Gao et al.
Progress in Earth and Planetary Science(2024) 11:20
https://doi.org/10.1186/s40645-024-00622-1

RESEARCH ARTICLE

Progress in Earth andPlanetary Science

Open Access

Phase diagram and density of ­SiO2–H2O fluid
across critical conditions
Hui Gao1,2, Guoge Li1,2 and Zhigang Zhang1,2*   

Abstract
The ­SiO2–H2O binary system serves as a basis for understanding complex silicate-water systems. In this study,
based on limited existing experimental data of solubility, we propose a new thermodynamic model for ­SiO2–H2O
fluid by modifying the traditional non-random two-liquid model with a simplified polymerization reaction. Thismodel is applicable from 773 K to the anhydrous quartz melting temperature and from 0.5 GPa to at least 2 GPa
across the critical conditions. It can predict solid–liquid equilibrium and vapor–liquid equilibrium in good agreementwith available experiments. The upper critical endpoint of the ­SiO2–H2O system is predicted to be at ~ 1.14 ± 0.18 GPa
and 1344 ± 87 K. With the new model, we obtain a quantitative three-dimensional pressure–temperature–composition phase diagram of the ­SiO2–H2O fluid, which greatly facilitates the understanding of the complex phase behaviorof this binary around the upper critical endpoint. In addition, since the model is based on the Gibbs free energy
foundation, we further discuss the derived density variations of ­SiO2–H2O fluid along with its complex phase changes
in typical geochemical processes.
Keywords SiO2–H2O, Supercritical fluid, Solubility, Upper critical endpoint, Vapor–liquid equilibrium, Density
1 Introduction
The Earth’s unique plate tectonic processes often involvea certain amount of water, making water–rock interactions in the Earth’s interiors important. Silicates are thedominant minerals in the crust and mantle, so thermodynamics of mixing silicate and water at high temperatures and pressures is critical to understanding deep
fluid-rock interactions (Hack et al. 2007a, 2007b; Manning 2004; Tian et al. 2019). Unlike solution in water suchas alkali salts, silicates generally have negligible solubilities in water at ambient conditions. With increase of
temperature and pressure, however, their solubilities inwater increase markedly, and so does the solubility ofwater in silicate melts. At sufficiently high temperature*Correspondence:
Zhigang Zhang
zgzhang@mail.iggcas.ac.cn
1
Key Laboratory of Earth and Planetary Physics, Institute of Geology
and Geophysics, Chinese Academy of Sciences, Beijing 100029, China
2
College of Earth and Planetary Sciences, University of Chinese Academyof Sciences, Beijing 100049, China

and pressure, aqueous fluids (labeled as V phase below)
and hydrous silicate melts (labeled as L phase below) can
be completely miscible and become supercritical fluids(SCF) (Boettcher and Wyllie 1969; Kennedy et al. 1962;
Ni et al. 2017). Therefore, magma rising from the deep
interiors would develop significantly altered geochemical and physical properties during their ascent, which
may influence many geological processes such as volcanic venting and pegmatite formation (Mibe et al. 2002;
Sowerby and Keppler 2002; Thomas 2000; Thomas andDavidson 2016).
As a prototype binary for the silicate-water systems,
Fig. 1 outlines schematically the complex phase behavior in the ­SiO2–H2O system. From Fig. 1a, as the temperature rises, the concentration of S
­ iO2 in the aqueous
fluid phase gradually increases. When the temperaturereaches the fluid-saturated solidus (T1), quartz begins
to melt. With a sufficient amount of ­H2O, hydrous melt
and aqueous fluid coexist in vapor–liquid equilibrium(VLE). In cases of insufficient ­H2O, quartz undergoes
partial melting, resulting in solid–liquid equilibrium© The Author(s) 2024. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which
permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to theoriginal author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images orother third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line
to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory
regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of thislicence, visit http://creativecommons.org/licenses/by/4.0/.

Gao et al. Progress in Earth and Planetary Science(2024) 11:20

Page 2 of 14

Fig. 1 Schematic phase relations of ­SiO2–H2O binary system, a T–X phase diagram. The solid and dashed curves represent P1 and P2 conditions,
respectively, with P1 < P2. Green curve represents the solubility of quartz in aqueous fluid (①); red curve represents the solubility of ­H2O in ­SiO2 melt(②); ③ represents the V–L–S three-phase coexistence; ④ and ⑤ represent the vapor–liquid equilibrium curves. b Critical curve, hydrous melting
curve and the upper critical endpoint (UCEP). V: aqueous fluid; L: hydrous silica melt; S: solid (quartz); F: single phase fluids(SLE). The SLE ends at the anhydrous melting temperature (T3). Between T1 and T2, aqueous fluid andhydrous melt coexist with increasing mutual solubility
till their final miscibility.
Pressure has important effects on the phase boundaries. As the pressure increases, from P1 to P2 in Fig. 1a,the temperatures of the consolute point (T2) and thehydrous melting point (T1) decrease. The T–P variations of the consolute point and the hydrous melting
point are schematically illustrated in Fig. 1b. These two
curves eventually intersect at a point, which is called
the upper critical endpoint (UCEP) (Hack et al. 2007b).
Data from existing experimental solubility measurements of ­SiO2–H2O can be roughly divided into two categories: SLE (the green ① and red ② curves in Fig. 1a)and VLE (the purple ④ and orange ⑤ curves in Fig. 1a).
Available experimental data, with 875 data points, aresummarized in Table 1. Most of these determinations
were focused on the quartz solubility in aqueous solutions. Experimental data for hydrous silicate melts atpressures higher than 0.5 GPa are scarce. Only Holtzet al. (2000) measured several data points for water solubility in S
­ iO2-rich liquid, and there are no experimental
data constraining the coexisting vapor phase. Experimental studies on the UCEP of the ­SiO2–H2O system have
proven challenging. Kennedy et al. (1962) estimated it to
be around 0.97 GPa and 1353 K, while some other studies
concluded that the UCEP should be located at a pressure
higher than 1.0 GPa (Mysen 1998; Stewart 1967). Nakamura (1974) discovered that ­SiO2–H2O fluids exhibit
supercritical behavior at 1.5 GPa, thereby establishing theupper limit of the UCEP.

