Structural and stability characteristics of agglomerated clusters

We investigated the cohesion of agglomerates formed by sticking two fractal clusters, each cluster having been previously generated by particle aggregation on a 3D lattice. The degree of cohesion of an agglomerate of a given configuration was defined by the number of connections simultaneously established on the two stuck clusters. All the possible nonoverlapping configurations were investigated and the corresponding porosity and brittleness as well as the pore volume and connection frequencies were determined. The numerical study showed the greater internal cohesion of agglomerates issued from sticking of reaction-limited aggregation (RLA) clusters compared to that of diffusion-limited-aggregation (DLA) clusters. DLA and RLA agglomerates presented continuously decreasing pore volume frequency curves, the latter agglomerates being characterised by a greater frequency of large pores. Comparison with typical controlled fragmentation experiments showed the number of connections to be the prevailing factor in the cohesion of aggregates formed in aqueous suspensions under various conditions. Received: 25 January 2001 Accepted: 16 May 2001     

Computer simulation of selective aggregation in binary colloids

The morphology of clusters formed by selective aggregation of binary colloids is studied in a two-dimensional Monte Carlo simulation for a large range of number fractions (200:1, 100:1, 10:1, 2:1). We find remarkable similarity in morphology to those observed in experiments, from the formation of closed "micelles" to large branched clusters. Quantitative studies of the fractal dimension, kinetics, and cluster size distribution are also carried out and compared with diffusion-limited cluster aggregation and reaction-limited cluster aggregation models.     

Destabilization of polystyrene latex particles induced by adsorption of polyvinylamine: mass, size and structure characteristics of the growing aggregates

The obviously visible aggregation of suspended colloidal particles resulting from the addition of polyvinylamine to the aqueous dispersion of polystyrene latex particles bearing surface sulfate groups set in with a delay of 24 h. The aggregation mechanisms and the fractal dimension of the aggregates were derived from the variations with time of the weight and number averaged masses of the aggregates as well as of the weight averaged harmonic mean diameter of the size distribution. Since the establishment of starved layers was determined to be relatively fast and to leave the liquid phase free of polymer, the delay for the obvious destabilization was attributed to the reconformation of adsorbed macromolecules that was expected to be extremely slow. This reconformation promoted the emergence of the diffusion-limited aggregation process that accompanies the permanent reaction-limited aggregation process. The fractal dimension of the latex particles/polyvinylamine aggregates was determined to be 2.12.     

Detailed model of the aggregation event between two fractal clusters

A model has been developed for describing the aggregation process of two fractal clusters under quiescent conditions. The model uses the approach originally proposed by Smoluchowski for the diffusion-limited aggregation of two spherical particles but accounts for the possibility of interpenetration between the fractal clusters. It is assumed that when a cluster diffuses toward a reference cluster their center-to-center distance can be smaller than the sum of their radii, and their aggregation process is modeled using a diffusion-reaction equation. The reactivity of the clusters is assumed to depend on the reactivity and number of their particles involved in the aggregation event. The model can be applied to evaluate the aggregation rate constant as a function of the prevailing operating conditions by simply changing the value of the particle stability ratio, without any a priori specification of a diffusion-limited cluster aggregation, reaction-limited cluster aggregation, or transition regime. Furthermore, the model allows one to estimate the structure properties of the formed cluster after the aggregation, based on the computed distance between the aggregating clusters in the final cluster.     

Orthokinetic Aggregation in Two Dimensions of Monodisperse and Bidisperse Colloidal Systems

Orthokinetic aggregation of colloids trapped at the air–liquid interface was studied by direct imaging in a couette cell. This method allowed us to follow the temporal evolution of both the cluster-mass distribution and the cluster structure at a shear rate where Brownian aggregation is suppressed. The interactions between the monodisperse latex particles floating at the air–liquid interface were controlled either by varying the electrolyte concentration or by creating a bidisperse system through the addition of small particles. The results show that the clusters in all of the systems are characterized by a high fractal dimension, indicating that the clusters are rearranged and densified by the shear. Kinetic analysis suggests that aggregation of monodisperse systems mainly proceeds through homogeneous aggregation, i.e., large clusters sticking to other large clusters. The bidisperse system, finally, with a size ratio around 10, favored a more heterogeneous aggregation among small and large clusters throughout the aggregation process; a slightly lower fractal dimension was observed compared to the strongly aggregated monodisperse system.     

