Ind. Eng. Chem. Res. 2004, 43, 8233-8238
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Prediction of Droplet Diameter for Microchannel Emulsification: Prediction Model for Complicated Microchannel Geometries Shinji Sugiura,† Mitsutoshi Nakajima,*,† and Minoru Seki‡ National Food Research Institute, 2-1-12 Kannondai, Tsukuba, Ibaraki 305-8642, Japan, and Graduate School of Engineering, Osaka Prefecture University, 1-1 Gakuen-Cho, Sakai, Osaka 599-8531 Japan
Microchannel (MC) emulsification is a novel technique for producing monodisperse emulsions. We previously proposed a droplet diameter prediction model for simple MC structures consisting of a narrow channel and a slitlike terrace. In this study, the prediction model is expanded to MCs with partition walls. The geometry of an MC with a partition wall is defined by five parameters: channel length, channel width, terrace length, terrace width, and MC depth. The droplet diameter prediction models are proposed under the simple assumption that the portion of the dispersed phase within the detachment length from the terrace end detaches and forms a discrete droplet. The volumes of the detaching dispersed-phase droplets are calculated, and the calculated values are compared with the experimental data. The mean percentage deviation of the calculated values from the experimental results is 5.7%, and all of the data have deviations within 11%. 1. Introduction Emulsions have been utilized in various industries, including the food, cosmetics, and pharmaceutical industries. Many important emulsion properties are determined by the size of the droplets the emulsions contain.1-3 Therefore, a method for reliably predicting and controlling the size distribution of emulsions is important. Monodisperse emulsions are useful for fundamental studies because the interpretation of experimental results is simpler for monodisperse than polydisperse emulsions.2 Monodisperse emulsions are also useful in measuring important properties of emulsions. Monodisperse emulsions can be obtained by fractionation from polydisperse emulsions, but repeated operations are required.4,5 Membrane emulsification, in which a pressurized dispersed phase is passed through a microporous membrane and forms emulsion droplets, is a promising technique for producing quasi-monodisperse emulsions with a coefficient of variation of approximately 10%.6-8 The emulsion droplet size is controlled by the membrane pore size. This technique can be used to produce emulsions without strong mechanical stress.9 Another method for producing quasi-monodisperse emulsions has been proposed in which shearinduced rupturing is applied to a Couette flow.10-12 In earlier work, we proposed a method for making monodisperse emulsions with a coefficient of variation of less than 5%, in which emulsion droplets are formed from a microfabricated channel array.13,14 This technique, called microchannel (MC) emulsification, is successful in preparing emulsions with droplet sizes ranging from 3 to 100 µm.15,16 It is promising for preparing not only monodisperse emulsions, but also monodisperse microspheres composed of various materials. We have applied it in preparing several types of oil-in-water, * To whom correspondence should be addressed. Tel.: +81298-38-8014. Fax: +81-298-38-8122. E-mail: mnaka@ nfri.affrc.go.jp. † National Food Research Institute. ‡ Osaka Prefecture University.
water-in-oil,17 and water-in-oil-in-water emulsions,18 as well as polymer microparticles.19 MC emulsification exploits the interfacial tension, the dominant force on the micrometer scale, as the driving force for droplet formation.14 Therefore, interfacial tension affects the dynamic behavior of droplet formation.20 The energy input for MC emulsification is low compared to that for the conventional emulsification technique because droplet formation from the MC is based on spontaneous transformation.14 Recently, a straight-through MC was devised for scaling up of the process, and the droplet formation volume rate reached 6.5 mL of dispersed phase per hour per MC plate.21 Further scaling up is underway. In MC emulsification, the MC geometry significantly affects the diameter of the droplets formed.22,23 Droplet formation from MC is a hydrodynamic phenomenon, and it can be modeled as the solution of the Navier-Stokes equation, along with the proper boundary conditions. However, mathematical complexities of the equation, considering the complicated MC geometry, intricately shaped interface, and wetting phenomena, preclude this approach. We previously proposed an empirical model for predicting the droplet diameter for MCs with simple geometries composed of a channel part and a terrace part.22 This model enables us to predict the droplet diameter in a simple way. It assumed that the portion of the dispersed phase that was within the detachment length (A) from the terrace end detached and formed a droplet. The droplet volume was estimated from the volume of the dispersed phase that detached from the terrace during this process. The values of A were correlated with MC depth by regression analysis. A prediction curve expressed in terms of the terrace length and MC depth was proposed. Although the model was found to provide agreement with the experimental data, it is applicable only to MCs with simple geometries. Previous studies used MCs with complicated geometries that have partition walls between each MC.14,15,17-19,24 MCs with partition walls are useful for preparing emulsions with dispersed phases that easily wet the MC surfaces because the partition wall disturbs the contact
10.1021/ie0494770 CCC: $27.50 © 2004 American Chemical Society Published on Web 11/16/2004
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Figure 1. Schematic of the structure of an MC plate with a partition wall.
