Approach to the limit of counterion condensation

According to counterion condensation theory, one of the contributions to the polyelectrolyte free energy is a pairwise sum of Debye-Hückel potentials between polymer charges that are reduced by condensed counterions. When the polyion model is taken as an infinitely long and uniformly spaced line of charges, a simple closed expression for the summation, combined with entropy-derived mixing contributions, leads to the central result of the theory, a condensed fraction of counterions dependent only on the linear charge density of the polyion and the valence of the counterion, stable against increases of salt up to concentrations in excess of 0.1 M. Here we evaluate the sum numerically for B-DNA models other than the infinite line of B-DNA charges. For a finite-length line there are end effects at low salt. The condensation limit is reached as a flat plateau by increasing the salt concentration. At a fixed salt concentration the condensation limit is reached by increasing the length of the line. At moderate salt even very short B-DNA line-model oligomers have condensed fractions not far from the infinite polymer limit. For a long double-helical array with charge coordinates at the phosphates of B-DNA, the limiting condensed fraction appears to be approached at low salt. In contrast to the results for the line of charges, however, the computed condensed fraction varies strongly with salt in the range of experimentally typical concentrations. Salt invariance is restored, in agreement with both the line model and experimental data, when dielectric saturation is considered by means of a distance-dependent dielectric function. For sufficiently long B-DNA line and helical models, as typical salt concentrations, the counterion binding fraction approaches the polymer limit as a linear function of 1/P, where P is the number of phosphate groups of B-DNA.     

Polyelectrolyte properties of biopolymers: conductivity and secondary structure of polyriboadenylic acid and its salts in solutions

Polyriboadenylates of alkali metals were obtained from (1) K(+)-poly(A) (salts 1) and (2) H(+)-poly(A) (salts II) by the ion-exchange method. The conductivity of these salts as well as of H(+)-poly(A) were studied. Salts I and II of the same counterion were shown to have significantly different conductivity coefficients (f) and polyion conductances (lambda 0p). the charge density parameter (xi) was 1.3 and 2.5, respectively, with lambda 0p equal to 44 and 83 ohm-1 cm2 mole-1 for poly(A)-I and poly(A)-II salts, respectively. This is credited to the difference in the conformations of corresponding polyions. The linear dependence of equivalent conductivity on the square root of polymer concentration (Kohlrausch coordinates), earlier obtained for DNA, is also satisfied for the studied polynucleotides. A comparison of the slopes of straight lines in Kohlrausch coordinates for poly(A), simple electrolytes, and for earlier studied polyribouridylic acid salts lends credence to the concepts, developed by a number of authors, that DNA can act as a "buffer" against the ion-ion interaction in concentrated electrolyte solutions. Using the approximation that the polyion conductance is independent of the counterion nature, parameter f (agreeing in this case with Eisenberg parameter phi) has been shown to decrease as the polynucleotide concentration is increased; the decrease is caused by the relaxation effect. The transference numbers of counterions, which have negative values in poly (A)-II solutions, grow with the increase in polymer concentration; the higher the xi, the more apparent is this increase. This is explained by the increase in the fraction of conductivity along the polyion chains ("surface" conductivity) with the growth of polyelectrolyte concentration.     

Polyion-counterion interaction behavior for sodium carboxymethylcellulose in methanol-water mixed solvent media

Precise measurements on the electrical conductivities have been reported for solutions of sodium carboxymethylcellulose in methanol-water mixed solvent media. The conductivity vs. concentration data have been analyzed on the basis of the scaling theory approach for semidilute polyelectrolyte conductivity. The effects of the temperature, the medium, and the polymer concentration on the fractions of uncondensed counterions, the polyion conductivities, the standard state free energies of counterion association, and the coefficients of friction between the polyion and the solvent have also been investigated and the results have been interpreted from the viewpoints of polyion-countreion interactions, effective charge on the polyion, solvation of counterions and the polyionic sites, and counterion dissociation.     

Radii of gyration of sodium carboxymethylcellulose in aqueous and mixed solvent media from viscosity measurement

The viscosities of three sodium carboxymethylcellulose samples with molecular weights of 90,000 [degree of substitution (DS): 0.7], 250,000 (DS: 0.9), and 700,000 (DS: 0.9) have been reported in water and methanol–water mixtures in salt-free and salt-containing solutions at 35 °C. The results were analyzed in terms of a phenomenological approach for the viscosity of polymer solutions to determine the intrinsic viscosities [η] of the polyelectrolyte samples. This contribution presents a new and convenient method for the determination of the root-mean-square radii of gyration of the polyion chains using the [η] values obtained as a function of the added salt concentration. The polyion coils are found to expand at low ionic strength and these collapse drastically with increasing ionic strength. Addition of methanol to the medium in which these samples are dissolved causes a contraction of the polyion chains, although this influence is less pronounced than that of the added salt.     

