CFD Simulation and Analysis of Emulsion Droplet Formation from

We recently proposed a technique for preparing monodisperse emulsions with a coefficient of variation below 5% from a silicon array of micrometer-size...
0 downloads 0 Views 871KB Size
Download PDF
9868

Langmuir 2004, 20, 9868-9877

CFD Simulation and Analysis of Emulsion Droplet Formation from Straight-Through Microchannels Isao Kobayashi,†,‡,§ Sukekuni Mukataka,‡ and Mitsutoshi Nakajima*,† National Food Research Institute, 2-1-12 Kannondai, Tsukuba, Ibaraki 305-8642, Japan, Graduate School of Life and Environmental Sciences, University of Tsukuba, 1-1-1 Tennoudai, Tsukuba, Ibaraki 305-8572, Japan, and Japan Society for the Promotion of Science, 6 Ichibancho, Chiyoda-ku, Tokyo 102-8471, Japan Received May 20, 2004. In Final Form: July 29, 2004 We recently proposed a technique for preparing monodisperse emulsions with a coefficient of variation below 5% from a silicon array of micrometer-sized channels perpendicular to the plate surface, named a straight-through microchannel (MC). This study involved three-dimensional computational fluid dynamics (CFD) simulations to calculate the formation of an oil-in-water (O/W) emulsion droplet from straightthrough MCs with circular and elliptic cross sections. The CFD results demonstrated that the oil phase that passed through the elliptic MCs exceeding a threshold aspect ratio between 3 and 3.5 was cut off spontaneously into a small droplet with a diameter of ∼40 µm. Sufficient space for water at the channel exit had to be maintained for successful droplet formation. The formation and shrinkage of a neck inside the channel caused an increased pressure difference inside the channel and an increased velocity value near the neck. The pressure and velocity values at the neck drastically changed, and the neck was cut off instantaneously just before the completion of droplet formation. This process was triggered by a gradually increased pressure difference between the circular neck and inflating oil phase. The findings obtained in this paper provide useful numerical and visual information about the droplet formation phenomena from the straight-through MCs. The CFD results were verified by the experimental results, showing that the CFD approach can help design a suitable channel structure.

Introduction Emulsification is an important process in various industries such as the food, pharmaceutical, cosmetics, and chemical industries. Monodisperse emulsions can improve emulsion stability, and their physicochemical and organoleptic properties can be analyzed and controlled more clearly than those of polydisperse emulsions.1 Their features have received a great deal of attention in both the scientific and the industrial fields. Practical applications of monodisperse emulsions include valuable materials such as multiple emulsions and microcapsules for drug delivery systems2,3 and monodisperse polymer and inorganic microparticles.4,5 Several emulsification techniques, developed in the past decade, have allowed the direct production of monodisperse emulsions. Mason and Bibette reported that shear-rupturing of polydisperse viscous droplets in viscoelastic complex fluids can yield monodisperse emulsions.6 Umbanhowar et al. developed a technique for preparing monodisperse emulsions by forcing the to-be-dispersed phase into the coflowing continuous phase through a tapered capillary.7 Several research groups have shown that monodisperse droplets were formed by shear force due to * To whom correspondence should be addressed. Phone: +81298-38-8014. Fax: +81-298-38-8122. E-mail: [email protected]. † National Food Research Institute. ‡ University of Tsukuba. § Japan Society for the Promotion of Science. (1) McClements, D. J. Food Emulsions: Principles, Practice and Techniques; CRC Press: Boca Raton, FL, 1999; Chapter 1. (2) Higashi, S.; Tabata, N.; Nakashima, T.; Iwata, K.; Uchiyama, F.; Tamura, S.; Setoguti, T. Cancer 1995, 75, 1245. (3) O’Donnell, P. B.; McGinity, J. W. Adv. Drug Delivery Rev. 1997, 28, 25. (4) Omi, S.; Katami, K.; Yamamoto, A.; Iso, M. J. Appl. Polym. Sci. 1995, 57, 1013. (5) Nakashima, T.; Shimizu, M.; Kukizaki, M. Adv. Drug Delivery Rev. 2000, 45, 47. (6) Mason, T. G.; Bibette, J. Langmuir 1997, 13, 4600.

