A comprehensive review on effective medium theories to find effective dielectric constant of composites

References

• Stölzle S, Enders A, Nimtz G. Numerical simulation of random composite dielectrics. J Phys I. 1992;2:401–408.
   Google Scholar
• Maxwell Garnett JC. Colours in metal glasses and in metallic films. Phil Trans R Soc Lond A. 1904;203:385–420.
   Google Scholar
• Stroud D, Pan FP. Effect of isolated inhomogeneities on the galvanomagnetic properties of solids. Phys Rev B. 1976;13:1434–1438.
   Web of Science ®Google Scholar
• Bergman DJ, Stroud D. Physical properties of macroscopically inhomogeneous media. Solid State Phys. 1992;46:147–269.
   Web of Science ®Google Scholar
• Levy O, Stroud D. Maxwell Garnett theory for mixtures of anisotropic inclusions: application to conducting polymers. Phys Rev B. 1997;56:8035–8046.
   Web of Science ®Google Scholar
• Bruggeman DAG. Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen. Ann Phys. 1935;416:636–679.
   Google Scholar
• Mackay TG, Lakhtakia A. Modern analytical electromagnetic homogenization with mathematica®. Bristol: IOP Publishing; 2020.
   Google Scholar
• Slovick BA, Yu ZG, Krishnamurthy S. Generalized effective-medium theory for metamaterials. Phys Rev B. 2014;89:155118–1–5.
   Web of Science ®Google Scholar
• Koschny T, Kafesaki M, Economou EN, et al. Effective medium theory of left-handed materials. Phys Rev Lett. 2004;93:107402–1–4.
   Web of Science ®Google Scholar
• Jin J, Liu S, Lin Z, et al. Effective-medium theory for anisotropic magnetic metamaterials. Phys Rev B. 2009;80:115101–1–6.
   Web of Science ®Google Scholar
• Van Beek LKH. Dielectric behaviour of heterogeneous systems. Prog Dielectr. 1967;7:113.
   Google Scholar
• Kim J, Kim T, Kim C. Simulation method for complex permittivities of carbon black/epoxy composites at microwave frequency band. J Appl Polym Sci. 2006;100:2189–2195.
   Web of Science ®Google Scholar
• Ragossnig H, Feltz A. Characterization of dielectric powders by a new defined form factor. J Eur Ceram Soc. 1998;18:429–444.
   Web of Science ®Google Scholar
• Looyenga H. Dielectric constants of heterogeneous mixtures. Physica. 1965;31:401–406.
   Google Scholar
• Rayleigh L. On the influence of obstacles arranged in rectanglular order on the properties of a medium. Phil Mag. 1892;34:481–502.
   Google Scholar
• Hakki BW, Coleman PD. A dielectric resonator method of measuring inductive capacities in the millimeter range. IRE Trans Microw Theory Tech. 1960;8:402–410.
   Google Scholar
• Lou J, Hatton TA, Laibinis PE. Effective dielectric properties of solvent mixtures at microwave frequencies. J Phys Chem A. 1997;101:5262–5268.
   Web of Science ®Google Scholar
• Böttcher CJF, Brown WF. Theory of electric polarisation. Phys Today. 1953;6:17.
   Google Scholar
• Tinga WR, Voss WAG, Blossey DF. Generalized approach to multiphase dielectric mixture theory. J Appl Phys. 1973;44:3897–3902.
   Web of Science ®Google Scholar
• Dryden JS, Meakins RJ. Examples of the Maxwell-Wagner type of dielectric absorption using wool wax-water mixtures. Proc Phys Soc Sect B. 1957;70:427–430.
   Google Scholar
• Hanai T. Dielectric theory on the interfacial polarization for two-phase mixtures. Bull Inst Chem Res. 1962;39:341–367.
   Google Scholar
• Achour ME, Malhi ME, Miane JL, et al. Microwave properties of carbon black–epoxy resin composites and their simulation by means of mixture laws. J Appl Polym Sci. 1999;73:969–973.
   Web of Science ®Google Scholar
• Böttcher CJF, Van Belle OC, Bordewijk P, et al. Theory of electric polarization. J Electrochem Soc. 1974;121:211C.
