Flow-Induced Aggregation and Breakup of Particle Clusters Controlled

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Flow-Induced Aggregation and Breakup of Particle Clusters Controlled by Surface Nanoroughness Amgad S. Moussa,†,§ Marco Lattuada,†,∥ Breanndán Ó . Conchúir,‡ Alessio Zaccone,‡ Massimo Morbidelli,† and Miroslav Soos*,† †

Institute for Chemical and Bioengineering, Department of Chemistry and Applied Biosciences, ETH Zurich, 8093 Zurich, Switzerland ‡ Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, CB3 0HE Cambridge, United Kingdom S Supporting Information *

ABSTRACT: Interactions between colloidal particles are strongly affected by the particle surface chemistry and composition of the liquid phase. Further complexity is introduced when particles are exposed to shear flow, often leading to broad variation of the final properties of formed clusters. Here we discover a new dynamical effect arising in shear-induced aggregation where repeated aggregation and breakup events cause the particle surface roughness to irreversibly increase with time, thus decreasing the bond adhesive energy and the resistance of the aggregates to breakup. This leads to a pronounced overshoot in the time evolution of the aggregate size, which can only be explained with the proposed mechanism. This is demonstrated by good agreement between time evolution of measured light-scattering data and those calculated with a population-balance model taking into account the increase in the primary particle nanoroughness caused by repeated breakup events resulting in the decrease of bond adhesive energy as a function of time. Thus, the proposed model is able to reproduce the overshoot phenomenon by taking into account the physicochemical parameters, such as pH, till now not considered in the literature. Overall, this new effect could be exploited in the future to achieve better control over the flow-induced assembly of nanoparticles.

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INTRODUCTION Flow-induced aggregation and breakage are two phenomena frequently encountered in systems comprising of flowing colloidal dispersion. Flocculation,1 flotation,2 crystallizations,3,4 and emulsion and suspension polymerization5,6 are examples of engineering processes where these phenomena play an important role. Despite the large effort of researchers the majority of these studies focused on physical parameters such as the shear rate, solid volume fraction, primary particle size, and equipment geometry.7−14 Despite this effort, very little amount of work was dedicated to the effect of surface chemistry on the aggregation process,15−18 even though it is known that surface chemistry plays a paramount role in controlling the interactions among colloidal particles.19,20 Therefore, it is expected to greatly influence the flow-induced aggregation and breakage by modification of the liquid phase composition. This deficiency in the literature is due to the conditions often probed during flow-induced aggregation in aqueous media, where ionic strength values above the critical coagulation concentration (CCC) are used. Under these conditions, no energy barrier exists between aggregating particles, and therefore van der Waals attractive forces are bringing about rapid aggregation down to a distance of few angstroms imposed by Born repulsion. Thus, the outcome of aggregation above the CCC should be universal at least for the same bulk material.21 However, this view is based on observations of aggregation © 2013 American Chemical Society

under quiescent conditions and is valid for systems where DLVO theory20 properly describes the interparticle interactions. The validity of this extrapolation to aggregation in the presence of flow was never put to test. Furthermore, the structure of the double layer, that is, depth and coverage of the counterions, co-ions, and hydration water molecules adjacent to the surface, can still influence the interparticle forces within the aggregate above the CCC through various mechanisms such as: hydration repulsion,22−25 hydrophobic attraction,26 steric hindrance, and ion bridging,27 which might influence flowinduced breakage of aggregates, a phenomenon that is absent in quiescent systems. Clearly overlooking the role of surface chemistry might confound the results of studies investigating a specific parameter, for example, primary particle size or shear rate, using latexes that are not identical in their surface chemistry. The objective of this work is to elucidate the possible effects that a change in surface chemistry might induce on the aggregation and breakage of clusters composed of polystyrene particles exposed to turbulent flow under fully destabilized conditions. The aggregation of the particles has been induced by means of MgCl2 well above the critical coagulation Received: August 28, 2013 Revised: October 21, 2013 Published: October 24, 2013 14386

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I(0). Both these moments were extracted from the measured scattered intensity in the Guinier region (i.e., for q⟨Rg⟩ up to about unity) using slope and intercept of the linear relation between ln (I(q)/P(q)) and q2/3

concentration in combination with the appropriate amounts of HCl or NaOH to adjust final pH. The first part of the work is dedicated to elucidating the effect of the protonation of surface carboxyl groups stabilizing latex particles, while the second part is dedicated to the effect of surface charge density on the aggregation and breakup process. The gained knowledge about the controlling mechanisms was in the last part used to develop a mathematical model to describe the observed phenomenon. It is demonstrated that the interplay between hydrodynamic interactions and surface adhesion forces among primary particles can lead to situations where a plastic deformation of the particles occurs, resulting in a progressive increase in the particles surface roughness. The phenomenon has been quantitatively described by means of a mathematical model based on population balance equations with an ad-hoc breakage kernel that explicitly accounts for surface roughness effects. The obtained results clearly indicate that surface chemistry has to be taken into consideration when dealing with aggregation of fully destabilized latex particles under flow conditions.

