- Email: [email protected]
Asymmetric breakup of a droplet in an axisymmetric extensional flow
Asymmetric breakup of a droplet in an axisymmetric extensional flow Dongming Yu, Manman Zheng, Taoming Jin, Jingtao Wang PII: DOI: Reference:
S1004-9541(15)00263-3 doi: 10.1016/j.cjche.2015.07.016 CJCHE 342
To appear in: Received date: Revised date: Accepted date:
16 November 2014 30 May 2015 30 May 2015
Please cite this article as: Dongming Yu, Manman Zheng, Taoming Jin, Jingtao Wang, Asymmetric breakup of a droplet in an axisymmetric extensional flow, (2015), doi: 10.1016/j.cjche.2015.07.016
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Fluid Dynamics and Transport Phenomena
Asymmetric breakup of a droplet in an axisymmetric extensional flow*1 Dongming YU（余冬明）1,2, Manman ZHENG (郑嫚嫚)1, Taoming JIN (靳涛鸣)1, Jingtao WANG (王靖涛)1,** 1 School of Chemical Engineering and Technology, Tianjin University, Tianjin 300072, China 2 Sinopec Shanghai Engineering Co., Ltd., Shanghai 200120, China
MA
NU
SC R
IP
T
Abstract The asymmetric breakups of a droplet in an axisymmetric cross-like microfluidic device are investigated by using a three-dimensional volume of fluid (VOF) multiphase numerical model. Two kinds of asymmetries (droplet location deviation from the symmetric geometry center and different flow rates at two symmetric outlets) generate asymmetric flow fields near the droplet, which results in the asymmetric breakup of the latter. Four typical breakup regimes (no breakup, one-side breakup, retraction breakup and direct breakup) have been observed. Two regime maps are plotted to describe the transition from one regime to another for the two types of different asymmetries, respectively. A power law model, which is based on the three critical factors (the capillary number, the asymmetry of flow fields and the initial volume ratio), is employed to predict the volume ratio of the two unequal daughter droplets generated in the direct breakup. The influences of capillary numbers and the asymmetries have been studied systematically in this paper. The larger the asymmetry is, the bigger the one-side breakup zone is. The larger the capillary number is, the more possible the breakup is in the direct breakup zone. When the radius of the initial droplet is 20μm, the critical capillary numbers are 0.122, 0.128, 0.145, 0.165, 0.192 and 0.226 for flow asymmetry factor AS=0.05, 0.1, 0.2, 0.3, 0.4 and 0.5, respectively, in the flow system whose asymmetry is generated by location deviations. In the flow system whose asymmetry is generated by two different flow rates at two outlets, the critical capillary numbers are 0.121, 0.133, 0.145, 0.156 and 0.167 for AS=1/21, 3/23, 1/5, 7/27 and 9/29, respectively. Keywords Asymmetric breakup, Droplet, Volume of fluid method，Axisymmetric extensional flow
AC
CE P
TE
D
1 Introduction In the dispersion process of immiscible liquids, the breakup of drops plays a critical role to determine the size distribution of drops. This issue is of major importance in heterogeneous reactions, liquid-liquid extractions, oil-water separation, emulsion polymerization and other applications that require multiphase mixing [1-3].Thus, a better understanding of the physical mechanisms of drops breakup would be very important for these unit operations in chemical industries. It will be helpful to solve the problems involving mass transfer and heat transfer between different phases, especially, it is critical to predict the drop size distribution. Since the pioneer work by Taylor [4] who invented the four-roll mill to generate flow fields to study the rheology of a single droplet, extensive researches have been done to investigate, theoretically, experimentally and numerically, the deformation and breakup of droplets under the shears of external flows both in infinite media and in confined geometries [5-10]Among these work, infinite extensional flows [5-7] and hyperbolic flows in cross hydrodynamic traps [8] are employed to study the effects of viscosity ratios, interfacial tensions and capillary numbers Ca on the deformation and breakup since droplets trapped at the stagnant points are naturally deformed by the flow shears. When Ca is below a critical value Cac, the droplet will finally reach an equilibrium shape; when Ca > Cac, the droplet will break up into two daughter droplets. However, most of the works concentrate on symmetric deformation and breakup [5,6,11]. Actually, asymmetric deformation and breakup always occur naturally in the macroscopic dispersion of immiscible liquids in a stirred tank. Even in an experiment which particularly studies the rheology of a single droplet in a symmetric flow field, it is very hard to locate (keep) the droplet at the exact balance point [4,12]. Recently, droplet-based microfluidics (also called digital microfluidics) technology has been developed rapidly, whose major advantage is to generate droplets with monodispersity [13]. Based on these droplets, diverse precise manipulations could be done in order to achieve various purposes, such as the fabrication of nonspherical polymer particles, microcapsules, spherical photonic crystals, and so on [14,15]. In 2004, Link and co-workers[9] successfully broke the droplet into two daughter droplets with unequal sizes through T-junctions with two side arms of unequal lengths, which verified very well that the asymmetric geometry of microchannels could generate an asymmetric flow field and result in the asymmetric deformation and breakup of droplets. After that, various devices have been designed to study the asymmetric rheology behaviors of droplets in confined geometries, such as bifurcating channel [16], asymmetric T junction [17], Y junctions [16], obstacles [18,19] etc. In 2013, Salkin and co-workers[20] investigated the fragmentation of isolated slugs against rectangular obstacles and asymmetric loops. They found that a critical capillary number (Cac) was needed to break a slug. Only above Cac, the slug could be broken into two small daughter slugs. A mean-field approximation was introduced to predict the volume ratio of the two daughter slugs. Along with the rapid development of computer technology, numerical simulation has become a powerful tool to study the complex fluid dynamics of droplet-based microfluidics over the last few decades[21-25]. Among various numerical methods, volume of fluid (VOF) model has been widely employed to simulate the generation and breakup of droplets in microchannels Received 2014-11-16, revised 2015-05-30, accepted 2015-05-30. *Supported by Major State Basic Research Development Program of China ( 2012CB720305), and the National Natural Science Foundation of China (21376162). **Corresponding author, E-mail address: [email protected]
ACCEPTED MANUSCRIPT
SC R
IP
T
due to the simple but accurate way for dealing with topological changes of the interface in the calculation of three-dimensional flows [26,27]. Sang and co-workers [26] developed a numerical (VOF) and an analytical model to study the viscosity effect of the continuous phase on drop formation in a T junction in 2008. The prediction of drop sizes by using the analytical method shows a satisfactory agreement to the numerical result. In 2011, Bedram and Moosavi [28] successfully employed VOF model to investigate the breakup of mother drops to generate unequal daughter droplets in an asymmetric T junction with arms of different cross-sections. The numerical results were verified by comparing with an analytical theory in the limit of thin-film approximation. Up to now, the asymmetric breakup of a droplet has not been studied systematically, especially the deep understanding of physical mechanisms. Recently, Wang and Yu [29] proposed a power law model to predict the volume ratio of the two unequal daughter droplets. Thus, based on their works, the Volume of Fluid (VOF) model in the commercial software FLUENT is used to investigate the asymmetric breakup of a droplet suspended in an asymmetric and axisymmetric flow field generated in a 3-dimensional (3D) cross-like micro-device in this paper. The asymmetry of the flow system is produced by putting the droplet at a position deviating from the balance point or by controlling the volume flow rate of the left and right exit. 2 Mathematical Formulations 2.1 Numerical model The governing equations that include the equation of continuity, the equation of motion and the volume fraction equation for two-phase flows are written as follow:
NU
ρ ρv 0 t ( ρv ) ρvv p μ v v T ρg F t ad v ad 0 t
(1)
MA
(2) (3)
D
In these equations, ρ is the volume averaged density defined by Eq. (4), v is the velocity, μ is the volume averaged viscosity defined by Eq. (5), and F is the continuum surface force (CSF) [30] which is adopted to include the surface tension in the momentum equation and defined by Eq. (6).
μ ad μd 1 ad μc ρkd ad
(5)
CE P
F σ
(4)
TE
ρ ad ρd 1 ad ρc
(6)
1 ρd ρc 2 n ad kd n n ad
AC
(7)
where ρd and ρc are densities of the dispersed and continuous phase, ad is the volume fraction of the dispersed phase, μd and μc are viscosity of the dispersed and continuous phase, σ is the interfacial tension and kd is curvature calculated from the divergence of the unit surface normal (Eq. 7). A transient, three-dimensional multiphase model in the commercial software FLUENT is employed. The PISO (Pressure-Implicit with Splitting of Operations) algorithm is used for pressure-velocity coupling. Spatial discretization terms were set as follows: the PRESTO! method for the pressure term, second order upwind for the momentum equation, and the Geo-Reconstruction scheme for the volume fraction. 2.2 Validation of the numerical method The numerical model is first verified for Newtonian fluids by comparing the predicted results about the deformation of an initially spherical droplet in an axisymmetric extensional flow to those obtained by Stone and Leal [31]. The axisymmetric extensional flow is defined by
1 0 0 Ca u 0 1 0 x 2 0 0 2
(8)
where the capillary number Ca=Gaμc/σ. G is the shear rate, a is the radius of the initially droplet, σ is the stress tensor. Deformation parameter D=(L-S)/(L+S) is employed to describe the deformation of a droplet, where L and S are the half-length and half-width of the droplet, respectively. In addition, viscosity ratio: λ=μd/μc, was defiend. As shown in Fig. 1, the predicted results (circles and stars) are very consistent to those obtained by Stone and Leal[31] for both λ=1 and λ=0.1, which verified the validity of the numerical model.
