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Initial growth kinetics and structure of colloidal aggregates in a turbulent coagulator
Powder Technology 156 (2005) 226 – 234 www.elsevier.com/locate/powtec
Initial growth kinetics and structure of colloidal aggregates in a turbulent coagulator Maurice H. Waldner, Jan Sefcik, Miroslav Soos, Massimo Morbidelli* Institut fu¨r Chemie- und Bioingenieurswissenschaften, Swiss Federal Institute of Technology Zu¨rich, ETH Ho¨nggerberg/HCI, CH-8093 Zu¨rich, Switzerland Available online 28 June 2005
Abstract Aggregation kinetics of colloidal polymeric particles in diluted dispersions was monitored in a stirred tank coagulator using small angle static light scattering. Experimental data were evaluated to obtain two independent moments of the cluster mass distribution along the aggregation process. We applied a population balance modeling approach to early stages of aggregate growth, where breakage is not yet significant compared to aggregation, in order to simultaneously describe both measured moments of the cluster mass distribution. This approach allowed us to uniquely determine both the aggregation rate constant and the effective fractal dimension for aggregates produced during the initial growth regime in the coagulator. We also showed that, similarly as observed in quiescent aggregating systems, the power law region analysis of static light scattering data underestimates the fractal dimension of aggregates early in the course of aggregation. D 2005 Elsevier B.V. All rights reserved. Keywords: Aggregation kinetics; Stirred tank; Static light scattering; Structure factor; Fractal dimension
1. Introduction Aggregation of colloidal particles is an important step involved in processing of particulate materials for numerous applications including polymer manufacturing, food processing or water treatment. Colloidal particles aggregate due to attractive dispersive forces when repulsive interactions between particles are sufficiently weak and can be overcome by their relative motion. Aggregation can be driven by Brownian motion or by fluid velocity gradients or fluctuations. Smoluchowski derived the second order rate constant for aggregation of spherical particles driven by undisturbed simple shear flow in the absence of interparticle interactions. Under these assumptions the aggregation rate constant for two particles with radii R 1 and R 2 in the simple shear flow with the shear rate G is K 12 = a ssG(R 1 + R 2)3, where a ss = 4/3 [17]. A similar expression was derived for turbulent flows, with a prefactor a tu of 1.294 [15], where the shear rate is
* Corresponding author. Tel.: +41 1 6323033; fax: +41 1 6321082. E-mail address: [email protected] (M. Morbidelli). 0032-5910/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2005.04.014
related to the local turbulent energy dissipation rate e and pffiffiffiffiffiffiffi the kinematic viscosity m through G ¼ e=m. However, aggregation rate prefactors obtained from experiments [1,5] at fully destabilized conditions (in the absence of repulsive interactions) are lower than the theoretical values provided by these simple models, mainly because of fluid flow effects at close separations between two aggregating particles. The rate constant of aggregation between primary particles is one of basic characteristics of particles in a given flow and can be estimated the same way as in quiescent systems, for example from light scattering measurements [9], provided that the aggregation of primary particles is slow enough so that the kinetics of doublet formation can be conveniently observed. When we consider shear aggregation beyond the doublet formation, even for spherical primary particles, we need to take into account that aggregating entities are not necessarily spherical. Then we have to use effective collision radii for clusters instead of simple radii for spheres. Moreover, aggregation rate prefactors for irregular clusters may in general be different from those for spheres with mass equivalent radii. Thus the aggregation rate constant for two
M.H. Waldner et al. / Powder Technology 156 (2005) 226 – 234
aggregates with collision radii R c,1 and R c,2 can be written as K 12 = aG(R c,1 + R c,2)3, where the aggregation rate prefactor a can in general depend on flow characteristics, the collision radii and the aggregate structure. Since the structure of aggregates influences the fluid flow around them as well as the strength and the extent of intercluster interactions, kinetics of shear driven aggregation is expected to be strongly influenced by the aggregate structure. Aggregation kinetics is customarily modeled with population balance equations [14]. It is convenient to take mass as the internal coordinate, because mass is the primary conserved quantity in aggregation and breakage events. However, other quantities have been used, such as volume (e.g. for coalescence) or particle length (e.g. for crystallization). Of course, if density is size independent, then the aggregate volume is simply proportional to the aggregate mass. If, in addition, also the shape habit is size independent, then length is proportional to the cube root of the aggregate mass. However, if the aggregate density does depend on size, as has been well established by computer simulations as well as by experimental observations for cluster – cluster aggregation, we need to use mass as the internal coordinate. Quantitative relationships between aggregate mass and size are thus among critical components of aggregation kinetic models. How can we characterize and quantify structure of aggregates produced by shear driven aggregation? There have been numerous studies on structure of aggregates produced in diffusion or ballistic driven regimes using light scattering and microscopy as well as Monte Carlo simulations. In these conditions aggregates are fractal-like and there is a well documented power law scaling relationship between mass and size, usually presented in terms of the radius of gyration; m å R gd f, where d f is the fractal dimension. It was found that for irreversible cluster – cluster aggregation, in the absence of subsequent aggregate restructuring, fractal dimensions around 2 are obtained (from 1.7 to 1.8 for the diffusion limited regime to 2.0 for the ballistic regime to 2.1 for the reaction limited regime). Electron microscopy was used to determine sizes and masses of aggregates with fractal dimensions up to 2, where a planar projection is an accurate representation of a three dimensional object. Static light scattering can be used to determine fractal dimensions, with appropriate light scattering models for aggregate structure factors. However, it has been observed that the apparent fractal dimension increases during early stages of aggregate growth [6,10], approaching asymptotically the true fractal dimension of growing aggregates. The slope of the power law region underestimates the true fractal dimension early in the course of aggregation, due to a significant contribution to scattering from smaller aggregates, which do not have fully developed fractal scattering behavior. Therefore light scattering measurements taken alone are in general not suitable for determination of the fractal dimension of aggregates produces in early stages of aggregate growth. Several approaches have been used to determine fractal dimension of aggregates from turbulent shear conditions,
227
including settling and image analysis, light scattering and kinetic modeling. Direct characterization techniques for measurement of aggregate fractal dimension are discussed in a recent review [2]. Determination of aggregate fractal dimension from settling velocities is limited by effects of polydispersity, aggregate heterogeneity and shape. Although direct imaging and subsequent image analysis seems in principle to be the most reliable way to quantify relationships between aggregate size and mass, collection of representative data sets is not straightforward. More work is desired to evaluate a practical utility of this technique in order to relate aggregate structure to processing conditions. The kinetic modeling approach is based on fitting of measured integral quantities (such as the total number of particles, scattered intensity or average size) with a population balance model with an adjustable parameter being the fractal dimension d f in the relation between the cluster mass and the collision radius, m å R cd f. There are two critical issues related to a proper application of the kinetic modeling approach. First, one needs to establish a quantitative correspondence between the measured and calculated quantities, so that the same thing is measured and calculated. Second, one should use sufficient experimental information in order to uniquely determine kinetic model parameters, in order to get a physically meaningful fractal dimension. Neither of these issues has been satisfactorily addressed in the previous literature, although this approach is in principle the most straightforward one to determine fractal dimension in early stages of aggregate growth. Here we report small angle static light scattering measurements on time evolution of aggregate population during turbulent shear aggregation of polystyrene latex particles in a stirred vessel. We apply the kinetic modeling approach to the early stages of aggregate growth, where breakage is not yet significant compared to aggregation, in order to simultaneously describe two different moments of the cluster mass distribution. This approach allows us to uniquely determine both the aggregation rate prefactor and the fractal dimension for small aggregates produced in the initial growth regime in a turbulent coagulator.