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⎛ I(q) ⎞ q2 ln⎜ ⎟ = ln(I(0)) − ⟨R g2⟩S(q) 3 ⎝ P(q) ⎠

The root-mean-square radius of gyration of a population of aggregates was calculated according to ⟨R2g⟩ = ⟨R2g⟩S(q) + ⟨R2g,p⟩ using the radius of gyration of the primary particles, Rg,p = (3/5)1/2Rp. Because of the large size of primary particles light scattering cannot be used to properly determine aggregates internal structure,7 so an alternative method using analysis of aggregate images was employed. In particular, a log−log plot of aggregates area (A) versus their perimeter (P) was used to determine aggregates’ perimeter fractal dimension, dpf, according to the following scaling8 A ∝ P 2/ dpf

The specifications of the latexes used in this work are the following. The main latex used was a carboxyl polystyrene white latex purchased from Interfacial Dynamics Corporation (IDC, Portland, OR U.S.A.) (product no., 7-300; CV, 3.2%; batch no., 2440; solid % = 4.3). The diameter of primary particles was found to be 300 nm with a narrow size distribution. The surface charge density was equal to 9.8 μC/cm2 with a corresponding area occupied by a single carboxyl group equal to 162 Å2. To compare the effect of primary particle size on the size of the formed aggregates carboxyl polystyrene white latex (product no., 7-900; CV, 3.6%; batch no., 1761; solid % = 3.9) was used. The primary particles diameter was equal to 860 nm. The surface charge density was equal to 12.4 μC/cm2 and the corresponding area occupied by a single carboxyl group was equal to 130 Å2. The last studied latex was the surfactant-free sulfate polystyrene white latex (product no., 1−300; CV,:5.6%; batch no., 1919; solid % = 8.8) with a low surface charge density of 1.5 μC/cm2, corresponding to an area per charge group equal to 1073 Å2 and a narrowly distributed primary particles with diameter equal to 320 nm. All experiments were carried out in a stirred tank with a volume of 2.5L, as previously described.7,8 Shortly, the original latex was diluted using deionized water to the desired concentration and fed to the stirred tank. After filling the vessel, the desired rotation speed was set and an aggregation was initiated by injecting 60 mL of 4.9 molar aqueous MgCl2 solution into the tank, along with the necessary amount of HCl or NaOH to obtain the desired pH. It is worth noting that in all experiments the resulting salt concentration in the tank was always above the CCC ensuring complete destabilization of primary particles. An online light scattering measurement device, Mastersizer 2000 (Malvern, U.K.), was used to measure the intensity of scattered light as a function of time. Characterization of Aggregates by Light Scattering and Image Analysis. The dependency of the scattered light intensity I(q) as a function of the scattering vector amplitude q can be expressed as28

dNm 1 = dt 2

n ⎛⎜ θ ⎞⎟ sin λ ⎝2⎠

i+j=m

∑

∞ A B K ijANN i j − Nm ∑ K imNi − K mNm

i,j=1

i=1

∞

∑

+

i=m+1

K iBGimNi

(5)

where Nm is the concentration of clusters with mass m, KAij is the aggregation kernel between two clusters with masses i and j, respectively, KBi is the breakage rate constant of a cluster with mass i and Gim is a fraction of fragments with mass m produced by a breakage of a cluster with mass i. Aggregation Kernel. Considering that the two mechanisms driving the aggregation process are Brownian diffusion and shear aggregation, the aggregation kernel takes the form31 2kBT 1/ df 4 + j1/ df )(i−1/ df + j−1/ df ) + G(i1/ df + j1/ df )3 (i 3μ 3 (6) This first term of the aggregation kernel is the classical diffusionlimited rate, first proposed by Smoluchowski and later extended to fractal clusters. The second term represents instead the rate of aggregation due to shear. The additive-flux assumption in eq 6 is an approximation to the full solution of the convective-diffusion equation that works reasonably well in the limit of completely screened electrostatic repulsion,32 which has been confirmed also by more detailed calculations.33 One of the assumptions of PBEs is that the concentration of clusters is homogeneous in the system, and therefore the shear rate used in eq 6 is the average shear rate of the system, estimated from CFD calculations (see Table SI 1 in Supporting Information). Breakage Kernel. The breakage kernel used in eq 5 for the shearinduced breakup rate of a colloidal aggregate of radius Rg and fractal dimension df is based on the work of Ó Conchúir and Zaccone.34 In particular it is based on the solution of a diffusion-drift equation in a flow field for two mutually bonded colloidal particles (dimer) and K ijA =

(1)

where I(0) is the zero-angle intensity, P(q) is the form factor (due to primary particles), S(q) is the structure factor (due to the arrangement of primary particles within the aggregates), and q is defined as

q = 4π

(4)

The value of dpf for 2D projections of aggregates varies between 1 (corresponding to an Euclidean aggregate), and 2 (corresponding to a linear aggregate). Typical values of dpf for aggregates produced under turbulent conditions are in the range from 1.1 to 1.4.8,13 Aggregates area and perimeter were evaluated using the image analysis software ImageJ v1.34s (http://rsb.info.nih.gov/ij/). The correlation between dpf and df developed by Ehrl et al.29 was used to convert obtained dpf to the mass-based fractal dimension df. Mathematical Model. Population Balance Equations. The simulations of shear aggregation carried out in this work have been performed using population balance equations (PBEs). PBEs are material balances for the entire population of clusters produced during the aggregation process. In the presence of shear forces, both aggregation and breakage phenomena are occurring, which means that PBEs take the form30

MATERIALS AND METHODS

I(q) = I(0)P(q)S(q)

(3)