NU
SC R
IP
T
ACCEPTED MANUSCRIPT
MA
Figure 1. Drop deformation in a steady axisymmetric extensional flow: D versus Ca. The open circle symbols and solid line represent our results through VOF method and those of Stone [31] for viscosity ratio λ=1, respectively. The stars and dashed line represent our results through VOF method and those of Stone for viscosity ratio λ=0.1, respectively.
AC
CE P
TE
D
2.3 Parameters setup In order to generate an axisymmetric extensional flow and to investigate the asymmetric breakup of a droplet trapped near the stagnant point, the micro-device shown in Fig. 2 is designed. As shown in Fig. 2(b), the continuous phase is injected into the device through the gap of two parallel circle plates, and flows out through the left and right exit. Thus, near the center between two parallel plates, an axisymmetric extensional flow is generated and the droplet will put at or near the stagnant point of the flow. Fig 2(a) shows the details of the geometry from the front view, in which the unit is μm.
Figure 2. The geometry of the channel from the front view (a). Sketch for the 3D channel (b).Illustration of the mesh of the cross-like channel from the left and front view (c). In order to improve the mesh quality, we drew the mesh through O-grid method in ICEM CFD. Fig 2(c) is the illustration of the mesh, in which hexahedral elements are employed. The grid size around the axis of the device from the left to the right exit is about 2μm, where the grid is densest. In order to testify the accuracy of the chosen grid, a finer grid with the interval size of 1Error! Reference source not found. has been employed to calculate the asymmetric breakup of a droplet. It is found that the
ACCEPTED MANUSCRIPT difference of the results from the two different grids is less than 2%. The droplet (oil) is initially located at the center or near the center of the device, and the continuous phase (CP, water) is injected into the device through the circular inlet between two parallel plates. The velocity profile at the inlet is 2
A z U in G [1 ] 2 A
Error!
Reference
source
not
found.,
(9) Where G is the shear rate at the wall of inlet, A is the half width of the inlet. The velocity profile of the outlet is Poiseuille flow and is calculated based on the mass conservation
T
r2 ), R2
(10)
IP
U out U max (1
SC R
Where Umax is the maximum velocity, R is the radius of the circular outlet. For simplification, we set the densities of the droplet and CP equal in order to neglect the influence of the buoyancy-driven velocity in a typical oil-water microfluidic system [32]. The surface tension is 1 mN∙m-1. The droplet and CP are both Newtonian fluids, whose viscosities are 0.0005 and 0.001 Pa·s, respectively.
AC
CE P
TE
D
MA
NU
3 Numerical results and discussions Asymmetric breakup of a droplet could be achieved due to the asymmetry of its surrounding flow fields or due to the unevenness of its interfacial properties such as non-uniform interface tensions caused by uneven distribution of surfactants. The symmetric flow is shown in Fig.3a, based on which the asymmetry of flow fields could be generated by many ways such as the asymmetric geometry of the channels, the uneven adjustment of the pressures at different exits, the asymmetric effect of external force fields like the magnetic field, and the control of the initial location of the droplet. In this paper, we will focus on the asymmetric breakup of a droplet in asymmetric flow fields generated by the deviation of the initial location of the droplet from the balance point of the flow field (case I, Fig.3b), and by changing the ratio of the volume flow rates at two exits (case II, Fig.3c).
Figure 3. Illustrations of a droplet suspended in symmetric and asymmetric flow fields. SP is the stagnation point, O and O' is the center of the flow field and droplet, respectively. l and s are the distance of the longest and the shortest point from the center of the device, respectively. QL and QR are the volume flow rates at the left and right exit, respectively. During the past decades, the symmetric breakup of a droplet has been studied extensively. In the published work, the droplet is located exactly at the balance point of the symmetric flow field. Deformation parameter (D) and capillary number have been widely used in order to characterize the rheology behaviors of the droplet due to the shear of external flows [4, 6]. However, in the investigation of an asymmetric breakup, a new parameter which defines the asymmetry of the flow field, must be introduced. In this paper, AS, which is a general symbol to describe asymmetry of the flow system and could have many expressions according to various causes of the asymmetry origination, has been employed. For instance, the asymmetry parameter AS is proportional to the location deviation and can be expressed as location asymmetry AS=(l-s)/(l+s)=xcen/a for case I (Fig.3b), and proportional to the volume flow rates at two exits and can be expressed as flow rates asymmetry AS=(QR-QL)/(QR+QL) for case II (Fig.3c). QR and QL are the volume flow rates of the right and left outlet, respectively. xcen is
ACCEPTED MANUSCRIPT
T
the distance between the initial location of the droplet center and the center of the flow field. a is the radius of the initial spherical droplet. In order to describe more asymmetric situations, the definition of AS could be generalized to AS=(XB-XS)/(XB+XS), where X indicates the physical quantity whose unevenness leads to the asymmetry of the flow field, the subscripts B and S stand for the bigger and smaller value of the quantity, respectively. The range of AS varies from 0 to 1. When AS= 0, it means the physical quantity is even and the flow system is symmetric. The extreme value 1 of AS means that XS=0 and the droplet could not be broken at all and will move out of the flow field from one side due to the huge asymmetry. Capillary number Ca is important for both symmetric and asymmetric rheology behaviors of a droplet, especially for the critical capillary number Cac. Cac is the parameter to determine whether the droplet will break up or not. AS for the symmetric case, its value is mainly dependent on the properties of the droplet such as the viscosity. However, for asymmetric case, it is severely influenced by the asymmetry of the flow system.
SC R
IP
3.1 Effects of location deviations from the balance point In this section, droplets will initially be placed along the axis direction at the distance of 1, 2, 4, 6, 8 and 10 μm from stagnation point of the flow field (center of the device), respectively. For each location deviation, the capillary number is varied over a broad range by changing the inlet flow rate so that the different deformation and breakup could be observed.
AC
CE P
TE
D
MA
NU
3.1.1 Droplet breakup regimes As for the symmetric deformation and breakup of a droplet, when CaCac, the droplet will breakup into two equal daughter droplets. However, as for asymmetric case, hydrodynamic regimes are more complicated as shown in Fig. 4 and 5. The rheological behaviors of a droplet for various Ca in the axisymmetric extensional flow with AS= 0.3 are shown in Fig.4. When Ca is very small (Ca=0.158) and AS is fixed to 0.3, mother droplet can't breakup and will move out from one side of the cross-like device (Fig. 4a, NB). With the increase of the Ca (Ca=0.164), as the elongated tail of the mother droplet is too thin to retract, the mother droplet will split from the drop similar to end-pitching fragmentation. This phenomenon is called one-side breakup (Fig. 4b, OB) because all of the breakup droplet will move into the same side of the device. A special kind of breakup so-called retraction breakup (Fig. 4c, RB) is observed when Ca is increased a little continually (Ca=0.166). When Ca is increased to 0.180, the droplet will be extended continuously until it breaks into two daughter droplets moving in the opposite direction (Fig. 4d, DB), which is similar to the breakup process of the symmetric case. The difference between RB and DB is that the left end of the elongated droplet will retract a little during the process of the RB.
Figure 4. Four different hydrodynamic patterns. (a) NB (no breakup), (b) OB (one-side breakup), (c) RB (retraction breakup), (d) DB (direct breakup). AS=0.3, Ca=0.158, 0.164, 0.166, and 0.180, respectively. The four regimes of the asymmetric breakup of a droplet as function of Ca and AS are shown in Fig. 5, in there the radius of
ACCEPTED MANUSCRIPT
AC
CE P
TE
D
MA
NU
SC R
IP
T
the initial droplet placed in the device is 20 μm. Different regimes are marked by different symbols. In this figure, for a fixed As, the transitions from NB to OB, RB and DB are observed when Ca is increased gradually. When Ca is fixed, the process of the whole transition from NB to OB, RB and DB occurs along with the decrease of the asymmetry. Thus, in the regime (NB zone marked by squares) with low Ca or high AS, the droplet is squeezed firstly near the balance point, then moves into the right arms of the device and relaxes there. In this process, the droplet could not be broken up due to the relatively strong surface tension or due to the quick movement of the droplet caused by the high asymmetry of the flow. Along with the increase of Ca or the decrease of AS, the regime transits into the OB zone (marked by open circles) and the boundary is drawn by the dash line. In this regime, as the shear action of the flow becomes a little stronger or the time taken by the droplet moving into one side of the arms gets a bit longer, the tail of the droplet is cut down and becomes a very small daughter droplet which follows the main droplet and moves out of the device from one side of the arms. From the above description, it could say, whether the droplet could be broken asymmetrically or not is the result of the competition between the cutting action of flow and the oriented displacement of the droplet due to the asymmetry. After the breakup, the small daughter droplet and the main droplet move forward together in the right side of two arms and finally merge into one because of the parabolic velocity profile of Poiseuille flow. As shown in Fig. 5, the width of this regime increases gradually along with the increment of the asymmetry. The third regime (RB zone marked by diamonds) is very narrow. It is also a special process in which a retraction occurs during the droplet breakup. Along with the increase of Ca or decrease of AS, the shear action gets a bit stronger and the displacement of the droplet gets correspondingly. The mother droplet is broken into two daughter droplets which finally move out of the device from two different sides of the arms. The way to differentiate RB from DB zone is that there is a small retraction at the left end of the elongated droplet during the process of RB. This zone is so thin that it could be treated as the starting point at which the direct breakup regime begins. DB occurs when the capillary number Ca is big enough or AS is very small. In this process, the mother droplet breaks up directly into two daughter droplets moving in the opposite direction. The droplet breakups occurring in this regime are mainly affected by the strong shear force of the extensional flows in the device. When AS= 0, it is a symmetric flow system and the regime is simply divided into two zones, NB and DB, by the critical point Cac. Here, Cac is defined as the capillary number above which the droplet will breakup into two daughter droplets moving in the opposite direction and is depicted by the solid curve in Fig 5. The width of RB regime is very narrow, which means that RB is a critical phenomenon. As the AS decreases, the value of the critical capillary number decreases.