   Google Scholar
• Polder D, Van Santeen JH. The effective permeability of mixtures of solids. Physica. 1946;12:257–271.
   Google Scholar
• Chaudhari A, Chaudhari H, Mehrotra S. Dielectric properties for the binary mixture of dimethylsuphoxide and dimethylacetamide with 2-nitrotoluene at microwave frequencies. Fluid Phase Equilib. 2002;201:107–118.
   Web of Science ®Google Scholar
• Wakino K. A new proposal on mixing rule of the dielectric constant of mixture. Proceedings of 1994 IEEE International Symposium on Applications of Ferroelectrics. University Park (PA): IEEE; 1994. p. 33–38.
   Google Scholar
• Bittar A, Berthier S, Lafait J. Non metal-metal transition in Bruggeman optical theory for inhomogeneous media. J Phys. 1984;45:623–631.
   Google Scholar
• Neelakanta PS. Handbook of electromagnetic materials: monolithic and composite versions and their applications. Boca Raton, FL: CRC Press; 1995.
   Google Scholar
• Lakhtakia A. Selected papers on linear optical composite materials. Bellingham: SPIE Optical Engineering Press; 1996.
   Google Scholar
• Sihvola AH. Electromagnetic mixing formulas and applications. London: IET; 1999.
   Google Scholar
• Milton GW. The theory of composites (Cambridge monographs on applied and computational mathematics). Cambridge: Cambridge University Press; 2002.
   Google Scholar
• Mackay T. Linear and nonlinear homogenized composite mediums as metamaterials. Electromagnetics. 2005;25:461–481.
   Web of Science ®Google Scholar
• Choy TC. Effective medium theory: principles and applications. Oxford: Oxford University Press; 2015.
   Google Scholar
• Mackay T, Lakhtakia A. Electromagnetic anisotropy and bianisotropy: a field guide. Singapore: World Scientific Publishing; 2019.
   Google Scholar
• Faxén H. Der Zusammenhang zwischen den Maxwellschen Gleichungen für Dielektrika und den atomistischen Ansätzen von H. A. Lorentz u. a. Zeitschrift für Phys. 1920;2:218–229.
   Google Scholar
• Lakhtakia A. Size-dependent Maxwell-Garnett formula from an integral equation formalism. Optik. 1992;91:134–137.
   Web of Science ®Google Scholar
• Prinkey MT, Lakhtakia A, Shanker B. On the extended Maxwell-Garnett and the extended Burggeman approaches for dielectric-in-dielectric composites. Optik. 1994;96:25–30.
   Web of Science ®Google Scholar
• Lakhtakia A. Frequency-dependent continuum electromagnetic properties of a gas of scattering centers. Adv Chem Phys. 2009;85:311–359.
   Google Scholar
• Mackay TG, Weiglhofer WS. Homogenization of biaxial composite materials: bianisotropic properties. J Opt A Pure Appl Opt. 2001;3:45–52.
   Google Scholar
• Weiglhofer WS, Lakhtakia A, Michel B. Maxwell Garnett and Bruggeman formalisms for a particulate composite with bianisotropic host medium. Microw Opt Technol Lett. 1997;15:263–266.
   Web of Science ®Google Scholar
• Weiglhofer WS, Lakhtakia A, Monzon JC. Maxwell-Garnett model for composites of electrically small uniaxial objects. Microw Opt Technol Lett. 1993;6:681–684.
   Web of Science ®Google Scholar
• Xu G, Tazawa M, Jin P, et al. Wavelength tuning of surface plasmon resonance using dielectric layers on silver island films. Appl Phys Lett. 2003;82:3811–3813.
   Web of Science ®Google Scholar
• Lakhtakia A. Incremental Maxwell Garnett formalism for homogenizing particulate composite media. Microw Opt Technol Lett. 1998;17:276–279.
   Web of Science ®Google Scholar
• Michel B, Lakhtakia A, Weiglhofer WS, et al. Incremental and differential Maxwell Garnett formalisms for bi-anisotropic composites. Compos Sci Technol. 2001;61:13–18.
