Effective medium approximations

Short description: Method of approximating the properties of a composite material

In materials science, effective medium approximations (EMA) or effective medium theory (EMT) pertain to analytical or theoretical modeling that describes the macroscopic properties of composite materials. EMAs or EMTs are developed from averaging the multiple values of the constituents that directly make up the composite material. At the constituent level, the values of the materials vary and are inhomogeneous. Precise calculation of the many constituent values is nearly impossible. However, theories have been developed that can produce acceptable approximations which in turn describe useful parameters including the effective permittivity and permeability of the materials as a whole. In this sense, effective medium approximations are descriptions of a medium (composite material) based on the properties and the relative fractions of its components and are derived from calculations,[1][2] and effective medium theory.[3] There are two widely used formulae.[4]

Effective permittivity and permeability are averaged dielectric and magnetic characteristics of a microinhomogeneous medium. They both were derived in quasi-static approximation when the electric field inside a mixture particle may be considered as homogeneous. So, these formulae can not describe the particle size effect. Many attempts were undertaken to improve these formulae.

## Contents

* 1 Applications
* 2 Bruggeman's model
* 2.1 Formulas
* 2.1.1 Circular and spherical inclusions
* 2.1.2 Elliptical and ellipsoidal inclusions
* 2.2 Derivation
* 2.3 Modeling of percolating systems
* 3 Maxwell Garnett equation
* 3.1 Formula
* 3.2 Derivation
* 3.3 Validity
* 4 Generalization of the Maxwell Garnett Equation to describe the nanoparticle size distribution
* 4.1 Generalization to include shape distribution of nanoparticles
* 5 Formula describing size effect
* 6 Effective permeability formula
* 7 Effective medium theory for resistor networks
* 8 See also
* 9 References
* 10 Further reading

## Applications

There are many different effective medium approximations,[5] each of them being more or less accurate in distinct conditions. Nevertheless, they all assume that the macroscopic system is homogeneous and, typical of all mean field theories, they fail to predict the properties of a multiphase medium close to the percolation threshold due to the absence of long-range correlations or critical fluctuations in the theory.

The properties under consideration are usually the conductivity σ or the dielectric constant ε[6] of the medium. These parameters are interchangeable in the formulas in a whole range of models due to the wide applicability of the Laplace equation. The problems that fall outside of this class are mainly in the field of elasticity and hydrodynamics, due to the higher order tensorial character of the effective medium constants.

## Bruggeman's model

For a mixture of two materials with permittivities εm and εd with corresponding volume fractions cm and cd, D.A.G. Bruggeman proposed a formula of the following form:[7]

εeff=Hb+Hb2+8εmεd4, with Hb=(3cd−1)εd+(3cm−1)εm.

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(3)

Here the positive sign before the square root must be altered to a negative sign in some cases in order to get the correct imaginary part of effective complex permittivity which is related with electromagnetic wave attenuation. The formula is symmetric with respect to swapping the 'd' and 'm' roles. This formula is based on the equality

ΔΦ=∬εr(𝐫)En(𝐫)ds−εeff∬E0ds=0,

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(4)

where ΔΦ is the jump of electric displacement flux all over the integration surface, En(𝐫) is the component of microscopic electric field normal to the integration surface, εr(𝐫) is the local relative complex permittivity which takes the value εm inside the picked metal particle, the value εd inside the picked dielectric particle and the value εeff outside the picked particle, E0 is the normal component of the macroscopic electric field. Formula (4) comes out of Maxwell's equality div⁡(εr𝐄)=0. Thus only one picked particle is considered in Bruggeman's approach. The interaction with all the other particles is taken into account only in a mean field approximation described by εeff. Formula (3) gives a reasonable resonant curve for plasmon excitations in metal nanoparticles if their size is 10 nm or smaller. However, it is unable to describe the size dependence for the resonant frequency of plasmon excitations that are observed in experiments [8]

### Formulas

Without any loss of generality, we shall consider the study of the effective conductivity (which can be either dc or ac) for a system made up of spherical multicomponent inclusions with different arbitrary conductivities. Then the Bruggeman formula takes the form:

#### Circular and spherical inclusions

∑iδiσi−σeσi+(n−1)σe=0

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(1)

In a system of Euclidean spatial dimension n that has an arbitrary number of components,[9] the sum is made over all the constituents. δi and σi are respectively the fraction and the conductivity of each component, and σe is the effective conductivity of the medium. (The sum over the δi's is unity.)

