Reduction of Surface Charges during Coalescence of Elastomer

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J. Phys. Chem. B 2010, 114, 8838–8845

Reduction of Surface Charges during Coalescence of Elastomer Particles Cornelius Gauer,† Hua Wu, and Massimo Morbidelli* Institute for Chemical and Bioengineering, Department of Chemistry and Applied Biosciences, ETH Zurich, 8093 Zurich, Switzerland ReceiVed: January 19, 2010; ReVised Manuscript ReceiVed: June 2, 2010

Reaction-limited aggregation of soft elastomer particles has been studied with specific attention to the fate of surface charges during coalescence. The employed system is an aqueous dispersion of fluoroelastomer particles, which are known to coalesce completely at 70 °C. In contrast to diffusion-limited conditions, under reactionlimited conditions the stability of the system is expected to change during aggregation because of surface reduction and charge accumulation resulting from coalescence. This allows investigating the mechanism of charge relocation during cluster coalescence. For particles stabilized by ionic surfactants, it has been found that the charges are mobile (i.e., they redistribute between aqueous solution and particle surface according to their adsorption equilibrium) (Gauer, C.; Jia, Z.; Wu, H.; Morbidelli, M. Langmuir 2009, 25, 9703). In this work, we consider the case of fixed charges, as those given by charged polymer end groups covalently bound to the particle surface. We demonstrate that a loss of fixed surface charges occurs during the coalescence and strongly affects the time evolution and the shape of the resulting cluster mass distribution. 1. Introduction 1.1. Motivation. A large variety of polymers is produced in emulsion either because they are applied as such (e.g., latex paints) or because of manufacturing advantages (e.g., molding resins).1 The processing properties of the obtained particle-inliquid dispersion (so-called latex) depend largely on the functional groups at the particle surface, which are fixed if they originate from initiator, comonomer, etc., or mobile if stabilizer (surfactant) is used during the polymerization process. Understanding the fate and behavior of functional surface groups during particle fusion or coalescence2,3 is important in determining colloidal stability, polymer adhesion, film wettability, and other relevant end use properties of a latex. It has been shown in a previous work4 that mobile charge groups (i.e., ionic surfactants) redistribute during cluster coalescence according to their adsorption equilibrium. On the other hand, the behavior of functional groups fixed to the particle surface remains an open issue. A question naturally arises whether during the coalescence some of the surface fixed charges are lost (e.g., entrapped and neutralized inside the cluster) or all the charges are accumulated on the cluster surface. In this work, the fate of surface fixed charges during the coalescence has been investigated by monitoring the colloidal stability along the reaction-limited cluster-cluster aggregation (RLCA) for rubbery particles in aqueous dispersion. In particular, when coalescence occurs, clusters exhibit different colloidal stability depending on their size, i.e., the number of incorporated, fused primary particles. In principle, one would expect that the charge density increases during cluster coalescence as a result of surface reduction, leading to stronger repulsion, thus larger values of the corresponding Fuchs stability ratio.5-9 Some examples under such consideration have been reported recently in the literature.4,10-12 Our approach is to postulate how the surface charge density changes with the cluster * To whom correspondence should be addressed. E-mail: morbidelli@ chem.ethz.ch; Tel: 0041 44 632 30 34. † Current address: Novartis Pharma AG, 4002 Basel, Switzerland.

size, and then to model the colloidal stability for all different cluster sizes involved in the aggregation process. From this, we can compute the time evolution of the RLCA process and compare it with experimental findings in terms of the size evolution of the average hydrodynamic and gyration radii measured through dynamic (DLS) and static light scattering (SLS), respectively. From such a comparison we can derive a quantitative understanding of the behavior of fixed charges during cluster coalescence. 1.2. Introduction of the PBE Model for Coalescence. Let us now introduce the basic concept of the kinetic model used in this work. One established approach to model aggregation processes is to formulate mass or population balance equations (PBEs), as originally proposed by von Smoluchowski.13 Assuming only binary collisions, irreversible aggregation of particles or clusters is expressed by a set of second-order rate equations that can be written in discrete form as13-16 ∞ i-1 dNi 1 K N N Ki,jNiNj + )dt 2 j)1 i-j,j i-j j j)1

∑

∑

(1)

According to the right-hand side of eq 1, aggregation leads to depletion or accumulation of clusters composed of i primary particles, which changes the cluster mass distribution (CMD) or number concentration Ni in time. The matrix of aggregation rate constants or kernel, Ki, j, comprises all physical and chemical effects influencing the aggregation rate. Looking at available expressions for Ki, j, it turns out that the so-called product kernel,

