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A ternary model for double-emulsion formation in a capillary microfluidic device{ Jang Min Parkab and Patrick D. Anderson*a Received 12th December 2011, Accepted 30th March 2012 DOI: 10.1039/c2lc21235h To predict double-emulsion formation in a capillary microfluidic device, a ternary diffuse-interface model is presented. The formation of double emulsions involves complex interfacial phenomena of a three-phase fluid system, where each component can have different physical properties. We use the Navier–Stokes/Cahn–Hilliard model for a general ternary system, where the hydrodynamics is coupled with the thermodynamics of the phase field variables. Our model predicts important features of the double-emulsion formation which was observed experimentally by Utada et al. [Utada et al., Science, 2005, 308, 537]. In particular, our model predicts both the dripping and jetting regimes as well as the transition between those two regimes by changing the flow rate conditions. We also demonstrate that a double emulsion having multiple inner drops can be formed when the outer interface is more stable than the inner interface.
1 Introduction A double emulsion is a particular type of emulsion which has smaller drops inside a larger drop. Because of the intermediate phase in between the smaller inner drops and the surrounding media, the double emulsion is of high potential for various applications, such as drug delivery and controlled drug release.1,2 Recently, several kinds of microfluidic device have been developed for a continuous formation of double emulsions with tailored properties.3–6 The main advantage of using microfluidics in the double-emulsion formation is the precise control over the droplet size and its size distribution, which could not be achieved by conventional processes.7–9 Double-emulsion formation in the microfluidic device is a complex process which involves hydrodynamics and interfacial phenomena of a three-phase fluid system, and there are many processing variables which can affect the final property of the double emulsion. In general, the size of drops, the number of smaller drops inside each larger drop and the frequency of drop formation are all determined by the physical properties of the fluids, the imposed flow rate conditions and the micro channel geometry. For instance, Utada et al.5 could control the drop formation mechanism and the size of the drops by changing the flow rate conditions (see Fig. 3 of Utada et al.5). In the dripping regime, the drop size decreases monotonically as the outer flow rate increases, while in the jetting regime, the drop size increases a Department of Mechanical Engineering, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands. E-mail:
[email protected] b Institute for Complex Molecular Systems, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands { Electronic supplementary information (ESI) available. See DOI: 10.1039/c2lc21235h
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and then decreases with the outer flow rate. In addition, they could produce double emulsions having multiple inner drops, where the number of inner drops can be precisely controlled by changing the flow rate conditions. Several experimental studies have provided empirical knowledge on the double-emulsion formation mechanisms.3–6 However, the experimental data alone would not be sufficient for design or optimization of the process. For instance, there are various kinds of materials, such as polymers, hydrogels and biofluids, which are potential candidates to be employed in this process for various applications. As each material behaves in a different way, the structure of the resulting double emulsion will vary significantly depending on the material, which cannot be predicted easily. Or one may need to optimize the channel geometry to achieve a more efficient production of double emulsions, which generally requires a parametric study of the process. In this regard, it would be desirable to develop theoretical and numerical models which can describe and predict the details of flow and interface evolution of the ternary fluid system during the double-emulsion formation. Also the usefulness of such a numerical model for design and characterization of the microfluidic devices can be observed in many previous studies, particularly for micromixers inducing complex flow fields.10–14 There are many kinds of theoretical and numerical studies regarding the droplet behavior,15 for example, droplet deformation and breakup,16,17 droplet coalescence and interaction,18,19 drop deformation in confined geometries,20 and droplet formation in microchannels.21–23 Among those studies, the work of Zhou et al. (2006) is the only one regarding the double-emulsion formation in a microfluidic device,23 to the best of our knowledge. They could reproduce some experimental results of This journal is ß The Royal Society of Chemistry 2012
Utada et al. (2005),5 and they also studied the effects of flow conditions, viscosity ratio and viscoelasticity. However, the numerical method was based on a two-phase model, thus they inevitably had to assume that the innermost drop fluid and outermost fluid are identical. As a result, they could not reproduce important phenomena such as the dripping and the transition between the dripping and jetting regime, and also material properties of each fluid could not be varied independently, which limits the practicability of the method. This motivates a more general modeling of the process, which can handle a general three-phase fluid system. This present study builds on a numerical simulation of doubleemulsion formation using a ternary diffuse-interface method.24,25 The Navier–Stokes/Cahn–Hilliard (NS/CH) model, which couples hydrodynamics and thermodynamics of the phase field variable, is solved by a finite element method in an axisymmetric flow focusing geometry. In section 2, we describe the theoretical model and numerical methods applied. In section 3, the numerical simulation results are presented. In particular, numerical studies were carried out similar to the experiments of Utada et al. (2005),5 and we could obtain consistent results as shown in Fig. 3 of Utada et al.5 Important issues and limitations of the present simulations are also discussed for future studies. In section 4, we make a brief summary with conclusions.
