Dependence of Aggregate Strength, Structure ... - ACS Publications

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Dependence of Aggregate Strength, Structure, and Light Scattering Properties on Primary Particle Size under Turbulent Conditions in Stirred Tank Lyonel Ehrl, Miroslav Soos, and Massimo Morbidelli* Institute for Chemical and Bioengineering, Department of Chemistry and Applied Biosciences, ETH Zurich, 8093 Zurich, Switzerland ReceiVed October 17, 2007. In Final Form: December 5, 2007 The steady-state size and structure of aggregates produced under turbulent conditions in stirred tank, for primary particle diameter, dp, equal to 420 nm and 120 nm, were studied experimentally for various values of the volume average shear rate, 〈G〉, and solid volume fraction, φ, and compared with data for dp ) 810 nm. To exclusively investigate the effect of dp, polystyrene latexes with same type and similar density of surface charge groups (sulfate) were used. The mass fractal dimension, df, obtained by image analysis, was found to be invariant of dp and 〈G〉, with a value equal to 2.64 ( 0.18. Small-angle static light scattering was used to characterize the cluster mass distributions by means of the root-mean-square radius of gyration, 〈Rg〉, and the zero-angle intensity of scattered light, I(0), whose steady-state values proved to be fully reversible with respect to 〈G〉. The absolute values of 〈Rg〉 obtained for similar φ and 〈G〉 proved to be independent of dp, and for all studied conditions, 〈Rg〉 was proportional to 〈G〉-1/2. At very low φ, a critical aggregate size for breakage was obtained and used to evaluate the aggregate cohesive force, as a characteristic for the aggregate strength. The aggregate cohesive force was found to be independent of aggregate size, with similar values for the investigated dp. Due to large dp and high df, the effect of multiple light scattering within the aggregates was found to be present, and by relating the scaling of 〈Rg〉 with I(0) to df, the corresponding correction factors were evaluated. By combination of the independently measured aggregate size and structure, it is possible to experimentally determine the relation between the maximum stable aggregate mass and the hydrodynamic stresses independent of the multiple light scattering present for large dp and compact aggregates.

1. Introduction Many solid-liquid separation processes, such as settlement, flotation, filtration, or water removal, are preceded by an aggregation step to improve the solid-phase characteristics for better removal.1 Coagulation or flocculation are traditionally used when the dispersed phase comprises particles in the colloidal size range, e.g., in the postprocessing of polymer latexes prepared by emulsion polymerization and in purification and wastewater treatment.2 If properly controlled, aggregation can be used to produce aggregates with well-defined size distribution and morphology3 and to provide desired characteristics of powders, such as flowability or the avoidance of fines, as it is often requested in the polymer and paint industries. In order to achieve this, it is necessary to understand and quantify the physical mechanisms which control the aggregation process. Studies on the aggregation of colloidal systems typically include the destabilization of a stable dispersion of primary particles through the addition of a coagulant, so as to fully screen the repulsive forces between the primary particles and, therefore, to initiate the aggregation process (diffusion-limited cluster aggregation (DLCA)).3-21 Depending on the physical properties of * Corresponding author. E-mail: [email protected]; Phone: +41-44-63-23033; Fax: +41-44-63-21082. (1) Letterman, R. D.; Amirtharajah, A.; O’Melia, C. R. Coagulation and Flocculation. In Water Quality and Treatment-A Handbook of Community Water Supplies, 5th ed.; Letterman, R. D., Ed.; McGraw-Hill: New York, 1999; pp 6.1-6.66. (2) Elimelech, M.; Gregory, J.; Jia, X.; Williams, R. A. Particle Deposition and Aggregation-Measurement, Modelling and Simulation; Butterworth-Heinemann: Woburn, 1998. (3) Flesch, J. C.; Spicer, P. T.; Pratsinis, S. E. AIChE J. 1999, 45, 1114-1124. (4) Reich, I.; Vold, R. D. J. Phys. Chem. 1959, 63, 1497-1501. (5) Kusters, K. A. The Influence of Turbulence on Aggregation of Small Particles in Agitated Vessel. Ph.D. Thesis, Eindhoven University of Technology, Netherlands, Eindhoven, 1991.

the dispersed particles and the suspending fluid, the aggregation process can be driven by several mechanisms, such as Brownian motion,22-24 shear,25-27 or differential settling.28 We note that due to the vast body of literature on these topics the references (6) Oles, V. J. Colloid Interface Sci. 1992, 154, 351-358. (7) Spicer, P. T.; Keller, W.; Pratsinis, S. E. J. Colloid Interface Sci. 1996, 184, 112-122. (8) Spicer, P. T.; Pratsinis, S. E. Water Res. 1996, 30, 1049-1056. (9) Kusters, K. A.; Wijers, J. G.; Thoenes, D. Chem. Eng. Sci. 1997, 52, 107-121. (10) Serra, T.; Colomer, J.; Casamitjana, X. J. Colloid Interface Sci. 1997, 187, 466-473. (11) Spicer, P. T.; Pratsinis, S. E.; Raper, J. A.; Amal, R.; Bushell, G.; Meesters, G. Powder Technol. 1998, 97, 26-34. (12) Ducoste, J. J.; Clark, M. M. EnViron. Eng. Sci. 1998, 15, 215-224. (13) Thill, A.; Lambert, S.; Moustier, S.; Ginestet, P.; Audic, J. M.; Bottero, J. Y. J. Colloid Interface Sci. 2000, 228, 386-392. (14) Biggs, C. A.; Lant, P. A. Water Res. 2000, 34, 2542-2550. (15) Selomulya, C.; Amal, R.; Bushell, G.; Waite, T. D. J. Colloid Interface Sci. 2001, 236, 67-77. (16) Bouyer, D.; Line´, A.; Cockx, A.; Do-Quang, Z. Chem. Eng. Res. Des. 2001, 79, 1017-1024. (17) Selomulya, C.; Bushell, G.; Amal, R.; Waite, T. D. Langmuir 2002, 18, 1974-1984. (18) Bouyer, D.; Line´, A.; Do-Quang, Z. AIChE J. 2004, 50, 2064-2081. (19) Wang, L.; Vigil, R. D.; Fox, R. O. J. Colloid Interface Sci. 2005, 285, 167-178. (20) Bouyer, D.; Coufort, C.; Line´, A.; Do-Quang, Z. J. Colloid Interface Sci. 2005, 292, 413-428. (21) Coufort, C.; Bouyer, D.; Line´, A. Chem. Eng. Sci. 2005, 60, 2179-2192. (22) Lin, M.; Lindsay, H. M.; Weitz, D. A.; Ball, R. C.; Klein, R.; Meakin, P. J. Phys.: Condens. Matter 1990, 2, 3093-3113. (23) Wu, H.; Lattuada, M.; Sandku¨hler, P.; Sefcik, J.; Morbidelli, M. Langmuir 2003, 19, 10710-10718. (24) Lattuada, M.; Wu, H.; Sandku¨hler, P.; Sefcik, J.; Morbidelli, M. Chem. Eng. Sci. 2004, 59, 1783-1793. (25) von Smoluchowski, M. Z. Phys. Chem. 1917, 92, 129-168. (26) Saffman, P. G.; Turner, J. S. J. Fluid Mech. 1956, 1, 16-30. (27) Waldner, M. H.; Sefcik, J.; Soos, M.; Morbidelli, M. Powder Tech. 2005, 156, 226-234. (28) Friedlander, S. K., Smoke, Dust and Haze, 1st ed.; John Wiley & Sons: New York, 1977.

