Counterion adsorption theory of dilute polyelectrolyte solutions: apparent molecular weight, second virial coefficient, and intermolecular structure factor

Polyelectrolyte chains are well known to be strongly correlated even in extremely dilute solutions in the absence of additional strong electrolytes. Such correlations result in severe difficulties in interpreting light scattering measurements in the determination of the molecular weight, radius of gyration, and the second virial coefficient of charged macromolecules at lower ionic strengths from added strong electrolytes. By accounting for charge-regularization of the polyelectrolyte by the counterions, we present a theory of the apparent molecular weight, second virial coefficient, and the intermolecular structure factor in dilute polyelectrolyte solutions in terms of concentrations of the polymer and the added strong electrolyte. The counterion adsorption of the polyelectrolyte chains to differing levels at different concentrations of the strong electrolyte can lead to even an order of magnitude discrepancy in the molecular weight inferred from light scattering measurements. Based on counterion-mediated charge regularization, the second virial coefficient of the polyelectrolyte and the interchain structure factor are derived self-consistently. The effect of the interchain correlations, dominating at lower salt concentrations, on the inference of the radius of gyration and on molecular weight is derived. Conditions for the onset of nonmonotonic scattering wave vector dependence of scattered intensity upon lowering the electrolyte concentration and interpretation of the apparent radius of gyration are derived in terms of the counterion adsorption mechanism.     

Rheology of dilute polyelectrolyte solutions in narrow channels

The shear flow of dilute polyelectrolyte solutions bounded by either neutral or repulsive walls is modeled using a nonlinear dumbbell with conformation-dependent friction. Assuming that the configurational probability density function depends on the internal coordinates (r) and the distance of the center of mass of the molecule to the walls, coupled differential equations for the tensor moments <rr> are obtained. Coulombic repulsion between beads is considered to simulate the charge repulsion between ionized sites distributed along the backbone of a real polyelectrolyte. The repulsive interaction between the polyelectrolyte molecule and the charged walls is that of the DLVO model and the molecule is considered to be a charged sphere. Numerical solutions for the components of the tensor <rr> are worked out with the preaverage approach, and only when neutral walls considered are exact solutions obtained. Viscosity results show that in the limit of very wide channels, the corresponding viscosity in the bulk is obtained. The wall repulsion on the charged molecules produces migration of molecules towards the center of the channel resulting in a depleted layer with lower viscosity next to the walls. The calculated slip phenomenon using the method employed by Grisafi and Brunn is dependent on the beads repulsion and the shear rate. The slip velocity obtained with the Mooney method shows similarities with available experimental results for polyelectrolyte solutions. Birefringence calculations are performed in narrow and wide channels for different bead repulsions, with interesting results for both flexible and rigid molecules. Received: 26 September 1998 Accepted in revised form: 11 March 1999     

Brownian dynamics simulation of polyelectrolyte dilute solutions: Relaxation time and elongational flow

The authors used the bead‐and‐spring model and the Brownian dynamics simulation technique including hydrodynamic interaction to study the behavior of dilute polyelectrolyte solutions under elongational flow. First they carried out simulations to determine the longest relaxation time of a polyelectrolyte, finding that the relaxation time depends on the ionic strength of the solution. Then, they studied the coil‐stretch transition of polyelectrolyte molecules in elongational flow and determined the critical value of the elongational rate necessary in order this transition to occur. In this way, they could compute the value of the Deborah number at which coil‐stretch transition sets in for polyelectrolyte dilute solutions. Finally, they studied the power law relationship that relates the critical elongational rate with the molecular weight of the polyelectrolyte. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 714–722, 2007     

Simulation of nonlinear shear rheology of dilute salt-free polyelectrolyte solutions

Brownian dynamics simulations are used to conduct a systematic analysis of the nonlinear shear rheology of dilute polyelectrolyte solutions, exploring its relationship to shear rate, Bjerrum length, and concentration. A simple coarse-grained bead-spring chain model that incorporates explicit counterions is used. It is found that the polyelectrolyte chains exhibit a shear thinning behavior at high shear rate (as characterized by bead Peclet number Pe) that is independent of the electrostatic strength due to the stripping of ions from close proximity to the chain caused by the flow. In contrast, at low values of Pe, the viscosity increases monotonically with increasing Bjerrum length over the range studied here, in contrast to the nonmonotonic trend displayed by the chain size. Furthermore, at fixed Bjerrum length, the reduced viscosity increases monotonically with concentration. The mechanism underlying these observations is essentially the primary electroviscous effect; the ion cloud surrounding a polyelectrolyte chain deforms in flow, causing a significant increase in viscosity as concentration increases. Finally, the authors have also considered the role of hydrodynamic interactions in these simulations, finding that for low concentration studies in shear flow, these do not qualitatively affect the results.     

