# A new mixing rule based on a spectral function approach  Alternative to mixing rules, properties of composites can be considered in terms of the Bergman-Milton (BM) spectral function theory (Bergman, 1978; Stroud et al., 1986; Bergman & Stroud, 1992; Milton, 2001). A spectral function is an unambiguous and universal characteristic of a composite (Bergman & Stroud, 1992). It describes the statistical distribution of inclusions with respect to their form factors. The spectral function can take into account variations of the effective form factor due to interactions within the mixture, as well as the spread of inclusion form factors. This is an important and favorable feature of the BM theory, as opposed to the existing mixing rules and effective medium theory. The effective normalized susceptibility of the composite can be found as

where B(n)is the spectral function. The sums rules for a spectral function (Bergman, 1978; Fuchs, 1978),

jB(n)dn = 1 and jnB(n)dn = —— , (10)

о о D

provide an agreement with the LLL theory. The parameter D is the dimensionality of the composite, e. g., D=2 for the case of infinite cylinders, and D=3 for arbitrary-shaped inclusions of a bulk isotropic composite.

All the known mixing rules are particular cases of the general BM theory with their own spectral functions. Examples of spectral functions for a few different mixing theories are presented in Fig. 4. For convenience of plotting, the functions nB(n) are presented instead of just B(n). The calculations are made for the mean form factor of n=1/3, and the volume concentration of inclusions of p=0.25. Each spectral function "peak", or continuous region, corresponds to some frequency dispersion range.

The spectral function of the MG theory is the S —function, concentrated at the average form factor n, which is around 1/3,   The spectral function for the effective medium BSR is otherwise

In (12), n1 and n2 are the range of the possible form factors of inclusions, and D is the composite dimensionality. The BSR spectral function is a semi-circle in terms of nB(n). It can be seen that the BAR has an asymmetric spectral function nB(n) with a higher spectral density at a lower form factors n. Sheng’s cluster theory (Sheng, 1980) results in a few separate regions of the nB(n) function. However, there is no physically meaningful explanation, why the distribution with respect to form factors is not continuous in the cluster theory.

Though the BM spectral function approach is general for taking into account physically existing spread of inclusion parameters, it is not widely applied to analyze actual experimental data. The main reason for this is that obtaining the spectral function is not a straightforward procedure. Equation (9) is the integral relation for the intrinsic and effective parameters, and a simple algebraic representation is not always possible.

One of the most practically useful approaches that apply the notion of the BM spectral function is the Ghosh-Fuchs formulation (Ghosh & Fuchs, 1988),

B (n )= Iе ( – n1 )1-A (n2 – n)B / for n1 < n < n2. (13)

[ 0, otherwise In this theory, the spectral function is taken the same as in the BSR, but with five fitting parameters: A, B, C, n2, and n2. C, n2, and n2 determine the amplitude, position, and the "peak" width of the spectral function. As a result, the width of the spectral function may vary over some range to take into account the actual spread of the inclusion form factors. The critical exponents A and B are introduced to get agreement with percolation theory. If A=B=1/2 in (13), and C = D/(47t), then the spectral function will be the same as in the BSR (12).

The formulation (13), in contrast to the other mixing rules, agrees very well with experiments for both permittivity and permeability of composites containing ferromagnetic inclusions (Rozanov et al., 2009). However, (13) is not convenient for practical use because of cumbersome integral representation for the spectral function, and the necessity of using many (five) fitting parameters. Hence, it is appealing to have a simplified formalism dealing with algebraic operations.  The first step in the development of a simplified model is to set the critical exponents A=B=1/2 in the spectral function (13). This assumption is consistent with the BSR, but is different from the classical percolation theory. The reason is that the classical percolation behavior of material parameters is observed only in soot-filled (carbon black) polymers. As for the other materials, it is difficult to make them with concentration close to percolation, and even if it is possible, frequency dependence of permittivity may differ significantly from predicted by the percolation theory. One of the reasons may be imperfect contacts between particles in conducting clusters, since cluster conductivity depends mainly on these contacts, but not geometry of clusters, as the percolation theory assumes. Hence, it is reasonable to exclude the critical indices different from 1/2 from further consideration. This assumption would allow for further representing the formulation (13) in a simple algebraic form, analogous to the BSR. Also, the scaling coefficient is set C=1, as follows from the sums rule (10). It is also important that two limits – the LLL at high frequencies

and Odelevsky’s static case will be satisfied,

a>^ 0, a » 1: в = 1 + P —P^~ . (15)

n pc – p

Since the spectral function chosen is a particular case of the BSR formulation, which is basically a quadratic equation, as follows from (4), the effective medium solution can be written as

в = Q(R + a±j(a-S)2 – T), (16)

where Q, R,S, and T are unknown coefficients, but they can be found uniquely using the limits (14) and (15). These coefficients depend on the physical parameters: composite dimensionality, inclusion concentration, average form factor, and the percolation threshold. In the BSR and in (16), the percolation threshold is determined by switching the sign (+) in the solution of the quadratic equation for an effective parameter. This is a consequence of the material passivity, i. e., energy can dissipate, but not generate in the material.