2. Methodology 2.1. Experimental We used a white sulfate polystyrene latex supplied by Interfacial Dynamics Corporation (IDC), Portland, OR (USA) (Product-No: 1-800, CV: 2%, Batch-No: 642,4, solid% = 8.1). The declared mean diameter of latex particles was 810 nm, in agreement with our small angle light scattering measurements. The particle size distribution was very narrow and can be well approximated as monodisperse. Aggregation kinetics was studied in a 2.5 L stirred tank coagulator (Table 1, Fig. 1) as a function of the solid volume
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Table 1 Specifications of the coagulator Coagulator dimensions Volume Diameter Height Baffles Impeller Ports
2.5 L 150 mm 150 mm 4 cylinders, diameter 12 mm, 55 mm from the center Rushton Turbine, 50 mm from bottom Located at the bottom, 40 mm from center
fraction / s and the energy input controlled by the stirring speed (expressed in terms of rpm). The initial suspension at the desired solid volume fraction / s was obtained by diluting the original latex with a predetermined volume of doubly distilled water. In order to break possible aggregates, the solution was put into an ultrasonic bath for 15 min. The initial suspension was then pumped from the storage tank into the previously cleaned coagulator. After filling, a care was taken to remove any air bubbles inside the coagulator by alternating pumping and stirring. A small reservoir (V = 100 mL) was positioned 1 m above the coagulator in order to allow replenishing of liquid in the coagulator during sampling. This helped to ensure that air does not enter the coagulator at any point during the process. The suspension was first stirred for 15 min at 1000 rpm, then for 10 min at the desired stirring speed for subsequent coagulation before a destabilizing salt solution was added to screen the electrostatic repulsive interactions between latex particles. To start the process of aggregation, 20 mL of Al(NO3)3 solution at the weight fraction of 0.363 was injected into the coagulator. The resulting salt concentration was well above the critical coagulation concentration (CCC) for the given system, i.e. the particles were completely destabilized. The
injection procedure was carried out with a Lambda Vit-Fit programmable syringe pump, set to the maximum speed, corresponding to the injection time of approximately 15 s. A typical mixing time of the injected solution was about 5 s at 200 rpm (tested by dye measurement), an order of magnitude below the characteristic time of shear aggregation, given by (G/ s) 1, at all conditions considered here. Two process parameters are available to control the coagulation process of given primary particles in a specific stirred vessel: the solid volume fraction / s and the stirring speed. For our experiments, we chose five different rotational speeds (200 – 800 rpm) and three different solid volume fractions / s (2 10 5, 4 10 5 and 8 10 5). This resulted in two series of experiments, Fig. 2: (i) one series varying the stirring speed (and therefore the average energy dissipation rate e) while keeping the volume fraction / s constant at 4 10 5, and (ii) another series varying the volume fraction / s, while keeping the stirring speed constant at 200 rpm. The overlap between these two series of experiments is at / s = 4 10 5 and 200 rpm. Each experiment was carried out several times to assess reproducibility of the data. The process of aggregation was monitored with small angle light scattering. Small angle light scattering measurements were performed with the Mastersizer 2000 of Malvern, Inc. (UK). For offline measurements, 10 mL samples were withdrawn from the coagulator into a syringe filled with 50 mL of polyvinylalcohol (PVA) solution (2.5 mg L 1) in order to stabilize clusters and prevent their further aggregation [8]. The diluted and stabilized sample was then injected into a light scattering measurement cell, where data were collected in four measurement cycles, with a cycle time of 20 s. In order to rule out artifacts due to sampling and measuring offline, we performed also online measurements for several coagulation runs. These were
Fig. 1. Diagram of the coagulator with specified dimensions (in mm).
M.H. Waldner et al. / Powder Technology 156 (2005) 226 – 234
800
rpm
600
400 300 200
2x10-5
4x10-5
8x10-5
φs Fig. 2. Region where kinetic measurements were carried out.
done with a continuous recycling from the coagulator through a plastic tube to the measurement cell and back into the coagulator. The flow in the measurement loop was driven with a membrane pump positioned after the measurement cell. The flow rate was set so that the average shear rate G in the tube was close to the average shear rate in the coagulator to prevent excessive breakage in the loop. On the other hand, the dimensionless residence time in the loop s l = t lG/ s was kept below 0.05 so that aggregation during external recycling was negligible compared to that in the coagulator. 2.2. Modeling of aggregation kinetics 2.2.1. Population balance equation Growth of aggregates can be described using population balance equations (PBE). There are two principal processes influencing the overall growth kinetics in a stirred tank coagulator: aggregation and breakage. The corresponding PBE can be written as [14]: V X dNk ðt Þ 1 X A ¼ Kij Ni ðt ÞNj ðt Þ Nk ðt Þ KikA Ni ðt Þ dt 2 iþj¼k i¼1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Aggregation
þ
V X
KmB Cmk Nm ðt Þ KkB Nk ðt Þ
ð1Þ
m¼kþ1
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Breakage
where N k (t) is the number concentration of clusters composed of k primary particles, i.e. with dimensionless mass k. In an aggregation event, a cluster with mass i collides efficiently with a cluster of mass j to form an aggregate with mass k = i + j. In a breakage event, a cluster with mass m is split into a collection of fragments with a distribution density function C mk in terms of fragment mass k. Aggregation follows the second order kinetics in concentrations of aggregating clusters as described by the corresponding rate expression K ijAN i (t)N j (t), while breakage is the first order process, with the rate expression K kBN k (t). Unless the system is very dense, the rate
229
constants should be independent of particle concentrations. The matrix of the rate constants (kernel) K ijA and KBi for aggregation and breakage, respectively, then contains all the kinetic information and that is where all the kinetic modeling really goes into. See below for a description of the kernels used in this work, Section 2.2.2. The PBE are solved numerically using the fixed pivot technique of Kumar and Ramkrishna [7]. This procedure is numerically efficient and allows accurate simulations for very broad cluster mass distributions. We used a combined linear/ geometric pivot grid for the cluster mass distribution, where the linear grid with the unit step was used for masses between 1 and 5, followed by a geometric grid with the grid factor of 1.2. It is instructive to rewrite the PBE (1) in a dimensionless form using dimensionless time s ¼ tG/s
ð2Þ
and dimensionless concentration xi ð sÞ ¼
Ni ðsÞV0 /s
ð3Þ
where G is the shear rate, / s is the solid volume fraction and V 0 is the primary particle volume. Note that / s = N 0V 0, where N 0 is the initial number concentration of primary particles. The dimensionless PBE then takes the following form: V X dxk ðsÞ 1 X KijA KikA ¼ xi ðsÞxj ðsÞ xk ðsÞ x i ð sÞ ds 2 iþj¼k V0 G V0 G i¼1
þ
V X Cmk KmB KB xm ðsÞ k xk ðsÞ: /s G /s G m¼kþ1
ð4Þ
The dimensionless form of PBE (4) provides guidance on how to assess the importance of various terms in the PBE model. For example, as far as the evolution of cluster mass distribution in terms of dimensionless time s is independent of the initial solid volume fraction / s, one can observe that the effect of breakage on the aggregate growth kinetics is not significant compared to that of aggregation. As was mentioned before the main objective of this work is to analyze the initial stages of aggregation where breakage is not significant, and therefore discussion concerning breakage will be omitted from further analysis. 2.2.2. Aggregation kernels Aggregation occurs when clusters collide and remain attached to each other. For dilute system in turbulent flows we can assume then only binary collisions are significant. There are two relevant driving forces for aggregation to be considered here: Brownian motion and turbulent shear. Since both Brownian motion driven and shear driven aggregation act simultaneously, one needs to consider their mutual interplay, which can be nontrivial, especially in the case of repulsive particle interactions [4,12]. In the case of fully destabilized conditions considered here it has
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been shown that the full aggregation kernel is well approximated by the sum of the Brownian and the shear kernel [12]: KijA
¼
KijBr
þ
KijSh :
ð5Þ
The respective kernels are given in a general form [3]: KijBr ¼
4p
Di þ Dj Rc;i þ Rc;j W
KijSh ¼ aG Rc;i þ Rc;j
3
dimension d f. The same scaling relationship applies for the collision radius R c,i : Rc;i ¼ Rp i1=df
as well as for the hydrodynamic radius R h,i , which is related to the diffusion coefficient D i through the Stokes – Einstein equation, so that
ð6Þ Di ¼ ð7Þ
where W is the stability ratio of primary particles (near unity for fully destabilized conditions), and G is the shear rate, which can be related to the turbulent energy dissipation rate pffiffiffiffiffiffi ffi e as G ¼ e=m at lengthscales below the Kolmogorov microscale g=(m 3/e)1/4. Here m is the kinematic viscosity of the suspending fluid. The value of prefactor a depends on operating conditions, type of flow and properties of primary particles and has numerical values on the order of 0.1 estimated from experimental measurements, e.g. [1,5], compared to theoretical values on the order of unity derived from simple models not accounting for particle interactions [15,17]. Therefore in this study we treat the prefactor a as an adjustable parameter which is to be obtained by fitting the experimental data. We should note that typical devices used in studies of the aggregation processes in turbulent flows, such as tanks, tubes or slits, exhibit spatial heterogeneity of the shear rate. Therefore we need to consider effects of such heterogeneity on the analysis presented here. Since we consider dilute conditions in small well-mixed tank where the time scale of aggregation (minutes) is much larger than the time scale of mixing (seconds), spatial heterogeneity of concentration is negligible. Thus we can do the simple volume averaging of the aggregation kernel. Since the kernel is linear in G, we get the resulting volume averaged kernel with the volume averaged shear rate G. Therefore in the following analysis we use the volume averaged shear rate G obtained from computational fluid dynamics calculation (see Section 2.3 for more details). 2.2.3. Fractal structure of aggregates For colloidal systems it has been well established that cluster – cluster aggregation results in random aggregates that are described by the fractal scaling relationship [18]. The structure of aggregates is accounted for through a power law relationship between the aggregate dimensionless mass i (the number of primary particles contained in the aggregate) and the radius of gyration R g,i : Rg;i df i ¼ kg ð8Þ Rp with the prefactor k g close to unity. Here R p is the radius of primary particles. The exponent of this scaling is the fractal