of the dispersed phase passing through each MC. In this study, the prediction model is expanded to modified models that are applicable to MCs with partition walls. The expanded model is correlated with the experimental data. The calculated values are compared with experimentally obtained droplet diameters using MCs with different geometrical parameters. 2. Theory Figure 1 shows that the structure of an MC plate with a partition wall consists of a channel part and a terrace part. Over the end of the terrace part is a deeply etched well part. The structure is defined by five parameters: channel length (LC), channel width (WC), terrace length (LT), terrace width (WT), and MC depth (H). LT is defined as the length from the exit of the channel and the end of the terrace. The droplet diameter is estimated from the volume of the dispersed phase that flows into the well during the detachment of the dispersed phase. The diameter of the formed droplet is calculated as
D)
(6Vπ )
1/3
(1)
where D is the droplet diameter and V is the volume of the detaching dispersed phase. The volume of the detaching dispersed phase is estimated from the volume of the dispersed phase on the terrace using the detachment length (A). In this model, the portion of the dispersed phase within a length of A from the terrace end detaches and forms a discrete droplet. In other words, A corrects for the portion of the liquid volume that remains on the terrace during detachment. In the previous study, A was expressed as a function of H and was independent of LT.22 The relationship between A and H was determined by regression analysis and is written as follows22
A ) 6.88HC
(2)
where HC (µm) is the corrected MC depth, which is described in terms of H (µm) as
HC ) H + 0.626
(3)
The constant term corrects for the space between the
Figure 2. Possible detachment processes for various MC shapes.
MC plate and the glass plate, even though this space fluctuated during different runs of the experiment. In the previous study, the volumes of the detaching dispersed-phase droplets were calculated under the assumption that the dispersed-phase droplets had circular-disk shapes on the terrace. The shapes of the dispersed-phase droplets on the terrace for MCs with partition walls are more complicated because five parameters of the MC shape are considered. The detachment behaviors differ for different ratios of these parameters. In this study, the detachment processes are grouped into two classes according to the relationship between WT and LT, these two classes can be further divided into a total of five cases. The model proposed in the previous study is expanded into these five cases under the simple assumption that the portion of the dispersed phase within a length of A from the terrace end detaches and forms a discrete droplet. For derivation of the droplet diameter modeling equation, it was assumed that eqs 2 and 3 are applicable for the other geometries, which is indicated by the experimental observation of droplet formation behavior (data not shown). To derive the droplet diameter prediction equations, we classified the droplet detachment process into five cases according to the relationship between the MC geometry and length of A. Figure 2 illustrates the possible detachment processes for various MC shapes. First, we divided the droplet diameter prediction model
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into two groups according to the relationship between WT and LT. If WT is larger than LT, the dispersed phase passing through the channel can inflate on the terrace with a circular disklike shape (A-1 and A-2 in Figure 2). If WT is smaller than LT, the dispersed phase passing through the channel inflates on the terrace with a complicated shape consisting of a rectangle and a halfcircular disk (B-1, B-2, and B-3 in Figure 2). The detachment processes of the first group (WT > LT) are further divided into two cases according to the relationship between A and LT. If A is smaller than LT, the dispersed phase within a length of A from the terrace end detaches and forms a droplet (A-1). The volume of the detaching dispersed phase (VA-1), represented by the dark color in Figure 2, is calculated according to the following equation, assuming that the dispersed phase has a circular-disk shape on the terrace
[
LT2φ1 LT(LT - 2A) VA-1 ) HC sin φ1 4 4
]
LT - 2A cos φ1 ) LT
{
(
)
2 LT - 13.76H - 8.61 6(H + 0.626) LT cos-1 π 4 LT
LT(LT - 13.76H - 8.61) 4
[ (
sin cos-1
)]})
(
)]
R1 - LT 2
(7)
where R1 is the radius of the circle of the dispersed phase on the terrace and φ2 is the angle depicted in Figure 2. R1 is defined as 2
R1 )
2
4LT + WC 8LT
cos φ2 )
WC2 + 4LT2
)
WC2 - 4LT2
WC2 + 4LT2
(
-
)]})
WC2 - 4LT2 16LT
1/3
(10)
The second class (WT < LT) is further divided into three cases according to the relationship between A, LT, and WT (B-1, B-2, and B-3 in Figure 2). If A is smaller than WT/2, the detachment process is similar to the case of A-1. The dispersed phase within a length of A from the terrace end detaches and forms a droplet (B-1). The volume of the detaching dispersed phase (VB-1) is calculated as
