Association of Percolation Theory with Princen's Approach To Model

Research Note pubs.acs.org/IECR

Association of Percolation Theory with Princen’s Approach To Model the Storage Modulus of Highly Concentrated Emulsions Emilio Paruta-Tuarez and Philippe Marchal* Université de Lorraine, LRGP (GEMICO), UMR7274-CNRS, Nancy, F-54000, France ABSTRACT: An approach based on percolation theory is associated to the rigorousness of the two-dimensional (2D) model of close-packed, monodisperse, cylindrical emulsions established by Princen, to obtain an equation to model the dispersed-phase volume fraction (ϕ) dependence of the storage modulus (G′) of highly concentrated emulsions. A first-order Taylor expansion of this general percolation model leads to an expression similar to the three-dimensional (3D) model proposed by Princen and Kiss for real polydisperse emulsions, with a more satisfactory derivation of the function E(ϕ). Finally, the robustness of this model is tested against different experimental data collected from the literature.

1. INTRODUCTION The objective of this work was to deepen our understanding of how to model the dispersed-phase volume fraction (ϕ) dependence of the storage modulus (G′) of highly concentrated emulsions. During the past decade, a big effort has been made to elucidate this problem and many models have been proposed in the literature to describe the dependence of G′ with ϕ.1−19 We can particularly single out the classical publications of Mason, Bibette and Weitz,6,7 as well as the work of Lacasse et al.8 By using well-controlled emulsions consisting of droplets of a single size, Mason et al. related the dispersed-phase volume fraction to the packing of identical spheres. Therefore, in this approach, all the droplets have the same Laplace pressure. They found that the elastic shear modulus exhibits a universal dependence on dispersed-phase volume fraction (ϕ) when scaled by the Laplace pressure of droplets (σ/r), and they demonstrated that the volume fraction dependencies of the yield strain and scaled yield stress are independent of droplet size. From these observations, they concluded that the precise nature of the droplet packing and the Laplace pressure scale (σ/a) are the important microscopic properties that determine the yield behavior in emulsions.6,7 Lacasse et al. presented the first three-dimensional (3D) 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. They used the SE software to calculate the excess surface as the confinement is increased, under the constraint of a fixed droplet volume, and 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 that replaces the droplets with soft spheres that interact with their nearest neighbors through central-force potentials that reflect the behavior of the facets.8,20 In a previous research note,17 we have already reported and compared different models found in the literature, and we described, with more detail, the models proposed by Mougel et al.,14 Masalova and Malkin,15 and Foudazi et al.16 More specifically, we carefully analyzed the Princen and Kiss model, © 2013 American Chemical Society

since it is a rigorous detailed model that is derived from pressure/ volume surface relationships, leading, in particular, to an expression relating the geometric and thermodynamic properties of highly concentrated monodisperse emulsions to their elastic modulus.3,5 However, a phenomenological function E(ϕ) had to be introduced to accommodate experimental deviations. As mentioned by Princen and Kiss themselves, there is no satisfactory theoretical derivation of E(ϕ).5 To compensate, to some extent, for the lack of theoretical arguments, we will show that it is possible to derive this function from a general approach based on percolation theory. In fact, while percolation theory is one of the most popular approaches used to describe gelation as a critical phenomena in polymeric systems and concentrated suspensions, it has only been mentioned in a few papers in the case of emulsions.21−24 So, in this work, we want specifically to show that, similar to viscosity−concentration relationships, the percolation theory could provide a general theoretical framework in order to develop elasticity−concentration relationships to model the rheological behavior of highly concentrated emulsions. In this spirit, we obtained a general expression describing the ϕ dependence of G′ and leading to a satisfactory derivation of E(ϕ). Finally, we compared this general model with some experimental data as in a previous research note.17

2. THEORETICAL BACKGROUND As mentioned in the Introduction, percolation theory has been widely studied in polymer systems and concentrated suspensions to describe gelation as a critical phenomena. Critical phenomena are those which occur exactly at the so-called “sol−gel transition” or asymptotically to it. This sol−gel transition is also known as the percolation threshold, since gelation is a transition from a liquid-like state to gel-like state involving the formation of an infinite network (see Figure 1). Received: Revised: Accepted: Published: 11787

May 3, 2013 July 22, 2013 August 1, 2013 August 1, 2013 dx.doi.org/10.1021/ie401414u | Ind. Eng. Chem. Res. 2013, 52, 11787−11791

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Figure 1. Schematic representation of some of the basic definitions of percolation theory.

