Power-law scaling of thermal conductivity of highly porous ceramics

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ScienceDirect Journal of the European Ceramic Society 35 (2015) 1933–1941

Power-law scaling of thermal conductivity of highly porous ceramics Ch. Pichler ∗ , R. Traxl, R. Lackner Material Technology Innsbruck (MTI), University of Innsbruck, Technikerstraße 13, A-6020 Innsbruck, Austria Received 27 August 2014; received in revised form 15 November 2014; accepted 2 December 2014 Available online 13 January 2015

Abstract In this paper (i) literature data on the thermal conductivity of (highly) porous ceramics and (ii) related analytical modelling schemes based on the effective media theory are reviewed. The latter do not seem to satisfactorily represent the experimentally observed quadratic power-law scaling of effective thermal conductivity with respect to the volume fraction of the solid material phase (or relative density). In the analytical/modelling part, two microstructural configurations of the conducting and the non-conducting material phase, respectively, are investigated: (i) thin, interconnected spherical shells of the conducting material in a simple cubic arrangement and (ii) regular and statistically random packing of solid, conducting spheres. These microstructural configurations differ from the arrangement used for derivation of classical homogenization schemes based on the effective media theory, i.e. non-conducting pores in a conducting solid matrix. © 2014 Elsevier Ltd. All rights reserved. Keywords: Heat transfer; Porous media; Homogenization; Unit-cell modelling

1. Introduction

relevant in the high temperature regime) are not accounted for and Fourier’s law of heat conduction,

The proper understanding of morphology vs. property relations of porous materials is vital for their optimization and application-based design. Here we focus on the thermal conductivity of porous ceramics. These materials are generally characterized by a large contrast between conductivity of the solid material matrix (with volume fraction fm and (isotropic) thermal conductivity km in the order of 1×100 –1×101 W/(mK)) and the gas saturated porous space (volume fraction φ, hence fm = 1 − φ, thermal conductivity kφ ≈2.5×10−2 W/(mK) for air).a Thus, as regards modelling, the conductivity of the gaseous phase kφ is usually neglected, i.e. set to zero, see e.g.1 We further limit our considerations to conductive heat transfer, hence convection (which is relevant for characteristic pore sizes of >∼10 mm) and radiative heat transfer (what may be

q = −k∇T,

∗

Corresponding author. Tel.: +43 512 507 63527; fax: +43 512 507 63502. E-mail address: [email protected] (Ch. Pichler). a In the literature f is frequently termed “relative density”, f = ρ /ρ , with m m eff m ρm denoting the density of the solid material matrix and ρeff the bulk density of the porous material. http://dx.doi.org/10.1016/j.jeurceramsoc.2014.12.004 0955-2219/© 2014 Elsevier Ltd. All rights reserved.

(1)

with q [W/m2 ] denoting the heat flow vector and ∇T [K/m] the temperature gradient, is employed in the theoretical part of this paper. 1.1. Modelling of effective behavior of porous ceramic materials As regards determination of the effective thermal conductivity keff (characterizing ceramic behavior at the engineering scale of observation) based on the intrinsic properties of the constituents and a microstructural representation, numerical methods have been employed (allowing a realistic representation, e.g. based on micrographs, see e.g.2 ) On the other hand, analytical methods have been used widely, e.g. fractal representation of the porous space, see e.g.3–5 or schemes falling in the class of effective media theory – continuum micromechanics. An excellent review of the latter methods, specialized for thermal conduction in porous materials, can be found in Ref. 1. Due to analogies in the underlying field and constitutive equations, the application of continuum micromechanics schemes

