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Acta Materialia 60 (2012) 3380–3392

Anisotropic elastic and thermal properties of the doubleperovskite slab–rock salt layer Ln2SrAl2O7 (Ln = La, Nd, Sm, Eu,

Gd or Dy) natural superlattice structure

Jing Feng a,b, Bing Xiao c, Rong Zhou b, Wei Pan a,⇑, David R. Clarke d,⇑

a State Key Laboratory of New Ceramics and Fine Processing, Department of Materials Science and Engineering, Tsinghua University,

Beijing 100084, People’s Republic of Chinab Key Laboratory of Advanced Materials of Precious-Nonferrous Metals, Education Ministry of China, Kunming University of Science and Technology,

Kunming 650093, People’s Republic of Chinac Department of Physics School of Science and Engineering, Tulane University, New Orleans, LA 70118, USA

d School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA

Received 15 September 2011; received in revised form 25 February 2012; accepted 2 March 2012Available online 6 April 2012

Abstract

The anisotropic elastic and thermal properties of layered compounds in the series Ln2SrAl2O7 (Ln = La, Nd, Sm, Eu, Gd or Dy) arecalculated from first principles using density functional theory combined with the Debye quasi-harmonic approximation. The polycrys-talline values of the elastic constants and bulk, shear and Young’s moduli are consistent with those determined experimentally. All com-pounds in the compositional series have weakly anisotropic elastic and thermal properties. For instance, thermal expansion in the [001]direction of the tetragonal unit cell is slightly larger than along the [100] or [010] directions for most Ln2SrAl2O7 compounds and thecalculated in-plane thermal conductivity is always larger than that along the c-axis, parallel to the layer stacking direction.� 2012 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Ceramics; Elastic behavior; Thermal properties; First principles electron theory

1. Introduction

Mechanical and thermal properties, such as the elasticmoduli, hardness, heat capacity, thermal expansion coeffi-cients and thermal conductivity are all fundamental prop-erties that must be determined before new materials canbe considered for many of today’s technological applica-tions. This is especially so of materials for applicationssuch as thermal barrier coatings and thermoelectrics, wherea combination of outstanding properties are required [1,2].For instance, new thermal barrier coatings will have to notonly have a lower thermal conductivity than the currently

1359-6454/$36.00 � 2012 Acta Materialia Inc. Published by Elsevier Ltd. All

http://dx.doi.org/10.1016/j.actamat.2012.03.004

⇑ Corresponding authors. Tel.: +86 01062772858; fax: +86 01062771160(W. Pan), tel.: +1 6174954140 (D.R. Clarke).

E-mail addresses: panw@mail.tsinghua.edu.cn (W. Pan), clarke@seas.harvard.edu (D.R. Clarke).

used 6–8 mol.% yttria-stabilized zirconia (6–8YSZ) butalso higher thermal expansion (to minimize the thermalexpansion mismatch with the underlying nickel-basedsuperalloy) and superior sintering resistance [3].

In searching for materials with lower thermal conductiv-ity there are a number of guidelines for identifying candi-date compounds, for instance a large mean atomic mass, alarge number of atoms per unit cell, site disorder of ionson multiple sub-lattices, weak atomic bonding and a highconcentration of defects and a distorted crystal structure[4,5]. Using these guidelines a number of layered com-pounds have been identified and shown to have very lowthermal conductivities, between 1 and 2 W mK�1 [6–9].One of these compounds is an unusual layered perovskite,in essence a natural superlattice, consisting of alternatinglayers of a double perovskite block with a rock salt block.The compound, a rare earth strontium aluminate, forms a

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J. Feng et al. / Acta Materialia 60 (2012) 3380–3392 3381

compositional series with the rare earth ions from La to Dy,and has the nominal formula Ln2SrAl2O7 [10]. Measure-ments indicate that the thermal conductivity depends onthe atomic number of the rare earth, as well as on the unitcell volume [8]. Single crystals of these very refractorycompounds are not known and so only a number of thepolycrystalline properties have been measured and there isno information about their crystalline anisotropy. In thiscontribution we compute the anisotropic elastic and ther-mal properties of these compounds, as well as their depen-dence on the rare earth ion. It is anticipated that this willguide selection of the appropriate rare earth member ofthe series for different applications and guide futureexperiments.

2. Methods and details

The crystal structure of Ln2SrAl2O7 (Ln = La, Nd, Sm,Eu, Gd or Dy) is illustrated in Fig. 1. As described else-

Fig. 1. The crystal structure of Ln2SrAl2O7 (Ln = La, Nd, Sm, Eu, Gd orDy) indicating the layered sequence of a double perovskite and a rock saltblock.

where, it is a tetragonal structure and consists of a periodicstacking of double perovskite blocks, formed from AlO6

octahedra, and a rock salt block, based on LnO5 polyhedra[10]. The space group is determined to be I4/mmm (No.139). The pertinent features of the structure for the compu-tations is that the rare earth and strontium atoms occupythe 4e (0,0,0.318 ± 0.008) and 2b (0, 0,0.5) Wyckoff sites,respectively. In addition, the aluminum atoms occupy the4e (0, 0,0.094 ± 0.008) sites, and the oxygen atoms are sit-uated at the 2a (0,0,1), 8 g (0, 0.5,0.097 ± 0.008), and 4e(0,0,0.199 ± 0.008) sites.

The calculations in this work were implemented usingthe first principles, density functional perturbation theory(DFPT) method [11]. This method combines density func-tional theory [12] with the electron density linear responsemethod, which have been successfully applied to calculatethe thermal properties of many materials with simple crys-tal structures. The calculations were supplemented usingthe quasi-harmonic approximation (QHA) [13] in orderto compute the thermal expansion and Gruneisen anisotro-pies. In implementing the density functional calculationsthe spin polarized local density approximation (LSDA)was employed to approximate the exchange correlationenergy [14]. The ultrasoft pseudo-potentials in theCASTEP [15] database were used for Ln, Sr, Al, and Oatoms. For different atomic species the valence shells andelectronic configurations for pseudo-atoms are O 2s2 2p4,Al 3s2 3p1, Sr 4s2 4p6 5s2, La 5s2 5p6 5d1 6s2, Nd 4f4 5s2

5p6 6s2, Sm 4f6 5s2 5p6 6s2, Eu 4f7 5s2 5p6 6s2, Gd 4f7 5s2

5p6 5d1 6s2 and Dy 4f10 5s2 5p6 6s2. The change in totalenergy was reduced to 1.0 � 10�6 eV per atom. The Monk-horst–Pack scheme was used for k point sampling(10 � 10 � 10) in the first irreducible Brillouin zone (BZ).The kinetic energy cut-off was set to 500.0 eV [16].

