Size and Structure of Clusters Formed by Shear Induced Coagulation

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Size and Structure of Clusters Formed by Shear Induced Coagulation: Modeling by Discrete Element Method Martin Kroupa, Michal Vonka, Miroslav Soos, and Juraj Kosek Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.5b01046 • Publication Date (Web): 23 Jun 2015 Downloaded from http://pubs.acs.org on July 3, 2015

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Size and Structure of Clusters Formed by Shear Induced Coagulation: Modeling by Discrete Element Method Martin Kroupa, Michal Vonka, Miroslav Soos, and Juraj Kosek∗ University of Chemistry and Technology Prague, Department of Chemical Engineering, Technicka 5, 16628 Prague 6, Czech Republic E-mail: [email protected]

Abstract Coagulation process has a dramatic impact on the properties of dispersions of colloidal particles including the change of optical, rheological, as well as texture properties. We model the behavior of a colloidal dispersion with moderate particle volume fraction, i.e., 5 wt%, subjected to high shear rates employing the time-dependent Discrete Element Method (DEM) in three spatial dimensions. The Derjaguin-Landau-VerweyOverbeek (DLVO) theory was used to model non-contact inter-particle interactions, while contact mechanics was described by the Johnson-Kendall-Roberts (JKR) theory of adhesion. Obtained results demonstrate that the steady-state size of the produced clusters is a strong function of the applied shear rate, primary particle size and the surface energy of the particles. Furthermore, it was found that the cluster size is determined by the maximum adhesion force between the primary particles and not the adhesion energy. This observation is in agreement with several simulation studies ∗

To whom correspondence should be addressed

1

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and is valid for the case when the particle-particle contact is elastic and no plastic deformation occurs. These results are of major importance, especially for the emulsion polymerization process, during which the fouling of reactors and piping causes significant financial losses.

Introduction The shear induced coagulation of colloidal dispersions is an important phenomenon in natural (spider’s silk formation), as well as artificial systems (gelation and fouling in polymer production). Depending on the application, coagulation can be either a desired or undesired process. In both cases, the key factor is the knowledge of the coagulation mechanism, allowing us to control its existence and the dynamics of the process. While the coagulation mechanism is well documented in the literature for systems under quiescent conditions; 1 in real applications, however, dispersions are rarely quiescent. In the presence of shear, the kinetics of coagulation depends heavily on the primary particle size and colloidal stability. In the case of completely destabilized particles, the aggregation process starts immediately upon primary particle contact by the formation of doublets and larger aggregates. As the cluster size increases, another mechanism, the breakage, becomes important. In the case of a low particle volume fraction, when the coagulation is balanced by the breakage, the average cluster size is no longer changing and a steady state is reached. However, already for moderate particle volume fractions, formed clusters start to percolate, which results in a rapid increase of viscosity and the gelation occurs. 2 Several techniques can be employed to monitor the processes of coagulation and breakage. One of these techniques, the light scattering, monitors the evolution of both gyration radius (Rg ) and the internal cluster structure characterized by the fractal dimension (df ). 3 For higher particle volume fraction, rheological properties could provide a valuable information about the behavior of a system undergoing aggregation. From the above mentioned it becomes clear that the size to which the clusters can grow 2


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is influenced by many factors, among which the most studied is the applied shear rate (G). For diluted colloidal particle dispersions, Soos et al. (2008) reported that the mean radius of gyration hRg i of clusters in a stirred tank was a decreasing power law function of G. However, since the conditions in the tank were highly turbulent, which resulted in a large variation of the shear rate in the reactor, it was possible to determine only the scaling of hRg i with hGi, and not the breakage criterion as a function of the local shear rate. As an attempt to identify the maximum value of shear rate corresponding to the maximum measured cluster’s Rg , Soos et al. (2013) performed detailed numerical simulations of the turbulent flow in the stirred vessel. 5 They found the maximum stable size of the clusters to be determined by the maximum shear rate present in the system. In parallel to experimental investigations, various simulation methods are used to model the coagulation in colloidal dispersions. This includes Stokesian Dynamics (SD), which is based on the hydrodynamic interactions of particles and has been successfully used for the modeling of breakage and restructuring of clusters. 6–11 It captures very accurately the hydrodynamics of multi-particle system, but due to the necessity of the inversion of the resistance matrix, the number of particles in simulations is limited to several hundreds. Another tool, which was used for the modeling of breakage is the Discrete Element Method (DEM). This method is capable of performing simulations with much larger number of primary particles than SD; however, the hydrodynamic interactions are often neglected and the so-called permeable aggregate model is used. 12,13 In contrary to these studies, which consider laminar flow, Derksen (2012) performed direct numerical simulation of coagulation in isotropic turbulence using the Lattice Boltzmann Method. 14,15 The inter-particle potential, however, was greatly simplified and reduced to the square-well model. Based on the theoretical description of the cluster internal strength, Zaccone et al. (2009) provided an analytical expression for the power-law dependence of hRg i on G during the single cluster breakage. 16 According to this theory, the power-law coefficient (p) is a function of the fractal dimension df . Using the data from Stokesian dynamic simulations of the breakup

