J. Fluid Mech. (2008), vol. 595, pp. 141–161.
`doi:10.1017/S002211200700910X Printed in the United Kingdom
`
`c(cid:1) 2008 Cambridge University Press
`
`141
`
`Transition from squeezing to dripping in a
`microfluidic T-shaped junction
`
`M. D E M E N E C H1, P. G A R S T E C K I2, F. J O U S S E3
`A N D H. A. S T O N E4
`1Max–Planck Institute for the Physics of Complex Systems, N¨othnitzer Str. 38,
`01187, Dresden, Germany
`2Institute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52,
`01-224, Warsaw, Poland
`3Unilever Corporate Research, Colworth House, Sharnbrook, Bedfordshire, MK44 1LQ, UK
`4School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA
`
`(Received 12 May 2006 and in revised form 5 September 2007)
`
`We describe the results of a numerical investigation of the dynamics of breakup of
`streams of immiscible fluids in the confined geometry of a microfluidic T-junction.
`We identify three distinct regimes of formation of droplets: squeezing, dripping and
`jetting, providing a unifying picture of emulsification processes typical for microfluidic
`systems. The squeezing mechanism of breakup is particular to microfluidic systems,
`since the physical confinement of the fluids has pronounced effects on the interfacial
`dynamics. In this regime, the breakup process is driven chiefly by the buildup of
`pressure upstream of an emerging droplet and both the dynamics of breakup and
`the scaling of the sizes of droplets are influenced only very weakly by the value
`of the capillary number. The dripping regime, while apparently homologous to the
`unbounded case, is also significantly influenced by the constrained geometry; these
`effects modify the scaling law for the size of the droplets derived from the balance
`of interfacial and viscous stresses. Finally, the jetting regime sets in only at very high
`flow rates, or with low interfacial tension, i.e. higher values of the capillary number,
`similar to the unbounded case.
`
`1. Introduction
`Because of the small size of the microchannels (widths of the order of 10 to 100 µm)
`−1), flows in microfluidic systems are generally dominated
`and typical flow rates (1 µl s
`by viscous effects. Two characteristics of microflows, i.e. the laminar flow and the
`typically large values of the P´eclet number (measuring the ratio of the convective
`to diffusive transport), allow for an extensive control both in space and time over
`the transport of chemical substances (Kenis, Ismagilov & Whitesides 1999; Stone,
`Strook & Ajdari 2004; Squires & Quake 2005). This control, in conjunction with the
`ease of fabrication of the microfluidic devices (Duffy et al. 1998; McDonald et al.
`2000), is one of the main features driving the interest in microfluidic systems for
`engineering and research applications.
`The laminar flow of a single Newtonian fluid at low Reynolds numbers, which is
`described by linear equations and boundary conditions, is to be contrasted with a wide
`class of nonlinear phenomena which have been uncovered with the first experiments
`on two-phase flows in microfluidic systems (Thorsen et al. 2001; Gan ´an-Calvo &
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`M. De Menech, P. Garstecki, F. Jousse and H. A. Stone
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`Application
`
`Reference
`
`Chemical processing
`Micromixing inside drops
`Drops and slugs as mixing elements
`High-throughput screening
`Kinetic analyses
`Organic chemistry
`Bioanalysis
`Diagnostic assays
`Handling and/or analysis of living cells
`Molecular evolution
`Material science
`Anisotropic particles
`
`Song et al. (2003)
`Gunther et al. (2004); Garstecki et al. (2005a)
`Zheng et al. (2003); Zheng & Ismagilov (2005)
`Song & Ismagilov (2003)
`Cygan et al. (2005)
`
`Sia et al. (2004)
`He et al. (2005)
`Cornish (2006)
`
`Jeong et al. (2004, 2005); Nisisako et al. (2004);
`Dendukuri et al. (2005); Xu et al. (2005)
`Microcapsules
`Takeuchi et al. (2005)
`Colloidal shells
`Subramaniam et al. (2005)
`Gunther & Jensen (2006); Song et al. (2006)
`Review articles
`Table 1. Examples of applications of multiphase flow in microfluidic devices.
`
`Gordillo 2001; Anna, Bontoux & Stone 2003; Dreyfus, Tabeling & Willaime 2003).
