Electrostatic Persistence Length Revisited. I. Theory
The problem of single-chain conformations of polyelectrolytes in salt-added dilute solutions, and the associated concept of the electrostatic persistence length $\textbf{l}_\mathrm{e}$, has remained unresolved for decades. To address this challenge, we develop a comprehensive scaling theory and corroborate it with simulations. A unified scaling diagram is constructed that encompasses both flexible and intrinsically semiflexible/stiff polyelectrolytes. Nine distinct scaling regimes are identified, each characterized by different conformational statistics. The concept of the Gaussian electrostatic blob $\xi_\mathrm{e}$ used in the description of flexible polyelectrolytes is extended to the semiflexible case, and the new characteristic length $\xi_\mathrm{e}^{|}$ referred to as the rodlike electrostatic blob is introduced. This enables delineating the important crossovers and showcasing an analogy between semiflexible and flexible chains. The concept of electrostatic excluded volume is also reconsidered and generalized. Upon increasing the salt concentration, i.e., decreasing the Debye length $r_\mathrm{D}$, chain conformations evolve from (i) rodlike stretches to (ii) Gaussian and then (iii) swollen coils with local electrostatic stiffening characterized by $\textbf{l}_\mathrm{e}$, followed by (iv) swollen coils without stiffening but with electrostatic exclude volume, and finally to (v) quasi-neutral Gaussian coils. In regimes of type (ii) and (iii), the electrostatic persistence length scales quadratically with $r_\mathrm{D}$, in agreement with the OSF and KK predictions for semiflexible and flexible chains, respectively: $\textbf{l}_\mathrm{\mathrm{OSF}} \simeq r_\mathrm{D}^{2}/\xi_\mathrm{e}^{|}$ and $\textbf{l}_\mathrm{\mathrm{KK}} \simeq r_\mathrm{D}^{2}/\xi_\mathrm{e}$. These asymptotic scalings are confirmed by coarse-grained simulations presented in the accompanying article.