The VoigtReussHill (VRH) Approximation and the Elastic Moduli of PolycrystallineZnO, TiO2 (Rutile), and Al2O3D. H. Chung and W. R. Buessem

Citation: Journal of Applied Physics 39, 2777 (1968); doi: 10.1063/1.1656672 View online: http://dx.doi.org/10.1063/1.1656672 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/39/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Native point defects and dangling bonds in -Al2O3 J. Appl. Phys. 113, 044501 (2013); 10.1063/1.4784114 Ultraviolet dosimetric properties of Al2O3 crystals Appl. Phys. Lett. 63, 894 (1993); 10.1063/1.109893 Raman scattering and xray diffractometry studies of epitaxial TiO2 and VO2 thin films and multilayers on Al2O3(1120) J. Appl. Phys. 73, 2841 (1993); 10.1063/1.353036 A comparison of Pt overlayers on Al2O3(0001), ZnO(0001)Zn, and ZnO(0001)O J. Chem. Phys. 93, 5337 (1990); 10.1063/1.459653 The VoigtReussHill Approximation and Elastic Moduli of Polycrystalline MgO, CaF2, ZnS, ZnSe, and CdTe J. Appl. Phys. 38, 2535 (1967); 10.1063/1.1709944

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JOURNAL OF APPLIED PHYSICS VOLUME 39, NUMBER 6 MAY 1968

The Voigt-Reuss-Hill (VRH) Approximation and the Elastic Moduli of Polycrystalline ZnO, Ti02 (Rutile), and a-AI20 a

D. H. CHUNG* AND W. R. BUESSEM

Materials Research Laboratory, The Pennsylvania State University, University Park, Pennsylvania (Received 22 May 1967; in final form 15 January 1968)

The Voigt-Reuss-Hill (VRH) approximation, a useful scheme by which one calculates isotropic poly­crystalline elastic moduli in terms of the anisotropic single-crystal elastic constants, is shown to be accurately applicable for ZnO, TiO. (rutile), and a-AbO" which are the simplest examples of oxide materials with the hexagonal, tetragonal, and trigonal s)rmmetries, respectively. An experimental proof is established by measuring isotropic elastic moduli on dense-formed polycrystalline specimens. The recommended values of the polycrystalline isotropic elastic moduli of these oxides at room temperature are as follows: for ZnO, G=4.56(±0.04) and E=12.35(±0.09); for TiO., G=11.15(±0.09) and E=28.42(±0.14); and for a-AbO" G=16.35(±0.12) and E=40.39(±0.31), where G and E are the isotropic shear and Young's moduli, respectively, and they are in units of lOu dyn/cm2•

1. INTRODUCTION

The Voigt-Reuss-Hill (VRH) approximation is a useful scheme for calculating isotropic polycrystalline elastic moduli from the corresponding single-crystal elastic constants.1- 5 In our earlier communication1

dealing with aggregates made of moderately anisotropic single-phase cubic crystals, it was shown that the measured polycrystalline isotropic elastic moduli were in good agreement with the VRH moduli calculated from the corresponding single-crystal data and con­cluded that the VRH approximation gives realistic values of the polycrystalline elastic moduli in terms of the corresponding single-crystal properties.

The purpose of the present paper is to extend this study of the VRH approximation to an aggregate containing crystals of a lower symmetry and to examine to what extent the VRH approximation is useful for calculating the isotropic elastic module from the corre­sponding single-crystal data of the lower symmetry crystals. The materials to be considered in the present work are hexagonal ZnO (point group 6 mm) , tetragonal Ti02 (point group 4/mnm), and trigonal a-A120 a (point group :3 m). These materials were chosen for three reasons: (i) For these oxides, the anisotropic single-crystal elastic constants are accurately known and thus the VRH approximation can be readily ap­plied; (ii) Polycrystalline aggregate specimens for these oxides have been successfully fabricated by means of pressure-sintering methods and characterized so that measurements of isotropic elastic moduli on these specimens provide empirical data that test the validity of the VRH approximation resulting from (i); and finally, (iii) These oxides represent the simplest exam­ples for polycrystalline oxide materials of hexagonal,

tetragonal, and trigonal crystals that are of technical importance.

