Bicriticality in the Polymerization of Chains and Rings

The equilibrium polymerization of chains and rings can be viewed as a bicritical point in a suitably augmented parameter space. The polymerization of chains in equilibrium with rings is described by the cross-over exponent, $\phi$, for quadratic anisotropy in the $\mathrm{O}\mathrm{}\left(n\right)\mathrm{}$ vector model with $n\to 1$. While the total mass fraction of polymerized material is described by the Ising exponent $\alpha$, the mass fraction of chains separately (and therefore also of rings) is described by the exponent ${\alpha }_c=\phi -1+\alpha$.