`The existence of an interface and the influence of interfacial tension introduce strong
`nonlinearities in the flow, which are responsible for the appearance of a range of
`novel effects, some of which are particular to microfluidic systems when the interfacial
`dynamics is strongly influenced by the confinement of the fluids by the walls of the
`channel (Garstecki et al. 2005b,c,d , 2006; Guillot & Colin 2005).
`The interest in detailed understanding of the emulsification processes in microfluidic
`systems is additionally motivated by the wide range of work on applications of
`microfluidic multiphase flows (see table 1). For example, microfluidics offers new
`routes and better control to chemistry, including chemistry inside small containers
`(e.g. Li, Zheng & Harris 2000; Song, Chen & Ismagilov 2006), control of dispersion
`(e.g. Pedersen & Horvath 1981), etc., approaches that have a long history in chemical
`engineering. Most of the applications require precise control over the process of
`formation of droplets or bubbles (e.g. Basaran 2002), and characterization or,
`preferably, understanding of the scaling laws that describe the volume of the bubbles
`or droplets formed in the devices as a function of the material (e.g. viscosities,
`interfacial tension) and flow parameters (e.g. pressures or rates-of-flow applied to the
`system). Understanding the flows is closely related to more classical studies of drop
`breakup and emulsification in sheared unbounded fluid systems (Rallison 1984; Stone
`1994). Nevertheless, as we shall discuss, the confinement that naturally accompanies
`flow in small devices has significant qualitative and quantitative effects on the drop
`dynamics and breakup (Garstecki et al. 2005d , 2006; Guillot & Colin 2005).
`Several methods of formation of both bubbles and droplets have already been
`described. Table 2 provides references to several recent experimental reports.
`Numerical simulations of breakup in microfluidic geometries have also been
`conducted, see for example studies of axisymmetric geometries by Jensen, Stone &
`Bruus (2006), Suryo & Basaran (2006) and Zhou, Yue & Feng (2006). For a thorough
`up-to-date review of drop formation in microfluidic devices see Christopher & Anna
`(2007). Here we report fully three-dimensional numerical simulations to understand
`and characterize drop formation in a microfluidic T-junction geometry (figure 1)
`which was first introduced for the controlled formation of water-in-oil dispersions by
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`Planar geometries
`Review article
`Planar geometries
`Flow-focusing
`
`Crossflow
`
`Diverging flow T-junction
`Axisymmetric geometries
`
`Transition from squeezing to dripping in a microfluidic T-shaped junction
`
`143
`
`Subject
`
`Reference
`
`Christopher & Anna (2007)
`
`Anna et al. (2003); Dreyfus et al. (2003); Cubaud & Ho (2004);
`Garstecki et al. (2004); Xu & Nakajima (2004);
`Ward et al. (2005)
`Blackmore et al. (2001); Thorsen et al. (2001); Song & Ismagilov
`(2003); Okushima et al. (2004); Zheng et al. (2003); Gerdts et al.
`(2004); Tice et al. (2004); Dendukuri et al. (2005);
`Guillot & Colin (2005); Garstecki et al. (2006)
`Link et al. (2004); Engl et al. (2005)
`Gan ´an-Calvo & Gordillo (2001); Jeong et al. (2004, 2005);
`Takeuchi et al. (2005); Utada et al. (2005)
`Sugiura et al. (2001, 2005)
`Nonplanar geometries
`Table 2. Examples of droplet and bubble formation in microfluidic devices. Given the rapid
`growth in the number of such two-phase-flow studies, the above references are representative
`of the kinds of studies that have been performed.
`
`Vd
`
`µ
`d
`
`Vc
`
`µ
`c
`
`L
`
`Figure 1. Diagram of a T-junction with crossflow. The channels have square cross-section
`with side L. vc and vd are the mean flow velocities of the continuous and dispersed phases,
`while µc and µd are the corresponding shear viscosities.
`
`Thorsen et al. (2001). The authors made the reasonable suggestion that the dynamics
`of droplet formation is dominated by the balance of tangential shear stresses and
`interfacial tension (i.e. the capillary number) as expected in unbounded shear flows, via
`an analogy to breakup processes in shear and extensional flows (Taylor 1934; Rallison
`1984; Stone 1994). A detailed experimental study of the T-junction configuration
`(Garstecki et al. 2006) identified a different (squeezing) mechanism that is directly
`connected to the confined geometry in which the drop is formed (see also Guillot &
`Colin 2005). It was proposed that when the capillary number is sufficiently small,
`the dominant contribution to the dynamics of breakup arises from the buildup of
`pressure upstream of the emerging droplet (Garstecki et al. 2005d , 2006). This model
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`M. De Menech, P. Garstecki, F. Jousse and H. A. Stone
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`results in a scaling law for the size of the droplets that is independent of the value
`of the capillary number and includes only the ratio of the rates of flow of the two
`immiscible fluids.
