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GEOPHYSICAL RESEARCH LETTERS,VOL. 17, NO. 6, PAGES689-692,

MAY 1990

THERMAL EXPANSIONOF MANTLEMINERALSAT HIGH PRESSES - A THEORETICAL STUDY

BrunoReynard andGeoffreyD. Price

Geological Sciences, University College, London, GreatBritain. Abstract. Recentexperimental workhasshownthatthepressure

dependence of thethermal expansion coefficient canbeexpressed as: (a/ao)= (P/Po)'"'

description of the methodology of our computer simulations, anda discussion of our predictedvaluesfor 5r, which are in excellent

agreement with what experimental damare available.We finally

(1)

discuss thediscrepancy betweenourcalculated damandthoseobtained from seisinology.

wk•.re3r, the Anderson-Gruneisen parameter,is assumedto be

independent of pressure, andforthematerials studied hasa valuethat

Measurements and estimates of 5r

liesbetween 4 and 6. Calculationof 5r from seismicdata,however,

appears to suggest a contradictory valueof between 2 and3 for

TheAnderson-Gruneisen parameter haspreviously beeninferred for severalminerals(see Table 1) from variousexperimen•

mantle-forming phases.Using an atomistic model based on our

previously successful many-body interatomic potential set(THB1),we haveperformed calculations to obtainvaluesof 5r for four major mantle-forming minerals. Ourmodelresultsarein excellent agreement with experimental data, yielding valuesof between4 and 6 for forsterite and MgO, and valuesin the samerange for MgSiO3perovateand¾-Mg2SiO4. Moreover,the calculations confu'mthat5r is indeed constant with pressure up to thecore-mantle boundary. The apparent conflict between thevaluesof 5r predicted fromseismic data and thoseobtainedfrom experiment,and now from theory, is Introduction

An understanding of the effect of pressureon the thermal expansion of silicatesis of great geophysicalimportance,as it is required in orderto calculate mineraldensities undermanfiepressure andtemperature conditions independently from the adiabaticgradient

approaches, including: 1) measurements of theirthermalexpansion coefficient, isothermal bulkmodulus andtheirrespective temperature dependence, thatallow br to be retrieved fromequation (2) (Isaaket al.,1989a,b).

2) directmeasurement of thermalexpansion at highpressure, that allows3r to be measured directlyCYagiet al., 1987;Boehleret al., m89).

3) spectroscopic measurements at variouspressures that allow, by meansof vibrationalspectramodelling(e.g. Kieffer, 1979), the calculation of the pressure dependence of thermodynamic properties, whichChopelas (1989a,b) hasrecentlyusedto estimated the thermal expansion pressure dependence of forsteritcandMgO. D.L. Anderson(1987) has developedan independent estimation of 5r underlowermanfieconditions,from lateralvariationsof seismic wave velocities.Assumingan adiabaticgradientfor the lower manfie, andthatthelateral(i.e. isobaric)variationsof Vs andVp areonly due to temperature, he definedthe parametersg and 15s:

assumption. O.L.Anderson (1967)firstshowed thatthermalexpansion canbe relatedto densityby the equation(1). 5r is the so-called Anderson43runeisen parameter,and is definedas:

g = (alnG/alnph, = -( 8• = OlnKdalnp h, = -(1/K•)(aKdaT)•

(4)

(5)

% = (alnlQ/alnp),, whichareequivalents of the Anderson-Gruneisen parameter•r for G = -(]ac,a)(aK4aT)•

(2)

and Ks respectively.Given the observed values of lateral wave

velocityvariations(Hageret al., 1985) and the assumptions above, he obtainedvaluesof g = 5.8 to 7.0 and õs = 1.0 to 1.8. From the relationK. = Kd(l+eqr), it can be shownthat:

andis assumed to be constant withpressure. From basic thermodynamic relations,it .can be shown that (Anderson andYamamoto,1987):

(6)

5r = -01nct/91np),

(3) where7, the Gmneisen parameter, is takento be unity thusyielding

Assuming that5r is constant withpressure, andseparating variables, yieldsequation (1). Despiteits importance, thereis considerable uncertainty both concerning the valuesof the Anderson-Gruneisen parameters for manfie minerals, andconcerning the validityof the assumption that

bris essentially pressure independent. Indeed, thereis a conflict between thevalues of 5r obtained fromexperiments (seeChopelas andBoehler, 1989fora review)andthoseobtained fromseismic data

