Main

The local conformation of ring or cyclic chains is substantially the same as that of their linear analogs if the chain length is sufficiently longer than the Kuhn segment length λ−1, which is the chain stiffness parameter of the Kratky–Porod worm-like chain.1 The discrepancy between cyclic and linear polymers may become significant upon shortening or stiffening the main chain because the difference in the curvature distribution becomes prominent owing to the topological constraint. Little is known, however, about the chain stiffness (or length) -dependent local conformational change, except for the case of the super-helical structure of cyclic DNA,2 although abundant studies have examined the physical properties of rather flexible cyclic polymers including polystyrene,3, 4, 5, 6, 7, 8, 9 polydimethylsiloxane10, 11 and polysaccharides.12, 13, 14, 15 This is likely because cyclic polymers are synthesized by end-linking or ring expansion methods. A variety of novel cyclic polymers have recently been reported not only for fundamental synthetic studies, but also for building blocks of higher order structures consisting of amphiphilic block copolymers.16, 17, 18, 19 Considering that dense macrocyclic comb polymers that should have relatively stiff main chains form tube-like complexes,20 the chain stiffness effect on the dimensional properties and intermolecular interactions should be an important topic.

Semi-flexible and rigid cyclic polymers were recently obtained by means of ‘stiffening’ the main chain of rather flexible cyclic chains. Cyclic amylose tris(phenylcarbamate) (cATPC)21 and cyclic amylose tris(n-butylcarbamate) (cATBC)22 prepared from enzymatically synthesized cyclic amylose23 behave as semi-flexible and/or rigid cyclic macromolecules in solution. The molar mass dependence of the mean square radius of gyration 〈S2〉 for cATPC and cATBC in various solvents is fairly explained by current theories for the cyclic worm-like chain24 with the parameters, λ−1 and the helix pitch (or helix rise) h per unit chain length, determined for the corresponding linear chains.21, 22 The experimental values for cATPC in some ketones and esters are, however, appreciably smaller than the theoretical values calculated with the parameters so determined.25 This suggests that both the local helical structure and the chain stiffness depend on the molar mass of the cyclic polymer chains. Although the particle scattering function P(q) was determined as a function of the magnitude q of the scattering vector for the two cyclic polymers in different solvents,21, 22 appropriate theories or simulation data were not yet available for semi-rigid cyclic chains. Recently, Ida and colleagues26, 27 calculated P(q) for worm-like rings with a variety of reduced chain lengths, that is, the ratio of the contour length L to λ−1 or the Kuhn segment number NK, allowing us to compare them with experimental P(q) data. We thus reanalyzed the previously published P(q) data to discuss the difference between the molecular structures of linear and cyclic amylose derivatives in solution.

The previously reported P(q) data21, 22 for cATPC in 1,4-dioxane (DIOX), 2-ethoxyethanol (2EE), methyl acetate (MEA), ethyl acetate (EA), and 4-methyl-2-pentanone (MIBK) and cATBC in methanol (MeOH), 2-propanol (2PrOH), and tetrahydrofuran (THF) were determined at the BL40B2 beamline in SPring-8 and the BL-10C beamline in KEK-PF. They were obtained for five cATPC and nine cATBC samples ranging in weight-average molar mass Mw between 1.25 × 104 g mol−1 and 1.49 × 105 g mol−1 for cATPC and 1.60 × 104 g mol−1 and 1.11 × 105 g mol−1 for cATBC, corresponding to numbers of saccharide units between 24 and 290 for the former polymer, and between 35 and 240 for the latter. The dispersity index Ð defined as the ratio of Mw to the number-average molar mass was estimated to be 1.05–1.23.

Monte Carlo simulations were performed using the method reported by Ida et al.27, 28 to calculate the particle scattering function P0(q) of worm-like rings without chain thickness as a function of NK (≡λL). A discrete worm-like chain model originally proposed by Frank-Kamenetskii et al.29 was used with the bond number being 200. For each NK, 105 configurations were generated to obtain an ensemble average with the appropriate monitoring steps. According to our previous study,21, 22 the chain thickness significantly affects P(q) at a relatively high q range. The relationship can be considered by means of the touched bead model30, 31 as follows:

Since actual cATPC and cATBC samples have finite molar mass distributions, the z-average particle scattering function Pz(q) with a log-normal distribution was calculated numerically to compare the experimental data.

