A novel dichotomy model based on the traditional CCA

Abstract

The Cluster-Cluster Aggregation model (CCA) was often used to simulate the aggregation of colloids, in which the rapid and slow aggregation processes of colloids can be well characterized. In practical system, the movement of particles or clusters occurs in parallel. In the CCA model, in order to facilitate computer simulation, the movement of particles or clusters is processed serially by the method of Metropolis sampling. That is, randomly selecting one particle or cluster to move randomly, colliding with other cluster in the movement, and aggregating according to certain conditions, such aggregation would usually occur between a moving cluster and a static cluster, and this process is essentially a dimerization process. Based on this idea, this study proposes a cluster dichotomy model, in which the dispersion process can be regarded as the inverse process of cluster aggregation, that is, the randomly selected cluster always breaks up into two small clusters. This kind of dispersion is called dichotomy. Based on this model, the rapid and slow dispersion processes of colloidal clusters can be simulated by setting the relevant parameters. The dichotomy model proposed in this paper can be used to study colloidal stability by coupling with the traditional CCA model.

Introduction

The Diffusion-Limited Aggregation (DLA) proposed by Witten and Sander [1] has the following three contributions: Firstly, the model uses extremely simple algorithms to capture the key components of a wide range of natural phenomena even though its detailed physical internals are not clear [2]. Secondly, through simple kinematic and dynamic processes, it can generate self-similar fractal structures with scale invariance. As a result, it can bridge the fractal theory and laboratory experiments [3]. Thirdly, it enables the interface to have a complex shape and instability [4]. Considering it is not in accordance with the objective reality to set the immovable core in the DLA model, Meakin modified the DLA model and proposed the Cluster-Cluster Aggregation (CCA) model [5]. The CCA model assumes that initially a number of equal-sized individual particles are randomly setting in a box at low concentration and then these particles are allowed to move in Brownian motion, to meet each other, and to form new clusters [6]. Then, two regimes of colloid aggregation have been identified by computer simulation with the CCA model [7], to reflect the rapid and slow aggregation of colloid, which have long been recognized in colloid science. One regime is the Diffusion-Limited Cluster Aggregation (DLCA) corresponding to the rapid aggregation. Another is the Reaction-Limited Cluster Aggregation (RLCA) corresponding to the slow aggregation. Lin [8] found that the basic process of colloidal aggregation is unified, which is also realized by the three basic processes: movement, collision and aggregation of particle or cluster. In brief, it has been proved that the CCA model is universal in the case of studying the aggregation process of colloids [8], [9].

All particles are in motion and have chances to collide and aggregate in realistic situation. In order to facilitate computer simulation, the Metropolis sampling method is often used to serialize the movement of particles or clusters [10]. Accordingly, a particle or cluster is selected to carry out random movements according to some predefined rules. The particle or cluster encounters other particle or cluster, then the aggregation would occur according to a certain probability [11], [12]. By the serialization, the aggregation process of CCA model can be regarded as the aggregation process with a moving particle or cluster collides with a static particle or cluster [13]. Based on the idea of dimerization of CCA model, the dichotomy model is proposed to describe the dispersion process simulation of fractal cluster, which is actually the inverse process of the CCA model.

Colloidal stability is one of the critical research topics in colloidal chemistry, which has a wide application in many fields such as biology, chemical industry, and ecological environment [14]. Especially, the stability of colloidal system is determined by the Van der Waals force and electrostatic repulsion between colloidal particles, which can be described by the DLVO (Derjaguin and Landau, Verwey and Overbeek) theory [15], [16]. According to the theory of double electric layer, there is a repulsive barrier between colloidal particles. Then, if the Van der Waals force between colloidal particles is large enough, the aggregation between colloidal particles will occur [17]. However, when the electrostatic repulsion force between colloidal particles is greater than the Van der Waals force, the cluster would break up into small clusters from one or more special positions [18]. The electrostatic repulsion and the Van der Waals force between particles would change once the factors of system temperature, pH value, electrolyte type or concentration, the surface potential of particles undergo changes [16], [19]. Therefore, not only the aggregation but also the dispersion may occur in the colloidal system. The dichotomy model proposed in this paper is derived from the CCA model, thus it is also possible to couple both of them in the study of colloidal stability.

Some researchers have put forward cluster dispersion models based on the discrete element method mainly considering the shear forces acting on particles. These kinds of methods are suitable for the case of erosion dispersion simulation [20], [21]. Marco Vanni studied the cluster fracture when rigid colloidal clusters are exposed to the fluid in low density system [22]. Bbler M.U. et al. studied the dispersion when clusters suspending in turbulence [23]. Most of these models are based on the effect of external shear forces. Actually, there are two types of dispersion regimes, namely erosion dispersion and collapse dispersion. In terms of the dispersion position, the former characterized with “from outside to inside”, while the latter characterized with “from inside to outside”. Obviously, the previous dispersion models based on the shear force cannot be used to simulate the collapse dispersion process. By referring to the idea of dimerization process of the CCA model, the dichotomy model proposed in this paper can integrate the erosion dispersion and the collapse dispersion. In which, the control parameter is finally related to the selection of dispersion position. In this study, the dichotomy probability is calculated by referring to the number of the effective connection points. In the past, the CCA model was often used to simulate rapid aggregation and slow aggregation of colloid. Similar to these situations, the dichotomy model can also be used to simulate the processes of rapid dispersion and slow dispersion. In brief, in terms of dispersion type, this model can simulate both erosion-type dispersion and collapse-type dispersion. In terms of dispersion speed, this model can simulate both fast dispersion and slow dispersion. Thus, it can be said that the dichotomy model proposed in this article is a sister model of the classical CCA model, and is a beneficial extension of the CCA model.

In addition, “dichotomy” itself is a serialization of the process of “one divided into many”, that is, the process of splitting a large cluster into several small clusters simultaneously can be regarded as the superposition of several “dichotomy” processes. Therefore, it is completely reasonable to simulate the process of cluster dispersion by treating the dichotomy as the inverse process of the CCA models dimerization. The dichotomy model, the dichotomy probability and the dichotomy position probability are discussed in detail in the following parts.