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Nomenclature (2a) A cm cair d D Deff fm h k keff km kφ M n nDS nMT nexp N q q Q R (2R) r s t t1/2 T ∇T T V z α, β δ φ ξ ξ ρ air ρeff ρm

diameter of contact region [m]; cross sectional area of ringlike cross section of thin shell [m2 ]; heat capacity of solid material phase [J/(kg K)]; heat capacity of air [J/(kg K)]; sample thickness [m]; thermal diffusivity [m2 /s]; thermal diffusivity of porous material [m2 /s]; volume fraction of solid material phase [–]; shell thickness [m]; thermal conductivity [W/(mK)]; effective thermal conductivity of porous material [W/(mK)]; thermal conductivity of solid material phase [W/(mK)]; thermal conductivity of gaseous material phase [W/(mK)]; molar mass [kg/mol]; power-law exponent [–]; power-law exponent related to differential scheme [–]; power-law exponent related to Mori-Tanaka scheme [–]; power-law exponent determined experimentally [–]; coordination number, number of contact points [–]; heat flow vector [W/m2 ]; heat flux in circumferential direction [W/m2 ]; total heat flowing through unit cell [W]; radius of middle surface of thin shell [m]; diameter of solid sphere [m]; radius of ringlike cross section of thin shell [m]; arc coordinate, arc length [m]; time [s]; half time in laser-flash measurements [s]; temperature [K]; temperature gradient [K/m]; temperature difference [K]; volume of unit cell [m3 ]; spatial coordinate, direction of macroscopic heat flow [m]; angles in unit-cell model [rad]; parameter in analysis of laser-flash measurements [–]; porosity, volume fraction of gaseous material phase [–]; dimensionless spatial coordinate [–]; dimensionless spatial coordinate of constricting section of shell [–]; density of air [kg/m3 ]; bulk density of porous material [kg/m3 ]; density of solid material phase [kg/m3 ];

(ρc)

volume heat capacity [J/(m3 K)]; thermal resistance of one contact point [K/W].

is not restricted to determination of mechanical properties of composite materials (see e.g.,6–8 for application to ceramics see e.g.9,10 ), they are also applicable (and have indeed been developed earlier) for determination of various effective transport properties: heat conduction, see e.g.,11 ionic diffusion in porous media, see e.g.,12,13 , or electric conduction.14 . In this paper, the following classical homogenization schemes/bounds will be employed, with the listed equations giving the respective specialization for matrix/spherical pore morphology and zero thermal conductivity in the pore space, see also1 : • Maxwell-Euken expression15 ≡ Hashin-Shtrikman upper bound16 ≡ Mori-Tanaka (MT) scheme17 for spherical inclusions; may be employed for matrix/inclusion-type morphologies with moderate values for the inclusion (pore) volume fraction of <∼0.2, i.e. fm > ∼0.8, with the effective thermal conductivity given as keff = km

2fm 3 − fm

(2)

• The differential scheme (DS)7,12,18–21 represents an infinitesimal formulation of the dilute distribution estimation, with the latter representing the situation where the pores are diluted in the matrix material and their interaction can be neglected. Starting with the homogeneous matrix material, the inclusion phase (spherical pores) is embedded into the matrix material in infinitesimal steps. After each of these steps, the behavior of the matrix phase is updated based on the dilute distribution estimation, finally leading to keff = km fm3/2 .

(3)

As opposed to the MT scheme, the DS has no restrictions as regards the pore volume fraction, hence, it may be a suitable homogenization scheme for highly porous materials. • Landauer’s expression14,18 (effective medium percolation theory) ≡ self-consistent scheme (SCS)22,23 : best-suited for microstructures with no distinct matrix/inclusion (or pore) morphology, e.g., for polycrystals; the model configuration is characterized by the consideration of the different material phases as inclusions embedded in the homogenized medium; 3fm − 1 for fm ≥1/3; 2 keff = 0 for fm < 1/3.

keff = km

(4)

Note that the SCS predicts nonzero effective thermal conductivity only above a percolation threshold of fm = 0.3, i.e. is not suitable for highly porous materials. A rather recent analytical modelling approach is the so-called functional equation approach,1,24–26 characterized by a virtual

