## 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

- McNaught, A.D.; McNaught, A.D. Compendium of Chemical Terminology; Blackwell Science Oxford: Oxford, UK, 1997; Volume 1669. [Google Scholar]
- Dobrynin, A.V.; Colby, R.H.; Rubinstein, M. Scaling theory of polyelectrolyte solutions. Macromolecules
**1995**, 28, 1859–1871. [Google Scholar] [CrossRef] - Dobrynin, A.V.; Rubinstein, M. Theory of polyelectrolytes in solutions and at surfaces. Prog. Polym. Sci.
**2005**, 30, 1049–1118. [Google Scholar] [CrossRef] - De Gennes, P.G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, USA, 1979. [Google Scholar]
- Doi, M.; Edwards, S.F. The Theory of Polymer Dynamics; Oxford University Press: Oxford, UK, 1988. [Google Scholar]
- Boroudjerdi, H.; Kim, Y.W.; Naji, A.; Netz, R.R.; Schlagberger, X.; Serr, A. Statics and dynamics of strongly charged soft matter. Phys. Rep.
**2005**, 416, 129–199. [Google Scholar] [CrossRef] - Slater, G.W.; Holm, C.; Chubynsky, M.V.; de Haan, H.W.; Dube, A.; Grass, K.; Hickey, O.A.; Kingsburry, C.; Sean, D.; Shendruk, T.N.; et al. Modeling the separation of macromolecules: A review of current computer simulation methods. Electrophoresis
**2009**, 30, 792–818. [Google Scholar] [CrossRef] - Pagonabarraga, I.; Rotenberg, B.; Frenkel, D. Recent advances in the modelling and simulation of electrokinetic effects: Bridging the gap between atomistic and macroscopic descriptions. Phys. Chem. Chem. Phys.
**2010**, 12, 9566–9580. [Google Scholar] [CrossRef] - Streek, M.; Schmid, F.; Duong, T.T.; Ros, A. Mechanisms of DNA separation in entropic trap arrays: A Brownian dynamics simulation. J. Biotechnol.
**2004**, 112, 79–89. [Google Scholar] [CrossRef] - Frank, S.; Winkler, R.G. Mesoscale hydrodynamic simulation of short polyelectrolytes in electric fields. J. Chem. Phys.
**2009**, 131, 234905. [Google Scholar] [CrossRef] - Grass, K.; Böhme, U.; Scheler, U.; Cottet, H.; Holm, C. Importance of hydrodynamic shielding for the dynamic behavior of short polyelectrolyte chains. Phys. Rev. Lett.
**2008**, 100, 096104. [Google Scholar] [CrossRef] - Smiatek, J.; Schmid, F. Mesoscopic simulations of electroosmotic flow and electrophoresis in nanochannels. Comput. Phys. Commun.
**2011**, 182, 1941–1944. [Google Scholar] [CrossRef] - Smiatek, J.; Schmid, F. Polyelectrolyte electrophoresis in nanochannels: A dissipative particle dynamics simulation. J. Phys. Chem. B
**2010**, 114, 6266–6272. [Google Scholar] [CrossRef] - Muthukumar, M. Dynamics of polyelectrolyte solutions. J. Chem. Phys.
**1997**, 107, 2619–2635. [Google Scholar] [CrossRef] - Krishnamoorthy, A.N.; Holm, C.; Smiatek, J. Specific ion effects for polyelectrolytes in aqueous and non-aqueous media: The importance of the ion solvation behavior. Soft Matter
**2018**, 14, 6243–6255. [Google Scholar] [CrossRef] [PubMed] - Smiatek, J.; Holm, C. From the atomistic to the macromolecular scale: Distinct simulation approaches for polyelectrolyte solutions. In Handbook of Materials Modeling; Springer International Publishing: Heidelberg, Germany, 2018. [Google Scholar]
- Smiatek, J.; Harishchandra, R.K.; Rubner, O.; Galla, H.J.; Heuer, A. Properties of compatible solutes in aqueous solution. Biophys. Chem.
**2012**, 160, 62–68. [Google Scholar] [CrossRef] [PubMed] - Smiatek, J.; Wohlfarth, A.; Holm, C. The solvation and ion condensation properties for sulfonated polyelectrolytes in different solvents—A computational study. New J. Phys.
