CALCIJLATION OF ELASTIC PROPERTIES FROM ...craig/publish/pdf/...The elastic properties of the Earth's interior-i.e. density and elastic wave velocities-are functions of composition,

CALCIJLATION OF ELASTIC : PROPERTIES FROM

THERMODYNAMIC EQUATION OF STATE PRINCIPLES

Craig R. Bina

Department of Geological Sciences, north wester^^ University, Evanston, Illinois 60208

George R. Helffriclz

Department of Terrestrial Magnetism, Carnegie Institution of Washington, Washington, DC 20015

KEY WORDS: seismic velocities, mantle structure, mantle composition, elastic moduli, finite strain theory, Mie-Griineisen theory, Debye theory, lattice dynamics

INTRODUCTION

The analysis of seismic velocities at depth within the Earth has a long history and attracts an increasingly broad spectrum of researchers. Almost as soon as seismic travel times were available, Williamson & Adams (1923)

I used them to obtain velocity profiles and check their consistency with the densities that would result from self-compression of a uniform material. Birch (1939) addressed the same question of homogeneity and composition by incorporating the effect of pressure in his extrapolation of the shallow Earth's density and wave speeds to mantle depths. This traditional use of seismological data to investigate mantle properties and phenomena continues since seismic waves (in a broad sense that includes normal modes as well) constitute our primary probe of Earth structure. For example, core-diffracted P and S waves (Wysession & Olcal 1988, 1989) carry

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528 BINA & HELFFRICH

information concerning the properties of the U" layer at the core-mantle boundary (Bullen 1949, Lay & Helmberger 1983, Lay 1986) that can be related to temperature and composition (Wysession 1991). Slabs of subducted oceanic lithosphere influence the travel times and waveforms of seismic waves (Oliver & Isacks 1967; Toksoz et a1 19'11; Sleep 1973; Engdahl et a1 197'7; Jordan 1977; Suyehiro & Sacks 1979; Creager & Jordan 1984, 1986; Silver & Chan 1986; Vidale 1987; Fischer et a1 1988; Cormier 1989) which can be related to their thermal structure and composition. Seismic waves reflected and refracted at the slab-mantle interface also provide constraints on the properties of this interface that can be inter- preted by computing seismic velocities in slab mineralogies (Helffrich et a1 1989). In a similar fashion, the Earth's velocity structure and discontinuities provide information concerning the composition of the mantle (Birch 1952, Lees et al 1983, Anderson & Bass 1984, Bass & Anderson 1984, Bina & Wood 1984, Weidner 1985, Bina & Wood 1987, Akaogi et a1 1989, Uuffy & Anderson 1989). Finally, the issue of the composition of the lower mantle can be addressed by comparing computed densities of the candidate mineralogies with density profiles of the lower mantle (Watt & O'Connell 1978, Jackson 1983, Anderson 1987, Chopelas & Boehler 1989, Bina & Silver 1990, Fei et al 1991).

The variety of approaches involved in these studies demonstrates the need for a review of the methodology used to compute elastic wave speeds, in the spirit of an earlier review by Anderson et a1 (1968). Three broad areas bear on this topic: thermodynamic analysis, continuum mechanics, and solid state physics. Experiments provide the raw data and some important rules of thumb that make i t possible to compute elastic velocities inside the Earth. Our intent is to combine theory and data into a practical method that will facilitate calculations of this type and to indicate some of the failings and alternative approaches that may be pursued.

The elastic properties of the Earth's interior-i.e. density and elastic wave velocities-are functions of composition, mineralogy, pressure, and temperature. The dependence of the elastic properties on these factors has been measured for numerous minerals over a range of conditions. Compositional variations traditionally have been treated through empiri- cal systematics based on the mineral structure and the mean atomic weight of a mineral (e.g. Birch 1961; Anderson & Nafe 1965; Anderson 1967, 1987). On the other hand, temperature and pressure effects have been investigated through laboratory measurements (e.g. Christensen 1984). These results have given rise to the view that pressure and temperature derivatives of wave speeds, assumed to be constant and applied to speeds measured at room conditions, yield elastic wave speeds under mantle conditions. This process requires extensive extrapolations, by factors of

up to -- 20 in temperature and 100 in pressure. In general, however, these derivatives are not constant throughout the extrapolation interval, and this assumptioll is likely to lead to incorrect results. When large extrapolations away from measured properties are necessary, they should be guided by a theoretical framework that controls the functional form. The purpose of this paper is to review the relations that provide a conceptual basis for bridging the gap between experimental and mantle conditions. We deal first with the relevant thermodynamic relations and then briefly with the continuum mechanical and lattice dynamical theories that bear on elastic properties at mantle conditions.

