【文章內(nèi)容簡(jiǎn)介】 on effects. ?Taylor(1965) put Polder and Van Santen’s work on the sound basis of generalized boundary conditions obtaining a theory valid for plex ?. So, what’s the problem? They prove that ellipsoidal particles of high aspect ratio have a much stronger effect on the material’s properties than an equivalent volume of spheres. ? For this very reason they have a limited range of validity: f 21. ? They can contribute to misleading conclusions: “There is no doubt that, given a choice of conductivity and shape, we can produce a loss of any magnitude at any frequency with as small a quantity of conducting medium as we please...” Sillars And, for all the labor involved in these formulations, none of them meet all three of Fricke’s requirements. ? The effective coating models meet requirement number 2, giving us fitting parameters, but they have no guidelines on the range of validity for shape and volume fraction. ? The mean field models meet requirement number 1, correctly accounting for shape, but that leaves no fitting parameters. Plus, their limited range of validity is a drawback. ? The classic dilemma between rigor and practicality… ? If this is so, why are these theories so prevalent today? The promise of something for (almost) nothing ?Sillars quote… A small amount of slender particles (slivers) can have a large effect on the material properties. ? i ? h F o r f s m a l l a n d ? i ? h (= 1 ) ? eff ~ 1 +3 f ? i ? h F o r f s m a l l a n d ? i ? h (= 1 ) ? eff ~ 1 + f ? i /3 The ideal material: a useful susceptibility from a vanishingly small addition of slivers ?This is misleading because in the case of conductors ?i can be virtually infinite, s/w?01, so that f can be arbitrarily small. ?But if we try to apply the same reasoning to obtaining magic susceptibility from noninfinite mi we find the limitation to f 21 extremely restrictive. ?Take Nickel, with mi at microwave frequencies of the order of 30. At f =2%, the maximum effective permeability that high aspect ratio slivers can give is meff =. ?And even f =2% is pushing the limit of validity... A random mixture of any kind of particle has a finite probability of generating chains and clusters ?Chains of spheres can behave like high aspect ratio objects (Percolation theory Lagarkov et al 1992) ?Under dynamic mixing, the lowest energy configuration of slivers may be spherical clusters (Doyle and Jacobs 1992 ) And so the discussion of the most useful artificial material reaches a surprising conclusion ?Even though the morphology of the individual constituent is a crucial factor, it is the internal morphology of the mixture that controls the material’s final property. ?And this transition from individual to collective must be a function of the particle shape and f. 0 0 . 5 11101001 103RMiP r o la t eiO b l a t eiCMifi ?Fricke’s model recognized this: At low f, the particles contribute their full enhancement, as in the RM limit. But at high f, they switch over to the CM limit. The collective behavior has its own characterizable morphology ?Isolated slivers resemble an early percolating system ?Clustered slivers resemble a late percolating system RM CM So we must change our focus from the morphology of the particle to the morphology of the mixture ?Which is unknown… The first approach to guess is Percolation Theory, regularized to dielectric particles (Sihvola et al (1994), McLachlan et al (1992)) ?Merrill(1996) following Lagarkov, where pc is the percolation threshold. ? ? ? ?he f fhi hihe f fhe f f he f f f ???????? ???????????? ??? 22222 )( xahxb ???212123)1(1,)/1(11,/hpphppbhpphahcccccih ????????????ccppp ???ie f f x???The second alternative is the spherical cluster theory of Doyle and Jacobs (1990) generalized to dielectrics (Diaz et al 1998) ?Let p be the volume fraction of particles and pc be the percolation limit. Assume that of the p particles, f are clustered into percolated spheres and (1f ) are isolated. ?Then the average polarizability is ?And the effective permittivity is ?Similarly, for nonspherical particles, we start with the b for metal particles derived in DJ, JAP 71(8),3926, 1992, and proceed in the same way. ? ?? ?? ? 22922)1( hichiihhihihippff