#### Elliptical and ellipsoidal inclusions

1nδα+(1−δ)(σm−σe)σm+(n−1)σe=0

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(2)

This is a generalization of Eq. (1) to a biphasic system with ellipsoidal inclusions of conductivity σ into a matrix of conductivity σm.[10] The fraction of inclusions is δ and the system is n dimensional. For randomly oriented inclusions,

α=1n∑j=1nσ−σeσe+Lj(σ−σe)

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(3)

where the Lj's denote the appropriate doublet/triplet of depolarization factors which is governed by the ratios between the axis of the ellipse/ellipsoid. For example: in the case of a circle (L1=1/2, L2=1/2) and in the case of a sphere (L1=1/3, L2=1/3, L3=1/3). (The sum over the Lj 's is unity.)

The most general case to which the Bruggeman approach has been applied involves bianisotropic ellipsoidal inclusions.[11]

### Derivation

The figure illustrates a two-component medium.[9] Consider the cross-hatched volume of conductivity σ1, take it as a sphere of volume V and assume it is embedded in a uniform medium with an effective conductivity σe. If the electric field far from the inclusion is E0‾ then elementary considerations lead to a dipole moment associated with the volume

p‾∝Vσ1−σeσ1+2σeE0‾.

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(4)

This polarization produces a deviation from E0‾. If the average deviation is to vanish, the total polarization summed over the two types of inclusion must vanish. Thus

δ1σ1−σeσ1+2σe+δ2σ2−σeσ2+2σe=0

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(5)

where δ1 and δ2 are respectively the volume fraction of material 1 and 2. This can be easily extended to a system of dimension n that has an arbitrary number of components. All cases can be combined to yield Eq. (1).

Eq. (1) can also be obtained by requiring the deviation in current to vanish.[12] [13] It has been derived here from the assumption that the inclusions are spherical and it can be modified for shapes with other depolarization factors; leading to Eq. (2).

A more general derivation applicable to bianisotropic materials is also available.[11]

### Modeling of percolating systems

The main approximation is that all the domains are located in an equivalent mean field. Unfortunately, it is not the case close to the percolation threshold where the system is governed by the largest cluster of conductors, which is a fractal, and long-range correlations that are totally absent from Bruggeman's simple formula. The threshold values are in general not correctly predicted. It is 33% in the EMA, in three dimensions, far from the 16% expected from percolation theory and observed in experiments. However, in two dimensions, the EMA gives a threshold of 50% and has been proven to model percolation relatively well.[14][15][16]

## Maxwell Garnett equation

In the Maxwell Garnett approximation,[17] the effective medium consists of a matrix medium with εm and inclusions with εi. Maxwell Garnett was the son of physicist William Garnett, and was named after Garnett's friend, James Clerk Maxwell. He proposed his formula to explain colored pictures that are observed in glasses doped with metal nanoparticles. His formula has a form[18]

εeff=εd[1+3cmεm−εdεm+2εd−cm(εm−εd)],

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(1)

where εeff is effective relative complex permittivity of the mixture, εd is relative complex permittivity of the background medium containing small spherical inclusions of relative permittivity εm with volume fraction of cm≪1. This formula is based on the equality

εeff=εd+cmpmε0E,

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(2)

where ε0 is the absolute permittivity of free space and pm is electric dipole moment of a single inclusion induced by the external electric field E. However this equality is good only for homogeneous medium and εd=1. Moreover, the formula (1) ignores the interaction between single inclusions. Because of these circumstances, formula (1) gives too narrow and too high resonant curve for plasmon excitations in metal nanoparticles of the mixture.[19]

### Formula

The Maxwell Garnett equation reads:[20]

(εeff−εmεeff+2εm)=δi(εi−εmεi+2εm),

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(6)

where εeff is the effective dielectric constant of the medium, εi of the inclusions, and εm of the matrix; δi is the volume fraction of the inclusions.

