2: Development of nonlinear flow.
figure 2

a, b, Simulated flow in a channel with obstacles (open circles), showing no recirculation for low Re (a) and noticeable recirculation near the obstacles for larger Re (b). c, d, Experimentally observed flows around the obstacles (filled circles), visualized using pictures of fluorescent particles (marked in pink). The particle tracks trace the underlying flow structure, confirming the development of recirculation regions (white areas) as Re is increased from low (c) to moderate values (d). e, f, Experimentally measured relation between pressure loss and Re for a channel with (e; red curve) and without (f; blue curve) obstacles. The dashed line in e is a reference to guide the eye and indicates an approximately quadratic relation between pressure loss and flow rate.

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The nonlinearity we observe in the relation between ΔP and Q conforms to the Forchheimer effect in porous media, which characterizes flow through many interconnected microchannels when local inertial effects at the points of interconnection are non-negligible, even for laminar flow33,34,35. We use the Forchheimer equation to derive a relation between ΔP and Re for the channel with obstacles, given by

$$-\Delta P=\frac{\alpha {\mu }^{2}L}{2\rho w}{\rm{R}}{\rm{e}}+\frac{\beta {\mu }^{2}L}{4\rho {w}^{2}}{{\rm{R}}{\rm{e}}}^{2}$$
(2)

where α is the reciprocal permeability and β is the non-Darcy flow coefficient, both depending solely on the system geometry (Methods).

The physical mechanism giving rise to this nonlinearity is the increase in flow recirculation and velocity gradients for larger Re, as evidenced in Fig. 2a, b for Re = 1 and 220. To test the impact of the inertial effects in realistic systems, we perform experiments using microchannels fabricated from stiff PDMS (hardened by curing). Figure 2c, d shows experimental evidence of the increase in the number and size of the recirculation regions with Re, in agreement with our simulations. An approximately linear relation between −ΔP/Re and Re (and thus an approximately quadratic relation between −ΔP and Q) for a channel containing 20 obstacles is shown in Fig. 2e, which contrasts with the constant relation measured for a channel without obstacles in Fig. 2f.

Switching and Braess’s paradox

We incorporate the channel segment with obstacles characterized above into a network by considering the microfluidic system presented in Fig. 1a. We take the common static pressure Pin at the inlets to be the controlled variable in the system. The total flow rate through the network is now simply the sum of the flows at the outlets, (Q4 + Q5). In Fig. 3, we present results for this system from direct simulations of the steady-state solutions of the Navier–Stokes equations. As Pin is increased from zero, the flow rate through the linking channel Q3 is initially positive before changing direction and becoming negative once a critical pressure, defined as \({P}_{{\rm{in}}}^{\ast }\), is reached (Fig. 3). This flow switch results from the nonlinear change in pressure loss along the channel segment containing obstacles, which causes a switch in the sign of the pressure difference along the linking channel ΔP21 (approximately P2 − P1) as the flow rate through the system increases with Pin. We define QC to be the total flow rate for the connected system configuration and QD to be the total flow rate for the disconnected system configuration, where both are regarded as functions of Pin.

Fig. 3: Braess’s paradox and flow switching.
figure 3

Simulation results for the connected and disconnected configurations of the system for a range of inlet pressures Pin. The flow rates are presented as a percentage of the total flow rate through the connected system, QC, where we adopt the sign convention for the flow directions as defined in Fig. 1a. The flow through the linking channel switches direction at the critical pressure \({P}_{{\rm{in}}}={P}_{{\rm{in}}}^{\ast }\), which coincides with the onset of negative \(\Delta Q\) that marks the occurrence of Braess’s paradox.

