Introduction

Aqueous solutions rich in dissolved rock-forming sili-cates constitute one of the most important types of fluid inthe Earthrsquos crust and upper mantle for the latter specifi-cally in or near subduction zones (eg Manning 2004) Atdepth such fluids can approach and attain the compositionand the properties of hydrous melts (eg Shen amp Keppler1997 Audetat amp Keppler 2004) Until recently systematicexperimental studies on the solubilities of minerals and

other materials relevant to natural rocks have been limitedmainly due to experimental constraints to pressures below~ 5 kbar Such data have been used in various compilationsfor instance in a series of many articles extending overmore than 30 years by Helgeson and coworkers (egWalther amp Helgeson 1977 Helgeson et al 1981Sverjensky et al 1997) for modeling and predicting phaseequilibria and thermodynamic properties of aqueousspecies However as discussed in detail by Manning(1994) these models do not allow rigorous extrapolation todepths beyond those corresponding to 5 kbar More recent

Eur J Mineral This paper was presented at the EMPG X2005 17 269-283 meeting in FrankfurtMain Germany (April 2004)

Thermodynamic modeling of solubility and speciation of silica in H2O-SiO2fluid up to 1300degC and 20 kbar based on the chain reaction formalism

TARAS V GERYA12 WALTER V MARESCH3 MICHAEL BURCHARD3 VITALI ZAKHARTCHOUK3NIKOS L DOLTSINIS4 and THOMAS FOCKENBERG3

1Geologisches Institut ETH - Zuumlrich CH-8092 Zuumlrich Switzerland2Institute of Experimental Mineralogy Russian Academy of Sciences Chernogolovka Moscow district 142432 Russia

3Institut fuumlr Geologie Mineralogie und Geophysik Ruhr-Universitaumlt Bochum D-44780 Bochum Germany4Lehrstuhl fuumlr Theoretische Chemie Ruhr-Universitaumlt Bochum D-44780 Bochum Germany

Abstract Recent systematic studies of mineral solubilities in water to high pressures up to 50 kbar call for a suitable thermody-namic formalism to allow realistic fitting of the experimental data and the establishment of an internally consistent data base Thevery extensive low-pressure (lt 5 kbar) experimental data set on the solubility of SiO2 in H2O has in the last few years beenextended to 20 kbar and 1300degC providing an excellent experimental basis for testing new approaches In addition solubilityexperiments with different SiO2-buffering phase assemblages and in situ determinations of Raman spectra for H2O-SiO2 fluidshave provided both qualitative and quantitative constraints on the stoichiometry and quantities of dissolved silica species Wepropose a thermodynamic formalism for modeling both absolute silica solubility and speciation of dissolved silica using a combi-nation of the chain reaction approach and a new Gibbs free energy equation of water based on a homogeneous reaction formalismFor a given SiO2-buffer (eg quartz) and the coexisting H2O-SiO2 fluid both solubility and speciation of silica can be describedby the following two reactions- monomer-forming standard reaction SiO2(s) + 2(H2O)L = (SiO2)bull(H2O)2 (A)- polymer-forming chain reaction (SiO2)n-1bull(H2O)n + (SiO2)bull(H2O)2 = (SiO2)nbull(H2O)n+1 + (H2O)L (B)where 2 le n le infin and (H2O)L stands for ldquoliquid-likerdquo (associated clustered) water molecules in the aqueous fluid We show that reac-tions (A) and (B) lead to the simplified relationships ∆Gdeg(mono)rPT = ∆Hdeg(mono)r ndash T∆Sdeg(mono)r + ∆Cpdeg(mono)r [T ndash 29815 ndash Tln(T29815)] +∆Vdeg(mono)r(P ndash 1) and ∆Gdeg(poly)rPT = ∆Hdeg(poly)r ndash T∆Sdeg(poly)r + ∆Vdeg(poly)r (P ndash 1)(where the ∆GdegrPT are the standard molar Gibbs free energy changes in reactions (A) and (B) as a function of pressure P andtemperature T the ∆Hdegr ∆Sdegr ∆Cpdegr and ∆Vdegr are standard molar enthalpy entropy isobaric heat capacity and volume changesrespectively in reactions (A) and (B) at reference temperature To = 29815 K and pressure Po = 1 bar) that provide excellentdescriptions of the available H2O-SiO2 data set in terms of both SiO2 solubility and silica speciation Discrepancies betweendirectly determined solubility data and data obtained from in situ Raman spectra are ascribed to (i) possible experimental prob-lems of equilibration and (ii) inherent difficulties of interpreting Raman spectra of dilute H2O-SiO2 solutions In agreement withrecent findings our model indicates that dissolved silica in quartz-buffered aqueous solutions is considerably polymerizedexceeding 20-25 at all temperatures above 400degC

Key-words silica solubility silica speciation chain reaction high pressure thermodynamics

0935-1221050017-0269 $ 675copy 2005 E Schweizerbartrsquosche Verlagsbuchhandlung D-70176 StuttgartDOI 1011270935-122120050017-0269

E-mail tarasgeryaerdwethzch

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg270

Table 1 Summary of thermodynamic variables

Variable Definition Unitsφ parameter characterizing attractive forces between all H2O molecules barγi activity coefficient of fluid component imicroi chemical potential (partial molar Gibbs free energy) of fluid component i Jmola1 activity of monomersaH2O activity of H2Oai activity of fluid component iaL activity of liquid-like H2Oan activity of polymers of dimension naSiO2bullnH2O(aq) activity of SiO2 bull nH2O(aq)c1 c2 empirical parameters for the vibrational part of the Gibbs free energy of water∆CPdegLiq-Gas difference in standard isobaric molar heat capacity of H2O in liquid-like and gas-like states at J(K mol)

reference pressure (Po =1 bar) and temperature (To = 29815 K)∆Cpdeg(mono)r standard molar isobaric heat capacity change for the monomer-forming reaction at reference

pressure (Po =1 bar) and temperature (To = 29815 K)f H2Odeg fugacity of pure H2O at given temperature and pressure∆Gdeg(1)rPT standard molar Gibbs free energy change for reaction 1 at given pressure and temperature Jmol∆Go

(Buffer-Quartz)PT standard molar Gibbs free energy change for the quartz buffer at given pressure and temperature JmolGH2O molar Gibbs free energy of pure H2O Jmol

Gm the part of the Gibbs free energy of pure H2O related to the association (clustering) of molecules Jmol∆Gdeg(mono)rPT standard molar Gibbs free energy change for the monomer- and polymer-forming reactions at given Jmol∆Gdeg(poly)rPT pressure and temperatureGo

Qtz standard molar Gibbs free energy of quartz JmolGs standard molar Gibbs free energy of water in the pure liquid-like state JmolH2981 standard enthalpy of formation from elements of pure liquid-like water at reference pressure Jmol

(Po = 1 bar) and temperature (To = 29815 K)∆HdegLiq-Gas difference in standard molar enthalpy of H2O in liquid-like and gas-like states at reference Jmol

pressure (Po = 1 bar) and temperature (To = 29815 K)∆Hdeg(mono)r standard molar enthalpy change for the monomer- and polymer-forming reactions at reference Jmol∆Hdeg(poly)r pressure (Po = 1 bar) and temperature (To = 29815 K)∆Hs1

o empirical parameter for the vibrational part of the Gibbs free energy of water J(H2O)G gas-like unassociated free water molecules in fluid(H2O)L liquid-like associated clustered water molecules in fluidK(mono) K(poly) equilibrium constant of the monomer- and polymer-forming reactionsmSiO2(aq) molality of dissolved SiO2 in H2O molkgn dimension of polymerP pressure barPo reference pressure (1 bar) barS2981 standard entropy of pure liquid-like water at reference pressure (Po = 1 bar) and temperature J(K mol)

(To = 29815 K)∆SdegLiq-Gas difference in standard molar entropy of H2O in liquid-like and gas-like states at reference J(K mol)

pressure (Po = 1 bar) and temperature (To = 29815 K)∆Sdeg(mono)r standard molar entropy change for the monomer- and polymer-forming reactions at reference J(K mol) ∆Sdeg(poly)r pressure (Po = 1 bar) and temperature (To = 29815 K)SiO2(s) silica in solid formSiO2 bull nH2O(aq) silica in dissolved form in waterT temperature KTo reference temperature (29815 K) K∆Vdeg(mono)r standard molar volume change for the monomer- and polymer-forming reactions at reference J(bar mol)∆Vdeg(poly)r pressure P = 1 bar and temperature T = 29815 K∆Vdegs1 empirical parameter for the vibrational part of the Gibbs free energy of water JbarWCp WH WS heat capacity enthalpy and entropy Margules parameters for the interaction of ldquoliquid-likerdquo and J(K mol)

ldquogas-likerdquo water moleculesXG mole fraction of gas-like H2O in H2O-SiO2 fluidXGas mole fraction of gas-like H2O in pure H2OX1 mole fraction of monomer in H2O-SiO2 fluidXL mole fraction of liquid-like H2O in H2O-SiO2 fluidXLiq mole fraction of liquid-like H2O in pure H2OXn mole fraction of polymer of dimension n in H2O-SiO2 fluidXSS sum of the mole fractions of all dissolved silica species in H2O-SiO2 fluidYSiO2

bulk concentration of SiO2 in H2O-SiO2 fluid ie SiO2(H2O+SiO2)

systematic studies on mineral solubilities to considerablyhigher pressures (eg Manning 1994 2004 Zhang ampFrantz 2000 Newton amp Manning 2000 2002 2003Fockenberg et al 2004) have raised the necessity ofseeking a thermodynamic formulation for aqueous fluidsup to at least 50 kbar

Manning (1994) Zhang amp Frantz (2000) and Newton ampManning (2002 2003) have extended the available veryextensive low-pressure experimental data set on the solu-bility of SiO2 in H2O to 20 kbar and 1300degC thus providingan excellent data base for developing and testing alternativeapproaches to the problem Manning (1994) also derived anempirical polynomial equation that describes the solubilityof quartz in aqueous fluid well from 1 bar and 25degC up to20 kbar and 900degC

log(mSiO2(aq)) = 42620 ndash 57642T + 17513 times 106T 2 ndash22869 times 108T 3 + [28454 ndash 10069T + 35689 times 105T 2]times log(ρH2O)

where mSiO2(aq) is the molality of SiO2 in the fluid molkgρH2O is the density of pure water in gcm3 at given pressureand temperature (T K) Definitions of the thermodynamicvariables used in this paper are summarized in Table 1

Experimental studies on the solubility of quartz in fluidmixtures (Walther amp Orville 1983) on the diffusion ofdissolved silica (Applin 1987) and in situ determinationsof the Raman spectra of dissolved silica species in H2O-SiO2 fluids (Zotov amp Keppler 2000 2002) up to 900degC and14 kbar provide both qualitative and quantitativeconstraints on the stoichiometry and quantities of dissolvedsilica species However at present no precise generallyapplicable thermodynamic treatment describing both thebulk solubility and the speciation of silica in aqueoussilica-rich fluid has been developed In the present paperwe derive and test such a formalism for the SiO2 ndash H2Osystem using a combination of the chain reaction approachand a new Gibbs free energy equation of water based on ahomogeneous reaction formalism (Gerya amp Perchuk 1997Gerya et al 2004)

Thermodynamic formalism

As suggested by Dove amp Rimstidt (1994) silicadissolves in water by the following general reaction

SiO2(s) + n H2O = SiO2 bull nH2O(aq) (1)

Accordingly the thermodynamic equilibrium conditionfor the dissolution of quartz is

∆Gdeg(1)rPT + RTln[aSiO2bullnH2O(aq)] ndash n times ln(aH2O) ndashn times ln(fH2Odeg) = 0 (2)

where ∆Gdeg(1)rPT is the standard molar Gibbs free energychange of reaction (1) as a function of temperature T andpressure P ai are activities of fluid components and fH2Odegis the fugacity of pure H2O at given pressure and tempera-ture (eg Holland amp Powell 1998) In the case of low bulksilica concentrations (YSiO2

lt 005 or mSiO2(aq) lt 1 where YSiO2

= SiO2(SiO2+H2O) and m is molality in molkg) aH2O asymp 1and aSiO2bullnH2O(aq) asymp YSiO2

Then equation (2) can be rewritten as

ln(YSiO2) = n times ln(fdegH2O) ndash ∆Gdeg(1)rPTRT (3)

implying an apparently simple thermodynamic treatmentof this system However the available experimental data(Fig 1a) indicate that the correlation of ln(fH2O

o) andln(YSiO2) along different isotherms is actually verynonlinear Our test calculations have shown that equations(1)-(3) fail to provide a precise continuous description ofexperimental data on quartz solubility at both low and highpressure from 1 bar to 20 kbar with a reasonably smallnumber of empirical parameters for the representation of∆Gdeg(1)rPT (eg Walther amp Helgeson 1977) On the otherhand Manning (1994) has demonstrated a strong positivecorrelation between the logarithm of silica concentration inquartz-buffered H2O-SiO2 fluid and the logarithm of thedensity of pure H2O fluid along different isotherms (Fig1b) Our calculations have also shown a similar strongpositive correlation between isothermal silica solubilityand the degree of association or clustering of H2Omolecules in pure H2O (Fig 1c) which increases withincreasing density (eg Gorbaty amp Kalinichev 1995Luck 1980 Gerya amp Perchuk 1997 Gerya et al 2004)This strongly suggests that the dissolution of silica isdirectly related to the actual proportion of associated (ieclustered ldquoliquid-likerdquo) H2O molecules in the aqueousfluid and offers the possibility of deriving a general ther-modynamic description for silica solubility in water

It has been shown recently (Gerya amp Perchuk 1997Gerya et al 2004) that from the viewpoint of thermody-namics pure H2O fluid (and many other gases) can be effec-tively treated as a mixture of molecules in two distinct states(1) a ldquoliquid-likerdquo (associated clustered) state (defined as(H2O)L in this paper) and (2) a ldquogas-likerdquo (unassociatedfree) state (defined as (H2O)G) This phenomenologicalapproach uses the concept of homogeneous reactions of H2Omolecules leading to their liquid-like linkage (for a detaileddiscussion of the thermodynamic background and derivationof the associationclustering reactions see Gerya amp Perchuk1997 Gerya et al 2004) Such an approach is theoreticallywell developed for describing the hydrogen bonding insupercritical fluids (eg Luck 1980 Gupta et al 1992)and its potential applicability to any type of fluid has beendemonstrated (Barelko et al 1994 Gerya amp Perchuk1997) This approach is also in accord with the detailedRaman spectroscopic studies of Lindner (1970) and Kohl etal (1991) who concluded on the basis of their data thatwater can be treated as a solution of liquid-like and gas-liketypes of H2O molecules According to this formalism themolar Gibbs free energy equation of pure H2O valid for awide range of P-T conditions takes the form (see Gerya ampPerchuk 1997 Gerya et al 2004 for details of derivation)

GH2O = Gs + Gm (4)

where

Gs = H2981 ndash TS2981 + VdegsΨ + RT[c1ln(1 ndash e1) + c2ln(1 ndash e2)]ndash (c1 + c2)[∆Hs1

o(1 ndash TTo) eo(1 ndash eo) + RTln(1 ndash eo)] (5)

Gm = RT[XLiqln(XLiq) + XGasln(XGas)] + XGasRTln(P + φ XLiq2) ndash

XGas∆HdegLiq-Gas ndash T∆SdegLiq-Gas + ∆CPdegLiq-Gas[T ndash 29815 ndash Tln(T 29815)]+ WH ndash TWS + WCp[T ndash 29815 ndash Tln(T29815)]XLiqXGas (6)

Modeling of silica solubility and speciation 271

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

and e1 = exp[ndash (∆Hdegs1 + ∆Vdegs1Ψ)RT)] e2 = exp[ndash ∆Hdegs1RT)]eo = exp[ndash ∆Hdegs1R29815)] Ψ = 54(1 + φ)15[(P + φ)45 ndash(1 + φ)45] XLiq and XGas are the mole fractions (or morespecifically the thermodynamic probabilities in this

dynamic system) of molecules in ldquoliquid-likerdquo and ldquogas-likerdquo states respectively and XLiq + XGas = 1 Gs is the stan-dard molar Gibbs free energy of water in the pureldquoliquid-likerdquo state (XLiq = 1) and Gm is that part of the

272

Fig 1 Experimental data on the solubility ofquartz in aqueous fluid YSiO2

= SiO2(SiO2 +H2O) as a function of various thermodynamicparameters (a) fugacity of pure H2O (b) densityof pure H2O (c) mole fraction (thermodynamicprobability) of ldquoliquid-likerdquo (associated clus-tered) aqueous molecules (XLiq) in pure H2Ocalculated according to equations (4)-(9) Thedifferent symbols represent experimental datafrom the literature sources indicated Theisotherms have been obtained from the empiricalpolynomial fit suggested by Manning (1994)

Gibbs free energy of H2O related to the association (clus-tering) of molecules ∆HdegLiq-Gas = ndash 44839 J ∆SdegLiq-Gas =ndash 1224 JK and ∆CPdegLiq-Gas = 215 JK are the differencesin standard molar enthalpy entropy and heat capacity ofH2O in ldquoliquid-likerdquo and ldquogas-likerdquo states at reference pres-sure (Po = 1 bar) and temperature (To = 29815 K) WH =ndash28793 J WS = ndash117 JK and WCp = 51 JK are enthalpyentropy and heat capacity Margules parameters for theinteraction of ldquoliquid-likerdquo and ldquogas-likerdquo water moleculesat Po and To φ = 6209 bar is a parameter characterizing theattractive forces between all H2O molecules H2981 =ndash 286832 J S2981 = 652 JK are respectively the standardmolar enthalpy of formation from the elements and themolar entropy of ldquoliquid-likerdquo water molecules at Po andTo Vdegs = 1714 Jbar is the standard molar volume ofldquoliquid-likerdquo water molecules at T = 0 K and Po c1 = 7236c2 = 0315 ∆Hdegs1 = 4586 J and ∆Vdegs1 = 00431 Jbar areempirical parameters for the vibrational part of the Gibbspotential

By using the Gibbs-Duhem relation chemical poten-tials of ldquoliquid-likerdquo (microLiq) and ldquogas-likerdquo (microGas) moleculescan be calculated from equations (4)-(6) as follows

microLiq = Gs + RTln(XLiq) + 2RTXLiqXGas2φ(P + φ XLiq

2) + WH ndashTWS + WCp[T ndash 29815 ndash Tln(T29815)]XGas

2 (7)

microGas = Gs + RTln(XGas) + RTln(P + φ XLiq2) ndash 2RTXLiq

2XGasφ(P+ φ XLiq

2) ndash ∆HdegLiq-Gas ndash T∆SdegLiq-Gas + ∆CPdegLiq-Gas[T ndash 29815ndash Tln(T29815)] + WH ndash TWS + WCp[T ndash 29815 ndash Tln(T29815)]XLiq

2 (8)

The equilibrium values for XLiq and XGas in a pure H2Ofluid at given pressure and temperature can then be calcu-lated according to the standard relation

Gm = min ie microLiq = microGas (9)

The software for calculating the equilibrium values ofXLiq according to equation (9) is available by request fromthe corresponding author

Furthermore it is well-known (eg Applin 1987 Doveamp Rimstidt 1994 Zotov amp Keppler 2000 2002) thatdissolved silica can occur in the form of various types ofspecies Although (SiO2)bull(H2O)2-monomers dominate inthe fluid at low silica concentrations in equilibrium withquartz at 25oC and 1 bar (eg Walther amp Helgeson 1977Cary et al 1982 Applin 1987) (SiO2)2bull(H2O)3-dimersand possibly higher polymers become important at highersilica concentrations particularly with increasing pressureand temperature (eg Zotov amp Keppler 2000 2002Newton amp Manning 2002 2003) In the following there-fore we propose that H2O-SiO2-fluids can be treated as amixture of ldquoliquid-likerdquo (H2O)L and ldquogas-likerdquo (H2O)Gwater molecules with (SiO2)bull(H2O)2-monomers and allhigher (SiO2)nbull(H2O)n+1-polymers (2 le n le infin) such that

XL + XG + X1 +Σn=2

infin Xn= 1 (10)

where XL and XG are the mole fractions of ldquoliquid-likerdquo(H2O)L and ldquogas-likerdquo (H2O)G molecules respectively X1

is the mole fraction of (SiO2)bull(H2O)2-monomers and Xn isthe mole fraction of silica polymers with dimension ofn ge 2

On the basis of the above we can describe both the solu-bility and the speciation of silica in an aqueous fluid by thefollowing two types of reactions

a) the monomer-forming standard reaction

SiO2(s) + 2(H2O)L = (SiO2)bull(H2O)2 (11)

b) the polymer-forming chain reaction

(SiO2)n-1bull(H2O)n + (SiO2)bull(H2O)2 = (SiO2)nbull(H2O)n+1 + (H2O)L (12)

where 2 le n le infin The equilibrium conditions for these tworeactions can be written as∆Gdeg(mono)rPT + RTln(K(mono)) = 0 (13)

∆Gdeg(poly)rPT + RTln(K(poly)) = 0 (14)

where K(mono) = a1(aL)2 and K(poly) = anaL(an-1a1) are theequilibrium constants of reactions (11) and (12) respec-tively aL is the activity of ldquoliquid-likerdquo water molecules(H2O)L a1 is the activity of silica monomers and an as wellas an-1 are the activities of silica polymers with dimensionsof n and n-1 respectively ∆Gdeg(mono)rPT and ∆Gdeg(poly)rPT arethe standard Gibbs free energy changes of reactions (11)and (12) as functions of P and T respectively

In order to obtain analytical expressions for silica solu-bility we first assume that ∆Gdeg(poly)rPT is independent of nAccording to equations (7) and (8) the activity of ldquoliquid-likerdquo water molecules differs from the mole fraction andvaries with pressure and temperature For the followingderivations we therefore assume that the activities of thesilica species are also different from their mole fractionsand vary with pressure and temperature in such a way thatthe ratios of the activity coefficients (but not the absolutevalues of these coefficients) for the equilibrium constantsof reactions (11) and (12) are always equal to one

γ1(γL)2 = 1 (15)

γnγL(γn-1γ1) = 1 (16)

where γi = aiXi ne 1The chemical potentials for all species in H2O-SiO2 fluidscan then be formulated according to reactions (11) (12)and equations (7) (8) as follows(H2O)G - standard state XG = 1 XGas = 1 XLiq = 0

micro[(H2O)G] = RTln(XG) + Gs + RTln(P + φ XLiq2) ndash

2RTXLiq2XGasφ(P + φ XLiq

2) ndash ∆HdegLiq-Gas ndash T∆SdegLiq-Gas +∆CPdegLiq-Gas[T ndash 29815 ndash Tln(T29815)] + WH ndash TWS +WCp[T ndash 29815 ndash Tln(T29815)]XLiq

2 (17)

(H2O)L ndash standard state XL =1 XLiq = 1 XGas = 0

micro[(H2O)L] = RTln(XL) + Gs + 2RTXLiqXGas2φ(P + φ XLiq

2)+ WH ndash TWS + WCp[T ndash 29815 ndash Tln(T29815)]XGas

2 (18)

(SiO2)bull(H2O)2 ndash standard state X1 = 1 XLiq = 1 XGas = 0

Modeling of silica solubility and speciation 273

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

micro[(SiO2)bull(H2O)2] = RTln(X1) + ∆Go(mono)rPT + Go

Qtz + 2Gs+ 4RTXLiqXGas

2φ(P + φ XLiq2) + 2WH ndash TWS + WCp[T ndash

29815 ndash Tln(T29815)]XGas2 (19)

(SiO2)nbull(H2O)n+1 ndash standard state Xn = 1 XLiq = 1 XGas = 0

micro[(SiO2)nbull(H2O)n+1] = RTln(Xn) + n∆Gdeg(mono)rPT + (n ndash1)∆Gdeg(poly)rPT + nGdegQtz + (n + 1)Gs + (n +1)2RTXLiqXGas

2φ(P + φ XLiq2) + (n + 1)WH ndash TWS +

WCp[T ndash 29815 ndash Tln(T29815)]XGas2 (20)

where GdegQtz is the standard molar Gibbs free energy ofquartz (eg Holland amp Powell 1998 Gerya et al 19982004) XLiq and XGas are respectively equilibrium molefractions (thermodynamic probabilities) of ldquoliquid-likerdquoand ldquogas-likerdquo water molecules calculated for pure H2Ofluid at given pressure and temperature according to equa-tions (7)-(9) The presence of XLiq- and XGas-dependentterms in equations (19) (20) means that according to ourmodel the thermodynamic properties of silica speciesstrongly depend on the degree of association of moleculesin the aqueous fluid These XLiq- and XGas-dependent termsare similar to those of ldquoliquid-likerdquo rather than ldquogas-likerdquowater molecules (cf equations (17)-(20))

For the following derivations we assume that the pres-ence of silica species affects the absolute values but doesnot change the ratio between the mole fractions of ldquoliquid-likerdquo and ldquogas-likerdquo water molecules in SiO2-H2O-fluidsThis ratio then always corresponds to the conditions ofequilibrium in a pure water fluid at the same pressure andtemperature

XLXG = XLiqXGas (21)

The following analytical expressions can then beobtained for the concentrations of different fluid species

K(mono) = exp(ndash ∆Gdeg(mono)rPT RT) (22)

K(poly) = exp(ndash ∆Gdeg(poly)rPT RT) (23)

XL = [4XLiqK(mono)(XLiq ndash K(poly)) + (1 + XLiqK(mono)K(poly))2]05

ndash (1 + XLiqK(mono)K(poly))[2K(mono)(XLiq ndash K(poly))] (24)

X1 = K(mono)(XL)2 (25)

Xn = X1(K(poly)X1XL)n-1 (26)

XG = XGasXLXLiq (27)

The bulk concentration of dissolved silica in the fluid YSiO2

= SiO2(SiO2+H2O) and the relative proportion ofdissolved silica in monomers YSiO2(mono) = SiO2(mono) SiO2(total)as well as in higher polymers YSiO2(n-poly) = SiO2(n-poly)SiO2(total) can then be calculated according to

YSiO2= (Σ

infin

i=l Xii)[Σ

infin

i=l Xi (2i + 1) + XL + XG]

= (1 ndash XL ndash XG)2[2(1 ndash XL ndash XG)2 + X1](28)

YSiO2(mono) = X1(Σinfin

i=l Xii) = (X1)2(1 ndash XL ndash XG)2 (29)

YSiO2(n-poly) = nXn(Σinfin

i=l Xii) = nX1Xn(1 ndash XL ndash XG)2 (30)

To derive equations (28)-(30) we have used equation (26)

the condition Σinfin

i=l Xi+ XL + XG = 1 (cf equation (10)) and

the standard mathematical relations Σinfin

i=l Zi-1 = 1(1 ndash Z) and

Σinfin

i=l Zi-1i = 1(1 ndash Z)2 where 0 lt Z lt 1 in order to obtain

Σinfin

i=l Xi = X1(1 ndash K(poly)X1XL) = 1 ndash XL ndash XG Σ

infin

i=l Xii = X1(1 ndash

K(poly)X1XL)2 = (1 ndash XL ndash XG)2X1

Application to available experimental data

Our test calculations for the H2O-SiO2 system in a wideregion of pressure (1-20000 bar) and temperature (25-1300oC) have shown that the standard molar Gibbs freeenergy change for reactions (11) and (12) can be repre-sented well by the following relatively simple standardformulae

∆Gdeg(mono)rPT = ∆Hdeg(mono)r ndash T∆Sdeg(mono)r + ∆Cpdeg(mono)r

[T ndash To ndash Tln(TTo)] + ∆Vdeg(mono)r(PndashPo) (31)

∆Gdeg(poly)rPT = ∆Hdeg(poly)r ndash T∆Sdeg(poly)r + ∆Vdeg(poly)r(P ndash Po) (32)

where To = 29815 K and Po = 1 bar are standard pressureand temperature respectively ∆Hdegr ∆Sdegr ∆Cpdegr and ∆Vdegrare respectively standard molar enthalpy entropy isobaricheat capacity and volume changes at To and Po in reactions(11) and (12)

In the case where additional SiO2-buffering mineralassemblages are equilibrated with aqueous fluid (egZhang amp Frantz 2000 Newton amp Manning 2003) theGibbs free energy change of the monomer-forming reac-tion (11) can be obtained as follows

∆Gdeg(mono)rPT = ∆Go(Buffer-Quartz)PT + ∆Hdeg(mono)r ndash

T∆Sdeg(mono)r + ∆Cpdeg(mono)r [T ndash To ndash Tln(TTo)] +∆Vdeg(mono)r(P ndash Po) (33)

where ∆Go(Buffer-Quartz)PT is the difference between the stan-

dard molar Gibbs free energies of the given SiO2-bufferingmineral assemblage and quartz in Jmol which is a func-tion of P and T (Newton amp Manning 2003) For calculating∆Go

(Buffer-Quartz)PT we used the internally consistent thermo-dynamic database of Holland amp Powell (1998) As shownby Zhang amp Frantz (2000) and Newton amp Manning (2003)experimental data on silica solubility in the presence ofvarious SiO2-buffering mineral assemblages allow accuratemodeling of silica speciation at elevated pressure andtemperature

In order to quantify the seven parameters in equations(31) and (32) we have used a set of 439 experimental datapoints (Table 2) retrieved from the literature on the bulksilica solubility in the presence of quartz (Anderson ampBurnham 1965 Walther amp Orville 1983 Hemley et al1980 Kennedy 1950 Morey et al 1962 Manning 1994

274

Modeling of silica solubility and speciation 275

Ref

eren

ces

SiO

2-bu

ffer

No

ofP

kba

rT

degC

YSi

O2

Stat

isti

cal

Mea

n sq

uare

∆∆ YSi

O2

data

wei

ght

for

erro

rmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

poin

tsea

ch d

ata

σσ (Y

SiO

2)

min

max

poin

t

And

erso

n amp

Bur

nham

(19

65)

quar

tz52

1-9

8559

6-90

00

0008

-00

31

12 (

16)

-6 (

-3)

+35

(+

41)

Wal

ther

amp O

rvil

le (

1983

)qu

artz

131-

235

0-55

00

0004

-00

021

6 (8

)-1

2 (-

15)

+9

(+16

)H

emle

y et

al

198

0qu

artz

141-

220

0-50

00

0001

-00

011

7 (9

)-1

4 (-

14+

3 (+

4)K

enne

dy (

1950

)qu

artz

870

01-1

75

182-

610

000

005-

000

11

13 (

11)

-11

(-42

)+

38 (

+12

)M

orey

et

al (

1962

)qu

artz

281

0145

-300

000

0006

-00

003

124

(19

)-3

7 (-

31)

-3 (

-6)

Man

ning

(19

94)

exp

qu

artz

515-

2050

0-90

00

002-

004

18

(10)

-17

(-21

)+

17 (

+22

)M

anni

ng (

1994

) c

alc

quar

tz91

000

1-20

25-9

000

0000

02-0

04

39

-16

+14

Wat

son

amp W

ark

(199

7)qu

artz

310

570-

810

000

5-0

021

14 (

17)

+7

(+15

)+

18 (

+18

)N

ewto

n amp

Man

ning

(20

02)

quar

tz e

nsta

tite

-for

ster

ite

22

1-14

700-

900

000

05-0

05

18

(5)

-14

(-13

)+

15 (

0)N

ewto

n amp

Man

ning

(20

03)

quar

tz e

nsta

tite

-for

ster

ite

612

800

000

1-0

021

10-9

+22

kyan

ite-

coru

ndum

fors

teri

te-r

util

e-ge

ikie

lite

Woo

dlan

d amp

Wal

ther

(19

87)

quar

tz53

095

-27

134

5-50

00

0004

-00

021

7 (9

)-2

0 (-

19)

+11

(+

18)

Wal

ther

amp W

oodl

and

(199

3)qu

artz

151

5-2

400-

600

000

09-0

002

113

(13

)-1

9 (-

19)

+17

(+

24)

Hem

ley

et a

l(1

977a

b)

talc

-chr

ysot

ile-

wat

er t

alc-

751-

220

0-71

50

0000

1-0

001

122

(-32

)+

70fo

rste

rite

-wat

er t

alc-

anti

gori

te-

wat

er t

alc-

anth

ophy

llit

e-w

ater

ta

lc-e

nsta

tite

-wat

er

anth

ophy

llit

e-fo

rste

rite

-wat

erH

emle

y et

al

(197

7b)

enst

atit

e-fo

rste

rite

71

680-

720

000

047-

000

053

110

(9)

-14

(-16

)+

15 (

+4)

Zha

ng amp

Fra

ntz

(200

0) e

xp

enst

atit

e-fo

rste

rite

610

-20

900-

1200

000

6-0

041

34 (

62)

+16

(0)

+52

(+

103)

Zha

ng amp

Fra

ntz

(200

0) c

alc

enst

atit

e-fo

rste

rite

710

700-

1300

000

2-0

091

31 (

60)

-2 (

-19)

+51

(+

103)

Dov

e amp

Rim

stid

t (1

994)

quar

tz1

000

125

000

0002

100

000

(-9)

--

Tabl

e 2

Exp

erim

enta

l da

ta u

sed

for

the

cali

brat

ion

of e

quat

ions

(31

)-(3

3)

va

lues

in

brac

kets

are

cal

cula

ted

on t

he b

asis

of

the

poly

nom

ial

fits

of

Man

ning

(19

94)

and

New

ton

amp M

anni

ng (

2002

)

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

Walther amp Woodland 1993 Woodland amp Walther 1987Watson amp Wark 1997 Newton amp Manning 2002) andother SiO2-buffering mineral assemblages (Zhang ampFrantz 2000 Newton amp Manning 2002 2003 Hemley etal 1977ab) as well as silica diffusion (Applin 1987) andNMR data (Cary et al 1982) at 25degC and 1 bar for esti-mating both the bulk mole fraction of silica and the fractionof silica in monomeric form In situ measurements ofRaman spectra for H2O-SiO2 fluids coexisting with quartz(Zotov amp Keppler 2000 2002) provide qualitativeconstraints in that the relative amount of silica in polymericform must increase with an isochoric increase of tempera-ture for T = 600-900degC and P = 56-14 kbar

As discussed further below it was not possible to inte-grate the numerical results of Zotov amp Keppler (2002) whoderived quantitative data on the mole fractions of silicamonomers and dimers from in situ Raman spectra ofaqueous fluid equilibrated with quartz at pressure from56-14 kbar and temperatures from 700 to 900degC Theseresults have been excluded from our database for tworeasons (i) our calculations show that these data are inlarge part inconsistent with the other studies consideredhere requiring a much more pronounced dependence ofsilica speciation on the density of the fluid at a giventemperature and (ii) it was not possible to resolve thediscrepancies in the paper by Zotov amp Keppler (2002)between the formalism presented in their equations (9) and(10) and the derived data on monomer and dimer quantitiespresented in their Table 2

For the calculation of thermodynamic parameters weused a non-linear least square method in order to minimizethe relative error of reproducing the bulk silica concentra-tion which is defined as∆YSiO2

() = 100[YSiO2(calculated) ndash

YSiO2(experimental)]YSiO2

(experimental)We have used a uniform statistical weight of 1 for all exper-imental data points (Table 2) thus making no attempt togauge the relative quality of the various data subsets intheir accuracy and precision (see discussion by Manning

1994) However because of the very irregular distributionof data points in pressure and temperature we found itnecessary to balance our database by adding 91 pointsinterpolated from the polynomial equation of Manning(1994) within the P-T range of experiments used for thecalibration of this equation These ldquosyntheticrdquo points wereregularly distributed with pressure (1 kbar steps) along 9isotherms (100 to 900degC with 100degC steps) A higherstatistical weight of 3 was assigned to these data points Inorder to insure an exact reproduction of the concentrationand the speciation of silica at the reference conditions of To= 25degC and Po = 1 bar (Morey et al 1962 Applin 1987Dove amp Rimstidt 1994 Cary et al 1982) we assigned avery high statistical weight of 10000 to the single datapoint representing these conditions The resulting valuesfor the thermodynamic parameters of equations (31) and(32) are∆Hdeg(mono)r = 24014 plusmn 1781 J∆Sdeg(mono)r = ndash 2810 plusmn 423 JK∆Vdeg(mono)r = ndash 02354plusmn00749 Jbar∆Cpdeg(mono)r = 3196 plusmn 645 JK∆Hdeg(poly)r = ndash 21059 plusmn 3589 J∆Sdeg(poly)r = 966 plusmn 1166 JK∆Vdeg(poly)r = 03161 plusmn 01798 Jbar

Table 2 and Fig 2 3 show that the data set is fitted wellby the above parameters The mean square error in repro-ducing 439 data points on bulk silica solubility is 142 relative No systematic variations of calculated errors withtemperature pressure bulk silica concentration and XLiqhave been found (Fig 3) In quartz-buffered systems rela-tively large deviations (gt 20 ) are indicated for the exper-imental data of Morey et al (1962) and Kennedy (1950) atpressures below 1 kbar and temperatures below 200degC Ourtest calculations show that these data are in part inconsis-tent with the bulk silica concentrations at the referenceconditions of To = 25degC and Po = 1 bar (Dove amp Rimstidt1994) requiring either notably higher (Morey et al 1962)or lower (Kennedy 1550) values These discrepancies canin part be explained by the problems of reaching equilib-

276

Fig 2 Results of modeling bulk molality ofdissolved silica in aqueous fluid in the presence ofquartz based on equations (21)-(30) (solid lines)The different symbols represent experimental datafrom the sources indicated Dashed lines are calcu-lations on the basis of the empirical equationsuggested by Manning (1994)

Modeling of silica solubility and speciation 277

Fig 3 Distribution of relative error ∆YSiO2() = 100[YSiO2

(calculated)-YSiO2(experimental)]YSiO2

(experimental) of reproducing experi-mental data with temperature (a) pressure (b) bulk silica concentration (c) and degree of association (XLiq) of pure water fluid (d) Thedifferent symbols represent experimental data from the sources indicated

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

rium at low temperatures (eg Walther amp Helgeson 1977)Indeed the overall accuracy of reproducing these experi-ments is comparable (Table 2) with the polynomial equa-

tion suggested by Manning (1994) Larger errors weredetected in silica-undersaturated systems (Table 2 Fig 3)for the experimental data of Hemley et al (1977a b) and

278

Fig 4 Relative concentration of silica inmonomers (a) dimers (b) and higher silicapolymers (c) dissolved in aqueous fluid in thepresence of quartz at P = 0001-20 kbar and T= 25-90degC Calculations are based on equa-tions (21)-(30) Thin grey lines are isobarsBlack lines are isotherms The solid squarerepresents the experimental data of Cary et al(1982) and Applin (1987) at 25degC and 1 bar

Zhang amp Frantz (2000) This may be related to the rela-tively large experimental uncertainties (Zhang amp Frantz2000 Newton amp Manning 2002) and to possible errorsincurred during the calculation of ∆Go

(Buffer-Quartz)PT forSiO2-buffering reactions involving water at low pressureand temperature (Hemley et al 1977a b) in combinationwith the database of Holland amp Powell (1998) Obviouslyat T = 25degC and P = 1 bar the highly weighted referencedata on the concentration of silica in the presence of quartz(Dove amp Rimstidt 1994) and on the distribution of silica

between (SiO2)bull(H2O)2-monomers and (SiO2)2bull(H2O)3-dimers (Cary et al 1982 Applin 1987) are accuratelyreproduced with a relative error of less then 02 (Table 2Fig 3 4a) Figure 2 shows that the results of our thermo-dynamic modeling coincide well with the empiricalapproximation suggested by Manning (1994) for the bulksolubility of silica in water in the presence of quartz Table2 also compares our numerical values (Table 2) with thoseobtained from the empirical equations of Manning (1994)for the quartz-buffered system and of Manning amp Newton

Modeling of silica solubility and speciation 279

Fig 5 Results of modeling bulk concentration (a)(b) and speciation (c) of dissolved silica in aqueousfluid in the presence of different buffering mineralassemblages The symbols represent data from thesources indicated (a) experimental data of Newtonamp Manning (2002) and empirical calibration ofZhang amp Frantz (2000) based on original andpublished (Nakamura amp Kushiro 1974 Manningamp Boettcher 1994) experimental data (b) experi-mental data of Newton amp Manning (2003) (c)monomer-dimer speciation model of Newton ampManning (2003)

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

(2002) for the enstatite-forsterite-buffered system For the358 compared experimental data points (Table 2) ourmodel yields a mean square error of 134 versus 165 for the equations of Manning (1994) and Manning ampNewton (2002)

As noted above our model was constrained to accu-rately reproduce experimental data on silica partitioningbetween monomers and larger clusters at 25degC and 1 bar(Cary et al 1982 Applin 1987) The silica species at theseconditions (Fig 4) are represented by (SiO2)bull(H2O)2-monomers (94 of total dissolved silica) and(SiO2)2bull(H2O)3-dimers (6 of total dissolved silica) witha negligible amount of higher polymers (lt 03 of totaldissolved silica) Cary et al (1982) detected the presenceof (SiO2)2bull(H2O)3-dimers by NMR in solutions containing16 mM SiO2 at a pH of 72 and estimated their concentra-tion to be about 6 of the total dissolved silica No addi-tional polymerized forms were detected which isconsistent with our calculations for higher polymers (Fig4c) However the lack of such accurate experimental datafor speciation over a wider range of pressures and temper-atures (eg Watson amp Wark 1997) leads to the relativelylarge uncertainties in ∆Sdeg(poly)r and ∆Vdeg(poly)r ie for chainreaction (12) It should be noted that reaction (12) can alsobe used for the calculation of silica speciation at an arbi-trary bulk concentration of dissolved silica YSiO2

=SiO2(SiO2 + H2O) Equations characterizing speciationthen become

X1 = XLiqXSS(1 ndash XSS)[XSSK(poly) + (1 ndash XSS)XLiq] (34)

XL = XLiq(1 ndash XSS) (35)

XSS = [4XLiqYSiO2(K(poly) ndash XLiq)(1 ndash 2YSiO2

) + (XLiq)2(1 ndashYSiO2

)2]05 ndash XLiq(1 ndash YSiO2) [2(K(poly) ndash XLiq)(1 ndash 2YSiO2

)] (36)

where XSS = X1 + Σinfin

n=2 Xn is the sum of the mole fractions of

all dissolved silica species (ie XSS + XL + XG = 1) Otherquantities are as given in equations (21)-(27)

Figure 4 shows SiO2 speciation in the presence ofquartz as a function of pressure temperature and bulksilica content as predicted from our model calculations Itis seen that the relative amount of silica in the form ofmonomers is most strongly dependent on bulk silicaconcentration and generally decreases with increasingSiO2 content of the solution (Fig 4a) Significant amountsof dissolved silica (10-30 ) in dimeric form are calculatedfor temperatures gt 300degC (Fig 4b) The relative proportionof dimeric silica shows a maximum (Fig 4b) at YSiO2

asymp001 At higher YSiO2

trimers and higher polymers becomeimportant (Fig 4c) consuming 10 to 50 of the dissolvedsilica at temperatures gt 500degC Because increasing temper-ature leads to higher bulk silica there is also a similarcorrelation with temperature The dependence on pressureis least pronounced as can be expected from the competingeffects of reactions (11) and (12) (H2O)L is consumedduring the monomer-forming reaction (11) and liberatedduring the polymer-forming reaction (12) thereforedictating opposite signs of ∆Vdegr of the reactions Increasingpressure should lead to increased bulk silica solubility (dueto reaction (11) ∆Vdeg(mono)r lt 0) and therefore increasing

polymerization on the one hand (Fig 4a) whereasincreasing pressure should also favor depolymerizationaccording to reaction (12) (∆Vdeg(poly)r gt 0) on the other

Figure 5 shows silica solubility and speciation inaqueous fluid coexisting with different SiO2-bufferingmineral assemblages It is seen that our model accuratelyreproduces the available experimental data (Zhang ampFrantz 2000 Newton amp Manning 2003) on bulk silicasolubility (Fig 5ab) The calculated distribution of silicabetween monomers and higher polymers as a function ofbulk SiO2(SiO2 + H2O) at constant pressure and tempera-ture (Fig 5c) is very similar to the results of Newton ampManning (2003) which are based on a somewhat simplermonomer-dimer model At constant pressure and tempera-ture the amount of monomeric silica decreases non-linearlywith decreasing bulk silica solubility

Discussion

The semi-empirical thermodynamic formalism devel-oped in this paper allows accurate reproduction of existingdata on silica concentration and silica speciation inaqueous fluid coexisting with quartz and other silica-buffering assemblages up to 20 kbar and 1300degC Incomparison to Mannings (1994) polynomial fit ourapproach also predicts the speciation of silica in theaqueous fluid without increasing the number of empiricalparameters in the model In addition our chain-reactionmodel allows an excellent data fit with full consideration ofall silica polymers (Fig 4) rather than only monomers anddimers (cf Newton amp Manning 2003) Figure 6 shows thepositions of isopleths of bulk silica concentration calcu-lated for both silica-saturated (Fig 6a) and the silica-under-saturated enstatite-forsterite-buffered (Fig 6b) systems atelevated temperature and pressure It is seen that our resultsare in general compatible with the polynomial equationssuggested by Manning (1994) and Newton amp Manning(2002) with the largest discrepancies found in the high-temperature (gt 900degC) high-YSiO2

(gt 005) region Itshould be emphasized here that our present model has beendeveloped for a bulk silica concentration (YSiO2

) in the fluidof less than 01 (Table 2) Therefore this model mayrequire modification (eg by incorporating terms for non-ideality of the aqueous solution) in order to account prop-erly for the thermodynamics of silica-rich hydroussolutions (eg Shen amp Keppler 1997 Audetat amp Keppler2004) in proximity to the second critical point in the H2O-SiO2 system

