Predictions of real surface properties for radiation

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Although the radiative properties for many materials have been measured, there are still many others for which no data are available. For some classes of opaque materials, it is possible to predict the spectral, directional, total and hemispherical properties through the use of electromagnetic theory. These predictions have limitations in accuracy and applicability, but serve in the absence of other data.

Maxwell's Equations can be used to predict the amplitude of an electromagnetic wave that is reflected when it interacts with a surface. Because the energy carried by the wave is proportional to the square of the wave amplitude, this allows prediction of the reflectivity of a surface. Invoking Kirchhoff's Law then allows prediction of the other properties as well.

Electromagnetic theory predictions depend on certain approximations. There are terms in the mathematically exact solutions that decay rapidly with distance (a few wavelengths), and these are often neglected to simplify the solutions. Neglect of these "near-field" terms to obtain simplified solutions result in "far-field" solutions, and are used in the results and predictions presented in this section. When nanoscale interactions between EM waves and materials are considered, the additional terms must be retained, and these "near-field" solutions are more complex.

Accurate far-field solutions require that the features of the reflecting material be much smaller than the wavelength of the incident EM wave. Such surfaces are called optically smooth, and have the directional characteristics of a mirror. These surfaces in radiative transfer are often called specular surfaces, not to be confused with spectral (wavelength-dependent) surfaces. The radiative properties of optically smooth surfaces as derived from EM theory are presented here. Again, emphasis is on the wavelength dependence of the materials rather than their directional dependence, although some directional characteristics that lead to useful applications are discussed.

Figure 1: Interaction of EM wave with a dielectric surface.

Derivation of the results and more complete treatments including directional effects are in Siegel and Howell (2002), Modest (2003), and others.

Properties of Dielectric Materials. Dielectric materials as treated by EM theory are considered to be perfect insulators, so that no coupling occurs between the EM field and electrons in the material. In this case, EM theory predicts that the EM wave is not attenuated in the material, and that the reflective properties can be found in terms of other measurable properties of the material, including the simple index of refraction, $n$ , the dielectric constant, or the electrical permittivity. An EM wave propagating through dielectric material 1 and incident on an interface with dielectric material 2 is shown in Figure 1. In most engineering applications, radiation is incident through air or a vacuum; for either of these cases, the refractive index $n 1$ is taken as unity.

The EM wave will reflect at an angle of reflection equal to the angle of incidence, $θ$ . A portion of the wave from material 1 will pass through the interface and be refracted at angle $χ$ in material 2. The EM theory predicts that the angle of refraction is given by the familiar Snell's Law:

$\sin \chi = \frac{{{n_1}}}{{{n_2}}}\sin \theta \qquad \qquad(1)$

The refractive indices can be directly measured or can be computed from other properties of a dielectric. In terms of the dielectric constant $K$ (also called the electric permittivity $γ$ ) the relations are

$\frac{{{n_1}}}{{{n_2}}} = \frac{{\sqrt {{K_1}} }}{{\sqrt {{K_2}} }} = \frac{{\sqrt {{\gamma _1}} }}{{\sqrt {{\gamma _2}} }} \qquad \qquad(2)$

The reflectivity for unpolarized incident radiation is

${\rho _\lambda }\left( {\lambda ,\theta } \right) = \frac{1}{2}\frac{{{{\sin }^2}\left( {\theta - \chi } \right)}}{{{{\sin }^2}\left( {\theta + \chi } \right)}}\left[ {1 + \frac{{{{\cos }^2}\left( {\theta + \chi } \right)}}{{{{\cos }^2}\left( {\theta - \chi } \right)}}} \right] \qquad \qquad(3)$

If the radiation is polarized, the reflectivity for the components of the electric field $E$ that are parallel and perpendicular to the surface are, after substituting eq. (1) to eliminate $χ$ ,

${\rho _\lambda }_{,\parallel }\left( {\lambda ,\theta } \right) = {\left\{ {\frac{{{{\left( {\frac{{{n_2}}}{{{n_1}}}} \right)}^2}\cos \theta - {{\left[ {{{\left( {\frac{{{n_2}}}{{{n_1}}}} \right)}^2} - {{\sin }^2}\theta } \right]}^{1/2}}}}{{{{\left( {\frac{{{n_2}}}{{{n_1}}}} \right)}^2}\cos \theta + {{\left[ {{{\left( {\frac{{{n_2}}}{{{n_1}}}} \right)}^2} - {{\sin }^2}\theta } \right]}^{1/2}}}}} \right\}^2}\qquad \qquad(4a)$ ${\rho _{\lambda , \bot }}\left( {\lambda ,\theta } \right) = {\left\{ {\frac{{{{\left[ {{{\left( {\frac{{{n_2}}}{{{n_1}}}} \right)}^2} - {{\sin }^2}\theta } \right]}^{1/2}} - \cos \theta }}{{{{\left[ {{{\left( {\frac{{{n_2}}}{{{n_1}}}} \right)}^2} - {{\sin }^2}\theta } \right]}^{1/2}} + \cos \theta }}} \right\}^2}\qquad \qquad(4b)$

