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Morphological Change in Drop Structure with Time for Abnormal Polymer/Water/Surfactant Dispersions S. Sajjadi, M. Zerfa, and B. W. Brooks* Department of Chemical Engineering, Loughborough University, Loughborough, Leicestershire LE11 3TU, United Kingdom Received March 31, 2000. In Final Form: September 12, 2000 The morphological variation of complex drops of oil/water/oil (O/W/O) in unstable polymer/water/surfactant systems was investigated. A new, but simple, technique was developed to measure the internal phase ratio of multiple drops. The morphology change occurred because of inclusion of oil droplets into the dispersed water drops which contained water-soluble surfactant. The variation in morphology can lead to a delayed phase inversion if a substantial variation in the effective volume fraction of the dispersed phase occurs. The time sequence of the morphology change of complex drops is as follows: In the early stage of stirring, most water drops that are formed contain no internal oil droplets but some contain large internal oil droplets. As time proceeds, the relative size of the internal oil droplets to that of surrounding water drops is appreciably reduced, while the water drops become richer in the continuous phase. The progressive increase in the internal phase ratio of the dispersed phase will continuously increase the effective volume fraction of the dispersed phase and hence enlarge the water drop size. Eventually a balance might be reached between drop inclusion and escape, to give a steady-state, at which the drop morphology and size reach a constant value. If the rate of inclusion exceeds that of escape, so that a steady-state is not reached, a phase inversion will occur. The extent of variation in drop morphology with time was found to depend on the size of drops, which is highly influenced by the volume fraction of the dispersed phase, and the surfactant concentration in the system, which enables the water drops to entrain a larger volume of the continuous phase. The large drops are able to entrain a larger volume of the internal oil phase and, thus, contribute more to increasing the effective volume fraction of the dispersed phase and to inducing a phase inversion.
Introduction Dispersions, or emulsions, are usually regarded as mixtures of two immiscible fluids with one fluid dispersed as drops in a second fluid which forms a continuous phase. Such dispersions are normally unstable and break as soon as stirring is ceased. If an appropriate surfactant is present in the system, a stable emulsion can be formed. Bancroft1,2 was the first to address the stability of emulsions. According to Bancroft, the phase in which the surfactant is predominantly dissolved tends to be the continuous phase. Griffin3 developed an idea to deal with the solubility of surfactants. He suggested an empirical hydrophilic-lipophilic balance (HLB) which characterizes the tendency of emulsifiers to form W/O or O/W emulsions. He showed that surfactants with high HLB values tend to stabilize O/W emulsions, while those with low HLB values stabilize W/O emulsions. However, the natural tendency of surfactants to form corresponding emulsions can be counteracted by using different start-up procedures. There are two such procedures: first by immersing the impeller in the phase intended to be the continuous phase and second by starting stirring from the continuous phase and then adding the dispersed phase. Obviously, surfactants can be added to the dispersed phase in such emulsions. Salager used the term abnormal emulsion for such an emulsion.4,5 Brooks and Richmond defined it as an unstable emulsion.6 Obviously, this definition only * To whom correspondence should be addressed. (1) Bancroft, W. D. J. Phys. Chem. 1913, 17, 501. (2) Bancroft, W. D. J. Phys. Chem. 1915, 19, 275. (3) Griffin, W. C. J. Soc. Cosmet. Chem. 1954, 5, 249. (4) Salager, J. L.; Minanaperez, M.; Perezsanchez, M.; Ramirezgouveia, M.; Rojas, C. I. J. Dispersion Sci. Technol. 1983, 4, 313-329. (5) Salager, J. L. In Encyclopaedia of Emulsion Science; Becher, P., Ed.; Marcel Dekker: New York, 1988; Vol. 3, p 79.
