Droplet traffic in microfluidic networks: a simple model for understanding and designing

Flow pattern maps of double emulsions transporting through bifurcation microchannels

The transportation behaviors of compound droplets in confined channels are widespread phenomena while the physical mechanisms are far from being completely unraveled. In this work, behaviors of double emulsions flowing through bifurcation microchannels are experimentally studied with the aim of building universal flow pattern maps. Three flow patterns are categorized according to different features of daughter droplets in terms of size, uniformity, and shell thickness. A detailed analysis of the dynamics of interfacial evolutions in different patterns is carried out and the coupling interaction between interfaces is found to affect the minimum tail distance during transportation. It is feasible to obtain the threshold of the occurrence of the coupling interaction, due to the different variation tendencies in the two states, which relies on three dimensionless parameters, i.e. droplet length, length ratio, and capillary number. Furthermore, a novel physical model is proposed to build the flow pattern map, with the two transition boundaries being expressed as different relationships in terms of the three identified parameters. The physical mechanisms are summarized with the aid of force analysis. An excellent agreement is shown between the model and experimental results in different liquid systems and bifurcation structures, indicating the generality of the proposed model.

Multi-scale model of clogging in microfluidic devices with grid-like geometries

We propose a coarse-grained theoretical model to capture the ageing of microfluidic devices under different conditions including constant applied flow rate and constant applied pressure gradient. Microfluidic devices that sort cells by their deformability hold significant promise for medical applications. However, clogging in these microfluidic systems causes their properties to change over time and potentially limits their reliability. We compare the results of the coarse-grained model with those of stochastic simulations and with existing theoretical studies. Lastly, we apply the model to experimental data on the clogging of sickle red blood cells and discuss its wider applicability.

Nonlinear Phenomena in Microfluidics

This review focuses on experimental work on nonlinear phenomena in microfluidics, which for the most part are phenomena for which the velocity of a fluid flowing through a microfluidic channel does not scale proportionately with the pressure drop. Examples include oscillations, flow-switching behaviors, and bifurcations. These phenomena are qualitatively distinct from laminar, diffusion-limited flows that are often associated with microfluidics. We explore the nonlinear behaviors of bubbles or droplets when they travel alone or in trains through a microfluidic network or when they assemble into either one- or two-dimensional crystals. We consider the nonlinearities that can be induced by the geometry of channels, such as their curvature or the bas-relief patterning of their base. By casting posts, barriers, or membranes─situated inside channels─from stimuli-responsive or flexible materials, the shape, size, or configuration of these elements can be altered by flowing fluids, which may enable autonomous flow control. We also highlight some of the nonlinearities that arise from operating devices at intermediate Reynolds numbers or from using non-Newtonian fluids or liquid metals. We include a brief discussion of relevant practical applications, including flow gating, mixing, and particle separations.

Non-Newtonian droplet-based microfluidics logic gates

Droplet-based microfluidic logic gates have many applications in diagnostic assays and biosciences due to their automation and the ability to be cascaded. In spite of many bio-fluids, such as blood exhibit non-Newtonian characteristics, all the previous studies have been concerned with the Newtonian fluids. Moreover, none of the previous studies has investigated the operating regions of the logic gates. In this research, we consider a typical AND/OR logic gate with a power-law fluid. We study the effects of important parameters such as the power-law index, the droplet length, the capillary number, and the geometrical parameters of the microfluidic system on the operating regions of the system. The results indicate that AND/OR states mechanism function in opposite directions. By increasing the droplet length, the capillary number and the power-law index, the operating region of AND state increases while the operating region of OR state reduces. Increasing the channel width will decrease the operating region of AND state while it increases the operating region of OR state. For proper operation of the logic gate, it should work in both AND/OR states appropriately. By combining the operating regions of these two states, the overall operating region of the logic gate is achieved.

