Colloidal Aggregation Induced by Long Range Attractions - Langmuir

Langmuir 2004, 20, 9861-9867

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Colloidal Aggregation Induced by Long Range Attractions Antonio M. Puertas,*,† Antonio Fernańdez-Barbero,† F. Javier de las Nieves,† and Luis F. Rull‡ Group of Complex Fluids Physics, Department of Applied Physics, University of Almerıá, 04120 Almerıá, Spain, and Departamento de Fı´sica Atomıća, Molecular y Nuclear, AÄ rea de Fı´sica Teo´ rica, Universidad de Sevilla, Aptdo 1065, Sevilla 41080, Spain Received May 31, 2004. In Final Form: August 2, 2004 The structure of colloidal clusters formed by long-range attractive interactions under diluted conditions is studied by means of Monte Carlo simulations. For a not-too-long attraction range, clusters show selfsimilar internal structure with lower density than that typical for diffusive aggregation. For long-range interactions, low κ, nonfractal clusters are formed (dense at short scales but open at long ones). The dependence on the volume fraction shows that more-compact clusters are grown the higher the colloidal density for diffusive aggregation and attraction-driven aggregation in the fractal regime. The whole trend is explained in terms of the interpenetration among aggregates. In attraction-driven aggregations, the interpenetration of clusters competes with aggregation in the tips of the clusters, causing low-density clusters.

I. Introduction Aggregation in colloids is a phenomenon with no analogue in other systems (simple liquids or granular materials) and is caused by a strong short-range attraction.1,2 Despite the fact that it has received increasing attention over the past few years, a complete theory is not available yet.3 It has long been known that cluster structure is self-similar and can be described using fractal geometry.4 Two universal aggregation regimes become apparent depending on the limiting factor for the aggregation: diffusion-limited cluster aggregation (DLCA) and reaction-limited cluster aggregation (RLCA).5-11 For DLCA, the aggregation is led by cluster diffusion, showing a fractal dimension in the range df ≈1.7-1.8. Particles under RLCA repel each other, growing more-compact aggregates with a limit fractal dimension of df ≈ 2.0-2.1. A continuous transition between both regimes has also been reported, e.g., charged systems as a function of the range of the interaction.12-14 The colloid volume fraction * Author to whom correspondence should be addressed. E-mail: [email protected]. † University of Almeria. ‡ Universidad de Sevilla. (1) Elimelech, M.; Gregory, J.; Jia, X.; Williams, R. Particle Deposition and Aggregation; Butterworth Heinemann, Ltd: Oxford, 1995. (2) Sonntag, H.; Strenge, K. Coagulation Kinetics and Structure formation; Plenum Press: New York, 1987. (3) Kroy, K.; Cates, M. E.; Poon, W. C. K. Phys. Rev. Lett. 2004, 92, 148302. (4) Vicsek, T. Fractal Growth Phenomena; World Scientific: Singapore, 1989. (5) Weitz, D. A.; Oliveria, M. Phys. Rev. Lett. 1984, 52, 1433. (6) Weitz, D. A.; Huang, J. S.; Lin, M. Y.; Sung, J. Phys. Rev. Lett. 1985, 54, 1416. (7) Lin, M. Y.; Lindsay, H. M.; Weitz, D. A.; Ball, R. C.; Klein, R.; Meakin, P. Proc. R. Soc. London A 1989, 423, 71. (8) Lin, M. Y.; Lindsay, H. M.; Weitz, D. A.; Ball, R. C.; Klein, R.; Meakin, P. Phys. Rev. A 1990, 41, 2005. (9) Carpineti, M.; Ferri, F.; Giglio, M. Phys. Rev. A 1990, 42, 7347. (10) Gonza´lez, A. E. Phys. Rev. Lett. 1993, 71, 2248. (11) Poon, W. C. K.; Haw, M. D. Adv. Colloid Interface Sci. 1997, 73, 71. (12) Asnaghi, D.; Carpineti, M.; Giglio, M.; Sozzi, M. Phys. Rev. A 1992, 45, 1018. (13) Odriozola, G.; Tirado-Miranda, M.; Schmitt, A.; Lope´z, F. M.; Callejas-Fernańdez, J.; Martıńez-Garcıá, R.; Hidalgo-Alvarez, R. J. Colloid Interface Sci. 2001, 90, 240. (14) Kim, A. Y.; Berg, J. C. Langmuir 2000, 16, 2101.

