Experimental and Modeling Study of Breakage and ... - ACS Publications

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Experimental and Modeling Study of Breakage and Restructuring of Open and Dense Colloidal Aggregates Yogesh M. Harshe, Marco Lattuada, and Miroslav Soos* Department of Chemistry and Applied Biosciences, Institute for Chemical and Bioengineering, ETH Zurich, Zurich 8093, Switzerland

bS Supporting Information ABSTRACT:

In this work we present experimental and simulation analysis of the breakage and restructuring of colloidal aggregates in dilute conditions under shear. In order to cover a broad range of hydrodynamic and interparticle forces, aggregates composed of primary particles with two sizes, dp = 90 and 810 nm, were generated. Moreover, to understand the dependence of breakage and restructuring on the cluster structure, aggregates grown under stagnant and turbulent conditions, having substantially different initial internal structures with fractal dimension df equal to 1.7 and 2.7, respectively, were used. The aggregates were broken by exposing them to a well-defined elongational flow produced in a nozzle positioned between two syringes. To investigate the evolution of aggregate size and morphology, respectively, the mean radius of gyration, ÆRgæ, and df were monitored during the breakup process using light scattering and confocal laser scanning microscopy. It was found that the evolution of aggregates’ fractal dimension during breakage is solely controlled by their initial structure and is independent of the primary particles size. Similarly, the scaling of the steady-state ÆRgæ vs the applied hydrodynamic stress is independent of primary particle size, however, depends on the history of aggregate structure. To quantitatively explain these observations, the breakage process was modeled using Stokesian dynamics simulations incorporating DLVO and contact interactions among particles. The required flow-field for these simulations was obtained from computational fluid dynamics. The complex flow pattern was simplified by considering a characteristic stream line passing through the zone with the highest hydrodynamic stress inside the nozzle, this being the most critical flow condition experienced by the clusters. As the flow-field along this streamline was found to be neither pure simple shear nor pure extensional flow, the real flow was approximated as an elongational flow followed by a simple shear flow, with a stepwise transition between them. Using this approach, very good agreement between the measured and simulated aggregate size values and structure evolution was obtained. The results of this study show that the process of cluster breakup is very complex and strongly depends on the initial aggregate structure and flowfield conditions.

1. INTRODUCTION It is well-known that the properties of colloidal aggregates and their suspensions can be modified by controlling the size, size distribution, and the morphology of the aggregates. Colloidal aggregates are usually exposed to different flow fields for their further processing, e.g., in polymer manufacturing, pharmaceutical industry, coagulation processes,1 wastewater treatment,2,3 and formation of marine snow.4 The final size of the produced aggregates is a strong function of the primary particle size, initial aggregate size and structure, applied flow field, interparticle forces, and so forth.5
9 In spite of many experimental and modeling efforts published in the literature, the current understanding in this r 2011 American Chemical Society

area is still not satisfactory. The complications in understanding the breakage process are due to the simultaneous occurrence of three phenomena, i.e., aggregation, breakage, and restructuring, which are affected by different physical processes, some of them acting at the macroscale and others at the nanoscale level. Examples include the macroscopic shear rate heterogeneity at the scale of the process vessel, the mesoscopic hydrodynamic interactions between approaching particles or clusters, and at the scale of primary particles Received: November 23, 2010 Revised: March 3, 2011 Published: April 20, 2011 5739

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Langmuir the interaction of changed groups with the surrounding solvent molecules. The aggregation mechanism, which leads to an increase in the cluster mass and size, has been thoroughly studied in the literature, and hence, it is well-understood.10
14 However, the evolution of aggregate size and internal structure during breakage and restructuring is so far inadequately characterized. Furthermore, the interplay between breakup and restructuring is almost an untouched subject of study. A wide variety of flow devices encompassing laminar as well as turbulent conditions have been used in the literature to experimentally investigate the processes mentioned above. In particular, turbulent jets15 and stirred tanks with different impeller geometries16
19 are typical configurations used for studying the breakage of colloidal aggregates under turbulent flow conditions. Although practically relevant, these configurations are inappropriate for fundamental studies due to heterogeneous distribution of the velocity field, even locally, caused by the presence of turbulence. Under these conditions, the total hydrodynamic force acting on the suspended aggregates is difficult to quantify. Due to this reason, laminar flow, for which the velocity field can be easily characterized (in many situations even using analytical expressions), is more interesting and often used in performing fundamental studies. In the literature different devices are used to generate different types of laminar flows, such as rheometers generating simple shear flow,20
22 contracting nozzles23
26 for extensional flow, or four-roll mills27 generating two-dimensional straining flow. From the modeling point of view, it is essential to evaluate the total force acting on each particle within an aggregate to understand the process of aggregate breakup and restructuring. Moreover, in order to be able to model the time evolution of an aggregate size and shape, the trajectory of each constitutive particle must be followed, which requires the estimation of dynamic evolution of forces on each individual particle in the cluster. The estimation of forces requires two types of interactions to be considered, i.e., particle
fluid
particle interaction, also called hydrodynamic interactions, and all the other interparticle interactions. It must be noted that the former are both short- and long-range, whereas the latter are always short-range. In order to evaluate the total force acting on each particle, different levels of approximation have been used in the literature to account for these two types of interactions, leading to two approaches. The first approach is the so-called free-draining approximation, where the hydrodynamic force on each particle is calculated as if all other particles were not present, but the interparticle interactions are treated in detail.28,29 The second approach estimates the hydrodynamic interactions accurately but uses reduced interparticle interactions.30 There are only a few works, such as that of Higashitani et al.6 and Harada et al.,31 where both types of interactions are accounted for; however, none of them had incorporated both contact forces and rigorous hydrodynamic interactions, which could limit the interpretation of the results. There is a scarcity of literature presenting a systematic comparison between experimental and modeling results for the breakage of colloidal aggregates. The aim of this work is to fill this gap. First of all, due to the lack of suitable experimental data, a complete set of data on aggregates’ breakup in elongational flow is presented. A broad range of experimental parameters was investigated covering (i) the initial morphology of aggregate, for which aggregates produced both under stagnant and under turbulent conditions were used; (ii) primary particle size, where aggregates composed of two sizes of primary particles, differing

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by an order of magnitude, were used; and (iii) different magnitudes of shear rates were imposed in the nozzle. The evolution of aggregate size and internal morphology during breakage in welldefined elongational flow26 was characterized through light scattering and confocal microscopy. To quantitatively understand and explain the observed experimental results, we have also developed a mathematical model wherein we have adopted the Stokesian dynamics approach32 to account for the hydrodynamic effects imposed by the fluid on a single aggregate. The normal interparticle forces were defined by using the usual DLVO theory, which include the van der Waals attraction and Born repulsion. In addition to the DLVO interactions, the tangential contact forces, which have been proved to be present in colloidal aggregates and responsible for an increase in the aggregate rigidity,7,33 between touching particles in a cluster were incorporated. The flow field required to simulate the motion of an aggregate was estimated through the computational fluid dynamics (CFD) analysis. Using the developed methodology, dynamic simulations were performed to follow the experimentally observed evolution of cluster properties, such as the average radius of gyration, ÆRgæ; the fractal dimension, df; and the steady-state cluster size dependence on the applied shear rate and initial cluster morphology.

