Infrared Physics & Technology 77 (2016) 162–170

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Infrared Physics & Technology

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Near-infrared optical properties of a porous alumina ceramics producedby hydrothermal oxidation of aluminum

http://dx.doi.org/10.1016/j.infrared.2016.05.0281350-4495/� 2016 Elsevier B.V. All rights reserved.

⇑ Corresponding author.E-mail address: ldombr@yandex.ru (L.A. Dombrovsky).

Aleksey V. Lisitsyn a, Leonid A. Dombrovsky b,⇑, Vladimir Ya. Mendeleyev a, Anatoly V. Grigorenko a,Mikhail S. Vlaskin a, Andrey Z. Zhuk a

a Joint Institute for High Temperatures, NCHMT, Krasnokazarmennaya 17A, Moscow 111116, Russiab Joint Institute for High Temperatures, Izhorskaya 13-2, Moscow 625003, Russia

h i g h l i g h t s

� The transport scattering albedo is retrieved from the normal-reflectance measurements.� The known data for absorption coefficient of alumina in a wide temperature range are used.� The absorption coefficient is obtained using the additive hypothesis for porous ceramics.� The Mie theory for grains in porous ceramics is used to estimate near-infrared scattering.

a r t i c l e i n f o

Article history:Received 29 April 2016Revised 28 May 2016Accepted 30 May 2016Available online 31 May 2016

Keywords:Alumina ceramicsReflectanceTransport approximationMie scatteringRadiative conductivity

a b s t r a c t

The measured spectral normal–hemispherical reflectance of an optically thick sample of porous aluminaceramics produced by hydrothermal oxidation of aluminum and subsequent high-temperature treatmentof boehmite (or böhmite), AlO(OH), is used in a combination with the published data for absorptioncoefficient of alumina to retrieve the near-infrared optical properties of the alumina ceramics at bothroom and elevated temperatures. The spectral emissivity of porous alumina ceramics is also determined.An approximate model based on the Mie theory for single grains is suggested to estimate the transportscattering coefficient and the radiative conductivity of the material at high temperatures.

� 2016 Elsevier B.V. All rights reserved.

1. Introduction

Porous alumina ceramics under investigation was producedusing a specific procedure based on the hydrothermal oxidationof aluminum powder [1–5]. It was shown that aluminum powderwith average particle size of about 10 lm is fully oxidized duringfew seconds in water steam at temperature about 300 �C. The pro-duct of this oxidation is boehmite, AlO(OH), in the form of singlecrystals with sizes from 10 to 200 nm, and the primary crystalsare then agglomerated into the particles with the size about10 lm. The complete process including the described first stagewas realized in experimental plant with the use of a continuousflow reactor [6]. The second stage is a high-temperature treatmentof boehmite at first in muffle furnace at 600 �C for crystallizedwater removal and at then in vacuum furnace at 1600 �C for

a-Al2O3 obtaining. This phase of pure alumina is widely used dueto its advantages as compared to other phases [7,8]. The near-infrared properties of porous alumina are important for combinedheat transfer analysis at the high-temperature stage of the process.

The particular objective of the present paper is twofold: (1) toobtain near-infrared radiative properties of a porous aluminaceramics in a wide range of temperatures using the suggested com-bined experimental and theoretical approach and (2) to develop anapproximate theoretical model for both the scattering propertiesand radiative conductivity of the alumina ceramics.

2. Microscale morphology and average porosity of aluminaceramics

Surface morphology of a solid sample of alumina ceramics wasstudied with the use of JEOL JSM-7401F scanning electron micro-scope (SEM). A carbon substrate was used to support the samplesin themicroscope. The shootingwas carried out at 1 kV accelerating

Nomenclature

a grain or pore radiusd distancef fraction or concentrationG irradiationI radiation intensityJ diffuse component of radiation intensitym complex index of refractionM massn index of refractionp porosityq radiative fluxQ efficiency factor~r radius-vectorR reflectanceS relative areaT temperatureV volumex diffraction parameter

Greek symbolsa, b absorption and extinction coefficientsc coefficient introduced by Eq. (9b)d relative radius of cavity in particlese emissivityh polar anglej index of absorptionk wavelength of radiationl cosine of polar angle

�l asymmetry factor of scatteringn coefficient in Eq. (18)r scattering coefficientr0 Stefan–Boltzmann constantq density of aluminas optical thicknessu, w functions introduced by Eq. (8)v eigenvalue introduced by Eq. (9b)U scattering phase functionx albedo~X the unit vector of direction