`The numerical results that we present here provide a unifying picture of the
`dynamics of formation of droplets in microfluidic T-junction geometries that includes
`both of the aforementioned squeezing (Garstecki et al. 2005d , 2006) and shear-driven
`(Thorsen et al. 2001) types of breakup; for another three-dimensional simulation
`of drop formation at a T-junction, see van der Graaf et al. (2006). We confirm
`the existence of the ‘rate-of-flow controlled’ or ‘squeezing’ breakup mechanism at
`low values of the capillary number Ca. We provide the details of the dynamics,
`which includes fluctuations of pressure upstream of the immiscible tip postulated
`by Garstecki et al. (2006). We identify a critical value of the capillary number
`at which the system transits into a shear-dominated or dripping mechanism of
`droplet formation. Also, we indicate the differences in drop formation in confined and
`unbounded systems. Finally, similarly to breakup into an unbounded fluid, we observe
`a transition from dripping to jetting at larger values of the capillary number. We
`note that although jetting refers to the formation of long threads prior to formation
`of a drop, which is usually associated with inertial effects of the internal phase, here
`we have a case where shear in the external phase drives a jetting transition at low to
`modest Reynolds numbers (e.g. Utada et al. 2007).
`In the following section, we define the geometry and the parameters of the system
`
`that we study and introduce the important dimensionless quantities. In § 3, we describe
`the numerical methods employed in our work. In § 4, we detail the results, both for the
`low- and high-capillary-number regimes, and we summarize our observations in § 5.
`
`2. Description of the system
`For a planar geometry, the characteristic dimensions of the T-shaped junction are
`the height h, and the widths of the main and side channels. We will consider the
`simplest case in which the widths of both ducts equal L, and the channels have a
`square cross-section h = L (figure 1). The dispersed phase is injected into the main
`channel from the side inlet. For simplicity, we set the densities of both phases equal;
`we expect that this choice has negligible influence on the results, since in most
`microfluidic configurations buoyancy-driven speeds are much smaller than the actual
`flow speeds. Besides the width L, which is constant, the problem is fully described
`by six parameters characterizing the flow and material properties of the fluids. These
`parameters are the mean speeds of the continuous and dispersed phases, vc and vd
`respectively, the viscosities of the two fluids µc and µd , the interfacial tension γ , and
`the density ρ. We will assume perfect wetting for the continuous phase, while the
`dispersed fluid does not wet the walls. The rescaled volume V = Vd /L3 of the droplets
`formed in the device is the eighth physical quantity, and following the Buckingham-Π
`theorem, it can be described as a function of four dimensionless parameters. We
`chose the following groups: the capillary number calculated for the continuous phase,
`Ca = µcvc/γ , the Reynolds number Re = ρvcL/µc, the viscosity ratio λ = µd /µc, and
`the flow rate ratio Q = vd /vc = Qd /Qc, where Qd = vd L2 and Qc = vcL2 are the flow
`rates at the two inlets. The pressure will be rescaled by the typical viscous shear stress
`µcvc/L, while the time unit is L/vc.
`For the flow regimes under consideration, the Reynolds number is small (Re < 1),
`and does not influence the droplet size, which leaves us with the three governing
`parameters: Ca, λ and Q. Our main focus is the discussion of the transition
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`from the interfacial-tension dominated to the shear-dominated regime, which is best
`characterized by considering the effects of the capillary number on the droplet size.
`Within this framework, we will also consider the influence of λ and Q.
`
`3. The numerical model
`We use a phase-field model (De Menech 2006) to simulate numerically the flow of
`the two immiscible fluids at the T-junction. In common experimental configurations,
`low capillary numbers characterize the microfluidic flows, hence surface tension
`stresses are large in comparison to viscous stresses. These flows can be effectively
`tackled numerically with the diffuse interface method that we employ here, which
`models the phase boundary separating the two fluids as a diffuse region. The
`equilibrium properties of the mixture, including phase behaviour, wetting properties
`and the concentration profile in the interface region, are derived from a generalized
`free-energy functional, which also determines the diffusive and capillary forces in the
`transport equations. The transport equations are solved on a three-dimensional grid,
`and the method has been tested successfully in the case of droplet breakup in a
`microfluidic T-junction with diverging flow (De Menech 2006).
