Particle-volume distribution for coagulation and fragmentation phenomena

An exact analytical solution to the unsteady Smoluchowski coagulation equation complicated by particle fragmentation is derived for constant coagulation and fragmentation kernels. The particle-volume and particle-radius distributions as well as the large-time asymptotics and steady-state solution of the coagulation–fragmentation equation are found. It is shown that the unsteady distribution function approaches the steady-state distribution with increasing time. As this takes place, the distribution function approaches the steady-state one from above/below with increasing time if particle coagulation/fragmentation predominates. The analytical solution obtained has the limiting transition to the case of pure particle coagulation. The particle-volume/radius distribution function for large times is in agreement with the numerical calculations and experimental data. Namely, this function represents a bell-shaped curve whose maximum is lower and shifted to the left side compared to the Lifshitz–Slyozov (LS) distribution function for pure Ostwald ripening. In addition, its right branch is above the corresponding branch of the LS distribution. The theory under consideration can also be used for averaged coagulation and fragmentation kernels when dealing with a particular mechanism of particle merging/disintegration.