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process of a single fractal aggregate, 7,8 Conchuir et al. (2014) unified the above mentioned measurements with the theoretical considerations and predicted the size of clusters as a function of their df and the applied shear rate. 17,18 Majority of the aforementioned studies are limited to diluted systems with particle volume fraction φ ≤ 1 × 10−5 , in which breakage is the dominant mechanism determining cluster size. However, in concentrated dispersions (up to φ = 0.5) the effect of coagulation must be taken into account. Moreover, the adhesive properties of the primary particles significantly affect the inter-particle interactions and, thus, the size of the clusters. All these require (i) the usage of models with appropriate inter-particle forces, including the tangential forces; and (ii) sufficiently large number of particles in the simulations to capture both coagulation and breakage, while still resolving the hydrodynamic interactions. This issue has not yet been addressed in the literature. We present a model based on the Discrete Element Method with the interaction between particles described by the connection of Derjaguin-Landau-Verwey-Overbeek (DLVO) and Johnson-Kendall-Roberts (JKR) theories. 19 This enables us to study the effect of the elasticity and adhesion of the primary particles on the properties of the clusters. The model is spatially three-dimensional and we implemented the two-way coupling between particles and flow to capture the hydrodynamic interactions. The predicted hydrodynamic properties of clusters agree well with the literature data in terms of the settling speed ratio or hydrodynamic radius. In our simulations, the particle volume fraction φ is equal to 0.05, which is a sufficiently concentrated system to capture the simultaneous effect of coagulation and breakage. Unlike other modeling approaches presented in the literature, our model is able to simulate the collective behavior of many clusters, not just a single one, which increases the statistical relevance of the results. By comparison with literature experimental data we show that our model is able to correctly predict the kinetics of shear-induced coagulation. 20,21 We demonstrate that the final size of the produced clusters is a function of the applied shear rate and the surface

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energy, which is the measure of particle adhesion. Moreover, we show that the cluster size is determined by the balance between the hydrodynamic force acting on the cluster and the maximum adhesion force between the primary particles.

Mathematical model The model is composed of two parts. The first part is based on the Discrete Element Method and is used to compute the particle trajectories and rotation rates based on the forces and torques acting on them. The second part serves for the computation of the flow field in the domain. It employs the Finite Volume Method for the discretization of the Navier-Stokes equations and for this purpose the OpenFOAM software 22 was used. These two parts are coupled together by the particle hydrodynamic force and torque by the so-called two-way coupling technique. 23 In the following, we first introduce the overall framework of the Discrete Element Method, then we describe the computation of the flow field including the connection with DEM. The description of interparticle forces including both elastic and adhesive interactions is given afterwards and finally we introduce the particle-wall interactions. Particles in the Discrete Element Method are described as discrete elements. Each element is characterized by its mass (mi ), position (xi ), velocity (vi ) and rotation rate (Ωi ). We assume the primary particles to be mono-disperse with radius Rp and having the same density ρp . The governing equation for the translational motion is the Newton’s second law: d2 xi Fi = , 2 dt mi

(1)

where Fi represents the sum of all forces acting on the discrete element i. Since the rotation of particles plays an important role in coagulation, 23 it must be resolved in the model. The temporal change in the rotation rate (Ωi ) is expressed as follows: dΩi Mi = , dt Ii 5


(2)

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where Mi is the sum of all torques acting on particle i and Ii is the particle momentum of inertia. For homogeneous solid sphere we have Ii = 25 mi Rp2 (particle moment of inertia around its center) and mi = 43 πRp3 ρp .

Interactions of the particles with the flow Particles can interact with the surrounding fluid by various forces and torques. These include drag, buoyancy, gravity and others. The large applied shear rates cause the Peclet number to be very high and the effect of the Brownian motion thus can be neglected. This section introduces the forces and torques relevant for our model and their implementation. For small particles, the drag force (Fd,i ) is usually one of the most important forces in the particle-fluid interaction. Since the small size of particles also implies their low Reynolds number, Fd,i can be expressed by the Stokes’ law:

Fd,i = 6πηf Rp (vif − vip ),

(3)

where ηf is the fluid dynamic viscosity, Rp is the radius of the particle and vif and vip are the velocities of the fluid and the particle, respectively. Buoyancy and gravity forces can be neglected, since we set the particles and the fluid to have the same density, which is a common assumption in emulsion polymerization. The torque on a particle caused by the local rotation of the fluid can be computed as: 24

Mfi

= πηf (2Rp )

3

1 ω i − Ωi , 2

(4)

where ω i is the angular velocity of the fluid at the position of the particle center. The particles in non-diluted dispersions also exert a body force on the fluid flow. 23 To account for this body force, we used the so-called two-way coupling between particles and fluid. We followed the approach suggested by Marshall (2009) and used the modified NavierStokes equations in the following form: 23 6


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∂ (cv ρf ) + ∇ · (cv ρf u) = 0 ∂t

(5)

∂ (cv ρf u) + (u · ∇)(cv ρf u) = cv η∇2 u − cv ∇p − b, ∂t

(6)

where cv = 1 − c is the fraction of voids, ρf is the fluid density, u is the fluid velocity, p is the pressure and b is the body force exerted by particles. For the computation of hydrodynamics we used the program OpenFOAM, 22 which was coupled with the DEM model. Equations 5 and 6 were discretized using the Finite Volume Method (FVM). The computational grid was chosen to be regular with cubic grid cells of size a (length of the edge) and volume V . When particles and fluid have the same density, then the body force bn,i , which the particle i exerts on the node n can be expressed as follows:

bn,i = −

Vn Fd,i , V

(7)

where Vn is the volume of the sub-grid cell opposite to node n (see Figure S2 in the Supporting Information for explanation), V is the total grid cell volume and Fd,i is the drag force acting on the i-th particle. In Equation 7, the so-called volume partitioning is used to distribute the body force of the particle onto the grid. Details of this approach were described by Marshall (2009). 23 We tested the hydrodynamic behavior of the model by simulating the forced flow around a conglomerate of 55 primary particles and around a fractal cluster composed of 313 primary particles (see Figure S3 in the Supporting Information). From the results of these simulations we can conclude that with proper settings of the parameters (i.e., grid cell size, domain size, velocity), the model is able to reproduce the literature data for the settling speed ratio and hydrodynamic radius. 20,21 Based on these results, we chose the grid cell size to be equal to κ = 3Rp resulting in 24 cubic cells in each spatial direction (13 824 cells overall). The effect 7


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of lubrication forces on the results of a coagulation simulation is shown in Figure S10 in SI. Due to the insignificance of this effect and the large increase in computational complexity, these forces were not included in the model.