2. MATERIAL AND METHODS 2.1. Experimental Setup. To investigate the effect of primary particles’ size on the steady-state aggregate size and morphology, two primary particle sizes were used in the present study. Hence, sulfate polystyrene latex (product no. 1
800, coefficient of variation = 2.0%, batch no. 642.4, solid % = 8.1, and surface charge density = 5.2 μC/cm2) with particle diameter dp = 810 nm and carboxyl polystyrene latex (product no. 7
80, coefficient of variation = 11.8%, batch no. BM603, solid % = 4.3, and surface charge density = 7.2 μC/cm2) with dp = 90 nm, both purchased from Interfacial Dynamics Corp. (IDC), Portland, OR, were used in this study. Aggregates with substantially different morphologies were produced from these primary particles as follows. The first approach uses aggregation of primary particles under stagnant diffusion-limited conditions. By applying this method, aggregates with very open structure characterized by low fractal dimension, df, around 1.75, can be produced.34 Aggregates were prepared by mixing latex solution with D2O containing an appropriate amount of salt [1.2 mL of 20 wt % Al(NO3)3 solution in 100 mL of total solution] to completely destabilize the suspended primary particles.35 The solid volume fraction of the final dispersion was equal to 1  10
5. In order to minimize sedimentation of aggregates during Brownian aggregation, the suspending liquid density was prematched with that of the aggregates by adjusting the proportion of water to D2O.35 In order to continuously monitor structure and size evolution of the produced aggregates, the aggregation process was performed, directly inside the light scattering instrument (Mastersizer 2000, Malvern, UK). To reach the required size of the produced aggregates (typically 30
60 μm for ÆRgæ) the aggregation was performed over a period of 12 or 72 h for 90 and 810 nm primary particles, respectively. It must be noted that the aggregates produced under these conditions were large enough to be broken even under the modest shear rates reached in the nozzle used for this study. Consequently, 20 mL of this aggregate suspension taken from the Mastersizer was pumped out into 100 mL of pure water through a nozzle of defined diameter. This passage was considered as the first pass of the solution. Diluting the aggregate suspension to solid volume fraction equal to 1.7  10
6 was required to suppress any further aggregation over a period of several hours.36 In contrast, the second approach permits the generation of very dense aggregates with fractal dimension around 2.7,17,18 which is much higher 5740

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Other information that can be extracted from the light scattering signal is the cluster fractal dimension, df, which characterizes the internal structure of the formed aggregates. According to the Rayleigh
Debye
Gans (RDG) theory,37
39 the average structure factor ÆS(q)æ scales with q according to ÆSðqÞæ  q
df Figure 1. Sketch of the breakage equipment with dimensions. than that of the aggregates produced under stagnant conditions. In particular, aggregates composed of 90 nm primary particles were prepared in a 225 mL stirred tank equipped with Rushton turbine using an initial solid volume fraction of 1  10
5. Aggregation was initiated by injecting 2.8 mL of 20 wt % Al(NO3)3 salt solution into a stirred tank completely filled with polymer latex solution. An overflow tube was used to prevent air from entering the dispersion during aggregation. After approximately 1 h at 200 rpm, when the aggregate size reached the steady-state value, samples were gently withdrawn and consequently diluted with pure water to stop further aggregation. It is worth noting that light scattering measurements of samples before and after the dilution step validated that dilution did not affect the size and structure of the cluster population. In both cases of open and dense aggregates, the breakage experiments were carried out by applying converging flow inside the nozzle mounted between two syringes (see Figure 1).26 A syringe pump was used to provide a constant flow rate of the fluid through the nozzle. To ensure that the measured size of the whole fragments population reached a steady state, the aggregate suspension was pumped through the nozzle repeatedly several hundred times. Light scattering and confocal microscopy were used to monitor the evolution of average aggregate size and internal morphology, characterized by the root-mean-square radius of gyration, ÆRgæ, and the fractal dimension, df, respectively. It is worth pointing out that the duration of a single pass through the nozzle is in the range of miliseconds, and therefore, the contracting nozzle setup represents an ideal tool to investigate very fast processes controlled by hydrodynamics, such as aggregate breakage. 2.2. Light Scattering Measurements. The cluster mass distribution (CMD) of the produced aggregates was characterized by measuring the intensity of scattered light, I(q), which is a function of the aggregate size and structure and the size of primary particles constituting an aggregate. It can be expressed as37 IðqÞ ¼ Ið0Þ PðqÞ ÆSðqÞæ

ð1Þ

where I(0) is the zero-angle intensity, P(q) is the form factor (due to primary particles), ÆS(q)æ is the average structure factor (due to the arrangement of primary particles within the aggregates), and q is the scattering vector amplitude defined as n ð2Þ q ¼ 4π sinðθ=2Þ λ Here θ stands for the scattering angle, n is the refractive index of the dispersing fluid, and λ is the laser wavelength in vacuum. The Guinier approximation of the structure factor ÆS(q)æ is expressed as ÆSðqÞæ ¼

IðqÞ Ið0Þ PðqÞ

q2 ÆRg 2 æSðqÞ ¼ exp
3

! qÆRg 2 æSðqÞ < 1

ð3Þ

was used to evaluate the root-mean square radius of gyration of a population of aggregates according to ÆRg2æ = ÆRg2æS(q) þ ÆRg,p2æ using the radius of gyration of the primary particles, Rg,p = (3/5)1/2Rp, where Rp is the radius of the primary particle.

for

1=ÆRg æ < q < 1=Rp

ð4Þ

Therefore, in the given q range, plotting ÆS(q)æ vs q in a double logarithmic plot results in a straight line with slope equal to
df. It is worth noting that this approach can be applied only in the case of large aggregates with large ratio of cluster to primary particle size, i.e., in our case for primary particles with a diameter of 90 nm. In the case of large primary particles (dp = 810 nm), due to the small ratio of cluster size compared to the primary particle size, the determination of df from the light scattering data leads to substantial underestimation of the actual value.40 In this case, a different method, which is described in the following subsection, was used to characterize the aggregates internal structure. 2.3. Confocal Laser Microscopy. An alternative approach to quantify the cluster morphology in terms of df is to use the 2D projections of aggregates. In this way, the structure of aggregates is characterized by the perimeter fractal dimension, dpf, obtained from the projected surface area, A, and the perimeter, P, of the binary image of an aggregate:41
43 A  P2=dpf

ð5Þ

The value of dpf for 2D projections of an aggregate varies between 1 (corresponding to an Euclidean aggregate) and 2 (corresponding to a linear aggregate). Typical values of dpf for aggregates produced under turbulent conditions are in the range of 1.1
1.4.44
46 Having determined the perimeter fractal dimension, dpf, the mass fractal dimension, df, can be estimated using a correlation proposed by Ehrl et al.43 This method was adopted for aggregates produced with 810 nm primary particles, where 2D aggregates projections were taken with a Zeiss Axiovert 100 confocal laser-scanning microscope (CLSM). The image analysis software ImageJ v1.34s (http://rsb.info.nih.gov/ij/) was used to perform the analysis of all 2D aggregate images.

