Estimation of Critical Point in Branching Reactions: Further Examination of Gel Point Formula

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The theory of gel point in real polymer solutions is examined with the empirical correlation between the reciprocal of the percolation threshold and the coordination number given by the percolation theory. Applying a larger value of the relative frequency of cyclization, an excellent agreement is obtained between the present theory and the percolation result. This suggest that while the ring distribution on lattices is similar to that in real systems, ring production is more frequent in the lattice model than in real systems. To confirm this conjecture, we derive the ring distribution function of the lattice model as a limiting case of d → ∞, and show that the solution is in fact identical to the asymptotic formula of C → ∞ in real systems except for the coefficient 𝒸, which has a maximum at d = 5, in support of the above conjecture. To examine the validity of the asymptotic solution for the lattice model, we apply it to the critical point problem of the percolation theory, showing that the solution works well in high dimensions greater than six.

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