Role of the Secondary Minimum on the ... - ACS Publications

Langmuir 2005, 21, 6675-6687

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Role of the Secondary Minimum on the Flocculation Rate of Nondeformable Droplets German Urbina-Villalba* and Ma´ximo Garcıá-Sucre Instituto Venezolano de Investigaciones Cientı´ficas (IVIC), Centro de Fı´sica, Laboratorio de Fisicoquı´mica de Coloides, Aptdo. 21827, Caracas 1020-A, Venezuela Received January 5, 2005. In Final Form: May 5, 2005 The kinetic stability of suspensions is usually associated with a decrease in the flux of flocculating particles due to the action of a repulsive potential. However, previous calculations on bitumen drops suggest the possible occurrence of relatively fast aggregation rates in systems with large electrostatic barriers for primary minimum flocculation. This indicates a strong effect of the secondary minimum in the process of aggregation. Here, emulsion stability simulations (ESS) are used to study the aggregation behavior of 11 systems showing different depths of the secondary minimum and three particle sizes. Micron size drops (as those of Bitumen emulsions) usually exhibit deep secondary minima, which rarely occur between nanometer size particles. At high surfactant concentrations, these drops do not coalesce but can still show fast aggregation rates caused by irreversible secondary-minimum flocculation. On the other hand, the extent of coalescence in nanometer-size systems markedly depends on the height of the repulsive barrier. Furthermore, the secondary minimum of these smaller particles is usually shallow, causing reversible aggregation or no aggregation at all. In this article, the consequences of the referred behaviors on the magnitude of the stability ratio are discussed.

1. Introduction Despite significant advances in the understanding of the coalescence mechanism of deformable droplets,1-9 the phenomenon of Ostwald ripening,10-13 and the scaling behavior of flocculating suspensions,14-17 the prediction of emulsion stability is still very difficult. This limitation is mostly related to the concurrent occurrence of distinct destabilization processes and the fact that they all depend on the surfactant concentration and chemical structure. Realistic simulations of emulsion stability require the solution of a large number of technical problems that are not present in typical molecular dynamics simulations. This is true even in the case in which stability considerations are circumscribed to the processes of flocculation * To whom correspondence should be addressed. E-mail: [email protected]. (1) Ivanov, I. B.; Dimitrov, D. S. In Thin Liquid Films; Ivanov, I. B., Ed.; Marcel Dekker: New York, 1988. (2) Danov, K. D. In Fluid Mechanics of Surfactant and Polymer Solutions; Starov, V. M., Ivanov, I. B., Eds.; Springer: New York, 2004. (3) Velikov, K. P.; Velev, O. D.; Marinova, K. G.; Constantinides, G. N. J. Chem. Soc., Faraday Trans. 1997, 93, 2069. (4) Danov, K. D. Denkov, N. D.; Petsev, D. N.; Ivanov, I. B.; Borwankar, R. Langmuir 1993, 9, 1731. (5) Petsev, D. N. In Encyclopedia of Surface Science and Colloid Science; Hubbard, A., Ed.; Marcel Dekker: New York, 2002. (6) Petsev, D. N. Langmuir 2000, 16, 2093. (7) Nikolova, A.; Exerowa, D. Colloids Surf., A 1999, 149, 185. (8) Kashchiev, D.; Exerowa, D. J. Colloid Interface Sci. 1980, 77, 501. (9) Exerowa, D.; Balinov, B.; Kashchiev, D. J. Colloid Interface Sci. 1983, 94, 45. (10) Schmitt, V.; Leal-Calderon, F. Europhys. Lett. 2004, 67, 662. (11) Schmitt, V.; Cattelet, C.; Leal-Calderon, F. Langmuir 2004, 20, 46. (12) Schmitt, V.; Arditty, S.; Leal-Calderon, F. In Emulsions: Structure, Stability and Interactions; Petsev, D. N., Ed.; Elsevier: Amsterdam, 2004; Chapter 15, p 607. (13) Kabalnov, A. S.; Makarov, K. N.; Pertsov, A. V.; Shchukin, E. D. J. Colloid Interface Sci. 1990, 138, 98. (14) Sandku¨hler, P.; Sefcik, J.; Morbidelli, M. Langmuir 2005, 21, 2062. (15) Berka, M.; Rice, J. A. Langmuir 2005, 21, 1223. (16) Lin, M. Y.; Lindsay, H. M.; Weitz, D. A.; Ball, R. C.; Klein, R.; Meakin, P. Nature 1989, 339, 360. (17) Weitz, D. A.; Oliveria, M. Phys. Rev. Lett. 1984, 52, 1433.

and coalescence only. Emulsions are polydispersed systems in which neither the size of the particles nor their number is constant. Drops are massive deformable entities, which cannot be regarded as point particles or hard spheres. Simulations require periodic boundary conditions, which consider the coalescence process between real and imaginary particles as the drops move outside the simulation box. The code also needs to account for changes in the interaction potential during the course of the simulation. The potential is a function of the size of the drops and the degree of surfactant adsorption18 that change as a function of time.19 In the case of ionic surfactants, this often requires the numeric calculation of the surface potential at each time step. In the case of nonionic surfactants, it implies the frequent evaluation of the volume of overlap between the adsorbed surfactant layers of flocculated drops. In either case, the total interparticle potential varies sensibly in a distance of a few nanometers. This requires small time steps, making emulsion stability simulations (ESS) very expensive in computer time. Further complications arise from the fact that the movement of drops of submicron and micron size depends on the momentum exchange with the molecules of the solvent and their hydrodynamic interactions with other suspended particles. These two effects are not independent, and they are also difficult to incorporate in conventional Brownian dynamics (BD) simulations. However, they influence the structure of the aggregates,14 their flocculation rate,15 and the coalescence probability of aggregated drops. During the past decade, there has been significant progress in the solution of some of the problems outlined above.20-29 Of all remaining limitations, the consideration (18) Deminiere, B.; Stora, T.; Colin, A.; Leal-Calderon, F.; Bibette, J. Langmuir 1999, 15, 2246. (19) Ward, A. F. M.; Tordai, L. J. Chem. Phys. 1946, 14, 453. (20) Urbina-Villalba, G.; Toro-Mendoza, J.; Loszań, A.; Garcıá-Sucre, M. In Emulsions: Structure, Stability and Interactions; Petsev, D. N., Ed.; Elsevier: Amsterdam, 2004; Chapter 17, p 677. (21) Urbina-Villalba, G.; Garcıá-Sucre, M. Langmuir 2000, 16, 7975.

10.1021/la050024p CCC: $30.25 © 2005 American Chemical Society Published on Web 06/14/2005

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of the coalescence mechanism in the case of deformable droplets is by far the most serious difficulty. The thinning of the film between large drops (Ri > 5 µm, where Ri is the radius of the drop) involves at least six different steps.1-3 These include the formation of a dimple, the generation of a plane-parallel film, the appearance and enhancement of surface oscillations, and either the final coalescence of the drops or the formation of long lasting films. Surface-active molecules markedly affect each of these steps. Thus, surfactants (a) regulate the flux of the liquids in the vicinity of the interface,1 (b) facilitate the deformation of drops by decreasing the interfacial tension,4 (c) attenuate tension gradients produced by surface deformation,30,31 and (d) determine the rate of rupture of black films.7-9 It is generally accepted that the viscoelastic properties of the interface are closely connected to the coalescence process.32-34 However, the role of the surfactant molecules in the drainage of the intervening liquid between flocculated drops is still under debate.1,30-31,34-35 This problem was revisited recently, taking into account the dependence of the surface diffusion coefficient of the adsorbed surfactant molecules on the density of the adsorption layer. Following this procedure, Stoyanov and Denkov demonstrated that the equations describing the drainage and hydrodynamic stability of thin liquid films do not depend on the Gibbs elasticity.35 Instead, film thinning relies on the surface excess and on the friction coefficient of the adsorbed molecules. This allows a much simpler account of surfactant effects in the simulation of coalescence between deformable drops. Due to the simultaneous occurrence of several destabilization processes, the direct measurement of flocculation and coalescence rates in concentrated emulsions is not yet possible. The evaluation of flocculation rates is typically restricted to the short-time behavior of solid suspensions.36 The coalescence rates in oil/water emulsions are generally approximated by the coalescence times of oil droplets with a planar oil-water interface.37 Recently, a laborious procedure for the direct evaluation of coalescence kinetics using video-enhanced microscopy was implemented.38-41 It relies on the direct observation of coalescence and fragmentation times in dilute emulsions. Analytical expressions for the total number of aggregates and particles were obtained as a function of characteristics times for flocculation (in the primary and (22) Urbina-Villalba, G.; Garcıá-Sucre, M. Interciencia 2000, 25, 415. (23) Urbina-Villalba, G.; Garcıá-Sucre, M. Colloids Surf., A 2001, 190, 111. (24) Urbina-Villalba, G.; Garcıá-Sucre, M. Mol. Simul. 2001, 27, 75. (25) Urbina-Villalba, G.; Garcıá-Sucre, M. Toro Mendoza, J. Mol. Simul. 2003, 29, 393. (26) Urbina-Villalba, G.; Garcıá-Sucre, M.; Toro-Mendoza, J. Phys. Rev. E 2003, 68, 061408. (27) Urbina-Villalba, G. Langmuir 2004, 20, 3872. (28) Urbina-Villalba, G.; Toro-Mendoza, J.; Lozsań, A.; Garcıá-Sucre, M. J. Phys. Chem. 2004, 118, 5416. (29) Urbina-Villalba G.; Toro-Mendoza, J.; Garcıá-Sucre, M. Langmuir 2005, 21, 1719. (30) Marangoni, C. G. M. Ann. Phys. Chem. (Poggendorf) 1871, 143, 337. (31) Scriven, L. E.; Sternling, C. V. Nature 1960, 187, 186. (32) Yeung, A.; Moran, K.; Masliyah, J.; Czrnecki, J. J. Colloid Interface Sci. 2003, 265, 439. (33) Wasan, D. T.; Shah, S. M.; Aderangi, N.; Chan, M. S.; McNamara, J. J. Soc. Pet. Eng. J. 1978, 409. (34) Malhotra, A. K.; Wasan, D. T. In Thin Liquids Films: Fundamentals and Applications; Ivanov, I. B., Ed.; Marcel Dekker: New York, 1988; Chapter 12, pp 829-890. (35) Stoyanov, S. D.; Denkov, N. D. Langmuir 2001, 17, 1150. (36) Puertas, A. M.; de las Nieves, F. J. J. Phys.: Condensed Matter 1997, 9, 3313. (37) Dickinson, E.; Murray, B. S.; Stainsby, G. J. Chem. Soc., Faraday Trans. 1 1988, 84, 871.

