Cluster-cluster Aggregation Model Research Articles

Consolidation of collodal suspensions

A key step in the processing of ceramics is the consolidation of powders into engineered shapes. Colloidal processing uses solvents (usually water) and dispersants to break up powder agglomerates in suspension and thereby reduce the pore size in a consolidatd compact. However, agglomeration and particle rearrangement leading to pore enlargement can still occur during drying. Therefore, it is beneficial to consolidate the compact asdensely as possible during the suspension stage. The consolidation techniques of pressure filtration and centrifugation were studied and the results are reported in this paper. In particular, the steady-state pressure-density relationship was studied, and information was obtained regarding the consolidation process, the microstructure, and the average density profile of consolidated cakes. We found that the compaction processes in these two consolidation methods are quite different. In general, a consolidated cake is a particulate network made up of many structural units which are fractal objects formed during aggregation in the suspension. In pressure filtration, compaction is a process of breaking up the fractal structural units in the particulate network by applied pressure; the resulting partie rearrangement is produced by overcoming energetic barriers which are related to tee packing density of the compact. Recently, we performed Monte Carlo simulations on a cluster-cluster aggregation model with restructuring,1 and found the exponential relationship between pressure and density is indeed the result of the breaking up of the fractal structural units.2 On the other hand, in centrifugation, compaction involves the rearrangement of the fractal structural units without breaking them so that the self-similar nature of the aggregates is preserved. Furthermore, we calculated density profiles fom the bottom to the top of the cakes by solving the local static force balance equation n the continuum particulate network. In pressure filtration of alumina and boehmite, the cakes are predicted have uniform density. The results of γ-ray densitometry3 on a pressure-filtrated alumina cake confirmed this prediction. In contrast, in centrifugation, the density profiles are predicted to show significant variation for cakes on the order of one centimeter high. Moreover, the pressure-filtered boehmite cakes showed no cracking during drying. This indicated that pressure filtration is a good consolidation technique for nanometer-sized particles such as boehmite. The improved drying property is probably a result of a minimal shrinkage due to the formation of higher packing densities during filtration.

The growth of compact layers at the electrode interface: Part IV. Fractal character of uracil films at the mercury-aqueous electrolyte interface

Compact layers of uracil films grown on the mercury electrode-aqueous solution interface were studied by means of time-resolved FFT impedance spectroscopy. The films are characterized by the fractal dimension which evolves with time. The results are discussed in terms of the cluster-cluster aggregation model and the hole formation mechanism proceeding in the compact layer.

Fractal structure of gas evaporated metal aggregates

We analyze electron micrographs which show projections of three-dimensional fractal aggregates. Shadowing and finite size effects in the projection are treated by calculations for two model systems, namely a single and a connected aggregate. The calculations allow us to obtain the fractal dimension of the aggregates from the measured effective dimensions on the micrographs. Aggregates of metal particles were produced by gas evaporation and studied by electron microscopy. We measured the effective dimensions of the specimens and obtained the fractal dimensions from the connected aggregate model. The results are in good agreement with cluster-cluster aggregation models

Projection effects in electron micrographs of three-dimensional fractal aggregates: theory and application to gas-evaporated specimens

The authors analyse electron micrographs which show projections of three-dimensional fractal aggregates. Shadowing and finite size effects in the projection are treated by calculations for two model systems, namely a single aggregate and a connected aggregate. The calculations allow the authors to obtain the fractal dimension of the aggregates from the measured effective dimensions on the micrographs. Aggregates of metal particles were produced by gas evaporation and studied by electron microscopy. The authors analysed the structure of the specimens by the connected-aggregate model. Fractal dimensions in the range 1.9 to 2.0 were found. This is in good agreement with the cluster-cluster aggregation model if the aggregating particles follow linear trajectories.

Simple models for the restructuring of three-dimensional ballistic aggregates

The effects of some simple restructuring processes in which pairs of rigid clusters can readjust their positions with respect to each other after collision have been investigated using the three-dimensional hierarchical cluster-cluster aggregation model. In these models we assume that contacts between pairs of particles once formed cannot be broken but may move on the surface of particles in one cluster which contact particles in a different cluster immediately after contact has been made. Such structural readjustment processes increase the fractal dimensionality from a value of about 1.89 without readjustment by an amount which depends on model details. The highest fractal dimensionality obtained in the models investigated here was about 2.13. For ballistic aggregation with a zero impact parameter the fractal dimensionality can be raised from 2.04 to 2.21.

