Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Original Article
  • Published:

Cluster growth from a dilute system in a percolation process

Polymer Journal volume 52, pages 289–297 (2020)Cite this article

Subjects

Abstract

The gelation process has been explained in the framework of percolation models. Although the percolation models accurately predict the gelation behaviors above the overlap concentration of prepolymers (c*0), they do not predict the gelation behaviors below c*0. We measured the osmotic pressure during the gelation of a series of model polymer gels, namely, tetra-polyethylene glycol (PEG) gels. The osmotic pressure decreased during the gelation reaction and was constant after the sol–gel transition. These results suggest that the clusters grow and fill the system at the sol–gel transition point. As a result, the gels behaved as semidilute systems regardless of the initial polymer concentration. This representation of the sol–gel transition at the overlap condition of the critical clusters corresponds well to the aggregation process predictrion. These results will help better understanding of general percolation problems in the dilute regime.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Last BJ, Thouless DJ. Percolation theory and electrical conductivity. Phys Rev Lett. 1971;27:1719–21.

    CAS  Google Scholar 

  2. Watson BP, Leath PL. Conductivity in the two-dimensional-site percolation problem. Phys Rev B. 1974;9:4893–6.

    Google Scholar 

  3. Southern BW, Young AP, Pfeuty P. Effects of percolation on spin glass order. J Phys C. 1979;12:683–94.

    Google Scholar 

  4. de Arcangelis L, Coniglio A, Peruggi F. Percolation transition in spin glasses. 1991. iopscience.iop.org.

    Google Scholar 

  5. Moore C, Newman MEJ. Epidemics and percolation in small-world networks. Phys Rev E. 2000;61:5678–82.

    CAS  Google Scholar 

  6. Ren J, Zhang L, Siegmund S. How inhomogeneous site percolation works on bethe lattices: theory and application. Sci Rep. 2016;6:22420.

    CAS  PubMed  PubMed Central  Google Scholar 

  7. Antonelli C. Localized knowledge percolation processes and information networks. J Evol Econ. 1996;6:281–95.

    Google Scholar 

  8. Duffie D, Manso G. Information percolation in large markets. Am Econ Rev. 2007. pubs.aeaweb.org.

  9. Flory PJ. molecular size distribution in three dimensional polymers. I. Gelation1. J Am Chem Soc. 1941;63:3083–90.

    CAS  Google Scholar 

  10. Nakagawa H, Suzuki M, Hanabusa K. Physical gelation by amides derived from trans-1,2-diaminocyclohexane and their tetrasiloxane-based gelators. Polym J. 2017;49:439–47.

    CAS  Google Scholar 

  11. Takata K, Kawahara K, Yoshida Y, Kuzuya A, Ohya Y. Analysis of the sol-to-gel transition behavior of temperature-responsive injectable polymer systems by fluorescence resonance energy transfer. Polym J. 2017;49:677–84.

    CAS  Google Scholar 

  12. Izumi A, Shudo Y, Shibayama M. Network structure evolution of a hexamethylenetetramine-cured phenolic resin. Polym J. 2019;51:155–60.

    CAS  Google Scholar 

  13. Zhang P, Bin Y, Zhang R, Matsuo M. Average gap distance between adjacent conductive fillers in polyimide matrix calculated using impedance extrapolated to zero frequency in terms of a thermal fluctuation-induced tunneling effect. Polym J. 2017;49:839–50.

    CAS  Google Scholar 

  14. Wang J, Zhou Z, Zhang W, Garoni TM, Deng Y. Bond and site percolation in three dimensions. Phys Rev E. 2013;87:052107.

    Google Scholar 

  15. Xu X, Wang J, Lv J-P, Deng Y. Simultaneous analysis of three-dimensional percolation models. Front Phys. 2014;9:113–9.

    Google Scholar 

  16. Stauffer D, Coniglio A, Adam M. Gelation and critical phenomena. Adv Polym Sci. 1982;44:103–58.

    CAS  Google Scholar 

  17. Martin JE, Wilcoxon J, Adolf D. Critical exponents for the sol-gel transition. Phys Rev A. 1987;36:1803–10.

    CAS  Google Scholar 

  18. Katashima T, Sakurai H, Chung U-I, Sakai T. Dilution effect on the cluster growth near the gelation threshold. Nihon Reoroji Gakkaishi. 2019;47:61–66.

    CAS  Google Scholar 

  19. Takigawa T, Urayama K, Masuda T. Critical-behavior of the intrinsic-viscosity of poly(vinylalcohol) solutions near the gelation point. J Chem Phys. 1990;93:7310–3.

    CAS  Google Scholar 

  20. Zheng H, Zhang Q, Jiang K, Zhang H, Wang J. Critical behavior of viscosity for alginate solutions near the gelation threshold induced by cupric ions. J Chem Phys. 1996;105:7746–52.

    CAS  Google Scholar 

  21. Takigawa T, Urayama K, Masuda T. Critical-behavior of the specific viscosity of poly(vinyl alcohol) solutions near the gelation threshold. Chem Phys Lett. 1990;174:259–62.

