Model evaporation of FeO-bearing liquids: Application to chondrules

Geochimica et Cosmochimica Acta, Vol. 69, No. 12, pp. 3183–3193, 2005 Copyright © 2005 Elsevier Ltd Printed in the USA. All rights reserved 0016-7037/05 $30.00 ⫹ .00

doi:10.1016/j.gca.2005.02.008

Model evaporation of FeO-bearing liquids: Application to chondrules DENTON S. EBEL* Dept. of Earth and Planetary Sciences, American Museum of Natural History, Central Park West at 79th St., New York, NY 10024, USA (Received March 29, 2004; accepted in revised form February 10, 2005)

Abstract—Models for thermodynamic behavior of FeO-bearing liquids are required for understanding the separate roles of evaporation, condensation and crystallization in the formation of free-floating silicate liquid droplets in the early solar nebula. These droplets, frozen as chondrules, are common in chondritic meteorites. Evaporation coefficients for Fe and FeO of ⬃0.2 are calculated here from existing data using silicate liquid activity models. These models, used to describe gas-liquid-solid equilibria and to constrain kinetic processes, are compared and found similar, and the effects of liquid non-ideality are assessed. A general approach is presented for predicting the evaporation behavior of FeO-bearing Al2O3-CaO-SiO2-MgO liquids in H2-rich gas above 1400 K at low total pressure. Results are vapor pressure curves for Fe, FeO and other gas species above typical chondrule liquids, suitable for predicting compositional trajectories of residual liquids evaporating in a hydrogen-dominated vapor. These predictions are consistent with chondrule formation in the protoplanetary disk in heating events of short duration, such as those expected from shock wave or current sheet models. Copyright © 2005 Elsevier Ltd MELTS or CMAS liquid model, however, these processes differ in their sensitivity to activity models. Here, the two models and the effects of non-ideal parameters are compared, and a method is presented for calculating saturation vapor pressures above liquids for which the MELTS model is more suitable, such as FeO-rich type II chondrule liquids. The results of evaporation from MELTS model liquids are compared with results obtained using the method (Ebel et al., 2000a) and the parameters (Richter et al., 2002) used previously to calculate evaporative flux from CMAS liquids using the Berman (1983) model (Grossman et al., 2000). Evaporation coefficients for Fe evaporation are derived from the sparse data available. Equilibrium partial pressures of CMAS and Fe-bearing gaseous species above various chondrule liquids in pure H2 are calculated using the MELTS model, for ranges of P(H2) and temperature. These partial pressures must be used with caution in considering evaporation phenomena, due to the poor constraints on evaporation coefficients, which should stimulate vigorous experimental work in the near future.

1. INTRODUCTION

Of the Ca-, Al-rich inclusions (CAIs) and chondrules found in carbonaceous and ordinary chondrites, only a subset of the CAIs exhibits enrichment in the heavy isotopes of Si and Mg (Clayton et al., 1988). Other elements, such as K (Humayun and Clayton, 1995; Alexander et al., 2000), Fe (Alexander and Wang, 2001; Zhu et al., 2001; Kehm et al., 2003; Mullane et al., 2003), Mg (Galy et al., 2000; Nguyen et al., 2000), and Si (Clayton et al., 1991) show no such enrichment in chondrules. There have been many attempts to address this evidence by modeling the evaporation of presumed molten chondrule and CAI precursors (e.g., Nagahara and Ozawa, 2000; Alexander, 2001; Grossman et al., 2000, 2002). Here, a method is presented to address FeO-bearing liquids, which is consistent with the classical formalism and with the parameterization of Richter et al. (2002), extended to larger chemical systems. At present, there exist exactly two well-tested models describing the thermodynamic activities of oxide components in silicate liquids, suitable for modeling crystal-liquid equilibria in magmatic systems. Both use a classical non-ideal thermodynamic formalism to describe the chemical potential energy of silicate liquids, and are calibrated against internally consistent data sets describing the thermodynamic behavior of coexisting mineral phases. One, the Berman (1983) model for CaO-MgOAl2O3-SiO2 (CMAS) liquids, does not address TiO2 or FeO. The other, the MELTS model of Ghiorso and Sack (1995), includes components containing these simple oxides, but because of the choice of stoichiometric end-member components, it cannot address CMAS⫹FeO liquids where molar SiO2 is less than (½ MgO ⫹ ½ FeO ⫹ CaO). In the absence of a universal, tested liquid model, one must use either one model or the other, depending upon the composition region of interest. The outcomes of equilibrium gas-liquid condensation, or kinetic evaporation of liquids can be predicted using the

2. EVAPORATION CALCULATION

The instantaneous evaporative flux of element i from a molten liquid surface (Ji, mol cm⫺2 sec⫺1) can be expressed using the Hertz-Knudsen equation in SI units (Hirth and Pound, 1963) Ji ⫽

n

␣ j␯ij P j,sat

j⫽1

兹2 ⫻ 10⫺5␲m jRT

兺

,

(1)

as the summation over n gaseous species j of molecular weight mj (g/mol), containing vij mols i per mol of j, with partial pressures Pj,sat (bar) at saturation, where ␣j is the evaporation coefficient of species j, R is the gas constant (8.314 Jmol⫺1K⫺1), and T the absolute temperature (K). Eqn. 1 describes free evaporation where the partial pressure of j in the ambient gas far from the evaporating surface is set to zero, as would be true for a molten droplet evaporating

