# Polymerization and Collision in High Concentrations for Brownian Coagulation
## 1 Introduction  
The coagulation of nanoparticles in the air or industrial processes is an inevitable outcome; in such a process, the total particle number decreases but the mean size of the particles increases [1, 2, 3]. In 1927, [PERSON] established a governing equation for describing the coagulation process, which was later called the Smoluchowski equation (SE) or coagulation equation [4]. SE is suitable for a system where one spherical particle forms once two particles collide and the volume fraction of the system, \(\phi\), is limited (\(\phi<0.01\)) [5]. The basis of this SE is [PERSON]'s theory of particle diffusion in the flow field, which focuses on many assumptions and restrictive conditions in the process of [PERSON]'s steady-state force on particles and [PERSON]'s calculation of osmotic pressure. However, when the multiphase system is far away from the dilute phase conditions (\(\phi<0.01\)), the distance between adjacent particles will decrease accordingly. When the distance between particles reaches the magnitude of the particle-free path, the correlation between particles cannot be ignored--that is, the SE is no longer applicable [6]. The coagulation of nanoparticles in dense conditions happens in many industrial processes. For dealing with the ultrafine particles emitted from exhaust pipes [7], dust particle agglomerations formed of clusters [8], emulsions [9], and particle coagulations, the aerosol process has been described by the Smoluchowski theory of Brownian aggregation, which refers to the random motion in a fluid [10]. It is believed that the process of particle collision mostly depends on the grain size, the concentration of particles, and the transport coefficients in the mechanism. Early examples of research into Brownian aggregation include research on the effect of bulk stress [11]; fluid shear involving polymerization [10]; the influence of transport on the coagulation rate, such as diffusion consistency [7] and fluid viscosity shear [12]; transition regime; and the mass transfer function of the Knudsen number [13]. Such approaches, however, have failed to address the dense particulate system (effect fraction \(\phi>0.01\)). Simultaneously, a high solid concentration is common in industrial production--for instance, flame, plasma, and laser material synthetic technology works on high particle concentrations [14], the aerosol of aggregating clusters [15], and TiO\({}_{2}\) aerosol agglomerate at high solid concentrations [16]. The research results of [PERSON] and [PERSON] [15] confirmed that the classical Smoluchowski theory is limited to cases where the particle volume concentration is less than 0.1%. When the particle volume concentration is higher than this value, the collision frequency between particles is higher than the predicted value given by the Smoluchowski theory, and the collision frequency of the particles in the system is a function of the volume rate of the particles in the system. The larger the particle volume ratio is, the higher the collision frequency of the system particles is and the greater the deviation between the real situation and the predicted value of the classical Smoluchowski theory is. Based on the numerical simulation results of Langevin dynamics (LD), [PERSON] and [PERSON] [15] and [PERSON] et al. [6] give fitting expressions for the collision frequency between particles in the dense phase state. However, although the expressions they gave are quite different in form, they do not deviate from the framework of classical Smoluchowski theory. They are all modified expressions of the classical theory, and the fitting expressions cannot be interpreted in a strictly physical meaning. It should be pointed out that [PERSON] et al. (2006) studied monodisperse systems, while [PERSON] and [PERSON] studied polydisperse systems, but the LD simulation gave very similar results. Both [PERSON] and [PERSON] [15] and [PERSON] et al. [6] deal with spherical particles.