# Optical Properties of Oxide Films Dispersed with Nanometal Particles

## Abstract

Solid materials reveal some special behaviors like quantum effects in semiconductors and surface-enhanced effects in metals by decreasing their diameters. In this review, the enhancement of the optical response due to the electric field of the light is reviewed as the recent active field of plasmonics. The production methods of various metal nanoparticles are summarized for the bared state and for the embedded state within the dielectric medium. The features of the optical properties of these nanoparticles are reviewed, and typical formula to reproduce the absorption spectra due to the surface plasmon resonance is summarized. Several applications of these systems are shortly introduced.

## Keywords

Surface Plasmon Resonance Metal Nanoparticles Dielectric Function Effective Permittivity Polyol Process## 1 Definition of the Topic

Solid materials reveal some special behaviors like quantum effects in semiconductors and surface-enhanced effects in metals by decreasing their diameters. In this review, the enhancement of the optical response due to the electric field of the light is reviewed as the recent active field of plasmonics. The production methods of various metal nanoparticles are summarized for the bared state and for the embedded state within the dielectric medium. The features of the optical properties of these nanoparticles are reviewed, and typical formula to reproduce the absorption spectra due to the surface plasmon resonance is summarized. Several applications of these systems are shortly introduced.

## 2 Overview

The optical responses of metal nanoparticles are different to those of the conventional bulk metals and are actively studied as the field of plasmonics. The interesting optical behavior comes from the enhanced surface effect and called surface-enhanced plasmon resonance (SPR). The basic researches and application studies are introduced. The synthesis methods for the bared nanoparticles and the composite films dispersed with metal nanoparticles are briefly introduced. Basic optical theories describing the optical properties of metal nanoparticles at both bared and embedded states are reviewed. The detailed optical study of the composition dependence of the silver nanoparticles within a ZrO_{2} matrix is introduced, and the applicability of the Maxwell–Garnett and Bruggeman theories is discussed. Finally, recent application researches of metal nanoparticles are briefly summarized.

## 3 Introduction

^{−9}m). The dimensions of these groups of materials are much larger than those of metal complexes or clusters and smaller than so-called fine particles with the size of micrometer order (∼10

^{−6}μm) (Fig. 8.1). As an example of gold nanoparticle, the minimum one is consisted of 13 atoms and the diameter is 0.86 nm. The particle becomes larger as the number of the atoms increases. The particles with the diameter of 1.44 nm, 2.02 nm, and 2.6 nm contain the number of 55, 147, and 309 atoms, respectively.

- 1.
These materials belong to the interdisciplinary region between bulk materials, and atoms/molecules and have been few chances to be the target of research.

- 2.
Means to observe objects with these dimensions like TEM were not popular.

- 3.
It was difficult to obtain a large quantity of nanoparticles with a same size because of the lack of needs for applications.

As a result, few reports have been submitted by several physicists and researchers of a colloid interface. Recently, researches on metal nanoparticles have expanded rapidly as the basic materials for the nanotechnology in accordance with the establishment of the synthesis method with simple and reproducible ways using a wet process and also with the progress of the observation techniques possible to characterize down to nanoscale order using high-resolution TEM, SEM, and surface probe microscopes like AFM.

The nanotechnology is considered as the innovative technology to bring about an industrial revolution in this era and is actively studied through the world. Especially, metal nanoparticles attract interests as the basic materials in the nanotechnology. The metal nanoparticles have many features, and the most of them are related with a surface or an interface. The surface area (total surface area) of the particles per unit weight becomes larger as the size of the particle becomes smaller. For instance, the specific surface area of the particle with the diameter of 5 nm is two million times larger compared with that with the diameter of 1 cm. A single atom has 100 % of surface area. Surface positions are occupied by 12 atoms (92 % of total atoms) in the nanoparticle consisted of 13 atoms and occupied by 42 atoms (76 %) in the nanoparticle consisted of 55 atoms.

- (a)
Confinement of light within the nanoscale region beyond the diffraction limit

- (b)
Enhancement of the local electric field on the vicinity of the object inducing a SPR

- (c)
High sensibility of the resonance condition to the state of the surface of the metal nanoparticle

Applications in the fields of nonlinear optics and near-field optics have been developed according to these effects, and the researches on the second harmonic generation (SHG), optical switching, optical logic circuits, etc., are actively carried out. Especially, the noble metals like gold and silver have the resonance wavelength in the visible region, and various applications for optical devices can be expected. A gold nanoparticle with the diameter of around 20 nm exhibits a red color, and it becomes a dark brownish color by reducing the diameter further [1, 2]. The origin of the red color ascribed to the absorption caused by the SPR and the effect has been utilized for the red color in the stained glass. Faraday first succeeded to obtain the red-colored gold nanoparticles in the dispersed colloidal system 150 years ago by controlled adding of yellow phosphor to the tetrachloroaurate [3, 4]. He also found out that such gold sol becomes stabilized by adding a protective colloid like a gelatin. Turkevich et al. observed directly these gold nanoparticles in 1951 using a SEM after 50 years of Faraday’s works [5]. They also tried to make clear the generation mechanism of nanoparticles by preparing various sizes of gold nanoparticles [6]. Further, they succeeded to synthesize alloy nanoparticles like Pt–Au, Pt–Pa, etc., without protective colloids [7]. On the other hand, Nord et al. reported the preparation of nanoparticles with protective polymers like polyvinyl alcohol [8, 9, 10].

The nanoparticles which exhibit a plasmon absorption in the visible region are typically gold, silver, copper, and other metal nanoparticles that generally show stronger absorptions in the longer wavelength region giving a brownish color. The features brought by reducing metals to a nanoscale size are not only colors but also catalytic behaviors coming from the extremely large specific surface area [11, 12, 13]. Haruta et al. recently discovered the function of gold nanoparticles as an effective catalyst which have been known inactive as a catalyst [14]. Furthermore, it was found that nanoparticles of Pt group work as the catalyst for an effective hydrogenation and also as the catalyst to extract hydrogen from water [15, 16, 17].

The review articles on the topics of metal nanoparticles have been issued recently reflecting the active researching field of plasmonics [18, 19, 20, 21, 22, 23, 24, 25, 26]. Most of the reviews are discussed about the synthesis and optical behaviors of metal nanoparticles. There are few reviews commented on the composite materials between dielectric materials and metal nanoparticles [27]. In these reviews, the optical behaviors of the composite materials consisting of oxide materials and noble metal nanoparticles are especially reviewed minutely using various mixing models for the effective permittivity of the composite materials.

