Derivation of Lichtenecker's Logarithmic Mixture ... - IEEE Xplore

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 58, NO. 3, MARCH 2010

545

Derivation of Lichtenecker’s Logarithmic Mixture Formula From Maxwell’s Equations Ray Simpkin, Member, IEEE

Abstract—Lichtenecker’s logarithmic mixture formula for dielectrics has proven to be a useful practical formulation for determining the effective permittivity of homogenized dielectric mixtures. However, the formula itself has long been regarded as semiempirical in nature without firm theoretical justification. This paper shows that Lichtenecker’s formula can be derived by applying Maxwell’s equations and the principle of charge conservation to a mixture for which the spatial distribution of shapes and orientations of the components is randomly distributed. Furthermore, the relationship between the Lichtenecker formula and the well-known mixture formulas due to Maxwell-Garnett is demonstrated. Similarly, it is shown that the symmetric mixture formula of Bruggeman can also be obtained from the Lichtenecker formula. Index Terms—Dielectrics, electromagnetic theory, mixture formulas.

I. INTRODUCTION ICHTENECKER’S logarithmic mixture formula for dielectrics [1], [2] has proven to be a useful practical formulation for determining the effective permittivity of homogenized dielectric mixtures. The formula has been applied to a wide range of mixtures including biological materials such as human blood [3], geophysical applications involving dry rocks, and minerals [4], and in plastic-based composite materials using titanium dioxide loading [5]. However, the formula itself has long been regarded as semiempirical in nature without firm theoretical justification. The comments of Reynolds and Hough [6] in their 1957 paper were particularly critical of Lichtenecker and Rother’s justification for the logarithmic formulation. Since then, there has been a small body of published research aimed at providing a more robust theoretical basis for Lichtenecker’s formula. Neelakantaswamy et al. [7] have proposed a modification to the Lichtenecker formula aimed at addressing the concerns expressed by Reynolds and Hough in [6]. Zakri et al. [8] derived the formula by firstly considering the general case of a beta distribution for the geometrical shapes of particles embedded in a matrix medium. When considering the special case of a uniform distribution, the analysis given in [8] results in the Lichtenecker formula. Most recently, Goncharenko et al. [9] considered the spectral density function of the Lichtenecker formula and related this to a particular topology. In contrast to the above

L

Manuscript received June 11, 2009; revised November 02, 2009. First published February 17, 2010; current version published March 12, 2010. This work was supported by the Foundation for Research, Science and Technology (FRST), New Zealand under Contract C08X0414. The author is with Industrial Research Limited, Auckland 1140, New Zealand (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2010.2040406

approaches, the aim of this paper is to present a derivation of Lichtenecker’s logarithmic mixture formula starting from the fundamental physics embodied in Maxwell’s equations and the principle of charge conservation. The limitation of the derived formula is discussed in relation to the topology of the embedded dielectric inclusions. In addition, for the case of two-part mixtures, it will be shown that the well-known Maxwell-Garnett formula [10], and its inverse form, plus the symmetric formula due to Bruggeman [11] can be derived from the Lichtenecker logarithmic formula within certain limits imposed on the ratio of the component permittivities and the associated volume fractions. Whilst these other mixture formulas are discussed in relation to Lichtenecker’s formula, the purpose of this paper is not to compare predictions of effective permittivity using different mixture models. The purpose is to provide a more rigorous electromagnetic justification for the Lichtenecker result. II. THEORY Consider the Maxwell curl equation for the magnetic field for time-harmonic dependence of the form (1) where, in (1), the electric field vector is denoted by , the angular frequency by , and the materials’ parameters are defined by electrical conductivity and complex permittivity . Both electrical conductivity and complex permittivity are included in (1) to account for dissipative losses in the medium due to conduction and dielectric relaxation. The right-hand side of (1) can be put into a more compact form as follows: (2) is the electric displacement vector given by where, in (2), with the constitutive parameters represented by the total complex permittivity . The medium is assumed to be linear, isotropic, and homogeneous so that and are scalars. Taking the divergence of both sides of (2) leads to (3) which is a statement of the continuity of free electric charge flowing through a volume. Integrating (3) over the volume gives

0018-9480/$26.00 © 2010 IEEE

Constant

(4)

