Brownian Dynamics Simulation of Emulsion Stability - ACS Publications

Langmuir 2000, 16, 7975-7985

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Brownian Dynamics Simulation of Emulsion Stability German Urbina-Villalba* and Ma´ximo Garcıá-Sucre Centro de Fı´sica, Laboratorio de AÄ tomos, Mole´ culas y Campos, Instituto Venezolano de Investigaciones Cientı´ficas (IVIC), Aptdo. 21827, Caracas, Venezuela Received March 15, 2000. In Final Form: July 3, 2000

To simulate the evolution of an oil-in-water emulsion toward flocculation and coalescence, a modification of a standard Brownian dynamics algorithm was made. The resulting program takes into account the effects of surfactant diffusion and interfacial adsorption on the drop-drop interaction potential. Different realizations of the possible surfactant distributions are considered. In this work, the evolution of a small 64-particle system in the presence of a surfactant concentration gradient is studied. These results are compared with the predictions of well-known analytical formulas, which do not account for non-homogeneous surfactant distributions or a time-dependent surfactant adsorption. The particles are assumed to interact through a DLVO potential, which changed with surfactant concentration. The variation of the total number of particles with time follows the analytical predictions of Borwankar et al. for initial and intermediate steps of the flocculation/coalescence process. However, significant differences in the drop size distribution were found for longer times.

1. Introduction The adsorption of surface-active agents to oil/water interfaces plays a major role in emulsion stability. During emulsification, surfactants adsorb to the newly created interfacial area, decreasing the surface tension, and allowing partial mixing of the otherwise immiscible phases.1-2 As soon as the first drops are formed, the former emulsion starts to change due to several time-dependent processes, among which creaming, sedimentation, Ostwald ripening, flocculation, and coalescence appear to be the most important.3 It is well-known that surfactant adsorption generates a repulsive barrier toward flocculation4 that is usually ionic or steric in nature depending on the surfactant structure. However, even in the cases where flocculation is unavoidable, coalescence is still found to depend on the viscosity of the interfacial layer, the elasticity of the surfactant film, and the dynamics of drainage of the thin liquid film that separates flocculated drops.5-10 Each of those factors depends on surfactant adsorption and chemical structure, which are difficult to relate theoretically to the coalescence rate. Experimental data on surfactant adsorption to air/water interfaces are available for a large number of systems.11 * To whom correspondence should be addressed. E-mail: guv@ pion.ivic.ve. (1) Isaacs, E. E.; Chow, R. S. In Emulsions. Fundamentals and Applications in the Petroleum Industry; Schramm, L. L., Ed.; American Chemical Society: Washington, DC, 1992; Chapter 2. (2) Tadros, Th. F. Emulsion Stability Course; INTEVEP: Venezuela, 1994. (3) Adamson, A. W. Physical Chemistry of Surfaces, 5th ed.; John Wiley & Sons: New York, 1990; Chapter XIV. (4) Verwey, E. J. W.; Overbeek, J. Th. G. The Theory of Liophobic Colloids; Elsevier: New York, 1948. (5) Campanelli, J. R.; Cooper, D. G. J. Chem. Eng. 1989, 67, 851. (6) Wasan, D. T.; McNamara, J. J.; Shah, S. M.; Sampath, K.; Aderangi, N. J. Rheol. 1979, 23, 181. (7) Biswas, B.; Haydon, D. A. Proc. R. Soc. London 1963, A271, 296. (8) Jain, R. K.; Ivanov, I. B. J. Chem. Soc., Faraday Trans. 2 1980, 76, 250. (9) Ivanov, I. B., Dimitrov, D. S., Somasundaran, P., Jain, R. K, Chem. Eng. Sci. 1985, 44, 137. (10) Danov, K. D.; Petsev, D. N.; Denkov, N. D.; Borwankar, R. J. Chem. Phys. 1993, 99, 7179. (11) Rosen, M. J. Surfactants and Interfacial Phenomena, 2nd ed.; John Wiley & Sons: New York, 1989; Chapter 2.

The corresponding adsorption isotherms can be understood15-17 as particular cases of the Gibbs adsorption isotherm,18 which roughly applies to the vast majority of systems. The behavior of surfactants at oil/water interfaces is more complex. Anionic surfactants generally adsorb to oil/water interfaces according to Gibbs. This is the case of sodium nonyl sulfate, sodium lauryl sulfate, sodium undecyl sulfonate, potassium laurate, and sodium undecyl carboxylate at mineral-oil/water and heptane/water interfaces.19 Similar results had been reported by Rehfeld,20 for the case of sodium dodecyl sulfate at water/hexane, water/octane, water/nonane, water/decane, water/heptadecane, water/cyclohexane, water/benzene, and water/ carbon tetrachloride interfaces. Nonionic surfactants on the other hand do not follow the Gibbs adsorption model at oil/water interfaces due to their considerable solubility in the oil phase. This was shown by Becher,21 who measured the interfacial tension of six solutions of polyoxyethylene nonyl phenols against a total of twentynine oil phases, and five solutions of polyoxyethylene lauryl alcohols against eleven oil phases. For surfactant concentrations that ranged between 10 and 100 times the critical micellar concentration (CMC), hydrocarbon/water interfaces show a minimum of interfacial tension with respect to surfactant concentration, while aromatichydrocarbon/water systems present a monotonic decrease. Surfactant adsorption to fluid/liquid interfaces is also known to depend on time. Adsorption times may vary from fractions of a second to several minutes depending on the system and the surfactant concentration.22-31 In (12) Langmuir, I. J. Am. Chem. Soc. 1917, 39, 1848. (13) Szyszkowski, B., Z. Phys. Chem. 1908, 64, 385. (14) Frumkin, A. Z. Phys. Chem. 1925, 116, 466. (15) Lucassen-Reynders, E. H. Prog. Surf. Membr. Sci. 1976, 10, 253. (16) Lucassen-Reynders, E. H. J. Phys. Chem. 1966, 70, 1777. (17) Butler, J. A. V. Proc. R. Soc. London 1932, A135, 348. (18) Gibbs, J. W. The Collected Works of J. W. Gibbs; Longmans: London, 1928; Vol. 1. (19) van Voorst Vader, F. Trans. Faraday Soc. 1960, 56, 1067. (20) Rehfeld, S. J. J. Phys. Chem. 1967, 71, 738. (21) Becher, P. J. Colloid Interface Sci. 1963, 18, 665. (22) Addison, C. C.; Hutchinson, S. K. J. Chem. Soc. 1949, 3387. (23) Kloubek, J. J. Colloid Interface Sci. 1972, 41, 17. (24) Delahay, P.; Fike, C. T. J. Am. Chem. Soc. 1958, 80, 2628.