Comparison of Poisson-Boltzmann and condensation model expression for the colligative properties of cylindrical polyions

Closed-form expression have been derived for the polyelectrolyte contribution to the colligative properties of solutions containing rodlike polyions in the presence of excess added salt. The derivations are based on: the conventional Poisson-Boltzmann equation for cylindrical symmetry; the thermodynamics of the cell model developed by Marcus [J. Chem. Phys. 23 , 1057–1068 (1955)]; and an equation derived from the cylindrical Poisson-Boltzmann cell model by Anderson and Record [Biophys. Chem. 11 , 353–360 (1980)]. Subject to the inherent limitations of the Poisson-Boltzmann approximation [Fixman (1979) J. Chem. Phys. 70 , 4995–5005], the resulting expressions are nevertheless applicable outside the “limit of infinite dilution.” They conform over a range of salt concentrations to the limiting laws deduced by Manning from the hypothesis of counterion condensation [J. Chem. Phys. 51 , 924–933 (1969)]. This hypothesis is found to be compatible with the Poisson-Boltzmann cell model but is not required in the derivation of the thermodynamic coefficients presented here. It is demonstrated that the magnitude of the polyion axial charge density plays a critical role in determining the low-salt limiting forms of the colligative properties obtained from the Poisson-Boltzmann equation, in close analogy with Manning's model.     

Evaluation of the counterion condensation theory of polyelectrolytes.

We compare free energies of counterion distributions in polyelectrolyte solutions predicted from the cylindrical Poisson-Boltzmann (PB) model and from the counterion condensation theories of Manning: CC1 (Manning, 1969a, b), which assumes an infinitely thin region of condensed counterions, and CC2 (Manning, 1977), which assumes a region of finite thickness. We consider rods of finite radius with the linear charge density of B-DNA in 1-1 valent and 2-2 valent salt solutions. We find that under all conditions considered here the free energy of the CC1 and the CC2 models is higher than that of the PB model. We argue that counterion condensation theory imposes nonphysical constraints and is, therefore, a poorer approximation to the underlying physics based on continuum dielectrics, point-charge small ions, Poisson electrostatics, and Boltzmann distributions. The errors in counterion condensation theory diminish with increasing distance from, or radius of, the polyion.     

Ultrasonic study of the kinetics of counterion site binding in aqueous solutions of polyelectrolytes

The excess ultrasonic absorption due to counterion binding has been studied as a function of frequency for a series of polysalts in the range 1–150 MHz. All the relaxation spectra can be represented by a relaxation equation with two relaxation terms. The relaxation frequencies appear concentration independent and the relaxation amplitudes seem proportional to concentration. The low frequency relaxation process appears to depend mainly on the nature of the counterion while the high frequency relaxation process seems to be mostly dependent on the nature of the polyion. These results are quite similar to those obtained in ultrasonic studies of ion-pairing in solutions of divalent sulfates. The kinetic model used for the quantitative analysis of these results has been modified for polysalts through introducing the concept of“counterion condensation”. In this modified model the excess absorption is assigned to the perturbation by the ultrasonic waves of the equilibria between the three states of hydration of ths complex formed by a counterion and that part of the polyion where it is bound. Analytical expressions of the relaxation amplitudes have been derived using classical procedures for this modified kinetic model. In the case of cobalt-polyphosphate (Co-PP), the ultrasonic data together with the results of NMR measurements on either Co²⁺ or Co-PP have been used for the evaluation of the volume changes, the rate constants and the fractions of counterions in the three states of hydration involved in the binding equilibria. The volume changes obtained in this manner depend only slightly on the method of calculation and appear to be consistent with volume changes for outer-sphere and inner-sphere complex formation. These results are discussed.     

