We have measured the vapor pressure of solid SiO as a function of temperature over ... Using the new vapor pressure curve makes a significant difference in the.
The Astrophysical Journal, 649:1178Y1183, 2006 October 1 # 2006. The American Astronomical Society. All rights reserved. Printed in U.S.A.
SILICATES DO NUCLEATE IN OXYGEN-RICH CIRCUMSTELLAR OUTFLOWS: NEW VAPOR PRESSURE DATA FOR SiO Joseph A. Nuth III1 and Frank T. Ferguson1,2 Received 2006 January 27; accepted 2006 May 25
ABSTRACT We have measured the vapor pressure of solid SiO as a function of temperature over the range from 1325 up to 1785 K in vacuo using a modified Thermo-Cahn thermogravimetric system. Although an extrapolation of the current vapor pressure data to 2200 K is close to that predicted from the work of Schick under reducing conditions, the vapor pressures measured at successively lower temperatures diverge significantly from such predictions and are several orders of magnitude lower than predicted at 1200 K. This new vapor pressure data has been inserted into a simple model for the gas expanding from a late-stage star. Using the new vapor pressure curve makes a significant difference in the temperature and stellar radius at which SiO gas becomes supersaturated, although SiO still becomes supersaturated at temperatures that are too low to be consistent with observations. We have therefore also explored the effects of vibrational disequilibrium (as explored by Nuth & Donn) of SiO in the expanding shell on the conditions under which nucleation occurs. These calculations are much more interesting in that supersaturation now occurs at much higher kinetic temperatures. We note, however, that both vibrational disequilibrium and the new vapor pressure curve are required to induce SiO supersaturation in stellar outflows at temperatures above 1000 K. Subject headinggs: astrochemistry — methods: laboratory — stars: mass loss
1. INTRODUCTION
the gas in the outflows to induce even a barely acceptable rate of grain formation (Draine 1979). Sedlmayr and his colleagues (Gail et al. 1984; Gail & Sedlmayr 1986; 1998) have been modeling the nucleation of silicate grains for many years and have made considerable effort to include the most realistic physical processes in their models. Despite their efforts to model the formation of circumstellar silicates via homogeneous nucleation under the conditions in which these materials are observed to form, they have been unsuccessful and have now turned to other explanations for the formation of silicate dust, such as heterogeneous condensation of silicates on titania nuclei. Observations of dust shells around oxygen-rich, AGB stars by Danchi et al. (1994) indicate the presence of silicate grains at temperatures of approximately 1000 K, placing an important constraint on the homogeneous nucleation of these grains; if these silicate grains do not form via homogeneous nucleation above this temperature, then they likely form via heterogeneous growth on preexisting seed nuclei. Jeong et al. (2003) examined the viability of homogeneous nucleation for grain formation for five of the most abundant species in these oxygen-rich, AGB outflows: Fe, SiO, SiO2, Al2O3, and MgO. Both SiO and Fe would seem to be likely condensates, but the peak nucleation flux for SiO does not occur until the gas cools to approximately 600 K, while a significant nucleation flux for Fe only occurs below 1000 K. Al2O3 has been suggested as a possible initial condensate because of its extremely low vapor pressure, but thermodynamic modeling by Patzer et al. (1999) has shown that Al2O3 is highly unstable in the gas phase, making formation of Al2O3 grains via homogeneous nucleation very unlikely. Jeong et al. (2003) have also shown that the nucleation flux of the remaining primary candidates, SiO2 and MgO, are quite negligible. In contrast, circumstellar Al2O3 grains are relatively abundant in meteorites (Nittler et al. 1997), although perhaps not as abundant as would be expected if all of the aluminum condensed in these grains (Alexander 1997, 2001). Observations of two Al2O3 grains, one corundum and one amorphous, show no evidence of heterogeneous accretion (Stroud et al. 2004; Stroud 2005).