2 Diffuse-interface method We consider a three-phase fluid where each component is an incompressible Newtonian fluid. We use the ternary diffuseinterface model by Kim and Lowengrub (2005)25 which follows from the two-phase model by Lowengrub and Truskinovsky (1998).24 In the diffuse-interface model, the fluid–fluid interface has a finite width which is determined by the molecular force balance.26 Since the diffuse-interface method can describe the multi-phase phenomena involving complex topological transitions without singularity problems, it has been widely accepted to study the flow and structure development in many applications.27,28 2.1 Governing equations We briefly summarize the governing equations for the ternary NS/CH model, while the details of the model can be found in the references.24,25 In this study, we consider the density-matched fluid, thus the mass balance equation can be written as
The composition equation can be written as r(
Lci zu:+ci )~+:(Mi (+mi {+mN )), i~1,2 Lt
(3)
where Mi is the mobility parameter of component i. The chemical potential difference is defined as mi {mN ~(
N{1 X Lf Lf { ){ei +2 ci {eN +2 ck , Lci LcN k~1
i~1,2
(4)
The specific free energy can be written as the sum of the homogeneous part (f0) and gradient contribution part as f ~f0 (c1 , ,cN )z
N 1X ei j+ci j2 2 i~1
(5)
where ei is the gradient energy parameter. For the homogeneous part of the specific ternary free energy, we use the following form29 1 f0 ~ ½ac21 (1{c1 )2 zbc22 (1{c2 )2 zcc23 (1{c3 )2 z3Vc21 c22 c23 2
(6)
where a, b, c and V are material parameters. It is assumed that the thickness of three kinds of the interface between two components are equal, thus rffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffi e1 ze2 e2 ze3 e3 ze1 ~ ~ ~j (7) azb bzc cza where j is the diffuse-interface thickness. In this ternary model, there can be three kinds of interfaces between two different phases, and the interface property is represented by the free-energy parameters (a, b, c, V) and the gradient-energy parameters (ei,i = 1,2). The contact angle of the interface with respect to the others will be determined by the freeenergy parameters, and the surface tension of each interface is determined by the free-energy and gradient-energy parameters.29 In order to obtain those parameters experimentally, a phase diagram of the ternary mixture system and surface tension of each interface should be measured by appropriate experimental methods.30,31 The viscosity of the mixture, which depends on the composition, can be assumed as32,33 g~
N X
gi ci
(8)
i~1
+?u = 0
(1)
where u is the mass-averaged velocity. The momentum balance equation is written as r(
N X Lu zu:+u)~+:(2gD){r+gz r(mi {mN )+ci Lt i~1
(2)
where r is the constant density, g is the viscosity, 1 D~ (+uz(+u)T ) is the rate-of-strain tensor, g = p/r + f is the 2 Gibbs free energy with f the specific free energy, mi is the chemical potential of component i, and ci is the mass fraction of component i. In this work, the total number of components is N = 3. The last term in the right hand side comes from the surface tension contribution. This journal is ß The Royal Society of Chemistry 2012
where gi is the viscosity of component i as a pure substance. There are important dimensionless numbers, namely capillary number Ca, Reynolds number Re and Cahn number C, which are defined as Ca~
g u viscous force ~ c surface tension
(9)
Re~
r u L inertial force ~ viscous force g
(10)
j interface thickness ~ L characteristic length
(11)
C~
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where an asterisk indicates the characteristic value. In the following discussion, the minimum radius of the capillary (namely, R0) is selected as a characteristic length (L*), and the characteristic velocity (u*) is defined as u* = Q/2pR20 where Q is the total flow rate. 2.2 Numerical method The velocity–pressure formulation is used to solve the flow problem. The weak form for the mass and momentum balance equations can be written as (v,r
Lu zru:+u)z(+vT ,2gD){(+:v,rg) Lt N X (v,r(mi {mN )+ci )z(v,t)CN ~
(12)
(13)
Fig. 1 Axisymmetric capillary microfluidic device where dashed-lines indicate computational domain (top) and the computational domain with initial interface contours (bottom).