10.1021/la7032302 CCC: $40.75 © 2008 American Chemical Society Published on Web 02/27/2008

The Steady-State Size and Structure of Aggregates

do not claim completeness. Especially, when referring to shear aggregation, the focus is mainly on work obtained under welldeveloped turbulent conditions. Shear driven aggregation typically results in a sigmoid-like evolution of the aggregates size, due to a self-accelerating growth, followed by a steadily slowing relaxation to a steady-state aggregate size.3-10,12,14-17,19,27,29 It was found that the initial time evolution of the cluster mass distribution (CMD), if plotted in terms of a certain characteristic size and in appropriate dimensionless time, follows a single curve.30,31 In this phase, the process is controlled by pure aggregation, with the value of the mass fractal dimension, df, around 1.9.32 After this initial phase of self-accelerating aggregate growth, the kinetics gradually slows down as breakage sets in and eventually reaches the steady state. This steady state predominantly depends on the size and concentration as well as surface and bulk properties of the primary particles, composition of the liquid phase and type of coagulant, and average value and distribution of the shear rate in the vessel. For systems in which coagulation is induced solely by screening the electrostatic double-layer repulsion through addition of nonprecipitating salts, it was shown that the steady state is controlled by a balance between aggregation and breakage, with df ≈ 2.6, independent of the applied history of solid volume fraction30 or shear rate.5,31 There exists only little work on the effect of primary particle size on the aggregates characteristics, and the change in particle size is often accompanied by a change in the surface chemistry and/or solid volume fraction, impeding a direct comparison. Kusters5 coagulated polystyrene particles with diameters equal to 560, 880, and 1100 nm, where the solid volume fraction and surface chemistry varied in all cases. Serra et al.10 obtained comparable aggregate sizes using polystyrene particles with diameters equal to 2 and 5 µm and same solid volume fraction. Selomulya et al.17 studied the aggregation of polystyrene particles with varying surface charge density and diameters equal to 810, 380, and 60 nm and obtained comparable results for the first two latexes (whereas for the 60 nm latex the solid volume fraction differed from that of the former ones and steady state was not reached until the end of the experiment). Ivanauskas et al.33 and Muhle and Domasch34 used glass beads with diameters ranging from 2 to 8 µm and obtained varying relations of the aggregate size on the primary particle diameter, depending on the hydrodynamic conditions. The goal of this work is to investigate the effect of primary particle size on the steady-state aggregate size and structure for various values of the shear rate (via both batch experiments and dynamic experiments with step changes in the stirring speed31) and solid volume fraction (using continuous dilution30,31). For this, two polystyrene latexes with similar surface chemistry, so that the primary particle diameter, dp, is the only changing parameter, were aggregated under turbulent conditions in a stirred tank, and the results are compared to previous data obtained under identical conditions for dp ) 810 nm.31 The CMD has been characterized by two independent quantities, namely, the root-mean-square (rms) radius of gyration (29) Wang, L.; Marchisio, D. L.; Vigil, R. D.; Fox, R. O. J. Colloid Interface Sci. 2005, 282, 380-396. (30) Moussa, A. S.; Soos, M.; Sefcik, J.; Morbidelli, M. Langmuir 2007, 23, 1664-1673. (31) Soos, M.; Moussa, A. S.; Ehrl, L.; Sefcik, J.; Wu, H.; Morbidelli, M. J. Colloid Interface Sci., 2008, 319, 577-589. (32) Soos, M.; Sefcik, J.; Morbidelli, M. Ind. Eng. Chem. Res. 2007, 46, 17091720. (33) Ivanauskas, A.; Neesse, T.; Seifert, G.; Graichen, K. Chem. Tech. 1987, 39, 60-63. (34) Muhle, K.; Domasch, K. Chem. Eng. Process. 1991, 29, 1-8.

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and the zero-angle intensity of scattered light obtained by smallangle static light scattering (SASLS). Soos et al.31 showed that for primary particles with an average size comparable to or larger than the laser wavelength, or for systems with a considerable amount of small nonfractal aggregates, using the power-law region of the structure factor obtained by SASLS measurements can lead to an erroneous characterization of the aggregates structure. Therefore, the aggregates structure was independently characterized by image analysis of two-dimensional images of aggregates obtained by confocal microscopy.31 Since for primary particles with a diameter equal to 810 nm, destabilized by a strong electrolyte, complete reversibility of the steady-state characteristics with respect to shear rate was observed,31 this effect was also investigated for smaller sizes of primary particles (dp equal to 420 nm and 120 nm). The aggregate cohesive force, as a characteristic for the aggregate strength, was evaluated, considering a balance between the hydrodynamic forces exerted on the aggregates causing their rupture and the cohesive forces strengthening the aggregates.35-39 To estimate these hydrodynamic forces, knowledge of the maximum stable aggregate size which cannot be further broken by the flow is necessary. Since the steady-state aggregate size results from a balance between aggregation and breakage, this maximum stable (or critical) aggregate size has to be obtained under conditions in which aggregation can be safely neglected.30 The obtained values for the aggregate cohesive force are compared with available data in the literature,40-48 and the effect of primary particle size on the aggregate cohesive force is discussed. From the critical aggregate size for breakage and the independently measured aggregate structure, a critical aggregate mass for breakage is evaluated, and its dependency on the hydrodynamic stresses, which can serve as a breakage criterion when modeling aggregation kinetics, is compared with experimental49,50 and numerical data.51-53 Finally, the effect of multiple light scattering on the obtained scaling of the steady-state properties of the CMD, due to large primary particle sizes and dense aggregate structure, is discussed, and experimentally obtained correction factors to the Rayleigh-Debye-Gans (RDG) theory are given. 2. Materials, Methodology, and Measurement Methods 2.1. Materials and Methodology. In the experiments two white sulfate polystyrene latexes were used, both supplied by Interfacial Dynamics Corporation (Portland, OR) (product no. 1-400, coefficient of variation ) 4.0%, batch no. 1034,2, solids (%) ) 8.0, surface (35) Thomas, D. G. AIChE J. 1964, 10, 517-523. (36) Higashitani, K.; Inada, N.; Ochi, T. Colloids Surf. 1991, 56, 13-23. (37) Blaser, S. The hydrodynamical effect of vorticity and strain on the mechanical stability of flocs. Ph.D. Thesis, No. 12851, ETH Zurich, Switzerland, Zurich, 1998. (38) Kobayashi, M.; Adachi, Y.; Ooi, S. Langmuir 1999, 15, 4351-4356. (39) Coufort, C.; Line´, A. Chem. Eng. Res. Des. 2003, 81, 1206-1211. (40) Francois, R. J. Water Res. 1987, 21, 1023-1030. (41) Yeung, A. K. C.; Pelton, R. J. Colloid Interface Sci. 1996, 184, 579-585. (42) Boller, M.; Blaser, S. Water Sci. Technol. 1998, 37, 9-29. (43) Blaser, S. J. Colloid Interface Sci. 2000, 225, 273-284. (44) Blaser, S. Colloids Surf., A 2000, 166, 215-223. (45) Hodges, C. S.; Cleaver, J. A. S.; Ghadiri, M.; Jones, R.; Pollock, H. M. Langmuir 2002, 18, 5741-5748. (46) Hodges, C. S.; Looi, L.; Cleaver, J. A. S.; Ghadiri, M. Langmuir 2004, 20, 9571-9576. (47) Kobayashi, M. Colloids Surf., A 2004, 235, 73-78. (48) Kobayashi, M. Water Res. 2005, 39, 3273-3278. (49) Sonntag, R. C.; Russel, W. B. J. Colloid Interface Sci. 1986, 113, 399413. (50) Sonntag, R. C.; Russel, W. B. J. Colloid Interface Sci. 1987, 115, 390395. (51) Potanin, A. A. J. Colloid Interface Sci. 1993, 157, 399-410. (52) Higashitani, K.; Iimura, K.; Sanda, H. Chem. Eng. Sci. 2001, 56, 29272938. (53) Harada, S.; Tanaka, R.; Nogami, H.; Sawada, M. J. Colloid Interface Sci. 2006, 301, 123-129.

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Ehrl et al. Pe )

