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2424 J. Opt. Soc. Am. A/Vol. 22, No. 11 /November 2005 Liu et al.

Calculation of emittance of a coating layer withthe Kubelka–Munk theory and the

Mie-scattering model

Lingyun Liu, Rongzhou Gong, Dexiu Huang, Yan Nie, and Changhui Liu

Department of Electronic Science and Technology, Huazhong University of Science and Technology, Wuhan,Hubei 430074, China

Received October 14, 2004; revised manuscript received March 11, 2005; accepted April 7, 2005

The radiative transfer equation in a coating layer with a not too high concentration of pigment particles, whichis attached to a substrate, is resolved in terms of Kubelka–Munk theory and a Mie-scattering model. In thecase of a coating layer constituted with aluminum spherical particles, the dependence of emittance of the coat-ing layer on particle radius and thickness of the layer are studied. The optimum radius of a pigment particleis suggested as well. © 2005 Optical Society of America

OCIS codes: 290.5850, 290.4020, 290.7050.

wcpKNtprdaidss

mftiar

2FCWncmbpteaat

. INTRODUCTIONow to control the thermal signatures of various objects

s a formidable task in either civil industry or defensiveechnology. In order to obtain higher energy efficiency, weave to make the solar energy collectors able to absorbhe most energy incident on the collectors and emit theeast resulting from their intrinsic thermal emission.herefore selectively emissive power of the collectors,amely, the high absorbance or high emittance in mostpectra of sunlight but low emittance in the thermal in-rared band, is desired. To solve this problem we alwaysttach a layer of special paint that lends selectively spec-ral emissive power to the surface of the collector. In de-ensive technology, the low infrared signatures of somearts of the furnishments are expected so as to break offheir profiles in the thermal infrared band. We also oftenaint the furnishments with low infrared emittance pig-ents to achieve low thermal infrared emittance.In both situations many particles embedded randomly

n a matrix are concerned. Usually the particles selec-ively scatter the radiation energy emitted by the sub-trate, and the matrix binds the pigment particles. Radi-nt energy may be envisioned as being transported eithery electromagnetic waves or by photons. Two theoriesave corresponded to the two viewpoints mentionedbove: multiple-scattering theory and transport theory.1

he multiple-scattering theory started with Maxell’squations, which are mathematically rigorous, but solu-ions were not obtainable that included all conditions, andhe approximate solution was difficult as well. The trans-ort theory, which is not as mathematically rigorous ashe Maxwell’s equations, was more tractable. This theoryas first proposed by Schuster2 and was developed thor-

ughly by Chandrasekhar.3 In this paper we resolve theadiation transfer equation with the approximate theoryf transport theory, i.e., the well-known Kubelka–MunkK-M) theory,1,4 and deal with the determination of the co-fficients by use of electromagnetic-wave theory.

1084-7529/05/112424-6/$15.00 © 2

The K-M theory is mathematically tractable comparedith the original radiative transfer equation1,3 but was

onsidered phenomenological in the past because inter-retation of the physics meaning of the coefficients in the-M model was absent. Maheu et al.5 and Vargas andiklasson6–8 established the relation of those coefficients

o the parameters of a single particle by introducing twoarameters called, respectively, average path-length pa-ameter (APP) and forward-scattering ratio for diffuse ra-iation (FSRD). We take advantage of their investigationsnd Mie-scattering theory9–11 to figure out the coefficientsn the K-M model by an appropriate program. The depen-ence of the emittance of a coating layer attached to aubstrate on particle radius and thickness of the layer istudied in terms of the computation results.

This paper is organized as follows: In Section 2 weake use of the K-M model to resolve the radiation trans-

er equation and obtain the formula of emittance. In Sec-ion 3 we discuss the determination of various parametersn the formula of emittance. We perform the calculationnd discuss the results in Section 4 and give concludingemarks in Section 5.

. KUBELKA–MUNK THEORY ANDORMULA OF EMITTANCE OF THEOATING LAYERe consider a painted plate. As shown in Fig. 1, the thick-

ess of the coating layer is equal to d. For mathematicalonvenience, we assume that the layer consists of one orore types of pigment particle immersed in a continuous

inder that extends for a small distance in front of theigment particles and also behind them. We assume thathe area of the plate is so large that we can ignore thedge effect. According to the conditions of K-M theory welso assume that the light is everywhere perfectly diffusend the clearances of the particles are large enough thathe dependent scattering can be dismissed. The radiative

005 Optical Society of America

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Liu et al. Vol. 22, No. 11 /November 2005 /J. Opt. Soc. Am. A 2425

ransfer in the coating layer can be described in terms ofhe two-flux model of Kubelka and Munk (we chose theymbols used by Ishimaru1 to distinguish between fluxnd intensity3).The radiative transfer equations can be written as two

ifferential equations:1

dF+= − �K + S�F+ + SF−,

Fig. 1. K-M model of radiative transfer in the coating layer.

dz l

R

waftt

ed

dF−

dz= �K + S�F− − SF+, �1�

here F+ and F− are, respectively, the ingoing and outgo-ng diffuse flux perpendicular to the surface of the coatingayer. K and S are the absorption coefficient and back-cattering coefficient within the layer, respectively.

