Abstract
This paper studies growth, percolation, and correlations in disordered fiber networks. We start by introducing a 2D continuum deposition model with effective fiber-fiber interactions represented by a parameterp which controls the degree of clustering. Forp=1 the deposited network is uniformly random, while forp=0 only a single connected cluster can grow. Forp=0 we first derive the growth law for the average size of the cluster as well as a formula for its mass density profile. Forp>0 we carry out extensive simulations on fibers, and also needles and disks, to study the dependence of the percolation threshold onp. We also derive a mean-field theory for the threshold nearp=0 andp=1 and find good qualitative agreement with the simulations. The fiber networks produced by the model display nontrivial density correlations forp<1. We study these by deriving an approximate expression for the pair distribution function of the model that reduces to the exactly known case of a uniformly random network. We also show that the two-point mass density correlation function of the model has a nontrivial form, and discuss our results in view of recent experimental data on mss density correlations in paper sheets.
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References
J. W. Evans,Rev. Mod. Phys. 65:1281 (1993).
V. Privman, InAnnual Reviews in Computational Physics, Vol. 3, D. Stauffer, ed. (World Scientific, Singapore, 1995).
G. Y. Onoda and E. G. Liniger,Phys. Rev. A 33:715 (1986); A. Schmit, R. Varoqui, S. Uniyal, J. L. Brash, and C. Pusiner,J. Colloid Interface Sci. 92:25 (1983); J. Feder and I. Giaever,J. Colloid Interface Sci. 78:144 (1980).
M. Deng and C. T. J. Dodson,Paper: An Engineered Stochastic Structure (Tappi Press, 1994).
K. J. Niskanen and M. J. Alava,Phys. Rev. Lett. 73:3475 (1994).
P. Nielaba and V. Privman,Phys. Rev. E 51:2022 (1995).
N. Ryde, H. Kihira, and E. Matijevic,J. Colloid Interface Sci. 151:421 (1992).
A.-L. Barabasi and H. E. Stanley,Fractal Concepts in Surface Growth (Cambridge University Press, Cambridge, 1995).
C. A. Murray and D. G. Grier,Am. Sci. 83:238 (1995).
M. L. Kurnaz and J. V. Maher,Phys. Rev. E 53:978 (1996).
S. Schwarzer,Phys. Rev. E. 52:6461 (1995).
N. Provatas, M. J. Alava, and T. Ala-Nissila,Phys. Rev. E 54 (1996). N. Provatas, M. J. Alava, and T. Ala-Nissila, Unpublished (1996).
G. E. Pike and C. H. Seager,Phys. Rev. B 10:1421 (1974).
I. Balberg and N. Binenbaum,Phys. Rev. B 28:3799 (1983).
P. C. Robinson,J. Phys. A 16:605 (1983).
I. Balberg, C. H. Anderson, S. Alexander, and N. Wagner,Phys. Rev. B 30:3933 (1984).
P. C. Robinson,J. Phys. A 17:2823 (1984).
A. L. R. Bug, S. A. Safran, and G. S. Grest,Phys. Rev. Lett. 55:1896 (1985).
I. Balberg,Phil. Mag. B 56:991 (1987).
D. Laría and F. Vericat,Phys. Rev. B 40:353 (1989).
S. B. Lee and S. Torquato,Phys. Rev. A 41:5338 (1990).
U. Alon, I. Balberg, and A. Drory,Phys. Rev. Lett. 66:2879 (1991).
C. Vanneste, A. Gilabert, and D. Sornette,Phys. Lett. A 155:174 (1991).
A. Drory, I. Balberg, and B. Berkowitz.Phys. Rev. E 49:949 (1994).
H. S. Choi, J. Talbot, G. Tarjus, and P. Viot,Phys. Rev. E 51:1353 (1995).
M. D. Rintoul and S. Torquato,Phys. Rev. E 52:2635 (1995).
W. J. Boudville and T. C. McGill,Phys. Rev. B 39:369 (1980).
J. Aström, Pro Gradu avhandling, Abo Akademi, Unpublished (1989).
N. Provatas, T. Ala-Nissila, and M. J. Alava,Phys. Rev. Lett. 75:3556 (1995).
D. Stauffer and A. Aharony,Introduction to Percolation Theory (Taylor and Francis. London. 1994).
J. Hoshen and R. Kopelman,Phys. Rev. B 14:3438 (1976).
P. J. Reynolds, H. E. Stanley, and W. Klein,Phys. Rev. B 21:1223 (1980).
A. Weinrib.Phys. Rev. B 29:387 (1984).
O. J. Kallmes and H. Corte,Tappi J. 43:737 (1960).
B. Ghosh,Calcutta Math. Soc. 43(1):17 (1951).
D. E. Sanders and J. W. Evans,Phys. Rev. A 38:4186 (1988).
L. Haugland, B. Norman, and D. Wahren, Svensk Papperstidning no. 10, 362 (1974).
J.-P. Hansen and I. R. McDonald,Theory of Simple Liquids (Academic Press, New York, 1986).
R. J. Kerekes and C. J. Shell,J. Pulp Paper Sci. 18:132 (1992).
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Provatas, N., Haataja, M., Seppälä, E. et al. Growth, percolation, and correlations in disordered fiber networks. J Stat Phys 87, 385–413 (1997). https://doi.org/10.1007/BF02181493
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DOI: https://doi.org/10.1007/BF02181493