ð9Þ
kT 1=df i : 6plRp
ð10Þ
After substituting the fractal scaling formulas above to aggregation kernel expressions from the previous section we get the aggregation kernel in the following form, cf. [3,11]:
2kT 1=df i Kij ¼ þ j1=df i1=df þ j1=df 3l
3 þ aGR3p i1=df þ j1=df : ð11Þ In order to accurately describe the radius of gyration of aggregates, which will be compared to light scattering measurements (see below), we use explicit expressions for the first three clusters [9]: rffiffiffi 3 Rg;1 ¼ ð12Þ Rp 5 Rg;2
Rg;3
rffiffiffi 8 Rp ¼ 5
ð13Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 107 40 cos ð2p=3Þ Rp ¼ 45
ð14Þ
where R p is the primary particle radius. For aggregates with 4 and more primary particles we use the scaling relationship in Eq. (8) with k g = 1. 2.2.4. Small angle static light scattering The measurables obtained from the small angle light scattering instrument were exported as raw data in the form of the intensity of the scattered light I as a function of the scattering vector amplitude q. The scattered light intensity can be expressed as [18] I ðqÞ ¼ I ð0ÞPðqÞS ðqÞ
ð15Þ
where I(0) is the zero angle intensity, P( q) is the form factor (due to primary particles) and S( q) is the structure factor (due to an arrangement of primary particles within aggregates), and q is the scattering angle amplitude n q ¼ 4p sinðh=2Þ k
ð16Þ
Here h is the scattering angle, n the refractive index of the dispersant and k the wave length in vacuo of laser light used. One can use, for example, the Zimm plot [19] to obtain the root mean square radius of gyration hR gi=hR g2i1/2 and
M.H. Waldner et al. / Powder Technology 156 (2005) 226 – 234
the zero angle intensity I(0). The scattered intensity I( q) is expressed in terms of q as I ð 0Þ 1 D 2E 2 ¼1þ R q : ð17Þ I ð qÞ 3 g This equation can be rearranged to D E 2 2 1 1 q Rg ¼ þ I ð qÞ I ð 0Þ 3 I ð0Þ
ð18Þ
and I(0), hR gi are obtained from the intercept and slope estimated from the linear regression of 1/I(q) vs. q 2 in the region where q R g 1. If the aggregates are fractals, then one can estimate their fractal dimension from the slope of the power law region in the log –log plot of S( q) as a function of q. If the primary particles are sufficiently small, so that P( q) is very close to unity within the power law region, then one could simply use the intensity I( q) instead of the structure factor S( q), since they would be proportional. Validity of this assumption is easily checked by calculating the appropriate form factor from the Mie theory or simply by measuring the experimental form factor P( q). In our case it turns out that the primary particles are large enough so that their form factor cannot be neglected. Moreover, the intensity of the scattered light depends on the detector location respective to the plane in which the incident light is polarized. Since it is not known to us whether in our instrument all the detectors are located in the same plane with respect to the polarization plane, we factor out this effect by measuring the form factor P( q) separately and then dividing the measured intensity I( q) by P( q) to get the structure factor S( q). Contributions to the average squared radius of gyration hR g2i from both the form factor P( q) the structure factor S( q) are additive, so that D E D E þ R2g;1 ð19Þ R2g ¼ R2g S ðqÞ
We can determine hR g2iS( q) from S( q) using the Zimm plot as above: D E R2g 2 PðqÞ 1 q S ðqÞ ¼ þ : ð20Þ I ð qÞ I ð 0Þ 3 I ð 0Þ Let us now address the validity of light scattering models used here, considering that our primary particles are larger than the wavelength of the incident light used in light scattering experiments. The value of zero angle intensity measured by light scattering is by definition independent of the scattering wave vector q and therefore does not depend on the laser light wavelength. Although the Zimm plot procedure was originally derived within the Rayleigh – Debye– Gans approximation, we have checked its applicability for larger particles as follows. We used the exact Mie theory to calculate the dependence of scattered intensity on q for spherical particles of diameters from the primary particle size to the largest average size detected in our
231
experiments. Then we applied the Zimm procedure as outlined above to estimate the radius of gyration. The comparison with the true theoretical value of the radius of gyration showed that the difference was never larger than 10%, which is comparable to typical experimental errors due to light scattering measurements reported here. 2.2.5. Relation of cluster mass distribution to static light scattering measurables By solving the PBE with appropriate kernels we get the cluster mass distribution as a function of time. Then from a given cluster mass distribution at any time one can evaluate appropriate moments of the cluster mass distribution corresponding to quantities measured in light scattering experiments [9]. The two moments considered here are the root mean square radius of gyration hR gi=hR g2i1/2, where X D E N i2 R2g;i 2 i i Rg ¼ X ð21Þ N i2 i i and the zero angle intensity I(0) relative to its value at the beginning of aggregation I(0)t=0 X Ni i2 I ð0Þ ¼ X i : ð22Þ I ð0Þt¼0 Ni ð0Þi2 i
2.3. Computational fluid dynamics The fluid dynamics in the coagulator was calculated at steady state for various stirring speeds with a commercial computational fluid dynamics (CFD) software Fluent using a standard k –e model combined with the standard wall function. Due to the symmetry of the vessel it was possible to define the calculation domain as one half of the vessel using periodic boundary conditions. In order to calculate the steady state flow field in the vessel with a rotating impeller, it is recommended to use two different coordinate systems to describe the fluid movement in the vessel. The coordinate system rotating with the impeller was used for a section of the fluid around the impeller. The fixed coordinate system was used for the rest of the vessel. The initial round of CFD calculations involved a mesh with approximately 150,000 computational cells. Since this initial mesh was relatively coarse, we used an adaptation technique based on the dimensionless wall distance and the velocity gradient to improve the accuracy of fluid flow description. The tolerance for the numerical convergence was set to 10 6. Since dispersions considered here were very dilute, one could neglect the effect of solid particles on the fluid flow and therefore the calculations were performed for single phase system. The final result of CFD calculations provided a steady state spatial distribution of the turbulent energy dissipation rate e and the corresponding shear rate G. The corresponding values of the volume averaged shear rate G computed by CFD are shown in Table 2. The width of the
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M.H. Waldner et al. / Powder Technology 156 (2005) 226 – 234
Table 2 Volume averaged shear rate G for various stirring speeds rpm
G/s 1
200 300 400 600 800
108 199 306 564 866
shear rate distribution is the same for all rotation speeds. The minimum and maximum values are equal 0.067 times G and 63 times G, respectively.
3. Results and discussion Results from light scattering measurements on early aggregation kinetics at various solid volume fractions / s and stirring speeds are all shown together in Fig. 3 in terms of the relative mean radius of gyration hR gi/R g,1 and the relative zero angle intensity I(0)/I(0)t=0 against dimensionless time s. The two measured moments of the cluster mass distribution are separate quantitative measures of aggregate growth. It can be seen in Fig. 3 that we get a single master curve for each of the two moments in dimensionless time s up to at least 1.5 for all volume fractions and stirring speeds. Since the breakage terms in the dimensionless form of PBE (4) depend on the solid volume fraction, we can
〈Rg〉 / Rg,1
102
101
conclude that in early stages of aggregate growth (s up to 1.5) breakage is not significantly affecting the aggregate growth kinetics in our system. Moreover, since the early aggregate growth also follows the same kinetics in dimensionless time regardless of the average shear rate G, we can conclude that the contribution from the Brownian motion to the aggregation kernel is only minor and so the aggregation kernel divided by G, see Eq. (4), is approximately constant, as expected for shear driven aggregation. There are only few sets of data in the literature which demonstrate the effect of the solid volume fraction on the coagulation process [13,16]. Results by both of them show that at the beginning the aggregation kinetics, if properly scaled, collapse on one line in agreement with our observations. 3.1. Fitting two moments from light scattering with PBE model Since breakage is not significant for early kinetics of aggregate growth, we can use PBE to model the kinetics up to a certain dimensionless time s considering just the aggregation. The aggregation kernel has two parameters which need to be specified for a given system: the fractal dimension d f and the aggregation rate prefactor a. For any given values of these parameters we can compute time evolution of the cluster mass distribution and from that the two measured moments through Eqs. (21) and (22). Let us first see how well the kinetic data summarized in Fig. 3 can be fitted when considering each of the two moments separately. It turns out that either of the two moments can be fitted separately for dimensionless times s 1.5 for any value of the fractal dimension from 1.8 up to 3. In Fig. 4 we show the aggregation rate prefactor a required to fit the experimental data as a function of the assumed fractal dimension d f. The fitted aggregation rate
φ = 4 × 10-5
0.6 〈Rg〉 / Rg,1 I(0) / I(0)t=0
200 rpm φ = 2 × 10-5 φ = 4 × 10-5 φ = 8 × 10-5
102
I(0) / I(0)t=0
300 rpm 600 rpm
Aggregation rate prefactor
200 rpm, 400 rpm, 800 rpm
100
101
100 0
2
τ
4
6
Fig. 3. Evolution of dimensionless radius of gyration hR gi/R g,1 and zero angle intensity I(0)/I(0)t=0 versus dimensionless time s for various stirring speeds and solid volume fractions.