At the front and back interfaces, the boundary condi-ions are

F+�0� = �1 − Re�F+e + RiF−�0�,

F−�d� = RgF+�d�, �2�

here Re denotes the external diffuse reflectance at theront interface and Ri is the internal diffuse reflectance athe same location, namely, z=0. Rg denotes the diffuse re-ectance at the interface between the coating layer andhe substrate and is generally different from that of theubstrate without an infrared coating layer. Applying thebove boundary conditions to Eqs. (1), we obtain the ex-licit expression of the apparent reflectance of the coating

ayer:

R =F−e

F+e= Re +

�1 − Re��1 − Ri�R��1 − R�Rg� − �1 − Re��1 − Ri��R� − Rg�exp�− 2�d�

�1 − RiR���1 − R�Rg� − �R� − Ri��R� − Rg�exp�− 2�d�, �3�

here R� and � are defined as

�R� = 1 +K

S− ��K

S�2

+ 2�K

S��1/2

� = K2 + 2KS� . �4�

Therefore we can work out the apparent emittance ofhe coating system from the reflectance in terms of the so-alled Kirchhoff ’s law for opaque substances:

� = 1 − R. �5�

. DETERMINATION OF THE PARAMETERS. Reflectancesor simplicity we assume that the front surface of the pig-ent particles/binder medium is optically smooth in the

hermal infrared window, i.e., 8–14 �m band. The quan-ities Re can be computed from the index of the binder byntegration of the Fresnel equation over a hemisphere.

e assume for simplicity that the binder is not dissipa-ive; on the other hand we select the resin that is trans-arent as possible in the thermal infrared band. So the re-ectance of an optically smooth surface for completelyiffuse radiation incident from vacuum or air can be writ-en as

e

=

0

2�

d�0

�/2 1

2� sin2�� − ��

sin2�� + ��+

tan2�� − ��

tan2�� + ���sin � cos �d�

0

2�

d�0

�/2

sin � cos �d�

,

�6�

here � is the angle of incidence or polar angle, � is thezimuth angle, and � is the angle of refraction. If the re-ractive index of the binder is a known quantity m2, andhe index of refraction of air is always set to be unity, thewo angles are related by Snell’s law:

m2 sin � = sin �. �7�

Equation (6) has been evaluated by Walsh to give12

Re =1

2+

�m2 − 1��3m2 − 1�

6�m2 + 1�2 +m2

2�m22 − 1�2

�m22 + 1�3

ln�m2 − 1�

�m2 + 1�

− 2m23

�m22 + 2m2 − 1�

�m22 + 1��m2

4 − 1�+

8m24�m2

4 + 1�

�m22 + 1��m2

4 − 1�2ln m2.

�8�

In the diffuse incidence case, the relation between thexternal reflectance Re and the internal reflectance Ri isifferent from that of the collimated incidence case. For

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2426 J. Opt. Soc. Am. A/Vol. 22, No. 11 /November 2005 Liu et al.

erfectly diffuse radiation incident on the interface fromithin the layer of index m2, all the radiant energy that is

ncident at angles greater than the critical angle, which isqual to arcsin 1/m2, will be totally reflected. And for theux that hits the surface from within at angles less thanhe critical angle, the reflected fraction is equal to Re ac-ording to the Fresnel equation. Similarly, by use of inte-ration we can readily obtain the relation

Ri = 1 −1 − Re

m22 . �9�

. Absorption Coefficient K and Backscatteringoefficient S

. Derivation of K and Ss pointed out by Ishimaru1 and Maheu et al.,5 the coef-cients of absorption and scattering in their models wereonsidered as purely phenomenological and should be de-ermined by experiments. Some researchers attempted tostablish experimentally the relationship between thoseoefficients and the parameters of a single particle.udgett and Richards13 found that the absorption coeffi-

ient K always approaches 2nCabs, and the backscatteringoefficient S is approximately n�30-1� /4, where n de-otes the number density of the particles, Cabs is the ab-orption cross section of the single particle incident by alane wave, and 0 is the albedo of a single particle; 1 ishe first coefficient of the expansion of the single-particlehase function in terms of Legendre polynomials.Maheu et al.5 and Vargas and Niklasson6–8 made use of

he APP and forward FSRD to uncover the relation be-ween those coefficients and the parameters of a singlearticle. The definition of APP is as follows. At a certainosition z inside the coating layer, a light beam travels aistance L at an angle � with respect to the z axis, whichs perpendicular to the interface as shown in Fig. 1.herefore � can be expressed as

