By continuing to browse this site you agree to us using cookies as described in About Cookies
Notice: Wiley Online Library will be unavailable on Saturday 7th Oct from 03.00 EDT / 08:00 BST / 12:30 IST / 15.00 SGT to 08.00 EDT / 13.00 BST / 17:30 IST / 20.00 SGT and Sunday 8th Oct from 03.00 EDT / 08:00 BST / 12:30 IST / 15.00 SGT to 06.00 EDT / 11.00 BST / 15:30 IST / 18.00 SGT for essential maintenance. Apologies for the inconvenience.
Abstract— The role of oxygen isotope exchange during evaporation and condensation of silicate melt is quantitatively evaluated. Silicate dusts instantaneously heated above liquidus temperature are assumed to cool in gas and experience partial evaporation and subsequent recondensation. The results show that isotopic exchange effectively suppresses mass-dependent O-isotope fractionation even if the degree of evaporation is large, which is the fundamental difference from the case without isotopic exchange. The final composition of silicate melt strongly depends on the initial abundance of oxygen in the ambient gas relative to that in silicate dust, but not on the cooling rate of the system. The model was applied to O-isotope evolution of silicate melts in isotopically distinct gas of the protoplanetary disk. It was found that deviation from a straight mixing line toward the δ18O-rich side on the three-oxygen isotope diagram is inevitable when mass-dependent fractionation and isotopic exchange take place simultaneously; the degree of deviation depends on the abundance of oxygen in an ambient gas and isotopic exchange efficiency. The model is applied to explain O-isotopic compositions of igneous CAIs and chondrules.
Calcium-aluminum-rich inclusions (CAIs) and chondrules show wide variability in chemical composition and texture, suggesting notable chemical fractionation before and/or during their formation. CAIs often show evidence for evaporation and/or condensation, whereas very a limited fraction of them shows notable mass-dependent isotopic fractionation. An exception is FUN (fractionated with unknown Nuclear effects) inclusions, which show significant mass-dependent oxygen isotopic fractionation with associated Mg and other elemental isotopic fractionation (Clayton and Mayeda 1977; Wasserburg et al. 1977). The inclusions are thought to have suffered evaporation during melting.
The temperature of chondrule formation was above 1600 °C or more, at which extensive evaporation of silicate melt and accompanying mass-dependent isotopic fractionation are expected if chondrules were heated in the gas of the protoplanetary disk under low total pressure (Hewins et al. 1997). There is, however, little evidence for coupled chemical and isotopic fractionation in chondrules caused by evaporation. The lack of mass-dependent isotopic fractionation is the case even for volatile elements: Fe and K are depleted in elemental abundances but do not show evidence for mass-dependent isotopic fractionation (Alexander et al. 2000; Alexander and Wang 2001), which is interpreted as a result of chondrule formation in a highly dust-enriched environment. An extreme model as proposed by Alexander et al. (2008) and other workers invokes chondrule formation in close association with planetesimal formation in a gravitational field. Formation of chondrules in a dust-enriched environment has also been supported by thermochemical consideration; chemical equilibrium calculation showed a dust-enriched environment is required to stabilize silicate melt at low total pressure of the protoplanetary disk (Ebel and Grossman 2000).
It is important to investigate the relationship between chemical and isotopic fractionations through evaporation and recondensation during CAI and chondrule formation to understand the physical and chemical conditions that resulted in the insignificant mass-dependent isotopic fractionation (Nagahara et al. 2008). Ozawa and Nagahara (2001) explored a model for a silicate melt sphere that was heated instantaneously to an above solidus temperature and then cooled in a closed system. They showed that elemental and isotopic fractionations during evaporation in vacuum are coupled when evaporation Péclet number, the ratio of time scales of diffusion and evaporation, is less than about 100, but are decoupled if the ratio is more than 100. They examined decoupling between elemental and isotopic fractionations for K in chondrules under various dust-enriched environments. Alexander (2004) investigated chemical and isotopic fractionation of major elements in a silicate melt sphere that was instantaneously heated to evaporate and recondensed in a closed system. He showed that the isotopic composition of the melt sphere becomes heavier due to evaporation and comes back to the initial value by isotopic exchange. The model is, however, isothermal, so that the effect of cooling during chondrule formation was not properly considered.
There is a crucial aspect that should be considered to understand the behavior of isotopic fractionation of residual melts (igneous CAIs or chondrules) but that was overlooked in previous studies. Previous models are based on net-transfer reactions of appropriate components between chondrule melt and gas assuming coupling of an element and stoichiometric oxygen. Evaporation and condensation of metallic element M are assumed to take place through a reaction
This equation is appropriate to examine elemental fractionation between melt and gas. Isotope fractionation is affected by the following exchange reactions,
where pM implies p isotope of the element M. Isotopic exchange can take place between specific gas species (e.g., O) and corresponding elements exposed on the surface of the condensed phase (oxygen bonding to Si or Mg). As pointed out by Yu et al. (1995), O-isotopic exchange between silicate melt and ambient gas is significant during chondrule formation. Therefore, it is important to quantitatively investigate the role of isotopic exchange to understand the isotopic composition of CAIs and chondrules. This is of particular importance for the O-isotopic composition, because oxygen is one of the major elements forming abundant gas species. It is easily imagined that isotopic exchange would be more effective for oxygen as compared with metallic elements, because oxygen is more abundant in the ambient gas than metallic elements, particularly at low temperature before high temperature processing.
In this article, we quantitatively examine the role of exchange reactions in mass-dependent O-isotope fractionation during evaporation and recondensation of flash-heated dust aggregates to form melt spheres in a gas initially in isotopic equilibrium with the dust as well as in a gas with isotopically distinct compositions. The results are applied to explain O-isotope compositions of natural igneous CAIs and chondrules by considering separation of evaporated gas and reaction with isotopically distinct gas if necessary.
Evaporation flux from silicate melts into vacuum is given by the Hertz–Knudsen equation (Hirth and Pound 1963),
where Ji is the evaporation flux of elemental or isotopic gas species i; is the evaporation coefficient for i, which represents the kinetic barrier for evaporation; is the equilibrium vapor pressure of the gas species i; mi is the molecular weight of i; k is the Boltzmann’s constant; NA is the Avogadro’s number; and T is temperature. Equation 4 states that the evaporation flux is controlled by thermodynamic driving force and kinetic barrier. The kinetic barrier , which includes surface atomistic processes such as break off of chemical bonding, surface diffusion, or detachment from the liquid surface, needs experimental determination (Nagahara and Ozawa 2000).
