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Prediction of Droplet Diameter for Microchannel Emulsification Shinji Sugiura,†,‡ Mitsutoshi Nakajima,*,† and Minoru Seki‡ National Food Research Institute, Kannondai 2-1-12, Tsukuba, Ibaraki 305-8642, Japan, and Department of Chemistry and Biotechnology, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-8656, Japan Received January 28, 2002. In Final Form: March 14, 2002 Recently, we proposed microchannel (MC) emulsification, a novel method for making monodisperse emulsions using a microfabricated channel array. The previous study demonstrated that droplet size is affected by MC geometry. This study proposes a model for the prediction of droplet diameter based on the droplet formation mechanism and on experimental observation. The MC structure used in this study is composed of a narrow channel and a terrace. The terrace is a microfabricated slitlike shape, on which the dispersed phase inflates to a disklike shape. The MC geometry is defined in terms of two variables, terrace length (L) and MC depth (H). First, the relationship between droplet diameter and MC geometry was investigated experimentally. Experimental observation suggests that the dispersed phase, which is within the detachment length (A) from the terrace end, detaches and forms a droplet. The droplet volume was estimated from the volume of the dispersed phase that detaches from the terrace during this process. This volume was calculated using a detachment length parameter, A, assuming the dispersed phase on the terrace to be disk-shaped. Experimental observation and regression analysis indicate that A is independent of L. The prediction curves were fitted by regression analysis as functions of L, using fitting parameters A for each H. The values of A obtained by regression analysis were linearly correlated with H. The prediction curve, which is expressed by two variables L and H, was obtained. The prediction model was correlated with the experimental data. The mean percentage deviation of the calculated values from experimental results was 5.4%. The prediction curve was corrected using the corrected MC depth. The final form of the corrected prediction curve shows a mean percentage deviation from experimental results of 4.6%.
1. Introduction Emulsions have been utilized in various industries, including the food, cosmetics, and pharmaceutical industries. Many of the most important properties of emulsionbased products (e.g., shelf life, appearance, texture, and flavor) are determined by the size of the droplets they contain.1 Stability, rheology, appearance, chemical reactivity, and physical properties depend on their size and size distribution.1-3 The resistance of emulsions to creaming and Ostwald ripening is influenced by their size and size distribution.2,3 Colloidal interactions between emulsion droplets are affected by the droplet size. Consequently, it is important to reliably predict and control the size and size distribution of emulsions. Monodisperse emulsions are useful for fundamental studies because the interpretation of experimental results is much simpler than that for polydisperse emulsions.2 They can also serve as useful systems for measuring important properties of emulsions. For example, the stability of droplets can be monitored very simply, since changes in the droplet size of monodisperse droplets are easily studied. Monodisperse emulsions can greatly reduce Ostwald ripening by reducing the effective Laplace pressure difference because the droplets are identical in * Corresponding author. Phone: +81-298-38-7997. Fax: +81298-38-8122. E-mail:
[email protected]. † National Food Research Institute. ‡ The University of Tokyo. (1) Dickinson, E. An Introduction to Food Colloids; Oxford University Press: Oxford, 1992. (2) McClements, D. J. Food Emulsions: Principles, Practice, and Techniques; CRC Press: Boca Raton, FL, 1999; Chapter 1. (3) Mason, T. G.; Krall, A. H.; Gang, H.; Bibette, J.; Weitz, D. A. In Encyclopedia of emulsion technology; Becher, P., Ed.; Marcel Dekker: New York, 1996; Vol. 4, Chapter 6.