For unpolarized incident radiation, the reflectivity is the average of the reflectivities for the two components, or

$\begin{array}{l} {\rho _\lambda }\left( {\lambda ,\theta } \right) = \frac{1}{2}\left[ {{\rho _\lambda },\parallel \left( {\lambda ,\theta } \right) + {\rho _{\lambda , \bot }}\left( {\lambda ,\theta } \right)} \right] \\ = \frac{1}{2}\left( {{{\left\{ {\frac{{{{\left( {\frac{{{n_2}}}{{{n_1}}}} \right)}^2}\cos \theta - {{\left[ {{{\left( {\frac{{{n_2}}}{{{n_1}}}} \right)}^2} - {{\sin }^2}\theta } \right]}^{1/2}}}}{{{{\left( {\frac{{{n_2}}}{{{n_1}}}} \right)}^2}\cos \theta + {{\left[ {{{\left( {\frac{{{n_2}}}{{{n_1}}}} \right)}^2} - {{\sin }^2}\theta } \right]}^{1/2}}}}} \right\}}^2} + {{\left\{ {\frac{{{{\left[ {{{\left( {\frac{{{n_2}}}{{{n_1}}}} \right)}^2} - {{\sin }^2}\theta } \right]}^{1/2}} - \cos \theta }}{{{{\left[ {{{\left( {\frac{{{n_2}}}{{{n_1}}}} \right)}^2} - {{\sin }^2}\theta } \right]}^{1/2}} + \cos \theta }}} \right\}}^2}} \right) \\ \end{array} \qquad \qquad(5)$

Table 1 Optical Property Values and Normal Spectral Reflectivity of
Various Dielectric Materials at $T$ = 300 K and $λ$ = 0.589 $μ$ m; Data from Lide (2008).

PROPERTY	$n$ Refractive index	$K$ Dielectric constant	$ρ$ _{$λ$ ,n} eq. (9.38)
MATERIAL
SiO₂ (glass)	1.458	4.42	0.035
SiO₂ (fused quartz)	1.544	3.75	0.046
NaCl	1.5441	5.9	0.046
KCl	1.4902	4.86	0.039
H₂O (liquid)	1.332	77.78	0.020
H₂O (ice, 0°C)	1.309	91.6	0.018
vacuum	1.000	1.000	0.000

In eq. (4a), the reflectivity for the parallel component becomes equal to zero at the particular incident (and therefore reflected) angle of

$\theta = {\tan ^{ - 1}}\left( {{n_2}/{n_1}} \right)$ .

This angle is called Brewster's angle. Reflected energy at this angle from a mirror-like surface is thus all perpendicularly polarized, and this fact is the basis for polarized glasses. Lenses in polarized glasses filter out the perpendicular component of reflected radiation, eliminating glare from highly reflecting surfaces. For the special case of $θ$ = 0 (normal incidence), eq. (5) reduces to

${\rho _{\lambda ,n}}(\lambda ) = {\left( {\frac{{{n_2} - {n_1}}}{{{n_2} + {n_1}}}} \right)^2}\qquad \qquad(6)$

The spectral dependence for reflectivity enters because of the wavelength dependence of the refractive index $n$ .

Kirchhoff's Law may now be invoked to find the emissivity of dielectrics. Using eqs. $α λ + ρ λ = 1$ from Opaque Surface Property Definitions and (6), the normal spectral emissivity is

