Proceedings of the 7th International Symposium on Radiative Transfer, RAD-13June 2–8, 2013, Kuşadasi, Turkey

RAD-13-040

SPECTRAL RADIATIVE PROPERTIES OF THREE-DIMENSIONALLYORDERED MACROPOROUS CERIA PARTICLES

Vincent M. Wheeler∗, Jaona Randrianalisoa∗∗, Kumar K. Tamma∗, and Wojciech Lipiński∗,1

∗Department of Mechanical Engineering, University of MinnesotaChurch Street 111 S.E., Minneapolis, MN 55455, USA

∗∗GREPSI, University of ReimsCampus du Moulin de la Housse - BP 1039

51687 Reims Cedex 2, Reims, France

ABSTRACT. Radiative properties of spherical heterogeneous particles consisting of three-dimensionally ordered macroporous (3DOM) cerium dioxide (ceria) are numerically predictedin the spectral range 0.3–10µm. The particles are 1µm in diameter, with interconnected poresof diameter 330 nm and a face-centered cubic lattice arrangement. Predictions are obtainedby solving macroscopic Maxwell’s equations using the discrete dipole approximation and thefinite element method as a complementary means of numerical prediction. The scattering andabsorption efficiency factors as well as the asymmetry factor are determined as a function ofthe particle orientation relative to the direction of the incident plane wave. The scatteringand absorption efficiency factors show significant dependence on the particle orientation in thespectral range equal to particle diameter to 560 nm. Compared to homogeneous ceria particles,3DOM particles of identical size tend to cancel the wave extinction for wavelength greater than560 nm. Approximating the 3DOM particles as a homogeneous sphere with properties calcu-lated from an effective medium theory is also considered. This approach is shown to be validonly for wavelengths much greater than the pore size, demonstrating that a detailed geomet-rical representation of the internal particle structure is essential to obtain accurate radiativecharacteristics of nano-structured particles.

NOMENCLATUREap lattice parameter, mA dipole moment coefficient matrixd dipole lattice parameter, mDp pore diameter, m~E electric field, N C−1g asymmetry factor~H magnetic field, A m−1k complex component of refractive indexm complex refractive indexn real component of refractive indexQ efficiency factorp porosity~P dipole moment vector, A m−1rp particle radius, m~r location in space, m~S time-averaged Poynting vector, W m−2V integration volume, m3

Greek symbols

α polarizabilityΓ integration surface, m2� permitivity, F m−1η wavenumber, m−1θ particle orientation angleλ wavelength, mσ electrical conductivity, S m−1φ particle orientation angleω angular frequencyΩ solid angle, sr

Subscripts and superscriptsabs absorptioneff effectiveext extinctioninc incidentrel relativesca scatteringtot total0 vaccuum

1Corresponding author. Tel.: +1 612 626 0875; fax: +1 612 626 1854. E-mail address: lipinski@umn.edu(W. Lipiński)

INTRODUCTION

Cerium dioxide (ceria) has been proposed as a novel reactive material to realize solar-driventhermochemical cycles to split water and carbon dioxide for production of hydrogen and carbonmonoxide [1–4]. Ceria forms oxygen vacancies in its lattice structure in response to changes inphysical conditions, such as temperature and oxygen partial pressure, making the material suit-able for non-stoichiometric redox chemical reactions. Three-dimensionally ordered macroporous(3DOM) ceria structures offer high porosity and specific surface area due to their nano-porousstructure. Faster chemical kinetics was observed for packed beds of 3DOM structures in com-parison to the kinetics measured for sintered ceria structures [5]. Synthesis techniques haveresulted in improved structural stability, as well as retention of the 3DOM structure when thematerial undergoes thermochemical cycling making the 3DOM structure more desirable thanconventional micro-structured porous ceramics [6]. Radiative properties are needed to deter-mine medium temperature and the reaction rates. The characteristics of the reactive medium,which can be tailored by modifying medium morphology and composition, should simultane-ously allow for (i) efficient absorption of incident concentrated solar radiation, (ii) rapid heattransfer between the absorption and reaction sites, (iii) confinement of the emitted thermalradiation in the close vicinity of the reaction site, (iv) minimum heat losses from the reactingmedium by conduction and convection, and (v) rapid chemical reaction. High specific surfacearea and porosity as well as varying levels of semi-transparency in the visible and infraredspectral ranges are a desired combination of morphological and optical characteristics to satisfythe above criteria for optimizing reactive media for solar thermochemical applications.