size. The effect of change in the droplet size on physical properties of emulsions can be studied directly. They are also applicable to valuable materials, such as microcapsules for drug delivery vehicles4-6 and monodisperse microparticles.7 There are several research methods for producing monodisperse emulsions. Membrane emulsification, in which the pressurized dispersed phase permeates a microporous membrane and forms emulsion droplets, enables the production of monodisperse emulsions with a coefficient of variation greater than 10%.8-10 The emulsion droplet size is controlled by the membrane pore size. This technique can be used to produce emulsions without high mechanical stress.11 Mason and Bibette proposed that shear rupturing in couette flow is applicable to the production of monodisperse emulsions.12,13 Recently, we proposed a novel method for making monodisperse emulsion droplets using a microfabricated channel array.14-17 This emulsification technique is called (4) O’Donnell, P. B.; McGinity, J. W. Adv. Drug Delivery Rev. 1997, 28, 25-42. (5) Yadav, S. K.; Khilar, K. C.; Suresh, A. K. J. Membr. Sci. 1997, 125, 213-218. (6) Poncelet, D.; Lencki, R.; Beaulieu, C.; Halle, J. P.; Neufeld, R. J.; Fournier, A. Appl. Microbiol. Biotechnol. 1992, 38, 39-45. (7) Omi, S.; Katami, K.; Taguchi, T.; Kaneko, K.; Iso, M. J. Appl. Polym. Sci. 1995, 57, 1013-1024. (8) Nakashima, T.; Shimizu, M.; Kukizaki, M. Key Eng. Mater. 1991, 61&62, 513-516. (9) Joscelyne, S. M.; Tra¨gårdh, G. J. Membr. Sci. 2000, 169, 107117. (10) Abrahamse, A. J.; van der Padt, A.; Boom, R. M.; de Heij, W. B. C. AIChE J. 2001, 47, 1285-1291. (11) Schro¨der, V.; Schubert, H. Colloids Surf., A 1999, 152, 103109. (12) Mason, T. G.; Bibette, J. Langmuir 1997, 13, 4600-4613. (13) Mabille, C.; Schmitt, V.; Gorria, Ph.; Calderon, F. L.; Faye, V.; Deminie`re, B.; Bibette, J. Langmuir 2000, 16, 422-429.
10.1021/la0255830 CCC: $22.00 © 2002 American Chemical Society Published on Web 04/18/2002
Droplet Diameter for Microchannel Emulsification
microchannel (MC) emulsification. Emulsions with droplet sizes ranging from several micrometers to 100 µm, with a coefficient of variation of approximately 5%, have been successfully prepared by applying this technique.15,18,19 This emulsification technique exploits interfacial tension as a driving force for droplet formation, this being the dominant force on a micrometer scale.15 During droplet formation, the distorted dispersed phase is spontaneously transformed into spherical droplets by interfacial tension. The energy input for MC emulsification is very low compared to the conventional emulsification technique, because droplet formation from MC is based on spontaneous transformation.15 This technique is promising, not only for preparing emulsions but also for various systems. We have applied it to the preparation of several types of oil-in-water emulsions,20 water-in-oil emulsions,21 lipid microparticles,22 polymer microparticles,23 and microcapsules. Monodisperse emulsions prepared by MC emulsification have also been used for fundamental studies.24 The previous study demonstrated that the droplet size is affected by MC geometry.22,25 However, there is no quantitative study of the relationship between MC geometry and droplet diameter. An analysis of the droplet formation process should involve the solution of the Navier-Stokes equation, along with the proper boundary conditions. However, mathematical complexities of the equation, considering the intricately shaped interface and wetting phenomena, preclude this approach. As an alternative, an overall mass balance during the droplet formation is used in this study. First, the relationship between MC geometry and droplet diameter was investigated experimentally. On the basis of the observed experimental results and the reported droplet formation mechanism,15 a model for predicting the droplet diameter was proposed. The model was correlated with the experimental data. 2. The Droplet Formation Process in MC Emulsification The schematic of MC emulsification and the droplet formation process is depicted in Figure 1. MC is constituted of a channel and a terrace. Monodisperse droplets were formed