10.1021/la000405x CCC: $19.00 © 2000 American Chemical Society Published on Web 09/23/2000

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the limit of very short times, and under diffusion-controlled adsorption,32 theory predicts a square root dependence of the surface coverage on time (t), as well as a t-1/2 variation for very long times. Modifications of this former treatment33-41 include the effects of surface diffusion,36 micellization,26,41 surfactant mixtures,37 double-layer formation,25,27,28,34,40 oil-phase composition,34,38,42 etc. Along with the diffusion-limited regime, current treatments usually take into account the possible slowing of the adsorption rate due to the generation of some kind of repulsive potential between the surfactant molecules already adsorbed and those diffusing to the interface (kinetically limited adsorption).28,43 Since the repulsive interaction between emulsion drops markedly depends on surfactant adsorption, surfactant concentration and diffusion are expected to play a prominent role in emulsion stability. If the drops of the internal phase are only stabilized when they achieve certain surfactant coverage, the initial drop-size distribution resulting from emulsification is likely to evolve with time until the excess interfacial area is suppressed. On the other hand, if there is enough surfactant in the system to completely stabilize the initial drop-size distribution (DSD), such stabilization should dramatically depend on surfactant diffusion and adsorption times. Thus, it is not surprising that the prediction of the initial DSD based on the knowledge of the unstirred system and the emulsification parameters is extremely difficult. During emulsification, the breakage of drops is favored by the action of the impeller, while their coalescence is enhanced due to convection. In batch mixing tanks, the coalescence rate approximately varies as φaNb (where φ is the volume fraction of the disperse phase, N is the impeller speed, and a and b are constants, 0.5 < a < 1.1, 1.5 < b < 3.3).44 The initial drops’ diameters depend on the density of the phases, the volume fraction of the disperse phase, the interfacial tension, and the impeller’s diameter and speed.44,45 For mineral-oil/water mixtures stirred between 160 and 278 rpm, Ross, Verhoff, and Curl46,47 reported average coalescence rates between 3 and 200 min.47 However, a dramatic variation of the mixing rate as a (25) Hua, X. Y.; Rosen, M. J. Colloid Interface Sci. 1991, 141, 180. (26) Makievski, A. V.; Fainerman, V. B.; Joos, P. J. Colloid Interface Sci. 1994, 166, 6. (27) Lin, S.; McKeigue, K.; Maldarelli, C. AiChE J. 1990, 36, 12. (28) Bonfillon, A.; Sicoli, F.; Langevin, D. J. Colloid Interface Sci. 1994, 168, 497. (29) van den Bogaert, R.; Joos, P. J. Phys. Chem. 1980, 84, 190. (30) Pollard, M. L.; Pan, R.; Steiner, C.; Maldarelli, C. Langmuir 1998, 14, 7222. (31) Borwankar, R. P.; Wasam, D. T. Chem. Eng. Sci. 1983, 38, 1637. (32) Ward, A. F. H.; Tordai, L. J. Chem. Phys. 1946, 14, 453. (33) Miller, R.; Kretzchmar, G. Adv. Colloid Interface Sci. 1991, 37, 97. (34) Borwankar, R. P.; Wasam, D. T. Chem. Eng. Sci. 1988, 43, 1323. (35) Chang, C. H.; Franses, E. I. Colloids Surf. 1992, 69, 189. (36) Good, P. A.; Schechter, R. S. J. Colloid Interface Sci. 1972, 40, 99. (37) Ariel, G.; Diamant, H.; Andelman, D. Langmuir 1999, 15, 3574. (38) Kim, Y.; Koczo, K.; Wasan, D. T. J. Colloid Interface Sci. 1997, 187, 29. (39) Hsu, C.; Chang, C.; Lin, S. Langmuir 1999, 15, 1952. (40) Radke, C. J.; MacLeod, C. A. Langmuir 1994, 10, 3555. (41) Dushkin, C. D.; Ivanov, I. B.; Kralchevsky, P. A. Colloids Surf. 1991, 235. (42) Hansen, F. K.; Fagerheim, H. Colloids Surf., A 1998, 137, 217. (43) Diamant, H.; Andelman, D. Europhys. Lett. 1996, 34, 575. (44) Sovova, H. Chem. Eng. Sci. 1981, 36, 1567. (45) Mendiboure, B.; Dicharry, C.; Marion, G.; Morel, G.; Salager, J. L.; Lachaise, J. Prog. Colloid Polym. Sci. 1993, 93, 307. (46) Verhoff, F. H.; Ross, S. L.; Curl, R. L. Ind. Eng. Chem. Fundam. 1997, 16, 371. (47) Ross, S. L.; Verhoff, F. H.; Curl, R. L. Ind. Eng. Chem. Fundam. 1978, 17, 101.