Ch. Pichler et al. / Journal of the European Ceramic Society 35 (2015) 1933–1941

Fig. 1. Thermal conductivity of porous mullite ceramics at room temperature, comparison with models from effective media theory, data taken from Ref. 30 (circle, mullite ceramics), 31 (box, mullite ceramics, conductivity measurements at room temperature, laser flash method with thickness correction), 32 (star, mullite/corundum ceramics); for micromechanical modelling km was set to 5 W/(mK).

decomposition of total porosity φ into partial porosities, which are added in subsequent steps. For a spherical pore morphology this leads to an exponential equation   3/2(1 − fm ) . keff = km exp − (5) fm This paper is organized as follows: in the following section, experimental results for highly porous ceramics from the open literature are reviewed and compared to the prediction of the aforementioned classical homogenization schemes. In the analytical/modelling part (Section 3) two microstructural configurations for the arrangement of the conducting and the non-conducting material phase, respectively, differing from the arrangement used for derivation of the classical homogenization schemes (i.e. closed, spherical, non-conducting pores in a conducting solid matrix for the MT scheme and the DS) are investigated: 1. thin, interconnected spherical shells made of the conducting material phase in a simple cubic arrangement; 2. regular and statistically random packing of solid, conducting spheres employing analytical models derived in Refs. 27–29. 2. Review of experimental results Figs. 1–3 show a compilation of experimental data from the open literature focusing on the thermal conductivity of highly porous ceramics. E.g., when plotting results of porous mullite ceramics in a double-logarithmic diagram (Fig. 1), none of the

Fig. 2. Thermal conductivity of porous ceramics at room temperature. Data taken from Refs. 33–37.

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Fig. 3. Thermal conductivity of porous alumina ceramics, figure modified from Ref. 1, functional equation approach developed in Refs. 1,24–26.

classical homogenization schemes listed in Section 1 (MT, SCS, DS) is capable to successfully represent the data. For moderate porosity values, i.e. for fm > 0.4, the functional equation approach1,24–26 shows the best agreement with the experimental data. In this range, the functional equation approach constitutes a viable alternative to the SCS. As already pointed out in Ref. 26, experimental data on porous ceramic materials is usually found below the model prediction of the DS (power law with exponent 3/2). Considering the whole data range, 0.1 < fm ≤ 1, the data may be approximated by a quadratic power-law dependence with keff ∝ fm2 . km

(6)

As regards modelling, non-porous mullite conductivity was set to km =5 W/(mK), which compares well to measured room temperature values, e.g. to 4.7 W/(mK) reported in Ref. 31. For porous mullite ceramics, the proportionality factor in Eq. (6) is approximated as one, hence keff ≈ fm2 . km

(7)

This is also the case for porous zirconia data shown in Fig. 2, with km ≈2 W/(mK) as a representative value for the non-porous material. On the other hand, the latter amounts to km ≈ 33 W/(mK) for non-porous alumina Al2 O3 . With an approximation of the data for porous alumina as keff = 12fm2 [W/(mK)] (see Fig. 2), the proportionality factor in Eq. (6) follows as 12/33 = 0.36. The latter may be explained by the relatively large scatter of the (room temperature) test data and a possible difference between a single crystal value of km ≈ 35 W/(mK) and a polycyrstal value of ≈ 15 W/(mK) as suggested in Ref. 33, with the decrease caused by the interfacial thermal resistance of grain boundaries.33 When assigning the polycrystal value to km , the proportionality factor in Eq. (6) is close to one, also for highly porous alumina ceramics. As regards this so-called grain size effect on the value of km , the reader is referred to25 and references therein. The model response for the DS is characterized by a power-law dependence with an exponent of 3/2 (see Eq. (3)), being smaller than the one observed experimentally, i.e. nDS = 3/2 < nexp ≈ 2. For fm → 0, the response of the MT

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Ch. Pichler et al. / Journal of the European Ceramic Society 35 (2015) 1933–1941

Fig. 4. Laser-flash method for assessment of thermal diffusivity. Table 1 XRF spectroscopy results of raw material and corundum/glass ceramic samples, oxides in mass%. Oxide

