Cluster–cluster aggregation controlled by the number of intercluster connections: kinetics of aggregation and cluster mass frequency
Introduction
Despite the fact that aggregation (and fragmentation) of colloidal particles has been the subject of extended and diverse experimental, numerical, and theoretical investigations [1], [2], [3], [4], the aggregation process controlled by the internal cohesion of the agglomerate has only been invoked to interpret recent experimental results [5]. This special aspect and its effect on the kinetics of aggregation and on the cluster-mass-frequency function were taken into account in a numerical investigation. In our opinion, looking at the control of the internal cohesion of newly forming clusters may address various experimental situations, and especially the two following cases.
The first case corresponds to aggregation in the presence of hydrodynamic forces, which is very important from a practical standpoint. As pointed out by Mason [6] and Gregory [7], two particles approaching each other in a shear field form a doublet that can be either transient or permanent. In the former case, the doublets rotate through a certain angle before fluid forces separate the particles. This image may be adapted to two colliding clusters. One may imagine that the breakup issue is relatively reduced when the encounter between the two clusters immediately induces a great number of intercluster connections. Conversely, when the two clusters are only connected by a single bond, the sticking certainly will be transient [8]. Regarding the given conformation of the stuck clusters, the knowledge of the number of intercluster connections is thus of major interest for the determination of the evolution of the sticking toward rupture or permanent consolidation.
The second case is relative to systems of marginal instability such as hairy or hydrated colloids and colloidal materials for which intercolloid attraction forces are greatly reduced due to the small value of their Hamaker constant, such as low-density biological materials [5], [9], [10]. One may imagine that permanent aggregation requires the formation of multiple interparticle connections between the aggregates.
Obviously, this supplementary constraint may complete the already existing conditions imposed to interpret the slow aggregation affecting colloids suspended in aqueous media of small ionic strength (Fuchs) [11]. Similar conditions were expressed in the algorithm of the standard reaction-limited aggregation process [12], [13], [14], [15], [16], where W(i,j) expresses the sticking probability of colliding clusters of mass i and j and W11 the probability of doublet formation. The exponent σ constitutes the usual variable in the numerical investigation. Below or above the critical σc, the mass frequency function—the number concentration c(n) of colloids composed of n particles as a function of the aggregate mass n—displays a bell-shaped or a continuously decreasing curve, respectively. Only results relative to W11=0.01 and σ=0.4 greater than σc are presented here, insofar as the objective of the investigation is to determine to what extent the connection constraint modifies the standard reaction-limited aggregation process.
In order to fix a realistic constraint on the fulfillment of the permanent sticking of two clusters, the constraint was elaborated according to the irregular fracture model of Horwatt et al. [17]. This model assumed that the force binding the fragment to the parent cluster depended on the number of bonds that must be severed in the cluster breakup. Therefore, this model is briefly presented first and the expression of the constraints relative to the number of intercluster connection and the relative sizes of the cluster and the fragment are given in the second section.
In the third section, we detailed the algorithm of the complete aggregation process and the results relative to the kinetics of aggregation and the mass-frequency function of the aggregates. Finally, we present some experimental results of the aggregation of hydrated polystyrene latex [5], which display modes of cluster growth and mass frequency curves similar to those provided by the present numerical simulation.
Section snippets
Theoretical
Investigation of the aggregate breakup in the presence of hydrodynamic forces requires modeling of both the hydrodynamic forces FH acting on the aggregate and the forces FC responsible for the aggregate cohesion [17]. First, the hydrodynamic forces and the mechanical characteristics of the aggregates (considered as nonpermeable spheres) are briefly developed according to the presentation of Horwatt et al., and second, the model of the aggregate cohesion is developed according to the model
Numerical methods
First of all, clusters of various masses were synthesized employing the algorithms of the standard diffusion- and reaction-limited aggregation processes [20], [21] in order to diversify the fractal dimension. In the second step, clusters were agglomerated two by two and the number v of connections between adjacent particles was determined for each agglomerate conformation. Although off-lattice models are more realistic concerning the short-range structure of 3D systems, the cubic lattice model
The connection frequency and the number v of intercluster connections
Fig. 3 shows schematic 2D configurations for agglomeration of two clusters of masses 34 and 41. Conformation (a) is secured by seven intercluster connections and thus resists fragmentation when the threshold value A(v,i,j) of permanent sticking was set to 10 in the simulation run (Eq. (6)). Conversely, sticking with conformations (b) or (c) is transient since the corresponding A(v,i,j) values are equal to 15.5 and 31, respectively.
Fig. 4, Fig. 5 correspond to agglomeration of clusters generated
Conclusion
The present numerical study had the objective to lay the foundations of a new aggregation process taking into account the connection constraints that may exert the major role when hydrodynamic forces additionally restrict the establishment of permanent sticking. The investigation was developed on 2D and 3D systems, but without other indications, all presented results concerned 3D systems that are of major interest in most areas. In the conclusion of Ref. [5], we noticed that the outlined
Acknowledgements
This work was carried out with the support of the Centre National de la Recherche Scientifique (CNRS, France). S. Stoll is warmly acknowledged for helpful discussions. The Université Louis Pasteur (Strasbourg) is acknowledged for giving access to the computers of the CURRI. The reviewers are warmly acknowledged for improving the presentation of the results.
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Present address: Département de physique, Université d'Ottawa, 150 Louis Pasteur, Ottawa, Ontario K1N 6N5, Canada.