A few solubility models have been developed for S­ iO2–
H2O fluids. Sverjensky et al. (2014) proposed the Deep
Earth Water (DEW) model by refining the dielectric constant model at high pressures based on a well-behaved
equation of state for water (Zhang and Duan 2005). The
DEW model extends the temperature and pressure of theelectrostatic model to 6 GPa and 1473 K. Other, empirical models based on linear relations with pure water density, have been proposed by a number of studies (Denget al. 2021; Fournier and Potter 1982; Manning 1994; Shiet al. 2019; Wei et al. 2012), but these models are generally applied to aqueous fluids at low temperatures. Huntand Manning (2012) developed a model (HM2012 model)
based on the excess mixing properties in the ­SiO2–H2O
fluids, which is applicable from 773 K to the anhydrous
quartz melting temperature and from 0.5 to 2.0 GPa.
Acknowledging the potential disparity between local
and overall mole fractions in the system, Wilson (1964)
introduced the concept of local composition to the excess
Gibbs free energy. This refinement facilitates characterizing equilibrium within polymer solutions.
The non-random two-liquid (NRTL) model introducesa correction factor based on the concept of local composition, which is advantageous in depicting the immiscibility in liquid systems (Renon and Prausnitz 1968).
However, the traditional NRTL model is commonlyused to study systems with organic compounds at low
temperatures and pressures, and the interaction parameters are assumed to be pressure independent. To extend
this model to the deep Earth environment, we modify
its interaction parameters as a function of temperatureand pressure. In addition, the NRTL model requires

Gao et al. Progress in Earth and Planetary Science(2024) 11:20

Page 3 of 14

Table 1 Summary of experimental data on the mutual solubility of ­SiO2–H2OSolubility studies

T/K

P/GPa

n

Types

Kennedy (1950)

433.15–883.15

0.00062–0.175

106

1Vanlier et al. (1960)

333.15–373.15

0.00016

1Kennedy et al. (1962)#

1316.15–1488.15

0.2–0.95

42

3

Morey et al. (1962)

298.15–573.15

0.00003–0.1013

139

1

Siever (1962)

298.15–413.15

0.000114

1Weill and Fyfe (1964)

673.15–898.15

0.1–0.4

48

1

Anderson and Burnham (1965)

773.15–1173.15

0.1–0.985

79

1

Anderson and Burnham (1967)

973.150.4

2

1Sommerfeld (1967)

673.15–773.15

0.1

24

1

Crerar and Anderson (1971)

392.15–598.15

0.000131

1Mackenzi and Gees (1971)

293.150.0001

1

1

Nakamura (1974)#

1171.276–1674.15

1.5

29

5

Novgorodov (1977)

973.150.15

1

1

Hemley et al. (1980)

473.15–773.15

0.00155–0.2

181

Ragnarsdottir and Walther (1983)

523.150.025–0.1

71

Walther and Orville (1983)

623.15–835.15

0.1–0.2

20

1

Xie and Walther (1993)

685.15–846.15

0.2–0.2005

7

1

Manning (1994)*

773.15–1173.15

0.5–2

52

1,5

Manning and Boettcher (1994)*

773.15–873.15

0.5–1.25

3

1

Rimstidt (1997)

294.15–369.15

0.0001

10

1Watson and Wark (1997)#

802.15–1141.15

1

4

1

Holtz et al. (2000)#

1473.15–1623.15

0.101–0.6

16

4

Gunnarsson and Arnorsson (2000)

281.15–583.15

0.0001–0.0098.6

221

Newton and Manning (2000)#

873.15–1123.15

0.2–1

8

1

Shmulovich et al. (2001)*

773.15–1173.15

0.5, 0.9

61Wang et al. (2004)

399.15–763.15

0.08–0.89

128

1Shmulovich et al. (2006)#

673.15–1073.15

0.1–0.9

19

1

Newton and Manning (2008)*

1023.15–1403.15

1

20

2

Hunt and Manning (2012)*

1173.15–1373.15

0.5–2

13

2,5

n is the number of data points; Types in the last column: 1: quartz solubility in H
­ 2O; 2: solubility of ­H2O in quartz-saturated melt; 3: solubility of solid(quartz)–V–L
coexistence; 4: solubility of ­H2O in hydrous silicate melts when the two liquid phases are in equilibrium; 5: quartz solubility in supercritical fluid. The superscripts * and# indicate the entire or partial experimental data sets used in the parameter regressions or validation, respectively

accounting for the interactions and local orderings
between every two different species in the liquid. Therefore, the application of the traditional NRTL model to thesilicate-water fluid systems seems to be unrealistic withtoo many parameters for fitting. Based on this, we propose a two-step model: in the first step, we quantify themixing of ­SiO2 and ­H2O binary (i.e., no polymerization)with the traditional NRTL model; in the second step,using polymerization reactions to describe the generation of other new species. The two-step method has theadvantage of greatly reducing the number of parametersthat need to be fitted. It can accurately predict both SLE
and VLE properties of ­SiO2–H2O and constrain the density that is still lacking from experimental measurements,
thus paving the way to quantify various more complex
silicate-water systems.

2 Methods
2.1 Fundamental two‑step model for the fluid

We model the mixing of S
­ iO2 and H­ 2O in two steps. In
the first step: we consider a binary system of N moles­SiO2–H2O fluid with xs mole fraction of S
­ iO2 and 1−xsmole fraction of ­H2O. In this step, we apply the traditional NRTL model (Additional file 1: Appendix A) andex
obtain the excess molar Gibbs free energy gNRTL
as follows: (Renon and Prausnitz 1968):ex= RTxs (1 − xs )gNRTL

τ12 G12
τ21 G21+xs + (1 − xs )G21
(1 − xs ) + xs G12(1)
where R is the universal gas constant, τ12 and τ21 aredimensionless parameters to characterize the difference in interaction energy between different molecules pairs in the local composition, which are related

Gao et al. Progress in Earth and Planetary Science(2024) 11:20

Page 4 of 14

to the interaction energy parameters G12 and G21 by:
G12 = exp(−α12 τ12 ) and G21 = exp(−α21 τ21 ), where α12and α21 (α12 = α21 = α) are the characteristic constantsof the non-random mixture with an appropriate value
between 0.2 and 0.47 (Prausnitz et al. 1998).
There is numerous experimental and computational
evidence leading to the suggestion that S­ iO2–H2O system is dominated by monomers (Si (OH)4) and dimers­(Si2O(OH)6) at low temperatures and pressures (Mysen
2010; Zotov and Keppler 2000, 2002). However, withincreasing temperature and pressure, other higher
polymers may be present in significant concentrations(Doltsinis et al. 2007). It is possible to extend Eq. (1)and derive a multi-component NRTL model, but itwould involve too many parameters. Alternatively, following the strategy by Hunt and Manning (2012), we
further amend the free energy with a second step byconsidering the exchange of three types of oxygen (oxygen in free water molecules ­(H2O), bridging oxygen(Obr ), and terminal hydroxyl oxygen of silanol groups(OHtm )) in a simplified polymerization reaction (Geryaet al. 2005; Hunt and Manning 2012; Plyasunov andZyubina 2021):

1
1
H2 O + Obr = OHtm
2
2

(2)