* Author to whom correspondence should be addressed (debel@ amnh.org). 3183

3184

D. S. Ebel

into an infinite H2 reservoir, or for experiments run at very low total pressure. A prerequisite in applying the Hertz-Knudsen equation is to determine the partial pressures of the gaseous species in a vapor with which a liquid of fixed composition is in chemical equilibrium. That is, the composition of the vapor is buffered by that of the liquid for all elements present in the liquid. Ebel et al. (2000a) described a method for calculation of partial pressures of gaseous species over a CMAS liquid in restricted circumstances, namely, at either an imposed total pressure (Ptot) or at an imposed P(H2). In the latter case, hydrogen pressure is assumed to be buffered by the vapor, and all other elements are buffered by the liquid. This method applies only to Ca-Mg-Al-Si-O-H systems. A more general method is presented here, suitable for calculation of partial pressures over any silicate liquid for which activities may be calculated, in any vapor. Although, for example, SiO is the dominant Si-bearing gaseous species at high temperature in an oxidizing vapor, it is sufficient to write only one evaporation reaction for each thermodynamic component in the liquid, so long as speciation among all possible gaseous species is accounted for. It is convenient, also, to involve only the monatomic gaseous species. For example, for the SiO2 component of either liquid model, a simple oxide: SiO2(l) ⫽ Si(g) ⫹ 2 · O(g)

(2a)

and for the MgO-bearing component of the MELTS liquid model, a complex oxide: Mg2SiO4(l) ⫽ 2 · Mg(g) ⫹ Si(g) ⫹ 4 · O(g)

(2b)

Such reactions may be added and subtracted, such that for a given partial pressure of monatomic oxygen, P(O), a single evaporation reaction defines the partial pressure of each liquidforming cation in the gas phase. For each such reaction, known standard state Gibbs energies may be combined in the usual way to obtain equilibrium constants written in terms of the activities of liquid components and partial pressures of elements. Given P(O), each resulting equation defines the partial pressure of a cationic element (e.g., Si, Mg, Fe) above the liquid. With these partial pressures in hand, the partial pressures of all other gaseous species may be calculated at once, from standard thermodynamic data, and the value of P(O) can then be adjusted by iteration to suit the imposed conditions. If all oxygen in the gas is assumed to result from evaporation of the liquid, then for each Si evaporated into the gas, two O must come from the liquid, and similarly for cations Mg, Ca, etc. Then the sum of the oxygen present in all n gas species can be set equal to the sum of all oxygen contributed by each of m evaporated simple oxide components of the liquid, which have contributed elements to the gas: n

兺v j⫽1

O, j

Pj ⫽

m

兺 q⫽1

冉 冊兺 vO,oxide q

n

vk,oxide q

j⫽1

␯kj P j

(3)

where the partial pressures Pj are summed over n gas species containing oxygen, on the left, and, on the right, over all gaseous species containing cation k, for each of m simple oxides q containing cations k, where each cation k occurs in

only one simple oxide component. Here, expressions vab represent the stoichiometric coefficient of element a in a formula unit of b, so for Al2O3 the term in parentheses in Eqn. 3 is 3/2. Saturation vapor pressures obtained this way are used in Eqn. 1, with experimentally determined coefficients ␣, to model evaporation of molten liquid droplets into H2 gas, where evaporated molecules are assumed instantaneously removed from the ambient gas. Suppose, however, that one desires to consider other elements in the vapor. There are several classes of elements in this problem. First, inert gaseous elements, which influence only the total pressure. Second, elements which combine with elements present in the liquid to make gas species, but are not themselves present in the liquid, such as Cl, S, C. For each such element, a partial pressure must be defined (buffered) by the gas. When considering evaporation into a speciated gas of, for example, near-solar elemental composition, it is convenient to define these partial pressures in relation to a single element. For example, one can constrain the ratio P(S)/P(H2) to remain at its predicted value in a gas of solar composition, calculated with or in the absence of condensates, at the T and Ptot of interest. Alternatively, the partial pressures of monatomic gas species imposed by the external gas can be calculated at a different temperature than that of the liquid. This would approximate disequilibrium such as may occur between distal gas and melt droplets in a current sheet or shock heating scenario. It is the elements in a third class which are problematic. These are oxygen, and the cationic elements Si, Mg, etc., present in, and buffered by, the liquid. One might postulate that P(Si), P(Mg), etc. are buffered by the silicate liquid, but that the P(O2) imposed by surrounding vapor exceeds the P(O2) predicted for equilibrium between the liquid and hydrogen gas, so Eqn. 3 becomes superfluous. Yet it is desirable to obtain saturation vapor pressures for modeling evaporation where oxygen is initially present in the vapor, but does not buffer the vapor pressure of the melt. One way to do this is to subtract some “initial oxygen” (PO,ini) from the left hand side of Eqn. 3, in either a fixed quantity or in some ratio dependent upon another gaseous species, such as by fixing a constant ratio PO,ini/ P(H2). Similarly, suppose we wish to model evaporation of a CAI liquid into an otherwise solar gas from which 90% of the Mg and only 45% of the Si have been removed. This can be accomplished by subtraction of some PMg,ini and PSiO,ini from the right hand side of Eqn. 3, and subtraction of appropriate PO,ini on the left. These initial P may be set by fixing ratios PMg,ini/P(H2) and PSiO,ini/P(H2), taken from a calculation of equilibrium vapor pressures in a gas of solar composition from which the requisite Mg and SiO have been subtracted, at the appropriate temperature and Ptot. This approach is facilitated by the fact that the budgets of the cationic elements of interest (Mg, Si) are dominated by a single gas species, but it is easily generalized. In the discussion below, only evaporation into pure H2 will be considered. Nagahara and Ozawa (1996) and Tsuchiyama et al. (1999) showed that P(H2) is the most important variable for many evaporation calculations of interest. This should hold equally in the evaporation of FeO-bearing liquids. It is assumed throughout that Fe⫹3 is negligible in the melt.

