Hiromi Yamakawa

Transport Properties of Polymer Chains in Dilute Solution: Hydrodynamic Interaction

Several problems related to hydrodynamic interaction in flexible chain polymers without excluded volume are investigated. First, a correction to the Oseen hydrodynamic interaction tensor for a system of many spheres is derived, taking into account the finite volume of the spheres. The new hydrodynamic interaction tensor gives the positive definite diffusion tensor identical with that of Rotne and Prager. Second, the possible effects of the correction term in the hydrodynamic interaction tensor on the translational diffusion coefficient and intrinsic viscosity are examined when preaveraging the hydrodynamic interaction tensor is avoided following the procedure of Pyun and Fixman. If the Rouse free‐draining normal coordinates together with the diagonal approximation are used, the effects are shown to be negligibly small in the case of flexible chains. Third, however, the intrinsic viscosity and diffusion coefficient are re‐evaluated, taking the lower limit of | i − j | as unity in the evaluation of sums ove...

Statistical mechanics of helical worm‐like chains. XV. Excluded‐volume effects

The expansion factors for the mean‐square end‐to‐end distance and radius of gyration are studied for the helical worm‐like (HW) chain with the excluded‐volume effects incorporated in the Yamakawa–Stockmayer scheme. In this scheme, approximate closed expressions for them are expressed in terms of the excluded‐volume parameter z and the first‐order perturbation coefficient K(L) as a function of the total contour length L of the chain. The ring‐closure probabilities necessary for the evaluation of K(L) are evaluated by a slight modification of the method previously developed for the Kratky–Porod (KP) worm‐like chain. The actual evaluation is carried out for the values of the HW model parameters in their limited ranges, and also for the KP chain. It is then numerically shown that K(L) for the HW chain is approximately equal to that for the KP chain. From a simple analysis, this conclusion may be expected to be generally correct, so that the expansion factors for the HW chain may be expressed in terms of the s...

Second Virial Coefficient of Linear Polymer Molecules

A variation theory is developed for obtaining a closed expression for the second virial coefficient A 2 of linear polymer molecules, which corresponds to the Fixman theory of the excluded volume effect. It is shown that the second virial coefficient of the polymer in good solvents is effectively identical with that of a nonpenetrating sphere whose radius is proportional to the root‐mean‐square statistical radius 〈S 2〉½. In extremely good solvents,A 2 becomes a constant independent of molecular weight M. A graphical method is proposed for separate determination of the effective bond length and the polymer—solvent interactions; the method consists of plotting A 2 M ½ against M ½. Applications of this method are illustrated by four examples, polystyrene in toluene, isotactic polypropylene in tetralin and in α‐chloronaphthalene, and nitrocellulose in acetone. The effective bond lengths obtained are in good agreement with those previously evaluated from the molecular weight dependence of intrinsic viscosity [η], but the polymer—solvent interactions are not. The theoretical value of the ratio A 2 M/[η] is about 60 in good solvents, which is approximately a half of the ordinary experimental values. A modification of the variation theory is proposed, which corresponds to Ptitsyns modification of the Fixman theory and leads to a more plausible value 110 for A 2 M/[η]. The triple contact term in the perturbation expansion series of A 2 is also evaluated. It is found that the expansion of a molecular coil due to the volume effect is somewhat suppressed by intermolecular interactions in the proximity of the second molecule.

Statistical mechanics of helical wormlike chains. I. Differential equations and moments

A statistical‐mechanical theory of wormlike chains with helical conformations arising from bending and torsional energies is developed. A differential equation for the trivariate distribution function of the end‐to‐end distance, the unit tangent vector, and the unit curvature vector is derived from the path integral formulation, and several moments are evaluated. The results show that the characteristic ratio for the end‐to‐end distance or the radius of gyration as a function of chain length t exhibits a maximum with a swelling at some t greater than the maximum point under certain conditions. It is applied to atactic and syndiotactic poly(methylmethacrylate) chains, and their helix parameters are determined reasonably. The theory is also applied to the helix–coil transition in polypeptide chains.

Excluded‐Volume Effects in Linear Polymer Chains: A Hierarchy of Differential Equations

A hierarchy of differential equations for the expansion factor of a linear polymer chain is derived in a purely formal way by successive differentiation of the general equation of Fixman with respect to the binary‐cluster integral for a pair of chain segments. The hierarchy is truncated by a physically reasonable assumption to obtain an approximate solution. The assumption introduced is discussed in detail. In order to obtain numerical results, it is necessary to develop at least the third‐order perturbation theory, and the third expansion coefficient is evaluated to be 6.459 for the mean‐square end‐to‐end distance. Assuming that the third coefficients are the same for the end‐to‐end distance and the radius of gyration, there is obtained the equation for the expansion factor αS for the latter, αS2=0.541+0.459(1+6.04z)0.46, where z is the well‐known excluded‐volume parameter. Although this equation has the asymptotic form αS4.35=constant z at large z, a definite conclusion concerning the value of the expon...

