Role of local geometry on droplet formation in axisymmetric microfluidics

Role of local geometry on droplet formation in axisymmetric microfluidics

Chemical Engineering Science 163 (2017) 56–67 Contents lists available at ScienceDirect Chemical Engineering Science journal homepage: www.elsevier...

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Chemical Engineering Science 163 (2017) 56–67

Contents lists available at ScienceDirect

Chemical Engineering Science journal homepage: www.elsevier.com/locate/ces

Role of local geometry on droplet formation in axisymmetric microfluidics Liangyu Wu a,b, Xiangdong Liu a, Yuanjin Zhao c, Yongping Chen a,d,⇑ a

School of Hydraulic, Energy and Power Engineering, Yangzhou University, Yangzhou, Jiangsu 225127, PR China Key Laboratory of Energy Thermal Conversion and Control of Ministry of Education, School of Energy and Environment, Southeast University, Nanjing, Jiangsu 210096, PR China c State Key Laboratory of Bioelectronics, Southeast University, Nanjing, Jiangsu 210096, PR China d Jiangsu Key Laboratory of Micro and Nano Heat Fluid Flow Technology and Energy Application, School of Environmental Science and Engineering, Suzhou University of Science and Technology, Suzhou, Jiangsu 215009, PR China b

h i g h l i g h t s  A model of droplet formation in microfluidics is developed, analyzed and verified.  Dripping, dripping-jetting transition and jetting are observed in microfluidics.  Droplet formation is facilitated by focusing orifice in microfluidics.  Orifice radius is significant in droplet formation rather than orifice length.

a r t i c l e

i n f o

Article history: Received 24 August 2016 Received in revised form 31 December 2016 Accepted 10 January 2017 Available online 12 January 2017 Keywords: Droplet Microfluidics Dripping Jetting

a b s t r a c t An unsteady model of droplet formation in co-flow and flow-focusing microfluidics is developed and numerically analyzed to investigate the dynamic behaviors of droplet formation in axisymmetric microfluidics with a focus on the role of local geometry. The effects of capillary number and local geometry on the droplet formation regimes, droplet sizes and monodispersity as well as droplet generation frequency are examined and analyzed. Once identified, a drop formation regime diagram is provided to quantitatively describe the respective regime of dripping, dripping-jetting transition, and jetting in axisymmetric microfluidics, depending on the Capillary number and orifice radius. The results indicate that, the existence of focusing orifice induces a strong hydrodynamic focusing effect, causing the droplet formation behaviors in flow-focusing microfluidics depart from the co-flow one. The dripping-jetting transition regime occurs at a smaller Capillary number in flow-focusing microfluidics, and the droplets produced by flow-focusing microfluidics are smaller than those in co-flow system with wider size distribution and higher frequency. Interestingly, the droplet formation in flow-focusing microfluidics is significantly affected by the orifice radius while it is insensitive to the orifice length. In addition, when the orifice radius is sufficiently small or the Capillary number is larger than 0.3, only the jetting regime is observed. As the orifice radius increases, the regions of Capillary number for both the dripping-jetting transition regime and dripping regime turn to be larger. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Emulsions are of significance in a variety of applications such as chemistry, food industry, pharmaceutics and environmental science (McClements and Li, 2010; Zhao et al., 2015; Chen et al., 2011; Wang et al., 2011; Sander et al., 2012; Wan, 2012). Manipulation of monodispersed emulsions has become a crucial ⇑ Corresponding author at: School of Hydraulic, Energy and Power Engineering, Yangzhou University, Yangzhou, Jiangsu 225127, PR China. E-mail address: [email protected] (Y. Chen). http://dx.doi.org/10.1016/j.ces.2017.01.022 0009-2509/Ó 2017 Elsevier Ltd. All rights reserved.

imperative motivated by the upsurge in fast and inexpensive analysis of biological and chemical samples (Gañán-Calvo and Gordillo, 2001; Herrada et al., 2010; Gañán-Calvo, 1998; Anna et al., 2003; Song et al., 2006; Kim et al., 2008). Among the possible approaches, the axisymmetric microfluidics system (Gañán-Calvo and Gordillo, 2001; Herrada et al., 2010) provides promising routes for producing highly monodispersed emulsions. To optimally design the microfluidic system and precisely manipulate the droplet production, it is of considerable importance to fully understand the dynamic behaviors of droplet formation in axisymmetric microfluidics.