is another key parameter influencing the aggregation, both cluster morphology and system kinetics. Aggregation in concentrated suspensions yields more-compact clusters in two and three dimensions than those obtained from lower volume fractions.15-19 Experiments have recently shown that aggregates grown in a 50/50 mixture of positive and negative particles show low-density fractal structures compared to those obtained from the universal regimes, df ≈1.2-1.4.20-22 Nevertheless, the mechanism leading to this structure has not been established yet; it is not clear whether the low density is caused by the attractive interactions or by the interplay of attractions and repulsions.21 The aim of this work is to establish if clusters growing only under attractive interactions reproduce the essentials of those formed under both interactions. It is not straightforward to find experimental systems with such long-ranged attractions; therefore, we have used computer simulations to study aggregation driven by longrange attractions. Long-range attractive hydrophobic interactions have been observed,23,25 but their origin is still a matter of debate24 and their range is not easily tuned. Also, magnetic colloids present long-range attractive interactions, but they also present a strong angular dependence. Aggregation in magnetic systems has indeed shown lower fractal dimension than DLCA when no (15) Dietler, G.; Aubert, C.; Cannell, D. S.; Wiltzius, P. Phys. Rev. Lett. 1986, 57, 3117. (16) Ferri, F.; Frisken, B. J.; Cannell, D. S. Phys. Rev. Lett. 1991, 67, 3626. (17) Lach-hab, M.; Gonza´lez, A. E.; Blastein-Barojas, E. Phys. Rev. E 1996, 54, 5456. (18) Gonza´lez, A. E.; Martıńez-Lo´pez, F.; Moncho-Jorda´, A.; HidalgoA Ä lvarez, R. J. Colloid Interface Sci. 2002, 246, 227. (19) Gonza´lez, A. E.; Martıńez-Lo´pez, F.; Moncho-Jorda´, A.; HidalgoA Ä lvarez, R. Physica A 2002, 314, 235. (20) Puertas, A. M.; Fernańdez-Barbero, A.; de las Nieves, F. J. J. Chem. Phys. 2001, 115, 5662. (21) Kim, A. Y.; Hauch, K. D.; Berg, J. C.; Martin, J. E.; Anderson, R. A. J. Colloid Interface Sci. 2003, 260, 149. (22) Puertas, A. M.; Fernańdez-Barbero, A.; de las Nieves, F. J. Colloid Interface Sci. 2003, 265, 36. (23) Craig, V. S. J.; Ninham, B. W.; Pashley, R. M. Langmuir 1998, 14, 3326. (24) Senden, T. J. Curr. Opin. Colloid Interface Sci. 2001, 6, 95. (25) Eriksson, J. C.; Ljunggren, S.; Claesson, P. M. J. Chem. Soc., Faraday Trans. 2 1989, 85, 163.

