# Recent Progress in Gel Theory: Ring, Excluded Volume, and Dimension

• Kazumi Suematsu
Chapter
Part of the Advances in Polymer Science book series (POLYMER, volume 156)

## Abstract

Recent developments in gel research are reviewed with emphasis on the gel point problem. We will describe in due course how the gel point equation can be deduced from first principles. First we review briefly the industrial development of gel science in Japan (Sect. 1) and a central aspect of the classical theory of gelation (Sect. 2). In Sect. 3, we survey the progress on the excluded volume problem from the author’s point of view. In all respects, this theme is, now, too biased to physics and hence beyond the scope of this review; while it is an essential subject to understand the nature of the gel point. Regarding the excluded volume problem, a recent interesting idea is the screening effect. This notion of screening is a different interpretation of the Flory excluded volume theory, but takes us into advanced physics. For instance, the behavior of a branched molecule in the melt becomes comprehensible in a natural fashion. In Sect. 4, we mention cyclization in branching media. Like the problem of volume exclusion, the cyclization problem has not been solved rigorously. The most troublesome aspect with cyclization is that there is no way to enumerate the combinatorial number of branched molecules with rings. On the other hand, the mathematical framework for the general solution has already been given. In this article we will mention the limiting solutions of C→∞ for real systems and of d→∞ for the lattice model. What is important is that these limiting solutions are by no means useless, fictitious entities, but have real meanings. By analogy with the f = 2 case, we can put forward the general relation, [Γ]|≅constant for gelation conditions, where [Γ] represents the total ring concentration; this is the basic premise of the gel point theory developed in Sect. 6. Through these analyses, essential differences between real gelations and the percolation model are brought into sharp relief (Sect. 5). The gel point theory starts from the obvious equality: D c = D(inter)+D(ring), where D c represents the gel point, D(inter) the extent of the intermolecular reaction alone at the gel point and D(ring) the corresponding quantity of cyclization. Then, according to some definitions, fundamental equalities for gel points can be deduced for all the models of real systems and the percolation model. The problem of seeking a gel point for a given system thus reduces to the problem of finding a solution for the corresponding fundamental equality. To solve the equalities, we introduce two main assumptions: (1) random distribution of cyclic bonds, and (2) that the ring distribution functions can be expanded about D c =D co , where D co is the gel point for the ideal tree model. Under these assumptions, we can derive analytical expressions for gel points as functions of γ (= 1/C: the reciprocal of an initial monomer concentration), κ (mole ratio of B-type functional units to A-type functional units), and d (space dimension); that is, D c = G(γ, κ,d). In Sect. 7, the theoretical equations thus obtained are compared with experiments. The result shows that the theory recovers well the points observed by Flory, Weil, and Gordon in all the regimes of κ = 1 ~ 2. The corresponding expression for the percolation model is found to agree well with simulation experiments in high dimensions, but fails in low dimensions. The discrepancy in low dimensions is analyzed in light of the critical dimension concept. One possible explanation is that the above-mentioned assumptions (1) and (2) do not work below. d c = 8.

## Keywords

Branching process Gelation Cyclization Excluded volume effects Dimension Concentration invariant High concentration expansion High dimension expansion Gel point Percolation threshold Critical dimension

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