Interactions between the particles An essential part of the inter-particle interactions are the normal forces, because they reflect the properties of the particles. We need to describe particles that are elastic, adhesive and also interact through non-contact forces. In our previous work, we proposed a model based on the DLVO theory for non-contact interactions and the JKR theory for solid-body contact. 19 The same model was used also in this work. We outline the main features of the model in the following paragraphs and refer to the previous paper 19 for more details. We also provide the description of tangential forces in the Supporting Information. Normal Interactions As mentioned above, our model of normal interactions (taken from the previous work 19 ) is based on the DLVO and JKR theories. In this work, we considered destabilized particles, where the effect of Electrostatic Double Layer is negligible. Therefore, for the non-contact interaction only the van der Waals potential energy (VvdW ) was used in the following form: 1

VvdW

AH =− 6

(

" 2 #) 2Rp2 2Rp2 2Rp , + ln 1 − + h2 + 4hRp (h + 2Rp )2 h + 2Rp

(8)

where h is the separation distance such that h = d − 2Rp , where d is the distance between centers of the two particles. The quantity AH is the Hamaker constant, which characterizes the material of the particles and the surrounding medium. The interaction force (FvdW ) acting on the particles was obtained as the negative derivative of VvdW with respect to vdW separation distance (h), i.e., FvdW = − dVdh .

For the interaction of adhesive elastic solid bodies in contact, we used the JKR theory

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originally proposed by Johnson et al. (1971). 25 The particles were assumed to be equal-sized. When no external force is acting on the particles and the force equilibrium is reached, we can define the equilibrium radius of the contact area (a0 ) as: 23

a0 =

9πγR2 E

1/3

,

where the effective radius for equal-sized particles is R =

(9) Rp , 2

the quantity γ denotes the

surface energy (a material property that can be found in textbooks 1,26 ) and E is the effective Young’s modulus given by: 1 1 − ν2 =2 , E EY

(10)

where EY is the Young’s modulus and ν is the Poisson’s ratio of the particles. The equation for the (non-equilibrium) radius of the contact area (a) is the following: 23 " 1/2 # 2 a 4 a h = 61/3 δC 2 − , a0 3 a0

(11)

where the critical overlap (δC ) is given by the following equation:

δC =

a20 . 2(6)1/3 R

(12)

Finally, the normal force (Fne ) between two colliding particles can be obtained from: Fne =4 FC

a a0

3

−4

a a0

3/2

,

(13)

where the critical force (FC ) is given by:

FC = 3πγR.

(14)

The quantity FC corresponds to the maximum adhesive force, which can occur between

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particles described by the JKR theory. Therefore, it was used for the connection with the DLVO theory to constrain the DLVO force-distance curve, which was cut at the point, where FC occurs. However, the separation distance at which FC occurs is predicted differently by each of the models due to different assumptions being considered in their derivation. This issue was solved by shifting of the JKR curve in the direction of x-axis (separation distance), such that the maximum adhesion force occurs exactly at the crossing with the DLVO curve. The distance of the shift ∆h is always small compared to the particle radius (∆h/Rp < 0.01), under which condition the effect of the shift on the results is negligible. Details of this procedure are described in our previous paper. 19 The advantage of this procedure is that it provides continuous potential-distance and force-distance curves with the well-defined effect of the model parameters on the properties of the curves. This can be seen in Figure 1, where typical force-distance and potential-distance curves are shown demonstrating the effect of E and γ on their shape. For each consecutive curve the value of the parameter is doubled. Two particles are considered as connected, if their separation distance is smaller than the distance, at which the DLVO and JKR curves are connected. The viscoelasticity of the material can be introduced into the inter-particle model by the addition of the dissipation force. We tested the model regarding the sensitivity to the presence of this dissipation force and and found this effect to be negligible. It is well known that for small (sub-micron) particles immersed in a viscous fluid, the drag and lubrication forces are responsible for the major part of energy dissipation. 23 Our observation agrees with this fact and therefore the dissipation force during solid-body contact was not included in the model. All normal forces acting between two particles are finally summed up and projected to the normal direction as follows: Fn = (FvdW + Fne ) n, where n is the unit normal vector (a tangent to the line connecting particle centers).

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(b)

(a) 100 E force (F) / nN

potential energy (V) / kBT

2

0 í í í í

1

E

0 í í í

0 10 separation distance (h) / nm

í

(c)

0 separation distance (h) / nm

10

(d)

1

0 force (F) / nN

potential energy (V) / kBT

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í í

0 í í í

í í í

0

í í

5 10 separation distance (h) / nm

0

4 6 8 10 separation distance (h) / nm

Figure 1: The effect of Young’s modulus ((a), (b)) and surface energy ((c), (d)) on the potential-distance and force-distance curves. The values of parameters were: Rp = 405 nm, γ = 0.3 − 3.0 mJ m−2 , E = 20 − 1600 MPa, AH = 1.3 × 10−20 J.

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Tangential interactions There are several different types of tangential interactions of particles described in the literature. 23 In this work we employed the models of the resistance to sliding, rolling and twisting. The formulas for these models are given in the Supporting Information.

