Breakage Rate of Colloidal Aggregates in Shear Flow through

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Breakage Rate of Colloidal Aggregates in Shear Flow through Stokesian Dynamics Yogesh M. Harshe and Marco Lattuada* Institute for Chemical and Bioengineering, Department of Chemistry and Applied Biosciences, ETH Zurich, Zurich 8093, Switzerland

bS Supporting Information ABSTRACT: We study the first breakage event of colloidal aggregates exposed to shear flow by detailed numerical analysis of the process. We have formulated a model, which uses Stokesian dynamics to estimate the hydrodynamic interactions among the particles in a cluster, van der Waals interactions and Born repulsion to describe the normal interparticle interactions, and the tangential interactions through discrete element method to account for contact forces. Fractal clusters composed of monodisperse spherical particles were generated using different Monte Carlo methods, covering a wide range of cluster masses (Nsphere = 30
215) and fractal dimensions (df = 1.8
3.0). The breakup process of these clusters was quantified for various flow magnitudes (γ), under both simple shear and extensional flow conditions, in terms of breakage rate constant (KB), mass distribution of the produced fragments (FMD, fm,k), and critical stable aggregate mass (Nc), defined as the largest cluster mass that does not break under defined flow conditions. The breakage rate KB showed a power law dependence on the product of the aggregate size and the applied stress, with values of the corresponding exponents depending only on the aggregate fractal dimension and the type of flow field, whereas the prefactor of the power law relation also depends on the size of the primary particles comprising a cluster. The FMD was fitted by Schultz
Zimm distribution, and the parameter values showed an analogous dependence on the product of the aggregate size and the applied stress similar to the rate constant. Finally, a power law relation between the applied stress and corresponding largest stable aggregate mass was found, with an exponent value depending on the aggregate fractal dimension. This unique and detailed analysis of the breakage process can be directly utilized to formulate a breakage kernel used in solving population balance equations.

I. INTRODUCTION Suspensions of colloidal aggregates are encountered in various chemical, pharmaceutical, and materials processes, and are important to a variety of industrial sectors such as polymer manufacturing via emulsion polymerization, food processing, and others. The control over the morphology and size of colloidal aggregates is a crucial aspect of processing, as many suspension properties, such as the rheological, optical, and electrical, are governed by their structure, size, and size distribution.1
5 Clusters are usually created by destabilizing colloidal dispersions of charged primary particles by the addition of electrolytes. This bottom up approach allows one to control the properties of the colloidal aggregates especially because, in the majority of industrially relevant processes, the formation of colloidal aggregates occurs in the presence of fluid flows, typically created by stirring. The interactions of such colloidal aggregates with flow fields is especially complex, as it depends on the flow field properties (i.e., the magnitude of shear rate experienced by a particle and the flow profile, i.e., simple shear, extensional or a combination of both), on the cluster structure, and on the interparticle interactions. When colloidal aggregates are exposed to fluid flows, the constitutive individual particles experience various types of interactions, such as hydrodynamic and colloidal forces. The balance r 2011 American Chemical Society

between them decides whether a bond between already connected particles will break, leading to an aggregate breakup, or whether the particles will just roll and slide leading to restructuring.6 Practically occurring aggregation processes involve a large number of particles. Population balance equations (PBEs), i.e., mass balances for the entire cluster mass distribution, can be conveniently used to quantify the process kinetics and to follow the time evolution of distribution.7 These PBEs for particulate systems can account for both aggregation and breakage of clusters, which are described as second- and first-order kinetics processes, respectively. In order to make PBE calculations, the rate constants for both aggregation and breakage process, called kernels, are required. Kernels contain all the physics of the involved processes, and are functions of the cluster size, primary particles, structure of the cluster, applied shear rate and interparticle interactions.8,9 Determining these kernels requires a thorough understanding of the physics underlying the corresponding processes. In this respect, the process of colloidal Received: September 30, 2011 Revised: November 25, 2011 Published: November 28, 2011 283