Ki,j )

KB (i1/Df + j1/Df)(i-1/Df + j-1/Df) λ (ij) W1,1 4

(2)

is the most suitable kernel to reproduce the RLCA kinetics for fractal clusters,17-24 as long as no crossover from RLCA to diffusion-limited cluster-clsuter aggregation (DLCA) occurs.17,21

10.1021/jp100504k  2010 American Chemical Society Published on Web 06/24/2010

Surface Charges of Elastomer Particles during Coalescence In eq 2, the so-called Smoluchowski rate constant KB quantifies the frequency of cluster collision due to Brownian motion or diffusion,8,13

KB )

8kT 3µl

(3)

which is defined by the thermal energy, kT, and the dynamic viscosity of the dispersant, µl. The reduction of aggregation efficiency, due to colloidal interactions on the primary particle level, is accounted for through the Fuchs stability ratio, W1,1.5,8 The influence of the cluster size and structure on its collision cross section and diffusivity is included in the second term of eq 2 through the fractal dimension Df. Further, the product term (ij)λ empirically accounts for the increase of cluster reactivity with size, typical of RLCA conditions. Geometrical considerations, which allow one to compute the number of primary particles on the cluster surface, indicate that the exponent λ should take a value of about 0.5.18,24 This term does not exist in the case of fast and complete coalescence, since a newly formed cluster, before being involved in a subsequent aggregation event, relaxes to spherical form where original primary particles loose their identity. Accordingly, since mass and size scale with an exponent of 3 (i.e., M ∝ L3) in this case, the fractal dimension is a priori fixed to Df ) 3. In addition, the colloidal stability is not anymore determined by the interaction of primary particles as in the case of fractal aggregates discussed above.18,24 Instead, we need to account individually for the interaction of spherical clusters with size i and j. Therefore, the classical Fuchs stability ratio, reflecting primary particle interaction (W1,1 in eq 2), turns into a pair stability ratio, Wi,j, and we obtain

Ki,j )

KB (i1/3 + j1/3)(i-1/3 + j-1/3) Wi,j 4

(4)

As mentioned already, under RLCA conditions, the coalescence process affects the charge density on the cluster surface and therefore the colloidal interaction. The respective simulation of the aggregation process and the coalescence model are discussed in detail later on, after the techniques used to monitor the aggregation experiments are introduced. 2. Experimental Section 2.1. Colloid Aggregation. Elastomer Colloid. The investigated colloid is an aqueous dispersion of fluoroelastomer particles, provided by Solvay Solexis (Italy) in a surfactantfree form. The latex was used as such after cleaning by centrifugation. Resulting from dissociation of polymer chain end groups at the surface, the particles carry a negative fixed charge, which is sufficient to achieve colloidal stability for medium time storage. The polymer phase, consisting of vinylidene fluoride and hexafluoropropylene units, has a glass transition temperature of Tg ≈ -20 °C, a density equal to 1.8 kg/L, a refractive index of 1.37, and the Mooney viscosity, µM in terms of ML(1 + 10) at 121 °C measured according to the standard test method ASTM D1646, is 10 MU. The same type of colloid has been found to coalesce completely above 55 °C,25 forming spherelike clusters (typical examples are shown in Figure 4 of ref 26). Evaluation of the hydrodynamic and gyration radii from DLS and SLS indicates some minor polydispersity of the initial latex, i.e., Rh,0 ≈ 83 nm and Rg,0 ≈ 75 nm, respectively.