Evaporation of FeO-bearing liquids

a

-5

a(SiO2) Berman83

0.25

SiO2, MELTS

P(SiO) MELTS

lower limit of MELTS composition

Al2O3, Berman83 SiO2, Berman83

Al2O3, MELTS

28

initial Al2O3

0

2

4

6

8

10

0.20

5x10

-6

0.15

0.10

0.05

0.0

0.00 0

0.00001

5

10

evaporation step

15

20

SiO

(Richter et al. 2002)

P(H2)=10

-5

12

MgO (wt%) in Residual Liquid

B133 liquid

1x10

activity of SiO2

P(SiO) Berman83

38

18

c

1.5x10

a(SiO2) MELTS

partial pressure of SiO

Oxide in Liquid (wt%)

b 0.30

initial SiO2

gMg/gSiO = 1 2073 K

3185

-10

O2

bar

partial pressure (bar)

0.000001

Mg

0.0000001

Berman83 (CMAS) model dotted MELTS model solid

0.00000001 1773

1873

1973

2073

Temperature (K)

Fig. 1. (a) Evaporation of composition B-133 of Richter et al. (2002) at 2073 K, P(H2) ⫽ 10⫺10 bar. Experimental residual liquid compositions are filled squares. Calculated trajectories use models of Berman (1983, open symbols), and MELTS (filled symbols), with liquid losing 1% of its initial MgO in each step from right to left with increasing time. (b) Partial pressures and activities of SiO2 calculated using both models, for liquids on the first 20 of 100 steps of the evaporation trajectory of (a), calculated using the Berman (1983) model. (c) Partial pressures over initial B133 liquid at various temperatures, calculated using both the Berman (1983) model (open symbols) and the MELTS (filled symbols) model (Ghiorso and Sack, 1995).

3. EVAPORATION COEFFICIENTS

3.1. CMAS Components in the MELTS Liquid Model Grossman et al. (2000) and Richter et al. (2002) calculated evaporation coefficients suitable for modeling the sequential compositional changes of residual CMAS liquids produced by evaporation at high temperature. The latter observed that without the use of the solution model of Berman (1983), “all the deficiencies of the activity-composition relationships will complicate the dependence of the evaporation coefficients on composition.” A striking example of this model’s effectiveness is illustrated in Figure 1a, drawn in part from Richter et al. (2002, their Fig. 2). Here, a CAI-like composition (B133) is evaporated at 2073 K, P(H2) ⫽ 10⫺10 bar, using the method described above, with evaporation coefficients ␣SiO ⫽ ␣Mg, and the CMAS liquid model of Berman (1983). The evaporated portion of the liquid is considered to be infinitely diluted into the H2 vapor, with no reservoir effects. Richter et al. (2002) drew upon a substantial database for evaporation of SiO and Mg from Fe-free liquids, and concluded that the Berman model gave “a good representation of the relative activities of SiO2 and MgO” in their sample liquids. Also shown in Figure 1a is a very different trajectory calcu-

lated by substituting the MELTS liquid model of Ghiorso and Sack (1995) for that of Berman (1983). Although the B133 data set contains one point in a composition region addressable by both liquid models, the difficulty in applying the MELTS model is apparent from the dashed line in Figure 1a, which defines the limit below which that model cannot address silicate liquids with the Al2O3/CaO ratio of B133 (see also Fig. 2). This limit defines an asymptote for the calculation of evaporation using the MELTS model, because the calculated P(SiO) approaches zero at this limit (Fig. 1b). The single experimental time-step on the B133 trajectory to which MELTS can be applied yields ␣Mg ⫽ 0.09, ␣SiO ⫽ 0.14, in close agreement with the evaporation coefficients 0.10 and 0.11 calculated by Richter et al. (2002, their Table 1, B-133-19) for the same point using the Berman (1983) model. In Figure 1c are shown the partial pressures of Mg, SiO, and O2 of vapor in equilibrium with the initial B133 liquid as functions of temperature, calculated using both models. The pressures of other Si- and Mgbearing species are more than an order of magnitude lower than P(SiO) and P(Mg), respectively, and P(Al) and P(Ca) are of order 10⫺12 and 10⫺10 bar, respectively. From this limited data it can be concluded that the MELTS model provides an equally good representation for liquid activities in CMAS melts as does

3186

D. S. Ebel

average values over the composition range of each experimental interval. Evaporation coefficients for Mg and SiO calculated in this way for B133 data are within 17 and 61%, respectively, of those reported by Richter et al. (2002, Table 1) for the first four time-steps. Table 1 is also available as electronic annex EA-1. In the cases of Mg and Si, there is a single dominant gas species. Both Fe and FeO gas species, however, have significant calculated saturation vapor pressures, although PFe is almost an order of magnitude larger than PFeO for most of the experiments (Table 1). This is consistent with recent results by Dauphas et al. (2004). There is no experimental evidence or strong physical argument for apportioning the total Fe flux tot (JFe ) between these two species. If it is assumed that flux is proportional to each species’ share of (PFe⫹PFeO), then the flux of Fe is: tot JFe ⫽ JFeO

Fig. 2. Experimental residual liquid paths during evaporation, for FeO-free B133 series (open diamonds) of Richter et al. (2002), FeObearing SC-1800 series (squares) of Wang et al. (2001), and FeObearing 1700, 1800, and 1900 C series (open triangles) of Hashimoto (1983). Dashed (Richter et al. and Wang et al. data) and dash-dot (Hashimoto data) lines show limits below which MELTS has too little SiO2, for the Al2O3/CaO fixed by the initial experimental compositions. The * are initial compositions of experiments, arrows indicate trajectories dominated by evaporative loss of MgO and SiO2 (open diamonds and squares), and of FeO (open triangles and filled squares).

PFe,sat

共PFe,sat ⫹ PFeO,sat兲

PFe,sat PFeO,sat ⫽

␣FePFe,sat

兹2 ⫻ 10⫺5␲mFeRT

(5)

with an analogous expression for JFeO. Combining these expressions and canceling yields a relation between the evaporation coefficients,

␣Fe ␣FeO

the Berman (1983) model, but only at some remove from the boundaries of its accessible composition region.

