Introduction

Interaction between polymer chains, arising only from the chain connectivity (which inhibits chains from crossing each other), is important to understand not only the dynamical properties of concentrated solutions or melts but also static properties of solutions of ring polymers.1, 2 For ring polymers, this interaction is usually called a topological interaction (TI) because it works to conserve a given link type between a pair of ring polymers—it is a topological invariant. Consequently, a repulsive force, in the sense of the potential of mean force, results from the TI between unlinked ring polymers; therefore, the second virial coefficient A2 remains positive even for the ideal rings without excluded volume, as explicitly shown in the Monte Carlo (MC) study made by Frank-Kamenetskii et al. using a lattice model.3, 4 Their pioneer work on A2 of rings was followed by the theoretical studies of Iwata,5, 6 des Cloizeaux7 and Tanaka,8 and by the MC study of Deguchi and Tsurusaki9 based on the Gaussian chain model, which is valid for very long, flexible ring polymers. We note that des Cloizeaux also calculated A2 for the rigid ring.7 Experimentally, positive values of A2 were observed for ring atactic polystyrene (a-PS) in cyclohexane at the Θ temperature (34.5 or 35 °C) by Roovers and Toporowski,10 by Huang et al.11 and very recently by Takano et al.12 in the range of weight-average molecular weight Mw from 1 × 104 to 6 × 105.

The effective volume VE excluded to one ring by the presence of another is defined from A2 by

where NA is the Avogadro constant and M is the molecular weight. The effective excluded volume may be considered proportional to the cube of the root-mean-square radius of gyration 〈S2〉1/2 in both the rigid-ring and random-coil limits. Therefore, VE∝L3 and L3v in the respective limits, where L is the total contour length of the ring and v is the exponent in the asymptotic relation 〈S2〉1/2∝Lv in the random-coil limit. The exponent v is 1/2 for an ensemble constituted of rings of all kinds of knots with the Boltzmann weight and is considered ∼0.6 for one constituted of rings of the trivial knot (unknotted rings).13, 14 Thus, A2 is proportional to M in the rigid-ring limit and to M3v−2 (M−1/2 or M−0.2) in the random-coil limit, and must necessarily have a maximum in the range of the crossover from the rigid ring to the random coil. The purpose of this paper is to clarify such behavior of A2 of the ideal ring in the crossover or, in other words, to examine the effect of chain stiffness on A2 on the basis of the Kratky–Porod (KP) worm-like chain model.15, 16

As analytical treatment of the TI is complicated even in the case of the Gaussian chain model,5, 6, 7, 8 we resorted to an MC approach using a model for semiflexible rings composed of infinitely thin bonds with a harmonic bending energy between successive bonds. We generated two kinds of statistical ensembles, that is, one composed of rings of all kinds of knots with the Boltzmann weight and the other composed of those only of the trivial knot, and compared the results for 〈S2〉 and A2.

Materials and methods

Model

The MC model used in this study was essentially the same as that used by Frank-Kamenetskii et al.,17 that is, a ring composed of infinitely thin n bonds of length l with a harmonic bending energy between successive bonds. The n joints in the ring were numbered 1, 2, ⋯, n from an arbitrary joint, and the ith bond vector from the ith joint to the (i+1)th was denoted by li (i=1, 2, ⋯, n−1); ln was the nth bond vector from the nth joint to the first. The configuration of the ring could then be specified by the set {ln}=[l1, l2, ⋯, ln−1, (ln)], apart from its position in an external Cartesian coordinate system. Note that ln is a dependent variable for the ring. Let θi (i=2, 3, ⋯, n) be the angle between li−1 and li and θ1 be the angle between ln and l1. The total potential energy U of the ring may be written in terms of θi as follows:

where α is the bending force constant. The MC model so defined becomes identical with the KP ring of total contour length L in the continuous limit n → ∞, l → 0, and 〈cos θi〉 → 1 under the conditions nl=L and

where λ−1 is the stiffness parameter of the KP model and 〈cos θ〉 is defined as

with kB the Boltzmann constant and T the absolute temperature.17, 18 Note that the MC model reduces to the freely jointed chain in the limit of α → 0.