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aggregation has been studied in detail in the literature both through modeling10
15 and experiments,16,17 and it can be well described by the theoretical models developed so far, with results usually found in very good agreement with experiments.16
20 On the other hand, the breakup of colloidal aggregates in shear flows is much less understood due to the complexity of the process itself.21,22 Even though the process of colloidal aggregate breakage has been studied in the literature by many researchers through both experimental and modeling approaches,22
33 many questions remain unanswered, both related to the breakage rate9 and to the evolution of an isolated cluster exposed to a shear flow. It is evident that experiments to understand the breakage of colloidal aggregates are generally difficult to perform.30
32,34,35 One of the reasons is because it is very difficult to experimentally obtain a monodisperse population of clusters with well-defined properties, and all experiments involve a population of aggregates with a range of masses, and the experimental techniques to analyze such population of clusters are nontrivial.21 It is also wellknown that the breakup of colloidal aggregates is a very fast process30,36 (on the order of miliseconds), almost instantaneous on the time scale of the whole experiment. Hence, quantities such as the characteristic time of cluster breakup and the corresponding fragment mass distribution, as well as their dependence on the initial cluster properties and on the applied hydrodynamic stress are difficult to access through experiments. For this reason, different modeling approaches, such as discrete element simulations,22,37 finite element methods,37 Stokesian dynamics,38
40 and so forth, have been proposed in the literature to study colloidal aggregates’ breakup in sheared flows. Such modeling approaches solve the motion equation for all particles in the clusters, and can account for all interactions, specifically, particle
fluid, referred to as the hydrodynamic interactions, and all the other particle
particle interactions that are called as interparticle interactions. These simulation methods have both advantages and disadvantages. These simulations are computationally very expensive since very small time steps are required in order to avoid unphysical situations due to rapid change in magnitude of these interactions with the interparticle distance. To reduce the computational efforts, most works performed in the literature accurately account for one type of interaction only in a detailed manner, either hydrodynamic or interparticle, while the other type of interaction is dealt with only in a more approximated manner.41 The purpose of this work is to make accurate calculations of all the quantities required to quantify the kinetics of breakage with the purpose of producing a breakage rate kernel and a fragment mass distribution to be used within the framework of PBEs. In this respect, our study is substantially different from what has been proposed or presented in the literature. The focus has been so far typically to follow the breakage process of an individual cluster until a steady state has been reached. This usually involves cluster fragmentation together with restructuring. Following this process provides very useful information, but it is not suitable to obtain a real rate of breakage. In fact, it would make sense only in the case where the time scale of aggregation is much longer than that of breakage. This might be acceptable only under sufficiently dilute conditions, since breakage process follows a first-order kinetics, while aggregation follows a second order kinetics. From the point of view of PBEs, instead, it makes more sense to consider the first breakage event, by definition a binary event, which is well-defined under any conditions.

The majority of the fundamental studies are carried out under laminar flow, as under turbulent conditions the total hydrodynamic force acting on aggregates is difficult to quantify.42 Conversely, breakup of colloidal clusters under laminar flow conditions is well characterized, and the forces acting on each particle can be precisely estimated. For this reason, laminar conditions have been studied in the present work. The simulations have been performed through detailed particle motion modeling to study the on-set of breakup of an aggregate composed of uniform sized spherical primary particles. A model has been developed that incorporates Stokesian dynamics38 to account for the hydrodynamic interactions, interparticle normal forces have been implemented by accounting for attractive van der Waals and shortrange repulsive Born forces,43,44 and finally tangential forces capable of supporting bending moments have been estimated via the discrete element method (DEM).45 The dependence of the breakage rate of a cluster and of the mass distribution of the produced fragments on the initial aggregate mass, fractal dimension, strength of interparticle interactions, and applied fluid flow has been established. To achieve this goal, we have performed breakup simulations of hundreds of aggregates having the same mass and fractal dimension, over a wide variety of aggregate masses (Nsphere = 30
215) and fractal dimensions (df = 1.8
3.0) and under a broad range of shear rates (γ = 2000
100 000 1/s), in simple shear and extensional flow has been studied. Both of these flow fields are of relevance even for modeling situations under turbulent conditions, because below the Kolomogorov length scale, the flow field can be represented as a combination of both types of flows.46

II. THEORETICAL BACKGROUND The dynamic evolution of the cluster mass distribution can be described by using PBE, which can be written as follows:7

where Nk(t) is the number of clusters per unit volume with mass k at time t. The aggregation terms in the equation above, starting from the left after the equality sign, represent (i) events where clusters with masses i and j aggregate with rate constant KAij to produce a cluster of mass k, whereas the second term corresponds to aggregation of a cluster with mass k with clusters of any mass to produce a larger cluster. The two terms related to breakup correspond to (i) events where any cluster with mass m greater than k breaks with a breakage rate constant KBm and produces a fragment with mass k characterized with a probability expressed by the fragment mass distribution function (FMD) Θm,k, and (ii) the case when a cluster with mass k breaks into smaller fragments with the rate of breakage KBk . As the aim of the present work is to compute the rate of breakage, we will concentrate on the breakage kernel (KBm) and the FMD (Θm,k) as a function of the number of particles in a cluster, the cluster fractal dimensions, the applied stress, and the fluid flow type. The breakage rate has been defined as the inverse of the time required (τbreak) to break a cluster, whereas the fragment distribution describes how likely it is for a cluster of a 284

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given mass and fractal dimension to break into fragments of a given mass.

