## Radiative Properties of Semi-Transparent Spherical Particles

* Following from: *The Mie solution for spherical particles

* Leading to: *Radiative properties of water droplets in the near-infrared spectral range, Spectral radiative properties of diesel fuel droplets, Radiative properties of gas bubbles in semi-transparent medium, Near-infrared properties of droplets of aluminum oxide melt

### 1 INTRODUCTION

Many substances are semi-transparent in the visible and infrared spectral ranges. The well-known example is pure water, which is practically transparent in the visible and semi-transparent in the short-wave part of the near-infrared range (Hale and Querry, 1973; Zolotarev and Dyomin, 1977). The majority of metal oxides and some other substances have spectral “windows” of semi-transparency in the visible and near-infrared (Brewster and Kunitomo, 1984; Cabannes and Billard, 1987; Parry and Brewster, 1991; Brewster, 1992; Chernyshev et al., 1993; Palik, 1998). In this article, we consider weakly absorbing materials (κ << 1) since this particular case appears to be important in numerous high-temperature applications. A good example is fly ash, which is semi-transparent in the spectral range of significant thermal radiation in combustion (Gupta et al., 1983; Gupta and Wall, 1985; Boothroyd and Jones, 1986; Goodwin and Mitchner, 1989; Im and Ahluwalia, 1993; Neubronner and Vortmeyer, 1994; Bahador and Sundén, 2007; Ruan et al., 2007).

### 2 HOMOGENEOUS PARTICLES

Consider the properties of homogeneous particles with an index of refraction in the limits of 1.3 ≤ *n* ≤ 2 and an index of absorption of 10^{-6} ≤ κ ≤ 0.05. The more interesting region of the diffraction parameter is 1 ≤ *x* ≤ 30. The typical dependences of the usual (not transport) efficiency factors of scattering and extinction on the diffraction parameter and also on the curve of the asymmetry factor of scattering are plotted in Fig. 1.

**Figure 1. Efficiency factors of scattering and extinction and asymmetry factor of scattering for spherical particles as functions of the diffraction parameter.**

The curves *Q*_{s}(*x*), *Q*_{t}(*x*), and μ(*x*) have the main oscillations with a period of

(1) |

and the secondary oscillations of a small period and amplitude. The main oscillations are produced by interference of transmitted and diffracted radiation, and the period of these oscillations is well predicted by the anomalous diffraction approximation. The typical nature of the secondary oscillations, occurring mainly as a result of interference from the surface waves and diffracted radiation, has been analyzed by van de Hulst (1957, 1981), Irvine (1965), and Dombrovsky (1974a). The intensity of both the main and secondary oscillations increases with the index of refraction, which corresponds to the transfer from the anomalous diffraction limit to the general Mie solution. The general result of *Q*_{t} → ∞ always takes place in the limit of *x* → ∞. The difference between the efficiency factors of extinction and scattering (explained by absorption) has a general tendency of increasing with the diffraction parameter. One can see that the asymmetry factor of scattering is always positive for particles of weakly absorbing substance. For large particles (*x* >> 1), the asymmetry factor decreases with the index of refraction because of the increasing role of radiation reflection from the particle surface.

Consider the main features of the angular distribution of the initially unpolarized radiation scattered by a particle. In the Rayleigh region, the scattering function is symmetric. With an increase of the diffraction parameter, the scattering function is deformed and one can observe a predominant forward scattering (see Fig. 2).

**Figure 2. Scattering functions of spherical particles at n = 1.5: 1, κ = 0.005; 2, κ = 0.2.**

Some extrema therewith occur, placed approximately uniformly in μ = cosθ, the number of extrema corresponds to the diffraction parameter value. The scattering function outside the Rayleigh region depends on both the index of refraction and the index of absorption, but their effect is negligible for large particles at small scattering angles, where diffraction predominates. One can see in Fig. 2 that increasing the index of absorption results in a considerable decreasing the backward scattering. The effect of the absorption index on the asymmetry of scattering is also illustrated in Fig. 3, where the typical curves μ(*x*) are plotted.