where v and q are weighting functions. The chemical potential is treated as a separate unknown instead of being substituted in the composition equation.34 The weak form for the composition equation and the chemical potential can be written as
3 3 5 the direction of the z-axis through 0ƒrv r0 , r0 ƒrv r0 and 8 8 8 5 r0 ƒrvr0 , respectively, where r0 is the radius of the inlet. Then 8 they flow through the flow focusing geometry with the diameter ratio of 4 : 1 : 2. As for the initial condition of the composition, we apply
i~1
2(q,+?u) = 0
(ri ,r
Lci zru:+ci )zMi (+ri ,+mi )~0, Lt
(si ,Pi ci ){ei (+si ,+ci ){
N{1 X
i~1,2
eN (+si ,+ck )
k~1
z(si ,mi {mN )~0,
(14)
(15)
i~1,2
where ri and si are weighting functions, and Pici includes nonlinear terms from Lf0 =Lci and f. The mass fraction c3 is obtained XN from the relation c ~1. A first-order implicit Euler scheme is i~1 i used for the temporal discretization with a time step of 5 6 1024, and the non-linear terms are linearized by a standard Picard method in each time step. A standard finite element method is used for a spatial discretization, where bi-quadratic interpolation is used for the velocity u, concentration ci and chemical potential mi, and bi-linear interpolation is used for the Gibbs free energy g. At each time step, the mass fraction ci and the chemical potential mi are calculated first, which are then used to solve the flow problem.
3 Results and discussion 3.1 Problem description Fig. 1 shows a schematic diagram of the capillary microfluidic device. As the geometry is axisymmetric, we have selected a twodimensional domain inside the dashed-lines as a computational domain and appropriate axisymmetric conditions are imposed in the model, similar to the previous work by Zhou et al. (2006).23 Since the model solves the axisymmetric problem, the results are three-dimensional (see ESI{), though we will show only the cross-sectional view of the capillary in the following results. There are four boundaries of symmetric (Cs), wall (Cw), inlet (Ci) and outlet (Co) in the computational domain in Fig. 1. At the inlet Ci where z = 0, inner, middle and outer fluids are injected in 2674 | Lab Chip, 2012, 12, 2672–2677
1 c1 jt~0 ~ ( tanh (r1 )z1) 2 1 c2 jt~0 ~max½ ( tanh (r2 ){ tanh (r1 )),0 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 { 2z2 zr2 z r0 8 pffiffiffi r1 ~ 2j pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 { 2z2 zr2 z r0 8 pffiffiffi r2 ~ 2j
(16)
of which the interface contours are shown in Fig. 1. The boundary condition for the Navier–Stokes equation is uz ~u0 (r), tz ~0, uz ~0,
ur ~0, ur ~0, ur ~0,
on Ci on Co |Cs on Cw
(17)
where u0(r) is specified as a constant velocity at the inlet of each fluid as 3 u0 (0ƒrv r0 )~u1 8 3 5 (18) u0 ( r0 ƒrv r0 )~u2 8 8 5 u0 ( r0 ƒrvr0 )~u3 8 The flow rate of each fluid is determined by u0(r). The boundary condition for the Cahn–Hilliard model is Lci =Ln~0, ci ~ci jt~0 ,
Lmi =Ln~0, mi ~0,
on Cs |Cw |Co on Ci , i~1,2
(19)
where ci jt~0 is given by eqn (16), and n is the outward pointing surface normal vector. This journal is ß The Royal Society of Chemistry 2012
In this study, we assume that the three components are immiscible with each other, but the model is general and accounts for mutual diffusion. Particularly, the free energy parameters are selected as a = 3, b = 21, c = 3 and V = 7, so that the middle fluid can spread in between the inner and outer fluids.29 Those parameters are selected rather intuitively since corresponding experimental data could not be found at this moment. Actually, the simulation result is significantly dependent on the free energy function and parameters, which deserves further studies. Before the actual numerical studies, we performed a mesh convergence analysis to select a proper spatial discretization which can provide accurate enough solutions within manageable computational resources and time. Five meshes denoted as M1– M5 are used for the test, and Table 1 summarizes the mesh information. In this test, all components have the same viscosity, and the inner–middle–outer flow rate ratio is 4 : 1 : 5.88. The non-dimensional parameters are Ca = 1.22 6 1021, Re = 9.8 and C = 5 6 1022. According to the mesh convergence analysis result, M4 mesh provides an almost converged solution, thus it is used hereinafter. In this numerical condition, we obtained the jetting of inner and middle fluids as shown in Fig. 2. As the middle fluid covers the innermost drop, the double emulsion could be formed successfully. The present numerical model is based on the three-phase model, thus the each phase field can be treated independently (see Fig. 2). For instance, we can change the density or viscosity of each component independently.