There has been a lapse of several years between the time of performance of the work in 19616 and its sub­mission for publication. Several measurements of polycrystalline elastic moduli have been recorded in the literature for some of these materials. Spriggs et aU reported Young's modulus measured statically on porous alumina specimens. Schreiber and Anderson8

reported isotropic elastic moduli of an alumina sample (",99% pure AbOa). Since the present paper was submitted for publication, polycrystalline elastic moduli of a ZnO sample have appeared in the meantime.9

2. SPECIMENS

2.1. Polycrystalline ZnO

Ten specimens, having a size in the neighborhood of 0.50 cm in height, 0.95 cm in width, and 7.62 cm in length, used in the present work are polycrystalline aggregates made of well-characterized zincite crystals.

Starting with a powdered form of zinc carbonate, the powder was decomposed at about 500°C, and then the resulting conglomerate was ground to give an average particle size of about 10 }.l. The resulting "white" powder had the chemical composition ZnO, as confirmed by x-ray studies. A typical spectrochemical analysis showed that the chemical purity of this powder was 99.9% ZnO. The minor constituents detected were traces of Cu, Fe, As, Cd, Sb, and about 0.001 % Pb. A resistant hot-pressing technique as described by Crandall et al.w was used as the means of fabricating a bar-shaped rectangular prism. The polycrystalline aggregates thus produced have bulk densities approaching the x-ray

* Present address: Department of Geology and Geophysics, 6 D. H. Chung, in Ref. 3, p. 912. Massachusetts Institute of Technology, Cambridge, Mass. 7 R. M. Spriggs, J. B. Mitchell, and T. Vasilos, J. Am. Ceram.

1 D. H. Chung and W. R. Buessem, J. App!. Phys. 38, 2535 Soc. 47,323 (1964). (1967). 8 E. Schreiber and O. L. Anderson, J. Am. Ceram. Soc. 38,184

• G. Simmons, J. Grad. Res. Center (SMU) 34, 1 (1965). (1966). 3 o. L. Anderson, J. Phys. Chem. Solids 24,909 (1963). 9 N. Soga and O. L. Anderson, J. App!. Phys. 38, 2967 (1967). 4 F. Birch, J. Geophys. Res. 66, 2199 (1961). 10 w. B. Crandall, D. H. Chung, and T. J. Gray, Mechanical o H. B. Huntington, Solid State Physics, F. Seitz and D. Properties of Engineering Ceramics, W. Kriegel and H. Palmour,

Turnbull, Eds. (Academic Press Inc., New York, 1958), Vol. 7. III, Eds. (Interscience Publishers, Inc., New York, 1961). 2777

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2778 D. H. CHUNG AND W. R. BUESSEM

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density value. Variable porosity in the aggregate was attained by varying the hot-pressing parameters; in the present work, the porosity ranges from about 1 % to 33%. This allows us to study a pprosity dependence on the elastic modulus of this material. The faces of the formed aggregates were ground and polished to give long rectangular prisms whose sides and ends are square and parallel to ±0.OO2 cm. The density values of each specimen were obtained by measuring the dimensions and weight at 25°C. X-ray density of ZnO at 25°C is 5.676 g/cm3, and this value is used in the evaluation of the elastic moduli of this material at the zero-porosity point.

2.2. Polycrystalline Ti02 (Rutile)

Ten specimens used in this work are polycrystalline aggregates made of high-purity rutile crystals, and these were produced by a method similar to one described in 2.1. The size of these specimens were in the neighbor­hood of 0.50 cm in height, 0.95 cm in width, and 7.62 cm in length, sufficiently large enough to make accurate resonance measurements.

Titanium dioxide precipitated from sulfate or chloride solutions at an ambient temperature was amorphous even after drying at 100°-150°C. A hydrolysis by a direct boiling of chloride solutions produced rutile as the initial crystalline product. The resulting con­glomerate was high-purity Ti02 and composed of both rutile and anatase as revealed by x-ray studies. The conglomerate was ground to give an average particle size of about 10 p., and the resulting powder was heated to 900°-950°C in a positive oxygen pressure for 12 h. The prepared "white" powder had an x-ray pattern

corresponding to that of rutile. A typical spectro­chemical analysis showed that the chemical purity of this powder was better than 99.9% Ti02, and the minor constituents detected were 0.03% Pb, 0.02% Fe, 0.01 % Zn, and others less than 0.01 %. The powder was pres­sure-sintered in a resistance hot press using a graphite die lined with molybdenum plates. The polycrystalline aggregates thus produced had bulk densities approach­ing the x-ray density of 4.250 g/cm3• Variable porosity in the aggregate was attained by varying the hot­pressing parameters, and in the present work the porosity ranged from 0.6% to 6%. A wide spread of the porosity thus enabled us to study a porosity dependence of the elastic moduli of this material. All the faces of the aggregate specimens were ground and polished to give long rectangular prisms whose sides and ends are square and parallel to ±0.OO2 cm.

