Thermodynamics of deep geophysical media
V. Pankov, W. Ullmann, R. Heinrich, and D. Kracke


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 ), (partial KS/partial P)T, (partiala/partial T)P, (partial CP/partial T)P, and (partial KS/partial T)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 KT, in the range of dKprime0/dTsim (0-4)cdot 10-4 K -1. The assignment of a value about 2cdot 10-4 K -1 for this derivative gives a in the lower mantle to be close to those by the (exponential) laws of O. 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 dT, KT, Kprime, and CV 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, KTsim Vb leads to Kprime = constant, KT = KT(V), dT = Kprime = constant, the Murnaghan EOS (41), and for CV = 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 CP 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 < Q, the intrinsic anharmonicty competely prevails, but at T > Q, when (partial CP/partial T)P is small, its contribution is only 15-30%.

(5) The difference between the thermal pressure model of O. Anderson and the Mie-Grüneisen 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 > Q, the nonlinear terms in Pth contribute no more than 1-3%. In total, we refer to the existence of, at least, four models of thermal EOS: the Mie-Grüneisen (or more general anharmonic lattice) EOS, a model with various forms of the reference isotherm P(V, T0) 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-Grüneisen parameters dS and dT are analyzed in more detail. An explicit expression for dT(V) at T (or S ) = constant was derived from the generalized formula of Birch. We find that, for dKprime0/dT = 2.3cdot 10-4 K -1 (see the derivation of (3)), dT 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 dKprime0/dT. Altogether, this parameter for various geomaterials is estimated by a value of the order of (1-3)cdot 10-4 K -1.

From the analyzed temperature behavior of bulk moduli, we infer that the dVT = Kprime - dT and dVS 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 dVT and from 0 to 1.5 for dVS. The approximation dTVapprox 0 ( dTapprox Kprime 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: dTV 0.2, q 0.8, g 1.1, dT 3-3.3, and dS 1.9-2.2 (provided that the thermal interpretation of these data is true).

The KS values at high temperature, evaluated by the power law with dS = 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 O. Anderson [1995] method is quite efficient.

(8) A number of identities for the Grüneisen parameter g and its logarithmic derivative q = (partial lng/partial ln V)T were given in section 11. They show that the conditions CV = constant or CV = CV(T) lead to g = g(V) or g = f(V)/CV(T), respectively. Both these cases are compatible to the O. Anderson thermal pressure model, with t = aKT = constant or t = t(V). Any of the indicated conditions for CV also gives q = q(V) or q = constant; moreover, from the inequality 0le qle 1, it follows that 0le Kprime - dTle 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 (partialg/partial T)P, the intrinsic anharmonicity prevails on the whole, suggesting a significant dependence of g on temperature. In addition to many known expressions for g(V), we derived a new one based on the parameter lequiv 1 - (partial ln gT/partial ln V)S 1. The l = 0 case is reduced to the Rice [1965] formula. Variation in l in the interval of 0-1 (accordingly, dVSequiv l(1 + agT)-1 ranges approximately over the same interval for agTll 1 ) appreciably affects the g values at high compression.

(9) The identities and approximations for the adiabatic temperature gradient tSequiv (partial T/partial P)S were systematized. Our thermodynamic estimates of the Boehler parameter n = (partial lntS/partial ln 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 (partialtS/partial T)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-Grüneisen EOS used to evaluate a, CP, t, 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.


We thank U. Walzer (Jena University), T. Spohn (University of Münster), A. Kalachnikov (the Moscow United Institute of Physics of the Earth), O. Kuskov (the Moscow Institute of Geochemistry), and K. Heide (Jena University) for useful discussion of this work. Thanks are also due to Mrs. H. Köttitz 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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