Relative Viscosity of Non-Newtonian Concentrated Emulsions of

Ind. Eng. Chem. Res. 2000, 39, 4933-4943

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Relative Viscosity of Non-Newtonian Concentrated Emulsions of Noncolloidal Droplets Rajinder Pal* Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

The relative viscosity of non-Newtonian concentrated emulsions of nearly spherical, nonBrownian droplets, in the absence of Coulombic and van der Waals interactions, is a function of four variables: the viscosity ratio (K), the volume fraction of dispersed phase (φ), the particle Reynolds number (NRe,p), and the maximum packing volume fraction (φm). The maximum packing volume fraction depends on the droplet size distribution. Based on five sets of data for monomodal and bimodal emulsions, a novel correlation is proposed to describe the relative viscosity of nonNewtonian concentrated emulsions. The proposed correlation can be used to predict the relative viscosity-Reynolds number behavior of non-Newtonian concentrated emulsions from the knowledge of the volume fraction of the dispersed phase, the viscosity ratio, and the maximum packing volume fraction of dispersed phase. Introduction

Dimensional Analysis

Emulsions are dispersions of two immiscible liquids, such as oil and water. The flow properties of concentrated emulsions are of interest in many industrial applications.1 For example, the choice of mixing equipment and the power requirements for producing an emulsion are dependent on the emulsion rheological properties. The prediction of the transport behavior of emulsions through pipelines requires a knowledge of possible variations of viscosity with shearing rate as well as with dispersed-phase concentration. A vast body of published literature exists on the rheology of dispersed systems.1-10 A majority of the published articles, however, are restricted to solids-inliquid suspensions. For concentrated solids-in-liquid suspensions, a number of viscosity equations have been proposed in the literature. Of the equations proposed for suspension viscosity, two have become widely used. The equations are those of Mooney11 and Krieger and Dougherty.12 There are two serious limitations of such suspension viscosity equations: (1) they cannot be generally applied to emulsions (only when the ratio of the dispersed-phase viscosity to the continuous-phase viscosity is very high is the application of suspension viscosity equations to emulsions appropriate), and (2) they are generally restricted to Newtonian systems where the shear rate effect is absent. To our knowledge, no commonly accepted theoretical or empirical expression has been developed to correlate the viscosity of non-Newtonian concentrated emulsions. In this paper, a new correlation is developed to correlate the viscosity data of non-Newtonian concentrated emulsions of noncolloidal droplets. The proposed correlation takes into account the effects of viscosity ratio, volume fraction of the dispersed phase, shear rate, droplet size, and droplet size distribution on the viscosity of emulsions.

In the absence of Coulombic and van der Waals interactions, the viscosity of dispersions of monodisperse hard spheres is expected to be a function of eight variables,13 namely, shear rate (γ˘ ), time (t), continuousphase viscosity (ηc), continuous-phase density (Fc), dispersed-phase density (Fd), particle radius (R), concentration of particles, i.e., number density (n), and thermal energy (kT). For dispersed systems consisting of fluid particles (such as emulsions), additional variables must be present in the rheological equation of state. These include the dispersed phase viscosity (ηd) and the interfacial tension (σ) (assuming that the interface can be described by a constant interfacial tension with negligible interfacial shear and dilational viscosities). In situations where Coulombic and van der Waals interactions are present, one must also consider the following variables: zeta potential (ξ), Hamaker constant (A), Debye length (κ-1), and permittivity of the medium (∈). Thus,

η ) f(γ˘ , t, ηc, ηd, Fc, Fd, R, n, kT, σ, ξ, A, ∈, κ-1)

(1)

The total number of basic dimensions involved in expressing all the variables in the above equation is 4, i.e., mass, length, time, and electric charge. Because there are 15 variables in the equation, the total number of independent dimensionless groups required to describe the problem is 15 - 4 ) 11. These dimensionless groups are defined as follows: relative viscosity ηr ) η/ηc, reduced time tr ) t/(ηcR3/kT), relativity density Fr ) Fd/Fc, viscosity ratio K ) ηd/ηc, volume fraction of dispersed phase φ ) 4πnR3/3, Peclet number NPe ) ηcγ˘ R3/kT, capillary number NCa ) ηcγ˘ R/σ, particle Reynolds number Fcγ˘ R2/ηc, shear-repulsion number14 NSR ) ηcγ˘ R2/ξ2, shear-attraction number14 NSA ) ηcγ˘ R3/A, and relative double-layer thickness κR. Thus

ηr ) f(tr, Fr, K, φ, NPe, NCa, NRe,p, NSR, NSA, κR) (2) * Phone: (519) 885-1211, ext. 2985. Fax: (519) 746-4979. E-mail: [email protected]

As the densities of oil and water are similar, Fr f 1.0.