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Figure 3 shows \(\Delta Q\equiv {Q}_{{\rm{C}}}-{Q}_{{\rm{D}}}\) for a range of applied pressures Pin. Intuition may suggest that \(\Delta Q\) is positive for all values of Pin because the linking channel in the disconnected system can be considered to have an infinite fluidic resistance, while for the connected system configuration the resistance of the linking channel is finite. Hence, reducing the resistance of any component of the system may seem to imply that the total flow rate should increase for fixed Pin. We observe, however, that \(\Delta Q\) becomes negative for Pin above the critical pressure that marks the flow switch, \({P}_{{\rm{in}}}^{\ast }\), meaning that an open linking channel between the parallel channels results in a lower total flow rate. Figure 3 also shows that the flow rate through the channel segment with obstacles, Q4, remains largely unchanged between the two configurations. Therefore, the difference in the total flow rate exists primarily in the difference in Q5, and Q3 acts as a controlling variable of Q5.

The observation of a lower total flow rate for the connected configuration compared to the disconnected configuration for fixed Pin is a manifestation of a fluid analogue of Braess’s paradox. Indeed, if we consider the disconnected system driven by an inlet pressure \({P}_{{\rm{in}}} > {P}_{{\rm{in}}}^{\ast }\), the addition of the linking channel can result in a decrease in the total steady-state flow rate (as large as 10% in our simulations). The value of the critical pressure \({P}_{{\rm{in}}}^{\ast }\) depends, of course, on the dimensions of the channels, but we find that the onset of Braess’s paradox and the flow switch always occur at the same pressure for the range of parameters investigated. We obtain similar results for Braess’s paradox and flow switching when instead the total pressure is controlled at the inlets (Supplementary Information section S3.4). Our observation of Braess’s paradox and flow switching also has the potential to lead to additional control features when existing microfluidic components are integrated into our system. For example, by incorporating an offset fluidic diode36 in the linking channel, the system can undergo negative (and positive) conductance transitions, where an increase in Pin leads to an abrupt decrease in the total flow rate (Supplementary Information section S4).

Experimental results

We performed experiments to validate our predictions of flow switching and Braess’s paradox in a network with dimensions typical of microfluidics. A schematic of the experimental apparatus is presented in Fig. 4a, where an open/close valve is used to implement the addition/removal of the linking channel (Methods). With the valve open, a flow switch is observed at a critical driving pressure \({P}_{{\rm{in}}}^{\ast }\) in the range 5–10 kPa, as demonstrated in Fig. 4a by images of the flows through the channel junctions at the end points of this pressure range. (The switching behaviour has no reliance on the valve, as explicitly shown in Supplementary Fig. 11.)

Fig. 4: Experimental observation of flow switch and Braess’s paradox.
figure 4

a, Experimental setup of the system presented in Fig. 1a, with flow tracking images (insets) at the junctions. An air-pressure pump is used to equally pressurize two vials containing red and blue dyed water, where each vial is connected to one of the system inlets. The linking channel is equipped with an open/close valve and channel 4 contains 20 obstacles. Images of the dyed flows through the junctions are shown for Pin below (5 kPa; left insets) and above (10 kPa; right insets) the flow switching pressure \({P}_{{\rm{in}}}^{\ast }\), where the flow directions are indicated by the arrows. b, Total flow rate (Q4 + Q5) when the linking channel valve is ‘open’ or ‘closed’ (see diagrams at right) for two different driving pressures above \({P}_{{\rm{in}}}^{\ast }\). c, d, Breakdown of the total flow rate into Q4 (c) and Q5 (d) for the two states of the valve. The plotted flow rates are averages derived from time series data, and the error bars indicate one standard deviation. The observed increase in the total flow rate when the valve is closed is direct evidence of Braess’s paradox.

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A confirmation of Braess’s paradox in this system is shown in Fig. 4b for driving pressures above \({P}_{{\rm{in}}}^{\ast }\), as observed in our simulations. The measured total flow rate is higher when the linking channel valve is closed than when it is open, thus demonstrating the paradox, and the magnitude of the paradox is observed to be larger for higher driving pressures. A breakdown of how the flow rate changes in channel segments 4 and 5 individually is shown in Fig. 4c, d. Closing the valve causes the flow rates through both channels to increase, which is in agreement with direct simulations and is yet another striking aspect of Braess’s paradox in this system; it would be, at first, intuitive to expect that Q5 would decrease when the in-flow from the linking channel is switched off. Time series of the flow rates measured as the linking channel is sequentially opened and closed further illustrate the transitions underlying the paradox (as shown in Supplementary Fig. 12).