## 4 Experimental and Analytical Methodology

### 4.1 Fabrication Methods of Metal Nanoparticles

Classification of synthesis methods of nanoparticles by chemical process

Reduction | Applicable metals | Capping |
---|---|---|

Alcohol | Noble metals | Water-soluble polymer |

Polyol | Transition metals, alloy | Polymer |

Amino alcohol | Gold, silver | Polymer |

Diborane | Gold, palladium, alloy | Phosphine ligand |

Hydrogen | Noble metals, transition metals | Amphiphilic molecule, long-chain amine |

Citric acidAscorbic acid | Gold, silver, palladium, platinum | Polymer, metal ligand |

Hydrazine | Noble metals, copper | Amphiphilic molecule, polymer |

Borohydride | Noble metals, transition metals | Metal ligand, amphiphilic molecule |

Light | Noble metals | Amphiphilic molecule, polymer |

Gamma ray | Noble metals, transition metals | Amphiphilic molecule, polymer |

Supersonic wave | Noble metals | Polymer |

Laser | Noble metals | Amphiphilic molecule |

Thermal decomposition | Gold, silver | Amphiphilic molecule, metal ligand |

Electrochemical | Metals | Amphiphilic molecule |

Supercritical fluid | Metals | Polymer |

In the wet method, metal atoms (0 valence) are obtained by introducing the precursors like metal ions or metal complexes into the solution followed by the thermal decomposition of metal complex (0 valence) or the reduction of metal ions. Metals with 0 valence are generally derived from the complexes by a thermal decomposition [28]. On the other hand, the reagents like alcohol [29, 30, 31], polyol [32, 33, 34, 35], aldehyde, citric acid and its salt, ascorbic acid and its salt, hydrazine [36], hydrogen [37, 38], diborane [39], boron hydride, alkylammonium salt [40, 41, 42, 43], alcohol amines, phosphor, etc., are used as the agents to reduce metal ions. The physical energy like γ-ray [44, 45, 46, 47], X-ray, ultraviolet and visible light [48, 49, 50], heat, ultrasonic wave, etc., are also used (Table 8.1).

### 4.2 Fabrication Methods of Dielectric Films Dispersed with Metal Nanoparticles

The typical methods to synthesize composite materials of metal nanoparticles and dielectric materials as a matrix are a melt and quench method [51, 52, 53], a sputtering method [54, 55, 56], an ion implantation method [57, 58, 59], etc.

The melt and quench method has been known well since olden time and applied to stained glasses according to their brilliant purple-red colors. The method is still used widely for color filters. In the method, the mixtures of SiO_{2}, Al_{2}O_{3}, and Na_{2}O together with the trace of metal (Au, Ag, or Cu) are put into a Pt crucible and melted at over 1,000 °C. High-quality glasses with less content of impurities and small departure from the stoichiometry are synthesized. After melting, the melt is poured among the twin rollers and quenched to the glassy state. After the process, metal ions melted within the glass are condensed to form aggregates of metal nanoparticles by thermal annealing at the temperature around the softening temperature of the glass to produce composite materials. The size and the density of the metal nanoparticles strongly depend on the composition of the glass, the temperature, and the time of thermal annealing. Large homogeneous glasses dispersed with metal nanoparticles are possible to fabricate at a low cost. However, there are several problems that the maximum content of the metal nanoparticles is limited below the solubility limit of about 0.1 vol.%, and it is also difficult to prepare the glass dispersed with high density of metal elements because metal elements are evaporated out during the high temperature process.

The sol–gel method is well known as the technique to synthesize glasses and ceramics by a hydrolysis reaction of a metal alchoxide solution [60, 61]. By using this technique, glass fibers, antireflection films, and contact lenses hybridized with a polymer are fabricated industrially. It is possible to synthesize bulk, fiber, and thin-film materials by this method. Composite materials dispersed with metal or semiconductor nanoparticles are possible to fabricate by the sol–gel method.

The fabrication process of SiO_{2} films dispersed with gold nanoparticles is shortly explained as a typical example. TEOS and metal salt as source materials, ethanol and water as solvents, and acid as a catalyst are mixed to form a sol solution. By using the sol solution, a gel-like thin film is coated on a substrate by the dip or the spin-coating method. Metal atoms dissolved within the glass are precipitated to form metal nanoparticles by heating the gel film. The thickness of the film is controlled by the withdrawal speed of the substrate during the dip coating, and the size of the metal nanoparticle is controlled by the annealing temperature. Both bulk-like and film-like glasses dispersed with metal nanoparticles are possible to fabricate with the relatively simple system.

An ion implantation technique is mainly used for the doping of impurities or the surface modification of solids in the semiconductor industry. Colloidal metal nanoparticles are possible to be precipitated within a glass by a higher doping than a conventional doping density. By using this method, it is possible to disperse nanoparticles into various matrix materials and control the size of nanoparticles by the thermal annealing after the implantation. However, strains induced by the damage are remained in the substrate due to the high doping (∼10^{17}dose), and very expensive systems are required.

The sputtering method is widely used for many fields as a thin-film coating technique on various types of substrates. Especially, insulators and materials with high melting temperatures are easily deposited by using a high-frequency sputtering. By the method, the dielectric matrix films dispersed with metal nanoparticles at high density are possible to deposit.

As other methods, film depositions by vacuum evaporation using resistive heating, by laser ablation, etc., have been reported.

### 4.3 Basic Theory on the Optical Properties of Nanoparticles

The optical responses of materials are described by using dielectric functions. The dielectric functions depend on the frequency of light and take specific values depending on the materials. The major particles to respond in the visible wavelength region are electrons. In view of this response, the related materials are classified into insulators or conductors which are featured by the motions of bound or free electrons, respectively. The optical responses of these materials are described using Maxwell’s equations.

The size and shape of the material strongly affect the optical response. Such effects are reflected in the boundary conditions on solving Maxwell’s equations. However, the quantum effects appear by reducing the size of the material to a nanoscale, and the dielectric function varies largely due to the discrete electronic states. Furthermore, the surface effects are enhanced by reducing the volume and peculiar optical responses which are not observed when bulk materials appear.

#### Drude Model for Free Electrons in Metal

*and –*

**r***e*are the displacement and charge of the electron. This equation is solved as follows assuming an incident light with the electric field

*=*

**E**

**E**_{ 0 }e

^{-iωt}of the angular frequency ω:

*e*

*, and the dipole moment per unit volume (polarization*

**r***P*) is given as follows using the number of electrons

*N*per unit volume:

*ε*(

*ω*) is given as

*ω*

_{ p }is the plasma frequency defined as

The plasma frequency corresponds to the frequency of the free oscillation mode of the longitudinal wave in a homogeneous metal. The wave transmits as the coupled mode between the polarization of the free electron and the depolarization field induced by the polarization. The restoration force of the oscillation becomes stronger as the density of the free electrons becomes larger. Free electrons within the metal are possible to respond for light at lower frequency than the plasma frequency which gives the high reflectivity of light. However, free electrons cannot respond to light at higher frequency than the plasma frequency, and light can penetrate into the metal.