Subscriptsa absorptionav averageb blackbodyc criticaldef defecth hemisphericaln normalopen open poresr radiativeR Rosselands scattering, surfacesample sampletr transportv volume

A.V. Lisitsyn et al. / Infrared Physics & Technology 77 (2016) 162–170 163

voltage. Several typical images of high resolution are presented inFig. 1. The sample morphology is really very complex, and onecan observe the grains of quite different shape and size. There areseveral big and relatively dense agglomerates with size about 30–50 lm (Fig. 1a). At the same time, one can see in Fig. 1b a lot of sep-arate small particles with typical size of 3–5 lm with numerousorifices of diameter at the level of 1 lm and also many sphericalparticles with diameter about 1 lm. Obviously, a contribution ofvarious particles to the near-infrared scattering is quite different,and the main attention should be focused on homogeneous orhollow small particles.

The laboratory measurements were also made to determineboth the total volumetric porosity, p, and the volume fraction ofopen pores, popen, of the sample of alumina ceramics. The totalporosity is defined as a ratio of the sample mass to the mass ofimaginary bulk material:

p ¼ Msample=ðqV sampleÞ ð1ÞThe accurate measurements showed the value the value of

p ¼ 0:638. The value of popen was also obtained experimentallyusing various liquids to fill the open pores. The repeating experi-ments showed that popen ¼ 0:386 ¼ 0:605p. This result is qualita-tively clear from the SEM image presented in Fig. 1b. A lot ofclosed cavities in alumina grains were produced in the specificprocess of the hydrothermal oxidation of aluminum.

3. Measurements of the normal–hemispherical reflectance

The spectral measurements of normal–hemispherical reflec-tance and transmittance of flat samples is a widely-used traditionalprocedure to get the data for subsequent identification of the mainabsorption and scattering characteristics of dispersed materials[9–11]. In some cases, as discussed in [12–14], porous materials

are highly scattering in the near-infrared. As a result, normal–hemispherical transmittance is low, on the order of 1% for samplesof geometric thickness of about 1 mm, due to very large opticalthickness. Hence, use of commercial spectrometers to measurenormal–hemispherical transmittance is challenging, due to theirlack of adequate signal-to-noise characteristics to measure suchlow values of transmittance. The measurements were performedin the wavelength range of 0:2 < k < 2:5 lm. The relative uncer-tainty of the measurements was no more than 0.5% at k < 2:2 lmand reached 2% at k > 2:2 lm.

Fortunately, there are reliable data in the literature for the spec-tral absorption coefficient of bulk alumina. Therefore, the measure-ments of normal–hemispherical reflectance, Rn�h, for opticallythick samples are sufficient. The value of Rn�h was measured usingspectrophotometer Cary 500 produced by the firm Varian andequipped by integrating sphere DRA-CA-5500 with internal diam-eter of 150 mm. This integrating sphere was produced by the firmLabsphere. The beam diameter at the sample surface was equal to10 mm. The schematic of the optical installation is traditional, andit is not presented in the paper. The results of measurements pre-sented in Fig. 2 indicate that the value of Rn�h is relatively low inthe visible range and increases up to about 0.88–0.9 in the near-infrared. We also found good agreement between the spectraldependences of Rn�h shown in Fig. 2 and similar values for theother local area of the sample surface. This result indicated thatoptical properties of the sample are homogeneous, and goodrepeatability of the Rn�h data is confirmed.

4. Spectral optical properties of alumina in the near-infrared

There are two optical constants of every substance, and thesequantities are usually treated as the complex index of refraction,m ¼ n� ij, where n is the index of refraction and j is the index

Fig. 1. SEM images of the sample surface: (a) more general view and (b) detailedmorphology.