**2014**, 16, 025001. [Google Scholar] [CrossRef] - Krishnamoorthy, A.N.; Zeman, J.; Holm, C.; Smiatek, J. Preferential solvation and ion association properties in aqueous dimethyl sulfoxide solutions. Phys. Chem. Chem. Phys.
**2016**, 18, 31312–31322. [Google Scholar] [CrossRef] - Krishnamoorthy, A.N.; Holm, C.; Smiatek, J. The influence of co-solutes on the chemical equilibrium—A Kirkwood-Buff theory for ion pair association-dissociation processes in ternary electrolyte solutions. J. Phys. Chem. C
**2018**, 122, 10293–10392. [Google Scholar] [CrossRef] - Oprzeska-Zingrebe, E.A.; Smiatek, J. Some Notes on the Thermodynamic Accuracy of Coarse-Grained Models. Front. Mol. Biosci.
**2019**, 6, 87. [Google Scholar] [CrossRef] - Guenza, M.; Dinpajooh, M.; McCarty, J.; Lyubimov, I. Accuracy, transferability, and efficiency of coarse-grained models of molecular liquids. J. Phys. Chem. B
**2018**, 122, 10257–10278. [Google Scholar] [CrossRef] - Onufriev, A.V.; Case, D.A. Generalized Born Implicit Solvent Models for Biomolecules. Annu. Rev. Biophys.
**2019**, 48, 275–296. [Google Scholar] [CrossRef] - Landsgesell, J.; Holm, C.; Smiatek, J. Simulation of weak polyelectrolytes: A comparison between the constant pH and the reaction ensemble method. Eur. Phys. J. Spec. Top.
**2017**, 226, 725–736. [Google Scholar] [CrossRef] - Landsgesell, J.; Holm, C.; Smiatek, J. Wang–Landau Reaction Ensemble Method: Simulation of Weak Polyelectrolytes and General Acid–Base Reactions. J. Chem. Theory Comput.
**2017**, 13, 852–862. [Google Scholar] [CrossRef] [PubMed] - Fahrenberger, F.; Hickey, O.A.; Smiatek, J.; Holm, C. The influence of charged-induced variations in the local permittivity on the static and dynamic properties of polyelectrolyte solutions. J. Chem. Phys.
**2015**, 143, 243140. [Google Scholar] [CrossRef] [PubMed] - Mukhopadhyay, A.; Fenley, A.T.; Tolokh, I.S.; Onufriev, A.V. Charge hydration asymmetry: The basic principle and how to use it to test and improve water models. J. Phys. Chem. B
**2012**, 116, 9776–9783. [Google Scholar] [CrossRef] [PubMed] - Weyman, A.; Bier, M.; Holm, C.; Smiatek, J. Microphase separation and the formation of ion conductivity channels in poly (ionic liquid) s: A coarse-grained molecular dynamics study. J. Chem. Phys.
**2018**, 148, 193824. [Google Scholar] [CrossRef] - Holm, C.; Limbach, H.; Kremer, K. Poor-solvent polyelectrolytes. J. Phys. Condens. Matter
**2002**, 15, S205. [Google Scholar] [CrossRef] - Limbach, H.; Sayar, M.; Holm, C. Polyelectrolyte bundles. J. Phys. Condens. Matter
**2004**, 16, S2135. [Google Scholar] [CrossRef] - Dormidontova, E.E.; Erukhimovich, I.Y.; Khokhlov, A.R. Microphase separation in poor-solvent polyelectrolyte solutions: Phase diagram. Macromol. Theory Simul.
**1994**, 3, 661–675. [Google Scholar] [CrossRef] - Cerda, J.J.; Qiao, B.; Holm, C. Understanding polyelectrolyte multilayers: An open challenge for simulations. Soft Matter
**2009**, 5, 4412–4425. [Google Scholar] [CrossRef] - Vögele, M.; Holm, C.; Smiatek, J. Coarse-grained simulations of polyelectrolyte complexes: MARTINI models for poly(styrene sulfonate) and poly(diallyldimethylammonium). J. Chem. Phys.