    CAS  Google Scholar 

  22. Martin JE, Adolf D. The sol-gel transition in chemical gels. Annu Rev Phys Chem. 1991;42:311–39.

    CAS  Google Scholar 

  23. Nishi K, Noguchi H, Sakai T, Shibayama M. Rubber elasticity for percolation network consisting of Gaussian chains. J Chem Phys. 2015;143:184905.

    PubMed  Google Scholar 

  24. Nishi K, Chijiishi M, Katsumoto Y, Nakao T, Fujii K, Chung U, et al. Rubber elasticity for incomplete polymer networks. J Chem Phys. 2012;137:224903.

    PubMed  Google Scholar 

  25. Winter HH. In: Borsali R, Pecora R, editors. Structure and dynamics of polymer and colloidal systems. Netherlands: Springer; 2002. p. 439–70. https://doi.org/10.1007/978-94-010-0442-8_14.

    Google Scholar 

  26. Sakai T, Katashima T, Matsushita T, Chung U-I. Sol-gel transition behavior near critical concentration and connectivity. Polym J. 2016;48:629–34.

    CAS  Google Scholar 

  27. Shibayama M, Norisuye T. Gel formation analyses by dynamic light scattering. Bull Chem Soc Jpn. 2002;75:641–59.

    CAS  Google Scholar 

  28. Takeda M, Norisuye T, Shibayama M. Critical dynamics of cross-linked polymer chains near the gelation threshold. Macromolecules. 2000;33:2909–15.

    CAS  Google Scholar 

  29. Jokinen M, Györvary E, Rosenholm JB. Viscoelastic characterization of three different sol–gel derived silica gels. Colloids Surf A. 1998;141:205–16.

    CAS  Google Scholar 

  30. Winter H, Mours M. In: Neutron spin echo spectroscopy viscoelasticity rheology. Berlin, Heidelberg: Springer; 1997. 134, p. 165–234.

  31. Miura T, Okumoto H, Ichijo H. Concentration dependence of the sol-gel transition point and the network formation of polymer gels. Phys Rev E. 1996;54:6596–602.

    CAS  Google Scholar 

  32. 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. 1994;101:798–804.

    CAS  Google Scholar 

  33. Izuka A, Winter HH, Hashimoto T. Molecular weight dependence of viscoelasticity of polycaprolactone critical gels. Macromolecules. 1992;25:2422–8.

    CAS  Google Scholar 

  34. Adam M. Growth-process of polymers near the gelation threshold. Makromol Chem-M Symp. 1991;45:1–9.

    CAS  Google Scholar 

  35. Fang L, Brown W, Konak C. Dynamic light scattering study of the sol-gel transition. Macromolecules. 1991;24:6839–42.

    CAS  Google Scholar 

  36. Martin JE, Wilcoxon J, Odinek J. Decay of density fluctuations in gels. Phys Rev A. 1991;43:858–72.

    CAS  PubMed  Google Scholar 

  37. Martin JE, Adolf D, Wilcoxon JP. Viscoelasticity near the sol-gel transition. Phys Rev A. 1989;39:1325–32.

    CAS  Google Scholar 

  38. Martin JE, Wilcoxon JP. Critical dynamics of the sol-gel transition. Phys Rev Lett. 1988;61:373–6.

    CAS  PubMed  Google Scholar 

  39. Vicsek T. Fractal growth phenomena. World Scientific; 1992.

  40. Sorensen CM, Chakrabarti A. The sol to gel transition in irreversible particulate systems. Soft Matter. 2011;7:2284–96.

    CAS  Google Scholar 

  41. Kolb M, Botet R, Jullien R. Scaling of kinetically growing clusters. Phys Rev Lett. 1983;51:1123–6.

    Google Scholar 

  42. Fry D, Sintes T, Chakrabarti A, Sorensen CM. Enhanced kinetics and free-volume universality in dense aggregating systems. Phys Rev Lett. 2002;89:148301.

    CAS  PubMed  Google Scholar 

  43. Sakai T, Matsunaga T, Yamamoto Y, Ito C. Design and fabrication of a high-strength hydrogel with ideally homogeneous network structure from tetrahedron-like macromonomers. ACS Publ. 2008;41:5379–84.

    CAS  Google Scholar 

  44. Sakai T. Experimental verification of homogeneity in polymer gels. Polym J. 2014;46:517–23.

    CAS  Google Scholar 

  45. Sakai T. Gelation mechanism and mechanical properties of Tetra-PEG gel. React Funct Polym. 2013;73:898–903.

    CAS  Google Scholar 

  46. Yoshikawa Y, Sakumichi N, Chung U-I, Sakai T. Connectivity dependence of gelation and elasticity in AB-type polymerization: an experimental comparison of the dynamic process and stoichiometrically imbalanced mixing. Soft Matter. 2019;15:5017–25.

    CAS  PubMed  Google Scholar 

  47. Horkay Ferenc, Ichiji Tasaki A, Basser PJ. Osmotic swelling of polyacrylate hydrogels in physiological salt solutions. Biomacromolecules. 2000;1:84–90.