Consider now the more important characteristics of single particles: the efficiency factor of absorption and the transport efficiency factor of scattering. The absorption curves *Q*_{a}(*x*) for the case of *n* = 1.5 are presented in Fig. 4. We give the results for only one value of the refraction index because the absorption is not sensitive to the value of *n*.

**Figure 3. Effect of the absorption index on the asymmetry factor of scattering: 1, κ = 0.001; 2, κ = 0.01; 3, κ = 0.05.**

**Figure 4. Efficiency factor of absorption at different values of the absorption index.**

One can see in Fig. 4 that the absorption curves are very different at small and moderate values of the absorption index. At κ ≤ 10^{-3}, the absorption curve looks like a linear function with a number of very strong local peaks. These peaks correspond to the above-mentioned secondary oscillations of the extinction. With the increase of the index of absorption, the secondary oscillations of absorption become smaller and more symmetric, especially at κ *>* 10^{-2} (see, also, Dombrovsky, 1996; Dombrovsky and Baillis, 2010). It is interesting that *Q*_{a} can reach values greater than unity. Of course, this can be observed only in the so-called Mie region (i.e., at moderate values of the diffraction parameter). Strictly speaking, the range κ *>* 10^{-2} cannot be treated as a case of weakly absorbing substance. One can even use the definition of weakly absorbing substance based on the condition of *Q*_{a} *<* 1 for particles of this substance in the Mie region.

The transport efficiency factor of scattering *Q*_{s}^{tr} is not as sensitive to the index of absorption when κ ≤ 10^{-2} (see Fig. 5). The value considered is defined as *Q*_{s}^{tr} = *Q*_{s} · (1 - μ), but the curves *Q*_{s}^{tr}(*x*) have no numerous main oscillations because the above-discussed main oscillations of *Q*_{s}(*x*) and μ(*x*) are in phase. The only main maximum of *Q*_{s}^{tr} corresponds to the first main maximum of extinction. The maximum value of *Q*_{s}^{tr} depends on the index of refraction: it increases more than twice by variation of *n* from 1.5 to 2. The increase of the index of absorption always results in the decrease of the transport scattering efficiency factor and the flattening of the secondary oscillations of the curves *Q*_{s}^{tr}(*x*).

**Figure 5. Transport efficiency factor of scattering.**

It is known that the so-called transport approximation is applicable in many radiation heat transfer problems. In this approximation, there are only two spectral characteristics of a medium in the radiative transfer equation: the absorption coefficient and the transport scattering coefficient. On this basis, it is sufficient to analyze the corresponding characteristics of single particles.

Calculations of efficiency factors *Q*_{a} and *Q*_{s}^{tr} by using the exact Mie theory are very complicated, especially for particles with large a diffraction parameter. The computational time for numerous spectral calculations of polydisperse systems may be too large. This restriction is more important in the combined heat transfer problems. On the other hand, the usual particle shape deviations from the ideal sphere, insufficient accuracy of the experimental size distribution, and the presence of admixtures, which affect the complex index of refraction, make the high accuracy of calculations of functions *Q*_{a}(*m, x*), *Q*_{s}^{tr}(*m, x*) unnecessary. Several attempts of approximation of various absorption and scattering characteristics of semi-transparent spherical particles have been made during the past four decades. Various asymptotic solutions have been used as the initial approximations: Rayleigh scattering, anomalous diffraction, and the geometrical optics approximation (van de Hulst, 1957, 1981; Smirnov and Mysev, 1964; Zelmanovich, 1970; Pinchuk and Romanov, 1977). In the monograph by van de Hulst (1957, 1981), the approximate expressions were proposed for the extinction and absorption efficiency factors. More accurate approximations were given by Pinchuk and Romanov (1977). One can also remember more recent papers (Dombrovsky, 1990; Caldas and Semião, 2001a,b) concerning some suggestions on approximation of the particle characteristics in particular cases.

