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Solving eqs (11 ) and (12 ) for Ll a nd L 2 , we get :L1J e(I - Pe)[ (1 {J)- Ps(l-{J) ][(1 /J) - Pi( I - /J )][ (1 /J) - Ps(1 - /J ) ]e uD [ (1 - iJ) - Pi(1 iJ) ][ (1-{J) - Ps(1 /J)] e uD- J e(I - Pe)[ (1-/J)- Ps(1 m]L 2 [(1 /J) - Pi(I -{J) J[ (1 /J )- Ps(1 - /J ) ]e uD [(1-/J) - Pi( 1 (J) J[ (1- {J) - Ps(1 m]e uD'(13)(14)The equation s can b e simplified by m aking th e following substi tutions :LetM N 0 p (1 5)(1 /J)- Ps(I - iJ)(1 /J)- Pi( I -iJ)(1-/J)- Ps(1 iJ)(1-/J)- Pi(I ,8 ).(16)(17)(18)Then eqs (13) and (14) become(19)(20)The overall specular plus diffuse r eflectance, R, of the specimen is defined as the reflectedflux divided by the incident flux. The r eflected flux will include th e fraction of J e tha t is specularly r efl ected at the coating-air in terface. That is(21)From eqs (5), (19 ), and (20 ),I J e(l - Pe)[(1- iJ)MeuD- (1 /J) Oe- uD]D-M N euD- OPe uD'(22)hen ce(23)R in eq (23) is the r eflect an ce of th e composite specimen under conditions of completelydiffuse illumination and hemispherical viewing. The equation has b een wTitten in terms ofth e coating parameters ,8, cr, Pe, Pi, Ps, and the thickness of the coating, D. The values ofth ese par ameters vary wi th wavelength . The equation applies only to sp ectral r eflectance,R x, under th e given geometric condi tions, and spectral values of th e coating and coatingsubstrate par ameters, ,8x. crx, PeX, PiX, and PsX ar e required . The subscrip t A indicates that th esymbol applies to th e spectral value at wavelength A.When tr ansmi ttance is zero , as it is in the case of the coated specimens un der consid eration ,th e emittance is equal to one minus the refl ectance, as is indicated by eqs (1) and (2) , if consisten t geometric conditions of incident, refl ected, and emit ted flux are used . R has b een defin edas the spectr al Tefl ectance under condi tions of diffuse illumination a nd hemispherical viewing.The emittance corresponding to R is E H , the hemispherical spectr al emi t tance. H ence wecan wTiteE 1H("\'- (1- )(1- .) (1-{J)M euD-(1 m Oe-uD} .PePcP.Ll!lNeuD- OPe "D220(24)

J L is legitimaLe also to apply eqs (1) and (2) to evaluate the normal spectral emittance,E.v, if, and only if, th e geometric conditions of irradiation and viewing under which the externalreflecta nce, Pc, is mca ured are (1) normal incidence and hemispherical viewing or (2) perfectlydiffuse incidence a nd normal v iewin g.If we consider that the intel'l1al refl ecta nce of t h e coatin g plus substrate, neglecting theeffect of specular reflecta nce at Lhe coaLin g-ail' in terface, is R i, then with diffu se illumina tionand hemispherical VlCWlOg(25)R Pe (1- Pe) R 1 ( 1- Pt).Equation (25 ) is another way 0[' wri ting eq (24 ), with R i replacin g the i'racLion co ntainingexponential terms.(26)For normal illumination and h emispherical viewing, the equ ation b eco mesR N PN ( l - PN) R t ( l - Pi)(27)in which R i is agitin Lhe internalrellectance, n eglecting the effect of specuht1' reflectance at thecoat in g-ail' interface, 0[' the coating plu substrate. If R t is the sa me for normit] and. din'usein ciden t Hux/ we ca n wri te(2 )which is the desired eq ua Lion.Sub tituting for )JI, N, 0, and P from eqs (15), (16), (17 ), and (1 ), a nd maki ng usc of th erelationship2 sinh x e"-e-"(29 )and(3 0)2 cos h x e" e-"eq (28 ) reduces toE V (1-PN) l - (1 -p;) .\.[( 1- p, )-,82( 1 ps)] sinh .rJD 2p s,8 cosh rJD} .