Figure 1. Schematic drawing of the experimental setup. charge density ) 4.8 µmC/cm2; product no. 1-100, coefficient of variation ) 8.4%, batch no. 1614,1, solids (%) ) 4.1, surface charge density ) 6.7 µmC/cm2). The mean particle diameter values of the two latexes were 420 and 120 nm, as measured by SASLS, which were in agreement with the particle sizes declared by the producer. The measured particle size distributions are very narrow, and the latexes can be considered as monodisperse. The latexes are chosen in a way to ensure similar surface chemistry (nature and number of surface charge groups per surface area) to that of the latex used in our previous work31 (product no. 1-800, dp ) 810 nm, coefficient of variation ) 2.0%, batch no. 642,4, solids (%) ) 8.1, surface charge density ) 5.2 µmC/cm2). Hence, the primary particle diameter is the only parameter that is different, and therefore, its effect can be studied independently from other phenomena that might be induced by a variation in the surface or bulk chemistry of the particles. All experiments were performed in a 2.5 L stirred tank coagulator, shown schematically in Figure 1, operated with stirring speeds ranging from 200 to 1073 rpm. More details about the geometry of the coagulation unit as well as the characterization of the flow field through computational fluid dynamics can be found in reports by Waldner et al.27 and Soos et al.,31 respectively. Each experiment consisted of two parts. In the first part, a suspension of primary particles was destabilized by a salt solution and aggregated under turbulent conditions until the system reached steady state with respect to the measured light scattering quantities. In the second part, the steady-state response of the system on either a variation of the shear rate or a variation of the solid volume fraction was investigated. The initial suspension of primary particles with a solid volume fraction equal to 2 × 10-5 for all primary particles sizes was prepared by diluting the original latex with an appropriate amount of deionized water and was subsequently pumped from a storage tank into the coagulator. An overflow tube was used as a small reservoir of about 100 mL in order to allow replenishing of the coagulator with liquid during sampling and to prevent air from entering the coagulator (see Figure 1). Then the aggregation process was started by adding a coagulant solution (30 mL of 20%, w/w, Al(NO3)3 in water) using a syringe (see Figure 1). The resulting salt concentration was well above the critical coagulation concentration for the given system, i.e., the electrostatic repulsive forces between the particles were fully screened, and the particles were completely destabilized (i.e., DLCA regime). In order to achieve a good reproducibility of the initial aggregation kinetics, for the salt injection, a programmable syringe pump, Vit-Fit (Lambda, Czech Republic), was used at maximum speed, corresponding to an injection time of approximately 15 s. The typical mixing time of the injected salt solution is about 5 s at 200 rpm (tested by dye measurements), which is an order of magnitude below the characteristic time of aggregation at all conditions considered in this work (for more details see Moussa et al.30). In order to estimate which aggregation mechanism determines the system behavior, we consider the Pećlet number, defined as the ratio of the timescale of diffusive transport over the timescale of convective transport caused by shear (for the given particle sizes and water as suspending phase settling effect can be neglected):

6πη〈G〉r3 kBT

(1)

where η is the dynamic viscosity, 〈G〉 is the volume average shear rate, r is the characteristic length scale (particle radius or aggregate radius), kB is the Boltzmann constant, and T is the absolute temperature. In the initial stage of the aggregation process, the primary particle Pećlet number will provide information on whether the aggregation will be determined by Brownian motion (Pe , 1) or by shear convection (Pe . 1). For T ) 298K and 〈G〉 ranging from 108 to 1353 s-1, the primary particle Pećlet numbers range from 15 to 183, from 2 to 26, and from 0.048 to 0.6, for primary particle diameter equal to 810, 420, and 120 nm, respectively. It can be seen that for larger primary particle diameters (810 nm and 420 nm) shear convection dominates the initial aggregation process over the entire range of shear rates applied and that for the smaller primary particle diameter (120 nm) and low shear rates Brownian motion governs the initial aggregation. On the other hand, due to the third power in r (using the aggregate radius), the Pećlet number, defined by eq 1, increases fast with aggregate growth, and for the steady-state aggregate sizes, which alltogether lie far above 1 µm, we have Pe . 1 and, hence, shear aggregation dominates. After the CMD reached the steady state, the second part of the experiment was carried out, in which the dynamic response of the system to either a stepwise variation of the stirring speed (shear experiment) or a continuous change of the solid volume fraction via dilution (dilution experiment) was investigated. For dilution experiments, a particle-free salt solution with the same salt concentration as that in the tank was pumped at a constant flow rate into the tank. In this way, the steady-state response of the system was investigated for solid volume fractions ranging from 2 × 10-5 down to below 1 × 10-6. 2.2. Small-Angle Static Light Scattering (SASLS) and Image Analysis. A SASLS instrument, Mastersizer 2000 (Malvern, U.K.), was used in all experiments for on-line characterization of the CMDs in the stirred tank in terms of the angle-dependent intensity of scattered light, which can be expressed as:54 I(q) ) I(0)P(q)S(q)

(2)

where I(0) is the zero-angle intensity of scattered light, P(q) is the form factor (due to primary particles), and S(q) is the structure factor (due to the arrangement of primary particles within the aggregates), where the scattering vector amplitude, q, is defined as: n q ) 4π sin(θ/2) λ

(3)

where θ is the scattering angle, n is the refractive index of the dispersing fluid, and λ is the laser wavelength in vacuum. Analysis of the measured scattered intensity, I(q), in the Guinier region (for qRg up to about unity) allows one to extract certain characteristic quantities of the CMD, namely, the rms radius of gyration, 〈Rg〉, and the zero-angle intensity, I(0), as described in detail by Moussa et al.30 Within the limits of the RDG theory (rigorously valid within the following constraints |m - 1| e 1 and (4πRp/λ)|m - 1| e 1, where m is the relative refractive index), I(0) scales with the second power of the scatterer mass and constitutes a second-order moment of the CMD.54-57 Outside the limits of the RDG theory, i.e., for primary particles of a size comparable to or larger than the laser wavelength58,59 or in the case of very dense aggregates, where multiple light scattering within the aggregate is (54) Sorensen, C. M. Aerosol Sci. Technol. 2001, 35, 648-687. (55) Kerker, M. The Scattering of Light and Other Electromagnetic Radiation; Academic Press: New York, 1969; Vol. 16, p 666, Chapter XV. (56) Farias, T. L.; Koÿlu¨, O ¨.U ¨ .; Carvalho, M. G. Appl. Opt. 1996, 35, 65606567. (57) Jones, A. R. Prog. Energy Combust. Sci. 1999, 25, 1-53. (58) Lambert, S.; Thill, A.; Ginestet, P.; Audic, J. M.; Bottero, J. Y. J. Colloid Interface Sci. 2000, 228, 379-385. (59) Tishkovets, V. P.; Petrova, E. V.; Jockers, K. J. Quant. Spectrosc. Radiat. Transfer 2004, 86, 241-265.


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present,58,60 the intensity of the forward scattered light will scale with the mass of the scatterer to a power smaller than 2, and therefore, I(0) will not anymore correspond to the second-order moment of the CMD. Further, for the scattering theory of fractal aggregates54 to be applicable, the aggregates size has to be large enough to exhibit a well developed fractal scaling.61 If the primary particle diameter is within the limits of the RDG theory and the obtained aggregates are large enough to exhibit a well developed fractal scaling, the mass fractal dimension of the aggregates, df, is an additional quantity, which can be obtained from the slope of the power-law region of the log-log plot of S(q). However, this is only explicitly valid within the limits of the RDG theory.54-57 For systems where the primary particle size is comparable to or larger than the laser wavelength,58,59 for dense aggregates, where multiple light scattering within the aggregates occurs,58,60 for small relative aggregate sizes,61 or for polydisperse populations of aggregates,62 the slope of the power-law region of the log-log plot of S(q) vs q cannot be directly interpreted as df. Therefore, we will denote this slope as the scaling exponent, SE, which is not necessarily equal to df. Due to the given reasons, the aggregate structure was independently charaterized using image analysis of two-dimensional images of aggregates, which were taken from samples withdrawn for off-line measurements at various operating conditions. In this way, the structure of aggregates can be characterized by the perimeter fractal dimension, dpf, obtained from the scaling of the projected surface area, A, vs the perimeter, P, of the binary images of aggregates,7,8,31,63,64 A ∝ P2/dpf

(4)

Subsequently, the perimeter fractal dimension was used to evaluate the mass fractal dimension according to the correlation presented by Lee and Kramer.65 Lee and Kramer applied a computational optical imaging analysis technique to populations of artifically generated aggregates with varying size and structure in order to derive a relation between their two-dimensional perimeter fractal dimension and their three-dimensional mass fractal dimension. They obtained df ) 4.6-1.628 × dpf with a Pearson product-moment correlation coefficient equal to 0.83 and a standard error of the estimate equal to 0.15, evaluated in the range of df ) [1.55, 2.45] and dpf ) [1.35, 1.8]. For the purpose of this work, this relation was extrapolated to the region of obtained dpf values. The authors are aware of further uncertainties that might be associated with an experimental imaging technique,66,67 but a quantification of such possible errors cannot be given. A detailed description of the experimental methodology, the used measurement techniques, and measurement data analysis can be found in Moussa et al.30 (dilution experiment) and Soos et al.31 (shear experiment, image analysis). 2.3. Characterization of Aggregate Strength by the Aggregate Cohesive Force. In a simplified manner, the breakup of an aggregate in liquid suspension can be conceived to occur when the hydrodynamic force, Fhyd, acting on the aggregate overcomes the aggregate cohesive force, Fcoh, holding the aggregate together, which is a combined result of, on one hand, the aggregate strength, the product of the attractive interparticle forces and the aggregate structure and, on the other hand, the aggregate size. In order to relate the aggregate cohesive force (extensive quantity) to the aggregate strength (intensive quantitiy), one requires knowledge on the breakup mode and a (60) Petrova, E. V.; Tishkovets, V. P.; Jockers, K. Solar Sysr. Res. 2004, 38, 309-324. (61) Lattuada, M.; Wu, H.; Morbidelli, M. J. Colloid Interface Sci. 2003, 268, 106-120. (62) Soos, M.; Sefcik, J.; Morbidelli, M. Chem. Eng. Sci. 2006, 61, 23492363. (63) Mandelbrot, B. B.; Passoja, D. E.; Paullay, A. J. Nature 1984, 308, 721722. (64) Serra, T.; Casamitjana, X. J. Colloid Interface Sci. 1998, 206, 505-511. (65) Lee, C.; Kramer, T. A. AdV. Colloid Interface Sci. 2004, 112, 49-57. (66) Baveye, P.; Boast, C. W.; Ogawa, S.; Parlange, J. Y.; Steenhuis, T. Water Res. 1998, 34, 2783-2796. (67) Chakraborti, R. K.; Atkinson, J. F. J. Water Supply: Res. Technol.s AQUA 2006, 55, 439-451.

