˜o M. Caldas and V. Semia

Vol. 18, No. 4 / April 2001 / J. Opt. Soc. Am. A

831

Radiative properties of small particles: extension of the Penndorf model ˜ o* Miguel Caldas and Viriato Semia Department of Mechanical Engineering, Instituto Superior Te´cnico, Technical University of Lisbon, Avenida Rovisco Pais, 1049-001 Lisboa, Portugal Received May 2, 2000; revised manuscript received September 22, 2000; accepted October 9, 2000 Explicit analytical expressions to represent the radiative properties of spherical particles in the small sizeparameter range were obtained. These expressions were deduced following Penndorf ’s approach of expanding the Mie coefficients in power series on the size parameter. However, in opposition to Penndorf ’s original work—in which some errors were found and corrected—the Mie coefficients were expanded to the eighth power of the size parameter, which results in a five-term approximation to the extinction and scattering efficiencies. Also, expressions for the evaluation of the asymmetry factor and both polarized and unpolarized phase functions were deduced and presented. The results so obtained have proved to be very accurate, even for size parameters beyond the limit of validity of the approach utilized. © 2001 Optical Society of America OCIS codes: 000.4430, 290.5850, 290.2200, 290.5870, 290.4020, 260.5430.

1. INTRODUCTION Thermal radiation scattering from dispersed particles plays an important role in many physical phenomena, such as industrial high-temperature combustors, optical diagnostic applications, and atmospheric scattering processes, just to name a few. In engineering applications detailed knowledge of this mechanism may become particularly crucial and useful, either in the prediction of radiative heat transfer (see, e.g., Ref. 1) or in the use of optical diagnostic methods. For both purposes it is necessary to know the radiative properties of particulate matter, the most important being the extinction and scattering efficiency factors and the polarized components of the scattering phase function, which allow for determination of the scattered intensities. Also useful is the knowledge of the asymmetry factor, since this parameter roughly describes the directional scattering behavior of particles. In fact, the asymmetry factor suffices for generating a linear approximation to the phase function by means of expanding it in Legendre polynomials. This parameter also plays a determinant role in a phase-function approximation model developed recently.2 Natural scatterers are generally nonspherical and present a distribution in size, shape, and orientation. It is recognized that nonsphericity has important effects in scattering processes.3 Some of the most-studied nonspherical particles are spheroids,4–7 owing to the simplicity of their geometry. Because of the sharp rectangular edges of finite circular cylinders, their scattering behavior has also been investigated.8 In spite of the importance granted to nonsphericity effects on the particles’ extinction and scattering properties, it is recognized that averaging over random orientations, over shape distribution, and over size distribution will smooth the nonspherical features of the scattering properties of particles.6 This smoothing is particularly pronounced if particles are small in comparison with the wavelength of the incident radiation.6,7 Therefore in the 0740-3232/2001/040831-08$15.00

range of small particles, which is the focus of this paper, it is quite reasonable to treat particles as volume-equivalent spheres.6 Needless to say, this assumption will greatly simplify the analytical expressions to be dealt with. If particles are assumed to be spherical and homogeneous, classical Mie theory provides a way of computing these properties. However, Mie theory is mathematically complex and difficult to manipulate. Even more difficult is to grasp the physical essence hiding behind the mathematics. Therefore a more accessible approach is required, particularly for numerical modeling purposes (see, e.g., Ref. 9). Frequently in the previously mentioned applications the ratio of the particles’ perimeter to the incident radiation wavelength, known as the particles’ size parameter, x, is small; i.e., particles are small in comparison with the incident radiation wavelength. Note that this is the wavelength in the medium surrounding the particles; i.e., x ⫽ ␲ DN/␭ 0 , where ␭ 0 is the free-space wavelength of the incoming radiation and N is the real refractive index of the medium supporting the particle. Since it is assumed here that the medium surrounding the particles is nonparticipating N ⫽ 1.0, the size parameter becomes simply x ⫽ ␲ D/␭. When this occurs it is possible to use limiting expressions to calculate the radiative properties of the particles. The main advantage of this procedure does not lie in a decrease in computational effort, since in the small-particle range, Mie computations are efficient enough, but it yields a considerable increase in the analytical manageability of the resulting expressions. In fact, it is frequently more convenient to use explicit analytical expressions rather than discrete values. Among the limiting expressions applicable to the calculation of the extinction and scattering efficiency factors in the range of small particle size parameter, probably the best known is the Rayleigh limit. This is a one-term polynomial approximation and it is usually assumed to be accurate for values of the size parameters up to 0.3. An© 2001 Optical Society of America

