CC BY
25
RUSSIAN JOURNAL OF EARTH SCIENCES, English Translation, VOL 1, NO 1, DECEMBER 1998
Russian Edition: JULY 1998
Thermodynamics of deep geophysical media
V. L. Pankov
Schmidt United Institute of Physics of the Earth, Russian Academy of Sciences, Bol’shaya Gruzinskaya ul. 10, Moscow, 123810 Russia
, R. Heinrich, and D. Kracke
Geoscience Institute, Friedrich Schiller University of Jena, Burweg 11, 07749 Jena, Germany
Abstract. Analysis of thermodynamic properties of geomaterials at high pressures and temperatures existing in the Earth’s interior is presented. The presentation includes a summary on the determination of equations of state based on measured properties of minerals, as well as thermodynamic identities and approximate relations between thermodynamic parameters of the second, third, and in some instances, fourth orders. New expressions were derived for the volume dependences of the coefficient of thermal expansion, the Griineisen parameter, and the Anderson-Griineisen parameter. Attention is given to the preparation of the database on mineral properties. Several geophysical estimates, including the lower-mantle properties, were obtained. It was shown that the thermal expansivity decreases 4-5 times along the “hot” mantle adiabat as the pressure increases from 0 to 1.4 Mbar. Under the same conditions, the heat capacity Cp drops about 10-15%. The thermal pressure at T > 0 is linear in temperature, with an accuracy of 1-3%. The parameter d2Kj/dPdT at P = 0 for mantle minerals was estimated to be (1 — 3) • 10-4 K-1. The acceptable ranges of other lower-mantle parameters are 8y = K' — 8j > 0.2, q < 0.8, 7 > 1.1, 8j < 3 — 3.3, and 8s < 1.9 — 2.2. Deviations from the Mie-Griineisen equation of state are discussed in relation to the volume-and temperature-dependent Griineisen parameter.
W. Ullmann
1. Introduction
Thermodynamically, the Earth is a heat engine described by a variety of parameters that can be determined from equations of state (EOS) and models of condensed media. A great progress has been achieved in the development of such models [e.g., Jeanloz, 1983; Hem-ley et al., 1985, 1987; Wall et al., 1986; Catti, 1986; Cohen, 1987a, 1987b; Dovesi et al., 1987; Wolf and Bukowmski, 1987, 1988; Wall and Price, 1988; Mat-sm et al, 1987; Matsm, 1988, 1989; Price et al, 1989; Catlow and Price, 1990; Isaak et al., 1990; Reynard and Price, 1990; Agnon and Bukowmski, 1990a; Matsui and Price, 1991; D’Arco et al., 1991; Walzer, 1992; Silvi et al., 1993; Catti et al., 1993; Boison and Gibbs, 1993]. Nevertheless, practical studies in geophysics are based,
©1998 Russian Journal of Earth Sciences.
Paper No. TJE98002.
Online version of this paper was published on July 20 1998. URL: http://eos.wdcb.rssi.ru/tjes/TJE98002/TJE98002.htm
to a large extent, on the use of semi-empirical EOS’s [Birch, 1952, 1986; 0. Anderson, 1966b, 1995; Pankov and Ullmann, 1979a, 1979b; D. Anderson, 1967, 1987, 1989; Stacey, 1981; Lelwa-Kopystynski, 1991; Bma and Helffrtch, 1992; Wall et al., 1993]. The properties of geomaterials directly determined from laboratory measurements at high pressures and temperatures are necessary for solving many geophysical problems and provide important constraints on the EOS structure.
Since the fundamental paper of Birch [1952], a great deal of information has been accumulated on the properties of geomaterials and their geophysical implication [e.g., Stacey, 1977a, 1977b, 1992, 1994; Jeanloz and Thompson, 1983; Brown and Shankland, 1981; Zharkov and Kalinin, 1971; Zharkov, 1986; Jeanloz and Kmttle, 1989; 0. Anderson et al., 1992a, 1992b; 1993; Kuskov and Panferov, 1991; D. Anderson, 1989; 0. Anderson, 1988, 1995].
This paper is devoted to the review of relationships between the basic thermodynamic characteristics and of their variation with pressure and temperature. First, we deal with eight parameters of the second order. We
11
emphasize their self-consistent determination and the relations to EOS’s and give a summary of approaches used to find the empirically based EOS’s. An example of the thermodynamically consistent database for mantle minerals is presented. Then, each of the second-order parameters is treated separately: the identities involving their P — T derivatives (third-order parameters) are established and practically useful approximations are analyzed, including some explicit P — T dependences of the second-order parameters. Some estimates for the fourth-order parameters are also given. The relations between various quantities are represented in the form convenient for practical use of experimental data and for theoretical analysis. Finally, a number of estimates are given for the low-mantle properties. Our analysis serves as an addition to the reviews of 0. Anderson [1995] and Stacey [1994].
2. Basic Thermodynamic Relations
In classical thermodynamics, simple systems experiencing reversable changes of state are described by a variety of parameters including the hydrostatic pressure P, temperature T, volume V (or density p), and entropy S. The starting point of thermodynamic analysis is the standard expresions for the total thermodynamic differentials [e.g., Callen, 1960; Morse, 1969; Kelly, 1973]
Kp = —V
dE = TdS - PdV,
(1)
a = V-
CP = Cy =
dE
~dT
dE
~dT
dV_
~dT
m ,'ds
~ dT
= T{dS
dr
(5)
(6)
dP
dV
Ks = -V
dP
dV
7 =
ocKj' V
dP dT
ts =
dT
~dP
(7)
(8) (9)
(10)
By equating the cross derivatives of the four thermodynamic potentials, we obtain the Maxwell relations [see, e.g., Stacey, 1977a]
dT
dV
dS_
dV
dP
~dS
dP
dT
T
v““7CV:
= aK
T,
dS
dV
\dp)T [dTjp aV’
dT
~dP
dV_
~dS
7T
Ks
aVT
~c7’
(11)
(12)
(13)
(14)
s /p
Moreover, it is easily shown that the second derivative of each of these potentials can be expressed in terms of the above parameters or coefficients; i.e., we may write the matrix
dF = —SdT - PdV, F = E-TS, (2)
dG = -SdT + VdP, G = F + PV, (3)
dE = TdS + VdP, E = E + PV, (4)
where E is the internal energy, F is the free energy (Helmholtz potential), G is the free enthalpy (Gibbs potential), and H is the enthalpy.
Eight second-order parameters are largely used in geophysics: the volume coefficient of thermal expansion a, the isobaric Cp and isochoric Cy heat capacities, the isothermal Kp and adiabatic Ks bulk moduli, the thermal pressure coefficient r, and the adiabatic pressure derivative of temperature (adiabatic temperature gradient in pressure) ts- The respective definitions of these parameters are
/ /d2E\ d2E
\dV2J ,
fd2E\
\dP2)s
f d2F\
\dv2)T fd2c\
\ \dP2 L dPdT
dVdS
d2E
dPdS
d2F
dVdT
d2G
\ \
d2E
( d2E \dS2
f d2F\ \dT2)v
fd2G\ \dT2)P
(15)
I<s
V
V Ks
Kp
~v~
V Kp
T
'v
TS
— T —
aF -
T
Cv
T
~Cp
Cv
T~
cP
T )
T
s
s
s
T
p
p
p
The number of independent second-order parameters is obviously three, and consequently, the eight second-order parameters introduced above must satisfy live relations. Four of them are (11)—(14), and the fifth can be derived by changing from one pair of characteristic variables to another; specifically,
VT „ 2 Ks = Kt + ~p, (аКт) Cv
Cp = C'v + VTa2Kx-
(16)
(17)
Thus, if the parameters a, Cp, and Ks (or Kt), as it usually is, are determined experimentally, then the remaining five parameters can be found from the identities
7 =
aKsV Cp _ Ks _ „
CP ’ Cv KT 7 ’
IT
(18)
r = aKT, ts = —
A.s
dP
dT\ 7Tg
= TSpg =
s
dl
Ф
(20)
3. Equations of State and Caloric Functions
The fundamental equation (1) relates five variables two of which are independent. A simple system can therefore be completely described, given knowledge of its thermal P(V,T) and caloric E(V,T) EOS’s. The thermal EOS relates the experimental P — T and theoretical V — T variables and is necessary for transforming these variables in analysis of any thermodynamic property [Zharkov and Kalinin, 1971]. The parameters determined by this EOS kind are termed thermal, whereas the quantities derived either from only the caloric EOS or from both thermal and caloric EOS’s are thermed caloric. The latter, in particular, include Ks, 7, Cp, and Ts-
The two EOS kinds are related by the equation
m)T = -p+oK*T
(21)
whose integral form is
We recall two examples of using the thermodynamic relations in geophysics. The first concerns the Williamson-Adams-Birch equation for the density gradient within the Earth [Birch, 1952; D. Anderson, 1989]. We quote this equation in the form
^=k~°‘PT=h1~ lpCpt) ’ (19)
where p and P are the density and pressure in the Earth’s interior, respectively, $ = Ks/p is the seismic parameter, and r = dT/dP — ts is the superadiabatic temperature gradient. Equation (19) is easily obtained from
E = E(T, V(P, T))
P'V{T, P')
At
- aTV(T, P')
dP' + E(T),
where the transformation
dE
=-V-4<T\^
dV
dP
is used, the integration constant is
with reference to (5), (16) and (14) or (18).
Another example is the adiabatic temperature gradient in depth I within the Earth (see (14)) [e.g., Quarem and Mulargia, 1989]
E(T) = E(T, V(T, 0)),
and the integral is taken along an isotherm. In view of (21), the caloric EOS, E(V, T), is completely determined by the given thermal EOS and function E(T) or H(T) at P = 0. It is clear that any of the caloric functions H(T), S(T), C(T), and Cp(T) at P = 0 can used for the same purpose, since the following identities take place
where g is the gravitational acceleration, and furthermore, the mechanical equalibrium equation dP/dl = pg is used.
According to the PREM model [D. Anderson, 1989], Ф = 50, 80, and 117 km2/s2 at I = 400, 1071, and 2740 km, respectively. Assuming that 7 = 1-1.5 and T = 1700, 2200, and 3000 K sequently at the indicated depths [D. Anderson, 1989; Pankov, 1989], (20) yields (dT/dl)s = 0.3 — 0.5 K/km, the value usually cited in geophysical literature.
1
H{T) = J CPdT + const,
0
T
S(T) = j ^dT + const,
(22)
(23)
C{T) = H (T) — TS(T).
The latter formula can be written in another useful form
T
C{T) = C(T*) - JS(T')dT'
T*
P
T
T
T / T'
where T* is a fixed temperature.
For a mineral whose composition can be expressed by a sum of oxides (component), the Gibbs energy is formulated in difference terms
G(T) = AHf(T*) — TASf (T*)
T / t1
-I {I
T* \T*
where AHf = H — i?ox, ASj = S — Sox, and ACp = Cp — C'pox are the differences of enthalpy, entropy and heat capacity between the mineral and oxide sum, respectively (with allowance for the stoichiometric coefficients). Expressions of type (25) are often used in calculating phase equilibria [e.g., Navrotsky and Akaogi, 1984; Kuskov and GaUmzyanov, 1986; Kuskov et al., 1989; Fabmchnaya and Kuskov, 1991; Fei and Saxena, 1986; Fei et at, 1990; Sobolev and Babeiko, 1989]. Some authors use an approximation AC'p = 0 (or const^ 0; the functions AC'p(T) are sometimes found from empirical formulas of type (67)). In any case, the term AF[f(T*) in (25) implicitly contains an arbitrary normalizing constant [Kalinin et al., 1991].
Integrating (3) gives p
G(P,T) = j V(P',T) dP' + G(T), (26)
o
where G(T) = G(T, 0) is defined by (24) or (25) and can be written in the reduced form
G(T)=AHf(T)-TASf(T) + Gox(T). (27)
Methods for determining EOS’s in geophysics can be classified as follows.
(1) The macroscopic approach suggested by Mur-naghan [1951] and Birch [1952] gives the volume dependence of pressure at T (or S) = const in the form [Ullmann and Pankov, 1976]
P = K0f(x; Kq, KoKq , A02A'",...), (28)
Hereafter, the values with the subscript 0, unless otherwise specified, are taken at P = 0 and an arbitrary temperature, the moduli Kq = Kto,K'0 = (dKp/dP)To,--
are material parameters, and x = V/Vq = po/p is the compression ratio parameter. Most data for the material parameter values were obtained at room temperature [e.g., Summo and 0. Anderson, 1984]. Among the last experimental achievements are ultrasonic measurements at high pressures [e.g., Fujisawa, 1987; Webb, 1989; Yoneda, 1990; Liebermann et al., 1993], X-ray data of high pressures and high temperatures [e.g., Yagi et al., 1987; Mao et al., 1991; Fei et al., 1992a, 1992b; Boehler et al., 1989], spectoscopic observations of minerals [e.g., Chopelas, 1990a, 1990b, 1991a, 1991b, 1993; Ftofmeister, 1987, 1991a], and high-temperature P = 0 measurements of elastic constants by the rectangular parallelepiped technique [0. Anderson et al., 1992a; O. Anderson, 1995].
An explicit form of function / in (28) (the volume dependence of pressure) was considered by Murnaghan [1951], Birch [1952, 1968, 1978, 1986] and others [Thomsen, 1970, 1971; Ahrens and Thomsen, 1972; Davies, 1973; Ullmann and Pankov, 1976, 1980; Pankov and Ullmann, 1979a; Stacey, 1981; Aidun et al., 1984; Jean-loz, 1989; Btna and Helffrich, 1992; Isaak et al., 1992; Wall et al., 1993]. The most widely used equation of this type is the Birch-Murnaghan EOS.