Mathematical modeling and computational analysis of centrifugal microfluidic platforms: a review

Centrifugal microfluidic platforms or lab-on-discs (LODs) have evolved into a popular technology for automating chemical and biological assays. LODs today enable scientists to implement and integrate different operational units, including fluid mixing, droplet generation, cell-sorting, gene amplification, analyte detection, and so forth. For an efficient design and cost-effective implementation of any microfluidic device, including LODs, theoretical analysis and considerations should play a more important role than they currently do. The theoretical analysis we will show is especially essential to the investigation of detailed phenomena at the small length scales and high-speed typical for LODs where a wide range of forces may be involved. Previous LOD review papers presented mostly experimental results with theory as an afterthought. Hence, a review paper focused on the theoretical aspects, and associated computational studies of LOD devices is an urgent need. In the present review paper, all previous computational studies on LOD devices are categorized as single-phase flows, two-phase flows, network simulation, and solids. For each of these categories, the governing equations and important formulas are presented and explained. Moreover, a handy scaling analysis is introduced to aid scientists when comparing different competing forces in LOD devices. We hope that by surveying and contrasting various theoretical LOD studies, we shed some light on existing controversies and reveal where additional theoretical work is needed.

Microvscular networks with uniform flow

Within animals, oxygen exchange occurs within vascular transport networks containing potentially billions of microvessels that are distributed throughout the body. By comparison, large blood vessels are theorized to minimize transport costs, leading to tree-like networks that satisfy Murray's law. We know very little about the principles underlying the organization of healthy micro-vascular networks. Indeed capillary networks must also perfuse tissues with oxygen, and efficient perfusion may be incompatible with minimization of transport costs. While networks that minimize transport costs have been well-studied, other optimization principles have received much less scrutiny. In this work we derive the morphology of networks that uniformize blood flow distribution, inspired by the zebrafish trunk micro-vascular network. To find uniform flow networks, we devise a gradient descent algorithm able to optimize arbitrary differentiable objective functions on transport networks, while exactly respecting arbitrary differentiable constraint functions. We prove that in a class of networks that we call stackable, which includes a model capillary bed, the uniform flow network will have the same flow as a uniform conductance network, i.e., in which all edges have the same conductance. This result agrees with uniform flow capillary bed network found by the algorithm. We also show that the uniform flow completely explains the observed radii within the zebrafish trunk vasculature. In addition to deriving new results on optimization of uniform flow in micro-vascular networks, our algorithm provides a general method for testing hypotheses about possible optimization principles underlying real microvascular networks, including exposing tradeoffs between flow uniformity and transport cost.

Simulation before fabrication: a case study on the utilization of simulators for the design of droplet microfluidic networks

The functional performance of passively operated droplet microfluidics is sensitive with respect to the dimensions of the channel network, the fabrication precision as well as the applied pressure because the entire network is coupled together. Especially, the local and global hydrodynamic resistance changes caused by droplets make the task to develop a robust microfluidic design challenging as plenty of interdependencies which all affect the intended behavior have to be considered by the designer. After the design, its functionality is usually validated by fabricating a prototype and testing it with physical experiments. In case that the functionality is not implemented as desired, the designer has to go back, revise the design, and repeat the fabrication as well as experiments. This current design process based on multiple iterations of refining and testing the design produces high costs (financially as well as in terms of time). In this work, we show how a significant amount of those costs can be avoided when applying simulation before fabrication. To this end, we demonstrate how simulations on the 1D circuit analysis model can help in the design process by means of a case study. Therefore, we compare the design process with and without using simulation. As a case study, we use a microfluidic network which is capable of trapping and merging droplets with different content on demand. The case study demonstrates how simulation can help to validate the derived design by considering all local and global hydrodynamic resistance changes. Moreover, the simulations even allow further exploration of different designs which have not been considered before due to the high costs.

Simulating microfluidic networks allows to check a design even before first prototypes are realized.

Optimal occlusion uniformly partitions red blood cells fluxes within a microvascular network

In animals, gas exchange between blood and tissues occurs in narrow vessels, whose diameter is comparable to that of a red blood cell. Red blood cells must deform to squeeze through these narrow vessels, transiently blocking or occluding the vessels they pass through. Although the dynamics of vessel occlusion have been studied extensively, it remains an open question why microvessels need to be so narrow. We study occlusive dynamics within a model microvascular network: the embryonic zebrafish trunk. We show that pressure feedbacks created when red blood cells enter the finest vessels of the trunk act together to uniformly partition red blood cells through the microvasculature. Using mathematical models as well as direct observation, we show that these occlusive feedbacks are tuned throughout the trunk network to prevent the vessels closest to the heart from short-circuiting the network. Thus occlusion is linked with another open question of microvascular function: how are red blood cells delivered at the same rate to each micro-vessel? Our analysis shows that tuning of occlusive feedbacks increase the total dissipation within the network by a factor of 11, showing that uniformity of flows rather than minimization of transport costs may be prioritized by the microvascular network.