For more than a quarter of a century infrared observations of circumstellar outflows around low-mass red giants and oxygenrich asymptotic giant branch (AGB) stars have revealed the distinctive absorption features near 10 and 20 m associated with silicate grains. Temperatures of these circumstellar materials vary from as high as 1000 K down to only a few hundred kelvins. In some circumstellar outflows the crystalline silicate minerals olivine and enstatite have been observed, and in each case the crystalline mineral seen is the nearly pure magnesian end member of the solid solution series (Waters et al. 1996; Molster 2000). No observations of crystalline iron-rich silicates have ever been reported in a circumstellar outflow, although both crystalline and amorphous iron and magnesium silicates are seen in primitive meteorites. Yet in spite of all that we do know about the grains found in oxygen-rich outflows, we do not understand how these grains form under the conditions where they are observed. Early attempts to model the formation of circumstellar silicates (Draine & Salpeter 1977, 1979; Yamamoto & Hasegawa 1977; Draine 1979) relied on classical nucleation theory (Becker & Doring 1935), which had originally been developed to explain the formation of small water droplets in the terrestrial atmosphere but had been suggested as a good method for modeling grain formation in circumstellar outflows (Donn et al. 1968). Unfortunately, many of the physical parameters required for the application of classical nucleation theory to the condensation of silicate grains were highly uncertain, and conditions in the outflows themselves violated some of the assumptions made in the derivation of the model (Donn & Nuth 1985). The results of these early models were usually disappointing, requiring considerable clumping of 1 Astrochemistry Laboratory, Code 691, NASA’s Goddard Space Flight Center, Greenbelt, MD 20771. 2 Chemistry Department, Catholic University of America, Washington, DC 20064.
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SILICATES IN OXYGEN-RICH CIRCUMSTELLAR OUTFLOWS However, TiO2 in solid solution may help the crystalline rather than the amorphous Al2O3 form. Hibonite (CaAl12O19) grains show no evidence for heterogeneous nucleation (Stroud 2005; Stroud et al. 2005) either, although there may now be one example (of more than 100 known circumstellar silicate grains) of a heterogeneously nucleated silicate grain with an iron-bearing spinel core (Vollmer et al. 2006). The only grains that consistently show evidence for heterogeneous nucleation are circumstellar graphite grains (Croat et al. 2005; Stroud 2005). This lack of primary condensates prompted Jeong et al. (2003) to look for less abundant but highly refractory species that may serve as seed nuclei for heterogeneous silicate growth. In particular, they have recently proposed that silicates form as coatings over more refractory titanium dioxide cores that can form deeper in the stellar atmosphere than can the silicates themselves (Patzer 2004). Although the formation of silicates via homogenous nucleation may require large supersaturations, the driving force for the heterogeneous growth of these species on refractory cores requires a much lower degree of supersaturation and can therefore occur at much higher temperatures. Unfortunately, this converts the problem to one that models the nucleation of tiny titania cores and the growth of a much larger amorphous iron-magnesium silicate shell. Although titania is more refractory than silica and will therefore survive at higher temperatures, it is much less abundant, and the basic physical parameters required by nucleation theory are even less certain. However, circumstellar TiO2 grains have been found in meteorites (Nittler & Alexander 1999; Stroud 2005). Application of classical nucleation theory to the condensation of these cores is therefore a very poorly constrained problem. In addition, there is no evidence that any significant fraction of circumstellar silicate grains extracted from meteorites contain more refractory cores (Messenger et al. 2003). Because there was some doubt concerning the ultimate stability of amorphous silicate grains even under the physical conditions where they had been observed to exist, we decided to measure the vapor pressure of SiO, the most important vapor-phase constituent of silicate dust. To this end we purchased a commercial thermogravimetric system (CAHN TG 2171), then modified the system to ensure that no oxygen could diffuse through microcracks that might develop in the alumina tube of the vacuum furnace as we cycled through measurements at temperatures approaching 2000 K. This was accomplished by enclosing the vacuum furnace in a box that was continuously purged by nitrogen gas. Measurement of the vapor pressure was carried out using the Knudsen cell technique, wherein the vapor over a solid or liquid sample of interest is lost through a small hole in an otherwise closed container maintained at the temperature of interest. With a sufficiently small hole, one can approach arbitrarily close to equilibrium vapor pressure within the cell. Knowing the size of the hole, the vapor pressure can be calculated from the mass-loss rate using the ideal gas law collision rate. Corrections to this massloss rate must be made, however, because the vapor must exit the cell through a ‘‘pipe’’ rather than through a hole in an infinitely thin wall, and because this mass loss creates a small (3%) disturbance to equilibrium conditions within the cell. A detailed description of the experiment and the techniques used to analyze the data produced can be found in Ferguson & Nuth (2006). As a calibration of our system, these same techniques were used to measure the vapor pressure of palladium metal, a very well behaved noble metal with a well-known vapor pressure. The vapor pressure of palladium metal determined in our experimental system is shown in Figure 1 and is in excellent agreement with the literature value (Arblaster 1995). These palladium measurements validate the use of our equipment and of our data reduction methods
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Fig. 1.—Vapor pressure measurements of palladium metal as a function of temperature over the range from 1400 to 2000 K. The symbols represent individual measurements under different experimental conditions, while the solid line represents the best available vapor pressure estimate for palladium metal (Arblaster 1995). The dashed vertical line is the melting point. These data are presented as a validation of our experimental technique.
