Chapter 3 AGGREGATION

Aggregation is a fundamental non-equilibrium process in which reactive clusters join together irreversibly when they meet (Fig. 3.1) so that the typical mass of a collection of aggregates grows monotonically with time. Developing an understanding aggregation is important both because of its ubiquitous applications| such as the gelation of jello, the curdling of milk, the coagulation of blood, and the formation of stars by gravitational accretion|and because aggregation is an ideal setting to illustrate theoretical analysis tools. We schematically write aggregation as Ai +Aj Kij!Ai+j: That is, a cluster of mass i+ j is created at an intrinsic rate Kij by the aggregation of a cluster of mass i and a cluster of mass j. The fundamental observables of the system are ck(t), the concentration of clusters of mass k at time t. j i+j i Figure 3.1: Schematic representation of aggregation. A cluster of mass i and mass j react to create a cluster of mass i+ j. The primary goal of this chapter is to elucidate the basic features of the mass distribution ck(t) and to understand which features of the reaction rate Kij in uence this distribution. In pursuit of this goal, we will write the master equations that describe the evolution of the cluster mass distribution in an innite system and discuss some of its elementary properties. Next, we work out exact solutions for specic examples to illustrate both the wide range of phenomenology and the many useful techniques for analyzing master equations. We will then discuss the reaction rate in general terms and show how to calculate this rate. This discussion will naturally lead to the notion of dynamical scaling, which provides a simple way to obtain a general understanding of aggregation. Finally, we will discuss several important extensions of aggregation that exhibit rich kinetic properties. The Master Equations The traditional starting point for treating aggregation is the innite set of equations that describe how the cluster mass distribution changes with time. For a general reaction rate Kij, the master equations are