In order to investigate the anisotropic mechanical prop-erties we calculated the elastic stiffness and other mechani-cal properties (including second order elastic constants,bulk modulus (B), shear modulus (G) and Young’s modu-lus (E)). The second order elastic constants were deter-mined by means of linear fitting of the stress–straincurves obtained from first principles calculations [17,18].To illustrate the procedure in Fig. 2 we show the deformedunit cell of a Ln2SrAl2O7 compound after applying one ofthe monoclinic strain modes. The deformed lattice vectorsare

aia ¼ a0ia þ g

Pb

T aba0ib ð1Þ

where aia and a0ia are the initial and deformed lattice vec-

tors, respectively and T ab is a matrix which characterizesthe symmetry of the strain mode. For the distorted crystalstructures the lattice constants and the shape of the cell arefixed and so only the atomic positions were optimized be-fore calculating the total energy and Cauchy stress tensor.Once the elastic constants had been computed the othermechanical properties were evaluated by standard proce-dures [19,20].

Fig. 2. The unit cell of Ln2SrAl2O7 deformed by monoclinic mode strain.The second order elastic constants are determined by means of linearfitting of the stress–strain curves obtained from first principles calcula-tions. One usually applies several different strain patterns to the optimizedcrystal structure and, by varying the strain amplitude for a specific strainpattern, the Cauchy stress tensor can be evaluated as a function of strain.

3382 J. Feng et al. / Acta Materialia 60 (2012) 3380–3392

3. Results

3.1. Elastic constants and polycrystalline moduli

The calculated elastic constants of the Ln2SrAl2O7

(Ln = La, Nd, Sm, Eu, Gd or Dy) series of compoundsare listed in Table 1, and the elastic compliance matricescalculated directly from them are listed in Table 2. Fortetragonal crystals such as the rare earth strontium alumi-

Table 1The calculated independent elastic constants of Ln2SrAl2O7 (Ln = La, Nd, Smbeen used to calculate the elastic coefficients of a-Al2O3 and SrO).

Method Cij

C11 C33

La2SrAl2O7 Calculateda 252.7 272.4Nd2SrAl2O7 Calculateda 292 276.5Sm2SrAl2O7 Calculateda 283.7 275.1Eu2SrAl2O7 Calculateda 288.4 273Gd2SrAl2O7 Calculateda 298.3 285.3Dy2SrAl2O7 Calculateda 286.4 261.8a-Al2O3 Calculateda 489.7 491

Experimentalb 497 501SrO Calculateda 170.7

Experimentalc 160.1

a Calculated in this work.b From Kittel [22].c From Grimvall [21].

nates C11 ¼ C22 – C33, and the difference between them is ameasure of the anisotropic properties of mechanical moduliin the stacking plane and in the perpendicular [001] direc-tion. Examination of the results in Table 1 indicates thatthese compounds have relatively weak anisotropic elasticproperties with only a weak directional dependence of themoduli, a point we will return to later. Notably, though,the values of C11 and C66 are slightly larger than those ofC33 and C44, respectively, implying that the intra-layerchemical bonds are stronger than those between the layers.Also shown in Table 1 are the elastic constants of Al2O3

and SrO for comparison. As expected, the elastic constantsare intermediate between those of Al2O3 and SrO [21,22].

From the values of the elastic constants the intrinsicmechanical stability of the structure can be checked, sincethe condition for mechanical stability of tetragonal crystalsis usually characterized by the conditions [23].

C11 > 0;C33 > 0; C44 > 0;C66 > 0;

ðC11 � C12Þ > 0; ðC11 þ C33 � 2C13Þ > 0;

½2ðC11 þ C12Þ þ C33 þ 4C13� > 0

ð2Þ

As indicated in Table 1, the calculated elastic constantsof all the Ln2SrAl2O7 compounds do satisfy these criteria,indicating that they are intrinsically stable. This is consis-tent with the results of the energy calculations, for instancethe formation enthalpy and cohesive energy [24].

As mentioned in the Introduction, no single crystals ofthese compounds are known to exist and so for comparisonwith the available polycrystalline data the polycrystallinemoduli have been calculated from the single crystal elasticconstants. The Voigt–Ruess–Hill approximation, an arith-metic average of the Voigt and Reuss moduli [19,20], hasbeen used.

BVRH ¼1

2ðBV þ BRÞ ð3Þ

GVRH ¼1

2ðGV þ GRÞ ð4Þ

, Eu, Gd or Dy) using the strain–stress method (the same methods have

C44 C66 C12 C13

117.7 137.6 130.1 141.6120.2 131.2 113.4 131.9123.2 123.8 102.8 129118.9 124.4 103.9 121.3122.1 120.2 109.4 126.6120.1 132.3 102.6 134.8138.05 142.6 126.6147 163 11656.9 44.159 43.5

Table 2The elastic compliance matrix of Ln2SrAl2O7 (Ln = La, Nd, Sm, Eu, Gd or Dy) calculated from the elastic coefficients using the strain–stress method.

Sij

S11 S33 S44 S66 S12 S13

La2SrAl2O7 0.006200 0.005965 0.008496 0.007267 �0.00196 �0.00221Nd2SrAl2O7 0.004588 0.005245 0.008319 0.007622 �0.00101 �0.00171Sm2SrAl2O7 0.004647 0.005291 0.008117 0.008078 �0.00088 �0.00177Eu2SrAl2O7 0.004467 0.005051 0.00841 0.008039 �0.00095 �0.00156Gd2SrAl2O7 0.004334 0.004838 0.008190 0.008319 �0.00095 �0.00156Dy2SrAl2O7 0.004719 0.005939 0.008326 0.007559 �0.00072 �0.00206

J. Feng et al. / Acta Materialia 60 (2012) 3380–3392 3383

E ¼ 9BVRHGVRH=ð3BVRH þ GVRHÞ ð5ÞIn the above equations B, G and E are the bulk modulus,shear modulus and Young’s modulus, respectively andVRH refers to the Voigt–Reuss–Hill approximation. Cal-culation of the Voigt and Reuss moduli are straightfor-ward using the obtained elastic constants, the formulafor which is