refers to the external emulsion. The morphology of these emulsions is nonpreferred, and so they are very unstable in nature. Phase behavior maps developed for some oil/ water/surfactant systems clearly show that abnormal emulsions cannot exist in a wide range of conditions and that they can only be prepared for the low range of dispersed phase ratio. However, the boundaries of abnormal emulsions were found to be greatly influenced by the hydrodynamics of the system and the mixing conditions.6-8 The boundaries of abnormal emulsions can be crossed by catastrophic inversions, which are induced by changes in the emulsion’s phase ratio.7,8 For normal emulsions, phase inversion is hampered by emulsifier molecules at the water-oil interface, which makes the coalescence of the drops very difficult. As a result, phase inversion can be delayed up to a very high dispersed phase ratio. A dispersed phase ratio as large as 96.0% has been reported previously for the polyisobutylene (PIB)/water/polyoxyethylene nonylphenyl ether (NPE) system at the conditions similar to those used here.8 For abnormal emulsions, the surfactant molecules at the interface do not contribute appreciably to the stability of the emulsions, and hence phase inversion can occur at a lower dispersed phase ratio. These catastrophic inversions have been called difficult and easy inversions, respectively.9 The morphology of abnormal emulsions is quite complex. The dispersed phase can entrain internal droplets of different sizes. The dispersed drops are very unstable and show a high tendency for coalescence, leading to an easy breakdown upon cessation of stirring. Because phase separation is almost instantaneous, the observation of (6) Brooks, B. W.; Richmond, H. N. Chem. Eng. Sci. 1994, 49, 1065. (7) Brooks, B. W.; Richmond, H. N. Colloids Surf. 1991, 58, 131. (8) Zerfa, M.; Sajjadi, S.; Brooks, B. W. Colloids Surf., in press.
10.1021/la0004808 CCC: $19.00 © 2000 American Chemical Society Published on Web 11/22/2000
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complex drops by using a microscope becomes extremely difficult. It is well-known that viscous media can hinder drop coalescence and hence can increase emulsion stability. Thus, to quantify drop structures in the present research, polyisobutylene (PIB) was used as the continuous phase so that drop visualization and measurement by microscope was possible with a good accuracy. The aim of this paper is to study the morphological variation of multiple drops with time in abnormal emulsions. To date, very little attention has been paid to this subject in the open literature. The effects of dispersed phase volume fraction and surfactant concentration on the time evolution of drop morphology and delayed phase inversion were also studied. Experimental Section Preparation. The experiments were performed using a standard baffled 1-L jacketed glass vessel and a conventional four-flat-blade turbine agitator connected to a digital variablespeed motor. The stirrer speed was set at 500 rpm.The phase inversion point was detected by following the variations in electrical conductivity of the continuous phase. The oil phase was low molecular weight polyisobutylene (PIB), supplied by British Petroleum with Trade name Hyvis07, having a molecular weight number average of 440, a density of 0.871 (at 15 °C), and a viscosity of 13 Cst at 100 °C. A water-soluble grade of polyoxyethylene nonylphenyl ether (NPE) with a polyoxyethylene chain length of 12 and a corresponding HLB of 14.2 (Igepal 720 supplied by Aldrich) was used as an emulsifier. Thus, the preferred morphology for the equilibrated state of the system under study is Winsor type I (O/W). At a HLB value of around 10.50, the system approaches Winsor III phase behavior.8 All experiments were carried out at the temperature 60.0 ( 0.5 °C. The experiments were started by stirring of 200 g of the oil phase in the vessel. A specified amount of distilled water, as the dispersed phase containing 0.50 wt % potassium chloride to facilitate detection of phase inversion by conductivity, was added to the oil. In some cases an induced phase inversion of type W/O to O/W occurred. For all experiments water and oil phases contained the same concentration of emulsifier. Samples were taken from the emulsions at specified time intervals, and the variations in drop structures and drop sizes were recorded. Drop Size Measurement. The sizing of drops was carried out by using a video camera connected to an optical microscope. Sizing of multiple drops and internal oil droplets was carried out according to the common procedure. The measurements of multiple water drops were carried out using one slide of glass on which a drop of emulsion was placed. For size measurement of internal oil droplets, another slide of glass was placed on the drop. This will facilitate the drop eruption, as explained below. Measurements were carried out by taking pictures from a zone in which the internal oil droplets, released from multiple drops, concentrate. Internal Phase Ratio of the Dispersed Phase. The sizing of internal droplets contained in multiple drops, for calculation of the internal phase ratio, was carried out as follows: the size measurement for multiple drops containing one, or a few, large oil droplets can be carried out in the usual way. For multiple drops containing a large number of small internal oil droplets, the resolution of the internal oil droplets becomes impossible. Groeneweg et al.10 did not measure directly the internal phase ratio of drops but investigated the variation in effective volume fraction of the dispersed phase with time by using a correlation for volume fraction of the dispersed phase in terms of the conductivities of the emulsion and the continuous phase. Davis and Burbage11 used a freeze-etching electron microscopy (9) Vaessen, G. E. J.; Visschers, M.; Stein, H. N. Langmuir 1996, 12, 875-882. (10) Groeneweg, F.; Agterof, W. G. M.; Jaeger, P.; Janssen, J. J. M.; Wieringa, J. A.; Klahn, J. K. Trans. Ichem, 1998, 76, Part A, 55. (11) Davis, S. S.; Burbage, A. S. J. Colloid Interface Sci. 1977, 62, 361.
