Letter Cite This: ACS Macro Lett. 2019, 8, 700−704
pubs.acs.org/macroletters
Direct Determination of Cross-Link Density and Its Correlation with the Elastic Modulus of a Gel with Slidable Cross-Links Kazuaki Kato,*,†,‡ Yuta Ikeda,† and Kohzo Ito*,† †
Department of Advanced Materials Science, Graduate School of Frontier Sciences, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8561, Japan ‡ Data-driven Polymer Design Group, Research and Services Division of Materials Data and Integrated System, National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan
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S Supporting Information *
ABSTRACT: The universal relationship between the elastic modulus and the cross-link density of a conventional rubber/ gel has been demonstrated experimentally to be inapplicable to gels with slidable cross-links. Herein, we describe the synthesis of slide-ring (SR) gel networks devoid of intramolecular cross-links by the cross-coupling of two differently functionalized polyrotaxanes. The cross-link density was determined from the characteristic UV absorption attributed to the asymmetric cross-linked moiety. The cross-link density was shown to correlate considerably more weakly with the Young’s modulus than conventional gels and rubbers that follow a universal proportional dependence. In addition, even at a similar cross-link density, the modulus appeared to be lower due to a lower density of cyclic components along the threading chain, i.e., the “coverage”, though the data were limited in the narrow cross-link density range. These results might suggest a considerably lower contribution from the conformational entropy of chains associated with sliding through the cross-links and the counteracting entropy attributed to ring arrangement, though effects of the different persistence length due to the coverage difference could affect the modulus.
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ineffective cross-links can depend on cCL. Moreover, recent experimental and theoretical studies have shown that the entropy associated with the arrangement of confined rings possibly contributes to the stress.19−22 Interactions between the ring and conformational entropies of polymer chains potentially impact the universal relationship between the elastic modulus and ν. Unless elucidating the relationship and creating more ideal sliding networks, it is unlikely to advance deeper understanding of the sliding effects on various mechanical uniqueness of slide-ring gels. In this study, we designed and synthesized a polyrotaxane network devoid of intramolecular cross-links and whose intermolecular cross-links are spectroscopically detectable, as shown in Figure 1b. Based on the relationship between the experimentally observed crosslink density and the elastic modulus, we discuss the impact of chain sliding on rubber elasticity. SR gels whose network is much closer to ideal compared to conventional ones were realized by the Cu(I)-catalyzed selective Huisgen cycloadditions23 of azide- and alkynylfunctionalized polyrotaxanes. Because the same types of ring are never bound to each other, one type of polyrotaxane is
he elastic modulus of a rubber or gel is clearly correlated to its cross-link density due to the definite entropic origin of its elasticity. As classical rubber elasticity theories predict, the origin of this correlation is the conformational entropy of polymer chains; hence the elastic modulus is proportional to the number density, ν, of network strands between crosslinks.1−3 Often referred to as the “cross-link density”, ν can be determined from the elastic modulus of the material.4 On the other hand, this universal correlation may not necessarily be applicable to a new class of gel and elastomer whose cross-links can slide along polymer chains.5−9 The most intensively studied of these materials is the slide-ring (SR) gel prepared by cross-linking a polyrotaxane,6,10 which consists of topologically constrained cyclic molecules and a threading polymer chain. Many unique properties, particularly mechanical properties, have been reported to date for these materials, highlighting the effectiveness of stress relaxation due to chain sliding through the cross-links.11−17 Although the effects of added cross-linker concentration, cCL, have usually been discussed, they remain qualitative because of the unknown value of ν. In addition, conventional SR gels unavoidably contain undesired intramolecular cross-links in the polyrotaxane, as shown in Figure 1a. Intramolecular cross-links form loops that do not contribute to stress or tubes that increase the persistence length and the stress;18 hence, the ratio between effective and © XXXX American Chemical Society
Received: April 1, 2019 Accepted: May 16, 2019
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DOI: 10.1021/acsmacrolett.9b00238 ACS Macro Lett. 2019, 8, 700−704
Letter
ACS Macro Letters