**2015**, 143, 243151. [Google Scholar] [CrossRef] - Smiatek, J.; Heuer, A.; Winter, M. Properties of Ion Complexes and their Impact on Charge Transport in Organic Solvent–based Electrolyte Solutions for Lithium Batteries: Insights from a Theoretical Perspective. Batteries
**2018**, 4, 62. [Google Scholar] [CrossRef] - Andreev, M.; Prabhu, V.M.; Douglas, J.F.; Tirrell, M.; de Pablo, J.J. Complex coacervation in polyelectrolytes from a coarse-grained model. Macromolecules
**2018**, 51, 6717–6723. [Google Scholar] [CrossRef] - Andelman, D. Electrostatic properties of membranes: The Poisson-Boltzmann theory. In Handbook of Biological Physics; Elsevier: Amsterdam, The Netherland, 1995; Volume 1, pp. 603–642. [Google Scholar]
- Grochowski, P.; Trylska, J. Continuum molecular electrostatics, salt effects, and counterion binding—A review of the Poisson–Boltzmann theory and its modifications. Biopolymers
**2008**, 89, 93–113. [Google Scholar] [CrossRef] [PubMed] - Israelachvili, J.N. Intermolecular and Surface Forces; Academic Press: Cambridge, MA, USA, 2011. [Google Scholar]
- Hickey, O.A.; Shendruk, T.N.; Harden, J.L.; Slater, G.W. Simulations of free-solution electrophoresis of polyelectrolytes with a finite Debye length using the Debye-Hückel approximation. Phys. Rev. Lett.
**2012**, 109, 098302. [Google Scholar] [CrossRef] [PubMed] - Hickey, O.A.; Holm, C.; Smiatek, J. Lattice-Boltzmann simulations of the electrophoretic stretching of polyelectrolytes: The importance of hydrodynamic interactions. J. Chem. Phys.
**2014**, 140, 164904. [Google Scholar] [CrossRef] - Roy, T.; Szuttor, K.; Smiatek, J.; Holm, C.; Hardt, S. Stretching of surface-tethered polymers in pressure-driven flow under confinement. Soft Matter
**2017**, 13, 6189–6196. [Google Scholar] [CrossRef] - Szuttor, K.; Roy, T.; Hardt, S.; Holm, C.; Smiatek, J. The stretching force on a tethered polymer in pressure-driven flow. J. Chem. Phys.
**2017**, 147, 034902. [Google Scholar] [CrossRef] - Manning, G. Limiting Laws and Counterion Condensation in Polyelectrolyte Solutions I. Colligative Properties. J. Chem. Phys.
**1969**, 51, 924–933. [Google Scholar] [CrossRef] - Manning, G.S.; Ray, J. Counterion condensation revisited. J. Biomol. Struct. Dyn.
**1998**, 16, 461–476. [Google Scholar] [CrossRef] - Oosawa, F. Polyelectrolytes; Marcel Dekker: New York, NY, USA, 1971. [Google Scholar]
- Muthukumar, M. Theory of counter-ion condensation on flexible polyelectrolytes: Adsorption mechanism. J. Chem. Phys.
**2004**, 120, 9343–9350. [Google Scholar] [CrossRef] - Dobrynin, A.V.; Rubinstein, M. Counterion condensation and phase separation in solutions of hydrophobic polyelectrolytes. Macromolecules
**2001**, 34, 1964–1972. [Google Scholar] [CrossRef] - Dobrynin, A.V. Effect of counterion condensation on rigidity of semiflexible polyelectrolytes. Macromolecules
**2006**, 39, 9519–9527. [Google Scholar] [CrossRef] - Manning, G.S. Counterion condensation on charged spheres, cylinders, and planes. J. Phys. Chem. B
**2007**, 111, 8554–8559. [Google Scholar] [CrossRef] [PubMed] - Marcus, Y. Ions in Solution and Their Solvation; John Wiley & Sons: Hoboken, NJ, USA, 2015. [Google Scholar]
- Manning, G.S. Counterion condensation theory constructed from different models. Phys. A
**1996**, 231, 236–253. [Google Scholar] [CrossRef] - Deserno, M.; Holm, C.; May, S. Fraction of condensed counterions around a charged rod: Comparison of Poisson-Boltzmann theory and computer simulations. Macromolecules
**2000**, 33, 199–206. [Google Scholar] [CrossRef] - Deserno, M.; Holm, C. Cell-model and Poisson-Boltzmann-theory: A brief introduction. In Electrostatic Effects in Soft Matter and Biophysics; Holm, C., Kékicheff, P., Podgornik, R., Eds.; NATO Science Series II—Mathematics, Physics and Chemistry; Kluwer Academic Publishers: Dordrecht, The Netherlands, 2001; Volume 46, pp. 27–50. [Google Scholar]
- Heyda, J.; Dzubiella, J. Ion-specific counterion condensation on charged peptides: Poisson–Boltzmann vs. atomistic simulations. Soft Matter
**2012**, 8, 9338–9344. [Google Scholar] [CrossRef] - Batys, P.; Luukkonen, S.; Sammalkorpi, M. Ability of Poisson–Boltzmann Equation to Capture Molecular Dynamics Predicted Ion Distribution around Polyelectrolytes. Phys. Chem. Chem. Phys.