Elastic moduli and sound velocities in minerals depends first of all on the composition, crystalline structure, pressure, and temperature. Data on these dependences are generalyzed and interpreted in terms of empirical laws such as the Birch’s law, the seismic EOS, the law of corresponding states, and a universal EOS [Birch, 1961; 0. Anderson and Nafe, 1965; D. Anderson, 1967, 1987; Chung, 1973; Davies, 1976; 0. Anderson, 1973; D. Anderson and 0. Anderson, 1970; Mao, 1974; Kalinin, 1972; Schankland and Chung, 1974; Campbell and Ffemz, 1992]. These laws enable us to estimate the parameters Ko and, to a lesser accuracy, K'0 for unmeasured minerals [D. Anderson, 1988; Duffy and D. Anderson, 1989].
(2) Statistical physics describing the vibrations of atoms in crystals provides the background for microscopic EOS theory including the Mie-Griineisen EOS [Griineisen, 1926; Born and Ftuang, 1954; Leibfried and Ludwig, 1961; Knopoff, 1963; Knopoff and Shapiro, 1969; Zharkov and Kalinin, 1971; Wallace, 1972; Mula-rgia, 1977; Mulargia and Boschi, 1980; Ftardy, 1980; 0. Andrrson, 1980; Gillet et al., 1989, 1990, 1991; Richet et al., 1992; Reynard et al., 1992]. This approach also uses the lattice or vibrational Griineisen parameters, as well as either semiempirical potentials of atomic interactions or the reference (isothermal or adiabatic) P — V relations derived from continuum mechanics [e.g., Al’tshuler, 1965; Zharkov and Kalinin, 1971; Ahrens and Thomsen, 1972; McQueen, 1991]. The material parameters in these cases are determined using static and dynamic compression data, elastic constant
A Cp
rpil
dT" dT' + Gox{T),
(25)
measurements, caloric functions, and vibrational spec-trums.
(3) Integrating (12) yields the pressure as a sum of two terms: a reference isotherm and the thermal pressure increment APth- This thermodynamic approach based on experimental data has been developed by 0. Anderson [1979a, 1979b, 1979c, 1980, 1982, 1984, 1988, 1995] and was used to describe the X-ray and resonance data for a set of minerals [0. Anderson et al., 1982, 1992a; 0. Anderson and Yamamoto, 1987; 0. Anderson and Zou, 1989; Mao et at, 1991; Fei et at, 1992a, 1992b].
(4) More intricate theoretical EOS models are derived from ab initio calculation using the Hartree-Fock and Thomas-Fermi-Dirak methods, as well as pseudopotential theory, many-term contributions in semiempir-ical potentials, and molecular dynamics [Hemley et al. 1985, 1987; Isaak et al., 1990; Wolf and Bukowmcki 1987, 1988; Wall and Price, 1988; Wall et al., 1986 D’Arco et al., 1991; Price et al., 1989; Matsm et al. 1987; Matsm, 1988, 1989; Reynard and Price, 1990 Agnon and Bukowmski, 1990a; Walzer, 1992; Cohen 1987a; Dovesi et al., 1987; Catlow and Price, 1990 Boisen and Gibbs, 1993; Silvi et al., 1993; Catti et al., 1993; Barton and Stacey, 1985].
As mentioned above, the complete description of a simple system requires knowledge of either any of its thermodynamic potentials or its thermal EOS and one of the caloric functions (at P = 0). Table 1 lists various approaches to the determination of EOS’s, showing which functions must be found from theory or experiment so as to provide such a complete description. These approaches can also be formulated in the form of partial differential equations with appropriately chosen boundary conditions.
4. Thermodynamic Parameters of the Third and Higher Orders
The order of a thermodynamic parameter (characteristic of a matter) is defined by the maximum order of the thermodynamic potential derivative involved to define the thermodynamic parameter. To find all of the third-order parameters (P, V, T, or S derivatives of the second-order parameters), whose total number for the potentials in (l)-(4) is 16, it is sufficient to know four independent and appropriately chosen third-order parameters, in addition to knowledge of the lower-order parameters. Specifically, experiments often provide information on the derivatives (dKs/ dP)p (or
8Kt
~dP
= A
( dKs
V dT
(dCP\ V dT J
a"d 1 %
To extrapolate data on thermodynamic properties to high pressures and temperatures, the power volume dependence is often applied stating that the logarithmic volume derivative of the parameter considered is a constant [Zharkov, 1986; D. Anderson, 1988, 1989]. The temperature derivative of any parameter A at P = const is represented in the dimensionless form
1 ( d In A
dT
1 /9 In A
dT
d In A d In V
d In A d In V
The relationships of these derivatives to other parameters are further discussed in later sections.
> (29)
y V 111 V- / T
where the first term characterizes the so-called intrinsic anharmonicity and the second is a parameter of the extrinsic anharmonicity related to thermal expansion [Jones, 1976; Smith and Cam, 1980]. Parameter A can be any physical property, such as the transport coefficients or mode Griineisen parameters [Reynard et al., 1992; Gillet et al., 1989].
5. Thermodynamic Database
The database on properties of minerals, required for geophysical analysis and EOS construction, must include first of all their density and the second and third order thermodynamic parameters. An example of such database for three mantle minerals is given in Tables 2 and 3, and the database for 25 mantle minerals, including their high-pressure phases (and some fictive phases), is presented in Internet [Pankov et al., 1997]. The parameter values in these tables refer to the conditions P = 0, T = 300 K or P = 0 and the temperature indicated. Apart from the second-order thermodynamic parameters, Table 2 includes the molar mass M, mean atomic weight p, density p, the melting temperature Tm, the Debye temperatures O (Oa is the acoustic temperature, ©„ is from fitting the Mie-Griineisen EOS to data on a [Suzuki, 1975a, 1975b], and Oth is our estimate from data on specific heat), the classical value C'v = 3R/fj (R is the gas constant), the enthalpy AHj and entropy ASj of mineral formation from oxides, and the estimated thermal pressure Pth & Q.haKp. It is important to have mutually consistent values of the second (and higher) order parameters: here, the calculations are based on the input values of Ks (or Kt), a, and Cp. At high temperatures, T > 300 K, the a and Cp values were found by the empirical formulas from Fei and Saxena [1987] and Fei et al. [1990, 1991], and for Ks, we give either experimental values or our estimates through the Anderson-Griineisen parameter 6s (at 300 K), which is assumed to be a constant (see Table 3 and sections 9 and 10). The values listed in Table 3 are based on the input values of the derivatives (dKs / dP)p, (dKs/dT)p (or (5s),
p
p
T
p
p
Table 1. Examples of complete thermodynamic description of a system
Version Given functions
1 P(V,T) and the temperature dependence of any of the caloric functions G, H, S, Cp, Ks, 7-5,7, C'v at P = 0
2 F(V,T) from statistical physics, resulting in a quasiharmonic or anharmonic EOS with 7(V,T) or the Mie-Griineisen EOS with ^(V)
3 P(V, T0) and E(V, T) for T0 > 0
4 P(V, T0),Cv(V, T) and E(V) for T = 0, T0 > 0
5 * P(V,To),t(V,T) and the temperature dependence of any of the caloric functions mentioned in version 1 at P = 0
6 P(V/Vo,Ko,K'0,...),Vo(T),Ko(T),K'0(T),... and the temperature dependence of any of the caloric functions mentioned in version 1
7 P(V, To), Ts(P, T), and any of the functions CP(T), a(T), H(T), S(T), and G(T) at P = 0
8 P — V Hugoniot, 7(V), and the Mie-Griineisen EOS form
9 KS(P, T),a(T), and CP(T) at P = 0
* The thermal EOS can also be found given the pairs of functions P(V, To) and a(P,T), P(V,T0) and KT(V,T), or V(0,T) and KT(P,T).
(dCp/3T)p, and a = a~2(da/dT)p, as well as on the second-order parameter values given in Table 2. The high-temperature values of the third-order parameters were evaluated using the condition (OKs/ dP)p & const. Finally, in Table 2 are given the references to sources of thermodynamic data for each of the minerals.
6. The Volume Coefficient of Thermal Expansion
6.1. P—T derivatives
It follows from the identity 32V/dTdP = d2V/dPdT that [Birch, 1952]
da
~dP
T
dKp
~dT
(30)
p
1
aK7
dK
dT
where the isothermal Anderson-Griineisen parameter St is introduced [0. Anderson, 1966a, 1967; Barron, 1979].
The variation of a with temperature at P = const is characterized by the parameter a Fiirth, 1944; 0. Anderson, 1966b; Birch, 1986; 0. Anderson et al, 1993]
1
da
dT
da
dT
p
d In a d In V
(32)
The term
da
dT
can be related to the derivatives of C'v and Kt , by making use of the identity d2S/dVdT = d2S/dTdV, which leads to
This fundamental relation is written in the dimension-less form
1
da
dT
_____l_ (dCv\_______l_
dKj
12VT \dP )T aI<T V dT
d In a d In V
d In Kt 9 In p
(31)
1
(dCv\ PVT \dP ),
+ (St — Kr)
p
p
T
p
Parameter MgO, periclase AI2O3, corundum Mg2SiC>4, forsterite
300 К 1800 К 300 К 1800 К 300 К 1700 К
M, g/mole 40.32 101.96 140.69
fi, g/mole 20.16 20.39 20.10
p, g/cm3 3.585 [1] 3.354 [1] 3.982 [1] 3.831 [1] 3.222 [1] 3.055 [1]
Tm, к 3125 [2] 2345 [2] 2163 [2]
Qa, К 945 [1] 811 [1] 1034 [1] 922 [1] 763 [1] 668 [1]
Qth, к* 761 706 966 871 887 763
Qa, К 942 [45] 1031 [45] 732 [45]
S, J/К mole 26.94 [2] 113.14 [2] 50.92 [2] 256.68 [2] 94.110 [53] 372.48
H, kJ/mole** 5.166 [2] 80.28 [2] 10.016 [2] 192.45 [2] -43.392*** 194.89
ASf, J/К mole 0 0 0 0 -1.23 -
AHf, kJ/mole 0 0 0 0 -60.64 -
Ks, GPa 163.9 [1] 132.7 [1] 253.7 [15] 221.8 [15] 128.7 [1] 103.8 [1]
a, 10-5 К”1 3.12 [1] 5.13 [1] 1.62 [1] 3.25 [1] 2.72 [1] 4.62 [1]
Cp, J/gK 0.928 [1] 1.358 [1] 0.771 [1] 1.318 [1] 0.840 [1] 1.370 [1]
KT, GPa 161.6 116.6 252.2 204.7 127.4 95.3
7 1.54 1.50 1.34 1.43 1.29 1.14
Cv, J/g К 0.915 1.193 0.766 1.216 0.831 1.258
CclaSS; J/g K 1.237 1.223 1.241
a^/T 0.014 0.138 0.006 0.084 0.011 0.089
т = аКт, MPa/K 5.04 6.00 4.08 6.65 3.46 4.40
rs, K/GPa 2.82 20.0 1.58 11.6 3.00 18.7
Pth, ГПа 0.717 [1] 9.95 [1] 0.6 4.13 [1] 0.429 [46] 6.30 [1]
3-12, 32, 35, 46, 49, 12-15, 74, 2, И, 12, 22, 32, 45, 66, 69, 74, 83,
Reference 53, 68-71, 73-78, 83, 97, 79-83, 118, 211 89, 95, 97, ИЗ, 126, 128, 129, 178,
124, 131, 193, 202, 211 190, 191, 194, 201, 204, 206-211
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2. Robie et al. [1978]
3. Isaak et al. 1989a]
4. Isaak et al. 1990]
5. Jackson and Niesler [1982]
6. 0. Anderson and Zou [1989]
7. 0. Anderson et al. [1993]
8. Chopelas [1990b]
9. Hemley et al. [1985]
10. Summo et al. [1983]
11. 0. Anderson and Suzuki [1983]
12. Summo and 0. Anderson [1984]
13. White and Roberts [1983]
14. Furukawa et al. [1968]
15. Goto et al. [1989]
22. Jeanloz and Thomsen [1983]
32. Richet et al. [1989]
35. M. Liu and L. Liu [1987]
45. Duffy and D. Anderson [1989]
49.
53.
Fei et Fei et
1991
1990
66. Kuskov and Gahmzyanov [1986] 124
68. Duffy and Ahrens [1993] 126
69. Fei and Saxena [1986] 128
70. Carter et al. [1971] 129
71. Vassihou and Ahrens [1981] 131
73. Boehler [1982] 178
74. Berman [1988] 188
75. Spetzler [1970] 190
76. Mao and Bell [1979]
77. Yoneda [1990] 191
78. Chopelas and Nicol [1982] 193
79. Richet et al. [1988] 194
80. Mao et al. [1986] 201
81. Gupta [1983] 202
82. Fmger and Hazen [1978] 204
83. Ullmann and Pankov [1976] 206
89. Chopelas [1990a] 207
95. O. Anderson and Goto [1989] 208
97. Suzuki [1975a] 209
113. Hofmeister [1987] 210
118. McMillan and Ross [1987] 211
RM,
Matsm [1989]
Gillet et al. [1991] Hofmeister et al. 1989] Chopelas [1991b]
Wolf and Bukowmski [1988] Graham et al. [1982] Weidner and Hamaya [1983] Yeganeh-Haeri and Vaugham, [1984]
Webb [1989]
O. Anderson et al. [1992b] Reynard et al. [1992]
Saxena [1988]
Saxena and Zhang [1990] Hazen and Fmger [1989] Fiquet et al. [1992]
Will et al. [1986] hshi [1978]
Hazen [1976]
Webb and Jackson [1985] Richet and Fiquet [1991]
* ®th from the Debye Cy at T = 300 K and from the classical approximation S = ---------------(4 — 3ln@/T) for T > ©.
** It is assumed that H = 0 at T = OK for quartz, periclase, and coesite; for other minerals, H{T) is normalized relative to the sum oxides at T = 300 K.
