Aggregation dynamics in systems of coalescing non-deformable droplets

Journal of Molecular Structure: THEOCHEM 769 (2006) 171–181 www.elsevier.com/locate/theochem

Aggregation dynamics in systems of coalescing non-deformable droplets German Urbina-Villalba *, Aileen Lozsa´n, Jhoan Toro-Mendoza, Kareem Rahn, Ma´ximo Garcı´a-Sucre Laboratorio de Fisicoquı´mica de Coloides, Instituto Venezolano de Investigaciones Cientı´ficas (IVIC), Centro de Fı´sica, Apartado 21827, Caracas 1020-A, Venezuela Received 10 March 2006; received in revised form 12 March 2006; accepted 24 April 2006 Available online 5 May 2006

Abstract Emulsion stability simulations (ESS) are used to study the aggregation behavior of oil-in-water (O/W) emulsions composed of non-deformable drops. In the absence of a strong repulsive force (low surfactant concentration), the number of aggregates decreases along with the number of drops following the dynamics predicted by Smoluchowski for irreversible flocculation. During this initial stage, the drops coalesce as soon as they make contact and no stable flocs are formed. The polydispersity of the system increases while its interfacial area decreases. This behavior disfavors the formation of aggregates with the characteristic fractal dimensions exhibit by suspensions under diffusion limited cluster aggregation (DLCA) and reaction limited cluster aggregation (RLCA) regimes. Moreover, the increase of the surfactant surface excess due to the redistribution of surfactant molecules among the available interfaces progressively augments the repulsive force between the remaining drops. In the absence of surface deformation, coalescence stops once the total interfacial area of the emulsion can be stabilized with the available surfactant concentration. At this point, aggregation occurs. In this terminal stage the aggregation behavior is likely to be determined by the characteristics of the secondary minimum of the interaction potential. A high surfactant concentration in the initial system leads to very different results. It prevents coalescence making the drops behave as solid particles. In this case clusters with the fractal dimensions corresponding to the DLCA and RLCA regimes are observed. q 2006 Elsevier B.V. All rights reserved. Keywords: Aggregation; Coalescence; Emulsions; Drops; Simulations

1. Introduction During the last two decades significant advances in the comprehension of the coagulation process of solid/liquid dispersions had occurred [1,2]. Those studies led to the establishment of a general form for the flocculation rate [1] that results from the careful consideration of the collision process and the formulation of suitable population balance equations for the case of irreversible aggregation. In the absence of a strong repulsive force, solid particles coagulate in aqueous media as soon as they collide. The clusters formed are loose, open structures that exhibit a fractal dimension (Df) between 1.7 and 1.8 [3–5]. In this regime, known as diffusion limited cluster aggregation (DLCA), the aggregation behavior is controlled by the thermal movement of the particles and is basically regulated * Corresponding author. Tel.: C58 2125041542; fax: C58 2125041148. E-mail address: [email protected] (G. Urbina-Villalba).

0166-1280/$ - see front matter q 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.theochem.2006.04.020

by their mean free path. The average radius of the flocs (R) increases with time (t) following a power-law behavior: RZ R0 ð1C k1 n0 tÞ1=Df , where R0, n0, k1 are the radius of the particles, the initial number of particles per unit volume, and the flocculation rate of the system, respectively [5,6]. This expression can be derived from Smoluchowski’s formalism [6] assuming a constant kernel (k1) and a scaling behavior of the aggregates formed. According to Smoluchowski, the variation of number of aggregates (na) as a function of time is equal to: na Z

n0 1 C k 1 n0 t

(1)

In Eq. (1), na accounts for aggregates of all sizes including the number of single particles (naZSnm). The product (k1n0)K1 stands for the mean time between collisions and it is also equal to the average time required to decrease the total number of aggregates from n0 to n0/2. In the case of suspensions, this formalism has been validated repeatedly using stopped flow microscopy, spectrophotometric measurements, and dynamic light scattering [7–12].

172

G. Urbina-Villalba et al. / Journal of Molecular Structure: THEOCHEM 769 (2006) 171–181

In the most general case particles interact with an attractive potential Va(r)-which is always present-, and a repulsive force that depends on the characteristics of the particles’ surfaces. According to Derjaguin–Landau–Verwey–Overbeek (DLVO) theory [13], the total potential between the particles V(r)Z Va(r)CVe(r), is the result of the attractive van der Waals potential, and a repulsive electrostatic one, Ve(r). These contributions give rise to two minima separated by a repulsive barrier. A high repulsive barrier causes a decrease in the frequency of collisions that leads to irreversible (primary) minimum flocculation. Hence, the aggregation rate of a stabilized system k2, turns out to be W times slower (WZk1/k2) than the one displayed by the same particles in the absence of the repulsive barrier [14]. Repulsive forces delay the flocculation of particles, and prevent every encounter from being effective (leading to irreversible flocculation). Multiple explorations of the particle’s surface prior to coagulation and the existence of small sticking probabilities [15–19] might be caused by several factors, including repulsive forces, the thermal interaction with the solvent or hydrodynamic fluxes. It generates compact structures with a fractal dimension around 2.1–2.2 and for the particular case of a sum kernel (kijZC(iCj), where C is a constant) it leads to an average radius of the clusters that varies exponentially with time: RZR0exp(k2n0t/Df) [12,13]. Due to the wider variety of destabilization processes (Ostwald ripening, coalescence, phase inversion, etc.) the development of similar expressions for the case of liquid/liquid dispersions has not been possible. These phenomena often occur simultaneously and are mediated by the partition of the surfactant molecules between the immiscible phases. Among other difficulties, the inherent polydispersity of emulsions and the surface deformation of drops necessarily lead to a wider variety of structures during the course of aggregation. Liquid drops usually deform during aggregation due to hydrodynamic and interaction forces [21,22]. In this case, there is no clear limit between flocculation and coalescence. The thinning of the intervening film between flocculating drops involves at least six different steps. These include the formation of a dimple, the generation of a plane-parallel film, the appearance and enhancements of surface oscillations, and either the final coalescence of drops or the formation of longlasting films [23,24]. However, there are liquid dispersions that exhibit non-deformable droplets due to the high viscosity of the internal phase [25], the small size of the drops [26], their high surface charge [27], or other intrinsic properties of the oil/water interface (see, for example, Figs. 2 and 8 in Ref. [28]). For those cases, analytic expressions based on the Smoluchowski formalism but including the process of coalescence in an average way had been forwarded [29–31]. Those equations are reasonably successful in describing the variation of the total number of aggregates with time for several systems including soybean oil-in-water emulsions stabilized with ethoxylated monoglycerides [30], and ethyl oleate-in-water emulsions stabilized with Emulgin O10 [32]. The role of surfactant molecules in emulsions can be very involved as demonstrated by the work of Bibette and