BV ¼ 19½2ðC11þC12ÞþC33þ4C13�

GV ¼ 130ðMþ3C11�3C12þ12C44þ6C66Þ

BR¼C2=M ;GR¼ 15fð18BV =C2Þþ½6=ðC11�C12Þ�þð6=C44Þþð3=C66Þg

M ¼C11þC12þ2C33�4C13;C2¼ðC11þC12ÞC33�2C2

13

8>>>><>>>>:

ð6ÞThese are listed in Table 3. Comparison of the tabulated

results indicates only a weak dependence of the mechanicalproperties of Ln2SrAl2O7 compounds on the radius of theLn3+ ion. Furthermore, the numerical values are very sim-ilar. In contrast, the experimental values of B and E

increase slightly with decreasing ionic radius of the Ln3+

Table 3The bulk modulus (GPa), shear modulus (GPa), Young’s modulus (GPa), Poisusing the elastic constants, with the Voigt–Reuss–Hill approximations applied

Method B G

BV BR BH GV GR

La2SrAl2O7 Calculateda 178.3 177.7 178 98.9 87.3Experimentalb 137 100

Nd2SrAl2O7 Calculateda 179.4 179.4 179.4 106.5 101.2Experimentalb 140 104

Sm2SrAl2O7 Calculateda 173.8 173.6 173.7 106.1 100.7Experimentalb 159.8 109.1

Eu2SrAl2O7 Calculateda 171.4 171.4 171.4 106 102.2Experimentalb 160 102

Gd2SrAl2O7 Calculateda 188.6 188 188.3 109.5 108.3Experimentalb 174.5 108.9

Dy2SrAl2O7 Calculateda 184.3 178.9 181.6 108.7 106.3Experimentalb 188 109

a-Al2O3 Calculateda 251.3 251.3 251.3 161.6 159.1Experimentalc

SrO Calculateda 86.1 86.1 86.1 59.3 59.3Experimentald 82.37 82.37 82.37 58.72 58.72

a Calculated in this work.b From Wan et al. [8].c From Kittel [22].d From Grimvall [21].

ion, especially for the Young’s modulus. Since the Young’smodulus characterizes the resistance to uniaxial deforma-tion it is closely related to the values of C11 and C33 and,interestingly, the calculated C11 and C33 also increase fromLa2SrAl2O7 to Gd2SrAl2O7. On the other hand, the shearmodulus does not change significantly with the radius ofthe Ln3+ ion, because C44 is almost a constant for all thecompounds in the series. The calculated bulk moduli forall of the Ln2SrAl2O7 compounds are less than 200 Gpa,with the largest being that of Gd2SrAl2O7, at 188GPa, and the smallest being that of Eu2SrAl2O7, at172 GPa. Similar results were found for the shear moduli.

In Table 3 we also show the Young’s modulus in differ-ent crystallographic directions. Interestingly, the modulusin the [001] direction, perpendicular to the layer planes,is smaller than that in the basal plane (in the [100] and[010] directions) for all the compounds in the rare earthseries. The experimental results for polycrystalline samplesare also listed in Table 3. The theoretical results are quali-tatively in agreement with the experimental results [8]. Theexperimental values are sensitive to porosity.

son ratio (m) of Ln2SrAl2O7 (Ln = La, Nd, Sm, Eu, Gd or Dy) calculatedfor the evaluation of mechanical moduli.

BV/GV (B/G) E m

GH Ex = Ey Ez mxy mxz mzx

93.1 1.803 161.2 167.6 0.215 0.356 0.371.37 166.4 0.203

103.9 1.685 218 190.7 0.22 0.372 0.3251.346 186 0.199

103.4 1.638 251.2 188.9 0.19 0.38 0.3341.465 252.3 0.223

104.1 1.617 223.8 197.9 0.213 0.35 0.3091.569 240 0.237

108.9 1.722 264.8 205.9 0.211 0.346 0.3251.602 257.9 0.243

107.5 1.695 245.9 198.6 0.179 0.306 0.3561.725 230.4 0.257

160.3 1.555 430.6 440.2 0.241 0.196 0.20.23

59.3 1.452 151.8 151.8 0.206 0.206 0.20658.72 1.403 0.21

4200 4400 4600 4800 5000 5200

4200

4400

4600

4800

5000

5200

V m [0

01] (

m.s-1

)

La2SrAl2O7 Nd2SrAl2O7 Sm2SrAl2O7 Eu2SrAl2O7 Gd2SrAl2O7 Dy2SrAl2O7

Vm [100] (m.s -1)

[100] = [010]

[001] [110]

Fig. 4. The average sound velocities (mm) in the [100], [001] and [110]directions of Ln2SrAl2O7 (Ln = La, Nd, Sm, Eu, Gd or Dy) compoundscalculated by first principles. For most of the compounds the averagesound velocities (mm) along the [110] direction are smaller than those in the[100] and [001] directions.

3384 J. Feng et al. / Acta Materialia 60 (2012) 3380–3392

3.2. Anisotropy of acoustic velocities

We have calculated the phase velocities of pure trans-verse and longitudinal modes of the Ln2SrAl2O7 seriesfrom the single crystal elastic constants following the pro-cedure of Brugger [25]. The tetragonal symmetry of thecompounds dictates that pure transverse and longitudinalmodes can only exist for the symmetry directions of type[001] (or [010]), [001] and [110]. In all other directionsthe propagating waves are either quasi-transverse orquasi-longitudinal. In the principal directions the acousticvelocities can be simply written as:

for ½100� ¼ ½010�ml ¼ffiffiffiffiffiffiffiffiffiffiffiffiC11=q

p; ½00 1�mt1

¼ffiffiffiffiffiffiffiffiffiffiffiffiC44=q

p; ½010�mt2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiC66=q

p; for ½001�ml

¼ffiffiffiffiffiffiffiffiffiffiffiffiC33=q

p; ½100�mt1 ¼ ½010�mt2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiC66=q

pfor ½110�ml ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðC11 þ C12 þ 2C66Þ=2q

p; ½00 1�mt1

¼ffiffiffiffiffiffiffiffiffiffiffiffiC44=q

p; ½1�10�mt2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðC11 � C12Þ=2q

pThe correlations between the calculated longitudinal,

transverse and average sound velocities (mm) of Ln2SrAl2O7

(Ln = La, Nd, Sm, Eu, Gd or Dy) compounds and theradii of the rare earth atoms are shown in Figs. 3 and 4.In the same figures we also compare the experimental val-ues for the sound velocities with the theoreticalcalculations.