Our model predicts basic similarities between the ther-modynamic properties of dissolved silica species and thoseof associated clustered ldquoliquid-likerdquo water molecules (egequations (18)-(20)) This is consistent with the conclu-sions of Bockris (1949) and Walther amp Orville (1983) whosuggested that hydrous monomeric silica species _ Si(OH)4_ are associated with two more water dipoles to form anaqueous silica complex _ Si(OH)4bull2H2O _ by hydrogenbonding This mechanism of association of molecules is infact identical to the association (clustering) of molecules ofpure water (eg Luck 1980 Gorbaty amp Kalinichev 1995)

280

Modeling of silica solubility and speciation 281

Fig 6 Isopleths of bulk silica concentration (solid lines)calculated according to equations (21)-(27) and (31)-(33)for silica-saturated (a) and silica-undersaturated enstatite-forsterite buffered (b) system at elevated temperature andpressure Dashed lines show isopleths calculated with theequations suggested by Manning (1994) and Newton ampManning (2002) Phase boundaries in (a) are calculatedfrom the database of Holland amp Powell (1998) Meltingrelations in the presence of water are not shown

Fig 7 Results of modeling silica speciation in aqueous fluidin the presence of quartz along the two isochores presented byZotov amp Keppler (2002) ρ = 078 gcm3 P = 56-68 kbar andρ = 094 gcm3 P = 106-140 kbar The solid squares repre-sent the results of Zotov amp Keppler (2002) based on the inter-pretation of in situ Raman spectra (a) dependence of silicaspeciation on temperature (b) dependence of silica speciationon bulk silica content

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

and is directly accounted for by equations (4)-(6) (for adiscussion see Gerya amp Perchuk 1997 Gerya et al2004) The formation of similar complexes with dimersand higher polymers can also be expected thus supportingour thermodynamic formalism

An important unresolved question is the discrepancybetween the results obtained from our data set which isbased mainly on direct solubility measurements and dataobtained from Raman spectra (Zotov amp Keppler 20002002) This is particularly evident in the pressure effect onsilica polymerization The positive ∆Vdegr of reaction (12)suggests that at constant temperature and bulk silica anyincrease in pressure (ie in fluid density) should lead to adecrease in the degree of polymerization This is alsosupported by the ab initio molecular dynamics simulationsof Doltsinis et al (2004) who calculated higher effectivepressures for monomeric-dimeric systems in comparisonwith purely monomeric systems at T = 900 K and constantfluid density In the presence of quartz bulk silica contentincreases with pressure and the competing effects of reac-tions (11) and (12) should lead to only small changes inthe degree of polymerization Figure 7 indicates the trendspredicted from our model In contrast the results of Zotovamp Keppler (2002) suggest strong increase in the degree ofpolymerization with pressure (ie fluid density) requiringa notably negative ∆Vo

r of polymer-forming reaction (12)Our test calculations show that no consistent set of ther-modynamic parameters for reactions (11) and (12) canprovide an accurate description of both the silica solubilitymeasurements and the changes of degree of polymeriza-tion with fluid density calculated by Zotov amp Keppler(2002)

We suggest that an explanation for the above discrep-ancies must be sought in (i) the possible problems of equi-libration in the relatively short experiments in thediamond-anvil cell performed by Zotov amp Keppler (20002002) and (ii) the difficult task of quantifying the Ramanspectra of aqueous silica solutions The interpretation ofRaman spectra is based mainly on theoretical spectraobtained using empirical interaction potentials (Catlow1980 Price amp Parker 1984) or on single molecule abinitio calculations as in the case of Zotov amp Keppler(2000 2002) The problem with the latter is that they donot include crucial temperature- and pressure-dependentsolvent effects To test these environmental effects wehave initiated a series of both ab initio molecular dynamics(AIMD) simulations and static gas phase ab initio calcula-tions (Doltsinis et al 2004) Our results to date suggestthat while the strong feature at ca 780-800 cm-1 can beascribed to monomers with some confidence there arestill considerable uncertainties in interpreting otherfeatures of the Raman spectra in terms of monomersandor dimers Considering the low amounts of total silicainvolved and the increasingly floppy nature of largerpolymers leading to increasingly broadened spectralfeatures it appears that the recognition and identificationof trimers and higher polymers can be problematical Ifhidden trimers are present then an interpretation basedsolely on monomerdimer ratios will obviously be flawedThus while the results of Zotov amp Keppler (2000 2002)

represent a valiant first attempt to tackle the problem andprovide valuable qualitative data direct comparisons withclassical solubility results should not be overextended Wesuggest that further Raman spectroscopic measurementson well-defined solutions with given bulk silica contents atdifferent pressures and temperatures rather than quartz-buffered solutions only will be a valuable step toward abetter understanding of such spectra

The approach outlined in this paper has been success-fully used to fit and describe experimental data on bulkspinel (MgAl2O4) solubility up to 30 kbar byZakhartchouk et al (2004) despite the lack of detailedinformation on the actual speciation of the dissolved spinelcomponents in that system Given additional independentdata on the detailed structure of condensed phase water inthe future (eg Marx 1999 and in press) and on solutespeciation at various temperatures and pressures (seeDoltsinis et al 2004) the formalism described here canbe used to provide an increasingly complete thermody-namic description of aqueous solutions at depths corre-sponding to the lower crust and upper mantle

AcknowledgementsThis work was supported by ETHResearch Grant TH -1204-1 by RFBR grant 03-05-64633 by the RF President Program ldquoLeading ScientificSchool of Russiardquo (grant 0351645) to TVG and by theDeutsche Forschungsgemeinschaft within the scope ofSonderforschungsbereich 526 at the Ruhr-University ofBochum The helpful reviews of C Manning J Waltherand an anonymous reviewer as well as the editorialhandling of A Woodland are greatly appreciated

References

Anderson GM amp Burnham CW (1965) The solubility of quartzin supercritical water Am J Sci 263 494-511

Applin KR (1987) The diffusion of dissolved silica in diluteaqueous solution Geochim Cosmochim Acta 51 2147-2151

Audetat A amp Keppler H (2004) Viscosity of fluids in subduc-tion zones Science 303 513-516

Barelko VV Zarikov IV Pechatnikov EL (1994) Effect of theorigin of molecular clusters in one-component gas mixtures onthe equation of state Khim Fiz 13 42-50

Bockris JOrsquoM (1949) Ionic solutions Chem Soc LondonQuarterly Rev 3 173-180

Cary LW De Jong BHWS Dibble WEJr (1982) A 29Si NMRstudy of silica species in dilute aqueous solution GeochimCosmochim Acta 46 1317-1320

Catlow CRA (1980) Computer modelling of ionic crystals JPhysics 41 C6-53

Doltsinis NL Burchard M Maresch WV (2004) Ab initiomolecular dynamics simulations as a tool for interpreting vibra-tional spectra in aqueous fluids EMPG-X SymposiumAbstracts Lithos supplement to v 73 S28

Dove PM amp Rimstidt JD (1994) Silica physical behaviorgeochemistry and materials applications in ldquoSilica-water inter-actionsrdquo Reviews in Mineral 29 259-amp

Fockenberg T Burchard M Maresch WV (2004) Solubilities ofcalcium silicates at high pressures and temperatures EMPG-XSymposium Abstracts Lithos supplement to v 73 S37

282

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg270

Table 1 Summary of thermodynamic variables

Variable Definition Unitsφ parameter characterizing attractive forces between all H2O molecules barγi activity coefficient of fluid component imicroi chemical potential (partial molar Gibbs free energy) of fluid component i Jmola1 activity of monomersaH2O activity of H2Oai activity of fluid component iaL activity of liquid-like H2Oan activity of polymers of dimension naSiO2bullnH2O(aq) activity of SiO2 bull nH2O(aq)c1 c2 empirical parameters for the vibrational part of the Gibbs free energy of water∆CPdegLiq-Gas difference in standard isobaric molar heat capacity of H2O in liquid-like and gas-like states at J(K mol)

reference pressure (Po =1 bar) and temperature (To = 29815 K)∆Cpdeg(mono)r standard molar isobaric heat capacity change for the monomer-forming reaction at reference

pressure (Po =1 bar) and temperature (To = 29815 K)f H2Odeg fugacity of pure H2O at given temperature and pressure∆Gdeg(1)rPT standard molar Gibbs free energy change for reaction 1 at given pressure and temperature Jmol∆Go

(Buffer-Quartz)PT standard molar Gibbs free energy change for the quartz buffer at given pressure and temperature JmolGH2O molar Gibbs free energy of pure H2O Jmol

Gm the part of the Gibbs free energy of pure H2O related to the association (clustering) of molecules Jmol∆Gdeg(mono)rPT standard molar Gibbs free energy change for the monomer- and polymer-forming reactions at given Jmol∆Gdeg(poly)rPT pressure and temperatureGo

Qtz standard molar Gibbs free energy of quartz JmolGs standard molar Gibbs free energy of water in the pure liquid-like state JmolH2981 standard enthalpy of formation from elements of pure liquid-like water at reference pressure Jmol

(Po = 1 bar) and temperature (To = 29815 K)∆HdegLiq-Gas difference in standard molar enthalpy of H2O in liquid-like and gas-like states at reference Jmol

pressure (Po = 1 bar) and temperature (To = 29815 K)∆Hdeg(mono)r standard molar enthalpy change for the monomer- and polymer-forming reactions at reference Jmol∆Hdeg(poly)r pressure (Po = 1 bar) and temperature (To = 29815 K)∆Hs1

o empirical parameter for the vibrational part of the Gibbs free energy of water J(H2O)G gas-like unassociated free water molecules in fluid(H2O)L liquid-like associated clustered water molecules in fluidK(mono) K(poly) equilibrium constant of the monomer- and polymer-forming reactionsmSiO2(aq) molality of dissolved SiO2 in H2O molkgn dimension of polymerP pressure barPo reference pressure (1 bar) barS2981 standard entropy of pure liquid-like water at reference pressure (Po = 1 bar) and temperature J(K mol)

(To = 29815 K)∆SdegLiq-Gas difference in standard molar entropy of H2O in liquid-like and gas-like states at reference J(K mol)

pressure (Po = 1 bar) and temperature (To = 29815 K)∆Sdeg(mono)r standard molar entropy change for the monomer- and polymer-forming reactions at reference J(K mol) ∆Sdeg(poly)r pressure (Po = 1 bar) and temperature (To = 29815 K)SiO2(s) silica in solid formSiO2 bull nH2O(aq) silica in dissolved form in waterT temperature KTo reference temperature (29815 K) K∆Vdeg(mono)r standard molar volume change for the monomer- and polymer-forming reactions at reference J(bar mol)∆Vdeg(poly)r pressure P = 1 bar and temperature T = 29815 K∆Vdegs1 empirical parameter for the vibrational part of the Gibbs free energy of water JbarWCp WH WS heat capacity enthalpy and entropy Margules parameters for the interaction of ldquoliquid-likerdquo and J(K mol)

ldquogas-likerdquo water moleculesXG mole fraction of gas-like H2O in H2O-SiO2 fluidXGas mole fraction of gas-like H2O in pure H2OX1 mole fraction of monomer in H2O-SiO2 fluidXL mole fraction of liquid-like H2O in H2O-SiO2 fluidXLiq mole fraction of liquid-like H2O in pure H2OXn mole fraction of polymer of dimension n in H2O-SiO2 fluidXSS sum of the mole fractions of all dissolved silica species in H2O-SiO2 fluidYSiO2

bulk concentration of SiO2 in H2O-SiO2 fluid ie SiO2(H2O+SiO2)

systematic studies on mineral solubilities to considerablyhigher pressures (eg Manning 1994 2004 Zhang ampFrantz 2000 Newton amp Manning 2000 2002 2003Fockenberg et al 2004) have raised the necessity ofseeking a thermodynamic formulation for aqueous fluidsup to at least 50 kbar

Manning (1994) Zhang amp Frantz (2000) and Newton ampManning (2002 2003) have extended the available veryextensive low-pressure experimental data set on the solu-bility of SiO2 in H2O to 20 kbar and 1300degC thus providingan excellent data base for developing and testing alternativeapproaches to the problem Manning (1994) also derived anempirical polynomial equation that describes the solubilityof quartz in aqueous fluid well from 1 bar and 25degC up to20 kbar and 900degC

log(mSiO2(aq)) = 42620 ndash 57642T + 17513 times 106T 2 ndash22869 times 108T 3 + [28454 ndash 10069T + 35689 times 105T 2]times log(ρH2O)

where mSiO2(aq) is the molality of SiO2 in the fluid molkgρH2O is the density of pure water in gcm3 at given pressureand temperature (T K) Definitions of the thermodynamicvariables used in this paper are summarized in Table 1

Experimental studies on the solubility of quartz in fluidmixtures (Walther amp Orville 1983) on the diffusion ofdissolved silica (Applin 1987) and in situ determinationsof the Raman spectra of dissolved silica species in H2O-SiO2 fluids (Zotov amp Keppler 2000 2002) up to 900degC and14 kbar provide both qualitative and quantitativeconstraints on the stoichiometry and quantities of dissolvedsilica species However at present no precise generallyapplicable thermodynamic treatment describing both thebulk solubility and the speciation of silica in aqueoussilica-rich fluid has been developed In the present paperwe derive and test such a formalism for the SiO2 ndash H2Osystem using a combination of the chain reaction approachand a new Gibbs free energy equation of water based on ahomogeneous reaction formalism (Gerya amp Perchuk 1997Gerya et al 2004)

Thermodynamic formalism

As suggested by Dove amp Rimstidt (1994) silicadissolves in water by the following general reaction

SiO2(s) + n H2O = SiO2 bull nH2O(aq) (1)

Accordingly the thermodynamic equilibrium conditionfor the dissolution of quartz is

∆Gdeg(1)rPT + RTln[aSiO2bullnH2O(aq)] ndash n times ln(aH2O) ndashn times ln(fH2Odeg) = 0 (2)

where ∆Gdeg(1)rPT is the standard molar Gibbs free energychange of reaction (1) as a function of temperature T andpressure P ai are activities of fluid components and fH2Odegis the fugacity of pure H2O at given pressure and tempera-ture (eg Holland amp Powell 1998) In the case of low bulksilica concentrations (YSiO2

lt 005 or mSiO2(aq) lt 1 where YSiO2

= SiO2(SiO2+H2O) and m is molality in molkg) aH2O asymp 1and aSiO2bullnH2O(aq) asymp YSiO2

Then equation (2) can be rewritten as

ln(YSiO2) = n times ln(fdegH2O) ndash ∆Gdeg(1)rPTRT (3)

implying an apparently simple thermodynamic treatmentof this system However the available experimental data(Fig 1a) indicate that the correlation of ln(fH2O

o) andln(YSiO2) along different isotherms is actually verynonlinear Our test calculations have shown that equations(1)-(3) fail to provide a precise continuous description ofexperimental data on quartz solubility at both low and highpressure from 1 bar to 20 kbar with a reasonably smallnumber of empirical parameters for the representation of∆Gdeg(1)rPT (eg Walther amp Helgeson 1977) On the otherhand Manning (1994) has demonstrated a strong positivecorrelation between the logarithm of silica concentration inquartz-buffered H2O-SiO2 fluid and the logarithm of thedensity of pure H2O fluid along different isotherms (Fig1b) Our calculations have also shown a similar strongpositive correlation between isothermal silica solubilityand the degree of association or clustering of H2Omolecules in pure H2O (Fig 1c) which increases withincreasing density (eg Gorbaty amp Kalinichev 1995Luck 1980 Gerya amp Perchuk 1997 Gerya et al 2004)This strongly suggests that the dissolution of silica isdirectly related to the actual proportion of associated (ieclustered ldquoliquid-likerdquo) H2O molecules in the aqueousfluid and offers the possibility of deriving a general ther-modynamic description for silica solubility in water

It has been shown recently (Gerya amp Perchuk 1997Gerya et al 2004) that from the viewpoint of thermody-namics pure H2O fluid (and many other gases) can be effec-tively treated as a mixture of molecules in two distinct states(1) a ldquoliquid-likerdquo (associated clustered) state (defined as(H2O)L in this paper) and (2) a ldquogas-likerdquo (unassociatedfree) state (defined as (H2O)G) This phenomenologicalapproach uses the concept of homogeneous reactions of H2Omolecules leading to their liquid-like linkage (for a detaileddiscussion of the thermodynamic background and derivationof the associationclustering reactions see Gerya amp Perchuk1997 Gerya et al 2004) Such an approach is theoreticallywell developed for describing the hydrogen bonding insupercritical fluids (eg Luck 1980 Gupta et al 1992)and its potential applicability to any type of fluid has beendemonstrated (Barelko et al 1994 Gerya amp Perchuk1997) This approach is also in accord with the detailedRaman spectroscopic studies of Lindner (1970) and Kohl etal (1991) who concluded on the basis of their data thatwater can be treated as a solution of liquid-like and gas-liketypes of H2O molecules According to this formalism themolar Gibbs free energy equation of pure H2O valid for awide range of P-T conditions takes the form (see Gerya ampPerchuk 1997 Gerya et al 2004 for details of derivation)

GH2O = Gs + Gm (4)

where

Gs = H2981 ndash TS2981 + VdegsΨ + RT[c1ln(1 ndash e1) + c2ln(1 ndash e2)]ndash (c1 + c2)[∆Hs1

o(1 ndash TTo) eo(1 ndash eo) + RTln(1 ndash eo)] (5)

Gm = RT[XLiqln(XLiq) + XGasln(XGas)] + XGasRTln(P + φ XLiq2) ndash

XGas∆HdegLiq-Gas ndash T∆SdegLiq-Gas + ∆CPdegLiq-Gas[T ndash 29815 ndash Tln(T 29815)]+ WH ndash TWS + WCp[T ndash 29815 ndash Tln(T29815)]XLiqXGas (6)

Modeling of silica solubility and speciation 271

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

and e1 = exp[ndash (∆Hdegs1 + ∆Vdegs1Ψ)RT)] e2 = exp[ndash ∆Hdegs1RT)]eo = exp[ndash ∆Hdegs1R29815)] Ψ = 54(1 + φ)15[(P + φ)45 ndash(1 + φ)45] XLiq and XGas are the mole fractions (or morespecifically the thermodynamic probabilities in this

dynamic system) of molecules in ldquoliquid-likerdquo and ldquogas-likerdquo states respectively and XLiq + XGas = 1 Gs is the stan-dard molar Gibbs free energy of water in the pureldquoliquid-likerdquo state (XLiq = 1) and Gm is that part of the

272

Fig 1 Experimental data on the solubility ofquartz in aqueous fluid YSiO2

= SiO2(SiO2 +H2O) as a function of various thermodynamicparameters (a) fugacity of pure H2O (b) densityof pure H2O (c) mole fraction (thermodynamicprobability) of ldquoliquid-likerdquo (associated clus-tered) aqueous molecules (XLiq) in pure H2Ocalculated according to equations (4)-(9) Thedifferent symbols represent experimental datafrom the literature sources indicated Theisotherms have been obtained from the empiricalpolynomial fit suggested by Manning (1994)

Gibbs free energy of H2O related to the association (clus-tering) of molecules ∆HdegLiq-Gas = ndash 44839 J ∆SdegLiq-Gas =ndash 1224 JK and ∆CPdegLiq-Gas = 215 JK are the differencesin standard molar enthalpy entropy and heat capacity ofH2O in ldquoliquid-likerdquo and ldquogas-likerdquo states at reference pres-sure (Po = 1 bar) and temperature (To = 29815 K) WH =ndash28793 J WS = ndash117 JK and WCp = 51 JK are enthalpyentropy and heat capacity Margules parameters for theinteraction of ldquoliquid-likerdquo and ldquogas-likerdquo water moleculesat Po and To φ = 6209 bar is a parameter characterizing theattractive forces between all H2O molecules H2981 =ndash 286832 J S2981 = 652 JK are respectively the standardmolar enthalpy of formation from the elements and themolar entropy of ldquoliquid-likerdquo water molecules at Po andTo Vdegs = 1714 Jbar is the standard molar volume ofldquoliquid-likerdquo water molecules at T = 0 K and Po c1 = 7236c2 = 0315 ∆Hdegs1 = 4586 J and ∆Vdegs1 = 00431 Jbar areempirical parameters for the vibrational part of the Gibbspotential

By using the Gibbs-Duhem relation chemical poten-tials of ldquoliquid-likerdquo (microLiq) and ldquogas-likerdquo (microGas) moleculescan be calculated from equations (4)-(6) as follows

microLiq = Gs + RTln(XLiq) + 2RTXLiqXGas2φ(P + φ XLiq

2) + WH ndashTWS + WCp[T ndash 29815 ndash Tln(T29815)]XGas

2 (7)

microGas = Gs + RTln(XGas) + RTln(P + φ XLiq2) ndash 2RTXLiq

2XGasφ(P+ φ XLiq

2) ndash ∆HdegLiq-Gas ndash T∆SdegLiq-Gas + ∆CPdegLiq-Gas[T ndash 29815ndash Tln(T29815)] + WH ndash TWS + WCp[T ndash 29815 ndash Tln(T29815)]XLiq

2 (8)

The equilibrium values for XLiq and XGas in a pure H2Ofluid at given pressure and temperature can then be calcu-lated according to the standard relation

Gm = min ie microLiq = microGas (9)

The software for calculating the equilibrium values ofXLiq according to equation (9) is available by request fromthe corresponding author

Furthermore it is well-known (eg Applin 1987 Doveamp Rimstidt 1994 Zotov amp Keppler 2000 2002) thatdissolved silica can occur in the form of various types ofspecies Although (SiO2)bull(H2O)2-monomers dominate inthe fluid at low silica concentrations in equilibrium withquartz at 25oC and 1 bar (eg Walther amp Helgeson 1977Cary et al 1982 Applin 1987) (SiO2)2bull(H2O)3-dimersand possibly higher polymers become important at highersilica concentrations particularly with increasing pressureand temperature (eg Zotov amp Keppler 2000 2002Newton amp Manning 2002 2003) In the following there-fore we propose that H2O-SiO2-fluids can be treated as amixture of ldquoliquid-likerdquo (H2O)L and ldquogas-likerdquo (H2O)Gwater molecules with (SiO2)bull(H2O)2-monomers and allhigher (SiO2)nbull(H2O)n+1-polymers (2 le n le infin) such that

XL + XG + X1 +Σn=2

infin Xn= 1 (10)

where XL and XG are the mole fractions of ldquoliquid-likerdquo(H2O)L and ldquogas-likerdquo (H2O)G molecules respectively X1

is the mole fraction of (SiO2)bull(H2O)2-monomers and Xn isthe mole fraction of silica polymers with dimension ofn ge 2

On the basis of the above we can describe both the solu-bility and the speciation of silica in an aqueous fluid by thefollowing two types of reactions

a) the monomer-forming standard reaction

SiO2(s) + 2(H2O)L = (SiO2)bull(H2O)2 (11)

b) the polymer-forming chain reaction

(SiO2)n-1bull(H2O)n + (SiO2)bull(H2O)2 = (SiO2)nbull(H2O)n+1 + (H2O)L (12)

where 2 le n le infin The equilibrium conditions for these tworeactions can be written as∆Gdeg(mono)rPT + RTln(K(mono)) = 0 (13)

∆Gdeg(poly)rPT + RTln(K(poly)) = 0 (14)

where K(mono) = a1(aL)2 and K(poly) = anaL(an-1a1) are theequilibrium constants of reactions (11) and (12) respec-tively aL is the activity of ldquoliquid-likerdquo water molecules(H2O)L a1 is the activity of silica monomers and an as wellas an-1 are the activities of silica polymers with dimensionsof n and n-1 respectively ∆Gdeg(mono)rPT and ∆Gdeg(poly)rPT arethe standard Gibbs free energy changes of reactions (11)and (12) as functions of P and T respectively

In order to obtain analytical expressions for silica solu-bility we first assume that ∆Gdeg(poly)rPT is independent of nAccording to equations (7) and (8) the activity of ldquoliquid-likerdquo water molecules differs from the mole fraction andvaries with pressure and temperature For the followingderivations we therefore assume that the activities of thesilica species are also different from their mole fractionsand vary with pressure and temperature in such a way thatthe ratios of the activity coefficients (but not the absolutevalues of these coefficients) for the equilibrium constantsof reactions (11) and (12) are always equal to one

γ1(γL)2 = 1 (15)

γnγL(γn-1γ1) = 1 (16)

where γi = aiXi ne 1The chemical potentials for all species in H2O-SiO2 fluidscan then be formulated according to reactions (11) (12)and equations (7) (8) as follows(H2O)G - standard state XG = 1 XGas = 1 XLiq = 0

micro[(H2O)G] = RTln(XG) + Gs + RTln(P + φ XLiq2) ndash

2RTXLiq2XGasφ(P + φ XLiq

2) ndash ∆HdegLiq-Gas ndash T∆SdegLiq-Gas +∆CPdegLiq-Gas[T ndash 29815 ndash Tln(T29815)] + WH ndash TWS +WCp[T ndash 29815 ndash Tln(T29815)]XLiq

2 (17)

(H2O)L ndash standard state XL =1 XLiq = 1 XGas = 0

micro[(H2O)L] = RTln(XL) + Gs + 2RTXLiqXGas2φ(P + φ XLiq

2)+ WH ndash TWS + WCp[T ndash 29815 ndash Tln(T29815)]XGas

2 (18)

(SiO2)bull(H2O)2 ndash standard state X1 = 1 XLiq = 1 XGas = 0

Modeling of silica solubility and speciation 273

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

micro[(SiO2)bull(H2O)2] = RTln(X1) + ∆Go(mono)rPT + Go

Qtz + 2Gs+ 4RTXLiqXGas

2φ(P + φ XLiq2) + 2WH ndash TWS + WCp[T ndash

29815 ndash Tln(T29815)]XGas2 (19)

(SiO2)nbull(H2O)n+1 ndash standard state Xn = 1 XLiq = 1 XGas = 0

micro[(SiO2)nbull(H2O)n+1] = RTln(Xn) + n∆Gdeg(mono)rPT + (n ndash1)∆Gdeg(poly)rPT + nGdegQtz + (n + 1)Gs + (n +1)2RTXLiqXGas

2φ(P + φ XLiq2) + (n + 1)WH ndash TWS +

WCp[T ndash 29815 ndash Tln(T29815)]XGas2 (20)

where GdegQtz is the standard molar Gibbs free energy ofquartz (eg Holland amp Powell 1998 Gerya et al 19982004) XLiq and XGas are respectively equilibrium molefractions (thermodynamic probabilities) of ldquoliquid-likerdquoand ldquogas-likerdquo water molecules calculated for pure H2Ofluid at given pressure and temperature according to equa-tions (7)-(9) The presence of XLiq- and XGas-dependentterms in equations (19) (20) means that according to ourmodel the thermodynamic properties of silica speciesstrongly depend on the degree of association of moleculesin the aqueous fluid These XLiq- and XGas-dependent termsare similar to those of ldquoliquid-likerdquo rather than ldquogas-likerdquowater molecules (cf equations (17)-(20))

For the following derivations we assume that the pres-ence of silica species affects the absolute values but doesnot change the ratio between the mole fractions of ldquoliquid-likerdquo and ldquogas-likerdquo water molecules in SiO2-H2O-fluidsThis ratio then always corresponds to the conditions ofequilibrium in a pure water fluid at the same pressure andtemperature

XLXG = XLiqXGas (21)

The following analytical expressions can then beobtained for the concentrations of different fluid species

K(mono) = exp(ndash ∆Gdeg(mono)rPT RT) (22)

K(poly) = exp(ndash ∆Gdeg(poly)rPT RT) (23)

XL = [4XLiqK(mono)(XLiq ndash K(poly)) + (1 + XLiqK(mono)K(poly))2]05

ndash (1 + XLiqK(mono)K(poly))[2K(mono)(XLiq ndash K(poly))] (24)

X1 = K(mono)(XL)2 (25)

Xn = X1(K(poly)X1XL)n-1 (26)

XG = XGasXLXLiq (27)

The bulk concentration of dissolved silica in the fluid YSiO2

= SiO2(SiO2+H2O) and the relative proportion ofdissolved silica in monomers YSiO2(mono) = SiO2(mono) SiO2(total)as well as in higher polymers YSiO2(n-poly) = SiO2(n-poly)SiO2(total) can then be calculated according to

YSiO2= (Σ

infin

i=l Xii)[Σ

infin

i=l Xi (2i + 1) + XL + XG]

= (1 ndash XL ndash XG)2[2(1 ndash XL ndash XG)2 + X1](28)

YSiO2(mono) = X1(Σinfin

i=l Xii) = (X1)2(1 ndash XL ndash XG)2 (29)

YSiO2(n-poly) = nXn(Σinfin

i=l Xii) = nX1Xn(1 ndash XL ndash XG)2 (30)

To derive equations (28)-(30) we have used equation (26)

the condition Σinfin

i=l Xi+ XL + XG = 1 (cf equation (10)) and

the standard mathematical relations Σinfin

i=l Zi-1 = 1(1 ndash Z) and

Σinfin

i=l Zi-1i = 1(1 ndash Z)2 where 0 lt Z lt 1 in order to obtain

Σinfin

i=l Xi = X1(1 ndash K(poly)X1XL) = 1 ndash XL ndash XG Σ

infin

i=l Xii = X1(1 ndash

K(poly)X1XL)2 = (1 ndash XL ndash XG)2X1

Application to available experimental data

Our test calculations for the H2O-SiO2 system in a wideregion of pressure (1-20000 bar) and temperature (25-1300oC) have shown that the standard molar Gibbs freeenergy change for reactions (11) and (12) can be repre-sented well by the following relatively simple standardformulae

∆Gdeg(mono)rPT = ∆Hdeg(mono)r ndash T∆Sdeg(mono)r + ∆Cpdeg(mono)r

[T ndash To ndash Tln(TTo)] + ∆Vdeg(mono)r(PndashPo) (31)

∆Gdeg(poly)rPT = ∆Hdeg(poly)r ndash T∆Sdeg(poly)r + ∆Vdeg(poly)r(P ndash Po) (32)

where To = 29815 K and Po = 1 bar are standard pressureand temperature respectively ∆Hdegr ∆Sdegr ∆Cpdegr and ∆Vdegrare respectively standard molar enthalpy entropy isobaricheat capacity and volume changes at To and Po in reactions(11) and (12)

In the case where additional SiO2-buffering mineralassemblages are equilibrated with aqueous fluid (egZhang amp Frantz 2000 Newton amp Manning 2003) theGibbs free energy change of the monomer-forming reac-tion (11) can be obtained as follows

∆Gdeg(mono)rPT = ∆Go(Buffer-Quartz)PT + ∆Hdeg(mono)r ndash

T∆Sdeg(mono)r + ∆Cpdeg(mono)r [T ndash To ndash Tln(TTo)] +∆Vdeg(mono)r(P ndash Po) (33)

where ∆Go(Buffer-Quartz)PT is the difference between the stan-

dard molar Gibbs free energies of the given SiO2-bufferingmineral assemblage and quartz in Jmol which is a func-tion of P and T (Newton amp Manning 2003) For calculating∆Go

(Buffer-Quartz)PT we used the internally consistent thermo-dynamic database of Holland amp Powell (1998) As shownby Zhang amp Frantz (2000) and Newton amp Manning (2003)experimental data on silica solubility in the presence ofvarious SiO2-buffering mineral assemblages allow accuratemodeling of silica speciation at elevated pressure andtemperature

In order to quantify the seven parameters in equations(31) and (32) we have used a set of 439 experimental datapoints (Table 2) retrieved from the literature on the bulksilica solubility in the presence of quartz (Anderson ampBurnham 1965 Walther amp Orville 1983 Hemley et al1980 Kennedy 1950 Morey et al 1962 Manning 1994

274

Modeling of silica solubility and speciation 275

Ref

eren

ces

SiO

2-bu

ffer

No

ofP

kba

rT

degC

YSi

O2

Stat

isti

cal

Mea

n sq

uare

∆∆ YSi

O2

data

wei

ght

for

erro

rmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

poin

tsea

ch d

ata

σσ (Y

SiO

2)

min

max

poin

t

And

erso

n amp

Bur

nham

(19

65)

quar

tz52

1-9

8559

6-90

00

0008

-00

31

12 (

16)

-6 (

-3)

+35

(+

41)

Wal

ther

amp O

rvil

le (

1983

)qu

artz

131-

235

0-55

00

0004

-00

021

6 (8

)-1

2 (-

15)

+9

(+16

)H

emle

y et

al

198

0qu

artz

141-

220

0-50

00

0001

-00

011

7 (9

)-1

4 (-

14+

3 (+

4)K

enne

dy (

1950

)qu

artz

870

01-1

75

182-

610

000

005-

000

11

13 (

11)

-11

(-42

)+

38 (

+12

)M

orey

et

al (

1962

)qu

artz

281

0145

-300

000

0006

-00

003

124

(19

)-3

7 (-

31)

-3 (

-6)

Man

ning

(19

94)

exp

qu

artz

515-

2050

0-90

00

002-

004

18

(10)

-17

(-21

)+

17 (

+22

)M

anni

ng (

1994

) c

alc

quar

tz91

000

1-20

25-9

000

0000

02-0

04

39

-16

+14

Wat

son

amp W

ark

(199

7)qu

artz

310

570-

810

000

5-0

021

14 (

17)

+7

(+15

)+

18 (

+18

)N

ewto

n amp

Man

ning

(20

02)

quar

tz e

nsta

tite

-for

ster

ite

22

1-14

700-

900

000

05-0

05

18

(5)

-14

(-13

)+

15 (

0)N

ewto

n amp

Man

ning

(20

03)

quar

tz e

nsta

tite

-for

ster

ite

612

800

000

1-0

021

10-9

+22

kyan

ite-

coru

ndum

fors

teri

te-r

util

e-ge

ikie

lite

Woo

dlan

d amp

Wal

ther

(19

87)

quar

tz53

095

-27

134

5-50

00

0004

-00

021

7 (9

)-2

0 (-

19)

+11

(+

18)

Wal

ther

amp W

oodl

and

(199

3)qu

artz

151

5-2

400-

600

000

09-0

002

113

(13

)-1

9 (-

19)

+17

(+

24)

Hem

ley

et a

l(1

977a

b)

talc

-chr

ysot

ile-

wat

er t

alc-

751-

220

0-71

50

0000

1-0

001

122

(-32

)+

70fo

rste

rite

-wat

er t

alc-

anti

gori

te-

wat

er t

alc-

anth

ophy

llit

e-w

ater

ta

lc-e

nsta

tite

-wat

er

anth

ophy

llit

e-fo

rste

rite

-wat

erH

emle

y et

al

(197

7b)

enst

atit

e-fo

rste

rite

71

680-

720

000

047-

000

053

110

(9)

-14

(-16

)+

15 (

+4)

Zha

ng amp

Fra

ntz

(200

0) e

xp

enst

atit

e-fo

rste

rite

610

-20

900-

1200

000

6-0

041

34 (

62)

+16

(0)

+52

(+

103)

Zha

ng amp

Fra

ntz

(200

0) c

alc

enst

atit

e-fo

rste

rite

710

700-

1300

000

2-0

091

31 (

60)

-2 (

-19)

+51

(+

103)

Dov

e amp

Rim

stid

t (1

994)

quar

tz1

000

125

000

0002

100

000

(-9)

--

Tabl

e 2

Exp

erim

enta

l da

ta u

sed

for

the

cali

brat

ion

of e

quat

ions

(31

)-(3

3)

va

lues

in

brac

kets

are

cal

cula

ted

on t

he b

asis

of

the

poly

nom

ial

fits

of

Man

ning

(19

94)

and

New

ton

amp M

anni

ng (

2002

)

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

Walther amp Woodland 1993 Woodland amp Walther 1987Watson amp Wark 1997 Newton amp Manning 2002) andother SiO2-buffering mineral assemblages (Zhang ampFrantz 2000 Newton amp Manning 2002 2003 Hemley etal 1977ab) as well as silica diffusion (Applin 1987) andNMR data (Cary et al 1982) at 25degC and 1 bar for esti-mating both the bulk mole fraction of silica and the fractionof silica in monomeric form In situ measurements ofRaman spectra for H2O-SiO2 fluids coexisting with quartz(Zotov amp Keppler 2000 2002) provide qualitativeconstraints in that the relative amount of silica in polymericform must increase with an isochoric increase of tempera-ture for T = 600-900degC and P = 56-14 kbar

As discussed further below it was not possible to inte-grate the numerical results of Zotov amp Keppler (2002) whoderived quantitative data on the mole fractions of silicamonomers and dimers from in situ Raman spectra ofaqueous fluid equilibrated with quartz at pressure from56-14 kbar and temperatures from 700 to 900degC Theseresults have been excluded from our database for tworeasons (i) our calculations show that these data are inlarge part inconsistent with the other studies consideredhere requiring a much more pronounced dependence ofsilica speciation on the density of the fluid at a giventemperature and (ii) it was not possible to resolve thediscrepancies in the paper by Zotov amp Keppler (2002)between the formalism presented in their equations (9) and(10) and the derived data on monomer and dimer quantitiespresented in their Table 2

For the calculation of thermodynamic parameters weused a non-linear least square method in order to minimizethe relative error of reproducing the bulk silica concentra-tion which is defined as∆YSiO2

() = 100[YSiO2(calculated) ndash

YSiO2(experimental)]YSiO2

(experimental)We have used a uniform statistical weight of 1 for all exper-imental data points (Table 2) thus making no attempt togauge the relative quality of the various data subsets intheir accuracy and precision (see discussion by Manning

1994) However because of the very irregular distributionof data points in pressure and temperature we found itnecessary to balance our database by adding 91 pointsinterpolated from the polynomial equation of Manning(1994) within the P-T range of experiments used for thecalibration of this equation These ldquosyntheticrdquo points wereregularly distributed with pressure (1 kbar steps) along 9isotherms (100 to 900degC with 100degC steps) A higherstatistical weight of 3 was assigned to these data points Inorder to insure an exact reproduction of the concentrationand the speciation of silica at the reference conditions of To= 25degC and Po = 1 bar (Morey et al 1962 Applin 1987Dove amp Rimstidt 1994 Cary et al 1982) we assigned avery high statistical weight of 10000 to the single datapoint representing these conditions The resulting valuesfor the thermodynamic parameters of equations (31) and(32) are∆Hdeg(mono)r = 24014 plusmn 1781 J∆Sdeg(mono)r = ndash 2810 plusmn 423 JK∆Vdeg(mono)r = ndash 02354plusmn00749 Jbar∆Cpdeg(mono)r = 3196 plusmn 645 JK∆Hdeg(poly)r = ndash 21059 plusmn 3589 J∆Sdeg(poly)r = 966 plusmn 1166 JK∆Vdeg(poly)r = 03161 plusmn 01798 Jbar

Table 2 and Fig 2 3 show that the data set is fitted wellby the above parameters The mean square error in repro-ducing 439 data points on bulk silica solubility is 142 relative No systematic variations of calculated errors withtemperature pressure bulk silica concentration and XLiqhave been found (Fig 3) In quartz-buffered systems rela-tively large deviations (gt 20 ) are indicated for the exper-imental data of Morey et al (1962) and Kennedy (1950) atpressures below 1 kbar and temperatures below 200degC Ourtest calculations show that these data are in part inconsis-tent with the bulk silica concentrations at the referenceconditions of To = 25degC and Po = 1 bar (Dove amp Rimstidt1994) requiring either notably higher (Morey et al 1962)or lower (Kennedy 1550) values These discrepancies canin part be explained by the problems of reaching equilib-

276

Fig 2 Results of modeling bulk molality ofdissolved silica in aqueous fluid in the presence ofquartz based on equations (21)-(30) (solid lines)The different symbols represent experimental datafrom the sources indicated Dashed lines are calcu-lations on the basis of the empirical equationsuggested by Manning (1994)

Modeling of silica solubility and speciation 277

Fig 3 Distribution of relative error ∆YSiO2() = 100[YSiO2

(calculated)-YSiO2(experimental)]YSiO2

(experimental) of reproducing experi-mental data with temperature (a) pressure (b) bulk silica concentration (c) and degree of association (XLiq) of pure water fluid (d) Thedifferent symbols represent experimental data from the sources indicated

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

rium at low temperatures (eg Walther amp Helgeson 1977)Indeed the overall accuracy of reproducing these experi-ments is comparable (Table 2) with the polynomial equa-

tion suggested by Manning (1994) Larger errors weredetected in silica-undersaturated systems (Table 2 Fig 3)for the experimental data of Hemley et al (1977a b) and

278

Fig 4 Relative concentration of silica inmonomers (a) dimers (b) and higher silicapolymers (c) dissolved in aqueous fluid in thepresence of quartz at P = 0001-20 kbar and T= 25-90degC Calculations are based on equa-tions (21)-(30) Thin grey lines are isobarsBlack lines are isotherms The solid squarerepresents the experimental data of Cary et al(1982) and Applin (1987) at 25degC and 1 bar

Zhang amp Frantz (2000) This may be related to the rela-tively large experimental uncertainties (Zhang amp Frantz2000 Newton amp Manning 2002) and to possible errorsincurred during the calculation of ∆Go

(Buffer-Quartz)PT forSiO2-buffering reactions involving water at low pressureand temperature (Hemley et al 1977a b) in combinationwith the database of Holland amp Powell (1998) Obviouslyat T = 25degC and P = 1 bar the highly weighted referencedata on the concentration of silica in the presence of quartz(Dove amp Rimstidt 1994) and on the distribution of silica

between (SiO2)bull(H2O)2-monomers and (SiO2)2bull(H2O)3-dimers (Cary et al 1982 Applin 1987) are accuratelyreproduced with a relative error of less then 02 (Table 2Fig 3 4a) Figure 2 shows that the results of our thermo-dynamic modeling coincide well with the empiricalapproximation suggested by Manning (1994) for the bulksolubility of silica in water in the presence of quartz Table2 also compares our numerical values (Table 2) with thoseobtained from the empirical equations of Manning (1994)for the quartz-buffered system and of Manning amp Newton

Modeling of silica solubility and speciation 279

Fig 5 Results of modeling bulk concentration (a)(b) and speciation (c) of dissolved silica in aqueousfluid in the presence of different buffering mineralassemblages The symbols represent data from thesources indicated (a) experimental data of Newtonamp Manning (2002) and empirical calibration ofZhang amp Frantz (2000) based on original andpublished (Nakamura amp Kushiro 1974 Manningamp Boettcher 1994) experimental data (b) experi-mental data of Newton amp Manning (2003) (c)monomer-dimer speciation model of Newton ampManning (2003)

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

(2002) for the enstatite-forsterite-buffered system For the358 compared experimental data points (Table 2) ourmodel yields a mean square error of 134 versus 165 for the equations of Manning (1994) and Manning ampNewton (2002)

As noted above our model was constrained to accu-rately reproduce experimental data on silica partitioningbetween monomers and larger clusters at 25degC and 1 bar(Cary et al 1982 Applin 1987) The silica species at theseconditions (Fig 4) are represented by (SiO2)bull(H2O)2-monomers (94 of total dissolved silica) and(SiO2)2bull(H2O)3-dimers (6 of total dissolved silica) witha negligible amount of higher polymers (lt 03 of totaldissolved silica) Cary et al (1982) detected the presenceof (SiO2)2bull(H2O)3-dimers by NMR in solutions containing16 mM SiO2 at a pH of 72 and estimated their concentra-tion to be about 6 of the total dissolved silica No addi-tional polymerized forms were detected which isconsistent with our calculations for higher polymers (Fig4c) However the lack of such accurate experimental datafor speciation over a wider range of pressures and temper-atures (eg Watson amp Wark 1997) leads to the relativelylarge uncertainties in ∆Sdeg(poly)r and ∆Vdeg(poly)r ie for chainreaction (12) It should be noted that reaction (12) can alsobe used for the calculation of silica speciation at an arbi-trary bulk concentration of dissolved silica YSiO2

=SiO2(SiO2 + H2O) Equations characterizing speciationthen become

X1 = XLiqXSS(1 ndash XSS)[XSSK(poly) + (1 ndash XSS)XLiq] (34)

XL = XLiq(1 ndash XSS) (35)

XSS = [4XLiqYSiO2(K(poly) ndash XLiq)(1 ndash 2YSiO2

) + (XLiq)2(1 ndashYSiO2

)2]05 ndash XLiq(1 ndash YSiO2) [2(K(poly) ndash XLiq)(1 ndash 2YSiO2

)] (36)

where XSS = X1 + Σinfin

n=2 Xn is the sum of the mole fractions of

all dissolved silica species (ie XSS + XL + XG = 1) Otherquantities are as given in equations (21)-(27)

Figure 4 shows SiO2 speciation in the presence ofquartz as a function of pressure temperature and bulksilica content as predicted from our model calculations Itis seen that the relative amount of silica in the form ofmonomers is most strongly dependent on bulk silicaconcentration and generally decreases with increasingSiO2 content of the solution (Fig 4a) Significant amountsof dissolved silica (10-30 ) in dimeric form are calculatedfor temperatures gt 300degC (Fig 4b) The relative proportionof dimeric silica shows a maximum (Fig 4b) at YSiO2

asymp001 At higher YSiO2

trimers and higher polymers becomeimportant (Fig 4c) consuming 10 to 50 of the dissolvedsilica at temperatures gt 500degC Because increasing temper-ature leads to higher bulk silica there is also a similarcorrelation with temperature The dependence on pressureis least pronounced as can be expected from the competingeffects of reactions (11) and (12) (H2O)L is consumedduring the monomer-forming reaction (11) and liberatedduring the polymer-forming reaction (12) thereforedictating opposite signs of ∆Vdegr of the reactions Increasingpressure should lead to increased bulk silica solubility (dueto reaction (11) ∆Vdeg(mono)r lt 0) and therefore increasing

polymerization on the one hand (Fig 4a) whereasincreasing pressure should also favor depolymerizationaccording to reaction (12) (∆Vdeg(poly)r gt 0) on the other

