OLIN-B 75-0291
Compression and bonding of ice VII and an empirical linear
expression for the isothermal compression of solids·
Bart Olinger and P. M. Halleck
Los Alamos Scientific Laboratory, University of California, Los Alamos, New Mexico 87544
(Received 14 June 1974)
The volume of ice VII was measured between 3.0 and 8.0 GPa at 298 K using a high-pressure x-ray
diffraction technique. The specific volume (0.708, +0.023, -0.012 cmJ/g), the isothermal bulk
modulus (12.S4±0.27 GPa), and the modulus' pressure derivative (S.S6±0.14) for ice VII at 298 K
and zero pressure were determined using an empirical equation for isothermal compression. That
equation is [P V~/(Vo - V)]I /2 = Cr +S,!, [P(Vo - V)]'f2 , where Cr and Sr are constants,
V 0 is the ambient volume, and V is the volume at pressure P . This linear relation, which describes
the state of nonporous materials along their Hugoniots, is shown to charact!!rize the isothermal
compression of solids as well as does the Murnaghan equation. The zero pressure, 298 K
oxygen-oxygen distance in ice VII extrapolated from the present data and a simple bonding model
for the hydrogen-bonded oxygen atoms strongly support Kamb's description of the ice VII structure
as two interpenetrating ice Ic frameworks.
INTRODUCTION
Properties of H20 in its liquid and many solid forms
are of fundamental interest to many branches of science.
Two recent reviews on the phase stability and the structures
of the various forms of ice are by Kamb1 and von Hippel
and Farre1. 2 Of these phases, ice vn has one of the
more simple structures, yields good x-ray diffraction
patterns, and has a large pressure range of stability.
The phase limits of ice vn at and above room temperature were determined up to 17 GPa (IGPa = 10 kbar) by \
Pistorius e( al. 3 At 298 K two high pressure phases of
ice are known: water transforms to ice VI at 0.96 GPa4 ;
ice VI transforms 'to ice VII at 2. 14 GPa3 and remains
stable to the limit of the study by Pistorius et al. The
precise molecular arrangement in ice vn is uncertain;
oxygen atoms are in a body-centered cubic structure,
while the locations of hydrogen atoms are not definitely
known. 5 Kamb 1 proposes that among the eight nearest
neighbor oxygen atoms, each oxygen atom is hydrogen
bonded to four others in tetrahedral coordination and is
in repulsive contact with the other four. The space
group of ice VII in this case would be .Pn3m. This cubic
structure could be described as two interpenetrating
frameworks of ice Ic. (Ice Ic forms when water vapor
is condensed below 190 K; the oxygen atoms are arranged
in a diamond structure with hydrogen bonds in the same
arrangement as the diamond covalent bonds.) Further
information about the hydrogen bonding in ice vn can be
gained from a study of the compression of the ice vn
structure . Here we report the results of such a study.
EXPERIMENT
The experimental techniques utilized here are described
in detail elsewhere. 6 ,7 Briefly, distilled water and aluminum powder were placed in a O. 3-mm-diam hole
drilled through the center of a O. 3-mm thick, 3-mmdiam Be disk. The disk was pressed between two tungsten-carbide anvils, one driven by a hydraulic ram.
A collimated CuKa x-ray beam was passed thro\lgh the
sample and the subsequent diffraction pattern recorded
on film in a 114. 6-mm-diam powder camera surrounding
the sample. Typical exposure times were from 8 to 12 h.
94
The diffraction patterns for all exposures exhibited the
111, 200, 220, and 311 diffraction lines of aluminum and
the 110, 200, and 211 lines from the body-centered cubic
framework of the oxygen atoms of ice VII. Based on
three initial diffraction patterns of the powdered aluminum
taken at ambient conditions, we found that no corrections
for sample pOSition or self-absorption were necessary.
The pressure in the sample region was deduced from
the compression of the aluminum powder. The isothermal compression of Al listed in Table I was calculated
from the bulk sound s~ed for aluminum, 8 the shock
Hugoniot fit for Al 1100 (99% pure), 9 and the other thermodynamiC data also listed in Table 110 using methods
described elsewhere. 11 No effects due to localized nonhydrostatic stresses in the sample mixture or to pressure gradients across the sample region could be detected. This observation is supported by the results of
Piermarini et al. 12; they found that H20 in the ice vn
phase region supported a localized nonhydrostatic stress
of only 0.6 GPa at 9.2 GPa and virtually no pressure
gradient 'up to 10 GPa.