J. Phys. Chem. B, Vol. 114, No. 27, 2010 8839 Aggregation Experiments. Reaction-limited aggregation experiments were run at a temperature of T ) 70 °C to ensure complete and very fast coalescence of the particles upon contact, while the particle volume fraction was fixed at φ ) 5.0 × 10-3. Since the refractive index of elastomer particles (1.37) is very close to that of water (1.33), in most cases adding salts would increase the refractive index of water and reduce the contrast, leading to poor light scattering signal. It is known27 that nitric acid (HNO3) does not significantly change the refractive index of water in a wide concentration range. Therefore, in this work analytical-grade HNO3 was used as the coagulant at three different concentrations, 0.12, 0.14, and 0.16 mol/L, respectively, which are roughly one decade below the critical coagulant concentration for the onset of diffusion-limited aggregation (CCC ≈ 1.1 mol/L28). It should be noted that, although adding HNO3 changes the pH of the systems, it acts mainly as an electrical double layer screening electrolytes, instead of protonation (e.g., for carboxylic charge groups). In fact, the CCC value of the latex for HNO3 is comparable to that for NaCl (1.3 mol/L28). The experiment was started by pouring 57 mL of prediluted HNO3 into 30 mL of elastomer latex (φ ) 1.45 × 10-2), both volumes were previously kept at T ) 70 °C overnight, and then the container was placed in a thermostat. Since the experiments lasted several weeks, possible cluster sedimentation was prevented by carefully turning the reaction container up and down once per day, so as not to disturb the perikinetic conditions of the aggregation experiments. In order to measure the size evolution by light scattering, small samples were drawn from the aggregating systems and diluted to φ ) 5.0 × 10-5. Dilution was carried out with deionized water that was additionally purified through a Millipore Simpak 2 column and filtered through 0.1 µm Acrodisc syringe filters (Pall, U.K.). 2.2. Colloid Characterization. Light Scattering. During aggregation, the CMD evolves with time. A full characterization of the CMD is rarely available; however, moments of the CMD, such as the average hydrodynamic and gyration radii, 〈Rh〉 and 〈Rg〉, can be reliably measured by DLS and SLS, respectively. Light scattering measurements were carried out using a BI200SM goniometer system (Brookhaven Instruments, U.S.A.), equipped with a solid-state laser, Ventus LP532 (Laser Quantum, U.K.) of wavelength λ0 ) 532 nm. In DLS, the selected angle was θ ) 90°, which corresponds to a scattering vector of q ) 4πn0/λ0 × sin(θ/2) ) 0.0222 1/nm, with n0 representing the refractive index of the dispersant. The measured intensity-time fluctuations were converted into the intensity autocorrelation function by the digital correlator BI-9000AT from Brookhaven Instruments to estimate the average translational diffusion coefficient 〈D〉. Using the Stokes-Einstein relation,29-31 the average hydrodynamic radius 〈Rh〉 is calculated according to

〈Rh〉 )

kT 6πµ1〈D〉

(5)

SLS measurements were carried out with the same setup described above, but employing angles in the range of θ ) 16-150°. The recorded normalized intensity curve of scattered light, I(q)/I(0), was used to estimate the average or root-meansquare radius of gyration, 〈Rg〉 () 〈Rg2〉1/2), by applying the Guinier plot:31,32 2

I(q) [ I(0) ] ) q3 〈R

-ln

〉,

2

g

q〈Rg〉 < 1

(6)

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Colloidal Stability. In systems where, due to the presence of a repulsive barrier, only a fraction of collisions between particles/clusters result in the formation of bigger clusters (i.e., RLCA conditions), quantification of the colloidal stability is crucial for a successful simulation of the aggregation process. Initially, the formation of doublets from primary particles is dominant, and accordingly, the kinetics can be treated based on one simple rate equation:

and obtain W1,1 from the average K1,1. It is worth noting that, in the case of coalescing particles stabilized by ionic surfactants, this procedure has to be modified because redistribution of the mobile charges upon surface reduction of coalescing doublets increases the charge of the primary particles already in the initial stage where eq 7 still holds.4,11,12 Then, K1, 1 is found to decrease with time and has to be extrapolated back to t ) 0 in order to estimate its initial value.

dN1 ) K1,1N12 dt

3. Simulation 3.1. Calculation of Colloidal Stability. As mentioned above, in order to solve the PBE accounting for coalescence, we need to evaluate the stability ratio for individual particle pairs (i, j). In general, two spherical clusters of size i and j have different surface charge densities, therefore the Fuchs stability ratio is written for the pair as4,5,7,8

(7)

After integration and introduction of the conversion of primary particle to doublets, x ) 1 - N1/N1,0, where N1 and N1,0 () 3φ/[4πRp3]) represent the primary particle number concentration at times t and t ) 0, respectively, eq 7 leads to

K1,1

1 x ) 1 - x N1,0t

(8)

In this work, the rate constant for primary particle aggregation, K1,1, is estimated from measurements of the primary particle conversions x during the initial aggregation stages where eq 7 applies. The x values are measured through light scattering. Since this methodology has been presented elsewhere,28 only the main points are summarized here. The hydrodynamic radius 〈Rh〉, measured by DLS, is used to calculate x according to

x) 〈Rh〉 - Rh,1

〈Rh〉 - Rh,1 〈Rh〉 P2(q) + 2Rh,1 1P1(q) Rh,2

(

)