0.5 1.0 1.5 2.0 2.5 3.01.7

1.75

1.8

1.85

1.9

λ, μm

n a

-1

164 A.V. Lisitsyn et al. / Infrared Physics & Technology 77 (2016) 162–170

of absorption [15]. Both n and j and many other quantities consid-ered in the paper depend on radiation wavelength, but the corre-sponding subscript is omitted hereafter for brevity. Thetemperature dependence of n for solid alumina is very weak.Therefore, it is sufficient to use an approximation based on thethree-term Sellmeier dispersion equation suggested in [16] forsynthetic sapphire at room temperature:

0.5 1.0 1.5 2.0 2.5

0.30

0.35

0.80

0.85

0.90

Rn-h

λ, μm

Fig. 2. Spectral dependence of normal–hemispherical reflectance for the opticallythick sample of porous alumina ceramics.

n2 � 1 ¼ 1:024k2

k2 � 0:003776þ 1:058k2

k2 � 0:01225þ 5:281k2

k2 � 321:4ð2Þ

where k is expressed in microns. Possible corrections which aresometimes used to take into account a monotonic increase of therefractive index with temperature can be found in [17,18]. The val-ues of dn=dT are from about 2 � 10�5 to 3 � 10�5 K�1, and this effectcan be neglected. The dependence of nðkÞ calculated by Eq. (2) isshown in Fig. 3a. The index of refraction decreases monotonicallywith the wavelength. This variation is relatively small in the near-infrared range.

The near-infrared absorption of alumina has been studied indetail. The dimensional value of spectral absorption coefficientdefined as [19]:

a ¼ 4pj=k ð3Þ

was usually determined. The experimental results of the researchgroup at the Institute for High Temperatures (Moscow) and alsodetailed overview of the early published data have been reportedin [19]. The recommended values of a for alumina in semi-transparency range of 0:5 < k < 7 lm were tabulated in [19] takinginto account early papers [20–24]. A short-wave part of the tabu-lated data is plotted in Fig. 3b. The absorption increases with tem-perature over the spectrum, but this effect is not strong. Note thatmore recent studies [25] are in good agreement with the data ofFig. 3b.

0.5 1.0 1.5 2.0 2.5 3.01E-3

0.01

0.1

500300

700900

11001300

15001700

19002100

λ, μm

α, c

m

T = 2300 Kb

Fig. 3. Spectral optical properties of alumina: (a) index of refraction and (b)coefficient absorption.

A.V. Lisitsyn et al. / Infrared Physics & Technology 77 (2016) 162–170 165

5. Approximate analytical solution for the normal–hemispherical reflectance

According to the continuous approach, the radiative transfer inan absorbing, refracting, and scattering medium is described by theradiative transfer equation (RTE) [26–28]:

~XrIð~r; ~XÞ þ bIð~r; ~XÞ ¼ r4p

Zð4pÞ

Ið~r; ~X0ÞUð~X0~XÞd~X0 þ an2Ib½Tð~rÞ� ð4Þ

The physical meaning of Eq. (4) is evident: variation of the spec-

tral radiation intensity in direction ~X takes place due to self-emission of thermal radiation (the last term), extinction by absorp-tion and also by scattering in other directions, as well as due toscattering from other directions (the integral term). The absorptioncoefficient, a, the scattering coefficient, r, the extinction coeffi-cient, b ¼ aþ r, and the scattering phase function, U, depend onthe coordinate ~r. For simplicity, Eq. (4) is written for the case ofan isotropic mediumwhen the coefficients do not depend on direc-tion. An accurate solution to the RTE in scattering media is a verycomplicated task. One can find a number of studies in the literatureon specific numerical methods developed to obtain more and moredetailed spatial and angular characteristics of the radiation field.Several modifications of the discrete ordinates method (DOM)and statistical Monte Carlo (MC) methods are the most populartools employed by many authors [26–28].

Fortunately, the transport approximation appears to be highlysuccessful method to solve many applied problems characterizedby multiple scattering [27,29–33]. According to this approxima-tion, the scattering phase function is replaced by a sum of the iso-tropic component and the term describing the peak of forwardscattering:

Uðl0Þ ¼ ð1� �lÞ þ 2�ldð1� l0Þ ð5ÞWith the use of transport approximation, the RTE can be written inthe same way as that for isotropic scattering, i.e., with Uk � 1:

~XrIð~r; ~XÞ þ btrIð~r; ~XÞ ¼ rtr

4pGð~rÞ þ an2IbðTÞ

Gð~rÞ ¼Zð4pÞ

Ið~r; ~XÞd~X ð6Þ

where the transport scattering and extinction coefficients aredefined as follows:

rtr ¼ rð1� �lÞ btr ¼ aþ rtr ¼ b� r�l ð7ÞThe complete mathematical formulation of radiative transfer

problems is very complicated even for the simplest approximationof scattering phase function. The main difficulty is an angulardependence of the radiation intensity. At the same time, this angu-lar dependence appears to be simple in many applied problems. Itenables one to use this property of solution to derive relativelysimple but fairly accurate differential approximations. Theprogress in computer engineering and numerical methods makesit possible to obtain more accurate solutions. Nevertheless,simple and physically clear differential approximations are widelyused at present for solving the radiative transfer problems in scat-tering media, particularly in combined heat transfer problems[27,34–36]. All the differential approximations for RTE are basedon some assumptions concerning the angular dependence of thespectral radiation intensity. These assumptions enable us to turnto the system of the ordinary differential equations.

