Marshall Fixman

Intrinsic Viscosity of Polymer Chains

The intrinsic viscosity [η] of polymer chains is recalculated by methods similar to those of Kirkwood and Zimm. The theory is based on the preceding paper by Fixman, with the adoption of a smoothed excluded‐volume potential and the following first‐order approximation to the operator L which determines the segment coordinate distribution function ψ; for L acting on the nonequilibrium part of ψ, only the diagonal part of L in the free‐draining basis representation is used. This corresponds qualitatively to the Zimm theory in Hearsts version. However, the avoidance of preaveraging the Oseen hydrodynamic interaction tensor in the present work leads to larger values for the relaxation times of various normal modes when the hydrodynamic shielding effect is not negligible. The effect is more pronounced for lower modes, amounting to about eight percent increase in the lowest mode excited by the velocity gradient in the nondraining limit. The value of [η] in the same limit is found to be [η]=2.68×1023 L3/Mw (L: t...

Viscosity of Critical Mixtures

A theory is given of the sharp viscosity rise in mixtures in the critical mixing region. Good agreement with the experimental dependence of viscosity on composition and temperature is found. The method involves a calculation of the entropy production through diffusion which results when a mixture in a state of composition fluctuation is caused to have a velocity gradient. The long wavelength part of the spectrum of composition fluctuations is intense and very easily distorted by a velocity gradient in the critical region. The return to uniform composition through diffusion dissipates energy, and the loss is interpreted as an excess viscosity. It is shown that this method, which requires no knowledge of the intermolecular potential but only of the long‐range part of the radial distribution function, correctly gives the viscosity of a dilute 1–1 electrolyte.

Radius of Gyration of Polymer Chains. II. Segment Density and Excluded Volume Effects

The segment density ρ(r) at a distance r from the center of mass is evaluated for chains constrained to have a large specified radius of gyration R. It is found that ρ(r)≅(N/2π2)[r2(2R2−r2)12]−1H(2R2−r2), where N is the number of segments and H( ) is the step function. It is argued that this conditional distribution takes better account of large excluded volume effects than a simple scaling of the random flight ρ(r).

Radius of Gyration of Polymer Chains

The probability distribution of the radius of gyration of unbranched polymer chains is discussed. The characteristic function of the distribution is presented in closed form, and calculations are the made of the semi‐invariants, moments, and behavior of the probability at very large and very small values of the radius of gyration.

Dynamics of Polymer Chains

A general formalism is presented to serve as a basis for the calculation of dynamic properties of chain‐polymer solutions. The polymer model which is adopted restricts the applicability of the formalism to dilute or moderately concentrated solutions.

Absorption and Dispersion of Sound in Critical Mixtures

The entropy previously associated with the long wavelength spectrum of composition fluctuations in the critical mixing region is studied under the supposition that the temperature varies harmonically with time. Relaxation effects in the response of the long range part of the radial distribution function to the temperature variation produce a complex heat capacity. The consequent sound absorption rises to a maximum at the critical point, in good quantitative agreement with experiment. The equation of motion for the radial distributions function is the modified diffusion equation previously used in a theory of the anomalous viscosity rise in the critical mixing region.

Heat Capacity of Critical Mixtures

A theory is given of the heat capacity of liquid mixtures in the critical region. The entropy density is taken to be a function of the local composition and is expanded to quadratic terms in the composition fluctuation. The anomalous heat capacity computed from the average entropy fluctuation is very large because of the extreme sensitivity of the large composition fluctuations to the temperature. An application to a cubic lattice shows that the theory is equivalent to a ring diagram or random phase approximation in the theory of the Ising lattice. The heat capacity goes to infinity as (T—Tc)—½ in the critical region, for a mixture at the critical composition. Fair agreement with experiment is obtained; the one parameter which enters can be obtained, in principle, from x‐ray or light scattering.

Polyelectrolytes : A Fuzzy Sphere Model

The expansion factor α of polyelectrolytes is calculated on the assumption that at least one member of each interacting pair of charged segments is immersed in a segment cloud having the average sement density, here given by a uniform distribution inside a sphere. The counterion concentration is assumed to be much larger inside the sphere than the byion concentration, and a binding parameter is introduced via an effective dielectric constant. As a consequence α3 is predicted to be linear in the reciprocal square root of ionic strength. The dominant source of expansion is repulsion between segments such that one member of an interacting pair is inside and the other outside the background cloud. This same repulsion makes the free energy a minimum when the polymer shape is spherical.

Rate of unwinding of DNA

The mean time of unwinding of a DNA molecule is calculated. The unwinding is assumed to begin at one end and to proceed under the combined forces of diffusion and a driving torque of arbitrary magnitude. The standard deviation of the unwinding time is calculated approximately. If the free energy decrease on unwinding one turn, in units of kT , times the number of turns in the molecule is a number much greater than unity, diffusion plays a negligible role in the unwinding.

Sound Absorption in Gases in the Critical Region

A theory is based on a dynamic heat capacity associated with long‐wavelength density fluctuations in the critical region. Evaluation of the dynamic heat capacity requires the time‐dependent response of the long‐range radial distribution function to a uniform harmonic variation of temperature with time. This response is calculated on the assumption that the motion of matter in the neighborhood of a fixed molecule is determined by macroscopic laws of momentum and energy transport, except for the addition of a familiar critical contribution to the pressure. Relaxation of the important Fourier space components of the radial distribution function is not hindered appreciably by fluid inertia or viscous resistance. Rather the relaxation follows a generalized heat diffusion law. The calculations are applied to data on Xe and HCl, and satisfactory agreement between theory and experiment is found.