Figure 1. Typical pictures from (a) a multiple water drop before eruption and (b) internal oil droplets released to the water layer immediately after eruption. technique to obtain the size distribution of internal droplets. Matsomato et al,12 Florence and Whitehill,13 and Ohtake et al.14 and many others used optical microscopy. However, in all research works reported in the literature, the study of the internal structure of multiple drops at a high internal phase ratio of the dispersed phase was hindered by overlapping of internal oil droplets (see for example ref 13). In this research, a simple technique was developed to measure the internal phase ratio of multiple drops. A drop of emulsion was placed on a glass slide, and then another thin slide of glass was gently placed above it. Since the emulsion is quite unstable, water drops wet the surface as soon as they touch the glass. Upon the contact of other multiple water drops with this zone, they erupt and the internal oil droplets are released into the water phase. Since oil droplets in the aqueous phase are quite stable at the conditions studied, their size distribution did not vary during the water drop eruption and their release to the water layer. Several pictures were snapped at the moment of eruption from multiple water drops with different sizes and analyzed for the calculation of internal phase ratio. Parts a and b of Figure 1 show typical pictures for a multiple water drop before eruption and for the internal oil droplets released to the water layer after the surrounding multiple drop erupted, respectively. Analysis of Drop Sizes and Structures. The surface average drop diameters of both multiple water drops and internal oil droplets were calculated using the following equation: (12) Matsumoto, S.; Inoue, T.; Kohda, M.; Ikura, K. J. Colloid Interface Sci. 1980, 77, 555. (13) Florence, A. T.; Whitehill, D. J. Colloid Interface Sci. 1981, 79, 243. (14) Ohtake, T.; Hano, T.; Takagi, K.; Nakashio, F. J. Chem. Eng. Jpn. 1988, 21, 272.
Abnormal Polymer/Water/Surfactant Dispersions Ds )
∑n D /∑n D 3
i
i
i
2
i
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where ni is the number of drops with diameter Di. Ds is equivalent to the Sauter mean diameter generally used to analyze liquidliquid dispersion systems. To study the morphology of drops, we used the ratio d/D as the ratio of the Sauter mean diameter of internal oil droplets to that of water drops. For individual drops, another morphology characterizing parameter is the internal phase ratio of the dispersed drops. φj stands for the internal phase ratio of water drops within the size class of j. For the multiple drops containing one or a few internal oil droplets, φj was calculated directly by sizing the internal droplets within the multiple drops. For drops containing a larger number of oil droplets, pictures from the eruption of between 4 and 10 multiple drops within the diameter range of Dj were analyzed. The internal phase ratio of multiple drops was calculated using the following equation:
di3
i)n
φj )
∑n D ij
0
3
(2)
j
where nij is the number of internal droplets having diameter di contained in a multiple drop of diameter Dj. The overall internal phase ratio of the dispersed phase (phase ratio of oil droplets in the water drops) is calculated by taking an average of φj over all ranges of multiple water drops in the dispersion as j)m
φ)
∑φ f
j j
(3)
0
where φj is the internal phase ratio of multiple drops having diameter Dj and fj is the volume fraction of multiple drops having diameter j in the sample, given by
fj )
njDj3
(4)
j)m
∑n D j
3
j
0
nj is the number of complex drops having diameter Dj. The error bars for the data points of internal structure were substantially large. This is because of the wide disparity in the internal structures of drops. Within any size range, a number of multiple drops contained a large number of internal oil droplets, whereas some others contained a few internal oil droplets. It was found that the disparity increases with decreasing drop size so that large error bars were obtained for the smaller water multiple drops. Obviously, to obtain statistically reliable data, a large number of drops should be analyzed for drop structures. However, we think that, for the current study, the precision of the data provided is satisfactory for a descriptive discussion. The overall mass balance equations are
Vtot ) Vo + Vw
(5)
Vd ) Vw + φVd
(6)
where Vo, Vw, Vd, and Vtot are volumes of oil, water, dispersed phase, and total, respectively. φ is the volume fraction of the dispersed phase occupied by the continuous phase. From eqs 5 and 6, the volume fraction of the dispersed phase fd is obtained as
fd )
fw 1-φ
(7)
where fw is the volume fraction of the water phase (Vw/Vtot).
