When can flexible weak polyelectrolytes be treated as effective rigid objects?

Conformational and ionization equilibria of flexible weak polyelectrolytes (PEs) are, in general, strongly coupled. In this article, we analyze the effect of averaging over (or "contracting") the conformational degrees of freedom so that the original flexible molecule is replaced by an effective rigid object with the same ionization properties. As a result, one obtains the so-called Site Binding (SB) model, much easier to treat both theoretically and computationally, and extensively used to characterize the ionization properties of PE. The conformational averages can be performed in a systematic way by means of the Conformational Contraction Equations (CCEs), which relate the SB parameters to the underlying conformational equilibrium. The conditions for the convergence of the CCE are evaluated in the presence of both Short Range (SR) and Long Range (LR) electrostatic interactions. Two analytically solvable models based on the Freely Jointed Chain (FJC), involving only SR interactions, are analyzed at first. Despite the large chain flexibility, the resulting SB model reproduces the ionization properties with high accuracy. In the case of independent bonds, a very flexible chain can be exactly replaced by an effective rigid object with neighboring pairwise interactions. In general, however, triplet and higher order interactions emerge at the SB level. When LR electrostatic interactions are introduced and combined with the FJC large chain flexibility, the convergence of the CCE for long chains becomes problematic since the SB free energy must be truncated. Similar conclusions are reached for the freely rotating chain and rotational isomeric state models.