# Polymerization and Collision in High Concentrations for Brownian Coagulation
## 3 Results and Discussion
### Particle Collisions at Different Volume Fractions  
The number of particle clusters with the occurrence of collisions is shown in Figure 5. \(N_{0}\) is the initial number of particles (\(N_{0}\) = 100,000); \(N_{i}\) represents the population of particle clusters, and the initial monomers are regarded as one cluster (\(N_{i}\leq N_{0}\)). In addition, highly concentrated fractal-like aggregates undergoing coagulation develops SPSDs only temporarily, and they approach gelation at an effective agglomerate concentration of about 0.15 [10]. All the clusters are monomers before the collision. In Figure 5a, the scaled number concentration of clusters is \(N_{i}/N_{0}=1\) when the time is at 0 ns. The number of clusters decreases rapidly in an earlier period, but finally it stabilizes with the evolution of time. Figure 5a also shows the larger volume fraction in the initial condition, the faster decay rate of the aggregate number. In the figure, the evolution of particle numbers with time is also displayed by implementing classical steady-state Smoluchowski model using Brownian Dynamics Simulations (BDS) in the volume fraction, \(\phi=0.005,\ 0.0005\)[36]. It is clear that the decay of particle numbers in the BDS is much slower than that in the present simulation, which indicates that the collision rate in dense conditions is much larger than that in the dilute condition. Figure 5b shows all the aggregates concentrations in the cases that are more than 15% so that all the systems can be developed into SPSDs. Figure 4: The CPU velocity and total walk time in the same running range. Figure 5: Agglomeration concentration with the change in particle numbers: (**a**) the evolution of particle numbers with time; (**b**) the effective agglomerate concentration in the system. Figure 6 shows the static structural \"snapshot\" of aggregate systems at the same time point, \(t=100\) ns, from the different volume fractions (\(\phi=0.05,\ 0.1,\ 0.2,\ 0.3\)). For the convenience of the visualization and analysis of the simulation data, the calculated data in the present study are rendered according to cluster size (number based) by OVITO [38]. In order to note the form of aggregates clearly, the clusters with only one monomer are restricted to transparent states. Among all the investigated cases, the system with initial volume fraction \(\phi=0.05\), has the least probability of occurrence of aggregates; in contrast, the system with the initial volume fraction \(\phi=0.3\) has the greatest probability. From the static structural \"snapshot\" shown in Figure 6, it can also be found that at the larger volume fractions, the aggregates are composed of more monomers and the fractal-like form is more obvious. This is consistent with the finding that the coagulation kinetic in the dense condition is faster than those at dilute concentrations [10]. Figure 7a is the normalized particle number size distributions at the same time \(t=100\) ns, generated from various solid volume fractions. In the figure, \(d_{i}\) represents the number-based size of aggregates \(i\), and the monomer seems as \(d_{i}=1\); \(d_{ave}\) is the number based geometric mean diameter. The distribution is closer to dilute concentration when the volume fraction is 0.05. As compared to the dilute particle system, the system with a higher volume fraction shows a wider distribution of numbers. It implies that under dense conditions, particles have more probability to collide, as shown in Equation (7). Figure 7b is the number-based average geometric standard deviations as the function of different volume fractions, \(\sigma_{g}\) as the function of different volume fractions, \(\phi\). As a comparison, the calculation data from [PERSON] and [PERSON] [10] is displayed. When \(\phi<0.1\), the present LD simulation and [PERSON] and [PERSON]'s work match for the geometric standard deviations (\(\sigma_{n}=1.52\)).