(e.g.D.L.Anderson, 1987).Thus,in thisstudywe haveusedan atomistic computer simulation approach, basedon our previously successful potential models(e.g.Priceet al., 1987a)andfree-energy

minim'..zza• tioncodePARAPOCS (Parker andPrice,!989),topredict thehighpressure andtemperature densities of fourmajormantleforming phases. Specifically, we haveattempted (1) to verifythat

valuesof br-- 2 to 3. Thus, on the one hand,there is a great consistency in the 5r

valuesfrom structural, elasticand spectroscopic measuremenu, but on the otherhand,the estimateof 5r for the lower manfie seismic data is a factor of two smaller than the experimentaldam. D.L. Anderson(1987, 1989) concludedthat the results obtained under laboratoryconditions(i.e. moderateP and T relative to the lower mantle) can not be extrapolatedto the extreme conditionsof the mantle.Thus, in an attemptto establishwhethersuch extrapolation of laboratorydata is indeedinvalid, we have performedatomistic computersimulations to calculatethe pressuredependence of thermal expansionfor major mantle-formingmineralsat temperatures and pressures directlyappropriate to mantleconditions. The atomisfic model

equation (!) holdsfor the wholemantle(i.e. that 5r is indeed

relatively insensitive topressure), and(2) to faxsome bounds onthe In order to performreliable computersimulationsof the free value of 5rforMgO,forsteritc, ¾-Mg2SiO4, andMgSiO3-perovskite. energyof a system,an accuratedescriptionof both the staticand In thefollowing, we present firsta summary of thedifferent vibrational behaviourof atomsin mineralsumctutes is required(Price values of 5r inferred fromvarious experimental measurements and et al., 1987a,b; Wolf and Bukowinski,1987). We use an approach fromtheseismic equation of state.Thisis followed by a brief

basedupontheBornmodelof solidsin whichtheinteraction between atomsis describedby a potentialof the form (Born and Huang, 1954):

Copyright1990 by the AmericanGeophysicalUnion. Paper number 89GL03736

0094-8276 / 90/ 89GL-03736503.00

wherethe first term represents the Coulombicenergy(with e the 689

69 0

Reynard andPrice:Thermal expansion at highpressures THB1 hasbeenfoundto be successful in describing thestructure and

TABLE 1. Experimental valuesof br for variousminerals.

latticedynamics of forsteritc andits high-pressure polymorphs (Price eta!., 1987a, b).

Compound

br

Someof thecalculated roomtemperature andpressure properties

6.0

5A

6.5

5.5

MgO

5.3

4.7 5.5

(4)

of the mineralsconsidered in this study,namelythe forsterite and spinelpolymorphs of Mg•SiO•,MgO and MgSiO•-perovskite, are reportedin Table3, andarecompared with the observed values. One can see that the thermal expansioncoefficientsas well as the Gruneisen parameters aresystematically underestimated Coyupto40% for a) as pointedout by Price et al. (1987b), becauseof some shortcomings in the potentialandbecauseof the inherentlimitations of the quasi-harmonic approximation usedin the energyminimization

Garnets

7.0

6.0

(2)

which is adoptedin the calculationof a. Expedmen•

T0D (D (4)

(1)

procedure. Thisis alsoa consequence of the assumption that),• =

measurements of )

0.95

obs

524.6

1.86(c) 1.84(d) 3.42

1.27(c)

calc

74.9

1.94

1.41

obs

73.8

3.12(e) 1.62(0 5.05

calc

161.3 1.35

obs

162.5

2.32

3.37

4.50

4.55

1.71(g) 2A7(h) 4.22

1.52(0

1.32 1.600)

a, quoted in Isaaket al., 1989a; b, Isaaket al., 1989a; c, Suzuki etal., 1979;d, Weidneret al., 1984;e, Suzuki,1975;f, Isaaket al., 1989b; g,

Rossand Hazen, 1989; h, Kudohet al., !987; i, estimated.

