Calculation of the Average Coefficient of Thermal Expansion inOriented Cordierite Polycrystals

Giovanni Brunow

Corning SAS, CETC, F-77210 Avon, France

Sven Vogel

LANSCE, LANL, Los Alamos, 87545 New Mexico

In this work, the calculation of the average value of a physicalquantity in a textured polycrystal is presented. The method isapplied to the coefficient of thermal expansion in cordieritesamples, presenting domain and crystal preferred orientation,and compared with experimental data. The knowledge of theexperimental or simulated texture intensity function is requiredto calculate the orientation distribution function. Then, a sumover all oriented crystals, weighted by their population, iscarried out. It is shown that this sum must be carried outdifferently, if different components of the physical quantity(usually a tensor) must be calculated. Results show a verygood agreement between the model and the experimental dataobtained (a) by neutron diffraction as a polycrystalline averageand (b) by dilatometry on real cordierite materials used as dieselparticulate filters. Although the method is resting on thepossibility of having a simple analytical form of the textureintensity, its numerical implementation does not present anyproblem.

I. Introduction

THE calculation of the average value of any physical quantityf(c, y, j) in textured polycrystals is a subject already stud-

ied by several authors. In particular, Popa1,2 and Wang3 haveassessed in a rigorous way that if a polycrystal possesses an ori-entation distribution function (ODF, as defined by Bunge4)f(c,y,j) in Euler’s space, the correct way to calculate this aver-age is simply

fh i ¼Z Z Z

f c; y;jð Þf c; y;jð Þsiny dc dy dj (1a)

Equation (1a) supposes the normalization of the ODF to 1and that the polar angle is y, counted from the z (or 3)-axis.

In some previous works,5,6 the calculation of the residualstress in a textured polycrystalline metal–matrix composite wascarried out by means of a simplified, but novel, method. Thesimplifications introduced by Bruno and colleagues addressedthe physics of the system: invariance of the stress tensor withrespect to the reinforcement long axis and equi-orientation of allreinforcement particles. This method put forward a sensibleidea: the average of each component of a physical quantity (thestress tensor in those works) should have been calculated by thesum of the values of all crystals (or particles), weighted over

their number (population). This implied (a) the knowledge ofa normalized distribution of orientations, which could providethe weight for summation and (b) the presence of an invariantquantity for each crystallite. In that case, the tensor of the ther-mal mismatch strain in a SiC whisker embedded in the Al matrixwas indeed the same for each particle and was calculated by anEshelby model. The best way for the evaluation of this distri-bution was supposed to be the experimental determinationof the crystallographic texture, naturally by X-ray or neutrondiffraction. If the oriented objects can be described by a singleorientation (in the case of Fernandez et al.5 and Bruno andFernandez,6 whiskers with the long axis oriented along the/111Sdirection), a simple pole figure of this lattice directionwould give the appropriate weight.

In this work, this method has been applied to the determina-tion of the average coefficient of thermal expansion (CTE) ofcordierite-extruded polycrystals, which are clustering in prolatedomains with their long axis along the (001) axis of the ortho-rhombic cell. Neutron diffraction texture measurements havebeen used to evaluate the orientation distribution. This tech-nique has the enormous advantage of being virtually nonde-structive and anyway applicable to bulk components, thanksto the high penetration depth of neutrons into most materials.Therefore, the interior of pellet samples can be probed, avoidingthe problem of possible orientation gradients or surface effects.

The method described below and in the quoted referencescould be applied, because indeed the CTE of cordierite singlecrystallites can roughly approximate that of domains or grains(see Fig. 1). Moreover, if an average over directions a and b ofthe orthorhombic variant or alternatively the hexagonal variant(indialite) is taken, the CTE tensor in the grain coordinatesystem can be written as

a ¼aa 0 00 aa 00 0 aa

0@

1A (1b)

α 25-800(a) = +38.7 x 10−7 °C−1

α25-800(b) = +25.8 x 10−7 °C−1

α25-800 (c) = −12.9 x 10−7 °C−1

Fig. 1. Average values of a, b, and c thermal expansions in single-crys-tal cordierite between 251 and 8001C, as reported by Taylor (1988), and aschematic representation of the crystallites.

K. Bowman—contributing editor

This work was financially supported by the Lujan Neutron Scattering Center at LAN-SCE, which is funded by the U.S. Department of Energy’s Office of Basic Energy Sciences.

wAuthor to whom correspondence should be addressed. e-mail: brunog@corning.com

Manuscript No. 24362. Received February 29, 2008; approved April 6, 2008.

