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Ind. Eng. Chem. Res. 2001, 40, 5666-5674
Single-Parameter and Two-Parameter Rheological Equations of State for Nondilute Emulsions Rajinder Pal* Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
The existing viscosity-concentration equations for concentrated emulsions of nearly spherical noncolloidal droplets are first reviewed. The equations of emulsion relative viscosity ηr vs volume fraction of dispersed-phase φ existing in the literature are characterized as single-parameter equations because viscosity ratio (ratio of dispersed-phase viscosity to continuous-phase viscosity) is the only parameter in the equations. The equations are evaluated in light of a large body of experimental data for concentrated emulsions, covering a broad range of viscosity ratios. The existing equations generally give good predictions of relative viscosity only at low values of φ (φ < 0.2). Several new equations are proposed and evaluated with experimental data. The proposed ηr vs φ equations are characterized as two-parameter equations because they contain two parameters, namely, the viscosity ratio and φm, the maximum packing volume fraction of droplets. The proposed equations generally show good agreement with the experimental data. 1. Introduction
2. Background
Emulsions are a class of dispersed systems consisting of two immiscible liquids, one is the dispersed-phase and the other is the continuous phase. The droplets of emulsions are generally larger than 0.5 µm in diameter. Emulsions also usually contain an emulsifying agent or emulsifier, which has two principal functions: (1) to decrease the interfacial tension between the two liquid phases (usually oil and water), thereby enabling easier formation of the emulsion, and (2) to stabilize the dispersed-phase against coalescence once it is formed. Emulsions play an important role in a number of household and industrial applications. The use of emulsions covers a broad field, ranging from lubrication and cooling of equipment in metal-working processes to more delicate use like cosmetics. Some of the industries where emulsions are of considerable importance are paint, food, polymer, petroleum, agriculture, pharmaceutical, textile, paper, leather, polish, and printing.1,2 The rheological behavior of emulsions is of interest in the many industrial applications just mentioned. Knowledge of the rheological properties of emulsions is required for the design, selection, and operation of the equipment involved in the mixing, storage, and pumping of emulsions. In particular, it is important to be able to predict the viscosity of the emulsion η as a function of the volume fraction of the dispersed-phase φ. A vast amount of published literature exists on the viscosity behavior of solids-in-liquid suspensions.2-14 However, emulsions of two immiscible liquids have received less attention. In this work, the existing viscosity equations for emulsions are first reviewed. Several new equations are then proposed and evaluated in light of a large body of experimental data for concentrated emulsions, covering a broad range of dispersed-phase to continuous-phase viscosity ratios.
For dilute and moderately concentrated dispersions of rigid, noncolloidal particles (non-Brownian hard spheres), the relative viscosity ηr can be written as
* To whom correspondence should be addressed. Phone: (519) 888-4567, ext. 2985. Fax: (519) 746-4979. E-mail:
[email protected].
