Abstract
For the critical exponents near the sol-gel phase transition, classical theories like those of Flory and Stockmayer predict one set of exponents, whereas scaling theories based on lattice percolation predict different exponents. The two groups of theories differ in their treatment of intramolecular loops, space dimensionality and excluded volume effects. In this article, the differences and similarities between the results of the competing theories are reviewed. For example, a gel fraction like (p-pc)β vanishes for conversion factors p very close to the gel point pc, the weight average molecular weight diverges as (pc-p)−γ for p very slightly below pc, and the radius of macromolecules at the gel point p=pc varies as the ϱ-th power of the number of monomers in that macromolecule. Classical theories predict β=γ=1 and ϱ=1/4 whereas the percolation theory gives β ≃ 0.45, γ ≃ 1.74 and ϱ ≃ 0.40. We also generalize the percolation concept to include interaction effects and concentration fluctuations; in this case the sol-gel phase transition may be connected with a phase separation.
Some experimental results are reviewed to check whether the percolation theory agrees with reality; no clear answer has been found so far, due to experimental difficulties. For instance, for the viscosity a power law (pc − p)−0.8, which agrees with one of the percolation ideas has been established in several experiments; the shear modulus of the gel vanishes roughly as (p − pc)3 in some experiments, which agrees better with the classical theory.
Preview
Unable to display preview. Download preview PDF.
I. References
Gels, Gelling Processes (conf. proc.): Faraday Disc. 57, 1 (1974)
de Gennes, P. G.: J. Phys. (Paris) 37, L1 (1976)
Stauffer, D.: J. Chem. Soc. Faraday Trans. II 72, 1354 (1976)
Flory, P. J.: J. Am. Chem. Soc. 63, 3083, 3091, 3096 (1941); Principles of Polymer Chemistry, Ithaca (NY), Cornell University Press 1953
Stockmayer, W. H.: J. Chem. Phys. 11, 45 (1943); 12, 125 (1944)
de Gennes, P. G.: Scaling Properties in Polymer Physics, Ithaca (NY), Cornell University Press 1979
Stauffer, D.: Pure Appl. Chem. 53, 1479 (1981); see also Physica 106 A, 177 (1981)
Essam, J. W., Gaunt, D. S., Guttmann, A. J.: J. Phys. A 11, 1983 (1978)
Gordon, M., Ross-Murphy, S. B.: Pure Appl. Chem. 43, 1 (1975); Dušek, K., Prins, W.: Adv. Polym. Sci. 6, 1 (1969); Dušek, K.: Makromol. Chem. Suppl. 2, 35 (1979)
Nakanishi, H., Stanley, H. E.: J. Phys. A 14, 693 (1981); Phys. Rev. B 22, 2466 (1980) with more literature on dimensionality effects
Gordon, M., Torkington, J. A.: Pure Appl. Chem. 53, 1461 (1981); Ferroelectrics 30, 237 (1980); Irvine, P., Gordon, M.: Macromol. 13, 761 (1980)
Peniche-Covas, C. et al.; Ref. 1, page 165
Aharony, A.: Phys. Rev. B 22, 400 (1980)
Brauner, U.: Makromol. Chem. 180, 251 (1979)