With the computed transverse and longitudinal acousticvelocities it is straightforward to calculate the Debye tem-perature HD using the equation [22]:

HD ¼hkB

3n4p

N AqM

� �� �1=3

mm ð7Þ

where h and kB are the Planck and Boltzmann constants,respectively, NA is Avogadro’s number, n is the total

0.122 0.120 0.118 0.116 0.114 0.112 0.110 0.108

4000

4800

5600

6400

7200

Sm3+ Dy3+Nd3+ Gd3+Eu3+

Soun

d ve

loci

ties

(m.s

-1)

Radius Ln3+ (10-9 .m)

vl cal.

vl exp.

vtcal.

vtexp.

vmcal.

vmexp.

La3+

Fig. 3. The correlation between the longitudinal, transverse and averagesound velocities (mm) of Ln2SrAl2O7 (Ln = La, Nd, Sm, Eu, Gd or Dy)compounds and the radius of rare earth atoms.

number of atoms in the formula unit, M is the meanmolecular weight, and q is the density. The average soundvelocity mm is given by [17]

mm ¼1

3

1

m3l

þ 2

m3t

� �� ��13

ð8Þ

ml ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðBþ 4

3GÞ=q

qmt ¼

ffiffiffiffiffiffiffiffiffiG=q

p8<: ð9Þ

where B and G are isothermal bulk and shear moduli, vl isthe longitudinal sound velocity and vt the transverse soundvelocity. These are tabulated, together with the calculatedDebye temperature, in Table 4. Since the elastic modulido not vary much with the atomic number of the rare earthelement, the small variation in average acoustic velocityand Debye temperature is not unexpected.

3.3. Angular variation in elastic moduli

Although the calculations suggest that the elastic prop-erties of the series of rare earth strontium aluminates arerather weakly anisotropic, the complexity of the structuresuggests that it is still useful to characterize the elasticanisotropy. A number of metrics, including the universalanisotropic index (AU) [26], percent anisotropy and shearanisotropic factors have been proposed. Their definitionsare given in Appendix A and the numerical values listedin Table 5 for the different rare earth members of the series.In many respects, however, the most straightforward wayto illustrate the mechanical anisotropic properties is as aplot the mechanical moduli in three dimensions as a func-tion of crystallographic orientation. The procedure is

Tab

le4

So

un

dve

loci

ties

alo

ng

diff

eren

td

irec

tio

ns

and

the

aver

age

velo

citi

esfo

rL

n2S

rAl 2

O7

(Ln

=L

a,N

d,

Sm

,E

u,

Gd

or

Dy)

calc

ula

ted

by

firs

tp

rin

cip

les

and

mea

sure

dex

per

imen

tall

y.

Met

ho

dq

[10

0]v l

[00

1]v t

1[0

10]

v t2

[00

1]v l

[10

0]v t

1[0

10]

v t2

[11

0]v l

[00

1]v t

1[1

–110

]vt2

m lm t

m mH

D

La 2

SrA

l 2O

7C

alcu

late

da

5.87

96.

556

4.47

44.

838

6.80

74.

838

4.83

87.

481

4.47

43.

229

7.73

54.

102

4.58

458

6E

xper

imen

talb

5.71

14.

898

4.18

54.

376

554

Nd

2S

rAl 2

O7

Cal

cula

ted

a6.

331

6.79

14.

357

4.55

26.

609

4.55

24.

552

7.26

24.

357

3.75

67.

226

4.10

14.

560

594

Exp

erim

enta

lb5.

914

4.86

54.

193

4.37

655

7S

m2S

rAl 2

O7

Cal

cula

ted

a6.

583

6.56

54.

326

4.33

76.

464

4.33

74.

337

6.94

04.

326

3.70

76.

871

4.01

54.

452

583

Exp

erim

enta

lb6.

082

5.12

64.

235

4.46

356

9E

u2S

rAl 2

O7

Cal

cula

ted

a6.

655

6.58

34.

227

4.32

46.

405

4.32

44.

324

6.94

04.

227

3.72

36.

824

3.99

14.

426

581

Exp

erim

enta

lb6.

193

5.08

34.

058

4.30

755

2G

d2S

rAl 2

O7

Cal

cula

ted

a6.

936

6.55

84.

196

4.16

36.

414

4.16

34.

163

6.83

54.

196

3.69

06.

944

3.97

34.

415

584

Exp

erim

enta

lb6.

343

5.24

54.

143

4.40

756

6D

y 2S

rAl 2

O7

Cal

cula

ted

a7.

006

6.39

44.

140

4.34

67.

186

4.34

64.

346

6.83

04.

140

3.62

26.

770

3.93

94.

370

576

Exp

erim

enta

lb6.

508

5.37

54.

093

4.38

356

4

Th

eu

nit

of

velo

city

isk

ms�

1,

of

den

sity

isg

cm�

3an

dth

eD

ebye

tem

per

atu

reis

inK

.a

Cal

cula

ted

inth

isw

ork

.b

Fro

mW

anet

al.

[8].

Table 5The calculated universal anisotropic index (AU), percent anisotropy (AG

and AB) and shear anisotropic factors (A1 and A2) of Ln2SrAl2O7

(Ln = La, Nd, Sm, Eu, Gd or Dy).

Species AU AB AG A1 A2

La2SrAl2O7 0.667 0.002 0.062 1.946 2.078Nd2SrAl2O7 0.262 0 0.025 1.578 1.535Sm2SrAl2O7 0.269 0.001 0.026 1.638 1.402Eu2SrAl2O7 0.186 0 0.018 1.492 1.407Gd2SrAl2O7 0.059 0.002 0.006 1.478 1.318Dy2SrAl2O7 0.143 0.015 0.011 1.724 1.543

J. Feng et al. / Acta Materialia 60 (2012) 3380–3392 3385

described in Appendix B and the results are shown in Figs.5 and 6 for the bulk and Young’s moduli, respectively.

3.4. Thermal expansion anisotropy

The thermal expansion coefficients and their tempera-ture dependence are of importance in estimating the ther-mal expansion mismatch with potential substratematerials. Rather than compute them from first principlesat different temperatures we have computed them fromthe specific heats and elastic constants using the approachdescribed below [27]. This then enables us to estimate theanisotropic Gruneisen parameters and thermal conductiv-ity anisotropy at very high temperatures in the next sub-section.

