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Surfaces of percolation systems in lattice problems.

P S Grinchuk1, O S Rabinovich

  • 1A V Luikov Heat and Mass Transfer Institute, National Academy of Sciences of Belarus, 15 Petrus Brovka Street, Minsk 220072, Belarus. gps@hmti.ac.by

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|June 6, 2003
PubMed
Summary

The study reveals that percolation cluster surface areas peak at specific site or bond occupation levels. Analytical and numerical methods were used to investigate these surface properties in various lattice systems.

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Area of Science:

  • Physics
  • Materials Science
  • Statistical Mechanics

Background:

  • Percolation theory describes the formation of connected clusters in random systems.
  • Understanding the surface area of these clusters is crucial for various physical phenomena.
  • Previous studies have focused on cluster size and connectivity, with less emphasis on surface area characteristics.

Purpose of the Study:

  • To analytically and numerically investigate the internal, external, and full surface areas of percolation clusters and systems.
  • To determine the conditions under which these surface areas exhibit maxima.
  • To establish relationships between percolation cluster surface area and density.

Main Methods:

  • Monte Carlo simulations on square and simple cubic lattices for site and bond percolation problems.

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  • Analytical derivations using a probabilistic approach.
  • Detailed analysis of surface area behavior across different lattice types.

    • Main Results:

    • Both external and full surface areas of percolation clusters, and the full surface area of the percolation system, show maxima at specific site/bond occupation thresholds.
    • Analytical expressions were derived to link percolation cluster surface area with its density.
    • The behavior of surface area was analyzed for various lattices, revealing distinct patterns.

    Conclusions:

    • The surface area of percolation clusters is a critical parameter with non-monotonic behavior, exhibiting maxima at intermediate densities.
    • The derived analytical expressions provide a theoretical framework for understanding surface area-density relationships.
    • The findings have implications for technological processes like fuel cell current generation and high-temperature synthesis.