Effect of Dynamic Surfactant Adsorption on Emulsion Stability

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Effect of Dynamic Surfactant Adsorption on Emulsion Stability German Urbina-Villalba* Centro de Fı´sica, Laboratorio de Fisicoquı´mica de Coloides, Instituto Venezolano de Investigaciones Cientı´ficas (IVIC), Aptdo. 21827, Caracas, Venezuela Received August 6, 2003. In Final Form: October 31, 2003 The effect of dynamic surfactant adsorption on the stability of concentrated oil in water emulsions is studied. For this purpose, a modification of the standard Brownian dynamics algorithm (Ermak, D.; McCammon, J. A. J. Chem. Phys. 1978, 69, 1352) previously used to study the behavior of bitumen emulsions assuming instantaneous adsorption (Urbina-Villalba, G.; Garcıá-Sucre, M. Langmuir 2000, 16, 7975) was employed. In the present case, dynamic adsorption (DA) was accounted for through a time-dependent electrostatic repulsion between the drops, a function of the surfactant surface excess. The surface excess was allowed to evolve with time according to well-established analytical expressions which depend parametrically on the surfactant diffusion constant (Ds) and the total surfactant concentration (C). The investigation required appropriate incorporation of hydrodynamic interactions in concentrated systems. This was achieved through a novel methodology, which expresses the diffusion constant of each particle as a function of its local concentration and the shortest distance of separation between nearest neighbors. In model systems, the variation of the number of drops as a function of time was followed for different magnitudes of the apparent diffusion constant Dapp of the surfactant. For each of these values, the effect of C and the volume fraction of internal phase (φ) was considered. DA was found to influence emulsion stability appreciably at moderately high φ. In this case, the average collision time between drops is comparable to the time required for the occurrence of a substantial surfactant adsorption, but the interdrop separation is sufficiently large to prevent a considerable slowdown of particle movement due to hydrodynamic interactions.

Introduction The drop size distribution (DSD) resulting from the emulsification of oil in water (O/W) is a function of the properties of the liquids, the concentration of dissolved solutes, and the geometry, material of construction, and size of the mixer.3-4 The influence of some of these factors can be often summarized in empirical relations such as the so-called Kolmogorov relationship:4

[

d32 ) 0.058

]

0.6

γ 2

FVimp Dimp

3

(1 + 5.4φ)Dimp

(1)

where d32 is Sauter mean diameter, γ is the interfacial tension, F is the fluid density of the continuous phase, Vimp is the impeller speed, and Dimp is the impeller diameter. While the effect of mechanical factors can be empirically quantified, chemical effects (like those resulting from surfactant adsorption) are more difficult to account for. During emulsification, the DSD results from a competition between the breakage of the liquid mixture and the coalescence of drops. In agitated mixing, these two processes depend on the impeller speed and the volume fraction of oil.3 Surfactant adsorption influences the breakage of the liquid in an average way, through its effect on the interfacial tension (see eq 1). Coalescence, on the other hand, drastically depends on the repulsive barrier created as a consequence of surfactant adsorption. Furthermore, it also depends on the drainage of the interven* E-mail: [email protected]. (1) Ermak, D.; McCammon, J. A. J. Chem. Phys. 1978, 69, 1352. (2) Urbina-Villalba, G.; Garcıá-Sucre, M. Langmuir 2000, 16, 7975. (3) Verhoff, F. H.; Ross, S. L.; Curl, R. L. Ind. Eng. Chem. Fundam. 1977, 16, 371. (4) Mlynek, Y.; Resnick, R. AIChE J. 1972, 18, 122.

ing liquid between colliding drops, a process that is strikingly affected by surfactant behavior at the O/W interface.5 The importance of dynamic surfactant adsorption in colloid systems is well-recognized.6-9 Surfactant adsorption is a time-dependent process, markedly dependent on the mechanism of dynamic adsorption. The extent of this process is generally quantified through the variation of the interfacial tension. However, the change of the tension as a function of time is in most cases substantially slower than the one expected from diffusion-controlled adsorption. A 0.1 g/L solution of sodium dodecyl sulfate (SDS), for example, causes a drop of 15 mN/m in the interfacial tension of a dodecane/water system in period of 100 s. However, a much more dilute solution of SDS (C ) 0.014 g/L) achieves the same drop in a period of only 6 s in the presence of 0.1 M NaCl. Such behavior is typical of ionic surfactants. They are known to produce an electrostatic barrier, which slows down further adsorption. During this stage, the surfactant accumulates at the subsurface until its local concentration is large enough to overcome the potential barrier. Thereafter, adsorption accelerates until equilibrium is reached. In the presence of salt, the barrier is considerably lowered and the adsorption is basically diffusion-controlled.6 The consequences of dynamic adsorption (DA) on emulsion stability are very difficult to study both theoretically and experimentally. To advance in the comprehension of the problem, we focus here on a simple scenario (5) Verwey, E. J. W.; Overbeek, J. Th. G. The Theory of Liophobic Colloids; Dover: New York, 1999. (6) Bonfillon, A.; Sicoli, F.; Langevin, D. J. Colloid Interface Sci. 1994, 168, 497. (7) Urbina-Villalba, G.; Garcıá-Sucre, M. Colloids Surf., A 2001, 190, 111. (8) Urbina-Villalba, G.; Garcıá-Sucre, M. Mol. Simul. 2001, 27, 75. (9) Urbina-Villalba, G.; Garcıá-Sucre, M. Interciencia 2000, 25, 415.