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The co-flow microfluidic device (Utada et al., 2007; Umbanhowar et al., 2000; Suryo and Basaran, 2006; CastroHernández et al., 2011; Shum et al., 2012) and flow-focusing microfluidic device (Gañán-Calvo and Gordillo, 2001; Utada et al., 2005; Takeuchi et al., 2005; Martin-Banderas et al., 2005) are two typical axisymmetric microfluidics, in which the dispersed phase flow and continuous phase flow are coaxial. The major advantage of axisymmetric devices is that there is no wetting problem (Liu and Yobas, 2015; Abate et al., 2010; Chu et al., 2007) which may damage the droplets (Takeuchi et al., 2005). The configurations and geometries of the two aforementioned devices are similar, but the orifice in the flow-focusing device brings in the hydrodynamic focusing effect (Martin-Banderas et al., 2005). This distinctive feature introduces special fluidic dynamic behaviors of droplet formation in axisymmetric microfluidics. To provide guidance for the optimization design of microfluidic system, several experimental attempts have been conducted to investigate the droplet formation with a focus on the monodisperse emulsion generation. Umbanhowar et al. (2000) developed a simple technique based on capillary tube that produces highly monosispersed emulsions with precise control over droplet size. Utada et al. (2007) experimentally investigated the transitions from dripping to jetting in co-flow liquid streams and proposed the criterion of transition based on the competition between the interfacial tension, viscous force and the inertia force. Gañán-Calvo and Gordillo (2001) proposed efficient mass production of micron size gas bubbles resorted on self-excited breakup phenomenon in a flowfocusing system. The physics was described with the bubble diameter expressed as a function of the fluid properties, geometry and flow parameters. Takeuchi et al. (2005) produced highly monodispersed polymer-coated droplets with controlled size distribution in a flow-focusing microfluidics in poly (dimethylsiloxane) (PDMS). Wu and Chen (2014) investigated the hydrodynamics of droplet formation visually based on a co-flow system of steel needle inserted into a rectangular PMMA channel. In comparison with experimental studies, there is comparatively less numerical work to explore the droplet dynamic behaviors in axisymmetric microfluidics. In the simulation, particular attention is paid to uncover the underlying physics of droplet growth, deformation and breakup, especially the velocity and pressure distributions at the interface during droplet formation. Zhang and Stone (1997) numerically investigated the dynamic behaviors of droplet formation in a co-flow capillary tube using the boundary integral method. The dependence of the volume of the primary droplet as a function of the capillary number, Bond number, and viscosity ratio are determined. Zhou et al. (2006) simulated the jet breakup and droplet formation in a flow-focusing system by the use of finite element with adaptive meshing. The flow regimes as well as influence factors were also studied numerically. Chen et al. (2013) numerically investigated the droplet formation in an axisymmetric co-flow microchannel via the volume of fluid (VOF) method, in which the droplet breakup difference between dripping regime and jetting regime was elucidated and the dependence of drop formation on Capillary number and Weber number was presented. While several experimental and numerical efforts have been implemented to investigate the droplet formation either in coflow device (Utada et al., 2007; Umbanhowar et al., 2000; Suryo and Basaran, 2006; Castro-Hernández et al., 2011; Shum et al., 2012) or in flow-focusing device, a comprehensive comparative study on these two axisymmetric microfluidics is less available. Up to know, it remains unclear how the local geometry affects the droplet formation process in axisymmetric microfluidics. In addition, the transient interfacial interaction and the flow field evolution during droplet formation in axisymmetric microfluidics are less understood. Particularly, the optimization design of

