Dielectric increment and dielectric dispersion of solutions containing simple charged linear macromolecules. II. Experimental results with synthetic polyelectrolytes

The electric permittivity of aqueous solutions of different synthetic polyelectrolytes have been measured as a function of frequency in the range 5 kHz up to 100 MHz in the absence of added salt. Solutions of polymethacrylic acid and polyacrylic acid of different degrees of polymerization, both partially neutralized with NaOH, were investigated as well as solutions of Na-polystyrenesulphonate at different concentrations.For all systems a dispersion profile with two separated dispersion regions was obtained with a molecular weight dependent value of the static electric permittivity. The low frequency dispersion region was found to be characterized by a molecular weight dependent mean relaxation time while for the high frequency dispersion region both the mean relaxation time and the dielectric increment are molecular weight independent. It is shown that the reciprocal values of the specific increments and of the relaxation times depend linearly on the macromolecular concentration. Extrapolation of the corresponding quantities to infinite dilution was found to be possible. A comparison of these extrapolated values with calculated ones according to the previously derived theory also applicable to flexible macromolecules establishes that this theory describes satisfactorily the dielectric behaviour of the systems investigated.The conclusion is reached that the high frequency dispersion and relaxation can be attributed to fluctuations in the distribution of bound counterions along limited parts of the macromolecule. The relaxation time of the low frequency dispersion region seems to be essentially determined by the rotation of the complete molecule and the static electric permittivity can he explained in terms of fluctuations in the counterion density extending over the whole macromolecule.     

Quasielastic light scattering by biopolymers. VI. Diffusion of mononucleosomes and oligonucleosomes in the presence of static and sinusoidal electric fields

Quasielastic light scattering and electrophoretic light scattering studies were carried out on mononucleosome and oligonucleosome systems. The electrophoretic light scattering experiments employed static and sinusoidal electric fields. Data are presented that suggest at least two relaxation modes. It is proposed that the small amplitude sinusoidal field effectively polarizes the ion atmosphere about the polyion, thus leading to an induced dipole moment that varies sinusoidally in time. This model is, in essence, an extension of the current interpretation of low-frequency dielectric dispersion data on DNA as being due to fluctuations of counterions along the polyion.     

Counterion fluctuation and dielectric dispersion in linear polyelectrolytes

The thermal fluctuation in the concentration of counterions bound to a rodlike polyion was analyzed by expanding the fluctuation in a Fourier series along the rod. The amplitude and the relaxation time of fluctuations of various wave lengths were obtained as functions of the charge density and the length of the polyion. From these results the real and imaginary parts of the dielectric constant of the polyelectrolyte solution were derived as the sum of contributions of fluctuations of different modes. The dielectric dispersion curve or the Cole-Cole plot obtained was found to be in good agreement with experimental data.     

Dielectric properties of polyelectrolytes. III. Effect of divalent cations on dielectric increment of polyacids

Dielectric dispersion of polyacrylic acid (PAA) and polystyrene sulfonic acid (PSS) was measured in the presence of divalent cations. Effects of divalent ions were studied by neutralization with varying ratios of sodium hydroxide and divalent base concentration, addition of salts of divalent cations, and neutralization with divalent bases only. Two dispersion regions were observed in all cases, i.e., low-frequency dispersion (10²–10⁴ Hz) and high-frequency dispersion (10⁵–10⁶ Hz). The dielectric increment increases in the presence of sodium and alkaline earth metal ions together, but not with sodium and transition metal ions. This is due to the increment of low-frequency dispersion and is attributable to the fluctuation of bound counterions which is explained by our theory previously reported.¹ In the case of PAA neutralized with large fractions of divalent ions, or with divalent ions only, the increment is very small because of reduction of the fluctuation by interaction between bound ions at the neighboring sites and reduction of the effective length of polyion probably due to chelation by divalent ions. There are some differences among the effects of Mg⁺⁺, Ca⁺⁺, and Ba⁺⁺ on dielectric increment which may result from affinity or chelating ability of these ions.     