structural model.34,35,68 We note that with our experimental setup we obtain information on extensive quantities only, and therefore, the issue of relating the aggregate cohesive force to the aggregate strength will not be treated in this work. The aggregate strength and, therefore, the aggregate cohesive force is affected by the material and size of the primary particles, their surface properties, the composition of the liquid phase and type of coagulant, and the aggregate structure. As the maximum stable aggregate size for the given hydrodynamic conditions is determined by the aggregate cohesive force, its knowledge is of fundamental importance. As in our previous work,31 we will evaluate the aggregate cohesive force via the maximum stable aggregate size that determines Fhyd.18,21,36,38,42,47,69 In the limit of observable aggregation kinetics, the steady-state CMD, and with it also the maximum aggregate size, results from the balance between aggregation and breakage and, therefore, depends on the solid volume fraction.30 For this reason, the aggregate cohesive force will be extracted from the critical aggregate size for breakage obtained at infinite dilution conditions, at which aggregation can be safely neglected and the aggregate size is controlled by breakage only.31 With the assumption that the maximum stable aggregate size results from a balance of forces acting on the aggregate, the criterion for breakage reads as: Fhyd > Fcoh

(5)

The aggregate sizes in all our experiments are comparable to or smaller than the Kolmogorov microscale. Therefore, we assume that breakage occurs within the viscous subrange. Further, due to the compact aggregate structure,31 the magnitude of the hydrodynamic forces comprising the force dipole pulling the aggregate apart can be approximated18,20,21 as the hydrodynamic force acting on the hemisphere of a sphere of comparable size in a simple shear flow39,70 5 2 Fvs hyd ) πηd G 8

(6)

where η is the dynamic viscosity, d is the characteristic size (which is assumed to be adequately represented by 2〈Rg〉 in the case of an aggregate), and G is the local shear rate.

3. Results In the presentation of the experimental results and the following discussion, we will focus on the steady-state response of destabilized suspensions of primary particles with varying diameter (120, 420, and 810 nm31) to variations in either the shear rate or the solid volume fraction investigated under turbulent conditions in a stirred tank. Information on average sizes and structural properties of the CMDs, evaluated from the intensity curve of scattered light as a function of the scattering wave vector amplitude, I(q), will serve as a basis of the discussion. The intensity curves are obtained by inloop SASLS measurements, and the evaluated properties are 〈Rg〉, I(0), and SE, where the mass fractal dimension, df, was additionally characterized by image analysis. 3.1. Dependency of Steady-State CMD Properties on Shear Rate and Solid Volume Fraction. To obtain steady-state properties of the CMD as a function of the shear rate, on one hand, one can run aggregation experiments at a fixed primary particle concentration and for various shear rates. On the other hand, if the process is fully reversible with respect to shear rate, where the steady-state values of the CMD are a result of equilibrium between aggregation and breakage, one can obtain the same equilibrium properties of the CMD by applying step changes in stirring speed. As this was observed for large primary particles (dp ≈ 1 µm),5,31 it is likely that similar behavior prevails (68) Firth, B. A.; Hunter, R. J. J. Colloid Interface Sci. 1976, 57, 248-256. (69) Tambo, N.; Hozumi, H. Water Res. 1979, 13, 421-427. (70) Blaser, S. Chem. Eng. Sci. 2002, 57, 515-526.

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Figure 2. Examples of the time evolution of (a) the rms radius of gyration, 〈Rg〉, and (b) the absolute zero-angle intensity, I(0) (in arbitrary units, au), during step changes of the volume average shear rate, 〈G〉, through a change in rotation speed measured for a solid volume fraction equal to 2 × 10-5 and a primary particle diameter of 420 nm. The volume average shear rate was changed in the following manner: (O,3) 〈G〉 ) 106-287-517-1097-106 s-1; (9) 〈G〉 ) 517 s-1. (O,3,9) Loop measurement; (solid left triangle) validation measurements (syringe).

for primary particle diameter equal to 420 and 120 nm when keeping the particle material, the nature and amount of the surface groups, and type and amount of the coagulant the same. An example of an experiment aiming to investigate this issue is shown in Figure 2 for a primary particle diameter equal to 420 nm. As can be seen, complete reversibility of the steady-state values, characterized by 〈Rg〉 and I(0), with respect to the variation of the shear rate was observed. Similar findings were made for the 120 nm primary particles (Figure SI1 in Supporting Information). Moussa et al.30 proposed a dynamic experiment to investigate the effect of solid volume fraction on the steady state, in which, using continuous dilution of the content of the tank, a critical size for breakage can be obtained. In particular, once the aggregation system reaches steady state, the content of the vessel is continuously diluted with a particle-free salt solution. To keep a uniform concentration profile in the tank during dilution as well as to ensure that a dynamic equilibrium between aggregation and breakage at steady state is achieved at every single instant of time, the dilution rate has to be selected in a way that the characteristic time of dilution is larger than both the characteristic time of aggregation and the characteristic time of macromixing, 30 i.e., dilution has to be the slowest of all processes. An example of the time evolution of two moment ratios of the CMD, 〈Rg〉 and I(0)/I(0)t)0, during a dilution experiment for dp ) 420 nm and 〈G〉 ) 106 s-1, where the dilution starts from steady-state conditions corresponding to an initial solid volume fraction, φ0,

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Figure 3. Example of the time evolution of (a) the rms radius of gyration, 〈Rg〉, and (b) the normalized zero-angle intensity, I(0)/ I(0)t)0, for a primary particle diameter of 420 nm during a dilution experiment measured for a volume average shear rate equal to 106 s-1 and an initial solid volume fraction equal to 2 × 10-5: (O) loop measurement, (solid left triangle) validation measurements (syringe).

equal to 2 × 10-5, is shown in Figure 3. Knowing the rate of dilution, one can calculate the actual solid volume fraction, φ, at any time. Since I(0) is proportional to the mass of the particles (represented by the solid volume fraction, φ) in the reactor, in order to compare results obtained at different values of φ, the value of I(0) is divided by the corresponding scattering signal φ of primary particles, I(0)t)0 ) I(0)t)0 × φ/φt)0, to obtain a φ , constiφ-normalized zero-angle intensity, I(0)φn ) I(0)/I(0)t)0 30,31 The rms radius of gyration tuting a moment ratio of the CMD. and the φ-normalized zero-angle intensity, already presented in Figure 3, are plotted as a function of the solid volume fraction in the vessel in Figure 4. It is seen that in the region of low solid volume fractions both 〈Rg〉 and I(0)φn start to level off and approach a certain plateau value. This clearly indicates the existence of a critical aggregate size below which the effect of breakage is negligible (as the aggregates cannot be further broken by the hydrodynamic stresses exerted on them, i.e., Fhyd ) Fcoh), with its value only depending on the applied shear rate.31 Similar findings were made for the 120 nm primary particles (Figure SI2 and SI3 in Supporting Information). 3.2. Steady-State Aggregates Structure. The aggregate structure is a key factor affecting the aggregation kinetics (as well as its steady-state size) and the scattering behavior of the aggregates. Therefore, we measure the dependency of the steadystate aggregates structure on applied operating conditions, i.e., for various solid volume fractions and shear rates. In Figure 5, a comparison of the structure factors versus qRg, obtained at φ ) 2 × 10-5 and at the end of the dilution experiment, is shown for dp equal to 120, 420, and 810 nm31 and 〈G〉 ) 106 s-1. As a first observation, one can see that the slopes of the structure


