A new constraint for chondrule formation: condition for the rim formation of barred-olivine textures
© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB. 2011
Received: 18 October 2010
Accepted: 3 June 2011
Published: 2 February 2012
A barred-olivine (BO) chondrule usually has an olivine rim that covers the chondrule surface. Numerous experiments have been carried out to reproduce the BO texture. However, the rim structure could be reproduced only in a few studies reported in the literature. The difficulty in reproducing the rim structure lies in the fact that its formation condition has not been constrained experimentally or theoretically. In the present paper, we have carried out numerical simulations of crystal growth of a highly-supercooled melt droplet of pure forsteritic composition (Mg2SiO4), and succeeded in reproducing the double structure, i.e. the rim and the dendrite. The droplet cools from the surface, the temperature of which should be cooler than the center of the droplet. Since a crystal grows faster along the cooler surface than across the hotter center, the rim was found to be formed when the temperature difference between the center of the droplet and its surface is large enough. From our results, both from numerical simulations and analytical consideration, we found that the double structure of rim and the dendrite could be formed only when the cooling rate is within a narrow range, which depends upon the degree of supercooling. Our results, for the first time, could explain why the formation of rim of BO texture was hardly reproduced in the previous experiments reported in the literature to date.
Chondrules are millimeter-sized, spherical-shaped grains containing olivine, pyroxene, metal, sulfide, and glass with igneous textures. They are considered to have been formed from molten droplets about 4.6 billion years ago in the solar nebula (Amelin et al., 2002); it is believed that they melted and cooled again to solidify in a short period of time (Sorby, 1877; Nelson et al., 1972; Tsuchiyama et al., 1980; Lofgren and Russell, 1986; Jones and Lofgren, 1993; Osada and Tsuchiyama, 2001; Tsuchiyama et al., 2004). Since chondrules possess about 80 vol.% of chondritic meteorites in the most abundant cases (Jones et al., 2000), they must carry information about the early history of our solar system. Chondrules have various textures commonly described as porphyritic, barred, and radial textures; all of these textures appear for the same chondrule bulk composition (Lux et al., 1981; Lofgren and Russell, 1986). Many authors have carried out dynamic crystallization experiments to constrain the formation condition of each texture (Hewins et al., 2005 and references therein), however, the formation mechanism is not yet fully understood.
Barred-olivine (BO) chondrules are characterized by parallel set(s) of olivine bars in a thin section (Weisberg, 1987). It has been considered that olivine bar crystals are actually platy in three-dimensions (Tsuchiyama et al., 2000; Noguchi, 2002). A BO chondrule usually has an olivine rim that covers the chondrule surface. This olivine rim has the same crystallographic orientation as inner olivine platelets. The olivine rim has not been reproduced in early dynamic crystallization experiments except in a limited number of runs (Lofgren and Lanier, 1990; Radomsky and Hewins, 1990). Tsukamoto et al. (1999, 2001) succeeded in reproducing the rim structure from a forsterite melt droplet in a container-less crystallization experiment using an aero-acoustic levitation technique. They found from their in-situ observation that the droplet cooled very rapidly at a rate of Rcool ≈ 100−1000 K s−1, and then crystallized at a large supercooling of Δ T ≈ 600 K within a very short period of time (less than 1 s). However, they reproduced the rim structure only in a few cases, so the formation condition was not constrained from their experiments. On the other hand, Tsuchiyama et al. (2004) also succeeded in reproducing the rim structure by evaporation in vacuum. The cooling rate was Rcool = 1000 K hr−1, which is much slower than that of the container-less experiments by about three orders of magnitude. They did not observe the solidification process of the sample in-situ, so detailed information such as the timing of nucleation, the solidification timescale, and crystal growth pattern were not available. They considered that the rim was formed by the rapid crystal growth along the droplet surface, which should become cooler than the interior because of the latent heat of evaporation. However, their hypothesis has not been verified yet.
