pubs.acs.org/Langmuir © 2010 American Chemical Society
Structure and Kinetics of Shear Aggregation in Turbulent Flows. I. Early Stage of Aggregation Matth€aus U. B€abler,† Amgad S. Moussa,‡ Miroslav Soos,‡ and Massimo Morbidelli*,‡ †
Institute of Process Engineering and ‡Institute for Chemical and Bioengineering, ETH Zurich, CH-8093 Zurich, Switzerland Received April 19, 2010. Revised Manuscript Received June 22, 2010
Aggregation of rigid colloidal particles leads to fractal-like structures that are characterized by a fractal dimension df which is a key parameter for describing aggregation processes. This is particularly true in shear aggregation where df strongly influences aggregation kinetics. Direct measurement of df in the early stages of shear aggregation is however difficult, as the aggregates are small and few in number. An alternative method for determining df is to use an aggregation model that when fitted to the time evolution of the cluster mass distribution allows for estimating df. Here, we explore three such models, two of which are based on an effective collision sphere and one which directly incorporates the permeable structure of the aggregates, and we apply them for interpreting the initial aggregate growth measured experimentally in a turbulent stirred tank reactor. For the latter, three polystyrene latexes were used that differed only in the size of the primary particles (dp = 420, 600, and 810 nm). It was found that all three models describe initial aggregation kinetics reasonably well using, however, substantially different values for df. To discriminate among the models, we therefore also studied the regrowth of preformed aggregates where df was experimentally accessible. It was found that only the model that directly incorporates the permeable structure of the aggregates is able to predict correctly this second type of experiments. Applying this model to the initial aggregation kinetics, we conclude that the actual initial fractal dimension is df = 2.07 ( 0.04 as found from this model.
1. Introduction Aggregation of colloidal and micrometer-sized particles in turbulent flows is encountered in many technological and natural processes, e.g., solid-liquid separations in the polymer, food, and pharmaceutical industry,1 mineral recovery, (waste)water treatment, and formation of marine snow.2 Its quantitative description, however, is still a challenging problem even in the simplest case where the particles are fully destabilized and the suspension is diluted. Difficulties arise in particular from the hydrodynamic and colloidal interactions of the colliding particles, which depend heavily on their structure, i.e., primary particles exhibit rather different interactions compared to large and permeable fractal aggregates. Accounting for collisions among the latter is, however, essential for describing aggregation dynamics of a particle population. Due to their larger size, aggregates exhibit a substantially larger aggregation rate compared to primary particles. This causes rapid formation of a few large aggregates that later act as collectors for primary particles. Typically, the aggregate structure is characterized through a fractal dimension, which therefore is a key parameter in the description of aggregation processes. Few studies focused on the aggregate structure in the early stages of shear aggregation in turbulent flows. Selomulya et al.3 studied the aggregation of different fully destabilized polystyrene latexes (dp = 60, 380, and 810 nm) in a Taylor-Couette flow. The fractal dimension in the early stage of the process was determined *To whom correspondence should be addressed. E-mail: massimo.morbidelli@ chem.ethz.ch. Phone þ41 44 632 3034. Fax þ41 44 632 1082. (1) Hunter, R. J. Foundations of Colloid Science, 2nd ed.; Oxford University Press: Oxford, 2001. (2) Flocculation in Natural and Engineered Environmental Systems; Droppo, I. G., Leppard, G. G., Liss, S. N., Milligan, T. G., Eds.; CRC Press: Boca Raton, 2005. (3) Selomulya, C.; Bushell, G.; Amal, R.; Waite, T. D. Langmuir 2002, 18, 1974–1984.
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through light scattering as df = 2.6, 2.0, and 1.8 ((0.2 for each) for the three latexes, respectively. Flesch et al.4 studied the aggregation of a polystyrene latex (dp = 0.87 μm) in a stirred tank. The fractal dimension was estimated from the obscuration of the light scattering signal, thus obtaining df = 2.05 ( 0.05. However, it has been shown that the interpretation of light scattering data using Rayleigh-Debye-Gans (RGD) theory, as done in these works, is crucial for primary particles whose size exceeds the laser wavelength .5-7 Moreover, in the early stages of the process there are only few small aggregates whose structure is even more difficult to measure directly. An alternative method to determine the fractal dimension in the early stage of the process is to fit the initial time evolution of cluster mass distribution (CMD) to a kinetic model that depends on the fractal dimension. Kuster et al.8 studied the aggregation of different fully destabilized polystyrene latexes (dp = 0.480.8 μm) in a stirred tank. Fitting the obtained kinetic data to their model gave df = 2.40-2.55. Polystyrene particles (dp = 810 nm) in a stirred tank were also studied by Walder et al.9 who found df = 1.8 from fitting the data to their model. It is clear that the latter approach requires a comprehensive aggregation model that accounts for the specific interactions between the aggregates in relation to their structure. The aim of this work is to compare three such models and to evaluate their capability for describing aggregation kinetics. The three aggregation models are two versions of the effective collision sphere model (ECS) of Kuster et al.8 and the uniformly permeable sphere (4) Flesch, J. C.; Spicer, P. T.; Pratsinis, S. E. AIChE J. 1999, 427, 241–247. (5) Ehrl, L.; Soos, M.; Morbidelli, M. Langmuir 2008, 24, 3070–3081. (6) Lattuada, M.; Ehrl, L. J. Phys. Chem. B. 2009, 113, 5938–5950. (7) Sorensen, C. M. Aerosol Sci. Technol. 2001, 35, 648–687. (8) Kusters, K. A.; Wijers, J. G.; Thoenes, D. Chem. Eng. Sci. 1997, 52, 107–121. (9) Waldner, M. H.; Sefcik, J.; Soos, M.; Morbidelli, M. Powder Technol. 2005, 156, 226–234.
Published on Web 07/14/2010
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model (UPS) of B€abler et al.10,11 For this, we studied the aggregation of three fully destabilized diluted polystyrene latexes over a wide range of stirring speeds in a stirred tank reactor operated under turbulent conditions. The latexes with identical surface chemistry and primary particle sizes of dp = 420 nm, 600 nm, and 810 nm were thereby chosen to be large enough to minimize the effect of Brownian aggregation and small enough so as the aggregates can grow sufficiently large before breakage sets in. Two types of experiments were conducted. On one hand, these were initial growth experiments where we started from a suspension of primary particles and measured the rate of aggregate formation upon the addition of an electrolyte (i.e., a salt to screen the electrostatic repulsion of the particles). Experiments of this first type were conducted for all three latexes used in this work. On the other hand, in the regrowth experiments we started from a suspension of previously formed aggregates and we measured the regrowth upon a sudden reduction of the stirring speed. Experiments of this second type were conducted for the largest primary particles, as well as for a fourth latex with a primary particle size of dp = 100 nm. The difference between these two types of experiments is that in the first one the initial aggregating particles are essentially impermeable solid spheres, whereas in the second one, the initial aggregating particles are porous aggregates of primary particles. For the interpretation of the measured aggregation kinetics, a population balance equation (PBE) model is formulated that is combined with the three different aggregation models. The paper is organized as follows. In section 2, the three aggregation models are described. The PBE model and the flow field heterogeneity present in stirred tanks are discussed in section 3. The experimental methods are presented in section 4. Results and conclusions follow thereafter in sections 5 and 6, respectively.
