18 June 2012 Bruggeman formalism versus "Bruggeman formalism": particulate composite materials comprising oriented ellipsoidal particles
Author Affiliations +
J. of Nanophotonics, 6(1), 069501 (2012). doi:10.1117/1.JNP.6.069501
Abstract
Two different formalisms for the homogenization of composite materials containing oriented ellipsoidal particles of isotropic dielectric materials are being named after Bruggeman. Numerical studies reveal clear differences between the two formalisms which may be exacerbated: (i) if the component particles become more aspherical, (ii) at mid-range values of the volume fractions, and (iii) if the homogenized component material is dissipative. Only the correct Bruggeman formalism uses the correct polarizability density dyadics of the component particles in the homogenized composite material (HCM) and is directly derivable from the frequency-domain Maxwell postulates specialized for the HCM.
Mackay and Lakhtakia: Bruggeman formalism versus “Bruggeman formalism”: particulate composite materials comprising oriented ellipsoidal particles

The Bruggeman formalism provides a well-established technique for estimating the effective constitutive parameters of homogenized composite materials (HCMs).12.3 The scope of its applicability is not restricted to dilute composite materials and it is easy to implement numerically, both of which contribute to its enduring popularity.

The Bruggeman formalism was originally devised for isotropic dielectric HCMs, comprising two (or more) isotropic dielectric component materials distributed randomly as electrically small spherical particles.4 Generalizations of the Bruggeman formalism which accommodate anisotropic and bianisotropic HCMs have been developed.5 A rigorous basis for the Bruggeman formalism—for isotropic dielectric,6 anisotropic dielectric,7,8 and bianisotropic9,10 HCMs—is provided by the strong-property-fluctuation theory, whose lowest-order formulation is the Bruggeman formalism. (In Ref. 7, the formula of Polder and van Santen, for the effective relative permittivity of an isotropic dielectric HCM, yields the same results as the formula of Bruggeman.4)

Our focus in this letter is on HCMs arising from two isotropic dielectric component materials, labeled a and b. Their relative permittivities are ϵa and ϵb, while their volume fractions are fa and fb1fa. Both component materials are assumed to be randomly distributed as electrically small ellipsoidal particles. For simplicity, all component particles have the same shape and orientation. The surface of each ellipsoid, relative to its centroid, may be represented by the vector

(1)

r̲e(θ,ϕ)=ηU̲̲r̲^(θ,ϕ),
with r̲^ being the radial unit vector from the ellipsoid’s centroid, specified by the spherical polar coordinates θ and ϕ. The linear dimensions of each ellipsoid, as determined by the parameter η, are assumed to be small relative to the electromagnetic wavelength(s). Let us choose our coordinate system to be such that the Cartesian axes are aligned with the principal axes of the ellipsoids. Then the ellipsoidal shape is captured by the dyadic

(2)

U̲̲=Uxx̲^x̲^+Uyy̲^y̲^+Uzz̲^z̲^,
wherein the shape parameters Ux,y,z>0 and {x̲^,y̲^,z̲^} are unit vectors aligned with the Cartesian axes.

The ellipsoidal shape of the component particles results in the corresponding HCM being an orthorhombic biaxial dielectric material. That is, the Bruggeman estimate of the HCM relative permittivity dyadic has the form

(3)

ϵ̲̲Br1=ϵxBr1x̲^x̲^+ϵyBr1y̲^y̲^+ϵzBr1z̲^z̲^.
The relative permittivity parameters ϵx,y,zBr1 are given implicitly by the three coupled equations11

(4)

ϵaϵBr11+D(ϵaϵBr1)fa+ϵbϵBr11+D(ϵbϵBr1)fb=0,({x,y,z}).
Herein D are components of the depolarization dyadic

(5)

D̲̲=Dxx̲^x̲^+Dyy̲^y̲^+Dzz̲^z̲^,
specified by the double integrals5

(6)

Dx=14π02πdϕ0πdθsin3θcos2ϕUx2ρDy=14π02πdϕ0πdθsin3θsin2ϕUy2ρDz=14π02πdϕ0πdθsinθcos2θUz2ρ},
which involve the scalar parameter

(7)

ρ=sin2θcos2ϕUx2ϵxBr1+sin2θsin2ϕUy2ϵyBr1+cos2θUz2ϵzBr1.
The coupled nature of the three equations in Eq. (4) means that numerical methods are generally needed to extract the relative permittivity parameters ϵx,y,zBr1 from them.

