Linear Polyelectrolyte Solutions Research Articles

Counterion condensation theory for finite polyelectrolyte and salt concentrations

In the present work we analyze the physical fundamentals of Manning’s counterion condensation using his charged line model in a simple salt solution. We extend the theory for the cases of finite saline concentration and polymeric concentration tending to zero and the case of both finite concentrations. To find the equilibrium between the phases of free and condensed counterions, besides minimizing the free energy, we deduce an auxiliary equation to determine the two characteristic parameters of the theory, the fraction of condensed counterions and the volume of condensation. We compare the obtained results in the present work for only one infinite charged line with the ones of counterion condensation theory by Schurr and Fujimoto. We find that the linear density of critical charge depends on the concentration of added salt and takes values higher than one, instead of the unitary value predicted by Manning. We obtain the equations by the activity and osmotic coefficients in function of the critical charge density. We compare them with the corresponding equations by Manning for these parameters. We extend the counterion condensation theory to solutions of linear polyelectrolytes for finite saline and polymeric concentrations using a cell model. We modify the electrostatic contribution to the Gibbs energy adding, to the traditional one calculated by Manning, the energy excess due to the macroion present in a cylindrical cell. We apply the theory to obtain the osmotic coefficient and we compare our results with experimental data of DNA osmotic coefficient and with theoretical adjustment using the Poisson–Boltzmann equation.

Divalent metal‐ion distribution around linear polyelectrolyte chains by continuous diafiltration: comparison of counterion condensation cell models

AbstractConcepts proposed in the two‐state model (Manning), two‐zone model (Deshkovski–Obukhov–Rubinstein) and extensions of these models based on Donnan equilibria around polyelectrolyte phases (Marinsky) were compared and discussed in terms of retention data obtained by continuous diafiltration of various linear polyelectrolyte solutions that were in contact with copper and cadmium salt solutions at pH = 6.0. Concentration and retention profiles were investigated and the metal‐ion retention values of the polymers (Rp) were compared with the respective values calculated using the theoretical assumptions of the models. An adequate connection was not observed between the two‐state model and diafiltration experiments when a null interaction between states 1 and 2 was assumed. If a Donnan equilibrium between states 1 and 2 was assumed, the metal‐ion fraction in state 1 was smaller than the Rp values obtained from the retention profiles for all systems. In contrast, a good correspondence between Rp values and the two‐zone model combined with Donnan's equilibrium was obtained by adding the weakly bound fraction of counterions around the polyelectrolyte chains. Copyright © 2010 Society of Chemical Industry

Ionic correlations in the inhomogeneous atmosphere surrounding cylindrical polyions: Catalytic effects of polyions

The structural properties of linear polyelectrolyte solutions in the presence of a salt as evidenced through ionic correlations in the inhomogeneous atmosphere around a polyion and their consequence such as the catalytic potential are studied by using Monte Carlo simulation techniques. The simulations are performed on the cylindrical cell model where a uniformly charged hard cylinder mimics the linear polyion, which is caged in its own cylindrical cell containing counterions and salt. The cell (volume) average of the interionic correlations is presented as a function of the polyion and salt concentrations and ion radius. These results are utilized to study the catalytic effects of polyions as manifested through the changes in the collision frequency between ions in the double layer surrounding the polyion relative to that in the pure electrolyte solution. The reported results suggest a strong influence of the added salt/polyelectrolyte concentration ratio on the structural properties of the solution and hence on ion-ion collision frequency. The machine simulations are supplemented by nonlinear Poisson-Boltzmann results. Fair agreement between two different theoretical methods of calculating the collision frequency is obtained.

Viscoelastic properties of aggrecan aggregate solutions: Dependence on aggrecan concentration and ionic strength

Aggrecan is the self-assembling proteoglycan complex whose physiological function is to provide a hydrated gel in cartilage that stabilizes the spatial distribution of the collagen fibers, and absorbs and resists compressive loads. The linear and nonlinear viscoelastic behavior of aggrecan solutions was studied as a function of aggrecan concentration and ionic strength. At physiological ionic strength, I=0.15, a sol-to-gel transition occurs at an aggrecan concentration just above the overlap value. Concentrated solutions exhibit a reversible yield point similar to that predicted and observed for close-packed dispersions of soft spheres. In contrast to the behavior of linear polyelectrolyte solutions, the storage modulus increases with ionic strength, until I=0.75, above which the modulus decreases. In light of the dense polyelectrolyte brush structure of aggrecan, we suggest that this behavior is a manifestation of the crossover from “osmotic brush” to “salted brush” conditions.

