CRITICAL REVIEW
`
`www.rsc.org/loc | Lab on a Chip
`
`Dynamics of microfluidic droplets
`
`Charles N. Baroud,*a Francois Gallaireb and Remi Danglaa
`
`Received 19th January 2010, Accepted 28th April 2010
`DOI: 10.1039/c001191f
`
`This critical review discusses the current understanding of the formation, transport, and merging of
`drops in microfluidics. We focus on the physical ingredients which determine the flow of drops in
`microchannels and recall classical results of fluid dynamics which help explain the observed behaviour.
`We begin by introducing the main physical ingredients that differentiate droplet microfluidics from
`single-phase microfluidics, namely the modifications to the flow and pressure fields that are introduced
`by the presence of interfacial tension. Then three practical aspects are studied in detail: (i) The
`formation of drops and the dominant interactions depending on the geometry in which they are
`formed. (ii) The transport of drops, namely the evaluation of drop velocity, the pressure-velocity
`relationships, and the flow field induced by the presence of the drop. (iii) The fusion of two drops,
`including different methods of bridging the liquid film between them which enables their merging.
`
`I.
`
`Introduction
`
`Interest in manipulating droplets in microchannels has emerged
`from two distinct but complementary motivations. The first
`results from the desire to produce well calibrated droplets for
`material science applications, for example in the pharmaceutical
`or food industries. In this context, microfluidics provides a way
`for producing such droplets in a controlled and reproducible
`manner, also allowing complex combinations to be designed and
`
`aLadHyX and Department of Mechanics, Ecole Polytechnique, CNRS,
`91128 Palaiseau cedex, France. E-mail: baroud@ladhyx.polytechnique.fr
`bInstitut de genie mecanique, Sciences et techniques de l’ingenieur, Ecole
`Polytechnique Federale de Lausanne, Switzerland
`
`explored.1,2 A second motivation originates in lab on a chip
`applications where drops are viewed as micro-reactors, in which
`samples are confined, and which offer a way to manipulate small
`volumes.3 The idea of performing chemical or biochemical
`reactions in droplets had already been explored, before the
`microfluidics era, through the use of emulsions in order to
`‘‘compartmentalize’’
`reactions
`inside many small parallel
`volumes.4,5 The introduction of microfluidics tools again acts to
`facilitate the production and manipulation of these compart-
`ments.
`By the same token, the use of drops addresses one of the most
`fundamental problems encountered in single-phase microfluidics
`by providing control over dispersion and mixing of chemicals,
`through the encapsulation of the analytes within the drop.3 The
`
`Charles Baroud is Professeur
`Charge de Cours at Ecole Poly-
`technique in France, where he
`founded and leads the micro-
`fluidics research group at the
`Laboratoire d’Hydrodynamique
`(LadHyX). He studied at MIT
`then at the University of Texas
`at Austin, before doing a post-
`Ecole
`doc
`at
`Normale
`Superieure in Paris. Since 2002,
`his
`microfluidics
`research
`focuses on multiphase flows in
`complex geometries and on the
`control of droplet microfluidics
`for lab on a chip applications.
`
`Franc¸ois Gallaire gained his
`PhD from the Hydrodynamics
`Laboratory (LadHyX), Ecole
`Polytechnique, Paris. A CNRS
`research fellow at the J.A. Die-
`udonne
`Laboratory,
`Nice
`Sophia-Antipolis
`University
`until 2009 he left to join the
`Ecole Polytechnique Federale de
`Lausanne (EPFL) as founding
`professor of the Laboratory of
`Fluid Mechanics and Instabil-
`ities (LFMI). His research into
`the basic physical mechanisms
`governing fluid dynamics
`is
`guided by real applications. He
`designs active flow control strategies to dampen instabilities in
`archetypical detached boundary layer and swirling flows or to
`enhance them in confined wakes. His contributions to the field of
`microfluidics include modeling laser manipulation of a droplet in
`a micro-canal using a depth-averaged set of equations to account
`for thermocapillary action at the interface.
`
`Francois Gallaire
`
`Charles N: Baroud
`
`2032 | Lab Chip, 2010, 10, 2032–2045
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`This journal is ª The Royal Society of Chemistry 2010
`
`

`
`tension. This new physical ingredient can be thought of in two
`complementary ways, either of which can be used depending on
`the point of view to be taken.