H = AH
Table 3. Third-order thermodynamic parameters of minerals at normal pressure and two values of temperature*
Parameter MgO, periclase 300 K 1800 K AI2O3, corundum 300 K 1800 K Mg2Si04, forsterite 300 K 1700 K
(dKs/dP)T 4.13 [5]*** 4.13 4.28 [12] 4.28 5.1 [36]** 5.1
(.dI<T/dP)T = K’ 4.16 4.42 4.30 4.37 5.12 5.23
(dKs/dP)s 4.09 3.70 4.26 4.05 5.05 4.75
Ss 2.83 [1] 3.12 [1] 3.83 [1] 2.68 [1] 4.45 3.96 [1]
St 4.96 4.83 5.99 4.42 5.89 5.27
ST °v -0.80 -0.42 -1.69 -0.05 -0.77 -0.04
Ss °v 1.24 0.51 0.42 1.27 0.59 0.72
a 46 [1] 3.5 [1] 140 [1] 4.9 [1] 42 [1] 5.9 [1]
{dlnCp/dlnT)p 0.43 [1] 0.15 [1] 0.67 [1] 0.16 [1] 0.53 [1] 0.23 [1]
(dlnC'p/dlnT)v 0.42 0.10 0.67 0.13 0.53 0.20
(dlnC'p/dln V)T 0.65 0.55 0.84 0.46 0.47 0.44
(9 In Cv/d In T)p 0.41 -0.01 0.66 0.06 0.52 0.13
(9 In CV/9 In TV 0.41 0.01 0.66 0.06 0.51 0.14
(91nCV/91nV)T 0.56 -0.24 0.79 0.04 0.39 -0.08
CP fda/Cp\ a2 V dT ) p 0.06 1.88 2.14 2.16 -23.0 1.47
(9 In 7/9 In V)t = q 1.24 1.66 1.90 1.01 1.38 1.12
a-1(9 ln7/dT)v -2.96 -1.90 -2.5 -0.52 -30.0 -2.60
(9 In Ts/d In V)t = n 5.31 5.29 6.15 4.96 6.43 4.74
a-1 (9 In ts/8T)p QO O 1—1 137 209 20.3 99.6 15.2
* Under the assumption of (dK5/dP)j^ = constant. Placed in brackets are the references (see Table 2). ** 4.9—5.3 [12]
*** 3.85-4.49 [12]
Similarly, the identity
a = —1 —
d2s
d2s
dPdT dTdP
1 (dCP\ 12VT V OP JT
yields
= -1 +
1 + a^T f 9 In Cp
a^T
9 In V
(34)
where the convenient dimensionless product is used T _ \=Cp 1
^ Kt Cy
Birch [1952] pointed out that parameter St for various materials usually lies between 4 and 8 (at normal conditions), which was borne out by subsequent studies [see, e.g., Summo and 0. Anderson, 1984; 0. Anderson et al., 1992a] some exclusions are also encountered: e.g., St 1 for KMnF3 and St 77 for Re2C>3.
The parameter a values at normal conditions are commonly greater than the St values [Birch, 1952; 0. Anderson et at, 1992a]. The data and estimates listed in
the tables of Pankov et al. [1997] for 25 minerals fall into the range 10^ci^270. However, at high temperatures (T > 0), the St and a values become closer to each other, and their concidence would mean that a were dependent only on volume (i.e., the intrinsic an-harmonicity were suppressed).
The following three assumptions and their consequences are of interest:
(1) The specific heat C'v is independent of pressure,
i.e., C'v = C'v(T), as in the Van der Vaals or Hildebrand EOS’s [0. Anderson, 1979a], or alternatively, C'v = const, as in the classical limit at T > 0. Then, from (32) and (33),
^(w)v = St~ki'
= 2 St — Kl
(35)
At P = 0, St > K’ is the common case.
(2) Kt depends only on volume; the Kt(V) approximation is often warranted at T > 0 [0. Anderson, 1982; D. Anderson, 1988, 1989]. Then, (3Kt/8T)v = 0, and from (29) and (31), we have K’ = St] i.e., St either
T
depends only on volume or is a constant (leading to the Murnaghan EOS (44)).
(3) If both conditions (1) and (2) take place, then a = a(V) and K' = St = a (that is either volume-dependent or a constant).
6.2. Explicit volume (pressure) dependences of a at T = const
In most of the interiror of the Earth, T > O and a minimally depends on temperature. Consider a few approximate relations for evaluating the isothermal or adiabatic variation of a.
6.2.1. By using the general EOS form of (28) and the formula (12), we find the expansion for a
o
“ 1 * P , K'
— — 1 — Stq——I------------—
Ct o -ft-T CXqi\T
df
dK'o
df d(KoK")
dK'Q dT ' d(KoK") dT ' J '
If only two first terms are retained in (36), then the formula of Birch [1952, 1968] derives. It should be noted that a in this formula changes its sign at Kt / P = Sto (the condition that may be achieved in the lower mantle at Kt/P = 4.7). Moreover, in this case, St given by most of the type (28) EOS’s increases with pressure instead of its usual decrease (see section 9). It was shown, however, that such a change in the a sign is forbidden thermodynamically [Pankov, 1992].
As an example of using (36), we calculated a(P/Ko) with the help of the EOS form proposed by UUmann and Pankov [1976, 1980], for which
f(x, K'0) = -yy', y=-(xu- 1)
1 ,0 / dV
“ = 3 ( “) ’ y=di:
so that
KT = Kqx (y2 + yy") .
setting dK'0/dT = ±2 • 10-3 K-1 (the values that we estimated for NaCl from data discussed by Birch [Birch, 1978]). The value 2 • 10-3 K-1 is not realistic since it results in the increase of a with pressure (Figures 1 and 3). On the other hand, the negative value —2 • 10-3 K-1 is too small, since it considerably lessens the pressure at which the condition a < 0 mentioned above is reached.
The approximations K' = St (see above) and St & constant at T > O imply a very weak temperature dependence of K'. The lattice dynamics models show that K' for MgSiOs perovskite varies less than 10% in the temperature range of 300-2000 K (dK'0/dT < 2 • 10-4 K-1). The theoretical PIB model for MgO [Isaak et al, 1990; O. Anderson et al, 1993] shows that dK'0/dT somewhat increases with temperature in the same interval of 300-2000 K, with the values ranging, on average, from 2.8 • 10-4 to 4.2 • 10-4 K-1. Values of a similar order follow from the approximation dinK'0/dhip = — 1 indicated by D. Anderson [1989] for PREM.
The derivative dK'0/dT can also be estimated by the approximation Kt = Kt(V) (i.e., K' = St and (70) are allowed for) noted above
fdK'
9 ( '3KT'
dT \ , dP y
d / 8Kt
: dP 1 dT
I dT
(39)
(37)
(38)
First, we set dK'0/dT = 0 and neglect terms containing K" , K"', .... The a/ao versus P/Ko curves, obtained for resonable values K'0 = 3 and 4 and Sto = 2,4, and
6, are shown in Figure 1. For change from P to volume, Figure 2 gives the variation of P/Ko with x. We see, in particular, that the Sto values significantly affect the estimated a under lower-mantle conditions (at the base of the mantle compressed along its “hot” adiabat, x & 0.7 and P/Ko & 0.70 for Ko & 1.9 — 2.0 Mbar and Kq = 3.8 — 4.1 [D. Anderson, 1989]). Furthermore, the approximation used for a may lead to the nonrealistic result a < 0 within the lower mantle.
To illustrate the influence of the non-zero dK'0/dT values, now we allow for the third term in (36), with the
where the primes indicate pressure derivatives. With a Pd 3-10-5 K-1 and —KtK" k, 5 — 10 [e.g., Pankov and Ullmann, 1979a; Hofmeister, 1991b], we find dK'0/dT & (1 — 3) • 10-4 K-1, which is close to the estimates found above.
Finally, we can use identity (91) from section 9, which of course leads to (39) for K' = St (see (71)). Although the terms in (91) are close to each other, the reasons given below justify the inequality (8St/0P)t&0, and consequently,
dK'
~dT
^ aST (St — Kr).
(40)
Substituting the parameter values from Tables 2 and 3 (see also tables in Pankov et al. [1997]) into the right side of the above, we find
dKo
dT
^(i^-io^k-1
It should be emphasized that the correction to a in (36) related to this derivative enables us to avoid negative or increased values of a at high pressures. The effect of the value dKydT = 2 • 10-4 K-1 on the a curve for Kf0 = Sto = 4 is shown in Figure 1.
T
p
— —a KtK
p
p
a/a„
- Lower mantle
a = a0x5r , from Anderson [1967]
a from Birch formula
dK0 dT
a from (44), * 0
to
o
0.2 0.4
Figure 1. a/ao versus P/Kq. (1) The vertical bars show the dependence a = aoxST, where St = 2,4, and 6; the upper and lower ends of the bars are for K'0 = 3 and 4, respectively. (2) The hatched bands are a by the Birch formula (36), where only two first terms are taken into account, i.e., dK'0/dT = 0; Sto = 2,4, and 6, respectively, and the upper and lower ends of the bands are for K'0 = 4 and 3, respectively. (3) The dot-and-dash lines are a by the generalized Birch formula (36) with allowance for the parameter dK'0/aT, whose values are indicated on the plot («o = 3 • 10-5 K-1 is assumed). The pressure and volume in all these cases are related by equation (37) (see Figure 2).
PANKOV ET AL.: THERMODYNAMICS OF DEEP GEOPHYSICAL MEDIA
0.7 0.8 0.9 X = V/V0
Figure 2. Relative pressure versus relative volume by equation (37) [UUmann and Pankov, 1976, 1980]
6.2.2. 0. Anderson [1967] derived the power law
a = aoxSTO, (41)
by integrating (30) and (8), provided that
(8C\
V dP
8CV
dP
0,
which yields
St = K' — 1 + q,
where parameter q is defined as
_ / 8 In 7 ^ ~ V 8 In V
(42)
(43)
Moreover, he assumed that St & K' K, const. The constancy of K' (or alternatively, K' = K'(T)) leads to the Murnaghan EOS (of a type of (28))
P=ISl(x-K_l
Kt = Knx
-k'
(44)
It is clear that (41) simply follows from the definition of St by (31) on the condition that St is either
only temperature-dependent, St = St(T), or is a constant; moreover, (44) can be replaced with any suitable approximation to the isotherm P(V,To).
The two functions a(V, T) and P(V, T), however, cannot be picked independently. For example, the EOS can apparently be defined by specifying a(V,T) and P(V,To). At this point, it is appropriate to discuss the following generalization of the results mentioned in various papers [Birch, 1968; Clark, 1969; O. Anderson, 1986; D. Anderson, 1989]. Consider four statements:
(1) the Murnaghan EOS (44) is valid, where it is assumed that Ko is a function of temperature, Ko(T), and K'0 is either a constant or depends only on temperature;
(2) (8St/8P)t = 0 (i.e., St = const or St = St(T));
(3) (8K1 /8T)p = 0; (4) St = K' that, according to (71), is equivalent to Kt = Kt(V) (i.e., t(T) = o:Kt or r = const).
Then, by making use of identities (44), (71), (91), and (92), it can be proved that, if any two (except the pair (1) and (5)) of the four statements above hold true, then the other two statements are also valid. Moreover, then St = K' = const and (44) always takes place. If, in addition to these two statements, it is assumed that C'v = const or C'v = C'v(T), then we have a = a(V),
T
s
T
a/a0| T
Figure 3. a/«o versus V/Vq at T = 1800 K. (1) From (36)—(38) with allowance for the three first terms in (36) and with MgO parameters at 1800 K: «o = 5.13 • 10-5 K_1, Ko = 117 GPa and Sto = 4.66 from [0. Anderson et al., 1992a] and K'0 = 4.41 (estimated under the condition that dlnK'/dlnV = 1 [D. Anderson, 1989]); shown at the curves are values of dK'0/dT. (2) From (48) for Sto = 5 and k = 1.31 [0. Anderson et al., 1992b]. (3) a = uqxSt , where St = 4.66 [0. Anderson, 1967]. (4) From (47) for Sto = 4.66 [Chopelas and Boehler, 1992]. The lower-mantle compression region is between x = 0.85 and x = 0.70 (P/Ko & 0.7 at the base of the mantle, for Ko & 190 — 200 GPa and K'0 = 4). Periclase (with Ko = 117 GPa and K'0 = 4.41) is more compressible than the lower mantle, so that the third-order Birch-Murnagjan EOS for periclase yields x Pd 0.65 and P/Kq & 1.16 for P = 136 GPa and T = 1800 K.
K' = St = a. = const and r = const (see section 6.1 and (69) and (71)).
It is clear that such statements place constraints on the EOS formulation. For example, when an equation of type (28) is accepted instead of the Murnaghan EOS, only one of the above statements can strictly be true. Specifically, the concurrent use of the Birch-Murnaghan EOS and the assumption St = St (T) (or St = const) is incompatible with the condition St = K' or K' = K'{P).
Another not obvious inference is that the Murnaghan equation (44) uniquely follows from the assumptions Kt = f(P) + o,T (a being a constant) and r = t(T) or const.
It is interesting to consider the use of the Murnaghan formula as the potential (lattice) part of the P — V — T
EOS in the classical high-temperature approximation. In general, for C'v = const, we have a linear dependence of Kt (V, T) on T (see (69) and (70)), and only for j/x = const (q = 1) and the Murnaghan potential, we obtain
Kt = aP + b + cT,
where a, b, and c are constants (K1 = const). Hence, it is seen that all isotherms (T > 0) are also represented by formula (44), but for a nonlinear j(x) behavior, this is, strictly speaking, not the case.
The consideration presented above concerns also the Birch’s law, which for minerals with the mean atomic mass of fi = 20-22 g/mole can be written in the power form Kt = aVb (where a and b are constants and the distinction between K$ and Kt is neglected). Since here K' = const and Kt = Kt(V) and (44) is used,
this law gives rise to St = K' = const, formula (41), and for C'v = const, a = a(V).
Figure 1 shows the behavior a determined by (41), where the EOS is found by (37) with K'0 = 3 and 4 (the respective a curves pass through the ends of the bars in Figure 1).