⫽ JFeO

⫽

冑

mFe mFeO

⫽ 0.8817.

(6)

Alternatively, the simpler assumption that ␣Fe ⫽ ␣FeO results in a mass-weighted relation between the total Fe flux and that of each species:

3.2. FeO-bearing Liquids The experiments of Hashimoto (1983), and Wang et al. (2001), yield the best available data to constrain the evaporation coefficients for Fe(g) and FeO(g) species using the MELTS model (Alexander, 2002). Figure 2 shows where all these experiments fall relative to the accessible MELTS composition space, and the same data for the B133 series. Richter et al. (2002, their Fig. 2) calculated evaporation coefficients for their B133 series using “the average saturation vapor pressure over the duration of each experiment calculated along appropriate segments of compositional trajectories” matching those of the experimental residua. The flux data are average values integrated over time intervals of the experiments. Here, partial pressures are calculated using the method presented above (Eqn. 3), over liquid compositions corresponding to the endpoints of successive time intervals (Table 1). Mass, volume, surface area calculations were done using equations from Lange and Carmichael (1987). For each pair of successive residual liquids, the average of two vapor pressures is combined with the reported flux between the pair, yielding evaporation coefficients calculated using, with units as in Eqn. 1, 2 ⫻ 10⫺5␲miRT ␣i ⫽ Ji 兹 , Pi,sat

(4)

where recondensation is implicitly ignored due to the extremely low total P of the experiments. Resulting coefficients represent

tot JFe ⫽ JFeO

PFe,sat

共P

Fe,sat

⫹ PFeO,sat兹mFe ⁄ mFeO兲 tot ⫽ JFeO

,

and

JFeO

PFeO,sat

共P 兹m Fe,sat

FeO

⁄ mFe⫹PFeO,sat兲

.

(7)

Within error, the two assumptions yield the same result. The assumption that ␣Fe ⫽ ␣FeO will be made here, allowing the parameter ␣ to reach unity, in principle, in the case of no inhibition to evaporation of either species. Evaporation coefficients are, then: tot ␣Fe ⫽ ␣FeO ⫽ JFe

兹2 ⫻ 10⫺5␲RT , 共PFe,sat Ⲑ 兹mFe ⫹ PFeO,sat Ⲑ 兹mFeO兲

(8)

Results are illustrated in Figure 3. On the basis of this comparison, it seems appropriate to choose evaporation coefficients for Fe and FeO species, for use with the MELTS model, based on the results shown in Table 1 and Figure 3. Although there are likely to be significant effect of composition on the evaporation coefficients, the data suggest that Fe and FeO behave very much like SiO and Mg. Averaging the ␣Fe in boldface in Table 1 (n ⫽ 18) yields ␣Fe ⫽ 0.19 ⫾ 0.06. The data do not permit determination of temperature dependence, although ␣ values seem to increase slightly with temperature (Fig. 3). This value may not apply to systems where the PFe/PFeO ratio deviates strongly from the experimental range (3–10). Because

Evaporation of FeO-bearing liquids

3187

Table 1. Evaporation coefficients calculated using the MELTS liquid model, P(H2) ⫽ 10⫺10 bar. Flux data (J ⫽ mol cm⫺2 sec⫺1) exp

%evap

T (K)

W-SC5 W-SC13 W-SC4 W-SC14 W-SC1 W-SC16 Hinitial H17C3-2 H17C5-av H17C7 H17C9 H17D2 Hinitial H18B6-2 H18B8-2 H18C1-av H18C3-av H18C5-av Hinitial H19B2 H19B4 H19B6 H19B8 H19C1 Hinitial H20B2 H20B4 H20B6

7.6 21.8 36.7 52.5 57.3 72.1 0 10.8 19.15 27.2 35.5 43.3 0 12.3 15.8 28.15 41.7 47.2 0 12.2 17.2 26.3 38.7 49.3 0 27 40.5 52.5

2073 2073 2073 2073 2073 2073 1973 1973 1973 1973 1973 1973 2073 2073 2073 2073 2073 2073 2173 2173 2173 2173 2173 2173 2273 2273 2273 2273

Partial P (bar) over liquid

Evaporation coefficients

J(Fe)

J(Si)

J(Mg)

Fe

FeO

SiO

Mg

1.56e-6 9.02e-7 2.37e-7 1.50e-8

3.75e-7 3.36e-7 2.34e-7 1.51e-7 1.47e-7 9.24e-8

1.91e-7 1.38e-7 1.13e-7 1.41e-7 1.87e-7 1.61e-7

2.88e-7 3.10e-7 1.88e-7 1.02e-7 4.38e-8

1.89e-8 6.02e-8 2.73e-8 4.11e-8 4.29e-8

1.86e-8 3.58e-8 2.42e-8 1.25e-8 1.43e-8

1.33e-6 6.70e-7 1.24e-6 4.68e-7 2.38e-7

2.29e-7 1.66e-7 3.09e-7 4.48e-7 1.22e-7

1.90e-7 2.88e-8 8.10e-8 3.96e-7 ⫺5.8e-8

3.43e-6 2.55e-6 2.15e-6 1.86e-6 7.12e-7

9.17e-7 3.41e-7 1.08e-6 8.66e-7 7.22e-7

6.06e-7 ⫺8.1e-8 2.10e-7 5.46e-7 4.21e-7

7.25e-6 4.87e-6 2.48e-6

2.41e-6 3.24e-6 2.45e-6

8.70e-7 2.19e-6 7.92e-7

4.93e-5 2.33e-5 1.56e-5 4.81e-6 1.74e-6 9.26e-7 2.02e-5 8.06e-6 6.50e-6 5.13e-6 3.88e-6 2.47e-6 7.00e-5 2.85e-5 2.63e-5 1.84e-5 1.46e-5 9.53e-6 2.15e-4 9.73e-5 7.75e-5 6.65e-5 4.66e-5 3.11e-5 5.94e-4 1.81e-4 1.36e-4 8.31e-5