Interactions between the particle and the wall The description of the particle-wall interactions is complex and involves most of the interaction types and theories mentioned above. In the present work, we focused on the behavior of the colloidal particles in the bulk. Therefore, we decided to minimize the influence of the wall on the system behavior, assuming the particle-wall interaction to be purely repulsive. Mathematically, this can be conveniently done using either a power-law or exponential function. The corresponding physical models are the Born repulsion and hydration force, respectively. In order to avoid the stiffness of the short-ranged Born repulsion, we decided to adopt the exponential decay for the description of the particle-wall repulsive force and for this purpose we utilized the formulation of the hydration (or solvation) force proposed in the literature. 26 In terms of the interaction potential energy (Vh ) this repulsion can be phenomenologically described as the following exponential dependence: 27

Vh = πRp F0 δ02 exp(−H/δ0 ),

(15)

where H is the particle-wall separation distance, F0 is the hydration force constant and δ0 is the characteristic decay length. We adopted the values of the constants used by Wu et al. (2010) as they satisfactorily describe the behavior of the force for colloidal particles. 28 The values of the constants used in our model for the particle-wall interactions were: F0 = 2 × 108 N m−2 and δ0 = 3 × 10−10 m. Keeping in mind the complexity of the particle-wall interaction, we believe that this simplification is justified by the purpose which is to focus on the bulk behavior. The detailed study of the interaction of particles and clusters with

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walls and the resulting fouling behavior is the subject of our on-going research.

Structure of the computational domain Simulations were performed in spatially three-dimensional (3D) domain. A box with dimensions L × L × L was considered with L = 30 µm for all simulations. A total number of N = 5000 primary particles was randomly placed into the domain yielding the particle volume fraction φ = 0.05. The initial positions of primary particles were chosen such that no solid body interaction occurred at the beginning of simulations. We used the simple shear model of flow in the domain (see Figure S1 in the Supporting Information). In this setup, the upper and lower boundary were set to represent a solid wall. The upper wall was moving with the velocity U and this created a linear velocity profile in the domain. All the remaining boundaries were considered periodic.

Characterization of the cluster size and structure The size of aggregates produced in our simulations was characterized by their radius of gyration (Rg ). We followed the procedure suggested by Lattuada et al. (2004), which requires to calculate the cluster center of mass (COM) as the first step. 29 The position of COM (xCOM ) is computed as follows:

xCOM

np 1 X = xi , np i=1

(16)

where np is the number of particles in the cluster and xi is the position of i-th particle. The determination of xCOM is, however, more complicated in a system with periodic boundaries. Therefore, we adopted the approach of Bai and Breen (2008), which consists in the transformation of the particle positions onto a circle and then computing COM from the transformed positions. 30 The position of COM is subsequently transformed back to the original domain and this procedure is done for each direction in which the periodic boundary is employed.

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Once xCOM is known, we can compute the radius of gyration of the cluster as follows:

Rg2 where Rg,p =

q

3 R 5 p

=

2 Rg,p

np 1 X (xi − xCOM )2 , + np i=1

(17)

is the radius of gyration of a primary particle. To suppress the effect

of the finite size of the domain, simulations with cluster size comparable to the domain size (Rg approaching L/2) were excluded from further analysis. The description of cluster structure is generally a complex task and many approaches including the particle-particle correlation function 31 and the radial density distribution 29 were used in the literature. The assumption of cluster fractality simplifies greatly this description and reduces it to a single quantity, the fractal dimension df , defined by the following equation:

np = k

Rg Rp

d f

,

(18)

where k is the prefactor. To obtain df for an ensemble of clusters, we used the Ordinary Least Square method leaving both k and df as adjustable parameters.

Breakage and restructuring of the resulting clusters Due to high particle volume fraction and fully destabilized conditions, formed fragments will undergo reaggregation. Therefore, to investigate the stability of a single cluster under given shear rate, we performed the breakage simulations in the following way: 1. Each cluster from the end of an aggregation simulation was placed into a new domain (of the same size) without other clusters and with an unperturbed flow-field. 2. The breakage simulation was run until any breakage event occurred. 3. In the case of breakage, the largest fragment was again isolated into a new domain and the breakage simulation was performed. 14


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4. When no breakage occurred for a sufficiently long period of time (i.e., more than 20 times longer than the typical breakage time 8 ), the cluster was considered as stable.

Numerical details The model used in this work consists of two parts, the particle motion part and the fluid dynamics part. The dynamics in these parts occurs on different time-scales and therefore different time-steps are required for each part. Marshall (2009) suggested guidelines for these time-step settings. 23 Following these guidelines, we obtain the characteristic particle collision time in our system TC = 2Rp (ρ2p /E 2 U )1/5 ≈ 2.5 × 10−9 s. The actual time-step of the simulation must be always smaller than this characteristic time, therefore the time-step for the particle motion was set to ∆tp = 2 × 10−10 s. The characteristic fluid time-scale is obtained as follows: TF = L/U and depends therefore inversely on the upper-wall velocity U , which can be computed from the applied shear rate G. Finally, the time-step for the fluid dynamics was set as ∆tF = 0.01TF . The computational time of the simulations depends on the inter-particle potential, since for steeper inter-particle potential the system of ODE will be stiffer resulting in a longer computational time. Lower shear rate implies slower coagulation dynamics, while the particlecollision time depends on G (through U ) only very weakly. Therefore ∆tp remains the same for all simulations and more time-steps are needed for lower G, leading again to longer computational time. The computational time thus varies from one day to several weeks depending on the system properties. All simulations were carried out with a sequential code on a workstation equipped with Intel Xeon E5 v3 processors.