Previous pertinent studies of radiative characteristics of ceria ceramics and packed beds aregiven in [7–11]. Overall transmittance of ceria with average porosities of 0.08 and 0.72 wereexperimentally found in [7] for the spectral range 0.3–1.1µm. Both samples were found to behighly opaque up to 400 nm. Using the same materials for the spectral range 0.9–1.7µm, it wasfound in [8] that the mean radiation penetration length is shorter in higher porosity samplessuggesting higher scattering. Using the Monte Carlo ray tracing technique along with experi-mental transmittance data, the transport scattering coefficient of porous ceria was obtained in[11] and found to be in agreement with theoretical estimates based on Mie theory.

Heterogeneous particles as well as their groups were radiatively characterized in the studies[12–16]. Of particular interest to solar thermochemical applications are the effects of internalparticle structure on macroscopic radiative characteristics. The effect of porosity on absorptioncharacteristics was previously studied in [12] using the discrete dipole approximation (DDA)for spherical composite particles. It was found that a shift in the inclusion volume fractioncorresponded to a shift in the absorption peak of the particle. Results obtained using theDDA for composite particles were in good agreement with observed interstellar extinctionefficiency factors [13]. Also using the DDA, Voschinnikov et al. [14] concluded that porosity ofparticles has only a slight effect on optical properties for porosities less than 0.5. The use ofan effective medium theory with exact solutions on approximate geometry, such as the Lorenz–Mie theory, to reproduce scattering characteristics obtained with the DDA as well as the finiteelement method (FEM) was examined in [15–17]. It was concluded in [15] that effective mediumtheories agree well with numerical methods directly discretizing the geometry for a wide rangeof porosities and particle size parameters as long as the effective medium theory assumptionsare upheld: statistical uniformity and small inclusions compared to wavelength. Porosities upto 90% were found to be accurately modeled when the inclusions are in the Rayleigh limit [17].

In the present paper, radiative properties of 3DOM ceria particles are studied in the spectralrange 0.29–10µm for particles with a diameter of 1µm. The DDA is employed to compute

2

(a) (b)

Figure 1: Scanning electron microscope images of 3DOM ceria with (a) 300 times magnificationand (b) 10,000 times magnification.

the radiative properties in the entire spectral range for four particle orientations. Orientation-averaged values are computed for 25 particle orientations in the spectral range 0.38–0.8µm. TheFEM is also applied to solve macroscopic Maxwell’s equations to provide a reference numericalsolution, particularly where DDA has been shown to be inaccurate. The FEM/DDA resultsare compared to those obtained using the Lorenz–Mie in conjunction with effective mediumtheories.

PROBLEM STATEMENT

3DOM ceria structure consists of a face-centered cubic (FCC) lattice of overlapping pores in acontinuous matrix of cerium dioxide as shown in Figure 1. This geometry can be described bytwo parameters, the lattice constant ap and the pore diameter Dp, as shown in Figure 2. In thisstudy, we consider a 3DOM structure with ap = 440 nm and Dp = 330 nm, for which the widthof the interconnecting struts is approximately 90 nm and the porosity is p = 0.85. We consideran idealization of the non-uniform particle morphology seen in Figure 1b. This study examinesa single particle of 3DOM ceria under the following assumptions: (i) the particle is sphericalwith a diameter of 1µm with a uniform pore structure (ii) the pores are vacuous (iii) theelectromagnetic behavior is sufficiently described by a continuous complex index of refraction,

Figure 2: Illustration of target orientation angles and incident wavevector direction as well asa unit cell of 3DOM ceria

3

Table 1: Particle orientations considered in this study. Cases 1, 3, and 4 represent a plane wavetraveling along a major symmetry plane of the lattice resulting in transparent windows in theparticle.

Angle (◦) Case 1 Case 2 Case 3 Case 4θ 0 22.5 45 45φ 0 0 0 45

m = n− ik. Since ceria is non-magnetic and the smallest feature size of the structure is greaterthan 10 nm, the last assumption should be valid [18].