${\varepsilon _{\lambda ,n}}(\lambda ) = {\alpha _{\lambda ,n}}(\lambda ) = 1 - {\rho _{\lambda ,n}}(\lambda ) = 1 - {\left( {\frac{{{n_2} - {n_1}}}{{{n_2} + {n_1}}}} \right)^2} = \frac{{4\left( {\frac{{{n_2}}}{{{n_1}}}} \right)}}{{{{\left[ {\left( {\frac{{{n_2}}}{{{n_1}}}} \right) + 1} \right]}^2}}} = \frac{{4n}}{{{{\left( {n + 1} \right)}^2}}}\qquad \qquad(7)$

where $n$ = $n 2 / n 1$ has been substituted. Because most engineering properties are used for emission into air, for which $n 1$ ≈ 1, the $n$ in this and following equations is taken as equal to the refractive index of the emitting material, $n 2$ . The hemispherical spectral emissivity ${\varepsilon _\lambda }(\lambda )$ is found by using eq. (5) and Kirchhoff's Law, and then integrating the result over all angles of the hemisphere. The final form is

$\begin{array}{l} {\varepsilon _\lambda }(\lambda ) = \frac{1}{2} - \frac{{\left( {3n + 1} \right)\left( {n - 1} \right)}}{{6{{\left( {n + 1} \right)}^2}}} - \frac{{{n^2}{{\left( {{n^2} - 1} \right)}^2}}}{{{{\left( {{n^2} + 1} \right)}^3}}}\ln \left( {\frac{{n - 1}}{{n + 1}}} \right) \\ {\rm{ }} + \frac{{2{n^3}\left( {{n^2} + 2n - 1} \right)}}{{\left( {{n^2} + 1} \right)\left( {{n^4} - 1} \right)}} - \frac{{8{n^4}\left( {{n^4} + 1} \right)}}{{\left( {{n^2} + 1} \right){{\left( {{n^4} - 1} \right)}^2}}}\ln n \\ \end{array}\qquad \qquad(8)$

Properties of Electrical Conductors. Metals are characterized by having a high concentration of electrons, resulting in large electrical and thermal conductivities. A radiative EM field can interact with the electrons, causing rapid attenuation of the EM wave. In this case, solutions of Maxwell's Equations depend on the complex refractive index $\bar n$ of the absorbing material, which is defined here as

$\bar n = n - i\kappa \qquad \qquad(9)$

where $n$ is the simple refractive index, $i$ is the imaginary $i = \sqrt { - 1}$ , and $κ$ is the dimensionless extinction coefficient for the wave. (Some references such as Bohren and Huffman (1983) define the complex index as $\bar n = n + i\kappa$ . In Lide (2008), the definition varies from table-to-table.) For the particular case of metals, $κ$ is very large, and the polarized components of spectral reflectivity are (assuming incidence is through a nonabsorbing medium with $n 1$ = 1 and ${\bar n_2} = n - i\kappa$ )

${\rho _\lambda }_{,\parallel }\left( {\lambda ,\theta } \right) = \frac{{{{\left( {n\cos \theta - 1} \right)}^2} + {{\left( {\kappa \cos \theta } \right)}^2}}}{{{{\left( {n\cos \theta + 1} \right)}^2} + {{\left( {\kappa \cos \theta } \right)}^2}}}\qquad \qquad(10a)$ ${\rho _{\lambda , \bot }}\left( {\lambda ,\theta } \right) = \frac{{{{\left( {n - \cos \theta } \right)}^2} + {\kappa ^2}}}{{{{\left( {n + \cos \theta } \right)}^2} + {\kappa ^2}}}\qquad \qquad(10b)$

and, for unpolarized incident radiation,

${\rho _\lambda }\left( {\lambda ,\theta } \right) = \frac{1}{2}\left[ {\frac{{{{\left( {n\cos \theta - 1} \right)}^2} + {{\left( {\kappa \cos \theta } \right)}^2}}}{{{{\left( {n\cos \theta + 1} \right)}^2} + {{\left( {\kappa \cos \theta } \right)}^2}}} + \frac{{{{\left( {n - \cos \theta } \right)}^2} + {\kappa ^2}}}{{{{\left( {n + \cos \theta } \right)}^2} + {\kappa ^2}}}} \right]\qquad \qquad(11)$

For radiation incident at $θ$ = 0, this reduces to

${\rho _\lambda }\left( {\lambda ,\theta = 0} \right) = {\rho _{\lambda ,n}}\left( \lambda \right) = \frac{{{{\left( {n - 1} \right)}^2} + {\kappa ^2}}}{{{{\left( {n + 1} \right)}^2} + {\kappa ^2}}}\qquad \qquad(12)$

Invoking Kirchhoff's Law and using eqs. $α λ + ρ λ = 1$ from Opaque Surface Property Definitions and (12), the normal emissivity for metals is predicted to be

${\varepsilon _{\lambda ,n}} = {\alpha _{\lambda ,n}} = 1 - {\rho _{\lambda ,n}}\left( \lambda \right) = \frac{{4n}}{{{{\left( {n + 1} \right)}^2} + {\kappa ^2}}}\qquad \qquad(13)$

