Manning's Counterion Condensation Research Articles

Stretchable polyvinyl alcohol and sodium alginate double network ionic hydrogels for low-grade heat harvesting with ultrahigh thermopower

Recently, employing ionic thermoelectric (i-TE) materials is regarded as a promising strategy to harness low-grade waste heat due to their remarkable ionic Seebeck coefficient. By blending polyvinyl alcohol (PVA), sodium alginate (SA), and polyethylene glycol (PEG) and additional freeze-thaw method, PVA/SA/PEG hydrogel is developed. Via immersing the hydrogel in NaBF4 solutions with different concentrations, the TE and mechanical properties could be adjusted. PVA/SA/PEG/NaBF4-1.5 M hydrogel demonstrates exceptional mechanical properties, with tensile stress and strain up to 69 kPa and 114%, respectively. Moreover, PVA/SA/PEG/NaBF4-1.5 M hydrogel exhibits a high ionic conductivity of 31.4 mS/cm, a maximum ionic Seebeck coefficient of 66.7 mV/K, and an impressive power factor of 13.96 mW/m/K2. These outstanding performances originate from the synergistic effect of Manning's counterion condensation facilitated by SA and the crystal PVA chains. The prototype application of the PVA/SA/PEG/NaBF4-1.5 M hydrogel is demonstrated by a flexible ionic thermoelectric supercapacitor. With the external load resistance is 90 kΩ, the energy collected in one thermal cycle could achieve 4 mJ. This study introduces the exceptional stretchable PVA/SA/PEG/NaBF4 hydrogels with a record-high ionic Seebeck coefficient as promising i-TE materials for future TE application in low-grade waste heat.

Extended Donnan-Manning theory for selective ion partition and transport in ion exchange membrane

The use of ion exchange membranes (IEMs) for electrochemical ion-ion separation leverages the selectivity of the IEMs toward like-charged species via valence difference and/or other ion-membrane interactions. A mechanistic model that relates selectivity with membrane structural and chemical properties is lacking in the literature. Here, we extend the Manning's counter-ion condensation model for describing ion partition and ion mobility inside IEMs to mixed salts scenarios. We evaluate the extended Donnan-Manning model against experimental data from literature and compare the performance of the Donnan-Manning model to that of the ideal Donnan model and the Donnan-Affinity model. Our analysis shows that, despite its structural complexity, the Donnan-Manning model has less fitting parameters than the Donnan-Affinity model and generally outperforms the two other models in predicting counter-ion and co-ion partition. With the assumption of a higher mobility of condensed ions than that of uncondensed ions, the generalized Manning's model can also predict counter-ion mobility selectivity for cation exchange membranes, but its performance for predicting mobility selectivity for anion exchange membranes is still unsatisfactory.

Effect of Dielectric Saturation on Ion Activity Coefficients in Ion Exchange Membranes.

Polymeric ion exchange membranes are used in water purification processes to separate ions from water. The distribution and transport of ionic species through these membranes depend on a variety of factors, including membrane charge density, morphology, chemical structure, and the specific ionic species present in the fluid. The electrical potential distribution between membranes and solutions is typically described using models based on Donnan theory. An extension of the original theory is proposed to account for the nonideal behavior of ions both in the fluid and in the membrane as well to provide a more robust description of interactions of solutes with fixed charge groups on the polymer backbone. In this study, the variation in dielectric permittivity in the membrane medium with electric field strength is taken into account in a model based on Gouy–Chapman double-layer theory to provide a more accurate description of ion activity coefficients in an ion exchange membrane. A semianalytical model is presented that accounts for the variation in dielectric permittivity of water in a charged polymer membrane. A comparison of this model with Manning’s counterion condensation model clearly demonstrates that by incorporating changes in water dielectric permittivity with electric field strength, much better agreement with experiments can be obtained over a range of salt concentrations for different ions.

Mutual effects between single-stranded DNA conformation and Na+-Mg2+ ion competition in mixed salt solutions.