An interesting aspect of optical characterization of highly-ordered nano-structured materials isthe potentially strong dependence of properties on particle orientation due to the anisotropyof the pore arrangement. Such dependence is expected to be most pronounced for orientationscorresponding to transparent windows in the direction of electromagnetic wave propagation.The windows exist along the major symmetry planes of the FCC lattice. In this work, theparticle orientation with respect to a fixed reference frame is described by two angles θ andφ shown in Figure 2. θ is the angle between the particle main axis ζ̂ and the x axis of thefixed reference. φ is the rotation angle of the ζ̂ axis around x axis, which is taken equal to theincident wavevector direction. Particle orientations considered for the full 0.29–10µm spectralrange considered in this study are given in Table 1.

The complex refractive index of ceria at 950 ◦C is taken from Patsalas et al. [19] for the spectralrange 0.29–1.5µm. It is observed from this data that ceria is non-absorbing in the near to farinfrared spectral ranges, and strongly absorbing for wavelengths less than λ ≈ 700 nm. Thereal part of the complex refractive index is wavelength independent in the near to far infraredrange. This is consistent with numerous experimental data reported for ceria such as in [20, 21].

GOVERNING EQUATIONS

The electromagnetic theory is applied. Assuming linear constitutive models, and the relativepermeability equal to unity, Maxwell’s equations are given as

∇×∇× ~E− η20m2~E = 0 (1)∇×∇× ~H− η20m2 ~H = 0 (2)

where ~E and ~H are complex-valued electric and magnetic field vectors, respectively, and η0 isthe vacuum wave number. The complex refractive index is given bym2 = (n−ik)2 = �rel − iσ/ω�0,where �rel is the relative permittivity of the material, ω is the angular frequency of radiation, σis the electrical conductivity of the material, and �0 is the vacuum permittivity. We assume atime-harmonic (or quasi-steady) field of constant frequency such that ~E(~r, t) = ~E(~r) exp(iωt).At an interface between two materials, indicated by subscripts 1 and 2, boundary conditionsenforce the normal and tangential components of the fields to be equal.

n̂× (~E1 − ~E2) = 0 (3)n̂ · (~E1 − ~E2) = 0 (4)

n̂× (~H1 − ~H2) = 0 (5)n̂ · (~H1 − ~H2) = 0 (6)

where we have assumed the surface charge density and surface current density are zero. Ra-

4

diative properties of the particle will be related to the time-averaged Poynting vector givenby

~S = 12(~E x ~H∗

)(7)

where ~H∗ is the complex conjugate of the magnetic field. Equation (7) represents the time-averaged flux of electromagnetic energy in W m−2. The domain is excited by a plane wavepropagating in the x-direction with the electric field only having a z-component (perpendicu-larly polarized) given by

~E = Ezẑ exp(−η0ix) (8)where Ez and ẑ are the electric field component and the unit vector in the z-direction, respec-tively.

SOLUTION METHODS

The DDA and FEM are used to solve Maxwell’s equations, (1)–(2), and accurately account forthe complex 3DOM structure of the ceria particle. The DDA is known to produce inaccurateresults for scattering targets with large |m| as shown in the work of Yurkin et. al [22]. Ceriareaches a maximum |m| ≈ 3.4 in the spectral range considered in the present study, resultingin an anticipated relative error in Qext as high as 105%. Therefore, We employ the FEM toprovide a complementary solution. Given the expense and complexity associated with theDDA and FEM solutions, we also consider the approximation of 3DOM ceria particles ashomogeneous spheres, with effective properties given by volume averaging theory, and applyLorenz–Mie theory as a computationally inexpensive approach to obtaining radiative propertiesof the 3DOM ceria particle. Details on each numerical method and expressions used to recoverspectral radiative properties are given in the following text.

