Analysis of the Princen and Kiss Equation To Model the Storage

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Analysis of the Princen and Kiss Equation To Model the Storage Modulus of Highly Concentrated Emulsions Emilio Paruta-Tuarez,† Philippe Marchal,† Veronique Sadtler,*,† and Lionel Choplin† †

Laboratoire Reactions et Genie des Procedes (LRGP), UPR 3349 CNRS, 1 Rue Grandville BP 20451, 54001 Nancy Cedex, France ABSTRACT: The universality of the model proposed by Princen and Kiss is analyzed in the case of highly concentrated water-in-oil emulsions containing dispersed-phase volume fractions (j) ranging from 0.89 to 0.97. Although Princen and Kiss equation has been rigorously established for a two-dimensional system and involves no adjustable parameter, the model for a tridimensional system, which is an extrapolation of the 2D model, requires the introduction of a phenomenological linear function E(j) to account for experimental deviations. As mentioned by Princen and Kiss themselves, there is no satisfactory theoretical derivation of E(j). Indeed, in this paper we point out that the linear dependence in j of E(j) is a consequence of the particular set of experimental data exploited by Princen and Kiss. Another choice of experimental data could have led to propose other mathematical functions since at very high volume fraction, some experimental data found in the literature show a more rapid increase of the storage modulus G0 with j than predicted by Princen and Kiss equation, which tends to underestimate the values of G0 . Finally, for the studied highly concentrated emulsions, the dispersed-phase volume fraction dependence of storage modulus is discussed.

1. INTRODUCTION Emulsions are dispersions of two immiscible liquids, such as oil and aqueous phase, stabilized by a surfactant. When the dispersed-phase volume fraction (j) is larger than the maximum packing volume fraction (jo) of about 0.74, the emulsion is referred to as a highly concentrated emulsion. This last fraction corresponds to the most compact arrangement of monodisperse hard spheres, reached for a face-centered cubic packing.1,2 In these emulsions, the drops are no longer present as spheres but are distorted into polyhedral shape. Hence, and due to this deformable nature, it is possible to formulate emulsions with dispersed-phase volume fractions up to around 0.99. Highly concentrated emulsions are also called high internal phase ratio emulsions (HIPRE),1,3 biliquid foams,4 aphrons,5 hydrocarbon gels,6 and gel emulsions.7,8 In the past decade, the potential of highly concentrated emulsions in pharmaceuticals, cosmetics, food, explosives and other applications has been investigated and several studies have been realized in order to understand the influence of preparation processes and formulation on the structure and properties of the resulting emulsions. Such highly concentrated emulsions are expected to exhibit complex rheological properties. Hence, a particular emphasis has been put on the rheological behavior of these emulsions. In this paper, we first analyze the way of how the Princen and Kiss equation was developed. It is worth remarking that the expression proposed by Princen and Kiss in 1986 is considered as a well-established model. However, we point out that if this declaration is not debatable for the two-dimensional (2D) model of close-packed, monodisperse, cylindrical emulsions, for real polydisperse emulsions, the three-dimensional (3D) model is an extrapolation of the 2D model, followed by an empirical adaptation accounting for experimental observations.9 That is why, we reproduce in a concise manner the Princen and Kiss arguments to obtain the proposed model. Then, we adjust our experimental r 2011 American Chemical Society

data and those of other authors using the same approach of Princen and Kiss. Finally, we compare our experimental data with the models found in the literature and we discuss the dispersedphase volume fraction dependence of the storage modulus.

2. THEORETICAL BACKGROUND Princen was the first to carry out a theoretical and experimental analysis of the rheology of highly concentrated emulsions.10 He studied the relationship between the shear stress and the shear strain in the elastic region as a function of the dispersedphase volume fraction (j). For the first time, he established the expressions for these rheological properties using a two-dimensional model of close-packed, monodisperse, cylindrical emulsions, i.e., emulsions that consist of infinitely long, parallel, cylindrical drops.10 He made a complete and detailed analysis, and he proposed the following relations for the yield stress (τo) and the shear modulus (G0 ):10 τo ¼ 1:050

