GEOPHYSICAL
RESEARCH
LETTERS, VOL. 16, NO. 12, PAGES1403-1406, DECEMBER
1989
COMPRESSION OF TETRAHEDRALLY BONDED SIO2 LIQUID AND
SILICATE LIQUID-CRYSTAL DENSITY INVERSION
Lars Stixrude and Mark S. T. Bukowinski
Department
of GeologyandGeophysics,
Universityof California,Berkeley
Abstract.We haveinvestigated
theresponse
to pressureof
liquid SiO2 by performinga quantitativelyrealisticMonte
Carlo simulation. The modelliquid was restrictedto at most
four-foldSi-O coordination
by theeffectiveimposition
of an
infinite potentialbarrierto a fifth bond. We thusobtainedan
unambiguous
comparisonof the compression
mechanisms
of
solidand liquid tetrahedralnetworks. In spiteof thisrestriction,thedensityof the simulatedliquidexceedsthatof thecorrespondingmodelsof quartz,coesiteand cristobaliteat high
pressure.The efficientcompression
of the liquidresultsfrom
a continuousrestructuringof the networkthat leavesthe mean
Si-Si distancevirtuallyunchanged
anddoesnotrequirean increasein the coordinationnumber. The restructuringis effectedby localbreakingandreconnecting
of bonds,a mechanismthatis not availableto a perfectcrystal.
Introduction
The evolutionof theterrestrialplanetsdepends
to a largeextenton thethermodynamic
propertiesof silicateliquids,which
mediatechemicaldifferentiationprocesses.In particular,the
recentsuggestion
thatthe densityof silicateliquidsmay surpassthat of coexistingcrystalsat pressuresof 5-20 GPa has
importantconsequences
for terrestrialchemical evolution
(Rigdenet al., 1984). For instance,magmaformedbelow a
few hundredkm in the Earth would never reach the surface,
andchemicaldifferentiationwouldproceedby the sinkingof
magmato greaterdepths.
It is a commonassumption
that the mechanismresponsible
for the densityinversionsis a pressure-induced
increasein the
A1-O andSi-O coordination(e.g., Rigdenet al., 1984). There
is nowgoodevidencethatsuchcoordination
changes
do occur
(Xue et al., 1989). However,the flexibility of liquids'structuressuggests
thatliquidscontainingtetrahedral
networkscan
respondto pressurefar moreeffectivelythancrystalsby simply rearrangingthe connectivityof the networks,evenwhen
no coordination
changesoccur. The r.ole of thisnetworkrestructuring
in liquid-crystal
densityinversions
hasnotyetbeen
demonstrates
that, at high pressures,liquids can adoptinherentlydensernetworkstructuresthan crystalsof the same
Si-O coordination.
The Model
Monte Carlo (and moleculardynamics)simulationsof condensedmatter are generallybasedon a modelof interatomic
forces. How faithfully the model describsreal interatomic
forcesdetermineshow faithfully the simulationsreproduce
knownpropertiesof the real material. Previoussimulationsof
silicateliquidshavemostlyusedsimpleionicmodels(Kubicki
and Lasaga, 1988; Erikson and Hostetler, 1987, and
referencestherein). Despitethe demonstrated
utility of these
simulations,
the potentialsusedare not suitablefor our purposessincethey havenot beenshownto reproduceavailable
dataon the corresponding
crystals. The bulk moduliof SiO2
crystalspredictedby ionicmodelsaretoolargeby asmuchas
an orderof magnitude(EfiksonandHostetler,1987).
We basedthe simulations
on a recentlydevelopedcovalent
modelof tetrahedralSi-O bonding(StixrudeandBukowinski,
1988). The modelincorporates
essentialphysics,including
theshortrangeanddirectionaldependence
of thecovalentSi-O
bond,the dichotomybetweenstrongintra- andweak inter-te-
trahedralforces,anda formof theinter-tetrahedral
forcesuggestedby molecular orbital calculations. Furthermore,the
modelis quantitativelyaccurate:thesearethefarstsimulations
of a liquid silicateto usea potentialmodelwhichreproduces
the equationof state,structure,compression
mechanisms
and
phasestabilityof the corresponding
tetrahedralcrystalline
phases
(Stixrude
andBukowinskiii•:•'O88).
