The elastic moduli and their pressure and temperature derivatives for calcium oxide

1. Phys. Chrm. Solids, 1977, Vol. 38. pp. 705-710.

Pergamon Press.

Printed in Great Britain

THE ELASTIC MODULI AND THEIR PRESSURE AND TEMPERATURE DERIVATIVES FOR CALCIUM OXIDE ALAN L. DRAGOO? Institute for Materials Research, National Bureau of Standards, Washington, DC 20234,U.S.A.

IAN L. SPAIN Department of Chemical Engineering, University of Maryland, College Park, MD 20742,U.S.A. (Received 20 April 1976; accepted in revised form I5 October 1976)

Alt&act-The room temperature adiabatic elastic moduli of CaO were obtained using a pulse superposition technique to measure the ultrasonic velocities. The elastic moduli were found to be (in GPa) at 298K. Cl1

226.2f 0.9

Cl2

CU

62.420.9 80.6kO.3

The pressure derivatives of the elastic moduli, which were measured at and below room temperature, decrease with decreasing temperature. The temperature derivatives of the elastic moduli were also obtained for several temperatures below room temperature. 1. INTRODUCTION

Values of the room temperature adiabatic elastic moduli of CaO reported by previous investigators[l, 21, see Table 1, do not agree. Moreover, there is little agreement between the values of the isothermal bulk modulus, Br, calculated from these two sets of elastic moduli and the other empirical values of Br for CaO. In addition, the uncertainties of most of the measurements are high. Son and Bartels[l] and Hite and Kearney[2] measured the transit time by pulse-echo methods to obtain the elastic moduli. Their differing densities for CaO alone do not account for the different values of the elastic moduli which they reported. These differences appear to be associated with different observed values of the acoustic transit time: using the specimen lengths and acoustic modes of Hite and Keamey, their transit times show delays ranging from 0.08 to 0.19 ps with respect to transit times calculated from the results of Son and Bartels. These delays are nearly the same as the delays which are observed in pulse superposition measurements if the radio-frequency crests in successive pulses are shifted by one-period (radio-frequency, j, - 10 MHz). The precision of transit times obtained from pulse superposition measurements[3] can be on the order of 0.02%, or 0.001 ps for transit times on the order of 5 ps. If the correct in-phase condition can be identified in pulse superposition measurements, then significant improvements in the values of the elastic moduli for CaO can be realized. The adiabatic elastic moduli can be used to calculate tWork based in part on Ph.D. thesis in the Chemical Physics Program, University of Maryland (1975).

Br by means of a thermodynamic identity. The calculated values of Br obtained from the elastic moduli of Son and Bartels and from those of Hite and Kearney are compared in Table 1 with a value calculated from Soga’s[4] measurement of the adiabatic bulk modulus, I&, and with Bridgman’s[5] and Weir’s[6] measurements of Br. The result of Son and Bartels is in agreement with that of Weir. The value of BT obtained for Soga’s work is lower than the two values calculated from the elastic moduli and may represent the problem of selecting the correct in-phase condition. The anomalously low value of Br reported by Bridgman is apparently due to measurements of the compression of compacted powder slugs, without a contining liquid, so that much of the observed compression resulted from the shrinkage of voids between the particles. In this work, more precise values of the room temperature adiabatic elastic moduli (co, clz, c,.,) of CaO were obtained: ambiguities in the determination of the in-phase condition were resolved, and transit times with uncertainties of 0.08%, and better, were achieved. Son and Bartels reported values for the pressure derivatives of the elastic moduli obtained at room temperature. These measurements have been extended in the present work to below room temperature in an attempt to observe the change in the pressure derivatives with temperature. From measurements of the acoustic transit time over the range of temperatures from 298 to 195K, Bartels and VetterD] obtained the temperature derivatives of the elastic moduli. These measurements have been extended here, to as low as 73 K in the case of the combined modulus cl,. 705

706

A. L. Dwcoo and I. L.

SPAIN

Table 1. Previous determinations of the elastic moduli of cat) at 298K Principal Moduli (GPa? Investigators