the continuous phase flow at the junction of T-shaped microfludic channels.8,9 Membrane emulsification, developed by Nakashima et al., can yield monodisperse emulsions by forcing the to-be-dispersed phase into the continuous phase through a microporous membrane of a narrow pore size distribution.10,11 Size-controlled emulsions with minimum coefficients of variation of ∼10% have been prepared using microporous glass and ceramic membranes.10,12-14 Kawakatsu et al. proposed microchannel (MC) emulsification for preparing monodisperse emulsions using a grooved MC array, which consists of uniformly sized channels and a slitlike terrace microfabricated on a silicon plate.15,16 This technique has successfully yielded emulsions with coefficients of variation below 5% by forcing the to-be-dispersed phase into the continuous phase through the MC.17-19 The resultant droplet size can be predicted by the MC geometry.20 The droplet formation (7) Umbanhowar, P. B.; Prasad, V.; Weitz, D. A. Langmuir 2000, 16, 347. (8) Thorsen, T.; Roberts, E. W.; Arnold, F. H.; Quake, S. R. Phys. Rev. Lett. 2001, 86, 4163. (9) Nishisako, T.; Torii, T.; Higuchi, T. Lab Chip 2002, 2, 24. (10) Nakashima, T.; Shimizu, M.; Kukizaki, M. Key Eng. Mater. 1991, 61/62, 513. (11) Joscelyne, S. M.; Tra¨gårdh, G. J. Membr. Sci. 2000, 169, 107. (12) Williams, R. A.; Peng, S. J.; Wheeler, D. A.; Morley, N. C.; Taylor, D.; Whalley, M.; Houldsworth, D. W. Trans. Inst. Chem. Eng. 1998, 76 (A), 902. (13) Vladisavljevic, G. T.; Schubert, H. Desalination 2002, 144, 167. (14) Vladisavljevic, G. T.; Schubert, H. J. Dispersion Sci. Technol. 2003, 24, 833. (15) Kawakatsu, T.; Kikuchi, Y.; Nakajima, M. J. Am. Oil Chem. Soc. 1997, 74, 317. (16) Kawakatsu, T.; Komori, H.; Nakajima, M.; Kikuchi, Y.; Yonemoto, T. J. Chem. Eng. Jpn. 1999, 32, 313. (17) Kobayashi, I.; Nakajima, M.; Nabetani, H.; Kikuchi, Y.; Shohno, A.; Satoh, K. J. Am. Oil Chem. Soc. 2001, 78, 797. (18) Sugiura, S.; Nakajima, M.; Seki, M. J. Am. Oil Chem. Soc. 2002, 79, 515. (19) Tong, J.; Nakajima, M.; Nabetani, H.; Kikuchi, Y. J. Surfactant Deterg. 2000, 3, 285.

10.1021/la0487489 CCC: $27.50 © 2004 American Chemical Society Published on Web 09/24/2004

Emulsion Droplet Formation from Straight-Through MCs

Langmuir, Vol. 20, No. 22, 2004 9869 Table 1. Geometry Modeled in This Study and the Number of Grid Cells in the Geometry

Figure 1. Schematic illustration of a silicon straight-through MC plate and the droplet formation process from a channel.

process in MC emulsification requires no mechanical stress at very low energy input.21 We recently proposed a technique for preparing monodisperse emulsions using an array of micro-through-holes fabricated on a silicon plate, which we call a straightthrough MC (Figure 1).22 Using this straight-through MC emulsification, emulsions with droplet diameters of 3050 µm and coefficients of variation below 5% have been successfully prepared.22-24 The current emulsification instrument has a throughput capacity of monodisperse emulsion droplets of 1-10 mL/h, which corresponds to a to-be-dispersed phase flux of 10-100 L/(m2 h). This instrument may also be scaled up by enlarging the channel area and by increasing the number of the straight-through MC plates. The droplet formation behavior in straightthrough MC emulsification is greatly affected by the aspect ratio of the channel cross section. Straight-through MCs exceeding a threshold aspect ratio were suitable for preparing monodisperse emulsions without mechanical stress.22,25 The channel geometry has been empirically designed on the basis of experimental results. To enable a better design, it is necessary to comprehend the shapes of the to-be-dispersed phase in the region over the channel exit and the oil-water interface inside the channel during droplet formation. Computational fluid dynamics (CFD) is a numerical simulation technique for the solution of fluid-flow and heat-transfer problems in three dimensions. The present CFD methods allow modeling of the fluid dynamics of two fluids with arbitrary immiscible fluid-fluid interfaces. Several CFD research studies have been performed to simulate the emulsification process. Agterof et al. predicted the droplet size of emulsions prepared in a stirring vessel.26 Abrahamse et al. simulated the shape and size of an emulsion droplet forming at a single cylindrical membrane pore in a laminar cross-flow.27 In this paper, we calculate and analyze the formation of an emulsion droplet from a single channel with CFD. We also describe the effect of the aspect ratio of the channel cross section on the droplet formation behavior using the (20) Sugiura, S.; Nakajima, M.; Seki, M. Langmuir 2002, 18, 3854. (21) Sugiura, S.; Nakajima, M.; Iwamoto, S.; Seki, M. Langmuir 2001, 17, 5562. (22) Kobayashi, I.; Nakajima, M.; Chun, K.; Kikuchi, Y.; Fujita, H. AIChE J. 2002, 48, 1639. (23) Kobayashi, I.; Nakajima, M. Eur. J. Lipid Sci. Technol. 2002, 104, 720. (24) Kobayashi, I.; Nakajima, M.; Mukataka, S. Colloids Surf., A 2003, 229, 33. (25) Kobayashi, I.; Mukataka, S.; Nakajima, M. J. Colloid Interface Sci. 2004, 279, 277. (26) Agterof, W. G. M.; Vaessen, G. E. J.; Haagh, G. A. A. V.; Klahn, J. K.; Janssen, J. J. M. Colloids Surf., B 2003, 31, 141. (27) Abrahamse, A. J.; van der Padt, A.; Boom, R. M.; de Heiji, W. B. C. AIChE J. 2001, 47, 1285.