`First of all, it is a force per unit length which pulls the interface
`with a magnitude g (N m1). As such, any spatial imbalance in
`the value of g will lead to a flow along the interface from the low
`to the high interfacial tension regions, a phenomenon known as
`Marangoni flow. Since the value of the surface tension varies
`with temperature and with the contamination of the interface by
`surfactant molecules, either of these can lead to a Marangoni
`flow, which is then referred to as thermocapillary or soluto-
`capillary flow, respectively.
`Interfacial tension can also be thought of as an energy per unit
`area (J m2) which acts to minimise the total surface area so as to
`reduce the free energy of the interface. The minimum area for
`a given volume is a sphere, which is the shape taken by an iso-
`lated droplet or bubble. Confined drops on the other hand must
`adapt their shape to the presence of walls, while still curving their
`interface. The curvature introduces a pressure jump, known as
`the Laplace pressure, between the inside and the outside of the
`droplet. It is written as DP ¼ g(1/R1 + 1/R2), where R1 and R2 are
`the two principal radii of curvature of the interface. The pressure
`jump is determined locally at each position of the interface; since
`R1 and R2 can vary in space, this can induce pressure variations
`within a droplet. These supplementary pressure variations will
`play a major role in determining the flow conditions as we shall
`see further.
`From a modeling point of view, the presence of droplets also
`introduces new kinematic and dynamic boundary conditions on
`the fluid flow, since the immiscible fluids cannot cross the inter-
`face. The first new boundary condition states that the local
`normal component of the velocities in each fluid must be equal to
`the interface velocity. Second, the velocities tangent to the
`interface must also be equal inside and outside the droplet. Third,
`the tangential shear stresses must also be balanced at the inter-
`face when it is clean of contaminants. This means that the vari-
`ation of the tangential velocity (uk) with respect to the normal
`direction r, inside (vuk/vr|in) and outside (vuk/vr|out) the drop,
`must balance
`
`:
`
`out
`
`(1)
`
`

`
`

`
`min
`
`vuk
`vr
`
`¼ mout
`
`vuk
`vr
`
`in
`
`manipulation of small volumes is also simplified: Indeed, drops
`provide new physical and chemical contrasts with the outer
`medium, such as the dielectric constant or interfacial tension,
`which can be used to manipulate the minute volumes on-chip
`while bypassing large lab machines.6 Moreover, they reduce the
`sensitivity of the devices to the surface properties of the micro-
`channel, since the fluid of interest is isolated from the walls by the
`carrier phase.
`All these advantages however come at the price of raising
`a new set of fluid dynamical problems that appear due to the
`deformable interface of the droplets, the need to take into
`account interfacial tension and its variations, and the complexity
`of singular events such as merging or splitting of drops. In the
`physicist’s vocabulary, drops introduce nonlinear laws into the
`otherwise linear Stokes flows. Evidence of this nonlinearity can
`be found, for example, by considering that different flow regimes
`can appear in the same channel and under the same forcing
`conditions.7 Moreover, small variations of the driving conditions
`can lead to transitions between the production of drops or of
`stable jets, a classical signature of nonlinear instabilities.8,9 These
`transitions between widely different behaviours are possible
`because modifications in the drop geometry couple back to the
`flow profiles and amplify initially small variations.
`A large body of work has recently attempted to tackle these
`fluid dynamical questions, leading along the way to creative new
`designs for microfluidic systems and new physical approaches to
`control the behaviour of drops. Below we will discuss this body
`of literature while concentrating on drops in microfluidic chan-
`nels. We will avoid any comparison between the behaviour of
`droplets within closed microchannels and on open patterned
`surfaces, an approach sometimes called ‘‘digital microfluidics’’.
`For a comparative study of these two approaches, the reader is
`referred to the review article by Darhuber and Troian.10 We will
`further limit our review to three fundamental aspects of droplet
`microfluidics: production of droplets, their transport, and their
`merging. We begin by considering the underlying physical
`ingredients, before moving on to specific considerations for each
`operation.
