Confidence Limits for Silicate Perovskite Equations of State Craig R

Phys Chem Minerals (1995) 22:375-382

Confidence Limits for Silicate Perovskite Equations of State Craig R. Bina Departrnelit of Geological Sc~ences.Northnestern Un~iers~tj.1817 Sher~danRoad. E~anston.IL 60208. USA (Fax i1-708-191-8060. Ernall cra~gfrrearth nwu edu)

Recelled January IS. 1995 Revlsed. accepted Aprll 13. 1995

Abstract. We examine thermoelastic equations of state for element composition from a pyrolite-like upper mantle silicate perovskite based on data from recent static com- (Jackson 1983; Knittle et al. 1986: Bina and Silver 1990: pressiori studies. We analyze trade-offs among the fitting Zhao and Anderson 1994). However, (Mg, Fe)SiO, per- parameters arid examine data sets for possible effects of ovskite is thermodynamically stable only at high pres- metastability and of Mg-Fe solid solution. Significant sures (> 24 GPa), and lriost experiinental studies of this differences are found between equations of' state based on phase have been perforrried outside of its stability field, low pressure measurements obtained for perovskite out- thus necessitating significant extrapolation to lower side of its stability field. I~icreasinglyconsistent results rriantle conditions. Despite experimental difficulties in are obtained when higller pressure data are used, despite studying perovskite, increasingly accurate rneasuremerits differences in the zero-pressure pararneters used to de- have been performed in recent years, and there is now scribe the equatio~~sof state. The results highligllt the good agreement alriong various groups 011 several of its inlportance of measuring the thermoelastic properties of ttierrnoelastic properties (i.e., KT,,K;,. and to a lesser perovskite at high pressure, specifically within its stabili- extent r,, the tliermal expansivity under moderate pres- ty field, and the potential problerns associated with large sures of 20-30 GPa) (Jeanloz and Hemley 1994). extrapolatioris of equations of state in the analysis of Continued experimental study of the tlrermoelastic seismic data. properties of silicate perovskites is needed over a wider rarige of P- T conditions. The significant pressure dependence of thermal expansivity determined by Mao Introduction et al. (1991) results in lower mantle densities for the perovskite which are close to those determined seismologi- Constraints upon the composition and milleralogy of cally, a result that suggests silica (and possibly iron) en- the Earth's mantle can be obtained by comparing the richment in the lower mantle (Stixrude et al. 1992; Hem- profiles of density and velocity which result from seis~no- ley et al. 1992: cf. Knittle et al. 1986; Jeanloz and Knittle logical studies with the elastic properties of candidate 1989). Iri contrast, other rneasurernents carried out at mineral assemblages under mantle conditions (Birch lower pressures find a loiver thermal expa~~sivityat low 1952). Measurements of elastic properties of minerals car1 pressures and a very weak pressure dependence, suggest- now be performed over an increasingly large P - T range ing that a pyrolite-like upper rrialitle composition fits of the Earth's interior. For determining properties at the lower mantle density profile (e.g., Wang et al. 19911 very high P- T conditions, measured elastic properties 1994). Here we examine this issue by analysis of these of mirierals must be modeled with suitable equations and more recent data (Funarnori and Yagi 1993: Yagi of state (Bina and Helffricli 1992). et al. 1993; Fei, unpublished). We study the trade-offs The dosriinant minerals in the lower mantle are in the best-fit parameters and use the results to assess believed to be ferromagnesian silicate perovskite the validity of various measurements. We show explicitly (Mg, Fe)SiO, arid magnesiowustite (Mg, Fe)O. The equa- the relationship arnong the pararneters used in different tion of' state for magnesiowustite is reasonably well con- approaches (e.g., q and 8,). strained (Fei et al. 1992; Hemley et al. 1992). in part because this phase is stable from ambient to lower mantle conditions. The behavior of perovskite at high P and Method T is of particular interest. since it is this behavior that determines whether or not seismic velocity arid density We are interested in an equation of state which relates profiles require the lower mantle to be distinct in major pressure P, volume and temperature T (Zharkov and Kalinill 1971). 7he pressure can be expressed as a fuiic- whele 11 is the nuniber of atoms per form~ilaunit and tion of vol~lnieand teniperatule sirnply by e\iparidirig R is the gas constant. The Debye temperature arid GI