Urbina-Villalba and Garcıá-Sucre

secondary minima), coalescence, and fragmentation (separation of the doublet into singlets).42 However, this technique and its theory are restricted to singlet-doublet emulsions. Thus, the theory reproduces the behavior of singlet-doublet emulsions with success, but some features of this behavior, like the stabilization of the average number of particles with time, are necessarily a consequence of the singlet-doublet equilibrium. Curiously, there are complex systems such as bitumen dispersions, which in some respects resemble the behavior of suspensions and are easier to simulate. Natural bitumen is a mixture of aromatic and aliphatic hydrocarbons that contain asphaltenic and resinous components with interfacial activity. Venezuelan Bitumen tapped at the Orinoco oil belt is typically composed of 14% saturates, 47% aromatics, 22% resins, and 17% asphaltenes. It has a density of 1.01 g/cm3 and a viscosity of 105 cP at 30 C.43-46 Thus, bitumen drops are neutrally buoyant, stable with respect to Ostwald ripening (due to their insoluble components), and nondeformable in the absence of surfactant molecules. According to colloidal particle scattering studies, bitumen drops behave as solid particles up to the moment of coalescence.47 For these experiments, natural bitumen from Athabasca oil sands47-49 was used to produce neutraly buoyant drops when dispersed in D2O. The deformation of the drops (determined by their crushed spherical segment) was estimated to be less than 0.01 nm for an average radius of 3 µm. It was observed that the drops were not perfectly spherical and exhibited protrusions in the range of 50-100 nm, which increase their repulsive force during collisions. These bumps affected the flocculation behavior producing either coagulation or elastic collisions in a 0.1 M KCl solution. However, the dynamics of the collision process was reproduced supposing that the effect of the protrusions was similar to that of a concentric disk around each drop. Thus, the use of an effective radius allowed the confirmation that the behavior of the drops conformed to the classical DLVO theory.50 Drops with zeta potentials of -85 mV coalesced in 1 M KCl solution maintaining their spherical shape almost until contact. In this case, the typical stages of film formation and thinning were not observed. Asphaltic bitumen emulsions behave similarly to oilsand bitumen emulsions. Asphalt is the residue from the distillation of petroleum widely used for road construction and surfacing. Like natural bitumen, asphalt has a density (38) Dukhin, S. S.; Mishchuk, N. A.; Loglio, G.; Liggieri, L.; Miller, R. Adv. Colloid Interface Sci. 2003, 100, 47. (39) Holt, O.; Saether, O.; Sjo¨blom, J.; Dukhin, S. S.; Mishchuk, N. A. Colloids Surf., A 1997, 123-124, 195. (40) Saether, O.; Sjo¨blom, J.; Verbich, S. V.; Mishchuk, N. A.; Dukhin, S. S. Colloids Surf., A 1998, 142, 189. (41) Holt, O.; Saether, O.; Sjo¨blom, J.; Dukhin, S. S.; Mishchuk, N. A. Colloids Surf., A 1998, 141, 269. (42) Mishchuk, N. O. In Emulsions: Structure, Stability and Interactions; Petsev, D. N., Eds.; Elsevier: Amsterdam, 2004; p 351. (43) Environment Canada, Orimulsion 400. A comparative Study; Emergencies Science Division, Environmental Technology Centre: Ontario, 1998; p 27. (44) Mohammed, R.; Di Lorenzo, M.; Marinõ, J.; Cohen, J. J. Colloid Interface Sci. 1997, 191, 517. (45) Rivas, H.; Nun˜ez, G.; Dalas, C. Visio´ n Tecnol. 1993, 1, 18. (46) Some physical properties of Venezuelan Bitumen can be found at these web sites: www.sovereign-publications.com/bitumene.htm, and www.worldenergy.com (47) Wu, X.; Czarnecki, J.; Hamza, N.; Masliyah, J. Langmuir 1999, 15, 5244. (See also www.ualberta.ca/∼masliyah/library/movies.htm). (48) Laroche, I.; Wu, X.; Masliyah, J. H.; Czarnecki, J. J. Colloid Interface Sci. 2002, 250, 316. (49) Wu, X.; Laroche, I.; Masliyah, J.; Czarnecki, J.; Dabros, T. Colloids Surf., A 2000, 174, 133. (50) Verwey, E. J. W.; Overbeek, J. Th. G. The Theory of Lyophobic Colloids, 1st ed.; Dover: New York, 1999; p 165.

Role of the Secondary Minimum

similar to that of water (1.02 at 25 C51,53) and a viscosity higher than 105 cP (7000 Pa s51-53). Their drops are usually of the order of microns and exhibit very large zeta potentials.52,53 These characteristics disfavor droplet deformation in bitumen/water dispersions.1 However, the use of stabilizers in bitumen emulsions is known to lower the interfacial tension considerably,44,55 increasing the probability of surface distortions. In previous papers, we used spherical nondeformable droplets in order to simulate the behavior of bitumen/ water emulsions.20-29 Unlike solid particles, these drops coalesce as soon as their interfaces touch. Because drops are not deformable, there is no transition between primary minimum flocculation and coalescence. Thus, drops can stay at the secondary minimum but they necessarily coalesce if they are able to jump over the repulsive potential barrier. This is a rough approximation to the coalescence mechanism, partially sustained on the physical properties of bitumen. Still, the simulations not only reproduce the thermal limit of Smoluchowski at very dilute concentration28,56 but also generate flocculation rates comparable in magnitude to the ones reported for distinct oil/water systems with similar volume fractions of the internal phase.29,57 This is possible using a very small number of particles when appropriate boundary conditions are implemented. Surprisingly, the flocculation behavior exposed by nondeformable droplets is not trivial. Among other features, these systems can exhibit fast flocculation rates even in the presence of high repulsive barriers for primary minimum aggregation.29 “Fast” flocculation meaning, in this context, that the kinetic rate is of the same order of magnitude than the one obtained in the absence of the repulsive barrier. Such phenomena evidence a strong influence of the secondary minimum in the process of flocculation and illustrate a lack of connection between the height of the repulsive barrier and the flocculation rate. This behavior is very different from the one reported for suspensions of nanometer-size particles where the depth of the secondary minimum is not profound. This paper studies the effect of the secondary minimum in the process of flocculation using drops of three different sizes. These include nanometer size particles like the ones employed in previous experimental studies of suspensions and micron size drops similar to the ones exhibited by bitumen emulsions. To quantify the effect of the potential on the flocculation rate, we follow the variation of the number of aggregates in the systems as a function of time and calculate the stability ratio using a similar procedure to that reported in ref 29 (see below). 2. Magnitude of the Stability Ratio W and its Relation to the Height of the Repulsive Potential between Drops In the absence of a repulsive barrier solid particles coagulate as soon as they collide. The clusters formed are loose, open structures that exhibit a fractal dimension (Df) between 1.7 and 1.8.15-17 In this regime, known as diffusion-limited cluster aggregation (DLCA), the sys(51) Leal-Calderon, F.; Biais, J.; Bibette, J. Colloids Surf., A 1993, 74, 303. (52) Salou, M.; Siffert, B.; Jada, A. Colloids Surf., A 1998, 142, 9. (53) Rodrı´guez-Valverde, M. A.; Cabrerizo-Vı´lchez, M. A.; PaézDuenãs, A.; Hidalgo-Alvarez, R. Colloids Surf., A 2003, 222, 233. (54) Al-Sabagh, A. M. Colloids Surf., A 2002, 204, 73. (55) Di Lorenzo, M.; Vinagre, H. T. M.; Joseph, D. D. Colloids Surf., A 2001, 180, 121. (56) von Smoluchowski, M. Z. Phys. Chem. 1917, 92, 129. (57) Borwankar, R. P.; Lobo, L. A.; Wasan, D. T. Colloids Surf. 1992, 69, 135.