The modified two-flux approximation developed in [37,38] canbe used to find a relation between the normal–hemisphericalreflectance of optically thick samples and transport scatteringalbedo, xtr ¼ rtr=btr, of a dispersed material. In this approach, theradiation intensity is presented as a sum of the collimated and

diffuse components, and the following approximate angulardependence of the diffuse component is assumed:

�Jðstr;lÞ ¼u�ðstrÞ;�1 6 l < �lc

wðstrÞ;�lc < l < lc

uþðstrÞ;lc < l 6 1lc ¼ ð1� 1=n2Þ1=2

8><>: ð8Þ

where str and l ¼ cos h are the current transport optical thicknessand angular coordinate [27], n is the index of refraction of the hostmedium. This approximation takes into account the effect of totalinternal reflection at n > 1. Note that the case of lc ¼ 0 (at n ¼ 1)corresponds to the traditional two-flux model. The intermediateangle interval of �lc < l < lc gives no contribution to the radia-tion flux and the words ‘‘two-flux” are applicable to the modifiedapproximation as well. In the limiting case of optically thick sam-ples, the following analytical solution can be derived [38]:

Rn�h ¼ R1 þ ð1� R1Þ c2n2

xtr

1�xtr

v2

ð1þ vÞ½2c=ð1þ lcÞ þ v� ð9aÞ

v2 ¼ 4

ð1þ lcÞ21�xtr

1�xtrlcc ¼ 1� R1

1þ R1lc ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 1=n2

qð9bÞ

The normal interface reflectivity can be used in Eqs. (9a) and(9b) as was done in [37,38]:

R1 ¼ ðn� 1Þ2=ðnþ 1Þ2 ð10ÞA more accurate expression for R1 from [39] can be also

employed as suggested in [12,13]. This expression was obtainedby averaging the Fresnel reflectivity over the hemisphere:

R1 ¼ 12þ ð3nþ 1Þðn� 1Þ

6ðnþ 1Þ2þ n2ðn2 � 1Þ2

ðn2 þ 1Þ3ln

n� 1nþ 1

� 2n3ðn2 þ 2n� 1Þðn2 þ 1Þðn4 � 1Þ þ 8n4ðn4 þ 1Þ

ðn2 þ 1Þðn4 � 1Þ2lnn ð11Þ

The effect of using Eq. (11) instead of Eq. (10) for the normalreflectivity was quantified in paper [40]. Eqs. (9a), (9b) and (11)can be employed to retrieve transport scattering albedo from themeasured normal–hemispherical reflectance.

It is interesting that the same solution for Rn�h can be easilyobtained using the two-step analytical solution for normal emis-sivity en [40] and the following relation follows from Kirchhoff’slaw in the limiting case of an optically thick sample:

en ¼ 1� Rn�h ð12ÞThe analytical solution for en derived in [40] is also based on the

modified two-flux approximation, but with subsequent analyticalintegration of the RTE along the ray. This integration was doneinstead of a separate consideration of the collimated and diffusecomponents of the radiation intensity. However, it is to be notedthat these two approaches are equivalent. The accuracy of the ana-lytical solution for normal emissivity has been examined in [40] bycomparison with both DOM and MC methods including effect ofrealistic scattering phase functions. It was shown that an error ofthis solution is very small. This implies that the above analyticalsolution is also sufficiently accurate and applicable to the problemunder consideration.

It should be recalled that there are two physically different sit-uations in the cases of relatively dense and highly-porous dis-persed materials. The above relations refer to the more complexcase of dense porous materials when p � 0:2 or less. One can con-sider such a material as a solid refractive matrix with some pores.This leads to appearance of coefficient n2 in the last term of Eq. (4)and also to Eq. (8) taking into account the effect of total internalreflection at the sample surface. Only few defects are usually

0.5 1.0 1.5 2.0 2.50.75

0.80

0.985

0.990

0.995

ωtr

λ, μm

a

0.5 1.0 1.5 2.0 2.50.10

0.15

0.20

0.25bσ tr

, cm-1

λ, μm

Fig. 5. The retrieved spectral properties of a porous alumina ceramics at roomtemperature: (a) the transport albedo and (b) the transport scattering coefficient.