If we assume that the reaction (2) produces M
molesOHtm . Then the reaction involves 0.5M moles H­ 2O
and 0.5M molesObr. This system contains N(1−xs)−0.5M
moles ­H2O and 2Nxs−0.5M molesObr. Therefore, for theeffectively ternary system of H
­ 2O, Obr andOH, the system contains N(1 + xs) moles oxygen-containing components and the concentrations of H
­ 2O, Obr and OHtm can
Ms )−0.5M 2Nxs −0.5M
be expressed as:N (1−xN (1+xs ) , N (1+xs ) ,N (1+xs ) , respecM
tively. If we denote yOH = N (1+x, then ­H2O and Obrs)sconcentration can be simplified as 1−x
1+xs − 0.5yOH and2xs
1+xs − 0.5yOH , respectively. Assuming that all the nonideality has been accounted for with the NRTL model,
the equilibrium constant K of Eq. (2) can be expressed as:◦−grec
K = exp
RT





= 

yOH


1−xs
1+xs − 0.5yOH

2xs
1+xs − 0.5yOH

(3)◦ is the standard molar Gibbs free energy
where grec

change of Eq. (2) at any temperature and pressure.
Then, the molar excess Gibbs free energy of this S­ iO2–
H2O binary can be expressed as:
ex
ex
◦
gtot
= gNRTL− (1 + xs )yOH �grec(4)Note that the yOH in Eq. (4) can be further expressed as◦ by solving Eq. (3). Now we havea function of xs and grec

established a fundamental model for the ­SiO2–H2O fluid◦ ).
with only four hyper-parameters (α, τ12 , τ21 and�grec
2.2 Phase equilibrium calculations and the model
regression

The key property for the phase equilibrium is the activity
coefficient, which can be derived fromexcess Gibbs freeexenergy by RT lnγi = g ex + (1 − xi ) ∂g
(Levine2008; Ma∂xi
2001). From Eq. (4), we obtain the following expressions:2
RT lnγSiO2 =RTxH
2O2
τ21 G21

τ12 G12τ21 G21

2 + 
2
xH2 O + xs G12


∂yOH
◦
− �grecyOH (1 + xs ) + xH2 O(1 + xs ) + yOH
∂xs




xs + xH2 O G21



RT lnγH2 O =RTxs2 (5)2
τ12 G12

2 + 

2
xH2 O + xs G12
xs + xH2 O G21



∂yOH
◦
− �grecyOH (1 + xs ) − xs
(1 + xs ) + yOH∂xs(6)yOH can be solved by Eq. (3) and calculating its derivative
to xs, the ∂y∂xOH
in Eqs. (5–6) can be further expressed as as
◦ .
function of xs and grec
For the solid–liquid equilibrium (SLE), the chemical
potential of quartz and the liquid-phase ­SiO2 are equal:µ◦quartz = µ◦SiO2 ,liq + RT lnXs + RT lnγSiO2(7)where µ◦quartz is the standard Gibbs free energy of quartz(Berman 1988), µ◦SiO2 ,liq is the standard Gibbs free energyof ­SiO2 in the liquid phase (Mao et al. 2001), Xs is thedetermined solubility of ­SiO2 in mole fraction.
For the vapor–liquid equilibrium (VLE) between aqueous fluid and hydrous silica melt, we haveV
L
XsV γSiO= XsL γSiO
2
2

(8)

XHV2 O γHV2 O = XHL 2 O γHL2 O(9)where XsV + XHV2 O = 1 and XsL + XHL 2 O = 1.
To describe phase equilibria over a broad range of temperature and pressure, while α is assumed to be constant,◦ , τ , τ ) in
the remaining three hyper-parameters (�grec
12 21
the fundamental model are modelled as functions of temperature and pressure:τ12 = b1 +

b2
b3 P
+
T
T

(10)

τ21 = b4 +

b5
b6 P
+
T
T

(11)

Gao et al. Progress in Earth and Planetary Science(2024) 11:20

Page 5 of 14(12)
where T is temperature in Kelvin, P is pressure in bars,◦ is the standard molar Gibbs free energy change
and grec
in J/mol.

From the available experimental solubility data, we
obtained parameters of the equilibrium model with nonlinear least squares regressions and Bayesian optimization (Wang and Yang 2023). In this process, we tested
the statistical significance of the fitting parameters, eliminated one parameter that was statistically insignificant,and finally obtained the fitting values of 13 independent
parameters (Table 2).

Table 2 The regressed parameters for the new model

3 Results

α12 = α21 = α

0.1961 (0.0086)

3.1 Comparison with experimental datab1

− 0.3846 (0.0111)

b2 (K)

1665.6239 (37.2928)

b3 (K/bar)− 0.0537 (0.0006)◦
grec= b7 + b8 T + b9 P + b10 P 2 + b11 T 2 + b12 PlnT +

b13 P
T

b4

− 0.2359 (0.0063)

b5 (K)

3061.3542 (151.5954)

b6 (K/bar)0.0399 (0.0054)

b7 (J/mol)

12,126.8110 (50.6106)

b8 (J/mol/K)− 23.6453 (0.5641)

b9 (J/mol/bar)− 23.2747 (0.5593)

b10 (× ­106) (J/mol/bar2)0b11(× ­103) (J/mol/K2)

9.1411 (0.2880)

b12 (J/mol/K/bar)

3.0663 (0.0230)

b13 (J*K/mol/bar)

2302.3663 (93.7457)

Numbers in the parentheses are the uncertainties for the regressed parameters
b10 was dismissed after the parameter significance test

As shown in Table 1, there are 875 experimental solubility data points available for the ­SiO2–H2O system, covering a wide range of temperature and pressure ranges(0.00003–2 GPa, 281.15–1674.15 K). Most of the data arein the low pressure-low temperature region (T < 773 K
or P < 0.5 GPa). However, the purpose of our model is tocharacterize the fluid properties near the upper critical
endpoint, so we focused on a more appropriate regime
with pressure higher than 0.5 GPa. A total of 204 available experimental data points were collected (temperature from 773 K to quartz dry melting temperature andpressure from 0.5 to 2.0 GPa). It should be noted thatcertain experimental data points may not be reliable. For
instance, Manning (1994) demonstrated that the dataof Anderson and Burnham (1965) had quench-related
issues leading to underestimation of quartz solubility,
so our model does not consider these data. There wereTable 3 Experimental data used for model calibration and verification
References

T (K)

P (GPa)n

AARD (%)

AAD

Experimental data used for model calibration
Kennedy et al. (1962)

1316.15–1488.15

0.2–0.95

10

28.9530.081

Manning (1994)

773.15–1173.15

0.5–1.25

41

13.6380.002

Holtz et al. (2000)

1473.15–1623.15

0.101–0.6

1

35.257

0.273

Newton and Manning (2000)