   Web of Science ®Google Scholar
• Fossheim K, Tuset ED, Ebbesen TW, et al. Enhanced flux pinning in Bi2Sr2CaCu2O8+x superconductor with embedded carbon nanotubes. Phys C Supercond. 1995;248:195–202.
   Google Scholar
• Rinzler AG, Hafner JH, Nikolaev P, et al. Unraveling nanotubes: field emission from an atomic wire. Science (80-). 1995;269:1550–1553.
   PubMed Web of Science ®Google Scholar
• De Heer WA, Chatelain A, Ugarte D. A carbon nanotube field-emission electron source. Science (80-). 1995;270:1179–1180.
   Web of Science ®Google Scholar
• Garcia-Vidal FJ, Pitarke JM, Pendry JB. Effective medium theory of the optical properties of aligned carbon nanotubes. Phys Rev Lett. 1997;78:4289–4292.
   Web of Science ®Google Scholar
• Lin M, Shung KW. Optical and magneto-optical properties of carbon nanotube bundles. J Phys Soc Japan. 1997;66:3294–3302.
   Google Scholar
• Lakhtakia A, Slepyan GY, Maksimenko SA, et al. Effective medium theory of the microwave and the infrared properties of composites with carbon nanotube inclusions. Carbon. 1998;36:1833–1839.
   Web of Science ®Google Scholar
• Mackay TG, Weiglhofer WS. Homogenization of biaxial composite materials: nondissipative dielectric properties. Electromagnetics. 2001;21:15–25.
   Web of Science ®Google Scholar
• Mackay TG, Weiglhofer WS. Homogenization of biaxial composite materials: dissipative anisotropic properties. J Opt A Pure Appl Opt. 2000;2:426–432.
   Google Scholar
• Sihvola A. Homogenization of a dielectric mixture with anisotropic spheres in anisotropic background. Electromagnetics. 1997;17:269–286.
   Web of Science ®Google Scholar
• Doyle WT. Optical properties of a suspension of metal spheres. Phys Rev B. 1989;39:9852–9858.
   PubMed Web of Science ®Google Scholar
• Lewin L. The electrical constants of a material loaded with spherical particles. J Inst Electr Eng III Radio Commun Eng. 1947;94:65–68.
   Google Scholar
• Mackay TG. Lewin’s homogenization formula revisited for nanocomposite materials. J Nanophotonics. 2008;2:29503–1–4.
   Web of Science ®Google Scholar
• Smith GB, Granqvist CG. Green nanotechnology: solutions for sustainability and energy in the built environment. J Nanophotonics. 2011;5:050201–1–2.
   Google Scholar
• Mackay TG, Lakhtakia A. Bruggeman formalism versus Bruggeman formalism: particular composite materials comprising oriented ellipsoidal particles. J Nanophotonics. 2012;6:69501–1–6.
   Web of Science ®Google Scholar
• Lakhtakia A. Enhancement of optical activity of chiral sculptured thin films by suitable infiltration of void regions. Optik. 2001;112:145–148.
   Web of Science ®Google Scholar
• Bohren CF, Battan LJ. Radar backscattering by inhomogeneous precipitation particles. J Atmos Sci. 1980;37:1821–1827.
   Web of Science ®Google Scholar
• Böttcher CJF. The dielectric constant of crystalline powders. Recl des Trav Chim des Pays-Bas. 1945;64:47–51.
   Google Scholar
• Tsang L, Kong JA. Scattering of electromagnetic waves from random media with strong permittivity fluctuations. Radio Sci. 1981;16:303–320.
   Web of Science ®Google Scholar
• Michel B, Lakhtakia A. Strong-property-fluctuation theory for homogenizing chiral particulate composites. Phys Rev E. 1995;51:5701–5707.
   PubMed Web of Science ®Google Scholar
• Weiglhofer WS, Lakhtakia A, Michel B. Correction to “Maxwell Garnett and Bruggeman formalisms for a particulate composite with bianisotropic host medium. Microw Opt Technol Lett. 1999;22:221–221.
   Web of Science ®Google Scholar
• Genchev ZD. Anisotropic and gyrotropic version of Polder and van Santen’s mixing formula. Waves Random Media. 1992;2:99–110.