from a precisely fabricated MC by pressurizing the dispersed phase into the continuous phase through the MC. The droplet formation mechanism was described in detail in a previous article.15 In this study, the droplet formation process is divided into two parts to simplify the prediction model. The first process is the inflation process, in which the dispersed phase, having passed through the (14) Kawakatsu, T.; Kikuchi, Y.; Nakajima, M. J. Am. Oil Chem. Soc. 1997, 74, 317-321. (15) Sugiura, S.; Nakajima, M.; Iwamoto, S.; Seki, M. Langmuir 2001, 17, 5562-5566. (16) Kawakatsu, T.; Komori, H.; Nakajima, M.; Kikuchi, Y.; Yonemoto, T. J. Chem. Eng. Jpn. 1999, 32, 241-244. (17) Kawakatsu, T.; Tra¨gårdh, G.; Kikuchi, Y.; Nakajima, M.; Komori, H.; Yonemoto, T. J. Surfactants Deterg. 2000, 3, 295-302. (18) Sugiura, S.; Nakajima, M.; Seki, M. J. Am. Oil Chem. Soc. 2002, in press. (19) Kobayashi, I.; Nakajima, M.; Nabetani, H.; Kikuchi, Y.; Shohno, A.; Satoh, K. J. Am. Oil Chem. Soc. 2001, 78, 797-802. (20) Tong, J.; Nakajima, M.; Nabetani, H.; Kikuchi, Y. J. Surfactants Deterg. 2000, 3, 285-293. (21) Kawakatsu, T.; Tra¨gårdh, G.; Tra¨gårdh, C.; Nakajima, M.; Oda, N.; Yonemoto, T. Colloids Surf., A 2001, 179, 29-37. (22) Sugiura, S.; Nakajima, M.; Tong, J.; Nabetani, H.; Seki, M. J. Colloid Interface Sci. 2000, 227, 95-103. (23) Sugiura, S.; Nakajima, M.; Itou, H.; Seki, M. Macromol. Rapid Commun. 2001, 22, 773-778. (24) Liu, X. Q.; Nakajima, M.; Nabetani, H.; Xu, Q. Y.; Ichikawa, S.; Sano, Y. J. Colloid Interface Sci. 2001, 233, 23-30. (25) Kawakatsu, T.; Tra¨gårdh, G.; Kikuchi, Y.; Nakajima, M.; Komori, H.; Yonemoto, T. J. Surfactants Deterg. 2000, 3, 295-302.
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Figure 1. Schematic of MC emulsification and droplet formation process.
channel, inflates on the terrace in a disklike shape. This distorted disklike shape is the essential point for spontaneous transformation in the second step, because the disklike shape has a larger interface area than a spherical shape, resulting in instability from the viewpoint of the interface free energy. The second process is the detachment process. When the dispersed phase reaches the end of the terrace, it flows into the well, detaches from the terrace, and spontaneously transforms into spherical droplets. The droplet diameter is determined by the volume of the dispersed phase that flows into the well during the detachment process. 3. Materials and Methods 3.1. 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) was purchased from Wako Pure Chemical Ind. (Osaka, Japan) and used as the surfactant for emulsification, dissolved in the continuous phase at 1% concentration. 3.2. Measurement and Analytical Method. The droplet diameters and MC geometries were determined from pictures obtained with the microscope video system described below. Winroof (Mitani Corp., Fukui, Japan) software was used to analyze the captured pictures. The number-average diameters and coefficients of variation of the prepared emulsions were determined from pictures taken with the microscope video system described above. The coefficient of variation is defined as follows:
CV ) (σ/Da) × 100
(1)
where CV is the coefficient of variation [%], Da is the numberaverage diameter [µm], and σ is the standard deviation of the diameter [µm]. 3.3. MC Emulsification. The laboratory scale apparatus for MC emulsification has been described previously.16 Figure 2 shows the experimental setup with an MC plate used in this study. The silicon MC plate was tightly covered with a flat glass plate and O-ring. The emulsification behavior was observed through the glass plate using a microscope video system. In this study, a high-speed camera (FASTCAM ultima 1024; Photoron Ltd., Tokyo, Japan), which can capture 16 000 frames/s, was
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Figure 4. Microscope photograph of the prepared emulsion using MC7-48. The applied pressure was 1.1 kPa. The average diameter and coefficient of variation of the prepared emulsion were 31.7 µm and 1.4%. Table 1. Dimensions of the MC Plates Used in This Study MC plate
Figure 2. Experimental setup and schematic flows in MC module.