#### Surface Wave Mode

_{1}(ω) and ε

_{2}(ω) satisfy following conditions for the electric vector parallel and vertical to the interface as shown in Fig. 8.5:

_{1}or ε

_{2}must take a negative value, which implies that there is an exponentially damping mode perpendicular to the interface. In other words, this condition is required to activate the eigenmodes existing at the interface. As mentioned previously, the permittivity of metal takes a negative value below the plasma frequency. By using the conservation of wavenumber at respective material, following relations are derived:

Either ε_{1} or ε_{2} is negative, and ε_{1} + ε_{2} is also requested to be negative to satisfy the equation.

*ε*(

*ω*) =

*1–ω*

_{ p }

^{ 2 }

*/ω*

^{ 2 }

*.*The surface plasmon mode exists in the region \( \omega < {\omega_p}/\sqrt {2} \), which corresponds to the condition of negative value of ε

_{1}+ ε

_{2}. The wavenumber of the surface plasmon becomes larger as

*ω*closes to

*ω*

_{ p }. The dispersion curve of light in vacuum does not cross the dispersion curve of the surface wave, which means the surface waves cannot be excited by irradiating light upon the metal surface from vacuum. As the excitation method of the surface plasmon, the irradiation of light from the dielectric layer in the three-layer system consisting of the dielectric, metal, and air layers is often used. The wavenumber of light within the dielectric medium becomes larger according to the refractive index of the medium, and the steepness in the dispersion curve becomes smaller to cross the dispersion curve of surface plasmon as illustrated in Fig. 8.7.

#### Point Dipole

*by an oscillating point dipole moment*

**r***located at the original point is given as*

**p***and*

**n***are the unit vector toward*

**k***direction and wave vector, respectively. The first term corresponds to the field induced by the accelerating movement of the charge and damps slowly as 1/r form the dipole. This component of the radiation field extends to long distance and can be observed by our eyes. The energy distribution (Poynting vector) by this component emitted from the electric dipole moment is shown in Fig. 8.8. The second term called an inductive electric field is induced from the movement of the charge and damps rapidly as 1/r*

**r**^{2}. The third term corresponds to the quasistatic electric field generated from the static charge and damps very rapidly as 1/r

^{3}and can be neglected at far field from the point dipole. The electric field distribution induced by the dipole is illustrated in Fig. 8.9. As seen in the figure, the induced electric field along the dipole moment has the same direction with the dipole moment vector, and the electric field perpendicular to the moment has the reverse direction to the moment, which is the most important factor to treat the interaction between particles and light in the nanoscale region.

#### Dipole Moment of Nanoparticle

The optical response of the metal particle changes largely depending on the size, shape, and surrounding medium in the frequency region where the dielectric function takes negative values in the lower frequency region than the plasma frequency. Apart from the point dipole model, it is required to consider the model which takes the shape and environment of the particle into consideration to understand such phenomena.

_{0}as shown in Fig. 8.10. The surface charges are induced on the surface of the dielectrics with the charge density of σ given as

*and*

**P***are the density of the polarized dipole moment and the unit vector perpendicular to the surface, respectively. The surface charges depend on the shape of the particle and surrounding medium and play an essential role in the optical response of the nanoparticle. The internal field*

**n***is different from the external field*

**E**

**E**_{ 0 }and composed of the summation of the contribution from all dipoles as shown in Fig. 8.11. The summation is proved equal to the electric field

**E**_{ 1 }induced by the surface charge σ in vacuum, and the electric field

**E**_{1}is called a depolarization electric field as the direction is reverse to the external field E

_{0}. As a result, the internal field

*within a dielectric material polarized homogeneously is given as*

**E**

**E**_{1i}is given as

*P*

_{i}(i =

*x*,

*y*,

*z*) is the component of the polarization along the principal axes of the ellipsoid. N

_{i}is called a depolarization factor and has the rule taking a constant value of the summation (N

_{x}+ N

_{y}+ N

_{z}= 4π). The depolarization factors for the particle with the typical shapes like a sphere, rod, and disk are shown in Fig. 8.12.

*is related with the internal field*

**P***using a dielectric susceptibility χ as*

**E***is written as follows if the external field*

**E**

**E**_{0}is parallel to the principal axis of the ellipsoid and the depolarization factor is

*N*:

*is expressed by the external field*

**P**

**E**_{0}as

*of the particle with the volume*

**p***V*is expressed as follows using the dielectric function:

*is given as*

**p**#### Surface Plasmon Resonance (SPR)

*ε*(

*ω*) = −2. As a result, the electric field around the particle is enhanced. In the case of a metal, such resonant oscillation of the electric polarization is called a localized surface plasmon. The dielectric function of the metal takes negative values at the lower frequency than the plasma frequency as shown in Fig. 8.4. The resonance phenomenon occurs in the visible wavelength region in the case of Au and Ag, because the plasma frequencies of these metals locate near the visible region. The condition where the denominator of the Eq. 8.23 becomes 0 in the Drude model gives the following relation between the resonance frequency and depolarization factor:

*ω*becomes lower as the depolarization factor becomes smaller as illustrated in Fig. 8.13.

The result implies this resonance phenomenon is sensitive to the shape of the particle and also the surrounding medium.

## 5 Key Research Findings

### 5.1 Shape Control Techniques of Nanoparticles

The nanostructure materials of noble metals are expected for wide range of applications in the fields of catalytic reactions, electronics, surface plasmon resonance (SPR), surface-enhanced Raman scattering (SERS), and biological and medical researches [62, 63, 64]. A specific shape of the nanostructure is required to optimize the performance for the application in the field. For example, of the application for catalysts, it is required to disperse the noble metal nanostructure on the ceramics with the large surface area. Nanowires are required for the observation of electric transport properties and the connection among electronic devices. On the other hand, the shape of the structure becomes the parameters to enhance and control the sensibility of the performance of noble metal nanostructures. The shapes of the nanostructure of Au or Ag determine the SPR properties and also become the important factors in the SERS measurement [65]. It is quite important to control the shape of the nanostructure of noble metals to improve the performance in various application fields.

In most case, the relative free energy of respective crystal plane changes by introducing a capping agent with high selectivity according to the intense reaction with a specific crystal plane. Such process is the important means to control the relative growth rate of respective crystal plane. This type of chemical adsorption or surface capping strongly affects the final morphology of the metal nanostructure. For instance, polyvinylpyrrolidone (PVP) is a polymer capping agent in which oxygen atoms strongly couple with the (100) plane of Ag and the formation of Ag nanowire or nanocube is enhanced [67, 68].

#### Polyol Process in Nanostructures

Polyol can decompose many metal salts, and its reduction ability depends on the temperature. As a result, the process is appropriate to synthesize noble metal nanostructures like Ag, Au, Pd, Pt, Rh, Ru, and Ir.