**2017**, 19, 24583–24593. [Google Scholar] [CrossRef] - Zeman, J.; Holm, C.; Smiatek, J. The Effect of Small Organic Cosolutes on Water Structure and Dynamics. J. Chem. Eng. Data
**2019**. [Google Scholar] [CrossRef] - Smiatek, J. Osmolyte effects: Impact on the aqueous solution around charged and neutral spheres. J. Phys. Chem. B
**2014**, 118, 771–782. [Google Scholar] [CrossRef] - Schröder, C.; Haberler, M.; Steinhauser, O. On the computation and contribution of conductivity in molecular ionic liquids. J. Chem. Phys.
**2008**, 128, 134501. [Google Scholar] [CrossRef] - Michalowsky, J.; Zeman, J.; Holm, C.; Smiatek, J. A polarizable MARTINI model for monovalent ions in aqueous solution. J. Chem. Phys.
**2018**, 149, 163319. [Google Scholar] [CrossRef] - Neumann, M. Dipole moment fluctuation formulas in computer simulations of polar systems. Mol. Phys.
**1983**, 50, 841–858. [Google Scholar] [CrossRef] - Caillol, J.; Levesque, D.; Weis, J. Theoretical calculation of ionic solution properties. J. Chem. Phys.
**1986**, 85, 6645–6657. [Google Scholar] [CrossRef] - Caillol, J.M.; Levesque, D.; Weis, J.J. Electrical properties of polarizable ionic solutions. I. Theoretical aspects. J. Chem. Phys.
**1989**, 91, 5544–5554. [Google Scholar] [CrossRef] - Bonthuis, D.J.; Gekle, S.; Netz, R.R. Dielectric profile of interfacial water and its effect on double-layer capacitance. Phys. Rev. Lett.
**2011**, 107, 166102. [Google Scholar] [CrossRef] - Gekle, S.; Netz, R.R. Anisotropy in the dielectric spectrum of hydration water and its relation to water dynamics. J. Chem. Phys.
**2012**, 137, 104704. [Google Scholar] [CrossRef] - Fahrenberger, F.; Hickey, O.A.; Smiatek, J.; Holm, C. Importance of varying permittivity on the conductivity of polyelectrolyte solutions. Phys. Rev. Lett.
**2015**, 115, 118301. [Google Scholar] [CrossRef] - Vögele, M.; Holm, C.; Smiatek, J. Properties of the polarizable MARTINI water model: A comparative study for aqueous electrolyte solutions. J. Mol. Liquids
**2015**, 212, 103. [Google Scholar] [CrossRef] - Michalowsky, J.; Schäfer, L.V.; Holm, C.; Smiatek, J. A refined polarizable water model for the coarse-grained MARTINI force field with long-range electrostatic interactions. J. Chem. Phys.
**2017**, 146, 054501. [Google Scholar] [CrossRef] - Hahn, M.B.; Solomun, T.; Wellhausen, R.; Hermann, S.; Seitz, H.; Meyer, S.; Kunte, H.J.; Zeman, J.; Uhlig, F.; Smiatek, J.; et al. Influence of the Compatible Solute Ectoine on the Local Water Structure: Implications for the Binding of the Protein G5P to DNA. J. Phys. Chem. B
**2015**, 119, 15212–15220. [Google Scholar] [CrossRef] - Hess, B.; van der Vegt, N.F. Cation specific binding with protein surface charges. Proc. Natl. Acad. Sci. USA
**2009**, 106, 13296–13300. [Google Scholar] [CrossRef] - Chremos, A.; Douglas, J.F. Communication: Counter-ion solvation and anomalous low-angle scattering in salt-free polyelectrolyte solutions. J. Chem. Phys.