BASIC PARAMETERS

Thermodynamic relations link the responses of substances to changes in their ambient conditions. A number of equally valid formalisnls may be invoked to describe these relations (see Appendix), but we live in an environment that exposes us to the responses of substances to changes in ten~perature and composition at constant pressure. Not only do these conditions shape our intuition, they affect the relative difficulty of certain types of measurements. This, in turn, affects the type of experimental information available for co~nputing elastic properties.

In the laboratory, the most easily controlled parameters are pressure, temperature, and bulk composition. For this reason the Gibbs potential G is often used to characterize such systems. In this formulation the natural variables are P (pressure), T (temperature), and the N, (compasitional quantities), and the equations of state (see Appendix and Table Al) are

where S, V, and p, are the entropy, volume, and che~nical potentials, respectively.

As elastic waves pass through a substance, they deform it and may perturb its volume. These volume perturbations coniprise one of the links between elasticity and thermodynamic analysis because through them many thermodynamic constraints may be brought to bear on seismic wave propagation. Another important factor is that elastic waves travel faster than heat diffuses so the therlnodyriamic system may be viewed as adiabatic (closed to heat transfer). The deformation, moreover, may often be viewed as a constant-entropy (isentropic) process, since any such deformation is essentially reversible in the absence of significant elastic attenuation. Since any entropy changes are negligible or poorly characterized, we focus on the effect of volume perturbations.

An obvious, measurable property of a substance is its volume. Its tern-


perature dependence is given by the volume coeflcient oj'tlzermal expansion u, where

a measure of the volurrie increase upon heating. a is generally positive but may be zero or negative for sorrie substances, and is 0 (lo-' K-') for most minerals at room conditions. Similarly, the pressure dependence of the volume is given by a measure of the volume decrease upon compression, the isotlzernzal compressibility pT, where

or by its reciprocal, the isotlzerrnal btillc modulus or isothermal irzconz- pressibility defined by

an 0 (lo8 bars) quantity. (Seisrriologists regrettably use cx and P for the P- and S-wave speeds; here these will be designated up and v,.)

The entropy S is a less intuitive property closely related [from the first of Equations (I)] to a directly measurable quantity, the isobaric heat capacitj, or specijic lzeat at constant pressure Cp, defined by

a measure of the heat dq absorbed by one mole of the substance per unit temperature change.

The parameters defined in Equations (2)-(5) above are also functions of pressure and temperature and thus possess their own temperature and pressure derivatives. Of these, the pressure derivative of KT

figures prominently in finite strain equations. Another useful quantity is the temperature dependence of K,, given by the isothermal Anderson- Gru~eisen ratio (Barron 1979) or the isotlzermal second Griineisenparameter 6, (Anderson et a1 1968):

ELASTIC PROPERTY CAL.CULATION 53 1

Since S and V are state functions, their mixed partial derivatives are equal

leading to additional relatiolls for the pressure dependence of u and (J,

Elastic waves are assumed to propagate adiabatically/isentropically, yet the relations given thus far are isothermal or isobaric. Various expressions relate these to isentropic conditions. Temperature and pressure under such conditions are related by the adiubatic gradient

The defined quantities that follow are analogous to the isothermal relations: the adiabatic con~pressibility Ps and the adiabatic bullc 17lodulus or adiubatic inco~npressihility Ks

the isoclzoric heat capacitji or specific Izerrt at constant volt~nze &

and the isothermal pressure derivative of Ks

The temperature dependence of Ks is given by the adiabatic A~zderso~z- Griilzeise~z ratio or the adiabatic second Grii~zeisen parameter iis:


The adiabatic arid isothermal parameters are related via the thernzal Griirzeisen ratio or first Griineisen paranzeter y,,,:

Implicit in y,,, is a deep connection between thermodynamic properties and the atomic structure of solids. Lattice dynamical theories relate macro- scopic solid properties to the effects of a lattice of oscillating atoms inter- acting through their mutual interatomic forces (Born & Huang 1954, Leibfried & Ludwig 1961). Griirieisen-like parameters characterize the volume dependence of the lattice's oscillator frequencies and are important parts of'the Mie-Griineisen equations of state which figure prominently in shock wave studies (Courant & Friedrichs 1948, McQueen et al 1967, Rigden et al 1988, Anderson & Zou 1989, Jeanloz 1989) and in thermal equations of state in general (Jeanloz & Knittle 1989). y , , is an averaged measure of such volume dependence of frequency, and characterizes anharmonic lattice behavior (Slates 1939, Ashcroft & Merrnin 1976). It also functions to interconvert adiabatically defined quantities and isothermally defined ones through