Themodel
potential
energy
is•i':(/en
by:
Nsi 4
U=i=l5•j=l5•fc(rij)D{exp[2[•(rij-r0)]-2exp
Nsi
4 5•
4fc[max(rij,rik)]
+Z
Z
12kct
•r•
(0tijk
-ct0)2
i=l j=l k>j
examined.
We reporthere the resultsof a Monte Carlo simulationof
liquidsilicabasedon an accuratecovalentpotentialmodelof
tetrahedrally
coordinated
SiO2phases(StixrudeandBukowinski, 1988). We find thattheliquid'sdensityovertakesthatof
all the crystalsby 50 GPa. Althoughthe comparison
is not
with thermodyn-amically
coexistingcrystals,it nevertheless
No
2fc[max(rij,rik)]•kL(rjk-L
1
+Y.
•2 Y.
0)2
i-1 j=l k>j
No No
+i=lZ j>iZ Aexp(-rij/b
)
(1)
and
fc={1+exp[y(rij
-rc)]}
-1
(2)
whereNsi andNOarethenumberof siliconandoxygenatoms
Copyright
Paper
1989 by the American
number
89GL03439.
0094-8276 / 89 / 89GL-03439503.00
Geophysical
Union.
respectively,
rij is thedistance
between
atomsi andj, ctijkis
the intra-tetrahedralO-Si-O angle,and the multiple summations of the first three terms indicate that at most the four near-
estoxygensabouta siliconand the two nearestsiliconsabout
1403
1404
Stixrude
andBukowinski:
Compression
of SiO2Liquid
an oxygen are included. The model is identical to the ORM
modelof Stixrudeand Bukowinski(1988) exceptfor the
function
fc(rij),whichintroduces
a smooth
cutoffof thethree
covalent forces as a function of Si-O distance. The values of
rcandT (2.5)• and20 A-l, respectively)
arechosen
sothat
fc(r)doesnotaffecttheproperties
of crystalline
phases.The
otherparameters
in themodelwereconstrained
by molecular
orbitalcalculations
on silicatemolecules
andtheequation
of
stateof (x-quartz. For a completediscussionof the model's
construction
seeStixrudeandBukowinski(1988).
Themodelis intentionally
restricted
to describe
onlytetrahedralSi-O bonding.Thisrestriction
manifests
itselfnotonlyin
therequirement
that at mostfouroxygensbe bondedto a siliconbutalsoin thefunctionalformandparameters
of thepotential:theO-Si-Oanglebendingterm,for instance,
favorsthe
ideal tetrahedralangle (it0 = 109.47ø). X-ray diffraction
(Spackman
et al., 1987)andmolecular
orbitalstudies
(Newton
andGibbs,1980)showthattetrahedral
bonding
(controlled
by
simulations,
we expectanyeffectsof metastability
to produce
volumesthatexceedthoseof theequilibriumliquid. Therefore,we tooktheresults
of thesimulation
whichproduced
the
smaller2,000K volumeasthebestestimate
of theequilibrium
liquidvolume.We assume
theuncertainty
in the2,000K volumesif half thedifferencebetweenthetwo coolinghistories.
Statisticaluncertainties
(AdamsandMcDonald,1974) in the
calculated
volumesweremuchsmaller,notexceeding
0.2%.
Crystalcomputations
are describedfully in Stixrudeand
Bukowinski(1988). Briefly,WMIN (Busing,1981)is used
tocalculate
staticcrystalproperties.
Debyemodelthermalcorrections
areaddedtoaccount
fortheeffects
of temperature.
Results and Discussions
The calculated2,000 K equationof stateof liquid SiO2
agreeswell withavailableexperimental
data,includingthezero
pressurevolume: 27.81+ 0.6 cc/molvs 26.9-28.2 cc/molexhighlydirectional
sp3 hybrids)
is substantially
different
from perimental(Langeand Carmichael,1987) and zeropressure
octahedral
bonding
(controlled
by morespherically
symmetric bulk modulus: 11.2 + 3 GPa vs. 13 GPa experimental(Krol
sp3d
2 hybrids)
whichoccurs
in highpressure
silicate
poly- et al., 1986). Figure 1 showsthat the simulatedstructureof
the liquid at 2,000 K, as measuredby the radial distribution
morphssuchas stishovite.While a completedescription
of
function,is in goodagreementwith experiment.The calcuSi-O bondingmustalsoincludeoctahedral
bonding,we concentratehereon our simplermodel,whichis a qualitatively latedroomtemperatureequationsof stateof a-quartz,coesite
andquantitatively
realisticdescription
of tetrahedral
bonding. and tt-cristobaliteagreewell with experimentas shownin
Althoughwecannotrealistically
simulate
octahedrally
coordi- Figure2. Our modelalsoaccuratelypredictsthe structureand
natedphasesor four- to six-foldcoordination
changes,our
modelallows us to examinecompression
mechanisms
in a
simplifiedsilicateliquidwhichretainsfour-foldbondedcoordinationat all pressures
andto address
thecentralquestion
of
compressionmechanismsof (x-quartzand coesiteand the
quartzto coesitephasetransition(Stixrudeand Bukowinski,
1988).
