880 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 40, NO. 4, APRIL 2002 How Strict are Theoretical Bounds for Dielectric Properties of Mixtures? Ari H. Sihvola, Senior Member, IEEE Abstract—This report discusses bounds for mixing rules. Sev- eral theoretical bounds and limits for the effective permittivity of heterogeneous media have been derived. However, it may happen that the bounds are trespassed in the real-life applications. Exam- ples of such behavior are given for snow, random media simula- tions, lossy heterogeneous materials, and magnetic and magneto- electric materials. Reasons for the bound-violation phenomenon are discussed. Index Terms—Bounds, effective permittivity, limits, mixing rules, mixtures, percolation. I. INTRODUCTION T HE EFFECTIVE permittivity of a mixture, a sample of heterogeneous material, is a problem in classical mathe- matical physics, in particular electromagnetics. Inhomogeneous media can be deterministic or random, but very often a need exists to characterize the material parameters in an averaged, homogenized way. Especially in the case of dealing with nat- ural geophysical materials, or in larger-scale engineering appli- cations, where it is not very important to be aware of all small details of the object, the number of degrees of freedom can be re- duced drastically. In the extreme case, the whole dielectric com- plicatedness of the material is shrunk to one single number, the (complex) permittivity. An extensive literature exists for the problem of calculating the effective macroscopic permittivity for a given heterogeneous material sample as a function of its structure and the geomet- rical and material characteristics of its constituent components [1]. Because of the randomness in the structure of such media, several different ways to calculate the effective parameters have been proposed and can be successfully applied for various types of media. No single mixing rule can be claimed to be universal. Major players on this arena of effective characterization of mix- tures have been the Maxwell Garnett mixing rule [2] (1) and the Bruggeman mixing rule [3] (2) for the effective permittivity of a mixture where spherical inclusions with permittivity occupy a volume fraction in a host material . Manuscript received July 2, 2001; revised December 8, 2001. The author is with the Electromagnetics and Acoustics Laboratory, Swiss Federal Institute of Technology, DE-EPFL, CH-1015 Lausanne, Switzerland. He is also with the Electromagnetics Laboratory, Helsinki University of Tech- nology, FIN-02015 HUT, Finland (e-mail: ari.sihvola@hut.fi). Publisher Item Identifier S 0196-2892(02)04593-X. These are special cases of a general family of mixing rules defined by the following formula [1]: (3) This formula contains a dimensionless parameter . For dif- ferent choices of , the previous mixing rules are recovered: 0 gives the Maxwell Garnett rule (1), 2 gives the Bruggeman formula (2), and 3 gives the coherent potential approximation [4]. In remote sensing studies, the Bruggeman formula is perhaps better known with the name Polder–van Santen mixing formula [5]–[7]. But there are many more mixing formulas [1], [8]. For ex- ample, if the inclusions are not spherical, their average polar- izabilities are higher and the mixing rules have to be modified accordingly. Also, many semi-empirical and heuristic formulas exist that are in use in dielectric modeling studies. Which of the formulas is closest to reality? This is a difficult question to answer. One way to try to find a winner in the com- petition is to solve the electromagnetic problem numerically. In [9] and [10], numerical simulations have been used to calculate the effective permittivity of two- and three-dimensional mix- tures with the result that when clustering effects are allowed, the Bruggeman prediction is closer to the simulations, whereas if the inclusions are all separate spheres, the results are better in agreement with the Maxwell Garnett model. But another approach can be taken to deal with the various mixing rules that are used and create order in them. This is to try to find upper and lower bounds for of a mixture sample, when we know the permittivities and volume fractions of the components. It is the intention of this paper to study the va- lidity of such bounds and limitations and to advise the user of mixing rules against a blind respect for theoretical bounds. The discussion is restricted to mixtures with two components, both of which are dielectrically isotropic. Note, however, that the mi- croscopic texture of the mixture need not necessary be spherical. Cubes or randomly oriented elongated inclusions fall as well within the treatment of this paper. II. BOUNDS FOR THE EFFECTIVE PERMITTIVITY OF A MIXTURE Bounds for the effective permittivity are often formulas like any mixing rule: functions of the component permittivities, the volume-fractional constitution but not the inner structure. To begin with, a very intuitive limitation for is given by the “absolute bounds” that one would expect from a mixture: no matter what the internal geometry of the mixture is, the effective permittivity is limited from above and below by the component permittivities: (4) 0196-2892/02$17.00 © 2002 IEEE