10.1021/ie000616x CCC: $19.00 © 2000 American Chemical Society Published on Web 11/03/2000

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Ind. Eng. Chem. Res., Vol. 39, No. 12, 2000

Table 1. Details of Emulsions Investigated set

emulsion type

1

O/W

petroleum oil

2

O/W

3

oil type

φ

comments

petroleum oil

0.628, 0.676, 0.701, 0.722 0.745

O/W

mineral oil

0.71

4

O/W

mineral oil

0.63

5

O/W

odorless kerosene

0.70

Monomodal emulsions of different average droplet sizes were prepared. The Sauter mean diameter was varied from 4.6 to 21.4 µm. Bimodal emulsions were prepared by mixing together the fine and coarse monomodal emulsions in various proportions. The Sauter mean diameters of fine and coarse emulsions were: 6.3 µm and 20 µm, respectively. Bimodal emulsions were prepared by mixing together the fine and coarse monomodal emulsions in various proportions. The Sauter mean diameters of fine and coarse emulsions were: 9 µm and 65 µm, respectively. Bimodal emulsions were prepared by mixing together the fine and coarse monomodal emulsions in various proportions. The Sauter mean diameters of fine and coarse emulsions were: 6.52 µm and 32.3 µm, respectively. Bimodal emulsions were prepared by mixing together the fine and coarse monomodal emulsions in various proportions. The Sauter mean diameters of fine and coarse emulsions were: 7.4 µm and 22 µm, respectively.

Figure 1. Viscosity versus shear stress plots of monomodal emulsions of set 1 at different values of φ. The figure also shows the Sauter mean diameters of emulsions.

Therefore, under steady-state conditions (tr f ∞)

ηr ) f(K, φ, NPe, NCa, NRe,p, NSR, NSA, κR)

(3)

For many practical emulsions, NPe f ∞, NSR f ∞, NSA f ∞, and κR f ∞, as the emulsion droplets are generally

large (.1 µm). Therefore, the effects of Brownian motion and Coulombic and van der Waals interactions on the viscosity of emulsions can be neglected. Thus

ηr ) f(K, φ, NCa, NRe,p)

(4)

This rheological equation of state is applicable to both

Ind. Eng. Chem. Res., Vol. 39, No. 12, 2000 4935

Figure 2. Viscosity versus shear stress plots of bimodal emulsions of sets 2-5. The bimodal emulsions were prepared by mixing together the fine and coarse monomodal emulsions in various proportions.

oil-in-water (O/W) and water-in-oil (W/O) emulsions. In the case of oil-in-water emulsions, however, this equation can be simplified further. The capillary number for oil-in-water emulsions is usually very small (NCa f 0). Therefore, the droplets of the oil-in-water emulsions generally do not suffer much deformation even at high rates of shear, and the droplets remain almost spherical. In such cases where the effect of capillary number can be neglected, the relative viscosity can be expressed as

ηr ) f(K, φ, NRe,p)

(5)

This equation is valid only for monodisperse emulsions of nearly spherical droplets. The relative viscosity of emulsions is expected to be a function of droplet size distribution in addition to the variables present in eq 5. The polydispersity of emulsions can be accounted for by φm, the maximum packing volume fraction of dispersed phase without any significant deformation of droplets. Thus

ηr ) f(K, φ, φm, NRe,p)

(6)