166 A.V. Lisitsyn et al. / Infrared Physics & Technology 77 (2016) 162–170

observed at the polished surface of a low-porosity material, andthese defects can be ignored.

It is important to clarify the physical sense of estimate p 6 0:2.Let imagine that a material contains many spherical bubbles of thesame radius a with the volume fraction of bubbles equal to f v � p.In the case of numerous bubbles, their surface concentration for anarbitrary cut of the sample does not depend on the possible orderin the spatial distribution of the bubbles [37]. For this reason, onecan consider the simplest cubic structure and the following for-mula which is true for the uniform volume distribution of bubbles:

p ¼ ð4p=3Þða=dÞ3 ð13Þ

where d is the distance between neighboring bubbles (step of thecubic structure). When one cuts the sample by planes, which areparallel to one of the sides of the cubic structure, the probabilitythat a bubble is located at the sample surface is equal to 2a=d,and the average surface concentration of bubbles is determined as:

f s ¼ 2a=d3 ð14ÞComparison of Eqs. (13) and (14) gives the following relations

between this value and the values of porosity and relative area ofsurface defects, Sdef :

f s ¼ 1:5p=ðpa2Þ Sdef ¼ f spr2av ¼ f spa2=2 ¼ 0:75p ð15ÞIn the case of p ¼ 0:2, we obtain Sdef ¼ 15%. This is the quanti-

tative basis of the above conventional criteria of low-porositymaterial.

In the case of highly-porous materials (p � 0:2), there is no flatsurface of the sample. It means that the internal reflection effect atthe interface does not take place and the ordinary two-flux methodis applicable. The resulting expression for Rn�h and the inverse rela-tion are:

Rn�h ¼ xtr=ð1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�xtr

pÞ2 xtr ¼ 4Rn�h=ð1þ Rn�hÞ2 ð16Þ

To illustrate the difference between the dense and highly-porousalumina ceramics, one can compare the functions Rn�hðxtrÞ asshown in Fig. 4. The black curve (n ¼ 1) corresponds to thehighly-porous ceramics at two curves at values of n typical for alu-mina in the near-infrared are plotted for the case of a dense ceram-ics. One can see that effect of porosity is significant whereas smallerrors in measurements of Rn�h for a weakly-absorbing mediumwillnot lead to considerable errors in the value of xtr.

0.88 0.90 0.92 0.94 0.96 0.98 1.00

0.4

0.6

0.8

1.0

n = 1 1.7 1.8

Rn-h

ωtr

Fig. 4. Effect of the medium transport albedo on the normal–hemisphericalreflectance of a semi-transparent highly-scattering medium in optically thick limit.

6. Spectral dependences of transport values of albedo andscattering coefficient

The transport scattering albedo of the highly-porous aluminaceramics at room temperature can be determined using Eq. (16)and the measured spectrum of Rn�hðkÞ (Fig. 2). The obtained spec-tral dependences ofxtrðkÞ are presented in Fig. 5a. One can see thatxtr is not small even in the visible spectral range and increases upto 0.994–0.996 in the near-infrared, i.e. the scattering is highly pre-dominant as compared with absorption in this spectral range.

One needs a physical model to make the next step and deter-mine the values of both the absorption coefficient and transportscattering coefficient for porous alumina ceramics. An experiencein studying infrared properties of advanced porous materials suchas fibrous and foam-like thermal insulations, micro-porous ceram-ics and other dispersed materials allowed formulating a simplephysical approach applicable to many semi-transparent materialsof different nature. This approach is based on the following princi-ples [41]:

1. The absorption coefficient is practically independent of thematerial morphology and directly proportional to the volumefraction of the absorbing substance:

a ¼ a0ð1� pÞ ð17Þwhere a0 is the absorption coefficient of the bulk material of thesame chemical composition.2. The characteristics of scattering are insensitive to a weak

absorption and can be predicted by an analysis of the materialmorphology. In doing so, it is usually important to take intoaccount a contribution of grains, pores, and cracks comparablein size with the radiation wavelength.