10.1021/la030327o CCC: $27.50 © 2004 American Chemical Society Published on Web 04/03/2004

Effect of Surfactant Adsorption on Emulsions

Langmuir, Vol. 20, No. 10, 2004 3873

in which a given distribution of drops (previously produced by the emulsification of oil in water) is suddenly put into contact with a surfactant solution. This procedure is usually implemented to freeze a given DSD for ulterior observation. Furthermore, it was used in the past to study bivariate distributions of drop volume and a tracer dye in a continuous-flow agitated dispersion.3 The question we wish to address in this paper is, under what experimental conditions (of φ, C, and Ds) is dynamic adsorption relevant for the evolution of a DSD? Emulsion Stability Simulations. As shown in previous communications,2,7-10 Brownian dynamics (BD) simulations can be particularly useful to simulate the behavior of colloidal systems. According to this algorithm, the movement of a suspended particle results from its explicit interactions with its neighbors and an average interaction with the solvent molecules. The mean effect of the solvent molecules is determined from the statistical sum (coarse graining) of its degrees of freedom. The result is a Langevin type equation of motion in which the suspending medium is incorporated through the diffusion tensor of the suspended particles and a set of random displacements. One of the most widely used algorithms of BD was formulated by Ermak and McCammon1 based on a Fokker-Planck description of the evolution of a system of dispersed particles in the phase space. According to this formalism, the equation of motion of a Brownian particle k is equal to

rk )

r0k

+

∑l

∂D0kl ∆t + ∂rl

∑l

D0kl F0l ∆t + RG(∆t) kT

(2)

where rk is the position of the particle, and Fl is the total force acting on particle k in direction l. Superscript 0 indicates that the value of the variable is evaluated at the beginning of the time step (∆t), RG(∆t) is a random displacement with a Gaussian distribution and variance 2Dkl∆t, and subscripts k and l span all directions (x,y,z) and particle labels. The complexity of eq 2 is due to the coupling of the diffusion tensor of the set of particles and their random displacements. In the case of spherical drops and in the absence of hydrodynamic interactions (HI), the diffusion tensor matrix takes a diagonal form. Its elements become independent of particle coordinates (diffusion constant), and more important, they get decoupled from the random displacements, yielding a much simpler equation of motion:

ri ) r0i +

DiF0 ∆t + R′G(∆t) kT

(3)

According to eq 3, the movement of a particle results from a diffusive term which depends on the interparticle potential and a random “kick” consequence of millions of collisions of the solvent molecules with the suspended particle surface. The prime on the last term indicates that the random contribution of particle i is now calculated with its diffusion constant Di alone. The reliability of this equation depends on the quality of the diffusion constant used. In emulsion stability simulations (ESS), the initial coordinates of a set of particles of radius Ri are generated at random. The particles are included in a threedimensional cell with periodic boundary conditions. Their (10) Urbina-Villalba, G.; Garcıá-Sucre, M.; Toro Mendoza, J. Mol. Simul. 2003, 29, 393.

movement is determined by the characteristics of the interparticle potential through the force F (eq 3). The particles are allowed to flocculate and coalesce depending on this force. Coalescence occurs whenever the distance between the centers of each particle becomes smaller than the sum of their radii. In this case, a new particle is created at the center of mass of the colliding particles, preserving the initial volume of internal phase. For the case of oil drops suspended in water (O/W emulsion) and interacting with a Derjaguin-LandauVerwey-Overbeek (DLVO) potential, force F contains two terms: a van der Waals attraction essentially resulting from the interaction between their oil molecules and an electrostatic contribution, which depends on the amount and charge of the surfactant adsorbed. The effective charge of the surfactant is an input of the BD simulation and can be calculated from the electrophoretic movement of a drop completely covered by surfactant molecules. The typical interfacial area of the surfactant at maximum adsorption (usually of the order of 50 Å2)8 allows one to apportion the total charge of a drop between the adsorbed surfactant molecules. In ESS, the amount of surfactant adsorbed at the interface of each drop depends on the surfactant distribution strategy. These schemes are a set of routines purposely designed to mimic the experimental behavior of surfactants, without explicit account of surfactant movement. In the case of dynamic adsorption, for instance, the amount of surfactant adsorbed can be calculated from the ratio of the total interfacial area of a drop and the surfactant interfacial area. The surfactant area is assumed to change with time in a way compatible to the experimental behavior of the surfactant surface excess. Thus, the quotient of the total area of the drop and the surfactant interfacial area increases with time. In this report, the findings of Rosen11 along with the equation of Ward and Tordai12 are used to relate an apparent surfactant diffusion constant to the experimental surface excess. As a result, the amount of surfactant adsorbed upon each drop (and consequently, the charge of each drop) at time t changes as a function of the surfactant diffusion constant, the total surfactant concentration, and the interfacial area of the emulsion at that time. The details of this procedure are specified in the following sections. Account of Dynamic Adsorption in Emulsion Stability Simulations. To analyze the effects of dynamics of adsorption on emulsion stability, we chose a system that we had previously studied assuming instantaneous surfactant adsorption: a bitumen/water emulsion stabilized by a cationic surfactant.2,7-10 In the present report, a modification2 of the BD algorithm1 is used to simulate the evolution of a small number of bitumen drops interacting through DLVO forces.5 The attractive potential between drops can be estimated employing the Hamaker formula for particles of unequal size:13

V)-

[

y y A + + 12 x2 + xy + x x2 + xy + x + y x2 + xy + x 2 ln 2 x + xy + x + y

(

)]

(4)

where A is the Hamaker constant (equal to 1.24 × 10-19 (11) Rosen, M. J.; Hua, X. Y. J. Colloid Interface Sci. 1990, 139, 397. (12) Ward, A. F. H.; Tordai, L. J. Chem. Phys. 1946, 14, 453. (13) Hamaker, H. C. Physica (Amsterdam), IV 1937, 1058.