The starting point is that the temperature dependence ofthermal expansion is related to the isothermal bulk modu-lus and the difference between the specific heats at constantpressure and constant volume, CP and CV, respectively, as[21]

CP � CV ¼ b2V ðT ÞTB0 ð10Þwhere b is the volume expansion coefficient and V(T) rep-resents the cell volume at temperature T, calculated asV ðT Þ ¼ ð1þ bT ÞV 0, and B0 is the isothermal bulk modulusat the equilibrium cell volume (V0). The difference betweenthe two specific heats also enables direct calculation of theisotropic Gruneisen parameter cG [21]:

CP � CV ¼ bcGCV T ð11ÞThe specific heat at constant pressure Cp was calculated

using the CASTEP code, which determines it automaticallyby analyzing the phonon frequencies of the crystal struc-ture. The specific heat at constant volume Cv cannot be cal-culated directly from the same code and so, instead, wascalculated from using the Debye QHA [22]:

cV ðT Þ ¼ 9nNAkBT

HD

� �3 Z HD=T

0

x4ex

ðex � 1Þ2dx ð12Þ

In this equation n is the total number of atoms in theunit cell and the other parameters and constants have theirusual meaning. Using the Debye temperatures already cal-culated the difference in specific heat for each compound inthe series is shown in Fig. 7 and the calculated volumetricthermal expansion coefficients are shown in Fig. 8. The vol-

-500

-250

0

250

500

-500

-250

0

250

500

-500

0

500

Bz

-500

-250

0

250Bx

00

-250

0

250By

-500-250

0

250

500

-500

-250

0

250500

-500

-250

0

250

500

Bz

-500-250

0

250Bx

500

-250

0

250By

-500-250

0

250

500

-500

-250

0

250

500

-500

-250

0

250

500

Bz

-500-250

0

250Bx

00

-250

0

250By

(a) La2SrAl2O7 (b) Nd2SrAl2O7 (c) Sm2SrAl2O7

-500-250

0

250

500

-500

-250

0

250

500

-500

-250

0

250

500

Bz

-500-250

0

250Bx

00

-250

0

250By

-500-250

0

250

500

-500

-250

0

250500

-500

-250

0

250

500

Bz

-500-250

0

250Bx

500

-250

0

250By

-500-250

0

250

500

-500

-250

0

250500

-500

-250

0

250

500

Bz

-500-250

0

250Bx

500

-250

0

250By

(d) Eu2SrAl2O7 (e) Gd2SrAl2O7 (f) Dy2SrAl2O7

La2SrAl2O7

Nd2SrAl2O7

Sm2SrAl2O7

Eu2SrAl2O7

Gd2SrAl2O7

Dy2SrAl2O7

-750

-500

-250

0

250

500

750 [010]

[100]

(001)GPa

-750 -500 -250 0 250 500 750 -750 -500 -250 0 250 500 750

-750

-500

-250

0

250

500

750 La2SrAl2O7

Nd2SrAl2O7

Sm2SrAl2O7

Eu2SrAl2O7

Gd2SrAl2O7

Dy2SrAl2O7

[010]

[001]

(100)

GPa

(g) the (001) crystal plane (h) the (100) crystal plane

Fig. 5. (a–f) The surface construction of the bulk modulus of Ln2SrAl2O7 compounds. The anisotropy of the bulk modulus of Ln2SrAl2O7 compounds isweak. (g and h) The (001) and (100) planar projections, respectively, of the bulk modulus for the Ln2SrAl2O7 compounds.

3386 J. Feng et al. / Acta Materialia 60 (2012) 3380–3392

umetric expansion coefficients vary between 2.3 � 10�5 and3.6 � 10�5 K�1, with La2SrAl2O7 having the smallest coef-ficients and Sm2SrAl2O7 having the largest values. The

results shown in Fig. 8 seem to indicate that the volumetriccoefficients of different structures are not correlated withthe radius of the Ln3+ ion.

-200-100

0

100

200

-200

-100

0

100200

-200

-100

0

100

200

Ez

-200-100

0

100Ex

00

-100

0

100Ey

-200-100

0100

200

-200

-100

0100

200

-200

-100

0

100

200

Ez

-200-100

0100Ex

-200

-100

0100Ey

-200-100

0100

200

-200

-100

0100

200

-200

-100

0

100

200

Ez

-200-100

0100Ex

-200

-100

0100Ey

(a) La2SrAl2O7 (b) Nd2SrAl2O7 (c) Sm2SrAl2O7

-200-100

0100

200

-200

-100

0100

200

-200

-100

0

100

200

Ez

-200-100

0100

200Ex

-200

-100

0100Ey

-200-100

0100

200

-200

-100

0100

200

-200

-100

0

100

200

Ez

-200-100

0100

200Ex

-200

-100

0100Ey

-200-100

0100

200

-200

-100

0100

200

-200

-100

0

100

200

Ez

-200-100

0100Ex

-200

-100

0100Ey

(d) Eu2SrAl2O7 (e) Gd2SrAl2O7 (f) Dy2SrAl2O7

-400

-300

-200

-100

0

100

200

300

400 La2SrAl2O7

Nd2SrAl2O7

Sm2SrAl2O7

Eu2SrAl2O7

Gd2SrAl2O7

Dy2SrAl2O7

[100]

[010] (001)GPa

-400 -300 -200 -100 0 100 200 300 400 -400 -300 -200 -100 0 100 200 300 400

-400

-300

-200

-100

0

100

200

300

400 La2SrAl2O7

Nd2SrAl2O7

Sm2SrAl2O7

Eu2SrAl2O7

Gd2SrAl2O7

Dy2SrAl2O7

[010]

[001](100) GPa

(g) 001 crystal plane (h) 100 crystal plane

Fig. 6. (a–f) The directional dependence of the Young’s modulus of Ln2SrAl2O7 structures (from La to Dy). (g and h) The (001) and (100) planarprojections, respectively, of the Young’s modulus for the Ln2SrAl2O7 compounds.

J. Feng et al. / Acta Materialia 60 (2012) 3380–3392 3387

Having determined the volumetric thermal expansion,the thermal expansion along different directions can be cal-culated from the linear compressibilities. By definition, the

volumetric thermal expansion is the trace of the thermalexpansion tensor. For a tetragonal crystal only the diago-nal elements are non-zero and, moreover, due to symmetry

400 600 800 1000 12000

5

10

15

20

25 La2SrAl2O7 Nd2SrAl2O7 Sm2SrAl2O7

Cp -

Cv

heat

cap

acity

(J.m

ol-1.K

-1 )

Temperature (K)

Eu2SrAl2O7 Gd2SrAl2O7 Dy2SrAl2O7

Fig. 7. The calculated difference in specific heat as a function oftemperature for each Ln2SrAl2O7 (Ln = La, Nd, Sm, Eu, Gd or Dy)compound. The large difference between CP and CV leads to large thermalexpansion coefficients.