If there is ambient gas, a certain degree of back reaction, recondensation, takes place. Net evaporation flux is rewritten as
where Pi is the partial pressure of i in the ambient gas and is the condensation coefficient, efficiency for condensation. Although the efficiencies for evaporation and condensation may be different, it is difficult to separate the two roles in experiments and therefore we assume
An application of Equation 5 to evaporation and condensation of metallic elements is straightforward. For example, evaporation or condensation of metallic iron is simply shown as,
where Mg, SiO, and O (or O2) are presumed common gas species actually evaporating or condensing, but Mg2SiO4 is a component in the condensed phase. The evaporation flux of forsterite is represented by the evaporation flux of one of gas species according to the stoichiometric constraint. The evaporation flux of forsterite is expressed by that of SiO,
The evaporation flux of Mg is twice of that of SiO, and that of O is three times of SiO,
In this case, all the reaction processes even partial pressure, surface diffusion, and surface reaction of Mg and O or O2 gas species are lumped into αSiO. This is also the case for an oxide component in silicate melt.
Evaporation and condensation of Mg isotopes from silicate melt is described by the following net-transfer reactions,
For 26Mg, net-transfer flux is expressed by
Dividing Equation 11 by the same equation for the major isotope 24Mg, we obtain the following isotopic fractionation equation,
where is kinetic isotopic fractionation factor, which should be distinguished from the equilibrium fractionation factor (see Appendix Equation A23). The experimentally determined kinetic isotopic fractionation factor for crystals or silicate melt is always closer to unity than that expected from (e.g., Davis et al. 1990; Wang et al. 1999; Richter et al. 2002; Yamada et al. 2006), indicating the presence of kinetic barrier for isotopic fractionation. Young et al. (2002) discussed that the kinetic isotopic fractionation is controlled by the reduced mass of isotopologues that govern the evaporation, which was experimentally supported for evaporation of forsterite (Yamada et al. 2006).
In addition to Equation 10, there are isotope exchange reactions between melt and gas,
The isotopic exchange flux for 26Mg in Equation 13 is expressed as follows,
where is the kinetic efficiency for exchange reaction between p in the gas and q in the melt and is the fraction of p in the melt. Total isotopic evaporation flux of 26Mg is the sum of net evaporation flux and exchange flux,
The same approach is applied to oxygen isotopes with some complications owing to stoichiometric constraints for evaporation of oxide components, which is described in detail in the Appendix.
We examine the role of oxygen isotopic exchange for silicate melt and ambient gas during evaporation and recondensation in a closed system that monotonically cools from a high temperature. Silicate dust aggregates are assumed to be instantaneously heated to an above liquidus temperature and melted, where the oxygen isotopic compositions of dusts and gas are identical. The concept of the model is illustrated in Fig. 1a.
The basic equations are Equation 8 for evaporation/recondensation of elements, which is modified for oxide components in silicate melt, Equation 11 for that of isotopes through a net-transfer reaction, and Equation 14 for isotope exchange. There are three free parameters: (1) the efficiency of the isotope exchange, (2) the cooling rate of the system, and (3) the amount of initial oxygen in the ambient gas. The efficiency of isotope exchange is critical to evaluate the role of isotope exchange reactions, and we do not know the plausible value for evaporation and condensation under the proto-solar disk conditions. The cooling rate should be evaluated because of the time-dependent nature of all the processes including evaporation, condensation, isotope fractionation, and isotope exchange, all of which have their own time constants. Variability in the amount of initial oxygen in gas should be examined, because the abundance of oxygen-bearing gas species in the proto-solar disk can be affected by the various degree of dust enrichment and the amounts of water ice and organic materials.
All the parameters, temperature, size of melt sphere, time, and evaporation/condensation rates are nondimensionalized. The initial oxygen abundance in gas is normalized by that in the initial dust, temperature by the initial temperature, size by the initial size, time by that required for total evaporation of a liquid sphere consisting of a chosen component (MgO in this study), and the evaporation/condensation rates by the initial evaporation rate. Cooling rate is an exponential function.
Oxygen bound in the dust is approximately 25% of the total oxygen for the solar composition, and the normalized initial oxygen abundance in gas is approximately 5. If the dust is 100 times richer than the solar composition with keeping the gas abundance constant, oxygen bound in the solid is 95% and that in the gas is 5% of the total oxygen, which corresponds to the normalized initial oxygen abundance of about 0.05. If the dust abundance is 0.01 times poorer than the solar composition, the normalized initial oxygen abundance in gas becomes about 500. We thus vary the initial oxygen abundance in the ambient gas from 0.1 to 100.
Silicate melt and ambient gas are assumed to be in thermal equilibrium. A silicate melt sphere is assumed to be homogeneous all through the process by rapid diffusion in the melt, and crystallization is assumed not to affect evaporation/condensation reactions. The system consists of Na-Mg-Al-Si-K-Ca-Ti-Fe-O. In this study, hydrogen is not considered and thus H2O is ignored as an oxygen-bearing gas species. As this article is aiming to examine the effect of exchange reaction, ignorance of H2O does not alter the essence of the results with regard to isotope compositions. The mass-dependent isotopic exchange reaction between oxygen in H2O gas molecule and oxygen in melt may be governed by the mass difference among H216O, H217O, and H218O, which are similar to that among 16O, 17O, and 18O so far as the cases examined in this study are concerned. The exact treatment of gas species of oxygen is therefore critical only when the results are converted to real dimensions, such as time scale of isotopic exchange, time scale of evaporation, or cooling rate. In a hydrogen-free system, the normalization time scale can be calculated from the initial temperature and the radius of melt sphere as shown in Fig. A1. In the real nebular environment with hydrogen, the time scale is probably much shorter than that obtained from this diagram because of the notable enhancement of melt evaporation by hydrogen (Richter et al. 2002). Discussions are therefore valid as far as we do not discuss the real time scale of the processes, and the maximum time scale may be estimates from Fig. A1.
In this article, temporal variation of mass-dependent isotopic fractionation in a closed-system evaporation and condensation is examined at first, for which the efficiency of isotope exchange, cooling rate of the system, and the initial oxygen abundance in gas are evaluated as critical parameters. The model is applied to evaporation and recondensation in a closed system with isotopically distinct O-bearing gas and dusts lying along the CCAM line. We chose the CCMA line as a reference from many mixing lines suggested by various literatures, and thus the essential results of this article are equally applicable to other mixing lines. In most model calculations, the oxygen isotopic composition of dust is assumed to be δ17O = δ18O = −50‰ and that of the initial gas δ17O = 5‰ and δ18O = 10‰, which cover the whole range of O-isotope variation along the CCAM line. We also examine the isotopic fractionation during evaporation/recondensation of silicate melt with heavy oxygen isotopic composition (δ17O = 5‰ and δ18O = 10‰) in isotopically light gas (δ17O = δ18O = −50‰) along the CCAM line, by considering the recent discussion on a possibility of initially light oxygen isotopic composition of the nebular gas (Krot et al. 2010; McKeegan et al. 2011). Then, we will examine evaporation and recondensation of silicate melt in open-system processes. The separation efficiency of gas and dust (melt) is introduced as an additional parameter to represent the degree of openness of the system.
