It was discovered that the isothermal equation of state for solids in compression is a simple, universal form. This single form accurately describes the pressure and bulk modulus as a function of volume for tonic, metallic, covalent, and rare gas solids

Ferrante, J.

Rose, J. H.

Smith, J. R.

Vinet, P.

English

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NASA Technical Memorandum 87303
Universality in the Compressive
Behavior of Solids
^
(NASA-TM-87303) GNIVEBSALITY IN THE N86-26775
COMPRESSIVE BiHAVIOR OF SCUDS (NASA) 11 p
HC A02/MF A01 CSCL 08M
Unclas
v G3/77 43382
Pascal Vinet and John Ferrante
Lewis Research Center
Cleveland, Ohio
John R. Smith
General Motors Research Laboratories
Warren, Michigan
James H. Rose
Ames Laboratory—USDOE
Ames, Iowa
April 1986
NASA
https://ntrs.nasa.gov/search.jsp?R=19860019303 2020-03-20T14:45:16+00:00Z
UNIVERSALITY IN THE COMPRESSIVE BEHAVIOR OF SOLIDS
Pascal V1net* and John Ferrante
National Aeronautics and Space Administration
Lewis Research Center
Cleveland, Ohio 44135
John R. Smith
General Motors Research Laboratories
Warren, Michigan 48090
and
James H. Rose
Ames Laboratory - USOOE
Iowa State University
Ames, Iowa 50011
SUMMARY
We have discovered that the Isothermal equation of state for sol Ids 1n
compression 1s of a simple, universal form. This single form accurately des-
cribes the pressure and bulk modulus as a function of volume for Ionic, meta-1-
Hc, covalent, and rare gas solids.
THEORETICAL ASPECTS
The various classes of solids - covalent, metallic, Ionic, and rare gas -
can be distinguished by their rather different types of Interatomic Inter-
actions. These differences would appear to preclude a common description of
the energetics of these systems.
:
 V1 We have found, however, that equations of state (P, V, T, relations) for
these classes of solids 1n compression are of a universal nature. In the fol-
lowing we will first discuss reasons why one might expect to find a universal
form for the equation of state (EOS) of solids. Next we will propose a uni-,
versal expression. Finally, the considerable EOS data available will be used
to test our proposed universal form.
EOS-are valuable not only for prediction of thermodynamlcal properties
but also to obtain Insight Into the nature of solid state and molecular
theories (refs. 1 and 2). In reference 3 a general zero temperature EOS was
reported which describes well the compression of metals (in the absence of
phase transitions). This was derived from a cohesive energy which was found
essentially to depend exponentially on the Interatomic spacing. That is, it
depends on the deviation of the Wigner-Seltz radius from the equilibrium
value, rys - r^se- (Here rw$ is defined 1n terms of the volume V per atom
as V = 4irr£s/3.)
*Visiting. Scientist from Laboratolre de Technologic des Surfaces, Ecole
Centrale de Lyon, B.P. 163, 69131 Ecully Cedex, France.
Wave-function-overlap Interactions such as those occurring in metals are
also found 1n varying degrees 1n covalent, 1on1c, and rare gas solids under
high pressures. There are additional types of Interactions Important In
determining the EOS of 1on1c and rare gas solids, however. Attractive cou-
lombic Interactions and attractive fluctuating .dlpole Interactions are also
present. In addition, 1f we now consider nonzero temperatures, there Is a
contribution to the pressure due to the variation of entropy. It 1s of the
form T(3S/3V)T for the Isothermal EOS.
The pressure contribution due to electron overlap varies essentially
exponentially with r^s - rw$e. All other contributions to the pressure for
solids on compression vary much less rapidly as a function of r^s- For
example, the entropy pressure Is T(3S/dV)j = aBT, where a 1s the thermal
expansion coefficient and B Is the Isothermal bulk modulus. The product <xB
1s known (ref. 4) to be essentially Independent of V for T > OQ, where QQ