Integrating the volulrie derivative first? at a reference in terrrls of their values Q,, arid ;,, at the reference ~ol- tenlperature To. gives the "cold" or roO~ll-temperatLll'e ume If, al1d the value of tile additiollal parameter q. corriponent of the pressure. and evaluating the tempera- Implicit ill the quasi-~larsllos~icapproximatioll [Eqs. (5)- ture derivative second, at the volulrie of interest I/: gives (7)] is that ?, (alld hence 8,) is illdependelit of Second- the "thermal'. pressure: ary assumptiolls not fuildamental to the theory. implicit in Eq. (S)?are that ;!, has a Vdependence similar in f'or~n to that of 0, and that q is independent of I/: Alternati~es to these secondary assumptions are the subject of'ongo- irig study. Pco~d + cherrnal . (2) Given a set of experirriental data points (c. I<. Ti). \vhere is the isotllerrnal bulk rnodLllus.;! is the Grun- we co~~structa best-tit equation of state by ernployilia eiseli parameter. C,. is the isochoric heat capacity. and the do\vnhill simplex method (Press et al. 1986) to solve I/; is the reference volulne corresponding to the reference for soille subset of tlie parameters {KT,, K;,. 0,". yDi,. te,nperatLlre To at the pressure P=o (Note that tile inte- q) which minimizes tlie mean-square misfit between the grand for the thermal pressure terrri can be written equi- calculated and measured pressures. The thermal pressure is most sensitive to 7, [Eq. (31. valently as rK,. where :! is the volurrie coefficient of thermal expansion.) Si~icewe wish to use our equation and hence to ;,, and q [Eq. (S)]. While previous studies of state for estrapolatioll as well as for interpolation, have shown that P- If- T data alone do not tightly con- we lll~st fullctiollal forlns that reflect ~trai"OD, (Hemley et al" 1992): as collfirmed by the tra"- the underlyirig microscopic physics of the materials. off analyses described below, additioiial constraints upon For the roorn-temperature isothern~al equation of Q,, often can be obtained from vibrational. thermochem- state. we employ the third-order. Birch-Murnaghan equa- ical, or phase equilibriun~data. tion from Eulerian finite strain theory (e.g., Birch 1952): Thermal equations of state are sonletirnes expressed in terms Pcold=3KT0j'(l+2 f')' '(1-< f). of the thermal expansivity 2 and the Anderson-Griiiieisen pilrame- (') tcr 8, (giving the pressure dependence of 2). rather than in terms