200 rpm 300 rpm 400 rpm
0.5 0.4 0.3 0.2 0.1 1.6
1.8
2.0
2.2
2.4
df Fig. 4. Combinations of fractal dimension d f and aggregation rate prefactor a that can separately fit each of the measured moments: hR gi/R g,1 (solid lines) and I(0)/I(0)t=0 (dashed lines).
M.H. Waldner et al. / Powder Technology 156 (2005) 226 – 234
〈Rg〉 / Rg,1
102
101
100
100
10-1
S(q)
prefactor does not significantly depend on the stirring speed for the range of values considered here. However, it can be seen that different aggregation rate prefactors are required to fit the two moments for the assumed fractal dimension d f 1.9, i.e. for such fractal dimension the two measured moments cannot be fitted simultaneously with our model. This indicates that such fractal dimension is not suitable for description of early aggregation kinetics in our system. On the other hand, for fractal dimensions d f 1.8 it is possible to fit both moments simultaneously up to a certain dimensionless time s, which becomes shorter with decreasing value of d f. The fitting of both moments is illustrated in Fig. 5. In the top part of Fig. 5 we show the measured radius of gyration together with modeling results for three different values of d f, 1.7, 1.8 and 2.2, respectively, with the corresponding prefactors a from Fig. 4. In all cases a good fit for short times (up to dimensionless time at least 1.5) is obtained. In the bottom part of Fig. 5 we show the measured zero angle intensity together with modeling results for the same set of parameters that were used to fit the measured radius of gyration. The measured zero angle intensity is well described with the model up to s = 1.5 for d f = 1.7 (T 0.1), while for larger d f the intensity increases too fast. We do not consider smaller values of d f to be physically reasonable,
233
10-2
10-3
10-4 10-5
10-4
10-3
q/
10-2
nm-1
Fig. 6. Example of power law region analysis of experimental structure factor S( q) for early stages of aggregation (s = 1.8, circles) and for steady state (s = 16, triangles). The negative slope of the linear part in log – log plot of S( q) vs. q represents the apparent fractal dimension. Conditions: / s 4 10 5 at 200 rpm.
since we are not aware of any experimental or theoretical support for values of d f 1.6 in neither static nor turbulent flow conditions in 3 dimensions. To summarize, we can simultaneously describe both measured moments at early stages of aggregate growth in our system, with the fractal dimension d f = 1.7 (T0.1) and the corresponding aggregation rate prefactor a = 0.2. The aggregation rate prefactor estimated here is similar to previously estimated values for primary particles [1,5]. In addition, the fractal dimension estimated here is close to values of fractal dimension expected for cluster – cluster aggregation in fully destabilized conditions [11], which is the case considered here. We note, however, that it is expected that small aggregates produced in early stages of aggregation process examined here would not have fully developed fractal structure [10]. Nevertheless, the meaning of the fractal dimension in this narrower context is that it is the exponent in the scaling relationship between the mass 2.8 2.4 2.0
101
SE
I(0) / I(0)t=0
102
1.6 1.2 0.8
100 0
2
τ
4
6
Fig. 5. Early growth (small s) of aggregates can be well described in terms of dimensionless radius of gyration hR gi/R g,1 for any fractal dimension d f with appropriate values of prefactor a (top graph, symbols represent experimental data). For example, we show calculations for d f equal to 1.7 (solid line), 1.8 (dashed line) and 2.2 (dotted line). However, only calculations with d f 1.8 can describe both moments simultaneously for s up to 1.5. Conditions: / s = 4 10 5 at 200 rpm.
0.4 0.0
0
3
6
τ
9
12
15
Fig. 7. Time evolution of apparent fractal dimension from power law region analysis of experimental structure factor. Data corresponding to structure factors from Fig. 6 are indicated by arrows. Conditions: / s = 4 10 5 at 200 rpm.
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and the collision radius, which is needed in the shear aggregation kernel. 3.2. Apparent fractal dimension from power law region of static structure factor Let us now look what additional information might be available from static light scattering measurements at early stages of aggregate growth in turbulent shear. Analysis of the power law region of the static structure factor can yield aggregate fractal dimension provided that certain conditions are satisfied, most importantly that aggregates are sufficiently large [10]. Obviously, this assumption is not fulfilled for early times while aggregates are still quite small. Nevertheless, we analyzed the power law region for static structure factors collected from very early times and extracted the corresponding apparent fractal dimension. In Fig. 6 we illustrate this analysis by plotting the measured structure factor S( q) for an early time (s = 1.8) near the inflection point at the growth kinetic curve and for a long time (s = 16) at the steady state. The apparent fractal dimension estimated at s = 1.8 is around 0.9, while the plateau value reached at long times is near 2.3, similar to values for other conditions examined here. The time evolution of the apparent fractal dimension is shown in Fig. 7. As has been observed in light scattering measurements on aggregation in quiescent conditions, the slope of the power law region underestimates the true fractal dimension early in the course of aggregation. This is due to the presence of smaller aggregates which scatter light differently than the larger ones with fully developed fractal scattering behavior. Therefore the measured apparent fractal dimension from the power law region analysis should not be taken as necessarily equal to the real fractal dimension of aggregates, except perhaps at later times, provided that the aggregates formed are sufficiently large.
4. Conclusions Simultaneous fitting of multiple moments of the cluster mass distribution can provide a stringent test for aggregation and breakage kernels used in population balance models. Parameters obtained from such fits should be more robust and reliable than those obtained from fitting a single moment with additional external assumptions. We illustrated this approach for early stages of aggregate growth in a turbulent coagulator, where we simultaneously fitted two measured moments of the cluster mass distribution: the mean radius of gyration and the zero angle intensity. Unlike in fitting just a single moment, we were able to uniquely determine both the aggregation rate prefactor and the fractal dimension for small aggregates produced in the initial growth regime. We also note that one cannot take literally the time evolution of the fractal dimension determined from
the power law region of the static light scattering data, unless it is verified that the aggregates formed are sufficiently large. In general, when the cluster mass distribution is not a narrow one, polydispersity can significantly influence simple relationships between various moments and the apparent fractal dimension estimated from the power law region analysis. In such case one should use a full population balance approach to properly evaluate respective quantities.
