Strength of magnesium oxide under high pressure:

evidence for the grain-size dependence

A.K. Singha,*, H.P. Liermannb, S.K. Saxenab

aMaterials Science Division, National Aerospace Laboratories, Bangalore 560 017, IndiabCeSMEC, Florida International University, Miami, FL 33199, USA

Received 23 August 2004; accepted 6 September 2004 by C.N.R. Rao

Available online 7 October 2004

Abstract

X-ray diffraction patterns from magnesium oxide compressed in a diamond anvil cell up to 55 GPa have been recorded and

the differential stress (a measure of compressive strength) and grain-size (crystallite size) determined as a function of pressure

from the line-width analysis. The strength agrees well with the uniaxial stress component (another measure of compressive

strength) derived earlier from the line-shift data. The strength increases while the crystallite size decreases steeply as the

pressure is raised from ambient to w10 GPa. The increase in strength is much smaller at higher pressures. The strength-pressure

data are explained by combining the grain-size dependence of strength and the shear-modulus scaling law. The dependence of

strength on grain-size has not been considered in the past in the discussion of high-pressure strength data.

q 2004 Elsevier Ltd. All rights reserved.

PACS: 61.10.Nz; 62.20.Dc; 62.25.Cg; 62.50.Cp

Keywords: A. Magnesium oxide; D. Compressive strength; D. Elasticity; D. X-ray diffraction; E. High pressure

The stress state in a solid sample compressed in a

diamond anvil cell (DAC) is non-hydrostatic. Modeling of

the stress state [1] and its effect on the shifts of diffraction

lines [2–10] has enabled the estimation under pressure of the

compressive strength, single-crystal elastic moduli, and

volume strains corresponding to hydrostatic pressure from

powder diffraction data under non-hydrostatic compression.

The analysis of line-width data [11,12] gives differential

stress, an independent measure of strength. The pressure

dependence of strength is of relevance to geophysics [11,

13–19]. In particular, the strength of polycrystalline MgO

under pressure has been studied extensively. The analyses of

both diffraction line-shift [16,17,20] and line-width [21]

data have been used to derive strength as a function of

pressure. The strength measurements have been carried out

0038-1098/$ - see front matter q 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ssc.2004.09.050

* Corresponding author.

E-mail address: aksingh@css.nal.res.in (A.K. Singh).

also on single-crystals [22,23]. In general, the strength of

solids increases under pressure and the rate of increase is

much larger than that predicted by the shear-modulus

scaling law [24]. We have compressed MgO powder

sample in a DAC and recorded X-ray diffraction

patterns up to 55 GPa. The diffraction-line widths are

analyzed to determine the grain-size (crystallite size) and

strength as a function of pressure. The pressure-strength

data are successfully explained by introducing a grain-size-

dependent factor in addition to the shear-modulus scaling

law.

High purity MgO powder with an average crystallite size

of 60(3) nm was used in the present study. In a typical

experiment, the powder sample contained in stainless steel

gasket (thickness, 400 mm; thickness of the indented region,

40 mm; and hole diameter, 125 mm) was pressurized in a

DAC with 250 mm anvil face. Non-hydrostatic component

of stress in the sample was maximized by not using any

pressure-transmitting medium. The X-ray diffraction

Solid State Communications 132 (2004) 795–798

www.elsevier.com/locate/ssc

A.K. Singh et al. / Solid State Communications 132 (2004) 795–798796

patterns were recorded on an image plate using 10 mm

incident X-ray beam (wavelength lZ0.03678 nm) from

HPCAT synchrotron beam line at the Advanced Photon

Source (APS), Chicago. The first nine diffraction lines from

MgO were recorded in each run. Typical profiles are shown

in Fig. 1. The positions and widths (full width at half

maximum, FWHM) of the diffraction peaks were obtained

by fitting a four-parameter pseudo-Voigt function to each

peak. The pressures were computed from the Birch–

Murnaghan equation using the measured volume com-

pressions, and 160.2 GPa and 3.99 for the bulk modulus and

pressure derivative of the bulk modulus, respectively.

The micro-strains and changes in the grain-size arising

from the deformation of the sample during pressurization

cause broadening of the diffraction lines. We used the

method suggested earlier [25] as modified for high-pressure

conditions [12] to determine the grain-size and the strength

from the line widths. The modified equation describing the

effect of grain-size and micro-stresses on the line width of a

Fig. 1. Typical background-corrected diffraction-line profiles of the first si

profiles are shifted arbitrarily along the intensity axis.

reflection is given by,

ð2whkl cos qhklÞ2 Z

l

d

� �2

Cns

Ehkl

� �2

sin2qhkl (1)

2whkl denotes the FWHM on 2q-scale of the reflection (hkl)

and d is the grain-size. Ehkl is the single-crystal Young’s

modulus along the direction [hkl] at the relevant pressure

and n is a constant. s is a measure of the compressive

strength. Following an earlier suggestion [11], we use nZ2

in the present analysis.

