Introduction

In a previous study,1 effects of chain stiffness on the second virial coefficient A2 for ideal ring polymers without excluded volume were investigated by Monte Carlo (MC) simulation using a discrete version of the Kratky–Porod (KP) wormlike chain.2, 3 The topological interaction between a pair of ideal rings to keep their linking number Lk zero causes an effective volume VE excluded to one ring by the presence of another, and therefore makes A2 (proportional to VE) positive. The behavior of A2 was examined as a function of the reduced total contour length λL as defined as the total contour length L of the KP ring divided by the stiffness parameter3 λ−1. A comparison was also made of the MC results with available literature data4, 5, 6 for ring atactic polystyrene (a-PS) in cyclohexane at Θ (34.5 or 35 °C) in the range of the weight-average molecular weight Mw from 1 × 104 to 6 × 105. Although agreement between the MC and experimental data is fairly well, the former is somewhat (order 10−5 cm3 mol g−2) larger than the latter. As the MC values are exact for the ideal KP ring, this minor discrepancy may be regarded as arising from the fact that real unperturbed ring polymers in the Θ state cannot fully be described by the ideal KP ring. The purpose of the present study is to consider a possible source of the discrepancy, that is, effects of three-segment interactions on A2 for the real unperturbed ring polymers.

If the ternary-cluster integral β3 representing the three-segment interaction is taken into account in addition to the binary-cluster integral β2 representing the two-segment interaction7, 8, 9 in the perturbation theory10 of the mean-square end-to-end distance 〈R2〉 and A2, then the first-order perturbation terms in 〈R2〉 and A2 are proportional to the effective binary-cluster integral β=β2+const. × β3 in the limit of infinitely large molecular weight M, and the Θ temperature is defined as the temperature at which β but not β2 vanishes. Note that β3 is usually positive, so that β2 is negative at Θ. Strictly, the first-order perturbation term in A2 has the residual contribution proportional to −β3M−1/2, so that at finite M, A2 remains small negative (order 10−5 cm3 mol g−2) for small M even at Θ. It means that the three-segment contact probability between a pair of linear polymers decreases faster than the two-segment contact probability as M (or λL) is decreased and then the attractive effect due to β2 (<0) exceeds the repulsive effect due to β3 (>0). The two effects balance out in the limit of M→∞. If the situation is also the case with the real unperturbed ring polymer, then the residual contribution seems to make its A2 smaller than that for the ideal ring.

In practice, we carry out the first-order perturbation calculation of A2 for the Gaussian and KP rings with consideration of the three-segment interactions in addition to the two-segment ones. In the calculation, we must evaluate an integration of the series expansion of A2 in terms of the χ function defined by Equation (13.2) of Ref. 10, which corresponds to the Mayer f-function,11 to the first order over the configuration space of a pair of rings under the topological constraint of Lk=0. Unfortunately, however, the necessary integrals of the χ function and its triple product for a pair of rings cannot simply be related to β2 and β3, respectively, defined for linear chains because of the topological constraint, as explained later in some detail. We then resort to a calculation using a pair of phantom rings without the topological constraint in order to utilize β2 and β3 also for the ring chains.

Materials and methods

In the first-order perturbation calculation of A2 for a pair of rings, we take into account the two- and three-segment interactions, the former arising from the contact between two segments (two-body contact) on each of the pair and the latter from that among three segments (three-body contact), two of them on either of the pair and the rest on the other. As easily seen from Figure 1, where the two-body (a) and three-body (b) contacts between a pair of rings with Lk=0 and 1 are schematically depicted as examples, the values of the binary- and ternary-cluster integrals resulting from the integrations of the χ function and its triple product, respectively, over the configuration space for the pair of rings with Lk=0 are naturally different from those for a pair of linear chains, the latter being obtained by integrations over the full configuration space. Strictly speaking, we must further take account of possible effects of knots. Note that all the rings depicted in Figure 1 are of the trivial knot.

Figure 1
figure 1

Illustrations of the two-body (a) and three-body (b) contacts between a pair of rings with Lk=0 and 1.

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Unfortunately, however, analytical treatment of the inter-12, 13, 14, 15 and intramolecular topological constraints may seem to be impossible even in the case of the Gaussian ring. We therefore adopt phantom rings without the constraints in the evaluation of the residual contribution of the ternary-cluster integral to A2, for convenience, as mentioned above. As a result, we use β2 and β3 introduced for the linear chains also for the binary- and ternary-cluster integrals, respectively, for the rings.

Gaussian ring

For the (phantom) Gaussian ring composed of n identical beads with the binary- and ternary-cluster integrals β2 and β3 connected by the Gaussian bonds with root-mean-square length a, the first-order perturbation theory of A2 may be given by (see Appendix)

where NA is the Avogadro constant and β is the effective binary-cluster integral defined by

It is important to note that the definition of β for the Gaussian ring is identical to that for the linear Gaussian chain,7 and further that the residual contribution, the second term in the square brackets on the right-hand side of Equation (1), is proportional to n−1β3 in contrast to the case of the linear Gaussian chain for which the residual contribution is proportional to n−1/2β3.9 The implication is that the relative contributions (or probabilities) of the two- and three-body contacts to A2 for the Gaussian ring are identical with those for the linear Gaussian chain in the limit of n→∞ but the residual contributions are different from each other.