qM•

+2.0

BM•. oA

0.2945

qa

+4.0

B•.o A

0.3205

qo•,ua

-2.848

BooA

0.1490

qo•,•

+0.848

Co.oeVA6

27.88

A•.o eV

1428.5

C.• eVA•

10.66

Aao eV

!283.9

ks eVA'2

74.92

A.o eV

22764.3

ka eVra&

2.09

Chopelas, personal communication), showsit to be systematically lowerthanthethermalGmneisen parameter, ¾,•,by between 10and 20%,andtherefore givevirtually indentical results to those predicteal

fromTHB1.However, bothof theseGmneisen parameters am-by

definition related to thefrequency shiftsof thevibrational modes. As THB1 hasproved successful in modelling boththeambient (e,g.Price et al., 1987a)andhighpressure (unpublished results) lattice dynamics of mantleminerals, we aretherefore confident thatthispotential will allowaccurate prediction of thevariation withpressure of boththe Gruneisen parameter and thermalexpansion.

Predicted thermalexpansion pressure derivatives

Thestructural andthermodynamic properties of thefourquoted minerals werecalculated for different pressures overtheirstab•ity structural,elastic,and thermodynamic properties, in association with phononfrequencies, are calculatedat varioussimulated pressures and temperatures usingenergyminimization proc•ures availablevia the codePARAPOCS(ParkerandPrice,!989). Of particularrelevanceto this study, the thermal expansiona is calculatedself-consistently duringthe energyminimisationprocedurefrom: a = T•Cv/K,V

(•0)

whereTa is estimatedfrom the averagemodeGmneisenparameter:

= •(Cv•)/Cv

(!!).

fieldsalongtwoisotherms, (1) at room-temperature (i.e.below the Debye temperature), and(2)at 1200K forMg2SiO• polymorphs and 2000KforMgOandMgSiO•-perovskite, in order toremain farabove theirrespective Debye temperatures evenat veryhighpressure. At suchtemperatures, intrinsic anharmonic effects areimportant atlow

pressures butaremostly supressed at highpressure (Hardy, 1980). Therefore, wemayexpect thequasi-harmonic approximation tobe validunder such highpressure conditions. Thecalculated ratios of od• to V/Voarereported onfigure 1 andderived values of•r for the four simulated minerals are listed in Table 4.

Thepredicted values arein verygoodagreement withthe experimental data forforsteritc andMgO.Wefindthesame trends

Reynard andPrice: Thermal expansion athigh pressures

,

,

.3 !

.2

.

!

SiO4 ß

g-Mg2SiO 4 MgO $5

p-!VlgSiO 3 ß

691

widerange oftemperature (Ross andHazen, 1989). Furthermore, using an approach basedon the modifiedelectron-gas theory,Woff and

Bukowinski (1987)havepresented the simulated prope•esof MgSiO•-perovskite at highpressures and temperatures. Froman analysis of theirdata,weinferthattheircalculations predict a similar Anderson-Gruneisen parameter, confirming that in spiteof some apparently minorshort-comings anditsempirical nature, thepotential setTHB1appears to be remarkably robustandhigMyversitile in modelling a widerangeof perfect(e.gPriceet al., 1987a,b) and defect properties (e.g.WallandPrice,1989;WrightandPrice,!989) of mineaais at highpressures andtemperatures. Discussion and Conclusion

Chope!as andBoehler (1989)havestressed that,fromexperimental evidence, thepressure dependence of thermal expansion, expressed as

.9O

the Anderson-Gruneisen parameter fir, lies between4 and6 for oxide

mineralsunder manfie conditions.Using atomisticsimulation techniques, we haveconfirmed theseresults forforsteritc andMgO, and shownthat •'-Mg:SiO4and MgSiO3-perovskite, for which experimental valueswerelacking,behavesimilarly. Ourcalculations

showthat•r canindeed beassumed independent of pressure, upto at leastthecore-manfie boundary conditions, and•s by about

.85.

one abovethe DebyetemperaS, thusremainingin the observed

experimental rangeof 4 to 6 overthewholemanfie. Theimplications for thelowermantlecomposition of • beingin therange4-6 have beendiscussed by Chopelas andBoehler(1989),whoshowed thatfor sucha situationa pyroliticcomposition manfie(2 perovskite+ 1 magnesiowustite) couldaccountfor the densitiesof the PREM model

.80'