σ cos θ 1=2 ~ j F max R

ð1Þ

G0 ¼ 0:525

σ cos θ 1=2 j R

ð2Þ

where σ is the interfacial tension of the fluidfluid interface, θ is the contact angle, R is the drop radius, j is the dispersed-phase volume fraction, and F~max is a dimensionless term that corresponds to the contribution of each drop to the yield stress. The term F~max can be evaluated graphically and depends on both j and θ. On the basis of simple analogy, he said to expect that the Received: February 1, 2011 Accepted: August 2, 2011 Revised: May 24, 2011 Published: August 02, 2011 10359

dx.doi.org/10.1021/ie200237c | Ind. Eng. Chem. Res. 2011, 50, 10359–10365

Industrial & Engineering Chemistry Research

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Figure 1. Adjustment of the function E(j) as a function of the dispersed-phase volume fraction (j) for the experimental data of Princen and Kiss.

corresponding expressions for real, monodisperse, spherical emulsions would be expressed as follows:10 τ o ¼ C1

σ cos θ 1=3 ~ j F max R

ð3Þ

G 0 ¼ C2

σ cos θ 1=3 j R

ð4Þ

0

where C1 and C2 are unknown numerical constants. However, on the basis of experimental evidence, in a subsequent work Princen in collaboration with Kiss concluded that eq 4 is untenable. Furthermore, they affirmed that the equation implies that the variation of the shear modulus is only 10% for a dispersed-phase volume fraction ranging from 0.74 to 1. Hence, Princen and Kiss9 proposed a corrected version: σ 1=3 G0 ¼ j EðjÞ ð5Þ R32 where E(j) is a function of j, which was arbitrarily introduced to take into account the experimental dependence of storage modulus (G0 ) with the dispersed-phase volume fraction (j). That is why we said in the introduction that for real polydisperse emulsions the model proposed was simply speculated. In fact, Princen and Kiss mentioned that the derivation of the function E(j) is not satisfactory. This function, E(j) = G0 R32/(σj1/3), was plotted as a function of j (Figure 1), and a least-squares fit was applied to the experimental data (seven data points). The studied highly concentrated oil-in-water emulsions were constituted of water with 20% of a commercial anionic surfactant (Alipal CD128, 58% active, ex GAF Corp.) and paraffin oil. The function was adjusted as EðjÞ ¼ 1:769j 1:259 ¼ 1:769ðj 0:712Þ

Ap and jo are 1.769 and 0.712, respectively.9 It is worth remarking that after working out the value of 1.769, they found that the value of 0.712 was reasonably close to 0.74. So, they related this term to the volume fraction of closed-packed monodisperse spheres.9 Mason et al. proposed key observations on the dispersedphase volume fraction (j) dependence of the elastic shear modulus of monodisperse emulsions.11 By using well-controlled emulsions consisting of droplets of a single size, the dispersedphase volume fraction was simply related to the packing of identical spheres. So, in this approach, all the droplets have the same Laplace pressure. The studied system was a silicone oilin-water emulsion stabilized by an ionic surfactant, sodium dodecylsulphate (SDS), at a concentration of 0.01 M. Basic polydisperse emulsions were first prepared by the classical inversion method, and a crystallization fractionation technique was used to purify the emulsions and to obtain highly monodisperse emulsions after about five fractionation steps, where the remaining polydispersity is less than 10% of the radius. They found that the elastic shear modulus exhibits a universal dependence on the dispersed-phase volume fraction (j) when scaled by the Laplace pressure of droplets (σ/r), which can be described by

ð6Þ

In this way, a combination of the eqs 5 and 6 leads to the wellknown Princen and Kiss model that relates the static shear modulus (G) to the dispersed-phase volume fraction (j), the surface-volume mean drop radius (R32), and the interfacial tension (σ): σ G ¼ 1:769 j1=3 ðj 0:712Þ ð7Þ R32 where Ap and jo are adjustment parameters, determined by a least-squares fit of GR32/σj1/3 vs j. The values of parameters