VB-1 ) HC
[
]
WT2φ3 WT(WT - 2A) sin φ3 4 4
(11)
where φ3 is the angle defined in the following equation, as depicted in Figure 2
cos φ3 )
WT - 2A WT
(12)
Upon substitution of eqs 2, 3, 11, and 12 into eq 1, the droplet diameter (DB-1) can be calculated as
DB-1 )
(
{
[ (
WT2 cos-1
)]
WT - 13.76H - 8.61 WT 4
WT(WT - 13.76H - 8.61) 4
})
)]
[ (
sin cos-1
WT - 13.76H - 8.61 WT
-
1/3
(13)
If A is larger than WT/2 and smaller than LT, the dispersed phase on the terrace, consisting of a half-circle plus a rectangular shape, detaches and forms the droplet (B-2). The volume of the detaching dispersed phase (VB-2) is calculated as
[ (
VB-2 ) HC WT A -
)
WT π + WT2 2 8
]
(14)
Upon substitution of eqs 2, 3, and 14 into eq 1, the droplet diameter (DB-2) can be calculated as
(8) DB-2 )
and φ2 is defined as
WC2 - 4LT2
) ( cos-1
WC
(6)
If A is larger than LT, the dispersed phase on the terrace plus the part of the dispersed phase in the channel detaches and forms droplets (A-2). The volume of the detaching dispersed phase (VA-2) is calculated according to the following equation, assuming that the dispersed phase has a circular shape on the terrace and a rectangular shape in the channel
[
(
2
4LT2 + WC2 8LT
6(H + 0.626) π
1/3
LT - 13.76H - 8.61 LT
VA-2 ) HC WC(A - LT) + R12φ2 - WC
{[
6(H + 0.626) WC(6.88H - LT + 4.31) + π
(5)
Substituting eqs 2-5 into eq 1, one can write an expression for the droplet diameter (DA-1) as
(
(
(4)
where φ1 is the angle defined in the following equation, as shown in Figure 2
DA-1 )
DA-2 )
{
6(H + 0.626) WT(6.88H + 4.31) + π π 1 WT2 8 2
[
( )]}
1/3
(15)
(9)
Upon substitution of eqs 2, 3, and 7-9 into eq 1, the droplet diameter (DA-2) is expressed as
If A is larger than LT, the dispersed phase on the terrace and the part of the dispersed phase in the channel detach and form droplets (B-3). The volume of the detaching dispersed phase (VB-3) is calculated
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according to the following equation, assuming that the dispersed phase has a circular shape plus a rectangular shape on the terrace and a rectangular shape in the channel
[
]
π VB-3 ) HC WC(A - LT) + WT(LT - WT) + WT2 (16) 4 Upon substitution of eqs 2, 3, and 16 into eq 1, the droplet diameter (DB-3) is calculated as
DB-3 )
{
6(H + 0.626) WC(6.88H - LT + 4.31) + π 1/3 π (17) WT(LT - WT) + WT2 4
[
]}
As the previous study demonstrated, the volume flowing from the channel during the detachment process is negligible because the detachment process is much faster than the inflation process (see Figure 1). The previous study also demonstrated that the droplet diameter does not change below the critical flow velocity of the dispersed phase,25 at which the dispersed phase begins to flow out continuously, but that larger droplets form above the critical velocity. Therefore, eqs 6, 10, 13, 15, and 17 are valid below the critical velocity. Materials and Methods Materials. Triolein (purity >90%) obtained from Nippon Lever B.V. (Tokyo, Japan) was used as the dispersed oil phase. MilliQ water was used as the continuous water phase. Sodium dodecyl sulfate (SDS) purchased from Wako Pure Chemical Industries (Osaka, Japan) was used as the surfactant for emulsification, dissolved in the continuous phase at 1% concentration. Measurement and Analytical Method. The droplet diameter is determined from pictures obtained with the microscope video system described below. Winroof (Mitani Corporation, Fukui, Japan) software was used to analyze the captured pictures. MC Emulsification. The laboratory-scale apparatus for MC emulsification was described previously.13 Figure 3 depicts the experimental setup and the MC plate used in this study. The emulsification behavior was observed through the glass plate using a microscope video system. We observed the droplet formation from a channel at the center of the terrace line and measured the droplet diameter in the well near this channel. Figure 1 schematically illustrates the silicon MC plate, which was fabricated by photolithography and orientationdependent etching.26 The present study used MC plates with different LT, WT, and H values to confirm the adequacy of the models. The MC plates listed in Table 1 have MCs with different LTs and similar WTs, although they have slightly different WT values. The MC plates listed in Table 2 have MCs with different WTs and similar LTs, although they have slightly different LT values. The small differences in WT in Table 1 and LT in Table 2 are inevitable because of the fabrication process used in this study. If the MC depth is different, the MC geometry will change because of the orientationdependent etching process. For the MC plates listed in Table 1, samples 2-L5 and 4-L7 have smaller WT values than the other MC plates for the same reason. Tables 1 and 2 also indicate the values of A calculated according to eqs 2 and 3 and the possible detachment process in Figure 2 for each MC plate.