Figure 2 shows that, depending on the dispersed-phase volume fraction (ϕ), two categories of rheological models can be pointed

We also notice in Figure 2 that the viscosity (η) of the sol and the elasticity (G) of the gel diverge or vanish, respectively, in the so-called “sol−gel transition”, according to the following relations:27 η ∝ (pc − p)−k p → pc−

(1)

G ∝ (p − pc )t p → pc+

(2)

where η is the zero shear rate viscosity, G is the static shear modulus, and p is a conversion factor that represents the fraction of occupied sites by the particles or the fraction of bonds that have been formed between the molecules of the system (i.e., the ratio of the actual number of bonds at the given moment to the maximally possible number of such bonds).27 The critical point (pc) is the percolation threshold, or the “gel point”, where an infinite cluster starts to appear; k and t are the so-called critical exponents. In order to extend this theory to emulsions, the fraction of occupied sites (p) has been considered as the dispersed-phase volume fraction in emulsions. Therefore, one has ϕ ≡ p and ϕc ≡ pc or, at least, ϕ ∝ p and ϕc ∝ pc. A simple rewriting of eq 1 leads to

Figure 2. Schematic representation of viscosity and elasticity as a function of the dispersed-phase volume fraction for dilute, semidilute, concentrated, and highly concentrated dispersions.

−k

η ∝ (pc − p)

out: viscosity−concentration relationships from dilute to concentrated regimes and elasticity−concentration relationships from concentrated to highly concentrated regime.25 The concentrated regime is characterized, in particular, by the apparition of shear thinning, time dependence of material functions, thixotropy, divergence of the viscosity, apparition of yield stress and elasticity. More specifically, in this regime, a sol− gel transition from viscous to elastic behavior is typically observed at a critical dispersed-phase volume fraction (ϕc), at which the droplets form an interconnected network. The value of ϕc depends on the droplets size distribution and on the formulation of the emulsion, but typical values range from the random loose packing fraction (ϕrlc ≅ 0.56) to the maximum packing volume fraction (ϕfcc = 0.74), reached for a face-centered cubic packing of monodisperse hard spheres.1,24 In comparison, a value of ϕc ≅ 0.58 represents the case of monodispersed hardsphere suspensions.26

−k

∝ (ϕc − ϕ)

−k ⎛ ϕ⎞ ∝ ⎜⎜1 − ⎟⎟ ϕc ⎠ ⎝

(3)

With ϕc ≡ ϕm (the maximum packing volume fraction), eq 3 is nothing but a Krieger−Dougherty or Quemada-type equation describing the evolution of the relative viscosity ηr as a function of the volume fraction in the Newtonian limit: ηr =

−k ⎛ ϕ⎞ ⎟⎟ = ⎜⎜1 − ηs ϕm ⎠ ⎝

ηo

(4)

where ηo is the zero shear viscosity of the dispersion and ηs the viscosity of the suspending medium (i.e., the continuous phase). Originally introduced in 1951 by Mooney28 and modified in 1959 by Krieger and Dougherty29 to take into account excluded volume effect through the free volume Vf = V(1 − ϕ/ϕm), eq 4 is well-established and widely used to describe the viscosity of dispersions.30,31 In the specific case of emulsions, in 1989, Pal and Rhodes proposed a particular form of eq 4 based on effective 11788