`
`II. Physical ingredients
`
`The main modification that droplets bring to single phase
`microfluidic flows comes through the introduction of interfacial
`
`Remi Dangla is a PhD student
`Ecole Poly-
`at LadHyX,
`technique, under the supervision
`of Prof. Charles N. Baroud. His
`research focuses on droplet
`dynamics and manipulations in
`microchannels of high aspect
`ratio. He received his Masters
`degree in Fluid Mechanics and
`his Diplome d’Ingenieur from
`Ecole Polytechnique, France.
`
`Remi Dangla
`
`Eqn (1) introduces the importance of the viscosity ratio l ¼
`min/mout, which plays a determining role for the flow fields inside
`and outside a moving drop or bubble. Fourth, the jump in
`normal stress at the interface leads to a generalization of
`Laplace’s law taking into account the viscous normal stress in
`addition to the pressure contribution.
`
`A. Dimensionless numbers
`
`As always in fluid dynamics, the fluid behaviour will depend on
`the values taken by some important dimensionless numbers
`which compare different physical ingredients. In what follows we
`will limit ourselves to inertia-less fluid mechanics, meaning that
`we will consider small Reynolds number regimes. The Weber
`number (We ¼ rU2l /g where U represents a characteristic
`velocity scale), which compares inertia to interfacial tension, will
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`Lab Chip, 2010, 10, 2032–2045 | 2033
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`

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`the chemical kinetics. Finally, any change in the shape of a drop
`will lead to local contraction or expansion of the interface, which
`lead to an increase or a decrease, respectively, of surface
`concentration.
`All of the above mechanisms can lead to variations of inter-
`facial tension along the drop surface, which will couple back with
`the drop formation and motion,
`in addition to influencing
`droplet
`fusion. Since different
`surfactant molecules have
`different characteristics, changing surfactants can have a major
`impact in drop behaviour regarding the areas covered in this
`review. In this regard, stationary model experiments, such as the
`pendant drop method for measuring surface tension, can help
`guide the physical understanding. Practical microfluidics situa-
`tions however often involve a complex interplay between several
`effects which cannot be simply described in intuitive terms.
`
`III. Droplet production in microchannels
`
`The first step in the microfluidic life cycle of a droplet is its
`production. Besides a few implementations of the drop-on-
`demand technique based on the control of integrated micro-
`valves, the majority of microfluidic methods produce droplet
`volumes ranging from femtolitres to nanolitres. This is achieved
`through passive techniques which generate a uniform, evenly
`spaced, continuous stream.17 These strategies take advantage of
`the flow field to deform the interface and promote the natural
`growth of interfacial instabilities, thus avoiding local external
`actuation. Droplet polydispersity in these streams, defined as the
`standard deviation of the size distribution divided by the mean
`droplet size, can be as small as 1–3%.
`Not only should devices for making drops produce a regular
`and stable monodisperse droplet stream, they also need to be
`flexible enough to provide droplets of prescribed volume at
`a prescribed rate. To this end, three main approaches have
`emerged based on different physical mechanisms; they are best
`described by the flow field topology in the vicinity of the drop
`production zone: (i) breakup in co-flowing streams (Fig. 1), (ii)
`breakup in cross-flowing streams (Fig. 2) and (iii) breakup in
`elongational strained flows (Fig. 3).
`In all three cases, the phase to be dispersed is driven into
`a microchannel, where it encounters the immiscible carrier fluid
`which is driven independently. The junction where the two fluids
`meet is designed to optimize the reproducibility of droplet
`production. Indeed, the geometry of the junction, together with
`the flow rates and the physical properties of the fluids (interfacial
`tension, viscosities) determine the local flow field, which in turn
`deforms the interface and eventually leads to drop/bubble pinch
`off. The size of the droplet is set by a competition between the
`
`Fig. 1 Example of droplet production in a co-axial injection device. The
`inner flow is produced by a thin round capillary and enters into a square
`capillary.