C. Aggregate Library. Colloidal aggregates exhibit a special property called fractal scaling. Through this scaling, the number of particles (Nsphere) in an aggregate can be related to its size (for example, to its radius of gyration, Rg) through a power law, where the exponent is termed fractal dimension, df, which reflects the aggregate morphology:57 !df Rg Nsphere ¼ kf ð5Þ Rp

III. NUMERICAL RECIPE As mentioned in the introduction, to estimate the breakage kernel parameters, the aggregate breakup process has to be modeled by accounting for all the interactions as accurately as possible. In the following subsections, we describe how each type of interactions has been modeled, along with the list of parameters used in the simulations performed in the work. Moreover it must be noted here that throughout this work the terms “number of particles” in an aggregate and “aggregate mass” are interchangeably used, as we are considering aggregates made of uniform sized identical rigid hard spherical particles.

here kf is a prefactor usually close to unity. Low values of the fractal dimension indicate very open structures and are typical of clusters obtained under stagnant conditions, whereas high values of the fractal dimension are instead usually encountered in sheardriven aggregation processes.42 In order to investigate the effect of the cluster fractal dimension on the breakage kinetics, an aggregate library containing clusters with a wide range of fractal dimensions and size was generated using a combination of various Monte Carlo methods. The details are briefed in the Supporting Information, and further details can also be found in the work of Ehrl et al.58 D. Flow Field. For all simulations, we have imposed simple shear flow with the velocity gradient in the x
y directions and the vorticity in the z direction, whereas for extensional flow, compression in the y and z directions and elongation in the x direction have been used. The schematics of both fluid flows are shown in Figure SM1. Breakup of colloidal aggregates over a range of shear rates (γ = 2000
100 000 1/s) has been investigated. The imposed hydrodynamic stress in terms of the applied shear rate is found as σ = ηfγ. For each simulation, the initial center of mass of an aggregate is chosen as the reference point for the coordinate system. E. Simulation Methodology and Parameters. Using the developed model, we performed simulations for various cluster masses and fractal dimension values. Before starting any simulation, each cluster was equilibrated assuming that only the interparticle interactions were acting on the cluster, i.e., in quiescent fluid, to begin simulations with an energetically stable structure. After equilibration of the cluster, the actual simulation was performed by introducing the flow field. As the interparticle forces and hydrodynamic forces change magnitudes rapidly when particles approach each other, very small time steps (10
9
10
8 s) are required in order to prevent particles from overlapping each other. At each simulation time-step, knowing the position of all particles, all interparticle forces and hydrodynamic forces acting on them were estimated. Thus, with eq 4, the velocities of all particles were determined. The velocities were integrated to find the displacement of each particle, and new particle
particle separations were found. If the distance between any pair of particles, which was connected at the beginning of the simulation, was greater than a critical distance for a bond breakage, chose to be equal to (sbreak = 2 nm), resulting in the formation of two distinct fragments, then the cluster was considered to be broken and the simulation stopped. It is important to note here that the critical distance for considering particle
particle contact to be broken has been selected as the distance where the van der Waals force is less than 2% of its maximum value (please refer to Figure SM2 for the relative magnitude of van der Waals attractive, Born repulsive, and total force as a function of particle
particle separation distance). The time τbreak required for the first breakup event leading to two fragments was recorded. After breakup, the number of particles in both fragments is calculated. For the same mass and fractal dimension, many cluster

A. Particle
Fluid
Particle Interactions. Hydrodynamic Interactions. In laminar flow, where particle Reynolds number

Rep , 1, defined as,
 dp up Ff Rep ¼ ηf

ð2Þ

where dp and up are the particle diameter and velocity, and ηf and Ff are the fluid viscosity and density, respectively, a spherical particle follows the Stokes law, which relates the drag force F acting on the particle to its velocity up through a linear relationship. F̅ ¼ 3πηf dp u̅ p

ð3Þ

However, an exact extension of eq 3 to systems consisting of many hydrodynamically interacting particles is available only for up to two spheres, as it is not straightforward to incorporate many-body interactions for multiparticle systems.47
49 Hence, we have used Stokesian dynamics, proposed by Brady and Bossis,38 to account for the hydrodynamic interactions among particles. 2 3 2 3 F̅ n U̅ n
U̅ ∞ ∞7 6 6 6 T̅ 7 7 6 7 ð4Þ 4 n 5 ¼ ηf R 3 4 ̅Ωn
̅Ω 5 ∞ Sn
E where the forces (F n), torques (T n), and stresslets (S n) acting on each particle are linearly proportional to the relative translational ̅ n
Ω ̅ ∞) velocity of all (U n
U ∞) and the relative angular (Ω particles with respect to the undisturbed fluid velocity computed at the centers of the particles, and to the rate of strain (E ∞) of the applied fluid flow through a grand resistance matrix (R), which accounts for the hydrodynamic interactions between particles. For more details about the method, the reader is referred to the Supporting Information and to the existing literature (Brady and Bossis38 and Durlofsky et al.47). B. Particle
Particle Interactions. Particles dispersed in a liquid interact via complex colloidal forces.50 In an aggregate there are several particle
particle bonds present that can withstand stresses and transmit forces.51 Hence all these interactions should be accounted for in order to accurately model the important physical phenomena involved at the particle level. In the present work we have considered two types of interparticle interactions: normal interactions (attractive van der Waals44,50,52,53 and repulsive Born54) and contact tangential55,56 interactions. The latter ones were introduced when the interparticle distance was around 3 Ao. The interparticle interactions along with respective equations are detailed in the Supporting Information. 285