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microfluidics and the detailed understanding of droplet dynamic behaviors require a unified view of flow regimes in microfluidics. To achieve these aims, we herein carried out a comprehensive numerical study on droplet formation in axisymmetric co-flow and flow-focusing microfluidics with a focus on the role of local geometry. 2. Mathematic model The co-flow and flow-focusing microfluidics are two typical axisymmetric microfluidics. The co-flow microfluidics comprises of two coaxially aligned capillary tubes, in which the inner capillary tube is located at the upstream of the outer capillary tube. Both the inner and outer capillary tube is cylindrical with corresponding radius Rin and Rout. Once a co-flow microchannel is configured and coaxially aligned with a focusing orifice at the downstream of the outlet for an inner capillary tube, this is another type microfluidic device called as flow-focusing microfluidics. The focusing orifice is the essential feature of flow-focusing microfluidics and it can be characterized by non-dimensional orifice radius (Rori = Rori/Rout, Rori is the orifice radius) and non-dimensional orifice length (Lori = Lori/Rout, Lori is the orifice length). The co-flow microfluidics can be regarded as a special case of the flowfocusing microfluidics, in which the orifice radius of flowfocusing microfluidics (i.e. Rori) is equal to the radius of the outer capillary tube (i.e. Rout). In order to save computational cost, both the co-flow and flowfocusing microfluidics are simplified as a two-dimensional (2D) axisymmetric structure, as shown in Fig. 1. Subsequently, an unsteady model of the Newtonian, incompressible and immiscible fluids flow in these two simplified microfluidics are developed. Driven by the external force, the dispersed phase fluid (density qd, viscosity ld) injects into a continuous phase fluid (density qc, viscosity lc) at the outlet of the inner capillary tube. In the simulation, the constant fluid velocity for both the continuous phase fluid and the dispersed phase fluid is imposed at the channel inlet as inlet boundary condition; the no-slip velocity boundary condition is applied at the channel wall; and the pressure boundary condition is applied at the channel outlet. In this study, the oil is assumed as the continuous phase while the water is assumed as the dispersed phase. The radius of inner and outer cylindrical capillary tube is assumed to be Rin = 400 lm and Rout = 800 lm, and the length of these two capillary tube is assumed to be L1 = 400 lm and L = 5 mm. 2.1. Governing equations Apparently, the droplet formation in these two typical axisymmetric microfluidics involves complicated interface evolutions (e.g. movement, deformation and breakup). Therefore, the most important issue in developing the numerical models is reasonably representing the interface behaviors. Generally, the categories for numerically representing interface evolutions in the available model can be classified into two kinds, the interface tracking method and the interface capturing method (Cristini and Tan, 2004; Worner, 2012). Among two typical categories for numerically representing the interface behaviors, i.e. interface tracking method (e.g. boundary integral method (Zhang and Stone, 1997) and finite element method (Zhou et al., 2006)) and interface capturing method (e.g. VOF method (Chen et al., 2013), level set method (Lu et al., 2010) and front-tracking method (Homma et al., 2006)), VOF method is a widely used interface capturing method with good mass conservation property and simple treatment of the interface topology. Therefore, it has been demonstrated to be an effective method in simulating various classes of

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Fig. 1. Schematic and simulation domain of droplet formation in co-flow and flow-focusing microfluidics.

multiphase flow including droplet formation, collision and break up (Li et al., 2000; Chen et al., 2013; Wang et al., 2016; Zhang et al., 2016). Therefore, herein, the position and motions of the liquid-liquid interfaces are captured by using the VOF method in which the portion of each fluid is represented by its volume fraction function, a. The value of a is 0 or 1 for either fluid and varies from 0 to 1 continuously for the interface regions. In the simulation, it is defined that the volume fraction of dispersed phase fluid ad = 1 represents the cell that is full of dispersed phase fluid, while ad = 0 represents the cell that is empty of dispersed phase fluid. The volume fraction of continuous phase fluid in one computational cell, ac = 1  ad. From a Lagrangian point of view, the volume fraction function is advected by the interface velocity, which is governed by a volume fraction equation

@ ad ! þ U rad ¼ 0 @t

ð1Þ

The fluid flow for both the dispersed phase and continuous phase are governed by the continuity equation and the momentum equation !

rU ¼0

ð2Þ

!

! ! ! ! @U rp l þ r  ½r U þrU T  þ F þ r  ðU U Þ ¼  @t q q !

ð3Þ

where U is the fluid velocity, p is the fluid pressure. The density, q, and viscosity, l, are obtained by linear interpolation of the corresponding physical property of the dispersed phase fluid and continuous phase fluid. The source term, F, includes the interfacial tension force, Fs, and gravitational force, g. Since the characteristic length of the channel is in the order of micrometers, the gravity can be considered negligible, i.e. F = Fs. The interfacial tension force is modeled

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as a body force in the momentum equation by the continuum surface force (CSF) method (Hirt and Nichols, 1981; Gueyffier et al., 1999; Li et al., 2000; Renardy et al., 2001; Brackbill et al., 1992; Yokoi, 2014)

^ ds F s ¼ rjn

ð4Þ

where r denotes the tension coefficient, j is the interface curvature, ^ is the unit normal to the interface and ds denotes the delta n function. Capillary number of the continuous phase fluid is the main dimensionless number that governs the droplet formation dynamics in axisymmetric microfluidics, which is defined by

Ca ¼

lc  U c r

ð5Þ

The behaviors of droplet formation in co-flow and flow-focusing microfluidics can be evaluated by the radius of generated droplet, Rdr, and the droplet generation frequency, f. The nondimensionalized droplet radius is defined as