[U ,82)(1 PiPs)-( 1-,82)(pt Ps)] smh rJD 2,8(1 - PiPs) cosh rJD(3 1)The specuht1' refi ectal1ces at the coaLing-air in terface, Pc a nd P I, are fo r diffuse illumin ationa nd hemispherical viewing. The value to be used for Ps must be determined experimentally,because the coating-substrate interface is normally rough , and th e r eilectan ce may be itfl'ectedby interaction between coating and substrate at th e interface. For an optically smoo th interface , which may b e approximated by the coating-air interface for some coatin gs at some wavelengths, Pe and Pi can be computed from the index of refraction of the coating 5 b y integrationof the Fresnel equ ittion over a hemisphere. In its general form , the equa tion for dielectricscan be written(32)in which Pd is t h e direction al specular reflectance for u npolarized parallel flux , is the angle ofincidence, and 8 is the angle of refracLion, both a ngles JneasUl'ed from the normal to th e in terface. The angles and 8 are related to in dex of rcl'ntction , 11, b y Sn ell's law. In its mostge neral form , this is 111 sin n 2 sin 8, in which 1h is the index of refraction of the medium on the Th is conditio n will be approximated if the inc ident fl u x is com pletel y diffused after t r aversing a small thickness of th c coating.s ) [ost opticall y inhomoge neous coatings consist of opacifyin g particles dispersed in a co ntinuous medium or vehicle. The scatterin g is due tothe di aCfcnce in i ndi cC's of re fraction of the opacific r fi nd vehi cle. I"or glossy coatings, the opacifying parti cles arc n ot u S llall ' exposed at the surfnce of the coatin g, and the cITrcli\'C' index of re fraction for spec ular rc nectio n is that of the vehicle.221

side of the interface from which the flux is incident, and n2 is that on the opposite side of theinterface. In the case of flux incident from vacuum, nl 1. In the case of air, no serious erroris introduced by considering nl l (nl 1.0003, approximately, for air under normalconditions).The reflectance of an optically smooth surface of index of refraction n, for completely diffuseflux incident from vacuum. (or air) can be expressed asr ,,/2 .JoPe sm cp cos cp{ sin 2 (cp-O)sin 2 (cp O)r ,,/22 J0tan 2 (cp - O) } tan2 (cp O)dcp(33)sin cp cos cp dcpThis expression has been integrated by Walsh [8] to give(34)For completely diffuse flu x incident on the interface from within the material of index n, allof the radiant energy that is incident at angles gr eater than the critical angle. 'Pc , will be totallyThe critical angle is that at which 0 -rr/2, or sin CPc L The fraction of flux incidentnat angles from sin- 1 1. to -rr/2 can be obtained by integration and divided by the total flux tonreflected.give the fraction 1- of the incident flux that is totally reflected. Hence only of the totalnnflux will b e incident at angles at which it is refracted, and a fraction Pe of that flux will be internally refl ected, as indicated in eq (33) . H ence we can write(35)The expression for Pe given in eq (34) is somewhat complex. Judd [10] gives computedvalues of Pe and P i for indices of refraction from 1.00 to 1.60 in increments of 0.01.The geometric distribution of the emitted flux will be determined by the directional reflectance at the coating-air interface, and can be computed from the refractive index by meansof the Fresnel equation. For normally incident parallel flux, the Fresnel equation reduces toPN (nn 1Y1(36)and1-4n1) 2PN (n (37)where PN is specular reflectance for normally incident flux .The normal spectral emittance, EN, of any specimen can be computed [rom the hemispherical spectral emittance by use of the Fresnel equations for r eflectance of internally incidentnormal and diffuse flux . This conversion is not affected by the coefficient of scatter of thecoating.4. DiscussionIn the case of paper, layers of powder and unglazed porous ceramic materials, air may beconsidered the continuous phase, and there will be no specular reflection at the interfaces. If,in addition, there is no substrate, or if the reflectance of the substrate is zero, the conditions will222