Figure 3. Experimental setup and schematic flows in the MC module.
The MC module was initially filled with the continuous phase. The dispersed phase was pressurized and fed into the module by lifting the liquid chamber containing it. The dispersed phase supplied from the liquid chamber entered the space between the silicon MC plate and the glass plate, and droplets were formed from the MC. The prepared emulsion could be recovered by a continuous phase flow. Results and Discussion We investigated the adequacy of the theoretical models by correlation with the experimental data. MC emulsification was performed, and the values of D were measured using MCs with different geometries. The geometry of an MC plate is defined by three parameters: LT, WT, and H. To understand the effect of each geometrical parameter, we used MC plates with different LTs and similar WTs (Table 1), and MC plates with different WTs and similar LTs (Table 2). Triolein was used as the dispersed phase, and 1% SDS aqueous solution was used as the continuous phase. The applied pressures were slightly higher than the breakthrough pressures at which droplet production began. Figures 4 and 5 show the experimentally obtained and calculated values of D for MCs with different geometries. Figure 4 shows the results for MCs with different LTs and similar WTs. The applicable equations (possible detachment process) depend on the relationships among LT, A, and WT, presented in Table 1 and depicted in Figure 2. Equations 6, 10, and 13 were used to calculate D for four types of MCs with a depth of 2 µm (H ) 2 µm). Equations 6, 10, and 15 were used to calculate D for four types of MCs with a depth of 4 µm (H ) 4 µm). Equations 10, 15, and 17 were used to calculate D for four types of MCs with a depth of 7 µm (H ) 7 µm). Figure 4 demonstrates that the experimentally obtained values of D were comparable to the values calculated
Ind. Eng. Chem. Res., Vol. 43, No. 26, 2004 8237 Table 1. Dimensions of the MC Plates with Different Terrace Lengths and Similar Terrace Widths structure of channel
structure of terrace
MC plate
MC depth H (µm)
length LC (µm)
width WC (µm)
length LT (µm)
width WT (µm)
detachment length A (µm)
detachment process
applied equation
2-L5 2-L24 2-L54 2-L96 4-L7 4-L34 4-L57 4-L100 7-L15 7-L39 7-L68 7-L107 7-L195
2 2 2 2 4 4 4 4 7 7 7 7 7
80.7 82.4 83.7 82.8 73.2 71.2 72.3 72.4 58.0 53.1 57.5 58.0 53.5
14.4 14.7 14.7 14.7 12.7 14.4 13.9 13.5 11.3 11.6 11.7 11.8 10.8
5.3 23.9 53.8 96.0 7.4 33.7 56.9 99.9 14.7 39.3 67.6 106.5 195.1
18.8 45.9 45.8 45.9 19.6 44.4 43.9 44.5 33.0 38.0 39.5 40.3 37.9
18.1 18.1 18.1 18.1 31.8 31.8 31.8 31.8 52.4 52.4 52.4 52.4 52.4
A-2 A-1 B-1 B-1 A-2 A-1 B-2 B-2 A-2 B-3 B-2 B-2 B-2
eq 10 eq 6 eq 13 eq 13 eq 10 eq 6 eq 15 eq 15 eq 10 eq 17 eq 15 eq 15 eq 15
Table 2. Dimensions of the MC Plates with Different Terrace Widths and Similar Terrace Lengths structure of channel
structure of terrace
MC plate
MC depth H (µm)
length LC (µm)
width WC (µm)
length LT (µm)
width WT (µm)
detachment length A (µm)
detachment process
applied equation
4-W32 4-W42 4-W58 4-W82 7-W27 7-W37 7-W56 7-W76
4 4 4 4 7 7 7 7
48.1 47.2 47.6 47.6 30.1 30.3 29.3 30.0
23.4 22.7 23.0 23.0 21.2 20.8 19.6 19.3
69.8 71.3 70.6 70.6 82.2 79.9 82.8 81.3
32.4 41.8 58.2 82.4 27.1 37.3 56.3 76.3
31.8 31.8 31.8 31.8 52.4 52.4 52.4 52.4
B-2 B-2 B-2 A-1 B-2 B-2 B-2 A-1
eq 15 eq 15 eq 15 eq 6 eq 15 eq 15 eq 15 eq 6
according to eqs 6, 10, 13, 15, and 17. Figure 5 shows the results for MCs with different WTs and similar LTs. Equations 6 and 15 were used to calculate D for different MC geometries as shown in Table 2. Figures 4 and 5 indicate that the calculated values had good agreement with the experimental results and the droplet diameters can be successfully predicted using eqs 6, 10, 13, 15, and 17. The mean percentage deviation of the calculated values from the experimental results was 5.7%, and all of the calculated values were within an 11% difference from the experimental results. These results confirm that the prediction model proposed in the previous study can be expanded to accommodate MCs with partition walls, with the simple assumption that the portion of the dispersed phase within a length of A from the terrace end detach and form discrete droplets. The calculated values exhibited a good correlation with the experimental data even for different values of H. The prediction model should be valid below the critical velocity and under conditions