(9)

In the case of complete coalescence, the effective hydrodynamic radii are Rh,1 ) Rp and Rh,2 ) 21/3Rp, where Rp denotes the primary particle radius. The light scattering form factors of primary particles and spherical doublets, P1(q) and P2(q), are given by the Rayleigh-Debye-Gans theory.31,33 Similarly, by using the 〈Rg〉 value evaluated from SLS, one can write

x)

〈Rg〉2 - Rg,12 2Rg,22 - Rg,12 - 〈Rg〉2

(11)

It should be pointed out that, since in our current system the charges are fixed to the particle surface, the surface charge density of the primary particles is always the same. Accordingly, their interaction is constant, thus K1,1 is time invariant. This fact is confirmed by the scattering of the K1,1 values estimated at different times using eq 8 around a constant value, as shown in Supporting Information SI-Figure 1. To ensure aggregation dominated by doublet formation, we determine the x and the corresponding K1,1 values typically up to x ≈ 20-30%,4,23,35,36

∫2

( kTU ) dl

exp

(12)

Gl2

In that case, the total interaction energy U and the hydrodynamic resistance function G depend not only on the dimensionless distance between the particle centers (l ) r/a) but also on the two cluster sizes i and j. The calculation of U and G follows the approach recently presented4 and is briefly summarized in the following. Accounting for DerjaguinLandau-Verwey-Overbeek (DLVO)6 as well as non-DLVO interactions,9,37 the total interaction energy is given by the sum of van der Waals UA, electrostatic UR, and hydration Uhyd interaction energies, i.e., U ) UA + UR + Uhyd. Introducing the average particle size, a ) (Ri + Rj)/2, and the ratio of collision radii, ω ) Rj/Ri, the Hamaker relation can be written as follows:8,9

UA ) -

{ [

]

AH 8ω 1 1 + + 6 (1 + ω)2 l2 - 4 1-ω2 2 l -4 1+ω l2 - 4 ln (13) 1-ω2 l2 - 4 1+ω

(10)

where the gyration radii of the primary particles and spherical doublets are Rg,1 ) (3/5)1/2Rp and Rg,2 ) (3/5)1/2 × 21/3Rp.33,34 Eventually, we use the measured K1,1 values to quantify the colloidal stability in terms of the stability ratio for the primary particles, W1,1, based on the definition5,7,8,23

W1,1 ) KB /K1,1

Wi,j ) 2

∞

[

(

)

(

)

]}

For fluoroelastomer particles in aqueous dispersion, the Hamaker constant can be assumed to take a value of AH ) 3.0 × 10-21 J (similar to polytetrafluoroethylene),37 leading to mild attraction. Electrostatic repulsion is calculated using the modified HoggHealy-Fuersteneau approximation,38 valid for moderate surface potentials as typical for RLCA conditions: UR )

4πε0εrωaψs,i2 (1 + ω)2l

× {(1 + Ψ)2 ln(1 + exp[-ka(l - 2)]) + (1 - Ψ)2 ln(1 - exp[-ka(l - 2)])}

(14)

where Ψ ) ψs,j/ψs,i is the ratio of the surface potentials of the two particles. Both ψs,i and ψs,j are computed from the effective charges on the particle surface applying the generalized stability model,36 which can account simultaneously for the equilibria among colloidal interaction, ion association, and surfactant adsorption. The influence of ionic strength on the repulsive barrier is computed through the Debye-Hu¨ckel parameter,8,9 κ ) [(e2NAjzj2Cbj )/(ε0εrkT)]1/2, where NA is the Avogadro number,

Surface Charges of Elastomer Particles during Coalescence

J. Phys. Chem. B, Vol. 114, No. 27, 2010 8841

Cbj and zj are the bulk concentration and the valency of ionic species j, e is the electron charge, and ε0εr is the permittivity of the dispersant. Additional repulsion due to hydration interactions is considered by an empirical exponential relation:36,37,39

Uhdy )

[

]

4πωa a F0δ02 exp - (l - 2) δ0 (1 + ω)2

(15)

where the hydration force constant and decay length are fixed to F0 ) 1 × 106 N/m2 and δ0 ) 0.6 nm, respectively, as discussed elsewhere.28 Finally, estimation of the hydrodynamic resistance function G, accounting for the reduction in mutual diffusivity upon particle approach compared to the undisturbed ∞ , was reported in the literature.4,7 diffusivity, i.e., Di,j(l)/Di,j 3.2. Calculation of the Experimentally Measured Quantities. From the experiments, we measure the time evolutions of 〈Rh〉 and 〈Rg〉, which represent two independent moments of the CMD and are used to verify the CMD computed from the PBE model. To this aim we calculate the hydrodynamic and gyration radii as30,32