Results and Discussion The results presented in this section were obtained by running the coagulation simulations for fully destabilized primary colloidal particles. The parameters of the system were chosen

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to resemble a polystyrene latex produced by emulsion polymerization (see Table 1). Table 1: Default values of model parameters used in the simulations. Quantity

Value

Name

Rp

405 nm

Radius of particles

AH

1.3 × 10−20 J

Hamaker constant

E

1.6 GPa

Young’s modulus

ν

0.2

Poisson’s ratio

ρp

1000 kg m−3

Particle density

µeff

0.5

Effective friction coefficient

crit

π/4

θ

Critical rolling angle

γ

0.3 mN m

ηF

1 × 10−3 Pa s

Fluid dynamic viscosity

ρF

1000 kg m−3

Fluid density

T

293.15 K

Absolute temperature

−1

Surface energy of particles

The time-development of the number of particles in aggregates (Np ) and the number of aggregates (aN ) is shown in Figure 3 for the system consisting of N = 5000 primary particles (see Figure 2 for snapshots). Initially, the particles aggregate very quickly forming large number of aggregates (mostly doublets), which tend to aggregate further resulting in a formation of large aggregates. This is the reason why aN decreases reaching a steady-state value. This value is given by the dynamic equilibrium between the aggregation and breakage, which also controls the cluster size.

Breakage and restructuring of the resulting clusters Before going into the analysis of the coagulation model, let us first set the results of the model in the context of previous studies. To be able to compare our results with the literature data, we performed simulations of a single cluster breakage and the snapshots from the beginning and end of a typical simulation are presented in Figure 4. In Figure 5a,b, we show the temporal development of Rg and df of the cluster and its fragments for the same 16


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in an aggregate (np ) and the normalized radius of gyration (Rg /Rp ). This plot is shown in Figure 5c for all clusters present in the system at the end of the aggregation simulation. After the breakage and restructuring processes, the distribution in Figure 5c shifts to the left combined with the simultaneous change of the scaling. The slope of the fitted line in the double logarithmic plot represents in this case the average fractal dimension of the whole ensemble. The second fitting parameter, the prefactor k, was close to unity for both cases, which is in agreement with the literature results. 32 It is clear that df is increased by the breakage and restructuring and the resulting clusters are more compact (also visible in Figure 4). The final value of the fractal dimension agrees well with the literature data for shear-induced coagulation, since values of df around 2.60 were reported for comparable systems. 4,7,17 The change caused by breakage is also clearly visible in the plot relating the maximum radius of gyration with the applied shear rate, as shown in Figure 5d. The slope of the fitted line is now p = −0.48, which is in close agreement with p = −0.45 obtained from model by Zaccone et al. (2009) for df = 2.51.

Effect of shear rate Getting back to the coagulation simulations in a moderately concentrated system, there are two important quantities characterizing a cluster size distribution: the root mean square radius of gyration hRg i; and the maximum radius of gyration Rg,max . Figure 6 shows the time development of hRg i and Rg,max for different shear rates. At the very beginning, both these quantities show fast increase, which differs only slightly for different shear rates when plotted against the dimensionless time. This behavior is typical for systems where aggregation is dominating. 4 Once the breakage starts to be more important, the cluster size starts to vary with the shear rate. Finally, both hRg i and Rg,max reach steady-state values. These values result from the balance between the aggregation and breakage and they clearly show the dependence on the shear rate. 3,4 19


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(b)

(a)

8

fractal dimension (df)

9

1 2

7 6

6 7 8 9 í

10

í

3

2

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1

10

10

1

Rg/Rp

6 7 8 9 í

10

(c)

df = 2.51

0

2

10

before breakage after breakage df = 2.13

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10 time (t / µs)

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max radius of gyration (Rg,max / µm)

radius of gyration (Rg / µm)

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np

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í

10 time (t / µs)

í

10

1

10

(d)

0

after breakage

10

4

10

5

10 shear rate (G / sí )

6

10

Figure 5: (a) Radius of gyration and (b) fractal dimension as a function of the dimensionless time for the breakage simulations of a cluster formed by the aggregation. The numbers in the legend indicate the order of simulations. (c) Number of particles in an aggregate as a function of the normalized radius of gyration. (d) Steady state maximum radius of gyration as a function of the shear rate after breakage. The exponent p of the power-law fit is: −0.48. The values of parameters were: G = 20 000 s−1 , Rp = 405 nm, AH = 1.3 × 10−20 J, E = 1.6 GPa.

20


rms radius of gyration ( Rg« / µm)

1

10

(a)

G = 40 000 sí G = 60 000 sí G = 80 000 sí G = 120 000 sí G = 200 000 sí í G = 300 000 s G = 400 000 sí

«

0

10

0

2

time (t G φ)

4

6

G = 40 000 sí G = 80 000 sí G = 200 000 sí G = 400 000 sí

(b)

1

10

0

10

0

2

time (t G φ)

4

6

Figure 6: (a) Root mean square and (b) maximum radius of gyration as a function of the dimensionless time. The values of parameters were: Rp = 405 nm, γ = 0.3 mJ m−2 , G is in s−1 .

An advantage of the Discrete Element model is that it provides not only hRg i, but also the entire cluster size distributions (CSD). An example of the CSD for three different values of shear rate is shown in Figure 7. These distributions were obtained by averaging the instantaneous distributions over a period of time in the steady state, i.e., from the time when the steady state is reached till the end of the simulation. It can be seen that upon decreasing the shear rate, the CSD broadens and moves towards larger cluster sizes. Similar trend was observed also by Harshe and Lattuada 2012 using Stokesian dynamics simulations. 8 1.5

fraction / µm

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max radius of gyration (Rg,max / µm)

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G = 60 000 s−1 G = 80 000 s−1 G = 120 000 s−1

1

0.5

0 0

1

2

3 Rg / µm

4

5

6

Figure 7: Averaged Cluster Size Distributions for the steady state parts of the curves in Fig. 6. The values of parameters were: Rp = 405 nm, γ = 0.3 mJ m−2 .