To clarify the formation condition of the rim, the crystallization process inside the chondrule melt droplet should be investigated. We carried out numerical simulations of crystallization of a highly-supercooled melt droplet with a pure forsteritic (Mg2SiO4) composition (Miura et al., 2010). We considered the situation that a tiny crystal seeded at the droplet surface triggers crystallization of the droplet. We found that the rapid crystal growth along the droplet surface occurs when the cooling rate is very large. However, we did not investigate over a wide range of supercooling of the droplet in the previous paper.
The aim of this paper is to clarify the formation condition of the rim structure. We carry out numerical simulations for a wide range of supercooling and cooling rate by using the phase-field method, which is one of the most effective numerical methods to simulate crystal growth in a supercooled liquid. As a first step, we consider the situation of container-less crystallization experiments using levitation methods (Tsukamoto et al., 1999, 2001; Nagashima et al., 2006, 2008; Srivastava et al., 2010) because in these experiments the thermal profile of the droplet and the crystal growth pattern were observed in-situ, so we can verify the results of our numerical simulations. For comparison, we consider a droplet of pure forsteritic (Mg2SiO4) composition, which was adopted in these experiments.
We describe the basic equation of the phase-field method in Section 2, where the basic concept of the phase-field method is summarized in Appendix A. In Section 3, we show the results of the calculation, in which various crystal growth patterns appear inside the droplet. In Section 4, we discuss the conditions for rapid crystal growth along the droplet surface and propose a new constraint for chondrule formation, especially, for barred-olivine chondrules. We present some conclusions in Section 5.
2.1 Basic equations
Although one could introduce anisotropies in μ and σ to express an anisotropic interface and to simulate facet formation (McFadden et al., 1993; Uehara and Sekerka, 2003), for simplicity we do not take anisotropy into account.
The governing equation of phase φ, Eq. (1), was derived from an entropy function to satisfy the entropy production being positive in any small volume element (Wang et al., 1993). The details of the derivation will be found in the reference. We briefly describe the physical meaning of Eq. (1) in Appendix A.
2.2 Numerical scheme
Chondrules are three-dimensional spherules, so a three-dimensional calculation is desirable. However, this requires an unrealistically large number of computational nodes for a chondrule-sized object (~109μ m3). To reduce the computational cost, we consider a two-dimensional disk in the xy-plane (circle but not a sphere), assume uniformity in the z-direction and drop the derivative by z. Figure 1 shows a schematic picture of the square mesh used for computation. We adopt a computational domain of −250 ≤ x ≤ 250 μ m and −250 ≤ y ≤ 250 μ m to consider a droplet of 250 μ m in radius, and discretized by a 1000 × 1000 square mesh (mesh sizes for x- and y-directions are Δ x = Δ y = 0.5 μ m). Other numerical procedures are the same as Miura et al. (2010).
We carried out a convergence test with smaller mesh sizes of Δ x = Δ y = 0.25 μ m, and found that the calculation result was not modified except for minor changes in the shapes of dendrite tips.
We put a seed crystal at the droplet surface to trigger the crystal growth of the supercooled droplet. At the beginning of the calculation, we substitute φ = 0 only at the seeding point without any change in temperature (see Fig. 2(a), for example). The radius of the seed crystal is assumed to be 10 μ m.
2.3 Initial temperature profile
2.4 Physical properties of forsterite sample
In this study, we consider the sample of pure forsteritic composition. The physical properties we adopted are as follows; c = 5.7 × 107 erg cm −3 K −1, L = 2.4 × 1010 erg cm −3, TM = 2163 K (Nagashima et al., 2006, 2008), κC = 5.0 × 105 erg cm −1 K −1 s−1 (Pertermann and Hofmeister, 2006), κl = 2.0 × 105 erg cm −1 K −1 s−1 (Moriya, 1963), σ = 620 erg cm −2 (Tanaka et al., 2008), μ = 0.4 cm s−1 K −1 (Tsukamoto et al., 1999, 2001; Nagashima et al., 2006), and W = 0.25 μ m (Murray et al., 1995), where κC is the thermal conductivity of the crystal phase. These parameters are the same as those adopted in Miura et al. (2010).