Arterial trees shuttle red blood cells from the heart to billions of capillaries distributed throughout the body. These trees have long been thought to be organized to minimize transport costs. Yet red blood cells are tightly squeezed within the finest vessels, meaning that these vessels account for as much as half of the total transport costs within the arterial network. It is unclear why vessel diameters and red blood cell diameters are so closely matched in a network that is presumed to optimize transport. Here, we use mathematical modeling and direct observations of red blood cell movements in embryonic zebrafish to show that occlusive feedbacks—the pressure feedbacks that alter the flows into a vessel when it is nearly blocked by a red blood cell—can optimally distribute red blood cells through microvessels. In addition to revealing an adaptive function for the matching of vessel and red blood cell diameters, this work shows that uniformity of red blood cell fluxes can be a unifying principle for understanding the elegant hydraulic organization of microvascular networks.

System-level network simulation for robust centrifugal-microfluidic lab-on-a-chip systems

Centrifugal microfluidics shows a clear trend towards a higher degree of integration and parallelization. This trend leads to an increase in the number and density of integrated microfluidic unit operations. The fact that all unit operations are processed by the same common spin protocol turns higher integration into higher complexity. To allow for efficient development anyhow, we introduce advanced lumped models for network simulations in centrifugal microfluidics. These models consider the interplay of centrifugal and Euler pressures, viscous dissipation, capillary pressures and pneumatic pressures. The simulations are fast and simple to set up and allow for the precise prediction of flow rates as well as switching and valving events. During development, channel and chamber geometry variations due to manufacturing tolerances can be taken into account as well as pipetting errors, variations of contact angles, compliant chamber walls and temperature variations in the processing device. As an example of considering these parameters during development, we demonstrate simulation based robustness analysis for pneumatic siphon valving in centrifugal microfluidics. Subsequently, the influence of liquid properties on pumping and valving is studied for four liquids relevant for biochemical analysis, namely, water (large surface tension), blood plasma (large contact angle hysteresis), ethanol/water (highly wetting) and glycerine/water (highly viscous). In a second example, we derive a spin protocol to attain a constant flow rate under varying pressure conditions. Both examples show excellent agreement with experimental validations.

Dense bubble traffic in microfluidic loops: Selection rules and clogging

We study the repartition of monodisperse bubbles at the inlet node of an asymmetric microfluidic loop for low to high bubble densities. In large loops, we evidence a new regime. Contrary to the classical belief, we point out that bubbles are directed not towards the arm having the higher total flow rate but towards the arm with the higher water flow rate at low and moderate relative gas flow rates. At higher rates, they enter the longer arm when they reach close packing in the shorter arm. In small loops, we evidence a clogging regime at high relative gas flow rates. Collisions between bubbles coming from the two arms at the outlet clog the longer arm. We propose a comprehensive analysis allowing us to explain these results.

Coalescence of drops in a 2D microchannel: critical transitions to autocatalytic behaviour

A single coalescence event in a 2D concentrated emulsion in a microchannel can trigger an avalanche of similar events that can destabilize the entire assembly of drops. The sensitive dependence of the process on numerous parameters makes the propagation dynamics appear probabilistic. In this article, a stochastic simulation framework is proposed to understand this collective behavior in a system employing a large number of drops. We discover that the coalescence propagation dynamics exhibit a critical behavior where two outcomes are favored: no avalanche and large avalanches. Our analysis reveals that this behavior is a result of the inherent autocatalytic nature of the process. The effect of the aspect ratio of the drop assembly on the propagation dynamics is studied. We generate a parametric plot that shows the region of the parameter space where the propagation, averaged over the ensemble, is autocatalytic: where the possibility of near destabilization of the drop assembly appears.

Between giant oscillations and uniform distribution of droplets: The role of varying lumen of channels in microfluidic networks

The simplest microfluidic network (a loop) comprises two parallel channels with a common inlet and a common outlet. Recent studies that assumed a constant cross section of the channels along their length have shown that the sequence of droplets entering the left (L) or right (R) arm of the loop can present either a uniform distribution of choices (e.g., RLRLRL...) or long sequences of repeated choices (RRR...LLL), with all the intermediate permutations being dynamically equivalent and virtually equally probable to be observed. We use experiments and computer simulations to show that even small variation of the cross section along channels completely shifts the dynamics either into the strong preference for highly grouped patterns (RRR...LLL) that generate system-size oscillations in flow or just the opposite-to patterns that distribute the droplets homogeneously between the arms of the loop. We also show the importance of noise in the process of self-organization of the spatiotemporal patterns of droplets. Our results provide guidelines for rational design of systems that reproducibly produce either grouped or homogeneous sequences of droplets flowing in microfluidic networks.