Nature of Specimens

Method

Moduli Determineda

=11

c12

c44

Isothermal Bulk Modulus BT (CPa)

Son and Bartels [l]

Single crysta1s; p = 3345 kg/m3 (x-ray); 99.0 to 99.9% pure; three specimew

Pulse echo (pe)

cil, c', c44

223

59

81

112 . gc

Hite and Kearney [21

Single crysta1; p = 3295 kg/m3; purity? One sample

Pe

=ll'

‘ii,

202

61

74

107

saga I41

Vacuum hotpressed; p = 3285 kg/m3

pulse superposition

BS, CS

_-

_-

--

104.92

Bridgman [51

Compacted Slug

compression

--

_-

__

21.5

Weir [6]

Compacted slug

Hydrostatic BT compreasion

‘44

BT

114 f 5d

a BS = (cll+2c12)/3; Gil = (cl1 + ~12 + 2~44)/2; C' = (c,. - ~,,)/2. LI

b1

I‘

-2 GPa = 109 Pa = 10 Mbar.

c Propagated error. d

Standard deviation. 2. -AL.

DETAILS

(a) Samples The CaO single crystal specimens were provided by R. Rice of the Naval Research Laboratory, who obtained them uncut from W. & C. Spicer, Ltd., Cheltenham, England. One specimen with a [lOO]orientation and one with a [llO] orientation were used. Each specimen was repolished several times during the course of the work so that measurements were made with a series of specimen lengths in the range of 9.6-12.3 mm. The specimens were light amber in color and transparent. Semiquantitative spectrochemical analysis showed that the material contained the following major impurities: Si, ~0.005%; Al, ~0.03%; Fe, <0.04%; Mg, 0.4%; Na, 0.01%; K, CO.07; Ti,
one another. The final polish was performed with 6 micron diamond paste. Coaxially plated quartz transducers were used: 10 Mhz X-cut for the longitudinal modes; 7 MHz Y-cut for the transverse modes. Nonaq stopcock grease (Fisher Scientific Co.)t was used for most of the bonds between specimen and transducer;salol was used for a few of the bonds. (b) Measurement of the elastic moduli Ultrasonic apparatus similar to that described by McSkimin[3] was used to measure the time, T, between superimposed pulses and echoes. For this technique, a pulsed rf-signal with frequency f is applied to a transducer bonded to one end of the specimen. The period, 7, of the pulses is precisely controlled with a stable L-C oscillator whose frequency, Y= UT, is measured with a frequency counter. The correct in-phase condition is realized when both the pulse-echo envelopes and similar d-peaks are made to coincide. McSkimin gave a procedure using a variation of the d-frequency to determine the pulsing frequency at which the d-peaks were correctly superimposed. This in-phase condition is called the n = 0 condition, where n is an integer which denotes the number of cycles by which similar #peaks in two successive pulses are shifted in the superposition of pulses and echoes: n = 0 for the correct superposition, n 0 if successive pulses lag the echoes. For the present measurements McSkimin’s procedure did not distinguish the n = 0 from the n = 1 condition in some instances. Consequently, a new procedure was de-

707

Elastic moduli

mic derivative relationship between Y and c is:

veloped using two specimens with the same sonic orientation but with different pathlengths. The details of this procedure are given elsewhere[9]. This procedure begins with the same fundamental transit-time condition use by McSkimin, and its results agreed with those of McSkimins’s procedure where the latter gave conclusive results. Moreover, it also conclusively identified the n = 0 condition when McSkimin’s procedure gave ambiguous results. To compute the sonic velocity, u, in the specimen, the specimen length, L, was combined with the round-trip delay time, 8, obtained from the analysis of the n = 0 condition. Since the bond phase-angle, y, was very small, the contribution ($36Ofl of the bond to the observed delay time for one round-trip of a sonic pulse was negligible with the result that u = 2Lv(n = 0). For a crystal with cubic symmetry the formulas relating the elastic moduli to the sonic velocities for the longitudinal (L) and transverse (T) modes for [ 1001and [ 1lo] sound waves are well known; see, for example, eqn (1) in Ref.