Figure 5. Droplet breakup regimes as a function of capillary number Ca and AS(□: NB, ○: OB, ◇: RB and △:DB). The dash line and the solid curve are employed to distinguish the different regimes. The solid curve is the critical capillary number (Cac) as a function of the asymmetry. The horizontal solid line at AS=0 shows the symmetric breakup of a droplet and the solid circle (●) is the critical capillary number in symmetric case. 3.1.2 The power law model As the main purpose to break a droplet asymmetrically is to collect droplets with different volumes, the prediction of the volume ratio rS=VB/VS of daughter droplets will be focused on the regime of DB. As for the symmetric breakup, the droplet never moves. The value of Ca does not change rS which is always unit, but will change the time taken to break the droplet. The larger Ca is, the larger the cutting speed of the flow should be. While, the situation will be different for an asymmetric breakup as the droplet will shift due to the asymmetry in the direction perpendicular to that of the cutting. The translation speed should be proportional to AS, and the translation distance is proportional to the product of the speed and the time needed to break the droplet, i.e., AS= (a/Ca). Thus, rS must be inversely proportional to Ca, but proportional to AS. When Ca is extremely large, the droplet will be broken instantly, which means that the droplet has no time to shift. Thus, rS of the final droplets should be very close to the initial volume ratio r0. As shown in Fig. 3b, SPs are the stagnation point and regarded as the initial cutting point. The solid line connecting two SPs divides the droplets into two parts. r0 is the volume ratio of the right to the left part. Thus, three major factors, which are critical to the process of asymmetric breakups when the initial droplet size is fixed, are the capillary number Ca determining the cutting speed, the asymmetry AS of flow fields which causes translation of the droplet, and the initial volume ratio r0. Based on above analyses, a formula to calculate rS as a function of Ca and AS is given
rS
ACCEPTED MANUSCRIPT
e ( AS )
Ca Cac
r0 ,
(11)
Where e, α and β are parameters fitted from data of numerical calculations. Cac are 0.122, 0.128 0.145, 0.165, 0.192 and 0.226 for different asymmetries, respectively, which are obtained from Fig. 5. r0 is a function of AS and could be calculated by
(AS )3 3 AS 2 (AS )3 3 AS 2
.
1 3 AS 4.5( AS ) 5.75( AS ) 7.125( AS ) 8.4375( AS ) ... Rn ( AS ) 2
3
4
(12)
5
T
r0
rS
2.346 (AS )3.920
Ca Cac
1.188
[1 3 AS 4.5( AS )2 5.75( AS )3 ] .
SC R
IP
The data and their corresponding fitted lines [by Eqs. (11) and 12)] are shown in Fig. 6. When Ca approaches Cac, rS will goes to infinity, and rS approaches a constant along with the increment of Ca. This is consistent to our qualitative analysis in the first paragraph of this section. The fitted parameters e, α and β are 2.346, 3.920, and 1.188, respectively. Substituting them into Eq. (11) and truncating Taylor expansion of r0 to the term of third order, we have an approximate equation (13)
TE
D
MA
NU
Eq. (13) could be employed to describe the breakup data with negligible errors when the asymmetry is small.
CE P
Figure 6. The volume ratios of two unequal daughter droplets versus Ca
AC
3.1.3 The effect of the droplet volume In the previous section, the volume of the droplet was fixed. However, the effect of the droplet volume is also important in rheological behaviors of droplets. Thus, we investigated the asymmetric breakup of droplets with different sizes including a=15, 20, and 25 μm in this section. As the asymmetry AS=0.05 is fixed, the droplets are initially placed in this device with xcen= AS×.a =0.75, 1, and 1.25 μm, respectively. The final results including the simulation data and their fitted lines are shown in Fig.7. As the initial volume ratio r0 is only a function of the asymmetry AS, all these three curves eventually merge into one. The only difference for the three cases is the different critical capillary number. The bigger the droplet is, the smaller the critical capillary number is, which means that the larger droplet breaks more easily. The corresponding Cac are 0.124, 0.122 and 0.109 for a=15, 20, and 25 μm, respectively, which could be read from Fig. 7.
Figure.7. The volume ratios of the two daughter droplets versus Ca at AS=0.05. 3.2 Effects of the different volume flow rates at the two outlets Besides the location deviation, the asymmetry can also be obtained by varying the volume flow rates at two exits. The
ACCEPTED MANUSCRIPT
T
difference of flow rates at exits will form an asymmetric flow field shown in Fig. 3c. The droplet is initially placed in the center of the device, i.e. the mass center of the droplet is the geometry center of the device. As shown in Fig. 3c, the connection line of the two stagnation points does not pass the mass center of the droplet, which causes the asymmetric breakup of the droplet in this part. Compared to the asymmetric breakup of a droplet caused by location deviation, that caused by the different volume flow rates at two exits is much more complex. For instance, the streamlines become irregular near the center part of the device when AS= (QR-QL)/(QR+QL) is over 0.3 due to the constraint of the device wall. Thus, our analysis might not be appropriate when AS>0.3. However, when AS is lower than 0.3, the streamlines are regular and the results show a quite good agreement to the power law model (Eq.11).
AC
CE P
TE
D
MA
NU
SC R
IP
3.2.1 Droplet breakup regimes Similar to those shown in Fig. 4, four typical breakup hydrodynamic patterns also occur at different AS and Ca for this case. Fig. 8 displays the four regimes of the droplet breakup and the transitions from one regime to another. The results are also similar to those shown in Fig. 5. The width of the OB regime decreases with the decrease of AS. Also, the curve of the critical capillary number, across which the transitions from OB to RB and DB occur, decreases with the decrease of AS. When AS is equal to zero, the asymmetric flow field becomes the symmetric case, and the dash line and the solid curve converge into one point. This means that there are only NB and DB zones in the symmetric breakup of a droplet. Comparing Fig. 8 to Fig. 5, the map for the case whose asymmetry is generated by two different volume flow rates at the two outlets (case II) has a wider OB zone than that for the case whose asymmetry is generated by location deviations from the balance point (case I). For case I, the flow rates at left and right exits are equal, and for case II, the flow rate at the right exit is larger than the left one. Thus, the flow rate at right exit for case II is larger than that for case I when the total flow rates are the same for both two cases. Furthermore, the stronger flow at the right side of device makes the one-side breakup occur at smaller Ca and makes DB occur at larger Ca. Therefore, case II has a wider OB zone.
Figure 8. Droplet breakup regimes as a function of capillary number Ca and AS(□: NB, ○: OB, ◇: RB and △:DB). The dash line and the solid curve are employed to distinguish the different regimes. The solid curve is the critical capillary number (Cac) as a function of the asymmetry. The horizontal solid line at AS=0 shows the symmetric breakup of a droplet and the solid circle (●) is the critical capillary number in symmetric case. 3.2.2 The influence of Ca and AS Eq. (11) based on the power law could also be employed to predict the volume ratio of the two daughter droplets in this section. In order to calculate r0 in Eq. (11), the asymmetry generated by the difference of the volume flow rates at exits should be converted to xcen in the location deviation case. After a careful analysis, the relation between the asymmetry employed in this section and xcen is given in Eq. (14) xcen 8.712 AS . (14) Thus, the initial volume ratio r0 could be calculated by combining Eq. (12) and (14),
r0
(8.712AS /a )3 3 (8.712AS /a ) 2 (8.712AS /a )3 3 (8.712AS /a ) 2
.
(15)
1 1.3068 AS 0.8539( AS )2 0.4753( AS )3 0.2565( AS )4 0.1323( AS )5 ... Rn ( AS ) Fig. 9 shows the data and the fitted lines [by Eqs. (11) and (15)] of rS as a function of Ca at AS=1/21 (○), 3/23 (□), 1/5 (◇), 7/27 (△), and 9/29 (☆), respectively. Cac are 0.121, 0.133, 0.145, 0.156 and 0.167 for different asymmetries, respectively, which are obtained from Fig. 8. These results show a quite good agreement between the data and the model. The fitted parameters e, α and β are 0.1794, 1.505, and 1.098, respectively. Substituting the fitted parameters e, α and β into Eq. (11) and truncating Taylor expansion of r0 in Eq. (15) to the terms of second order, we have
rS
0.1794 A
ACCEPTED MANUSCRIPT
1.505 S 1.0983
Ca Cac
[1 1.3068 AS 0.8539( AS ) 2 ] .