Results and Discussion Delayed Catastrophic Phase Inversion. Delayed phase inversion for stirred vessels has been reported before
Figure 2. Time evolution of the Sauter mean diameter of multiple water drops and internal oil droplets for a water volume fraction of 0.17.
in the literature.15-17 Pacek et al.17 described the delay time as the time required for sufficient droplets of the continuous phase to be incorporated into the dispersed phase so that the critical packing fraction or required effective volume fraction of the dispersed phase is reached. Groeneweg et al.10 investigated the variation in effective volume fraction of the dispersed phase with time. They showed that, for a 74 vol % O/W emulsion, a steady state at an effective volume fraction of 90% was reached, whereas, for an O/W emulsion of 78 vol %, the effective volume fraction increased with time and an inversion eventually occurred after 12 h. The results obtained in the present work confirm similar observations to those reported by Pacek et al.17 and Groeneweg et al.10 For the three water phase fractions used at the surfactant loading of 5 wt %, the critical effective volume fraction of the dispersed phase, necessary for the inversion, was only reached with fw ) 0.17 within the experiment time. Figure 2 shows the time evolution of the Sauter mean diameter of the internal oil droplets and water drops for the volume fraction of the dispersed phase of 0.17 and at a surfactant loading of 5.0 wt %. Different patterns of size variation with time can be observed for water drops and internal oil droplets. For internal oil droplets, the drop size is dramatically reduced until it reaches the steady-state value; thereafter, the size stays practically at a constant value.18 The size of multiple (water) drops initially decreases with time but rises again after reaching a minimum which eventually resulted in a phase inversion from a W/O emulsion to a O/W emulsion. The occurrence of a minimum in the time evolution of multiple water drops can be explained by the balance between drop breakup and drop coalescence due to the increasing effective volume fraction of the dispersed phase.18 For lower fw values of 0.07 and 0.12, drop sizes eventually leveled off and practically constant steady-state drop sizes were reached (not shown). It can be easily shown that the steady-state drop size is reached within the same time as the steady-state φ. This confirms that drop structures are interrelated with drop coalescence and breakup and, at steady-state conditions, neither drop structures nor drop sizes vary, or slightly vary, with time. The internal structure of water drops has a substantial effect on the delayed phase inversion, because it influences the effective volume fraction of the dispersed phase. The internal structure of the dispersed phase can be altered (15) Gilchrist, A.; Dyster, K. N.; Moore, I. P. T.; Nienow, A. W.; Car, K. J. Chem. Eng. Sci. 1989, 44, 2381. (16) Kato, S.; Nakayama, E.; Kawasaki, J. Can. J. Chem. Eng. 1991, 69, 222. (17) Pacek, A. W.; Nienow, A. W.; Moore, I. P. T. Chem. Eng. Sci. 1994, 49, 3485. (18) Sajjadi, S.; Zerfa, M.; Brooks, B. W. Submitted for publication.
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Figure 3. Variation in the internal phase ratio of water multiple drops of different sizes with time.