Raw material

Ceramic

SiO2 Al2 O3 Fe2 O3 TiO2 MgO K2 O Na2 O Ignition loss

18.1 70.1 0.7 0.4 1.9 1.2 1.4 6.2

19.1 74.7 0.9 0.4 2.0 1.3 1.6 –

schemeb is also of power-law type with keff /km ∝ fm , i.e. nMT = 1. As regards the model response for the limit of small porosities, the slope in a double-logarithmic diagram (see Fig. 1) for all homogenization schemes discussed in the Introduction (MT, DS, SCS, functional equation approach)c is given as 3/2 for fm → 1. 2.1. Laser flash analysis of corundum-glass ceramic samples In addition to the experimental results taken from the literature, the thermal conductivity of porous ceramic samples was investigated (Steka-Werke Technische Keramik, Innsbruck). Hereby, disc-shaped specimens were obtained by compression molding. XRF spectroscopy results of the employed raw materials and the resulting ceramic after sintering at 1320 degC are given in Table 1. XRD gave access to mineral phases of the raw material, with corundum Al2 O3 as main phase, further quartz SiO2 , talc Mg3 Si4 O10 (OH)2 , kaolinite Al2 Si2 O5 (OH)4 , microcline/orthoclase KAlSi3 O8 , albite NaAlSi3 O8 , nepheline Na0.75 K0.25 Al(SiO4 ), dolomite CaMg(CO3 )2 , and diaoyudaoite NaAl11 O17 . The main phases in the sintered material, in the following referred to as corundum/glass ceramic, are corundum Al2 O3 and a glassy phase, further traces of quartz SiO2 and spinel MgAl2 O4 .

b

In a double-logarithmic diagram, the slope for the highly porous material with fm → 0 is given as ∂[ln keff ]/∂[ln fm ] = 3/(3 − fm )|fm →0 = 1. Hence, the asymptotic behavior for fm → 0 can be written as keff ∝ km fm . c ∂[ln k ]/∂[ln f ] for the MT scheme = 3/(3 − f )| eff m m fm →1 = 3/2, for the DS = 3/2, for the SCS = 3fm /(3fm − 1)|fm →1 = 3/2, and for the functional equation approach = 3/(2fm )|fm →1 = 3/2.

An increased porosity in the material (in addition to the inherent porosity of the corundum/glass ceramic) was established by the addition of cocoa powder (16 vol.%) prior to sintering. Laser flash analyzes were conducted for determination of thermal diffusivity D [m2 /s], relating thermal conductivity k [W/(mK)] to (volume) heat capacity (ρc) [J/(m3 K)] of a solid material as D=

k . ρc

(8)

The laser flash method is characterized by exposing one side of a disc-shaped plane-parallel sample with thickness d [m] to an energy pulse from a light source (laser or xenon flash lamp) and measuring the temperature history on the other side of the sample38–40 (see Fig. 4). From the solution of the one-dimensional, adiabatic heat-transport problem follows the thermal diffusivity as D=δ

d2 t1/2

with

δ = 0.139,

(9)

where t1/2 denotes the time till the temperature reaches the half of the final temperature rise. The higher D, the faster the supplied energy is dispersed in the sample. The samples had a diameter of 13.6 × 10−3 m and a thickness d =4.1×10−3 m, the densities ρeff are given in Table 2. The density of the matrix material (corundum/glass compound) was determined pycnometrically as ρm = 3084 kg/m3 (compare to density of corundum of 3950 kg/m3 ), giving the volume fraction of the matrix material as fm = ρeff /ρm . With the heat capacity of the matrix material being set to the one of corundum as cm = 760 J/(kg K) (see Ref. 41: value for T = 300K = 27◦ C; molar mass of Al2 O3 M = 0.102 kg/mol; 78 J/(mol K)/0.102 kg/mol = 760 J/(kg K)), the thermal conductivity of the porous sample is determined from Eq. (8) Table 2 Laser flash measurements on corundum/glass ceramic samples, ρm = 3084 kg/m3 determined pycnometrically, cm = 760 J/(kg K)41 (indices “0%” and “16%” refer to zero and 16 vol.% cocoa powder addition prior to sintering).