Gp ≈jef f ðjef f jc Þ

σ r

ð8Þ

where jc ≈ 0.635 is the volume fraction of randomly close packed spheres, and r is the radius of the undeformed droplet. G0p is the storage modulus at the inflection point from the graph of the frequency (ω) dependence of the storage modulus (G0 ). jeff is the effective volume fraction that includes the film thickness between the droplet interfaces (h) and the volume of the droplets, and is expressed as 3h jef f ≈j 1 þ ð9Þ 2r This expression was obtained by linear interpolation between the values: h ≈ 175 Å (for low j), and h ≈ 50 Å (at the highest j). The h value at the highest j was estimated by comparison to measurements of the film thickness at similar osmotic pressures. Next, they measured the yield transition of monodisperse emulsions as a function of the volume fraction (j) and droplet radius (a). They demonstrated that the volume fraction dependencies of the yield strain and scaled yield stress are independent of droplet size. They concluded of these observations that the precise nature of the droplet packing and the Laplace pressure scale, σ/a, are the important microscopic properties which determine the yield behavior in emulsions.12 Lacasse et al. presented the first three-dimensional computer simulation modeling the elasticity of a disordered emulsion. They also proposed a new, more realistic interdroplet potential, based on numerical results obtained by calculating the change in surface energy of a single droplet as it is compressed using Brakke’s surface evolver (SE) software.13,14 They used the SE software to calculate the excess surface as the confinement is increased, under the constraint of a fixed droplet volume, and the data were well described by !R R 3 E1 j R ¼C 1 ¼C 1 ð10Þ jc h n 10360

dx.doi.org/10.1021/ie200237c |Ind. Eng. Chem. Res. 2011, 50, 10359–10365


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They concluded that the response of a droplet to compression is a nonlocal phenomenon and depends on the number of planes used to compress the droplet. To describe the elastic properties of a disordered droplet packing, they used a model which replaces the droplets by soft spheres that interact with their nearest neighbors through central-force potentials that reflect the behavior of the facets. They used the SE results to define a repulsive, centralforce interdroplet potential, 8 " #R 3 > > < 2C 2R 1 ðd < 2RÞ d ð11Þ UðdÞ ¼ > > :0 ðd g 2RÞ

eq 14 to experimental data gave A = 3.66 and jc = 0.96, taking Do = 30 nm as the micelle diameter.16 In the work of Mougel et al., the emulsions consisted of water droplets dispersed in oil phase containing sorbitan monooleate (Span 80). The surfactant/oil weight ratio was 0.81. The particularity of this model is the prediction of a divergence of G0 when j approaches the unity, unlike other models. Masalova and Malkin have proposed a model based on either geometrical or dimensional arguments.18 Both approaches lead to analogous equations. Geometrical and energetic approach: δ σðj jÞ G¼k ð2RÞ2

where d = 2h is the distance between the droplets centers, R is the undeformed soft sphere radius, and the factor 2 accounts for the two facets on the interacting pair. The coefficient C and the exponent R depend on n, which represents the number of facets. The shear modulus (G) is obtained from the excess energy density as a function of the extension ratio (λ), EN EN 0 1 ¼ þ Gðλ2 þ 2=λ 3Þ 2 V V EN ¼

N

∑ Uðdij Þ i, j > i

ð12Þ ð13Þ

where E°N is the excess energy of the unstrained system, dij is the distance between point particles i and j. Good agreement with the experimental data was obtained with the model proposed. Mason et al. also presented experimental measurements of the osmotic pressure and complex shear modulus of crude monodisperse emulsions of polydimethylsiloxane (PDMS) silicone oil droplets in water compressed to different volume fractions.15 Polydisperse emulsions were stabilized by an ionic surfactant, sodium dodecylsulphate (SDS), at a concentration of 10 mM, and fractionated using the crystallization fractionation technique based on a droplet-size-dependent depletion attraction. They observed an excellent agreement of the simulation based on the three-dimensional computer model with the experiments results. However, despite the success of their model for describing the static elastic modulus and osmotic pressure of compressed emulsions, it cannot predict the full ω dependence of the viscoelastic moduli, since it does not consider dissipative mechanisms. They found that the frequency dependence of the viscoelastic moduli exhibits the characteristic rheological features of a colloidal glass: a plateau in storage modulus (G0 ), a minimum in loss modulus (G00 ), and a frequency associated with minimum ωm which exhibits a cusp at the glass transition volume fraction. In the past years, some research groups have also shown a squared average drop-size-dependence of the storage modulus.3,1618 To our knowledge, two models have been proposed in the literature to predict such a R2 dependence. Mougel et al. have proposed a model based on dominant van der Waals interactions.16 On the basis of dimensional arguments accounting for such dominant interaction, they obtained: G0 ¼