by changing the affinity of the dispersed phase toward the continuous phase. This can be done, for example, by varying the concentration of the surfactant. When 1 wt % surfactant loading is used, there is no minimum in the Ds-t curves.18 The lack of tendency of the dispersed phase to entrain the continuous phase to induce phase inversion can be compensated by increasing the volume fraction of the dispersed phase. When the water volume fraction was increased to 0.40 at a constant surfactant loading of 1 wt %, the size of multiple drops continuously increased until phase inversion occurred within 12 min. Figure 3 depicts the time evolution of the internal phase ratio of the multiple water drops of different size ranges at fw ) 0.12. This graph was prepared using eq 2 for the distinct class sizes of water drops as follows: 8.3, 16.7, 25.0, 33.0, 50.0, 66.5 µm (the last two are not shown). The water drops initially do not contain oil droplets, but gradually they entrain oil droplets from the continuous phase. The large water drops contain a larger volume of oil droplets, compared to the case for small drops. Since most of the drop volume in an emulsion is attributed to its large drop components, it can be stated that the occurrence of large drops is the main reason the effective volume fraction of the dispersed phase increases with time. There are some contradictory reports, however, in the literature regarding the variation of internal phase ratio of drops with size. Ohtake et al.19 showed that the water uptake of oil drops in W/O/W emulsions increases with stirring speed, while the multiple drop size decreased with increasing stirring speed. They reported that the extent of entrainment can be correlated well with interfacial area. To explain the occurrence of phase inversion of O/W to W/O dispersion upon reducing the stirring speed, Kumar20 proposed that the phase ratio of internal oil droplets in water drops increases with decreasing stirring speed, equivalent to increasing drop size, because of a large restoring force of small droplets. He explained that W/O dispersions go through inversions at a decreased stirring speed due to an increase in the effective volume of the dispersed phase. The results obtained here, which are consistent with those reported by Kumar, are further supported by the observations of Brooks and Richmond.7 They reported that the catastrophic phase inversion for cyclohexane/water/NPE systems occurs at a smaller value of fw with decreasing stirring speed. This result can now be explained in terms of the new finding of increasing the effective volume fraction with drop diameter. As the stirring speed decreases, the size of multiple drops increases. Large drops can entrain a larger volume of the internal droplets, which results in an increased effective (19) Ohtake, T.; Hano, T.; Takagi, K. J. Chem. Eng. Jpn. 1987, 20, 443. (20) Kumar, S. Chem. Eng. Sci. 1996, 51, 831-834.
Sajjadi et al.
Figure 4. Variation in the average internal phase ratio of the dispersed phase and the average dispersed phase ratio with time for a water volume fraction of 0.12.
Figure 5. Variation in the ratio of the Sauter mean diameter of the internal oil droplets to that of water multiple drops with time for the three water volume fractions of 0.06, 0.12, and 0.17.
volume fraction. This will reduce the fw value at which phase inversion occurs. Structures of Multiple Drops. The examination of samples at different times revealed that, for fixed conditions, the structure of the complex drops is a function of time, and size of drops. Drop structures are influenced by drop breakup and coalescence, as well as inclusion and escape. The inclusion events are opposed by the escape events.10 The escape events will cause the internal oil droplets to exit the water drops and join the continuous phase. The internal phase ratio gradually increased with time and eventually reached a practically constant value. Knowing the overall internal phase ratio, φ, it is possible to estimate fd by using eq 7. The time variation of fd is also shown in Figure 4, which indicates that the dispersed phase ratio increases with time until it reaches a practically constant value. It can be easily shown that the number of oil droplets for each class of water drops increases with time. Accordingly, larger water drops contained more internal oil droplets. It can be inferred from Figure 5 that d/D is continuously decreasing with time, especially in the early stage of stirring. The sequence of time variation in the structure of the multiple drops is believed to be similar to those shown in Figure 6. The three classes of multiple drops are defined as A, B, and C. In the early stage of mixing a large d/D is obtained. Later, the large internal droplets are broken into smaller droplets. Eventually, a point might be reached where the multiple drops contain a large number of internal oil droplets. Interestingly, these three classes of multiple drop structures are identical to those defined by Florence and Whitehill15 in their investigation of stability of multiple drops. By calculation of the reduction in the free energy, Florence and Whitehill suggested that the order of stability is B > C > A (Although, some of their experimental results showed that C > B > A). It has also been reported that the escape process is
Abnormal Polymer/Water/Surfactant Dispersions
Figure 6. Three types of multiple drops in terms of internal structure.