400 600 800 1000 1200

2.4

2.8

3.2

3.6

La2SrAl2O7 Nd2SrAl2O7 Sm2SrAl2O7

Volu

me

ther

mal

exp

ansi

on c

oeffi

cien

t(1

0-5. K

-1 )

Temperature (K)

Eu2SrAl2O7 Gd2SrAl2O7 Dy2SrAl2O7

Fig. 8. The volumetric thermal expansion coefficient as a function oftemperature for each Ln2SrAl2O7 compound.

3388 J. Feng et al. / Acta Materialia 60 (2012) 3380–3392

aaa = abb – acc. Consequently, the relationship between thevolumetric coefficient b and aij is given by [21]

b ¼ 2aa þ ac ð13ÞIn order to determine the individual linear expansion

coefficients an additional relationship between them isrequired. This is given by the ratio of the linear compress-ibilities in the a- and c-directions [21,22].

ra

rc¼ aa

acð14Þ

The linear compressibilities, ra = �d lna/dP andrc = �d lnc/dP, are related to aa and ac along the [100]and [001] directions and can be calculated from the elasticconstants [28,29].

ra ¼C33 � C13

ðC11 þ C12ÞC33 � 2C213

ð15Þ

rc ¼C11 þ C12 � 2C13

ðC11 þ C12ÞC33 � 2C213

ð16Þ

The anisotropic thermal expansion coefficients shown inFig. 9 were calculated by substituting in the values for theelastic constants in Table 1.

Knowledge of the thermal expansion anisotropy, inturn, enables an estimate to be made of the anisotropicGruneisen parameters cG,i using as a generalization Eq.(17), using the Voigt’s notation for the thermal expansioncoefficient aj [21],

cG;i ¼VCP

P6j¼1

ðcijÞaj ð17Þ

For tetragonal crystal structures symmetry dictates thatthere are only two independent Gruneisen parameters, in-plane and out of plane, (cG)a and (cG)c, respectively, givenby:

ðcGÞa ¼ ðV =CP Þ½ðC11 þ C12Þ � aa þ C13 � ac� ð18ÞðcGÞc ¼ ðV =CP Þð2C13 � aa þ C33 � acÞ ð19Þ

The calculated values are shown in Fig. 10 for eachmember of the series. For each composition (cG)c is largerthan (cG)a, indicating that there is greater anharmonicityalong the [001] direction than in the stacking plane. Thisis consistent with our previous calculations of the electronicstructures and bond populations of the Ln2SrAl2O7 struc-ture, where it was found that the chemical bonds are differ-ent in the rock salt and double perovskite layers, leading toa natural anisotropy of bonding [24].

3.5. Minimum thermal conductivity and anisotropy

The calculation from first principles of the thermal con-ductivity and its anisotropy for large unit cells is presentlycomputationally prohibitive, since the full phonon spec-trum needs to be determined as an essential part of the cal-culation. However, for high temperature applications it isthe value of the minimum conductivity that is of practicalimportance, since the conductivity decreases with increas-ing temperature to a limiting value known as the minimumthermal conductivity [4,6,7,9]. In this section we calculatethe minimum thermal conductivity and its value paralleland perpendicular to the stacking plane.

According to Cahill, the minimum thermal conductivitycan be computed from the transverse and longitudinalsound velocities and the number of atoms per mole ofthe compound [30]:

jmin ¼kB

2:48n2=3ðml þ mt1 þ mt2Þ ð20Þ

Using the acoustic velocities listed in Table 4, the mini-mum thermal conductivities parallel and perpendicular tothe block layers can be evaluated. These are listed in Table6 and compared with the high temperature values extrapo-lated from the measured, polycrystalline values [8]. Clearly,Cahill’s model underestimates the thermal conductivity bynearly 1 W mK�1. In Fig. 11 the directional dependence ofminimum thermal conductivity is plotted for a bulk

400 600 8006

8

10

12

La2SrAl2O7 Nd2SrAl2O7 Sm2SrAl2O7

α a th

erm

al e

xpan

sion

coe

ffici

ent (

10 -6

. K-1)

Temperature (K)

Eu2SrAl2O7 Gd2SrAl2O7 Dy2SrAl2O7

400 600 8001000 1200 1000 12008

10

12

14 La2SrAl2O7 Nd2SrAl2O7 Sm2SrAl2O7

α c th

erm

al e

xpan

sion

coe

ffici

ent (

10 -6

. K-1

)

Temperature (K)

Eu2SrAl2O7 Gd2SrAl2O7 Dy2SrAl2O7

(a) (b)

Fig. 9. The calculated linear thermal expansion coefficients aa and ac along the [100] and [001] directions for Ln2SrAl2O7 compounds as a function oftemperature. (a) aa along the [100] direction; (b) ac along the [001] direction. Obviously, thermal expansion in the [001] direction is slightly larger thanthat in the [100] or [010] direction for most Ln2SrAl2O7 compounds.

0.124 0.122 0.120 0.118 0.116 0.114 0.112 0.110 0.108 0.1061.40

1.45

1.50

1.55

1.60

1.65

1.70

1.75

.

Dy3+

Gd3+

Eu3+

Sm3+

Nd3+

Gru

neis

en p

aram

eter

s

Radius Ln3+ (10-9 .m)

(γG)a(γG)c

La3+

.

Fig. 10. Dependence of the Gruneisen parameters on the radii of rareearth ions in Ln2SrAl2O7 compounds. Obviously cc is larger than ca,indicating that the anharmonicity along the [001] direction is strongerthan the basal plane and the bonds are relatively weak along the c-axis formost Ln2SrAl2O7 compounds.

J. Feng et al. / Acta Materialia 60 (2012) 3380–3392 3389

polycrystalline material without any texture, along with theminimum thermal conductivities in the [001], [110] and[001] directions, given in Table 6.

Table 6The minimum thermal conductivity of Ln2SrAl2O7 (Ln = La, Nd, Sm, Eu, G

Species n (1028) ½100� kMincalc: ½001� kMin

calc:

(W mK�1) (W mK�1)

La2SrAl2O7 7.33 1.55 1.61Nd2SrAl2O7 7.74 1.59 1.5Sm2SrAl2O7 7.87 1.56 1.55Eu2SrAl2O7 7.91 1.55 1.54Gd2SrAl2O7 8.09 1.55 1.54Dy2SrAl2O7 8.02 1.54 1.64

a From Wan et al. [8].