**2017**, 147, 241103. [Google Scholar] [CrossRef] [PubMed] - Chremos, A.; Douglas, J.F. Polyelectrolyte association and solvation. J. Chem. Phys.
**2018**, 149, 163305. [Google Scholar] [CrossRef] [PubMed] - Chremos, A.; Douglas, J.F. The influence of polymer and ion solvation on the conformational properties of flexible polyelectrolytes. Gels
**2018**, 4, 20. [Google Scholar] [CrossRef] [PubMed] - Wohlfarth, A.; Smiatek, J.; Kreuer, K.D.; Takamuku, S.; Jannasch, P.; Meier, J. Proton dissociation of sulfonated polysulfones: Influence of molecular structure and conformation. Macromolecules
**2015**, 48, 1134–1143. [Google Scholar] [CrossRef] - Krishnamoorthy, A.N.; Oldiges, K.; Heuer, A.; Winter, M.; Cekic-Laskovic, I.; Holm, C.; Smiatek, J. Electrolyte solvents for high voltage lithium ion batteries: Ion pairing mechanisms, ionic conductivity, and specific anion effects in adiponitrile. Phys. Chem. Chem. Phys.
**2018**. [Google Scholar] [CrossRef] - Borodin, O.; Smith, G.D. LiTFSI structure and transport in ethylene carbonate from molecular dynamics simulations. J. Phys. Chem. B
**2006**, 110, 4971–4977. [Google Scholar] [CrossRef] - Lesch, V.; Heuer, A.; Holm, C.; Smiatek, J. Properties of Apolar Solutes in Alkyl Imidazolium-Based Ionic Liquids: The Importance of Local Interactions. ChemPhysChem
**2016**, 17, 387–394. [Google Scholar] [CrossRef] - Nandy, A.; Smiatek, J. Mixtures of LiTFSI and urea: Ideal thermodynamic behavior as key to the formation of deep eutectic solvents? Phys. Chem. Chem. Phys.
**2019**, 21, 12279–12287. [Google Scholar] [CrossRef] - Gutmann, V. Empirical parameters for donor and acceptor properties of solvents. Electrochim. Acta
**1976**, 21, 661–670. [Google Scholar] [CrossRef] - Reichardt, C.; Welton, T. Solvents and Solvent Effects in Organic Chemistry; John Wiley & Sons: Hoboken, NJ, USA, 2011. [Google Scholar]
- Smiatek, J. Enthalpic contributions to solvent–solute and solvent–ion interactions: Electronic perturbation as key to the understanding of molecular attraction. J. Chem. Phys.
**2019**, 150, 174112. [Google Scholar] [CrossRef] - Smiatek, J. Specific Ion Effects and the Law of Matching Solvent Affinities: A Conceptual Density Functional Theory Approach. J. Phys. Chem. B
**2020**, 124, 2191–2197. [Google Scholar] [CrossRef] [PubMed] - Parr, R.G.; Pearson, R.G. Absolute hardness: Companion parameter to absolute electronegativity. J. Am. Chem. Soc.
**1983**, 105, 7512–7516. [Google Scholar] [CrossRef] - Chattaraj, P.K.; Giri, S. Electrophilicity index within a conceptual DFT framework. Ann. Rep. Phys. Chem. C
**2009**, 105, 13–39. [Google Scholar] [CrossRef] - Geerlings, P.; De Proft, F.; Langenaeker, W. Conceptual density functional theory. Chem. Rev.
**2003**, 103, 1793–1873. [Google Scholar] [CrossRef] - Chen, W.; Morrow, B.H.; Shi, C.; Shen, J.K. Recent development and application of constant pH molecular dynamics. Mol. Simul.
**2014**, 40, 830–838. [Google Scholar] [CrossRef] - Reed, C.E.; Reed, W.F. Monte Carlo study of titration of linear polyelectrolytes. J. Chem. Phys.
**1992**, 96, 1609–1620. [Google Scholar] [CrossRef] - Mongan, J.; Case, D.A.; McCammon, J.A. Constant pH molecular dynamics in generalized Born implicit solvent. J. Comput. Chem.
**2004**, 25, 2038–2048. [Google Scholar] [CrossRef] - Heath Turner, C.; Brennan, J.K.; Lisal, M.; Smith, W.R.; Karl Johnson, J.; Gubbins, K.E. Simulation of chemical reaction equilibria by the reaction ensemble Monte Carlo method: A review. Mol. Simul.