multiplied by Ki,j(0) gives the actual aggregation rate function Ki,j.14-18 A common strategy to derive such an aggregation efficiency for flow-induced aggregation is based on the analysis of the relative trajectories between a pair of particles. This method has been extensively applied to study aggregation events between primary particles,19-25 and rate expressions derived from these studies have been successful in describing the consumption of primary particles due to aggregation.26,27 However, these models cannot be directly extended to the whole aggregation process: A typical aggregation process includes many different particulate species, i.e., primary particles and aggregates of different mass. Due to the porous and permeable structure of the latter, their interactions differ substantially from those of primary particles,10,28-30 which has to be accounted for in order to predict the entire aggregation process. The three aggregation models investigated in this work address this issue and consider the specific interactions among aggregates. 2.2. Aggregation Efficiency Models. In this work, the concept of fractal aggregates is adopted which implies that, on average, the number of primary particles in an aggregate ( i) scales with the aggregate radius according to i ∼ Rdf where df is a fractal dimension. For df = 3, the aggregates have a regular Euclidian structure which is typically the case for dense aggregates; more open aggregates assume smaller values of df. Here, the fractal scaling rule is used in the common form i ¼ kg ðRg, i =Rp Þdf
ð2Þ
where Rg,i is the radius of gyration of aggregate i, Rp is the radius of the primary particle, and kg is a prefactor which has been shown to be close to unity.31 Hence, for simplicity kg = 1 is used hereafter. Equation 2 is in particular used to estimate the collision radius of an aggregate
2. Aggregation Models 2.1. Aggregation Rate and Efficiency. The basis for a quantitative understanding of aggregation processes in flows was established by Smoluchowski12 and Saffman and Turner13 who proposed that an aggregation event is the result of a collision between two particles that stick together thereafter. Considering a diluted suspension of small particles subject to a simple shear flow, Smoluchowski derived that the aggregation rate between particle i and particle j (later, i and j refer to the number of primary particles in the aggregates, that is, the dimensionless aggregate mass) is given by Ki,jNiNj, where Nl (l = i, j) is the concentration of particles (or clusters) of mass l and Ki,j is the aggregation rate function. Assuming that there are no interactions among the particles except sticking upon contact, the latter was derived as ð0Þ
Ki, j ¼ k0 GðRc, i þRc, j Þ3
ð1Þ
where G is the shear rate, k0 is a prefactor that depends on the flow structure at the particle scale (k0 = 4/3 for simple shear flow), and Rc,l is the collision radius of particle l. Considering instead an isotropic homogeneous turbulent flow, Saffman and Turner13 derived a similar expression for Ki,j(0) where k0 ≈ 1.29 and G = (Æεæ/ν)1/2 is the turbulent shear rate. To account for interactions between the colliding particles, it is convenient to introduce an aggregation efficiency fi,j that when (10) B€abler, M. U.; Sefcik, J.; Morbidelli, M.; Bazdyga, J. Phys. Fluids 2006, 18, 013302. (11) B€abler, M. U. AIChE J. 2008, 54, 1748–1760. (12) Smoluchowski, M. Z. Phys. Chem. 1917, 92, 129–168. (13) Saffman, P. G.; Turner, J. S. J. Fluid Mech. 1956, 1, 16–30.
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Rc, i ¼ Rp i1=df
ð3Þ
Hence, decreasing df increases the collision radius and therefore the aggregation rate function. The three aggregation models considered in this work assume that the particle (or aggregate) motion is driven by the fluid flow and by interparticle forces acting between the particles. Let v denote the relative velocity between the two particles: we have10 v ¼ vsh þ vint
ð4Þ
(14) Patil, D. P.; Andrews, J. R. G.; Uhlherr, P. H. T. Int. J. Mineral Proc. 2001, 61, 171–188. (15) Thill, A.; Moustier, S.; Aziz, J.; Wiesner, M. R.; Bottero, J. Y. J. Colloid Interface Sci. 2001, 243, 171–182. (16) Selomulya, C.; Bushell, G.; Amal, R.; Waite, T. D. Chem. Eng. Sci. 2003, 58, 327–338. (17) Ding, A.; Hounslow, M. J.; Biggs, C. A. Chem. Eng. Sci. 2006, 61, 63–74. (18) Soos, M.; Wang, L.; Fox, R. O.; Sefcik, J.; Morbidelli, M. J. Colloid Interface Sci. 2007, 307, 433–446. (19) Zeichner, G. R.; Schowalter, W. R. AIChE J. 1977, 23, 243–254. (20) Van de Ven, T. G. M.; Mason, S. G. Colloid Polym. Sci. 1977, 255, 468–479. (21) Adler, P. M. J. Colloid Interface Sci. 1981, 83, 106–115. (22) Adler, P. M. J. Colloid Interface Sci. 1981, 84, 461–473. (23) Greene, M. R.; Hammer, D. A.; Olbricht, W. L. J. Colloid Interface Sci. 1994, 167, 232–246. (24) Brunk, B. K.; Koch, D. L.; Lion, L. W. J. Fluid Mech. 1998, 364, 81–113. (25) Vanni, M.; Baldi, G. Adv. Colloid Interface Sci. 2002, 97, 151–177. (26) Higashitani, K.; Yamauchi, K.; Matsuno, Y.; Hosokawa, G. J. Chem. Eng. Jpn. 1983, 16, 299–304. (27) Brunk, B. K.; Koch, D. L.; Lion, L. W. J. Fluid Mech. 1998, 371, 81–107. (28) Li, X.; Logan, B. E. Environ. Sci. Technol. 1997, 31, 1229–1236. (29) Kim, A. S.; Stolzenbach, K. D. J. Colloid Interface Sci. 2002, 253, 315–328. (30) Veerapaneni, S.; Wiesner, M. R. J. Colloid Interface Sci. 1996, 177, 45–57. (31) Lattuada, M.; Wu, H.; Morbidelli, M. J. Colloid Interface Sci. 2003, 268, 106–120.
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where the two terms on the right-hand side refer to the particle motion induced by the fluid flow and by interparticle forces, respectively. In all models, Brownian motion of the particles is neglected. This is correct only when the Peclet number, which is the ratio between the convective aggregate motion and the Brownian aggregate motion, and which is proportional to the cube of the aggregate size, approaches infinity. To account for diffusion of finite-sized aggregates, Brownian aggregation is incorporated at the level of the PBE model as discussed in section 3.1. Equation 4 is the basis for the determination of the aggregation efficiency: Let x be the position of an aggregate at time t; integrating x_ = v gives the relative trajectory between the two aggregates from which the aggregation efficiency can be deduced (for details, see ref 10). Scaling analysis of the two velocities in eq 4 gives vsh ∼ GR and vint = Fint/fdrag ∼ (AH/R)/(μR) where Fint and fdrag are the interparticle force and the drag coefficient, respectively, and AH is the Hamaker constant.10 In dimensionless form, eq 4 contains therefore a dimensionless interaction parameter NA ¼
AH μR3p G
ð5Þ
that gives the ratio between the relative particle velocity due to interparticle forces and that due to fluid flow (note that NA is defined using the radius of the primary particle Rp rather than the aggregate radius that was used in ref 10). Besides the fractal dimension and the two aggregate masses (i and j), NA is the fourth parameter that enters the aggregation efficiency models discussed in this work. A fifth parameter appears in one of the models (i.e., UPS) to account for the retardation of the van der Waals interaction.1 The corresponding retardation parameter is given by NL ¼ λL =Rp
ð6Þ
where λL is the London wavelength, which is typically λL ≈ 100 nm. In the remainder of this section, the three aggregation efficiency models are described. The level of details given thereby shall account for the modular character of these models, which is particularly important for the ECS model for which several modifications have been proposed over the years.16,32 2.3. Effective Collision Sphere (ECS) Model. The ECS model assumes that the collision between two porous aggregates resembles that between two impermeable spheres. Explicitly, it is assumed that the results of the trajectory analysis of impermeable spheres are also applicable to permeable aggregates provided that the parameters of the former are reformulated such as to account for the properties of the aggregates, i.e., their permeable structure. For this, let us first consider two impermeable particles of radius R1 and R2. Given an undisturbed linear flow, the collision between these two particles (in the absence of repulsive forces) is fully characterized by two parameters, namely, NA0 = AH/(μGR13), i.e., NA with the primary particle radius substituted for R1, and the particle size ratio λ = R2/R1 E 1. In the ECS model, these two parameters are formulated as functions of the aggregate properties. Accordingly, the ECS model can be written as fi, j ¼ fimp N 0 A ði, j, df , NA Þ, λði, j, df , NA Þ
ð7Þ
where fimp = fimp(NA0 , λ) is the collision efficiency for two impermeable spheres. (32) Ducoste, J. Chem. Eng. Sci. 2002, 57, 2157–2168.