Urbina-Villalba and Garcıá-Sucre

function of the impeller speed was found, while moderate changes in the drop sizes were observed. The wide variety of possible DSDs resulting from emulsification is well illustrated in the work of Sańchez et al.48 They studied the evolution of 75 wt % sunflower oil in water emulsions, as a function of surfactant concentration and structure. Poly(ethylene glycol) ethers (Triton N-101, Triton X-100), sucrose laurates (L1695), and polyoxyethylene sorbitan monolaurate (Tween-20) were used. First, temperature was found to play a major role in the final form of the DSD. For temperatures lower than 25 °C, the DSDs are unimodal and the droplet size decreases as the emulsification time increases. At higher temperatures, the DSDs are unimodal at intermediate times but tend to be bimodal at larger times. Second, the larger average drop sizes were obtained for the bimodal DSDs of Tween-20 and L1695, while smaller droplet diameters and much narrower unimodal DSDs were obtained for the rest of the surfactants. Finally, the resulting dispersions were found to depend on the emulsification process: emulsions prepared with agitation speeds between 1.7 and 2.5 s-1 underwent phase separation after 4-10 days, while those prepared at 3.3-5.0 s-1 did not show phase separation after one month of aging. It is clear from the above paragraph that the process of emulsification may determine not only the initial characteristics of the DSD, but also its stability. Polydisperse DSDs and homogeneous surfactant distributions favor surfactant diffusion from one phase to another and from the bulk phases to the interface. If the surfactant adsorption is slow, the drops will collide before the surfactants reach the available interfaces, and the DSD will shift toward higher average radii. This could happen even in the case where the surfactant concentration is high enough to stabilize the total interfacial area of the initial DSD. Hence, experimentalists preequilibrate the oil/water/surfactant mixture before stirring49,50 to achieve an acceptable reproducibility. Since the surfactant is usually dissolved in one of the bulk phases, a certain amount of it has to diffuse to the other phase to equalize the chemical potentials. Thus, preequilibration decreases the times of adsorption, allowing the surfactant to adsorb from both bulk phases to the interface. However, it cannot prevent surfactant redistribution due to the creation of “fresh” interfacial area during mixing. The rich experimental behavior outlined prevents the elaboration of simple codes for the simulation of emulsion stability toward coalescence. Computer simulations on floc formation were used as early as 196351 to explain the large sediment volume of silica sols in organic media. Those calculations employed a simple model of successive random addition of individual spherical particles around a fixed position. More sophisticated two-dimensional simulations in which randomly selected particles were aggregated and their aggregates “compacted” through subsequent rotations had proved successful in reproducing Smoluchowski’s variation of the number of N-particle aggregates with time.52 Models as simple as hard spheres and pearlnecklace polymers with a sticking probability had proved useful for the calculation of the fractal properties of (48) Sańchez, M. C.; Berjano, M.; Guerrero, A.; Brito, E.; Gallegos, C. Can. J. Chem. Eng. 1998, 76, 479. (49) Salager, J. L. Emulsionacioń. Ediciones del Laboratorio ULAFIRP; Cuaderno 232: Me´rida, Venezuela, 1993. (50) Moreno, N. Emulsionacioń en sistemas no equilibrados. Br. Thesis, Me´rida, 1983. (51) Vold, M. J. J. Colloid Interface Sci. 1963, 18, 684. (52) Aoki, Y.; Yamada, Y.; Danjo, K.; Yonezawa, Y.; Sunada, H. Chem. Pharm. Bull. 1995, 43, 1197.

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aggregates formed by the complex phenomena of bridging flocculation.53,54 More elaborate models for colloidal systems including the explicit movement of particles as a function of time are also published. Computer simulations of impact flocculation between wet rotating particles interacting through capillary forces under the action of gravity are available.55 Mitchell et al. reported Brownian dynamics (BD) simulations of near-hard spheres under shear (V ) (σ/r)n, n ) 36).56 Their calculations used 108, 122, 256, 500, 1372, and even 4000 particles to evaluate the theoretical structure factor of sheared systems, and even the intensity spectra of the resulting particle distribution. The purpose of this paper is to describe a computational technique based on Brownian dynamics, which we found suitable for understanding the evolution of a DSD with time, in terms of surfactant partitioning, interfacial adsorption, and chemical structure. 2. Emulsion Stability Simulations Brownian dynamics simulations of monodispersed distributions of 864 Lennard-Jones particles had been successfully used for studying the gas/liquid-phase separation and its similarities with the process of gel formation.57 Segre et al.58 had included hydrodynamic interactions in the movement of hard-sphere particles by coupling Brownian dynamics with lattice Boltzmann techniques. Those calculations are probably the state of the art in the detailed mesoscopic description of the solvent influence upon Brownian movement. Several algorithms for Brownian dynamic simulations are available.59-61 According to Ermak and McCammon,61 the movement of N particles in a solvent can be simulated using a Langevin-type equation of motion for each particle. In the absence of hydrodynamic interactions, the position of a particle at time i + 1 along the X coordinate is given by

Xi+1 ) Xi + (DF/kT)∆t + Ran (2D∆t)1/2

(1)

where D is the diffusion constant, F the driving force, k the Boltzmann constant, T the temperature, ∆t the time step, and Ran a Gaussian random function with zero mean and unit variance. The force F includes interparticle, gravitational, electrostatic, and any other relevant external force. Hence, the movement of each particle is the result of monotonic driving force plus a random “kick”, which mimics the resulting thermal momentum, infringed by the solvent molecules upon the particle surface. For the purpose of simulating emulsion stability, DLVO,4,62-64 gravitational, hydrodynamic,61,65 and some kind of empirical potential that takes into account the (53) Stoll, S.; Buffle, J. J. Colloid Interface Sci. 1996, 180, 548. (54) Stoll, S.; Buffle, J. J. Colloid Interface Sci. 1998, 205, 290. (55) Lian, G.; Thornton, C.; Adams, M. J. Chem. Eng. Sci. 1998, 53, 3381. (56) Mitchell, P. J.; Heyes, D. M. J. Chem. Soc., Faraday Trans. 1995, 91, 1975. (57) Felicity, J.; Lodge, M.; Heyes, D. M. J. Chem. Soc., Faraday Trans. 1997, 93, 437. (58) Segre, P. N.; Behrend, O. P.; Pusey, P. N. Phys. Rev. E 1995, 52, 5070. (59) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids, 8th ed.; Oxford University Press: New York, 1997; Chapter 9. (60) Van Gunsteren, W. F.; Berendsen, H. J. C. Mol. Phys. 1982, 45, 637. (61) Ermak, D.; McCammon, J. A. J. Chem. Phys. 1978, 69, 1352. (62) Verwey, E. J. W.; Overbeek, J. Th. G. Trans. Faraday Soc. 1946, 42B, 117. (63) Melik, D. H.; Scott Fogler, H. In Encyclopedia of Emulsion Technology; Becher, P., Ed.; Marcel Dekker: New York, 1988; Vol. 3, Chapter 1.

effects of film drainage upon coalescence8-10 (hydrodynamic interactions of type I) are required. The present code only includes DLVO, gravitational, and a specific kind of the referred empirical potential that we shall call “elastic”. The effects of particle movement upon the diffusion tensor (type II hydrodynamic corrections) are not considered here, though it is known from experiments on dilute suspensions of monodisperse 500 nm polystyrene spheres that the hydrodynamic terms slow the coagulation rate66 due to fluid-mediated interactions. Since the present calculations are meant to serve as our standards for more involved computations, only DLVO forces will be included in F (eq 1), although two additional simulations showing the effects of gravity and thermal motion on the interparticle potential will also be presented. An excellent review on the fundamentals of colloidal stability in quiescent media is given in ref 63. Analysis of experimental data on a DLVO basis is common67-69 but is almost invariably restricted to a comparison of the timeindependent potential barriers between equally sized drops at a given salt concentration. According to DLVO theory, the interaction between two colloidal particles is basically the result of dispersion and electrostatic forces. The dispersion forces are assumed to be always attractive, and are the result of van der Waals interactions between the molecules of each particle. Electrostatic interactions depend on each particle’s charge and the distribution of counterions around it. The result of these two contributions gives a potential curve with two minima, the principal minimum associated with coalescence and the secondary minimum with flocculation. According to Liftshitz theory70,71 the assumption of attractive dispersion interactions is correct for identical interacting bodies in a medium in the absence of retardation effects. Retardation effects may cause a change in sign of the dispersion interactions of small hydrocarbons in water at intermediate separations (>5 nm).70,72 In these calculations, the behavior of bitumen-in-water emulsions will be simulated so that a change in sign of the dispersion interaction at intermediate distances is not expected. For the attractive component dispersion interaction between two spheres, we have employed the simplified version of the complete Hammaker formula:73