of time to obtain essentially the same gel. The gels in the UV cell were directly subjected to UV−vis spectroscopy, while those in the Teflon molds were removed and subjected to uniaxial tensile testing to obtain their Young’s moduli, E. It is notable that these Young’s moduli did not exceed 30 kPa, even after longer curing times (see SI 6), whereas much harder SR gels have been reported.18,19,22 It has been suggested that the tube structure (see Figure 1a) produced by the intramolecular cross-links increases the persistence length of the network strands in order to increase the modulus.18 It is also known that some of the SR gel characteristics disappear at the higher moduli associated with higher cross-linker concentrations.15,22,24,25 Therefore, the uncharacteristic properties of hard SR gels are attributable to nonideal network structures caused by intramolecular cross-links. The cross-link density was determined from the characteristic UV absorption of the cross-link structure that contains 1,2,3-triazole moieties; an absorption peak at λ = 381 nm was observed only after the cross-linking reaction had progressed. To correlate UV absorbance with the cross-link concentration, a model cross-linked compound was synthesized from azideand alkynyl-functionalized CDs, instead of the polyrotaxanes (see details in SI 7). Because the threading polymer is absent, these CDs remain dissolved in solvents even after the completion of the cross-linking reaction; hence, the concentration can be obtained unambiguously by 1H NMR spectroscopy. Figure 2a shows the 1H NMR spectrum of the reaction mixture after the cross-linking of these two CDs, with each peak assigned. Although the model cross-linked product was not isolated and the reaction mixture contained unreacted azide- and alkynyl-CDs, the molar ratio between the reacted and unreacted CDs can be determined from the ratios of integrated NMR peaks, such as B′/B or C′/C. Since the initial concentrations of the CDs are known, the concentration of the model cross-link, ν′, can be calculated. The UV absorption spectrum was also acquired. As was observed for the SR gels, the characteristic UV absorption peak at λ = 381 nm was only evident after cross-linking, as shown in Figure 2b. In order to subtract the absorptions of moieties other than the formed cross-links, which are mainly due to the azide-functionalized CD, the absorbance at λ = 381 nm was recorded immediately after the reagents were mixed, but prior to heating. The net peak absorbance obtained in this manner, normalized against the optical path length, A381, was plotted against the NMR-determined values of ν′ to produce the calibration curve displayed in Figure 2c. The cross-link densities in SR gels, ν, were determined from their UV absorptions using the correlation curve constructed for the model cross-link. The thickness of each gel was controlled to avoid excessive absorption; hence, absorbance was normalized against gel thickness (see details in the SI 8). The normalized spectra of seven SR gels are shown in Figure 2d. Similar peaks at around 381 nm were observed at different extents of cross-linking. The net peak absorbance, A381, was converted into ν′ using the calibration curve; ν′ is equal to ν because the chemical structures of the cross-links in the gels are identical to those of the model cross-link. As a result, ν was found to lie in the (0.97−7.6) × 1024 m−3 range, which corresponds to states in which 5−36% of the CDs are bound as cross-links (see details in the SI 8). At the experimentally obtained lowest value of ν, a single polyrotaxane has only four cross-links on average since only ∼90 CDs are threaded on a single polyrotaxane on average (see details in the SI 5). Since
Figure 1. (a) Depicting the conventional cross-linking of polyrotaxanes to form a slidable network with unavoidable loops and tubes that result from intramolecular cross-linking and (b) a more ideal network formed by the click reaction of two differently functionalized polyrotaxanes, in which the concentration of effective cross-links can be determined from their characteristic UV absorption. HMDI: hexamethylene diisocyanate.
always connected to the other type. By using a binary mixture of these polyrotaxanes, we could obtain SR gels whose crosslinks connect only the different types of polyrotaxanes. In addition, the asymmetric structure formed between different types of ring has a characteristic UV absorption. Based on the absorbance, the cross-link concentration can be determined. We should note that loop structures formed by intermolecular cross-link should exist, which has UV absorption but has no contribution to the stress. This causes overestimation of the cross-link concentration. However, the number of the intermolecular loop should be much less than that of the intramolecular one whose formation is significantly accelerated by the neighboring loop formation. Each polyrotaxane derivative was synthesized commonly from a polyrotaxane composed of polyethylene glycol (PEG, Mn = 32 000, Mw = 35 000) and α-cyclodextrin (CD) with a coverage, ϕ, of 25%, which represents the density of rings along the threading chain (see details in Supporting Information (SI) 3). The azide- and alkynyl-functionalized polyrotaxanes have degrees of modification of 5 and 6%, respectively, which correspond to about one functional group per CD. A series of gels were prepared by employing different curing times (see details in SI 4). Each pregel solution containing both polyrotaxane derivatives and CuI in DMSO was split and transferred separately to an UV cell and a Teflon mold. The separated solutions were simultaneously cured in the same oven for the same amount 701
DOI: 10.1021/acsmacrolett.9b00238 ACS Macro Lett. 2019, 8, 700−704
Letter
ACS Macro Letters
Figure 3. Relationship between cross-link density and the Young’s modulus of the SR gels without intramolecular cross-links (red). The solid line is the dependence predicted by classical rubber theory. The blue data points were obtained from gels whose coverages were reduced to one-fifth.