Rdr ¼

Rdr Rout

t U d =Rout

2.3. Case validation Based on a co-flow microfluidic device, which is fabricated by a steel needle inserted into the axis of a square channel, the emulsification of deionized water in silicone oil is conducted and recorded by a high speed camera in our experiment. To validate the present model, Fig. 2 compares the droplet morphology evolution in a co-flow system between the experimental data and simulation result under both dripping (We = qd2Rinl2d/r = 0.27, Ca = 0.57) and jetting regimes (We = 0.90, Ca = 4.44). In the plot, the two main dimensionless parameters, minimum radius of the neck (Rmin = Rmin/Rout, in which Rmin is the radius of the neck measured at the narrowest position) and the corresponding length (Lmin = lmin/Rout, where lmin is the distance from the inlet to the position of Rmin, and the subscripts sim and exp denote the results from simulation and experiment) are applied to analyze the droplet morphology evolution. As seen from the figure, the agreement of droplet morphology evolution between the numerical simulations and experimental data verify that the proposed model is capable to predict the droplet formation dynamics in axisymmetric microfluidic devices.

ð6Þ 3. Results and discussions

where the subscript c and f denotes the co-flow and flow-focusing, respectively. The non-dimensional time t⁄ is defined as the ratio of the real time t and the flow characteristic time,

t ¼

59

ð7Þ

2.2. Numerical solution The finite volume method based on quadrilateral meshes is utilized to numerically solve the partial differential equation [Eqs. (1)–(3)]. Laminar model is applied to characterize the fluid flow of both dispersed phase and continuous phase in co-flow and flow-focusing microfluidics since the Reynolds numbers of all cases are below 10. A numerical discretization method, called as secondorder upwind scheme, is employed to discretize the momentum equation. The explicit scheme is used for the VOF equation, Eq. (1). The pressure interpolation achieved by using the body force weighted approach while the pressure-velocity coupling are achieved by the semi-implicit method for pressure linked equations (SIMPLE) algorithm. In the simulation, fixed time step of Dt = 1  108 s is used for the first 200 time steps to circumvent the diffusive problems. Then Dt is automatically adjusted in term of the criterion that the global Courant number is smaller than 0.5 so as to ensure an adequate solution within acceptable computation consumptions. Furthermore, in every time step, the numerical solution is regarded as convergence not only by monitoring the relative residuals of the variables for velocity and pressure less than 0.1%, but also checking the mass balances. The resulting linear algebraic equations are numerically solved via the Gauss-Seidal iteration with underrelaxation factors to guarantee the convergence and improve the convergence time. A piecewise linear interface construction method is employed for the advection of volume fraction function. Grid independent test is performed by the use of three mesh size of 2.5 lm, 1.25 lm and 0.625 lm in the flow-focusing system. The results based on the meshes with resolution of 0.625 lm and 1.25 lm do not generate a difference of the time-dependent droplet morphology under the same parameters and operating conditions. In order to obtain acceptable results within reasonable computational cost, the grid resolution of 1.25 lm was used.

Based on the above mathematical model of droplet formation in co-flow and flow-focusing microfluidics, the continuous phase fluid is adjusted in a wide range of inlet velocity (Uc) while the inlet velocity of dispersed phase (Ud) is unvaried to observe the droplet flow regimes in the axisymmetric microfluidics. A wide range of variations in orifice radius and orifice length for flow-focusing microfluidics are also examined. In the simulation, a small Weber number of dispersed phase fluid (We = 1.10  104) is considered for all cases. The simulation results indicate that, with increasing Capillary number of continuous phase fluid, the droplet formation regimes sequentially undergoes the dripping regimes, drippingjetting transition and finally the jetting regime for the co-flow and flow-focusing microfluidics. In addition, decreases in orifice radius lead to a stronger hydrodynamic focusing effect, which could transform the droplet formation regime from dripping regime to dripping-jetting transition regime and even to the jetting regime. 3.1. Flow regimes 3.1.1. Dripping regime In the microfluidic systems, dripping occurs under moderate flow rates of the fluids and is characterized by droplet formation not far away from the inlets with strict periodicity (see Fig. 3). The process of droplet formation is dominated by interfacial tension and can be divided into two stages, growing and detaching, based on the competition between interfacial tension and viscous shear. In the co-flow system, the detachment of droplet happens spontaneously when the droplet reaches a critical volume that the interfacial tension cannot maintain semi-spherical interface and hold the interface attached to the inlet against the viscous shear from the continuous phase. The pressure inside the interface increases with the contraction of the neck and the fluid is squeezed towards the two sides of the high pressure region following with droplet detaching from the neck. Several satellite droplets are formed, governed by the amplitude of the perturbation and the wavelength-to-diameter ratio of the perturbation on the neck (Pimbley and Lee, 1977). In the flow-focusing system, the hydrodynamic focusing effect causes the bended stream line through the orifice and introduces the confinement to the interface evolution as shown in Fig. 3b.