`
`in
`
`out
`
`where Vk indicates the derivative along the tangent to the inter-
`face at every point. For clean and isothermal interfaces, one
`recovers eqn (1). The relation between g and the local surfactant
`concentration is nonlinear, sometimes modelled through the so-
`called ‘‘Langmuir model’’.14
`A complete description of surfactant transport is beyond the
`scope of this review but one can readily see that these molecules
`can be transported either by the hydrodynamic flow (advection),
`or through molecular diffusion, either in the bulk or along the
`interface.15,16 In addition to their transport, surfactants are
`characterised by several physico-chemical constants: (i) the
`partition coefficient, which measures the relative bulk and
`surface concentrations at equilibrium, as well as (ii)
`their
`adsorption and desorption rates on the interface, which measure
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`2034 | Lab Chip, 2010, 10, 2032–2045
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`This journal is ª The Royal Society of Chemistry 2010
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`in microfluidics. Note however that
`also generally be small
`inertial effects can come into play in certain situations of high-
`speed flows, for example for high throughput or droplet breakup
`situations. Finally, we will ignore the effects of gravity, which can
`be quantified by taking the Bond number, which compares
`gravity to interfacial tension, to be small: Bo ¼ Drgl2/g  1,
`where Dr is the difference in fluid densities, g is the acceleration
`of gravity, and l a characteristic length scale.
`This leaves interfacial tension and viscosity in competition
`with each other, since both tend to become important at small
`scales. The relative strength of the two is expressed by the
`Capillary number Ca ¼ mU/g, where m is generally the larger
`viscosity acting in the system. A low value of Ca indicates that
`the stresses due to interfacial tension are strong compared to
`viscous stresses. Drops flowing under such a condition nearly
`minimise their surface area by producing spherical ends. In the
`opposite situation of high Ca, viscous effects dominate and one
`can observe large deformations of the drops and asymmetric
`shapes.
`In some cases of interest the velocity varies over a length scale
`different from the radius of the drop, for example when the
`channel geometry expands or contracts. In this case, a new
`capillary number emerges, based on the characteristic magnitude
`of the shear stress inherent to the flow mdU/ds, where s represents
`a spatial direction. These stresses must still be compensanted by
`the Laplace pressure, which yields Cas ¼ m(dU/ds)R/g. This
`capillary number describes the magnitude of deformation
`observed on a drop due to variations in velocity,11 for example as
`a drop enters a bifurcating microchannel.12,13
`
`B. Surfactant effects
`
`The value of interfacial tension displays a strong dependence on
`the local surface coverage with surfactant molecules. These
`molecules are often added on purpose, in order to facilitate the
`creation and transport of drops, but can also appear as impuri-
`ties in the fluids or as by-products of chemical reactions. As such,
`the value of interfacial tension can vary spatially if the surface
`concentration displays spatial variations. This has an important
`consequence as it introduces a tangential stress jump in eqn (1),
`called Marangoni stress,
`
`þ Vkg
`
`(2)
`
`

`
`

`
`min
`
`vuk
`vr
`
`¼ mout
`
`vuk
`vr
`
`

`
`aligned with a square or rectangular outer channel, with the two
`streams flowing in parallel near the nozzle. It was first imple-
`mented in the context of microfluidics by Cramer et al.,18 who
`inserted a micro-capillary into a rectangular flow cell. They
`showed that the breakup of the liquid stream into droplets could
`be separated into two distinct regimes: dripping,
`in which
`droplets pinch off near the capillary tube’s tip, and jetting in
`which droplets pinch off from an extended thread downstream of
`the tube tip. The transition from dripping to jetting occurs when
`the continuous phase velocity increases above a critical value,
`U*. They found that the value of U* decreases as the flow rate of
`the dispersed phase increases. U* was also found to depend on
`the viscosities of the inner and outer phases, as well as on the
`interfacial tension.
`The trends from ref. 18 were confirmed simultaneously by
`Utada et al.19,20 and Guillot et al.,8,21 through stability analyses
`of viscous threads confined within a viscous outer liquid in
`a microchannel. Both groups interpreted the transition from
`dripping to jetting as a transition from an absolute to
`a convective instability, a terminology which refers to the
`ability of perturbations to grow and withstand the mean
`advection: Absolute instabilities grow faster than they are
`advected, contaminate the whole domain and yield a self-sus-
`tained well-tuned oscillation. In contrast, convective instabil-
`ities are characterised by a dominating advection of
`the
`perturbations and behave as amplifiers of the noise that may
`exist in the system.9 In co-axial injection devices, an absolutely
`unstable configuration is expected to result in a self-sustained
`formation of droplets close to the device inlet, while a con-
`vectively unstable flow is expected to result in droplets which
`form a finite distance downstream, only after the instability has
`had space to grow.