be identical to those po Lubted by Klein [7].it reduces toH zero is s ubstituted for p., p" and Pi in eq (23 ),R-(1 (32) s inh aD- (1 (32) sinh oD 2(3 cos h uD'(38)which is identical to Klein's [7] eq (46).4.1. Properties of Nonabsorbing, Nonscattering LayersIt may be of interest to consider how the emittance and reflectance of a cO iLLin g will v,tryas 8 and K approach zero. Obviously, if both 8 and K ar e zero , the material will have pOI·rec ltransmittance, and both reflectance and absorptance (or emittance) will be zero . Thi s is tru efor a perfect vacuum, and is closely approximated by a gas at those wavelen gths aL whi c h noabsorption occ urs .a. Properties of Non scattering LayersIf' there is no scatterin g, 8 0, and eqs (3) and (4) b ecomedI/dx - KI(39)dJ /dx KJ(40)I (x jx) I xe- Ktlr,(41)andhencewhere I x represents the flux density at a level x wi thin the m,tLeri,tl , and J(x jI) r epresen ts theflux den sity a fter traversing a thickness 6x of the m aterial. Equation (41) dill"ers frol11 th efamiliaT Bougu er 's law only in that the attenuation of co mpletely diffu se flu x is twice as rap idas that o/" uniclirec Lion al flu x:.If th e scatterin g coeffi cien t, 8 , is zero, which will be approached by oplically homogeneousmaterials, su ch as optical glass and many si ngle cr ys L,tls, t hen u 1 . a nd (3 1, h ence (24)reduces Lo(l -Pi) pse-KD }(42)Ell (l - Pe) { 1- KDKD'e - pipseH o wever , when 8 b eco m es small, th e intemally r efl ecLed flU-,( will no L be redifrused . ifreIlectocl from oplically s mooth smfaces. Thu s (42 ) is only valid if t he coatin g-subsLraLeinterface is so rough that p erJectly diffuse r efl ection occurs. For th e more general cnse, wherethis inlerface ap pro ach es opLi cal smoothness, a tlrree-dimensional a nalys is o[ the typ e used b yGardon [5] is more n earl y vaJid.b. Properties of Nonabsorbing LayersIf the absorption coefficient, K, is zero, as will be a pproximated at some wavelen gths by afreshly smoked layer of magnesium oxide or by freshly fallen snow, both a and (3 b ecom e zero,and eq (24) becomes indeterminate, which would be expected for an opaque coating of thesp ecified materials, but not for a composite specimen. Howev er , under these conditions, eqs(3) and (4 ) become(43)dI/dx - S (J - I ) dJ/dx.This equ ation can b e solved by a procedure similar to those used for eqs (3) and (4 ) to giveIJJ e[Sx(l - Pe)(l - Ps) Ps(l - Pe)]S D (l - p;)(l - ps) I - piPS(44)J efSx(l - Pe)(l - Ps) (1-P e)]SD(l - Pi) ( 1- Ps) 1- PiPS(45)223

Substi tuting eq (44) in eq (21) we getSD(l - ps) PsR Pe (1- p,) (1- Pe) SD(l - p.) (l - ps) (1- p,Ps)(46)If Pe, Pi, and Ps are zero, as they were under the conditions postulated by Klein [7 J, eq (46 )reduces toSD(47 )R SD 1'which is identical to K lein's eq (48 ).Under the conditions postulated for the derivation of eq (47 ), the emittance will approachzero and the reflectance will approach one as the coating approaches a thickness at which it iscompletely opaque, which will only occur at infinite thickness. Since K O, the absorptanceis zero by definition, and the transmittance is one minus the reflectance. Hence we can wTite1(48)T SD 1'which is identical to Klein's eq (a7).4.2. Example of Computed EmittanceFigure 1 is a plot of emittance, computed from eq (31 ), as a function of thickness for composite specimens with optically smooth surfaces comprised of a coating having an index ofrefraction of 1.40 (from which Pe 0.028, Pi 0.53 ), a value of {3 of 0.8, and values of (]' of 40,20, 13.3, and 10.0 mm- I , respectively, applied over a mat substrate having a reflectance intothe coating of 0.9. (The values of Sand K corresponding to a {3 of 0.8 and (]' of 40, 20, 13.3,and 10 mm- I are as follows: S 8.0, 4 .0, 2.65, and 2.0 nUll- 1 and K 28.44 , 14.22, 9.42, and7.11 111m-I, respectively. ) These conditions might be approximated at wavelengths in thevisible by a glossy gray paint applied over a white backing. All of the coating materials havethe same emissivity (about 0.918 ), but the coating with a (]' value of 40 mm- I r eaches 99 percent of this