Percolation of spheres in continuum

M. M. Buzmakova

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 12 (2012) no. 2, pp. 48-56.

Résumé

The model of the continuum percolation of hard spheres with permeable shells, which describes phase transition sol-gel, has been investigate. Spheres have hard parts in radii $r$, which can't be blocked with each other, and permeable shells in width $d$, which can be blocked. Such spheres of the equal size have been randomly packing in the cub with linear size $L$. The probability of joining the spheres in a cluster is proportional to the volume of overlapping of permeable shells. Spheres belong to a cluster, if a communication between spheres arises. The percolation cluster is the cluster connecting bottom and top sides of the cube. The packing fraction, at which probability of occurrence of the percolation cluster is 0.5, is called as the percolation threshold. The percolation threshold corresponds to the gel point. The dependency of the percolation threshold of the hard spheres with permeable shells from a thickness of the shell has been obtained.

Bibliographie

[1] Savina L. V., Kristalloskopicheskie struktury syvorotki krovi zdorovogo i bolnogo cheloveka, Sov. Kuban, Krasnodar, 1999, 96 pp.

[2] Shabalin V. N., Shatokhina S. N., Morfologiya biologicheskikh zhidkostei cheloveka, Khrizostom, M., 2001, 304 pp.

[3] Rapis E. G., Belok i zhizn (samoorganizatsiya, samosborka i simmetriya nanostruktur belka), Filobiblon, Ierusalim; Milta-PKP GIT, M., 2002, 257 pp.

[4] Pauchard L., Parisse F., Allain C., “Influence of salt content on crack patterns formed through colloidal suspension desiccation”, Phys. Rev. E, 59:3 (1999), 3737–3740 | DOI

[5] Yakhno T. A., Yakhno V. G., Sanin A. G., Sanina O. A., Pelyushenko A. S., “Belok i sol: prostranstvenno-vremennye sobytiya v vysykhayuschei kaple”, Zhurn. tekhn. fiziki, 74:8 (2004), 100–108

[6] Yakhno T. A., Yakhno V. G., “Osnovy strukturnoi evolyutsii vysykhayuschikh kapel biologicheskikh zhidkostei”, Zhurn. tekhnicheskoi fiziki, 79:8 (2009), 133–141

[7] Stauffer D., Aharony A., Introduction to Percolation Theory, Taylor Francis, L., 1992, 181 pp.

[8] Sahimi M., Application of Percolation Theory, Taylor Francis, L., 1994, 258 pp.

[9] Zaiman D., Modeli besporyadka. Teoreticheskaya fizika odnorodno neuporyadochennykh sistem, Mir, M., 1982, 591 pp.

[10] Feder E., Fraktaly, Mir, M., 1991, 254 pp. | MR

[11] Ohira K., Sato M., Kohmoto M., “Fluctuations in chemical gelation”, Phys. Rev. E, 75:4 (2007), 041402 | DOI

[12] Gado E., Fierro A., Arcangelis L., Coniglio A., “Slow dynamics in gelation phenomena: From chemical gels to colloidal glasses”, Phys. Rev. E, 69:5 (2004), 051103 | DOI

[13] Jespersen S., “Cluster diffusion at the gelation point”, Phys. Rev. E, 66:3 (2002), 031502 | DOI

[14] Vernon D., Plischke M., “Viscoelasticity near the gel point: A molecular dynamics study”, Phys. Rev. E, 64:3 (2001), 031505 | DOI

[15] Plischke M., Vernon D., Joós B., “Model for gelation with explicit solvent effects: Structure and dynamics”, Phys. Rev. E, 67:1 (2003), 011401 | DOI

[16] Monkos K., “Determination of some hydrodynamic parameters of ovine serum albumin solutions using viscometric measurements”, J. of Biological Phys., 31 (2005), 219–232 | DOI

[17] Rottereau M., Gimel J., Nicolai T., Durand D., “3d Monte Carlo simulation of site-bond continuum percolation of spheres”, The European Physical J. E: Soft Matter and Biological Physics, 11 (2003), 61–64 | DOI

[18] Johner N., Grimaldi C., Balberg I., Ryser P., “Transport exponent in a three-dimensional continuum tunneling-percolation model”, Phys. Rev. B, 77:17 (2008), 174204 | DOI

[19] Matsumoto M., “Mersenne twister: A 623-dimensionally equidistributed uniform pseudorandom number generator”, ACM Trans. on Modeling and Computer Simulations, 8:1 (1998), 3–30 | DOI | Zbl

[20] Hoshen J., Kopelman R., “Percolation and cluster distribution. I. Cluster multiple labeling technique and critical concentration algorithm”, Phys. Rev. B, 14:8 (1976), 3438–3445 | DOI

[21] Rubin F., “The Lee Path Connection Algorithm”, IEEE Transactions on Computers, 23 (1974), 907–914 | DOI | MR | Zbl

[22] Teilor D., Vvedenie v teoriyu oshibok, per. s angl., Mir, M., 1985, 272 pp.

[23] Tarasevich Yu. Yu., Perkolyatsiya: teoriya, prilozheniya, algoritmy, Editorial URSS, M., 2002, 112 pp.

[24] Balberg I., Binenbaum N., “Invariant properties of the percolation thresholds in the soft-core-hard-core transition”, Phys. Rev. A, 35:12 (1987), 5174–5177 | DOI

[25] Efros A. L., Fizika i geometriya besporyadka, Nauka, M., 1982, 260 pp. | MR

[26] Zhydkov V., “3D continuum percolation approach and its application to lava-like fuel-containing materials behavior forecast”, Condensed Matter Phys., 12:2 (2009), 193–203 | DOI