A.V. Lisitsyn et al. / Infrared Physics & Technology 77 (2016) 162–170 167

A relative mutual independence of absorption and scattering isnot a specific property of semi-transparent porous materials. Thisis a general behavior of disperse systems containing the so-calledoptically soft particles satisfying the conditions of the Rayleigh–Gans theory [27,42–44]. Of course, the spatially uniform absorp-tion cannot lead to the radiation scattering. On the contrary,appearance of strongly absorbing regions comparable in size withthe wavelength may lead to an additional scattering. It wasdemonstrated in experimental and theoretical studies of infraredproperties of a nanoporous silica material [45] and also morpho-logical changes arising in optical properties of polypropylene afterthermal treatment [46] that there are some cases when scatteringdepends on absorption.

In all cases, the different physical nature of absorption and scat-tering makes reasonable a separate analysis of the experimentaldata for absorption coefficient and transport scattering coefficient.The use of the sum of these values and single-scattering albedo ofthe medium is usually not convenient for the physical analysis. Theabove formulated approach is confirmed by direct spectral mea-surements in various engineering problems and can be used tostudy the radiative properties of semi-transparent dispersed mate-rials at elevated temperatures [27,41].

The transport scattering coefficient calculated using the spectraldependences ofxtr and a and also relation (17) is shown in Fig. 5b.Ignoring some local peaks, one can say that spectral behavior of rtr

is characterized by a decrease with the wavelength. This is typicalfor disperse systems containing small particles which radius is lessor comparable with the wavelength [27]. The latter will be used tosuggest an approximate model for scattering properties of the alu-mina ceramics under investigation.

Following [47,48], it is assumed that scattering is weakly sensi-tive to temperature because of negligible temperature dependenceof both the index of refraction of alumina and the morphology ofporous ceramics. The latter is explained by a small thermal expan-sion of alumina and negligible effect of high-temperature chemicalpurification on the material density. As a result, the spectral depen-dences of transport scattering coefficient obtained at room temper-ature can be also used at elevated temperatures.

0.60 0.65 0.70 0.75 0.80 0.85 0.900.0

0.2

0.4

0.6

0.8

c

b

123

Q str /x

1/x

a

Fig. 6. The ratio of Q trs =x for non-absorbing homogeneous and hollow spherical

particles: (a) d ¼ 0, (b) 0:5, (c) 0:7; (1) n ¼ 1:726, (2) 1:74, (3) 1:76.

7. Approximate theoretical model for near-infrared scattering

The relatively simple approximate theoretical models suggestedduring last two decades to understand and describe scatteringproperties of porous ceramics [12,13,49–54] are based on the Mietheory for single spherical pores (in the case of dense ceramics) orsingle grains (in the opposite case of highly-porous ceramics) byignoring possible electromagnetic interaction of closely positionedpores or grains. The latter assumption is known as the hypothesis ofindependent scattering [55–57]. According to this widely usedapproach, each particle is assumed to absorb and scatter the radia-tion in exactly the samemanner as if other particles did not exist. Inaddition, there is no systematic phase relation between partialwaves scattered by individual particles, so that the intensities ofthe partial waves can be added without regard to phase. In otherwords, each particle is in the far-field zones of all other particles,and scattering by individual particles is incoherent.

There are also some recent studies which do not use thehypothesis of independent scattering and traditional Mie theoryto develop a model for optical properties of porous materials. Inpaper [58], the shape of voids in a dense ceramics was assumedto be oblate spheroids, and the effect of their aspect ratio, size,and orientation on radiative properties of ceramics have beeninvestigated using a numerical technique based on the discretedipole approximation [59]. A study using the finite-differencetime-domain (FDTD) method was carried for the generated

microstructure in recent paper [60]. At the same time, the simpletraditional modeling of spectral radiative properties of porousceramics with the use of independent-scattering Mie theoryremains to be attractive for engineering applications.

It is known fromMie theory calculations for particles of weakly-absorbing materials that a decrease in transport scattering coeffi-cient with the wavelength takes place only in a narrow range ofthe diffraction parameter x ¼ 2pa=k [27]. Note that there is noeffect of small absorption index of alumina on value of Q tr

s , andthe index of refraction of alumina decreases slightly in the near-infrared from n ¼ 1:76 at k ¼ 0:8 lm to n ¼ 1:726 at k ¼ 2:5 lm.The dimensionless values of Q tr