873.15–1123.15

0.2–1

4

8.239

0.001

Shmulovich et al. (2001)

773.15–1173.15

0.5, 0.9

6

11.5890.001

Shmulovich et al. (2006)

673.15–1073.15

0.1–0.9

11

14.341

0.001

Newton and Manning (2008)

1023.15–1403.15

1

14

14.168

0.038

Experimental data used for model verification
Manning (1994)

773.15–973.15

1.5–2

11

13.8600.001

Nakamura (1974)

1171.276–1674.15

1.5

16

19.2460.072

Manning and Boettcher (1994)

773.15–873.15

0.5–1.25

3

16.2840.001

Watson and Wark (1997)

802.15–1141.15

13

13.676

0.002

Hunt and Manning (2012)

1173.15–1373.15

1.5–2

13

Overall

17.2770.027

15.9300.024

n: Number of experimental data points
AAD and AARD stand for average absolute deviation and average relative deviation respectively between the experimental data and that calculated with the model
proposed in this study,which are expressed as follows:N  xscali −xsexp
i 

i=1 

N
 cal exp exp N (xs −xsi )/xsi  × 100%
AARD = i=1  i

N
AAD =

Gao et al. Progress in Earth and Planetary Science(2024) 11:20

finally 133 experimental data used to build our model,
some of the experimental data were utilized for parameter regression, while the remaining data were used to validate the fitting results (Table 3). Table 3 also summarizes
the averaged absolute deviations (AAD) and averaged
relative deviations (AARD) of the model predictions. Figure 2 illustrates the modeled solubilities and the experimental data. The overall AAD and AARD are 0.02 and
16%, respectively. Furthermore, except for several outlier
data by Kennedy et al. (1962) and Holtz et al. (2000), themodel gives good predictions of the measured solubilities
accurate to 19%.
Figure 3 presents comparison of our and previous
models for quartz solubility under various temperatureor pressure conditions. The selected models include
widely used density models or empirical models, namely
F1982 (Fournier and Potter 1982), M1994 (Manning1994), WDM2012 (Wei et al. 2012), SMH2019 (Shi et al.
2019), DWP2021 (Deng et al. 2021), G2005 (Gerya et al.
2005), DM2010 (Dolejs and Manning 2010); the electrostatic model DEW2014 (Sverjensky et al. 2014) andthe solute–solvent interaction model HM2012 (Huntand Manning 2012). Under constant temperature conditions, F1982 agrees well with the experimental data

Page 6 of 14

at low pressures. However, as the pressure increases,
it deviates significantly from the experimental data
and underestimates the solubility of quartz in water. At
873 K, G2005 agrees well with experimental data at low
pressure, but it overestimates the solubility of quartz as
the pressure increases (Fig. 3a). By contrast, both M1994
and DEW2014 models tend to overestimate the solubility of quartz. Additionally, models that demonstrate
good performance in low-temperature regions (such as
G2005, DM2010, WDM2012, DWP2021 and SMH2019)no longer maintain their accuracy when the temperatureextends beyond 1173 K. Unlike density and electrostatic
models, models based on interactions between molecules
are easier to scale to higher temperatures. Therefore, overa wide range of temperatures and pressures, the HM2012
model and the two-step model are in good agreementwith the experimental data (Fig. 3d). While these twomodels indeed show quite similar accuracies, as shownin Fig. 4, slight differences can be discriminated: from
Fig. 4a, when the pressure is 1 GPa, HM2012 predicts
that the system is already supercritical, while our model
predicts that the miscible gap still exists (green line in
Fig. 4a). When the pressure is 1.5 GPa and 2 GPa, thepredicted results of the two models agree well with theFig. 2 Comparison of solubility between this model and experiments. The predicted solubilities of this model were calculatedwith the experimental conditions as input. Since most of the experimental data is concentrated in the lower left corner, an enlarged view of thisregion is provided in the inner plot

Gao et al. Progress in Earth and Planetary Science(2024) 11:20

Page 7 of 14

Fig. 3 Comparison of quartz solubilities in water at different T–P conditions. FP1982: Fournier and Potter (1982); M1994: Manning (1994); G2005:
Gerya et al. (2005); DM2010: Dolejs and Maning (2010); HM2012: Hunt and Manning (2012); WDM2012: Wei et al. (2012); DEW2014: Sverjensky et al.
(2014); SMH2019: Shi et al. (2019); DWP2021: Deng et al. (2021)

available experimental data (Fig. 4b, c), the fluid is supercritical fluid, and the miscibility gap disappears.
As shown in Fig. 5a, we estimate the location of theupper critical endpoint to be around 1.14 ± 0.18 GPa and1344 ± 87 K, while it is predicted by HM2012 to be about
0.93 GPa, 1340 K. The predictions of these two models
for the upper critical endpoint as well as the hydrous
melting curve temperatures agree with the measurements of Kennedy et al. (1962) (approximately 0.97 GPaand 1353 K). Besides these similarities, it is evident thatthe miscibility gap predicted by HM2012 is larger thanthat evaluated in the present study. A comparison of the

X–T relationships (Fig. 5b) suggests that the two models
show some differences in VLE: HM2012 tends to be more
asymmetric with the critical curve leaning towards lower
concentration of silicate components. As mentioned
previously, experimental measurements of hydrous meltsolubility for VLE are scarce, with only one Holtz et al.
(2000) measurement collected at 0.6 GPa (Fig. 5b).
3.2 Phase diagram of the ­SiO2–H2O system

Our thermodynamic model leads to a self-consistent
phase diagram (Fig. 6). Figure 5a, b corresponds to the
T–P and X–T projections, respectively. With the P–X

Gao et al. Progress in Earth and Planetary Science(2024) 11:20

Page 8 of 14hydrous melting curve often involves small amount ofsilicate solute. The solubility of S
­ iO2 is generally less than0.1. Nevertheless, in the vicinity of the hydrous melting
curve, particularly at high pressures, the solubility of ­SiO2
increases remarkably. On the other side of the hydrous
melting curve, the situation is more complex with several
phase regimes overlapping from this point of view. As
indicated by the thin black solid lines, the concentrationof ­SiO2 in the fluid could be high when it is equilibrated
with quartz.
The vapor–liquid equilibrium (VLE) regime between
the hydrous melting curve and the critical curve greatly
redefines the compositional range of the fluid. This VLE
regime determines the lower limit of ­SiO2 in the hydrous
melt (gray isopleths), on the other hand, at the same pressure it accommodates much higher concentration of ­SiO2
in the aqueous fluid (dashed isopleths) as compared withthat at lower temperature (left side of the hydrous melting curve). As will be discussed in the next section, thevariations of ­SiO2 solubility would significantly affect thedensity of the fluid and therefore are important for thetransportations of ­SiO2–H2O fluids.