The Maxwell Garnett equation is solved by:[21][22]

εeff=εm2δi(εi−εm)+εi+2εm2εm+εi−δi(εi−εm),

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(7)

so long as the denominator does not vanish. A simple MATLAB calculator using this formula is as follows.

% This simple MATLAB calculator computes the effective dielectric
% constant of a mixture of an inclusion material in a base medium
% according to the Maxwell-Garnett theory.
% INPUTS:
%     eps_base: dielectric constant of base material;
%     eps_incl: dielectric constant of inclusion material;
%     vol_incl: volume portion of inclusion material;
% OUTPUT:
%     eps_mean: effective dielectric constant of the mixture.

function eps_mean = MaxwellGarnettFormula(eps_base, eps_incl, vol_incl)

small_number_cutoff = 1e-6;

if vol_incl < 0 || vol_incl > 1
disp('WARNING: volume portion of inclusion material is out of range!');
end
factor_numer = 2 * (1 - vol_incl) * eps_base + (1 + 2 * vol_incl) * eps_incl;
factor_denom = (2 + vol_incl) * eps_base + (1 - vol_incl) * eps_incl;
if abs(factor_denom) < small_number_cutoff
disp('WARNING: the effective medium is singular!');
eps_mean = 0;
else
eps_mean = eps_base * factor_numer / factor_denom;
end
end

### Derivation

For the derivation of the Maxwell Garnett equation we start with an array of polarizable particles. By using the Lorentz local field concept, we obtain the Clausius-Mossotti relation: ε−1ε+2=4π3∑jNjαj Where Nj is the number of particles per unit volume. By using elementary electrostatics, we get for a spherical inclusion with dielectric constant εi and a radius a a polarisability α: α=(εi−1εi+2)a3 If we combine α with the Clausius Mosotti equation, we get: (εeff−1εeff+2)=δi(εi−1εi+2) Where εeff is the effective dielectric constant of the medium, εi of the inclusions; δi is the volume fraction of the inclusions.
As the model of Maxwell Garnett is a composition of a matrix medium with inclusions we enhance the equation:

(εeff−εmεeff+2εm)=δi(εi−εmεi+2εm)

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(8)

### Validity

In general terms, the Maxwell Garnett EMA is expected to be valid at low volume fractions δi, since it is assumed that the domains are spatially separated and electrostatic interaction between the chosen inclusions and all other neighbouring inclusions is neglected.[23] The Maxwell Garnett formula, in contrast to Bruggeman formula, ceases to be correct when the inclusions become resonant. In the case of plasmon resonance, the Maxwell Garnett formula is correct only at volume fraction of the inclusions δi<10−5.[24] The applicability of effective medium approximation for dielectric multilayers [25] and metal-dielectric multilayers [26] have been studied, showing that there are certain cases where the effective medium approximation does not hold and one needs to be cautious in application of the theory.

## Generalization of the Maxwell Garnett Equation to describe the nanoparticle size distribution

Maxwell Garnett Equation describes optical properties of nanocomposites which consist in a collection of perfectly spherical nanoparticles. All these nanoparticles must have the same size. However, due to confinement effect, the optical properties can be influenced by the nanoparticles size distribution. As shown by Battie et al.,[27] the Maxwell Garnett equation can be generalized to take into account this distribution.

(εeff−εm)εeff−2εm=3iλ316π2εm1.5fRm3∫P(R)a1(R)dR

R and P(R) are the nanoparticle radius and size distribution, respectively. Rm and f are the mean radius and the volume fraction of the nanoparticles, respectively. a1 is the first electric Mie coefficient. This equation reveals that the classical Maxwell Garnett equation gives a false estimation of the volume fraction nanoparticles when the size distribution cannot be neglected.

