Brownian coagulation at high particle concentrations

The process of Brownian coagulation, whereby particles are brought together by thermal motion and grow by collisions, is one of the most fundamental processes influencing the final properties of particulate matter in a variety of technically important systems. It is of importance in colloids, emulsions, flocculation, air pollution, soot formation, materials manufacture and growth of interstellar dust, to name a few of its applications. With continuous progress in particulate matter processing there is a constant trend to increase particle loadings in technical applications for efficiency reasons. The classical coagulation theory, which assumes that the collision kernel is independent from the particle concentration, is well proven for the diluted particle systems, however, this basic assumption does not necessarily hold any more once the particle system becomes highly concentrated. In this dissertation the consequences of coagulation at high concentrations are investigated theoretically, by computer simulations and by comparison with available experimental data. In chapter 1 the classical theory of Brownian coagulation is critically reviewed and forms the background to the rest of the work. In chapter 2 a novel simulation algorithm is developed to determine the Brownian coagulation kernels based on direct simulation of particle motion and collisions, while at the same time maintaining particle size and concentration as constant. The basic premise of the proposed method is to use first principle Langevin dynamics simulation to track the motion of particles placed in a cubic box endowed with periodic boundary conditions until a collision event occurs. The number of particles and their size is kept constant during the simulation by redistributing the collided particles back into the simulation box. The redistribution step requires special care to avoid introducing spurious effects and to preserve statistical independence of collisions. This is accomplished by choosing the positions of the redistributed particles in such a way, so as to preserve the nearest neighbour distance distribution naturally developing during the simulation. The more technical details of efficient practical implementation of this redistribution algorithm are relegated to appendix A. In chapter 3 the new simulation method is used to investigate Brownian coagulation at high particle volume fractions and over the entire size spectrum from the free-molecule to continuum regime. It is shown that the coagulation kernel is in general concentration dependent and that the classical concentration-independent values are recovered only in the limit of vanishing particle volumetric concentration. In the free-molecule regime (ballistic motion), the concentration effect is of a moderate magnitude, while the contrary is true in the opposite limit of the continuum regime (diffusional motion), where much stronger dependence on the particle concentration is observed. For conditions intermediate between the free-molecule and the continuum regime all data reduce to one master curve if plotted versus a ratio of coagulation kernels calculated for the two limiting cases. This master curve is well approximated by the classical interpolation formulae for the transition regime provided that the limiting coagulation kernels in this formula are corrected for the concentration dependence. The focus of chapter 4 shifts from the geometrical crowding effects towards the coagulation at particle number densities that are large in comparison to that of the background gas. With an increasing rarefaction of the suspending gas, while keeping the particle concentration fixed, there will be a point, when the particle relaxation time becomes longer than the characteristic coagulation time. At that point the average velocity of particles will decrease due to mutual inelastic collisions and will fall below the value predicted by the energy equipartition theorem for there is insufficient time between the collisions to restore the thermal equilibrium. This reduced particle velocity will in turn lead to a lower coagulation rate and this effect is shown through first principle Langevin dynamics simulations. A model quantifying the effect of the coagulation rate suppression is formulated by introducing the concept of a thermalization number, which is defined as the reduced mean kinetic energy of particle motion. An expression for this quantity is derived and used to extend the classical coagulation theory to this new non-equilibrium regime of coagulation. A closed-form, analytical formula for the coagulation kernel in this regime is obtained and its excellent agreement with the first principle simulations is also demonstrated. In the final chapter 5 the problem of accelerating kinetics in sol-gel transition is analysed. This process can be considered to be universal with respect to the particle volume fraction (or more specifically, to the free volume available for particle motion). A simple monodisperse model of agglomeration with the coagulation kernel dependent on the volumetric concentration is formulated and used to find a non-dimensional transformation that is required to obtain a universal solution for the kinetics of this process. By exploring this concept of free-volume universality of coagulation further, a coagulation kernel enhancement function established for a hard-sphere system is applied to the problem of agglomeration through an appropriate definition of the agglomerate volume fraction. It is shown through comparison with computational modelling and experimental data that the kinetics of agglomerate growth are adequately captured in this model before the agglomerates become interdigitated. The apparent increase in coagulation kernel homogeneity observed in cluster-cluster agglomeration, but not in coalescent coagulation, is identified as originating from the increase of the effective volume fraction of growing agglomerates stemming from their fractal-like structure.