4 Discussion
4.1 Density of the ­SiO2–H2O fluid

Density is the key property that controls the physicalstate and transportation of fluid. Experimental evidence
indicates that the density of ­SiO2–H2O fluids depends
on the ­SiO2 concentration and is correlated with species behavior (Mysen 2010). However, directly measuring fluid density in experiments is challenging. With themodel proposed in this study, it is possible to constrain
the density of the ­SiO2–H2O binary.
By calculating the partial derivative of the excess free
ex in Eq. (4) to the pressure, we obtain the excessenergy gtot
volume Vex:

(

ex
∂gtot
)
= V ex∂P T ,xs(13)

Then the volume of ­SiO2–H2O fluid can be expressedas:Fig. 4 Comparison with experimental solubility data of quartz
across the UCEP. We show a short green line for the predicted VLE
curve with the two-step model

projection in Fig. 6, the VLE boundary changes at different temperatures and pressures also can be quantified.
Isopleths of ­SiO2 solubility in the fluid across the critical region around the upper critical endpoint (UCEP)
is shown in Fig. 7. As noticed by Hack et al (2007a, b),
the aqueous fluid that resides on the most left side of the◦V = (1 − xs )VH◦ 2 O + xs VSiO
+ V ex2(14)◦where VH◦ 2 O and VSiO
are the standard molar volumes of2
liquid ­H2O and liquid ­SiO2 as given by Zhang and Duan(2005) and Mao et al. (2001), respectively. The fluid density can be easily obtained from the volume:ρ=xs MSiO2 + (1 − xs )MH2 O
V(15)

where MH2 O and MSiO2 are the standard molar mass ofliquid ­H2O and liquid ­SiO2.

Gao et al. Progress in Earth and Planetary Science(2024) 11:20

Page 9 of 14

Fig. 5 Comparison of T–P and X–T projections of this model and the HM2012 model. a The predicted hydrous melting curve and dry melting curveof quartz vs. the experimental constraints. Critical curve (dashed line) is included to show the predicted miscibility gap between aqueous fluidand hydrous melt. b comparison of VLE predicted by this study and HM2012 model (Solid lines represent this study, dotted lines represent HM2012
predictions)Fig. 6 Three-dimensional phase diagram of the ­SiO2–H2O system
predicted by the new model. The black solid line represents
the hydrous melting curve, and the red point represents the upper
critical endpoint. Projected phase boundaries on three planes (T–X,
T–P, P–X) are shown with labels marked in the single or coexisting
phase regimes. The colored contours are the derived densityof saturated fluid under different conditions

However, experimental errors and model parameter errors may affect density predictions. We used theMonte Carlo method by deploying multiple samplingstatistics within experimental and parameter uncertainty, yielding an uncertainty for the density of thefluid. The maximum uncertainty is 0.06 g/cm3. In Fig. 6,
we show the predicted density of quartz-saturated andFig. 7 Solubility of ­SiO2 in the fluid predicted by the new model
as a function of temperature and pressure. The red line represents
the critical curve, the thick black line represents the hydrous
melting curve; the thin black lines are the isopleths of ­SiO2
solubility under SLE; the black dashed and gray solid lines represent
the isopleths of ­SiO2 solubility in aqueous fluids and hydrous silicate
melts under VLE, respectively

vapor-saturated ­SiO2–H2O fluid with colored contours. Obviously, the density of the fluid is mainly controlled by the concentration of dissolved ­SiO2 (Xs). The
effects of temperature and pressure seem to be minor,
but they may indirectly regulate the density by changing the composition of the fluid, which may result incounterintuitive variation of density with temperature or pressure. This can be understood by relating it

Gao et al. Progress in Earth and Planetary Science(2024) 11:20

with the solubility profile as shown in Fig. 7: the aqueous fluid would dissolve much more solute as the temperature approaching the UCEP and this would greatlycompensate the intrinsic decreasing density effect oftemperature.
Figure 8 shows the comparison and analysis of thefluid density predicted by our two-step model (solidline) and the HM2012 model (dashed line). While thedensity of aqueous fluids is increased with pressure as
expected, the abovementioned reverse trend of temperature can be observed at higher temperatures and pressures (Fig. 8a, 1223 K). For quartz-saturated hydrous
melts, the two-step model is the opposite of what theHM2012 model predicts (dashed lines) showing that its
density decreases with increasing pressure. All thesefeatures can also be found in Fig. 8b, which reveals the

Page 10 of 14

variation of density along various isobars. While thosefor quartz-saturated aqueous fluids reveal minor disparity, which explains the resulted close agreements
between the two models as for the densities of aqueous
fluids, the volume of saturated hydrous silicate melts
show noticeable deviations especially under higher
pressures. The HM2012 model predicts a more rapid
decline in the volume of hydrous silicate melts withpressure, thus compensating for the decreased mass
with less silica in the fluid at higher pressures (Fig. 8c,d). The two-step model proposed in this study, on theother hand, predicts a less perturbed volume and thedensity of the melts is therefore decreased at higher
pressures. In summary, experimental density data forsilicate solutions are urgently needed to clarify thedeviations among the existing thermodynamic models.

Fig. 8 Comparison of the density, volume and solubility of quartz-saturated S­ iO2–H2O fluids predicted by the two-step model and the HM2012
model. a At isothermal and b isobaric the density of quartz-saturated fluids predicted by the two models; c solubility and d volumeof quartz-saturated fluids under isothermal conditions predicted by the two models (solid and dashed lines represent the two-step modeland the HM2012 model, respectively, and the lighter solid line in (b) represents the density of the coexisting VLE fluid)

Gao et al. Progress in Earth and Planetary Science(2024) 11:20

4.2 Thermodynamics of water exsolution from the risingmagma

The exsolution of hydrothermal fluids from magma controls the many aspects of eruption behavior of volcanoes. In the deep Earth, the immiscible transition state(transition process) fluid may cause deep source earthquakes (Wech et al. 2020). This fluid also is an important

Page 11 of 14

medium for the formation of many metallic mineraldeposits (Hedenquist and Lowenstern 1994). However,
the mechanism of magma differentiation of immiscible fluids (often named as “boiling”) is still unclear. This
process occurred in the Earth’s interior is often classified
into two types (Fig. 9a): (1) The pressure in the magma
chamber is suddenly reduced due to tectonic collapse or