832

˜o M. Caldas and V. Semia

J. Opt. Soc. Am. A / Vol. 18, No. 4 / April 2001

other frequently used approximation is the Penndorf model, a three-term polynomial approximation.10 It seems rather peculiar that although Penndorf ’s work allows for the use of a four-term approximation that would broaden the range of application of this model, all authors having recourse to that work, including Penndorf himself, have used only a three-term approximation, at least to the extent of the present authors’ knowledge. The reason for this may lie in the existence of some errors in Penndorf ’s original work, as hinted by Selamet and Arpaci,11 that will be addressed later on in this work. When the Penndorf model is used, accuracy is usually considered ensured up to x ⫽ 0.7, although Ku and Felske12 provided some more-rigorous expressions, involving both x and the complex refractive index (m ⫽ n ⫺ ik) to establish the limits of accuracy of both the Rayleigh and the Penndorf efficiency factors. However, the above-mentioned models exhibit some limitations. In fact, the accuracy of the Rayleigh approximation breaks down for relatively small values of the size parameter, making it unsuitable for a significant number of heat transfer and optical diagnostic applications (see, e.g., Ref. 11). As for the Penndorf model, its flaw stems from its application to the calculation of the extinction and scattering coefficients only; i.e., no results are available for both the polarized phase functions and the asymmetry factor. Moreover, there are some optical diagnostic and high-temperature heat-transfer problems that involve size parameters outside the Penndorf model range of validity.11 It is therefore most convenient to have available some explicit analytical expressions that represent all the important radiative properties of particles that have larger size parameters. In the present paper such analytical expressions are obtained by extending Penndorf ’s analysis, in reference both to the number of terms retained, as discussed below, and to the properties that are dealt with, which include the polarized phase functions and the asymmetry factor. As in Penndorf ’s work, the first five Mie coefficients are expanded in powers of the size parameter, but, since all other Mie coefficients are of the ninth order, they are expanded up to the eighth order rather than up to the seventh, as done by Penndorf. Then the amplitude functions and the scattering intensities are calculated up to the eighth and the eleventh orders, respectively. Note that it would be useless to use a higher order to calculate the intensities, since the terms in the Mie coefficients of the ninth order and higher would influence the result, and those terms are not known. With these results, all the relevant radiative properties of particles can be expressed in a power series manner, the size parameter being the variable, and with all the coefficients depending on the complex refractive index.

1

Qs ⫽

2



1

4i 共 ␮ 兲

⫺1

x2

d␮ .

(2)

In the above equations x is the particle size parameter, ␮ is the cosine of the scattering angle, and S(1) is the amplitude function in the forward direction. Note that there are two amplitude functions, S 1 ( ␮ ) and S 2 ( ␮ ), that represent the scattering amplitudes in the perpendicular and the parallel polarization, respectively. However, in the forward direction, these two functions assume an identical value that is represented by S(1). In Eq. (2), i( ␮ ) represents the unpolarized scattering intensity, whose definition is given by Eqs. (3), where subscripts 1 and 2 refer to perpendicular and parallel polarization, respectively: i共 ␮ 兲 ⫽

i 1共 ␮ 兲 ⫹ i 2共 ␮ 兲 2

,

(3a)

i 1共 ␮ 兲 ⫽ 兩 S 1共 ␮ 兲兩 2,

(3b)

i 2共 ␮ 兲 ⫽ 兩 S 2共 ␮ 兲兩 .

(3c)

2

The asymmetry factor g and the unpolarized phase function P( ␮ ) are given by Eqs. (4) and (5a), respectively. In turn, it is possible to define polarized phase functions, as in Eqs. (5b) and (5c). These represent the angular distribution of the radiative energy that is scattered with a given polarization. Obviously, the unpolarized phase function is related to the polarized phase functions as expressed in Eq. (5d). g⫽

P共 ␮ 兲 ⫽ P 1共 ␮ 兲 ⫽ P 2共 ␮ 兲 ⫽ P共 ␮ 兲 ⫽

1 1 Qs 2



1

4i 共 ␮ 兲

⫺1

x2

1 4i 共 ␮ 兲 Qs

x2

,

1 4i 1 共 ␮ 兲 Qs

x2

1 4i 2 共 ␮ 兲 Qs

x2

␮ d␮ ,

(4)

(5a)

,

(5b)

,

(5c)

P 1共 ␮ 兲 ⫹ P 2共 ␮ 兲 2

.

(5d)

Now it is possible to use Mie theory (e.g., Refs. 13 and 14) to evaluate the scattering amplitude functions: ⬁

S 1共 ␮ 兲 ⫽



n⫽1 ⬁

S 2共 ␮ 兲 ⫽

2n ⫹ 1 n共 n ⫹ 1 兲 2n ⫹ 1

关 a n ␲ n 共 ␮ 兲 ⫹ b n ␶ n 共 ␮ 兲兴 ,

兺 n共 n ⫹ 1 兲 关 b

n␲ n共 ␮ 兲

⫹ a n ␶ n 共 ␮ 兲兴 .

(6a)

(6b)

n⫽1

2. ANALYSIS AND MODELING As known from the general scattering theory for homogeneous isolated spherical particles (e.g., Refs. 13 and 14), the extinction and scattering efficiencies are given by Qe ⫽

4 x2

Re关 S 共 1 兲兴 ,

(1)

In Eqs. (6a) and (6b), a n and b n are the Mie coefficients that depend both on the size parameter x and on the complex index of refraction m (notice that this should be understood as the index of refraction of the particle relative to the surrounding medium), and ␶ n and ␲ n are directiondependent functions that can be found in the work of Van de Hulst13 and Kerker.14

˜o M. Caldas and V. Semia

Vol. 18, No. 4 / April 2001 / J. Opt. Soc. Am. A

Assuming that particles are small, it is possible to expand the Mie coefficients in power series, but in order to have a valid expansion it is required that both x ⭐ 1 and x 兩 m 兩 ⭐ 1 (see Refs. 10 and 12). In the present work, such expansion for the Mie coefficients was performed up to x 8 terms, as opposed to Penndorf ’s original work,10 where terms only up to x 7 were considered. It is important to notice that the increase in complexity brought about by this option is irrelevant. The coefficients of the power series expansion for the Mie coefficients are depicted in Table 1. Notice that only the first five Mie coefficients were considered since all the others are of order higher than the eighth. For further analytical development, it is convenient to write the scattering amplitude functions as follows: S 1 ⫽ S 13x 3 ⫹ S 15x 5 ⫹ S 16x 6 ⫹ S 17x 7 ⫹ S 18x 8 ,

(7a)

S 2 ⫽ S 23x 3 ⫹ S 25x 5 ⫹ S 26x 6 ⫹ S 27x 7 ⫹ S 28x 8 .