Udispersion ) -

Aij RiRj 6d Ri + Rj

(2)

where Rk is the radius of particle k, Aij is the Hammaker constant, d is the distance between the particles’ surfaces (d ) rij - Ri - Rj), and rij is the distance between the centers of mass of particles i and j. As shown by Lu et al.,74 the singular behavior of eq 2 at very short distances can be avoided considering finite molecular sizes in the (64) Evans, F.; Wennerstro¨m, H. The Colloidal Domain: Where Physics, Chemistry, Biology and Technology meet, 1st ed.; VCH Publishers: New York, 1994. (65) Rotne, J.; Prager, S. J. Chem. Phys. 1969, 50, 4831. (66) Zeichner, G. R.; Schowalter, W. R. J. Colloid Interface Sci. 1979, 71, 237. (67) Rios, G.; Pazos, C.; Coca, J. J. Dispersion Sci. Technol. 1998, 19, 661. (68) Salou, M.; Siffert, B.; Jada, A. Colloids Surf., A 1998, 142, 9. (69) Hunter, R. J. Introduction to Modern Colloid Science, 1st ed.; Oxford University Press: Oxford, 1993; Chapter 2. (70) Israelachvilli, J. Intermolecular and Surface Forces, 2nd Ed.; Academic Press: London, 1992; Chapters 6 and 11. (71) Liftshitz, E. M. Sov. Phys. JETP 1956, 2, 73. (72) Mahanty J.; Ninham, B. W. Dispersion Forces 1st Ed.; Academic Press: New York, 1976. (73) Hammaker, H. C. Physica (Amsterdam) 1937, 4, 1058. (74) Lu, J. X.; Marlow, W. H.; Arunachalam, V. J. Colloid Interface Sci. 1996, 181, 429.

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computation of the pairwise additive dispersion interaction.72 Since the nonsingular interaction formula between two spheres contains 17 terms (which are integrals or series on the radii of the spheres and separation distance), we have employed the usual analytical expression (eq 2). This assumption is justified whenever the Hammaker constant (Aij) has been deduced from critical coagulation concentration data employing eq 2. For other cases, the nonsingular corrections and the additional geometric terms reported by Hammaker for the case where d g Ri + Rj should be included. It is important to remark that geometric corrections of eq 2 for the case of slightly deformed droplets at flocculation distances are also available.75,76 In these initial calculations, however, we do not consider the process of film drainage in detail, and the drops are assumed to be rigid to avoid geometrical corrections of eq 2. The electrostatic interaction was modeled with the analytical formulas recently provided by Sader77 for the surface charge density and electrostatic potential of a particle immersed in a 1:1 electrolyte. This is a solution of the Poisson-Boltzmann equation for the far-field electrostatic potential. It is supposed to be valid for all values of κa (where κ stands for the inverse of the Debye length and a is the particle radius), keeping a good degree of accuracy for all surface potentials of practical interest (tf)Γsat-1 exp

)

[t - tf] ln C2-1 (14) tf,2 - tf

where C is a positive large number (∼1000), t0 the time at which surfactant diffusion starts, and tf the time needed for a surfactant to achieve the transition state. S(t1 > t2) is a Heaviside-type function (S(t1 > t2) ) 0 for t1 < t2; S(t1 > t2) ) 1 for t1 > t2). In eq 14, tf is the time required for the attainment of the transition state, Γsat-1 the corresponding interfacial area of the surfactant in such an intermediate condition, tf,2 the time required for the final saturation of the interface, and C2 a positive constant, whose value is determined by the final area of the surfactant at the end of the adsorption process, Γsat,2-1:

Γsat,2-1

) lim

tftf,2

Γsat-1

(

)

[t - tf] ln C2-1 exp tf,2 - tf

(15)

Thus

C2 ) Γsat-1/Γsat,2-1

(16)

It is clear that a technique similar to that described for the simulation of surfactant adsorption can be employed for processes as complex as Ostwald ripening. In that case, the transfer of matter from small to large drops can be simulated, increasing the size of the big drops at the expense of the small drops as a function of time. However, as Blinks et al.86 had deduced from interesting experiments of employing n-dodecyl octaoxyethylene glycol, and mixed drops of squalene and decane in water, Ostwald ripening increases with the solubility of the oil. Thus, for heavyoil-in-water emulsions such as those simulated in this paper, Ostwald ripening can be disregarded. The current version of the program incorporates several “strategies” for distributing the surfactant molecules among the available interfacial area. These strategies are closely connected with the temporal variation of the surface excess. They intend to simulate the cases where (a) the surfactant distribution resulting after mixing is either homogeneous or non-homogeneous, (b) the surfactant molecules adsorb either fast or slow to the interface, and (c) the surfactant adsorption is either reversible or irreversible. (86) Binks, B. P.; Clint, J. H.; Fletcher, P. D. I.; Rippon, S. Langmuir 1998, 14, 5402.