In our experiments, the total pressure is controlled at the inlets and the experimental results are in full qualitative agreement with simulations performed under the same pressure boundary conditions (Supplementary Information section S3.4). This illustrates the robustness of the phenomenon, given that our simulations are in two dimensions and three-dimensional effects are expected to be present in the experiments. We note that different aspects of the paradox have been considered in fluid networks, but only for macroscopic (that is, non-microfluidic) systems and while modelled by ad hoc flow equations37,38,39. Analogues of the paradox have also been studied in several other areas, including electrical, mechanical, biological, and contemporary traffic networks40,41,42,43,44. These examples show that Braess’s paradox is a potentially general network phenomenon, which has remained unexplored in microfluidic networks.

Network model

To characterize the microfluidic system in Fig. 1a, we construct an analytic model that captures the flow properties observed in our simulations and experiments. The model consists of pressure–flow relations for each channel segment and, crucially, includes the most dominant term resulting from minor pressure losses at the channel junctions45,46 (Methods). We model the contribution of the latter as an additive term K(Q3/Q1)f(Q5) to the pressure–flow equation for channel segment 5, where the scaling factor f and the coefficient K are increasing functions for Pin ≥ 0 such that f(0) = K(0) = 0. Several results are obtained from this model for Pin > 0, as assumed throughout. First, if β = 0 (that is, the quadratic term is zero in equation (2)) when the static pressure is controlled or the dynamic pressure is negligible, then flow switching does not occur, in agreement with direct simulations (Supplementary Information section S3.2). Second, when β > 0, a steady-state solution can be found satisfying Q3 = 0 provided that the following geometric condition is satisfied:

$${L}_{1} < \frac{12{L}_{2}{L}_{5}}{\alpha {w}^{2}{L}_{4}}={L}^{\ast }$$
(3)

This solution identifies the critical pressure \({P}_{{\rm{in}}}^{\ast }\). Third, for flow rates in the linking channel, the model predicts that a variation δQ3 is negatively related to a variation δPin around \({P}_{{\rm{in}}}^{\ast }\). This indicates that Pin above (below) \({P}_{{\rm{in}}}^{\ast }\) results in a negative (positive) flow rate through the linking channel. The first result implies that, in our experiments, the Forchheimer effect is necessary to achieve a flow switch. The second and third results, which hold even for when dynamic pressure is non-negligible, show that this model captures the flow switching behaviour observed in the simulations and experiments. Importantly, we validate the flow-switching condition in equation (3) by demonstrating quantitative agreement between the model and simulations both when the static pressure and when the total pressure is controlled (Supplementary Information section S3.2).

The model also predicts Braess’s paradox as observed in our experiments and simulations. Specifically, under the condition that equation (3) is satisfied and dynamic pressure is small (or static pressure is controlled), the model predicts the paradox to occur for δPin > 0 if and only if

$$K^{\prime} (0)\beta f\left(\frac{a}{\beta }\right) > c$$
(4)

where a and c are positive parameters and prime denotes derivative. If total pressure is controlled and dynamic pressure terms are included, the paradox is also predicted for δPin > 0 provided that a relation similar to equation (4) is satisfied (details for both cases are presented in Supplementary Information section S2). The dependence of condition (4) on β and K′(0) underlines the crucial roles of nonlinearity and minor losses in giving rise to Braess’s paradox in our experiments, and shows in particular that minor losses have to be sufficiently large. Indeed, if the effect of minor losses is neglected, a manifestation of Braess’s paradox is still predicted to occur, but with much smaller magnitude and only for δPin < 0, which is inconsistent with our simulations and experiments (Supplementary Information section S2.3).

