Radiative properties of single particles and fibers: the hypothesis of independent scattering and the Mie theory

DOI: 10.1615/thermopedia.000136


Following from: Spectral radiative properties of disperse systems

Leading to: The Mie solution for spherical particles; The scattering problem for cylindrical particles

We focus on the interaction of radiation with single particles and fibers assuming that the results obtained for single particles can be used to predict the radiative properties of various disperse systems. This means that we use the so-called independent scattering hypothesis when each particle is assumed to absorb and scatter the radiation in exactly the same manner as if all the other particles did not exist. In addition, there is no systematic phase relation between partial waves scattered by individual particles during the observation time interval, so that the intensities of the partial waves can be added without regard to phase. In other words, each particle is in the far-field zones of all the other particles, and scattering by individual particles is incoherent.

Obviously, the assumption of independent scattering is correct for rarefied disperse systems such as gas or liquid with suspended particles when the particles are placed far from each other and randomly positioned in space. In the case of relatively dense disperse systems, when distances between the neighboring particles are comparable to the radiation wavelength and particle size, one can observe the near field-dependent scattering effects. In the ordered disperse systems, as well as at some angles of observation, one can also find the far-field interference of the radiation scattered by single particles. The dependent-scattering problem as applied to radiation heat transfer has been discussed in some detail in the known paper by Tien and Drolen (1987). Many research papers deal with theoretical analysis of dependent-scattering effects in dense disperse systems such as packed particulate beds. The reader can be addressed to the monographs by Tsang and Kong (2001), Kokhanovsky (2004), and Mishchenko et al. (2006), and to some recent journal papers (Baillis and Sacadura, 2000; Vargas, 2003; Coquard and Baillis, 2004; Tishkovets et al., 2004; Mishchenko, 2006).

It is important that the analysis based on the independent-scattering hypothesis appears to be applicable even in some cases when particles are very close to each other or the average distance between the particles is much less than the wavelength. Several examples have been discussed in early papers by the author (Dombrovsky, 1979; Dombrovsky and Mironov, 1997). The other point is that complex amplitude functions of single particles can be used in the determination of the properties of dense disperse systems characterized by strong dependent-scattering effects. One can remember the well-known Maxwell-Garnett theory (Maxwell-Garnett, 1904) and some advanced models of effective permittivity based on properties of single particles immersed in a dielectric matrix (Sihvola, 1999; Choy, 1999). The intention of the detailed analysis of absorption and scattering characteristics of single particles is also supported by the positive experience in application of these characteristics to the identification of the main radiative properties of some composite materials and to combined heat transfer calculations in thermal engineering.

Absorption and scattering characteristics of individual particles can be determined by the analysis of the interaction of a plane electromagnetic wave with a macroscopic particle. This statement contains two assumptions that are usually correct for thermal engineering applications. The assumption of a plane-wave illumination is not correct in the case of a laser source when transverse dimensions of the beam are comparable with the particle size. This more complicated problem is considered in some detail in the review by Gouesbet and Gréhan (2000). The term “macroscopic particle” reminds us about another limitation of the traditional approach. It is well known that optical properties of bulk substances in solid or liquid phase are qualitatively different from those of their constituent atoms and molecules when the latter are isolated. This may cause a problem when one applies the concept of bulk optical constants to a very small particle because either the optical constants determined for the bulk matter provide an inaccurate estimate or the particle is so small that the entire concept of optical constants loses its validity. We will therefore assume that the particles are sufficiently large so that they can still be characterized by optical constants appropriate to bulk matter. It is assumed that the scattering of a wave by a particle takes place without frequency variation and can be described in terms of Maxwell’s equations.

Absorption and scattering of radiation by a particle depend on the alignment of particle geometrical parameters and wavelength, as well as on the complex index of refraction of the particle material m = n - iκ, where n is the index of refraction and κ is the index of absorption. For a homogeneous spherical particle, the only geometrical characteristic is the size parameter (diffraction parameter) x = 2πa/λ, where a is the particle radius and λ is the radiation wavelength. The rigorous solution of the scattering problem for a spherical particle of arbitrary substance is known as the Lorenz-Mie theory or simply the Mie theory with reference to the fundamental work by Gustav Mie (1908). One can see the survey by Logan (1965) for historical details and the early studies of scattering and absorption of plane waves by a sphere. Due to computational difficulties, only a small amount of quantitative information based on the general analytical solution could be obtained for a long period after the publication of the work by Mie. With appearance of high-speed computers and the increase of interest in some scattering theory applications, mainly in atmospheric optics, colloidal chemistry, and astrophysics, the number of numerical investigations based on Mie theory was significantly increased. Many papers with numerical data concerning the optical properties of various particles have been published during the second half of the last century. The reader can refer to monographs by Kerker (1969), Deirmendjian (1969), van de Hulst (1957), Bohren and Huffman (1983), Prishivalko et al. (1984), Dombrovsky (1996), and Dombrovsky and Baillis (2010), where one can find not only theoretical and computational analysis of radiation interaction with individual particles, but also the necessary bibliographies.