of ;% and (1. The relationship between these t\vo alternative represen- Here the strain variable f and the coefticie~lt< are given tations is gi\,en by: YK,I' -. ------19) ~E$[(V~/I/)'~-~],5- -$(K;,-4). (4) c,. where KT, and K;, denote the zero-pressure isotliermal bulk modulus and its pressure derivative. Equation of State Analysis For the thermal pressure, we ernploy the Mie-Griill- eisen-Debpe equation based on quasi-harmonic lattice We first employ the ibr the room-tempera- vibrations (Zharkov and Kalinin 1971). This approxinla- ture eqLlatioll of state [Eqs. (3H4)1to solve for KT, tion includes contributions from optical inodes as ail K;,. using recent static data at To=300K effective high-frequency cutoff for the acoustic modes. (Fig. (Kllittle and Jeanloz 1987: et al. 1991: Frlll- Although experiments and theory show that the density amori and Yagi 1993: Yagi al. 1993: Fei, of vibrational modes of perovskite is nlallifestly non- The are shon;n in Fig" 2a>R,hich illustrates the Deb~e-like(Hemley 'ohell 1992), the thermoelastic wel~-kllowlltrade-off (e.g.? Bell et al. 1987) betwees1 the properties of many such niaterials nonetheless can be best-fit values of StatisticalF-tests well represented over a wide range of temperatures with one- tvvo-parameter fits (Lirldgren et al. 1978) dem- such a model ie.g.. Jeallloz and Knittie 1989: He~nley ollstrate that K;, is illdistillguishable horn its null-lly- et al 1992: Anderson et a1 1992: Stixrude and Hemley. pothesis of at 95% confidence leuel. Fixing unpublished). I11 this approsirnation: we write K;, at 4 and erriploying either. all of the To experimental data or a variety of subsets of the data yields best-fit KT, values which range fiom 261 to 263 GPa: in good (') agreement with independent Brilloilin spectroscopy mea- surernents (Weidner et al. 1993). Here the therinal energy E,,, is given by: In contrast to this agreement fbr the room-temperature equation of state, different experimental results have (6) been reported for the thermal expansion at low pressures (cf: Jeariloz and Hemlep 1994). In particular, the data 120. - , , , , I , , 111111'1111'- - KJ - - M - 100 - LF - - FY - - YFU - - WWL - 80 ------a0 - - 0 - - - - 5 60-- - m - - E a - - GO - - - XXXXXX XXXXXXXXX XXX - - - Fig. 1. Experiinental ddta sets 111 pressure-tempera- - - ture space References are KJ (Knlttle and Jeanloz 20 - 0~~080g~~0~~00000~o~ - 1987). M (Mao et al 1991). FY (Funamor1 and Yagi - 000000000 - 9 000000000 1993). YFU (Yagl et al 1993). F (Fa. unpublished). - :,4" gg*Oi&&pb&:;$ Q f 2 * - g LVWL (\\rang et a1 1994) L)ortetl 11r1eshons ~lmenlte- - ot Q 4 4 ~q o2o*oao 2. .* * - [email protected]/IiII~IIII~lIIlpero\shlte phase boundar) 0 500 1000 1500 2000 Temperature (K)

Table 1. Best-fit thermal equatlon of state parameters for p\-(Mg. Fe)S103 Isother- Line Data Sets Poiilts tlDo (K) s Dri (1 mal equatlon of state parameters h,,= 261 GPa and h;,=4 Data set references KJ (Kruttle and Jearlloz 1987). M (hlao et al 1991). FY (Funamor1 and Yap1 1993). YFU (Yag~et al 1993). F (Fel. un- (Fe-free samples) publ~shed).LVWL (\\Jang et al 19941 FY +YFU FY +YFU +\\/WL