(17)
NU
SC R
IP
T
The approximate equation [Eq. (17)] could be employed to describe the breakup with negligible errors when the asymmetry is small.
Figure 9. The volume ratios of two unequal daughter droplets versus Ca
MA
As shown in Fig. 6 and Fig. 9, rS decreases rapidly and approaches the constant r0 along with the increase of Ca. This means that the effect of Ca is only significant near the critical point. Beyond that, the volume ratio of the two unequal daughter droplets is only a function of the asymmetry AS.
AC
CE P
TE
D
4. Conclusions By investigating the asymmetric breakup of a droplet in two different ways to generate asymmetries through a 3-dimensional VOF numerical model, it is observed that there are four typical regimes of the droplet rheological behaviors as a function of the capillary number Ca and the asymmetric parameter AS. The regime map is plotted to describe the transition from no breakup (NB), to one-side breakup (OB), to retraction breakup (RB), and to direct breakup (DB). When the asymmetry of the flow system increases, the width of OB zone enlarges continuously. Also, the critical capillary number of the direct breakup increases along with the increase of the asymmetry. The effects of the capillary number and the asymmetry parameter are studied in details in the DB regime. Three major factors critical to the asymmetric breakups in DB regime, which are the capillary number Ca determining the cutting speed, the asymmetry AS of flow fields which causes translation of the droplet, and the initial volume ratio r0 , are pointed out in this paper. Based on these qualitative analyses, a power law model is employed to predict the volume ratios of the two unequal daughter droplets for both two cases. The volume ratios rS is dependent on Ca only in the region very near the critical capillary number. When Ca is large, rS is not a function of Ca, but only depends on the asymmetry AS. As for the critical capillary numbers Cac, they vary with the different asymmetries and could be obtained in the droplet breakup regimes. When the radius of the initial spherical droplet is 20μm, the critical capillary numbers are 0.122, 0.128, 0.145, 0.165, 0.192 and 0.226 for AS=0.05, 0.1, 0.2, 0.3, 0.4 and 0.5, respectively, in the flow system whose asymmetry is generated by location deviations from the balance point (case I). Similarly, in the flow system whose asymmetry is generated by two different volume flow rates at the two outlets (case II), the critical capillary numbers also increase along with the increment of AS, and are 0.121, 0.133, 0.145, 0.156 and 0.167 for AS=1/21, 3/23, 1/5, 7/27 and 9/29, respectively. In addition, it is found that the larger droplet has a smaller critical capillary number and breaks more easily. When the radii of the droplet are 15, 20 and 25 μm, the corresponding Cac are 0.124, 0.122 and 0.109, respectively, for case I and AS=0.05. Comparing the maps for case I and case II, the width of OB zone for case II is wider as the flow at the right side of the device is stronger in case II than that in case I. The information provided in this paper might be useful to understand the physics of droplet breakups and might be helpful in the design of droplet-based microfluidic devices. Nomenclature A half width of the inlet, μm AS asymmetry parameter a radius of the initially droplet, μm ad volume fraction of the dispersed phase Ca capillary number（Ca=Gaμc/σ） Cac critical capillary number D deformation parameter e parameters fitted from date of numerical calculations F surface force, N G shear rate, m-1 kd curvature L half-length of the droplet
ACCEPTED MANUSCRIPT
IP
T
distance of the longest point from the center of the device, μm radius of the circular outlet, μm ratios of the volume of two daughter droplets initial volume ratio half-width of the droplet distance of the shortest point from the center of the device, μm velocity, m∙s-1 physical quantity distance between the initial location of the droplet center and the center of the flow field, μm parameters fitted from date of numerical calculations parameters fitted from date of numerical calculations viscosity ratio viscosity, Pa∙s volume average density, kg∙m-3 interfacial tension, mN∙m-1
SC R
l R rS r0 S s U X xcen α β λ μ ρ σ
MA
NU
Subscripts B bigger d dispersed phase c continuous phase L left side R right side S smaller
AC
CE P
TE
D
References 1 Zerfa, M.,Brooks, B. W., "Prediction of vinyl chloride drop sizes in stabilised liquid-liquid agitated dispersion", Chemical Engineering Science. 51(12), 3223-3233(1996). 2 Wioletta, P., "Modelling of high viscosity oil drop breakage process in intermittent turbulence", Chemical Engineering Science, 61(9), 2986-2993(2006). 3 Podgórska, W., "Influence of dispersed phase viscosity on drop coalescence in turbulent flow", Chemical Engineering Research and Design, 85(5), 721-729(2007). 4 Taylor, G. I., "The formation of emulsions in definable fields of flow. Proceedings of the Royal Society of London", Series A. 146(858), 501-523(1934). 5 Bentley, B.,Leal, L., "An experimental investigation of drop deformation and breakup in steady, two-dimensional linear flows", Journal of Fluid Mechanics, 167, 241-283(1986). 6 Stone, H. A., "Dynamics of drop deformation and breakup in viscous fluids", Annual Review of Fluid Mechanics, 26(1), 65-102(1994). 7 Cristini, V., Guido, S., Alfani, A., Bławzdziewicz, J.,Loewenberg, M., "Drop breakup and fragment size distribution in shear flow" ,Journal of Rheology, 47, 1283-1298(2003) 8 Janssen, J., Boon, A.,Agterof, W.," Influence of dynamic interfacial properties on droplet breakup in plane hyperbolic flow", AIChE Journal, 43(6), 1436-1447(1997). 9 Link, D., Anna, S., Weitz, D.,Stone, H.,"Geometrically mediated breakup of drops in microfluidic devices", Physical Review Letters, 92(5), 054503-054506(2004). 10 Cubaud, T., "Deformation and breakup of high-viscosity droplets with symmetric microfluidic cross flows", Physical Review E, 80(2), 026307-026310(2009). 11 Navot, Y., "Critical behavior of drop breakup in axisymmetric viscous flow", Physics of Fluids, 11(5), 990-996(1999). 12 Rumscheidt, F.,Mason, S., "Break-up of stationary liquid threads", Journal of Colloid Science, 17(3), 260-269(1962). 13 Fair, R.,"Digital microfluidics: is a true lab-on-a-chip possible?", Microfluidics and Nanofluidics, 3(3), 245-281(2007). 14 Teh, S.-Y., Lin, R., Hung, L.-H.,Lee, A. P., "Droplet microfluidics", Lab on a Chip, 8(2), 198-220(2008). 15 Wang, J.-T., Wang, J.,Han, J.-J., " Fabrication of advanced particles and particle-based materials assisted by droplet based microfluidics", Small, 7(13), 1728-1754(2011). 16 Ménétrier-Deremble, L.,Tabeling, P., "Droplet breakup in microfluidic junctions of arbitrary angles", Physical Review E, 74(3), 035303-035306(2006). 17 Samie, M., Salari, A.,Shafii, M. B., "Breakup of microdroplets in asymmetric T junctions", Physical Review E, 87(5), 053003-053010(2013). 18 Chung, C., Lee, M., Char, K., Ahn, K. H.,Lee, S. J., "Droplet dynamics passing through obstructions in confined microchannel flow", Microfluidics and Nanofluidics, 9(6), 1151-1163(2010). 19 Protière, S., Bazant, M. Z., Weitz, D. A.,Stone, H. A., "Droplet breakup in flow past an obstacle: A capillary instability due to permeability variations", Europhysics Letters, 92(5), 54002-54007(2010). 20 Salkin, L., Schmit, A., Courbin, L.,Panizza, P., "Passive breakups of isolated drops and one-dimensional assemblies of drops in microfluidic geometries: experiments and models", Lab on a Chip, 13(15), 3022-3032(2013). 21 Liu, J.,Nguyen, N.-T., "Numerical simulation of droplet-based microfluidics", Micro and Nanosystes, 2(3), 1-8(2010).
ACCEPTED MANUSCRIPT
AC
CE P
TE
D
MA
NU
SC R
IP
T
22 Wörner, M., "Numerical modeling of multiphase flows in microfluidics and micro process engineering: a review of methods and applications", Microfluidics and Nanofluidics, 12(6), 841-886(2012). 23 Wang, J., Liu, J., Han, J.,Guan, J., "Effects of complex internal structures on rheology of multiple emulsions particles in 2D from a boundary integral method", Physical Review Letters, 110(6), 066001-066005(2013). 24 Tao, J., Song, X., Liu, J.,Wang, J., "Microfluidic rheology of the multiple-emulsion globule transiting in a contraction tube through a boundary element method", Chemical Engineering Science, 97, 328-336(2013). 25 Wang, J., Liu, J., Han, J.,Guan, J.,"Rheology investigation of the globule of multiple emulsions with complex internal structures through a boundary element method”, Chemical Engineering Science. 96, 87-97(2013). 26 Sang, L., Hong, Y.,Wang, F., "Investigation of viscosity effect on droplet formation in T-shaped microchannels by numerical and analytical methods", Microfluidics and Nanofluidics, 6(5), 621-635(2008). 27 Afkhami, S., Leshansky, A. M.,Renardy, Y., "Numerical investigation of elongated drops in a microfluidic T-junction", Physics of Fluids, 23(2), 022002-022015(2011). 28 Bedram, A.,Moosavi, A.,"Droplet breakup in an asymmetric microfluidic T junction", The European Physical Journal E, 34(8), 34-45(2011). 29 Wang J, Yu D. "Asymmetry of flow fields and asymmetric breakup of a droplet", Microfluidics and Nanofluidics, 18,709-715(2015). 30 Brackbill, J. U., Kothe, D. B.,Zemach, C.,"A continuum method for modeling surface tension", Journal of Computional Physics, 100, 335-354(1992). 31 Stone, H. A.,Leal, L. G., "Relaxation and breakup of an initially extended drop in an otherwise quiescent fluid", Journal of Fluid Mechanics, 198, 399-427(1989). 32 De Menech, M., Garstecki, P., Jousse, F., Stone, H.,"Transition from squeezing to dripping in a microfluidic T-shaped junction", Journal of Fluid Mechanic, 595, 141-161(2008).