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To derive the constituting functions of eq 7, the following considerations are made. The interaction parameter NA0 is the ratio between the characteristic relative velocity due to interparticle forces and that due to the fluid flow. For impermeable particles, the interparticle forces are induced by the bulk body of the particles, i.e., Fint ∼ AH/R1. On the other hand, for aggregates only the forces acting between the primary particles in the two aggregates that are the nearest are relevant33 such that the above becomes Fint ∼ AH/Rp. Thus, to mimic collisions of aggregates the interaction parameter (N0 A in eq 7) is multiplied by the ratio (Rp/R1) to account for reduced interparticle forces. Furthermore, both characteristic velocities that enter the interaction parameter are subject to hydrodynamic interactions. For impermeable particles, the characteristic length scale of these interactions is R1. For permeable aggregates, on the other hand, this length scale is an effective hydrodynamic radius of an aggregate, Rh. Hence, what is R1 in the case of impermeable spheres becomes Rh in the case of permeable aggregates. An effective hydrodynamic radius can be derived by modeling the aggregates as permeable spheres where the flow inside the spheres is described by Brinkman’s equation. The original expression proposed by Kuster et al.8 uses the limiting trajectory of an isolated permeable sphere in a simple shear flow.34 Here, instead, we estimate the hydrodynamic radius of an aggregate from its settling behavior.30,32 This results in34 ! 2ξ2i 1 - tanhðξi Þ=ξi ð8Þ Rh, i ¼ Rc, i 2ξ2i þ 3 1 - tanhðξi Þ=ξi √ where ξi = Rc,i/ κi is the dimensionless shielding length, κi is the permeability, and Rc,i is given by eq 3. To estimate the aggregate permeability, Happel’s equation is used:8 9 1=3 9 5=3 3 - ji þ ji - 3j2i 2 2 Ki ¼ 5=3 9ji ð3 þ 2ji ÞCs
ð9Þ
where ji = i(1-3/df) is the solid volume fraction of an aggregate i and Cs is a coefficient accounting for the contacts between the primary particles within the aggregate:8 Cs = 2 for i = 2 and Cs= 0.5 for i>2. Hence, the interaction parameter that enters eq 7 and that accounts for the properties of the aggregates is obtained by multiplying NA0 by (Rp/R1) and by substituting R1 by Rh thereafter. In what follows, these considerations are combined with two different explicit expressions for fimp(NA0 , λ), resulting in two versions of the ECS model. Note that the expressions for fimp are essentially fitting correlations to results from trajectory analysis calculations. 2.3.1. Lumped ECS Model (LECS). Van de Ven and Mason20 performed trajectory analysis calculations of impermeable particles of equal size in a simple shear flow. In the absence of repulsive forces, it was found that fimp = Cr(NA0 )0.18 where the prefactor Cr depends on the retardation parameter NL. For the smallest particles investigated (dp = 1 μm), a value Cr = 0.95 is reported, which is the value used within this work. To use this result for approximating collisions of arbitrary aggregates, the interaction parameter was modified as described above and, additionally, also the particle size ratio was incorporated.8 The (33) Torres, F. E.; Russel, W. B.; Schowalter, W. R. J. Colloid Interface Sci. 1991, 142, 554–574. (34) Adler, P. M. J. Colloid Interface Sci. 1981, 81, 531–535.
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resulting model, hereafter referred to as lumped ECS model (LECS), reads as8 ðLECSÞ
00
¼ Cr ðN A Þ0:18
fi, j
00
NA
1 4NA Rp ¼ 36π λh ð1þλh Þ2 Rh, i
!4 ð10Þ
where NA and Rh,i are given by eqs 5 and 8, respectively, and λh= Rh,j/Rh;,i. The fractal dimension enters the model through Rh,i given by eqs 8 and 9, respectively. For this model, k0 = 4/3 because the undisturbed fluid flow is assumed to be a simple shear flow (the factor 36π in the second of eq 10 as well as 144π in eq 12 are included to recover the original expressions for impermeable spheres given in refs 20 and 21, respectively). 2.3.2. Detailed ECS Model (DECS). Adler21 performed trajectory analysis calculations of impermeable particles with different particle size ratios in a simple shear flow. A parametrization of these calculations is given by Han and Lawler35 ðDECSÞ fi, j
¼
8 ð1þλh Þ3
10
ða þ bλh þ cλ2h
þ dλ3h Þ
ð11Þ
where in the original formulation of Han and Lawler, a, b, c, and d are functions of the interaction parameter for impermeable particles. Reformulating the latter following the approach described above, we obtain 000
NA
NA Rp ¼ 144π Rh, i
!4 ð12Þ 000
and a, b, c and d become functions of NA, that depends on df through Rh,i given by eqs 8 and 9, respectively. Table S2 in the Supporting Information lists these functions for certain values of 000 given in ref 35. Linear interpolation was used for values NA as 000 of NA in between those listed in Supporting Information Table S1. In addition, for this model k0 = 4/3 because the undisturbed fluid flow is assumed to be a simple shear flow. 2.4. Uniform Permeable Sphere Model (UPS). A higher level of detail is included in the UPS model of B€abler et al.10,11 In the first part,10 the authors considered the hydrodynamic interaction of two permeable aggregates in a linear flow. A tractable hydrodynamic problem was formulated by modeling the aggregates as permeable spheres where Brinkman’s equation was adopted to describe the intra-aggregate flow. In the second part,11 the previous result was used to derive the aggregation efficiency. For this, an empirical relation for the effective permeability of an aggregate was developed that relates permeability to the aggregate mass and the fractal dimension. Interparticle forces were included by considering van der Waals forces acting between the primary particles that are the nearest to each other. The turbulent fluid motions on the length scale of the aggregates (that is, below the Kolmogorov length scale) were approximated through a steady axisymmetric extensional flow with the rate of extension given by E ≈ 0.15G36 which leads to k0 = 0.36 for this model. B€abler11 ran a series of trajectory calculations to determine the aggregation efficiency for a wide range of the relevant parameter values. The results of these calculations were summarized (35) Han, M. Y.; Lawler, D. F. J. Am. Water Works Assoc. 1992, 84, 79–91. (36) Batchelor, G. K. J. Fluid Mech. 1980, 98, 609–623.
