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We give a contemporary and direct derivation of a classical, but insufficiently familiar, result in the theory of linear elasticity-a representation for the energy of a stressed elastic rod with central axis that intrinsically takes the shape of a general space curve. We show that the geometric torsion of the space curve, while playing a crucial role in the bending energy, is physically unrelated to the elastic twist. We prove that the twist energy vanishes in the lowest-energy states of a rod subject to constraints that do not restrict the twist. The stretching and contraction energies of a free helical spring are computed. There are local high-energy minima. We show the possibility of using the spring to model the chirality of DNA. We then compare our results with an available atomic level energy simulation that was performed on DNA unconstrained in the same sense as the free spring. We find some possible reflections of springlike behavior in the mechanics of DNA, but, unsurprisingly, the base pairs lend a material substance to the core of DNA that a spring does not capture.

Counterions bound to a polyion are polarized by an applied electric field. A polyelectrolyte theory for this effect is constructed by placing the Mandel polarization model within the framework of counterion condensation theory. A central feature of the Mandel model, retained here, is equilibration with the field of the local concentration of condensed counterions along the length of the polyion. Briefly, the bound counterions are assumed to polarize but not to conduct. In qualitative agreement with observations, the theoretical induced dipole moment saturates; condensed counterions respond to increases of the field strength by piling up at one end of the polyion until further polarization is not possible. The value of the saturated dipole when polyelectrolyte effects are considered is the same as in the Mandel theory, but the onset of saturation is pushed to much higher field strengths. Massive quantitative discrepancies between theory and high‐field data point to reasons for dipole saturation other than ...

In an ongoing effort to model the mechanical behavior of the nucleosome core particle in terms of elasticity theory, we study the bending fluctuations of an elastic line (nucleosomal DNA) confined to a curved surface (outside surface of the histone octamer). We define what is meant by a bending fluctuation by generalizing from the plane, where a relaxed elastic line is a segment of a straight line, a fluctuation carries the straight segment to an arc of a circle, and the measure of the extent of bending is the total curvature of the arc. It is then natural to measure the extent of a bending fluctuation on a general surface by the total geodesic curvature of the arc into which the elastic line is carried by the fluctuation. The path of the bent elastic line must satisfy a differential equation on the surface, which is derived by variational methods. The bending energy per unit length scales with the square of the extent of bending (Hooke’s law) if the surface is a plane or a sphere, but not generally. For ...

Experimental data on well documented global polymer transitions can be correlated with the linear charge density critical for counterion condensation. Other data exhibit abrupt behavior at the counterion condensation point and may consistently be interpreted in terms of a transition between locally folded and extended structures. We suggest that the underlying cause of these observations is the onset of mechanical instability of locally folded structures at the counterion condensation point.

The dipole moment induced in the ion atmosphere of a macroion by application of a steady electrical field is calculated when both the applied field and the field of the macroion are small. The macroion is modeled as a rigid array of point charges in general configuration. Linearized transport and equilibrium equations are then solved analytically by Fourier transform analysis. In the absence of relative motion between macroion and solvent the moment is isotropic and proportional to the number of charges on the macroion. It varies inversely as the concentration of supporting salt. For a symmetrical salt (same charge but for sign on counterion and co‐ion), the dipole moment vanishes, as the salt ions of one sign prevent polarization of the ions of opposite sign. For a general salt the moment does not vanish. If the counterion charge is the larger of the two salt ions, the induced moment is positive. It is negative if the co‐ion has the larger charge. There is a clear sense in which the polarization of co‐io...

To a fixed‐charge lattice model previously employed to investigate ionic mobility in a system with no bulk flow, we have added crude hydrodynamic boundary conditions which make possible a solution of the appropriate Navier—Stokes equation in the presence of an external electric field. The resulting flow, electro‐osmosis, is strongly dependent on the concentration of neutral salt present, attaining a maximum in the absence of salts. The gradual transition from Poisseuille parabolic flow to ``plug'' flow as salt concentration or fixed‐charge density increases is explicitly demonstrated for the model.

The interaction with point ions of a line charge supplemented by a distance of closest approach is studied by means of the Mayer theory of ionic solutions. A number of points involved in a recent theory of polyelectrolyte solutions (Papers I and II of this series) are thereby clarified. It is demonstrated that the counterion condensation phenomenon occurs in the limit of zero concentration, that the Debye–Huckel approximation is applicable to the coions and uncondensed counterions, and that the relevant dielectric constant is that of the pure bulk solvent.