dlnv ldlnc -_=----___ dX 2 dX

ldlnp 2 dX

ldInp+d= 2 dP dP

dT

c’

[1101 T2

c’

[1101

Tl

11101

L

=44

‘ii

s.~ALREWLTs (a) Room temperature elastic moduli To avoid the anomalous transducer phase shifts, the room temperature elastic moduli were obtained only from sonic velocities measured for the n = 0 condition. These moduli are given in Table 2. A specimen density of 334Okg/m’ was used in these calculations. The temperature derivatives (dc/dT) measured by Bartels and Vetter[7] were used to adjust the values of the elastic moduli to 298 K. The elastic moduli given in the final

T, K

Expt'l Value at T, GPs

291.4

81.8f.Zb

81.8t.3=

Adj. values at 298 K GPs

298.2

82.0?.2

a1.9t.3

296.8

80.6t.2

80.5f.3

298.4

225.1t.2

224.8k.3

Derived

values 226.2t.P

=11

62.4k.9

=12

117.ot.7

%

s In addition to adjusting 298 K, small corrections and bond thickness. b

Propagated

error

'Error includes note lls@'.

based

the experimental values of the moduli to were applied for the effects of diffraction

on standard

uncertainties

06.

where a is the volume thermal expansion coefficient. The derivative (dc/dX) can be obtained using eqn (1) and the value of c which can be found by a reiterative method for conditions other than P = 0 and room temperature. Determinations of (d In u/dP) and (d In v/d T) may be made with n = 1 or n = 2 superpositions since the systematic error incurred may be less than other experimental errors and since the advantage of working with n = 1 or n = 2 echoes, which may be stronger than n = 0 echoes, may outweigh this error. For n = 1, the error amounts to 3 to 4%; for n = 2, 6 to 8%.

Table 2. The room temperature adiabatic elastic moduli of CaO

T2

(2)

ldlnpl dlnL__ 1 a, 2 dT

derivatives A two-stage high-pressure apparatus with argon (neon, for temperatures below 15OK) as the working-fluid was used to subject the specimens to pressures up to 600MPa. The pressures were read from a calibrated Bourdon gauge. The high-pressure cell was located in a cryostat for temperature control, and the temperature was monitored with a calibrated silicon diode located in a well at the top of the cell. The derivatives (dc/dP) and (dc/dT) of the elastic modulus c can be determined directly from the change in Y with pressure, P, or with temperature, T, respectively. For n = 0 and negligible bond phase-angle the logarith-

[llol

1 6 B,‘; 0

and for X = Z’,

(c) Measurement of the pressure and temperature

Elastic Modulus

(1)

where X = P or T. For X = P

PI.

Mode

dlnL dX

deviations.

in the corrections

mentioned

in foot-

A. L. Da~oooand I. L. SPAIN

708

column also were adjusted for the excess velocity due to diffraction and to bond thickness. The diffraction effect [lo] was estimated to contribute about 0.1% to the observed elastic moduli for all modes. For the phase angle effect due to a finite bond thickness, the increase of the moduli was estimated to be 0 to 0.06% for the [llO] L-mode and 0 to 0.8% for the [llO] Tl- and TZmodes. Median values of 0.03% and 0.04%, respectively, were used to adjust the results for the bond thickness effect. The error in the density determination contributes about two-thirds of the known error in the values of c:,, c’ and c,. The remainder of the error is ascribed to traction effects, bond thickness and small variable phase shifts arising from the drift of the &oscillator. (b) Pressure and temperature derivatives of the elastic moduli For each pressure or temperature run a linear regression analysis[l l] was used to obtain a slope from In v vs P or In v vs T data, respectively. The standard deviation of the slope was obtained by proportioning the standard deviation of the fit between the slope and the intercept. The standard deviations of the slope were in the range 5-80% of the observed value of a pressure derivative and 4-15% of a tempe~ture derivative. The logarithmic pressure derivatives’of Y are plotted against temperature in Fig. 1 for the [lo01 L-mode, the [l lo] L-mode ‘and the [l LO] TZ-mode. Least-squares fitted lines are included in the figure to indicate the trends of the data for the three modes. With the exception of the [ 1lo] L-mode, (d In v/dP) increases with increasing temperature. For the [IlO] L-mode the pressure derivative appears to be nearly independent of

32

!

