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Random walks, trapping and reactions in quasi-one dimensional lattices

Published online by Cambridge University Press:  10 February 2011

Panos Argyrakis ,
Anna L. Lin  and
Raoul Kopelman
Panos Argyrakis
Affiliation:
Department of Physics, University of Thessaloniki, 54OO6 Thessaloniki Greece., E-mail:, [email protected]
Anna L. Lin
Affiliation:
Department of Chemistry, University of Michigan, AnnArbor, Michigan 48109–1055, E-mail:, [email protected], [email protected]
Raoul Kopelman
Affiliation:
Department of Chemistry, University of Michigan, AnnArbor, Michigan 48109–1055, E-mail:, [email protected], [email protected]
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Abstract

We performed Monte Carlo simulations of diffusion processes on baguette-like lattices, which have very small width and height, but long length, in effect providing quasi-one dimensional systems. This is done by investigating random walk properties of single particles, and also the well known model bimolecular reactions A + A and A + B. We monitor the number of distinct sites visited, as a function of time, and also the survival probability in the presence of static traps. For the reaction systems we monitor, as usual, the decay of the particle density. The expected one-dimensional behavior is recovered, in the long time limit, for all cases studied. Our interest here is in the crossover time, from 3- (or 2-)dimensionaI behavior (early time) to one-dimensional (long time). We find that this crossover time scales with respect to the baguette's short dimension. However, this scaling deviates significantly from a mean square displacement law, and it is specific to both tube dimensionality and reaction type (e.g. A + A or A + B). Specifically, instead of an expected power of 2, the exponents range between 1 and 4. The densities of the A + B reactions at the dimensional crossover are compared to the densities at the segregation crossover in regular lattices at all three dimesnions. As expected, the time evolution of the A + A reaction parallels the behavior of the average number of distinct sites visited.

Type
Research Article
Information
MRS Online Proceedings Library (OPL) , Volume 464: Symposium FF – Dynamics in Small Confining Systems III , 1996 , 243
Copyright
Copyright © Materials Research Society 1997

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References

REFERENCES

[1] Molecular Dynamics in Restricted Geometries, eds. Klafter, J., Drake, J. M., John Wiley and Sons (1989).Google Scholar
[2] Chs. 20 and 21, Nonequilibrium Statistical Mechanics in One Dimension, ed. Privman, V., Cambridge University Press, (in press).Google Scholar
[3] Kopelman, R., Science 41, 1620 (1988).Google Scholar
[4] Prasad, J. and Kopelman, R., Chem. Phys. Lett. 157, 535 (1989).Google Scholar
[5] Prasad, J. and Kopelman, R., J. Phys. Chem. 91, 265 (1987).Google Scholar
[6] Kopelman, R., Parus, S. J., and Prasad, J., in Excited State Relaxation and Transport Phenomena in Solids, eds. Skinner, J. L. and Payer, M. D., special issue of Chem. Phys., 128, 209 (1988).Google Scholar
[7] Kopelman, R., Parus, S., Prasad, J., Phys. Rev. Lett., 56, 1742 (1986).Google Scholar
[8] Kopelman, R., Parus, S. J., and Prasad, J., Chem. Phys., 128, 209 (1988).Google Scholar
[9] Kopelman, R., Shi, Z.-Y. and Li, C. S., J. Lumin. 48 and 49, 143 (1991).Google Scholar
[10] Parus, S. J. and Kopelman, R., Phys. Rev. B 39, 889 (1989).Google Scholar
[11] Doering, C. R. and Ben-Avraham, D., Phys. Rev. A 38, 3035 (1988).Google Scholar
[12] Doering, C. R. and Ben-Avraham, D., Phys. Rev. Lett. 62, 2563 (1989).Google Scholar
[13] Ovchinnikov, A. A. and Zeldovich, Y. G., Chem. Phys. B, 28, 215 (1978).Google Scholar
[14] Toussaint, D. and Wilczek, F., J. Chem. Phys. 78, 2642 (1983).Google Scholar
[15] Lin, A., Kopelman, R., and Argyrakis, P., Phys. Rev. E., 53, 1502 (1996).Google Scholar
[16] Anacker, L. W. and Kopelman, R., Phys. Rev. Lett. 58, 289 (1987).Google Scholar
[17] Lindenberg, K., West, B. J., and Kopelman, R., Phys. Rev. Lett. 60, 1777 (1988).Google Scholar
[18] Argyrakis, P. and Kopelman, R., J. Phys. Chem., 91, 2699 (1987).Google Scholar
[19] Argyrakis, P., Kopelman, R., and Lindenberg, K., Chem. Phys. 177, 693 (1993).Google Scholar
[20] Kang, K. and Redner, S., Phys. Rev. Lett. 52, 955 (1984).Google Scholar
[21] Kang, K. and Redner, S., Phys. Rev. A, 32, 435 (1985).Google Scholar
[22] Argyrakis, P., Computers in Physics, 6, 525 (1992).Google Scholar
[23] Montroll, E. W. and Weiss, G. H., Math, J.. Phys. 6, 167 (1965).Google Scholar
[24] Henyey, F. S. and Seshadri, V., J. Chem. Phys. 76, 5530 (1982).Google Scholar
[25] Kopelman, R. and Argyrakis, P., J. Chem. Phys., 72, 3050 (1980).Google Scholar
[26] de Gennes, P. G., Chem. R. Acad. Sci. Paris 296, 881 (1983).Google Scholar
[27] Leyvraz, F. and Redner, S., Phys. Rev. A, 46, 3132 (1992).Google Scholar
[28] Weiss, G.H., Aspects and Applications of the Random Walk Elsevier, Amsterdam (1994).Google Scholar
[29] Anlauf, J.K., Phys. Rev. Lett., 52, 1845 (1984).Google Scholar
[30] Gallos, L.K. and Argyrakis, P., in Proceedings of Surface diffusion: atomistic and collective processes, Rhodes, Greece (1996).Google Scholar