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Urbina-Villalba

J for the bitumen emulsion under consideration14), x ) d/2R1, y ) R2/R1, Rk is the radius of particle k, and d is the shortest distance between the particles’ surfaces. In the case where ionic surfactants are used to stabilize dispersions of nonpolar oils in water, the surface charge density (σ) of the drops is essentially the result of surfactant adsorption. In this case,

σ)

Zi

(5)

4πRi2

Here Zi and Ri are the total charge and radius of drop i. The total charge Zi is equal to

Zi ) AizsΓ(t)

(6)

where Ai is the total area of drop i, zs is the effective charge of a surfactant molecule at the O/W interface, and Γ(t) is the surfactant surface excess at time t. The value of zs has to be deduced from experimental measurements. It can be calculated from ξ-potential measurements,2,7-10 and the equilibrium value of the excess, Γeq, evaluated from the slope of a γ versus log(C) plot. In most cases, 1/Γeq falls in the range 20-100 Å2. In the present work, it was taken as 50 Å2,15 an average value for alkylammonium surfactants in 0.1 M NaCl solutions. The average radius of the drops and their ξ-potential correspond to the experimental data of Salou et al. for an “E3” emulsion.14 This cationicstabilized emulsion was prepared from hot bitumen (150 °C) and an acidified water solution, stirring the mixture with an ultra-Turrax turbine for 5 min. The resulting emulsion has an average drop radius of 3.9 µm, an electrostatic potential of +115 mV at pH 2.9, and a very high ionic strength, κa ) 1536 (where κ stands for the inverse of the Debye length, and a is the average particle radius). Once the value of Γ(t) is estimated, the repulsive potential between drops can be calculated from Zi, using the Poisson-Boltzmann equation.16 In past simulations, we have found it convenient to use the approximate formulation of Sader.17 This is a solution of the PoissonBoltzmann equation which is valid for most values of κa, keeping a good degree of accuracy for all surface potentials of practical interest ( 0.10), the effect of HI is large. Consequently, a reliable simulation of emulsion stability with surfactant adsorption effects also requires the suitable incorporation of HI between drops. The general form of the diffusion matrix for a system of suspended particles is27-28

Dii ) D0Iˆ + D0

∑

{As(rij)rˆ ijrˆ ij + Bs(rij)[Iˆ - rˆ ijrˆ ij]}

j)1,j*1

(16)

Dij ) D0{Ac(rij)rˆ ijrˆ ij + Bc(rij)[Iˆ - rˆ ijrˆ ij]}

(17)

Subscripts i and j in eqs 16-17 are particle labels, rˆ ij ) rij/rij, rij ) ri - rj, and D0 is the diffusion (Stokes) coefficient of a solid particle at infinite dilution:

D0 )

kT 6πηRi

(18)

where η is the solvent viscosity. The scalar functions As, Bs, Ac, and Bc in eqs 16-17 are mobility functions, and their usual analytical expressions are calculated supposing pairwise additive interactions.27,29-31 As a consequence, these expressions commonly disregard the screening of HI that occurs in concentrated dispersions or flocs. This was first pointed out by Bacon et al.32 and recently discussed by Heyes.33 According to these authors,32 pairwise additivity can even (25) Velegol, D.; Feick, J. D.; Collins, L. R. J. Colloid Interface Sci. 2000, 230, 114. (26) Velegol, D.; Catana, S.; Anderson, J. L.; Garoff, S. Phys. Rev. Lett. 1999, 83, 1243. (27) Dhont, J. K. G. An Introduction to Dynamics of Colloids; Elsevier Science B.V.: Amsterdam, 1996; Chapters 3 and 5. (28) van de Ven, T. G. M. Colloidal Hydrodynamics; Academic Press Ltd.:, Padstow, 1989; Chapters 1-2. (29) Rotne, J.; Prager, S. J. Chem. Phys. 1969, 50, 4831. (30) Batchelor, G. K. J. Fluid Mech. 1982, 119, 379. (31) Batchelor, G. K. J. Fluid Mech. 1976, 74, 1. (32) Bacon, J.; Dickinson, E.; Parker, R. Faraday Discuss. Chem. Soc. 1983, 76, 165. (33) Heyes, D. M. Mol. Phys. 1996, 87, 287.

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three-body contributions (+1.80φ2). For φ < 0.30, the second-order (two-body) expansion of the diffusion constant appears to be more reliable. At higher volume fractions, the trend is adequate but sizable differences are observed.38 From the form of eq 19, it is clear that whenever φ ∼ 1.0, the diffusion constant tends to a minimum value of ∼0.2D0. However, Honig et al.35 showed that the diffusion constant could be reduced to less than 0.005D0 during the course of a binary collision. Such strong effects only occur at very close separations, within a distance of one particle radius. Thus, the value of the diffusion constant changes if there is at least one particle within the inner region of a given central particle (d e Rint). In this case, the diffusion constant is calculated according to Honig et al.’s expression:

D(u) )

D0 β(u)

(20)

where d/Ri ) u, and

β(u) ) Figure 1. To calculate the effect of HI on the movement of a given particle C, internal and external radii are defined. Particles beyond Rext do not influence the movement of particle C. Particles between Rint and Rext contribute to the calculation of a local volume fraction φ. This allows the evaluation of D(φ,d) from eq 19 unless there is at least one particle within the internal radii (rij e Rint) previously defined. In this case, eqs 20 and 21 are used to calculate D(φ,d) employing the distance of closest approach to calculate β (eqs 20 and 21).

generate negative diffusion constants above a critical volume fraction of internal phase. It was also shown that this overestimation of HI favors multiple collisions in BD calculations, abnormally increasing the value of the flocculation rate.34 To circumvent this problem, we had recently proposed a convenient methodology for the calculation of an effective diffusion constant in concentrated systems.34 It is based on the geometrical evaluation of the local volume fraction around each particle and the use of an analytical expression for the diffusion constant provided by Honig et al.35 for the case of binary collision. The methodology for the calculation of the local volume fraction is illustrated in Figure 1. First, internal (Rint) and external (Rext) radii are defined around each particle. These radii divide the space in three sections. Particles in the outermost region d > Rext (d ) rij - Ri - Rj) do not contribute to the HI of a given central particle. Particles that fall in the intermediate region (Rint < d e Rext) contribute with the fraction of their volume inside that region to the calculation of a local volume fraction around the central particle. Once the local volume fraction φ is calculated, the diffusion constant of the central particle is estimated as

D(φ) ) D0(1 - 1.73φ - 0.93φ2 + 1.80φ2 + ...)

(19)

Equation 19 was formerly deduced from the average of mobility functions over two and three particle distribution functions36-37 and was later confirmed by experiment.38-40 The φ2 term in eq 19 contains two distinct contributions due to two-body hydrodynamic interactions (-0.93φ2) and (34) Urbina-Villalba, G.; Garcıá-Sucre, M.; Toro-Mendoza, J. Phys. Rev. E 2003, 68, 061408. (35) Honig, E. P.; Roebersen, G. J.; Wiersema, P. H. J. Colloid Interface Sci. 1971, 36, 97.