(2)

where θ is the scattering angle, n is the refractive index of the dispersing fluid, and λ is the vacuum laser wavelength. The measured CMDs are characterized by two moments, namely the root-mean-square radius of gyration, ⟨R2g⟩1/2 (hereafter, for convenience, denoted simply by ⟨Rg⟩), and the zero-angle intensity, 14387

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where δ is the range of the attraction, k is the elastic constant, and Δ represents the adhesion energy (and contains various contributions such as van der Waals, hydrophobic, etc.). In our time-dependent approach for the overshoot, Δ is the timedependent quantity which sets the time-dependence of the kernel. Using the Persson theory42 Δ can be related to the surface roughness and with good accuracy it can be approximated by the applying the following interpolation formula42

subsequently including the structure-dependent effects of stresstransmission and many-body hydrodynamic interactions from the other particles in a semiempirical way. This kernel is the only firstprinciples kernel currently available that accounts for all the underlying physics, from the cluster structure, to the colloidal interaction and the hydrodynamics. The latter is a superposition of the direct effect of the flow field at the dimer level and the transmission of shear stresses throughout the rigid assembly by a force chain ensemble. This acts as an additional, structure-dependent drift term in the diffusion-drift equation governing the rupture of a bond in the aggregate’s interior.34 The breakup mechanism is controlled by the breakup kinetics of a bond inside the aggregate. Since colloidal aggregates are marginally stable from the point of view of rigidity percolation,35,36 as they restructure to be just able to withstand the shear, it is enough that few bonds in the aggregate interior break up to cause a global loss of rigidity, and the aggregate collapses when the average number of bonds per particle falls below 2.4 (which represents the rigidity threshold of disordered structures with bond-bending interactions35). The breakup kinetics of a single bond is a Kramers-type activatedescape process36,37 controlled by the competition between the interparticle bonding and the shear stress collectively transmitted down the particle chains into the aggregate.34 The collectively transmitted shear, as we will show below, has a strong dependence on the fractal structure through the stress-transmission efficiency which also depends on the many-body hydrodynamics. The breakup rate kernel accounting for all these effects has the following expression K iB =

− UT″(rmax )UT″(rmin) 2πb

⎛ U (r ) − UT(rmin) ⎞ exp⎜− T max ⎟ kBT ⎝ ⎠

Δ = 3.4723κ 3 − 5.3524κ 2 + 0.9378κ + 0.9449 Δ*

which is valid for the surface roughness κ in the range 0.05 nm < κ < 0.9 nm, and parameter Δ* is the adhesion energy of a perfectly smooth surface (κ = 0 nm). Assuming that the surface roughness sets the interaction range, we have δ=

2Δ ≃ 2κ k

(11)

from which we obtain

k=

Δ 2κ 2

(12)

For a perfectly smooth surface, the curvature of the parabola is infinite and this treatment is consistent with the Johnson-KendallRoberts limit where the adhesion energy is concentrated in a point on the radial axis.20 We assume that the roughness of the particle surface at the start of the experiment is equal to the radius of a styrene monomer

(7)

where UT(r) is the total or combined interparticle interaction energy defined as

⎛ r ⎞ UT(r ) = Uattr(r ) + Us,pair(r ) + Us,coll(r ) − 2kBT ln⎜⎜ ⎟⎟ ⎝ Rp ⎠

(10)

k0 = (8)

Δ 2κ02

(13)

where κ0 ≈ 0.2 nm as for styrene. Therefore, at the beginning of the experiment (t = 0) the adhesion potential is given by

It composes of an attractive part Uattr(r) due to the colloidal bonding and of contributions due to the shear stresses at the twoparticle level, Us,pair(r), and also transmitted collectively through the structure, Us,coll(r). UT(r) for a typical colloidal interactions has a maximum at UT(rmax) and a minimum at UT(rmin). The difference UT(rmax) − UT(rmin) represents the energy barrier that a colloidal particle has to cross over in order to break the bond inside the aggregate resulting in the global vanishing of rigidity and collapse. The various contributions are discussed more in detail below. The last logarithmic term is a metric term emanating from the 3D spherical coordinate system.34 It should be reminded that the energy terms due to flow stresses are treated on the same footing of conservative interaction forces because of mathematical convenience and notation economy, but they actually arise from nonconservative drift terms in the convective-diffusion equation, as is evident from the full mathematical derivation of eq 7 reported in ref 34. Because of the variation of separation distance between primary particles by a combination of electrostatic charge screening using salt and neutralization of the surface charge groups using an acid, and consequent action of the hydrodynamic force that could lead to the significant deformation of the primary particle surface by exceeding the full plasticity limit of a polystyrene,20,38−41 particular attention was taken when describing the interparticle interaction. Attractive (Adhesive) Potential. Uattr represents the binding potential between two particles due to interface adhesion. When considering h = r − 2Rp and approximating Uattr(h) by a standard harmonic function we can write

Uattr(h) = −Δ0 +

1 Δ0 2 h 2 2κ02

for 0 ≤ h < 2κ0

(14)

where Δ0 can be evaluated from eq 10 using Δ* as the interfacial adhesion energy polystyrene-water (available in the handbooks or from contact angle data). As time progresses across the overshoot, the potential becomes:

Uattr(h) = −Δ[κ(t )] +

1 Δ·[κ(t )] 2 h 2 2[κ(t )2 ]