Table 1 Mesh information
No. elems. No. nodes Min. elem. size (61022)
M1
M2
M3
M4
M5
684 2223 3.8
1174 3747 3.0
1804 5691 2.3
2602 8143 1.9
3137 9780 1.7
Fig. 2 Distributions of c1, c2 and c3, and their interfaces (from top to bottom) after the pinch-off of middle fluid.
This journal is ß The Royal Society of Chemistry 2012
Fig. 3 Snapshots of dripping (t = 66.5, 67.25, 68 and 68.75 from top to bottom). The flow rate ratio of inner, middle and outer fluid is 4 : 1 : 2.88.
In contrast to the previous two-phase simulation,23 we could also reproduce the dripping regime by changing the material or flow conditions. For instance, for the flow rate ratio of 4 : 1 : 2.88, Ca = 4.43 6 1022, Re = 1.41 6 101, C = 5 6 1022 and viscosity-matched system, the pinch-off of inner and middle fluids takes place close to the expansion region of the capillary as shown in Fig. 3. We consider this dripping case as a standard condition, and in the following we study the effect of the flow conditions by varying the outer flow rate and the effect of viscosity by varying the outer fluid viscosity. 3.2 Effect of flow conditions Similar to the experiments of Utada et al. (2005),5 the flow rates of the inner and middle fluids are fixed with the ratio of 4 : 1 and only the outer flow rate is varied from the standard conditions. Fig. 4 shows the interfaces right after the pinch-off of the middle fluid for three different flow rate conditions. As the outer flow rate increases, the jet becomes longer and the size of the drops changes. The transition between dripping and jetting is well predicted by changing the outer flow rate, which is consistent with the experimental results. The radii of coaxial jet and drops are measured for various outer flow rates, and are plotted in Fig. 5. Following the same notation of Utada et al. (2005),5 QOF is the flow rate of the outer flow and Qsum is the sum of inner and middle flow rates. R0 is the minimum radius of the capillary. As the outer flow rate increases, the drop radius decreases monotonically for QOF/Qsum ¡ 1.2, which is the dripping regime. Also a monodisperse emulsion could be obtained (for instance, see the first figure in Fig. 4), where the size of each drop is very uniform. In contrast, in the jetting regime, the radii of drops increases (1.2 ¡ QOF/Qsum ¡ 1.5) and then decreases (1.5 ¡ QOF/Qsum) as the outer flow rate increases. Particularly, we could see that the size of the drop tends to become polydisperse in the jetting regime, which was also mentioned by Utada et al. (2005).5 For instance, when QOF/Qsum = 1.55, the Lab Chip, 2012, 12, 2672–2677 | 2675
formation, which makes the system more complicated. Another issue is that the C is very small in the real phenomena. In order to resolve the high gradient of the diffuse-interface for C % 1, one needs to use a very fine spatial discretization at the interface, which requires special techniques23 and additional computational costs. Because of those limitations, we could not obtain reasonable results particularly when the viscosity ratio of inner–middle– outer fluids becomes very large, for instance, 1 : 1 : 10 as in the experiment of Utada et al. (2005).5 Instead, we study the effect of outer fluid viscosity in the following. 3.3 Effect of viscosity
Fig. 4 Snapshots after the pinch-off of middle fluid for different flow rate conditions. Flow rate ratios of inner, middle and outer fluids are 4 : 1 : 4.22 (top), 4 : 1 : 6.44 (middle) and 4 : 1 : 7.22 (bottom). The flow rates of inner and middle fluids are fixed.