References [1] B.K. Brunk, D.L. Koch, L.W. Lion, Observation of coagulation in isotropic turbulence, J. Fluid Mech. 371 (1998) 81 – 107. [2] G. Bushell, Y.D. Yan, D. Woodfield, J.A. Raper, R. Amal, On techniques for the measurement of the mass fractal dimension of aggregates, Adv. Colloid Interface Sci. 95 (2002) 1 – 50. [3] M. Elimelech, J. Gregory, X. Jia, R.A. Williams, Particle Deposition and Aggregation, Measurement, Modelling and Simulation, Butterworth-Heinemann, Woburn, 1995. [4] D.L. Feke, W.R. Schowalter, The effect of Brownian diffusion on shear-induced coagulation of colloidal dispersions, J. Fluid Mech. 133 (1983) 17 – 35. [5] K. Higashitani, K. Yamauchi, Y. Matsuno, G. Hosokawa, Turbulent coagulation of particles dispersed in a viscous fluid, J. Chem. Eng. Japan 16 (1983) 299 – 304. [6] R. Klein, D.A. Weitz, M.Y. Lin, H.M. Lindsay, R.C. Ball, P. Meakin, Theory of scattering from colloidal aggregates, Prog. Colloid Polym. Sci. 81 (1990) 161 – 168. [7] S. Kumar, D. Ramkrishna, On the solution of population balance equations by discretization: I. A fixed pivot technique, AIChE J. 51 (1996) 1311 – 1332. [8] K.A. Kusters, PhD thesis, Eindhoven University of Technology, Netherlands, 1991. [9] M. Lattuada, P. Sandku¨hler, H. Wu, J. Sefcik, M. Morbidelli, Aggregation kinetics of polymer colloids in reaction limited regime: experiments and simulations, Adv. Colloid Interface Sci. 103 (2003) 33 – 56. [10] M. Lattuada, H. Wu, P. Sandku¨hler, J. Sefcik, M. Morbidelli, Modelling of aggregation kinetics of colloidal systems and its validation by light scattering measurements, Chem. Eng. Sci. 59 (2004) 1783 – 1793. [11] P. Meakin, Fractal aggregates, Adv. Colloid Interface Sci. 28 (1988) 249 – 331. [12] S. Melis, M. Verduyn, G. Storti, M. Morbidelli, J. Baldyga, Effect of fluid motion on the aggregation of small particles subject to interaction forces, AIChE J. 45 (1999) 1383 – 1393. [13] V. Oles, Shear-induced aggregation and breakup of polystyrene latex particles, J. Colloid Interface Sci. 154 (1992) 351 – 358. [14] D. Ramkrishna, Population Balances, Academic Press, San Diego, 2000. [15] P. Saffman, J. Turner, On the collision of drops in turbulent clouds, J. Fluid Mech. 1 (1956) 16. [16] T. Serra, J. Colomer, X. Casamitjana, Aggregation and breakage of particles in a shear flow, J. Colloid Interface Sci. 187 (1997) 466 – 473. [17] M. Smoluchowski, Versuch einer mathematischen theorie der koagulations-kinetic kolloider losungen, Z. Phys. Chem. 92 (1917) 129. [18] C.M. Sorensen, Light scattering by fractal aggregates: a review, Aerosol Sci. Technol. 35 (2001) 648 – 687. [19] B.H. Zimm, Apparatus and methods for measurement and interpretation of the angular variation of light-scattering, preliminary results on polystyrene solutions, J. Chem. Phys. 16 (1948) 1099.
Initial growth kinetics and structure of colloidal aggregates in a turbulent coagulator Maurice H. Waldner, Jan Sefcik, Miroslav Soos, Massimo Morbidelli* Institut fu¨r Chemie- und Bioingenieurswissenschaften, Swiss Federal Institute of Technology Zu¨rich, ETH Ho¨nggerberg/HCI, CH-8093 Zu¨rich, Switzerland Available online 28 June 2005
Abstract Aggregation kinetics of colloidal polymeric particles in diluted dispersions was monitored in a stirred tank coagulator using small angle static light scattering. Experimental data were evaluated to obtain two independent moments of the cluster mass distribution along the aggregation process. We applied a population balance modeling approach to early stages of aggregate growth, where breakage is not yet significant compared to aggregation, in order to simultaneously describe both measured moments of the cluster mass distribution. This approach allowed us to uniquely determine both the aggregation rate constant and the effective fractal dimension for aggregates produced during the initial growth regime in the coagulator. We also showed that, similarly as observed in quiescent aggregating systems, the power law region analysis of static light scattering data underestimates the fractal dimension of aggregates early in the course of aggregation. D 2005 Elsevier B.V. All rights reserved. Keywords: Aggregation kinetics; Stirred tank; Static light scattering; Structure factor; Fractal dimension
1. Introduction Aggregation of colloidal particles is an important step involved in processing of particulate materials for numerous applications including polymer manufacturing, food processing or water treatment. Colloidal particles aggregate due to attractive dispersive forces when repulsive interactions between particles are sufficiently weak and can be overcome by their relative motion. Aggregation can be driven by Brownian motion or by fluid velocity gradients or fluctuations. Smoluchowski derived the second order rate constant for aggregation of spherical particles driven by undisturbed simple shear flow in the absence of interparticle interactions. Under these assumptions the aggregation rate constant for two particles with radii R 1 and R 2 in the simple shear flow with the shear rate G is K 12 = a ssG(R 1 + R 2)3, where a ss = 4/3 [17]. A similar expression was derived for turbulent flows, with a prefactor a tu of 1.294 [15], where the shear rate is
* Corresponding author. Tel.: +41 1 6323033; fax: +41 1 6321082. E-mail address: [email protected] (M. Morbidelli). 0032-5910/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2005.04.014
related to the local turbulent energy dissipation rate e and pffiffiffiffiffiffiffi the kinematic viscosity m through G ¼ e=m. However, aggregation rate prefactors obtained from experiments [1,5] at fully destabilized conditions (in the absence of repulsive interactions) are lower than the theoretical values provided by these simple models, mainly because of fluid flow effects at close separations between two aggregating particles. The rate constant of aggregation between primary particles is one of basic characteristics of particles in a given flow and can be estimated the same way as in quiescent systems, for example from light scattering measurements [9], provided that the aggregation of primary particles is slow enough so that the kinetics of doublet formation can be conveniently observed. When we consider shear aggregation beyond the doublet formation, even for spherical primary particles, we need to take into account that aggregating entities are not necessarily spherical. Then we have to use effective collision radii for clusters instead of simple radii for spheres. Moreover, aggregation rate prefactors for irregular clusters may in general be different from those for spheres with mass equivalent radii. Thus the aggregation rate constant for two
M.H. Waldner et al. / Powder Technology 156 (2005) 226 – 234
aggregates with collision radii R c,1 and R c,2 can be written as K 12 = aG(R c,1 + R c,2)3, where the aggregation rate prefactor a can in general depend on flow characteristics, the collision radii and the aggregate structure. Since the structure of aggregates influences the fluid flow around them as well as the strength and the extent of intercluster interactions, kinetics of shear driven aggregation is expected to be strongly influenced by the aggregate structure. Aggregation kinetics is customarily modeled with population balance equations [14]. It is convenient to take mass as the internal coordinate, because mass is the primary conserved quantity in aggregation and breakage events. However, other quantities have been used, such as volume (e.g. for coalescence) or particle length (e.g. for crystallization). Of course, if density is size independent, then the aggregate volume is simply proportional to the aggregate mass. If, in addition, also the shape habit is size independent, then length is proportional to the cube root of the aggregate mass. However, if the aggregate density does depend on size, as has been well established by computer simulations as well as by experimental observations for cluster – cluster aggregation, we need to use mass as the internal coordinate. Quantitative relationships between aggregate mass and size are thus among critical components of aggregation kinetic models. How can we characterize and quantify structure of aggregates produced by shear driven aggregation? There have been numerous studies on structure of aggregates produced in diffusion or ballistic driven regimes using light scattering and microscopy as well as Monte Carlo simulations. In these conditions aggregates are fractal-like and there is a well documented power law scaling relationship between mass and size, usually presented in terms of the radius of gyration; m å R gd f, where d f is the fractal dimension. It was found that for irreversible cluster – cluster aggregation, in the absence of subsequent aggregate restructuring, fractal dimensions around 2 are obtained (from 1.7 to 1.8 for the diffusion limited regime to 2.0 for the ballistic regime to 2.1 for the reaction limited regime). Electron microscopy was used to determine sizes and masses of aggregates with fractal dimensions up to 2, where a planar projection is an accurate representation of a three dimensional object. Static light scattering can be used to determine fractal dimensions, with appropriate light scattering models for aggregate structure factors. However, it has been observed that the apparent fractal dimension increases during early stages of aggregate growth [6,10], approaching asymptotically the true fractal dimension of growing aggregates. The slope of the power law region underestimates the true fractal dimension early in the course of aggregation, due to a significant contribution to scattering from smaller aggregates, which do not have fully developed fractal scattering behavior. Therefore light scattering measurements taken alone are in general not suitable for determination of the fractal dimension of aggregates produces in early stages of aggregate growth. Several approaches have been used to determine fractal dimension of aggregates from turbulent shear conditions,
227
including settling and image analysis, light scattering and kinetic modeling. Direct characterization techniques for measurement of aggregate fractal dimension are discussed in a recent review [2]. Determination of aggregate fractal dimension from settling velocities is limited by effects of polydispersity, aggregate heterogeneity and shape. Although direct imaging and subsequent image analysis seems in principle to be the most reliable way to quantify relationships between aggregate size and mass, collection of representative data sets is not straightforward. More work is desired to evaluate a practical utility of this technique in order to relate aggregate structure to processing conditions. The kinetic modeling approach is based on fitting of measured integral quantities (such as the total number of particles, scattered intensity or average size) with a population balance model with an adjustable parameter being the fractal dimension d f in the relation between the cluster mass and the collision radius, m å R cd f. There are two critical issues related to a proper application of the kinetic modeling approach. First, one needs to establish a quantitative correspondence between the measured and calculated quantities, so that the same thing is measured and calculated. Second, one should use sufficient experimental information in order to uniquely determine kinetic model parameters, in order to get a physically meaningful fractal dimension. Neither of these issues has been satisfactorily addressed in the previous literature, although this approach is in principle the most straightforward one to determine fractal dimension in early stages of aggregate growth. Here we report small angle static light scattering measurements on time evolution of aggregate population during turbulent shear aggregation of polystyrene latex particles in a stirred vessel. We apply the kinetic modeling approach to the early stages of aggregate growth, where breakage is not yet significant compared to aggregation, in order to simultaneously describe two different moments of the cluster mass distribution. This approach allows us to uniquely determine both the aggregation rate prefactor and the fractal dimension for small aggregates produced in the initial growth regime in a turbulent coagulator.