546


where, in (4), is the total free charge contained within the volume . This can, of course, be equal to 0, but the general case is considered here. Consider now a matrix material defined by the complex permittivity occupying a volume with electric field vector distributed throughout this volume. Consider further a number of different components added to this matrix to form a multicomponent mixture. Let the permittivity of the th component occupying a total volume with electric be defined by . The total volume of the mixture is denoted by field vector , where is the total number of different components in the mixture. For convenience, the component for which is defined as the matrix material with all other values of representing the different types of inclusion embedded within the matrix medium. It will be shown later, however, that the final expression for the effective permittivity of the mixture is independent of which component is designated as the matrix medium. The aim is to determine an expression for the permittivity of the mixture in terms of an equivalent homogeneous medium as a function of the mixture pawith effective permittivity rameters. The approach adopted here is to enforce charge conservation between the heterogeneous mixture and the equivalent homogeneous medium. This is done firstly by equating the free charge in the mixture to that of its effective medium representation. Later on in the derivation, the conservation of bound charge is indirectly enforced in a similar fashion by enforcing total charge conservation (bound plus free charge). Charge conservation is the fundamental principle used in this paper in deriving a dielectric mixture formula. It will be shown that the resulting formula is consistent with the logarithmic mixture law suggested by Lichtenecker. It should be noted that the size of the inclusions is assumed to be much smaller than the wavelength of the electromagnetic field in order for the effective medium model to be used. This is consistent with the notion of a quasielectrostatic representation common to all of the well-known mixture formulas. In the mixture, it is assumed that the dielectric materials are linear and isotropic and are homogeneous within occupied by each of the components. Therethe volume fore, for permittivity , and electric field , the electric displacement vector has the well-known form

the effective permittivity of the mixture lowing:

given by the fol-

(7) The partial derivatives in (7) can be determined from (6) and are given by (8)

Equation (8) follows from (6) since variations in the electric field vectors appear as second-order differential terms of the in the integrands and can, therefore, be neglected in form . Substituting in (7) for the partial derivathe limit as tives using (8) and then dividing by (6) eliminates the integral in. This gives the following equation for the fractional volving change in the effective permittivity in terms of small changes to the permittivity of each component

(9)

Equation (9) can be put into a more useful form by applying the divergence theorem and then enforcing the continuity of the normal component of the electric displacement vector at each dielectric interface. This is described in the following. At each dielectric interface between the th component and the matrix material , the normal component of the electric displacement vector must be continuous at the boundary enclosing the volume in order to satisfy the resurface quired electromagnetic boundary condition. That is, on (10) where, in (10), is a unit normal vector on the boundary surenclosing the volume pointing into the matrix mafaces terial. The divergence theorem is now applied to the volume integral appearing in (9) to give

(5) (11) The conservation of free charge requires that the total free charge in the mixture is equal to that in the effective medium so that, using (4) and (5),

The integrand of the surface integral in (11) can be cast in terms of the electric field in the matrix material using (10), namely,

(6)

(12)

Consider now a small perturbation to the permittivity of the th with the permittivity component such that of all other components held constant. There will be a corresponding perturbation to each of the electric field vectors such and . Applying simthat ilar perturbations to all other components results in a change to

By applying the divergence theorem once more to the field , the surface integral in (12) can be recast back into a volume integral to give (13)

SIMPKIN: DERIVATION OF LICHTENECKER’S LOGARITHMIC MIXTURE FORMULA FROM MAXWELL’S EQUATIONS

Equation (13) enables the volume integral involving the field to be expressed in terms of the permittivity ratio and an integral involving the divergence of the matrix material field . Substituting (13) into (9) gives the following equation for the fractional change in effective permittivity of the mixture:

in terms of deviations about the mean value (17) into (16) then gives

547

. Substituting

(18)

(14)

Note that, in the denominator of (14), the integral now spans the total volume of the mixture, denoted by . From Maxwell’s equation for the divergence of , one has , where is the total charge density (bound plus free charge) in the matrix medium and is the permittivity of , where is the total free space. Similarly, charge density within the th component of the mixture. Using these relations, (14) can be simplified by enforcing the conservation of total charge in the mixture as follows. The total charge (bound plus free) contained in the components of the mixture must be equal to that in the same volume of its effective permittivity representation, i.e.,