The release of monovalent counterions by addition of divalent counterions in coulombic interaction system

The additivity rule of counterion activity or osmotic pressure in rodlike polyelectrolyte solutions has been discussed on the basis of the Fokker-Planck and Poisson equations in relation to the fluctuation of counterion distribution. This new theory has concluded that the additivity rule of counterion activity is less applicable than that of osmotic pressure due to the electric expansion force acting on the free-volume surface resulting from the fluctuation of counterion distribution. The theory has introduced an approximate relation between the counterion activities in the mixture solution of divalent and monovalent counterions, such that Deltaa+ = DeltaC++ - Deltaa++, in which Deltaa+ represents the increase of activity of monovalent counter-ions resulting from the addition of divalent counterionsDeltaC++, (in molar) to the solution, and Deltaa++ means the increase of the divalent counterion activity (in molar) in this process. This relation has been experimentally examined for Na-PSS solutions in the process of Cu2+ ion addition by the use of Na+ and Cu2+ sensitive electrodes, and it has been turned out that the relation is established in the low charge state of polyion.     

Counter-Ion Condensation and System Dimensionality

Abstract

We review some of the characteristic properties of the structure of polyelectrolyte solutions: the condensed layer of counterions that forms abruptly at a critical threshold charge density on the polymer chain; the more diffuse Debye-Hückel cloud, which is spatially distinct from the condensed layer; and the entropie release of counterions from the condensed layer as a driving force for the binding of oppositely charged ligands. We present a reminder of the basis of our current understanding in a variety of experiments, simulations, and theories; and we attempt as well to clarify some misunderstandings. We present a new analysis of a lattice model that suggests why the limiting laws for polyelectrolyte thermodynamics have proved to be accurate despite the neglect of polymer-polymer interactions in their original derivation. We sketch recent progress in constructing a potential between counterion and polyion. A counterion located in the interface between condensed layer and Debye cloud is repelled from the polyion, creating a sharp boundary between the two counterion populations.     

Effect of counterion concentration on the dielectric behavior of a polypeptidic chain in the helix–coil transition

On the basis of the two-state model of a polyelectrolyte solution, the ion concentration in the polymer domain has been calculated by using the spherical Poisson–Boltzmann equation. The ion accumulation in the neighboring of the polyion influences, on different time scales, various electrical properties of the solution, in particular the low-frequency electrical conductivity and the high-frequency dielectric dispersion. These predictions have been compared with recent dielectric measurements on poly (L -glutamic acid) aqueous solutions during the conformational transition from the α-helix to random coil, and a satisfactory agreement has been found. This finding suggests that counterion distribution plays a different role in determining the electrical properties of charged polymer solutions, causing a electrophoretic contribution of the polymer domain to the electrical conductivity and influencing the high-frequency dielectric dispersion. © 1993 John Wiley & Sons, Inc.     

The charge effects of the self-diffusion constant of bovine mercaptalbumin

The charge effect on the translational self-diffusion constant, D, of polyelectrolytes has been quantitatively analyzed based on dynamic light scattering experiments. Perfectly monodisperse bovine mercaptalbumin has been used at low pH as a positively charged polyelectrolyte sample. Completely linear plots of log{g2(t)-1} vs. time t have been obtained for uncharged states of the protein, for the cor relation function of the scattered light intensity, g2(t). The plots deviate from linearity as polyions bear the charges. The D values for various ionic states, obtained from the initial slopes of the plots, have been analyzed using the simple theory of Imai and Mandel (N. Imai and M. Mandel, Macromolecules 15 (1982) 1562) derived based on the Onsager-Navier-Stokes equation for solvent flow with counterion distribution around a polyion. It has turned out that the experimental D values coincide well with the theory and that the characteristic nature of D can be elucidated principally from the charge effect.     

Dielectric increment in polyion solutions due to the distortion of counterion distribution

A theory of DC dielectric increment, delta, has been constructed based on the Fokker- Planck and Poisson equations on the mechanism of the distortion of spatial counterion distribution around a rodlike polyion under the applied electric field, E, perpendicular to the polyion lod. The DC dielectric constant of polyion solutions can be obtained analytically in the both cases of absence and presence of counterion-flow. The nonlinearity of the Poisson-Boltzmann equation in the absence of E has given the interesting behaviors of delta. The calculation in the absence of counterion-flow has shown the delta-values having the same order of magnitude as the experimental data at the higher frequency range (around 100 kHz), and the importance of this mechanism has been pointed out.     

Application of polyelectrolyte theories for analysis of DNA melting in the presence of Na+ and Mg2+ ions.

