Equilibrium polymerization of chains and rings: A bicritical phenomenon

The equilibrium polymerization of chains and rings together is a ($n=1\mathrm{}$) critical phenomenon with aspects of bicriticality, governed by exponents not found in any symmetric $\mathrm{O}\mathrm{}\left(n\right)\mathrm{}$ model. The bicritical nature of the critical point is the result of a competition between a transition to form long-chain polymers and a transition to form an infinite-rings condensate. As a result the fraction of monomers incorporated in chains and in rings varies with temperature according to a different power law than that for the total fraction of polymerized material and is governed by an exponent that depends on the crossover exponent $\phi$ for quadratic anisotropy for the $\mathrm{O}\mathrm{}\left(n\right)\mathrm{}$ vector model of magnetism in the limit $n\to 1$. Moreover, the fraction of monomers in polymeric rings is found to decrease with temperature with infinite slope just beyond the transition in the polymerized phase. Similarly, the geometry of large polymers is described by a power law $R\sim N^{\frac{\nu }{\phi }}$, where $R$ is the distance spanned by a polymer segment of $N$ monomers and where the exponent is not the de Gennes result $\nu \mathrm{}(n=0\mathrm{})\mathrm{}$ for chains alone, but rather also depends on the crossover exponent $\phi$ for $n=1\mathrm{}$. We study the equilibrium polymerization of chains and rings together using a variety of techniques, including simple equilibrium theory, a lattice model, field-theoretic correspondence to magnetism and a direct renormalization-group calculation on a polymer model. An explicit parametric form for the equation of state is presented to lowest order in $\epsilon$. Results are obtained for the fraction of polymeric material in chains and rings and for various correlation functions of interest in considering the geometry and distribution of chains and rings.