s =x calculated in a part of the abovementioned specific range of x for homogeneous and hollow spher-ical particles for three values of relative radius, d, of concentric cav-ity are shown in Fig. 6. The results for intermediate value ofn ¼ 1:74 are also shown in this figure, where the inverse value of1=x on the abscissa is used to make obvious a qualitative similarityof the curves in Figs. 5b and 6. The difference between three vari-ants of refractive index is insignificant, and the value of n ¼ 1:74will be used in further calculations. It should be noted that effectof cavity on the value of Q tr

s =x is significant. Additional calculationsshowed that the effect of cavity on the important value ofQ tr

s =½xð1� d3Þ� [27] remains considerable. It is problematic to esti-mate an average value of d, but it is not an important difficultyin developing an approximate physical model of scattering forthe alumina ceramics under consideration. Note that the range ofdiffraction parameter in Fig. 6 corresponds approximately to thewavelength range of 0:6 < k < 1 lm at a ¼ 0:16 lm and1:5 < k < 2:5 lm at a ¼ 0:4 lm. In other words, polydisperse alu-mina particles with radius about 0.2–0.4 lm are responsible forthe observed scattering of alumina ceramics. The volume fractionof these small particles is much less than the measured porosityof alumina ceramics (see Fig. 1).

The choice between homogeneous and hollow particles is notimportant because of approximate similarity of the resulting spec-tral curves. At the same time, the calculations for homogeneous alu-mina particles aremuch simpler [27]. Therefore, the simplestmodelis used below. Note that a contribution of larger particles withradius about 1 lm should be also taken into account. The calcula-tions showed that one can use the simplest bimodal distributionof particles to suggest an approximate model for the observed scat-tering of alumina ceramics. For simplicity, it is assumed that parti-cles of two sizes are present. In this case, the following equation forthe transport scattering coefficient can be used [27]:

rtr ¼ 0:75 f vQ tr

s ða1Þa21 þ nQ trs ða2Þa22

a31 þ na32ð18Þ

0.5 1.0 1.5 2.0 2.5 3.00.10

0.15

0.20

0.2512

σ tr, c

m-1

λ, μm

Fig. 7. Comparison of (1) experimental data and (2) analytical estimates for thetransport scattering coefficient.

168 A.V. Lisitsyn et al. / Infrared Physics & Technology 77 (2016) 162–170

The parameters a1 ¼ 0:24 lm, a2 ¼ 0:85 lm, n ¼ 0:06, andf v ¼ 1:2 � 10�5 give good agreement of Eq. (18) with experimentaldata (see Fig. 7). Eq. (18) has an obvious advantage as comparedto the experimental points because of possible extrapolation of rtr

to the range of k > 2:5 lm.

8. Near-infrared emissivity and radiative heat transfer in porousalumina ceramics

In the case of optically thick layer of alumina ceramics, there is avery simple relation (12) between the spectral normal emissivity,en and the measured value of Rn�h. In heat transfer calculations,the spectral hemispherical emissivity, eh, is usually used. In theoptically thick limit, one can use an analytical relation for eh forhighly-porous alumina ceramics. The accurate relation can beobtained using the combined two-step iterative method, but thetwo-flux approximation is sufficient to obtain the following ratheraccurate expression [27]:

0.5 1.0 1.5 2.0 2.50.0

0.2

0.4

0.6

εh

λ, μm

a

1600 1800 2000 2200120

160

200

240

280

k r, W

m-1K

-1

T, K

b

Fig. 8. The radiative properties of porous alumina ceramics: (a) the spectralemissivity and (b) the radiative conductivity.

eh ¼ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�xtr

p=ð1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�xtr

pÞ ð19Þ

Note that Eq. (19) gives exactly the same values of eh as Eqs.(12) and (16): eh � en. The near-infrared emissivity of porous alu-mina ceramics is about 0.12–0.14, in contrast to the short-wavevisible radiation characterized by much greater values of eh (seeFig. 8a).

A variation of the spectral radiative flux in alumina ceramicswith the distance from the interface at arbitrary temperature pro-file can be obtained using the two-flux approximation [27]. At thesame time, a very simple approach is often used in the engineeringpractice to estimate the local integral radiative flux in nonisother-mal semi-transparent materials. The so-called radiative conductiv-ity, kr, is introduced in this approach which is sometimesapplicable to heat transfer calculations inside the optically thickmedia [26–28]:

kr ¼ 163

r0T3

bRtrðTÞ

bRtrðTÞ ¼

4r0T3

pR10

1btr

@IbðTÞ@T dk

ð20Þ

where bRtr is the Rosseland mean transport extinction coefficient. Eq.