The calculated Pz(q) or P(q) values with the best fit parameters λ−1 and h for cATBC in three solvents are plotted in Figure 1. The calculated values of Pz(q) with appropriate Ð (1.05 (blue) and 1.20 (red)) well explain the behavior of the experimental data, while those of P(q) for the monodisperse ring (green) slightly deviate downward in the middle-q range from the experimental values for the lower-Mw samples. Good agreement was observed between the simulation and experimental data for cATPC in six solvent systems (see Supplementary Figure S1 in the Supplementary Information). We note that the values of λ−1 may be determined without ambiguity only for the samples with high Mw, for which the theoretical values for the rigid ring calculated with the appropriate h value (black dot-dashed curves) underestimate P(q) at approximately q=0.2 nm−1. We then adopted the λ−1 value determined for the samples with high Mw to calculate Pz(q) or P(q) for the samples with low Mw. We also note that Pz(q) and P(q) become insensitive to changes in λ−1 with decreasing Mw. The other two parameters, h and db, were uniquely determined for more samples, except for the two lowest Mw samples, for both cATBC and cATPC. The latter parameter db was consistent with those for the corresponding linear chain within ±11% if we estimate db for the linear polymer from the literature chain diameter d values32, 33, 34, 35 for the cylinder model by means of the known relationship db=1.118 d.36 The negligible difference between the P(q) data for the discrete worm-like chain and those for the continuous rigid cyclic chain in the high q region supports that the current simulation results are substantially the same as those for the continuous chain.

Figure 1
figure 1

Comparison between the experimental P(q) data22 (unfilled circles) and the theoretical values for the indicated cATBC samples in tetrahydrofuran (THF) at 25 °C (a), in 2-propanol (2PrOH) at 35 °C (b) and in methanol (MeOH) at 25 °C (c). Dashed, solid and dot-dashed curves are the simulation values for the worm-like ring model with the dispersity index Ð=1.00, 1.05 and 1.20, respectively. The double-dot-dashed black curve represents the theoretical values for the rigid ring (Ð=1.2). The ordinate values are shifted by A for clarity. A full color version of this figure is available at Polymer Journal online.

The determined λ−1 and h values are plotted against Mw in Figure 2. The former parameter λ−1 for cATBC in THF is much smaller than that for the linear ATBC. This difference is clearly recognized from the fact that the calculated values of Pz(q) and P(q) for cATBC in THF with the parameters determined for linear ATBC (h=0.28 nm and λ−1=75 nm) are appreciably smaller than the experimental values, as shown in Supplementary Figure S2 in the Supplementary Information. On the other hand, the theoretical values of 〈S2〉 for worm-like rings calculated from the Shimada–Yamakawa equation24 with the parameters determined currently for cATBC (h=0.33 nm and λ−1=20 nm) and those calculated using the previously estimated values (h=0.28 nm and λ−1=75 nm)22 assuming λ−1 for linear ATBC18 are indistinguishable in the investigated Mw range, as shown in Supplementary Figure S3 in the Supplementary Information. This indicates that the worm-like chain parameters for this system cannot be determined only from the reported 〈S2〉 data. When we consider that the worm-like chain parameters of infinitely long cyclic chains should be the same as those for the corresponding linear chains, as mentioned in the ‘Introduction,’ it is expected that a molar mass-dependent chain stiffness could be observed in the higher molar mass range. In the case of h, its values for most cases of cyclic chains are fairly identical to those of the corresponding linear chains, except for cATPC in MIBK, cATPC in EA, cATBC in THF and cATBC in 2PrOH. The former two polymer-solvent systems are rather specific because hydrogen bonding solvent molecules extend and stiffen the linear ATPC, while in the latter cases, the rigid helical main chain of ATBC in THF (or 2PrOH) is extended by the cyclization similarly to the case with bended helical springs. This local structural change also reduces the chain stiffness of the main chain of cATBC. It is, however, noted that this local structural change is insensitive to the solution infrared absorption, which reflects the intramolecular hydrogen bonds.22