Figure 5 shows silica solubility and speciation inaqueous fluid coexisting with different SiO2-bufferingmineral assemblages It is seen that our model accuratelyreproduces the available experimental data (Zhang ampFrantz 2000 Newton amp Manning 2003) on bulk silicasolubility (Fig 5ab) The calculated distribution of silicabetween monomers and higher polymers as a function ofbulk SiO2(SiO2 + H2O) at constant pressure and tempera-ture (Fig 5c) is very similar to the results of Newton ampManning (2003) which are based on a somewhat simplermonomer-dimer model At constant pressure and tempera-ture the amount of monomeric silica decreases non-linearlywith decreasing bulk silica solubility

Discussion

The semi-empirical thermodynamic formalism devel-oped in this paper allows accurate reproduction of existingdata on silica concentration and silica speciation inaqueous fluid coexisting with quartz and other silica-buffering assemblages up to 20 kbar and 1300degC Incomparison to Mannings (1994) polynomial fit ourapproach also predicts the speciation of silica in theaqueous fluid without increasing the number of empiricalparameters in the model In addition our chain-reactionmodel allows an excellent data fit with full consideration ofall silica polymers (Fig 4) rather than only monomers anddimers (cf Newton amp Manning 2003) Figure 6 shows thepositions of isopleths of bulk silica concentration calcu-lated for both silica-saturated (Fig 6a) and the silica-under-saturated enstatite-forsterite-buffered (Fig 6b) systems atelevated temperature and pressure It is seen that our resultsare in general compatible with the polynomial equationssuggested by Manning (1994) and Newton amp Manning(2002) with the largest discrepancies found in the high-temperature (gt 900degC) high-YSiO2

(gt 005) region Itshould be emphasized here that our present model has beendeveloped for a bulk silica concentration (YSiO2

) in the fluidof less than 01 (Table 2) Therefore this model mayrequire modification (eg by incorporating terms for non-ideality of the aqueous solution) in order to account prop-erly for the thermodynamics of silica-rich hydroussolutions (eg Shen amp Keppler 1997 Audetat amp Keppler2004) in proximity to the second critical point in the H2O-SiO2 system

Our model predicts basic similarities between the ther-modynamic properties of dissolved silica species and thoseof associated clustered ldquoliquid-likerdquo water molecules (egequations (18)-(20)) This is consistent with the conclu-sions of Bockris (1949) and Walther amp Orville (1983) whosuggested that hydrous monomeric silica species _ Si(OH)4_ are associated with two more water dipoles to form anaqueous silica complex _ Si(OH)4bull2H2O _ by hydrogenbonding This mechanism of association of molecules is infact identical to the association (clustering) of molecules ofpure water (eg Luck 1980 Gorbaty amp Kalinichev 1995)

280

Modeling of silica solubility and speciation 281

Fig 6 Isopleths of bulk silica concentration (solid lines)calculated according to equations (21)-(27) and (31)-(33)for silica-saturated (a) and silica-undersaturated enstatite-forsterite buffered (b) system at elevated temperature andpressure Dashed lines show isopleths calculated with theequations suggested by Manning (1994) and Newton ampManning (2002) Phase boundaries in (a) are calculatedfrom the database of Holland amp Powell (1998) Meltingrelations in the presence of water are not shown

Fig 7 Results of modeling silica speciation in aqueous fluidin the presence of quartz along the two isochores presented byZotov amp Keppler (2002) ρ = 078 gcm3 P = 56-68 kbar andρ = 094 gcm3 P = 106-140 kbar The solid squares repre-sent the results of Zotov amp Keppler (2002) based on the inter-pretation of in situ Raman spectra (a) dependence of silicaspeciation on temperature (b) dependence of silica speciationon bulk silica content

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

and is directly accounted for by equations (4)-(6) (for adiscussion see Gerya amp Perchuk 1997 Gerya et al2004) The formation of similar complexes with dimersand higher polymers can also be expected thus supportingour thermodynamic formalism

An important unresolved question is the discrepancybetween the results obtained from our data set which isbased mainly on direct solubility measurements and dataobtained from Raman spectra (Zotov amp Keppler 20002002) This is particularly evident in the pressure effect onsilica polymerization The positive ∆Vdegr of reaction (12)suggests that at constant temperature and bulk silica anyincrease in pressure (ie in fluid density) should lead to adecrease in the degree of polymerization This is alsosupported by the ab initio molecular dynamics simulationsof Doltsinis et al (2004) who calculated higher effectivepressures for monomeric-dimeric systems in comparisonwith purely monomeric systems at T = 900 K and constantfluid density In the presence of quartz bulk silica contentincreases with pressure and the competing effects of reac-tions (11) and (12) should lead to only small changes inthe degree of polymerization Figure 7 indicates the trendspredicted from our model In contrast the results of Zotovamp Keppler (2002) suggest strong increase in the degree ofpolymerization with pressure (ie fluid density) requiringa notably negative ∆Vo

r of polymer-forming reaction (12)Our test calculations show that no consistent set of ther-modynamic parameters for reactions (11) and (12) canprovide an accurate description of both the silica solubilitymeasurements and the changes of degree of polymeriza-tion with fluid density calculated by Zotov amp Keppler(2002)

We suggest that an explanation for the above discrep-ancies must be sought in (i) the possible problems of equi-libration in the relatively short experiments in thediamond-anvil cell performed by Zotov amp Keppler (20002002) and (ii) the difficult task of quantifying the Ramanspectra of aqueous silica solutions The interpretation ofRaman spectra is based mainly on theoretical spectraobtained using empirical interaction potentials (Catlow1980 Price amp Parker 1984) or on single molecule abinitio calculations as in the case of Zotov amp Keppler(2000 2002) The problem with the latter is that they donot include crucial temperature- and pressure-dependentsolvent effects To test these environmental effects wehave initiated a series of both ab initio molecular dynamics(AIMD) simulations and static gas phase ab initio calcula-tions (Doltsinis et al 2004) Our results to date suggestthat while the strong feature at ca 780-800 cm-1 can beascribed to monomers with some confidence there arestill considerable uncertainties in interpreting otherfeatures of the Raman spectra in terms of monomersandor dimers Considering the low amounts of total silicainvolved and the increasingly floppy nature of largerpolymers leading to increasingly broadened spectralfeatures it appears that the recognition and identificationof trimers and higher polymers can be problematical Ifhidden trimers are present then an interpretation basedsolely on monomerdimer ratios will obviously be flawedThus while the results of Zotov amp Keppler (2000 2002)

represent a valiant first attempt to tackle the problem andprovide valuable qualitative data direct comparisons withclassical solubility results should not be overextended Wesuggest that further Raman spectroscopic measurementson well-defined solutions with given bulk silica contents atdifferent pressures and temperatures rather than quartz-buffered solutions only will be a valuable step toward abetter understanding of such spectra

The approach outlined in this paper has been success-fully used to fit and describe experimental data on bulkspinel (MgAl2O4) solubility up to 30 kbar byZakhartchouk et al (2004) despite the lack of detailedinformation on the actual speciation of the dissolved spinelcomponents in that system Given additional independentdata on the detailed structure of condensed phase water inthe future (eg Marx 1999 and in press) and on solutespeciation at various temperatures and pressures (seeDoltsinis et al 2004) the formalism described here canbe used to provide an increasingly complete thermody-namic description of aqueous solutions at depths corre-sponding to the lower crust and upper mantle

AcknowledgementsThis work was supported by ETHResearch Grant TH -1204-1 by RFBR grant 03-05-64633 by the RF President Program ldquoLeading ScientificSchool of Russiardquo (grant 0351645) to TVG and by theDeutsche Forschungsgemeinschaft within the scope ofSonderforschungsbereich 526 at the Ruhr-University ofBochum The helpful reviews of C Manning J Waltherand an anonymous reviewer as well as the editorialhandling of A Woodland are greatly appreciated

References

Anderson GM amp Burnham CW (1965) The solubility of quartzin supercritical water Am J Sci 263 494-511

Applin KR (1987) The diffusion of dissolved silica in diluteaqueous solution Geochim Cosmochim Acta 51 2147-2151

Audetat A amp Keppler H (2004) Viscosity of fluids in subduc-tion zones Science 303 513-516

Barelko VV Zarikov IV Pechatnikov EL (1994) Effect of theorigin of molecular clusters in one-component gas mixtures onthe equation of state Khim Fiz 13 42-50

Bockris JOrsquoM (1949) Ionic solutions Chem Soc LondonQuarterly Rev 3 173-180

Cary LW De Jong BHWS Dibble WEJr (1982) A 29Si NMRstudy of silica species in dilute aqueous solution GeochimCosmochim Acta 46 1317-1320

Catlow CRA (1980) Computer modelling of ionic crystals JPhysics 41 C6-53

Doltsinis NL Burchard M Maresch WV (2004) Ab initiomolecular dynamics simulations as a tool for interpreting vibra-tional spectra in aqueous fluids EMPG-X SymposiumAbstracts Lithos supplement to v 73 S28

Dove PM amp Rimstidt JD (1994) Silica physical behaviorgeochemistry and materials applications in ldquoSilica-water inter-actionsrdquo Reviews in Mineral 29 259-amp

Fockenberg T Burchard M Maresch WV (2004) Solubilities ofcalcium silicates at high pressures and temperatures EMPG-XSymposium Abstracts Lithos supplement to v 73 S37

282

systematic studies on mineral solubilities to considerablyhigher pressures (eg Manning 1994 2004 Zhang ampFrantz 2000 Newton amp Manning 2000 2002 2003Fockenberg et al 2004) have raised the necessity ofseeking a thermodynamic formulation for aqueous fluidsup to at least 50 kbar

Manning (1994) Zhang amp Frantz (2000) and Newton ampManning (2002 2003) have extended the available veryextensive low-pressure experimental data set on the solu-bility of SiO2 in H2O to 20 kbar and 1300degC thus providingan excellent data base for developing and testing alternativeapproaches to the problem Manning (1994) also derived anempirical polynomial equation that describes the solubilityof quartz in aqueous fluid well from 1 bar and 25degC up to20 kbar and 900degC

log(mSiO2(aq)) = 42620 ndash 57642T + 17513 times 106T 2 ndash22869 times 108T 3 + [28454 ndash 10069T + 35689 times 105T 2]times log(ρH2O)

where mSiO2(aq) is the molality of SiO2 in the fluid molkgρH2O is the density of pure water in gcm3 at given pressureand temperature (T K) Definitions of the thermodynamicvariables used in this paper are summarized in Table 1

Experimental studies on the solubility of quartz in fluidmixtures (Walther amp Orville 1983) on the diffusion ofdissolved silica (Applin 1987) and in situ determinationsof the Raman spectra of dissolved silica species in H2O-SiO2 fluids (Zotov amp Keppler 2000 2002) up to 900degC and14 kbar provide both qualitative and quantitativeconstraints on the stoichiometry and quantities of dissolvedsilica species However at present no precise generallyapplicable thermodynamic treatment describing both thebulk solubility and the speciation of silica in aqueoussilica-rich fluid has been developed In the present paperwe derive and test such a formalism for the SiO2 ndash H2Osystem using a combination of the chain reaction approachand a new Gibbs free energy equation of water based on ahomogeneous reaction formalism (Gerya amp Perchuk 1997Gerya et al 2004)

Thermodynamic formalism

As suggested by Dove amp Rimstidt (1994) silicadissolves in water by the following general reaction

SiO2(s) + n H2O = SiO2 bull nH2O(aq) (1)

Accordingly the thermodynamic equilibrium conditionfor the dissolution of quartz is

∆Gdeg(1)rPT + RTln[aSiO2bullnH2O(aq)] ndash n times ln(aH2O) ndashn times ln(fH2Odeg) = 0 (2)

where ∆Gdeg(1)rPT is the standard molar Gibbs free energychange of reaction (1) as a function of temperature T andpressure P ai are activities of fluid components and fH2Odegis the fugacity of pure H2O at given pressure and tempera-ture (eg Holland amp Powell 1998) In the case of low bulksilica concentrations (YSiO2

lt 005 or mSiO2(aq) lt 1 where YSiO2

= SiO2(SiO2+H2O) and m is molality in molkg) aH2O asymp 1and aSiO2bullnH2O(aq) asymp YSiO2

Then equation (2) can be rewritten as

ln(YSiO2) = n times ln(fdegH2O) ndash ∆Gdeg(1)rPTRT (3)

implying an apparently simple thermodynamic treatmentof this system However the available experimental data(Fig 1a) indicate that the correlation of ln(fH2O

o) andln(YSiO2) along different isotherms is actually verynonlinear Our test calculations have shown that equations(1)-(3) fail to provide a precise continuous description ofexperimental data on quartz solubility at both low and highpressure from 1 bar to 20 kbar with a reasonably smallnumber of empirical parameters for the representation of∆Gdeg(1)rPT (eg Walther amp Helgeson 1977) On the otherhand Manning (1994) has demonstrated a strong positivecorrelation between the logarithm of silica concentration inquartz-buffered H2O-SiO2 fluid and the logarithm of thedensity of pure H2O fluid along different isotherms (Fig1b) Our calculations have also shown a similar strongpositive correlation between isothermal silica solubilityand the degree of association or clustering of H2Omolecules in pure H2O (Fig 1c) which increases withincreasing density (eg Gorbaty amp Kalinichev 1995Luck 1980 Gerya amp Perchuk 1997 Gerya et al 2004)This strongly suggests that the dissolution of silica isdirectly related to the actual proportion of associated (ieclustered ldquoliquid-likerdquo) H2O molecules in the aqueousfluid and offers the possibility of deriving a general ther-modynamic description for silica solubility in water

It has been shown recently (Gerya amp Perchuk 1997Gerya et al 2004) that from the viewpoint of thermody-namics pure H2O fluid (and many other gases) can be effec-tively treated as a mixture of molecules in two distinct states(1) a ldquoliquid-likerdquo (associated clustered) state (defined as(H2O)L in this paper) and (2) a ldquogas-likerdquo (unassociatedfree) state (defined as (H2O)G) This phenomenologicalapproach uses the concept of homogeneous reactions of H2Omolecules leading to their liquid-like linkage (for a detaileddiscussion of the thermodynamic background and derivationof the associationclustering reactions see Gerya amp Perchuk1997 Gerya et al 2004) Such an approach is theoreticallywell developed for describing the hydrogen bonding insupercritical fluids (eg Luck 1980 Gupta et al 1992)and its potential applicability to any type of fluid has beendemonstrated (Barelko et al 1994 Gerya amp Perchuk1997) This approach is also in accord with the detailedRaman spectroscopic studies of Lindner (1970) and Kohl etal (1991) who concluded on the basis of their data thatwater can be treated as a solution of liquid-like and gas-liketypes of H2O molecules According to this formalism themolar Gibbs free energy equation of pure H2O valid for awide range of P-T conditions takes the form (see Gerya ampPerchuk 1997 Gerya et al 2004 for details of derivation)

GH2O = Gs + Gm (4)

where

Gs = H2981 ndash TS2981 + VdegsΨ + RT[c1ln(1 ndash e1) + c2ln(1 ndash e2)]ndash (c1 + c2)[∆Hs1

o(1 ndash TTo) eo(1 ndash eo) + RTln(1 ndash eo)] (5)

Gm = RT[XLiqln(XLiq) + XGasln(XGas)] + XGasRTln(P + φ XLiq2) ndash

XGas∆HdegLiq-Gas ndash T∆SdegLiq-Gas + ∆CPdegLiq-Gas[T ndash 29815 ndash Tln(T 29815)]+ WH ndash TWS + WCp[T ndash 29815 ndash Tln(T29815)]XLiqXGas (6)

Modeling of silica solubility and speciation 271

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

and e1 = exp[ndash (∆Hdegs1 + ∆Vdegs1Ψ)RT)] e2 = exp[ndash ∆Hdegs1RT)]eo = exp[ndash ∆Hdegs1R29815)] Ψ = 54(1 + φ)15[(P + φ)45 ndash(1 + φ)45] XLiq and XGas are the mole fractions (or morespecifically the thermodynamic probabilities in this

dynamic system) of molecules in ldquoliquid-likerdquo and ldquogas-likerdquo states respectively and XLiq + XGas = 1 Gs is the stan-dard molar Gibbs free energy of water in the pureldquoliquid-likerdquo state (XLiq = 1) and Gm is that part of the

272

Fig 1 Experimental data on the solubility ofquartz in aqueous fluid YSiO2

= SiO2(SiO2 +H2O) as a function of various thermodynamicparameters (a) fugacity of pure H2O (b) densityof pure H2O (c) mole fraction (thermodynamicprobability) of ldquoliquid-likerdquo (associated clus-tered) aqueous molecules (XLiq) in pure H2Ocalculated according to equations (4)-(9) Thedifferent symbols represent experimental datafrom the literature sources indicated Theisotherms have been obtained from the empiricalpolynomial fit suggested by Manning (1994)

Gibbs free energy of H2O related to the association (clus-tering) of molecules ∆HdegLiq-Gas = ndash 44839 J ∆SdegLiq-Gas =ndash 1224 JK and ∆CPdegLiq-Gas = 215 JK are the differencesin standard molar enthalpy entropy and heat capacity ofH2O in ldquoliquid-likerdquo and ldquogas-likerdquo states at reference pres-sure (Po = 1 bar) and temperature (To = 29815 K) WH =ndash28793 J WS = ndash117 JK and WCp = 51 JK are enthalpyentropy and heat capacity Margules parameters for theinteraction of ldquoliquid-likerdquo and ldquogas-likerdquo water moleculesat Po and To φ = 6209 bar is a parameter characterizing theattractive forces between all H2O molecules H2981 =ndash 286832 J S2981 = 652 JK are respectively the standardmolar enthalpy of formation from the elements and themolar entropy of ldquoliquid-likerdquo water molecules at Po andTo Vdegs = 1714 Jbar is the standard molar volume ofldquoliquid-likerdquo water molecules at T = 0 K and Po c1 = 7236c2 = 0315 ∆Hdegs1 = 4586 J and ∆Vdegs1 = 00431 Jbar areempirical parameters for the vibrational part of the Gibbspotential

By using the Gibbs-Duhem relation chemical poten-tials of ldquoliquid-likerdquo (microLiq) and ldquogas-likerdquo (microGas) moleculescan be calculated from equations (4)-(6) as follows

microLiq = Gs + RTln(XLiq) + 2RTXLiqXGas2φ(P + φ XLiq

2) + WH ndashTWS + WCp[T ndash 29815 ndash Tln(T29815)]XGas

2 (7)

microGas = Gs + RTln(XGas) + RTln(P + φ XLiq2) ndash 2RTXLiq

2XGasφ(P+ φ XLiq

2) ndash ∆HdegLiq-Gas ndash T∆SdegLiq-Gas + ∆CPdegLiq-Gas[T ndash 29815ndash Tln(T29815)] + WH ndash TWS + WCp[T ndash 29815 ndash Tln(T29815)]XLiq

2 (8)

The equilibrium values for XLiq and XGas in a pure H2Ofluid at given pressure and temperature can then be calcu-lated according to the standard relation

Gm = min ie microLiq = microGas (9)

The software for calculating the equilibrium values ofXLiq according to equation (9) is available by request fromthe corresponding author

Furthermore it is well-known (eg Applin 1987 Doveamp Rimstidt 1994 Zotov amp Keppler 2000 2002) thatdissolved silica can occur in the form of various types ofspecies Although (SiO2)bull(H2O)2-monomers dominate inthe fluid at low silica concentrations in equilibrium withquartz at 25oC and 1 bar (eg Walther amp Helgeson 1977Cary et al 1982 Applin 1987) (SiO2)2bull(H2O)3-dimersand possibly higher polymers become important at highersilica concentrations particularly with increasing pressureand temperature (eg Zotov amp Keppler 2000 2002Newton amp Manning 2002 2003) In the following there-fore we propose that H2O-SiO2-fluids can be treated as amixture of ldquoliquid-likerdquo (H2O)L and ldquogas-likerdquo (H2O)Gwater molecules with (SiO2)bull(H2O)2-monomers and allhigher (SiO2)nbull(H2O)n+1-polymers (2 le n le infin) such that

XL + XG + X1 +Σn=2

infin Xn= 1 (10)

where XL and XG are the mole fractions of ldquoliquid-likerdquo(H2O)L and ldquogas-likerdquo (H2O)G molecules respectively X1

is the mole fraction of (SiO2)bull(H2O)2-monomers and Xn isthe mole fraction of silica polymers with dimension ofn ge 2

On the basis of the above we can describe both the solu-bility and the speciation of silica in an aqueous fluid by thefollowing two types of reactions

a) the monomer-forming standard reaction

SiO2(s) + 2(H2O)L = (SiO2)bull(H2O)2 (11)

b) the polymer-forming chain reaction

(SiO2)n-1bull(H2O)n + (SiO2)bull(H2O)2 = (SiO2)nbull(H2O)n+1 + (H2O)L (12)

where 2 le n le infin The equilibrium conditions for these tworeactions can be written as∆Gdeg(mono)rPT + RTln(K(mono)) = 0 (13)

∆Gdeg(poly)rPT + RTln(K(poly)) = 0 (14)

where K(mono) = a1(aL)2 and K(poly) = anaL(an-1a1) are theequilibrium constants of reactions (11) and (12) respec-tively aL is the activity of ldquoliquid-likerdquo water molecules(H2O)L a1 is the activity of silica monomers and an as wellas an-1 are the activities of silica polymers with dimensionsof n and n-1 respectively ∆Gdeg(mono)rPT and ∆Gdeg(poly)rPT arethe standard Gibbs free energy changes of reactions (11)and (12) as functions of P and T respectively

In order to obtain analytical expressions for silica solu-bility we first assume that ∆Gdeg(poly)rPT is independent of nAccording to equations (7) and (8) the activity of ldquoliquid-likerdquo water molecules differs from the mole fraction andvaries with pressure and temperature For the followingderivations we therefore assume that the activities of thesilica species are also different from their mole fractionsand vary with pressure and temperature in such a way thatthe ratios of the activity coefficients (but not the absolutevalues of these coefficients) for the equilibrium constantsof reactions (11) and (12) are always equal to one

γ1(γL)2 = 1 (15)

γnγL(γn-1γ1) = 1 (16)

where γi = aiXi ne 1The chemical potentials for all species in H2O-SiO2 fluidscan then be formulated according to reactions (11) (12)and equations (7) (8) as follows(H2O)G - standard state XG = 1 XGas = 1 XLiq = 0

micro[(H2O)G] = RTln(XG) + Gs + RTln(P + φ XLiq2) ndash

2RTXLiq2XGasφ(P + φ XLiq

2) ndash ∆HdegLiq-Gas ndash T∆SdegLiq-Gas +∆CPdegLiq-Gas[T ndash 29815 ndash Tln(T29815)] + WH ndash TWS +WCp[T ndash 29815 ndash Tln(T29815)]XLiq

2 (17)

(H2O)L ndash standard state XL =1 XLiq = 1 XGas = 0

micro[(H2O)L] = RTln(XL) + Gs + 2RTXLiqXGas2φ(P + φ XLiq

2)+ WH ndash TWS + WCp[T ndash 29815 ndash Tln(T29815)]XGas

2 (18)

(SiO2)bull(H2O)2 ndash standard state X1 = 1 XLiq = 1 XGas = 0

Modeling of silica solubility and speciation 273

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

micro[(SiO2)bull(H2O)2] = RTln(X1) + ∆Go(mono)rPT + Go

Qtz + 2Gs+ 4RTXLiqXGas

2φ(P + φ XLiq2) + 2WH ndash TWS + WCp[T ndash

29815 ndash Tln(T29815)]XGas2 (19)

(SiO2)nbull(H2O)n+1 ndash standard state Xn = 1 XLiq = 1 XGas = 0

micro[(SiO2)nbull(H2O)n+1] = RTln(Xn) + n∆Gdeg(mono)rPT + (n ndash1)∆Gdeg(poly)rPT + nGdegQtz + (n + 1)Gs + (n +1)2RTXLiqXGas

2φ(P + φ XLiq2) + (n + 1)WH ndash TWS +

WCp[T ndash 29815 ndash Tln(T29815)]XGas2 (20)

where GdegQtz is the standard molar Gibbs free energy ofquartz (eg Holland amp Powell 1998 Gerya et al 19982004) XLiq and XGas are respectively equilibrium molefractions (thermodynamic probabilities) of ldquoliquid-likerdquoand ldquogas-likerdquo water molecules calculated for pure H2Ofluid at given pressure and temperature according to equa-tions (7)-(9) The presence of XLiq- and XGas-dependentterms in equations (19) (20) means that according to ourmodel the thermodynamic properties of silica speciesstrongly depend on the degree of association of moleculesin the aqueous fluid These XLiq- and XGas-dependent termsare similar to those of ldquoliquid-likerdquo rather than ldquogas-likerdquowater molecules (cf equations (17)-(20))

For the following derivations we assume that the pres-ence of silica species affects the absolute values but doesnot change the ratio between the mole fractions of ldquoliquid-likerdquo and ldquogas-likerdquo water molecules in SiO2-H2O-fluidsThis ratio then always corresponds to the conditions ofequilibrium in a pure water fluid at the same pressure andtemperature

XLXG = XLiqXGas (21)

The following analytical expressions can then beobtained for the concentrations of different fluid species

K(mono) = exp(ndash ∆Gdeg(mono)rPT RT) (22)

K(poly) = exp(ndash ∆Gdeg(poly)rPT RT) (23)

XL = [4XLiqK(mono)(XLiq ndash K(poly)) + (1 + XLiqK(mono)K(poly))2]05

ndash (1 + XLiqK(mono)K(poly))[2K(mono)(XLiq ndash K(poly))] (24)

X1 = K(mono)(XL)2 (25)

Xn = X1(K(poly)X1XL)n-1 (26)

XG = XGasXLXLiq (27)

The bulk concentration of dissolved silica in the fluid YSiO2

= SiO2(SiO2+H2O) and the relative proportion ofdissolved silica in monomers YSiO2(mono) = SiO2(mono) SiO2(total)as well as in higher polymers YSiO2(n-poly) = SiO2(n-poly)SiO2(total) can then be calculated according to

YSiO2= (Σ

infin

i=l Xii)[Σ

infin

i=l Xi (2i + 1) + XL + XG]

= (1 ndash XL ndash XG)2[2(1 ndash XL ndash XG)2 + X1](28)

YSiO2(mono) = X1(Σinfin

i=l Xii) = (X1)2(1 ndash XL ndash XG)2 (29)

YSiO2(n-poly) = nXn(Σinfin

i=l Xii) = nX1Xn(1 ndash XL ndash XG)2 (30)

To derive equations (28)-(30) we have used equation (26)

the condition Σinfin

i=l Xi+ XL + XG = 1 (cf equation (10)) and

the standard mathematical relations Σinfin

i=l Zi-1 = 1(1 ndash Z) and

Σinfin

i=l Zi-1i = 1(1 ndash Z)2 where 0 lt Z lt 1 in order to obtain

Σinfin

i=l Xi = X1(1 ndash K(poly)X1XL) = 1 ndash XL ndash XG Σ

infin

i=l Xii = X1(1 ndash

K(poly)X1XL)2 = (1 ndash XL ndash XG)2X1

Application to available experimental data

Our test calculations for the H2O-SiO2 system in a wideregion of pressure (1-20000 bar) and temperature (25-1300oC) have shown that the standard molar Gibbs freeenergy change for reactions (11) and (12) can be repre-sented well by the following relatively simple standardformulae

∆Gdeg(mono)rPT = ∆Hdeg(mono)r ndash T∆Sdeg(mono)r + ∆Cpdeg(mono)r

[T ndash To ndash Tln(TTo)] + ∆Vdeg(mono)r(PndashPo) (31)

∆Gdeg(poly)rPT = ∆Hdeg(poly)r ndash T∆Sdeg(poly)r + ∆Vdeg(poly)r(P ndash Po) (32)

where To = 29815 K and Po = 1 bar are standard pressureand temperature respectively ∆Hdegr ∆Sdegr ∆Cpdegr and ∆Vdegrare respectively standard molar enthalpy entropy isobaricheat capacity and volume changes at To and Po in reactions(11) and (12)

In the case where additional SiO2-buffering mineralassemblages are equilibrated with aqueous fluid (egZhang amp Frantz 2000 Newton amp Manning 2003) theGibbs free energy change of the monomer-forming reac-tion (11) can be obtained as follows

∆Gdeg(mono)rPT = ∆Go(Buffer-Quartz)PT + ∆Hdeg(mono)r ndash

T∆Sdeg(mono)r + ∆Cpdeg(mono)r [T ndash To ndash Tln(TTo)] +∆Vdeg(mono)r(P ndash Po) (33)

where ∆Go(Buffer-Quartz)PT is the difference between the stan-

dard molar Gibbs free energies of the given SiO2-bufferingmineral assemblage and quartz in Jmol which is a func-tion of P and T (Newton amp Manning 2003) For calculating∆Go

(Buffer-Quartz)PT we used the internally consistent thermo-dynamic database of Holland amp Powell (1998) As shownby Zhang amp Frantz (2000) and Newton amp Manning (2003)experimental data on silica solubility in the presence ofvarious SiO2-buffering mineral assemblages allow accuratemodeling of silica speciation at elevated pressure andtemperature

In order to quantify the seven parameters in equations(31) and (32) we have used a set of 439 experimental datapoints (Table 2) retrieved from the literature on the bulksilica solubility in the presence of quartz (Anderson ampBurnham 1965 Walther amp Orville 1983 Hemley et al1980 Kennedy 1950 Morey et al 1962 Manning 1994

274

Modeling of silica solubility and speciation 275

Ref

eren

ces

SiO

2-bu

ffer

No

ofP

kba

rT

degC

YSi

O2

Stat

isti

cal

Mea

n sq

uare

∆∆ YSi

O2

data

wei

ght

for

erro

rmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

poin

tsea

ch d

ata

σσ (Y

SiO

2)

min

max

poin

t

And

erso

n amp

Bur

nham

(19

65)

quar

tz52

1-9

8559

6-90

00

0008

-00

31

12 (

16)

-6 (

-3)

+35

(+

41)

Wal

ther

amp O

rvil

le (

1983

)qu

artz

131-

235

0-55

00

0004

-00

021

6 (8

)-1

2 (-

15)

+9

(+16

)H

emle

y et

al

198

0qu

artz

141-

220

0-50

00

0001

-00

011

7 (9

)-1

4 (-

14+

3 (+

4)K

enne

dy (

1950

)qu

artz

870

01-1

75

182-

610

000

005-

000

11

13 (

11)

-11

(-42

)+

38 (

+12

)M

orey

et

al (

1962

)qu

artz

281

0145

-300

000

0006

-00

003

124

(19

)-3

7 (-

31)

-3 (

-6)

Man

ning

(19

94)

exp

qu

artz

515-

2050

0-90

00

002-

004

18

(10)

-17

(-21

)+

17 (

+22

)M

anni

ng (

1994

) c

alc

quar

tz91

000

1-20

25-9

000

0000

02-0

04

39

-16

+14

Wat

son

amp W

ark

(199

7)qu

artz

310

570-

810

000

5-0

021

14 (

17)

+7

(+15

)+

18 (

+18

)N

ewto

n amp

Man

ning

(20

02)

quar

tz e

nsta

tite

-for

ster

ite

22

1-14

700-

900

000

05-0

05

18

(5)

-14

(-13

)+

15 (

0)N

ewto

n amp

Man

ning

(20

03)

quar

tz e

nsta

tite

-for

ster

ite

612

800

000

1-0

021

10-9

+22

kyan

ite-

coru

ndum

fors

teri

te-r

util

e-ge

ikie

lite

Woo

dlan

d amp

Wal

ther

(19

87)

quar

tz53

095

-27

134

5-50

00

0004

-00

021

7 (9

)-2

0 (-

19)

+11

(+

18)

Wal

ther

amp W

oodl

and

(199

3)qu

artz

151

5-2

400-

600

000

09-0

002

113

(13

)-1

9 (-

19)

+17

(+

24)

Hem

ley

et a

l(1

977a

b)

talc

-chr

ysot

ile-

wat

er t

alc-

751-

220

0-71

50

0000

1-0

001

122

(-32

)+

70fo

rste

rite

-wat

er t

alc-

anti

gori

te-

wat

er t

alc-

anth

ophy

llit

e-w

ater

ta

lc-e

nsta

tite

-wat

er

anth

ophy

llit

e-fo

rste

rite

-wat

erH

emle

y et

al

(197

7b)

enst

atit

e-fo

rste

rite

71

680-

720

000

047-

000

053

110

(9)

-14

(-16

)+

15 (

+4)

Zha

ng amp

Fra

ntz

(200

0) e

xp

enst

atit

e-fo

rste

rite

610

-20

900-

1200

000

6-0

041

34 (

62)

+16

(0)

+52

(+

103)

Zha

ng amp

Fra

ntz

(200

0) c

alc

enst

atit

e-fo

rste

rite

710

700-

1300

000

2-0

091

31 (

60)

-2 (

-19)

+51

(+

103)

Dov

e amp

Rim

stid

t (1

994)

quar

tz1

000

125

000

0002

100

000

(-9)

--

Tabl

e 2

Exp

erim

enta

l da

ta u

sed

for

the

cali

brat

ion

of e

quat

ions

(31

)-(3

3)

va

lues

in

brac

kets

are

cal

cula

ted

on t

he b

asis

of

the

poly

nom

ial

fits

of

Man

ning

(19

94)

and

New

ton

amp M

anni

ng (

2002

)

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

Walther amp Woodland 1993 Woodland amp Walther 1987Watson amp Wark 1997 Newton amp Manning 2002) andother SiO2-buffering mineral assemblages (Zhang ampFrantz 2000 Newton amp Manning 2002 2003 Hemley etal 1977ab) as well as silica diffusion (Applin 1987) andNMR data (Cary et al 1982) at 25degC and 1 bar for esti-mating both the bulk mole fraction of silica and the fractionof silica in monomeric form In situ measurements ofRaman spectra for H2O-SiO2 fluids coexisting with quartz(Zotov amp Keppler 2000 2002) provide qualitativeconstraints in that the relative amount of silica in polymericform must increase with an isochoric increase of tempera-ture for T = 600-900degC and P = 56-14 kbar

As discussed further below it was not possible to inte-grate the numerical results of Zotov amp Keppler (2002) whoderived quantitative data on the mole fractions of silicamonomers and dimers from in situ Raman spectra ofaqueous fluid equilibrated with quartz at pressure from56-14 kbar and temperatures from 700 to 900degC Theseresults have been excluded from our database for tworeasons (i) our calculations show that these data are inlarge part inconsistent with the other studies consideredhere requiring a much more pronounced dependence ofsilica speciation on the density of the fluid at a giventemperature and (ii) it was not possible to resolve thediscrepancies in the paper by Zotov amp Keppler (2002)between the formalism presented in their equations (9) and(10) and the derived data on monomer and dimer quantitiespresented in their Table 2

For the calculation of thermodynamic parameters weused a non-linear least square method in order to minimizethe relative error of reproducing the bulk silica concentra-tion which is defined as∆YSiO2

() = 100[YSiO2(calculated) ndash

YSiO2(experimental)]YSiO2

(experimental)We have used a uniform statistical weight of 1 for all exper-imental data points (Table 2) thus making no attempt togauge the relative quality of the various data subsets intheir accuracy and precision (see discussion by Manning

1994) However because of the very irregular distributionof data points in pressure and temperature we found itnecessary to balance our database by adding 91 pointsinterpolated from the polynomial equation of Manning(1994) within the P-T range of experiments used for thecalibration of this equation These ldquosyntheticrdquo points wereregularly distributed with pressure (1 kbar steps) along 9isotherms (100 to 900degC with 100degC steps) A higherstatistical weight of 3 was assigned to these data points Inorder to insure an exact reproduction of the concentrationand the speciation of silica at the reference conditions of To= 25degC and Po = 1 bar (Morey et al 1962 Applin 1987Dove amp Rimstidt 1994 Cary et al 1982) we assigned avery high statistical weight of 10000 to the single datapoint representing these conditions The resulting valuesfor the thermodynamic parameters of equations (31) and(32) are∆Hdeg(mono)r = 24014 plusmn 1781 J∆Sdeg(mono)r = ndash 2810 plusmn 423 JK∆Vdeg(mono)r = ndash 02354plusmn00749 Jbar∆Cpdeg(mono)r = 3196 plusmn 645 JK∆Hdeg(poly)r = ndash 21059 plusmn 3589 J∆Sdeg(poly)r = 966 plusmn 1166 JK∆Vdeg(poly)r = 03161 plusmn 01798 Jbar

Table 2 and Fig 2 3 show that the data set is fitted wellby the above parameters The mean square error in repro-ducing 439 data points on bulk silica solubility is 142 relative No systematic variations of calculated errors withtemperature pressure bulk silica concentration and XLiqhave been found (Fig 3) In quartz-buffered systems rela-tively large deviations (gt 20 ) are indicated for the exper-imental data of Morey et al (1962) and Kennedy (1950) atpressures below 1 kbar and temperatures below 200degC Ourtest calculations show that these data are in part inconsis-tent with the bulk silica concentrations at the referenceconditions of To = 25degC and Po = 1 bar (Dove amp Rimstidt1994) requiring either notably higher (Morey et al 1962)or lower (Kennedy 1550) values These discrepancies canin part be explained by the problems of reaching equilib-

276

Fig 2 Results of modeling bulk molality ofdissolved silica in aqueous fluid in the presence ofquartz based on equations (21)-(30) (solid lines)The different symbols represent experimental datafrom the sources indicated Dashed lines are calcu-lations on the basis of the empirical equationsuggested by Manning (1994)

Modeling of silica solubility and speciation 277

Fig 3 Distribution of relative error ∆YSiO2() = 100[YSiO2

(calculated)-YSiO2(experimental)]YSiO2

(experimental) of reproducing experi-mental data with temperature (a) pressure (b) bulk silica concentration (c) and degree of association (XLiq) of pure water fluid (d) Thedifferent symbols represent experimental data from the sources indicated

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

rium at low temperatures (eg Walther amp Helgeson 1977)Indeed the overall accuracy of reproducing these experi-ments is comparable (Table 2) with the polynomial equa-

tion suggested by Manning (1994) Larger errors weredetected in silica-undersaturated systems (Table 2 Fig 3)for the experimental data of Hemley et al (1977a b) and

278

Fig 4 Relative concentration of silica inmonomers (a) dimers (b) and higher silicapolymers (c) dissolved in aqueous fluid in thepresence of quartz at P = 0001-20 kbar and T= 25-90degC Calculations are based on equa-tions (21)-(30) Thin grey lines are isobarsBlack lines are isotherms The solid squarerepresents the experimental data of Cary et al(1982) and Applin (1987) at 25degC and 1 bar

Zhang amp Frantz (2000) This may be related to the rela-tively large experimental uncertainties (Zhang amp Frantz2000 Newton amp Manning 2002) and to possible errorsincurred during the calculation of ∆Go

(Buffer-Quartz)PT forSiO2-buffering reactions involving water at low pressureand temperature (Hemley et al 1977a b) in combinationwith the database of Holland amp Powell (1998) Obviouslyat T = 25degC and P = 1 bar the highly weighted referencedata on the concentration of silica in the presence of quartz(Dove amp Rimstidt 1994) and on the distribution of silica

between (SiO2)bull(H2O)2-monomers and (SiO2)2bull(H2O)3-dimers (Cary et al 1982 Applin 1987) are accuratelyreproduced with a relative error of less then 02 (Table 2Fig 3 4a) Figure 2 shows that the results of our thermo-dynamic modeling coincide well with the empiricalapproximation suggested by Manning (1994) for the bulksolubility of silica in water in the presence of quartz Table2 also compares our numerical values (Table 2) with thoseobtained from the empirical equations of Manning (1994)for the quartz-buffered system and of Manning amp Newton

Modeling of silica solubility and speciation 279

Fig 5 Results of modeling bulk concentration (a)(b) and speciation (c) of dissolved silica in aqueousfluid in the presence of different buffering mineralassemblages The symbols represent data from thesources indicated (a) experimental data of Newtonamp Manning (2002) and empirical calibration ofZhang amp Frantz (2000) based on original andpublished (Nakamura amp Kushiro 1974 Manningamp Boettcher 1994) experimental data (b) experi-mental data of Newton amp Manning (2003) (c)monomer-dimer speciation model of Newton ampManning (2003)

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

(2002) for the enstatite-forsterite-buffered system For the358 compared experimental data points (Table 2) ourmodel yields a mean square error of 134 versus 165 for the equations of Manning (1994) and Manning ampNewton (2002)

As noted above our model was constrained to accu-rately reproduce experimental data on silica partitioningbetween monomers and larger clusters at 25degC and 1 bar(Cary et al 1982 Applin 1987) The silica species at theseconditions (Fig 4) are represented by (SiO2)bull(H2O)2-monomers (94 of total dissolved silica) and(SiO2)2bull(H2O)3-dimers (6 of total dissolved silica) witha negligible amount of higher polymers (lt 03 of totaldissolved silica) Cary et al (1982) detected the presenceof (SiO2)2bull(H2O)3-dimers by NMR in solutions containing16 mM SiO2 at a pH of 72 and estimated their concentra-tion to be about 6 of the total dissolved silica No addi-tional polymerized forms were detected which isconsistent with our calculations for higher polymers (Fig4c) However the lack of such accurate experimental datafor speciation over a wider range of pressures and temper-atures (eg Watson amp Wark 1997) leads to the relativelylarge uncertainties in ∆Sdeg(poly)r and ∆Vdeg(poly)r ie for chainreaction (12) It should be noted that reaction (12) can alsobe used for the calculation of silica speciation at an arbi-trary bulk concentration of dissolved silica YSiO2

=SiO2(SiO2 + H2O) Equations characterizing speciationthen become

X1 = XLiqXSS(1 ndash XSS)[XSSK(poly) + (1 ndash XSS)XLiq] (34)

XL = XLiq(1 ndash XSS) (35)

XSS = [4XLiqYSiO2(K(poly) ndash XLiq)(1 ndash 2YSiO2

) + (XLiq)2(1 ndashYSiO2

)2]05 ndash XLiq(1 ndash YSiO2) [2(K(poly) ndash XLiq)(1 ndash 2YSiO2

)] (36)

where XSS = X1 + Σinfin

n=2 Xn is the sum of the mole fractions of

all dissolved silica species (ie XSS + XL + XG = 1) Otherquantities are as given in equations (21)-(27)

Figure 4 shows SiO2 speciation in the presence ofquartz as a function of pressure temperature and bulksilica content as predicted from our model calculations Itis seen that the relative amount of silica in the form ofmonomers is most strongly dependent on bulk silicaconcentration and generally decreases with increasingSiO2 content of the solution (Fig 4a) Significant amountsof dissolved silica (10-30 ) in dimeric form are calculatedfor temperatures gt 300degC (Fig 4b) The relative proportionof dimeric silica shows a maximum (Fig 4b) at YSiO2

asymp001 At higher YSiO2

trimers and higher polymers becomeimportant (Fig 4c) consuming 10 to 50 of the dissolvedsilica at temperatures gt 500degC Because increasing temper-ature leads to higher bulk silica there is also a similarcorrelation with temperature The dependence on pressureis least pronounced as can be expected from the competingeffects of reactions (11) and (12) (H2O)L is consumedduring the monomer-forming reaction (11) and liberatedduring the polymer-forming reaction (12) thereforedictating opposite signs of ∆Vdegr of the reactions Increasingpressure should lead to increased bulk silica solubility (dueto reaction (11) ∆Vdeg(mono)r lt 0) and therefore increasing

polymerization on the one hand (Fig 4a) whereasincreasing pressure should also favor depolymerizationaccording to reaction (12) (∆Vdeg(poly)r gt 0) on the other

Figure 5 shows silica solubility and speciation inaqueous fluid coexisting with different SiO2-bufferingmineral assemblages It is seen that our model accuratelyreproduces the available experimental data (Zhang ampFrantz 2000 Newton amp Manning 2003) on bulk silicasolubility (Fig 5ab) The calculated distribution of silicabetween monomers and higher polymers as a function ofbulk SiO2(SiO2 + H2O) at constant pressure and tempera-ture (Fig 5c) is very similar to the results of Newton ampManning (2003) which are based on a somewhat simplermonomer-dimer model At constant pressure and tempera-ture the amount of monomeric silica decreases non-linearlywith decreasing bulk silica solubility

Discussion

The semi-empirical thermodynamic formalism devel-oped in this paper allows accurate reproduction of existingdata on silica concentration and silica speciation inaqueous fluid coexisting with quartz and other silica-buffering assemblages up to 20 kbar and 1300degC Incomparison to Mannings (1994) polynomial fit ourapproach also predicts the speciation of silica in theaqueous fluid without increasing the number of empiricalparameters in the model In addition our chain-reactionmodel allows an excellent data fit with full consideration ofall silica polymers (Fig 4) rather than only monomers anddimers (cf Newton amp Manning 2003) Figure 6 shows thepositions of isopleths of bulk silica concentration calcu-lated for both silica-saturated (Fig 6a) and the silica-under-saturated enstatite-forsterite-buffered (Fig 6b) systems atelevated temperature and pressure It is seen that our resultsare in general compatible with the polynomial equationssuggested by Manning (1994) and Newton amp Manning(2002) with the largest discrepancies found in the high-temperature (gt 900degC) high-YSiO2

(gt 005) region Itshould be emphasized here that our present model has beendeveloped for a bulk silica concentration (YSiO2

) in the fluidof less than 01 (Table 2) Therefore this model mayrequire modification (eg by incorporating terms for non-ideality of the aqueous solution) in order to account prop-erly for the thermodynamics of silica-rich hydroussolutions (eg Shen amp Keppler 1997 Audetat amp Keppler2004) in proximity to the second critical point in the H2O-SiO2 system

