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Fractal Aggregation: Scaling of Fractal Dimension with Stability Ratio Anthony Y. Kim and John C. Berg* Department of Chemical Engineering, University of Washington, Box 351750, Seattle, Washington 98195-1750 Received June 29, 1999. In Final Form: October 28, 1999 A scaling relation between fractal dimension and stability ratio is demonstrated for a charged polystyrene colloid undergoing aggregation. The stability ratio, i.e., the inverse collision efficiency, or inverse sticking probability, is systematically decreased by reducing the electrostatic barrier to aggregation. Fractal dimensions are measured by static light scattering, while stability ratios are evaluated from early-time aggregation kinetics (t < 10t1/2) using dynamic light scattering. The fractal dimension for the model hydrophobic colloid investigated increases monotonically with stability ratio, reflecting a continuous increase in particle packing density as the sticking probability is reduced.
Introduction Aggregation of colloidal materials determines the success of a variety of industrial processes, including the making of water-based paint coatings, wastewater clarification, and membrane filter production. There is, therefore, practical as well as fundamental interest in understanding the relationship between aggregation kinetics and structure. Since the development of fractal geometry1 and the discovery that aggregation of “iron smoke” produces fractal aggregates,2 there has been considerable theoretical and experimental study of fractal aggregation. A qualitative connection between fractal dimension (Df) and aggregation kinetics has been recognized for about a decade. Simulations3,4 and experimental studies5-7 indicate that fast aggregation, or diffusionlimited cluster aggregation (DLCA), produces aggregates with Df ) 1.75-1.8, while slow aggregation, or reactionlimited cluster aggregation (RLCA), gives Df ) 2.0-2.1. More recently, intermediate fractal dimensions have been observed for intermediate aggregation rates. Zhou and Chu8 investigated aggregation of a 39 nm diameter polystyrene at initial number density No ) 1.5 × 1012 cm-3 and found that the fractal dimension steadily decreased from 2.15 to 1.72 as NaCl concentration was increased from 0.3 to 1.5 M. Although aggregation rate constants were not reported, their plots of hydrodynamic radius versus time clearly suggested that the aggregation rate increased with salt concentration, at least for the range 0.3-0.75 M. Unfortunately, the extremely short half-life for their system (t1/2 ) 3µ/4kTN0 ) 0.1 s) precluded measurement of fast aggregation kinetics at higher salt concentrations. Burns et al.9 studied aggregation of a * To whom correspondence should be addressed. (1) Mandelbrot, B. B. Fractals: Form, Chance, and Dimension; W. H. Freeman and Company: San Francisco, CA, 1977. (2) Forrest, S. R.; Witten, T. A. J. Phys. A 1979, 12, L109. (3) Botet, R.; Kolb, M.; Jullien, R. In Physics of Finely Divided Matter; Boccara, N., Daoud, M., Eds.; Springer-Verlag: New York, 1985; p 231. (4) Meakin, P. In Annual Review of Physical Chemistry; Strauss, H. L., Babcock, G. T., Moore, C. B., Eds.; Annual Reviews: Palo Alto, CA, 1988; Vol. 39, p 237. (5) Majolino, D.; Mallamace, F.; Migliardo, P.; Micali, N.; Vasi, C. Phys. Rev. A 1989, 40, 4665. (6) Martin, J. E. Phys. Rev. A 1987, 36, 3415. (7) Martin, J. E.; Wilcoxon, J. P.; Schaefer, D.; Odinek, J. Phys. Rev. A 1990, 41, 4379. (8) Zhou, Z.; Chu, B. J. Colloid Interface Sci. 1991, 143, 356.