(3 + η)2νkBT, although the modulus decreases according to the degree of sliding at the cross-links, η.26−28 Although these studies considered the effects of cross-link or network-strandlength fluctuations by sliding, until recently they barely discussed how the degree of sliding is determined. As mentioned above, on the basis of recent experimental studies, the degree of sliding was hypothesized to be determined by the counteracting entropies of the cyclic components. Because sliding increases the conformational entropy of chains but simultaneously decreases the ring-arrangement entropy, the modulus is determined by these two competing entropies. A theory that took these counteracting entropies into account predicted a decrease in the exponent with increasing ν, based on the assumption that the rings lose arrangement entropy once they become cross-links.19 This assumption is equivalent to decreasing of effective ring density as the cross-linking reactions progress and suggests that the modulus decreases with decreasing coverage. To experimentally elucidate the contribution of ring entropy, we synthesized another series of SR gels from a polyrotaxane with a coverage of 5% (ϕ = 0.05), which is one-fifth that of the previously prepared gels (see SI 9 for characterization details). The obtained relationship between E and ν is shown in blue in Figure 3. Despite many trials under various cross-linking conditions, the gels were formed with a narrow range of ν values because of their considerably reduced CD densities. Thus, we believe that ν dependence should not be discussed based on the limited data in the ν narrow range. However, the difference in the modulus at similar ν, ∼1024 m−3, might be worth comparing; the modulus appears to decrease with decreasing ϕ, which is attributable to a change in the balance of the two competing entropies. Stress relaxation by chain sliding can be facilitated by a reduction in the counteracting ring entropy. Such effects, facilitated by a reduction in ϕ, have also been suggested under large deformation;22 however, we need to remember that a decrease in coverage simultaneously reduces the persistence length. Threading with CDs is known to dramatically change the polymer conformation,29 with the persistence length of PEG observed to increase by a factor of about three at ϕ = 0.26.30 Hence, until we can control the
Figure 2. (a) 1H NMR spectrum (400 MHz, DMSO-d6) of the model cross-link with each peak assigned and (b) its UV−vis spectrum. (c) Calibration curve of the model cross-link density, ν′, determined by NMR spectroscopy as a function of net peak absorbance at λmax = 381 nm per millimeter of optical path length, A381. (d) UV−vis spectra of SR gels without intramolecular cross-links normalized against optical path length.
more than two cross-links in each polymer on average are necessary for gel formation, this state is close to the minimum cross-link density for gel formation. At the highest value of ν that we were able to produce, two-thirds of the CDs most likely no longer encounter different types of CDs because of the constraints of the already formed network. In this way, the cross-link densities determined in this study provide plausible values of ν, and the prepared seven gels appear to cover almost the entire range of realizable cross-link densities. Finally, the relationship between the Young’s modulus, E, and the cross-link density, ν, was examined, as shown in red in Figure 3. The dependence is clearly weaker than that predicted by classical rubber elasticity theories and is close to E ∝ ν0.5. Even if stress relaxation by chain sliding decreases the modulus, as qualitatively suggested by several experiments, the universal proportional dependence, namely E ∝ ν, of a conventional polymer network would hold, as long as elasticity originates from the conformational entropy associated with the polymer chains. Indeed, theoretical and simulation studies have predicted the proportional dependence to follow: E = 3/ 702
DOI: 10.1021/acsmacrolett.9b00238 ACS Macro Lett. 2019, 8, 700−704
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ACS Macro Letters Notes
persistence length, for example, through the use of different threading polymers, we are unlikely to be able to experimentally extract the effect of ring entropy alone. In addition, the data range at ϕ = 0.05 should be dramatically expanded in order to clarify the effects of coverage on the relationship between E and ν. To establish elasticity theories for SR gels requires both the E−ν and E−ϕ dependences to be systematically determined. Much longer threading polymers and ways of increasing the cross-linking reaction rates are required to produce gels with low ϕ over a wide range of ν. These technical difficulties are expected to be overcome by simulation studies. Herein, we experimentally revealed the relationship between the elastic modulus and the cross-link density of gels with slidable cross-links for the first time. The cross-link density was quantitatively determined from the characteristic UV absorptions of the cross-links in newly designed gels devoid of intramolecular cross-links, though the density could be overestimated due to loops formed by the intermolecular cross-links. Surprisingly, the dependence of the modulus on the cross-link density was found to be considerably weaker than the universal proportional dependence that operates in conventional gels and rubbers. The observed power dependence, namely E ∝ ν0.5, suggests a low contribution from chain entropy, which is probably due to chain sliding counteracting the entropy of ring arrangement. The reduced coverage appeared to lower the counteracting ring entropy as suggested by a reduction in modulus, although the data were limited in narrow ν range and the effect of changes in the persistence length needs to be considered. To date, the universal proportional relationship (E ∝ ν) supported by classical rubber elasticity theories has been implicitly accepted as also being applicable to slidable polymer networks. However, this study suggests that risks potentially exist when quantitatively discussing the effects of cross-link density and when directly comparing different coverages. In the meantime, the suggested contribution of ring entropy indicates that various material properties can possibly be controlled not only through ν but also using the unique structural parameter ϕ. Simulation studies that take into account the contributions of ring components promise to reveal the complex relationships between E, ν, and ϕ and establish a molecular theory of elasticity for slidable polymer networks.