Fig. 9 First and second boiling processes with fluid water exsolution from the rising magma. a Schematic geological settings. b Thermodynamic
paths in the 3D phase diagram for the boiling processes. The green surface represents VLE, the black grid represents SLE. The dark red, light red
and gray lines (group marked with ①) indicate different initial conditions for the first boiling process (dark red line: T = 1389 K; xs = 0.5; light red line:
T = 1370 K; xs = 0.5; grey line: T = 1389 K, xs = 0.45). The purple line marked with ② represents an illustrative path for the second boiling process. In(c), (d) and (e), we show the density changes of the fluid evolving along the four paths in (b). M: unsaturated melts before SLE and VLE; L: hydrous
silicate melts; V: aqueous fluids

Gao et al. Progress in Earth and Planetary Science(2024) 11:20

the rapid rising of magma, so that the unsaturated fluidat depth quickly decreases the solubility of solute during
the ascending path and causes the exsolution. This typeof process is called “first boiling”, which involves essentially invariant temperature. (2) The unsaturated magma
is temporarily stored at a certain depth and thus the pressure keeps unchanged. As the temperature decreases, themelt gradually crystallizes and the water concentration in
the remaining melt gradually increases, eventually reaching the saturation of water and causing exsolution. This
type of process is known as the “second boiling” (Candela 1997; Heinrich 2007). Considering that the S­ iO2–
H2O system is too basic compared to the real complex
silicate-water system, we only try to discuss the changes
in the properties of the fluids due to the first and second
boiling.
The schematic thermodynamic paths for the first boiling process is illustrated in Fig. 9b (dark red, light red andgray lines marked as ①, representing three different initial conditions as discussed below) and the second boiling process (purple line marked as ②).
For a fluid-unsaturated melt (dark red line in Fig. 9b),
its density would gradually decrease as the pressure
decreases before the boiling process begins (blue solidline in Fig. 9c), when the melt reaches the fluid saturation and begins to exsolve aqueous fluid (Fig. 9b as thepath touches the VLE surface). With further pressure
decrease, more dilute aqueous fluid is exsolved and theremaining melt becomes more concentrated in ­SiO2. This
process results in decreasing density for the exsolved
aqueous fluid and increasing density for the residual melt
respectively (green and orange curves in Fig. 9c). Whenthe pressure is further reduced to the hydrous melting
curve, the quartz begins to crystallize and the first boilingstops.
Magma is usually thought to be stored at its neutral buoyancy level, where the density of the magma isequivalent to that of the host rock (Ryan 1987). However, another hypothesis suggests that the magma storage depth is influenced by the water content (Annen et al.
2006; Zellmer et al. 2016). As shown in Fig. 9b, when theinitial magma at the same temperature and pressure hasa higher water content (the gray line in Fig. 9b), it willreach the VLE surface and start the boiling process at
higher pressure (Fig. 9c). Therefore, wetter magma would
boil more readily than dry magma at depth. Rasmussen et al, (2022) found magma storage depths consistent
with predicted water degassing depths. Although the wet
magma contains higher water content and has a lower
density, the degassing process leads to an increase in viscosity, allowing it to stay deeper during ascent.
When the magma of the same composition and depth
initially is slightly cooler (the light red line in Fig. 9b), thePage 12 of 14

boiling process will be started at a higher pressure. In
addition, as more water enters the aqueous fluid, resulting in a larger density difference between the exsolved
aqueous fluid and the residual melt, this would drive theboiling process (Fig. 9d). With further decreasing of thepressure till the remaining magma reaches the hydrous
melting curve, the boiling process would stop more
quickly. Therefore, the rising magma with lower initial
temperature would also finish the boiling process at a
deeper location.
The purple line in Fig. 9b shows the second boilingprocess. The temporarily stored magma becomes denser
as the temperature gradually decreases (blue curve in
Fig. 9e). When the temperature reaches the liquidus
curve (quartz-saturated), the melt begins to crystallize.
As the temperature decreases further, more and more
solids would crystallize and the water content in theresidual melt increases gradually. This process results indecreasing of density (orange curve in Fig. 9e). When thetemperature decreases to the hydrous melting curve, thesecond boiling starts until most of the ­SiO2 in the fluid
has crystallized (grey curve in Fig. 9e). Obviously, when
the fluid becomes saturated during this process, thevariation of its density should be similar to the trend as
revealed in Fig. 9b.

5 Conclusions and remarks
The ­SiO2–H2O binary system is the basis for understanding complex silicate-water systems, but relevanthigh temperature and pressure and density data remains
scarce. This study considers the vapor–liquid equilibrium
region using the NRTL model as the basis and modifies itwith aggregated reactions to establish a two-step modelthat can simultaneously predict SLE and VLE. The twostep model is fairly accurate and applicable from 773 K to
the anhydrous quartz melting temperature and from 0.5
GPa to at least 2 GPa. Based on the model proposed inthis study, we provide a quantitative 3D phase diagram of­SiO2–H2O, which greatly helps our understanding of thecomplex phase relations of this binary across the criticalconditions. The upper critical endpoint is predicted to bearound 1.14 ± 0.18 GPa and 1344 ± 87 K.
Because the two-step model quantifies the Gibbs freeenergy of the S­iO2–H2O fluid under different conditions, we observe nontrivial different trends of density
along various P–T paths. The temperature and pressure
are vital for the density through indirectly regulating the­SiO2 dissolved in the fluid. By further considering thecomplex VLE, such as the boiling processes when thevolatile exsolution occurs in the rising magma, we finddistinctive thermodynamics would be involved along different paths.

Gao et al. Progress in Earth and Planetary Science(2024) 11:20

Finally, it should be noted that thermodynamic models inevitably rely on the experimental data. For the­SiO2–H2O fluids concerned in this study, experimental
determination of VLE and density of ­SiO2–H2O fluid
would be needed in the future.
Abbreviations
NRTL	Non-random two-liquidV	Aqueous fluids
L	Hydrous silicate melts
S	Solid (quartz)
SLE	Solid–liquid equilibrium
VLE	Vapor–liquid equilibrium
SCF	Supercritical fluids
UCEP	Upper critical endpoint
AAD	Averaged absolute deviations
AARD	Averaged relative deviations

Supplementary Information
The online version contains supplementary material available at https://​doi.​
org/​10.​1186/​s40645-​024-​00622-1.
Additional file 1: Appendix A. Introduction to the NRTL model. Appen‑
dix B. Workflow and derivation of the two-step model.
Acknowledgements
We are grateful to Prof. Mysen for his invitation of this paper as a contribution to the special collection of PEPS. We appreciate the kind help from Prof.
Manning by sharing his well-coded Excel spreadsheet for implementing theHM2012 model. We thank the reviewers for their constructive comments. This
work benefits from the helpful discussions with Mr. Zhiming Zhang and Mr.
Xiaoguang Niu.
Author contributions
ZGZ proposed the topic, conceived and designed the study. HG constructed
the model and was a major contributor in writing the manuscript. All authors
read and approved the final manuscript.
Funding
This study was supported by the National Key R&D Program of China(2018YFA0702700) and the key research program of IGGCAS (IGGCAS-202204).
Availability of data and material
The data used in this study is available on request.