`
`3.1. Modelling multiphase flow
`The key issue in the numerical modelling of droplet formation and breakup in
`microfluidic devices is the requirement of a consistent and robust description of
`the effects related to the large interfacial tensions, which dominate over inertial
`and viscous stresses. As for the broader context of multiphase flow simulation,
`we can choose from the two main approaches: interface tracking and interface
`capturing methods (for an introductory review focused on microfluidics, see for
`example Cristini & Tan 2004).
`In interface tracking methods, the displacement of the boundary surface is followed
`explicitly by an adapting mesh, which represents the discrete approximation of the
`classical sharp interface limit. The main advantage of this approach is the accurate
`description of the interface, which comes at the expense of a greater complexity of the
`algorithms that are required to manage the motion and addition of mesh nodes. The
`handling of the singularities associated with surface merging and breakup represents
`a difficult technical problem, since the remeshing procedure has to cope with the
`changes in topology (see, e.g. Cristini, Balwzdziewicz & Loewenberg 1998; Wilkes,
`Phillips & Basaran 1999). In this framework, a few examples are available for the
`description of the dynamics of droplets pinned at or sliding on flat surfaces, both
`in two- and three-dimensional studies (Feng & Basaran 1994; Li & Pozrikidis 1996;
`Schleizer & Bonnecaze 1999).
`In interface capturing methods, the computational mesh remains fixed and the
`boundary discontinuities are smeared out over the finite width ξ of a diffuse interface,
`whose location is reconstructed from the gradients of an auxiliary scalar field. Surface
`stresses are included in the Navier–Stokes equation as volume forces that depend on
`the spatial derivatives of the auxiliary scalar field, such that the momentum transport
`equation remains consistent with the single-phase equation in the bulk regions. The
`lower bound to the manageable interface thickness ξ is determined by the mesh
`spacing, and the discontinuity of the sharp interface limit behaviour is recovered
`asymptotically as ξ becomes negligible with respect to the characteristic length of the
`flow L. Broadly speaking, we can distinguish two mainstream approaches in the class
`of interface capturing methods, based on the physical interpretation of the auxiliary
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`M. De Menech, P. Garstecki, F. Jousse and H. A. Stone
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`scalar field. In phase-field models (also called diffuse interface models; see Anderson,
`McFadden & Wheeler 1998), the behaviour of the scalar field in the transition region
`strictly follows from a variational principle applied to a generalized free-energy
`functional, which determines the equilibrium and non-equilibrium properties of the
`system. On the other hand, in the case of popular techniques such as the volume of
`fluid (VOF) method (Scardovelli & Zaleski 1999), or the level-set method (Osher &
`Fedkiw 2001), the scalar field (also known as a colour function in VOF methods) is
`merely an indicator of the different phases; its profile across the transition region has
`no meaning, other than serving to define the position of the phase boundary.
`Phase-field methods were developed to investigate phase transition phenomena, such
`as nucleation, evaporation and coarsening (Cahn 1965), where the diffuse interface
`provides a mean field description of the concentration or density profile. The phase-
`field version of the Navier–Stokes equations converges to the classical sharp interface
`behaviour as the interface thickness is reduced to zero along with the diffusivity
`(Jacqmin 1999). In the same manner, contact line dynamics can be related to that of
`immiscible fluids (Jacqmin 2000).
`In this paper, we will be using the phase-field model in which the capillary stresses
`at the interfaces are represented as volume forces, and are distributed over the
`characteristic thickness ξ of the diffuse phase boundary. This important feature of
`the model allows the treatment of multiphase flows with relatively coarse grids,
`since typically only a few mesh points are required in order to resolve the smooth
`variation of the order parameter across the interface. The interface thickness should
`be compared to a characteristic length of the system, which could be the radius of
`a droplet or the size of the domain (here L) in the case of confined flow, leading to
`the definition of the Cahn number C = ξ /L. The droplet dynamics of two immiscible
`
`fluids, described classically in the sharp interface limit ξ → 0, is exactly recovered as
`
`C and the diffusivity goes to zero (Jacqmin 1999).
`There are two possible ways to interpret the results of the phase-field model.