On the equilibrium properties of polyelectrolyte solutions: isothermal-calorimetry

Study of the solution properties of natural and synthetic polyelectrolytes covers a relatively large field of both fundamental and application-oriented interest.This paper is concerned with equilibrium properties and offers examples of the usefulness of isothermal microcalorimetry investigations for a better understanding of the typical processes taking place in dilute aqueous solutions of ionic polymers, namely, ionization of weak polyacids, interactions between polysaccharidic macroions and divalent counterions, and the binding of certain dyes and drugs by DNA.     

Self-consistent mode-coupling theory for the viscosity of rodlike polyelectrolyte solutions

A self-consistent mode-coupling theory is presented for the viscosity of solutions of charged rodlike polymers. The static structure factor used in the theory is obtained from polymer integral equation theory; the Debye-Huckel approximation is inadequate even at low concentrations. The theory predicts a nonmonotonic dependence of the reduced excess viscosity eta(R) on concentration from the behavior of the static structure factor in polyelectrolyte solutions. The theory predicts that the peak in eta(R) occurs at concentrations slightly lower than the overlap threshold concentration, c*. The peak height increases dramatically with increasing molecular weight and decreases with increased concentrations of added salt. The position of the peak, as a function of concentration divided by c*, is independent of salt concentration or molecular weight. The predictions can be tested experimentally.     

A Brownian dynamics study on the self-diffusion of charged tracers in dilute polyelectrolyte solutions

Brownian dynamics simulations with hydrodynamic interactions are conducted to investigate the self-diffusion of charged tracer particles in a dilute solution of charged polymers, which are modeled by bead-spring chains. The Debye-Hückel approximation is used for the electrostatic interactions. The hydrodynamic interactions are implemented by the Ewald summation of the Rotne-Prager tensor. Our simulations find that the difference in short- and long-time diffusivities is very slight in uncharged short-chain solutions. For charged systems, to the contrary, the difference becomes considerable. The short-time diffusivity is found to increase with increasing chain length, while an opposite behavior is obtained for the long-time diffusivity. The former is attributed to the hydrodynamic screening among beads in a same chain due to the bead connectivity. The latter is explained by the memory effect arising from the electrostatic repulsion and chain length. The incorporation of hydrodynamic interactions improves the agreement between the simulation prediction and the experimental result.     

Clusters in strong polyelectrolyte solutions in the condensation theory approach

The interaction free energy of parallel clusters of like-charged rod polyelectrolytes in solution is calculated in the framework of the extended condensation theory. For sufficiently high linear charge density of the polyelectrolyte, clustering takes place. The greater is the number of polyelectrolytes participating to the cluster, the smaller is the equilibrium interpolyelectrolyte distance, and the deeper is the corresponding free energy minimum. It is a counterintuitive organization due to the increasing of the counterion condensed charge and condensation volume, taking place as the polyelectyrolytes approach each other.     

Dielectric scaling in polyelectrolyte solutions with different solvent quality in the dilute concentration regime

In this note, we present a set of radiowave dielectric spectroscopy measurements of two dilute, differently-charged polyelectrolyte solutions, under different solvent conditions. We have found that both the dielectric strength, Delta epsilon, and the relaxation time, tau(ion), of the dielectric relaxation process associated with the counterion polarization along a length scale of the order of the correlation length obey the scaling laws with the polyion concentration, according to the Ito model. This is verified with good accuracy independently of the quality of the solvent, which has been varied from poor to good solvent conditions. This finding supports evidence to the fact that, in dilute solutions, the counterion polarization is independent of the polyion concentration, in spite of what occurs at the semi-dilute concentrations.     

Development of the theory of a self-consistent field for polyelectrolyte solutions

A new theoretical approach to calculating the thermodynamic and structure functions of polyelectrolyte solutions is proposed, based on the method of Gaussian equivalent representation for calculating the functional integrals. Formulas for the mean-force potential, osmotic pressure, and complete monomer-monomer pair distribution functions are presented. A sodium polystyrene sulfonate solution with NaCl additives is considered as an example.     

On the phase behavior and structural properties of polyelectrolyte solutions

Correlations between structural properties and phase behavior of polyelectrolyte solutions were discussed along the line of the work reported by Châtellier and Joanny [Joanny JF, Châtellier X. J Phys France II 1996;6:1669]. A multicomponent system made of polyions, salt ions and counterions was considered under poor solvent conditions. Unlike this reference, partial structure factors were derived from the celebrated Zimm’s formula written in the matrix form including the effects of finite chain length. These effects were found to generate significant shifts in phase diagrams and qualitative changes in structural properties. The presence of a charged solid surface was briefly discussed. Here also, the phase diagram was found to shift with an increasing amount as the polyion chain length decreased.     