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tems follow Smoluchowskian dynamics.15,56 According to Smoluchowski,56 the number concentration of m-particle aggregates at time t in an irreversibly flocculating suspension, nm(t), is equal to the difference between the number of these clusters created by flocculation (i + j ) m) and lost by coagulation with aggregates of other sizes

dnm dt

)

1i)m-1 2

∑ i)1

∞

kimni ∑ i)1

kijninj - nm

j)m-i

m ) 1, 2, 3, ... (1)

In eq 1, the referred contributions are supposed to be proportional to the number of collisions between the aggregates (ni nj). Subscripts (i and j) identify the size of the aggregates, which is given by the number of primary particles in each cluster. Proportionality coefficients (kij) constitute the kernel of the system of differential equations given by eq 1. They are scalar variables for spherical particles that measure the flocculation rate between the aggregates. Their general analytic form can be calculated solving Fick law for the case of a fixed spherical collector (i) subject to an incoming flux of diffusing neighbors (j): kij ) 4πRijDij, where Rij and Dij stand for the collision radius and the relative diffusion constant between i and j. Smoluchowski obtained a generic form for an average flocculation constant using several assumptions. He approximated Rij ∼ Ri + Rj, and Dij ∼ Di + Dj (where Ri and Di are the radius and diffusion constant of particle i), and made corrections for the double counting of aggregates. Additionally, Smoluchowski assumed that the collision frequency was the same for all spherical aggregates: kij/2 ) kf ∼ 8πR0D0 (where R0 and D0 are the radius and diffusion constant of the primary particles in the suspension). This allows the sum of all members of eq 1 with different values of m (na ) ∑nm), producing one ordinary differential equation for the total number of aggregates (na). This procedure leads to a simple formula for the variation number of aggregates (na) as a function of time (t)

na )

n0 1 + kfn0t

(2)

Here n0 is the number of aggregates per unit volume at time t ) 0, na(t ) 0). The number of aggregates na also includes the number of single (primary) particles. In eq 2 the product (kfn0) -1 stands for the meantime between collisions and it is also equal to the average time required to decrease the total number of aggregates from n0 to n0/2. In the absence of interaction forces the flocculation rate only depends on the thermal movement of the particles, and is equal to

kf )

4kBT 3η

(3)

Here kB, T, and η stand for the value of the Boltzmann constant, the absolute temperature, and the viscosity of the suspending medium. Smoluchowski also provided explicit solutions for ni56

ni )

n0[kfn0t]i-1 [1 + kfn0t]i+1

(4)

Use of eq 4 allows calculating the average size of a cluster of “i” primary particles, as well as the mean number of particle per cluster. If the radius of the aggregates is

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supposed to grow linearly with the number of particles, Ri ) iRo, the referred quantities are found to be proportional to (1 + kfn0t). On the other hand, a power law dependence of the average radius with time is obtained if the original equations (eq 1) are expressed in terms of nondimensional quantities and the average radius Ri, is supposed to scale with Ro. The presence of a significant repulsive barrier between the particles (reaction limited cluster aggregation (RLCA) regime) produces very different results. It generates compact structures with a fractal dimension around of 2.1-2.2. The higher value of Df is usually associated with rotations and lateral movement of the particles during flocculation. In this case, the average radius of the clusters is found to vary exponentially with time. Equation 2 has been repeatedly validated for the case of dilute suspensions of attractive spheres, using stopped flow microscopy, spectrophotometric measurements, and dynamic light scattering.36,50,58-63 On the other hand, the value of the flocculation rate at infinite dilution, is still found to deviate significantly from the number suggested by eq 3.58,61-62 It is currently accepted that, in the case of solid/liquid dispersions, kij can be usually separated into four contributions64

kij )

kf B P W ij ij

(5)

Here Bij is a geometric correction related to the exact value of the product RijDij, and Pij is the probability of collision between particles i and j. The stability ratio W quantifies the effect of the interaction potential on the magnitude of the flocculation rate. It is equal to50,61-63

∫

∫

f(r)

( )

V(r) exp dr 2 2Ri kBT r W) Va(r) ∞ f(r) exp dr 2 2Ri kBT r ∞

( )

(6)

Here r is the distance between the particles. Function f(r) ) 6πηRiβ(r/Ri) stands for the friction between the particles and the solvent. It is related to the effective diffusion coefficient of the particles in the dispersion (Di f(r) ) kBT). Dimensionless function β(r/Ri) is a correction for hydrodynamic interactions. The total potential between the particles V(r) ) Va(r) + Ve(r), is assumed to be the result of two contributions including an attractive van der Waals interaction Va(r), and an electrostatic repulsive one, Ve(r).50 These contributions give rise to two minima separated by a repulsive barrier. A high repulsive barrier causes a decrease in the frequency of collisions that leads to primary minimum flocculation. Hence, the aggregation rate of a ) stabilized system turns out to be W times slower (kslow f than the one displayed by the same particles in the absence (58) Sonntag, H.; Strenge, K. Coagulation kinetics and structure formation, 1st ed.; VEB Deutscher Verlag der Wissenschaften: Berlin, 1987; Chapters 1-4. (59) Holthoff, H.; Schmitt, A.; Fernańdez-Barbero, A.; Borkovec, M.; Cabrerı´zo-Vı´lchez, M. A.; Schurtenberger, P.; Hidalgo-Alvarez, R. J. Colloid Interface. Sci. 1997, 192, 463. (60) Holthoff, H.; Egelhaaf, S. U.; Borkovec, M.; Schurtenberger, P.; Sticher, H. Langmuir 1996, 12, 5541. (61) Sonntag, H.; Shilov, V.; Gedan, H.; Lichtenfeld, H.; Du¨rr, C. Colloids Surf., A 1986, 20, 303. (62) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, U.K., 1989. (63) Fuchs, N. Z. Physik 1936, 89, 736. (64) Lattuada, M.; Sandku¨hler, P.; Wu, H.; Sefcik, J.; Morbidelli, M. Adv. Colloid Interface Sci. 2003, 103, 33.

of the repulsive barrier, kfast f

W)

kfast f kslow f

(7)

Current experimental techniques allow the evaluation of W in dilute solid/liquid dispersions based on the initial flocculation rate for doublet formation.36,60,64 However, it is not yet possible to discriminate between primary and secondary minimum flocculation by experimental means. It is also unfeasible to discern between the number of primary particles and the number of disaggregated particles at a given time. Thus, the experimental evaluation of kslow and kfast may generate values of W that can f f erroneously be ascribed to the height of a repulsive potential. Further limitations concern the exact knowledge of Va(r) and Ve(r) in eq 6. The shape of these potentials sensibly depend on the value of the Hamaker constant (AH) and the surface potential (φ) of the particles, respectively. As a consequence, substantially different potentials can be used to fit the same experimental data depending on these parameters and the exact analytical form of the repulsive potential.65 Reerink and Overbeek,66 suggested a quadratic fitting of the repulsive barrier in order to evaluate the integral of the numerator in eq 6. A similar parabolic approximation allowed Ruckenstein and Prieve62,67-68 to formulate a convenient relationship between the value of W and the height of the repulsive potential

ln W ) 0.40(Vmax/kBT)

(8)

Coefficient 0.40 was obtained fitting the linear portion of the variation of log W with Vmax. This parameter is related to the value of the second derivative of the repulsive potential at its maximum height. Its average value results from a large set of calculations spanning a wide range of experimental conditions. As remarked by Ruckenstein and Prieve,67 eq 8 is similar to an Arrhenius equation where the maximum height of the interaction potential can be understood as an energy barrier for particle flocculation. Recently, the coalescence frequency per unit area of a concentrated emulsion was also found to depend on an activation barrier through an Arrhenius type equation.10-12 This result suggests that primary minimum flocculation could be considered to be equivalent to coalescence in the population balance of a liquid/liquid dispersion. First theoretical evaluations of W showed values as high as 107 and even higher.50,58,69-70 Using a simplified expression which disregards the effect of the attractive potential (Va(r)∼ 0), and the hydrodynamic interaction between particles f(r) ≈ 6πηRi, Verwey and Overbeek50 estimated that a barrier height (Vmax) of at least of 25 kBT was necessary for a 7-day stabilization of a concentrated emulsion (n0 ∼ 1014 m-3). This corresponds to a stability ratio of 109. Use of eq 8 suggests a value of W g 7.9 × 107,7 Vmax/kBT g 18, for preserving a suspension of 0.1-µm polystyrene spheres with a volume fraction of φ ) 0.1 (n0 (65) Maroto, J. A.; de las Nieves, F. J. Prog. Colloid Polym. Sci. 1995, 98, 89. (66) Reerink, H.; Overbeek, J. Th. G. Faraday Discuss. Chem. Soc. 1954, 18, 74. (67) Ruckenstein, E.; Prieve, D. C. J. Chem. Soc., Faraday II 1973, 69, 1522. (68) Prieve, D. C.; Ruckenstein, E. J. Colloid Interface Sci. 1980, 73, 539. (69) Kruyt, H. R.; van Arkel, A. E. Rec. Trav. Chim. 1920, 39, 656. (70) Westgreen, A.; Reiststo¨tter, J. Z. Phys. Chem. 1918, 92, 750.