no.

channel cross section

TMC-1 TMC-2 TMC-3 TMC-3,5 TMC-3.75 TMC-4

circular elliptic elliptic elliptic elliptic elliptic

dimensions of channela (µm)

channel aspect ratio

domain over channel exitb (µm)

total no. of grid cells

10 × 10 × 200 20 × 10 × 200 30 × 10 × 200 35 × 10 × 200 37.5 × 10 × 200 40 × 10 × 200

1 2 3 3.5 3.75 4

20 × 20 × 70 60 × 40 × 70 60 × 40 × 70 60 × 40 × 70 60 × 40 × 70 60 × 40 × 70

36 036 40 082 40 082 40 082 40 082 40 082

a Major axis diameter (µm) × minor axis diameter (µm) × depth (µm). b Major axis diameter (µm) × minor axis diameter (µm) × height (µm).

CFD models of the circular and elliptic straight-through MCs. The goal of this paper is to provide insight into the droplet formation phenomena from the straight-through MCs and to clarify the design points for suitable channel geometry. CFD Simulations The simulations were performed using the CFD software package CFD-ACE+, version 2003 (CFD Research Co., U.S.A.), with a finite volume code. Droplet formation from a single channel in a straight-through MC was modeled as a three-dimensional multiphase system with laminar flow. An oil phase flowed through the channel, and a water phase filled the space over the channel exit without forced flow. The movement of an interface between two fluids is tracked on the basis of the distribution of a scalar variable, F, which specifies the volume fraction of fluid 2 in each computational cell.28,29 F takes a value of 1 in cells that contain only fluid 2 and a value of 0 in cells that contain only fluid 1. A cell that contains a liquid-liquid interface would have a value of F between 0 and 1. The volume fraction, F, is determined by solving the continuum equation:

∂F + (U‚∇)F ) 0 ∂t

(1)

where U is the fluid velocity. The piecewise linear interface construction (PLIC) method30 is applied to reconstruct the interface, which is assumed to be planar and to take any orientation within the cell. The numerical results presented in this paper are based on the solution of the mass and momentum conservation equations for incompressible fluids:

∇‚U ) 0

(2)

1 η 1 ∂U + ∇‚(UU) ) - ∇p + g + ∇2U + Fiv ∂t F F F

(3)

where F and η are the fluid density and viscosity, respectively, g is the acceleration due to gravity, and Fiv is the volumetric interfacial tension force. Geometry and Computational Grid. The channel structure modeled in this study had a minor axis diameter of 10 µm and major axis diameters between 10 and 40 µm (Table 1). All the channels had a depth of 200 µm. Figure 2 depicts the geometry modeled for simulating droplet formation from the elliptic MC with an aspect ratio (major (28) Hirt, C. W.; Nichols, B. D. J. Comput. Phys. 1981, 39, 201. (29) Rider, W. J.; Kothe, D. B.; Mosso, S. J.; Cerrutti, J. H.; Hochstein, J. I. AIAA Paper, 33rd AIAA Aerospace Science Meeting, 1995, 950699. (30) Youngs, D. L. In Numerical Methods for Fluid Dynamics; Morton, K. W., Banies M. J., Eds.; Academic Press: New York, 1982; p 273.