`Using a lubrication approximation, Guillot et al.8 analysed the
`transition in detail as a function of the viscosity ratio, the
`capillary number and the equilibrium confinement parameter x,
`defined as the ratio of the equilibrium jet radius to the effective
`radius of the square outer channel. For a given confinement
`parameter, absolute instability was found to exist below a critical
`value of the capillary number, which is assumed to determine the
`transition from dripping to jetting. The critical value decreases as
`the confinement parameter increases and the transition thresh-
`olds agree well with the experimental observations, making the
`interpretation of the dripping/jetting transition as an absolute/
`convective instability transition appealing. However, to date no
`experimental verification has been made of the frequency and
`wavelength selection that follows from the theoretical analysis.
`Such quantitative comparison would be useful to confirm the
`stability analysis interpretation.
`The theory mentioned above was developed for co-axial
`streams flowing in a circular cylindrical geometry. However, the
`authors also considered the influence of the geometry of the outer
`channel and showed that the instability was suppressed as soon
`as the inner jet radius increased beyond the smallest half-side of
`rectangular channels. The stabilization mechanism relies on the
`fact that a cylindrical thread can decrease its surface area when
`subjected to a varicose perturbation, while a squeezed, quasi two-
`dimensional thread always increases its surface when perturbed.
`This was first observed within the microfluidic context by
`Migler22 and further analyzed and applied by Humphry et al.,23
`
`Fig. 2 Example of droplet production in a T-junction. The dispersed
`phase and the carrier phase meet at 90 degrees in a T-shaped junction.
`
`Fig. 3 Example of droplet production in a flow-focusing device. The
`dispersed phase is squeezed by two counter-streaming flows of the carrier
`phase, forcing drops to detach.
`
`pressure due to the external flow and viscous shear stresses, on
`the one hand, and the capillary pressure resisting deformation on
`the other.
`Among all dimensionless numbers, the most important is
`therefore the capillary number Ca based on the mean continuous
`phase velocity, which compares the relative importance of the
`viscous stresses with respect to the capillary pressure. This
`number ranges between 103 and 101 in most microfluidic droplet
`formation devices. Additional dimensionless parameters are
`the ratio of flow-rates q ¼ Qin/Qout, viscosities l ¼ min/mout,
`and the geometric ratios, typically the ratio of channel widths
`x ¼ win/wout.
`Below, we review the current understanding regarding the
`mechanisms at play in each of the three geometries that have
`come to dominate droplet production. While the physics at the
`origin of droplet production in co-axial injectors is easily iden-
`tified as related to the Rayleigh-Plateau instability, the cylin-
`drical geometry of the injector is a serious obstacle to its
`implementation in soft lithography Lab on the Chip devices. In
`contrast, the two alternative geometries of T-junction and flow
`focusing are well suited to planar geometries but present more
`complex fluid dynamics, as detailed below.
`
`A. Co-flowing streams
`
`A typical example representing the geometry of co-flow devices is
`shown in Fig. 1. It corresponds to a cylindrical glass tube that is
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`Lab Chip, 2010, 10, 2032–2045 | 2035
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`

`
`among others. More recently, Utada et al.20 have generalized
`these results by relaxing first the lubrication assumption and then
`the creeping flow limit, thus considering inertial effects that
`become significant at large capillary numbers.
`
`B. T-junctions
`
`Droplet formation in a T-shaped device was first reported by
`Thorsen et al.,24 who used pressure controlled flows in micro-
`channels to generate droplets of water in a variety of different
`oils. A typical example of a T-junction is depicted in Fig. 2, which
`shows the two phases flowing through two orthogonal channels
`and forming drops when they meet.