`The first one is to relateξ
`to the physical width of the interface for real fluids,
`which is of the order of 1 nm or larger. In this approach, our simulations should,
`within the continuum model, reflect the actual flow of fluids at the nano-scale,
`with typical channel widths of the order of 10 nm. Another interpretation is to
`consider the diffuse interface model as an approximation of the flow of immiscible
`fluids at the microscale, which is justified when the interface thickness ξ is small
`compared with the characteristic length of the flow L. Despite ξ not being the interface
`thickness separating real bulk fluids (e.g. since we typically choose C = 1/20 for
`
`numerical convenience, then ξ ∼ 5 µm when simulating drops in 100 µm channels), the
`
`experimentally observed droplet dynamics in microfluidic devices can be reproduced
`effectively (De Menech 2006).
`De Menech (2006) showed that the numerical model, despite the limitations related
`to the coarse description of the interface description, is capable of capturing the
`quantitative details of the dynamics of droplet breakup in a T-junction, in agreement
`with the experimental results for water droplets in a continuous oil phase. The results
`that we describe in this report, which are expressed in terms of the non-dimensional
`groups, also match closely the experimental observation of flow of droplets and
`bubble formation in microfluidic T-junctions (Garstecki et al. 2006).
`
`3.2. The phase-field model
`Phase-field models and a range of related numerical methods based on the generalized
`free-energy functional approach (lattice Boltzmann) have already been applied to
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`the modelling of droplet dynamics in unbounded flow or in the presence of solid
`boundaries, including droplet deformation and breakup (Wagner & Yeomans 1997;
`Badalassi, Ceniceros & Banerjee 2003; Wagner, Wilson & Cates 2003; Kim 2005),
`droplet formation (Kuksenok et al. 2003), spreading on patterned surfaces (Dupuis &
`Yeomans 2004), and droplet formation in axially symmetric flow-focusing devices
`(Zhou et al. 2006). In this section, we summarize the main features of the phase-field
`model we used. A more complete description is given in De Menech (2006).
`We consider a mixture of two fluids, A and B, with the Cahn–Hilliard–van der
`Waals form for the free energy
`
`(cid:1)
`
`(cid:2)
`
`(cid:3)
`|∇ϕ|2 + nW (ϕ) + f (n)
`of component B, κ determines the interface thickness, ξ ∝ √
`tension, γ ∝ κ/ξ (De Menech 2006). In (3.1), f is the sum of the free-energy densities
`
`F [n, ϕ] =
`
`dr
`
`κn
`2
`
`,
`
`(3.1)
`
`where n = nA + nB is the total particle number density, ϕ = nB /n is the molar fraction
`κ, and the interfacial
`
`of the pure components, which for simplicity is assumed to depend only on the total
`density. Also, W is the free energy of mixing, having the characteristic double-well
`shape below the critical temperature, which therefore determines the coexistence of
`the A-rich and B-rich separate phases, with corresponding concentrations ϕA and ϕB.
`We will consider the incompressible limit, n = const, such that the equilibrium
`concentration satisfies the Euler–Lagrange equation
`(cid:7)
`
`µch (r) =− κ∇2ϕ(r) +W
`where µch (r) =δF /δϕ (r) is the chemical potential difference, and µcoex is the Lagrange
`multiplier which fixes the total molar fraction. In the framework of the Cahn–Hilliard
`theory of diffuse interfaces, the interaction of the fluid components with a wall is
`introduced by adding to the functional (3.1) a surface energy term (Cahn 1977), which
`fixes the relative affinity of the two components, A or B, for the solid boundaries. The
`surface energy therefore determines the wetting properties of the two phases, which
`can be expressed in terms of the equilibrium wetting angle. We choose the surface free
`energy such that the A component wets the walls, while the B component is repelled.
`In non-equilibrium conditions, local imbalances of the chemical potential µch (r)
`will generate diffusion currents which tend to restore the configuration satisfying (3.2).
`The advection–diffusion equation for the phase field is therefore
`
`(ϕ(r)) = µcoex ,
`
`(3.2)
`
`=−∇·(ϕ v) +Λ∇ 2[−κ∇2ϕ + W
`
`(cid:7)
`
`(ϕ)]
`
`∂ϕ
`
`∂t
`
`(3.3)
`
`where Λ is the mobility coefficient. Since the flow is considered incompressible,
`
`∇·v = 0. The expression for the pressure tensor, which includes the capillary forces
`
`generated by the presence of the interface, follows directly from the free-energy
`functional (3.1), based on the Gibbs–Duhem equation or on Noethers theorem
`(Anderson et al. 1998; De Menech 2006). The fluid momentum equation, besides the
`nonlinear advection term, contains both reactive and dissipative forces, depending,
`respectively, on the pressure tensor and the shear viscosity µ, which depends on ϕ.