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Table 1. Simulation Parameters Rp = 1 μm

Fp = 1050 kg/m3

sbreak = 2 nm

Ff = 1000 kg/m3

ηf = 1  10
3 Pa.s

Kt = 6.65  10
4 N/m

Nborn = 10


23

δmin = 3 Å

Ah = 1.33  10
20 J

realizations (at least 50 and up to 500) were used to perform identical simulations and obtain good statistics. For each cluster, the time required for the on-set of breakup or the first breakup event and the masses of the produced fragments were determined. For the same cluster mass and fractal dimension, τbreak for all cluster realizations was averaged to determine the average time required to break a cluster, which was then inverted to find KB. Similarly, the produced fragments for all cluster realizations were used to generate the average FMD resulting from the breakup of aggregates of certain mass and fractal dimension. The set of all simulation parameters, unless and otherwise mentioned, are reported in Table 1.

Figure 1. Effect of critical particle
particle separation distance sbreak to consider a cluster as broken on the breakage rate at various shear stresses for Nsphere = 60 and df = 1.8.

connected to each other through normal van der Waals interactions and also experience tangential contact interactions that impart bending rigidity to the structure. The magnitude of these forces as opposed to the externally applied hydrodynamic forces, defines the strength of an aggregate. The stronger the attractive forces are, the more difficult it is to break an aggregate. When shear forces are applied to particles, these change their relative positions, resulting in evolution of their interactions with time. Hence the strength of an aggregate determines the time required to break it. Therefore, the criteria to define the breakage of a bond should be carefully chosen. Bond breakage has been defined when the distance between two particles overcomes a critical threshold, which has been chosen as the distance that leads to a reduction of van der Waals forces by 98% with respect to its maximum value. We would like to start by validating the applicability of our criterion by performing simulations for a few critical distances. To do so, in Figure 1 breakage rates for Nsphere = 60 and df = 1.8 are plotted against different applied stresses for three different critical particle separations, namely 1, 2, 10, and 20 nm. It can be seen from Figure 1 that the breakage rate does not depend on the critical separation distance at low shear rate and depends very weakly at higher shear rates. At lower shear rates, the time required to break the cluster is fairly large, hence no difference has been observed as the value of sbreak is increased. However, at higher shear rates, the dynamics of the process is faster and hence the critical particle
particle distance plays a certain role, since the time required for the fragments to travel apart a distance equal to the critical one becomes comparable to the breakage process time. Nevertheless, we have never observed broken bonds that have reformed. In fact, when a cluster breaks, usually the flow field pulls the two fragments apart from each other. This confirms the validity of our approach for considering a cluster broken when the critical particle
particle distance is chosen to be equal to sbreak = 2 nm. We have performed Stokesian dynamic simulation for a range of cluster masses and fractal dimensions at various shear rates under simple shear flow to study the breakup of colloidal aggregates. In order to express the dependence of the breakage rate on these parameters, two functional forms of the breakage rate have been proposed in the literature, namely, exponential kernel59
61 and power law kernel.60,62 In the present work we use a power law form of the breakage rate kernel:

IV. RESULTS AND DISCUSSION As mentioned in a previous section, the aim of our work is to find a suitable form for the breakage rate of fractal clusters as a function of their mass and fractal dimension. Therefore, in the present simulation study, we have focused on finding the dependence of two quantities, namely, the breakage time and the fragment mass distribution generated by a breakage event, on various parameters. To achieve this purpose, we decided to focus only on the very first breakage event of a cluster. With this assumption, breakage is always a binary process leading to the formation of only two fragments, as the simultaneous formation of more than two fragments is a very unlikely event. This choice was made for several reasons. Within the framework of PBEs, a cluster of a given mass ceases to exist as soon as it is broken. The alternative would be to follow the evolution of an individual cluster until an arbitrary time, if possible until a steady state has been reached. However, reaching the steady state can take a long time, and would lead to meaningful definition of breakage rate only under very dilute conditions. Under more concentrated conditions, aggregation events involving a cluster might occur much before a steady state has been reached. Since the first breakage event is often a very fast process, our choice could lead to a definition of breakage rate constant to be used even in simulations under more concentrated conditions. Finally, in reaching equilibrium conditions, the cluster can often undergo substantial restructuring, which changes its structure and makes the interpretation of the results more challenging. In addition, we also provided an estimate of the critical aggregate mass for a given fractal dimension that is stable under a given fluid flow condition by performing simulations at progressively smaller shear rates, looking for the first conditions where a cluster does not break any longer. This is important, since it is known that clusters with a size lower than a critical value will never break when exposed to sufficiently small shear stresses. The dependence of the breakage rate constant and fragment mass distribution on the initial fractal dimension, size of the cluster and applied fluid flow is discussed in the following. This is then followed by a discussion on the critical aggregate mass of stable under a given value of shear rate. Breakage Rate (KB). According to the particle
particle interactions scheme used in the present work, in a cluster, particles are