The viscosity of the outer fluid is increased so that the viscosity ratio of inner–middle–outer fluids is 1 : 1 : 1.4, while other material parameters and flow conditions are kept the same as the standard conditions. In the standard conditions (where the viscosity ratio is 1 : 1 : 1), we have observed the dripping of double emulsion (see Fig. 3). Fig. 6 shows the snapshots when the viscosity ratio is changed to 1 : 1 : 1.4. As the viscosity of the outer fluid increases, the viscous stress on the outer interface becomes larger which suppresses the instability of the outer interface. Also the velocity profile in the coaxial jet becomes different from that of the outer flow as the outer viscosity increases.5 As a result, the inner and middle fluids form a long jet. However, as the outer interface is more stable than the inner interface due to the higher viscosity of the outer fluid, the pinch-off of the inner fluid takes place more quickly than the middle fluid, resulting in two innermost drops (see Fig. 6). In our simulations, a further increase in the outer fluid viscosity results in a very long jetting which cannot be captured within the current computational domain.
Fig. 5 Radii of jet, inner drop and outer drop versus outer flow rate.
smaller and the larger drops are produced in an alternative manner, which is indicated by the large error bar in Fig. 5. However, this could be due to a transient effect in the simulation, and a long-time simulation might be required in order to obtain conclusive results. The radius of the coaxial jet decreases monotonically as the outer flow rate increases irrespective of the regime. The overall numerical results agree qualitatively well with the experimental results except for the discontinuous change in the drop radius at the transition between dripping and jetting (see Fig. 3 in Utada et al.5 and Fig. 5 in this paper). In order to achieve a quantitative agreement between simulation and experiment, more information is required about the material properties. The most important one is the free energy function and its parameters, which characterize the thermodynamics of the phase field variables. According to our study, the numerical results are significantly affected by the free energy function and parameters, and finding a proper set of free energy parameter is very important for a reasonable prediction of double-emulsion formation. For instance, when we use a fully symmetric free energy with a = b = c = 1 and V = 0, we could never observe the double-emulsion formation. Also in the real experiments, surfactants are commonly used to stabilize the drop 2676 | Lab Chip, 2012, 12, 2672–2677
Fig. 6 Snapshots of double-emulsion formation (t = 63.5, 66.5, 69.5 and 72.5 from top to bottom). The viscosity ratio of inner, middle and outer fluid is 1 : 1 : 1.4 and the flow rate ratio of those is 4 : 1 : 2.88.
This journal is ß The Royal Society of Chemistry 2012
4 Conclusions In this study, we presented a model for double-emulsion formation in a capillary microfluidic flow focusing device. The ternary NS/CH model, coupling the hydrodynamics and thermodynamics of a three-phase fluid system, is solved by the finite element method in an axisymmetric domain. We could reproduce the drop formation either by dripping and jetting, and predict the transition between those two cases by changing the outer flow rate, which is consistent with the experiments. In particular, the effect of the outer flow rate on the radii of coaxial jet and drops could be reasonably predicted except for the discontinuous change in the drop radius at the transition regime. The overall numerical results agree well with the experimental results, which suggests the usefulness of the ternary NS/CH model. According to the numerical studies, several issues have arisen which motivates further studies in order to improve the predictability of the simulations. The most important one is to obtain suitable material parameters, especially the free energy function and its parameters. Also many materials used in the microfluidic device are of biological nature and hence nonNewtonian, the current model using Newtonian rheology for each fluid might not be accurate enough depending on the material and flow regime. The model could be improved further in this respect by employing, for instance, a viscoelastic constitutive model for more rigorous modeling of the material characteristics. In addition, surfactants are present in practice, which are known to have a large impact on drop dynamics; future work will be devoted to add surfactants as an additional component.
Acknowledgements This research forms part of the Project P1.03 PENT of the research program of the BioMedical Materials Institute, cofunded by the Dutch Ministry of Economic Affairs, Agriculture and Innovation. The financial contribution of the Nederlandse Hartstichting is gratefully acknowledged. The authors also thank Bram Pape and Patricia Dankers from the ICMS Institute for Complex Molecular Systems in Eindhoven for stimulating discussions.
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