6u2 + 13u + 2 6u2 + 4u

(21)

Tensorial expressions as well as average diffusion constants successfully reproduce the behavior of experimental data in dilute systems.35,41-45 As shown in ref 34, the described methodology reproduces the behavior of the exact diffusion tensor30,31 whenever φ e 0.1. At higher concentrations, the model overcomes the overestimation of HI producing reasonable flocculation rates at all volume fractions studied (10-6 e φ e 0.40). To our knowledge, the experimental evaluation of flocculation rates in highly concentrated systems is still not possible. Using the approximations of the previous section, eq 2 simplifies considerably. The resulting equation of motion has an analytical form similar to the one used for BD in the absence of HI:

ri(t + ∆t) ) ri(t) +

Di(φ,d)F∆t + RG(Di(φ,d)) kT

(22)

Here ri(t) is the position of particle i at time t, F is the sum of interparticle and external forces acting on i, D(φ,d) is the average diffusion constant of the particle (which depends on the local volume fraction of particles and the minimum distance of approach between particle i and its surrounding neighbors), and RG(Di(φ,d)) is a random Gaussian function with zero mean and variance 6Di(φ,d)∆t. Computational Details. As pointed out in the Introduction, the present calculations simulate the process in which oil and water are first stirred in a mixer without surfactants, and the resulting DSD is subsequently “put into contact” with a surfactant solution. This final stage can be achieved either by (a) pouring the initial O/W (36) Beenakker, C. W. J.; Mazur, P. Phys. Lett. A 1982, 91, 290. (37) Beenakker, C. W. J.; Mazur, P. Physica A 1984, 126, 349. (38) van Veluwen, A.; Lekkerkerker, H. N. W.; de Kruif, C. G.; Vrij, A. Faraday Discuss. Chem. Soc. 1987, 83, 59. (39) van Megen, W.; Underwood, S. M.; Ottewill, R. H.; Williams, N. St. J.; Pusey, P. N. Faraday Discuss. Chem. Soc. 1987, 83, 47. (40) Pusey, P. N.; van Megen, W. J. Phys. 1983, 44, 285. (41) Meiners, J.; Quake, S. R. Phys. Rev. Lett. 1999, 82, 2211. (42) Kollmann, M.; Na¨gele, G. Europhys. Lett. 2000, 52, 474. (43) Crocker, J. C. J. Chem. Phys. 1997, 106, 2837. (44) Bartlett, P.; Henderson, S. I.; Mitchell, S. J. Philos. Trans. R. Soc. London, Ser. A 2001, 359, 883. (45) 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.


emulsion into a surfactant solution or (b) opening a connection between the mixing tank and a concentrated surfactant solution (in a way that prevents convection). To simulate the last situation, four cubic cells of length L ) 17.36a, 13.78a, 12.10a, and 10.94a (where a ) 3.9 µm) were built. For 125 particles, these correspond to φ ) 0.10, 0.20, 0.30, and 0.40. Random configurations of particles were used for φ e 0.30, while a cubic arrangement was necessary for φ ) 0.40. At the start of the simulations, the drops do not contain any surfactants attached. Their diffusion constant is calculated from the hydrodynamic model previously described (see ref 34). The drops start moving according to eq 22, due to the van der Waals attraction (eq 4) and the action of the random force. When the first iteration finishes, t ) ∆t (1.36 × 10-6 s e ∆t e 3.40 × 10-5 s). From this time on, Γ(t) is calculated according to eq 14 given the total surfactant concentration and the apparent diffusion constant of the surfactant. Dapp and C are introduced as input. In these simulations, 10-12 m2/s e Dapp e 10-9 m2/s and 1 × 10-4 M e C e 5 × 10-4 M. The ionic strength was equal to 0.014 M. Having Γ(t), the electrostatic potential was calculated at each time step from eqs 5-11. In particular, eq 11 was solved numerically at every time step using the bisection method. It was known from previous works2,7-10,34 that a time step of 1.36 × 10-6 s was short enough to sample the present electrostatic potential appropriately. This was found empirically in trial calculations where neither HI nor random noise was introduced. Two bitumen drops completely covered by surfactant show an electrostatic potential barrier higher than 1000 kT.14,46 Thus, an adequate time step should preserve the number of particles. However, the selected time step is short for the BD simulation of dilute systems (φ < 0.20). To obtain reasonable computing times, we implemented an algorithm with two time steps. A variable time step was previously suggested in ref 47 in order to sample a shortrange potential appropriately. In the present version, the range of the interaction potential is used as input. The high ionic strength of the simulated emulsion (0.014 M) required a short width of 50 nm. Accordingly, the short time step was set equal to 1.36 × 10-6 s ()∆ts). The long time step could be set arbitrarily high, but in the present simulations it did not exceed 3.40 × 10-5 s (g∆tL). Once the long (∆tL) and short (∆ts) time steps are selected, a double time step calculation proceeds as follows: At the beginning of the simulation, all particles move at ∆tL. The minimum separation between the particles is calculated at every iteration. If this distance is smaller than twice the preselected potential width, all particles are returned to their previous positions, and the shorter time step is used instead. Following, the particles move at this lower time step for ∆tL/∆ts iterations. When this inner cycle finishes, the particles have moved for a space of ∆tL s, going back in phase with the longer time step formerly used. The coalescence of droplets can only occur in the inner cycle, where the interacting potential is properly sampled. The calculation proceeds in this way, entering the inner cycle from time to time whenever required. At the beginning of the calculations, the DSD is monodisperse. In each cell, the initial number of drops and their radii are the same independently of the volume fraction of internal phase. Thus, in these simulations the interfacial area is the same when the volume fraction (46) Rodrı´guez-Valverde, M. A.; Cabrerizo-Vı´lchez, M. A.; PaézDuenãs, A.; Hidalgo-Alvarez, R. Colloids Surf., A 2003, 222, 233.