2.5 Input parameters
2.5.1 Heat flux at droplet surface qs
Let us estimate the heat flux qs at the droplet surface from container-less crystallization experiments (Tsukamoto et al., 1999; Nagashima et al., 2006, 2008). Samples were melted during levitation using a CO2 laser and then quenched by turning off the laser (Tsukamoto et al., 1999). In this case, the droplet cools at a maximum rate due to the thermal radiation and the heat conduction by the gas-jet (Nagashima et al., 2008). The cooling rate can be controlled to be slower by adjusting the output power of the laser (Nagashima et al., 2006).
Cooling rate Rcool of molten droplet in container-less crystallization experiments using levitation methods. We read Rcool from each cooling curve obtained at each experiment just before the droplet crystallized. We calculate the heat flux qs and the initial temperature difference between the center and surface of the droplet δTc − s,0 by using Eqs. (5) and (4), respectively. Refs:  Tsukamoto et al. (2001),  Nagashima et al. (2006),  Nagashima et al. (2008).
Cooling rate Rcool [K s−1]
Heat flux qs [erg cm−2 s−1]
Temp. diff. δTc− s,0 [K]
≈6.7 × 108
≈1.0 × 108
≈1.3 × 108
2.5.2 Supercooling at droplet surface ΔTs,0
The molten droplet becomes largely supercooled by a few hundred K or more in the container-less environment (Nelson et al., 1972; Tsukamoto et al., 1999; Nagashima et al., 2006, 2008; Tanaka et al., 2008). Tsukamoto et al. (1999) carried out container-less crystallization experiments by using an aero-acoustic levitator and observed a forsterite melt droplet being supercooled by Δ T ≈ 600 K at a cooling rate of Rcool ≈ 400 K s−1. Nagashima et al. (2006) adopted a gas-jet levitator and observed a forsterite melt droplet being supercooled by Δ T ≈ 1000 K at a cooling rate of Rcool ≈ 100 K s−1. Tanaka et al. (2008) modeled the homogeneous nu-cleation and sequential crystal growth inside a supercooled melt droplet based on classical nucleation theory and found that a forsterite melt droplet should be supercooled by about 1000 K even if it cools at a much smaller cooling rate, e.g., Rcool = 10−1 K s−1.
In this paper, we suppose that a forsterite melt droplet never nucleates homogeneously at a relatively low supercooling (ΔT < 600 K), so the crystallization is triggered by a collision with a micron-sized crystal (seeding). Depending on the timing of the seeding, crystallization occurs at various values of supercooling. In this paper, we adopt values of the supercooling at the droplet surface when the seeding occurs of ΔTs,0 = 200 K, 300 K, 400 K, 500 K, and 600 K.
3.1 Dendritic growth
Figure 2 shows a result of phase-field simulation for ΔTs,0 = 200K(Ts,0 = 1963 K)and qs = 5 × 108 ergcm−2 s−1. This is a gray contour map of the phase value ranging from φ = 0 (solid phase, black) to φ = 1 (liquid phase, white). The transition layer between the solid and liquid is too thin to distinguish in this map, meaning that the solid-liquid interface is very sharp. At the beginning, there is a tiny seed crystal on the right hand side at the droplet surface (panel (a)). At 0.02 s after seeding, a dendritic crystal grows from the seed crystal (panel (b)). The temperature distribution is shown by isothermal lines in terms of T − TM. Crystal growth is accompanied by release of the latent heat of crystallization, so the temperature around the dendritic crystal increases up to almost the melting point. On the other hand, the supercooling far from the dendritic crystal is still large, resulting in a steep temperature gradient ahead of the growing dendrite tips. The dendritic crystal is spreading into the entire region inside the droplet (panel (c)). Around 0.06 s after seeding, the rapid dendritic crystal growth comes to a stop because the temperature inside the droplet increases to the melting point (panel (d)). It is found that a liquid phase still remains at the gap between dendrite tips. Finally, the remaining liquid solidifies completely as the latent heat of crystallization is removed by the surface cooling. It took 0.42 s to solidify completely. The luminosity of the droplet increases rapidly from panel (a) to (d), and then decreases gradually (Miura et al., 2010).