The result in equation (4) also highlights a fundamental difference between microfluidic and electronic circuits, namely that minor losses (that is, energy losses associated with interactions between circuit components) do not have direct analogues in common electronics. Given the central role played by such losses in equation (4), we posit that this difference might be the reason why no equivalent of the Braess paradox effect we present has been observed in electrical networks, even though aspects of it have40. We further investigated the impact of interactions between channel segments by varying the junction angles to show that the paradox can be further enhanced by manipulating the minor losses (Supplementary Information section S3.3).

Networks with multiple programmed switches

The system considered thus far can be generalized to create larger microfluidic networks with multiple flow switches—that is, networks with multiple disjoint channel segments in which the flow initially in one direction can be individually ‘switched’ to move in the opposite direction through the manipulation of one driving pressure alone. In our design, the linking channel plays the role of a switch (and can be referred to as such). Figure 1b shows the multiswitch generalization of the network in Fig. 1a, which incorporates multiple linking channels and a subset of channel segments with obstacles. We experimentally demonstrate an instance of a six-switch network that exhibits flow switching in all linking channels (as presented in Supplementary Information section S6.2). Multiswitch networks can be designed by extending the network model presented above.

One such network with ten linking channels is presented in Fig. 5a. By marking each inlet flow with a different colour, we show that a variety of patterns can form in the outlet flows (coloured circles in Fig. 5). The specific pattern at an outlet depends on the order in which the flow switches occur as Pin is varied. The network model for larger systems is constructed by combining pressure–flow relations for each channel segment with flow rate conservation equations for each junction. Using this model, we can design a network for which each flow switch occurs near a target value of Pin by optimizing the dimensions of the channel segments (Methods).

Fig. 5: Flow patterns in a multiswitch network.
figure 5

a, Schematic of ten-switch network. Fluids of different colours are driven to each inlet by a common static pressure source, Pin. The outlets are labelled by O1–O6 and the linking channels by 1–10. The arrows indicate the flow direction through each linking channel and multicoloured circles schematically indicate the fluid composition at each outlet for an initially low Pin. The segment lengths are denoted by ai, bi, ci, d, e and f, where a common length is assumed for all linking channels and  the segments with obstacles are marked with filled grey circles. b, Patterns of outlet flows for the network programmed with a chosen switching sequence as Pin is increased. Each column of coloured circles denotes the outlet flows after the corresponding flow switch occurs, where mixing between different coloured fluids is assumed to occur when passing through the same channel segment. c, d, Model predictions (c) and simulation results of the Navier–Stokes equations (d) for the flow rate through each linking channel for a network designed to exhibit the switching sequence in b. The flow rates are labelled according to the channels in a and are divided into two sets (top and bottom panels) for clarity. Positive flow rates correspond to flow in the upward direction in a, and each flow switch occurs when the corresponding curve crosses the horizontal axis. The segment dimensions that give rise to the particular switching order in b–d are reported in Supplementary Table 1. All 21 possible outlet flow colour combinations are realized between the switching sequence presented here and those in Supplementary Fig. 13.

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As illustrated in Fig. 5, a set of 11 different internal flow states and 17 unique colour combinations at the outlets are possible for the switching sequence realized in Fig. 5b. Figure 5c, d shows the agreement between the model predictions of these flow states and results from direct simulations of the Navier–Stokes equations. This variety of states (and output patterns) is achieved with only three channel segments containing obstacles and is parameterized by a single control variable—the driving pressure Pin. Moreover, the switching is implemented solely through the working fluid, which differs from existing approaches that rely on flexible valves and additional control flows15. Thus, multiswitch networks exhibit several properties exploitable in the design of new controllable microfluidic systems.

More generally, for a multiswitch network with nc horizontal channels interconnected by nl linking channels, the number of possible internal flow states is nl + 1 if each linking channel exhibits a flow switch. In addition, the possible number of unique colour combinations in the outlet flows is nc(nc + 1)/2 if each inlet flow is marked with a different colour. All colour combinations can be realized over the set of all switching sequences, provided that there exist flow paths allowing mixing of every set of k adjacent colours for k ranging from 1 to nc. The myriad states possible in such multiswitch networks underlie their ability to process inputs into multiple outputs and thus to support various applications, including implementing different mixing orders of chemical reagents and devising schemes for the parallel generation of mixtures with tunable concentrations.

