pubs.acs.org/Langmuir © 2009 American Chemical Society
Aggregation Kinetics of Coalescing Polymer Colloids Cornelius Gauer, Zichen Jia,† Hua Wu, and Massimo Morbidelli* Institute for Chemical and Bioengineering, Department of Chemistry and Applied Biosciences, ETH Zurich, 8093 Zurich, Switzerland. † Current address: Novartis Pharma AG, WSJ-145.8.51, Novartis Campus, Forum 1, 4056 Basel, Switzerland Received March 19, 2009. Revised Manuscript Received May 8, 2009 The aggregation behavior of a soft, rubbery colloidal system with a relatively low glass transition temperature, Tg ≈ -20 C, has been investigated. It is found that the average gyration and hydrodynamic radii, ÆRgæ and ÆRhæ, measured by light scattering techniques, evolve in time in parallel, without exhibiting the crossover typical of rigid particle aggregation. Cryogenic scanning electron microscopy (cryo-SEM) images reveal sphere-like clusters, indicating that complete coalescence between particles occurs during aggregation. Since coalescence leads to a reduction in the total colloidal surface area, the surfactant adsorption equilibrium, and thus the colloidal stability, change in the course of aggregation. It is found that to simulate the observed kinetic behavior based on the population balance equations, it is necessary to assume that all the clusters are spherical and to account for variations in the colloidal stability of each aggregating particle pair with time. This indicates that, for the given system, the coalescence is very fast, i.e., its time scale is much smaller than that of the aggregation.
1. Introduction Aggregation of colloidal particles is a common phenomenon involved in various industrial practices such as in ceramics, food, polymer and pharmaceutical processing. It has been widely studied both experimentally and theoretically as reported in reviews and textbooks.1-5 Typical colloidal particles used in the aggregation studies are rigid under ambient conditions, such as polystyrene, gold, and silica. In this case, the clusters formed during the aggregation are considered as fractal objects,1,2,4,6,7 and the aggregation kinetics and cluster structure follow certain so-called universal behaviors.8-11 For fast, diffusion-limited cluster-cluster aggregation (DLCA), the aggregation kinetics follows the power-law, and the fractal dimension of the clusters, Df, is around 1.7 to 1.8. Instead, for slow, reaction-limited cluster-cluster aggregation (RLCA), the average cluster size evolves with time exponentially, and Df ≈ 2.1. Moreover, starting from monodisperse primary particles, where the radius of gyration is smaller than the hydrodynamic radius,12 when the aggregation reaches a certain stage, crossover occurs under both *To whom correspondence should be addressed. E-mail: morbidelli@ chem.ethz.ch. Tel: 0041-44-6323034. (1) Jullien, R.; Botet, R. Aggregation and Fractal Aggregates; World Scientific: Singapore, 1987. (2) Meakin, P. Adv. Colloid Interface Sci. 1988, 28, 249–331. (3) Elimelech, M.; Gregory, J.; Jia, X.; Williams, R. A. Particle Deposition and Aggregation; Butterworth-Heinemann: Woburn, MA, 1995. (4) Friedlander, S. K. Smoke, Dust, and Haze; Oxford University Press: New York, 2000. (5) Bushell, G. C.; Yan, Y. D.; Woodfield, D.; Raper, J.; Amal, R. Adv. Colloid Interface Sci. 2002, 95, 1–50. (6) Forrest, S. R.; Witten, T. A. J. Phys. A: Math. Gen. 1979, 12, L109–L117. (7) Mandelbrot, B. B. The Fractal Geometry of Nature; Freeman: New York, 1982. (8) Lin, M. Y.; Lindsay, H. M.; Weitz, D. A.; Ball, R. C.; Klein, R.; Meakin, P. Nature 1989, 339, 360–362. (9) Lin, M. Y.; Lindsay, H. M.; Weitz, D. A.; Ball, R. C.; Klein, R.; Meakin, P. Phys. Rev. A 1990, 41, 2005–2020. (10) Lin, M. Y.; Lindsay, H. M.; Weitz, D. A.; Klein, R.; Ball, R. C.; Meakin, P. J. Phys.: Condens. Matter 1990, 2, 3093–3113. (11) Sandkuhler, P.; Lattuada, L.; Wu, H.; Sefcik, J.; Morbidelli, M. Adv. Colloid Interface Sci. 2005, 113, 65–83. (12) Kerker, M. The Scattering of Light; Academic Press: New York, 1969.
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DLCA and RLCA conditions, i.e., the average radius of gyration becomes larger than the hydrodynamic radius.13,14 However, for soft polymer colloids (e.g., rubbery particles), although these materials are produced in large quantities worldwide,15 little information can be found in the literature about their aggregation kinetics and cluster structure. Therefore, in this work we explore the aggregation behavior of soft latex particles. The key difference to rigid particles is that rubbery particles upon physical contact can deform or even coalesce (fuse) as a result of polymer diffusion or viscous flow,16,17 leading to the loss of particle identity within a cluster. The degree of coalescence between particles is related to the extent of polymer interdiffusion across particle-particle boundaries, which is controlled by various material properties such as its glass transition temperature Tg, degree of cross-linking, molecular weight, and hydrophilicity. At temperatures sufficiently above Tg, coalescence of polymer particles is assumed to occur by viscous flow,16 similar to the case of aerosols near their melting point.4 Particle coalescence can certainly affect the aggregation kinetics, but systematic studies about such effects are lacking in the literature. In the present work, aggregation experiments are carried out under RLCA conditions. We investigate not only the aggregation kinetics, but also the dynamics of colloidal stability during aggregation, which is related to shrinkage of the total colloidal surface area, a result of cluster coalescence, leading to redistribution of surfactant (mobile charges) in the system. The cluster structure is characterized by both light scattering techniques and cryogenic scanning electron microscopy (cryo-SEM). Experimental results are then simulated using the cluster mass distribution (CMD) generated by the solution of population balance (13) Lattuada, M.; Sandkuhler, P.; Wu, H.; Sefcik, J.; Morbidelli, M. Adv. Colloid Interface Sci. 2003, 103, 33–56. (14) Jia, Z.; Wu, H.; Xie, J. J.; Morbidelli, M. Langmuir 2007, 23, 10323–10332. (15) Fitch, R. M. Polymer Colloids; Academic Press: San Diego, CA, 1997. (16) Mazur, S. Coalescence of polymer particles. In Polymer Powder Technology; Narkis, M., Rosenzweig, N., Eds.; Wiley: Chichester, U.K., 1995; Chapter 8, pp 157-216. (17) Keddie, J. L. Mater. Sci. Eng., R 1997, 21, 101–170.
Published on Web 06/03/2009
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equations (PBE), where the dynamics of the colloidal stability is implemented. It is found from both experiments and simulations that the aggregation kinetics of a coalescing system is fundamentally different from that of colloidal systems with rigid particles. Strong cluster coalescence associated with surface reduction leads to surfactant accumulation and therefore increased colloidal stability as the aggregation proceeds.