for measurement of the vapor pressure of refractory samples over the temperature range of interest for SiO. Our measurement of the vapor pressure of SiO is shown in Figure 2 together with the vapor pressure curve generally used in modeling the nucleation of silicate grains in circumstellar outflows (Schick 1960). We also show a second vapor pressure curve from Schick (1960) that had been identified as applicable to more oxidizing conditions. There are a few other measurements of the vapor pressure of pure SiO over more restrictive temperature ranges in the literature (Gel’d & Kochnev 1948; Guenther 1958; Kubaschewski et al. 1967; Rocabois et al. 1992), and these are also plotted in Figure 2. It is obvious that the additional measurements agree quite well with our new measurements and with the ‘‘second’’ vapor pressure equation suggested by Schick (1960). Although SiO is commercially available and is used extensively in the preparation of optical coatings, there is no precise thermodynamic or structural data available for this material. Some previous measurements of the SiO vapor pressure used mixtures of Si metal and silica as starting materials for vapor pressure measurements rather than pure SiO. These experiments are more complex and are fully described in Ferguson & Nuth (2006). Our
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Fig. 3.—(a) Contour plots of log (Cux/Hatom ) as a function of total pressure and temperature. The flux is based on classical nucleation theory. The box in each panel represents the range of pressure and temperature conditions where grains are expected to form (Jeong et al. 2003). The vapor pressure of SiO was calculated using the equation of Schick (1960). (b) Same as (a), but for the vapor pressure of SiO calculated using the new equation reported here. (c) Same as (b), but for the value for the SiO supersaturation used in the nucleation calculations determined by the SiO vibrational temperature, Tv , rather than by the kinetic temperature of the gas. (d ) Same as (a), but for the value for the SiO supersaturation used in the nucleation calculations determined by the SiO vibrational temperature, Tv , rather than by the kinetic temperature of the gas.
Fig. 2.—Vapor pressure measurements of SiO are presented as a function of temperature over the range from 1220 to 1800 K. The thin solid line is the thermodynamic vapor pressure curve predicted by Schick (1960), while the dashed line is another equation for the vapor pressure of SiO, also from Schick (1960), said to apply to more oxidizing environments. The squares represent measurements of the vapor pressure of SiO synthesized by reacting silicon metal with SiO2 before the start of the experiments by Gunther (1958). Triangles are data obtained by Gel’d & Kochnev (1948) on solid SiO. The thick solid line is a more recent assessment of the properties of SiO by Kubaschewski et al. (1967), while the thick dashed line is a best-fit line to measurements made by Rocabois et al. (1992) to assess the difference in vapor pressure between SiO solid and a mixture of Si metal and silica.