Models for Colloidal Aggregation

Computer simulations have been used for more than a quarter of a century to develop a better understanding of non equilibrium growth and aggrega­ tion processes. Important early examples include VoId's ballistic deposi­ tion model for aggregation and sedimentation (1, 2), Sutherland's ballistic cluster-cluster aggregation model for floc formation (3), and Eden's surface growth model for the generation of cell colonies (4). Because of limited computer resources these early simulations were carried out on a small scale using simple models. Since these early pioneering efforts computer speed, storage capabilities, and availability have increased enormously and the task of writing computer programs has become much easier and more reliable. These developments have allowed two almost divergent research directions to develop. On the one hand, models have been developed that incorporate as much as possible of what is known about the physics and chemistry of aggregation processes, and on the other hand, a variety of very simple models have been introduced that allow us to generate the very large structures needed to investigate asymptotic (large size limit) geometric scaling relationships. The exploration of very simple aggregation models has been stimulated by the realization (5-10) that real aggregation processes and simple aggregation models frequently lead to structures that can be described in terms of the concepts of fractal geometry (11). In particular, the recent interest in nonequilibrium growth and aggregation models was initiated by the discovery by Witten & Sander (6) that a simple diffusion-limited aggregation (DLA) model in which particles are added, one at a time, to a growing cluster or aggregate of particles leads to

Fluctuations and distributions in random aggregates.

We have measured the distribution of mass ${P}_{s}$(r) within a distance r measured from occupied sites in the screened-growth model (${d}_{f}$=1.25, 1.5, and 1.75, where ${d}_{f}$ is the fractal dimension), off-lattice diffusion-limited cluster-cluster aggregates (${d}_{f}$=1.43), and off-lattice diffusion-limited aggregation (DLA) clusters. Here ${P}_{s}$(r) is the probability that a circle of radius r centered on an occupied site will contain S occupied sites or particles. For the screened-growth and cluster-cluster aggregation models, the distributions ${P}_{s}$(r) can be described in terms of the scaling ). The dependence of the moments 〈${S}^{n}$〉/〈S${〉}^{n}$ on the distance r has also been measured. For the screened-growth and cluster-cluster aggregation models these quantities are essentially independent of r for n\ensuremath{\ge}1. For DLA the simple scaling form for ${P}_{s}$(r) does not describe the mass distribution and (1/n)ln(〈${S}^{n}$〉/〈S${〉}^{n}$), with n=2--10, has a linear dependence on ln(r) with negative slopes which are different for different values of n for small values of r. Since 〈${S}^{n}$〉/〈S${〉}^{n}$ has a lower bound of 1, this cannot be the true asymptotic behavior for off-lattice DLA. It suggests that DLA has a more complex structure than either the screened-growth model or cluster-cluster aggregation.

Cluster-cluster aggregation with dipolar interactions

The hierarchical cluster-cluster aggregation model is extended in the presence of dipolar interactions between the magnetic dipoles attached to each individual particle. It is found that the fractal dimension of the resulting clusters decreases when the intensity of the momenta increases, as observed in recent experiments.

Computer simulations of cluster–cluster aggregation

We briefly introduce a simple extension of the diffusion-limited aggregation model of Witten and Sander, the cluster-cluster aggregation model, in which all clusters diffuse and grow by sticking at their first contact. This model is particularly well adapted to describe aggregation experiments on colloids or aerosols. Three recent extensions of the model are emphasized which take into account possible restructuring, cluster polarizability or indirect aggregation by polymers. In these three cases the computer simulations are compared with experiment.

Cluster-cluster aggregation in two-monomer systems

The authors have investigated the structure and growth kinetics of clusters formed in dilute solutions from two kinds of polyfunctional monomers. This was done by computer simulations within the framework of the cluster-cluster aggregation model with mass-independent cluster diffusion constants. The systems considered were of the type A2+Bf and Af+Bf, where A2, Af and Bf are different monomers. The monomer functionality f is equal to the coordination number of the lattice used in the simulations. The results of simulations obtained at different relative monomer concentrations are described and several novel effects are reported. Large clusters always have a fractal-like geometry but their effective fractal dimensionality, D, appears to depend on the composition of the mixture. Only in a restricted composition range does the aggregation process lead to a single large cluster. There, the cluster size distribution function Ns(t) can be described by the scaling form s-2f(s/tz). Outside this range, aggregation results in small clusters (oligomers) whose structure is governed by 'geometric selection rules'. Only certain non-consecutive oligomer weights are observed. The two boundary points of the interval in which kinetic scaling holds are shown to possess the properties of tricritical points.