3. NUMERICAL SCHEME 3.1. Hydrodynamic Interactions. The Stokesian dynamics (SD) method was developed by Brady and Bossis.32 The method was formulated by Durlofsky et al.47 in two forms, namely, the FT approach, which accounts for the forces
torques acting on particles when translational
angular velocities of particles are known, and the FTS approach, which along with the previous terms accounts for the stresslet acting on the particles and the rate of strain of the imposed fluid flow. The latter approach is more accurate, as the hydrodynamic interactions are approximated with higher order terms, and it also accounts for the effect of fluid flow on the motion of suspended particles. The SD method is based on the assumption that particles are spheres of identical size, which is consistent with our experimental conditions, since we used latex solutions containing highly monodisperse spherical particles. Moreover, the SD method is valid for very small particle Reynolds number (Rep = FfRp2γ*/ηf , 1, i.e., under laminar flow conditions and constant rate of strain of the flow, where Ff is the fluid density, γ* is the magnitude of the shear rate, and ηf is the viscosity of the fluid). It has been proven that the SD method is accurate even for very high volume fractions of the suspended particles and can be virtually used for any number of suspended particles. So we adopted the FTS formulation of the Stokesian dynamics method for estimating the hydrodynamic 5741

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interactions among particles, which in matrix form can be presented as follows: 2 3 2 3 ¥ Un
U F_n ¥7 6 6 6T 7 7 6 7 ð6Þ 4 n 5 ¼ R 3 4 Ωn
Ω 5 ¥ Sn
E In the equation above, Fn, Tn, and S n are force, torque and stresslet acting on each particle, whereas Un and Ωn are translational and angular velocities of each particle, respectively. The fluid translational and angular velocity at the center of each particle are U¥ and Ω¥, respectively, and E ¥ represents the rate of strain tensor imposed on the fluid; hence, γ* = |E ¥|. Here R stands for the grand resistance matrix, which is a 11Nsphere  11Nsphere symmetric, positive definite matrix with Nsphere defining the number of particles in the simulation domain, while Fn, Tn, Un, Ωn are all 3Nsphere vectors, and S n and E ¥ are 5Nsphere vectors. Henceforth vectors in all equations will be denoted by a line on the variable. The resistance matrix to account for multibody short- and long-range interactions is approximated in the following form32 ¥ R ¼ ðM¥ Þ
1 þ R lub 2B
R 2B

ð7Þ

where M¥ is the far-field mobility approximation, obtained by pairwise additivity of mobility of each particle, Rlub 2B represents the exact two-body lubrication interactions when particles are nearly touching, and R¥ 2B is subtracted as the two-body (2B) farfield interactions are already accounted for in M¥. Although the mobility is expressed by pairwise addition, its inverse, the mobility inverse (M¥)
1, accounts for the many-body interactions. Thus, once the grand resistance matrix is estimated and the forces and torques acting on each particle are known, the velocity and hence the trajectory of each particle can be tracked. There are two length scales in the grand resistance matrix, namely, the one related to the near-field lubrication interactions in Rlub 2B and the other one related to the far-field interactions through M¥, as described by Durlofsky et al.47 As mentioned above, the grand resistance matrix is composed of two contributions, i.e., the far-field hydrodynamic interactions and near-field lubrication interaction. The first one, the grand mobility matrix, M¥, and also its inversion, was calculated only when the distance between any two particles changed by an amount equal to the primary particle size.47 In contrast, as the lubrication forces are short-range forces and change very rapidly when the particle distances change even slightly, the lubrication matrix Rlub 2B was estimated when the particle
particle distance for any pair changed by more than 1% of the primary particle size. 3.2. Particle
Particle Interactions. Apart from the hydrodynamic interactions, which are transmitted through the fluid in which the particles are suspended, the particles experience different attractive and repulsive forces. These particle
particle interactions are usually satisfactorily accounted through the DLVO theory. In the present work, we have expressed the attractive forces through van der Waals interactions, whereas Born repulsion was introduced to prevent overlap between closely separated particles. Due to the large excess of salt used in our experiments, which results in complete screening of any other electrostatic repulsion otherwise present, this contribution has been neglected in the present modeling work. The expressions for the van der Waals (Vvdw) and Born repulsion (Vbom)

Table 1. Model Parameters Ah = 1.33  10
20 J ηf = 1  10


3

Fp = 1050 kg/m3

Pa s

Ff = 1000 kg/m3


23

Nborn = 1  10

δmin = 3 Å

Kt = 6.65  10
4 N/m

ξmax = 24 nm

potential are expressed as follows " !# Ah 2 2 s2
4 þ þ ln Vvdw ðsÞ ¼
s2 6 s2
4 s2 " # Ah Nborn s2
14s þ 54 60
2s s2 þ 14s þ 54 þ þ V born ðsÞ ¼ s s ðs
2Þ7 ðs þ 2Þ7 ð8Þ where Ah is the Hamaker constant, Nbom is the constant for Born repulsion, and the center
center distance, R~, between any particle pair is normalized by the primary particle radius, Rp, to yield the dimensionless center
center distance between the particles as s = R ~/Rp. Recently, Pantina and Furst7 have experimentally demonstrated the presence of tangential interactions between closely separated or almost touching particles, which are responsible for the restructuring and the bending strength of the aggregate. To take this phenomena into account, we have followed the methodology developed by Becker and Briesen33 to introduce the tangential forces between particles when they are separated by very small distance δmin. The tangential force (Fj) and critical bending moment (Tj) experienced by the jth particle are estimated as follows _ F j ¼ Kt ðξij
ξji ÞT j ¼ 2Rp Kt nij ξij ð9Þ where Kt is tangential rigidity, ξij is spring elongation, which is the tangential displacement vector, initiated between the two connected particles i and j, and nij is the unit normal vector between the centers of particles i and j . The spring is considered to be broken when the spring elongation, ξ, exceeds its critical value ξmax, estimated from the critical bending moment, and then sliding occurs. The SD equations were solved using the Runge
Kutta fourthorder integration approach. To estimate the time evolution of the tangential forces and of the relative tangential displacement between particles, the discrete element method (DEM) approach of Cundall and Strack48 was used. Due to very shortrange interactions, such as lubrication forces, van der Waals attractions, Born repulsion, and tangential interactions, careful selection of time steps was required. The use of very small time steps (10
9
10
8 s) ensured the accuracy and stability of the numerical method. At each time step the particle
particle interactions were updated and contacts were identified. At each time step the number of aggregates was computed from the number and distribution of bonds and the average root-meansquare radius of gyration of each cluster was estimated. The simulation parameters obtained from the literature are listed in Table 1 and were used without any fine-tuning. 3.3. Cluster Generation. To compare the simulation results with experimental data, clusters generated by Monte Carlo method composed of various numbers of spherical particles and having df equal to 1.8 and 2.7 were used in the SD simulations. Specifically, clusters with df equal to 1.8 were generated by off-lattice 5742

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Table 2. Flow Field Parameters for Flow Inside Contracting Nozzle Q σmax

γ*max

(Pa)

(1/s)

sr

rnozzle

(mL/

unozzle

no.

