The elastic properties of monoclinic ZrO2

Physica B 150 (1988) 230-233 North-Holland, Amsterdam

THE ELASTIC PROPERTIES OF MONOCLINIC ZrO2* M.V. NEVITT, S.-K. CHAN, J.Z. LIU,* M.H. GRIMSDITCH and Y. FANG* Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA

We have made ultrasonic and Brillouin scattering measurements at ambient temperature on small single crystals of monoclinic ZrO v Using these data, we have established various features of the angular anisotropy in the acoustic longitudinal wave and shear wave velocities, and we have computed the elastic stiffness and compliance moduli. We observe shallow minima in the transverse shear wave velocity in directions parallel to the a and c crystallographic axes. We anticipate that acoustic shear wave softening will be observed in these directions when measurements are made at temperatures close to the monoclinic-tetragonal transformation.

1. Introduction

Clarifying the microscopic origins of phase stability and transformation characteristics in Z r O 2 is a goal of major proportions from both scientific and technological points of view. The weakly first-order tetragonal-monoclinic transformation represents an excellent subject for the study of the basic physics of the partial softening of an acoustic mode (possibly driven by a coupled optic mode), occurring premonitorily to a martensitic shear transformation. From the technological perspective, the martensitic transformation in ZrO2, including its nucleation characteristics, plays a central role in the transformation toughening process employed to strengthen ZrO2-containing ceramics. On cooling, the transformation appears to be driven by the approach to condensation of symmetry-breaking, long wavelength phonon modes in the tetragonal crystal lattice. Further details concerning the soft mode identity and behavior are provided in other papers at this conference [1, 2]. Single-crystal elastic properties and anomalies are therefore important in determining interphase crystallographic orientations and symmetry relationships involved in the transformation. * Work supported by U.S. Department of Energy, Basic Energy Sciences-Materials Science, under contract W-31109-Eng-38. * Visiting Scholars from the People's Republic of China.

Moreover, the elastic properties are essential to the development of a general theory of nucleation in systems undergoing a first-order displacive transformation, a theory that can be applied to other materials as well as to ZrO 2. We are attempting to measure and interpret the elastic properties of tetragonal and monoclinic single-crystal ZrO 2, thereby filling a gap that has seriously impeded a quantitative understanding of the transformation process. Thus far we have confined our effort to growing monoclinic crystals, measuring sound velocities in various directions at ambient temperature and calculating the thirteen monoclinic elastic stiffness and compliance moduli.

2. Experimental methods

2.1. Monoclinic single crystal growth Two of us (JZL and YF) have developed a flux-growth method for making optically clear monoclinic single crystals that have maximum dimensions of 7 to 10mm on a side and are either completely or almost completely twin free. Fluxes of PbF2-KF and PbF2-B20 3 have been used with more or less equal success. Under the combined influences of evaporation and slow cooling, growth takes place over a period of about 2 weeks in a cylindrical platinum crucible with a (maximum) top temperature of 1060°C

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M.V. Nevitt et al. I Elastic properties of monoclinic Z r O and a vertical t e m p e r a t u r e gradient of - 4 to - 5 ° C / c m . The yield is of the order of one hundred crystals of varying size. This technique results in the inclusion of 1 - 2 atomic percent Pb, which we believe is in solid solution in the crystals. The crystals used in the study were carefully chosen and cut on the basis of microscopic and Laue-pattern examinations. To the extent feasible, the selected crystals contained no discernable internal twin boundaries. W h e n suitably sized crystals that were totally free of twinning could not be found, the following selection criterion was used: a crystal chosen for ultrasonic m e a s u r e m e n t s consisted of at least 99 volume percent of a single orientation, estimated visually, with the other twin orientation(s) occuring as thin (sub-micron) platelets randomly dispersed roughly a millimeter apart within the crystal volume. We believe that this selectivity was adequate to assure that we were sampling a velocity characteristic of a single, unique orientation. In the Brillouin scattering m e a s u r e m e n t s this precaution is also sufficient, since the laser spot is typically 100 microns in diameter.

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and compliance moduli. In calculating p V 2 values we used the density of monoclinic Z r O 2 derived from X-ray measurements, p = 5.836 g / c m 3 [4].