### Generalization to include shape distribution of nanoparticles

The Maxwell Garnett equation only describes the optical properties of a collection of perfectly spherical nanoparticles. However, the optical properties of nanocomposites are sensitive to the nanoparticles shape distribution. To overcome this limit, Y. Battie et al.[28] have developed the shape distributed effective medium theory (SDEMT). This effective medium theory enables to calculate the effective dielectric function of a nanocomposite which consists in a collection of ellipsoïdal nanoparticles distributed in shape.

εeff=(1−f)εm+fβεi1−f+fβ

with β=13∬P(L1,L2)∑i=13εmεm+Li(εi−εm)dL1dL2

The depolarization factors (L1,L2,L3) only depend on the shape of nanoparticles. P(L1,L2) is the distribution of depolarization factors. f is the volume fraction of the nanoparticles.

The SDEMT theory was used to extract the shape distribution of nanoparticles from absorption [29] or ellipsometric spectra.[30][31]

## Formula describing size effect

A new formula describing size effect was proposed.[19] This formula has a form εeff=14(Hε+i−Hε2−8εmεdJ(kma)),

Hε=(2−3cm)εd−(1−3cm)εmJ(kma),

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(5)

J(x)=21−xcot⁡(x)x2+xcot⁡(x)−1, where a is the nanoparticle radius and km=εmμmω/c is wave number. It is supposed here that the time dependence of the electromagnetic field is given by the factor exp(−iωt). In this paper Bruggeman's approach was used, but electromagnetic field for electric-dipole oscillation mode inside the picked particle was computed without applying quasi-static approximation. Thus the function J(kma) is due to the field nonuniformity inside the picked particle. In quasi-static region (kma≪1, i.e. a≤10nm for Ag) this function becomes constant J(kma)=1 and formula (5) becomes identical with Bruggeman's formula.

## Effective permeability formula

Formula for effective permeability of mixtures has a form [19]

μeff=14(Hμ+i−Hμ2−8μmμdJ(kma)),

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(6)

Hμ=(2−3cm)μd−(1−3cm)μmJ(kma).

Here μeff is effective relative complex permeability of the mixture, μd is relative complex permeability of the background medium containing small spherical inclusions of relative permeability μm with volume fraction of cm≪1. This formula was derived in dipole approximation. Magnetic octupole mode and all other magnetic oscillation modes of odd orders were neglected here. When μm=μd and kma≪1 this formula has a simple form [19]

μeff=μd(1+cm10ω2c2a2εm).

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(7)

## Effective medium theory for resistor networks

For a network consisting of a high density of random resistors, an exact solution for each individual element may be impractical or impossible. In such case, a random resistor network can be considered as a two-dimensional graph and the effective resistance can be modelled in terms of graph measures and geometrical properties of networks.[32] Assuming, edge length is much less than electrode spacing and edges to be uniformly distributed, the potential can be considered to drop uniformly from one electrode to another. Sheet resistance of such a random network (Rsn) can be written in terms of edge (wire) density (NE), resistivity (ρ), width (w) and thickness (t) of edges (wires) as:

Rsn=π2ρwtNE

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(9)