2. Experimental Section 2.1. Aggregation Experiments. The Elastomer Colloid. The investigated colloidal system is a latex of fluorinated elastomer particles synthesized through emulsion polymerization and supplied by Solvay Solexis (Italy). The radius of primary particles determined from dynamic (DLS) and static light scattering (SLS) is Rp = 60 nm. The Tg of the polymer material is approximately -20 C. Thus, the particles are rather soft at room temperature. According to the supplier, on the particle surface there are negligible fixed charges originating from the polymer chain end groups, and the stabilization against coagulation is realized by adsorption of an ionic surfactant, which is a mixture of bicomponent perfluoropolyether (PFPE)-based carboxylates similar to that reported in ref 18 referred to as E1 and E2, respectively, with an average molecular weight of 580 g/mol and sodium as the counterion. The original latex has pH ≈ 3. The RLCA Experiments. The aggregation experiments were conducted at T = 25 C. Since the charge groups of the surfactant are carboxylic, to effectively destabilize the system, sulfuric acid (H2SO4) diluted with deionized water was used. All the aggregation processes were operated under RLCA (partially destabilized) conditions. Each experiment was started by pouring 9 parts prediluted acid onto 1 part prediluted latex to end up with a particle volume fraction of φ = 5.0 10-3. The H2SO4 concentrations were in the range from 0.04 to 0.055 mol/L, and the total concentrations of E1 and E2 were each about 8 10-5 mol/L. The chosen range of the acid concentration allows us not only to monitor the aggregation kinetics in a large time interval but also to properly measure the initial aggregation rates for determining the colloidal stability. It should be mentioned that, since the density of the polymer (1.8 kg/L) is much larger than that of water, when the clusters grow to a certain size, sedimentation may occur, which not only leads to system inhomogeneity but also affects the aggregation kinetics.13,19 To prevent sedimentation, we turned the aggregation container upside down every couple of hours.
2.2. Characterization of the Aggregation Process. Characterization Using Light Scattering. To monitor the aggregation process, we determined, off-line, two moments of the CMD of the system: the average gyration and hydrodynamic radii, using SLS and DLS techniques, respectively. To avoid multiple light scattering and ensure negligible correlations among the particles during the light scattering measurements,20,21 small volumes of the aggregating system were diluted to φ = 5.0 10-5, using aqueous solution of sulfuric acid at pH ≈ 3. All the SLS measurements were performed using a goniometer, BI-200SM (Brookhaven Instruments, USA) covering angles from θ = 16 to 150. A solid-state laser, Ventus LP532 (Laser Quantum, UK) with a wavelength λ0 = 532 nm, was used as the light source. The average radius of gyration, ÆRgæ [= (ÆRg2æ)1/2], was estimated using the Guinier plot of the intensity (18) Mele, S.; Chittofrati, A.; Ninham, B. W.; Monduzzi, M. J. Phys. Chem. B 2004, 108, 8201–8207. (19) Wu, H.; Lattuada, M.; Sandkuhler, P.; Sefcik, J.; Morbidelli, M. Langmuir 2003, 19, 10710–10718. (20) Finsy, R. Adv. Colloid Interface Sci. 1994, 52, 79–143. (21) Neutrons, X-rays and Light: Scattering Methods Applied to Soft Condensed Matter; Lindner, P., Zemb, T., Eds.; Elsevier: Amsterdam, The Netherlands, 2002. (22) Sorensen, C. M. Aerosol Sci. Technol. 2001, 35, 648–687.
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Figure 1. Time evolutions of the average gyration and hydrodynamic radii, ÆRgæ (open symbols) and ÆRhæ (filled symbols), for the aggregation of rubbery particles at φ = 5 10-3, destabilized at CH2SO4 = 0.045 (Δ, 2), 0.05 (O, b) and 0.055 mol/L (0, 9), respectively. curve I(q):21,22 IðqÞ 3 ¼ 2 ÆRg2 æ, -ln Ið0Þ q
qÆRg æ < 1
ð1Þ
where I (0) is the intensity at zero angle, and q is the scattering wave vector, defined as q ¼
4πn0 θ sin 2 λ0
ð2Þ
with λ0 being the wavelength of incident light, and n0 being the refractive index of the dispersant. The DLS measurements were carried out at θ = 90 using a digital correlator, BI-9000AT. The average hydrodynamic radius, ÆRhæ, was computed using the measured average diffusion coefficient ÆDæ via the Stokes-Einstein relation:20,21,23 ÆRh æ ¼
kT 6πμl ÆDæ
ð3Þ
where kT is the thermal energy, and μ1 is the dynamic viscosity of the dispersion medium. Figure 1 shows the time evolutions of the measured ÆRgæ and ÆRhæ values at three levels of the H2SO4 concentration. It is found that both ÆRgæ and ÆRhæ increase linearly with logarithmic time and evolve rather parallel. This behavior is very different from the commonly observed kinetics in the case of rigid primary particles either in the RLCA regime (exponential growth)9 or in the DLCA limit (power law growth),10 for which, when the primary particles are monodisperse, starting from Rh/Rg = (5/3)1/2,12 the evolutions of ÆRgæ and ÆRhæ with time lead to a crossover with ÆRhæ smaller than ÆRgæ.13,14 We can simulate such distinct time evolutions of ÆRgæ and ÆRhæ using the time evolution of the CMD computed from PBE. However, before simulating the kinetics, we need to have information about the structure of the clusters. Cluster Structure and CMD from Cryo-SEM Images. Due to the softness of the elastomer particles, the conventional SEM technique with sample drying does not reproduce the correct cluster structure and size, e.g., the identity of primary particles is lost and drop-like structures (or films) are formed along dispersant evaporation, as shown in Figure 2. Thus, SEM has to be performed in cryogenic conditions (cryo-SEM).24 In particular, samples of φ = 9.0 10-5 with 8 wt % ethanol as an ice protective (23) Koppel, D. E. J. Chem. Phys. 1972, 57, 4814–4820. (24) Echlin, P. Low-Temperature Microscopy and Analysis; Plenum Press: New York, 1992.
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Article Moreover, it is well-known that, to validate a CMD generated from the solution of PBE, one needs to simulate a larger number of moments of the CMD.26 In this work, the light scattering techniques give only two moments of the CMD, ÆRgæ and ÆRhæ. Although there is high probability for the CMD to be correct if one can simulate both ÆRgæ and ÆRhæ, to further validate the CMD from PBE, we decided to perform image analysis of cryo-SEM pictures of selected reference samples. This becomes feasible because of the spherical form of the clusters resulting from coalescence. In this case, the CMD can be determined simply by measuring the particle diameter and then converting it into mass. For each sample, a number of particles corresponding to a total mass of about 200 primary particles has been considered. It is evident that this kind of sample preparation is very elaborate and costly, and we have determined the CMD only for the experiment at CH2SO4 = 0.05 mol/L, at times t = 5, 30, 200, 1350 min. Note that the particle size obtained by cryo-SEM differs typically by 10 to 20% from the light scattering value.27,28 For example, the average radius of the primary particles obtained from the cryo-SEM is 53.5 nm, while it is 60 nm according to both SLS and DLS measurements.
3. The Modeling Approach
Figure 2. Spreading (film formation) of coalescing rubber latex particles as a result of the drying process in conventional SEM.