(15)

where Δ[κ(t)] is evaluated through eq 10 at every time point with κ(t) being the unique fitting parameter. Clearly, from this analysis we get the roughness of the surface as a function of time, in terms of κ(t) curve obtained from the fitting. Two-Body Shear Stress Term. Two bound spherical particles in an external flow field are subjected to a hydrodynamic shear-induced radial force Fs,pair given by Fs,pair = bvr where b = 6πμRp is the Stokes friction coefficient, μ is the viscosity, vr = λG(h + 2Rp)[1 − A(h)] is the radial component of the relative velocity between both particles, λ is a geometric coefficient, G is the shear rate, and A(h) is a hydrodynamic screening function. Concentrating only on shear flow orientations that work to dissociate the particles and averaging them over the azimuthal and polar angles, we recover the (extensional) shear term in the interparticle energy Us,pair34

Us,pair = −

b G 3 3

∫ (h + 2R p)[1 − A(h)]dh

(16)

Collectively Transmitted Stress. In a rigid aggregate, the hydrodynamic stress imparted onto each bond is transmitted to all other bonds in the cluster. Thus the total force that ideally would be transmitted to an internal particle is zFshNi/2, where z is the 14388

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coordination number and Ni signifies the number of particles in the assembly.34 Introducing a semiempirical transmission coefficient Γi previously calibrated on experimental data34 to account for various dissipation processes as well as for many-body hydrodynamic screening effects, the collective stress contribution to the interparticle energy becomes

Us,coll = −

ΓiNi G 3 3 2 b

∫ (h + 2R p)[1 − A(h)]dh

(17)

α(df)

where Γi = (Rg,i/Rp) . The structure-dependent exponent α governing the stress transmission in the aggregate is given via the empirical relation34

α = − 2.06491df −

0.0180344 + 4.98585 3 − df

(18)

The relative importance of Us,pair and Us,coll controls the relative importance of erosion with respect to fragmentation. One can deduce that surface erosion, or the dissociation of a single particle on the aggregate surface controlled by Us,pair, dominates in the regime where NiΓi < 2 whereas the complete breakup of the cluster into two segments of comparable size occurs when NiΓi > 2.34 In this work we confine our investigation to the latter regime. Numerical Solution. The numerical solution of the PBEs has been obtained using the Kumar-Ramkrishna method.43 A broad range of cluster masses (up to 109 particles per cluster) has been covered by means of a hybrid grid with the first 16 bins linearly spaced and covering the clusters with up 16 particles, while the remaining mass interval has been covered by a logarithmic grid. The fractal dimension of the clusters has been set equal to 2.63 and kept constant throughout the entire simulation time. To account for final size of aggregates which undergo breakup7,8 it was assumed that clusters with a mass smaller than 1000 particles do not undergo breakage. The clusters radius of gyration has been computed as previously described.44 A mean-field T-Matrix scattering approach has been utilized to calculate the clusters scattering structure factors.45 Once the cluster mass distribution has been computed as a function of time, all average properties, such as the average radius of gyration and the normalized average scattered intensity at zero angle can be computed as follows45 ⟨R g2⟩ =

Figure 1. Time evolution of the average radius of gyration ⟨Rg⟩ (a) and the normalized zero angle intensity I(0)norm (b) measured during aggregation of 300 nm carboxyl latex in batch experiments using a solid volume fraction of 2 × 10−5, a stirring speed of 200 rpm, and various pH values; (+) pH 9, (▷) pH 8, (◀) pH 7, (◊) pH 6.8, (▼) pH 6, (△) pH 5, (●) pH 4, and (□) pH 3.

(see Figure 1b), where substantially higher values were measured at lower pH values in comparison to those obtained at high pH values. Furthermore, as can be seen from Figure 1, there is a significant change in the shape of the time evolution of both measured moments of the CMD. In particular, for pH equal to or larger than 7 the shape is sigmoidal, similarly to that observed in our earlier work.7,8 By lowering the pH, a hump appears in the time evolution curves, which becomes a clear peak measured during experiments at pH values below 5. To prove that such change in time evolution of the aggregate size is not controlled by the aggregation acting alone at the beginning of the process we performed an aggregation experiment covering the same range of pH as that presented in Figure 1, however under stagnant conditions. As can be seen from Supporting Information Figure SI 2 the same time evolution of ⟨Rg⟩ and I(0)norm was measured for various combinations of MgCl2 and pH values. Furthermore, to test whether the size of cation or the amount of surrounding water molecules would affect the observed overshoot phenomena the same aggregation experiments as that reported in Figure 1 was repeated in the presence of two other salts in particular LiCl and CsCl. By selecting these two salts, the diameter of a cation increases from approximately 0.13 to 0.34 nm while degree of cation hydration increases as Cs+ < Li+ < Mg2+ (see Supporting Information Figure SI 3). Also in this case it was found that independent of the cation used the obtained time evolution and aggregate steady states are only a function of the pH and not of the cation type. These results clearly indicate that aggregation alone, controlled by the van der Waals attraction force, is not responsible for the formation of an overshoot observed for both moments of the CMD measured under turbulent conditions (see Figure 1). As proposed by Selomulya et al.,10 an overshoot on the aggregate size could be explained by the process of restructuring once aggregates grow large enough to be affected by the flow. Since a rather large size of primary particles was used in this study, which could result in the underestimation of df from the light scattering data (see ref 7 and Supporting

∑i Nii 2R g,2 iIi0 ∑i Nii 2Ii0

I(0)norm =

(19)

