Abstract
None of the 40+ equations that have been proposed to describe material properties at the pressures of the Earth’s core and mantle have escaped serious criticism. In this paper, some basic algebraic and thermodynamic constraints are reviewed, with the conclusion that the next step should be a re-examination of the relationship between the dependence of the bulk modulus, K, on pressure, P, that is K ′ ≡ d K / d P , and the normalized (dimensionless) pressure, P / K . A linear relationship between 1 / K ′ and P / K terminating at the infinite pressure asymptote, at which these quantities become equal, has been used for analysing properties at extreme pressure, but may be inadequate for calculations requiring precise derivatives of an equation of state. A new analysis indicates that d ( 1 / K ′ ) / d ( P / K ) increases with compression (or P / K ), but there are, at present, no reliable equations representing this. Relationships between higher derivatives of K and the thermodynamic Grüneisen parameter offer the prospect of a resolution of the problem and hence a new generation of fundamentally-based equations of state. Although an earlier conclusion that a completely general ‘universal’ equation is not possible, in principle, is confirmed in this study, the fundamental relationships present strong constraints for the forms of other proposed equations.
Highlights
Relationships between higher derivatives of K and the thermodynamic Grüneisen parameter offer the prospect of a resolution of the problem and a new generation of fundamentally-based equations of state
Of the physical properties of geological materials that are central to studies of the lower mantle and core, a basic one is the bulk modulus, K = ρdP/dρ = dP/dlnρ, representing the variation of pressure, P, with density, ρ
This reduces Equation (13) to 1 = 1, a symptom of a circular argument. It means that the algebraic identities involved in the manipulations cause the indeterminacy of a Taylor expansion using these derivatives, reinforcing the conclusion that there can be no universal form for equations of state
Summary
Stacey and Hodgkinson [6] introduced another fundamental constraint using volume derivatives of the thermodynamic Grüneisen parameter, γ, which is, with its thermodynamic definition and an equation used for calculating it from K(P), γ = αKT /ρCV = [K0 /2 − ( f /3) (1 − P/3K)]/[1 − (2/3) f P/K] The use of this equation has been limited by the fact that f is a parameter that depends on statistical details of atomic thermal vibration and is assigned different values in alternative theories. Equation (2) has an advantage for calculations of the kind considered here because, if f can have any value and any compression dependence, the equation is completely general and, through f, incorporates the effect of the rigidity modulus This generality extends to the infinite pressure extrapolation, in which f is canceled out by Equations (1) and (2), reducing it to γ∞ = K∞.
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