Functions *Q*_{s}(*x*) and μ(*x*) have numerous intensive oscillations, and a separate approximation of these functions is a very difficult task. As long as curves *Q*_{s}^{tr}(*x*) have no any main oscillations, which take place for *Q*_{s} and μ, the approximation problem for *Q*_{s}^{tr} is essentially simpler. If one takes into account the specific importance of the range 1.5 ≤ *n* ≤ 2, 10^{-3} ≤ κ ≤ 2 · 10^{-2}, the satisfactory approximate equations for *Q*_{a} and *Q*_{s}^{tr} can be obtained in the more important region of the diffraction parameter. The following simple expressions were suggested by Dombrovsky (1990):

(2a) |

(2b) |

where

(2c) |

Equations (2a)-(2c) take into account the main features of the curves *Q*_{a}(*x*) and *Q*_{s}^{tr}(*x*) without their correct asymptotic description. Considerably more complex expressions for the absorption efficiency factor at 1 *< n <* 1.5, 10^{-5} ≤ κ ≤ 10^{-1} derived by Pinchuk and Romanov (1977) are based on the geometrical optics approximation and include the resonance absorption corrections. The calculations showed that for the above-specified small values of κ, expressions (2a)-(2c) can be used up to *n* = 2. The fine structure of the absorption spectrum, including the narrow peaks, may be important in some problems. It was shown in a recent study by Zender and Talamantes (2006) that neglecting the Mie resonances for water droplets in clouds may cause substantial biases in radiance-based retrievals from sensor channels where atmospheric absorption is particle dominated. Of course, the unrepresented absorption is negligible for the global climate. As long as we focus on heat transfer applications here, it is sufficient to use simple expression (2a), which gives practically the same integrated results for absorption as the more detailed approximation suggested by Pinchuk and Romanov (1977). Additionally, it is worth mentioning the approximation error of Eq. (2b) for the transport efficiency factor of scattering.

Alternative approximations for large semi-transparent particles have been suggested by Dombrovsky et al. (2001). These approximations were modified by Dombrovsky (2002) to the following:

(3a) |

(3b) |

An example of the calculations, using approximations (2a)-(2c), (3a), and (3b), is shown in Fig. 6. One can see that Eq. (3a) and (3b) yield better results for the particles with a large diffraction parameter. The latter appears to be important for many engineering applications.

**Figure 6. Comparison of various approximations with the exact Mie solution: 1, Mie solution; 2, approximations (2a)-(2c); 3, approximations (3a) and (3b).**

The earlier discussed dependences of the particle radiative properties on the diffraction parameter at fixed optical constants of the particle substance are clear and convenient in the physical treatment. At the same time, one should take into account spectral dependence *m*(λ), so that the spectral variation of the radiative properties, even in the monodisperse case, cannot be reduced to the effect of the diffraction parameter variation. In the subsequent sections below we consider the definite substances, keeping in mind the known spectral variation of the complex index of refraction.

### 3 HOLLOW PARTICLES

Scattering and absorption of electromagnetic waves by optically inhomogeneous particles are of interest both in the scattering theory and for many problems of atmospheric optics, astrophysics, colloid optics, as well as in engineering applications concerned with radiation transfer in disperse systems. The simplest optically inhomogeneous particles are hollow and two-layer spheres. Many inhomogeneous particles may be classified as these types of particles. It is important that hollow and two-layer spherical particles can be treated as physical models for the optical properties investigation of some more complex particles.