∑i Nii 2Ii0 ∑i Ni(0)i 2Ii0

(20)

where I0i is the correction factor for a cluster of mass i accounting for the intracluster multiple scattering and deviations from the simplified Rayleigh-Gans-Debye theory.45

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RESULTS AND DISCUSSION Let us examine first the series of experiments carried out at various pH values using the polystyrene latex stabilized with carboxyl surface groups with a size of 300 nm. To investigate exclusively the effect of pH, covering the range from 3 to 9, in all experiments the same particle solid volume fraction equal to 2 × 10−5 combined with constant stirring speed of 200 rpm, and same MgCl2 concentration was used. The measured time evolution of ⟨Rg⟩ and I(0)norm = I(0)/I(0)t=0, corresponding to two moments of the cluster mass distribution (CMD), is presented in Figure 1a,b, respectively.46 It can be seen that, while the initial aggregation kinetics is very comparable for all investigated pH values, there is a substantial difference in the steady state aggregate sizes. In the case of pH lower that 5, ⟨Rg⟩ reached values around 30 μm while substantially smaller aggregates were produced for pH above 7 with ⟨Rg⟩ equal to around 10 μm. A similar trend was also observed for I(0)norm 14389

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Figure 2. Example of aggregate images obtained for 300 nm carboxyl latex measured at (a) 10 min, (b) 30 min, and (c) 90 min. Applied stirring speed was equal to 200 rpm and pH was equal to 5. For comparison, an example of an aggregate composed of 860 nm carboxyl latex taken at steady state using 200 rpm and pH 2 is also presented in (d).

Information Figure SI 1), in what follows internal structure of aggregates was determined using image analysis of the aggregate images.29 An example of aggregate images obtained at three different times, that is, before and after the overshoot, and at the steady state, is presented in Figure 2a−c. It can be seen that in all cases the formed aggregates are slightly elongated with very compact structure, independent of the time when they were withdrawn from the stirred tank. Furthermore, the dpf evaluated for all three sampling times, 10, 30, and 90 min, was the same and equal to 1.18 ± 0.03 (see Figure 3). This value is comparable to those determined by other authors for aggregates generated under turbulent conditions.7,13 The correlation between dpf and df developed by Ehrl et al.29 was used to convert obtained dpf to mass-based fractal dimension df, which for this case was equal to 2.63 ± 0.04. This value is in close agreement to df values obtained previously by our group using image analysis or light scattering for aggregates produced under turbulent conditions and composed of primary particles of various sizes.7,47,48 These results clearly indicate that the observed overshoot in both moments of the CMD is not caused by aggregates restructuring, but another mechanism connected to the aggregate breakup is responsible for its appearance. As discussed by Israelachvili,20 in the case of surface groups that can be protonated, such as the carboxyl groups used in our study, when using a combination of salt and acid there will be a competition between protonation of the surface charge groups

Figure 3. Area versus perimeter obtained from aggregates images collected by confocal scanning microscope at three time instance, (□) 10 min, (●) 30 min, and (△) 90 min, during aggregation of 300 nm carboxyl latex in batch experiments using solid volume fraction of 2 × 10−5, stirring speed of 200 rpm and pH 5. For comparison area versus perimeter obtained from aggregates composed of 860 nm carboxyl latex particles aggregated at pH 2 are also presented (◊).

by acid protons and charge screening effect by adsorption of salt cations on the particle surface. In fact, by measuring the ζpotential for used primary particles it was found that there is very sharp decrease of the ζ-potential from approximately −60 to 0 mV when the pH decreases from 7 down to 5 (see Figure 4). When comparing this sharp change in ζ-potential with the appearance of an overshoot presented in Figure 1, it can be seen that both phenomena occur in the same pH range. 14390

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Figure 4. Comparison of the ζ-potential measured for PS primary particles with diameter 300 nm (●) and 860 nm (□).

Consequently, in the case of surface charge screening, that is, aggregation at high pH values, the primary particles will be attracted by van der Waals forces, but due to Born repulsion generated by the water molecules surrounding salt cations, there will be a small but not negligible separation distance between them. Furthermore, due to the small but finite distance between primary particles, there will be a free exchange of molecules between the electric double layer and the surrounding bulk solution. Therefore, when changing composition of the solution, for example, by decreasing the pH, there will be a change in the separation distance between primary particles, which will also affect the aggregate size. On the other hand, when aggregation is carried out at low pH values, due to complete protonation of carboxyl groups primary particles will come to much closer contact as compared to the previous case. This could result in the presence of additional forces acting between them, for example, adhesion or hydrogen bonding. Such forces would substantially increase the strength of the contact, which transformed to our experimental system would result in the higher mechanical resistance of a cluster composed of several primary particles. In such case, by increasing the pH of the solution, it is expected that not all contacting particles will increase their separation distance due to the establishment of the double layer leading to an aggregate strength higher than the one expected for high pH values. As a result, larger aggregates would be expected in this case compared to aggregation experiments performed by keeping high pH values from the very beginning. To test this hypothesis we performed aggregation experiments during which the pH of the solution was changed once the aggregates’ size reached a steady state value. Two limiting situations were selected, that is, aggregation experiment starting at pH 5 where the pH was increased to 8 after reaching steady state, and the opposite situation where aggregation started at pH 8 and the pH was lowered to 5 when steady state was reached. The results of these experiments are presented in Figure 5. It can be seen that in both cases, after the pH change, the systems immediately responded by changing the aggregate size, which indicates that there exists a dynamic competition between protonation of the surface carboxyl groups by acid protons and charge screening by Mg cations. In the case of pH reduction, that is, from 8 to 5, the observed overshoot in both moments of the CMD as well as their steady state values are very similar to those measured for experiments when the pH was set to 5 from the very beginning of the aggregation process (see Figure 5). In contrast, when aggregation experiment was started with low pH equal to 5, even though after an increase in pH to 8 there is an observed decrease of both moments of CMD to a new steady state, with ⟨Rg⟩ equal to 20 μm

