Reduced third virial coefficient for linear flexible polymers in good solvents

Abstract

Effects of chain stiffness and three-segment interactions on the reduced third virial coefficient g (= / • 2 ) for linear flexible chains in good solvents are theoretically investigated to explain the recent experimental finding that, in contrast to the two-parameter theory prediction, g remains positive when the radius expansion factor as approaches unity by lowering the polymer molecular weight M in a given good solvent. Here, A¡ and A3 are the second and third virial coefficients, respectively. The stiffness effect on g is evaluated in the binary cluster approximation first by applying perturbatively Yamakawa and Stockmayer's wormlike bead model and then by combining the first-order result with the Stockmayer-Casassa theory of g for flexible chains. The excluded-volume parameter is transformed to as3 using the combination of the Yamakawa-Stockmayer-Shimada theory and the Domb-Barrett equation for as. Finally, the effect of threesegment interactions is incorporated in a first-order perturbation approximation. The theoretical g thus calculated as a function of as3 is found to agree satisfactorily with the published data for polystyrene and polyisobutylene in good solvents for as down to near unity.

Second and Third Virial Coefficients for Polyisobutylene in Heptane, an Intermediate Solvent

“…This discrepancy confirms the previous conclusion 5 (derived for good solvent systems) that the two-parameter theory breaks down for rx8 3 below 1.7. The failure of the theory in this rx/ region for heptane solutions is attributable primarily to the neglect of ternary cluster interactions, since the g2 vs. rx/ relation with the effect of chain stiffness (not shown in the figure) comes only slightly above the dot-dash line for rx/ < 2; the stiffness effects on g 2 and rx/ contribute toward lowering and raising the g 2 vs. rx 8 3 curve, respectively, 5 and these opposite contributions, along with the small B 2 value (see Figure 2 of ref 5), lead to the less pronounced stiffness effect on the curve for heptane. The essentially constant values of experimental g in heptane (for rx/ < 1.7) may be explained as the result of a compensation of an increase in g 3 and a decrease in g 2 with decreasing rx/.…”

“…12, 1994 data for PIB in Figure 9 also conform to the predictions, lending additional support to the validity of the Yamakawa theory. In other words, the stiffness effect on the 'P vs. a/ relation is remarkable even at a 5 3 as large as 5. It should be pointed out, however, that in order to fit the theory to the 'P data in heptane and CH, we must choose B 2 values about three times as large as those determined above from a/.…”

“…In previous work, Norisuye et al 5 introduced approximately effects of chain stiffness and ternary cluster interactions into the two-parameter theory of Stockmayer and Casassa 2 7 for g. The former effect was evaluated for the wormlike bead chain in the binary cluster approximation and the latter effect, in the first-order perturbation approximation. The result is expressed by a sum of the binary cluster contribution g 2 and the ternary cluster contribution g 3 , i.e., (12) Here, B 3 is the dimensionless parameter defined by {3 3 J 3 // 3 with /3 3 the ternary cluster integral for beads and C is a known function of ).,L and a cutoff parameter (see Figure 1 of ref 5).…”

“…In the coil limit, C becomes equal to 2.22, so that z reduces to z 2 /rx/. As in our previous work, 5 we take the cutoff parameter to be 0.3.…”

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

“…This is consistent with the prediction from the two-parameter theory, except that the values of g were almost constant for 10 4 tM w t10 5 . Norisuye et al 10 examined g in the low-molecular-weight range and found that three-segment interaction has a more important role than chain stiffness. Later, Osa et al 11 reached essentially the same conclusion from a study on poly (a-methylstyrene) in toluene.…”