(7b)

After some algebra it is possible to write the forwardscattering amplitude and the scattering intensities as in Eqs. (8) and (9), respectively, where the overbar denotes the complex conjugate. In Eqs. (9) the terms of twelfth and higher orders were neglected. The reason for this procedure is that the terms appearing in Mie coefficients of ninth order and higher would contribute to this result. Since such terms were not considered in the Miecoefficient expansion itself, the result obtained for the scattering intensities with terms of order higher than or equal to the twelfth would be inconsistent. S共 1 兲 ⫽

x3 2

关 3a 13 ⫹ 共 3a 15 ⫹ 3b 15 ⫹ 5a 25兲 x 2 ⫹ 3a 16x 3

⫹ 共 3a 17 ⫹ 3b 17 ⫹ 5a 27 ⫹ 5b 27 ⫹ 7a 37兲 x

(8)

i 1 共 ␮ 兲 ⫽ 兩 S 13兩 x ⫹ 2 Re共 S 13S 15兲 x ⫹ 2 Re共 S 13S 16兲 x 8

⫹ 2 Re共 S 13S 18兲兴 x 11,

(9a)

i 2 共 ␮ 兲 ⫽ 兩 S 23兩 x ⫹ 2 Re共 S 23S 25兲 x ⫹ 2 Re共 S 23S 26兲 x 8

⫹ 关 兩 S 25兩 ⫹ 2 Re共 S 23S 27兲兴 x 2

10

⫹ 2 Re共 S 23S 28兲兴 x ,

x 6 ⫹ Re共 S 13S 15 ⫹ S 23S 25兲 x 8

⫹ Re共 S 13S 16 ⫹ S 23S 26兲 x 9 ⫹



兩 S 15兩 2 ⫹ 兩 S 25兩 2

2



⫹ Re共 S 13S 17 ⫹ S 23S 27兲 x 10

⫹ 关 Re共 S 16S 15 ⫹ S 26S 25兲 ⫹ Re共 S 13S 18 ⫹ S 23S 28兲兴 x 11.

(9c)

Next, it is necessary to evaluate the power series coefficients resulting from the expansion of the Mie coefficients in terms of the complex refractive index of the particles. This is achieved through the use of a McLaurin series expansion. The results are presented in Table 2, where i is the imaginary unit and P, Q, R, S, T, U, V, and W are parameters that depend only on the refractive index m. The expressions for the evaluation of these last m dependent parameters are as follows: P⫽

Q⫽

R⫽

S⫽

T⫽

U⫽

m2 ⫺ 1 m2 ⫹ 2 m2 ⫺ 2 m2 ⫹ 2

,

(10a)

,

(10b)

m 6 ⫹ 20m 4 ⫺ 200m 2 ⫹ 200 共 m2 ⫹ 2 兲2

m2 ⫺ 1 2m 2 ⫹ 3

,

共 2m 2 ⫹ 3 兲 2

3m 2 ⫹ 4

(10c)

(10d)

m2 ⫺ 1

m2 ⫺ 1

,

,

(10e)

,

(10f)

V ⫽ m 2 ⫺ 1,

(10g)

W ⫽ 共 m 2 ⫺ 1 兲共 2m 2 ⫺ 5 兲 .

(10h)

9

⫹ 关 2 Re共 S 26S 25兲

11

2

9

⫹ 兵 兩 S 15兩 2 ⫹ 2 Re共 S 13S 17兲 其 x 10 ⫹ 关 2 Re共 S 16S 15兲

2 6

兩 S 13兩 2 ⫹ 兩 S 23兩 2

4

⫹ 3a 18x 5 兴 , 2 6

i共 ␮ 兲 ⫽

833

(9b)

Table 1. Coefficients Appearing in the Power Series Expansion to the Eighth Power of the First Five Mie Coefficients (a 1 , b 1 , a 2 , b 2 and a 3 ) Power Series Mie Coefficient

x3

x5

x6

x7

x8

a1 b1 a2 b2 a3

a 13 – – – –

a 15 b 15 a 25 – –

a 16 – – – –

a 17 b 17 a 27 b 27 a 37

a 18 – – – –

Note that in Penndorf ’s work10 the coefficient a 37 exhibited a value that differs by a factor of 4 from our value. Also in error were both the R and the W coefficients. These errors may explain why, to the authors’ knowledge, no other author to this date, not even Penndorf himself, has used an expansion with more than three terms, even though the four-term expansion has been available. Using the previous results together with the simplifying definitions listed below, one can arrive at the results for the scattering intensities expressed by Eqs. (11): Q ⬘ ⫽ Q,

R ⬘ ⫽ 18R,

T ⬘ ⫽ 375T/P, W ⬘ ⫽ 5W/P.