K B ¼ mÆRg æα σ β 286

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Table 2. Breakage Rate Parameters for Different Conditions simple shear flow df

m

α

β 2.42

1.8

0.4

1.2

2.1

0.31

1.21

2.53

2.3

3.2  10
3

2.65

3.01

2.5

1.75  10
3

3.0

3.0

2.7

2.38  10
5

3.75

4.81

3.0

1.45  10
6

4.75

4.13

extensional flow

Figure 2. Estimation of power law breakage rate parameters for various fractal dimensions and numbers of constitute particles in aggregate.

where m, α, and β are interpreted as fitting parameters. One of the objectives of this work is to determine these parameters and their dependence on the initial aggregate properties and the fluid flow. We observed that for a given aggregate fractal dimension, when KB is plotted on a double logarithmic plot against ÆRgæ and σ, separately, for different Nsphere the data points showed a linear trend (please refer to Figures SM3 and SM4 for such dependence on σ for one specific case of df = 1.8), except at the smallest shear rate and smallest cluster size. The deviations from the linear behavior under these conditions are due to small aggregates being more prone to restructure at low shear rates, thus becoming more resistant to breakup. However, the values of the slopes of these lines were found to be different for different aggregate morphologies. This indicated that for a given cluster fractal dimension, by adjusting the value of m, all data points could be expressed in terms of the product ÆRgæασβ . This is shown in Figure 2, where the breakage rate of clusters with different number of constitutive particles and morphologies is plotted against ÆRgæασβ on a double logarithmic-scale plot, with different values of α and β for different aggregate fractal dimensions. It can be seen that for a given fractal dimension the data for all cluster masses studied in the present work collapse together on a mastercurve by using a given set of values for m, α, and β. In addition to this, it can be observed from the same figure that for each fractal dimension a different mastercurve is obtained. It must also be noted that for different df values the values of m, α, and β are different (please refer to Table 2). Additionally it can be seen that for a given cluster mass and shear rate the breakage rate decreases as the fractal dimension increases. This is due to three factors: (i) the connectivity, i.e., the number of interparticle bonds is higher in denser clusters; (ii) the total hydrodynamic force acting on the aggregate and hence the average hydrodynamic force acting on individual particles in a dense cluster is much smaller than in an open cluster due the smaller size of dense clusters; (iii) the hydrodynamic shielding effect on the particles located in the core of a cluster due to those on the outer surface of the cluster, which is much higher in the case of dense cluster compared to open clusters as the particles are packed more closely. Hence the time required to break an open aggregate is much smaller than for a dense cluster. This is consistent with the findings of Higashitani et al.,25 who observed that for the same mass, dense clusters are much more resistant to shear than open ones. Similar estimations were performed for extensional flow for two different cluster masses (Nsphere = 30 and 60) and for two

df

m

α

β

1.8 2.5

553 337

0.95 1.05

1.37 1.53

3.0

157

1.1

1.68

Figure 3. Effect of flow type (simple shear [SS] and extensional flow [EF]) on the breakage rate and power law kernel parameters at various shear stresses for two fractal dimensions (Nsphere = 60).

different cluster morphologies, df = 1.8 and 2.5. After evaluating the set of breakage rate fitting parameters (m, α, and β), in Figure 3 the breakage rates in simple shear and extensional flow have been compared for the same cluster mass (Nsphere = 60). It can be seen from the figure that for a given cluster mass the breakage rate in extensional flow is much higher than that in simple shear, for the same shear rate, meaning that the mastercurves are also dependent on the flow type. This can be explained by considering that in simple shear conditions, flow clusters can rotate and change orientation with respect to the flow filed. In this manner, they constantly change the side exposed to the directions of maximum shear, and bonds can be rearranged. In extensional flow, instead, cluster rotation is almost absent, leaving them exposed to the flow without the possibility to reorient and rearrange. Moreover, as opposed to the simple shear flow case, aggregates show no dependence of the breakage rate parameters on the aggregate morphologies in the case of extensional flow. The fitting parameters for the breakage rate kernel in eq 6 for all cluster morphologies in simple shear and extensional flow are listed in Table 2. Using the set of parameters from the table, the rate of breakage of a cluster of given mass and fractal dimension for a given fluid flow can be determined, which can be directly used in eq 1 for the solution of PBE. 287