#### Silver Nanoparticles

_{2}to the typical polyol reduction of AgNO

_{3}using EG [70, 71]. Cu(II) ions are reduced to Cu(I) ions by EG, and both ions play an important role in this synthesis. The mechanism is shortly introduced below. Oxygen adsorbed on the metal surface is consumed in the oxidation from Cu(I) to Cu(II), and the oxygen concentration around the surface decreases, which prevent multiply twinned crystals to dissolve by oxidation etching to increase the sites for Ag to deposit and promote the growth of wire. Ethylene glycol makes recycle of Cu(II) to Cu(I), and adding of small quantity of Cu(II) ions is required for the adjustment of the oxygen concentration. Further, chloride ions control the isolated Ag

^{+}concentration in the solution by forming AgCl, which makes slow the reduction rate and promotes the preferential growth of Ag at the high-energy twinned boundary on the top of the wire. In the typical synthesis, PVP plays a role to protect preferentially (100) side plane of nanowire. Other trace ion species are important to control the morphology of Ag nanostructures. In the polyol reduction of AgNO

_{3}, silver nanocubes are possible to synthesize homogeneously in large quantities by adding a small quantity of sulfates in the form of either Na

_{2}S or NaHS [10, 11, 12]. The SEM images of Ag nanorods and nanocubes synthesized by polyol process are shown in Figs. 8.16 and 8.17, respectively. UV–Vis–NIR absorption spectra of the reaction mixture at various reaction times during the synthesis of Ag nanocubes are shown in Fig. 8.18 [71]. The resonance peaks show the red shift depending on the reaction time.

#### Gold Nanoparticles

Gold nanostructures attract the intentions according to their excellent chemical stability, biological inactivity, SPR/SERS properties, and unique catalytic performance. Many methods have been tried to control the morphology of the Au nanostructures [72, 73, 74, 75]. As mentioned for the Ag system previously, the selection of reducing or stabilizing agents and reaction temperatures are important to form a specific morphology. In the Au system, the polyol reduction also promotes to form thermodynamically preferable polygon of Au nanostructure using strong reducing agents. But there are several different points compared with Ag. It is considered that the bonding of PVP to Au is not strong enough to promote the formation of (100) plane in contrast to the Ag system. As a result, the single crystal and multiply twinned polygon surrounded by (111) planes like octahedron, decahedron, and icosahedron are the preferable nanocrystals of Au in the polyol process. For example, Au octahedrons are synthesized at high yields by the improved polyol process using polyethylene glycol 600 as the reducing agent [76]. Polyethylene glycol 600 is used also as a solvent in this synthesis. The small addition of NaBH_{4} before adding an AuCl_{3} water solution is the key to obtain homogeneous Au octahedrons. The added NaBH_{4} works as a strong reducing agent and rapid reduction of Au precursor proceeds. On the other hand, multiply twinned nanostructures of Au can be synthesized by increasing the PVP concentration in the polyol reducing process of HAuCl_{4} using diethylene glycol [77].

#### Palladium Nanoparticles

Palladium is an important catalyst used in the various reactions like hydrogenation and dehydrogenation and also used in the reactions to produce C–C bond like Suzuki coupling, Heck coupling, and Stille coupling. The major morphology of Pd nanostructure in the polyol reduction of Na_{2}PdCl_{4} using EG takes an octahedron [78]. The morphologies of Pd nanostructures are controlled by the introduction of a specific capping agent like Br ions or by the oxidation etching. Pd nanobars are possible to synthesize by adding Br ions to the polyol reduction of Na_{2}PdCl_{4} [79, 80]. Bromide ions bond strongly to the (100) plane of Pd, and seed crystals of cubic Pd are formed at the nucleation stage. In this process, the oxidation etching occurs locally on one of the six planes of the cubic seed crystal, and a part of Br ions is removed, which promotes the selective growth of the cubic seed crystal along one direction to form Pd nanobars surrounded with (100) planes.

As an another example, Pd triangle nanoplates are formed by introducing Fe(III) species during the polyol process, which is due to the large decrease of reducing rate by using Fe(III) species and O_{2}/Cl pairs as the wet etching liquid for Pd(0) [81]. These Pd nanoplates exhibit a SPR peak in the visible region as the size becomes larger and are possible to use as the SERS active substrate owing to their sharp corners and edges.

_{3}and 5 mM HCl are shown in Fig. 8.23 [81]. The molar ratio of PVP to Pd precursor is 5 where the concentration given in the figure caption is the final value in the reaction solution. UV–Vis extinction spectra of as-prepared triangular and hexagonal nanoplates are shown in Fig. 8.24a [81]. The extinction, absorption, and scattering coefficients of a triangular nanoplate calculated using the DDA method are shown in Fig. 8.24b [81], where all random configurations of the plate with respect to the incident light were averaged. The optical coefficients were defined as C/πα

_{eff}

^{2}where C was the cross sections obtained directly from DDA calculation and α

_{eff}was defined through the concept of an effective volume equal to 4πα

_{eff}

^{3}/3 for the plate.

### 5.2 Spectral Analysis on the Absorption Spectra of Dielectric Films Dispersed with Metal Nanoparticles

#### Spectral Analysis on the Absorption Spectra by Surface Plasmon Using Mie Theory

The charges within a sphere are forced to oscillate when light incidents on the sphere. Such movement of charge emits light inside and outside of the sphere. As a result, the electromagnetic field at any point is the superposition of electromagnetic fields of incident and emitted light, and so-called diffraction occurs. In olden time, this phenomenon gave the basic theory for the Rayleigh scattering of solar ray by air molecules, which was the proof of the blue sky. In recent time, it gives the basic theory for the optical properties of the photonic crystal composed of ordered spheres.

*E*

_{i}can be expanded by vector spherical wave functions assuming a liner polarized plane wave traveling to z direction described as follows:

_{x}is the x component of the amplitude vector g. The amplitude vector component of the magnetic field

*H*

_{i}is given as g

_{y}because the direction of the magnetic field is perpendicular to that of the electric field. Utilizing Eq. 8.27,

*H*

_{i}is given as

When light incidents into the sphere, two types of components of light occur. One is departing from the sphere, and the other is trapped within the sphere after the reflection and transmission. The former component should vanish apart enough from the sphere and expressed by spherical Bessel function of the third kind, that is, Hankel function. On the other hand, the component within the sphere should be finite at the center of the sphere and described as the same expression with the incident electromagnetic field.