ηr ) f(φ)
(1)
where ηr is defined as the ratio of dispersion viscosity η to continuous-phase viscosity ηc, and φ is the volume fraction of the dispersed-phase. According to eq 1, the relative viscosity of different dispersed systems is the same when comparison is made at the same volume fraction of the dispersed-phase. One example of such zero-parameter rheological equations of state is the celebrated Einstein equation12,13 for the relative viscosity of very dilute, non-Brownian dispersions of hard spheres. For concentrated dispersions of non-Brownian hard spheres, the relative viscosity is not only a function of φ, it also depends on φm, the maximum packing volume fraction of particles. Thus,
ηr ) f(φ,φm)
(2)
One example of such single-parameter rheological equations of state is the Krieger and Dougherty equation14 for the relative viscosity of concentrated, non-Brownian dispersions of hard spheres. The rheological behavior of emulsions is more complicated than that of solids-in-liquid suspensions as the dispersed particles of emulsions are deformable in nature. For example, at low concentrations of the dispersed-phase, suspensions of spherical rigid particles exhibit Newtonian behavior, whereas emulsions of deformable particles exhibit shear-thinning and viscoelastic properties, even at very low values of the dispersed-phase concentration.15-18 Under a steady macroscopic shear flow, the droplets of emulsions are subjected to two opposing effects (1): (a) a viscous stress of magnitude ηcγ˘ that tends to elongate the droplet and (b) a stress of magnitude σ/R that tends to minimize
10.1021/ie010266u CCC: $20.00 © 2001 American Chemical Society Published on Web 10/16/2001
Ind. Eng. Chem. Res., Vol. 40, No. 23, 2001 5667
the surface energy and hence tends to maintain the droplet in a spherical shape. Therefore, the equilibrium shape of the droplet is governed by the ratio of viscous stress to σ/R; this ratio is referred to as the capillary number (NCa):
NCa ) ηcγ˘ R/σ
(3)
where γ˘ is the shear rate, R is the droplet radius, and σ is the interfacial tension. When the capillary number is small (NCa f 0), the deformation of the droplets is negligible and the droplets can be treated as spherical. However, at high capillary numbers, the deformation of droplets can be quite significant. 2.1. Single-Parameter Viscosity-Concentration Equations for Emulsions. At low capillary numbers (NCa f 0), where the emulsion droplets are nearly spherical, and under creeping flow conditions (particle Reynolds number f 0), the relative viscosity of dilute and moderately concentrated emulsions of noncolloidal droplets can be written as19
ηr ) f(K,φ)
(4)
where K is a parameter, defined as the ratio of dispersedphase viscosity ηd to continuous-phase viscosity. According to eq 4, the relative viscosity of different emulsion systems is the same when comparison is made at the same volume fraction of dispersed-phase and the same viscosity ratio K. One example of such singleparameter rheological equations of state is the celebrated Taylor equation20 for the relative viscosity of very dilute emulsions of nearly spherical noncolloidal droplets:
[
5K + 2 φ [2K + 2]
ηr )
[
]
10(K + 1) + 3φ(5K + 2) 10(K + 1) - 2φ(5K + 2)
]
(6)
In the limit φ f 0, this equation reduces to the Taylor equation (eq 5). Palierne6 recently developed a theory for the linear viscoelastic behavior of concentrated emulsions of two immiscible liquids. The interaction between neighboring drops was taken into account using the Lorentz sphere method which is equivalent to the effective medium method utilized by Oldroyd.15 For emulsions of two immiscible Newtonian liquids with uniform droplet size, Palierne’s analysis gives the following expression for the complex shear viscosity:
[
3 1 + φH 2 η* ) ηc* 1 - φH
]
(7)
where η* is the complex shear viscosity of emulsion, ηc* is complex shear viscosity of continuous phase, and H is given by
H)
2[jω(ηd - ηc)(19ηd + 16ηc) + (4σ/R)(5ηd + 2ηc)] [jω(2ηd + 3ηc)(19ηd + 16ηc) + (40σ/R)(ηd + ηc)] (8)
In eq 8, ω is the frequency of oscillation. In the limit ω f 0, eq 8 gives
H)
ηc + 2.5ηd η φ ηr ) ) 1 + ηc ηc + ηd )1+
Oldroyd’s analysis gives the following expression for the zero-shear relative viscosity of emulsions:
5K + 2 5(K + 1)
(9)
Equations 7 and 9 lead to the following equation for zero-shear relative viscosity of emulsion:
(5)
In the limit of K f ∞, eq 5 reduces to a zero-parameter equation of Einstein.12,13 Schowalter et al.21 and Frankel and Acrivos17 derived constitutive equations for dilute emulsions composed of two immiscible Newtonian liquids. The constitutive equations of Schowalter et al. and Frankel and Acrivos clearly demonstrate that even very dilute emulsions of deformable particles exhibit shear-thinning and viscoelastic properties (such as normal stresses). However, in the limit NCa f 0, where the emulsion droplets are spherical, these constitutive equations reduce to the Taylor equation (eq 5). Oldroyd15 utilized an effective medium approach to extend Taylor’s analysis to higher volume fractions of dispersed-phase (beyond first order in φ). According to the approach utilized by Oldroyd, the emulsion is first treated as an equivalent “effective medium” which is homogeneous and has the same viscosity as that of the emulsion (that is, η). Then a small portion of the effective homogeneous medium is replaced by the actual components of the emulsion. The properties of the effective medium are then determined by insisting that if a small portion of the effective homogeneous medium is replaced by the actual components of the emulsion, no difference in rheological behavior could be detected by macroscopic observations.