(mm)

min)

(m/s)

Renozzle

I

0.25

88

7.47

3735

621

248 000

II

0.375

88

3.32

2490

172

70 300

III

0.5

88

1.87

1867

68.21

IV

0.75

88

0.83

1245

18

V

1

88

0.47

934

6.97

VI

0.5

55

1.17

1167

40.73

16 295

VII

1

55

0.29

584

4.16

1 666

27 284 7 187 2 789

Monte Carlo cluster
cluster aggregation algorithm with a sticking probability of 1, corresponding to fully destabilized diffusion-limited conditions, as described by Lattuada et al.49 On the other hand, clusters with df equal to 2.7 were generated by applying a densification algorithm based on Voronoi tessellation to clusters with a df equal to 2.5, generated by a tunable fractal dimension algorithm, as described by Ehrl et al.43 3.4. Flow-Field Characterization with CFD. The complete set of flow-field conditions used experimentally is reported in Table 2. It must be noted from the table that the flow inside the nozzle varies from purely laminar, corresponding to very small Renozzle to transitional and turbulent flow at large Renozzle. A unified approach to calculate the stress imposed on an aggregate in both situations was developed. In doing so the maximum positive eigenvalue of the strain rate tensor was calculated. At small Renozzle the velocity gradient has components in both axial and radial directions. Hence, the analysis of the complete velocity gradient tensor is necessary. The symmetric part of this tensor corresponds to the rate of strain tensor E¥, which can be decomposed in to three eigenvalues RL > βL > γL, whereas the antisymmetric part governs the vorticity tensor Ω¥ responsible for aggregate rotation. Inside the nozzle different types of flow exist, namely, axisymmetric extensional flow at the center and simple shear near the nozzle wall. Depending on the type of flow, the magnitudes of the eigenvalues change. In either case, the maximum governing hydrodynamic stress (σmax) acting on an aggregate is dominated by the largest positive eigenvalue and is given by50 5 σmax ¼ ηf RL ð10Þ 2 On the other hand, for large values of Renozzle, the flow becomes turbulent and the hydrodynamic stress experienced by an aggregate is due to the local velocity of the fluid, which has two contributions, namely, the mean fluid velocity and its fluctuating part. The hydrodynamic stress imposed by the turbulent fluctuations depends on the relative size of the aggregate (dagg), with respect to the smallest eddies in the flow characterized by the Kolmogrov microscale, νk = (ν3/ε)1/4, where ν is the kinematic viscosity and ε is the energy dissipation rate, and it is expressed as follows for two extremes26 σVS

rffiffiffiffiffi 5 ε ¼ ηf 2 6ν

σ IS  Ff ðεdagg Þ2=3

νk > dagg νk < dagg

viscous subrange ðVSÞ

ð11Þ

inertial subrange ðISÞ ð12Þ

Figure 2. Structure factor measured by light scattering for different passes through nozzle at Renozzle = 3735 for (a) statically generated aggregates and (b) flow generated aggregates, both with 90 nm primary particle.

where, the turbulent energy dissipation rate was evaluated in terms of strain-rate tensor uij ! 1 Dui Duj ε ¼ 2νÆuij uij æ uij ¼ þ ð13Þ 2 Dxj Dxi The flow field in the nozzle was computed using full 3D, single liquid phase, time-dependent simulations of the Navier
Stokes equation through the CFD software Fluent v6.2. As already reported in our previous publication,26 the hydrodynamic stress in the contracting region (lnozzle region in Figure 1) is significantly larger than the stress experienced by an aggregate upstream and downstream of the nozzle, so the CFD simulations were focused on the region in the vicinity of the nozzle. In all CFD simulations the following geometrical parameters were used: lentry = lexit = 50 mm and rentry = rexit = 2.7 mm. More details about the CFD simulations can be found in Soos et al.26 and in the Supporting Information provided therein. As the CFD calculations indicate, all clusters further away from the wall of the nozzle experience much weaker shear forces, which are ultimately leading to weaker breakage or restructuring compared to near the wall of the nozzle. Therefore, the SD simulations have been performed using a flow field evaluated along a trajectory close to the nozzle wall.

4. RESULTS AND DISCUSSION 4.1. Breakup and Restructuring of Aggregates in a Nozzle. Let us start with the analysis of the light scattering data, measured for the different number of passes through the nozzle for both open and dense aggregates composed of 90 nm primary particles, as shown in Figure 2a,b. It can be seen that in both cases the bending part of the S(q), i.e., the Guinier region (plot showing the selection of this region is shown in Figure SM1 from the 5743

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Figure 3. Root-mean-square radius of gyration ÆRgæ as a function of the number of passes through different nozzle diameters for aggregates composed of 90 nm primary particles prepared (a) under stagnant conditions and (b) under turbulent conditions. Symbols indicate various flow conditions: Renozzle = 3735 (squares), 1867 (circles), and 1245 (pluses).

Figure 4. The evolution of fractal dimension estimated with structure factor measured by light scattering for different passes through the nozzle at Renozzle = 3735 for (a) statically generated aggregates and (b) shear-induced aggregates with 90 nm primary particles.

Supporting Information), shifts to the right as the number of passes increases, indicating reduction in the size of the produced aggregates. The corresponding evolution of ÆRgæ obtained at various values of Renozzle extracted from S(q) are plotted in Figure 3a,b. It can be seen that under all investigated conditions the average aggregate size reduces with the number of passes through nozzle. As already observed from S(q) data, the decrease of the average aggregate size is more pronounced at the beginning of the process with faster decay measured for open aggregates produced under stagnant conditions compared to the dense aggregates generated under turbulent conditions (see first 10 passes through the nozzle in Figure 3a,b). In contrast, to reach the steady state, independent of the initial cluster morphology (open or dense) of aggregates, approximately 100 passes through nozzle are required. Such an evolution can be explained as follows: at the beginning of the breakage experiment, aggregates are rather large and therefore more prone to breakup; however, as their size decreases with increasing number of passes very high hydrodynamic stress, which is located close to the nozzle wall, is required to break the aggregates further down to reach smaller sizes.26 As previously shown by Soos et al.,26 since the fraction of liquid passing through this high-shear region is very small, many passes are necessary until each aggregate passes through this zone and is subsequently broken to its steady state size. In agreement with our previous work,26 also in this study the obtained steady-state aggregate size decreases with the increase in applied Renozzle. This is in accordance with the previous works in the literature; see for example the work of Higashitani et al.,6 Harada et al.,31 Moussa et al.,36 which show that for each applied hydrodynamic stress there is a critical stable size below which the aggregate behaves as a rigid body and does not break further.