S ⬘ ⫽ 5S/P,

U ⬘ ⫽ 28U/P,

V ⬘ ⫽ V/P,

834

˜o M. Caldas and V. Semia

J. Opt. Soc. Am. A / Vol. 18, No. 4 / April 2001

Table 2. Functional Form of the Power Series Coefficients for the First Five Mie Coefficients Introduced in Table 1 (P, Q, R, S, T, U, V and W Are Parameters That Depend on m) Power Series Mie Coefficient

x3

a1

a13 ⫽

b1

45

a25 ⫽





⫽ 4兩P兩2x 4 1 ⫹ 4

1 15

1

Im共 P 兲 x 3 ⫹

3

6300 2

4i 共 ␮ 兲 x

2

– –

1 iV

1575



关 Re共 Q ⬘ 兲 ⫹ Re共 V ⬘ ⫹ S ⬘ 兲 ␮ 兴 x 2



945



8 2 P Q 15

iW

1 a 27 ⫽ ⫺ iT 21

iS



a 18 ⫽

1

b 17 ⫽ –

x8

1 iPR 175





4 1575

iU



1

⫽ 2兩P兩2x 4 共 1 ⫹ ␮ 2 兲 ⫹

15

关 Re共 Q ⬘ 兲共 1 ⫹ ␮ 2 兲

⫹ 2 Re共 V ⬘ 兲 ␮ ⫹ 2 Re共 S ⬘ 兲 ␮ 3 兴 x 2

关 7 兩 Q ⬘ 兩 2 ⫹ 4 Re共 R ⬘ 兲

¯ ⬘兴␮ ⫹ 7 共 兩 V ⬘ 兩 ⫹ 兩 S ⬘ 兩 兲 ␮ ⫹ 14 Re关共 V ⬘ ⫹ S ⬘ 兲 Q 2



4 3

Im共 P 兲共 1 ⫹ ␮ 2 兲 x 3 ⫹

1 6300

关 7 兩 Q ⬘兩 2共 1 ⫹ ␮ 2 兲

¯ ⬘兲␮2 ⫹ 4 Re共 W ⬘ ⫺ T ⬘ 兲 ␮ ⫹ 14 Re共 S ⬘ V

⫹ 4 Re共 R ⬘ 兲共 1 ⫹ ␮ 2 兲 ⫹ 7 共 兩 V ⬘ 兩 2 ⫹ 兩 S ⬘ 兩 2 兲

⫹ 20 Re共 V ⬘ 兲共 2 ␮ 2 ⫺ 1 兲 ⫹ Re共 U ⬘ 兲共 5 ␮ 2 ⫺ 1 兲兴 x 4

⫻ 共 1 ⫹ ␮ 2 兲 ⫹ 28兩 S ⬘ 兩 2 共 ␮ 4 ⫺ ␮ 2 兲



2 45

¯ ⬘ 兲 ␮ ⫹ 8 Re共 W ⬘ 兲 ␮ ⫹ 28 Re共 V ⬘ Q

¯ 兲兴 ⫺ Im关共 V ⬘ 兵 Im关 Q ⬘ 共 P ⫺ P

¯ ⬘ 兲 ␮ 3 ⫺ 8 Re共 T ⬘ 兲 ␮ 3 ⫹ 14 Re共 S ⬘ V ¯ ⬘兲 ⫹ 28 Re共 S ⬘ Q



¯ 兴␮其x5 , ⫹ S⬘兲P

x

a 17 ⫽

a37 ⫽

2

2

4 2 P 9

b27 ⫽



4i 2 共 ␮ 兲

x7

iV

1

15 –



a 16 ⫽

1



a3

x

3

x6

2 iPQ 5

a 15 ⫽

iP

b15 ⫽

b2

2

2



a2

4i 1 共 ␮ 兲

x5



⫽ 4兩P兩2x 4 ␮ 2 ⫹

(11a)

1 15

关 Re共 Q ⬘ 兲 ␮ 2 ⫹ Re共 V ⬘ ⫺ S ⬘ 兲 ␮

⫹ 2 Re共 S ⬘ 兲 ␮ 3 兴 x 2 ⫹ ⫹

1 6300

4 3

⫹ 兩 S ⬘ 兩 2 兲 ⫹ 28兩 S ⬘ 兩 2 共 ␮ 4 ⫺ ␮ 2 兲 ⫹ 14 Re关共 V ⬘

¯ ⬘ 兲 ␮ 3 ⫺ 8 Re共 T ⬘ 兲 ␮ 3 ⫹ 14 Re共 SV ¯ ⬘兲 ⫹ 28 Re共 S ⬘ Q ⫻ 共 2 ␮ 2 ⫺ 1 兲 ⫹ 20 Re共 V ⬘ 兲 ␮ 2 ⫹ Re共 U ⬘ 兲共 15␮ 4

45

¯ 兲兴 ␮ 2 兵 Im关 Q ⬘ 共 P ⫺ P

¯ 兴 ␮ ⫺ 2 Im共 S ⬘ P ¯ 兲␮ 3其 x 5 ⫺ Im关共 V ⬘ ⫺ S ⬘ 兲 P

2 45

¯ 兲兴 兵 Im关 Q ⬘ 共 P ⫺ P



¯ 兲␮ 3其 x 5 . ⫺ 2 Im共 S ⬘ P

(11c)

The final resulting expressions for the extinction and scattering efficiencies Q e and Q s are as follows:

¯ ⬘ 兴 ␮ ⫹ 4 Re共 W ⬘ ⫹ T ⬘ 兲 ␮ ⫺ S⬘兲Q

2

⫻共 15␮ 4 ⫺ 6 ␮ 2 ⫺ 1 兲兴 x 4 ⫹ ¯ 兲␮ ⫻共 1 ⫹ ␮ 2 兲 ⫺ 2 Im共 V ⬘ P

Im共 P 兲 ␮ 2 x 3

兵 7 兩 Q ⬘ 兩 2 ␮ 2 ⫹ 4 Re共 R ⬘ 兲 ␮ 2 ⫹ 7 共 兩 V ⬘ 兩 2

⫺ 11␮ 2 兲 其 x 4 ⫹

⫻共 3 ␮ 2 ⫺ 1 兲 ⫹ 20 Re共 V ⬘ 兲共 3 ␮ 2 ⫺ 1 兲 ⫹ Re共 U ⬘ 兲





Q e ⫽ x ⫺4 Im共 P 兲 ⫺ ⫹

(11b)

8 3

2 15

Re共 P 2 兲 x 3 ⫺

Im关 P 共 Q ⬘ ⫹ V ⬘ ⫹ S ⬘ 兲兴 x 2 2

1575

⫹ 5V ⬘ ⫹ U ⬘ 兲兴 x 4 ⫹

8 45

Im关 P 共 R ⬘ ⫹ W ⬘ ⫺ T ⬘



Re共 P 2 Q ⬘ 兲 x 5 ,

(12)

˜o M. Caldas and V. Semia

Qs ⫽

8 3



兩P兩2x 4 1 ⫹

1



31500

Vol. 18, No. 4 / April 2001 / J. Opt. Soc. Am. A

1 15

Re共 Q ⬘ 兲 x 2 ⫹

4 3

P 2共 ␮ 兲 ⫽

Im共 P 兲 x 3

3 2

关 35兩 Q ⬘ 兩 2 ⫹ 20 Re共 R ⬘ 兲 ⫹ 35兩 V ⬘ 兩 2

⫹ 21兩 S ⬘ 兩 2 兴 x 4 ⫹

2 45

16



3

2 15



¯ 兲兴 x 5 . Im关 Q ⬘ 共 P ⫺ P

Qs



关 Re共 V ⬘ ⫺ S ⬘ 兲 ␮

1 2100

兵 5 Re共 U ⬘ 兲共 15␮ 4

⫺ 11␮ 2 兲 ⫹ 7 兩 S ⬘ 兩 2 共 20␮ 4 ⫺ 23␮ 2 ⫹ 5 兲

(13)

⫺ 35兩 V ⬘ 兩 2 共 ␮ 2 ⫺ 1 兲 ⫹ 100 Re共 V ⬘ 兲 ␮ 2 ¯ ⬘ 兲共 2 ␮ 2 ⫺ 1 兲 ⫹ 70 Re关共 V ⬘ ⫹ 70 Re共 S ⬘ V ¯ ⬘ 兴 ␮ ⫹ 20 Re共 W ⬘ ⫹ T ⬘ 兲 ␮ ⫺ S⬘兲Q ¯ ⬘ 兲 ␮ 3 ⫺ 40 Re共 T ⬘ 兲 ␮ 3 其 x 2 ⫹ 140 Re共 S ⬘ Q ⫺

Im关 P 共 Q ⬘ ⫹ V ⬘ ⫹ S ⬘ 兲兴 x 2 P共 ␮ 兲 ⫽

关 Im共 P 兲兴 2 x 3 ⫺

⫹ 5V ⬘ ⫹ U ⬘ 兲兴 x 4 ⫺

2 1575

2 3



¯ 兴 ␮ ⫹ 2 Im共 S ⬘ P ¯ 兲␮ 3其 x 3 , 兵 Im关共 V ⬘ ⫺ S ⬘ 兲 P

16 45

4 225 ⫹



Im共 P 兲 Im共 PQ ⬘ 兲 x 5 .

(14)



210

⫹ 10 Re共 W ⬘ 兲 ⫺ 6 Re共 T ⬘ 兲兴 x ⫺



2



4 兩P兩2x 6 15 1

2100

Qs

2

2100

3



¯ ⬘ 兲 ␮ 3 ⫺ 40 Re共 T ⬘ 兲 ␮ 3 兴 x 2 ⫹ 140 Re共 S ⬘ Q ⫺

4 3



¯ 兲 ␮ ⫹ Im共 S ⬘ P ¯ 兲␮3兴x3 . 关 Im共 V ⬘ P

(16c)



A nP n共 ␮ 兲 .

By comparing Eq. (16c) and Eq. (17), we can derive explicit expressions for the first four coefficients in this expansion; they are presented below. Note that although the formula for A 1 is explicitly presented, an expression that is probably more convenient to use is A 1 ⫽ 3g; A 0 ⫽ 1,

兵 5 Re共 U ⬘ 兲共 5 ␮ 2 ⫺ 1 兲 ⫹ 7 兩 S ⬘ 兩 2 共 5 ␮ 2

A1 ⫽

12 兩 P 兩 2 x 6 225 ⫹

Qs 1 210



(18a)

关 5 Re共 V ⬘ 兲 ⫹ 3 Re共 S ⬘ 兲兴

¯ ⬘ 兲 ⫹ 21 Re共 S ⬘ Q ¯ ⬘ 兲 ⫹ 10 Re共 W ⬘ 兲 关 35 Re共 V ⬘ Q

⫺ 6 Re共 T ⬘ 兲兴 x 2 ⫺ (16a)