S1004-9541(15)00263-3 doi: 10.1016/j.cjche.2015.07.016 CJCHE 342
To appear in: Received date: Revised date: Accepted date:
16 November 2014 30 May 2015 30 May 2015
Please cite this article as: Dongming Yu, Manman Zheng, Taoming Jin, Jingtao Wang, Asymmetric breakup of a droplet in an axisymmetric extensional flow, (2015), doi: 10.1016/j.cjche.2015.07.016
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Fluid Dynamics and Transport Phenomena
Asymmetric breakup of a droplet in an axisymmetric extensional flow*1 Dongming YU（余冬明）1,2, Manman ZHENG (郑嫚嫚)1, Taoming JIN (靳涛鸣)1, Jingtao WANG (王靖涛)1,** 1 School of Chemical Engineering and Technology, Tianjin University, Tianjin 300072, China 2 Sinopec Shanghai Engineering Co., Ltd., Shanghai 200120, China
MA
NU
SC R
IP
T
Abstract The asymmetric breakups of a droplet in an axisymmetric cross-like microfluidic device are investigated by using a three-dimensional volume of fluid (VOF) multiphase numerical model. Two kinds of asymmetries (droplet location deviation from the symmetric geometry center and different flow rates at two symmetric outlets) generate asymmetric flow fields near the droplet, which results in the asymmetric breakup of the latter. Four typical breakup regimes (no breakup, one-side breakup, retraction breakup and direct breakup) have been observed. Two regime maps are plotted to describe the transition from one regime to another for the two types of different asymmetries, respectively. A power law model, which is based on the three critical factors (the capillary number, the asymmetry of flow fields and the initial volume ratio), is employed to predict the volume ratio of the two unequal daughter droplets generated in the direct breakup. The influences of capillary numbers and the asymmetries have been studied systematically in this paper. The larger the asymmetry is, the bigger the one-side breakup zone is. The larger the capillary number is, the more possible the breakup is in the direct breakup zone. When the radius of the initial droplet is 20μm, the critical capillary numbers are 0.122, 0.128, 0.145, 0.165, 0.192 and 0.226 for flow asymmetry factor AS=0.05, 0.1, 0.2, 0.3, 0.4 and 0.5, respectively, in the flow system whose asymmetry is generated by location deviations. In the flow system whose asymmetry is generated by two different flow rates at two outlets, the critical capillary numbers are 0.121, 0.133, 0.145, 0.156 and 0.167 for AS=1/21, 3/23, 1/5, 7/27 and 9/29, respectively. Keywords Asymmetric breakup, Droplet, Volume of fluid method，Axisymmetric extensional flow
AC
CE P
TE
D
1 Introduction In the dispersion process of immiscible liquids, the breakup of drops plays a critical role to determine the size distribution of drops. This issue is of major importance in heterogeneous reactions, liquid-liquid extractions, oil-water separation, emulsion polymerization and other applications that require multiphase mixing [1-3].Thus, a better understanding of the physical mechanisms of drops breakup would be very important for these unit operations in chemical industries. It will be helpful to solve the problems involving mass transfer and heat transfer between different phases, especially, it is critical to predict the drop size distribution. Since the pioneer work by Taylor [4] who invented the four-roll mill to generate flow fields to study the rheology of a single droplet, extensive researches have been done to investigate, theoretically, experimentally and numerically, the deformation and breakup of droplets under the shears of external flows both in infinite media and in confined geometries [5-10]Among these work, infinite extensional flows [5-7] and hyperbolic flows in cross hydrodynamic traps [8] are employed to study the effects of viscosity ratios, interfacial tensions and capillary numbers Ca on the deformation and breakup since droplets trapped at the stagnant points are naturally deformed by the flow shears. When Ca is below a critical value Cac, the droplet will finally reach an equilibrium shape; when Ca > Cac, the droplet will break up into two daughter droplets. However, most of the works concentrate on symmetric deformation and breakup [5,6,11]. Actually, asymmetric deformation and breakup always occur naturally in the macroscopic dispersion of immiscible liquids in a stirred tank. Even in an experiment which particularly studies the rheology of a single droplet in a symmetric flow field, it is very hard to locate (keep) the droplet at the exact balance point [4,12]. Recently, droplet-based microfluidics (also called digital microfluidics) technology has been developed rapidly, whose major advantage is to generate droplets with monodispersity [13]. Based on these droplets, diverse precise manipulations could be done in order to achieve various purposes, such as the fabrication of nonspherical polymer particles, microcapsules, spherical photonic crystals, and so on [14,15]. In 2004, Link and co-workers[9] successfully broke the droplet into two daughter droplets with unequal sizes through T-junctions with two side arms of unequal lengths, which verified very well that the asymmetric geometry of microchannels could generate an asymmetric flow field and result in the asymmetric deformation and breakup of droplets. After that, various devices have been designed to study the asymmetric rheology behaviors of droplets in confined geometries, such as bifurcating channel [16], asymmetric T junction [17], Y junctions [16], obstacles [18,19] etc. In 2013, Salkin and co-workers[20] investigated the fragmentation of isolated slugs against rectangular obstacles and asymmetric loops. They found that a critical capillary number (Cac) was needed to break a slug. Only above Cac, the slug could be broken into two small daughter slugs. A mean-field approximation was introduced to predict the volume ratio of the two daughter slugs. Along with the rapid development of computer technology, numerical simulation has become a powerful tool to study the complex fluid dynamics of droplet-based microfluidics over the last few decades[21-25]. Among various numerical methods, volume of fluid (VOF) model has been widely employed to simulate the generation and breakup of droplets in microchannels Received 2014-11-16, revised 2015-05-30, accepted 2015-05-30. *Supported by Major State Basic Research Development Program of China ( 2012CB720305), and the National Natural Science Foundation of China (21376162). **Corresponding author, E-mail address: [email protected]
ACCEPTED MANUSCRIPT
SC R
IP
T
due to the simple but accurate way for dealing with topological changes of the interface in the calculation of three-dimensional flows [26,27]. Sang and co-workers [26] developed a numerical (VOF) and an analytical model to study the viscosity effect of the continuous phase on drop formation in a T junction in 2008. The prediction of drop sizes by using the analytical method shows a satisfactory agreement to the numerical result. In 2011, Bedram and Moosavi [28] successfully employed VOF model to investigate the breakup of mother drops to generate unequal daughter droplets in an asymmetric T junction with arms of different cross-sections. The numerical results were verified by comparing with an analytical theory in the limit of thin-film approximation. Up to now, the asymmetric breakup of a droplet has not been studied systematically, especially the deep understanding of physical mechanisms. Recently, Wang and Yu [29] proposed a power law model to predict the volume ratio of the two unequal daughter droplets. Thus, based on their works, the Volume of Fluid (VOF) model in the commercial software FLUENT is used to investigate the asymmetric breakup of a droplet suspended in an asymmetric and axisymmetric flow field generated in a 3-dimensional (3D) cross-like micro-device in this paper. The asymmetry of the flow system is produced by putting the droplet at a position deviating from the balance point or by controlling the volume flow rate of the left and right exit. 2 Mathematical Formulations 2.1 Numerical model The governing equations that include the equation of continuity, the equation of motion and the volume fraction equation for two-phase flows are written as follow:
NU
ρ ρv 0 t ( ρv ) ρvv p μ v v T ρg F t ad v ad 0 t
(1)
MA
(2) (3)
D
In these equations, ρ is the volume averaged density defined by Eq. (4), v is the velocity, μ is the volume averaged viscosity defined by Eq. (5), and F is the continuum surface force (CSF) [30] which is adopted to include the surface tension in the momentum equation and defined by Eq. (6).
μ ad μd 1 ad μc ρkd ad
(5)
CE P
F σ
(4)
TE
ρ ad ρd 1 ad ρc
(6)
1 ρd ρc 2 n ad kd n n ad
AC
(7)
where ρd and ρc are densities of the dispersed and continuous phase, ad is the volume fraction of the dispersed phase, μd and μc are viscosity of the dispersed and continuous phase, σ is the interfacial tension and kd is curvature calculated from the divergence of the unit surface normal (Eq. 7). A transient, three-dimensional multiphase model in the commercial software FLUENT is employed. The PISO (Pressure-Implicit with Splitting of Operations) algorithm is used for pressure-velocity coupling. Spatial discretization terms were set as follows: the PRESTO! method for the pressure term, second order upwind for the momentum equation, and the Geo-Reconstruction scheme for the volume fraction. 2.2 Validation of the numerical method The numerical model is first verified for Newtonian fluids by comparing the predicted results about the deformation of an initially spherical droplet in an axisymmetric extensional flow to those obtained by Stone and Leal [31]. The axisymmetric extensional flow is defined by
1 0 0 Ca u 0 1 0 x 2 0 0 2
(8)
where the capillary number Ca=Gaμc/σ. G is the shear rate, a is the radius of the initially droplet, σ is the stress tensor. Deformation parameter D=(L-S)/(L+S) is employed to describe the deformation of a droplet, where L and S are the half-length and half-width of the droplet, respectively. In addition, viscosity ratio: λ=μd/μc, was defiend. As shown in Fig. 1, the predicted results (circles and stars) are very consistent to those obtained by Stone and Leal[31] for both λ=1 and λ=0.1, which verified the validity of the numerical model.
