### Abstract

Rigorous upper and lower bounds on the effective electrical conductivity σ* of a two-phase material composed of equi-sized spheres distributed with an arbitrary degree of impenetrability in a matrix are obtained and studied. In general, the bounds depend upon, among other quantities, the point/n-particle distribution functions G_{n}^{(i)}, which are probability density functions associated with finding a point in phase i and a particular configuration of n spheres. The G_{n}^{(i)} are shown to be related to the ρ_{n}, the probability density functions associated with finding a particular configuration of n partially penetrable spheres in a matrix. General asymptotic and bounding properties of the G _{n}^{(i)} are given. New results for the G_{n} ^{(i)} are presented for totally impenetrable spheres, fully penetrable spheres (i.e., randomly centered spheres), and sphere distributions between these latter two extremes. The so-called first-order cluster bounds on σ* derived here are given exactly through second order in the sphere volume fraction for arbitrary λ (where λ is the impenetrability or hardness parameter) for two different interpenetrable-sphere models. Comparison of these low-density bounds on σ* to an approximate low-density expansion of σ* derived here for interpenetrable-sphere models, reveals that the bounds can provide accurate estimates of the second-order coefficient for a fairly wide range of λ and phase conductivities. The results of this study suggest that general bounds derived by Beran, for dispersions of spheres distributed with arbitrary λ and through all orders in φ_{2}, are more restrictive than the first-order cluster bounds for 0≤γ<1; with the two sets of bounds being identical for the case of totally impenetrable spheres (λ=1). For most values of λ in the range 0≤λ≤1, however, the numerical differences between the Beran and cluster bounds should be small; the greatest difference occurring when λ=0. The analysis also indicates that the cluster bounds will be easier to compute than the Beran bounds for dispersions of partially penetrable spheres.

Original language | English (US) |
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Pages (from-to) | 6345-6359 |

Number of pages | 15 |

Journal | The Journal of chemical physics |

Volume | 84 |

Issue number | 11 |

DOIs | |

State | Published - Jan 1 1986 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Physics and Astronomy(all)
- Physical and Theoretical Chemistry