2.3, Polycrystalline a-Al20 a

Two types of polycrystalline alumina specimens were used in the present program. One type includes those specimens of polycrystalline a-AbOa which were pro­duced by hot-pressing high-purity, fine-grained alumina powders and used in the earlier work.6,lI The other type is "Lucalox" alumina specimens obtained from the General Electric Company. The Lucalox alumina is a high-density polycrystalline aluminum oxide available on a commercial scale. It is 99% AbOa and contains small amounts of MgO as the major impurity. The density of the Lucalox specimens was 3.971 (±0.OO3) / g/cm3 at 298°K, and this may be compared with an x-ray density of 3.986 g/cma for a-AbOa.

The chemical purity of the hot-pressed alumina

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11 D. H. Chung, Bull. Ceram. Res. 297,1 (1961).

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VRH APPROXIMATION FOR ZnO, TiO., AND a-AltO. 2779

specimens was better than 99.9% AbOg in all cases, and among the minor constituents 0.004% Si, 0.002% Na, and 0.002% Mg were predominant. All the specimens were composed of a-Ab03, which is the most stable form of aluminum oxide. The porosity in the specimens varied from about 0.1 % to 6% from one specimen to another, and this spread of porosity over a wide range allowed one to study a porosity-dependence of the elastic moduli of this material. Seventeen such speci­mens were considered in the present program.

2.4. Isotropy Tests on Polycrystalline Specimens

Every specimen considered in the present work has been subjected to an isotropy test. Using ultrasonic velocity measurements, each specimen was checked for the elastic isotropy by placing the transducer crystal on the specimen surfaces (thus changing the direction of wave propagation). Most specimens with relatively low porosity were found to be isotropic for both the longitudinal and shear waves, and variations in sound velocities were within 1 %. These variations are in the same order of magnitude as the expected experimental error. For the specimens with the porosity greater than 4%-5%, the isotropy test was disturbed by uncer­tainties arising from variations in the specimen micro­structure. The deviations in the elastic modulus, due to these variations, were significant; frequently, the uncertainties arising from these variations amount to 2%-3%, depending on the specimen. The important source of these deviations is directly associated with effects of pores, for which no allowance is made in the usual application of the classical theory of elasticity to homogeneous elastic bodies.

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3. EXPERIMENTAL METHODS

The present work utilized primarily a modified Forster-type resonance method to measure resonance frequencies of a bar-shaped specimen.11- 13 As a com­plimentary method, an ultrasonic pulse-superposition technique14 was used to measure sound velocities of those specimens with the low porosity. Since the litera­ture dealing with the experimental elasticity describes both of these methods, a detailed description of these methods will not be repeated here.

4. EXPERIMENTAL RESULTS

Figures 1--6 are plots of the measured isotropic moduli on polycrystalline ZnO (Figs. 1 and 2), Ti02 (Figs. 3 and 4), and a-Ab03 (Figs. 5 and 6), as a function of specimen densities. These plots enable one to observe the porosity dependence of the elastic moduli. In all cases, the measured elastic moduli were reproducible within 1.5% of the value entered in the figures. The scatter, as indicated by a flagmark, shows variations in the elastic modulus when it was measured as the specimen was rotated along its length. Two probable reasons for the observed scatter are mentioned: (1) The density of the aggregate specimen may not be homo­geneous from one point to another, and (2) There may be preferred orientations of the crystal grains within the specimen. The effect of (1) is evident, and it is very serious if the specimen concerned is highly inhomo­geneous. The effect that may be introduced by (2) is difficult to estimate, mainly because the influences of

12 J. B. Wachtman, Jr. and D. G. Lam, Jr., J. Am. Ceram. Soc. 42, 254 (1959).

13 S. Spinner and W. E. Tefft, Proc. ASTM 61, 1221 (1961). 14 H. J. McSkimin, J. Acoust. Soc. Am. 37, 864 (1965).

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TABLE 1. Literature values of single-crystal elastic constants of ZnO, TiO:! (rutile), and a-AhO,.