Relation between aggregation kinetics and the structure of kaolinite aggregates

Dynamic and static light scattering were applied to the determination of the stability ratio and fractal dimension of kaolinite (KGa-2) at different kaolinite or/and electrolyte concentrations at pH 9.5. Dynamic light scattering was used to measure the kinetics of early stage aggregation to determine the stability ratio, W, as well as the cluster sizes which determine the fractal regime. Static light scattering was used to measure the fractal dimension, D(f). Results show that the two classes of "universality" (Lin et al. Nature 1989, 339, 360) characterizing the diffusion- and reaction-limited regimes of cluster-cluster aggregation do apply to colloidal kaolinite as limit cases when W approximately 1 or W > 100, respectively. In the intermediate regime where 5 < W < 100, the growth of the aggregate radius showed a power-law behavior similar to diffusion-limited cluster aggregation. For the intermediate aggregation regime, a scaling relation between fractal dimension and stability ratio, reflecting a continuous increase in particle packing density in the aggregate as the sticking probability of particles was reduced, was demonstrated.     

Coagulation of antibody-sensitized latexes in the presence of antigen

The paper describes a study of the kinetics and mechanism of the coagulation of two types of immunoassays using sensitized latexes. The positive response to the first test is based on the aggregation of the gamma globulin (IgG)-coated polystyrene latexes in the presence of IgM rheumatoid antigen. The second test is relative to the heteroaggregation of two types of sensitized latexes induced by the presence of human chorionic gonadotropin (HCG). In the latter test, two identical polystyrene latexes bearing carboxylic acid surface groups were sensitized by covalent coupling of monoclonal antibodies specific for the αHCG determinant on one type of latex and for the βHCG determinant on the other type. Using the Coulter Counter method, the aggregate size distribution c(n) was determined as a function of the number n of elementary constituents, thus enabling calculation of the number N(t) and weight S(t) average sizes of the aggregates. The temporal variations of the average sizes were compared with typical situtions of reaction-limited aggregation processes in order to characterize the mechanism of aggregation induced by antibody–antigen reactions.     

Irreversible versus reversible aggregation: mean field theory and experiments

Colloidal aggregation processes arising at different electrolyte concentrations were studied by means of experiments and confronted with theoretical predictions of different kinetic aggregation models. For this purpose, aqueous dispersions of relatively large polystyrene microspheres were chosen as experimental systems. Aggregation was induced by adding KBr electrolyte to the initially stable particle dispersions. During the aggregation processes, the cluster-size distribution was monitored by means of single cluster light scattering. Analyzing the time evolution of the monomer concentration, we found that the processes arising even at moderate electrolyte concentrations cannot be described by pure time-independent irreversible aggregation models. Hence, alternative models such as time-dependent irreversible aggregation and several reversible aggregation models were also tested. The model that considers a time-dependent sticking probability was found to fit the data quite satisfactorily. Nevertheless, the fitted was so slow that it seems not very likely to find such a behavior in real systems. The aggregation-fragmentation models reported in the literature were unable to reproduce the experimental observations. Hence, a more realistic reversible aggregation model was developed. This model accounts also for reenforced or double bonds between the constituent particles. The corresponding fit improved significantly and reached the same quality as the time-dependent model. Moreover, the obtained fitting parameters were in qualitative agreement with the DLVO predictions and so, reversible aggregation seems to be a more reasonable explanation for the experimental data than time-dependent irreversible aggregation. However, no definite statement on the possible secondary bond fragmentation mechanism may be made since both the applied shear stress in the measuring cell and thermal fluctuations can cause weaker bonds to break.     

Population balance modeling of aggregation and breakage in turbulent Taylor-Couette flow

An experimental and computational study of aggregation and breakage processes for fully destabilized polystyrene latex particles under turbulent-flow conditions in a Taylor-Couette apparatus is presented. To monitor the aggregation and breakage processes, an in situ optical imaging technique was used. Consequently, a computational study using a population balance model was carried out to test the various parameters in the aggregation and breakage models. Very good agreement was found between the time evolution of the cluster size distribution (CSD) calculated with the model and that obtained from experiment. In order to correctly model the left-hand side of the CSD (small clusters), it was necessary to use a highly unsymmetric fragment-distribution function for breakage. As another test of the model, measurements with different solid volume fractions were performed. Within the range of the solid volume fractions considered here, the steady-state CSD was not significantly affected. In order to correctly capture the right-hand side of the CSD (large aggregates) at the higher solid volume fraction, a modified aggregation rate prefactor was used in the population balance model.     