value at a thickness of 0.05 mm, while the coating with a (]' value or 10 mm- Ir equires a thickness of 0.2 mm to reach the same value.The plotted emittance at zero thickness of the coating is greater than the emittance(0.10 ') of the substrate because the coating has a lower index of refraction than the substrate,and bence reduces reflection at the interface. The plotted value is what would be obtainedwith a very thin, peri'ectly transparent glossy coating having an index of rei';raction of 1.40 .1. Com pu ted normal s pectral emittance ,plotted as a func tion of coating thickn ess, for acom posite s pecimen comprised of partiall y transmitting coatings with opticall y smooth surfaceshaving an index of Tefraction of 1.40, {3 value of0.80, and values of 40, 20, 13.3, and 10 1nm- 1 ,respectively, applied over a substrate haviny areflectance of 0. 90, plotted as a func tion of coatinythickness . T hese conditions would be appToximated by a ylossy yray paint applied ova apolished metal 01' white substrate .FI GU R E(JNote t hat the omittance at zero coating thickness is n ot that ofthe su hstra tc.It was assllmod that even at zero thickn ess, there0. 2would be an eHect 01 the index olrelraction 01the coating. Actually, such eHect would be produced by a trans parent coatin g 010. 1index of refraction of 1.40. 'l"'he coati ng h avi ng a rr val ue of40 rnrn - 1 becomes essentially opaque at abou t 0.05 rnm , an d theother coatin gs at success ive ly greater th icknesses.TH IC KNE SS224OFCO ATI NG. 'n m

4.3. Limitations to EquationsThere are a number of condition encountered in practice t hat deviate from the conditionspostulated in th e derivation of eq (24 ). There will always be a thermal gradient normal tothe surface in any si t ufttio n where a body is heated internally and is dissipftting radiant energy,and the effect of such a gradient h as been omitted in the derivat,ion. In t he case of thick cem miccoatings (typically 0.5 mm or more), significftn t errors may be introduced by ignoring the effectof thermal gradients, especially at high temperatures. The gradients will be negligibly smallwhen the specimen and surroundings are at room temperature or below, and the effect of thegradients in thin coatings, typically less than 0.1 mm, is considered to be negligibly smftll evenat high temperatures .During firing of porcelain enamels and some other types of ceramic coatings, on somem etals, there is appreciable chemical reaction at the coating-metal interface, and some of thereaction products diffuse into the coating for measurable distance. The reaction products maychan ge the reflectance of the substrate into the coatin g, and the reaction products diffusingin to the coating may significantly change its optical properties near the in terface . The extenta nd importance of these effects depend upon the particular materials involved, their thicknesses, the firing , ftnd the service conditions . vVhen the total effect of these factors is large,appropri aLe adjustments in t he values substitu ted into the equation are required for its usefulappli cation. i\10st types of organic and flame-sprayed ceramic coatings, however, will beesse n tially fr ee from these eD'ects.In th e case of some paints, there may be segregation of the pigment particles within Lhevehicle during dryin g, which will also tend to invalidate the equation. For mat coatin gs, wherethe coating-air interface is not optically s mooth even on a micro scale, it may be necessary tom easure Pe and P i experim en tally, instead of computing them from the index of refraction.5 . SummaryThe thermal radiation properties of a composite specimen, comprised of a partially transmitting coatin g applied over an opaque substrate, usually var y significantly with the thicknessof th e coating. An equa tion was derived relating these properties to the thickness of thecoating, the reflectance