Our compression data for ice VII are listed in Table
TABLE 1. Compression of a luminum at 298 0 K. a
P
viVo
0.98675
0.97441
0.96285
0.95198
0.94173
0.93201
0.92278
0.91400
0.90562
.0.89760
P
(GPa)
VIVo
(GPa)
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
0.88992
0.88255
O. 87546
0.86864
0.86206
0.85571
0.84958
0.84364
0.83790
0.83233
11. 0
12.0
13.0
14.0
15.0
16.0
17.0
18.0
19.0
20.0
aorhe Hugoniot and thermodynamic quantities used to calculate
the isothermal compression of aluminum (see Ref. 10). Us
=5 .304 + 1. 352 Up (km/s) , Po = 4/(n x a~) = 2.6985 g/cm 3 , 3nk
=924.45 J/kg 'K, C y = Cp -9a 2 C~ T =862. 74 J /kg 'K, aE/8P y
=(Poy)-1=0.173 cm3/ g, where Y=QlC~/Cp, 8D =352. 9 K, derived from C y /3nk.
The Journal of Chemical Physics, Vol. 62, No. 1, 1 January 1975
Copyright © 1975 American I nstitute of Physics
B. Olinger and P. M. Halleck: Compression and bonding of ice VII
95
TABLE II. Simultaneous compression .of ice VII and Al and the deduced pressures .
aa(ice VII)
(nm)
VOce VII)
(cmS/g)
~V X 104 b
aC(AJ)
~(V/ VO)b
P
(cmS/g)
(nm)
V/Vo(AJ)
x 104
(GPa)
0.33280
0.33060
0 . 32923
0.32776
0.32640
0.32514
0.32412
0.32252
0.32218
0.32130
0.32080
0.31921
0.31847
0. 31853
0.31833
0.31814
0.6164
0.6042
0.5963
0.5889
0.5815
0.5748
0.5695
0.5610
0.5593
0.5547
0.5518
0.5440
0.5402
0.5405
0.5395
0.5385
13
5
4
8
1
4
2
7
2
7
3
3
9
7
3
7
0.40041
0.39945
0.39890
0.39807
0.39760
0.39692
0.39649
0 . 39549
0.39542
0.39491
0.39440
0.39338
0.39298
0.39304
0.39297
0.39279
0.9672
0.9602
0.9563
0.9514
0.9470
0.9421
0.9390
0.9320
0.9315
0.9279
0.9250
0.9171
0.9150
0.9147
0. 9143
0.9130
4
15
7
10
4
6
9
10
2
1
7
7
7
4
8
14
2. 60 ± O. 03
3. 22 ± 0.13
3 . 56 ± 0.16
4. 04 ± O. 08
4.45 ± O. 03
5. 05 ± 0. 05
5.27 ± 0.07
6. OO ± 0.08
6.07 ± 0 . 02
6.46 ± O. 01
6 . 78 ± O. 07
7.67 ± 0.07
7. 92 ± 0. 08
7.93 ± 0.06
7.98 ± 0.09
8.15 ± 0.18
""Cell parameter of the resolved simple body centered cubic cell formed by the oxygen
atoms.
b l> represents the standard deviation.
Cao(Al) = 0.40489 nm.
II. The data are presented in Fig. 1 along with
Bridgman's4 compression data, the volume of ice VII
at 2.5 GPa, 298 K, determined by Weir et al. , is and
the volume of ice vrn at 2.5 GPa, 223 K, determined
by Kamb and Davis. 5 Since the volume difference between ice VII and ice VIII across the phase boundary at
2.5 GPa (-273 K) is not detectable14 and the estimated
thermal expansion of ice vm at 2. 5 GPa is small, 15
the two specific volumes should be nearly equal at 2.5
GPa, 298 K. Although the magnitude of our volume for
ice VII is in poor agreement with Bridgman's,4 we
appear to agree well with the slope of his compression
over his entire pressure range. Our data are in more
serious disagreement with the volume of ice VII at 2.5
GPa measured by Weir et al. is The Kamb and Davis 5
ice vrn volume agrees well with our value for ice VII;
after considering thermal expansion,15 Kamb and Davis'
value at 2.5 GPa, 298 K, is O. {j19 cms/g while our one
data point at 2.6 GPa at 298 K is 0.616 cms/g.