1s the Debye temperature. Thus, the entropy contribution to the pressure 1s
expected to be nearly Independent of r^St at least for 1 > QQ. Also the
1on1c or fluctuating dlpole Interactions depend only on powers of r^s- Thus,
these Ionic or fluctuating dlpole Interactions provide a pressure component
which 1s also relatively slowly varying with r^s compared to electron over
lap Interactions.
Consequently, one may expect the shape of the EOS in compression to be
determined essentially by the exponential wave function overlap. Note this 1s
not true 1n expansion. For example, as the Interatomk separation becomes
significantly larger than the equilibrium separation for 1on1c solids, the
wave-function overlap becomes small and the electrostatic Interaction between
the Ions becomes dominant. This 1s quite different from metals, where 1t has
been demonstrated (ref. 3) that the exponential interaction remains dominant
well Into the attractive region.
A NEW EQUATION OF STATE FOR SOLIDS
• From the foregoing discussion, we propose that an exponential expression
similar to that found (ref. 3) for metals applies to all classes of solids in
compression over a range of temperatures excluding phase transitions. This
expression has the form
P(X.-f) = 3BT[(1 - X)/X2]exp[nT(l - X)] (1)
where By is the Isothermal bulk modulus at zero pressure, X = (V/Vp,)1/3
(V0 = 4irr$se/3 is the equilibrium atomic volume), and
nT -- 3[(3B/3P)T - l]/2 (2)
Equation (2) follows from equation (1), where (3B/3P)-| is evaluated at zero
pressure. To test this, we first rearrange terms:
H(X,1) = ln[PX2/3(l - X)] = InB-j + n-,0 - X) (3)
If our expression were universal, then one would expect isothermal plots of
experimental H(X,T) to be linear in 1 - X, with intercept and slope yielding
2
By and (3B/3P)T, respectively. The effects of entropy and forces other,than
those due to wave-function overlap enter only through these two parameters and
V
We have tested equation (3) against high pressure data from all classes
of solids, excluding materials which have phase transitions within the experi-
mental pressure range. We found 1t to be remarkably successful. In figure 1
we have examples from each class of solid at a variety of temperatures: solid
hydrogen (4.2 K data, ref. 5), solid xenon (150 K, ref. 6), cesium (295 K,
ref. 7), molybdenum (293 K, ref. 8), sodium chloride (298 K, ref. 9, see also
ref. 10), and magnesium oxide (293 K, ref. 8).
Despite the fact that all classes of solids are represented from a vari-
ety of experimental methods, the data are linear. 1n (1 - X) as predicted by
equation (3). This graphically Illustrates that EOS are, 1n fact, universal
1n form, our primary result.
One may also find this way of plotting the data convenient for analyzing .,
1t. The values of B-f and (3B/3P)j obtained from Intercept and slope of
these plots as listed 1n the caption of figure 1 are 1n very good agreement
with experimental values available 1n the literature (see, e.g., refs. 1, 5,
and 11). •
One need only know two zero pressure equilibrium numbers, B-f and
(3B/3P)-|, at each temperature to obtain the pressure-volume curve through
equation (3). Many of these equilibrium values are available 1n the litera-
ture. For example, one can use ultrasonic values of By and (3B/3P)7 to
predict P(V) through equation (1) and compare 1t with Independent high pres-
sure ..experimental results. This 1s shown 1n figure 2 for NaCl. NaCl was .cho-
sen because of the large amount of experimental data available which allows 1t
to be used as a high pressure reference scale (refs. 1 and 10). Values of
By = 2.34xl010 Pa and (3B/3P)f = 5.35 were taken from ultrasonic experiments
(ref. 1). The agreement between the predicted pressures (eq. (1), solid
curve), and the experimental data 1s very good over a broad range of V/VQ.
Another gauge of the accuracy of equation (1) 1s found from a comparison
