## Radiative Properties of Metal Particles in Infrared and Microwave Spectral Ranges

* Following from: *The Mie solution for spherical particles

* Leading to: *Liquid droplet radiator for space applications

The optical properties of metal particles differ in considerable degree from those of other particles due to a great number of free electrons in metals. The part of the physical optics dealing with the metal properties is known as a separate branch-the optics of metals (Sokolov, 1961). In the middle and far-infrared ranges (10 μm < λ < 1 mm) and in the microwave range (1 mm < λ < 1 m), the optical properties of many metals can be described in terms of the Drude theory, according to which the dielectric constant value is determined by free electrons. In this case, *n* and κ are approximately equal to each other and increase with the wavelength up to large values. There are no other materials that have as large values of *n* and κ in the visible as the metals in the middle and far infrared. The large value of |*m*| results in the extraordinary radiative properties of metal particles in the long-wave range.

In this article, we consider some typical metal particles in two spectral regions as they apply to known engineering problems. Consider first metal particles in the infrared and (for special cases) in the visible spectral range. The choice of particle material is connected mainly to two practical problems: shielding of walls in power installations by use of refractory metal particles and the radiative cooling of space power systems by use of a liquid droplet radiator. In these applications, tungsten particles are considered for the first problem and aluminum or alkaline metal particles are considered for the second problem. Some properties of aluminum particles in the microwave range are discussed in the second part of this section. This problem is of interest, for example, in determination of the microwave radiation absorption in aluminized solid propellants.

For calculation of the tungsten particle radiative properties at temperatures greater than 2000 K, it is necessary to know the high-temperature optical constants in the wavelength range from 0.5 μm to 2 μm. The corresponding data in the visible (up to 0.7 μm) have been reported by Tingwaldt et al. (1965), in the near infrared (0.9 < λ < 1.7 μm) by Martin et al. (1965), and in the whole spectral range of interest by Barnes (1966). The experimental data in these studies considerably differ from each other, particularly regarding the absorption index data. However, this difference is not as important for optical properties of particles, and the spectral dependences of optical constants at *T* > 2000 K can be approximated by the following simple expressions:

(1) |

where λ is expressed in microns. This approximation for the tungsten complex index of refraction can be applied to the group of refractory metals (Barnes, 1966). The calculated absorption and scattering characteristics of tungsten particles in the visible and near infrared are shown in Fig. 1. Both the curves of absorption *Q*_{a}(*x*) and curves of scattering *Q*_{s}^{tr}(*x*) have a maximum at *x* ≈ 1. In the visible, absorption by both small particles (*x* << 1) is and large particles (*x* >> 1) absorb greater than scattering. In the infrared, at a large absorption index, the scattering by large tungsten particles is considerably greater than absorption. The spectral dependences of the radiative properties of particles with radii from 1 to 3 μm appear to be near linear in the wavelength region 0.2-2 μm. One can see that *Q*_{a} > *Q*_{s}^{tr} in the visible but *Q*_{s}^{tr} > *Q*_{a} in the near infrared.

**Figure 1. Efficiency factor of absorption and transport efficiency factor of scattering for tungsten particles in the visible and near infrared: (a) as functions of diffraction parameter (1, λ = 0.5 μm; 2, λ = 1 μm; 3, λ = 2 μm); (b) as functions of wavelength (1, a = 1 μm; 2, a = 2 μm; 3, a = 3 μm).**

It is seen from the data presented in Fig. 1 that tungsten particles with diffraction parameter *x* ≈ 1 are more effective for surface shielding from the visible and near-infrared radiation (see the computational results reported by Eroshenko and Mos’yakov, 1981). The absorption and scattering coefficients for such particles reach a maximum. For the spectral region considered, these coefficients have comparable values and one should take into account both absorption and scattering in combined heat transfer calculations.

Let us consider the radiative properties of aluminum particles for liquid droplet radiators at a temperature about 950 K. For the radiator calculations, we should know the optical constants of aluminum in near-infrared spectral region 2.5 < λ < 8 μm (about 70% of the blackbody radiation at the referenced temperature takes place in this range). A comparison with the experimental data obtained by Beattie (1957), Golovashkin et al. (1960), Lenham and Treherne (1966) (see the table in Brennon and Goldstein, 1970), and table data from review by Ordal et al. (1983), shows a near linear increase of *n* and κ with the wavelength. In the above-mentioned spectral range, one can use the following approximate expressions for optical constants to estimate the spectral properties of aluminum particles:

(2) |

where λ is expressed in microns. The corresponding computational results are presented in Fig. 2. One can see that despite the variation of the complex refractive index from *m* = 4.4 - 32*i* at λ = 3 μm up to *m* = 11.9 - 56*i* at λ = 6 μm, the values of *Q*_{a} and *Q*_{s}^{tr} at a fixed diffraction parameter are almost invariable. The maximum values of *Q*_{a} and *Q*_{s}^{tr}, as for tungsten particles, take place at *x* ≈ 1, but aluminum particles absorb radiation more weakly, which is their deficiency when being employed in the droplet radiator. At low temperature, the alkaline metals can be used in the droplet radiator (at *T* ~ 300 K, the eutectic Na-K; at *T* ~ 500 K, Li) (Mattick and Hertzberg, 1985). According to the data reported by Althoff and Hertz (1967), at λ = 10.15 μm we have *m* = 6.43 - 44.5*i* for Na and *m* = 4.77 - 28.2*i* for K. These values of the complex index of refraction are practically the same as those for aluminum in the wavelength range from 3 to 4 μm. There are reasons to believe that the optical constants of metals considered in the spectral range of interest do not vary strongly while melting.

**Figure 2. Optical properties of aluminum particles in the near infrared.**

Consider the simpler limiting case of *n* = κ >> 1 for which the variation of particle radiative properties with *n* (or κ) is quite evident (see Fig. 3). The absorption of radiation by particle decreases by transfer to the limit of perfectly reflecting particles (|*m*| → ∞). This variation is very slow. For instance, large particles with *m* = 10 - 10*i* absorb less than at *m* = 5 - 5*i* only by two times. The corresponding variation of the scattering characteristics appear to be even less than the absorption variation, except in the region *x* << 1 where transfer from the Rayleigh to the magnetic dipole scattering takes place (Shifrin, 1968). A weak sensitivity of the radiative properties of the metal particles considered for optical constant variations (at least, within the limits of the experimental data uncertainty) makes these particles an interesting object for some optical measurements. We can refer to the study done by Dombrovsky and Zhiravov (1980), in which spherical iron particles were used in laser measurements of the particle drag coefficient in a two-phase supersonic flow.

**Figure 3. Transport efficiency factor of scattering, efficiency factor of absorption, and asymmetry factor of scattering for particles with n = κ: 1, n = 3; 2, n = 5; 3, n = 10; 4, n → ∞.**

In the discussion of the radiative properties of metal particles, it is worth considering the interesting case of the low index of refraction (less than unity). This takes place for alkaline metals at λ < 4 μm and, for example, for aluminum near the wavelength 1.2 μm. The minimum values of the refractive index for these metals are realized in the visible: according to Hass and Waylonis (1961) and El Naby (1963), we have *m* ≈ 0.13 - 1.3*i* for K at λ = 0.5 μm and *m* ≈ 0.1 - 2.35*i* for Al at λ = 0.2 μm. The relative index of refraction, which is in the Mie theory, may be even smaller if metal particles are immersed in a medium with a large refractive index. Some calculations of the extinction efficiency factor *Q*_{t} for particles with a low refractive index have been conducted by Latyshev et al. (1968). It was shown that curves *Q*_{t}(*x*) at *n <<* 1 and κ ~ 1 have a number of intensive oscillations and a large maximum at a small refractive index. For instance, the value *Q*_{t} ≈ 19 was obtained at *m* = 0.05 - 5*i* near the point *x* = 0.3. The results of similar calculations obtained by Dombrovsky (1996) and Dombrovsky and Baillis (2010) are given in Fig. 4.

**Figure 4. Transport efficiency factor of scattering, efficiency factor of absorption, and asymmetry factor of scattering for metal particles with a low index of refraction.**

More evident feature of dependences *Q*_{a}(*x*), *Q*_{s}^{tr}(*x*), and μ(*x*) is a variation of the curve shapes with an increase of the absorption index from κ = 1 to κ = 2. In the first case, curves *Q*_{a}(*x*), *Q*_{s}^{tr}(*x*), and μ(*x*) have no oscillations, but for κ = 2 there are oscillations of *Q*_{a} and μ with period Δ*x* ≈ 0.83. It is interesting to note that the variation of the refractive index from 0.05 to 0.1 makes practically no difference in the radiation scattering by particles, whereas the absorption increases by two or three times. Note that the maximum scattering is characterized by the zero asymmetry factor.