1.16e-5 4.22e-6 2.56e-6 6.23e-7 1.99e-7 9.46e-8 5.00e-6 1.34e-6 1.01e-6 7.52e-7 5.27e-7 2.99e-7 1.95e-5 5.44e-6 4.90e-6 3.16e-6 2.38e-6 1.40e-6 6.64e-5 2.15e-5 1.63e-5 1.34e-5 8.74e-6 5.32e-6 2.01e-4 4.13e-5 2.91e-5 1.54e-5

1.23e-6 7.48e-6 9.87e-6 9.82e-6 8.56e-6 3.94e-6 8.92e-8 1.32e-6 1.80e-6 2.26e-6 2.46e-6 2.49e-6 4.36e-7 5.90e-6 6.44e-6 9.43e-6 1.04e-5 1.03e-5 1.84e-6 1.81e-5 2.74e-5 2.99e-5 3.64e-5 3.61e-5 6.79e-6 1.03e-4 1.14e-4 9.02e-5

2.46e-6 1.57e-6 1.75e-6 2.86e-6 3.73e-6 6.54e-6 6.70e-7 3.34e-7 3.29e-7 3.38e-7 3.89e-7 4.90e-7 2.97e-6 1.49e-6 1.52e-6 1.55e-6 1.68e-6 2.16e-6 1.13e-5 6.25e-6 5.68e-6 6.20e-6 6.73e-6 8.37e-6 3.81e-5 2.06e-5 2.30e-5 3.62e-5

Liquid residua compositions (wt%)

Fe

SiO

Mg

0.162 0.082 0.010

0.533 0.186 0.106 0.111 0.102

0.176 0.175 0.157 0.146 0.080

0.129 0.282 0.216 0.153 0.094

0.194 0.279 0.097 0.126 0.125

0.199 0.580 0.390 0.185 0.174

0.172 0.163 0.371 0.192 0.135

0.523 0.195 0.282 0.327 0.085

0.459 0.103 0.284 1.320

0.140 0.195 0.201 0.223 0.126

0.666 0.108 0.274 0.189 0.144

0.370 0.190 0.454 0.300

0.119 0.209 0.156

0.318 0.216 0.174

0.159 0.540 0.144

MELTS component activities, density of Ti-free liquids (g/cc)

exp

MgO

Al2O3

SiO2

CaO

FeO

TiO2

SiO2

Al2O3

Mg2SiO4

Fe2SiO4

CaSiO3

TiO2

W-SC5 W-SC13 W-SC4 W-SC14 W-SC1 W-SC16 Hinitial H17C3-2 H17C5-av H17C7 H17C9 H17D2 Hinitial H18B6-2 H18B8-2 H18C1-av H18C3-av H18C5-av Hinitial H19B2 H19B4 H19B6 H19B8 H19C1 Hinitial H20B2 H20B4 H20B6

26.1 29.7 35.2 42.7 43.2 42.7 23.84 26.32 28.49 30.97 34.34 37.81 23.84 26.26 27.26 31.45 34.14 38.79 23.84 26.09 27.76 30.74 34.77 38.94 23.80 30.80 34.60 41.10

2.90 3.61 4.00 5.60 6.75 8.65 3.16 3.57 3.97 4.49 4.92 5.48 3.16 3.55 3.88 4.46 4.80 5.24 3.16 3.55 3.74 4.02 5.03 5.88 3.16 4.14 5.40 6.24

36.2 40.5 44.9 46.7 45.0 41.7 35.43 39.09 41.74 45.24 48.04 49.03 35.43 38.73 39.55 43.53 45.86 47.20 35.43 37.94 39.64 41.09 44.23 45.56 35.40 40.90 43.00 43.70

2.27 2.58 3.09 4.00 4.79 6.59 2.53 3.09 3.53 3.78 4.14 4.80 2.53 3.07 3.53 3.82 4.31 4.76 2.53 3.29 3.27 3.56 4.48 5.07 2.53 3.72 4.76 5.77

32.40 23.50 12.60 0.81 0.07 0.01 35.04 27.92 22.28 15.53 8.56 2.88 35.04 28.40 25.78 16.74 10.90 4.02 35.04 29.14 25.60 20.59 11.49 4.55 35.00 20.50 12.30 3.15

0.106 0.141 0.173 0.218 0.252 0.325

0.0567 0.2655 0.3185 0.2494 0.1925 0.0790 0.0252 0.2496 0.3196 0.3770 0.3820 0.3453 0.0239 0.2209 0.2355 0.3174 0.3321 0.2955 0.0227 0.1601 0.2301 0.2414 0.2735 0.2471 0.0217 0.2220 0.2300 0.1580

1.19e-2 1.05e-2 8.89e-3 1.13e-2 1.49e-2 2.30e-2 1.29e-2 1.05e-2 9.75e-3 9.01e-3 8.62e-3 9.17e-3 1.38e-2 1.16e-2 1.20e-2 1.08e-2 1.02e-2 1.04e-2 1.46e-2 1.33e-2 1.24e-2 1.22e-2 1.27e-2 1.38e-2 1.54e-2 1.36e-2 1.57e-2 1.71e-2

0.3344 0.3781 0.4675 0.6068 0.6220 0.6240 0.2127 0.1500 0.1103 0.0720 0.0358 0.0104 0.2191 0.1576 0.1364 0.0765 0.0456 0.0139 0.2251 0.1669 0.1373 0.0979 0.0470 0.0157 0.2310 0.0996 0.0512 0.0098