Hollow particles are produced in volcanic eruptions of certain types by the escaping gases from hardened particles. Some hollow particles can be formed by spraying solutions of solid substances in volatile liquids or various slurry compounds (Leong, 1981; Baumgärtner and Schauer, 1989; McDonald and Devon, 2002; Roy et al., 2005; Yeo and Kiran, 2005). One can observe gas bubbles or mirovoids in organic coatings (Ramaiah and Rao, 1983). Hollow spheres also have been found among hail particles (Kokhanovsky, 2004). As for the engineering problems of radiation heat transfer, one is reminded of the hollow spheres present in large fly ash particles in the combustion products of coal (Raask, 1968; Kutchko and Kim, 2006), as well as the hollow aluminum oxide particles produced by combustion of aluminized propellants (Gossé et al., 2003, 2006). In the last decade, numerous potential applications stimulate the production of hollow microspheres from very different substances (Cochran, 1998; McDonald and Devon, 2002; Chou et al., 2003; Schmidt and Roessling, 2006; Gelei et al., 2006; Wang et al., 2007). Note that hollow particles of aluminum oxide are used in high-temperature insulation of space vehicles (Dombrovsky, 1974b, 2004; Moiseev et al., 2004; Dozhdikov et al., 2007). Hollow glass microspheres are used in paint coatings to decrease the heat losses due to the thermal radiation of building walls (Budov, 1994; German and Grinchuk, 2002; Dombrovsky, 2005, Dombrovsky et al., 2007). The radiative properties of hollow microspheres determine the infrared radiative properties of composite coatings.

Hollow particles of metal oxides, fly ash particles, glass microspheres, and some atmospheric aerosols are semi-transparent in the visible and near infrared, and the typical values of the optical constants of the particle substance are *n* = 1.5 - 2 and κ ≤ 0.01. The corresponding typical dependences *Q*_{s}(*x*''), μ(*x*''), and *Q*_{s}^{tr}(*x*'') of nonabsorbing hollow particles are shown in Fig. 7 (for more detailed numerical data, see Dombrovsky, 1974a, 1996; Dombrovsky and Baillis, 2010).

**Figure 7. Scattering characteristics for hollow nonabsorbing spherical particles at n = 1.5: 1, δ = 0; 2, δ = 0.5; 3, δ = 0.9.**

With an increase of the relative cavity radius δ, the curves stretch along the abscissa, and for δ → 1 we find some changes in the main oscillations. One can see the increase in the asymmetry factor, which practically reaches unity at δ = 0.9 at some values of diffraction parameter *x*''. The corresponding values of the transport efficiency factor of scattering *Q*_{s}^{tr} are very small. There are large-scale oscillations of the function *Q*_{s}^{tr}(*x*''), which take no place for homogeneous particles. Thus, the deformation of curve *Q*_{s}^{tr}(*x*'') by increasing the relative cavity radius is not reduced to the curve stretching along the abscissa and diminishing the main maximum amplitude. The evolution of function *Q*_{s}^{tr}(*x*'') at large δ also cannot be described by some average index of refraction. It is worth noting that the positions of the ripple structure extrema on curves *Q*_{s}^{tr}(*x*'') for spherical shells that are not too thin remain the same. This fact confirms the conclusion that the ripple structure is the result of the interference of surface waves and diffracted radiation.

The typical radiative characteristics of absorbing hollow particles are presented in Fig. 8. One can see that the efficiency factor of absorption *Q*_{a} decreases with increasing the relative cavity radius approximately proportional to the absorbing material volume. Of course, this statement is valid only for particles of a weakly absorbing material. Dependences *Q*_{s}^{tr}(*x*'') at κ = 0.01 differ slightly from the corresponding dependences for nonabsorbing particles but the secondary oscillations diminishing make the second main maximum more visible.

**Figure 8. Efficiency factor of absorption and transport efficiency factor of scattering for hollow spherical particles at m = 1.5 - 0.01i: 1, δ = 0.25; 2, δ = 0.5; 3, δ = 0.75; 4, δ = 0.9.**

More essential special features of the hollow particles take place at a large relative radius of the cavity. For this reason, it is of interest also to consider the limiting case of radiation scattering by a very thin nonabsorbing spherical shell. It goes without saying that the main extrema of curves *Q*_{s}(*x*'') and μ(*x*'') disappear due to the absence of a phase difference between the transmitted and diffracted radiation. In the limit of δ → 1, naturally, *Q*_{s} → 0, and the scattering function does not depend on δ. It appears that the scattering function also does not depend on the index of refraction when (*n* 1)(1 - δ) → 0. The curves μ(*x*'') for thin-wall spherical bubbles (δ ≥ 0.999) are shown in Fig. 9. One can see the exact periodic sinusoidal variation of the asymmetry factor of scattering. This result is explained by the corresponding variation of the backscattering intensity: the value of *g*(π) is periodically equal to zero (for the complete scattering functions, see Dombrovsky, 1996). This makes clear the physical cause of the oscillations: the interference of waves, reflected from the front and back surfaces of the particle. Indeed, the backward scattering will be equal to zero when 4*a*'' = *k*λ at *x*'' = π*k*/2, *k* = 1,2*,...*, for example. This condition corresponds exactly to the computational data. At the same time, the forward scattering *g*(0) increases monotonically with diffraction parameter *x*'', and oscillations μ(*x*'') are shifted in phase relative to the backscattering oscillations.