[ [
] ]
3 5K + 2 1+ φ 2 5(K + 1) ηr ) 5K + 2 1-φ 5(K + 1)
(10)
Equation 10 is the same as Oldroyd’s equation eq 6. It should also be noted that eq 6 or eq 10 when expanded in terms of power series in φ gives the coefficient of φ2 to be 2.5 in the limit K f ∞. This value of 2.5 is about one-half the actual theoretical value determined by Batchelor and Green.22 Therefore, the Oldroyd equation (eq 6) is expected to underpredict the values of ηr. Yaron and Gal-Or23 and Choi and Schowalter18 used the “cell-model” approach to derive viscosity equations for concentrated emulsions of spherical droplets (NCa f 0). In the cell model approach, the droplets of emulsion are envisioned to reside in well-defined unit cells in which the flow behavior is the same. The radius of the unit cell (spherical in shape) is chosen to give the actual volume fraction φ of the dispersed-phase, i.e., Rcell ) R/φ1/3. The flow behavior in a unit cell is determined by solving Stokes equations in and around a reference droplet surrounded by the continuous-phase fluid. The interaction between the reference droplet and the droplets in the neighboring cells is taken into account solely through the boundary conditions at the cell boundary. Yaron and Gal-Or23 and Choi and Schow-
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alter18 have used different boundary conditions to solve this problem. Yaron and Gal-Or’s analysis gives the following expression for the relative viscosity of concentrated emulsions: ηr ) 1 + φ
{
}
5.5[4φ7/3 + 10 - (84/11)φ2/3 + (4/K)(1 - φ7/3)]
10(1 - φ10/3) - 25φ(1 - φ4/3) + (10/K)(1 - φ)(1 - φ7/3) (11)
In the limit K f ∞, eq 11 reduces to the Happel equation.24 Choi and Schowalter’s analysis leads to the following for the viscosity of concentrated emulsions: ηr ) 1 + φ
{
}
2[(5K + 2) - 5(K - 1)φ7/3]
4(K + 1) - 5(5K + 2)φ + 42Kφ5/3 - 5(5K - 2)φ7/3 + 4(K - 1)φ10/3 (12)
In the limit K f ∞, eq 12 reduces to the Simha equation.25 It is interesting to note that in the limit φ f 0 Choi and Schowalter’s equation (eq 12) reduces to the Taylor equation (eq 5). However, the Yaron and Gal-Or equation (eq 11) does not reduce to the Taylor equation (eq 5) in the limit φ f 0. Using the differential scheme originally proposed by Brinkman26 and Roscoe,27 Phan-Thien and Pham28 developed the following equation for the relative viscosity of concentrated emulsions:
η2/5 r
[
]
2ηr + 5K 2 + 5K
3/5
) (1 - φ)-1
[
]
ηd + 0.4η η dφ ηd + η
(14)
This equation is separable, since we can write it as
[
]
ηd + η dη ) 2.5dφ ηd + 0.4η η
(15)
Upon integration with the limits η f ηc at φ ) 0 and η f η at φ ) φ, eq 15 gives
ηr
[
]
2ηr + 5K 2 + 5K
3/2
) exp(2.5φ)
ηr ) ηr(K,φm,φ)
(16)
This new single-parameter viscosity-concentration equation for concentrated emulsions reduces to the wellknown Arrhenius equation29 for solids-in-liquid suspensions in the limit K f ∞.