Microfluidic plug steering using surface acoustic waves

Digital microfluidic systems, in which isolated droplets are dispersed in a carrier medium, offer a method to study biological assays and chemical reactions highly efficiently. However, it's challenging to manipulate these droplets in closed microchannel devices. Here, we present a method to selectively steer plugs (droplets with diameters larger than the channel's width) at a specially designed Y-junction within a microfluidic chip. The method makes use of surface acoustic waves (SAWs) impinging on a multiphase interface in which an acoustic contrast is present. As a result, the liquid-liquid interface is subjected to acoustic radiation forces. These forces are exploited to steer plugs into selected branches of the Y-junction. Furthermore, the input power can be finely tuned to split a plug into two uneven plugs. The steering of plugs as a whole, based on plug volume and velocity is thoroughly characterized. The results indicate that there is a threshold plug volume after which the steering requires elevated electrical energy input. This plug steering method can easily be integrated to existing lab-on-a-chip devices and it offers a robust and active plug manipulation technique in closed microchannels.

Collective dynamics of non-coalescing and coalescing droplets in microfluidic parking networks

We study the complex collective dynamics mediated by flow resistance interactions when trains of non-coalescing and coalescing confined drops are introduced into a microfluidic parking network (MPN). The MPN consists of serially connected loops capable of parking arrays of drops. We define parking modes based on whether drops park without breakage or drop fragments are parked subsequent to breakage or drops park after coalescence. With both non-coalescing and coalescing drops, we map the occurrence of these parking modes in MPNs as a function of system parameters including drop volume, drop spacing and capillary number. We find that the non-coalescing drops can either park or break in the network, producing highly polydisperse arrays. We further show that parking due to collision induced droplet break-up is the main cause of polydispersity. We discover that collisions occur due to a crowding instability, which is a natural outcome of the network topology. In striking contrast, with coalescing drops we show that the ability of drops to coalesce rectifies the volume of parked polydisperse drops, despite drops breaking in the network. We find that several parking modes act in concert during this hydrodynamic self-rectification mechanism, producing highly monodisperse drop arrays over a wide operating parameter space. We demonstrate that the rectification mechanism can be harnessed to produce two-dimensional arrays of microfluidic drops with highly tunable surface-to-volume ratios, paving the way for fundamental investigations of interfacial phenomena in emulsions.

Cooperative breakups induced by drop-to-drop interactions in one-dimensional flows of drops against micro-obstacles

Depending on the capillary number at play and the parameters of the flow geometry, a drop may or may not break when colliding with an obstacle in a microdevice. Modeling the flow of one-dimensional trains of monodisperse drops impacting a micro-obstacle, we show numerically that complex dynamics may arise through drop-to-drop hydrodynamic interactions: we observe sequences of breakup events in which the size of the daughter drops created upon breaking mother ones becomes a periodic function of time. We demonstrate the existence of numerous bifurcations between periodic breakup regimes and we establish diagrams mapping the possible breakup dynamics as a function of the governing (physicochemical, hydrodynamic, and geometric) parameters. Microfluidic experiments validate our model as they concur very well with predictions.

Observations of spontaneous oscillations in simple two-fluid networks

We investigate the laminar flow of two-fluid mixtures inside a simple network of interconnected tubes. The fluid system is composed of two miscible Newtonian fluids of different viscosity which do not mix and remain as nearly distinct phases. Downstream of a diverging network junction the two fluids do not necessarily split in equal fraction and thus heterogeneity is introduced into network. We find that in the simplest network, a single loop with one inlet and one outlet, under steady inlet conditions, the flow rates and distribution of the two fluids within the network loop can undergo persistent spontaneous oscillations. We develop a simple model which highlights the basic mechanism of the instability and we demonstrate that the model can predict the region of parameter space where oscillations exist. The model predictions are in good agreement with experimental observations.