larger, 330 nm, diameter polystyrene at lower No ((1.1 to 3.7) × 109 cm-3). The half-life for this system was about 100 s, opening the possibility of measuring the early-time kinetics for fast aggregation. In fact, for the highest salt concentration, they measured a mean diameter over 10 µm during the first minutes of reaction. In light of a numerical integration of the Smoluchowski equations,10 a 30-fold increase in diameter over the course of a few half-lives is too large to be explained by diffusion-limited aggregation alone. Although intermediate fractal dimensions have been observed for aggregation kinetics between DLCA and RLCA, the aggregation kinetics have not been adequately quantified, particularly for DLCA. A scaling relationship between fractal dimension and aggregation kinetics encompassing the theoretical limits of DLCA and RLCA remains to be established. The goal of this work is to quantify the early-time aggregation kinetics (t < 10t1/2) for a range of conditions, including DLCA, in terms of the stability ratio, W, and to identify a quantitative relation, if any, between fractal dimension and W. Aggregation kinetics are often quantified in terms of the stability ratio, defined here as the ratio of the rate constant for Smoluchowski (diffusion-limited) aggregation to the experimentally determined rate constant for doublet formation, k11: W ) kSmol/k11, where kSmol ) 8kT/3µ. A theoretical W can also be calculated by integration of an assumed total interaction potential (usually consisting of attractive van der Waals and repulsive electrostatic components). The stability ratio may be considered as the inverse collision efficiency, or inverse sticking probability, for two colliding particles to permanently stick together on contact. The mass fractal dimension of an aggregate is the exponent characterizing the change in mass, M, with linear dimension, R, according to M ∝ RDf, where R is the distance from a suitable point within the aggregate to the surface of a circumscribing sphere and Df is less than the dimension of space (Df < 3).11,12 Further reference to a (9) Burns, J. L.; Yan, Y.; Jameson, G. J.; Biggs, S. Langmuir 1997, 13, 6413. (10) Cametti, C.; Codastefano, P.; Tartaglia, P. Phys. Rev. A 1987, 36, 4916. (11) Martin, J. M.; Hurd, A. J. J. Appl. Cryst. 1987, 20, 61. (12) Hiemenz, P. C.; Rajagopalan, R. Principles of Colloid and Surface Chemistry, 3rd ed.; Marcel Dekker: New York, 1997; Chapter 1.
10.1021/la990841n CCC: $19.00 © 2000 American Chemical Society Published on Web 12/16/1999
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fractal dimension will imply mass fractal dimension. The fractal scaling relation applies over a limited range of R, approximately defined by the radius of the primary particle, r, and the radius of the aggregate, Ragg. The average density of a fractal aggregate scales as RDf-3, resulting in a decreasing density for increasing distance from the center of the aggregate. The fractal dimension thus measures the openness of the aggregate, with low Df corresponding to an open architecture. Experimentally, the fractal dimension of an ensemble of aggregates is determined from the slope of the scattered light intensity I(Q) versus the magnitude of the scattering vector (Q) on log-log coordinates, I(Q) ∝ Q-Df, where Q ) (4πn/λ0) sin(θ/2), n ) refractive index of the solvent, θ ) scattering angle, and λ0 ) incident wavelength. Q-1, having units of length, is the “yardstick” of the experiment.13 The fractal scaling of the scattered light intensity requires that Ragg > Q-1 > r0, where r0 is the primary particle radius. In other words, Q must probe length scales between Ragg and r0 in order to neglect scattering from the surfaces of the aggregate and primary particles. Under these conditions, I(Q) measures the interparticle structure factor11,14 or alternatively the intraaggregate structure factor.13 Experiment The polystyrene latex (IDC 1-100, batch 629,1) was supplied by Interfacial Dynamics Corp., Portland, OR. According to the manufacturer, the mean diameter is 98 ( 5.3 nm (1 standard deviation), and charge density 1.2 µC/cm2 (13 nm2/SO4). IDC 1-100 was chosen for this study, because it is the smallest latex prepared without surfactant by the supplier. Latexes less than about 100 nm in diameter contain sodium dodecyl sulfate at approximately the critical micelle concentration. Preliminary work with a 23 nm diameter IDC 1-20 indicated that this latex partially aggregates upon dilution. Sample cells were of borosilicate glass, 27 mm in diameter, with Teflon-lined screw caps. Cells were cleaned using sulfuric acid and rinsed with 0.2 µm filtered, deionized, doubly distilled water. All solutions and suspensions were prepared using 0.2 µm filtered water and under continuously flowing, filtered