If each component of the mixture is considered to be a random distribution of particle shapes and orientations, then the mean value of the charge density perturbation function, denoted by , will be zero for each of the components. This follows since there can be no preferred orientation for the dielectric polarization vector for such a random distribution. This means that each integral in (18) can be set to zero since

(19) Therefore, from the results of (17) and (19), the volume integrals in (14) can be written as

(15) (20a)

which is equivalent to and similarly

(20b) The left-hand side of (15) is the total charge contained within the volume when considered as a macroscopically isotropic and . Therehomogeneous medium with effective permittivity must be a constant and (15) becomes fore, the charge density the following:

(16)

is numerically equal to the mean This expression shows that charge density of the heterogeneous mixture. Consider now the charge density as a function of position within the th component of the mixture. This can be written in the following form: (17) In (17), is a position vector, and the charge density is expressed as the sum of the mean charge density and a . The latter describes the spatial perturbation function distribution of charge densities throughout the th component

It is important to note that the results of (20a) and (20b) are independent of the dielectric properties of the constituents and are proportional to the volume over which the integral is evaluated. Therefore, under the assumption of a random distribution of shapes and orientations for each component of the mixture, identical results to (20a) and (20b) are obtained whatever choice ). Applying of medium is designated as the matrix (index is a constant then (20a) and (20b) to (14), and noting that gives the following result: (21) where, in (21), the factor is the volume fraction of the th component . Integrating both sides of (21) then gives

548


so that

here, the designation of which component is regarded as the matrix medium and which is regarded as the inclusion has no effect on the end result. This is consistent with a mixture formula that is symmetric with respect to its constituents. IV. RELATIONSHIP BETWEEN THE LICHTENECKER LOGARITHMIC MIXTURE FORMULA AND OTHER MIXTURE FORMULAS

leading to (22) Equation (22) is Lichtenecker’s logarithmic mixture law. III. DISCUSSION Equation (22) was derived directly from Maxwell’s equations and by enforcing the conservation of charge between the heterogeneous mixture and its homogeneous effective medium representation. The formula is not restricted to lossless dielectric materials since the constituents were assumed to have complex permittivity values at the outset. The assumption made in obtaining (22) was that of a random distribution of particle shapes and orientations for each component such that the charge density within any component can be replaced by the mean charge density of the mixture. This allows charge fractions to be replaced by volume fractions in (14). The derivation given in Section II did not specify a priori the shapes or orientations of the inclusions. This does not prevent the boundary condition on the electric displacement vector being rigorously enforced at the material interfaces. However, the Lichtenecker formula can only be obtained when the spatial distribution of shapes and orientations of the inclusions is treated as statistically random. The success (or failure) of the Lichtenecker formula in predicting the effective permittivity of dielectric mixtures is most likely allied to how well this condition is satisfied. It is also worth commenting on the symmetry of the Lichtenecker formula. For the familiar case of a two-part mixture obeying (22), the Lichtenecker formula becomes

A. Relationship to the Formulas of Maxwell-Garnett Consider the familiar case of a simple two-part mixture for which the Lichtenecker logarithmic mixture formula is that of (23). It can be shown that the dielectric mixture formula due to Maxwell-Garnett [10] can be obtained from (23) under certain conditions. The analysis is outlined as follows. The variable is defined as follows for permittivities and : (24) which is the well-known Clausius-Mossotti factor [12]. Physically, this factor defines the magnitude of the polarization field (when multiplied by the applied electric field vector) for an electrically small homogeneous sphere of permittivity embedded in a homogeneous medium of permittivity . The form given in (24) requires that the matrix medium is defined with permittivity and is that of the spherical dielectric inclusion. Equation in (24) can be rearranged to give the permittivity ratio terms of , i.e., (25) Substituting (25) into (23) for the permittivity ratio gives (26) Noting that the volume fraction of the inclusion is always less than or equal to unity, the numerator of (26) can be expanded as a binomial series to give