〈Rh〉 )

∑ Nii2Pi(q) ∑ Nii2Pi(q)Rh,i-1

(16)

∑ Nii2Rg,i2 ∑ Nii2

(17)

〈Rg2〉 )

where the CMD, i.e., Ni(t), is obtained by solving discrete PBE,16,23,40 and Pi(q) is the form factor of the coalesced, spherical clusters calculated from Lorenz-Mie theory,33,41 with the individual radii: Rh,i ) i1/3Rp and Rg i ) (3/5)1/2 × i1/3Rp. In solving the PBE model (eqs 1, 4, and 12), the discretization was linear for the first 10 integer sizes and then logarithmic. Convergence of the numerical solution has been verified previously.40 The calculation was started assuming monodisperse particles with Rp ) 83 nm, which corresponds to the measured, initial hydrodynamic radius Rh, 0. In order to account for the polydispersity of the real latex, i.e., Rh 0/Rg,0 ≈ 1.1 instead of the theoretical value of (5/3)1/2 for monodisperse spheres,33 the calculated average gyration radius has been corrected as follows:

3 〈Rg2〉 ) 〈Rg2〉calculated - Rp2 + Rg,02 5

(18)

This correction is based on the idea of splitting the gyration radius into a structure and a form part of which the latter can be changed so as to adapt the simulated to the measured initial value of 〈Rg〉.42 4. Results and Discussion 4.1. Experimental Observations. The measured time evolutions of 〈Rh〉 and 〈Rg〉 are shown in Figure 1 for the conditions φ ) 5 × 10-3, T ) 70 °C, and three HNO3 concentrations: 0.12, 0.14, and 0.16 mol/L. From Figure 1a it is seen that as the acid concentration increases the clusters grow faster. This is expected because a higher coagulant concentration (ionic strength) screens the electrostatic repulsion between particles more, leading to a higher aggregation rate. For the same reason, the colloidal stability measured during the initial aggregation

Figure 1. Measured size evolutions of the average hydrodynamic and gyration radii, 〈Rh〉 (filled symbols) and 〈Rg〉 (open symbols), for the aggregation experiments at φ ) 5 × 10-3, T ) 70 °C, and three HNO3 concentrations: 0.12 (0,9), 0.14 (O,b), and 0.16 mol/L (4,2). The real time dependence is shown in panel a, while dimensionless scaling is used in panel b.

phase (i.e., W1,1) decreases with increasing HNO3 concentration, as shown in Supporting Information SI-Figure 2. When one looks at the later times in Figure 1, two interesting features appear for the coalescing system with fixed surface charges. The first one is that the growth rate of 〈Rh〉 and 〈Rg〉 as a function of time (which is in logarithmic scale in the figure) is relatively low compared to that of corresponding noncoalescing latex at 25 °C, where exponential or power-law trends are found.28 This is evidently ascribed to charge accumulation due to cluster coalescence. The second feature refers to the fact that, at longer times, 〈Rg〉 approaches 〈Rh〉, and in the cases of 0.14 and 0.16 mol/L HNO3, a temporary crossover occurs. This feature, which has never been discussed previously in the literature, will actually be confirmed by simulation results in the following. Let us now consider whether the trends under different coagulant concentrations are universal when represented as a function of the dimensionless time, τ ) tN1,0KB/W1,1,23,43 as we found previously for several noncoalescing latex systems.23,26,28,42,44-47 For this, we report in Figure 1b the dimensionless sizes, 〈Rh〉/Rp and 〈Rg〉/ Rp, as a function of τ, where the initial number concentration