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masses of individual fragments depend on the location of the weakest bond in the cluster, which depends on the number of contacts and the magnitude of the hydrodynamic force acting on each particle in the aggregate. For a given cluster mass and fractal dimension, a great number of representative structures have been tested, leading to a large number of possibilities in which a cluster of given mass and fractal dimension would break. This leads to the FMD (Θm,k) describing the probability that a fragment of a certain mass would be produced after a cluster breaks. However, when aggregates with the same fractal dimension but with different mass (number of particles) are broken, the mass of the produced fragments would be different. Hence, in order to obtain an FMD independent of the initial aggregate mass, we have normalized the mass of the produced fragments by the initial aggregate mass, and we will refer to this quantity as mass fraction Nfr = N/N0, where N is the fragment mass and N0 is initial cluster mass. The resulting FMD function is denoted by fm,k. It must be noted that knowing the initial cluster mass Θm,k and fm,k are always related by multiplying the x-axis of fm,k function by the corresponding initial aggregate mass, N0 . Due to mass conservation, when two clusters are produced from the breakup of an aggregate, the knowledge of the mass of one of the fragments univocally determines the mass of the second fragment. Therefore, we have focused only on the distribution of smaller fragments produced when an aggregate is broken. In order to have a functional form of the FMD, we used Schultz
Zimm distribution—often used to fit particle size distribution—to fit the data. The distribution of smaller fragment was suitable to be fitted by means of Schultz
Zimm distribution, because it shows as a peak on the left side of the distribution. The Schultz
Zimm distribution takes the following form:64

Figure 4. Effect of primary particle size on the aggregate breakage rate and power law kernel parameters.

To investigate the generality of the breakage rate kernel parameters reported in Table 2, we performed simulations using different primary particle sizes. The magnitude of interparticle interactions between particles is governed by their size and the distance between them. Accordingly, the strength of the aggregate constituted by these particles will be affected. This could affect the breakage rate parameters found in the aforementioned discussion. Thus by changing the primary particle size we could also modify the interparticle interactions to explore their effects on the breakage rate and on the breakage rate kernel parameters. In earlier studies concerned with fragmentation of clusters in fluid flow, Fragmentation number (Fa) has been used for characterizing the process.9,63 The Fragmentation number is defined as the ratio of the hydrodynamic stress (σ = ηfγ) to the cohesive strength of the aggregate (Ah/R3p) Fa ¼

σRp3 Ah

fm, k ¼ ðaÞz þ 1 ðNfsr Þz exp½
ða 3 Nfsr Þ

ð7Þ

1 Γðz þ 1Þ

ð8Þ

where a = (z + 1)/ÆNsfræ and z are the fitting parameters. The average mass fraction of the whole FMD of smaller fragments is given by ÆNsfræ = ÆNsæ/N0 with ÆNsæ being the average mass of the smaller fragment, whereas z is the parameter determining the width of the FMD. The standard deviation (SD) of the FMD can be related to polydispersity index F as described by Kotlarchyk et al.,64

In order to make a fair evaluation of the effect of the primary particle size on the breakage process, we kept Fa constant by adjusting the shear rate to counteract the change in the primary particle size in order to derive meaningful conclusions. The effect of primary particle size on the breakage rate in simple shear flow for two different aggregate morphologies is shown in Figure 4. It can be seen that if the x-axis is scaled using the cube of the primary particle size (in micrometers), all data points for different primary particle sizes fall on the same mastercurve. Moreover, the breakage rate kernel exponent parameters, α and β, are the same, irrespective of the cluster primary particle size for a given fractal dimension. This indicates that apart from the parameter m, which needs to be scaled by R3p, all the other breakage rate kernel parameters are independent of the primary particle size. This dependence on the primary particle size is not surprising, since by writing the equations of motion of clusters in dimensionless form, the natural characteristic time of particle motion is proportional to the inverse of the cube of the particle size. This analysis helps generalize the parameter values reported in Table 2. Breakage Fragment Mass Distribution Function (fm,k). So far we have discussed how fast a cluster breaks. However, another piece of information required to fully quantify the breakage process and use the information in the solution of PBEs is the distribution of fragments produced after a cluster is broken. The mass of the produced fragments is completely governed by the stress distribution in the aggregate. More precisely, the

1 SD ¼ FÆNfrs æ with F ¼ pffiffiffiffiffiffiffiffiffiffiffiffi z þ 1

ð9Þ

In Figure 5 the FMD, number fraction (nfr(k) = fm,k) as a function of mass fraction (Nsfr(k)), obtained by analyzing the breakup of 200 aggregate realizations with Nsphere = 100 and df = 1.8 and 2.5 at various shear rates is presented by points, whereas the corresponding Schultz
Zimm distribution (eq 8), by fitting the parameters a and z, is presented by lines. It is important to point out that since the FMD refers only to the smaller fragments produced, the x-axis scale ranges from 0 to 0.5. Furthermore, it can be seen from the figure that using the fitted parameters for the Schulz
Zimm distribution, the actual FMD can be reproduced very well. Second, these distributions show that at higher shear rates for both open and dense clusters, the FMD shifts to smaller fragments and becomes narrower, indicating that the breakage process tends to become more an erosion type of mechanism. Additionally, it looks like at high shear rates, aggregates with different structures tend to break approximately in the same manner. 288

