Download calcijlation of elastic properties from thermodynamic equation of

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
I
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)
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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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 interpreted 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 it 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 empirical 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~peratureand 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~nputingelastic 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 A l ) are
where S, V , and p, are the entropy, volume, and che~nicalpotentials,
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-
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BINA & HELFFRICH
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 irzconzpressibility defined by
an 0 (lo8 bars) quantity. (Seisrriologists regrettably use cx and P for the Pand 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 AndersonGru~eisenratio (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~pressibilityPs and the adiabatic bullc 17lodulus
or adiubatic inco~npressihilityKs
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~zGriilzeise~zratio or the adiabatic second Grii~zeisenparameter iis:
532
BINA & HELFFRICH
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 macroscopic solid properties to the effects of a lattice of oscillating atoms interacting 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 compressions. 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 thirdorder truncation yields the Birch-Murnaghan equation (Birch 1938, 1939,
1952, 1978). TJse of this fourth-order expansion may be considered optimistic 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):
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BINA & HELFFRICH
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 K b 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~oximatethem as KF~'
and p; ' or ignore j 2 terms completely (Duffy & Anderson 1989, Gwanmesia et a1 1990).
'
THERMAL PARAMETERIZATION
Some important approximations greatly simplify elastic property calculatioris. The first concerns the pressure dependence of y,,. To a good
approximation (McQueen et a1 1967)
implying that
JYtho)
(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
( ~ t
= ( V/ Vo)"
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)pis constant [as well as (dp/aT)p].
I11 this sanle limit, 6, in nlany materials is observationally constant (Anderson et a1 1968, Anderson & Goto 1989, Isaak et a1 1989). Since aKTappears
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 F a r e 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~icalsimulation 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 recommends 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 deformations 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.
536
BINA & HELFFRICH
a at pressure may be obtained by straightforward integration of (9)
but this requires characterization of 6, at pressure. Some simple dependences 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,
Y vib
' = - - 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 CVwith (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
n
rn
o
.c
x
.--
1
Y
a
60
55
50
45
40
35
30
25
20
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
= 4.47 was contputed, and Sumino &
for MgO are from Fei et a1 (1991), from which 67,12,X,hl
Anderson (1984), whence g,, = 1 52
538
BINA & HELFFRICH
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 (18). 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~nately133
I<band 1500°C (Bina & Wood 1987). To calculate these values we use the
elastic data in Table 1. Mantle olivines are taken to be of (Mg, ,Fe, ,)2Si04
composition (Jeanioz & Thompson 1983), so their properties must be
interpolated fronl the tabulated end-member values. The standard practice
with elastic properties is to average on the mole fractions of the cornponents in the phase (Weidner et a1 1984). At 133 kb and 1500"C, values
~ , lc~nS- I,
for p, up, US,and u,,, in (Mg, ,Fe, ,),SiO, are 3.491 g ~ m - 8.92
4.73 1cn1 s-I, and 7.05 1cn1 s - ' for the a phase and 3.693 g ~ m - 9.62
~ , km
s- ', 5.17 lcrn s- I, and 7.55 km s- for the P phase. Comparing with the
nlantle Av,,, value of 4.9% from Bina & Wood (1987), the computed Au,~
value suggests that the mantle is 70% olivine by volume (Figure 2). The
b~ilksound velocity v,, provides the most reliable comparison of the three
velocities since it is computed entirely from volumetric material properties
that are better constrained by compression measurements. The shear
n~odulusinAuences both up and us and is known only from elastic constant
or sound speed measurements. The mutually contradictory results from
colnparison of up and v, illustrate this difference. Comparing the mantle
4.5-6% Aup and 4.6% Avs values reported in Bina & Wood (1987) with
those calculated above leads to a -- 55-75% mantle olivine content based
on matching Aup yet a -,50% olivine content based on Avs. Thus, a
'
-
8
19.20
3-71
0
0
24.72
3.347
26
33.27
29.96
3.998
84
11
17
12
d &df
xlo9 K-2
15
3.276
16.12
3.194
22.1
20.63
3.474
4.424
26.20
4.404
%WK
x106 K-l
26.1 1
P3m
&m3
3.366
KT
1.700
1.118
1.417
1.030
1.054
1.859
1.726
1.359
Mbar
1.278
4.46
2.30
2.40
2.36
2.36
4.73
4.73
5.2
5.1
&I/@
0.927
0.670
0.840
0.750
0.750
0.730
1.140
0.510
Mbar
0.810
1.47
2.30
2.40
2.36
2.36
0.62
1.82
0.62
1.82
dp@
87
154
84
180
180
146
228
130
bar K-I
309
d
5.94
6.76
1.o
7.33
4.46
4.48
5.51
4.27
5.45
3.96
~(IU)OK:
1.1
1.3
1.1
1.1
1.1
1.1
1.1
1.2
ylh
" Helffrich et a1 (l989), (f}t/(lPvalues corrected.
300 K reference temperature for all properties. Opx, orthopyroxene; Cpx, clinopyroxene. Except as noted, source of elastic dala is Bina & Wood (1987).
