Double Asymptotic Structures of Topologically Interlocked Molecules

The mean square size of topologically interlocked molecules (TIMs) is presented as a linear combination of contributions from the backbone and subcomponents. Using scaling analyses and extensive molecular dynamics simulations of polycatenanes, as a typical example of TIMs, we show that the effective exponent $\nu(m)$ for the size dependence of the backbone on the monomer number of subcomponent $m$ is asymptotic to a value $\nu$ (approximately 0.588 in good solvents) with a correction of $m^{-0.47}$, which is the same as for the covalently linked polymer. However, the effective exponent for the size dependence of subcomponents on $m$ is asymptotic to the same value $\nu$ but with a new correction of $m^{-1.0}$. The different corrections to the scaling on the backbone and subcomponent structure induce a surprising double asymptotic behavior for the architecture of the TIMs. The scaling model that takes into account the double asymptotic behavior is in good quantitative agreement with the simulation result that the effective exponent for the size dependence of TIMs on $m$ increases with the subcomponent number $n$. The full scaling functional form of the size dependence on $m$ and $n$ for polycatenanes in a good solvent is well described by a simple sum of two limiting behaviors with different corrections.