`The Navier–Stokes equation is
`
`ρ
`
`∂v
`∂t
`
`=−ρ∇· (vv) − ∇p +∇· (µ[∇v + (∇v)T ]) +
`
`∇· (∇ϕ∇ϕ),
`
`γ ξ
`
`Ω
`
`(3.4)
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`where the last term comes from the non-isotropic part of the pressure tensor, as
`first introduced by Korteweg, and Ω is a dimensionless number that depends on the
`choice of the mixing potential W (De Menech 2006).
`Finally, we remark that in the case of contact line dynamics, the results of a phase-
`field model match those of immiscible fluids in the far-field region, i.e. at a distance
`from the contact line which is larger than the characteristic interface thickness ξ
`(Jacqmin 2000). Similarly to the case of droplet breakup and interface merging, the
`dynamics of diffuse interface models in the near-field region have the significant
`advantage of removing the singularities that cause the breakdown of the sharp
`interface limit description of these phenomena. In our three-dimensional simulations,
`it is not strictly possible to identify a far-field region, since the pipe diameter is
`comparable with the interface width for the values of the Cahn number we have
`used. In other words, the adoption of the phase-field model for the description of two
`immiscible fluids is not justified a priori. On the other hand, as for the case of droplet
`breakup in a T-junction (De Menech 2006), we will show that the agreement with
`the experiments corroborates the validity of our approach, which in essence aims at
`classifying the variety of patterns observed in multiphase flows in microfluidic devices.
`
`3.3. The numerical method
`The transport equations (3.3) and (3.4) are discretized on a uniform three-dimensional
`Cartesian grid with staggered velocities; the molar fraction and pressure fields are
`collocated at the centres of the cubic cells, while the velocity components are placed
`on the faces, and the boundaries of the simulated domain always coincide with a
`face of a grid cell, be it a wall, an inlet or a pressure outlet. The time evolution is
`implemented with a fully implicit Euler scheme, which is first order in time (see De
`Menech 2006). The boundary condition (BC) for the molar fraction ϕ is of Neumann
`
`type at the outlet, n·∇ϕ = 0, where n is the vector normal to the outlet. A Dirichlet BC
`
`is set at the inlets, ϕ = ϕA at the main channel inlet and ϕ = ϕB at the side inlet. The
`BCs for the fluid velocity v are of Dirichlet and Neumann type at the inlets and outlet,
`respectively, with the inlet velocity profile set equal to the fully developed Poiseuille
`flow in a square pipe. The BCs for the pressure field p are of Neumann and Dirichlet
`type at the inlet and outlet, respectively. Finally, on all boundaries it is assumed that
`the chemical potential gradient is zero, so that locally there are no contributions to a
`flux of mole fraction. With respect to the actual discretization the typical grid spacing
` x/L was equal to C , which implies 20 grid points across the tube diameter. For the
`smallest domain constructed, with a tube length of 6L and a side inlet of length L,
`we have therefore 56 000 grid points in the three-dimensional domain.
`Given the stress tensor T in the Navier–Stokes equation (3.4), which is the sum
`
`of the viscous stress tensor µ[∇v + (∇v)T ] and the negative of the pressure tensor,
`P = pI + (γ ξ /Ω)∇ϕ∇ϕ, then at the outlet we haveT zz = Pzz = p = p0 (where z is normal
`
`to the boundary); at the outlet, the pressure is constant and equal to the isotropic
`part of the pressure tensor. This set of boundary conditions at the outlet is equivalent
`to the assumption that the flow is fully developed at the end of the tube, and that
`there is no net flux of the two components A and B (note that at the inlet there is
`still a net influx because of the Dirichlet BC).