Langmuir, Vol. 20, No. 10, 2004 3877

increases from φ ) 0.10 to φ ) 0.40. The increase of the volume fraction is attained decreasing the dimension of the cubic cell from L ) 17.36a to 10.94a. This situation differs from previous calculations in which the interfacial area was varied at fixed cell length, changing the DSD and/or the number of particles of the simulation.2,7-10 Each calculation resembles the case in which oil and water phases are emulsified in the absence of surfactants, and the resulting emulsion is then poured into a surfactant solution. Since the interfacial area was chosen equal in all simulation cells regardless of the volume fraction of oil, the specification of the surfactant concentration and its variation with the volume fraction of oil deserve some special consideration. In typical experiments, an ionic surfactant is generally dissolved in the water phase prior to emulsification. Since the content of the water phase increases as the volume fraction of oil decreases, use of the same surfactant solution to produce emulsions of different φ will generate dispersions with different amounts of surfactant molecules. Thus, if the total number of surfactant molecules is distributed homogeneously between all drops and instantaneous adsorption occurs, the highest surface potential will be observed for the case in which φ ) 0.10, and the lowest one for φ ) 0.40. This will happen whenever the number of surfactant molecules is not enough to attain maximum coverage for all the values of φ considered. On the other hand, a set of surfactant solutions of different surfactant concentrations could also be prepared in order to produce emulsions of equal surfactant concentrations at different φ. In this latter case, the surface potential of the drops will be the same independently of the volume fraction of internal phase. If computational conditions are chosen in a way that corresponds to the former situation above, the number of surfactant molecules will decrease in the cells as the volume fraction of oil increases. In the latter case instead, the number of surfactant molecules will be equal in all cells regardless of the oil content. To simplify the calculations and keep track of the surfactant concentration in a convenient way, we have adopted this last convention in all our previous works. In practical terms, the number of surfactant molecules available is calculated at the beginning of the simulation multiplying the nominal surfactant concentration by the total volume of the simulation cell. To allow the comparison of the present results with former simulations in which surfactant adsorption is assumed to occur instantaneously, the same method of evaluation of the number of surfactant molecules was adopted here. However, in the present case the size of the cell is changed. As a result, the number of surfactant molecules does increase as φ decreases from 0.40 to 0.10. These computational conditions produce the same effect as that of the former case described above: as the volume fraction of oil increases, the number of surfactant molecules available decreases. The drops in the simulation box were allowed to coalesce with other real drops or its periodic boundary images. In either case, a new particle was created at the center of mass of the colliding drops. Its corresponding area was computed from the conservation of mass. The diffusion constant at infinite dilution was calculated from the Stokes formula (eq 18). In the present work, we are solely concerned with the variation of the initial DSD as a function of time. In this regard, the possible outcomes of the simulations are as follows: (a) The number of particles is preserved. In this case, the conditions imposed by the values of Dapp and C are

3878


Urbina-Villalba

such that surfactant adsorption is fast enough to preserve the initial number of drops. (b) The number of particles diminishes and then stabilizes. This means that some drops may collide before the surfactants cover the available interfaces. However, as time passes, surfactant adsorption increases until a point in which it is high enough to build up a considerable barrier that prevents coalescence. (c) The number of particles keeps decreasing during the length of the simulation. In this case, the surfactant concentration might be too low to generate a sizable repulsive barrier between drops, or the surfactants might be too slow to reach the available interfaces before the drops collide. To quantify the effect of DA on emulsion stability, the Smoluchowski expression48 was employed:

n)

n0 1 + kfn0t

(23)

Here n is the number of particles per unit volume, n0 is the initial particle density, t is the time, and kf is a flocculation constant. In the absence of interaction forces,

kf-1 ) n0tf )

1 3η ) 8πaD0 4kT

(24)

The value of the flocculation constant increases with the volume fraction and attractive interactions. In the absence of interaction forces and taking water as the dispersing medium, kf is equal to 5.49 × 10-18 m3/s at T ) 298 K. Whenever interaction forces are considered, the value of kf changes but can still be calculated as the slope of a 1/n versus t plot, whenever the simulation data follow eq 23. Results and Discussion As shown in refs 14 and 46, the drops of bitumen emulsions usually present a very high surface potential (∼115 mV), caused by the amount of surfactant adsorbed and the activation of surface-active groups of natural origin. Hence, a surfactant concentration of C ) 10-4 M produces a repulsive barrier of V/kT ) 2330 between every pair of drops at φ ) 0.10, if instantaneous surfactant adsorption occurs. Figure 2 shows the total potential of interaction (Vt) between two drops along with its electrostatic component (Ve), for the case in which φ ) 0.10, C ) 1 × 10-4 M, and Dapp ) 1 × 10-10 m2/s. At the beginning of the simulations, each cell has a total interfacial area of 1571a2 and 4.03 × 10+14 m-3 particles per unit volume. The surfactant distributes among these particles, increasing their electrostatic repulsion as time evolves. This trend is illustrated in Figure 2 for t ) 0.034 s, t ) 0.068 s, and t ) 0.120 s. It can be observed that the electrostatic component and the total interaction potential between two drops increase with time. At φ ) 0.10, the referred surfactant concentration produces a barrier of V/kT ) 2330 after a sufficiently long time. This time is equal to 0.87, 8.7, and 867 s for Dapp ) 10-9, 10-10, and 10-12 m2/s. Thus, the fastest surfactant adsorbs first, conferring stability to the receptor drop. A quantitative description of the results is given in Table 1. The interfacial area and the number of drops per unit volume at the end of the simulation (nf) are given in the (47) Romero-Cano, M.; Puertas, A. M.; de las Nieves, F. J. Colloidal aggregation under steric interactions: Simulation and experiments. J. Chem. Phys. 2000, 112, 8654. (48) von Smoluchowski, M. Z. Phys. Chem. 1917, 92, 129.