10.1021/la0486640 CCC: $27.50 © 2004 American Chemical Society Published on Web 09/21/2004

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external magnetic field is applied, and quasi-linear clusters form when an external magnetic field is present. However, these clusters have low-density structures, probably because the interaction is nonisotropic. In this work, we use the Monte Carlo technique to simulate a system of hard spheres with an attractive interaction, similar to the electrostatic interaction between colloidal particles of opposite charge. In such a way, we isolate the effect of the attractions in a simple system. The structure of the clusters formed in the simulations is studied by their pair distribution function and by the radius of gyration of the aggregates. The aggregates are shown to have lower fractal dimension than that of DLCA, implying that the fractal dimensions in charge heteroaggregation are caused by the attractive interactions. At extremely low κa, nonfractal aggregates are formed with open structures made of compact clusters of particles. The effect of the volume fraction is studied next. The fractal dimension increases with increasing colloid concentration, caused by the interpenetration of clusters. An interpenetration parameter is defined that successfully accounts for the increase of the fractal dimension with volume fraction in diffusive aggregation. For attractive aggregation, the clusters are very open and interpentration is more important than in the case of diffusive aggregation. In the next section, we explain the computational method used in this work and give the details of the simulation. In Section III, the results are presented; first, the structure of clusters formed in attractive aggregation is studied, and the effects of the interaction range and colloid concentration is discussed afterward. Finally, we present the relevant conclusions of this study. II. Computational Method and Details Monte Carlo (MC) simulations in the canonical ensemble were used to obtain the clusters in a system of monodisperse hard spheres. To mimic the short-range van der Waals interaction, which causes colloidal aggregation, a narrow square well of infinite depth was added to the potential in the particle surface. The width of this well was set to 0.1a, where a is the particle radius. Different values of the well width have been studied, and no effect on the cluster structure was found for narrow enough wells. The long-range attraction was modeled by an exponential potential:

V(r) ) V0 exp{-κ(r - 2a)}

(1)

where κ sets the interaction range and V0 ) - 5kBT. This potential has the form derived by Derjaguin and Landau28 and Verwey and Overbeek29 for the electrostatic interaction between similarly charged particles, with κ being the inverse Debye length. In our case, V0 is negative, yielding an attractive interaction, which mimics the attraction between oppositely charged particles.1,30 The value of V0 has been selected low enough (in absolute value) to prevent ballistic aggregation. Diffusive aggregation is studied with V0 ) 0 (the square well being present in all cases). In this work, the unit length will be the particle radius, a, and kBT will be the unit of energy. The density will be reported as volume fraction, φc ) 4/3 πa3F, where F is the number density. The simulation box is cubic, with the size ranging from L ) 355a to L ) 234a, depending on the volume fraction, and with periodic (26) Helgesen, G.; Skjeltorp, A. T.; Mors, P. M.; Botet, R.; Jullien, R. Phys. Rev. Lett. 1988, 61, 1736. (27) Morimoto, H.; Maekawa, T. Int. J. Modern Phys. B 2001, 15, 912. (28) Derjaguin, B. V.; Landau, L. Acta Physicochim. (USRR) 1941, 14, 633. (29) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (30) Hogg, R.; Healy, T. W.; Fuersteanu, D. W. Trans. Faraday Soc. 1966, 62, 1638.

Figure 1. Normalized number of clusters during the simulation for diffusive aggregation (black lines) and attractive interaction with κa ) 1 (grey lines). From top to bottom, total number of clusters (dashed lines) and clusters with one, two, three, four, and five particles (solid lines). boundary conditions. The number of particles is N ) 10 648 in all cases studied in this work. During the simulation, particles or clusters perform standard MC steps of maximal length, ∆. When a particle falls in the well of another one, an irreversible bond is established between them, and thereafter, they belong to the same cluster. Two clusters aggregate when a particle of one cluster forms a bond with a particle of the other cluster. A cluster is moved with the same procedure as single particles, and no restructuring ocurrs, i.e., internal motions are not allowed. The maximal displacement of clusters is set to ∆ ) 0.25a, low enough to prevent hopping of particles, which would lead to artificial filling of voids inside the aggregates. Because the maximal displacement of clusters is so small and the density is very low, the ratio of accepted movements is very close to 1 throughout the simulation. For each Monte Carlo cycle, every cluster randomly attempts a displacement (single particles are clusters with just one particle). Simulations start with N single particles at random biased positions, forcing the center-to-center distance to be larger than 3a. The number of clusters is monitored during the simulation, as well as the energy, the acceptance ratio, and other parameters to be dicussed below. The evolution of the number of the smaller clusters and the total number of clusters is presented in Figure 1 as a function of the number of cycles for a diffusive aggregation and an attractive aggregation (κa ) 1), both of them at a volume fraction φc ) 0.0025. In all cases, the curves follow the usual bell-shaped behavior for aggregation kinetics; light species aggregate to form big clusters, the velocity of this reaction being proportional to the concentrations of the aggregating clusters. Due to the attractive interaction, the evolution of the clusters in a diffusive aggregation (black lines) is much slower than that of the attractive aggregation (grey lines). The goal of our work is the study of the inner structure of the aggregates. For this purpose, we construct the pair distribution function, g(r), of the system when big clusters have formed. According to fractal theory, g(r) for a fractal cluster follows4