2πσADo j R 2 ðjc jÞ

ð14Þ

where A is a proportionality coefficient of the order of unity, Do an intermolecular distance, jc is a critical dispersed-phase volume fraction, and σ is the interfacial tension. The fitting of

Dimensional approach: !m δσ G¼k ðj jÞn ð2RÞ2

ð15Þ

ð16Þ

where k is a factor that reflects the transition from the diagonal elongation to the simple shear, δ is the width of a thin surface layer, R is the average drop radius, σ is the interfacial tension, j is the dispersed-phase volume fraction, and j* is a characteristic volume fraction of the dense packing; m and n are scaling factors. According to eq 15, the fitting of G(2R)2/σ vs j to experimental data leads to kδ = 1.73 104 m and j* = 0.71.18 The dispersed phase consisted of overcooled concentrated aqueous solution of inorganic salts, while the continuous phase was a paraffin oil with 15% of surfactant constituted of derivatives of poly(isobutylene) succinic anhydride. Recently, Foudazi et al. studied the interdroplet interactions in highly concentrated water-in-oil explosive emulsions. Moreover, the osmotic pressure and shear modulus of highly concentrated emulsions were modeled by considering both interfacial energy and interdroplet interaction, and by optimization and approximation methods of predicting film thickness.19 To explain the reciprocal squared diameter dependency of elastic modulus and to analyze the interdroplet interactions, the disjoining pressure, which takes into account the interaction between the surfaces of droplets, was included to build a complete model: σ G ¼ ½1:3jef f 2 ðjef f jÞ0:4 þ 0:105 þ fdis ðhÞ R 0:863jef f 8:0

ð17Þ

where jeff is the effective volume fraction of a dispersed phase with zero film thickness, j* is the effective volume fraction of close-packed spherical droplets with zero film thickness, and fdis(h) is a term that takes into account the disjoining pressure effect. In this model, it was assumed that interfacial and disjoining pressure contributions are addable. Furthermore, it was shown that the contributions of micellar, steric, and van der Waals forces to the elasticity are negligible, but an electrostatic interaction of surface layers containing adsorbed NH4+ ions may explain this unusual elasticity.19 As mentioned in the introduction, one of the objectives of the present paper is to discuss about the dispersed-phase volume fraction dependence on storage modulus of highly concentrated emulsions. It is worth remarking that we use an approach in which the physicochemical formulation is changed in terms of HLB number of surfactant or surfactant mixture. Hence, highly 10361



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Figure 3. Adjustment of the function E(j) as a function of the dispersed-phase volume fraction (j) for the experimental data of Pal.3 The nonlinear fits are so close to each other that the different curves cannot be distinguished on the graph.

Figure 2. (a) E(j) as a function of the dispersed-phase volume fraction (j) for all our experimental data. (b) Adjustment of the function E(j) for highly concentrated emulsions at HLB = 5.6. The linear fit of experimental data does not take into account the point corresponding to the dispersed phase volume fraction j = 0.97.

concentrated emulsions are prepared at different interfacial tension and average drop size, as described in the next section.

3. MATERIALS AND METHODS 3.1. Materials. Dodecane was obtained from Aldrich (Reagent Plus, 99% Purum) and used as received. Water was deionized and Milli-Q filtrated with a Millipore apparatus. Three nonionic surfactants were used as received: sorbitan monooleate (Span 80, HLB = 4.3), sorbitan monolaurate (Span 20, HLB = 8.6), and polyethoxylated (20EO) sorbitan trioleate (Tween 85, HLB = 11) supplied by Sigma and Fluka. The total concentration of surfactant or surfactant mixture (1 wt %) was maintained constant in all emulsions. Dodecane/surfactant mixture and water were used as the continuous and dispersed phase, respectively. 3.2. Preparation of the Highly Concentrated Inverse Emulsions. A 100 g amount of highly concentrated water-in-oil emulsions was prepared using the same semibatch process described in a previous work.20 In this study, the surfactant was incorporated into the oil phase and the aqueous phase was added at a flow rate of 15 g/min and an agitation speed of 500 rpm. Three formulations, in terms of HLB number, were prepared: 5.6, 7.7, and 10. They were obtained by using a mixture of two

