On the theory of the second virial coefficient for polymer chains

Abstract

The theory of the second virial coefficient A2 for flexible polymers is reconsidered in order to explain two observations inconsistent with the two-parameter theory prediction. One is the decrease in the interpenetration function with increasing molecular weight M in good solvents even in the ordinary range of M in which the unperturbed mean-square radius of gyration is proportional to M. The other is the nonvanishing of A2 for small M at the temperature. It is shown that the first observation may be explained by considering effects of chain stiffness on and also on the expansion factor as, which remain rather large even in the range of M above, although vanishingly small for very large M, according to the Yamakawa-Stockmayer-Shimada theory for the wormlike and helical wormlike chains. Therefore the theory also predicts that the increase in with increasing solvent power depends in general on M. The second may be explained by considering possible effects of chain ends. Both effects are then analyzed by a comparison with experiment. Thus it is emphasized that the two-parameter theory should be considered a limiting law for extremely large M.

Repulsive and Attractive Interactions between Polystyrene Chains in a Poor Solvent

“…On the other hand, A 2 can be formulated also by the cluster expansion theory for the helical wormlike beads model. 3,22,23 Retaining the first order terms of the binary and ternary cluster integrals, we can write…”

Fine characterization of oligomers and polymers in dilute solution

“…The conventional TP theory [10] of A 2 predicts the interpenetration function C appearing in A 2 to be a function only of z, as in the cases of the expansion factors. On the other hand, the theory [1,27,28] of A 2 based on the HW chain model predicts that C is not a function only of z norz because of chain stiffness, and also that the dependence of A 2 on M is remarkably affected, especially for small M, by a chemical difference of the chain ends from the inner parts of the chain. The main purpose of this subsection is to confirm the validity of the HW theory of A 2 for a-PaMS and a-PS.…”

Second and third virial coefficients of low-molecular-weight polystyrene with a benzyl end in toluene

“…3 The importance of the effects of chain stiffness on A 2 was discussed by Yamakawa. 4 Einaga et al 5 later showed that the effects of chain ends, as well as chain stiffness, must be considered to obtain a quantitative agreement between the theoretical and observed values of A 2 for polystyrene (PS) in toluene (a good solvent) for M w o10 4 , where M w denotes the weight-average molecular weight. The PS samples used by Einaga et al 5 had a butyl group at one end of the chain.…”

“…The positive A 2 of butyl-PS increasing with decreasing M w is attributed to effects from the chain ends. 4 On the other hand, values of A 2 for benzyl-PS became negative and decreased with decreasing M w . 6 It was considered that the negative values of A 2 occurred because of the effects of the three-segment interactions and that effects from chain ends were negligibly small.…”