9870

Langmuir, Vol. 20, No. 22, 2004

Kobayashi et al. Table 3. CFD Results of Droplet Formation from Straight-Through MCs no.

channel cross section

droplet diameter (µm)

droplet formation time (s)

TMC-1 TMC-2 TMC-3 TMC-3.5 TMC-3.75 TMC-4

circular elliptic elliptic elliptic elliptic elliptic

a a a 46.9 39.3 39.8

a a a 0.0872 0.0485 0.0456

a The continuous inflation of the oil phase from the channel exit.

Figure 2. Schematic illustration of a modeled geometry including a channel with an aspect ratio of 4.

Figure 3. Grid and boundary conditions of the geometry: (a) x-y plane (z ) 0); (b) y-z plane (x ) 0); (c) x-z plane (y ) 0). Table 2. Physical Properties of the Liquids under Simulation Conditions

a

property

value

water density oil density water viscosity oil viscosity interfacial tension wall contact angle for water

997 kg m-3 a 920 kg m-3 a 9.1 × 10-4 Pa sa 5.0 × 10-2 Pa sa 2.5 × 10-2 N m-1 a 0°

The values at 25 °C were used in the simulations.

axis diameter/minor axis diameter) of 4. Only one-fourth of the actual geometry was modeled, since it is symmetrical in the planes x ) 0 and y ) 0. The domain over the channel exit had a height of 70 µm, a minor axis of 40 µm, and a major axis of 60 µm. Nonuniform grid systems in the x-, y-, and z-directions were used in the simulation (Figure 3). The modeled geometry was divided into 40 082 cells,

Figure 4. (a-c) Continuous outflow of the oil phase that passed through the TMC-1. (d-f) Shape of an oil-water interface (contour lines) and velocity vectors of both phases in the region around the channel exit as a function of time.

with 16 562 cells in the channel and 23 520 cells in the domain over the channel exit. The previous study22 predicted that droplet formation takes place near the channel exit. We also expected that the velocity of both phases and the pressure would change fastest there. Hence, the grid was made finest in that region. Boundary, Initial, and Volume Conditions. Refined soybean oil was selected as the oil phase, and water was selected as the water phase in the simulations. The physical properties of the liquids under simulation conditions are given in Table 2. The wall contact angle for water

Emulsion Droplet Formation from Straight-Through MCs

Langmuir, Vol. 20, No. 22, 2004 9871

Figure 5. (a-d) Continuous outflow of the oil phase that passed through the TMC-3. (e-h) Shape of an oil-water interface and velocity vectors of both phases in the region around the channel exit as a function of time.

was set at 0°, the same as that used in a CFD study of membrane emulsification.27 A symmetric boundary condition was applied at the planes x ) 0 and y ) 0. A no-slip boundary condition was applied at the plane wall outside the channel exit and at the curved wall inside the channel. The pressures at the outlet planes of the domain over the channel exit were set at 1.0 × 105 Pa. The water phase was not forced to flow in the domain over the channel exit. The average oil phase velocity at the channel entrance in the z-direction was set at 1.0 mm/s; the oil phase velocity in the center of the channel entrance was 2.0 mm/s. At time t ) 0 s, the domain over the channel exit was filled with the water phase and the channel was completely filled with the oil phase. The interface between the oil and water phases was flat at t ) 0 s. Calculation Procedure and Analysis. The CFDACE+ code adopts the SIMPLEC algorithm31 for the pressure-velocity coupling scheme. The initial time step was set at 1 × 10-6 s. The maximum number of iterations per time step was 25 in the simulations. Except for the initial time step, each time step was automatically adjusted in order to try to match the target Courant-

Friedrichs-Lewy (CFL) number. The CFL number was set at 0.2, which means that the interface can cross a maximum of 20% of the width of a cell during each time step. The droplet formation from the straight-through MCs was simulated for 0.05-0.25 s, which took 5880-35 200 steps. This required 7-14 days of central processing unit (CPU) time on a Windows (XP Professional) personal computer (PC) with a Pentium IV processor of 2.53 GHz and an internal memory of 4*512 MB. The droplet formation behavior from the channel, calculated by CFD, was analyzed from images visualized by the postprocessing application in the CFD-ACE+ software package. An equivalent droplet diameter was calculated from the resultant droplet volume:

ddr,eq )

( ) 6Vdr π

1/3

(4)

where ddr,eq is the equivalent droplet diameter and Vdr is (31) Van Doormal, J. P.; Raithby, G. D. Numer. Heat Transfer, Part B 1984, 7, 147.

