An important problem in describing the transport properties of random multi-phase materials is the prediction of percolation thresholds as a function of volume fraction [1,2], interparticle interaction [3], shape [4,5,6,7,8] and orientation of the component phases or particles of the random material [9]. In many practical applications, the structure of composite materials evolves in time by chemical reaction so that the percolation transition occurs after an "ageing time" (e.g., cement-based materials, gels) [10,11,12,13,14,15,16]. These applications motivate further study of ordinary geometrical percolation theory, which provides insights into these kinds of complex kinetic processes.
An idealized model of percolation is that of completely permeable objects, whose free overlap as more and more objects are randomly added to a matrix eventually results in a geometrically connected phase. There are important materials science applications of this idealized model. For example, we could imagine a material that develops multiple cracks, which eventually percolate geometrically. Since cracks can interpenetrate, the percolation model of randomly overlapping objects is physically appropriate. A second example is the random growth of a microstructure such that an isolated phase becomes geometrically continuous or a continuous phase becomes geometrically isolated. Particular realizations of this random growth include the disconnection of the pore phase in sintering ceramic powders and hydrating cement-based materials [15,17], and the liquid-to-solid transition in sol-gel materials and cement-based materials [10,15]. The gradual build-up of a connected phase via overlap of permeable particles, although idealized, is much like these examples of chemical growth in real materials.
Ideas from percolation theory are commonly applied to the properties of suspensions and composites of impenetrable particles, where a "percolation threshold" is identified with the asymptotic variation of some material property P near a characteristic concentration φ*,
where φ is the volume fraction of the suspended "particles" and δ is a "critical exponent" describing the often rapid variation of P near the threshold concentration φ*. Although this phenomenological approach to describing the properties P of complex random materials is often successful in summarizing experimental observations, the identification of the "percolation threshold" φ* , obtained by fitting experimental data to Eq. (1), with the geometrical threshold pc should be made with caution. Even in the well-understood case of the conductivity of a suspension of particles [18,19], there is only a simple relation between the apparent percolation threshold φ* and pc in the limit where the suspended particles have conductivities extremely different from the suspending medium. Otherwise, when the ratio of the particle and medium conductivities is not so large, the experimental estimate φ* for the conductivity percolation threshold can differ from the geometrical quantity pc. The use of eq. (1) for other properties, where theory is more limited, is evidently even more suspect, but the practical utility of this approach is undeniable. These concentration thresholds are often reported in the physical literature and the estimation of these parameters is a matter of practical interest.
Eq. (1) is often used to successfully describe the electrical and thermal conductivity [6,20,21,22], dielectric constant [23], and shear modulus [5] of composites, the permeability of porous media [24], and transport properties, like the viscosity, of fluid suspensions of rigid particles [25, 26,27]. It is known that φ* varies significantly with particle asymmetry [25,26] and qualitatively similar variations of φ* with shape are found for these various transport properties. This phenomenology suggests that the dependence of pc on particle shape should give some insight into observed variations of φ* . This possibility remains to be checked through more quantitative comparisons between φ* and pc obtained from numerical calculations where the mixture geometry is precisely specified.
In the present paper, we compute the geometrical percolation threshold, pc, in a model two-phase material in which objects are randomly placed without regard to overlap. The value of pc is defined by a transition in the connectivity of the randomly-placed objects from a disconnected to a connected state. In general, pc is much easier to compute than φ*, since determination of the quantities in eq. (1) requires a full solution of the appropriate hydrodynamic equations (Laplace, Navier-Stokes, etc.) for a very complicated geometry and general boundary conditions.
There have been many previous efforts to obtain the variation of pc with particle asymmetry [28,29,30,31,32,33,34,35,36,37,38]. Significant progress especially has been made in two dimensions where a fairly general understanding of pc in terms of the leading order virial coefficient for the electrical conductivity has been obtained for overlapping elliptical particles and particles of more general shape [29]. Results in 3-D are more fragmentary. The present paper is intended to fill this gap by performing numerical computations of pc over a wide range of aspect ratios, from the extreme oblate limit to the extreme prolate limit.
To obtain further insight into the shape dependence of pc for non-spherical particles we follow the previous successful 2-D approach [29] and theoretical arguments by Balberg [28] and others that attempt to relate pc to other more analytically and numerically tractable measures of object shape. Balberg [38], for example, has derived rough bounds of pc for different objects in terms of the "excluded volume" between different objects. From this work and the well-known contribution of Scher and Zallen [39], which phenomenologically relates the on-lattice and off-lattice site percolation thresholds of non- overlapping spherical particles, we seek an "invariant" ratio of pc and other particle properties of the ellipsoid that universally summarizes the shape dependence of pc for ellipsoids and ultimately for particles having more general shapes. We approach this goal by calculating explicitly a large range of functionals of particle shape for ellipsoids (surface area, mean radius of curvature, radius of gyration, electrostatic capacity, excluded volume, and intrinsic conductivity) and forming ratios with pc. In Section 2 we begin our investigation with a review of important functionals of particle shape and explicit analytic results for ellipsoidal particles that are scattered throughout the mathematical and physical literature. Section 3 develops a rough perturbative estimate of the percolation threshold φ* for the conductivity and reviews past efforts to describe the shape dependence of pc through shape functionals. We proceed to the numerical determination of pc for ellipsoids and a brief description of the algorithm used in Section 4. Numerical results for pc are given separately in Section 5 and fitted by a Pade-type approximant. We finally return to a discussion of these numerical results in terms of shape functionals.