g(r) ≈ rdf-dexp{-r/ξ}

(2)

where df and d are the fractal dimension and the dimension of the space (in our case d ) 3), respectively, and ξ is a correlation length, which can be identified with the cluster size for diluted clusters.15,16 Therefore, only the inner structure of clusters is fractal, and the power-law behavior, g(r) ≈ rdf-d, is expected only well inside the aggregate, but for distances larger than a few particle radii. At long distances, larger than the cluster size, g(r) must be equal to 1. Alternatively, the fractal dimension of the clusters can be extracted from the dependency of the cluster size on the number of particles in the aggregate. As a measure of the size, the radius

Colloidal Aggregation Induced by Long-Range Attractions

Figure 2. Pair distribution function of the system for different stages of aggregation at φc ) 0.0025 and κa ) 0.5. From left to right, the total number of clusters in the system is: 1024, 512, 256, 128, 64, and 32. The gray line is a power law with an exponent of -1.35, corresponding to a fractal dimension of df ) 1.65.

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Figure 3. Radius of gyration of the aggregates as a function of the number of particles in the cluster for diffusive aggregation (open circles) and attractive aggregation with κa ) 0.5 (filled circles). In both cases, φc ) 0.0025.

The first aim of this work is to study the structure of the clusters formed under attractive interactions and to compare it with that of the clusters obtained in aggregation of oppositely charged particles. In charge heteroaggregation, it was observed that the inner structure of the clusters was indeed fractal and the dimension characterizing it was lower than that of DLCA.20-22 The pair distribution function in an attraction-driven aggregation is presented in Figure 2 at different stages of the aggregation for a volume fraction φc ) 0.0025 and a potential range κa ) 0.5. At early stages, the clusters are too small to exhibit fractal structure and the behavior of g(r) is dominated by the surface of the cluster. The power-law behavior becomes apparent only well inside the aggregate, implying that only the inner structure is self-similar. The oscillations in the pair-distribution function at short distances are caused by the finite size of the particles, and the minimum in g(r) for distances bigger than the aggregate size reflects that the clusters