Figure 5. Response of average radius of gyration ⟨Rg⟩ (a) and normalized zero angle intensity I(0)norm (b) to step changes in a pH during aggregation of 300 nm carboxyl latex using a solid volume fraction of 2 × 10−5 and a stirring speed of 200 rpm; (▷) pH 8 → pH 5, (△) pH 5 → pH 8.

compared to ⟨Rg⟩ equal to 10 μm measured when pH 8 was used from the beginning of the aggregation experiment (see Figure 5). We believe that these results present an indirect proof that an additional mechanism such as the adhesion between primary particles composing an aggregate or hydrogen bonding are present at low pH and acting together or individually are responsible for the measured differences in the aggregates sizes at the same final pH values. To discriminate between these two mechanisms detailed analysis of the forces acting between two primary particles was performed (see Supporting Information). It was found that the corresponding onsets of fully plastic and elastic deformation of polystyrene primary particles are equal to Pplastic of 85 nN and Pelastic of 0.49 nN, respectively (see eqs 2 and 3 of the Supporting Information). For comparison, the surface force Fsurface due to the work of adhesion of primary particles without any external force will be equal to 57 nN (as estimated by eq 5 in Supporting Information), indicating that primary particles can undergo elastic deformation. This would explain the observed difference in steady states measured at various pH values, since at low pH particles have a finite contact area due to their adhesion, which increases substantially the strength of the formed aggregates. From the evaluation of the hydrodynamic force by which flow is acting on an aggregate (see Supporting Information), it was found for the highest aggregates sizes Fhydro is equal to approximately 78 nN, while much lower values, in the range from 22 to 3 nN, were found when the steady state aggregate sizes measured at pH values below 5 and above 7 were used. Considering these values it was found that for pH values equal to or lower than 5, where a significant overshoot in the aggregate sizes was found, the sum of adhesion and hydrodynamic forces is above the onset of plastic deformation (see Figure SI 4 presented in Supporting Information), while for all steady state values we are below this limit. This indicates that for low pH values primary particles at the point of contact could deform plastically, that is, there would be interpenetration between these particles but not enough to significantly alter their shape and hence the spherical nature 14391

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of the interparticle potential. When comparing the free energy change due to the adhesion of two particles (eq 6 of Supporting Information) with that of hydrogen bonding from Table SI 2 in Supporting Information, we can see that contribution of adhesion is higher favoring adhesion to be responsible for the observed phenomena. Because of the dynamic equilibrium between aggregation and breakup these aggregates undergo breakup and some of these connections will be broken. As illustrated by Quesnel et al.,49,50 when there is plastic deformation between a particle and a substrate at the point of contact during their separation, which is a situation comparable to our breakup event, some of the material from one particle is transferred to the surface of the other particle, resulting in an increase in the surface roughness of the contacting particles. Because of the dynamic equilibrium between aggregation and breakup, many of these events will occur until a steady state value is reached, so there will be an increase of the surface roughness of the particles located on the outer periphery of the clusters with time. As documented by several authors,20,51,52 even a small increase in the surface roughness results in a substantial reduction of the surface force between contacting surfaces. By referring this to our experimental observation, the reduced aggregates strength will result in an increased aggregate breakup for given hydrodynamic conditions, thus leading to progressively smaller aggregates sizes (see Figure 1). The same phenomenon of reduction in attractive force between two surfaces with corresponding increase in the number of contact-detachments was observed by Schmitt et al.53 and Du et al.54 using polystyrene and gold, respectively. Even in this case it was attributed to the ductile separation and to the resulting increase in surface roughness. Because both Fsurface and Fhydrodynamic are a function of the primary particle size and of the applied shear rate (see eqs 5 and 7 in the Supporting Information), it is expected that both would have an effect on the size of formed aggregates as well as on the time evolution of measured moments of the CMD. A set of experiments performed at different stirring speeds, that is, different shear rates (see Table SI 1 in Supporting Information), is presented in Figure 6. It can be seen that when a higher stirring speed is applied, that is, 400 rpm, resulting in more frequent aggregation and breakup events compared to 200 rpm, the relative size of the formed aggregates at the overshoot becomes smaller than that measured at 200 rpm. Furthermore, when aggregation is initially carried out at 400 rpm and shortly after the overshoot the stirring speed is reduced to 200 rpm, the formed aggregates do not reach the same size obtained when a constant stirring speed of 200 rpm is used from the beginning (see Figure 6). This difference can be attributed to more frequent aggregate breakup events at 400 rpm, resulting in higher surface roughness when the stirring speed changes, as compared to the case when 200 rpm are used from the beginning, leading to lower aggregates strength, and smaller size (see open circles versus solid square at 30 min after first reduction of stirring speed in Figure 6). When the stirring speed change is applied several times, it can be seen that the measured steady states for both stirring speeds could be recovered and is completely reversible, that is, the steady-state aggregate size is independent of the applied stirring speed history. A second way to affect the plastic deformation of primary particles is to change their size, so an additional experiment was performed with polystyrene latex particles with 860 nm in

