The classical Rotational Isomeric State (RIS) model, originally proposed by Flory, has been used to rationalize a wide range of physicochemical properties of neutral polymers. However, many weak polyelectrolytes of interest are able to regulate their charge depending on the conformational state of the bonds. Recently, it has been shown that the RIS model can be coupled with the Site Binding (SB) model, for which the ionizable sites can adopt two states: protonated or deprotonated. The resulting combined scheme, the SBRIS model, allows for analyzing ionization and conformational equilibria on the same foot. In the present work, this approach is extended to include pH-dependent electrostatic Long-Range (LR) interactions, ubiquitous in weak polyelectrolytes at moderate and low ionic strengths. With this aim, the original LR interactions are taken into account by defining effective Short-Range (SR) and pH-dependent parameters, such as effective microscopic protonation constants and rotational bond energies. The new parameters are systematically calculated using variational methods. The machinery of statistical mechanics for SR interactions, including the powerful and fast transfer matrix methods, can then be applied. The resulting technique, which we will refer to as the Local Effective Interaction Parameters (LEIP) method, is illustrated with a minimal model of a flexible linear polyelectrolyte containing only one type of rotating bond. LEIP reproduces very well the pH dependence of the degree of protonation and bond probabilities obtained by semi-grand canonical Monte Carlo simulations, where LR interactions are explicitly taken into account. The reduction in the computational time in several orders of magnitude suggests that the LEIP technique could be useful in a range of areas involving linear weak polyelectrolytes, allowing direct fitting of the relevant physical parameters to the experimental quantities.
Keywords: Debye–Hückel interactions; Ising models; Monte Carlo simulations; charge regulation; conformational equilibria; long-range interactions; polyelectrolytes; semi-grand canonical ensemble; transfer matrix; variational methods.