We have considered the movement of an ion under the influence of Brownian motion, an externally applied constant electric field, and the spatially inhomogeneous potential set up by an assemblage of fixed charges. The resulting equation of motion is solved when the lattice of fixed charges is periodic, and when the fixed‐charge potential is small. Expressions are obtained for the macroscopic friction constant of the mobile ions as a function of the fixed‐charge potential, and, equivalently, for the relaxation field exerted by the fixed charges. The Einstein relation is used to obtain the macroscopic diffusion constant. The fixed‐charge potential is approximated by a periodic solution of the Poisson—Boltzmann equation in linearized form. The results are markedly dependent on the geometry of the fixed‐charge lattice. By varying the lattice parameters we present models for both polyelectrolyte solutions and simple ion‐exchange membranes, which give, in spite of the approximation of small fixed‐charge potential, semiquantitative agreement with data from self‐diffusion experiments on these systems.

The relations between the first‐order derivatives of the thermodynamic properties of three‐dimensional systems have been evaluated. The method of Jacobians which was used previously for simple or two‐dimensional systems has been extended, and the results presented in tabular form.

Formulas are derived for the osmotic coefficient, the Donnan salt‐exclusion factor, and the mobile‐ion activity coefficients in a polyelectrolyte solution with or without added sample salt. The formulas, which contain no adjustable parameters, are based on the (theoretical) observation by several workers that counterions will “condense” on the polyion until the charge density on the polyion is reduced below a certain critical value. The uncondensed mobile ions are treated in the Debye–Huckel approximation. In a restricted sense, the formulas are “limiting laws,” and this aspect is discussed at length. Detailed comparison with experimental data in the literature is given; agreement of the theory with experiment is usually found to be quantitative.

The relations between second‐order derivatives of the thermodynamic properties of three‐dimensional systems have been evaluated. The method of Jacobians which was used previously for first‐order systems has been extended, and results are presented in tabular form.

The concepts developed in Part I of this series are applied to the self‐diffusion of counterions and co‐ions. The condensed counterions are assumed to have no mobility, while the uncondensed counterions and all co‐ions execute Brownian motion subject to the locally inhomogeneous electric field created by the polyelectrolyte molecules. The polyelectrolyte field is assumed to have small amplitude, in analogy to the Debye–Huckel treatment of uncondensed mobile ions in Part I. The self‐diffusion coefficients of the counterion and the co‐ion are both less than the coresponding values in the absence of polyelectrolyte, but the coefficient for the counterion is much less than that for the co‐ion. The results for the self‐diffusion coefficient of the counterion in a salt‐free polyelectrolyte solution are in good agreement with experimental data.

A previous theory of counterion self‐diffusion in a polyelectrolyte solution is extended to describe the counterion flow in the presence of an applied electric field. A term proportional to the velocity of the polyion is obtained in a natural way, with no assumptions concerning ``bound'' counterions. The previous expression for the counterion self‐diffusion coefficient is regained as a special case. The significance of the formal agreement with the ion binding model is discussed at some length.

We have applied the ring‐diagram approximation of the Mayer ionic solution theory to solutions containing bolaform ions and to solutions containing dipolar ions. For the former case we derive a simple expression for the activity coefficient of the counterion to the bolaform ion. This expression has the property of increasing with the length of the bolaform ion from the Debye—Huckel limiting value for a uni‐divalent salt (zero length) to the limiting value for a uni‐univalent salt (infinite length). For dipolar ions we recover as a limiting case an expression derived by Kirkwood by classical means. We also derive a term accounting for dipolar ion—dipolar ion interactions, which is comparable to one obtained by Fuoss for a different model.

We apply the results of the previous paper describing the free energy of a polyelectrolyte‐added salt solution to the derivation of a relation between the osmotic coefficient and the activity coefficients of the mobile ions. From this relation, observed to be in excellent agreement with experiment, we obtain expressions resembling the empirical additivity rules. We conclude that the concept of ion binding may not be the appropriate way to interpret these rules. We also obtain an expression for the Donnan distribution of mobile ions and discuss the difficulties involved in the calculation of the Donnan osmotic pressure.