ISOTHERMAL EQUATION OF STATE OF ICE VII
In order to deduce the nature of the bonding in ice VII
in relation to other ice structures, it is necessary to
know the interatomic spacings in the various polymorphs
under common conditions. While several ice structures
can be studied metastably at 1 atm, this is not the case
for ice VII and we must deduce the information from
available data. Attempts to calculate the specific volume of ice VII from tabulated O-H-O distances 16 fail
1. e., O-H-O distance"" O. 250 nm and O-H-O angle
=1T together yield a specifiC volume of 0.402 cms/g
which is much too small). We must, then, extrapolate
the present compression data from 2.6 GPa down to
zero pressure.
It is desirable that the equation of state chosen for
this purpose yield reliable zero-pressure values for the
volume, Vo, the isothermal bulk modulus Bon and its
pressure derivative, aBT/aPh,p=o' Rather than use
one of the existing isothermal compression functions,
we make use of an empirical observation which allows
us to constrain a form of our data to lie on a straight
line. It will be shown that this empirical isothermal
compression equation is nearly equivalent to the wellknown Murnaghan finite strain analysis.
When a nonporous material is shock compressed to
various internal energy states, it has been observed that
\
\
9 .0
'8.0
if
7.0
\
\
\
'.
.,
c
0
d
e
\
'-.\
-., 6 .0
on
on
b
0
0
\
(!)
~
~:-:.-:-~
0
\
a
+
+
\
\
5 .0
,
...
\
ct 4 .0
o
o
,
o
,
0
" 0.,
3 .0
"
2 .0
0
0
'.",.""
0
,,\-....
'".,.
. :--.:...:
1.0
.
'~'.~-=:.
0 .55
0 .60
0.65
0 .70
v(p)(cm 3 /g)
FIG. 1. Comparison of the volumes of ice VII and ice VIII
measured at high pressure by various investigators. (a) Data
for present study listed in Table II. The size of the cross
arms indicate the magnitude of the errors. (b) Fit to the same
data in Table II calculated from Eq. (8) for Vo = 0.708 cms/g,
CT ~ 2.979 km/s, and ST =1.641. The dotted lines indicate uncertainties. (c) Bridgman's compression of ice VII at 298 K
(Ref. 4). (d) Ice VIII volume at 2.5 GPa, 223 K, determined
by Kamb and Davis (Ref. 5). (e) Ice VII volume at 2 . 5 GPa,
298 K, determined by Weir et al. (Ref. 13).
J . Chern. Phys., Vol. 62, No.1, 1 January 1975
B. Olinger and P. M. Halleck: Compression and bonding of ice VII
96
can show that
a linear relationship exists between the shock wave velocity, Us, and the material velocity, Up, behind the
shock,
(1)
Us = Co+SUp ,
where Co corresponds to the zero pressure isentropic
bulk sound speed. When this velocity equation is converted to pressure-volume space using the Hugoniot
relations, Eq. (1) becomes
=Co + S[(P -
[(p - Po)Vo 2/(VO - V)]1/2
p_
(7)
Equation (6) can then be arranged to give an expression
for the compression; the root of the quadratic equation
is chosen so vivo has a value between 0 and 1,
The zero subscript parameters represent the initial
state of the material. Plots of [PV0 2/(Vo- V)J1/2 and
[pC Vo - V) ]1/2, where the P, V values are taken from any
of the isothermal compression curves calculated from
shock compressions ll ,17-19 as well as dilatometric compression data 20-23 show clearly that a linear equation
similar to Eq. (2) is equally suited for representing the
isothermal compression of solids, in many cases to
pressures as high as 50 or 100 GPa.
~ __ {-.!.
~[
ST + 2S~ 1 -
Vo - 1
(3)
V/Vo=[1+(B~T/BoT)P]-1IBoT .
Thus, the isothermal bulk modulus at zero pressure can
be related to C T as follows:
(5)
If Eq. (3) is rearranged and combined with Eq. (5), we
TABLE III. Comparison of compressions calculated from the Murnaghan equation and from
Eq. (8) with isothermal compressions calculated from shock compressions.
viVo Shock
ViVo Murn.,
V!Vo Eq. (8)
5 GPa
vivo
Shock
V!Vo Murn.
ViVo Eq. (8)
10 GPa
V!Vo Shock
viVo Murn.