Extent of Evaporation/Condensation and Chemical Fractionation with Time
Temporal change in evaporated fraction of a silicate melt sphere cooling in a gas after instantaneous heating in a closed system (Fig. 1a) was calculated at four different cooling rates (Fig. 1b). The change of evaporated fraction of the silicate melt is shown in Fig. 1c, where the sphere experiences evaporation at first, although temperature is decreasing, and then recondenses with the progress of time. When the system cools very slowly (cooling rate of 0.001), the melt sphere evaporates almost entirely (the evaporated fraction approaches to about 1) at the time of 1 (normalization time scale) to 10, and then, starts to recondense due to the accumulation of metallic element-containing gas species. Metallic elements once evaporated from the melt fully condense back after a long time (at time >100) to the mass almost equal to the initial one (evaporated fraction approximately 0) at the cooling rate of 0.001. If the system cools rapidly (cooling rates of 0.1), the maximum evaporated fraction is smaller because of rapid temperature decrease that depresses evaporation. It is worth noting that the evaporated fraction does not come back to zero in the case of rapid cooling, that is the final mass is smaller than the initial mass and a certain fraction of metallic elements evaporated from the melt kept remaining in the gas due to large kinetic barrier for recondensation at low temperature. For very rapid cooling (cooling rate of 1.0), melt evaporates to some extent but almost no recondensation takes place, and the system is quenched.
These behaviors of evaporated fraction are faithfully manifested by chemical fractionation as shown in Fig. A2, which shows temporal change of two chemical parameters, (Al + Ca)/(Al + Ca + Mg + Si) and Mg/(Mg + Si), for the model illustrated in Fig. 1. With increasing cooling rate, the overall chemical fractionation becomes more suppressed, the time to attain maximum fractionation decreases, the frozen chemical fractionation relative to the maximum increases (Fig. A2).
The Role of Isotope Exchange
The change in O-isotope composition of silicate melt cooling from an above liquidus temperature in ambient gas in a closed system was calculated. The examined evaporation and recondensation behavior of this system is shown by the solid line in Fig. 1b and 1c (cooling rate of 0.01), where the melt evaporates significantly with time, almost by >80% at the time of about 1, which then undergoes recondensation to recover almost the initial mass at the time of >30. Figure 2 compares the evolution of O-isotope compositions in the cases with (three thick lines) and without (thin broken line) isotope exchange for melt (Fig. 2a) and for ambient gas (Fig. 2b). The overall patterns of evaporation/recondensation of melt are similar in that the melt once becomes isotopically heavier and then lighter toward the initial value after passing a minimum degree of fractionation (Fig. 1a). The degree of maximum fractionation, the time to attain the maxima, degree of minimum fractionation, and time to resume the initial value decrease with the isotope exchange efficiency. For the isotope exchange efficiency of 1.0, the melt recovers the original isotope composition without becoming negatively fractionated. Most importantly, the original isotope composition is exactly resumed in all the cases. When no isotope exchange is available, however, the system undergoes the most extensive isotopic mass fractionation. The melt then becomes enriched in light isotopes, significantly lighter than the initial composition, and does not return to the initial isotopic composition.
The gas follows the opposite path, which once becomes lighter and then heavier than the initial composition. The degree of isotope fractionation is much greater for the melt than the gas owing to the higher abundance of oxygen in the gas in this case, ten times of that in the initial dust. The lighter isotope composition of the melt and heavier isotope composition for the gas after a long time for the case without isotope exchange is due to combination of significant suppression of evaporation flux owing to cooling and greater condensation flux of light isotope gas species than heavy ones.
The most important role of isotope exchange is the effective isotopic homogenization of the system. The effect of isotopic exchange for metallic elements is discussed in the Appendix, and is trivial in the model presented in Fig. 1 owing to the presence of lesser amount of metallic gas species in the ambient gas than O-bearing gas species (see Fig. A3). When compared with chemical fractionation, the O-isotope fractionation is more decoupled with the chemical fractionation with increase in isotope exchange efficiency.
The Role of Cooling Rate on Oxygen Isotope Fractionation
The role of cooling rate on O-isotope fractionation during evaporation and recondensation of silicate melt and gas in a closed system is evaluated in Fig. 3 for isotopic exchange efficiency of 1.0. Four cooling rates shown in Fig. 1c are applied, and corresponding chemical fractionation and isotopic fractionation of metallic elements are shown in Figs. A2 and A3. With a decrease in cooling rate, the maximum degree of O-isotope fractionation becomes more pronounced, the time to reach the peak value of fractionation is more delayed, and the time to reach the isotopic equilibrium is more prolonged. The same tendency is true for the O-isotopic composition of the gas with the opposite sign of fractionation to the melt, that is, the gas becomes lighter before reaching negligible fractionation. Although the path is different, the final O-isotopic composition of the silicate melt is the same regardless of the cooling rates for the isotopic exchange efficiency of unity. This is not the case if the parameter is smaller than 0.001.
The Role of Ambient Gas Abundance on Oxygen Isotope Fractionation
The effect of oxygen abundance in ambient gas on O-isotope fractionation of silicate melt cooled in a closed system (Fig. 1) is evaluated in Fig. 4, where four values of oxygen abundance are compared with other parameters fixed (cooling rate = 0.01 and isotopic exchange efficiency = 1.0). The abundance of 10 (solid curve) corresponds to the solid curves in Figs. 1–3. As mentioned above, the normalized oxygen abundance in the initial gas is about 5 for the solar elemental abundance, which is between the curves for 10 and 1.0 in Fig. 4. Isotope homogenization is very effective when the ambient gas contains abundant oxygen; the degree of fractionation during the course of evaporation and recondensation is small and the isotopic fractionations of melt and gas quickly diminish at the oxygen abundance greater than 1. If oxygen is present in the initial ambient gas by factor of 100 (about 20 times of the average solar nebula), O-isotope fractionation is almost negligible during the course of evaporation and recondensation.