Declarations
Competing interests
We declare that we have no competing interests.
Received: 22 December 2023 Accepted: 29 March 2024

References
Anderson GM, Burnham CW (1965) The solubility of quartz in supercriticalwater. Am J Sci 263:494–511
Anderson GM, Burnham CW (1967) Reactions of quartz and corundum withaqueous chloride and hydroxide solutions at high temperatures andpressures. Am J Sci 265:12–27
Annen C, Blundy JD, Sparks RSJ (2006) The genesis of intermediate andsilicic magmas in deep crustal hot zones. J Petrol 47:505–539

Page 13 of 14

Berman RG (1988) Internally consistent thermodynamic data for minerals inthe system N­ a2O-K2O-CaO-MgO-FeO-Fe2O3-Al2O3-SiO2-TiO2-H2O-CO2. J
Petrol 29:445–522
Boettcher AL, Wyllie PJ (1969) Phase relationships in system ­NaAlSiO4-SiO2-H2O
to 35 kilobars pressure. Am J Sci 267:875–909
Candela PA (1997) A review of shallow, ore-related granites: textures, volatiles,
and ore metals. J Petrol 38:1619–1633
Crerar DA, Anderson GM (1971) Solubility and solvation reactions of quartz indilute hydrothermal solutions. Chem Geol 8:107–122
Deng WP, Wei Q, Liu X (2021) A new density model of quartz solubility in­H2O-CO2-NaCl ternary systems up to high temperatures and high pressures. Geofluids 2021(4):1–23
Dolejs D, Manning CE (2010) Thermodynamic model for mineral solubility inaqueous fluids: theory, calibration and application to model fluid-flow
systems. Geofluids 10:20–40
Doltsinis NL, Burchard M, Maresch WV, Boese AD, Fockenberg T (2007) Ab initio
molecular dynamics study of dissolved S­ iO2 in supercritical water. J Theor
Comput Chem 6:49–62
Fournier RO, Potter RW (1982) An equation correlating the solubility of quartz
in water from 25° to 900°C at pressures up to 10,000 bars. Geochim
Cosmochim Acta 46:1969–1973
Gerya TV, Maresch WV, Burchard M, Zakhartchouk V, Doltsinis NL, Fockenberg
T (2005) Thermodynamic modeling of solubility and speciation of silicain ­H2O-SiO2 fluid up to 1300°C and 20 kbar based on the chain reaction
formalism. Eur J Mineral 17:269–283
Gunnarsson I, Arnorsson S (2000) Amorphous silica solubility and the thermodynamic properties of ­H4SiO4 degrees in the range of 0 °C to 350°C at
P-sat. Geochim Cosmochim Acta 64:2295–2307
Hack AC, Hermann J, Mavrogenes JA (2007a) Mineral solubility and hydrous
melting relations in the deep earth: analysis of some binary A-H2O systempressure-temperature-composition topologies. Am J Sci 307:833–855
Hack AC, Thompson AB, Aerts M (2007b) Phase relations involving hydrous
silicate melts, aqueous fluids, and minerals. Rev Mineralal Geochem
65:129–185
Hedenquist JW, Lowenstern JB (1994) The role of magmas in the formation ofhydrothermal ore-deposits. Nature 370:519–527
Heinrich CA (2007) Fluid-fluid interactions in magmatic-hydrothermal ore
formation, Fluid-Fluid Interactions, 363–387
Hemley JJ, Montoya JW, Marinenko JW, Luce RW (1980) Equilibria in thesystem ­Al2O3-SiO2-H2O and some general implications for alteration/mineralisation processes. Econ Geol 75:210–228
Holtz F, Roux J, Behrens H, Pichavant M (2000) Water solubility in silica andquartzofeldspathic melts. Am Mineral 85:682–686
Hunt JD, Manning CE (2012) A thermodynamic model for the system S­ iO2-H2O
near the upper critical end point based on quartz solubility experiments at 500–1100 degrees C and 5–20 kbar. Geochim Cosmochim Acta
86:196–213
Kennedy GC (1950) A portion of the system silica-water. Econ Geol 45:629–653
Kennedy GC, Wasserburg GJ, Heard HC, Newton RC (1962) The upper threephase region in the system ­SiO2–H2O. Am J Sci 260:501–521
Levine IN (2008) Physical chemistry, 6th edn. McGraw-Hill Education, New YorkMa ZT (2001) Thermodynamic description for concentrated metallic solutions
using interaction parameters. Metall Mater Trans B Process Metall Mater
Process Sci 32:87–103
Mackenzi FT, Gees R (1971) Quartz: synthesis at earth-surface conditions. Science 173:533–535
Manning CE (1994) The solubility of quartz in ­H2O in the lower crust and upper
mantle. Geochim Cosmochim Acta 58:4831–4839
Manning CE (2004) The chemistry of subduction-zone fluids. Earth Planet Sci
Lett 223:1–16
Manning CE, Boettcher SL (1994) Rapid-quench hydrothermal experiments at
mantle pressures and temperatures. Am Mineral 79:1153–1158
Mao HH, Sundman B, Wang ZW, Saxena SK (2001) Volumetric properties andphase relations of silica—thermodynamic assessment. J Alloy Compd
327:253–262
Mibe K, Fujii T, Yasuda A (2002) Composition of aqueous fluid coexisting withmantle minerals at high pressure and its bearing on the differentiation of
the Earth’s mantle. Geochim Cosmochim Acta 66:2273–2285
Morey GW, Fournier RO, Rowe JJ (1962) The solubility of quartz in water inthe temperature interval from 25° to 300° C. Geochim Cosmochim Acta
26:1029–1043