In what follows, we set l=1 for simplicity and carried out MC simulations for the rings with values of α/kBT given in the first column of Table 1. The values of 〈cos θ〉 calculated from Equation (4) and those of λ−1 of the corresponding KP model calculated from Equation (3) with l=1 are given in the second and third columns, respectively.

Table 1 Values of 〈cos θ〉 and λ−1
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MC sampling

For the initial configuration {ln}, we adopted an n-sided regular polygon of unit side length, which is the most stable configuration, and sequentially deformed it by the virtual motion introduced by Deutsch.19 We let v be the unit vector along the vector distance between a pair of joints randomly chosen under the condition that they not be next to each other. If the ith and jth joints (i<j) were chosen, v was along the vector sum . As illustrated in Figure 1, a trial configuration {l′n} was generated by rotating the shorter part of the ring around v by an angle φ randomly chosen in the range of [−π, π). The bond vectors li, li+1, ⋯, lj−1 were rotated if j−i⩽n/2 and the rest otherwise. If the bond vector lk underwent the rotation, l′k could be given by20

where I is the unit matrix and the rotation matrix R(v; φ) is given by

with vx, vy, vz the Cartesian components of v in an external system. With this rotation, l′k was renormalized to l′k (corr) so that ∣l′k (corr)∣=1, that is,

This was carried out to suppress roundoff errors characteristic of computer work. (Note that ∣l′k−l′k (corr)∣≪1.) If the bond vector lk did not undergo the rotation, on the other hand, we had l′k=lk.

Figure 1
figure 1

Illustration of an elementary step in MC simulations.

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Then, the adoption of the next trial configuration {l′n} was determined by the Metropolis method of importance sampling21 on the basis of the total potential energies given by Equation (2) for {l′n} and {ln}. That is, {l′n} was adopted as the next configuration with the (transition) probability τ({l′n}∣{ln}) defined as

with ΔU given by

where θ′i (i=2, 3, ⋯, n) is the angle between l′i−1 and l′i and θ′1 the angle between l′n and l′1. If {l′n} was discarded, {ln} was again adopted as the next configuration.

Through this MC algorithm, we sampled one configuration at every Mnom (nominal) steps and Ns configurations in total after an equilibration of 104 × Mnom steps. As each MC step did not conserve a knot type, the ensemble so obtained was composed of Ns configurations of all kinds of knots with the Boltzmann weight. We call this ensemble the mixed ensemble. Following the procedure of Vologodskii et al.22 to distinguish the trivial knot from the others using the Alexander polynomial,23 we extracted configurations of the trivial knot from the mixed ensemble and evaluated the ratio ft.k. of the number of the configurations to Ns. Unfortunately, however, this procedure could not exclude all the nontrivial knots, for example, the Kinoshita–Terasaka knot having 11 crossings.24 Nevertheless, we accepted the values of ft.k. so evaluated, considering that the number of residual nontrivial knots could be very small, if any existed.

To make a trivial-knot ensemble, in addition to the mixed ensemble, Ns configurations of the trivial knot were extracted from many mixed ensembles by the above procedure. Although this trivial-knot ensemble inevitably included residual nontrivial knots, we ignored their contribution as in the case of the evaluation of ft.k..

It should be noted here that a possible roundoff error in numerical processes might violate the ring-closure condition,

Therefore, we confirmed that the absolute value of the vector sum on the left-hand side of Equation (10) did not exceed 10−9 for every configuration.

All numerical work was carried out using a personal computer with an Intel Core i7-860 CPU (Intel, Santa Clara, CA, USA). A source program coded in C was compiled by the GNU C compiler version 4.5.0 (Free Software Foundation, Inc., Boston, MA, USA) with real variables of double precision. For the generation of pseudorandom numbers, the subroutine package MT19937 supplied by Matsumoto and Nishimura25 was used instead of the subroutine RAND included in the standard C library.