Counterion diffusion in heparin solutions

The physiological importance of heparin is due to its strong interaction with bivalent counterions, especially Ca2+. A diffusional approach of this property is presented in this article: the observable is the self-diffusion coefficient of the counterions, as a function of the ratio of the polyelectrolyte over the added salt concentrations. All the results are in agreement with a simple "quasi-chemical model" in which two different states are assumed for the counterions: "free" or "bound." The proportions of these two types of ions are calculated according to the distribution function of the counterions around the polyion. We assume that those of counterions located at a distance closer than a, the characteristic distance, are bound; the others are free. The ionic distribution function is evaluated by a numerical integration of a cell model Poisson-Boltzmann equation. Finally, this model leads to a very good agreement with the experimental results, if the radius of heparin polyion is assumed to be 6 and 10 A.     

Dielectric behavior of DNA in water–organic co-solvent mixtures

The radiowave dielectric dispersions of DNA in different water–organic co-solvent mixtures have been measured in the frequency range from 100 kHz to 100 MHz, where the polarization mechanism is generally attributed to the confinement of counterions within some specific lengths, either along tangential or perpendicular to the polyion chain. The dielectric dispersions have been analyzed on the basis of two partially different dielectric models, a continuum counterion fluctuation model proposed by Mandel and a discrete charged site model, proposed by Minakata. The influence of the quality of the solvent on the dielectric parameters has been investigated in water–methanol and water–glycerol mixtures at different composition, by varying the permittivity ?_m and the viscosity η of the solvent phase. The analysis of the dielectric spectra in solvents where electrostatic and hydrodynamic interactions vary with the solvent composition suggests that both the two models are able, in principle, to account for the observed high-frequency dielectric behavior. However, while some certain assumptions are necessary about the polyion structure within the Mandel model, no structural prerequisite is needed within the Minakata model, where the polarization mechanism invoked considers a radial counterion exchange with the outer medium, which is largely independent of the local polyion conformation.     

Dielectric properties of polyelectrolytes. II. A theory of dielectric increment due to ion fluctuation by a matrix method

The mean square of dipole moment of a linear macromolecule which is responsible for dielectric increment of aqueous polyelectrolyte solutions is calculated by means of a matrix method in which ion binding at discrete sites and the nearest-neighbor interaction are taken into account. On the basis of the relationship between polarization of poly-ion and fluctuation of bound counterions the present theory indicates that the loosely bound ions result in larger increment and otherwise smaller increment. Also, the theory predicts that the dielectric increment has a maximum at an intermediate monovalent–divalent ion ratio when both species coexist. These results are consistent with experiments on polyacrylic acid neutralized with NaOH and Ca(OH)₂. At large contents of divalent ions the effect of chelation is also discussed.     

Conductometric analysis of the competition between monovalent and divalent counterions in their interaction with polyelectrolytes

A procedure is described for the analysis of the conductivity of solutions of anionic polyelectrolytes in which both mono- and divalent counterions are present. The method is based on analysis of the relation between the overall conductivity of the system and the conductivity of the individual monovalent cations which are only electrostatically (non-specifically) bound. The system is described in terms of the two-state approach, implying that the counterions are considered to be either fully bound to the polyion or completely free. The potentialities of the proposed method are explored by studying solutions of alkali polyacrylates with and without added zinc nitrate at several alkali nitrate concentrations. The results give a picture of the composition of the counterionic atmosphere around the polyion in systems with both mono- and divalent counterions present. To a certain degree, the divalent ion Zn(II) was found to be bound quantitatively by the polyion. The composition of the counterionic atmosphere around the polyion was largely independent of alkali nitrate concentration when the latter was present in not too large an excess with respect to both Zn(II) and the charged monomers.     

Electric permittivity of alfalfa mosaic virus in aqueous solutions

The electric permittivity of alfalfa virus particles in buffer solutions of three different concentrations at pH 7 was studied between 10 kHz and 100 MHz. The experimental results could be described with one single dispersion curve of the Cole-Cole type characterized by a concentration independent specific dielectric increment and mean relaxation time. The results were interpreted semi-quantitatively in terms of counterions–atmosphere polarizability, neglecting counterion repulsion.     