air. KNO3 solutions were filtered again (0.2 µm) prior to use. D2O (99.9%) was obtained from Cambridge Isotope Laboratories. It was found to be dust-free and did not require filtration. For each experiment, IDC 1-100 suspension, D2O, and water were mixed in the sample cell and pre-equilibrated in the temperature-controlled cell holder. D2O was used to match the density of the final salt solution with that of polystyrene. Density matching eliminated any orthokinetic effects due to sedimentation. It also allowed repeated static light scattering measurements to be taken on relatively large aggregates without loss of signal. Aggregation was induced by adding KNO3 solution to the latex, followed by gentle swirling. Primary particle number density was 3 × 109 cm-3, and KNO3 concentration was varied from 0.05 to 1.5 M. The temperature for all experiments was 25 ( 0.5 °C. Dynamic light scattering was used to follow the hydrodynamic radius during the initial stages of aggregation. At later stages, after the aggregates had grown larger than about 2 µm in radius, static light scattering data were collected to evaluate the mass fractal dimension. Both dynamic and static measurements were taken using a Brookhaven Instruments 72-channel model BI2030 correlator, BI-200SM goniometer, BI-DS photomultiplier, and Spectra Physics model 124B He-Ne laser operating at 633 nm and 22 mW. An 85 mm diameter temperature-controlled vat held the sample cell on the goniometer. Alignment of the system was checked by measuring scattered light intensities from a Rayleigh scatterer (Decalin) over 9 to 105° scattering angle. Error in scattered intensity, after correction for scattering volume, was less than 2% relative to the intensity at 90° scattering angle. Prior to each experiment, the index-matching fluid, Decalin, was (13) ref 12, Chapter 5. (14) Teixeira, J. J. Appl. Crystallogr. 1988, 21, 781.
Kim and Berg
Figure 1. Hydrodynamic diameter of polystyrene aggregates (100 nm primary particle diameter) as a function of time and KNO3 concentration. The aggregation rate increases with increasing salt concentration. No increase is observed beyond the rapid aggregation condition (about 0.3 M). filtered of dust by pumping it through a 0.2 µm filter. Dynamic light scattering measurements were taken at 90° scattering angle, and mean diameters were calculated by the method of cumulants.13 Static scattering data were analyzed over scattering angles of 9-45°, corresponding to a Q range of 2.1 × 10-3 to 1.0 × 10-2 nm-1. Scattered intensities were corrected for dark counts, dead time, and scattering volume.
Results and Discussion Aggregation kinetics, as quantified by the change in hydrodynamic particle size, provide an indirect measure of the net interparticle interactions. Figure 1 shows the early-time kinetics (t < 10t1/2) as a function of salt concentration. At 50 mM KNO3, the mean diameter increased slowly. Electrostatic repulsive forces opposing particle-particle contact play a dominant role in limiting the rate of aggregation; only a small fraction of the particles have thermal energies sufficient to overcome the electrostatic barrier. As salt concentration is increased, the height of the barrier is reduced, and the aggregation rate increases. At a concentration of 300 mM, the electrostatic barrier was effectively removed (Debye screening length, κ-1 ) 1.8 nm), and the aggregation rate reached a maximum. Further increases in salt concentration had little effect on the aggregation rate, since the attractive van der Waals and hydrophobic forces are but weakly affected by the intervening electrolyte.15 Stability ratios were determined from the slope of the mean hydrodynamic particle radius with time using the method of Virden and Berg.16 Slopes were calculated using the first 200 s (3t1/2), except at 50 and 70 mM KNO3. At these two concentrations, the relatively small increase in aggregate size required that the slopes be calculated over a longer period (600 s, or 10t1/2). Viscosity was corrected for D2O concentration using a mass-averaged viscosity based on the amount of H2O and D2O. Viscosity was also corrected for KNO3 concentration assuming the same functional dependence as water on KNO3 concentration. The functional dependence was obtained by correlation of literature data.17 Table 1 summarizes the measured stability ratios, and Figure 2 shows the stability ratio as a function of KNO3 concentration. The shape of the curve is typical of colloids stabilized by electrostatic repulsions.18 (15) Israelachvili, J. Intermolecular and Surface Forces, 2nd ed., Academic Press: San Diego, CA, 1992. (16) Virden, J. W.; Berg, J. C. J. Colloid Interface Sci. 1992, 149, 528. (17) Handbook of Chemistry and Physics, 78th ed.; Lide, D. R., Ed.; CRC Press: Boca Raton, FL, 1997; Chapter 8.
