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A theory of percolation in liquids

J. Chem. Phys. 85, 391 (1986); doi:10.1063/1.451615

Issue Date: 1 July 1986

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Tony DeSimone, Sophia Demoulini, and Richard M. Stratt
Department of Chemistry, Brown University, Providence, Rhode Island 02912
Problems involving percolation in liquids (i.e., involving connectivity of some sort) range from the metal–insulator transition in liquid metals to the properties of supercooled water. A common theme, however, is that connectivity can be distinguished from interaction and that one should not be slighted in order to describe the other. In this paper we suggest a model for percolation in liquids—the model of extended spheres—which permits connectivity to be studied in the context of, but independently from, liquid structure. This model is solved exactly in the Percus–Yevick approximation, revealing the existence of an optimum liquid structure for percolation. We analyze this behavior by first deriving an explicit diagrammatic representation of the Percus–Yevick theory for connectivity and then studying how the various diagrams contribute. The predictions are in excellent qualitative agreement with recent Monte Carlo calculations. The Journal of Chemical Physics is copyrighted by The American Institute of Physics.
History: Received 23 December 1985; accepted 21 March 1986
Permalink: http://link.aip.org/link/?JCPSA6/85/391/1
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KEYWORDS and PACS

Keywords
PACS
  • 61.20.Gy
    Structure of liquids and solids; crystallography Classical, semiclassical, and quantum theories of liquid structure Statistical theories of liquid structure
  • 05.20.-y
    Statistical physics and thermodynamics Statistical mechanics
  • 71.30.+h
    Electron states Metalinsulator transitions
  • 72.15.Cz
    Electronic transport in condensed matter Electronic conduction in metals and alloys Electrical and thermal conduction in amorphous and liquid metals and alloys
  • YEAR: 1986

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ISSN:
0021-9606 (print)   1089-7690 (online)
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REFERENCES (37)
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  35. Besides implying a value of the critical density, rhoc, Percus-Yevick theory also predicts critical exponents for percolation. In particular, the cluster size and the connectedness length are predicted to diverge as (rhoc−rho)−l to the powers of gamma = 2 and nu = 1, respectively. These results are not exact, though they are somewhat better than the mean field results of gamma = 1 and gamma = (1/2). Note, in particular, that the mean field theory discussed in the text, Eq. (4.2), gives these last exponents. See G. Stell and J. S. Hoye, J. Phys. A 18, L951 (1985), and references therein.
  36. A complete analysis of the relative errors of the Percus-Yevick theories for correlation and connectedness is actually a bit more subtle than the discussion in the text would indicate. The structure of a hard sphere fluid is determined by the values of the correlation function diagrams outside the hard core. Here the approximate cancellation of order rho2 diagrams is indeed quite good [G. Stell, Physica 29, 517 (1963)].
    37. However, the percolation threshold is determined by the integral of c‡ inside the core (r<d). Indeed it turns out that the extra factor of (1/2) in connectedness theory noticeably improves the cancellation inside the core—just not to the degree experienced by correlation Percus-Yevick theory outside the core. (T. DeSimone, Ph.D. thesis, Brown University, 1985). Unfortunately, we have no analogous information about the level of cancellation for diagrams beyond those of order rho2.
  37. H. C. Andersen in Statistical Mechanics, Part A, edited by B. J. Berne (Plenum, New York, 1977), p. 1.
  38. W. E. Boyce and R. C. DiPrima, Elementary Differential Equations, 2nd ed. (Wiley, New York, 1965), Sect. 7.9.

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