grow by depleting their surroundings and leads to the well-known low-angle peak in the structure factor.33-35 The fractal dimension of the clusters in this aggregation is df ) 1.65, as indicated in the figure, lower than the typical values for diffusive aggregation, reproducing the results observed in charge heteroaggregation. We have tested this result calculating the fractal dimension of the aggregates by means of their radius of gyration. The size of the aggregates produced throughout four different simulations is plotted in Figure 3 vs the number of particles of the aggregate (only clusters with more than 50 particles are considered). The linear trends in the log-log plot indicate again the fractal structure of these objects, and the fractal dimensions calculated from the slopes are fully consistent with the DLCA result and the previous determination from the pair distribution function (df ) 1.665 ( 0.025 and df ) 1.794 ( 0.024, for attractive and diffusive aggregation, respectively). These results show that attractive interactions lead to colloidal clusters with lower density than DLCA. We may thus conclude that the more-open structures that are observed in aggregation of oppositely charged systems are caused by the attractive interactions and not because of the interplay of attractions and repulsions. Lower fractal dimensions in attraction-driven aggregations and charge heteroaggregation are obtained because clusters aggregate in their tips since the attraction makes them touch as soon as they are within the interaction range. Big spaces are thus left unfilled inside the newborn cluster, causing branched structures with low fractal dimension. A. Effect of the Interaction Range. We study now the dependency of the fractal dimension on the range of the interaction. At high κa, the attraction is expected to have no effect and the aggregation proceeds in the DLCA regime; a transition between attraction-driven aggregation and DLCA is expected. For a long interaction range, the aggregation will be dominated by the attraction and, eventually, an attraction-driven regime can be reached. The inner structure of the aggregates is analyzed for different values of κa in Figure 4, where the pair distribution function, g(r) is multiplied by r3-df, where df is fitted for every value of κa using eq 2. The fractal structure is recognized as a horizontal trace in the plot.

(31) Sorensen, C. M.; Roberts, G. C. J. Colloid Interface Sci. 1997, 186, 447. (32) Tirado-Miranda, M.; Schmitt, A.; Callejas-Fernańdez, J.; Fernańdez-Barbero, A. Langmuir 2000, 16, 7541.

(33) Carpineti, M.; Giglio, M. Phys. Rev. Lett. 1992, 68, 3327. (34) Carpineti, M.; Giglio, M. Phys. Rev. Lett. 1993, 70, 3828. (35) Gonza´lez, A. E.; Ramı´rez-Santiago, G. Phys. Rev. Lett. 1995, 74, 1238.

of gyration of the aggregate, Rg, is usually chosen, and for a fractal object, it follows4,31

Rg ) R0n1/df

(3)

where R0 is a constant of the order of the particle radius31,32 and n is the number of particles in the aggregate. The fractal dimensions reported in this work are calculated from the pair distribution function of the system when only 32 clusters remain in the system, implying an average of 333 particles per aggregate, with cluster sizes much smaller than the simulation box. Although the simulation may be made to proceed until only one cluster is formed, this is an unphysical situation and may induce undesirable effects in the structure of the final aggregate. In some cases, we also report the fractal dimension from the radius of gyration, calculated for aggregates with more than 50 particles, where the internal structure is indeed uniform and a reliable fractal dimension can be obtained. The results for the fractal dimension reported in the next section are averages over four different simulations, and the errors are the statistical deviation in the averaging.

III. Results and Discussion

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Figure 4. Pair distribution function, g(r), multiplied by r3-df for different values of κa ) 0.1563, 0.1953, 0.2441, 0.3815, 0.5960, 09313, 1.4552, 2.2737, and 3.5527 from top to bottom. The vertical dashed line marks where fitting starts. φc ) 2.5 × 10-3. The slanted dashed lines are r3-df for every value of κa.

Figure 5. Fractal dimension of the clusters as a function of the range of the interaction: κa fitting expression 2 (black circles), fitting a simple power law (grey circles), and by means of the radius of gyration (open circles). The horizontal solid line is the diffusive value and the dashed lines mark its estimated error. φc ) 2.5 × 10-3.

At intermediate and high κa, fractal structures are indeed observed in the inner parts of the aggregates, whereas the clusters formed within the long-range interaction range are not self-similar. The fractal dimension used in Figure 4 is presented in Figure 5 as a function of κa (filled circles). At intermediate κa, the fractal dimension grows until it reaches a plateau at high κa, with a value typical of diffusive aggregation. We must report, though, that the attraction still speeds up the aggregation, and it is much faster than diffusive

Puertas et al.