In summary, isotope exchange reaction plays a critical role in evolution of O-isotopes of evaporating and condensing silicate melt. The isotopic composition comes back to the initial value if the exchange reaction takes place, whereas it does not necessarily if the exchange reaction is not available. Previous theoretical studies on isotope fractionation for CAI and chondrule formation have a priori assumed that evaporation and condensation proceed keeping stoichiometry such as MO(l) → M(g) + O(g) and M(g) + O(g) → MO(l), which resulted in generating considerable degree of O-isotope mass fractionation in the liquid (thin dashed curve in Fig. 2). This is, however, not true if isotope exchange reactions are effective (thick curves in Fig. 2). The present work also shows that the timing of isotope fractionation is generally decoupled with chemical fractionation, and the degree of decoupling is dependent on abundance of oxygen in the ambient gas, on isotope exchange efficiency, and weakly on cooling rate of the system. The average proto-solar disk, where oxygen in the gas is approximately five times of that in silicates, is an environment where O-isotopes between coexisting melt and gas are easily homogenized.
Evolution of O-isotopic composition of silicate melt accompanying isotope exchange with ambient gas shown above assumes the same O-isotopic composition between the initially present dust and gas regardless of the relative abundance. Oxygen isotopes of chondrites, however, show evidence for heterogeneity between gas and solid. Oxygen isotopic compositions of CAIs and chondrules from carbonaceous chondrites are distributed along the CCAM line (Clayton et al. 1983), which have been proved by later numerous analyses with secondary ion-microprobe mass spectrometry (SIMS) (e.g., Yurimoto et al. 2008). The CCAM line has been interpreted to be a mixing line between 16O-rich solid and 17O and 18O-rich gas, although the origin of the isotopically light and heavy components have been a matter of debate. Recent detection of the solar wind by the Genesis mission has proposed a possibility that the O-isotopic composition of the Sun can be δ17O ∼ δ18O ∼ −60‰ and the proto-solar gas should have this composition (McKeegan et al. 2011), which further requires the primitive solids with the heavier O-isotopic composition (Krot et al. 2010). The two models are totally conflicting in terms of O-isotope composition of initial dust and gas, which will be clarified in future work. In the following discussion, a conventional scenario assuming that the initial solids are isotopically lighter than the ambient gas is adopted, and the isotopic evolution during instantaneous heating and subsequent cooling will be examined. We further evaluate a scenario assuming that the initial silicate dust has heavier O-isotope composition than the ambient gas by taking the new model into account (Krot et al. 2010).
Isotopic Evolution of Silicate Melt in Gas with Distinct Oxygen Isotope Compositions
Oxygen isotopic evolution of silicate melt cooling in gas with different O-isotopic compositions in a closed-system environment was investigated by assuming δ17O ∼ δ18O ∼ −50‰ for solids and δ18O ∼ 10‰ and δ17O ∼ 5‰ for the ambient gas (Figs. 5, 6, and 7a). The initial oxygen abundance in the ambient gas is assumed to be 10 times of that in the initial liquid, and the isotope exchange efficiency of 1.0, which are identical to the parameters applied in Fig. 3. Four cooling paths shown in Fig. 1b are applied. Figure 5 shows the trajectory of O-isotope composition in the three-oxygen isotope plot. The composition of the liquid starting from δ17O ∼ δ18O ∼ −50‰ is strongly controlled by the isotope exchange with ambient gas roughly following the CCAM line. The degree of deviation from the CCAM line is larger for smaller cooling rates. It should be noted that the final composition of the melt and gas is about δ17O ∼ −1‰ and δ18O ∼ 3‰ regardless of the trajectory. It is important that the O-isotope composition of silicate melt becomes close to that of the ambient gas during evaporation/recondensation for the dust/gas ratio of 10 (Fig. 5), although the bulk chemical composition is fractionated to various degrees owing to incomplete recondensation in rapid cooling (Fig. 1c).
The effect of initial oxygen abundance in the gas on the evolution of O-isotope composition of melt is shown in Fig. 6. The parameters are identical to those applied to Fig. 4, the evaporation path is shown in Fig. 1c, and the solid curve in Fig. 6 corresponds to that in Fig. 5. The evolution paths in Fig. 6 once tend to deviate from the CCAM line owing to mass-dependent isotopic fractionation, but approach the CCAM line in the end owing to isotope exchange with the ambient gas. The deviation from the CCAM line depends on the initial oxygen abundance in the gas; smaller initial oxygen abundance in the gas follows a larger degree of mass-dependent isotopic fractionation. This is the same tendency as the case with the initially same O-isotopic composition for melt and gas (Fig. 4); the greater abundance of initial oxygen in the ambient gas depresses mass-dependent isotopic fractionation due to extensive isotope exchange. The final O-isotopic composition of melt is close to the initial melt composition when the initial oxygen abundance in the ambient gas is small (oxygen abundance of 0.1), but is heavier with increasing initial oxygen abundance in the gas. The O-isotope composition of the melt becomes the same as that of the initial ambient gas when the abundance of oxygen in the initial gas is as much as 100 times of that in melt (oxygen abundance of 100), where the trajectory of O-isotopic composition of the melt follows the CCAM line. So far as the initial gas and dust lie on the CCAM line, effectiveness of isotope exchange forces the melt reacted with isotopically distinct gas stay on the CCAM line unless the oxygen abundance is lower than 1.0 to promote significant mass-dependent isotopic fractionation. In other words, a straight mixing line reflects the high abundance of oxygen in the ambient gas relative to dusts. A slight and temporal deviation to the left hand side (smaller δ18O side) of the CCAM lines occurs at low oxygen abundance in the initial gas, corresponds to an overstepping in the recovery of the initial value from fractionated state as observed in the closed-system cases (Figs. 2–4).
The temporal change of O-isotope composition and final composition through evaporation and recondensation of silicate melt in gas with distinct initial O-isotope composition are dependent on the isotope exchange efficiency. Figure 7a compares the evolutionary trend of O-isotope compositions for four different isotopic exchange efficiencies, where deviation from the CCAM line is larger with the smaller isotopic exchange efficiency, which is consistent with the results for the same initial O-isotopic composition for melt and gas (Fig. 2). Although the isotope fractionation path varies with the isotopic exchange efficiency, the final O-isotope composition is the same as far as the initial abundance of oxygen in the ambient gas is almost the same except for the no exchange cases. This is the same tendency as the case with the same initial O-isotopic compositions for melt and gas (Fig. 2). Figure 7b shows the isotope fractionation path for silicate melt with initially heavy O-isotopic composition (δ17Ο = 5‰, δ18Ο = 10‰) cooled in gas with initially light O-isotopic composition (δ17Ο = δ18Ο = −50‰) in a closed system. This model takes the recent observational results of the O-isotopic composition of the solar wind (Krot et al. 2010; McKeegan et al. 2011) into account. The conditions for cooling rate and initial dust/gas ratio are the same as those applied in Fig. 7a. The isotopic fractionation tendency is similar to that in Fig. 7a, but is not exactly the same; mass-dependent fractionation is more distinct than that in Fig. 7a because of the vectors of exchange reaction and mass fractionation are opposite. It should be noted that at small values of isotope exchange efficiency (<0.1), a slight and temporal deviation to the left hand side of the CCAM lines occurs, particularly for the no exchange case. They correspond to the negative fractionation observed in Fig. 2a in the case of the isotopically homogeneous initial condition.