Mean-square radius of gyration

The mean-square radius of gyration 〈S2〉, that is, the ensemble average of the square radius of gyration S2 as a function of {ln}, could be evaluated from

where the sum is taken over Ns configurations in a given ensemble and Si is the vector distance from the center of mass of the ring to the ith joint given by

with δij the Kronecker delta. In what follows, 〈S2〉mix and 〈S2〉t.k. denote 〈S2〉's calculated from Equations (11) and (12) using the mixed and trivial-knot ensembles, respectively.

Second virial coefficient

To evaluate the intermolecular potential energy resulting from the TI between two unlinked rings, a proper method was needed to determine whether the two rings were linked (nontrivial link) or not (trivial link). Two methods are available. One is based on the Alexander polynomial for links of two components,26 which is more reliable but less feasible and was adopted by Frank-Kamenetskii et al.3, 4 and Deguchi and Tsurusaki.9 The other is based on the Gauss linking number Lk,26 which is more feasible but less reliable and was adopted by Iwata,5, 6 des Cloizeaux7 and Tanaka.8 We note that Lk of some kinds of nontrivial links, including the so-called Whitehead link,26 vanishes as in the case of the trivial link.26 In this study, we adopted the latter method to save computation time.

We considered a pair of rings 1 and 2 and let be the vector position of the contour point on the ipth bond vector (ip=1, 2, ⋯, n) of ring p (p=1, 2), with the contour distance from the ipth joint being xp (0⩽xp<1). The vector is given by

where , in turn, is given by Equation (12) and rc.m.,p is the vector position of the center of mass of ring p. The Gauss linking number Lk for the pair of rings 1 and 2 could then be written in the form,

where is defined by

Integration over x1 and x2 led to27

with F(x, y) given by

where is the angle between and and a1, a2 and a3 in Equations (16) and (17) are given by

with r=rc.m.,2–rc.m.,1 the vector distance between the two centers of mass.

The potential energy U12 between rings 1 and 2 is explicitly defined, using the McMillan–Mayer symbolism,28, 29 as

and the averaged intermolecular potential (potential of mean force) (r) as a function of the distance r=∣r∣ between the centers of mass of rings 1 and 2 is defined by

In Equation (20), 〈⋯〉r indicates the conditional equilibrium average over the configurations of the two rings with r fixed using the single-ring distribution function for each and the bending potential energy given by Equation (2). The second virial coefficient A2 may be written in terms of (r) as follows,

In practice, exp[− (r)/kBT], that is, the conditional average on the right-hand side of Equation (20), was evaluated as a function of r for each ensemble constituted of Ns configurations. First, we randomly chose a pair of configurations (rings 1 and 2) from the Ns configurations and randomized their orientations in the external coordinate system. To obtain exp[− (1, 2)/kBT]=δ0,Lk, we then calculated Lk for the pair at given r from Equation (14) with Equations (16)–(18). Finally, we determined the value of exp[− (r)/kBT] to be Nt.1./Np, where Np is the total number of sample pairs and Nt.1. is the number of the trivial links (Lk=0) included in the Np pairs.

With the values of exp[− (r)/kBT] so obtained for various values of r, the effective volume VE could then be calculated from Equation (1) with Equation (21) by numerical integration applying the trapezoidal rule formula. As in the case of 〈S2〉, VE's obtained for the mixed and trivial-knot ensembles are denoted by VE,mix and VE,t.k., respectively.

Results and discussion

We carried out MC simulations for polymer rings with the values α/kBT given in the first column of Table 1 and with n=10, 20, 50, 100 and 200. Extra MC simulations were carried out for the rings with α/kBT=0 (freely jointed chain) and for n=500 and 1000. To keep the mean number of (real) configurational changes at every Mnom (nominal) steps nearly equal to n, we set Mnom=n for α/kBT=0, Mnom≃2n for α/kBT=0.3 and 1, Mnom≃5n for α/kBT=3 and 10, and Mnom≃10n for α/kBT=30 and 100. Five mixed and five trivial-knot ensembles were then constructed for each case of α/kBT and n, each constituted of 105 (=Ns) configurations except for α/kBT=0 and n=1000. For that case, each ensemble was constituted of 104 configurations. We note that many mixed ensembles were constructed to extract 105 (or 104) configurations of the trivial knot from them. To determine A2 [or exp(− /kBT)], 106 (=Np) pairs of configurations were chosen from each ensemble.