Figure 6. Snapshots of the systems at a late stage of aggregation for κa ) 0.1563 (a) and DLCA (b).

aggregation (see Figure 1). A similar decoupling between aggregation kinetics and cluster structure was observed in charge heteroaggregation.20 At low κa, however, the parameter df increases again with decreasing κa. The increase in df corresponds to the values of κa where non-self-similar aggregates were observed, in Figure 4. The value of df, thus, is not a real fractal dimension but a dimension characterizing the structure at very short distances. At short scales, the structure is rather compact, whereas at longer distances, the aggregate is more open, leaving voids (as observed by the slope of g(r) in Figure 4). Figure 6 shows a snapshot of the system at a late stage of aggregation for an attraction-driven aggregation with low κ (a) and for diffusive aggregation (b). It can be seen in the figure that, indeed, the structure of the low-κ aggregation is not self-similar; the clusters are formed by thick stems and branches. These high-density cores can be formed at the beginning of the simulation, when the long-range attraction causes quasi-ballistic aggregation, forming high-density structures.36 The system is then depleted of particles and the ratio of the range the interaction to the mean cluster distance decreases, leading to an increase of the effect of diffusion on aggregation. It is important to note, though, that these nonfractal clusters are formed at extremely low κ, κa < 0.3, which, for a 1:1 electrolyte, corresponds to a concentration below 10-3mM (taking a ) 100 nm). The results of the fractal dimension have been tested by determining df from the radius of gyration of the clusters, included in Figure 5 as open circles. The fractal dimensions so obtained agree with the fractal dimensions from the pair distribution function for intermediate and (36) Meakin, P. Phys. Rev. A 1984, 29, 997.


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Figure 7. Fractal dimension of the clusters as a function of the range of the interaction, κa, for φc ) 10-3 (circles), φc ) 1.75 × 10-3 (squares), φc ) 2.5 × 10-3 (diamonds), and φc ) 3.5 × 10-3 (triangles). The horizontal lines mark the diffusive fractal dimension for φc ) 10-3 (black line) and φc ) 3.5 × 10-3 (grey line).

Figure 8. Fractal dimension of the clusters as a function of the volume fraction, φc, for different interaction ranges: κa ) 0.4768 (black circles), κa ) 0.7450 (open squares), κa ) 1.1642 (black diamonds), κa ) 3.552 (leftwards triangles), and diffusive aggregation (grey circles).

high κa. However, at long interaction range, low κa, important differences are noticed between the two methods. If a simple power law is used in the determination of df, good agreement is found at all κa with the fractal dimensions from the radius of gyration, within the numerical accuracy (grey circles in the figure). Using the power law, one assumes that the internal structure of the aggregates is fractal at all κa, as one would interpret from the analysis of the radius of gyration. The value of df at low κa is an average dimension but not an actual fractal dimension. B. Volume Fraction Effects. We will study next the structure of the clusters formed in aggregations at different volume fractions in the low-density range: φc ) 10-3-5 × 10-3. The pair distribution functions show that the structure of the aggregates is fractal for κa larger than ∼0.3 at all volume fractions studied. Thus, we restrict our analysis to those values of κa. In Figure 7, the values of df are presented as a function of the interaction range, κa, for different volume fractions, calculating from a power-law fiffing to g(r). The fractal dimension decreases continuously with decreasing κ and φc. The DLCA regime is reached at the same κa for all volume fractions, κa ≈1.5, although the fractal dimension of DLCA is shifted upward as the volume fraction raises. This effect has been reported for diffusive aggregations in two and three dimensions15-19 and is observed not only in the cluster morphology but also in the aggregation kinetics. The effect of the volume fraction is dealt with in Figure 8 by means of iso-κ lines. For high κ, the interaction is screened and the fractal dimension is similar to that of DLCA, with a similar effect of φc. At all the ranges studied in the figure, the trends of the fractal dimension are roughly parallel, although the absolute values are different. At low κa, where the structure of the clusters is not fractal, the value of df obtained by fitting expression 2 also increases with the volume fraction (not shown in the figure), as well as the average df obtained from fitting a simple power law or the radius of gyration. The increase in this case is much steeper. Because the volume-fraction dependence of the fractal dimension in attraction-driven aggregation is similar to that of DLCA, we will study the latter in more detail. This