Two mechanisms, namely, breakage and restructuring, could lead to reduction in ÆRgæ. The former breaks an aggregate into smaller fragments, thus reducing the average aggregate size, whereas the latter process densifies an aggregate, hence reducing its size by changing its morphology. This process is certainly affected by the type of flow and the interparticle forces, including tangential interactions. To investigate the governing mechanism responsible for the observed reduction in the size, the aggregate internal structure should be characterized. Due to the small size of primary particles, this information can be extracted from the same light scattering data37,40 already presented in Figure 2a,b. To better compare the slopes of the power law region of the S(q) plot, the data from Figure 2a,b are plotted as a function of qÆRgæ (see Figure 4a,b). For each data set presented in the figure the corresponding ÆRgæ value obtained from Figure 3a,b was used to generate Figure 4a,b. In this way the shift of S(q) toward smaller q values is eliminated and the measured values of df obtained from the power-law region of the S(q) vs qÆRgæ plot are clearly seen. As it can be seen from Figure 4a for aggregates produced under stagnant conditions, there is a strong increase of the fractal dimension from the initial value of 1.7 to 2.4 over the first five passes through the nozzle. However, there is no further increase of df above 2.4, even after hundreds of additional passes though the nozzle (see Figure 4a). Surprisingly, completely different evolution has been observed during the breakage of dense clusters, where the average fractal dimension remains constant with df around 2.65, as shown in Figure 4b. This clearly indicates that very open aggregates, such as those produced under stagnant conditions, when exposed to high shear stress, undergo simultaneous breakup and restructuring. The former process is caused by the applied hydrodynamic stress on the aggregate exceeding the strength of the aggregate, whereas 5744

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Figure 5. Structure factor at zero passes (black squares) compared with the steady-state values (200 passes) for (a) statically generated and (b) shear-induced aggregates with 90 nm primary particles as a function of Renozzle = 3735 (circles), 1870 (triangles), and 934 (diamonds).

Figure 6. Structure factor as function of scattering wave vector for initially dense aggregates with 810 nm primary particles at Renozzle = 3735 and different number of passes: original sample corresponding to zero passes (squares), three passes (circles), 15 passes (triangles), 201 passes (triangles pointing down), 301 passes (diamonds), and 501 passes (pluses).

the latter mechanism is due to the effect of the flow that moves the particles in an aggregate. This increases the number of bonds among the particles and consequently the density and rigidity of the cluster. On the other hand, in the case of dense aggregates that are produced under turbulent conditions, further densification is not possible as indicated by constant df values, so that size reduction is solely controlled by the breakage mechanism. It is worth noting that similar results in terms of the evolution and the steady-state values (see Figure 5) of the fractal dimension were obtained also for the other values of applied Renozzle and, therefore, for the hydrodynamic stress investigated in this work. To investigate the effect of primary particle size on the results presented above, similar measurements have been performed for

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Figure 7. Root-mean-square radius of gyration ÆRgæ as a function of number of passes through different nozzle diameters for (a) DLCA aggregates and (b) dense aggregates composed of 810 nm primary particles with symbols indicating Renozzle = 3735 (squares), 2490 (triangles pointing left), 1867 (circles), 1167 (diamonds), 934 (red triangles), and 584 (triangles pointing down) as obtained from experiments.

aggregates composed of 810 nm primary particles. As an example, in Figure 6 the S(q) has been plotted as a function of the scattering wave vector q for different number of passes through nozzle at Renozzle = 3735 for aggregates composed of 810 nm primary particles and having initial df = 2.7. In agreement with the results reported for aggregates composed of 90 nm primary particles, with increasing number of passes S(q) shifts toward the right, indicating reduction in the average aggregate size. The summary of the estimated ÆRgæ values for various values of Renozzle and different number of passes is presented in Figure 7a,b for initially open and dense aggregates, respectively. Similar to aggregates composed of primary particles with diameter of 90 nm, the reduction of ÆRgæ is faster and more pronounced for open aggregates compared to dense aggregates. Moreover, as can be seen from Figure 7a,b, also in this case, independent of the initial value of df, a few hundred (approximately 150) passes are required until the CMD attains the steady-state. Additionally, the steady state average aggregate size is found to decrease rapidly with increasing the applied Renozzle. Comparing the results plotted in Figures 3 and 7, it can be seen that the steady-state fragment sizes measured for aggregates composed of 90 nm primary particles obtained for the same Renozzle (i.e., hydrodynamic stress) are substantially larger than those of fragments composed of 810 nm primary particles. This difference is most probably due to different surface charges present on the primary particles, specifically, a carboxyl group for 90 nm and a sulfate group in the case of 810 nm primary particles. It is worth noting that, when the same surface charge type and density are present, no differences in the steady-state aggregate sizes were reported by Ehrl et al.18 5745

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Figure 10. Evolution of the fractal dimension as a function of the number of passes through the nozzle for statically generated aggregates (open symbols) and shear-induced aggregates (filled symbols) for aggregates with 90 nm particles at Renozzle = 3735 (circles), 1867 (squares), 1245 (triangles) and for 810 nm particles at Renozzle = 3735 using statically generated (exes) and shear generated (pluses) aggregates.

Figure 8. Image analysis pictures from confocal laser-scanning microscopy for 810 nm primary particle statically generated aggregates for different nozzle passes, Npass: (a) 0, (b) 1, (c) 5, and (d) 110.

Figure 9. Image analysis pictures from confocal laser-scanning microscopy for 810 nm primary particle shear generated aggregates for different nozzle passes, Npass: (a) 0, (b) 1, (c) 10, and (d) 200.