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Table 1. Values of the Stability Ratio W Commonly Found in the Literature system

average diameter of the particles (nm)

stability factor, W

ref

AgI polystyrene “Dow latices” polystyrene polystyrene rutile latex latex, hematite sulfonated latex sulfate latex functionalized latexessulfate, carboxyl, aldehyde, sulfonated sulfate latex + triton polystyrene latex + PEO

41, 104, 400 60, 103, 243, 368 357, 600 500 86 240 175, 1 104, 309, 122 (h) 195, 196 297 352, 287, 281, 324 361, 413 191

1 e W e 104 1 e W e 102 1.7 e W e 102 1 e W e 102 1 e W e 103 1 e W e 80 1eWe3 1 e W e 103 1 e W e 316 1 e W e 10 1 e W e 18 1 e W e 25 1 e W e 11

66 74 75 76 77 78 79 80 81,82 65 83 84,85 86

∼ 1019 m-3) the same period of time. Experimental measurements on sols (Se, Au, and AgI66,69-70,71-73) appear to confirm these large magnitudes for W. Nonetheless, former evaluations used eq 3 as an approximation for kslow , disregarding the acceleration produced by the f attractive potential between the particles. More recent determinations involving suspensions of larger particles, usually lead to considerably lower values of the stability ratio (1-103, see Table 1). Ruckenstein87 studied the effect of reversible coagulation when the interaction potential has primary and secondary minimum of comparable depths. Quasi steady state was assumed for the movement of the particles between the bulk and the secondary minimum and also for the transfer between the secondary minimum and the primary one. The kinetic rates for primary and secondary minimum flocculation also showed exponential forms, depending on the energy difference between the maximum and each minimum of the potential curve.88,89 Furthermore, the coefficients of proportionality were found to depend on the product (geometric average) of the curvatures of the potential at its zenith and at each minimum. Recently, Beherens and Borkovec90 revisited the problem of irreversible dimer formation supposing a two-step process in that charged colloidal particles first aggregate (71) Kruyt, H. R.; van Arkel, E. Rec. Trav. Chim. Pays-Bas 1921, 40, 169. (72) Tuorilla, A. Kolloidchem. Beihefte 1926, 22, 193. (73) Kruyt, H. R.; Troelstra, S. A. Kolloidchem. Beihefte 1943, 54, 225. (74) Ottewill, R. H.; Shaw, J. N. Faraday Discuss. Chem. Soc.1966, 42, 154. (75) Matthews, B. A.; Rhodes, C. T. J. Pharm. Sci. 1967, 57, 558. (76) Zeichner, G. R.; Schowalter, W. R. J. Colloid Interface Sci. 1979, 71, 237. (77) Einarson, M. B.; Berg, J. C. J. Colloids Interface Sci. 1993, 155, 165. (78) McGown, D. N. L.; Parfitt, G. D. Faraday Discuss. Chem. Soc. 1966, 42, 225. (79) Watillon, A.; Joseph-Petit, A. M. Faraday Discuss. Chem. Soc. 1966, 42, 143. (80) Behrens, S. H.; Borkovec, M.; Schurtenberger, P. Langmuir 1998, 14, 1951. (81) Peula-Garcıá, J. M.; Hidalgo-Alvarez, R.; de las Nieves, F. J. Colloids Surf., A 1997, 127, 19. (82) Peula-Garcıá, J. M.; Fernańdez-Barbero, A.; Hidalgo-Alvarez, R.; de las Nieves, F. J. Langmuir 1997, 13, 3938. (83) Bastos-Gonza´lez, D.; de las Nieves, F. J. Prog. Colloid Polym. Sci. 1995, 98, 1. (84) Romero-Cano, M. S.; Martıń-Rodrı´guez, A.; Chauveteau, G.; de las Nieves, F. J. J. Colloid Interface Sci. 1998, 198, 266. (85) Romero-Cano, M. S.; Martıń-Rodrı´guez, A.; Chauveteau, G.; de las Nieves, F. J. J. Colloid Interface Sci. 1998, 198, 273. (86) Cowell, C.; Vincent, B. J. Colloid Interface Sci. 1983, 95, 573. (87) Ruckenstein, E. J. Colloid Interface Sci. 1978, 66, 531. (88) Wang, Q. J. Colloid Interface Sci. 1991, 145, 99. (89) Marmur, A. J. Colloid Interface Sci. 1979, 72, 41. (90) Behrens, S. H.; Borkovec, M. J. Colloid Interface Sci. 2000, 225, 460.

reversibly in the secondary minimum before they can cross the energy barrier. The secondary minimum was found to have a pronounced effect in the regime of slow aggregation if either the ionic strength of the solution is high or the particles are relatively large. Under these conditions, their calculations predict a transient period of fast aggregation into the secondary minimum followed by slow primary minimum flocculation. A year ago, Tufenkji and Elimelech reported deviations from the classical colloid filtration theory (CFT) in the presence of repulsive DLVO interactions.91 In particular, the deposition rates obtained from microbial particles on porous columns only reproduce the theoretical estimations in the absence of a repulsive barrier. Because the dynamics of the aggregation process are similar to those of a deposition process, DLVO theory was used in the past for the calculation of deposition rates in porous media92 and solid collector surfaces.93 According to Tufenkji and Elimelech the referred behavior was caused by the combined effect of favorable and unfavorable colloidal interactions. Moreover, these authors related fast deposition rates to the influence of deep secondary minimum in the aggregation process and proposed a dual-deposition model that includes slow and fast aggregation rates. Those previous results were confirmed by a systematic experimental study on the deposition of uniform polystyrene latex colloids in columns packed with spherical glass beads.94 It was concluded that both secondary minimum deposition and surface charge heterogeneities contribute significantly to the deviations observed from the predictions of CFT. It was also shown that it is not the existence of repulsive conditions, which causes these discrepancies, but rather the combined occurrence of slow and fast deposition rates. 3. Previous Results from ESS of Bitumen/Water Emulsions Until recently, most ESS were devoted to the study of bitumen/water systems stabilized with ionic surfactants. The parameters of the potential used were taken from the experiments of Salou et al.52 They corresponded to a bitumen-in-water emulsion stabilized with a cationic surfactant (emulsion E3 in ref 52). The drops of this emulsion exhibit a surface potential of 115 mV, which produces very large repulsive barriers (Vmax ∼10 000kBT) for drops of 3.9 µm. This characteristic, along with an (91) Tufenkji, N.; Elimelech, M. Langmuir 2004, 20, 10818. (92) Grolimund, D.; Elimelech, M.; Borkovec, M. Colloids Surf. A 2001, 191, 179. (93) Prieve, D.; Ruckenstein, E. J. Colloid Interface Sci. 1978, 63, 317. (94) Tufenkji, N.; Elimelech, M. Langmuir 2005, 21, 841.