(23) and volume where, in (23), component 2 has permittivity and volume fraction , and component 1 has permittivity . It is well known that the Lichtenecker formula fraction is symmetric with respect to the permittivities of the components for a two-part mixture. That is, interchanging the permitand and putting in (23) leads to the tivities same formula. At first glance, it may seem that the designation in the derivation presented of a specific matrix medium in Section II is inconsistent with the symmetry of the Lichtenecker result (which does not distinguish the roles of matrix and inclusion). However, the same result could have been obtained regardless of which component had been labeled as the matrix medium. Equation (14) could, for example, just as easily . This folhave been cast in terms of integrals involving lows from the arguments given previously in obtaining (20a) and (20b), which are independent of the choice of matrix medium. Therefore, in the derivation of the Lichtenecker formula given

and similarly for the denominator of (26)

Provided that only linear terms in are retained in the above binomial expansions, the following approximation to the Lichtenecker formula is obtained: (27) Substituting for

in (27) using (24) then gives (28)

Equation (28) is the well-known Maxwell-Garnett mixture formula [10] for the case when is regarded as the matrix has been used in (28) to identify medium. The superscript

SIMPKIN: DERIVATION OF LICHTENECKER’S LOGARITHMIC MIXTURE FORMULA FROM MAXWELL’S EQUATIONS

it as such. The above analysis shows that this formula can be regarded as an approximation to Lichtenecker’s logarithmic is small. mixture law when the parameter It is well known that the Maxwell-Garnett formula is not symmetric with respect to the permittivities and . Interchanging the volume fractions and permittivities in (28) leads to the inverted form of the Maxwell-Garnett formula given in the following: (29) In (29), is the volume fraction pertaining to permittivity , has been used to deas it is in (28), and the superscript note the inverted form of the Maxwell-Garnett formula. Equation (29) can also be derived directly from the Lichtenecker formula of (23) using a similar binomial expansion procedure as outlined above for the Maxwell-Garnett formulation. Thus, by interchanging the permittivities in (24), one can define the Clausius-Mossotti factor as (30)

549

spection of (34) indicates the presence of the following Clauand : sius-Mossotti factors defined by and

(35)

The expression for in (35) determines the strength of the polarization field for an electrically small homogeneous sphere of permittivity embedded in a homogeneous medium of effec. The expression for in (35) is similarly tive permittivity defined, but for an electrically small homogeneous sphere of permittivity embedded in the same effective medium. Rearranging (35) gives and

(36)

The two-part Lichtenecker formula (23) can then be written in terms of the above factors as follows:

which gives

Rearranging (30) gives

(37) (31)

Substituting the expressions from (36) into (37) results in the following form of the Lichtenecker formula:

The Lichtenecker formula in (23) can then be written as follows in terms of (31): (32)

(38) For small absolute values of and , the denominator factors in (38) can be approximated as follows:

Retaining just the first few terms of a binomial expansion of the numerator and denominator in (32) gives

and Therefore, an approximate expression for (38) becomes (39)

(33) which is equivalent to the result of (29), the inverted form of the Maxwell-Garnett formula. The quantity must remain small in absolute magnitude to obtain (33) from (32). B. Relationship to the Symmetric Formula of Bruggeman The symmetric mixture formula due to Bruggeman [11] is given as follows for the case of a two-part mixture: (34) In (34), is the volume fraction of component 2 with permittivity . Component 1 has permittivity and is the effective permittivity of the mixture. In common with the two-part Lichtenecker formula of (23), the expression in (34) is symmetric with respect to the permittivities of the components. In-

A binomial expansion of each factor in (39) then gives

and

Substituting the above expressions into (39) and retaining only and gives the linear terms in (40) which is identical to the symmetric Bruggeman formula given in (34). The result in (40) was obtained from the Lichtenecker formula under the assumption that the Clausius-Mossotti factors of (35) are small enough in magnitude for the dependence and to be taken as linear. This is equivalent to considon ering only first-order interactions between each component of the mixture embedded in a homogeneous effective medium.