trend is attributed to the interpenetration of clusters as they grow.18,19 The clusters, being ramified, leave spaces between their branches or sub-branches, where other clusters or branches of other clusters can be present. Aggregation of these clusters would result in a morecompact one. This effect is more important the higher the volume fraction, resulting in an increase of the fractal dimension with φc. An interpenatration parameter is defined in order to test this possibility. Two radial densities are calculated from the center of mass of every cluster; one of them, Fin(r), measures the density of particles belonging to the cluster, whereas the other, Fout(r), accounts for the particles of other clusters. The single-cluster interpenetration parameter, φ, is thus defined as

φ)

a2 F

∫0∞ Fin(r)Fout(r)dr

(4)

which adimensionally measures the overlap between both densities, i.e., the interpenetration of clusters (F is the overall number density of particles). For isolated (nonpenetrating) clusters, φ ) 0, whereas φ grows proportional to the volume of the cluster (thus growing with time) for fully interpenetrated clusters. In Figure 9, we present the normalized densities and their product, as well as its integral (whose limit at r f ∞ is φ/F) for an attractiondriven aggregation at φc ) 0.0025. For presentation purposes, the curves in this figure have been averaged for all of the aggregates produced during the simulation of sizes between 50 and 100 particles. The final value for the interpenetration parameter in the whole system, Φ, will be: Φ ) 〈〈φ〉〉, where the brackets indicate double averaging. At every cycle, the average of the single cluster interpenetration parameter is performed over all of the aggregates in the system. Second, this parameter is averaged over a number of cycles (100 in our case) to get a smooth curve and to allow for comparison between different states. In this way, Φ is obtained throughout the simulation at intervals given by the second averaging. In Figure 10, this parameter is presented for different volume fractions in diffusive aggregation as a function of the number of aggregates in the simulation. This type of presentation was chosen instead of as a function of the number of cycles to permit comparison between aggregations with different growth rates (see Figure 1).

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Figure 9. Normalized densities of particles averaged for clusters from an attraction-driven aggregation of sizes between 50 and 100: Fin(r) (dotted line), Fout(r) (dashed line), Fin(r)Fout(r) (grey line), and the integral of the latter (black line). The limit r f ∞ is φ/F.

Figure 10. Interpenetration parameter, Φ, as a function of the total number of clusters in the system for diffusive aggregation at: φc ) 10-3 (solid black line), φc ) 2.5 × 10-3 (solid gray line), and φc ) 4.8 × 10-3 (dashed black line).

In the figure, the interpenetration parameter has a slightly decreasing trend as the aggregation evolves since the clusters merge but the lonely ones survive. However, it can be seen that the values of Φ grow with the volume fraction, though the data is very noisy. This trend indicates that the clusters are more interpenetrated at higher densities than at lower ones, which confirms the prediction by Gonza´lez et al.18,19 We have then rationalized the behavior of diffusive aggregation, as well as the observations by other authors.17-19 We study now the interpenetration of clusters in attraction-driven aggregation. Since these aggregates are more open than those formed in DLCA, the effect of cluster interpenetration is expected to be larger. It must be remembered, however, that the decrease in the fractal dimension is attributed to the clusters aggregating by their tips, leaving big voids inside them. In Figure 11, Φ is presented as a function of the total number of aggregates for different values of κa and for two extreme volume fractions: φc ) 10-3 and φc ) 3.5 × 10-3. In all cases, Φ decreases as the aggregation proceeds, in the same way as the diffusive aggregation data in Figure 10 (note that the Φ-range is much larger in the DLCA case). As expected, the interpenetration is more important for highly branched clusters that form at low κa. The data

Puertas et al.

