Abstract
The first-order perturbation calculation is carried out of the second virial coefficient A2 of the phantom Gaussian and Kratky–Porod (KP) wormlike rings without inter- and intramolecular topological constraints with consideration of the ternary-cluster integral β3 in addition to the binary-cluster integral β2. The behavior of the residual contribution of β3 to A2 of the KP rings is examined as a function of the reduced total contour length λL as defined as the total contour length L divided by the stiffness parameter λ−1. From a comparison of the present theoretical result with experimental data, it is found that the residual contribution of β3 to A2 is negligibly small for ring atactic polystyrene in cyclohexane at Θ in the range of the molecular weight from 1 × 104 to 6 × 105.
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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.
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.
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.
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.
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Appendix
Appendix
FIRST-ORDER PERTURBATION THEORY
In this appendix, we derive the first-order perturbation theories of A2 for the Gaussian and KP rings with consideration of β3 in addition to β2.
Gaussian ring
In the same manner as in the case of the first-order perturbation theory of A2 for the linear Gaussian chain,7 A2 for the Gaussian ring under consideration may be expanded in the form
where the n beads composing the Gaussian ring are serially numbered 1, 2, ⋯, n from an arbitrary bead and P(0i1i2) represents the probability of the contact between the ith and jth beads with P(Rij) being the unperturbed distribution function of the vector distance Rij between them. The function P(Rij) may be given by10
with Rij=|Rij| and μij=(j−i)[1−(j−i)/n]. Substitution of Equation (A2) into Equation (A1) and conversion of the sums to integrals leads to
where the additional cutoff parameter9 appearing in the integrations has been set equal to unity as in the case of the linear Gaussian chain.16 The result may then be rewritten in Equation (1) with Equation (2).
Wormlike ring
In the same manner as in the case of the first-order perturbation theory of A2 for the linear KP (or HW) chain,16 A2 for the KP ring of total contour length L under consideration may be expanded in the form
with P(0; s2−s1, L) the probability of the contact between the contour points s1 and s2 (0≤s1<s2<L) on the KP ring separated by the contour distance s2−s1 (or L−s2+s1). In what follows, for simplicity, all lengths are measured in units of λ−1 unless otherwise noted, so that, for instance, λL is replaced by (reduced) L. Carrying out the integration in the second term in the square brackets on the right-hand side of Equation (A4) over s1, s2 and s3 with t=s2−s1 fixed, we obtain
with I(L) the dimensionless factor as a function of (reduced) L defined by
Using the relation P(0;L−t, L)=P(0;t, L) which naturally holds for the KP ring, Equation (A6) reduces to
The conditional distribution function P(R, u|u0; t, L) of both the vector distance R between the points s1 and s2 and the unit tangent vector u at s2 with the unit tangent vector u0 at s1 fixed may be given by3, 18
where G(R, u|u0; L) is the conditional distribution function of the end-to-end vector R of the linear KP chain of contour length L and the unit tangent vector u at its terminal end with the unit tangent vector u0 at its initial end fixed3 and G(0, u0|u0; L) represents the probability that the linear KP chain forms a ring. Integration of both sides of Equation (A8) over u and u0 leads to
using the relation G(R,−u|−u0; L)=G(R,u0|u; L) and the fact that G(0,u0|u0; L) is independent of u0.
The conditional (or angle-dependent) ring-closure probability3 G(0, u|u0; t) for the linear KP chain appearing in Equation (A9) may be expanded in terms of the normalized spherical harmonics3 Ylm as follows,19
where hl(t) is the expansion coefficient and u=(1,θ,φ) and u0=(1,θ0,φ0) in spherical polar coordinates. We note that hl(t) is identical to (3/2π)3/2gl(t)/(2l+1) with gl(t) defined in Ref. 19 and also to hl00(t) given by Equation (8.13) of Ref. 3. For l=0 and 1, interpolation formulas for hl(t) are given by20
with Δ=t−3.075, which have been constructed from the Daniels approximation for large t and a solution for small t with consideration of small thermal fluctuations in the configuration of the KP ring around its most probable one.21 As for l≥2, the relation hl(t)=O(t−l−3/2) for large t can be obtained from Equation (8.13) with Equation (4.177) of Ref. 3 in the Daniels approximation.
Substituting Equation (A10) into Equation (A9) and carrying out the integrations over u0 and u, we obtain
Substitution of Equation (A12) into Equation (A7) leads to
with
In the same manner as in the cases of h0(t) and h1(t), an interpolation formula has been constructed for G(L)=G(0,u0|u0; L),21 which is given by
with Δ1=L−1.9.
From the asymptotic behavior of hl(t)hl(L−t) (in the range of 0≤t ≤L/2) and G(L) in the limit of L→∞, it can be shown that Il(L)=O(L−l) in the limit. We then have
where we have used the asymptotic form, h0(L−t)/4πG(L)=1, in the limit of L→∞. Considering the fact that the (angle-independent) ring-closure probability3 G(0; t) is the integral of G(0, u|u0; t) given by Equation (A10) over u and therefore identical to h0(t), I(∞) for the KP ring is identical to that for the linear KP chain given by Equation (A4) of Ref. 16 with c∞=1, so that I(∞)=1.465. Although I(L) for the KP ring becomes identical to I(L) for the linear KP chain in the limits of L→0 and ∞, the behavior of the former as a function of L is different from that of the latter.
Figure 4 shows plots of I0(L) and I1(L) against the logarithm of L. It is seen that I0(L) increases monotonically from 0 to 1.465 with increasing L, while I1(L) first increases from 0 then decreases to 0 after passing through a very small maximum with increasing L. Since the relative magnitude of I1(L) to I0(L) is 2.5% at most, the contributions of Il(L) with l≥2 to I(L) may be considered to be very small if any. We therefore put I(L)≃I0(L)+I1(L) with omission of Il(L) with l≥2 in Equation (A13) and construct an interpolation formula for I(L) on the basis of the values of I0(L) and I1(L) obtained by numerical integration of the right-hand side of Equation (A14), the formula being given by Equation (5).
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Ida, D., Yoshizaki, T. Effects of three-segment interactions on the second virial coefficient of ring polymers in the Θ state. Polym J 48, 883–887 (2016). https://doi.org/10.1038/pj.2016.48
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DOI: https://doi.org/10.1038/pj.2016.48