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Fig. 2. Comparison of droplet formation between simulation and experiment in co-flow microfluidics under (a) dripping regime (We = 0.27, Ca = 0.57) and (b) jetting regime (We = 0.90, Ca = 4.44) (scale bar = 200 lm).

The fluid of continuous and dispersed phases in the vicinity of orifice is accelerated and consequently larger viscous force arising from the continuous phase is imposed onto the interface when compared with the co-flow system. Besides, due to the confinement of orifice, a more curved front interface and a higher pressure inside the interface are observed as compared with the co-flow system. As the dispersed phase flow into the flow-focusing system, the front interface enters the downstream of the orifice and transforms into a bulb-end under the combination of interfacial tension and deceleration of the fluids. This implies that the droplet formation process enters into the detaching process. Noteworthy, under the same inlet velocities of continuous and dispersed phases, a droplet formation circle in flow-focusing system under dripping regime is shorter than that in the co-flow system which also produces smaller droplets. This attributes to the larger viscous shear during the growing stage, resulting in the smaller critical volumes of the detaching droplets. Hence, it can be concluded that, the interface breakup is enhanced in the flow-focusing system. 3.1.2. Jetting regime Characterized by long jet that emits downstream, jetting occurs when the viscous shear force from the continuous phase dominates as shown in Fig. 4. Jetting flow regime in the co-flow and flowfocusing systems are similar, in which the droplet is formed via the mechanism of Rayleigh-Plateau instability and the droplet size is proportional to the wavelength of the fastest growing instability (Rayleigh, 1878; Utada et al., 2008). The jetting in the flow-

focusing system is more irregular for the interface breakup and the resulting droplet size distribution with respect to the co-flow system. In the co-flow system, the diameter of the jet decreases along the flow direction into an almost constant value and the fluctuation of the interface happen at the tip of the interface that breaks the jet into droplets. However, in the flow-focusing system, at the downstream of the orifice, the interface inflated slightly due to a velocity decrement after the fluid pass the orifice and the pressure difference between the inlets and the outlets is much higher. The variation of the velocity upstream of the jet contributes to the fluctuation of the interfaces adding irregularity. The droplet formation circle takes less time under jetting regimes in flow-focusing system as well. 3.1.3. Dripping-jetting transition Dripping-jetting transition is also an interesting topic in droplet formation in microfluidics happens when viscous shear force from the continuous fluid is about to exceed the interfacial tension force. With characteristics of both dripping and jetting, the transitional regime indicates the transition of dominate force in the process of droplet formation. Tapered interfaces are formed in co-flow and flow-focusing systems [see t⁄ = 0.09 in Fig. 5a and t⁄ = 0.03 in Fig. 5b], indicating that the viscous force are exceeding the interfacial tension and the necks cannot be retracted back to the inlet after the primary droplets detaching. Long stretched necks can be observed at earlier stage of droplet formation and these necks

L. Wu et al. / Chemical Engineering Science 163 (2017) 56–67

t* = 0

61

t* = 0 (c-1)

t*= 1.99

t* = 0.18

t* = 2.69

t* = 0.22

t*= 2.76

t* = 0.23 (c-2)

t* = 2.82

t* = 0.24

(c-2) (c-1)

satellite droplets

(a) Co-flowing microfluidics

(b) Flow-focusing microfluidics

(c) Local flow field

Fig. 3. Dripping regimes in axisymmetric microfluidics (Ca = 0.028, top halves of the figures are the pressure contours and bottom halves are the streamlines).

can be stabilized into jet if the flow rate of the continuous phase increases. Unlike jetting flow, the neck experiences several times of breakup, and the droplets are sheared off the neck with sizes smaller than the previous ones. The viscous shear force and interfacial tension force dominate the process alternatively before and after the breakup of the droplets due to the large variation in the area of interfaces. After detachment of strings of droplets and satellite droplets, decreases in interface area bring in smaller viscous shear force from the continuous phase, which in turn the interfacial tension force takes in control. Finally, the interfaces are retracted back to the inlets after the last droplet sheared off the front interface. The existence of the flow-focusing orifice induces the variation of local flow field in microfluidic devices, which finally produces different behaviors of the dripping-jetting transition in these two devices. On account of strong viscous shear force by the acceleration of continuous phase in the orifice, long stretched neck occurs more easily in the flow-focusing microfluidics under the same Ca. As a result, the transitional regime occurs under a lower Ca in the flow-focusing microfluidics. When the continuous phase flows out of the orifice, its velocity experiences an obvious deceleration due to the sudden expansion of the channel. The front produced droplets are slowed down by this deceleration, and thus the following produced droplets can catch up with them and coalesce with them. For example, two following smaller droplets crash into the primary one while a series of progressively smaller droplets are formed behind [t⁄ = 0.10–0.13 in Fig. 5(b)] since the primary droplet is decelerated. A bigger primary droplet followed by two satellite droplets are observed at the downstream of the channel. In