The relationship between the poisson-boltzmann model and the condensation hypothesis: an analysis based on the low salt form of the Donnan coefficient

Two common models for the interaction of counterions with cylindrical polyions are considered in the context of the Donnan membrane equilibrium. General analytic expressions are obtained from the Poisson-Boltzmann equation for the Donnan coefficient in terms of the potential at the surface of the polyion or the local concentration of unbound ions at the surface. Analysis based on these expressions shows that if, and only if, the polyion charge density exceeds a certain critical value a large local concentration of ions will persist near the polyion surface at low ionic strengths. We therefore conclude that this principal hypothesis of the condensation model is consistent with the characteristics of the Poisson-Boltzmann potential at the surface of the polyion.     

Density functional theory for the nonspecific binding of salt to polyelectrolytes: thermodynamic properties

The thermodynamics of the nonspecific binding of salt to a polyelectrolyte molecule is studied using a density functional approach. The polyelectrolyte molecule is modeled as an infinite, inflexible, and impenetrable charged cylinder and the counterions and co-ions are modeled as charged hard spheres of equal diameter. The density functional theory is based on a hybrid approach where the hard-sphere contribution to the one-particle correlation function is evaluated nonperturbatively and the ionic contribution to the one-particle correlation function is evaluated perturbatively. The advantage of the approach is that analytical expressions are available for all the correlation functions. The calculated single ion preferential interaction coefficients, excess free energy, and activity coefficients show a nonmonotonic variation as a function of polyion charge in the presence of divalent ions. These properties display considerable departure from the predictions of the nonlinear Poisson-Boltzmann (NLPB) equation, with qualitative differences in some cases, which may be attributed to correlation effects neglected in the NLPB theory.     

Dielectric studies of electrolytic poly(α-amino acid)s in aqueous solutions

The dependence of the dielectric constant and dielectric loss of aqueous solutions of poly-ε, N-succinyl-L -lysine on its degree of polymerization, degree of neutralization, concentration of the polymer, and counterion type was studied in a frequency range from 300 Hz to 5 MHz. Regardless of the conformation, a low-frequency dispersion in a frequency range lower than 10 kHz and a high-frequency dispersion in a range higher than 100 kHz were found. The large value of the dielectric increment, its nonlinear dependence on concentration, its remarkable dependence on counterion type, and its dependence on the degree of polymerization suggest that the low-frequency dispersion is mainly due to the polarization of loosely bound counterions. These data were found for both the helical and coiled forms. The rotational motion of the electric dipole on the molecule could not have been primarily responsible for these results. On the other hand, the high-frequency dispersions may be attributable to the Maxwell–Wagner-type effect. The results were compared with the dispersions of poly(L -glutamic acid), poly(L -lysine), and their salts reported previously.     

A comparative dielectric study of human serum low density lipoprotein before and after partial digestion by trypsin

The relative permittivity of aqueous solutions of human serum low density lipoprotein (LDL) and partially trypsin digested lipoprotein (T-LDL) has been determined for various concentrations at 20°C over the frequency range 0.15–100 MHz. Comparison of the dielectric dispersion curves for the digested lipoprotein with those for the native preparation revealed a larger low-frequency dielectric increment, which may be attributed to an increase in the number of counterions moving over the surface of the molecule. An explanation of this observation is an elevation of 70% in the net negative charge on the surface of the trypsin-treated particle as compared to its native counterpart.     

Estimation of bound quaternary ammoniumion to a rod-like polyion with a large alkyl residue

The amount of quaternary ammonium ion (Bu4N+), which is believed not to be bound to general carboxylpolyelectrolytes, bound to poly(iso BVE-co-MA) was estimated by conductivity measurements. In the region of the density of polyion charge in which the polyions are thought to take a free drain'ng conformation, it has been confirmed that the activity coefficient of Bu4N+ ions is less than 0.5 in the presence of a small amount of Bu4NCl, showing that the force between the counterions and the polyion is probably due to the hydrophobic interaction. Moreover, from the electrophoretic mobility Up of the polyion observed from the data of conductivity, it has been ascertained that Up of this polyion is two times larger than PAA, and the behavior of the quantity e/xiP with changing degree of ionization corresponds to that of the viscosity.     