Using eq. (11) with Kirchhoff's Law to find $ε$ _$λ$( $λ$ , $θ$ ) and then integrating over all angles of a hemisphere and applying some approximations gives the hemispherical spectral emissivity as

$\begin{array}{l} {\varepsilon _\lambda }\left( \lambda \right) = 4n - 4{n^2}\ln \left( {\frac{{1 + 2n + {n^2} + {\kappa ^2}}}{{{n^2} + {\kappa ^2}}}} \right) + \frac{{4n\left( {{n^2} - {\kappa ^2}} \right)}}{\kappa }{\tan ^{ - 1}}\left( {\frac{\kappa }{{n + {n^2} + {\kappa ^2}}}} \right) \\ {\rm{ + }}\frac{{4n}}{{{n^2} + {\kappa ^2}}} - \frac{{4{n^2}}}{{{{\left( {{n^2} + {\kappa ^2}} \right)}^2}}}\ln \left( {1 + 2n + {n^2} + {\kappa ^2}} \right) - \frac{{4n\left( {{\kappa ^2} - {n^2}} \right)}}{{\kappa {{\left( {{n^2} + {\kappa ^2}} \right)}^2}}}{\tan ^{ - 1}}\left( {\frac{\kappa }{{1 + n}}} \right) \\ \end{array}\qquad \qquad(14)$

Measured spectral normal emissivity of polished platinum at 1400K (Harrison et al., 1961) compared with the prediction of eq. (16) using re (T = 1400K) = 47.3x10-6 Ohm-cm

Figure 2: Measured spectral normal emissivity of polished platinum at 1400K (Harrison et al., 1961) compared with the prediction of eq. (16) using $r e$ (T = 1400K) = 47.3x10^-6 Ohm-cm.

As before, the spectral dependence of emissivity enters through the spectral dependence of $n$ and $κ$ . Equation (14) is said to be accurate within a few percent for most highly polished pure metals (Siegel and Howell, 2002).

At long wavelengths ( $λ$ > about 5 $μ$ m), Maxwell's Equations predict that $n$ and $κ$ become equal for metals, and in turn are related to the electrical resistivity of the metal, $r e$ , through the relation

$n = \kappa = \left( {\frac{{0.003{\lambda _o}}}{{{r_e}}}} \right)\qquad \qquad(15)$

In eq. (15), $λ$ _o is the wavelength in a vacuum in $μ$ m, and $r$ _e is in ohm-cm. The relation is the Hagen-Rubens Equation.

Substituting $n$ = $κ$ into eq. (13), expanding the result in a series, and then substituting eq. (15) after retaining only two terms in the series (and making a small adjustment to account for the truncation) results in the Davisson-Weeks emissivity equation as modified by Schmidt and Eckert (1935).

${\varepsilon _{\lambda ,n}}\left( \lambda \right) = 36.5{\left( {\frac{{{r_e}}}{{{\lambda _o}}}} \right)^{1/2}} - 464\frac{{{r_e}}}{{{\lambda _o}}}\qquad \qquad(16)$

Using the resistivity of platinum at 1400K, $ρ$ _e = 47.3x10^-6 Ohm-cm (extrapolated from data for pure platinum), the spectral normal emissivity of polished platinum can be predicted using eq. (16). The result is compared with measured values in Fig. 2.

The predictions at shorter wavelengths usually become worse, and are often in greater error for other metals, especially for surfaces that are not highly polished or are oxidized. However, if no measured data is available, the EM theory predictions provide a useful resource, and can also be used to extend limited data. The total properties from EM theory are found by integrating the spectral properties, which incorporates some further approximations. The results are

${\varepsilon _n}(T) = 0.578{({r_e}T)^{1/2}} - 0.178{r_e}T + 0.0584{({r_e}T)^{3/2}}\qquad \qquad(17)$ $\varepsilon (T) = 0.766{({r_e}T)^{1/2}} - \left[ {0.309 - 0.0889\ln \left( {{r_e}T} \right)} \right]{r_e}T - 0.0175{({r_e}T)^{3/2}}\qquad \qquad(18)$

Comparisons of predictions of these two equations with experimental data for ten or more metals show good agreement (Siegel and Howell, 2002).

Table 2 gives data for the various optical properties at $T$ = 300K for a variety of metals along with comparisons of the spectral normal reflectivity, $ρ$ _{$λ$ ,n}, calculated from the EM theory relations in this section with measured values.