The microwave properties of metal particles are briefly discussed below. Consider spherical aluminum particles with radii from 1 to 100 μm in the wavelength range from 1 to 100 mm. One can see that the diffraction parameter is less than unity even for large particles. This makes the problem sufficiently simpler: the approximate description for *x* << 1 can be used by retaining only those Mie series terms that correspond to the partial waves of low order. The complex index of refraction of aluminum can be calculated by the Hagen-Rubens formula, which yields

(3) |

where σ_{e} = 3.72 × 10^{7} S/m is the electric conductivity, ε_{0} = 8.86 × 10^{-12} A·s/(V·m) is the dielectric constant, and *c* = 3 × 10^{8} m/s is the velocity of light. In the spectral range of interest here, the value of |*m*| = *n*√2 varies from 1.5 × 10^{3} to 1.5 × 10^{4}. Therefore, a majority of the particles under consideration do not satisfy the condition |*m*|*x* << 1 and the Rayleigh theory is not applicable. The magnetic dipole terms (and, presumably, some higher-order terms) must be retained in the Mie solution. Retaining only the electric dipole (*a*_{1}) and magnetic (*b*_{1}) terms greatly simplifies the formulas based on the Mie theory:

(4) |

In the limit *x* << 1, approximate expressions for the lowest-order Mie coefficients can be written as follows:

(5a) |

(5b) |

In the electric-dipole (Rayleigh) approximation, *b*_{1} = 0. The computational results reported by Dombrovsky and Mironov (1997) have demonstrated that the Rayleigh approximation is totally inapplicable to the problem under consideration and the magnetic partial wave is almost entirely responsible for absorption and, to a considerable extent, for the scattering of microwave radiation by aluminum particles.

Some results of the calculations presented in Eqs. (4), (5a), and (5b) are given in Fig. 5. Even for particles of radius 100 μm, the scattering is important only in the shortwave range λ < 5 mm. At longer wavelengths, absorption is significantly greater than scattering.

**Figure 5. Efficiency factors of absorption and scattering for aluminum particles: 1, a = 1 μm; 2, a = 5 μm; 3, a = 50 μm; 4, a = 100 μm.**

One can easily make sure that the efficiency factor of scattering can be calculated with high accuracy by the formula

(6) |

which is exact in the limit |*m*| → ∞ (van de Hulst, 1957, 1981). If we proceed to the limit |*m*| → ∞ in the Rayleigh equation [Eq. (8)] from the article Rayleigh Scattering, the result is *Q*_{s} = 8*x*^{4}/3. This makes it evident that the Rayleigh theory is inapplicable. The absorption efficiency factor depends on the diffraction parameter in a more complex manner, but its dependence on the particle radius is rather weak at *a* > 5 mm (see Fig. 5). It this range, one can use the approximate formula

(7) |

Note that the calculated efficiency of absorption by small particles (*a* < 2 mm) is substantially lower in the centimeter wave range, where the following wavelength dependence is obtained:

(8) |

We see that as the wavelength is increased an increase in *Q*_{s}^{(1)}/*x* is followed by a decrease. The maximum ratio *Q*_{s}^{(1)}/*x* ≈ 0.71 does not depend on the particle size and corresponds to *nx* ≈ 2.4. This is a rather general result because it applies to the particles of any metal (not only aluminum).

Taking into account the first-order partial waves in the limit *x* << 1, we obtain the following scattering function for randomly polarized radiation:

(9) |

This expression agrees with the scattering function for small particles in the limit |*m*| → ∞ (van de Hulst, 1957, 1981). The first term in Eq. (9) is the Rayleigh function; the second term is responsible for the scattering asymmetry: the greater part of radiation is scattered in the back hemisphere, and the asymmetry factor of scattering is μ^{(1)} = -0.4.

In the millimeter-wave range, radiation effectively interacts with the largest particles at *x* ~ 1. Therefore, the applicability of *Q*_{s}, *Q*_{t}, and μ as approximate expressions for *Q*_{s}^{(1)}, *Q*_{t}^{(1)}, and μ^{(1)} should be verified. To do this, one can perform the calculations based on the general Mie theory, where the Mie coefficients corresponding to the second-order (quadrupole) partial waves, *a*_{2} and *b*_{2}, were taken into account. Table 1 shows some results of such calculations at *a* = 100 μm for the spectral band where the contribution of these coefficients reaches its maximum. Table 1 also shows the values of *Q*_{s}^{(1)}, *Q*_{t}^{(1)}, and *1*^{(1)} calculated with the exact values of *a*_{1} and *b*_{1} used instead of approximate expressions (5a) and (5b), which are valid only for *x* << 1. One can see that the contribution of the second-order partial waves to absorption and scattering by particles of radii *a* ≤ 100 μm can be neglected even in the millimeter-wave range, and approximate equations (4), (5a), and (5b) can be applied.