9872

Langmuir, Vol. 20, No. 22, 2004

Kobayashi et al.

Figure 6. (a-d) Formation of an expanded oil droplet from the TMC-3.5. (e-h) Shape of an oil-water interface and velocity vectors of both phases in the region around the channel exit as a function of time.

the droplet volume. The calculated pressure and velocity, extracted from the output data files, were used to analyze the pressure and flow conditions inside the channel during droplet formation. We took the preceding values in the grid cell at the centers of the channel entrance and the neck (channel exit). Results and Discussion CFD Results. Table 3 lists the CFD results of droplet formation from straight-through MCs. The dimensions of the geometry including the circular and elliptic MCs used in this study are presented in Table 1. Figure 4a-c depicts the shapes of the oil-water interface inside and outside a circular TMC-1 at three steps in time. The five contour lines at volume fractions of 0.1, 0.25, 0.5, 0.75, and 0.9 are depicted in Figure 4d. The oil-water interface is estimated to be somewhat broad (about two grid cells), whereas it is assumed that the interfacial tension force is correctly calculated. The approximate interface shape between the two fluids can be reconstructed on the basis of an interpolation to determine the isosurface corresponding to a volume

fraction of 0.5 in CFD-ACE+ with the free surface module based on the volume of fluid (VOF) method.28,29 The calculated oil-water interfaces in the other figures are therefore expressed as the contour line at a volume fraction of 0.5. The oil phase that passed through the channel started to inflate into the water phase. The diameter of the oil phase gradually increased and reached 35 µm at t ) 0.250 s (Figure 4c); thus, continuous outflow of the oil phase was observed. Figure 4d-f depicts the velocity vector profiles of both phases at three steps in time. The formation of a neck near the channel exit and the intrusion of the water phase into the channel did not occur during the calculation period. This result confirmed that it is difficult to form a small oil droplet from the circular MC without the forced water phase flow. Figure 5a-d depicts the shapes of the oil-water interface inside and outside an elliptic TMC-3 at four steps in time. Figure 5e-h depicts the velocity vector profiles of both phases at four steps in time. A neck was formed inside the channel at a distance of ∼10 µm below the channel exit, as the oil phase inflated from the TMC-3 (e.g., Figure 5b). The elliptic neck shrunk slowly, whereas

Emulsion Droplet Formation from Straight-Through MCs

Langmuir, Vol. 20, No. 22, 2004 9873

Figure 7. (a-d) Successful formation of a small droplet from the TMC-4. (e-h) Shape of an oil-water interface and velocity vectors of both phases in the region around the channel exit as a function of time.

the circular neck did not form (Figure 5a-c). The velocity vector profile in Figure 5f indicates that water entered the channel. However, the oil phase enlarged and closed the whole channel exit at t ) 0.072 s (Figure 5c,g), hindering the further intrusion of water into the channel. After that, the oil phase flowed out continuously without the further shrinkage of the neck (Figure 5d,h). While the CFD results of an elliptic TMC-2 also exhibited a slight intrusion of water into the channel, the whole channel exit was closed by the oil phase within 0.018 s. A small oil droplet did not form from the above-mentioned elliptic MCs during the calculation period of 0.100 s. Figure 6a-d illustrates the calculated droplet formation process for an elliptic TMC-3.5 at four steps in time. Figure 6e-h depicts the velocity vector profiles of both phases at four steps in time. A neck formed inside the channel shrunk slowly as the oil phase inflated from the TMC-3.5 (Figure 6a,b). Although the space for water at the channel exit narrowed, as indicated in Figure 6f, the oil phase did not close the whole channel exit, suggesting that water can continuously enter the channel. This behavior was confirmed by the velocity vector profiles in Figure 6e-h. The contour of the originally elliptic neck

became almost circlar at t ) 0.085 s (Figure 6b). The process by which the shape of the channel cross section transforms from elliptic into circular was named the neck transformation process. Further shrinkage and cutoff of the circular neck were then initiated (Figure 6b-d). This neck cutoff process was completed within 0.003 s, and an oil droplet with a diameter of 46.9 µm was formed (Figure 6d). Figure 7a-d depicts the calculated droplet formation process for an elliptic TMC-4 at four steps in time. Figure 7e-h illustrates the velocity vector profiles of both phases at four steps in time. Rapid shrinkage of a neck formed inside the channel was observed during the neck transformation process (Figure 7a,b). Sufficient space for water at the channel exit, as depicted in Figure 7e,f, promoted the intrusion of water into the channel and allowed rapid shrinkage of the neck. The neck transformation process time continued for the first 0.041 s (Figure 7b), which took half of the process time in Figure 6b. It took ∼0.004 s to complete the neck cutoff process (Figure 7b-d), and a small oil droplet with a diameter of 39.8 µm was formed (Figure 7d). The calculated droplet formation process for an elliptic TMC-3.75 exhibited the neck transformation