Hydrodynamic radius of fractal clusters

The Kirkwood-Riseman theory has been used to derive an analytical formula for the evaluation of the hydrodynamic radii of fractal clusters. The proposed relation is based on knowledge of the particle-particle correlation function and can be applied to clusters containing any number of particles larger than 4. The calculated values of the hydrodynamic radius are compared with the results of other theoretical approaches proposed in the literature, as well as several experimental data. Finally, the developed relation has been used in connection with a population balance equation model that computes the clusters' mass distribution to estimate the average hydrodynamic radius for a population of fractal clusters of different sizes. The obtained results have been compared to suitable experimental data for a silica colloidal suspension aggregating under reaction-limited conditions.     

Effect of polymer adsorption on PEO-coated latex particles during salt-induced aggregation

The salt-induced aggregation of polystyrene particles in dilute aqueous solutions has been studied by means of dynamic light scattering measurements and the hydrodynamic radius of the resulting aggregates has been evaluated during the time evolution of the whole process. Poly(ethylene oxide) (PEO) polymer adsorbed on the particle surface at different amounts has been used to modify the inter-particle interactions resulting in the formation of clusters of increasing size or in the stabilization of the suspension, depending on the polymer molecular weight. The aggregation regime, i.e. a diffusion limited cluster aggregation (DLCA) occurring in the polymer-free latex suspension, is partially modified according to the polymer percentage adsorbed on the particle surface. At high polymer content, the polystyrene latex undergoes a complete steric stabilization. The deviation from a DLCA regime has been observed for different polymer contents and for polymers of different molecular weights, from 1.5 to 2000 kD. The alterations of the aggregation rates, induced by the polymer interactions, are presented and briefly discussed.     

Further insights into the universality of colloidal aggregation

Dynamic light scattering (DLS) performed at various scattering wave vectors provides detailed information about the aggregation kinetics and the cluster mass distribution (CMD) in colloidal dispersions. Detailed modeling of the aggregation kinetics with population balance equations requires a quantitative connection between the CMD and measurable quantities such as the angle dependent hydrodynamic radii obtained by DLS. For this purpose we evaluate and compare various models for the structure factor of fractal aggregates. Additionally, we introduce a simple scattering model that accounts for the contribution of internal cluster dynamics of fractal clusters to the first cumulant of the dynamic structure factor. We show that this contribution allows to quantitatively describe previously measured experimental data on the scattering wave vector dependence of the hydrodynamic radius in diffusion limited cluster-cluster aggregation (DLCA), which was shown to exhibit some kind of universality behavior (master curve). Using the same scattering model, we analyze a similar set of experimental data but in reaction limited cluster-cluster aggregation (RLCA). We find that in this case the crossover from RLCA to DLCA and gravitational settling both have a significant influence on the CMD and consequently on the scattering wave vector dependent DLS data. Only when accounting for both these effects they temporarily compensate each other and a satisfactory representation of the aggregation master curve is possible for the RLCA data at longer times. Indeed, we find that either crossover from RLCA to DLCA or gravitational settling, when present individually, causes the loss of a master curve for aggregation.     

CFD simulation of shear-induced aggregation and breakage in turbulent Taylor-Couette flow

An experimental and computational investigation of the effects of local fluid shear rate on the aggregation and breakage of approximately 10 microm latex spheres suspended in an aqueous solution undergoing turbulent Taylor-Couette flow was carried out. First, computational fluid dynamics (CFD) simulations were performed and the flow field predictions were validated with data from particle image velocimetry experiments. Subsequently, the quadrature method of moments (QMOM) was implemented into the CFD code to obtain predictions for mean particle size that account for the effects of local shear rate on the aggregation and breakage. These predictions were then compared with experimental data for latex sphere aggregates (using an in situ optical imaging method). Excellent agreement between the CFD-QMOM and experimental results was observed for two Reynolds numbers in the turbulent-flow regime.     

CFD simulation of aggregation and breakage processes in laminar Taylor-Couette flow

An experimental and computational investigation of the effects of local fluid shear rate on the aggregation and breakage of approximately 10 microm latex spheres suspended in an aqueous solution undergoing laminar Taylor-Couette flow was carried out according to the following program. First, computational fluid dynamics (CFD) simulations were performed and the flow field predictions were validated with data from particle image velocimetry experiments. Subsequently, the quadrature method of moments (QMOM) was implemented into the CFD code to obtain predictions for mean particle size that account for the effects of local shear rate on the aggregation and breakage. These predictions were then compared with experimental data for latex sphere aggregates (using an in situ optical imaging method) and with predictions using spatial average shear rates. The mean particle size evolution predicted by CFD and QMOM using appropriate kinetic expressions that incorporate information concerning the particle morphology (fractal dimension) and the local fluid viscous effects on aggregation collision efficiency match well with the experimental data.     