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Figure 2. Stability ratio of 100 nm diameter polystyrene particles as a function of KNO3 concentration. The CCC is 160 mM.
Figure 3. Static scattering curves for polystyrene aggregates at three salt concentrations. Fractal dimension (slope of curve) increases as salt concentration is decreased.
Table 1. Measured Stability Ratio as a Function of Salt Concentration
Table 2. Fractal Dimension as a Function of Stability Ratio
KNO3 (M)
av stability ratio
no. of measmts
1.5 0.93 0.75 0.5 0.3 0.15 0.1 0.08 0.07 0.05
4.4 4.2 4.6 5.3 4.8 6.6 8.1 7.7 20.0 28.5
3 1 1 1 2 1 3 1 1 2
3 std devs 0.8
0.475
As the salt concentration is increased, the aggregation rate increases (W decreases), and above a critical salt concentration (the critical coagulation concentration, CCC), the aggregation rate approaches a constant value. The CCC for this latex was found to be 160 mM. At high salt, where the electrostatic repulsive barrier is removed, the aggregation rate is termed “rapid”. For particles as small as those of the present study, any influence of shear fields is negligible. In the absence of viscous drainage effects, rapid aggregation is determined by Brownian motion alone, and theoretical W (incorporating any weak van der Waals interaction) would be close to unity. Experiment yields Wrapid ≈ 4.5, a value explained, in part, by drainage effects as the particles approach one another. Spielman has shown that the stability ratio for Brownian coagulation can be as high as 10 when viscous interactions are taken into account, with the highest factors associated with low Hamaker constant, high Stern potential, and thick double layers.19 In this study, the Hamaker constant is about 2.0 × 10-20 J, and the surface potentials are estimated to vary from -4 to -23 mV depending on KNO3 concentration. For polystyrene colloids, one may also expect an attractive hydrophobic contribution to the total interaction potential, partially offsetting the viscous term. Any remaining difference between theoretical and experimental Wrapid may be due to systematic underestimation of k11, since the slopes were calculated for times longer than 0.5t1/2.16 Nonetheless, the stability ratio as determined provides a useful means to quantify the aggregation kinetics and is a convenient indicator for the effective interparticle potential and inverse sticking probability. (18) Prieve, D. C.; Ruckenstein, E. J. Colloid Interface Sci. 1980, 73, 539. (19) Spielman, L. A. J. Colloid Interface Sci. 1970, 33, 562.
stability ratio
av fractal dimension
no. of measmts
3 std devs
4.1 4.2 4.6 5.3 4.7 4.9 8.0 8.3 7.7 20.0 29.9 27.1
1.64 1.72 1.65 1.68 1.74 1.68 1.87 1.8 1.81 1.98 1.92 2.01
3 3 3 6 3 5 5 5 4 6 3 5
0.24 0.22 0.18 0.09 0.11 0.09 0.15 0.2 0.07 0.17 0.15 0.1
The fractal dimension, as determined from the slope of scattered light intensity vs Q, quantifies the openness of the aggregate, with low fractal dimensions associated with an open structure. Since intensity is a function of aggregate size and mass concentration, care must be excercised in using the slope to calculate the fractal dimension, particularly when using a goniometer system to collect the scattered light. Matching the density of the solvent with that of the latex minimized sedimentation, so that the latex concentration could be taken as a constant. For a given latex concentration, intensity increases with time as aggregation proceeds. Intensity becomes independent of time once the aggregates have grown large enough such that QRagg . 1.11 Figure 3 shows static scattering curves obtained for aggregates of the 100 nm latex at three salt concentrations. Angular scans were collected after Ragg reached about 2 µm, and each scan required 5 min to collect. Repeated measurements indicated no change in light intensities within experimental error. For each set of data, the line shown indicates the best power-law fit, with the negative slope giving the fractal dimension. Rapid aggregation (300 mM KNO3 for this polystyrene) yielded open aggregates with Df ) 1.7, in good agreement with other investigations of polystyrene5,8,20 and the DLCA model.4,5 The fractal dimension was found to increase continuously as the salt concentration was decreased. For slow aggregation (50 mM KNO3), dense aggregates formed with Df ) 2.0, consistent with other studies5,20 and the RLCA model.4,5 An increase in fractal dimension for decreasing salt was also observed by Zhou and Chu.8 They considered (1) changes in the height of the potential energy (20) Asnaghi, D.; Carpineti, M.; Giglio, M.; Sozzi, M. Phys. Rev. A 1992, 45, 1018.
