Second Virial Coefficient of Linear Polymer Molecules

Abstract

A variation theory is developed for obtaining a closed expression for the second virial coefficient A2 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 (S2 )t. In extremely good solvents, A2 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 2M' against Mt. Applications of this method are illustrated by four examples, polystyrene in toluene, isotactic polypropylene in tetralin and in a-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 ['1J, but the polymer-solvent interactions are not. The theoretical value of the ratio A 2M/['1J 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 Ptitsyn's modification of the Fixman theory and leads to a more plausible value 110 for A 2M/ ['1} The triple contact term in the perturbation expansion series of A2 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.

Studies of Polyelectrolyte Solutions III. Correlations between the Second and the Third Virial Coefficient

“…One of the first investigations was done by Flory and Krigbaum28 which simply assumed that A3 has the same relation to A2 as in the case of rigid sphere molecules; that is (27) Subsequently, smoothed-density theories of A3 were developed by Stockmayer and Casassa,29 and by Koyama. 30 An approximate closed expression for A3 was firstly given by Yamakawa 31 ; that is M w)=4/3 (l-exp( -1.664' z» (28) with z=z/ri 3 , where ri is again the "intermolecular" expansion factor.…”

“…Up to now there exists no theory of A3 for polyelectrolytes. Therefore, we assume that eq 27 to 30 are applicable to polyelectrolytes if the parameter Z is replaced by its electrostatic part Zel' This idea is in analogy to the theories of a and P. However, it must the proved in detail.…”

Light Scattering and Viscometric Study of Cellulose in Aqueous Lithium Hydroxide

“…According to Kurata-Fukatsu-SotobayashiYamakawa (KFSF-I) theory, zh 0 (z) is related to z through the relation 15 zho(£)=(1/5.047){1-(1 +0.683£)-7 · 39 } (9) and in the smali rx. region, we can employ the relation, 16 (10) Putting the experimental data on A 2 , Mw, and (S 2 )!j 2 into eq 7-10, rx.…”

Further Test of the Two-Parameter Theory of Dilute Polymer Solutions: Poly(p-bromostyrene)

“…Next we test approximate theories of as with respect to the second criterion, i.e., the linearity between z and Mw 1 (22) and also the theoretical curve obtained by Fujita and Norisuye (FN). 27 The values of z thus obtained are plotted against Mw 112 in Figure 4.…”