**2008**, 34, 119–146. [Google Scholar] [CrossRef] - Smith, P.E. Cosolvent interactions with biomolecules:Relating computer simulation data to experimental thermodynamic data. J. Phys. Chem. B
**2004**, 108, 18716–18724. [Google Scholar] [CrossRef] - Mazzini, V.; Craig, V.S. Specific-ion effects in non-aqueous systems. Curr. Opin. Colloid Int. Sci.
**2016**, 23, 82–93. [Google Scholar] [CrossRef] - Mazzini, V.; Craig, V.S. What is the fundamental ion-specific series for anions and cations? Ion specificity in standard partial molar volumes of electrolytes and electrostriction in water and non-aqueous solvents. Chem. Sci.
**2017**, 8, 7052–7065. [Google Scholar] [CrossRef] [PubMed] - Mazzini, V.; Liu, G.; Craig, V.S. Probing the Hofmeister series beyond water: Specific-ion effects in non-aqueous solvents. J. Chem. Phys.
**2018**, 148, 222805. [Google Scholar] [CrossRef] [PubMed] - Collins, K.D. Charge density-dependent strength of hydration and biological structure. Biophys. J.
**1997**, 72, 65. [Google Scholar] [CrossRef] - Salis, A.; Ninham, B.W. Models and mechanisms of Hofmeister effects in electrolyte solutions, and colloid and protein systems revisited. Chem. Soc. Rev.
**2014**, 43, 7358–7377. [Google Scholar] [CrossRef] - Mazzini, V.; Craig, V.S. Volcano Plots Emerge from a Sea of Nonaqueous Solvents: The Law of Matching Water Affinities Extends to All Solvents. ACS Cent. Sci.
**2018**, 4, 1056–1064. [Google Scholar] [CrossRef] - Lytle, T.K.; Chang, L.W.; Markiewicz, N.; Perry, S.L.; Sing, C.E. Designing electrostatic interactions via polyelectrolyte monomer sequence. ACS Cent. Sci.
**2019**, 5, 709–718. [Google Scholar] [CrossRef] - Sing, C.E. Development of the modern theory of polymeric complex coacervation. Adv. Colloid Interface Sci.
**2017**, 239, 2–16. [Google Scholar] [CrossRef] - Yigit, C.; Heyda, J.; Ballauff, M.; Dzubiella, J. Like-charged protein-polyelectrolyte complexation driven by charge patches. J. Chem. Phys.
**2015**, 143, 064905. [Google Scholar] [CrossRef] - Chudoba, R.; Heyda, J.; Dzubiella, J. Tuning the collapse transition of weakly charged polymers by ion-specific screening and adsorption. Soft Matter
**2018**, 14, 9631–9642. [Google Scholar] [CrossRef] - Solis, F.J.; De La Cruz, M.O. Collapse of flexible polyelectrolytes in multivalent salt solutions. J. Chem. Phys.
**2000**, 112, 2030–2035. [Google Scholar] [CrossRef] - Antila, H.S.; Van Tassel, P.R.; Sammalkorpi, M. Repulsion between oppositely charged rod-shaped macromolecules: Role of overcharging and ionic confinement. J. Chem. Phys.
**2017**, 147, 124901. [Google Scholar] [CrossRef] [PubMed] - Nguyen, T.D.; Olvera de la Cruz, M. Manipulation of confined polyelectrolyte conformations through dielectric mismatch. ACS Nano
**2019**, 13, 9298–9305. [Google Scholar] [CrossRef] [PubMed] - Smiatek, J. Aqueous ionic liquids and their influence on protein conformations: An overview on recent theoretical and experimental insights. J. Phys. Condens. Matter
**2017**, 29, 233001. [Google Scholar] [CrossRef] [PubMed] - Ben-Naim, A.Y. Statistical Thermodynamics for Chemists and Biochemists; Springer: Berlin, Germany, 1992. [Google Scholar]
- Pierce, V.; Kang, M.; Aburi, M.; Weerasinghe, S.; Smith, P.E. Recent Applications of Kirkwood-Buff Theory to Biological Systems. Cell. Biochem. Biophys.
**2008**, 50, 1–22. [Google Scholar] [CrossRef] [PubMed]

**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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