with the Murnaghan EOS:
[ -3(3B/3P) 1
P(X,T) = [B1/(3B/3P)T] |^X - ij (4)
Equation (4) 1s obtained by Integrating the linear approximation
B(P,T) = BT + (3B/3P)-,P (5)
so that equation (4) has Increasing accuracy as P -» 0. The Murnaghan EOS
1s perhaps the most extensively used EOS for solids and has been argued (refs.
1 and 2) to be one of the "best" two-parameter EOS available. The dashed
curve 1n figure 2 1s a plot of the Murnaghan EOS (eq. (4)). While good agree-
ment with the data and equation (1) 1s obtained at low pressures, the Murnaghan
expression deviates from both at larger pressures. This 1s a critical test of
the universal expression equation (1), because 1t shows that the exponential
form correctly describes the nonlinear pressure contributions which are not
Included 1n equation (5).
3
A more severe test of nonlinear pressure effects Is shown In figure 3, ,
where hydrogen data 1s plotted for relative compressions of over 50 percent in
volume. The data are due to Anderson and Swenson (ref. 5). The predicted
pressures from the universal expression, (eq. (1)), are shown 1n figure 3(a)
as the solid curve while the Murnaghan prediction, (eq. (4)), 1s the dashed
curve. The Input values used 1n both predictions are By = 1.7x10^ Pa and
(3B/3P)j = 7.0, as listed 1n reference 5. Note these Input values are quite
close to those obtained from the Intercept and slope, respectively, 1n
figure 1. Again, the agreement between equation (1) and experiment is good,
while at the higher pressures the Murnaghan prediction deviates signifi-
cantly. Bulk modulus data are also given in reference 5, which provides
another test. From equation (1) we have
B(X,T) = (BT/X)[2 (nT - X - - X) ] (6)
Figure 3(b) compares the experimental data (circles) with the predictions of
equation (6) (solid curve) and of Murnaghan (eq. (5), dashed curve). Again,
the prediction from the universal expression, (eq. (6)), agrees well with the
data, while the linear (Murnaghan) prediction deviates at high pressures.
We conclude that the EOS of all classes of solids in compression is,, to a
good approximation, a universal function. It can be represented by a simple
exponential form which yields at each temperature accurate predictions of
pressure and bulk modulus versus volume.
ACKNOWLEDGMENT
One of us (P.V.) would like to thank the French "Ministere des relations
exterieures" for providing him a grant. The Ames Laboratory 1s operated for
the U.S. Department of Energy by Iowa State University under Contract number
W-7405-ENG-82 and this work was supported by the Director of Energy Research,
U.S. Office of Basic Energy Sciences. We would also like to thank Professor
C.S. Smith for helpful discussions.
REFERENCES
1. A.L. Ruoff and L.C. Chhabildas, 1n High Pressure Science and Technology.
Sixth AIRAPT Conference. Vol. 1, edited by K.D. fimmerhaus and M.S. Barber
(Plenum Press, New York, 1978), pp. 19-32.
2. J. Ross Macdonald, Rev. Mod. Phys. 41_, 316 (1969).
3. J.H. Rose, J.R. Smith, F. Guinea, and J. Ferrante, Phys. Rev. B 29, 2963
(1984).
4. C.A. Swenson, J. Phys. Chem. Solids 29. 1337 (1968).
5. M.S. Anderson and C.A. Swenson, Phys. Rev. B 1J), 5184 (1984).
6. J.R. Packard and C.A. Swenson, 3. Phys. Chem. Solids 24, 1405 (19b3).
7. M.S. Anderson and C.A. Swenson, Phys. Rev. B 31_, 668 (1985).
4
8. W.J. Carter, S.P. Marsh, J.N. Fritz, and R.G. McQueen, 1n Accurate
Characterization of the High Pressure Environment. Natl. Bur. Std. (U.S.)
Special Publication No. 326, edited by E.G. Lloyd (Natl. Bur. Std.,
Washington, D.C., 1971), p. 147.
9. O.L. Decker, J. Appl . Phys. 42, 3239 (1971).
10. S.D. Wood and V.E. Bean, High Temp. -High Pressures 1_5, 715 (1983).
11. Z.P. Chang and G.R. Barsch, J. Geophys. Res. 7_4, 3291 (1969).
12. E.A. Perez-Albuerne and H.G. Drlckamer, J. Chem. Phys. 43, 1381 (1965).
13. S.N. Valdya and G.C. Kennedy, J. Phys. Che.m. Solids 32, 951 (1971).