Millifluidics as a simple tool to optimize droplet networks: Case study on drop traffic in a bifurcated loop

We report that modular millifluidic networks are simpler, more cost-effective alternatives to traditional microfluidic networks, and they can be rapidly generated and altered to optimize designs. Droplet traffic can also be studied more conveniently and inexpensively at the millimeter scale, as droplets are readily visible to the naked eye. Bifurcated loops, ladder networks, and parking networks were made using only Tygon(®) tubing and plastic T-junction fittings and visualized using an iPod(®) camera. As a case study, droplet traffic experiments through a millifluidic bifurcated loop were conducted, and the periodicity of drop spacing at the outlet was mapped over a wide range of inlet drop spacing. We observed periodic, intermittent, and aperiodic behaviors depending on the inlet drop spacing. The experimentally observed periodic behaviors were in good agreement with numerical simulations based on the simple network model. Our experiments further identified three main sources of intermittency between different periodic and/or aperiodic behaviors: (1) simultaneous entering and exiting events, (2) channel defects, and (3) equal or nearly equal hydrodynamic resistances in both sides of the bifurcated loop. In cases of simultaneous events and/or channel defects, the range of input spacings where intermittent behaviors are observed depends on the degree of inherent variation in input spacing. Finally, using a time scale analysis of syringe pump fluctuations and experiment observation times, we find that in most cases, more consistent results can be generated in experiments conducted at the millimeter scale than those conducted at the micrometer scale. Thus, millifluidic networks offer a simple means to probe collective interactions due to drop traffic and optimize network geometry to engineer passive devices for biological and material analysis.

Coalescing drops in microfluidic parking networks: A multifunctional platform for drop-based microfluidics

Multiwell plate and pipette systems have revolutionized modern biological analysis; however, they have disadvantages because testing in the submicroliter range is challenging, and increasing the number of samples is expensive. We propose a new microfluidic methodology that delivers the functionality of multiwell plates and pipettes at the nanoliter scale by utilizing drop coalescence and confinement-guided breakup in microfluidic parking networks (MPNs). Highly monodisperse arrays of drops obtained using a hydrodynamic self-rectification process are parked at prescribed locations in the device, and our method allows subsequent drop manipulations such as fine-gradation dilutions, reactant addition, and fluid replacement while retaining microparticles contained in the sample. Our devices operate in a quasistatic regime where drop shapes are determined primarily by the channel geometry. Thus, the behavior of parked drops is insensitive to flow conditions. This insensitivity enables highly parallelized manipulation of drop arrays of different composition, without a need for fine-tuning the flow conditions and other system parameters. We also find that drop coalescence can be switched off above a critical capillary number, enabling individual addressability of drops in complex MPNs. The platform demonstrated here is a promising candidate for conducting multistep biological assays in a highly multiplexed manner, using thousands of submicroliter samples.

Synchronized reinjection and coalescence of droplets in microfluidics

Coalescence of two kinds of pre-processed droplets is necessary to perform chemical and biological assays in droplet-based microfluidics. However, a robust technique to accomplish this does not exist. Here we present a microfluidic device to synchronize the reinjection of two different kinds of droplets and coalesce them, using hydrostatic pressure in conjunction with a conventional syringe pump. We use a device consisting of two opposing T-junctions for reinjecting two kinds of droplets and control the flows of the droplets by applying gravity-driven hydrostatic pressure. The hydrostatic-pressure operation facilitates balancing the droplet reinjection rates and allows us to synchronize the reinjection. Furthermore, we present a simple but robust module to coalesce two droplets that sequentially come into the module, regardless of their arrival times. These re-injection and coalescence techniques might be used in lab-on-chip applications requiring droplets with controlled numbers of solid materials, which can be made by coalescing two pre-processed droplets that are formed and sorted in devices.

Origin of periodic and chaotic dynamics due to drops moving in a microfluidic loop device

Droplets moving in a microfluidic loop device exhibit both periodic and chaotic behaviors based on the inlet droplet spacing. We observe that the periodic behavior is an outcome of carrier phase mass conservation principle, which translates into a droplet spacing quantization rule. This rule implies that the summation of exit spacing is equal to an integral multiple of inlet spacing. This principle also enables identification of periodicity in experimental systems with input scatter. We find that the origin of chaotic behavior is through intermittency, which arises when drops enter and leave the junctions at the same time. We derive an analytical expression to estimate the occurrence of these chaotic regions as a function of system parameters. We provide experimental, simulation, and analytical results to validate the origin of periodic and chaotic behavior.