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of primary particles N1,0 and the initial colloidal stability in terms of W1,1 have been measured independently. In general, all experimental data merge to form single curves with respect to 〈Rh〉 and 〈Rg〉 at small τ values. For τ > 102, however, significant deviations between the trends for different HNO3 concentrations are indeed observed. This indicates that the existence of universal master-curves is no longer true for coalescing systems, neither for fixed nor for mobile28 surface charges. 4.2. Simulation Results. As a starting point for the simulations, we need to know the surface charge density of the primary particles, σs,1, under experimental conditions to calculate the electrostatic repulsion. By applying the generalized stability model mentioned before,36 we can estimate σs,1 from the measured stability ratio for primary particles W1,1. The best fit to the experimental data, reported in Supporting Information SI-Figure 2, is obtained by assuming an effective charge density of σs,1 ) -1.5 × 10-2 C/m2. Note that, because of different ionic strengths or κ values for the three HNO3 concentrations of 0.12, 0.14, and 0.16 mol/L, the corresponding primary particle surface potentials are also different, i.e., ψs,1 ) -19.9, -18.5, and -17.3 mV, respectively. It is worth noting that these values are moderately low, which justifies the use of the HoggHealy-Fuersteneau approximation in eq 14.38 With the known σs,1 value, the next step is to compute the charge density of a cluster constituted by i g 2 primary particles, σs,i. This can be done through geometrical considerations and making different hypotheses about particle coalescence and charge entrapment as discussed in the next section. Once this is done, using eqs 13-15, one can compute the pair stability ratio Wi,j in eq 12 and then the aggregation kernel Ki,j in eq 4. Let us now analyze different scenarios with respect to relocation of the charge groups, but always assuming full coalescence (i.e., spherical clusters). Accordingly, in all calculations the cluster surface area is Si ) i2/3S1, with S1 ) 4πRp2 for primary particles. The Case of Full Charge AWailability. As a first approach, let us imagine a coalescence process where all charged surface sites are preserved during cluster coalescence and located at the cluster surface, thus being fully available for colloidal stabilization. In this case, the overall surface charge density of a cluster with mass i is given by

σs,i ) σs,1i1/3

(19)

The same relation also holds for the surface potential ψs,i, considering the linear approximation σs,i ) ε0εrκψs,i.8,9 Using the σs,i value given by eq 19 to calculate the pair stability ratio Wi,j (eqs 12-15), we have simulated the experimental data in Figure 1b, and the results are reported in Figure 2. For small times (i.e., τ e 100), the experimental 〈Rh〉 and 〈Rg〉 values are adequately reproduced, as it is expected since this initial stage involves mainly the aggregation of primary particles. At later times, the simulations clearly deviate from the experimental data. The predicted leveling off for τ g 101 results from a strongly increasing colloidal stability with cluster size. For example, according to eq 19, the surface charge density of a coalesced doublet (i ) 2) increases by approximately 26%, but the stability ratio is several orders larger, i.e., W2,2/W1,1 ≈ 105. Consequently, in the simulations, primary particles are consumed, and the formation of bigger clusters (i g 2) is suppressed. However, the clear mismatch between simulation and experimental results demonstrates that the assumption of full charge availability overpredicts the surface charge density of coalesced clusters, and thus, does not represent the experimental reality.

Figure 2. Comparison of the experimental 〈Rh〉 and 〈Rg〉 values in Figure 1 with the model predictions assuming full charge availability after cluster coalescence. Solid and dashed curves are 〈Rh〉 and 〈Rg〉 simulations; symbols are the same as those in Figure 1.

Figure 3. Schematic shape evolution of a coalescing doublet.

The Case of Partial Charge Loss. In order to model the surface charge density for clusters of size i g 2, let us assume geometrical conditions during coalescence as proposed in a modification of Frenkel’s model for particle sintering,48,49 shown in Figure 3. In this simple model, the extent of particle fusion is characterized by the neck radius y. We can envision the coalescence process in Figure 3 as given by an initial deformation of the two attached spheres at the point of contact (a), leading to a circular region (b) where a surface area of 2πy2 is lost. We assume that all charges located in this region are lost, for example, through entrapping and neutralized by counterions, i.e., H+ in this case, and consequently do not contribute anymore to the particle stability. After stage b in Figure 3, the process proceeds to the final spherical geometry (c) where charge redistribution without further entrapment takes place. In this frame, we can use y as a parameter to calculate how many charged sites are left at the end of the coalescence process. For example, in the case of two coalescing primary particles with radius R1, the neck radius y is constrained between zero and R2 () 21/3R1). Thus, for a given y value, the balance of initial, lost, and remaining surface charges returns the value of the surface charge density σs,2 after full doublet coalescence. This calculation can be easily extended to the coalescence of any equal sized clusters with radius Ri/2 and final radius Ri (i g 2). The obtained results are shown in Figure 4 (open circles) in terms of the relative surface charge density, σs,i/σs,1, as a function of the resulting cluster mass i. The solid line represents a power-law fit to these data, i.e., σs,i/σs,1 ) in, where n is an adjustable parameter that depends on the chosen y value. In the case where the two coalescing clusters have different sizes i and j, the same procedure could in principle be adopted. However, this would significantly complicate the model since the charge density would then become a function of the cluster growth history. Therefore, we assume that the cluster of mass i + j, formed through aggregation of a cluster i and a