[kg/m3 ]

ρeff fm = ρeff /ρm [–] φ = 1 − fm [–] Deff [m2 /s] (laser-flash result) keff = Deff ρeff cm [W/(mK)]

Corundum/glass0%

Corundum/glass16%

2838 0.920 0.080 2.7×10−6 5.82

2298 0.745 0.255 1.6×10−6 2.79

Ch. Pichler et al. / Journal of the European Ceramic Society 35 (2015) 1933–1941

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Fig. 5. Thermal conductivity of porous corundum ceramics, data taken from Refs. 33,36 and of porous mullite ceramics, data taken from Ref. 30–32.

[neglecting the heat capacity of the air-filled porous space] as (see Table 2) keff = Deff (fm ρm cm + φρ air cair ) ρeff ρm cm = Deff ρeff cm , ≈ Deff ρm

|q| = q = −km (10)

fitting well within literature data on porous corundum and mullite ceramics (see Fig. 5). 3. Analysis

Contrary to the model configuration used for derivation of the homogenization schemes listed in Section 1 – closed, spherical pores embedded in a solid matrix – we will investigate a different configuration of the conducting and the non-conducting material phase, respectively: inter-connecting spherical shells in a simple cubic arrangement.d Hereby, consider the two connected spherical half-shells depicted in Fig. 6 as a unit-cell. With a volume of the latter of V = (2R)3 = 8R3 , the volume fraction of the solid material phase is given as 4πR2 h πh = , 3 2R (2R)

∂T , ∂s

(12)

with s denoting the arc coordinate. In a steady state situation, the total heat flowing circumferentially through the spherical shell is constant and given as |Q| = |q(s)|A(s) = km

∂T 2rπh = const. ∂s

(13)

In Eq. (13), r(s) denotes the radius of the considered ringlike cross section A(s) = 2r(s)πh. The total heat passing through the shell has to be equal to the total heat passing through the effective material,

3.1. Spherical-shell based unit-cell modelling

fm =

law (Eq. (1)) is specialized for axisymmetric (with respect to the z-axis), circumferential transport

(11)

where the volume of the two thin half-shells (with h R) is determined from their surface 2 × 2πR2 . Note that the resulting relation of type fm ∝ h/R is valid only for the considered configuration – interconnected thin spherical shells, with the anticipated outcome from the related unit-cell modelling being valid in the high porosity regime. As the thermal conductivity of the porous space is set to zero, no heat flux occurs perpendicular to the shell, Fourier’s

|Q| = keff

T (2R)2 = const., R

(14)

with 2T denoting the temperature difference between the upper and lower surface of the effective material (see Fig. 6(b). Hence, with a height of the unit cell of 2R the temperature gradient in the effective material is given as ∇T|eff = 2T/(2R) = T/R (with respect to the z-axis). The same temperature difference 2T is imposed in the unit-cell model. Writing z as a function of arc length s s = βR;

sin β = z/R;

→

z = R sin β = R sin

s R

(15)

allows rewriting the temperature gradient in the shell with respect to the z-axis, ∂T ∂T ∂z ∂T s ∂T ∂T r = = cos = cos β = . ∂s ∂z ∂s ∂z R ∂z ∂z R

(16)

Inserting Eq. (16) into Eq. (13) and replacing r2 by R2 − z2 gives r2 R 2 − z2 ∂T ∂T 2πh = km 2πh ∂z R ∂z R   2 ∂T z = km 2πhR 1 − 2 . ∂z R

|Q| = km d We have already employed such thin-spherical shell based unit cells for determination of effective material properties of highly porous materials (see42 for the modelling of elastic properties of closed-cell foams and43 for the modelling of plateau strength of closed-cell foams with elastoplastic behavior of the solid phase).