The original assumption of 0. Anderson [1967] St = const was justified by the ultrasonic and shock-wave data of that time and was seemingly corroborated by later ultrasonic and resonance measurements [0. Anderson et al., 1990; Chopelas and Boehler, 1992]. In particular, based on data for seven minerals, the value of St = 4-6 was recommended to be representative of the lower mantle. However, analyzing seismological and geoid data, D. Anderson [1987, 1989] found St = 2-3 for in situ lower-mantle conditions. The assumption can therefore be made that St must decrease with pressure. Evidence for this can also be found in shock-wave and static compression data [Birch, 1986; 0. Anderson et al, 1993].
Ab initio calculations for MgO by Reynard and Price [1990] give a constant value in the range 0.7 < x < 1.0. Another ab initio results [Isaak et al., 1990] reveal, however, that St actually decreases by decreasing x.
To determine the a(x) more accurately than given by the power law (41), Chopelas and Boehler [1992] used data on the adiabatic pressure gradient ts and specific heat Cp. From the Maxwell relation (14), they derived
St =
(9 In CP d In V
- 1,
(45)
where n = (9 In Ts/<9 In V)t- Then, they set n = mx, where m = 6 ± 1 from measurements for weakly compressible materials, and (d In Cp/d In V)t = 1 or 0 for T < O or T > O, respectively (compare with the data listed in Table 3 and the paper by Pankov et al. [1997]). Thus, the Chopelas and Boehler’ formula for a can be written in the form
2/_ — £<5to(>-1) «o
— = — e<-ST0+1',<-x 1), St = mx — 1, for T > O. (47) «o x
0. Anderson et al. [1992a, 1992b, 1993] favored the power law St = STo%k (for T<>0) that yields
St = mx, for T < O,
(46)
k\
— = e k «o
(48)
with only small deviation from values by (47). The value of k = 1.1-1.4 in (48) was inferred from the theoretical PIB model of Isaak et al. [1990].
Applying (47) or (48) to the lower mantle, we find that a decreases 4-5 times along the “hot” lower-mantle adiabat, from the state P = 0 and T Pd 1700-2000 K to the base of the mantle. The power law (41) with St = 5-6 gives a greater decrease in a (6-8 times), and the same law with St = 2-3 results in a smaller decrease of a (2-3 times). Although approximations (47) and (48) are more preferable than (41), they require additional confirmation and information on parameters m, k, and (d In Cp/d In V)t- For comparison, we note that the estimation of a in the lower mantle by (36) and (37), as described in section 6.2.1, with dK'0/dT = 2.3 • 10-4 K-1, gives the results close to those derived from (47) or (48) (Figure 3).
Similar results for a with St decreasing under compression were obtained by Zharkov [1997] from his analysis of EOS’s at extremely high pressures. Still earlier, Zharkov [1959] showed that the lower mantle thermodynamics quantified on the basis of the Debye model and seismic data gives the 4-5-fold decrease in a at the mantle base compared to the value at P = 0.
6.2.3. To this point, considering a at high compression, we have not applied to the Griineisen parameter 7. However, the problem of thermal expansivity at high pressures and temperatures is intimately related to the problem of a similar variation of the Griineisen parameter. D. Anderson [1987, 1989] characterized the lower-mantle thermodynamics by using the acoustic or Bril-loin 7. For the adiabatic lower mantle, he found from PREM that 70 = 1.4 and «o = 3.8 • 10-5 K-1 at P = 0 and T = 1700 K; the value of 7 was determined by the thermodynamic relation (8) for C'v = const (T > O). Given function j(V), the variation of a with volume in the classical temperature range can be evaluated by the formula derived from (8)
a
«o
7/i'c
7o Kt x
(49)
Note that the thermodynamic parameter 7, generally speaking, is different from the so-called lattice Griineisen parameter [e.g., Mulargia, 1977; D. Anderson, 1989;
O. Anderson, 1968, 1979b, 1980]. However, assuming that the latter depends only on volume, both parameters were found to coincide (the same inference follows from the quasiharmonic atomistic model of EOS at high temperatures, when, on the other hand, we come up with the purely thermodynamic consequence 7 = *j(V) for C'v = const (see section 11).
The three most familiar formulas for the lattice 7 can be written in the general form [Zharkov and Kalinin, 1971]
9K' + 2mP/I<T — 6 m — 3 7 =------6(3-2 mP/KT)--------------------' (5tl1
where m = 0, 1, or 2 gives the formulas of Slater,
n
T
Dugdale-Macdonald, and Zubarev-Vashcheno (or Irvine and Stacey [1975]), respectively. The latter of these formulas appears to be the most favored, at least at T > 0, for high symmetry crystals. Following Leibfried and Ludwig, 1961], 7 can approximately be expressed in terms of the root-mean-square frequency of atomic oscillations. For cubic crystals with the central interaction, when only the nearest neighbors are allowed for, this approximation also leads to the Zubarev-Vashchenko formula [Pankov, 1983; Hofmeister, 1991a].
Calculation by (50) requires knowledge of the P(V) dependence at T = 0 K, but the replacement of the T = 0 K isotherm by any isotherm at T > 0 K is not significant for this case. Using the EOS from (37) at K'o = 4 and determining 7 by (50) at m = 2 and then a by (49), we obtain ao/a = 1.7 for x = 0.7 (that is approximately at the mantle base). Such a small decrease in a compared to the 4-5-fold decrease found above is due to the fact that the EOS by (37), like many other P(V) relationships [Pankov and UUmann, 1979b], results in a low value of the slope
0 - 0.5,
data for MgO, CaO, CaMgSi206 and e-Fe at pressures to P > 140 GPa. By using (49) and (51) with q = const and Kt/Kq from the PIB model for MgO [Isaak et at, 1990], they found q = 0.5 ± 0.5 that is smaller than q = 0.83-1.26 in the compression range x = 0.67-1.0 along the PIB isotherm. Periclase is more compressible than the lower mantle matter and has x = 0.67 at P = 134 GPa near the mantle base (according to the PIB 2000 К isotherm of MgO, P/I<0 = 1-047, KT/K0 = 4.699, and K’ = 4.74). With these values, the shock wave results of Duffy and Ahrens for MgO give ao/a = 3.1-4.7 at the mantle base (x = 0.67), i.e., the value 11.6 times less than ao/a by (47) and (48) at the typical value of St0 = 5 ± 1 (if the value St0 = 4 is used in (47) and (48), the resulting ao/a value will be closer to the shock-wave estimate above). These results can be viewed as an argument for the decrease of both St and q under compression.
In analysis of the volume dependence of 7, 0. Anderson et al. [1993] proposed the power law
q = qox
(52)
calculated from (50). This either tells us that a more flexible EOS involving the independent parameter KoK” = K2 (such as in model 2 by UUmann and Pankov [1980] or the Birch-Murnaghan fourth-order EOS) must be introduced, or some amendments to (50) are required. The Zubarev-Vashchenko formula was somewhat improved by Stacey [1981, 1992], but nevertheless, the slope q for most two-parametric (Ko and K'0) EOS’s appears to remain low).
Another useful approximation for 7 is the empirical power law [e.g., 0. Anderson, 1968, 1974; McQueen et al, 1970]
7 = loxq, (51)
where q is often assumed to be one, according to shockwave data [McQueen, 1991] or studies of the mantle [0. Anderson, 1979b; D. Anderson, 1989]. From (49) with q = 1, we find Birch’s formula oiKt = aoKo = const, which gives ao/a = 3.5-4.0 at the mantle base (for * = 0.7, K'o = 4, KT/Ko = 3.51, and T = 20003000 K). The value of q = 1.5-2.0 may be more favored for the mantle perovskite [Pankov et al., 1998], yielding, however, ao/a = 4.2-5.0 that is close to the result obtained from (47) and (48).
Note that, according to (49), the assumption of the power laws for a (41) and 7 (51) again gives the Mur-naghan EOS (44). Since the latter fits data well over a range of P/ Ko&0.3 (*^0.82), we expect (41) to be a sufficient approximation for a in the same compression range.
Duffy and Ahrens [1993] estimated a from shock wave
(53)
which, similarly to (48), yields
7 )
— = e V .
7o
Setting qo = 1.5-2.0 and v = 1 (as for MgO, according to 0. Anderson et al. [1992b]), we find from (49) that ao/a = 4-4.5 at x = 0.7. Note that qo for MgO descends from 1.72 to 1.26 as temperature increases from 300 K to 2000 K [0. Anderson et al., 1993].
In total, many estimates of ao/a using various methods described above consistently show that the thermal expansion coefficient in the lower mantle decreases 4-5 times along the hot low-mantle adiabat as the pressure increases from zero to the base of the mantle. Nevertheless, the complete consensus on all the parameter values related to these estimates (e.g., for q and St) has yet not been achieved.
6.3. Temperature dependence of a at P= 0
6.3.1. Most data on thermal expansion refers to the dependence a(T) = a0 at P = 0. The value of ao is necessary, in particular, to extrapolate the thermal expansivity data to higher pressures in the mantle. The typical behavior of a(T) is illustrated in Figures 4 and 5. Usually, the data at P = 0 are fitted using the empirical formula [e.g., Fei et al, 1990, 1991]
a = ao + a\T + агТ
(54)
which we used to calculate a presented in Table 2 (and in Pankov et al [1997]). Note that the applicability
of (54) can also be justified by calculations of phase diagrams [Fei et al., 1990, 1991].
A theoretically based approach to calculating a(T) was developed by Suzuki [1975a, 1975b], who used the Mie-Griineisen EOS yielding
dEt
dT
i Et Krjt Q +
kEt
Q
l -
kEt
Q
-1
where Et is the Debye thermal energy
(55)
2
Et = —D(z), z = Q/T, D(z) = 4 f № z J
V3dy
ey -1’
k = —(K'0 — 1), Q = ^° , and D(z) is the Debye
2 7
function. Here, it is assumed that 7 = constant, and parameters Ko and K'0 are defined at T = 0. The fitted parameters are Q, k, and O. Formula (55) is derived by expanding the potential pressure in V and truncating at only two first terms. The O values obtained from this method are given in Table 2 (see also Pankov et al. [1997]).
0. Anderson et al. [1992a] extrapolated ao(T) from a fixed value at T* > O to higher temperature using the relation
1
«0 (T) _ ________________________
a0(T*) ~ 1 -a0(T*)-ST -(T-T*)'
(56)
where St = a. = constant (see (32)). Formula (56) is easily derived from the condition that «o at P = 0 varies with density by the power law. Note that (56) has an asymptote close to which a dramatically increase with temperature (reflecting to some extent the fact that the potential energy has an inflection point).
6.3.2. For a more complete consideration of the temperature behavior of a, we calculated a(T) for three minerals from the Mie-Griineisen EOS (with the Debye model), in which, unlike the Suzuki method, 7(2;) was found by (51) with q = 0,1, and 2, and room-temperature isotherms were represented by equations (28) and (37). The material parameters of the EOS’s were found from values of p, Ks, a, Cp, and (OKs/9P)t at normal conditions (Tables 2 and 3).
The results of the computations are shown by the solid lines in Figures 4 and 5. We see that the curves for periclase and particularly forsterite systematically deviate from the experimental points at high temperatures, although there is a considerable uncertainty in data for a at high temperatures. Nevertheless, such deviations can be caused by the fact that the temperature dependence of 7 (at V = const) is not accounted for in the Mie-Griineisen EOS [Mulargia, 1977; Mulargia and Boschi, 1980; Mulargia et al., 1984; O. Anderson
et al., 1992a; Molodets, 1998]. To gain a better insight into the quality of the Mie-Griineisen EOS and to construct a self-consistent database on EOS parameters, it is very important to measure the thermal expansivity of minerals at high temperatures, up to their melting points. This conclusion was emphasized by many authors [e.g., Saxena, 1988, 1989; Goto et al., 1989; Isaak et al., 1989b; Gillet et al., 1991; Richet et al., 1992].
7. Specific Heat
The lattice specific heat of minerals at T^IOOO K is close to the classic limit 3Rn = 3RM / p (the molar value, where R is the gas constant and n is the number of atoms in chemical formula). From calorimetry, we have information on the isobaric heat capacity Cp, which exceeds Cy by 1-3% at 300 K and 10-15% at T > © (Table 2 and Pankov et al., 1997]. Since p 20-22 g/mole for mantle minerals, the classic value of C'v for them is 1.13-1.25 J/g K. Depending on mineral, the high-temperature anharmonic corrections to C'v become singnificant either near the melting point or even at room temperature (sometimes, at T equal 1/6 of the melting point) [Mulargia and Boschi, 1980; Quarem and Mulargia, 1988; Reynard et al., 1992; Fiquet et al., 1992].
7.1. PVT derivatives of specific heat
From the identities d2E/dVdT = d2E/dTdV and d2S/dPdT = d2S/dTdP, using the Maxwell relations, we find
fdCv\ VT fdaKT V dP )T ~~ KT V dT
(jf)t = -T (§p)P = -"‘'l'T|1+“1'
(57)
(58)
Note that (57) and (58) are the alternate forms of (33) and (34), repectively. The logarithmic volume derivatives at T = constant can be expressed as follows:
91nCV\ dlnV )r
= «7T [a — 2St + K'\
(9 In CP <9 In V
= ajT (1 + ajT) 1 (1 + a)
(59)
(60)
These identities were used to compute the derivative values given in Tables 3 and in Pankov et al. [1997]. O. Anderson et al. [1993] noted that, for the Debye model, (dC'v/9P)t^0 and therefore a ^ 2St — K'.
The difference Cp — C'v satisfies the identity
1 (d{Cp - Cv)
t2VT \
<9 P
= K' - 28T - 1,
(61)
P
T
T
0 1000 2000 3000
Temperature , K
Figure 4. Temperature dependence of the thermal expansion coefficient at P = 0 for MgO and AI2O3. The solid lines are from the Mie-Griineisen EOS with q = 1; the bars show deviations for q = 0 and 2, respectively. A simple linear extrapolation of a is shown by points. The dashed line is a by (56) for To = 1000 K and Sp0 = 4.84. The dot-and-dash line is a by (54) [Fet et al., 1990]. The experimental points (circles for MgO and crosses for AI2O3) are from 0. Anderson et al. [1992a],
i.e., it decreases with pressure for usual values of K' and St (see Table 3 and Pankov et al. [1997]).
