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

equation1
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where ck is the concentration of particles with mass k at time t  . The symmetric matrix Kij (aggregation kernel) specifies rate constants for aggregation of i-mers with j  -mers. Similarly, Fij is a symmetric matrix of rate constants for breakage of i+j-mers into i-mers and j-mers. The long-time behavior of (1) therefore will depend upon the choice of Kij and Fij, and possibly also on initial conditions. For aggregation and breakage rate kernels that obey the homogeneity relations
equation2
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several investigators have used scaling arguments to show that a necessary condition for the system to always (irrespective of initial conditions) attain a steady-state solution is given by [1], [2], [3] and [4]:
equation3
β-λ+2>0.
Eq. (3) is useful for anticipating the existence or absence of steady-state solutions for arbitrary homogeneous kernels Kij and Fij without the need to analytically or numerically solve Eq. (1), but it does not provide information concerning the nature of any steady-state solutions produced, such as whether or not they are also equilibrium solutions.

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

equation4
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As has been observed many times previously (e.g. [5], [6] and [7]), Eq. (4) will be satisfied if the steady-state solution View the MathML source also obeys
equation5
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for all pairs of particle masses i and j. The condition (5) is a statement of microscopic detailed balance that requires each aggregation or breakage pathway to be reversible and in dynamic equilibrium at steady state. Therefore, solutions of Eq. (4) that satisfy Eq. (5) are equilibrium steady states. For a given set of aggregation and breakage kernels, however, there may exist steady-state solutions that satisfy Eq. (4) but not Eq. (5), and it follows that any such steady-state solutions are non-equilibrium solutions.