Statistical Mechanics of High Polymer Solutions

The configurational partition function for random mixtures containing any numbers of components which can consist of simple, simple chain, branched chain, or closed ring molecules is examined. Using a general statistical method a set of partial differential equations is obtained for the appropriate combinatory factor. The integrability of this set of equations is examined. This provides a criterion by which it can be decided in what cases a mathematically precise value for the combinatory factor can be obtained rigorously by solving the set of partial differential equations. In these cases general formulae are given for the combinatory factor. It is shown that precise formulae can be deduced rigorously (a) for all binary mixtures whether the components consist of simple, simple chain, branched chain, or closed ring molecules ; (b) mixtures of any number of components containing not more than one high polymer species, which can consist of closed ring equally well as of simple chain or branched chain molecules ; and (c) mixtures of any number of components containing more than one high polymer species provided these consist only of simple chain or branched chain molecules. It is also shown that even in the case in which the condition of integrability is not satisfied, an approximation, which involves negligible error for high polymer molecules, indicates that the general formula must still provide a good approximation practically. The approximations inherent in the physical model are also considered.