Figures 5–7 give important clues on the origin of CAIs and chondrules in carbonaceous chondrites. Evaporation and recondensation of silicate melt, even in gas with different O-isotopic composition, causes mass-dependent O-isotopic fractionation that has a slope 1/2 on a three O-isotope plot, and the O-isotopic fractionation path should be deviated to the right side of a slope 1 mixing line. The degree of deviation is larger with a decrease in initial oxygen abundance in the ambient gas and with a decrease in efficiency of isotope exchange. The deviation is more enhanced if an isotopically heavier dust is heated in a lighter gas (Fig. 7b). Deviation to the left side of a straight mixing line by a few ‰ may occur temporarily if the initial abundance is smaller than 0.1 or if the isotope exchange efficiency is smaller than 0.1 (Figs. 6 and 7). However, more than a few ‰ deviation to the left is never realized during high-temperature processes under rapid cooling. If cooling rate is slow to be able to attain equilibrium fractionation between melt and ambient gas before the solidification of the melt, deviation from the isotopic composition of gas toward the left hand side of mixing line like the CCAM line by a few ‰ (Kita et al. 2010) may occur.
Calcium-aluminum-rich inclusions show a diversity in O-isotope composition not only among different CAIs but also among constituent minerals in a particular inclusion, suggesting that the isotopic homogenization between gas and melt was not achieved and that fractionation or isotopic exchange reactions were disrupted during cooling. The present study shows that the isotopic equilibration (<±0.2‰) between silicate melt and the ambient gas during evaporation and recondensation takes place in a period as short as 0.1–1 (<∼3 h for initial temperature of 1800 °C and melt radius of 1 mm, if isotopic exchange efficiency does not depend on hydrogen pressure; Fig. A1) irrespective of cooling rate (Fig. 3), if the amount of oxygen in the initial ambient gas is higher than that of the solar composition (initial oxygen abundance approximately 5). We cannot specify exact time scale, but the reported mineral-specific O-isotope fractionation suggests that a responsible reaction process was very short. Although the variation of O-isotope fractionation among the entire CAIs can be explained by the gas/dust ration if the isotopic composition of the gas was constant as shown in Fig. 6, kinetics should be involved to explain the intra-inclusion diversity. Because cooling rate does not play an important role in producing the diversity in frozen isotopic fractionation (Figs. 3 and 5), separation of melt from the ambient gas during cooling (i.e., formation in an open system) must be invoked.
If this is the case, the variation of O-isotopic compositions of many CAIs and their minerals should have been achieved by a rapid cooling not to homogenize the inclusion, in high oxygen abundance in gas to reproduce the full compositional range, and high isotopic exchange efficiency not to cause a large deviation toward 18O-rich composition from the CCAM line. Separation of silicate melt from cooling gas generates CAIs with O-isotopic composition enriched in 18O but not in 17O, being plotted on the right hand side of a slope 1 mixing line like the CCAM line in the three O-isotope plot (Figs. 5–7). Equilibrium fractionation, which can cause deviation from the CCAM line to the left hand side by a few ‰, becomes effective only near equilibrium conditions (Fig. A4), and may not play an important role in the above scenario.
Oxygen Isotopic Fractionation of FUN Inclusions
Oxygen isotope compositions of most CAIs plot along the CCAM line, but FUN inclusions are exceptional in that they show notable mass-dependent O-isotopic fractionation (Clayton and Mayeda 1977; Lee et al. 1980; Ireland et al. 1992). Recent data (Ushikubo et al. 2007; Thrane et al. 2008; Krot et al. 2010) also support the involvement of mass-dependent O-isotope fractionation (Fig. 8a). The mass-dependent fractionation trend of FUN inclusions extends from δ17O ∼ δ18O ∼ −50‰ to δ17O ∼ 20‰ and δ18O ∼ 10‰ including HAL and similar objects, which further extends toward the terrestrial fractionation line (TF) resulting in triangular distribution on three-isotope oxygen plot (Fig. 8a). It is easily expected from Figs. 5–7 that silicate melt with the initial composition of δ17O ∼ δ18O ∼ −50‰ cooling from above liquidus temperature in gas with the composition of δ17O ∼ 5‰ and δ18O ∼ 10‰ changes the composition almost along the CCAM line with small deviation to the 18O-rich composition if the system is closed. The significantly large mass-dependent O-isotopic fractionation in FUNs is hardly expected from Figs. 5–7. A possible condition for the very large mass-dependent isotope fractionation was investigated with the present evaporation and recondensation model by considering open processes.
We have examined many cases of mass-dependent O-isotope fractionation and isotope exchange, and found that the mass-dependent fractionation in FUNs as large as approximately 40‰ in δ18O requires evaporation in near vacuum if isotope exchange efficiency of unity is adopted. Such near vacuum condition is realized by continuous separation of gas from melt in the proto-solar disk, that is the solid is extracted from the gas being open in terms of gas. To represent the degree of openness of the system, “gas separation factor” was introduced to the present model as an additional parameter, which is defined as a fraction of gas separated from the system. To explain the data plotted between the slope 1/2 line starting from δ17O ∼ δ18O ∼ −50‰ and the slope 1 CCAM line (Fig. 8a), mixing with a gas on the intersection of the TF and CCAM lines, δ17O ∼ 5‰ and 18O ∼ 10‰, from a certain time during evaporation was also considered. Details of the modeling method of open processes are described in Appendix (Equations A8–9).
Results of evolution of O-isotope compositions during evaporation of melt sphere with δ17O = δ18O = −50‰ (solid circle) under near vacuum conditions followed by mixing with the gas on the TF line (solid square) are presented in Fig. 8 for isotope exchange efficiency of unity and four values of mixing oxygen abundance (oxygen abundance relative to that in the dust = 100, 10, 1.0, and 0.1). Stars on an end of the evolutional trends represent O-isotope compositions frozen in a melt sphere during cooling. Fig. 8a is for rapid cooling, moderate gas separation, and slow mixing; 8b for slow cooling, moderate gas separation, and rapid mixing; 8c for slow cooling, moderate gas separation, and slow mixing; and 8d for slow cooling, intensive gas separation, and slow mixing.