dx.doi.org/10.1021/ie401414u | Ind. Eng. Chem. Res. 2013, 52, 11787−11791


Research Note

medium theory.32 Following another approach based on critical phenomena, Douglas and Garboczi (1995)33 and Bicerano, Douglas, and Brune (1999)34 obtained an expression equivalent to eq 3, in the general case of dispersions, with a critical exponent of k = 2. The same value of k is obtained by applying the principle of minimization of energy dissipated by viscous effect.35 In 2006, Thivilliers et al. used percolation theory to study the kinetic evolution of the bulk elastic modulus (G′) of a bicontinuous oilin-water emulsion obtained by a gelation process based on the unrelaxed coalescence of partially crystallized droplets.23 Finally, viscosity−concentration relationships obtained both via effective medium theory and the critical phenomena approach have been compared and applied to emulsions by Bullard et al. (2009), from the dilute regime to the concentrated regime.24 Beyond the concentrated regime, a sol−gel transition occurs and the pertinent coefficient of transport is no longer the viscosity but rather the elasticity (see Figure 2). In the concentrated regime, the most popular study is, without doubt, the work of Princen and Kiss, which leads to the following expression for the elastic shear modulus: ⎛ σ ⎞ 1/3 G = αP⎜ ⎟ϕ (ϕ − ϕP) ⎝ R32 ⎠ ⎛ σ ⎞ 1/3 = 1.769⎜ ⎟ϕ (ϕ − 0.712) ⎝ R32 ⎠

Figure 3. Adjustment of the function E(ϕ) as a function of the dispersed-phase volume fraction (ϕ) for the experimental data of Princen and Kiss.5

3. RESULTS AND DISCUSSION 3.1. Development of the Princen and Kiss Equation from a Percolation Theory Model. In the same way as for eq 1, eq 2 can be expressed considering the fraction of occupied sites (p) as the dispersed-phase volume fraction in emulsions, i.e., ϕ ≡ p and ϕc ≡ pc or, at least, ϕ ∝ p and ϕc ∝ pc. So, we can rewrite eq 2 as

with ϕ > ϕP (5)

G′ ∝ (p − pc )t ∝ (ϕ − ϕc)t

where σ is the interfacial tension of the fluid−fluid interface and R32 is the surface-volume mean drop radius. αP and ϕP are parameters of the model whose values are discussed below (αP = 1.769 and ϕP = 0.712). In two dimensions, this model has been established on the basis of rigorous physical arguments and involves no adjustable parameters. However, the 3D model is a simple extrapolation of the two-dimensional (2D) model, which requires the introduction of a phenomenological function E(ϕ) to take into account the experimental deviations, leading to G=

σ 1/3 ϕ E(ϕ) R32

⇔

E(ϕ) =

⎛ σ ⎞ G = α⎜ ⎟(ϕ − ϕc)t ⎝ R32 ⎠

(6)

(9)

where α is a dimensionless adjustable parameter. A Taylor expansion of this general expression describing the ϕ dependence of G′ (eq 9) lead to

The exponent “ /3” is the result of a generalization to three dimensions of the exponent “1/2” of the 2D model. We have shown in a previous paper17 that the linear dependence of E on ϕ (E(ϕ)) is a consequence of the particular set of experimental data exploited by Princen and Kiss. As mentioned by Princen and Kiss themselves,5 there is no satisfactory theoretical derivation of E(ϕ). Retaining the upper seven data points in Figure 3, they obtained, via a least-squares fit, 1

⎛ σ ⎞ G = α⎜ ⎟(ϕ − ϕc)t ⎝ R32 ⎠ t ϕc ⎞ σ t⎛ ϕ ⎜1 − ⎟ =α ϕ⎠ R32 ⎝

E(ϕ) = αP(ϕ − ϕP)

2 ⎞ ⎛ σ ⎞ t⎛ ϕ t(t − 1) ϕc ⎟⎟ − ··· = α⎜ ⎟ϕ ⎜⎜1 − t c + ϕ 2! ϕ2 ⎝ R32 ⎠ ⎝ ⎠

= 1.769(ϕ − 0.712) = 1.769ϕ − 1.259

(8)

According to the rigorous physical arguments of the twodimensional (2D) model of close-packed, monodisperse, cylindrical emulsions established by Princen, the shear modulus (G) is a function of the interfacial tension of the fluid (σ) and the drop radius (R32): G ∝ σ/R32.3 Therefore, eq 8 can be expressed as

GR32 σϕ1/3

with ϕ > ϕc

(7)