��+� =

0

1

I�z,��d�

0

1

I�z,���d�

, ��−� =

−1

0

I�z,��d�

0

1

I�z,���d�

, �10�

n which ��+����−�� denotes the forward (backward) APP,nd I�z, �� is the intensity of radiation; �=cos �. Vargasnd Niklasson7 devised a multiple-scattering approachhat was introduced by Hartel14 to solve the radiativeransfer equation in a particulate medium. They specifiedhe angular distribution of radiation of every scatteringvent by the so-called generalized phase function fk���nd obtained a set of coupled linear differential equationsy use of a radiative transfer equation. They solved thequations and obtained the explicit expressions of APP�±� and �d

�±�, the forward-scattering ratio for diffuseadiation:7,15

��+� = 2 1 ± �n=1

�gnc̄n�±�

1 ± 2c̄1�±�/3 + 2�

n=2

�hnc̄n�±�� , �11a�

�d�±� =

�d�i� ±

1

2�n=1

�gnc̄n�±��1 + �nn̄n� +

1

2�n=1

�gn̄n �m=2

��nmc̄m�±�

1 ± �n=1

�gnc̄n�±�

,

�11b�

here

cn�±��z� = 2n+1

4� �k=1

Qk�±��z�

k! � ̄n

2n+1�k,

�nm =0

1

Pn���Pm���d�, c̄n�±� = cn

�±�/c0�±�,

�d�i� = 1

2�1 + �n=1

gn

2n¯ /0� ,

gn = �0n, hn = �1n,

nd n is the corresponding coefficient of expansion of aingle-particle phase in terms of Legendre polynomials,

¯ n=� /0, and Qk�±� denotes the relative amounts of radia-

ion that arise from k scattering events and go into a for-ard (backward) hemisphere. The absorption coefficientand backscattering coefficient S can be expressed as

K = �̄nCabs, S = �̄n�1 − �̄d�Csca, �12�

here �̄= ���+���−��1/2 , �̄d= ��d�+�+�d

�−�� /2.The K-M theory included two assumptions, i.e., the

ymmetry and semi-isotropy assumptions. The symmetryssumption implies that �̄�+�=��−� and �d̄=�d

�+�=�d�−�, and

he another condition leads to �̄=2. In this work the inci-ent radiation is the thermal radiation of the object,hose angular distribution differs markedly from that of

ollimated radiation and can be approximated to isotropyo some extent. In the research of Vargas andiklasson6–8 they extended the Maheu–Letoulouzan–ouesbet four-flux model5 and found even for collimated

ncident radiation that the APP will deviate from 1 andpproach a saturation value after the radiation passesome optical depth (usually greater than 5). For simplic-ty we adopted the assumptions of standard K-M theorynd Maheu et al.5 and therefore set �̄=2 and �̄d=�d

�+�

�d�−�=�c, where �c is the forward-scattering ratio for col-

imated radiation and will be determined by Mie theory.

. Computation of K and S with the Mie-scatteringodele assume that the pigment particles embedded inside

he binder are all spherical. According to Mie theory9–11

he extinction efficiency of a sphere is usually written as

a

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wtic

i

s

w+p

o

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cuistcsatbcdt�

Fsrsmltotbce

Fv

Liu et al. Vol. 22, No. 11 /November 2005 /J. Opt. Soc. Am. A 2427

Qext =2

x2�n=1

�2n + 1�Re�an + bn�, �13�

nd the scattering efficiency is

Qsca =2

x2�n=1

�2n + 1�� an 2 + bn 2�, �14�

here x=2�m2a /� is the size parameter of the particle (as the radius of the particle, � is the wavelength inacuum). The coefficients of the scattering functions annd bn can be expressed as

an =m�n�mx��n��x� − �n�x��n��mx�

m�n�mx��n��x� − �n�x��n��mx�,

bn =�n�mx��n��x� − m�n�x��n��mx�

�n�mx��n��x� − m�n�x��n��mx�, �15�

here �n and �n are the Riccati–Bessel functions and m ishe refractive index of the particle relative to the binder,.e., m=m1 /m2 (where m1 and m2 are the refractive indi-es of the particle and the matrix).