Our model predicts basic similarities between the ther-modynamic properties of dissolved silica species and thoseof associated clustered ldquoliquid-likerdquo water molecules (egequations (18)-(20)) This is consistent with the conclu-sions of Bockris (1949) and Walther amp Orville (1983) whosuggested that hydrous monomeric silica species _ Si(OH)4_ are associated with two more water dipoles to form anaqueous silica complex _ Si(OH)4bull2H2O _ by hydrogenbonding This mechanism of association of molecules is infact identical to the association (clustering) of molecules ofpure water (eg Luck 1980 Gorbaty amp Kalinichev 1995)

280

Modeling of silica solubility and speciation 281

Fig 6 Isopleths of bulk silica concentration (solid lines)calculated according to equations (21)-(27) and (31)-(33)for silica-saturated (a) and silica-undersaturated enstatite-forsterite buffered (b) system at elevated temperature andpressure Dashed lines show isopleths calculated with theequations suggested by Manning (1994) and Newton ampManning (2002) Phase boundaries in (a) are calculatedfrom the database of Holland amp Powell (1998) Meltingrelations in the presence of water are not shown

Fig 7 Results of modeling silica speciation in aqueous fluidin the presence of quartz along the two isochores presented byZotov amp Keppler (2002) ρ = 078 gcm3 P = 56-68 kbar andρ = 094 gcm3 P = 106-140 kbar The solid squares repre-sent the results of Zotov amp Keppler (2002) based on the inter-pretation of in situ Raman spectra (a) dependence of silicaspeciation on temperature (b) dependence of silica speciationon bulk silica content

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

and is directly accounted for by equations (4)-(6) (for adiscussion see Gerya amp Perchuk 1997 Gerya et al2004) The formation of similar complexes with dimersand higher polymers can also be expected thus supportingour thermodynamic formalism

An important unresolved question is the discrepancybetween the results obtained from our data set which isbased mainly on direct solubility measurements and dataobtained from Raman spectra (Zotov amp Keppler 20002002) This is particularly evident in the pressure effect onsilica polymerization The positive ∆Vdegr of reaction (12)suggests that at constant temperature and bulk silica anyincrease in pressure (ie in fluid density) should lead to adecrease in the degree of polymerization This is alsosupported by the ab initio molecular dynamics simulationsof Doltsinis et al (2004) who calculated higher effectivepressures for monomeric-dimeric systems in comparisonwith purely monomeric systems at T = 900 K and constantfluid density In the presence of quartz bulk silica contentincreases with pressure and the competing effects of reac-tions (11) and (12) should lead to only small changes inthe degree of polymerization Figure 7 indicates the trendspredicted from our model In contrast the results of Zotovamp Keppler (2002) suggest strong increase in the degree ofpolymerization with pressure (ie fluid density) requiringa notably negative ∆Vo

r of polymer-forming reaction (12)Our test calculations show that no consistent set of ther-modynamic parameters for reactions (11) and (12) canprovide an accurate description of both the silica solubilitymeasurements and the changes of degree of polymeriza-tion with fluid density calculated by Zotov amp Keppler(2002)

We suggest that an explanation for the above discrep-ancies must be sought in (i) the possible problems of equi-libration in the relatively short experiments in thediamond-anvil cell performed by Zotov amp Keppler (20002002) and (ii) the difficult task of quantifying the Ramanspectra of aqueous silica solutions The interpretation ofRaman spectra is based mainly on theoretical spectraobtained using empirical interaction potentials (Catlow1980 Price amp Parker 1984) or on single molecule abinitio calculations as in the case of Zotov amp Keppler(2000 2002) The problem with the latter is that they donot include crucial temperature- and pressure-dependentsolvent effects To test these environmental effects wehave initiated a series of both ab initio molecular dynamics(AIMD) simulations and static gas phase ab initio calcula-tions (Doltsinis et al 2004) Our results to date suggestthat while the strong feature at ca 780-800 cm-1 can beascribed to monomers with some confidence there arestill considerable uncertainties in interpreting otherfeatures of the Raman spectra in terms of monomersandor dimers Considering the low amounts of total silicainvolved and the increasingly floppy nature of largerpolymers leading to increasingly broadened spectralfeatures it appears that the recognition and identificationof trimers and higher polymers can be problematical Ifhidden trimers are present then an interpretation basedsolely on monomerdimer ratios will obviously be flawedThus while the results of Zotov amp Keppler (2000 2002)

represent a valiant first attempt to tackle the problem andprovide valuable qualitative data direct comparisons withclassical solubility results should not be overextended Wesuggest that further Raman spectroscopic measurementson well-defined solutions with given bulk silica contents atdifferent pressures and temperatures rather than quartz-buffered solutions only will be a valuable step toward abetter understanding of such spectra

The approach outlined in this paper has been success-fully used to fit and describe experimental data on bulkspinel (MgAl2O4) solubility up to 30 kbar byZakhartchouk et al (2004) despite the lack of detailedinformation on the actual speciation of the dissolved spinelcomponents in that system Given additional independentdata on the detailed structure of condensed phase water inthe future (eg Marx 1999 and in press) and on solutespeciation at various temperatures and pressures (seeDoltsinis et al 2004) the formalism described here canbe used to provide an increasingly complete thermody-namic description of aqueous solutions at depths corre-sponding to the lower crust and upper mantle

AcknowledgementsThis work was supported by ETHResearch Grant TH -1204-1 by RFBR grant 03-05-64633 by the RF President Program ldquoLeading ScientificSchool of Russiardquo (grant 0351645) to TVG and by theDeutsche Forschungsgemeinschaft within the scope ofSonderforschungsbereich 526 at the Ruhr-University ofBochum The helpful reviews of C Manning J Waltherand an anonymous reviewer as well as the editorialhandling of A Woodland are greatly appreciated

References

Anderson GM amp Burnham CW (1965) The solubility of quartzin supercritical water Am J Sci 263 494-511

Applin KR (1987) The diffusion of dissolved silica in diluteaqueous solution Geochim Cosmochim Acta 51 2147-2151

Audetat A amp Keppler H (2004) Viscosity of fluids in subduc-tion zones Science 303 513-516

Barelko VV Zarikov IV Pechatnikov EL (1994) Effect of theorigin of molecular clusters in one-component gas mixtures onthe equation of state Khim Fiz 13 42-50

Bockris JOrsquoM (1949) Ionic solutions Chem Soc LondonQuarterly Rev 3 173-180

Cary LW De Jong BHWS Dibble WEJr (1982) A 29Si NMRstudy of silica species in dilute aqueous solution GeochimCosmochim Acta 46 1317-1320

Catlow CRA (1980) Computer modelling of ionic crystals JPhysics 41 C6-53

Doltsinis NL Burchard M Maresch WV (2004) Ab initiomolecular dynamics simulations as a tool for interpreting vibra-tional spectra in aqueous fluids EMPG-X SymposiumAbstracts Lithos supplement to v 73 S28

Dove PM amp Rimstidt JD (1994) Silica physical behaviorgeochemistry and materials applications in ldquoSilica-water inter-actionsrdquo Reviews in Mineral 29 259-amp

Fockenberg T Burchard M Maresch WV (2004) Solubilities ofcalcium silicates at high pressures and temperatures EMPG-XSymposium Abstracts Lithos supplement to v 73 S37

282

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

and e1 = exp[ndash (∆Hdegs1 + ∆Vdegs1Ψ)RT)] e2 = exp[ndash ∆Hdegs1RT)]eo = exp[ndash ∆Hdegs1R29815)] Ψ = 54(1 + φ)15[(P + φ)45 ndash(1 + φ)45] XLiq and XGas are the mole fractions (or morespecifically the thermodynamic probabilities in this

dynamic system) of molecules in ldquoliquid-likerdquo and ldquogas-likerdquo states respectively and XLiq + XGas = 1 Gs is the stan-dard molar Gibbs free energy of water in the pureldquoliquid-likerdquo state (XLiq = 1) and Gm is that part of the

272

Fig 1 Experimental data on the solubility ofquartz in aqueous fluid YSiO2

= SiO2(SiO2 +H2O) as a function of various thermodynamicparameters (a) fugacity of pure H2O (b) densityof pure H2O (c) mole fraction (thermodynamicprobability) of ldquoliquid-likerdquo (associated clus-tered) aqueous molecules (XLiq) in pure H2Ocalculated according to equations (4)-(9) Thedifferent symbols represent experimental datafrom the literature sources indicated Theisotherms have been obtained from the empiricalpolynomial fit suggested by Manning (1994)

Gibbs free energy of H2O related to the association (clus-tering) of molecules ∆HdegLiq-Gas = ndash 44839 J ∆SdegLiq-Gas =ndash 1224 JK and ∆CPdegLiq-Gas = 215 JK are the differencesin standard molar enthalpy entropy and heat capacity ofH2O in ldquoliquid-likerdquo and ldquogas-likerdquo states at reference pres-sure (Po = 1 bar) and temperature (To = 29815 K) WH =ndash28793 J WS = ndash117 JK and WCp = 51 JK are enthalpyentropy and heat capacity Margules parameters for theinteraction of ldquoliquid-likerdquo and ldquogas-likerdquo water moleculesat Po and To φ = 6209 bar is a parameter characterizing theattractive forces between all H2O molecules H2981 =ndash 286832 J S2981 = 652 JK are respectively the standardmolar enthalpy of formation from the elements and themolar entropy of ldquoliquid-likerdquo water molecules at Po andTo Vdegs = 1714 Jbar is the standard molar volume ofldquoliquid-likerdquo water molecules at T = 0 K and Po c1 = 7236c2 = 0315 ∆Hdegs1 = 4586 J and ∆Vdegs1 = 00431 Jbar areempirical parameters for the vibrational part of the Gibbspotential

By using the Gibbs-Duhem relation chemical poten-tials of ldquoliquid-likerdquo (microLiq) and ldquogas-likerdquo (microGas) moleculescan be calculated from equations (4)-(6) as follows

microLiq = Gs + RTln(XLiq) + 2RTXLiqXGas2φ(P + φ XLiq

2) + WH ndashTWS + WCp[T ndash 29815 ndash Tln(T29815)]XGas

2 (7)

microGas = Gs + RTln(XGas) + RTln(P + φ XLiq2) ndash 2RTXLiq

2XGasφ(P+ φ XLiq

2) ndash ∆HdegLiq-Gas ndash T∆SdegLiq-Gas + ∆CPdegLiq-Gas[T ndash 29815ndash Tln(T29815)] + WH ndash TWS + WCp[T ndash 29815 ndash Tln(T29815)]XLiq

2 (8)

The equilibrium values for XLiq and XGas in a pure H2Ofluid at given pressure and temperature can then be calcu-lated according to the standard relation

Gm = min ie microLiq = microGas (9)

The software for calculating the equilibrium values ofXLiq according to equation (9) is available by request fromthe corresponding author

Furthermore it is well-known (eg Applin 1987 Doveamp Rimstidt 1994 Zotov amp Keppler 2000 2002) thatdissolved silica can occur in the form of various types ofspecies Although (SiO2)bull(H2O)2-monomers dominate inthe fluid at low silica concentrations in equilibrium withquartz at 25oC and 1 bar (eg Walther amp Helgeson 1977Cary et al 1982 Applin 1987) (SiO2)2bull(H2O)3-dimersand possibly higher polymers become important at highersilica concentrations particularly with increasing pressureand temperature (eg Zotov amp Keppler 2000 2002Newton amp Manning 2002 2003) In the following there-fore we propose that H2O-SiO2-fluids can be treated as amixture of ldquoliquid-likerdquo (H2O)L and ldquogas-likerdquo (H2O)Gwater molecules with (SiO2)bull(H2O)2-monomers and allhigher (SiO2)nbull(H2O)n+1-polymers (2 le n le infin) such that

XL + XG + X1 +Σn=2

infin Xn= 1 (10)

where XL and XG are the mole fractions of ldquoliquid-likerdquo(H2O)L and ldquogas-likerdquo (H2O)G molecules respectively X1

is the mole fraction of (SiO2)bull(H2O)2-monomers and Xn isthe mole fraction of silica polymers with dimension ofn ge 2

On the basis of the above we can describe both the solu-bility and the speciation of silica in an aqueous fluid by thefollowing two types of reactions

a) the monomer-forming standard reaction

SiO2(s) + 2(H2O)L = (SiO2)bull(H2O)2 (11)

b) the polymer-forming chain reaction

(SiO2)n-1bull(H2O)n + (SiO2)bull(H2O)2 = (SiO2)nbull(H2O)n+1 + (H2O)L (12)

where 2 le n le infin The equilibrium conditions for these tworeactions can be written as∆Gdeg(mono)rPT + RTln(K(mono)) = 0 (13)

∆Gdeg(poly)rPT + RTln(K(poly)) = 0 (14)

where K(mono) = a1(aL)2 and K(poly) = anaL(an-1a1) are theequilibrium constants of reactions (11) and (12) respec-tively aL is the activity of ldquoliquid-likerdquo water molecules(H2O)L a1 is the activity of silica monomers and an as wellas an-1 are the activities of silica polymers with dimensionsof n and n-1 respectively ∆Gdeg(mono)rPT and ∆Gdeg(poly)rPT arethe standard Gibbs free energy changes of reactions (11)and (12) as functions of P and T respectively

In order to obtain analytical expressions for silica solu-bility we first assume that ∆Gdeg(poly)rPT is independent of nAccording to equations (7) and (8) the activity of ldquoliquid-likerdquo water molecules differs from the mole fraction andvaries with pressure and temperature For the followingderivations we therefore assume that the activities of thesilica species are also different from their mole fractionsand vary with pressure and temperature in such a way thatthe ratios of the activity coefficients (but not the absolutevalues of these coefficients) for the equilibrium constantsof reactions (11) and (12) are always equal to one

γ1(γL)2 = 1 (15)

γnγL(γn-1γ1) = 1 (16)

where γi = aiXi ne 1The chemical potentials for all species in H2O-SiO2 fluidscan then be formulated according to reactions (11) (12)and equations (7) (8) as follows(H2O)G - standard state XG = 1 XGas = 1 XLiq = 0

micro[(H2O)G] = RTln(XG) + Gs + RTln(P + φ XLiq2) ndash

2RTXLiq2XGasφ(P + φ XLiq

2) ndash ∆HdegLiq-Gas ndash T∆SdegLiq-Gas +∆CPdegLiq-Gas[T ndash 29815 ndash Tln(T29815)] + WH ndash TWS +WCp[T ndash 29815 ndash Tln(T29815)]XLiq

2 (17)

(H2O)L ndash standard state XL =1 XLiq = 1 XGas = 0

micro[(H2O)L] = RTln(XL) + Gs + 2RTXLiqXGas2φ(P + φ XLiq

2)+ WH ndash TWS + WCp[T ndash 29815 ndash Tln(T29815)]XGas

2 (18)

(SiO2)bull(H2O)2 ndash standard state X1 = 1 XLiq = 1 XGas = 0

Modeling of silica solubility and speciation 273

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

micro[(SiO2)bull(H2O)2] = RTln(X1) + ∆Go(mono)rPT + Go

Qtz + 2Gs+ 4RTXLiqXGas

2φ(P + φ XLiq2) + 2WH ndash TWS + WCp[T ndash

29815 ndash Tln(T29815)]XGas2 (19)

(SiO2)nbull(H2O)n+1 ndash standard state Xn = 1 XLiq = 1 XGas = 0

micro[(SiO2)nbull(H2O)n+1] = RTln(Xn) + n∆Gdeg(mono)rPT + (n ndash1)∆Gdeg(poly)rPT + nGdegQtz + (n + 1)Gs + (n +1)2RTXLiqXGas

2φ(P + φ XLiq2) + (n + 1)WH ndash TWS +

WCp[T ndash 29815 ndash Tln(T29815)]XGas2 (20)

where GdegQtz is the standard molar Gibbs free energy ofquartz (eg Holland amp Powell 1998 Gerya et al 19982004) XLiq and XGas are respectively equilibrium molefractions (thermodynamic probabilities) of ldquoliquid-likerdquoand ldquogas-likerdquo water molecules calculated for pure H2Ofluid at given pressure and temperature according to equa-tions (7)-(9) The presence of XLiq- and XGas-dependentterms in equations (19) (20) means that according to ourmodel the thermodynamic properties of silica speciesstrongly depend on the degree of association of moleculesin the aqueous fluid These XLiq- and XGas-dependent termsare similar to those of ldquoliquid-likerdquo rather than ldquogas-likerdquowater molecules (cf equations (17)-(20))

For the following derivations we assume that the pres-ence of silica species affects the absolute values but doesnot change the ratio between the mole fractions of ldquoliquid-likerdquo and ldquogas-likerdquo water molecules in SiO2-H2O-fluidsThis ratio then always corresponds to the conditions ofequilibrium in a pure water fluid at the same pressure andtemperature

XLXG = XLiqXGas (21)

The following analytical expressions can then beobtained for the concentrations of different fluid species

K(mono) = exp(ndash ∆Gdeg(mono)rPT RT) (22)

K(poly) = exp(ndash ∆Gdeg(poly)rPT RT) (23)

XL = [4XLiqK(mono)(XLiq ndash K(poly)) + (1 + XLiqK(mono)K(poly))2]05

ndash (1 + XLiqK(mono)K(poly))[2K(mono)(XLiq ndash K(poly))] (24)

X1 = K(mono)(XL)2 (25)

Xn = X1(K(poly)X1XL)n-1 (26)

XG = XGasXLXLiq (27)

The bulk concentration of dissolved silica in the fluid YSiO2

= SiO2(SiO2+H2O) and the relative proportion ofdissolved silica in monomers YSiO2(mono) = SiO2(mono) SiO2(total)as well as in higher polymers YSiO2(n-poly) = SiO2(n-poly)SiO2(total) can then be calculated according to

YSiO2= (Σ

infin

i=l Xii)[Σ

infin

i=l Xi (2i + 1) + XL + XG]

= (1 ndash XL ndash XG)2[2(1 ndash XL ndash XG)2 + X1](28)

YSiO2(mono) = X1(Σinfin

i=l Xii) = (X1)2(1 ndash XL ndash XG)2 (29)

YSiO2(n-poly) = nXn(Σinfin

i=l Xii) = nX1Xn(1 ndash XL ndash XG)2 (30)

To derive equations (28)-(30) we have used equation (26)

the condition Σinfin

i=l Xi+ XL + XG = 1 (cf equation (10)) and

the standard mathematical relations Σinfin

i=l Zi-1 = 1(1 ndash Z) and

Σinfin

i=l Zi-1i = 1(1 ndash Z)2 where 0 lt Z lt 1 in order to obtain

Σinfin

i=l Xi = X1(1 ndash K(poly)X1XL) = 1 ndash XL ndash XG Σ

infin

i=l Xii = X1(1 ndash

K(poly)X1XL)2 = (1 ndash XL ndash XG)2X1

Application to available experimental data

Our test calculations for the H2O-SiO2 system in a wideregion of pressure (1-20000 bar) and temperature (25-1300oC) have shown that the standard molar Gibbs freeenergy change for reactions (11) and (12) can be repre-sented well by the following relatively simple standardformulae

∆Gdeg(mono)rPT = ∆Hdeg(mono)r ndash T∆Sdeg(mono)r + ∆Cpdeg(mono)r

[T ndash To ndash Tln(TTo)] + ∆Vdeg(mono)r(PndashPo) (31)

∆Gdeg(poly)rPT = ∆Hdeg(poly)r ndash T∆Sdeg(poly)r + ∆Vdeg(poly)r(P ndash Po) (32)

where To = 29815 K and Po = 1 bar are standard pressureand temperature respectively ∆Hdegr ∆Sdegr ∆Cpdegr and ∆Vdegrare respectively standard molar enthalpy entropy isobaricheat capacity and volume changes at To and Po in reactions(11) and (12)

In the case where additional SiO2-buffering mineralassemblages are equilibrated with aqueous fluid (egZhang amp Frantz 2000 Newton amp Manning 2003) theGibbs free energy change of the monomer-forming reac-tion (11) can be obtained as follows

∆Gdeg(mono)rPT = ∆Go(Buffer-Quartz)PT + ∆Hdeg(mono)r ndash

T∆Sdeg(mono)r + ∆Cpdeg(mono)r [T ndash To ndash Tln(TTo)] +∆Vdeg(mono)r(P ndash Po) (33)

where ∆Go(Buffer-Quartz)PT is the difference between the stan-

dard molar Gibbs free energies of the given SiO2-bufferingmineral assemblage and quartz in Jmol which is a func-tion of P and T (Newton amp Manning 2003) For calculating∆Go

(Buffer-Quartz)PT we used the internally consistent thermo-dynamic database of Holland amp Powell (1998) As shownby Zhang amp Frantz (2000) and Newton amp Manning (2003)experimental data on silica solubility in the presence ofvarious SiO2-buffering mineral assemblages allow accuratemodeling of silica speciation at elevated pressure andtemperature

In order to quantify the seven parameters in equations(31) and (32) we have used a set of 439 experimental datapoints (Table 2) retrieved from the literature on the bulksilica solubility in the presence of quartz (Anderson ampBurnham 1965 Walther amp Orville 1983 Hemley et al1980 Kennedy 1950 Morey et al 1962 Manning 1994

274

Modeling of silica solubility and speciation 275

Ref

eren

ces

SiO

2-bu

ffer

No

ofP

kba

rT

degC

YSi

O2

Stat

isti

cal

Mea

n sq

uare

∆∆ YSi

O2

data

wei

ght

for

erro

rmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

poin

tsea

ch d

ata

σσ (Y

SiO

2)

min

max

poin

t

And

erso

n amp

Bur

nham

(19

65)

quar

tz52

1-9

8559

6-90

00

0008

-00

31

12 (

16)

-6 (

-3)

+35

(+

41)

Wal

ther

amp O

rvil

le (

1983

)qu

artz

131-

235

0-55

00

0004

-00

021

6 (8

)-1

2 (-

15)

+9

(+16

)H

emle

y et

al

198

0qu

artz

141-

220

0-50

00

0001

-00

011

7 (9

)-1

4 (-

14+

3 (+

4)K

enne

dy (

1950

)qu

artz

870

01-1

75

182-

610

000

005-

000

11

13 (

11)

-11

(-42

)+

38 (

+12

)M

orey

et

al (

1962

)qu

artz

281

0145

-300

000

0006

-00

003

124

(19

)-3

7 (-

31)

-3 (

-6)

Man

ning

(19

94)

exp

qu

artz

515-

2050

0-90

00

002-

004

18

(10)

-17

(-21

)+

17 (

+22

)M

anni

ng (

1994

) c

alc

quar

tz91

000

1-20

25-9

000

0000

02-0

04

39

-16

+14

Wat

son

amp W

ark

(199

7)qu

artz

310

570-

810

000

5-0

021

14 (

17)

+7

(+15

)+

18 (

+18

)N

ewto

n amp

Man

ning

(20

02)

quar

tz e

nsta

tite

-for

ster

ite

22

1-14

700-

900

000

05-0

05

18

(5)

-14

(-13

)+

15 (

0)N

ewto

n amp

Man

ning

(20

03)

quar

tz e

nsta

tite

-for

ster

ite

612

800

000

1-0

021

10-9

+22

kyan

ite-

coru

ndum

fors

teri

te-r

util

e-ge

ikie

lite

Woo

dlan

d amp

Wal

ther

(19

87)

quar

tz53

095

-27

134

5-50

00

0004

-00

021

7 (9

)-2

0 (-

19)

+11

(+

18)

Wal

ther

amp W

oodl

and

(199

3)qu

artz

151

5-2

400-

600

000

09-0

002

113

(13

)-1

9 (-

19)

+17

(+

24)

Hem

ley

et a

l(1

977a

b)

talc

-chr

ysot

ile-

wat

er t

alc-

751-

220

0-71

50

0000

1-0

001

122

(-32

)+

70fo

rste

rite

-wat

er t

alc-

anti

gori

te-

wat

er t

alc-

anth

ophy

llit

e-w

ater

ta

lc-e

nsta

tite

-wat

er

anth

ophy

llit

e-fo

rste

rite

-wat

erH

emle

y et

al

(197

7b)

enst

atit

e-fo

rste

rite

71

680-

720

000

047-

000

053

110

(9)

-14

(-16

)+

15 (

+4)

Zha

ng amp

Fra

ntz

(200

0) e

xp

enst

atit

e-fo

rste

rite

610

-20

900-

1200

000

6-0

041

34 (

62)

+16

(0)

+52

(+

103)

Zha

ng amp

Fra

ntz

(200

0) c

alc

enst

atit

e-fo

rste

rite

710

700-

1300

000

2-0

091

31 (

60)

-2 (

-19)

+51

(+

103)

Dov

e amp

Rim

stid

t (1

994)

quar

tz1

000

125

000

0002

100

000

(-9)

--

Tabl

e 2

Exp

erim

enta

l da

ta u

sed

for

the

cali

brat

ion

of e

quat

ions

(31

)-(3

3)

va

lues

in

brac

kets

are

cal

cula

ted

on t

he b

asis

of

the

poly

nom

ial

fits

of

Man

ning

(19

94)

and

New

ton

amp M

anni

ng (

2002

)

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

Walther amp Woodland 1993 Woodland amp Walther 1987Watson amp Wark 1997 Newton amp Manning 2002) andother SiO2-buffering mineral assemblages (Zhang ampFrantz 2000 Newton amp Manning 2002 2003 Hemley etal 1977ab) as well as silica diffusion (Applin 1987) andNMR data (Cary et al 1982) at 25degC and 1 bar for esti-mating both the bulk mole fraction of silica and the fractionof silica in monomeric form In situ measurements ofRaman spectra for H2O-SiO2 fluids coexisting with quartz(Zotov amp Keppler 2000 2002) provide qualitativeconstraints in that the relative amount of silica in polymericform must increase with an isochoric increase of tempera-ture for T = 600-900degC and P = 56-14 kbar

As discussed further below it was not possible to inte-grate the numerical results of Zotov amp Keppler (2002) whoderived quantitative data on the mole fractions of silicamonomers and dimers from in situ Raman spectra ofaqueous fluid equilibrated with quartz at pressure from56-14 kbar and temperatures from 700 to 900degC Theseresults have been excluded from our database for tworeasons (i) our calculations show that these data are inlarge part inconsistent with the other studies consideredhere requiring a much more pronounced dependence ofsilica speciation on the density of the fluid at a giventemperature and (ii) it was not possible to resolve thediscrepancies in the paper by Zotov amp Keppler (2002)between the formalism presented in their equations (9) and(10) and the derived data on monomer and dimer quantitiespresented in their Table 2

For the calculation of thermodynamic parameters weused a non-linear least square method in order to minimizethe relative error of reproducing the bulk silica concentra-tion which is defined as∆YSiO2

() = 100[YSiO2(calculated) ndash

YSiO2(experimental)]YSiO2

(experimental)We have used a uniform statistical weight of 1 for all exper-imental data points (Table 2) thus making no attempt togauge the relative quality of the various data subsets intheir accuracy and precision (see discussion by Manning

1994) However because of the very irregular distributionof data points in pressure and temperature we found itnecessary to balance our database by adding 91 pointsinterpolated from the polynomial equation of Manning(1994) within the P-T range of experiments used for thecalibration of this equation These ldquosyntheticrdquo points wereregularly distributed with pressure (1 kbar steps) along 9isotherms (100 to 900degC with 100degC steps) A higherstatistical weight of 3 was assigned to these data points Inorder to insure an exact reproduction of the concentrationand the speciation of silica at the reference conditions of To= 25degC and Po = 1 bar (Morey et al 1962 Applin 1987Dove amp Rimstidt 1994 Cary et al 1982) we assigned avery high statistical weight of 10000 to the single datapoint representing these conditions The resulting valuesfor the thermodynamic parameters of equations (31) and(32) are∆Hdeg(mono)r = 24014 plusmn 1781 J∆Sdeg(mono)r = ndash 2810 plusmn 423 JK∆Vdeg(mono)r = ndash 02354plusmn00749 Jbar∆Cpdeg(mono)r = 3196 plusmn 645 JK∆Hdeg(poly)r = ndash 21059 plusmn 3589 J∆Sdeg(poly)r = 966 plusmn 1166 JK∆Vdeg(poly)r = 03161 plusmn 01798 Jbar

Table 2 and Fig 2 3 show that the data set is fitted wellby the above parameters The mean square error in repro-ducing 439 data points on bulk silica solubility is 142 relative No systematic variations of calculated errors withtemperature pressure bulk silica concentration and XLiqhave been found (Fig 3) In quartz-buffered systems rela-tively large deviations (gt 20 ) are indicated for the exper-imental data of Morey et al (1962) and Kennedy (1950) atpressures below 1 kbar and temperatures below 200degC Ourtest calculations show that these data are in part inconsis-tent with the bulk silica concentrations at the referenceconditions of To = 25degC and Po = 1 bar (Dove amp Rimstidt1994) requiring either notably higher (Morey et al 1962)or lower (Kennedy 1550) values These discrepancies canin part be explained by the problems of reaching equilib-

276

Fig 2 Results of modeling bulk molality ofdissolved silica in aqueous fluid in the presence ofquartz based on equations (21)-(30) (solid lines)The different symbols represent experimental datafrom the sources indicated Dashed lines are calcu-lations on the basis of the empirical equationsuggested by Manning (1994)

Modeling of silica solubility and speciation 277

Fig 3 Distribution of relative error ∆YSiO2() = 100[YSiO2

(calculated)-YSiO2(experimental)]YSiO2

(experimental) of reproducing experi-mental data with temperature (a) pressure (b) bulk silica concentration (c) and degree of association (XLiq) of pure water fluid (d) Thedifferent symbols represent experimental data from the sources indicated

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

rium at low temperatures (eg Walther amp Helgeson 1977)Indeed the overall accuracy of reproducing these experi-ments is comparable (Table 2) with the polynomial equa-

tion suggested by Manning (1994) Larger errors weredetected in silica-undersaturated systems (Table 2 Fig 3)for the experimental data of Hemley et al (1977a b) and

278

Fig 4 Relative concentration of silica inmonomers (a) dimers (b) and higher silicapolymers (c) dissolved in aqueous fluid in thepresence of quartz at P = 0001-20 kbar and T= 25-90degC Calculations are based on equa-tions (21)-(30) Thin grey lines are isobarsBlack lines are isotherms The solid squarerepresents the experimental data of Cary et al(1982) and Applin (1987) at 25degC and 1 bar

Zhang amp Frantz (2000) This may be related to the rela-tively large experimental uncertainties (Zhang amp Frantz2000 Newton amp Manning 2002) and to possible errorsincurred during the calculation of ∆Go

(Buffer-Quartz)PT forSiO2-buffering reactions involving water at low pressureand temperature (Hemley et al 1977a b) in combinationwith the database of Holland amp Powell (1998) Obviouslyat T = 25degC and P = 1 bar the highly weighted referencedata on the concentration of silica in the presence of quartz(Dove amp Rimstidt 1994) and on the distribution of silica

between (SiO2)bull(H2O)2-monomers and (SiO2)2bull(H2O)3-dimers (Cary et al 1982 Applin 1987) are accuratelyreproduced with a relative error of less then 02 (Table 2Fig 3 4a) Figure 2 shows that the results of our thermo-dynamic modeling coincide well with the empiricalapproximation suggested by Manning (1994) for the bulksolubility of silica in water in the presence of quartz Table2 also compares our numerical values (Table 2) with thoseobtained from the empirical equations of Manning (1994)for the quartz-buffered system and of Manning amp Newton

Modeling of silica solubility and speciation 279

Fig 5 Results of modeling bulk concentration (a)(b) and speciation (c) of dissolved silica in aqueousfluid in the presence of different buffering mineralassemblages The symbols represent data from thesources indicated (a) experimental data of Newtonamp Manning (2002) and empirical calibration ofZhang amp Frantz (2000) based on original andpublished (Nakamura amp Kushiro 1974 Manningamp Boettcher 1994) experimental data (b) experi-mental data of Newton amp Manning (2003) (c)monomer-dimer speciation model of Newton ampManning (2003)

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

(2002) for the enstatite-forsterite-buffered system For the358 compared experimental data points (Table 2) ourmodel yields a mean square error of 134 versus 165 for the equations of Manning (1994) and Manning ampNewton (2002)

As noted above our model was constrained to accu-rately reproduce experimental data on silica partitioningbetween monomers and larger clusters at 25degC and 1 bar(Cary et al 1982 Applin 1987) The silica species at theseconditions (Fig 4) are represented by (SiO2)bull(H2O)2-monomers (94 of total dissolved silica) and(SiO2)2bull(H2O)3-dimers (6 of total dissolved silica) witha negligible amount of higher polymers (lt 03 of totaldissolved silica) Cary et al (1982) detected the presenceof (SiO2)2bull(H2O)3-dimers by NMR in solutions containing16 mM SiO2 at a pH of 72 and estimated their concentra-tion to be about 6 of the total dissolved silica No addi-tional polymerized forms were detected which isconsistent with our calculations for higher polymers (Fig4c) However the lack of such accurate experimental datafor speciation over a wider range of pressures and temper-atures (eg Watson amp Wark 1997) leads to the relativelylarge uncertainties in ∆Sdeg(poly)r and ∆Vdeg(poly)r ie for chainreaction (12) It should be noted that reaction (12) can alsobe used for the calculation of silica speciation at an arbi-trary bulk concentration of dissolved silica YSiO2

=SiO2(SiO2 + H2O) Equations characterizing speciationthen become

X1 = XLiqXSS(1 ndash XSS)[XSSK(poly) + (1 ndash XSS)XLiq] (34)

XL = XLiq(1 ndash XSS) (35)

XSS = [4XLiqYSiO2(K(poly) ndash XLiq)(1 ndash 2YSiO2

) + (XLiq)2(1 ndashYSiO2

)2]05 ndash XLiq(1 ndash YSiO2) [2(K(poly) ndash XLiq)(1 ndash 2YSiO2

)] (36)

where XSS = X1 + Σinfin

n=2 Xn is the sum of the mole fractions of

all dissolved silica species (ie XSS + XL + XG = 1) Otherquantities are as given in equations (21)-(27)

Figure 4 shows SiO2 speciation in the presence ofquartz as a function of pressure temperature and bulksilica content as predicted from our model calculations Itis seen that the relative amount of silica in the form ofmonomers is most strongly dependent on bulk silicaconcentration and generally decreases with increasingSiO2 content of the solution (Fig 4a) Significant amountsof dissolved silica (10-30 ) in dimeric form are calculatedfor temperatures gt 300degC (Fig 4b) The relative proportionof dimeric silica shows a maximum (Fig 4b) at YSiO2

asymp001 At higher YSiO2

trimers and higher polymers becomeimportant (Fig 4c) consuming 10 to 50 of the dissolvedsilica at temperatures gt 500degC Because increasing temper-ature leads to higher bulk silica there is also a similarcorrelation with temperature The dependence on pressureis least pronounced as can be expected from the competingeffects of reactions (11) and (12) (H2O)L is consumedduring the monomer-forming reaction (11) and liberatedduring the polymer-forming reaction (12) thereforedictating opposite signs of ∆Vdegr of the reactions Increasingpressure should lead to increased bulk silica solubility (dueto reaction (11) ∆Vdeg(mono)r lt 0) and therefore increasing

polymerization on the one hand (Fig 4a) whereasincreasing pressure should also favor depolymerizationaccording to reaction (12) (∆Vdeg(poly)r gt 0) on the other

Figure 5 shows silica solubility and speciation inaqueous fluid coexisting with different SiO2-bufferingmineral assemblages It is seen that our model accuratelyreproduces the available experimental data (Zhang ampFrantz 2000 Newton amp Manning 2003) on bulk silicasolubility (Fig 5ab) The calculated distribution of silicabetween monomers and higher polymers as a function ofbulk SiO2(SiO2 + H2O) at constant pressure and tempera-ture (Fig 5c) is very similar to the results of Newton ampManning (2003) which are based on a somewhat simplermonomer-dimer model At constant pressure and tempera-ture the amount of monomeric silica decreases non-linearlywith decreasing bulk silica solubility

Discussion

The semi-empirical thermodynamic formalism devel-oped in this paper allows accurate reproduction of existingdata on silica concentration and silica speciation inaqueous fluid coexisting with quartz and other silica-buffering assemblages up to 20 kbar and 1300degC Incomparison to Mannings (1994) polynomial fit ourapproach also predicts the speciation of silica in theaqueous fluid without increasing the number of empiricalparameters in the model In addition our chain-reactionmodel allows an excellent data fit with full consideration ofall silica polymers (Fig 4) rather than only monomers anddimers (cf Newton amp Manning 2003) Figure 6 shows thepositions of isopleths of bulk silica concentration calcu-lated for both silica-saturated (Fig 6a) and the silica-under-saturated enstatite-forsterite-buffered (Fig 6b) systems atelevated temperature and pressure It is seen that our resultsare in general compatible with the polynomial equationssuggested by Manning (1994) and Newton amp Manning(2002) with the largest discrepancies found in the high-temperature (gt 900degC) high-YSiO2

(gt 005) region Itshould be emphasized here that our present model has beendeveloped for a bulk silica concentration (YSiO2

) in the fluidof less than 01 (Table 2) Therefore this model mayrequire modification (eg by incorporating terms for non-ideality of the aqueous solution) in order to account prop-erly for the thermodynamics of silica-rich hydroussolutions (eg Shen amp Keppler 1997 Audetat amp Keppler2004) in proximity to the second critical point in the H2O-SiO2 system

Our model predicts basic similarities between the ther-modynamic properties of dissolved silica species and thoseof associated clustered ldquoliquid-likerdquo water molecules (egequations (18)-(20)) This is consistent with the conclu-sions of Bockris (1949) and Walther amp Orville (1983) whosuggested that hydrous monomeric silica species _ Si(OH)4_ are associated with two more water dipoles to form anaqueous silica complex _ Si(OH)4bull2H2O _ by hydrogenbonding This mechanism of association of molecules is infact identical to the association (clustering) of molecules ofpure water (eg Luck 1980 Gorbaty amp Kalinichev 1995)

280

Modeling of silica solubility and speciation 281

Fig 6 Isopleths of bulk silica concentration (solid lines)calculated according to equations (21)-(27) and (31)-(33)for silica-saturated (a) and silica-undersaturated enstatite-forsterite buffered (b) system at elevated temperature andpressure Dashed lines show isopleths calculated with theequations suggested by Manning (1994) and Newton ampManning (2002) Phase boundaries in (a) are calculatedfrom the database of Holland amp Powell (1998) Meltingrelations in the presence of water are not shown

Fig 7 Results of modeling silica speciation in aqueous fluidin the presence of quartz along the two isochores presented byZotov amp Keppler (2002) ρ = 078 gcm3 P = 56-68 kbar andρ = 094 gcm3 P = 106-140 kbar The solid squares repre-sent the results of Zotov amp Keppler (2002) based on the inter-pretation of in situ Raman spectra (a) dependence of silicaspeciation on temperature (b) dependence of silica speciationon bulk silica content

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

and is directly accounted for by equations (4)-(6) (for adiscussion see Gerya amp Perchuk 1997 Gerya et al2004) The formation of similar complexes with dimersand higher polymers can also be expected thus supportingour thermodynamic formalism

An important unresolved question is the discrepancybetween the results obtained from our data set which isbased mainly on direct solubility measurements and dataobtained from Raman spectra (Zotov amp Keppler 20002002) This is particularly evident in the pressure effect onsilica polymerization The positive ∆Vdegr of reaction (12)suggests that at constant temperature and bulk silica anyincrease in pressure (ie in fluid density) should lead to adecrease in the degree of polymerization This is alsosupported by the ab initio molecular dynamics simulationsof Doltsinis et al (2004) who calculated higher effectivepressures for monomeric-dimeric systems in comparisonwith purely monomeric systems at T = 900 K and constantfluid density In the presence of quartz bulk silica contentincreases with pressure and the competing effects of reac-tions (11) and (12) should lead to only small changes inthe degree of polymerization Figure 7 indicates the trendspredicted from our model In contrast the results of Zotovamp Keppler (2002) suggest strong increase in the degree ofpolymerization with pressure (ie fluid density) requiringa notably negative ∆Vo

r of polymer-forming reaction (12)Our test calculations show that no consistent set of ther-modynamic parameters for reactions (11) and (12) canprovide an accurate description of both the silica solubilitymeasurements and the changes of degree of polymeriza-tion with fluid density calculated by Zotov amp Keppler(2002)

We suggest that an explanation for the above discrep-ancies must be sought in (i) the possible problems of equi-libration in the relatively short experiments in thediamond-anvil cell performed by Zotov amp Keppler (20002002) and (ii) the difficult task of quantifying the Ramanspectra of aqueous silica solutions The interpretation ofRaman spectra is based mainly on theoretical spectraobtained using empirical interaction potentials (Catlow1980 Price amp Parker 1984) or on single molecule abinitio calculations as in the case of Zotov amp Keppler(2000 2002) The problem with the latter is that they donot include crucial temperature- and pressure-dependentsolvent effects To test these environmental effects wehave initiated a series of both ab initio molecular dynamics(AIMD) simulations and static gas phase ab initio calcula-tions (Doltsinis et al 2004) Our results to date suggestthat while the strong feature at ca 780-800 cm-1 can beascribed to monomers with some confidence there arestill considerable uncertainties in interpreting otherfeatures of the Raman spectra in terms of monomersandor dimers Considering the low amounts of total silicainvolved and the increasingly floppy nature of largerpolymers leading to increasingly broadened spectralfeatures it appears that the recognition and identificationof trimers and higher polymers can be problematical Ifhidden trimers are present then an interpretation basedsolely on monomerdimer ratios will obviously be flawedThus while the results of Zotov amp Keppler (2000 2002)

represent a valiant first attempt to tackle the problem andprovide valuable qualitative data direct comparisons withclassical solubility results should not be overextended Wesuggest that further Raman spectroscopic measurementson well-defined solutions with given bulk silica contents atdifferent pressures and temperatures rather than quartz-buffered solutions only will be a valuable step toward abetter understanding of such spectra

The approach outlined in this paper has been success-fully used to fit and describe experimental data on bulkspinel (MgAl2O4) solubility up to 30 kbar byZakhartchouk et al (2004) despite the lack of detailedinformation on the actual speciation of the dissolved spinelcomponents in that system Given additional independentdata on the detailed structure of condensed phase water inthe future (eg Marx 1999 and in press) and on solutespeciation at various temperatures and pressures (seeDoltsinis et al 2004) the formalism described here canbe used to provide an increasingly complete thermody-namic description of aqueous solutions at depths corre-sponding to the lower crust and upper mantle

AcknowledgementsThis work was supported by ETHResearch Grant TH -1204-1 by RFBR grant 03-05-64633 by the RF President Program ldquoLeading ScientificSchool of Russiardquo (grant 0351645) to TVG and by theDeutsche Forschungsgemeinschaft within the scope ofSonderforschungsbereich 526 at the Ruhr-University ofBochum The helpful reviews of C Manning J Waltherand an anonymous reviewer as well as the editorialhandling of A Woodland are greatly appreciated

References

Anderson GM amp Burnham CW (1965) The solubility of quartzin supercritical water Am J Sci 263 494-511

Applin KR (1987) The diffusion of dissolved silica in diluteaqueous solution Geochim Cosmochim Acta 51 2147-2151

Audetat A amp Keppler H (2004) Viscosity of fluids in subduc-tion zones Science 303 513-516

Barelko VV Zarikov IV Pechatnikov EL (1994) Effect of theorigin of molecular clusters in one-component gas mixtures onthe equation of state Khim Fiz 13 42-50

Bockris JOrsquoM (1949) Ionic solutions Chem Soc LondonQuarterly Rev 3 173-180

Cary LW De Jong BHWS Dibble WEJr (1982) A 29Si NMRstudy of silica species in dilute aqueous solution GeochimCosmochim Acta 46 1317-1320

Catlow CRA (1980) Computer modelling of ionic crystals JPhysics 41 C6-53

Doltsinis NL Burchard M Maresch WV (2004) Ab initiomolecular dynamics simulations as a tool for interpreting vibra-tional spectra in aqueous fluids EMPG-X SymposiumAbstracts Lithos supplement to v 73 S28

Dove PM amp Rimstidt JD (1994) Silica physical behaviorgeochemistry and materials applications in ldquoSilica-water inter-actionsrdquo Reviews in Mineral 29 259-amp

Fockenberg T Burchard M Maresch WV (2004) Solubilities ofcalcium silicates at high pressures and temperatures EMPG-XSymposium Abstracts Lithos supplement to v 73 S37

282

Gibbs free energy of H2O related to the association (clus-tering) of molecules ∆HdegLiq-Gas = ndash 44839 J ∆SdegLiq-Gas =ndash 1224 JK and ∆CPdegLiq-Gas = 215 JK are the differencesin standard molar enthalpy entropy and heat capacity ofH2O in ldquoliquid-likerdquo and ldquogas-likerdquo states at reference pres-sure (Po = 1 bar) and temperature (To = 29815 K) WH =ndash28793 J WS = ndash117 JK and WCp = 51 JK are enthalpyentropy and heat capacity Margules parameters for theinteraction of ldquoliquid-likerdquo and ldquogas-likerdquo water moleculesat Po and To φ = 6209 bar is a parameter characterizing theattractive forces between all H2O molecules H2981 =ndash 286832 J S2981 = 652 JK are respectively the standardmolar enthalpy of formation from the elements and themolar entropy of ldquoliquid-likerdquo water molecules at Po andTo Vdegs = 1714 Jbar is the standard molar volume ofldquoliquid-likerdquo water molecules at T = 0 K and Po c1 = 7236c2 = 0315 ∆Hdegs1 = 4586 J and ∆Vdegs1 = 00431 Jbar areempirical parameters for the vibrational part of the Gibbspotential