At any given value of φ, ηr is expected to be a function of the viscosity ratio K, the maximum packing concentration φm, and the particle Reynolds number NRe,p,

where NRe,p is calculated on the basis of the mean droplet radius (Sauter mean radius). Experimental Work Materials. The emulsions of oil-in-water type were prepared from three different oils: petroleum oil of viscosity 5.52-5.8 mPa s (depending on the batch) at 25 °C, mineral oil of viscosity 29.2 mPa s at 22 °C, and odorless kerosene of viscosity at 2.2 mPa s at 22 °C. The water used throughout the experiments was deionized. Triton X-100 was used as a surfactant for the preparation of emulsions. Triton X-100 is an octylphenol ethoxylate with an average of 9-10 molecules of ethylene oxide. It is water-soluble and has a high HLB value of 13.5. Procedures. The emulsions (O/W type) were prepared by shearing together the known amounts of oil and aqueous surfactant solution (approximately 2% by weight Triton X-100) in a variable-speed homogenizer (Gifford-Wood model 1-L). Five sets of monomodal and bimodal emulsions were prepared (see Table 1). In set 1, the volume fraction of the dispersed phase (oil) was varied from 0.628 to 0.722. At any given volume fraction of dispersed phase (φ), monomodal emulsions of different

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Figure 3. Viscosity as a function of fine emulsion content of the bimodal emulsion.

average droplet sizes were prepared by varying the speed and duration of shearing in the homogenizer. In set 2, the volume fraction of oil was kept constant at 0.745. Two monomodal emulsions of different average droplet sizes, that is, fine and coarse emulsions, were first prepared. The fine and coarse emulsions were then mixed together in various proportions to produce bimodal emulsions. A similar procedure was followed in the case of the sets 3-5 emulsions in that the fine and coarse monomodal emulsions were mixed in various proportions to produce bimodal emulsions. The volume fractions of oil were as follows: set 3, φ ) 0.71; set 4, φ ) 0.63; and set 5, φ ) 0.70. Table 1 provides further information about the droplet size of emulsions. The rheological measurements in most cases (except set 4) were carried out in a Bohlin controlled-stress rheometer (Bohlin CS-50) using either the double-gap cylindrical geometry or the cone-and-plate geometry. In the case of the set 4 emulsions, a Fann coaxial cylinder viscometer having a gap width of 1.17 mm was used. The rheological data were collected at a temperature of 22-25 °C. The droplet sizes of emulsions were determined by taking photomicrographs. The samples were diluted with the same continuous phase before the photomicrographs were taken. The photomicrographs

were taken with a Zeiss optical microscope equipped with a camera. Results and Discussion Figure 1 shows viscosity versus shear stress plots of monomodal emulsions of set 1 at different values of φ. The data are shown for different average droplet sizes (Sauter mean diameters). The emulsions exhibit shear-thinning behavior in that the viscosity decreases with the increase in shear stress. Emulsions with smaller average droplet diameters are significantly more viscous than those with larger average droplet diameters. Also, the effect of droplet size decreases with the decrease in the volume fraction of the dispersed phase. The viscosity versus shear stress plots of bimodal emulsions of sets 2-5 are shown in Figure 2. For any set, the fine emulsion exhibits a much higher viscosity than the coarse emulsion. When the fine emulsion is mixed with the coarse emulsion, the resulting bimodal emulsion exhibits viscosities even lower than that of the coarse emulsion. This is especially true at low shear stresses and at low values of fine emulsion content of the mixed emulsion. The plots of viscosity as a function


given by

φm )

∑D3i fi 1 ∑(Di ∼ Dh )3fi + β ∑[(Di + Dh )3 - (Di ∼ Dh )3] fi

(7)

where

β)1+

4 (8φ°m - 1)D h 13

[

(3/8)D h

]

∑(Di + Dh )2 1 - (Di + Dh ) fi

D h )

∑[D3i - (Di ∼ Dh )3]fi

∑Difi

(8) (9)

Here φ°m is the maximum packing concentration of monodisperse emulsion, fi is the number fraction of h is the number-average droplets of diameter Di, D diameter of the emulsion, and the abbreviation (Di ∼ D h ) is defined as Figure 4. Variation of φm with fine emulsion content of the bimodal emulsion.

of the fine emulsion content of the mixed emulsion are shown in Figure 3 for sets 2-5. Clearly, the viscosity of the emulsion goes through a minimum value at a certain proportion of the fine emulsion. The decrease in viscosity of bimodal emulsions with the addition of fine emulsion to a coarse emulsion (at constant φ) can be explained in terms of the maximum packing concentration (φm) of particles. It is well-known that the higher the φm, the lower the emulsion viscosity. The maximum packing concentration is expected to increase upon the addition of a fine emulsion to a coarse one at fixed φ as fine droplets can easily fit into the voids between the large droplets. The maximum packing concentration of emulsions was calculated from the Ouchiyama and Tanaka15-19 equation using the experimental droplet size distribution data. According to Ouchiyama and Tanaka theory,15-19 the maximum packing concentration is