550


V. CONCLUSIONS It has been shown that Lichtenecker’s logarithmic mixture formula can be derived from Maxwell’s equations when the conservation of charge is applied to the mixture and its components. The formula was obtained by further assuming a random distribution of shapes and orientations for each component enabling the charge density at any position to be replaced by the mean charge density of the mixture. This allows the charge fraction of each component to be replaced by its volume fraction. The random distribution of shapes and orientations for dielectric inclusions results in there being no preferred direction for the polarization vector within each constituent when averaging over the volume of that constituent. This will result in the effective permittivity remaining a scalar quantity, as assumed throughout this paper. It was also shown that both variants of the Maxwell-Garnett mixture formula and the symmetrical Bruggeman formula can be derived directly from the Lichtenecker logarithmic mixture law when restricting the magnitude of the volume fraction and appropriate Clausius-Mossotti factor. This suggests that the Maxwell-Garnett and Bruggeman formulas can be regarded as valid for first-order interactions between the mixture components, but are, in fact, approximations to the Lichtenecker result. Both the Maxwell-Garnett and Bruggeman formulas were originally derived by considering electrically small homogeneous dielectric spheres embedded in a host medium. As such, due to the simplifying assumption of spherical inclusions, the Maxwell-Garnett and symmetric Bruggeman formulas should be regarded as approximations for predicting the effective permittivity of a mixture containing particles that conform to a random distribution of shapes and orientations. The derivation of the Lichtenecker formula given in this paper made no assumptions about the shape or orientation of inclusions, only that their spatial distribution should be random. The derivation of the Lichtenecker logarithmic mixture formula from Maxwell’s equations and the principle of charge conservation suggests that the formula may be of a more fundamental nature than previously regarded.

REFERENCES [1] K. Lichtenecker, “Die Dielektrizitätskonstante natürlicher und künstlicher Mischkörper,” Phys. Zeitschr., vol. XXVII, pp. 115–158, 1926. [2] K. Lichtenecker and K. Rother, “Die Herleitung des logarithmischen Mischungs-gesetzes aus allegemeinen Prinzipien der stationaren Stromung,” Phys. Zeitschr., vol. XXXII, pp. 255–260, 1931. [3] P. S. Neelakantaswamy, K. F. Aspar, A. Rajaratnam, and N. P. Das, “A dielectric model of the human blood,” Biomed. Tech., vol. 28, no. 1–2, pp. 18–22, 1983. [4] Y. Zheng, S. Wang, J. Feng, Z. Ouyang, and X. Li, “Measurement of the complex permittivity of dry rocks and minerals: application of polythene dilution method and Lichtenecker’s mixture formulas,” Geophys. J. Int., vol. 163, pp. 1195–1202, 2005. [5] A. Büchner, , Wissenschaftliche Veröffentlichungen aus den SiemensWerken. Berlin, Germany: Siemens-Werken, 1939, vol. 18, p. 84. [6] J. A. Reynolds and J. M. Hough, “Formulas for dielectric constant mixtures,” Proc. Phys. Soc., vol. LXX, pp. 769–775, 1957. [7] P. S. Neelakantaswamy, R. I. Turkman, and T. K. Sarkar, “Complex permittivity of a dielectric mixture corrected version of Lichtenecker’s logarithmic law of mixing,” Electron. Lett., vol. 21, no. 7, pp. 270–271, Mar. 1985. [8] T. Zakri, J.-P. Laurent, and M. Vauclin, “Theoretical evidence of Lichtenecker’s mixture formulas based on the effective medium theory,” J. Appl. Phys., vol. 31, pp. 1589–1594, 1998. [9] A. V. Goncharenko, V. Z. Lozovski, and E. F. Venger, “Lichtenecker’s equation: Applicability and limitations,” Opt. Commun., vol. 174, pp. 19–32, Jan. 2000. [10] J. C. Maxwell-Garnett, “Colours in metal glasses and metal films,” Trans. Royal Soc., vol. CCIII, pp. 385–420, 1904. [11] D. A. G. Bruggeman, “Calculation of various physical constants of heterogeneous substances, part I: Dielectric constant and conductivity of mixtures of isotropic substances,” Annal. Phys., ser. 5, vol. 24, pp. 636–679, 1935. [12] A. R. von Hippel, Dielectrics and Waves. New York: Wiley, 1954. Ray Simpkin (M’96) was born in St. Albans, U.K., on October 29, 1960. He received the B.Sc. degree in physics (with first-class honors) and Ph.D. degree in electromagnetics from Imperial College, London, U.K., in 1982 and 1996, respectively. From 1983 to 2001, he worked within the U.K. aerospace industry as a Microwave Engineer and Research Manager. In 2001, he became a Research Scientist with Industrial Research Limited (IRL), Auckland, New Zeland. His research interests include antennas, radomes, frequency-selective surfaces, electromagnetic scattering, dielectric materials, microwave imaging for breast cancer detection, and inverse scattering methods.