General Becker–Döring equations: effect of dimer interactions

Abstract

The effect of dimer (two-particle) interactions on the Becker–Doring model of nucleation is investigated. Initially we consider the problem with size-independent aggregation and fragmentation coefficients and a constant monomer concentration. Either an equilibrium or a steady-state solution is found: the former when fragmentation is stronger than aggregation, the latter otherwise. By employing asymptotic techniques, the manner in which the system reaches these states is examined. The dimer interaction is found to accelerate the system towards the equilibrium solution, but has little impact on the relaxation time to the steady-state solution. In cases where aggregation is dominant, the steady-state cluster size distribution can only be determined consistently when the manner of approach to steady state is also known. In the terminology of asymptotic methods, one needs to know the first correction term in order to deduce the leading-order solution. We show how this can be derived and so at steady state we find a flux of matter to larger aggregation numbers due to monomer interactions, with a small and decreasing reverse flux due to dimer interactions. We then consider the case of constant density, that is allowing the monomer concentration to vary, and investigate the effect of a strong dimer interaction on the convergence to equilibrium. Two timescales are present and each one is investigated. We determine the intermediate meta-stable state, the final state and the timescales over which the system relaxes into these states. All results are shown to agree with numerical simulations.