(Di ∼ D h ) ) 0 for Di e D h

(10)

h for Di > D h ) Di - D

(11)

To calculate φm for polydisperse emulsions using the Ouchiyama and Tanaka theory (eqs 7-11), the value of the maximum packing concentration of the monodisperse emulsion (φ°m) is required. For solid-in-liquid suspensions, φ°m is often taken to be about 0.637,4,18 corresponding to random close packing of hard spheres. The relative viscosity of monodisperse suspensions of hard spheres approaches infinity at φ°m = 0.637.4 However, in the case of emulsions, the situation is somewhat different. The relative viscosity of a monodisperse (random close emulsion does not diverge at φ ) φRCP m packing concentration), although the rate of increase of viscosity with dispersed-phase concentration (φ) shows a significant rise near φ ) φRCP m . Emulsions generally exhibit solidlike behavior20-22 characterized by high values of yield stress and storage modulus only when φ

Figure 5. Comparison of relative viscosity-Reynolds number curves for monomodal emulsions of set 1 (Sauter mean diameter of 4.6 µm) at different values of φ.

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Figure 6. Critical Reynolds number, (NRe,p)critical, as a function of φ for monomodal emulsions of set 1 (Sauter mean diameter of 4.6 µm).

> φHCP (hexagonal close packing concentration). Thus, it is more appropriate to take φ°m equal to 0.74, corresponding to hexagonal close packing of spheres. Figure 4 shows the plots of the maximum packing concentration for mixed fine and coarse emulsions (bimodal emulsions) as a function of volume fraction of fine emulsions. In calculating the values of φm from the Ouchiyama and Tanaka theory, the value of φ°m was taken to be 0.74. As can be seen from Figure 4, the plots of maximum packing concentration versus volume fraction of fine emulsion exhibit a maximum value. Interestingly, the maximum in φm occurs almost at the same concentration of the fine emulsion where a minimum in viscosity is observed (see Figure 3). Also, the magnitude of the maximum value of φm increases with the decrease in the ratio of Sauter mean diameters of fine to coarse emulsions (d/D).

Figure 8. Critical Reynolds number, (NRe,p)critical, as a function of fine emulsion content for bimodal emulsions of sets 2-5.

According to eq 6, the relative viscosity of an emulsion for fixed values of φ, φm, and K is a function of the particle Reynolds number alone. Furthermore, the ηr versus NRe,p data can be represented very well by the following equation:

ηr - ηr∞ 1 ) ηro - ηr∞ 1 + A(N

Re,p)

N

(12)

where ηr∞ is the high-shear limiting relative viscosity, ηro is the zero-shear limiting relative viscosity, and A and N are constants. Figure 5 compares the ηr versus NRe,p curves for monomodal emulsions of set 1 (Sauter mean diameter of 4.6 µm) at different volume fractions of the dispersed phase (φ). These curves represent the best fit of eq 12 through the experimental data. The relative viscosity

Figure 7. Comparison of relative viscosity-Reynolds number curves for bimodal emulsions of set 2 having different values of fine emulsion content.


Figure 9. {[(ηr - ηr∞)/(ηro - ηr∞)]-1 - 1} as a function of NRe,p and NRe,p/(NRe,p)critical for monomodal emulsions of set 1 (Sauter mean diameter of 4.6 µm).

increases with the increase in φ. At low values of particle Reynolds number, the increase in relative viscosity with the increase in φ is quite large. The gap between different ηr versus NRe,p curves becomes smaller with the increase in NRe,p because of the shear-thinning effect. The critical Reynolds number, (NRe,p)critical, where (ηr - ηr∞)/(ηro - ηr∞) becomes 0.5, also varies with the volume fraction of the dispersed phase. With the increase in φ, (NRe,p)critical for monomodal emulsions decreases linearly, as can be seen in Figure 6. Figure 7 compares the ηr versus NRe,p curves for bimodal emulsions of set 2 with different values of fine emulsion content. These curves represent the best fit of eq 12 through the experimental data. Clearly, the zero-shear limiting relative viscosity varies with the fine emulsion content (droplet size distribution). The zeroshear limiting relative viscosity exhibits a minimum at a fine emulsion content of 35% (35f/65c). The high-shear limiting relative viscosity, however, is independent of the fine emulsion content (droplet size distribution). A