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Figure 1. Aggregation efficiency according to UPS (solid curve), LECS (dashed curve), and DECS (dashed-dotted curve). (a) to (c) refer to a mass ratio of the colliding aggregates equal to 1, 10, and 100, respectively. In all models, df = 2.5 and NA = 1 (and NL f ¥ in UPS).
through a set of fitting correlations that are not reported here. These relations give the aggregation efficiency f i,j (UPS) as a function of the masses of the colliding aggregates, the fractal dimension, and two interaction parameters NA and NL (note that the relations in ref 11 were formulated for an interaction parameter NF ≈ NA/0.15 that is based on the underlying extensional flow used in this model). 2.5. Comparison of Aggregation Efficiencies. A comparison of the three aggregation efficiency models discussed above is presented in Figure 1, where the aggregation efficiency as a function of the mass of the larger aggregate is shown. The three panels (a-c) refer to an aggregate mass ratio of i/j = 1, 10, and 100, respectively. The three models exhibit a rather different behavior. For the given parameter values, i.e., df = 2.5 and NA = 1 (and NL f ¥, i.e., negligible retardation), UPS (solid curves) assumes a maximum at i = j = 3 (Figure 1a), whereas for larger aggregates, it rapidly converges to a constant as seen in all three panels. On the other hand, LECS (dashed curves) and DECS (dashed-dotted curves) exhibit in all cases a power-law decay. For aggregates of equal size (Figure 1a), the two models predict similar values for fii, which is expected because in this case the models are based on identical assumptions; the difference between LECS and DECS seen in Figure 1a stems entirely from the different forms of parametrization of fimp.20,35 The two models deviate however for larger aggregate mass ratios (Figure 1b and c) where DECS is more strongly decreasing with increasing i/j. The strong decay of LECS and DECS in comparison to UPS is caused by the strong reduction of the interparticle force (∼Rp) forwith respect to the hydrodynamic resistance (∼Rh,i) in the 000 mer two models. This causes an increase of NA00 and NA with increasing aggregate mass (eq 10 and eq 12, respectively) which, accordingly, decreases fi,j. UPS, on the other hand, incorporates the hydrodynamic resistance directly into the trajectory analysis which results in a substantially different behavior. It is the subject of the present work to assess this different behavior.
3. PBE Modeling of Aggregation Processes 3.1. Governing Equations. In a spatially uniform domain, the time evolution of the CMD is described through a PBE i-1 ¥ X dNi 1X Kj , i - j Nj Ni - j - Ni Ki, j Nj ¼ 2 j¼1 dt j¼1
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The aggregation rate function Ki,j is assumed to be a linear combination of shear aggregation and Brownian aggregation ðShÞ
ðBrÞ
Ki , j ¼ Ki , j þ Ki , j
ð14Þ
Hence, the Brownian motion of the aggregates that was neglected in the trajectory analysis enters the model in a linear way through eq 14. This choice still presents the most common way of incorporating Brownian aggregation into a PBE model.16,37-39 Its applicability received support from the theoretical study40 of Melis et al. who included Brownian motion in the study of the aggregation event. It was found that when electrostatic repulsion is negligible, i.e., when the particles are fully destabilized, the two aggregation mechanisms add linearly as given by eq 14. The two constituting functions of eq 14 read as ðShÞ
Ki , j ðBrÞ
Ki , j
¼
¼ k0 fi, j GðRc, i þRc, j Þ3
2kT - 1 Rc, i þ Rc-, j1 Rc, i þ Rc, j 3μWi, j
ð15Þ
ð16Þ
where Wi,j is the Fuchs stability ratio,1 that can be considered as the inverse aggregation efficiency for Brownian aggregation. Honig et al.41 derived a relation for the stability ratio for primary particles, taking into account the hydrodynamic and colloidal interactions of the approaching particles. For polystyrene in water, i.e., the experimental system studied in this work, for which AH ≈ 1 10-20 J, they report W1,1 ≈ 1.7. This suggests that for fully destabilized colloidal particles the correction due to particle interactions is small. If one recalls that the interactions become weaker with increasing i and j, it appears reasonable to assume Wi,j 1, i,j G1, which is applied throughout this work. 3.2. Flow Field Heterogeneity. The turbulent flow in a stirred tank exhibits substantial variations in spatial coordinates. This is most prominently expressed in the shear rate that assumes large values close to the impeller, whereas smaller values are found in the bulk of the flow. Typical values of the shear rate near the impeller are reported to be 4.5 to 10 times the volume average shear rate,42-44 whereas even larger values are found at the tips of the impeller blades. To treat this flow field heterogeneity, we apply volume averaging to the PBE (eq 13). This leads to i-1 X
dÆNi æ 1 ¼ ÆKj, i - j Nj Ni - j æ dt 2 j¼1
¥ X
ÆKi, j Ni Nj æ
ð17Þ
j¼1
where, here Æ.æ refers to volume average quantities. Marchisio et al.45 found that, when macromixing is much faster than the rate of change of the CMD, the correlations in eq 17 reduce to ÆKi, j Ni Nj æ ÆKi, j æÆNi æÆNj æ
ð18Þ
(37) B€abler, M. U.; Morbidelli, M. J. Colloid Interface Sci. 2007, 316, 428–441. (38) Li, X. Y.; Zhang, J. J. J. Colloid Interface Sci. 2003, 262, 149–161. (39) Soos, M.; Sefcik, J.; Morbidelli, M. Chem. Eng. Sci. 2006, 61, 2349–2363. (40) Melis, S.; Verduyn, M.; Storti, G.; Morbidelli, M.; Bazdyga, J. AIChE J. 1999, 45, 1383–1393. (41) Honig, E. P.; Roebersen, G. J.; Wiersema, P. H. J. Colloid Interface Sci. 1971, 36, 97–109. (42) Bourne, J. R.; Yu, S. Ind. Eng. Chem. Res. 1994, 33, 41–55. (43) Sharp, K. V.; Adrian, R. J. AIChE J. 2001, 47, 766–778. (44) Baldi, S.; Yianneskis, M. Chem. Eng. Sci. 2004, 59, 2659–2671. (45) Marchisio, D. L.; Soos, M.; Sefcik, J.; Morbidelli, M. AIChE J. 2006, 52, 158–173.
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The characteristic time for macromixing in a stirred tank can be approximated through the circulation time46 tc = 5.9T(ÆεæD)-1/3 where T and D are tank and impeller diameter, respectively. Estimating the characteristic time for the change of the CMD as ta ≈ (Gφ)-1 we arrive at the following condition for the applicability of eq 18 1 Dη2 φ, 5:9 T 3
!1=3 ð19Þ
where we used G = (Æεæ/ν)1/2. Applied to the most severe condition investigated in this study (ω = 1000 rpm) where η ≈ 36 μm, eq 19 leads to φ , 4 10-4 (T = 0.15 m and D = 0.06 m). Hence, the maximum solid volume fraction investigated in this study (φ = 4 10-5) is well within the aggregation controlled regime. Given that eq 18 holds true, a dimensionless form of eq 17 is readily obtained by dividing the latter by (ÆGæRp3N1,02) where N1,0 is the initial number of primary particles. This results in i-1 ¥ X 4π dni 1X kj, i - j nj ni - j - ni k i , j nj ¼ 3 dτ 2 j¼1 j¼1
ð20Þ
where ni = ÆNiæ/N1,0 and τ = tÆGæφ. The factor (4/3)π on the lefthand side is introduced to recover φ = (4/3)πRp3N1,0. Furthermore, the dimensionless aggregation rate function reads as 3 ki, j ¼ k0 Æfi, j γ^ æ i1=df þ j 1=df þ
4π - 1=df i þ j - 1=df i1=df þ j 1=df Pep
ð21Þ
where Pep is the primary particle Peclet number Pep ¼
6πμR3p ÆGæ kT
ð22Þ
The aggregation efficiency averaged over the tank volume Æfi,jγˆ æ depends solely on the variation of the dimensionless shear rate γˆ G/ÆGæ within the stirred tank. Characterizing the latter through a probability density function, it follows45 Z
¥
Æfi, j γ^ æ ¼ 0
fi, j γ^ pγ^ ð^γ Þ d^γ
ð23Þ
where pγˆ (γˆ ) = ÆGæpG(G) is the distribution of the (dimensionless) shear rate and pG(G) dG is the volume fraction of the stirred tank assuming a shear rate ∈(G, G þ dG). Figure S1 in Supporting Information shows pG(G) for the stirred tank used in this study for 200 and 600 rpm as obtained from CFD. Details of the CFD calculations are also given there. The distribution is found to be self-similar with respect to ÆGæ, and it exhibits wide tails. Calculations shown hereafter all include flow field heterogeneity. To numerically solve eq 20, the fixed pivot method of Kumar and Ramakrishna47was used. The coarseness and width of the applied grid were adjusted to properly resolve the growth kinetics. For all investigated cases, the grid consisted of 220 grid points with the first 25 distributed linearly followed by a geometric grid with a geometric factor of 1.04. (46) Nienow, A. W. Chem. Eng. Sci. 1997, 52, 2557–2565. (47) Kumar, S.; Ramkrishna, D. Chem. Eng. Sci. 1996, 51, 1311–1332.