favored when the size of enclosed droplets becomes comparable to the size of the surrounding drops,10 that is, d/D ≈ 1.0. This means that the rate of escape is appreciable at the initial stage of mixing when d/D is large. This might be the reason that a plateau is observed for φj in the early stage of stirring for all drop sizes, as shown in Figure 3, and correspondingly for φ in the early stage, as shown in Figure 4. The overall phase ratio of internal oil droplets in water drops depicted in Figure 4 was estimated using the whole size distribution of water drops at fw ) 0.12 and eqs 3 and 4. It should be noted that, at the initial stage, only a small fraction of water drops contains a single large droplet, or a few large droplets, and the average φ is very low. This can explain why the error margin in the calculation of φ is appreciable for this region, as mentioned in the Experimental Section. The size of internal droplets is probably reduced through drop breakup as they pass by the impeller region. The chance of escape is diminished as the size of internal droplets is reduced. The second factor which controls escape is the phase fraction of internal oil droplets within the water drops, φ. It is inferred from Figures 3 and 4 that the rate of inclusion (dφ/dt) is very low at the initial stage, increases at a later stage, but decreases as the steady-state φ is reached. For the emulsion under study, no escape was experimentally observed when drops were at rest. However, escapes can occur when the water drops are exposed to shear flow or energetic eddies. Any deformation of the water drops will lead to compression of internal oil droplets that are near the surface of the host water drops. Hou et al.21 showed that, for a W/O/W multiple emulsion, the convex-convex interaction (droplet-droplet) leads to more stable drops than those of the convex-concave (droplet-continuous phase) interaction. Since in the context of this research work the interfacial film between O/W drops is stable, while that between O/W and W/O is unstable, it seems that such a compression of internal droplets will most likely lead to escape events. In the initial stage of mixing, the frequency of escape is controlled by the size ratio of internal droplets over multiple drops and, hence, decreases with time. With the fewer number of internal oil droplets entrained in the water drop, the chance of bringing together O/W and W/O interfaces is reduced. After a while, the phase ratio of the internal phase becomes predominant and controls the escape events so that escape begins to increase with time. As φ increases with time, the frequency of escape will increase until it eventually equals the rate of inclusion.10 At the steady state, φ stays at a practically constant value. This can be seen from Figures 3 and 4, which show that the steady value for φ is reached approximately 10-20 min after start up. There are no (21) Hou, W.; Papadopoulos, K. D. Chem. Eng. Sci. 1996, 51, 5043.
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minima in φ, indicating that water drops become continuously richer in oil droplets, despite their size, until they reach a steady-state size. It should be noted, however, that the variation in φ at the presumably steady state might not be reduced to zero but becomes trivial so that it cannot be detected within the experiment time. Even a very low rate of increase in φ can eventually lead to a phase inversion at a finite time. It is possible that drop breakup and escape occur simultaneously at a high φ, because both are induced by drop deformation.22 Similarly, internal oil droplets might experience coalescence during drop deformation when they are compressed against each other. This emphasizes that more in depth information on the dynamics of drop behaviors can only be obtained by using on-line techniques for drop visualization and measurement. Conclusions In agitated liquid-liquid dispersions, catastrophic phase inversion can be induced as the stirring time increases. The growth of water drops, due to inclusion events, is responsible for delayed phase inversion. The structure of the multiple drops is a function of time, and size of drops. The large drops are capable of accommodating a larger volume of internal droplets. In the context of this research, the diameter ratio of the internal oil droplet to that of external water drops is a decreasing function of time until a steady state is reached. The internal phase ratio of the dispersed phase increases slightly in the early minutes of mixing but increases appreciably in the later stage until a steady state is reached. Beyond the steady state no appreciable variation in the size of drops and structures of drops was observed within the experiment time. If a continuous increase in effective volume fraction of the dispersed phase occurs, a phase inversion might be induced. A high surfactant concentration in the dispersed phase and also a high phase ratio of the dispersed (water) phase will result in a large build up of internal droplets, and that increases the chance of inducing a phase inversion. Acknowledgment. The authors thank the EPSRC for financial support [Ref GR/K 78249]. Notation d ) Sauter mean diameter of internal oil droplets di ) diameter of internal droplets contained in a multiple water drop D ) Sauter mean diameter of multiple drops Di ) drop diameter of fraction i, nm Ds ) surface average diameter, nm fd ) volume fraction of the dispersed phase fj ) volume fraction of the multiple drops having diameter Dj fw ) volume fraction of the water ni ) number of drops with diameter Di nij ) number of internal droplets having diameter di in a multiple drop with diameter Dj Vd ) volume of the dispersed phase Vo ) volume of the oil phase Vtot ) total volume of the dispersion Vw ) volume of the water phase Greek Letters φ ) overall internal phase ratio of the dispersed phase φj ) internal phase ratio of multiple drops having diameter Dj LA0004808 (22) Sajjadi, S.; Zerfa, M.; Jahanzad, F.; Brooks, B. W. In preparation.