Fraction of the trivial knots

The ratio ft.k. of the number of configurations of the trivial knot included in a given mixed ensemble to the total number Ns of configurations in the ensemble was evaluated. The values of ft.k. and its statistical error are given in the second column of Table 2 as the mean and s.d., respectively, of five independent MC results for given values of α/kBT and n.

Table 2 Values of ft.k., 〈S2〉/n, and VE/n2
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Figure 2 shows plots of ft.k. against the logarithm of the reduced total contour length λL, that is, the total contour length L=n divided by the stiffness parameter λ−1. The open circles represent the MC values for α/kBT=0 (pip up), 0.3 (pip right-up), 1 (pip right), 3 (pip right-down), 10 (pip down), 30 (pip left-down) and 100 (pip left). For comparison, the MC values (dot) obtained by Moore et al.14 are shown for rings with α/kBT=0 in the range of n(=λL) from 15 to 3000. We note that Moore et al.14 adopted the procedure for extracting configurations of the trivial knot proposed by Deguchi and Tsurusaki9 using not only the Alexander polynomial but also the Vassiliev invariants30 of degree 2 and 3. The data points for various values of α/kBT along with those of Moore et al. seem to form a single-composite curve, indicating that ft.k. is a function only of λL. It is interesting to note that ft.k. is almost equal to unity up to λL≃10 and then monotonically decreases to zero with increasing λL, and finally seems to vanish as predicted by Diao et al.31, 32

Figure 2
figure 2

Plots of ft.k. against log λL. The open circles represent the present MC values, with various directions of pips indicating different values of α/kBT: pip up, 0; successive 45° rotations clockwise correspond to 0.3, 1, 3, 10, 30 and 100, respectively. The dots represent the values obtained by Moore et al.14

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The agreement of ft.k. as a function of λL between the present MC results and those of Moore et al. indicates that the present MC method for constructing mixed ensembles and the procedure for extracting configurations of the trivial knot from the ensembles works satisfactorily.

Mean-square radius of gyration

The mean-square radii of gyration 〈S2〉mix and 〈S2〉t.k were calculated from Equation (11) with Equation (12) for all mixed and trivial-knot ensembles, respectively. The values of 〈S2〉mix/n and its statistical error are given in the third column of Table 2 as the mean and s.d., respectively, of five independent MC results for given values of α/kBT and n. The fifth column gives values of 〈S2〉t.k/n and its statistical error evaluated in the same manner.

Figure 3 shows double-logarithmic plots of λ〈S2〉/L (=λ〈S2〉/n) against λL. The open and closed circles represent λ〈S2〉mix/L and λ〈S2〉t.k./L, respectively; the various directions of pips carry the same meaning as those in Figure 2. The solid curve represents the theoretical values of λ〈S2〉mix/L for the KP ring calculated from the following:16, 18, 33

The KP theory of 〈S2〉mix is for the so-called phantom chain, and its value becomes L2/4π2 in the rigid-ring limit and λ−1L/12 in the random-coil limit.29, 34, 35 All MC values of λ〈S2〉mix/L, except those for α/kBT⩽1 and n⩽20, seem to form a single-composite curve and to agree almost completely with the KP theory values. This is an indication that the present MC method for constructing mixed ensembles works satisfactorily. The values for α/kBT⩽1 and n⩽20 are somewhat dispersed because of chain discreteness. It is interesting to see that λ〈S2〉t.k./L deviates upward gradually from λ〈S2〉mix/L as λL is increased from ∼10, where ft.k. begins to decrease from unity. Note that the dimensions of the ring of the trivial knot ought to be larger than those of nontrivial knots.

Figure 3
figure 3

Double-logarithmic plots of λ〈S2〉/L against λL. The open and closed circles represent the values of λ〈S2〉mix/L and λ〈S2〉t.k./L, respectively, and the solid curve represents the KP theory values.16, 18, 33 Various directions of pips attached to the circles carry the same meaning as those in Figure 2.