HIGH PRESSURE PARAMETERIZATION

The pressures of the Earth's interior lead to considerable material com- pressions. Continuum mechanical theory provides the link with thermodynamic properties by first postulating the existence of a strain energy function and then identifying it with one of the thermodynamic potentials (see Appendix). Under constant temperature conditions the natural choice is the Helniholz free energy F because P = - (BFIdV),, where a V clearly relates to the strain imposed by material compression. If instead constant entropy conditions are imposed, the choice is the internal energy U since P = - (aU/a V),. Either of these imposed conditions leads to identical equations in which the appropriate modulus, either adiabatic or isothermal, must be used. The choice of F as the strain energy function leads to the well-known Birch-Murnaghan equation that employs KT. Application of the thermodynamics of F to the analysis of silicate melting relationships at high pressures is demonstrated by Stixrude & Bukowinski (1990).

In these functional relations, volume (strain) depends upon pressure in

EL.ASTIC PROPERTY CAL.CULATION 533

a rather awkward implicit fashion. Expanding the elastic potential energy arising from an applied hydrostatic finite strain in a Taylor series in strain derivatives and truncating to fourth order in strain leads to (Davies 1973, 1974; Davies & Dziewonski 1975)

Here

where V,, KT,, K;", and K f , are all evaluated at zero pressure. A third- order truncation yields the Birch-Murnaghan equation (Birch 1938, 1939, 1952, 1978). TJse of this fourth-order expansion may be considered opti- mistic since the higher derivatives of the bulk modulus are difficult to reliably measure (Bell et al 1987, Jeanloz 1981) and mantle strains are adequately modeled to third order [but not strains in the core or those encountered in shock wave studies (Jeanloz 1989)]. We include it to show the strain order at which these derivatives become significant and for consistency with the pressure dependence of the elastic moduli that follow.

Similar methods yield the pressure dependence of the individual isotropic elastic moduli. KT follows from differentiation of (IS), but to ensure a consistent truncation at fourth order we use the Davies & Dziewonski (1975) formulation

The pressure dependence of the shear modulus p is derived with greater difficulty but may be cast in a similar form (Birch 1939; Davies 1973, 1974; Davies & Dziewonski 1975):


Bracketed terms which are second order in f involve the higher-order pressure derivatives K," = (d2KT/aP2), arid p" = (d2p/dP2),. Their iriclusion is not warranted by compression data available to date, but they are given here to show where these higher-order strain parameters enter the expressions. K';', and p';, appear to be of order Kb l and p; ' (Davies & Dziewonski 1975, Col.len 198'1, Isaak et al 1990, Yoneda 1990) and so make O(1) contributions to the bracketed f' terms in (20) and (21). In the absence of reliable K;, and p';, values, one may app~oximate them as KF~' and p; ' or ignore j 2 terms completely (Duffy & Anderson 1989, Gwan- mesia et a1 1990).

THERMAL PARAMETERIZATION

Some important approximations greatly simplify elastic property cal- culatioris. The first concerns the pressure dependence of y,,. To a good approximation (McQueen et a1 1967)

implying that

( ~ t JY tho) = ( V/ Vo)" (23) with q = 1. Departures from this approximation are known (e.g. Jeanloz & Ahrens 1980a,b) and the approximation has been refined (Irvine & Stacey 1975), but the error introduced by this approximation when used to calculate seismic velocities is small. The exponent q is related to other thermodynamic properties (Bassett et al 1968, Anderson 1984, Anderson & Yamamoto 1987) through

If q is constant, a relation among these is implied, leading to the second important approximation. Anderson (1984) has drawn attention to the fact that aK, is esseritially constant. While his coriclusions are based or1 metal and alkali halide measurements, they also hold for olivine (Isaak et a1 1989), and we assume this to be true for other mantle materials. The last term of (24) is therefore zero, and along with the q = I approximation this leads to

J1 = K;,. (25)

This is a good rule of thumb and provides a consistency check between compressiori measurements (yielding K,, and K;,) and a through (7).