9874

Langmuir, Vol. 20, No. 22, 2004

Figure 8. Oil phase pressure and velocity at the channel entrance and exit and the pressure difference between the channel entrance and exit for the TMC-1 as a function of time.

process within 0.047 s and the subsequent neck cutoff process within 0.002 s. The resultant small oil droplet had a diameter of 39.3 µm. There was no significant change in the diameter of the formed droplets between the TMC3.75 and TMC-4. The CFD results obtained in this section demonstrate that an oil phase that passes through the elliptic MCs exceeding a threshold aspect ratio transforms spontaneously into a small droplet without forced water phase flow. We also found that sufficient space for water at the exit of these MCs contributes to successful droplet formation, as presented in Figure 7. Comparison between the CFD and Experimental Results. The CFD results obtained in the previous section were compared with the experimental results of droplet formation from silicon straight-through MCs in our earlier papers.22,25 The straight-through MCs used in the experimental studies were one circular MC with a diameter of 10 µm and four oblong MCs with minor axis diameters of ∼10 µm and aspect ratios between 1.9 and 3.8. Their depth was 200 µm. Experimental observation for the circular MC revealed the continuous outflow of the oil phase without the forced water phase flow. Moreover, water did not enter the channel when the diameter of the oil phase that inflated from the channel reached that of the channel. These results validate that the CFD results for the TMC-1 (Figure 4) are valid. The oil phase that passed through the oblong MC with an aspect ratio of 1.9 also flowed out continuously.25 It was difficult to observe directly in the experiments whether the formation of a neck near the channel exit and the coverage of the oil phase at the channel exit occurred or not. We believe that the CFD results of the TMC-2 (Table 3) can explain the experimental results. In the preceding experimental studies, large oil droplets with diameters exceeding 100 µm were formed by applying shear stress due to the water phase flowing across the channel.22,25 The oil phase that inflated from the oblong MCs exceeding a threshold aspect ratio of ∼3 transformed spontaneously into uniformly sized droplets,22,25 which we regarded as successful droplet formation. Monodisperse oil droplets with an average diameter of 41.9 µm were formed using the oblong MC with an aspect ratio of 3.8. The formed droplets exhibited a size relatively similar to

Kobayashi et al.

Figure 9. Oil phase pressure and velocity at the channel entrance and the neck and the pressure difference between the channel entrance and the neck for the TMC-3 as a function of time. The neck was determined at z ) -10.14 µm.

Figure 10. Oil phase pressure and velocity at the channel entrance and the neck and the pressure difference between the channel entrance and the neck for the TMC-4 as a function of time. The neck was determined at z ) -5.82 µm.

that of the calculated droplets for the TMC-3.75 and TMC-4 (Table 3). We experimentally observed that water remained in the space at the corners of the exit of the oblong MC after the oil phase had passed through the channel exit. However, directly observing the neck transformation and cutoff processes was difficult. We therefore considered that the calculated droplet formation process in Figure 7 gives important information as to why droplets form successfully using the straight-through MCs exceeding a threshold aspect ratio. When the oblong straight-through MC with an aspect ratio of 2.7 was used, large droplets with diameters over 350 µm were formed in some channels. In contrast, the oil phase that grew from other working channels was cut off spontaneously into small droplets with diameters of ∼50 µm. A polydisperse emulsion was obtained from the

Emulsion Droplet Formation from Straight-Through MCs

Langmuir, Vol. 20, No. 22, 2004 9875

Figure 11. Shape of the oil-water interface, pressure distribution of the oil phase, and velocity distribution of both phases in the region around the channel exit during the neck cutoff process as a function of time.