## See also

* Constitutive equation
* Percolation threshold

## References

1. ↑ Wenshan, Cai; Shalaev, Vladimir (November 2009). Optical Metamaterials: Fundamentals and Applications. Springer. pp. Chapter 2.4. ISBN 978-1-4419-1150-6. https://books.google.com/books?id=q8gDF2pbKXsC&q=artificial+dielectrics&pg=PA59.
2. ↑ Wang, M; Pan, N (2008). "Predictions of effective physical properties of complex multiphase materials" (Free PDF download). Materials Science and Engineering: R: Reports 63: 1–30. doi:10.1016/j.mser.2008.07.001. http://ningpan.net/publications/151-200/156.pdf.
3. ↑ T.C. Choy, "Effective Medium Theory", Oxford University Press, (2016) 241 p.
4. ↑ M. Scheller, C. Jansen, M. Koch, "Applications of Effective Medium Theories in the Terahertz Regime" in Recent Optical and Photonic Technologies, ed. by K.Y. Kim, Intech, Croatia, Vukovar (2010), p. 231.
5. ↑ Tinga, W. R.; Voss, W. A. G.; Blossey, D. F. (1973). "Generalized approach to multiphase dielectric mixture theory". J. Appl. Phys. 44 (9): 3897. doi:10.1063/1.1662868. Bibcode: 1973JAP....44.3897T. http://link.aip.org/link/?JAPIAU/44/3897/1. Retrieved 2019-04-24.
6. ↑ Lova, Paola; Megahd, Heba; Stagnaro, Paola; Alloisio, Marina; Patrini, Maddalena; Comoretto, Davide (2020-06-15). "Strategies for Dielectric Contrast Enhancement in 1D Planar Polymeric Photonic Crystals" (in en). Applied Sciences 10 (12): 4122. doi:10.3390/app10124122. ISSN 2076-3417.
7. ↑ Bruggeman, D. A. G. (1935). "Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen" (in de). Annalen der Physik 416 (7): 636–664. doi:10.1002/andp.19354160705. ISSN 0003-3804. Bibcode: 1935AnP...416..636B.
8. ↑ S.J. Oldenburg. "Silver Nanoparticles: Properties and Applications". Sigma Aldrich. https://www.sigmaaldrich.com/technical-documents/articles/materials-science/nanomaterials/silver-nanoparticles.html.
9. ↑ 9.0 9.1 Landauer, Rolf (April 1978). "Electrical conductivity in inhomogeneous media". 40. American Institute of Physics. pp. 2–45. doi:10.1063/1.31150. http://link.aip.org/link/?APCPCS/40/2/1. Retrieved 2010-02-07.
10. ↑ Granqvist, C. G.; Hunderi, O. (1978). "Conductivity of inhomogeneous materials: Effective-medium theory with dipole-dipole interaction". Phys. Rev. B 18 (4): 1554–1561. doi:10.1103/PhysRevB.18.1554. Bibcode: 1978PhRvB..18.1554G.
11. ↑ 11.0 11.1 Weiglhofer, W. S.; Lakhtakia, A.; Michel, B. (1998). "Maxwell Garnett and Bruggeman formalisms for a particulate composite with bianisotropic host medium". Microw. Opt. Technol. Lett. 15 (4): 263–266. doi:10.1002/(SICI)1098-2760(199707)15:4<263::AID-MOP19>3.0.CO;2-8. http://www3.interscience.wiley.com/journal/53983/abstract?CRETRY=1&SRETRY=0.
12. ↑ Stroud, D. (1975). "Generalized effective-medium approach to the conductivity of an inhomogeneous material". Phys. Rev. B 12 (8): 3368–3373. doi:10.1103/PhysRevB.12.3368. Bibcode: 1975PhRvB..12.3368S.
13. ↑ Davidson, A.; Tinkham, M. (1976). "Phenomenological equations for the electrical conductivity of microscopically inhomogeneous materials". Phys. Rev. B 13 (8): 3261–3267. doi:10.1103/PhysRevB.13.3261. Bibcode: 1976PhRvB..13.3261D.
14. ↑ Kirkpatrick, Scott (1973). "Percolation and conduction". Rev. Mod. Phys. 45 (4): 574–588. doi:10.1103/RevModPhys.45.574. Bibcode: 1973RvMP...45..574K.