(17)
where φm is a function of droplet size distribution. Starting from the Taylor equation (eq 5) and using the concept of effective medium, Pal30 recently derived the following new two-parameter rheological equations of state for concentrated emulsions of spherical droplets:
(13)
In the limit K f ∞, eq 13 reduces to the equation proposed by Brinkman26 and Roscoe27 for solids-in-liquid suspensions. I will now take the liberty of suggesting a new singleparameter rheological equation of state for concentrated emulsions. Consider an emulsion of viscosity η(φ). If the droplet concentration of this emulsion is increased by dφ, the increment increase in viscosity dη resulting from the addition of the new droplets can be calculated from the Taylor equation (eq 5) by treating the emulsion into which the new droplets are added as an equivalent homogeneous medium of viscosity η(φ). Thus,
dη ) 2.5
In summary, the single-parameter viscosity-concentration equations for emulsions predict that the relative viscosity is a function of viscosity ratio K and volume fraction of the dispersed-phase φ. 2.2. Two-Parameter Viscosity-Concentration Equations for Emulsions. One serious limitation of the single-parameter viscosity-concentration equations is that they predict the viscosity of emulsion to be independent of the droplet size distribution. The droplet size/droplet size distribution does not appear in any of the equations discussed in the preceding section. Furthermore, the single-parameter equations either predict finite viscosities over the entire φ range or predict divergence of viscosity only at φ ) 1.0. This is in contradiction with a vast amount of experimental data available on solids-in-liquid suspensions (K f ∞) which indicates that the divergence of viscosity occurs at φ ) φm, the maximum packing volume fraction of dispersedphase. For random close packing of monosized spherical particles, φm is 0.637.5 To account for the effects of droplet size distribution and divergence of viscosity, the correct form of the rheological equation of state for emulsions is a twoparameter equation:
ηr
[ [
ηr
] [ ] ] ( )
2ηr + 5K 2 + 5K
3/2
2ηr + 5K 2 + 5K
3/2
) exp
2.5φ 1 - φ/φm
) 1-
φ φm
(18)
-2.5φm
(19)
In the limit K f ∞, these equations reduce to the wellknown formulas for solids-in-liquid suspensions; eq 18 reduces to the Mooney equation31 and eq 19 reduces to the Krieger and Dougherty equation.14 The salient features of the two-parameter viscosityconcentration equations (eqs 18 and 19) are (a) they take into account the effect of viscosity ratio, (b) they exhibit divergence of viscosity at φ ) φm, and (c) they take into account the effect of droplet size distribution through the parameter φm. Figure 1 shows the effect of droplet size distribution on the relative viscosity of emulsions at two values of K: K ) 0 and K ) ∞. Emulsions with bimodal droplet size distribution are considered. The relative viscosities of mixtures of large and small particles are plotted as a function of the volume fraction of small particles, with total volume fraction as a parameter. The particle size ratio is 5:1. The viscosities are calculated from eq 19 using φm values predicted from the Ouchiyama and Tanaka formula:32-34
φm )
∑D3i fi 1 ∑(Di∼Dh )3fi + β ∑[(Di + Dh )3 - (Di∼Dh )3]fi
where
(20)
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Figure 1. The effect of droplet size distribution on the relative viscosity of bimodal emulsions. The particle size ratio between large and small particles is 5:1.
β)1+
4 h (8φo - 1)D 13 m
∑
D h )
[
(Di + D h )2 1 -
∑
[D3i
(3/8)D h
]
(Di + D h) 3
fi
- (Di∼D h ) ]fi
∑Di fi.