CaMgSi20g
(Opx* Cpx)
py73AlJ'Il16u~6a
(Garnet)
(CPX>
Mg2Si20k
(Opx* Cpx)
Fc$i206a
(OPX, CPX)
NaA1Si206a
(P)
F%Si04
(P)
Mg2Si04
(a)
F%Si04
(a)
Mg2Si04
Component
Table 1 Elastic data used in calculations
2
T!
?f
E;;
i-3
R.
k?P
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~ologicallyobserved
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 (displacement) 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 HashinShtrikman (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 proportion 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 M H ) ,where
M represents either K or p, are:
Those for the HS moduli KHskand pHs+are as follows:
543
EL.ASTIC PROPERTY CAL.CULATION
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 shapedependent 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~riputedfrom 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<- for P and -0.34 m s- K - for
S. Between 1000°C and 500°C the P figure drops to -0.44 m s- K- but
the temperature dependence of S is the same [see (38)l. The temperature
'
'
'
'
'
Table 2 Volunletric proportions of minerals in pyrolitic lnantle composilion at 96 kb"
1500°C
Phase
CPX
OPx
Garnet
Olivine
bulk P
Averages
HS
VRH
Bounds
HSHSc
Voigt
Reuss
--
-
1000°C
500°C
vol%
p
up
v,
v,,,
vol%
p
v,
v,
v,
vol%
p
3.6
22.3
15.0
59.1
3.419
3.329
3.810
3.439
3.469
8.36
7.96
8.95
8.49
4.79
4.51
4.94
4.60
6.27
6.03
6.90
6.62
3.6
221
15.1
59.2
3.470
3.415
3.845
3.488
3.525
8.66
8.31
9.16
8.75
4.96
4.73
5.04
4.78
6.50
6.26
7.08
6.79
3.6
220
15.2
59.2
3.513
3.482
3.877
3.531
3.572
up
vs
v,,
8.92
8.60
9.36
8.99
5.12
4.95
5.13
4.95
6.69
6.43
7.24
6.94
8.42 4.64 6.50
8.43 4.64 6.51
8.70 4.81 6.70
8.71 4.81 6.70
8.95 4.98 6.85
8.95 4.98 6.85
8.42
8.43
8.46
8.40
8.70
8.70
8.73
8.68
8.95
8.95
8.97
8.93
4.64
4.64
4.65
4.63
6.50
6.51
6.53
6.48
4.81
4.81
4.82
4.81
6.69
6.70
6.72
6.68
4.98
4.98
4.99
4.98
6.85
6.85
6.87
6.84
" Elastic data given in Table I . Zero pressure volumetric proportions and compositions of phases: 3 5 vol % Cpx
(Jd,,Diz,Fs,Enz,), 22 1 vol % Opx (En,,, ,Di,, ,Fe, ,), 15 0 vol ?hgarnet (Py ,,Aim,,), 59.0 vol % olivine (Fo,,Fa ,)
(Helffrich el at 1989).
544
BINA & HELFFRICH
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 510% 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 ternperattires. 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 , ~ irequires direct elastic constant or
elastic velocity measurements (e.g. Gwanmesia et a1 1990, Weidner &
Car leton 1977, Anderson & Goto 1989) that are technically more demanding 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 firstorder function of the other extensive parameters:
potential by analogy with the potentials of
and is called a therr~zod~~nanzic
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
546
BINA & HELFFRICH
(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
T
= - U+
P
Pi
V- CT
N,
T
,
-
(AS)
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~zsof 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).
equilibriunz is given by the entropy
Finally, the state of tlzernzodj~nar~zic
maxir~zlimprirzciple:
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
548
BINA & HELFFRICH
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
Summary of'some common thermodynamic potentials"
Table A1
Entropy: S = S(U,V,mi})
Equilibrium maximizes S at constant U.
Internal Energy: U = U(S,V,CNi})
iw, = TdS - PdV
U = TS-PV+CpjNi,
+ &dNi
0 = SdT-VdP+mid&
.
T = T(S,V,{Ni}) , P = P(S,V,Wil)
Pi = &(S,V,Wil)
Equilibrium minimizes U at constant S.
Helmholtz Potential: F
F = -PV+CHN,,
-
U - TS = F(T,V,Wi})
O = SdT-VdP+midpi
i
S = sV,v,{Nil) , P = P(T,V,{Ni}) , I L ~= l*;(T,V,gUi})
Equilibrium
- minimizes F at constant T.
Enthalpy: H
=
+
[z]~,~,?
-
U + PV = H(S;P,{Ni})
z
+
dNi = T ~ +
S VdP + &mi
F [%]st.m,,
i
Equilibrium minimizes H at constant P .
I
Gibbs Potential: G
dG *
[%]Rml)d~
+
U - TS +PV = H - TS = G(T,P,Wi})
[%]T,wl)dP
+
F
dNi = -SdT
[%]Tt.wl,l>
+ VdP + &mi
i
G = C k N i , 0 = SdT- VdP + mid&
i
S = S(TJ',{NiD , V = v(TJ',{NiU
Pi = &(TJ'*Wil)
Equilibrium minimizes G at constant P and T .
9
-
"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
I
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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