Table 2: Optical Property Values and Normal Spectral Reflectivity
of Various Metals at $T$ = 300 K and $λ$ = 0.589 and 10 $μ$ m

PROPERTY	Refractive index¹ $n$	Extinction coefficient¹ $κ$	Electrical resistivity¹ $r e$	Normal spectral reflectivity $ρ λ, n$
MATERIAL			(Ohm-cm)	eq. (12)	eq. (16)	Measured²
Aluminum (0.589 $μ$ m)	1.15	7.147	2.73 x10^-6	0.917	0.923	0.91
(10 $μ$ m)	25.4	90.0	"	0.988	0.981	0.988
Chromium (0.589 $μ$ m)	3.33	4.39	12.7 x10^-6	0.650	0.841	0.57
(10 $μ$ m)	13.9	27.7	"	0.944	0.960	0.946
Copper (0.589 $μ$ m)	0.47	2.81	1.72x10^-6	0.813	0.939	0.76
(10 $μ$ m)	28	68	"	0.980	0.985	0.985
Gold (0.589 $μ$ m)	0.18	2.84	2.27x10^-6	0.924	0.930	0.85
(10 $μ$ m)	6.7	73	"	0.995	0.983	0.994
Iron (0.589 $μ$ m)	2.80	3.34	9.98x10^-6	0.562	0.858	0.517
(10 $μ$ m)	6.34	28.2	"	0.970	0.964	0.953
Nickel (0.589 $μ$ m)	1.85	3.48	7.20x10^-6	0.634	0.878	0.655
(10 $μ$ m)	7.59	38.5	"	0.980	0.969	0.965
Platinum (0.589 $μ$ m)	2.23	3.92	10.8 x10^-6	0.654	0.852	0.673
(10 $μ$ m)	10.8	38.2	"	0.973	0.963	0.970
Silver (0.589 $μ$ m)	0.26	3.96	1.63 x10^-6	0.940	0.941	0.95
(10 $μ$ m)	8.22	79	"	0.995	0.985	0.995
Titanium (0.589 $μ$ m)	2.01	2.77	39 x10^-6	0.520	0.734	0.490
(10 $μ$ m)	4.1	19.7	"	0.960	0.930	0.970
Tungsten (0.589 $μ$ m)	3.54	2.84	5.44 x10^-6	0.506	0.893	0.51
(10 $μ$ m)	11.6	48.4	"	0.981	0.973	0.97

¹Lide (2008); ²Measured data is the largest reflectivity value from multiple references cited in Touloukian (1970)

Note the relatively poor prediction of normal spectral reflectivity at the short wavelength (0.589 $μ$ m) using eq. (16), which is based on the electrical resistivity $r e$ , and the much improved predictions at 10 $μ$ m. The assumptions used in the derivation of eq. (16) are clearly better satisfied at the longer wavelength. However, the assumption that $n$ ≈ $κ$ is not met even at 10 $μ$ m. The predictions of eq. (12), based on the components of the complex refractive index $n$ and $κ$ , are better at both wavelengths. However, these properties (especially $κ$ ) are often unavailable, whereas the electrical resistivity is often in handbooks or easily measured.

References

Bohren, C.F. and Huffman D.R., 1983, Absorption and Scattering of Light by Small Particles, John Wiley & Sons, New York, NY.

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

Harrison, W.N., Richmond, J.C., Plyler, E.K., Stair, R. and Skramstad, H.K., 1961, Standardization of Thermal Emittance Measurements, WADC-TR-59-510 (Pt.2), 21 pp.

Lide, D.R. (ed.), Handbook of Chemistry and Physics, 89^th ed., CRC Press, Boca Raton, 2008.

Modest, M.F., 2003, Radiative Heat Transfer, 2^nd ed., Academic Press, Sand Diego, CA.

Schmidt, E. and Eckert, E.R.G., 1935, Über die Richtungsverteilung der Wärmestrahlung von Oberflächen, Forschung Geb. D. Ingenieurwes., Vol. 6, pp.175-183.

Siegel, R. and Howell, J.R., 2002, Thermal Radiation Heat Transfer, 4^th ed., Taylor and Francis, New York, NY.

Touloukian, Y.S. and DeWitt, D.P., (eds.), Thermophysical Properties of Matter, vol. 7, Thermal radiative properties: metallic elements and alloys, and v. 8. Thermal radiative properties: non-metallic solids, IFI/Plenum, New York, 1970-[79].

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Predictions of real surface properties for radiation

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