(21) (22)
Here φom 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
h) ) 0 (Di ∼ D ) Di - D h
for Di e D h for Di > D h
(23) (24)
The maximum packing concentration for monodisperse emulsions (φom) is taken to be 0.637. The Ouchiyama and Tanaka formula has been used by several investigators35-37 to predict the value of φm. For example, Gupta and Seshadri35 utilized Ouchiyama and Tanaka’s expression to predict the maximum packing concentration for solids-in-liquid suspensions. The experimentally measured values of φm were found to agree reasonably well with those predicted from the Ouchiyama and Tanaka formula. The following points should be noted from Figure 1: (a) at high values of φ (total volume fraction of dispersedphase), a large reduction in relative viscosity occurs when a monodisperse emulsion is changed to a bimodal emulsion. This effect is negligible when φ < 0.55; (b) at high values of φ, the plots of relative viscosity vs volume fraction of fine particles exhibit a minimum at a fine particle volume fraction of 0.1; and (c) with the increase in viscosity ratio K, the relative viscosity of emulsion becomes more sensitive to droplet size distribution. This can be seen more clearly in Figure 2. At K ) ∞, a large reduction in relative viscosity occurs when a monodisperse emulsion is changed to a bimodal one. However,
at K ) 0 the reduction in relative viscosity is much smaller at the same value of φ. A large reduction in relative viscosity observed when a monodisperse emulsion is changed to a bimodal emulsion at high values of φ can be explained as follows: when a fine monodisperse emulsion is added to a coarse monodisperse emulsion while keeping φ constant, the large droplets are replaced by the fine droplets. As the fine droplets can easily fit into the voids between the large droplets, the net effect of adding the fine droplets is to separate the large ones, and consequently the droplets are no longer closely packed. Because of the increased mobility of the droplets, a large reduction in viscosity takes place. Also, note that when a monodisperse emulsion is changed to a bimodal emulsion, an increase in φm (maximum packing volume fraction of dispersed-phase) is expected. 3. Further Development of Two-Parameter Viscosity-Concentration Equations for Emulsions Equations 18 and 19 can be written in a more general form as
ηr
[
]
2ηr + 5K 2 + 5K
3/2
) f(φ,φm)
(25)
This equation is valid over the full range of the viscosity ratio K; that is, 0 e K e ∞. In the limit K f ∞, the lefthand side term of eq 25 becomes ηr. Consequently, eq 25 reduces to eq 2:
ηr ) f(φ,φm)
(2)
An important point to note is that the function f(φ,φm) is the same as that in eqs 2 and 25. This implies that one can transform relative viscosity-concentration equations for solids-in-liquid suspensions to corresponding equations for emulsions by replacing ηr with ηr[(2ηr + 5K)/(2 + 5K)]3/2. On the basis of this reasoning, the following new two-parameter equations for emulsion
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Figure 2. The effect of viscosity ratio K on the relative viscosity of bimodal emulsions.
viscosity are proposed:
ηr
]
3/2
[ [
] [ ] [
) 1+
2ηr + 5K 2 + 5K
3/2
2ηr + 5K 2 + 5K
3/2
ηr
[
[
[
2ηr + 5K 2 + 5K
ηr
ηr
ηr
[
) 1-
φ φm
) 1-
φ φm
] [
2ηr + 5K 2 + 5K
3/2
)
] [
2ηr + 5K 2 + 5K
]
1.25φ φ 1φm
] ]
2
-2.5
) 1+
(27)
4. Comparison of Experimental Data with Predictions of Viscosity Equations
(28)
Seventeen sets of viscosity-concentration data on emulsions covering a viscosity ratio K range of 3.9 × 10-4 to 1.17 × 103 are considered to evaluate the viscosity equations. These data were selected on the basis of the following criteria: (a) the capillary number is small (NCa f 0) so that deformation of droplets can be neglected; (b) the emulsions are stable (unflocculated), consisting of noncolloidal droplets with negligible colloidal interactions; and (c) the dispersed-phase volume fraction φ is less than 0.64, corresponding to random close packing of uniform spheres. For φ > 0.64, the droplets of emulsions are expected to undergo some deformation even when the capillary number is small. For non-Newtonian emulsions, only the zero-shear viscosity data are considered. Note that emulsions
-2.0
1/3 9 (φ/φm) 8 1 - (φ/φ )1/3 m
3/2
(26)
to the corresponding suspension equations: eq 26 reduces to the Eilers38 equation, eq 27 reduces to the Roscoe27 equation, eq 28 reduces to the Maron and Pierce39 equation, eq 29 reduces to the Frankel and Acrivos40 equation, and eq 30 reduces to the Chong et al.7 equation.
