Abstract
A polymer cross-linking system undergoes a phase transition from liquid to solid at a critical point, which is called the sol-gel transition. The sol-gel transition is often understood in the context of the lattice-based percolation model. Two parameters govern the sol-gel transition including the connectivity and polymer concentration. In this study, we independently tuned these parameters and experimentally accessed the sol-gel transition point as a function of connectivity and concentration using a model gel system (tetra-polyethylene glycol gel). The connectivity required to percolate the system continuously increased as the polymer concentration decreased, which is completely different from that predicted by the site-bond percolation model. The viscoelastic behavior at the critical points indicates that the fractal dimension of the percolation clusters deviated from the prediction of the lattice-based percolation model as the polymer concentration decreased. These results indicate that the lattice assumption cannot be applied for a gelling system prepared far below the overlapping concentration.
Similar content being viewed by others
Log in or create a free account to read this content
Gain free access to this article, as well as selected content from this journal and more on nature.com
or
References
Adam, M. Growth-process of polymers near the gelation threshold. Makromol. Chem. Macromol. Symp. 45, 1–9 (1991).
Fang, L., Brown, W. & Konak, C. Dynamic light scattering study of the sol-gel transition. Macromolecules 24, 6839–6842 (1991).
Lang, P. & Burchard, W. Dynamic light scattering at the gel point. Macromolecules 24, 814–815 (1991).
Martin, J. E., Wilcoxon, J. & Adolf, D. Critical exponents for the sol-gel transition. Phys. Rev. A 36, 1803–1810 (1987).
Martin, J. E., Wilcoxon, J. & Odinek, J. Decay of density fluctuations in gels. Phys. Rev. A 43, 858–872 (1991).
Martin, J. E. & Wilcoxon, J. P. Critical dynamics of the sol-gel transition. Phys. Rev. Lett 61, 373–376 (1988).
Ren, S. Z., Shi, W. F., Zhang, W. B. & Sorensen, C. M. Anomalous diffusion in aqueous solutions of gelatin. Phys. Rev. A 45, 2416–2422 (1992).
Adam, M., Delsanti, M. & Durand, D. Mechanical measurements in the reaction bath during the polycondensation reaction, near the gelation threshold. Macromolecules 18, 2285–2290 (1985).
Adam, M., Delsanti, M., Durand, D., Hild, G. & Munch, J. P. Mechanical-properties near gelation threshold, comparison with classical and 3d percolation theories. Pure Appl. Chem. 53, 1489–1494 (1981).
Takigawa, T., Kasihara, H., Urayama, K. & Masuda, T. Critical-behavior of modulus of poly(vinylalcohol) gels near the gelation point. J. Phys. Soc. Jpn. 59, 2598–2599 (1990).
Takigawa, T., Urayama, K. & Masuda, T. Critical-behavior of the intrinsic-viscosity of poly(vinylalcohol) solutions near the gelation point. J. Chem. Phys. 93, 7310–7313 (1990).
Takigawa, T., Urayama, K. & Masuda, T. Critical-behavior of the specific viscosity of poly(vinyl alcohol) solutions near the gelation threshold. Chem. Phys. Lett 174, 259–262 (1990).
Chambon, F. & Winter, H. H. Linear viscoelasticity at the gel point of a crosslinking PDMS with imbalanced stoichiometry. J. Rheol. 31, 683–697 (1987).
Martin, J. E., Adolf, D. & Wilcoxon, J. P. Viscoelasticity near the sol-gel transition. Phys. Rev. A 39, 1325–1332 (1989).
Winter, H. & Mours, M. Advances in Polymer Science Vol. 134, Ch. 3, 165–234 (Springer, Berlin Heidelberg, 1997).
Winter, H. H. in Progress in Colloid & Polymer Science Vol. 75, Ch. 13, 104–110 (Springer, Berlin Heidelberg, 1987).
Winter, H. H. & Chambon, F. Analysis of linear viscoelasticity of a crosslinking polymer at the gel point. J. Rheol. 30, 367–382 (1986).