Low- and High-Frequency Electric Birefringence Relaxations in Linear Polyelectrolyte Solutions

To clarify the molecular mechanism of the counterion fluctuations in linear polyelectrolyte solutions, we have studied the dependence of the low-frequency electric birefringence relaxation on the degree of polymerization N and the monomer concentration C p of sodium polystyrenesulfonate (NaPSS) in aqueous solution and the dependence of the low-frequency (LF) and high-frequency (HF) relaxations on the neutralization ratio a of polystyrenesulfonic acid (HPSS) in aqueous solution by the frequency-domain electric birefringence (FEB) method. The experimental results of the N and C p dependence of the LF relaxation definitely show that the LF relaxation strongly depends on N but is weakly dependent on or almost independent of C p , which indicates that the LF relaxation arises from the counterion fluctuation along the polyion. Also, it turns out from the experimental results of the α dependence of the LF and HF relaxations that the LF relaxation is ascribed to tightly bound counterions in the vicinity of a polyion and that the HF one is due to loosely bound counterions away from the polyion. These results support the molecular mechanism that the LF and HF relaxations come from the fluctuation of the tightly bound counterions along the polyion axis and from the fluctuation of the loosely bound counterions perpendicular to the polyion axis, respectively.

Low- and high-frequency relaxations in linear polyelectrolyte solutions with different counter-ion species

We have studied the low- and high-frequency relaxations in polystyrene sulfonate solutions with different counter-ion species [X +=H +, Li +, Na +, K +, (C 4H 9) 4N +], from the frequency-domain electric birefringence (FEB) and the dielectric relaxation spectroscopies. The high-frequency (HF) relaxation frequency f H is proportional to the free intrinsic mobility, μ 0, of the counter-ion, whereas the low-frequency (LF) relaxation frequency f L is not proportional to μ 0. These different dependencies of f L and f H on the counter-ion species, support the notion that the LF relaxation arises from the fluctuation of counter-ions tightly bound in the vicinity of a poly-ion, in the direction parallel to the poly-ion long axis, and that the HF relaxation arises from the motion of counter-ions loosely bound away from the poly-ion in the direction perpendicular to the poly-ion axis.

Polyelectrolyte solutions containing mixed valency ions in the cell model: A simulation and modified Poisson–Boltzmann study

Monte Carlo simulations of linear polyelectrolyte solutions containing mixed valency simple ions in the cylindrical cell model are reported. The equilibrium distributions of the simple ions and the osmotic pressure of the solution are calculated at various concentrations of the monomer units of the polyelectrolyte. Specifically, the following systems are studied—monovalent counterions with added 2:2 salt, divalent counterions with added 1:1 salt, and systems containing mono- and divalent counterions only, and mono- and trivalent counterions only. The simulation results are compared with the corresponding predictions from the Poisson–Boltzmann and modified Poisson–Boltzmann theories applied to the cell model. It is seen that upto moderate concentrations of the polyion, the modified Poisson–Boltzmann theory provides a very good description of the systems with deviations occurring at higher concentrations. The theory also reproduces the charge reversal observed in the simulations when strongly correlated counterions overscreen the charge of the polyion. On the other hand, the classical Poisson–Boltzmann results begin to show discrepencies from the Monte Carlo results at relatively lower concentrations. Comparisons of the simulated osmotic pressures with available experimental results confirm the validity of the cell model in a spectrum of practical situations of interest.