An alternative formalism for the homogenization of the same composite material as in the foregoing paragraph is also referred to as the Bruggeman formalism.1213.14.15 Let us write the estimate of the HCM’s relative permittivity dyadic provided by this alternative formalism as

(8)

ϵ̲̲Br2=ϵxBr2x̲^x̲^+ϵyBr2y̲^y̲^+ϵzBr2z̲^z̲^.
The relative permittivity parameters ϵx,y,zBr2 are given by the three equations

(9)

ϵaϵBr2ϵBr2+L(ϵaϵBr2)fa+ϵbϵBr2ϵBr2+L(ϵbϵBr2)fb=0,({x,y,z}),
wherein the depolarization factors16

(10)

L=UxUyUz20ds1(s+U2)(s+Ux2)(s+Uy2)(s+Uz2),({x,y,z})
are components of the depolarization dyadic

(11)

L̲̲=Lxx̲^x̲^+Lyy̲^y̲^+Lzz̲^z̲^.
Each of the three equations in Eq. (9) is a quadratic equation in ϵBr2 whose solution may be explicitly expressed as

(12)

ϵBr2=β±β24αγ2α,({x,y,z}),
with α=L1, β=ϵa(faL)+ϵb(fbL), and γ=Lϵaϵb. The sign of the square root term in the solution seen in Eq. (12) may be determined by appealing to the anisotropic dielectric generalization of the Hashin–Shtrikman bounds,17 for example.

Let us illustrate the differences between the estimates ϵ̲̲Br1 and ϵ̲̲Br2 by means of some representative numerical results. The two estimates are identical for the limiting case represented by Ux=Uy=Uz=1 (i.e., for isotropic dielectric HCMs), but differences emerge as the asphericity of the component particles intensifies. Suppose that the shape parameters describing the component ellipsoids have the form Ux=1, Uy=1+(Δ/3), and Uz=1+2Δ. Thus, the asphericity of the ellipsoids is governed by the scalar parameter Δ. We begin with the nondissipative scenario wherein ϵa{0.5,1.5} and ϵb=12. Also, we fix fa=0.5. Plots of the relative permittivity parameters ϵx,y,zBr1,Br2 versus the asphericity parameter Δ are presented in Fig. 1. The difference between ϵxBr1 and ϵxBr2 grows steadily as Δ increases, reaches a maximum for 1<Δ<2, and then slowly shrinks as Δ increases beyond 2. The difference between ϵzBr1 and ϵzBr2 follows a similar pattern. However, in the case of ϵyBr1 and ϵyBr2, the difference increases uniformly as Δ increases. The differences between ϵx,y,zBr1 and ϵx,y,zBr2 are generally greater for ϵa=0.5 than for ϵa=1.5. In the former case the maximum difference is approximately 15%, whereas in the latter case it is approximately 5%.

Fig. 1

The estimates ϵxBr1,2 (blue, dashed curves), ϵyBr1,2 (green, solid curves), and ϵzBr1,2 (red, broken dashed curves) plotted versus the asphericity parameter Δ(0,4.5). The ϵx,y,zBr1 estimates are represented by thick curves and the ϵx,y,zBr2 estimates are represented by thin curves. The ellipsoidal shapes of the component material particles are described by shape parameters Ux=1, Uy=1+(Δ/3), and Uz=1+2Δ. The relative permittivities of the component materials are ϵa{0.5,1.5} and ϵb=12; and the volume fraction fa=0.5.

JNP_6_1_069501_f001.png

We turn now to the effect of volume fraction. The calculations of Fig. 1 are repeated for Fig. 2 except that here the relative permittivity parameters ϵx,y,zBr1,Br2 are plotted versus the volume fraction fa, while the asphericity parameter is fixed at Δ=4.5. The differences between the estimates of the two formalisms are clearly greatest at mid-range values of fa, and they are generally greater for ϵa=0.5 than for ϵa=1.5.

Fig. 2

As Fig. 1 except that Δ=4.5 and the estimates ϵx,y,zBr1,2 are plotted versus the volume fraction fa(0,1).