**Table 1. Absorption and scattering characteristics of aluminum particles of radius 100 μm**

λ (mm) |
Q_{a}^{(1)} (approximate/exact) |
Q_{a} |
Q_{s}^{(1)} (approximate/exact) |
Q_{s} |
μ^{(1)} (approximate/exact) |
μ |

1 | 0.0039/0.0035 | 0.0037 | 0.508/0.537 | 0.534 | -0.40/-0.32 | -0.30 |

2 | 0.0022/0.0020 | 0.0021 | 0.0324/0.0332 | 0.0332 | -0.40/-0.38 | -0.39 |

3 | 0.0017/0.0016 | 0.0017 | 0.0065/0.0065 | 0.0065 | -0.40/-0.39 | -0.39 |

The properties of the metal particles considered above were calculated under the assumption that the particles are embedded in a medium that neither refracts nor absorbs radiation. If the complex index of refraction of the matrix is not unity (*m*_{e} ≠ 1), the Mie equations should be modified, as was discussed in the article Radiative properties of gas bubbles in semi-transparent medium. Since |*m _{}* - 1| << 1 for microwaves propagating in a dielectric medium, the results obtained remain valid almost without modification.

Assuming that dependent scattering is negligible, one can calculate a specific absorption coefficient of a monodisperse system of aluminum particles as follows:

(10) |

where ρ_{s} is the aluminum density. The dependences *E*_{a}(*a*) calculated at several wavelengths and ρ_{s} = 2700 kg/m^{3} are presented in Fig. 6. The point of maximum absorption can be determined by combining Eq. (7) with Eq. (8). It is easy to see that the corresponding value of the particle radius is proportional to √λ. It should be noted that this range of wavelengths and particle sizes is of primary interest in practical applications.

**Figure 6. Specific spectral absorption coefficient for a monodisperse system of spherical aluminum particles: 1, λ = 5 mm; 2, λ = 10 mm; 3, λ = 20 mm; 4, λ = 40 mm.**

An experimental study of the microwave radiation absorption by aluminum particles in a dielectric matrix was been performed by Dombrovsky and Mironov (1997). A microwave source at wavelength λ = 45 mm was used to attain the radiant-energy surface density *W* = 10 kJ/cm^{2} in a 10 cm^{2} spot in the focal plane. The experimental specimens were 31 mm × 24mm × 7mm parallelepipeds of a hardened epoxy-resin matrix containing aluminum powder. The aluminum mass fraction was varied from 5 to 20%. The details of the two powder disperse compositions can be found in the study done by Dombrovsky and Mironov (1997). Here, we will give only the more reliable data for a powder with a volume-to-surface radius of aluminum particles *a*_{32} in the range from 7.5 to 8.5 μm. Note that the relatively large value of *a*_{32} enables us to use a monodisperse approximation with *a* = *a*_{32} in the computational analysis (see also the article Radiative properties of polydisperse systems of independent particles). A specimen was placed in a microwave-transparent, thermally insulating container. The irradiated specimen was immediately put into a calorimeter together with the container. Omitting the details, we should note that the heat absorbed by the specimen was measured with an estimated error of about 10% (Dombrovsky and Mironov, 1997). Figure 7 compares the energy *Q* absorbed by aluminum-containing specimens, measured as a function of the powder mass *M*, with the results calculated according to the physical model described above. It should be noted that the experimental dependences *Q*(*M*) are linear for *M* < 0.5 g; i.e., for particle volume fraction *f*_{v} = 3.7%. This volume fraction corresponds to a mean particle spacing approximately equal to five times the particle radius: *d* ≈ 5*a*.

**Figure 7. Comparison of calculated (1-3) and measured (vertical bars) values of microwave radiation energy absorbed by aluminum powder in a dielectric matrix: 1, a = 7.5 μm; 2, a = 8.5 μm; 3, a = 10 μm.**

It is of interest to compare this result with the estimate based on the Maxwell-Garnett theory and valid under the Rayleigh scattering conditions. In that theory, small metal particles embedded in a dielectric material are treated as dipoles, and the complex permittivity of a disperse system is calculated in terms of the particle polarizability by applying the Lorentz-Lorenz formula. In this case, the resulting dependence is linear when

(11) |

i.e., even at particle concentrations substantially higher than those used in the experiments described. However, when condition (11) is violated, the character of the dependence is similar to that obtained in this study: in the vicinity of its linear portion, the electromagnetic interaction between particles leads to a more efficient absorption of radiation by the disperse system. Away from the linear portion, the particle-concentration dependence is obviously nonmonotonic, as predicted by the classical Maxwell-Garnett theory (Choy, 1999; Sihvola, 1999; Ruppin, 2000).

The calculated results presented in Fig. 7 were obtained by using the simple formula

(12) |

which predicts a linear dependence of the absorbed energy on the total mass of particles. In essence, the calculation is reduced to determining the specific absorption coefficient. Figure 7 shows that the calculated and measured results are in good agreement; a minor discrepancy may be attributed to a 10% error in the value of the mean particle radius. This result confirms the hypothesis that the contribution of dependent scattering is small, up to the volume fraction of aluminum particles of about 4%.

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