this paper: whether mechanismsother than coordination
changes
contribute
significantly
to liquid-crystal
densityinversions.
•
Computations
Ns,-0(r)
14
11
''
50
OPa
10
Sl-O
Monte Carlo simulationswere performedin the constant
pressure,temperatureand particle number (NPT) ensemble
(AdamsandMcDonald,1974). The simulated
liquidconsisted
of 192 atomsrepeatedwith periodicboundaryconditionsto
minimizefinitesizeeffects.Althoughlargersystems
werenot
tested,
theprimarycellwaslargeenough
(15)• on a sideat
zeropressure)sothat the radial distributionfunctionfluctuated
lessthan2% aboutits longrangelimitingvalue(unity)at half
thecell size. AlthoughSi-O bondswereallowedto breakand
newbondsto form, Si-O coordinations
greaterthanfourwere
prohibited. The maximum allowablevolume fluctuationand
particledisplacement
werechosensothatapproximately
50%
of theconfigurations
wereaccepted.Thevolumewasallowed
to changewith everyconfiguration.Typicalrunlengthswere
from 8 to 12 millionconfigurations.
The simulations
were startedby meltingideal [5-cristobalite
at 10,000K and 10 GPa. The liquidwasthencooledto differentpressures
on the 6,000 K isotherm(0-50 GPa). Each
6,000 K point was then isobaricallycooledto 4,000 K. At
2,000 K, liquidscooledisobaricallyfrom 4,000 K differed
somewhatfrom thosecooleddirectlyfrom 6,000 K (volumes
differedby 7% at 0 GPa, but lessthan2% at all otherpressures).Sincethesimulations
wereinitializedwithhighertemperature(largervolume)configurations,
andsincethevolumes
decreased
steadilytowardstheirfinal stablevaluesduringthe
8
,•
sio2
'• •
LIQUID
0 GPo 2000K
2
I",t
o
I ',
0
1
2
"\
3
'"•""'"*
4
5
6
,-
Fig.1. Theradialdistribution
functions
go(r)(Si-O,O-O,Si-
Si) computedfrom Monte Carlo simulationsof our tetrahedrallybondedSiO2liquidat 0 GPaand2,000K. Alsoplotted
isthecumulative
Si-Ocoordination
number(Nsi-o(r),proportionalto the areaundergsi-o(r))asa functionof Si-O distance
at 0 GPa(boldline) and50 GPa(thinline). Thepositions
of
thefirst sixpeaks(Si-O, O-O, Si-Si,2ndSi-O,2ndO-O, 2nd
Si-Si) areconsistent
with experiment
(WasedaandToguri,
1977)exceptthe first Si-Si peakwhichoccursat a distance
0.15 A smallerthantheobserved
value.TheSi-O,O-O and
Si-Sicoordination
numbers
andalsoconsistent
withexperiment (Waseda and Toguri, 1977). The plateaus in
Nsi-o(r)at around2 )• showthattheSi-O coordination
numberis well definedanddoesnotexceedfourup to 50 GPa.
StixrudeandBukowinsld:Compression
of SiO2Liquid
,,o
l q05
ILl QUID, CRYSTALS
I•.
20
I•.
20
JIj COESITE
• •-QUARTZ
SHOVlTE
•
o, , , ,
v (cc/mo)
Fig. 2. Roomtemperature
experimentaldata(symbols)for aquartz (Levien et al., 1980; Hemlcy et al., 1988), coesite
(LevienandPrewitt, 1981;Hemley et al., 1988) anda-cristobalite(Peacor,1973) comparedwith theoreticalequations
of
state(solidlines)basedon the samepotentialmodelthatwas
usedin the liquid simulations.The size of the symbolsis
comparable
to uncertainties
in thehighpressure
data. The experimentalstishovitecurve (dashedline) (Basset al., 1982) is
shownfor comparison.