oblong straight-through MC. In addition, the average diameter of the emulsion became much larger than that of the emulsion obtained from the oblong straight-through MC with an aspect ratio of 3.8.25 The elliptic TMC-3.5 yielded an expanded oil droplet, as depicted in Figure 6d. The CFD and preceding experimental results exhibited the increase in the resultant droplet size for the straightthrough MCs just below threshold aspect ratios. This suggested transient droplet formation behavior between the continuous outflow of the oil phase and successful droplet formation. The CFD results obtained in the previous section were almost validated by the experimental results for the silicon straight-through MCs regardless of the aspect ratio. Additionally, the CFD approach can predict the size of the droplets formed from the straight-through MCs to some extent. Thus, the CFD models developed in this study were found to be useful for simulating the droplet formation from the straight-through MCs. Analysis of the CFD Results. We first analyzed the pressure and flow conditions in the channel during droplet formation. The output data of the oil phase pressure and velocity in the grid cell at the centers of the channel entrance (z ) -200 µm) and the neck (or channel exit) were used. The pressure difference between the channel entrance and the neck (or channel exit) was calculated by subtracting the pressure value at the channel entrance from that at the neck (or channel exit). Figure 8 gives the oil phase pressure and velocity values at the channel entrance and exit for the circular TMC-1 and the pressure difference between the channel

entrance and exit in time. We took the pressure and velocity values in the grid cell at the center of the plane of z ) -1.28 µm as the channel exit. This distance was selected to prevent effects from the outflow of the oil phase from the channel into the water phase on the laminar flow in the channel. The pressure at the channel exit almost corresponded to the Laplace pressure of the oil phase that inflated from the channel. This Laplace pressure could be calculated by the following Young-Laplace equation:32

∆PLap )

4γ d

(5)

where ∆PLap is the Laplace pressure of the oil phase that inflated from the channel, γ is the interfacial tension, and d is the diameter of the inflating oil phase. This equation explains the gradual decrease of pressure at the channel exit with an increase in the diameter of the inflating oil phase. The pressure difference in Figure 8a hardly changed during the calculation period of 0.250 s. Moreover, the velocity value at the channel exit remained nearly constant and differed only slightly from that at the channel entrance. This numerical data verified the continuous outflow of the oil phase without the formation of a neck inside the channel and the intrusion of water into the channel, as depicted in Figure 4. Figure 9 plots the oil phase pressure and velocity at the channel entrance and the neck for the elliptic TMC-3 as (32) Lyklema, J. Fundamentals of Interface and Colloid Science, Volume III: Liquid-Fluid Interfaces; Academic Press: San Diego, CA, 2000; Chapter 1.

9876

Langmuir, Vol. 20, No. 22, 2004

well as the pressure difference between the channel entrance and the neck in time. A continuous decrease in the pressure at the neck demonstrates the continuous outflow of the oil phase (Figure 5). The pressure difference and the velocity at the neck gradually increased by 20 and 90%, respectively, for the first ∼0.03 s. Additionally, the oil phase velocity in the region near the neck became higher than that in the other region (Figure 5f). The formation and shrinkage of the neck inside the channel were observed in Figure 5a,b. The pressure difference showed little change and the velocity at the neck decreased somewhat for the next ∼0.07 s. Only slight additional shrinkage of the neck occurred during this period (Figure 5b-d). We therefore found that the formation and shrinkage of the neck inside the channel caused a slight increase of the pressure difference and the velocity at the neck. Figure 10 plots the oil phase pressure and velocity at the channel entrance and the neck for the elliptic TMC-4 as well as the pressure difference between the channel entrance and the neck in time. The pressure difference and the velocity at the neck greatly increased until the circular neck was formed at t ) ∼0.036 s. The pressure at the circular neck remained almost constant and then started to increase at t ) ∼0.042 s. A drastic increase in the pressure and an instantaneous decrease in the velocity at the neck were observed at t ) ∼0.045 s. Besides the pressure and velocity values at the neck, we expected drastic changes in the shape of the neck to occur in this case. Figure 11 depicts the shape of the oil-water interface and the pressure and velocity distribution data in the region around the channel exit during the neck cutoff process. At t values between 0.045 and 0.046 s, the circular neck inside the channel was cut off instantaneously, and then, droplet formation was complete. The oil phase at the bottom of the formed droplet moved rapidly toward the channel exit, and that at the top of the oil phase inside the channel moved rapidly toward the channel entrance just after a droplet was formed (Figure 11h). The oil phase pressure inside the channel in Figure 11a gradually decreased as the dispersed phase approached the channel exit. The oil phase velocity in the region near the neck in Figure 11e became higher than that in the other region. These trends were observed for 0.0444 s. The oil phase pressure in the region near the neck drastically increased at t ) ∼0.045 s (Figure 11b,c). The oil phase in the region above the neck flowed rapidly toward the channel exit (Figure 11f,g). In contrast, the oil phase in the region below the neck flowed rapidly toward the channel entrance (Figure 11g), differing from its original flow direction in Figure 11e. The oil phase at the neck began to flow a little (Figure 11g). Additionally, the water phase flowed rapidly toward the neck (Figure 11g). Figures 10 and 11 demonstrate that drastic changes occurred in the pressure and velocity distributions in the region near the neck and the neck was instantaneously cut off just before droplet formation was complete. We next analyzed the mechanism that triggered the instantaneous cutoff of the neck. The output data of the pressures in the centers of the neck and the oil phase that inflated from the channel were used. The minor axis diameter of the neck was determined using the visualized output data. Figure 12a,b illustrates the oil phase pressures at the neck and inflating oil phase for the elliptic TMC-4 and the pressure difference between the neck and the inflating oil phase in time. Figure 12c shows the minor axis diameter of the neck in time. The pressure at the circular neck and its minor axis diameter remained almost constant for t values between 0.036 and 0.042 s. In