3.1. Kinetic Model. The evolution of CMD along the aggregation process can be conveniently described by the mass or PBEs, which are based on the Smoluchowski kinetic model,29 assuming that the aggregation follows an irreversible secondorder rate law. The PBE in discrete form can be written as26,29-31 i -1 max X dNi 1X ¼ Ki, j Ni Nj þ Ki -j, j Ni -j Nj 2 j ¼1 dt j ¼1 j
Figure 3. Cryo-SEM images of (a) primary particles and (b) clusters after 1350 min aggregation under CH2SO4 = 0.05 mol/L. were frozen at about -160 C in the high-pressure freezing machine HPM 010 from BAL-TEC (Liechtenstein). Sample storage and fracture under liquid nitrogen were followed by approximately 40 min of freeze-drying and coating with tungsten (e3 nm) under high vacuum at -80 C in BAL-TEC’s freeze-etch system BAF 060. After transfer via the airlock shuttle system VCT 100 (BAL-TEC),25 images were taken on a Zeiss (Germany) Gemini 1530 FEG scanning electron microscope equipped with a cold stage. As a general result, it is found from all the cryo-SEM images that clusters of the present elastomer particles always have sphere-like shape. Figure 3 illustrates the typical appearance of primary particles and clusters. It should be mentioned that when taking a sample from the aggregating system, a certain time is needed to freeze the sample, similar to the situation of measuring the intensity curve in SLS. In this time period, if the coalescence between clusters was not yet complete, it could continue. Then, the obtained cryo-SEM picture would not represent the cluster structure during aggregation. To verify this, we have monitored the time evolution of ÆRhæ through DLS measurements for samples after taking them from the aggregating system. It was found that the ÆRhæ value is independent of time, which indicates that the coalescence of generated clusters can be regarded as instantaneous with respect to aggregation. The cryo-SEM evidence of complete coalescence supplies additional information for interpreting the kinetic data in Figure 1. (25) Ritter, M.; Henry, D.; Wiesner, S.; Pfeiffer, S.; Wepf, R. A versatile highvacuum cryo-transfer for cryo-FESEM, cryo-SPM and other imaging techniques. Microscopy and Microanalysis; New York, 1999; Vol. 5, Suppl. 2, pp 424-425.
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ð4Þ
where Ni is the number concentration of clusters with dimensionless mass i (i.e., composed of i primary particles) at time t, and Ki,j is the aggregation rate constant (or kernel) between two clusters with masses i and j, respectively. A widely accepted form of Ki,j for reaction-limited aggregation of rigid particles in perikinetic conditions is the so-called product kernel, given by13,32-38 Ki , j ¼
KB ði1=Df þ j 1=Df Þði -1=Df þ j -1=Df Þ λ ðijÞ 4 W
ð5Þ
where KB = 8kT/(3 μl) is the Brownian-motion-controlled collision rate constant for primary particles. The cluster structure is taken into account by the fractal dimension Df. The product term, (ij)λ, accounts for the fact that the reactivity of fractal clusters increases with the cluster mass in the RLCA regime. An (26) Ramkrishna, D. Population Balances; Academic Press: San Diego, CA, 2000. (27) Bootz, A.; Vogel, V.; Schubert, D.; Kreuter, J. Eur. J. Pharm. Biopharm. 2004, 57, 369–375. (28) Geze, A.; Putaux, J. L.; Choisnard, L.; Jehan, P.; Wouessidjewe, D. J. Microencapsul. 2004, 21, 607–613. (29) von Smoluchowski, M. Z. Phys. Chem. 1917, 92, 129–168. (30) Ziff, R. M.; McGrady, E. D.; Meakin, P. J. Chem. Phys. 1985, 82, 5269– 5274. (31) Meakin, P.; Chen, Z. Y.; Deutch, J. M. J. Chem. Phys. 1985, 82, 3786–3789. (32) Family, F.; Meakin, P.; Vicsek, T. J. Chem. Phys. 1985, 83, 4144–4150. (33) Axford, S. D. T. J. Chem. Soc., Faraday Trans. 1997, 93, 303–311. (34) Schmitt, A.; Odriozola, G.; Moncho-Jorda, A.; Callejas-Fernandez, J.; Martinez-Garcia, R.; Hidalgo-Alvarez, R. Phys. Rev. E 2000, 62, 8335–8343. (35) Odriozola, G.; Tirado-Miranda, M.; Schmitt, A.; Lopez, F. M.; CallejasFernandez, J.; Martinez-Garcia, R.; Hidalgo-Alvarez, R. J. Colloid Interface Sci. 2001, 240, 90–96. (36) Odriozola, G.; Moncho-Jorda, A.; Schmitt, A.; Callejas-Fernandez, J.; Martinez-Garcia, R.; Hidalgo-Alvarez, R. Europhys. Lett. 2001, 53, 797–803. (37) Sandkuhler, P.; Sefcik, J.; Lattuada, M.; Wu, H.; Morbidelli, M. AIChE J. 2003, 49, 1542–1555. (38) Lattuada, M.; Wu, H.; Sefcik, J.; Morbidelli, M. J. Phys. Chem. B 2006, 110, 6574–6586.
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interpretation of this term is that the aggregation between two clusters can be described as a process controlled by the collision of two primary particles belonging to each cluster, and that the rate of aggregation of two clusters is proportional to the product of the number of primary particles of the two clusters that can participate in the collision process.33,38 For the present system, where contacting particles coalesce very fast and all the clusters are basically spheres, i.e., the identity of the primary particles is lost, there is no reason to include this product term in the aggregation kernel. Moreover, in the case of full coalescence the fractal dimension has to be set Df = 3. Thus, the aggregation kernel used in this work becomes Ki , j ¼
KB ði1=3 þ j 1=3 Þði -1=3 þ j -1=3 Þ 4 W
ð6Þ
The quantity W in eqs 5 and 6 is the Fuchs stability ratio, which is the ratio of the aggregation rate in the presence of repulsive particle interactions to that in the absence of repulsive particle interactions and is expressed as3,39,40 Z
¥
W ¼2 2
U exp kT dl GðlÞl 2
U ¼ UA þ UR þ Uhyd ð7Þ
where U is the total interaction energy between two particles (clusters), l is the dimensionless center-to-center distance, normalized by the particle radius, and G(l) accounts for the hydrodynamic resistance resulting from squeezing of the fluid during the particle approach. In the case of aggregation between rigid particles or clusters, W is considered to be constant, independent of cluster size and structure, and determined by the interactions between two primary particles. In the present case, due to cluster coalescence, the total surface of the dispersed phase decreases as the aggregation proceeds, and redistribution of the surfactant on the particle surface leads to changing W values with time as well as with size of two colliding entities. A similar situation has been observed recently in the case of aggregation of polyelectrolyte complex nanoparticles,41 where, although no coalescence occurs, the W value also depends on the cluster size and structure, and it follows that such changes in W with time have to be integrated into the kinetic modeling. In the present case, integrating these effects into the kinetic model requires one to correctly describe the time evolution of the colloidal stability of the coalescing system. 3.2. Stability Model. Colloidal stability is affected by various factors. In the case where ionic surfactants stabilize the particles, the stability involves strong interplay among various physicochemical processes such as colloidal interactions, surfactant adsorption equilibria, and association equilibria of surface charge groups with counterions. To correctly describe the stability of the system, one has to account for such interplays among all the processes. For this, we have proposed a generalized model for the stability of colloidal systems in a previous work,42 in which the total interaction energy in the W expression (eq 7) is obtained by simultaneously accounting for the interplay among three processes: interaction forces, surfactant adsorption equilibria, and association equilibria of surface charge groups with counterions. Moreover, all the equilibria are considered at the particle-liquid interface. For details of the stability model, the reader is referred to the original work.42 (39) Fuchs, N. Z. Phys. 1934, 89, 736–743. (40) Spielman, L. A. J. Colloid Interface Sci. 1970, 33, 562–571. (41) Starchenko, V.; Muller, M.; Lebovka, N. J. Phys. Chem. C 2008, 112, 8863– 8869. (42) Jia, Z. C.; Gauer, C.; Wu, H.; Morbidelli, M.; Chittofrati, A.; Apostolo, M. J. Colloid Interface Sci. 2006, 302, 187–202.