surfactants. The surfactant mixture HLB number was calculated according to a linear mixing rule on a weight basis.21 As the total surfactant concentration was maintained constant, the surfactant quantity in all the samples was 1 g. The dispersed-phase volume fraction (j) ranged from 0.89 to 0.97. 3.3. Storage Modulus. The storage modulus (G0 ) of the emulsions was determined with a stress-controlled rheometer (AR 2000, TA Instruments) equipped with a plate/plate geometry of 40 mm diameter (coated aluminum) at a gap of 1.5 mm. The values of G0 were determined via small amplitude oscillatory tests at a frequency of 10 rad/s, into the linear viscoelastic domain of the samples, one hour after the end of the emulsion preparation process. Elastic modulus of the prepared emulsions is essentially independent of the frequency (ω), as discussed in greater detail in a previous publication.22 3.4. Interfacial Tension. The interfacial tensions were determined with a spinning drop tensiometer model 300, from the University of Texas. This instrument was modified to control the sample temperature. Aqueous phase and continuous phase (oil and surfactants) were the external media and internal media, respectively. 3.5. Average Drop Size. The average drop sizes (R) of emulsions were determined using the analysis of incoherent polarized steady light transport (AIPSLT) and the geometrical model proposed in a previous work.20

4. RESULTS AND DISCUSSION 4.1. Determination of the Princen and Kiss Equation Using Different Experimental Data. For our experimental data, we

apply the approach used by Princen and Kiss, that is, the function E(j) = G0 R32/(σj1/3) was represented as a function of j for the different highly concentrated water-in-dodecane emulsions prepared (see Figure 2). For the three formulations studied (HLB = 5.6, 7.7, and 10), it is obvious (Figure 2a) that the evolution of the function E(j) cannot be represented by a linear function. In our case, a corrected adjustment could be obtained using a nonlinear fit, for example, power-law fit, exponential fit, or rational fit. For example, in Figure 2b, we show the adjustment of the function E(j) for the highly concentrated emulsions at HLB = 5.6. So, we can conclude that, in the approach used by Princen and Kiss, the fit of the function E(j) depends on the system studied. It is worth remembering that, for real polydisperse highly 10362



Figure 4. Adjustment of the function E(j) as a function of the dispersed-phase volume fraction (j) for the experimental data of Pal.3,23 The nonlinear fits are so close to each other that the different curves cannot be distinguished on the graph.

concentrated emulsions, the model proposed by Princen and Kiss (eq 7) was extrapolated, and the function E(j) was introduced from empirical observations, as discussed in section 2. In consequence, this model cannot be used in a general form. To confirm this statement, we have analyzed the experimental data collected from the works of Pal.3,23 It is worth remarking that the average drop size is constant in the works of Pal and Princen and Kiss. Contrarily, the average drop size of our highly concentrated emulsions depends on formulation and process conditions. So, we have chosen Pal’s experimental data, since he used an emulsion preparation procedure similar of the Princen and Kiss one. We have applied once more the approach used by Princen and Kiss to adjust and represent the experimental data. In Figure 3, the experimental data of Pal’s work are plotted.3 The studied system consists of kerosene, deionoized water, and Triton X-100 (a commercially available nonionic surfactant). In this study, a series of eight O/W emulsions of identical droplet size was prepared by dilution of a parent emulsion (j = 0.8961).3 In 2006, the oil phase was changed by a light mineral oil. To study the influence of polydispersity on the rheology of highly concentrated emulsions, two emulsion series of identical droplet size were prepared by dilution of parent emulsions. The series were referred to as fine and coarse emulsions, with a droplet size radius of 1.28 and 1.95 μm, respectively. Then, the parent emulsions of the fine and coarse emulsion series were mixed in different proportions to prepare seven mixed fine-and-coarse emulsions of different average droplet size. Figure 4 presents the experimental data of the emulsion series referred to as fine emulsions.23 In both studies (1999 and 2006), Pal used the model proposed by Princen and Kiss to interpret the storage modulus of their highly concentrated oil-in-water emulsions. After comparison of experimental values of storage modulus with the prediction of Princen and Kiss equation, he found that the model underpredicts modulus values. He attributes the underprediction to the fact that the effective volume fraction of the dispersed-phase may be significantly higher than the actual volume fraction because of the finite interdroplet film thickness. For the system, he showed that even if eq 7 is modified and an effective volume fraction is introduced, at high values of j, the agreement between his experimental data and the Princen and Kiss equation is not satisfactory. His explanation for this deviation was that the parameters of eq 7