Kobayashi et al.

Figure 12. (a and b) Oil phase pressure at the neck and in the center of the inflating oil phase and the pressure difference between the neck and the center of the inflating oil phase for the TMC-4 as a function of time. (c) Minor axis diameter of the neck as a function of time (the shape of the neck is depicted).

contrast, the pressure at the inflating oil phase decreased continuously during that period. Thus, the pressure difference between the neck and the inflating oil phase increased after the circular neck was formed at t ) ∼0.036 s. This pressure difference achieved over 800 Pa at t ) ∼0.042 s then began to increase drastically. The diameter of the circular neck also started to decrease drastically at that moment; that is, instantaneous cutoff of the neck was initiated. These results indicate that the mechanism triggering instantaneous cutoff of the neck can be explained as follows. The increase in the pressure difference between the neck and the inflating oil phase promotes the flow of the oil phase near the neck into the inflating oil phase. When the slowly increasing pressure difference reaches a critical value, the flow of the oil phase near the neck into the inflating oil phase becomes too large for the oil near the neck to flow without shrinkage of the circular neck. We consider that instantaneous cutoff of the neck is triggered by an imbalance between the inflow and outflow of the oil phase at the circular neck. Summary and Conclusions The three-dimensional CFD models developed in this study were successfully used to simulate emulsion droplet formation from straight-through MCs with different aspect ratios. The CFD results demonstrated that droplet formation considerably depends on the aspect ratio of the modeled channel. Continuous outflow of the oil phase was observed for the MCs below a threshold aspect ratio. The oil phase that inflated from the channel exit transformed spontaneously into a small oil droplet with a diameter of ∼40 µm for the MCs exceeding a threshold aspect ratio. Sufficient space for water at the channel exit was

Emulsion Droplet Formation from Straight-Through MCs

Langmuir, Vol. 20, No. 22, 2004 9877

maintained during the droplet formation from the MCs exceeding a threshold aspect ratio. This allowed the rapid shrinkage and cutoff of the neck inside the channel, yielding a small droplet without the forced continuous phase flow. The CFD results were in relatively good agreement with the experimental results,22,25 with both showing the continuous outflow of the oil phase for the MCs below a threshold aspect ratio and successful droplet formation for those exceeding a threshold aspect ratio. The oil phase pressure and velocity inside the channel were also helpful in analyzing the droplet formation from the channel. The formation and shrinkage of the neck inside the channel were confirmed by the increased pressure difference between the channel entrance and the neck and the velocity at the neck. The drastic increase in the pressure and drastic decrease in velocity at the neck of the TMC-4 confirmed that the neck was cut off instantaneously just before the moment of droplet formation. In addition, the oil phase in the regions above and below the neck began to flow in opposite directions at that moment. An analysis of this instantaneous cutoff process

revealed that the gradually increased pressure difference between the circular neck and the oil phase that inflated from the channel exit triggered the cutoff of the circular neck. The findings obtained in this study provide useful information about the droplet formation phenomena from the straight-through MC, particularly in the region around the neck and channel exit. We expect that the use of CFD will facilitate the design of channel geometry tailor-made for the high-throughput formation of monodisperse emulsion droplets. A clear strategy for designing an optimized channel geometry is to make the aspect ratio of the channel cross section sufficiently large. Acknowledgment. This work was supported by the Nanotechnology Project of the Ministry of Agriculture, Forestry and Fisheries of Japan and a Grant-in-Aid for Science Research from the Ministry of Education, Culture, Sports, Science, and Technology. I.K. gratefully acknowledges the Japan Society for the Promotion of Science for its financial support. LA0487489