EL.ASTIC PROPERTY CALCULATION 535

6, is an expression of the temperature behavior of the elastic constants. Quasi-harmonic lattice dynamics theory predicts a linear dependence of the elastic constants on temperature in the high-temperature limit (Born & Huang 1954; Leibfried & Ludwig 1961; Davies 1973, 1974; Garber & Granato 1975a,b), implying that (dKT/dT)p is constant [as well as (dp/aT)p]. I11 this sanle limit, 6, in nlany materials is observationally constant (Ander- son et a1 1968, Anderson & Goto 1989, Isaak et a1 1989). Since aKT appears constant as well as (dK,/aT),, 6, should also be constant by (7). We thus have a consistent picture of mantle material behavior within the fralnework of the two approximations.

Recent high-quality elastic constant measurements agree with the linear temperature derivatives predicted by theory (Anderson & Goto 1989, Goto et a1 1989, Isaak et a1 1989). What of the temperature dependence of K; and p'? Relation (25) and the constancy of 6, at high teniperatures imply that the temperature dependence of K; is weak. More generally, Davies (1973) notes that the thermal contributions to Fare O(J- ' ) where f is the strain. Since K; and Kr appear in third- and fourth-order strairi terms, their temperature dependence is negligible compared to the temperature dependence of V (with a first-order dependence on strain) and KT (where the dependence is second order). The T dependence of p' and p" can be neglected by a sinlilar argument. Lattice dynan~ical simulation supports this argument (Hemley et a1 1989) by showing less than 10% variation in I(; in MgSiO, perovskite over a 300-2000 K temperature range.

Integration of (7) gives the temperature dependence of KT:

The large body of a(T) measurements at room pressure strongly recom- mends 1-bar evaluation of this integral. 6, is only weakly tenlperature dependent and can be taken as constant, but the high-temperature value [or its approximation (25)] should be used if possible. Another useful simplification is that a is generally linear with T at high temperature so the integral can be analytically transformed to a quadratic polynomial in T.

The adiabatic moduli are needed to conipute elastic wave speeds while KT, not Ks, is the modulus measured by static compression. Shear defor- mations preserve volume, so the adiabatic and isothermal shear moduli are identical. Given KT, (17) gives Ks but the conversion must be done at the conditions of interest if (18) and (20) are used since the up-P path they imply is isothermal. This introduces the need for the pressure dependence of a.


a at pressure may be obtained by straightforward integration of (9)

but this requires characterization of 6, at pressure. Some simple depen- dences are now becoming clear (Chopelas & Boehler 1989, Anderson et al 1990), suggesting that

If borne out by further experimental and theoretical work, this relation will greatly simplify calculation of u(P). Relations (28) and (27) together imply 8, is independent of V.

A more involved u pressure dependence can be obtained as follows. Rearranging (16) following Jeanloz & Ahrens (1980b), differentiating and invoking approximation (22), we obtain

Moving the focus to the pressure dependence of C;, note that Debye theory parameterizes Cv as a function of Tand a characteristic temperature 8, summarizing the lattice vibration frequency distribution (Slater 1939; Ashcroft & Mermin 1976; Kieffer 1979, 1982). The vibrational Griineisen parameter yvib is related to the averaged frequency dependence of lattice vibrations on the lattice volume

dln 8, vib Y ' =- - din V '

and is one member of an entire family of Griineisen-type parameters of differing theoretical heritages (Quareni & Mulariga 1988). Since Cv = CV(T,8,), and y , , = yvi, in the quasi-harmonic approximation, we can use the volume dependence of y,,, to parameterize the pressure dependence of Cv following Kieffer (1982). Writing

we note that each of the terms above is known: aCv/d8, is calculable from its definition (Slater 1939)

E,LASTlC PROPERTY CALCULATION 537

(where rz is the number of atoms per molecular formula unit); (18) and (20) give T/ and KT; arid y,,,/V is approximately constant (22). The terms involving 0 , explicitly read:

Rearranging (29) and (31) and eliminating CV with (16) yields a differential equation for x(P)

that readily yields to numerical solution as an initial value problem (e.g. Press et a1 1987). Note that (34) is equivalent, to first order, to substituting (25) into (27).

We show in Figure 1 the approximate a pl,essure dependence for MgO.

a pressure dependence 60 55 5 0

n 45 rn o 40 .c

x 35 .-- 30 1 25 Y - 20 a 15

10

0 1 00 200 300 400 500 600 700 800 900 1000 1 100 1200

P (Kb) F i e I Pressure dependence of thermal expansion cr in MgO a t 1500°C. Equation (28) denotes method of Anderson et a1 (1990) where (aIncc/8In V), = 6,. Equation (34) denotes method whereby (aajap), I S obtained via the pressure dependence of the heat capacity Data for MgO are from Fei et a1 (1991), from which 67,12,X,hl = 4.47 was contputed, and Sumino & Anderson (1984), whence g,, = 1 52


The P dependence for each formulation is similar, and the a values given by (28) and (34) differ by about 20% at core-mantle boundary pressures. This leads to about a 0.5% difference in 1 + Tay,,, which can be considered negligible when compared to the errors in the various moduli measurements themselves.