(17)

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Fig. 6. Spherical-shell based unit-cell model (simple cubic arrangement) for thermal conductivity.

Introducing the dimensionless coordinate ξ = z/R and ∂T/∂z = ∂T/∂ξ ∂ξ/∂z = ∂T/∂ξ 1/R gives |Q| = km

ξ=

∂T 2πh(1 − ξ 2 ), ∂ξ

(18)

what allows determination of the temperature difference along half of the height of the unit cell  0

ξ

∂T |Q| 1 dξ = T = ∂ξ km 2πh |Q| 1 artanhξ = km 2πh

with

 0

ξ

1 dξ 1 − ξ2 (19)

R cos α R 1 = cos α = = R R + h/2 1 + h/(2R)

(20)

referring to the cross section at the junction of the two half shells at r = r, see Fig. 6 (labelled “constricting section of shell”). Note, as opposed to the temperature distribution in the effective material, T(z) is nonlinear, with T(z) = |Q|/km 1/(2πh)artanh(z/R) (plus a constant, see Fig. 6). Comparing Eq. (19) with Eq. (14) gives keff πh/(2R) = . km artanh(1/(1 + h/(2R)))

(21)

Ch. Pichler et al. / Journal of the European Ceramic Society 35 (2015) 1933–1941

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Fig. 7. Prediction of unit-cell model. Table 3 Solid volume fraction fm , coordination number N and (dimensionless) effective thermal conductivity of regular isotropic packings of equal sized spheres with diameter (2R)28 .

Fig. 8. Dimensionless effective thermal conductivity of regular packings of equal sized spheres as a function of fm .

Finally, h/(2R) is replaced by the associated volume fraction fm , with h/(2R) = fm /π (see Eq. (11)), giving

localized in the contact region of the spheres.f In Ref. 28 these packings of equal sized spheres where modelled by a network of thermal resistances (contact points/regions between spheres) with the nodes connecting the resistances corresponding to the solid spheres. For the case of kφ = 0, i.e. the assumption considered for modelling in this paper, and contact points between solid spheres, =∞ in Eq. (23) gives keff = 0. However, when considering a circular contact region with diameter (2a) the heat-affected zone is still localized for a R and the network model of discrete resistances can be applied28 with

keff fm = . km artanh(π/(π + fm ))

=

Regular packing Tetrahedral Simple cubic Body-centered cubic Face-centered cubic

fm [–] √ π 3/16=0.340 π/6=0.524 √ π√3/8=0.680 π 2/6=0.740

N [–] 4 6 8 12

2R keff √ 3/4 = 0.433 1√ 3 = 1.732 √ 2 2 = 2.828

(22)

In Fig. 7 the prediction of the unit-cell model is compared to the prediction of the DS. In the highly porous range, e.g. for fm = 0.1, the unit cell response may be approximated by a power-law function keff /km ∝ fmn with an exponent of 1.237 and a proportionality factor of 0.830, i.e., keff /km ≈ 0.830 fm1.237 (see thick gray curve in Fig. 7).e The power-law exponent determined analytically significantly differs from the one observed experimentally (nexp ≈ 2). Hence, the shell-based unit cell model is not capable to capture the experimentally observed behavior. 3.2. Packing of solid, conducting spheres According to Ref. 28 the effective thermal conductivity of statistically isotropic random packing of solid spheres with diameter (2R) (but also of the regular packings listed in Table 3) is given as keff

fm N , = (2R)π

(23)

where N is the coordination number, i.e. the (average) number of contact points per solid sphere, [K/W] the thermal resistance of one contact point, and fm the volume fraction of the solid material phase. Eq. (23) was derived for materials with km  kφ , where heat flow in the gas phase is strongly nonuniform and

e In a double-logarithmic diagram, the slope is given as ∂[ln keff ]/∂[ln fm ] = 1 + π/[(2π + fm )artanh(π/(fm + π))]. Hence, for fm = 0 this gives 1; for fm = 0.1 the exponent in the power law approximation is 1.237.