2 ⎞ ⎛ σ ⎞ t − 1⎛ t(t − 1) ϕc − ···⎟⎟ = α⎜ ⎟ϕ ⎜⎜ϕ − tϕc + 2! ϕ ⎝ R32 ⎠ ⎝ ⎠

It is worth remembering that, Princen and Kiss decided not to take into account the first experimental point corresponding to the most dilute emulsion in order to obtain the function E(ϕ). They considered that the value of G is overestimated, because the effect of gravitational drainage cannot be neglected at low volume fraction. That is why they retained only the next seven points, leading to the customary values αP = 1.769 and ϕP = 0.712 in eq 7. Therefore, the objective of the next section is to demonstrate that the empirical function E(ϕ) can be obtained from a general model established in the framework of percolation theory.

⎛ σ ⎞ t−1 = α⎜ ⎟ϕ (ϕ − tϕc + ε(ϕ)) ⎝ R32 ⎠

(10)

where ε(ϕ) = 11789

2 3 t(t − 1) ϕc t(t − 1)(t − 2) ϕc + ··· − 2! 3! ϕ ϕ2 dx.doi.org/10.1021/ie401414u | Ind. Eng. Chem. Res. 2013, 52, 11787−11791


Research Note

that is not constant, and depends on formulation and process conditions. In Figure 4, the function E(ϕ) was represented as a

It leads to E(ϕ) =

GR32 σϕ1/3

= αϕ−1/3(ϕ − ϕc)t = αϕt − 4/3(ϕ − tϕc + ε(ϕ)) = αperco(ϕ − ϕperco)

(11)

with αperco = αϕt − 4/3

and

ϕperco = tϕc − ε(ϕ)

Thus, considering a first-order expansion and letting α ≅ αP, t ≅ 4/3 and tϕc − ε(ϕ) ≅ ϕP, eq 10 can be expressed as ⎛ σ ⎞ G = α⎜ ⎟(ϕ − ϕc)t ⎝ R32 ⎠

Figure 4. Adjustment of the function E(ϕ) as a function of the dispersed-phase volume fraction (ϕ) for the experimental data of (⧫) Pal (1999), (▲) Pal (2006), and (■) Paruta-Tuarez (2011).9,13,17

⎛ σ ⎞ t−1 = α⎜ ⎟ϕ (ϕ − tϕc + ε(ϕ)) ⎝ R32 ⎠ ⎛ σ ⎞ 1/3 ≅ αP⎜ ⎟ϕ (ϕ − ϕP) ⎝ R32 ⎠

function of ϕ for the data collected from the literature. The experimental data represented by solid diamonds (⧫) correspond to Pal’s 1999 work (values of G′/(σ/R) and ϕ collected f rom Figure 9 in ref 9): kerosene in deionized water emulsions of identical droplet size stabilized with Triton X-100, and prepared by dilution of a parent emulsion (ϕ = 0.8961).9 The experimental data represented by solid triangles (▲) correspond to Pal’s 2006 work (values of G′/(σ/R) and ϕ collected f rom Figure 10 in ref 13): a series of light mineral oil in deionized water emulsions of identical droplet size stabilized with Triton X-100, referred as fine emulsions with a droplet size radius of 1.28 μm.13 The experimental data represented with solid squares (■) correspond to Paruta-Tuarez’s 2011 work: a series of deionized water in dodecane emulsions stabilized with a mixture of nonionic surfactants at HLB = 5.6.17 Thus, by fitting the experimental data with eq 14, we obtain α = 12.99, ϕc = 0.560, and t = 2.44 for the 1999 experimental data of Pal9 [⧫], α = 6.196, ϕc = 0.560, and t = 2.44 for the 2006 data from Pal13 [▲], and α = 34160, ϕc = 0.80, and t = 6.183 for the 2011 data from Paruta-Tuarez17 [■]. We notice that the model based on percolation theory fits all experimental data collected from literature fairly well (see Figure 4). Hence, this model based on percolation theory could be exploited in order to fit the G′ of highly concentrated emulsions as function of ϕ.