0.1840 0.1141 5.04e-2 2.34e-3 1.85e-4 1.71e-5 0.2822 0.3093 0.3396 0.3753 0.4326 0.4962 0.2894 0.3167 0.3320 0.3962 0.4390 0.5250 0.2961 0.3246 0.3481 0.4012 0.4629 0.5373 0.3020 0.4090 0.4690 0.5940

2.81e-2 2.54e-2 2.49e-2 2.58e-2 3.06e-2 4.38e-2 3.26e-2 3.32e-2 3.39e-2 3.22e-2 3.17e-2 3.38e-2 3.39e-2 3.49e-2 3.86e-2 3.47e-2 3.58e-2 3.51e-2 3.51e-2 4.03e-2 3.66e-2 3.62e-2 3.93e-2 3.95e-2 3.62e-2 3.95e-2 4.45e-2 4.59e-2

2.98e-3 3.36e-3 3.52e-3 3.56e-3 3.82e-3 4.36e-3 3.11 2.99 2.90 2.80 2.72 2.66 3.08 2.97 2.93 2.81 2.74 2.67 3.05 2.96 2.91 2.85 2.73 2.66 3.02 2.82 2.73 2.64

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Values of ␣Mg and ␣SiO are consistent within 1␴ with those derived by Richter et al. (2002, their Table 1). Clearly, given the assumptions involved, there is a need for much more experimental work to constrain these parameters. Alexander (2002, 2004) also used the Berman (1983) CMAS and MELTS models to obtain liquid activities, using a model conformable to that of Richter et al. (2002). From the data of Hashimoto (1983), he calculated temperature-dependent evaporation coefficients for an equilibrium model (EQR), at 2073 K: ␣MgO ⫽ 0.278, ␣SiO2 ⫽ 0.174, ␣Fe ⫽ 0.265. He then modeled the evaporation experiments of Wang et al. (2001) using those coefficients, switching between the Berman (1983) CMAS model and the MELTS model as required. Relative to other uncertainties (e.g., H2 pressure) in astrophysical environments controlling nebular evaporation, the differences in ␣ of Figure 3, and between the results of Alexander (2002) and this work, can be considered insignificant. Fig. 3. Evaporation coefficients calculated using the MELTS model for FeO-bearing 2073 K series (solid squares) of Wang et al. (2001, SC-1800 series) and 1973 (open diamonds), 2073 (open squares), 2173 (open triangles), and 2273 (open circles) series of Hashimoto (1983).

4.1. Equilibrium Condensation vs. Evaporation

the composition trajectories are similar in all the experiments (Fig. 2), composition dependence is difficult to assess, however, ␣Fe tends to decrease with increasing wt% MgO. Results for Mg and SiO are very scattered, due to the small changes in MgO and SiO2 in residua over the course of the experiments.

To first order, the temperatures at which elements condense in equilibrium from the gas phase as either solid assemblages or liquid-solid mixtures depend on the standard Gibbs free energies of formation of the solid and liquid components, relative to the gas. Second order effects in condensation are due to variations in the Gibbs free energy of the ensemble, introduced by

4. DISCUSSION

Fig. 4. Effects of liquid activity model on (a) phase stability and (b) proportions of condensed oxides. Non-ideal liquid models are B83-CMAS (Berman, 1983), and MELTS (Ghiorso and Sack, 1995). Condensation calculation is for a solar gas enriched 1000x in a CI composition dust at Ptot ⫽ 10⫺3 bar (Ebel and Grossman, 2000), varying only the liquid allowed to condense. Compositional limits of models affect results more than differences in calculated activities.

Evaporation of FeO-bearing liquids

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Table 2. Selected properties calculated for liquids of Fig 4. Temperature (K): moles Berman (1983) CaO MgO Al2O3 SiO2 MELTS (Sack and Ghiorso, 1995) SiO2 TiO2 Al2O3 Mg2SiO4 Fe2SiO4 CaSiO3

1990 activity

1990 std G*

1900 activity

1900 std G*

1.258e-3 1.094e-2 8.771e-4 7.294e-3

2.79e-6 0.01224 0.05887 0.12615

⫺496643 ⫺430105 ⫺1618740 ⫺1002511

3.38e-5 0.02639 0.01054 0.26562

⫺476330 ⫺409096 ⫺1556891 ⫺963093

8.564e-4 4.947e-5 8.811e-4 5.070e-5 4.970e-3 1.260e-3

0.14116 0.00550 0.02120 0.00198 0.55478 0.04788

⫺1004740 ⫺1065827 ⫺1623079 ⫺1865038 ⫺1983411 ⫺1642982

0.29968 0.00671 0.01788 0.02305 0.38739 0.05480

⫺965262 ⫺1029522 ⫺1560082 ⫺1783137 ⫺1899012 ⫺1582752

1990 wt%

1900 wt%

log[P(O2)]: SiO2 Al2O3 MgO CaO log[P(O2)]:

⫺8.071 42.18 8.607 42.42 6.792 ⫺8.074

⫺8.855 47.921 9.3265 35.57 7.1831 ⫺8.865

SiO2 TiO2 Al2O3 FeO MgO CaO others:

42.82 0.395 8.971 0.727 40.01 7.054 0.019

45.201 0.4073 9.2081 5.7605 32.129 7.1022 0.1915

MELTS components present at ⬍ 0.2 wt% are not shown here. * standard Gibbs energy of formation (Joules) from monatomic gaseous elements at T, 1 bar.