I

I

temperature. A relationship between the temperature coefficients of (din v/dP) and (d In cfdP) can be obtained by further differentiations with respect to T, both of eqn (l), with X = P, and of eqn (2). Since (dB,ldT) < 0, it follows from the present results that (din c,,/dP), (d In c{,/dP) and (d In c’/dP) have positive temperature coefficients. Pressure derivatives of the elastic moduli are listed in Table 3 as (d In cldP) in column 2 and as (dc/dP) in column 3. For each mode the room temperature values were calculated after averaging the room temperature values of (d in YldP) weighted with respect to the number of data points in the co~es~n~ng pressure run. The value of BS given in Table 2 was used in eqn (2) to calculate (d In c/dP) at room temperature since & equals BT to within the precision of the pressure derivatives. For the calculation of the low temperature values of (d In c/dP) and (dc/dP) estimates of the low temperature values of & and of the elastic moduli were used by linearly extrapolating the respective room temperature quantities to low temperatures using the temperature derivatives of Bartel and Vetter[7]. The room temperature values of (dc/dP) agree’ well with those of Son and Ba~elsrl] which are also listed in the table. In addition, the value of (dc,,ldP) calculated from the pressure derivatives of c;,, c’ and c, obtained in this work is 12+ 3 in agreement with the value obtained from the [ lOO]L-mode measurements. Estimates of the temperature derivatives are given in Table 4 along with values obtained by Bartels and Vetter [7]. In calculating (d In c/dT), the room temperature value of the volume thermal expansion coefficient[l2] was used in eqn (5). To calculate (dc/dT) from (d In c/dT), the room temperature value of the elastic modulus was

I

I

I T’

28

-

125

f

150

175

200

225

250

275

300

325

T,K

Fig. 1. (dln ddP) vs T: [MO] L-mode--(A), salol bond; (0), Nonaq bond; (-), least-square tine; IL.tu) L-mode--(@)),Nonaq bond; (-.-), least-squares line; [ 1101TZmode--(Ok Nonaq bond; 6---k least-Wares Ilne. 1TE’a= 10Mbar.

709

Elastic moduii Table 3. Isothermalpressurederivativesof the adiabaticelastic moduli

T

T.-L. 299 K

Elastic Modulus

c11 *I v,

206 r.t.

(dLnc/dP) Ref.

(dc/dP)

TPa-1

1

10.5 49+ba

11?2a

34*14

823

It

7.7

1

%

this work *1

295

,I

3028

7t2

201

,I

32C4

7.220.8

I,

*.

164

1,

30+3

6.9tO.6

II

r*

c44 1,

711

0.6fO.l

293

12+6

1.oto.5

r.t.

c'

42ta

3.4?0.8

47?6

3.eo.5

24t2

2.ozo.z

r.t.

II

292

,I

136 %opagated

this work

1 this work 1 this work 1r

1,

ermr based bn standard deviations.

Table 4. Isobarictemperaturederivativesof the adiibaGcelastic moduli Temp. Range. K

204-217 195-298

Elastic Modulus

%1

249-296

'il II

197-224

,I

162-167

,I

112-139

II

73-146

I,

195-298

(dtnc/dT) 1O-5 K-l

(dc/dT) -1

MPa K

-18.5t0.ba

-12.1+0.3

Ref.