]
]
0.75(φ/φm) 1 - (φ/φm)
(29) 2
(30)
Equations 26-30 exhibit features similar to those of eqs 18 and 19. The relative viscosity increases with the increases in viscosity ratio K and volume fraction of dispersed-phase φ, and a divergence in relative viscosity occurs at φ ) φm. In the limit K f ∞, eqs 26-30 reduce
Table 1. Summary of Various Emulsion Systems Considered in the Present Work set no. 1a 1b 2 3 4a 4b 4c 4d 5a 5b 5c 5d 6 7 8 9 10
range of φ 0-0.60 0-0.60 0-0.60 0-0.5961 0-0.6284 0-0.5972 0-0.6149 0-0.6282 0-0.343 0-0.230 0-0.397 0-0.218 0-0.635 0-0.64 0-0.60 0-0.51 0-0.60
viscosity ratio (K) 10-3
4.15 × 1.12 × 10-2 5.82 × 10-2 2.57 5.52 5.52 5.52 5.52 5.573 12.35 21.74 29.41 1.17 × 103 64.064 7.127 3.906 × 10-4 142.7
ref 41 41 42 43 44 44 44 44 45 45 45 45 unpublished data 46 47 48 49
comments polymer-thickened oil-in-water emulsions polymer-thickened oil-in-water emulsions polymer-thickened oil-in-water emulsions mineral oil-in-water emulsions oil-in-water emulsions of sets 4a-d were prepared from the same oil (petroleum oil) and aqueous phase. however, the sauter mean diameter was different for different sets; set 4a, 21.4 µm; set 4b, 9.12 µm; set 4c, 8.1 µm; set 4d, 4.6 µm. oil-in-water emulsions containing milk fat. the viscosity measurements were made at 64 °c. the continuous phase consisted of skim milk, diluted skim milk, and concentrated milk. heavy oil-in-water emulsions mineral oil-in-water emulsions petroleum oil-in-water emulsions polymer-thickened oil-in-water emulsions mineral oil-in-water emulsions
Ind. Eng. Chem. Res., Vol. 40, No. 23, 2001 5671
Figure 3. (a, b) Comparison between the experimental data and the predictions of the various single-parameter viscosity-concentration equations for emulsions.
behave as non-Newtonian shear-thinning fluid when either the dispersed-phase volume fraction is high (φ >
0.5) or the continuous-phase itself is non-Newtonian (as in the case of polymer-thickened emulsions). Table 1
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Figure 4. (a-d) Comparison between the experimental data and the predictions of the two-parameter viscosity-concentration equations for emulsions.
gives further details on the various emulsion systems considered in the present work. 4.1. Comparison of Experimental Data with Single-Parameter Viscosity-Concentration Equations. Figure 3a,b show comparison between the experimental data and predictions of the various singleparameter viscosity-concentration equations. The ratio of predicted relative viscosity to experimental relative viscosity is plotted as a function of the dispersed-phase volume. The equations of Taylor (eq 5), Oldroyd/Palierne (eq 6), Phan-Thien and Pham (eq 13), and Pal (eq 16) all give good predictions of relative viscosity only at low values of φ (φ < 0.2). When φ > 0.2, eqs 5, 6, 13, and 16 underpredict the relative viscosity and the deviation between the experimental and predicted values increases with the increase in φ. The Yaron and Gal-Or equation (eq 11) gives reasonable predictions of relative viscosity for φ < 0.55: the predicted values are somewhat higher than the experimental values but the deviation is not very large. For φ > 0.55, the Yaron and Gal-Or equation tends to underpredict the relative viscosity by a large amount. The Choi and Schowalter equation (eq 12) gives good predictions of relative viscosity when φ < 0.2. At higher values of φ, eq 12 shows large deviation from experimental data. Thus, it can be concluded that the Yaron and Gal-Or equation (eq 11) represents the experimental data for concentrated emulsions more accurately as compared with the other single-parameter viscosity-concentration equations for emulsions provided that φ is less than 0.55. 4.2. Comparison of Experimental Data with Two-Parameter Viscosity-Concentration Equations. Figure 4a-d shows comparisons between the