(17)

n⫽0

关 Re共 V ⬘ ⫹ S ⬘ 兲 ␮ 兴



关 5 Re共 U ⬘ 兲共 15␮ 4 ⫺ 6 ␮ 2

¯ ⬘ 兲 ␮ ⫹ 40 Re共 W ⬘ 兲 ␮ ⫹ 140 Re共 V ⬘ Q

P共 ␮ 兲 ⫽

(15)

¯ 兴␮其x3 , 兵 Im关共 V ⬘ ⫹ S ⬘ 兲 P

2 关 Re共 V ⬘ 兲 ␮

¯ ⬘ 兲共 3 ␮ 2 ⫺ 1 兲 ⫻ 共 3 ␮ 2 ⫺ 1 兲 ⫹ 70 Re共 S ⬘ V

¯兲 关 5 Im共 V ⬘ P

¯ ⬘ 兴 ␮ ⫹ 20 Re共 W ⬘ ⫺ T ⬘ 兲 ␮ 其 x 2 ⫹ S⬘兲Q

3

1



N max

¯ ⬘ 兲 ␮ 2 ⫹ 70 Re关共 V ⬘ ⫺ 1 兲 ⫹ 70 Re共 S ⬘ V



Qs

⫺ 1 兲 ⫹ 14兩 S ⬘ 兩 2 共 10␮ 4 ⫺ 9 ␮ 2 ⫹ 1 兲 ⫹ 100 Re共 V ⬘ 兲

⫺ 3 兲 ⫹ 35兩 V ⬘ 兩 2 共 ␮ 2 ⫺ 1 兲 ⫹ 100 Re共 V ⬘ 兲共 2 ␮ 2

2

15

As a final remark, note that it is frequent in radiative transfer computations to expand the phase function in Legendre polynomials:

¯ 兲兴 x 3 , ⫹ 3 Im共 S ⬘ P



2 兩P兩2x 6

¯ ⬘ 兲 ⫹ 21 Re共 S ⬘ Q ¯ ⬘兲 关 35 Re共 V ⬘ Q 2

3

4

共1 ⫹ ␮2兲 ⫹

⫹ Re共 S ⬘ 兲 ␮ 3 兴 ⫹

兩 P 兩 2 x 6 关 5 Re共 V ⬘ 兲 ⫹ 3 Re共 S ⬘ 兲兴

1

3

Im关 P 共 R ⬘ ⫹ W ⬘ ⫺ T ⬘

Finally, the expressions that allow for the calculation of the asymmetry factor and of the polarized and unpolarized phase functions are presented in Eqs. (15) and (16). Although these expressions appear to be intricate and mathematically involved, they are analytically much more manageable than Mie theory:

P 1共 ␮ 兲 ⫽

15

(16b)

Q a ⫽ x ⫺4 Im共 P 兲 ⫺

gQ s ⫽

4 兩P兩2x 6

⫹ 2 Re共 S ⬘ 兲 ␮ 3 兴 ⫹

From these two last equations it is possible to calculate the absorption efficiency factor Q a explicitly, although in some cases it is preferable to obtain it by numerical subtraction. The analytical result for Q a is presented in Eq. (14), which shows that if the refraction index is a real number the absorption efficiency factor vanishes, as it should:



␮2 ⫹

835

2 3



¯ 兲 ⫹ 3 Im共 S ⬘ P ¯ 兲兴 x 3 , 关 5 Im共 V ⬘ P (18b)

836

˜o M. Caldas and V. Semia

J. Opt. Soc. Am. A / Vol. 18, No. 4 / April 2001

A2 ⫽

1 2



2

兩P兩2x 8

55125

Qs

¯ ⬘兲 关 350 Re共 V ⬘ 兲 ⫹ 245 Re共 S ⬘ V

⫹ 40 Re共 U ⬘ 兲 ⫺ 7 兩 S ⬘ 兩 2 兴 , A3 ⫽

24 兩 P 兩 2 x 6 225

Qs



Re共 S ⬘ 兲 ⫹

⫺ 2 Re共 T ⬘ 兲兴 x 2 ⫺ A4 ⫽

4

兩P兩2x 8

55125

Qs

2 3

1 210

(18c) ¯ ⬘兲 关 7 Re共 S ⬘ Q



¯ 兲x3 , Im共 S ⬘ P

(18d) Fig. 1. Relative error, in comparison with Mie theory results, of extinction efficiencies Q e from the Penndorf model and the present model.

关 28兩 S ⬘ 兩 2 ⫹ 15 Re共 U ⬘ 兲兴 .

(18e)