The effects of reorganization processes on two-dimensional cluster-cluster aggregation

The effects of two simple reorganization mechanisms in which parts of two-dimensional clusters, formed by the cluster-cluster aggregation model, are allowed to rotate about their contact points until additional contact is made and loops are formed have been investigated. Both the ordinary fractal dimensionality, D, and the intrinsic dimensionality, D l , which measures the mass contained within a minimum path length l measured on the fractal clusters were determined. Both D and D l are changed very little, if at all, by these reorganization processes. However, there is some evidence that the effective values of these dimensionalities are increased slightly on large length scales. In these models loops are considered to be stable once formed and the final state consists of a system of interconnected loops.

Fractal geometry of vapor-phase aggregates.

Static and dynamic light scattering and small-angle x-ray scattering measurements are reported for vapor-phase aggregates of silica. From the static scattering data the fractal dimension is found to be approx.1.84 +- 0.08, in agreement with the prediction of the cluster-cluster aggregation model. The Rayleigh linewidth is found to vary as the 2.7 power of the momuntum transfer for large momentum transfer.

Scaling in steady-state cluster-cluster aggregation.

The diffusion-limited cluster-cluster aggregation model is investigated under conditions which for long times lead to steady-state coagulation. Single particles are added to the system at a constant rate and the larger clusters appearing as a result of the aggregation process are removed according to various rules. Our results show that the dependence of the number of clusters, N(t), on the feed rate \ensuremath{\kappa} in a unit volume and the time t can be well represented by a scaling form N(t)\ensuremath{\sim}${\ensuremath{\kappa}}^{\ensuremath{\alpha}}$f(${\ensuremath{\kappa}}^{\ensuremath{\beta}}$t) with a scaling function f(x)\ensuremath{\sim}x for x\ensuremath{\ll}1 and f(x)=1 for x\ensuremath{\gg}1. The exponents \ensuremath{\alpha} and \ensuremath{\beta} are found to depend on the spatial dimension d, of the system, but within the statistical errors they always satisfy the relation \ensuremath{\alpha}+\ensuremath{\beta}=1 in accordance with the prediction of a generalized rate equation discussed by R\'acz (see the companion paper). The values we have obtained for \ensuremath{\alpha} and \ensuremath{\beta} are consistent in two and three dimensions with the corresponding results of the Smoluchowski equation approach but inconsistent in one dimension. This can be considered as an indication of the fact that the upper critical dimension for the kinetics of the diffusion-limited cluster-cluster aggregation model is 2.

Clustering of clusters processes above their upper critical dimensionalities

In contrast with the diffusion limited particle-cluster aggregation (P-Cl), the cluster-cluster aggregation model (Cl-Cl) has a finite upper critical dimensionality above which the fractal dimension of the clusters is trivial. This exponent is derived analytically for a whole set of models, including P-Cl and Cl-Cl. The results clarify the physical differences between these two models.

Diffusion-limited aggregation in three dimensions: Results from a new cluster-cluster aggregation model

Diffusion-limited aggregation has been investigated using three-dimensional lattice models in which both particles and clusters can move and stick together. The final state consists of a single cluster or network of occupied lattice sites with a fractal dimensionality of about 1.75. Two versions of the model have been investigated. In Model 1, the cluster diffusion coefficient is independent of cluster size, and in Model 2, only the smallest clusters are allowed to “diffuse.” Both models give final states which are identical within the accuracy of the simulations. However, the cluster size dispersion at intermediate stages is distinctly different for Model 1 and Model 2. The results taken together with more extensive simulations in two dimensions indicate that a fractal dimensionality of ≈ 1.75 will be obtained for all versions of the model in which the diffusion coefficient of small clusters is equal to or greater than the diffusion coefficient for larger clusters. A number of the geometric properties which characterize final state and intermediate clusters are analyzed. Some of these properties should be amenable to measurement in real systems. The results of the simulations are in good agreement with measurements of the fractal dimensionality of metal particle aggregates.

Some universality properties associated with the cluster-cluster aggregation model

Simulations have been carried out to investigate the effects of sticking probabilities less than 1.0 and the effects of lattice details (lattice versus nonlattice simulations) on the fractal dimensionality D of clusters generated using a new cluster-cluster aggregation model. We find that D is insensitive to both the sticking probability and lattice details.

Diffusion-controlled aggregation on two-dimensional square lattices: Results from a new cluster-cluster aggregation model

Diffusion-controlled aggregation on two-dimensional square lattices: Results from a new cluster-cluster aggregation model