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.     

Kerr effect of charged dipolar macromolecules without condensed counterions in conducting solution.

The theoretical treatment of the Kerr constant of rigid, dipolar, conducting ellipsoidal macromolecules of O'Konski and Krause (1970. J. Phys. Chem. 74:3243) has been extended to very low ionic strength solutions for charged macromolecules. The O'Konski and Krause theoretical treatment postulated a surface conductivity directly on the surface of each macromolecule. For charged macromolecules, this surface conductivity was generally assumed to be caused by movement of condensed counterions on the macromolecules. In the present work, it has been assumed that, at very low ionic strength, the average counterion is at the Debye characteristic distance from the surface of each charged macromolecule and contributes to surface conductivity at that distance, with no additional surface conductivity on the true surface of the macromolecule. Essentially, these considerations change the calculated interaction energy of the macromolecule with an externally applied electric field via a change in both the internal field components and in the reaction field of the macromolecular dipole. The new interaction energy is used to calculate the orientation distribution function of the macromolecules in solution and this distribution function can, in principle, be used to calculate the steady state electric linear or circular dichroism, electric light scattering, anisotropy of conductivity, etc., using the appropriate theoretical treatment for each of these quantities.     

Generalized stern models of the electric double layer considering the spatial variation of permittvity and finite size of ions in saturation regime

The interaction between a charged metal implant surface and a surrounding body fluid (electrolyte solution) leads to ion redistribution and thus to formation of an electrical double layer (EDL). The physical properties of the EDL contribute essentially to the formation of the complex implant-biosystem interface. Study of the EDL began in 1879 by Hermann von Helmholtz and still today remains a scientific challenge. The present mini review is focused on introducing the generalized Stern theory of an EDL, which takes into account the orientational ordering of water molecules. To ascertain the plausibility of the generalized Stern models described, we follow the classical model of Stern and introduce two Langevin models for spatial variation of the relative permittivity for point-like and finite sized ions. We attempt to uncover the subtle interplay between water ordering and finite sized ions and their impact on the electric potential near the charged implant surface. Two complementary effects appear to account for the spatial dependency of the relative permittivity near the charged implant surface — the dipole moment vectors of water molecules are predominantly oriented towards the surface and water molecules are depleted due to the accumulation of counterions. At the end the expressions for relative permittivity in both Langevin models were generalized by also taking into account the cavity and reaction field.     

Effect of the external electrostatic field on the conformation of a macromolecule containing charged groups

A method for calculating the free energy of a macromolecule containing charged groups in electrostatic field in aqueous solution was proposed. The non-electrostatic component of free energy was calculated with consideration of van der Waals interactions between uncharged parts of the macromolecule. The electrostatic component of free energy was calculated with regard for the interactions of charged groups of the macromolecule with each other and with water molecules. It was found that, depending on the strength of external electric field, the free energy of the system passes through a minimum, whereas the internal energy passes through a maximum. By minimizing the free energy, relative changes in the mean radius 'r' and the distance between the termini of the macromolecule 'h' were calculated. It was found that, at some values of field strength, both 'r' and 'h' decrease. An increase in strength led to an increase in 'r' and 'h'. The regularities observed depend on the charge of the macromolecule and the spatial redistribution of macromolecules and counterions.     

Site binding as a local equilibrium. Mn2+ binding to poly(acrylic acid) as a function of the degree of ionization

Mn2+ binding to poly(acrylic acid) at different degrees of ionization, alpha, has been studied from the frequency dependence of the water protons' relaxation rates T1(-1) and T2(-1). Site binding is treated as an equilibrium with the concentration of free ions at the immediate vicinity (CIV) of the polyion. The CIV is calculated as the solution of the Poisson-Boltzmann equation at the surface of the cylindrical polyion. A single value of K is shown to fit the results at all values of alpha. The amount of site binding is higher than the total amount of condensed divalent counterions predicted for a finite polyion concentration in the presence of monovalent counterions by Manning's theory.