Birch [1952] estimated the decrease in Cp with pressure in the lower mantle, setting a ~ 4 and a-/T ~ 0.1. By the power law for Cp, this gives a 13-16% decrease in C'p along an isotherm, for x descending from 1.0 to 0.7. According to Birch, the maximum decrease in C'p in the mantle does not appear to exceed 20%.
The adiabatic volume derivative of C'p is
(9 In CP <9 In V
= -7
9 In CP <9 In T
= (1 + ajT)
(9 In CP <9 In V
<9 In CP <9 In T
(62)
0.1, and (<9 In Cp/<9 In V)t ~ 0.5 (for a ~ 4), we find (<91nCp/<91n V")s Pd 0.3 — 0.4. Consequently, the power law for C'p yields a 8-13% decrease of this value along the mantle adiabat (to x ~ 0.7).
By using (29), the temperature derivative of C'p can be represented in the form
<9 In CP <9 In T
<9 In CP <9 In T
ajT f <9 In Cp <9 In V
Substituting (<9 In Cp/d In T)p ~ 0.15 (the typical value for minerals for T ^ 1000 K), 7 ~ 1 — 1.5, ajT ~
, (63)
v 7 \ o' In 1/ j T
The second term arising from the extrinsic anharmonic-ity can be estimated by making use of (58), so that (63) in conjuction with data for (dCp/dT)p allows the first term coming from the intrinsic anharmonicity to be evaluated. At room temperature, (<9 In Cp/d In V)t
p
s
s
T
p
Figure 5. Temperature dependence of the thermal expansion coefficient at P = 0 for forsterite. For notation, see Figure 4.
is on the order of (<9 In Cp/d In T)p, but nevertheless, the contribution of the second term to the sum (63) is small because of the small factor ajT. At high temperatures, this contribution generally increases to 15-30% (perhaps, 60-70% for ilmenite and perovskite, according to our estimates) [Pankov et at, 1997]. In the classical limit, C'v = constant, assuming that a Pd a(V) and 7 Pd 7(V"), we find (dCp/dT)v & a^C'v •
Further, from (17) and (18), it is easy to obtain the identity
91nCV <9 InT
—ay T —
*iTf 1
T fdCp CV \ dT
(1 + 2a — Sp)
<9 In Cv <9 In T
<9 In Cv <9 In T
ayT f d In Cy <9 In V
7
An explicit dependence C'v(V, T) can be derived from models and measurements of the vibrational spectra of solids (e.g., Pitzer and Brewer, 1961]):
Cv = 3nk
eyy2
(ev - 1)
7g(v)db
(66)
(64)
which we used to estimate the values of this derivative presented in Table 3 and Pankov et al. [1997]. Then, with the help of the identity
(65)
it is possible to compute ((<9 In CV)/(<9 In T))y, provided that the second term in (65) is given by (59).
where k is the Boltzman constant, y = hv/kT, g(v) is the spectrum density, and v are the lattice frequencies including optic and acoustic modes [Kieffer, 1979a, 1979b, 1979c, 1980; Hofmeister, 1991a, 1991b; Richet et at, 1992]. This method provides information on the inadequacy of the Debye theory and the related approximation 7 = —dlnQ/dlnV. The characteristic temperature 0 in the Debye theory is usually estimated from acoustic data, but 0 found from data for C'v at T > 300 K (labelled Oth), on average, exceeds the acoustic 0 (labelled ©a) by about 20% (larger deviations are common to quartz and coesite, see Table 2 and Pankov et al. [1997] and Watanabe [1982]). Chopelas [1990b] found, however, good agreement of the Debye model with the spectrum data for MgO at pressures to 200 kbar, provided that 0 Pd V1, i.e., q = 0.
Spectroscopic measurements at high pressures allow us to estimate the derivative (dC'v/9P)p. For example,
p
p
p
T
the data of Chopelas [1990a, 1990b] to 200 kbar show that C'v linearly decreases with P, so that the gradient ~{dCv/dP)T is 17.6 • 10“3 (T = 300 K) and 0.91 • 10“3 (T = 1800 K) for MgO and 49.8 • 10“3 (T = 300 K) and 3.4 • 10-3 (T = 1800 K) J/(mole K kbar) for forsterite. With these values, C'v being extrapolated (by the power volume dependence) to the maximum pressure P = 1357 kbar in the mantle will decrease 40-60, 8-12, and 2-3% on the 300, 1000, and 1700 K isotherms, respectively. Comparing these results with the decrease in C'p estimated above, we verify that the difference C'p — C'v in the lower mantle must be exceedingly small. From the same Chopelas’ data, using also the Kp and C'v values from Table 2, we find (9 In Cv/d In V)p = 0.77 (T = 300 K) and 0.022 (T = 1800 K) for MgO and 0.54 (T = 300 K) and 0.022 (T = 1700 K) for forsterite. These results are comparable to our estimates of this derivative from thermodynamic data (Table 3).
7.2. An explicit temperature dependence of Cp at P = 0
8. Thermal Pressure
8.1. PVT derivatives of the thermal pressure coefficient
The thermal pressure coefficient defined by (9) or (12) is the basic characteristic of the thermal pressure and can also be defined as r = (dPth/dT)v ■ Note that r has also the meaning of the latent heat of expansion per 1 K. O. Anderson and his co-workers [0. Anderson, 1982, 1984, 1988; 0. Anderson and Summo, 1980;
0. Anderson and Goto, 1989; 0. Anderson et al., 1982, 1991, 1992a] paid special attention to this parameter, in particular, in relation to their development of the rectangular parallelepiped resonance technique for measuring elastic properties of minerals at high temperatures.
The basic identities for the derivatives of r can easily be derived from those given in sections 6 and 7. The following identities are especially suitable [Brennan and Stacey, 1979; Birch, 1978; 0. Anderson and Yamamoto, 1987]:
Calorimetric data for C'p versus temperature at P = 0 are commonly fitted to various empirical expressions [e.g., Fei and Saxena, 1987; Berman, 1988; Saxena, 1989; Richet and Fiquet, 1991]:
Cp = Co + c\t + c2t2 + C3T-1/2 + C4T~2, Cp = Co + CiT-1/2 + C2T~2 + C3T~3, Cp = Co + CiT-1 + C2T~2 + C3T~3 + CAT, (67) Cp = Co + C2T~2 + C4T~4 + c6T~6, C'p = C'o + CiT-1 + C'2T~2 + C'3T~3 + CA In T.
Richet and Fiquet [1991] showed that the last of the above formalas are favored but no one of them provides an accurate description of C'p over a wide temperature range.
In addition, Figure 6 compares C'p(T) found by the simpler formula used by Watanabe [1982] to fit the measurement in the temperature interval of 350-700 K. An example of MgO shows that the extrapolation by this formula can lead to series errors.
As well as in the analysis of thermal expansivity in section 6.3, we calculated C'p(T) (Figure 6) from the same Mie-Griineisen EOS as was used to compute a. One can see that the theoretical curves can be reconciled with the data shown by varying parameter q in the limits 1-2. In so doing, we find q Pi 0 for MgO, q = 1-2 for AI2O3, and q Pi 0.5 for forsterite. However, these values of q are not always consistent to data on a (see section 6.3), and this fact also suggests a certain inaccuracy of the Mie-Griineisen EOS model.
daKq
dT
= Kp ( 7^ ) = a2 Kp (a — Sj
fdaKT\ _ 1 fdCv\ V dT )V-T{~dv):
= a2Kp (a + K' — 2Sp) ,
(68)
daKq
dV )T V V dT y v
(69)
(70)
/ daKp I dP
1 / dKp ~Kt I dT
= a{K'-ST). (71)
Formula (68) is obtained by expanding the derivative at the left side and then by using (30); (69) is a consequence of (57), and (70) is easily derived by equating the second derivatives of P with respect to V and T taken in one order or another; finally, (71) follows from (31) and (70).
Combining (68) and (69), we have
V dT
According to (71), for the common inequality Sp > K', parameter r decreases with pressure along an isotherm. However, at high pressure, the decrease can change to an increase, as, e.g., in the PIB model for MgO [0. Anderson et al., 1993]. It is clear from (68)
p
T
Figure 6. Temperature dependence of the specific heat at P = 0 for forsterite, corundum, and periclase. (1) Cp from [0. Anderson et al., 1992a]. (2) C'v from [Chopelas, 1990a, 1990b]. The solid lines are from the Mie-Griineisen EOS (with the same parameter values for respective minerals as in Figures 4 and 5). The dashed lines are the polynomials from [Watanabe, 1982] descrbing his data in the range 350-700 K. The arrows indicate the classical values of C'v-
that r increases with temperature at constant pressure, at least for T < O.
As follows from (29), the logarithmic temperature derivative of r at P = constant is represented as
d In r \ dlnTJ
dT
d In r d In V
dT
uKt V dT
da
dT
= (K1 — St) + (a — St)
and the second term in (73), in view of (70) and (71), is
d In r d In V
= St — K'
(73)
1 f dK-T
ocKt V dT
= -4
(75)
where the first, intrinsic anharmonic term is positive due to (69) and can be written as the sum of two terms
1 / d In r
(74)
Here, the symbol Sy is introduced for convenience (see a further analysis in section 9). Although due to large values of a, the intrinsic anharmonic term in (73) is dominant in value at T < O, the sign of (73) at high temperatures can be either positive or negative.
The temperature behavior of r resembles that of C'v, so that at T > O, r tends to be independent of temperature [0. Anderson, 1984]. Accordingly, the thermal pressure Pth tends to a linear dependence on T. Our estimates of r (Table 2 and Pankov et al. [1997]) show that the nonlinear terms in Pth versus T makes a con-
T
T
tribution not greater than 1-3% at the highest temperatures indicated in Table 3 and Pankov et al. [1997]. The linear temperature behavior of Pth is considered to be the universal property of solids at high temperatures [0. Anderson et al., 1992a]. Unlike the temperature dependence, the extent to which r depends on volume at T = constant varies from one type of solid to another. According to 0. Anderson et al. [1992a], the earth minerals fall into an intermediate group between materials with significant (e.g., gold) and relatively weak (e.g., sodium chloride, alkali metals, noble elements) volume dependences of r. It is important that these inferences are based on both P — V — T data (analysis of Pth{V, T)) and high-temperature data for Kt and aKp at P = 0 (analysis of (70)).
At first glance, the observed regularity in t(T) at T > 0 is explained by the fact that C'v & constant in this temperature range, where, in view of (8) and (69), 7 is therefore independent of temperature. In other words, the quasiharmonic Mie-Griineisen EOS is seemingly justified at high temperatures. However, this is not quite true to be the general case when we start with the condition r = constant or r = t(V) which are compatible with the case of C'v(T) (see (69)) and therefore with a dependence of 7 on both volume and temperature [0. Anderson and Yamamoto, 1987]. Theoretically, the departure of C'v from the Dulong and Petit law [e,g., Mulargia and Broccio, 1983; 0. Anderson and Suzuki, 1983; Gillet et al., 1991; Reynard et al., 1992] is in part related to the intrinsic anharmonicity described by the third and higher order terms in the lattice Hamiltonian expansion. In describing experimental data, the thermal EOS generally requires smaller number of terms in this expansion than the caloric EOS [Leibfried and Ludwig, 1961; Wallace, 1972; Davies, 1973]. Thus, when anharmonicity in Pth versus T and the temperature dependence of 7 are not observed, this may suggest that either the quasiharmonic limit for the vabrational 7 has not yet been achieved or the higher order terms in the thermal part of the EOS are mutually cancelled [0. Anderson et al., 1982].
In conclusion to the above analysis of r, we formulate the following important assumptions and their consequences that can easily be verified: (1) Let C'v be independent of V, i.e., either C'v = C'v(T) or C'v = constant. (2) Assume that Kt = Kt{V) that is equivalent to St = K'. From (1), it follows that either r = t(V) or r = constant, and in addition, Kt is either a linear function of T or Kt(V), which leads to q = q(V) (or q = 1) and either 7 = Vt(V)/CV(T) or 7 = y{V). The statement (2) is equivalent to either r = t(T) or r = constant. If both (1) and (2) statements are valid (but C'v 7^ constant), then 7 = const • V/C'v(T) and <7=1. The conditions C'v = constant and Kt = Kt{V) yield 7 = const • V.
8.2. Thermal pressure model
The thermal EOS resulting directly from integrating (12) is of the form
T T
P = f(V) + I aKTdT = f(V) + 11^-dT, (76)
0 0
where f(V) is the static lattice pressure plus the zero oscillation pressure. The second term in (76) is the total thermal pressure Pth accounting for all anharmonic contributions. This EOS can be rewritten in the form
T
P = P(V,T0) + j aKTdT, (77)
To
where, for example, To = 300 K. In accordance with the behavior of r described above, the thermal pressure can be approximated as [0. Anderson, 1984, 1988]
Ti T
Pth = J aKTdT + J aKTdT
0 Ti
= a(V)+b(V)(T-T1), (78)
where T > Ti > 0. As already noted, the variation with volume in (78) is insignificant for some solids.
In the Mie-Griineisen EOS, the thermal pressure is however defined as
T
Pth = ^ j CvdT, (79)
0
where the quasiharmonic approximation for C'v is used. Thus, here, at high temperatures T > T2 > 0, when C'v & constant,
Pth = a*(V) + b*(V)(T-T2). (80)
Even when b(V) = b*(V) in some temperature range, the distinction between (78) and (80) is retained since, in the general case, 7 = 7(V,T) in (78) and a* ^ a. In practice, for certain minerals and for the present accuracy of measurements, Pth from (78) and the Mie-Griineisen theory can be indistinguishable [Fei et al., 1992a, 1992b; Mao et al., 1991], especially for minerals with low Debye Temperature.