Figure 6. Response of average radius of gyration ⟨Rg⟩ (a) and normalized zero angle intensity I(0)norm (b) to step changes in the stirring speed during aggregation of 300 nm carboxyl latex in batch experiments using a solid volume fraction of 2 × 10−5, pH 5 and various stirring speeds. (■) 200 rpm, (○) 400−200−400−200−400− 200 rpm, (▲) 400 rpm,.

diameter. Also in this case, it is worth noting that primary particles are stabilized by carboxyl groups with a surface charge density comparable to that of the 300 nm primary particles investigated so far (see Figure 4 for a comparison of ζ-potential values measured for both particles). As it can be seen from Figure 7, during aggregation experiments at two different pH

Figure 7. Time evolution of average radius of gyration ⟨Rg⟩ (a) and normalized zero angle intensity I(0)norm (b) during aggregation of 860 nm carboxyl latex in batch experiments using a solid volume fraction of 2 × 10−5, a stirring speed of 200 rpm, and two pH values; (○) pH 9, (■) pH 2.

values, that is, 2 and 9, the evolution of both moments of the CMD is having sigmoidal shape without any trace of an overshoot. However, larger aggregates were produced at pH equal to 2 as compared to those formed at pH equal to 9. Even in this case, the aggregates are very dense with a df of about 2.63 (see Figure 2d and open diamonds in Figure 3). When 14392

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evaluating the contribution of surface and hydrodynamic forces for these two conditions, it was found that Fsurface was equal to 162 nN while Fhydrodynamic evaluated for measured two pH values was around 0.5 and 2.0 nN, respectively. Comparing these values with the onset of elastic and plastic deformation, which were equal to 4 and 697 nN, respectively, it can be seen that primary particles under both conditions deform elastically. This finding is in agreement with the observation of Wang et al.,41 who showed that larger particles deform less compared to smaller particles when applying comparable external loads. To demonstrate that overshoot phenomenon can be observed also for other surface charge group an additional experiment was carried out using primary particles with diameter of 320 nm stabilized with sulfate charged groups having approximately 10 time lower surface charge density compare to those discussed above. As can be seen from Supporting Information Figure SI 5 also, in this case clear overshoot comparable to that shown in Figure 1 is visible. To quantitatively describe the observed overshoot the aggregation kinetics data shown in Figure 1 were simulated using PBEs. In order to quantify the phenomenon, it is necessary to have a breakage kernel that is able to explicitly account for the energy of interparticle bond, as well as for its effect on the structural resistance of a cluster. As discussed in the derivation of the breakage kernel eq 7, an increase of the surface roughness results in larger interparticle separation and therefore lower binding energy between primary particles. The kernel predicts an exponential increase of the breakage rate as the binding energy decreases. To implement the breakage kernel eq 7 into the PBE a semiempirical relationship that describes the increase of particle surface roughness as a function of time was defined. The following expression has been used in this work

Figure 8. Comparison between the time evolution of the average radius of gyration ⟨Rg⟩ (a) and normalized zero angle intensity I(0)norm (b) measured during aggregation of 300 nm carboxyl latex in batch experiments using a solid volume fraction of 2 × 10−5, a stirring speed of 200 rpm, and three pH values: pH = 3 (□), pH = 6 (▼), pH = 7 (◀). Lines represent the corresponding PBEs simulations results. Simulations were performed using an asymmetric breakage daughter distribution function.

values of the order of a few angstroms are sufficient to explain the presence of the overshoot even at pH 3. To better visualize this phenomenon, it is instructive to take a look for all three simulated pH values at the breakage rate for a given cluster mass, chosen to be equal to 106 particles. This mass has been chosen because it corresponds approximately to the average asymptotic mass of the clusters in the experiments. As can be seen from Figure 9 for pH 6 and pH 3, that the breakage rate

κ∞

κ= 1+

(

κ∞ κ0

)

− 1 e−t/ τ

(21)

where κ0 and κ∞ are the surface roughness values at time zero and after an infinite time, respectively, and τ is a time constant defining the rate of surface roughness evolution. The value of τ has been set equal to 2000 s for all the simulations, while the values of the initial and final surface roughness have been chosen as a function of pH, which determines the extent of surface deformation of the overshoot. The other parameter that has been fixed is the adhesion energy of particles in the limit of infinite smooth surfaces Δ*, which has been kept equal to 50kBT. This value is consistent with van der Waals interaction energy for two PS particles with the same size as the primary particles used in the experiments computed at a subnanometer separation distance. Three selected pH values equal to 3, 6, and 7, respectively, were used in the simulations. They correspond to the case of no-overshoot, mild overshoot, and large overshoot, respectively (see Figure 1). The results of the simulations are shown in Figure 8a,b for the average radius of gyration and the average scattered intensity at zero angle, respectively. It can be observed that the model is able to quantitatively predict very well all the data sets with minor offsets in the predicted values of ⟨I(0)⟩ for pH 3 and pH 6. In order to obtain this very satisfactory agreement, the following values of κ0 and κ∞ have been used: at pH 7, κ0 = κ∞ = 0.6 nm, at pH 6, κ0 = 0.55 nm and κ∞ = 0.585 nm, while at pH 3 κ0 = 0.45 nm and κ∞ = 0.57 nm. As it can be seen from the obtained results, even very small variations in the surface roughness