4. Discussion and conclusions

The results of the calculations clearly indicate that theelastic and thermal properties of the rare earth strontiumaluminate Ln2SrAl2O7 series of compounds are crystallo-graphically anisotropic, but weakly so. Reflecting the crys-tal structure of the unit cell, which consists of a periodicstacking of a double perovskite block and a rock salt block,the properties perpendicular to the stacking plane are dif-ferent from those in the stacking plane. The anisotropiesare found to be relatively insensitive to the particular rareearth ion in the compound, despite systematic variationsin the lattice parameter and unit cell volume [10]. The elas-tic constants and elastic moduli are typical of many oxides,such as zirconia, but the thermal expansion coefficients arelarger. The bonding anisotropy, reflected in the elastic con-stant anisotropy, leads to anisotropy of the thermal expan-sion coefficients and, in turn, leads to anisotropy in theGruneisen parameter and to anisotropy in the high temper-ature limit of thermal conductivity.

The high temperature stability of the rare earth stron-tium aluminates suggests a comparison between their cal-culated properties with those of yttria-stabilized zirconia(8YSZ), the current material of choice for thermal barrier

d or Dy) evaluated by Cahill’s method.

½110� kMincalc: kMin

calc: kMinexp: kExp

1273Ka

(W mK�1) (W mK�1) (W mK�1) (W mK�1)

1.48 1.51 1.49 3.11.55 1.55 1.54 2.51.53 1.53 1.59 2.61.53 1.52 1.54 2.41.53 1.55 1.60 2.31.51 1.53 1.59 2.3

1.40 1.45 1.50 1.55 1.60 1.651.40

1.45

1.50

1.55

1.60

1.65

kmin (exp.)

k min

[001

] (W

.m-1

. K-1

)

kmin [100] kmin [001] kmin [110] kmin (cal.) kmin (exp.)

kmin [110]

[100] = [010]

[001]

kmin (cal.)

La2SrAl2O7

1.40

1.45

1.50

1.55

1.60

1.65

kmin (exp.)

kmin [100] kmin [001] kmin [110] kmin (cal.) kmin (exp.)

kmin [110]

[100] = [010]

[001]

kmin (cal.)

Nd2SrAl2O7

1.40

1.45

1.50

1.55

1.60

1.65

kmin (cal.)

kmin [100] kmin [001] kmin [110] kmin (cal.) kmin (exp.)

kmin [110]

[100] = [010]

[001]

kmin (exp.)

Sm2SrAl2O7

1.40

1.45

1.50

1.55

1.60

1.65

kmin (cal.)

kmin [100] kmin [001] kmin [110] kmin (cal.) kmin (exp.)

kmin [110]

[100] = [010]

[001]

kmin (exp.)

Eu2SrAl2O7

1.40

1.45

1.50

1.55

1.60

1.65

kmin (cal.)

kmin [100] kmin [001] kmin [110] kmin (cal.) kmin (exp.)

kmin [110]

[100] = [010]

[001]

kmin (exp.)

Gd2SrAl2O7

1.40

1.45

1.50

1.55

1.60

1.65

kmin (cal.)

kmin [100] kmin [001] kmin [110] kmin (cal.) kmin (exp.)

kmin [110]

[100] = [010]

[001]

kmin (exp.)

Dy2SrAl2O7

k min

[001

] (W

.m-1

. K-1

)

k min

[001

] (W

.m-1

. K-1

)k m

in [0

01] (

W.m

-1. K

-1)

k min

[001

] (W

.m-1

. K-1

)k m

in [0

01] (

W.m

-1. K

-1)

kmin [100] (W.m-1. K-1) kmin [100] (W.m-1. K-1) kmin [100] (W.m-1. K-1)

kmin [100] (W.m-1. K-1)kmin [100] (W.m-1. K-1)kmin [100] (W.m-1. K-1)

1.40 1.45 1.50 1.55 1.60 1.65 1.40 1.45 1.50 1.55 1.60 1.65

1.40 1.45 1.50 1.55 1.60 1.651.40 1.45 1.50 1.55 1.60 1.651.40 1.45 1.50 1.55 1.60 1.65

Fig. 11. The minimum thermal conductivities along different principle directions of Ln2SrAl2O7 (Ln = La, Nd, Sm, Eu, Gd or Dy) evaluated usingCahill’s equation are compared with the experimental results measured using an ultrasonic instrument (ultrasonic pulser/receiver model 5900 PR,Panametrics, Waltham, MA).

3390 J. Feng et al. / Acta Materialia 60 (2012) 3380–3392

coatings. The calculated and measured thermal conductiv-ities of all the members of the rare earth strontium alumi-nate series are comparable with that of fully dense 8YSZ,which is approximately 2.3 W mK�1 [4,31]. On the otherhand, the volumetric thermal expansion coefficients areall larger than that of 8YSZ, which averages 1.06 � 10�5 K�1 between room temperature and 1073 K. This is asignificant difference, since the thermal expansion mis-match between 8YSZ and single crystal nickel-based super-alloys is a major factor limiting the life of thermal barriercoatings. The thermal expansion coefficients of second gen-eration superalloys increases from 1.2 � 10�5 K�1 at roomtemperature up to 1.8 � 10�5 K�1 at 1400 K, so the largerthermal expansion of the Sm2SrAl2O7 compound makes itparticularly attractive, especially as it is amongst the leastanisotropic.

Acknowledgements

The authors are grateful to the National Natural ScienceFoundation of China (Grants Nos. 50232020, 50990302,51171074 and 50572042) and the Foundation ofKunming University of Science and Technology (GrantZDS2010020D) for financial support. The collaborationwith Harvard University was supported by the USNational Science Foundation through a World MaterialsNetwork Grant (DMR-0710523).

