(117e) Stochastic Calculation of Collision Kernels for Brownian Coagulation in Dense Systems

Brownian coagulation is one of the most fundamental processes shaping the properties of particle ensembles. Its dynamics is determined by a rate coefficient, the so called coagulation or collision kernel, which was traditionally derived from the analyses of an isolated two-particle system. This of course raises a question if and under what conditions the derived formulas can be used to describe many-particle systems. In order to investigate this problem in some more detail we have developed a special algorithm that allows us to determine coagulation kernels directly from the simulation of motion and collisions among any number of particles and for arbitrary concentration and Knudsen number. In the proposed method particles move in a cubic box endowed with periodic boundary conditions and their trajectories are calculated from Langevin equations of motion. Collisions between particles are counted and after each of them one of the collided particles is redistributed randomly back into the simulation box. In this way the particle system can be restored (on average) to a state from before the collision enabling us to sustain quasi-stationary conditions during the whole simulation. It is important, however, that the new positions of redistributed particles are chosen carefully if statistical independence of successive collisions is to be achieved. We have accomplished this by placing particles consistently with the nearest neighbour distance distribution that develops (and is sampled) during the simulation. The simulation is finally stopped after a sufficient number of collisions has been reached that will guarantee satisfactory statistical accuracy. The coagulation kernel is then determined straightforwardly from the mean time between collisions. Our main observation from a number of conducted simulations is that the coagulation kernel is in general concentration dependent, contrary to what has been traditionally assumed, and classical concentration-independent values are recovered only in the limit of zero particle volume fraction. For large Knudsen number i.e. in the free-molecular regime (ballistic motion), however, concentration influence is fairly weak, especially in comparison with the opposite limit of continuum regime (diffusional motion). In this latter case a strong increase of coagulation kernel is observed with increasing particle volume fraction. For conditions intermediate between free-molecular and continuum regime all data can be reduced to one master curve if plotted versus a ratio of coagulation kernels calculated from the two limiting cases. This master curve is well approximated by a Dahneke's interpolation formula, traditionally introduced to calculate coagulation kernels for arbitrary Knudsen numbers, provided that limiting coagulation kernels in this formula are corrected for concentration dependence.