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Application of this stability model to the present work becomes essential in describing the dynamics of the colloidal stability during aggregation, where the surfactant adsorption equilibria vary continuously with time as a result of shrinkage of the total particle surface area. However, let us first of all constitute all the elements involved in the generalized stability model. Colloidal Interaction Model. The first element for the stability model are the colloidal interaction forces. The classical Derjaguin-Landau-Verwey-Overbeek (DLVO) interactions43 naturally need to be included, which combine the attractive van der Waals and repulsive electrostatic forces. However, nonDLVO interactions are observed in various systems.44,45 For example, in aqueous dispersions, additional repulsion may arise as a result of the structuring of water molecules near the solidliquid interface, which is referred to as hydration force.44,46 Its inclusion often becomes necessary to achieve a correct description of the colloidal stability.42,47,48 Therefore, in this work, the total interaction energy is considered to include the van der Waals (UA), electrostatic (UR) and hydration (Uhyd) interaction energies: ð8Þ
During the aggregation, since small particles form larger particles as a result of cluster coalescence, we need to compute interaction energies between particles of different sizes. Let us consider two spherical particles with masses i and j, and the corresponding radii Ri and Rj, respectively. Then, UA in eq 8 can be expressed, using the Hamaker relation,3,45 as ( AH 2Ri Rj 2Ri Rj þ þ UA ¼ 2 2 2 6 r -ðRi þ Rj Þ r -ðRi -Rj Þ2 " #) r2 -ðRi þ Rj Þ2 ln ð9Þ r2 -ðRi -Rj Þ2 where r is the center-to-center distance between two particles, and AH is the Hamaker constant. Since the polymer composition is similar to that of polytetrafluoroethylene (PTFE), the AH value of PTFE (3.0 10-21 J44) has been used in this work. By defining the following quantities: a ¼
Ri þ Rj , 2
r Rj l ¼ ; and ω ¼ a Ri
we can transform eq 9 into 8 2 > < AH 8ω 6 1 þ UA ¼ 4 > 6 :ð1 þ ωÞ2 l 2 -4 2
ð10Þ
3 1
7 2 5 þ
l 2 -4 11 -ω þω
39 > = 6 l -4 7 ln4 2 5 > ; l 2 -4 1 -ω 2
ð11Þ
1 þω
(43) Verwey, E. J.W.; Overbeek, J. T. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, The Netherlands, 1948. (44) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: London, 1992. (45) Hunter, R. J. Foundations of Colloid Science; Oxford University Press: New York, 2001. (46) Israelachvili, J. N.; McGuiggan, P. M. Science 1988, 241, 795–800. (47) Thompson, D. W.; Collins, I. R. J. Colloid Interface Sci. 1994, 163, 347– 354. (48) Runkana, V.; Somasundaran, P.; Kapur, P. C. AIChE J. 2005, 51, 1233– 1245.
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For moderate surface potentials (i.e., ψs e 50 mV), the repulsive interaction energy UR can be computed through the modified Hogg-Healy-Fuersteneau approximation,49 which, for particles of unequal size, has the form UR ¼
4πε0 εr ωaψ2s, i 2
ð1 þ ωÞ l
fð1 þ ΨÞ2 lnð1 þ exp½ -Kaðl -2ÞÞ þ ð1 -ΨÞ2 lnð1 -exp½ -Kaðl -2ÞÞg
ð12Þ
where Ψ = ψs,j /ψs,i is the ratio of surface potentials of the two particles. Both ψs,j and ψs,j are computed from the effective charges on the particle surface, which are related to surfactant adsorption and counterion association to be discussed in the following. The Debye-Huckel parameter κ in eq 12 is given by3,45 !1=2 P e2 NA j z2j Cjb K ¼ ε0 εr kT
ð13Þ
Figure 4. Relative hydrodynamic resistance, G, as a function of the dimensionless center-to-center distance, l = 2 + h/a, between two nonequal sized spheres approaching by diffusion; exact solution40 (open symbols), equal sphere approximation51 (f), and fittings for ω = 1, 2, 5, 10 (solid lines).
with NA being the Avogadro number, Cjb and zj the bulk concentration and the valency of ionic species j, e the electron charge, and ε0εr the permittivity of the dispersant. For the hydration force, which is rather short-range, we adopt the typical empirical exponential relation44 h Fhyd ¼ F0 exp δ0
ð14Þ
where F0 is the hydration force constant and δ0 the decay length. Application of the Derjaguin approximation (valid for surface separations h , a or l - 2 , 1)50 yields the hydration interaction energy for sphere-sphere geometry:42,48 Uhyd ¼
4πωa ð1 þ ωÞ2
F0 δ 0
2
a exp - ðl -2Þ δ0
ð15Þ
It is reported that the value of the decay length δ0 can be found between 0.2 and 1.5 nm,46,47 and we set δ0 = 0.6 nm.42,48 The force constant F0 can be estimated from the initial W values, measured from doublet formation kinetics. The quantity G(l) [= Di,j(l)/D¥ i,j] in the W expression (eq 7) accounts for the reduction in the mutual diffusion Di,j(l), compared to the undisturbed diffusion at infinite distance D¥ i,j, as two particles approach. Instead of elaborately computing the G(l) value for pairs of unequally sized spheres,40 we use the approximation for identical spheres by Honig et al.,51 but with the coefficients Rn (n = 1, ... ,5) being functions of the ratio of radii ω: GðlÞ ¼
R1 l 2 þ R2 l þ R3 R4 l 2 þ R5 l
ð16Þ
All the Rn expressions were obtained by fitting the exact G(l) solution according to Spielman,40 and are listed in the Appendix, valid for 1 e ω e 10. In Figure 4 are compared the predictions of eq 16 with the exact solution. As can be seen, the dependence of
(49) Sader, J. E.; Carnie, S. L.; Chan, D. Y. C. J. Colloid Interface Sci. 1995, 171, 46–54. (50) Deryaguin, B. V. Kolloid Z. 1934, 69, 155. (51) Honig, E. P.; Roebersen, G. J.; Wiersema, P. H. J. Colloid Interface Sci. 1971, 36, 97–109. (52) Baldyga, J.; Jasinska, M.; Krasinksi, A.; Rozen, A. Chem. Eng. Technol. 2004, 27, 315–323.
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Figure 5. Fit of Langmuir isotherms to the experimental adsorption data of PFPE surfactants E1 and E2 on PTFE particles.