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The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by a JSPS KAKENHI Grant (16H06050) and the ImPACT Program of the Council for Science, Technology, and Innovation (Cabinet Office), Government of Japan.
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsmacrolett.9b00238.
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REFERENCES
(1) Flory, P. J. Principle of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953. (2) Treloar, L. R. G. The Physics of Rubber Elasticity; Oxford University Press: Oxford, 2005. (3) Mark, J. E.; Erman, B. Rubberlike Elasticity: A Molecular Primer; Cambridge University Press: Cambridge, 2007. (4) Flory, P. J.; Rehner, J., Jr. Statistical Mechanics of Cross-Linked Polymer Networks II. Swelling. J. Chem. Phys. 1943, 11, 521−526. (5) de Gennes, P.-G. Sliding Gels. Physica A 1999, 271, 231−237. (6) Okumura, Y.; Ito, K. The Polyrotaxane Gel: A Topological Gel by Figure-of-Eight Cross-links. Adv. Mater. 2001, 13, 485−487. (7) Kubo, M.; Hibino, T.; Tamura, M.; Uno, T.; Itoh, T. Synthesis and Copolymerization of Cyclic Macromonomer Based on Cyclic Polystyrene: Gel Formation via Chain Threading. Macromolecules 2002, 35, 5816−5820. (8) Tan, S.; Blencowe, A.; Ladewig, K.; Qiao, G. G. A Novel OnePot Approach Towards Dynamically Cross-Linked Hydrogels. Soft Matter 2013, 9, 5239−5250. (9) Murakami, T.; Schmidt, B. V. K. J.; Brown, H. R.; Hawker, C. J. One-Pot “Click” Fabrication of Slide-Ring Gels. Macromolecules 2015, 48, 7774−7781. (10) Ito, K.; Mayumi, K.; Kato, K. Polyrotaxane and Slide-Ring Materials; The Royal Society of Chemistry: Cambridge, 2016. (11) Fleury, G.; Schlatter, G.; Brochon, C.; Hadziioannou, G. From High Molecular Weight Precursor Polyrotaxanes to Supramolecular Sliding Networks. The ‘Sliding Gels’. Polymer 2005, 46, 8494−8501. (12) Ito, K. Novel Cross-Linking Concept of Polymer Network: Synthesis, Structure, and Properties of Slide-Ring Gels with Freely Movable Junctions. Polym. J. 2007, 39, 489−499. (13) Murata, N.; Konda, A.; Urayama, K.; Takigawa, T.; Kidowaki, M.; Ito, K. Anomaly in Stretching-Induced Swelling of Slide-Ring Gels with Movable Cross-Links. Macromolecules 2009, 42, 8485−8491. (14) Kato, K.; Ito, K. Dynamic Transition Between Rubber and Sliding States Attributed to Slidable Cross-Links. Soft Matter 2011, 7, 8737−8740. (15) Bitoh, Y.; Akuzawa, N.; Urayama, K.; Takigawa, T.; Kidowaki, M.; Ito, K. Peculiar Nonlinear Elasticity of Polyrotaxane Gels with Movable Cross-Links Revealed by Multiaxial Stretching. Macromolecules 2011, 44, 8661−8667. (16) Katsuno, C.; Konda, A.; Urayama, K.; Takigawa, T.; Kidowaki, M.; Ito, K. Pressure-Responsive Polymer Membranes of Slide-Ring Gels with Movable Cross-Link. Adv. Mater. 2013, 25, 4636−4640. (17) Bin Imran, A. B.; Esaki, K.; Gotoh, H.; Seki, T.; Ito, K.; Sakai, Y.; Takeoka, Y. Extremely Stretchable Thermosensitive Hydrogels by Introducing Slide-Ring Polyrotaxane Cross-Linkers and Ionic Groups into the Polymer Network. Nat. Commun. 