As mentioned before,40 due to the small ratio between the size of aggregates and that of primary particles, light scattering measurements cannot be accurately used to determine the internal structure of aggregates composed of 810 nm primary particles. Hence, an alternative approach, i.e., image analysis of 2D pictures of aggregates captured by confocal laser scanning

microscopy, described in detail elsewhere,26,43 was used to determine the morphology of the aggregates. To capture the evolution of df during the breakage process, images were taken at different numbers of passes through the nozzle. For each pass, approximately 50
70 images were taken to ensure that the obtained average internal structure was statistically representative of the whole cluster population. Figure 8a
d shows microscopic images of some of the representative aggregates, taken along the breakage process and starting from very open clusters with initial df = 1.7. The images taken at different numbers of passes clearly indicate the change in the aggregate size and morphology. It can be concluded that the aggregates are severely broken down to smaller sizes with increasing number of passes along with densification, as evident from their compact structures. To quantify the aggregate morphology in terms of dpf, the 2D images of aggregates obtained at different numbers of passes were analyzed according to the image analysis described above (see section 2.3). The correlation between df and dpf, developed by Ehrl et al.,43 was used to evaluate the mass fractal dimension df of the evolving aggregates. Similarly, image analysis was performed on aggregates composed of 810 nm primary particles with initial df = 2.7. Representative pictures of these aggregates along the breakage process are shown in Figure 9a
d, as measured by Soos et al.26 It can be seen that during the breakage process dense aggregates preserve their morphology. A summarizing comparison of the evolution of df along the breakage process for various Renozzle obtained for dp = 90 and 810 nm and aggregates with different initial fractal dimension, df = 1.7 and 2.7, is presented in Figure 10. For completeness, experimental values of df measured by Soos et al.,26 for aggregates composed of 810 nm primary particles generated under turbulent conditions characterized by df = 2.7, are included as well. The figure shows that independently of the primary particle size the initially open aggregates become compact very fast, with df increasing from 1.7 to 2.4. After such a fast increase, in about five passes the average aggregate fractal dimension df remains unchanged with a value around 2.45 ( 0.05 for any further number of passes through the nozzle. In contrast, for aggregates generated under turbulent conditions, there is no significant change in the cluster’s morphology, as indicated by almost constant df with a value around 2.75 ( 0.1, as measured by both 5746

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Figure 11. (a) Hydrodynamic stress σ(Pa) in the contracting nozzle evaluated by CFD for Renozzle = 2490. Black line represent the closest stream line to the wall along which aggregate can pass the nozzle and where the hydrodynamic stress is the highest. Variation of the rate of stress tensor’s eigenvalues (short-dash line - RL, dash line - βL, dash-dot line - γL) along this streamline (b), with the solid line indicating the throat area of the nozzle with total length equal to lnozzle, together with variation of the angle between maximum positive eigenvalues, RL, and the velocity vector (c), and the angular velocity components (dash line - ωx, solid line - ωy, dotted line - ωz) (d).

light scattering and image analysis for aggregates with dp= 90 and 810 nm, respectively. 4.2. Modeling of Aggregate Breakup and Restructuring. Although the presented experimental data clearly indicate that the breakup of open aggregates is accompanied by aggregate restructuring, leading to densification, whereas for the dense aggregates only breakage process prevails, these results alone are not able to explain the difference in the steady-state value of df measured for initially open and dense aggregates. To investigate the parameters controlling the transition between these two mechanisms, and also to provide more insight into the breakage kinetics of a single aggregate, a mathematical model based on Stokesian dynamic32 (SD) was developed. The model includes both short- and long-range interactions along with the multibody effects through the SD formulation. Thus, it accurately accounts for the hydrodynamic force acting on each particle in the simulation domain. In addition, all other relevant interparticle interactions have been incorporated in the model, including van der Waals, Born repulsion, and tangential forces. All required simulation parameters are listed in Table 1 and were obtained from the literature7,33 without any fine-tuning. To model the effect of hydrodynamic stress acting on a cluster during the passage through the contracting nozzle, the flow field experienced by an aggregate in its journey through the contracting nozzle was approximated from CFD analysis. An example of the hydrodynamic stress contour plot calculated for Renozzle = 2490 is presented in Figure 11a, while further results obtained for low Renozzle = 584 can be found in the Supporting Information (see Figure SM 2). In agreement with the work of Higashitani et al.24 and also with our previous work,26 the highest values of the hydrodynamic stress are located at the nozzle entrance, while

negligibly small stress values are observed in the upstream and downstream regions. In addition to the radial variation of the hydrodynamic stress at the nozzle entrance, which covers approximately 1 order of magnitude (see Soos et al.26 and Figures SM 3, 5, and 6 in Supporting Information), high stress zone can also be found in the nozzle body and in the cloudlike region located at the nozzle exit (see Figure 11a). Henceforth, we will refer to the combination of these three high-stress zones as the breakage zone. Even though breakup of aggregates could also occur at the other radial locations (especially in the early stages of the breakup process), the observed slow decay of the average aggregate size with the number of passes measured at the later stage of the breakup process (as presented in Figures 4 and 7) indicates that each aggregate from the population has to pass at least once through the breakage zone to be exposed to the highest hydrodynamic stress, so that the whole population will approach the steady state. To characterize the highest possible hydrodynamic stress that aggregate could experience, a streamline passing through this breakage zone (see black line in Figure 11a) and having distance from the wall at the nozzle entrance equal to the aggregate diameter at steady state, i.e., ÆDgæ = 2ÆRgæ, was used to characterize flow field for SD simulations. To further characterize the flow type, the variation of three eigenvalues of the rate of strain tensor along the streamline passing through the breakage zone is presented in Figure 11b. It can be seen that two eigenvalues, the maximum positive and the maximum negative, have very similar magnitude and differ only in sign, whereas the third, intermediate eigenvalue, is negligibly small. As it is well-known20,27 that there exist two types of flows which fulfill this constraint: a two-dimensional straining flow and simple shear flow. To uniquely identify the type of flow to which 5747

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Figure 12. Evolution of size ÆRgæ and morphology df as a function of the number of passes through a nozzle by modeling open clusters (df = 1.8) (a, b), and dense clusters (df = 2.7) (c, d) for dp = 810 nm at two different shear rates γ*max = 70 000 1/s (triangles) and 250 000 1/s (circles) obtained by model simulations.