the activity models used to describe non-ideal mixing of endmembers in solid and liquid solutions. Models describing kinetic processes such as evaporation are, by contrast, highly sensitive to the choice of activity model for the evaporating liquid. 4.1.1. Non-ideal Liquid Models in Equilibrium Condensation Ebel and Grossman (2000, their Fig. 3), compared liquid compositions calculated to condense at equilibrium from a gas of solar composition enriched 100⫻ in a dust of CI chondrite composition, cooling at Ptot ⫽ 10⫺3 bar, in separate calculations, otherwise identical and swapping only the Berman (1983) and MELTS liquid models (Ebel et al. 2000b). Such comparisons yield curves for liquid composition which, although they differ in detail, are quite similar near the particular temperature (Tol) at which model olivine (Sack and Ghiorso, 1989) crystallizes, ⬃10 K lower with respect to the Berman (1983) liquid model than to the MELTS model. Not far above Tol, the MELTS liquid SiO2 content diverges to much higher values than the Berman liquid, due to the requirement that SiO2 ⬎ (1/2 (MgO-Cr2O3) ⫹ ½ FeO ⫹ (CaO-3 P2O5) ⫹ Na2O ⫹ ½ K2O) imposed by the MELTS liquid components. A similar comparison at 1000⫻ dust enrichment, illustrates the same effects (Fig. 4). In the absence of liquid at high temperatures, the assemblage of Ca-, Al-rich solids does not predict smooth changes in the proportions of CMAS oxides, because the bulk compositions of the stoichiometrically constrained solids cannot vary as smoothly as liquid, as seen in Figure 4b. Nevertheless, first order features such as the absolute amounts of matter condensed, and the proportions of oxides, are very weakly sensitive to the liquid model choice near the condensation temperature of olivine. Several tens of degrees below Tol, the liquid compositions begin to diverge only when significant amounts of FeO become stable in the MELTS liquid. In general, differences in results are caused more by model limitations on composition, and depend only weakly on the large differences in calculated activities of liquid components shown in Table 2. An interesting comparison in this regard is to condense the

MELTS liquid as if it were ideal (all activity coefficients equal unity), with results shown in Figure 5. In the non-ideal case, the liquid becomes stable at ⬃2230 K, because negative deviations from ideality favor its stability over competing calcium aluminosilicates. In both cases, it is the Gibbs energies of the liquid components which grossly determine their condensation temperatures and relative proportions. The non-ideal activity model modifies this primary effect. In Figure 5b, non-ideality changes the relative abundances of olivine and liquid, not their existence as condensates. If valid endmember data are used, ideal silicate liquid models are adequate for capturing first order effects in many condensation scenarios. 4.1.2. Activity Models Constrain Evaporation Trajectories Equilibrium addresses the relative thermodynamic stability of elements and oxides in liquids and solids, relative to the vapor. In modeling equilibrium of an evolving system, each step is, in principle, independent of the one before it, even though in practice the previous result may be used as an initial state for calculation of equilibrium at the next set of conditions. In the present approach to modeling evaporation, a sequential series of flux calculations is performed, with each step of the series being dependent upon the preceding step. The composition trajectory calculated for a particular initial condition is therefore highly dependent upon the activity model chosen for the liquid and used to obtain saturation vapor pressures. This dependence is amplified by the choice of evaporation coefficients, which depend in turn upon the activity model chosen to derive them. It is important that the Berman (1983) and MELTS model give very similar predictions of saturation vapor pressures in the compositional region where they overlap (e.g., Fig. 1). In the course of exploring the evaporation of CMAS liquids (Grossman et al., 2000; Richter et al. 2002), it has been observed that liquid compositions evolve to a “valley of stability” in MgO-SiO2 composition space. Examples are illustrated in Figure 6 for two arbitrary liquid compositions (s1 and s2), using the Berman (1983) liquid activity model and assuming two different evaporation coefficient relations. For both s1 and

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Fig. 5. Effect of non-ideality of MELTS liquid model on (a) liquid composition, and (b) phase stability. Non-ideal (solid lines) and ideal (dashed lines) results calculated as in Figure 4, varying only the liquid.

s2, Al2O3 and CaO are added to yield the Al/Ca ratio of a gas of solar composition (Anders and Grevesse, 1989). These curves illustrate the strong dependence of evaporation trajectories upon even small variations in the calculated evaporation coefficients.

Fig. 6. Effect of evaporation coefficients on calculated evolution of composition trajectories using the CMAS liquid activity model (Berman, 1983). Evaporating liquids s1 and s2 (solid squares ⫽ initial composition) reach ‘valleys of stability’ as MgO and SiO2 evaporate for two different choices of evaporation coefficients (dark and light symbol lines), at Ptot ⫽ 1 bar, 2073 K. Dashed line indicates the minimum SiO2 for feasibility of the MELTS model, for a solar Al/Ca ratio in the liquids.

4.2. Evaporation of Chondrule Liquids Liquid droplet chondrules were at one time, if only briefly, immersed in high temperature nebular vapor, with which they would have tended toward a state of chemical equilibrium. It will be some time before experimental data are available with which to completely constrain the tem-

Fig. 7. Partial pressures of major gaseous species above a molten Type IA chondrule (Table 3) with fixed P(H2), calculated at 2073 K using the MELTS liquid activity model. Filled symbols are: SiO (diamond), Fe (square), Mg (triangle), AlO (circle).

Evaporation of FeO-bearing liquids

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Fig. 8. Partial pressures of major gaseous species above molten bulk chondrules listed in Table 3, calculated for P(H2) ⫽ 10⫺10 bar. Symbols as in Figure 7.

perature and composition dependence of evaporation coefficients for FeO-bearing liquids. The mean evaporation coefficients derived here can be applied to chondrule formation models, given values for saturation vapor pressures. It therefore seems useful to report the partial pressures calculated for dominant gaseous species in H2-rich systems, buffered by chondrule liquids. The partial pressures graphed in Figures 7 and 8 are calculated using the MELTS liquid model. Chondrule compositions (Table 3) are averages of n bulk oxide data taken from the reports by Jones and colleagues, as noted in Table 3. Base data for averages, and also for glassy and Al-rich chondrules (Bischoff and Keil, 1984; Krot and Rubin, 1994), are presented in EA-2. Calculated vapor pressures above chondrules are tabulated in EA-3.