The ion-dependence of single-stranded DNA (ssDNA) conformational changes has attracted growing attention because of its biological and technological importance. Although single-species ion effects have been extensively explored, it is challenging to study the ssDNA conformational properties under mixed monovalent/divalent ion conditions due to the complications of ssDNA flexibility and ion-ion competition. In this study, we apply Langevin dynamics simulations to investigate mixed Na+/Mg2+ ion-dependent ssDNA conformations. The ssDNA structure is described using a coarse-grained model, in which the phosphate, base, and sugar of each nucleotide are represented by three different beads. A novel improvement in our simulation model is that mixed-salt-related electrostatic interactions are computed via combining Manning counterion condensation (MCC) theory with the Monte Carlo tightly bound ion (MCTBI) model. Based on this MCC-MCTBI combination, we report new empirical functions to describe the ion-concentration-dependent and ssDNA conformation/structure-dependent electrostatic effects. The calculation results relating to the ion binding properties and the simulation results relating to the ssDNA conformational properties are validated against experimental results. In addition, our simulation results suggest a quantitative relationship between the ssDNA conformation and Na+-Mg2+ competition; this in turn reveals their mutual impact in the ion atmosphere.

Aqueous ion partitioning in Nafion: Applicability of Manning's counter-ion condensation theory

Nafion is a well-studied polymer of significant interest as a fuel cell membrane due to rapid transport of protons, which is often ascribed, at least in part, to its phase-separated morphology. The literature detailing ion sorption and transport in Nafion diverges considerably from that of water purification membranes, with contrasting analyses and models. To help bridge these gaps and facilitate comparisons of membranes with various structures, this study presents measurements of ion activity coefficients and sorption in Nafion 117 equilibrated with aqueous solutions of NaCl, MgCl2, and Na2SO4, which are relevant to water purification. The experimental data are compared to a theoretical framework based on Manning's counter-ion condensation theory and Donnan theory. Good agreement between the model and data is observed for NaCl and MgCl2, using no adjustable parameters. However, the model significantly underpredicts Na2SO4 solubility, which is attributed to ion pairing effects. The results highlight the general applicability of counter-ion condensation theory to highly charged polymers, including those exhibiting nanoscale ordering, while revealing fundamental limitations related to ion pairing.

Assessment of anion exchange membrane selectivity with ionic membrane conductivity, revised with Manning's theory or the Kohlrausch's law

The evaluation of anion exchange membrane (AEM) selectivity towards different anions requires facile characterization methods. The anion selectivity of membranes could principally, but practically not easy to, be extracted from the measured anion transport selectivity in electrodialysis (ED), which is a parameter closely related to the process conditions. According to the classical solution-diffusion model, the membrane anion selectivity could be decoupled into two parts: the partitioning selectivity factor of anions between the membrane and the solution phase; the mobility ratio of anions in the membrane. Alternatively, based on the relationship between the anion composition in the membrane and the mixed-ionic membrane conductivities, these two parts can be calculated from the volumetric swelling ratios and ionic conductivities when the AEMs are in pure and mixed-anion forms, respectively. This relationship is proved to be linear in two commercial AEMs (ASE and ACS) for the NO3−/Cl− and SO42−/Cl− pairs. Further, it is possible to obtain the anion selectivity directly for the ASE membrane from corresponding membrane ionic conductivities, with Manning's counter-ion condensation theory to account for the anion activity coefficients in the membrane. While for the NO3−/Cl− pair in the monovalent-anion selective ACS membrane, the selectivity thus calculated is very different from that obtained in ED. For the nonlinear anion composition-conductivity relationship observed in the commercial FAA membrane, the Kohlrausch's parameter (σ) is successfully introduced to describe the mutual influence of anions in the membrane. In this case, it is also possible to obtain the anion selectivity with the proposed approach. This work provides a different perspective to study the selectivity and conductivity of AEMs.

Combining Manning's theory and the ionic conductivity experimental approach to characterize selectivity of cation exchange membranes