Conclusions and outlook

The flow switch, conductance transitions and Braess’s paradox established in this study are all emergent behaviours of common origin resulting from nonlinearity and interactions between different parts of the system. The nonlinearity is directly determined by fluid inertia effects, which can be enhanced and manipulated through the placement of obstacles and has the advantage of not being reliant on flexible components, fluid compressibility or dedicated control flows. The onset of Braess’s paradox is marked by the flow-switching pressure, above which the increased resistance of the nonlinear channel causes the flow to be routed in the negative direction through the linking channel. When constrained by a diode, the switch in flow direction also enables  negative conductance transitions. Our results demonstrate an approach for routing and switching in microfluidic networks through control mechanisms that are coded into the network structure, thus responding to the call for design strategies that allow diverse microfluidic systems to be assembled from a small set of core components2,47.

Here we considered the scenario in which the inlets and the outlets are (separately) held at the same pressure, rendering the network a two-terminal system in all cases, because this is the most stringent scenario for flow manipulation. If a multi-terminal system is configured, by allowing the pressures at each of the inlets (and/or outlets) to be varied independently, then the effects that we presented may be further enhanced. Finally, although we focused on boundary conditions in which the inlet pressures are controlled, it would be natural to explore in future research the scenario in which the controlled variables are the inlet flow rates. We anticipate, for example, that the negative conductance transitions would then be converted into pressure amplification (pressure release) transitions in which the inlet–outlet pressure difference increases (decreases) abruptly at the transition point. Accordingly, Braess’s paradox is also expected to take a complementary form in which closing the linking channel causes the inlet–outlet pressure difference to drop. Incidentally, it is this complementary form of Braess’s paradox that has been previously established for electrical circuits40, thus suggesting an additional  correspondence between electronic and microfluidic circuits.

Methods

Navier–Stokes simulations

The numerical simulations were performed using48 OpenFOAM version 4.1. We used meshes with an average cell area ranging from 10 μm2 to 340 μm2, where the finest meshing was applied near the obstacles. All meshes were generated using Gmsh49. The two-dimensional solutions were found using the simpleFoam solver within OpenFOAM, employing second-order numerical schemes, where a fixed static pressure of zero was set for the boundary conditions at the outlets. At the inlets, the static (total) pressure was fixed for the static (total) pressure controlled cases. For simulations of the multiswitch network in Fig. 5, the same geometry and dimensions were used as for the model predictions, provided in Supplementary Table 1, and equal driving pressures were applied at each of the six inlets.

Reynolds numbers

The characteristic length scale used in defining the Reynolds number of the flow is the hydraulic diameter of the channels, defined as 4A/P, where A is the area and P is the perimeter of the channel cross-section (common to all segments). The hydraulic diameter in two and three dimensions is 2w and 2wh/(w + h), respectively, where h is the height of the channels in the three-dimensional case. The characteristic velocity used in two and three dimensions is Q/w and Q/wh, respectively. Therefore, we define Re = 2ρQ/μ for our simulations in two dimensions and Re = 2ρQ/μ(w + h) for our experiments in three dimensions. The undeclared ranges of Re for the channel segment with obstacles considered in the presented data are: 21–385 (Fig. 3), 12–121 (Fig. 4), 1–220 (Fig. 5), 1–380 (Supplementary Fig. 2), 4–111 (Supplementary Fig. 4), 40–385 (Supplementary Fig. 7), 20–400 (Supplementary Fig. 8), 2–10 (Supplementary Fig. 11b), 75–85 (Supplementary Fig. 11c), 76–89 (Supplementary Fig. 12), 10–20 (Supplementary Fig. 14b) and 110–120 (Supplementary Fig. 14c).