3. RESULTS AND DISCUSSION As far as the extinction and scattering efficiencies and the asymmetry factor are concerned, the results presented in this paper were assessed by their error relative to Mie theory results and comparing the error with the error of Penndorf ’s three-term approximation. This test was performed for a vast number of cases with size parameters and refractive indices in the ranges of 0 ⭐ x ⭐ 1, 1 ⭐ n ⭐ 5, and 10⫺6 ⭐ k ⭐ 4. More concretely, 1331 cases were tested, with the following values of x, n, and k: x 苸 兵 0;0.1;0.2;0.3;0.4;0.5;0.6;0.7;0.8;0.9;1.0其 ; n 苸 兵 1.0;1.4;1.8;2.2;2.6;3.0;3.4;3.8;4.2;4.6;5.0其 ; k 苸 兵 10⫺6 ;0.4;0.8;1.2;1.6;2.0;2.4;2.8;3.2;3.6;4.0其 . Among these cases, only those respecting the condition x 兩 m 兩 ⭐ 1 were considered, since otherwise the series expansion used is not valid. Note that the accuracy of these polynomial approximations increases with the increase in 兩m兩 while respecting the condition x 兩 m 兩 ⭐ 1 (Ref. 12), so it was unnecessary to consider higher values of the refractive index since the more disadvantageous cases are already being considered. It was concluded, during the present research effort, that the parameter that determines the accuracy of the studied approximations is the product of the size parameter and the absolute value of the refractive index, x 兩 m 兩 . This is in agreement with the results presented in Ref. 12 for the x and m limits that ensure an accuracy with an error of 1% for Penndorf ’s approximation to the extinction efficiency. Therefore the relative errors of the approximations considered were plotted against x 兩 m 兩 (with 0 ⭐ x 兩 m 兩 ⭐ 1) for the various cases tested. In Fig. 1 the case of the extinction efficiency factor can be observed. As can be seen, the present model presents a much lower dispersion of results than the Penndorf model. Also, the maximum error presented by the present model (5.8%) is lower than the one presented by Penndorf ’s expression (7.4%). The root-mean-square error of these approximations is shown in Table 3, where it can be seen that the accuracy of the present approximation is nearly triple that of Penndorf ’s. Referring now to the scattering efficiency factor depicted in Fig. 2, it can be seen that, although both approximations are less accurate than in the previous case, the present approximation is superior to that of Penndorf. In fact, the maximum error of the present approximation is 12.3%, as opposed to the 26.3% error attained by

Table 3. Root-Mean-Square Errors Relative to Mie Theory of Three-, Four-, and Five-Term Approximations in the Range 0 Ï x Ï1, 0 Ï x 円 m 円 Ï 1 a

冉兺 N

rms ⫽ 100

共 ␨ App / ␨ Mie ⫺ 1 兲 2 /N



1/2

Extinction and Scattering Efficiency and Asymmetry Factor

Three-Term

Four-Term

Present Model: Five-Term

Qe Qs gQ s

1.55 2.07 133

1.28 1.99 10.8

0.547 1.15 1.67

a

i⫽1

Penndorf Model

␨ App , approximated value; ␨ Mie , Mie theory value.

Fig. 2. Relative error, in comparison with Mie theory results, of scattering efficiencies Q s from the Penndorf model and the present model.

Fig. 3. Relative error, in comparison with Mie theory results, of the product of the asymmetry factor and the scattering efficiency based on the Penndorf model and the present model.

Penndorf ’s expression. In Table 3 one can see by the values of the root-mean-square error that the accuracy of the present approximation is almost twice of that of Penndorf ’s model.

˜o M. Caldas and V. Semia

Vol. 18, No. 4 / April 2001 / J. Opt. Soc. Am. A

837

As far as the asymmetry factor is concerned, although Penndorf has not presented any result for this parameter, it is possible to deduce a one-term approximation from his work, i.e., gQ s ⫽ 4 兩 P 兩 2 x 8 关 5 Re(V⬘) ⫹ 3Re(S⬘)兴/225. The results of this one-term approximation are compared with the present model results in Fig. 3. For this case the improvement in accuracy is remarkable, since the maximum error is decreased from 107% to 12.3% and the root-meansquare error decreases more than a hundredfold, as can be seen from Table 3. The values displayed in Table 3 also make clear the advantage of using a five-term approximation over a fourterm approximation. The increase in accuracy attained by the present model is quite significant. This conclusion is reinforced by the results presented in Table 4, where the maximum values that can be assumed by the paramTable 4. Maximum Allowable Value of x 円 m 円 to Ensure a 1% Maximum Relative Error with Use of a Three-, Four-, or Five-Term Approximation Extinction and Scattering Efficiency Penndorf Model and Asymmetry Present Model: Factor Three-Term Four-Term Five-Term Qe Qs gQ s

0.360 0.539 0.189

0.360 0.640 0.278

0.768 0.640 0.556

Fig. 4. Polarized and unpolarized phase functions for soot (m ⫽ 2.2 ⫺ i1.122, x ⫽ 0.75).

Fig. 5. Polarized and unpolarized phase functions for fly ash (m ⫽ 1.5 ⫺ i0.02, x ⫽ 1.0).

eter x 兩 m 兩 in order to ensure an error less than 1% are shown. Notice that the result presented by Ku and Felske12 is in good agreement with the values that are presented here. Referring now to the scattering phase function, the expressions presented here were tested for two types of particles that are important in combustion systems: soot and fly ash. The values of the refractive indices of these two kinds of particles were taken from Ref. 15. Figure 4 shows the present model’s results of the polarized and unpolarized phase functions for a soot particle with x ⫽ 0.75 compared with those obtained from Mie theory. As can be seen, the agreement attained is very good, in spite of the fact that this case lies outside the range of validity of the present approach, since x 兩 m 兩 ⬇ 1.85. This shows us that the requirement that x ⭐ 1/兩 m 兩 is somewhat conservative since, for some values of the refractive index, the present expressions remain accurate for larger values of the size parameter. Naturally, for values of the size parameter that respect the condition x ⭐ 1/兩 m 兩 , the results would be much more accurate. In fact, the maximum error present in the unpolarized phase function for the presented value of x is 1.7%, but for the case x 兩 m 兩 ⫽ 1 the maximum error exhibited by the present approach is only 0.1%. Similar results for fly ash can be seen in Fig. 5. Again the agreement is very good, although x 兩 m 兩 ⬇ 1.5. In this case the size parameter assumes the value of x ⫽ 1, and the approximation remains accurate for values of the size parameter up to x ⬇ 1.2. It is therefore possible to infer

838

˜o M. Caldas and V. Semia

J. Opt. Soc. Am. A / Vol. 18, No. 4 / April 2001

that even the condition x ⭐ 1 is not very stringent, since for some refractive indexes it can be broken without compromising accuracy.