The transport selectivity of different cations through cation exchange membranes (CEMs) could be estimated with the partitioning selectivity factor (Kji) and the cation mobility ratio in the membrane (uim/ujm), which in turn can be related to corresponding membrane conductivity and dimensional swelling ratio data [Journal of Membrane Science, 2020, 597, 117645]. This method has been validated in two hydrocarbon polymer-based CEMs, and the obtained K+/Na+ selectivity equals to the one obtained with conventional electrodialysis (ED) method. However, the K+/Na+ selectivity of perfluorosulfonic acid (PFSA) membranes, and the bi-/monovalent cation (Mg2+/Na+) selectivity of all three types of CEMs estimated with this ionic conductivity experimental approach deviate noticeably from corresponding values obtained with ED. In this work, it is proved that this deviation is partly due to the simplification of cation activity coefficients in the membrane. Here, the cation activity coefficients in three types of CEMs are calculated according to Manning's counter-ion condensation model. In this model, the Manning parameter (ξ) characterizing the dimensionless linear charge density is determined by the average distance between two adjacent fixed sulfonate groups (b) and the permittivity of hydrated membranes (ε). In hydrocarbon polymer-based CEMs, the average distance between fixed sulfonate groups can be estimated by assuming homogeneous distribution of the fixed groups, while in PFSA membranes three representative structure models are employed to estimate this average distance. After accounting for the cation activity coefficients in the membrane, the Mg2+/Na+ cation transport selectivity obtained with the ionic conductivity experimental approach gets closer to, but is still smaller than the selectivity obtained with the ED method. This work shows the importance of cation activity coefficients in the membrane phase in interpreting the membrane transport properties, and complements the proposed conductivity approach to characterize the counter-ion selectivity of ion exchange membranes.

Modeling and validation of concentration dependence of ion exchange membrane permselectivity: Significance of convection and Manning's counter-ion condensation theory

Electrodialysis, reverse electrodialysis, and related electrochemical processes are increasingly important technologies for water purification and renewable energy generation and storage. The electrical efficiency of these processes is directly related to the permselectivity of the ion exchange membranes (IEMs) – defined as the extent to which the membrane permits the passage of counter-ions (ions of opposite charge to the membrane, e.g., cations for a cation exchange membrane) while blocking passage of co-ions. Permselectivity is not a material constant, but rather depends on the concentration and composition of the electrolyte solutions in contact with the IEM. Thus, even though permselectivity is routinely measured at standardized conditions (usually 0.5 M/0.1 M NaCl or KCl), the practical utility of such data is limited because we lack an accurate, quantitative way of using it to predict permselectivity under relevant process conditions. Moreover, the concentration dependence of IEM permselectivity has historically been studied primarily by evaluating the performance of (reverse) electrodialysis stacks rather than individual membranes, which has made it difficult to relate the concentration dependence of permselectivity to specific membrane characteristics. In this study, we measured the permselectivity of four commercial IEMs in six different concentration gradients employing 4 M and 0.5 M NaCl as the high salt concentration. We then constructed a predictive model of membrane permselectivity based on the extended Nernst-Planck equation and investigated how accounting for convection and electrostatic effects (via Manning's counter-ion condensation theory) affected model accuracy. We demonstrate that accurate, quantitative predictions of IEM permselectivity as a function of external salt concentrations are possible and require knowledge of only four easily measured membrane properties: water uptake, water permeability, charge, and thickness.

Structure-Based Model of RNA Pseudoknot Captures Magnesium-Dependent Folding Thermodynamics.

We develop a simple, coarse-grained approach for simulating the folding of the Beet Western Yellow Virus (BWYV) pseudoknot toward the goal of creating a transferable model that can be used to study other small RNA molecules. This approach combines a structure-based model (SBM) of RNA with an electrostatic scheme that has previously been shown to correctly reproduce ionic condensation in the native basin. Mg2+ ions are represented explicitly, directly incorporating ion-ion correlations into the system, and K+ is represented implicitly, through the mean-field generalized Manning counterion condensation theory. Combining the electrostatic scheme with a SBM enables the electrostatic scheme to be tested beyond the native basin. We calibrate the SBM to reproduce experimental BWYV unfolding data by eliminating overstabilizing backbone interactions from the molecular contact map and by strengthening base pairing and stacking contacts relative to other native contacts, consistent with the experimental observation that relative helical stabilities are central determinants of the RNA unfolding sequence. We find that this approach quantitatively captures the Mg2+ dependence of the folding temperature and generates intermediate states that better approximate those revealed by experiment. Finally, we examine how our model captures Mg2+ condensation about the BWYV pseudoknot and a U-tail variant, for which the nine 3' end nucleotides are replaced with uracils, and find our results to be consistent with experimental condensation measurements. This approach can be easily transferred to other RNA molecules by eliminating and strengthening the same classes of contacts in the SBM and including generalized Manning counterion condensation.

Separation and Characterization of Highly Charged Polyelectrolytes Using Free-Solution Capillary Electrophoresis.