By using the Gibbs-Duhem relation chemical poten-tials of ldquoliquid-likerdquo (microLiq) and ldquogas-likerdquo (microGas) moleculescan be calculated from equations (4)-(6) as follows

microLiq = Gs + RTln(XLiq) + 2RTXLiqXGas2φ(P + φ XLiq

2) + WH ndashTWS + WCp[T ndash 29815 ndash Tln(T29815)]XGas

2 (7)

microGas = Gs + RTln(XGas) + RTln(P + φ XLiq2) ndash 2RTXLiq

2XGasφ(P+ φ XLiq

2) ndash ∆HdegLiq-Gas ndash T∆SdegLiq-Gas + ∆CPdegLiq-Gas[T ndash 29815ndash Tln(T29815)] + WH ndash TWS + WCp[T ndash 29815 ndash Tln(T29815)]XLiq

2 (8)

The equilibrium values for XLiq and XGas in a pure H2Ofluid at given pressure and temperature can then be calcu-lated according to the standard relation

Gm = min ie microLiq = microGas (9)

The software for calculating the equilibrium values ofXLiq according to equation (9) is available by request fromthe corresponding author

Furthermore it is well-known (eg Applin 1987 Doveamp Rimstidt 1994 Zotov amp Keppler 2000 2002) thatdissolved silica can occur in the form of various types ofspecies Although (SiO2)bull(H2O)2-monomers dominate inthe fluid at low silica concentrations in equilibrium withquartz at 25oC and 1 bar (eg Walther amp Helgeson 1977Cary et al 1982 Applin 1987) (SiO2)2bull(H2O)3-dimersand possibly higher polymers become important at highersilica concentrations particularly with increasing pressureand temperature (eg Zotov amp Keppler 2000 2002Newton amp Manning 2002 2003) In the following there-fore we propose that H2O-SiO2-fluids can be treated as amixture of ldquoliquid-likerdquo (H2O)L and ldquogas-likerdquo (H2O)Gwater molecules with (SiO2)bull(H2O)2-monomers and allhigher (SiO2)nbull(H2O)n+1-polymers (2 le n le infin) such that

XL + XG + X1 +Σn=2

infin Xn= 1 (10)

where XL and XG are the mole fractions of ldquoliquid-likerdquo(H2O)L and ldquogas-likerdquo (H2O)G molecules respectively X1

is the mole fraction of (SiO2)bull(H2O)2-monomers and Xn isthe mole fraction of silica polymers with dimension ofn ge 2

On the basis of the above we can describe both the solu-bility and the speciation of silica in an aqueous fluid by thefollowing two types of reactions

a) the monomer-forming standard reaction

SiO2(s) + 2(H2O)L = (SiO2)bull(H2O)2 (11)

b) the polymer-forming chain reaction

(SiO2)n-1bull(H2O)n + (SiO2)bull(H2O)2 = (SiO2)nbull(H2O)n+1 + (H2O)L (12)

where 2 le n le infin The equilibrium conditions for these tworeactions can be written as∆Gdeg(mono)rPT + RTln(K(mono)) = 0 (13)

∆Gdeg(poly)rPT + RTln(K(poly)) = 0 (14)

where K(mono) = a1(aL)2 and K(poly) = anaL(an-1a1) are theequilibrium constants of reactions (11) and (12) respec-tively aL is the activity of ldquoliquid-likerdquo water molecules(H2O)L a1 is the activity of silica monomers and an as wellas an-1 are the activities of silica polymers with dimensionsof n and n-1 respectively ∆Gdeg(mono)rPT and ∆Gdeg(poly)rPT arethe standard Gibbs free energy changes of reactions (11)and (12) as functions of P and T respectively

In order to obtain analytical expressions for silica solu-bility we first assume that ∆Gdeg(poly)rPT is independent of nAccording to equations (7) and (8) the activity of ldquoliquid-likerdquo water molecules differs from the mole fraction andvaries with pressure and temperature For the followingderivations we therefore assume that the activities of thesilica species are also different from their mole fractionsand vary with pressure and temperature in such a way thatthe ratios of the activity coefficients (but not the absolutevalues of these coefficients) for the equilibrium constantsof reactions (11) and (12) are always equal to one

γ1(γL)2 = 1 (15)

γnγL(γn-1γ1) = 1 (16)

where γi = aiXi ne 1The chemical potentials for all species in H2O-SiO2 fluidscan then be formulated according to reactions (11) (12)and equations (7) (8) as follows(H2O)G - standard state XG = 1 XGas = 1 XLiq = 0

micro[(H2O)G] = RTln(XG) + Gs + RTln(P + φ XLiq2) ndash

2RTXLiq2XGasφ(P + φ XLiq

2) ndash ∆HdegLiq-Gas ndash T∆SdegLiq-Gas +∆CPdegLiq-Gas[T ndash 29815 ndash Tln(T29815)] + WH ndash TWS +WCp[T ndash 29815 ndash Tln(T29815)]XLiq

2 (17)

(H2O)L ndash standard state XL =1 XLiq = 1 XGas = 0

micro[(H2O)L] = RTln(XL) + Gs + 2RTXLiqXGas2φ(P + φ XLiq

2)+ WH ndash TWS + WCp[T ndash 29815 ndash Tln(T29815)]XGas

2 (18)

(SiO2)bull(H2O)2 ndash standard state X1 = 1 XLiq = 1 XGas = 0

Modeling of silica solubility and speciation 273

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

micro[(SiO2)bull(H2O)2] = RTln(X1) + ∆Go(mono)rPT + Go

Qtz + 2Gs+ 4RTXLiqXGas

2φ(P + φ XLiq2) + 2WH ndash TWS + WCp[T ndash

29815 ndash Tln(T29815)]XGas2 (19)

(SiO2)nbull(H2O)n+1 ndash standard state Xn = 1 XLiq = 1 XGas = 0

micro[(SiO2)nbull(H2O)n+1] = RTln(Xn) + n∆Gdeg(mono)rPT + (n ndash1)∆Gdeg(poly)rPT + nGdegQtz + (n + 1)Gs + (n +1)2RTXLiqXGas

2φ(P + φ XLiq2) + (n + 1)WH ndash TWS +

WCp[T ndash 29815 ndash Tln(T29815)]XGas2 (20)

where GdegQtz is the standard molar Gibbs free energy ofquartz (eg Holland amp Powell 1998 Gerya et al 19982004) XLiq and XGas are respectively equilibrium molefractions (thermodynamic probabilities) of ldquoliquid-likerdquoand ldquogas-likerdquo water molecules calculated for pure H2Ofluid at given pressure and temperature according to equa-tions (7)-(9) The presence of XLiq- and XGas-dependentterms in equations (19) (20) means that according to ourmodel the thermodynamic properties of silica speciesstrongly depend on the degree of association of moleculesin the aqueous fluid These XLiq- and XGas-dependent termsare similar to those of ldquoliquid-likerdquo rather than ldquogas-likerdquowater molecules (cf equations (17)-(20))

For the following derivations we assume that the pres-ence of silica species affects the absolute values but doesnot change the ratio between the mole fractions of ldquoliquid-likerdquo and ldquogas-likerdquo water molecules in SiO2-H2O-fluidsThis ratio then always corresponds to the conditions ofequilibrium in a pure water fluid at the same pressure andtemperature

XLXG = XLiqXGas (21)

The following analytical expressions can then beobtained for the concentrations of different fluid species

K(mono) = exp(ndash ∆Gdeg(mono)rPT RT) (22)

K(poly) = exp(ndash ∆Gdeg(poly)rPT RT) (23)

XL = [4XLiqK(mono)(XLiq ndash K(poly)) + (1 + XLiqK(mono)K(poly))2]05

ndash (1 + XLiqK(mono)K(poly))[2K(mono)(XLiq ndash K(poly))] (24)

X1 = K(mono)(XL)2 (25)

Xn = X1(K(poly)X1XL)n-1 (26)

XG = XGasXLXLiq (27)

The bulk concentration of dissolved silica in the fluid YSiO2

= SiO2(SiO2+H2O) and the relative proportion ofdissolved silica in monomers YSiO2(mono) = SiO2(mono) SiO2(total)as well as in higher polymers YSiO2(n-poly) = SiO2(n-poly)SiO2(total) can then be calculated according to

YSiO2= (Σ

infin

i=l Xii)[Σ

infin

i=l Xi (2i + 1) + XL + XG]

= (1 ndash XL ndash XG)2[2(1 ndash XL ndash XG)2 + X1](28)

YSiO2(mono) = X1(Σinfin

i=l Xii) = (X1)2(1 ndash XL ndash XG)2 (29)

YSiO2(n-poly) = nXn(Σinfin

i=l Xii) = nX1Xn(1 ndash XL ndash XG)2 (30)

To derive equations (28)-(30) we have used equation (26)

the condition Σinfin

i=l Xi+ XL + XG = 1 (cf equation (10)) and

the standard mathematical relations Σinfin

i=l Zi-1 = 1(1 ndash Z) and

Σinfin

i=l Zi-1i = 1(1 ndash Z)2 where 0 lt Z lt 1 in order to obtain

Σinfin

i=l Xi = X1(1 ndash K(poly)X1XL) = 1 ndash XL ndash XG Σ

infin

i=l Xii = X1(1 ndash

K(poly)X1XL)2 = (1 ndash XL ndash XG)2X1

Application to available experimental data

Our test calculations for the H2O-SiO2 system in a wideregion of pressure (1-20000 bar) and temperature (25-1300oC) have shown that the standard molar Gibbs freeenergy change for reactions (11) and (12) can be repre-sented well by the following relatively simple standardformulae

∆Gdeg(mono)rPT = ∆Hdeg(mono)r ndash T∆Sdeg(mono)r + ∆Cpdeg(mono)r

[T ndash To ndash Tln(TTo)] + ∆Vdeg(mono)r(PndashPo) (31)

∆Gdeg(poly)rPT = ∆Hdeg(poly)r ndash T∆Sdeg(poly)r + ∆Vdeg(poly)r(P ndash Po) (32)

where To = 29815 K and Po = 1 bar are standard pressureand temperature respectively ∆Hdegr ∆Sdegr ∆Cpdegr and ∆Vdegrare respectively standard molar enthalpy entropy isobaricheat capacity and volume changes at To and Po in reactions(11) and (12)

In the case where additional SiO2-buffering mineralassemblages are equilibrated with aqueous fluid (egZhang amp Frantz 2000 Newton amp Manning 2003) theGibbs free energy change of the monomer-forming reac-tion (11) can be obtained as follows

∆Gdeg(mono)rPT = ∆Go(Buffer-Quartz)PT + ∆Hdeg(mono)r ndash

T∆Sdeg(mono)r + ∆Cpdeg(mono)r [T ndash To ndash Tln(TTo)] +∆Vdeg(mono)r(P ndash Po) (33)

where ∆Go(Buffer-Quartz)PT is the difference between the stan-

dard molar Gibbs free energies of the given SiO2-bufferingmineral assemblage and quartz in Jmol which is a func-tion of P and T (Newton amp Manning 2003) For calculating∆Go

(Buffer-Quartz)PT we used the internally consistent thermo-dynamic database of Holland amp Powell (1998) As shownby Zhang amp Frantz (2000) and Newton amp Manning (2003)experimental data on silica solubility in the presence ofvarious SiO2-buffering mineral assemblages allow accuratemodeling of silica speciation at elevated pressure andtemperature

In order to quantify the seven parameters in equations(31) and (32) we have used a set of 439 experimental datapoints (Table 2) retrieved from the literature on the bulksilica solubility in the presence of quartz (Anderson ampBurnham 1965 Walther amp Orville 1983 Hemley et al1980 Kennedy 1950 Morey et al 1962 Manning 1994

274

Modeling of silica solubility and speciation 275

Ref

eren

ces

SiO

2-bu

ffer

No

ofP

kba

rT

degC

YSi

O2

Stat

isti

cal

Mea

n sq

uare

∆∆ YSi

O2

data

wei

ght

for

erro

rmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

poin

tsea

ch d

ata

σσ (Y

SiO

2)

min

max

poin

t

And

erso

n amp

Bur

nham

(19

65)

quar

tz52

1-9

8559

6-90

00

0008

-00

31

12 (

16)

-6 (

-3)

+35

(+

41)

Wal

ther

amp O

rvil

le (

1983

)qu

artz

131-

235

0-55

00

0004

-00

021

6 (8

)-1

2 (-

15)

+9

(+16

)H

emle

y et

al

198

0qu

artz

141-

220

0-50

00

0001

-00

011

7 (9

)-1

4 (-

14+

3 (+

4)K

enne

dy (

1950

)qu

artz

870

01-1

75

182-

610

000

005-

000

11

13 (

11)

-11

(-42

)+

38 (

+12

)M

orey

et

al (

1962

)qu

artz

281

0145

-300

000

0006

-00

003

124

(19

)-3

7 (-

31)

-3 (

-6)

Man

ning

(19

94)

exp

qu

artz

515-

2050

0-90

00

002-

004

18

(10)

-17

(-21

)+

17 (

+22

)M

anni

ng (

1994

) c

alc

quar

tz91

000

1-20

25-9

000

0000

02-0

04

39

-16

+14

Wat

son

amp W

ark

(199

7)qu

artz

310

570-

810

000

5-0

021

14 (

17)

+7

(+15

)+

18 (

+18

)N

ewto

n amp

Man

ning

(20

02)

quar

tz e

nsta

tite

-for

ster

ite

22

1-14

700-

900

000

05-0

05

18

(5)

-14

(-13

)+

15 (

0)N

ewto

n amp

Man

ning

(20

03)

quar

tz e

nsta

tite

-for

ster

ite

612

800

000

1-0

021

10-9

+22

kyan

ite-

coru

ndum

fors

teri

te-r

util

e-ge

ikie

lite

Woo

dlan

d amp

Wal

ther

(19

87)

quar

tz53

095

-27

134

5-50

00

0004

-00

021

7 (9

)-2

0 (-

19)

+11

(+

18)

Wal

ther

amp W

oodl

and

(199

3)qu

artz

151

5-2

400-

600

000

09-0

002

113

(13

)-1

9 (-

19)

+17

(+

24)

Hem

ley

et a

l(1

977a

b)

talc

-chr

ysot

ile-

wat

er t

alc-

751-

220

0-71

50

0000

1-0

001

122

(-32

)+

70fo

rste

rite

-wat

er t

alc-

anti

gori

te-

wat

er t

alc-

anth

ophy

llit

e-w

ater

ta

lc-e

nsta

tite

-wat

er

anth

ophy

llit

e-fo

rste

rite

-wat

erH

emle

y et

al

(197

7b)

enst

atit

e-fo

rste

rite

71

680-

720

000

047-

000

053

110

(9)

-14

(-16

)+

15 (

+4)

Zha

ng amp

Fra

ntz

(200

0) e

xp

enst

atit

e-fo

rste

rite

610

-20

900-

1200

000

6-0

041

34 (

62)

+16

(0)

+52

(+

103)

Zha

ng amp

Fra

ntz

(200

0) c

alc

enst

atit

e-fo

rste

rite

710

700-

1300

000

2-0

091

31 (

60)

-2 (

-19)

+51

(+

103)

Dov

e amp

Rim

stid

t (1

994)

quar

tz1

000

125

000

0002

100

000

(-9)

--

Tabl

e 2

Exp

erim

enta

l da

ta u

sed

for

the

cali

brat

ion

of e

quat

ions

(31

)-(3

3)

va

lues

in

brac

kets

are

cal

cula

ted

on t

he b

asis

of

the

poly

nom

ial

fits

of

Man

ning

(19

94)

and

New

ton

amp M

anni

ng (

2002

)

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

Walther amp Woodland 1993 Woodland amp Walther 1987Watson amp Wark 1997 Newton amp Manning 2002) andother SiO2-buffering mineral assemblages (Zhang ampFrantz 2000 Newton amp Manning 2002 2003 Hemley etal 1977ab) as well as silica diffusion (Applin 1987) andNMR data (Cary et al 1982) at 25degC and 1 bar for esti-mating both the bulk mole fraction of silica and the fractionof silica in monomeric form In situ measurements ofRaman spectra for H2O-SiO2 fluids coexisting with quartz(Zotov amp Keppler 2000 2002) provide qualitativeconstraints in that the relative amount of silica in polymericform must increase with an isochoric increase of tempera-ture for T = 600-900degC and P = 56-14 kbar

As discussed further below it was not possible to inte-grate the numerical results of Zotov amp Keppler (2002) whoderived quantitative data on the mole fractions of silicamonomers and dimers from in situ Raman spectra ofaqueous fluid equilibrated with quartz at pressure from56-14 kbar and temperatures from 700 to 900degC Theseresults have been excluded from our database for tworeasons (i) our calculations show that these data are inlarge part inconsistent with the other studies consideredhere requiring a much more pronounced dependence ofsilica speciation on the density of the fluid at a giventemperature and (ii) it was not possible to resolve thediscrepancies in the paper by Zotov amp Keppler (2002)between the formalism presented in their equations (9) and(10) and the derived data on monomer and dimer quantitiespresented in their Table 2

For the calculation of thermodynamic parameters weused a non-linear least square method in order to minimizethe relative error of reproducing the bulk silica concentra-tion which is defined as∆YSiO2

() = 100[YSiO2(calculated) ndash

YSiO2(experimental)]YSiO2

(experimental)We have used a uniform statistical weight of 1 for all exper-imental data points (Table 2) thus making no attempt togauge the relative quality of the various data subsets intheir accuracy and precision (see discussion by Manning

1994) However because of the very irregular distributionof data points in pressure and temperature we found itnecessary to balance our database by adding 91 pointsinterpolated from the polynomial equation of Manning(1994) within the P-T range of experiments used for thecalibration of this equation These ldquosyntheticrdquo points wereregularly distributed with pressure (1 kbar steps) along 9isotherms (100 to 900degC with 100degC steps) A higherstatistical weight of 3 was assigned to these data points Inorder to insure an exact reproduction of the concentrationand the speciation of silica at the reference conditions of To= 25degC and Po = 1 bar (Morey et al 1962 Applin 1987Dove amp Rimstidt 1994 Cary et al 1982) we assigned avery high statistical weight of 10000 to the single datapoint representing these conditions The resulting valuesfor the thermodynamic parameters of equations (31) and(32) are∆Hdeg(mono)r = 24014 plusmn 1781 J∆Sdeg(mono)r = ndash 2810 plusmn 423 JK∆Vdeg(mono)r = ndash 02354plusmn00749 Jbar∆Cpdeg(mono)r = 3196 plusmn 645 JK∆Hdeg(poly)r = ndash 21059 plusmn 3589 J∆Sdeg(poly)r = 966 plusmn 1166 JK∆Vdeg(poly)r = 03161 plusmn 01798 Jbar

Table 2 and Fig 2 3 show that the data set is fitted wellby the above parameters The mean square error in repro-ducing 439 data points on bulk silica solubility is 142 relative No systematic variations of calculated errors withtemperature pressure bulk silica concentration and XLiqhave been found (Fig 3) In quartz-buffered systems rela-tively large deviations (gt 20 ) are indicated for the exper-imental data of Morey et al (1962) and Kennedy (1950) atpressures below 1 kbar and temperatures below 200degC Ourtest calculations show that these data are in part inconsis-tent with the bulk silica concentrations at the referenceconditions of To = 25degC and Po = 1 bar (Dove amp Rimstidt1994) requiring either notably higher (Morey et al 1962)or lower (Kennedy 1550) values These discrepancies canin part be explained by the problems of reaching equilib-

276

Fig 2 Results of modeling bulk molality ofdissolved silica in aqueous fluid in the presence ofquartz based on equations (21)-(30) (solid lines)The different symbols represent experimental datafrom the sources indicated Dashed lines are calcu-lations on the basis of the empirical equationsuggested by Manning (1994)

Modeling of silica solubility and speciation 277

Fig 3 Distribution of relative error ∆YSiO2() = 100[YSiO2

(calculated)-YSiO2(experimental)]YSiO2

(experimental) of reproducing experi-mental data with temperature (a) pressure (b) bulk silica concentration (c) and degree of association (XLiq) of pure water fluid (d) Thedifferent symbols represent experimental data from the sources indicated

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

rium at low temperatures (eg Walther amp Helgeson 1977)Indeed the overall accuracy of reproducing these experi-ments is comparable (Table 2) with the polynomial equa-

tion suggested by Manning (1994) Larger errors weredetected in silica-undersaturated systems (Table 2 Fig 3)for the experimental data of Hemley et al (1977a b) and

278

Fig 4 Relative concentration of silica inmonomers (a) dimers (b) and higher silicapolymers (c) dissolved in aqueous fluid in thepresence of quartz at P = 0001-20 kbar and T= 25-90degC Calculations are based on equa-tions (21)-(30) Thin grey lines are isobarsBlack lines are isotherms The solid squarerepresents the experimental data of Cary et al(1982) and Applin (1987) at 25degC and 1 bar

Zhang amp Frantz (2000) This may be related to the rela-tively large experimental uncertainties (Zhang amp Frantz2000 Newton amp Manning 2002) and to possible errorsincurred during the calculation of ∆Go

(Buffer-Quartz)PT forSiO2-buffering reactions involving water at low pressureand temperature (Hemley et al 1977a b) in combinationwith the database of Holland amp Powell (1998) Obviouslyat T = 25degC and P = 1 bar the highly weighted referencedata on the concentration of silica in the presence of quartz(Dove amp Rimstidt 1994) and on the distribution of silica

between (SiO2)bull(H2O)2-monomers and (SiO2)2bull(H2O)3-dimers (Cary et al 1982 Applin 1987) are accuratelyreproduced with a relative error of less then 02 (Table 2Fig 3 4a) Figure 2 shows that the results of our thermo-dynamic modeling coincide well with the empiricalapproximation suggested by Manning (1994) for the bulksolubility of silica in water in the presence of quartz Table2 also compares our numerical values (Table 2) with thoseobtained from the empirical equations of Manning (1994)for the quartz-buffered system and of Manning amp Newton

Modeling of silica solubility and speciation 279

Fig 5 Results of modeling bulk concentration (a)(b) and speciation (c) of dissolved silica in aqueousfluid in the presence of different buffering mineralassemblages The symbols represent data from thesources indicated (a) experimental data of Newtonamp Manning (2002) and empirical calibration ofZhang amp Frantz (2000) based on original andpublished (Nakamura amp Kushiro 1974 Manningamp Boettcher 1994) experimental data (b) experi-mental data of Newton amp Manning (2003) (c)monomer-dimer speciation model of Newton ampManning (2003)

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

(2002) for the enstatite-forsterite-buffered system For the358 compared experimental data points (Table 2) ourmodel yields a mean square error of 134 versus 165 for the equations of Manning (1994) and Manning ampNewton (2002)

As noted above our model was constrained to accu-rately reproduce experimental data on silica partitioningbetween monomers and larger clusters at 25degC and 1 bar(Cary et al 1982 Applin 1987) The silica species at theseconditions (Fig 4) are represented by (SiO2)bull(H2O)2-monomers (94 of total dissolved silica) and(SiO2)2bull(H2O)3-dimers (6 of total dissolved silica) witha negligible amount of higher polymers (lt 03 of totaldissolved silica) Cary et al (1982) detected the presenceof (SiO2)2bull(H2O)3-dimers by NMR in solutions containing16 mM SiO2 at a pH of 72 and estimated their concentra-tion to be about 6 of the total dissolved silica No addi-tional polymerized forms were detected which isconsistent with our calculations for higher polymers (Fig4c) However the lack of such accurate experimental datafor speciation over a wider range of pressures and temper-atures (eg Watson amp Wark 1997) leads to the relativelylarge uncertainties in ∆Sdeg(poly)r and ∆Vdeg(poly)r ie for chainreaction (12) It should be noted that reaction (12) can alsobe used for the calculation of silica speciation at an arbi-trary bulk concentration of dissolved silica YSiO2

=SiO2(SiO2 + H2O) Equations characterizing speciationthen become

X1 = XLiqXSS(1 ndash XSS)[XSSK(poly) + (1 ndash XSS)XLiq] (34)

XL = XLiq(1 ndash XSS) (35)

XSS = [4XLiqYSiO2(K(poly) ndash XLiq)(1 ndash 2YSiO2

) + (XLiq)2(1 ndashYSiO2

)2]05 ndash XLiq(1 ndash YSiO2) [2(K(poly) ndash XLiq)(1 ndash 2YSiO2

)] (36)

where XSS = X1 + Σinfin

n=2 Xn is the sum of the mole fractions of

all dissolved silica species (ie XSS + XL + XG = 1) Otherquantities are as given in equations (21)-(27)

Figure 4 shows SiO2 speciation in the presence ofquartz as a function of pressure temperature and bulksilica content as predicted from our model calculations Itis seen that the relative amount of silica in the form ofmonomers is most strongly dependent on bulk silicaconcentration and generally decreases with increasingSiO2 content of the solution (Fig 4a) Significant amountsof dissolved silica (10-30 ) in dimeric form are calculatedfor temperatures gt 300degC (Fig 4b) The relative proportionof dimeric silica shows a maximum (Fig 4b) at YSiO2

asymp001 At higher YSiO2

trimers and higher polymers becomeimportant (Fig 4c) consuming 10 to 50 of the dissolvedsilica at temperatures gt 500degC Because increasing temper-ature leads to higher bulk silica there is also a similarcorrelation with temperature The dependence on pressureis least pronounced as can be expected from the competingeffects of reactions (11) and (12) (H2O)L is consumedduring the monomer-forming reaction (11) and liberatedduring the polymer-forming reaction (12) thereforedictating opposite signs of ∆Vdegr of the reactions Increasingpressure should lead to increased bulk silica solubility (dueto reaction (11) ∆Vdeg(mono)r lt 0) and therefore increasing

polymerization on the one hand (Fig 4a) whereasincreasing pressure should also favor depolymerizationaccording to reaction (12) (∆Vdeg(poly)r gt 0) on the other

Figure 5 shows silica solubility and speciation inaqueous fluid coexisting with different SiO2-bufferingmineral assemblages It is seen that our model accuratelyreproduces the available experimental data (Zhang ampFrantz 2000 Newton amp Manning 2003) on bulk silicasolubility (Fig 5ab) The calculated distribution of silicabetween monomers and higher polymers as a function ofbulk SiO2(SiO2 + H2O) at constant pressure and tempera-ture (Fig 5c) is very similar to the results of Newton ampManning (2003) which are based on a somewhat simplermonomer-dimer model At constant pressure and tempera-ture the amount of monomeric silica decreases non-linearlywith decreasing bulk silica solubility

Discussion

The semi-empirical thermodynamic formalism devel-oped in this paper allows accurate reproduction of existingdata on silica concentration and silica speciation inaqueous fluid coexisting with quartz and other silica-buffering assemblages up to 20 kbar and 1300degC Incomparison to Mannings (1994) polynomial fit ourapproach also predicts the speciation of silica in theaqueous fluid without increasing the number of empiricalparameters in the model In addition our chain-reactionmodel allows an excellent data fit with full consideration ofall silica polymers (Fig 4) rather than only monomers anddimers (cf Newton amp Manning 2003) Figure 6 shows thepositions of isopleths of bulk silica concentration calcu-lated for both silica-saturated (Fig 6a) and the silica-under-saturated enstatite-forsterite-buffered (Fig 6b) systems atelevated temperature and pressure It is seen that our resultsare in general compatible with the polynomial equationssuggested by Manning (1994) and Newton amp Manning(2002) with the largest discrepancies found in the high-temperature (gt 900degC) high-YSiO2

(gt 005) region Itshould be emphasized here that our present model has beendeveloped for a bulk silica concentration (YSiO2

) in the fluidof less than 01 (Table 2) Therefore this model mayrequire modification (eg by incorporating terms for non-ideality of the aqueous solution) in order to account prop-erly for the thermodynamics of silica-rich hydroussolutions (eg Shen amp Keppler 1997 Audetat amp Keppler2004) in proximity to the second critical point in the H2O-SiO2 system

Our model predicts basic similarities between the ther-modynamic properties of dissolved silica species and thoseof associated clustered ldquoliquid-likerdquo water molecules (egequations (18)-(20)) This is consistent with the conclu-sions of Bockris (1949) and Walther amp Orville (1983) whosuggested that hydrous monomeric silica species _ Si(OH)4_ are associated with two more water dipoles to form anaqueous silica complex _ Si(OH)4bull2H2O _ by hydrogenbonding This mechanism of association of molecules is infact identical to the association (clustering) of molecules ofpure water (eg Luck 1980 Gorbaty amp Kalinichev 1995)

280

Modeling of silica solubility and speciation 281

Fig 6 Isopleths of bulk silica concentration (solid lines)calculated according to equations (21)-(27) and (31)-(33)for silica-saturated (a) and silica-undersaturated enstatite-forsterite buffered (b) system at elevated temperature andpressure Dashed lines show isopleths calculated with theequations suggested by Manning (1994) and Newton ampManning (2002) Phase boundaries in (a) are calculatedfrom the database of Holland amp Powell (1998) Meltingrelations in the presence of water are not shown

Fig 7 Results of modeling silica speciation in aqueous fluidin the presence of quartz along the two isochores presented byZotov amp Keppler (2002) ρ = 078 gcm3 P = 56-68 kbar andρ = 094 gcm3 P = 106-140 kbar The solid squares repre-sent the results of Zotov amp Keppler (2002) based on the inter-pretation of in situ Raman spectra (a) dependence of silicaspeciation on temperature (b) dependence of silica speciationon bulk silica content

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

and is directly accounted for by equations (4)-(6) (for adiscussion see Gerya amp Perchuk 1997 Gerya et al2004) The formation of similar complexes with dimersand higher polymers can also be expected thus supportingour thermodynamic formalism

An important unresolved question is the discrepancybetween the results obtained from our data set which isbased mainly on direct solubility measurements and dataobtained from Raman spectra (Zotov amp Keppler 20002002) This is particularly evident in the pressure effect onsilica polymerization The positive ∆Vdegr of reaction (12)suggests that at constant temperature and bulk silica anyincrease in pressure (ie in fluid density) should lead to adecrease in the degree of polymerization This is alsosupported by the ab initio molecular dynamics simulationsof Doltsinis et al (2004) who calculated higher effectivepressures for monomeric-dimeric systems in comparisonwith purely monomeric systems at T = 900 K and constantfluid density In the presence of quartz bulk silica contentincreases with pressure and the competing effects of reac-tions (11) and (12) should lead to only small changes inthe degree of polymerization Figure 7 indicates the trendspredicted from our model In contrast the results of Zotovamp Keppler (2002) suggest strong increase in the degree ofpolymerization with pressure (ie fluid density) requiringa notably negative ∆Vo

r of polymer-forming reaction (12)Our test calculations show that no consistent set of ther-modynamic parameters for reactions (11) and (12) canprovide an accurate description of both the silica solubilitymeasurements and the changes of degree of polymeriza-tion with fluid density calculated by Zotov amp Keppler(2002)

We suggest that an explanation for the above discrep-ancies must be sought in (i) the possible problems of equi-libration in the relatively short experiments in thediamond-anvil cell performed by Zotov amp Keppler (20002002) and (ii) the difficult task of quantifying the Ramanspectra of aqueous silica solutions The interpretation ofRaman spectra is based mainly on theoretical spectraobtained using empirical interaction potentials (Catlow1980 Price amp Parker 1984) or on single molecule abinitio calculations as in the case of Zotov amp Keppler(2000 2002) The problem with the latter is that they donot include crucial temperature- and pressure-dependentsolvent effects To test these environmental effects wehave initiated a series of both ab initio molecular dynamics(AIMD) simulations and static gas phase ab initio calcula-tions (Doltsinis et al 2004) Our results to date suggestthat while the strong feature at ca 780-800 cm-1 can beascribed to monomers with some confidence there arestill considerable uncertainties in interpreting otherfeatures of the Raman spectra in terms of monomersandor dimers Considering the low amounts of total silicainvolved and the increasingly floppy nature of largerpolymers leading to increasingly broadened spectralfeatures it appears that the recognition and identificationof trimers and higher polymers can be problematical Ifhidden trimers are present then an interpretation basedsolely on monomerdimer ratios will obviously be flawedThus while the results of Zotov amp Keppler (2000 2002)

represent a valiant first attempt to tackle the problem andprovide valuable qualitative data direct comparisons withclassical solubility results should not be overextended Wesuggest that further Raman spectroscopic measurementson well-defined solutions with given bulk silica contents atdifferent pressures and temperatures rather than quartz-buffered solutions only will be a valuable step toward abetter understanding of such spectra

The approach outlined in this paper has been success-fully used to fit and describe experimental data on bulkspinel (MgAl2O4) solubility up to 30 kbar byZakhartchouk et al (2004) despite the lack of detailedinformation on the actual speciation of the dissolved spinelcomponents in that system Given additional independentdata on the detailed structure of condensed phase water inthe future (eg Marx 1999 and in press) and on solutespeciation at various temperatures and pressures (seeDoltsinis et al 2004) the formalism described here canbe used to provide an increasingly complete thermody-namic description of aqueous solutions at depths corre-sponding to the lower crust and upper mantle

AcknowledgementsThis work was supported by ETHResearch Grant TH -1204-1 by RFBR grant 03-05-64633 by the RF President Program ldquoLeading ScientificSchool of Russiardquo (grant 0351645) to TVG and by theDeutsche Forschungsgemeinschaft within the scope ofSonderforschungsbereich 526 at the Ruhr-University ofBochum The helpful reviews of C Manning J Waltherand an anonymous reviewer as well as the editorialhandling of A Woodland are greatly appreciated

References

Anderson GM amp Burnham CW (1965) The solubility of quartzin supercritical water Am J Sci 263 494-511

Applin KR (1987) The diffusion of dissolved silica in diluteaqueous solution Geochim Cosmochim Acta 51 2147-2151

Audetat A amp Keppler H (2004) Viscosity of fluids in subduc-tion zones Science 303 513-516

Barelko VV Zarikov IV Pechatnikov EL (1994) Effect of theorigin of molecular clusters in one-component gas mixtures onthe equation of state Khim Fiz 13 42-50

Bockris JOrsquoM (1949) Ionic solutions Chem Soc LondonQuarterly Rev 3 173-180

Cary LW De Jong BHWS Dibble WEJr (1982) A 29Si NMRstudy of silica species in dilute aqueous solution GeochimCosmochim Acta 46 1317-1320

Catlow CRA (1980) Computer modelling of ionic crystals JPhysics 41 C6-53

Doltsinis NL Burchard M Maresch WV (2004) Ab initiomolecular dynamics simulations as a tool for interpreting vibra-tional spectra in aqueous fluids EMPG-X SymposiumAbstracts Lithos supplement to v 73 S28

Dove PM amp Rimstidt JD (1994) Silica physical behaviorgeochemistry and materials applications in ldquoSilica-water inter-actionsrdquo Reviews in Mineral 29 259-amp

Fockenberg T Burchard M Maresch WV (2004) Solubilities ofcalcium silicates at high pressures and temperatures EMPG-XSymposium Abstracts Lithos supplement to v 73 S37

282

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

micro[(SiO2)bull(H2O)2] = RTln(X1) + ∆Go(mono)rPT + Go

Qtz + 2Gs+ 4RTXLiqXGas

2φ(P + φ XLiq2) + 2WH ndash TWS + WCp[T ndash

29815 ndash Tln(T29815)]XGas2 (19)

(SiO2)nbull(H2O)n+1 ndash standard state Xn = 1 XLiq = 1 XGas = 0

micro[(SiO2)nbull(H2O)n+1] = RTln(Xn) + n∆Gdeg(mono)rPT + (n ndash1)∆Gdeg(poly)rPT + nGdegQtz + (n + 1)Gs + (n +1)2RTXLiqXGas

2φ(P + φ XLiq2) + (n + 1)WH ndash TWS +

WCp[T ndash 29815 ndash Tln(T29815)]XGas2 (20)

where GdegQtz is the standard molar Gibbs free energy ofquartz (eg Holland amp Powell 1998 Gerya et al 19982004) XLiq and XGas are respectively equilibrium molefractions (thermodynamic probabilities) of ldquoliquid-likerdquoand ldquogas-likerdquo water molecules calculated for pure H2Ofluid at given pressure and temperature according to equa-tions (7)-(9) The presence of XLiq- and XGas-dependentterms in equations (19) (20) means that according to ourmodel the thermodynamic properties of silica speciesstrongly depend on the degree of association of moleculesin the aqueous fluid These XLiq- and XGas-dependent termsare similar to those of ldquoliquid-likerdquo rather than ldquogas-likerdquowater molecules (cf equations (17)-(20))

For the following derivations we assume that the pres-ence of silica species affects the absolute values but doesnot change the ratio between the mole fractions of ldquoliquid-likerdquo and ldquogas-likerdquo water molecules in SiO2-H2O-fluidsThis ratio then always corresponds to the conditions ofequilibrium in a pure water fluid at the same pressure andtemperature

XLXG = XLiqXGas (21)

The following analytical expressions can then beobtained for the concentrations of different fluid species

K(mono) = exp(ndash ∆Gdeg(mono)rPT RT) (22)

K(poly) = exp(ndash ∆Gdeg(poly)rPT RT) (23)

XL = [4XLiqK(mono)(XLiq ndash K(poly)) + (1 + XLiqK(mono)K(poly))2]05

ndash (1 + XLiqK(mono)K(poly))[2K(mono)(XLiq ndash K(poly))] (24)

X1 = K(mono)(XL)2 (25)

Xn = X1(K(poly)X1XL)n-1 (26)

XG = XGasXLXLiq (27)

The bulk concentration of dissolved silica in the fluid YSiO2

= SiO2(SiO2+H2O) and the relative proportion ofdissolved silica in monomers YSiO2(mono) = SiO2(mono) SiO2(total)as well as in higher polymers YSiO2(n-poly) = SiO2(n-poly)SiO2(total) can then be calculated according to

YSiO2= (Σ

infin

i=l Xii)[Σ

infin

i=l Xi (2i + 1) + XL + XG]

= (1 ndash XL ndash XG)2[2(1 ndash XL ndash XG)2 + X1](28)

YSiO2(mono) = X1(Σinfin

i=l Xii) = (X1)2(1 ndash XL ndash XG)2 (29)

YSiO2(n-poly) = nXn(Σinfin

i=l Xii) = nX1Xn(1 ndash XL ndash XG)2 (30)

To derive equations (28)-(30) we have used equation (26)

the condition Σinfin

i=l Xi+ XL + XG = 1 (cf equation (10)) and

the standard mathematical relations Σinfin

i=l Zi-1 = 1(1 ndash Z) and

Σinfin

i=l Zi-1i = 1(1 ndash Z)2 where 0 lt Z lt 1 in order to obtain

Σinfin

i=l Xi = X1(1 ndash K(poly)X1XL) = 1 ndash XL ndash XG Σ

infin

i=l Xii = X1(1 ndash

K(poly)X1XL)2 = (1 ndash XL ndash XG)2X1

Application to available experimental data

Our test calculations for the H2O-SiO2 system in a wideregion of pressure (1-20000 bar) and temperature (25-1300oC) have shown that the standard molar Gibbs freeenergy change for reactions (11) and (12) can be repre-sented well by the following relatively simple standardformulae

∆Gdeg(mono)rPT = ∆Hdeg(mono)r ndash T∆Sdeg(mono)r + ∆Cpdeg(mono)r

[T ndash To ndash Tln(TTo)] + ∆Vdeg(mono)r(PndashPo) (31)

∆Gdeg(poly)rPT = ∆Hdeg(poly)r ndash T∆Sdeg(poly)r + ∆Vdeg(poly)r(P ndash Po) (32)

where To = 29815 K and Po = 1 bar are standard pressureand temperature respectively ∆Hdegr ∆Sdegr ∆Cpdegr and ∆Vdegrare respectively standard molar enthalpy entropy isobaricheat capacity and volume changes at To and Po in reactions(11) and (12)

In the case where additional SiO2-buffering mineralassemblages are equilibrated with aqueous fluid (egZhang amp Frantz 2000 Newton amp Manning 2003) theGibbs free energy change of the monomer-forming reac-tion (11) can be obtained as follows

∆Gdeg(mono)rPT = ∆Go(Buffer-Quartz)PT + ∆Hdeg(mono)r ndash

T∆Sdeg(mono)r + ∆Cpdeg(mono)r [T ndash To ndash Tln(TTo)] +∆Vdeg(mono)r(P ndash Po) (33)

where ∆Go(Buffer-Quartz)PT is the difference between the stan-

dard molar Gibbs free energies of the given SiO2-bufferingmineral assemblage and quartz in Jmol which is a func-tion of P and T (Newton amp Manning 2003) For calculating∆Go

(Buffer-Quartz)PT we used the internally consistent thermo-dynamic database of Holland amp Powell (1998) As shownby Zhang amp Frantz (2000) and Newton amp Manning (2003)experimental data on silica solubility in the presence ofvarious SiO2-buffering mineral assemblages allow accuratemodeling of silica speciation at elevated pressure andtemperature

In order to quantify the seven parameters in equations(31) and (32) we have used a set of 439 experimental datapoints (Table 2) retrieved from the literature on the bulksilica solubility in the presence of quartz (Anderson ampBurnham 1965 Walther amp Orville 1983 Hemley et al1980 Kennedy 1950 Morey et al 1962 Manning 1994

274

Modeling of silica solubility and speciation 275

Ref

eren

ces

SiO

2-bu

ffer

No

ofP

kba

rT

degC

YSi

O2

Stat

isti

cal

Mea

n sq

uare

∆∆ YSi

O2

data

wei

ght

for

erro

rmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

poin

tsea

ch d

ata

σσ (Y

SiO

2)

min

max

poin

t

And

erso

n amp

Bur

nham

(19

65)

quar

tz52

1-9

8559

6-90

00

0008

-00

31

12 (

16)

-6 (

-3)

+35

(+

41)

Wal

ther

amp O

rvil

le (

1983

)qu

artz

131-

235

0-55

00

0004

-00

021

6 (8

)-1

2 (-

15)

+9

(+16

)H

emle

y et

al

198

0qu

artz

141-

220

0-50

00

0001

-00

011

7 (9

)-1

4 (-

14+

3 (+

4)K

enne

dy (

1950

)qu

artz

870

01-1

75

182-

610

000

005-

000

11

13 (

11)

-11

(-42

)+

38 (

+12

)M

orey

et

al (

1962

)qu

artz

281

0145

-300

000

0006

-00

003

124

(19

)-3

7 (-

31)

-3 (

-6)

Man

ning

(19

94)

exp

qu

artz

515-

2050

0-90

00

002-

004

18

(10)

-17

(-21

)+

17 (

+22

)M

anni

ng (

1994

) c

alc

quar

tz91

000

1-20

25-9

000

0000

02-0

04

39

-16

+14

Wat

son

amp W

ark

(199

7)qu

artz

310

570-

810

000

5-0

021

14 (

17)

+7

(+15

)+

18 (

+18

)N

ewto

n amp

Man

ning

(20

02)

quar

tz e

nsta

tite

-for

ster

ite

22

1-14

700-

900

000

05-0

05

18

(5)

-14

(-13

)+

15 (

0)N

ewto

n amp

Man

ning

(20

03)

quar

tz e

nsta

tite

-for

ster

ite

612

800

000

1-0

021

10-9

+22

kyan

ite-

coru

ndum

fors

teri

te-r

util

e-ge

ikie

lite

Woo

dlan

d amp

Wal

ther

(19

87)

quar

tz53

095

-27

134

5-50

00

0004

-00

021

7 (9

)-2

0 (-

19)

+11

(+

18)

Wal

ther

amp W

oodl

and

(199

3)qu

artz

151

5-2

400-

600

000

09-0

002

113

(13

)-1

9 (-

19)

+17

(+

24)

Hem

ley

et a

l(1

977a

b)

talc

-chr

ysot

ile-

wat

er t

alc-

751-

220

0-71

50

0000

1-0

001

122

(-32

)+

70fo

rste

rite

-wat

er t

alc-

anti

gori

te-

wat

er t

alc-

anth

ophy

llit

e-w

ater

ta

lc-e

nsta

tite

-wat

er

anth

ophy

llit

e-fo

rste

rite

-wat

erH

emle

y et

al

(197

7b)

enst

atit

e-fo

rste

rite

71

680-

720

000

047-

000

053

110

(9)

-14

(-16

)+

15 (

+4)

Zha

ng amp

Fra

ntz

(200

0) e

xp

enst

atit

e-fo

rste

rite

610

-20

900-

1200

000

6-0

041

34 (

62)

+16

(0)

+52

(+

103)

Zha

ng amp

Fra

ntz

(200

0) c

alc

enst

atit

e-fo

rste

rite

710

700-

1300

000

2-0

091

31 (

60)

-2 (

-19)

+51

(+

103)

Dov

e amp

Rim

stid

t (1

994)

quar

tz1

000

125

000

0002

100

000

(-9)

--

Tabl

e 2

Exp

erim

enta

l da

ta u

sed

for

the

cali

brat

ion

of e

quat

ions

(31

)-(3

3)

va

lues

in

brac

kets

are

cal

cula

ted

on t

he b

asis

of

the

poly

nom

ial

fits

of

Man

ning

(19

94)

and

New

ton

amp M

anni

ng (

2002

)

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

Walther amp Woodland 1993 Woodland amp Walther 1987Watson amp Wark 1997 Newton amp Manning 2002) andother SiO2-buffering mineral assemblages (Zhang ampFrantz 2000 Newton amp Manning 2002 2003 Hemley etal 1977ab) as well as silica diffusion (Applin 1987) andNMR data (Cary et al 1982) at 25degC and 1 bar for esti-mating both the bulk mole fraction of silica and the fractionof silica in monomeric form In situ measurements ofRaman spectra for H2O-SiO2 fluids coexisting with quartz(Zotov amp Keppler 2000 2002) provide qualitativeconstraints in that the relative amount of silica in polymericform must increase with an isochoric increase of tempera-ture for T = 600-900degC and P = 56-14 kbar

