On equilibrium solutions of aggregation–fragmentation problems
Abstract
A characteristic feature of particulate systems that evolve due to competition between aggregation and breakage is that they sometimes produce non-trivial steady-state particle size distributions. If such solutions satisfy detailed balance conditions, then they are equilibrium solutions. The conditions that must be satisfied by aggregation and fragmentation rate kernels in order for equilibrium solutions to be produced are elaborated, and it is shown that the rate kernels are uniquely determined by the aggregation and breakage rate constants for the reactions involving monomers. Consequently, for equilibrium systems there is a significant reduction in the amount of information needed in order to infer the general form for aggregation or breakage kernels, and we explore implications for constructing rate kernels by using atomistic simulations such as molecular dynamics.
Graphical abstract
Equilibrium aggregation–fragmentation solutions depend only on reactions involving monomers, and this result simplifies the construction of rate kernels using atomistic simulations.
Keywords
- Reversible aggregation;
- Aggregation–fragmentation;
- Population balance equations;
- Equilibrium solutions;
- Detailed balance
1. Introduction
Population balance equations (PBEs) describing processes in which particles undergo growth and decay through aggregation and breakage have long been studied in connection with a variety of physical phenomena, such as polymerization and the growth of colloidal particles. Not surprisingly, these aggregation–fragmentation PBEs can exhibit the same kinetic behavior that is displayed by more restrictive population balance models of either irreversible aggregation or breakage, such as gelation, shattering, and self-preserving size distributions [1]. However, aggregation–fragmentation PBEs differ fundamentally from irreversible aggregation or irreversible fragmentation PBEs in at least one important respect. Namely, the possibility exists that non-trivial steady-state particle size distributions can be produced by a balance in the competition between growth of particles by aggregation and degradation due to breakage.
The possibility that steady-state solutions are produced by competition between aggregation and breakage leads naturally to the question of what conditions must be satisfied in order for this to occur. For example, consider a system of well-mixed particles that undergoes aggregation and binary fragmentation (two fragments produced when a cluster undergoes fission). If the system is closed to mass exchange, the relevant population balance equation can be expressed as

The contrast between equilibrium and non-equilibrium steady states can be explained as follows. Time-independent solutions of Eq. (1), , must satisfy