VELOCITY CALCULATIONS

The density p and bulk sound velocity v,, of a material are functions of the volume and the adiabatic bulk modulus

where M is the molar mass of the material. The isotropic P- and S-wave velocities, v, and us, are additionally functions of the shear modulus

Given P and T conditions where up and us are desired, we may choose any P-T path along which to evaluate them since V, Ks, and p are all state functions. The choice is one of convenience. Volumetric thermal expansivity and heat capacity data are widely available for minerals at room pressure (Skinner 1966; Touloukian 196'1; Helgeson et a1 1978; Robie et a1 1979; Taylor 1986; Fei & Saxena 1986; Berman 1988; Fei et a1 1990, 1991) but are scarce at high pressure. This suggests that the preferred route to the high P-T state is a single step up-temperature IT,,f-+ T at room pressure PI,[ followed by a second step up-pressure Pnf -+ P at high temperature T. Since most volume measurements are made at room conditions arid thermal expansion measurements are made at room pressure, 298 K and 1 bar is a natural reference state. For simplicity, the 1 bas pressure reference may be taken as zero bars for solids.

To evaluate the volume at Pr,, and T, integrate (2)

Similarly, KT may be evaluated at T from its value at IT;,, by (26), and C,(T) may be obtained from data fitted to polynomials. I11 integrating the polynomial formulation for a(T) in (26) and (37), it is helpful to note that a increases nearly linearly with Tabove intermediate temperatures (Jeanloz & Ahrens 1980b, Saxena 1989). This simple behavior suggests a linear

ELASTIC PROPERTY CALCULATION 539

parameterization of a from a high-temperature reference state such as 1000 K.

To evaluate V(P, T) we insert V(P,,, T), KT(PrCf, T), K;(P,,), and P into Equation (1 8). This equation is implicit for Vas a function of P, so iterative numerical techniques must be used to find V at a corresponding P; the Newton-Raphson method converges rapidly on the solution (e.g. Press et a1 1987). KT(P, T) is obtained from Equation (20) using the f value from (18). Then a(P, T) is obtained from a(P,,, T) using (28) or (34), at which poirlt Ks(P, T) can be computed from KT(P, T) via (17).

To obtain p(P, T) from its value at the reference state, we first account for the telnperature dependence, linear in the high temperature limit, via

and we than account for the pressure dependence via (21). Finally, we insert values for V, Ks, and p at P and T into (35) and (36) to obtain p, v,,,, UP, and 0s.

EXAMPLE

As an example of the use of this method, we calculate the velocity difference between the olivine a and p phases at the mantle conditions of the 400-km discontinuity. The pressure and temperature here are approxi~nately 133 I

Tab

le 1

E

last

ic d

ata

used

in

calc

ulat

ions

k? P

P3

m

%WK

d &

df

KT

&I/@

d

p@

d

ylh

~(I

U)O

K:

Com

pone

nt

R.

&m

3

x106

K-l

xl

o9

K-2

M

bar

Mba

r ba

r K-I

M

g2S

i04

3.36

6 26

.1 1

15

1.

278

5.1

0.81

0 1.

82

309

1.2

3.96

i-3

(a)

?f E;;

F%

Si0

4 4.

404

26.2

0 12

1.

359

5.2

0.51

0 0.

62

130

1.1

5.45

T!

(a)

Mg2

Si0

4 3.

474

20.6

3 17

1.

726

4.73

1.14

0 1.

82

228

1.1

4.27

2 (P)

F

%S

i04

4.42

4 22

.1

11

1.85

9 4.

73

0.73

0 0.

62

146

1.1

5.51

(P)

M

g2S

i20k

3.

194

16.1

2 84

1.

054

2.36

0.

750

2.36

18

0 1.

1 4.

48

(Opx

* Cpx

) Fc

$i20

6a

3.99

8 29

.96

26

1.03

0 2.

36

0.75

0 2.

36

180

1.1

4.46

(O

PX

, CP

X)

NaA

1Si2

06a

3.34

7 24

.72

0 1.