(highest-P only) F Y of Wang et al. (1994) show an u~lusuallyweak pressure tions in Fig. 2 are all show11with the same contour levels. dependence of the thermal expansivity in comparison it is evident that the trade-off between OD, and q in to the results of Mao et al. (1991) and Fu~lamoriand Fig. 2c is the least well co~lstrained.Employi~lg these Yagi (1993). Accordingly, we have chose11 to perfhrrri esperime~italdata yields best-fit values of 1480 K for O,,, several separate analyses in order to esplore the differ- 2.3 for ;,,. and 4.9 for q (Table 1. line 1). Stisrude et al. ences in the equation-of-state fits. Note tliat differences (1992) and Herriley et al. (1992) fit a li~llitedset of these among the various data sets are not strictly associated data: using inforrnatio11 on phase boundaries as an addi- with rand0111 errors: instead: there appear to be irnpor- tional constraint they obtained OD, 2 1100 K but similar- tarit systematic dif'fere~lcesthat have bee11 diff'icult to es- ly high values for ;!,, and q. These uiiusually large values tablish experimentally. Hence. these fits may provide in- for y,, and q translate [Eq. (9)] illto a strongly pressure- sight into possible systematic errors. dependent volume coefficient of' thermal expansion r Adopting a 300 K isothermal equation of state cliar- (Fig. 3a) or. equivalently. a large Anderson-Griineisen acterized by KT, of 261 GPa and K;, of 4: we incorpor- parameter 6, (Fig. 3 b). ate the thermal pressure fbrmulatio~l[Eqs. (5)-(S)] to We perforrried an additional set of' fits illcludir~gthe sol\;e for OD,, y,,. and q. using recent static compressio~l data of Wang et al. (1994). I~lclusior~of this data set data at a \;ariety of temperatures (Fig. 1) (Knittle and yields best-fit values of 1060 K for OD,, 1.5 for ;:,,: arid Jeanloz 1987: Mao et al. 1991: Funarnori and Yagi 1993: 1.9 fbr q (Table 1. line 2). However, statistical F-tests be- Yagi et al. 1993: Fei. unpublished). ?'he results are shown tween two- and three-parameter fits demonstrate that in Figs. 2 b-d, wliich illustrate the trade-offs between OD, q is now indistinguishable from its null-hypothesis value and y,,. between OD, and q. and between ;',, and q. of' 1 at the 95% corlfidence level. As shown in Fig. 4c, Statistical F-tests between two- and three-parameter fits the primary effect of incorporating this data set is to demonstrate that q differs from its null-liypothesis value deform the trade-off ellipsoids between OD, and q toward of 1 at the 99% confidence level. Since the ellipsoid sec- s~nallervalues of q, thus sir~~ultaneouslyyielding srrlaller 500 1000 1500 2000 2500 1 2 3 c Bo(K)- d YO- 1 ' 1 Fig. 2a-d. Contours of RMS misfit showing sections through five- is given by - 7 - (JE'G~)' for d &=&"@'(I/:. Ti)-~c"'c(l<. Ti) dimensional hyper-ellipsoid about best-fit equation of state (Ta- !V iz2 ble 1, line 1) Four panels give conlpletc illustration of parameter with G~ arbitrarily fixed at 0.1 GPa trade-offs. Contour values arc of' where the normal estimator

values of 7," through the trade-off betweell yDo arid q (1994) data alone is too Ilalrow to constrain simulta- slio\vn in Fig. 4d. Fixing q at 1 and employing all of' neously OD,, and q. However, this data set can be the experinlental data now yields best-fit values of 950 K cornbi~iednit11 those of Funamori and Yagi (1993) and for OD, and 1.4 for y,,, (Table 1. line 3). 'These sn~aller Yagi et al (1993). as discussed in the analysis of Fe-free values for ;,, and q translate [Eq. (9)] into a less strongly pe~ovskitebelow. to yield a similar low-;,, equation of pressure-dependent volume coefficient of thermal expan- state (Table 1. line 5). sion r (Fig. 3c) or. equivalently. a snialler Andersoii- Griineisen parameter 6, (Fig. 3d) closer to a value of -4. The significant change in equation of state parame- Discussion ters resulting from incorporation of the Warig et al. (1994) data invites arialysis of this data set in isolation. We begin by considel.i~lgpossible effects of metastability but the range of pressures spanned by the Wang et al. upon measurements carried out at low pressures. First, Lirie 1 pv-MgSi03 EoS Line 1 pv- MgSi03 EoS

4.5 :

4.0

3.5 P=5OGPa 7.5 :

6.5 6.0 c 5.5 : - 5.0 - 4.5 f

0 ~"""""""""~~4"~-~""""""~~"~~~' Line 2 pv-MgSi03 EoS Line 2 pv-MgSi03 EoS ~.O.~-T(~I~I~I*I~I~I~I*I~:~1~1~1~161~1*~~~~'"'~ - C d - 9.5 : 4.5 r 9.0 I 4.0 : 8.5 3.5 r ...- 3.0 :

6.5

1.0 : - P=~~~~~----P:~~GP~"=~,LoGP~- - L~=50GPa-~z5~~~a 0.5 , 4.5 = ,l,l,ltlll~~l~~~ll~l~ 0 "'"""""''1'1'1 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 'Temperature (K) Temperature (K)