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Figure 5. Storage modulus as a function of formulation (HLB number of surfactant or surfactant mixture) and dispersed-phase volume fraction.

Table 1. Average Drop Size and Interfacial Tension as a Function of Formulation and Dispersed-Phase Volume Fraction R (μm) HLB

σ (mN/m)

j = 0.97

j = 0.93

j = 0.89

5.6

5.7

8.1

4.6

3.0

7.7

4.8

6.5

2.6

1.7

1.6

5.9

2.3

1.5

10

are not universal constants and may vary from one system to another.3 In fact, the problem is more dramatic because of the strong nonlinear evolution of G0 at very high dispersed-phase volume fractions: It is evidently not possible to fit accurately a nonlinear set of experimental data with a linear function, since a straight line remains a straight line even by changing the values of the parameters. In Figures 3 and 4, we show the equations of the adjustment of the function E(j) using a power-law fit, an exponential fit, or rational fit. By introducing these functions in eq 5, we can have an idea of how the Princen and Kiss equation would be if the experimental data of Pal were used. 4.2. Relationship between Storage Modulus, Average Drop Size, and Interfacial Tension. Figure 5 shows the variation of storage modulus (G0 ) as a function of the dispersed-phase volume fraction (j) for the different formulations studied (HLB = 5.6, 7.7, 8.6, and 10). In a first time, for all formulations, we can notice that the storage modulus increases with the dispersedphase volume fraction, as reported in the literature.3,24,25 In a second time, we can clearly observe a distinctive behavior of emulsions at HLB = 10. For this formulation, G0 values are always smaller than for the other HLB numbers. This behavior is due to the influence of the formulation, particularly to the proximity of the so-called optimum formulation, that bring a decrease of interfacial tension (see Table 1), as discussed with more details in a previous work.22 To analyze the dispersed-phase volume fraction (j) dependence of the storage modulus of our highly concentrated emulsions, the different models proposed in the literature were tested against our experimental data. From all models, the best fit of our experimental data is obtained with the equation proposed by Mougel et al. (eq 14). The particularity of this equation is the prediction of a divergence of G0 when j approaches unity, unlike other models. That is why, we carefully revised the demonstration of this model. It is worth 10363



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parameter, and that they depend on the formulation. Nevertheless, it is important to remember that this parameter is an average one, related to an intermolecular distance (Do) and a distance between the surfaces of two droplets (D), which are not well-defined, especially for complex systems.

Figure 6. Adjustment of experimental data (G0 R2/σ vs j) to the model of Mougel et al., modified (eq 19) for all the formulations studied.

noting that, in this approach based on dominant van der Waals interactions, it was supposed that the distance between the surfaces of two droplets (D) was independent of j, and it was assumed that this distance was of the order of the intermolecular distance (Do). However, except in particular cases, these distances are not generally of the same order of magnitude. So, we consider that the approximation D ≈ Do should not be made in general and that the model should be expressed as G0 ¼

2πσADo 2 j DR 2 ðjc jÞ

ð18Þ

Moreover, as pointed out by Israelachvili,26 in a complex system it is not at all obvious to clearly define Do and D. So, we can write eq 5 as G0 ¼