14. J.S. Weaver, T. Takahashi, W.A. Basset, 1n A^ jjr^ e_CJijLrjc^ ter_1^ a_tlqn of
the High Pressure Environment. Natl. Bur. Std. (U.S.) Special Publication
No. 326, edited by E.C. Lloyd (Natl. Bur. Std., Washington, D.C., 1971),
p. 189.
15. J.N. Fritz, S.P. Marsh, W.J. Carter, and R.G. McQueen, 1n Accurate
High Pressure Environment. Natl. Bur". Std. (U.S.)
_ _ _
Special Pub. No. "326, edited by E.C. Lloyd (Natl. Bur. Std., Washington,
D.C., (1971), p. 201.
16. L. Thomsen, J. Phys. Chem. Solids 31_, 2003 (1970).
17. P.W. Bridgman, Proc. Am. Acad. Arts Sci. 76, 1 (1945).
210x10"
200
190
HYDROGEN
0 5 10 15 20 25x10"'
220X10-1 XENON
215
1 2 3 4 5 6 7 8 9 lOxlO'2
218x10
216
214
1
CESIUM
268x10"
266
264
— MOLYBDENUM
0 2 4 6 8 10 12 14 16xlO"2 0 2 4 6 8 10 12xlO"2
245x10-1 262x10"C"t^,
240
0
..••"
NaCI
 x.-"''
• ••"
""'''i 1 1 1 1 1 1
2 4 6 8 1 0 1 2 14xlO"2
260
258
0
MgO
. •
< ,
2 4
. •
1 1 1
6 8 lOxlO"2
- (V/V0)
1/3
Figure 1. - Experimental values of H(X,T) = In CPX2/3U - X)] versus 1 - X, where x = (v/v0)1/3and v0 is the equilibrium
volume oer atom. The source of the data and values of BT and (dB/dP)T as obtained from the intercept and slope of the
plots of figure 1 are respectively: hydrogen, ref. 5, 1.66xl08Pa. 7.33; xenon, ref. 6, 1.63xlOvPa. 8.87; cesium,
ref. 7. L68xl09Pa, 407; molybdenum, ref. 8, 2.66xlOn Pa, 3.99; sodium chloride, ref. 9, 2.35xl010 Pa, 5.15;
and magnesium oxide, ref. 8, 1.52X1011 Pa, 481.
NaCI
UNIVERSAL
RELATION
MURNAGHAN
EXPRESSION
I
EXPERIMENTAL DATA
PEREZ-ALBUERNE
AND DRICKAMER (12)
VAIDYA AND KENNEDY (13)
WEAVER et al. (14)
FRITZ etal. (15)
THOMSEN (16)
BRIDGMAN (17)
.6 .7
.REDUCED VOLUME, VA/0
.9 1.0
Figure 2. - Pressure versus reduced volume for NaCI. Solid curve, Eq.. (1).
Dashed curve, Murnaghan expression, Eq. (4).
20
10
SOLID HYDROGEN
PRESENT MODEL
MURNAGHAN EXPRESSION
EXPERIMENTAL DATA
100
50
10
.4 .6 .8 1
REDUCED VOLUME. V /Vg
Figure 3. - Pressure (3a) and isothermal bulk rr.odulus (3b)
versus reduced volume for solid hydrogen. Experimental
data is from ref. 5. Present n.odel is Eqs. (1) and (6).
(V.urnaghan expressions are Eqs. (4) and (5).
1. Report No.
NASA TM-87303
2. Government Accession No. 3. Recipient's Catalog No.
4. Title and Subtitle
Universality 1n the Compressive Behavior of Solids
5. Report Date
April 1986
6. Performing Organization Code
506-53-12
7. Author(s)
Pascal V1net, John Ferrante, John R. Smith, and
James H. Rose
8. Performing Organization Report No.
E-2702-1
10. Work Unit No.
9. Performing Organization Name and Address
National Aeronautics and Space Administration
Lewis Research Center
Cleveland, Ohio 44135
11. Contract or Grant No.
12. Sponsoring Agency Name and Address
National Aeronautics and Space Administration
Washington, D.C. 20546
13. Type of Report and Period Covered
Technical Memorandum
14. Sponsoring Agency Code
15. Supplementary Notes
Pascal V1net, Visiting scientist from Laboratolre de Technologic des Surfaces,
Ecole Centrale de Lyon, B.P. 163, 69131 Ecully Cedex, France; John Ferrante,
NASA Lew1s\Research Center; John R. Smith, General Motors Research Laboratories,
Warren, Michigan 48090; James H. Rose, AmeTs Laboratory - USDOE, Iowa State
University, Ames, Iowa 50011. /\
16. Abstract
We have discovered that the IsotherffiaV equation of state for solids in compres
slon 1s of a simple, universal form. Ih1s single form accurately describes the
pressure and bulk modulus as a. function of volume for tonic, metallic, covalent,
and rare gas solids.
17. Key Words (Suggested by Author(s))
Equation of state
Bulk modulus
Compressibility
18. Distribution Statement
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Unclassified
20. Security Classif. (of this page)
Unclassified
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Universality in the compressive behavior of solids

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