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Effect of surface energy The steady-state size of the clusters produced in our simulations is a result of the interplay between the hydrodynamic stress acting on the particles and the inter-particle attractive forces. Intuitively, more adhesive primary particles should produce larger clusters under a given shear rate. The results presented in Figure 8a are in agreement with this expectation, as they indicate the increase of the cluster size upon increasing the surface energy (γ) of particles. Note that, according to the JKR theory, the increase of γ leads to the increase of both the adhesion energy (Eadh ) and the maximum adhesion force (Fc ), this effect will be further explained later. By taking the steady state values of hRg i, we can obtain the scaling of the average aggregate size with the shear rate as a function of applied values of γ (see Figure 8b). This relation is often published in the literature as the slope (p) of the fitted line in a double logarithmic plot. In the case of obtained results producing aggregates with df around 2.1 the coefficient p is equal to −0.65 ± 0.075. When comparing the obtained scaling with those published in the literature we found that our values are higher approximately by a factor of 3. 4,7–9,12 This can be explained by the fact that available scalings in the literature were obtained for very diluted systems where system was dominated by breakup. This is however not the case of presented simulations where the effect of aggregation is significant. The steady state root mean square radius of gyration hRg i can be also plotted against the surface energy (see Figure 8c). In that case it produces a well-developed power-law scaling with the slope equal to 0.63 ± 0.033. Another important quantity characterizing the coagulation process is the number of particles in a single aggregate np . When plotted against the shear rate, the steady state root mean squared quantity hnp i also follows the power-law scaling (see Figure 8d). For low values of γ the degree of adhesion is low and the inter-particle potential curve (see Figure 1) resembles the one for fully destabilized, non-adhesive particles, which is commonly used in the literature. 8 Harshe and Lattuada (2012) used this model and reported values around −1.1 22


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for the slope of hnp i as a function of G, which corresponds to our results for γ = 0.1 mJ m−2 . The slopes of the fitted lines in Figure 8d clearly follow the trend of the increasing slope with increasing surface energy. A similar trend was observed in the literature with increasing fractal dimension of the clusters in the simulations of single-cluster breakage. 6,8,33 However, as the df was varying in our simulations only weakly around df ≈ 2.1 the reason for observed scaling in Figure 8d has to be different. In particular, by the increase of the surface potential, the maximum adhesive force between primary particles is increased, which increases also the internal strength of the aggregate, an effect comparable to the increase of df .

Effect of primary particle size The effect of Rp can be shown either for absolute hRg i or relative hRg i/Rp cluster size. When plotted against the primary particle radius, the steady state hRg i follows a power-law scaling with the slope equal to 0.43 ± 0.0603 (see Figure 9a). On the other hand, Figure 9b shows that the relative size of the clusters (hRg i /Rp ) is reduced for larger primary particles. It means that smaller particles tend to form larger clusters relative to their size, which is in agreement with experimental studies; 2,3 however, in the absolute size the clusters are smaller, due to the prevailing effect of the primary particle size.

Effect of the Young’s modulus So far we presented that the relative size of the clusters (hRg i /Rp ) increases with increasing γ and decreases with increasing G and Rp . These observations are in agreement with our expectations as well as with the literature results. However, rather surprising results were obtained for the effect the of Young’s modulus E. By varying E in the range from 10 MPa to 1.6 GPa literally no effect on the cluster size was observed. Deeper look in the literature revealed that the same results were obtained by Marshall (2007) for aerosol particles in a channel flow and by Eggersdorfer et al. (2010) for the breakup and restructuring of colloidal particles in a simple shear flow. 12,34 In both papers, a similar model of elastic and adhesive 23


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γ = 0.3 mJ/m2 γ = 0.6 mJ/m2 γ = 1.0 mJ/m2 2

γ = 3.0 mJ/m 0

10

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2 3 time (t G φ)

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(c)

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3 2 1.5 1 0.8 0.6 0.4 4 10

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10

«

G = 60 000 G = 80 000 G = 120 000 G = 200 000 G G

1 0.8

10

(b)

γ = 0.1 γ = 0.3 γ = 0.6 γ = 1.0 γ = 3.0 5

1

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10 4 10

í

10 surface energy (γ / mJmí )

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3

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í

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np «

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rms radius of gyration ( Rg « / µm)


γ = 0.1 mJ/m2

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rms radius of gyration ( Rg« / µm)

1

10

«

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γ = 0.1 γ = 0.3 γ = 0.6 γ = 1.0 γ = 3.0 p í p í p í p í p í 5


Figure 8: (a) Root mean square radius of gyration as a function of the dimensionless time (G = 200 000 s−1 ); (b) steady state root mean square radius of gyration as a function of the shear rate; (c) steady state root mean square radius of gyration as a function of the surface energy (G in s−1 ); and (d) steady state root mean squared number of particles per aggregate as a function of shear rate for different values of the surface energy γ (mJ m−2 ). Primary particle size was: Rp = 405 nm.

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í

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rms radius of gyration ( Rg« / Rp)

0

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G = 80 000 sí G = 100 000 sí G = 200 000 sí G = 400 000 sí

1

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Rp = 100 nm R = 200 nm p

Rp = 300 nm

«

«

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(a) 2

Rp = 405 nm

(b)

0

10

5

10

10 particle size (Rp / nm)

6

í

10

Shear rate (G / s )

Figure 9: Steady state root mean square radius of gyration as a function of (a) the radius of primary particles for different shear rates; and (b) the shear rate for different radii of primary particles. The surface energy was: γ = 0.3 mJ m−2 .

particles as employed in this work was used. It is important to clarify here the role of E in the JKR model. The model considers both elastic repulsion and adhesive attraction acting simultaneously within the contact region of two spheres. Therefore, by reducing the Young’s modulus, the depth of the potential well is increased (see Figure 1). For the minimum and maximum value of E used in this work, the increase of the well depth is approximately three times. If the cluster size was determined by the well depth, then such an increase should lead to much larger clusters. Clearly, in our simulations this was not the case.