G(l) on ω is rather significant.40,52 Note that in Figure 4 the G(l) values at ω = 10 correspond to the particle mass i = 1000. Surfactant Adsorption Equilibria. As mentioned before, the particle surface potentials in eq 12 are computed from the effective surface charges originating from the adsorbed surfactant molecules. The surfactant used for stabilizing the particles is composed of two components, E1 and E2, respectively. The adsorption isotherms reported by the latex supplier are shown in Figure 5.53 These data can be well fitted using the Langmuir adsorption isotherm:45 bCEi j Γ ¼ Γ¥ 1 þ bCEi j
ð17Þ
where b is the adsorption rate constant, and Γ and Γ¥ are the equilibrium and the saturation surfactant densities on the surface, respectively. It should be noted that, in the stability model, the relevant surfactant concentration is the one at the particle-liquid interface denoted by CiEj, instead of the one in the liquid bulk. The estimated values for b and Γ¥ from fitting the experimental data in Figure 5 are 0.87 103 L/mol and 4.0 10-6 mol/m2 for E1, and 13.4 103 L/mol and 6.0 10-6 mol/m2 for E2. Counterion Association Equilibria. Computing the effective surface charges requires accounting for counterion association of the surfactant charge groups (i.e., -COO-). There are two types (53) Chittofrati, A. Self-association of model perfluoropolyether carboxylic salts. Internal presentation, Solvay Solexis, 2001.
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of counterions in the system, Na+ from the surfactant itself and H+ from the destabilizer H2SO4. Since the concentration of H+ is much larger than that of Na+, and the association of -COOwith H+ is much stronger, we neglect the association of -COOwith Na+. Thus, on the basis of the mass action law, the association of E1 and E2 with protons H+ can be written as KHE ¼
½ -COOH ½ -COO - ½H þ
ð18Þ
where [-COO-] gives the effective surface charge for computing the surface potentials in the generalized stability model.42 Note that we assume equal association constants for E1 and E2, and KHE = 30.54 The corresponding pKa is ∼1.5, which is well in the range of expected values for perfluorooctanoic acid and analo+ gous compounds.55 For the association between SO24 and H , the association constant is KHSO-4 = 97.56 Weak association + between HSO4 and H is neglected. 3.3. Simulation of the Aggregation Process. As mentioned in the experimental section, for each aggregating system, we have determined the time evolutions of the average gyration and hydrodynamic radii, ÆRgæ and ÆRhæ, as given in Figure 1. Both radii can be considered as two independent moments of the CMD that are evolving with time, i.e., Ni(t), and can be expressed as20,22 P 2 2 Ni i Rg, i ÆRg 2 æ ¼ P 2 Ni i P 2 Ni i Pi ðqÞ ÆRh æ ¼ P 2 Ni i Pi ðqÞRh-1 ,i
ð19Þ
4. Results and Discussion 4.1. Doublet Formation Kinetics. An effective approach to characterize the stability of an aggregating colloid is to measure the absolute aggregation rate for conversion of primary particles to doublets. The corresponding doublet formation rate constant K1,1 defines the stability ratio W for primary particle aggregation, which, from eq 5, is given by W ¼ KB =K1, 1
ð21Þ
dN1 ¼ -K1, 1 N1 2 dt
9708 DOI: 10.1021/la900963f
ð23Þ
Integration of this equation and introduction of conversion of primary particles to doublets, x = 1 - N1/N1,0, with N1,0 (= φ/[4/3 πRp3]) the number concentration of primary particles at t = 0 and N1 the number concentration of primary particles at time t, yield K1, 1 ¼
x 1 1 -x N1, 0 t
ð24Þ
Thus, for each given aggregation time t, from eq 24 we can compute the K1,1 value if we know the conversion value x. We determine the x value using the SLS and DLS techniques.13,42,58-60 At the initial aggregation stage where only primary particles and doublets are present, the normalized scattered light intensity is given by IðqÞ N1 P1 ðqÞ þ 4N2 P2 ðqÞ ¼ Ið0Þ N1 þ 4N2
ð25Þ
where P1(q) and P2(q) are the form factors of primary particles and doublets, respectively, and N1 and N2 are their number concentrations. Let us apply the RDG expression for the form factors,12,21 valid for small particles, which reads for primary particles, "
(54) D’Aprile, F.; Geniram, G.; Chittofrati, A. Costante di dissociazione di acidi carbossilici Galden “monopicco-via estere”: Prima stima a temperatura ambiente. Internal report, Solvay Solexis, 1997. (55) Goss, K. U. Environ. Sci. Technol. 2008, 42, 456–458. (56) CRC Handbook of Chemistry and Physics, 88th ed. (Internet Version 2008); Lide, D. R., Ed.; CRC Press/Taylor and Francis: Boca Raton, FL, 2008. (57) Wiscombe, W. J. Mie Scattering Calculations: Advances in Technique and Fast, Vector-Speed Computer Codes; NCAR Technical Note, NCAR/TN140+STR; National Center for Atmospheric Research: Boulder, CO, 1979 (1996 revised).
ð22Þ
From doublet formation kinetics, one can already observe the difference between coalescing and noncoalescing systems. In addition, by fitting the W values for primary particle aggregation, it is possible to estimate the hydration force constant F0, in the stability model. Estimation of K1,1. At the very initial stage of the aggregation, one can assume that the system contains only primary particles and doublets. On the basis of the calculated CMD of the coalescing system, such an assumption is valid as long as the conversion of primary particles to doublets is x < 30%, as can be compared in Figure 6. In this case, the PBE, eq 4, reduce to
ð20Þ
where Pi (q), Rh,i and Rg,i are the form factor, the effective hydrodynamic radius, and the radius of gyration of the individual cluster with mass i, respectively. Justified by the spherical cluster structure, Pi(q) is calculated using the Lorenz-Mie theory,12,57 Rh,i = i1/3Rp, and Rg,i = (3/5)1/2 i1/3Rp. The main objective of the simulations is to compute the time evolution of the CMD using the kinetic model (PBE) with implemented stability model for estimating the time evolution of W, as discussed in previous subsections. Then, applying eqs 19 and 20 to the computed CMD predicts the measured time evolutions of ÆRgæ and ÆRhæ. It will be seen that, because of the cluster coalescence, the interplay between the aggregation kinetics and the colloidal stability becomes especially important with respect to noncoalescence systems, because evolution of the CMD leads to a decrease in the total surface area Stotal/V, which changes the surfactant adsorption equilibria, and thus the surface potential, and then alters the aggregation kinetics through W. The relation between the total surface area and CMD is given by Stotal X ¼ Ni Si V
where Si = i2/3Sp is the surface area of the sphere with mass i and Sp = 4πRp2 is the surface area of a primary particle. In solving the PBE,26 the CMD grid is structured linear for the first 10 integer sizes and then logarithmic. The computation starts from (monodisperse) primary particles of radius Rp = 60 nm, as measured by light scattering.
P1 ðqÞ ¼ 9
#2 sinðqRp Þ -ðqRp Þ cosðqRp Þ ðqRp Þ3
ð26Þ
(58) Herrington, T. M.; Midmore, B. R. J. Chem. Soc. Farad. T. 1 1989, 85, 3529–3536. (59) van Zanten, J. H.; Elimelech, M. J. Colloid Interface Sci. 1992, 154, 1–7. (60) Holthoff, H.; Egelhaaf, S. U.; Borkovec, M.; Schurtenberger, P.; Sticher, H. Langmuir 1996, 12, 5541–5549.
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Figure 6. Computed CMD at low conversions, x, of primary particles to doublets. It is seen that for x < 30% practically no clusters i g 3 exist. (The lines serve only for guiding the eye.)