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fields, by fitting parameters (a and z) for each distribution. Using these values, the corresponding average mass fraction (ÆNsfræ) of the smaller fragment and the standard deviation of mass fraction distribution (SD) were obtained. We have plotted these values in Figure 6a
d against the same parameter ÆRgæασβ used for the breakage rate kernel for simple shear and extensional flow. The corresponding α and β for all cluster masses and fractal dimensions studied in the present work from Table 2 were used. In Figure 6a,b, this dependence is shown for ÆNsfræ and z, respectively, for breakup of clusters in shear flow. From the figures it can be seen that irrespective of the cluster mass and aggregate fractal dimension, a linear dependence is found. Both ÆNsfræ and z show inverse power law dependence on ÆRgæασβ with slopes of
0.2 and
0.6, respectively. Moreover, for the same aggregate fractal dimension, irrespective of the initial cluster mass, all data points collapse on the same mastercurve. In Figure 6c,d such dependence has been presented for the breakup of aggregates in extensional flow. As opposed to simple shear flow, no linear dependence of both ÆNsfræ and z on ÆRgæασβ is found. Rather, both FMD parameters are a very weak functions of ÆRgæα σβ. This once again indicates that, for a give shear rate, extensional flow is inherently more severe than simple shear flow and shows no significant dependence of average aggregate mass fraction and its distribution on the shear rate over the entire shear rate range investigated in the present work. We can therefore conclude that, similarly to what was observed for the rate of breakage, the parameters of the fragment mass distribution follow a master curve when plotted against the product ÆRgæασβ, with only a dependence on the fractal dimension and cluster fractal dimension and flow filed conditions, but not on the cluster mass. It must be noted that using these plots the fitting parameters values necessary to compute the full Schultz
Zimm distribution and hence the FMD can be found, which can then be directly used in the PBEs.

We have performed the same analysis using Schultz
Zimm distribution for various aggregate masses, morphologies, and flow

Figure 5. FMD in terms of cluster mass normalized by initial cluster mass versus number fraction for Nsphere = 100 and (a) df = 1.8 and (b) df = 2.5 at various hydrodynamic stresses [σ = 25 (0, 9), 50 (O,b), 75 (4,2),100 (3,1)] in simple shear flow. Symbols indicate data obtained from SD, while the lines represent the fitting of the data using Schultz-Zimm distribution (eq 8).

Figure 6. Average fractional cluster mass as estimated by eq 9 for the breakup of aggregates in simple shear flow (a,b) and extensional flow (c,d) with Nsphere = 60. 289

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Figure 8. Stable critical aggregate mass (Nc)as a function of critical hydrodynamic stress (σc) for different aggregate morphologies.

Table 3. Critical Aggregate Mass and Critical Applied Stress Relation Parameters for Simple Shear Flow for eq 10

Figure 7. Dependence of breakage rate on stress to find critical stress in simple shear flow for (a) aggregates having two different morphologies for Nsphere = 100 and (b) aggregates with different masses and the same fractal dimension for df = 1.8.

df

ψ

p

1.8 2.3

190.5 340

0.901 1.1

2.5

853

1.42

2.7

5366.35

1.925

value of σc, as indicated by vertical lines, is different. The critical aggregate stress decreases with increasing the aggregate mass. This can be explained by means of the fractal scaling law: for the same aggregate fractal dimension an increase in the aggregate mass leads to an increase in the aggregate size, and the total hydrodynamic force acting on the aggregate, which scales as σ(Rg)2, increases as well. This suggests that the critical stress found might also scale as (Rg)2. A similar behavior was observed for the critical stress dependence on aggregate mass for other aggregate morphologies. We extended this analysis for various aggregate morphologies to generate a master curve representing the critical aggregate mass as a function of applied hydrodynamic stress, and this is shown in Figure 8. Here the aggregate mass that survives the breakage process, which we refer to as the critical aggregate mass (Nc), has been presented on a double-logarithmic scale plot against the corresponding critical applied stress (σc) for various aggregate fractal dimensions. For all fractal dimensions, a linear dependence can be clearly seen, which indicates that a power law type relation between the two, which could be expressed as follows,