2πHL σj R 2 ðjc jÞ ADo2/D

ð19Þ

where HL = is an average distance accounting for the different molecular interactions. The value of HL can be determined by adjusting eq 19 to experimental data. Furthermore, from a phenomenological point of view, we suggest that the divergence occurs at a critical dispersed-phase volume fraction (jc), beyond which it becomes impossible to prepare the emulsion. Hence, we could explore the idea that the storage modulus is related to this jc (upper limit). In this sense, even if in some cases it is possible to prepare highly concentrated emulsions up to j = 0.99, an upper limit jc should exist, above which it is not possible to incorporate more dispersed phase. Therefore, if this upper limit is taken into account, it becomes possible to describe accurately experimental data on the whole range of volume fractions as shown in Figure 6. The value of jc was experimentally determined by preparing highly concentrated emulsions for different dispersed-phase fractions approaching unity, on a weight basis for practical purposes. For all formulations studied, emulsions were obtained up to a dispersed-phase weight fraction fw = 0.98, which corresponds to j = 0.974. For fw = 0.99, corresponding to j = 0.987, it was not possible to prepare any emulsions with the studied system at the established process conditions. This suggests that the value of jc is between j = 0.974 and j = 0.987. As a consequence, we chose the average of these two values, leading jc = 0.981. G0 , R, σ, and jc being known, it is easy to determine the values of parameter HL. So, by adjustment of experimental data, we obtain 6.72, 3.85, and 3.27 nm, for HLB = 5.6, 7.7, and 10, respectively. We can notice that the HL values are typical of molecular distance, which is in agreement with the definition given to this

5. CONCLUSIONS Even if the Princen and Kiss equation associated with the linear form of the function E(j) has shown, to a large extent, a successful applicability to describe the rheological behavior of concentrated emulsions, its applicability is not universal. We showed that the function E(j) depends on the system studied. For many highly concentrated emulsions, the storage modulus increases more rapidly with dispersed-phase volume fraction, than predicted by Princen and Kiss model. It means that we have to make use of nonlinear increasing functions to account for the divergence of the storage modulus at very high dispersed-phase volume fractions. From the comparison of our experimental values with the different models in the literature, we have found a good agreement with the model proposed by Mougel et al. that takes into account the divergence of G0 when j approaches the unity, contrary to other models. We carefully analyzed this model, and we pointed out some modifications. In particular, we have suggested that, the divergence of G0 is related to a critical dispersed-phase volume fraction (jc), which is considered as an upper limit, beyond which it becomes impossible to prepare the emulsion. Finally, we can notice the robustness of this model which fits fairly well the experimental data for several physicochemical formulations leading to different average drop sizes and interfacial tensions. ’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. Tel: +33383175079.

’ ACKNOWLEDGMENT We thank the Venezuelan Ministry of Science and Technology for financial help through a FUNDAYACUCHO scholarship for E.P.-T and the Postgraduate Cooperation Program PCP between FONACIT-Venezuela and MRE-France. We acknowledge M. J. Stebe of the “Laboratoire de Physico-Chimie des Collo€ides at Universite Henri Poincare” (Nancy, France) for making available the tensiometer. We gratefully acknowledge the discussion and support of Christophe Baravian of the “Laboratoire d’Energetique et de Mecanique Theorique et Appliquee” (Vandoeuvre Cedex, France). ’ REFERENCES (1) Lissant, K. J. The geometry of high-internal-phase-ratio emulsions. J. Colloid Interface Sci. 1966, 22, 462–468. (2) Princen, H. M. Highly concentrated emulsions. I. Cylindrical systems. J. Colloid Interface Sci. 1979, 71, 55–66. (3) Pal, R. Yield stress and viscoelastic properties of high internal phase ratio emulsions. Colloid Polym. Sci. 1999, 277, 583–588. (4) Sebba, F. Biliquid foamsA preliminary report. J. Colloid Interface Sci. 1972, 40, 468–474. (5) Sebba, F. Foams and biliquid foams: Aphrons; Wiley: New York, 1987. (6) Ebert, G.; Platz, G.; Rehage, H. Elastic and rheological properties of hydrocarbon gels. Ber. Bunsen-Ges. 1988, 92, 1158–1164. 10364