1 . (2a)km

(24)

Inserting Eq. (24) into Eq. (23) gives keff a fm N = . km R π

(25)

As already noted in Ref. 28, a correlation between N and fm is probable. Fig. 8 shows the dimensionless effective thermal conductivity 2R keff = keff /km R/a = fm N/π as a function of fm . Considering the tetrahedral, simple cubic, and body-centered cubic packing (solid volume fraction from 0.340 to 0.680), leastsquares regression gives fm N/π ≈ 3.79fm2.04 (see dashed lineg in Fig. 8). Introducing this correlation in Eq. (25), a quadratic power law scaling relation is obtained: keff R fm N = ≈ 3.8 fm2 ; km a π

(26)

compare to keff /km ∝ fm2 , the scaling law observed experimentally (see Section 2).

f According to Ref. 28 for k /k > 700 −−1000, what is the case for metallic m φ (and many) ceramic powders in air, the overall heat transfer is controlled by the properties of the gas phase only, i.e. among others by kφ . For a detailed discussion on the dependence of on the adiabatic exponent of the gas and the Knudsen number see28 . g Linear scaling of N with f as introduced in Fig. 8 has also been obtained m by Ref. 44: the mean coordination number in random packings which are characterized by deviation from the regular simple cubic packing can be determined from φ = 1 − fm = 1 − π/36 N, see Fig. 2 in Ref. 44. Hence, N = 36/πfm = 11.5 fm . This is close to the scaling encountered in Fig. 8 and used in Eq. (26), i.e., N ≈ 3.8πfm = 11.9 fm .

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4. Concluding remarks In this paper, different microstructure-based models for porous materials with large contrast between thermal conductivity of the solid material phase and the porous space, respectively, were investigated. Many porous ceramics fall into this class of materials. For the experimental data reviewed in this paper, a quadratic power-law dependence of effective thermal conductivity with respect to the volume fraction of the solid materials phase was identified, i.e. a power-law exponent of nexp ≈ 2. As regards modelling, two microstructural configurations for the arrangement of the conducting and the non-conducting material phase, respectively, differing from the arrangement used for derivation of classical homogenization schemes based on the effective media theory (i.e. closed, spherical, non-conducting pores in a conducting solid matrix) were considered: i. A unit-cell model incorporating thin, interconnected spherical shells of the conducting material in a simple cubic arrangement. ii. An adapted version of the analytical model for regular and statistically random packing of solid, conducting spheres developed in Ref. 28. Both configurations lead to a power-law scaling relation in the high-porosity regime, with n ≈ 1.2 for the shell-based unit cell model and n ≈ 2 for the packed-spheres model. The differential scheme from the effective media theory, underlying a microstructure characterized by closed pores embedded in a continuous solid matrix, predicts a power-law exponent of n = 1.5. Hence, among the model configurations considered, the packedspheres model represents the experimentally observed behavior best. A syllogism when regarding raw material and manufacturing of ceramic materials, powders and sintering, respectively. Acknowledgments Financial support by the Austrian Research Promotion Agency Ltd. (FFG) for project “High-performance ceramics” (project number 838965) is gratefully acknowledged. The authors thank Markus Dax, Anne Reinisch and Martin Hörtnagl (Steka-Werke Technische Keramik, Innsbruck, Austria) and Heinrich Berthold (formerly at Steka-Werke, now at CoorsTek Advanced Materials, Lauf, Germany) for fruitful discussions within this research project and for providing ceramic samples. We further thank (i) Marco Baratieri, Francesco Patuzzi and Allesandro Prada (Free University of Bozen, Italy) for conducting laser flash measurements on these samples and (ii) Anja Diekamp and Andreas Saxer (University of Innsbruck) for the XRF and XRD analyses contained in this paper. References 1. Zivcova Z, Gregorova E, Pabst W, Smith DS, Michot A, Poulier C. Thermal conductivity of porous alumina ceramics prepared using starch as a poreforming agent. J Eur Ceram Soc 2009;29:347–53.

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