_{n}(k

_{2}R) in the expression of m

^{(3)}

_{e1n},

_{o1n}and n

^{(3)}

_{e1n},

_{o1n}is replaced by the Hankel function h

^{(1)}n(k

_{2}R). On the other hand, the electromagnetic field within the sphere (R < a) is written as

_{ien}and b

_{ien}. The values of a

_{ien}and b

_{ien}are determined by using the boundary conditions which claims that the tangential components of the electromagnetic fields at the surface of the sphere should be continuous:

_{1}a = Nρ is obtained by putting k

_{1}≡ Nk

_{2}and ρ ≡ k

_{2}a, and following relations are derived from Eq. 8.32:

The values a_{ien} and b_{ien} increase largely due to the resonance effect as the *ω* of incident light becomes close to the angular frequency of the WG mode. The light scattering intensity from the sphere becomes enhanced by exciting the WG mode.

Gustav Mie was the first to rigorously explain the colors exhibited by metal colloids using Maxwell’s equations. However, exact solutions to Maxwell’s equations are known only for spheres, shells, spheroids, and infinite cylinders, and an approximation is required to solve the equations for other geometries. The approximation of choice is called the discrete dipole approximation (DDA) [70, 71, 72, 82, 83]. In a DDA calculation, the nanoparticle is discretized into a cubic array of N polarizable points, with each point representing the polarizability of a discrete volume of material. The presence of an electromagnetic field (light) gives these points a dipole moment. In the steady state limit, each dipole is polarized by an electric field that includes contributions from incident light and every other dipole in the array. The polarization at each point (P_{j}) must thus be calculated through an iterative procedure, which can be accelerated with FFT methods (thus the periodicity of the array). The polarization P_{j} can be plugged into an equation to determine the scattering and absorption cross sections of the particle, which in turn are divided by the cross-sectional area to give the dimensionless optical coefficients. As the coefficients can be solved exactly for this N-point array, the only approximation is in the number of points. A greater N increases the accuracy of the calculation at the expense of computation time.

#### Spectral Analysis on the Absorption Spectra by Surface Plasmon for the Dielectric Materials Dispersed with Metal Nanoparticles Using Effective Medium Approximation (EMA) Theory

The interaction between light and a metal sphere is treated by the Mie theory. However, real metal nanoparticles take various shapes, and the theories for other shapes like ellipsoids are required. The interaction between nanoparticles is also related for the high concentration of nanoparticles. The process parameters with which one can control the macroscopic properties of the composite (dielectric materials dispersed with metal nanoparticles) are related to the individual material phases, their relative fractional volumes, and the shapes of the inclusions. The effective medium theories to treat the composite media dispersed with nanoparticles with extended shape parameters are reviewed.

The EMA theory is the approximation which assumes the composite material as a homogeneous medium with an effective permittivity. The theoretical model to mix the components of the composite is featured by the effective permittivity. Classical mixing formulae, like the Maxwell–Garnett and Bruggeman [84, 85], have been widely used to calculate the effective permittivity of such mixtures. These formulae work very well in some fields of applications. For example, in the area of the remote sensing, the contrasts of the permittivity among various phases of the heterogeneous structures are not very large. In the case, both Maxwell–Garnett and Bruggeman mixing formulae predict reasonably accurate results.

However, in the case of applications to the material science, the contrast of the permittivity between the matrix and the inclusion is often very large. In some cases, the composites are required to be constructed using a polymer matrix because of desired mechanical properties. The contrast of the electrical conductivity between the matrix and the metal nanoparticles may be extremely large, even several thousands. In addition, the volume filling ratio of inclusions changes over wide range. The combination of highly packed inclusions with the high electrical contrast leads to the situation where two mixing equations should be used simultaneously. The choice of the mixing theory to give the desired effective permittivity of the composite is very difficult when one must find out general principles to choose the best material combination for the given application. Another shortcoming of the semiempirical models is that the parameters do not correspond to separate physical properties of the mixture. For example, if one parameter is responsible for the percolation threshold of the mixture, a change in this parameter also changes the volume filling ratio properties of the mixture. The effective permittivity of the composite with very low filling ratios of spheres is dominated by the shape of the inclusions. As a result, the percolation parameter is also dependent on the shape of the inclusions.

##### Classical Mixing Formulae for the Composite Materials

_{i}and the volume filling fraction f is dispersed in the matrix material with the permittivity

*ε*

_{ e }and volume fraction

*1-f*. The basic idea behind all classical mixing formulae for the composite materials is to try to estimate and define the effective permittivity

*ε*

_{ eff }for the mixture with the assumption that the dimension of inhomogeneity within the composite is much smaller than the wavelength of the incident light into the material.

*ε*

_{ i }is the permittivity of the inclusions, and

*ε*

_{ e }is the permittivity of the matrix. The depolarization factor of the dispersed ellipsoids is denoted by N

_{x,y,z}. The factor of the spheres is N

_{x}= N

_{y}= N

_{z}= 1/3. The depolarization factors for general ellipsoids contain elliptic integrals, but closed-form expression can be written for spheroids.

General mixing rules based on EMA theories

1. Conventional effective medium theories |

(a) Separated-grain topology |

(a-1) Maxwell–Garnett model [84] \( \hskip 30pt\displaystyle\frac{{\epsilon - {\epsilon_m}}}{{\epsilon + 2{\epsilon_m}}} = {f_i}\frac{{{\epsilon_i} - {\epsilon_m}}}{{{\epsilon_i} + 2{\epsilon_m}}}. \) |

(a-2) Inverse Maxwell–Garnett model [84] \( \hskip 30pt\displaystyle\frac{{\epsilon - {\epsilon_i}}}{{\epsilon + 2{\epsilon_i}}} = {f_m}\frac{{{\epsilon_m} - {\epsilon_i}}}{{{\epsilon_m} + 2{\epsilon_i}}}. \) |

(a-3) Coherent potential model [86] \( \hskip 30pt\displaystyle\frac{{\epsilon - {\epsilon_m}}}{{4\epsilon + 2{\epsilon_m}}} = {f_i}\frac{{{\epsilon_i} - {\epsilon_m}}}{{{\epsilon_i} + 3\epsilon - {\epsilon_m}}} \) |

(a-4) Inverse coherent potential model [86] \( \hskip 30pt\displaystyle\frac{{\epsilon - {\epsilon_i}}}{{4\epsilon + 2{\epsilon_i}}} = {f_m}\frac{{{\epsilon_m} - {\epsilon_i}}}{{{\epsilon_m} + 3\epsilon - {\epsilon_i}}} \) |

(b) Aggregated-grain topology |

(b-1) Bruggeman model [85] \( \hskip 34pt\displaystyle{f_i}\frac{{{\epsilon_i} - \epsilon }}{{\epsilon i + 2{\epsilon_i}}} + {f_m}\frac{{{\epsilon_m} - \epsilon }}{{{\epsilon_m} + 2\epsilon }} = 0 \) |