that the viscosity ratio of the two phases is similar to that used in the present study. The good correlation indicates adequacy of the assumption that eqs 2 and 3 are applicable for the complicated MC plate geometries investigated and that A is independent of terrace width (WT) and length (LT). Even though the classification of possible cases and the mathematical equations are complicated, the ability to predict droplet diameters without experimentation is important because the fabrication of MCs is expensive and time-consuming. Computational fluid dynamics (CFD) should be another approach for predicting droplet diameters. However, complicated MC geometries and intricately shaped interfaces would require massive computations. Our empirical model enables us to predict the droplet diameter in a simple way. This model will be useful in designing microchannels and developing applications using this emulsification method. In this study, we investigated only the effects of MC geometry on the prepared droplet diameter. The previous study revealed that droplet diameter was also
Figure 4. Experimentally measured and calculated droplet diameters (D) for MCs with different LTs and similar WTs. The MC depth (H) is 7 µm (O and b), 4 µm (0 and 9), and 2 µm (4 and 2). Open symbols represent experimental results; solid symbols represent calculated values. Equations used to calculate D for each MC are listed in Table 1.
Figure 5. Experimentally measured and calculated droplet diameters (D) for MCs with different WTs and similar LTs. The MC depth (H) is 7 µm (O and b) and 4 µm (0 and 9). Open symbols represent experimental results; solid symbols represent calculated values. Equations used to calculate the D for each MC are listed in Table 2.
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affected by the viscosities of the dispersed and continuous phases.17 Such a dependency is reasonable because droplet formation is a hydrodynamic phenomenon. Therefore, the prediction model presented in this study is valid only for cases in which the viscosity ratio of the two phases is similar to that used in the present study. To explain the effect of viscosity, the equations proposed in this study should be expanded to a form that includes a viscosity term. This study assumed that A is a function of H based on the previous research. For predicting the droplets for phases with different viscosities, A should be expressed as a function of H and the viscosities of both phases. We will address this problem in future work. Conclusion Droplet diameter prediction models for MCs with partition walls were proposed under the simple assumption that the portions of the dispersed phase within a length of A from the terrace end detach and form discrete droplets. The possible detachment processes were classified into five cases. The volumes of the detaching dispersed-phase droplets were calculated by estimating the volume of dispersed phase within a length of A from the terrace end. The calculated values were compared with the experimentally obtained droplet diameters using MCs with different LT, WT, and H. The mean percentage deviation of the calculated values from experimental results was 5.7%, and all of the data had deviations within 11%. Acknowledgment This work was supported by the Nanotechnology Project, Ministry of Agriculture, Forestry and Fisheries, Japan. Nomenclature A ) detachment length (µm) D ) droplet diameter (µm) H ) MC depth (µm) L ) length (µm) V ) volume of the detaching dispersed phase (µm3) W ) width (µm) Greek Letter φ ) angle defined in the text Subscripts A-1, A-2, B-1, B-2, and B-3 ) symbols for classification of detachment processes C ) channel T ) terrace
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Received for review June 16, 2004 Revised manuscript received September 18, 2004 Accepted October 4, 2004 IE0494770
