Conformational properties of strictly two-dimensional equilibrium polymers

Two-dimensional monodisperse linear polymer chains are known to adopt for sufficiently large chain lengths N and surface fractions $\varphi$ compact configurations with fractal perimeters. We show here by means of Monte Carlo simulations of reversibly connected hard disks (without branching, ring formation and chain intersection) that polydisperse self-assembled equilibrium polymers with a finite scission energy E are characterized by the same universal exponents as their monodisperse peers. Consistently with a Flory-Huggins mean-field approximation, the polydispersity is characterized by a Schulz-Zimm distribution with a susceptibility exponent $\gamma =19/16$ for all not dilute systems and the average chain length $\left(N\right)\propto exp\left(\delta E\right){\varphi }^{\alpha }$ thus increases with an exponent $\delta =16/35$ . Moreover, it is shown that $\alpha =3/5$ for semidilute solutions and $\alpha \approx 1$ for larger densities. The intermolecular form factor F(q) reveals for sufficiently large $\left(N\right)$ a generalized Porod scattering with $F\left(q\right)\propto 1/q^{11/4}$ for intermediate wavenumbers q consistently with a fractal perimeter dimension $d_s=5/4$ .