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(7) Kunieda, H.; Fukui, Y.; Uchiyama, H.; Solans, C. Spontaneous formation of highly concentrated water-in-oil emulsions (gel-emulsions) Langmuir 1996, 12, 2136–2140. (8) Ravey, J. C.; Stebe, M. J.; Sauvage, S. Water in fluorocarbon gel emulsions: Structures and rheology. Colloids Surf., A 1994, 91, 237–257. (9) Princen, H. M.; Kiss, A. D. Rheology of foams and highly concentrated emulsions: III. Static shear modulus. J. Colloid Interface Sci. 1986, 112, 427–437. (10) Princen, H. M. Rheology of foams and highly concentrated emulsions: I. Elastic properties and yield stress of a cylindrical model system. J. Colloid Interface Sci. 1983, 91, 160–175. (11) Mason, T. G.; Bibette, J.; Weitz, D. A. Elasticity of Compressed Emulsions. Phys. Rev. Lett. 1995, 75, 2051–2054. (12) Mason, T. G.; Bibette, J.; Weitz, D. A. Yielding and flow of monodisperse emulsions. J. Colloid Interface Sci. 1996, 179, 439–448. (13) Lacasse, M.-D.; Grest, G. S.; Levine, D.; Mason, T. G.; Weitz, D. A. Model for the elasticity of compressed emulsions. Phys. Rev. Lett. 1996, 76, 3448. (14) Brakke, K. The surface evolver. Experiment. Math. 1992, 1, 141–165. (15) Mason, T. G.; Lacasse, M.-D.; Grest, G. S.; Levine, D.; Bibette, J.; Weitz, D. A. Osmotic pressure and viscoelastic shear moduli of concentrated emulsions. Phys. Rev. E 1997, 56, 3150. (16) Mougel, J.; Alvarez, O.; Baravian, C.; Caton, F.; Marchal, P.; Stebe, M.-J.; Choplin, L. Aging of an unstable w/o gel emulsion with a nonionic surfactant. Rheol. Acta 2006, 45, 555–560. (17) Malkin, A. Y.; Masalova, I.; Slatter, P.; Wilson, K. Effect of droplet size on the rheological properties of highly-concentrated w/o emulsions. Rheol. Acta 2004, 43, 584–591. (18) Masalova, I.; Malkin, A. Y. Rheology of highly concentrated emulsionsConcentration and droplet size dependencies. Appl. Rheol. 2007, 17. (19) Foudazi, R.; Masalova, I.; Malkin, A. Y. The role of interdroplet interaction in the physics of highly concentrated emulsions. Colloid J. 2010, 72, 74–92. (20) Paruta-Tuarez, E.; Fersadou, H.; Sadtler, V.; Marchal, P.; Choplin, L.; Baravian, C.; Castel, C. Highly concentrated emulsions: 1. Average drop size determination by analysis of incoherent polarized steady light transport. J. Colloid Interface Sci. 2010, 346, 136–142. (21) Salager, J. L. In Encyclopedia of Emulsion Technology; Becher, P., Ed.; M. Dekker: New York, 1988; Vol. 3. (22) Paruta-Tuarez, E.; Sadtler, V.; Marchal, P.; Choplin, L.; Salager, J.-L. Making use of the formulationcomposition map to prepare highly concentrated emulsions with particular rheological properties. Ind. Eng. Chem. Res. 2011, 50, 2380–2387. (23) Pal, R. Rheology of high internal phase ratio emulsions. Food Hydrocolloids 2006, 20, 997–1005. (24) Pons, R.; Erra, P.; Solans, C.; Ravey, J.-C.; Stebe, M.-J. Viscoelastic properties of gel-emulsions: Their relationship with structure and equilibrium properties. J. Phys. Chem. 1993, 97, 12320–12324. (25) Taylor, P. The effect of an anionic surfactant on the rheology and stability of high volume fraction O/W emulsion stabilized by PVA. Colloid Polym. Sci. 1996, 274, 1061–1071. (26) Israelachvili, J. N. Van der Waals dispersion force contribution to works of adhesion and contact angles on the basis of macroscopic theory. J. Chem. Soc., Faraday Trans. 2 1973, 69, 1729–1738.

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Analysis of the Princen and Kiss Equation To Model the Storage

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