   Web of Science ®Google Scholar
• Zhuck NP. Strong-fluctuation theory for a mean electromagnetic field in a statistically homogeneous random medium with arbitrary anisotropy of electrical and statistical properties. Phys Rev B. 1994;50:15636–15645.
   Web of Science ®Google Scholar
• Mackay TG, Lakhtakia A, Weiglhofer WS. Strong-property-fluctuation theory for homogenization of bianisotropic composites: formulation. Phys Rev E. 2000;62:6052–6064.
   Web of Science ®Google Scholar
• Mackay TG, Lakhtakia A, Weiglhofer WS. Erratum: strong-property-fluctuation theory for homogenization of bianisotropic composites: formulation [Phys. Rev. E 62, 6052 (2000)]. Phys Rev E. 2001;63:49901.
   Web of Science ®Google Scholar
• Mackay TG, Lakhtakia A, Weiglhofer WS. Third-order implementation and convergence of the strong-property-fluctuation theory in electromagnetic homogenization. Phys Rev E. 2001;64:66616–1–9.
   Web of Science ®Google Scholar
• Lakhtakia A. Bruggeman formalism for uniaxial dielectric-magnetic composites. Microw Opt Technol Lett. 1996;11:290–291.
   Web of Science ®Google Scholar
• Michel B, Lakhtakia A, Weiglhofer WS. Homogenization of linear bianisotropic particulate composite media-numerical studies. Int J Appl Electromagn Mech. 1998;9:167–178.
   Web of Science ®Google Scholar
• Lakhtakia A, Michel B, Weiglhofer WS. The role of anisotropy in the Maxwell Garnett and Bruggeman formalisms for uniaxial particulate composite media. J Phys D Appl Phys. 1997;30:230–240.
   Web of Science ®Google Scholar
• Bragg WL, Pippard AB. The form birefringence of macromolecules. Acta Crystallogr. 1953;6:865–867.
   Web of Science ®Google Scholar
• Sherwin JA, Lakhtakia A. Bragg–Pippard formalism for bianisotropic particulate composites. Microw Opt Technol Lett. 2002;33:40–44.
   Web of Science ®Google Scholar
• Perutz MF. Polarization dichroism, form birefringence, and molecular orientation in crystalline haemoglobins. Acta Crystallogr. 1953;6:859–864.
   Google Scholar
• Wiener O. Die Theorie des Mischkörpers für das Feld der Stationären Strömung. Erste Abhandlung: Die Mittelwertsätze für Kraft, Polarisation und Energie. Abhandlungen der Math Klasse der Königl Sächsischen Gesellschaft der Wissenschaften. 1912;32:507–604.
   Google Scholar
• Mackay TG, Lakhtakia A, Weiglhofer WS. Ellipsoidal topology, orientation diversity and correlation length in bianisotropic composite mediums. AEU Int J Electron Commun. 2001;55:243–251.
   Web of Science ®Google Scholar
• Lakhtakia A, Mackay TG. Size-dependent Bruggeman approach for dielectric–magnetic composite materials. AEU Int J Electron Commun. 2005;59:348–351.
   Web of Science ®Google Scholar
• Mackay TG. Depolarization volume and correlation length in the homogenization of anisotropic dielectric composites. Waves Random Media. 2004;14:485–498.
   Web of Science ®Google Scholar
• Bohren CF. Applicability of effective-medium theories to problems of scattering and absorption by nonhomogeneous atmospheric particles. J Atmos Sci. 1986;43:468–475.
   Web of Science ®Google Scholar
• Cui J, Mackay TG. Depolarization regions of nonzero volume in bianisotropic homogenized composites. Waves Random Complex Media. 2007;17:269–281.
   Web of Science ®Google Scholar
• Shanker B, Lakhtakia A. Extended Maxwell Garnett model for chiral-in-chiral composites. J Phys D Appl Phys. 1993;26:1746–1758.
   Web of Science ®Google Scholar
• Shanker B. The extended Bruggeman approach for chiral-in-chiral mixtures. J Phys D Appl Phys. 1996;29:281–288.