Passive breakups of isolated drops and one-dimensional assemblies of drops in microfluidic geometries: experiments and models

Using two different geometries, rectangular obstacles and asymmetric loops, we investigate the breakup dynamics of deformable objects, such as drops and bubbles, confined in microfluidic devices. We thoroughly study two distinct flow configurations that depend on whether object-to-object hydrodynamic interactions are allowed. When such interactions are introduced, we find that the volumes of the daughter objects created after breakup solely depend on the geometrical features of the devices and are not affected by the hydrodynamic and physicochemical variables; these results are in sharp contrast with those obtained for non-interacting objects. For both configurations, we provide simple phenomenological models that capture well the experimental findings and predict the evolution of the volumes of the daughter objects with the controlling dimensionless quantities that are identified. We introduce a mean-field approximation, which permits accounting for the interactions between objects during breakup and we discuss its conditions of validity.

Droplet sorting in a loop of flat microfluidic channels

Motivated by recent experiments, we numerically study the droplet traffic in microfluidic channels forming an asymmetric loop with a long and a short arm. The loop is connected to an inlet and an outlet channel by two right angled T-junctions. Assuming flat channels, we employ the boundary element method (BEM) to numerically solve the two-dimensional Darcy equation that governs two phase flow in the Hele-Shaw limit. The occurrence of different sorting regimes is summarized in sorting diagrams in terms of droplet size, distance between consecutive droplets in the inlet channel, and loop asymmetry for mobility ratios of the liquid phases larger and smaller than one. For large droplet distances, the traffic is regulated by the ratio of the total hydraulic resistances of the long and short arms. At high droplet densities and below a critical droplet size, droplet-droplet collisions are observed for both mobility ratios.

Path selection rules for droplet trains in single-lane microfluidic networks

We investigate the transport of periodic trains of droplets through microfluidic networks having one inlet, one outlet, and nodes consisting of T junctions. Variations of the dilution of the trains, i.e., the distance between drops, reveal the existence of various hydrodynamic regimes characterized by the number of preferential paths taken by the drops. As the dilution increases, this number continuously decreases until only one path remains explored. Building on a continuous approach used to treat droplet traffic through a single asymmetric loop, we determine selection rules for the paths taken by the drops and we predict the variations of the fraction of droplets taking these paths with the parameters at play including the dilution. Our results show that as dilution decreases, the paths are selected according to the ascending order of their hydrodynamic resistance in the absence of droplets. The dynamics of these systems controlled by time-delayed feedback is complex: We observe a succession of periodic regimes separated by a wealth of bifurcations as the dilution is varied. In contrast to droplet traffic in single asymmetric loops, the dynamical behavior in networks of loops is sensitive to initial conditions because of extra degrees of freedom.

Bistability in droplet traffic at asymmetric microfluidic junctions

We present the first experimental demonstration of confined microfluidic droplets acting as discrete negative resistors, wherein the effective hydrodynamic resistance to flow in a microchannel is reduced by the presence of a droplet. The implications of this hitherto unexplored regime in the traffic of droplets in microfluidic networks are highlighted by demonstrating bistable filtering into either arm of symmetric and asymmetric microfluidic loops, and programming oscillatory droplet routing therein.

Building droplet-based microfluidic systems for biological analysis

In the present paper, we review and discuss current developments and challenges in the field of droplet-based microfluidics. This discussion includes an assessment of the basic fluid dynamics of segmented flows, material requirements, fundamental unit operations and how integration of functional components can be applied to specific biological problems.

Filtering microfluidic bubble trains at a symmetric junction

We report how a nominally symmetric microfluidic junction can be used to sort all bubbles of an incoming train exclusively into one of its arms. The existence of this "filter" regime is unexpected, given that the junction is symmetric. We analyze this behavior by quantifying how bubbles modulate the hydrodynamic resistance in microchannels and show how speeding up a bubble train whilst preserving its spatial periodicity can lead to filtering at a nominally symmetric junction. We further show how such an asymmetric traffic of bubble trains can be triggered in symmetric geometries by identifying conditions wherein the resistance to flow decreases with an increase in the number of bubbles in the microchannel and derive an exact criterion to predict the same.

