Maxwell-Garnett model

This chapter is devoted to describe the impact of metallic nanosphere to the multi-photon excitation fluorescence of Tryptophan, and little further consideration to multi-photon absorption process will be given, as the reader can find several studies in [11-14]. In section II, the nonlinear light-matter interaction in composite materials is discussed through the mechanism of nonlinear susceptibilities. In section III, experimental results of fluorescence induced by multi-photon absorption in Tryptophan are reported and analyzed. Section IV described the main results of this chapter, which is the effect of metallic nanoparticles on the fluorescent emission of the Tryptophan excited by a multi-photon process. Influence of nanoparticle concentration on the Tryptophan-silver colloids is observed and discussed based coi a nonlinear generalization of the Maxwell Garnett model, introduced in section II. The main conclusion of the chapter is given in secticHi IV. 

Sipe, J.E., Boyd, R.W. Nonlinear susceptibility of composite optical materials in the Maxwell Garnett model. Phys. Rev. A 43, 1614-1629 (1992)... 

A key issue in nanostructured materials is the dipole coupling between nanocrystals which will cause the optical properties of a nanocrystal ensemble to become like those of the bulk material. There has been extensive investigation of the interactions between particles embedded within media for a range of boundary conditions. We have found that the effective dielectric function given by Eq. (10), based on the Maxwell-Garnett model [1] is very accurate for quite dense nanocrystal arrays. In practice, one measures the transmittance of a thin film of the dense nanoparticle based film. Conventional solutions are simply... 

Here, Cp is a capacitance measured, f is a thickness of a film, vo is the electric constant and S is an electrode area. The dielectric constants of various films, i.e., cast films (non particles films), layered films of nonporous NPs and layered films of porous NPs, were summarized in Table 2. The dielectric constants of the cast films consisted of BPDA-PDA, 6FDA-ODA and CBDA-TFMB were 3.17, 2.76 and 2.52, respectively. The dielectric constants of multilayer films of nonporous PI NPs were somewhat lower, resulting from generation of air voids between NPs. The porosity of the porous films was 15 22%, calculated by using the Maxwell-Garnett model, the equation (3) [8]. 

For example, the Maxwell-Garnett model just sets the first material which has the largest volume fraction to be the host material ( h= i) and then the EMA equation reduces to ... 

The TEM data have been used to simulate, in the frame of the Mie theory and Maxwell-Garnett effective medium approximation [15], the optical absorption spectra of the sample implanted with 5 x lO Au /cm. The results are reported in Figure 8(c). In the first model used to describe... 

We apply simple effective medium models in an attempt to understand the diffusion process in the complex pore network of a porous SiC sample. There is an analogy between the quantities involved in the electrostatics problem and the steady state diffusion problem for a uniform external diffusion flux impinging on a coated sphere. Kalnin etal. [17] provide the details of such a calculation for the Maxwell Garnett (MG) model [18]. The quantity involved in the averaging is the product of the diffusion constant and the porosity for each component of the composite medium. The effective medium approach does not take into account possible effects due to charges on the molecules and/or pore surfaces, details in the size and shape of the protein molecules, fouling (shown to be negligible in porous SiC), and potentially important features of the microstructure such as bottlenecks. 

The understanding of factors that lead to enhanced band intensities and dispersive band shapes is of central interest in studies with nanostructured electrodes. Effective medium theory has often been employed to identify mechanisms for enhanced infrared absorption [28, 128, 172, 174, 175]. Osawa and coworkers applied Maxwell-Garnett and Bruggeman effective medium models in early SEIRAS work [28, 128]. Recently, Ross and Aroca overviewed effective medium theory and discussed the advantages and disadvantages of different models for predicting characteristics of SEIRAS spectra [174]. When infrared measurements on nanostructured electrodes are performed by ATR sampling, as is typically the case in SEIRAS experiments, band intensity enhancements occur, but the band shapes are usually not obviously distorted. In contrast, external... 

The first exact expression of this type was derived by Maxwell [1881] for the dc conductivity of a dispersion of spheres in a continnons medinm. Maxwell Garnett [1904] derived a similar expression for dielectric and optical properties. Wagner [1914] extended Maxwell s model to the complex domain and this model has thereafter been known as the Maxwell-Wagner model. It gives the following expression for complex conductivity ... 

A collection of nanoparticles embedded in a dielectric medium is modelled by effective medium theories such as the Maxwell-Garnett theory where each nanoparticle is treated as a dipole, and the medium is treated as homogeneous with effective dielectric properties. This model provides qualitative agreement with experimental absorption spectra, but applications such as sensing and catalysis demand greater agreement between theoretical predictions and experimental results. 

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