The term APth in (77) can be approximated in various ways. For example, with given volume dependences of K' and St, by integrating (75), we can find t(V) at T = constant [0. Anderson et al., 1992a, 1993]. The temperature dependence of r is derived from data on
3
300 500 1000 2000 Temperature, K
Figure 7. Thermal pressure coefficient versus temperature at P = 0. The solid lines are from the Mie-Griineisen EOS (as in Figures 4-6). The experimental points are from [O. Anderson et al., 1992a].
a(T) and Kt{T) at P = 0. Another possibility to explicitly approximate the thermal pressure is given by the power law for t(V) on an isotherm.
Fei et al. [1992a, 1992b] and Mao et al. [1991] used a number of models for Pth in order to describe P — V — T X-ray data. Specifically, they assumed {дКт/дТ)у = constant. Then, (70) yields
r(P,T) = r(0,T) +
дКт
~dT
i„EI
к 1'(0,Г) :
cases of specific heat (section 7) and thermal expansivity (section 6). A better accuracy of the EOS is undoubtedly required than that of the Mie-Griineisen EOS in order to describe experimental data, to reliably predict unmeasured properties, and in particular, to calculate the phase diagrams at high pressures (when a 10% error in Pth can substantially affect the estimated phase boundary slopes and positions).
(81) 9. Anderson—Gruneisen Parameters
where t(0,T) is determined from data on a provided that (dKx/dT)p = constant, the condition assumed over the entire P — V — T range of measurements. From the assumption that both temperature derivatives of Kt are constant, it follows that tK' = const. However, in such a case, K' will increase with pressure along an isotherm and decrease with temperature along an isobar—the behavior that disagrees with the usual properties of this parameters (see section 6). Note that the estimated St values can be very sensitive to the adopted model of Pth [e.g., Mao et al., 1991], although the Pth values themselves from various models can be close.
Finally, in Figure 7, we illustrate the temperature dependences of r at P = 0, calculated from the Mie-Griineisen EOS for periclase, corundum, and forsterite, described in section 6, with various values of q in the interval 0-2. Although this type EOS gives correct orders of magnitude and the correct regularities in the P — T variations of r, it is difficult to achieve the complete consistency for all of the data given, as well as in the
Here, we consider the two useful Anderson-Griineisen parameters in more detail [Griineisen, 1926; 0. Anderson, 1966a, 1967]: the isothermal St parameter introduced above (see (31)) and the adiabatic Ss parameter defined as
Ss = -
aKs V 9T
d In Кs 9 In p
(82)
Both parameters are used in geophysical and physical studies [Chung, 1973; Barron, 1979]. The parameter St is largely applied in analyzing the P — T behavior of a, Kt , and r, and Ss is used to estimate the temperature dependence of Ks and to treat the relations between elastic properties (elastic wave velocities) and thermodynamic data [D. Anderson, 1987; 0. Anderson et al., 1987; Isaak et al., 1992; Duffy and Ahrens, 1992a, 1992b; Agnon and Bukowmski, 1990b].
As the temperature decreases in the range T& 300 K at P = constant, both parameters Ss and St sharply increase due to decreasing a, but at high temperatures,
p
p
T > 0, they become more or less constant [0. Anderson et al., 1992a]. From (16), we derive the identity relating St and Ss [Birch, 1952]
St = Ss + 7 — «7 T
St — 1 — 2a +
1
fdCP\
(83)
= ayT
a — St + 1 +
= ajT(l + ay T)
aCy \ dT J p
ayT
L + a
1 + ay T
(84)
7
— St + 1
1 + ay T
where, for convenience, the notation
_ CP (da/Cp
±J ~
t2 V dT
Sp рі a + 1.
£
K' - 1.
dy 9 In T
/ ayT
St = Ss + 7 —
ayT 1 + o^T
1
(8S|
If the last term in (88) is small, then [0. Anderson et al., 1992a]
St Pi Ss + 7. (89)
This approximation is recommended for evaluating of the Anderson-Griineisen parameters at high temperatures, when there are no sufficient data for applying (83) or (88).
In addition to the analysis of the parameter ST(P,T) described in sections 6 and 8, we consider the following features in the behavior of St- 1) If y = l(V), then q = q(V), but generally speaking, St = St{V,T) since C'y = Cy(V,T) and K' = K'(V,T). 2) If C'y = C'y(T), then (42) holds true, and moreover, y = f(V)/C'y(T), q = q(V), although, generally speaking, K' = K'(V,T) and St = St{V,T). Combining the former of these assumptions with the condition Kt = Kt{V) (i.e., St = K'), we find
aCP V dT J pJ
which was used to calculate the St values listed in Table 3 and Pankov et al. [1997]. Deriving (83), we find in passing that
T 1*1
дт
<9 In су d In V
= i - q,
(90)
and C'y therefore takes the form
Cv(V,T) =
MV)
В{ту
(85)
is introduced.
The values of L and (dy/dT)p at P = 0, calculated by (84) and (85), are also presented in Table 3 and Pankov et al. [1997].
Data for some minerals [0. Anderson et al., 1992a] show that (dy/dT)p=o Pi 0 over a wide temperature range. Assuming that (dy/dT)p = 0 for C'y = constant (T > 0), (84) gives
In section 6.2.2, the arguments were given for decreasing St with pressure. The same behavior of this parameter follows from the approximation (42) since both K' and q decrease with pressure. The exact relations for the P and T derivatives of St result from the definition of St by (31)
dr \ ( dK''
.dPL +
fdST\ __1 \dPJT~ т
St
(86)
8St
~dT
1
St
дт
dT
V дТ
д2І<т
дТ2
(91)
Moreover, since C'y = constant, this case results in q = 0, and consequently, according to (35) and (42),
(87)
However, the estimation of St by (42) for g ~ 1 is more accurate than the values from (87). Then, one might expect that in (84)
for C'y —?> constant.
Using (83), identity (84) can be rearranged to the form
• (92)
p \ ^ / pj If the first term in (91) prevails, we have an unusual case ((dSp)/(8P))t > 0. Neglecting the second temperature derivative in (92) (at least at room temperature), we find that (dST/9T)P < 0. However, the approximation St = K' (more realistic at T > 0) gives, by constast, (SSt/dT)p > 0 because of (dK'/dT)p > 0. Experimental data indicate that ((d2Kp)/(dT2))p for T > 0 is negative and small in value [0. Anderson et al., 1992a].
Finally, the EOS of type (28), which we used to calculate the thermal expansion coefficient by (36), allows us to determine the explicit pressure (or volume) dependence of Kt along an isotherm. Retaining in (36) only the terms involving dK'0/dT, we obtain
St St o
p
т
p
p
0.6 0.7 0.8 0.9 X = V/V0 10
Figure 8. Parameter St versus x at T = constant. The solid lines are by (93), corresponding to the a curves in Figure 3. The crosses are by the power law corresponding to (48) [0. Anderson et al., 1993] or St = &x — 1 from (47) [Chopelas and Boehler, 1992]. The dashed line is the lower limit St = K' — 1 calculated by (28) and (37) (under the condition that C'v = constant and ■!>«)
1-A-f
J\T
Kn
dK'0
(XqStqKt dT \dK[
df
dK'n
1 -
P
Kt
TO
K0 df dK'Q «oKt dK'0 dT
m
where f denote the derivative of / with respect to x. This approximation generalizes the similar Birch formula that follows from (93) for dK'0/dT = 0 [Birch, 1968]. However, when using usual EOS types, St from the Birch formula increases with P (except for the Mur-naghan EOS for which the behavior of St depends on the sign of the difference K' — Sto). Again, we convince ourselves that the term with derivative dK'0/dT is important in analyzing the thermal expansion by (28).
Figure 8 shows several curves of St(x) calculated using (93) and EOS (37), which correspond to the curves of a in Figures 1 and 3 (the straight line St = 6x — 1 by (47) is also drawn for comparison). The favored value of dK'0/dT is 2 • 10-3 K-1 (see section 6.2), and deviations from it substantially affect the St values at compressions in the lower mantle. At high temperatures, according to (42), we have also the lower limit for St , St > K' — 1 [O. Anderson et al., 1992a].
10. Bulk Moduli
10.1. PVT derivatives
Elastic moduli and their P — V — T derivatives are the characteristics constituting the basis of the Earth’s interior thermodynamics [Birch, 1952, 1961; Sumino and O. Anderson, 1984; D. Anderson, 1967, 1987, 1989; O. Anderson et al., 1992a; Duffy and D. Anderson, 1989; Bma and Helffrich, 1992; Duffy and Ahrens, 1992a, 1992b]. These quantities also serve as parameters of EOS’s. The simplest estimates of adiabatic and isothermal bulk moduli at high pressure are given by their linear pressure dependences, which, however, begin to overestimate the bulk modulus at a compression of about x < 0.85. The P — T variation of the pressure derivative was considered in many papers devoted to EOS’s (see section 3). In order to assess the applicability of empirical EOS’s, one often uses a relation between the first K' = (BKt/8R)t and second K" = (d2Kt/0P2)t pressure derivatives at P = 0 [Pankov and Ullmann, 1979a; Jeanloz, 1989; 0. Anderson, 1986; Hofmeister, 1991b]. Values of K' and
KK" at P = 0 generally lie in the intervals 4-6 and — (5 — 10), respectively. The uncertainty in (dKs/9P)t (measured by ultrasonic or Brillouin scattering methods) can reach 1-5% (with allowance for data from various laboratories) or, in some cases, 20% and even 50%. Anomalous values of K' and K" are sometimes reported (see references in tables of Pankov et al. [1997]): for example, K' = 5 — 7 (garnet), K' = 9 — 14 (pyroxene), and KK" = —60 (spinel), which are assumed to take on more usual values as pressure increases.
Let us turn our attention to the relations between the adiabatic-isothermal derivatives of Ks and Kt- Changing the variables P and S to P and T, we find
dKs
dP
9KS
dP
- Ssa~/T,
8Kf
dP
= K'(l + ajT)
+ KtT
dy
~dP
■7
da
~dP
9KS
dP
= K' + ajT (2K' - 2ST - 1)
dKs
dP
= K' + a1T(2K' - iST - 1 + 7)
+ («7T)2 K' - UT + 3a + 1 -
1
(dCP\ aCP I dT )
dKs
dP
<
dKs
dP
< K'
dKs
dP
> K'Ks/Kt-
For our high-temperature estimates given in Table 3 and Pankov et al. [1997], it was arbitrarily assumed that
dKs
dP
= const (or d2Ks/dPdT&10 K )
(94)
S v KJ± / T where 8s is defined by (82). The derivative (dKs/dP)s characterizes the curvature of an adiabatic P — V or a Hugoniot curve. Further, from (16)
(95)
T / TJ
Eliminating (dj/dP)T with the help of (8) and (17) (or (122)) and using (58), we find
Hence, using (94)—(97), we found substantial differences (up to 10-30%) between the pressure derivatives of Ks and Kt at high temperatures. In fact, the differences are of the same order of magnitude as the derivative increments due to increasing temperature. Specifically, the estimated d2Kt/dPdT values are 3.5 • 10-4 (stishovite), 2 -10—4 (ilmenite), 3 -10—4 (Mg-perovskite), and 1.7 • 10-4 (MgO) and do not exceed 1 • 10-4 for other minerals (although some estimates appear to be negative).
Isaak [1993] estimated d2Kt/dPdT using an identity of type (91) and Boehler’s data on the adiabatic temperature gradient (see also section 12.2). He found d2KT jdPdT = (3.9 ± 1.0) • 10“4 and (3.3 ± 0.9) • 10“4 K-1 for MgO and olivine, respectively. Furthermore, he showed this derivative to decrease 30% as the pressure increases isothermally to 100 GPa. A similar order of magnitude was found from shock wave data to be a lower limit for this derivative value [Duffy and Ahrens, 1992a] (see also sections 6 and 9).
In addition to the analysis of the mixed derivatives, we give the following identity
\ ( d (dKs
+ (ajT)2 (K1 - 28T + a) ■ (96)
Substituting (94) for (dKs /dP)T and (93) for Ss, we arrive at the Birch [1952] formula
a8s \dT \ dP
s/ p
( dK‘
V dP
d In Ss d In p
+ 8s(l + a~fT), (98)
(97)
The difference between the adiabatic-isothermal derivatives of Ks and Kt at normal conditions are generally small (1-2% for mantle minerals). Data and estimations by (94)-(97) show that we usually have
which we derived from (94), using (127) and (dj/dT)p from (88). Note that a similar relation of Bukowmski and Wolf [1990] is different from (98) (because of either a reprint or mistake). To give an example, we substitute in (98) the values typical of the lower mantle: (dKs/dP)s = 4, «7T = 0.1, ((51nJs)/(51n/o))s = -1, and Ss = 3 for x = 1 and Ss&2 for x = 0.7. Then,
fd(dI<s/dP)s V dT
£ 0.8 • 10
-4
s \ / T
(except for FeO for which (dKs /dP)T is poorly known [Pankov et al., 1997]); however, D. Anderson [1989] indicated the inverse inequality
which is in agreement with the preceding estimates.
When considering the temperature behavior of Ks and Kt, the Anderson-Griineisen parameters Ss and St are represented in the form of (29) [D. Anderson, 1987; Duffy and D. Anderson, 1989; 0. Anderson et al., 1992a], which can be rewritten as
Ss = ~
1 f dKs aKs V dT j v
( dK‘
V dP
Kt
Ks
T
T
T
s
s
T
p
T
T
d In Ks 9 In p
= -si
f dKs \dP ,T
(99)
Ay
—Sy + K',
aKn
St =
dKq
dT
d In Kt d In p
+ K' = -Si + K'.
and additionally, in view of (89) and (99), Sy Pi Sy + 7 Pd 7.
