Nakano and Hashimoto Progress in Earth and Planetary Science
https://doi.org/10.1186/s40645-020-00335-1(2020) 7:47

Progress in Earth andPlanetary Science

RESEARCH ARTICLE

Open Access

Bubbles to Chondrites-I. Evaporationand condensation experiments, andformation of chondrules
Yuki Nakano1*

and Akihiko Hashimoto2

Abstract
We propose a simple model of chondrule formation that is supported by our new experiments. With a laser-heating
and inert-gas-cooling technique, we obtained evaporation and condensation pathways starting with chondritic
compositions till ends, and extracted ‘relative volatilities’ of elements from them. Above boiling points, we observed
numerous silicate droplets being ejected from collapsed cavities of vapor bubbles on the surface of molten sample,
known as jet-droplets. We postulate jet-droplets as origin of chondrules. The formation mechanism of jet-droplets
requires a dense and large solid body (>3 cm across), named ‘duston’, for chondrule precursors. Our chondrule
formation model presumes dustons having CI-like composition. Upon boiling, a duston ejects jet-droplets from its
molten surface and simultaneously forms an adiabatically expanding vapor cloud around it. The jet-droplets
supercool and incorporate the supersaturated vapor and fine condensates while they travel through the cloud, thus
completing their makeup as chondrules. The compositions and the mixing ratio of the three components (jet-droplet,vapor and condensate) can be exactly predicted by using relative volatilities of elements, given the chondrulecomposition to be fitted and the conditions: vaporization degree (VD) and redox state (fs ) of the duston. We attempt
to reproduce bulk compositions of chondrules in total of 600. About 75% chondrules are successfully matched withspecific combinations of VD and fs for each chondrule. The model altogether explains 3.5 features of chondrules:
maximum size and size-frequency distribution; chemical variety; and textural variety.
Keywords: Evaporation, Condensation, Experiment, Chondrule, Chemical fractionation, Boiling, Jet-droplet,
Chondrule fromation, Protoplanetary disk, Solar system

Introduction
The present (I) and companion (II) (Hashimoto andNakano 2020) papers in tandem present a new formation mechanism of chondrules and chondrites, that begins
with bursting of silicate vapor bubbles and ends withasteroid formation as chondritic rubble piles: Bubbles to
Chondrites.
Chondrules are compact, mm-sized silicate spheruleswith igneous textures. They are major constituents, 20-80
vol.%, of chondritic meteorites, fragments of bodies in theasteroid belt (e.g., Wood 1988).
*Correspondence: yudaruma@ep.sci.hokudai.ac.jp
Department of Cosmosciences, Hokkaido University, N10 W8, Sapporo
060-0810, Japan
Full list of author information is available at the end of the article
1

A model of chondrule formation must comply with 3.5
fundamental features that characterize chondrules, apartfrom the fact that they were once molten. 1) They have amaximum size of 1-5 mm in diameter. In addition, theirsize frequency distributions roughly obey log-normal distributions (e.g., Friedrich et al. 2015). 2) Their chemicalcompositions are highly variable (e.g., Jones et al. 2005).
3) They exhibit various petrographic textures that may
be related to their thermal histories even amongst chondrules with similar compositions (e.g., Hewins 1997).
Chondrule formation models advocated so far (many references in Boss 1996; Desch et al. 2012; Connolly andJones 2016), are mostly concerned with heating mechanism to melt preexisting solids, but do not generally
address the above features.

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Nakano and Hashimoto Progress in Earth and Planetary Science(2020) 7:47