co-workers [27,33]. Using surfactant concentrations well above the critical micelle concentration (CMC), they found that sodium dodecyl sulphate (SDS) micelles could generate the flocculation of a silicon oil-in-water emulsion due to the depletion of micellar structures around the surfaces of the drops. Similarly, the decrease of the total interfacial area of an emulsion due to the coalescence of drops may generate appreciable changes the interaction potential of the remaining particles [34]. The increase in the charge of micron-size drops due to the augment of the surfactant adsorption increases the repulsive barrier of the potential, but also enhances the depth of its secondary minimum favoring the occurrence of fast flocculation rates. Schmitt et al. [35,36] studied the effect of the molecular transfer of oil from small drops to big drops (Ostwald ripening) for the case of alkane-in-water emulsions stabilized with infralan and SDS. They identified two regimes in the evolution of a highly concentrated emulsion characterized by the occurrence of Ostwald ripening and coalescence, respectively. The degree of Ostwald ripening depends on the solubility of the oil and therefore it sensitively decreases from heptane to hexadecane. The referred regimes were preceded by a small period of time in which the uniformity of the emulsions diminished while the average radius increased. However, this brief stage was not analyzed in detail. On the other hand, Katsumoto et al. [37–39] carried on a very meticulous study on water/hexadecane/ Mergital (C12EO7) and brine/ethyl oleate/Eumulgin O10 (C16EO8CC16EO10) systems. They also found two regimes of destabilization characterized by linear variations between the cube of the average radius and the time. The characteristic times for destabilization of the referred systems (t) were found to be much more closer to the ones predicted by the Lifshitz– Slezov–Wagner theory of Ostwald ripening (104 s) than those calculated for flocculation by means of Smoluchowski theory (10K6 s). On the other hand, the destabilization times implied that the aggregation process should occur first. Use of relative scattering intensity measurements and a relationship between the osmotic compressibility (P) and the volume fraction of oil f demonstrated that the first regime (12 days) corresponds to Smoluchowski theory for diffusive coagulation, except for the fact that t was of the order of 2.4 days. In order to study oil-in-water dispersions, emulsion stability simulations (ESS) can be very valuable. In previous studies, we only focused on the variation of the total number of aggregates per unit volume as a function of time (na vs. t) [20,34–43]. Those studies evidenced that at least for the case of nondeformable droplets and in the absence of a substantial repulsive force, the emulsions follow Smoluchowski’s dynamics. The application of this flocculation theory to the combined processes of flocculation and coalescence required that single drops were regarded as single-particle aggregates independently of their actual radii. On the other hand, the adsorption of surfactant to the drops causes an appreciable change in the curves of na vs. t which cannot be described by Eq. (1). As will be shown below, the aggregation behavior of coalescing non-deformable droplets can be very rich, even in the absence of other destabilization phenomena.

G. Urbina-Villalba et al. / Journal of Molecular Structure: THEOCHEM 769 (2006) 171–181

It is important to notice that the number of aggregates in emulsions strongly depends on the volume fraction of oil. In a previous paper, we showed that the flocculation time, tfZ (k2n0)K1, diminishes as a function of f [42]. In the limit of highly concentrated emulsions (fw0.7–0.9) the drops deform appreciably leading to the formation of oil/water/oil (o/w/o) films. The mean free path between the drops becomes negligible, favoring a temporal variation of the number of drops per unit volume (n) that is only proportional to the number of ruptures of the films [35,36]. Thus vn K Z uA vt

(2)

where u is the frequency of coalescence events (ruptures) per unit area, and A is the total area of the emulsion. The present paper deals with the intermediate regime of concentration (0.01%f%0.4) where the processes of flocculation and coalescence significantly contribute to the change in the number of drops [34]. Under these conditions, a low surfactant concentration favors coalescence leading to Smoluchowskian dynamics. On the other hand, a high surfactant concentration prevents coalescence leading to a substantial degree of aggregation. The aggregates formed show fractal coefficients similar to the ones exhibited by solid particles in suspensions. Their fractal dimensions are used here to characterize their geometry and relate it to the velocity of flocculation [44,45]. Finally, we will show how moderate surfactant concentrations produce a variety of intermediate results. In these cases the variation of number of aggregates as a function of time (na vs. t) can be described using average formulae [34]. 2. Emulsion stability simulations

is able to correct the pathologies originated from the screening of hydrodynamic forces in concentrated systems. In the present calculations only van der Waals and electrostatic forces are included. As usual, the sum of these contributions gives rise to a Derjaguin–Landau–Verwey–Overbeek (DLVO) potential. This potential shows to minima separated by a repulsive barrier. The primary minimum corresponds to the irreversible flocculation of drops. In the case of deformable droplets, flocculation in the primary minimum does not imply coalescence. The potential of surface deformation and the friction of the intervening liquid between flocculated drops act as effective repulsive forces [21,22]. However, non-deformable drops unavoidably coalesce if they are able to jump over the repulsive barrier. Thus, a decrease in the initial number of particles (Np) in these calculations is a distinctive indication of primary minimum ‘flocculation’. All simulations presented in this article correspond to dispersions of non-deformable drops. The van der Waals term of the DLVO potential is implemented here with the expression of Hamaker for spheres of different radii [45]. The formulation of Sader is employed for the electrostatic interaction [46–48]. This expression connects the limiting forms of the surface potential and surface charge for small and large values of kRi (where k is the Debye length, and Ri the radius of the particle). In ESS the electrostatic potential of the drops is supposed to be the result of surfactant adsorption. The program contains several routines to handle the surfactant population. These are aimed to resemble experimental conditions including timedependent adsorption [49], non-homogeneous distributions due to incomplete mixing [50], etc. Hence, the total charge of a drop (Zi) depends on the surfactant surface excess G(t): Zi Z Ai zs eGðtÞ