Stanley, H. E.: Introduction to Phase Transitions and Critical Phenomena, Oxford, Clarendon Press 1971; Domb, C., Green, M. S. (eds.): Phase Transitions and Critical Phenomena, vols. 1 to 6, London, Academic since 1972; Ahlers, G. in: The Physics of Liquid and Solid Helium, Ketterson, J. B., Bennemann, K. H., (eds.), Vol. I, Chap. II, New York, Wiley 1976
Schulthess, G. K. V., Benedek, G. B., de Blois, R. W.: Macromolecules 13, 939 (1980)
Family, F., Coniglio, A.: J. Phys. A 13, L403 (1980); Harris, A. B., Lubensky, T. C.: Phys. Rev. B 24, 2656 (1981); Family, F., Reynolds, P. J.: Z. Phys. B, 45, 123 (1981)
Parisi, G., Sourlas, N.: Phys. Rev. Lett. 46, 871 (1981)
Cotton, J. P.: J. Phys. (Paris) 41 L231 (1980); Kremer, K., Baumgärtner, A., Binder, K.: Z. Phys. B 40, 331 (1980) with further literature on self-avoiding walks
Isaacson, J., Lubensky, T. C.: J. Phys. (Paris) 41, L469 (1980); de Gennes, P. G., Comptes Rend. Acad. Sci. Paris 291, 17 (1980)
Schmidt, W., Burchard, W.: Macromolecules 14, 370 (1981)
Stauffer, D.: Phys. Reps. 54, 1 (1979) and in: Int. Conf. Disordered Systems and Localization, di Castro, C. (ed.), Springer Lecture Notes in Physics 149, 9 (1981)
Essam, J. W.: Repts. Progr. Phys. 43, 833 (1980)
Fisher, M. E.: Physics 3, 255 (1967)
Essam, J. W., Gwilym, K. M.: J. Phys. C 4, L228 (1971)
Mandelbrot, B. B.: Fractals: Form, Chance and Dimension. San Francisco, Freeman 1977
Levelt Sengers, A., Hocken, R. J., Sengers, J. V.: Phys. Today 30, 42 (Dec. 1977); Levelt Sengers, J. M. H.: Physica 73, 73 (1974); 98 A, 363 (1979)
Aharony, A., Hohenberg, P. C.: Phys. Rev. B 13, 3081 (1976)
Baxter, R. J.: Phys. Rev. Lett. 26, 832 (1971)
Scholl, F., Binder, K.: Z. Phys. B 39, 239 (1980)
Haan, S. W., Zwanzig, R.: J. Phys. A 10, 1547 (1977); Vicsek, T., Kertesz, J.: J. Phys. A 14, L31 (1981); Gawlinsky, E. T., Stanley, H. E.: J. Phys. A 14, L291 (1981)
Domany, E., Kinzel, W.: Phys. Rev. Lett. 47, 5 (1981) and Ref. 104; Wu, F. Y., Stanley, H. E.: Phys. Rev. Lett. 49, 775 (1982); Klein, W.: J. Phys. A 15, in press
Stoll, E. P., Domb, C.: J. Phys. A 12, 1843 (1979) and Ref. 83
Lubensky, T. C., Isaacson, J.: Phys. Rev. Lett. 41, 829 (1978); 42, 410 (E) (1979); Phys. Rev. 20, 2130 (1979); J. Phys. (Paris) 42, 175, 1591 (1981)
Marro, J. J.: Phys. Lett. 59 A, 180 (1976)
Shante, V. K. S., Kirkpatrick, S.: Adv. Phys. 20, 325 (1971)
Fisher, M. E., Essam, J. W.: J. Math. Phys. 2, 609 (1961)
Heinrichs, J.: Phys. Rev. B 19, 3788 (1979)
Zimm, B. H., Stockmayer, W. H.: J. Chem. Phys. 17, 1301 (1949); Dobson, G. R., Gordon, M.: J. Chem. Phys. 41, 2389 (1964); de Gennes, P. G.: Biopolymers 6, 715 (1968), appendix
Dobson, G. R., Gordon, M.: J. Chem. Phys. 43, 705 (1965)
Gordon, M., Ross-Murphy, S. B.: J. Phys. A 12, L155 (1979)
Burchard, W.: Macromolecules 5, 604 (1972), Eq. (16b)