V!Vo Eq. (8)
V!VOShock
viVo Murn.
viVo Eq. (8)
25 GPa
50 GPa
Mg
Cu
MgO
LiCl
0.890
0.890
0.967
0.965
0.965
0.970
0.970
0.970
0.885
0.882
0.881
0.822
0.823
0.822
0.939
0.937
0.937
0.944
0.945
0.945
0.817
0.814
0.813
0.701
0.709
0.705
0.876
0.872
0.872
0.884
0.885
0.885
0.703
0.704
0.701
0.599
0.617
0.610
0.806
0.803
0.803
0.815
0.819
0.818
0.722
0.723
0.722
0.729
0.738
0.736
0.~91
VIVo Shock
VIVo Murn.
V!Vo Eq. (8)
100 GPa
o!v
Mg
Cu
MgO
Liel
(11K)
BT,p=o
(g/cm 3)
x 10 5
'Y
dCV)
dS v
Co
Ref.
(km/s)
S
(GPa)
8BT)
8P T,P=O
17
17
17
18
1. 740
8.930
3.585
2.075
7.17
4.89
2.61
13.4
1.43
1.96
1.32
1.81
0.063
0.101
0.780
0.118
4.492
3.940
6.597
3.915
1.263
1.489
1.369
1.410
34.07
130.0
154.4
29.67
3.97
4.74
4.47
4.42
Po
(10)
Since the Murnaghan equation has been shown to accurately represent the compression of solids to very
high pressure, 25 it is of interest to compare the isothermal compression values calculated from Eq. (8)
with those calculated from the Murnaghan equation. In
Table III this comparison is made for four solids having
a large range of compressibilities and various types of
chelnical bonding for pressures from 5 to 100 GPa,
. (4)
Pressure
(8)
If Eq. (9) is integrated with respect to pressure and
volume from Pi = 0 to P and Vi = Vo to V we have
where CT is the intercept constant and ST is the slope
constant. The subscript T indicates isothermal conditions. By letting P- 0, we see from Eq. (3) that C T is
the isothermal bulk sound speed,
BOT =- Vo(ap/aV)T, p-o=PoC~ •
(1 + ~)1/2J}
BOT
•
Equations (6) and (8), like the Murnaghan equation,
equates pressure and compression involving only two
constants, BOT and B~T which can be derived from ultrasonic data. The Murnaghan equation24 ,25 assumes the
equality
8P
B T(Pi)=-Vi
=BoT+B~TPi .
(9)
)
8 V T, P= Pi
For isothermal compression we can write the analog
to Eq. (2),
CT =[ - V~(8P/av)T, p_O]1/2.
(6)
From Eq. (6) the pressure derivative of the bulk modulus at zero pressure can be found in terms of the slope,
(2)
po)(Vo - V)]1/2.
B oT (l- ViVo)
- [1- ST(l- V/Vo))2
J. Chern. Phys., Vol. 62, No.1, 1 January 1975
B. Olinger and P. M. Halleck: Compression and bonding of ice VI I
TABLE IV. Least squares fits to data for various Vo in the
form [PV~/(VO - VI)JI/2 = C T +ST[P(VO - V 1)]1/2 +QT [P(V O- VI)]
•
Vo
(emS/g)
C T ± A 95%
(km/s)
ST ± A95 %
Q T± A 95%
(km/s)·t
0.6500
0.6700
0.6900
0.7079
0.7300
0.7500
0.7700
6.554 ± 0.258
4 .722 ± 0.154
3 .636±0.135
2. 981± 0.132
2 . 410 ± 0.130
2.037 ± 0.127
1. 752 ± O. 122
- 4 . 057± 0.842
-0.616 ± 0.444
+0 .924±0. 352
+ 1.639± 0. 31 8
+2.108 ± O. 290
+2 .3 27±0. 268
+2 .437 ± O. 244
+3 .188± O. 636
+1: U 6±0.298
+ 0.322 ±0 .216
0.000 ± 0.1 84
-0.182± 0.156
- 0. 255±0.136
-0.283± 0.118
Also listed are isothermal compression values for these
substances calculated from their shock wave HugoniotS. 17•18 The three sets of compression values are .
based on the linear fits to the shock compression data
for the four solids. 17 •18 The isothermal values from the
Hugoniots were calculated by methods described elsewhere l l and are labeled "V /Vo shock" in Table Ill. Compressions found from both the Murnaghan equation [Eq.