In order to study the phenomenon of aggregation in the referred systems, emulsion stability simulations (ESS) are used. In ESS the movement of the particles is similar to the one of Brownian dynamics simulations. The displacement of particle i during the time Dt, rði ðtC DtÞKðr i ðtÞ, is the result of two contributions: rði ðt C DtÞKðr i ðtÞ Z

Di ðfL ;dÞFð i Dt ð C R G ðDi ðfL ;dÞÞ kB T

(3)

The first term on the right hand side accounts for the diffusion of the particle under the action of interaction and external forces (F). This term supposes that each particle moves at a constant velocity vZDF/kBT during time Dt, where kB is the Boltzmann constant and T the absolute temperature of the dispersing medium. The diffusion constant modulates the velocity of the particle, which is primarily limited by the viscosity of the solvent. In our simulations an average diffusion coefficient Di(fL, d) is used [43]. It depends on the local volume fraction of particles (fL) and a minimum distance of approach between particle i and its neighbors (d). It assumes the limit value of Stokes law (D0) at infinite dilution and reproduces the predictions of the diffusion tensor of Batchelor in dilute systems. Furthermore, this methodology

173

(4)

Here Ai is the area of drop i, and zse is the charge of one surfactant molecule, and e is the elementary charge (1.6! 10K19 C). The electrostatic energy is expressed in terms of the surface potential fp, which can be easily computed from surface charge density (s): sZ

Z 4pR2i

(5)

In all systems studied here adsorption was assumed to occur instantaneously. That is, substantially faster than the collision of drops. Thus, the total surfactant population is evenly distributed among the number of drops at the beginning of the simulation, up to a saturation limit. That limit correspond to the maximum surface excess of the surfactant at the oil/water ˚ 2). Depending on the amount of surfacinterface (GK1w50 A tant adsorbed, the interaction potential may vary from totally attractive to highly repulsive. Coalescence occurs if the distance between the drops becomes smaller than the sum of their radii. In this case a new drop is created at the center of mass of the coalescing particles. Whenever coalescence occurs, the total interfacial area of the emulsion decreases. As a consequence, either a certain

174

G. Urbina-Villalba et al. / Journal of Molecular Structure: THEOCHEM 769 (2006) 171–181

Table 1 Parameters used in the first set of simulations corresponding to systems with a high surfactant concentration (systems s1 and b2 in Ref. [20,51]) System

Radius (m)

Hamaker constant AH (J)

zs

Total surfactant concentration (M)

Ionic strength (M)

Barrier height (kBT)

Depth of the secondary minimum (kBT)

Potential width (nm)

s1 b2

1.0!10K8 3.9!10K6

2.0!10K19 1.24!10K19

20.57 0.2057

0.01 0.01

0.0232 0.007

40.6 17,000

K0.005 K311

14.0 23.3

amount of surfactant dissolves in the continuous phase, or the resulting drops inherits the surfactant population of the parent drops. In either case the surface potential of the final drop increases. However, in the first case, the bulk surfactant concentration increases, and as a consequence, the surface excess of all drops may slightly increase. In the latter case, it is unlikely that one drop preserves a surfactant concentration substantially higher than the rest. If this occurs, that anomalous accumulation of surfactant should rapidly relax leading to the same result outlined above. Hence, in these calculations the total surfactant concentration of the system is evenly re-distributed among the surviving drops every time coalescence occurs up to the saturation limit referred above. As a consequence, the interaction force between the drops (F) changes during the course of the simulation, unless the initial number of drops is preserved. The last term on the right hand side of Eq. (3) simulates the thermal interaction of the suspended particle with the solvent, which is implemented using a random function Rð G ðDi ðfL ;dÞÞ, which has a Gaussian distribution with zero mean and variance 6Di(fL, d)Dt. In this work two sets of calculations are analyzed. The parameters of the simulations are given in Tables 1 and 2. The behavior of these systems was previously studied in terms of the variation of na vs. t. The first two emulsions correspond to systems s1 and b2 in Ref. [20,51]. They are initially composed of 125 particles with radius R0Z10 nm (s1) and 3.9 mm (b2). In these two systems the surfactant concentration is enough to completely cover the interface of all drops, generating a repulsive barrier that preserves the number of drops (Np). The second set of emulsions consists of systems nos.5, 8, 10 and 14 in Refs. [34,51,52]. These are polydispersed systems of 216 and 125 particles [52] in which the initial surfactant concentration is too low to generate a repulsive barrier. As a consequence, the initial number of particles decreases as the systems evolve, leading to the occurrence of flocculation and coalescence. Three-dimensional periodic boundary conditions (PBC) were used in all calculations.

