Statistical Mechanics and the Critically Branched State

THIS communication deals with controversies concerning a standard reference case for equilibria in polymer science, namely the equilibrium distribution for a random f-functional polycondensation system. On the assumption that all functionalities are equally reactive and that intramolecular reaction does not occur, this distribution was derived by different probability arguments by Flory1,2, Stockmayer3, Good4, Whittle5 and others: where wx is the weight fraction of x-mer and α the conversion. Recently, Masson et al.6 have claimed that Flory's derivation contains logical errors in the use of a priori probability and have denied that certain probabilistic operations used by Flory were meaningful. From an amended probability argument, they derived a new distribution which in the same notation takes the form Distributions (1) and (2) are both normalized and share the same number average DPn=〈x−1〉−1 . Distribution (1), however, describes correctly the unique dimerization equilibrium for f = 1, whereas distribution (2) obviously does not. Moreover, distributions (1) and (2) differ vitally as regards the weight average ( DPw〈x〉), which diverges for distribution (1) at This divergence was interpreted by Flory as the statistical basis of gelation at the gel point αc. The behaviour of matter in the critically branched state, that is, of systems near their gel point, has been rationalized on this basis. For instance, sol–gel analysis7 accounts for measurements on many different systems.