Design of pressure-driven microfluidic networks using electric circuit analogy

This article reviews the application of electric circuit methods for the analysis of pressure-driven microfluidic networks with an emphasis on concentration- and flow-dependent systems. The application of circuit methods to microfluidics is based on the analogous behaviour of hydraulic and electric circuits with correlations of pressure to voltage, volumetric flow rate to current, and hydraulic to electric resistance. Circuit analysis enables rapid predictions of pressure-driven laminar flow in microchannels and is very useful for designing complex microfluidic networks in advance of fabrication. This article provides a comprehensive overview of the physics of pressure-driven laminar flow, the formal analogy between electric and hydraulic circuits, applications of circuit theory to microfluidic network-based devices, recent development and applications of concentration- and flow-dependent microfluidic networks, and promising future applications. The lab-on-a-chip (LOC) and microfluidics community will gain insightful ideas and practical design strategies for developing unique microfluidic network-based devices to address a broad range of biological, chemical, pharmaceutical, and other scientific and technical challenges.

Hamiltonian traffic dynamics in microfluidic-loop networks

Recent microfluidic experiments revealed that large particles advected in a fluidic loop display long-range hydrodynamic interactions. However, the consequences of such couplings on the traffic dynamics in more complex networks remain poorly understood. In this Letter, we focus on the transport of a finite number of particles in one-dimensional loop networks. By combining numerical, theoretical, and experimental efforts, we evidence that this collective process offers a unique example of Hamiltonian dynamics for hydrodynamically interacting particles. In addition, we show that the asymptotic trajectories are necessarily reciprocal despite the microscopic traffic rules explicitly break the time-reversal symmetry. We exploit these two remarkable properties to account for the salient features of the effective three-particle interaction induced by the exploration of fluidic loops.

Droplet formation in microfluidic T-junction generators operating in the transitional regime. II. Modeling

This is the second part of a two-part study on the generation of droplets at a microfluidic T-junction operating in the transition regime. In the preceding paper [Phys. Rev. E 85, 016322 (2012)], we presented our experimental observations of droplet formation and decomposed the process into three sequential stages defined as the lag, filling, and necking stages. Here we develop a model that describes the performance of microfluidic T-junction generators working in the squeezing to transition regimes where confinement of the droplet dominates the formation process. The model incorporates a detailed geometric description of the drop shape during the formation process combined with a force balance and necking criteria to define the droplet size, production rate, and spacing. The model inherently captures the influence of the intersection geometry, including the channel width ratio and height-to-width ratio, capillary number, and flow ratio, on the performance of the generator. The model is validated by comparing it to speed videos of the formation process for several T-junction geometries across a range of capillary numbers and viscosity ratios.

Bubbles navigating through networks of microchannels

This paper describes the behavior of bubbles suspended in a carrier liquid and moving within microfluidic networks of different connectivities. A single-phase continuum fluid, when flowing in a network of channels, partitions itself among all possible paths connecting the inlet and outlet. The flow rates along different paths are determined by the interaction between the fluid and the global structure of the network. That is, the distribution of flows depends on the fluidic resistances of all channels of the network. The movement of bubbles of gas, or droplets of liquid, suspended in a liquid can be quite different from the movement of a single-phase liquid, especially when they have sizes slightly larger than the channels, so that the bubbles (or droplets) contribute to the fluidic resistance of a channel when they are transiting it. This paper examines bubbles in this size range; in the size range examined, the bubbles are discrete and do not divide at junctions. As a consequence, a single bubble traverses only one of the possible paths through the network, and makes a sequence of binary choices ("left" or "right") at each branching intersection it encounters. We designed networks so that, at each junction, a bubble enters the channel into which the volumetric flow rate of the carrier liquid is highest. When there is only a single bubble inside a network at a time, the path taken by the bubble is, counter-intuitively, not necessarily the shortest or the fastest connecting the inlet and outlet. When a small number of bubbles move simultaneously through a network, they interact with one another by modifying fluidic resistances and flows in a time dependent manner; such groups of bubbles show very complex behaviors. When a large number of bubbles (sufficiently large that the volume of the bubbles occupies a significant fraction of the volume of the network) flow simultaneously through a network, however, the collective behavior of bubbles-the fluxes of bubbles through different paths of the network-can resemble the distribution of flows of a single-phase fluid.