addition, owing to the additional flow resistance, the pressure drop through the focusing orifice is also larger in the flow-focusing microfluidics with respect to the corresponding co-flow microfluidics. 3.2. Role of local geometry on droplet behaviors 3.2.1. Comparison between co-flow and flow-focusing microfluidics Fig. 6 compares the interfaces morphologies during the droplet formation in these two devices with different Ca. As shown, although the droplet formation transits from the dripping regime to jetting regime as Ca increases in both the co-flow and flowfocusing devices, the detailed interfaces morphologies in the flow-focusing geometry is different from those in the co-flow geometry owing to the hydrodynamic focusing effect. It is seen that the hydrodynamic focusing effect triggers the appearance of dripping-jetting transition and jetting regimes under a smaller Ca, due to the enhancement of viscous shear force of continuous phase in the focusing orifice. Moreover, in a flow-focusing geometry, the drop coalescence between the front produced droplet and following ones is possible owing to the sudden variation in continuous phase velocity through the focusing orifice [see insets of Ca = 0.224–0.294 in Fig. 6a]. Notably, under high capillary number of the continuous phase fluid, the formation positions of droplets are far downstream from the focusing orifice, resulting in almost the same droplet detachment as those in co-flow device (see Ca = 0.560 in Fig. 6(a)). In order to gain further insight into the droplet formation behaviors in co-flow and flow-focusing microfluidic devices,

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t* = 0

t* = 0

t* = 0.0015

t* = 0.0005

(c-1)

(c-2) t* = 0.0029

t* = 0.0012

t* = 0.0036

t* = 0.0016

(a) Co-flowing microfluidics

(b) Flow-focusing microfluidics

(c-1)

(c-2)

(c) Local flow field Fig. 4. Jetting regimes in axisymmetric microfluidics (Ca = 0.34).

Fig. 6(b) and (c) quantify the droplet radius, R⁄dr, droplet polydispersity [error bar in Fig. 6b] as well as droplet generation frequency, f, in co-flow and flow-focusing microfluidic devices. Generally, the hydrodynamic flow-focusing effect produces a small jet and the velocity variation of continuous phase enhances the interface instability. As a result, the droplets produced in flowfocusing microfluidics are smaller with more polydispersed size than the co-flow device under the same flow conditions. Moreover, owing to the distinct droplet formation hydrodynamics, the dripping regime produces droplets with better monodispersity than the jetting regime, while the transition regime produces polydispersed droplets with multiple sizes. Corresponding to the consequent droplet size, the droplet generation frequency follows the opposite variation tendency since bigger droplets requires longer time to grow. Especially, discontinuities in statistical data of droplet radius and formation frequency are observed during transition regime, indicating that dripping-jetting transition is an unstable mode for producing the single-phase droplet. Furthermore, with increasing Ca, the viscous shear force turns to be dominate and the droplets formation location become far from the orifice, which cause the hydrodynamic focusing effect insignificant on the droplets formation under jetting regimes. Consequently, as Ca increases, the mechanisms of droplet formation both turn into Rayleigh-Plateau instability and the differences in droplet radius and formation frequency between these two microfluidic geometries become small.

3.2.2. Detailed flow-focusing geometry As stated above, the focusing orifice introduces a strong hydrodynamic focusing effect, causing the droplet formation regimes in a flow-focusing microfluidics depart from the co-flow one. In this situation, a question arises as to how the detailed flow-focusing geometry affects the droplet formation. Consequently, the effects of orifice radius and orifice length on the droplet formation behaviors are analyzed and discussed here based on the current numerical simulation. 3.2.2.1. Orifice radius. In order to quantify the effect of orifice radius on the droplets formation behaviors, we performed several numerical cases on droplets formation process under different orifice radius with fixed flow rates, as shown in Fig. 7. As seen from Fig. 7(a), the droplet formation regime is significantly affected by the non-dimensional orifice radius, Rori . For a smaller orifice radius, the hydrodynamic focusing effect is stronger and the local flow resistance is larger, which easily triggers a jetting and induces a larger pressure drop through the focusing orifice. In addition, two symmetrical vortexes are observed in both sides at the outlet of focusing orifice, and the vortex region turns to be enlarged by shrinking the orifice.Decreases in orifice radius makes the transformation of droplet formation regime from the dripping to drippingjetting transition and even to the jetting. Correspondingly, the transformation of droplet formation regime leads inevitably to the variations of droplet size, Rori , non-dimensional neck length