(b-2) Looyenga model [87] \( \hskip 33pt\displaystyle{\epsilon^{{1/3}}} = {f_i}\epsilon_i^{{1/3}} + {f_m}\epsilon_m^{{1/3}} \) |

(b-3) Lichtenecker model [88, 89] \( \hskip 31pt\displaystyle\log {\epsilon_{{eff}}} = f\log {\epsilon_i} + (1 - f)\log {\epsilon_e} \) |

2. Extended effective medium theories |

(a) Stroud and Pan model [90] \( \hskip 30pt\displaystyle\sum\limits_{{j = i,m}} {{f_i}\left( {\frac{{{\epsilon_j} - \epsilon }}{{{\epsilon_j} + 2\epsilon }} + \frac{{x_j^2}}{{30}}({\epsilon_j} - \epsilon )} \right) = 0} \) |

(b) Wachniewski and McClung model [91] \(\begin{array}{lll}\mathop \sum \limits_{j = i,m} {f_i}\left( {\epsilon \frac{{{\epsilon_j} - \epsilon }}{{{\epsilon_j} + 2\epsilon }} + \frac{{x_j^2}}{{10}}\left[ 6 \right.\epsilon (\epsilon + {\epsilon_j})\frac{{5{\epsilon_j} - 10\epsilon + ({\epsilon_j}/\epsilon ){\epsilon_j} + 4({\epsilon_j}/\epsilon )}}{{{{({\epsilon_j} + 2\epsilon )}^2}}}} \right.\\ - (\epsilon + {\epsilon_j})({\epsilon_j}/\epsilon + 1)\frac{{3\epsilon }}{{{\epsilon_j} + 2\epsilon }} - (\epsilon + {\varepsilon_j}) - (\epsilon + {\epsilon_j})\frac{{5\epsilon }}{{3\epsilon + 2{\epsilon_j}}} \\{\left. { +\ 2\epsilon \left( {\sqrt {{{\epsilon_j}/\epsilon }} + 1} \right) + 4\epsilon \sqrt {{{\epsilon_j}/\epsilon }} } \right)}= 0.\end{array} \) |

_{z}> a

_{x}= a

_{y}), the depolarization factors are written as

_{z}< a

_{x}= a

_{y}), the factors are written as

_{x}, a

_{y}, and a

_{z}are the three semiaxes of the ellipsoid. Depolarization factors for any ellipsoid satisfy N

_{x}+ N

_{y}+ N

_{z}= 1.

The first attempt to describe optical constants of inhomogeneous materials with effective refractive index dates back to 1904 (the so-called Maxwell–Garnett rule [84]). Bruggeman derived an independent mixing rule for the effective dielectric constant of inhomogeneous materials [85]. Both the Maxwell–Garnett and Bruggeman rules are based on the assumption of the Rayleigh size of the inhomogeneity but consider different internal structures of the mixtures. Maxwell and Garnett (M–G) initiated the study of nanocomposites as they investigated the optical properties of metal colloids with minute metal spheres embedded in an optically linear host material. The size of the inclusions is assumed to be much smaller than the wavelength of the incident light, and the composite can be treated as one homogeneous medium with the effective permittivity as shown in Fig. 8.25. In the classical Maxwell-Garnett mixing rule, it is assumed that the local electric field on each ellipsoid is the superposition of the average external field and the average field caused by other spheres. The result for the effective permittivity is also known as the Clausius–Mossotti equation. Unfortunately, the Maxwell-Garnett model is restricted to relatively small volume fraction of the inclusions because of the assumptions imposed on the model. The Maxwell–Garnett rule assumes that the composite material is composed of a matrix material with embedded inclusions. As a result, the effective refractive index depends on which material is considered as the matrix and which as the inclusions. The Bruggeman rule avoids this problem by considering the medium composed of a set of randomly distributed cells of different materials. For large volume fraction of the inclusions and for randomly intermixed constituents, Bruggeman (Polder–van Santen) derived the EMA model by considering the host material as an effective medium. It assumes asymmetry between nanoparticles and the matrix phase. The formula similar to the Maxwell-Garnett equation was derived for the small volume fractions of nanoparticles. These two mixing rules have been generalized to adapt to the multiphase mixtures including magnetic, anisotropic, and chiral components; polydispersions of inhomogeneity; nonspherical (ellipsoidal, cylindrical, cubic, Jakes, and rods) and stratified inclusions. Even though the Maxwell–Garnett and Bruggeman rules still remain as the most general mixing rules, a variety of other effective medium theories have been developed. One of the most essential developments of the effective medium theories was suggested by Stroud and Pan [90]. It allows the inhomogeneity to be modeled not as infinitely small like Rayleigh particles but as larger particles which size is limited only by the condition that the attenuation length (mean free path of photon) is large compared to the characteristic particle dimension.

Three mathematical approaches were used to derive the mixing rules: direct calculation of the average polarizability of the composite material (see, e.g., [92]), the T-matrix method [93], and the method based on the requirement that the complex forward scattering amplitude of the light passing through the mixture should be zero [94] when it is immersed in a medium of the effective refractive index. Classification of EMA theories based on the physical description of the composite materials looks more illustrative, and it is described below. The physical description depends on the properties of the inclusions and on their arrangement inside the composite material. The main property of the inclusions is their size. The majority of EMA theories were developed within the flamework of “quasistatic approximation.” They were derived at zero frequency, that is, inclusions are considered as very small like Rayleigh particles that can be treated as dipoles. As was mentioned above, the paper by Stroud and Pan suggested a new approach in the framework by which a number of EMA theories for inclusions of larger size have been developed. This method is known as the “dynamic approximation” or extended EMA theory [95] because it extends the effective medium approach to finite frequencies, using a full multipole expansion to treat scatterings from small particles. Within this method, the scattering by inclusions is represented by multipole coefficients obtained from the exact solution of the Maxwell equations for particles that have the same shape as the inclusions (spheres, spheroids, cylinders, etc.). The further classification of EMA theories as well as extended EMA theories can be done according to the number of components in the mixture, the shape and the size distributions of the inclusions, and other properties of the individual inclusions, including their structures (for instance, core–mantle inclusions can be considered), magnetic properties, anisotropy, chirality, etc.

Another important characteristic is the internal structure (topology) of the composite material. The main structures usually treated in EMA theories are aggregated- or separated-grain topologies. “Separated-grain” means that the composite material can be represented as the matrix of a main material with the inclusions inside that are separated from each other by the matrix material. One of the most popular separated-grain EMA theories is the Maxwell–Garnett mixing rule. The separated-grain EMA theories give different results depending on which component of the mixture is chosen to be the matrix. The aggregated-grain topology represents the mixture as a set of “cells” made of one or the other component of the mixture. In this case, mixing rules are symmetric relative to which component is called matrix and which is called inclusions. The most popular aggregated-grain EMA theory is the Bruggeman mixing rule. Recent development of EMA theories for the astronomical application resulted in some new versions of EMA theories based on the aggregated-grain topology. They allow more complicated fractal, percolated, radially varying internal structures of composite materials [96].