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As will be shown below, a non-homogeneous surfactant distribution can cause emulsion destabilization at higher surfactant concentrations than those required for complete stabilization of the droplets under homogeneous conditions. Whenever the surfactant adsorption is extremely fast, the effects of surfactant diffusion on coalescence should be negligible. This is also the case of small oil/ water volume-fraction-percentage emulsions (φoil/water), even if surfactant adsorption is not very fast. Minimum collision times of 11.9, 1.70, and 0.0017 s were found for 64-, 125-, and 216-particle calculations, departing from simple cubic arrangements (a ) 3.9 µm, D ) 5.6 × 10-14 m2/s, φoil/water ) 15%, 30%, and 51%, respectively). Slightly smaller times are expected for initial random arrangements which are only possible for the lower volume fractions, since φoil/water ) 52% already corresponds to maximum packing for spheres of equal size distributed in simple cubic arrangements.87 To study concentration effects separately from timedependent ones, instantaneous adsorption of the surfactant was assumed in the present φoil/water ) 15% simulations. This is the simplest case in which Asurf ) Γsat-1, and the available interfacial area is equal to the sum of the partial area of each drop. Once the surfactant is distributed between the available drops, the total charge of the drops can be computed as well as the E value (eq 9). The diffusion constant of the drops is either introduced as input or estimated according to Stokes law. In either case, when the radius of the drop changes due to coalescence, its diffusion constant is reevaluated employing an inverse dependence on the radius as predicted by Stokes. Following, the total force acting upon each particle can be computed, and the particles moved according to usual Brownian dynamics (eq 1). At each step, the program checks for overlap, in which case a new particle is formed at the center of mass of the colliding drops, evaluating the new radius from the conservation of mass. A new diffusion constant is then calculated, and the available surfactant redistributed according to the previously selected scheme. At that time, the program recomputes the forces, and is ready for another iteration. 3. Computational Details To test our program, we have employed the data of Salou et al.68 for the stability of their “E3” bitumen-in-water emulsion. Such an emulsion has an average drop radius of 3.9 µm, a very high ionic force (κa ) 1536), a Hammaker constant of 1.24 × 10-19 J, and an electrostatic ζ-potential of +115 mV at pH 2.9. Use of eq 7 allows the surfactant charge density to be calculated, which gives +115 mV for a drop of radius 3.9 µm, assuming a surfactant interfacial area of 50 Å2.11 The resulting DLVO potential between two drops of ri ) r/a ) 1.0 (a ) 3.9 µm) is shown in Figure 1. The potential presents the usual minima, which due to the high polarity of the medium are situated extremely close to the particle surface. As previously discussed, the elastic potential was set to negligible values (K2 ) 0 J‚m2, K1 ) 1 × 10-8 m2, n ) 2, a1 ) -6.95 × 10-37 J‚m2, a2 ) 1.04 × 10-49 J‚m4, ai ) 0 for all i > 2). For a fixed ionic force, the electrostatic repulsion diminishes as the surfactant concentration decreases (Figure 1). It should be noticed, however, that a small concentration might be enough to build a considerable energetic barrier toward coalescence. If the electric charge of the drop diminishes due to a deficiency in surfactant adsorption, the potential maximum lowers and moves toward smaller interdrop distances. This limits the time step of the simulation, which should be kept short enough to sample the total potential, but long enough to allow considerable movement (87) Ashcroft, N. W.; Mermin, D. Solid State Physics, 1st ed.; Holt, Rinehart & Winston: New York, 1976; Chapter 2.


Figure 1. Dependence of the DLVO potential on interdrop distance for several surfactant concentrations: (a) repulsion barrier between the second and the primary minima; (b) closer look of the secondary minimum as a function of the interdrop distance for different surfactant concentrations. of the particles. The time step in units of a2/D was set to 5 × 10-9. For 150 million iterations, this time step was equivalent to 3.49 min of real time. Although Salou’s data correspond to bitumen-in-water emulsions which contain a high amount of internal phase, only 64drop simulations for 15% v/v oil-in-water (O/W) emulsions are presented in this work. The total surfactant concentrations ranged from 5.0 × 10-5 to 1.3 × 10-3 M. Most simulations were carried out in the presence of a surfactant concentration gradient which increased from one of the bottom corners (x < 0, y < 0) of the simulation box to the diagonally opposite upper corner. In practical terms this means that the drops in the bottom of the simulation box are covered by surfactant before those located at the top. Additional simulations for the cases of homogeneous surfactant distributions, density differences between oil and water, and thermal motion effects were also calculated for the concentration of 1.0 × 10-4 M. Three-dimensional periodic boundary conditions were employed under the minimum image convention.59 Since emulsion drops cannot be considered as point particles, a considerable volume of a drop could lay outside the simulation box, while its center of mass is still inside. Thus, a real drop could overlap with the image of another drop. We let real particles coalesce with imageparticles in the same way they do with other real ones inside the simulation box. In either case, the resulting center of mass was computed with the coordinates of the colliding particles. If the new center of mass fell outside the simulation box, these coordinates corresponded to the image of the new real particle, and therefore, the real particle coordinates were determined, adding or subtracting the side length of the simulation box. Since the image particle involved in the collision was the image of a real particle, that real particle also disappeared after the collision. At the end of the process, two real drops disappeared, and one new real drop was created.

4. Results and Discussion Figure 2 shows the variation of the total number of particles with time for various surfactant concentrations.


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Figure 2. Variation of the number of drops as a function of scaled time for (]) 5.0 × 10 -4 M, (4) 3.5 × 10 -4 M, (0) 2.5 × 10 -4 M, (O) 1.0 × 10 -4 M, and (+) 5.0 × 10 -5 M.

Figure 4. X-Y projection of the final positions of the surviving drops after 150 million iterations for different surfactant concentrations. The number of remaining drops decreases when the surfactant concentration is lowered. Except for the most dilute cases in which only a few drops are leftover, a substantial degree of secondary minimum flocculation is observed. Figure 3. Surface potential (φp) as a function of surfactant concentration for a homogeneous surfactant distribution among 64 and 125 drops.

It is observed that when the total surfactant concentration is 5 × 10-4 M the initial number of drops is preserved during the whole simulation. In this case, the total surfactant concentration (c) is higher than a critical value required for complete coverage of the available interfacial area (c > cc(64) ) 3.87 × 10-4 M). Figure 3 shows the variation of the electrostatic surface potential of the drops as a function of surfactant concentration for the case in which the surfactant is homogeneously distributed among all drops. At c ) cc(64) the surface potential of each drop is +115 mV as reported by Salou et al.68 If instantaneous adsorption is assumed, the surfactant distribution strategy used to simulate the existence of a surfactant concentration gradient does not influence coalescence, since there is enough surfactant to cover all the drops. At this high concentration, the potential energy between drops (a ) 3.9 µm) shows a very high barrier toward coalescence (Figure 1), as well as a deep flocculation minimum (∼-600 kT). Figure 4 shows the X-Y projection of the particle positions at the end of the simulation. As expected, a considerable degree of (secondary-minimum) flocculation is clearly observed (see inset 2 in Figure 4). Table 1 presents the mean number of contacts between drops, defined as the average number of inter-drop distances lower than a certain multiple of the Debye length: Nc ) Mλc, where λc ) 2/κa ) 0.001 318 (for the highest surfactant concentration employed). According to van den Temple,88-90 the number of contacts in an (88) van den Temple, M. Recl. Trav. Chim. Pays-Bas 1953, 72, 419.