As discussed further below it was not possible to inte-grate the numerical results of Zotov amp Keppler (2002) whoderived quantitative data on the mole fractions of silicamonomers and dimers from in situ Raman spectra ofaqueous fluid equilibrated with quartz at pressure from56-14 kbar and temperatures from 700 to 900degC Theseresults have been excluded from our database for tworeasons (i) our calculations show that these data are inlarge part inconsistent with the other studies consideredhere requiring a much more pronounced dependence ofsilica speciation on the density of the fluid at a giventemperature and (ii) it was not possible to resolve thediscrepancies in the paper by Zotov amp Keppler (2002)between the formalism presented in their equations (9) and(10) and the derived data on monomer and dimer quantitiespresented in their Table 2

For the calculation of thermodynamic parameters weused a non-linear least square method in order to minimizethe relative error of reproducing the bulk silica concentra-tion which is defined as∆YSiO2

() = 100[YSiO2(calculated) ndash

YSiO2(experimental)]YSiO2

(experimental)We have used a uniform statistical weight of 1 for all exper-imental data points (Table 2) thus making no attempt togauge the relative quality of the various data subsets intheir accuracy and precision (see discussion by Manning

1994) However because of the very irregular distributionof data points in pressure and temperature we found itnecessary to balance our database by adding 91 pointsinterpolated from the polynomial equation of Manning(1994) within the P-T range of experiments used for thecalibration of this equation These ldquosyntheticrdquo points wereregularly distributed with pressure (1 kbar steps) along 9isotherms (100 to 900degC with 100degC steps) A higherstatistical weight of 3 was assigned to these data points Inorder to insure an exact reproduction of the concentrationand the speciation of silica at the reference conditions of To= 25degC and Po = 1 bar (Morey et al 1962 Applin 1987Dove amp Rimstidt 1994 Cary et al 1982) we assigned avery high statistical weight of 10000 to the single datapoint representing these conditions The resulting valuesfor the thermodynamic parameters of equations (31) and(32) are∆Hdeg(mono)r = 24014 plusmn 1781 J∆Sdeg(mono)r = ndash 2810 plusmn 423 JK∆Vdeg(mono)r = ndash 02354plusmn00749 Jbar∆Cpdeg(mono)r = 3196 plusmn 645 JK∆Hdeg(poly)r = ndash 21059 plusmn 3589 J∆Sdeg(poly)r = 966 plusmn 1166 JK∆Vdeg(poly)r = 03161 plusmn 01798 Jbar

Table 2 and Fig 2 3 show that the data set is fitted wellby the above parameters The mean square error in repro-ducing 439 data points on bulk silica solubility is 142 relative No systematic variations of calculated errors withtemperature pressure bulk silica concentration and XLiqhave been found (Fig 3) In quartz-buffered systems rela-tively large deviations (gt 20 ) are indicated for the exper-imental data of Morey et al (1962) and Kennedy (1950) atpressures below 1 kbar and temperatures below 200degC Ourtest calculations show that these data are in part inconsis-tent with the bulk silica concentrations at the referenceconditions of To = 25degC and Po = 1 bar (Dove amp Rimstidt1994) requiring either notably higher (Morey et al 1962)or lower (Kennedy 1550) values These discrepancies canin part be explained by the problems of reaching equilib-

276

Fig 2 Results of modeling bulk molality ofdissolved silica in aqueous fluid in the presence ofquartz based on equations (21)-(30) (solid lines)The different symbols represent experimental datafrom the sources indicated Dashed lines are calcu-lations on the basis of the empirical equationsuggested by Manning (1994)

Modeling of silica solubility and speciation 277

Fig 3 Distribution of relative error ∆YSiO2() = 100[YSiO2

(calculated)-YSiO2(experimental)]YSiO2

(experimental) of reproducing experi-mental data with temperature (a) pressure (b) bulk silica concentration (c) and degree of association (XLiq) of pure water fluid (d) Thedifferent symbols represent experimental data from the sources indicated

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

rium at low temperatures (eg Walther amp Helgeson 1977)Indeed the overall accuracy of reproducing these experi-ments is comparable (Table 2) with the polynomial equa-

tion suggested by Manning (1994) Larger errors weredetected in silica-undersaturated systems (Table 2 Fig 3)for the experimental data of Hemley et al (1977a b) and

278

Fig 4 Relative concentration of silica inmonomers (a) dimers (b) and higher silicapolymers (c) dissolved in aqueous fluid in thepresence of quartz at P = 0001-20 kbar and T= 25-90degC Calculations are based on equa-tions (21)-(30) Thin grey lines are isobarsBlack lines are isotherms The solid squarerepresents the experimental data of Cary et al(1982) and Applin (1987) at 25degC and 1 bar

Zhang amp Frantz (2000) This may be related to the rela-tively large experimental uncertainties (Zhang amp Frantz2000 Newton amp Manning 2002) and to possible errorsincurred during the calculation of ∆Go

(Buffer-Quartz)PT forSiO2-buffering reactions involving water at low pressureand temperature (Hemley et al 1977a b) in combinationwith the database of Holland amp Powell (1998) Obviouslyat T = 25degC and P = 1 bar the highly weighted referencedata on the concentration of silica in the presence of quartz(Dove amp Rimstidt 1994) and on the distribution of silica

between (SiO2)bull(H2O)2-monomers and (SiO2)2bull(H2O)3-dimers (Cary et al 1982 Applin 1987) are accuratelyreproduced with a relative error of less then 02 (Table 2Fig 3 4a) Figure 2 shows that the results of our thermo-dynamic modeling coincide well with the empiricalapproximation suggested by Manning (1994) for the bulksolubility of silica in water in the presence of quartz Table2 also compares our numerical values (Table 2) with thoseobtained from the empirical equations of Manning (1994)for the quartz-buffered system and of Manning amp Newton

Modeling of silica solubility and speciation 279

Fig 5 Results of modeling bulk concentration (a)(b) and speciation (c) of dissolved silica in aqueousfluid in the presence of different buffering mineralassemblages The symbols represent data from thesources indicated (a) experimental data of Newtonamp Manning (2002) and empirical calibration ofZhang amp Frantz (2000) based on original andpublished (Nakamura amp Kushiro 1974 Manningamp Boettcher 1994) experimental data (b) experi-mental data of Newton amp Manning (2003) (c)monomer-dimer speciation model of Newton ampManning (2003)

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

(2002) for the enstatite-forsterite-buffered system For the358 compared experimental data points (Table 2) ourmodel yields a mean square error of 134 versus 165 for the equations of Manning (1994) and Manning ampNewton (2002)

As noted above our model was constrained to accu-rately reproduce experimental data on silica partitioningbetween monomers and larger clusters at 25degC and 1 bar(Cary et al 1982 Applin 1987) The silica species at theseconditions (Fig 4) are represented by (SiO2)bull(H2O)2-monomers (94 of total dissolved silica) and(SiO2)2bull(H2O)3-dimers (6 of total dissolved silica) witha negligible amount of higher polymers (lt 03 of totaldissolved silica) Cary et al (1982) detected the presenceof (SiO2)2bull(H2O)3-dimers by NMR in solutions containing16 mM SiO2 at a pH of 72 and estimated their concentra-tion to be about 6 of the total dissolved silica No addi-tional polymerized forms were detected which isconsistent with our calculations for higher polymers (Fig4c) However the lack of such accurate experimental datafor speciation over a wider range of pressures and temper-atures (eg Watson amp Wark 1997) leads to the relativelylarge uncertainties in ∆Sdeg(poly)r and ∆Vdeg(poly)r ie for chainreaction (12) It should be noted that reaction (12) can alsobe used for the calculation of silica speciation at an arbi-trary bulk concentration of dissolved silica YSiO2

=SiO2(SiO2 + H2O) Equations characterizing speciationthen become

X1 = XLiqXSS(1 ndash XSS)[XSSK(poly) + (1 ndash XSS)XLiq] (34)

XL = XLiq(1 ndash XSS) (35)

XSS = [4XLiqYSiO2(K(poly) ndash XLiq)(1 ndash 2YSiO2

) + (XLiq)2(1 ndashYSiO2

)2]05 ndash XLiq(1 ndash YSiO2) [2(K(poly) ndash XLiq)(1 ndash 2YSiO2

)] (36)

where XSS = X1 + Σinfin

n=2 Xn is the sum of the mole fractions of

all dissolved silica species (ie XSS + XL + XG = 1) Otherquantities are as given in equations (21)-(27)

Figure 4 shows SiO2 speciation in the presence ofquartz as a function of pressure temperature and bulksilica content as predicted from our model calculations Itis seen that the relative amount of silica in the form ofmonomers is most strongly dependent on bulk silicaconcentration and generally decreases with increasingSiO2 content of the solution (Fig 4a) Significant amountsof dissolved silica (10-30 ) in dimeric form are calculatedfor temperatures gt 300degC (Fig 4b) The relative proportionof dimeric silica shows a maximum (Fig 4b) at YSiO2

asymp001 At higher YSiO2

trimers and higher polymers becomeimportant (Fig 4c) consuming 10 to 50 of the dissolvedsilica at temperatures gt 500degC Because increasing temper-ature leads to higher bulk silica there is also a similarcorrelation with temperature The dependence on pressureis least pronounced as can be expected from the competingeffects of reactions (11) and (12) (H2O)L is consumedduring the monomer-forming reaction (11) and liberatedduring the polymer-forming reaction (12) thereforedictating opposite signs of ∆Vdegr of the reactions Increasingpressure should lead to increased bulk silica solubility (dueto reaction (11) ∆Vdeg(mono)r lt 0) and therefore increasing

polymerization on the one hand (Fig 4a) whereasincreasing pressure should also favor depolymerizationaccording to reaction (12) (∆Vdeg(poly)r gt 0) on the other

Figure 5 shows silica solubility and speciation inaqueous fluid coexisting with different SiO2-bufferingmineral assemblages It is seen that our model accuratelyreproduces the available experimental data (Zhang ampFrantz 2000 Newton amp Manning 2003) on bulk silicasolubility (Fig 5ab) The calculated distribution of silicabetween monomers and higher polymers as a function ofbulk SiO2(SiO2 + H2O) at constant pressure and tempera-ture (Fig 5c) is very similar to the results of Newton ampManning (2003) which are based on a somewhat simplermonomer-dimer model At constant pressure and tempera-ture the amount of monomeric silica decreases non-linearlywith decreasing bulk silica solubility

Discussion

The semi-empirical thermodynamic formalism devel-oped in this paper allows accurate reproduction of existingdata on silica concentration and silica speciation inaqueous fluid coexisting with quartz and other silica-buffering assemblages up to 20 kbar and 1300degC Incomparison to Mannings (1994) polynomial fit ourapproach also predicts the speciation of silica in theaqueous fluid without increasing the number of empiricalparameters in the model In addition our chain-reactionmodel allows an excellent data fit with full consideration ofall silica polymers (Fig 4) rather than only monomers anddimers (cf Newton amp Manning 2003) Figure 6 shows thepositions of isopleths of bulk silica concentration calcu-lated for both silica-saturated (Fig 6a) and the silica-under-saturated enstatite-forsterite-buffered (Fig 6b) systems atelevated temperature and pressure It is seen that our resultsare in general compatible with the polynomial equationssuggested by Manning (1994) and Newton amp Manning(2002) with the largest discrepancies found in the high-temperature (gt 900degC) high-YSiO2

(gt 005) region Itshould be emphasized here that our present model has beendeveloped for a bulk silica concentration (YSiO2

) in the fluidof less than 01 (Table 2) Therefore this model mayrequire modification (eg by incorporating terms for non-ideality of the aqueous solution) in order to account prop-erly for the thermodynamics of silica-rich hydroussolutions (eg Shen amp Keppler 1997 Audetat amp Keppler2004) in proximity to the second critical point in the H2O-SiO2 system

Our model predicts basic similarities between the ther-modynamic properties of dissolved silica species and thoseof associated clustered ldquoliquid-likerdquo water molecules (egequations (18)-(20)) This is consistent with the conclu-sions of Bockris (1949) and Walther amp Orville (1983) whosuggested that hydrous monomeric silica species _ Si(OH)4_ are associated with two more water dipoles to form anaqueous silica complex _ Si(OH)4bull2H2O _ by hydrogenbonding This mechanism of association of molecules is infact identical to the association (clustering) of molecules ofpure water (eg Luck 1980 Gorbaty amp Kalinichev 1995)

280

Modeling of silica solubility and speciation 281

Fig 6 Isopleths of bulk silica concentration (solid lines)calculated according to equations (21)-(27) and (31)-(33)for silica-saturated (a) and silica-undersaturated enstatite-forsterite buffered (b) system at elevated temperature andpressure Dashed lines show isopleths calculated with theequations suggested by Manning (1994) and Newton ampManning (2002) Phase boundaries in (a) are calculatedfrom the database of Holland amp Powell (1998) Meltingrelations in the presence of water are not shown

Fig 7 Results of modeling silica speciation in aqueous fluidin the presence of quartz along the two isochores presented byZotov amp Keppler (2002) ρ = 078 gcm3 P = 56-68 kbar andρ = 094 gcm3 P = 106-140 kbar The solid squares repre-sent the results of Zotov amp Keppler (2002) based on the inter-pretation of in situ Raman spectra (a) dependence of silicaspeciation on temperature (b) dependence of silica speciationon bulk silica content

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

and is directly accounted for by equations (4)-(6) (for adiscussion see Gerya amp Perchuk 1997 Gerya et al2004) The formation of similar complexes with dimersand higher polymers can also be expected thus supportingour thermodynamic formalism

An important unresolved question is the discrepancybetween the results obtained from our data set which isbased mainly on direct solubility measurements and dataobtained from Raman spectra (Zotov amp Keppler 20002002) This is particularly evident in the pressure effect onsilica polymerization The positive ∆Vdegr of reaction (12)suggests that at constant temperature and bulk silica anyincrease in pressure (ie in fluid density) should lead to adecrease in the degree of polymerization This is alsosupported by the ab initio molecular dynamics simulationsof Doltsinis et al (2004) who calculated higher effectivepressures for monomeric-dimeric systems in comparisonwith purely monomeric systems at T = 900 K and constantfluid density In the presence of quartz bulk silica contentincreases with pressure and the competing effects of reac-tions (11) and (12) should lead to only small changes inthe degree of polymerization Figure 7 indicates the trendspredicted from our model In contrast the results of Zotovamp Keppler (2002) suggest strong increase in the degree ofpolymerization with pressure (ie fluid density) requiringa notably negative ∆Vo

r of polymer-forming reaction (12)Our test calculations show that no consistent set of ther-modynamic parameters for reactions (11) and (12) canprovide an accurate description of both the silica solubilitymeasurements and the changes of degree of polymeriza-tion with fluid density calculated by Zotov amp Keppler(2002)

We suggest that an explanation for the above discrep-ancies must be sought in (i) the possible problems of equi-libration in the relatively short experiments in thediamond-anvil cell performed by Zotov amp Keppler (20002002) and (ii) the difficult task of quantifying the Ramanspectra of aqueous silica solutions The interpretation ofRaman spectra is based mainly on theoretical spectraobtained using empirical interaction potentials (Catlow1980 Price amp Parker 1984) or on single molecule abinitio calculations as in the case of Zotov amp Keppler(2000 2002) The problem with the latter is that they donot include crucial temperature- and pressure-dependentsolvent effects To test these environmental effects wehave initiated a series of both ab initio molecular dynamics(AIMD) simulations and static gas phase ab initio calcula-tions (Doltsinis et al 2004) Our results to date suggestthat while the strong feature at ca 780-800 cm-1 can beascribed to monomers with some confidence there arestill considerable uncertainties in interpreting otherfeatures of the Raman spectra in terms of monomersandor dimers Considering the low amounts of total silicainvolved and the increasingly floppy nature of largerpolymers leading to increasingly broadened spectralfeatures it appears that the recognition and identificationof trimers and higher polymers can be problematical Ifhidden trimers are present then an interpretation basedsolely on monomerdimer ratios will obviously be flawedThus while the results of Zotov amp Keppler (2000 2002)

represent a valiant first attempt to tackle the problem andprovide valuable qualitative data direct comparisons withclassical solubility results should not be overextended Wesuggest that further Raman spectroscopic measurementson well-defined solutions with given bulk silica contents atdifferent pressures and temperatures rather than quartz-buffered solutions only will be a valuable step toward abetter understanding of such spectra

The approach outlined in this paper has been success-fully used to fit and describe experimental data on bulkspinel (MgAl2O4) solubility up to 30 kbar byZakhartchouk et al (2004) despite the lack of detailedinformation on the actual speciation of the dissolved spinelcomponents in that system Given additional independentdata on the detailed structure of condensed phase water inthe future (eg Marx 1999 and in press) and on solutespeciation at various temperatures and pressures (seeDoltsinis et al 2004) the formalism described here canbe used to provide an increasingly complete thermody-namic description of aqueous solutions at depths corre-sponding to the lower crust and upper mantle

AcknowledgementsThis work was supported by ETHResearch Grant TH -1204-1 by RFBR grant 03-05-64633 by the RF President Program ldquoLeading ScientificSchool of Russiardquo (grant 0351645) to TVG and by theDeutsche Forschungsgemeinschaft within the scope ofSonderforschungsbereich 526 at the Ruhr-University ofBochum The helpful reviews of C Manning J Waltherand an anonymous reviewer as well as the editorialhandling of A Woodland are greatly appreciated

References

Anderson GM amp Burnham CW (1965) The solubility of quartzin supercritical water Am J Sci 263 494-511

Applin KR (1987) The diffusion of dissolved silica in diluteaqueous solution Geochim Cosmochim Acta 51 2147-2151

Audetat A amp Keppler H (2004) Viscosity of fluids in subduc-tion zones Science 303 513-516

Barelko VV Zarikov IV Pechatnikov EL (1994) Effect of theorigin of molecular clusters in one-component gas mixtures onthe equation of state Khim Fiz 13 42-50

Bockris JOrsquoM (1949) Ionic solutions Chem Soc LondonQuarterly Rev 3 173-180

Cary LW De Jong BHWS Dibble WEJr (1982) A 29Si NMRstudy of silica species in dilute aqueous solution GeochimCosmochim Acta 46 1317-1320

Catlow CRA (1980) Computer modelling of ionic crystals JPhysics 41 C6-53

Doltsinis NL Burchard M Maresch WV (2004) Ab initiomolecular dynamics simulations as a tool for interpreting vibra-tional spectra in aqueous fluids EMPG-X SymposiumAbstracts Lithos supplement to v 73 S28

Dove PM amp Rimstidt JD (1994) Silica physical behaviorgeochemistry and materials applications in ldquoSilica-water inter-actionsrdquo Reviews in Mineral 29 259-amp

Fockenberg T Burchard M Maresch WV (2004) Solubilities ofcalcium silicates at high pressures and temperatures EMPG-XSymposium Abstracts Lithos supplement to v 73 S37

282

Modeling of silica solubility and speciation 275

Ref

eren

ces

SiO

2-bu

ffer

No

ofP

kba

rT

degC

YSi

O2

Stat

isti

cal

Mea

n sq

uare

∆∆ YSi

O2

data

wei

ght

for

erro

rmdash

mdashmdash

mdashmdash

mdashmdash

mdashmdash

poin

tsea

ch d

ata

σσ (Y

SiO

2)

min

max

poin

t

And

erso

n amp

Bur

nham

(19

65)

quar

tz52

1-9

8559

6-90

00

0008

-00

31

12 (

16)

-6 (

-3)

+35

(+

41)

Wal

ther

amp O

rvil

le (

1983

)qu

artz

131-

235

0-55

00

0004

-00

021

6 (8

)-1

2 (-

15)

+9

(+16

)H

emle

y et

al

198

0qu

artz

141-

220

0-50

00

0001

-00

011

7 (9

)-1

4 (-

14+

3 (+

4)K

enne

dy (

1950

)qu

artz

870

01-1

75

182-

610

000

005-

000

11

13 (

11)

-11

(-42

)+

38 (

+12

)M

orey

et

al (

1962

)qu

artz

281

0145

-300

000

0006

-00

003

124

(19

)-3

7 (-

31)

-3 (

-6)

Man

ning

(19

94)

exp

qu

artz

515-

2050

0-90

00

002-

004

18

(10)

-17

(-21

)+

17 (

+22

)M

anni

ng (

1994

) c

alc

quar

tz91

000

1-20

25-9

000

0000

02-0

04

39

-16

+14

Wat

son

amp W

ark

(199

7)qu

artz

310

570-

810

000

5-0

021

14 (

17)

+7

(+15

)+

18 (

+18

)N

ewto

n amp

Man

ning

(20

02)

quar

tz e

nsta

tite

-for

ster

ite

22

1-14

700-

900

000

05-0

05

18

(5)

-14

(-13

)+

15 (

0)N

ewto

n amp

Man

ning

(20

03)

quar

tz e

nsta

tite

-for

ster

ite

612

800

000

1-0

021

10-9

+22

kyan

ite-

coru

ndum

fors

teri

te-r

util

e-ge

ikie

lite

Woo

dlan

d amp

Wal

ther

(19

87)

quar

tz53

095

-27

134

5-50

00

0004

-00

021

7 (9

)-2

0 (-

19)

+11

(+

18)

Wal

ther

amp W

oodl

and

(199

3)qu

artz

151

5-2

400-

600

000

09-0

002

113

(13

)-1

9 (-

19)

+17

(+

24)

Hem

ley

et a

l(1

977a

b)

talc

-chr

ysot

ile-

wat

er t

alc-

751-

220

0-71

50

0000

1-0

001

122

(-32

)+

70fo

rste

rite

-wat

er t

alc-

anti

gori

te-

wat

er t

alc-

anth

ophy

llit

e-w

ater

ta

lc-e

nsta

tite

-wat

er

anth

ophy

llit

e-fo

rste

rite

-wat

erH

emle

y et

al

(197

7b)

enst

atit

e-fo

rste

rite

71

680-

720

000

047-

000

053

110

(9)

-14

(-16

)+

15 (

+4)

Zha

ng amp

Fra

ntz

(200

0) e

xp

enst

atit

e-fo

rste

rite

610

-20

900-

1200

000

6-0

041

34 (

62)

+16

(0)

+52

(+

103)

Zha

ng amp

Fra

ntz

(200

0) c

alc

enst

atit

e-fo

rste

rite

710

700-

1300

000

2-0

091

31 (

60)

-2 (

-19)

+51

(+

103)

Dov

e amp

Rim

stid

t (1

994)

quar

tz1

000

125

000

0002

100

000

(-9)

--

Tabl

e 2

Exp

erim

enta

l da

ta u

sed

for

the

cali

brat

ion

of e

quat

ions

(31

)-(3

3)

va

lues

in

brac

kets

are

cal

cula

ted

on t

he b

asis

of

the

poly

nom

ial

fits

of

Man

ning

(19

94)

and

New

ton

amp M

anni

ng (

2002

)

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

Walther amp Woodland 1993 Woodland amp Walther 1987Watson amp Wark 1997 Newton amp Manning 2002) andother SiO2-buffering mineral assemblages (Zhang ampFrantz 2000 Newton amp Manning 2002 2003 Hemley etal 1977ab) as well as silica diffusion (Applin 1987) andNMR data (Cary et al 1982) at 25degC and 1 bar for esti-mating both the bulk mole fraction of silica and the fractionof silica in monomeric form In situ measurements ofRaman spectra for H2O-SiO2 fluids coexisting with quartz(Zotov amp Keppler 2000 2002) provide qualitativeconstraints in that the relative amount of silica in polymericform must increase with an isochoric increase of tempera-ture for T = 600-900degC and P = 56-14 kbar

As discussed further below it was not possible to inte-grate the numerical results of Zotov amp Keppler (2002) whoderived quantitative data on the mole fractions of silicamonomers and dimers from in situ Raman spectra ofaqueous fluid equilibrated with quartz at pressure from56-14 kbar and temperatures from 700 to 900degC Theseresults have been excluded from our database for tworeasons (i) our calculations show that these data are inlarge part inconsistent with the other studies consideredhere requiring a much more pronounced dependence ofsilica speciation on the density of the fluid at a giventemperature and (ii) it was not possible to resolve thediscrepancies in the paper by Zotov amp Keppler (2002)between the formalism presented in their equations (9) and(10) and the derived data on monomer and dimer quantitiespresented in their Table 2

For the calculation of thermodynamic parameters weused a non-linear least square method in order to minimizethe relative error of reproducing the bulk silica concentra-tion which is defined as∆YSiO2

() = 100[YSiO2(calculated) ndash

YSiO2(experimental)]YSiO2

(experimental)We have used a uniform statistical weight of 1 for all exper-imental data points (Table 2) thus making no attempt togauge the relative quality of the various data subsets intheir accuracy and precision (see discussion by Manning

1994) However because of the very irregular distributionof data points in pressure and temperature we found itnecessary to balance our database by adding 91 pointsinterpolated from the polynomial equation of Manning(1994) within the P-T range of experiments used for thecalibration of this equation These ldquosyntheticrdquo points wereregularly distributed with pressure (1 kbar steps) along 9isotherms (100 to 900degC with 100degC steps) A higherstatistical weight of 3 was assigned to these data points Inorder to insure an exact reproduction of the concentrationand the speciation of silica at the reference conditions of To= 25degC and Po = 1 bar (Morey et al 1962 Applin 1987Dove amp Rimstidt 1994 Cary et al 1982) we assigned avery high statistical weight of 10000 to the single datapoint representing these conditions The resulting valuesfor the thermodynamic parameters of equations (31) and(32) are∆Hdeg(mono)r = 24014 plusmn 1781 J∆Sdeg(mono)r = ndash 2810 plusmn 423 JK∆Vdeg(mono)r = ndash 02354plusmn00749 Jbar∆Cpdeg(mono)r = 3196 plusmn 645 JK∆Hdeg(poly)r = ndash 21059 plusmn 3589 J∆Sdeg(poly)r = 966 plusmn 1166 JK∆Vdeg(poly)r = 03161 plusmn 01798 Jbar

Table 2 and Fig 2 3 show that the data set is fitted wellby the above parameters The mean square error in repro-ducing 439 data points on bulk silica solubility is 142 relative No systematic variations of calculated errors withtemperature pressure bulk silica concentration and XLiqhave been found (Fig 3) In quartz-buffered systems rela-tively large deviations (gt 20 ) are indicated for the exper-imental data of Morey et al (1962) and Kennedy (1950) atpressures below 1 kbar and temperatures below 200degC Ourtest calculations show that these data are in part inconsis-tent with the bulk silica concentrations at the referenceconditions of To = 25degC and Po = 1 bar (Dove amp Rimstidt1994) requiring either notably higher (Morey et al 1962)or lower (Kennedy 1550) values These discrepancies canin part be explained by the problems of reaching equilib-

276

Fig 2 Results of modeling bulk molality ofdissolved silica in aqueous fluid in the presence ofquartz based on equations (21)-(30) (solid lines)The different symbols represent experimental datafrom the sources indicated Dashed lines are calcu-lations on the basis of the empirical equationsuggested by Manning (1994)

Modeling of silica solubility and speciation 277

Fig 3 Distribution of relative error ∆YSiO2() = 100[YSiO2

(calculated)-YSiO2(experimental)]YSiO2

(experimental) of reproducing experi-mental data with temperature (a) pressure (b) bulk silica concentration (c) and degree of association (XLiq) of pure water fluid (d) Thedifferent symbols represent experimental data from the sources indicated

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

rium at low temperatures (eg Walther amp Helgeson 1977)Indeed the overall accuracy of reproducing these experi-ments is comparable (Table 2) with the polynomial equa-

tion suggested by Manning (1994) Larger errors weredetected in silica-undersaturated systems (Table 2 Fig 3)for the experimental data of Hemley et al (1977a b) and

278

Fig 4 Relative concentration of silica inmonomers (a) dimers (b) and higher silicapolymers (c) dissolved in aqueous fluid in thepresence of quartz at P = 0001-20 kbar and T= 25-90degC Calculations are based on equa-tions (21)-(30) Thin grey lines are isobarsBlack lines are isotherms The solid squarerepresents the experimental data of Cary et al(1982) and Applin (1987) at 25degC and 1 bar

Zhang amp Frantz (2000) This may be related to the rela-tively large experimental uncertainties (Zhang amp Frantz2000 Newton amp Manning 2002) and to possible errorsincurred during the calculation of ∆Go

(Buffer-Quartz)PT forSiO2-buffering reactions involving water at low pressureand temperature (Hemley et al 1977a b) in combinationwith the database of Holland amp Powell (1998) Obviouslyat T = 25degC and P = 1 bar the highly weighted referencedata on the concentration of silica in the presence of quartz(Dove amp Rimstidt 1994) and on the distribution of silica

between (SiO2)bull(H2O)2-monomers and (SiO2)2bull(H2O)3-dimers (Cary et al 1982 Applin 1987) are accuratelyreproduced with a relative error of less then 02 (Table 2Fig 3 4a) Figure 2 shows that the results of our thermo-dynamic modeling coincide well with the empiricalapproximation suggested by Manning (1994) for the bulksolubility of silica in water in the presence of quartz Table2 also compares our numerical values (Table 2) with thoseobtained from the empirical equations of Manning (1994)for the quartz-buffered system and of Manning amp Newton

Modeling of silica solubility and speciation 279

Fig 5 Results of modeling bulk concentration (a)(b) and speciation (c) of dissolved silica in aqueousfluid in the presence of different buffering mineralassemblages The symbols represent data from thesources indicated (a) experimental data of Newtonamp Manning (2002) and empirical calibration ofZhang amp Frantz (2000) based on original andpublished (Nakamura amp Kushiro 1974 Manningamp Boettcher 1994) experimental data (b) experi-mental data of Newton amp Manning (2003) (c)monomer-dimer speciation model of Newton ampManning (2003)

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

(2002) for the enstatite-forsterite-buffered system For the358 compared experimental data points (Table 2) ourmodel yields a mean square error of 134 versus 165 for the equations of Manning (1994) and Manning ampNewton (2002)

As noted above our model was constrained to accu-rately reproduce experimental data on silica partitioningbetween monomers and larger clusters at 25degC and 1 bar(Cary et al 1982 Applin 1987) The silica species at theseconditions (Fig 4) are represented by (SiO2)bull(H2O)2-monomers (94 of total dissolved silica) and(SiO2)2bull(H2O)3-dimers (6 of total dissolved silica) witha negligible amount of higher polymers (lt 03 of totaldissolved silica) Cary et al (1982) detected the presenceof (SiO2)2bull(H2O)3-dimers by NMR in solutions containing16 mM SiO2 at a pH of 72 and estimated their concentra-tion to be about 6 of the total dissolved silica No addi-tional polymerized forms were detected which isconsistent with our calculations for higher polymers (Fig4c) However the lack of such accurate experimental datafor speciation over a wider range of pressures and temper-atures (eg Watson amp Wark 1997) leads to the relativelylarge uncertainties in ∆Sdeg(poly)r and ∆Vdeg(poly)r ie for chainreaction (12) It should be noted that reaction (12) can alsobe used for the calculation of silica speciation at an arbi-trary bulk concentration of dissolved silica YSiO2

=SiO2(SiO2 + H2O) Equations characterizing speciationthen become

X1 = XLiqXSS(1 ndash XSS)[XSSK(poly) + (1 ndash XSS)XLiq] (34)

XL = XLiq(1 ndash XSS) (35)

XSS = [4XLiqYSiO2(K(poly) ndash XLiq)(1 ndash 2YSiO2

) + (XLiq)2(1 ndashYSiO2

)2]05 ndash XLiq(1 ndash YSiO2) [2(K(poly) ndash XLiq)(1 ndash 2YSiO2

)] (36)

where XSS = X1 + Σinfin

n=2 Xn is the sum of the mole fractions of

all dissolved silica species (ie XSS + XL + XG = 1) Otherquantities are as given in equations (21)-(27)

Figure 4 shows SiO2 speciation in the presence ofquartz as a function of pressure temperature and bulksilica content as predicted from our model calculations Itis seen that the relative amount of silica in the form ofmonomers is most strongly dependent on bulk silicaconcentration and generally decreases with increasingSiO2 content of the solution (Fig 4a) Significant amountsof dissolved silica (10-30 ) in dimeric form are calculatedfor temperatures gt 300degC (Fig 4b) The relative proportionof dimeric silica shows a maximum (Fig 4b) at YSiO2

asymp001 At higher YSiO2

trimers and higher polymers becomeimportant (Fig 4c) consuming 10 to 50 of the dissolvedsilica at temperatures gt 500degC Because increasing temper-ature leads to higher bulk silica there is also a similarcorrelation with temperature The dependence on pressureis least pronounced as can be expected from the competingeffects of reactions (11) and (12) (H2O)L is consumedduring the monomer-forming reaction (11) and liberatedduring the polymer-forming reaction (12) thereforedictating opposite signs of ∆Vdegr of the reactions Increasingpressure should lead to increased bulk silica solubility (dueto reaction (11) ∆Vdeg(mono)r lt 0) and therefore increasing

polymerization on the one hand (Fig 4a) whereasincreasing pressure should also favor depolymerizationaccording to reaction (12) (∆Vdeg(poly)r gt 0) on the other

Figure 5 shows silica solubility and speciation inaqueous fluid coexisting with different SiO2-bufferingmineral assemblages It is seen that our model accuratelyreproduces the available experimental data (Zhang ampFrantz 2000 Newton amp Manning 2003) on bulk silicasolubility (Fig 5ab) The calculated distribution of silicabetween monomers and higher polymers as a function ofbulk SiO2(SiO2 + H2O) at constant pressure and tempera-ture (Fig 5c) is very similar to the results of Newton ampManning (2003) which are based on a somewhat simplermonomer-dimer model At constant pressure and tempera-ture the amount of monomeric silica decreases non-linearlywith decreasing bulk silica solubility

Discussion

The semi-empirical thermodynamic formalism devel-oped in this paper allows accurate reproduction of existingdata on silica concentration and silica speciation inaqueous fluid coexisting with quartz and other silica-buffering assemblages up to 20 kbar and 1300degC Incomparison to Mannings (1994) polynomial fit ourapproach also predicts the speciation of silica in theaqueous fluid without increasing the number of empiricalparameters in the model In addition our chain-reactionmodel allows an excellent data fit with full consideration ofall silica polymers (Fig 4) rather than only monomers anddimers (cf Newton amp Manning 2003) Figure 6 shows thepositions of isopleths of bulk silica concentration calcu-lated for both silica-saturated (Fig 6a) and the silica-under-saturated enstatite-forsterite-buffered (Fig 6b) systems atelevated temperature and pressure It is seen that our resultsare in general compatible with the polynomial equationssuggested by Manning (1994) and Newton amp Manning(2002) with the largest discrepancies found in the high-temperature (gt 900degC) high-YSiO2

(gt 005) region Itshould be emphasized here that our present model has beendeveloped for a bulk silica concentration (YSiO2

) in the fluidof less than 01 (Table 2) Therefore this model mayrequire modification (eg by incorporating terms for non-ideality of the aqueous solution) in order to account prop-erly for the thermodynamics of silica-rich hydroussolutions (eg Shen amp Keppler 1997 Audetat amp Keppler2004) in proximity to the second critical point in the H2O-SiO2 system

Our model predicts basic similarities between the ther-modynamic properties of dissolved silica species and thoseof associated clustered ldquoliquid-likerdquo water molecules (egequations (18)-(20)) This is consistent with the conclu-sions of Bockris (1949) and Walther amp Orville (1983) whosuggested that hydrous monomeric silica species _ Si(OH)4_ are associated with two more water dipoles to form anaqueous silica complex _ Si(OH)4bull2H2O _ by hydrogenbonding This mechanism of association of molecules is infact identical to the association (clustering) of molecules ofpure water (eg Luck 1980 Gorbaty amp Kalinichev 1995)

280

Modeling of silica solubility and speciation 281

Fig 6 Isopleths of bulk silica concentration (solid lines)calculated according to equations (21)-(27) and (31)-(33)for silica-saturated (a) and silica-undersaturated enstatite-forsterite buffered (b) system at elevated temperature andpressure Dashed lines show isopleths calculated with theequations suggested by Manning (1994) and Newton ampManning (2002) Phase boundaries in (a) are calculatedfrom the database of Holland amp Powell (1998) Meltingrelations in the presence of water are not shown

Fig 7 Results of modeling silica speciation in aqueous fluidin the presence of quartz along the two isochores presented byZotov amp Keppler (2002) ρ = 078 gcm3 P = 56-68 kbar andρ = 094 gcm3 P = 106-140 kbar The solid squares repre-sent the results of Zotov amp Keppler (2002) based on the inter-pretation of in situ Raman spectra (a) dependence of silicaspeciation on temperature (b) dependence of silica speciationon bulk silica content

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

and is directly accounted for by equations (4)-(6) (for adiscussion see Gerya amp Perchuk 1997 Gerya et al2004) The formation of similar complexes with dimersand higher polymers can also be expected thus supportingour thermodynamic formalism

An important unresolved question is the discrepancybetween the results obtained from our data set which isbased mainly on direct solubility measurements and dataobtained from Raman spectra (Zotov amp Keppler 20002002) This is particularly evident in the pressure effect onsilica polymerization The positive ∆Vdegr of reaction (12)suggests that at constant temperature and bulk silica anyincrease in pressure (ie in fluid density) should lead to adecrease in the degree of polymerization This is alsosupported by the ab initio molecular dynamics simulationsof Doltsinis et al (2004) who calculated higher effectivepressures for monomeric-dimeric systems in comparisonwith purely monomeric systems at T = 900 K and constantfluid density In the presence of quartz bulk silica contentincreases with pressure and the competing effects of reac-tions (11) and (12) should lead to only small changes inthe degree of polymerization Figure 7 indicates the trendspredicted from our model In contrast the results of Zotovamp Keppler (2002) suggest strong increase in the degree ofpolymerization with pressure (ie fluid density) requiringa notably negative ∆Vo

r of polymer-forming reaction (12)Our test calculations show that no consistent set of ther-modynamic parameters for reactions (11) and (12) canprovide an accurate description of both the silica solubilitymeasurements and the changes of degree of polymeriza-tion with fluid density calculated by Zotov amp Keppler(2002)

We suggest that an explanation for the above discrep-ancies must be sought in (i) the possible problems of equi-libration in the relatively short experiments in thediamond-anvil cell performed by Zotov amp Keppler (20002002) and (ii) the difficult task of quantifying the Ramanspectra of aqueous silica solutions The interpretation ofRaman spectra is based mainly on theoretical spectraobtained using empirical interaction potentials (Catlow1980 Price amp Parker 1984) or on single molecule abinitio calculations as in the case of Zotov amp Keppler(2000 2002) The problem with the latter is that they donot include crucial temperature- and pressure-dependentsolvent effects To test these environmental effects wehave initiated a series of both ab initio molecular dynamics(AIMD) simulations and static gas phase ab initio calcula-tions (Doltsinis et al 2004) Our results to date suggestthat while the strong feature at ca 780-800 cm-1 can beascribed to monomers with some confidence there arestill considerable uncertainties in interpreting otherfeatures of the Raman spectra in terms of monomersandor dimers Considering the low amounts of total silicainvolved and the increasingly floppy nature of largerpolymers leading to increasingly broadened spectralfeatures it appears that the recognition and identificationof trimers and higher polymers can be problematical Ifhidden trimers are present then an interpretation basedsolely on monomerdimer ratios will obviously be flawedThus while the results of Zotov amp Keppler (2000 2002)

represent a valiant first attempt to tackle the problem andprovide valuable qualitative data direct comparisons withclassical solubility results should not be overextended Wesuggest that further Raman spectroscopic measurementson well-defined solutions with given bulk silica contents atdifferent pressures and temperatures rather than quartz-buffered solutions only will be a valuable step toward abetter understanding of such spectra

The approach outlined in this paper has been success-fully used to fit and describe experimental data on bulkspinel (MgAl2O4) solubility up to 30 kbar byZakhartchouk et al (2004) despite the lack of detailedinformation on the actual speciation of the dissolved spinelcomponents in that system Given additional independentdata on the detailed structure of condensed phase water inthe future (eg Marx 1999 and in press) and on solutespeciation at various temperatures and pressures (seeDoltsinis et al 2004) the formalism described here canbe used to provide an increasingly complete thermody-namic description of aqueous solutions at depths corre-sponding to the lower crust and upper mantle

AcknowledgementsThis work was supported by ETHResearch Grant TH -1204-1 by RFBR grant 03-05-64633 by the RF President Program ldquoLeading ScientificSchool of Russiardquo (grant 0351645) to TVG and by theDeutsche Forschungsgemeinschaft within the scope ofSonderforschungsbereich 526 at the Ruhr-University ofBochum The helpful reviews of C Manning J Waltherand an anonymous reviewer as well as the editorialhandling of A Woodland are greatly appreciated

References

Anderson GM amp Burnham CW (1965) The solubility of quartzin supercritical water Am J Sci 263 494-511

Applin KR (1987) The diffusion of dissolved silica in diluteaqueous solution Geochim Cosmochim Acta 51 2147-2151

Audetat A amp Keppler H (2004) Viscosity of fluids in subduc-tion zones Science 303 513-516

Barelko VV Zarikov IV Pechatnikov EL (1994) Effect of theorigin of molecular clusters in one-component gas mixtures onthe equation of state Khim Fiz 13 42-50

Bockris JOrsquoM (1949) Ionic solutions Chem Soc LondonQuarterly Rev 3 173-180

Cary LW De Jong BHWS Dibble WEJr (1982) A 29Si NMRstudy of silica species in dilute aqueous solution GeochimCosmochim Acta 46 1317-1320

Catlow CRA (1980) Computer modelling of ionic crystals JPhysics 41 C6-53

Doltsinis NL Burchard M Maresch WV (2004) Ab initiomolecular dynamics simulations as a tool for interpreting vibra-tional spectra in aqueous fluids EMPG-X SymposiumAbstracts Lithos supplement to v 73 S28

Dove PM amp Rimstidt JD (1994) Silica physical behaviorgeochemistry and materials applications in ldquoSilica-water inter-actionsrdquo Reviews in Mineral 29 259-amp

Fockenberg T Burchard M Maresch WV (2004) Solubilities ofcalcium silicates at high pressures and temperatures EMPG-XSymposium Abstracts Lithos supplement to v 73 S37

282

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

Walther amp Woodland 1993 Woodland amp Walther 1987Watson amp Wark 1997 Newton amp Manning 2002) andother SiO2-buffering mineral assemblages (Zhang ampFrantz 2000 Newton amp Manning 2002 2003 Hemley etal 1977ab) as well as silica diffusion (Applin 1987) andNMR data (Cary et al 1982) at 25degC and 1 bar for esti-mating both the bulk mole fraction of silica and the fractionof silica in monomeric form In situ measurements ofRaman spectra for H2O-SiO2 fluids coexisting with quartz(Zotov amp Keppler 2000 2002) provide qualitativeconstraints in that the relative amount of silica in polymericform must increase with an isochoric increase of tempera-ture for T = 600-900degC and P = 56-14 kbar

As discussed further below it was not possible to inte-grate the numerical results of Zotov amp Keppler (2002) whoderived quantitative data on the mole fractions of silicamonomers and dimers from in situ Raman spectra ofaqueous fluid equilibrated with quartz at pressure from56-14 kbar and temperatures from 700 to 900degC Theseresults have been excluded from our database for tworeasons (i) our calculations show that these data are inlarge part inconsistent with the other studies consideredhere requiring a much more pronounced dependence ofsilica speciation on the density of the fluid at a giventemperature and (ii) it was not possible to resolve thediscrepancies in the paper by Zotov amp Keppler (2002)between the formalism presented in their equations (9) and(10) and the derived data on monomer and dimer quantitiespresented in their Table 2

For the calculation of thermodynamic parameters weused a non-linear least square method in order to minimizethe relative error of reproducing the bulk silica concentra-tion which is defined as∆YSiO2

() = 100[YSiO2(calculated) ndash

YSiO2(experimental)]YSiO2

(experimental)We have used a uniform statistical weight of 1 for all exper-imental data points (Table 2) thus making no attempt togauge the relative quality of the various data subsets intheir accuracy and precision (see discussion by Manning

1994) However because of the very irregular distributionof data points in pressure and temperature we found itnecessary to balance our database by adding 91 pointsinterpolated from the polynomial equation of Manning(1994) within the P-T range of experiments used for thecalibration of this equation These ldquosyntheticrdquo points wereregularly distributed with pressure (1 kbar steps) along 9isotherms (100 to 900degC with 100degC steps) A higherstatistical weight of 3 was assigned to these data points Inorder to insure an exact reproduction of the concentrationand the speciation of silica at the reference conditions of To= 25degC and Po = 1 bar (Morey et al 1962 Applin 1987Dove amp Rimstidt 1994 Cary et al 1982) we assigned avery high statistical weight of 10000 to the single datapoint representing these conditions The resulting valuesfor the thermodynamic parameters of equations (31) and(32) are∆Hdeg(mono)r = 24014 plusmn 1781 J∆Sdeg(mono)r = ndash 2810 plusmn 423 JK∆Vdeg(mono)r = ndash 02354plusmn00749 Jbar∆Cpdeg(mono)r = 3196 plusmn 645 JK∆Hdeg(poly)r = ndash 21059 plusmn 3589 J∆Sdeg(poly)r = 966 plusmn 1166 JK∆Vdeg(poly)r = 03161 plusmn 01798 Jbar

Table 2 and Fig 2 3 show that the data set is fitted wellby the above parameters The mean square error in repro-ducing 439 data points on bulk silica solubility is 142 relative No systematic variations of calculated errors withtemperature pressure bulk silica concentration and XLiqhave been found (Fig 3) In quartz-buffered systems rela-tively large deviations (gt 20 ) are indicated for the exper-imental data of Morey et al (1962) and Kennedy (1950) atpressures below 1 kbar and temperatures below 200degC Ourtest calculations show that these data are in part inconsis-tent with the bulk silica concentrations at the referenceconditions of To = 25degC and Po = 1 bar (Dove amp Rimstidt1994) requiring either notably higher (Morey et al 1962)or lower (Kennedy 1550) values These discrepancies canin part be explained by the problems of reaching equilib-