Figure 2. Time dependence of the electrostatic (Ve) and total (Vt) interaction potentials as a function of time for C ) 1 × 10-4 M. The time in seconds is shown in parentheses. As time evolves, the repulsion increases up to a value limited by the surfactant concentration in the system and the total volume fraction of internal phase.

sixth and eighth columns of the table. These values are listed as a function of φ, Dapp, and C. As previously discussed, the combined effect of Dapp and C is neatly summarized in eq 14. The surface excess changes linearly with t1/2 with a proportionality coefficient equal to 2[Dapp/ π]1/2C. The values of this coefficient are given in the fourth column of Table 1. Higher values indicate a quicker increase of the number of surfactants per unit area, Γ(t), as a function of time. The time required for maximum adsorption (tc) is listed in column five. This value is inversely proportional to the coefficient 2[Dapp/π]1/2C. For the case in which φ ) 0.10, C ) 1 × 10-4 M, and Dapp ) 1 × 10-10 m2/s (14th row of Table 1), the coefficient 2[Dapp/ π]1/2C is reasonably high (6.79 × 10-3 Å2 s1/2), and the time required for maximum adsorption is tc ) 8.67 s. Consequently, the number of particles per unit volume suffers a slight decrease during the course of the simulation, going from 4.03 × 10+14 to 3.96 × 10+14 m-3. Under the referred chemical conditions (of C and Dapp), a small number of drops coalesced, decreasing the total interfacial area of the emulsion. The final value of the interfacial area (1560a2) corresponds to the sum of areas of the surviving drops. As shown in Figure 3, a surfactant concentration of C ) 1 × 10-4 M is not sufficient to generate a potential barrier between the initial number of drops at φ ) 0.20. As explained in the previous section, the number of surfactant molecules in the case of φ ) 0.20 is lower than that of φ ) 0.10. The constancy of the initial interfacial area in the simulations and the smaller size of the cell at φ ) 0.20 cause this decrease. Notice that a similar situation occurs in experiment if the same number of molecules is used to stabilize a monodisperse dispersion of oil droplets with φ ) 0.20 and φ ) 0.10. The electrostatic repulsion will be considerably lower in the former case due to its smaller number of surfactant molecules. At φ ) 0.20 and C ) 1 × 10-4 M, the total potential between any pair of drops is attractive even if instantaneous surfactant adsorption occurs. The total potential between two drops is similar to the one shown in Figure 3 for t ) 0.12 s. As a result, some drops will coalesce as time evolves and the initial DSD will consequently change. As soon as the drops coalesce, the total interfacial area decreases. The exceeding surfactant passes to the bulk and is available for the remaining drops, according to eq 14. Figure 3 also shows the potential generated between two drops of equal sizes when the number of drops


Langmuir, Vol. 20, No. 10, 2004 3879

Table 1. Summary of the Results from the Simulationsa φ × 100 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40

Dapp (m2/s) 10-9

1× 1 × 10-9 1 × 10-9 1 × 10-9 1 × 10-9 1 × 10-9 1 × 10-9 1 × 10-9 1 × 10-10 1 × 10-10 1 × 10-10 1 × 10-10 1 × 10-10 1 × 10-10 1 × 10-10 1 × 10-10 1 × 10-12 1 × 10-12 1 × 10-12 1 × 10-12 1 × 10-12 1 × 10-12 1 × 10-12 1 × 10-12

C (M) 10-4

5× 5 × 10-4 5 × 10-4 5 × 10-4 1 × 10-4 1 × 10-4 1 × 10-4 1 × 10-4 5 × 10-4 5 × 10-4 5 × 10-4 5 × 10-4 1 × 10-4 1 × 10-4 1 × 10-4 1 × 10-4 5 × 10-4 5 × 10-4 5 × 10-4 5 × 10-4 1 × 10-4 1 × 10-4 1 × 10-4 1 × 10-4 0 0 0 0

2[Dapp/π]1/2C 10-1

1.07 × 1.07 × 10-1 1.07 × 10-1 1.07 × 10-1 2.15 × 10-2 2.15 × 10-2 2.15 × 10-2 2.15 × 10-2 3.40 × 10-2 3.40 × 10-2 3.40 × 10-2 3.40 × 10-2 6.79 × 10-3 6.79 × 10-3 6.79 × 10-3 6.79 × 10-3 3.40 × 10-3 3.40 × 10-3 3.40 × 10-3 3.40 × 10-3 6.79 × 10-4 6.79 × 10-4 6.79 × 10-4 6.79 × 10-4

final interfacial area (a2)

tc 0.03 0.03 0.03 0.03 0.87 0.87 0.87 0.87 0.35 0.35 0.35 0.35 8.67 8.67 8.67 8.67 34.7 34.7 34.7 34.7 867 867 867 867 ∞ ∞ ∞ ∞

n0 (m-3)

nf (m-3)

10+14

4.03 × 8.05 × 10+14 1.19 × 10+15 1.61 × 10+15 4.03 × 10+14 8.05 × 10+14 1.19 × 10+15 1.61 × 10+15 4.03 × 10+14 8.05 × 10+14 1.19 × 10+15 1.61 × 10+15 4.03 × 10+14 8.05 × 10+14 1.19 × 10+15 1.61 × 10+15 4.03 × 10+14 8.05 × 10+14 1.19 × 10+15 1.61 × 10+15 4.03 × 10+14 8.05 × 10+14 1.19 × 10+15 1.61 × 10+15 4.03 × 10+14 8.05 × 10+14 1.19 × 10+15 1.61 × 10+15

1571 1571 1571 1571 1571 1477 986 671 1571 1571 1571 1571 1560 1466 971 732 1523 1250 892 669 1229 813 385 365 392 359 314 314

4.03 × 10+14 8.05 × 10+14 1.19 × 10+15 1.61 × 10+15 4.03 × 10+14 7.02 × 10+14 4.28 × 10+14 3.35 × 10+14 4.03 × 10+14 8.05 × 10+14 1.19 × 10+15 1.61 × 10+15 3.96 × 10+14 6.96 × 10+14 4.57 × 10+14 3.61 × 10+14 3.74 × 10+14 4.77 × 10+14 3.33 × 10+14 1.55 × 10+14 2.13 × 10+14 1.55 × 10+14 3.81 × 10+13 2.58 × 10+13 6.44 × 10+12 1.93 × 10+13 9.52 × 10+12 1.29 × 10+13

a The table shows the final number of particles per unit volume (n ) and the final interfacial area as a function of the volume fraction f (φ), the surfactant concentration (C), and the apparent diffusion constant of the surfactant (Dapp). The value of 2[Dapp/π]1/2C in molecules per (Å2 s1/2) and the critical time required for maximum adsorption (tc) are also shown (see eq 14).