Journal

J. Am. Ceram. Soc., 91 [8] 2646–2652 (2008)

DOI: 10.1111/j.1551-2916.2008.02485.x

r 2008 The American Ceramic Society

2646

II. Calculation Procedure

Consider a cordierite-extruded sample. It is well known that thecordierite grains (crystals) are organized in domains, which arecollections of nearly equiaxed crystallites, all oriented with theircrystallographic (001) (or c) axes parallel to each other. Thesedomains in turn have their common c-axes roughly alignedalong the extrusion axis. However, their orientation scatterwith respect to this direction creates the difference observed be-tween the axial and radial CTE of a polycrystal (as for instanceshown in Fig. 2, data courtesy of G. Merkel, Corning Inc.) andthe ‘‘pure’’ c- and a-axes CTE (Fig. 2, as extracted from7).

When c-axes of the grains or of the domains (in the followingthe term grains will be used in a loose sense) are not perfectlyaligned with the extrusion axis, the reference systems of the sam-ple and those of each grain, obviously, do not coincide. In orderto calculate the orientation distribution density of the crystals(assumed as prolate ellipsoids) along axis 3 (see Fig. 3), let us as-sume that they are distributed according to a pseudofiber texture,i.e., with a cylindrical symmetry typical of extruded materials.

The diffraction intensity distribution of the 001 reflectionwould therefore be a simple Gaussian (as found in Borregoet al.8 for a 1D case) with width Dy

I ¼ 2pAexpð�wy2Þ þ I0 (2)

where w ¼ 1=2Dy2, A is a normalization constant, 2p is the in-tegration factor over the azimuthal angle, and I0 is the randompopulation signal. This distribution is shown in Fig. 4 forDy5 201.

Obviously, if there is no axisymmetry around axis 3, the dis-tribution would be truly two-dimensional and would depend onanother angle j (see Fig. 5). Therefore, the treatment we presentcan be easily extended beyond the assumption we make here.

The density of oriented domains (or crystals) can be calcu-lated by multiplying the volume element in the spherical coor-dinates by the intensity (Eq. (2)) and normalizing over the wholespace:

dr y;jð Þ ¼ B expð�wy2Þsinydy dj (3)

where B is a normalization constant and the reference system (y,j) has been used (see Fig. 5).

It must be noted that the normalization must be done to thefraction of oriented crystals Vf, not to 1, i.e., the normalizationconstant B is calculated according to

1

4p

Zdr ¼ 2B

p

Z p2

0

dyZ p

2

0

dj sin y e�wy2 ¼ Vf (4)

If oriented and random populations coexist (say r and r0,respectively), then the normalization condition reads as

1

4p

Zd rþ r0ð Þ ¼ 2B

p

Z p2

0

dyZ p

2

0

dj sin y e�wy2 þ r0

¼ 1 (5)

Therefore, a knowledge of the fraction of oriented grains(domains or particles in our case) will allow calculationof the normalization constant. Vf can be obtained simply bythe evaluation of the area (or volume for a truly 3D texture in-tensity distribution) subtended by the angular-dependentdiffraction signal and by the constant signal (seen as a back-ground).

Once we have the density of orientations, we can simply state,following Bruno and Fernandez,6 that the total CTE will begiven by the sum of the CTE of all single domains, weightedover their density of orientations.

This simple statement needs to be clarified: as mentionedbefore, the CTE tensor of each domain (in its own system ofreference) is diagonal, and we assume that it is isotropic inthe basal plane (see Eq. (1b)). Of course, this needs to be re-ferred to the sample coordinate system. If now the domain isoriented along the coordinates (y, j), a rotation to bring itback to the sample coordinate system would imply a firstrotation (of �j) around axis 3 and then another (of �y) aroundaxis 2.

The total rotation is given by the tensor product:

T ¼ T2 �að ÞT3 �jð Þ

¼cos y 0 � sin y0 1 0

sin y 0 cos y

0@

1A �

cosj � sinj 0sinj cosj 00 0 1

0@

1A (6)

Using Eqs. (1) and (6), the CTE tensor, rotated from the grainto the sample coordinate system can be calculated by

−2000

−1000

0

1000

2000

0 100 200 300 400 500 600 700 800 900

Temperature, °C

Str

ain

, 10

−6

Cellular axialCellular radialLee & Pentecost- aLee & Pentecost- c

Fig. 2. The measured a and c thermal expansion in single crystalcordierite (Lee), and the macroscopic average obtained in a cellularbar (Merkel) in the extrusion axis (axial direction) and perpendicular toit (radial direction).