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It is seen from Figure 3 that the intramolecular topological constraint, which makes the ring preserve the trivial knot, works in the same manner as the intramolecular excluded-volume effect.7, 13 In light of this observation, we examined the behavior of a kind of expansion factor defined by 〈S2〉t.k/〈S2〉mix as a function of λL. Figure 4 shows double-logarithmic plots of 〈S2〉t.k/〈S2〉mix against λL. The closed circles represent the values 〈S2〉t.k/〈S2〉mix calculated from the MC values 〈S2〉mix/n and 〈S2〉t.k./n given in the third and fifth columns of Table 2; the various directions of pips carry the same meaning as those in Figure 2. For comparison, the MC values (dot) obtained by Moore et al.14 for rings with α/kBT=0 in the range of n(=λL) from 300 to 3000 are also shown. The data points for various values of α/kBT along with those of Moore et al. seem to form a single-composite curve, indicating that 〈S2〉t.k./〈S2〉mix is also a function only of λL. As a natural consequence of the behavior of ft.k. shown in Figure 2, 〈S2〉t.k./〈S2〉mix is almost equal to unity up to λL≃10, then monotonically increases with increasing λL. Although 〈S2〉t.k./〈S2〉mix is considered to become proportional to Lv (v≃0.2) in the random-coil limit,13, 14 it is difficult to derive a definite conclusion from only the present data for λL⩽103.

Figure 4
figure 4

Double-logarithmic plots of 〈S2〉t.k./〈S2〉mix against λL. The closed circles represent the MC values, with various directions of pips carrying the same meaning as those in Figure 2. The dots represent the values obtained by Moore et al.14

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The agreement of 〈S2〉t.k./〈S2〉mix as a function of λL between the present MC results and those of Moore et al. reconfirms the validity of the present MC method for constructing mixed ensembles and the procedure for extracting configurations of the trivial knot from the ensembles.

Second virial coefficient

We first examined the behavior of the average intermolecular potential as a function of the reduced distance ρ=r〈S2〉1/2 between the centers of mass of rings 1 and 2; r is the distance between the centers of mass of the two rings. For both the mixed and the trivial-knot ensembles with given values of α/kBT and n, /kBT was evaluated for various values of ρ. We note that the values of 〈S2〉mix and 〈S2〉t.k. given in Table 2 were used in the determination of ρ for the mixed and trivial-knot ensembles, respectively.

Figure 5 shows plots of (ρ)/kBT against ρ for pairs of rings with the indicated values of α/kBT and n. The solid and dashed line segments connect the values for the mixed and trivial-knot ensembles, respectively. For comparison, we also show the values for a pair of 200-sided regular polygons of unit side length (dotted line segments), which corresponds to the values in the limit of α/kBT → ∞ and which were obtained in the same manner as that for the pairs of rings with finite α/kBT.

Figure 5
figure 5

Plots of (ρ)/kBT against ρ for a pair of rings with the indicated values of α/kBT and n. The solid and dashed line segments connect the values for the mixed and trivial-knot ensembles, respectively, and the dotted line segments connect those for a pair of 200-sided regular polygons.

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For rings with (α/kBT, n)=(0, 1000), (0, 500), (0, 200) and (1, 200), whose λL were 1000, 500, 200 and ∼80, respectively, /kBT for a given mixed ensemble was larger than that for the corresponding trivial-knot ensemble; their values themselves and the difference between them become small with decreasing λL. Such behavior of (ρ)/kBT may be considered to reflect the density of the bonds constituting the ring around its center of mass. As for the rings with (α/kBT, n)=(10, 200) and (100, 200), whose λL were ∼10 and ∼1, respectively, /kBT for the two ensembles become almost identical with each other, because any configurations of nontrivial knots can hardly exist in the mixed ensemble for λL≲10, as seen from Figure 2.

