Theory of Solutions. III. Thermodynamics of Aggregation or Polymerization

Polymeric and colloidal systems are usually polydisperse, containing possibly hundreds or thousands or more, of subspecies differing only in degree of polymerization n. The required thermodynamic approach is that appropriate to a highly multicomponent system. Let P(n) be the fraction of polymer molecules of size n. The distribution function P(n) will depend on one or more parameters, for example, the mean n̄ and standard deviation σ in the case of a Gaussian distribution. The thermodynamic problem becomes tractable if we replace, as composition variables, the numbers of molecules of all the polymer subspecies by the total number of polymer molecules N and the parameters of P(n). The new set of variables may be only two, three, etc., in number. Our main interest in this paper is to investigate how the thermodynamic functions of the system change when the distribution function P(n) changes, as, for example, in a kinetic study in which P(n) evolves sufficiently slowly with time or in thermodynamic studies on different samples of the same polymer.