(20) in combination with the Fourier-type equation for the integral(over the spectrum) radiative flux:

~qr ¼ �krrT ð21Þare known as the Rosseland’s or radiative conduction approxima-tion. This approach works good only at large optical distances fromthe boundaries and also from the regions with strong variation oftemperature and medium properties [27,36]. To understand theselimitations, it is sufficient to recall that Eq. (21) gives zero radiativeflux in the case of an isothermal medium and may give high valuesof qr at strong temperature gradient even in the vicinity of highly-reflecting boundary surfaces, when the radiative flux is extremelysmall. It should be noted that one can make some corrections tothe value of kr as it was done in early paper [61]. A more sophisti-cated approach of this type based on a comparison with relativelyaccurate solutions to the radiative transfer problem for thermalboundary layers can be found in [27], where the optical propertiesof both the medium and boundary surfaces are taken into account.

Strictly speaking, the value of bRtr cannot be determined at

arbitrary temperatures on the basis of the data for thesemi-transparency range only. It is a serious limitation for lowtemperatures where a contribution of the middle-infrared range issignificant. The optical properties of alumina in the opacity rangehave been overviewed in paper [62] with reference to several earlypapers such as [21,63,64]. Note that the authors of [19] consideredthe wavelengths up to k ¼ 7 lm as a region of alumina semi-transparencywhere the absorption coefficient at room temperaturedoes not exceed the value of a ¼ 50:1 cm�1.

For alumina ceramics under investigation, the scattering isexpected to be greater than absorption up to the wavelength about4 lm. It means that the data of the present paper enables us toestimate the radiative conductivity at high temperatures only. Inthe case of weakly absorbing alumina ceramics, Eq. (20) can besimplified as follows:

kr ¼ 163

r0T3

rRtrðTÞ

rRtrðTÞ ¼

Z k2

k1

@IbðTÞ@T

dk�Z k2

k1

1rtr

@IbðTÞ@T

dk ð22Þ

The form of the last equations is preferable to minimize the errorsrelated with the limiting range of integration. One can use approx-imation (18) and the values of k1 ¼ 0:5 lm and k2 ¼ 4 lm to deter-mine rR

tr at temperatures T > 1600 K when the near-infraredradiation is more important. The calculated values of kr appearedto be very large (see Fig. 8b), and it is important for heat transfercalculations.

A.V. Lisitsyn et al. / Infrared Physics & Technology 77 (2016) 162–170 169

It should be recalled that the spectral values of both the absorp-tion coefficient and the transport scattering coefficient are suffi-cient to use more accurate approaches (not only differentialapproximations) to radiative heat transfer in porous aluminaceramics.

9. Conclusion

Absorption and scattering properties of a porous aluminaceramics produced by hydrothermal oxidation of aluminum havebeen retrieved in the visible and near-infrared on the basis of thesimplest methodology applicable to weakly-absorbing andhighly-scattering materials with the known spectral index ofrefraction and absorption coefficient of the non-scattering bulkmaterial. This methodology includes the following three stages:(1) the measurements of normal–hemispherical reflectance of anoptically thick sample at room temperature and the use of analyt-ical relation between the measured value and the transport albedoof the material, (2) the use of the hypothesis of additive absorptionand the measured sample porosity to determine the spectralabsorption coefficient at room and elevated temperatures, (3) thesimple calculation to obtain the transport scattering coefficientwhich is practically independent of temperature.

This procedure cannot be repeated for porous ceramics made ofa substance with not well-known optical properties, but it was agood surprise for the authors that optical properties of semi-transparent porous alumina ceramics at arbitrary temperaturescan be retrieved without significant experimental and computa-tional efforts.

The obtained spectral dependence of transport scattering coef-ficient was analyzed using approximate model based on thehypothesis of independent scattering and Mie solution for spheri-cal alumina grains. It was demonstrated that a bimodal size distri-bution with realistic sizes of particles is sufficient to suggest anapproximate relation for the near-infrared scattering. It wasdemonstrated that the data obtained can be used to estimate boththe near-infrared emissivity and the radiative transfer inside theporous alumina ceramics. The latter is important for heat transfercalculations in engineering applications.

Acknowledgments

This study was financially supported by the Ministry of Educa-tion and Science of the Russian Federation under Grant AgreementNo. 14.607.21.0082 (project RFMEFI60714X0082).

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