Figure 7. Doublet formation rate constant, K1,1, evaluated from the initial aggregation kinetics at CH2SO4 = 0.05 mol/L. The solid curve is a fit of the experimental data for extrapolating K1,1 to t = 0. For comparison, a horizontal (broken) line is drawn showing K1,1 for a system with constant stability.
and for spherical doublets with R2 = 21/3Rp, " P2 ðqÞ ¼ 9
1=3
sinð2
qRp Þ -ð2
1=3
qRp Þ cosð2
1=3
#2 qRp Þ
ð21=3 qRp Þ3
x ¼ ð27Þ
Introducing the above P1(q) and P2(q) expressions into eq 25 and considering N1 = (1 - x)N1,0 and N2 = xN1,0/2, we obtain through proper arrangement IðqÞ x FðqÞ ¼ 1Ið0ÞP1 ðqÞ 1 þx
ð28Þ
where " #2 1 sinð21=3 qRp Þ -ð21=3 qRp Þ cosð21=3 qRp Þ FðqÞ ¼ 2 ð29Þ 2 sinðqRp Þ -ðqRp ÞcosðqRp Þ It is clear from eq 28 that a plot of I (q)/[I(0)P1(q)] versus F(q) should yield a straight line, the slope of which gives the conversion x. Alternatively, the x value can be estimated from ÆRgæ, which is evaluated from the SLS data through the Guinier plot (eq 1). In particular, when the system contains only primary particles and doublets, from eq 19, ÆRgæ can be expressed as ÆR2g æ ¼
N1 Rg2, 1þ 4N2 Rg2, 2 N1 þ 4N2
ð30Þ
where Rg,1 = (3/5)1/2 Rp and Rg,2 = (3/5)1/2 21/3Rp are gyration radii for primary particles and spherical doublets, respectively. Then, introducing the relations of N1 and N2 with x, as shown above, we have x ¼
ÆRg2 æ -Rg2, 1 2 2Rg, 2 -Rg2, 1-ÆRg2 æ
¼
ÆRg2 æ - 35 Rp2 ð2
5=3
-1Þ 35 Rp2 -ÆRg2 æ
ð31Þ
Complementarily, one can also use ÆRhæ from the DLS data to estimate x, which from eq 20 can be written as ÆRh æ ¼
N1 P1 ðqÞ þ 4N2 P2 ðqÞ -1 N1 P1 ðqÞRh-1 , 1 þ 4N2 P2 ðqÞRh, 2
ð32Þ
with Rh,1 = Rp and Rh,2 = 21/3Rp. Introducing the relations of N1 and N2 with x leads to Langmuir 2009, 25(17), 9703–9713
ÆRh æ -Rp
ÆRh æ ÆRh æ -Rp þ 2Rp PP21 ðqÞ 1 1=3 ðqÞ 2 R
ð33Þ
p
Once the x value is obtained at each given time t, from eq 24, one can compute the corresponding K1,1 value. For noncoalescing systems, one generally observes that the computed K1,1 value is independent of x, i.e., each pair of the x and t values should always give the same K1,1 under the experimental errors. For coalescing systems, however, since the total surface area changes with the aggregation extent, leading to surface charge redistribution, the computed K1,1 value becomes dependent on the pair of x and t values. Then, the correct K1,1 value for eq 22 to estimate the W value is the one corresponding to t = 0. This can be realized by backward extrapolation of the K1,1 values at different t values to t = 0. Figure 7 reports the K1,1 values obtained from both the SLS and DLS data as a function of time, for the case where aggregation is initiated at CH2SO4 = 0.05 mol/L. It is obvious that the K1,1 value decreases monotonically as the aggregation proceeds in time. Such an observation is general for all aggregation systems reported in this work. It can be explained by the fact that complete cluster coalescence occurs during aggregation, such that the total colloidal surface area decreases with time, and redistribution of the surfactant leads to progressive increase in the surface charge density and, consequently, to decrease in the aggregation rate. On the other hand, the observed phenomenon can also be considered as a further confirmation of the coalescence of the elastomer particles. Extrapolating the K1,1 data in Figure 7 to t = 0, through, e.g., a second-order polynomial fitting, allows us to obtain the K1,1 values required for determining the initial W values from eq 22, which then can be used to evaluate the missing model parameter, F0. Interpretation of the Initial w Values Using the Generalized Stability Model. The initial W values obtained at four levels of the H2SO4 concentration are shown in Figure 8 (symbols). Now, let us fit these W values through the generalized stability model. As a first step, let us ignore the hydration interaction and consider only the DLVO interactions in the generalized stability model. In this case, there is no adjustable parameter in the model, and the predicted W values as a function of the acid concentration are shown in Figure 8 (dashed curve). Obviously, the predictions are several orders below the experimental data. We have tried to tune the Hamaker constant AH in order to improve the fitting, but DOI: 10.1021/la900963f
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Figure 8. Fuchs stability ratio, W, at the aggregation onset (t = 0) as evaluated from experiments (symbols) and as predicted by the stability model. Dashed curve: using the DLVO interactions; solid curve: using the DLVO plus hydration interactions.
the required AH value is 2.0 10-22 J, which is at least 1 order of magnitude smaller than the generally considered values for fluorinated polymers,44 and thus unrealistic. Then, the hydration interaction is included in the model calculation. Employing a least-squares optimization, the best fit to the experimental data is obtained at F0 = 1.1 106 N/m2, which is well within the reported range of 106 to 5 108 N/m2.47 Now it is seen from Figure 8 that the predictions including the hydration interaction (solid curve) are in excellent agreement with the experimental data. In addition, with the obtained F0 value, we have applied the generalized stability model to predict the critical coagulant concentration (CCC) for fast coagulation of the given colloid through H2SO4 addition. We define the CCC as the situation where the W versus CH2SO4 curve levels off by dW/dC e 10%, and the obtained CCC value is 0.4 mol/L, which corresponds to the CCC value measured experimentally. This further supports the presence of hydration interaction in our colloid. 4.2. Simulation of the Full Aggregation Process. With the F0 value estimated above, now the developed aggregation model is fully predictive. It should be mentioned, however, that in the generalized stability model, since the classical Gouy-Chapman theory45 has been applied, all concentrations of species at the particle-liquid interface, e.g., surfactants and ions, are the same independently of the particle (cluster) size. Then, the surfactant surface density, as well as the effective charge density, are identical for all particles and, consequently, so is the surface potential ψs. It follows that we have Ψ = 1 in eq 12. In this case, the variation of ψs is entirely due to the total colloidal surface reduction that leads to redistribution of the surfactant. As an example, Figure 9 shows the surface potential ψs as a function of Stotal/S0, where S0 is the initial total surface area for the aggregating system at CH2SO4 = 0.05 mol/L. It is seen that, in this case, the surface potential rises from an initial value of -17.6 mV to -19.0 mV when the total surface is reduced by 15%. The corresponding increase in the W value is 5-fold. Therefore, in all of the simulations, the ψs value, although changing with time or conversion, is the same for all particle sizes. The effect of particle size on the aggregation rate arises only from geometrical terms, i.e., the size effect present in diffusion rate, collision radius, and all the interaction terms. Predictions with Dynamic Pairwise Wi,j. In the calculation of the aggregation rate for each pair of particles i and j, the Fuchs stability ratio Wi,j was computed by integrating eq 7, where the U value results from eqs 8-16, accounting for the particle sizes and the ψs value at the given time. For the purpose of illustration, 9710 DOI: 10.1021/la900963f
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Figure 9. Surface potential, ψs, as a function of the total surface area, S, reduced by the initial total surface area, S0, computed through the stability model in the case of CH2SO4 = 0.05 mol/L.