Surface Charges of Elastomer Particles during Coalescence

Figure 4. Right scale: Surface charge density of cluster of mass i relative to the primary particle charge density, σs,i/σs,1. Discrete values (O) are calculated assuming entrapment and neutralization of charges from the area 2πy2 where y ) 1.005Ri/2 (i ) 2, 4, 8, 16, etc.). The solid line is a power-law fit, σs,i/σs,1 ) i-0.0865, to predict σs,i for clusters of arbitrary size. Left scale: Calculated pair stability ratios for equal sized clusters, Wi,i, relative to the stability ratio for primary particles, W1,1, for the used HNO3 concentrations: 0.12 (9), 0.14 (b), and 0.16 mol/L (2).

Figure 5. Dimensionless, experimental 〈Rh〉 and 〈Rg〉 time evolutions from Figure 1 (symbols as in Figure 1) compared to the simulation results using the charge entrapment and neutralization model with y ) 1.005Ri/2; solid and dashed curves correspond to 〈Rh〉 and 〈Rg〉, respectively.

cluster j, has a charge density equal to σs,i+j after its coalescence, where the σs,i+j value is taken from the power-law fit to the equal size cases, reported in Figure 4, i.e., σs,i+j/σs,1 ) (i + j)n. Our best guess for the parameter that describes the amount of lost surface charge groups during coalescence is y ) 1.005Ri/2. This value has been obtained by fitting the experimental values of 〈Rh〉 and 〈Rg〉 as a function of time. The obtained fit, as shown in Figure 5, is very good and includes even the temporary crossover between 〈Rh〉 and 〈Rg〉 in the cases of 0.14 and 0.16 mol/L HNO3. A deeper insight into the process can be obtained by looking at the results shown in Figure 4, which are obtained for y ) 1.005Ri/2. From the right scale, it is seen that as clusters increase in size i, their surface charge density σs, i decreases due to the charge entrapment during coalescence. However, when using these values in eqs 13-15 to compute the pair stability ratio Wi,j in eq 12, we find that Wi,j nevertheless increases with cluster size. This fact is illustrated in Figure 4 on the left scale for the case of clusters with equal size, i.e., Wi,i/W1,1. It is worth noting that these values are quantitatively different for the three HNO3 concentrations, thus producing different CMDs that actually correspond to a faster growth of 〈Rh〉 and 〈Rg〉 at larger HNO3 concentration, particularly at larger dimensionless times τ, as seen in Figure 5. Thus, despite having

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Figure 6. Dimensionless 〈Rh〉 (b) and 〈Rg〉 (O) values for the experiment at 0.14 mol/L HNO3, φ ) 5 × 10-3, and T ) 70 °C. Predictions for 〈Rh〉 and 〈Rg〉 using the charge entrapment and neutralization model are labeled as dash-dot and dotted curves in the case of y ) 0.91Ri/2, i.e., σs,i ≈ σs,1, and solid and dashed curves in the case of y ) 1.07Ri/2, i.e., Wi,i ≈ W1,1, respectively.

equal charge loss, i.e., the same value of y, the different evolution of Wi,j with ionic strength explains the experimentally observed differences in the size evolutions at τ > 102 for the three employed HNO3 concentrations. It should be mentioned that, in principle, the evolution of the surface charge density along the aggregation may be determined by in situ measurements of the zeta-potential. We have indeed tried to measure the time evolution of the zetapotential along the aggregation using a Zetasizer Nano (Malvern, U.K.) instrument in PALS (phase analysis light scattering) mode with the folded capillary cell (DTS1060). The main problem comes from the errors in the measured zeta-potential, which are typically in the range around 10-20% (even more when the ionic strength is high), which is the expected change of charges due to coalescence. Thus, it was very difficult to obtain a meaningful trend of the zeta-potential variation with time. A further difficulty arises from the sticky nature of the used particles, which leads to progressive coating of the electrodes. From the stability data in Figure 4, it can be readily concluded that aggregation of smaller clusters is preferred, leading to rather narrow CMDs, actually narrower than those in classical, fractal RLCA or even DLCA systems.15,43,50-52 A rather narrow CMD explains why, in Figure 5, 〈Rh〉 is larger than 〈Rg〉 for most of the time. The temporary crossover can be attributed to a kink in the 〈Rh〉 curve, characteristic of a CMD of spherical clusters where, due to the size dependency of the light scattering form factor Pi(q),31,33 certain particle classes preferentially contribute to the average, as apparent from eq 16. The Cases of Constant Charge Density and Constant Stability Ratio. We can now use the entrapment model to investigate two other interesting situations. In the first case, the parameter describing charge entrapment is set to y ) 0.91Ri/2. This corresponds to the situation in which the calculated surface charge densities are practically constant, i.e., σs,i ≈ σs,1. As a result, the pair stability ratio Wi,j increases with cluster size much stronger than in the case considered above in Figure 4. Accordingly, the experimental 〈Rh〉 and 〈Rg〉 values are clearly underpredicted, as can be seen by comparing them with the dashdot and dotted curves in Figure 6, respectively. In the second case, we have selected y ) 1.07Ri/2, i.e., a stronger charge entrapment with respect to the case y ) 1.005Ri/2, so as to keep basically constant the stability ratios for equal cluster sizes, i.e., Wi,i ≈ W1,1. As expected, in this situation the predicted 〈Rh〉 (solid) and 〈Rg〉 (dashed) curves largely overestimate the experimental data in Figure 6. On the other hand, in this case the