similar behavior was seen in the case of other bimodal emulsions (sets 3-5). The critical Reynolds number, (NRe,p)critical, of bimodal emulsions also varies with the fine emulsion content. Figure 8 shows the plots of the critical Reynolds number as a function of the volume fraction of fine emulsion. The critical Reynolds number exhibits a maximum value at almost the same concentration of the fine emulsion where a maximum in φm (Figure 4) or a minimum in viscosity (Figure 3) was observed. Correlation of Viscosity Data. According to eq 12

(

ηr - ηr∞ ηro - ηr∞

)

-1

- 1 ) A(NRe,p)N

(13)

Therefore, the plots of {[(ηr - ηr∞)/(ηro - ηr∞)]-1 - 1} versus NRe,p on a log-log scale are expected to be linear with a slope of N. Figure 9 shows such plots for monomodal emulsions of set 1 (Sauter mean diameter of 4.6 µm). At any given φ, the plot of {[(ηr - ηr∞)/(ηro -

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Figure 10. {[(ηr - ηr∞)/(ηro - ηr∞)]-1 - 1} as a function of NRe,p and NRe,p/(NRe,p)critical for bimodal emulsions of set 2.

ηr∞)]-1 - 1} versus NRe,p exhibits a nearly linear relationship. Furthermore, the value of the slope (N) is nearly the same for emulsions with different φ. If N is taken to be constant, then the data for all emulsions with different values of φ are expected to fall on the same line, provided that the data are plotted as {[(ηr ηr∞)/(ηro - ηr∞)]-1 - 1} versus NRe,p/(NRe,p)critical on a loglog scale. This follows from eq 12, which can be rewritten as

(

ηr - ηr∞ ηro - ηr∞

)

-1

- 1 ) [NRe,p/(NRe,p)critical]N

(14)

where (NRe,p)critical, defined earlier, is equal to A-1/N. Figure 9 shows the plot of {[(ηr - ηr∞)/(ηro - ηr∞)]-1 1} versus NRe,p/(NRe,p)critical on a log-log scale. Interestingly, all of the data for monomodal emulsions of set 1 (with different φ) fall on the same line, as expected from eq 14. For bimodal emulsions of set 2, the plots of {[(ηr - ηr∞)/(ηro - ηr∞)]-1 - 1} versus NRe,p are shown in Figure 10. The data exhibit almost a linear relationship at any given value of fine emulsion content. Further-

more, the plots at different values of fine emulsion content are nearly parallel to each other. When the data are replotted on an NRe,p/(NRe,p)critical basis, the data for different values of fine emulsion content almost fall on the same line. A similar behavior was seen in the case of other bimodal emulsions of sets 3-5. Figure 11 shows all of the data for monomodal and bimodal emulsions (all five sets). The data for different sets of emulsions all fall on the same linear relationship between {[(ηr - ηr∞)/(ηro - ηr∞)]-1 - 1} and NRe,p/ (NRe,p)critical. The slope of the line, that is, N, is 0.673. Thus, the relative viscosity of non-Newtonian emulsions can be described adequately by the following equation

ηr - ηr∞ ) ηro - ηr∞ 1 + [N

1 0.673 /(N Re,p Re,p)critical]

(15)

To predict the relative viscosity of emulsions from eq 15, the values of (NRe,p)critical, ηro, and ηr∞ are needed. In Figure 12, the critical Reynolds number data of monomodal and bimodal emulsions (all five sets) are plotted


Figure 11. {[(ηr - ηr∞)/(ηro - ηr∞)]-1 - 1} as a function of NRe,p/(NRe,p)critical for all five sets of monomodal and bimodal emulsions; the solid line represents eq 15.