(10) and Eq. (8) were calculated using values for BOT and
B~T derived from the Co and the S of the shock compression linear fit [Eq. (1)J. 17 •18 The Co and S were first converted to Bas and B~s values using analog relations of
Eqs. (5) and (7). These adiabatic moduli were then
transformed to BOT and B~T with the thermodynamic
parameters listed in the footnote of Table III. The
Murnaghan compres~ion values are labeled "V/Vo Murn"
in the table. From Table III it can be seen that the
three methods of calculating isothermal compressions
are in good agreement to very high pressures, and the
ability of Eqs. (3)-(8) to represent the isothermal compreSSion of solids is demonstrated.
The constraint of linearity in Eq. (3) is now used to
find ambient volume of ice vn and the phase's zero
pressure bulk modulus and modulus ' pressure derivative. To select the proper value for the ambient VOlume,
Vo. Eq. (3.) is first expanded to include a quadratic term,
[PV02/(VO- V)JI/2=CT+ST[P(VO- V)J1/2
+ QT[P(VO -
V)).
(11)
A search is -then conducted to find the value of Vo for
which the solution of QT, the quadratic coefficient in Eq.
(11), is zero. The values of the coefficients of Eq. (11)
are found u,sing regression analysis and the data in Table
II.
These coeffiCients for Va = 0.65 cm 3/ g to Va = O. 77
cm 3/ g are listed in Table IV with their 95% confidence
intervals. The value of Va for which QT =0 ~s 0.708
cm 3/ g. The largest and smallest Va values which have
QT values within the 95% confidence interval of QT =0
are 0.731 and 0.696 cm 3/ g. The other coefficients
associated with these Va values lie far outside the 95%
confidence intervals of the coefficients for Vo=0.708
cm 3/ g. Solutions for the values of the coefficients of
the linear equation [Eq. (3)] using regression analysis
and Va =0. 708 cm 3/ g are C T =2. 979 ± 0. 032 km/ sand
S T =1.641 ± O. 034. Again 95% confidence intervals are
given. Thus we conclude that for ice VII at P = 0, 298 K,
Vo =0.708, +0.023', -0.012cm 3/ g , BOT = 12.54±0.27
97
GPa, and B~T = 5. 56 ± O. 14. We emphasize that these
equation-of-state values and their associated errors are
based on the assertion that Eq. (3) accurately describes
the compression of ice VII. These values and Eq. (6)
were used to calculate the curve and errors shown in
Fig. 1.
Though the volume of ice VII has not yet been determined directly at P = 0, a determination of the volume of
ice VIII has been made 26 ,27 and was reported by Kamb 1
as 0.671 cm 3/g at 110 K, 0 GPa. As it was stated earlier , the volume difference of ice VII and ice VIII at 2.5
GPa, 273 K is not detectable. 14 Therefore, it would be
of interest to compare their volumes at 298 K , 0 GPa.
If we assume ice VIII has the same thermal expansion
as ice Ih (ordinary ice) , o!v =1. 7 x 1O-4/K, then the volume of ice VIII at 298 K, 0 GPa is O. 692 cm 3/ g , or
about 2% smaller than the volume calculated for ice VII.
Holzapfel and Drickamer 28 collected compreSSion data
on ice using a technique similar to ours in the pressure
range 2. 2-22 GPa. No absolute volume values were
given. Though the data were said to be that of ice VII,
most of the data were collected at 243 K in a phase region where ice VIII is considered stable. 14,3 Because of
Similarities of the structures of ice VII and ice VIII,
their results are probably valid. Centering a least
squares of their data at 2.2 GPa to a Murnaghan equation, Holzapfel and Drickamer report that BT (at 2.2
GPa) = 22. 7 GPa and that B; = 5. 3 for ice VII. From
Eq. (9) we find that their B OT = 11. 0 GPa and B~T= 5. 3,
both in good agreement with our results.
DISCUSSION
Kamb and Davis 5 have described the ice VII structure
as consisting of "two interpenetrating but not interconnecting frameworks of ice Ic type." Noting that the oxygen-oxygen distance is greater in the ice VII structure
than in the ice Ic structure, they further state "the increase is caused by the four repulsive contacts, which
have the same 0-0 distance as the four bonds." Figure 2
illustrates Kamb and Davis' proposed structure for ice
VII. Spheres in the figure represent oxygen atoms , connecting tubes represent the hydrogen bonds; filled tubes
identify one ice Ic sublattice; unfilled tubes identify the
other. The cell in Fig. 2 can be divided into eight cubes,
each cube having an oxygen atom at all eight corners and an
oxygen atom at its center. It is this smaller cell's lattice
parameter that is listed in Table II.