   Web of Science ®Google Scholar
• Lakhtakia A, Shanker B. Beltrami fields within continuous source regions, volume integral equations, scattering algorithms and the extended Maxwell-Garnett model. Int J Appl Electromagn Mater. 1993;4:65–82.
   Google Scholar
• Mallet P, Guérin C-A, Sentenac A. Maxwell-Garnett mixing rule in the presence of multiple scattering: derivation and accuracy. Phys Rev B. 2005;72:14205–1–9.
   Web of Science ®Google Scholar
• Soukoulis CM, Wegener M. Past achievements and future challenges in the development of three-dimensional photonic metamaterials. Nat Photonics. 2011;5:523–530.
   Web of Science ®Google Scholar
• Zheludev NI, Kivshar YS. From metamaterials to metadevices. Nat Mater. 2012;11:917–924.
   PubMed Web of Science ®Google Scholar
• Bohren CF, Huffman DR. Absorption and scattering of light by small particles. New York: Wiley; 1983.
   Google Scholar
• Veselago VG. Electrodynamics of substances with simultaneously negative electrical and magnetic permeabilities. Sov Phys Uspekhi. 1968;10:509–514.
   Google Scholar
• Pendry JB, Holden AJ, Stewart WJ, et al. Extremely low frequency plasmons in metallic mesostructures. Phys Rev Lett. 1996;76:4773–4776.
   PubMed Web of Science ®Google Scholar
• Pendry JB, Holden AJ, Robbins DJ, et al. Low frequency plasmons in thin-wire structures. J Phys Condens Matter. 1998;10:4785–4809.
   Web of Science ®Google Scholar
• Pendry JB, Holden AJ, Robbins DJ, et al. Magnetism from conductors and enhanced nonlinear phenomena. IEEE Trans Microw Theory Tech. 1999;47:2075–2084.
   Web of Science ®Google Scholar
• Zhang X, Wu Y. Effective medium theory for anisotropic metamaterials. Sci Rep. 2015;5:7892–1–7.
   Web of Science ®Google Scholar
• Sheng P. Introduction to wave scattering, localization and mesoscopic phenomena. Heidelberg: Springer Berlin; 2006.
   Google Scholar
• Lamb W, Wood DM, Ashcroft NW. Long-wavelength electromagnetic propagation in heterogeneous media. Phys Rev B. 1980;21:2248–2266.
   Web of Science ®Google Scholar
• Herpin A. Calcul du pouvoir réflecteur d’un système stratifié quelconque. Comptes Rendus Hebd des Séances L’Academie des Sci. 1947;225:182–183.
   Google Scholar
• Abelès F. Investigations on the propagation of sinusoidal electromagnetic waves in stratified media. application to thin films. Ann Phys. 1950;5:596–640.
   Google Scholar
• Epstein LI. The design of optical filters. JOSA. 1952;42:806–810.
   Web of Science ®Google Scholar
• Thelen A. Equivalent layers in multilayer filters. JOSA. 1966;56:1533–1538.
   Web of Science ®Google Scholar
• Macleod HA. A new approach to the design of metal-dielectric thin-film optical coatings. Opt Acta Int J Opt. 1978;25:93–106.
   Web of Science ®Google Scholar
• Lafait J, Yamaguchi T, Frigerio JM, et al. Effective medium equivalent to a symmetric multilayer at oblique incidence. Appl Opt. 1990;29:2460–2465.
   PubMed Web of Science ®Google Scholar
• Reese PS, Lakhtakia A. A periodic chiral arrangement of thin identical bianisotropic sheets: effective properties. Optik. 1990;86:47–50.
   Web of Science ®Google Scholar
• Ramakrishna SA, Lakhtakia A. Spectral shifts in the properties of a periodic multilayered stack due to isotropic chiral layers. J Opt A Pure Appl Opt. 2009;11:74001–1–11.
   Google Scholar
• Lakhtakia A. Constraints on effective constitutive parameters of certain bianisotropic laminated composite materials. Electromagnetics. 2009;29:508–514.
   Web of Science ®Google Scholar
• Biot J-B, Arago DFJ. Mémoire sur les affinités des corps pour la lumière, et particulièrement sur les forces réfringentes des différens gaz. Baudouin, imprimeur de l’Institut; 1806.