aggregate should be proportional to the rate of coalescence. This appealing assumption allowed the inclusion of coalescence rates within Smoluchowski theory. However, as shown in Table 1, the quantification of the number of contacts between aggregates of nondeformable droplets is not a reliable measurement of flocculation, due to the arbitrary geometrical definition of a floc. Although we did not endeavor a detailed study of the internal dynamics of flocs, such an investigation can be carried out with the use of time-dependent correlation functions. These ambiguities in the geometrical definition of a floc are easily distinguished by visual inspection whenever the drops change their shape upon flocculation. However, under the present computational conditions, nondeformable drops are expected to occur, since it has been demonstrated10 that, for the similar case of 5 µm size droplets, interacting through electrostatic (σ ) 10-50 mV), van der Waals (A ) 1 × 10-20 J), and surface deformation forces, the kinetic energy is generally not large enough to produce deformation and film formation whenever the particles are small and the electrostatic repulsion is large. Deformation energies can be suitably included within the elastic potential (eqs 8 and 9), but the explicit geometrical deformation of the drops is probably very expensive in terms of computation times. As shown in Figure 2, surfactant concentrations lower than cc(64) lead to coalescence. In these cases the curves present three distinct regions of variation of the number of particles with time: (a) a small region of constant number of particles at the beginning of the simulation (89) van den Temple, M. Recl. Trav. Chim. Pays-Bas 1953, 72, 433. (90) van den Temple, M. Recl. Trav. Chim. Pays-Bas 1953, 72, 443.

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Table 1. Geometrical Flocculation Data Evaluated as the Number of Contacts for the Final Configuration of Drops after 150 Million Iterations

surfactant concn (M)

av. interparticle distance (a)

no. of surviving particles

av. no. of contacts per particle for Nc < 1.10 λc



av. no. of contacts per particle for Nc < 20 λc

t ) 0 (all concns) 5.0 × 10-4 3.5 × 10-4 2.5 × 10-4 1.0 × 10-4 5.0 × 10-5 1.0 × 10-6

7.75 7.44 7.54 4.40 8.97 5.90 5.02

64 64 54 15 4 4 2

0.0000 0.7187 1.0000 0.1333 0.0000 0.0000 0.0000

0.0000 2.3125 1.9259 0.4000 0.0000 0.0000 0.0000

0.0000 3.3750 3.1852 1.2000 0.0000 0.0000 0.0000

4.5000 7.1481 7.1481 5.8667 0.0000 0.0000 0.0000

Table 2. Remaining Number of Drops and Total Interfacial Area Available at the End of the Simulation for Different Surfactant Concentrations

Figure 5. Variation of the number of particles as a function of time for a surfactant concentration of 1 × 10-4 M. Computational conditions similar to those of curve “O” in Figure 2, except for (a) the random Brownian step was duplicated in magnitude, (b) the gravity force was included assuming a density difference of 0.1 g/cm3, (c) the surfactant was distributed randomly among drops at each time step of the simulation, and the available surfactant interfacial area was exponentially decreased.

(region A), (b) a flocculation-coalescence zone where the number of particles decreases with time (region B), and finally (c) a region of constant number of particles toward the end of the simulation (region C). At the beginning of the simulation there is a certain time required for the first inelastic collision to happen (region A). As pointed out in the Introduction, the minimum collision time (tm) decreases as the drop volume fraction (φoil/water) increases. The magnitude of this time is not physically meaningful, since it depends on the initial configuration of the particles. As shown by unpublished calculations in which initial random configurations are used, tm lowers with the increase in randomness of the initial configuration. Totally random configurations are only possible for low volume fractions due to packing constraints. For t > tm, the total number of drops diminishes with time due to coalescence (region B). According to Verwey and Overbeek,62 coalescence is feasible whenever the DLVO barrier is lower than 25 kT. For the case in which the surfactant is evenly distributed among all drops, a barrier of 25 kT is obtained when c ) 8.1 × 10-5 M. At this concentration, the electrostatic surface potential of each drop is equal to 44.3 mV ()φp). According to the Salou et al. data,68,91,92 stable bitumen emulsions are obtained for high resin/asphalthene ratios, and φp ) 50 mV. Figure 5 shows the variation of the number of drops for a surfactant concentration of 1.0 10-4 M for which φp ) 50.7 mV (barrier (91) Salou, M.; Siffert, B.; Jada, A. Fuel 1998, 77, 343. (92) Salou, M.; Siffert, B.; Jada, A. Fuel 1998, 77, 339.

initial no. of drops

surfactant concn (M)

no. of remaining drops

final interfacial area in units of a2 (a ) 3.9 µm)

64 64 64 64 64 64

5.0 × 10-4 3.5 × 10-4 2.5 × 10-4 1.0 × 10-4 5.0 × 10-5 1 × 10-6

64 54 8 4 4 2

804.25 731.73 289.64 306.50 298.30 251.95

height 104 kT). As expected, the initial number of drops is preserved during the whole simulation. The equivalent calculation for the case in which the surfactant is not homogeneously distributed shows that the number of drops changes dramatically with time, until only a few drops are leftover. When a gradient of concentration exists, the drops near the most concentrated zones are completely covered by surfactant while the others remain partially or totally uncovered. Partially covered drops present a potential barrier toward coalescence, which is intermediate between the most repulsive barrier shown by completely covered drops and a totally attractive potential shown by completely uncovered drops; thus, some collisions lead to coalescence, while others do not. As the drops coalesce, the available interfacial area diminishes, and less surfactant is required to cover the remaining drops. At a certain time, the surfactant adsorbed is enough to generate a considerable repulsive barrier toward coalescence. From that point on, the number of drops remains constant because the repulsive potential between drops is maximum (region C). Drops flocculate but do not coalesce further (see Table 2). Whenever the surfactant concentration is high enough, the remaining oil/water interfacial area becomes equal to the sum of the surfactants’ interfacial areas (Ac ) ΣAs ) NsAs), and maximum drop-drop repulsion is achieved. The referred behavior has already been observed by Cornell et al.93 These researchers carried out a direct microscopic study on the motion of spherical polystyrene particles of 3.76 µm, at NaCl concentrations between 10-1 and 10-5 M. At the most dilute electrolyte concentration no change was observed in the number of single particles until 6.3 h. Between 10-3 and 10-2 M the percentage of single particles initially decreased and then became constant over an extended period of time, suggesting the possibility of a steady-state condition in the aggregation process. At higher electrolyte concentration, the initial rate of disappearance increased. About critical coagulation concentration (3 × 10-1 M), aggregation became very extensive after only 30 min. (93) Cornell, R. M.; Goodwin, J. W.; Ottewill, R. H. J. Colloid Interface Sci. 1979, 71, 254.