MC2-203 MC2-101 MC2-56 MC2-30 MC4-213 MC4-110 MC4-60 MC4-35 MC7-219 MC7-120 MC7-74 MC7-48 MC11-227 MC11-125 MC11-80 MC11-55 MC16-240 MC16-138 MC16-98
MC depth terrace length channel channel H [µm] L [µm] length [µm] width [µm] 2 2 2 2 4 4 4 4 7 7 7 7 11 11 11 11 16 16 16
203 101 56.3 30.6 213 110 60.0 34.7 219 120 73.9 48.1 227 125 79.5 54.7 240 138 98.0
118 138 119 112 168 125 109 100 155 107 91.2 81.4 141 93.2 74.8 66.9 115 66.6 51.2
38.5 38.5 38.3 37.9 36.1 35.9 36.3 36.8 32.5 32.4 32.7 32.8 32.7 32.2 32.5 32.7 28.7 29.1 28.7
4. Results and Discussion
Figure 3. Schematics of silicon MC plate. (A) Schematic of MC plate. (B) Enlargement of MC. (C) Microscope photograph of MC. used to observe the emulsification behavior. Figure 3 shows the silicon MC plate used in this study. The silicon MC plate was fabricated using a process of photolithography and orientationdependent etching.29 The MC structure consists of the narrow channel part and the terrace part. Over the terrace end, there is a deep etched well. The MC geometry is expressed in terms of two variables; MC depth (H) and terrace length (L). The value of L is defined as the length from the channel exit to the end of the terrace, as shown in Figure 3. In the present study, MC plates with different H and L were used for preparing emulsions with different sizes. The dimensions of the MC plates used in this study are shown in Table 1. MC plates were identified by their combinations of H and L values. The MC module was initially filled with the continuous phase. The dispersed phase was pressurized into the module by lifting the liquid chamber filled with the dispersed phase. 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 can be recovered by a continuous phase flow.
Monodispersity of the prepared emulsions was investigated. MC emulsification was carried out using MC748. Triolein was used as the dispersed phase, and 1% SDS aqueous solution was used as the continuous phase. Figure 4 shows microscope photographs of the resulting emulsion. The applied pressure was 1.1 kPa. The diameters of 100 droplets were measured. The emulsion had an average diameter of 31.7 µm and a coefficient of variation of 1.4%. The emulsions prepared using other MCs had similar monodispersity; their coefficients of variation were less than 3% (data not shown). Next, the relationships of droplet diameter (D) to MC depth (H) and terrace length (L) were investigated. MC emulsification was carried out, and the values of D were measured using MCs with different H and L as shown in Table 1. 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 begins. The measured values of D are shown in Figure 5. The values of D increased with H and L. For the quantitative analysis, detailed observations of the droplet detachment process were made using MCs of different H and L values. MC emulsification was carried out using MC7-48, MC7-120, MC7-219, MC2-101, and MC11-125. 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. Figure 6 shows the detachment process for MCs of different H and L values. MC7-48, MC7-120, and MC7-219 have MCs with the same H and different L. Therefore, comparing parts A, B, and C of Figure 6, one can see the effect of L on detachment
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Figure 5. Relationship between droplet diameter (D) and terrace length (L). The MC depth (H) is 16 µm (O), 11 µm (0), 7 µm (4), 4 µm (b), and 2 µm (9). The lines are fitting curves calculated by eq 5.
Figure 7. Model for predicting droplet diameter.
Figure 6. Microscope photographs of the droplet detachment process using MCs with different MC depth (H) and terrace length (L).
behavior. In the case of short L, most of the dispersed phase on the terrace flowed into the well and detached. In the case of long L, only the dispersed phase near the terrace end flowed into the well and detached; the dispersed phases near the channel exit remained on the terrace after detachment. The droplets produced were larger than those for shorter L, as shown in Figure 5. Close observation suggests that the dispersed phase, which
is within a certain length (detachment length, A) from the terrace end, detached and formed a droplet. Figure 6A,B,C indicates that the value of A seems to be independent of L. MC7-120, MC2-101, and MC11-125 have different H and similar L. Therefore, comparing parts B, D, and E of Figure 6, one can see the effect of H on detachment behavior. In the case of large H, the volume of the detaching dispersed phase is larger than those of smaller H. This indicates that values of A increased with H. The droplet volume is determined by the volume of the dispersed phase flowing into the well in the detachment process. On the basis of experimental observation, this volume was estimated from the volume of the dispersed phase inflated on the terrace using a parameter of detachment length, A [µm]. Figure 7 shows the model for estimating the volume of the detaching dispersed phase, that is, the volume of the droplet. It was assumed that the dispersed phase, which is within the length of A from the terrace end, detaches and forms the discrete droplet. The detaching dispersed phase and detached droplets are depicted in dark color in Figure 7. The microscope photographs shown in Figure 6 indicate that A depends on H and is independent of L. This mechanism seems similar to droplet formation from a circular nozzle, which is a classical problem.26-28 The parameter A is similar to the Harkins-Brown correlation factor, which is used for estimating the droplet volume formed from a circular nozzle. The parameter A corrects for the fraction of the liquid volume that remains on the terrace during MC emulsification. The Harkins-Brown correlation factor corrects for the fraction of the liquid (26) Scheele, G. F.; Meister, B. J. AIChE J. 1968, 14, 9-19. (27) de Chazal, L. E. M.; Ryan, J. T. AIChE J. 1971, 17, 1226-1229. (28) Lando, J. L.; Oakley, H. T. J. Colloid Interface Sci. 1967, 25, 526-530. (29) Kikuchi, Y.; Sato, K.; Ohki, H.; Kaneko, T. Microvasc. Res. 1992, 44, 226-240.
