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This study presents a new framework for extracting single ion diffusion coefficients in ion exchange membranes from experimental ion sorption, salt permeability, and ionic conductivity data. The framework was used to calculate cation and anion diffusion coefficients in a series of commercial ion exchange membranes contacted by aqueous NaCl solutions. Counterion diffusion coefficients were greater than co-ion diffusion coefficients for all membranes after accounting for inherent differences due to ion size. A model for ion diffusion coefficients in ion exchange membranes, incorporating ideas from counterion condensation theory, was proposed to interpret the experimental results. The model predicted co-ion diffusion coefficients reasonably well with no adjustable parameters, while a single adjustable parameter was required to accurately describe counterion diffusion coefficients. The results suggest that for cross-linked ion exchange membranes in which counterion condensation occurs condensed counterions mi...

We give approximate solutions for some problems in polyelectrolyte theory involving nonuniform charge distributions. Distributions of condensed counterions and electrostatic free energies are found for line-charge models of oligomers, end segments, junctions separating regions of different charge densities, and two semi-infinite line charges approaching each other end-on. The end-on potential of mean force in the latter case is screened by the Debye length. With that exception, we find little correlation between the Debye screening length and the much smaller distances marking the onset of significant end effects. We suggest a biological application to DNA unwinding by helicases.

In a previous paper, we have shown that counterions condense on charged spheres and planes as well as on the linear polyelectrolyte models of charged lines and cylinders, provided that the surface charge densities exceed a critical value. Here we give a more complete calculation resulting in the electrostatic free energies of all of these geometric shapes. We consider a macroion immersed in salt solution and also a macroion in the presence only of its own counterions. For a sphere, cylinder, and plane with equal surface charge densities, the sphere has the lowest electrostatic free energy and the plane the highest. We apply the results to the coil-to-globule transition and to the growth of spheres by fusion of two smaller spheres to form a larger one.

We extend the theory of counterion condensation from the standard representation of a locally stiff rodlike polyion as a line of discrete charges to a helical lattice of charges. The number of counterions condensing on a helix with given axial charge density is the same as on a line of the same charge density, but the electrostatic free energy is substantially less for the helix than for the line. In fact, the free energy of assembling the helical charge lattice becomes negative at higher salt concentrations, indicating electrostatic stabilization of the helix due to the mixing entropy of condensed counterions. The electrostatic persistence lengths of the line and helix subjected to locally elastic thermal bending fluctuations differ only slightly. They have the same κ -2 dependence as the Odijk-Skolnick-Fixman persistence length but a different prefactor. It is commonly believed that elastic bending models for the persistence length are applicable to double-helical DNA, but we point out that DNA apparently has unique mechanical properties that do not conform to the elastic model of bending. On the other hand, the helical model and its subsequent generalization to a double helix will serve as a basis for the electrostatic free energy of DNA conformational transitions.

We report further development of a pair potential for two identically charged rodlike polyions oriented in parallel. An attractive force mediated by condensed counterions of any valence, including univalent, is identified for separation distances between a Debye screening length and about a third of a Debye length. The characteristics of the potential are explored as a function of ionic strength, dielectric constant, counterion valence, and charge density of the polyions. The work required to assemble more than two polyions in parallel array is shown not to be pairwise additive. The free energy of assembling many parallel polyions in a hexagonally ordered cluster is calculated as a function of the number N of polyions in the cluster. The free energy has a minimum at a finite cluster size. Discrepancies between the theoretical structure of the stable cluster and experimental observations on polyelectrolyte clusters (with univalent counterions) are discussed, and it is tentatively concluded that the apparen...

We have computed the fraction of DNA phosphate charge neutralized by condensed counterions for parallel pairs of linear DNA segments, DNA circles, and representative closed-circular supercoiled DNA configurations ranging in length from 42 to 3000 base pairs. We find significant but small uptake of condensed counterions for the more compact structures relative to an isolated linear DNA segment.

We develop formulas for the counterion−polyion and coion−polyion potentials of mean force in the context of counterion condensation theory. For separation distances on the scale of a Debye screening length, both potentials conform to Debye−Huckel screening behavior. Close to the polyion, in the region defining the layer of condensed counterions, the counterion potential is attractive and unscreened. A coion in this region acts like a charged group on the polyion; its potential is repulsive, and it augments the ability of the polymer to condense counterions. There is an interfacial region between condensed layer and diffuse cloud from which counterions are expelled back into the cloud but which accumulates coions. The associated radial distribution functions exhibit two peaks for the counterion and a single peak for the coion.

A theory of attractive forces between identical highly charged polyions in a 1:1 supporting electrolyte, generated by the interaction of the layers of condensed counterions, is extended to a Z:Z‘ electrolyte and to rodlike polymer segments of any charge density. If the combined charge density of a polyion pair does not exceed the charge density critical for counterion condensation, then there is no attraction. But an attraction arises if the combined charge density does exceed the condensation threshold, even if the individual polymer charge densities lie below the critical value. The intensity of the attraction at a fixed ionic strength decreases with higher values of counterion valence. The effect of increased ionic strength is to decrease both the range and the intensity of the attractive force.