Abstract
We use the finite-size scaling method to estimate the critical exponent λ that characterizes the scaling behavior of conductivity and permeability anisotropy near the percolation thresholdp c . Here λ is defined by the scaling lawk l /k t −1∼(p−p c )λ, wherek t andk t are the conductivity or permeability of the system in the direction of the macroscopic potential gradient and perpendicular to this direction, respectively. The results are λ(d=2)≃0.819±0.011 and λ(d=3)≃0.518±0.001. We interpret these results in terms of the structure of percolation clusters and their chemical distance. We also compare our results with the predictions of a scaling theory for λ due to Straley, and propose that λ(d=2)=t-β B , wheret is the critical exponent of the conductivity or permeability of the system, and β B is the critical exponent of the backbone of percolation clusters.
This is a preview of subscription content, log in via an institution to check access.
References
D. Stauffer and A. Aharony,Introduction to Percolation Theory, 2nd ed., (Taylor and Francis, London, 1992).
M. Sahimi,Applications of Percolation Theory (Taylor and Francis, London, 1994).
M. Sahimi,Rev. Mod. Phys. 65:1393 (1993).
D. Stauffer, A. Coninglio, and M. Adam,Adv. Polymer Sci. 44:105 (1982).
M. Daoud and A. Lapp,J. Phys.: Condens. Matter 2:4021 (1990).
M. Sahimi,Mod. Phys. Lett. B 6:507 (1992).
M. Adam and D. Lairez,Fractals,1:149 (1993).
M. Sahimi,Physica A 186:160 (1992).
S. Kirkpatrick,Rev. Mod. Phys.,45:574 (1973).
J.-M. Normand, H. J. Herrmann, and M. Hajjar,J. Stat. Phys. 52:441 (1988).
D. B. Gingold and C. J. Lobb,Phys. Rev. B 42:8220 (1990).
T. P. Troadec, D. Bideau, and E. Guyon,J. Phys. C 14:4807 (1981).
K. S. Mendelson and F. G. Karioris,J. Phys. C 13:6197 (1980).
L. N. Smith and C. J. Lobb,Phys. Rev. B 20:3653 (1979).
I. Balberg and P. J. Zanzucchi,Appl. Phys. Lett. 40:1022 (1982).
M. A. J. Michels, J. C. M. Brokken-Zip, W. M. Groenewoud, and A. Knoester,Physica A 157:529 (1989).
J. P. Straley,J. Phys. C 13:L773 (1980).
B. I. Shklovskii,Phys. Stat. Sol. (b) 85:K111 (1978).
C. J. Lobb, D. J. Frank, and M. Tinkham,Phys. Rev. B 23:2262 (1981).
A. K. Sarychev and A. P. Vinogradoff,J. Phys. C 16:L1073 (1983).
J. Vannimenus and M. Knezevic,J. Phys. C 17:4927 (1984).
S. Mukhopadhyay and M. Sahimi,Water Resour. Res., to be published.
M. Sahimi and S. Mukhopadhyay,AIChE J., to be published.
M. Sahimi and S. Arbabi,J. Stat. Phys. 62:453 (1991).
E. Duering and H. E. Roman,J. Stat. Phys. 64:851 (1991).
H. E. Stanley,J. Phys. A 10:L211 (1977).
K. M. Middlemiss, S. G. Whittington, and D. S. Gaunt,J. Phys. A 13:1835 (1980).
Z. Alexandrowicz,Phys. Lett. A 80:284 (1980).
S. Havlin and R. Nossal,J. Phys. A 17:L427 (1984).
H. J. Herrmann and H. E. Stanley,J. Phys. A 21:L829 (1988).
M. Sahimi,J. Phys. A 17:L601 (1984).
A. Aharony and D. Stauffer,Phys. Rev. Lett.,52:2368 (1984).
M. D. Rintoul and H. Nakanishi,J. Phys. A 25:L945 (1992).
Author information
Authors and Affiliations
Additional information
Communicated by D. Stauffer
Rights and permissions
About this article
Cite this article
Mukhopadhyay, S., Sahimi, M. Scaling behavior of permeability and conductivity anisotropy near the percolation threshold. J Stat Phys 74, 1301–1308 (1994). https://doi.org/10.1007/BF02188233
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02188233