In order to study the dynamics of flocculation of the referred dispersions it is necessary to calculate the number of aggregates as a function of time. This is achieved using the data resulting from ESS and an auxiliary code [20,34]. The program uses as input the variation of the coordinates of the particles as a function of time, and a ‘flocculation distance’ which is set slightly larger than the position of the secondary minimum of the corresponding DLVO potential. A particle is assumed to form an aggregate with a surrounding neighbor if it can get within a distance equal or lower the flocculation distance. If the number of particles is preserved during the simulation (first set of calculations) several flocculation distances can be tested. The distance selected is the one that is closer to the secondary minimum and does not generate an appreciable variation in the number of aggregates when it is appreciably increased. The second set of simulations (systems nos.5, 8, 10 and 14 in Refs. [34,51]) corresponds to polydisperse systems with several interparticle potentials. Moreover, the number of particles in these systems is not preserved during the course of the simulation. As a result, the potential between particles changes during the calculation. In order to compare the aggregation behavior of these four systems, a fixed flocculation distance of 100 nm was selected. This is twice the distance of the secondary minimum of the potential between two 3.9 mmparticles completely covered by surfactant molecules. Once the number of aggregates has been identified for each set of coordinates, the variation of the number of particles, aggregates, particles in aggregates, average radius of the flocs, etc. can be obtained. 3. Results 3.1. Aggregation phenomena in systems of highly charged non-deformable droplets Fig. 1(a) shows the variation of the total number of aggregates N a(N aZnaV, where V is the volume of the system) as a function of time for a system composed of 125

Table 2 Parameters used in the second set of simulations (see Refs. [34], [51] and [52] for further details) System no.

Maximum and minimum radii R0Z3.9!10K6 m

Hamaker constant AH (J)

zs

Total surfactant concentration (M)

Ionic strength (M)

Initial barrier height (kBT)

Poly-dispersity index (PI)

5 8 10 14

0.4%R0%0.8 1.0%R0%0.1 1.0%R0%0.1 2.0%R0%0.1

1.24!10K19 1.24!10K19 1.24!10K19 1.24!10K19

0.2057 0.2057 0.2057 0.2057

5!10K5 8!10K5 4!10K5 8!10K5

0.0232 0.0232 0.0232 0.0232

– – – –

0.10 0.25 0.35 0.52

The surfactant concentration in these systems is not enough to preserve the initial number of drops. The polydispersity index (PI) is defined as: P
where NT is the total number of drops of the system, and R
the average radius. In this formula Ni stands for the number of particles of PIZ ð1=NT Þ i Ni jRi KRj, radius Ri.

G. Urbina-Villalba et al. / Journal of Molecular Structure: THEOCHEM 769 (2006) 171–181

175

is only K0.005kBT, a value that can be easily surpassed with the thermal interaction supplied by the solvent (last term on Eq. (3)). Fig. 2(a) shows a similar calculation for drops of R0Z 3.9 mm, AHZ1.24!10K19 J, zsZ0.2057, and i.s. Z0.007 M (system b2 [20,51]). As discussed in Ref. [20], micron-size particles completely covered by surfactant molecules, are likely to produce large repulsive barriers along with deep secondary minima. The volume fraction of particles (f) in this system is equal to 0.30. Due to the size dependence of the interaction potential the referred parameters generate an insurmountable repulsive barrier of 17,000kBT. As a consequence, Np does not change during the course of the simulation as in the former case. However, the aggregation behavior is strikingly different from the one depicted in Fig. 1. In system b2, the depth of the secondary minimum is K311kBT. Thus, the secondary minimum promotes irreversible flocculation. The number of aggregates and single particles (Na) decreases as a function of time, although it does not show a smooth Smoluchowskian slope. Nagg shows a maximum, and Nin agg increases non-monotonically as a function of time. Fig. 3 shows the evolution of the aggregates of different sizes as a function of time. In this figure Ni stands for an aggregate composed of i

Fig. 1. (a) Variation in the number of aggregates (Nagg), particles in aggregates (Ninagg), and single particle plus aggregates (Na) as a function of time for system s1 (barrier height 40kBT with no secondary minimum). (b) Behavior of the average number of particles per aggregates (Nper agg).

drops with radius R0Z10 nm. The repulsive barrier between the particles is equal to 40kBT (system s1). This barrier results from the homogeneous distribution of surfactant molecules of effective charge zse (where zsZ20.57 and e is the unit of electrostatic charge) among the surface of the drops [20,51]. The charge of each particle is equal to ZiZAzseG, where A is its area, and G is the maximum surface excess of the surfactant ˚ 2/mol). The ionic strength of the molecules (G K1w50 A solution (i.s.) is 0.0232 M, and the effective Hamaker constant (AH) between the drops is equal to 2.0!10K19 J. Under the above computational conditions the total number of particles of the system (Np) is preserved during the course of the simulation. Within this period, Na slightly oscillates around 125. The average number of aggregates excluding singleparticle clusters (Nagg), and the total number of particles in aggregates (Nin agg) are equal to zero. This means that the particles do not form stable clusters. This is confirmed by Fig. 1(b) where the average number of particles per aggregates is shown (Nper agg). Nper agg oscillates between 0 and 2, indicating that the short-lived doublets eventually formed are rapidly disunited. This behavior is consistent with the absence of a deep secondary minimum in the potential of interaction. In this particular case, the depth of the secondary minimum (Vsm)

Fig. 2. (a) Changes in the number of aggregates as a function of time for system b2 (large repulsive barrier and deep secondary minimum). The dotted lines remark the possible occurrence of a phase transition. (The nomenclature is equal to that of Fig. 1). (b) Nper agg vs. time (t).

176

G. Urbina-Villalba et al. / Journal of Molecular Structure: THEOCHEM 769 (2006) 171–181

1.79, 1.83, 2.00, 2.00, 1.99, 2.31, and 2.35 for: 13!t!27 s, 27!t!39 s, 39!t!76 s, 76!t!105 s, 105!t!162 s, 162!t!180 s, and 180!t!227 s. This evolution of Df indicates that the flocculation dynamics can change from fast to slow (DLCA to RLCA) at constant interparticle potential. A close inspection of Fig. 4 also indicates that aggregates of bigger sizes (ln(Rg/R0)O1) have a different fractal dimension than smaller ones. As shown in Fig. 4(d) larger aggregates have a substantially higher dimension. Notice also that in this simulation the stability ratio is fixed. However, the two characteristic values of the fractal dimension for fast and slow flocculation are observed. This apparently contradicts recent evidence, which supports a connection between the stability ratio W and Df [53]. However, it is known that only in the cases of scale invariance and cluster–cluster aggregation the fractal dimension is maintained [54,55]. Aggregates growing by particle-cluster flocculation are not scale-invariant becoming increasingly compact. Moreover, Gmachowski [54] demonstrated that this process of aggregation leads to the appearance of a pre-factor (structural coefficient) in a log–log plot of the mass–radius relation. In these cases, the plots do not pass through the origin (as is observed in Fig. 4).