-43i2a

this work

-35.3to.7

7

-27.4?0.8

this work

-2l.Ul.S

-49f3

0

-20.4t1.9

-47e

u

11

-11.410.7

-26?2

It

‘1

-7.7to.a

-la+2

I,

11

c'

-26.6fO.7

152-161

I,

-39.6S.8

-34?2

135-148

,I

-39.6S.3

-34*1

It

7 this work $0

II

%'ropagated error based on standard deviations.

extrapolated imearly to the midpoint of the temperature range using the room temperature value of (dc/dT) given by Bartels and Vetter. The resulting temperature derivatives are overestimated as the result of using room temperahue values of a and of (dcfdT) which are larger than the appropriate low-temperature values. The caicnlation of the temperature derivatives at low temperatures can be improved if a is estimated from the Gruneisen relation[l3], and if a reiterative method is used in the extrapolation of c. The largest overestimate of (dc/dT) occurs for CL in the lowest temperature range, for which the error in (dcl,/dT) is -2.33 MPa.K-‘. With the exception of the value of (dcl,/dT) obtained in tbe temperature range 24%!!X K, the rna~~de of the derivative for the fllO] L-mode decreases as the temperature decreases in accord with the ‘Wving over”

to flatness which is required of the elastic modulus vs ~rn~~~~ curve as 0 K is app~ac~[l4].

4. cmsIoNs

The elastic moduli obtained in this work are essentially in agreement with the values reported by Son and Bartels [ I]. However, the precision realized here is about a factor of 10 better than that in their work. A careful determmination of the II = 0 condition was necessary in the case of the pulse superposition method to attain this precision. Gur value for BT, 113.02 0.7 GPa, agrees with that measured by Weir[6]. The positive pressure derivatives, their decreasing value with decreasing temperature and the negative values for burns derivatives are all consistent with a crystal which stiRens as the interionic separations are

710

A. L. DRAOOO and 1. L. SPAIN

reduced either by compression or by thermal contraction. The pressure derivatives of cl,, c’ and c,, in Table 3 can be used to compute the difference (a~,,/@)(&,/aP) at room temperature. For a central-force model at T = OK this difference is 2 for a cubic ionic crystal[lS]. For CaO at room temperature the difference appears to be closer to 1; however the combined error of the pressure derivatives is about +3. The presence of noncentral forces is clearly suggested, however, by the fact that at zero pressure c,*-c~=

-2OGPa

rather than 0 for the difference of the approximate harmonic [ 141values of cIz and cU. Ackaowledgmenfs-Thanks are due to Dr. R. Rice of NRL for donating crystals; to Mr. J. Paauwe of NSWL, White Oak, for help with calibrating the Bourdon gauge; and to Drs. P. Bolsaitis and K. Ishizaki of the University of Maryland for use and assistance with the McSkimmin pulse-superposition equipment.

Financial aid for the experiments was provided by the Center of Materials Research, University of Maryland; for the computational work by the National Bureau of Standards. REFERENCES

1. Son P. R. and Bartels R. A., J. Phys. Chem. Solids 33, 819 (1972). 2. Hite H&E. and Kearney R. J., J. Appl. Phys. 38,5424 (1%7). 3. McSkimin H. J., J. Acoust. Sot. Am. 33, 12 (1%)). 4. Soga N., J. Geophys. Res. 73, 5385 (1%8). 5. Bridgman P. W., Proc. Am. Acad. Alts Sci. 67, 345 (1932). 6. Weir C. E., J. Res. Natl. Bur. Stand. 56, 187 (1956). 7. Bartels R. A. and Vetter V. H., J. Phys. Chem. So/ids 33, 1991(1972). 8. Ochs T., L Sci. Instr. (L Phys. E) 1, 1122(1%8). 9. Dragoo A. L., Ph.D. thesis, University of Maryland (1975). 10. McSkimin H. J., J. Acoust. Sot. Am. 32, 1401(1960). 11. Mandel J., The Statistical Analysis of ExperimentaL Data, Chap. 12. Wiley, New York (1964). 12. Rice R., J. Am. Ceram. Sot. 52, 428 (1%9). 13. Griineisen E., Handbuch der Physik (Edited by H. Geiger and K. Scheel), Vol. 10, p. 1. Julius Springer, Berlin (1926). 14. Leibfried G. and Ludwig W., Solid S&e physics (Edited by F. Seitz and D. Turnbull), Vol. 12. D. 275. Academic Press. New York (l%l). 15. Lazarus D., Phys. Rev. 76, 545 (1949).