Astrophysical observations have revealed that a protostar and a protoplanetary disk (PPD) evolve in concert andform from the same source materials, gas and dust of lowtemperature origin that evolved from interstellar medium
in accord with physical conditions in PPD (e.g., Drain
2003; Natta et al. 2007; Waters 2015). Macroscopic dust
aggregates that are random collections of lightly modified
interstellar dust grains, originally 0.02-1 μm in size (Kimet al. 1994), will be uniform in bulk chemical composition.
Thus, the composition of the source material for chondrules should be identical to the composition of the Sun(except gases) or that of CI chondrites.
Observation of chondrules suggests that four processes,
in combination, were involved in establishing the chondrule properties: melting, evaporation, condensation andcrystallization (e.g., Hewins 1997). As pointed out by
Sears et al. (1996) and Krot et al. (2004), evaporationand condensation are a requisite for bringing about thevariation in chondrule composition. Therefore, a study
on evaporation and condensation of silicate systems isa key to understanding detailed conditions of chondrule
formation.
Evaporative chemical fractionation of chondritic materials has been studied experimentally (Hashimoto et al.
1979; Hashimoto 1983; Floss et al. 1996; Wang et al. 2001;
Yu et al. 2003). These experiments provided evaporative
fractionation trends, but were of limited applications as
their experimental conditions were selective. Nevertheless, they have been extensively used to develop evaporation models that could be applied over a wider range ofconditions (e.g., Alexander 2001, 2002; Fedkin et al. 2006,
2012; Fedkin and Grossman 2013). Condensation of themulti-component silicate vapor has not been studied yet.
For a comprehensive understanding of chemical fractionations caused by evaporation and condensation, we havebuilt a new experimental apparatus that allows simultaneous investigations of evaporation and condensation.
In “Experimental methods” section, we describe hightemperature boiling experiments on chondritic materials
using a CO2 laser as a heat source. In “Experimental
results” section, we investigate two products of the experiments: numerous droplets ejected from the silicate liquids
and ultrafine vapor condensates. They define chemical
fractionation trajectories caused by evaporation and condensation, from which we extract a new quantity, ‘relative
volatilities’ of elements to be used in the following analyses. In “Jet-droplets as origin of chondrules” section, weshow that the droplets are jet-droplets that were ejected

Page 2 of 37

from collapsed vapor cavities, and propose jet-droplets
as a possible origin of chondrule nuclei. The presence ofdense and large (>3 cm across) dust aggregates, ‘dustons’,
in the solar PPD is required for the source of chondrules. In “Origin of chondrule chemistry” section, we
propose a new chondrule formation model that considers an interaction of chondrule nuclei with the vapor
cloud that enshrouds the duston, while they travel throughit. A quantitative analysis of model chondrules thus
produced explains real chondrule chemistry. “Origin ofother features of chondrules” section shows the consistency of the model with other features of chondrules.
“Conclusions” section summarize our works.

Experimental methods
Initial materials

Two initial materials are used in this study: Allende meteorite powder and a synthetic L silicate (hereafter Lsil ). Lsil
was prepared by mixing ten chemical reagents in oxide
forms (MgO, SiO2 , FeO, AlO1.5 , CaO, MnO, CrO1.5 , TiO2 ,NaO0.5 and KO0.5 ; purity 99 - 99.999%), in their proportions in the silicate phase of the averaged wet-chemical
analyses of 54 L chondrites (falls) compiled by Jarosewich(1990). Exceptions were made for KO0.5 , MnO and TiO2 ,whose concentrations were raised on purpose to 0.5-1.0
wt.% from their original concentrations in Jarosewich’s
compilation, for the sake of better precision in our chemical analyses. The chemical composition of Lsil thus produced is shown in Table 1.
From here on, the molar unit for oxides is taken in theform of MOx , where M is a metallic element and x represents the normal oxidation number as to M, as shown in
Table 1. When the mol%, molar concentration, or molar
fraction of the element M in a silicate such as Lsil is mentioned in this paper, they refer to those values of the oxide
MOx in the silicate.
A pulverized and homogenized powder of Allendemeteorite (<5 μm in grain size) of originally a 100 gchunk, prepared by A. H. in the laboratory of the late Dr.
Onuma, was used for experiments. We did not chemically analyze the powder and instead used the analysis by
Jarosewich (1990) as shown in Table 1, since 100 g areenough to average out possible indigenous heterogeneity
in the meteorite (e.g., Stracke et al. 2012).
The powders were pressed into cylindrical shapes andfired for 12 hours at two temperatures in a mixed gas: 1100
°C and H2 /CO2 = 1 for Lsil , and 1000 °C and H2 /CO2 =1/2 for Allende. The sintered samples were drilled halfway

Table 1 The bulk chemical compositions of starting materials (mol%)
Material

NaO0.5

MgO

AlO1.5

SiO2

KO0.5

CaO

TiO2

CrO1.5

MnO

FeO

Total

Allende

0.84

35.19

3.40

32.82

0.04

2.47

0.10

0.39

0.15

24.60

100

Lsil

1.82

37.29

2.88

40.98

0.68

2.22

0.78

0.41

0.46

12.48

100

Nakano and Hashimoto Progress in Earth and Planetary Science(2020) 7:47

through their central axes and adjusted for their shape andweight to ∼3 mm in height ∼2.3 mm in diameter, and ∼15
mg.
The choice of the two initial materials is explained.
Allende meteorite (CV3) has its bulk chemical composition close to that of CI chondrites, and is available in large
quantities for experiments. In the wet chemical analysesof chondrites by Jarosewich (1990), L chondrites are by
far the largest in the number of analyses (falls). Therefore,
we regarded that the average bulk composition of L chondrites was most reliable. Another and important reason
is that during the course of evaporation, Si is preferentially lost, and the Si/Mg ratio continues to decrease. The
L chondrite has the Si/Mg molar ratio higher than 1. By
adopting the L chondrite composition as the initial material, it is possible to investigate the chemical fractionation
in a wide range of the Si/Mg scale. The significance of
the Si/Mg scale will be clarified in “Experimental results”
section.
Using the two different, initial compositions in experiments, we should be able to extract a general rule fromtheir results.
Apparatus