Thedensityof themodelliquidsurpasses
thatof thetetrahedral crystallinephasesby 50 GPa (Figure3). Theseliquidcrystaldensityinversionsoccurwithoutan increasein Si-O
coordination
numberin the liquid: themodelguarantees
that
the bondedcoordinationof the liquid remainsfour-fold,and
Figure 1 showsthat the effectivecoordinationof the liquid
alsoremainsfour-fold,at leastup to 50 GPa. Eventheroom
temperature
densitiesof the tetrahedralcrystals(Figure2) are
surpassed
by the 2,000 K liquid at 24, 30 and 60 GPa for
coesite,a-quartzanda-cristobalite,
respectively.Further,the
reportedliquid densities,if theydiffer from trueequilibrium
values,are likely to err towardslower density,as discussed
above,thusleadingus to overestimate
the pressures
of liquidcrystaldensityinversion. Thus, uncertainties
in the calculationsdueto thermalcorrections
of thecrystallineequationsof
state(dueto inadequacies
of theDebyemodelat hightemperature) andpossiblemetastabilityin the liquid simulations,
do
not affectour conclusionthat,for our model,the liquidis the
densest
fourcoordinated
phaseat highpressure.
Althoughthesedensityinversionsdo notinvolvecoexisting
liquidsandcrystals(experimentally,octahedralstishoviteis
the thermodynamically
stablecrystalphaseabove12 GPa),
theydo illustratethe efficientdensification
of the tetrahedral
liquid and suggestliquid compression
mechanisms
very differentfrom thosein thecrystals.The structures
of liquidand
crystallinephasesconsistof a continuousnetworkof nearly
rigid,comer-sharing
SiO4tetrahedra.While compression
of a
crystalis accomplished
withoutbreakingbondsby simplyreducingthedistancebetweentetrahedra,
we find thatthemodel
liquidcompresses
by breakingandreformingbonds,thusrearrangingthe connectivityof the tetrahedralnetworkwithout
substantialchanges in the inter-tetrahedraldistance (L)
(Stixrudeand Bukowinski,to be published). The contrast
v (cc/mo)
Fig. 3. Resultsof 2,000 K Monte Carlo simulations
of liquid
SiO2(circlesandboldlines)comparedwith 2,000K equations
of stateof crystals(thin lines). The boldline is a Birch-Murnaghanfit to the simulated
liquidvolumes,denotedby thecircles;thesizeof thecirclesis comparable
to volumeuncertainties(seetext). The crystalequations
of stateamlabelledat the
pressure
wheretheliquid surpasses
theirdensity.For clarity,
only the high pressureportionof the cristobaliteequationof
stateis shown. Becauseof the unusualshapeof the cristobaliteequationof state(negativecurvatureat low pressures,
seeFigure2) it crosses
theliquidequationof statetwice:at 12
and50 GPa. The experimental
sfishovite
curve(thesameasin
Figure2) is shownfor comparison.
betweenliquid andcrystalcompression
mechanisms
is illustratedin Figure4. The largedifferences
betweenvolume(V)
andscaledvolume(V*) for thecrystalsat highpressure
reflect
large pressure-induced
decreasesin L. In contrast,the near
identityof the scaledand standardequationsof stateof the
liquidreflectsmallchanges
in L (V andV* for theliquiddiffer
by only9% at 50 GPacomparedwith 40% for a-quartz).
The compression
of the modelliquidis analogous
to pressure-induced
phasetransitionsin crystals. For instance,the
transitionfrom a-quartzto coesiteat 2 GPa,like compression
in the liquid, changesthe way the tetrahedraare connected
rather than the distance between them: both model and data
(Levien et al., 1980; Levien and Prewitt, 1981) showthat the
transitioncausesonly a 1% changein L (coesiteactuallyhasa
slightlylargerL thana-quartz)butan 8% increasein density.
The liquidis ultimatelythe densestphasebecause,unlikethe
crystals,it is not limited to structuralchangeat phasetransitionsbut is free to continuouslyadoptinherentlydenserarrangements
of itstewahedral
networkwith increasing
pressure.
