As already mentioned, the structural properties of polyeletrolytes are governed by a combination of electrostatic and short-range molecular interactions. With regard to this point, the influence of dissociated counterions as well as salt ions have to be considered as integral contribution to the observed polyelectrolyte behavior. Here, we discuss the theoretical description of electrostatic effects in terms of counterion condensation and charge screening mechanisms.

#### 2.1. Electrostatic Screening Effects

Charged species in solution are governed by electrostatic interactions. For two charged and small species in a solvent with dielectric constant

${\u03f5}_{r}$, the electrostatic Coulomb potential

$\mathsf{\Phi}\left(r\right)$ shows a long-range decay

where

r denotes the distance between the charges at a high dilution. At moderate and high ion concentration, the long-range decay of Equation (

1) changes significantly due to electrostatic screening effects. A suitable mathematical description for polyelectrolytes and ions of species

$\alpha $ with valency

${z}_{\alpha}$ and unit charge

e is given by the mean field Poisson-Boltzmann (PB) equation [

36,

37,

38]

with Boltzmann constant

${k}_{B}$, temperature

T, and vacuum dielectric constant

${\u03f5}_{0}$, where

${\rho}_{\alpha}^{\infty}$ corresponds to the ion density in bulk phase at

$\mathsf{\Phi}\left(r\right)=0$. A well-known approximation for Equation (

2) can be introduced for simple and diluted ions under the condition of charge neutrality

$({\sum}_{\alpha}{z}_{\alpha}{\rho}_{\alpha}^{\infty}=0)$, unit valency

z and moderate maximum electrostatic potential

$\mathsf{\Phi}\left({r}_{s}\right)={\mathsf{\Phi}}_{s}$ at the hydrodynamic boundary position

${r}_{s}$ with

${\mathsf{\Phi}}_{s}/{k}_{B}T\ll 1$. With regard to the latter condition, Equation (

2) can be linearized, which yields for the electrostatic potential around a spherical object in presence of monovalent ions

with the Debye-Hückel length

thereby highlighting a short-range decay of electrostatic interactions at finite salt concentration. Thus, it becomes clear that the ion density, the temperature and the dielectric constant as parameters of the solution have a significant influence on the polyelectrolyte behavior. In terms of a mechanistic explanation, an ion cloud around the polyelectrolyte forms due to attractive Coulomb interactions, which thus lowers the effective charge in terms of electrostatic screening effects. It was recently shown that the corresponding charge screening mechanisms also govern the dynamic properties of polyelectrolytes [

6,

39,

40,

41,

42]. In more detail, for a polyelectrolyte with mean radius of gyration

${R}_{g}$, a screening of hydrodynamic interactions can be observed for

${R}_{g}\gg 1/{\kappa}_{D}$ whereas a more standard polymer-like behavior becomes obvious for

${R}_{g}\le 1/{\kappa}_{D}$ [

10,

11]. The corresponding PB equation as well as the Debye-Hückel approach are typical examples of mean field theories. More specifically, all ion sizes and correlations as well as excluded volume effects are ignored such that only the most dominating conributions are taken into consideration by simplification.

#### 2.2. Counterion Condensation Theory

For strong polyelectrolytes with a high charge fraction, the full dissociation of counterions is reduced by electrostatic attraction in terms of condensed counterions. The Manning-Oosawa counterion condensation (MOCC) theory [

43,

44,

45] aims to estimate the number of condensed counterions by introducing a very long and charged cylinder which mimics the properties of an idealized polyelectrolyte. For the sake of mathematical simplicity, all ionic correlations, finite ion sizes as well as the presence of explicit solvent molecules are ignored. In principle, counterion condensation mechanisms rely on a combination of strong electrostatic interactions between the counterions and the polyelectrolyte and the translational entropy of the free counterions [

46,

47,

48,

49]. The loss of the translation entropy for the counterions upon condensation is compensated by electrostatic attraction which rationalizes the fact that only highly charged polyelectrolytes reveal counterion condensation behavior. Moreover, in very dilute polyelectrolyte solutions, the entropic penalty for counterion condensation is very high, such that free counterions dominate. However, with increasing polyelectrolyte concentration, the entropic loss for counterion condensation decreases, such that a finite number of condensed counterions can be observed [

48]. Noteworthy, one should also consider the solvation entropy and enthalpy between the counterions and the solvent but these contributions are usually neglected for the sake of simplicity. Further approaches also take the polyelectrolyte flexibility as well as the polarity into consideration [

46,

47,

48]. However, in terms of a straightforward mean field consideration which ignores all specific intra- and intermolecular interactions, the stable fraction of condensed counterions is determined at the threshold where the derivative of the resulting free energy with respect to the amount of condensed counterions vanishes [

49]. Hence, the central quantity in the MOCC theory is the so-called Manning parameter

with the Bjerrum length

and the contour charge length

b, which is the distance between two charged groups of the polyelectrolyte. The value of the Bjerrum length corresponds to the distance between the charges, where electrostatic interactions are of comparable magnitude as the thermal energy

${k}_{B}T$. In accordance with the theory [

44], strong counterion condensation sets in for values

$\xi \ge 1$, a condition which is met for polyelectrolytes with small

b and solvents with large

${\lambda}_{B}$ (Equation (

6)). Thus, even at very dilute or vanishing salt concentrations, electrostatic interactions between the ions can be ignored for distances

$r\ge {\lambda}_{B}$, as induced by

dielectric screening mechanisms of the surrounding solvent molecules [

34,

50].