NU
SC R
IP
T
ACCEPTED MANUSCRIPT
MA
Figure 1. Drop deformation in a steady axisymmetric extensional flow: D versus Ca. The open circle symbols and solid line represent our results through VOF method and those of Stone [31] for viscosity ratio λ=1, respectively. The stars and dashed line represent our results through VOF method and those of Stone for viscosity ratio λ=0.1, respectively.
AC
CE P
TE
D
2.3 Parameters setup In order to generate an axisymmetric extensional flow and to investigate the asymmetric breakup of a droplet trapped near the stagnant point, the micro-device shown in Fig. 2 is designed. As shown in Fig. 2(b), the continuous phase is injected into the device through the gap of two parallel circle plates, and flows out through the left and right exit. Thus, near the center between two parallel plates, an axisymmetric extensional flow is generated and the droplet will put at or near the stagnant point of the flow. Fig 2(a) shows the details of the geometry from the front view, in which the unit is μm.
Figure 2. The geometry of the channel from the front view (a). Sketch for the 3D channel (b).Illustration of the mesh of the cross-like channel from the left and front view (c). In order to improve the mesh quality, we drew the mesh through O-grid method in ICEM CFD. Fig 2(c) is the illustration of the mesh, in which hexahedral elements are employed. The grid size around the axis of the device from the left to the right exit is about 2μm, where the grid is densest. In order to testify the accuracy of the chosen grid, a finer grid with the interval size of 1Error! Reference source not found. has been employed to calculate the asymmetric breakup of a droplet. It is found that the
ACCEPTED MANUSCRIPT difference of the results from the two different grids is less than 2%. The droplet (oil) is initially located at the center or near the center of the device, and the continuous phase (CP, water) is injected into the device through the circular inlet between two parallel plates. The velocity profile at the inlet is 2
A z U in G [1 ] 2 A
Error!
Reference
source
not
found.,
(9) Where G is the shear rate at the wall of inlet, A is the half width of the inlet. The velocity profile of the outlet is Poiseuille flow and is calculated based on the mass conservation
T
r2 ), R2
(10)
IP
U out U max (1
SC R
Where Umax is the maximum velocity, R is the radius of the circular outlet. For simplification, we set the densities of the droplet and CP equal in order to neglect the influence of the buoyancy-driven velocity in a typical oil-water microfluidic system [32]. The surface tension is 1 mN∙m-1. The droplet and CP are both Newtonian fluids, whose viscosities are 0.0005 and 0.001 Pa·s, respectively.
AC
CE P
TE
D
MA
NU
3 Numerical results and discussions Asymmetric breakup of a droplet could be achieved due to the asymmetry of its surrounding flow fields or due to the unevenness of its interfacial properties such as non-uniform interface tensions caused by uneven distribution of surfactants. The symmetric flow is shown in Fig.3a, based on which the asymmetry of flow fields could be generated by many ways such as the asymmetric geometry of the channels, the uneven adjustment of the pressures at different exits, the asymmetric effect of external force fields like the magnetic field, and the control of the initial location of the droplet. In this paper, we will focus on the asymmetric breakup of a droplet in asymmetric flow fields generated by the deviation of the initial location of the droplet from the balance point of the flow field (case I, Fig.3b), and by changing the ratio of the volume flow rates at two exits (case II, Fig.3c).
Figure 3. Illustrations of a droplet suspended in symmetric and asymmetric flow fields. SP is the stagnation point, O and O' is the center of the flow field and droplet, respectively. l and s are the distance of the longest and the shortest point from the center of the device, respectively. QL and QR are the volume flow rates at the left and right exit, respectively. During the past decades, the symmetric breakup of a droplet has been studied extensively. In the published work, the droplet is located exactly at the balance point of the symmetric flow field. Deformation parameter (D) and capillary number have been widely used in order to characterize the rheology behaviors of the droplet due to the shear of external flows [4, 6]. However, in the investigation of an asymmetric breakup, a new parameter which defines the asymmetry of the flow field, must be introduced. In this paper, AS, which is a general symbol to describe asymmetry of the flow system and could have many expressions according to various causes of the asymmetry origination, has been employed. For instance, the asymmetry parameter AS is proportional to the location deviation and can be expressed as location asymmetry AS=(l-s)/(l+s)=xcen/a for case I (Fig.3b), and proportional to the volume flow rates at two exits and can be expressed as flow rates asymmetry AS=(QR-QL)/(QR+QL) for case II (Fig.3c). QR and QL are the volume flow rates of the right and left outlet, respectively. xcen is
ACCEPTED MANUSCRIPT
T
the distance between the initial location of the droplet center and the center of the flow field. a is the radius of the initial spherical droplet. In order to describe more asymmetric situations, the definition of AS could be generalized to AS=(XB-XS)/(XB+XS), where X indicates the physical quantity whose unevenness leads to the asymmetry of the flow field, the subscripts B and S stand for the bigger and smaller value of the quantity, respectively. The range of AS varies from 0 to 1. When AS= 0, it means the physical quantity is even and the flow system is symmetric. The extreme value 1 of AS means that XS=0 and the droplet could not be broken at all and will move out of the flow field from one side due to the huge asymmetry. Capillary number Ca is important for both symmetric and asymmetric rheology behaviors of a droplet, especially for the critical capillary number Cac. Cac is the parameter to determine whether the droplet will break up or not. AS for the symmetric case, its value is mainly dependent on the properties of the droplet such as the viscosity. However, for asymmetric case, it is severely influenced by the asymmetry of the flow system.
SC R
IP
3.1 Effects of location deviations from the balance point In this section, droplets will initially be placed along the axis direction at the distance of 1, 2, 4, 6, 8 and 10 μm from stagnation point of the flow field (center of the device), respectively. For each location deviation, the capillary number is varied over a broad range by changing the inlet flow rate so that the different deformation and breakup could be observed.
AC
CE P
TE
D
MA
NU
3.1.1 Droplet breakup regimes As for the symmetric deformation and breakup of a droplet, when Ca
Figure 4. Four different hydrodynamic patterns. (a) NB (no breakup), (b) OB (one-side breakup), (c) RB (retraction breakup), (d) DB (direct breakup). AS=0.3, Ca=0.158, 0.164, 0.166, and 0.180, respectively. The four regimes of the asymmetric breakup of a droplet as function of Ca and AS are shown in Fig. 5, in there the radius of
ACCEPTED MANUSCRIPT
AC
CE P
TE
D
MA
NU
SC R
IP
T
the initial droplet placed in the device is 20 μm. Different regimes are marked by different symbols. In this figure, for a fixed As, the transitions from NB to OB, RB and DB are observed when Ca is increased gradually. When Ca is fixed, the process of the whole transition from NB to OB, RB and DB occurs along with the decrease of the asymmetry. Thus, in the regime (NB zone marked by squares) with low Ca or high AS, the droplet is squeezed firstly near the balance point, then moves into the right arms of the device and relaxes there. In this process, the droplet could not be broken up due to the relatively strong surface tension or due to the quick movement of the droplet caused by the high asymmetry of the flow. Along with the increase of Ca or the decrease of AS, the regime transits into the OB zone (marked by open circles) and the boundary is drawn by the dash line. In this regime, as the shear action of the flow becomes a little stronger or the time taken by the droplet moving into one side of the arms gets a bit longer, the tail of the droplet is cut down and becomes a very small daughter droplet which follows the main droplet and moves out of the device from one side of the arms. From the above description, it could say, whether the droplet could be broken asymmetrically or not is the result of the competition between the cutting action of flow and the oriented displacement of the droplet due to the asymmetry. After the breakup, the small daughter droplet and the main droplet move forward together in the right side of two arms and finally merge into one because of the parabolic velocity profile of Poiseuille flow. As shown in Fig. 5, the width of this regime increases gradually along with the increment of the asymmetry. The third regime (RB zone marked by diamonds) is very narrow. It is also a special process in which a retraction occurs during the droplet breakup. Along with the increase of Ca or decrease of AS, the shear action gets a bit stronger and the displacement of the droplet gets correspondingly. The mother droplet is broken into two daughter droplets which finally move out of the device from two different sides of the arms. The way to differentiate RB from DB zone is that there is a small retraction at the left end of the elongated droplet during the process of RB. This zone is so thin that it could be treated as the starting point at which the direct breakup regime begins. DB occurs when the capillary number Ca is big enough or AS is very small. In this process, the mother droplet breaks up directly into two daughter droplets moving in the opposite direction. The droplet breakups occurring in this regime are mainly affected by the strong shear force of the extensional flows in the device. When AS= 0, it is a symmetric flow system and the regime is simply divided into two zones, NB and DB, by the critical point Cac. Here, Cac is defined as the capillary number above which the droplet will breakup into two daughter droplets moving in the opposite direction and is depicted by the solid curve in Fig 5. The width of RB regime is very narrow, which means that RB is a critical phenomenon. As the AS decreases, the value of the critical capillary number decreases.