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unusually high Hamaker constant of 1.24 × 10-19 J, generates deep secondary minima which favor fast aggregation rates. As shown in ref 28, the use of lower Hamaker constants (1.24 × 10-21 J) can lead to significantly lower magnitudes of the flocculation rates. This was confirmed by recent calculations on latex particles stabilized by nonionic surfactants.95 However, the absolute values of the flocculation rates mostly account for the effect of the Hamaker constant and the volume fraction of internal phase in the process of aggregation. Relative parameters as the stability ratio (eq 7) are needed to discriminate between fast and slow flocculation rates.29 In ESS simulations the dynamics of the systems are determined by the interaction potential between the drops. The attractive potential is assumed to depend on the composition of the oil. Hence, it is taken into account by the effective value of the Hamaker constant in the van der Waals potential.96 Drops coalesce if either the attractive potential or the thermal interaction provide enough energy to overcome the repulsive interaction. In this case, a new particle is created at the center of mass of the 3 coalescing drops (Ri ) xiRo). The coalescence process is assumed to occur instantaneously once the surface-tosurface distance becomes equal to zero. Clearly, the polydispersity of the emulsion increases as coalescence occurs. This leads to different magnitudes of the potentials during the course of the same simulation. Unless it is explicitly stated, simulations of equal number of particles begin from the same initial configuration. In this way, the effect of interaction parameters can be quantified independently of the initial mean free path. In the absence of repulsive potentials the variation of the total number of drops calculated by ESS follows the predictions of eq 221,26,28 if kf is regarded as an adjustable parameter. Notice that eq 2 was formulated for the phenomenon of aggregation of solid particles. However, if coalescence occurs instantaneously as soon as the surfaces of the particles make contact, particles cannot aggregate. In this case, the variation of the number of aggregates per unit volume (na) is equal to the change in the number of particles (np). Thus, the flocculation rate can be evaluated from the coalescence of drops. As expected, the flocculation rate is found to depend sensibly on the volume fraction of internal phase (φ) and the initial polydispersity of the system. The simulations also reproduce the thermal limit of Smoluchowski at very dilute concentration.28 In most simulations, it is assumed that the repulsive potential of the drops basically depends on the degree of surfactant adsorption to their surfaces. The nature of the repulsive force depends on the physical properties of the surfactant molecules. In the case of ionic surfactants, the surface charge of the drops results from the sum of the individual contribution of each surfactant molecule adsorbed.97 Notice that the initial surfactant distribution among the drops can vary very amply depending on the characteristics of the emulsification process and the method of addition of the surfactant molecules. Other complications arise from the fact that (a) the adsorption process is time dependent, (b) the repulsive potential between the drops increase with their radii, and (c) the interfacial area of emulsions decreases as a function of time. Due to the different size scales of the problem, it is not computationally feasible to simulate the movement of a (95) Lozsań, A. Doctoral Thesis, 2004. (96) Hamaker, H. C. Physica (Amsterdam) 1937, 4, 1058. (97) Sader, J. E. J. Colloid Interface Sci. 1997, 188, 508.


typical number of surfactant molecules along with that of the drops. Hence, ESS employs a number of routines to mimic the variation of the surfactant surface excess without moving the surfactant molecules explicitly. This can be done using different methods for surfactant distribution among the drops.21-24,27 Due to points (b) and (c), in the paragraph above, the repulsive potential between the drops cannot be constant as a function of time. This produces a wide variety of outcomes that are not limited to the appearance of compact and open structures as in the case of suspensions. Furthermore, because the interfacial area of two drops decreases when they coalesce, some special mechanism for the handling of the surfactant population is required. The ESS program contains several realizations of surfactant adsorption. However, in the vast majority of the simulations, the surfactant is assumed to adsorb extremely fast with respect to the collision of drops. Hence, the surfactant population is distributed instantaneously and homogeneously among the surfaces of the drops up to a saturation limit. Independently of the method of surfactant distribution chosen, ESS require an additional methodology for the handling of surfactant molecules during the process of coalescence. To our knowledge, there are very few experiments that allow estimating the variation of the surfactant surface excess during the process of coalescence.69 Because previous ESS concerned the behavior of bitumen/ water emulsions stabilized with cationic surfactants, the surfactant concentration inside the drops was supposed to be negligible. Furthermore, it was implicitly assumed that the surfactant excess at the interfaces of the drops was in equilibrium with the water solution. Hence, the local increase in the surfactant concentration produced by the decrease of the interfacial area during the process of coalescence should rapidly relax. Thus, the extra number of surfactant molecules resulting from the coalescence of drops should increase the total surfactant concentration of the external phase. This will promote an augment of the surfactant excess in all drops of the system that may be simulated by the homogeneous distribution of surfactant drops among the available interfaces. This procedure leads to a long-term stabilization of the average radius of the emulsion, as the one usually observed in concentrated systems stabilized by surfactant molecules. Notice that several alternative mechanisms of surfactant redistribution may also lead to stabilization of the emulsion. For example, if the resulting drop inherits the total surfactant population of the coalescing drops, its charge will increase along with its size. Thus, independent coalescence events followed by a local augment of the surface excess will also lead to stabilization. This case is likely to happen if the surfactant molecules are barely insoluble in both phases and stay at the interface after coalescence, similar to what happens in the experiment of Hu and Lips.98 Using an elaborate procedure, these authors divided a parent drop many times into smaller daughter drops. The fact that the interfacial tension of the daughter and parent drops was found to be the same was supposed to indicate that the surfactant molecules were being evenly distributed after the process of breakup. Curiously, Hu and Lips were also able to fit Langmuir, Frumkim, and Volmar isotherms to the variation of the interfacial tension with the surface excess of subsequent generation of drops. In our view, this appears to indicate some kind of equilibrium between the interface of each drop and the external phase. (98) Hu, Y. T.; Lips, A. Phys. Rev. Lett. 2003, 91, 44501.


Langmuir, Vol. 21, No. 15, 2005 6681

Former ESS assumed instantaneous irreversible adsorption of surfactant molecules during the process of mixing. The existence of homogeneous (H) and nonhomogeneous (NH) surfactant distributions among the initial number of drops was considered. Nonhomogeneous (NH) surfactant distributions were used to study the effects of imperfect mixing on emulsion stability. As expected, a quick destabilization of the emulsion was observed.21-24 Homogeneous (H) surfactant distributions generate a much efficient stabilization at lower surfactant concentrations.20,25-27 Time-dependent surfactant adsorption with homogeneous surfactant distributions, favor intermediate situations.27 It was also found that the initial slopes of the variation of np vs t found in ESS did not show a strong dependence on the surfactant concentration. Instead, np appears to follow the behavior of completely attractive drops until the surfactant concentration was enough to generate an appreciable repulsive barrier between the remaining drops. At this point, the slope of the curve drastically changes. The time at which this phenomenon occurs markedly depends on the total surfactant concentration of the system (Cs). Similarly, the amount of interfacial area preserved, the final particle size distribution (PSD), and the value of kf, estimated by a rough fitting the whole variation of np vs t, also depend on Cs. Clearly, the quality of the fitting of eq 2 to the simulation data diminishes with the increase of φ as well as with the augment of the repulsive potential. Although the simulations directly give the variation of np vs t, the change in the number of aggregates (na) requires further treatment of the simulation data. For this purpose a separate code was elaborated. It uses the coordinate file produced during ESS to calculate the number of flocs and their size. The program requires as input the maximum flocculation distance between two adjacent particles in a floc (df). df is close to the position of the secondary minimum and is necessarily smaller than two double-layer widths.29 The program analyzes each set of particle coordinates sequentially. For each set, it identifies the number of neighbors that stand at a distance less or equal than df from the first particle. Then it repeats the same procedure for the particle’s neighbors and its neighbors’ neighbors until no other new member of the floc appear. At this point, the number of particles in the first aggregate had been identified and can be stored. Following, the program erases the particles of this first floc from the set of coordinates, and repeats the whole procedure for the rest of the particles. As a result, the number of particles (np) and aggregates (na) is known as a function of time. It is important to remark that, regardless of its radius, one particle counts as one unit whether it had flocculated into a bigger aggregate or remains isolated. It is clear from the discussion above that the outcome of the simulations can vary amply. Surprisingly, it was recently found that the number of aggregates of a large number of simulations with distinct polydispersity indexes, surfactant concentrations, and volume fractions up to 30%, can be fitted to eq 9

[

na ) n0

A + Be-kc,fn0t 1 + kfn0t

]

(9)

Here, kf stands for the usual flocculation rate that measures the average frequency of collisions in the emulsion. kc,f is a second kinetic constant formerly ascribed to coalescence. A and B are constants that measure the relative importance of flocculation and coalescence during the whole destabilization process (A + B ) 1). Although

Table 2. Values of the Stability Ratio Calculated as the Slow a Quotients Wf ) kFast /kSlow and Wc,f ) kFast f f c,f /kc,f ID

Wf

Wc,f

ID

Wf

Wc,f

ID

Wf

Wc,f

1 2 3 4 5 6

0.3 3.6 9.0 1.0 2.6 0.4

5.5 1.2 0.5 19.6 2.5 17.0

7 8 9 10 11 12

0.7 0.8 0.6 2.2 0.9 0.7

3.8 8.2 0.7 2.0 68.8 1.4

13 14 15 16 17

0.5 4.1 0.4 1.4 0.8

38.2 4.1 13.1 10.0 10.0

a The kinetic rates presented were evaluated fitting the data from emulsion stability simulations of 17 different systems to eq 9. The identification number (ID) given in column 1 is the same one used to distinguish the simulations of refs 25 and 29. The specific particle size distribution corresponding to each system is given in ref 25. The computational details of the simulations are given in ref 29.