`
`The constant pressure condition and the condition n·∇ϕ = 0 at the outlet deforms
`
`the shape of droplets which touch the outlet, since the Laplace pressure difference
`here is forced to be zero and the curved interface is rapidly stretched and stabilized to
`become perpendicular to the boundary. Very long jets which touch the outlet are, in
`fact, stabilized, with their shape remaining stationary. These effects cannot be avoided
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`in a finite simulation domain. On the other hand, such effects play a role only when
`the droplet or the thread is very close to the boundary (a distance comparable to the
`interface thickness ξ ), and the pressure fluctuations associated with these events have
`very high frequency, and do not affect the slow dynamics of droplet formation inside
`the junction. These two facts still enable us to trust reliably the breakup dynamics
`observed far from the outlet, as for the squeezing and dripping regimes. We also
`performed a few tests to check that the size of droplets formed in the channel is not
`affected by changing the length of the pipe downstream from the T-junction.
`
`4. Results
`Before we present our results, we sketch the characteristics of the two main
`dynamical models for breakup in a microfluidic T-junction: shear-driven breakup
`(Thorsen et al. 2001), which is derived from the balance of shear and interfacial
`stresses, and the rate-of-flow-controlled breakup (Garstecki et al. 2005d , 2006), which
`was proposed for systems operating at low values of the capillary number and is
`based on the evolution of pressure in the continuous phase upstream of the tip of the
`dispersed phase.
`In shear-driven breakup, the volume of the droplet can be estimated from a balance
`of the viscous drag that the continuous fluid exerts on the emerging droplet and the
`interfacial force that opposes the elongation of the neck, which connects the reservoir
`of the dispersed fluid with the droplet (e.g. Umbanhowar, Prasad & Weitz 2000;
`Thorsen et al. 2001). This model leads to a relation in which the diameter of the
`droplet is inversely proportional to the capillary number calculated for the flow of the
`continuous liquid. Because the drag force exerted on the droplet depends only very
`weakly on the viscosity of the droplet, within the shear-driven regime, the viscosity
`of the dispersed phase does not influence the size of droplets appreciably (e.g. see
`discussion of flow past a drop in Batchelor 1967). This effect has been confirmed
`experimentally by Cramer, Fischer & Windhab (2004).
`Qualitatively, the rate-of-flow-controlled breakup (Garstecki et al. 2005d , 2006) can
`be described as follows (see figure 1): the tip of the stream of the immiscible fluid
`enters the main channel, and because interfacial stresses dominate the shear stresses,
`the tip blocks almost the entirety of the cross-section of the main channel; the shear
`stresses exerted on the tip by the continuous fluid are not strong enough to deform
`the tip significantly away from an area minimizing shape. As a result, the continuous
`phase fluid is confined to thin films between the tip of the other immiscible fluid and
`the walls of the device. Flow in these thin films is subject to an increased viscous
`resistance, which leads to a build-up of pressure in the continuous phase upstream of
`the tip (Stone 2005). This pressure is larger than the pressure in the immiscible tip,
`and the continuous fluid displaces the interface or squeezes the neck of the inner fluid,
`which leads to breakup and detachment of a droplet. Notably, within this regime the
`breakup is not driven by interfacial stresses, at least not until the very last stage; the
`speed at which the neck of the immiscible thread collapses is proportional to the rate
`of flow of the continuous fluid, and does not depend significantly on the value of
`the interfacial tension, or on the values of the viscosities of either of the two fluids
`(Garstecki et al. 2005d ). Within this regime, we can expect that the volume of the
`droplet will be a function of the ratio of the rates of flow of the two fluids, and will
`not depend strongly on the value of the capillary number (Garstecki et al. 2006).
`The qualitatively different predictions of each of the two models of breakup make
`it possible to distinguish between them by inspecting how the volume of the droplets
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`Fluidigm Exhibit 2022
`Page 9 of 21
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`M. De Menech, P. Garstecki, F. Jousse and H. A. Stone
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`depends on the material (viscosities and interfacial tension) and flow (rates of flow)
`parameters. In this case, an advantage of numerical simulations over the experiments
`is the control over the different parameters. In the experiments, for example, it is
`difficult to change the interfacial tension between the two fluids over a wide range of
`values without introducing dynamic interfacial tension effects (at low concentrations
`of surfactant), or changing the wetting properties of the two fluids. In contrast,
`in simulations, it is straightforward to change the value of any parameter without
`affecting any others. Here we present the results of three-dimensional simulations for
`a range of the values of Ca that is broad enough to observe all three mechanisms of
`breakup in the confined geometry of a T-junction: squeezing, dripping and jetting.
`4.1. Overview of the dynamics of the system
`As the dispersed phase ent