It is interesting to find a dip in (ρ)/kBT around ρ=0 for the ring with (α/kBT, n)=(100, 200) and also for the regular polygon. The dip may reflect the fact that the density of the bonds around the center of mass almost vanishes to allow another ring to enter the space. Such behavior of /kBT has also been found by Hirayama et al.36 for a self-avoiding polygon and by Bohn and Heermann37 for a self-avoiding closed path on a simple cubic lattice.

Now we proceed to achieve the purpose of this paper: to examine the behavior of the effective volume VE or the second virial coefficient A2 as a function of λL. We numerically calculated VE,mix and VE,t.k. from Equation (1) with Equation (19) with the values of /kBT for the mixed and trivial-knot ensembles, respectively. The values of VE,mix/n2 and VE,t.k./n2 are given in the fourth and sixth columns, respectively, of Table 2, along with their statistical errors. These values and their associated statistical error for given values of α/kBT and n are the mean and s.d., respectively, of five independent MC results.

Figure 6 shows double-logarithmic plots of λVE/L2 (=λVE/n2∝A2) against λL. The open and closed circles represent λVE,mix/L2 and λVE,t.k./L2, respectively; the various directions of pips carry the same meaning as those in Figure 2. The dotted straight line with unit slope represents the theoretical values for the rigid ring calculated from7

and the curve associated with the data points for λVE,mix/L2 represents the values calculated from an interpolation formula, which is given below.

Figure 6
figure 6

Double-logarithmic plots of λVE/L2 against λL. The open and closed circles represent the values of λVE,mix/L2 and λVE,t.k./L2, respectively, with various directions of pips carrying the same meaning as those in Figure 2. The dotted straight line with unit slope represents the theoretical values for the rigid ring7 and the solid curve represents the values calculated from the interpolation formula for λVE,mix/L2 of the KP ring.

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The data points for λVE,mix/L2 for various values of α/kBT seem to form a single-composite curve, which first increases along the dotted straight line in the range of λL≲0.1, then deviates downward progressively from the line with increasing λL, and finally decreases after passing through a maximum at λL≃5. Each data point for λVE,t.k./L2 almost completely agrees with the corresponding one for λVE,mix/L2 in the range of λL≲10, where ft.k.≃1 and 〈S2〉t.k./〈S2〉mix≃1, and then λVE,t.k./L2 gradually deviates upward from λVE,mix/L2 with increasing λL. Although λVE,mix/L2 and λVE,t.k./L2 are considered to become proportional to L−1/2 and L−0.2, respectively, in the random-coil limit, as mentioned in the introduction, it is difficult to confirm the validity of the exponents on the basis of the present data for λL⩽103.

From the results shown in Figure 6, λVE,mix/L2 (and also λVE,t.k./L2) for the (continuous) KP ring without excluded volume may be considered a function only of λL. For convenience, we constructed an interpolation formula for λVE,mix/L2 of the KP ring based on the MC values of VE,mix/n2 for n⩾100 given in the fourth column of Table 2 and the asymptotic form (23) in the rigid-ring limit and the asymptotic exponent −1/2 in the random-coil limit. We note that the MC values for n=10, 20 and 50 were not used to suppress possible effects of the chain discreteness. The interpolation formula for λL≲103 could be given by

where

with the coefficients Ci given by

The interpolation formula (25) with Equations (26) was accurate within 1%, and the solid curve in Figure 6 represents its values. An asymptotic relation λVE,mix/L2=0.082(λL)−1/2 in the random-coil limit may be obtained from the formula, although we are uncertain of the accuracy of the factor 0.082.

To illustrate the relation of VE to 〈S2〉, we considered the interpenetration function Ψ defined by29

which could be calculated from

with the values of 〈S2〉/n and VE/n2 given in Table 2. Figure 7 shows plots of Ψ against the logarithm of λL. The open and closed circles represent the present MC values for the mixed and trivial-knot ensembles, respectively; again, the various directions of pips carry the same meaning as those in Figure 2. We omitted data points for n=10 and 20 from Figure 7 because they were dispersed because of chain discreteness. The dotted horizontal line indicates the asymptotic value 1/3√π in the rigid-ring limit, which was calculated from Equation (28) with VE given by Equation (23) and with 〈S2〉=L2/4π2. The curve represents values calculated from Equation (28) using the interpolation formula for λVE,mix/L2 given by Equations (24)–(26) and λ〈S2〉mix/L of the KP ring given by Equations (22).