15. ↑ Zallen, Richard (1998). The Physics of Amorphous Solids. Wiley-Interscience. ISBN 978-0-471-29941-7.
16. ↑ Rozen, John; Lopez, René; Haglund, Richard F. Jr.; Feldman, Leonard C. (2006). "Two-dimensional current percolation in nanocrystalline vanadium dioxide films". Appl. Phys. Lett. 88 (8): 081902. doi:10.1063/1.2175490. Bibcode: 2006ApPhL..88h1902R. http://link.aip.org/link/?APPLAB/88/081902/1. Retrieved 2019-04-24.
17. ↑ Garnett, J. C. M. (1904). "Colours in Metal Glasses and in Metallic Films". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 203 (359–371): 385–420. doi:10.1098/rsta.1904.0024. ISSN 1364-503X. Bibcode: 1904RSPTA.203..385G.
18. ↑ Markel, Vadim A. (2016). "Introduction to the Maxwell Garnett approximation: tutorial". Journal of the Optical Society of America A 33 (7): 1244–1256. doi:10.1364/JOSAA.33.001244. PMID 27409680. Bibcode: 2016JOSAA..33.1244M. https://hal.archives-ouvertes.fr/hal-01282105v2/file/Paper.pdf.
19. ↑ 19.0 19.1 19.2 19.3 Belyaev, B. A.; Tyurnev, V. V. (2018). "Electrodynamic Calculation of Effective Electromagnetic Parameters of a Dielectric Medium with Metallic Nanoparticles of a Given Size". Journal of Experimental and Theoretical Physics 127 (4): 608–619. doi:10.1134/S1063776118100114. ISSN 1063-7761. Bibcode: 2018JETP..127..608B.
20. ↑ Choy, Tuck C. (1999). Effective Medium Theory. Oxford: Clarendon Press. ISBN 978-0-19-851892-1.
21. ↑ Levy, O., & Stroud, D. (1997). Maxwell Garnett theory for mixtures of anisotropic inclusions: Application to conducting polymers. Physical Review B, 56(13), 8035.
22. ↑ Liu, Tong, et al. "Microporous Co@ CoO nanoparticles with superior microwave absorption properties." Nanoscale 6.4 (2014): 2447-2454.
23. ↑ Jepsen, Peter Uhd; Fischer, Bernd M.; Thoman, Andreas; Helm, Hanspeter; Suh, J. Y.; Lopez, René; Haglund, R. F. Jr. (2006). "Metal-insulator phase transition in a VO2 thin film observed with terahertz spectroscopy". Phys. Rev. B 74 (20). doi:10.1103/PhysRevB.74.205103. Bibcode: 2006PhRvB..74t5103J. http://orbit.dtu.dk/en/publications/metalinsulator-phase-transition-in-a-vo2-thin-film-observed-with-terahertz-spectroscopy(b8474464-d43f-4a26-baa3-5777a1c34e7f).html.
24. ↑ Belyaev, B. A.; Tyurnev, V. V. (2018). "Electrodynamic calculation of effective electromagnetic parameters of a dielectric medium with metallic nanoparticles of a given size". Journal of Experimental and Theoretical Physics 127 (4): 608–619. doi:10.1134/S1063776118100114. Bibcode: 2018JETP..127..608B.
25. ↑ Zhukovsky, S. V.; Andryieuski, A., Takayama, O.; Shkondin, E., Malureanu, R.; Jensen, F., Lavrinenko, A. V. (2015). "Experimental demonstration of effective medium approximation breakdown in deeply subwavelength all-dielectric multilayers.". Physical Review Letters 115 (17). doi:10.1103/PhysRevLett.115.177402. PMID 26551143. Bibcode: 2015PhRvL.115q7402Z. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.115.177402.
26. ↑ Sukham, J.; Takayama, O., Mahmoodi, M.; Sychev, S., Bogdanov, A.; Hassan Tavassoli, S., Lavrinenko, A. V.; Malureanu R. (2019). "Investigation of effective media applicability for ultrathin multilayer structures.". Nanoscale 11 (26): 12582–12588. doi:10.1039/C9NR02471A. PMID 31231735. https://pubs.rsc.org/en/content/articlelanding/2019/nr/c9nr02471a.