Stauffer, D. & Aharony, A. Introduction To Percolation Theory, 2nd edn (Taylor & Francis, London, UK, 1994).
Stauffer, D., Coniglio, A. & Adam, M. Gelation and critical phenomena. Adv. Polym. Sci. 44, 103–158 (1982).
Coniglio, A., Stanley, H. E. & Klein, W. Site-bond correlated-percolation problem: a statistical mechanical model of polymer gelation. Phys. Rev. Lett. 42, 518–522 (1979).
Sakai, T. Gelation mechanism and mechanical properties of Tetra-PEG gel. React. Funct. Polym. 73, 898–903 (2013).
Sakai, T. Experimental verification of homogeneity in polymer gels. Polym. J. 46, 517–523 (2014).
Sakai, T., Akagi, Y., Matsunaga, T., Kurakazu, M., Chung, U. I. & Shibayama, M. Highly elastic and deformable hydrogel formed from tetra-arm polymers. Macromol. Rapid Commun. 31, 1954–1959 (2010).
Kondo, S., Chung, U.-I. & Sakai, T. Effect of prepolymer architecture on the network structure formed by AB-type crosslink-coupling. Polym. J. 46, 14–20 (2013).
Sakai, T., Matsunaga, T., Yamamoto, Y., Ito, C., Yoshida, R., Suzuki, S., Sasaki, N., Shibayama, M. & Chung, U. I. Design and fabrication of a high-strength hydrogel with ideally homogeneous network structure from tetrahedron-like macromonomers. Macromolecules 41, 5379–5384 (2008).
Nishi, K., Fujii, K., Chijiishi, M., Katsumoto, Y., Chung, U. I., Sakai, T. & Shibayama, M. Kinetic study for AB-type coupling reaction of tetra-arm polymers. Macromolecules 45, 1031–1036 (2012).
Nishi, K., Fujii, K., Katsumoto, Y., Sakai, T. & Shibayama, M. Kinetic aspect on gelation mechanism of tetra-PEG hydrogel. Macromolecules 47, 3274–3281 (2014).
Akagi, Y., Katashima, T., Katsumoto, Y., Fujii, K., Matsunaga, T., Chung, U. I., Shibayama, M. & Sakai, T. Examination of the theories of rubber elasticity using an ideal polymer network. Macromolecules 44, 5817–5821 (2011).
Akagi, Y., Gong, J. P., Chung, U.-I. & Sakai, T. Transition between phantom and affine network model observed in polymer gels with controlled network structure. Macromolecules 46, 1035–1040 (2013).
Matsunaga, T., Sakai, T., Akagi, Y., Chung, U. & Shibayama, M. Structure characterization of tetra-PEG gel by small-angle neutron scattering. Macromolecules 42, 1344–1351 (2009).
Ferry, J. D. Viscoelastic Properties of Polymers, 3rd edn (Wiley, New York, Chichester, Brisbane, Tronto and Singapore, 1980).
Sakai, T., Kurakazu, M., Akagi, Y., Shibayama, M. & Chung, U. Effect of swelling and deswelling on the elasticity of polymer networks in the dilute to semi-dilute region. Soft Matter 8, 2730–2736 (2012).
Takeda, M., Norisuye, T. & Shibayama, M. Critical dynamics of cross-linked polymer chains near the gelation threshold. Macromolecules 33, 2909–2915 (2000).
Miura, T., Okumoto, H. & Ichijo, H. Concentration dependence of the sol-gel transition point and the network formation of polymer gels. Phys. Rev. E 54, 6596–6602 (1996).
de Gennes, P. G. Scaling Concepts In Polymer Physics, 1st edn (Cornell University Press, Ithaca and London, 1979).
Yanuka, M. & Englman, R. Bond-site percolation: empirical representation of critical probabilities. J. Phys. A Gen. Phys. 23, L339 (1990).
