The equilibrium statistics of a clustering process in the uncondensed phase

Abstract

The paper could as well have been entitled. 'The equilibrium statistics of a polymerization process in the sol phase', because it will discuss a process in which particles aggregate into structured clusters, just as organic units aggregate into polymers, and for most of the paper the polymer terminology will be employed for concreteness. However, the title is intended to indicate that these processes, and our interest in them, is not confined to models of organic polymers. In $\S\S$1 and 2 reference is made to a previous paper (Whittle 1964, referred to as 'W.') in which it was shown that the complete statistics of a polymerization process could be determined from the expected numbers of the different types of polymer. We therefore work in terms of the generating function of these expected values, defined in equation (18), and denoted by M(x). In W. the function M(x) was determined by combinatorial considerations for the case of one type of unit. In the present paper we consider the case of several types of unit, and avoid combinatorial considerations altogether by using a different method, based upon the detailed balance relation. Under the assumption that the binding energy of a unit in a polymer depends only upon the numbers and types of unit with which it has direct bonds, it is shown in lemma 1 that a function C(y) immediately related to M(x) obeys the non-linear first-order partial difference equations (31). From this the determination (24) of M(x) given in theorem 1 is obtained. Theorem 1 also contains a number of immediate corollaries of this result: the partition (27), (28) of M(x) into 'unit-number' and 'bond-number' generating functions determination of the numbers of bonds of various types, and a criterion for change of phase. These results are illustrated numerically for some simple cases. These results are extended in $\S$5 to the case when two given units can form a common bond in more than one way. All this work is on the assumption of acyclic polymers, so that each polymer has the form of a tree. If this condition is removed entirely so that unrestricted intramolecular association is allowed, then equations (31) change to second-order linear equations (88), and the solution corresponding to (24) has the integral form (92). It is found that if branching is possible (as distinct from the building of simple chains and loops) then the generating functions diverge under all circumstance, indicating formation of high polymers, which may in some cases correspond to irreversible condensation.