Abstract
The theory of gel point in real polymer solutions is examined with the empirical correlation between the reciprocal of the percolation threshold and the coordination number given by the percolation theory. Applying a larger value of the relative frequency of cyclization, an excellent agreement is obtained between the present theory and the percolation result. This suggest that while the ring distribution on lattices is similar to that in real systems, ring production is more frequent in the lattice model than in real systems. To confirm this conjecture, we derive the ring distribution function of the lattice model as a limiting case of d→∞, and show that the solution is in fact identical to the asymptotic formula of C→∞ in real systems except for the coefficient C, which has a maximum at d = 5, in support of the above conjecture. To examine the validity of the asymptotic solution for the lattice model, we apply it to the critical point problem of the percolation theory, showing that the solution works well in high dimensions greater than six.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
P. J. Flory, J. Am. Chem. Soc. 63:3091 (1941). (b) P. J. Flory, Principles of Polymer Chemistry (Cornell University Press, Ithaca and London, 1953).
S. R. Broadbent and J. M. Hammersley, Proc. Camb. Philos. Soc. 53:629 (1957). (b) H. L. Frisch and J. M. Hammersley, J. Soc. Indust. Appl. Math. 11:894 (1963).
P. G. de Gennes, J. Phys. (Paris) 37:L1 (1976).
D. J. Stauffer, J. Chem. Soc. Faraday Trans. II72:1354 (1976); Adv. Polymer Sci. 44:103 (1982).
P. G. de Gennes, Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca and London, 1979).
D. Stauffer, Introduction to Percolation Theory (Taylor & Francis, London and Philadelphia, 1985).
V. A. Vyssotsky, S. B. Gordon, H. L. Frisch, and J. M. Hammersley, Phys. Rev. 123:1566 (1961). (b) M. F. Sykes and J. W. Essam, Phys. Rev. 133:A310 (1964). (c) R. Zallen, The Physics of Amorphous Solid (Wiley, New York, 1983); Phys. Rev. B 16:1426 (1977). (d) V. K. S. Shante and S. Kirkpatrick, Adv. Phys. 20:325 (1971).
S. Galam and A. Mauger, Physica A 205:502 (1944); Phys. Rev. E 53:2177 (1996); Eur. Phys. J. B 1:255 (1998). (b) S. C. van der Marck, Phys. Rev. E 55:1228 (1997). (c) S. Galam and A. Mauger, Phys. Rev. E 55:1230 (1997).
K. Suematsu and T. Okamoto, J. Stat. Phys. 66:661 (1992). (b) K. Suematsu, T. Okamoto, M. Kohno, and Y. Kawazoe, J. Chem. Soc., Faraday Trans. 89:4181 (1993). (c) K. Suematsu and Y. Kawazoe, J. Chem. Soc. Faraday Trans. 92:2417 (1996).
J. L. Spouge, J. Stat. Phys. 43:143 (1986).
C. Domb, Adv. Chem. Phys. 15:229 (1969); C. Domb and M. S. Green & C. Domb and J. L. Lebowitz, Phase Transitions and Critical Phenomena (Academic Press, New York, 1972).
J. L. Martin, M. F. Sykes, and F. T. Hioe, J. Chem. Phys. 46:3478 (1967).
N. A. Dotson, C. W. Macosko, and M. Tirrell, Synthesis, Characterization and Theory of Polymeric Networks and Gels, A. Aharoni, ed. (Plenum Press, New York, 1992). (b) F. R. Jones, L. E. Scales, and J. A. Semlyen, Polymer 15:738 (1974); D. R. Cooper and J. A. Semlyen, Polymer 14:185 (1973). (c) J. Somvarsky and K. Dusek, Polym. Bull. 33:369, 377 (1994). (d) K. Dusek and M. Ilavsky, J. Polym. Sci.: Symposium 53:57, 75 (1975). (e) K. Dusek, Development in Polymerization, Vol. 3, R. N. Haward, ed. (Elsevier Applied Science Publishers Ltd., London, 1982), p. 143.
W. Burchard, Advances in Polymer Sciences 1 (1983).
J. Hoshen, P. Klymko, and R. Kopelman, J. Stat. Phys. 21:583 (1979).
P. Agrawal, S. Redner, P. J. Reynolds, and H. E. Stanley, J. Phys. A: Math. Gen. 12:2073 (1979).
P. D. Gujrati, J. Chem. Phys. 98:1613 (1993); J. Chem. Phys. 107:22 (1997).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Suematsu, K., Kohno, M. Estimation of Critical Point in Branching Reactions: Further Examination of Gel Point Formula. Journal of Statistical Physics 93, 293–305 (1998). https://doi.org/10.1023/B:JOSS.0000026735.03546.a8
Issue Date:
DOI: https://doi.org/10.1023/B:JOSS.0000026735.03546.a8