2. Methodology 2.1. Experimental We used a white sulfate polystyrene latex supplied by Interfacial Dynamics Corporation (IDC), Portland, OR (USA) (Product-No: 1-800, CV: 2%, Batch-No: 642,4, solid% = 8.1). The declared mean diameter of latex particles was 810 nm, in agreement with our small angle light scattering measurements. The particle size distribution was very narrow and can be well approximated as monodisperse. Aggregation kinetics was studied in a 2.5 L stirred tank coagulator (Table 1, Fig. 1) as a function of the solid volume
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M.H. Waldner et al. / Powder Technology 156 (2005) 226 – 234
Table 1 Specifications of the coagulator Coagulator dimensions Volume Diameter Height Baffles Impeller Ports
2.5 L 150 mm 150 mm 4 cylinders, diameter 12 mm, 55 mm from the center Rushton Turbine, 50 mm from bottom Located at the bottom, 40 mm from center
fraction / s and the energy input controlled by the stirring speed (expressed in terms of rpm). The initial suspension at the desired solid volume fraction / s was obtained by diluting the original latex with a predetermined volume of doubly distilled water. In order to break possible aggregates, the solution was put into an ultrasonic bath for 15 min. The initial suspension was then pumped from the storage tank into the previously cleaned coagulator. After filling, a care was taken to remove any air bubbles inside the coagulator by alternating pumping and stirring. A small reservoir (V = 100 mL) was positioned 1 m above the coagulator in order to allow replenishing of liquid in the coagulator during sampling. This helped to ensure that air does not enter the coagulator at any point during the process. The suspension was first stirred for 15 min at 1000 rpm, then for 10 min at the desired stirring speed for subsequent coagulation before a destabilizing salt solution was added to screen the electrostatic repulsive interactions between latex particles. To start the process of aggregation, 20 mL of Al(NO3)3 solution at the weight fraction of 0.363 was injected into the coagulator. The resulting salt concentration was well above the critical coagulation concentration (CCC) for the given system, i.e. the particles were completely destabilized. The
injection procedure was carried out with a Lambda Vit-Fit programmable syringe pump, set to the maximum speed, corresponding to the injection time of approximately 15 s. A typical mixing time of the injected solution was about 5 s at 200 rpm (tested by dye measurement), an order of magnitude below the characteristic time of shear aggregation, given by (G/ s) 1, at all conditions considered here. Two process parameters are available to control the coagulation process of given primary particles in a specific stirred vessel: the solid volume fraction / s and the stirring speed. For our experiments, we chose five different rotational speeds (200 – 800 rpm) and three different solid volume fractions / s (2 10 5, 4 10 5 and 8 10 5). This resulted in two series of experiments, Fig. 2: (i) one series varying the stirring speed (and therefore the average energy dissipation rate e) while keeping the volume fraction / s constant at 4 10 5, and (ii) another series varying the volume fraction / s, while keeping the stirring speed constant at 200 rpm. The overlap between these two series of experiments is at / s = 4 10 5 and 200 rpm. Each experiment was carried out several times to assess reproducibility of the data. The process of aggregation was monitored with small angle light scattering. Small angle light scattering measurements were performed with the Mastersizer 2000 of Malvern, Inc. (UK). For offline measurements, 10 mL samples were withdrawn from the coagulator into a syringe filled with 50 mL of polyvinylalcohol (PVA) solution (2.5 mg L 1) in order to stabilize clusters and prevent their further aggregation [8]. The diluted and stabilized sample was then injected into a light scattering measurement cell, where data were collected in four measurement cycles, with a cycle time of 20 s. In order to rule out artifacts due to sampling and measuring offline, we performed also online measurements for several coagulation runs. These were
Fig. 1. Diagram of the coagulator with specified dimensions (in mm).
M.H. Waldner et al. / Powder Technology 156 (2005) 226 – 234
800
rpm
600
400 300 200
2x10-5
4x10-5
8x10-5
φs Fig. 2. Region where kinetic measurements were carried out.
done with a continuous recycling from the coagulator through a plastic tube to the measurement cell and back into the coagulator. The flow in the measurement loop was driven with a membrane pump positioned after the measurement cell. The flow rate was set so that the average shear rate G in the tube was close to the average shear rate in the coagulator to prevent excessive breakage in the loop. On the other hand, the dimensionless residence time in the loop s l = t lG/ s was kept below 0.05 so that aggregation during external recycling was negligible compared to that in the coagulator. 2.2. Modeling of aggregation kinetics 2.2.1. Population balance equation Growth of aggregates can be described using population balance equations (PBE). There are two principal processes influencing the overall growth kinetics in a stirred tank coagulator: aggregation and breakage. The corresponding PBE can be written as [14]: V X dNk ðt Þ 1 X A ¼ Kij Ni ðt ÞNj ðt Þ Nk ðt Þ KikA Ni ðt Þ dt 2 iþj¼k i¼1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Aggregation
þ
V X
KmB Cmk Nm ðt Þ KkB Nk ðt Þ
ð1Þ
m¼kþ1
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Breakage
where N k (t) is the number concentration of clusters composed of k primary particles, i.e. with dimensionless mass k. In an aggregation event, a cluster with mass i collides efficiently with a cluster of mass j to form an aggregate with mass k = i + j. In a breakage event, a cluster with mass m is split into a collection of fragments with a distribution density function C mk in terms of fragment mass k. Aggregation follows the second order kinetics in concentrations of aggregating clusters as described by the corresponding rate expression K ijAN i (t)N j (t), while breakage is the first order process, with the rate expression K kBN k (t). Unless the system is very dense, the rate
229
constants should be independent of particle concentrations. The matrix of the rate constants (kernel) K ijA and KBi for aggregation and breakage, respectively, then contains all the kinetic information and that is where all the kinetic modeling really goes into. See below for a description of the kernels used in this work, Section 2.2.2. The PBE are solved numerically using the fixed pivot technique of Kumar and Ramkrishna [7]. This procedure is numerically efficient and allows accurate simulations for very broad cluster mass distributions. We used a combined linear/ geometric pivot grid for the cluster mass distribution, where the linear grid with the unit step was used for masses between 1 and 5, followed by a geometric grid with the grid factor of 1.2. It is instructive to rewrite the PBE (1) in a dimensionless form using dimensionless time s ¼ tG/s
ð2Þ
and dimensionless concentration xi ð sÞ ¼
Ni ðsÞV0 /s
ð3Þ
where G is the shear rate, / s is the solid volume fraction and V 0 is the primary particle volume. Note that / s = N 0V 0, where N 0 is the initial number concentration of primary particles. The dimensionless PBE then takes the following form: V X dxk ðsÞ 1 X KijA KikA ¼ xi ðsÞxj ðsÞ xk ðsÞ x i ð sÞ ds 2 iþj¼k V0 G V0 G i¼1
þ
V X Cmk KmB KB xm ðsÞ k xk ðsÞ: /s G /s G m¼kþ1
ð4Þ
The dimensionless form of PBE (4) provides guidance on how to assess the importance of various terms in the PBE model. For example, as far as the evolution of cluster mass distribution in terms of dimensionless time s is independent of the initial solid volume fraction / s, one can observe that the effect of breakage on the aggregate growth kinetics is not significant compared to that of aggregation. As was mentioned before the main objective of this work is to analyze the initial stages of aggregation where breakage is not significant, and therefore discussion concerning breakage will be omitted from further analysis. 2.2.2. Aggregation kernels Aggregation occurs when clusters collide and remain attached to each other. For dilute system in turbulent flows we can assume then only binary collisions are significant. There are two relevant driving forces for aggregation to be considered here: Brownian motion and turbulent shear. Since both Brownian motion driven and shear driven aggregation act simultaneously, one needs to consider their mutual interplay, which can be nontrivial, especially in the case of repulsive particle interactions [4,12]. In the case of fully destabilized conditions considered here it has
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been shown that the full aggregation kernel is well approximated by the sum of the Brownian and the shear kernel [12]: KijA
¼
KijBr
þ
KijSh :
ð5Þ
The respective kernels are given in a general form [3]: KijBr ¼
4p
Di þ Dj Rc;i þ Rc;j W
KijSh ¼ aG Rc;i þ Rc;j
3
dimension d f. The same scaling relationship applies for the collision radius R c,i : Rc;i ¼ Rp i1=df
as well as for the hydrodynamic radius R h,i , which is related to the diffusion coefficient D i through the Stokes – Einstein equation, so that