Wormlike ring

For the (phantom) KP ring of contour length L on which n identical beads with the binary- and ternary-cluster integrals β2 and β3 are placed with interval a (L=na), the first-order perturbation theory of A2 may be given by (see Appendix)

with λ−1 the stiffness parameter and β the effective binary-cluster integral redefined by

The result so obtained for the KP ring is apparently equivalent to that obtained for the linear KP (or HW) chain given by Equation (34) with Equation (35) of Ref. 16 with c∞=1 except for the expression for the dimensionless function I(L) of (reduced) L which may be given by

with Δ=L−3.075. The function I(λL) approaches 0 and 1.465 in the limits of λL→0 and ∞, respectively, as in the case of the linear KP chain,16 and therefore β defined by Equation (4) becomes identical to that for the linear KP chain. As a result, the factor I(∞)−I(λL) on the right-hand side of Equation (3) approaches 1.465 and 0 in the limits of λL→0 and ∞, respectively, as in the case of the linear KP chain, although the asymptotic form 3(λL)−1 in the limit of λL→∞ is very different from 4(λL)−1/2 for the linear KP chain,16 the situation being consistent with the above-mentioned difference between the linear Gaussian chain and Gaussian ring.

Results and Discussion

Figure 2 shows plots of I(λL)−I(∞) against log λL. The heavy solid and dashed curves represent the theoretical values calculated from Equation (5) for the KP ring and from Equation (36) of Ref. 16 for the linear KP chain, respectively. For comparison, in the figure are also plotted values of the asymptotic forms I(λL)−I(∞) =−3(λL)−1 for the KP ring and I(λL)−I(∞) =−4(λL)−1/2 for the linear KP chain, represented by the light solid and dashed curves, respectively, which in principle correspond to the values for the Gaussian ring and linear chain, respectively. It is seen that I(λL)−I(∞) for the KP ring vanishes with increasing λL more rapidly than that for the linear KP chain because of the above-mentioned difference in the asymptotic form, that is, the former is proportional to (λL)−1 while the latter to (λL)−1/2.

Figure 2
figure 2

Plots of I(λL)−I(∞) against log λL. The heavy solid and dashed curves represent the theoretical values calculated from Equation (5) for the KP ring and from Equation (36) of Ref. 16 for the linear KP chain, respectively. The light solid and dashed curves represent the values of the asymptotic forms I(λL)−I(∞)=−3(λL)−1 for the KP ring and I(λL)−I(∞) =−4(λL)−1/2 for the linear KP chain, respectively.

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Now we proceed to we make a comparison of the present theoretical results with the experimental data for ring a-PS in cyclohexane at Θ obtained by Roovers and and Toporowski4 and by Takano et al.6 For this purpose, we simply assume that A2 for the KP ring at Θ (β=0) may be written as a sum of the contribution of the intermolecular topological interaction (Lk=0) given by Equation (29) with Equations (25) and (26) in Ref. 1 and the residual contribution of β3 given by Equation (3) with β=0 along with Equation (5). On this assumption, the values of A2 are calculated as a function of Mw, λL being converted to Mw by log Mw=log(λL)+log(λ−1ML) with ML the shift factor3 defined as the molecular weight per unit contour length of the KP ring. In the calculation, we use the relation a=M0/ML, where M0 is the molecular weight of repeat units and set equal to 104 for a-PS, and the values of the necessary parameters determined for linear a-PS in cyclohexane at 34.5 °C (Θ): λ−1=16.8 Å,16 ML=35.8 Å−1,16 and β3=4.5 × 10−45 cm6.17 We note that although the ring a-PS samples used in the literatures4, 6 might be of the trivial knot, the difference in A2 between the ring of the trivial knot and the phantom ring is negligibly small in the range of M where the experimental data exist as shown in figure 6 of Ref. 1.

Figure 3 shows double-logarithmic plots of A2 (in cm3 mol g−2) against Mw for ring a-PS in cyclohexane at 34.5 °C (Θ). The open circles and triangles represent the experimental values by Roovers and Toporowski4 with the correction for residual linear a-PS1 and by Takano et al.,6 respectively. The solid and dashed curves represent the theoretical values of A2 at Θ for the KP ring with and without the residual contribution of β3 to A2 so calculated. The theoretical values of A2 with the residual contribution of β3 deviate downward very slowly from those without the contribution with decreasing Mw, and the deviation is very small in the range of Mw where the experimental data exist. For comparison, there are also plotted the theoretical values for the KP ring with the residual contribution for the linear KP chain, the contribution being calculated from the right-hand side of Equation (34) with β=0 along with Equation (36) in Ref. 16 and with the above-mentioned values of λ−1, ML and β3 (and M0), represented by the dot-dashed curve. Although the downward deviation of the values of A2 with the residual contribution for the linear KP chain from those without the contribution is larger than that in the case of A2 with the contribution for the KP ring, the theoretical values are still appreciably larger than the experimental ones. It may then be concluded that the consideration of the residual contribution of β3 cannot compromise the difference between theory and experiment.

Figure 3
figure 3

Double-logarithmic plots of A2 (in cm3 mol g−2) against Mw. The open circles and triangles represent the experimental values for ring a-PS in cyclohexane at Θ by Roovers and Toporowski4 with the correction for residual linear a-PS1 and by Takano et al.,6 respectively. The solid and dashed curves represent the theoretical values for the KP ring with and without the residual contribution of β3 to A2. The dot-dashed curve represents the theoretical values for the KP ring with the residual contribution for the linear KP chain.

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Conclusion

We have carried out the first-order perturbation calculation of the second virial coefficient A2 of the phantom Gaussian and KP rings without the intra- and intermolecular topological constraints with consideration of the ternary-cluster integral β3 in addition to the binary-cluster one β2. It has been shown that the residual contribution of β3 to A2 of the KP rings as a function of the reduced contour length λL increases rapidly from a negative constant and vanishes in the limit of λL→∞ following the asymptotic relation A2 ∝−(λL)−1 in this limit. From a comparison between the present theoretical results and literature experimental data, it has been found that the residual contribution of β3 to A2 is negligibly small for ring a-PS in cyclohexane at Θ in the range of 1 × 104 ≲ Mw ≲6 × 105.