胶粒分形粒子簇反应控制聚集动态行为位垒影响的Monte Carlo模拟

通过对反应控制聚集过程的MonteCarlo模拟,从微观及介观层次上探讨了胶粒间相互作用位能曲线上位垒高度的变化对胶粒分形粒子簇大小分布和动态标度函数及聚集动力学行为的影响规律。     

Control over colloidal aggregation in monolayers of latex particles at the oil-water interface

The controlled generation of 2D aggregate networks is studied experimentally using micrometer-sized polystyrene latex particles attached to the oil-water interface. Starting from an initially crystalline monolayer, appropriate combinations of carefully added electrolyte and surfactant enable control over both the fractal dimension and the kinetics of aggregation. Remarkably, the colloidal crystals formed upon first spreading remain stable, even for days, when substantial amounts of electrolyte are added to the aqueous phase. Pressure-area isotherms reveal a slow time evolution of the electrostatic dipole-dipole interaction. When the electrostatic interaction has been sufficiently weakened, aggregation proceeds in well-defined, reproducible manner. The aggregation process is analyzed using quantitative video microscopy. The evolution of the cluster size distribution and its moments is characterized, and static and dynamic scaling exponents are derived to identify the nature of the aggregation process. In the range of concentrations studied here, the kinetics all agree with a "fast", diffusion-limited cluster type of aggregation. However, the fractal dimension strongly depends on the composition of the aqueous subphase. Rather dense structures are found when only electrolyte is used, whereas more open structures are obtained when even small amounts of surfactant are added. It is suggested that this structural dependency is related to the effect of surfactant on the contact angle and its consequences for the anisotropic nature of the capillary interactions.     

Kinetics of cluster growth by aggregation

The classical problem of the coalescence of isolated species to produce growing clusters/colloids/polymers by successive statistical encounters having the same rate constant, is revisited using numerical simulation for a maximum nuclearity value of a few 10³ units. The evolution with time of the abundance of clusters of a given nuclearity and of the total population, and the distribution of sizes at a given time are obtained and compared with models from the literature. A remarkable feature of these curves is that they exhibit parity effects for the nuclearity, even clusters being systematically more abundant than odd ones. For easier comparison with experiments, some simulated curves are presented in the form of an approximated analytical expression: kinetics of the total population, and of the monomer, dimer and higher oligomers populations, amplitudes at the maximum and delay for the maximum as functions of the nuclearity, size distribution at a given time. The validity of the approximations is discussed.     

A scaling theory for number-flux distributions generated during steady-state coagulation and settling and application to particles in Lake Zurich,Switzerland

In this study, we extend the established scaling theory for cluster size distributions generated during unsteady coagulation to number-flux distributions that arise during steady-state coagulation and settling in an unmixed water mass. The scaling theory predicts self-similar number-flux distributions and power-law decay of total number flux with depth. The shape of the number-flux distributions and the power-law exponent describing the decay of the total number flux are shown to depend on the homogeneity and small i/j limit of the coagulation kernel and the exponent kappa, which describes the variation in settling velocity with cluster volume. Particle field measurements from Lake Zurich, collected by U. Weilenmann and co-workers (Limnol. Oceanogr.34, 1 (1989)), are used to illustrate how the scaling predictions can be applied to a natural system. This effort indicates that within the mid-depth region of Lake Zurich, clusters of the same size preferentially interact and large clusters react with one another more quickly than small ones, indicative of clusters coagulating in a reaction-limited regime.     

A refined algorithm to simulate latex colloid agglomeration at high ionic strength

This study is focussed on the simulation of particle agglomeration at relatively high ionic strength using a refined stochastic algorithm developed in the context of parcel-tracking approaches. For that purpose, experimental data of both diffusion-limited and reaction-limited aggregation of latex particles were obtained using dynamic light scattering techniques for different initial particle sizes (diameters ranging from 24 to 495 nm) and at various chemical conditions (ionic strength between 0.5 and 2 M with NaCl or CaCl\(_2\) solutions). The experimental data collected have been compared to numerical results obtained with the refined parcel-tracking algorithm for particle agglomeration which has been developed. Results show that the evolution of the aggregate diameters over time can be properly captured by the present model with the value of the aggregate fractal dimension that is extracted from experimental data.