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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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).
Plots of Il(L) against log L for l=0 and 1. The solid curves represent the theoretical values obtained by numerical integrations of Equation (A14).
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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