the frequency of these processes changes as a function of time, one pair of (A, B) values was found to be enough to fit the total variation of na vs t during the whole simulation. Equation 9 was obtained considering separate reaction kinetics for the processes of flocculation and coalescence. As a matter of fact, eq 2 (first term on the right-hand side of eq 9) is the solution of a second order differential equation for the total number of aggregates in a dispersion.56 The exponential term of eq 9 is obtained supposing first-order coalescence kinetics, along with the assumption of the random occurrence of coalescence events.29 However, a more detailed analysis of the simulation data indicated that the exponential term was also able to fit the terminal flocculation rate shown by the surviving particles toward the end of the simulation. Interestingly, the first and second terms of eq 9 are similar to the variation of the number of aggregates predicted for suspensions under DLCA and RLCA regimes, respectively. However, eq 9 appears to be more general considering that only half of the calculations of ref 29 showed the occurrence of clusters in their final configuration. As shown in Tables 3 and 4 of ref 29 the flocculation rates obtained from the fitting of the simulation data to eq 9, are relatively fast. Table 2 shows the relatively low values of the stability ratio computed from the data of Ref. 29 using eq 7. The simulations calculated in the absence of the repulsive potential provided the data for and kfast the determination of kfast f c,f by fitting of eq 9. The pair of kinetic rates evaluated for the same systems in the presence of a repulsive barrier correspond to the values and kslow of kslow f c,f in eq 7. Notice that the stability ratios obtained are relatively low. They are similar in magnitude to the ones presented in the small compilation of Table 1, but they are considerably lower than those previously reported for colloids of nanometer size, several years ago. Moreover, eq 2 was found to fit the initial variation of na in those simulations, suggesting fast initial flocculation rates. That is, kinetic rates similar in magnitude to the ones calculated in the absence of a repulsive barrier between the drops. Interestingly, the flocculation rates obtained are comparable in magnitude to the ones previously reported for distinct oil/water systems with a similar volume fraction of the internal phase.29,33 4. Computational Approach To understand the effect of the secondary minimum on the flocculation rate, we studied 11 systems divided into two sets of particle sizes. Figure 1 illustrate the potentials corresponding to the big particle systems (b1 - b5). The potential employed by our group in previous simulations of bitumen/water emulsions,20-29 corresponds to the label “b1” in Figure 1 and Table 3.

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Table 3. Parameters of the Potentials Shown in Figures 1-3

system

radius (m)

Hamaker constant AH (J)

B1 B2 B3 B4 B5 B6 S1 S2 S3 S4 S5

3.9 × 10-6 3.9 × 10-6 3.9 × 10-6 3.9 × 10-6 3.9 × 10-6 3.9 × 10-6 1.0 × 10-8 1.0 × 10-7 1.0 × 10-8 1.0 × 10-8 1.0 × 10-8

1.24 × 10-19 1.24 × 10-19 1.24 × 10-19 1.24 × 10-19 1.24 × 10-21 1.24 × 10-19 2.0 × 10-19 1.0 × 10-19 2.0 × 10-19 2.0 × 10-19 2.0 × 10-19

zs

total surfactant concentration (M)

ionic strength (M)

6.9 × 10-4 6.9 × 10-4 6.9 × 10-4 6.9 × 10-4 6.9 × 10-4

0.01 0.01 0.01 0.00025 0.01

0.014 0.007 0.001 0.0001 0.001

6.9 × 10-2 7.2 × 10-5 6.9 × 10-4 6.9 × 10-3

0.01 0.01 0.01 0.01

0.0232 0.0009 0.0002 0.0232

To obtain secondary minima of different depths, the ionic strength, Hamaker constant, and total surfactant concentration of system b1 were modified. In all cases, the potential barriers corresponding to these big-particle systems are extremely large and should be insurmountable under standard conditions. The repulsive barrier is located very close to the surface (Figure 1a), producing secondary minima of several hundreds kBT (see Figure 1b and Table 3). Thus, the total potential curve closely follows the shape of the van der Waals potential (potential b6 in Figure 1b) prior to the appearance of the repulsive barrier. The interaction potentials and the secondary minimum corresponding to the small particle systems simulated are the ones shown in Figures 2 and 3. The potential shown in Figure 2 for system s2 (Table 3) was parametrized with data from a previous work of Verwey and Overbeek.99 The rest of the potentials were obtained after a significant modification of the ionic strength, effective surfactant charge, and the radius of system s2. Notice that the value of Vmax of system s1, varies between 20 and 40 kBT, whereas

Figure 1. Repulsive barrier (a) and secondary minimum (b) of the potential energy V(r), corresponding to systems b1 to b6. In all cases, the repulsive barrier is higher than 10 000 kBT, preventing primary minimum flocculation. In the case of nondeformable droplets, which coalesce “instantaneously” as soon as they touch, primary minimum flocculation is equivalent to coalescence.

barrier height (kBT) 10,668 17,000 27,716 11,313 38,491 40.6 22.1 1.8 0.3

depth of the secondary minimum (kBT)

potential width (nm)

-548 -311 -96 -23 -0.55

12.8 23.3 73.0 245.5 131.8

-0.005 -0.346 0 -0.223

14.0 50.8 32.7 4.1

the depth of the secondary minimum is lower than 1 kBT. These are typical values for nanometer size particles,50 where the depth of the secondary minimum rarely surpasses few kBT. Notice also that the value of AH in a typical sol is similar to the one previously reported for Bitumen emulsions.52 This value is 2 orders of magnitude higher than the one of latex dispersions. Still, a deep secondary minimum is not observed for nanometer-size particles, despite the considerable variations of the parameters of the potential (see Table 3). In ESS the movement of the particles is similar to that of Brownian Dynamics (BD) simulations.100 The displacement of particle i during the time ∆t, b ri(t + ∆t) - b ri(t), is the result of two contributions26,100

b r i(t + ∆t) - b r i(t) )

Di(φ,d)F Bi∆t +R B G(Di(φ,d)) kBT

(10)

The first term on the right-hand side accounts for the diffusion of the particle under the action of interaction and external forces (F). This term supposes that each particle moves at a constant velocity v ) DF/kBT during time ∆t. The diffusion constant modulates the velocity of the particle, which is limited by the viscosity of the solvent. In our simulations an average diffusion coefficient Di(φ,d) is used. It depends on the local volume fraction of particles and a minimum distance of approach between particle “i” and its neighbors, assuming the limit value of Stokes law (D0) at infinite dilution. The last term on the right-hand side of eq 10 simulates the thermal interaction of the suspended particle with the solvent. This is implemented using a random function R B G(Di(φ,d)), which has a Gaussian distribution with zero mean and variance 6Di(φ,d)∆t.26 The attractive van der Waals (vdW) interaction is represented by Hamaker expression for two drops of different size.95 This potential depends on the value of the Hamaker constant AH which does not vary during the course of the simulations. For electrostatic interactions we used the formulation of Sader97 developed for the farfield potential between two particles immersed in a symmetric z:z electrolyte solution. As other solutions of the nonlinear Poisson-Boltzmann equation, this electrostatic potential can only be evaluated if either the surface potential, φp, or the charge of the particles per unit area (σ), are specified. In ESS, the charge of the surfactant is introduced as input. This charge can be estimated theoretically using experimental values of σ or can also be calculated to reproduce an experimental value of φp. (99) Verwey, E. J. W.; Overbeek, J. Th. G. Trans. Faraday Soc. 1946, 42B, 117. (100) Ermak, D.; McCammon, J. A. J. Chem. Phys. 1978, 69, 1352.


Langmuir, Vol. 21, No. 15, 2005 6683

surfactant molecule, and e is the elementary charge (1.6 × 10-19 C). The surface potential φp is computed from surface charge density (σ) which can be calculated very simply once the surface excess is known

σ)

Figure 2. Total potential of interaction V(r) between two small drops as a function of the distance. The label in the parenthesis at the top of each graph identifies the type of interacting particles. To appreciate height of the repulsive barrier and the depth of the secondary minimum, the variation V(r) vs r was divided into two sections. The figures on the left-hand side (lhs) show the potential barrier for primary minimum flocculation. The figures on the right (rhs) illustrate the depth of the secondary minimum of each system. The two graphs on the top correspond to two drops of type s1, while the two graphs on the bottom belong to two s2 drops.

Figure 3. Total potential of interaction V(r) between two small drops as a function of the distance. The label in the parenthesis at the top of each graph identifies the type of interacting particles. In the case of system s3, the variation V(r) vs r was divided into two sections to illustrate the height of the repulsive barrier (lhs) and the absence of a secondary minimum (rhs). The relevant features of the potential (barrier and secondary minimum) between two s4 drops are clearly visible in only inset (bottom, lhs). The potential between two s5 drops (bottom, rhs) is completely attractive, and does not present a repulsive barrier.