According to Kamb and Davis' model, the center oxygen atom is hydrogen bonded to four of the corner oxygen
atoms in tetrahedral symmetry. The other four corner
oxygens have net repulsive forces interacting with the
central oxygen. In ice Ic the central oxygen atom is
still tetrahedrally hydrogen bonded to the four corner
oxygens, but the other four nonbonded oxygen atoms are
miSSing. With the extrapolated value for the volume of
ice VII at ambient conditions, we can now directly compare the oxygen-oxygen distances in ice VII with those
in ice Ic and see the effect of the 'repulsive oxygen contact.
The lattice parameter for ice Ic, which has the oxygen
atoms in a cubic diamond structure, is 0.6352 nm at
zero pressure and 143 K.l If it is assumed that the
J. Ch em. Phys., Vol. 62, No. 1, 1 Jan uary 1975
B. Olinger and P. M. Halleck : Compression and bonding of ice VII
98
coefficient of the repulsive term doubles, then the
equilibrium value for R, the 0-0 distance, will increase
from 0.2774 nm (ice Ic) to 0.3018 nm (ice VII). This
would mean a doubling of the coefficient ratio in Eq.
(14). The ratio
(e 50 R/R7)
(e 5OR o/Rb)
FIG. 2. The proposed structure of ice VII by Kamb and Davis
(Ref. 5). The spheres represent the oxygen atoms and the
connecting tubes represent the hydrogen bonds; the black tubes
identify one ice Ic sublattice, the open tubes the other.
thermal expansion of ice Ic is the same as for ice Ih
(ordinary ice), o!v =1. 7 x 10-4/K, then the lattice parameter at 298 K would be 0.6407 nm. This assumption
is probably good since the atomic arrangement and
atomic distances in the two are nearly identical, 1,2 and
in addition, the densities of the two phases have been
shown to be nearly the same at zero pressure, 110 K.
Therefore, at 298 K, the 0-0 distance in ice Ic is
taken to be 0.2774 nm. Using the value for Vo just
derived, it is found that the zero pressure lattice parameter for ice VII is 0.3486 nm at 298 K. From this,
the 0-0 distance in ice VII is found to be 0.3018 nm.
Using the Kamb-Davis model for ice VII, we shall attempt to quantitatively explain this increase in distance
between hydrogen-bonded oxygens between ice Ie and ice .
VII.
We assign two interacting forces between oxygen atoms
in ice VII and ice Ic structures. One is a directional
attractive force between hydrogen-bonded oxygens; the
other is a central repulsive force which is the same for
all oxygen atoms. Bringing four nonbonding oxygen atoms
into the four empty corners of the cube defined by the
tetrahedrally hydrogen bonded oxygens doubles the total
repulsive force in the cube. Following Reid29 we describe the attractive potential of hydrogen bonded oxygen
as being proportional to R-6 , where the R is the 0-0
distance; the repulsive potential we describe as being
proportional to e- CR , where C =50/nm. Thus the potential function of this hydrogen-bonded oxygen system is
given as
At equilibrium,
=0 =6aR- 7 -
Kamb30 introduced a more complete potential energy
model for ice VII based also on the two interpenetrating
ice Ic frameworks. He broke down the potential model
into an ice Ic framework expansion component, a van der
Waals interaction between frameworks, an electrostatic
interaction, and a repulsive interaction. In addition, a
vibrational component was calculated for the ice VII
structure. The potential model is conveniently expressed in terms of R(ice VII) / R(ice Ic), where R is
again the oxygen-oxygen distance. When the potential
energy components are transformed to forces by differentiating with respect to volume, it is found that the
equilibrium ratio, R (ice VII)/R (ice Ic), is 1.070 at
298 K, 0 GPa. For, R (ice Ic) = O. 2774 nm, we find that
R (ice VII) = O. 2968 nm, or about 1. 5% smaller than our
extrapolated value and slightly outside our error limits.
2.5
1-0
a::
"o
a::
2 .0
"-
1.5
- - - - -
bee-CR.
(13)
Therefore, at equilibrium, the ratio of the coefficients
of the potential function terms is
(14)
According to the model previously described, if the
- - - ---- --
o
If')
1-
a::
a::'
o
...
If')
1.0
0.28
(12)
<II = aR-il - be-CR.