Table 2. Average Collision Timesa t (s)/φm × 100 cubic arrangement Hu¨tter (φm ) 0.51) tf ) 1/kfn0

10

20

30

40

50

589.3 141.3 45.2

155.4 36.4 7.6

45.2 10.2 1.0

9.6 1.9 0.14

0.3 0.01 0.0012

a The values of the first two rows were calculated from t ) l2/D , 0 where l is the average distance between particles calculated from a cubic configuration (first row) or a random one (second row); see the text. Mean flocculation times (third row) were deduced from the simulation data obtained for C ) 0.

Figure 3. Variation of the total interaction potential between two drops as a function of time (φ ) 0.20). At t ) 0.12 s, the number of particles is still 125. At t ) 0.36, it had already decreased to 108 particles. Since the interfacial area diminished, the initial surfactant concentration (1 × 10-4 M) can now generate a repulsive barrier between the remaining drops (1.4 kT at t ) 0.36 s). As time evolves, the surface excess can now augment, and the repulsive barrier increases to 26.7 kT at t ) 10.15 s.

decreases from 125 to 108 (at t ) 0.36 s). Notice that once the interfacial area decreases, the same surfactant concentration (C ) 1 × 10-4 M) is now enough to generate a potential barrier of 1.4 kT at t ) 0.36 s, between two drops of equal radius a ) 3.9 µm. This barrier increases as a consequence of further surfactant adsorption, reaching a maximum of 26.7 kT after t ) 10.2 s. Thus, a surfactant concentration of C ) 1 × 10-4 M in a φ ) 0.20 emulsion (Figure 3) is not able to stabilize the initial number of particles per unit volume (n ) 8.05 × 1014) causing its decrease to n ) 6.96 × 1014 (Table 1). This last number did not present further changes during the rest of the simulation.

Table 2 shows order-of-magnitude estimations of the average time for the first collision (t1) between 3.9-µm drops in the present model systems. They were calculated as t ) l2/D0 where l is the average distance between the drops and D0 ) 5.6 × 10-14 m2/s (eq 18). In these evaluations, it is supposed that the particles only move as a result of the thermal interactions with the solvent. The values of the first row (labeled “cubic arrangement”) were calculated using the minimum separation distance between two drops in simple cubic configurations, employing the same dimensions and number of particles of the present simulations. The second row shows similar calculations in which the surface-to-surface separation between two nearest neighbors in a randomly distributed configuration was calculated according to eq 25:49

[x 3

l ) Ri

φm -1 φ

]

(25)

Here φm is the maximum volume fraction, and Ri is the radius of the particle. This equation was used previously to relate the dependence of the coagulation time of (49) Hu¨tter, M. Phys. Chem. Chem. Phys. 1999, 1, 4429.

3880


Figure 4. Time dependence of the total interaction potential (Vt) for C ) 5.0 × 10-4 M. D10 and D12 stand for a surfactant diffusion constant of Dapp ) 10-10 m2/s and Dapp ) 10-12 m2/s, respectively. The faster surfactant can build a repulsive barrier in a short time (t ) 0.027 s), while the slower one presents an effective attractive potential during the same period of time. As time evolves, the slower surfactant (Dapp ) 1 × 10-12 m2/s) manages to build up a very strong repulsive barrier. The barriers tend to the same limit at very long times (∼27.28 s).

suspensions of silica particles to their volume fraction of internal phase. In either case, the estimations are considerably longer (within 1 order of magnitude) than the average flocculation time (deduced from the fitting of eq 23 to the present data and calculated for the case in which only van der Waals attraction was present). Whichever the method of evaluation chosen, the collision time between drops comes out to be appreciably longer than the time it takes for a surfactant to travel a similar distance. So it would appear in principle that surfactant molecules should always arrive first to the O/W interfaces, with plenty of time to build up a repulsive barrier against drop collisions. However, a number of factors attempt against the validity of this statement: (1) The estimations of rows 2 and 3 in Table 2 disregard the effect of van der Waals attraction in the calculation of the collision time between two drops (term DF∆t/kT in eq 22). (2) The amount of surfactant adsorbed should be the only one that effectively contributes to the surface potential of a drop. This quantity changes with time according to the characteristics of the adsorption isotherm, and it is not necessarily diffusion-controlled. (3) As shown by eq 14, a small surfactant concentration increases the adsorption time, despite the value of Dapp. (4) The distance between some drops in a spatially nonhomogeneous dispersion, such as the one produced by vigorous stirring, might be appreciably lower than the one geometrically estimated. (5) As time evolves, the particles aggregate, shortening their average distance. Figure 4 illustrates the effect of the apparent diffusion constant of the surfactant for C ) 5 × 10-4 M. According to Hua and Rosen,22,23 this is the minimum surfactant concentration for which the surface tension attains a value that does not change much after 1 s, even if the surfactant concentration is appreciably increased. Since the value of the surface tension is an indirect measurement of the degree of surfactant adsorption, the referred concentration could be expected to be enough to stabilize an O/W emulsion. That would be possible whenever the following conditions are fulfilled: (1) the total interfacial area of the emulsion is small enough to be substantially covered by surfactant molecules at this concentration, and (2) the

Urbina-Villalba

Figure 5. Time dependence of the total interaction potential between two drops (Vt) as a function of surfactant concentration (C), apparent diffusion constant (Dapp), and time (t). According to eq 13, different combinations of the formulation variables can give rise to the same potential curve either at different times or at the same time depending on the values of C and Dapp (see the text).