**Figure 9. Asymmetry factor and backward intensity of scattering for thin-wall nonabsorbing spherical bubbles.**

In practice, it is often necessary to estimate the effect of a comparatively small cavity on the particle optical properties. It was shown by Dombrovsky (1974a) that one can use the homogeneous particle model with the following diffraction parameter:

(4) |

One can see from the above analysis that Eq. (4) does not describe the asymptotic behavior of the hollow particle radiative properties. Equation (4) was used by Dombrovsky (1974b) to analyze the computational results for the integral emissivity of disperse systems containing micron-size hollow alumina particles.

In some applications, it is important to estimate the radiative characteristics of relatively large thin-wall hollow particles of a weakly absorbing material. One can remember the near-infrared properties of hollow-microsphere ceramics. The typical radius of these particles *a*'' is about 20-70 μm, whereas the particle wall thickness δ_{w} is usually between 0.5 and 4 μm. In the near infrared, these particles have a very large diffraction parameter *x*'' = 2π*a*''/λ, but their wall thickness is comparable with the wavelength. It can be shown that the effect of absorption on the transport efficiency factor of extinction *Q*_{tr} is negligible in the case of κ *<* 10^{-2}. Therefore, we consider the case of a nonabsorbing particle when *Q*_{tr} depends on diffraction parameter *x*'', index of refraction *n*, and the relative thickness of the particle wall, Δ_{w} = δ_{w}/λ. The typical dependences *Q*_{tr}(Δ_{w}) are presented in Fig. 10.

**Figure 10. Transport efficiency factor of extinction for large hollow particles at n = 1.5: 1-3, calculations by the Mie theory (1, x'' = 50; 2, x'' = 100; 3, x'' = 200); 4, calculations based on the model of single small plates.**

One can see that the curves for various values of *x*'' differ insignificantly from one another. This result implies the degeneracy of the general solution and transfer to the region of anomalous diffraction. The large-scale oscillations of the *Q*_{tr}(Δ_{w}) curves are caused by the interference of radiation passing through the particle. These oscillations are approximately symmetric relative to some constant value of Q_{tr}. The results of additional calculations (Dombrovsky, 2004) showed that dependence Q_{tr}(*n*) can be well approximated by the simple formula

(5) |

Consider the physical sense of limiting dependence *Q*_{tr}(Δ_{w}) for large thin-wall spherical particles. Since the value of *Q*_{tr} does not depend on the particle radius, it is natural to assume that different elements of such particles scatter radiation independently of one another. In this case, a particle can be represented as a combination of a large number of separate small flat plates. The coefficient of reflection of unpolarized radiation for a single plate can be determined on the basis of the known solution (Born and Wolf, 1999):

(6a) |

(6b) |

(6c) |

(6d) |

where θ is the angle between the direction of incident radiation and the normal to the plate surface. Because the angle between the directions of incident and reflected radiation is π - 2θ, the transport efficiency factor of extinction of a single plate is calculated by the obvious relation:

(7) |

If we take into account the orientation of the plates in forming the shell of a hollow particle, the following relation can be derived (Dombrovsky, 2004):

(8) |

The insignificant shading of the back hemisphere is taken into account in the latter equation. The results of the calculations based on Eqs. (6)-(8) are also shown in Fig. 10. One can see that the model of the single plates gives quite an adequate picture of radiation scattering by large thin-wall spherical particles. This supports the assumption that different elements of such particles scatter radiation independently of one another. The result obtained appeared to be important for theoretical modeling of radiation scattering in hollow-microsphere ceramics. As was shown by Dombrovsky (2004) and Dozhdikov et al. (2007), the theoretical predictions based on the above-discussed models are in good agreement with the experimental data presented by Moiseev et al. (2004) and Dozhdikov et al. (2007).