Figure 2
figure 2

Weight-average molar mass Mw dependence of the worm-like chain parameters for cATBC and cATPC in solution. In (a, b), the unfilled circles, filled circles and triangles indicate the parameters for cATBC in THF at 25 °C, in 2PrOH at 33 °C and in MeOH at 25 °C, respectively. The solid lines are h or λ−1 for the corresponding linear chain. In (c, d), the unfilled circles, filled circles, unfilled triangles, filled triangles, unfilled inverted triangles and filled inverted triangles represent the parameters for cATPC in DIOX at 25 °C, in 2EE at 25 °C, in MEA at 25 °C, in EA at 33 °C, and in MIBK at 25 and 58 °C, respectively. The solid and dashed magenta lines in (d) are the λ−1 values for linear ATPC in MIBK at 25 and 58 °C, respectively. The ordinate values are shifted by A for clarity. A full color version of this figure is available at Polymer Journal online.

The Kuhn segment numbers NK,ring for cATBC and cATPC are plotted against that for the corresponding linear chain NK,linear in Figure 3. Note that NK,linear was calculated from the worm-like chain parameters reported for the corresponding linear chain with the same Mw. This number is well-known as a universal parameter that mainly reflects the chain conformation of unperturbed linear polymer chains in solution. Interestingly, in the range of NK,linear>1.5, NK,ring is fairly close to NK,linear, in contrast to the case of cATBC in THF in the range of NK,linear<1, for which λ−1 is clearly smaller than that of the linear ATBC (Figure 2b). The threshold value 1.0–1.5 of NK is close to the value at which the (angle-independent) ring closure probability for the (linear) worm-like chain significantly increases.37 It is, however, noted that NK,ring for ATPC in EA and MIBK is incidentally close to NK,linear, although h and λ−1 for cATPC are smaller than those for linear ATPC (Figure 2c and d). As mentioned above and in previous papers,17, 38 the origin of the chain stiffness of these systems is different from that of other systems. We may thus conclude that some strain effects for cyclic chains arising from the relaxation of the internal-rotation-angle distribution in their main chain become appreciable for shorter (stiffer) cyclic chains with small NK. In contrast, λ−1 for the cyclic chain is substantially the same as that for the corresponding linear chain when NK is larger, and the local helical structure of the cyclic chain is almost identical to that of the corresponding linear chain. Although such a threshold value of NK may depend on the polymer-solvent system and/or the origin of the chain stiffness, for example, differences in h and λ−1 between cyclic and linear chains were reported for ATPC in EA and MIBK (Figure 2c and d) in the range of NK from 0.42 to 6.6,25 such strain effects seem to make cyclic chains more flexible than linear chains as far as amylose carbamates are concerned. Unfortunately, however, it is still not clear whether rigid cyclic chains should have a smaller λ−1 compared to the corresponding linear chains. It is desirable to test other rigid cyclic chain systems to clarify this issue.

Figure 3
figure 3

Double logarithmic plots of NK,ring vs NK,linear for (a) cATBC and (b) cATPC in various solvents. The symbols are the same as those used in Figure 2. The solid lines indicate NK,ring=NK,linear. A full color version of this figure is available at Polymer Journal online.

We reanalyzed our recently published scattering function data for cATBC in three and cATPC in five solvent systems in terms of the cyclic worm-like chain to determine the contour length per residue h and the Kuhn segment length λ−1 (stiffness parameter). While there was no significant molar mass dependence of the worm-like chain parameters, a clearly small λ−1 was estimated for cATBC in THF, even though the difference cannot be distinguishable from the radius of gyration, showing that the analysis of the particle scattering function of the cyclic chain is important to determining the chain shape of the rigid cyclic chains in solution.