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shows the relationship between the obtained A and H. A linear relationship was found between A [µm] and H [µm], which is expressed by the following equation:
A ) 6.92H + 7.62
(6)
The correlation coefficient was 0.9999. This good correlation indicates adequacy of the assumption that A is independent of L. Substitution of eq 6 into eq 5 yields
Figure 8. Relationship between detachment length (A) and MC depth (H).
volume that remains attached to the nozzle after drop break-off, during droplet formation from a circular nozzle. The volume of the detaching dispersed phase is calculated by the following equations, assuming the dispersed phase on the terrace to be disk-shaped:
V)H
[
]
L2φ L(L - 2A) sin φ 4 4
(2)
where V is the volume of the detaching dispersed phase; φ is the angle defined by the following equation as shown in Figure 7:
cos φ )
L - 2A L
(3)
Equations 2 and 3 are applicable in the cases of both parts A and B of Figure 7. The diameter of the droplet formed is calculated as in the following equation:
D)
(6Vπ )
1/3
(4)
Substituting eqs 2 and 3, one can write the expression of the droplet diameter as
〈 {
6H L2 L - 2A cos-1 D) π 4 L L(L - 2A) L - 2A sin cos-1 4 L
(
)
[
(
)]}〉
1/3
(5)
The volume of flow from the channel during the detachment process can be negligible, because the detachment process is much faster than the inflation process as shown in Figure 1. The previous study also demonstrated that the droplet diameter does not change below the critical pressure,18,22 at which the dispersed phase begins to flow out continuously, but that larger droplets form above the critical pressure. Therefore, eq 5 is valid below the critical pressure. For simplicity of analysis, it was assumed that A depends on H and is independent of L. This assumption was supported by experimental observation (Figure 6). The values of A for each H were determined by regression analysis. The prediction curves calculated from eq 5 were fitted by regression analysis as functions of L, using fitting parameters A for each H. If the value of (L - 2A)/L in eq 3 was less than -1, π was used as the value of φ. The curves obtained from eq 5 are shown in Figure 5 and agree well with the experimental results. The values of A for each H were estimated by regression analysis. Figure 8
〈{
L2 L - 13.85H - 15.23 cos-1 4 L 6H L(L - 13.85H - 15.23) D) × π 4 L - 13.85H - 15.23 sin cos-1 L
(
[
)
)]
(
}〉
1/3
(7)
The droplet diameters (D [µm]) formed from MCs with various L [µm] and H [µm] can be calculated from this equation. The values calculated with eq 7 were compared with the experimental results. The mean percentage deviation of the value calculated with eq 7 from experimental results was 5.4%. This good correlation supports the prediction model and assumption that A is independent of L. The deviation from the calculated result was large for small H. The line of eq 6 has the intercept with the x-axis at H ) -1.10. These results indicate that there is a space between the MC plate and the glass plate, even though the MC plate and the glass plate are tightly fixed with an O-ring. In fact, it was experimentally observed that dust, smaller than 1 µm, which accidentally contaminated the experimental setup, permeated through the space between the MC plate and the glass plate. Therefore, for further good correlation, H should be corrected by considering the space between the MC plate and glass. The corrected H was calculated by repetitive regression analysis. The corrected value of H after n corrections (Hn) is estimated as in the following equation: n-1
Hn ) Hn-1 - ∆Hn-1 ) H0 -
∆Hi ∑ i)0
(8)
where ∆Hn-1 is the correction length for the nth correction. Equation 8 can be rewritten as
Hn ) H0 + Cn
(9)
where Cn is the corrected depth after n corrections and is defined as n-1
Cn ) -
∆Hi ∑ i)0
(10)
The values of H shown in Table 1 were used as the values of H0. The values of ∆Hn-1 were determined as the intercept with the x-axis, plotting H-A as in Figure 8; that is, ∆Hn-1 ) Hn-1 (A)0) plotted as H-A. For example, ∆H0 is calculated as -1.1 from eq 6. The obtained ∆Hn-1 was substituted into eq 8, and Hn was determined. The corrected Hn was used for the regression analysis. The values of Hn were substituted into eq 5 instead of H. The values of fitting parameter An instead of A for each Hn were obtained by regression analysis as in Figure 5, and the relationship between An and Hn was obtained as a
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eq 14 will be valid for different surfactant systems. On the other hand, the previous study revealed that viscosities of the dispersed phase and the continuous phase affect the droplet diameter. This is reasonable because of the hydrodynamic nature of the droplet formation process. To address this, eq 14 should be expanded to a form that includes a viscosity term. In this study, A is estimated as a function of H. To predict the droplet diameter of liquids of different viscosities, A should be expressed as a function of H and the viscosities of both phases. We would like to challenge this problem in future work. Figure 9. Repetitive regression analysis for correcting MC depth.