Coniglio, A., Stanley, H. E., Klein, W.: Phys. Rev. Lett. 42, 518 (1979) and Phys. Rev. B, in press
Burchard, W. et al.: Pure Appl. Chem. 53, 1519 (1981)
Domb, C., Stoll, E., Schneider, T.: Contemp. Phys. 20, 577 (1980)
Celik, T., Karsch, F., Satz, H.: Phys. Lett. 97 B, 128 (1980)
Chatzis, I. & Dullien, F. A. L.: Can. Petroleum Techn. J. 1977, 1 (Jan–March 1977)
Eschbach, P. D., Stauffer, D., Herrmann, H. J.: Phys. Rev. B 23, 422 (1981); Kopelman, R.: priv. commun. (May 1981); Moriarty, K. J. M., Stauffer, D.: in preparation
Blöte, H. W. J., Nightingale, M. P., Derrida, B.: J. Phys. A 14, L45 (1981); for Q=4 see also Herrmann, H. J.: Z. Phys. B 43, 55 (1981)
Kunz, H., Souillard, B.: J. Statist. Phys. 19, 77 (1978)
Lubensky, T. C., McKane, A. J.: J. Phys. A 14, L157 (1981)
den Nijs, M. P. M.: J. Phys. A 12, 1857 (1979); Nienhuis, B., Riedel, E. K., Schick, M.: J. Phys. A 13, L 189 (1980), Pearson, R. P.: Phys. Rev. B 22, 2579 (1980), Nienhuis, B.: J. Phys. A 15, 199 (1982)
Sykes, M. F., Essam, J. W.: J. Math. Phys. 5, 1117 (1964); Phys. Rev. 133 A, 310 (1964)
Kesten, H.: Comm. Math. Phys. 74, 41 (1980)
Le Guillou, J. C., Zinn-Justin: J. Phys. Rev. B 21, 3976 (1980)
Hoshen, J. et al.: J. Phys. A 12, 1285 (1979)
Stauffer, D., Coniglio, A.: Z. Phys. B 38, 267 (1980); but see Aharony, A., Stephen, M. J.: J. Phys. C 14, 1665 (1981); Aharony, A., Stauffer, D.: Z. Phys. B, in press (1982)
Gaunt, D. S., Whittington, S. G., Sykes, M. F.: J. Phys. A 14, L247 (1981)
de Gennes, P. G.: J. Phys. (Paris) 38, L355 (1977); Daoud, M.: J. Phys. (Paris) 40, L201 (1979)
Coniglio, A., Stauffer, D.: Lett. Nuovo Cim. 28, 33 (1980)
Broadbent, S. R., Hammersley, J. M.: Proc. Camb. Phil. Soc. 53, 629 (1957); Frisch, H. L., Hammersley, J. M.: J. Soc. Industr. Appl. Math. 11, 894 (1963)
Ord, G., Whittington, S. G.: J. Phys. A 15, L29 (1982)
Holl, K.: Staatsexamensarbeit, Cologne University 1980; Gould, H., Holl, K.: J. Phys. A 14, L443 (1981)
Levelt Sengers, J. M. H., Sengers, J. V., in: Perspectives in Statistical Physics, Raveché, H. J. (ed.), p. 339, Amsterdam, North Holland 1981. (For an extreme example see Liebmann, R., Schaub, B., Schauster, H. G.: Z. Phys. B 37, 69 (1980))
Heermann, D. W., Stauffer, D.: Z. Phys. B 44, 339 (1981)
Sievers, D.: J. Phys. (Paris) 41, L535 (1980)
de Gennes, P. G.: Compt. Rendus Acad. Sci. (Paris) 286 B, 131 (1980); J. Phys. (Paris) 40, L197
Batchelor, G. K.: J. Fluid Mech. 83, 97 (1977) and priv. commun. (1981)
Kertesz, J., Vicsek, T.: Z. Phys. B (1982), to be published
de Gennes, P. G., Guyon, E.: J. Mechanique 17, 403 (1978)
Vicsek, T.: Priv. commun. (Sept. 1981)
Kikuchi, R.: J. Chem. Phys. 53, 2713 (1970)
Stoll, E., Binder, K., Schneider, T.: Phys. Rev. B 6, 2777 (1972)
Müller-Krumbhaar, H.: Phys. Lett. 50 A, 27 (1974)
Coniglio, A.: J. Phys. A 8, 1773 (1975), Phys. Rev. B 13, 2194 (1976)