Nakanishi, H. & Reynolds, P. J. Site-bond percolation by position-space renormalization group. Phys. Lett. A 71, 252–254 (1979).
Xu, X., Wang, J., Lv, J.-P. & Deng, Y. Simultaneous analysis of three-dimensional percolation models. Front. Phys. 9, 113–119 (2014).
Chambon, F., Petrovic, Z. S., MacKnight, W. J. & Winter, H. H. Rheology of model polyurethanes at the gel point. Macromolecules 19, 2146–2149 (1986).
Takahashi, M., Yokoyama, K., Masuda, T. & Takigawa, T. Dynamic viscoelasticity and critical exponents in sol-gel transition of an end-linking polymer. J. Chem. Phys. 101, 798–804 (1994).
Hess, W., Vilgis, T. A. & Winter, H. H. Dynamical critical behavior during chemical gelation and vulcanization. Macromolecules 21, 2536–2542 (1988).
Muthukumar, M. Dynamics of polymeric fractals. J. Chem. Phys. 83, 3161–3168 (1985).
Muthukumar, M. Screening effect on viscoelasticity near the gel point. Macromolecules 22, 4656–4658 (1989).
Nicolai, T., Randrianantoandro, H., Prochazka, F. & Durand, D. Viscoelastic relaxation of polyurethane at different stages of the gel formation. 2. sol−gel transition dynamics. Macromolecules 30, 5897–5904 (1997).
Kolb, M., Botet, R. & Jullien, R. Scaling of kinetically growing clusters. Phys. Rev. Lett. 51, 1123–1126 (1983).
Meakin, P. Diffusion-controlled cluster formation in 2—6-dimensional space. Phys. Rev. A 27, 1495–1507 (1983).
Witten, T. A. & Sander, L. M. Diffusion-limited aggregation, a kinetic critical phenomenon. Phys. Rev. Lett. 47, 1400–1403 (1981).
Eden, M. in Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability Contributions to Biology and Problems of Medicine, vol. 4, 223–239 (University of California Press, Berkeley, CA, USA, 1961).
Akagi, Y., Matsunaga, T., Shibayama, M., Chung, U. & Sakai, T. Evaluation of topological defects in tetra-PEG gels. Macromolecules 43, 488–493 (2010).
Katashima, T., Kurakazu, M., Akagi, Y., Chung, U.-I. & Sakai, T. Relationships between mechanical properties of polymer gels and polymer volume fractions at preparation and at interested state. Nihon Reoroji Gakk. 42, 97–102 (2014).
Acknowledgements
This work was supported by the Japan Society for the Promotion of Science (JSPS) through the Grants-in-Aid for the Graduate Program for Leaders in Life Innovation (GPLLI); the International Core Research Center for Nanobio; Core-to-Core Program A, Advanced Research Networks; and the Grants-in-Aid for JSPS Fellows Grant Number 12J07980 to TK, for Young Scientists (A) Grant Number 23700555 to TS, and Scientific Research (A) Grant Number 24240069 to UC. This work was also supported by the Japan Science and Technology Agency (JST) through the S-innovation program (to UC) and PREST (to TS).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Competing interests
The authors declare no conflict of interest.
Rights and permissions
About this article
Cite this article
Sakai, T., Katashima, T., Matsushita, T. et al. Sol-gel transition behavior near critical concentration and connectivity. Polym J 48, 629–634 (2016). https://doi.org/10.1038/pj.2015.124
Received:
Revised:
Accepted:
Published:
Issue date:
DOI: https://doi.org/10.1038/pj.2015.124
This article is cited by
-
Molecular crystallization directed by polymer size and overlap under dilute and crowded macromolecular conditions
Polymer Journal (2021)
-
Cluster growth from a dilute system in a percolation process
Polymer Journal (2020)
-
Fast-forming hydrogel with ultralow polymeric content as an artificial vitreous body
Nature Biomedical Engineering (2017)
-
PJ ZEON Award for outstanding papers in Polymer Journal 2016
Polymer Journal (2017)