3

1 2

Fig. 3. Sketch of the distribution of the domains (indicated as blackprolate ellipsoids) in a cellular and in a rod cordierite sample.

a0 ¼ TTaT ¼

aa cos2 y cos2 jþ aa sin2 j

þac sin2 y cos2 j aa � acð Þ sin2 y sinj cosj ac � aað Þ sin y cos y cosj

aa � acð Þ sin2 y sinj cosjaa cos2 y sin

2 jþ aa cos2 jþac sin2 y sin2 j ac � aað Þ sin y cos y sinj

ac � aað Þ sin y cos y cosj ac � aað Þ sin y cos y sinj aa sin2 yþ ac cos2 y

0BBBBBBBB@

1CCCCCCCCA

ð7Þ

August 2008 Calculation of Average CTE in Oriented Cordierite Polycrystals 2647

where TT is the transposed rotation matrix, given by

TT ¼cos y cosj sinj sin y cosj� cos y sinj cosj � sin y sinj� sin y 0 cos y

0@

1A (8)

At this point, the average component of a0 along axis 3 can becalculated as a weighted average:

azh i ¼2B

pVf

Z p2

0

djZ p

2

0

dy aa sin2 yþ ac cos2 y

� �sin y e�wy

2

(9)

This integral can be discretized into a sum over a finite num-ber of angles yi:

azh i ¼B

Vf

Xi

Dyi aa sin2 yi þ ac cos2 yi

� �sin yie�wy

2i

h i

(10)

where the integral over j is just a factor and Dyi is the integra-tion step.

Analogously, the component of a0 along axis 1 can simplybe calculated using Eq. (7), i.e., the component of a0 alongaxis 1

axh i ¼B

2pVf

Z 2p

0

dfZ p

2

0

dyðaa cos2 y cos2 fþ aa sin2 f

þ ac sin2 y cos2 fÞ sin y � e�wy2

(11)

Although the domain/grain orientation distribution does nothave cylindrical symmetry around axis 1, the integral over thewhole space can be calculated taking the same system of refer-ence as in the preceding case.

The j dependence in integral (11) is purely a multiplying factor

for each term. Taking into account thatR 2p0 dj cos2 j=R 2p

0 dj sin2 j ¼p, the component along axis 1 will become

axh i ¼B

2Vf

Z p2

0

dyðaa cos2 yþ aa þ ac sin2 yÞ

� expð�wy2Þ sin y(12)

This integral can be discretized into

axh i ¼BDy2Vf

Xi

expð�wy2i Þ sin yi½aað1þ cos2 yiÞ

þ ac sin2 yi�

(13)

III. Neutron Diffraction Texture Measurements

Two samples underwent neutron diffraction texture measure-ments: a solid cordierite rod and a cellular bar. Both ofthem were cold extruded with the same processing parametersand then fired with the same firing schedule, i.e., they are basicallythe same material with different shapes and different apparentdensities.

The texture measurements were carried out on the instrumentHIPPO at LANSCE, Los Alamos, NM.9 See Fig. 6 for a sketchof the instrument.

HIPPO is a time-of-flight powder diffractometer that canachieve very high neutron count rates because of its vicinityto the high-intensity water moderator. Moreover, it features1360 3He detector tubes covering an area of 4.8 m2 and an an-gular range from 101 to 1501. This implies that a large d-spacingrange is available for the study of crystal orientation distribution(texture), with a minimum number of specimen orientations(typically four orientations are sufficient).

The texture measurements yielded the typical ND ToFspectra as shown in Fig. 7. The fit of each spectrum was carriedout using GSAS. The ordered phase (Indialite) as well asthe disordered phase (low cordierite) were found to be present,10

2

1

∆ ϕθ

θ

Fig. 5. The definition of the spherical coordinate system, to calculatethe density of oriented domains.

Fig. 4. The simulated diffraction intensity distribution for an extrudedmaterial. Based on the results shown below, it will be assumed that thisdistribution holds also for cellular samples.

Fig. 6. Sketch of the instrument HIPPO at LANSCE

2648 Journal of the American Ceramic Society—Bruno and Vogel Vol. 91, No. 8

but the former was neglected in the Rietveld fit, as its presenceled to no substantial improvement of R2. The complete polefigures were built assuming a 10-component Legendre polyno-mial for the spherical harmonics expansion, following VonDreele.11

The pole figures for the (002), (100), and (010) poles along theextrusion axis are shown in Figs. 8(a) and (b) for the rod and thecellular material.