measurements of the direct vaporization of commercially available, 99.9% pure, SiO solid are not complicated by other vapor-phase species such as Si metal or SiO2. We find that the SiO solid vaporizes completely from the alumina crucibles, leaving no residues or secondary reaction products. We have never seen a change in the slope of our mass-loss measurements at any given temperature as would occur if we were to begin evaporating a residue of Si metal or SiO2; we simply run out of material in the crucible and the mass-loss rate drops to zero. These observations support the hypothesis that SiO solid evaporates directly to SiO vapor. Our vaporization experiments indicate that solid SiO is more stable than previously thought (e.g., Schick 1960) and should therefore nucleate and grow much more readily at higher temperatures. We explore these differences below. 2. SiO VAPOR PRESSURE AND SILICATE NUCLEATION The equations for classical nucleation theory are well developed, as are the equations describing the evolution of the local tem-
perature, total pressure, and vapor pressure of SiO. The flux of droplets, J (number of nuclei cm3 s1), of a critical size sufficient to continue growing by the addition of monomers is given by " # 2 2 16 3 V 2 J¼ n V exp ; m 1 3ðkT Þ3 (ln S)2 where is the surface free energy of the developing cluster, m and V are the mass and volume of the condensing molecule, respectively, n1 is the number concentration of the condensing species, T is the condensation temperature, and S is the degree of supersaturation of the condensing species. The value of the surface free energy for SiO is unknown but has been estimated to be 350 ergs cm2 (Nuth & Donn 1982). Jeong et al. (2003) produced an extremely useful visual method to show the likelihood that silicate (or any other) vapor will nucleate in a circumstellar outflow, and we have adopted this method in Figure 3. Silicate grains are believed to form within a pressure range of 102 to 1 dynes cm2 and at temperatures from 1000 to 1200 K. Jeong et al. (2003) estimate that the density of dust grains in oxygen-rich, circumstellar dust shells per total number of hy13 require drogen atoms is approximately 10 and would therefore nucleation rates, log J /nhHi , to be approximately 1022Y1014. Contour plots are constructed out of the predicted nucleation rates, normalized to the total hydrogen number density, as a function of temperature and pressure to identify condensates with suitable nucleation rates that pass through this expected grain formation window. Such a plot (Fig. 3a) has been constructed
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using the ‘‘old’’ vapor pressure equation of Schick (1960); e.g., P ¼ (1 atm) expf12:81 ½29;475/T ðKÞg. The box near the center of the figure shows the conditions where silicate grains are believed to nucleate. Figure 3b is a similar plot run using our newly obtained vapor pressure equation, P ¼ (1 atm) expf17:56 ½40;760/T ðKÞg. One can see that although the new vapor pressure equation brings the box closer to stellar conditions in which nucleation is observed to occur, more is required. 3. VIBRATIONAL DISEQUILIBRIUM For strong, energetic vibrational transitions at low pressure in optically thin environments, collisional excitation and photon absorption cannot balance the spontaneous emission processes in molecules such as CO and SiO (Nuth & Donn 1981). In this case the population density in the upper vibrational levels drops dramatically. We assume that lower vibrational temperature, Tv , promotes the nucleation of SiO by offering a wider number of rotational-vibrational states that SiO clusters might occupy following reactive association. For SiO, Tv is given by the equation 1þ(C1;0 =A1;0 )þWQ ; Tv ¼ T = 1 þ T ln (C1;0 =A1;0 )þWQ exp (=T ) where T is the kinetic temperature of the gas, C1,0 is the rate of collisional deexcitation from energy level 1 to 0, and A1,0 is the Einstein A coefficient representing the rate of spontaneous transition from v ¼ 1 to the ground state. The term is the characteristic temperature of the transition given by ¼
hc! ; k
where h and k are Planck’s and Boltzmann’s constants, respectively, c is the speed of light, and ! is the energy (in wavenumbers) of the transition. The effect of radiative transitions on the vibrational temperature is included in the term Q, given by 1 1 Q ¼ exp : T In the expanding outflow, the amount of radiation received at any distance from the star will depend on the angle subtended by the star at that point. In this work the factor W has been introduced as a dilution factor to account for this diminishing radiative effect with distance from the star. It ranges from 0 at an infinite distance from the star to a maximum value, 0.5, at the stellar photosphere. The Einstein A value for SiO was taken from the work of Hedelund & Lambert (1972). The value for the collisional deexcitation