JNP_6_1_069501_f002.png

Lastly, the effects of dissipation are considered. We repeated the calculations of Fig. 1 but with Δ=4.5 and ϵa{0.5+iδ,1.5+iδ}. Here δ>0 governs the degree of dissipation exhibited by component material a. The real and imaginary parts of the relative permittivity parameters ϵx,y,zBr1,Br2 are plotted versus the dissipation parameter δ in Fig. 3. The differences between the real parts of the estimates ϵx,y,zBr1 and ϵx,y,zBr2 are largest when component material a is nondissipative and they decrease uniformly as δ increases. In contrast, the differences between the imaginary parts of the estimates ϵx,y,zBr1 and ϵx,y,zBr2 increase as δ increases. These differences in the imaginary parts generally reach a maximum for mid-range values of δ and thereafter decrease as δ increases. For both the real and imaginary parts of the estimates ϵx,y,zBr1 and ϵx,y,zBr2, generally larger differences arise for ϵa=0.5+iδ than for ϵa=1.5+iδ.

Fig. 3

As Fig. 1 except that Δ=4.5, ϵa{0.5+iδ,1.5+iδ}, and the real and imaginary parts of the estimates ϵx,y,zBr1,2 are plotted versus the dissipation parameter δ(0,2).

JNP_6_1_069501_f003.png

Thus, there are significant differences between the estimates ϵ̲̲Br1 and ϵ̲̲Br2 when ellipsoidal component particles are considered. These differences may be exacerbated: (i) if the component particles become more aspherical, (ii) at mid-range values of the volume fractions of the component materials, and (iii) if the HCM is dissipative. The differences between the two estimates may be further exacerbated if one of the component materials has a positive-valued relative permittivity which is less than unity (or a relative permittivity whose real part is positive-valued and less than unity). Relative permittivities in this range are associated with novel materials possessing engineered nanostructures; these artificial materials have been the subject of intense research lately.1819.20 (The parameter regime wherein one of the component materials has a positive-valued relative permittivity while the other has a negative-valued relative permittivity (or likewise for the real parts of the relative permittivities in the case of dissipative HCMs) is avoided here because the Bruggeman formalism can deliver estimates in this regime which are not physically plausible.21)

The differences between the two formalisms stem from the differences between the depolarization dyadics D̲̲ and L̲̲. The Bruggeman formalism conceptually employs an average-polarizability-density approach:22 Suppose the composite material has been homogenized into an HCM. Into this HCM, let additional aligned ellipsoidal particles of the two component materials be dispersed in such a way as to maintain the overall volume fractions of a and b. This dispersal cannot change the effective properties of the HCM—because the volume-fraction-average of the polarization density dyadics of two particles, one of each component material, embedded in the HCM with relative permittivity ϵ̲̲Br1 is exactly zero. In computing the polarizability density dyadic of each particle, it must therefore be assumed that the particle is surrounded by the HCM. This fact legitimizes the use of D̲̲, which contains the anisotropic HCM’s effective constitutive properties via the scalar ρ of Eq. (7). Indeed, Eq. (4) is directly derivable from the frequency-domain Maxwell postulates specialized for the actual HCM.67.8.9.10

On the other hand, use of L̲̲ to compute the polarizability density dyadic of a particle implies that it is surrounded by an isotropic HCM, which is clearly incorrect. The alternative formalism that delivers ϵ̲̲Br2 is a heuristic extrapolation of the Bruggeman formalism for isotropic dielectric HCMs,12,13 and it lacks the rigorous basis that underpins the estimate of ϵ̲̲Br1. The dispersal of aligned ellipsoidal particles of the two component materials without disturbing the overall volume fractions into the HCM with relative permittivity ϵ̲̲Br2 would change the effective properties of the HCM—because the volume-fraction-average of the polarization density dyadics of two particles, one of each component material, embedded in the HCM with relative permittivity ϵ̲̲Br2 would not be zero.

We have thus delineated the differences between the two formalisms and identified one of them as the correct Bruggeman formalism. We hope that this exposition will prevent confusion between the two formalisms from perpetuating.

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Tom G. Mackay, Akhlesh Lakhtakia, "Bruggeman formalism versus "Bruggeman formalism": particulate composite materials comprising oriented ellipsoidal particles," Journal of Nanophotonics 6(1), 069501 (18 June 2012). http://dx.doi.org/10.1117/1.JNP.6.069501
Submission: Received ; Accepted
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