of the substrate, and the optical properties of the coatin g material.If th e optical proper Lies of a cOttting and t h e l·eilectance of the s ubstrate are known as functionsof wavelength, the equation can be used to compu te (1) the normal spectral emittance (orreflectance) of a ny thickness of coating over the substrate or (2) the thickn ess of t he coatin gover the substrate required to give any normal spectral emittance (or reflectance) within anygiven wavelength interval in termed iate between th e emi ttance of th e substrate and of an in.finitely thick coating.The author gratefully acknowledges the assistance of Louis Joseph, of the Applied Mathematics Division, in checking the mathematics, and of Deane B. Judd, of t he Opticsand Metrology Division, for his advice and helpful suggestions.6. List of SymbolsA absorp tance.T transmittance.R reflectance.E emittance.I diffuse radiant flux proceeding outward from the interior of a specimen.J diffuse radiant flux proceeding inward toward the interior of a specimen.K absorption coefficient.225

-------S backsca ttering coefficient.x distance fro111 the coating-substrate interface to a point in the coating.L a constant that depends upon boundary conditions.u --/K(K 2S).(3 K/ (K 2S).Pe reflectance of the coating-ail' interface for externally-incident diffuse radiant flux.ps reflectance or the substrate for diffuse radiant flux incident on the coating-substrateinterface from within the coating.pi r eflectance oj' the coating-air interface for internally incident diffuse radiant flux .J e externally incident diffuse radiant flux.J o flux J at x O (at coating-substrate interface).I D flux I at x D (at coating-ail' interface).D thickness of coating.M (l (3) - Ps(l - (3).N (l (3) - Pi( l - (3).0 (l - (3 )- Ps(l (3).p (1 - (3) - Pi(l - (3 ).n index or refraction.nl index of refraction in medium from which flux is incident.n 2 index of refraction in medium into which flux is r efracted.'1' angle of incidence, from the normal.8 angle of refraction, from the normal.cp c critical angle of incidence for total internal reflectance.EH hemispherical spectral emittance.PN reflectance of the coating-air interface for normally incident parall el flux.E N normal spectral emittance.R i internal r eflectan ce or a coating. 7. References[1] Judd, D. B. , Optical specification of light -scatte ring matcrials, J. R es. :\lBS 19, 28 7 (1937) H.PIO:26.[2] ASTM Method C 347- 57, R e fl ectiv ity and Coe ffici ent of Scatte r of ViThi te Porcelai n Enamels, ]9 61 ASTMBook of Standard s, pp . 612- 615.[3] Kubelka, P., and F . Munk, Ein Beit rag zu r Optik d er Farbanstriche, Z. t ec h. Ph ys ik, 12, 593 (1931) .[4] Kubelka, P ., New con t ribution s to the optics of intensely light-scattering material s, P ar t 1. J. Opt. Soc.Am . 38, 448 (1948) .[5] Gardon , R , The e miss ivity of t ra nsparen t m aterials, J. Am. Cera m. Soc. 39 [8], 278- 285 (1956) .[6] Hamaker, H. C. , R a diation and heat conduction in light-scatterin g m aterial, Philips R es . Heport:;, 2,55- 67, 103- 111, 112- 125, 420- 425 (1947).[7] Klein , J. D ., H eat transfer by radiation in po\\-ders, Doctor's Disse rtation, Mass. Inst. of T ec hnology (1960) .[8] Wals h, J . W. 1'., The reflection fac tor of a polished glass surface for diffused li ght, D e pt. Sci. and Ind.Res. Illumin ation R esearch, T ech. P aper Ko. 2, p . 10 (1926).[9] Saunderson , J . L ., Calculation of the color of pigmented plastics, J. Opt. Soc . Am . 32 [12], 727- 736.[10] Judd , D. B. , Fresnel reflection of diffu sely incident light, J . R esear ch KBS 29, 329- 332 (1942) RP150.f .(Paper 67C3- 132)226

materials are sufficiently nonselective lo permit use of the pertinent equation with a single set of optical proper Lies applicable Lo all wfLvelengths. Hence the optical properties as fl, function of wavelength are required for computation of wide-range spec