The measured line widths were corrected for instru-

mental broadening and the ð2whkl cos qhklÞ2 versus ðsin qhkl

=EhklÞ2 plots were constructed for the first six diffraction

lines. The inclusion of the remaining lines resulted in

significantly larger scatter in the plots in many runs and

these lines were excluded from all the plots. Single-crystal

elastic moduli at high pressures required for the calculation

of Ehkl were estimated by using in Birch equations [26] the

single-crystal elastic constants and pressure derivatives at

x reflections at ambient pressure (a) and 48 GPa (b). For clarity, the

Fig. 2. Typical ð2whkl cos qhklÞ2 versus ðsin qhkl=EhklÞ

2 plots at

ambient pressure (a), 17 GPa (b), and 48 GPa (c).

Fig. 4. (a) Pressure dependence of compressive strength. Filled

circles and triangles show the present data and those from Ref. [17],

respectively. Solid line is computed from Eq. (2). (b) Strength

versus pressure data in the 0–10 GPa range. The dashed and solid

lines show the data computed from Eq. (2) with different sets of m

and k values (see the text).

A.K. Singh et al. / Solid State Communications 132 (2004) 795–798 797

ambient pressure determined from ultrasonic experiments

[27]. A few typical plots are shown in Fig. 2. The grain-size

and strength in each run were computed from the intercept

and slope, respectively.

The grain-sizes at different pressures are shown in Fig. 3.

It is found to decrease rapidly from 60(3) to 19(4) nm as the

pressure is raised from ambient to w10 GPa and remains

nearly constant at higher pressures. The measurements after

reducing the pressure from 55 to w0.2 GPa show that the

grain-size undergoes a permanent decrease during pressur-

ization. This decrease is caused by the process of

comminution and occurs mostly during pressurization up

to 10 GPa. The reduction in grain-size of the sample

material depends on the extent of deformation the sample

undergoes before a given pressure is established. It is

expected to depend also on the ductility of the material. In

this sense, the measured pressure dependence of grain-size

under non-hydrostatic compression is not an intrinsic

property of the sample material.

The strength-pressure data are shown in Fig. 4(a). Also

shown in this figure are the strength data obtained earlier

[17] from the analysis of peak-shifts. The strengths derived

Fig. 3. Pressure dependence of grain-size.

using two totally different approaches show excellent

agreement. The strength is found to increase from w1 to

w7 GPa as the pressure is raised from ambient to w10 GPa.

The increase in strength is much less pronounced at higher

pressures. The concomitance between the rapid increase in

strength and decrease in grain-size is indicative of the grain-

size dependence of strength. We propose the following

relation to describe the combined effect of pressure and

grain-size on strength:

sðp; dÞZ sð0; lÞ½GðpÞ=Gð0Þ�½1Ck=dm� (2)

s(0,l) is the compressive strength at zero-pressure (ambient

pressure) of the sample with large grain-size, G(0) and G(p)

are the zero-pressure and high-pressure shear-moduli,

respectively; d is the pressure-dependent grain-size, and k

and m are empirically determined constants. The second

term in Eq. (2) represents the shear-modulus scaling law

A.K. Singh et al. / Solid State Communications 132 (2004) 795–798798

[24]. For most materials, shear and Young’s modulus

scaling give numerically comparable results. The last term

describes the grain-size dependence of strength. With mZ0.5, it reduces to the Hall–Petch relation [28,29] that

describes the grain-size dependence of yield strength or

hardness measured at ambient pressure.

For clarity, the pressure-strength data in the 0–10 GPa

pressure range from this study are shown separately in Fig.

4(b). The parameters m and k were estimated by fitting Eq.

(2) to the strength versus grain-size data in this pressure

range. The G(p) values were calculated using Birch

equations [26] with the single-crystal elasticity data

obtained from ultrasonic experiments [27]. A value of

0.1 GPa was chosen for s(0,l). This value corresponds to the

yield strength of MgO single-crystal compressed parallel to

the cube axis [22]. In the first set of calculations, both m and

k were refined by the method of least-squares. This gave kZ1042(749) and mZ1.1(2) (nm)1.1. The strengths computed

from Eq. (2) using these values of m and k are shown by the

solid line in Fig. 4(b). In the second set of calculations, m

was fixed at 0.5 in accordance with the Hall–Petch relation.

A least-squares fit of Eq. (2) to the grain-size-strength data

gave kZ198(25) (nm)1/2. The strength data computed from

Eq. (2) with mZ0.5 and kZ198 (nm)1/2 are shown by the

dashed line in Fig. 4(b). The solid line is seen to fit the

measured data better than the dashed line, particularly near

ambient pressure. In the 10–55 GPa pressure range the

measured grain-size remains unchanged with an average

value of 19(4) nm. Eq. (2) suggests that the pressure

dependence of shear-modulus alone controls the pressure

dependence of strength in this case. The strength data

calculated from Eq. (2) with dZ19 nm, mZ0.5, and kZ198

(nm)1/2 are shown as a function of pressure by the solid line

in Fig. 4(a). The strength data calculated with mZ1.1 and

kZ1042 (nm)1.1 also fall close to this line. The computed

data are found to agree well with the measured strength data.