Excluded-Volume Effects

This chapter deals with the theory of the excluded-volume effects in dilute solution, such as various kinds of expansion factors and the second and third virial coefficients, developed on the basis of the perturbed HW chain which enables us to take account of both effects of excluded volume and chain stiffness. Necessarily, the derived theory is no longer the two-parameter (TP) theory [1], but it may give an explanation of experimental results [2] obtained in this field since the late 1970s, which all indicate that the TP theory breaks down. There are also some causes other than chain stiffness that lead to its breakdown. On the experimental side, it has for long been a difficult task to determine accurately the expansion factors since it is impossible to determine directly unperturbed chain dimensions in good solvents. However, this has proved possible by extending the measurement range to the oligomer region where the excluded-volume effect disappears. Thus an extensive comparison of the new non-TP theory with experiment is made mainly using such experimental data recently obtained for several flexible polymers. As for semiflexible polymers with small excluded volume, some remarks are made without a detailed analysis.

Experimental Test of the Two‐Parameter Theory of Dilute Polymer Solutions: Poly‐p‐methylstyrene

In order to test the two‐parameter theory of dilute polymer solutions, light‐scattering and intrinsic‐viscosity measurements were carried out on fractions of poly‐p‐methylstyrene in toluene, dichloroethane, cyclohexane, and methyl ethyl ketone at 30°C, and in diethyl succinate at temperatures ranging from 16 to 60°C. The theta temperature for this polymer in diethyl succinate was found to be 16.4°C. With the data for the statistical‐radius expansion factor αS and the interpenetration function Ψ appearing in the second virial coefficient, validity of a theory of these quantities was examined using the two criteria introduced previously: (1) consistency in the values of the excluded‐volume parameter z determined from αS and Ψ, and (2) linearity between z and the square root of the molecular weight. It was found that the Yamakawa–Tanaka theory of αS and the Kurata–Yamakawa theory of Ψ were a self‐consistent pair of intramolecular and intermolecular theories of interaction which satisfied both of the two crit...

Concentration Dependence of Polymer Chain Configurations in Solution

On the basis of the theory of fluids and of the fashion prevailing in the statistical thermodynamics of dilute chain polymer solutions, the segment distribution functions are formally derived as a power series in concentration. The mean‐square radius of gyration and end‐to‐end distance at finite concentrations are calculated by using the general equations derived and introducing the modified random flight model. Evaluation is carried out up to the linear term in concentration. The coefficients of the linear terms are obtained as a power series in the excluded volume parameter, and also appropriate closed forms for those are proposed, which might be properly applied to good solvent systems. The results show that the polymer chain dimension decreases with increasing concentration. Then the concentration‐dependent term in the intramolecular intensity function in light scattering is evaluated. It is pointed out that the separation of this term and the intermolecular correlation leads to the possibility of est...

Dynamics of spheroid‐cylindrical molecules in dilute solution

The dilute solution dynamics of spheroid–cylindrical molecules, i.e., straight cylinders with oblate, spherical, or prolate hemispheroid caps at the ends, is studied in detail. The translational and rotatory diffusion coefficients and the dynamic intrinsic viscosity are evaluated numerically for short cylinders by determining the frictional force by an orthodox method of classical hydrodynamics. For long cylinders, the Oseen–Burgers procedure is shown to be valid, and the results previously obtained by it are still useful. Thus, empirical interpolation formulas for the transport coefficients above are also constructed to be applied to spheroid–cylinders of arbitrary size. The end effects on the translational and rotatory diffusion coefficients are rather small, while the effect on the zero‐frequency intrinsic viscosity is remarkable, depending appreciably on the shape of the ends, though for relatively short cylinders. In general, for a rigid body of revolution having a plane of symmetry perpendicular to its axis, it is shown that the dynamic intrinsic viscosity may be expressed in terms of the zero‐frequency intrinsic viscosity, the rotatory diffusion coefficient about a principal axis in the symmetry plane, and a newly defined factor, which is also associated with the rotational motion about this axis.

Theory of Dilute Polymer Solution. I. Excluded Volume Effect

A statistical theory of the excluded volume effect in a polymer molecule is developed, making use of a pearl necklace model with a continuous medium approximation. For the mean square end‐to‐end distance 〈R2〉 and the mean square radius of gyration 〈S2〉, the theory leads to the same expressions as Zimm, Stockmayer, and Fixmans. Taking solutions of polystyrene in cyclohexane as an example, it is illustrated that, in the vicinity of the Flory temperature, the expression for 〈S2〉 enables to give a complete quantitative interpretation to light scattering data. The distribution function for distance Rkl between segments k and l, its 2mth moment 〈Rkl2m〉, and the mean square distance 〈Sk2〉 between segment k and the center of molecular mass are also calculated to make clear the nature of the excluded volume effect. Finally, the influence of the volume effect upon light scattering measurement is discussed somewhat in detail.