An explicit expression for the number of condensed counterions can be derived as follows [

44]. With regard to the Debye-Hückel approach (Equation (

3)), the electrostatic interactions and thus the electrostatic work to assemble a linear polyelectrolyte with

P monomers of charge

q reads [

51]

which corresponds to a very long and linear polyelectrolyte with fixed charge fraction. Thereby, a mean field description of the linearized PB equation is taken into consideration, which also points to the fact that all ionic correlations as well as excluded-volume and finite size effects of the ions are ignored. Due to the presence of counterions, the reduced charge of a monomer reads

with

$0\le \theta \le 1$ as a consequence of the fraction

$\theta $ of condensed counterions. The effective work overcoming the loss of translational entropy for the counterions required to assemble the effective charge on the polyelectrolyte thus reads [

51]

after insertion of Equation (

8). In presence of a monovalent and 1:1 electrolyte salt with bulk concentration

${c}_{s}$, the work required to transfer

$\theta $ counterions from bulk solution to the polyelectrolyte [

51] is given by

with the partition function

$Z\left(\theta \right)$ for the condensed counterions, and with the concentration of one uncondensed counterion

${c}_{1}$ as reference state. As a sum of both contributions, the total work required for the formation of the polyelectrolyte reads

$w={w}_{\mathrm{tr}}+{w}_{\mathrm{el}}$ which separates the work into ion- and polyelectrolyte-related contributions. If a steady equilibrium distribution of the ions around the polyelectrolyte is assumed, the fraction of condensed counterions can be calculated by

which yields

where

$f\left(\theta \right)$ does not depend on the salt concentration.

With regard to Equation (

12), one can define two limiting cases. For

$\xi \le 1$, it follows that the equilibrium state of minimum free energy is located at

$\theta =0$ which corresponds to free counterions only. In contrast for

$\xi >1$, it follows that Equation (

12) changes sign at

$\theta =1-{\xi}^{-1}$, which corresponds to the actual minimum free energy state and thus highlights the presence of non-vanishing condensed counterions. Hence, the actual stable fraction of condensed counterions reads

which implies that counterion condensation is initiated by small values of contour charge lengths as well as high values for the Bjerrum length in accordance with Equation (

5). In summary, all mean field approaches ignore electrostatic correlations between the ions as well as finite size and excluded volume effects for the sake of simplicity. Moreover, long range electrostatic interactions are replaced by short range counterion screening interactions which emphasizes the mean field characteristics of the previous approach. Despite all crucial approximations, the validity of the MOCC theory was demonstrated for coarse grained polyelectrolyte and counterions in a continuum solvent [

52].

A more advanced theory, focusing on an explicit expression for the radial counterion density around the polyelectrolyte is represented by the PB cell model approach [

52,

53]. Here, the electrostatic potential is computed around a charged rod, which corresponds to a very long cylinder with radius

${r}_{0}$, thereby resembling an idealized polyelectrolyte with uniform charge distribution. Moreover, it is assumed that the rod is embedded in a cylindrical cell with a fixed and finite radius. With regard to these considerations, the PB equation (Equation (

2)) is transformed to cylindrical coordinates in order to evaluate the charge distribution of monovalent counterions in terms of the Debye-Hückel approach (Equation (

4)) around the rod. In combination with boundary conditions on the derivative of the electrostatic potential, two equations can be obtained, which can be used to define an expression for the Manning radius

${R}_{M}$ and a prefactor

${\gamma}_{M}$. The Manning radius thus defines the largest distance for condensed counterions such that the radial fraction of condensed counterions is given by

which coincides for

$r={R}_{M}$ with Equation (

13).

Despite the reasonable assumptions of the MOCC theory and its modifications, recent atomistic MD simulations highlighted significant deviations to the cell model approach for short distances around polyelectrolytes [

15,

18,

54,

55]. In order to correct for these deviations, it was suggested [

54] to introduce a modified Poisson-Boltzmann equation with

${\psi}_{\infty}=0$ according to

with the additional potential

where the prefactor

${V}_{0}$ can be interpreted as an empirical hydration potential, with

${r}_{0}$ as the corresponding position of the first counterion shell around the polyelectrolyte and

${\sigma}_{s}$ as the range of ion-specific interactions [

54]. With regard to the contribution of the hydration potential, it becomes clear that the solvent has a significant influence on the counterion distribution as well as on the amount of condensed counterions [

15].

In summary, it can be concluded that mean field descriptions mainly rely on continuum solvent approaches with fixed values for the dielectric constant as well as crucial approximations for the ions and the polyelectrolyte, such that any molecular or local interactions between the species are ignored.