(102)
(100)
In section 9, we considered the principal regularites in the variation of Ss and St with pressure and temperature (some decrease of them with T in the vicinity of T = 300 K, the trend to constant values at T > 0, and the decrease with pressure). Now we dwell on the contributions of the intrinsic Sy and Sy and extrinsic K' anharmonic terms in (99) and (100).
D. Anderson [1988, 1989] pointed out that the temperature variation of the bulk modulus at P = constant mostly occurs by the variation in a; i.e., here, the extrinsic anharmonicity generally prevails and enhances with temperature (due to increasing K'). The estimates given in Table 3 and Pankov et al. [1997] show that, at T = 300 K, we have Sy < 0 (except very uncertain data for FeO); most of the estimates falls into an interval between —1 and —2 (although it was found Sy = —17, —5, and —3.9 for coesite, stishovite, and Active Fe-perovskite phase, respectively). According to D. Anderson [1989], the values of Sy are typically between —4 and —1 (his table 5, however, contains values outside this interval: 2.2 for orthopyroxene, —5.3 for SrTiOs, and —19 for CaCOs). The parameter Sy satisfies the inequality > 2 for 11 out of 54 minerals considered by D. Anderson (specifically, Sy = —4.1 (Ge02), 3.8 (orthopyroxene), —3.1 (SrTiOs), and —18 (CaCOs). At high temperatures, Sy can be either positive or negative (between —1 and 1), and is generally positive, lying in the interval 0-1.5 [Pankov et al., 1997]. Thus, as temperature increases, \Sy\, on average, decreases, but increases. The contribution of Sy /Ss to (99) is usually less than 10-30% at 300 K and does not exceed 15-60% at high temperatures. Accordingly, Sy contributes no more than 30-40% in (100) at 300 K and usually less than 10% at high temperatures. From this analysis, it follows that the extrinsic anharmonic term, although it generally dominates in the derivatives of Ks and Kt , decreases its contribution in the case of Ks and increases its contribution in the case of Kt ■ The decrease of Sy with temperature leads to the approximation
St Pi K1, Kt Pi Kt(V) (101)
Thus, at high temperatures, namely the isothermal, rather than adiabatic bulk modulus becomes depending mostly on volume (i.e., temperature-independent function).
10.2. Interpretation of seismic tomography data
In his analysis of the thermodynamic properties of the lower mantle, D. Anderson [1987, 1988, 1989] relies on seismic tomography and geoid data and assumes that the observed horizontal velocity anomalies are caused by temperature variations. Stacey [1992] showed, however, that it is not possible to explain the anomalies with a purely temperature effect, since in such a case, the geoid highs would be too great. Other hypotheses proposed to interpret the seismic anomalies were related to inhomogeneities of composiion, or partial melting, or even the presence of small amounts of fluids [Price et al., 1989; Duffy and Ahrens, 1992b; and others]. Nevertheless, following D. Anderson, below, we estimate the thermodynamic parameters for the lower mantle, considering the temperature effect formally as a limiting case.
Based on the PREM model, the formula for the acoustic Griineisen parameter, and tomography data, we have in the lower mantle (dKs/dP)s = 3 — 3.8, 7 = 1.2±0.1, and Ss = 1 — 1.8. Consequently, using (89), (94), (96), (99), and (100), we find St Pi Ss + 7 = 2.2 — 3.0, K' = 3 — 3.8 (a small correction can be introduced with the help of a = aoxST for a0 = 4 • 10-5 K-1 and T Pd2000-3000 K), S* Pi K' - Ss = 1.2 - 2.8, and Sy = Sy — 7 = —0.1 — 1.7. If a greater uncertainty is assumed for 7, say ±0.4, then St ~ 1.9 — 3.3 and Sy ~ —0.4 — 2.0. Thus, although the extrinsic anharmonic effects weaken with pressure, they still prevail under the lower mantle conditions (K' > or I^D-The intrinsic anharmonic term Sy significantly increases with pressure, but its isothermal analog Sy can either increase or decrease and reach zero. D. Anderson, by reference to experimental data, points out the case of Sy Pi 0, which yields Sy ^ 7 = 1.2.
However, this value of Sy disagrees with data for such a representative lower-mantle material as peri-clase. Using Sy = 0 and St = K' Pi 3.2 — 3.5 for x = 0.7 and the Birch-Murnaghan EOS for MgO, we find Ss ~ St — 7 = 2.2 — 2.5 for 7 = jox (q = 1, see (119)). These values of St and Ss are, on average, still exceed the results inferred from seismic models. 0. Anderson et al. [1992b] noted that data for MgO can be reconciled with seismic results by assuming that q < 1. In particular, our analysis leads to the following consistent sequence of values: S^^Z0.2, qi&0.8, 7 = 7o*9^l.l
p
p
T
(for x = 0.7), St = K' —Sy ^3 — 3.3, and Ss^ 1.9 —2.2. In any case, the consistency to seismic data could be found in this manner if the values derived from seismic tomography were explained by only horizontal variations of temperature.
10.3. Estimation of Ks at high temperature
10.3.1. The consideration of the temperature behavior of Ks at P = 0 will be added by the following two methods. One of them uses the condition Ss = S*s = constant [0. Anderson, 1988; Duffy and D. Anderson, 1989], and in view of (99), yields the power law
KS = K*s(p/p*)s
(103)
which we used to estimate the values listed in Table 2 and Pankov et al. [1997].
Another approach proposed by 0. Anderson [1989] and extended by 0. Anderson et al. [1992a] is based on data for enthalpy. We obtain the relation of Ks to enthalpy using a somewhat different procedure, namely, the formula 7 = aKsV/Cp from which the derivative (dKg/dH)p is found, and thus, approximately,
11. Thermodynamic Griineisen Parameter
The thermodynamic Griineisen parameter is defined by (8) or (14), which further lead to several useful identities
V(dS/dV)T _ Vd2F/dVdT V(dS/dT)v
T{d2F/dT2jv
\ dlnV, s
(105)
The typical values of 7 by (8) or (14) range from 1 to 2 (see, e.g., Table 2 and Pankov et al. [1997]). Of 54 minerals treated by D. Anderson [1989], only five have 7 greater than 2, and none has 7 over 3. Low values of 7 are seldom encountered: e.g., 7 = 0.4 for a-quartz, 0.3 for coesite, and even 7 < 0 for U2O, AgJ, and /3-quartz.
11.1. PVT derivatives of
7
The logarithmic derivatives of 7 with respect to V (or P) are characterized by the parameter q, for which from (8) and (100), we find
Ks = K*s - S*sj*p*(H — H*),
(104)
where the asterisk marks the values at a fixed temperature. By using the parameter values from Table 2 and Pankov et al. [1997], as well as data for enthalpy, we estimated Ks by (104) for a number of minerals (Table 4). One can see that the O. Anderson’s method is quite efficient: the uncertainty of the estimated values at high temperatures is 2-5%. It is also clear that both methods described above would give more accurate results when high-temperature values for Kg, p*, 7*, H*, and Sg are used in the respective formulas.
d In 7 d In V
= —Ki
= 1 -
7Cv V dT
= 1 + St — K' —
d In 7 d In P
<91nCV\ d\nV )r
SinCv_ d In V
= 1 - Si -
d\n Cv d In V
(106)
(107)
s
T
T
T
T
10.3.2. In conclusion to this analysis, we show the dependence of Ks versus T at P = 0 (Figure 9) calculated by the Mie-Griineisen EOS with 7 = 70 for three minerals considered in sections 6-8. Comparing the Ks curves for various values of q = 0 — 2 with experimental data, we see that it is possible to choose appropriate values of q consistent to the data. However, considering the results presented for the same EOS’s in sections 6-8, it is not always possible to find such values of q for which the EOS becomes consistent to data simultaneously for a, Cp, r, and Ks- Thus, as in sections 6-8, we conclude that the thermal part of the Mie-Griineisen EOS does not provide suffuciently high accuracy of all the thermodynamic parameters calculated from this EOS.
= 1 + ST - K' - ajT(a + K' - 2ST). (108)
As noted earlier (see (57) or (69) and (8)), in general, the Cv = constant case leads to 7 = and therefore,
q = q(V) or q = constant. If Cv is only temperature-dependent, there are three possibilities: (1) q = q(V),
(2) q = constant ^ 1 (i.e., Sy = constant ^ 0), and
(3) q = 1 (Sy = 0, K' = St(V), and r = uKt = constant). Thus, both Cv = constant and Cv = C'v(T) conditions result in the case that the two inequalities are equivalent:
0 < q < 1 and 0 < Si = K' - ST < 1 (109)
If we simply assume that 7 is only volume-dependent, then from (14), (33), and (106),
Minerals Temperature, K Ks (GPa) for (5S(300A') = 8s = const Ks (GPa) by the method of O. Anderson AKS ~ H Ks (GPa) from measurements [0. Anderson et at, 1992a]
MgO, periclase 1800 135.7 (2.3) 134.8 (1.5) 132.7
AI2O3, corundum 1800 218.3 (1.4) 217.1 (2.1) 221.8
Si02, a-quartz 1000 30.4 (5.5) 28.8 -
Si02, coesite 1000 82.8 (2.1) 81.1 -
Si02, stishovite 1800 222.4 (0.8) 220.6 —
CaO, lime 1200 97.4 (1.3) 97.6 (1.1) 98.7
FeO, wiistite 900 156.8 (0.4) 157.5 —
Mg2Si04, forsterite 1700 101.5 (2.2) 101.6 (2.6)** 103.8
Mg2Si04, /3-spinel 1700 139.5 (2.3) 136.4 -
Mg2Si04, 7-spinel 1700 152.6 (3.5) 147.4 -
MgSiOs, enstatite 1700 89.6 (4.4) 85.8 -
MgSiC>3, ilmenite 1700 179 (3.8) 186 -
MgSiOs, perovskite 1700 172 (4.9) 180.8 -
MgSiOs, garnet 1700 121 (8.0) 112 -
Fe2SiC>4, fayalite 1500 105.7 (4.3) 101.3 -
Fe2Si04, /3-spinel 1700 137 (1.0) 138.4 -
Fe2SiC>4, 7-spinel 1700 170 (2.4) 166.0 -
FeSiOs, ferrosilite 1700 78.5 (2.6) 76.6 -
FeSiOs, perovskite 1700 207 (7.6) 192.4 -
Grossular 1300 148 (3.0) - 152.6
Pyrope 1200 149.8 (2.2) - 153.2
Olivine, Fo 90 1500 103.3 (4.2) - 107.8
Olivine, Fo 92 1400 108.6 (3.8) - 112.9
* Given in parentheses are the deviations (in percentages) from the measured Kg', when the latter is not available, the difference (in percentages) between two indicated values of Kg ls given.
** Extrapolation from T = 400 K, using the O. Anderson et al. [1992a] data. The extrapolation from T = 300 K gives Kg — 97.4 GPa (6.2% deviation).
( 9K‘
V <9T
dKj
~8T
dKs
dP
- i-ss -j-ajT{L + 1), (112)
(1 + ajT) + KT
which, upon excluding L by (85), yields the important identity [Bassett et at, 1968]
jT + aj . (110)
Placing in (110) for (Ks/dT)v by identity (99),
fdK<,\
9 = 1 + 7 — y~gp~ J +(1 + «7 T)8S (111)
However, in the general case, 7 = 7(V,T), and from the formula for 7 in (18), we find
I<s ( dj_\
7 WJt
q = 8s(l + «7 T) + 1 -
( 9KS
1 +
<9 In 7 <9 InT
V dP
Vi
(113)
= «5s + 1 -
V dP
1 +
<9 In 7 <9 In T
For 7 = 7(V), this identity is reduced to (111).
In section 6, we have already referred to some data on values of q. In general, values of q can be inferred
T
T
s
Temperature, K
Figure 9. Adiabatic bulk modulus versus temperature at P = 0 for three minerals. The solid lines are from the Mie-Griineisen EOS (as in Figures 4-7). The cicles are data of 0. Anderson et al. [1992a].
from the following sources: 1) thermodynamic estimation by (108) or (111), 2) fit of the Mie-Griineisen type EOS to data on a(T), Cp(T), and Ks(T) at P = 0, 3) shock wave data [e.g., McQueen, 1991; Duffy and Ahrens, 1992a], 4) adiabatic temperature gradient measurements [Boehler, 1982, 1983], 5) spectroscopy of solids [e.g., Reynard et al., 1992; Williams et al., 1987],
6) theoretical EOS models [e.g., Isaak et al., 1990],
7) analysis of geophysical data [0. Anderson, 1979b; D. Anderson, 1989]. The values of q estimated by (108) and given in Table 3 and Pankov et al. [1997], fall into
the interval 0.5-2, except for the high values for co-esite (about 17), fayalite (2-3), and Fe-perovskite (4-5). Small negative values were also found for enstatite and FeO (probably, due to inaccurate input data). With increasing T at P = constant or with increasing P at T (or S = constant), q decreases (see also section 6).
For the temperature derivative of 7, we again have the expansion of type (29)
1
«7
dj
dT
= q + -
a
1 f d In 7
dT
(114)
P
where the intrinsic anharmonicity term can be evaluated using (18) and (85)
a^T
= 1-283-! +
fdKs V dP
— ajT 8s ■
= T
<9r\ V 7 /dCv\
dT
Cv Cv \dT J,
(9 In Cv <9 In V
<9 In Cv <9 In T
(9 In Cv <9 In V
7 =
7o*
1 - 7o (* - 1) ’ which is derived from (110) under the condition that
0KS <9 T
= 0 and 7 = ~l{V).
and Panov, 1972], but it also follows from the given thermodynamic consideration.
Equation (118) can be considered a partial case of the more general representation 7 = j(V, S). We introduce a parameter A defined as
(115)
_ 1 (d$
= V557
= <$(l + a7T)
This term is usually negative and completely prevails in (114) at T <0, but at high temperatures, its value is comparable to q. Thus, the frequently used assumption that 7 = *j{V) is unsatisfactory in the general case, and the temperature effect on the Griineisen parameter can serve as a measure of the validity of the Mie-Griineisen EOS [Molodets, 1998].