Scaling of the cluster size The fact that in our simulations the depth of the potential well was not important led us to the following idea: When an aggregated cluster of colloidal particles is exposed to an external flow, this flow provides energy for the cluster to break up. Because the attractive forces act on a short distance (order of nanometers), the actual distance which the particles must be moved apart is also short and it is necessary for the flow to act only for a short time in order to separate the particles - provided that the potential well is not too steep. This steepness is given by the attractive force and in fact the most important parameter is the 25


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maximum attractive force, in the JKR model given by FC . In order to test the aforementioned hypothesis, let us consider a simple example. If the quantity determining the cluster stability was the maximum attractive force (FC ) acting between two primary particles, then the cluster size would be given by the balance between FC and the hydrodynamic force acting on the cluster. This hydrodynamic force is for a sphere given by: 3 Fhyd = 5/8πηF d2 G, where d is the sphere diameter. Considering dense clusters, this expression is a reasonable approximation. By assuming FC ∝ Fhyd we propose the rough scaling of the approximated cluster size (d) with the system parameters:

d∝

s

Rp γ . ηF G

(19)

The dependence of cluster size on G has been studied extensively in the literature and it is known that for a single cluster it is a decreasing power-law function with the exponent p being a function of fractal dimension. 16 We have already shown that in our simulations the exponent was equal to −0.65 ± 0.075, which is close to the scaling predicted by Equation 19. Furthermore, Equation 19 predicts that d should grow with the square root of Rp ; again the dependence found by our simulation (i.e., the power-law coefficient equal to 0.43 ± 0.0603). Finally, the obtained dependence of the cluster size on the surface energy and the fluid viscosity with exponents equal to 0.63 ± 0.033 and −0.49 ± 0.120 (see Figure S11 in SI), respectively, are close to those predicted by Equation 19, thus confirming the used approach that FC ∝ Fhyd . The important implication of these results is that the cluster size in a semi-diluted dispersion under shear is given by the balance between the adhesive force between primary particles and the hydrodynamic force. The depth of the potential well, and therefore the adhesion energy, is not important (see the effects of E and γ in Figure 1). This result is valid for solid elastic primary particles. In the case of plastic deformations, a permanent change in the particle surface occurs, which leads to the increase in the maximum adhesive force, increase of the surface roughness and the behavior is therefore different. However, this issue 26


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is beyond the scope of this paper and will be addressed in our future work.

Conclusions Our model of shear-induced coagulation was successfully used for the simulations of coagulation in moderately concentrated destabilized dispersions. The predictions of the model in terms of hydrodynamics, coagulation dynamics and the resulting-cluster properties agree well with the literature data. In addition, we have shown that the size of the clusters is affected not only by the applied shear rate, but also by the degree of particle adhesion. In contrary to some literature data, our results suggest that the maximum adhesion force (FC ), and not the adhesion energy, is the quantity that actually determines the cluster size. There are two main consequences of this statement. First, the dependence of cluster size on primary particle size (Rp ) would be governed by the scaling of FC with Rp , which was confirmed by our simulations. The second consequence is that cluster size should be independent of the Young’s modulus of the particles, which is, once again, in agreement with our simulation results and also the available literature. 12,34 Once confirmed by further studies, the results of this work might be of huge importance for the operation of industrial units dealing with colloidal dispersions. Although the aforementioned consequences are valid only for the case when particle-particle interactions are elastic and no plastic deformation occurs, future work is already in progress on the incorporation of plastic deformations into our particle interaction model.

Acknowledgement The authors are grateful for support provided by EC FP7 project COOPOL (NMP2-SL2012-280827). Financial support from specific university research (MSMT No 20/2015) is gratefully acknowledged.

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Supporting Information Available The scheme of the computational domain; the mechanism of the volume partitioning; the detailed information to the particle-particle interactions; the testing of the hydrodynamic properties predicted by the model; the perturbed velocity field; the effect of lubrication forces; the effect of the fluid viscosity on the cluster size predicted by the model.

This

material is available free of charge via the Internet at http://pubs.acs.org/.

References (1) Hunter, R. J. Foundations of Colloid Science, 2nd Edition.; Oxford University Press, 2001. (2) Xie, D. L.; Wu, H.; Zaccone, A.; Braun, L.; Chen, H. Q.; Morbidelli, M. Criticality for shear-induced gelation of charge-stabilized colloids. Soft Matter 2010, 6, 2692–2698. (3) Ehrl, L.; Soos, M.; Morbidelli, M. Dependence of aggregate strength, structure, and light scattering properties on primary particle size under turbulent conditions in stirred tank. Langmuir 2008, 24, 3070–3081. (4) Soos, M.; Moussa, A. S.; Ehrl, L.; Sefcik, J.; Wu, H.; Morbidelli, M. Effect of shear rate on aggregate size and morphology investigated under turbulent conditions in stirred tank. J. Colloid Interface Sci. 2008, 319, 577–589. (5) Soos, M.; Kaufmann, R.; Winteler, R.; Kroupa, M.; Luthi, B. Determination of Maximum Turbulent Energy Dissipation Rate generated by a Rushton Impeller through Large Eddy Simulation. AIChE J. 2013, 59, 3642–3658. (6) Harada, S.; Tanaka, R.; Nogami, H.; Sawada, M. Dependence of fragmentation behavior of colloidal aggregates on their fractal structure. J. Colloid Interface Sci. 2006, 301, 123–129. (7) Harshe, Y. M.; Lattuada, M.; Soos, M. Experimental and Modeling Study of Breakage and Restructuring of Open and Dense Colloidal Aggregates. Langmuir 2011, 27, 5739–5752. (8) Harshe, Y. M.; Lattuada, M. Breakage Rate of Colloidal Aggregates in Shear Flow through Stokesian Dynamics. Langmuir 2012, 28, 283–292.

