J. Phys. Chem. B 2004, 108, 20105-20121
20105
Kinetics of Aggregation and Gel Formation in Concentrated Polystyrene Colloids Peter Sandku1 hler, Jan Sefcik, and Massimo Morbidelli* Swiss Federal Institute of Technology Zurich, Institut fu¨r Chemie- und Bioingenieurswissenschaften, ETH-Ho¨nggerberg/HCI, CH-8093 Zurich, Switzerland ReceiVed: August 6, 2004; In Final Form: September 29, 2004
We measure with dynamic and static light scattering the radius of gyration, the hydrodynamic radius at various angles, and the average structure factor and determine the kinetic behavior for aggregation processes of a polystyrene colloid at moderately concentrated solid volume fractions φ0 e 0.1 and various salt concentrations. In addition, we measure the gelation kinetics in these colloidal dispersions at volume factions φ0 g 0.05 using on-line oscillatory shear measurements. By modifying the surface of the sodium dodecyl sulfate stabilized polystyrene particles with Triton X-100, we show that the changed interaction potential is responsible for a clear change in the aggregation behavior and kinetics as well as in the gelation kinetics and gel properties. We demonstrate that the measured aggregation kinetics can be accurately described by population balance equations, i.e., the Smoluchowski aggregation equation, at all solid volume fractions investigated. The model for the aggregate structure allows us to distinguish between different fractal dimensions through comparison to the measured average structure factor. It is found that the structure of the aggregates is best described by a small fractal dimension of 1.8 and a power-law growth of the average radii. This is surprising for the relatively slow reaction-limited cluster aggregation regime with stability ratios of W ∼ 1 × 106. A kinetic scaling in the aggregation behavior, independent of the salt concentration and the volume fraction, is found through a dimensionless time derived from the rate equations. Remarkably, the time evolution of the gel elastic modulus could also be scaled by the dimensionless time derived from the aggregation model. This finding suggests that the kinetics of gel formation still can be described as a second-order rate process, like aggregation, even for very concentrated systems close to the gel point.
1. Introduction and Motivation Aggregation of colloidal particles, the corresponding kinetics, and the resulting aggregate structures have been the topic of intensive research over the last 15 years leading to an extensive literature (cf. refs 1-23). These works deal mostly with the kinetics of aggregation in very dilute systems. In these conditions, multiple scattering is negligible and light scattering techniques can be used to monitor in situ the time evolution of the system and the obtainable measurables can be quantitatively interpreted. Convenient mathematical tools, like the Smoluchowski aggregation equation, can be used to model the aggregation kinetics at sufficiently low volume fractions. Although denser systems are of great practical and industrial interest,24-29 only a limited number of investigations on the aggregation kinetics at higher solid volume fractions have been reported.30,31 The so far limited progress in the kinetic modeling of aggregation and gelation can be attributed to the fact that in these conditions both the experimental and the modeling analysis become more demanding.32-36 In addition, gel formation often occurs in more concentrated systems,37-50 which adds further complexity to the aggregation process itself. In recent work,51 experiments employing fiber-optics based dynamic light scattering in the backscattering geometry together with linearviscoelastic oscillatory rheology on slowly aggregating and gelling acrylic latexes at concentrated solid volume fractions of 17% and 23% have been reported. Through measurements of various power-law exponents, the analogy of colloidal * To whom correspondence should be addressed.
gelation to polymer gelation has been tested and some consistent features have been found. However, there is a lack of quantitative studies in order to understand when (in time) this liquid-solid transition in colloidal systems occurs and how the aggregation kinetics preceding the gel formation can be described. Several studies investigated gel transitions on a purely mathematical basis, the so-called mathematical gelation, in the frame of the Smoluchowski aggregation equation (cf. refs 52 and 53). However, the aggregation models typically required to produce such mathematical gels are in general not applicable to colloidal dispersions undergoing aggregation and gelation, as can be seen by comparison with appropriate experimental data.54 It is thus evident that quantitative and predictive kinetic models of aggregation and gelation would be very useful for many practical and industrial applications to produce aggregates and gels of desired size and structure.35,55-58 In our recent work, we have investigated the problem of modeling the kinetics of aggregation in nondilute dispersions, where aggregation might result in gel formation.54,59-61 In this work, we report and analyze new aggregation and gelation experiments in order to rationalize and quantify the kinetics of slow aggregation and gelation at higher solid volume fractions up to φ0 e 0.1, using a typical polystyrene colloid. In previous work,54 we have investigated the same colloid in more diluted conditions in the DLCA regime, where a behavior typical for DLCA was observed. Here, light scattering and oscillatory shear measurements are used to characterize the aggregation progress. In particular, several averages of the aggregate mass distribution
10.1021/jp046468w CCC: $27.50 © 2004 American Chemical Society Published on Web 11/26/2004
20106 J. Phys. Chem. B, Vol. 108, No. 52, 2004
Sandku¨hler et al.