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Table 1. Polystyrene Latexes Used in This Studya White sulfate polystyrene latex, dp = 810 nm Product-No: 1-800, c.v. 2%, Batch-No: 642,4, s.c. 8.1 g/100 mL White sulfate polystyrene latex, dp= 600 nm Product-No: 1-600, c.v. 5.2%, Batch-No: 2042,1, s.c. 8 g/100 mL White sulfate polystyrene latex, dp = 420 nm Product-No: 1-400, c.v. 4.0%, Batch-No: 1034,2, s.c. 8 g/100 mL White carboxyl modified polystyrene, dp = 100 nm Product-No: 2-100, c.v. 5.8%, Batch-No: 1909,1, s.c. 4 g/100 mL a c.v. = coefficient of variation, s.c. = solid content.
4. Experimental Section 4.1. Experimental Procedure and Material. All experiments were performed in a 2.5 L baffled stirred tank reactor. The tank was equipped with a lid and an overflow tube such that it could be operated without a gas-liquid interface. An external circulation loop was installed for online characterization of the CMD. Details of the experimental setup can be found in our previous works48,49 and in the Supporting Information. The initial suspension with the desired solid volume fraction was obtained by diluting the original latex with the appropriate amount of deionized water. The aggregation process was started by injecting the required amount of a coagulant solution (30 mL of 20% w/w Al(NO3)3 in water) to the coagulator using a syringe pump. The injection time was approximately 15 s which is at maximum onethird of the characteristic time for aggregation ta = (ÆGæφ)-1 (40 s at 1000 rpm and φ = 2 10-5). The resulting salt concentration was well above the critical coagulation concentration, i.e., the primary particles were completely destabilized by fully suppressing any electrostatic repulsion between them. Altogether, four different polystyrene latexes, all purchased from Interfacial Dynamics Corp. (IDC), Portland, OR, USA, were studied in the present work. An overview is given in Table 1. The first and third latexes were already used in our previous work,48-50 and results from these are used here. The last latex that exhibits a different surface chemistry was only used for the regrowth experiments (section 5.2). 4.2. Small-Angle Static Light Scattering. The CMD in our experiments was characterized through SASLS using a Mastersizer 2000 (Malvern, U.K.). This device measures the scattered light intensity I(q) as a function of the scattering vector amplitude q = 4πn sin(θ/2)/λ, where θ is the scattering angle, n is the refractive index of the dispersing fluid, and λ is the laser wavelength in vacuum. For the present system, λ = 633 nm and n = 1.33 (water). Typically, the scattered light intensity is expressed as7 IðqÞ ¼ Ið0ÞPðqÞSeff ðqÞ
ð24Þ
where I(0) is the scattering intensity extrapolated to zero angle, and P(q) and Seff(q) are the form factor and an average structure factor, respectively, that account for the scattering due to the primary particles and their arrangement within the aggregates composing the population. Accordingly, P(q), Seff(q) f 1 as q f 0. As described elsewhere48,49 from the analysis of I(q) in the Guinier region, that is, for qÆRgæ up to about unity, one can extract two moments of the CMD, namely, the mean radius of gyration ÆRgæ and the zero-angle intensity I(0). The relation of these two quantities to the CMD will be given shortly. Often, SASLS is also used to estimate the fractal dimension from the log-log slope of Seff(q) vs q, ÆRgæ-1 , q , Rp-1. This procedure is, however, only valid within the limits of the Rayleigh-DebyeGans (RDG) theory,6 that is, |m - 1| E 1 and (4πRp/λ)|m - 1| E 1 (48) Moussa, A. S.; Soos, M.; Sefcik, J.; Morbidelli, M. Langmuir 2007, 23, 1664–1673. (49) Soos, M.; Moussa, A. S.; Ehrl, L.; Sefcik, J.; Wu, H.; Morbidelli, M. J. Colloid Interface Sci. 2008, 319, 577–589. (50) Ehrl, L.; Soos, M.; Morbidelli, M.; B€abler, M. U. AIChE J. 2009, 55, 3076–3087.
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where m is the relative refractive index (m = 1.195 for polystyrene in water). For the two larger primary particles used in this work (dp =600 and 810 nm), these conditions are not fulfilled. Moreover, in the early stages of the aggregation process the CMD is constituted of nonfractal objects such as singlets, doublets, and triplets, which impedes a proper estimate of df from the scattering pattern even for the smaller primary particles investigated. 4.3. CMD Moments from SASLS. The intensity of scattered light, I(q), of an ensemble of aggregates is the sum of the individual contributions from all scatterers. Within the frame of RDG theory, the latter is expressed as IðqÞ ¼ k1 I0 Vp2 PðqÞ
¥ X
i2 Ni Si ðqÞFi ðq; dp , df Þ
ð25Þ
i¼1
where k1 is a constant depending on the instrumentation, I0 is the intensity of the incident light, Vp is the volume of the primary particle, and Si(q) is the structure factor of aggregates i according to RGD theory. Further, Fi(q; dp,df) is a correction factor accounting for the deviation from RDG theory, i.e., multiple scattering within dense aggregates and aggregates consisting of large primary particles. Recently, Lattuada and Ehrl6 calculated Fi(q; dp,df) using a mean-field T-matrix method and found that it depends in a nontrivial manner on the aggregate mass i, as well as on dp and df. Also, a dependency on the material refractive index was established, whereas the dependency on the scattering vector amplitude q was found to be weak. To keep the computational load for fitting df on a reasonable level, instead of including the detailed model for Fi(q; dp,df) in our model, we approximate the latter through a power law, i.e. Fi ðq; dp , df Þ Fi ð0; dp , df Þ k10 i - c
ð26Þ
where the scattering exponent c c(Rp, df) G 0 is used as second fitting parameter. Once having fitted both df and c to the experimental data, the detailed model is used to validate the fitted value of c and the approximation made in eq 26. This approach is reasonable, since we found that ÆRgæ is relatively insensitive to c justifying the use of two fitting parameters. From eq 25 and eq 26, the following relations are derived that connect the CMD to the light scattering measurements: 1 0¥ P 2 - c 2 1=2 N i R g, i C B C B i i C ÆRg æ ¼ B ¥ A @ P Ni i2 - c i
Ið0Þ ¼ k1 k10 I0 Vp2
¥ X
Ni i2 - c
ð27Þ
i
where Rg,i is the radius of gyration of aggregate i which for i >3 is given by eq 2. For i e 3, rigorous formulas are employed11 pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi Rg, 1 ¼ 3=5Rp ; Rg, 2 ¼ 8=5Rp ; Rg, 3 1:625Rp ð28Þ The factor (k1k10 I0Vp2) in the second part of eq 27 is eliminated by measuring the zero-angle scattering intensity of a suspension of primary particles, I(0)p, at the same solid volume fraction as the suspension having undergone aggregation. Thus, the normalized zero-angle intensity reads as ¥ P
Ið0Þ=Ið0Þp ¼
i
Ni i2 - c
¥ P
ð29Þ Ni i
i
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Figure 2. Overview of initial growth experiments. Lines indicate conditions along which the interaction parameter and the Peclet number are constant. Experimental conditions are given in Table 2. Having Ni as a function of time from solving eq 20, the time evolution of ÆRgæ and I(0)/I(0)p is readily obtained from eqs 27 and 29, respectively.