Figure 7
figure 7

Plots of Ψ against log λL. The open and closed circles represent the values for the mixed and trivial-knot ensembles, respectively, with various directions of pips carrying the same meaning as those in Figure 2. The curve represents the values calculated from the interpolation formula and the dotted horizontal line indicates the asymptotic value 1/3√π in the rigid-ring limit. The closed triangles represent the values obtained by Deguchi and Tsurusaki9 for trivial-knot rings.

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The quantity Ψ for the mixed ensemble deviates slightly downward from the dotted horizontal line with increasing λL for λL≲5 and then increases after passing through a minimum. The quantity Ψ for the trivial-knot ensemble is almost identical to that for the mixed ensemble for λL≲10, as a natural consequence of the behavior of λ〈S2〉/L and λVE/L2 shown in Figures 3 and 6; it is then found to gradually deviate downward from the latter with increasing λL. Both quantities may be considered to approach respective asymptotic values in the random-coil limit. Unfortunately, however, we cannot determine the asymptotic values from the MC data for λL≲103 but only suppose that the value for the mixed ensemble is larger than that for the trivial-knot ensemble, also in the random-coil limit. We note that an asymptotic value 0.61 of Ψ in the random-coil limit may be temporarily estimated for the mixed ensemble from the asymptotic relations λVE,mix/L2=0.082(λL)−1/2 and λ〈S2〉mix/L=1/12 mentioned above.

For comparison, Figure 7 shows literature MC data for trivial-knot ensembles (closed triangle) obtained by integrating over ρ of the interpolation formulas for (ρ)/kBT constructed by Deguchi and Tsurusaki9 on the basis of their MC values for rings of the trivial knot composed of n Gaussian springs (n=50, 100, 200 and 500). We note that they adopted more reliable criteria to distinguish knot and link types than ours, that is, the Alexander polynomial and the Vassiliev invariants of degree 2 and 3 used to select trivial-knot rings and the Alexander polynomial for links of two components used to select trivial-link pairs of rings. The present MC values agree with theirs within numerical error, indicating that the criteria adopted in this study work satisfactorily.

Comparison with experiment

Finally, we made a comparison of the present MC results with the experimental ones mentioned in the introduction, where f(λL) given by Equation (25) with Equation (26) as a function of λL is converted into A2 determined as a function of Mw by

where ML is the shift factor16 defined as the molecular weight per unit contour length of the corresponding KP ring.

Figure 8 shows double-logarithmic plots of A2 (in cm3 mol g−2) against Mw. The open circles, squares and triangles represent the experimental values for ring a-PS in cyclohexane at Θ determined from light scattering (LS) measurements by Roovers and Toporowski (at 34.5 °C),10 by Huang et al. (at 35 °C)11 and by Takano et al. (at 34.5 °C),12 respectively. The curve represents MC values of A2 calculated from Equation (29) with Equations (25) and (26), with the KP parameter values λ−1=16.8 Å and ML=35.8 Å−1 determined previously from data for 〈S2〉 for linear a-PS in cyclohexane at 34.5 °C (Θ).38 The experimental values obtained by Roovers and Toporowski and by Takano et al. agree fairly well with each other (within experimental error), as they stand, but the values obtained by Huang et al. are somewhat smaller. In any case, the experimental values are definitely smaller than the MC values, and, therefore, we tried to guess possible causes for the difference.

Figure 8
figure 8

Double-logarithmic plots of A2 (in cm3 mol g–2) against Mw. The open circles, squares and triangles represent the experimental values for ring a-PS in cyclohexane at Θ by Roovers and Toporowski,10 by Huang et al.11 and by Takano et al.,12 respectively; the closed symbols represent the corresponding corrected values. The curve represents the MC values calculated from the interpolation formula.