Structure functions of rod-like DNA fragment and polystyrenesulfonate solutions in the modified Poisson-Boltzmann theory

The partial structure functions of aqueous solutions of rod-like DNA fragments and polystyrenesulfonic acid are calculated in the modified Poisson-Boltzmann theory. The cylindrical cell model appropriate for linear polyelectrolyte solutions with monovalent counterions and without any added salt is utilized. The predicted results are compared with the corresponding results from the classical Poisson-Boltzmann theory, and experimental small angle neutron scattering data obtained in the monomer concentration range 0.05-0.2 mol/dm 3. It is seen that both the modified Poisson-Boltzmann and the Poisson-Boltzmann results lead to a very good fit to the experimental structure functions.

Employment of Gibbs-Donnan-based concepts for interpretation of the properties of linear polyelectrolyte solutions

Employment of Gibbs-Donnan-based concepts for interpretation of the properties of linear polyelectrolyte solutions

Revisit to the intrinsic viscosity-molecular weight relationship of ionic polymers: I. Viscosity behavior of dilute suspensions of ionic polymer latices

The viscosity of aqueous suspensions of ionic polymer latices was measured by using a variable shear viscometer, Ubbelohde viscometer, and a rotational viscometer. At variance with previous data, the reduced viscosity showed a strong shear rate dependence in the range between 10 and 1350 s −1 when the concentration of coexisting salt was low. At high-salt conditions, the shear rate dependence became smaller and the Einstein behavior was approached. For low-charge density latices at low- and high-salt conditions, the reduced viscosity ( η sp φ ) increased with increasing latex concentration in the low-concentration range. On the other hand, the η sp φ of highly charged samples decreased with increasing concentration at low-salt concentrations while it monotonically increased at high-salt concentration. The effective charge number of the latices, which was determined in a direct manner (by electric conductivity), was found to be rather insensitive toward the latex concentration in the same concentration range as employed in the viscosity measurements. By using the charge number thus found, we evaluated the first-order electroviscous effect from the viscosity and compared it with Booth's theory. A satisfactory agreement was found at relatively high-salt concentrations. Implications of these findings for latex systems in the viscosity properties of linear polyelectrolyte solutions are briefly discussed. The interpretation in terms of the Debye screening length is critically discussed.

Osmotic Pressure of Aqueous Rod‐Like Polyelectrolyte Solutions with Mono‐ and Di‐valent Counterions

AbstractMembrane osmometry has been used to measure the Donnan osmotic pressure of different salts of carboxymethylcellulose (Na+‐ Li+‐ NH4‐ Ca++‐ Mg++‐CMC) in aqueous solutions of simple chloride electrolytes. The results are analyzed in terms of cylindrical cell model for linear polyelectrolyte solutions, using the Poisson‐Boltzmann equation in order to calculate the counterion densities at the cell boundaries. Significant deviations with increasing polyion concentration have been found for all the counterions investigated.

Thermodynamic properties of rodlike polyelectrolyte solutions in the presence of salts

AbstractA theoretical foundation is given to the experimentally established relation called the additivity law, according to which the thermodynamic and transference properties of linear polyelectrolyte solutions with low molecular salts are described as a superposition of the effects of ions from polyelectrolytes and those from low molecular salts. By employing a cylindrical free volume model for a solution of thin and long rodlike polyelectrolytes and analyzing the Poisson‐Boltzmann equation it is found that the electrostatic free energy is expressed in a special form as a function of the geometry of the system. On the basis of this expression of the free energy, the additivity law is derived by a thermodynamic treatment without an approximate procedure to solve the Poisson‐Boltzmann equation. The additivity is a property not accidental but essential to rodlike polyelectrolytes. The same method is applicable to analysis of the interaction between two parallel rodlike polyions in the presence of low molecular salts.

Chain Model for Polyelectrolytes. VII. Potentiometric Titration and Ion Binding in Solutions of Linear Polyelectrolytes

The electrostatic potential about a rodlike polyion in an aqueous solution containing added simple electrolyte is calculated by numerical integration of the Poisson-Boltzmann equation. The results are applied to the calculation of the ionic distributions about the polyion and the potentiometric titration curves of polyweak acids. The nature of ion binding is discussed in terms of a molecular model and it is found that the degree of ion binding f* may be expressed as f*=1−(2Ne2/DkTL)−1,where the parameters are defined in the text.