Odagaki, T.: Progr. Theor. Phys. 54, 1067 (1975)
Coniglio, A., Klein, W.: J. Phys. A 13, 2775 (1980)
Napiorkowski, M., Hemmer, P. C.: Phys. Lett. 76 A, 359 (1980)
Domb, C., Stoll, E.: J. Phys. A 10, 1141 (1977)
Coniglio, A., Russo, L.: J. Phys. A 12, 545 (1979)
Fisher, M. E.: J. Math. Phys. 2, 620 (1961)
Coniglio, A. et al.: J. Phys. A 10, 205 (1977)
Coniglio, A., Lubensky, T. C.: J. Phys. A 13, 1783 (1980)
Stauffer, D.: J. Phys. (Paris) 42, L99 (1981); Roussenq, J.: J. Aerosol Sci. 12, 432 (1981) and in preparation; Ottavi, H.: Z. Phys. B, 44, 203 (1981); Delyon, F., Souillard, B., Stauffer, D.: J. Phys. A 14, L243 (1981); Klein, W., Stauffer, D.: J. Phys. A 14, L413 (1981); Kertesz, J.: priv. commun.; Chakrabarti, B. K., Kaski, K., Kertesz, J.: Phys. Lett. 85 A, 423 (1981)
Sykes, M. F., Gaunt, D. S.: J. Phys. A 9, 2131 (1976)
Stauffer, D.: J. Phys. A 10, L71 (1977)
Murata, K. K.: J. Phys. A 12, 81 (1979)
Penrose, O. et al.: J. Statist. Phys. 19, 243 (1978)
Coniglio, A., di Liberto, F., Monroy, G.: J. Phys. A 14, 3017 (1981)
Potts, R. B.: Proc. Camb. Phil. Soc. 48, 106 (1952); for a recent review see Wu, F. Y.: Rev. Mod. Phys. 54, 235 (1982)
Baxter, R. J.: J. Phys. C 6, L445 (1973)
Pytte, E.: Phys. Rev. B 22, 4450 (1980); Aharony, A., Pytte, E.: Phys. Rev. B 23, 362 (1981)
Zallen, R.: Phys. Rev. B 16, 1426 (1977); Halley, J. W., Holcomb, K.: Phys. Rev. Lett. 40, 1670 (1978)
Stanley, H. E.: J. Phys. A 12, L329 (1979); Stanley, H. E., Teixeira, J.: J. Chem. Phys. 73, 3404 (1980)
Coniglio, A., Peruggi, F.: J. Phys. A 15, in press (1982)
Blumberg, R. L., Shlifer, G., Stanley, H. E.: J. Phys. A 13, L147 (1980)
Gonzales, A. E., Reynolds, P. J.: Phys. Lett. 80 A, 357 (1980)
Duarte, J. A. M. S., Ruskin, H. J.: Physica, in press; Reich, G. R., Leath, P. L.: J. Statist. Phys. 19, 611 (1978)
Coniglio, A., Daoud, M.: J. Phys. A 12, L 259 (1979)
de Gennes, P. G.: J. Phys. (Paris) 36, 1049 (1975)
Gaunt, D. S., Guttmann, A. J., Whittington, S. G.: J. Phys. A 12, 75 (1979); Kertesz, J., Chakrabarti, B. K., Duarte, J. A. M. S.: J. Phys. A 15, L 13 (1982)
Chalupa, J., Leath, P. L., Reich, G. R.: J. Phys. C 12, L 31 (1979); Kogut, P. M., Leath, P. L.: J. Phys. C 14, 3187 (1981)
Gonzales, A., Daoud, M.: J. Phys. A 14, 2441 (1981)
Blease, J.: J. Phys. C 10, 917, 925, 3461 (1977); Obukhov, S. P.: Physica 101 A, 145 (1980); Kertesz, J., Vicsek, T.: J. Phys. C 13, L 343 (1980); Dhar, D., Barma, M.: J. Phys. A 14, L 5 (1981); Kinzel, W., Yeomans, J. M.: J. Phys. A 14, L 163 (1981); and Ref. 32; Redner, S., Coniglio, A.: J. Phys. A 15, in press (1982)
Redner, S., Brown, A. C.: J. Phys. A 14, L 285 (1981); Redner, S.: J. Phys. A 14, L 349 (1981); Reynolds, P. J.: preprint (1981)
Brown, E., Essam, J. W., Place, C. M.: J. Phys. C 8, 321 (1975); Agrawal, P. et al.: J. Phys. A 12, 2073 (1979); Nakanishi, H., Reynolds, P. J.: Phys. Lett. 71 A, 252 (1979); Hoshen, J., Klymko, P., Kopelman, R.: J. Statist. Phys. 21, 583 (1979)