Figure 10. Stability ratio of equal sized particles, Wi,i, as a function of the particle mass, i, calculated through the stability model at various aggregation times in the case of CH2SO4 = 0.05 mol/L.
the time evolution of Wi,j computed from the model is shown in Figure 10 for equally sized particles (i = j) in the cases of mass i = 1-10, for the case where CH2SO4 = 0.05 mol/L. Two observations can be readily drawn from Figure 10. First, the W value increases sharply as the mass of the particles i increases, which indicates that aggregation between small particles is preferred over large ones. Consequently, the system CMD should remain rather narrow with respect to noncoalescing systems. Second, for a particle of given mass, the stability ratio increases progressively with time, which means that the reactivity of the entire system decreases progressively with time or conversion. Time evolutions of ÆRhæ and ÆRgæ predicted by the model, are shown in Figure 11 (curves) and are compared with the experimental results (symbols). As can be seen, good agreement between experiments and predictions has been obtained in all cases. This confirms that fast, complete fusion (coalescence) between particles of a cluster indeed occurs during the aggregation, and the adopted colloidal stability model accounting for coalescence and surfactant redistribution reflects the evolution of the physical and chemical processes during aggregation. It is known9,10,13,14,19,61,62 that, for colloidal aggregations where no coalescence occurs, if the aggregation kinetics is universal, the evolutions of ÆRhæ or ÆRgæ observed at different particle concentrations, ionic strengths, particle sizes, temperatures, surfactant types, and so forth, would collapse to form a (61) Carpineti, M.; Ferri, F.; Giglio, M.; Paganini, E.; Perini, U. Phys. Rev. A 1990, 42, 7347–7354. (62) Sandkuhler, P.; Sefcik, J.; Morbidelli, M. J. Phys. Chem. B 2004, 108, 20105–20121.
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Figure 12. The stability ratio for primary particle aggregation, W1,1, as a function of the total surface area, S, where W(t = 0) and S0 are the initial values of the stability ratio and the total surface area at t = 0.
Figure 11. Comparisons between measured and simulated time evolutions of ÆRgæ (open symbols) and ÆRhæ (filled symbols) for the aggregation system at φ = 5 10-3 destabilized at CH2SO4 = 0.045 (Δ, 2), 0.05 (O, b), and 0.055 mol/L (0, 9), respectively. Dashed and solid lines correspond to simulations of ÆRgæ and ÆRhæ, respectively.
master-curve when they are plotted as a function of a dimensionless time, which can be defined as13 τ ¼ tN1, 0 KB =W
ð34Þ
This is due to the fact that, if one introduces the dimensionless particle number concentration, Xi = Ni/N1,0, the PBE (eq 4) reduces to the following dimensionless form:13 imax i -1 X dXi 1 X ¼ βi -j, j Xi -j Xj β i , j Xi Xj 2 j ¼1 dτ j ¼1
ð35Þ
where βi,j is the dimensionless aggregation kernel defined from eq 5 as: βi , j ¼
ði1=Df þ j 1=Df Þði -1=Df þ j -1=Df Þ λ ðijÞ 4
ð36Þ
Since βi,j is only a function of Df and λ, which depend only on the aggregation mechanism (DLCA or RLCA), the predicted dimensionless CMD, Xi, is independent of any operating conditions. This leads to the above-mentioned master-curve. Now, the following questions have to be addressed: What happens in the case of the coalescing system? Can we obtain a master-curve when a similar dimensionless time is defined? First of all, for a noncoalescing system, W in eq 5 is constant, while, for a coalescing system, W in eq 6 varies with time in reaction-limited conditions. A reasonable way to define the dimensionless time τ may be to apply the initial W(t = 0) value to eq 34. In this way, the Langmuir 2009, 25(17), 9703–9713
measured and simulated kinetic data, shown as a function of t in Figure 11a, have been replotted as a function of τ in Figure 11b. It is seen that, although all the measured and simulated ÆRhæ or ÆRgæ curves indeed shift closer, particularly at the initial stage, at later times they do not collapse to form a master-curve. Instead, they depend on the amount of acid used in the experiment. We can understand this when we consider that different acid concentrations lead not only to different ionic strength (screening effect) but also to different counterion association equilibria as well as different surfactant adsorption equilibria, and consequently to different time evolutions of ÆRhæ or ÆRgæ. To illustrate this fact, the time evolutions of W1,1 as a function of the surface area reduction are plotted in Figure 12 for the three conditions from Figure 11. Although all of them follow nearly a power law scaling, it would be difficult to find a simple relation to predict this functionality. Different slopes reveal different effects of the acid concentration on the evolution of colloidal stability, indicating very complex functionality with respect to surface reduction. All the above analyses clearly demonstrate the importance of the generalized stability model, implemented into the PBE modeling, to account for the interplay among the colloidal interactions, surfactant adsorption equilibria, and the counterion association equilibria during the aggregation process. Predictions with Constant W. As mentioned previously, for noncoalescing systems where aggregation leads to fractal clusters, the colloidal surface area does not change with time, and therefore it is expected that W is constant for all clusters, independent of size or aggregation time. In order to highlight the effect of coalescence on reaction-limited aggregation, let us simulate our coalescing system with a constant W, equal to the initial W(t = 0). The results are shown in Figure 13 as a function of the dimensionless time τ and compared with the experimental data. It is seen that all the numerical simulation results at different conditions (CH2SO4) collapse to form a single curve, and at a certain τ, crossover between the ÆRhæ and ÆRgæ curves occurs, resulting from a comparably broader CMD. However, the numerical simulations agree with the experiments at the very initial stage of the aggregation; but then, much faster growth of both ÆRhæ and ÆRgæ is predicted, indicating that a constant W is improper to represent the coalescing colloid. Predictions Assuming Fixed Charges. Using the developed model, we can see qualitatively how much the surfactant redistribution determines the aggregation behavior of the studied colloid. For this purpose we have repeated the simulation assuming that all initially adsorbed surfactant molecules remain DOI: 10.1021/la900963f
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Figure 13. Simulation of the aggregation kinetics reported in Figure 1, assuming constant W. The dashed line and open symbols correspond to ÆRgæ, whereas the solid line and filled symbols correspond to ÆRhæ.