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Figure 4. Fractal dimension as a function of stability ratio. Fractal dimension follows the scaling law: Df ) 1.45 + 0.373 log10(W).
barrier and resultant sticking probabilities and (2) the possibility of restructuring due to a finite energy minimum,21 concluding that mechanism 2 was most likely active in their system. In this work, mechanism 1 is explored further by measuring the sticking probability in terms of W and identifying a relation, if any, between W and Df. In fact, the fractal dimension increased systematically with increasing stability ratio. Table 2 summarizes the results, and Figure 4 plots the average fractal dimension as a function of stability ratio. Error bars ((3 standard deviations) are indicated for 50, 80, and 300 mM KNO3. Empirically, one finds
Df ) 1.45 + 0.373 log10(W)
(1)
as shown by the dashed line in Figure 4. As explained earlier in this section, a combination of factors (hydrophobic and viscous interactions, systematic underestimation of k11) result in W > 1 for rapid aggregation. Therefore, eq 1 should be viewed as a demonstration of the scaling relationship between fractal dimension and stability ratio, rather than as a general equation. To the authors' knowledge, this is the first quantitative demonstration of the link between kinetics and structure for a range of conditions encompassing DLCA and RLCA. Future work may prove this relationship to be applicable for similar
systems (100 nm diameter hydrophobic colloid, electrostatically stabilized, undergoing irreversible aggregation in water) when W is evaluated from the early-time kinetics (t < 10t1/2). However, if W is evaluated for t > 10t1/2, it will be high in value, and eq 1 will overpredict the fractal dimension. For example, a crude analysis of Figure 6 in Zhou and Chu, where t . 10t1/2, indicates W ) 800-1000 for the 40 nm diameter latex at 300 mM NaCl. Equation 1 predicts Df ) 2.5-2.6, while the experimental value is 2.1.8 It should also be noted that eq 1 is based on aggregation in the absence of sedimentation effects. Differential sedimentation, if significant, would impose a shearing effect and cause restructuring to more dense aggregates. An inherent assumption of correlating Df with W is that the early-time kinetics dictate the aggregate structure at later times. This is most likely the case for irreversible aggregation, where the depth of the primary minimum is much greater than kT, and restructuring does not occur. Polystyrene latexes with low surface charge density are perhaps the most suitable colloids for studying irreversible aggregation. It would be desirable to investigate a range of surface charge densities to determine if W is, in fact, the common currency for fractal aggregation. The empirical form of eq 1 admits Df values less than 1.75 (DLCA), particularly for W < 1. In fact, stability ratios less than 1 are expected in certain situations (for example, strong van der Waals interactions18 or aggregation between oppositely charged species22). More experimental work is necessary to refine the relationship between stability ratio and fractal dimension. Conclusions A scaling relation between stability ratio and fractal dimension was established for an electrostatically stabilized polystyrene latex undergoing irreversible aggregation. For the first time, the link between kinetics and structure for conditions encompassing DLCA and RLCA was quantified. The fractal dimension increases monotonically with stability ratio, reflecting a continuous increase in packing density as the sticking probability is decreased. A logarithmic equation provides a useful correlation of fractal dimension with stability ratio. Acknowledgment. This work was supported by the Center for Surfaces, Polymers, and Colloids at the University of Washington. LA990841N
(21) Shih, W. Y.; Aksay, I. A.; Kikuchi, R. Phys. Rev. A 1987, 36, 5015.
(22) Sunkel, J. M.; Berg, J. C. J. Colloid Interface Sci. 1996, 179, 618.