3. Results 3.1. Elastic anisotropy In figs. l(a) and l ( b ) we show data points and

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2.2. Measurement methods We used the McSkimin-Fisher phasecomparison method [3] to measure ultrasonic wave velocities in single 'crystals cut to provide path lengths of at least 1.5 m m in various directions relative to the monoclinic axes. The experimental method has b e e n extensively d o c u m e n t e d and its description will not be repeated here. We m a d e a total of fifty sets of velocity m e a s u r e m e n t s for the longitudinal and transverse branches in thirteen directions. We employed the Brillouin-scattering technique in back-scattering, 90 ° scattering and platelet geometries to determine velocity values and refine the angular dependence of the longitudinal and transverse acoustic waves, concentrating on velocities in the basal plane. Finally, we combined algebraic computation with a c o m p u t e r based nonlinear least-squares fit to the data, based on the usual Christoffel equation, to calculate values for the thirteen monoclinic stiffness

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Fig. 1. (a) Measured pV 2 values for the longitudinal branch in the monoclinic basal plane of ZrO 2. Solid line is the computer-generated fit to the data. (b) Measured pV 2 values for the two transverse branches. T 1 branch is associated with the quasi-shear wave with propagation direction in the basal plane and displacement vector in the perpendicular [010] direction. T 2 branch has the propagation direction and the displacement vector in the basal plane. Dashed and solid lines are, respectively, computer-generated fits to the T 1 and T 2 data.

232

M.V. Nevitt et al. / Elastic properties o f monoclinic ZrO 2

computer-generated p V 2 v s . 0 traces for the (010) monoclinic basal plane. Fig. l(a) shows the angular dependence of p V ~ , where VL is the velocity of the longitudinal wave. Fig. l(b) shows the corresponding p V 2 plots for the two transverse branches. T 1 (dashed line) is the quasishear wave having a propagation direction in the basal plane and a displacement vector in the perpendicular [010] direction. The T 2 branch has both the propagation direction and the displacement vector in the basal plane. The L and T I branches show angular dependences that can be associated primarily with the two-fold monoclinic symmetry. More noteworthy is the angular dependence of the T 2 branch, where there are p V 2 minima in directions that are coparallel with the a and c crystallographic axes. The symmetry relationships between the monoclinic and tetragonal forms suggest the likelihood that in these directions we will observe acoustic shear wave softening when these measurements are repeated at temperatures closer to the monoclinic-tetragonal transformation. We are now planning to look for this effect using doped single crystals to depress T c.

3.2. Elastic constants We list in table I the ambient-temperature elastic stiffness and compliance matrices for Z r O 2 determined in this study, and estimates of the uncertainty in each stiffness modulus. We are not aware of previous measurements with which to compare our values. It is of interest to compare the stiffness moduli with the calculated values of Cohen et al. [5]. Excellent to moderate agreement exists between the measured and calculated values of the diagonal moduli. Discrepancies in sign, as well as larger differences in magnitude, are observed among the off-diagonal moduli. Using the calculated moduli of ref. [5], we have generated p V 2 plots corresponding to those of fig. 1. The traces are qualitatively similar, but shifted in orientation. For example, the (somewhat shallower) minima in the pV~2 plot are displaced approximately 45 ° relative to the corresponding ones in fig. l(b).

Table I E l a s t i c c o n s t a n t s of m o n o c l i n i c Z r O 2 at a m b i e n t t e m p e r a t u r e Stiffness m o d u l i (1012 d y n / c m 2) C , = 3.58 -+ 0.45 (?22 = 4.26 -+ 0.11 C33 = 2.40 -4- 0.28 C44 = 0.991-4- 0.017 C55 = 0.787 -+ 0.10 (?66 = 1.30 -+ 0.021

1.44 --- 0.29 Cl3 = 0.670-+ 0.64 C15 = - 0 . 2 5 9 -+ 0.047 C23= 1.27-+0.25 C25 = 0.383-+ 0.077 C35 = - 0 . 2 3 3 -+ 0.047 C46 = - 0 . 3 8 8 ~'~ 0.10 C12 =