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t* = 0

t* = 0

t* = 0.09

t* = 0.03

t* = 0.11

t* = 0.05

t* = 0.13

t* = 0.09

t* = 0.14

t* = 0.10

t* = 0.16

t* = 0.13

(a) Co-flow microfluidics (Ca =0.28)

(b) Flow-focusing microfluidics (Ca = 0.067)

Fig. 5. Dripping-jetting transition in axisymmetric microfluidics.

Lb (=Lb/Rout) and droplet generation frequency, f, as indicated in Fig. 7(b)–(d). The droplet size mainly decreases with Rori , while the droplet detaching position moves towards downstream, i.e. Lb is increased. In a flow-focusing microfluidics with very small focusing orifice, the dispersed phase is focused into a narrow liquid thread along the symmetrical axis and then it expands downstream the focusing orifice. Interestingly, the droplet generation frequency does not simply increase or decrease with the change in orifice radius, which reaches the maximum value around Rori = 0.625. In addition, when orifice radius is small (e. g. Rori < 0.5), the jet is stabled by the continuous fluid accompanied with the detaching position moving far away from the orifice, and there is a tiny variation in the generated droplet size. The results indicate that, to facilitate the production of droplets under high frequency and also maintain the quality of the droplets within acceptable polydispersity, there is an optimized value of the orifice radius.

3.2.2.2. Orifice length. Our numerical simulation also represents the effect of orifice length on the droplet formation characteristics in both dripping and jetting regimes, as shown in Fig. 8. As indicated, the elongation of orifice raises the flow resistance through it, which enlarges the pressure droplet across the orifice. Although the interface morphologies during the droplet formation varies with orifice length under the same flow conditions, the resulting droplet sizes and the formation frequencies have a slight difference. Noteworthy, as discussed above and documented by the previous study (Utada et al., 2007), during the droplet formation in the flowfocusing device, the viscous force from the continuous phase determines the droplet size and the droplet formation regimes. Note that, when the orifice length varies with the constant orifice section size, the velocity of the continuous phase keeps unchanged under the same flow rates of all phases, implying little change in viscous force with the variation in orifice length. Therefore, within the orifice lengths investigated currently, the droplet formation

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Ca = 0.010

Ca = 0.028

Ca = 0.067

Ca = 0.087

Ca = 0.112

Ca = 0.224

Ca = 0.280

Ca = 0.294

Ca = 0.560 (a) Interface morphologies 0.9

*

R dr

0.6 0.3

105 4

10

f

Co-flow: dripping transitional jetting Flow-focusing: dripping transition jetting

Co-flow: dripping transition jetting Flow-focusing: dripping transition jetting

3

10

2

10

0.0 101

0.00

0.15

0.30

0.45

0.60

Ca (b) Droplet size and polydispersity

0.00

0.15

0.30

0.45

0.60

Ca (c) Droplet generation frequency

Fig. 6. Comparison of droplet formation in co-flow and flow-focusing microfluidics.

performance in flow-focusing microfluidics is insensitive to the orifice length. 3.3. Regime diagram It can be concluded from the above statement that the co-flow microfluidics exhibits superior performance in producing droplets with higher monodispersity. However, if the monodispersity is not strictly required, flow-focusing device can produce droplets at a much higher frequency. The flow-focusing device possesses larger flow resistance than the co-flow one. Particularly, by changing the Capillary number of continuous phase fluid and the orifice radius of the flow-focusing device, various flow regimes can be obtained with wanted droplet monodispersity and producing rate. Therefore, herein, we summarize the numerical simulation data and plot

a regime diagram of droplet formation in axisymmetric microfluidics (see Fig. 9), depending on orifice radius and Capillary number of continuous phase fluid. This plot quantitatively characterizes the flow regime of dripping, jetting, and dripping-jetting transition in axisymmetric microfluidics. Note that the cases shown in Fig. 9 are for the flow-focusing microfluidics expect the case of Rori ¼ 1 (co-flow microfluidics). This regime diagram is beneficial to the active control of flow condition and the optimal design of microfluidic device for the droplets production in the real application.As shown in Fig. 9, the orifice radius and Capillary number play a significant role in the droplet formation regime. When the focusing orifice is sufficiently small (e.g. Rori = 0.125) or Ca > 0.3, the dripping regime and dripping-jetting transition regime disappear and only the jetting regime is observed. Increase in orifice radius or decrease in Ca leads to the appearance of dripping regime and