Fig. 3a-d. Volume coefficient of tilerma1 expansion r and isother ere for the equation or state of Fig 2: panels c and d arc for that ma1 Anderson-Griineisen parameter dT versus pressure and temper- of Fig 4 ature as derived frorn best-fit equatiorls of state. Panels a and b it is slow well established that prolonged heating of per- elastic or optic mode) instabilit)? in structure in the meta- ovskite at pressures below those of its stability field re- stable state (Hemley and Cohen 1992). In this case, de- sults in degeneration of the sample. so that measured composition need not be invoked and the thermal expan- elastic properties in some experiments rnay be influenced sivity will be reversible: it will be high as a result of by partial amorphizatio~~.back-transformation to lower increasing anharmonicity analogous to the increasing pressure phases, or change in Mg/(Mg+Fe) ratio (cf. thermal expansivity of materials near melting (Ubbe- Hemley and Cohen 1992). While the therrllal expansivity lohde 1965). 'Third, if the phase is close to an intrinsic is reported to be reversible as a function of temperature instability in its thermodynamically metastable state, its in some studies. this does not prove that decomposition structure may be highly sensitive to non-hydrostatic of the perovskite has not occurred and has not signifi- stresses which, for example. could drive metastable (and cantly affected the measured expansivity. Second. the difficult to reproduce) crystalline-crystalline transitions. thermal expansivity could be anorrialously high at low The latter phenorne~lonwould conlpromise any determi- pressures because the phase is close to an intrinsic (e.g.. nation of the true thermal expansivity of the material the aggregate from the introduction of defects into the Conclusions perovskite or a lower therrnal expansivity fioln a com- I pressive stress exerted by the glass on the perovskite Metastability of' silicate perovskite at low pressures and (indeed, the latter is suggested by the reduction in zero- the effects of Mg- Fe solid solution appear to contribute pressure volurile after ternperatuse cycling (Wang et al. to the scatter in reported thermal expansi.irity data well 1991. 1994: Fei. unpublished). below its stability field. Resi~ltsfrom fitting the lowest Iii view of'the possible problerns with lower pressure pressure data suggest the need for rnore flexible thermal data. \ve explored the consistency of the data at higher equations of' state than those used in con~:entio~laltreat- pressure by dowriweighting individi~aldata poirits ac- uients. and care shoi~ldbe exercised in extrapolating such cordirig to their distance from the perovskite stability equations of state well beyond the measured data regime. field. We redetermined best-fit equations of state using One alternative would be to adopt a for~nulatio~lnfliich the origi~laldata set (Knittle and Jeanloz 1987: Mao allows (1 to fall from high to low values with increasi~ig et al. 1991 : Funanrlori and Yagi 1993: Yagi et al. 1993: cornpression (cf: Anderson et al. 1993: Stacey 1994). rath- Fei. ullpublished) but weighting each point by the recip- er that1 assun~ingconstant q. Another alternative would rocal of' 11o.cv fir the experimelltal pressure falls be lo\\^ be to employ a high pressure (instead of zero-pressure) the ilrnel~ite-perovskitephase boundary (Fig. 1) (Ito and reference state it1 existi~igequation-of-state f~inctions. Takahashi 1989). This approach results in values of I~icreasitiglycollsistent results for the thern~oelasticity 1190 K for @,,. 