Then the resultant equation corresponding to the new mixing rule can be written as

(1 – P)Рев2 – D(pa-^)_P(Pc + nfi)-npcfi~ = 0. (17)

A consequence of this new formulation is that only two fitting parameters (n = n and pc ) are used to approximate the spectral function, as opposed to five fitting parameters in the formulation (13). These fitting parameters can be found from the concentration dependence
of permittivity of the composite. The limiting case for this formulation coincides with the LLL mixing rule, and provides a unique equation for the effective material constant (either magnetic or dielectric susceptibility) as a function of inclusion concentration, percolation threshold, and dimensionality.

The correctness of (16) can be checked for a few particular cases. If concentration of inclusions p=1, then (16) leads to the result = a. If the average form factor is taken as n = 1/3, and the percolation threhold is not taken into account, i. e., allowed to be pc=1, then (16) converts to the MG formulation (3). If n = 1/3, but the percolation threshold is pc=1/3, then (16) transforms to (4), which is the Bruggeman effective medium theory, or the BSR.

The proposed above formulation is valid for only crumb-like inclusions in the composite, since it should satisfy the inequalities

1/4 < pc < 1 and .y/4 / 3 -1« 0.154 < n < 1 /3. (18)  The results of applying the above simplified algebraic formulation to reconstruct the experimentally obtained permeability of a polymer-based CIP composite with different volume concentrations of CIP are shown in Fig. 5 (Rozanov et al., 2009). As is seen from Fig. 5, the reconstruction of experimental results well achieved over the frequency range from 10 MHz to 3 GHz. The CIP inclusions had arbitrary crumb-like shape with a mean size of inclusions around 60 pm. Fitting according to the proposed model is provided at each frequency point.

(a) (b)

Fig. 5. Experimental (heavy grey) and modeled (thin black) curves for (a) real part of the permeability p’ and (b) imaginary part of permeability p11 of polymer-based CIP composite with different volume concentrations of CIP (Rozanov et al, 2009)

To reconstruct the experimental data, it is important to retrieve the intrinsic permeability of inclusions. The spectral function as the distribution of inclusions with respect to varying form factor is presented in Fig. 6 (a). As the volume concentration of inclusions increases, the spectral function broadens, and this is evidence of strong cooperative phenomena, such as cluster formation. On the other hand, the spectral function has a significant width even at
the lowest volume fraction under study (15%), which means that the spread in the shapes of powder particles significantly impacts the measured permeability. Fig. 6. Restoration of (a) spectral functions for different concentrations of CIP inclusions in the mixture (Rozanov et al, 2009), and (b) intrinsic perbeability of CIP inclusions

Fig. 6 (b) shows the frequency dependence of the intrinsic permeability obtained from different concentrations. It is seen that the intrinsic permeability obtained from different concentrations is almost identical. The reconstructed static permeability value for inclusions is Astatic ~ 30, while the Odelevsky formula results in about /J. static ~ 17. An additional check for the obtained result is made based on Snoek’s constant of the inclusions (Snoek, 1948). This value is known to depend on the composition only, basically, on the magnetic saturation 4nMs, according to Snoek’s law (Snoek, 1948; Liu, Y. et al., 2006),

(/W – l)-fres = 3 • Y • 4kMs, (19)

where the gyromagnetic ratio у ~ 2.8 GHz/kOe for iron and the majority of ferromagnetics, and fres is the resonance frequency. From the microwave performance, Snoek’s constant is estimated as the product of the static permeability and the ferromagnetic resonance frequency (5 = lus£a£ic • fres). At the ferromagnetic resonance frequency, the real part of permeability is unity, and the imaginary goes through zero. As is seen in Fig. 6(b), it is approximately 1.3 GHz, so Snoek’s constant is about S ~ 39 GHz, which is in a good agreement with the reference value for iron (4nMs = 21.5 kG=2.15 T), so S ~ 40 GHz (Liu, Y. et al., 2006). This confirms that the static permeability of inclusions is predicted correctly, while Odelevsky’s formula underestimates this value.