an aggregate is exposed to during its passage through the nozzle in Figure 11c, the angle θ between the eigenvector corresponding to the maximum positive eigenvalue (RL) responsible for the aggregate stretching and the velocity vector along the streamline in the direction of the flow has been plotted. It can be seen that except the nozzle entrance, where the flow is elongational as indicated by the value of angle dropping down to zero, everywhere else the flow can be well-represented as a simple shear flow characterized by the angle equal to 45. Hence, a combination of both flow types has been used in the present study. Moreover, to account for the rotational contribution of the flow, which is responsible for the rotation of aggregates and therefore for their densification,6 the fluid angular velocity has been evaluated along the streamline from Figure 11a. The obtained values of all three components of the rotation vector ωi are plotted in Figure 11d. As expected, only the component perpendicular to the shear plane is nonzero, with magnitude equal to that of the maximum positive eigenvalue RL. As the impact of two types of flow on the aggregate is different,6,31 the last important parameter required as an input to the SD simulation is the relative contribution of the elongational and simple shear flow to the total flow field in the breakage zone. In order to capture the switching of the flow from elongational to simple shear, sharp stepwise switch between the two regions was used in the SD simulations, starting with the elongational flow at the nozzle entrance followed by the simple shear flow in the nozzle body and in the cloudlike region at its exit, both indicated by the solid line in Figure 11b. According to the CFD simulation (Figure 11b), where the magnitude of the shear rate in the simple shear region is equal to one-half of that found in the elongation region, different shear rate magnitudes of elongational and simple shear flow were used in SD simulations (see the solid line in Figure 11b). It can be noted that for different

values of the shear rate (i.e., hydrodynamic stress) one pass of an aggregate through the nozzle would correspond to different absolute times. Hence, to uniquely represent the time evolution a dimensionless parameter τ describing this time scale has been used. This dimensionless time is defined as τ = tγ*, where t is the real time. In this way, each pass through the nozzle, covering the complete breakage zone, i.e., nozzle entrance, nozzle body, and cloudlike region at the nozzle exit, has been quantified as being approximately equal to 25 dimensionless times. More detailed analysis of the flow field inside the nozzle with CFD showed that the relative time spent by an aggregate in the elongational and simple shear flow can be well approximated by τ = (0,3) and τ = (3,25), respectively. For further details of the simulation conditions, the reader is referred to Table 2. The interparticle interactions involved the usual DLVO interactions found by eq 8 along with the tangential interactions estimated with eq 9 using the same set of parameters as reported by Becker and Briesen33 (Table 1). On the basis of the previous SD simulations by Harshe et al.31, where it was found that, within the computational model implemented in the present work, the interparticle forces compensate each other in a way that the only controlling parameter affecting breakage is the Peclet number Pe = (6 πηfγ*Rp3)/(kbT) where, kb is the Boltzmann constant, and T is the absolute temperature. This means that identical simulation results can be obtained for different primary particle sizes when the shear rate is changed in such a way that Pe remains constant. Therefore, all SD simulations were performed for aggregates composed of 810 nm primary particles. The major numerical limitation concerning the simulations is the time required to invert the grand resistance matrix, having (11Nsphere  11Nsphere) number of elements and, therefore, leading to a strong increase in the computational time 5748

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* Figure 13. Structures of produced aggregates for different numbers of passes starting with different initial geometries for γmax = 70 000 1/s.

with the number of particles within an aggregate. For each SD simulation, a single cluster with either open or dense morphology is randomly chosen from the cluster library. As simulations require inverting the grand mobility matrix, which is a (Nsphere)3 operation, the cluster mass that can be simulated in reasonable time was restricted to Nsphere e 100. To simulate multiple passes through the nozzle, the cluster population obtained after 25 dimensionless times was reintroduced in the flow field, and simulation continued for a further 25 dimensionless times to cover the subsequent pass, again imposing the same combination of elongational and simple shear flow described above. To determine the statistical variation of the obtained results, the SD simulation was repeated for five different clusters with the same mass (i.e., number of primary particles composed in an aggregate, Nsphere) and morphology (df). The dynamic simulations were performed at two shear rates, γ*max = 70 000 and 250 000 1/s, corresponding to conditions I and II from Table 2, respectively. The number of clusters produced at each time interval, the radius of gyration of an individual cluster Rg, and its fractal dimension df were determined. However, to obtain the relevant average properties of the produced cluster, the weight average Rg and df were calculated. Depending on the initial cluster morphology and the applied shear, 5
25 daughter aggregates were produced at the end of the simulation. The obtained results corresponding to five passes through the nozzle calculated for initially open and dense aggregates, i.e. with df = 1.8 and 2.7 respectively, at two different shear rates are shown in Figure 12a
d. The error bars in the figures represent the statistical variation of the results obtained from five different aggregates with the same initial mass and morphology. It can be seen from Figure 12a that the average radius of gyration ÆRgæ for initially open clusters (df = 1.8) decreases after five passes, thus reducing from approximately 5.5 μm initially to 1.35 and 0.814 μm at the end for γ*max equal to 70 000 and 250 000 1/s, respectively. It is worth noting that the observed fast decay is in accordance with that measured experimentally. Even more striking is the very good agreement between the calculated steady-state value of ÆRgæ with the experimentally measured values of ÆRgæ, which for the corresponding shear rates are equal to 1.4 and 0.753 μm, respectively (see Figure 7a). Similar simulations were also performed for the dense clusters (Figure 12b) with initial df = 2.7. For these aggregates after applying shear rate, γ*max, equal to 70 000 and 250 000 1/s, the average aggregate size ÆRgæ evaluated

after five passes reduced to a steady-state size equal to 2.11 and 1.57 μm, respectively. These results are again in good agreement with the experimentally measured values of ÆRgæ equal to 2.3 and 1.17 μm, for the respective conditions. It is important to note that due to the simplified flow profile used in the SD simulations, and because we have simulated a single aggregate breakup event, the evolution of ÆRgæ and df obtained from both simulations and experiments are not directly comparable. Therefore, only their steady-state values should be compared. In the following we would like to elucidate our choice of a single streamline used for the SD simulations with an example. Let us take flow conditions through the nozzle corresponding to Renozzle = 3735 for which γ*max = 250 000 1/s was used. Since this value of the shear rate represents the maximum possible value that an aggregate would experience, all the other streamlines away from the nozzle wall will have shear rate values smaller than 250 000 1/s. As it has been shown in Figure SM6 of the Supporting Information, there will be a radial distribution of the shear rate ranging from 25 000 1/s at the nozzle axis to 250 000 1/s close to the nozzle wall. Therefore, for the same Renozzle there will be a streamline with the shear rate equal to 70 000 1/s. As can be seen from Figure 12a,c, where results of aggregate breakup at different shear rates have been presented, one can immediately see that the breakup of aggregate is less severe for 70 000 1/s compared to the streamline used for this Renozzle, i.e., with the shear rate of 250 000 1/s. Hence, it is clear that when an aggregate has already been exposed to the shear rate of 250 000 1/s the produced fragments will be barely further broken when exposed to the lower values of the shear rate seen along the other streamlines away from the nozzle wall. On the other hand, when an aggregate is broken at a lower value of the shear rate, corresponding to a streamline away from the wall, and then passes through the selected breakage zone near the nozzle wall, with the shear rate of 250 000 1/s, it will undergo severe breakup, leading to fragments of an average size comparable to that measured at the steady state for the corresponding flow conditions. This further supports our choice of a single streamline instead of using the whole distribution of the shear rate in the nozzle. After comparing the evolution of the aggregate size, we focused on the evolution of the aggregate morphology, which is presented in Figure 12b,d, for open and dense aggregates, respectively. It can be seen that, in the case of initially open 5749