Numerical calculations, using Poisson-Boltzmann (PB) and counterion condensation (CC) polyelectrolyte theories, of the electrostatic free energy difference, DeltaGel, between single-stranded (coil) and double-helical DNA have been performed for solutions of NaDNA + NaCl with and without added MgCl2. Calculations have been made for conditions relevant to systems where experimental values of helix coil transition temperature (Tm) and other thermodynamic quantities have been measured. Comparison with experimental data has been possible by invoking values of Tm for solutions containing NaCl salt only. Resulting theoretical values of enthalpy, entropy, and heat capacity (for NaCl salt-containing solutions) and of Tm as a function of NaCl concentration in NaCl + MgCl2 solutions have thus been obtained. Qualitative and, to a large extent, quantitative reproduction of the experimental Tm, DeltaHm, DeltaSm, and DeltaCp values have been found from the results of polyelectrolyte theories. However, the quantitative resemblance of experimental data is considerably better for PB theory as compared to the CC model. Furthermore, some rather implausible qualitative conclusions are obtained within the CC results for DNA melting in NaCl + MgCl2 solutions. Our results argue in favor of the Poisson-Boltzmann theory, as compared to the counterion condensation theory.     

New evidence of the nonequilibrium nature of the "slow modes" of diffusion in polyelectrolyte solutions

The time-dependent behavior of the dissolution of polyelectrolyte powders in pure water and moderate ionic strength aqueous solvent was monitored by flowing dissolving material through an online filter, and then through a multiangle light scattering unit, a refractometer, and a capillary viscometer. When the polyelectrolytes were dissolved in solutions of moderate ionic strength, their dissolution behavior was similar to that of neutral polymers. When dissolved in pure water, however, there was consistently a small population of aggregates that appeared at the beginning of the dissolution process, which then rapidly diminished. For large pore filtration, the aggregates reached a final low level, and slowly disappeared over the span of many days, whereas for small pore filtration the aggregates disappeared completely over a scale of minutes. The real-time data, together with size exclusion chromatography analysis, shed light on previously unanswered questions concerning the nonequilibrium nature of this small population of polyelectrolyte aggregates in low ionic strength solutions, and its relation to the "extraordinary phase" of diffusion (or "slow modes"). Further evidence is also provided that both angular scattering maxima due to interpolyion correlations and the maximum of reduced viscosity vs polyion concentration ("electroviscous" effect) at low ionic strength are equilibrium properties that are unrelated to these aggregates.     

Dielectric properties of poly-L-glutamic acid in salt-free aqueous solutions

The electric permittivity of poly-L-glutamic acid (PGA) in salt-free aqueous solutions was measured in the frequency range 2.5 kHz – 100 MHz at different concentrations and degrees of ionization. Two samples of different molecular weight were investigated. The experimental results could under most circumstances be described by a superposition of two dispersion curves of the Cole-Cole type. The low-frequency dielectric parameters were strongly molecular weight dependent, the high-frequency ones not. Strong concentration effects were observed resulting in increasing specific dielectric increments and relaxation times with decreasing concentration. Using the theory proposed by Van der Touw and Mandel to interpret the experimental results these concentration effects could be ascribed to the influence of the polyion interactions on the average dimensions and the rigidity of the polyelectrolyte chains. The change in the total dielectric increment and low-frequency relaxation time with degree of ionization correctly reflects the helix-coil transition of PGA occurring in ths region α = 0.3–0.5. The effect of counterion size and charge on the dielectric behaviour was also found to be consistent with the theoretical model.     

Ion-binding and conformation of poly-L-methionine S-methylsulfonium salts in added salt systems

In order to study the type of ion binding and the conformation of several salts of poly-L -methionine S-methylsulfonium hydroxide, the viscosity, conductance, counterion activity, and optical rotatory dispersion of the polysalts were measured in systems with a small amount of added salt. It was shown that the ion binding of chloride and bromide salts was of a diffuse binding type due only to the electrostatic potential of the polyion, and that both polysalts underwent no conformational transition by the addition of a simple salt, NaCl for chloride salt and NaBr for bromide salt, and retained a random coil conformation. Iodide and thiocyanate salts showed a conformational change, probably from the random coil into the β form, with increasing concentrations of NaI and NaSCN, respectively. On the other hand, perchlorate salt existed in the α-helix conformation in part even in pure aqueous solution, and the fraction of α-helix increased on the addition of NaClO₄. On considering several possible situations, it is suggested that there is a specific and strong interaction between the polyion and the small counterion in iodide, thiocyanate, and perchlorate salts.     