Discrete dipole approximation The DDA can be viewed as discrete solution method of theintegral form of Maxwell’s equations. More precisely, the DDA subdivides the target into cubicsub-volumes and models each sub-volume as a dipole point. The points acquire dipole momentsin response to the local electric field. They interact with each other through the electric fields.As for all discrete approaches of a continuum problem, the accuracy of the DDA depends on thechoice of the discretization. The smaller the dipole spacing, the more accurate the results. TheDDA is attractive due to its ability to easily handle non-homogeneous and anisotropic targetsof arbitrary geometry. The method faces computational limits in the presence of targets withlarge relative refractive index, size, and/or irregular boundaries.

Several implementations of the DDA exist as reviewed by Penttilä et al. [23]. DDSCAT [24–26]and ADDA [22] are popular open-source programs. In this work, DDSCAT is employed sinceit has been demonstrated to be more accurate than ADDA even though it is computationallymore expensive than the latter [23]. Moreover, DDSCAT is highly portable and modifiable,able to automatically generate a number of standard target shapes, and offers the option ofsupplying a list of occupied lattice sites to describe any desired target geometry, making itattractive for non-homogeneous targets. The inputs to DDSCAT are the list of dipole locationsand the refractive index nj − ikj for each dipole j, j = 1 to N , as well as the parameters forcontrolling convergence, the target orientation or the incident wave direction, and the desiredoutput data such as the Mueller scattering matrix components.

A spherical particle consisting of dipoles arranged in a cubic lattice of parameter d is generatedfirst. Next, a cubic arrangement of spherical pores in the FCC lattice is obtained. The size ofthe cube is taken larger than the particle diameter to locate one of the pores at the basis ofthe FFC lattice at the center of the particle to ensure symmetry of the porous structure. Theresulting dipole representation of the 3DOM ceria particle can be seen in Figure 3.

5

(a) (b) (c) (d)

Figure 3: DDA geometrical representation of the model 3DOM ceria particle for (a) θ = 0 andφ = 0, (b) θ = 22.5 and φ = 0, (c) θ = 45 and φ = 0, and (d) θ = 45 and φ = 45.

The validity condition of the DDA (but not its accuracy) is now established. The discretedipole spacing should be small as compared to any structural length in the target geometry,and the radiation wavelength λ [25]. These criteria are satisfied if

|m|ηd < 0.5 (9)

where m is the relative complex refractive index of the target material with respect to the hostmedium. The dipole spacing is then obtained as the minimum in the entire spectral rangeconsidered in this study,

d <1

2max (|m|η) (10)

For ceria as the target material and air as the host medium, the dipole spacing is selected asd = 0.008µm, resulting in 125 discrete dipoles along the particle diameter.

By substituting the continuum target as a finite array of N dipoles, each one is located atposition ~rj where j ∈ {1, 2, 3, ..., N}, the solution to a scattering problem can be found bysolving the local electric field ~Ej for each dipole j [24]:

~Ej = ~Einc,j +∑m6=j

~Em (11)

where ~Einc,j is the incident electric field vector on the dipole j. The quantity ~Em = −Ajm~Pm isthe electric field vector from the dipole m, in which ~Pm is the dipole moment vector and Ajmis a 3 × 3 complex symmetric matrix constituted of the wavenumber η = 2π/λ and the spacevector separating the dipoles j and m, namely ~rjm = ~rj−~rm. It can be shown that the solutionto the scattering problem is reduced to solving a system of 3N complex linear equations with3N unknown dipole moments [24]:

N∑m=1

Ajm~Pm = ~Einc,j (12)

Once Eq. (12) is solved, the calculation of complete scattering quantities is straightforward

6

(a) (b) (c) (d)

Figure 4: FEM geometrical representation of the model 3DOM ceria particle for (a) θ = 0 andφ = 0, (b) θ = 22.5 and φ = 0, (c) θ = 45 and φ = 0, and (d) θ = 45 and φ = 45.