Figure 5. Droplet breakup regimes as a function of capillary number Ca and AS(□: NB, ○: OB, ◇: RB and △:DB). The dash line and the solid curve are employed to distinguish the different regimes. The solid curve is the critical capillary number (Cac) as a function of the asymmetry. The horizontal solid line at AS=0 shows the symmetric breakup of a droplet and the solid circle (●) is the critical capillary number in symmetric case. 3.1.2 The power law model As the main purpose to break a droplet asymmetrically is to collect droplets with different volumes, the prediction of the volume ratio rS=VB/VS of daughter droplets will be focused on the regime of DB. As for the symmetric breakup, the droplet never moves. The value of Ca does not change rS which is always unit, but will change the time taken to break the droplet. The larger Ca is, the larger the cutting speed of the flow should be. While, the situation will be different for an asymmetric breakup as the droplet will shift due to the asymmetry in the direction perpendicular to that of the cutting. The translation speed should be proportional to AS, and the translation distance is proportional to the product of the speed and the time needed to break the droplet, i.e., AS= (a/Ca). Thus, rS must be inversely proportional to Ca, but proportional to AS. When Ca is extremely large, the droplet will be broken instantly, which means that the droplet has no time to shift. Thus, rS of the final droplets should be very close to the initial volume ratio r0. As shown in Fig. 3b, SPs are the stagnation point and regarded as the initial cutting point. The solid line connecting two SPs divides the droplets into two parts. r0 is the volume ratio of the right to the left part. Thus, three major factors, which are critical to the process of asymmetric breakups when the initial droplet size is fixed, are the capillary number Ca determining the cutting speed, the asymmetry AS of flow fields which causes translation of the droplet, and the initial volume ratio r0. Based on above analyses, a formula to calculate rS as a function of Ca and AS is given
rS
ACCEPTED MANUSCRIPT
e ( AS )
Ca Cac
r0 ,
(11)
Where e, α and β are parameters fitted from data of numerical calculations. Cac are 0.122, 0.128 0.145, 0.165, 0.192 and 0.226 for different asymmetries, respectively, which are obtained from Fig. 5. r0 is a function of AS and could be calculated by
(AS )3 3 AS 2 (AS )3 3 AS 2
.
1 3 AS 4.5( AS ) 5.75( AS ) 7.125( AS ) 8.4375( AS ) ... Rn ( AS ) 2
3
4
(12)
5
T
r0
rS
2.346 (AS )3.920
Ca Cac
1.188
[1 3 AS 4.5( AS )2 5.75( AS )3 ] .
SC R
IP
The data and their corresponding fitted lines [by Eqs. (11) and 12)] are shown in Fig. 6. When Ca approaches Cac, rS will goes to infinity, and rS approaches a constant along with the increment of Ca. This is consistent to our qualitative analysis in the first paragraph of this section. The fitted parameters e, α and β are 2.346, 3.920, and 1.188, respectively. Substituting them into Eq. (11) and truncating Taylor expansion of r0 to the term of third order, we have an approximate equation (13)
TE
D
MA
NU
Eq. (13) could be employed to describe the breakup data with negligible errors when the asymmetry is small.
CE P
Figure 6. The volume ratios of two unequal daughter droplets versus Ca
AC
3.1.3 The effect of the droplet volume In the previous section, the volume of the droplet was fixed. However, the effect of the droplet volume is also important in rheological behaviors of droplets. Thus, we investigated the asymmetric breakup of droplets with different sizes including a=15, 20, and 25 μm in this section. As the asymmetry AS=0.05 is fixed, the droplets are initially placed in this device with xcen= AS×.a =0.75, 1, and 1.25 μm, respectively. The final results including the simulation data and their fitted lines are shown in Fig.7. As the initial volume ratio r0 is only a function of the asymmetry AS, all these three curves eventually merge into one. The only difference for the three cases is the different critical capillary number. The bigger the droplet is, the smaller the critical capillary number is, which means that the larger droplet breaks more easily. The corresponding Cac are 0.124, 0.122 and 0.109 for a=15, 20, and 25 μm, respectively, which could be read from Fig. 7.
Figure.7. The volume ratios of the two daughter droplets versus Ca at AS=0.05. 3.2 Effects of the different volume flow rates at the two outlets Besides the location deviation, the asymmetry can also be obtained by varying the volume flow rates at two exits. The
ACCEPTED MANUSCRIPT
T
difference of flow rates at exits will form an asymmetric flow field shown in Fig. 3c. The droplet is initially placed in the center of the device, i.e. the mass center of the droplet is the geometry center of the device. As shown in Fig. 3c, the connection line of the two stagnation points does not pass the mass center of the droplet, which causes the asymmetric breakup of the droplet in this part. Compared to the asymmetric breakup of a droplet caused by location deviation, that caused by the different volume flow rates at two exits is much more complex. For instance, the streamlines become irregular near the center part of the device when AS= (QR-QL)/(QR+QL) is over 0.3 due to the constraint of the device wall. Thus, our analysis might not be appropriate when AS>0.3. However, when AS is lower than 0.3, the streamlines are regular and the results show a quite good agreement to the power law model (Eq.11).
AC
CE P
TE
D
MA
NU
SC R
IP
3.2.1 Droplet breakup regimes Similar to those shown in Fig. 4, four typical breakup hydrodynamic patterns also occur at different AS and Ca for this case. Fig. 8 displays the four regimes of the droplet breakup and the transitions from one regime to another. The results are also similar to those shown in Fig. 5. The width of the OB regime decreases with the decrease of AS. Also, the curve of the critical capillary number, across which the transitions from OB to RB and DB occur, decreases with the decrease of AS. When AS is equal to zero, the asymmetric flow field becomes the symmetric case, and the dash line and the solid curve converge into one point. This means that there are only NB and DB zones in the symmetric breakup of a droplet. Comparing Fig. 8 to Fig. 5, the map for the case whose asymmetry is generated by two different volume flow rates at the two outlets (case II) has a wider OB zone than that for the case whose asymmetry is generated by location deviations from the balance point (case I). For case I, the flow rates at left and right exits are equal, and for case II, the flow rate at the right exit is larger than the left one. Thus, the flow rate at right exit for case II is larger than that for case I when the total flow rates are the same for both two cases. Furthermore, the stronger flow at the right side of device makes the one-side breakup occur at smaller Ca and makes DB occur at larger Ca. Therefore, case II has a wider OB zone.
Figure 8. Droplet breakup regimes as a function of capillary number Ca and AS(□: NB, ○: OB, ◇: RB and △:DB). The dash line and the solid curve are employed to distinguish the different regimes. The solid curve is the critical capillary number (Cac) as a function of the asymmetry. The horizontal solid line at AS=0 shows the symmetric breakup of a droplet and the solid circle (●) is the critical capillary number in symmetric case. 3.2.2 The influence of Ca and AS Eq. (11) based on the power law could also be employed to predict the volume ratio of the two daughter droplets in this section. In order to calculate r0 in Eq. (11), the asymmetry generated by the difference of the volume flow rates at exits should be converted to xcen in the location deviation case. After a careful analysis, the relation between the asymmetry employed in this section and xcen is given in Eq. (14) xcen 8.712 AS . (14) Thus, the initial volume ratio r0 could be calculated by combining Eq. (12) and (14),
r0
(8.712AS /a )3 3 (8.712AS /a ) 2 (8.712AS /a )3 3 (8.712AS /a ) 2
.
(15)
1 1.3068 AS 0.8539( AS )2 0.4753( AS )3 0.2565( AS )4 0.1323( AS )5 ... Rn ( AS ) Fig. 9 shows the data and the fitted lines [by Eqs. (11) and (15)] of rS as a function of Ca at AS=1/21 (○), 3/23 (□), 1/5 (◇), 7/27 (△), and 9/29 (☆), respectively. Cac are 0.121, 0.133, 0.145, 0.156 and 0.167 for different asymmetries, respectively, which are obtained from Fig. 8. These results show a quite good agreement between the data and the model. The fitted parameters e, α and β are 0.1794, 1.505, and 1.098, respectively. Substituting the fitted parameters e, α and β into Eq. (11) and truncating Taylor expansion of r0 in Eq. (15) to the terms of second order, we have
rS
0.1794 A
ACCEPTED MANUSCRIPT
1.505 S 1.0983
Ca Cac
[1 1.3068 AS 0.8539( AS ) 2 ] .
(17)
NU
SC R
IP
T
The approximate equation [Eq. (17)] could be employed to describe the breakup with negligible errors when the asymmetry is small.
Figure 9. The volume ratios of two unequal daughter droplets versus Ca
MA
As shown in Fig. 6 and Fig. 9, rS decreases rapidly and approaches the constant r0 along with the increase of Ca. This means that the effect of Ca is only significant near the critical point. Beyond that, the volume ratio of the two unequal daughter droplets is only a function of the asymmetry AS.
