### Abstract

Three-point bounds on the effective conductivity σ_{e} of isotropic two-phase composites, that improve upon the well-known two-point Hashin-Shtrikman bounds [J. Appl. Phys. 23, 779 (1962)], depend upon a key microstructural parameter ζ_{2}. A highly accurate approximation for σ_{e} developed by Torquato [J. Appl. Phys. 58, 3790 (1985)] also depends upon ζ_{2}. This paper reports a new and accurate algorithm to compute the three-point parameter ζ_{2} for dispersions of hard spheres by Monte Carlo simulation. Data are reported up to values of the sphere volume fraction φ_{2} near random close-packing and are used to assess the accuracy of previous analytical calculations of ζ_{2}. A major finding is that the exact expansion of ζ_{2} through second order in φ_{2} provides excellent agreement with the simulation data for the range 0≤φ_{2} ≤0.5, i.e., this low-volume-fraction expansion is virtually exact, even in the high-density region. For φ_{2} >0.5, this simple quadratic formula is still more accurate than other more sophisticated calculations of ζ_{2}. The linear term of the quadratic formula is the dominant one. Using our simulation data for ζ_{2}, we compute three-point bounds on the conductivity σ_{e} and Torquato's approximation for σ_{e}.

Original language | English (US) |
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Pages (from-to) | 5486-5493 |

Number of pages | 8 |

Journal | Journal of Applied Physics |

Volume | 68 |

Issue number | 11 |

DOIs | |

State | Published - Dec 1 1990 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Physics and Astronomy(all)

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## Cite this

*Journal of Applied Physics*,

*68*(11), 5486-5493. https://doi.org/10.1063/1.347007