417

2.40

0.

840

2.40

84

1.3

7.

33

(CPX

> C

aMgS

i20g

3.

276

33.2

7 0

1.11

8 2.

30

0.67

0 2.

30

154

1.1

5.94

(O

px* C

px)

py73

AlJ

'Il16

u~6a

3-

71

19.2

0 8

1.70

0 4.

46

0.92

7 1.

47

87

1 .o

6.76

(G

arne

t)

300

K r

efer

ence

tem

pera

ture

for

all

prop

ertie

s. O

px, o

rtho

pyro

xene

; Cpx

, clin

opyr

oxen

e. E

xcep

t as

note

d, s

ourc

e of

ela

stic

dal

a is

Bin

a &

Woo

d (1

987)

. " H

elff

rich

et

a1 (

l989

), (

f}t/

(lP

valu

es c

orre

cted

.

Mantle olivine content a t 400km discontinuity

Fi.gure 2 Olivine content at 400 km depth in the mantle cornputed from velocity contrast between cr and 11 (Mg,, ,Fe,, ,)2Si0, olivine. When compared with the seisn~ologically observed velocity jumps in bulk sound velocity o,,,, P velocity u,,, and S velocity us, different estimates of mantle olivine content are obtained.

pyrolitic upper mantle cornposition is suggested by both Av,,, and the upper Aup value but not by either AvS or the lower Avp value. One possible explanation for this systematic variation in compositional estimates as a function of velocity parameterization would be overestimation of (PP - P~)P,T.

MINERAL AGGREGATES

Four methods have been advanced for averaging elastic moduli in isotropic mineral aggregates. Two of these are the familiar Reuss and Voigt bounds which are derived by assuming either stress continuity or strain (dis- placement) continuity, respectively, at individual mineral interfaces. There is nothing to particularly favor either assumption, so the third method is to average the Reuss and Voigt bounds; this Hill average constitutes the well-known Voigt-Reuss-Hill (VRH) approximation. Finally, the Hashin- Shtrikman (HS) bounds seek to define the properties of the composite solid by a variational technique that incorporates shape effects, unlike the other averaging schemes. If only the elastic moduli and volumetric proportions of its constituents are known, the HS bounds appear to provide the most restrictive bounds for an isotropic composite (Watt & O'Connell 1978). These methods are all based upon the assumption

of l~omogeneous, isotropic mineral aggregates and thus are not directly applicable to studies of seismic anisotropy (e.g. Silver & Char1 1988).

All of these methods are described by Watt et a1 (1976), who also give examples of the averaging methods applied to sample mixtures. When all four bounds are computed for the same aggregate, the Reuss and Voigt bounds place broad upper and lower limits on the material properties. The HS bounds define similar but narrower limits outside of which the VRH average may occasionally lie-probably indicative of the lack of theoretical basis for the VRH average. Although the VRH average is easier to calculate, the HS bounds are not difficult to compute and should be the preferred averaging method. When the upper and lower HS bounds themselves are averaged, they give an approximate average like VRH.

All of these methods require the determination of the volumetric pro- portion 11, of each phase i of the N phases present in the aggregate:

where X, is the mole fraction of phase i in the assemblage. Formulae for tlie Reuss-, Voigt-, and Hill-averaged moduli (M,, Mv, and MH) , where M represents either K or p, are:

Those for the HS moduli KHsk and pHs+ are as follows:

EL.ASTIC PROPERTY CAL.CULATION 543

The HS formulae require sorting of the moduli: K1 and p , represent the srnallest values and KN and pN the largest. [See Watt et a1 (1976) for shape- dependent formulae for HS bounds, useful for modeling the properties of fluid-solid aggregates (e.g. Helffricl~ et a1 1989).]

For a sample calculation using this method, we will compute the temperature dependence of velocities in a pyrolitic bulk composition at 96 Icb and 1000°C. This is a useful exercise to show the magnitude of the P and S velocity increase anticipated within a slab of subducted lithosphere. A temperature decrease of about 500°C from the slab-mantle interface temperature of 1000°C is expected from thermal modeling (Helffrich et a1 1989). Table 2 lists the mineral densities, volumetric proportions, and velocities at 1 500°C, 1000°C, and 50O0C, co~riputed from the elastic data of Table 1. Note that the Hashin-Shtrilcman bounds are much tighter than the Voigt and Reuss bounds, though the VRH and HS averages are identical.