    CAS  PubMed  Google Scholar 

  48. Vink H. Precision measurements of osmotic pressure in concentrated polymer solutions. Eur Polym J. 1971;7:1411–9.

    CAS  Google Scholar 

  49. Xu X, Wang J, Lv J-P, Deng Y. Simultaneous analysis of three-dimensional percolation models. Front Phys. 2013;9:113–9.

    Google Scholar 

  50. Cloizeaux Des J, Noda I. Osmotic pressure of long polymers in good solvents at moderate concentrations: a comparison between experiments and theory. Macromolecules. 1982;15:1505–7.

    Google Scholar 

  51. Li X, Hirosawa K, Sakai T, Gilbert EP, Shibayama M. SANS Study on Critical Polymer Clusters of Tetra-Functional Polymers. Macromolecules. 2017;50:3655–61.

    CAS  Google Scholar 

  52. Oh C, Sorensen CM. The effect of overlap between monomers on the determination of fractal cluster morphology. J Colloid Inter Sci. 1997;193:17–25.

    CAS  Google Scholar 

  53. de Gennes PG. Scaling concepts in polymer physics. Cornell University Press; 1979.

  54. Urayama K, Kawamura T, Kohjiya S. Elastic modulus and equilibrium swelling of networks crosslinked by end-linking oligodimethylsiloxane at solution state. J Chem Phys. 1996;105:4833–40.

    CAS  Google Scholar 

  55. Sakai T, Kurakazu M, Akagi Y, Shibayama M, Chung U-I. Effect of swelling and deswelling on the elasticity of polymer networks in the dilute to semi-dilute region. Soft Matter. 2012;8:2730–6.

    CAS  Google Scholar 

  56. Katashima T, Chung U-I, Sakai T. Effect of swelling and deswelling on mechanical properties of polymer gels. Macromol Symp. 2015;358:128–39.

    CAS  Google Scholar 

  57. Kuhn W. Beziehungen zwischen Molekülgröße, statistischer Molekülgestalt und elastischen Eigenschaften hochpolymerer Stoffe. Kolloid-Z. 1936;76:258–71.

    CAS  Google Scholar 

  58. Flory PJ. Principles of polymer chemistry. Cornell University Press; 1953.

  59. Akagi Y, Matsunaga T, Shibayama M, Chung U, Sakai T. Evaluation of topological defects in tetra-PEG gels. Macromolecules. 2010;43:488–93.

    CAS  Google Scholar 

  60. Akagi Y, Katashima T, Katsumoto Y, Fujii K, Matsunaga T, Chung U, et al. Examination of the theories of rubber elasticity using an ideal polymer network. Macromolecules. 2011;44:5817–21.

    CAS  Google Scholar 

  61. Akagi Y, Gong JP, Chung U-I, Sakai T. Transition between phantom and affine network model observed in polymer gels with controlled network structure. Macromolecules. 2013;46:1035–40.

    CAS  Google Scholar 

  62. Obukhov SP, Rubinstein M, Colby RH. Network modulus and superelasticity. Macromolecules. 1994;27:3191–8.

    CAS  Google Scholar 

  63. 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 Gakkaishi. 2014;42:97–102.

    CAS  Google Scholar 

  64. Katashima T, Asai M, Urayama K, Chung U-I, Sakai T. Mechanical properties of tetra-PEG gels with supercoiled network structure. J Chem Phys. 2014;140:134906.

    Google Scholar 

Download references

Acknowledgements

This work was supported by the Japan Society for the Promotion of Science through the Grants-in-Aid for the Graduate Program for Leaders in Life Innovation, the International Core Research Center for Nanobio, Core-to-Core Program A. Advanced Research Networks, the Grants-in-Aid for Scientific Research (B) Grant Number 18H02027 to TS, and Scientific Research (S) Grant Number 16746899 to UC and by the Japan Science and Technology Agency through Center of Innovation.

Author information

Authors and Affiliations

  1. Department of Bioengineering, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan

    Ikuo Fujinaga, Takashi Yasuda, Ung-il Chung, Takuya Katashima & Takamasa Sakai

  2. Department of Mechanical Engineering, Keio University, 3-14-1, Hiyoshi, Kohoku-ku, Yokohama, Kanagawa, 223-8522, Japan

    Makoto Asai

  3. Department of Chemical Engineering, Columbia University, 116th St. and Broadway, New York, NY, 10027, USA

    Makoto Asai

Authors
  1. Ikuo Fujinaga
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Takashi Yasuda
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Makoto Asai
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ung-il Chung
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Takuya Katashima
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Takamasa Sakai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Makoto Asai or Takamasa Sakai.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fujinaga, I., Yasuda, T., Asai, M. et al. Cluster growth from a dilute system in a percolation process. Polym J 52, 289–297 (2020). https://doi.org/10.1038/s41428-019-0279-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41428-019-0279-z

This article is cited by

Search

Advanced search

Quick links