## 1 Introduction  
Flocculation is an important initial stage in colloidal gel formation. The incorporation of many functional chemicals during the initial stages of gelation and the eventual formation of compact structures are critical to establishing new material processes. In the case of water treatment, it can improve separation efficiency and reduce the amount of bulky sludge. When a flocculant is added to treated water containing suspended particles and the solution is agitated, the initially dispersed fine suspended particles collide and adhere to each other, inducing the development of large flocs that can be easily identified by the naked eye. At this point, it is necessary to develop the technology to form flocs that are easily separated according to the mechanism of solid-liquid separation (sedimentation, dewatering, membrane filtration) which will be carried out in a later stage of the operation. For this purpose, clarification of the physical properties of flocs, such as size and strength, in relation to flocculation conditions--going back to the floc formation process--is warranted, and many studies exist on the flocculation rate, floc formation process, sedimentation rate, and flow characteristics of turbid water containing flocs [1]. The kinetics of the initial stage of flocculation has been analyzed theoretically as a subject of fundamental physics long ago [2]; however, from an engineering standpoint, issues have been considerably addressed by the field of water treatment. One of the basic experiments was reported by [PERSON] et al. [3] using kaolinite suspension to flocculate with aluminum sulfate. The flocs were grown by rapid agitation and mixing followed by slow agitation, which mimics actual water treatment. The formed flocs were taken out in a single state and the settling behavior of the flocs was photographed as a function of floc diameter. By applying [PERSON] law, the authors recognized that the density of the floc effective sedimentation decreased with increasing floc diameter following a power-law relationship. The relation was named the floc density function. This means that there is a certain law of floc density as a function of floc diameter in the seemingly random structural formation of flocs. Furthermore, [PERSON] continued his experiments and organized the floc density function in terms of the ALT ratio, which is the ratio of the concentration of aluminum ions in the flocculant to the concentration of suspended particles, and pointed out that the floc morphology changes systematically according to chemical conditions [4]. On the other hand, [PERSON] [5] reported that differences in agitation intensity during floc formation affected the floc density function, pointing out the importance of agitation intensity in the formation of floc structure. However, the interrelation between the results of [PERSON] and [PERSON] has not been investigated. In the 1980s, many papers on the structural formation of flocs were published as a subject of fractal analysis based on the concept according to which flocs were formed by the binary collision of clusters [6]. By means of numerical methods, [PERSON] [7; 8] demonstrated occurrence of cluster rearrangement, the importance of which has also been demonstrated experimentally [9; 10; 11; 12; 13]. The prediction of floc size is important for the rational design of the separation process and has received substantial attention, alongside floc strength, since the early days of the research process [14] and many articles have been published which address, experimentally and numerically, the breakup of flocs in a simple system [15; 16; 17; 18; 19; 20; 21; 22; 23; 24] and real systems. There are two types of fracture mechanisms: large-scale fragmentation due to tensile force and surface erosion due to shear force [25]. However, the mechanisms of such destruction and the issue of incorporating inter-cluster rearrangements into the floc formation process remain unclear despite their importance [26; 27; 28]. [PERSON] et al.