(12)

Finally, from eq 12, one obtains the function E(ϕ): E(ϕ) =

GR32 σϕ1/3

≅ αP(ϕ − ϕP)

(13)

These last equations (eqs 12 and 13) are formally equivalent to those proposed by Princen and Kiss (eqs 5 and 6), and appears to be a first-order approximation of eq 9, obtained on the basis of a percolation approach and the rigorous physical arguments of the 2D model of Princen. The function E(ϕ) deduced from this general model, describing the ϕ dependence of G′ (eq 9), can be written as E(ϕ) =

GR32 1/3

σϕ

= αϕ−1/3(ϕ − ϕc)t

(14)

Figure 3 shows the adjustment of experimental data of Princen and Kiss (seven points) with eqs 13 and 14. The dotted line corresponds to the fitting of eq 13 with αP = 1.769 and ϕP = 0.712, while the solid line corresponds to the fitting of eq 14 to experimental points with α = 2.2825, ϕc = 0.660 and t = 1.333. We notice that a better fit of all experimental data (eight points) of the 1986 research by Princen and Kiss can be obtained with eq 14. As mentioned in the Introduction, eq 14 will be used to adjust some experimental data in the next section. In order to adjust the experimental data, we apply the approach used by Princen and Kiss, that is, the function E(ϕ) = G′R32/(σϕ1/3) was represented as a function of ϕ. 3.2. Adjustment of Some Experimental Data Using the Percolation Theory Model. In the same way as in a previous research note,17 we chose two types of experimental data collected from the works of Pal9,13 and Paruta et al.17 From Pal’s works, we collected the experimental data of highly concentrated emulsions with an average drop size constant and an emulsion preparation procedure similar to those used by Princen and Kiss. From the work of Paruta et al., we chosen the series of highly concentrated emulsions at HLB = 5.6, with an average drop size

4. CONCLUSIONS The association of a percolation approach and the rigorous physical arguments of the two-dimensional (2D) model of closepacked, monodisperse, cylindrical emulsions established by Princen, lead to a general model (eq 9) describing the ϕ dependence of G′. From this equation, we have demonstrated that a first-order Taylor expansion leads to expressions similar to those proposed by Princen and Kiss, and, more importantly, a more-satisfactory derivation of the function E(ϕ). Moreover, the advantage of this approach based on percolation theory is that it takes into account the fast increases of G′ when ϕ approaches unity, which it is not achieved with the Princen and Kiss equation, because of the linear ϕ dependence of E(ϕ). Finally, from the comparison of some experimental data collected in the literature with this model, we have found a fairly good fitting, easily applicable to other experimental data. 11790



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Research Note

(23) Thivilliers, F.; Drelon, N.; Schmitt, V.; Leal-Calderon, F. Bicontinuous emulsion gels induced by partial coalescence: Kinetics and mechanism. Europhys. Lett. 2006, 76, 332−338. (24) Bullard, J. W.; Pauli, A. T.; Garboczi, E. J.; Martys, N. S. A comparison of viscosity−concentration relationships for emulsions. J. Colloid Interface Sci. 2009, 330, 186−193. (25) Quemada, D.; Berli, C. Energy of interaction in colloids and its implications in rheological modeling. Adv. Colloid Interface Sci. 2002, 98, 51−85. (26) Vanmegen, W.; Underwood, S. The Glass-Transition in Colloidal Hard-Spheres. J. Phys. Condens. Matter 1994, 6, A181−A186. (27) Stauffer, D.; Coniglio, A.; Adam, M. Gelation and critical phenomena. In Polymer Networks; Dušek, K., Ed.; Advances in Polymer Science, Vol. 44; Springer: Berlin/Heidelberg, 1982; pp 103−158. (28) Mooney, M. The viscosity of a concentrated suspension of spherical particles. J. Colloid Sci. 1951, 6, 162−170. (29) Krieger, I. M.; Dougherty, T. J. A mechanism for non-newtonian flow in suspensions of rigid spheres. Trans. Soc. Rheol. 1959, 3, 137−152. (30) Heyes, D. M.; Sigurgeirsson, H. The Newtonian viscosity of concentrated stabilized dispersions: Comparisons with the hard sphere fluid. J. Rheol. 2004, 48, 223−248. (31) Quemada, D. Modélisation rhéologique structurelle: dispersions concentrées et fluides complexes; Éditépar Tecet DocLavoisier: Paris, 2006. (32) Pal, R.; Rhodes, E. Viscosity/concentration relationships for emulsions. J. Rheol. 1989, 33, 1021−1045. (33) Douglas, J. F.; Garboczi, E. J. Intrinsic viscosity and the polarizability of particles having a wide range of shapes. In Advances in Chemical Physics; Prigogine, I., Rice, S. A., Eds.; John Wiley & Sons, Inc.: Chichester, U.K., 2007; pp 85−153. (34) Bicerano, J.; Douglas, J. F.; Brune, D. A. Model for the viscosity of particle dispersions. J. Macromol. Sci., Polym. Rev. 1999, 39, 561−642. (35) Quemada, D. Rheology of concentrated disperse systems and minimum energy dissipation principle. Rheol. Acta 1977, 16, 82−94.