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simulated time evolutions exhibit a crossover between 〈Rh〉 and 〈Rg〉 in the interval 10° < τ < 101, which is typical of fractal aggregations.23,47 Consequently, for y ) 1.07Ri/2, a much broader CMD results compared to the cases y ) 0.91Ri/2 and y ) 1.005Ri/2. It is worth pointing out that the results shown in Figure 6 provide a clear indication of the strong sensitivity of the time evolution of the CMD with respect to rather small changes in the parameter y, i.e., in the extent of surface charge loss. 5. Conclusion Loss of fixed surface charges during coalescence of rubbery particles has been studied in this work under reaction-limited aggregation conditions at a temperature (T ) 70 °C) high enough to ensure full coalescence of formed clusters. Under those conditions, changes in the surface charge density with cluster size are reflected in the corresponding aggregation rates, which can be modeled by PBEs. Their solution allows one to predict the hydrodynamic and gyration radii, 〈Rh〉 and 〈Rg〉, measured in three experiments as a function of time. In the PBE model, different surface charge density of clusters with respect to their mass or size is accounted for by calculating the Fuchs stability ratio individually for all possible cluster pairs, i.e., the classical W turns into Wi,j. Prediction of the experimental data fails if one considers preservation of all charges at the cluster surface during coalescence. The obtained strong underprediction for the time evolutions of the experimental 〈Rh〉 and 〈Rg〉 clearly demonstrates that charges are lost during cluster coalescence. One possible mechanism is that the lost charges are entrapped inside the clusters and neutralized, thus playing no further role in stabilizing the clusters. Accordingly, we use a simple entrapment model that allows one to relate the amount of the lost charges to the surface area buried in the neck region of coalescing particles. Herein, the neck radius y is considered as a variable parameter to estimate the amount of lost surface charges, which then allows calculation of the surface charge densities σs,i and surface potentials ψs,i for clusters of arbitrary size. For the investigated experimental conditions, it was found that the value of y ) 1.005Ri/2 gives a very good prediction of the experimental data. This implies that a certain amount of surface charges is indeed lost during coalescence. As a result, the charge density σs,i decreases with cluster size i, but the pair stability ratio Wi,j still increases because of the stronger effect of the cluster size. An increasing Wi,j with cluster size leads to rather narrow CMDs, which is reflected by the fact that 〈Rh〉 and 〈Rg〉 evolve in time very closely to each other. On the other hand, the pair stability ratios Wi,j evolve with size quantitatively different for different coagulant concentrations. As a result, the CMDs also evolve differently in dimensionless time τ, and thus, the 〈Rh〉 and 〈Rg〉 growth curves do not perfectly collapse onto each other. This means that the process of aggregation and cluster coalescence is not universal with respect to changes in ionic strength. Finally, a sensitivity analysis based on the entrapment model demonstrates that small changes in the fraction of lost surface charges leads to strong changes in the time evolution of the CMD both in terms of the average and broadness, as is reflected through the different behaviors of the 〈Rh〉 and 〈Rg〉 growth curves. This finding might be interesting for the design of process conditions to obtain specific CMDs. Acknowledgment. The authors gratefully acknowledge latex supply by Solvay Solexis and funding by the Swiss National Science Foundation (Grant 200020-126487/1).

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