Therefore the absorption efficiency is written as

Qabs = Qext − Qsca. �16�

So the absorption coefficient K is

K = n�̄Cabs =3f

4a�̄Qabs, �17�

n which f denotes the volume fraction of the particles.The calculation of S can be achieved by use of the

ingle-particle phase function:

�c = �0

1

p���d���−1

1

p���d��−1

, �18�

here the phase function16 p���=2/x2Qext� S1��� 2 S2��� 2�, with S1��� and S2��� being the scattering am-litudes.Similarly we can readily obtain the explicit expression

f the backscattering coefficient S:

S =3f�̄�1 − �̄d�Qsca

4a. �19�

. COMPUTATION OF EMITTANCE ANDISCUSSIONe assume for simplicity that the matrix is not absorptive

nd the refractive index is equal to 1.8. We choose alumi-um spherical particles as the pigments and sample theavelengths in a 8–14 �m window. The complex refrac-

ive indices of Al at sampled wavelengths come from Ref.7.With an appropriate program we can figure out the val-

es of Qext, Qsca, Qabs, and �1−�c�Qsca for each sampledavelength. The computation results for different particle

adii are shown in Fig. 2(a).

Now we can readily figure out the emittance of theoating layer at different wavelength by applying the val-es of K and S to Eqs. (3) and (5). The results are shown

n Fig. 2(b). Concerning the validity of the independentcattering approximation, other researchers used to takehe independent or dependent scattering regime into ac-ount in terms of � /d, the ratio of center-to-center particlepacing to the diameter, or c /�, the ratio of particle clear-nce to wavelength. Hottel et al.18 found experimentallyhat dependent scattering effects were better correlatedy c /� than � /d. Furthermore, c /�=0.3 or 1/3 was alwaysonsidered as the demarcation of independent and depen-ent scattering.19 In a rhombohedral arrangement of par-icles, the correlation of f to c /� is given18 by x=��c /���f1/3� / �0.905− f1/3�. In Fig. 3 the demarcation line has

ig. 2. (a) Extinction efficiency, scattering efficiency, back-cattering efficiency �1−�c�Qsca, and absorption efficiency versusadius of pigment particle incident by 8, 11, and 14 �m light, re-pectively. (b) Emittance of coating layer versus radius of pig-ent particle. The particle volume fraction and thickness of the

ayer were 0.0275 and 2 mm. When a�0.7�m, independent scat-ering dominates, making our approach valid. The dependencen particle radius is shown by the solid curve. When a�0.7 �m,he dependent scattering is not negligible, but we still treat thisy the independent scattering approach and depict it by a dashedurve. In this region, the relation of � to a might deviate mark-dly from practice.

ig. 3. Independent and dependent scattering: size parameter xersus volume fraction f.

bwppddowemmi

dT8e

w�Fe=cst8ct

itols

1c

i�(

−Atd

srcmrrw

Ft

Fw

Fe

2428 J. Opt. Soc. Am. A/Vol. 22, No. 11 /November 2005 Liu et al.

een given according to c /�=0.3. Within our computation,e set f=0.0275, which is low enough to satisfy the inde-endent scattering condition when a�0.7 �m. With thearticle radius decreasing, from Fig. 3 we can see that theependent scattering dominates and scattering efficiencyescends. When the particle radius approaches the orderf electron mean free path, the permittivity of the particleill differ from that of bulk material due to a nanometerffect such as boundary scattering. Those two factorsight make the absorption and scattering properties re-arkably different from that of the independent scatter-

ng approximation.From Fig. 2(b) we can see that the emittance differs at

ifferent wavelengths even for the same particle radius.o study the thermal signatures of an object in an–14 �m window, we introduce a quantity called meanmittance that is defined as

�̄ =

8

14

�Mb�d�

8

14

Mb�d�

, �20�

here �̄ denotes mean emittance in an 8–14 �m window;� is the spectral emittance of the coating layer shown inig. 2(b), and Mb�=C1 /�5�exp�C2 /�T�−1�−1 is Planck’smissive power function of blackbody. Here C1�3.7415±0.0003��108�W m−2 �m4�, the first radiationonstant, and C2= �1.438 79±0.000 19��104��m K�, theecond radiation constant. We assume that the tempera-ure of the object is 323 K, so the peak radiation is–14 �m (�39% for blackbody or graybody). In practicalalculation we adopt a numerical summation to replacehe integration of the numerator as follows:

�̄ =

8

14

��Mb�d�

8

14

Mb�d�

=

�i

�i + �i+1

2

�i

�i+1

Mb�d�

8

14

Mb�d�

, �21�

n which �i denotes the emittance of the coating layer athe sample wavelength at 8–14 �m, and the integrationf Planck’s function over different bands is readily calcu-ated by various numerical methods in MATLAB. Theample of the wavelength is 8.0, 8.266, 8.856, 9.0, 9.537,

ig. 4. Mean emittance versus particle radius. d denotes thehickness of the layer.