Pressure boundary conditions

We consider two different boundary conditions for the driving pressure Pin at the system inlets. Under one condition, total pressure is controlled and the inlets open directly into a high-pressure reservoir. Under the other condition, static pressure is controlled and the inlets are connected to the reservoir by pressure regulators. Total pressure is the sum of static pressure and dynamic pressure, where dynamic pressure is defined as \(\frac{1}{2}\rho {v}^{2}\) for a fluid with density ρ and velocity v. The distinction between these boundary conditions is often neglected in the microfluidics literature when the Reynolds number is less than one50, but it can become important for larger Reynolds numbers (even though the flow remains laminar)51.

Pressure–flow relations

We use equation (1) to describe the pressure–flow relation for straight, obstacle-free channels; it is derived directly from the Navier–Stokes equations by assuming plane Poiseuille flow through a two-dimensional channel. To describe the nonlinear pressure–flow relation observed for the channel with obstacles, we refer to the Forchheimer equation: −ΔP = αμLV + βρLV2, where V is the average fluid velocity. In two dimensions, V = Q/w = μRe/2ρw and, thus, the Forchheimer equation can be written in the form of equation (2). In agreement with equation (2), we find an excellent linear fit between −ΔP/Re and Re for a channel with ten obstacles, and we validate the fit by predicting flows through the same channel for a fluid with a different viscosity (Supplementary Information section S3.1). We observe no unsteady flow through the channel with obstacles due to vortex shedding for Re of up to 400, as expected for systems with highly confined obstacles52, which permits the use of the steady-state relation in equation (2) over the range of Re considered here. We experimentally verify the source of nonlinearity in PDMS channels with obstacles, which were designed to have approximately square cross-sections to minimize deformation (which could lead to other forms of nonlinearity53,54). Through additional experiments, we confirmed that pressure–flow relations similar to those in Fig. 2e, f hold for channels constructed from materials with both higher rigidity (SU-8 photoresist) and lower rigidity (Flexdym) than the PDMS (Supplementary Information section S5 and Supplementary Fig. 10). We note that porous-like structures have been previously used both to study non-inertial effects in microfluidics, such as droplet formation55 and viscous fingering56, and to study inertial effects in larger systems57. In our system, inertial effects arise at the microfluidic scale even for a much smaller number of obstacles than the typical number in porous-like materials.

Network flow model construction

The analytic model used to describe the system in Fig. 1a is constructed as follows: (i) we consider the pressure at the inlets Pin to be in the vicinity of \({P}_{{\rm{in}}}^{\ast }\); (ii) we approximate the pressure–flow relation through the linking channel as Q3 = κ(γP1 − P2), where κ is the channel conductivity and γ is a free parameter allowing for an effective pressure difference; (iii) the flow equation for each other channel segment without obstacles is written as in equation (1), where −ΔP is the pressure drop along the segment and L is the segment length; (iv) for the channel segment with obstacles, we take the flow equation to be in the form of equation (2) (with Re expressed as 2ρQ/μ); (v) we include the most dominant term resulting from minor pressure losses at the channel junctions. Therefore, the model consists of five pressure–flow relations, in addition to two flow conservation equations at the junctions: Q3 + Q2 − Q4 = 0 and Q3 + Q5 − Q1 = 0. When the static pressure is controlled at the inlets, the only nonlinearity that exists in the model comes from the Forchheimer term due to the presence of obstacles and the minor loss term. The model can also be adapted for when total pressure is controlled by taking the static pressure at each inlet to be Pin − kρQ2/2w2, where Pin now denotes total pressure and the coefficient k is a constant of order unity that only depends on the shape of the inlet velocity profile (k ≈ 1 for a uniform velocity profile at the inlet, as considered here). However, the dynamic pressure term ρQ2/2w2 is often negligible in real microfluidic systems because of the high pressures needed to drive fluid though the channels. Indeed, in our experiments, the dynamic pressure near \({P}_{{\rm{in}}}^{\ast }\) was smaller than the static pressure by two orders of magnitude and smaller than the pressure loss due to the Forchheimer effect by one order of magnitude. This can also be seen in Fig. 2f, where a constant relation between Re and ΔP/Re is measured. Details of the model are presented in Supplementary Information section S1.