If cooling rate is as fast as 0.01, the maximum O-isotopic fractionation is only by about 20‰ in δ18O. A high rate of cooling results in a linear mixing trend along the CCAM line (Fig. 8a). A slow cooling (approximately 0.001) is necessary to reproduce the entire data clusters for FUN inclusions (cf. Fig. 8a and 8b). So far as the cooling rate is 0.001, gas mixing rate does not affect the final isotope composition (indicated by stars in Fig. 8), although evolution paths change depending on the mixing rate (cf. Fig. 8b and 8d). The abundance of oxygen in the gas also strongly controls the isotopic mixing trajectory. A linear mixing trend requires rapid mixing as well as oxygen abundance in gas as high as 100 times of that in melt, which corresponds to enrichment of oxygen by more than 20 times relative to the average solar nebula.
To reproduce extreme isotope fractionation reported by Ireland et al. (1992), intensive gas-dust separation as high as 0.9 is required (cf. Fig. 8c and 8d). Because of slow cooling, the final isotope compositions indicated by stars lie on an approximately linear trend tying between the gas composition and a certain point on the mass-fractionation line. The position of the final composition depends on the oxygen abundance in the gas. The frozen trends shown in Fig. 8b and 8c are fairly consistent with the trend reported by Lee et al. (1980) as well as the recent data by Krot et al. (2010). Krot et al. (2010) suggested a two stage model; the first stage of melting and isotope exchange of δ16O-poor dust with δ16O-rich nebular gas to generate the variation along the CCAM line and the second stage of melting and evaporation in vacuum producing mass-dependent fractionation trends with slope 1 in three-isotope oxygen diagram. This model may be good for the presence of a slope 1 trend other than that starting from δ17O = δ18O = −50 on the CCAM line. Our model employing a single flash heating and cooling event can equally well reproduce the observed FUN CAI variations. FUN inclusions underwent evaporation accompanying extreme efficiency of gas-dust separation during slow cooling after a flash-heating event, which produces the observed mass-dependent fractionation by up to 80‰ in δ18O. They have subsequently mixed with various amount of gas with the terrestrial O-isotopic composition in a later stage of the same cooling event.
Oxygen Isotopes in Chondrules
Oxygen isotope compositions of chondrules show a wide variation on the three-oxygen isotope plot. Chondrules in carbonaceous chondrites are along or near the CCAM line but heavier compositions than CAIs in average, which reach the TF line, and those of ordinary chondrites are slightly above the TF line with considerably large scatter (Clayton 1993). Previous SIMS data generally contain large errors, which makes it difficult to distinguish isotopic exchange effect from mass-dependent fractionation. However, recent data by Kita et al. (2010) showed a small degree of mass-dependent fractionation related to mineralogy for some chondrules in various groups of chondrites, though some others show a “mixing line.”
Chondrules from Allende form a trend extending from a composition on the CCAM line toward the TF line, among which barred olivine chondrules have heavier O-isotope compositions than porphyritic ones (Clayton 1993), suggesting effective isotopic exchange for totally melted chondrules. Recently, Rudraswami et al. (2012) examined chondrules in Allende with high-precision SIMS and revealed a wider variation in O-isotope ratios than reported by Clayton et al. (1983) but notable consistency among minerals in each chondrule except for some extremely 16O-rich olivine grains, suggesting that chondrule melts were very homogeneous. The new data show a bimodal distribution, from which Rudraswami et al. (2012) argue for formation of chondrules in the local dust-enriched protoplanetary disk.
The O-isotope composition of Allende chondrules after Clayton et al. (1983) is examined with the present model. Figure 9 shows an example of well-fitted O-isotopic composition change for dust with the initial composition on CCAM line heated in a gas with heavier composition on the TF line without gas/melt separation. To fit the data, the gas composition should have δ17O = 2 and δ18O = 4, which is lighter along the TF line than the gas considered in Fig. 8. The chondrule data are well fitted with cooling rate of 0.1–5 and oxygen abundance in gas of 40–100. The range of cooling rates is rapid enough not to recover all the once evaporated materials (Fig. 1c), and the gas abundance of 40–100 is much more abundant compared with the average proto-solar disk (approximately 5), suggesting a fairly high dust-gas ratio.
The most important consequence of the high efficiency of isotope exchange reaction is rapid homogenization of gas and chondrule melts under highly dust-enriched environment. This is consistent with the notable homogeneity of O-isotope ratios among minerals in each chondrule (Rudraswami et al. 2012). Although we need more experiments for accurate determination of isotope exchange efficiency, we argue that the diversity of O-isotope composition in chondrules represents significant variation of gas/dust ratio (Fig. 6) producing large isotopic heterogeneity among chondrules.
Oxygen isotope composition is one of the most intensively debated subjects in meteorite study since the first finding of “anomaly” in Allende CAIs. Numerous analyses for refractory inclusions, chondrules, and other constituent components of chondrites have been compiled, and the origin of isotopic fractionation has been discussed extensively. Evaporation/condensation process has been quantitatively investigated for the last 20 yr with the goal of reproducing the chemical composition and isotopic compositions of oxygen and metallic elements within CAIs and chondrules. On the contrary, mass-dependent isotopic fractionation of oxygen during evaporation/recondensation has not been quantitatively investigated. The abundance of oxygen in proto-solar disk can be variable by dust enrichment or evaporation of water ice, and the efficiency of isotope exchange has not been experimentally fully examined, which affects the O-isotopic compositions of chondritic components.
The present work developed a model incorporating exchange reactions for isotopes to simulate O-isotopic evolution during evaporation/recondensation of silicate melt monotonically cooled in a closed system, where O-isotope compositions of silicate and gas are the same or distinct. Effects of cooling rate of the system, initial abundance of oxygen in the ambient gas, and isotope exchange efficiency were quantitatively investigated.
We have obtained the following results.
1 Evolution of O-isotopic fractionation is decoupled from the chemical fractionation of silicate melt; O-isotopic fractionation proceeds more rapidly than chemical fractionation. The decoupling is more enhanced with increase in isotope exchange efficiency.
2 Mass-dependent O-isotopic fractionation of silicate melt is suppressed by isotopic exchange during evaporation/recondensation of silicate melt having the identical O-isotopic composition with the ambient gas. If isotopic exchange does not take place, the O-isotopic composition of silicate melt becomes notably lighter than the initial one during recondensation, which is attributed to combination of significant suppression of evaporation flux by cooling and higher condensation flux of light isotope gas species than heavy ones. Isotopic exchange effectively homogenizes the isotopic composition of coexisting silicate and gas and nullifies the negative fractionation.
3 Cooling rate of the system affects time of evolution of O-isotopic mass fractionation, but the final composition is not affected so far as the isotope exchange efficiency is higher than 0.01.