276

Fig 2 Results of modeling bulk molality ofdissolved silica in aqueous fluid in the presence ofquartz based on equations (21)-(30) (solid lines)The different symbols represent experimental datafrom the sources indicated Dashed lines are calcu-lations on the basis of the empirical equationsuggested by Manning (1994)

Modeling of silica solubility and speciation 277

Fig 3 Distribution of relative error ∆YSiO2() = 100[YSiO2

(calculated)-YSiO2(experimental)]YSiO2

(experimental) of reproducing experi-mental data with temperature (a) pressure (b) bulk silica concentration (c) and degree of association (XLiq) of pure water fluid (d) Thedifferent symbols represent experimental data from the sources indicated

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

rium at low temperatures (eg Walther amp Helgeson 1977)Indeed the overall accuracy of reproducing these experi-ments is comparable (Table 2) with the polynomial equa-

tion suggested by Manning (1994) Larger errors weredetected in silica-undersaturated systems (Table 2 Fig 3)for the experimental data of Hemley et al (1977a b) and

278

Fig 4 Relative concentration of silica inmonomers (a) dimers (b) and higher silicapolymers (c) dissolved in aqueous fluid in thepresence of quartz at P = 0001-20 kbar and T= 25-90degC Calculations are based on equa-tions (21)-(30) Thin grey lines are isobarsBlack lines are isotherms The solid squarerepresents the experimental data of Cary et al(1982) and Applin (1987) at 25degC and 1 bar

Zhang amp Frantz (2000) This may be related to the rela-tively large experimental uncertainties (Zhang amp Frantz2000 Newton amp Manning 2002) and to possible errorsincurred during the calculation of ∆Go

(Buffer-Quartz)PT forSiO2-buffering reactions involving water at low pressureand temperature (Hemley et al 1977a b) in combinationwith the database of Holland amp Powell (1998) Obviouslyat T = 25degC and P = 1 bar the highly weighted referencedata on the concentration of silica in the presence of quartz(Dove amp Rimstidt 1994) and on the distribution of silica

between (SiO2)bull(H2O)2-monomers and (SiO2)2bull(H2O)3-dimers (Cary et al 1982 Applin 1987) are accuratelyreproduced with a relative error of less then 02 (Table 2Fig 3 4a) Figure 2 shows that the results of our thermo-dynamic modeling coincide well with the empiricalapproximation suggested by Manning (1994) for the bulksolubility of silica in water in the presence of quartz Table2 also compares our numerical values (Table 2) with thoseobtained from the empirical equations of Manning (1994)for the quartz-buffered system and of Manning amp Newton

Modeling of silica solubility and speciation 279

Fig 5 Results of modeling bulk concentration (a)(b) and speciation (c) of dissolved silica in aqueousfluid in the presence of different buffering mineralassemblages The symbols represent data from thesources indicated (a) experimental data of Newtonamp Manning (2002) and empirical calibration ofZhang amp Frantz (2000) based on original andpublished (Nakamura amp Kushiro 1974 Manningamp Boettcher 1994) experimental data (b) experi-mental data of Newton amp Manning (2003) (c)monomer-dimer speciation model of Newton ampManning (2003)

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

(2002) for the enstatite-forsterite-buffered system For the358 compared experimental data points (Table 2) ourmodel yields a mean square error of 134 versus 165 for the equations of Manning (1994) and Manning ampNewton (2002)

As noted above our model was constrained to accu-rately reproduce experimental data on silica partitioningbetween monomers and larger clusters at 25degC and 1 bar(Cary et al 1982 Applin 1987) The silica species at theseconditions (Fig 4) are represented by (SiO2)bull(H2O)2-monomers (94 of total dissolved silica) and(SiO2)2bull(H2O)3-dimers (6 of total dissolved silica) witha negligible amount of higher polymers (lt 03 of totaldissolved silica) Cary et al (1982) detected the presenceof (SiO2)2bull(H2O)3-dimers by NMR in solutions containing16 mM SiO2 at a pH of 72 and estimated their concentra-tion to be about 6 of the total dissolved silica No addi-tional polymerized forms were detected which isconsistent with our calculations for higher polymers (Fig4c) However the lack of such accurate experimental datafor speciation over a wider range of pressures and temper-atures (eg Watson amp Wark 1997) leads to the relativelylarge uncertainties in ∆Sdeg(poly)r and ∆Vdeg(poly)r ie for chainreaction (12) It should be noted that reaction (12) can alsobe used for the calculation of silica speciation at an arbi-trary bulk concentration of dissolved silica YSiO2

=SiO2(SiO2 + H2O) Equations characterizing speciationthen become

X1 = XLiqXSS(1 ndash XSS)[XSSK(poly) + (1 ndash XSS)XLiq] (34)

XL = XLiq(1 ndash XSS) (35)

XSS = [4XLiqYSiO2(K(poly) ndash XLiq)(1 ndash 2YSiO2

) + (XLiq)2(1 ndashYSiO2

)2]05 ndash XLiq(1 ndash YSiO2) [2(K(poly) ndash XLiq)(1 ndash 2YSiO2

)] (36)

where XSS = X1 + Σinfin

n=2 Xn is the sum of the mole fractions of

all dissolved silica species (ie XSS + XL + XG = 1) Otherquantities are as given in equations (21)-(27)

Figure 4 shows SiO2 speciation in the presence ofquartz as a function of pressure temperature and bulksilica content as predicted from our model calculations Itis seen that the relative amount of silica in the form ofmonomers is most strongly dependent on bulk silicaconcentration and generally decreases with increasingSiO2 content of the solution (Fig 4a) Significant amountsof dissolved silica (10-30 ) in dimeric form are calculatedfor temperatures gt 300degC (Fig 4b) The relative proportionof dimeric silica shows a maximum (Fig 4b) at YSiO2

asymp001 At higher YSiO2

trimers and higher polymers becomeimportant (Fig 4c) consuming 10 to 50 of the dissolvedsilica at temperatures gt 500degC Because increasing temper-ature leads to higher bulk silica there is also a similarcorrelation with temperature The dependence on pressureis least pronounced as can be expected from the competingeffects of reactions (11) and (12) (H2O)L is consumedduring the monomer-forming reaction (11) and liberatedduring the polymer-forming reaction (12) thereforedictating opposite signs of ∆Vdegr of the reactions Increasingpressure should lead to increased bulk silica solubility (dueto reaction (11) ∆Vdeg(mono)r lt 0) and therefore increasing

polymerization on the one hand (Fig 4a) whereasincreasing pressure should also favor depolymerizationaccording to reaction (12) (∆Vdeg(poly)r gt 0) on the other

Figure 5 shows silica solubility and speciation inaqueous fluid coexisting with different SiO2-bufferingmineral assemblages It is seen that our model accuratelyreproduces the available experimental data (Zhang ampFrantz 2000 Newton amp Manning 2003) on bulk silicasolubility (Fig 5ab) The calculated distribution of silicabetween monomers and higher polymers as a function ofbulk SiO2(SiO2 + H2O) at constant pressure and tempera-ture (Fig 5c) is very similar to the results of Newton ampManning (2003) which are based on a somewhat simplermonomer-dimer model At constant pressure and tempera-ture the amount of monomeric silica decreases non-linearlywith decreasing bulk silica solubility

Discussion

The semi-empirical thermodynamic formalism devel-oped in this paper allows accurate reproduction of existingdata on silica concentration and silica speciation inaqueous fluid coexisting with quartz and other silica-buffering assemblages up to 20 kbar and 1300degC Incomparison to Mannings (1994) polynomial fit ourapproach also predicts the speciation of silica in theaqueous fluid without increasing the number of empiricalparameters in the model In addition our chain-reactionmodel allows an excellent data fit with full consideration ofall silica polymers (Fig 4) rather than only monomers anddimers (cf Newton amp Manning 2003) Figure 6 shows thepositions of isopleths of bulk silica concentration calcu-lated for both silica-saturated (Fig 6a) and the silica-under-saturated enstatite-forsterite-buffered (Fig 6b) systems atelevated temperature and pressure It is seen that our resultsare in general compatible with the polynomial equationssuggested by Manning (1994) and Newton amp Manning(2002) with the largest discrepancies found in the high-temperature (gt 900degC) high-YSiO2

(gt 005) region Itshould be emphasized here that our present model has beendeveloped for a bulk silica concentration (YSiO2

) in the fluidof less than 01 (Table 2) Therefore this model mayrequire modification (eg by incorporating terms for non-ideality of the aqueous solution) in order to account prop-erly for the thermodynamics of silica-rich hydroussolutions (eg Shen amp Keppler 1997 Audetat amp Keppler2004) in proximity to the second critical point in the H2O-SiO2 system

Our model predicts basic similarities between the ther-modynamic properties of dissolved silica species and thoseof associated clustered ldquoliquid-likerdquo water molecules (egequations (18)-(20)) This is consistent with the conclu-sions of Bockris (1949) and Walther amp Orville (1983) whosuggested that hydrous monomeric silica species _ Si(OH)4_ are associated with two more water dipoles to form anaqueous silica complex _ Si(OH)4bull2H2O _ by hydrogenbonding This mechanism of association of molecules is infact identical to the association (clustering) of molecules ofpure water (eg Luck 1980 Gorbaty amp Kalinichev 1995)

280

Modeling of silica solubility and speciation 281

Fig 6 Isopleths of bulk silica concentration (solid lines)calculated according to equations (21)-(27) and (31)-(33)for silica-saturated (a) and silica-undersaturated enstatite-forsterite buffered (b) system at elevated temperature andpressure Dashed lines show isopleths calculated with theequations suggested by Manning (1994) and Newton ampManning (2002) Phase boundaries in (a) are calculatedfrom the database of Holland amp Powell (1998) Meltingrelations in the presence of water are not shown

Fig 7 Results of modeling silica speciation in aqueous fluidin the presence of quartz along the two isochores presented byZotov amp Keppler (2002) ρ = 078 gcm3 P = 56-68 kbar andρ = 094 gcm3 P = 106-140 kbar The solid squares repre-sent the results of Zotov amp Keppler (2002) based on the inter-pretation of in situ Raman spectra (a) dependence of silicaspeciation on temperature (b) dependence of silica speciationon bulk silica content

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

and is directly accounted for by equations (4)-(6) (for adiscussion see Gerya amp Perchuk 1997 Gerya et al2004) The formation of similar complexes with dimersand higher polymers can also be expected thus supportingour thermodynamic formalism

An important unresolved question is the discrepancybetween the results obtained from our data set which isbased mainly on direct solubility measurements and dataobtained from Raman spectra (Zotov amp Keppler 20002002) This is particularly evident in the pressure effect onsilica polymerization The positive ∆Vdegr of reaction (12)suggests that at constant temperature and bulk silica anyincrease in pressure (ie in fluid density) should lead to adecrease in the degree of polymerization This is alsosupported by the ab initio molecular dynamics simulationsof Doltsinis et al (2004) who calculated higher effectivepressures for monomeric-dimeric systems in comparisonwith purely monomeric systems at T = 900 K and constantfluid density In the presence of quartz bulk silica contentincreases with pressure and the competing effects of reac-tions (11) and (12) should lead to only small changes inthe degree of polymerization Figure 7 indicates the trendspredicted from our model In contrast the results of Zotovamp Keppler (2002) suggest strong increase in the degree ofpolymerization with pressure (ie fluid density) requiringa notably negative ∆Vo

r of polymer-forming reaction (12)Our test calculations show that no consistent set of ther-modynamic parameters for reactions (11) and (12) canprovide an accurate description of both the silica solubilitymeasurements and the changes of degree of polymeriza-tion with fluid density calculated by Zotov amp Keppler(2002)

We suggest that an explanation for the above discrep-ancies must be sought in (i) the possible problems of equi-libration in the relatively short experiments in thediamond-anvil cell performed by Zotov amp Keppler (20002002) and (ii) the difficult task of quantifying the Ramanspectra of aqueous silica solutions The interpretation ofRaman spectra is based mainly on theoretical spectraobtained using empirical interaction potentials (Catlow1980 Price amp Parker 1984) or on single molecule abinitio calculations as in the case of Zotov amp Keppler(2000 2002) The problem with the latter is that they donot include crucial temperature- and pressure-dependentsolvent effects To test these environmental effects wehave initiated a series of both ab initio molecular dynamics(AIMD) simulations and static gas phase ab initio calcula-tions (Doltsinis et al 2004) Our results to date suggestthat while the strong feature at ca 780-800 cm-1 can beascribed to monomers with some confidence there arestill considerable uncertainties in interpreting otherfeatures of the Raman spectra in terms of monomersandor dimers Considering the low amounts of total silicainvolved and the increasingly floppy nature of largerpolymers leading to increasingly broadened spectralfeatures it appears that the recognition and identificationof trimers and higher polymers can be problematical Ifhidden trimers are present then an interpretation basedsolely on monomerdimer ratios will obviously be flawedThus while the results of Zotov amp Keppler (2000 2002)

represent a valiant first attempt to tackle the problem andprovide valuable qualitative data direct comparisons withclassical solubility results should not be overextended Wesuggest that further Raman spectroscopic measurementson well-defined solutions with given bulk silica contents atdifferent pressures and temperatures rather than quartz-buffered solutions only will be a valuable step toward abetter understanding of such spectra

The approach outlined in this paper has been success-fully used to fit and describe experimental data on bulkspinel (MgAl2O4) solubility up to 30 kbar byZakhartchouk et al (2004) despite the lack of detailedinformation on the actual speciation of the dissolved spinelcomponents in that system Given additional independentdata on the detailed structure of condensed phase water inthe future (eg Marx 1999 and in press) and on solutespeciation at various temperatures and pressures (seeDoltsinis et al 2004) the formalism described here canbe used to provide an increasingly complete thermody-namic description of aqueous solutions at depths corre-sponding to the lower crust and upper mantle

AcknowledgementsThis work was supported by ETHResearch Grant TH -1204-1 by RFBR grant 03-05-64633 by the RF President Program ldquoLeading ScientificSchool of Russiardquo (grant 0351645) to TVG and by theDeutsche Forschungsgemeinschaft within the scope ofSonderforschungsbereich 526 at the Ruhr-University ofBochum The helpful reviews of C Manning J Waltherand an anonymous reviewer as well as the editorialhandling of A Woodland are greatly appreciated

References

Anderson GM amp Burnham CW (1965) The solubility of quartzin supercritical water Am J Sci 263 494-511

Applin KR (1987) The diffusion of dissolved silica in diluteaqueous solution Geochim Cosmochim Acta 51 2147-2151

Audetat A amp Keppler H (2004) Viscosity of fluids in subduc-tion zones Science 303 513-516

Barelko VV Zarikov IV Pechatnikov EL (1994) Effect of theorigin of molecular clusters in one-component gas mixtures onthe equation of state Khim Fiz 13 42-50

Bockris JOrsquoM (1949) Ionic solutions Chem Soc LondonQuarterly Rev 3 173-180

Cary LW De Jong BHWS Dibble WEJr (1982) A 29Si NMRstudy of silica species in dilute aqueous solution GeochimCosmochim Acta 46 1317-1320

Catlow CRA (1980) Computer modelling of ionic crystals JPhysics 41 C6-53

Doltsinis NL Burchard M Maresch WV (2004) Ab initiomolecular dynamics simulations as a tool for interpreting vibra-tional spectra in aqueous fluids EMPG-X SymposiumAbstracts Lithos supplement to v 73 S28

Dove PM amp Rimstidt JD (1994) Silica physical behaviorgeochemistry and materials applications in ldquoSilica-water inter-actionsrdquo Reviews in Mineral 29 259-amp

Fockenberg T Burchard M Maresch WV (2004) Solubilities ofcalcium silicates at high pressures and temperatures EMPG-XSymposium Abstracts Lithos supplement to v 73 S37

282

Modeling of silica solubility and speciation 277

Fig 3 Distribution of relative error ∆YSiO2() = 100[YSiO2

(calculated)-YSiO2(experimental)]YSiO2

(experimental) of reproducing experi-mental data with temperature (a) pressure (b) bulk silica concentration (c) and degree of association (XLiq) of pure water fluid (d) Thedifferent symbols represent experimental data from the sources indicated

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

rium at low temperatures (eg Walther amp Helgeson 1977)Indeed the overall accuracy of reproducing these experi-ments is comparable (Table 2) with the polynomial equa-

tion suggested by Manning (1994) Larger errors weredetected in silica-undersaturated systems (Table 2 Fig 3)for the experimental data of Hemley et al (1977a b) and

278

Fig 4 Relative concentration of silica inmonomers (a) dimers (b) and higher silicapolymers (c) dissolved in aqueous fluid in thepresence of quartz at P = 0001-20 kbar and T= 25-90degC Calculations are based on equa-tions (21)-(30) Thin grey lines are isobarsBlack lines are isotherms The solid squarerepresents the experimental data of Cary et al(1982) and Applin (1987) at 25degC and 1 bar

Zhang amp Frantz (2000) This may be related to the rela-tively large experimental uncertainties (Zhang amp Frantz2000 Newton amp Manning 2002) and to possible errorsincurred during the calculation of ∆Go

(Buffer-Quartz)PT forSiO2-buffering reactions involving water at low pressureand temperature (Hemley et al 1977a b) in combinationwith the database of Holland amp Powell (1998) Obviouslyat T = 25degC and P = 1 bar the highly weighted referencedata on the concentration of silica in the presence of quartz(Dove amp Rimstidt 1994) and on the distribution of silica

between (SiO2)bull(H2O)2-monomers and (SiO2)2bull(H2O)3-dimers (Cary et al 1982 Applin 1987) are accuratelyreproduced with a relative error of less then 02 (Table 2Fig 3 4a) Figure 2 shows that the results of our thermo-dynamic modeling coincide well with the empiricalapproximation suggested by Manning (1994) for the bulksolubility of silica in water in the presence of quartz Table2 also compares our numerical values (Table 2) with thoseobtained from the empirical equations of Manning (1994)for the quartz-buffered system and of Manning amp Newton

Modeling of silica solubility and speciation 279

Fig 5 Results of modeling bulk concentration (a)(b) and speciation (c) of dissolved silica in aqueousfluid in the presence of different buffering mineralassemblages The symbols represent data from thesources indicated (a) experimental data of Newtonamp Manning (2002) and empirical calibration ofZhang amp Frantz (2000) based on original andpublished (Nakamura amp Kushiro 1974 Manningamp Boettcher 1994) experimental data (b) experi-mental data of Newton amp Manning (2003) (c)monomer-dimer speciation model of Newton ampManning (2003)

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

(2002) for the enstatite-forsterite-buffered system For the358 compared experimental data points (Table 2) ourmodel yields a mean square error of 134 versus 165 for the equations of Manning (1994) and Manning ampNewton (2002)

As noted above our model was constrained to accu-rately reproduce experimental data on silica partitioningbetween monomers and larger clusters at 25degC and 1 bar(Cary et al 1982 Applin 1987) The silica species at theseconditions (Fig 4) are represented by (SiO2)bull(H2O)2-monomers (94 of total dissolved silica) and(SiO2)2bull(H2O)3-dimers (6 of total dissolved silica) witha negligible amount of higher polymers (lt 03 of totaldissolved silica) Cary et al (1982) detected the presenceof (SiO2)2bull(H2O)3-dimers by NMR in solutions containing16 mM SiO2 at a pH of 72 and estimated their concentra-tion to be about 6 of the total dissolved silica No addi-tional polymerized forms were detected which isconsistent with our calculations for higher polymers (Fig4c) However the lack of such accurate experimental datafor speciation over a wider range of pressures and temper-atures (eg Watson amp Wark 1997) leads to the relativelylarge uncertainties in ∆Sdeg(poly)r and ∆Vdeg(poly)r ie for chainreaction (12) It should be noted that reaction (12) can alsobe used for the calculation of silica speciation at an arbi-trary bulk concentration of dissolved silica YSiO2

=SiO2(SiO2 + H2O) Equations characterizing speciationthen become

X1 = XLiqXSS(1 ndash XSS)[XSSK(poly) + (1 ndash XSS)XLiq] (34)

XL = XLiq(1 ndash XSS) (35)

XSS = [4XLiqYSiO2(K(poly) ndash XLiq)(1 ndash 2YSiO2

) + (XLiq)2(1 ndashYSiO2

)2]05 ndash XLiq(1 ndash YSiO2) [2(K(poly) ndash XLiq)(1 ndash 2YSiO2

)] (36)

where XSS = X1 + Σinfin

n=2 Xn is the sum of the mole fractions of

all dissolved silica species (ie XSS + XL + XG = 1) Otherquantities are as given in equations (21)-(27)

Figure 4 shows SiO2 speciation in the presence ofquartz as a function of pressure temperature and bulksilica content as predicted from our model calculations Itis seen that the relative amount of silica in the form ofmonomers is most strongly dependent on bulk silicaconcentration and generally decreases with increasingSiO2 content of the solution (Fig 4a) Significant amountsof dissolved silica (10-30 ) in dimeric form are calculatedfor temperatures gt 300degC (Fig 4b) The relative proportionof dimeric silica shows a maximum (Fig 4b) at YSiO2

asymp001 At higher YSiO2

trimers and higher polymers becomeimportant (Fig 4c) consuming 10 to 50 of the dissolvedsilica at temperatures gt 500degC Because increasing temper-ature leads to higher bulk silica there is also a similarcorrelation with temperature The dependence on pressureis least pronounced as can be expected from the competingeffects of reactions (11) and (12) (H2O)L is consumedduring the monomer-forming reaction (11) and liberatedduring the polymer-forming reaction (12) thereforedictating opposite signs of ∆Vdegr of the reactions Increasingpressure should lead to increased bulk silica solubility (dueto reaction (11) ∆Vdeg(mono)r lt 0) and therefore increasing

polymerization on the one hand (Fig 4a) whereasincreasing pressure should also favor depolymerizationaccording to reaction (12) (∆Vdeg(poly)r gt 0) on the other

Figure 5 shows silica solubility and speciation inaqueous fluid coexisting with different SiO2-bufferingmineral assemblages It is seen that our model accuratelyreproduces the available experimental data (Zhang ampFrantz 2000 Newton amp Manning 2003) on bulk silicasolubility (Fig 5ab) The calculated distribution of silicabetween monomers and higher polymers as a function ofbulk SiO2(SiO2 + H2O) at constant pressure and tempera-ture (Fig 5c) is very similar to the results of Newton ampManning (2003) which are based on a somewhat simplermonomer-dimer model At constant pressure and tempera-ture the amount of monomeric silica decreases non-linearlywith decreasing bulk silica solubility

Discussion

The semi-empirical thermodynamic formalism devel-oped in this paper allows accurate reproduction of existingdata on silica concentration and silica speciation inaqueous fluid coexisting with quartz and other silica-buffering assemblages up to 20 kbar and 1300degC Incomparison to Mannings (1994) polynomial fit ourapproach also predicts the speciation of silica in theaqueous fluid without increasing the number of empiricalparameters in the model In addition our chain-reactionmodel allows an excellent data fit with full consideration ofall silica polymers (Fig 4) rather than only monomers anddimers (cf Newton amp Manning 2003) Figure 6 shows thepositions of isopleths of bulk silica concentration calcu-lated for both silica-saturated (Fig 6a) and the silica-under-saturated enstatite-forsterite-buffered (Fig 6b) systems atelevated temperature and pressure It is seen that our resultsare in general compatible with the polynomial equationssuggested by Manning (1994) and Newton amp Manning(2002) with the largest discrepancies found in the high-temperature (gt 900degC) high-YSiO2

(gt 005) region Itshould be emphasized here that our present model has beendeveloped for a bulk silica concentration (YSiO2

) in the fluidof less than 01 (Table 2) Therefore this model mayrequire modification (eg by incorporating terms for non-ideality of the aqueous solution) in order to account prop-erly for the thermodynamics of silica-rich hydroussolutions (eg Shen amp Keppler 1997 Audetat amp Keppler2004) in proximity to the second critical point in the H2O-SiO2 system

Our model predicts basic similarities between the ther-modynamic properties of dissolved silica species and thoseof associated clustered ldquoliquid-likerdquo water molecules (egequations (18)-(20)) This is consistent with the conclu-sions of Bockris (1949) and Walther amp Orville (1983) whosuggested that hydrous monomeric silica species _ Si(OH)4_ are associated with two more water dipoles to form anaqueous silica complex _ Si(OH)4bull2H2O _ by hydrogenbonding This mechanism of association of molecules is infact identical to the association (clustering) of molecules ofpure water (eg Luck 1980 Gorbaty amp Kalinichev 1995)

280

Modeling of silica solubility and speciation 281

Fig 6 Isopleths of bulk silica concentration (solid lines)calculated according to equations (21)-(27) and (31)-(33)for silica-saturated (a) and silica-undersaturated enstatite-forsterite buffered (b) system at elevated temperature andpressure Dashed lines show isopleths calculated with theequations suggested by Manning (1994) and Newton ampManning (2002) Phase boundaries in (a) are calculatedfrom the database of Holland amp Powell (1998) Meltingrelations in the presence of water are not shown

Fig 7 Results of modeling silica speciation in aqueous fluidin the presence of quartz along the two isochores presented byZotov amp Keppler (2002) ρ = 078 gcm3 P = 56-68 kbar andρ = 094 gcm3 P = 106-140 kbar The solid squares repre-sent the results of Zotov amp Keppler (2002) based on the inter-pretation of in situ Raman spectra (a) dependence of silicaspeciation on temperature (b) dependence of silica speciationon bulk silica content

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

and is directly accounted for by equations (4)-(6) (for adiscussion see Gerya amp Perchuk 1997 Gerya et al2004) The formation of similar complexes with dimersand higher polymers can also be expected thus supportingour thermodynamic formalism

An important unresolved question is the discrepancybetween the results obtained from our data set which isbased mainly on direct solubility measurements and dataobtained from Raman spectra (Zotov amp Keppler 20002002) This is particularly evident in the pressure effect onsilica polymerization The positive ∆Vdegr of reaction (12)suggests that at constant temperature and bulk silica anyincrease in pressure (ie in fluid density) should lead to adecrease in the degree of polymerization This is alsosupported by the ab initio molecular dynamics simulationsof Doltsinis et al (2004) who calculated higher effectivepressures for monomeric-dimeric systems in comparisonwith purely monomeric systems at T = 900 K and constantfluid density In the presence of quartz bulk silica contentincreases with pressure and the competing effects of reac-tions (11) and (12) should lead to only small changes inthe degree of polymerization Figure 7 indicates the trendspredicted from our model In contrast the results of Zotovamp Keppler (2002) suggest strong increase in the degree ofpolymerization with pressure (ie fluid density) requiringa notably negative ∆Vo

r of polymer-forming reaction (12)Our test calculations show that no consistent set of ther-modynamic parameters for reactions (11) and (12) canprovide an accurate description of both the silica solubilitymeasurements and the changes of degree of polymeriza-tion with fluid density calculated by Zotov amp Keppler(2002)

We suggest that an explanation for the above discrep-ancies must be sought in (i) the possible problems of equi-libration in the relatively short experiments in thediamond-anvil cell performed by Zotov amp Keppler (20002002) and (ii) the difficult task of quantifying the Ramanspectra of aqueous silica solutions The interpretation ofRaman spectra is based mainly on theoretical spectraobtained using empirical interaction potentials (Catlow1980 Price amp Parker 1984) or on single molecule abinitio calculations as in the case of Zotov amp Keppler(2000 2002) The problem with the latter is that they donot include crucial temperature- and pressure-dependentsolvent effects To test these environmental effects wehave initiated a series of both ab initio molecular dynamics(AIMD) simulations and static gas phase ab initio calcula-tions (Doltsinis et al 2004) Our results to date suggestthat while the strong feature at ca 780-800 cm-1 can beascribed to monomers with some confidence there arestill considerable uncertainties in interpreting otherfeatures of the Raman spectra in terms of monomersandor dimers Considering the low amounts of total silicainvolved and the increasingly floppy nature of largerpolymers leading to increasingly broadened spectralfeatures it appears that the recognition and identificationof trimers and higher polymers can be problematical Ifhidden trimers are present then an interpretation basedsolely on monomerdimer ratios will obviously be flawedThus while the results of Zotov amp Keppler (2000 2002)

represent a valiant first attempt to tackle the problem andprovide valuable qualitative data direct comparisons withclassical solubility results should not be overextended Wesuggest that further Raman spectroscopic measurementson well-defined solutions with given bulk silica contents atdifferent pressures and temperatures rather than quartz-buffered solutions only will be a valuable step toward abetter understanding of such spectra

The approach outlined in this paper has been success-fully used to fit and describe experimental data on bulkspinel (MgAl2O4) solubility up to 30 kbar byZakhartchouk et al (2004) despite the lack of detailedinformation on the actual speciation of the dissolved spinelcomponents in that system Given additional independentdata on the detailed structure of condensed phase water inthe future (eg Marx 1999 and in press) and on solutespeciation at various temperatures and pressures (seeDoltsinis et al 2004) the formalism described here canbe used to provide an increasingly complete thermody-namic description of aqueous solutions at depths corre-sponding to the lower crust and upper mantle

AcknowledgementsThis work was supported by ETHResearch Grant TH -1204-1 by RFBR grant 03-05-64633 by the RF President Program ldquoLeading ScientificSchool of Russiardquo (grant 0351645) to TVG and by theDeutsche Forschungsgemeinschaft within the scope ofSonderforschungsbereich 526 at the Ruhr-University ofBochum The helpful reviews of C Manning J Waltherand an anonymous reviewer as well as the editorialhandling of A Woodland are greatly appreciated

References

Anderson GM amp Burnham CW (1965) The solubility of quartzin supercritical water Am J Sci 263 494-511

Applin KR (1987) The diffusion of dissolved silica in diluteaqueous solution Geochim Cosmochim Acta 51 2147-2151

Audetat A amp Keppler H (2004) Viscosity of fluids in subduc-tion zones Science 303 513-516

Barelko VV Zarikov IV Pechatnikov EL (1994) Effect of theorigin of molecular clusters in one-component gas mixtures onthe equation of state Khim Fiz 13 42-50

Bockris JOrsquoM (1949) Ionic solutions Chem Soc LondonQuarterly Rev 3 173-180

Cary LW De Jong BHWS Dibble WEJr (1982) A 29Si NMRstudy of silica species in dilute aqueous solution GeochimCosmochim Acta 46 1317-1320

Catlow CRA (1980) Computer modelling of ionic crystals JPhysics 41 C6-53

Doltsinis NL Burchard M Maresch WV (2004) Ab initiomolecular dynamics simulations as a tool for interpreting vibra-tional spectra in aqueous fluids EMPG-X SymposiumAbstracts Lithos supplement to v 73 S28

Dove PM amp Rimstidt JD (1994) Silica physical behaviorgeochemistry and materials applications in ldquoSilica-water inter-actionsrdquo Reviews in Mineral 29 259-amp

Fockenberg T Burchard M Maresch WV (2004) Solubilities ofcalcium silicates at high pressures and temperatures EMPG-XSymposium Abstracts Lithos supplement to v 73 S37

282

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

rium at low temperatures (eg Walther amp Helgeson 1977)Indeed the overall accuracy of reproducing these experi-ments is comparable (Table 2) with the polynomial equa-

tion suggested by Manning (1994) Larger errors weredetected in silica-undersaturated systems (Table 2 Fig 3)for the experimental data of Hemley et al (1977a b) and

278

Fig 4 Relative concentration of silica inmonomers (a) dimers (b) and higher silicapolymers (c) dissolved in aqueous fluid in thepresence of quartz at P = 0001-20 kbar and T= 25-90degC Calculations are based on equa-tions (21)-(30) Thin grey lines are isobarsBlack lines are isotherms The solid squarerepresents the experimental data of Cary et al(1982) and Applin (1987) at 25degC and 1 bar

Zhang amp Frantz (2000) This may be related to the rela-tively large experimental uncertainties (Zhang amp Frantz2000 Newton amp Manning 2002) and to possible errorsincurred during the calculation of ∆Go

(Buffer-Quartz)PT forSiO2-buffering reactions involving water at low pressureand temperature (Hemley et al 1977a b) in combinationwith the database of Holland amp Powell (1998) Obviouslyat T = 25degC and P = 1 bar the highly weighted referencedata on the concentration of silica in the presence of quartz(Dove amp Rimstidt 1994) and on the distribution of silica

between (SiO2)bull(H2O)2-monomers and (SiO2)2bull(H2O)3-dimers (Cary et al 1982 Applin 1987) are accuratelyreproduced with a relative error of less then 02 (Table 2Fig 3 4a) Figure 2 shows that the results of our thermo-dynamic modeling coincide well with the empiricalapproximation suggested by Manning (1994) for the bulksolubility of silica in water in the presence of quartz Table2 also compares our numerical values (Table 2) with thoseobtained from the empirical equations of Manning (1994)for the quartz-buffered system and of Manning amp Newton

Modeling of silica solubility and speciation 279

Fig 5 Results of modeling bulk concentration (a)(b) and speciation (c) of dissolved silica in aqueousfluid in the presence of different buffering mineralassemblages The symbols represent data from thesources indicated (a) experimental data of Newtonamp Manning (2002) and empirical calibration ofZhang amp Frantz (2000) based on original andpublished (Nakamura amp Kushiro 1974 Manningamp Boettcher 1994) experimental data (b) experi-mental data of Newton amp Manning (2003) (c)monomer-dimer speciation model of Newton ampManning (2003)

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

(2002) for the enstatite-forsterite-buffered system For the358 compared experimental data points (Table 2) ourmodel yields a mean square error of 134 versus 165 for the equations of Manning (1994) and Manning ampNewton (2002)

As noted above our model was constrained to accu-rately reproduce experimental data on silica partitioningbetween monomers and larger clusters at 25degC and 1 bar(Cary et al 1982 Applin 1987) The silica species at theseconditions (Fig 4) are represented by (SiO2)bull(H2O)2-monomers (94 of total dissolved silica) and(SiO2)2bull(H2O)3-dimers (6 of total dissolved silica) witha negligible amount of higher polymers (lt 03 of totaldissolved silica) Cary et al (1982) detected the presenceof (SiO2)2bull(H2O)3-dimers by NMR in solutions containing16 mM SiO2 at a pH of 72 and estimated their concentra-tion to be about 6 of the total dissolved silica No addi-tional polymerized forms were detected which isconsistent with our calculations for higher polymers (Fig4c) However the lack of such accurate experimental datafor speciation over a wider range of pressures and temper-atures (eg Watson amp Wark 1997) leads to the relativelylarge uncertainties in ∆Sdeg(poly)r and ∆Vdeg(poly)r ie for chainreaction (12) It should be noted that reaction (12) can alsobe used for the calculation of silica speciation at an arbi-trary bulk concentration of dissolved silica YSiO2

=SiO2(SiO2 + H2O) Equations characterizing speciationthen become

X1 = XLiqXSS(1 ndash XSS)[XSSK(poly) + (1 ndash XSS)XLiq] (34)

XL = XLiq(1 ndash XSS) (35)

XSS = [4XLiqYSiO2(K(poly) ndash XLiq)(1 ndash 2YSiO2

) + (XLiq)2(1 ndashYSiO2

)2]05 ndash XLiq(1 ndash YSiO2) [2(K(poly) ndash XLiq)(1 ndash 2YSiO2

)] (36)

where XSS = X1 + Σinfin

n=2 Xn is the sum of the mole fractions of

all dissolved silica species (ie XSS + XL + XG = 1) Otherquantities are as given in equations (21)-(27)

Figure 4 shows SiO2 speciation in the presence ofquartz as a function of pressure temperature and bulksilica content as predicted from our model calculations Itis seen that the relative amount of silica in the form ofmonomers is most strongly dependent on bulk silicaconcentration and generally decreases with increasingSiO2 content of the solution (Fig 4a) Significant amountsof dissolved silica (10-30 ) in dimeric form are calculatedfor temperatures gt 300degC (Fig 4b) The relative proportionof dimeric silica shows a maximum (Fig 4b) at YSiO2

asymp001 At higher YSiO2

trimers and higher polymers becomeimportant (Fig 4c) consuming 10 to 50 of the dissolvedsilica at temperatures gt 500degC Because increasing temper-ature leads to higher bulk silica there is also a similarcorrelation with temperature The dependence on pressureis least pronounced as can be expected from the competingeffects of reactions (11) and (12) (H2O)L is consumedduring the monomer-forming reaction (11) and liberatedduring the polymer-forming reaction (12) thereforedictating opposite signs of ∆Vdegr of the reactions Increasingpressure should lead to increased bulk silica solubility (dueto reaction (11) ∆Vdeg(mono)r lt 0) and therefore increasing

polymerization on the one hand (Fig 4a) whereasincreasing pressure should also favor depolymerizationaccording to reaction (12) (∆Vdeg(poly)r gt 0) on the other

Figure 5 shows silica solubility and speciation inaqueous fluid coexisting with different SiO2-bufferingmineral assemblages It is seen that our model accuratelyreproduces the available experimental data (Zhang ampFrantz 2000 Newton amp Manning 2003) on bulk silicasolubility (Fig 5ab) The calculated distribution of silicabetween monomers and higher polymers as a function ofbulk SiO2(SiO2 + H2O) at constant pressure and tempera-ture (Fig 5c) is very similar to the results of Newton ampManning (2003) which are based on a somewhat simplermonomer-dimer model At constant pressure and tempera-ture the amount of monomeric silica decreases non-linearlywith decreasing bulk silica solubility

Discussion

The semi-empirical thermodynamic formalism devel-oped in this paper allows accurate reproduction of existingdata on silica concentration and silica speciation inaqueous fluid coexisting with quartz and other silica-buffering assemblages up to 20 kbar and 1300degC Incomparison to Mannings (1994) polynomial fit ourapproach also predicts the speciation of silica in theaqueous fluid without increasing the number of empiricalparameters in the model In addition our chain-reactionmodel allows an excellent data fit with full consideration ofall silica polymers (Fig 4) rather than only monomers anddimers (cf Newton amp Manning 2003) Figure 6 shows thepositions of isopleths of bulk silica concentration calcu-lated for both silica-saturated (Fig 6a) and the silica-under-saturated enstatite-forsterite-buffered (Fig 6b) systems atelevated temperature and pressure It is seen that our resultsare in general compatible with the polynomial equationssuggested by Manning (1994) and Newton amp Manning(2002) with the largest discrepancies found in the high-temperature (gt 900degC) high-YSiO2

(gt 005) region Itshould be emphasized here that our present model has beendeveloped for a bulk silica concentration (YSiO2

) in the fluidof less than 01 (Table 2) Therefore this model mayrequire modification (eg by incorporating terms for non-ideality of the aqueous solution) in order to account prop-erly for the thermodynamics of silica-rich hydroussolutions (eg Shen amp Keppler 1997 Audetat amp Keppler2004) in proximity to the second critical point in the H2O-SiO2 system

Our model predicts basic similarities between the ther-modynamic properties of dissolved silica species and thoseof associated clustered ldquoliquid-likerdquo water molecules (egequations (18)-(20)) This is consistent with the conclu-sions of Bockris (1949) and Walther amp Orville (1983) whosuggested that hydrous monomeric silica species _ Si(OH)4_ are associated with two more water dipoles to form anaqueous silica complex _ Si(OH)4bull2H2O _ by hydrogenbonding This mechanism of association of molecules is infact identical to the association (clustering) of molecules ofpure water (eg Luck 1980 Gorbaty amp Kalinichev 1995)

280

Modeling of silica solubility and speciation 281

Fig 6 Isopleths of bulk silica concentration (solid lines)calculated according to equations (21)-(27) and (31)-(33)for silica-saturated (a) and silica-undersaturated enstatite-forsterite buffered (b) system at elevated temperature andpressure Dashed lines show isopleths calculated with theequations suggested by Manning (1994) and Newton ampManning (2002) Phase boundaries in (a) are calculatedfrom the database of Holland amp Powell (1998) Meltingrelations in the presence of water are not shown

Fig 7 Results of modeling silica speciation in aqueous fluidin the presence of quartz along the two isochores presented byZotov amp Keppler (2002) ρ = 078 gcm3 P = 56-68 kbar andρ = 094 gcm3 P = 106-140 kbar The solid squares repre-sent the results of Zotov amp Keppler (2002) based on the inter-pretation of in situ Raman spectra (a) dependence of silicaspeciation on temperature (b) dependence of silica speciationon bulk silica content

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

and is directly accounted for by equations (4)-(6) (for adiscussion see Gerya amp Perchuk 1997 Gerya et al2004) The formation of similar complexes with dimersand higher polymers can also be expected thus supportingour thermodynamic formalism

An important unresolved question is the discrepancybetween the results obtained from our data set which isbased mainly on direct solubility measurements and dataobtained from Raman spectra (Zotov amp Keppler 20002002) This is particularly evident in the pressure effect onsilica polymerization The positive ∆Vdegr of reaction (12)suggests that at constant temperature and bulk silica anyincrease in pressure (ie in fluid density) should lead to adecrease in the degree of polymerization This is alsosupported by the ab initio molecular dynamics simulationsof Doltsinis et al (2004) who calculated higher effectivepressures for monomeric-dimeric systems in comparisonwith purely monomeric systems at T = 900 K and constantfluid density In the presence of quartz bulk silica contentincreases with pressure and the competing effects of reac-tions (11) and (12) should lead to only small changes inthe degree of polymerization Figure 7 indicates the trendspredicted from our model In contrast the results of Zotovamp Keppler (2002) suggest strong increase in the degree ofpolymerization with pressure (ie fluid density) requiringa notably negative ∆Vo

r of polymer-forming reaction (12)Our test calculations show that no consistent set of ther-modynamic parameters for reactions (11) and (12) canprovide an accurate description of both the silica solubilitymeasurements and the changes of degree of polymeriza-tion with fluid density calculated by Zotov amp Keppler(2002)

We suggest that an explanation for the above discrep-ancies must be sought in (i) the possible problems of equi-libration in the relatively short experiments in thediamond-anvil cell performed by Zotov amp Keppler (20002002) and (ii) the difficult task of quantifying the Ramanspectra of aqueous silica solutions The interpretation ofRaman spectra is based mainly on theoretical spectraobtained using empirical interaction potentials (Catlow1980 Price amp Parker 1984) or on single molecule abinitio calculations as in the case of Zotov amp Keppler(2000 2002) The problem with the latter is that they donot include crucial temperature- and pressure-dependentsolvent effects To test these environmental effects wehave initiated a series of both ab initio molecular dynamics(AIMD) simulations and static gas phase ab initio calcula-tions (Doltsinis et al 2004) Our results to date suggestthat while the strong feature at ca 780-800 cm-1 can beascribed to monomers with some confidence there arestill considerable uncertainties in interpreting otherfeatures of the Raman spectra in terms of monomersandor dimers Considering the low amounts of total silicainvolved and the increasingly floppy nature of largerpolymers leading to increasingly broadened spectralfeatures it appears that the recognition and identificationof trimers and higher polymers can be problematical Ifhidden trimers are present then an interpretation basedsolely on monomerdimer ratios will obviously be flawedThus while the results of Zotov amp Keppler (2000 2002)

represent a valiant first attempt to tackle the problem andprovide valuable qualitative data direct comparisons withclassical solubility results should not be overextended Wesuggest that further Raman spectroscopic measurementson well-defined solutions with given bulk silica contents atdifferent pressures and temperatures rather than quartz-buffered solutions only will be a valuable step toward abetter understanding of such spectra

The approach outlined in this paper has been success-fully used to fit and describe experimental data on bulkspinel (MgAl2O4) solubility up to 30 kbar byZakhartchouk et al (2004) despite the lack of detailedinformation on the actual speciation of the dissolved spinelcomponents in that system Given additional independentdata on the detailed structure of condensed phase water inthe future (eg Marx 1999 and in press) and on solutespeciation at various temperatures and pressures (seeDoltsinis et al 2004) the formalism described here canbe used to provide an increasingly complete thermody-namic description of aqueous solutions at depths corre-sponding to the lower crust and upper mantle

AcknowledgementsThis work was supported by ETHResearch Grant TH -1204-1 by RFBR grant 03-05-64633 by the RF President Program ldquoLeading ScientificSchool of Russiardquo (grant 0351645) to TVG and by theDeutsche Forschungsgemeinschaft within the scope ofSonderforschungsbereich 526 at the Ruhr-University ofBochum The helpful reviews of C Manning J Waltherand an anonymous reviewer as well as the editorialhandling of A Woodland are greatly appreciated

References

Anderson GM amp Burnham CW (1965) The solubility of quartzin supercritical water Am J Sci 263 494-511

Applin KR (1987) The diffusion of dissolved silica in diluteaqueous solution Geochim Cosmochim Acta 51 2147-2151

Audetat A amp Keppler H (2004) Viscosity of fluids in subduc-tion zones Science 303 513-516

Barelko VV Zarikov IV Pechatnikov EL (1994) Effect of theorigin of molecular clusters in one-component gas mixtures onthe equation of state Khim Fiz 13 42-50

Bockris JOrsquoM (1949) Ionic solutions Chem Soc LondonQuarterly Rev 3 173-180

Cary LW De Jong BHWS Dibble WEJr (1982) A 29Si NMRstudy of silica species in dilute aqueous solution GeochimCosmochim Acta 46 1317-1320

Catlow CRA (1980) Computer modelling of ionic crystals JPhysics 41 C6-53

Doltsinis NL Burchard M Maresch WV (2004) Ab initiomolecular dynamics simulations as a tool for interpreting vibra-tional spectra in aqueous fluids EMPG-X SymposiumAbstracts Lithos supplement to v 73 S28

Dove PM amp Rimstidt JD (1994) Silica physical behaviorgeochemistry and materials applications in ldquoSilica-water inter-actionsrdquo Reviews in Mineral 29 259-amp