[24, 25, 27]. The extinction, absorption and scattering cross sections are obtained from:

Qext =4η

E2incr2p

N∑j=1

Im(E∗inc,jPj) (13)

Qabs =4η

E2incr2p

N∑j=1

{Im[Pj(α−1j )∗P ∗j ]−

23k

3P 2j

}(14)

and

Qsca =η4

E2incπr2p

∫4π

dΩ∣∣∣∣∣∣N∑j

[~Pj − n̂(n̂ · ~Pj)

]exp(−iηn̂ ·~rj)

∣∣∣∣∣∣2

(15)

where Einc is the amplitude of the incident electric field, αj is the polarizability of the dipolej, and the integration takes place over the solid angle dΩ corresponding to the unit vector n̂.Finally, the asymmetry parameter can be recovered by

g = η4

E2incQscaπr2p

∫4π

n̂ · x̂∣∣∣∣∣∣N∑j

[~Pj − n̂(n̂ · ~Pj)

]exp(−iηn̂ ·~rj)

∣∣∣∣∣∣2

dΩ (16)

where we have x̂ since the incident plane wave is in the x-direction

Finite element method The FEM is a powerful and general method of solving partialdifferential equations. Unlike DDA, FEM has no limitation to material parameters. Themethod is based on the spatial decomposition of the solution domain into small finite elements,allowing for precise geometry representation through computer-aided drafting software. FEMgeometrical representations of the model 3DOM ceria particle is shown in Figure 4.

For this study, we have employed a commercially available FEM package, COMSOL 4.3. FEMcan be computationally very expensive in the case of optical studies. Around 6 elements perwavelength in each spatial direction is recommended to properly resolve the wave solution [28].

The application of the non-reflecting, absorbing boundary conditions is important to the accu-racy of electromagnetic scattering calculations. The perfectly matched layer (PML) approach,developed by Berenger [29], allows for the convenient geometry representation and conservationof matrix sparsity in the finite element paradigm making it the proper choice for the proposedstudy. The perfectly matched layer introduces a layer of elements around the domain understudy. The layer was chosen to be 5–6 elements across as recommended in [30]. In addition toimplementing a spherical PML, we have applied a simple first-order absorbing condition [28]

7

given byn̂×

[∇×

(~Etot + ~Einc

)]− iη0n̂×

(~Etot × n̂

)= 0 (17)

This condition perfectly absorbs waves at normal incidence making the PML more effective.

In order to introduce an incident plane wave with a desired wavelength into the numericalmodel with the existence of this artificial layer, the field variables are split into relative andincident fields [28],

~Etot = ~Erel + ~Einc (18)where ~Etot is the total electric field, ~Einc is the incident plane wave propagating through amedium without scatterers, and ~Erel is the electric field resulting from interactions with scat-terers. Using the split field formulation, (18), the radiative properties are obtained in terms ofthe relative and total fields. The absorption efficiency factor is calculated by volume-integratingover a sphere encompassing the 3DOM ceria particle,

Qabs =1

Sincπr2p

∫VσE2tot dV (19)

where σ is the electrical conductivity of ceria and Sinc is the magnitude of the Poynting vectorof the incident radiation given by Sinc =

√�0/µ0E0/2.

The scattering efficiency factor is calculated by integrating over the surface of a sphere con-taining the particle,

Qsca =1

Sincπr2p

∫Γ~Srel · n̂ dΓ (20)

Since the PML is known to be a poor absorber of evanescent waves [31], the radius of theintegration sphere is chosen to be a wavelength larger than the radius of the model particle.The extinction efficiency factor is then obtained from

Qext = Qabs +Qsca (21)

The asymmetry factor is calculated by

g =∫

Γ~Srel · n̂ (n̂ · x̂) dΓ∫

Γ~Srel · n̂ dΓ

(22)

where the surface integral is now defined to be in the far-field zone of the scatterer.

Lorenz-Mie theory Lorenz-Mie theory is an exact analytical solution to Maxwell’s equationsfor the problem of a homogeneous sphere in a non-absorbing background subject to an impingingplane wave. The solution depends on the particle radius, wavelength of incident radiation, andthe complex index of refraction of the particle and background. For the properties under studyin this exposition, the expressions are given by [32],

Qsca =2x2

∞∑n=1

(2n+ 1)(|an|2 + |bn|2) (23)

Qext =2x2

∞∑n=1

(2n+ 1)R(an + bn) (24)

Qabs = Qext −Qsca (25)

g = 4x2Qsca

∞∑n=1

{n(n+ 2)n+ 1 R

(ana

∗n+1 + bnb∗n+1

)+ 2n+ 1n(n+ 1)R (anb

∗n)}

(26)

8

where

an =Ψ′(mx)Ψ(x)−mΨ(mx)Ψ′(x)Ψ′(mx)ζ(x)−mΨ(mx)ζ ′(x) (27)

bn =mΨ′(mx)Ψ(x)−Ψ(mx)Ψ′(x)mΨ′(mx)ζ(x)−Ψ(mx)ζ ′(x) (28)

The functions Ψn and ζn are Riccati-Bessel functions, R() denotes the real part of its argue-ments, and x is the size parameter defined as x = 2πrp/λ.