The characterization of statistical copolymers of various charge densities remains an important and challenging analytical issue. Indeed, the polyelectrolyte (PE) effective electrophoretic mobility tends to level off above a certain charge density, due to the occurrence of Manning counterion condensation. Surprisingly, we demonstrate in this work that it is possible to get highly resolutive separations of charged PE using free-solution capillary electrophoresis, even above the critical value predicted by the Manning counterion condensation theory. Full separation of nine statistical poly(acrylamide-co-2-acrylamido-2-methylpropanesulfonate) polymers of different charge densities varying between 3% and 100% was obtained by adjusting the ionic strength of the background electrolyte (BGE) in counter electroosmotic mode. Distributions of the chemical charge density could be obtained for the nine PE samples, showing a strong asymmetry of the distribution for the highest-charged PE. This asymmetry can be explained by the different reactivity ratios during the copolymerization. To shed more light on the separation mechanism, effective and apparent selectivities were determined by a systematic study and modeling of the electrophoretic mobility dependence according to the ionic strength. It is demonstrated that the increase in resolution with increasing BGE ionic strength is not only due to a closer matching of the electroosmotic flow magnitude with the PE electrophoretic effective mobility, but also to an increase of the dependence of the PE effective mobility according to the charge density.

Equilibrium ion partitioning between aqueous salt solutions and inhomogeneous ion exchange membranes

This study investigates the ability of a recently proposed thermodynamic framework (i.e., Donnan/Manning model) to quantify equilibrium ion partitioning between aqueous salt solutions and inhomogeneous (i.e., phase separated) ion exchange membranes. Two inhomogeneous cation exchange membranes (CEMs) and one homogeneous anion exchange membrane (AEM), which served as a basis for comparison, were synthesized via free radical polymerization. Surface characterization of the membranes via cryogenic scanning electron microscopy revealed the inhomogeneous structure of the CEMs and the reasonably homogeneous structure of the AEM. Equilibrium ion concentrations in membranes equilibrated with NaCl, KCl, or MgCl2 solutions were experimentally measured. The experimental results were interpreted via a thermodynamic model based on Donnan theory and Manning's counter-ion condensation theory. The framework predicted NaCl partitioning in the AEM, presumably due to its reasonably homogeneous structure, but failed to predict NaCl partitioning in the CEMs when the Manning parameter, ξ, was calculated assuming a homogeneous membrane. For the inhomogeneous CEMs, the Manning parameter was used as an adjustable constant, which was extracted from the NaCl sorption results. Remarkably, the value of ξ obtained from the NaCl sorption results accurately predicted KCl and MgCl2 sorption in the two CEMs over the entire salt concentration range considered.

Specific co-ion sorption and diffusion properties influence membrane permselectivity

The apparent permselectivity of three cation exchange membranes (CEMs) was measured using four monovalent strong electrolytes (sodium chloride, sodium bromide, sodium nitrate and sodium perchlorate). The salt (co-ion) sorption and diffusion coefficients of a crosslinked sulfonated polymer, based on the sodium salt form of 2-acrylamido-2-methylpropane sulfonic acid (AMPS), were measured to describe the influence of sorption and diffusion properties on apparent permselectivity. Three models were used (in conjunction with electroneutrality and thermodynamic equilibrium conditions) to probe the influence of electrostatic and dispersion energy forces on specific co-ion sorption. Dispersion forces appear to be important for capturing specific co-ion sorption properties. A model combining the Mackie and Meares hindered diffusion theory and Manning's counter-ion condensation theory was used to explore the influence of electrostatics and polymer tortuosity on co-ion diffusion. Ion-polymer complexation interactions between sodium perchlorate and the poly(ethylene oxide)-based crosslinker used in the sulfonated polymer significantly influenced sorption and diffusion properties. The occurrence of these ion-polymer complexes was supported by salt sorption measurements made on three uncharged, crosslinked poly(ethylene oxide)-containing polymers with 3, 10, or 13 ethylene oxide repeat units between crosslinks. Ion specific sorption, diffusion, and ion-polymer complexation phenomena influence apparent permselectivity properties of charged polymers, and the sorption and complexation phenomena appear to affect apparent permselectivity properties to a greater extent compared to ion specific diffusion effects other than size-based differences.