Salt effects on the cross-linking mechanism of cupric-induced sol—gel transition in alginate solutions

The viscosity behaviour of alginate-Cu²⁺-NaCl systems has been experimentally examined at various concentrations of cupric and sodium salts. Dependence of the intrinsic viscosity of alginate as a function of NaCl concentration is discussed to supplement the previous study which shows a similar behaviour to that found for other polyelectrolytes in aqueous solution in the presence of an added salt. The effects of sodium ions on the cupric association in cupric-induced alginate solutions were investigated by means of viscosity measurements. The mechanisms of complex formation in the presence of the simple added salt were studied. It was found that, at a given NaCl concentration, the viscosity of the mixture will pass through a maximum with increasing cupric concentration. The amounts of cupric cations corresponding to the maximum depends on the concentration of NaCl in the solution. Comparison of salt effects on the viscosity behaviour of alginate solutions during sol—gel transition reveals that an optimum NaCl concentration of 10⁻² mol 1⁻¹ exists where the viscosity of the mixture gives a maximum value at a certain cupric amount. This result indicates that salt effects play an important role in the sol—gel transition of the polyelectrolyte solutions. The observed phenomenon was interpreted in terms of conformational change of polyelectrolyte chain due to the addition of salt resulting in a different cross-linking mode in the system.     

Comparison of different approaches for calculation of polyelectrolyte free energy

We consider the problem of the mean field (Poisson-Boltzmann) calculation of the electrostatic free energy for a strongly charged polyelectrolyte such as DNA in a salt solution. We compare two approaches to calculate the free energy: (i) direct one starting from the statistical-mechanical expression for the electrostatic free energy and (ii) the polyion charge variation method. In the infinite dilution limit (in respect to polyion) and in excess salt (IDLES) the two approaches are fully equivalent. This is shown by straight forward algebra. We have performed specific calculations of the free energy difference for the case of B-Z transition in DNA as a function of ionic strength. As expected, the two approaches led to identical results. The ionic strength dependence of the B-to-Z free energy proves to be concaved up and as a result Z-DNA is stabilized at low ionic concentration as well as at high salt, in full agreement with our previous results (M.D.Frank-Kamenetskii et al., J. Biomol. Struct. Dyn. 3, 35-42 (1985]. Our data quantitatively agree with the results of Soumpasis (D.M.Soumpasis, J. Biomol. Struct. Dyn. 6, 563-574 (1988]. However, his claim about the absence of the effect of stabilization of Z-DNA at low salt proves to be groundless, and the criticism of our earlier approach seems to be irrelevant.     

Transport phenomena and ion binding in solutions of sodium polystyrenesulphonate

The results of transference and conductance measurements on solutions of sodium polystyrenesulpnonate, in the concentration range from 0.0025 to 0.075 monomolar, are presented. The fraction of free Counterions calculated from experimental data is found to he independent of molecular weight of the polyelectrolyte within the range of degree of polymerization from 200 to 2400. The experimental data, combined with the cell model, are used to calculate the electrostatic potential at a cylindrical surface Which surrounds the polyion and divides the counterions into bound and free. The same value for this potential is found for the whole concentration range investigated, which lends support to the assumption of cylindrical distribution of counterions in polyelectrolyte solutions.     

Nanomechanical behaviors of microcantilever-based single-stranded DNA chips induced by counterion osmotic effects

Experiments show that deflections of microcantilever-DNA chip can be induced by many factors, such as grafting density, hybridization efficiency, concentration, length and sequence of DNA molecules, buffer salt concentration, time, and temperature variation. However, there are few theoretical works on microcantilever-DNA chips. The present paper is aimed to study the influence of counterion effects of single-stranded DNA (ssDNA) polyelectrolyte solution on the nanomechanical behaviors of microcantilever-based ssDNA chips during packing process. First, the effect of osmotic pressure induced by ingress of counterions into DNA brush structures is studied with Hagan’s model for a cylindrical polyelectrolyte brush system on the basis of Poisson-Boltzmann distribution hypothesis. Second, Zhang’s two-variable method for a laminated cantilever is used to formulate a four-layer energy model for ssDNA chips with weak interactions. Third, the influence of grafting density, ssDNA chain length, and salt concentration on packing deflection is investigated using the principle of minimum energy. The predictive tendency is qualitatively similar to those observed in some related ssDNA chip experiments. The difference between the four-layer model and the simplified two-layer model for ssDNA chips is also discussed.