rate, C1,0, was calculated based on the empirical correlations for vibrational relaxation times developed by Millikan & White (1963) and later adopted for vibrational disequilibrium calculations by Thompson (1973). A more complete description of the vibrational temperature calculations for SiO is given in the work of Nuth & Donn (1981). In the following nucleation calculations, we use Tv to calculate the supersaturation, S, and use the kinetic temperature of the gas for all other calculations. The results of these calculations, using our newly determined vapor pressure, are shown in Figure 3c. Note that in this case, a very high nucleation rate occurs ‘‘within the box,‘‘ e.g., under exactly the conditions in which nucleation is expected to occur. Figure 3d presents the same calculations using the Schick (1960) vapor pressure equation. As can be seen in this figure, vibrational disequilibrium alone is not sufficient to cause SiO to nucleate in circumstellar outflows. Models of silicate
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nucleation based on our new vapor pressure equation therefore seem to require some other parameter or process, such as vibrational disequilibrium, to account for the onset of grain formation. 4. DISCUSSION We would prefer to calculate the rate of grain formation based on a reasonable estimate of the chemical reaction rates for the polymerization of SiO molecules or, even better, for the formation and growth of the amorphous iron and magnesium silicates that are observed to grow in circumstellar outflows. Unfortunately, none of the required reaction rates are known, and we have little confidence that they will be available any time in the near future. Classical nucleation theory was designed to handle just such situations in which only a limited amount of information is known about a specific condensate. However, Donn & Nuth (1985) have argued that nucleation theory might not be applicable to condensation in circumstellar shells. Classical nucleation theory was derived for use in terrestrial environments where certain critical assumptions hold true. Specifically, the theory assumes that a steady state distribution of clusters exists in the gas and that this distribution can be calculated based on the temperature and pressure of the gas, the ‘‘surface energy’’ of the condensate, and the binding energy of the thermodynamically stable solid or liquid. First, the surface energy for a solid, such as SiO, is poorly determined and may not be applicable to the formation of amorphous magnesium silicate and iron silicate grains. Second, the concept of ‘‘vibrational disequilibrium’’ provides a very significant level of complication to calculations of the relative stabilities of various SiO clusters, as the vibrational temperature of each specific cluster is likely to be somewhat different from that of any other cluster under the same conditions of temperature, pressure, and optical depth (Nuth et al. 1985). Third, the condensates themselves do not exist in the gas phase but instead must be formed by the reaction of SiO, Mg, Fe, O, and OH; because these constituents react to form amorphous grains, the binding energy of the condensate solid or liquid is also uncertain and may change as the reactions proceed. Finally, even if a stable precondensation cluster distribution does exist at some point in the stellar outflow, it is not clear that such an equilibrium distribution can be maintained as the gas expands. At some point in the outflow, the reactant population will freeze out as the gas cools faster than the rates of chemical reactions can adjust the population. Having listed some of the problems with the application of classical nucleation theory to circumstellar environments, we must also admit that no appropriate substitute is yet available. Until one does become available, we must continue to use the modified version of this theory in an attempt to understand the formation of silicates in circumstellar outflows. The most radical modification that we have made involves the use of the vibrational temperature to calculate the supersaturation of SiO. Using Tv to calculate S assumes that cooler monomers will indeed react more rapidly to form (SiO)n clusters and that these larger clusters will also be cooler than an equilibrium population. In fact, larger clusters will be closer to equilibrium than will the monomers, because energy can be fed into the rotationalvibrational populations through lower energy rotational transitions (Nuth et al. 1985), but they will still be less populated at the upper vibrational levels than at equilibrium. In addition, as the number of monomers in a cluster increases, vibrational disequilibrium is likely to become less important as a growth factor because large clusters will more easily partition the energy released by the addition of a single monomer among its many vibrational modes.