Figure 9. Comparison of the time evolution of the breakage rate constant for a cluster with a mass of 106 particles, with a primary particle equal to 300 nm, a fractal dimension equal to 2.63, computed using eq 7, assuming that the surface roughness has a time dependence given by eq 21 with initial and final roughness values equal to κ0 = κ∞ = 0.6 nm at pH 7, κ0 = 0.55 nm and κ∞ = 0.585 nm at pH 6, and κ0 = 0.45 nm and κ∞ = 0.57 nm at pH 3.

increases substantially as a function of time, in the first case by a factor ∼30, while in the second case by a factor ∼105. It is also interesting to note that a change in final surface roughness of about 0.3 angstroms is sufficient to produce a 17-fold change in the asymptotic breakage rate. One very important point in the simulations is the role played by the fragment distribution function. In fact, in order to achieve the results shown in Figure 8a,b it is necessary to use a 14393

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the main text is shown in Figure SI 1. A comparison of the time evolution of average radius of gyration ⟨Rg⟩ (a) and normalized zero angle intensity I(0)norm (b) during static aggregation applying various combinations of MgCl2 concentration and pH is shown in Figure SI 2. An effect of salt type on the time evolution of aggregate sizes is present in Figure SI 3. Detailed characterization of the forces acting on the aggregates and individual particles within an aggregate is presented in Figure SI 4 using size of aggregate as presented in Figure 1 combined with maximum shear rate present in the stirred tank (Table SI 1) and material properties of polystyrene particles (Table SI 2). A comparison of the time evolution of ⟨Rg⟩ obtained for primary particles bearing carboxyl and sulfate charge groups is presented in Figure SI 5. Simulations of the same experimental data shown in Figure 8a,b with PBEs under the conditions of symmetric breakage are shown in Figures SI 6. This material is available free of charge via the Internet at http://pubs.acs.org.

binary breakage with a sufficiently asymmetric fragment mass distribution with a mass ratio of formed fragments equal to onetenth. In fact, by using a symmetric breakage, that is, by assuming that each cluster breaks into two identical fragments, the agreement between simulations and experimental data is less satisfactory. This is demonstrated in Supporting Information Figures SI 5, which shows the same data of Figure 8a,b but this time simulated with symmetric breakage. The results indicate that while satisfactory agreement in terms of the average radius of gyration can still be found the agreement obtained in terms of ⟨I(0)⟩ is poorer, especially for the lower pH values, where the values of ⟨I(0)⟩ are overestimated. This suggests that the cluster mass distribution is not correctly captured in the case of symmetric breakage, because it is only possible to correctly simulate on moment of the CMD, that is, ⟨Rg⟩ but not a second one, that is, ⟨I(0)⟩. It also proves the importance of measuring and simulating more than one moment of the cluster mass distribution.

■

■

CONCLUSIONS In this work, we experimentally investigated the interparticle interaction between primary particles during flow-induced aggregation and breakup. To completely destabilize primary particles, a combination of salt and acid was used in all experiments. It was found that, while the initial stage of aggregation is independent of the applied pH, the steady state aggregates size is a strong function of the pH value. Furthermore, the appearance of an overshoot was observed at pH values below 5. ζ-potential measurements confirmed that at such low pH values carboxyl groups are completely protonated resulting in a zero charge on the primary particles. As a consequence much smaller interparticle distances can be reached at low pH values, resulting in the appearance of another force between contacting particles in addition to conventional DLVO forces, that is, particle adhesion. All experiments were performed in the presence of shear, resulting in strong hydrodynamic forces acting on primary particles inside the aggregates, thus further increasing particle adhesion. At low pH values, the overall force acting on two particles in contact can exceed the limit of plastic deformation of the material resulting in an increase of particle surface roughness. Several experiments with programmed variations of pH values, stirring speed, primary particle surface charge density, and size were performed to support this hypothesis. To quantitatively describe this phenomenon the aggregation behavior of particles showing overshoot at different pH values has been simulated using population balance equations, accounting for both shearinduced aggregation and breakage. The dependency of the breakage rate on the strength of the interparticle bonds was explicitly included in the model considering a time-dependent evolution of the particle surface roughness, which in turn leads to an increase in the breakage rate, and can therefore quantitatively account for the overshoot. The agreement between simulations and experimental data is very satisfactory, and pinpoints the importance and sensitivity of surface phenomena on the aggregation of colloidal particles under shear conditions.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +41 44 6334659. Fax: +41 44 6321082. Present Addresses §

Bayer Technology Services GmbH, BTS-PT-PD-Conceptual Design, Leverkusen, Germany. ∥ Adolphe Merkle Institute, University of Fribourg, Route de l’ancienne Papeterie CP 209, CH-1723 Marly 1, Switzerland. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors would like to thank Dr. Lyonel Ehrl and Dr. Matthäus U. Bäbler for helpful discussions and suggestions. The financial support of the Swiss National Science Foundation (Grant 200020-147137/1) is gratefully acknowledged. M.L. acknowledges financial support from the Swiss National Science Foundation (Grant PP00P2_133597). This study was also supported by the Winton Programme (B.O.C.) and the Ernest Oppenheimer Fellowship (A.Z.).

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REFERENCES

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ASSOCIATED CONTENT

S Supporting Information *

A time evolution of the S(q) measured for 300 nm primary particles using pH = 3 together with the corresponding transformation of the S(q) into a Guinier plot using eq 3 from 14394

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