Figure 12. Critical Reynolds number data for all five sets of monomodal and bimodal emulsions as a function of φ/φm.

as a function of φ/φm, where φm was calculated from eq 7, as discussed earlier. The data can be represented satisfactorily by the following equation:

[

(NRe,p)critical ) exp

]

-1.1φ/φm 1 - φ/φm

(16)

The solid curve shown in Figure 12 represents eq 16. The zero-shear relative viscosity (ηro) data are plotted in Figure 13 as (ηro)1/KI versus φ/φm , where KI is defined as

KI )

K + 0.4 K+1

(17)

and K is the viscosity ratio defined earlier. As can be seen in Figure 13, the zero-shear relative viscosity data

Figure 13. Zero-shear relative viscosity (ηro) data for all five sets of monomodal and bimodal emulsions, plotted as (ηro)1/KI versus φ/φm, where KI is defined in eq 17.

of monomodal and bimodal emulsions (all five sets) can be represented satisfactorily by the following equation (shown as the solid curve in Figure 13):

[

(ηro)1/KI ) exp

]

0.782φ/φm 1 - φ/φm

(18)

The high-shear limiting relative viscosity (ηr∞) of monomodal and bimodal emulsions (all five sets) is plotted in Figure 14 as (ηr∞)1/KI versus φ. The data can be represented reasonably well by the following equation (shown as the solid curve in Figure 14):

(ηr∞)1/KI ) exp

[1 -2.5φ 0.71φ]

(19)

A similar equation was proposed earlier by Pal23 to

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Figure 14. High-shear limiting relative viscosity (ηr∞) data for all five sets of monomodal and bimodal emulsions, plotted as (ηr∞)1/KI versus φ, where KI is defined in eq 17.

describe the high-shear viscosity data of polymerthickened emulsions. Predictive Ability of the Proposed Correlation. The proposed correlation (eq 15) can be used to predict the relative viscosity versus Reynolds number behavior of non-Newtonian emulsions. The values of (NRe,p)critical, ηro, and ηr∞ needed in this equation can be predicted from eqs 16, 18, and 19, respectively. The only information needed about the emulsion is the following: the volume fraction of dispersed phase (φ), the viscosity ratio (K), and the maximum packing concentration (φm). φm can be determined from the droplet size distribution using the Ouchiyama and Tanaka expression (eq 7). The correlation was tested for the following two systems: (a) an oil-in-water emulsion characterized by φ ) 0.70, K ) 6.4, Sauter mean diameter ) 24.61 µm, and φm ) 0.768; and (b) an oil-in-water emulsion characterized by φ ) 0.65, K ) 6.36, Sauter mean diameter ) 35.51 µm, φm ) 0.754. For system a, the zero-shear relative viscosity predicted from eq 18 is 1625, and the high-shear limiting relative viscosity predicted from eq 19 is 24.46. The

Figure 15. Comparison between the experimental ηr versus NRe,p data and the data predicted from the proposed correlation.


critical Reynolds number for this system predicted from eq 16 is 1.18 × 10-5. For system b, the zero-shear relative viscosity predicted from eq 18 is 86.71, and the high-shear limiting relative viscosity predicted from eq 19 is 15.98. The critical Reynolds number predicted from eq 16 is 1.06 × 10-3. Figure 15 shows the comparison between the predicted ηr versus NRe,p data and the experimental data. As can be seen, the proposed correlation (eq 15) gives a reasonably good prediction of ηr versus NRe,p behavior of emulsions. Conclusions A dimensional analysis of the viscous behavior of concentrated non-Newtonian emulsions of nearly spherical, non-Brownian droplets indicates that, in the absence of Coulombic and van der Waals interactions, the relative viscosity of emulsions is a function of four variables, namely, the viscosity ratio (K), the volume fraction of the dispersed phase (φ), the particle Reynolds number (NRe,p), and the maximum packing volume fraction of the dispersed phase (φm). Five sets of viscosity data for monomodal and bimodal oil-in-water emulsions were obtained experimentally. The bimodal emulsions were prepared by mixing together the fine and coarse monomodal emulsions in various proportions. The experimental results indicate that the relative viscosity of monomodal emulsions having different average droplet sizes but the same φ, φm, and K values can be scaled with the particle Reynolds number. However, the relative viscosity of different bimodal emulsions at the same φ and K cannot be scaled with the particle Reynolds number; the relative viscosity of bimodal emulsions at the same φ and K varies with the fine emulsion content in addition to the particle Reynolds number. A new correlation is proposed to describe the relative viscosity of non-Newtonian concentrated emulsions. The proposed correlation gives a reasonable prediction of relative viscosity versus Reynolds number behavior of non-Newtonian concentrated emulsions. To use the proposed correlation, the following information about the emulsion is needed: the volume fraction of dispersed phase, the viscosity ratio, and the maximum packing volume fraction of dispersed phase. Acknowledgment Financial support from the Natural Sciences and Engineering Research Council (NSERC) of Canada is greatly appreciated. Literature Cited (1) Pal, R. Rheology of Emulsions Containing Polymeric Liquids. In Encyclopedia of Emulsion Technology; Becher, P., Ed.; Dekker: New York, 1996; vol. 4, p 93.