Daoud, M., Coniglio, A.: J. Phys. A 14, L 108 (1981)
Tanaka, T., Swislow, G., Ohmine, A.: Phys. Rev. Lett. 42, 1556 (1979)
Joanny, J. F.: Polymer 21, 71 (1980)
Tanaka, Phys. Rev. A 17, 763 (1978); Phys. Rev. Lett. 40, 820 (1978); 43, 217 (1979)
Ziff, R. M., Stell, G.: J. Chem. Phys. 73, 3492 (1980); Ziff, R. M.: J. Statist. Phys. 23, 241 (1980) and Ref. 16 and Cohen, R. J., Benedek, G. B.: Preprint (1982)
Peters, H. P. et al.: Z. Phys. B 34, 399 (1979)
Kirkpatrick, S. in: I11-Condensed Matter. Balian, R., Maynard, R., Toulouse, G. (eds.), p. 321, Amsterdam, North Holland 1979
Manneville, P., de Seze, L., in: Numerical Methods in the Study of Critical Phenomena, Della Dora, J., Demongeot, J., Lacolle, B. (eds.), p. 116, Berlin — Heidelberg — New York, Springer 1981
Okasha, R., Hild, G., Rempp, P.: Eur. Polym. J. 15, 975 (1979); Kalal, J., Gordon, M., Devoy, C.: Makromol. Chem. 152, 233 (1972)
Yamakawa, H.: Modern Theory of Polymer Solutions, p. 220, New York, Harper and Row 1971
Gordon, M. et al.: Makromol. Chem. 176, 2413 (1975)
Whitney, R. S., Burchard, W.: Makromol. Chem. 181, 869 (1980)
Burchard, W., et al.: Macromolecules 6, 642 (1973)
Champetier, G.: Chimie Macromoleculaire, vol. 2, p. 200. Hermann, Paris
de Gennes, P. G.: Ref. 67 and priv. commun.
Bacri, J. C. et al.: J. Phys. (Paris) 41, L 369 (1980)
Valles, E. M., Macosco, M. W.: Macromolecules 12, 521 (1979), for example; and Ref. 12
Dumas, J., Bacri, J. C.: J. Phys. (Paris) 41, L 279 (1980)
Adam, M. et al.: J. Phys. (Paris) 40, L 539 (1980)
Adam, M. et al.: Pure Appl. Chem. 53, 1489 (1981)
Gordon, M., Roberts, K. R.: Polymer 20, 681 (1979)
Gautier-Manuel, B., Guyon, E.: J. Phys. (Paris) 41, L 503 (1980)
Joanny, J. F.: J. Phys. (Paris), to be published
Hoshen, J., Kopelman, R.: Phys. Rev. B 14, 3428 (1976)
Derrida, B., de Seze, L.: Preprint submitted to J. Phys. (Paris) (1981); and Family, F.: J. Phys. A 13, L 325 (1980)
Grassberger, P.: preprint submitted to Biopolymers (1982)
Author information
Authors and Affiliations
Editor information
Additional information
Dedicated to Prof. Manfred Gordon on the occasion of his 65th birthday, who discussed critical phenomena at the gel point as early as in 1974, see Ref. 1.
Rights and permissions
Copyright information
© 1982 Springer-Verlag
About this paper
Cite this paper
Stauffer, D., Coniglio, A., Adam, M. (1982). Gelation and critical phenomena. In: Dušek, K. (eds) Polymer Networks. Advances in Polymer Science, vol 44. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-11471-8_4
Download citation
DOI: https://doi.org/10.1007/3-540-11471-8_4
Received:
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-11471-0
Online ISBN: 978-3-540-39086-2
eBook Packages: Springer Book Archive