The general analytical solution to the electromagnetic scattering problem for concentric core-mantle spherical particles was derived by Aden and Kerker (1951). Practically simultaneously with (Aden and Kerker, 1951), similar solutions were published by Shifrin (1952) and Güttler (1952). Since the 1960s, a series of publications appeared in which the optical properties of radially inhomogeneous spherical particles with a more complex nonstepped variation of the refractive index were calculated. The properties of particles of optically anisotropic materials have also been investigated. We do not give a review of numerous papers concerning absorption and scattering characteristics of inhomogeneous or anisotropic spherical particles. Detailed bibliographies of early papers can be found in monographs by Prishivalko et al. (1984), Dombrovsky (1996), and Mishchenko et al. (2002). One can also read recent papers by Kai and Massoli (1994) and Perelman (1996), where the general case of the radially inhomogeneous spheres was considered.

Another classical problem of scattering is the interaction of an electromagnetic wave with an infinite circular cylinder. The analytical solution for a cylinder, along the normal to the axis, was derived for the first time by Lord Rayleigh (1881), i.e., earlier than the Mie solution for spheres. The solution to the scattering problem for cylinders at oblique illumination was derived by Wait (1955). The same problem for two-layer cylinders at normal and oblique illumination was solved considerably later. The general problem for the radially inhomogeneous infinite cylinder was solved in more recent papers by Barabas (1987) and Kai and D’Alessio (1995). Obviously, the scattering problem for finite cylinders is also of interest for many applications. Unfortunately, this problem appears to be much more complicated than that for infinite cylinders. The solution to the problem for dielectric and conducting finite cylinders can be found in papers by Uzunoglu et al. (1978) and Waterman and Pedersen (1992, 1995, 1998).

Only particles of simple geometrical shape (spheres and cylinders) are usually considered in engineering applications. Nevertheless, we give some important references that can be used for the study of radiation scattering properties of complex-shape particles. The latter may be important in many fields of science and engineering. In the limiting cases of very small or large particles, one can use approximate analysis based on the Rayleigh or geometrical optics approximations. But in the so-called resonance region of particle size parameters, these approximations are inapplicable, and numerical methods based on directly solving Maxwell’s equations are the most universal tool. Reviews of the earlier papers on this subject can be found in the books edited by Schuerman (1980) and Varadan and Varadan (1980). The more recent book edited by Mishchenko et al. (2000) can be recommended as a systematic source of information on calculations of electromagnetic scattering by nonspherical and heterogeneous particles before 2000. The review by Kahnert (2003) gives a complete description of the modern techniques for solving the electromagnetic scattering problem for nonspherical particles. This review is restricted to those techniques that are based on rigorous electromagnetic theory. The detailed discussion of the popular T-matrix approach is given in the review by Mishchenko et al. (1996). At present, Waterman’s T-matrix approach is one of the most powerful and widely used tools for rigorously computing light scattering by nonspherical particles, both single and aggregated. Note that in the case of the spherical particle shape, all formulas of the T-matrix approach become identical to the corresponding formulas of Mie theory. Therefore, the T-matrix approach can be considered as an extension of Mie theory to nonspherical particles. A comprehensive reference database on T-matrix computations reported up to the present time can be found in papers by Mishchenko et al. (2004, 2007).

One should remember the problem of absorption and scattering of radiation by particle clusters and agglomerates. The long-time interest in this problem is supported by the significant role of aggregated soot particles in the thermal radiation of combustion products (Viskanta, 2005). The agglomerates of aerosol particles are also observed in the atmosphere (Kondratyev et al., 2006). At a large concentration of pigment particles in paints, the particles can also form aggregates (Bentley and Turner, 1998). Some other applications of light scattering by complex particles in material science were discussed recently by Holoubek (2007). It should be noted that light scattering has proved to be one of the most powerful techniques for probing the properties of particulate systems. One can recommend a comprehensive review by Jones (1999) on this subject. The information on specific models employed in the analysis of such complex particles and discussion of the main results can be found in the literature. Only references to some important recent papers by Sorensen (2001), Kimura (2001), Auger et al. (2003), Mackowski (2006), and Yurkin and Hoekstra (2007) are given in this article.

In many practical problems, we have no detailed information on the shape and the composition of particles. Moreover, the presentation of a disperse system as a combination of separate particle may be problematic (Dombrovsky and Baillis, 2010). In this case, a reasonable approach to the theoretical description of the disperse system radiative properties should be based on the well-known solutions for particles of simple shape, namely, spheres and long cylinders. Only such particles are considered in subsequent articles of Thermopedia.


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