Studies of the validity of various effective medium theories in the radiative characterizationsof particles are widely reported in the literature [12, 16, 17, 33]. Existing results suggest thateffective medium theories can be valid for a wide range of porosities and size parameters whenthe particle inclusions are in the Rayleigh limit. Such a result does not include the entirespectral range under study. As such, we expect that this effective medium approach to givelarge errors in the optical and even near infrared range. However, we consider this approachfor the purpose of exploration of its applicability to highly-ordered porous particles as well asto provide a reference solution to highlight the change in particle radiative behavior due tothe existence of the ordered pores. Following the studies analyzing the application of effectivemedium theories to radiative characterization of ordered porous thin films [34–36], we applythe volume averaging theory (VAT) developed in [37, 38] to Maxwell’s equations, (1)–(2) toobtain the effective complex index of refraction meff = neff − keff,

n2eff =12(A+√A2 +B2

)(29)

k2eff =12(−A+

√A2 +B2

)(30)

where

A = p(n2pore − k2pore

)+ (1− p)

(n2CeO2 − k

2CeO2

)(31)

B = 2nporekporep+ 2nCeO2kCeO2 (1− p) (32)

The effective complex index of refraction is used as an input to the Lorenz–Mie calculationsto obtain the radiative efficiency factors to be compared to those obtained using the DDA andFEM calculations.

RESULTS

Simulations were run using DDA in the spectral range 290–6000 nm at 310 nm increments and6–10µm at 1000 nm increments. This corresponds to a range of particle size parameters from0.314 to 10.833. FEM calculations were carried out at four wavelengths (size parameters) ofinterest, the endpoints of the DDA study, 290 nm (0.314) and 10, 000 nm (10.833) as well as nearthe peak of the solar spectrum, 510 nm (6.160), and near the peak in the emission spectrumcorresponding to an expected typical temperature of a solar thermochemical reactor, 2000 nm(1.571). It is acknowledged that for a result capturing the potentially intricate fluctuatingbehavior typical for a scattering characteristics analysis, a higher resolution scan of the spectrumis necessary. However, even without a high spectral resolution, noteworthy conclusions aredrawn.

Orientation-averaged results The 4 particle orientations found in Table 1 where averagedto obtain the radiative properties shown in Figure 5. The 25-direction orientational averagingseen in Figure 5 is performed by considering 5 angular intervals between 0◦ to 45◦ for θ and 5

9

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

x

Qab

s

Lorenz-Mie, mVATLorenz-Mie, mCeO2

DDA Avg. (4 Orientations)DDA Avg. (25 Orientations)FEM Avg. (4 Orientations)

0.00

1.13

2.27

3.40

4.54

5.68

6.82

8.07

9.37

10.6

611

.9413

.4314

.9516

.6118

.6921

.0223

.8927

.5131

.7539

.2646

.6539

.2932

.47

2x|m− 1|

(a)

0 2 4 6 8 100

2

4

6

x

Qsca

0.00

1.13

2.27

3.40

4.54

5.68

6.82

8.07

9.37

10.6

611

.9413

.4314

.9516

.6118

.6921

.0223

.8927

.5131

.7539

.2646

.6539

.2932

.47

2x|m− 1|

(b)

0 2 4 6 8 100

2

4

6

x

Qext

0.00

1.13

2.27

3.40

4.54

5.68

6.82

8.07

9.37

10.6

611

.9413

.4314

.9516

.6118

.6921

.0223

.8927

.5131

.7539

.2646

.6539

.2932

.47

2x|m− 1|

(c)

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

x

g

0.00

1.13

2.27

3.40

4.54

5.68

6.82

8.07

9.37

10.6

611

.9413

.4314

.9516

.6118

.6921

.0223

.8927

.5131

.7539

.2646

.6539

.2932

.47

2x|m− 1|

(d)

Figure 5: Orientation-averaged spectral radiative properties of the 3DOM ceria particle: (a)absorption efficiency factor, (b) scattering efficiency factor, (c) extinction efficiency factor, and(d) asymmetry factor.

angular intervals between 0◦ to 45◦ for φ. Very small differences in the predicted properties areobserved for 4-orientation average and the higher angular resolution. At an example wavelengthof 383 nm, averaging with 5 and 9 angular intervals for φ and θ, respectively, leads to thedifference of less than 0.5%.