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Figure 4. (a) Root-mean-square radius of gyration, 〈Rg〉, and (b) φ-normalized zero-angle intensity, I(0)φn , as a function of the solid volume fraction, φ, for a primary particle diameter of 420 nm during a dilution experiment measured for a volume average shear rate equal to 106 s-1 and an initial solid volume fraction equal to 2 × 10-5.

factors at φ ) 2 × 10-5 and at the end of the dilution experiment coincide. As over the whole time period of dilution no change in SE is observed, according to our previous work,30 we can conclude that dilution does not affect the structure of the aggregates. Further, it is apparent that with increasing primary particle diameter the steady-state SE decreases from 2.55 for dp ) 120 nm to 2.25 for dp ) 810 nm. One possible explanation for this behavior of SE could be that for decreasing primary particles the resulting aggregates become more compact.17 Another explanation could be that for decreasing size of primary particles we are approaching the validity region of the RDG theory,54-57 where SE can be used to approximate df, if the aggregates contain enough primary particles to exhibit a scattering behavior close to that of fully developed fractal objects.61 The broadening of the power-law region (in q) from around one-half order of magnitude for dp ) 810 nm to around 2 orders of magnitude for dp ) 120 nm supports the second explanation. To distinguish between the two possible explanations for the decay in SE with increasing primary particle diameter and its variation with respect to shear rate, we obtained additional information about the aggregates structure using image analysis.31 In Figure 6 a-c, examples of confocal microscopy images of aggregates produced under similar experimental conditions (200 rpm, φ ) 2 × 10-5) for different sizes of primary particles are presented, and in Figure 6 d-f, aggregates formed under various shear conditions for one primary particle diameter (420 nm) are shown. It can be seen that independent of the shear rate or primary particle diameter the obtained aggregates are very similar, i.e., highly compact with slightly elongated shape.

Figure 5. Comparison of the steady-state structure factors at a solid volume fraction equal to 2 × 10-5 (symbols) and at the end of dilution experiment (solid lines) for primary particle diameter equal to (a) 120 nm, (b) 420 nm, and (c) 810 nm (from ref 31), where the dashed lines indicate the power-law scaling of S(q) measured for a volume average shear rate equal to 106 s-1.

The scaling of projected area vs perimeter of the aggregates, obtained by image analysis of approximately 50 images for each experiment, was used to characterize the structure of the aggregates. Results for four shear rates (106, 287, 517 , and 1097 s-1) and dp values equal to 120 and 420 nm are summarized in Figure 7. As can be seen, the data for dp values equal to 120 and 420 nm can be well approximated by a power-law scaling with a slope equal to 1.7 (in agreement with the data for 810 nm31). According to eq 4 this slope is equal to 2/dpf, and consequently the perimeter fractal dimension is approximately equal to 1.18 over the whole range of stirring speed and primary particle diameter investigated. This is in agreement with values of the perimeter fractal dimension measured by other authors for aggregates formed with polystyrene primary particles of different sizes.7,8,19,64,71

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Figure 6. (a-f) Images of aggregates composed of primary particles with different diameters for various values of the volume average shear rate obtained by confocal microscopy from samples taken at steady state with a solid volume fraction equal to 2 × 10-5. In particular, the images correspond to primary particles size equal to (a) 810 nm (from ref 31), (b) 420 nm, and (c) 120 nm at a volume average shear rate equal to 106 s-1 and for a primary particle diameter equal to 420 nm at volume average shear rate equal to (d) 287, (e) 517, and (f) 1097 s-1.

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Figure 8. Comparison between values of the scaling exponent, SE (symbols), obtained from the power-law region of the structure factor, S(q), with the range of fractal dimension, df (hatched bar), obtained by image analysis using perimeter fractal dimension and the relation developed by Lee and Kramer65 for a solid volume fraction equal to 2 × 10-5 and different primary particle diameters (810 nm from ref 31) against the volume average shear rate, 〈G〉.

Figure 7. Relationship between area and perimeter obtained by image analysis for a solid volume fraction equal to 2 × 10-5 and various values of the volume average shear rate for primary particle diameter equal to (a) 420 nm and (b) 120 nm: (B) 〈G〉 ) 106 s-1; (4) 〈G〉 ) 287 s-1; (solid left triangle) 〈G〉 ) 517 s-1; (0) 〈G〉 ) 1097 s-1.

the results obtained over the whole range of shear rate investigated in this work for dp equal to 120 and 420, and 810 nm.31 On the one hand, the steady-state values of df estimated from dpf stay rather constant and are equal to 2.64 ( 0.18 (hatched area in Figure 8), independent of primary particle diameter and shear rate, which are in agreement with available data published in the literature using various stirring devices.5,6,11,64 On the other hand, it can be seen that the values of SE for aggregates composed of primary particles with diameters equal to 810 and 420 nm are significantly smaller than the values of df obtained by image analysis over the entire range of stirring speed investigated. This suggests that the second hypothesis mentioned above is correct, i.e., for these primary particle diameters, the RDG theory is not anymore valid and the CMDs contain a large portion of small nonfractal aggregates, which leads to a decrease of the value of SE.31 For these systems, SE is not an appropriate estimate of df, and its use could lead to erroneous conclusions. This reasoning is further supported by the fact that similar values of SE and df are obtained for aggregates composed of primary particles with diameter equal to 120 nm. Another information that can be obtained by image analysis is the shape of the aggregates. To characterize this, we chose the aspect ratio of the aggregates, defined as the ratio between the major and minor axis of the best fitted ellipse. In Figure 9 is shown the aspect ratio distributions for three different primary particle diameters (120, 420, and 810 nm) obtained from aggregates produced at a shear rate of 106 s-1. As can be seen, in all cases, the shape of the aggregates is nonspherical, with aspect ratios covering the range from 1 to 2.6. This is in agreement with data published by Blaser,37,43,44 for aggregates generated from silica and styrene/acrylate copolymer primary particles as well as by Ivanauskas et al.72 for glass beads, yielding aspect ratios in the range from 1.6 to 2.86. We found that the aspect ratios for constant primary particle diameter and various shear rates were also similar (data not shown), which is in agreement with the data of our previous work31 and the results of Kobayashi et al.38

A comparison of the values of SE, obtained from S(q), with df, recalculated from the correlation between the dpf and df reported by Lee and Kramer,65 is summarized in Figure 8. This includes

(71) Soos, M.; Wang, L.; Fox, R. O.; Sefcik, J.; Morbidelli, M. J. Colloid Interface Sci. 2007, 307, 433-446. (72) Ivanauskas, A.; Domasch, K.; Neesse, T. Freiberg. Forschungsh. A 1985, 720, 47-62.


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Figure 10. Structure factor, S(q), for aggregates composed of primary particles with different diameters, i.e., (O) 120 nm, (solid left triangle) 420 nm, and (0) 810 nm (from ref 31) at a solid volume fraction equal to 2 × 10-5 and a volume average shear rate 106 s-1. The dashed line indicates the limit of the Guinier regime at qRg ) 1.

Figure 9. Comparison of the distributions of aggregates aspect ratio at steady state obtained from confocal microscopy images for a solid volume fraction equal to 2 × 10-5 and a volume average shear rate equal to 106 s-1 for various primary particle diameters: (a) 120 nm, (b) 420 nm, and (c) 810 nm (from ref 31).

4. Discussion 4.1. Scaling of Steady-State Aggregate Size with Shear Rate: Aggregate Cohesive Force. The rms radius of gyration, 〈Rg〉, used to characterize the aggregate size, was evaluated from the steady-state structure factors obtained by SASLS. As an example, in Figure 10 is plotted the steady-state structure factors (of Figure 5) vs q for CMDs produced from particles with dp equal to 120, 420, and 810 nm for batch experiments at 〈G〉 ) 106 s-1 and φ ) 2 × 10-5. It is seen that the Guinier regimes (the dashed line indicates its limit at qRg ) 1) of the structure factors are equal and, therefore, the obtained values of 〈Rg〉 are independent of the primary particle diameter. Similar observations were made for all other values of 〈G〉 investigated. In Figure 11a is plotted the steady-state values of the rms radius of gyration, 〈Rg〉, obtained from experiments with fixed stirring speed as well as by step variation of the stirring speed (Figure 2a), as a function

Figure 11. Scaling of the rms radius of gyration, 〈Rg〉, at (a) steady state for a solid volume fraction equal to 2 × 10-5 and (b) infinite dilution conditions for different primary particle diameters, i.e., (O) 120 nm, (solid left triangle) 420 nm, and (0) 810 nm (from ref 31) as a function of the volume average shear rate, 〈G〉. The solid line indicates a scaling with a slope equal to -1/2.