5. Results 5.1. Initial Aggregation Kinetics. Initial aggregate growth kinetics were measured for three different latexes (dp = 810, 600, and 420 nm) starting from a suspension of primary particles. Figure 2 gives an overview of the experimental conditions. The lines refer to conditions along which the interaction parameter NA, and hence, the Peclet numbers Pep are constant, as the two admit NA Pep = 6πAH/(kT) ≈ 46, which is constant for a given material at a given temperature. The experimental conditions are enumerated along increasing interaction parameter. Detailed specifications are given in Table 2. Figure 3 shows exp 1 (dp = 810 nm, ω = 635 rpm) where the upper and lower panels show the mean radius of gyration and the zero-angle scattering intensity obtained from light scattering measurements, respectively. The different symbols refer to independent repetitions of the same experiment, i.e., three repetitions were conducted and good reproducibility was found. As can be seen, starting from a suspension of primary particles we first observe a rapid exponential growth of the aggregates, whereas later, secondary phenomena, i.e., breakage and restructuring, set in which slow down the aggregate growth, and eventually, a steady state is reached where aggregation and breakage balance each other (the insets to Figure 3 show the long-term evolution of the two measured quantities). The solid volume fraction in this experiment had to be chosen relatively large (φ= 4.0 10-5) to ensure that the aggregates grew fast enough and large enough before substantial breakage and restructuring occurred. Running the experiment at a lower solid volume fraction resulted in slower aggregation (in real time) and gave smaller aggregates. Thus, they had more time to restructure,51 and they were more prone to breakage.52 For aggregates consisting of smaller primary particles, the influence of these secondary effects was less pronounced, and accordingly, experiments could be run at smaller φ. The lines in Figure 3 show UPS with df = 2.04, 2.08, and 2.12. The region enclosed by these lines describes the experimental data up to τ = tÆGæφ ≈ 3, and hence, they give an estimate in the uncertainty in estimating df in the early stage of the present experiment (a similar behavior is found for the other two models, however in another range of df as (51) Becker, V.; Schlauch, E.; Behr, M.; Briesen, H. J. Colloid Interface Sci. 2009, 339, 362–372. (52) B€abler, M. U.; Morbidelli, M.; Bazdyga, J. J. Fluid Mech. 2008, 612, 261–289.
13148 DOI: 10.1021/la1015516
shown in Supporting Information, S3). At later times, the model deviates from the experiments, as it does not include breakage and restructuring. Exp 2-8 are shown in Figures 4 and 5. Thereby, Exps 5 and 8 were repeated at two different solid volume fractions (φ = 1.0 10-5 and 2.0 10-5), whereas Exp 4 was repeated four times at the same solid volume fraction (φ = 4.0 10-5); the other experiments were not repeated. As can be seen, with increasing NA (increasing experimental index) aggregation in dimensionless coordinates becomes faster due to both the increase in interactions (characterized through NA) and Brownian aggregation (characterized through Pep). Notably, no trend with the primary particle size or with the stirring speed is observed indicating that NA and Pep properly characterize the aggregation process (in the absence of repulsive interactions).50 The two parts of Exp 5 exhibit a relatively large deviation such that they are analyzed separately (Exps 5a and 5b). For all other experiments, reproducibility was good. For each experiment and for each aggregation model, the fractal dimension and the light scattering exponent were fitted to the experimental data using a least-squares procedure. It was found that the different models describe the data only over a certain time interval that was determined based on a predefined threshold value for the error. Within this interval, the uncertainties of both df and c are relatively small. A general trend between the different models is observed. In the very early stage of the process, UPS (solid line) is closest to the experimental data, whereas LECS (dashed line) and DECS (dashed-dotted line) overestimate both ÆRgæ and I(0). The latter is thereby farther off. This very early stage lasts until τ ≈ 3.0 for Exp 1, and it becomes shorter for the later experiments. Later, a crossing between UPS and LECS and DECS occurs. From there on, LECS and DECS are in good agreement with the experiments, whereas UPS is rapidly increasing. At even later times, LECS and DECS are also diverging, as the PBE model does not include breakage or restructuring. The fitted model parameters are given in Table 2. The numbers in brackets give the range in τ where the models were fitted. It is seen that the different models predict similar values for df for the different experiments. With respect to UPS that predicts an average fractal dimension of df = 2.07 ( 0.04, LECS and DECS predict smaller values, i.e., df = 2.01 ( 0.05 and df = 1.71 ( 0.06, respectively. For Exp 5b, UPS predicts a relatively small fractal dimension (df = 2.02). However, predictions of df for the same experiment by LECS and DECS are well within the range of values found with these models, which is due to the different time intervals where the models were fitted. On the other hand, for Exp 3 all models predict relatively large fractal dimensions implying that the process is slow compared to the others. However, as this experiment refers to a stirring speed of ω = 1000 rpm the experimental time scales (in real time) are relatively small, which made it more difficult to resolve the aggregation kinetics. The scattering exponent c in most cases was found to depend only on the primary particle size. For UPS and LECS, we found c = 0.09, 0.14, 0.22 and 0.12, 0.20 and 0.22 for dp = 420, 600, 810 nm, respectively; whereas for DECS, we found c to vary around zero (Table 2). An exception was Exp 7 where for UPS and LECS we found c = 0.26 in combination with a relatively small fractal dimension of df = 2.03 and 1.96, respectively. However, using a fractal dimension in the range given above, i.e., df = 2.10 and 1.99 for UPS and LECS, respectively, we found that I(0) is compatible with the former values for c, whereas in this case, ÆRgæ was underestimated for τ ∈ (1.7, 2.5). This underestimation of ÆRgæ indicates a fast growth of the largest aggregates (which influences ÆRgæ more than I(0)), which might stem from insufficient mixing in Langmuir 2010, 26(16), 13142–13152
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Article Table 2. Experimental Conditionsa df
exp
dp (nm)
ω (rpm)
-6
φ (10 )
-1
ÆGæ (s )
ta (s)
NA (-)
PeP (-)
UPS
LECS
DECS
1 2 3 4 5a 5b 6 7 8
810 638 40 618 40 0.27 167 2.10 (0, 3.0) 2.02 (2.2, 3.8) 1.70 (2.1, 4.0) 600 638 40 618 40 0.67 68 2.06 (0, 2.8) 1.96 (0, 3.2) 1.65 (2.4, 3.2) 420 1000 20 1216 40 1.0 46 2.14 (0, 2.4) 2.12 (0, 3.4) 1.82 (0, 3.4) 810 201 40 108 230 1.6 29 2.04 (0, 2.8) 1.95 (0, 3.3) 1.63 (2.6, 3.2) 420 635 10 613 160 2.0 23 2.11 (0, 2.5) 2.05 (0, 2.6) 1.78 (0, 2.7) 420 635 20 613 80 2.0 23 2.02 (0, 2.1) 1.98 (0, 2.4) 1.69 (1.8, 2.7) 420 438 20 350 140 3.5 13 2.10 (0, 2.3) 2.03 (2.0, 2.6) 1.75 (2.2, 3.1) 600 201 40 108 230 3.9 12 2.03 (0, 2.3) 1.97 (1.7, 2.3) 1.69 (1.7, 2.3) 420 201 10, 20 108 930, 460 11 4.1 2.08 (0, 2.0) 1.99 (1.5, 2.3) 1.73 (1.7, 2.4) a AH = 1 10-20 J, kT = 4.11 10-21 J, μ = 8.9 10-4 kg (m s)-1, ta = (ÆGæ φ)-1. For the nine experiments: (UPS) c = 0.22, 0.14, 0.09, 0.22, 0.09, 0.04, 0.09, 0.26, 0.09; (DECS) c = 0.22, 0.20, 0.12, 0.22, 0.12, 0.09, 0.12, 0.26, 0.12; (LECS) c = 0.06, -0.01, -0.05, -0.06, -0.05, -0.09, -0.07, 0.08, -0.07.
