J. Phys. Soc. Jpn. 53, pp. 480-482 (1984) [3 Pages]

Percolation Theory and Elastic Modulus of Gel

+ Affiliations
1Department of Polymer Science, Faculty of Science2Institute of Dairy Science, Faculty of Agriculture, Hokkaido University

Dynamic shear modulus of casein gel was measured slightly above the gelation concentration as functions of frequency and concentration. The results show that the shear modulus G is expressed by G = C · ε t , where ε is the reduced concentration in reference to the gelation concentration. The exponent t is found to be constant at lower frequency region and is close to the value expected from the percolation theory.

©1984 The Physical Society of Japan


  • 1 For the review of percolation theory; D.Stauffer: Phys. Rep. 54 (1979) 1 and its application to gelation is; D.Stauffer, A.Coniglio and M.Adam: Adv. Polym. Sci. 44 (1982) 103, related references are listed in these reviews. CrossrefGoogle Scholar
  • 2 A.Geiger and H. E.Stanley: Phys. Rev. Lett. 49 (1982) 1895. CrossrefGoogle Scholar
  • 3 M.Gordon and J. A.Torkington: Pure and Appl. Chem. 53 (1981) 1461. CrossrefGoogle Scholar
  • 4 M.Adam, M.Delsanti, D.Durand, G.Hild and J. P.Munch: Pure and Appl. Chem. 53 (1981) 1489. CrossrefGoogle Scholar
  • 5 B.Gauthier-Manuel and E.Guyon: J. Phys. Lett. (France) 41 (1980) L503. CrossrefGoogle Scholar
  • 6 P. G.de Gennes: J. Phys. Lett. (France) 37 (1976) L1 and Crossref;, Google Scholaralso see P. G.de Gennes: Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca, 1979) Chap. V. Google Scholar
  • 7 C. V.Morr: in Functionality and Protein Structure, ed. A.Pour-E1, ACS Symposium Series 92 (1979) p. 72; Google ScholarV. A.Bloomfield and C. V.Morr: Neth. Milk Dairy J. 27 (1973) 103; D. G.Schmidt, P.Walstra and W.Buchheim: Neth. Milk Dairy J. 27 (1973) 128. Google ScholarFurther details of the structure of casein micelle and the chemical reaction of the milk clotting enzymes are mentioned in Fundamentals of Dairy Chemistry, ed. B. W.Webb and A. H.Johnson (AVI Publishing Company, 1974) and also in the review Developments in Dairy Chemistry-1, ed. D. G.Schmidt, P. F.Fox (Applied Science Publishers, 1982) Chap. 2. Google Scholar
  • 8 P. H.Stothart and D. J.Cebula: J. Mol. Biol. 160 (1982) 391. CrossrefGoogle Scholar
  • 9 M.Tokita, K.Hikichi, R.Niki and S.Arima: Biorheology 19 (1982) 695. CrossrefGoogle Scholar
  • 10 M.Tokita: Doctoral Dissertation of Hokkaido University (1982). Google Scholar
  • 11 K.Miki, K.Hikichi and M.Kaneko: Jpn. J. Appl. Phys. 6 (1967) 931. CrossrefGoogle Scholar
  • 12 The volume fraction of the casein micelle at the gelation point was calculated from the molecular weight and the radius of the casein micelle. The value was about 2∼20%(v/v). Thus, we thought that the gelation point of the casein micelle solution corresponds to the percolation threshold. Google Scholar
  • 13 In the percolation theory, typical observed parameter such as the percolation probability P(p) are expressed as a function of the occupied probability of sites (or bonds) p, hence, ε=(p-pc)/pc is chosen for the reduced variable in the vicinity of the percolation threshold. In the present study, only the concentration of casein was changed, thus, we simply defined ε=(φ-φg)/φg. casein micelle concentration of the present study corresponds to 5.62×10-2≤ε≤3.14. Google Scholar
  • 14 A. G.Dunn, J. W.Essam and D. S.Ritchie: J. Phys. C8 (1975) 4219. Google Scholar
  • 15 M. A. A.Cox and J. W.Essam: J. Phys. C9 (1976) 3985. Google Scholar
  • 16 Recent advanced studies seem to yield \(t{=}{\nu}\cdot(d-2)+\nu\cdot(\bar{d}/\tilde{d})(2-\tilde{d})\) where \(\bar{d}\cong 2.5\) and \(\tilde{d}\cong 4/3\) for d=3 implies that \(t{\cong}1.98\). The observed value of the exponent t is not too far from the above prediction. See for instance, S.Alexander and R.Orbach: J. Phys. Lett. (France) 43 (1982) L625; R.Rammal and G.Toulouse: J. Phys. Lett. (France) 44 (1983) L13. CrossrefGoogle Scholar