#### Spectral Analysis on the Absorption Spectra by Surface Plasmon of ZrO_{2} Films Dispersed with Ag Nanoparticles Synthesized by Sol–Gel Method

_{2}–Ag composite materials [97, 98, 99]. The silver nanoparticle/ZrO

_{2}thin-film composites were prepared by a sol–gel method with various silver fill fractions. The synthesized films were analyzed by a UV–Vis–NIR spectrophotometer, a transmission electron microscope (TEM) and an X-ray diffractometer (XRD). The absorption spectra due to the silver surface plasmon resonance were simulated using the dielectric functions reflecting the M–G and the Bruggeman mixture rules. In this study, the composite materials are consisted of metal nanoparticles and dielectric materials, and two typical mixing models of Maxwell–Garnett and Bruggeman are applied to check the applicability for the simulation of the optical spectra. ZrO

_{2}thin films dispersed with silver nanoparticles were synthesized by the sol–gel method. The starting solution was prepared from zirconium n-propoxide, acetylacetone, 1-propanol, 2-propanol, and distilled water. The silver solution was prepared from silver nitrate and diethylenetriamine. The resulting solutions were mixed with the following molar ratios: Zr:Ag = 90:10, 80:20, 70:30, 60:40, 50:50, 40:60, and 30:70. These densities of silver correspond to the volume fractions of 5.3 %, 11.1 %, 17.7 %, 25.0 %, 33.3 %, 42.9 %, and 53.9 %, respectively. X-ray diffraction measurements were performed in a 2θ scan configuration in the range of 10–80° using an X-ray diffractometer with Cu Kα radiation (MacScience, MXP18HF). The X-ray diffraction peaks were observed at 2θ of 38.1°, 44.3°, 64.5°, and 77.5° which were identified by JCPDS card as (1 1 1), (2 0 0), (2 2 0), and (3 1 1) planes of silver, respectively. A clear peak of ZrO

_{2}was not observed. It is supposed that the matrix material of ZrO

_{2}takes an amorphous structure. TEM images of silver nanocrystallites in the zirconia films are shown in Fig. 8.26 [97] for the nominal Ag to Zr molar ratio [Ag]/[Zr] of 0.25, 1.00, and 2.33, respectively. The silver nanoparticles can be clearly seen, embedded in the ZrO

_{2}matrix. For the nominal Ag to Zr molar ratio [Ag]/[Zr] = 0.25 (Fig. 8.26a [97]), the particles are well separated from each other and their shapes are basically spherical. For the nominal Ag to Zr molar ratio [Ag]/[Zr] = 2.33 (Fig. 8.26c [97]), the particles become coagulated and their shapes changed to an oval shape. For [Ag]/[Zr] = 1.00 (Fig. 8.26b [97]), the particles show both spherical and oval shapes, and the appearance of coagulation lies in the middle in Fig. 8.26 [97]. With the case of any silver density, silver took a particle phase but not matrix.

_{2}thin films doped with silver nanoparticles at various molar ratios are shown in Fig. 8.27 [98]. The films exhibit an absorption band centered at about 450 nm due to the surface plasmon resonance of silver nanoparticles. The absorption intensities become stronger as the densities of silver increase from 10 mol% to 50 mol%, while the peak wavelength remains constant. The FWHM of the peaks remains almost constant till the Ag density of 30 mol% and becomes relatively larger above the density. On the other hand, red shift of the absorption maximum to around 480 nm and the broadening of the peak were observed for the densities of silver above 60 %.

*ε*

_{ eff }was described using the permittivity of the inclusion

*ε*

_{ i }, the permittivity of matrix

*ε*

_{ e }, the volume fraction of inclusion f, and the depolarization factor N. The following dielectric equation is given from the M–G model:

*R*is the mean diameter of the nanoparticle for an ellipsoidal shape

*R*= (

*xyz*)

^{1/3}, in which

*x, y, z*represent the ellipsoid semiaxis, and

*β*is the parameter depending on the particle geometry related with the depolarization factor N which takes a value of 1/3 for spherical shapes. The parameter

*K*represents the ratio between the electric fields created at the particle position by the adjacent particles and created by the rest of the material.

_{2}matrix containing spherical silver nanoparticles by the Maxwell-Garnett model for different volume fractions f using the dielectric parameters of

*ε*

_{ m }= 3.57 (dielectric constant of ZrO

_{2}matrix),

*β*= 1/3 (spherical silver nanoparticles), and

*R*= 25 nm (mean diameter of Ag nanoparticles) as shown in Fig. 8.28 [97]. In the case of silver densities from 10 mol% to 40 mol%, the peak intensities become stronger keeping same spectral profile as the density of silver increases. This result shows that the dielectric property of the ZrO

_{2}–Ag composite does not depend on the silver density in the low volume fraction region. On the other hand, in the case of larger silver volume fraction region such as over 60 mol%, the absorption peak wavelength shows red shift from 450 to 480 nm and the spectral profile becomes broader, which cannot be simulated by the Maxwell-Garnett model.

_{2}–Ag composite with various shapes of silver nanoparticles ranging β parameter from 1/5 to 1/2 are shown in Fig. 8.29 [97]. These values of

*β*parameter define the particle aspect ratio from two times to half against the incident light direction, respectively. The absorption peak wavelength strongly depends on the

*β*value. The peak position shifts toward shorter wavelength from 580 to 380 nm as the parameter

*β*increases.

_{2}thin films containing 20 mol% of Ag nanoparticles were fitted. The particle diameter was fixed as 25 nm in the calculation from the TEM observation. The absorption spectrum calculated according to the M–G model is shown in Fig. 8.30a. The calculated absorption peak wavelength is well fitted to the experimental result except the difference in band width. The FWHM of the observed absorption band takes considerably larger value. This broadening of the band is considered caused by the distribution of the particle shape and size. Much better agreement with the experimental results was obtained when the particle shape distribution was considered as shown in Fig. 8.30a [97]. Excellent fit between the Maxwell-Garnett theory and experimental spectrum was found by assuming the particle geometry distribution corresponding to β parameters from 1/5 to 1/2 as shown in Fig. 8.30b [97]. The real distribution of the shape was observed for the synthesized composite system using a TEM, and the aspect ratio distribution of the particles from 1.0 to 2.75 was observed. This result supports the validity of the particle shape distribution used in the calculation.