`Three regimes could be distinguished as x ¼ win/wout, the ratio
`of the dispersed phase channel width to the carrier phase channel
`width, and the flow-rate ratio are varied. When x  1 and when
`the capillary number is large enough, the droplets are emitted
`before they can block the channel and their formation is entirely
`due to the action of shear-stress. In this regime, sometimes called
`the dripping regime, droplets break when the viscous shear stress
`overcomes the interfacial tension, analogous to spherical droplet
`breakup. A second regime, the squeezing regime, is observed for
`x of order 1 and when the capillary number is low enough, as
`described by Garstecki et al.25 In this case, the droplet obstructs
`the channel as it grows, restricting the flow of the continuous
`phase around it. This reduction in the gap through which
`the continuous phase can flow leads to a dramatic increase in the
`dynamic pressure upstream of the droplet, thus forcing the
`interface to neck and pinch off into a droplet. The combined
`influence of the Capillary number and the viscosity ratio on the
`transition to this second regime of droplet formation has been
`analyzed numerically by de Menech et al.26 The squeezing regime
`further evolves into the formation of stable parallel flowing
`streams when the dispersed phase flow rate becomes larger than
`the continuous phase flow rate.27 The critical dispersed phase
`velocity required for the transition from droplet formation to
`parallel flowing streams decreases with an increase in viscosity of
`the dispersed phase.
`With their analysis of the squeezing regime, Garstecki et al.
`predict that the drop length increases linearly with the flow-rate
`ratio25 and that the droplet length is independent of the contin-
`uous phase viscosity over a wide range of oil viscosities. On the
`other hand, more recent numerical studies28 and experimental
`work29,30 demonstrate that the viscosity ratio is indeed important
`for the droplet formation process in the intermediate regime
`(x < 1) where both shear stress and confinement strongly influ-
`ence the shape of the emerging droplet. Christopher et al.30
`further establish an extended scaling law which accounts for the
`influence of
`the viscosity and channel width ratios, also
`proposing scaling laws for the rate of production of droplets,
`which agree well with the experiments. Most recently Van Steijn
`et al.31 related the neck collapse to significant reverse flow in the
`corners between the phase to be dispersed and the channel walls.
`
`C. Flow focusing devices
`
`In the flow focusing geometry, first proposed by Anna et al.32 and
`Dreyfus et al.,33 the dispersed phase is squeezed by two counter-
`flowing streams of the continuous phase. Four main regimes can
`
`be identified as the parameters are varied: squeezing, dripping,
`jetting and thread formation. However, the large number of
`geometrical aspect ratios characterizing flow-focusing devices
`has prevented the determination of simple scaling laws to predict
`the droplet size, distribution and rate of emission as a function of
`the key parameters. Indeed, three new lengths are introduced in
`the problem in addition to win and wout, as seen in Fig. 3: the
`width of the aperture D and its length La, as well as the collector
`channel width w.
`Nevertheless, the mechanisms governing squeezing-dripping
`regime when the dispersed phase is a gas have been studied by
`Garstecki et al.34 and later by Dollet et al.35 In this squeezing
`regime, the droplet breakup proceeds in two distinct phases: The
`squeezed thread begins by thinning down quasi-statically through
`the effect of the hydrodynamic forcing34 and the duration of this
`first phase increases with the aspect ratio of the channel and is
`absent for square capillaries.35 Then, as the thread size becomes
`similar to the depth of channel, it adopts a cylindrical shape and
`rapidly becomes unstable due to the capillary (Rayleigh-Plateau)
`instability. The breakup then takes place as classical droplet
`pinch-off, governed by inertia and surface tension.35
`It is yet not clear if this scenario for gas threads operates in the
`same way for the viscous liquid jets described for instance by
`Cubaud et al.36 or Lee et al.37 In addition to the difference in the
`viscosity contrast in the two cases, liquid flows are generally
`forced by controlling the volumetric flow rates while constant
`pressure is typically used to control the flow of gas. As such,
`many of the physical arguments used in deriving the droplet
`scaling laws34,35 break down. Indeed, Ward et al.38 report a much
`higher sensitivity of the bubble size to flow rate variations when
`flow rate rather than pressure is controlled, even though the two
`parameters are linearly related to each other in a single-phase
`flow. The details of these differences are complex and not fully
`explained, although they are attributed to the nonlinearities
`introduced by surface tension.
`As already mentioned, there are no available clear-cut scaling
`laws for the transitions between various regimes nor for the size
`and rate of production of droplets. Recent velocity field
`measurements39 suggest that the squeezing phenomenon is gov-
`erned by the build up of a pressure difference as the advancing
`finger partially blocks the outlet channel, via a mechanism very
`similar to the one active in T-junctions. Other reports however
`state that squeezing/dripping droplet breakup depends solely on
`the upstream geometry and associated flow field, and not on the
`geometry of the channel downstream of the flow focusing
`orifice.37 By contrast, the elongation and breakup of the fine
`thread during the thread formation mode of breakup depends
`solely on the geometry and flow field in the downstream channel.