   Google Scholar
• Rumpf RC. Lecture on introduction to engineered materials. 2014.
   Google Scholar
• Lichtenecker K, Rother K. Deduction of the logarithmic mixture law from general principles. Phys Z. 1931;32:255–260.
   Google Scholar
• Simpkin R. Derivation of Lichtenecker’s logarithmic mixture formula from Maxwell’s equations. IEEE Trans Microw Theory Tech. 2010;58:545–550.
   Web of Science ®Google Scholar
• Lichtenecker K. Die Dielektrizitätskonstante natürlicher und künstlicher Mischkörper. Phys Z. 1926;27:115–158.
   Google Scholar
• Pan M, Bender BA. A bimodal grain size model for predicting the dielectric constant of calcium copper titanate ceramics. J Am Ceram Soc. 2005;88:2611–2614.
   Web of Science ®Google Scholar
• Ward L. The optical constants of bulk materials and films. Adam Hilger Series on Optics and Optoelectronics. 1988. p. 155.
   Google Scholar
• Nelson SO, You T-S. Relationships between microwave permittivities of solid and pulverised plastics. J Phys D Appl Phys. 1990;23:346–353.
   Web of Science ®Google Scholar
• Sheen J, Hong Z-W, Liu W, et al. Study of dielectric constants of binary composites at microwave frequency by mixture laws derived from three basic particle shapes. Eur Polym J. 2009;45:1316–1321.
   Web of Science ®Google Scholar
• Sheng P. Theory for the dielectric function of granular composite media. Phys Rev Lett. 1980;45:60–63.
   Web of Science ®Google Scholar
• Landauer R. The electrical resistance of binary metallic mixtures. J Appl Phys. 1952;23:779–784.
   Web of Science ®Google Scholar
• Sheng P. Pair-cluster theory for the dielectric constant of composite media. Phys Rev B. 1980;22:6364–6368.
   Web of Science ®Google Scholar
• Chang R, Chung H-Y, Chen C-W, et al. Optical effects of charges in colloidal solutions. Opt Mater. 2017;66:43–47.
   Web of Science ®Google Scholar
• Bohren CF, Hunt AJ. Scattering of electromagnetic waves by a charged sphere. Can J Phys. 1977;55:1930–1935.
   Web of Science ®Google Scholar
• Hui PM, Stroud D. Complex dielectric response of metal-particle clusters. Phys Rev B. 1986;33:2163–2169.
   Web of Science ®Google Scholar
• Harvey AH, Lemmon EW. Method for estimating the dielectric constant of natural gas mixtures. Int J Thermophys. 2005;26:31–46.
   Web of Science ®Google Scholar
• Kirkwood JG. The dielectric polarization of polar liquids. J Chem Phys. 1939;7:911–919.
   Web of Science ®Google Scholar
• Oster G. The dielectric properties of liquid mixtures. J Am Chem Soc. 1946;68:2036–2041.
   PubMed Web of Science ®Google Scholar
• Wang P, Anderko A. Computation of dielectric constants of solvent mixtures and electrolyte solutions. Fluid Phase Equilib. 2001;186:103–122.
   Web of Science ®Google Scholar
• Franks F, Reid DS. Water: a comprehensive treatise, Vol. 2. F Franks, editor. New York (NY): Plenum Press; 1973.
   Google Scholar
• Haggis GH, Hasted JB, Buchanan TJ. The dielectric properties of water in solutions. J Chem Phys. 1952;20:1452–1465.
   Web of Science ®Google Scholar
• Sigvartsen T, Gestblom B, Noreland E, et al. Conductometric and dielectric behaviour of solutions of tetrabutylammonium perchlorate in solvents of low and medium permittivity. Acta Chem Scand. 1989;43:103–115.
   Google Scholar
• Sigvartsen T, Songstad J, Gestblom B, et al. Dielectric properties of solutions of tetra-iso-pentylammonium nitrate in dioxane-water mixtures. J Solution Chem. 1991;20:565–582.
   Web of Science ®Google Scholar