Figure 11. Interpenetration parameter, Φ, as a function of the total number of clusters in the system for attractive aggregations for two volume fractions: φc ) 10-3 (black lines) and φc ) 3.5 × 10-3 (grey lines) and with different values of κ. From top to bottom, κa ) 0.3052, κa ) 0.7451, and DLCA (dashed lines).

at the higher κa is very similar to the diffusive values, as predicted from the behavior of df in Figure 8. At longer interaction ranges, when the clusters formed are not fractal, the interpenetration parameter is even higher, implying that the clusters are more interpenetrated in this case than in diffusive aggregation. At this low κ, the effect of the volume fraction is more important than at higher κ or in diffusive aggregation, as noticed by the differences in Figure 11 between the volume fractions studied, still with the same trend. Thus, two opposing factors control the aggregation in attractive systems: aggregation between the tips and interpenetration of the clusters. Whereas the former tends to form open structures, the latter is promoted in a system with such clusters and causes more compact aggregates. In addition to this complex situation, at low κ, the aggregates are not fractal but are formed by compact branches, though they are open at large scales. The role of interpenetration in these regime is not clear: Φ is larger than in the case of DLCA, reflecting overlapping on the large scales. Finally, it is important to note that this interpenetration mechanism can account for the difference between experimental fractal dimensions and simulated ones in attractive aggregation since the experiments are done at much lower densities than the simulations; experimentally, fractal dimensions as low as df ) 1.36 ( 0.0320 or 1.21 ( 0.1521 have been measured, with volume fractions on the order of φc ≈ 10-5. IV. Summary and Conclusions The structure of clusters formed in aggregation of colloidal systems induced by attractive interactions has been studied by means of Monte Carlo simulations. The clusters are shown to exhibit internal fractal structures for not-too-long interaction ranges, characterized by a fractal dimension which is lower than that typical in DLCA. Thus, we may conclude that the structure of the clusters formed in aggregation between oppositely charged colloids, which is less dense than the structure of DLCA aggregates, is caused mainly by the attractive interactions and not by the interplay of repulsions and attractions. The behavior of the fractal dimension as a function of the interaction range shows that df decreases with κ, but for too-low κ, the structure is no longer self-similar. The


Langmuir, Vol. 20, No. 22, 2004 9867

lowest fractal dimensions reported in this work are ∼1.6, whereas DLCA yields df ≈ 1.8. The clusters formed in the low-κ regime show thick high-density branches forming an open structure which might be fractal in longer scales. The apparent fractal dimension, determined from powerlaw fitting of g(r) or the radius of gyration describes a minimum with κa, reflecting the short scale compactness. This regime is to be found for extremely low salt concentrations, below 10-3 mM for a 1:1 electrolyte and a ) 100 nm. The fractal dimension increases with increasing volume fraction, in a similar fashion for DLCA, and attractive aggregation in the fractal-clusters regime. This behavior is rationalized in terms of an interpenetration parameter, which quantifies the overlapping of aggregates. The aggregates grow interpenetrating with each other, this effect being increased for higher colloidal density. In attraction-driven aggregation, a similar mechanism is found, but since the aggregates are more open in this case, the overlapping is more important. Aggregation can be

viewed as a competition between the clustering process, which leads to ramified structures with large void spaces inside the aggregates, and interpenetration of clusters in their growing, which tends to fill the voids inside each other. It would be encouraging to find these low fractal dimensions in a purely attractive system with tunable interaction range. It is predicted here that the fractal dimension decreases for longer attraction ranges until a nonfractality regime is found at extremely long ranges (on the order of several particle radii). Acknowledgment. This work was partially supported by the DGCYT under projects No. MAT2001-2767 and BQU2001-3615-C02-2. We wish to thank A. E. Gonza´lez for bringing to our attention the importance of the interpenetration. LA0486640

Colloidal Aggregation Induced by Long Range Attractions - Langmuir

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