As in previous simulations, we consider the charge of the drop (Zi) to be the result of surfactant adsorption exclusively. Hence, the total charge of a drop depends on the surfactant surface excess Γ(t)

Zi ) AizseΓ(t)

(11)

Here Ai is the area of drop i, and zse is the charge of one

Zi 4πRi2

(12)

As it was discussed in section 3, the value of Γ(t) is set through several routines for surfactant handling which were designed to resemble distinct experimental conditions.20,22-24 However, in the present study, we want to separate surfactant effects from the influence of the secondary minimum in the process of flocculation. For this reason, a fixed surface excess was used. It was equal to the inverse of the typical interfacial area per molecule of a cationic surfactant at an oil/water interface [Γ(t)]-1 ∼ [Γ(tf∞)]-1 ∼ 50 Å2/molec.101-103 As a result, the potentials of Figures 1-3 did not vary during the course of the simulations unless coalescence occurred. In this event, the same initial number of surfactant molecules per unit area was assigned to the resulting drop. Thus, both the attractive and repulsive forces increased steadily after coalescence as a result of the increase of the particle radius. This moderates the changes in the interaction potential as much as possible. It is important to notice that the charge of the surfactant molecule zse is an input of the simulations which is difficult to estimate. The maximum surface charge density of a solid particle is about 0.3 C/m2 that is equivalent to one electron per 50 Å2.104 The fact that most measured densities differ from this maximum value is usually associated to the chemical nature of the charging process. However, those deviations are also known to be the product of counterion adsorption.105,106 These ions screen the bare surface charge of the particle, giving rise to an effective charge. A theoretical approach for the calculation of effective charges was provided by Sanghiran and Schmitz.105 These authors used radial distribution functions in order to define a limit radius for the macroion -counterion double layer. This procedure allows computing an effective charge for the macroion, which is found to differ in more than 1 order of magnitude from that of the bare surface. Voltammetric determinations suggest that the effective charge of a 44-nm latex particle, Zi, is equal to 0.024 Zbare, where Zbare is the charge of the bare particle surface. Recently, Gu and Li107 measured the charge of silicone oil droplets in the presence of cetyltrimethylammonium bromide (CTAB), balancing the forces of gravity and buoyancy with an external field. This procedure allowed defining an electrostatic charge per surface area, which ranged between -215 and +680 µC/m2 as the surfactant concentration increased from 10-6 M to 10-3 M. Taking 50 Å2 as the typical interfacial area of one cationic surfactant molecule at maximum packing, a value of zs ) 2.1 × 10-3 results. The value of zs used in previous simulations (6.9 × 10-4) was obtained adjusting the surfactant charge in order to reproduce the experimental (101) Lin, S.; McKeigue, K.; Mardarelli, C. AIChE J. 1990, 36, 12. (102) Makievski, A. V.; Fainerman, V. B.; Joos, P. J. Colloid Interface Sci. 1994, 166, 6. (103) Rosen, M. J. Surfactants and Interfacial Phenomena, 2nd ed.; John Wiley & Sons: New York, 1989. (104) Israelachvilli, J. Intermolecular and Surface Forces, 2nd ed.; Academic Press: London, 1992. (105) Sanghiran, V.; Schmitz, K. Langmuir 2000, 16, 7566. (106) Roberts, J. M.; O’Dea, J. J.; Osteryoung, J. G. Anal. Chem. 1998, 70, 3667. (107) Gu, Y.; Li, D. Colloids Surf., A 1998, 139, 213.

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surface potential of a bitumen in water emulsion stabilized with a cationic surfactant, +115 mV.29 To illustrate the influence of surfactant effects on the flocculation rate, two additional simulations were run under the usual conditions of surfactant redistribution among the surviving drops. For this purpose we selected systems for systems s3 and s4 since they show small repulsive barriers which can be easily overcome by the thermal interaction with the solvent (last term on the right-hand side of eq 10). In these additional simulations, the total surfactant population was redistributed among the surviving drops after each coalescence event. As a result, the repulsion between the drops increased much sensibly than in the case of fixed surface excess, favoring stabilization of a system with a larger number of drops. As usual, the time step was selected to guarantee the appropriate sampling of the repulsive barrier. Due to the dependence of V(r) on AH, Ri, and φ, the time step of the simulations may change for each system. In the case of large potential barriers (as the ones of b-systems), the selection of the time step can be done using test simulations in which the random contribution (last term in eq 10) is switched off. A time step is adequate if the number of particles is preserved during the course of the test simulation. Large time steps produce long displacement of the particles. This generates an inadequate sampling of the repulsive barrier, favoring coalescence. Thus, a decrease in the number of particles in the test simulations indicates an incorrect selection of the time step. For the present calculation, the time step of the micron-size systems was chosen to be 3.0 × 10-8 s. In the case of small potential barriers, the test simulations are run under completely attractive conditions (switching off the repulsive potential along with the random force). These test simulations are used to estimate the length of the displacements of the particles in the region of influence of the attractive force. The final time step selected is the one that covers the width of the small potential barrier in at least 4 movements. This arbitrary convention guarantees that the repulsive potential is not overlooked. For system s2, the time step was equal to 5.0 × 10-8 s, whereas for the rest of small-size systems, it was set to 5.0 × 10-11 s. The simulations were run for 600 million iterations, or until the initial number of particles was reduced to only one drop. In all simulations, the initial number of particles was equal to 125. Despite this small number of particles, the simulations lasted for more than 8 months using a Dell workstation with two Xeon processors of 2.4 GHz. Initially, a cubic simulation box of length L ) 12.1Ri was selected for all calculations. This corresponds to a volume fraction of 0.30 that is independent of the actual size of the drops. The analysis of the simulation data obtained for systems s1 and s2 indicated that the drops did not tend to aggregate at this particle density. Instead, the average interparticle distance increased slightly at the beginning of the calculation reaching a plateau shortly afterward. It was then realized the selected computational conditions led to an initial interparticle separation of the order of the double layer length. This did not occur in the b-type systems where the same definition of the system size: L ) 12.1Ri led to interparticle distances significantly longer in the initial configuration. To avoid the problem, L was set equal to 80.0Ri (φ ) 0.001) for all small particle systems. This restriction does not allow a direct comparison between the flocculation rates of small and big systems, due to the dependence of the mean free path of the particles on the volume fraction of internal phase.28 However, the referred limitation does


Figure 4. Variation of na vs t for the small-particle systems (s systems).

not affect the magnitude of the stability ratios calculated. Those values are evaluated using eq 7. Hence, the fast flocculation rate was calculated for each system preserving the same computational conditions but switching the repulsive potential off. Thus, the quotient of eq 7 does not depend on the density of the systems. It just measures the decrease of the flocculation rate caused by the repulsive potential. 5. Results and Discussion 5.1. Small Particle Systems. Figure 4 illustrates the variation of the number of aggregates per unit volume (na) as a function of time for the small particles systems (s1, s3, s4, and s5). The behavior of emulsions s1 and s2 is very similar so only one of them is shown in this figure. In these two systems the repulsive barrier (∼40 and 20 kBT, respectively) is high enough to prevent primary minimum flocculation (coalescence). The threshold for coalescence between two drops of 200 nm is around 13 kBT according to recent simulations on sterically stabilized emulsions.95 Hence, the initial number of drops in systems s1 and s2 is preserved during the extent of the simulation. Moreover, the secondary minimum of these systems (s1, s2) is too shallow to promote any kind of aggregation. The small fluctuations in the number of aggregates evidence that the attempts of aggregation are counteracted by the repulsive potential. This was confirmed changing the distance of flocculation (df) appreciably. It was found that the number of aggregates in s1 and s2 was appreciably different from the number of particles only at 60 and 500 nm, respectively. Furthermore, the number of group of particles found at these additional distances of separation fluctuated erratically as a function of time, indicating that no perdurable clusters were formed. It is clear then that these two dispersions are stable with regard to both flocculation and coalescence. Consequently, the curves of na vs t do not display the typical monotonic decrease predicted by eq 2. Unlike emulsions s1 and s2, systems s3 and s4 display very small repulsive barriers (1.8 and 0.3 kBT, respectively). These barriers are unable to prevent coalescence. Hence, the particles flocculate and coalesce causing a monotonic decrease in the number of aggregates as a function of time. However, the variation of na(t) for dispersion s3 is considerably slower than the one of s4 (Figure 4). In fact, the values of na(t) are very similar to the ones s5, whose potential of interaction is completely attractive (see Figure 3). The flocculation rate of systems s3, s4, and s5 can be obtained, fitting the variation of na vs t to eq 2, or using the time required for the initial number of particles (n0) to drop to n0/2. The fast flocculation rate calculated for s5


Figure 5. Fittings of eq 9 to the simulation data of systems s3, s4, and s5, for additional simulations run at variable surface excess. In these simulations, the total number of surfactant molecules is redistributed among the surviving drops each time that coalescence occurs. The monotonic continuous lines represent the fittings in each case. The stepwise variation behind each fitting corresponds to the simulation data.