C o m p l i a n c e m o d u l i (10 -~2 c m 2 / d y n ) $11 = 0.345 S2z = 0.355 533 = 0.537 544 = 1.14 $55 = 1.53 566 = 0.873

$12 $13 $15 $23 $25 $35 $46

= = = = = = =

-0.128 -0.0116 0.173 -0.178 -0.268 0.242 0.342

3.3. Estimates o f errors The stiffness moduli in table I are reported uniformly to three places and corresponding estimates of error to two places. It is clear that the number of significant figures differs among the moduli. The estimated errors, which vary considerably, arise from two general sources, experimental and computational. These will be discussed briefly; a more detailed treatment will be published elsewhere. Experimental error sources: Ultrasonic. Uncertainties in measured velocities were estimated by a statistical analysis of individual data sets and by noting sample-to-sample variability. The probable relative error from this source was between 2 and 12%. An additional error arises from the estimated (from Laue patterns) 1° uncertainties in the angular orientation of the ultrasonic path, an error whose magnitude depends on the orientation involved. We made sample calculations to show that a p V 2 quantity that we associate with a particular direction does not have an uncertainty greater than ---1% arising from this source. Other uncertainties such as errors in path length and variations in crystal density were

M.V. Nevitt et al. / Elastic properties o f monoclinic Z r O 2

233

evaluated and d e e m e d to be insignificant relative to the foregoing errors.

4. Summary

E x p e r i m e n t a l error sources: Brillouin scattering. The relative error in a velocity measure-

We have demonstrated ambient temperature anisotropy in the elastic stiffness in the basal plane of the monoclinic Z r O 2 crystal. This anisotropy is principally seen in the form of shallow minima in the transverse shear wave velocity in directions parallel to the a and c axes. In these directions we can anticipate shear acoustic-wave softening at temperatures in proximity to the monoclinic/tetragonal transformation. We have determined with moderate precision the thirteen elastic stiffness moduli and corresponding compliance moduli for monoclinic Z r O 2.

ment was assumed to consist of a 5% random error and a 1% systematic error, the latter associated with physical limitations in orienting the crystal in the laser beam and with possible uncertainty in the refractive index. We assigned a relative error of -+12% to each of the p V 2 values derived from the Brillouin scattering measurements. C o m p u t a t i o n a l error sources. Moduli derived uniquely from a single velocity measurement (C22, Can, C66) have errors arising only from the aforementioned velocity-measurement source. Most of the remaining moduli (C~1, C33, C55, C13 , C15 , C35 , C46) were calculated from algebraic expressions based on the Christoffel equation in which two or more velocities and moduli appear. Relative errors of up to +--25% can be generated in these calculations, particularly where differences are taken between velocities or pairs of velocities. The remaining stiffness moduli (C12 , C23 , C25 ) were derived from a nonlinear least-squares fitting program based on the Christoffel equation and incorporating appropriate aggregates of the ultrasonic and Brillouin data. These moduli have errors arising from all of the sources described above, plus a fitting error, the components of which are not easily analyzed quantitatively. Somewhat arbitrarily, we assigned a -+25% relative error to these three moduli, assuming that their uncertainties are probably no larger or smaller than those of the other off-diagonal moduli. The compliance moduli, also reported to three places, were computer generated. The matrix inversion involved does not provide a corresponding error estimate for these moduli.

Acknowledgements We wish to express our thanks to Mr. J.B. Downey for technical assistance in the ultrasonic measurements, to Mr. J.-S. Pan, Visiting Scholar from the People's Republic of China, for computational assistance in crystal orientation and to Mr. R. Bhadra for assistance in some of the Brillouin experiments.

References [1] S.-K. Chan, Physica B 150 (1988), these Proceedings. [2] R.C. Garvie and S.-K. Chan, Physica B 150 (1988), these Proceedings. [3] E.S. Fisher and H.J. McSkimin, J. Appl. Phys. 29 (1958) 1473. See also H.J. McSkimin, Proc. Nat. Electronics Conf. XII, Chicago, IL (October 1956). [4] J. Adam and M.D. Rogers, Acta Crystallogr. 12 (1959) 951. [5] R.E. Cohen, M.J. Mehl and L.L. Boyer, B 150 (1988), these Proceedings.