Figure 5. Initial growth kinetics for Exps 5b-8. Symbols and lines have the same meaning as in Figure 4. Experimental conditions are given in Table 2.
Figure 3. Evolution of the mean radius of gyration (a) and the zero-angle scattering intensity (b) for dp = 810 nm, ω = 638 rpm, and φ = 4.0 10-5 (exp 1). Different symbols refer to independent repetitions. Lines refer to UPS with df = 2.04, 2.08, and 2.12 (from left to right), and c = 0.22, NA = 0.27, Pep = 167, NL = 0.25. The insets show the long-term evolution of the two quantities.
Figure 6. Scattering correction function Fi(q; dp, df) from the mean field T-matrix method for q = 0 and dp = 810 (circles), 600 (squares), and 420 nm (triangles). Main axis: df = 2.1. Inset: df= 1.7. The straight lines refer to power-law approximations Fi ∼ i-c with the exponents determined from fitting the experiments. Main axis: c = 0.22, 0.14, 0.09. Inset: c = 0.
Figure 4. Initial growth kinetics for Exps 2-5a. Different symbols refer to experiments performed at different solid volume fractions. Triangles: φ = 4.0 10-5. Squares and diamonds: φ = 2.0 10-5. Crosses: φ = 1.0 10-5. Solid line: UPS. Dashed line: LECS. Dashed-dotted line: DECS. Experimental conditions are given in Table 2.
this particular experiment. A further deviation of c was found for Exp 5b where c = 0.04 and 0.09 for UPS and LECS, respectively. Langmuir 2010, 26(16), 13142–13152
In Figure 6, we compare the obtained values of c against detailed calculations of the scattering correction function Fi(q; dp, df) using mean field T-matrix method.6 Figure 6 shows Fi(0; dp, df) as a function of the aggregate mass for df = 2.1 and dp = 810, 600, and 420 nm, where the former value of df corresponds to the one found from UPS. The straight lines represent the power-law approximation of Fi(0; dp, df) (eq 26) with the scattering exponents found from this model. Good agreement is found which justifies the power-law approximation and which validates the values DOI: 10.1021/la1015516
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B€ abler et al. Table 3. Regrowth Experimentsa
exp
dp (nm)
ω (rpm)
-6
φ (10 )
ÆGæ (s-1)
ÆRgæSt.St. (μm)
ÆRgæ¥ (μm)
df (-)
c (-)
810 1085 f 201 40 1376 f 108 5.8 11.9 2.50 0.45 100 638 f 419 40 618 f 327 5.1 2.70 0.22 a ÆRgæSt.St. = mean radius of gyration at steady state at the higher stirring speed. ÆRgæ¥ = mean radius of gyration at infinite dilution at the lower stirring speed. Model parameters. Exp Rl: NA = 157, Pep = 29, θ = 1.36 10-2, ζ = 0.16. Exp R2: NA = 275, Pep = 0.17, θ = 3.73 10-5, ζ = 0.22. R1 R2
found for c. A similar plot for df = 2.0 as found for LECS is shown in Supporting Information (S5). Notably, the agreement with the power-law approximation is less good in this case. The inset in Figure 6 shows Fi(0; dp, df) for df = 1.7, which is the value found for DECS. The straight lines refer to the power-law approximation with c = 0 as found from this model. A clear deviation can be observed in this case indicating a deficiency of DECS. 5.2. Regrowth Experiments. From the analysis of the initial growth experiments, we can conclude that all three models predict the initial aggregation kinetics reasonably well using, however, substantially different values of the fractal dimension. Since df is not accessible experimentally during this early stage of the experiment, a proper discrimination between the models from initial kinetics therefore cannot be made. Consequently, another kind of experiment is required. Such are provided by what we call regrowth experiments. In these experiments, we run the aggregation process similar to the previous experiments; this time, however, we left the aggregates to grow until the CMD reached a steady state. As this steady state is controlled by a balance between aggregation and breakage, lowering the stirring speed will reduce the breakage rate, and thus, it will cause the aggregates to undergo a regrowth step, until later breakage again stops the growth and a new steady state is reached. Measuring this regrowth step not only enables us to monitor aggregation events between clusters, but also the fractal dimension can be estimated with confidence as the aggregates in these experiments are large and many in number. The idea that the regrowth is initially dominated by aggregation is supported by comparing the limiting aggregate size at infinite dilution at the lower stirring speed and the steady-state size obtained at the higher stirring speed. The former corresponds to a critical aggregate size below which breakage is insignificant at the lower stirring speed.48,49 Hence, having the critical aggregate size at the low stirring speed larger than the steady-state aggregate size at the high stirring speed suggests that breakage can be ignored during the initial regrowth. Two sizes of primary particles were used for this type of experiment. In exp R1 (Table 3), dp = 810 nm, and the stirring speed was changed from 1085 to 201 rpm. Due to the non-RDG behavior of these particles, the determination of df through light scattering is inappropriate.6 To overcome this limitation, we applied image analysis of 2D projections of aggregates for determining df. Details of this method are given in ref 49. From this, we found df = 2.62 ( 0.18 independent of the stirring speed. This is substantially higher than the values found for the initial kinetics. Hence, there has occurred substantial restructuring16,39,51 once the CMD has reached steady state. The limiting aggregate size at infinite dilution at 201 rpm was measured for this latex by Moussa et al.48 who found ÆRgæ = 11.9 μm. Compared to the steady-state aggregate size at 1085 rpm, which in our experiment was ÆRgæ= 6.2 μm, we see that the former is twice as large, suggesting that breakage can initially be ignored. In exp R2, dp = 100 nm, and the stirring speed was changed from 638 to 419 rpm. For these particles, df could be estimated from the log-log slope of Seff(q) as shown in Figure 7. From this, we found df = 2.7 independent of the stirring speed. The limiting aggregate size at infinite dilution was not measured for this latex. However, detailed simulations 13150 DOI: 10.1021/la1015516
Figure 7. Structure factor as a function of the scattering vector amplitude at different times during the regrowth experiment R2 (dp = 100 nm). Squares: t = 25 min corresponding to the first steady state at high rpm. Triangles: t = 27.5 min corresponding to the regrowth period. Circles: t = 30 min corresponding to the second steady state at low rpm.