Another suitable representation of (dj/dT)y follows from (8) and (69) [Stacey, 1977b]
T 1^1
1 (9 T
= 1 + 7 -
<9 In 7 <9 In V
= 1 -
<9 In 7 T <9 In V
(119)
where $ = Ks/p. Assuming that A = A (S') or A = constant and using (119) and (105), we find by integration that
7 =
7o*
1 —A
1 -
7o 1 - A
K“A-1)
7 =
7o
1 — 70 In X
for A = 1
(120)
(116)
7 =
lox 1 - 7o (* - 1)
for A = 0,
If 7 = 7(Vr), then either Cy = Cy(S) or Cy= constant. The case C'y(S) results in
_(dlnCy/dlnV)T 1 (dlnCv/dlnT)v' 1 ’
Moreover, (117) leads to Cy(V,T) = Cy(Q/T) and 0/T = /(S), so that 7 is represented as 7 = —dlnQ/dlnV, where 0 is a characteristic temperature.
11.2. Some explicit volume dependences of 7
The frequently used volume dependences of the latice Griineisen parameter were given in section 6. The Rice [1965] formula is also of interest
where 70 = 7o(S) and Vo = Vo(S). These dependences of 7(2;) for various A are illustrated in Figure 10. One can see that they are quite sensitive to variations of A in the interval from 0 to 1.
12. Adiabatic Temperature Gradient
In geophysics, the conditions close to adiabatic are realized in the convecting mantle and core, as well as in seismic wave propagation. Furthermore, the state at the initial part of Hugoniot are close to adiabatic. Adia-bats of a given material form a one-parametric family of curves. In this case, the temperature and pressure are related by the adiabatic gradient ts, which, considering its definition by (10) and relations in section 2, can be written in the form
(118)
ts =
IT
ajT
A't(1 + a'jT) r(l + a'jT)
= -{1~
Cy
~Cp
(121)
The inequality (dKs/dT)y > 0 (see section 10 and D. Anderson [1989]) holds true of many materials and therefore gives a lower limit for their dependence j(V),
i.e., q < 1 + 7 and 7 > 7(2;) by (118). This limit was previously found from the Mie-Griineisen EOS [Kalinin
Typical values of ts found by (121) are given in Table 2 and Pankov et al. [1997].
Direct measurements of ts at high pressures and temperatures were made in a series of works [Dzhavadov, 1986; Boehler and Ranakrishnan, 1980; Boehler, 1982, 1983]. Chopelas and Boehler [1992] reported corrections to the Boehler [1982] initial results on ts
t
s
s
T
s
1 1 1 X = 2 1
- -
y/ -
. s'
A -
1 1 1 1
1.0
0.8
0.6
0.4
0.6
0.7
0.8
0.9
1.0
x = V/K
o
Figure 10. The Griineisen parameter versus volume calculated by (120) for jo = 1-
12.1. PVT derivatives of
ts
We consider the basic identities and approximaions for the derivatives. Denoting by n the logarithmic volume derivative of ts and using q by (106), we have
d In Ts d In V
= -K7
= q-
Kt
Ks
dKs
dP
= 1 + St + Kq
d In ts dP /T
5 In CP dP
(122)
K,
Kt and
hence,
9KS
dP
K'.
K'
(123)
= n — 7 T
= n( 1 + ajT) — 7
d In ts
dT
d In ts 5 In T
(124)
Approximation (89) and n = 1 + St give ns = n — 7. Writing the derivative of ts with respect to T in the form of (29),
(compare with (45)).
Formula (122) can be represented in various forms, using q by (113), (115), and (58). It is clear that n decreases by isothermal or adiabatic compression. The simplest estimate of n is given by assuming that
d In ts \ 1 / d In ts
5 In V ) p = a V dT
or after substituting (d In ts/dT)y by (124) d In ts
= - [n(l + «7T) - 1 - (5S] .
7
(125)
(126)
The typical values of q = 1-2 and K' = 4-5 yield n = 5 — 7. If we neglect the last term in (122) at T > 0, then n = 1 + St [Chopelas and Boehler, 1992].
Changing from variables (V, S) to (V,T), the adiabatic derivative with respect to volume takes the form
_ ( 9 In Ts ^
*i= w =1 + <s
The values of ((9 In r^) (9 In V))p and n (an extrinsic anharmonic contribution) calculated by (126) and (122) are given in Table 2 and Pankov et al. [1997]. They show that the intrinsic anharmonic term dominates in (i25).
Note that the ts parameter occurs in any expression when changing variables P, S to P, T: for example,
fdST\ _ fdSr\ (dSt\
V dP ,
dP ,
\dT
ts
(127)
which was used in deriving (98).
p
n —
T
T
T
n
p
T
12.2. Explicit volume dependences of ts
For a moderate compression, the volume dependence of Ts can be described by the power law
ts = tsox '
(128)
where n = n(T) or constant, tso = tso(T) and Vo = Vo (T). This formula was used to fit the measured ts values to P = 50 kbar and T = 1000 K [Boehler and Ramacmshnan, 1980; Boehler, 1982].
However, Chopelas and Boehler [1992], accounting for the variation of St with V (see (46) and (47)), found that the linear dependence of In ts on V(n = mx) better describes teir data on ts than the power law, and consequently,
_ _ mVn(x-l) /ion\
ts = TS0e °v (129)
where constant m is determined by the approximation
- ( 1 + ST -
(9 In Cv <9 In V
(130)
(see (122), where C'p is approximately C'y). Thus, on the condition that (<9 In C'y/8 In V)t is independent of V, the derivative (BSt /Bx)t can be found given knowledge of the m value. Isaak [1993] applied this method to evaluate the derivative 82Ks/dPdT with the help of (91).
12.3. EOS based on data for ts (P, T).
Measured values of Ts(P, T) allow us first to find the isobaric specific heat [Dzhavadov, 1986]
In
Cp(P S) T(P, S)
<9 ts\
<9 T
dP + In
CP(0,S) T(0,S) :
(131)
which is deduced from the identity d2T/dPdS = d2T/dSdP. The integral in (131) is taken over an adi-abat, and the specific heat versus temperature, C'p(T), for P = 0, is assumed to be known. Then, given a reference isotherm V(P,To), from (14), we can find the thermal EOS in the form
1
V(P,T) = I T^LdT+V(pTo).
(132)
Conversely, given the thermal EOS and ts(T) at P = 0, (14) gives Cp(T) at P = 0, and thus, the caloric EOS can be determined.
Conclusion
We have reviewed thermodynamic properties of geomaterials necessary to study the thermodynamics of the deep interior of the Earth.
(1) In sections 2-5, it was shown that the determination of all the second-order thermodynamic parameters requires knowledge of values of three such parameters. In relation to EOS’s, all the thermodynamic parameters were lumped into thermal and caloric types. A summary to finding of EOS’s from experimental data was presented. The approaches directly based on measured thermodynamic characteristics can be formulated in the form of partial differential equations. Of 16 third-order thermodynamic parameters, only four (appropriately chosen) are independent. Attention is given to the compilation of a self-consistent database for minerals, relying on input data for a, Ks (or Kt), (dKs/9P)t, {da/dT)P, (dCp/dT)p, and (dKs/dT)P.
(2) Each of the eight second-order parameters was analyzed separately, following the plan: the derivation of the identities between their P and T derivatives, the estimation of the intrinsic and extrinsic contributions to the temperature derivatives, useful simplifications of these relations and their consequences, and the explicit approximate dependences of the second-order parameters on pressure and temperature.
(3) In the analysis of thermal expansivity (section 6), the Birch formula for a = a(P) at T (or S) = constant is generalized. It was shown that a in the lower mantle, calculated by the generalized formula, is sensitive to assumed values of the mixed P — T derivative of the bulk modulus Kp, in the range of dK'0/dT ~ (0 — 4) • 10-4 K-1. The assignment of a value about 2 • 10-4 K-1 for this derivative gives a in the lower mantle to be close to those by the (exponential) laws of 0. Anderson et al. [1993] and Chopelas and Boehler [1992]. Based on these estimates and our analysis, we conclude that the coefficient of thermal expansion decreases along the hot lower-mantle adiabat (from P = 0 to P = 1.35 Mbar) by a factor of 4-5. Considering the O. Anderson power law for a, we stated strict conditions for the consistency of various assumptions regarding the EOS and parameters St , Kt , K', and C'y and cleared up the consequencies of these assumptions. In many cases, these conditions are useful for a self-consistent thermodynamic analysis. For example, the power form of the Birch law, Kt ~ Vb leads to K' = constant, Kt = Kt(V), St = K' = constant, the Murnaghan EOS (41), and for C'y = constant, a = a(V). Various extrapolations of a to high temperatures at P = 0 show a great uncertainty in the resulting thermal expansivity (to 30-50% at T^1500 — 2000 K), which indicates that high-temperature measurements of a are very neeeded to improve the knowledge of a.
(4) The isobaric specific heat C'p under the lower-mantle conditions (section 7) decreases approximately 10% along the hot adiabat, from P = 0 to P = 1.35 Mbar. At low temperatures T < O, the intrinsic an-harmonicty competely prevails, but at T > O, when (dC'p/dT)p is small, its contribution is only 15-30%.
m
x
T
p
T
(5) The difference between the thermal pressure model of O. Anderson and the Mie-Griineisen EOS is emphasized (section 8). This model has two specific features: in general, its thermal pressure is linear in temperature, but the volume dependence of thermal pressure depends on the kind of material. From our estimation, at T > O, the nonlinear terms in Pth contribute no more than 13%. In total, we refer to the existence of, at least, four models of thermal EOS: the Mie-Griineisen (or more general anharmonic lattice) EOS, a model with various forms of the reference isotherm P(V,To) and with a given a(P,T) dependence, the O. Anderson model mentioned above, and the formulation of type (28) with assumed temperature variations of the EOS parameters.
(6) In section 9, the Anderson-Griineisen parameters 6s and St are analyzed in more detail. An explicit expression for St(V) at T (or S) = constant was derived from the generalized formula of Birch. We find that, for dK'0/dT = 2.3 • 10-4 K-1 (see the derivation of (3)), St at the base of the mantle is almost half the value at P = 0.
(7) The adiabatic-isothermal transformation of bulk moduli are discussed in section 10. In addition to the previous considerations, the useful formula (98) was derived for the mixed derivative dK'0/dT. Altogether, this parameter for various geomaterials is estimated by a value of the order of (1 — 3) • 10-4 K-1.
From the analyzed temperature behavior of bulk moduli, we infer that the Sy = K' — 6t and values at room temperature fall mostly between —4 and —1 and between —1 and 1, respectively. However, their high-temperature values are in the range from —1 to 1 for Sy and from 0 to 1.5 for Sy. The approximation Sy & 0 (St ^ K' and Kt = Kt(V)) is justified for many but not all minerals.
In relation to the interpretation of seismic tomography data for the lower mantle, we found the following ranges of acceptible value for this largest layer of the Earth: 6^^0.2, 0.8, 7^1.1, St^3 — 3.3, and
(5s^l.9 — 2.2 (provided that the thermal interpretation of these data is true).
The Ks values at high temperature, evaluated by the power law with 8s = constant and by the O. Anderson enthalpy method have errors of the order of 2-6 and 1-3%, respectively. Thus, it is confirmed that the 0. Anderson [1995] method is quite efficient.
(8) A number of identities for the Griineisen parameter 7 and its logarithmic derivative q = (d In 7/9 In V)t were given in section 11. They show that the conditions C'v = constant or C'v = C'v(T) lead to 7 = y(V) or 7 = f(V)/C'v(T), respectively. Both these cases are compatible to the O. Anderson thermal pressure model, with r = oiKt = constant or r = t(V). Any of the indicated conditions for C'v also gives q = q(V) or q = constant; moreover, from the inequality 0 < q < 1, it
follows that 0 < K' — St < 1 and vice versa. Thermodynamically estimated q values fall largely into the interval 0.5-2. This parameter generally decreases with pressure and temperature. In the derivative (dy/8T)p, the intrinsic anharmonicity prevails on the whole, suggesting a significant dependence of 7 on temperature. In addition to many known expressions for y(V), we derived a new one based on the parameter A = 1 — (dlnjT/dhiV)s^- The A = 0 case is reduced to the Rice [1965] formula. Variation in A in the interval of 0-1 (accordingly, Sy = A(1 + ayT)-1 ranges approximately over the same interval for ayT <C 1) appreciably affects the 7 values at high compression.
(9) The identities and approximations for the adiabatic temperature gradient ts = (8T/dP)s were systematized. Our thermodynamic estimates of the Boehler parameter n = (d In Ts /d In V)t are close to his experimental results for olivine, quartz, and periclase. The uncertainty of the order of one in the estimated n is caused by errors in the used input thermodynamic data. In the derivative (drs/dT)p, the intrinsic anharmonic contribution was found to dominate. When determining the EOS from data for ts , an important role is played by the relation of this parameter to specific heat.
Finally, in sections 6-8 and 10, we checked on the validity of the Mie-Griineisen EOS used to evaluate a,Cp, r, and Ks- Qualitatively, this EOS model correctly describes the P — T behavior of the indicated parameters, but in general, it does not always provide a sufficiently high accuracy of the estimated values. For this reason (see also the inference (8) above), it is concluded that care must be exercised when applying this type of EOS in geophysics.
Acknowledgements. We thank U. Walzer (lena University), T. Spohn (University of Munster), A. Kalachnikov (the Moscow United Institute of Physics of the Earth),
O. Kuskov (the Moscow Institute of Geochemistry), and K. Heide (lena University) for useful discussion of this work. Thanks are also due to Mrs. H. Kottitz for her help in the preparation of this text. The work was supported by the German Research Society, Grant 436-RVS, by the German KAI 015936 and 015892 projects, and by the Russian Foundation for Basic Research, Grant No. 96-05-66247.
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(Received May 15 1998.)