Similar abrupt decreases in the flocculation rates have been reported for the case of bridging flocculation,94 but no evident connection appears to exist between these two cases. The attainment of a stable state at long times, different from phase separation, was predicted by Hall et al. in 1991, who introduced the concept of re-stabilization in coalescing emulsions.95,96 In these ingenious calculations, a matrix of surface potentials and drop radia was initially defined. Two drops (represented by their matrix positions) were randomly selected, and their total DLVO free energy was computed at contact. Whenever the total potential allowed it, the drops were supposed to coagulate, and the respective matrix position was increased by one. After some coagulation time, the matrix was found to remain constant, suggesting that a new stable equilibrium state was achieved. Unfortunately, these authors assumed that the charge of each new drop was the sum of the partial charges between the coalescing droplets. This led to an unphysical increase of the surface potential, since the interfacial area decreased as the drop charge increased. Thus, new drops were found to be more and more stable until a steady-state condition was achieved. Interestingly, however, our calculation also predicts a restabilization of the emulsion but through a different mechanism: as the interfacial area diminishes as a result of coalescence, more surfactant is left free to adsorb to less populated interfaces. Surfactant adsorption increases the surface potentials of the drops. When the amount of surfactant is large enough to generate a repulsive barrier between the most uncovered drops, the number of drops should not change any further. Naturally, there is a very high number of “final” dropsize distributions compatible with a given interfacial area, Ac. A specific distribution will be the result of the dynamics of flocculation and coalescence, which in turn depend on the details of surfactant partition among drops. For a given surfactant concentration, Ac might be expected to be similar among those steady-state distributions. As shown in Figure 5, a density difference of 0.10 g/cm3 may accelerate coalescence considerably, but the Ac value is almost equal to that of the nongravitational case (Figure 2). Experimental data may still show destabilization due to Ostwald ripening. The extent of Oswald ripening will be different depending on the characteristics of the dropsize distribution, and on the solubility of the dispersed phase. Bitumen emulsions are expected to be highly insoluble in water, so they are expected to show restabilization. However, even in the cases in which oil solubilization may favor Oswald ripening, this effect can be prevented, producing a very narrow drop-size distribution within a homogeneous surfactant solution. Two different kinds of region B behavior were found (see Figure 2). For the lower concentrations (1 × 10-6, 5 × 10-5, and 1 × 10-4 M) a monotonic decrease in the number of particles with time was observed. However, for 2.5 × 10-4 M, two regions of rapid variation of the number of drops with time were observed (t* ) 0.160 and 0.374). In both cases multiple collisions occurred. This calculation (2.5 × 10-4 M) was repeated three times employing different seeds for the random variable, and a similar behavior was found. Since no hydrodynamic corrections were made, this could possibly be an artifact of the (94) Matsumoto, T.; Adachi, Y. J. Colloid Interface Sci. 1998, 204, 328. (95) Hall, S. B.; Duffield, J. R.; Williams, D. R. J. Colloid Interface Sci. 1991, 143, 411. (96) Hall, S. B.; Duffield, J. R.; Williams, D. R. J. Colloid Interface Sci. 1991, 143, 416.

Langmuir, Vol. 16, No. 21, 2000 7983

calculation. In highly concentrated simulations,97 multiple collisions are expected and had been distinguishably observed. In such constant-volume simulations of concentrated systems, they are unavoidable due to geometric restrictions, even if diffusion constants with hydrodynamic corrections are employed. Again, this could also be an artifact of the simulation but caused by the constantvolume constraint. Curve c in Figure 5 shows that this behavior might be related to the mechanism of surfactant distribution/adsorption. In this case (c ) 1 × 10-4 M) the surfactant was randomly distributed at every time step of the simulation, and its surface area varied according to the diffusive term of eq 14. This elaborate mechanism was meant to illustrate the possible effects of a complex surfactant adsorption mechanism. The changing surfactant distribution simulated surfactant desorption and readsorption to the interface, while the variable surface area was meant to mimic diffusion-controlled adsorption. Two major “jumps” in the variation of the number of particles as a function of time can be observed. As before, those jumps are evidence of simultaneous collisions between particles, which were absent when the surfactant was homogeneously distributed among all drops (Figure 5), or a surfactant concentration gradient existed (Figure 2). The variation of the total number of drops as a function of time was compared with the analytical equations of Smoluchowski,98 van den Temple,88-90 and Borwankar et al.99 First, the theoretical flocculation rate constant of Smoluchowski was employed, and afterward both the flocculation and coalescence rate constants were fitted until the minimum average difference between the simulation data and each theoretical prediction was obtained. The theoretical flocculation constant formerly provided by Smoluchowski and included in the models of van den Temple and Borwankar et al. (kf ) 4kT/η ) 5.49 × 10-18 m3/s) produced the largest deviations, because it was formulated for the case of noninteracting particles which adhered irreversibly upon contact. Direct application of Fuch’s correction (k′ ) kf/W),100 employing Reering and Overbeek’s stability factor (W ) kf/2 exp(Gmax/kT)),101 was not possible since the magnitude of the maximum free energy barrier between drops (Gmax) continuously changed during the dynamics as a consequence of the unequal surfactant distribution among drops. Thus, both fast and slow flocculation may take place between different drops at the same time, which is probably a more realistic situation. In the present case, the considerable depth of the secondary minimum and the sizable height of the energy barrier for irreversible flocculation at high surface coverage should cause gathering of the stable drops at the end of the simulation, even if coalescence does not occur (see Figure 4(2)). Hence, the kf constant employed in the van den Temple model should be different from that found by Smoluchowski for intermediate surfactant concentrations. Figure 6 shows the four best analytical predictions for a surfactant concentration of 1.0 × 10-6 M. In this case, the surfactant concentration is too low to generate a barrier toward coalescence, and fast flocculation is expected. As expected for the case of fast flocculating particles interacting through an attractive potential, the flocculation (97) Urbina-Villalba, G.; Garcıá-Sucre, M. Molec. Simul. (98) Smoluchowski, M. Z. Phys. Chem. 1917, 92, 129. (99) Borwankar, R. P.; Lobo, L. A.; Wasan, D. T. Colloids Surf. 1992, 69, 135. (100) Fuchs, N. Z. Phys. 1936, 89, 736. (101) Reering, H.; Overbeek, J. Th. G. Discuss. Faraday Soc. 1954, 18, 74.