4. CONCLUSIONS In the present paper, analytical expressions to calculate the radiative properties of small spherical particles were obtained. These expressions have proved to be very accurate within their range of applicability, x ⭐ 1/兩 m 兩 . In fact, within this range, the maximum relative error compared with that of Mie theory results in any of the tested approximations was below 13%, and the root-meansquare error was always below 2%. The advantage of using the present five-term approximation for the extinction and scattering efficiencies over Penndorf ’s three-term approximation was well established, since a considerable increase in accuracy was thus attained. The expressions referring to the phase function, both polarized and unpolarized, were also tested and presented very good results, even for size parameters outside the approximation’s limit of validity. The present expressions are certainly useful in radiative transfer calculations and in optical diagnostic problems that concern particles in the small size-parameter range and that require analytically manageable expressions for the representation of the radiative and optical properties of the particles.

Superscripts

⬘ —

Subscripts 1 2 a e s

REFERENCES 1. 2. 3. 4.

5. APPENDIX A: NOMENCLATURE an , bn g i i( ␮ ) Im k m n P P( ␮ ) Q R Re S S( ␮ ) T U V x W

Mie coefficients Asymmetry factor Imaginary unit Scattering intensity Imaginary part of Imaginary part of the complex refractive index Complex refractive index Real part of the complex refractive index Parameter dependent on m Scattering phase function Efficiency factor Parameter dependent on m Parameter dependent on m Real part of Parameter dependent on m Scattering amplitude function Parameter dependent on m Parameter dependent on m Parameter dependent on m Size parameter Parameter dependent on m

Greek symbols

␮ Cosine of the scattering angle ␲ n , ␶ n Angle dependent functions

Perpendicular polarization Parallel polarization Absorption Extinction Scattering

˜ o can be *Corresponding author Viriato Semia reached at the address on the title page or by phone, 351-21-841-7378; fax, 351-21-847-5545; or e-mail, [email protected].

5.

Latin symbols

Modified value Complex conjugate

6.

7. 8.

9. 10. 11. 12.

13. 14. 15.

˜ o, ‘‘Modelling of optical properties M. Caldas and V. Semia for a polydispersion/gas mixture,’’ J. Quant. Spectrosc. Radiat. Transfer 62, 495–501 (1999). ˜ o, ‘‘A new approximate phase funcM. Caldas and V. Semia tion for isolated particles and polydispersions,’’ J. Quant. Spectrosc. Radiat. Transfer 68, 521–542 (2001). A. R. Jones, ‘‘Light scattering for particle characterization,’’ Prog. Energy Combust. Sci. 25, 1–53 (1999). F. M. Schulz, K. Stamnes, and J. J. Stamnes, ‘‘Scattering of electromagnetic waves by spheroidal particles: a novel approach exploiting the T matrix computed in spheroidal coordinates,’’ Appl. Opt. 37, 7875–7896 (1998). B. T. N. Evans and G. R. Fournier, ‘‘Analytic approximation to randomly oriented spheroid extinction,’’ Appl. Opt. 33, 5796–5804 (1994). M. I. Mishchenko, ‘‘Light scattering by size–shape distributions of randomly oriented axially symmetric particles of a size comparable to a wavelength,’’ Appl. Opt. 32, 4652– 4666 (1993). G. R. Fournier and B. T. N. Evans, ‘‘Bridging the gap between the Rayleigh and Thompson limits for spheres and spheroids,’’ Appl. Opt. 32, 6159–6166 (1993). Y. Liu, W. P. Arnott, and J. Hallett, ‘‘Anomalous diffraction theory for arbitrarily oriented finite circular cylinders and comparison with exact T-matrix results,’’ Appl. Opt. 37, 5019–5030 (1998). ˜ o, ‘‘Modelling of scattering and abM. Caldas and V. Semia sorption coefficients for a polydispersion,’’ Int. J. Heat Mass Transf. 42, 4535–4548 (1999). R. B. Penndorf, ‘‘Scattering and extinction coefficients for small absorbing and nonabsorbing aerosols,’’ J. Opt. Soc. Am. 52, 896–904 (1962). A. Selamet and V. S. Arpaci, ‘‘Rayleigh limit—Penndorf extension,’’ Int. J. Heat Mass Transfer 32, 1809–1820 (1989). J. C. Ku and J. D. Felske, ‘‘The range of validity of the Rayleigh limit for computing Mie scattering and extinction efficiencies,’’ J. Quant. Spectrosc. Radiat. Transfer 31, 569– 574 (1984). H. C. Van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957). M. Kerker, The Scattering of Light (Academic, San Diego, Calif., 1969). W. A. Fiveland, W. J. Oberjohn, and D. K. Cornelius, ‘‘COMO: A numerical model for predicting furnace performance in axisymmetric geometries,’’ Vol. 1, Technical Summary, Final Report DOE/PC/40265-9 (U.S. Department of Energy, Washington, D.C., 1985).

Radiative Properties of Small Particles

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