Material and Elastic constantS" reference Density

temperature (KO) (g/cm3) 11 12 13 14 33 44 66 References

ZnO(298) 5.676 Cij 20.97 (±0.02) 12.11 (±0.02) 10.51(±0.02) 21.09(±0.02) 4.247(±0.04) 4.429(±0.04) 62Blb

Sij 7.855 -3.431 -2.205 6.939 23.546 22.573

TiO:!(298) 4.260 Cij 24.8(±0.8) 20.0(±1.0) 14.0(±1.0) 45.2(±O.8) 12.0(±0.3) 16.0(±1.0) 6OV1e

S;,j 11.80 -9.03 -0.86 2.74 8.33 6.25

TiO:!(298) 4.264 Cij 27.3 17.6 14.9 48.4 12.5 19.4 60Bld

Si,j 6.55 -3.76 -0.86 2.59 8.00 5.16

TiO:!(298) 4.250 Cij 26.60(±0.66) 17.33(±0.71) 13.62(±0.81) 46.99(±0.81) 12.39(±0.07) 18.86(±0.50) 62W1·

$i; 6.788 -4.017 -0.799 2.592 8.072 5.302

a-AhOa(300) 3.986 eii 49.6 10.9 4.8 3.8 50.2 20.6 19.35 59Mlf

SiJ 2.18 -0.50 -0.16 -0.49 2.02 5.04 5.36

a-AI2<h(298) 3.986 CiJ 49.68(±0.18) 16.36(±0.18) 11.09(±0.22) -2.35(±0.03) 49.81 (±0.14) 14. 74 (±O. 02) 16.66(±0.4O) 60Wlg

Sii 2.353 -0.716 -0.364 0.489 2.170 6.940 6.138

a-AbO, (300) 3.986 Ci; 49.02 16.54 11.30 -2.32 49.02 14.54 16.24 63B1 h

Sid 2.404 -0.747 -0.382 0.503 2.216 7.038 6.302

a-AI2<h(300) 3.986 Cij 49.20 16.84 11.64 -2.34 49.20 14.68 16.19 63B1 h

Sii 2.408 -0.756 -0.391 0.504 2.218 6.973 6.328

a All eij are in units of 1011 dyn/em' and the $,j are in units of 10-13 em'/dyn. f 59Ml: W. G. Mayer and E. H. Hiedemann, Aeta Cryst. 12. 1 (1959); see also J. Aeoust. Soe. Am. b 62Bl: T. B. Bateman, J. Appl. Phys. 33, 3309 (1962). 30,756 (1958). • 6OVl: G. L. Viek and L. E. Hollander, J. Acoust. Soc. Am. 32, 947 (1960). c 60Wl: J. B. Wachtman, Jr .. W. E. Tefft. D. G. Lam. Jr., and R. P. Stinchfield, J. Res. Nat!. d 60Bl: F. Birch, J. Geophys. Res. 65, 3855 (1960). Bur. Std. 64A, 213 (1960). • 62Wl:J. B. Wachtman,Jr .. W.E. Tefft. and D.G.Lam,Jr .• J. Res. Nat!. Bur. Std. 66A, 465 (1962). h 63Bl: B. T. Bernstein. J. Appl. Phys. 34, 169 (1963).

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VRH APPROXIMATION FOR ZnO, Ti0 2 , AND a-AI 2 0. 2781

TABLE II. The Voigt-Reuss-Hill (VRH) moduli of ZnO, Ti02 (rutile), and a-Aba •. •

Material Referenceb K YRH GYRH EYRH Ky-KR Gy-GR Ey-ER

ZnO 62B1 14.358 4.555 12.359 0.005 0.046 0.114

Ti02 6OV1 20.909 9.014 23.520 0.582 4.211 9.752 Ti~ 6OB1 21.520 11.354 28.940 0.916 2.465 5.535 Ti02 62W1 20.666 11.171 28.363 0.741 2.530 5.607

a-Aba. 59M1 24.799 16.454 40.421 0.003 0.591 1.190 a-Aba. 60W1 25.115 16.336 40.274 0.048 0.535 1.098 a-Aba. 63B1 25.012 16.002 39.566 0.052 0.514 1.062 a-Aba. 63B1 25.290 16.024 39.687 0.051 0.500 1.036

• All moduli are in units of lO" dyn/cm'. b See Table I for the complete references .

the shape of grain boundaries and of pores ar-e unknown. However, it is conceivable that the scatter in the meas­ured elastic modulus might result from possible varia­tions in the local concentrations of stresses due to the pores and the elastic anisotropy of the constituting crystals. For each material, a graphical extrapolation of the measured moduli to the x-ray density point results in the following values of the isotropic poly­crystalline elastic moduli: For ZnO, G=4.56 (±0.04) and E= 12.35 (±0.09), for Ti02, G= 11.15 (±0.09) and E=28.42 (±0.14), and for a-Ah03, G=16.35 (±0.12) and E=40.39 (±0.31), where all the moduli are in units of 1011 dyn/ cm2• These values represent the polycrystalline isotropic elastic moduli of the materials, and they are to be compared with the Voigt-Reuss-Hill (VRH) moduli calculated from the corresponding single-crystal elastic constants.