4 In a system with initially different O-isotopic compositions between silicate and gas, deviation from a straight mixing line to the δ18O-rich side on a three-oxygen isotope plot is inevitable through evaporation even if isotope exchange is effective. The deviation increases with a decrease in cooling rate and with a decrease in exchange efficiency. A slight (<5‰) deviation to the opposite (δ18O-poor) side is expected if the isotope exchange efficiency is very low (<0.1) or if the initial oxygen abundance is as small as 0.1. A few ‰ deviation to the δ18O-poor side can also be explained by equilibrium fractionation factor larger than unity (Kita et al. 2010).
5 A straight mixing line such as CCAM line requires the presence of oxygen in the ambient gas more abundant than the average proto-solar disk. In a gas of the proto-solar disk with the average oxygen abundance about 5 relative to that in silicate, silicate melt acquires O-isotopic composition notably deviated from the gas.
6 The mass-dependent O-isotope fractionation of FUN inclusions by about 40‰ or more is not achieved by evaporation in a closed system, and near vacuum condition is required, which was possible by efficient and continuous separation of gas and silicate melt during evaporation. If FUN inclusions were formed from O-isotopically light solid, they have suffered a reaction between isotopically heavy gas following the near vacuum evaporation during the same cooling event, where the gas abundance must vary from almost zero to more than 20 times of the average proto-solar gas.
7 Oxygen isotope composition of Allende chondrules was well reproduced by evaporation/recondensation of chondrule melts under rapid cooling in a gas with terrestrial O-isotopic composition and O abundance 8–20 times that of the average proto-solar gas. Isotopic exchange with a gas having O-isotope composition of δ17O ∼ 2‰ and δ18O ∼ 4‰ is requisite, if it lies on the terrestrial fractionation line, to explain the trend of Allende chondrules plotting to the left of the CCAM line.
Acknowledgments— This article is dedicated to Roger R. Hewins, who made various pioneering contributions to the study of chondrules, among which experimental work on oxygen isotopic exchange between chondrule melt and ambient gas was specifically outstanding. The present study made theoretical work, which supports his conclusions. We are grateful to critical reviews by N. Kita, A. N. Krot, and the editor H. Connolly, who greatly improved the manuscript. This work was supported by the Grants-in-Aid for Scientific Research from JSPS, #22224010 (HN).
Editorial Handling— Dr. Harold Connolly Jr.
Model calculations are based on the formulation of Ozawa and Nagahara (2001). The model assumes constant volatility of a multi-component melt sphere, and an evaporation flux of component i from the melt sphere is expressed as:
where is the concentration of i in the condensed phase, and is the free evaporation flux from the pure end component i at a temperature T(t). The temperature is a function of time, t, and the following cooling function after a flash-heating event at t = 0 is assumed:
where s is the cooling rate, T0 is the temperature of the flash heating (initial temperature), and T is the minimum temperature.
The is given by the Hertz–Knudsen equation:
where is the equilibrium vapor pressure of the pure component i, is the evaporation coefficient, mi is the molecular mass of the gas species i, NA is the Avogadro number, and k is the Boltzmann constant. The ideal gas equation can be applied to relate the ambient pressure of the gas species i, Pi, to the gas composition Ni/V:
The number of the gas species i is expressed as follows:
where is the initial abundance of i in the gas, is the net evaporation flux of k, n is the number of melt spheres, Nc is the total number of components, r0 is the initial radius of the melt sphere, is the molar volume of the melt. The first term of the right hand side of Equation A5 is the amount of initially present gas species, and the second term represents the net contribution of evaporation and condensation of the melt sphere. The condensation flux is given by:
where ξ is a function of the partial pressure of i and evaporation coefficient, which is represented by the simplest form by using a single parameter l. This parameter represents pressure-dependence of kinetic barrier for condensation. If l is zero, , as assumed in this article. The composition of melt sphere, , obeys the following mass balance equation:
where r is the size of melt sphere and f is the evaporated fraction. In Equation A7 neither separation from nor influx of condensed phase into the relevant system is assumed. Similarly the composition of the gas, , obeys the following mass balance equation:
where f is the evaporated fraction, δg is the initial gas abundance, ηg is the amount of gas influxed into the system, ζg is the amount of gas separated, is the composition of the influxing gas, and dfi is the fraction of the component i evaporated from the melt sphere. Parameters δg, ηg, ζg, and f are all scaled relative to the mass of the initially present condensed phase. The efficiency of gas separation is defined as follows:
If γsep is 0, the system is closed to separation; if γsep is 1, the efficiency of gas separation is maximized (Rayleigh fractionation). The amount of the initially present ambient gas was varied by changing δg, and the amount of gas mixed with the evaporating/condensing melt spheres was varied by changing ηg.
Equations A1–A9 for elements and isotopes are combined to solve a set of ordinary differential equations in this article. One of the most important model parameters is , which was determined for each element as an exponential function of temperature by fitting the results of melt evaporation experiments in vacuum (Hashimoto 1983; Wang et al. 2001; Richter et al. 2002; and our unpublished data).
Nondimensionalized equations for partial pressure and evaporation flux of a major component (gas species) i ( and Ji, respectively) from a condensed phase, which can be obtained from Equations A1 and A3–A7, are given as follows after scaling the evaporation rate by the free evaporation rate of the end component i at the initial temperature and partial pressure by equilibrium vapor pressure of the end component i at the initial temperature :
where T ′ = T/T0 is the temperature normalized by the initial temperature T0, V ′ is the system volume normalized by the initial system volume V0, and is the evaporation coefficient normalized by that of pure i at the initial temperature and may depend on T ′ and surface composition. The parameter is l in Equation A6 normalized by the evaporation coefficient. Parameter θi is expressed as , where is the activation energy for evaporation of the component i. Parameter is the amount of the major component in the initial system relative to the amount that barely stabilizes the condensed phase with the end component and is expressed as follows:
This parameter corresponds to “dust enrichment factor” of Ebel and Grossman (2000), although it is not relative to the abundance of hydrogen but relative to the saturation limit. Parameter is the amount of the component i initially present in gaseous state relative to the amount present in the condensed phase and expressed as follows:
where is the concentration of i in the initial gas, and ω0 is the initial gas-dust ratio (Ozawa and Nagahara 2001). In Equation A10, time is scaled by the time required for complete evaporation of the end component i at the initial temperature and in vacuum, . This value depends on initial melt radius and temperature and is shown in Fig. A1 for i = MgO under hydrogen-free systems.
For another component l, nondimensionalized equation is given as follows:
where is the kinetic fractionation parameter for evaporation, is the equilibrium fractionation parameter (Ozawa and Nagahara 2001). Equations A13 were nondimensionalized by using the same scales used for the major component in Equations A10, which results in the appearance of the two fractionation parameters. By applying Equation A13, we calculated temporal changes of major element composition and isotopes of metallic elements in silicate melt evaporated and recondensed according to the model illustrated in Fig. 1a and 1b.