ð6Þ Di ¼ ð7Þ
where W is the stability ratio of primary particles (near unity for fully destabilized conditions), and G is the shear rate, which can be related to the turbulent energy dissipation rate pffiffiffiffiffiffi ffi e as G ¼ e=m at lengthscales below the Kolmogorov microscale g=(m 3/e)1/4. Here m is the kinematic viscosity of the suspending fluid. The value of prefactor a depends on operating conditions, type of flow and properties of primary particles and has numerical values on the order of 0.1 estimated from experimental measurements, e.g. [1,5], compared to theoretical values on the order of unity derived from simple models not accounting for particle interactions [15,17]. Therefore in this study we treat the prefactor a as an adjustable parameter which is to be obtained by fitting the experimental data. We should note that typical devices used in studies of the aggregation processes in turbulent flows, such as tanks, tubes or slits, exhibit spatial heterogeneity of the shear rate. Therefore we need to consider effects of such heterogeneity on the analysis presented here. Since we consider dilute conditions in small well-mixed tank where the time scale of aggregation (minutes) is much larger than the time scale of mixing (seconds), spatial heterogeneity of concentration is negligible. Thus we can do the simple volume averaging of the aggregation kernel. Since the kernel is linear in G, we get the resulting volume averaged kernel with the volume averaged shear rate G. Therefore in the following analysis we use the volume averaged shear rate G obtained from computational fluid dynamics calculation (see Section 2.3 for more details). 2.2.3. Fractal structure of aggregates For colloidal systems it has been well established that cluster – cluster aggregation results in random aggregates that are described by the fractal scaling relationship [18]. The structure of aggregates is accounted for through a power law relationship between the aggregate dimensionless mass i (the number of primary particles contained in the aggregate) and the radius of gyration R g,i : Rg;i df i ¼ kg ð8Þ Rp with the prefactor k g close to unity. Here R p is the radius of primary particles. The exponent of this scaling is the fractal
ð9Þ
kT 1=df i : 6plRp
ð10Þ
After substituting the fractal scaling formulas above to aggregation kernel expressions from the previous section we get the aggregation kernel in the following form, cf. [3,11]:
2kT 1=df i Kij ¼ þ j1=df i1=df þ j1=df 3l
3 þ aGR3p i1=df þ j1=df : ð11Þ In order to accurately describe the radius of gyration of aggregates, which will be compared to light scattering measurements (see below), we use explicit expressions for the first three clusters [9]: rffiffiffi 3 Rg;1 ¼ ð12Þ Rp 5 Rg;2
Rg;3
rffiffiffi 8 Rp ¼ 5
ð13Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 107 40 cos ð2p=3Þ Rp ¼ 45
ð14Þ
where R p is the primary particle radius. For aggregates with 4 and more primary particles we use the scaling relationship in Eq. (8) with k g = 1. 2.2.4. Small angle static light scattering The measurables obtained from the small angle light scattering instrument were exported as raw data in the form of the intensity of the scattered light I as a function of the scattering vector amplitude q. The scattered light intensity can be expressed as [18] I ðqÞ ¼ I ð0ÞPðqÞS ðqÞ
ð15Þ
where I(0) is the zero angle intensity, P( q) is the form factor (due to primary particles) and S( q) is the structure factor (due to an arrangement of primary particles within aggregates), and q is the scattering angle amplitude n q ¼ 4p sinðh=2Þ k
ð16Þ
Here h is the scattering angle, n the refractive index of the dispersant and k the wave length in vacuo of laser light used. One can use, for example, the Zimm plot [19] to obtain the root mean square radius of gyration hR gi=hR g2i1/2 and
M.H. Waldner et al. / Powder Technology 156 (2005) 226 – 234
the zero angle intensity I(0). The scattered intensity I( q) is expressed in terms of q as I ð 0Þ 1 D 2E 2 ¼1þ R q : ð17Þ I ð qÞ 3 g This equation can be rearranged to D E 2 2 1 1 q Rg ¼ þ I ð qÞ I ð 0Þ 3 I ð0Þ
ð18Þ
and I(0), hR gi are obtained from the intercept and slope estimated from the linear regression of 1/I(q) vs. q 2 in the region where q R g 1. If the aggregates are fractals, then one can estimate their fractal dimension from the slope of the power law region in the log –log plot of S( q) as a function of q. If the primary particles are sufficiently small, so that P( q) is very close to unity within the power law region, then one could simply use the intensity I( q) instead of the structure factor S( q), since they would be proportional. Validity of this assumption is easily checked by calculating the appropriate form factor from the Mie theory or simply by measuring the experimental form factor P( q). In our case it turns out that the primary particles are large enough so that their form factor cannot be neglected. Moreover, the intensity of the scattered light depends on the detector location respective to the plane in which the incident light is polarized. Since it is not known to us whether in our instrument all the detectors are located in the same plane with respect to the polarization plane, we factor out this effect by measuring the form factor P( q) separately and then dividing the measured intensity I( q) by P( q) to get the structure factor S( q). Contributions to the average squared radius of gyration hR g2i from both the form factor P( q) the structure factor S( q) are additive, so that D E D E þ R2g;1 ð19Þ R2g ¼ R2g S ðqÞ
We can determine hR g2iS( q) from S( q) using the Zimm plot as above: D E R2g 2 PðqÞ 1 q S ðqÞ ¼ þ : ð20Þ I ð qÞ I ð 0Þ 3 I ð 0Þ Let us now address the validity of light scattering models used here, considering that our primary particles are larger than the wavelength of the incident light used in light scattering experiments. The value of zero angle intensity measured by light scattering is by definition independent of the scattering wave vector q and therefore does not depend on the laser light wavelength. Although the Zimm plot procedure was originally derived within the Rayleigh – Debye– Gans approximation, we have checked its applicability for larger particles as follows. We used the exact Mie theory to calculate the dependence of scattered intensity on q for spherical particles of diameters from the primary particle size to the largest average size detected in our
231
experiments. Then we applied the Zimm procedure as outlined above to estimate the radius of gyration. The comparison with the true theoretical value of the radius of gyration showed that the difference was never larger than 10%, which is comparable to typical experimental errors due to light scattering measurements reported here. 2.2.5. Relation of cluster mass distribution to static light scattering measurables By solving the PBE with appropriate kernels we get the cluster mass distribution as a function of time. Then from a given cluster mass distribution at any time one can evaluate appropriate moments of the cluster mass distribution corresponding to quantities measured in light scattering experiments [9]. The two moments considered here are the root mean square radius of gyration hR gi=hR g2i1/2, where X D E N i2 R2g;i 2 i i Rg ¼ X ð21Þ N i2 i i and the zero angle intensity I(0) relative to its value at the beginning of aggregation I(0)t=0 X Ni i2 I ð0Þ ¼ X i : ð22Þ I ð0Þt¼0 Ni ð0Þi2 i
2.3. Computational fluid dynamics The fluid dynamics in the coagulator was calculated at steady state for various stirring speeds with a commercial computational fluid dynamics (CFD) software Fluent using a standard k –e model combined with the standard wall function. Due to the symmetry of the vessel it was possible to define the calculation domain as one half of the vessel using periodic boundary conditions. In order to calculate the steady state flow field in the vessel with a rotating impeller, it is recommended to use two different coordinate systems to describe the fluid movement in the vessel. The coordinate system rotating with the impeller was used for a section of the fluid around the impeller. The fixed coordinate system was used for the rest of the vessel. The initial round of CFD calculations involved a mesh with approximately 150,000 computational cells. Since this initial mesh was relatively coarse, we used an adaptation technique based on the dimensionless wall distance and the velocity gradient to improve the accuracy of fluid flow description. The tolerance for the numerical convergence was set to 10 6. Since dispersions considered here were very dilute, one could neglect the effect of solid particles on the fluid flow and therefore the calculations were performed for single phase system. The final result of CFD calculations provided a steady state spatial distribution of the turbulent energy dissipation rate e and the corresponding shear rate G. The corresponding values of the volume averaged shear rate G computed by CFD are shown in Table 2. The width of the
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Table 2 Volume averaged shear rate G for various stirring speeds rpm
G/s 1
200 300 400 600 800
108 199 306 564 866
shear rate distribution is the same for all rotation speeds. The minimum and maximum values are equal 0.067 times G and 63 times G, respectively.