Designing multiswitch networks

For a network with multiple switches and a given set of channel dimensions, the value of Pin for which a specific flow switch occurs can be determined through the addition of a constraint to the model that enforces the flow through the corresponding linking channel to be zero. Then, the dimensions of a chosen subset of channel segments may be varied through an optimization procedure in order to design a network for which each flow switch occurs near a target value of Pin. Depending on which dimensions are allowed to be adjusted, the desired relative order of the switches can be achieved exactly, and the final set of switching pressures can be very close to the target ones (often <5% difference), where the former is expected to be more important in applications. Further details on the design of multiswitch networks are presented in Supplementary Information section S6.1.

PDMS channel fabrication

The flow channels were assembled by sealing a patterned PDMS chip against a glass slide. The PDMS chip was made by pouring a mixture of PDMS oligomer and cross-linking curing agent (Sylgard 184) at a weight ratio of 10:1 into a mould after being degassed under vacuum. The mixture was cured at 74 °C for 1 h and then peeled off from the mould to yield the microchannel design. The dimensions of the channels in Figs. 2 and 4 were 200 μm (width) × 185 μm (height), and the diameter of the obstacles was 97 μm. After punching the holes for inlet and outlet connections, the PDMS chip was thermally aged at 200 °C for 12 h to reduce pressure-induced deformation58, yielding a chip with a Young’s modulus of59 approximately 3 MPa. Both the PDMS chip and the glass substrate were cleaned with isopropanol and treated by plasma for 90 s before bringing them into contact. Once the PDMS chip was sealed against the glass slide, the device was placed in an oven for 30 min at 74 °C to improve bonding quality.

The mould used was a silicon wafer containing microchannel patterns created by soft photolithography using a negative photoresist60,61. A 4-inch silicon wafer (test grade, University Wafer, Boston, MA) was cleaned with acetone and isopropanol and dried with nitrogen gas. The wafer was then coated with SU-8 50 negative photoresist (MicroChem Corp., Newton, MA) on a spin coater (Laurell Technologies Corp., North Wales, PA) operating at 600 rpm for 30 s. After a pre-exposure bake at 65 °C and subsequently at 95 °C, each for 60 min, the coated wafer was exposed to UV light (Autoflood 1000, Optical Associates, Milpitas, CA) through a negative transparent photomask that contained the desired channel design. Following a 3.5 min post-exposure bake at 95 °C, the wafer was developed in SU-8 developer (MicroChem Corp., Newton, MA) for 60 min to obtain the pattern.

Flexdym channel fabrication

Flexdym (Blackholelab Inc., Paris) is a thermoplastic elastomer (Young’s modulus of 1.18 MPa) with a rapid and easy moulding process for microfluidic devices62. After fabrication of the silicon wafer mould containing the channel designs, a sheet of Flexdym (6 cm × 4 cm) was placed directly above the mould with another sheet of unpatterned PDMS (about 1 mm thick) placed above the Flexdym for protection. The whole set was then placed on a heat press between two Teflon sheets. The plate on the heat press was heated to 175 °C before starting to mould the Flexdym. Once the target temperature was reached, the lever on the heat plate was locked down with a timer set for 5 min. After the process was finished, the lever was released and the Flexdym sheet was inspected visually to make sure that no bubbles were trapped around the channel. The chip was allowed to cool down for 5 min before unfolding the layers. The Flexdym was permanently sealed with a glass slide by following the same sealing procedure used for the PDMS channels. The dimensions of the cross-section of the channels were 201 μm (width) × 166 μm (height), and the diameter of the obstacles was 99 μm.

SU-8 photoresist channel fabrication

To make microfluidic channels directly from SU-8 photoresist, an inverse mask was designed and printed on transparency. The desired channel was printed on the inverse mask in black with transparent dots marking the obstacles, and the rest of the mask was left transparent. The same procedure used to make the silicon wafer master as described in Methods section ‘PDMS channel fabrication’ was followed to fabricate the channels on glass slides. The chip was then sealed by 3M VHB tape to another glass slide with holes for connections. The dimensions of the cross-section of the channels were 209 μm (width) × 196 μm (height), and the diameter of the obstacles was 90 μm. The Young’s modulus of SU-8 photoresist is 2 GPa (from the table of properties for SU-8 permanent photoresists, MicroChem Corp., Newton, MA, available at http://microchem.com/pdf/SU-8-table-of-properties.pdf).