Fig. 3. Aggregation of system b2 as a function of time. Ni stands for the number of i-particle aggregates.

particles, independently of the actual radius of the conforming particles. The number of single particles (N1) decreases slightly more sharply than the one of particles and aggregates Na (in Fig. 2(a)) as predicted by the theory of Smoluchowski. Single particles are lost not only by the formation of doublets (see N2 in Fig. 3(a)) but also through their incorporation to clusters of higher sizes. As shown in Fig. 3(b) the number of large clusters increases as time evolves for t!170 s. Aggregates of large sizes appear very early during the simulation for (t!25 s) due to the high volume fraction of internal phase. However, they do not necessarily appear consecutively as the function of size. The lack of a smooth variation in the number of aggregates is likely to be caused by the small number of particles of the system but the overall trend follows the prediction of Smoluchowski formalism qualitatively (Fig. 3(a) and (b)). The evolution of the fractal dimension of the clusters found in system b2 is shown in Fig. 4. This is studied using a log–log plot of Nin agg vs. (Rg/R0), where Rg is the radius of gyration of the aggregates. According to a recent communication [20], systems with profound secondary minimum are likely to undergo fast secondary-minimum flocculation, despite the large magnitude of the repulsive barrier. This is confirmed by the initial value of the fractal dimension (DfZ1.74) corresponding to 0 s !t!13 s. Curiously, the fractal dimension of the clusters increases as the time elapses. It shows values of

Fig. 4. Variation of the fractal dimension of the aggregates during the evolution of system b2. (a) DfZ1.74, 0 s!t!13 s; (b) DfZ1.99, 106 s!t!162 s; (c) DfZ2.31, 163 s!t!180 s; (d) DfZ2.87 (large aggregates only), 181 s!t! 227 s.

G. Urbina-Villalba et al. / Journal of Molecular Structure: THEOCHEM 769 (2006) 171–181

177

decreases as a consequence of aggregation (increasing Nagg) and also as a result of a decrease in Np. When coalescence occurs, the total interfacial area of the emulsion decreases. In the case of oil-in-water (O/W) emulsions stabilized with ionic surfactants, the surfactant concentration inside the drops is supposed to be negligible. Furthermore, it is implicitly assumed that the local increase in the surfactant concentration produced by the decrease of the total interfacial area should rapidly relax. Thus, the extra number of surfactant molecules resulting from the coalescence of drops should promote an augment of the surfactant excess of all drops in the system. This process can be mimicked by the homogeneous distribution of surfactant molecules among the available interfaces. Such procedure is carried on each time coalescence occurs. As a consequence the surface concentration of the drops increases. This mechanism leads to an augment of the repulsive force between the drops as a function of time. Furthermore, it leads to the long-term stabilization of the average radius of the emulsion as the one usually observed in concentrated systems. It was recently found [34] that the total number of aggregates per unit volume (na) in systems of coalescing

Fig. 4 (continued)

Further insight into the dynamics of the system is obtained by looking at the two marked fluctuations in Nagg, Na, Nin agg, and Nper agg which occur at tw160 and w198 s, respectively. The earlier fluctuation leads to a jump in Nper agg from a value of 9 to 30 particles. The latter one shows that almost all particles aggregate into a single cluster of 105 particles. Such behavior is typical of phase transitions and the formation of gels. At the end of the calculation, after the last transition occurs, the system is basically composed of a large floc, three doublets and 14 singlets. 3.2. Aggregation phenomena in systems of coalescing non-deformable droplets It is clear that the behavior of systems s1 and b2 (Figs. 1–4) resembles that of solid particles. It is likely to happen in emulsions with non-deformable drops at high surfactant concentrations. However, in the most common situation, the original interfacial area of the emulsion cannot be stabilized with the initial surfactant concentration of the system. In this case, the population of surfactant molecules cannot build a critical repulsive barrier of at least 13kBT between all drops ˚ 2/molec) [56]. As a result, the drops coalesce and (GK1[50 A the number of particles progressively decreases. Na now

Fig. 5. (a) Changes in the number of aggregates as a function of time for system no.14 of Ref. [34]. The surfactant concentration is unable to preserve the initial number of drops. The number of particles decreases initially but then stabilizes. (b) Nper agg vs. time (t) for system no.14.

178

G. Urbina-Villalba et al. / Journal of Molecular Structure: THEOCHEM 769 (2006) 171–181

non-deformable droplets, follows Eq. (6) for at least 34 different systems characterized by distinct polydispersity indexes (PI), surfactant concentrations (Cs), and a wide range of volume fractions fZ0.01–0.30:
 A Kkc;f n0 t C Be na Z n0 (6) 1 C k f n0 t Here, kf stands for the usual flocculation rate that measures the average frequency of collisions in the emulsion. kc,f is a second kinetic constant formerly ascribed to coalescence. A and B are constants that measure the relative importance of flocculation and coalescence during the whole destabilization process (ACBZ1). Although the frequency of these processes changes as a function of time, one pair of (A, B) values was found to be enough to fit the total variation of na vs. t during the whole simulation. The parameters of the potential used in the referred simulations were taken from the experiments of Salou et al. [57]. They correspond to a bitumen-in-water emulsion stabilized with a cationic surfactant. The drops of this emulsion exhibit a surface potential of 115 mV, which produces very large repulsive barriers (Vmaxw10,000kBT) for drops of 3.9 mm. As in the case of system b2 discussed above: zsZ

Fig. 6. Aggregation behavior of system no.14 (Ref. [34]). Ni stands for the number of i-particle aggregates. The maximum aggregate size observed was N12.