Approximate theory of the viscoelasticity of chain-molecule solutions not infinitely dilute

The theories of viscoelasticity of chain-molecule solutions, based on the bead-spring model, have been modified to take into account approximately the effect of interactions among chain molecules. The motion of a given chain molecule has been analyzed in detail while all other chain molecules have been treated as a background of uniformly distributed beads in which the given chain molecule is suspended. The inclusion of intermolecular interactions leads to a hydrodynamic interaction tensor, which after averaging, differs from that of Kirkwood by a concentration-dependent correction term. The modified theory correctly predicts the transition from the non-draining to the free-draining behavior which has been observed in chain-molecule solutions as their concentrations are increased.     

A theory of viscosity of dilute and moderately concentrated polymer solutions

A model theory of viscosity η for moderately concentrated polymer solutions is based on the assumption of a “local viscosity” effect and intermolecular hydrodynamic and thermodynamic interactions. It is shown that η is given by

$\text{η = η}{}_{\mathrm{o}}\text{{1 + γc[η]}}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{·}\text{exp}\text{H}{}_{\mathrm{o}}\text{RT}\text{aø}\text{1 ? aø}$

where γ is 0–0.4 and depends on the quality of the solvent, a varies between 0,4 and 0.8 and depends on the fraction of the “free volume” of the systems, H₀ is the activation energy of the solvent and π is the polymer volume concentration. The dependence of η and “activation energy” of π and T for various molecular weights and qualities of solvents is described quantitatively. Anomalous dependences of [η] and of η on M for low polymer are obtained. An expression for η is proposed:

$\text{η}\text{η}{}_{\mathrm{o}}{}^{\mathrm{<mtext>1\; ?\; 2</mtext><mtext>K</mtext>}}\text{= {1 + (1 ? 2K)c[η]}F(π)}$

where K is the Huggins-Martin coefficient and F(π) = 1 for most solutions when T is > T_g. For poor solvents the H vs c curve (where H is the activation energy of η of solution) has a minimum value at moderate concentrations. For good solvents, H depends slightly on the molecular weight according to an empirical equation:

$\text{H = H}{}_{\mathrm{o}}\text{+}\text{660}\text{α}{}^3\text{1}\text{n}\text{η}\text{η}{}_{\mathrm{o}}$

Expressions are given from the viscosities of solutions of miscible and also solutions of immiscible polymers.     

Comparison of theory and experiment for diffusion in dilute polymer solutions

Results from a number of theories for the concentration dependence of the mutual diffusion coefficient in dilute polymer solutions are examined, and clarifications are made as to what forms of the equations for these theories should be used in comparisons with experimental diffusivity data. An evaluation of the available theories for the concentration dependence of the diffusivity under theta conditions is carried out using experimental diffusivity data taken using sharp fractions of polystyrene. It is concluded that the Pyun—Fixman theory appears to provide the most promising method for estimating the concentration dependence of the mutual diffusion coefficient in dilute polymer solutions at the present time.     

On the binding of ethidium bromide in synthetic polyelectrolyte solutions

Results are reported for spectroscopic and calorimetric studies of ethidium bromide (EBr) in water and in aqueous solutions of three alternating maleic acid copolymers. The dimerization constant, dimerization enthalpy and the dimer spectrum of EBr are derived. The interaction of EBr with the polycarboxylates is characterized on the basis of spectral and microcalorimetric data according to two limiting mechanisms: at low dye-to-polymer concentration ratios, EBr would be bound essentially in monomeric form with small exothermic effects: at high concentration ratios, the observed comparatively large exothermic heat is attributed to bound dye stacking.     

On the attenuation of X-rays and γ-rays in dilute solutions

The theory of X-ray and γ-ray attenuation in solutions is developed. The rule of mixture for the calculation of mass and linear attenuation coefficients is elaborated in the general case as well as in the limit of extreme dilution. The validity of the latter approximation is illustrated by the attenuation of 17.443 keV X-rays in aqueous solutions of NaCl.     

Fermi resonance - the theory of isotropic and anisotropic Raman spectra in dilute solutions

Raman scattering from dilute solution was studied limiting considerations to one active molecule, in which the Fermi resonance occurs, interacting with the molecules of an inert solvent. For such a system, the isotropic and anisotropic correlation functions ⧸CFs⧸ for symmetric top molecule have been calculated and examples of Fermi resonance spectra ⧸FRS⧸ are presented.