# Theoretical and Computational Insight into Solvent and Specific Ion Effects for Polyelectrolytes: The Importance of Local Molecular Interactions

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background: Polyelectrolytes and Ions in Solution

#### 2.1. Electrostatic Screening Effects

#### 2.2. Counterion Condensation Theory

## 3. Solvent Effects

#### 3.1. Dielectric Decrement Effects

#### 3.2. Molecular Properties of the Solvent: Donor/Acceptor Numbers and Chemical Hardnesses

#### 3.3. Weak Polyelectrolytes: pH Value Effects

## 4. Specific Ion Effects

## 5. Co-Solute and Co-Solvent Effects

## 6. Summary and Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**Left**) Normalized ionic conductivity $\mathsf{\Lambda}/{\mathsf{\Lambda}}_{0}$ of polyelectrolyte solutions with varying salt concentrations ${C}^{1/2}$. The single dots denote the values of experimental outcomes. The straight solid red line shows the results of coarse-grained molecular dynamics simulations with a varying dielectric constant. The dashed green line highlights the corresponding results for a constant value of ${\u03f5}_{r}$. Snapshots of polyelectrolyte conformations for specific salt concentrations in combination with counterions are shown in the inset. (

**Right**) Fraction of condensed counterions ${f}_{\mathrm{cci}}$ around a highly charged polyelectrolyte for constant (blue circles) and varying values of the dielectric constant ${\u03f5}_{\mathrm{poly}}$ (red triangles). A foxed dielectric constant was set to a value of ${\u03f5}_{r}=56$ whereas the resulting outcomes in terms of dielectric decrement effects are denoted as black triangles. Figure reproduced from Ref. [65].

**Figure 2.**Simulation snapshots of sulfonated oligosulfonic acids with sodium counterions (blue spheres) in water (

**left side**), dimethyl sulfoxide (DMSO) (

**middle panel**), and chloroform (

**right side**). Solvent molecules are ignored for the sake of clarity. Figure reproduced from Ref. [18].

**Figure 3.**Fraction of condensed counterions around cylindric model polyelectrolytes with identical charge density in water (

**left side**), methanol (

**middle**) and dimethylacetamide (DMAc,

**right side**). The straight blue lines correspond to the predicted fraction of counterions from counterion condensation theory. Figure reproduced from Ref. [15].

**Figure 4.**Endothermic $\Delta {E}_{\mathrm{AB}}>0$ and exothermic $\Delta {E}_{\mathrm{AB}}<0$ regions for solvents with distinct hardnesses ${\eta}_{\mathrm{S}}$ and electronegativities ${\chi}_{\mathrm{S}}$ in combination with a cation (blue square) with arbitrary values of ${\eta}_{\mathrm{A}}=10$ eV and ${\chi}_{\mathrm{A}}=5$ eV and an anion with arbitrarily chosen values of ${\eta}_{\mathrm{B}}=2$ eV and ${\chi}_{\mathrm{B}}=1$ eV (red square). The red solid line denotes the maximum value for an endothermic reaction energy as defined for a solvent with ${\chi}_{\mathrm{S}}^{\mathrm{max}}$ (Equation (23)). The black solid lines denote the separatrices for values of $\Delta {E}_{\mathrm{AB}}=0$. Figure reproduced from Ref. [80].

**Figure 5.**Resulting Debye-Hückel lengths ${\lambda}_{D}$ and degree of association $\overline{n}$ (

**bottom**) for flexible weak polyelectrolytes with different pK${}_{\mathrm{a}}$ values and a fixed Bjerrum length as obtained by the reaction ensemble (RE) method and the constant pH method. The actual pH value of the solution is defined by the relation pK${}_{\mathrm{a}}$-pH. Figures reproduced from Ref. [24].

**Figure 6.**Fraction of condensed counterions $x\left(r\right)$ around polyglutamic acid (

**top left**), polyallylamine hydrochloride (

**top right**), polystyrene sulfonate (

**bottom left**) and polyacrylic acid (

**bottom right**) for various counterion species as denoted in the legend. The dashed black lines correspond to the fits of the modified PB equation (Equation (15). The effects of varying line charge density are studied for polyacrylic acid and polyallylamine hydrochloride on the right side. Figure adapted from Ref. [55].

**Figure 7.**MD simulation outcomes of the resulting dielectric constant ${\u03f5}_{r}$ for an aqueous DMSO solution with increasing mole fractions of DMSO ${x}_{\mathrm{DMSO}}$ in presence (blue triangles) and absence of low concentrated ion pairs (bue). The corresponding values for TIP3P water and DMSO are ${\u03f5}_{r}^{\mathrm{TIP}3\mathrm{P}}=95.32$ and ${\u03f5}_{r}^{\mathrm{DMSO}}=55.54$. The black squares correspond to experimental results. Figure reproduced from Ref. [19].

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Smiatek, J.
Theoretical and Computational Insight into Solvent and Specific Ion Effects for Polyelectrolytes: The Importance of Local Molecular Interactions. *Molecules* **2020**, *25*, 1661.
https://doi.org/10.3390/molecules25071661

**AMA Style**

Smiatek J.
Theoretical and Computational Insight into Solvent and Specific Ion Effects for Polyelectrolytes: The Importance of Local Molecular Interactions. *Molecules*. 2020; 25(7):1661.
https://doi.org/10.3390/molecules25071661

**Chicago/Turabian Style**

Smiatek, Jens.
2020. "Theoretical and Computational Insight into Solvent and Specific Ion Effects for Polyelectrolytes: The Importance of Local Molecular Interactions" *Molecules* 25, no. 7: 1661.
https://doi.org/10.3390/molecules25071661