0.2057, AHZ1.24!10K19 J, but i.s.Z0.014 M. The potential of interaction between the drops of systems nos.5, 8, 10 and 14 is completely attractive at the beginning of the calculation (Table 2). The final value of the repulsive barrier depends on the size of the particles, ranging from 3 to 180kBT for the referred systems. The larger barriers were observed in system no.14 (10–180kBT) while the lower ones corresponded to system no.10 (3–30kBT). The potential at the secondary minimum ranged from K15.1 to K2480kBT. Eq. (6) was deduced considering first and second order reaction kinetics for the processes of flocculation and coalescence, respectively, along with the random occurrence of coalescence events. However, a more detailed analysis of the simulation data indicated that the exponential term was also able to fit the terminal flocculation rate shown by the surviving particles towards the end of the simulation. It was also clear then that the first and second terms of Eq. (6) are similar to the variation of the number of aggregates with time predicted for suspensions under DLCA and RLCA regimes, respectively. Surprisingly, it was observed that only half of the calculations of Ref. [34] showed the occurrence of clusters during the evolution of the system and/or in their final configuration.

Fig. 7. (a) Change in the number of aggregates as a function of time for system no.8 of Ref. [34]. Coalescence occurs simultaneously in different regions of the system favoring the creation of a large number of independent flocs. (b) Nper agg vs. time (t).

G. Urbina-Villalba et al. / Journal of Molecular Structure: THEOCHEM 769 (2006) 171–181

179

Fig. 8. Aggregation behavior of system no.8 (Ref. [34]). The maximum aggregate size observed was N4.

This is an indication that in a considerable number of the cases, the nucleus of flocculation are well separated, and are not necessarily the result of the addition of individual particles to clusters of bigger size. These two possibilities are well illustrated by the aggregation behavior of asphaltene dispersions extracted from Furrial and Boscan crude oils [12].

Fig. 10. (a) Changes in the number of aggregates as a function of time for system no.10 of Ref. [34]. The surfactant concentration is too low to preserve the number of particles. The variation of the number of particles and aggregates as a function of time is the same and no clusters are formed during the course of the simulation. The system conforms to Smoluchowski’s dynamics.

Fig. 9. (a) Changes in the number of aggregates as a function of time for system no.5 of Ref. [34]. (b) Aggregation does not occur during the coalescence stage. Only singlets of different sizes are formed.

The next Figs. 5–10 illustrate the aggregation behavior shown by 4 of the 34 systems formerly studied in terms of the variation of na vs. t only (Eq. (3) [34]). In all these systems the simulation box is cubic cell with a side length LZ12.1R0 (R0Z3.9 mm). The calculations differ in the volume fraction of internal phase (f), the total interfacial area (IA), and the surfactant concentration employed. Fig. 5 corresponds to System no. 14 in Ref. [34]. This emulsion has a volume fraction fZ0.295, and a surfactant concentration CsZ8!10K5 M. This surfactant concentration is insufficient to stabilize the initial number of particles which decreases from NpZ125 to NpZ75 particles during the first 200 s. As in the case of system b2 referred above (Fig. 2), the curves of Na and Nin agg cross each other. However, a monotonic increase of Nper agg is observed. The increase in the polydispersity of the system does not lead to a phase transition although the final number of aggregates is 11 (NaggZ11, NaZ38) and the average number of particles per cluster is four. The actual distribution of aggregates is shown in Fig. 6. Aggregates of several sizes (Ni!N12) are progressively

180

G. Urbina-Villalba et al. / Journal of Molecular Structure: THEOCHEM 769 (2006) 171–181

created during the course of the simulation, but unlike those of system b2 (Fig. 2) most of them survive until 200 s. Fig. 7 shows the aggregation behavior of system no.8 in Ref. [34]. The surfactant concentration used in this calculation is equal to the one of system no.14: CsZ8!10K5 M. However, the volume fraction of the former system (0.15) is considerably lower, and its interfacial area is appreciably higher: 1027R20 vs. 938 R20 , respectively. This was achieved through a considerable increase in number of smaller particles (Np changed from 125 to 216). The particle size distribution (PSD) of system no.14 is bimodal with peaks at 1.5 and 0.1R0 [52]. In system no.8, the radii of the particles are uniformly distributed between 1.0 and 0.1R0. As a result, the polydispersity index (PI) of system no.8 (0.25) is appreciably lower than the one of system no.14 (0.52). It should be noticed that despite our detailed knowledge of the initial conditions of these two systems and the types of interaction forces involved, the prediction of stability is apparently difficult. Because the same surfactant concentration in used in both systems, the higher values of IA and Np in system no. 8 suggest a substantially weaker repulsive force between the particles. However, the particles of system no.14 are larger. Since, both the attractive and repulsive forces increase with the radius of the particles, a larger amount of surfactant molecules is required to stabilize these bigger particles. Moreover, the volume fraction of system no.14 is also higher. This means that the mean free path between the particles is lower, which favors the occurrence of fast flocculation rates. According to Fig. 7(a), Na now shows a very smooth decrease in curvature as a function of time. As in the case of system no.14 there is a pronounced decrease in the initial number of particles. The initial interfacial area of the system must decrease appreciably in order to stabilize the number of particles. It was shown in Ref. [34] that for the present value of ˚ 2 per surfactant molecule is zs a minimum density of 200 A required in order to stabilize the emulsion. Due its lower value of f, the decrease of Np occurs at a lower rate. Moreover, the larger mean free path between the particles favors the formation of a higher number of independent coalescing nuclei, substantially separated from one another. As the repulsive potential increases, the number of particles in aggregates smoothly increases along with the number of aggregates. However, the maximum number of particles per aggregates observed was four (quadruplets), and the average number of particles per aggregates at the end of the simulations was only two. Although the number of surviving particles in system no.8 (NpZ96) is considerably higher than the one of system no.14, a substantial formation of aggregates is not observed (Fig. 8). Both systems contain 11 clusters at the end of the simulation NaggZ11, but Nper agg is considerably larger in the case of system no.14. However, the number of singlets -of any size- in system nos. 8 is 70, while the one of system no.14 is only 27. As a result of all these differences, the flocculation and coalescence rates of system no.14 are one order of magnitude higher than the ones of system no. 8 [34]. Thus the aggregation behavior of system no. 8 is similar to the one of asphaltene aggregates from Furrial crude oil, while that of the system