of the volume average shear rate at a solid volume fraction equal to 2 × 10-5 and primary particle diameter equal to 120, 420, and 810 nm.31 It is worth noting that the three latexes compared in this study exhibit similar surface chemistry so that, for the same solid volume fraction, the primary particle diameter is the only changing parameter. The solid line in Figure 11a indicates that the rms radius of gyration is proportional to the volume average

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Table 1. Volume Averaged Values of the Shear Rate, 〈G〉, and Size of the Kolmogorov Eddies, 〈η〉, in the Stirred Tank for Various Values of the Stirring Speed, N, Data Reported by Soos et al.31 N/rpm

〈G〉/(1/s)a

〈η〉/µmb

200 417 635 854 1073

108 325 613 958 1353

120.6 69.3 50.4 40.3 34.0

a Maximum shear rate values obtained from CFD simulation are equal to 70 × 〈G〉. b Minimum Kolmogorov microscale evaluated with computational fluid dynamics is approximately equal to 〈η〉/10, which is valid for all stirring speeds.

shear rate according to 〈Rg〉 ∝ 〈G〉-1/2, which is consistent with the scaling of the maximum stable aggregate size obtained under the assumption of a constant aggregate cohesive force independent of the aggregate size, as presented by Potanin.51 In this work, it was shown that for systems in which attractive forces between primary particles depend only on their radial distance, without any resistance to tangential displacements, i.e., no friction with respect to rolling and sliding, the force required to break an aggregate is independent of the aggregate size, hence called constant-force approach, and proportional to the attractive force between two primary particles. The use of a constant aggregate cohesive force in the breakup criterion, eq 5, and replacement of the hydrodynamic force by the rhs of eq 6 results in the following proportionality of the maximum stable aggregate size to the applied shear rate:18,20,21

dmax ∝ G-1/2

(7)

On comparing the measured scaling between 〈Rg〉 and 〈G〉 with data published in the literature,3,5,9,15,17,18,20,21,38,47,48 we found good agreement. Similar observation of an absolute aggregate size independent of primary particle diameter can be found in the work of other authors who studied the aggregation of polystyrene latexes10,17 and glass beads,33,34 for the size of the aggregates within the viscous subrange. The values of the critical aggregate size for breakage obtained from dilution experiments (Figures 3a and 4a) for φ f 0 are shown in Figure 11b. By comparing them to the corresponding steady-state values obtained for a solid volume fraction equal to 2 × 10-5 (Figure 11a), one notes that the absolute aggregate sizes are shifted in parallel to lower values for φ f 0, and therefore, the same mechanism controls the maximum stable aggregate size preserving the scaling with respect to the volume average shear rate. As in the case with infinite dilution condition, the aggregates reach a critical size at which they are not further broken by the hydrodynamic forces exerted on them.30 This critical size for breakage comprises information on the aggregate strength31 and, therefore, will be used in the following analysis. The scaling between the maximum stable aggregate size and the volume average shear rate is equal to -1/2, which is in agreement with the concept of soft aggregates proposed by Potanin,51 where the cohesive force of these soft aggregates is directly proportional to the attractive force between primary particles but independent of the aggregate size (constant-force approach). In the following, we consider the constant-force approach to be valid for the primary particle diameters investigated in this work. Further, from the comparison of the aggregate sizes (using 2〈Rg〉 from Figure 11b) with values of the corresponding Kolmogorov microscale (see

Table 1), one can see that under all conditions investigated in this work, the maximum size of aggregates is within the limit of the viscous subrange. Therefore, eq 6 is applicable for the systems investigated in this work and will be used to calculate the absolute value of the aggregate-size-independent aggregate cohesive force. Since mixing at these stirring conditions as well as breakage are very fast,49,50 we assume that every aggregate experiences at least once the highest value of the shear rate in the vessel, and therefore, the maximum size of the aggregates is controlled by the highest value of the shear rate in the vessel. By use of the highest value of the shear rate and the aggregate diameter (2〈Rg〉) obtained by infinite dilution,31 the aggregate cohesive forces are equal to 3.5 ( 1 nN and 6.1 ( 1.5 nN for dp equal to 420 and 120 nm, respectively (and 6.2 ( 1 nN for dp ) 810 nm31). We note that no trend of the aggregate cohesive force with respect to primary particle diameter can be observed. Further, it is worth mentioning that the indicated variations have been calculated using the maximum observed deviation from the mean values for 〈Rg〉 equal 15%. On the basis of the scaling of 〈Rg〉 vs 〈G〉 presented in Figure 11, which is consistent with the argument proposed by Potanin,51 the force required to break an aggregate is proportional to the attractive force between primary particles. Since under conditions investigated in this work the salt concentration is far above the critical coagulation concentration and, therefore, the repulsive electrostatic forces are completely screened, only the attractive van der Waals force has to be considered. From definition, this is linearly proportional to the primary particle diameter, once the minimum separation distance is assumed to be constant.73 However, this is not in agreement with our results, where an aggregate cohesive force independent of the primary particle diameter was observed. As was shown by various authors measuring the pull-off force using atomic force microscopy (AFM) for particles of different nature and diameter (altogether covering a range from 2 to 250 µm), depending on the particles surface roughness, a pull-off force with either linear74-78 or no dependency45,46,79 on the particle size can be observed. Unfortunately, direct comparison of our results with data in the literature is not possible since only measurements of the adhesion force between particles and surfaces or between two particles with particles diameter bigger than 2 µm are reported in the literature. It may be noted that our estimated values fall in the range of the values reported in the literature for conditions closest to ours, i.e., the data by Hodges et al.,45,46 who measured the pull-off force between polystyrene particles in liquid phase with diameters ranging from 2 to 100 µm or the data of Sonntag and Russel49 and Kobayashi,38,47,48 who altogether measured values of the aggregate cohesive force in the range from 0.01 to 10 nN with no dependency on the aggregate size. As reported by several authors through both theoretical as well as experimental studies,45,46,79-88 one factor which strongly affects the behavior of adhesion forces is surface roughness. (73) Hunter, R. J. Foundations of Colloid Science, 2nd ed.; Oxford University Press: Oxford, 2001; p 816. (74) Lam, K. K.; Newton, J. M. Powder Technol. 1992, 73, 117-125. (75) Sharma, M. M.; Chamoun, H.; Sita, Rama, Sarma, D. S. H.; Schechter, R. S. J. Colloid Interface Sci. 1992, 149, 121-134. (76) Heim, L. O.; Blum, J.; Preuss, M.; Butt, H. J. Phys. ReV. Lett. 1999, 83, 3328-3331. (77) Cooper, K.; Ohler, N.; Gupta, A.; Beaudoin, S. J. Colloid Interface Sci. 2000, 222, 63-74. (78) Salazar-Banda, G. R.; Felicetti, M. A.; Goncalves, J. A. S.; Coury, J. R.; Aguiar, M. L. Powder Technol. 2007, 173, 107-117. (79) Heim, L. O.; Ecke, S.; Preuss, M.; Butt, H. J. J. Adhes. Sci. Technol. 2002, 16, 829-843. (80) Krupp, H. AdV. Colloid Interface Sci. 1967, 1, 111-239. (81) Fuller, K. N. G.; Tabor, D. Proc. R.. Soc. London, Ser. A 1975, 345, 327-342.


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Along this line, Hodges et al.45,46 elaborated a modified JohnsonKendall-Roberts model, which predicts a particle-size-independent adhesion force, once corrections regarding the surface energy of the cantilever tip and the measured surface roughness of the particles are considered. In this way, they related the size and number of asperities in the contact region, both decreasing with decreasing particle diameter as measured by AFM, to the discrepancy of their pull-off force data with simple linear-bondingforce models. Based on these observations, we think that the most reasonable cause for the obtained aggregate cohesive force being independent from the primary particle diameter is the surface roughness, which is in agreement with the findings of Hodges et al.45,46 Another common way to present data from numerical studies on aggregate breakage52,53 is to plot the maximum stable normalized aggregate mass, imax, as a function of the hydrodynamic stress, ηG. Since for large primary particles which are outside the RDG limit and for dense aggregates imax cannot be evaluated straightforward from I(0),54 we will evaluate imax from the independently measured size and structure of the aggregates. For the aggregate size, we will use 〈Rg〉, as it represents a higher order moment ratio of the CMD and, therefore, characterizes the largest aggregates in the system (confirmed also by image analysis), while for the aggregate structure, the mass fractal dimension obtained from image analysis will be used. According to this, it is possible to relate imax to the hydrodynamic stresses acting on the aggregates as follows.