`In light of these recent papers and despite the widespread use of
`flow-focusing devices, it is clear that the understanding of their
`detailed dynamics still warrants further research.
`
`D. Active control of droplet production
`
`Applications of droplet microfluidics to Lab on a Chip tech-
`nologies will eventually require finer and more local control of
`droplet production than what is allowed by passive techniques.
`When the fluids are driven with constant flow rates, the volume
`fraction of the dispersed vs. carrier phase is fixed by the ratio of
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`
`assumed to flow at the local velocity of the carrier fluid and will
`tend to follow the streamlines of the external phase. This implies
`that drops that are nearer to the channel centreline will flow
`faster than those close to the edges. Moreover, drops arriving at
`a bifurcation will take the path that is dictated by the local
`streamlines of the carrier fluid.46 In contrast, the second category
`is more interesting, from a hydrodynamics point of view, because
`the flow is strongly modified by capillary effects and by the
`deformability of the drop interfaces. This places the capillary
`number based on the velocity of the droplets Cad ¼ mVd/g at the
`centre of the discussion. A third case exists when the channel has
`a large width/height aspect ratio. This can lead to drops that are
`strongly confined in only one direction, a situation that has been
`studied extensively in classical fluid mechanics.47,48 The flow of
`such drops and bubbles is very different compared to the above
`cases. For simplicity, we will restrict our discussion to channels
`with aspect ratio near one.
`In this section we explain the different models for drop
`transport in microchannels. We assume for simplicity that the
`carrier fluid completely wets the channel walls, thereby avoiding
`discussions of contact line dynamics. We also distinguish flows in
`circular tubes from those in rectangular tubes, which are more
`relevant to microfluidic situations. Moreover, it is useful to keep
`in mind that the models of droplet transport can also be under-
`stood by focusing on the plugs that separate droplets, which may
`be easier to address in some cases. Below we concentrate on three
`aspects of drop transport: the deposition of lubrication films and
`its relationship to droplet velocity, the pressure drop vs. droplet
`velocity relationships, and the flow patterns that are induced by
`the immiscible interface.
`
`A. Lubrication films and droplet velocity
`
`Consider a large droplet that is transported in a microchannel,
`with a velocity Vd from left to right, as depicted in Fig. 4B. As the
`drop flows, a thin lubrication film of the continuous phase is
`deposited between the droplet and the channel walls,49,50
`a process that can be understood by balancing viscous entrain-
`ment by the channel walls against the capillary pressure in the
`drop. In the reference frame of the droplet, the channel walls
`move in the opposite direction with velocity –Vd. By viscous
`entrainment, they pull the carrier fluid from right to left,
`depositing a ‘‘coating film’’ between the droplet and the walls. On
`the other hand, the pressure in the droplet is larger than the
`outside because of the Laplace pressure jump at the interfaces. It
`therefore pushes against the walls and expels liquid from the
`deposited films into the bulk. The competition between the
`viscous drag and capillary pressure determines the thickness e of
`the lubrication films, which therefore depend on the capillary
`number Cad.
`Bretherton51 found a nonlinear law for e in the case of an
`inviscid bubble moving at small capillary number in a circular
`tube of diameter H
`
`(3)
`
`f Ca2=3
`
`d
`
`:
`
`e H
`
`Similar scaling results have been derived for moving foams and
`bubble trains,52 viscous drops,53 and extended for any polygonal
`cross section geometry in the case of a single bubble.54,55 These
`
`flow rates. The control of drop formation can therefore only
`change the frequency and size of drops simultaneously while
`respecting the volume fraction. In the case when the dispersed
`flow is controlled by a pressure source, one can block the
`production of drops for long times and thus vary independently
`the size and frequency of the droplets.
`Control mechanisms for droplet production that rely on
`integrated micro-valves have been proposed.40–42 Closer to the
`topic of this review, variations in drop generation can be
`produced by varying any of the physical or geometric parameters
`that enter into the stress balances described