Appendix A. Anisotropy parameters

The universal anisotropic index (AU) and percent aniso-tropies (AB and AG) can be written as

AU ¼ 5GV

GRþ BV

BR� 6 P 0 ðA:1Þ

AB ¼ BV �BRBV þBR

AG ¼ GV �GRGV þGR

(ðA:2Þ

For isotropic structures the Voigt and Reuss approxi-mations should give the same values for B and G, respec-tively. Thus all of the indices in Eqs. (A.1) and (A.2) arezero. Deviations from zero indicate anisotropy. In Xiaoet al. [23], the shear anisotropic factors A1 and A2 aredefined as

A1 ¼4C44

C11 þ C33 � 2C13

ðA:3Þ

A2 ¼4C66

C11 þ C22 � 2C12

ðA:4Þ

The results are shown in Table 5. It is found thatGd2SrAl2O7 has the smallest AU value among Ln2SrAl2O7

compounds. The elastic moduli of this compound are notstrongly dependent on different orientations, and the calcu-lated AG, A1 and A2 values support this conclusion. Thecalculated AG for Nd2SrAl2O7 and Eu2SrAl2O7 are zero,

J. Feng et al. / Acta Materialia 60 (2012) 3380–3392 3391

as expected, because the Voigt and Reuss approximationspredict the same values for the shear modulus. For theother Ln2SrAl2O7 compounds AG has a very small value,indicating that these compounds do not show stronganisotropy in shear modulus. It seems that the elasticanisotropy parameters of La2SrAl2O7 are slightly differentfrom those of other Ln2SrAl2O7 compounds; this behavioris possibly related to the empty 4f orbitals of La. AlthoughAG, A1 and A2 all determine the anisotropy of the shearmodulus, the values of A1 and A2 are quite different fromAG. The calculated A1 and A2 values seem to support thehypothesis that the shear modulus of Ln2SrAl2O7 com-pounds has a strong directional dependence. The universalanisotropic index (AU) is a better indicator than other indi-ces, which can provide unique and consistent results for themechanical anisotropic properties of Ln2SrAl2O7 com-pounds. This can be clearly seen in Table 5. When AU islarge the other calculated anisotropic indices are also large,and vice versa.

Appendix B. Graphical representation of the crystallographic

dependence of the elastic moduli

For tetragonal crystals [25] the elastic moduli are relatedto the direction cosines, li,

1

B¼ ðS11 þ S12 þ S13Þðl2

1 þ l22Þ þ ð2S13 þ S33Þl2

3 ðB:1Þ

1

E¼ S11ðl4

1 þ l42Þ þ ð2S13 þ S44Þðl2

1l23 þ l2

2l23Þ þ S33l4

3

þ ð2S12 þ S66Þl21l2

2 ðB:2Þ

Substituting the relationships of the direction cosines inspherical coordinates with respect to h and u into Eqs.(B.1) and (B.2) we obtain the equations used to plot theanisotropic mechanical properties in three dimensions.

1

B¼ ðS11 þ S12 þ S13Þ sin2 hþ ð2S13 þ S33Þ cos2 h ðB:3Þ

1

E¼ S11½sin4 hðcos4 uþ sin4 uÞ� þ ð2S13 þ S44Þ

� sin2 h cos2 hþ S33 cos4 hþ ð2S12 þ S66Þ� sin4 h cos2 u sin2 u ðB:4Þ

Two planar projections on the (100) and (001) crystalplanes are illustrated for the Young’s and bulk moduli ofLn2SrAl2O7 compounds, respectively. The analytical solu-tions of the projections of B and E for each compoundcan be derived from Eqs. (B.3) and (B.4). Some resultsare shown below.

B.1. The (0 01) plane

B½010� ¼ sin uðS11þS12þS13Þ

B½1 0 0� ¼ cos uðS11þS12þS13Þ

(ðB:5Þ

The contour is a circle and B[010] and B[100] satisfyðB½010�Þ2 þ ðB½100�Þ2 ¼ B2, consistent with the bulk modulibeing isotropic in the stacking plane. Similarly, for theYoung’s modulus the results are

E½100� ¼ cos uS11ðcos4 uþsin4 uÞþð2S12þS66Þ cos2 u sin2 u

E½010� ¼ sin uS11ðcos4 uþsin4 uÞþð2S12þS66Þ cos2 u sin2 u

(ðB:6Þ

B.2. The (100) plane

B½010� ¼ sin hðS11þS12þS13Þ sin2 hþð2S13þS33Þ cos2 h

B½001� ¼ cos hðS11þS12þS13Þ sin2 hþð2S13þS33Þ cos2 h

(ðB:7Þ

E½010� ¼ sin hS11 sin4 hþð2S13þS44Þ sin2 h cos2 hþS33 cos4 h

E½001� ¼ cos hS11 sin4 hþð2S13þS44Þ sin2 h cos2 hþS33 cos4 h

(ðB:8Þ

From Fig. 5 we can see that the bulk moduli ofLn2SrAl2O7 compounds show weak anisotropy. Other thanDy2SrAl2O7, the surface constructions of the Ln2SrAl2O7

compounds are close to a sphere. The projections on the(001) and (10 0) planes show more details about the aniso-tropic properties of the bulk modulus. On the basal plane(the (001) plane for tetragonal crystals) the bulk modulusof Ln2SrAl2O7 is isotropic, due to the symmetry of tetrag-onal crystals. On the other hand, on the (001) plane thebulk modulus of Ln2SrAl2O7 in the [110] direction is smal-ler than those in the [010] and [100] directions. Other thanfor La2SrAl2O7, the bulk moduli in the [100] and [010]directions on the (001) crystal plane are comparable witheach other. This is consistent with the calculated elasticconstants; for example, the difference between C11 andC33 is found to be small for these compounds. For La2S-rAl2O7 we find B½001� > B½010�, because C33 is larger thanC11 in this case. Projections of the Young’s modulus onthe (100) and (001) planes show more anisotropic featuresthan the bulk modulus. One can clearly see some butterflyshaped curves for the Young’s modulus projected on the(001) plane. Therefore, the Young’s modulus has a strongdirectional dependence on this plane, and the value alongthe [100] or [010] direction is significantly smaller thanthat along the [110] direction (Fig. 6). The differencebetween the two directions for the Young’s modulus is esti-mated to be 100 GPa for Ln2SrAl2O7. The longitudinalmodulus of the tetragonal crystal class in the [110] direc-tion is calculated as L½110� ¼ C11 þ C12 þ 2C66, and theYoung’s modulus E[110] can be related to the longitudinal

modulus using E ¼ ð1þmÞð1�2mÞð1�mÞ L, where m is the Poisson ratio

and L represents the longitudinal modulus along a specificdirection. For Ln2SrAl2O7 the calculated m is close to 0.3.The Young’s moduli in different directions are simply pro-portional to L, because L½110� > L½100� ¼ L½010�, thusE½110� > E½100� ¼ E½010� is obvious on the (001) plane. A sim-ilar analysis can be used to explain the results on the (100)plane.

3392 J. Feng et al. / Acta Materialia 60 (2012) 3380–3392

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