Critical Hydrodynamic Stress (σc). The balance of attractive and repulsive forces governs whether an aggregate breaks. If the net hydrodynamic force acting on an aggregate is smaller than a critical value, then the aggregate remains intact, maybe just undergoing simple restructuring. Thus, below a critical shear stress, an aggregate does not break, while above it the aggregate breaks, though after a very long time. We refer to this as the critical hydrodynamic shear stress(σc), a quantity that depends on the aggregate mass and fractal dimension and the applied stress. In solving PBEs, the knowledge of this critical shear stress is crucial, because it means that certain clusters will not break under given conditions. To obtain this piece of information, we performed simulations at decreasing shear rates and found out the corresponding breakage rate, under simple shear conditions. Such an analysis for Nsphere = 100 and for two different fractal dimensions is shown in Figure 7a. From the figure it can be seen that as the hydrodynamic stress decreases, the breakage rate decreases rapidly and after a certain stress the breakage rate drops to very small values. In the figure these limiting values are shown by lines; for a given aggregate, this is the critical hydrodynamic stress that it can withstand. From the same figure it can also be seen that the breakage rate of the dense aggregates (df = 2.5)is smaller than that of the open aggregates (df = 1.8) for the same hydrodynamic stress, and hence their critical stress is larger. Similarly we analyzed clusters having the same fractal dimension but different masses to understand the dependence of σc on the aggregate mass, and the results are presented in Figure 7b. It can be seen from Figure 7b that for all aggregate masses, the trend is the same, so that large breakage rates are found at higher stress values and gradually decreasing rates are found at smaller shear stresses. Moreover, as the critical stress value is reached, the breakage rate very rapidly drops down to zero. However, the

Nc ¼ ψðσc Þ
p

ð10Þ

A similar type of relation has been already reported in the literature both experimentally (Horwatt et al.65) and also through numerical investigations (Harada et al.41 and Zaccone et al.33). The power law exponent, p, referred to as the fracture exponent, and also the prefactor ψ showed dependence on the aggregate fractal dimension. The values of p and ψ for different fractal dimensions have been reported in Table 3. Even though these parameters depend on the type and magnitude of the interparticle interactions taken into account, the effect of which has not been investigated in this work, the exponent found in the present study is in good agreement with previously reported exponent values for different df (The comparison has been shown in Figure SM5). Nevertheless, the scaling presented 290

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’ REFERENCES (1) Batchelor, G. K. Transport properties of 2-phase materials with random structure. Annu. Rev. Fluid Mech. 1974, 6, 227–255. (2) Filippov, A. V.; Zurita, M.; Rosner, D. E. Fractal-like aggregates: relation between morphology and physical properties. J. Colloid Interface Sci. 2000, 229 (1), 261–273. (3) Brenner, H.; Condiff, D. W. Transport mechanics in systems of orientable particles 0.3. Arbitrary particles. J. Colloid Interface Sci. 1972, 41 (2), 228–. (4) Soos, M.; Morbidelli, M.; Wang, L.; Fox, R. O.; Sefcik, J. Population balance modeling of aggregation and breakage in turbulent Taylor
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V. CONCLUSIONS In the present work we have developed an innovative methodology to determine the breakage rate of colloidal aggregates, which can be used as kernels in PBEs calculation. This was accomplished by detailed simulations performed using Stokesian dynamics, accounting for all interparticle interactions in a rigorous manner, of the first breakage event of model clusters, generated by Monte Carlo methods. Breakup of aggregates covering a broad range of masses and fractal dimension values has been analyzed in a wide interval of hydrodynamic stress values, under both simple shear and extensional flow conditions. Several hundred aggregates having the same mass and fractal dimension were simulated until they were broken. The performed simulations were used to develop a breakage kernel to be used in PBEs. Three important variables, namely, the breakage rate, fragment mass distribution, and critical hydrodynamic stress, have been determined. A power law breakage model, expressing the breakage rate as a product of the applied stress to an exponent and of the aggregate size to a different exponent, has been used to interpret the results. It was found that a proper set of breakage rate parameters exists, which makes all the data for a given cluster fractal dimension and flow field type collapse on a mastercurve. Different fractal dimension and flow field conditions lead to different master curves. The fragment mass distribution, giving the probability of a fragment of certain mass to be produced as a result of breakage of a cluster of given mass and fractal dimension, has been effectively approximated by a Schultz
Zimm distribution. The corresponding fitting parameters have been calculated. It was found that their dependence on the initial cluster mass and shear stress is analogous to that of the breakage rate, leading to different mastercurves for different fractal dimension values and flow filed conditions. Finally, the critical stress below which an aggregate of certain mass and fractal dimension does not break under given flow conditions has been determined. For a given fractal dimension, the stable aggregate mass showed a power law dependence on the applied stress, with the exponent being a function of the aggregate fractal dimension. The presented work will help solving PBE by providing the first comprehensive breakage kernel, a quantity so far elusive and only indirectly obtained by analyzing unequivocal experimental data. ’ ASSOCIATED CONTENT

bS

Supporting Information. A brief description of the Stokesian dynamics method used in the present work along with the interparticle interactions considered is presented. Information about the cluster’s library utilized and the schematics of the flowfields is presented. In addition to this, relative magnitudes of different interparticle interactions and the dependence of the breakage rate on the applied hydrodynamic stress and aggregate size has been presented. This material is available free of charge via the Internet at http://pubs.acs.org.

’ ACKNOWLEDGMENT The authors are thankful to Prof. Dr. Massimo Morbidelli for his valuable suggestions and guidance in the work. The work was financially supported by the Swiss National Foundation (Grant No. 200020-126487/1). 291

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