### 4 TWO-LAYER PARTICLES

The interaction of the radiation with two-layer spherical particles is significantly more complicated than for homogeneous or hollow particles. This problem abounds with physical effects, which do not take place for earlier considered particles. A total analysis of all situations is not possible. In this section, we consider only some typical variants.

First of all, consider the effect of the difference between the core and mantle complex indices of refraction on the particle optical properties. The characteristics of such particles are presented in Figs. 11 and 12.

**Figure 11. Scattering and extinction of radiation by two-layer particles at κ' = κ'' = 0.01 and δ = 0.5: (a) n' = 1.5, n'' = 2; (b) n' = 2, n'' = 1.5.**

**Figure 12. Absorption efficiency factor for two-layer particles at n' = n'' = 1.5: (a) κ' = 0, κ'' = 0.01; (b) κ' = 0.01, κ'' = 0; 1, δ = 0.25; 2, δ = 0.5; 3, δ = 0.75; 4, δ = 0.9.**

A considerable difference from the homogeneous particles is observed in curves *Q*_{t}(*x*'') and μ(*x*''). The main oscillations vary significantly for different refractive indices: at *n*' *< n*'' their amplitude may be very large, and at *n*' *> n*'' the “double” main maxima occur. At the same time, curves *Q*_{s}^{tr}(*x*'') vary weakly, at least in the second case (at a weakly reflecting mantle). For inhomogeneous particles with radial variation of the index of absorption, functions *Q*_{a}(*x*'') remain generally the same as those for hollow particles or particles without the mantle, but there are no any secondary oscillations (surface wave absorption) at the small absorbing core.

Let us return to the effect of the main oscillation amplitude increasing in the case of a strongly reflecting mantle. The variant with a more pronounced effect is presented in Fig. 13. With the diffraction parameter variation from *x*''≈ 5 to 9, the value of *Q*_{s} decreases by more than 50 times. The transport efficiency factor of scattering varies just as strong, and dependences *Q*_{s}^{tr}(*x*'') have a number of pronounced main oscillations. The secondary oscillations in the example under consideration are very intensive, particularly at the small diffraction parameter. A discussion of this effect and some additional data regarding the scattering function are given by Dombrovsky (1996). The pronounced main oscillations of *Q*_{s}^{tr} in Fig. 13 can be treated as a wide-band spectral selectivity of disperse systems containing such particles.

**Figure 13. Scattering of radiation by two-layer nonabsorbing particles at n' = 1.2, n'' = 2, and δ = 0.8.**

Regarding the abnormally low scattering of radiation by some two-layer particles, one should review the early work done by Kerker et al. (1975), Chew and Kerker (1976), and Kerker (1977), in which the theoretical analysis of the scattering decrease for small nonabsorbing particles was given. It was shown that the electric dipole moment of a two-layer spherical particle with *n*' *>* 1 and *n*'' *<* 1 may be equal to zero, and scattering near the Rayleigh region decreases sharply. For instance, the decrease of *Q*_{s} in several orders of magnitude occurs at *n*' = 1.97, *n*'' = 0.66. These parameters correspond to the pigment containing titania particles (*n* = 2.97) with a concentric void (*n* = 1) in a resin matrix (*n* = 1.51) at wavelength λ = 0.546 μm. The computational data presented in Fig. 13 show that the sharp decrease of scattering at certain combinations of core and mantle optical constants also takes place for large particles, when the scattering is a result of the interaction of a great number of complex partial waves. A considerable decrease in radiation scattering by spherical particles having a core and a mantle with certain optical constants is not only of theoretical interest but also can be used in practice--e.g., in dispersive infrared filters (Prishivalko et al., 1984).

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