Figure 14. Simulation of the aggregation kinetics reported in Figure 1 at CH2SO4 = 0.05 mol/L, assuming fixed surface charges. The dashed line and open symbols correspond to ÆRgæ, whereas the solid line and filled symbols correspond to ÆRhæ.
on the particle surface, although the total surface area reduces in the course of aggregation. This is equivalent to assuming that the mobile charges of the adsorbed surfactant behave like fixed charges, permanently anchored to the particle surface, and therefore the surface charge density of a cluster with mass i, is given by σs, i ¼ σ s, p i1=3
ð37Þ
where σs,p is the surface charge density of primary particles, determined by the measured initial stability of the aggregating system. It is clear that, in this case, the surface charge density of each cluster becomes dependent only on its mass and, consequently, so does the surface potential. The advantage of this situation is that the Wi,j value for each pair of clusters is independent of time or conversion. The usual linear approximation, σs,i = ε0εrκψs,i,3,45 is then used to compute ψs,i and Ψ, to obtain the Wi,j value through the stability model. Figure 14 shows the corresponding time evolutions of ÆRhæ and ÆRgæ for the system aggregating at CH2SO4 = 0.05 mol/L. At the aggregation onset, where aggregation occurs mainly between primary particles, the predictions with the fixed charge assumption is rather close to the experimental results, even though ÆRhæ is slightly overpredicted. At later times, this fixed charge approach diverges substantially from the experiments, and so do the two time evolution curves, too. It is therefore concluded that this fixed charge approach is improper to describe the present coalescing system, and redistribution of the surfactant does occur during aggregation. However, it is interesting to analyze the peculiar shape of the curves in Figure 14, showing that ÆRhæ and ÆRgæ grow until reaching a plateau in the range of 10 j τ j 3000 and then start to grow again. From the analysis of the computed CMD evolution, it can be found that the first growth reaching the plateau is mainly due to aggregation of primary particles (forming doublets), while, as a result of the 5 orders of magnitude difference in W (W2,2 ≈ 1 109 compared to W1,1 ≈ 2 104), significant aggregation of doublets to tetramers can be observed only after a substantially longer time. Actually, another plateau, corresponding to tetramer formation, occurs at a τ value about 5 orders of magnitude later than the one for the doublet formation. Note also that, because of the substantial difference in W between two primary particles and between a primary particle and a doublet, the formation of doublets from primary particles is preferred with respect to the formation of trimers from a primary particle and a doublet. Such stepwise aggregation might have potential applications, but it requires specific material and surface 9712 DOI: 10.1021/la900963f
Figure 15. Comparison between the cumulative CMDs evaluated from the cryo-SEM image analysis (open symbols) and those computed from the PBE model (filled symbols), at times t = 0 (0, 9), 5 (O, b), 30 (Δ, 2), 200 (sideways 4, sideways 2), 1350 min (], [) for the aggregation system with CH2SO4 = 0.05 mol/L. (The lines between the computed points serve only for guiding the eye.)
properties of the colloid, so as to realize the (fixed) surface charge condition. 4.3. Comparison between Predicted and Measured CMD. Although the ÆRhæ and ÆRgæ values computed by the PBE as a function of time are in good agreement with the corresponding experimental data, as shown in Figure 11, these quantities are only two moments of the CMD. In this work, some attempts have been made to have a direct comparison between the predicted and measured CMD. To this aim, we have determined the CMD from cryo-SEM images, as described in the experimental section, for the elastomer aggregation at CH2SO4 = 0.05 mol/L, at times t = 5, 30, 200, and 1350 min, respectively. For these reference cases, we have found convincing agreement between simulated and measured CMDs, as given in Figure 15. Some discrepancies at small cluster mass i result from the initial distribution of the primary particles, which always involves a certain level of polydispersity, while the simulation starts from monodisperse primary particles (instead of the real distribution). Nevertheless, the observed discrepancies are acceptable without leaving doubts about the validity of the employed aggregation model. Interestingly, the simulated and measured CMDs yield very similar number average mass Æiæ or ÆRgæ, as reported in Table 1.
5. Concluding Remarks Aggregation of soft, rubbery colloidal particles under reactionlimited conditions has been investigated in this study. To monitor Langmuir 2009, 25(17), 9703–9713
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Table 1. Average Cluster Mass Æiæ and Average Radius of Gyration ÆRgæ As Given by the Simulated CMD, the CMD of Cryo-SEM Images, and SLS for the Aggregation Experiment with CH2SO4 = 0.05 mol/L PBE solution
image analysis
SLS
t/min
Æiæ
ÆRgæ/nm
Æiæ
ÆRgæ/nm
ÆRgæ/nm
0 5 30 200 1350
1 1.24 1.56 1.96 2.29
46.5 54.3 59.1 62.8 65.2
1 1.26 1.54 2.03 2.41
49.7 53.1 60.2 62.8 68.2
46.6 53.1 56.8 60.5 64.2
the aggregation kinetics, the time evolutions of two quantities, the average gyration and hydrodynamic radii, ÆRgæ and ÆRhæ, have been determined by SLS and DLS techniques. The obtained results are rather different from those typical for rigid colloidal particle aggregation, reported in the literature. The kinetic data are then simulated using the CMD generated by solving the PBE, and it is found that they can be well reconstructed only when the following two processes are accounted for. First, fast coalescence occurs during aggregation, i.e., the time scale of particle fusion in this case is substantially smaller than that of aggregation at the given temperature, T = Tg + 45 C. Thus, all the clusters can be assumed to be spheres in the simulations. Their complete coalescence is supported by cryoSEM images of frozen samples. Second, since coalescence leads to reduction of the total particle surface area, redistribution of the surfactant in the course of aggregation has to be accounted for in the modeling. As a consequence, the colloidal stability is found to increase as the aggregation proceeds. Thus, at each time step, the Fuchs stability ratio of each pair of particles, Wi,j, has to be recomputed for the PBE solution. It is found that the generalized stability model, developed in our previous work,42 is the most suitable tool to handle the stability dynamics, since it accounts for the interplay among colloidal interactions, surfactant adsorption equilibria and counterion association equilibria. A further support to complete cluster coalescence and surfactant redistribution as aggregation proceeds is that the CMDs, computed using the presented modeling approach, are in good agreement with those measured by cryo-SEM image analysis of samples taken at a few selected aggregation times. Moreover, unlike the aggregation of rigid particles, for the studied coalescing system, the time evolution curves of ÆRgæ or ÆRhæ do not collapse to yield a master-curve when they
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are plotted as a function of the considered dimensionless time. Thus, because of complex interplay among particle aggregation, cluster coalescence, and surfactant redistribution, the RLCA process of soft, rubbery colloidal particles has been found to be nonuniversal. Acknowledgment. We appreciate courtesy of Roger Wepf from Electron Microscopy Center of ETH Zurich (EMEZ) in generating electron micrographs by the cryogenic route as well as conventional SEM by Frank Krumeich. Latex supply by Solvay Solexis and financial support by the Swiss National Science Foundation (Grant 200020-113805/1) are gratefully acknowledged. We also thank Marco Lattuada and Lyonel Ehrl for beneficial discussions.
Appendix The following coefficients Rn have been obtained by fitting eq 16 in the range 1 e ω e 10 to the results computed from reference 40: R1 ¼ -0:0331ω2 þ 0:6028ω þ 5:5602
ðA 1Þ
R2 ¼ 0:0103ω2 þ 0:0414ω -20:089
ðA 2Þ
R3 ¼ -2ð2R1 þ R2 Þ
ðA 3Þ
R4 ¼ R1
ðA 4Þ
R5 ¼ 0:041ω2 -0:9289ω -10:088
ðA 5Þ
Supporting Information Available: To support the high reproducibility of the experiments, SI-Figure 1 reports the ÆRgæ and ÆRhæ values with error bars for the aggregation system with 0.045 mol/L H2SO4, which are obtained by repeating the same aggregation three times. The mean variation coefficient (or relative standard deviation) is around 1.5% over all the measured sizes. This material is available free of charge via the Internet at http://pubs. acs.org.
DOI: 10.1021/la900963f
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