imax ≈ (Rg,imax/Rg,p)df ∝ (ηG)γ

∝

Gγ

(8)

η)const

Using the values of the critical aggregate size for breakage obtained at infinite dilution conditions (Figure 11b) and a value for the mass fractal dimension equal to 2.64 ( 0.18 (as obtained by image analysis) and by applying eq 8, one can compute imax. As the aggregate size scales with the volume average shear rate to the power of -1/2, this leads to a scaling of the maximum stable normalized average aggregate mass with the volume average shear rate according to imax ∝ 〈G〉-1.32(0.09, where the exponent is simply the mass fractal dimension multiplied by -0.5, the scaling exponent of 〈Rg〉 vs 〈G〉. The so-obtained scaling can only be compared to the data of numerical studies published by Higashitani et al.,52 who simulated the break-up of aggregates in simple shear and elongational flow with slopes equal to -0.936 and -1.569, respectively, or by Harada et al.,53 who obtained slopes equal to -0.849 and -2.355 using open and dense aggregates, respectively, for the break-up in simple shear flow. We note that our experimentally obtained scaling lies within the given range. However, since the type of flow within the stirred tank used in this work results from a nontrivial combination of simple shear and elongational flow which is further combined with the effect of polydispersity of the CMD (which is different from the monodisperse systems used in the numerical studies52,53), we cannot precisely define the type of flow prevailing in the (82) Johnson, K. L. Proceedings of the 14th International Congress on Theoretical and Applied Mechanics; North-Holland: Delft, Netherlands, 1976; pp 133-143. (83) Das, S. K.; Schechter, R. S.; Sharma, M. M. J. Colloid Interface Sci. 1994, 164, 63-77. (84) Soltani, M.; Ahmadi, G. J. Adhes. 1995, 51, 105-123. (85) Bhattacharjee, S.; Ko, C. H.; Elimelech, M. Langmuir 1998, 14, 33653375. (86) Rabinovich, Y. I.; Adler, J. J.; Ata, A.; Singh, R. K.; Moudgil, B. M. J. Colloid Interface Sci. 2000, 232, 10-16. (87) Rabinovich, Y. I.; Adler, J. J.; Ata, A.; Singh, R. K.; Moudgil, B. M. J. Colloid Interface Sci. 2000, 232, 17-24. (88) Katainen, J.; Paajanen, M.; Ahtola, E.; Pore, V.; Lahtinen, J. J. Colloid Interface Sci. 2006, 304, 524-529.

Figure 12. Scaling of (a) the steady-state values of the absolute zero-angle intensity, I(0) (in arbitrary units, au), for a solid volume fraction equal to 2 × 10-5 and (b-d) the φ-normalized zero-angle intensity, I(0)φn , at infinite dilution conditions for different primary particle diameters, i.e., (O) 120 nm, (solid left triangle) 420 nm, and (0) 810 nm (from31) as a function of the volume average shear rate, 〈G〉. The solid lines in (b-d) correspond to the power-law fittings in (a) normalized by the corresponding zero-angle intensities of the initial dispersions of primary particles.

system. On the other hand, using the procedure developed in this work, it is possible to experimentally determine the relation between the maximum stable aggregate mass and the hydrodynamic stresses using any type of mixing unit, which is of importance for any type of modeling of the aggregates breakage under shear, since such a scaling determines the breakage kernel.89 4.2. Effect of Multiple Light Scattering on the Scaling of the Zero-Angle Intensity of Scattered Light. Also the second measured quantity of the CMD, the zero-angle intensity, presented in Figure 12a, exhibits a certain proportionality to the volume average shear rate, where the power-law slope decreases with increasing primary particle diameter and is equal to -0.69, -0.82, and -1.18 for primary particle diameter equal to 810, 420, and 120 nm, respectively. In the case of the zero-angle intensity, the measured scaling obtained from our data can be compared only with a few sets of data published by Sonntag and Russel,49,50 who produced open aggregates from fully destabilized primary particles under static conditions (DLCA) and subsequently broke them in laminar Couette and converging flow, and data published by Selomulya et al.,17 who studied aggregation of different size polystyrene latexes in a Taylor-Couette apparatus under turbulent conditions. In the case of Sonntag and Russel’s data,49,50 with a primary particle diameter equal to 140 nm, the obtained values of the slopes equal to -0.88 and -1.0 for shear and elongational flow, respectively, are in between the values for aggregates made from primary particle diameter equal to 120 and 420 nm measured in this work. On the other hand, data published by Selomulya et al.,17 using primary particles with a diameter equal to 810 nm, show similar scaling to that obtained by Soos et al.31 for the same primary particle diameter, with a slope equal to -0.65. Accordingly, one can conclude that the scaling of I(0) with the volume average shear rate is influenced by both the type of flow field, as it determines the polydispersity of the CMD and, therefore, the scaling of I(0) with the aggregate size, as well as the primary particle diameter, as it influences the scattering behavior of the aggregates. When comparing the values of I(0) (89) Ba¨bler, M. U. Modelling of Aggregation and Breakage of Colloidal Aggregates in Turbulent Flows. Ph.D. Thesis, ETH Zurich, Switzerland, Zurich, 2007.

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obtained at infinite dilution (presented in normalized form in Figure 12b-d) with the corresponding steady-state values obtained for a solid volume fraction equal to 2 × 10-5 (Figure 12a), one observes the same behavior as for 〈Rg〉, where the values of the zero-angle intensity, I(0), are shifted in parallel to lower values for the case of infinite dilution, and therefore, the scaling with respect to the volume average shear rate is preserved. The most important conclusion that can be drawn from these data is that the slope of the power-law scaling is different for different primary particle diameters. Tambo and Watanabe,90 Spicer and Pratsinis et al.,8,91,92 Kostoglou and Karabelas,93 and Ba¨bler et al.94 showed, by experiment and by modeling, that steady-state CMDs obtained by shear coagulation under turbulent conditions scale with the average aggregate size, i.e., they are self-preserving with respect to the shear rate. Furthermore, it was shown that the steady-state moment ratios of such self-preserving CMDs scale with respect to the shear rate. Since we find the same scaling for the steadystate moment ratios, we assume our CMDs to be self-similar with respect to shear rate. Moreover, we assume that the scaling of the zero-angle intensity of scattered light with the aggregate mass, i, for aggregates that are outside the RDG limit can be described by a modified power law95 using a correction factor, c, which is a function of both dp and df. For single aggregates this reads as follows

I(0) ∝ i2-c

(9)

and for a population of aggregates with mass, i, and an absolute number, Ni,

I(0) ∝

∑i Nii2-c

(10)

or in terms of a relative mass with respect to the total mass of the aggregates population, mi ) Mi/Mpop ) Nii/Mpop,

I(0) ∝ Mpop I(0)φn ∝

∑i mii1-c

∑i mii1-c

(11) (12)

We note that eqs 9-12 are valid for dp and df being constant with respect to i. Therefore, for a self-preserving CMD of aggregates obtained using the same size of primary particles and an aggregate structure independent from the applied shear rate, plotting the steady-state values of I(0) versus 〈Rg〉 (for constant Mpop) or I(0)φn versus 〈Rg〉 (for changing Mpop) one should obtain powerlaw scalings with slopes equal to df(1-c). Such graphs are shown in panels a and b-d of Figure 13 for a solid volume fraction equal to 2 × 10-5 and infinite dilution conditions, respectively. The obtained power-law slopes take the values 2.15, 1.50, and 1.24 for primary particle diameter equal to 120, 420, and 810 nm, respectively. By using a value of the mass fractal dimension equal to 2.64 ( 0.18, as obtained by image analysis, for φ ) 2 × 10-5 as well as for infinite dilution conditions, this results in (90) Tambo, N.; Watanabe, Y. Water Res. 1979, 13, 429-439. (91) Spicer, P. T.; Pratsinis, S. E.; Trennepohl, M. D.; Meesters, G. H. M. Ind. Eng. Chem. Res. 1996, 35, 3074-3080. (92) Spicer, P. T.; Pratsinis, S. E. AIChE J. 1996, 42, 1612. (93) Kostoglou, M.; Karabelas, A. J. J. Aerosol Sci. 1999, 30, 157-162. (94) Ba¨bler, M. U.; Morbidelli, M. J. Colloid Interface Sci., 2007, 316, 428441. (95) Soos, M.; Lattuada, M.; Sefcik, J.; Morbidelli, M. Proceedings of the International Congress on Particle Technology (in PARTEC); Nuremberg, Germany, 2007.