Comparing the present values with the literature data of other investigators, we note the following: For

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ZnO (p=5.621 g/cm3), G=4.417, E= 11.985, K8= 13.936 in units of 1011 dyn/cm2 (Ref. 9); for a-A120 3 (p=3.972 g/cm3), G=16.132, E=39.886, and K8= 25.206 in units of 1011 dyn/cm2 (Ref. 8). Although there are apparent differences in specimen densities and purities, the agreement in values is good.

5. SINGLE-CRYSTAL ELASTIC CONSTANTS AND CALCULATION OF THE VOIGT-REUSS-HILL

(VRH) MODULI

The single-crystal elastic constants of all the materials considered in the present work are known. For Ti02

and a-Ah03, there is more than one set of single-crystal elastic constants in the literature, and in some cases, they differ in values considerably from one author's result to another. This situation presents a problem of selecting the elastic constants realistic of the solid. Table I lists more reliable sets of single-crystal elastic constants.

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2782 D. H. CHUNG AND W. R. BUESSEM

TABLE III. Comparison of measured polycrystalline elastic moduli with the VRH moduli calculated from the single-crystal elastic cons tan ts.

Elastic modulus· Material and referenceb Young's shear Longitudinal

ZnO 12.359 4.555 20.431 Single-crystal (62B1) polycrystalline 12.35(±0.09) 4.56(±0.04) 20.39(±0.12)

28.940 11.354 36.658 28.363 11.171 35.563

Single-crystal (60B 1) Single-crystal (62W1)

polycrystalline 28.42 (±0.14) 11.15(±0.09) 35.92(±0.14)

a-AbOa 40.274 16.336 46.896 39.627 16.013 46.601

Single-crystal (60W1) Single-crystal (63B 1) c

p0lycrystalline 4O.39(±0.31) 16.35(±0.12) 46.80(±0.17)

• All values are at 298°K and they are in units of 10" dyn/cm'. b See Table I for the complete references.

Using these single-crystal data, the isotropic bulk and shear moduli are calculated through the Voigt­Reuss-Hill (VRH) approximation as follows:

( 1)

and

GvRH=G*= (Gy +Ga)/2, (2)

where the subscripts V and R designate the Voigt and the Reuss averaging schemes and they represent the maximum and minimum limits of the true polycrystal­line elastic modulUS The rigorous expressions for the bulk modulus K and the shear modulus G differ accord­ing to the crystal symmetry and these can be found from Hill's paper.

Table II lists isotropic bulk and shear moduli calcu­lated from the corresponding single-crystal data using the VRH approximation. Young's modulus E* was calculated from K* and G* using the relation of the isotropic elasticity, and the results have been entered in the table. Also entered are the differences in the limiting moduli, (Gy-GR ) and (Ey-ER ), in addition to (K y-K R), since they are a measure of the magnitude

16 R. Hill, Proc. Phys. Soc. (London) 65,349 (1952).

C These values are the average of two values given by Bernstein •

of the elastic anisotropy possessed by the crystal.16

Table III compares experimental polycrystalline elastic moduli with calculated values of isotropic elastic moduli for ZnO, Ti02, and a-AbOa.

6. DISCUSSION AND CONCLUSION

Examining Table III, it is clear that the experimental polycrystalline elastic moduli are in good agreement with the isotropic VRH moduli calculated from their single-crystal data. The differences observed between the experimental and calculated values are in every case within the scatters in the calculated VRH moduli resulting from the differences in the single-crystal elastic constants of one author to another. Such an agreement seen here indicates the success of the VRH approximation for hexagonal ZnO, tetragonal Ti02,

and trigonal a-AbOa.

ACKNOWLEDGMENTS

One of us (DHC) is grateful to W. B. Crandall, under whose direction this work was originally initiated. This work was in parts supported by the U.S. Office of Naval Research.

16 D. H. Chung and W. R. Buessem, Anisotropy in Single­Crystal Refractory Compounds (Plenum Press, Inc., New York, 1968) .

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