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We first measured the effect of contamination by linear residues in the ring a-PS samples synthesized by Roovers and Toporowski,10 who remarked that the ingredients of the sample with Mw=5.50 × 105, for which A2 was not determined, were 76 wt% ring polymer with molecular weight Mring=6.05 × 105, 7 wt% residual linear parent polymer with molecular weight Mring, and 17 wt% residual linear polymer with molecular weight Mring/2. Applying the LS theory for a solution of heterogeneous polymers,29, 39, 40 with the proper assumption that intermolecular interaction vanishes between the linear chains and also between the linear and ring chains, Mring and the second virial coefficient A2,ring for a solution only of ring a-PS may be related to observed Mw and A2 by Mring=1.09 Mw and A2,ring=1.45 A2, respectively (see Appendix). If the ingredients of all the samples shown in Figure 8 are assumed to be the same as those mentioned above, the data points represented by the open circles in Figure 8 may be replaced by the closed circles. For the samples synthesized by Takano et al.,12 the weight fraction w of residual linear polymer having the same Mw as that of the corresponding ring polymer was evaluated to be 1.0, 3.4, 1.0 and 2.0 wt% for samples with Mw=1.6 × 104, 4.17 × 104, 1.09 × 105 and 5.73 × 105, respectively. For the solutions of such samples, A2,ring may be related to observed A2 by A2,ring=w−2 A2, and Mw remains unchanged (see Appendix). The closed triangles in Figure 8 represent the corrected values. As for the samples synthesized by Huang et al.,11 no information was given about residual linear polymers. As seen from Figure 8, the corrected data points seem rather dispersed. Some experimental problems remain to be resolved.

Theoretically, on the other hand, it should be remarked that the present MC model takes account of only the TI between a pair of rings but not of the interaction between the segments constituting the real ring polymers (that is, the ordinary excluded-volume effect). The residual contribution of three-segment interactions41 on A2 remains even at Θ, where the effective binary-cluster integral and, therefore, A2 for linear polymer with very large M vanish;29 A2 remains slightly negative (up to order 10−5 cm3 mol g–2) at Θ if M is not very large.38, 42, 43 The residual contribution must, to some extent, decrease the theoretical value of A2. Furthermore, there is no information about knot types of ring polymers included in a given test sample (that is, about whether or not the configurations of the ring polymers in the sample are of all kinds of knots with the Boltzmann weight). If rather complicated knots happen to be preferred, 〈S2〉 and, therefore, A2 may become smaller than the respective values for the mixed ensemble.

Apart from the unresolved discrepancy in the A2 value of order 10−5 cm3 mol g–2, the present MC results in Figure 8 combined with the values of the KP model parameters previously determined may qualitatively explain the behavior of the experimental data. The most important implication of Figures 6, 7, 8 is that the ring a-PS in the range of 1 × 104≲Mw≲6 × 105 is still far from the random-coil limit.

Conclusion

The second virial coefficient A2 of the KP ring without excluded volume, resulting only from the TI, was evaluated based on the present MC results; also, its behavior was examined as a function of the reduced total contour length λL in the range of the crossover from the rigid ring to the random coil. The reduced quantity λVE/L2 proportional to A2 was shown to be a function only of λL, which first increases along the values of the rigid ring (∝L) and then decreases after passing through a maximum at λL≃5 as λL is increased. Although λVE/L2 was considered to become proportional to (λL)−1/2 for a mixed ensemble of configurations of all kinds of knots with the Boltzmann weight and to (λL)−0.2 for the trivial-knot ensemble, in the random-coil limit of λL → ∞, the range of λL⩽103 (where MC simulations were actually carried out) does not yet enter in the limit and cannot directly confirm the asymptotic relations. The present results with the values of the KP model parameters determined previously allow for a qualitative explanation of the behavior of the available literature data for ring a-PS in cyclohexane at Θ, and clearly show that the range of Mw≲6 × 105 (where the experiments were actually carried out) is still far from the random-coil limit. Quantitatively, however, there is a discrepancy in the A2 value of order 10−5 cm3 mol g–2, causes of which seem to lie on both the theoretical and the experimental sides.