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The effect of dimer (two-particle) interactions on the Becker-Döring 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.

Abstract. We investigate the Becker-Döring model of nucleation with three generalisations; an input of monomer, an input of inhibitor and finally, we allow the monomers to form two morphologies of cluster. We assume size-independent aggregation and fragmentation rates. Initially we consider the problem of constant monomer input and determine the steady-state solution approached in the large-time limit, and the manner in which it is approached. Secondly, in addition to a constant input of monomer we allow a constant input of inhibitor, which prevents clusters growing any larger and this removes them from the kinetics of the process; the inhibitor is consumed in the action of poisoning a cluster. We determine a critical ratio of poison to monomer input below which the cluster concentrations tend to a non-zero steady-state solution and the poison concentration tends to a finite value. Above the critical input ratio, the concentrations of all cluster sizes tend to zero and the poison concentration grows without limit. In both cases the solution in the large-time limit is determined. Finally we consider a model where monomers form two morphologies, but the inhibitor only acts on one morphology. Four cases are identified, depending on the relative poison to monomer input rates and the relative thermodynamic stability. In each case we determine the final cluster distribution and poison concentration. We find that poisoning the less stable cluster type can have a significant impact on the structure of the more stable cluster distribution; a counter-intuitive result. All results are shown to agree with numerical simulation.

Abstract. This paper is concerned with an analysis of the Becker-Döring equations which lie at the heart of a number of descriptions of non-equilibrium phase transitions and related complex dynamical processes. The Becker-Döring theory describes growth and fragmentation in terms of stepwise addition or removal of single particles to or from clusters of similar particles and has been applied to a wide range of problems of physicochemical and biological interest within recent years. Here we consider the case where the aggregation and fragmentation rates depend exponentially on cluster size. These choices of rate coefficients at least qualitatively correspond to physically realistic molecular clustering scenarios such as occur in, for example, simulations of simple fluids. New similarity solutions for the constant monomer Becker-Döring system are identified, and shown to be generic in the case of aggregation and fragmentation rates that depend exponentially on cluster size.

Recently it has been reported that the growth of vesicles is strongly influenced by the presence of pre-added vesicles. In particular, the size distribution of vesicles is sharply centered about the size of the pre-added vesicles:  the size-templating “matrix” effect. A description of the dynamics of the concentrations of cluster sizes has been proposed Bolton and Wattis (J. Phys. Chem. B 2003, 107, 7126), in terms of a nonlinear microscopic model that is valid far from equilibrium while agreeing with the equilibrium cluster size distribution data. In this paper we approximate this model by a coarse-graining reduction to a low-dimensional dynamical system, a process introducing two parameters on which the solution depends. We solve the system in two parameter regimes and demonstrate that there is a borderline case between the two types of solution that contain features of each. In each case matched asymptotics are used to derive the solution over a series of time scales, and we consider a range of initial conditions including experimentally relevant cases where a small concentration of vesicles is pre-added. We interpret these results on a phase-plane diagram to illustrate how the matrix effect manifests itself in this model.