AUTHOR INFORMATION

Corresponding Author

*Tel.: +33383175143. Fax: +33383175185. E-mail: philippe. [email protected]. Notes

The authors declare no competing financial interest.

■

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) 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. (4) Princen, H. M. Rheology of foams and highly concentrated emulsions: II. Experimental study of the yield stress and wall effects for concentrated oil-in-water emulsions. J. Colloid Interface Sci. 1985, 105, 150−171. (5) 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. (6) Mason, T. G.; Bibette, J.; Weitz, D. A. Elasticity of Compressed Emulsions. Phys. Rev. Lett. 1995, 75, 2051−2054. (7) Mason, T. G.; Bibette, J.; Weitz, D. A. Yielding and flow of monodisperse emulsions. J. Colloid Interface Sci. 1996, 179, 439−448. (8) 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. (9) Pal, R. Yield stress and viscoelastic properties of high internal phase ratio emulsions. Colloid Polym. Sci. 1999, 277, 583−588. (10) Ponton, A.; Clement, P.; Grossiord, J. L. Corroboration of Princen’s theory to cosmetic concentrated water-in-oil emulsions. J. Rheol. 2001, 45, 521−526. (11) Rocca, S.; Stébé, M. J. Mixed Concentrated Water/Oil Emulsions (Fluorinated/Hydrogenated): Formulation, Properties, and Structural Studies. J. Phys. Chem. B 2000, 104, 10490−10497. (12) 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. (13) Pal, R. Rheology of high internal phase ratio emulsions. Food Hydrocolloids 2006, 20, 997−1005. (14) Mougel, J.; Alvarez, O.; Baravian, C.; Caton, F.; Marchal, P.; Stébé, M.-J.; Choplin, L. Aging of an unstable w/o gel emulsion with a nonionic surfactant. Rheol. Acta 2006, 45, 555−560. (15) Masalova, I.; Malkin, A. Y. Rheology of highly concentrated emulsions−Concentration and droplet size dependencies. Appl. Rheol. 2007, 17. (16) 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. (17) Paruta-Tuarez, E.; Marchal, P.; Sadtler, V.; Choplin, L. Analysis of the Princen and Kiss equation to model the storage modulus of highly concentrated emulsions. Ind. Eng. Chem. Res. 2011, 50, 10359−10365. (18) Masalova, I.; Foudazi, R.; Malkin, A. Y. The rheology of highly concentrated emulsions stabilized with different surfactants. Colloids Surf., A 2011, 375, 76−86. (19) Trappe, V.; Prasad, V.; Cipelletti, L.; Segre, P. N.; Weitz, D. A. Jamming phase diagram for attractive particles. Nature 2001, 411, 772− 775. (20) Brakke, K. The surface evolver. Exp. Math. 1992, 1, 141−165. (21) Buzza, D. M. A.; Cates, M. E. Uniaxial elastic modulus of concentrated emulsions. Langmuir 1994, 10, 4503−4508. (22) Valdez, M. A.; Acedo-Carrillo, J. I.; Rosas-Durazo, A.; Lizardi, J.; Rinaudo, M.; Goycoolea, F. M. Small-deformation rheology of mesquite gum stabilized oil in water emulsions. Carbohydr. Polym. 2006, 64, 205− 211. 11791


Association of Percolation Theory with Princen's Approach To Model

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