# Polymerization and Collision in High Concentrations for Brownian Coagulation
## 3 Results and Discussion
### Cluster Analysis of Collision Particles  
The relationship between the radius of gyration and the cluster size of aggregates at different volume fractions is studied (Figure 8). When the volume fraction, \(\phi\), is higher, the radius of gyration is smaller with the same cluster size (number-based), which means the aggregates are in a tighter structure. When the monomers in aggregations are less than two, the radius of gyration of clusters is less than the initial radius (\(r_{p}=1.5\)) in all the volume fractions. An increasing particle concentration results in clusters with larger collision diameters and a higher degree of aggregate, which is consistent with Grass [39]. Assuming the \(k_{g}=1\), with fractal dimensions \(D_{f}=1.8\) and \(D_{f}=3\) according to Equation (12), the radius of gyration as the function of cluster sizes is displayed in the green and purple scatters. The results of different volume fractions are between 1.8 to 2.0 when the agglomerations appear in a fractal-like structure. Agglomerates containing about 10 to 30 primary particles on average attain their asymptotic fractal dimension, \(D_{f}\), of 1.91 or 1.78 by ballistic or diffusion-limited cluster-cluster agglomeration, corresponding to coagulation in the free molecular or continuum regimes, respectively [40]. The transition of \(D_{f}\) from three (initial monomers) to its asymptotic 1.78 or 1.91 occurs as particle coalescence upon collision slows down by increasing particle size. The aggregates have not rigorously attained a fractal-like structure when the volume fraction \(\phi=0.05\) because such structures have not reached sufficiently large sizes.  
Figure 8: Agglomeration compactness under different initial particle sizes.  
Figure 7: (**a**) Normalized particle size distributions; (**b**) geometric standard deviations with different volume fractions.  
The particle \(j\) coordination number in the range of \(d_{ij}\) of particle \(i\) with different volume fractions is shown in Figure 9. Here, \(Z(x)\) indicates the average number of contacts per monomer, and \(N_{d}\) is the count that indicates the number of monomers whose coordination number is \(Z(x)\). When the volume fractions are more than 0.1, the distributions of coordination numbers are similar, but the coordination numbers are more aggregated with single connected particles (value 1). The aggregation of monomers formed with a larger geometric diameter and wider distribution as the increase in the volume fraction, which means that the polydisperse of coagulation is higher with the denser system. [PERSON] and [PERSON] also found that the coordination number (i.e., the average number of contacts per monomer) increases with the compression of aggregates [41].  
Figure 10 shows the power-law relationship between the radius of gyrations and the number of monomers in each aggregate under stable conditions. The slope of the straight line is the fractal dimension of the cluster, and the length of the straight line is the size distribution width of the cluster [40]. With an increase in the volume fraction, the aggregates cluster tends to be a fractal-like structure and the fractal dimension in the range of 1.614-2.060. The aggregate structure and \(D_{f}\) both depend on the ratio of collision time. Under the restrictive condition, \(t_{c}>\Delta t\), monomers evolve into fractal-like aggregates immediately upon collision. Lower \(D_{f}\) exhibits an increased cross surface area, which leads to a faster collision among aggregates and monomers. Hence, the aggregate system achieves its SPSD faster with a decreasing \(D_{f}\)[42]. For agglomerates, the primary particles stick upon inter-particle collision, forming fractal-like agglomerates with \(D_{f}=1.9\)[42; 43; 44]. When the volume fractions are higher, the \(D_{f}\) of aggregates formed by collision is closer to 1.9.