3.2 Dendritic growth with rim structure
Figure 3 shows a result for ΔTs,0 = 300 K (Ts,0 = 1863 K) and qs = 2 × 109 erg cm−2 s−1. At the beginning, one can see concentric isothermal lines of T0 − TM = −250 K and −200 K (panel (a)), meaning that there is a large temperature difference between the center and surface of the droplet. The temperature difference calculated by Eq. (4) is δTc−s,0 = 125 K, so the temperature at the droplet center (T0 − TM = −175 K) is 175 K below the melting point. At 0.008 s after seeding, a dendritic crystal grows from a seed crystal as well, as shown in Fig. 2(b) (panel (b)). At 0.016 s after seeding, it is found that the dendritic crystal growing across the droplet center is suppressed compared with Fig. 2(c) (panel (c)). At 0.024 s after seeding, the crystal is growing rapidly along the droplet surface (panel (d)). The crystal along the droplet surface is one of the branches of the dendritic crystal, and it seems to be surrounding the droplet like the rim of a BO chondrule.
3.3 Rim growth
Figure 4 shows a result for ΔTs,0 = 400 K (Ts,0 = 1763 K) and qs = 1 × 1010 erg cm−2 s−1. One can see dense concentric isotherms at the beginning (panel (a)). The temperature difference between the center and surface is calculated to be δTc−s,0 = 625 K from Eq. (4). The inner most isotherm is T0 − TM = 200 K, indicating a temperature above the melting point. The crystal growth pattern is completely different from Figs. 2 and 3. No dendritic crystal grows inside the droplet (panel (b)). The crystal grows only along the droplet surface where the temperature is below the melting point. At 0.016 s after seeding, almost of all of the droplet surface is covered with the crystal (panel (c)). There is liquid remaining inside the crystal rim (panel (d)). The remaining liquid gradually solidifies from the outside to the inside as the latent heat of crystallization is removed through the shell crystal.
3.4 Parallel dendritic growth
3.5 Isotropic growth
4.1 Constraint on cooling rate for rim formation
Here, we derive the constraint on the cooling rate for rim formation based on the discussion of Miura et al. (2010).
It should be noted that the condition given by Eq. (11) or (12) is applicable only when τs < τcool, where τcool is the cooling timescale of the droplet (Miura et al., 2010). If not, the droplet temperature drops considerably during solidification, so Eq. (3) cannot be used in this analysis. We can apply the condition given by Eq. (11) or (12) to our results because the typical growth timescale of τs < 0.1 s is shorter than the cooling timescale of τcool > 0.1 s.
4.2 Condition to produce each growth pattern
The dashed curve is the parameter space below which the entire droplet is hypercooled. It is found that isotropic growth (see Section 3.5) occurred at conditions below the dashed curve.
A large star symbol indicates the condition of the container-less crystallization experiment by Tsukamoto et al. (1999), in which the rim surrounding parallel sets of olivine bars was successfully reproduced from a pure forsteritic composition melt droplet. In their experiment, the droplet crystallized at a supercooling of ΔTs,0 ≈ 600 K, which gives the normalized surface undercooling of Δθs,0 ≈ 1.4. The temperature difference inside the droplet was estimated as δTc−s,0 ≈ 170 K (see Table 1), which gives α ≈ 0.4. Since this experimental condition satisfied the condition for rim formation, their sample having the rim structure is a reasonable result.
