Genzo Tanaka

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...

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...

Dilute‐Solution Properties of Polar Polymers: Poly‐p‐chlorostyrene and Poly‐p‐bromostyrene

Light‐scattering and intrinsic‐viscosity measurements were carried out on fractions of poly‐p‐chlorostyrene and poly‐p‐bromostyrene in toluene at 30°C. Unperturbed dimensions were estimated from the Berry plots of second virial coefficients and also from the Stockmayer–Fixman plots of intrinsic viscosities. Values of the binary‐cluster integral β per monomeric unit were determined by the two methods proposed in the previous paper; one is based on the use of the Yamakawa–Tanaka equation for the statistical‐radius expansion factor, and the other the modified Stockmayer–Fixman plot. The results by the two methods were in excellent agreement with each other. The differences in β between poly‐p‐chlorostyrene and poly‐p‐methylstyrene, and between poly‐p‐bromostyrene and poly‐p‐methylstyrene in toluene, were analyzed using the theory of dilute solutions of polar polymers developed by Yamakawa and Rice on the basis of the cavity field and reaction field arguments. The agreement between theory and experiment was f...

Correlation function formalism for the intrinsic viscosity of polymers

A correlation function formalism for the intrinsic viscosity of polymers is studied. For this purpose, a Fokker‐Planck equation and the momentum flux equation in the full phase space of the polymer are derived by an application of the projection operator method to the whole system consisting of the polymer and solvent molecules. These relations are further reduced to Kirkwoods Fokker‐Planck equation and the corresponding flux in the polymer coordinate space. These steps give a molecular‐theoretical basis to the Kirkwood equation, and the reduced flux obtained is essentially equivalent to that suggested previously by Stockmayer and co‐workers. Some comments are made on related studies, particularly that of Doi and Okano, which is considered to be erroneous.

Intrinsic Viscosity of Polymer Chains with Small Excluded Volume

The intrinsic viscosity of polymer chains with small excluded volume is calculated on the basis of the Fixman—Pyun theory for unperturbed chains. The eigenfunctions of the unperturbed free‐draining time evolution operator are chosen as the basis set, and the formulation is made so as to take into account the coupling of the normal coordinates for excluded‐volume interactions but not for hydrodynamic interactions. An additional approximation introduced is the preaveraging of the Oseen hydrodynamic interaction tensor. Therefore, the present theory gives the same result as the Zimm theory in Hearsts version for vanishing excluded volume. Evaluation is carried out only in the nondraining limit, and the cubed viscosity—expansion factor is obtained as αη3= 1+1.06z –···, where z is the well‐known excluded‐volume parameter. This is consistent with the recent experimental results. Furthermore, it is emphasized that the present value for the coefficient of z is smaller than the corresponding value of 1.80 predicte...

Second‐Order Perturbation Theory of the Mean‐Square Radius of a Linear Polymer Molecule

A second‐order perturbation theory of the expansion factor αS for the radius of gyration of a linear polymer molecule is presented. The result is αS2=1+1.276z−2.082z2+···, where z is the well‐known excluded‐volume parameter. A Ptitsyn‐type closed expression for αS is also presented. Upon combining this and the corresponding expression for the expansion factor αR for the end‐to‐end distance, there is obtained the relation αR2/αS2=1.06 at large z. This is in excellent agreement with the Monte Carlo result obtained by Wall and Erpenbeck.

Some further comments on correlation‐function formulas for polymer intrinsic viscosity

A correlation‐function formalism for the intrinsic viscosity of polymers is reinvestigated. It is shown that the nonprojected part of the total momentum flux for the solution may be neglected for long enough chains and that the diffusion‐force term does not appear in the flux for the polymer in the diffusion limit.