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(9) Seto, R.; Botet, R.; Auernhammer, G. K.; Briesen, H. Restructuring of colloidal aggregates in shear flow Coupling interparticle contact models with Stokesian dynamics. Eur. Phys. J. E: Soft Matter Biol. Phys. 2012, 35, 128. (10) Vanni, M.; Gastaldi, A. Hydrodynamic Forces and Critical Stresses in Low-Density Aggregates under Shear Flow. Langmuir 2011, 27, 12822–12833. (11) Fellay, L. S.; Vanni, M. The effect of flow configuration on hydrodynamic stresses and dispersion of low density rigid aggregates. J. Colloid Interface Sci. 2012, 388, 47–55. (12) Eggersdorfer, M. L.; Kadau, D.; Herrmann, H. J.; Pratsinis, S. E. Fragmentation and restructuring of soft-agglomerates under shear. J. Colloid Interface Sci. 2010, 342, 261–268. (13) Heinson, W. R.; Sorensen, C. M.; Chakrabarti, A. Shear History Independence in Colloidal Aggregation. Langmuir 2012, 28, 11337–11342. (14) Derksen, J. J. Direct numerical simulations of aggregation of monosized spherical particles in homogeneous isotropic turbulence. AIChE J. 2012, 58, 2589–2600. (15) Derksen, J. J. Direct simulations of aggregates in homogeneous isotropic turbulence. Acta Mech 2013, 224, 2415–2424. (16) Zaccone, A.; Soos, M.; Lattuada, M.; Wu, H.; Baebler, M. U.; Morbidelli, M. Breakup of dense colloidal aggregates under hydrodynamic stresses. Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 2009, 79, 061401–1–061401–5. (17) Conchuir, B. O.; Harshe, Y. M.; Lattuada, M.; Zaccone, A. Analytical Model of Fractal Aggregate Stability and Restructuring in Shear Flows. Ind. Eng. Chem. Res. 2014, 53, 9109–9119. (18) Conchuir, B. O.; Zaccone, A. Mechanism of flow-induced biomolecular and colloidal aggregate breakup. Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 2013, 87, 032310. (19) Kroupa, M.; Vonka, M.; Kosek, J. Modeling the Mechanism of Coagulum Formation in Dispersions. Langmuir 2014, 30, 2693–2702. (20) Cichocki, B.; Hinsen, K. Stokes Drag On Conglomerates of Spheres. Phys. Fluids 1995, 7, 285–291. (21) Harshe, Y. M.; Ehrl, L.; Lattuada, M. Hydrodynamic properties of rigid fractal aggregates of arbitrary morphology. J. Colloid Interface Sci. 2010, 352, 87–98.

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(22) www.openfoam.org. (23) Marshall, J. S. Discrete-element modeling of particulate aerosol flows. J. Comput. Phys. 2009, 228, 1541–1561. (24) Crowe, C.; Sommerfeld, M.; Tsuji, Y. Multiphase Flows with Droplets and Particles; CRC Press, Boca Raton, 1998. (25) Johnson, K. L.; Kendall, K.; Roberts, A. D. Surface Energy and Contact of Elastic Solids. Proc. R. Soc. A 1971, 324, 301–313. (26) Israelachvili, J. Intermolecular and Surface Forces; Elsevier Science, 2010. (27) Cowley, A. C.; Fuller, N. L.; Rand, R. P.; Parsegian, V. A. Measurement of Repulsive Forces Between Charged Phospholipid Bilayers. Biochemistry 1978, 17, 3163–3168. (28) Wu, H.; Tsoutsoura, A.; Lattuada, M.; Zaccone, A.; Morbidelli, M. Effect of Temperature on High Shear-Induced Gelation of Charge-Stabilized Colloids without Adding Electrolytes. Langmuir 2010, 26, 2761–2768. (29) Lattuada, M.; Wu, H.; Morbidelli, M. Radial density distribution of fractal clusters. Chem. Eng. Sci. 2004, 59, 4401–4413. (30) Bai, L.; Breen, D. Calculating Center of Mass in an Unbounded 2D Environment. Journal of Graphics Tools 2008, 13, 53–60. (31) Lattuada, M.; Wu, H.; Morbidelli, M. A simple model for the structure of fractal aggregates. J. Colloid Interface Sci. 2003, 268, 106–120. (32) Ehrl, L.; Soos, M.; Lattuada, M. Generation and Geometrical Analysis of Dense Clusters with Variable Fractal Dimension. J. Phys. Chem. B 2009, 113, 10587–10599. (33) Saha, D.; Soos, M.; Luethi, B.; Holzner, M.; Liberzon, A.; Babler, M. U.; Kinzelbach, W. Experimental Characterization of Breakage Rate of Colloidal Aggregates in Axisymmetric Extensional Flow. Langmuir 2014, 30, 14385–14395. (34) Marshall, J. S. Particle aggregation and capture by walls in a particulate aerosol channel flow. J. Aerosol Sci. 2007, 38, 333–351.

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Size and Structure of Clusters Formed by Shear Induced Coagulation

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