Gao et al. Progress in Earth and Planetary Science(2024) 11:20

Mysen BO (1998) Interaction between aqueous fluid and silicate melt in thepressure and temperatures regime of the Earth’s crust and upper mantle.
Neues Jahrb Mineral Abh 172:227–244
Mysen BO (2010) Speciation and mixing behavior of silica-saturated aqueous
fluid at high temperature and pressure. Am Mineral 95:1807–1816
Nakamura Y (1974) The system S­ iO2-H2O-H2 at 15 kbar. Carnegie Institution of
Washington Year Book 73:259–263
Newton RC, Manning CE (2000) Quartz solubility in ­H2O-NaCl and H­ 2O-CO2
solutions at deep crust-upper mantle pressures and temperatures: 2–15
kbar and 500–900 °C. Geochim Cosmochim Acta 64:2993–3005
Newton RC, Manning CE (2008) Thermodynamics of ­SiO2-H2O fluid near theupper critical end point from quartz solubility measurements at 10 kbar.
Earth Planet Lett 274:241–249
Ni HW, Zhang L, Xiong XL, Mao Z, Wang JY (2017) Supercritical fluids at
subduction zones: evidence, formation condition, and physicochemical
properties. Earth Sci Rev 167:62–71
Novgorodov PG (1977) Solubility of quartz in mixture in of ­H2O-CO2
and ­H2O-NaCl and at 700°C and at pressure of 1.5 Kbar. Geokhimiy
8:1270–1273
Plyasunov AV, Zyubina TS (2021) Thermodynamic modeling of the solubilityof quartz in water up to high temperatures and pressures. Earth Space
Chem 5:941–959
Prausnitz JM, Lichtenthaler RN, de Azevedo EG (1998) Molecular thermodynamics of fluid-phase equilibria. 2nd Edn. Prentice-Hall, Upper Saddle
River, NJ, 1986
Ragnarsdottir KV, Walther JV (1983) Pressure sensitive “silica geothermometer”
determined from quartz solubility experiments at 250 °C. Geochim
Cosmochim Acta 47:941–946
Rasmussen DJ, Plank TA, Roman DC, Zimmer MM (2022) Magmatic watercontent controls the pre-eruptive depth of arc magmas. Science
375:1169–1172
Renon H, Prausnitz JM (1968) Local compositions in thermodynamic excess
functions for liquid mixtures. Aiche J 14:135–144
Rimstidt JD (1997) Quartz solubility at low temperatures. Geochim Cosmochim Acta 61:2553–2558
Ryan MP (1987) Neutral buoyancy and the mechanical evolution of magmatic
systems. The Geochemical Society, Special Publication No. 1, 259–287
Shi XL, Mao SD, Hu JW, Zhang J, Zheng JX (2019) An accurate model for thesolubilities of quartz in aqueous NaCl and/or C­ O2 solutions at temperatures up to 1273 K and pressures up to 20,000 bar. Chem Geol 513:73–87
Shmulovich K, Graham C, Yardley B (2001) Quartz, albite and diopside solubilities in ­H2O-NaCl and H­ 2O-CO2 fluids at 0.5-0.9 GPa. Contrib Mineral Petrol
141:95–108
Shmulovich KI et al (2006) Solubility of quartz in crustal fluids: experiments
and general equations for salt solutions and H­ 2O-CO2 mixtures at
400–800 °C and 0.1-0.9 GPa. Geofluids 6:154–167
Siever R (1962) Silica solubility, 0°C-200°C, and the diagenesis of siliceous sediments. J Geol 70:127–150
Sommerfeld RA (1967) Quartz solution reaction: 400°C–500°C, 1000 bars. J
Geophys Res 72:4253
Sowerby JR, Keppler H (2002) The effect of fluorine, boron and excess sodium
on the critical curve in the albite-H2O system. Contrib Mineral Petrol
143:32–37
Stewart DB (1967) Four-phase curve in the system ­CaAl2Si2O8-SiO2-H2O
between 1 and 10 kilobars. Schweiz Mineral Petrogr Mitt 47:35–39
Sverjensky DA, Harriaon B, Azzolini D (2014) Water in the deep Earth: thedielectric constant and the solubilities of quartz and corundum to 60 kb
and 1200°C. Geochim Cosmochim Acta 129:125–145
Thomas R (2000) Determination of water contents of granite melt inclusions by confocal laser Raman microprobe spectroscopy. Am Mineral
85:868–872
Thomas R, Davidson P (2016) Revisiting complete miscibility between silicate
melts and hydrous fluids, and the extreme enrichment of some elementsin the supercritical state—consequences for the formation of pegmatites
and ore deposits. Ore Geol Rev 72:1088–1101
Tian M, Katz RF, ReeJones DW (2019) Devolatilization of Subducting Slabs, Part
I: Thermodynamic parameterization and open system effects. Geochem
Geophys Geosyst 20:5667–5690
Vanlier JA, Debruyn PL, Overbeek JTHG (1960) The solubility quartz. J Phys
Chem 64:1675–1682

Page 14 of 14

Walther JV, Orville PM (1983) The extraction quench technique for determination of the thermodynamic properties of solute complexes: application
to quartz solubility in fluid mixtures. Am Mineral 68:731–741
Wang H, Yang K (2023) Bayesian optimization. In: Brockhoff D, Emmerich M,
Naujoks B, Purshouse R (eds) Many-criteria optimization and decision
analysis: state-of-the-art, present challenges, and future perspectives.
Springer International Publishing, Cham, 271–297
Wang HM, Henderson G, Brenan J (2004) Measuring quartz solubility by in situ
weight-loss determination using a hydrothermal diamond cell. Geochim
Cosmochim Acta 68:5197–5204
Watson EB, Wark DA (1997) Diffusion of dissolved ­SiO2 in ­H2O at 1 GPa, with
implications for mass transport in the crust and upper mantle. Contrib
Mineral Petrol 130:66–80
Wech AG, Thelen WA, Thomas HM (2020) Deep long-period earthquakes
generated by second boiling beneath Mauna Kea volcano. Science
368:775–779
Wei Q, Duan ZH, Mao SD (2012) A thermodynamic model of quartz solubilityin ­H2O-CO2-NaCl systems up to 1000 °C and 1.5GPa. Acta Petrol Sin
28:2656–2666
Weill DF, Fyfe WS (1964) The solubility of quartz in ­H2O in the range 1000–4000
bars and 400–550°C. Geochim Cosmochim Acta 28:1243–1255
Wilson GM (1964) Vapor-liquid equilibrium. X I. A new expression for theexcess free energy of mixing. J Am Chem Soc 86:127–130
Xie ZX, Walther JV (1993) Quartz solubilities in NaCl solutions with and without
wollastonite at elevated temperatures and pressures. Geochim Cosmochim Acta 57:1947–1955
Zellmer GF, Pistone M, Iizuka Y, Anderws BJ, Gómez-Tuena A, Straub SM,
Cottrell E (2016) Petrogenesis of antecryst-bearing arc basalts from theTrans-Mexican Volcanic Belt: insights into along-arc variations in magmamush ponding depths, ­H2O contents, and surface heat flux. Am Mineral
101:2405–2422
Zhang ZG, Duan ZH (2005) Prediction of the PVT properties of water over wide
range of temperatures and pressures from molecular dynamics simulation. Phys Earth Planet Inter 149:335–354
Zotov N, Keppler H (2000) In-situ Raman spectra of dissolved silica species inaqueous fluids to 900 °C and 14 kbar. Am Mineral 85:600–603
Zotov N, Keppler H (2002) Silica speciation in aqueous fluids at high pressuresand high temperatures. Chem Geol 184:71–82

Publisher’s NoteSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.