The temperature dependence based on the difference between the 1500°C and 1000°C values is - 0.56 n-~ s- ' I


dependence for P is at the low end or lower than the range used in some seismological studies (-0.5 to -0.9 m s - ' K- ' ) (e.g. Jacob 1972, Sleep 1973, Creager & Jordan 1984) but is compatible with others (e.g. Creager & Jordan 1986). These low temperature derivatives suggest that the 5- 10% Aa, changes implied by seismic waves reflected and converted at the slab-mantle interface are not entirely due to thermal effects (Helffrich et a1 1989).

CONCLUSIONS

Fundamental thermodynamic relations prescribe a method for the calculation of densities and seismic velocities at high pressures and tern- perattires. The mineralogical data required for these calculations continue to improve. Two challenges to the mineral physics and seismological communities are presented by the current state of affairs, however.

The seismological parameter that can be computed with most confidence from current mineralogical data is the bulk sound velocity v,~. This is simply due to the quality and availability of static compression data that give K , (and hence Ks). Seismologically, however, v,) is a relatively poorly determined property of the Earth, since it is not measured directly but is derived from v, and vs profiles. These are separately solved for (e.g. Jeffreys & Bullen 1988, Kennett & Engdahl1991) and may have different systematic biases leading to v,/, profiles of uncertain accuracy. Or1 the other hand, density profiles are obtained independently of body wave travel times but are directly related to v,/, since in the Earth v,; = (aP/dp), (Bullen 1963), so some consistency checking is feasible. Nonetheless, direct inversion for v,/, would supply the mineral physics community with a useful datum that could be compared with models of' the Earth's constitution.

Seisrnology already provides very useful data for mineral physics that have not been fully exploited to date: notably the us profiles, which depend upon the depth variation of a single elastic modulus, p, in addition to the density. Laboratory determination of , ~ i requires direct elastic constant or elastic velocity measurements (e.g. Gwanmesia et a1 1990, Weidner & Car leton 1977, Anderson & Goto 1989) that are technically more demand- ing than compression measurements. The biggest gap in our knowledge of p is the mattes of its pressure dependence. A concerted experimental and computational modeling effort (e.g. Cohen 1987, Isaak et a1 1990) could make us a much more valuable tool for research into Earth composition.

For the moment, the set of material properties at hand must be used, and it is much more complete than in 1968 (Anderson et a1 1968). Room pressule values of cc(T), C',(T), K, K', arid p are available for virtually all known mantle minerals. This permits v,, and p calculations to be made

ELASTIC PROPERTY CALCULATION 545

with some confidence. Perhaps the coming quarter-century will bring a further expansion in materials property knowledge that will improve our understanding of the Earth's interior as much as the first seismic wave travel time tables did for Adams arid Williamson.

C. Bina acknowledges the hospitality of Tokyo University's Geophysical Institute and the Carnegie Institution of Washington where portions of this review were prepared. Thanks also to G. Helffrich's colleagues at DTM for tolerating his closeted work habits during the final weeks of work on this manuscript and to Lars Stixrude for helpful comments.

APPENDIX: THERMODYNAMIC POTENTIALS AND FUNDAMENTAL. EQUATIONS

The thermodynamic state of a system call be described by a number of parameters (cf Callen 1960). The entropy S, internal energy U , volume V, and number of moles N , of each component i of the system are called extensive parar77eters; they scale linearly with the size of the system (is . amount of matter) under consideration. Entropy is a homogeneous first- order function of the other extensive parameters:

and is called a therr~zod~~nanzic potential by analogy with the potentials of physics. The differential quantities of the potential are its measurable properties. Equation (Al) is the fundamental equation of the thermodynamic potential and constitutes a cornplete thermodynamic description of a system. The thermodynamic potential is a state function; that is, the potential difference between any two states is independent of the path taken between those states. Given this fundamental equation, we may write the total differential of the entropy:

We now define three additional parameters for describing the thermodynamic state of a system:

where the pressure P, temperature T, and chemical poteritial p, of component i are called intensive parurizeters and do not scale with the size of the system. IJpon inserting these definitions (A.3) into the total differential


(A2), we obtain the following expression for the total differential of the entropy:

The significance of this result is that when the state of a system changes, the changes in the extensive parameters describing that system are not all independent. This provides a constraint on the mutual variation of the extensive parameters of the system, in this case volume, entropy, and composition.