In the discussions so far the parameters m and k are

considered purely empirical. Eq. (2) can acquire some

degree of predictive capability if the parameters are

constrained from other considerations. The value of s(0,l)

relevant to the present discussion can be obtained by directly

measuring the compressive strength on large-grained

polycrystalline compact of MgO under constant load

condition. At low pressures, often the sample does not

fully deform in a DAC and the maximum deviatoric stress

does not develop. This leads to an underestimation of

strength. If data points below 0.5 GPa are omitted from Fig.

4(b), then the Eq. (2) with mZ0.5 fits the observed data very

well. The choice mZ0.5 should be preferred as it amounts to

relying on the Hall–Petch relation that is supported by large

volume of data on widely different materials. This leaves

only k as the adjustable parameter.

In general, the diffraction experiments with DAC and

other high-pressure devices employ 5–50 mm diameter

incident X-ray beams. This necessitates the use of fine-

grained samples to avoid spotty diffraction patterns. The

grain-size reduces further on pressurization. The grain-size

changes during pressurization are expected to have

significant influence on the strength and should be

considered in the interpretation of the strength data

measured by diffraction techniques.

Acknowledgements

We thank M. Somayazulu and Y. Ding for rendering

technical help at HPCAT beam line and S. Usha Devi for

providing diffraction data for the characterization of MgO

powder samples.

References

[1] A.L. Ruoff, J. Appl. Phys. 46 (1975) 1389.

[2] A.K. Singh, J. Appl. Phys. 73 (1993) 4278.

[3] T. Uchida, N. Funamori, T. Yagi, J. Appl. Phys. 80 (1996) 739.

[4] A.K. Singh, C. Balasingh, H.K. Mao, J. Shu, R.J. Hemley,

Phys. Rev. Lett. 80 (1998) 2157.

[5] A.K. Singh, C. Balasingh, H.K. Mao, R.J. Hemley, J. Shu,

J. Appl. Phys. 83 (1998) 7567.

[6] A.K. Singh, K. Takemura, J. Appl. Phys. 90 (2001) 3269.

[7] Y. Akahama, H. Kawamura, A.K. Singh, J. Appl. Phys. 95

(2004) 4767.

[8] A.K. Singh, Philos. Mag. Lett. 67 (1993) 379.

[9] R. Jeanloz, B.K. Godwal, C. Meade, Nature 349 (1991) 687.

[10] T.S. Duffy, G. Shen, J. Shu, H.-K. Mao, R.J. Hemley,

A.K. Singh, J. Appl. Phys. 86 (1999) 6729.

[11] J. Chen, D.J. Weidner, M.T. Vaughan, Nature 419 (2002) 824.

[12] A.K. Singh, J. Phys. Chem. Solids 65 (2004) 1589.

[13] R.J. Hemley, H.-K. Mao, G. Shen, J. Badro, P. Gillet,

M. Hanfland, D. Hausermann, Science 276 (1997) 1242.

[14] S.R. Shieh, T.S. Duffy, Phys. Rev. Lett. 89 (2002) 255507

(Also pp. 1–4).

[15] S. Merkel, A.P. Jephcoat, J. Shu, H.-K. Mao, P. Gillet,

R.J. Hemley, Phys. Chem. Miner. 29 (2002) 1.

[16] T.S. Duffy, R.J. Hemley, H.K. Mao, Phys. Rev. Lett. 74 (1995)

1371.

[17] S. Merkel, H.R. Wenk, J. Badro, G. Montagnac, P. Gillet,

H.-K. Mao, R.J. Hemley, J. Geophys. Res. 107 (2002) 2271

(No B11).

[18] S.R. Shieh, T.S. Duffy, G. Shen, Phys. Earth Planet. Int. 143-

44 (2004) 93–105.

[19] S. Merkel, H.R. Wenk, J. Badro, G. Montagnac, P. Gillet,

H.K. Mao, R.J. Hemley, Earth Planet. Sci. Lett. 209 (2003) 351.

[20] G.L. Kinsland, W.A. Bassett, Rev. Sci. Instrum. 47 (1976)

130.

[21] D.J. Weidner, Y. Wang, M.T. Vaughan, Geophys. Res. Lett.

21 (1994) 753.

[22] C.W. Weaver, M.S. Patterson, J. Am. Ceram. Soc. 52 (1969)

293.

[23] C. Meade, R. Jeanloz, J. Geophys. Res. 93 (1988) 3261.

[24] J.O. Chua, A.L. Ruoff, J. Appl. Phys. 46 (1975) 4659.

[25] J.I. Langford, J. Appl. Crystallogr. 4 (1971) 164.

[26] F.J. Birch, J. Geophys. Res. 83 (1978) 1257.

[27] S.V. Sinogeikin, J.D. Bass, Phys. Rev. B 59 (1999) R14141.

[28] E.O. Hall, Proc. Phys. Soc. (London) B 64 (1951) 747.

[29] N.J. Petch, J. Iron Steel Inst. 174 (1953) 25.