- 8if>/8R
as a function of R, where Ro is the 0-0 distance for ice
Ic, is plotted in Fig. 3. It can be seen that for R = O. 3018
(the value extrapolated for ice VII) this ratio has a value
of 1. 9, indicating that the repulSion term nearly doubles
as predicted from the Kamb and Davis model of ice VII.
If we force a fit to the compression curve given in Fig.
1 and if we require a doubling of the repulsive potential
to expand the 0-0 distance from 0.2774 to 0.3018 nm,
we find the most suitable repulsive potential to be proportional to e-42 R and the attractive potential to be proportional to R-3.
0.30
R(nm)
FIG. 3. Plot of the ratio [(e5ORIR7Ie5OROIR~)1 as a function of
R, the distance between hydrogen bonded oxygen atoms. The
ratio indicates the magnitude of the repulsive term in the bond:
ing force relative to that in ice Ie. Ro is the value for R in ice
Ie, 0.2774 nm. The dashed lines indicate the correlation of a
ratio value of 1 to Ro and the value for R at which the ratio
equals 2. The arrow pOints to the value for R in ice VII calculated here for ambient conditions.
J. Chem. Phys., Vol. 62, No. 1, 1 January 1975
B. Olinger and P. M. Halleck: Compression and bonding of ice VII
The difference of the internal energies of ice Ic and
ice VII can be estimated from available information and
several assumptions. Kamb 30 calculated the difference
of internal energies of ice Ih at 273 K, 0 GPa, and ice
VII at 273 K, 2.5 GPa, to be 279 Jig based on the data
of Bridgman. 31 The first assumption is that this energy
difference is the same at 298 K. The change in the internal energy of ice VII at 298 K from 0 to 2. 5 GPa can
be calculated by integrating the following thermodynamic
relation with respect to pressure at constant temperature:
dE =TdS - PdV,
(15)
where
(S2 TdS= (P2 _ TV(P)cxv(P)dP
)SI
)Pl
(16)
and
v2
P2
( - PdV= ( pv(p)IBT(p)dP.
)Vl
)Pl
(17)
For V(P) , Eq. (8) is used, and Eq. (9) serves as a convenient expreSSion for BT(P). The thermal expansion
coefficient of ice VII is assumed to be the same as that
for ice Ih at 0 GPa (1. 7 x 10-4/K) and has the same coefficient at 2. 5 GPa as calculated by Kamb,30 1.0 x 10-41
K. It is further assumed that the thermal expansion
. coefficient decreases linearly with pressure.
From Eqs. (15)-(17), the preceding assumptions, and
the equation-of-state values for ice VII derived from the
data just presented, the difference in internal energy of
ice VII between 0 and 2.5 GPa is calculated to be 29 Jig.
Combining this difference with that of Kamb,30 we find
the difference of internal energies of ice Ih and ice VII
at 298 K, 0 GPa is 250 Jig. This would be the same between ice VII and ice Ic if the energy difference between
ice Ih and ice Ic is negligible. Evidence for a small
energy difference between ice Ih and ice Ic was found by
Bertie et al. 32 ; they could not detect a heat of transformation between the two phases at 200 K. Thus, we
estimate energy differences between ice Ic and ice VII
to be 250 Jig at 298 K, 0 GPa. Kamb30 estimated nearly
the same energy difference using an assumed volume and
compressibility for ice VII. It is of interest to note that
despite denSity and structural similarities between ice
VIII and ice VII, Bertie et al. 26 determined the heat of
transformation between ice VIII and ice Ic to be 130 Jig
at 125 K, 0 GPa. This is only half the value previously
determined for the ice VII-ice Ic internal energy difference.
ACKNOWLEDGMENTS
We wish to thank Joseph Fritz, M-6 at LASL for calculating the compreSSion of our Al pressure standard
and for his many helpful diSCUSSions. We also wish to
thank John C. Jamieson, University of Chicago, Robert
MCQueen, M-6, and Jerry Wackerle, WX-7, for their
suggestion and criticisms, and Howard Cady, WX-2, for
providing the facility at which we carried out our exPeriments.
"'Work performed under the auspices of the U.S. Atomic Energy
99
Commission.
lB. Kamb, Trans. Am. Crystallogr. Assoc. 5, 61 (1969).
2A. von Hippel and E. F. Farrell, Mat. Res. Bull. 8, 127
(1973).
3C . w. F. T. Pistorius, E. Rapoport, and J. B. Clark, J.
Chem. Phys. 48, 5509 (1968).
4p . W. Bridgman, Proc. Am. Acad. Arts Sci. 74, 399 (1942).