7984


Figure 6. Comparison between the prediction of analytical theories and the results of the simulation for 64-particle calculation at a surfactant concentration of 1 × 10-6 M.

rate was found to be considerably higher than 5.49 × 10-18 m3/s (between 1 and 3 orders of magnitude higher depending on the analytical equation). This result is also consistent with the significant increase of the effective diffusion constant found by Ermak102 in Brownian dynamics simulations of hard-sphere polyion particles immersed in a solution of counterions, byions, and a uniform liquid solvent. Average deviations of 1.12, 1.13, 1.48, and 1.78 particles were found for the theoretical formula of Borwankar et al., a simple exponential decay (e-Kt), Smoluchowski n1 formula, and van den Temple’s expression. For the cases of Borwankar and van den Temple, the best results were obtained with the theoretical expressions that were deduced assuming that the number of contacts in an aggregate was equal to its number of drops (m). As discussed by van den Temple, this assumption was expected to hold in concentrated emulsions, while an m - 1 dependence is likely to be found in more diluted systems. Whenever this assumption holds and the coalescence rate is very high, the resulting van den Temple equation reduces to Smoluchowski’s formula for the number of primary particles per unit volume (n1) that have not combined into aggregates (n1 ) n0/(1 + kfn0t)2), n0 being the initial number of particles per unit volume). Thus, the van den Temple and Smoluchowski n1 curves look similar in Figure 6. As shown by Danov et al.,103 Smoluchowski’s n1 expression also holds for the number of “single” particles whether they resulted from coagulation or had not collided at all. The Borwankar et al. series expression was kept up to fourth order, but under the present calculation conditions, no significant error was found when all terms except the first were eliminated. That term is a simple exponential decay (K ) 0.0205 s-1) also shown in Figure 6. As in the case of van den Temple’s formula, the greatest deviations occur at long times, when the simulation predicts the attainment of a steady state. It should be noticed that for 64-particle simulations an average deviation between 1 and 2 particles is fairly high. However, both the van den Temple and Borwankar et al. models can be fitted to a good degree of accuracy for small and intermediate times (t < 60 s). For longer times, a slow decrease in the number of particles is predicted by the analytical formulas, while a constant number of particles is suggested by the (102) Ermak, D. L. J. Chem. Phys. 1975, 62, 4197. (103) Danov, K. D.; Ivanov, I. B.; Gurkov, T. D.; Borwankar, R. J. Colloid Interface Sci. 1994, 167, 8.


simulations in the absence of Ostwald ripening (similar drop size and negligible oil solubility). It is known by experience that the DSD resulting from emulsification usually differs from the “steady-state” dropsize distribution which results after some time (hours, days) of aging. According to the present results, the initial drop size distribution should evolve until it achieves a total interfacial area which can be stabilized with the available surfactant concentration. That critical concentration is less than the concentration required for full close-packing coverage of the available interfaces. Under the present conditions, that concentration is equal to the amount of surfactant required to produce an electrostatic potential barrier of at least 25 kT (φp ) 44.3 mV) between any pair of emulsion drops. In this paper, we have compared our simulation results with well-known analytical formulas to validate the computational algorithm. In this initial publication, the hydrodynamic effects (types I and II) have not been included, and the consequences of a time-dependent surfactant adsorption on the variation of the flocculation rate are not fully understood. The validation of analytical formalisms in terms of the present simulations is likely to require appropriate consideration of these shortcomings. Whenever this is attempted, the findings of at least two other theoretical developments should be considered. (1) The work of Melik, Reddy, and Fogler,63,104,105 who forwarded a convective-diffusion equation for finding the particle loss rates corresponding to Brownian flocculation, creaming, and sedimentation flocculation, along with a population balance equation which gives the particle-size distribution as a function of position and time. In particular, the population balance equation is a nonlinear partial integrodifferential equation which cannot be solved analytically. (2) The paper of Danov et al.,103 who proposed a kinetic model for the simultaneous processes of flocculation and coalescence, paying special attention to the formation and rupture of thin liquid films between aggregated drops. These authors formulated a general set of populationbalance equations, which include possible fluxes of particles out of the system due to sedimentation or creaming. In this case the set of kinetic differential equations has to be solved numerically, except for some particular cases. In our view, BD simulations of emulsion stability provide a simpler approach to the emulsion stability problem without any loss of physical content. The present algorithm can be suitably complicated to treat the cases in which either an inhomogeneous surfactant distribution or a kinetically controlled surfactant adsorption plays a mayor role in shaping the “stable” drop-size distribution of an emulsion, and its evolution with time. 6. Conclusion A new technique for studying the influence of surfactant adsorption on emulsion stability was described. It is based on a modification of a well-known Brownian dynamics algorithm described in ref 61. It avoids explicit simulation of surfactant diffusion, but retains the important physical consequences of the adsorption phenomena upon emulsion stability. These preliminary results confirmed the dramatic influence of the mechanism of surfactant adsorption on emulsion stability. In the present bitumen-in-water (104) Reddy, S. R.; Melik, D. H.; Fogler, H. S. J. Colloid Interface Sci. 1981, 82, 116. (105) Reddy, S. R.; Fogler, H. S. J. Colloid Interface Sci. 1981, 82, 128.


emulsion, such a condition is achieved by the generation of an electrostatic surface potential of at least 44.3 mV. Thus, whenever the surfactant concentration is not high enough to stabilize the interfacial area corresponding to the initial drop-size distribution resulting from emulsification, coalescence will occur. This process will continue until the surfactant concentration of each drop is high enough to generate a considerable barrier toward coalescence. At this point, the coalescence rate shows an abrupt decrease, and a steady-state drop-size distribution is achieved.

Langmuir, Vol. 16, No. 21, 2000 7985

Acknowledgment. This research was partially supported by the National Council of Scientific and Technologic Research (CONICIT), through Grant S1-97001361. Dr. Matthew Oberholtzer’s communications regarding the details of Sader’s formalism are deeply appreciated. Former discussions with Dr. I. Szleifer and Dr. M. Carignano regarding the possible application of Brownian dynamics to the problem of protein adsorption on polymercovered surfaces are also acknowledged. LA000405X

Brownian Dynamics Simulation of Emulsion Stability - ACS Publications

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