The scattering and extinction efficiency factors increase monotonically for particle size parame-ters from 0.5 to 5.5, corresponding to increasing values of the parameter 2x|m−1| from 1.15 to14. For size parameters from 5.5 to 8.25 corresponding to 2x|m−1| from 14 to 25, the scatteringand extinction efficiency factors are maximum with some fluctuations known as the interferencestructures due to interference between diffracted waves and transmitted waves through the par-ticle. It is interesting to note that the 3DOM ceria particle reduces the range of the interferencestructure to values of 2x|m − 1| between 14 to 25 (here for wavelengths between 375 nm and550 nm, which are comparable to the pore size). Recall that the location of the interferencestructure for a homogeneous particle (shown in Figure 5) is generally for 2x|m − 1| from 4

10

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

x

Qab

sLorenz-Mie, mVATDDA, θ = 0, φ = 0DDA, θ = 45, φ = 0DDA, θ = 45, φ = 45DDA, θ = 22.5, φ = 0FEM, θ = 0, φ = 0FEM, θ = 45, φ = 0FEM, θ = 45, φ = 45FEM, θ = 22.5, φ = 0

(a)

0 2 4 6 8 100

1

2

3

x

Qsca

(b)

0 2 4 6 8 100

1

2

3

4

x

Qext

(c)

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

x

g

(d)

Figure 6: Spectral radiative properties of the 3DOM ceria for various particle orientations: (a)absorption efficiency factor, (b) scattering efficiency factor, (c) extinction efficiency factor, and(d) asymmetry factor.

(at wavelength 1800 nm here) to 24. Above this value, the interference structure disappearsbecause the particle becomes totally absorbing. More precisely, for 2x|m − 1| between 4 to14, the diffracted waves and waves crossing the particles either through pore channels and/orthrough solid interfere destructively. At the limit of large particle 2x|m− 1| > 30, typically forwavelengths less than 350 nm, the DDA and FEM results converge to the asymptotic result fora large particle corresponding to the radiative properties: Qext = 2, Qabs = 1 and Qsca = 1.

Effect of particle orientation The orientational dependence of the optical characteristicsis shown to be important in the range of the aforementioned interference structures as shownin Figure 6. This behavior could prove to be important for the particles being incorporatedinto a reactive flow as in a solar thermochemical reactor since the flow could cause a preferredorientation of the particles. Indeed, it has been shown that for groups non-spherical andnon-homogeneous particles, radiative properties can significantly differ from those of a volumeequivalent sphere, even if the particles are randomly oriented [39]. For values of 2x|m − 1|less than 3.6, the standard deviation between orientational values of all radiative propertiesconsidered in this study is less than 0.002.

The FEM predicted properties show satisfactory agreement with those obtained using the

11

2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,0000

0.1

0.2

λ, nm

Qext

Lorenz-Mie, mVATDDA Avg. (4 Orientations)FEM Avg. (4 Orientations)

3.57

2.85

2.38

2.04

1.78

1.58

1.43

1.30

1.19

1.10

1.02

0.95

0.89

0.84

0.79

0.75

2x|m− 1|

Figure 7: Extinction efficiency factor of 3DOM ceria particles in the near infrared range. It canbe seen that the predictions made using VAT with Lorenz-Mie theory converge to the DDAand FEM predictions in this range.

DDA. Asymmetry parameter calculations at short wavelengths demonstrate the computationalchallenges of the FEM approach as a very high spatial resolution is needed to obtain theproperties for even the homogeneous particle. The maximum relative error between DDA andFEM results of about 10% occurs in this spectral range. This error of DDA is expected tooriginate from the inaccurate modeling of pore edges, which impact the interference pattern,and can be addressed by reducing the dipole spacing.