TABLE 1: Expressions for the Structure Factor, Si(Q), the Radius of Gyration, Rg,i/Rp, and the Hydrodynamic Radius, Rh,i/Rp, of Primary Particles, Dimers, Trimers, and Tetramersa
aThe
values of sij ) rij/(2Rp) are given in Table 2.
TABLE 2: Values of the Average Normalized Interparticle Distances sij ) rij/(2Rp) for Tetramers, in the Case of df ) 2.05 and df ) 1.8563 df ) 1.85
df ) 2.05
1 1 1 1.3942 1.6798 2.0614
1 1 1 1.3756 1.6625 2.0347
s12 s13 s14 s23 s24 s34
TABLE 3: Values of the Parameters d, e, f, and m to Be Used in Equation 3, to Compute the Parameters of the Particle-Particle Correlation Function: Nnn i , ai, bi, and γi, as a Function of the Number of Particles Per Cluster for Both df ) 1.85 and df ) 2.0563 Fi Ninn ai bia γi
have been measured as a function of time in order to gain more detailed information on its shape and width. The analysis of the aggregation process is conducted using detailed population balance equation and light scattering models. 2. Theoretical Background 2.1. Dimensionless Structure and Properties of Single Aggregates. The fractal scaling concept can often be used to characterize the structure of clusters created through aggregation in quiescent suspensions. Therein, the dimensionless radius of gyration, Fg, i, of a fractal cluster is related to its dimensionless mass i through the fractal dimension df
i ) kf
( ) Rg,i Rp
df
≡ kf(Fg,i)df
(1)
where Rp is the primary particle radius, Rg,i the radius of gyration of an aggregate of mass i, and kf the fractal prefactor. However, the aggregates produced in concentrated systems are relatively small and contain less than 100 primary particles. In this case, the applicability of eq 1 and of the general concept of fractal geometry can be questioned. For example, it has been shown in the context of colloidal gels62 obtained at φ0 > 0.01 that the nonfractal part, due to the smaller aggregates, influences significantly the apparent fractal dimension estimated from the average structure factor 〈S(q)〉 measured by static light scattering. Since in this work we use light scattering measurements to characterize the mass distribution of the aggregate as well as their structure, we need to properly account for the deviations from the fractal behavior exhibited by the smaller clusters. Explicit dimensionless expressions for the individual cluster structure factor, the radius of gyration (Fg,i ) Rg,i/Rp) and the hydrodynamic radius (Fh,i ) Rh,i/Rp) for aggregates composed of less than five particles, are summarized in Tables 1 and 2.59 To describe the structure of aggregates containing five or more primary particles in more detail, the particle-particle correlation function g(r) can be used. A semiempirical model for g(r) has recently been developed using Monte Carlo
aggregation mechanism
d
e
f
m
df ) 1.85 df ) 2.05 df ) 1.85 df ) 2.05 df ) 1.85 df ) 2.05 df ) 1.85 df ) 2.05
2.0342 2.0415 0.0095 0.0138 0.6425 0.4857 2.1976 2.16
1.1477 1.1511 4.1292 2.7544 6.2352 9.6836 3.8377 0.1966
0.9997 1.0086 0.1997 4.1792 5.1747 11.6665 -0.1784 -3.5926
1 1 2 2 1 1 1 2
a b ) 0 for clusters with d ) 1.85 and i < 7 or with d ) 2.05 and i f f i < 10.
[
simulations for DLCA and RLCA aggregates of various aggregate sizes.63-66 Its dimensionless form is given by
0 if s < 2 Nnn i if s ) 2 (Rp)2δ(s-2) 16π 3 Fg,i(s,df)≡(Rp g(r))) aisbi if2