1.7 fbr and 2.4 for q (Table 1, line 6). of (Mg. Fe)SiO, perovskites are obtained when higher sit~iilarto values obtalned previously (Stixrude et al. pressure data are used. ?'he data of Mao et al. (1991) 1992: Hemley et al. 1992). Statistical F-tests between arid Funamori and Yagi (1993) together \\lit11 the highest two- and three-parameter fits indicate that q is distin- pressure data of' Wang et al. (1994) yield a consistent guishable fro~llits siull-hypothesis \!slue of 1 at the 95% high-pressure therlnal espansivity (despite difrere~lcesit1 but not the 990h confidence level, a marginal result. Fix- zero-pressure parameters) at pressures characteristic of ing q at 1 yields 1100 K fbr @,, and 1.5 for y,, (Table 1: the top of the lo~vernlantle (cf. Anderson arid Masuda line 7), compared to 1030 K for OD, and 1.5 for y,, upon 1994). 111 particular. the current data sets suggest r i~iclusionof the data of Wang et al. (1994) fbr which 2 2.0 x 10-j K-' at 20-30 GPa (cf. Jeanloz and Helnley q is indistinguishable from 1 at the 95% confidence level 1994) i11 the temperature range 700-1000 K (Fig. 3). (Table 1. line8). Thus. dov\lnweightillg of' ~lletastable While high-;!,,, equations of' state derived from per- measurelrients also yields low-;!,, equations of state, re- ovskite thermoelastic data suggest silica enrichment in sulting in convergence between equations derived for order to match to seismological density and velocity pro- data sets with and without the data of Wang et al. (1994). files at the top of the lower mantle. co~isideration of Alternatively. we can sirnultarieously minimize both nletastability ar~d,'orMg- Fe solid solution effects yields the effects of riletastability and the necessity for estrapo- lo~v-y,, equatiolls of state collsistent witli a pyrolite latio~iby co~lsideringonly those data points closest to lower tnantle (cf: A~ldersonet al. 1995). These results il- lower mantle conditions, those at the highest pressures lustrate the importance of making comparisons in re- (36 GPa) and temperatures ( > 1500 K) (Funamori and gions. such as the upper portion of the lower mantle. Yagi 1993). Although such a data set is too srnall to where rninirnal extrapolation is required and where constrain OD, (cf. Hemley et al. 1992) and q, values of problerns with ~netastabilitycan be avoided bp co~lduct- 1100 K and 1, respectively, yield 7," of 1.5, rising to 1.6 irlg ~xleasurementswithin the stability field of the perovs- for OD, of 1600 K (Table 1. line 9). This approach. too. kite. That such tech~liqueshave recently becolne avail- yields low-yDoequations of' state. able bodes well fbr irriproving cor~lpositionalconstraints Moreover. the latter analysis yields the density of the on the lower ~nantle. perovskite directly at pressures near 36 GPa and over ~I~~krio~~letl,oe~~i~'~~isI tliarlk Rus Hemley for helpful input tlirough- a temperature range relevant to the lower mantle. Corn- out this project I am grateful to Y. Fei. RC Liebermann. and parison to a seismological density profile (Uziewonski T Yagi for generously providing data prior to publication. I thank et al. 1975) at these pressures (assuming &cO = 2000 K 0. Anderson. U Isaak. I Jackson. A Na1,rotsk.. 'T. Shankland, at the top of'a lower mantle adiabat) yields optinrial Mg/ F. Stacey. L Stisrude. Y LVang. D. ilieidner. and an anonymous revie~verfor helpful discussioris and cornnients. I ackno\\,lcdge the (Mgi-Fe) of 0.89, requiring no Fe-enrichment relative support of National Science Foundation grant EAR-9158594. to a pyrolite upper mantle. On the other hand. constraints upon the Si/(Si + Mg + Fe) ratio require cornpar- ison to the bulk modulus or seisn~ologicalbulk sound References velocity (Kennett and Engdahl 1991). Such analyses I based on the high-?,, equations of' state predict Si/ Anderson OL. Masuda K (1994) A therniodynan~icmethod for (Si i-Mg + Fe) > 0.8: indicating Si-enrichment relative to computing thernial esparisivity. 2. versus 7' along isobars at pyrolite (Hemley et al. 1992: Stixrude et al. 1992). Com- various pressures for silicate perovskite Phys Earth Planet Inter positions closer to pyrolite, fbr which Si/ 85:227--236 Anderson OL. Oda H. Chopelas A. 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