27. ↑ Battie, Y.; Resano-Garcia, A., Chaoui, N.; Zhang, Y., En Naciri, A. (2014). "Extended Maxwell-Garnett-Mie formulation applied to size dispersion of metallic nanoparticles embedded in host liquid matrix.". Journal of Chemical Physics 140 (4): 044705. doi:10.1063/1.4862995. PMID 25669565. Bibcode: 2014JChPh.140d4705B. https://pubs.aip.org/aip/jcp/article-abstract/140/4/044705/72892/Extended-Maxwell-Garnett-Mie-formulation-applied?redirectedFrom=fulltext.
28. ↑ Resano-Garcia, A.; Battie, Y., En Naciri, A.; Akil, S., Chaoui, N. (2015). "Experimental and theoretical determination of the plasmonic responses and shape distribution of colloidal metallic nanoparticles.". Journal of Chemical Physics 142 (13): 134108. doi:10.1063/1.4916917. PMID 25854229. Bibcode: 2015JChPh.142m4108R. https://pubs.aip.org/aip/jcp/article-abstract/142/13/134108/901974/Experimental-and-theoretical-determination-of-the?redirectedFrom=fulltext.
29. ↑ Battie, Y.; Resano-Garcia, A., En Naciri, A.; Akil, S., Chaoui, N. (2015). "Determination of morphological characteristics of metallic nanoparticles based on modified Maxwell-Garnett fitting of optical responses.". Applied Physics Letters 107 (14): 143104. doi:10.1063/1.4932638. Bibcode: 2015ApPhL.107n3104B. https://pubs.aip.org/aip/apl/article-abstract/107/14/143104/29556/Determination-of-morphological-characteristics-of?redirectedFrom=fulltext.
30. ↑ Battie, Y.; Izquierdo-Lorenzo, I., Resano-Garcia, A.; En Naciri, A., Akil, S.; Adam, P.M., Jradi, S. (2016). "How to determine the morphology of plasmonic nanocrystals without transmission electron microscopy?". Journal of Nanoparticle Research 18 (8): 217. doi:10.1007/s11051-016-3533-8. Bibcode: 2016JNR....18..217B. https://link.springer.com/article/10.1007/s11051-016-3533-8.
31. ↑ Battie, Y.; Stchakovsky, M., En Naciri, A.; Akil, S., Chaoui, N. (2017). "Ellipsometry of Colloidal solutions: New experimental setup and application to metallic colloids.". Langmuir 33 (30): 7425–7434. doi:10.1021/acs.langmuir.7b00490. PMID 28727434. https://pubs.acs.org/doi/abs/10.1021/acs.langmuir.7b00490.
32. ↑ Kumar, Ankush; Vidhyadhiraja, N. S.; Kulkarni, G. U . (2017). "Current distribution in conducting nanowire networks". Journal of Applied Physics 122 (4): 045101. doi:10.1063/1.4985792. Bibcode: 2017JAP...122d5101K.

## Further reading

* Lakhtakia, A., ed (1996). Selected Papers on Linear Optical Composite Materials [Milestone Vol. 120]. Bellingham, WA, USA: SPIE Press. ISBN 978-0-8194-2152-4.
* Tuck, Choy (1999). Effective Medium Theory (1st ed.). Oxford: Oxford University Press. ISBN 978-0-19-851892-1.
* Lakhtakia (Ed.), A. (2000). Electromagnetic Fields in Unconventional Materials and Structures. New York: Wiley-Interscience. ISBN 978-0-471-36356-9.
* Weiglhofer (Ed.); Lakhtakia (Ed.), A. (2003). Introduction to Complex Mediums for Optics and Electromagnetics. Bellingham, WA, USA: SPIE Press. ISBN 978-0-8194-4947-4.
* Mackay, T. G.; Lakhtakia, A. (2010). Electromagnetic Anisotropy and Bianisotropy: A Field Guide (1st ed.). Singapore: World Scientific. ISBN 978-981-4289-61-0.

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