5B. Kamb and B. L. Davis, Proc. Nat!. Acad. Sci. 52, 1433
(1964).
Gp . M. Halleck and B. Olinger, Rev. Sci. Instr. 45, 1408
(1974).
7J. C. Jamieson, in Metallurgy at High Pressures and High
Temperatures, edited by K. A. Gschneidner, Jr., M. T.
Hepworth, and N. A. D. Parlee (Gordon & Breach, New York,
1964), p. 201.
8J . F. Thomas, Phys. Rev. 175, 955 (1968).
9The compreSSion of Al was calculated by Joseph Fritz (Group
M-6, Los Alamos Scientific Lab) using Hugoniot data collected
by M-S.
IOn (atomic mass): W. H. Johnson and A. O. Nier, in Handbook of Physics, edited by E. U. Cou"don and H. Odishaw
(McGraw-Hill, New York, 1967), p. 9-66. ao (cell edge):
R. W. G. Wyckoff, 9rystal Structure, Vol. I, 2nd ed. (Interscience, New York, 1965) p. 10. cp (heat capacity): JANAF
Thermochemical Tables, edited by D. R. Stull (Dow Chemical,
Midland, MI, 1965). CI! (thermal expansion): B. J. Skinner
in Handbook of Physical Constants, Memoir 97, edited by S.
P. Clark, Jr. (Geological Society of America, Inc., New
York, 1966), p. 78. Co (sound speed): Ref. 8 .
uR. G. McQueen, S. P. Marsh, J. W. Taylor, J. N. Fritz,
and W. J. Carte r, in High-Velocity Impact Phenomena, edited by R . Kinslow (Academic, New York, 1970), pp. 299302 and 530-568.
12G • J. Piermarini, S. Block, · and J. D. Barnett, J. Appl.
Phys. 44, 5377 (1973) .
l~C. Weir, S. Block, and G. Piermarini, J. Res. Nat!. Bur.
Stand. C 69, 275 (1965).
14A. J. Brown and E. Whalley, J. Chem. Phys. 45, 4360(1966).
15B. Kamb, J. Chem. Phys. 43, 3917 (1965).
IGW. C. Hamilton and J. A. Thers, Hydrogen Bonding in Solids
(Benjamin, New York, 1968), p. 260.
17W. J. Carter, S. P. Marsh, J. N. Fritz, and R. G.
McQueen, "Accurate Characterization of the High Pressure
Environment," (Natl. Bur. Std. Spec. Pub.) 326, 147 (1971).
18W. J. Carter, High Temp. High Press. 5, 313 (1973).
19J. N. Fritz, S. P. Marsh, W. J. Carter, and R. G.
McQueen, "Accurate Characterization of the High Pressure
Environment," (Nat!. Bur. Std. Spec. Pub.) 326, 201 (1971).
20S. N. Vaidya and G. C. Kennedy, J. Phys. Chem. Solids 31,
2329 (1970).
21S . N. Vaidya and G. C. Kennedy, J. Phys. Chem. Solids 32,
951 (1971).
22S . N. Vaidya, I. C. Getting, and G. C. Kennedy, J. Phys.
Chem. Solids 32, 2545 (1971).
23S. N. Vaidya, S. Bailey, T. Pasternack, and G. C. Kennedy,
J. Geophys. Res. 78, 6893 (1973) .
24F • D. Murnaghan, Proc. Natl. Acad. Sci. USA 30, 244(1944).
250 . L. Anderson, J. Phys. Chem. Solids 27, 547 (1966).
26J . E. Bertie, L. D. Calvert, and E. Whalley, Can. J. Chem.
42, 1373 (1964).
27Reference reported by Kamb (Ref. 1) is "B. Kamb and A.
Prakash, unpublished data (1969). "
'lilW. Holzapfel and H. G. Drickamer, J. Chem. Phys. 48,
4798 (1968).
2SC. Reid, J. Chem. Phys. 30, 182 (1959).
30B. Kamb, J. Chem. Phys. 43, 3917 (1965).
31p . W. Bridgman, Proc. Am. Acad. Arts Sci. 47,441(1912);
J. Chem. Phys. 3, 597 (1935); J. Chem. Phys. 5, 964 (1937).
32J . E. Bertie, L. D. Calvert, and E. Whalley, J. Chem.
Phys. 38, 840 (1963).
J. Chern. Phys .• Vol. 62. No. 1. 1 January 1975