Accounting for frame of reference and thermodynamic non-idealities when calculating salt diffusion coefficients in ion exchange membranes

Accurate evaluation of salt diffusion coefficients from transport rate data in ion exchange membranes requires accounting for frame of reference and non-ideal thermodynamic effects. Due to a lack of models and experimental data quantifying membrane ion activity coefficients, it has been impossible to evaluate the impact of non-ideal thermodynamic effects on observed salt diffusion coefficients. Here, a framework is presented that includes both frame of reference (i.e., convection) and non-ideal thermodynamic effects in calculating salt diffusion coefficients in ion exchange membranes. Effective concentration averaged NaCl diffusion coefficients were determined as a function of upstream NaCl concentration in commercial ion exchange membranes from NaCl permeability and sorption measurements via the solution-diffusion model. Frame of reference effects were evaluated using a version of Fick's law that accounts for convection. The factors necessary to account for non-ideal thermodynamic effects were developed using Manning's counter-ion condensation theory. At low upstream NaCl concentrations, frame of reference and non-ideal thermodynamic effects on diffusion coefficients were negligible. However, at higher upstream NaCl concentrations (e.g., >0.1M), both effects contribute measurably to NaCl diffusion coefficients. Correcting for frame of reference effects increased apparent NaCl diffusion coefficients. However, correcting for thermodynamic non-idealities of the ions sorbed into the membranes reduced apparent NaCl diffusion coefficients. Fortuitously, for the materials considered in this study, frame of reference and non-ideal thermodynamic effects nearly cancel each other.

Predicting Salt Permeability Coefficients in Highly Swollen, Highly Charged Ion Exchange Membranes.

This study presents a framework for predicting salt permeability coefficients in ion exchange membranes in contact with an aqueous salt solution. The model, based on the solution-diffusion mechanism, was tested using experimental salt permeability data for a series of commercial ion exchange membranes. Equilibrium salt partition coefficients were calculated using a thermodynamic framework (i.e., Donnan theory), incorporating Manning's counterion condensation theory to calculate ion activity coefficients in the membrane phase and the Pitzer model to calculate ion activity coefficients in the solution phase. The model predicted NaCl partition coefficients in a cation exchange membrane and two anion exchange membranes, as well as MgCl2 partition coefficients in a cation exchange membrane, remarkably well at higher external salt concentrations (>0.1 M) and reasonably well at lower external salt concentrations (<0.1 M) with no adjustable parameters. Membrane ion diffusion coefficients were calculated using a combination of the Mackie and Meares model, which assumes ion diffusion in water-swollen polymers is affected by a tortuosity factor, and a model developed by Manning to account for electrostatic effects. Agreement between experimental and predicted salt diffusion coefficients was good with no adjustable parameters. Calculated salt partition and diffusion coefficients were combined within the framework of the solution-diffusion model to predict salt permeability coefficients. Agreement between model and experimental data was remarkably good. Additionally, a simplified version of the model was used to elucidate connections between membrane structure (e.g., fixed charge group concentration) and salt transport properties.

INTERMOLECULAR ASSOCIATION OF AMPHIPATHIC POLYELECTROLYTES IN AQUEOUS SOLUTIONS

The intermolecular association of amphipathic polyelectrolytes is studied using fluorescence and conductivity methods. This type of interaction is evidenced by the decrease in the average distances between charges as the polyelectrolyte side chain increases in length. This modify the dissociation degree and consequently, the linear charge density parameter. These distances were calculated with the Manning counterion condensation theory for the conductivity of polyelectrolyte solutions. The determination of the ratio I1/I3 of the fluorescence bands of pyrene with polymer content reveals the formation of hydrophobic microdomains at very low concentrations, smaller than the required concentration to produce a significant change in the average distances between charges. These distances and also the I1/I3 ratio vary with polymer concentration and their values were dependent on the size of the side chain in the polyelectrolyte. Finally, the experimental behavior of the solution viscosity and electrical conductivity of polyelectrolytes, which increase drastically with dilution, can be explained as a continuous change in the average distances between charges which produce conformational changes.

Generalized Manning Condensation Model Captures the RNA Ion Atmosphere.

RNA is highly sensitive to the ionic environment and typically requires Mg(2+) to form compact structures. There is a need for models capable of describing the ion atmosphere surrounding RNA with quantitative accuracy. We present a model of RNA electrostatics and apply it within coarse-grained molecular dynamics simulation. The model treats Mg(2+) ions explicitly to account for ion-ion correlations neglected by mean-field theories. Since mean-field theories capture KCl well, it is treated implicitly by a generalized Manning counterion condensation model. The model extends Manning condensation to deal with arbitrary RNA conformations, nonlimiting KCl concentrations, and the ion inaccessible volume of RNA. The model is tested against experimental measurements of the excess Mg(2+) associated with the RNA, Γ(2+), because Γ(2+) is directly related to the Mg(2+)-RNA interaction free energy. The excellent agreement with experiment demonstrates that the model captures the ionic dependence of the RNA free energy landscape.