using the half-lifetime of the simulation (kf ) 1/(n0t1/2)) is equal to 1.2 × 10-17 m3/s. In the case of s3 and s4, the kinetic rates are equal to 2.9 × 10-18 and 8.4 × 10-18 m3/s, respectively. This suggests W values of the order of 4.0 and 1.4 for these two systems. Notice that due to the attractive van der Waals interaction, the values of the flocculation rates calculated can be faster than the one predicted by eq 3. Alternatively, eq 9 can be used in order to fit the data of systems s3, s4, and s5. Regarding the parameters obtained for s5 as the fast kinetic rates, stability ratios for flocculation and coalescence can be obtained, similar to those shown in Table 2. In this way, values of 2.1 (s3) and 1.5 (s4) for Wf, and 216.4 (s3) and 1.5 (s4) for Wc,f, are obtained. These results suggest that a small barrier of only 1.5 kBT can slow the rates of flocculation and coalescence sensibly but cannot prevent either one of these processes to occur. Figure 5 shows additional data on the variation of na vs t for systems s3 and s4. In this case, the total surfactant population is redistributed among the surviving drops each time coalescence occurs. As a result, the repulsive barrier between the drops increases more pronouncedly than in the case of constant surface excess (Figure 4). However, the shape of the curves is similar. Using Smoluchowski’s equation, stability ratios of 25.2 (s3) and 1.3 (s4) are obtained. The fittings of eq 9 to the simulation data correspond to the three monotonic curves of Figure 5. As discussed in ref 29, the coalescence rates calculated using eq 9 cannot be separated from the slow flocculation rate occurring when a substantial number of surfactant molecules cover the particles. Thus, the exponential term on the right-hand side of eq 9 includes both coalescence and a terminal flocculation rate. This is why this term predominates at long times even though the number of particles remains constant and no coalescence occurs. The terminal flocculation rate is substantially different from the initial flocculation rate corresponding to the simultaneous occurrence of flocculation and coalescence. Fitting of eq 9 to the simulation data leads to values of Wf of 1.4 (s3) and 0.8 (s4). The corresponding values of Wc,f are considerably higher: 8.3 × 102 (s3) and 2.9 × 106 (s4). The apparent lower magnitude of Wc,f for system s3 is the consequence of the method of evaluation of kc,f. The values Wc,f reported in the previous paragraphs do not include the data corresponding to the plateau of the curves of na vs t. The reason for this is that the extent of the plateau is arbitrary since it depends on the length of the simulation

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Figure 6. Variation of na vs t for the big-particle systems (bsystems) b1, b2, b3, and b6. The label vdW 21 stand for a simulation run with van der Waals attractive force with 1.24 × 10-21 J. System b6 corresponds to a completely attractive potential with AH ) 1.24 × 10-19 J.

once the number of aggregates stabilizes. Since system s3 is more stable than system s4, it preserves a larger number of particles, reaching a plateau in a shorter period of time. If all of the plateau data is included, the values of Wc,f corresponding to system s4 are not appreciably modified, whereas the ones of system s3 increase to 381.5 and 5.8 × 106 for the case of constant and variable surface excess, respectively. In any event, it is clear that the variation of the surface excess with the available interfacial area leads to higher values of the stability ratio than the ones obtained at constant surface excess. 5.2. Big Particle Systems. As expected from the large magnitudes of the repulsive barriers (Table 3), the number of particles was preserved in all micron-size particle systems except in the case of a purely attractive potential (b6). Figure 6 shows the variation of na vs t for those b systems with the most pronounced secondary minimum (b1, b2, and b3). The curves do not follow the smooth prediction of eq 2, although a decrease in the number of aggregates as a function of time is observed. Notice that this behavior is different from the one of systems s1 and s2. The deep secondary minimum of systems b1, b2, and b3 promotes flocculation, leading to a decreasing number of aggregates with time. This is not observed in systems s1and s2, although the number of particles is also preserved. Figure 6 also contains the results corresponding to system b6, as well as an additional calculation run with AH ) 1.24 × 10-21 J (labeled vdW 21). To obtain an approximate estimation of the stability factor, we used the values of t1/2 to calculate the kinetic constants. Values of W ) 146, 134, 47, and 17 were obtained for systems b1, b2, b3, and b4. Similar magnitudes are obtained fitting the data to eq 2 (162, 117, 31, and 14, respectively). Use of eq 9 leads to values of Wf equal to 27, 9, 11, and 12. The corresponding values of Wc,f are 1097, 441, 117, and 36, respectively. As previously discussed, these last numbers correspond to the terminal flocculation rate since the number of particles is preserved. In fact, it is possible to fit the data to an equation, which contains two flocculation terms (see eqs 9 and 2), instead of one flocculation term and a coalescence one.29 Despite the method of calculation, the stability factors shown in the above paragraph are considerably lower than the ones reported for sols several decades ago. Instead, they are similar in magnitude to those recently published for latex systems of different sorts (Table 1). Furthermore, except for the value of b4, the rest of the values of W calculated from the kinetic rates deduced from either eq 2 or the value of t1/2, decrease as the height of the repulsive

6686


Figure 7. Dependence of the stability ratio on the position of the secondary minimum for systems b1-b4. The circles correspond to Wf (b) and Wc,f (O), which were calculated from the flocculation rates obtained by fitting of the simulation data to eq 9. The triangles (4) correspond to W values calculated from eq 2. Diamonds identify those stability ratios calculated from the half-life time of the number of particles of each system: kf ) (n0t1/2)-1.

barrier increases. Moreover, the same data indicates that the stability factor decreases as the depth of the secondary minimum diminishes. This tendency is also followed by the values of Wc,f. The above results imply that the flocculation rates of the b systems are faster as the depth of the secondary minimum decreases. This was highly unexpected, especially if it was presumed that the relatively low magnitudes of the stability ratio observed (Table 1) were the result of the van der Waals attraction in the region r > rsm (where rsm is the position of the secondary minimum). As shown in Figure 1, the branch of V(r) corresponding to r > rsm closely follows the shape of the attractive potential Va(r) for systems b1 to b4, causing the overlap of the four potential curves at long distances. Thus, systems of particles with more profound depths of the secondary minimum should be subject to the fastest acceleration of the van der Waals potential, which occurs at shorter distances of separation. On the other hand, the overlap of the potential curves for r > rsm indicates that the acceleration of the particles should be similar prior to the appearance of the secondary minimum. Since rsm is longer as the well of secondary minimum decreases, particles flocculate in those systems at longer distances of approach. Because the initial position and the radii of the particles are the same in all b-systems, the particles will have to cover a longer trajectory in systems with deeper secondary minimum. Consequently, the flocculation rate will apparently increase as the depth of the secondary minimum decreases, producing lower values of the stability ratio. A plot of W as a function of rsm supports this hypothesis. As shown in Figure 7, a monotonic decrease of W with rsm is observed independently of the method of calculation selected for the kinetic rates. As comparison of Figures 6 and 8 indicates, the dynamics of the flocculation process in system b4 is remarkably different from that of systems b1, b2, and b3. In dispersion b4 the number of aggregates fluctuates much more frequently and pronouncedly as it experience a monotonic decrease. This occurs despite the fact that the depth of the b4 minimum is 23 kBT. This behavior might be related to the shallowness of the secondary minimum, and the fact that it occurs at a distance of 245 nm, more than three times farther away than those of systems b1, b2, and b3. As a result, there is an ample range of distances of flocculation, accompanied with a smaller tendency of the particles to aggregate.


Figure 8. Variation of na vs t for system b4.

Figure 9. Variation of na vs t for system b5.

The above hypothesis is partially supported by the behavior of system b5 (Figure 9). In this system, the secondary minimum depth is only 0.5 kBT. Again, the number of aggregates fluctuates randomly near n0 during the course of the simulation. However, in this case, the absence of a profound secondary minimum does not lead to aggregation as it occurs in system b4. Therefore, the system is stable with regard to flocculation and coalescence as the ones of s1 and s2. 6. Conclusions According to the present simulations, the classical analysis of colloidal stability based on the evaluation of the stability ratio by means of eq 6 only applies when the particles are able to surmount the repulsive potential barrier, as in the case of systems s3 and s4. Barrier heights larger than 20 kBT prevent primary-minimum flocculation giving rise aggregation dynamics that are mostly determined by the characteristics of the secondary minimum. Deep secondary minima are more likely to occur with micron-size particles than with smaller ones. Secondary minima of 90 kBT or more lead to irreversible aggregation. They might be displayed by highly charged drops as the ones of bitumen, if the repulsive potential is located near the particle surface. This can happen whenever the ionic strength or the Hamaker constant of the particles is large. In this case, the rate of flocculation is fast, similar in magnitude to the one displayed by attractive particles of the same size. For the big particle systems studied, the stability ratio predicted by eq 9 did not surpass the value of 1097. Considerably lower values were found using eq 2. These results are compatible with the findings of Behrens and Borkovec90 and could probably be related to the unexpected deposition behavior of latex particles in porous media.94 On the other hand, secondary minima between 0.2 kBT and 0.5 kBT promote reversible aggregation. “Reversible”


meaning, in this context, that the particles strive to form a floc, but their mutual repulsive force constantly rejects them. As a consequence, the number of aggregates calculated fluctuates continuously. Lower depths of the secondary minimum are unable to achieve any type of

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flocculation. In the presence of a high repulsive barrier, particles will remain disunited. Otherwise, they may flocculate in the primary minimum and coalesce. LA050024P

Role of the Secondary Minimum on the ... - ACS Publications

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