including the relaxation to the second steady state show that breakage is insignificant during the early stage of the regrowth.53 To model the regrowth experiments, we proceeded as follows. In a first step, an initial CMD was constructed that has the same values of ÆRgæ and I(0) as those measured experimentally before the regrowth. In a second step, this initial CMD was used as an initial condition to eq 20 to simulate the regrowth upon lowering the stirring speed. The initial CMD was obtained by including breakage in the PBE model and solving for the steady state.37 For this, we assumed that breakage is governed by a simple power-law rate function,54 KB,i = kBRc,iν, where kB is a prefactor that depends on the shear rate (typically described through a power-law) and ν is a breakage exponent. Here, ν = 3 is assumed, i.e., the breakage rate is proportional to the volume of the aggregate. Furthermore, the breakup of an aggregate is assumed to create two fragments with relative masses ζ and (1 - ζ). Denoting by pi, j the probability that the breakup of an aggregate j forms a fragment i, the former translates into37 pi,j = (1/2)δi,k þ (1/2)δi,j-k, k = round(ζ j). The dimensionless PBE including breakage then reads as ¥ X dni 2pi, j ðθj v=df Þnj ¼ ½rhs of eq 20 - θiv=df ni þ dτ j¼iþ1
ð30Þ
where θ = kB/(ÆGæRp3N1,0) is the dimensionless breakage coefficient. Although this breakage model has been criticized for being oversimplified52 and not flexible enough for describing breakage for an extended range of operation conditions,37 it is suitable for the current purpose of constructing a CMD approximating the experimental one. In particular, varying both θ and ζ allows us to describe both ÆRgæ and I(0) before the regrowth. The initial CMDs constructed in this way are shown in Supporting Information (S6). The parameters used in eq 30 to model these CMDs are given in Table 3 where, for simplicity, before the regrowth we used an aggregation efficiency of unity, i.e., fi, j 1 and k0 = 1.29. (53) Manuscript in preparation. (54) Spicer, P. T.; Pratsinis, S. E. AIChE J. 1996, 42, 1612–1620.
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fidence. Using the measured fractal dimension in the three models, we found that only UPS was able to correctly predict the aggregation kinetics of the regrowth whereas the other two models clearly underestimate the growth rate. Interestingly, the third model, i.e., DECS, that presents a rigorous implementation of the effective collision sphere model leads to the worst prediction, whereas LECS that presents a simplified implementation of the effective collision sphere model gives better predictions. With regard to the detailed behavior of the three models that is illustrated in Figure 1, we conclude that the relaxation of the aggregation efficiency to a plateau value, as seen for UPS (solid lines), is more reasonable than a monotonic decreasing efficiency as predicted by the two ECS models (dashed and dashed-dotted lines). Furthermore, having established the validity of the UPS model, we conclude that the actual fractal dimension in the initial stage of aggregation is the one found from this model, i.e., df = 2.07 ( 0.04. Figure 8. Regrowth experiments. (a,b) Exp R1: dp = 810 nm and ω = 1085 f 201 rpm at t* = 29 min. (c,d) Exp R2: dp = 100 nm and ω = 638 f 419 rpm at t* = 26.6 min. Solid lines: UPS. Dashed line: LECS. Dashed-dotted line: DECS. In (a,b), for all models df = 2.50 and c = 0.45. In (c,d), for all models df = 2.70 and c = 0.22.
Figure 8 shows the two regrowth experiments. The two left panels show exp R1 (dp = 810 nm), whereas the two right panels show exp R2 (dp = 100 nm). In the former, for all models df = 2.5, whereas in the latter, df = 2.7. Notably, both these values are within the range of the experimentally determined fractal dimensions. Furthermore, c = 0.45 and c = 0.22, which are close to the values determined by Ehrl et al.5 for dense aggregates made of 810 and 120 nm particles (c = 0.53 ( 0.03 and 0.18 ( 0.06, respectively). As can be seen from Figure 8, the difference between the three models is striking: The two ECS models (dashed = LECS, dashed-dotted = DECS) clearly underestimate the growth rate, whereas UPS leads to satisfactory agreement for both ÆRgæ and I(0). A fractal dimension as small as df = 2.2 and 2.4 would be required for describing the two experiments using LECS, and an even smaller one would be required for DECS, too far below the experimental values to be acceptable.
6. Conclusions We explored three different aggregation models to describe shear aggregation of fractal clusters in turbulent flows, and we applied them to interpret initial aggregation kinetics of three different fully destabilized polystyrene latexes measured in a stirred tank. The models account for hydrodynamic and colloidal interactions between the colliding particles and aggregates, and they have a single tunable parameter, which is the fractal dimension. In the initial stage of the process, the latter is not accessible to direct measurements, as the aggregates are small and few in numbers, and hence, df was used as a fitting parameter. It was found that all three models are able to describe the initial aggregation kinetics reasonably well using, however, substantially different values of the fractal dimension, i.e., df = 2.07 ( 0.04, 2.01 ( 0.05, and 1.71 ( 0.06 in the case of UPS, LECS, and DECS, respectively. These fractal dimensions are reasonable, as they roughly correspond to the values found under static conditions, i.e., RLCA, df ≈ 2.1; and DLCA, df ≈ 1.8. Moreover, the small variation in df over all experiments indirectly validates the linear additivity of Brownian and shear aggregation, at least for the fully destabilized systems studied here. To discriminate among the models, we run a second type of experiment where we studied the regrowth of preformed aggregates upon the sudden reduction of the stirring speed. Since the aggregates in these experiments are well-developed and many in number, the fractal dimension could be measured with conLangmuir 2010, 26(16), 13142–13152
Acknowledgment. This work was financially supported by the Swiss National Science Foundation (Grant No. 200021-121882 and 200020-113805/1). The authors thank Marco Mazzotti and Lyonel Ehrl (both ETH Zurich) for useful discussions and valuable input. Furthermore, we thank Marco Lattuada (ETH Zurich) for providing the mean field T-matrix code. Supporting Information Available: Additional tables and figures as described in the text. This material is available free of charge via the Internet at http://pubs.acs.org.
Glossary Notation AH D c df dp Fi(q; dp, df) fi,j G I(q) Ki,j k0 k Ni, (ni) NA NL Pep Rp Rc,i Rg,i Rh,i T
Hamaker function, J diffusion coefficient, m2 s-1 scattering exponent fractal dimension primary particle diameter, m scattering correction function aggregation efficiency shear rate, s-1 light scattering intensity, arb. units aggregation rate function, m3 s-1 prefactor to the aggregation rate function Boltzmann constant, J K-1 (dimensionless) number concentration of aggregates i, m-3 interaction parameter, eq 5 retardation parameter, eq 6 particle Peclet number, eq 22 primary particle radius, m collision radius of aggregate i, eq 3, m radius of gyration of aggregate i, eq 2 and eq 28, m hydrodynamic radius of aggregate i, m temperature, K
Greek Symbols γˆ δi,j ε η λ λL μ ν τ
dimensionless shear rate, γˆ = G/ÆGæ Kronecker function turbulent energy dissipation rate, m2 s-3 Kolmogorov length scale, η = (ν3/ε)1/4, m particle size ratio London wavelength, m dynamic viscosity, kg (m s)-1 kinematic viscosity, m2 s-1 dimensionless time, τ = tÆGæφ DOI: 10.1021/la1015516
13151
Article
φ ω
B€ abler et al.
solid volume fractions stirring speed, rpm
Acronyms CMD DECS
cluster mass distribution detailed ECS model
13152 DOI: 10.1021/la1015516
ECS LECS PBE RDG rpm SASLS UPS
effective collision sphere model lumped ECS model population balance equation Rayleigh-Debye-Gans (light scattering theory) revolutions per minute, 2π min-1 small angle static light scattering uniform permeable sphere model
Langmuir 2010, 26(16), 13142–13152