We consider the experimental condition of Tsuchiyama et al. (2004). In their experiment, the cooling rate of the droplet was Rcool = 1000 K hr−1 if we assume that it is the same as the cooling rate of the furnace. The temperature difference between the center and surface of the droplet can be estimated as δTc − s,0 ≈ 0.02 K for a sample radius of rd = 500 μ m and with typical thermodynamic properties of silicates (c = 3 × 107 erg K−1 cm−3 and κl = 2 × 105 erg cm−1 s−1 K−1, Murase and McBirney, 1973). From the condition of Eq. (11), the molten sample in their experiment must have crystallized at very low supercooling as ΔTs,0 < 0.1 K to form the rim. The nucleation at such very small supercooling might be due to the contact of the droplet to the carbon capsule. However, the exact value of the nucleation supercooling was not measured, so we cannot test our new constraint with their experiment. In-situ temperature measurement during droplet solidification is required to clarify the condition of rim formation.
4.3 Implication for chondrule solidification textures
The numerical model that we adopted in this study was used to simulate the crystal growth process in a supercooled melt droplet of a single composition, namely, the composition of the crystal phase is the same as that of the parent liquid phase. However, real chondrules are multi-component, namely, the composition of the crystal phase is not necessarily the same as the bulk composition of the parent liquid (e.g., Jones, 1990). Therefore, not only the temperature distribution inside the droplet, but also the partitioning and diffusion of elements near the crystal-liquid interface (Watson, 1996, 2004) should be taken into consideration. To elucidate the formation mechanism of chondrule solidification textures, it is required to model the crystallization process of a multi-component melt droplet at a largely undercooled state (e.g., Bi and Sekerka, 1998, 2002).
Before the multi-component modeling begins, we make some predictions for chondrule solidification textures based on our single-component phase-field calculations. In our model, the droplet after complete solidification is just a single crystal spherule, probably with no solidification texture remaining inside. However, if the droplet contains small amounts of incompatible elements in olivine, such as Ca and Al, these elements tend to partition into the remaining liquid phase during crystal growth (Libourel, 1999; Pack and Palme, 2003). The inhomogeneity in elemental composition should relate to the solidification texture. Therefore, the distribution of the remaining liquid phase inside the droplet has some implications for the formation mechanism of chondrule solidification textures.
The gray-scale map of τcry just visualized the difference of crystal growth timescale during solidification. The relationship to the chondrule solidification textures was not clear. However, we consider that the chondrule solidification textures reflect the crystal growth pattern inside the droplet. Our phase-field simulations are the first step to understand the formation mechanism of chondrule solidification textures.
The chondrule solidification texture reflects the crystal growth pattern inside the melt droplet at the time of formation in the early solar nebula. We numerically investigated the following two thermal effects on the crystal growth pattern; (i) the cooling at the droplet surface and (ii) the release of latent heat of crystallization. Surface cooling makes the droplet surface cooler than the center. Since the crystal growth rate is faster along the cooler surface than across the hotter center, the rim was formed when the cooling rate is large enough. The release of the latent heat of crystallization caused a rapid temperature increase near the growing rim, resulting in a very steep temperature gradient at the interface between the rim and the remaining liquid (melt) phase. This “reversed” temperature gradient led to den-drite formation due to the morphological instability. We found that the double structure of rim and dendrite could be formed only when the cooling rate is within a narrow range. An understanding of these thermal effects on chon-drule melt solidification is the first step to elucidate the formation mechanism of chondrule solidification textures.
We thank Dr. H. Kimura for his kind encouragement to submit this paper. We are grateful to Prof. Y. Inatomi for useful discussion on the results of our phase-field simulations. We acknowledge helpful comments from Prof. Joseph A. Nuth and an anonymous referee. H. M. was supported by Tohoku University Global COE Program “Global Education and Research Center for Earth and Planetary Dynamics”. This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (A), 22244066, 2010–2013.
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