Summary
We havecompareddensitiesof crystalsandliquid at pressurein a simplesilicatesystembasedon a realisticmodelof
tetrahedralSi-O bonding. Even thoughthe modelliquid remains four-fold coordinated, it becomesdenser than tetrahe-
dralcrystalsat highpressure.We attributethemoreefficient
compaction
of theliquidto itsability,unlikea crystal,to easily
rearrange
its structure
in response
to pressure.
1406
StixrudeandBukowinski:Compression
of SiO2Liquid
moleculesand crystalsin termsof potentialenergyfunctions,Oak RidgeNationalLaboratory,Oak Ridge,1981.
Erikson,R. L., and C. J. Hostetler,Applicationof empirical
ionicmodelsto SiO2liquid:Potentialmodelapproximations
and integrationof SiO2 polymorphdata, Geochim. Cosmochim. Acta, 51, 1209-1218, 1987.
Hemley, R. J., A. P. Jephcoat,H. K. Mao, L. C. Ming, and
M. H. Manghnani,Pressure-induced
amorphization
of crystalline silica, Nature, 334, 52-54, 1988.
Krol, D. M., K. B. Lyons, S. A. Brawer, and C. R.
Kurkjian, High-temperaturelight scatteringand the glass
transitionin vitreous silica, Phys. Rev. B, 33, 4196-4202,
1986.
Kubicki, J. D., andA. C. Lasaga,Moleculardynamicssimulationsof SiO2 melt and glass: Ionic and covalentmodels,
12
1•/, = V(L./L)
20 • (½½/mol)
24
28
Amer. Mineral., 73, 941-955, 1988.
Lange, R. A., and I. S. E. Cannichael,Densitiesof Na20K20-CaO-MgO-FeO-Fe203-A1203-TiO2-SiO2
liquids:New
measurementsand derived partial molar properties,
Fig. 4. 300 K crystalequationsof stateand 2,000 K liquid
equationof statescaledto the zero-pressure
inter-tetrahedral
Geochim. Cosmochim.Acta, 51, 2931-2946, 1987.
distance(L0). The plotexhibitstwo interesting
end-members:
Levien, L., C. T. Prewitt, and D. J. Weldnet, Structure and
the first, approximatingthe crystals,is homogeneous
comelasticpropertiesof quartzat pressure,Amer. Mineral., 65,
pression
(V scales
withL3) whichwouldappear
asa vertical
920-930, 1980.
line; the second,approximatingthe liquid, is inherentcomLevien,
L., andC. T. Prewitt,High pressurecrystalstructure
pression(involvingno changein L) whichwouldappearidenand compressibilityof coesite,Amer. Mineral., 66, 324ticalto the standard
equationof state(P vs. V). Valuesof L are
333, 1981.
structuralaverages.For theliquid,L wastakenastheposition
Newton,
M.D., andG. V. Gibbs,Ab initiocalculatedgeomeof the Si-Si peak in g(r) (see Figure 1). Also shownis a
tries
and
chargedistributionsfor H4SiO4 andH6Si207 comschematic
drawingof the Si207 bitetrahedron
(filled circles=
paredwith experimentalvaluesfor silicatesand siloxanes,
Si atoms,opencircles= O atoms),the largestunit that all the
Phys. Chem. Min., 6, 221-246, 1980.
tetrahedralstructuresshare,indicatingthe inter-tetrahedral
Peacot,D. R., High-temperaturesingle-crystalstudyof the
distance, L.
cristobaliteinversion, Zeit. Krist., 138, 274-298, 1973.
It is likely thatsilicateliquidsundergopressure-induced
coordinationincreases(Xue et al., 1989), andthat thesewill tend
to increasethe densityof liquidsrelativeto coexistingsolids.
However,althoughour simplifiedmodeldoesnot allow us to
draw finn conclusions
aboutcomplexnaturalmagmaticsystems,we suggestthat the efficient,purely tetrahedralliquid
compression
mechanism
illustrated
heremaybeequallyimportant in determiningliquid-crystaldensityinversionsin the
Earth.
Acknowledgments. The computersimulationswere done
with a Monte Carloprogramadaptedfrom a codegenerously
providedby Marvin Ross. This researchwas supported
by
NSF andtheInstituteof Geophysics
andPlanetaryPhysicsat
the Lawrence Livermore National Laboratory. The latter,
alongwith the SanDiego Supercomputer
Centerandthe UC
BerkeleyComputingCenter,providedgeneroussupercomputer support. We thankJ. Kubicki and an anonymousrefereefor constructive
commentson themanuscript.
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