# Polymerization and Collision in High Concentrations for Brownian Coagulation
## 2 Materials and Methods  
We consider particles as exclusively spheres; sphere-sphere collisions have occurred in the random motion. The force of particle \(j\) acting on particle \(i\) is defined by the Langevin Dynamic equation. \[\overset{\rightarrow}{F_{i}}=\left(F_{c}+F_{f}+F_{r}\right)F_{ij}, \tag{1}\]  
where \(\overset{\rightarrow}{F_{i}}\) is the conservative force of atom \(i\) given by atom \(j\) computed; \(F_{f}\) is the friction term proportional to the particle's velocity; \(F_{r}\) is the force due to implicit solvent atoms bumping into the particle, which represents the randomicity of Brownian motion; \(F_{ij}\) is the unit vector of direction of \(r_{i}\) to \(r_{j}\). Equation (1) is the motion of the particles in the simulating procedure and the MD simulation. \[F_{f}=-fv_{i}, \tag{2}\]  
\[f=-m\gamma, \tag{3}\]  
\[F_{r}=\alpha\sqrt{\frac{2k_{B}T\gamma}{\Delta t}}, \tag{4}\]  
where \(v_{i}\) is the velocity of particle \(i\); \(m\) is the mass of the particle; \(k_{B}\) is the Boltzmann constant; \(T\) is the temperature; \(\Delta t\) is the timestep size; \(\alpha\) is a Gaussian random number; \(\gamma\) is the friction rate between nanoparticles and implicit solvent fluid; \(f\) is the friction constant, which equals the mass of the particle multiplied by the friction rate [26]. The force corresponding to the potential function is:  
\[F_{c}=-\
abla u\big{(}a_{ij}\big{)}, \tag{5}\]  
\[u\big{(}a_{ij}\big{)}=4\epsilon\left[\left(\frac{\sigma}{a_{ij}}\right)^{12}- \left(\frac{\sigma}{a_{ij}}\right)^{6}\right]a_{ij}<r_{c}. \tag{6}\]  
The Lennard-Jones (L-J) potential equation describes both the attractive force and repulsive force [27; 28], where the former part of Equation (6) expresses the repulsive effect while the latter expresses the attraction. The aggregation in the L-D simulation is completed with the collision of pair particles (or multiple), occurring once two (or multiple) particles contact or overlap (Figure 1). In the initial state, all nanoparticles in the fluid environment are in a simple cubic distribution, and the distances between the particles are equal. It is not necessary to consider the effects of gravity coagulation and shear coagulation due to the small size of the nanoparticles [29]. In order to facilitate the calculation, all particles in the simulation system are divided into neighbor lists. A concentric circle in the cutoff distance, \(r_{c}\), is constructed to calculate the interaction force between particles. Figure 1: Neighbor list and collision progress. The simulation boundary condition is periodic and the neighbor list store with \(r_{c}\). The statistic of collision in unit time \(\Delta t\) of particle \(I\) could be described by:  
\[\beta_{c}(Collision)\Longleftrightarrow a_{ij}\leq r_{min}, \tag{7}\]  
where \(a_{ij}\) is the collision diameter with particle \(i\) and particle \(j\); \(r_{min}=d_{p}\). The particles were sampled under the steady state. In order to make the list effective in continuous time steps and increase the fault tolerance rate, bin skin is added out of the cutoff distance, which is generally set to 0.3 or 0.4. All particles in the neighbor list should be calculated at collision (Figure 1). Each particle is regarded as the center particle, \(i\), and the surrounding particles are regarded as the collision particle, \(j\). It is supposed that each particle has an effective cross-section when the collision particle contacts with the center particle, which is determined by the effect diameter \(a_{ij}=d_{p}\) (Figure 1). For analysis, the number of aggregations is counted once particles, excepting the center particle, are in the effective cross-section. The center particle is also in the flow state. As depicted in Figure 1, the random movement of particle \(j\) with radius \(r_{j}\) enters the range of influencing spheres. In this article, the definition for dilution and dense condition from the reference [2] is accepted, such that the volume fraction \(\phi<0.01\) is a dilute concentration while \(\phi>0.01\) is a high concentration.