3. Results and discussion Results from light scattering measurements on early aggregation kinetics at various solid volume fractions / s and stirring speeds are all shown together in Fig. 3 in terms of the relative mean radius of gyration hR gi/R g,1 and the relative zero angle intensity I(0)/I(0)t=0 against dimensionless time s. The two measured moments of the cluster mass distribution are separate quantitative measures of aggregate growth. It can be seen in Fig. 3 that we get a single master curve for each of the two moments in dimensionless time s up to at least 1.5 for all volume fractions and stirring speeds. Since the breakage terms in the dimensionless form of PBE (4) depend on the solid volume fraction, we can
〈Rg〉 / Rg,1
102
101
conclude that in early stages of aggregate growth (s up to 1.5) breakage is not significantly affecting the aggregate growth kinetics in our system. Moreover, since the early aggregate growth also follows the same kinetics in dimensionless time regardless of the average shear rate G, we can conclude that the contribution from the Brownian motion to the aggregation kernel is only minor and so the aggregation kernel divided by G, see Eq. (4), is approximately constant, as expected for shear driven aggregation. There are only few sets of data in the literature which demonstrate the effect of the solid volume fraction on the coagulation process [13,16]. Results by both of them show that at the beginning the aggregation kinetics, if properly scaled, collapse on one line in agreement with our observations. 3.1. Fitting two moments from light scattering with PBE model Since breakage is not significant for early kinetics of aggregate growth, we can use PBE to model the kinetics up to a certain dimensionless time s considering just the aggregation. The aggregation kernel has two parameters which need to be specified for a given system: the fractal dimension d f and the aggregation rate prefactor a. For any given values of these parameters we can compute time evolution of the cluster mass distribution and from that the two measured moments through Eqs. (21) and (22). Let us first see how well the kinetic data summarized in Fig. 3 can be fitted when considering each of the two moments separately. It turns out that either of the two moments can be fitted separately for dimensionless times s 1.5 for any value of the fractal dimension from 1.8 up to 3. In Fig. 4 we show the aggregation rate prefactor a required to fit the experimental data as a function of the assumed fractal dimension d f. The fitted aggregation rate
φ = 4 × 10-5
0.6 〈Rg〉 / Rg,1 I(0) / I(0)t=0
200 rpm φ = 2 × 10-5 φ = 4 × 10-5 φ = 8 × 10-5
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I(0) / I(0)t=0
300 rpm 600 rpm
Aggregation rate prefactor
200 rpm, 400 rpm, 800 rpm
100
101
100 0
2
τ
4
6
Fig. 3. Evolution of dimensionless radius of gyration hR gi/R g,1 and zero angle intensity I(0)/I(0)t=0 versus dimensionless time s for various stirring speeds and solid volume fractions.
200 rpm 300 rpm 400 rpm
0.5 0.4 0.3 0.2 0.1 1.6
1.8
2.0
2.2
2.4
df Fig. 4. Combinations of fractal dimension d f and aggregation rate prefactor a that can separately fit each of the measured moments: hR gi/R g,1 (solid lines) and I(0)/I(0)t=0 (dashed lines).
M.H. Waldner et al. / Powder Technology 156 (2005) 226 – 234
〈Rg〉 / Rg,1
102
101
100
100
10-1
S(q)
prefactor does not significantly depend on the stirring speed for the range of values considered here. However, it can be seen that different aggregation rate prefactors are required to fit the two moments for the assumed fractal dimension d f 1.9, i.e. for such fractal dimension the two measured moments cannot be fitted simultaneously with our model. This indicates that such fractal dimension is not suitable for description of early aggregation kinetics in our system. On the other hand, for fractal dimensions d f 1.8 it is possible to fit both moments simultaneously up to a certain dimensionless time s, which becomes shorter with decreasing value of d f. The fitting of both moments is illustrated in Fig. 5. In the top part of Fig. 5 we show the measured radius of gyration together with modeling results for three different values of d f, 1.7, 1.8 and 2.2, respectively, with the corresponding prefactors a from Fig. 4. In all cases a good fit for short times (up to dimensionless time at least 1.5) is obtained. In the bottom part of Fig. 5 we show the measured zero angle intensity together with modeling results for the same set of parameters that were used to fit the measured radius of gyration. The measured zero angle intensity is well described with the model up to s = 1.5 for d f = 1.7 (T 0.1), while for larger d f the intensity increases too fast. We do not consider smaller values of d f to be physically reasonable,
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10-2
10-3
10-4 10-5
10-4
10-3
q/
10-2
nm-1
Fig. 6. Example of power law region analysis of experimental structure factor S( q) for early stages of aggregation (s = 1.8, circles) and for steady state (s = 16, triangles). The negative slope of the linear part in log – log plot of S( q) vs. q represents the apparent fractal dimension. Conditions: / s 4 10 5 at 200 rpm.
since we are not aware of any experimental or theoretical support for values of d f 1.6 in neither static nor turbulent flow conditions in 3 dimensions. To summarize, we can simultaneously describe both measured moments at early stages of aggregate growth in our system, with the fractal dimension d f = 1.7 (T0.1) and the corresponding aggregation rate prefactor a = 0.2. The aggregation rate prefactor estimated here is similar to previously estimated values for primary particles [1,5]. In addition, the fractal dimension estimated here is close to values of fractal dimension expected for cluster – cluster aggregation in fully destabilized conditions [11], which is the case considered here. We note, however, that it is expected that small aggregates produced in early stages of aggregation process examined here would not have fully developed fractal structure [10]. Nevertheless, the meaning of the fractal dimension in this narrower context is that it is the exponent in the scaling relationship between the mass 2.8 2.4 2.0
101
SE
I(0) / I(0)t=0
102
1.6 1.2 0.8
100 0
2
τ
4
6
Fig. 5. Early growth (small s) of aggregates can be well described in terms of dimensionless radius of gyration hR gi/R g,1 for any fractal dimension d f with appropriate values of prefactor a (top graph, symbols represent experimental data). For example, we show calculations for d f equal to 1.7 (solid line), 1.8 (dashed line) and 2.2 (dotted line). However, only calculations with d f 1.8 can describe both moments simultaneously for s up to 1.5. Conditions: / s = 4 10 5 at 200 rpm.
0.4 0.0
0
3
6
τ
9
12
15
Fig. 7. Time evolution of apparent fractal dimension from power law region analysis of experimental structure factor. Data corresponding to structure factors from Fig. 6 are indicated by arrows. Conditions: / s = 4 10 5 at 200 rpm.
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and the collision radius, which is needed in the shear aggregation kernel. 3.2. Apparent fractal dimension from power law region of static structure factor Let us now look what additional information might be available from static light scattering measurements at early stages of aggregate growth in turbulent shear. Analysis of the power law region of the static structure factor can yield aggregate fractal dimension provided that certain conditions are satisfied, most importantly that aggregates are sufficiently large [10]. Obviously, this assumption is not fulfilled for early times while aggregates are still quite small. Nevertheless, we analyzed the power law region for static structure factors collected from very early times and extracted the corresponding apparent fractal dimension. In Fig. 6 we illustrate this analysis by plotting the measured structure factor S( q) for an early time (s = 1.8) near the inflection point at the growth kinetic curve and for a long time (s = 16) at the steady state. The apparent fractal dimension estimated at s = 1.8 is around 0.9, while the plateau value reached at long times is near 2.3, similar to values for other conditions examined here. The time evolution of the apparent fractal dimension is shown in Fig. 7. As has been observed in light scattering measurements on aggregation in quiescent conditions, the slope of the power law region underestimates the true fractal dimension early in the course of aggregation. This is due to the presence of smaller aggregates which scatter light differently than the larger ones with fully developed fractal scattering behavior. Therefore the measured apparent fractal dimension from the power law region analysis should not be taken as necessarily equal to the real fractal dimension of aggregates, except perhaps at later times, provided that the aggregates formed are sufficiently large.
4. Conclusions Simultaneous fitting of multiple moments of the cluster mass distribution can provide a stringent test for aggregation and breakage kernels used in population balance models. Parameters obtained from such fits should be more robust and reliable than those obtained from fitting a single moment with additional external assumptions. We illustrated this approach for early stages of aggregate growth in a turbulent coagulator, where we simultaneously fitted two measured moments of the cluster mass distribution: the mean radius of gyration and the zero angle intensity. Unlike in fitting just a single moment, we were able to uniquely determine both the aggregation rate prefactor and the fractal dimension for small aggregates produced in the initial growth regime. We also note that one cannot take literally the time evolution of the fractal dimension determined from
the power law region of the static light scattering data, unless it is verified that the aggregates formed are sufficiently large. In general, when the cluster mass distribution is not a narrow one, polydispersity can significantly influence simple relationships between various moments and the apparent fractal dimension estimated from the power law region analysis. In such case one should use a full population balance approach to properly evaluate respective quantities.
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