no.14 resembles the conduct of asphaltene aggregates from Boscan oil [12]. Fig. 9 shows another typical outcome of ESS that can also be fitted by Eq. (6). System no. 5 in Ref. [34] has a similar interfacial area (1023R20 ) as the one of system no. 8. Both systems have the same number of particles (216) and a similar volume fraction of internal phase: 0.15 (no. 8) vs. 0.13 (no. 5). The PI is equal to 0.10 as compared to 0.25 for system no. 8. However, the value of Cs employed in this case is considerably lower (5!10K5 M). As a result Na decreases very sharply. As a matter of fact, the variation of Np is equal to the one of Na during the first 100 s. In this period of time the particles do not aggregate but only coalesce. Moreover, only one doublet is formed, leading to 30 singlets out of 31 aggregates. Fig. 10 illustrate the aggregation behavior of system no.10 of Ref. [34]. This system has a similar volume fraction (0.15) and number of particles (216) as the ones of system no. 8. PI is equal 0.35 but the PSD is bimodal, centered at 1.5 and 0.1R0 [52] as in the case of system no.14. However, other parameters are very different (IAZ 916R20 ; Cs Z 4 !10K5 M). As can be seen (Fig. 10) Cs now plays a determining role. The low surfactant concentration employed is unable to stop the occurrence of coalescence during the whole course of the simulation. The variation of Na is equal to the one of Np up to tZ200 s. The curve reproduces the behavior of Eq. (1), and the flocculation rate can be evaluated from the variation of the number of particles. In other words, the flocculation rate can be calculated from the coalescence of drops, and Eq. (1) can be fitted to the data as well as Eq. (6). 4. Conclusions The aggregation behavior of an emulsion composed of coalescing non-deformable droplets can vary very amply depending on the interaction potential. In the absence of surfactant the behavior of the systems conforms to Smoluchowskian dynamics (Eq. (1)). In the presence of a high surfactant concentration, the emulsion drops behave as solid particles showing a dynamic behavior basically controlled by the characteristics of the secondary minimum of the interaction potential. Intermediate surfactant concentrations favor the occurrence of flocculation and coalescence. As shown in Ref. [34], Eq. (6) is able to fit the variation of the total number of aggregates as a function of time for a wide variety of systems and physicochemical conditions (Cs, PI, IA, i.s., zs, etc). This allows the calculation of flocculation and coalescence rates in the absence of other destabilization phenomena. However, the aggregation dynamics of these emulsions and their final states cannot be deduced from the simple parametrization of Eq. (6). This point was illustrated in this communication by a detailed analysis of the number of particles (Np) aggregates (Nagg), and average number of particles per aggregates (Nper agg) of four different systems, previously studied by means of Eq. (6). The present analysis of ESS data is likely to be very useful in order to establish general trends for the aggregation behavior in systems of non-deformable particles [58].

G. Urbina-Villalba et al. / Journal of Molecular Structure: THEOCHEM 769 (2006) 171–181

References [1] M. Lattuada, P. Sandku¨hler, H. Wu, J. Sefcik, M. Morbidelli, Adv. Colloids Interface Sci. 103 (2003) 33. [2] P. Sandku¨hler, M. Lattuada, H. Wu, J. Sefcik, M. Morbidelli, Adv. Colloids Interface Sci. 113 (2005) 65. [3] M.Y. Lin, H.M. Lindsay, D.A. Weitz, R.C. Ball, R. Klein, P. Meakin, Nature 339 (1989) 360. [4] D.A. Weitz, M. Oliveria, Phys. Rev. Lett. 52 (1984) 1433. [5] M. Berka, M. Rice, Langmuir 20 (2004) 6152. [6] M. von Smoluchowski, Z. Phys. Chem. 92 (1917) 129. [7] H. Sonntag, K. Strenge, Coagulation kinetics and structure formation, VEB Deutscher Verlag der Wissenschaften, Berlin, 1987 (Chapters 1–4). [8] H. Holthoff, A. Schmitt, A. Ferna´ndez-Barbero, M. Borkovec, M.A. Cabrerı´zo-Vı´lchez, P. Schurtenberger, R. Hidalgo-Alvarez, J. Colloid Interface. Sci. 192 (1997) 463. [9] H. Holthoff, S.U. Egelhaaf, M. Borkovec, P. Schurtenberger, H. Sticher, Langmuir 12 (1996) 5541. [10] H. Sonntag, V. Shilov, H. Gedan, H. Lichtenfeld, C. Du¨rr, Colloids Surf. A 20 (1986) 303. [11] W.B. Russel, D.A. Saville, W.R. Schowalter, Colloidal Dispersions, Cambridge University Press, Cambridge, 1989. [12] J. Hung, J. Castillo, A. Reyes, Energy Fuels 16 (2005) 198. [13] E.J.W. Verwey, J.Th.G. Overbeek, The Theory of Lyophobic Colloids, Dover, New York, 1999. p. 165. [14] N. Fuchs, Z. Physik 89 (1936) 736. [15] F. Family, P. Meakin, J.M. Deutch, Phys. Rev. Lett. 57 (1986) 727. [16] P. Meakin, M.H. Ernst, Phys. Rev. Lett. 60 (1988) 2503. [17] E. Pefferkorn, S. Stoll, J. Colloid Interface Sci. 138 (1990) 261. [18] E. Pefferkorn, J. Widmaier, Colloids Surf. 145 (1998) 25. [19] G. Odriozola, R. Leone, A. Schmitt, J. Callejas-Ferna´ndez, R. Martı´nez´ lvarez, R. Martı´nez-Garcı´a, R. Hidalgo-Alvarez, J. Chem. Phys. 121 A (2004) 5468. [20] G. Urbina-Villalba, M. Garcı´a-Sucre, Langmuir 21 (2005) 6675. [21] N.D. Denkov, D.N. Petsev, K.D. Danov, J. Colloid Interface Sci. 176 (1995) 189. [22] D.N. Petsev, N.D. Denkov, P.A. Kralchevsky, J. Colloid Interface Sci. 176 (1995) 201. [23] K.P. Velikov, O.D. Velvev, K.G. Marinova, G.N. Constantinides, J. Chem. Soc. Faraday Trans. 93 (1997) 2069. [24] K.D. Danov, N.D. Denkov, D.N. Petsev, I.B. Ivanov, R. Borwankar, Langmuir 9 (1993) 1731. [25] I.B. Ivanov, D.S. Dimitrov, in: I.B. Ivanov (Ed.), Thin Liquid Films, Marcel Dekker, New York, 1988. [26] X. Wu, J. Czarnecki, N. Hamza, J. Masliyah, Langmuir 15 (1999) 5244. [27] P. Izquierdo, J. Esquena, Th.F. Tadros, C. Dederen, M.M. Garcı´a, N. Azemar, C. Solans, Langmuir 18 (2002) 26. [28] J. Bibette, F. Leal-Calderon, P. Poulin, Rep. Prog. Phys. 62 (1999) 969.