0.0, 10.33, 11.0, 11.27, 12.0, 12.4, 13.0, 13.78, 14.0. Theomputational results are shown in Fig. 4.

The mean emittance descends quickly with d increas-ng, but approaches a saturation value when d

6000 �m. This can be explained easily in terms of Eq.3).

When d→ , R→ �Re+�1−Re��1−Ri�R��1−R�Rg�

�1−RiR���1−R�Rg� � and so �=1R→const. It is obvious in Fig. 4 that the radius range ofl particles, which we can make use of to constitute a low

hermal infrared emittance coating layer, is narrow andistributed around 0.9 �m.The dependence of K and S, K /S, and forward-

cattering ratro for collimated radiation �c on differentadii is illustrated in Figs. 5(a)–5(c). With the radius in-reasing, �c descends from the Rayleigh limit of 0.5 to theinimum and then increases. This means that when the

adius of a particle is greater than 0.3 �m, the larger theadius and the higher the intensity that is scattered for-ard, until the radius is much larger than 2 �m. In the

ig. 5. Various parameters versus particle radius. Incidentavelength is 11 �m, f=0.0275, and d=2 mm.

ig. 6. Dependence of optimum radius on wavelength at differ-nt layer thicknesses.

laprwtorroi

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AThc

R

1

1

1

1

1

1

1

1

1

1

2

2

Liu et al. Vol. 22, No. 11 /November 2005 /J. Opt. Soc. Am. A 2429

arge particle case �a�3 �m�, we can conclude that thebility that the coating layer constituted with Al sphericalarticles curtains off the radiation of the substrate lowersapidly. Combined with the dependence of K, S, and K /Sith respect to particle radius and other factors such as

hickness of the layer (so as to not harm the performancesf the substrate such as a microwave absorber, the infra-ed layer cannot be too thick), we believe that the optimaladius is 0.9 or 1.0 �m. We plotted the dependence of theptimum radius of Al particles on different wavelengthsn Fig. 6.

. CONCLUDING REMARKSe can draw a conclusion that if we choose aluminium

pherical particles as pigments to constitute a low-mittance coating layer, within the independent scatter-ng approximation, the optimal radius of pigment par-icles is around 0.9 �m. It seems that the lower the ratiof the forward-scattering cross section to that of absorp-ion, the greater the decrease in emittance. We can choosether materials with different complex refractive indicess pigment particles and compute them similarly. Bohrennd Huffman11 pointed out that there exists in generaluch a relation of scattering cross sections: CsphereCneedle�Cdisc. So the sphere is not the optimal shape of

igment particles that is needed to constitute a low-mittance coating layer. However, the quantitative treat-ent of other irregular-shaped particles is difficult be-

ause of the absence of analytical solutions of scatteringquations of most nonspherical particles. There are vari-us numerical methods such as the T-matrix method, theiscrete dipole approximation, and the finite-differenceime-domain method, which can all be used to deal withhe scattering of nonspherical particles. On the otherand, in practice the dependent scattering is often en-ountered in thermal radiation transfer and we have toace this problem. Some investigators used experimentsnd some used numerical methods20,21 to treat the many-ody scattering. Dependent scattering can be attributedo the perturbation of other particles on the internal fieldf the particle being studied in the near-field region andnterference of scattered waves of different particles inhe far-field region. When the number of particles be-omes large, as in practice, the accurate analytical treat-ent of dependent scattering is almost impossible. The

tatistic or asymptotic treatment of multiple-scatteringffect may help resolve the dependent scattering of many-ody scattering when the order of scattering eventsmong the particulate medium approaches infinity. Byeans of a different kind of method, we will compare the

fficiencies of various pigment particles to decide the op-imal formula of low infrared emittance paints.

CKNOWLEDGMENTShe authors are grateful to Z. S Zhao and Z. K. Feng forelpful discussions and to the reviewers for constructiveriticism, which helped us improve the paper.

Lingyun Liu’s e-mail address is frankee33@126.com.

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1. B. E. Barrowes, C. O. Ao, F. L. Teixeira, J. A. Kong, and L.Tsang, “Monte Carlo simulation of electromagnetic wavepropagation in dense random media with dielectricspheroids,” IEICE Trans. Electron. E83-C, 1797–1802(2000).