It is noteworthy that there is no strong distinction between the radiative properties of particlesoriented such that the window features of the 3DOM ceria are in line with the incident waveand the properties of particles oriented differently.

Lorenz-Mie theory For incident wavelengths much greater than pore diameter, the structureof the 3DOM ceria particle is anticipated to have little effect on the radiative properties obtainedusing the FEM and DDA approaches. In this case, the properties become consistent withthose obtained using VAT with Lorenz-Mie theory. For the absorption efficiency factor andasymmetry parameter, VAT gives accurate predictions up to particle size parameters of 7.8and 4 (λ = 400 nm and λ = 775 nm), respectively. This is a much larger range of validity thanexpected. The extinction and absorption efficiency factors were found to be accurately predictedusing VAT with Lorenz-Mie theory for wavelengths greater than 5000 nm. Convergence to theVAT prediction for the extinction efficiency factor is shown in Figure 7 where the results forthe extinction efficiency factor at long wavelengths are shown to detail the small size parameterregion shown in Figure 5c. The effective medium theory given by VAT, however, does notaccurately capture the scattering and absorption efficiency factors of 3DOM ceria particleswhen the interference structure between diffracted and transmitted waves through the particleis present. This is consistent with previous studies which have confirmed the validity of effectivemedium approximations only for pore size in the Rayleigh scattering limit. The Effectivemedium theory does, however, accurately predict the occurrence of the interference. That is tosay, Lorenz-Mie theory along with VAT does provide qualitative agreement with orientation-averaged radiative transfer quantities.

Numerical validation The DDA and FEM simulations were verified by comparing the scat-tering and absorption efficiency factors as well as the asymmetry parameter for the case of ahomogeneous sphere with those obtained using the exact solution given by Lorenz–Mie theory.A comparison at wavelengths of 300 nm, 500 nm, 1000 nm, and 10000 nm showed that (i) theFEM simulation match accurately the exact results confirming the correctness of its implemen-tation, and (ii) the DDA is satisfactory with a maximum error of approximately 3% at 500 nm.To check for convergence of the FEM simulations, solutions at successive mesh refinements werecompared. Relative errors of the predicted radiative properties were found be less than 5%.

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The iterative solver was considered converged when the relative tolerance reached 10−3. ForDDA this value was set at 10−5.

SUMMARY AND CONCLUSION

Radiative characteristics of 3DOM cerium dioxide particles have been computed using twonumerical approaches, the discrete dipole approximation and the finite element method. Theparticle characteristics were found to be strongly dependent upon the orientation at smallwavelengths comparable to the pore size at which (i) the solar radiation power is maximumand (ii) the role of interference phenomenon seems very important. Scattering efficiency wasfound to be over 5 times smaller than for the solid ceria particle case. The incorporation ofordered overlapped pores within the 1000 nm ceria particle cancels the incident wave extinctionfor wavelengths greater than 560 nm. A spherical particle made up of an effective medium basedon volume averaging theory was also considered as computationally economical alternative. Thescattering problem was then solved using the Lorenz–Mie theory and found to give excellentagreement in the scattering, absorption, and extinction efficiency factors for wavelengths greaterthan 5000 nm–five times the diameter of the particle. Poor quantitative predictions were foundat lower wavelengths but the approximation still exhibited behavior analogous to that calculatedfrom FEM and DDA.

Acknowledgements This work was partially supported by the University of Minnesota’sGrant-in-Aid of Research, Artistry and Scholarship Program. From the University of Min-nesota, we thank Professor Andreas Stein, Krithiga Ganesan, and Lance Wheeler for discus-sions of the 3DOM ceria fabrication process and properties, SEM images of ceria samples, andaiding in figure creation, respectively. We also thank Prof. Dominique Baillis from LaMCoS,INSA of Lyon for discussions of radiative property modeling for heterogeneous media. Thecomputer grants and technical support by the Minnesota Supercomputing Institute is great-fully acknowledged. Jaona Randrianalisoa is grateful for the partial financial support from theGRESPI Laboratory of the University of Reims.

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