Since the fundamental equation (Al) is a first-order homogeneous form, we have

1 P Pi = - U+ - V - C T N , (AS) T T ,

which is called the Euler equation for the potential S. Upon differentiating Equation (A5) and subtracting our expression (A4) for the total differential of S, we obtain

which is called the Gibbs-Dulzenz relation for the potential S. While the total differential (A4) expresses the relationships between changes in the extensive parameters describing a system, the Gibbs-Duhem relation (A6) expresses the relationships between changes in the intensive parameters of the system.

In the fundamental equation (Al) for the thermodynamic potential S, we have written the potential as a homogeneous first-order function of certain thermodynanlic parameters. Hence, the additional parameters (A3) introduced as coefficients in the total differential (A4) are homogeneous zero-order functions of these same thermodynamic parameters:

The functions (A7) are called the equatio~zs of state, and the arguments are called the natural variables of the functions. Knowledge of all three equations of state is equivalent, through the Euler equation, to complete knowledge of the fundamental equation of a system. While the equations

EL.ASTIC PROPERTY CALCUL,ATION 547

of state may be written as functions of other parameters, the natural variables have optimum informational content and are ttie only ones to permit a complete thermodynamic description of a system (Callen 1960).

Finally, the state of tlzernzodj~nar~zic equilibriunz is given by the entropy maxir~zlim prirzciple:

For example, solution of the first of Equations (A8) for equilibrium between two multi-component phases a and /j yields

which are the conditions of thermal, mechanical, and chemical equilibriunl, respectively. Solution of tlie second of Equations (A8) yields the tl~ermodynamic stahilit,y corzditions:

where dq represents the molar heat absorbed by a substance. In the above discussion, and in Equations (A])-(As), we have treated

the entropy as the thermodynamic potential, with the internal energy, volume, and mole numbers as the natural variables. However, we niay recast the entire discussion from an erztropy repr~eserztatio~z into an energy representatiorz:

The internal energy U is now the thermodynamic potential, and the total differential becomes

The equilibrium state, for example, is now given by the energy nzini~zur17 principle:

However, for both of the thermodynamic potentials S a n d TI, the natural variables are all extensive parameters. For a particular problem, it may be useful to have a potential whose natural variables include intensive


parameters. We may define numerous such therrriodynamic potentials through the technique of partial Legendse transformation. For example, we may define the Helmholtz potential F as the quantity U - TS so that the fundamental equation becomes

Table A1 Summary of'some common thermodynamic potentials"

Entropy: S = S(U,V,mi})

Equilibrium maximizes S at constant U.

Internal Energy: U = U(S,V,CNi})

iw, = TdS - PdV + &dNi

U = T S - P V + C p j N i , 0 = S d T - V d P + m i d &

T = T(S,V,{Ni}) , P = P(S,V,Wil) . Pi = &(S,V,Wil) Equilibrium minimizes U at constant S.

Helmholtz Potential: F - U - TS = F(T,V,Wi}) F = - P V + C H N , , O = S d T - V d P + m i d p i

i

S = sV,v,{Nil) , P = P(T,V,{Ni}) , IL~ = l*;(T,V,gUi}) Equilibrium - minimizes F at constant T.

Enthalpy: H z U + PV = H(S;P,{Ni})

= + [ z ] ~ , ~ , ? + F [%]st.m,, dNi = T ~ S + VdP + &mi i

Equilibrium minimizes H at constant P .

I Gibbs Potential: G - U - TS +PV = H - TS = G(T,P,Wi}) I dG * [%]Rml)d~ + [%]T,wl)dP + F [%]Tt.wl,l> dNi = -SdT + VdP + &mi i

G = C k N i , 0 = SdT- VdP + mid& i

S = S(TJ',{NiD , V = v(TJ',{NiU 9 Pi = &(TJ'*Wil) Equilibrium minimizes G at constant P and T . -

"For each potential: line I gives name, definition (if'applicable), and hndamental equation; line 2 gives total differential; line 3 gives Euler equation and Gibbs-Duhem relation; line 4 gives three equations of state associated with fundamental equation; line 5 gives equilibrium extremum principle

EL.ASTIC PROPERTY CALCULATION 549

Note that this transformation has replaced the entropy S with the temperature T as a natural variable. Similarly, we may define the enthalpy H which replaces V with P, or we may define the Gibbs potential G which replaces both S and V with T and P. Table A1 suinmarizes the name, fundamental equation, total differential, Euler equation, Gibbs-Duhem relation, equations of state, and equilibrium extremum principle for each of these common thermodynamic potentials.

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CALCIJLATION OF ELASTIC PROPERTIES FROM ...craig/publish/pdf/...The elastic properties of the Earth's interior-i.e. density and elastic wave velocities-are functions of composition,