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The nucleation rate is sensitive to Tv , and the calculation of Tv is extremely sensitive to the assumed value for the optical dilution coefficient, W. If we use a geometric method to calculate W, Tv plunges dramatically, even relatively close to the photosphere (where W ¼ 0:5). A more realistic approach would account specifically for the optical environment seen by the monomers and clusters at the wavelengths where they absorb. Of course, if we start with a system where no grains exist and nucleate silicate dust, one might expect that this would change the optical environment of the clusters, increase Tv , and thereby inhibit further nucleation. If the rate of nucleation and growth is very high, then the dust above the first point of nucleation could render the system optically thick in the infrared. Under these conditions W equals unity, the system is in LTE, and Tv equals the gas kinetic temperature. The system we have described in Figure 3c applies most closely to the initial formation of an optically thin dust cloud in a circumstellar outflow. As the dust mass in the shell increases, the dust formation rate must drop as Tv becomes closer to the gas kinetic temperature. In order to model more massive dust shells, a more realistic approach must be found for the calculation of the vibrational distribution of molecules in the SiO cluster distribution. Such complexities are beyond the scope of this work, as are complications that might be induced by assuming nonspherically symmetric outflows, stellar pulsations, or shocks. We note that in Figure 3c silicate nucleation begins at temperatures as high as 1200 K, although we believe that this is an overestimate due to the sensitivity of Tv to W. In a real stellar atmosphere the SiO molecules will not radiate directly to space without interference, especially after the nucleation process begins to produce a dusty atmosphere. Nevertheless, such a high temperature does imply that it may be possible to nucleate grains at a sufficiently high temperature to induce some degree of annealing before the initially formed grains reach the cooler environment of the outer shell. Thermal annealing of amorphous magnesium silicate grains occurs on timescales of minutes at temperatures near 1200 K (Hallenbeck et al. 1998, 2000) and hours at temperatures near 1100 K. If dust nucleation begins at high temperatures and continues to occur at continuously lower temperatures (because grain growth is sufficiently slow that the SiO monomer concentration is not depleted on a rapid timescale), then it is possible that these first formed grains could anneal to crystallinity, while the grains formed at lower temperatures would remain amorphous. In addition, we might also see amorphous silicate coatings forming over crystalline grains;
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although none have yet been observed, few presolar crystalline silicates have even been found. It may therefore be possible to test models for silicate nucleation by comparison of the predicted versus observed fraction of crystalline dust in the outflows of a variety of oxygen-rich stars. 5. CONCLUSIONS The new vapor pressure equation for SiO reported above greatly increases the level of supersaturation in outflows compared to the previously accepted expression. Even so, this correction is insufficient by itself to induce SiO nucleation in most circumstellar outflows; vibrational disequilibrium must also be invoked in order to nucleate refractory silicates. Accounting for vibrational disequilibrium while using the currently accepted vapor pressure equation does not induce nucleation. Using vibrational disequilibrium together with the new vapor pressure works, and classical nucleation theory predicts reasonably high nucleation rates at temperatures as high as 1200 K. This is in remarkable agreement with observational constraints given the uncertainties in classical nucleation theory. Future work aimed at better understanding radiative transfer in a dust-forming stellar outflow would be helpful, as the calculated nucleation rate is very sensitive to Tv , and Tv in turn is highly dependent on the optical dilution factor, W. Some degree of understanding of the effects of these choices might be obtained by comparison of the degree of crystallinity calculated to exist in models of the outflows of oxygen-rich AGB stars as a function of mass-loss rate versus the fraction of crystalline silicates actually observed in these outflows. We fully acknowledge that there are problems with both the physical parameters used, with the application of classical nucleation theory to silicate condensation, as well as with the potential violation of the assumptions essential to the derivation of the model when it is applied to circumstellar outflows. Unfortunately, there is no currently workable alternative model for this process. Until such time as we have available both measured rate constants and an appropriate reaction network leading to the production of silicate grains, some version of classical nucleation theory will be needed to provide a working model for silicate condensation. This working model will benefit from the use of the more accurate SiO vapor pressure equation presented above, as will simple thermochemical calculations of condensation in other astrophysical environments such as in protostellar nebulae.
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