AC
CE P
TE
D
4. Conclusions By investigating the asymmetric breakup of a droplet in two different ways to generate asymmetries through a 3-dimensional VOF numerical model, it is observed that there are four typical regimes of the droplet rheological behaviors as a function of the capillary number Ca and the asymmetric parameter AS. The regime map is plotted to describe the transition from no breakup (NB), to one-side breakup (OB), to retraction breakup (RB), and to direct breakup (DB). When the asymmetry of the flow system increases, the width of OB zone enlarges continuously. Also, the critical capillary number of the direct breakup increases along with the increase of the asymmetry. The effects of the capillary number and the asymmetry parameter are studied in details in the DB regime. Three major factors critical to the asymmetric breakups in DB regime, which are the capillary number Ca determining the cutting speed, the asymmetry AS of flow fields which causes translation of the droplet, and the initial volume ratio r0 , are pointed out in this paper. Based on these qualitative analyses, a power law model is employed to predict the volume ratios of the two unequal daughter droplets for both two cases. The volume ratios rS is dependent on Ca only in the region very near the critical capillary number. When Ca is large, rS is not a function of Ca, but only depends on the asymmetry AS. As for the critical capillary numbers Cac, they vary with the different asymmetries and could be obtained in the droplet breakup regimes. When the radius of the initial spherical droplet is 20μm, the critical capillary numbers are 0.122, 0.128, 0.145, 0.165, 0.192 and 0.226 for AS=0.05, 0.1, 0.2, 0.3, 0.4 and 0.5, respectively, in the flow system whose asymmetry is generated by location deviations from the balance point (case I). Similarly, in the flow system whose asymmetry is generated by two different volume flow rates at the two outlets (case II), the critical capillary numbers also increase along with the increment of AS, and are 0.121, 0.133, 0.145, 0.156 and 0.167 for AS=1/21, 3/23, 1/5, 7/27 and 9/29, respectively. In addition, it is found that the larger droplet has a smaller critical capillary number and breaks more easily. When the radii of the droplet are 15, 20 and 25 μm, the corresponding Cac are 0.124, 0.122 and 0.109, respectively, for case I and AS=0.05. Comparing the maps for case I and case II, the width of OB zone for case II is wider as the flow at the right side of the device is stronger in case II than that in case I. The information provided in this paper might be useful to understand the physics of droplet breakups and might be helpful in the design of droplet-based microfluidic devices. Nomenclature A half width of the inlet, μm AS asymmetry parameter a radius of the initially droplet, μm ad volume fraction of the dispersed phase Ca capillary number（Ca=Gaμc/σ） Cac critical capillary number D deformation parameter e parameters fitted from date of numerical calculations F surface force, N G shear rate, m-1 kd curvature L half-length of the droplet
ACCEPTED MANUSCRIPT
IP
T
distance of the longest point from the center of the device, μm radius of the circular outlet, μm ratios of the volume of two daughter droplets initial volume ratio half-width of the droplet distance of the shortest point from the center of the device, μm velocity, m∙s-1 physical quantity distance between the initial location of the droplet center and the center of the flow field, μm parameters fitted from date of numerical calculations parameters fitted from date of numerical calculations viscosity ratio viscosity, Pa∙s volume average density, kg∙m-3 interfacial tension, mN∙m-1
SC R
l R rS r0 S s U X xcen α β λ μ ρ σ
MA
NU
Subscripts B bigger d dispersed phase c continuous phase L left side R right side S smaller
AC
CE P
TE
D
References 1 Zerfa, M.,Brooks, B. W., "Prediction of vinyl chloride drop sizes in stabilised liquid-liquid agitated dispersion", Chemical Engineering Science. 51(12), 3223-3233(1996). 2 Wioletta, P., "Modelling of high viscosity oil drop breakage process in intermittent turbulence", Chemical Engineering Science, 61(9), 2986-2993(2006). 3 Podgórska, W., "Influence of dispersed phase viscosity on drop coalescence in turbulent flow", Chemical Engineering Research and Design, 85(5), 721-729(2007). 4 Taylor, G. I., "The formation of emulsions in definable fields of flow. Proceedings of the Royal Society of London", Series A. 146(858), 501-523(1934). 5 Bentley, B.,Leal, L., "An experimental investigation of drop deformation and breakup in steady, two-dimensional linear flows", Journal of Fluid Mechanics, 167, 241-283(1986). 6 Stone, H. A., "Dynamics of drop deformation and breakup in viscous fluids", Annual Review of Fluid Mechanics, 26(1), 65-102(1994). 7 Cristini, V., Guido, S., Alfani, A., Bławzdziewicz, J.,Loewenberg, M., "Drop breakup and fragment size distribution in shear flow" ,Journal of Rheology, 47, 1283-1298(2003) 8 Janssen, J., Boon, A.,Agterof, W.," Influence of dynamic interfacial properties on droplet breakup in plane hyperbolic flow", AIChE Journal, 43(6), 1436-1447(1997). 9 Link, D., Anna, S., Weitz, D.,Stone, H.,"Geometrically mediated breakup of drops in microfluidic devices", Physical Review Letters, 92(5), 054503-054506(2004). 10 Cubaud, T., "Deformation and breakup of high-viscosity droplets with symmetric microfluidic cross flows", Physical Review E, 80(2), 026307-026310(2009). 11 Navot, Y., "Critical behavior of drop breakup in axisymmetric viscous flow", Physics of Fluids, 11(5), 990-996(1999). 12 Rumscheidt, F.,Mason, S., "Break-up of stationary liquid threads", Journal of Colloid Science, 17(3), 260-269(1962). 13 Fair, R.,"Digital microfluidics: is a true lab-on-a-chip possible?", Microfluidics and Nanofluidics, 3(3), 245-281(2007). 14 Teh, S.-Y., Lin, R., Hung, L.-H.,Lee, A. P., "Droplet microfluidics", Lab on a Chip, 8(2), 198-220(2008). 15 Wang, J.-T., Wang, J.,Han, J.-J., " Fabrication of advanced particles and particle-based materials assisted by droplet based microfluidics", Small, 7(13), 1728-1754(2011). 16 Ménétrier-Deremble, L.,Tabeling, P., "Droplet breakup in microfluidic junctions of arbitrary angles", Physical Review E, 74(3), 035303-035306(2006). 17 Samie, M., Salari, A.,Shafii, M. B., "Breakup of microdroplets in asymmetric T junctions", Physical Review E, 87(5), 053003-053010(2013). 18 Chung, C., Lee, M., Char, K., Ahn, K. H.,Lee, S. J., "Droplet dynamics passing through obstructions in confined microchannel flow", Microfluidics and Nanofluidics, 9(6), 1151-1163(2010). 19 Protière, S., Bazant, M. Z., Weitz, D. A.,Stone, H. A., "Droplet breakup in flow past an obstacle: A capillary instability due to permeability variations", Europhysics Letters, 92(5), 54002-54007(2010). 20 Salkin, L., Schmit, A., Courbin, L.,Panizza, P., "Passive breakups of isolated drops and one-dimensional assemblies of drops in microfluidic geometries: experiments and models", Lab on a Chip, 13(15), 3022-3032(2013). 21 Liu, J.,Nguyen, N.-T., "Numerical simulation of droplet-based microfluidics", Micro and Nanosystes, 2(3), 1-8(2010).
ACCEPTED MANUSCRIPT
AC
CE P
TE
D
MA
NU
SC R
IP
T
22 Wörner, M., "Numerical modeling of multiphase flows in microfluidics and micro process engineering: a review of methods and applications", Microfluidics and Nanofluidics, 12(6), 841-886(2012). 23 Wang, J., Liu, J., Han, J.,Guan, J., "Effects of complex internal structures on rheology of multiple emulsions particles in 2D from a boundary integral method", Physical Review Letters, 110(6), 066001-066005(2013). 24 Tao, J., Song, X., Liu, J.,Wang, J., "Microfluidic rheology of the multiple-emulsion globule transiting in a contraction tube through a boundary element method", Chemical Engineering Science, 97, 328-336(2013). 25 Wang, J., Liu, J., Han, J.,Guan, J.,"Rheology investigation of the globule of multiple emulsions with complex internal structures through a boundary element method”, Chemical Engineering Science. 96, 87-97(2013). 26 Sang, L., Hong, Y.,Wang, F., "Investigation of viscosity effect on droplet formation in T-shaped microchannels by numerical and analytical methods", Microfluidics and Nanofluidics, 6(5), 621-635(2008). 27 Afkhami, S., Leshansky, A. M.,Renardy, Y., "Numerical investigation of elongated drops in a microfluidic T-junction", Physics of Fluids, 23(2), 022002-022015(2011). 28 Bedram, A.,Moosavi, A.,"Droplet breakup in an asymmetric microfluidic T junction", The European Physical Journal E, 34(8), 34-45(2011). 29 Wang J, Yu D. "Asymmetry of flow fields and asymmetric breakup of a droplet", Microfluidics and Nanofluidics, 18,709-715(2015). 30 Brackbill, J. U., Kothe, D. B.,Zemach, C.,"A continuum method for modeling surface tension", Journal of Computional Physics, 100, 335-354(1992). 31 Stone, H. A.,Leal, L. G., "Relaxation and breakup of an initially extended drop in an otherwise quiescent fluid", Journal of Fluid Mechanics, 198, 399-427(1989). 32 De Menech, M., Garstecki, P., Jousse, F., Stone, H.,"Transition from squeezing to dripping in a microfluidic T-shaped junction", Journal of Fluid Mechanic, 595, 141-161(2008).