Fig.I. Log-log plotof thecalculated •ch vs.V/V0ratios.Thelines correspond to differentvaluesof •r as indicated. The stippled area corresponds to 5, valuesinferredfromthe seismic equation of state of the lower manfie (D.L. Anderson,1987). Filled symbols:T0D

forthecalculations as for the experiments, namely,(1) fir is fairly independent of pressure, and (2) • decreases by a valueof oneto twoabovethe Debye temperature, but remainsin the experimental rangeof 4 to 6 under manfie conditions. In addition,we confn'mthat the Anderson-Gmneisen parameter

canbe assumed to be constantwith pressure. up to the core-mantle

boundary, so thatequation(1) holdsfor the wholemantle.Finally, THB1allowstheprediction of 8r for high-pressure polymorphs, such as •Mg:SiO4 and MgSiO•-perovskite,for which metastability

(especially for MgSiO•perovskite) precludes structural studies overa

TABLE4. Calculated valuesof the Anderson-Gruneisen parameter atvarious pressures andtemperatures. Fo T(K)

300

1200

MgO

Sp 300

1200

30O

20O0

Pv 30O

2000

7.6

6.5

7.5

6.3

6.3

5.9

100

7.6

6.4

7.9

6.1

6.2

5.6

8.0

7.0

200

7.8

6.2

8.1

6.0

6.1

5.3

8.0

7.0

400

....

5.9

5.0

8.1

6.6

5.8

4.7

8.7

6.3

5.9

4.6

10.1

6.5

(Dziewonski andAnderson, 1981)belowthe 6701can-discontinuity. Havingcontinnedthe findingsof the experimentists, it is now essential to try to findaninterpretation of D.L. Anderson's seismically inferredvaluesof theAnderson-Gmneisen parameter. It mustbe noted thattheestimate givenby D.L. Andersonis basedon theassumptions that(1) thegeothermal gradientis adiabaticin thelowermantle (i.e. no accountis givenfor possiblesuperadiabaticity), and (2) thelateral seismicvelocityvariationsare due only to temperauric variations.It is widelyheldthat in the uppermantle,the discrepancy betweenthe "seismic" andexperimental $v mightbe explained by partialmelting, whichstrongly affectstherigidity(i.e. V•) andweaklyaffetc.•the bulk modulus(i.e. V•,), yieldingan apparentlylow • The fact that in the lower mantlepartial melting is unlikely, led D.L Andersonto describethe Vs and Vp isobaricvariation as a temperatureeffect. Nevertheless, in additionto changingthe elasticpropertiesof the lower manfieminerals(pemvskiteand Mg-wusti•), temperature can alsoaffectthecomposition of the coexisting phases(differentFe4•g equilibrium constant) and/ortheirstruck. Variationsof FMg ratios have been shownto have a strongdiffeaentialeffect on the rigidity and bulk modulus for the (Mg,Fe)O solid solution (Sum• and Anderson,1984), and could thus explain the greater variationin rigidity (Vs) than in bulk modulus (Vv) in an isochemicallower manfie. Unfortunately,no data are available for the (Mg,Fe)SiO,perovskitesolid solution,and the temperaturedependenceof the equilibrium constant between(Mg,Fe)Oand(Mg,Fe)SiO,is notclearly elucidated(Guyot et al., 1989). An alternativeexplanationis that orthorhombicMgSiO3-txaovskiteundergoesa ferroelasticphase transitionat high-temperature, transformingto a higher symmetry tetmgonalor cubic polymorphCYeganeh-Haefi et al., 1987). This wouldalsoresultin a largevariationin rigidity,comparedwith bulk modulus,in the vicinity of the transitionboundary,leading to the

observedhigh OlnVdalnVu)• ratios. Further expeximentsand simulations are neededto establishwhethereither of theseproposals are a viablealternativeinterpretation of the apparentlyanomalous Vc V•, systematics of the lowermanfie. Acknowldgements. The authors wishto thankNancyRoss,RossAngel and RichardCatlowfor their helpfulcommentsand lively discussion. Dr Ann Chopelas is particularly thankedfor stimulating this line of research.Two anonymousreferees are aknowledgedfor fruitful comments.The supportof NERC grant GR3/6356 is gratefully acknowledged. References

1200

-

-

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(ReceivedOctober26, 1989; acceptedDecember18, 1989.)