experimental data and predictions of various twoparameter viscosity-concentration equations proposed in the paper. The experimental data are plotted as 0.6 vs volume fraction φ. The η0.4 r [(2ηr + 5K)/(2 + 5K)] value of the maximum packing volume fraction (φm) of droplets in the equations was taken as 0.637. According to several authors,50,51 the maximum packing volume fraction of monodisperse emulsions, where the droplets just touch each other without any significant deformation is approximately 0.637 (corresponding to random close packing of uniform spheres). Equation 18 gives good prediction of viscosities only when φ e 0.30. At higher values of φ, eq 18 overpredicts the viscosity of emulsions. Equation 19 gives good predictions of emulsion viscosity over the full range of φ covered. Equation 26 gives a reasonably good prediction of data over the full range of φ covered. Equation 27 gives good predictions only when φ < 0.30; at higher φ values, eq 27 overpredicts the viscosities. Equation 28 also slightly overpredicts the viscosities when φ > 0.40. Equation 29 gives good predictions of viscosities when φ g 0.05; at lower values of φ, eq 29 shows deviation from the experimental data. Note that eq 29 does not give ηr of unity as φ f 0. Equation 30 gives good prediction of emulsion viscosities over the full range of φ covered. It should be noted that some scatter of experimental data is expected as the emulsions were not monodisperse. Also, the droplet size distribution was not the same for different sets. 5. Conclusions The existing relative viscosity ηr vs volume fraction φ equations for concentrated emulsions of nearly spheri-
Ind. Eng. Chem. Res., Vol. 40, No. 23, 2001 5673
cal (small capillary number), noncolloidal droplets are first reviewed. Several new equations are then proposed and evaluated in light of a large body of experimental data for concentrated emulsions with small capillary numbers. The experimental data covers a broad range of viscosity ratio K. For non-Newtonian emulsions, limiting viscosities corresponding to zero shear rate are considered. On the basis of the analysis presented in the paper, the following conclusions are reached: • The existing ηr vs φ equations are single-parameter equations with viscosity ratio K as the only parameter. The single-parameter equations give good predictions of ηr only at low values of φ (φ < 0.2). A large deviation between the predicted and experimental values is seen at higher values of φ. • Among the single-parameter equations, the Yaron and Gal-Or23 equation is the best in terms of relative viscosity prediction of concentrated emulsions, provided that φ < 0.55. • The new ηr vs φ equations for concentrated emulsions proposed in the paper are two-parameter equations with viscosity ratio K and φm (maximum packing volume fraction of droplets) as the parameters. • Among the proposed two-parameter ηr vs φ equations for emulsions, the following equations give good predictions of emulsion viscosity:
ηr
[ [
ηr
ηr
3/2
2ηr + 5K 2 + 5K
3/2
[
]
2ηr + 5K 2 + 5K
ηr
] ( ] [
2ηr + 5K 2 + 5K
3/2
[
]
3/2
-2.5φm
φ φm
) 1+
1.25φ φ 1φm
1/3 9 (φ/φm) 8 φ 1/3 1φm
)
2ηr + 5K 2 + 5K
[
)
) 1-
[
( )
) 1+
]
]
2
φ > 0.05
]
0.75φ/φm φ 1φm
2
Acknowledgment The financial support for this work was provided by Natural Sciences and Engineering Research Council (NSERC) of Canada. Nomenclature D ) diameter of emulsion droplet D h ) number-average diameter of emulsion f ) number fraction of droplets H ) defined by eq 8 K ) viscosity ratio NCa ) capillary number R ) radius of emulsion droplet Greek Symbols β ) defined by eq 21 η ) viscosity of emulsion ηc ) viscosity of continuous-phase ηd ) viscosity of dispersed-phase ηr ) relative viscosity η* ) complex shear viscosity of emulsion ηc* ) complex shear viscosity of continuous phase φ ) volume fraction of dispersed-phase
φm ) maximum packing volume fraction φom ) maximum packing volume fraction of monodisperse system γ˘ ) shear rate ω ) frequency of oscillation σ ) interfacial tension
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Received for review March 23, 2001 Revised manuscript received August 15, 2001 Accepted August 15, 2001 IE010266U