The results are shown in Fig. A2 for major elements and in Fig. A3 for Fe, Si, Mg, and Ca isotopes. The effect of isotopic exchange distinctive for oxygen fractionation is trivial for metallic elements in the model presented in Fig. 1 as shown in Fig. A3, which compares cases of no isotope exchange and of high exchange efficiency of 1.0 for Fe, Si, Mg, and Ca isotopes for cooling rate of 0.01 (cf. thick solid and thin long-dashed curves). This is due to the presence of lesser amount of metallic gas species in the ambient gas than O-bearing gas species although metallic gas species are built up by evaporation.
The maximum degree of isotopic fractionation is reached at a much earlier stage than the chemical fractionation and extent of evaporation; the chemical fractionation and evaporation extent attain the maxima at approximately t = 1 (e.g., solid curves in Figs. 1c and A2), but the isotope fractionation with notable isotopic exchange attains the maximum at approximately t = 0.05 (solid curves in Figs. 2). This tendency is also noticed for the time to recover the initial values (Figs. 1c, 2, and A2); the O-isotope composition for cases of efficient isotope exchange goes back to the initial value after passing a maximum, when the total mass and the chemical composition of melt sphere is still on the way to recover the initial values. The O-isotope fractionation is more decoupled with the chemical fractionation and the extent of net-transfer reaction, as isotope exchange is more effective (cf. Figs. 1c, 2, and A2). The isotope exchange accelerates O-isotopic homogenization between silicate melt and ambient gas in a closed system even during early stage of evaporation.
In this study, hydrogen (as well as carbon and sulfur) is not considered and thus H2O and OH are ignored as O-bearing gas species. Since this article is aiming at examination of the effect of exchange reactions, ignorance of H2O and OH does not alter the essence of the results, although they definitely affect isotopic fractionation as a function of hydrogen pressure. Hydrogen enhances evaporation rates, but in our approach employing nondimensionalized equations the ignorance of H becomes critical only when the modeling results are converted to real dimensions. This approach has an advantage in that isotopic fractionation can be modeled appropriately being independent from unclear parameters, such as evaporation/condensation coefficients, pressure, and temperature. On the other hand, the approach has a demerit that we cannot say anything definite on the dimensional variables, such as time scale of isotopic exchange and cooling rate. By ignoring H2O and OH, oxygen-bearing gas species to be considered are O2, O, and SiO. Other O-bearing species are minor.
Evaporation flux of oxygen for the kth net-transfer evaporation/condensation reaction of oxide is given by:
where is the reaction stoichiometry for oxygen in the evaporation reaction of the component k, , multiplied by the fraction of evaporating oxygen molecule in all oxygen gaseous species, . Summing up A14 for all the components, the following equation is obtained
By adopting the Hertz–Knudsen equation and solving for oxygen partial pressure at equilibrium, we obtain the following equation:
where is the kinetic barrier for evaporation of oxygen in reaction involving the kth oxide component and is the kinetic fractionation factor of oxygen and component i. Equation A16 defines “equilibrium oxygen pressure,” which can be regarded as the partial pressure of oxygen molecule to maintain gas-condensed phase equilibrium.
Partial pressure of oxygen molecules normalized by “equilibrium oxygen pressure” defined by A16 is given by:
Parameter represents temperature and composition dependence of evaporation coefficient for oxygen, , and is assumed to be the same for all oxide components. Equations A14–A17 are similarly applicable to oxygen gas atom.
If net-transfer reactions of oxide components are considered in evaluation of evaporation flux, nondimensionalized evaporation flux of 16O17O is given by:
where is the fraction of 16O17O in O2 gas molecule and is the fraction of 17O in the oxygen of the condensed phase. Equation A18 and corresponding equation for 16O2 effectively cause isotopic exchange between the condensed phase and gas at equilibrium. However, it is not the case under a kinetic environment, which requires reaction paths for isotope exchange.
Exchange reactions of oxygen isotopes between gas and a condensed phase are represented by the following four reactions:
The first two equations represent an exchange of gaseous oxygen molecule and oxygen in the condensed phase, and the last two equations represent an exchange of gaseous oxygen atom and oxygen in the condensed phase. Many other reactions involving only oxygen isotopes are conceivable, but the above four dominate because the other reactions involve collisions of more than three isotopes at the exchange site on the surface of condensates or because of the negligible abundance of 17O and 18O.
Net evaporation flux of 16O17O via reaction (A19-1) is expressed as follows:
where Pi is the pressure of gas species i, represents the kinetic barrier for exchange reaction of p in the gas and q in the condensed phase. The first term represents evaporation flux of 17O in molecular oxygen via exchange reaction of 16O2 gas molecule with 17O in the condensed phase, and the second term represents condensation flux of 17O through exchange reaction of 16O17O gas molecule with 16O in the condensed phase. Evaporation flux of 16O18O (), 17O (), and 18O () through remaining three exchange reactions (A19-2–A19-4) can be similarly obtained. Equation A21 can be expressed as follows:
By using dimensionless pressure , defined in (A17), the above flux owing to exchange reaction is written as follows:
From the equilibrium condition for exchange reaction, , we can introduce equilibrium fractionation factor of the exchange reaction:
Adding A22 and A10 after nondimensionalization of A13 and using A23, flux equation for 16O17O is given by the following equation:
where is the magnitude of kinetic barrier for exchange reaction relative to the evaporation coefficient of oxygen, appeared in (A16) and is defined as follows:
where is assumed. In the same manner, nondimensionalized flux equation of 17O can be derived as follows:
Important parameters in Equation A26 are and , which are called isotope exchange efficiency in the main text and vary from 0 (no effect owing to exchange reactions) to 1.0 (similar efficiency as net-transfer reactions).
In model calculations, the contribution of SiO gas species on oxygen isotopic fractionation was taken into consideration. Although we have to deal with homogeneous isotopic reaction in the gas because of temporal change of isotopic composition of each gaseous species, we did not treat the reaction kinetics assuming complete isotopic equilibrium among different gaseous species (O, O2, and SiO) or no isotopic exchange among the species. The two cases limit the actual situation in homogeneous reaction in the gas. We only present the equilibrium model in this article, but the results are essentially the same. In this article, equilibrium fractionation factor for exchange reaction between melt and gas as defined in Equation A23 is assumed to be unity, and there is no isotopic fractionation at equilibrium. The effect of this parameter is shown in Fig. A4 for a flash-heating model with evaporation and recondensation as illustrated in Fig. 1a and 1b.