surfactant selected is able to build a sizable repulsive barrier against coalescence. As shown in Figure 4, a cationic surfactant concentration of C ) 5 × 10-4 M is able to build a considerable repulsive barrier (949.5 kT) in only 0.027 s, whenever its apparent diffusion constant is high enough (Dapp g 1 × 10-10 m2/s). At longer times (27.28 s), a huge barrier develops.14,46 Only in the case in which the apparent diffusion constant is considerably low (Dapp ) 1 × 10-12 m2/s) is the effective potential at 0.027 s attractive, and the large barrier is only present after 27 s. These findings are corroborated by the entries of Table 1. A concentration of C ) 5 × 10-4 M produces large values of 2[Dapp/π]1/2C. At this concentration, the total number of drops is preserved regardless of the volume fraction of oil, whenever Dapp g 1 × 10-10 m2/s (see rows 2-5 for example). Figure 5 shows an interesting property of eq 14. There exist combinations of Dapp and C that can produce the same surface excess at a given time. Conversely, the same surface excess can be also attained at different times depending on the values of Dapp and C. This is shown in this picture (Figure 5) for t ) 0.02728 s and t ) 2.7280 s in the cases in which (C ) 5 × 10-4 M, Dapp ) 1 × 10-10 m2/s) and (C ) 5 × 10-4 M, Dapp ) 1 × 10-12 m2/s), respectively. It follows that a slow diffusion constant can be compensated with a high surfactant concentration. It is clear, however, that a low surfactant concentration can never produce a large repulsive barrier. This is illustrated in Figure 5 for the case in which C ) 1 × 10-5 M. That concentration will not generate a barrier at long times (27.28 s) unless the total interfacial area diminishes due to coalescence. Table 1 shows a summary of the results. As expected, a high volume fraction produces shorter mean free paths between drops, and consequently, the effects of surfactant adsorption are more pronounced. Curiously, the largest volume fraction tested (φ ) 0.40) generates a considerable HI which decelerates the drops. As shown in Figure 6d, the hydrodynamic interaction generates a lag time prior to the first collision. This period favors surfactant adsorption similar to what happens in the much more dilute φ ) 0.10 case. Unlike other calculations, the one for φ ) 0.40 required an initial cubic arrangement between the drops. As shown by previous simulations and confirmed by the data of Table 2, ordered arrangements of particles always produce longer collision times than random ones. Figure 6 shows the variation of the number of drops per unit volume at φ ) 0.10, 0.20, 0.30, and 0.40, for typical


Figure 6. Changes of the particle concentration per unit volume (n) as a function of time, for D ) 1.0 × 10-10 m2/s and φ ) 0.10, 0.20, 0.30, and 0.40.

values of Dapp ) 1 × 10-10 m2/s and C ) 1 × 10-4 M. The decrease in n follows the van der Waals variation while the repulsive barrier is not strong enough. The flocculation rates corresponding to the van der Waals case are 5.49 × 10-17 m3/s (φ ) 0.10), 1.64 × 10-16 m3/s (φ ) 0.20), 8.09 × 10-16 m3/s (φ ) 0.30), and approximately 4.30 × 10-15 m3/s for φ ) 0.40. These were calculated from fitting of kf in eq 23. For φ ) 0.40, the lag time was discarded previous to the fitting process. When the number of drops decreases, the repulsive potential increases (Figure 3). However, the velocity of this increase might be lower than the collision rate, especially for concentrated emulsions. In this case, the number of drops lowers considerably prior to stabilization, causing appreciable variations of the DSD (Figure 6).

Langmuir, Vol. 20, No. 10, 2004 3881

Conclusions A continuous mass transfer is a characteristic feature of emulsions, which results from their kinetic stability. The behavior of these systems is difficult to simulate due to the time-dependent processes that simultaneously occur. One of those processes is surfactant adsorption. Surfactants are known to delay the flocculation of oil drops dispersed in water and can also prevent their coalescence. This is possible due to the generation of a repulsive barrier between particles, which depends on surfactant concentration and chemical structure. Since surfactant adsorption is time-dependent, the building of such barrier also is. When a pair of drops approach in a freshly prepared emulsion, their repulsive potential continuously increases due to surfactant adsorption. This process depends on several factors neatly summarized in eq 14. The adsorption is faster if the surfactant concentration is high, and the mechanism of adsorption is diffusion-controlled. The final DSD of the emulsion depends on a competition between the surfactant surface excess at O/W interfaces and the average mean free path between drops. This latter distance depends on the volume fraction of internal phase, the spatial distribution of the drops resulting from emulsification, and the effective value of the Hamaker constant between the drops. In the present calculations, we considered the consequences of dynamic surfactant adsorption on the stability of bitumen emulsions. We addressed this problem using a modification of the Brownian dynamic algorithm from Ermak and McCammon,1 which accounts for surfactant adsorption2 and hydrodynamic interactions between the drops.34 For typical surfactant molecules (Ds ∼ 10-10 m2/s), diffusion-controlled adsorption occurs very fast. Hence, its transient effects on the stability of dilute dispersions (volume fractions φ < 0.10) are negligible. On the other hand, the effect of dynamic adsorption on the coalescence rate of concentrated O/W emulsions could be critical depending on the volume fraction of oil, the drop size distribution of the emulsion, and the composition of the oil phase. The effects of dynamic adsorption were found to be larger when (a) the spatial distribution of drops is not homogeneous, (b) the drops attract each other with moderately high values of the Hamaker constant (A ∼ 10-19 J), and (c) the volume fraction of oil (φ) is relatively high. Under these conditions, the average time between collisions is short enough to prevent sizable surfactant adsorption prior to coalescence. At very high φ, however, hydrodynamic interactions between the drops may produce a lag time. During this lapse, the collision frequency is considerably low, allowing surfactant molecules to reach O/W interfaces as occurs in the dilute case. The present algorithm can be useful to study the effects of cationic surfactant adsorption upon negatively charged drops. It is also suitable to investigate the stability of food emulsions where monoglycerides and nonionic surfactants are known to displace milk proteins from the interface as a function of time.50-52 Furthermore, convenient modifications of the present methodology are expected to be advantageous in order to simulate the coalescence of deformable droplets, in which case interfacial concentration gradients are expected to occur. LA030327O (50) Chen, J.; Dickinson, E.; Iveson, G. Food Struct. 1993, 12, 135. (51) Dickinson, E. J. Colloid Interface Sci. 1998, 94, 1657. (52) Wijmans, C. M.; Dickinson, E. Langmuir 1999, 15, 8344.

Effect of Dynamic Surfactant Adsorption on Emulsion Stability

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