_{2}–Ag composite is calculated by using the following equation:

### 5.3 Applications of Metal Nanoparticles to Various Optoelectric Fields

#### Application to Immobilization by Light and Gilding

Giant particles of around several hundred nm grow by irradiating the near-infrared laser light of 1,064 nm upon the gold nanoparticles tailored with dodecanethiol (DT). During the process, immobilization phenomena proceed. By utilizing the phenomena, a new type of gilding is developed. Nanoparticles are immobilized on a glass substrate depending on the photomask by irradiating the pulsed laser light of 532 nm through the mask after dipping the glass substrate into the colloidal solution mentioned above. In case of immobilization of the catalyst within the flow channel to fabricate the electric line, it is required to fabricate before formation of flowing channel. However, the process becomes possible after forming the flow channel by using this technique [100, 101].

#### Photoelectric Conversion Technique Utilizing the Enhanced Electric Field

The aggregates of gold nanoparticles function as porous electrodes with the 3-dimensional structures under the condition electric conductivity are maintained. As a result, the surface area of the electrode increases, and the enhancement of the efficiency of molecular excitations by the localized enhanced electric field is expected. In case of the electrodes, the enhancement of photocurrent was observed in the wavelength region where the surface plasmon is excited by copying a single layer of gold nanoparticles onto the transparent electrode. The enhancement of florescence is also observed due to the increase of the excitation efficiency of dye molecules by the enhanced local electric field [102, 103].

#### Application for Optical Memories

The rod-like gold nanoparticles are known to exhibit two plasmon bands according to the difference in the resonance condition of the surface plasmon for longer and shorter axes [104]. One of the features of this gold nanorod is the deformation by the heat induced by the light. The development of application to optical memory is possible by utilizing this light-induced deformation within polymer. Polyvinyl alcohol dispersed with gold nanorods is used as a recording medium, and the plasmon band for a longer axis is excited selectively by irradiating the linear polarized laser light of 1,064 nm to induce the deformation of nanorods with the longer axis aligned to the direction of polarization. After the process, the spectra originating from the rods without the deformation are observed. On the other hand, the plasmon band for shorter axis is excited by irradiating the laser light of 532 nm, and the rods aligned vertically regarding to the excitation of 1,064 nm deform selectively. It is possible to write and read using the polarized light with two colors by this principle.

#### Applications for Sensing

The localized enhanced electric field is extremely effective to the surface-enhanced Raman scattering (SERS). The mechanism of SERS has not yet been made clear, but it is utilized as an ultrahigh sensitive measuring method because the normal Raman scattering intensity is enhanced by 10^{10}. But the degree of enhancement is quite sensitive to the nanostructure, and the reproducible fabrication method of a SERS chip has not been established. The researches of SERS utilizing the aggregates of gold nanoparticles and other spectroscopic methods are advancing steadily, and the transmission type of sensing of the localized plasmon resonance is proposed.

## 6 Conclusions and Future Perspective

It is requested to derive an easy to use mixing formula for material engineering applications, which combines the feature of low volume filling ratio properties in the Maxwell–Garnett mixing equation to that of percolation properties in the Bruggeman mixing equation. The equation should not involve any adjustable-free parameters, and the derivation is based only on the electromagnetic field analysis. The required parameters are the shape and volume fraction of inclusions and the permittivity of the matrix and inclusions. The new mixing equation can predict accurately the effective permittivity as a function of volume fraction of inclusions in the case where the shape of inclusions can be modeled as ellipsoids and their shape parameter is evenly distributed. With small volume filling fractions of inclusions, the Maxwell–Garnett mixing equation should be applied, because it takes the effect of the shape of inclusions correctly into account. When approaching the percolation threshold, the Bruggeman equation should be effective.

In this section, the main result of the new mixing equation [105] that combines the desired properties of both Maxwell–Garnett and Bruggeman models is introduced. The derivation of the Maxwell–Garnett equation is based on combining the polarizability of an isolated particle with the averaged polarization effect of the effective medium. The drawback of the Maxwell–Garnett model is that when the volume filling ratio of spheres increases and the inclusion becomes connected to each other, the Maxwell–Garnett model underestimates the effective permittivity. In the derivation of the Bruggeman model, the internal field of spheres is calculated by assuming that the sphere is surrounded by the effective permittivity *ε* _{ eff }, not by the permittivity of the environment phase *ε* _{ e }. As a result, the Bruggeman equation is in good agreement with the simulated and measured values of the effective permittivity when the volume fraction of the inclusions is above the percolation threshold. However, when the volume fraction is below the percolation threshold, the Bruggeman tends to overestimate the effective permittivity.

*ε*

_{ eff }can be calculated using the Taylor expansion when the original background permittivity is

*ε*

_{ e }and a differential amount of spheres is added:

*V*is the volume of inclusion sphere,

*n*is the number of spheres in the mixture, and Δ

*n*is the change in the number of spheres.

*α*and Δ

*α*are the polarizability and the change in the polarizability, respectively. Equation 8.42 can be written as

*f*=

*nV*is the volume fraction of spheres. The corresponding differential equation finally gives the resulting differential mixing rule as

*ε*

_{ eff }

*= ε*

_{ e }for the zero filling value

*f*= 0. This differential equation is easy to solve numerically by starting from

*ε*

_{ eff }

*= ε*

_{ e }with f = 0. For practical reasons, it is actually easier to solve the inverse function

*df/dε*

_{ eff }, especially if the electrical contrast between

*ε*

_{ i }and

*ε*

_{ e }is large. Even if no limiting value for

*f*= 1 is assumed, the limit for

*ε*

_{ eff }seems to be within

*ε*

_{ i }for all filling ratios. If the inverse

*df/dε*

_{ eff }is solved for ε

_{e}≤ ε

_{eff}≤ ε

_{i}, then volume filling ratios are always exactly in the range 0 ≤ f ≤ 1.

*ε*

_{ i }and

*ε*

_{ e }leads to the following equation:

*ε*

_{ i }= 10 and

*ε*

_{ e }= 1. On the left-hand side, the result of the differential equation (Diff.) is presented together with those of the Maxwell–Garnett (M–G), Bruggeman (BS), and coherent potential (CP) formulae and, on the right-hand side, with the Looyenga (LO), Bruggeman nonsymmetric (BN), and Sen–Scala–Cohen (SSC) formulae. The results of the same formulae are plotted for the higher contrast of the permittivity,

*ε*

_{ i }= 100 and

*ε*

_{ e }= 1 in Fig. 8.33. Note that the vertical axis is logarithmic in these figures.

Figures 8.32 and 8.33 show that the effective permittivity according to the differential mixing equation shifts smoothly from the Maxwell–Garnett model to the symmetric Bruggeman model as the volume fraction of spheres increases. Figure 8.33 shows clearly that the derivative of the effective permittivity in the limit f → 0 is the same for all models on the left-hand side, but on the right-hand side, it is very different for the Looyenga and Sen–Scala–Cohen models. It is required to check the validity of the new model using real materials and to make clear the limit of applications through further research.

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