2014, 5, 5124. (18) Fleury, G.; Schlatter, G.; Brochon, C.; Travelet, C.; Lapp, A.; Lindner, P.; Hadziioannou, G. Topological Polymer Networks with Sliding Cross-Link Points: The “Sliding Gels”. Relationship between Their Molecular Structure and the Viscoelastic as Well as the Swelling Properties. Macromolecules 2007, 40, 535−543. (19) Mayumi, K.; Tezuka, M.; Bando, A.; Ito, K. Mechanics of SlideRing Gels: Novel Entropic Elasticity of a Topological Network Formed by Ring and String. Soft Matter 2012, 8, 8179−8183. (20) Pinson, M. B.; Sevick, E. M.; Williams, D. R. M. Mobile Rings on a Polyrotaxane Lead to a Yield Force. Macromolecules 2013, 46, 4191−4197.
Synthesis and characterization of functionalized polyrotaxanes and CDs, uniaxial tensile-testing results, determining cross-link densities from UV−vis absorption measurements, and the synthesis and characterization of polyrotaxanes with lower coverages (PDF)
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Kazuaki Kato: 0000-0002-9997-8599 703
DOI: 10.1021/acsmacrolett.9b00238 ACS Macro Lett. 2019, 8, 700−704
Letter
ACS Macro Letters (21) Kato, K.; Yasuda, T.; Ito, K. Viscoelastic Properties of SlideRing Gels Reflecting Sliding Dynamics of Partial Chains and Entropy of Ring Components. Macromolecules 2013, 46, 310−316. (22) Kato, K.; Okabe, Y.; Okazumi, Y.; Ito, K. A Significant Impact of Host−Guest Stoichiometry on the Extensibility of Polyrotaxane Gels. Chem. Commun. 2015, 51, 16180−16183. (23) Rostovtsev, V. V.; Green, L. G.; Fokin, V. V.; Sharpless, K. B. A Stepwise Huisgen Cycloaddition Process: Copper(I)-Catalyzed Regioselective “Ligation” of Azides and Termina Alkynes. Angew. Chem., Int. Ed. 2002, 41, 2596−2599. (24) Karino, T.; Okumura, Y.; Zhao, C.; Kataoka, T.; Ito, K.; Shibayama, M. SANS Studies on Deformation Mechanism of SlideRing Gel. Macromolecules 2005, 38, 6161−6167. (25) Konda, A.; Mayumi, K.; Urayama, K.; Takigawa, T.; Ito, K. Influence of Structural Characteristics on Stretching-Driven Swelling of Polyrotaxane Gels with Movable Cross Links. Macromolecules 2012, 45, 6733−6740. (26) Ball, R. C.; Doi, M.; Edwards, S. F.; Warner, M. Elasticity of Entangled Networks. Polymer 1981, 22, 1010−1018. (27) Edwards, S. F.; Vilgis, T. The Effect of Entanglements in Rubber Elasticity. Polymer 1986, 27, 483−492. (28) Koga, T.; Tanaka, F. Elastic Properties of Polymer Networks with Sliding Junctions. Eur. Phys. J. E: Soft Matter Biol. Phys. 2005, 17, 225−229. (29) Fleury, G.; Brochon, C.; Schlatter, G.; Bonnet, G.; Lapp, A.; Hadziioannou, G. Synthesis and Characterization of High Molecular Weight Polyrotaxanes: Towards the Control over a Wide Range of Threaded a−Cyclodextrins. Soft Matter 2005, 1, 378−385. (30) Mayumi, K.; Osaka, N.; Endo, H.; Yokoyama, H.; Sakai, Y.; Shibayama, M.; Ito, K. Concentration-Induced Conformational Change in Linear Polymer Threaded into Cyclic Molecules. Macromolecules 2008, 41, 6480−6485.
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DOI: 10.1021/acsmacrolett.9b00238 ACS Macro Lett. 2019, 8, 700−704