linear relationship, as expressed in the following equation:
An ) anHn + bn
(11)
where an and bn are the constants obtained by regression analysis. The x-axis intercept gives the value of ∆Hn for the next correction. Figure 9 shows the change of Cn by repetitive correction. After correcting 10 times, Cn converged to 0.63. The relationship between A and H after correcting 10 times is written as
A10 ) 6.88H10 + 0.09
(12)
where A10 [µm] is the detachment length after 10 times correction, and H10 [µm] is the corrected MC depth after 10 times correction as expressed by following equation.
H10 ) H0 + 0.626
(13)
The constant term corrects for the space between the MC plate and the glass plate, even though it fluctuates during different runs of the experiment. Substitution of A10 and H10 expressed as eqs 12 and 13 into eq 5 instead of A and H yields
〈 {
2
L L - 13.76H - 8.61 cos-1 4 L 6(H + 0.626) L(L - 13.76H - 8.61) D) × π 4 L - 13.76H - 8.61 sin cos-1 L
(
[
)
(
)]
}〉
1/3
(14)
The droplet diameters (D [µm]) with various L [µm] and H [µm] can be calculated from this equation. The curves obtained from eq 14 did not exhibit the apparent difference in Figure 7 to the curves obtained from eq 7. The values calculated with eq 14 were compared with the experimental results. The mean percentage deviation of the value calculated with eq 14 from experimental results was 4.6%. The value predicted by eq 14 shows some modification compared to that from eq 7. This study investigated the relationship between droplet diameter and MC geometry. The previous studies demonstrated that the surfactant type and concentration have no significant effect on the droplet diameter.20 Therefore,
5. Conclusion The model for predicting droplet diameter was proposed, based on the droplet formation mechanism and on experimental observation. The droplet formation process during MC emulsification is divided into an inflation process and a detachment process. The droplet volume was estimated from the volume of the dispersed phase that detaches from the terrace during the detachment process. The volume was estimated using a parameter of detachment length, A, assuming the dispersed phase on the terrace to be disk-shaped. The experimental observation of the detachment process indicated that A depends on H and is independent of L. The model was correlated with the experimental data for different H and L. The prediction curves were fitted by regression analysis as functions of L, using fitting parameters A for each H. The values for A obtained by regression analysis were linearly correlated with H. The prediction curve for droplet diameter for MCs with various geometries was proposed and includes two variables, H and L. The mean percentage deviation of the calculated values from experimental results was 5.4%. The prediction curve was corrected using the corrected MC depth. The final form of the corrected prediction curve shows a slight modification from the original curve. The mean percentage deviation of the calculated value from experimental results was 4.6%. Acknowledgment. This work was supported by the Program for Promotion of Basic Research Activities for Innovative Biosciences (MS-Project). We thank Professor Shin-ichi Nakao, The University of Tokyo, for valuable comments on our research. Nomenclature A ) detachment length [µm] a, b ) constants defined in the text C ) corrected depth [µm] CV ) coefficient of variation [%] D ) droplet diameter [µm] H ) MC depth [µm] L ) terrace length [µm] V ) volume of the detaching dispersed phase [µm3] Greek Symbols φ ) angle defined in the text σ ) standard deviation on the diameter [µm] Subscripts a ) number-average n ) number of correction LA0255830