Figure 13. Scaling of (a) the steady-state values of the absolute zero-angle intensity, I(0) (in arbitrary units, au), for a solid volume fraction equal to 2 × 10-5 and (b-d) the φ-normalized zero-angle intensity, I(0)φn , at infinite dilution conditions for different primary particle diameters, i.e., (O) 120 nm, (solid left triangle) 420 nm, and (0) 810 nm (from ref 31) as a function of the rms radius of gyration, 〈Rg〉. The solid lines in (b-d) correspond to those of the power-law fittings from (a) normalized by the corresponding zero-angle intensities of the initial dispersions of primary particles. The experimental correction to the RDG theory due to multiple light scattering within the aggregates can be computed by applying eq 11.

experimentally obtained correction factors equal to 0.18 ( 0.06, 0.42 ( 0.04, and 0.53 ( 0.03 for primary particle sizes equal to 120, 420, and 810 nm, respectively. This increase in the value of the correction factor with increasing primary particle diameter can be explained by the fact that with increasing primary particle diameter the system deviates more and more from the validity region of the RDG approximation, leading to a scaling of I(0) with the scatterers mass smaller than 254-59, and therefore, according to eq 11, I(0) for a population of scatterers with constant total mass will scale with a power smaller than unity. An additional contribution could be that due to a change in the shape or polydispersity of the CMDs as a result of variation of the primary particle diameter a different scaling behavior of I(0) for a variation in 〈G〉 is observed. We note that even for aggregating systems which are clearly within the RDG limits, e.g., for a primary particle diameter equal to 120 nm, deviations from RDG can occur when the aggregates are very dense, as it is usually the case for aggregates produced under turbulent conditions. A quantitative analysis of the different contributions requires a more profound discussion of the light scattering properties of aggregates, similar to those formed under turbulent conditions, taking into account the nonvalidity of the RDG approximation (large primary particle diameter and multiple light scattering within the aggregates) as well as the nonfractal nature of small aggregates, which is outside the scope of this paper.

5. Conclusion The steady-state size and structure of aggregates produced from polystyrene latexes with primary particle diameter equal to 120 and 420 nm under turbulent conditions in a stirred tank were obtained for various values of the volume average shear rate and solid volume fraction and were compared to previous data for dp ) 810 nm.31 Applying step changes in stirring speed, it was found that the steady-state CMD is completely reversible with no dependency on shear rate history. Characterization of the steady-state aggregate structure via image analysis provided a mass fractal dimension equal to 2.64 ( 0.18, independent of


primary particle diameter and shear rate. This does not agree with the lower values of the scaling exponent obtained from light scattering data, which can be explained by the nonvalidity of the RDG theory combined with the presence of a large portion of small nonfractal aggregates, especially for the case of large primary particles and highest stirring speeds. Therefore, for these conditions, SE cannot be used as an approximation of df but another independent method needs to be used to obtain information about the aggregate structure. Both the steady-state aggregate size (shear experiments) and the critical aggregate size for breakage (dilution experiment) were characterized by two characteristic quantities of the CMD obtained by SASLS, i.e., rms radius of gyration and zero-angle intensity. The absolute values of rms radius of gyration obtained under similar solid volume fraction and hydrodynamic conditions proved to be independent of the primary particle diameter. The rms radius of gyration for both low concentration as well as infinite dilution conditions followed a power-law scaling with the volume average shear rate according to 〈Rg〉 ∝ 〈G〉- 1/2, which is in agreement with our previous data31 and data published in the literature.5,17 Due to the scaling behavior of 〈Rg〉 versus 〈G〉, the constant-force approach is assumed to be the most appropriate way to evaluate aggregate cohesive forces, which were evaluated from the critical aggregate size for breakage obtained at low solid volume fraction, where aggregation can be safely neglected. It was found to be equal to 3.5 ( 1 nN and 6.1 ( 1.5 nN for primary particle diameter equal to 420 and 120 nm, respectively (and 6.2 ( 1 nN for dp ) 810 nm31). This observation of an aggregate cohesive force independent of the primary particle diameter is explained by the nonideality of the particles surface, and in particular the effect of the surface roughness is held responsible, which is in agreement with AFM pull-off force measurements on micron sized polystyrene particles45,79 that provide absolute bonding forces of the same order. The second measured moment of the CMD, the zero-angle intensity of scattered light, showed a powerlaw scaling with the rms radius of gyration according to I(0) ∝ 〈Rg〉a, with the exponent depending on the primary particle diameter (values for a equal to 2.36, 1.64, and 1.38 for dp equal to 120, 420, and 810 nm, respectively), valid both at finite volume fractions as well as at infinite dilution conditions. These slopes do not agree with the expected scaling, within the limits of RDG theory for a monosized or self-preserving system of scatterers with constant mass fractal dimension a ) df. This deviation is explained by three reasons: first, the primary particle diameters become comparable or larger than the wavelength of the used laser and, therefore, the RDG theory is not valid anymore; second, due to a very dense aggregate structure, multiple light scattering within the aggregate occurs; and third, for larger primary particles, the relative aggregate size decreases, leaving a large portion of small nonfractal aggregates that scatter light not according to the RDG theory as well as a short power-law region. With an independently measured mass fractal dimension equal to 2.64 ( 0.18, as obtained by image analysis, this results in proposed correction factors equal to 0.18 ( 0.06, 0.42 ( 0.04, and 0.53 ( 0.03 for primary particles of average size equal to 120, 420, and 810 nm, respectively. We conclude that with the procedure developed in this work it is possible to experimentally determine, on one hand, the relation between the maximum stable aggregate mass and the hydrodynamic stresses using any type of mixing unit and, on the other hand, the correction factors for the light scattering behavior of

Langmuir, Vol. 24, No. 7, 2008 3081

compact aggregates, which can serve as a basis for comparison to simulated light scattering data. Nomenclature a ) scaling exponent for the power-law relation between I(0) and 〈Rg〉 c ) correction factor for scaling of I(0) with i, which is a function of dp and df d, nm or µm ) characteristic length scale (diameter) df ) mass fractal dimension dmax, µm ) maximum stable aggregate size dp, nm ) primary particle diameter dpf ) perimeter fractal dimension Fcoh, kg m s-2 ) cohesive force holding the aggregates together Fhyd, kg m s-2 ) hydrodynamic force exerted on the aggregates G,〈G〉, s-1 ) local shear rate, volume average shear rate i,imax ) normalized aggregate mass, maximum stable normalized aggregate mass I(0), au ) zero-angle intensity of scattered light I(0)φn ) φ-normalized zero-angle intensity of scattered light kB, kg2 m s-2 K-1 ) Boltzmann contstant (1.3806503 × 10-23 kg2 m s-2 K-1) m ) relative refractive index mi ) relative mass of an aggregate with normalized mass i (mi ) Mi/Mpop ) Nii/Mpop) Mi, au ) total mass of aggregates with normalized mass i Mpop, au ) total mass of aggregates population n ) refractive index Ni ) absolute number of aggregates with normalized mass i Pe ) Pećlet number, defined in eq 1 P(q) ) form factor of primary particles q, nm-1 ) scattering vector amplitude r, nm or µm ) characteristic length-scale (radius) Rg,i, Rg,p, µm ) radius of gyration of an aggregate with normalized mass i, radius of gyration of a primary particle 〈Rg〉, µm ) root-mean-square radius of gyration, 〈Rg〉 ) x〈R2g〉 SE ) scaling exponent of the power-law region of the intensity curve S(q) ) structure factor as a result of primary particle arrangement within the aggregate T, K ) absolute temperature (298K) Greek Letters η, Pas ) dynamic viscosity of water (0.001Pa‚s) θ, rad ) scattering angle φ,φ0 ) solid volume fraction, initial solid volume fraction Abbreviations AFM ) Atomic force microscopy CMD ) Cluster mass distribution DLCA ) Diffusion-limited cluster aggregation RDG ) Rayleigh-Debye-Gans SASLS ) Small-angle static light scattering

Acknowledgment. The authors thank Dr. Matthaüs Ulrich Ba¨bler, Dr. Marco Lattuada, Amgad Salah Moussa, and Dr. Hua Wu for useful discussions and their helpful suggestions. This work was financially supported by the Swiss National Science Foundation (Grant No. 200020-113805/1). Supporting Information Available: Additional illustrations for primary particles with a size equal to 120 nm, an example of the steady-state reversibility, and results of a dilution experiment for such primary particles. (Figures SI1-3 correspond to Figures 2-4 of the main paper, respectively.) This information is available free of charge via the Internet at http://pubs.acs.org. LA7032302

Dependence of Aggregate Strength, Structure ... - ACS Publications

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