181

[29] A. Koh, G. Gilles, J. Gore, B.R. Saunders, J. Colloid Interface Sci. 227 (2000) 390. [30] M. van den Temple, Recueil 71 (1953) 433. [31] R.P. Borwankar, L.A. Lobo, D.T. Wasan, Colloids Surf. 69 (1992) 135. [32] A.J.F. Sing, A. Graciaa, J. Lachaise, P. Brochette, J.L. Salager, Colloid Surf. A 152 (1999) 31. [33] J. Bibette, D. Roux, F. Nallet, Phys. Rev. Lett 65 (1990) 2470. [34] G. Urbina-Villalba, J. Toro-Mendoza, M. Garcı´a-Sucre, Langmuir 21 (2005) 1719. [35] V. Schmitt, C. Cattelet, F. Leal-Calderon, Langmuir 20 (2004) 46. [36] V. Schmitt, S. Arditty, F. Leal-Calderon, in: D.N. Petsev (Ed.), Emulsions: Structure, Stability and Interactions, Elsevier, Amsterdam, 2004, p. 607. (Chapter 15). [37] Y. Katsumoto, H. Ushiki, A. Graciaa, J. Lachaise, Colloid Polym. Sci. 279 (2001) 122. [38] Y. Katsumoto, H. Ushiki, B. Mendiboure, A. Graciaa, J. Lachaise, J. Phys.: Condens. Matter 12 (2000) 3569. [39] Y. Katsumoto, H. Ushiki, A. Graciaa, J. Lachaise, J. Phys.: Condens. Matter 12 (2000) 249. [40] G. Urbina-Villalba, M. Garcı´a-Sucre, Langmuir 16 (2000) 7975. [41] G. Urbina-Villalba, J. Toro-Mendoza, A. Losza´n, M. Garcı´a-Sucre, in: D. Petsev (Ed.), Emulsions: Structure, Stability and Interactions, Elsevier, Amsterdam, 2004 (Chapter 17). [42] G. Urbina-Villalba, J. Toro-Mendoza, A. Lozsa´n, M. Garcı´a-Sucre, J. Phys. Chem. 118 (2004) 5416. [43] G. Urbina-Villalba, M. Garcı´a-Sucre, J. Toro-Mendoza, Phys. Rev. E 68 (2003) 061408. [44] S.N. Yao, P.W. Zhu, D.H. Napper, J. Colloid Interface Sci. 174 (1995) 162. [45] M.Y. Lin, H.M. Lindsay, D.A. Weitz, R.C. Ball, R. Klein, P. Meakin, Nature 339 (1989) 360. [46] J.E. Sader, J. Colloid, Interface Sci. 188 (1997) 508. [47] M.R. Oberholtzer, N.J. Wagner, A.M. Lenhoff, J. Chem. Phys. 107 (1997) 9157. [48] M.R. Oberholtzer, J.M. Stankovich, S.L. Carnie, D.Y.C. Chan, A.M. Lenhoff, J. Colloid Interface Sci. 194 (1997) 138. [49] G. Urbina-Villalba, Langmuir 20 (2004) 3872. [50] G. Urbina-Villalba, M. Garcı´a-Sucre, Mol. Simul. 27 (2001) 75. [51] G. Urbina-Villalba, M. Garcı´a-Sucre, Langmuir 22 (2006) 5968. [52] G. Urbina-Villalba, M. Garcı´a-Sucre, J. Toro Mendoza, Mol. Simul. 29 (2003) 393. [53] M. Berka, A. Rice, Langmuir 21 (2005) 1223. [54] L. Gmachowski, Colloids Surf. A 211 (2002) 197. [55] B.B. Mandelbrot, Phys. A 191 (1992) 95. [56] A. Lozsa´n, M. Garcı´a-Sucre, G. Urbina-Villalba, J. Colloid Interface Sci. 299 (2006) 366. [57] M. Salou, B. Siffert, A. Jada, Colloids Surf. A 142 (1998) 9. [58] V.J. Anderson, H.N.W. Lekkerkerker, Nature 416 (2002) 811.