Fockenberg T Burchard M Maresch WV (2004) Solubilities ofcalcium silicates at high pressures and temperatures EMPG-XSymposium Abstracts Lithos supplement to v 73 S37

282

Zhang amp Frantz (2000) This may be related to the rela-tively large experimental uncertainties (Zhang amp Frantz2000 Newton amp Manning 2002) and to possible errorsincurred during the calculation of ∆Go

(Buffer-Quartz)PT forSiO2-buffering reactions involving water at low pressureand temperature (Hemley et al 1977a b) in combinationwith the database of Holland amp Powell (1998) Obviouslyat T = 25degC and P = 1 bar the highly weighted referencedata on the concentration of silica in the presence of quartz(Dove amp Rimstidt 1994) and on the distribution of silica

between (SiO2)bull(H2O)2-monomers and (SiO2)2bull(H2O)3-dimers (Cary et al 1982 Applin 1987) are accuratelyreproduced with a relative error of less then 02 (Table 2Fig 3 4a) Figure 2 shows that the results of our thermo-dynamic modeling coincide well with the empiricalapproximation suggested by Manning (1994) for the bulksolubility of silica in water in the presence of quartz Table2 also compares our numerical values (Table 2) with thoseobtained from the empirical equations of Manning (1994)for the quartz-buffered system and of Manning amp Newton

Modeling of silica solubility and speciation 279

Fig 5 Results of modeling bulk concentration (a)(b) and speciation (c) of dissolved silica in aqueousfluid in the presence of different buffering mineralassemblages The symbols represent data from thesources indicated (a) experimental data of Newtonamp Manning (2002) and empirical calibration ofZhang amp Frantz (2000) based on original andpublished (Nakamura amp Kushiro 1974 Manningamp Boettcher 1994) experimental data (b) experi-mental data of Newton amp Manning (2003) (c)monomer-dimer speciation model of Newton ampManning (2003)

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

(2002) for the enstatite-forsterite-buffered system For the358 compared experimental data points (Table 2) ourmodel yields a mean square error of 134 versus 165 for the equations of Manning (1994) and Manning ampNewton (2002)

As noted above our model was constrained to accu-rately reproduce experimental data on silica partitioningbetween monomers and larger clusters at 25degC and 1 bar(Cary et al 1982 Applin 1987) The silica species at theseconditions (Fig 4) are represented by (SiO2)bull(H2O)2-monomers (94 of total dissolved silica) and(SiO2)2bull(H2O)3-dimers (6 of total dissolved silica) witha negligible amount of higher polymers (lt 03 of totaldissolved silica) Cary et al (1982) detected the presenceof (SiO2)2bull(H2O)3-dimers by NMR in solutions containing16 mM SiO2 at a pH of 72 and estimated their concentra-tion to be about 6 of the total dissolved silica No addi-tional polymerized forms were detected which isconsistent with our calculations for higher polymers (Fig4c) However the lack of such accurate experimental datafor speciation over a wider range of pressures and temper-atures (eg Watson amp Wark 1997) leads to the relativelylarge uncertainties in ∆Sdeg(poly)r and ∆Vdeg(poly)r ie for chainreaction (12) It should be noted that reaction (12) can alsobe used for the calculation of silica speciation at an arbi-trary bulk concentration of dissolved silica YSiO2

=SiO2(SiO2 + H2O) Equations characterizing speciationthen become

X1 = XLiqXSS(1 ndash XSS)[XSSK(poly) + (1 ndash XSS)XLiq] (34)

XL = XLiq(1 ndash XSS) (35)

XSS = [4XLiqYSiO2(K(poly) ndash XLiq)(1 ndash 2YSiO2

) + (XLiq)2(1 ndashYSiO2

)2]05 ndash XLiq(1 ndash YSiO2) [2(K(poly) ndash XLiq)(1 ndash 2YSiO2

)] (36)

where XSS = X1 + Σinfin

n=2 Xn is the sum of the mole fractions of

all dissolved silica species (ie XSS + XL + XG = 1) Otherquantities are as given in equations (21)-(27)

Figure 4 shows SiO2 speciation in the presence ofquartz as a function of pressure temperature and bulksilica content as predicted from our model calculations Itis seen that the relative amount of silica in the form ofmonomers is most strongly dependent on bulk silicaconcentration and generally decreases with increasingSiO2 content of the solution (Fig 4a) Significant amountsof dissolved silica (10-30 ) in dimeric form are calculatedfor temperatures gt 300degC (Fig 4b) The relative proportionof dimeric silica shows a maximum (Fig 4b) at YSiO2

asymp001 At higher YSiO2

trimers and higher polymers becomeimportant (Fig 4c) consuming 10 to 50 of the dissolvedsilica at temperatures gt 500degC Because increasing temper-ature leads to higher bulk silica there is also a similarcorrelation with temperature The dependence on pressureis least pronounced as can be expected from the competingeffects of reactions (11) and (12) (H2O)L is consumedduring the monomer-forming reaction (11) and liberatedduring the polymer-forming reaction (12) thereforedictating opposite signs of ∆Vdegr of the reactions Increasingpressure should lead to increased bulk silica solubility (dueto reaction (11) ∆Vdeg(mono)r lt 0) and therefore increasing

polymerization on the one hand (Fig 4a) whereasincreasing pressure should also favor depolymerizationaccording to reaction (12) (∆Vdeg(poly)r gt 0) on the other

Figure 5 shows silica solubility and speciation inaqueous fluid coexisting with different SiO2-bufferingmineral assemblages It is seen that our model accuratelyreproduces the available experimental data (Zhang ampFrantz 2000 Newton amp Manning 2003) on bulk silicasolubility (Fig 5ab) The calculated distribution of silicabetween monomers and higher polymers as a function ofbulk SiO2(SiO2 + H2O) at constant pressure and tempera-ture (Fig 5c) is very similar to the results of Newton ampManning (2003) which are based on a somewhat simplermonomer-dimer model At constant pressure and tempera-ture the amount of monomeric silica decreases non-linearlywith decreasing bulk silica solubility

Discussion

The semi-empirical thermodynamic formalism devel-oped in this paper allows accurate reproduction of existingdata on silica concentration and silica speciation inaqueous fluid coexisting with quartz and other silica-buffering assemblages up to 20 kbar and 1300degC Incomparison to Mannings (1994) polynomial fit ourapproach also predicts the speciation of silica in theaqueous fluid without increasing the number of empiricalparameters in the model In addition our chain-reactionmodel allows an excellent data fit with full consideration ofall silica polymers (Fig 4) rather than only monomers anddimers (cf Newton amp Manning 2003) Figure 6 shows thepositions of isopleths of bulk silica concentration calcu-lated for both silica-saturated (Fig 6a) and the silica-under-saturated enstatite-forsterite-buffered (Fig 6b) systems atelevated temperature and pressure It is seen that our resultsare in general compatible with the polynomial equationssuggested by Manning (1994) and Newton amp Manning(2002) with the largest discrepancies found in the high-temperature (gt 900degC) high-YSiO2

(gt 005) region Itshould be emphasized here that our present model has beendeveloped for a bulk silica concentration (YSiO2

) in the fluidof less than 01 (Table 2) Therefore this model mayrequire modification (eg by incorporating terms for non-ideality of the aqueous solution) in order to account prop-erly for the thermodynamics of silica-rich hydroussolutions (eg Shen amp Keppler 1997 Audetat amp Keppler2004) in proximity to the second critical point in the H2O-SiO2 system

Our model predicts basic similarities between the ther-modynamic properties of dissolved silica species and thoseof associated clustered ldquoliquid-likerdquo water molecules (egequations (18)-(20)) This is consistent with the conclu-sions of Bockris (1949) and Walther amp Orville (1983) whosuggested that hydrous monomeric silica species _ Si(OH)4_ are associated with two more water dipoles to form anaqueous silica complex _ Si(OH)4bull2H2O _ by hydrogenbonding This mechanism of association of molecules is infact identical to the association (clustering) of molecules ofpure water (eg Luck 1980 Gorbaty amp Kalinichev 1995)

280

Modeling of silica solubility and speciation 281

Fig 6 Isopleths of bulk silica concentration (solid lines)calculated according to equations (21)-(27) and (31)-(33)for silica-saturated (a) and silica-undersaturated enstatite-forsterite buffered (b) system at elevated temperature andpressure Dashed lines show isopleths calculated with theequations suggested by Manning (1994) and Newton ampManning (2002) Phase boundaries in (a) are calculatedfrom the database of Holland amp Powell (1998) Meltingrelations in the presence of water are not shown

Fig 7 Results of modeling silica speciation in aqueous fluidin the presence of quartz along the two isochores presented byZotov amp Keppler (2002) ρ = 078 gcm3 P = 56-68 kbar andρ = 094 gcm3 P = 106-140 kbar The solid squares repre-sent the results of Zotov amp Keppler (2002) based on the inter-pretation of in situ Raman spectra (a) dependence of silicaspeciation on temperature (b) dependence of silica speciationon bulk silica content

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

and is directly accounted for by equations (4)-(6) (for adiscussion see Gerya amp Perchuk 1997 Gerya et al2004) The formation of similar complexes with dimersand higher polymers can also be expected thus supportingour thermodynamic formalism

An important unresolved question is the discrepancybetween the results obtained from our data set which isbased mainly on direct solubility measurements and dataobtained from Raman spectra (Zotov amp Keppler 20002002) This is particularly evident in the pressure effect onsilica polymerization The positive ∆Vdegr of reaction (12)suggests that at constant temperature and bulk silica anyincrease in pressure (ie in fluid density) should lead to adecrease in the degree of polymerization This is alsosupported by the ab initio molecular dynamics simulationsof Doltsinis et al (2004) who calculated higher effectivepressures for monomeric-dimeric systems in comparisonwith purely monomeric systems at T = 900 K and constantfluid density In the presence of quartz bulk silica contentincreases with pressure and the competing effects of reac-tions (11) and (12) should lead to only small changes inthe degree of polymerization Figure 7 indicates the trendspredicted from our model In contrast the results of Zotovamp Keppler (2002) suggest strong increase in the degree ofpolymerization with pressure (ie fluid density) requiringa notably negative ∆Vo

r of polymer-forming reaction (12)Our test calculations show that no consistent set of ther-modynamic parameters for reactions (11) and (12) canprovide an accurate description of both the silica solubilitymeasurements and the changes of degree of polymeriza-tion with fluid density calculated by Zotov amp Keppler(2002)

We suggest that an explanation for the above discrep-ancies must be sought in (i) the possible problems of equi-libration in the relatively short experiments in thediamond-anvil cell performed by Zotov amp Keppler (20002002) and (ii) the difficult task of quantifying the Ramanspectra of aqueous silica solutions The interpretation ofRaman spectra is based mainly on theoretical spectraobtained using empirical interaction potentials (Catlow1980 Price amp Parker 1984) or on single molecule abinitio calculations as in the case of Zotov amp Keppler(2000 2002) The problem with the latter is that they donot include crucial temperature- and pressure-dependentsolvent effects To test these environmental effects wehave initiated a series of both ab initio molecular dynamics(AIMD) simulations and static gas phase ab initio calcula-tions (Doltsinis et al 2004) Our results to date suggestthat while the strong feature at ca 780-800 cm-1 can beascribed to monomers with some confidence there arestill considerable uncertainties in interpreting otherfeatures of the Raman spectra in terms of monomersandor dimers Considering the low amounts of total silicainvolved and the increasingly floppy nature of largerpolymers leading to increasingly broadened spectralfeatures it appears that the recognition and identificationof trimers and higher polymers can be problematical Ifhidden trimers are present then an interpretation basedsolely on monomerdimer ratios will obviously be flawedThus while the results of Zotov amp Keppler (2000 2002)

represent a valiant first attempt to tackle the problem andprovide valuable qualitative data direct comparisons withclassical solubility results should not be overextended Wesuggest that further Raman spectroscopic measurementson well-defined solutions with given bulk silica contents atdifferent pressures and temperatures rather than quartz-buffered solutions only will be a valuable step toward abetter understanding of such spectra

The approach outlined in this paper has been success-fully used to fit and describe experimental data on bulkspinel (MgAl2O4) solubility up to 30 kbar byZakhartchouk et al (2004) despite the lack of detailedinformation on the actual speciation of the dissolved spinelcomponents in that system Given additional independentdata on the detailed structure of condensed phase water inthe future (eg Marx 1999 and in press) and on solutespeciation at various temperatures and pressures (seeDoltsinis et al 2004) the formalism described here canbe used to provide an increasingly complete thermody-namic description of aqueous solutions at depths corre-sponding to the lower crust and upper mantle

AcknowledgementsThis work was supported by ETHResearch Grant TH -1204-1 by RFBR grant 03-05-64633 by the RF President Program ldquoLeading ScientificSchool of Russiardquo (grant 0351645) to TVG and by theDeutsche Forschungsgemeinschaft within the scope ofSonderforschungsbereich 526 at the Ruhr-University ofBochum The helpful reviews of C Manning J Waltherand an anonymous reviewer as well as the editorialhandling of A Woodland are greatly appreciated

References

Anderson GM amp Burnham CW (1965) The solubility of quartzin supercritical water Am J Sci 263 494-511

Applin KR (1987) The diffusion of dissolved silica in diluteaqueous solution Geochim Cosmochim Acta 51 2147-2151

Audetat A amp Keppler H (2004) Viscosity of fluids in subduc-tion zones Science 303 513-516

Barelko VV Zarikov IV Pechatnikov EL (1994) Effect of theorigin of molecular clusters in one-component gas mixtures onthe equation of state Khim Fiz 13 42-50

Bockris JOrsquoM (1949) Ionic solutions Chem Soc LondonQuarterly Rev 3 173-180

Cary LW De Jong BHWS Dibble WEJr (1982) A 29Si NMRstudy of silica species in dilute aqueous solution GeochimCosmochim Acta 46 1317-1320

Catlow CRA (1980) Computer modelling of ionic crystals JPhysics 41 C6-53

Doltsinis NL Burchard M Maresch WV (2004) Ab initiomolecular dynamics simulations as a tool for interpreting vibra-tional spectra in aqueous fluids EMPG-X SymposiumAbstracts Lithos supplement to v 73 S28

Dove PM amp Rimstidt JD (1994) Silica physical behaviorgeochemistry and materials applications in ldquoSilica-water inter-actionsrdquo Reviews in Mineral 29 259-amp

Fockenberg T Burchard M Maresch WV (2004) Solubilities ofcalcium silicates at high pressures and temperatures EMPG-XSymposium Abstracts Lithos supplement to v 73 S37

282

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

(2002) for the enstatite-forsterite-buffered system For the358 compared experimental data points (Table 2) ourmodel yields a mean square error of 134 versus 165 for the equations of Manning (1994) and Manning ampNewton (2002)

As noted above our model was constrained to accu-rately reproduce experimental data on silica partitioningbetween monomers and larger clusters at 25degC and 1 bar(Cary et al 1982 Applin 1987) The silica species at theseconditions (Fig 4) are represented by (SiO2)bull(H2O)2-monomers (94 of total dissolved silica) and(SiO2)2bull(H2O)3-dimers (6 of total dissolved silica) witha negligible amount of higher polymers (lt 03 of totaldissolved silica) Cary et al (1982) detected the presenceof (SiO2)2bull(H2O)3-dimers by NMR in solutions containing16 mM SiO2 at a pH of 72 and estimated their concentra-tion to be about 6 of the total dissolved silica No addi-tional polymerized forms were detected which isconsistent with our calculations for higher polymers (Fig4c) However the lack of such accurate experimental datafor speciation over a wider range of pressures and temper-atures (eg Watson amp Wark 1997) leads to the relativelylarge uncertainties in ∆Sdeg(poly)r and ∆Vdeg(poly)r ie for chainreaction (12) It should be noted that reaction (12) can alsobe used for the calculation of silica speciation at an arbi-trary bulk concentration of dissolved silica YSiO2

=SiO2(SiO2 + H2O) Equations characterizing speciationthen become

X1 = XLiqXSS(1 ndash XSS)[XSSK(poly) + (1 ndash XSS)XLiq] (34)

XL = XLiq(1 ndash XSS) (35)

XSS = [4XLiqYSiO2(K(poly) ndash XLiq)(1 ndash 2YSiO2

) + (XLiq)2(1 ndashYSiO2

)2]05 ndash XLiq(1 ndash YSiO2) [2(K(poly) ndash XLiq)(1 ndash 2YSiO2

)] (36)

where XSS = X1 + Σinfin

n=2 Xn is the sum of the mole fractions of

all dissolved silica species (ie XSS + XL + XG = 1) Otherquantities are as given in equations (21)-(27)

Figure 4 shows SiO2 speciation in the presence ofquartz as a function of pressure temperature and bulksilica content as predicted from our model calculations Itis seen that the relative amount of silica in the form ofmonomers is most strongly dependent on bulk silicaconcentration and generally decreases with increasingSiO2 content of the solution (Fig 4a) Significant amountsof dissolved silica (10-30 ) in dimeric form are calculatedfor temperatures gt 300degC (Fig 4b) The relative proportionof dimeric silica shows a maximum (Fig 4b) at YSiO2

asymp001 At higher YSiO2

trimers and higher polymers becomeimportant (Fig 4c) consuming 10 to 50 of the dissolvedsilica at temperatures gt 500degC Because increasing temper-ature leads to higher bulk silica there is also a similarcorrelation with temperature The dependence on pressureis least pronounced as can be expected from the competingeffects of reactions (11) and (12) (H2O)L is consumedduring the monomer-forming reaction (11) and liberatedduring the polymer-forming reaction (12) thereforedictating opposite signs of ∆Vdegr of the reactions Increasingpressure should lead to increased bulk silica solubility (dueto reaction (11) ∆Vdeg(mono)r lt 0) and therefore increasing

polymerization on the one hand (Fig 4a) whereasincreasing pressure should also favor depolymerizationaccording to reaction (12) (∆Vdeg(poly)r gt 0) on the other

Figure 5 shows silica solubility and speciation inaqueous fluid coexisting with different SiO2-bufferingmineral assemblages It is seen that our model accuratelyreproduces the available experimental data (Zhang ampFrantz 2000 Newton amp Manning 2003) on bulk silicasolubility (Fig 5ab) The calculated distribution of silicabetween monomers and higher polymers as a function ofbulk SiO2(SiO2 + H2O) at constant pressure and tempera-ture (Fig 5c) is very similar to the results of Newton ampManning (2003) which are based on a somewhat simplermonomer-dimer model At constant pressure and tempera-ture the amount of monomeric silica decreases non-linearlywith decreasing bulk silica solubility

Discussion

The semi-empirical thermodynamic formalism devel-oped in this paper allows accurate reproduction of existingdata on silica concentration and silica speciation inaqueous fluid coexisting with quartz and other silica-buffering assemblages up to 20 kbar and 1300degC Incomparison to Mannings (1994) polynomial fit ourapproach also predicts the speciation of silica in theaqueous fluid without increasing the number of empiricalparameters in the model In addition our chain-reactionmodel allows an excellent data fit with full consideration ofall silica polymers (Fig 4) rather than only monomers anddimers (cf Newton amp Manning 2003) Figure 6 shows thepositions of isopleths of bulk silica concentration calcu-lated for both silica-saturated (Fig 6a) and the silica-under-saturated enstatite-forsterite-buffered (Fig 6b) systems atelevated temperature and pressure It is seen that our resultsare in general compatible with the polynomial equationssuggested by Manning (1994) and Newton amp Manning(2002) with the largest discrepancies found in the high-temperature (gt 900degC) high-YSiO2

(gt 005) region Itshould be emphasized here that our present model has beendeveloped for a bulk silica concentration (YSiO2

) in the fluidof less than 01 (Table 2) Therefore this model mayrequire modification (eg by incorporating terms for non-ideality of the aqueous solution) in order to account prop-erly for the thermodynamics of silica-rich hydroussolutions (eg Shen amp Keppler 1997 Audetat amp Keppler2004) in proximity to the second critical point in the H2O-SiO2 system

Our model predicts basic similarities between the ther-modynamic properties of dissolved silica species and thoseof associated clustered ldquoliquid-likerdquo water molecules (egequations (18)-(20)) This is consistent with the conclu-sions of Bockris (1949) and Walther amp Orville (1983) whosuggested that hydrous monomeric silica species _ Si(OH)4_ are associated with two more water dipoles to form anaqueous silica complex _ Si(OH)4bull2H2O _ by hydrogenbonding This mechanism of association of molecules is infact identical to the association (clustering) of molecules ofpure water (eg Luck 1980 Gorbaty amp Kalinichev 1995)

280

Modeling of silica solubility and speciation 281

Fig 6 Isopleths of bulk silica concentration (solid lines)calculated according to equations (21)-(27) and (31)-(33)for silica-saturated (a) and silica-undersaturated enstatite-forsterite buffered (b) system at elevated temperature andpressure Dashed lines show isopleths calculated with theequations suggested by Manning (1994) and Newton ampManning (2002) Phase boundaries in (a) are calculatedfrom the database of Holland amp Powell (1998) Meltingrelations in the presence of water are not shown

Fig 7 Results of modeling silica speciation in aqueous fluidin the presence of quartz along the two isochores presented byZotov amp Keppler (2002) ρ = 078 gcm3 P = 56-68 kbar andρ = 094 gcm3 P = 106-140 kbar The solid squares repre-sent the results of Zotov amp Keppler (2002) based on the inter-pretation of in situ Raman spectra (a) dependence of silicaspeciation on temperature (b) dependence of silica speciationon bulk silica content

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

and is directly accounted for by equations (4)-(6) (for adiscussion see Gerya amp Perchuk 1997 Gerya et al2004) The formation of similar complexes with dimersand higher polymers can also be expected thus supportingour thermodynamic formalism

An important unresolved question is the discrepancybetween the results obtained from our data set which isbased mainly on direct solubility measurements and dataobtained from Raman spectra (Zotov amp Keppler 20002002) This is particularly evident in the pressure effect onsilica polymerization The positive ∆Vdegr of reaction (12)suggests that at constant temperature and bulk silica anyincrease in pressure (ie in fluid density) should lead to adecrease in the degree of polymerization This is alsosupported by the ab initio molecular dynamics simulationsof Doltsinis et al (2004) who calculated higher effectivepressures for monomeric-dimeric systems in comparisonwith purely monomeric systems at T = 900 K and constantfluid density In the presence of quartz bulk silica contentincreases with pressure and the competing effects of reac-tions (11) and (12) should lead to only small changes inthe degree of polymerization Figure 7 indicates the trendspredicted from our model In contrast the results of Zotovamp Keppler (2002) suggest strong increase in the degree ofpolymerization with pressure (ie fluid density) requiringa notably negative ∆Vo

r of polymer-forming reaction (12)Our test calculations show that no consistent set of ther-modynamic parameters for reactions (11) and (12) canprovide an accurate description of both the silica solubilitymeasurements and the changes of degree of polymeriza-tion with fluid density calculated by Zotov amp Keppler(2002)

We suggest that an explanation for the above discrep-ancies must be sought in (i) the possible problems of equi-libration in the relatively short experiments in thediamond-anvil cell performed by Zotov amp Keppler (20002002) and (ii) the difficult task of quantifying the Ramanspectra of aqueous silica solutions The interpretation ofRaman spectra is based mainly on theoretical spectraobtained using empirical interaction potentials (Catlow1980 Price amp Parker 1984) or on single molecule abinitio calculations as in the case of Zotov amp Keppler(2000 2002) The problem with the latter is that they donot include crucial temperature- and pressure-dependentsolvent effects To test these environmental effects wehave initiated a series of both ab initio molecular dynamics(AIMD) simulations and static gas phase ab initio calcula-tions (Doltsinis et al 2004) Our results to date suggestthat while the strong feature at ca 780-800 cm-1 can beascribed to monomers with some confidence there arestill considerable uncertainties in interpreting otherfeatures of the Raman spectra in terms of monomersandor dimers Considering the low amounts of total silicainvolved and the increasingly floppy nature of largerpolymers leading to increasingly broadened spectralfeatures it appears that the recognition and identificationof trimers and higher polymers can be problematical Ifhidden trimers are present then an interpretation basedsolely on monomerdimer ratios will obviously be flawedThus while the results of Zotov amp Keppler (2000 2002)

represent a valiant first attempt to tackle the problem andprovide valuable qualitative data direct comparisons withclassical solubility results should not be overextended Wesuggest that further Raman spectroscopic measurementson well-defined solutions with given bulk silica contents atdifferent pressures and temperatures rather than quartz-buffered solutions only will be a valuable step toward abetter understanding of such spectra

The approach outlined in this paper has been success-fully used to fit and describe experimental data on bulkspinel (MgAl2O4) solubility up to 30 kbar byZakhartchouk et al (2004) despite the lack of detailedinformation on the actual speciation of the dissolved spinelcomponents in that system Given additional independentdata on the detailed structure of condensed phase water inthe future (eg Marx 1999 and in press) and on solutespeciation at various temperatures and pressures (seeDoltsinis et al 2004) the formalism described here canbe used to provide an increasingly complete thermody-namic description of aqueous solutions at depths corre-sponding to the lower crust and upper mantle

AcknowledgementsThis work was supported by ETHResearch Grant TH -1204-1 by RFBR grant 03-05-64633 by the RF President Program ldquoLeading ScientificSchool of Russiardquo (grant 0351645) to TVG and by theDeutsche Forschungsgemeinschaft within the scope ofSonderforschungsbereich 526 at the Ruhr-University ofBochum The helpful reviews of C Manning J Waltherand an anonymous reviewer as well as the editorialhandling of A Woodland are greatly appreciated

References

Anderson GM amp Burnham CW (1965) The solubility of quartzin supercritical water Am J Sci 263 494-511

Applin KR (1987) The diffusion of dissolved silica in diluteaqueous solution Geochim Cosmochim Acta 51 2147-2151

Audetat A amp Keppler H (2004) Viscosity of fluids in subduc-tion zones Science 303 513-516

Barelko VV Zarikov IV Pechatnikov EL (1994) Effect of theorigin of molecular clusters in one-component gas mixtures onthe equation of state Khim Fiz 13 42-50

Bockris JOrsquoM (1949) Ionic solutions Chem Soc LondonQuarterly Rev 3 173-180

Cary LW De Jong BHWS Dibble WEJr (1982) A 29Si NMRstudy of silica species in dilute aqueous solution GeochimCosmochim Acta 46 1317-1320

Catlow CRA (1980) Computer modelling of ionic crystals JPhysics 41 C6-53

Doltsinis NL Burchard M Maresch WV (2004) Ab initiomolecular dynamics simulations as a tool for interpreting vibra-tional spectra in aqueous fluids EMPG-X SymposiumAbstracts Lithos supplement to v 73 S28

Dove PM amp Rimstidt JD (1994) Silica physical behaviorgeochemistry and materials applications in ldquoSilica-water inter-actionsrdquo Reviews in Mineral 29 259-amp

Fockenberg T Burchard M Maresch WV (2004) Solubilities ofcalcium silicates at high pressures and temperatures EMPG-XSymposium Abstracts Lithos supplement to v 73 S37

282

Modeling of silica solubility and speciation 281

Fig 6 Isopleths of bulk silica concentration (solid lines)calculated according to equations (21)-(27) and (31)-(33)for silica-saturated (a) and silica-undersaturated enstatite-forsterite buffered (b) system at elevated temperature andpressure Dashed lines show isopleths calculated with theequations suggested by Manning (1994) and Newton ampManning (2002) Phase boundaries in (a) are calculatedfrom the database of Holland amp Powell (1998) Meltingrelations in the presence of water are not shown

Fig 7 Results of modeling silica speciation in aqueous fluidin the presence of quartz along the two isochores presented byZotov amp Keppler (2002) ρ = 078 gcm3 P = 56-68 kbar andρ = 094 gcm3 P = 106-140 kbar The solid squares repre-sent the results of Zotov amp Keppler (2002) based on the inter-pretation of in situ Raman spectra (a) dependence of silicaspeciation on temperature (b) dependence of silica speciationon bulk silica content

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

and is directly accounted for by equations (4)-(6) (for adiscussion see Gerya amp Perchuk 1997 Gerya et al2004) The formation of similar complexes with dimersand higher polymers can also be expected thus supportingour thermodynamic formalism

An important unresolved question is the discrepancybetween the results obtained from our data set which isbased mainly on direct solubility measurements and dataobtained from Raman spectra (Zotov amp Keppler 20002002) This is particularly evident in the pressure effect onsilica polymerization The positive ∆Vdegr of reaction (12)suggests that at constant temperature and bulk silica anyincrease in pressure (ie in fluid density) should lead to adecrease in the degree of polymerization This is alsosupported by the ab initio molecular dynamics simulationsof Doltsinis et al (2004) who calculated higher effectivepressures for monomeric-dimeric systems in comparisonwith purely monomeric systems at T = 900 K and constantfluid density In the presence of quartz bulk silica contentincreases with pressure and the competing effects of reac-tions (11) and (12) should lead to only small changes inthe degree of polymerization Figure 7 indicates the trendspredicted from our model In contrast the results of Zotovamp Keppler (2002) suggest strong increase in the degree ofpolymerization with pressure (ie fluid density) requiringa notably negative ∆Vo

r of polymer-forming reaction (12)Our test calculations show that no consistent set of ther-modynamic parameters for reactions (11) and (12) canprovide an accurate description of both the silica solubilitymeasurements and the changes of degree of polymeriza-tion with fluid density calculated by Zotov amp Keppler(2002)

We suggest that an explanation for the above discrep-ancies must be sought in (i) the possible problems of equi-libration in the relatively short experiments in thediamond-anvil cell performed by Zotov amp Keppler (20002002) and (ii) the difficult task of quantifying the Ramanspectra of aqueous silica solutions The interpretation ofRaman spectra is based mainly on theoretical spectraobtained using empirical interaction potentials (Catlow1980 Price amp Parker 1984) or on single molecule abinitio calculations as in the case of Zotov amp Keppler(2000 2002) The problem with the latter is that they donot include crucial temperature- and pressure-dependentsolvent effects To test these environmental effects wehave initiated a series of both ab initio molecular dynamics(AIMD) simulations and static gas phase ab initio calcula-tions (Doltsinis et al 2004) Our results to date suggestthat while the strong feature at ca 780-800 cm-1 can beascribed to monomers with some confidence there arestill considerable uncertainties in interpreting otherfeatures of the Raman spectra in terms of monomersandor dimers Considering the low amounts of total silicainvolved and the increasingly floppy nature of largerpolymers leading to increasingly broadened spectralfeatures it appears that the recognition and identificationof trimers and higher polymers can be problematical Ifhidden trimers are present then an interpretation basedsolely on monomerdimer ratios will obviously be flawedThus while the results of Zotov amp Keppler (2000 2002)

represent a valiant first attempt to tackle the problem andprovide valuable qualitative data direct comparisons withclassical solubility results should not be overextended Wesuggest that further Raman spectroscopic measurementson well-defined solutions with given bulk silica contents atdifferent pressures and temperatures rather than quartz-buffered solutions only will be a valuable step toward abetter understanding of such spectra

The approach outlined in this paper has been success-fully used to fit and describe experimental data on bulkspinel (MgAl2O4) solubility up to 30 kbar byZakhartchouk et al (2004) despite the lack of detailedinformation on the actual speciation of the dissolved spinelcomponents in that system Given additional independentdata on the detailed structure of condensed phase water inthe future (eg Marx 1999 and in press) and on solutespeciation at various temperatures and pressures (seeDoltsinis et al 2004) the formalism described here canbe used to provide an increasingly complete thermody-namic description of aqueous solutions at depths corre-sponding to the lower crust and upper mantle

AcknowledgementsThis work was supported by ETHResearch Grant TH -1204-1 by RFBR grant 03-05-64633 by the RF President Program ldquoLeading ScientificSchool of Russiardquo (grant 0351645) to TVG and by theDeutsche Forschungsgemeinschaft within the scope ofSonderforschungsbereich 526 at the Ruhr-University ofBochum The helpful reviews of C Manning J Waltherand an anonymous reviewer as well as the editorialhandling of A Woodland are greatly appreciated

References

Anderson GM amp Burnham CW (1965) The solubility of quartzin supercritical water Am J Sci 263 494-511

Applin KR (1987) The diffusion of dissolved silica in diluteaqueous solution Geochim Cosmochim Acta 51 2147-2151

Audetat A amp Keppler H (2004) Viscosity of fluids in subduc-tion zones Science 303 513-516

Barelko VV Zarikov IV Pechatnikov EL (1994) Effect of theorigin of molecular clusters in one-component gas mixtures onthe equation of state Khim Fiz 13 42-50

Bockris JOrsquoM (1949) Ionic solutions Chem Soc LondonQuarterly Rev 3 173-180

Cary LW De Jong BHWS Dibble WEJr (1982) A 29Si NMRstudy of silica species in dilute aqueous solution GeochimCosmochim Acta 46 1317-1320

Catlow CRA (1980) Computer modelling of ionic crystals JPhysics 41 C6-53

Doltsinis NL Burchard M Maresch WV (2004) Ab initiomolecular dynamics simulations as a tool for interpreting vibra-tional spectra in aqueous fluids EMPG-X SymposiumAbstracts Lithos supplement to v 73 S28

Dove PM amp Rimstidt JD (1994) Silica physical behaviorgeochemistry and materials applications in ldquoSilica-water inter-actionsrdquo Reviews in Mineral 29 259-amp

Fockenberg T Burchard M Maresch WV (2004) Solubilities ofcalcium silicates at high pressures and temperatures EMPG-XSymposium Abstracts Lithos supplement to v 73 S37

282

T V Gerya W V Maresch M Burchard V Zakhartchouk N L Doltsinis T Fockenberg

and is directly accounted for by equations (4)-(6) (for adiscussion see Gerya amp Perchuk 1997 Gerya et al2004) The formation of similar complexes with dimersand higher polymers can also be expected thus supportingour thermodynamic formalism

An important unresolved question is the discrepancybetween the results obtained from our data set which isbased mainly on direct solubility measurements and dataobtained from Raman spectra (Zotov amp Keppler 20002002) This is particularly evident in the pressure effect onsilica polymerization The positive ∆Vdegr of reaction (12)suggests that at constant temperature and bulk silica anyincrease in pressure (ie in fluid density) should lead to adecrease in the degree of polymerization This is alsosupported by the ab initio molecular dynamics simulationsof Doltsinis et al (2004) who calculated higher effectivepressures for monomeric-dimeric systems in comparisonwith purely monomeric systems at T = 900 K and constantfluid density In the presence of quartz bulk silica contentincreases with pressure and the competing effects of reac-tions (11) and (12) should lead to only small changes inthe degree of polymerization Figure 7 indicates the trendspredicted from our model In contrast the results of Zotovamp Keppler (2002) suggest strong increase in the degree ofpolymerization with pressure (ie fluid density) requiringa notably negative ∆Vo

r of polymer-forming reaction (12)Our test calculations show that no consistent set of ther-modynamic parameters for reactions (11) and (12) canprovide an accurate description of both the silica solubilitymeasurements and the changes of degree of polymeriza-tion with fluid density calculated by Zotov amp Keppler(2002)

We suggest that an explanation for the above discrep-ancies must be sought in (i) the possible problems of equi-libration in the relatively short experiments in thediamond-anvil cell performed by Zotov amp Keppler (20002002) and (ii) the difficult task of quantifying the Ramanspectra of aqueous silica solutions The interpretation ofRaman spectra is based mainly on theoretical spectraobtained using empirical interaction potentials (Catlow1980 Price amp Parker 1984) or on single molecule abinitio calculations as in the case of Zotov amp Keppler(2000 2002) The problem with the latter is that they donot include crucial temperature- and pressure-dependentsolvent effects To test these environmental effects wehave initiated a series of both ab initio molecular dynamics(AIMD) simulations and static gas phase ab initio calcula-tions (Doltsinis et al 2004) Our results to date suggestthat while the strong feature at ca 780-800 cm-1 can beascribed to monomers with some confidence there arestill considerable uncertainties in interpreting otherfeatures of the Raman spectra in terms of monomersandor dimers Considering the low amounts of total silicainvolved and the increasingly floppy nature of largerpolymers leading to increasingly broadened spectralfeatures it appears that the recognition and identificationof trimers and higher polymers can be problematical Ifhidden trimers are present then an interpretation basedsolely on monomerdimer ratios will obviously be flawedThus while the results of Zotov amp Keppler (2000 2002)

represent a valiant first attempt to tackle the problem andprovide valuable qualitative data direct comparisons withclassical solubility results should not be overextended Wesuggest that further Raman spectroscopic measurementson well-defined solutions with given bulk silica contents atdifferent pressures and temperatures rather than quartz-buffered solutions only will be a valuable step toward abetter understanding of such spectra

The approach outlined in this paper has been success-fully used to fit and describe experimental data on bulkspinel (MgAl2O4) solubility up to 30 kbar byZakhartchouk et al (2004) despite the lack of detailedinformation on the actual speciation of the dissolved spinelcomponents in that system Given additional independentdata on the detailed structure of condensed phase water inthe future (eg Marx 1999 and in press) and on solutespeciation at various temperatures and pressures (seeDoltsinis et al 2004) the formalism described here canbe used to provide an increasingly complete thermody-namic description of aqueous solutions at depths corre-sponding to the lower crust and upper mantle

AcknowledgementsThis work was supported by ETHResearch Grant TH -1204-1 by RFBR grant 03-05-64633 by the RF President Program ldquoLeading ScientificSchool of Russiardquo (grant 0351645) to TVG and by theDeutsche Forschungsgemeinschaft within the scope ofSonderforschungsbereich 526 at the Ruhr-University ofBochum The helpful reviews of C Manning J Waltherand an anonymous reviewer as well as the editorialhandling of A Woodland are greatly appreciated

References

Anderson GM amp Burnham CW (1965) The solubility of quartzin supercritical water Am J Sci 263 494-511

Applin KR (1987) The diffusion of dissolved silica in diluteaqueous solution Geochim Cosmochim Acta 51 2147-2151

Audetat A amp Keppler H (2004) Viscosity of fluids in subduc-tion zones Science 303 513-516

Barelko VV Zarikov IV Pechatnikov EL (1994) Effect of theorigin of molecular clusters in one-component gas mixtures onthe equation of state Khim Fiz 13 42-50

Bockris JOrsquoM (1949) Ionic solutions Chem Soc LondonQuarterly Rev 3 173-180

Cary LW De Jong BHWS Dibble WEJr (1982) A 29Si NMRstudy of silica species in dilute aqueous solution GeochimCosmochim Acta 46 1317-1320

Catlow CRA (1980) Computer modelling of ionic crystals JPhysics 41 C6-53

Doltsinis NL Burchard M Maresch WV (2004) Ab initiomolecular dynamics simulations as a tool for interpreting vibra-tional spectra in aqueous fluids EMPG-X SymposiumAbstracts Lithos supplement to v 73 S28

Dove PM amp Rimstidt JD (1994) Silica physical behaviorgeochemistry and materials applications in ldquoSilica-water inter-actionsrdquo Reviews in Mineral 29 259-amp

Fockenberg T Burchard M Maresch WV (2004) Solubilities ofcalcium silicates at high pressures and temperatures EMPG-XSymposium Abstracts Lithos supplement to v 73 S37

282

Gerya TV amp Perchuk LL (1997) Equations of state ofcompressed gases for thermodynamic databases used inpetrology Petrology 5 366-380

Gerya TV Podlesskii KK Perchuk LL Swamy V KosyakovaNA (1998) Equations of state of minerals for thermodynamicdatabases used in petrology Petrology 6 511-526

Gerya TV Podlesskii KK Perchuk LL Maresch WV (2004)Semi-empirical Gibbs free energy formulations for mineralsand fluids for use in thermodynamic databases of petrologicalinterest Phys Chem Minerals 31 429-455

Gorbaty YE amp Kalinichev AG (1995) Hydrogen-bonding insupercritical water 1Experimental results J Phys Chem 995336-5340

Gupta RB Panayiotou CG Sanchez IC Johnston KP (1992)Theory of hydrogen bonding in supercritical fluid AIChE J38 1243-1253

Helgeson HC Kirkham DH Flowers GC (1981) Theoreticalprediction of the thermodynamic behaviour of aqueous elec-trolytes at high pressures and temperatures IV Calculation ofactivity coefficients osmotic coefficients and apparent molaland standard and relative partial molal properties to 600degC and5 kb Am J Sci 281 1249-1516

Hemley JJ Montoya JW Christ CL Hostetler PB (1977a)Mineral equilibria in the MgO-SiO2-H2O system I Talc-antig-orite-forsterite-anthophyllite-enstatite stability relations andsome geologic implications in the system Am J Sci 277353-383

Hemley JJ Montoya JW Shaw DR Luce RW (1977b)Mineral equilibria in the MgO-SiO2-H2O system II Talc-chrysotile-forsterite-brucite stability relations Am J Sci 277322-351

Hemley JJ Montoya JW Marinenko JW Luce RW (1980)Equilibria in the system Al2O3-SiO2-H2O and some generalimplications for alterationmineralization processes EconGeol 75 210-228

Holland TJB amp Powell R (1998) Internally consistent thermody-namic data set for phases of petrlogical interest J metamorphicGeol 16 309-344

Kennedy GC (1950) A portion of the system silica-water EconGeol 45 629-653

Kohl W Lindner EU Franck EU (1991) Raman spectra ofwater to 400degC and 3000 bar Berichte BunsengesellschaftPhys Chem 9512 1586-1593

Lindner HA (1970) Ramanspektroskopische Untersuchungen anHDO geloumlst in H2O an HDO in waumlsserigen Kaliumjodidlouml-sungen und an reinem H2O bis 400degC und 5000 bar UnpublDoctorate Thesis Karlsruhe Technical University 80 pp

Luck WAP (1980) A model of hydrogen-bonded liquid AngewChem Int Ed Eng 19 28

Manning CE (1994) The solubility of quartz in H2O in the lowercrust and upper mantle Geochim Cosmochim Acta 584831-4839

mdash (2004) The chemistry of subduction-zone fluids Earth PlanetSci Let 223 1-16

Manning CE amp Boettcher SL (1994) Rapid-quenchhydrothermal experiments at mantle pressures and tempera-tures Am Mineral 79 4831-4839

Marx D (1999) Ab initio liquids Simulating liquids based onfirst principles In New Approaches to Problems in LiquidState Theory pp 439-457 Eds C Caccamo J-P Hansen GStell (Kluwer Dordrecht)

Marx D (in press) Wasser Eis und Protonen MitQuantensimulationen zum molekularen Verstaumlndnis von

Wasserstoffbruumlcken Protonentransfer und PhasenuumlbergaumlngenPhysik Journal

Morey GW Fournier RO Rowe JJ (1962) The solubility ofquartz in the temperature interval from 25degC to 300degCGeochim Cosmochim Acta 26 1029-1043

Nakamura Y amp Kushiro I (1974) Composition of the gase phasein MgSiO4-SiO2-H2O at 15 kbar Carnegie Institution ofWashington Year Book 73 255-258

Newton RC amp Manning CE (2000) Quartz solubility inconcentrated aqueous NaCl solutions at deep crust-uppermantle metamorphic conditions 2-15 kbar and 500-900degCGeochim Cosmochim Acta 64 2993-3005

Newton RC amp Manning CE (2002) Solubility of enstatite + for-sterite in H2O at deep crustupper mantle conditions 4 to 15kbar and 700 to 900degC Geochim Cosmochim Acta 664165-4176

mdash mdash (2003) Activity coefficient and polymerization of aqueoussilica at 800degC 12 kbar from solubility measurements on SiO2-buffering mineral assemblages Contrib Mineral Petrol 146135-143

Price GD amp Parker SC (1984) Computer simulations of thestructural and physical properties of the olivine and spinel poly-morphs of Mg2SiO4 Phys Chem Minerals 10 209-216

Shen AH amp Keppler H (1997) Direct observation of completemiscibility in the albite-H2O system Nature 385 710-712

Sverjensky DA Shock EL Helgeson HC (1997) Predictionof the thermodynamic properties of aqueous metal complexesto 1000 degrees C and 5 kb Geochim Cosmochim Acta 611359-1412

Walther JV amp Helgeson HC (1977) Calculation of the thermo-dynamic properties of aqueous silica and the solubility ofquartz and its polymorphs at high pressures and temperaturesAm J Sci 277 1315-1351

Walther JV amp Orville PM (1983) The extraction-quench tech-niques for determination of the thermodynamic properties ofsolute complexes application to quartz solubility in fluidmixtures Am Mineral 68 731-741

Walther JV amp Woodland AB (1993) Experimental determina-tion and interpretation of the solubility of the assemblagemicrocline muscovite and quartz in supercritical H2OGeochim Cosmochim Acta 57 2431-2437

Watson EB amp Wark DA (1997) Diffusion of dissolved SiO2 inH2O at 1 GPa with implications for mass transport in the crustand upper mantle Contrib Mineral Petrol 130 66-80

Woodland AB amp Walther JV (1987) Experimental determina-tion of the assemblage paragonite albite and quartz in super-critical H2O Geochim Cosmochim Acta 51 365-372

Zakhartchouk VV Burchard M Fockenberg T Maresch WV(2004) The solubility of natural spinel in supercritical waterExperimental determination EMPG-X Symposium AbstractsLithos supplement to v 73 S123

Zhang Y-G amp Frantz JD (2000) Enstatite-forsterite-water equi-libria at elevated temperatures and pressures Am Mineral 85918-925

Zotov N amp Keppler H (2000) In-situ Raman spectra of dissolvedsilica in aqueous fluids to 900degC and 14 kbar Am Mineral 85600-604

Zotov N amp Keppler H (2002) Silica speciation in aqueous fluids athigh pressures and high temperatures Chemical Geol 184 71-82

Received 16 May 2004Modified version received 5 October 2004Accepted 12 November 2004

Modeling of silica solubility and speciation 283