Although some noise is present, and the texture is weak, theyshow essentially a fiber texture, with the c-axis distributed sym-metrically around the sample z-axis, and the a- and b-axesalmost randomly perpendicular. This feature is surprisingly vis-ible for the cellular filter sample (Fig. 8(b)), which is not ex-pected to have fiber texture symmetry. In the following we willlimit the discussion to the (002) pole figure, without loss ofgenerality.

From Fig. 8, it can therefore be concluded that, within theexperimental uncertainty,

(1) the cellular bar and the rod have the same texture interms of intensity and distribution and

(2) the texture is pseudofiber around the extrusion axis.Therefore, an axisymmetric orientation density r(y) can be

used to calculate the /azS and /axSaverages. The use of theintegrated and normalized (002) pole figure is indeed correctto evaluate the weights of the different grain orientations. There-fore integration around the azimuthal axis can beperformed and the 1D function I(y) can be extracted as shownin Fig. 9. The parameters of the Gaussian fit can also becalculated, in particular the width DyB281 and the maximumintensity I051.5. This is a clear indication that the texture isweak.

IV. Computation of CTE

Bruno and colleagues have reported some values for the latticeexpansion in polycrystalline cordierite samples.12 Figure 10 givesan example of these data and shows the corresponding CTE, ascalculated assuming a parabolic temperature dependence of theexpansion in all axes.

Taylor13 has measured the single-crystal expansion of cordie-rite and determined the CTE values, as averaged between251 and 8001C: /aaS25�8005 3.87� 10�6 K�1 and /acS25�8005�1.29� 10�6 K�1. Some other pioneering works have beencarried out by Lee and Pentecost7 and Milberg and Blair14 onsingle crystals (see Fig. 10), and those values can be used asstarting points for our simulation, because they are measuredover a large temperature range. The sums (10) and (19) can becarried out and the average CTE can be calculated, weighted onthe texture of the samples. The results for the axial and theradial transverse average CTE and expansion are shown inFigs. 11(a) and (b) respectively. They compare well with theexperimental values, but coincide neither with the diffraction-averaged neutron data nor with the macroscopic values,measured on equivalent samples.

This discrepancy can be easily explained by the fact that thestarting points of the simulations are the single-crystal data ofLee and Pentecost, who certainly did not use the same cordieritecomposition as the present sample. The dependence of thethermal expansion of cordierite on the material compositionhas been shown to be sometimes radical.15

The calculations presented above allow one to plot a densityof contributions to the final polycrystalline CTE in differentsample orientations, following Eqs. (9) and (11):

azðyÞ ¼2B

pVfaa sin

2 yþ ac cos2 y� �

sin y e�wy2

(14)

axðyÞ ¼B

2Vfaa cos2 yþ aa þ ac sin

2 y� �

sin y e�wy2

(15)

This is shown in Fig. 12, where the radial contribution is axand the axial az. The functions of Eqs. (14) and (15) and thecurves in Fig. 12 must be interpreted as elementary contribu-tions, which cannot be separated one from the other. In otherwords, Fig. 12 implies that the populations of grains oriented atabout 201 from the extrusion axis yield the maximum contribu-tion, but each volume element contains the whole ‘‘spectrum’’shown in Fig. 12.

V. Discussion

The analysis above assumes that the texture does not change asa function of temperature and does not take into account anycomposition dependence of the CTE. High-temperature textureand single-crystal CTE measurements on compositions of inter-est would therefore be a natural extension of this work, althoughit is extremely difficult for the texture to change significantly as afunction of temperature below say 12001C.

Other approaches are possible and indeed used to calculatedaverage values. For instance Efremov16 used Turner’s relation17

to weight the contributions along the crystallographic axes a, b,and c over the elastic constants and some volume fractions ofcrystallite oriented along the main crystal direction. This anal-ysis started from an isostrain state across the sample and usedthe stress balance condition. Efremov used, however, an ap-proximate crystallographic texture using the simple ratio of thediffraction intensity for three different sample orientations.Therefore, the Turner–Efremov method indeed takes into ac-count the interaction between grains, but approximates the ori-entation distribution to three single values. Another way oftaking into account the interaction between grains was proposedby Hughes and Brittain18 and by Kerner.19 They both proposeadding an interaction term to the simple rule-of-mixtures. This

Cordierite filter Bank 6, 2-Theta 144.4, L-S cycle 333 Obsd. and Diff. Profiles

D-spacing, A

1.0 2.0 3.0 4.0

Cou

nts

/mus

ec.