In Figure 7, the variations of partial pressures of major gas species are shown as functions of imposed pressure of H2, for the average composition of 11 type IA chondrules given by Jones and Scott (1989). These results are qualitatively similar to the vapor pressure calculation of Grossman et al. (2000, their Fig. 4). Partial pressures of SiO2, MgO and FeO remain constant with P(H2) because they are directly related to P-independent activities of liquid components, aj, and daj/dP ⬃ 0. The liquid-buffered, ‘vacuum’ regime (P(H2) ⬍ 10⫺6 bar) is of greatest relevance to processes which might occur in an X-wind, during the sudden removal of molten chondrules from the nebular midplane. Results for other chondrule types, including glassy and Al-rich chondrules (Bischoff and Keil, 1984; Krot and Rubin, 1994), at

Table 3. Representative chondrule bulk compositions (wt%) used in calculations. Chondrule type Fe-poor ol IA px IB Fe-rich ol IIA px IIB

Reference Jones Jones Jones Jones

& Scott 1989 1994 1990 1996

n

SiO2

TiO2

Al2O3

FeO

MgO

CaO

11 3 11 11

47.54 58.92 47.02 56.19

0.20 0.11 0.12 0.13

4.14 2.41 2.79 3.62

1.25 2.82 15.49 10.42

43.16 33.98 32.61 27.41

3.71 1.76 1.97 2.22

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2073 and 1873 K, are presented in EA-4, and tabulated in EA-3. Because chondrules do not record Rayleigh fractionation of Fe isotopes (Alexander and Wang, 2001), models for the chondrule formation process must predict pressure—temperature— time paths such that either significant evaporation of FeO from precursors does not occur, or evaporation is followed by isotopic re-equilibration before chondrule-gas exchange ceases. Shock models for chondrule formation (e.g., Iida et al., 2001; Desch and Connolly, 2002; Ciesla and Hood, 2002), and the recently proposed current sheet model (Joung et al., 2004), all involve passage of chondrule precursors through steep pressure and temperature gradients. Correct prediction of chondrule evaporation is, therefore, useful as a test of these models. Conditions of very high H2 pressure in the nebula, for example the “solar composition dominated” regime of Richter et al. (2002, their Fig. 11), are likely to include high partial pressures of other elements (e.g., C, N, S). An infinite number of scenarios can be constructed involving these elements. Therefore, in Figure 8, a single P(H2) ⫽ 10⫺10 bar is assumed, and the results of partial pressure calculations are presented for a range of temperatures. Results for other chondrule types, at P(H2) ⫽ 10⫺10 and 10⫺4 bar, are presented in EA-4, and tabulated in EA-3. Chondrule formation is thought to involve planar astrophysical phenomena that rapidly heat chondrule precursors. Shock wave models “inevitably” (Desch and Connolly, 2002) predict that chondrules are heated above 1000 K for hours before they reach the shock front. Although the dynamical aspects of their model for current sheets are less rigorously developed, Joung et al. (2004, their Fig. 4) predict chondrule heating times an order of magnitude shorter (⬃10 –15 min), before dust melting (⬃1600 K). Both models predict cooling rates consistent with experiments comparing rates and chondrule textures (Lofgren, 1996). Predicted vapor pressures over a typical 1 mm diameter type IIA chondrule (Fig. 8c) can be used to calculate that less than one percent of liquid FeO will evaporate in the first hour, even at 1600 K, and only ⬃10% at 1800 K. Many tens of hours are required to evaporate significant FeO at 1400 K. It is not surprising, therefore, that chondrules show no evidence for FeO evaporation. Iron isotopic signatures cannot be used to distinguish between these two proposed chondrule-forming processes. 5. CONCLUSIONS

In the absence of a liquid model calibrated against all the available experimental data, and addressing all the composition space spanned by the simple oxides, a combination of the Berman (1983) CMAS model with the MELTS model of Ghiorso and Sack (1995) is sufficient to describe both equilibrium condensation of, and saturation vapor pressures over silicate liquids. Both liquid models give similar results when the melt has ⬍5% non-CMAS components, and ⬃5% more SiO2 than is necessary for the MELTS model to apply. First order effects in equilibrium condensation depend primarily upon the Gibbs free energies of solid and liquid components. Evaporation coefficients used in models of evaporation are, by contrast, strongly dependent upon the liquid model used to calculate saturation vapor pressures. Both Fe and FeO gaseous species must be considered im-

portant components of vapors in equilibrium with FeO-bearing silicate liquids. Although more experimental data are needed, existing results yield evaporation coefficient of ␣Fe ⫽ ␣FeO ⫽ 0.19 ⫾ 0.06, very similar to ␣ for gaseous SiO and Mg. Available data do not allow determination of temperature and composition dependence of these coefficients. Cooling rates of chondrules have been estimated from laboratory studies, and calculated from physical models of chondrule-formation (e.g., current sheets, shock waves). Given the evaporation coefficients calculated here, these cooling rates are sufficiently rapid that chondrules of representative types would not be expected to record significant evaporation of iron, if heated in a low-pressure H2 gas. Indeed, chondrules do not record Rayleigh fractionation of iron isotopes. In the presence of significant partial pressures of iron and oxygen, due to the local evaporation of submicron dust grains, iron evaporation from molten chondrules would be further suppressed.

Acknowledgments—This research has made use of NASA’s Astrophysics Data System Bibliographic Services. Work was supported by NASA Cosmochemistry grant NAG5-12855. I thank C. Alexander, A. Campbell, H. Nagahara, and N. Kita for very careful reviews and editorial care that significantly improved this work. Associate editor: N. Kita

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