21am2-F2 高分子電解質ブラシの粗視化シミュレーション

Course-grained Monte Carlo Brownian Dynamics simulations are used to simulate the nature of polyelectrolyte brushes. For the basic understanding, the effect of salt and surface charges on the structure of polyelectrolyte brushes are studied. Polyions are grafted to the surface of up to several tens of nanometers pitch. For flexible polyion model of 128 segments, bond and bond angle degree of freedom is adopted. Linear charge density parameter of the polyion is set to 0.5 to 4.0 in order to discuss the Manning's counterion condensation. Thermal equilibrium structure in room temperature of a brush system without salt and a system of added 1:1 salt are compared. The effect of salt caused the decrease of film thickness as have been detected in small angle neutron scattering experiments. The difference of the thickness at the salt-free system is discussed with the polarization of counterion atmosphere.

Transport, and thermodynamic investigations on sodium polystyrenesulfonate in ethylene glycol–water media

Polyion–counterion interactions in sodium polystyrenesulfonate dissolved in (ethylene glycol+water) mixed solvent media have been investigated conductometrically with special reference to their variations as functions of polyelectrolyte concentration, relative permittivity and temperature. Manning counterion condensation theory for polyelectrolyte solutions failed to describe the present experimental results. The data have, therefore, been analyzed using a new model for semidilute polyelectrolyte conductivity which takes into account the scaling arguments proposed by Dobrynin et al. The fractions of uncondensed counterions were found to depend on the polyelectrolyte concentration varying from 0.27 to 0.37, within the concentration range investigated here, indicating a strong interaction between counterions and polyion. A considerable fraction of the counterions is shown to migrate in the same direction as the polyions. The results further demonstrate that the monomer units experience more frictional resistance in solutions as the ethylene glycol content of the mixture increases or as the temperature decreases.

Studies of base pair sequence effects on DNA solvation based on all-atom molecular dynamics simulations

Detailed analyses of the sequence-dependent solvation and ion atmosphere of DNA are presented based on molecular dynamics (MD) simulations on all the 136 unique tetranucleotide steps obtained by the ABC consortium using the AMBER suite of programs. Significant sequence effects on solvation and ion localization were observed in these simulations. The results were compared to essentially all known experimental data on the subject. Proximity analysis was employed to highlight the sequence dependent differences in solvation and ion localization properties in the grooves of DNA. Comparison of the MD-calculated DNA structure with canonical A- and B-forms supports the idea that the G/C-rich sequences are closer to canonical A- than B-form structures, while the reverse is true for the poly A sequences, with the exception of the alternating ATAT sequence. Analysis of hydration density maps reveals that the flexibility of solute molecule has a significant effect on the nature of observed hydration. Energetic analysis of solute-solvent interactions based on proximity analysis of solvent reveals that the GC or CG base pairs interact more strongly with water molecules in the minor groove of DNA that the AT or TA base pairs, while the interactions of the AT or TA pairs in the major groove are stronger than those of the GC or CG pairs. Computation of solvent-accessible surface area of the nucleotide units in the simulated trajectories reveals that the similarity with results derived from analysis of a database of crystallographic structures is excellent. The MD trajectories tend to follow Manning's counterion condensation theory, presenting a region of condensed counterions within a radius of about 17 A from the DNA surface independent of sequence. The GC and CG pairs tend to associate with cations in the major groove of the DNA structure to a greater extent than the AT and TA pairs. Cation association is more frequent in the minor groove of AT than the GC pairs. In general, the observed water and ion atmosphere around the DNA sequences is the MD simulation is in good agreement with experimental observations.

Nonlinear screening of charged macromolecules

We present several aspects of the screening of charged macromolecules in an electrolyte. After a review of the basic mean field approach, based on the linear Debye-Hückel theory, we consider the case of highly charged macromolecules, where the linear approximation breaks down and the system is described by the full nonlinear Poisson-Boltzmann equation. Some analytical results for this nonlinear equation give some interesting insight on physical phenomena like the charge renormalization and the Manning counterion condensation.