Passive droplet trafficking at microfluidic junctions under geometric and flow asymmetries

When droplets enter a junction they sort to the channel with the highest flow rate at that instant. Transport is regulated by a discrete time-delayed feedback that results in a highly periodic behavior where specific patterns can continue to cycle indefinitely. Between these highly ordered regimes are chaotic structures where no pattern is evident. Here we develop a model that describes droplet sorting under various asymmetries: branch geometry (length, cross-section), droplet resistance and pressures. First, a model is developed based on the continuum assumption and then, with the assistance of numerical simulations, a discrete model is derived to predict the length and composition of the sorting pattern. Furthermore we derive all unique sequences that are possible for a given distribution and develop a preliminary estimation of why chaotic regimes form. The model is validated by comparing it to numerical simulations and results from microfluidic experiments in PDMS chips with good agreement.

Emergent behavior in particle-laden microfluidic systems informs strategies for improving cell and particle separations

Colloidal particles placed in an energy landscape interact with each other, giving rise to complex dynamic behavior that affects the ability to process and manipulate suspensions of these particles. Propagating across scales ranging from the local behavior of 10's of particles to non-local behavior encompassing >10(6) particles, these particle interactions are pervasive and challenging to describe quantitatively, especially in the confined environments typical of microfluidic devices. To better understand the effects of particle interactions in this context, we have performed experiments and simulations involving a simple microfluidic device in which hydrodynamic and electrostatic forces are leveraged to concentrate and separate particle mixtures. These investigations reveal the mechanisms underlying the dynamic patterns formed by micron-scale particles as they impinge on a dielectrophoretic force barrier: their tendency to aggregate and recirculate under constant operating conditions, and to reorganize when the operating conditions are changed. The emergent behaviors of these ensembles of interacting particles exhibit features of dynamical frustration and cooperativity that suggest non-intuitive strategies for concentrating and sorting suspensions. Finally, we present a simple analytic model based on hydrodynamic coupling that captures important features of strongly interacting particle suspensions.

Droplets formation and merging in two-phase flow microfluidics

Two-phase flow microfluidics is emerging as a popular technology for a wide range of applications involving high throughput such as encapsulation, chemical synthesis and biochemical assays. Within this platform, the formation and merging of droplets inside an immiscible carrier fluid are two key procedures: (i) the emulsification step should lead to a very well controlled drop size (distribution); and (ii) the use of droplet as micro-reactors requires a reliable merging. A novel trend within this field is the use of additional active means of control besides the commonly used hydrodynamic manipulation. Electric fields are especially suitable for this, due to quantitative control over the amplitude and time dependence of the signals, and the flexibility in designing micro-electrode geometries. With this, the formation and merging of droplets can be achieved on-demand and with high precision. In this review on two-phase flow microfluidics, particular emphasis is given on these aspects. Also recent innovations in microfabrication technologies used for this purpose will be discussed.

Transport of resistance through a long microfluidic channel

We introduce a continuous analytical model of propagation of resistance in pressure-driven flow of two-phase fluid in a single channel. This model can be used to predict and interpret experimental results in droplet microfluidics where the hydrodynamic resistance of a capillary comprises a constant part, specific to the channel and the viscosity of the continuous fluid, and a variable part, related to the presence and distribution of droplets. The continuous model is a convenient generalization of the discrete models as demonstrated via comparisons with discrete simulations and with experiments.

Complex dynamics of droplet traffic in a bifurcating microfluidic channel: periodicity, multistability, and selection rules

The binary path selection of droplets reaching a T junction is regulated by time-delayed feedback and nonlinear couplings. Such mechanisms result in complex dynamics of droplet partitioning: numerous discrete bifurcations between periodic regimes are observed. We introduce a model based on an approximation that makes this problem tractable. This allows us to derive analytical formulae that predict the occurrence of the bifurcations between consecutive regimes, establish selection rules for the period of a regime, and describe the evolutions of the period and complexity of droplet pattern in a cycle with the key parameters of the system. We discuss the validity and limitations of our model which describes semiquantitatively both numerical simulations and microfluidic experiments.

Simulation of droplet trains in microfluidic networks

We show that in a microfluidic network with low Reynolds numbers, a system can be irreversible due to hysteresis effects. We simulated a network of pipes that was used in a recent experiment. The network consists of one loop connected to input and output pipes. A train of droplets enters the system at a uniform rate, but the droplets may leave the system in a periodic or even a chaotic pattern. The output pattern depends on the time interval between incoming droplets as well as the network geometry. For some parameters, the system is not reversible.