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Figure 14. Scaling of the steady-state mean radius of gyration normalized by the primary particle radius, ÆRgænorm, as a function of the applied stress composing of two primary particle sizes and two different initial fractal dimensions: primary particle size of 810 nm with initial df = 1.7 (open triangles pointing down) and initial df = 2.7 (solid triangles) and primary particle size of 90 nm with initial df = 1.7 (open squares) and initial df = 2.7 (solid squares). SD simulation performed for aggregates composed of 810 nm primary particle with initial df = 1.7 (open circles) and initial df = 2.7 (solid circles). The lines represent scaling obtained according to Zaccone et al.52 with df = 2.45 (- - -,


) and df = 2.7 (—,
3
3 ) as measured at the end of the breakage experiment for aggregates with initial df = 1.7 and 2.7, respectively.

aggregates with df = 1.8, the fractal dimension increases with the number of passes and reaches the steady-state value of about 2.5 (Figure 12b). This result is in very close agreement with the measured experimental values, which for the steady state equals 2.5 ( 0.1. On the other hand, the values of df for dense clusters with df = 2.7 does not change with the number of passes (see Figure 12d) and remains almost constant at df = 2.7 ( 0.1, which is in close agreement with experiments. Both of these simulation results confirm our experimental findings that the breakage of initially open clusters involves breakage in conjunction with restructuring, while initially dense clusters exhibit only breakage. This also confirms that the detailed flow field information used and the interparticle interactions, such as tangential contact forces, incorporated in the model lead to realistic predictions. In addition to this, the simulations indicate that an individual cluster requires almost four passes to attain the steady-state size and morphology. The evolution of a typical cluster structure for different numbers of passes and starting with two different initial morphologies is shown in Figure 13. It can be seen that initially open clusters are broken into smaller fragments in the first pass, while structures become denser and denser as the number of passes increases, thus increasing the fractal dimension. Even though densification of the produced fragments takes place over a long period of time, the steady-state fractal dimension is limited to 2.5 ( 0.1. The aggregates are not able to reach higher values of fractal dimension or values as high as those produced under turbulent conditions. On the other hand, for initially dense clusters, the particles are rearranged and the geometry of the cluster is slightly modified. As expected, dense clusters are more rigid due to more contacts per particle resulting in greater aggregate strength, and they do not break apart in fragments as the open clusters do. Moreover, the fractal dimension of initially dense clusters remains constant. On the basis of these results, we believe that the reason for the difference in the steady-state values of df is due to a lack of aggregation during the breakage experiment compared to clusters produced under turbulent conditions, which

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are produced by combination of aggregation and breakage. While the densification of initially open clusters is driven by the combination of both rotational flow and tangential interactions, in the case of initially dense clusters, the structure leaves little room to additional restructuring. In Figure 14 comparison of the obtained data from both experiments and simulations for the scaling of the steady-state radius of gyration normalized by the primary particle size, ÆRgænorm = ÆRgæ/Rp, plotted as a function of applied hydrodynamic stress, σmax, is presented. The experimental values of the maximum applied stress for different conditions are summarized in Table 2. It can be seen that under all conditions investigated in this work the steady-state aggregate size follows a power law scaling when plotted as a function of the shear stress, i.e., ÆRgænorm  (σmax)
p, with p =
0.35 and
0.55. This is in agreement with the theoretical scaling obtained by Zaccone et al.51 with corresponding steady-state fractal dimensions df = 2.45 and 2.75 (see Figure 10) as well as with the values reported by other researchers.6,23,52,53 It can be concluded that the scaling exponent is independent of the primary particle size. However, it depends on the history of aggregate morphology.

5. CONCLUSIONS The process of breakage of very open and dense aggregates was studied experimentally for aggregates composed of polystyrene primary particles with diameter of 90 and 810 nm using elongational flow generated in a nozzle. Different hydrodynamic stresses were generated using various nozzle diameters and flow rates. The evolution of aggregate size and structure was followed by light scattering measurements and confocal laser-scanning microscopy, wherever possible. It was found that the root-meansquare radius of gyration decreases with the number of passes through the nozzle and reached a steady-state value after about 100 passes for all cases. In addition to the aggregate size, also the morphology of the resulting aggregates, quantified by the fractal dimension, showed an evolution, but only for aggregates with initially open structures. Starting with open aggregates, relatively dense structures were formed at the end of the experiment for both 90 and 810 nm primary particle aggregates. The mass fractal dimension evolved from 1.7 to 2.45 ( 0.05. Therefore, it can be concluded that the process involved both breakage and restructuring. On the other hand, dense clusters reduced their size as the number of passes was increased, while their morphology remained unchanged, as indicated by the constant value of df = 2.75 ( 0.1, suggesting that only breakage of aggregates was present. Thus, it can be concluded that the history of the aggregates governed the evolution of the final aggregate morphology. The steady-state value of ÆRgæ when plotted against the applied hydrodynamic stress showed power law scaling with an exponent equal to
0.35 and
0.55 for initially open and dense aggregates, respectively, independent of the primary particle size. The experimental results were supported by simulations obtained using the Stokesian dynamics method, incorporating DLVO and tangential interactions along with flow field profile extracted through a detailed CFD analysis of flow pattern inside the nozzle. Such a detailed modeling approach leads to very good quantitative and qualitative agreement with the experimental findings. The combination of experiments and model can be used to validate particle
particle interactions models and their magnitude and also to better understand the breakage process. This approach will help to improve and generalize scaling laws under 5750

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’ ASSOCIATED CONTENT

bS Supporting Information. A sample plot of ÆS(q)æ vs q showing the Guinier region selected for the estimation of ÆRgæ, together with detailed CFD analysis of the flow field inside the nozzle, showing the hydrodynamic stress along different streamlines and corresponding contour plots, and the radial distribution of the hydrodynamic stress in the nozzle. This material is available free of charge via the Internet at http://pubs.acs.org. ’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT The authors would like to thank Prof. Massimo Morbidelli for his guidance and support for the work. This work was financially supported by the Swiss National Foundation (Grant No. 200020-126487/1). ’ REFERENCES (1) Hunter, R. J. Foundations of Colloid Science, 2nd ed.; Oxford University Press: New York, 2001; p xii, 806 pp. (2) Wills, B. A. Mineral Processing Technology: An Introduction to the Practical Aspects of Ore Treatment and Mineral Recovery, 4th ed.; Pergamon Press: New York, 1988; p xvii, 785 pp. (3) Elimelech, M. Particle Deposition and Aggregation: Measurement, Modelling, and Simulation. Butterworth-Heinemann: Boston, 1995; p xv, 441 pp. (4) Droppo, I. G.; Leppard, G. G.; Liss, S. N.; Milligan, T. G. Opportunities, needs, and strategic direction for research on flocculation in natural and engineered systems. Flocculation in Natural and Engineered Environmental Systems; CRC Press: Boca Raton, FL, 2005, 407
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