Flow rate measurement

Experimental measurements in Figs. 2 and 4 were made with the system shown in Fig. 4a. When measuring the relation between pressure and flow rate, the linking channel valve was closed to allow separate measurement of the channel with and the channel without obstacles. Deionized (DI) water was pumped through each channel and a pressure scan from 0 to 100 kPa was performed using an Elveflow OB1 pressure controller. The flow rate was measured by an Elveflow MFS5 flow sensor (0.2–5 ml min−1). To verify Braess’s paradox, the same instruments were used and the pressure was set constant while recording the flow rate at each outlet. Red (3 g l−1, FD&C Red #40, Flavors & Colours) and blue (1.5 g l−1, FD&C Blue #1, Flavors & Colours) dyes were added into DI water to demonstrate the switching behaviour. The concentrations of the dyes were adjusted for similar flow rate under the same pressure. The flow rate measurements in Supplementary Fig. 10 were performed using isolated channels constructed from Flexdym and SU-8 photoresist, respectively.

Fluorescence imaging

Fluorescent polyethylene microspheres (10–20 μm) were suspended in Tween 80 solution (Cospheric LLC, Santa Barbara, CA) and pumped through a single microfluidic channel with obstacles by an Elveflow OB1 pressure controller. Two different pressures were applied, 3 kPa and 100 kPa, to demonstrate different flow profiles around the obstacles. Fluorescence images were captured with an Olympus BX51 microscope equipped with a NIBA filter through an Infinity 3 CCD camera.

Measured flow rate data and statistics

Savitsky–Golay filtering was applied to all flow rate data collected through experiments, using a window length of 11 data points and a second-order polynomial. For each of the fixed pressures presented in Fig. 4b–d, a 60 s time series of flow rate data was collected at each of the outlets with a sampling rate of 10 Hz. Over the 60 s interval, the linking channel valve was sequentially opened/closed every 15 s. For each time series, the 15 s intervals in which the valve was open (closed) were averaged to create a single 15 s time series for each outlet. The total flow rate (Q4 + Q5) was calculated when the valve is open and closed, respectively, by summing the 15 s time series for the two outlets point-by-point. The statistics presented in Fig. 4 are the average and standard deviation of the resulting series. For Supplementary Fig. 12, the flow rate at each of the two outlets was measured experimentally at a sampling rate of 100 Hz over a 180 s interval, during which the linking channel was sequentially opened/closed every 30 s. The total flow rate in Supplementary Fig. 12c was calculated by summing, point-by-point, the data in Supplementary Fig. 12a, b.

Parameters in simulations and experiments

In the simulations, we set ρ = 103 kg m−3, μ = 10−3 Pa s, ν = μ/ρ = 10−6 m2 s−1, w = 500 μm for the width of all channels, and r = 100 μm for the radius of all obstacles, unless otherwise noted. In all experiments, DI water was used as the working fluid. The other undeclared dimensions were as follows. In Fig. 2a, b, the length of the (partially shown) channel was 1.25 cm. In Fig. 2c–e, the channel length was 4.3 cm, and in Fig. 2f the channel length was 2.0 cm (see Methods section ‘PDMS channel fabrication’ for the remaining dimensions). In Fig. 3, L1 = 0.17 cm, L2 = 1.0 cm, L3 = 0.1 cm, L4 = 1.25 cm, and L5 = 1.0 cm. In Fig. 4, L1 = 0.6 cm, L2 = 2.9 cm, L4 = 1.4 cm and L5 = 1.4 cm. For the linking channel, the switch valve was connected to the two parallel channels through 15 cm of round tubing and 0.7 cm of microchannel on each side. Each inlet was connected to the pressurized vials through 62 cm of tubing, and each outlet was attached to 50 cm of tubing. The inner diameter of all tubing was 0.79 mm.