(2) Jeffrey, D. J.; Acrivos, A. The rheological properties of suspensions of rigid particles. AIChE J. 1976, 22, 417. (3) Metzner, A. B. Rheology of suspensions in polymeric liquids. J. Rheol. 1985, 29, 739. (4) Larson, R. G. The structure and rheology of complex fluids; Oxford University Press: New York, 1999. (5) Palierne, J. F. Linear rheology of viscoelastic emulsions with interfacial tension. Rheol. Acta 1990, 204. (6) Chong, J. S.; Christiansen, E. B.; Baer, A. D. Rheology of concentrated suspensions. J. Appl. Polym. Sci. 1971, 15, 2007. (7) Hoffman, R. L. Factors affecting the viscosity of unimodal and multimodal colloidal dispersions. J. Rheol. 1992, 36, 947. (8) Woutersen, A. T. J. M.; de Kruif, C. G. The viscosity of semidilute, bidisperse suspensions of hard spheres. J. Rheol. 1993, 37, 681. (9) D’Haene, P.; Mewis, J. Rheological characterization of bimodal colloidal dispersions. Rheol. Acta 1994, 33, 165. (10) Cheng, D. C. H.; Kruszewski, A. P.; Senior, J. R.; Roberts, T. A. The effect of particle size distribution on the rheology of an industrial suspension. J. Mater. Sci. 1990, 25, 353. (11) Mooney, M. The viscosity of a concentrated suspension of spherical particles. J. Colloid Sci. 1951, 6, 162. (12) Krieger, I. M.; Dougherty, T. J. A mechanism for nonNewtonian flow in suspensions of rigid particles. Trans. Soc. Rheol. 1959, 3, 137. (13) Krieger, I. M. Rheology of monodisperse latices. Adv. Colloid Interface Sci. 1972, 3, 111. (14) Probstein, R. F.; Sengun, M. Z. Dense slurry rheology with application to coal slurries. Phys. Chem. Hydrodyn. 1987, 9, 299. (15) Ouchiyama, N.; Tanaka, T. Estimation of the average number of contacts between randomly mixed solid particles. Ind. Eng. Chem. Fundam. 1980, 19, 338. (16) Ouchiyama, N.; Tanaka, T. Porosity of a mass of solid particles having a range of sizes. Ind. Eng. Chem. Fundam. 1981, 20, 66. (17) Ouchiyama, N.; Tanaka, T. Porosity estimation for random packings of spherical particles. Ind. Eng. Chem. Fundam. 1984, 23, 490. (18) Gupta, R. K.; Seshadri, S. G. Maximum loading levels in filled liquid systems. J. Rheol. 1986, 30, 503. (19) Poslinski, A. J.; Ryan, M. E.; Gupta, R. K.; Seshadri, S. G.; Frechette, F. J. Rheological behaviour of filled polymeric systems II. The effect of a bimodal size distribution of particles. J. Rheol. 1988, 32, 1988. (20) Pal, R. Yield stress and viscoelastic properties of high internal phase ratio emulsions. Colloid Polym. Sci. 1999, 277, 583. (21) Princen, H. M. The mechanical and flow properties of foams and highly concentrated emulsions. In Food Colloids; Bee, R. D., Richmond, P., Mirgins, J., Eds.; Royal Society of Chemistry: Cambridge, U.K., 1980; p 14. (22) Princen, H. M.; Kiss, A. D. Rheology of foams and highly concentrated emulsions. J. Colloid Interface Sci. 1989, 128, 176. (23) Pal, R. Rheology of polymer-thickened emulsions. J. Rheol. 1992, 36, 1245.

Received for review June 23, 2000 Revised manuscript received September 12, 2000 Accepted September 13, 2000 IE000616X

Relative Viscosity of Non-Newtonian Concentrated Emulsions of

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