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0.55 0.50 ori

0.45

= 0.125

R*dr

R

*

0.40 0.35

R

*

ori

0.30

= 0.250

0.25 0.0

Lb

0.2

R*ori = 0.375

0.4

0.6

R*ori

0.8

1.0

0.8

1.0

(b) Drop sizes 3.0

ori

= 0.500

2.4

L*b

R

*

1.8 1.2

R*ori = 0.625

0.6 0.0

0.2

0.4

0.6

R*ori

R*ori = 0.750

(c) Length of a jet 5x103

ori

= 0.875

f

R

4x103

*

3x103 2x103

R*ori = 1.000

1x103

0 0.0

(a) Interface morphologies

0.2

0.4

0.6

0.8

1.0

R*ori

(d) Drop formation frequency

Fig. 7. Effect of orifice radius on droplet formation in flow-focusing microfluidics (Ca = 0.112).

dripping-jetting transition regime. As the orifice radius increases, the regions of Ca for both the dripping-jetting transition regime and dripping regime turn to be larger.

4. Conclusions In this paper, the droplet formation processes in co-flow and flow-focusing microfluidic devices are investigated and compared numerically by CFD simulation to elucidate the role of local geometry on the droplet formation in axisymmetric microfluidics. The effects of capillary number and local geometry on the droplet formation regimes, droplet sizes and droplet generation frequency are examined and investigated. The major conclusions are summarized as follows: (1) With increasing Capillary number of continuous phase fluid, the droplet formation regimes sequentially undergoes the dripping regimes, dripping-jetting transition and finally the jetting regime for the co-flow and flow-focusing microfluidics. Droplets formed under dripping regime are dominated by the interfacial tension force, while jetting regimes occurs when the viscous force from the continuous phase dominates. The dripping-jetting transition regime is

characterized by primary droplets followed by several satellite smaller droplets, and it produces polydispersed droplets with multiple sizes, which is an unstable mode for producing droplets. (2) The existence of the focusing orifice induces a strong hydrodynamic focusing effect, causing the droplet formation behaviors in flow-focusing microfluidics depart from the co-flow one. The dripping-jetting transition regime occurs at a smaller Capillary number in flow-focusing microfluidics, and the droplets produced by flow-focusing microfluidics are smaller than those in co-flow microfludics with wider size distribution and higher frequency. Interestingly, the droplet formation in flow-focusing microfluidics is significantly affected by the orifice radius while it is insensitive to the orifice length. (3) A drop formation regime diagram quantitatively describes the flow regimes of dripping, dripping-jetting transition, and jetting in axisymmetric microfluidics, depending on the Capillary number and dimensionless orifice radius. When the orifice radius is sufficiently small or Capillary number is larger than 0.3, only the jetting regime is observed. As the orifice radius increases, the regions of Capillary number for both the dripping-jetting transition regime and dripping regime turn to be larger.

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L. Wu et al. / Chemical Engineering Science 163 (2017) 56–67

0.5

L*ori = 0.5

Dripping Jetting

R*dr

0.4 0.3 0.2

L*ori = 1.5

0.1 0.5

1.0

1.5 L*ori

2.0

2.5

(c) Drop sizes

L*ori = 2.5 1.6

L*ne

1.4

(a) Dripping regimes (Ca = 0.028)

Dripping Jetting

1.2 1.0 0.8 0.6

L

*

ori

0.5

= 0.5

1.0

1.5 L*ori

2.0

2.5

(d) Length of the necks when drops are detaching L*ori = 1.5

f

L

*

ori

Dripping Jetting

104

103

= 2.5 102

0.5

1.0

1.5

2.0

2.5

L*ori

(b) Jetting regimes (Ca = 0.112)

(e) Drop formation frequency

Fig. 8. Effect of the orifice length on droplet formation in flow-focusing microfluidics.

Jetting

Transition

Dripping

Acknowledgments The authors gratefully acknowledge the supports provided by National Natural Science Foundation of China-NSAF (No. U1530260), National Natural Science Foundation of China (No. 51406175) and Natural Science Foundation of Jiangsu Province (No. BK20140488).

0.3

Ca

0.2

References

0.1

0.0 0.0

0.2

0.4

0.6

0.8

1.0

*

R

ori

Fig. 9. Regime diagram of the droplet formation in axisymmetric microfluidics.

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