X10

E 2

−2.0

0.0

2.0

4.0

6.0

(a) Hist 30

1.0 2.0 3.0 4.0

Cordierite compacted rod Bank 6, 2-Theta 144.4, L-S cycle 298 Obsd. and Diff. Profiles

D-spacing, A

Cou

nts

/mus

ec.

X10

E 3

−0.5

0.0

0.5

1.0

1.5

2.0

(b) Hist 30

Fig. 7. Examples of neutron diffraction spectra as recorded on HIPPO:(a) Cellular material; (b) rod. The corresponding Rietveld fit and residueplot are shown.

August 2008 Calculation of Average CTE in Oriented Cordierite Polycrystals 2649

interaction term depends on the elastic constant of the constit-uent phases.

The method proposed in this work addresses the CTE of asingle crystal as a material property, and attempts to calculatethe average CTE of a polycrystal, assuming no interaction oc-curs between the grains, but taking into account the whole grainorientation distribution. In essence, an integral rule-of-mixturesis used, along the original work of Gurtler.20 The hypothesis ofnegligible grain interaction (strains) is supported by the presenceof pores and microcracks. Moreover, no substantial dilationdifference has been observed between fine powders and solidcordierite samples,12 thus implying that intergranular strains areindeed low and the grains have a great deal of freedom duringexpansion. A similar result has been obtained by Yu et al.21 on

polymer–ceramic composites: Turner’s model proved not to fitthe data as a more empiric Thomas model.

Comparison between the two methods (interaction vs. rule-of-mixtures) is however possible and almost mandatory to verifywhether or not samples present internal stresses.

The interest of the present model lies, therefore, not only inthe calculation of the average CTE curve from lattice data, to becompared with dilatometry data, but also in the assessmentof the influence of the crystallographic texture as a way oftailoring the CTE properties: if we could change the texturedistribution of the material, for a known material, we couldobtain different properties, which could match particular spec-ifications. This idea has been fully theoretically developed bySigmund and Torquato,22 addressing the issue of the morpho-

Fig. 8. Pole figures for the compact rod (a), texture index 1.3 and for the cellular material (b), texture index 1.2. The extrusion axis is in the center of thepole figures.

2650 Journal of the American Ceramic Society—Bruno and Vogel Vol. 91, No. 8

logical texture rather than the crystallographic orientationdistribution. It is obvious that the mechanical properties(in particular Young’s modulus and Poisson’s ratio, but alsomodulus of rupture, MOR) could change as a function oftexture and therefore should be monitored to check that thematerial does not depart from the design specifications.

VI. Summary and Conclusions

Starting from a previous work, the present paper has shown thatthe introduction of realistic microstructural parameters, such asthe texture of the material, allows calculation of the average mac-roscopic value of CTE. The main hypotheses in the calculation arethat the objects (domains or crystallites) have similar (002) orien-tation (i.e., share the same intrinsic physical property) and do notinteract with each other. Results do not match exactly with thepolycrystalline data measured in previous works or with the mac-roscopic values measured by dilatometry, but are in good agree-ment with them. The discrepancy can be explained by literaturesingle-crystal data, which do not belong to the same cordieritecomposition. The method has also allowed calculation of the dis-tribution of the density of CTE, i.e., the contribution of the differ-ent orientations to the final macroscopic value.

Acknowledgments

The authors thank Sasha Efremov (Corning Inc., St. Petersburg, Russia) forchecking the calculations, Greg Merkel (Corning Inc., Corning, NY) for providingsome data and Fig. 1, and to JimWebb (Corning Inc., Corning, NY) for providing

the samples. Los Alamos National Laboratory is operated by Los Alamos Na-tional Security LLC under DOE contract DE-AC52-06NA25396.

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−5.00

−4.00

−3.00

−2.00

−1.00

0.00

1.00

2.00

3.00

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0 100 200 300 400 500 600 700 800Temperature, °C

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−2000

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0 100 200 300 400 500 600 700 800

Temperature, °C

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, 10−6

−4.0

−3.0

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August 2008 Calculation of Average CTE in Oriented Cordierite Polycrystals 2651

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2652 Journal of the American Ceramic Society—Bruno and Vogel Vol. 91, No. 8