Figure 1 shows the apparatus and the details inside thevacuum chamber. A cw-CO2 laser (wave length 10.6 μm;
20-210 watts; NIIC model NC-3010-M) and a ZnSe lenswith focal distance of 127 mm are used to heat a sample

Page 3 of 37

charge with a spot diameter of 1 mm or less. The sample charge is set in the center of the chamber with a Re
wire (0.5 mm in diameter) inserted halfway through the
charge on the bottom, and backed up by a graphite disk
with a 0.5-1 mm space between them. With the aid of a
coaxial diode laser, the charge position is aligned beforehand for the CO2 laser to take aim at it. The Re wire’s role
is to pin down the melted charge, while the graphite disk
prevents the charge from slipping down along the wireby taking advantage of its poor wettability with silicate
liquids.
Temperature measurements are made on the laserirradiated sample surfaces with an automated pyrometer(IMPAC model IS-12; 0.7-1.1 μm; 700-3500 °C; effective
spot diameter ∼0.7 mm at the optimum focal length),whose photometric path is off the laser path by an apex
angle of 18°. A prism and a fused-quartz window in thephotometric path require their absorption corrections,and they are made beforehand.
The stainless-steel chamber (97 mm ID and 310 mm H)
is evacuated with a turbomolecular pump (50 Liter s−1 )and an oil-free roughing pump, which are connected tothe chamber at its rear side in Fig. 1. An Ar gas (99.9%
purity) and a H2 gas (99.99% purity) are introduced into
the chamber using precision valves and mass flow control(for H2 ) to the total pressure of 0.3 bar (Ar 0.3 bar; H2 0.01
bar). The ambient gas has three purposes. (1) It prevents
the sample charge from boiling until a high temperature

Fig. 1 Schematic diagram of the experimental apparatus. a Shows arrangement of the laser and pyrometer light paths. b Shows system forcollecting condensed particles. c Shows sample holding system. Evacuation line is omitted in the diagramNakano and Hashimoto Progress in Earth and Planetary Science(2020) 7:47

is reached. (2) It regulates the redox condition of the sample charge. (3) It immediately cools the generated vapor
and changes the latter into solid condensates, preventing
vapor deposition on the chamber wall. The vapor is 100%
converted to the solid condensates, that can be recovered at any time. By opening the valve to a vacuum pot,
which is connected to the chamber, the condensed particles are collected on a paper filter (7 μm mesh) set in astainless-steel feedthrough (Fig. 1b).
The ejected silicate droplets from the boiling sample
collide with Cu-plates that are set at five positions (top,bottom, front and two sides) inside the chamber, and also
collide with the chamber wall where no plate is attached.
The droplets are quenched as glasses upon collision with
the Cu plate owing to its high thermal conductivity. The
chamber wall acts as good as the Cu plate because it is
kept cool during the laser heating owing to its large mass.
Throughout the heating experiment, a video camera
records scenes covering the area 100 × 70 mm around thesample.
Experimental procedure

The initial masses of the sample charge and of the combined assemblage (sample + Re wire + graphite disk) are
weighed with a precision microbalance. The assemblage is
set inside the chamber. The vacuum chamber and the vacuum pot are evacuated through separate evacuation lines.
When vacuum is reached (10−5 Torr) inside the chamber, valves are closed. An H2 gas is introduced throughthe mass flow control up to 10−2 bar inside the chamber,then Ar is introduced up to 0.3 bar and valves are closed.
The pyrometer and video camera are switched on, and thelaser heating commences. With a 200W laser power, thesample charge melts completely in ∼0.2 s and starts to boil
in ∼0.5 s. After melting, the sample charge keeps its single
rounded liquid body; we name it ‘Mother Liquid Body’ or
MLB hereafter. Once boiling starts, numerous droplets areejected from the MLB in radial directions (“Characteristics of the droplets and the condensed particles” section).
The ejection continues until the laser is turned off, or until
the boiling stops. The video record indicates their ejection speeds to be 1-3 m s−1 , or possibly faster due to thelimitation of the conventional video camera.
Often boiling and droplet-ejection stop, as a result of
the MLB moving away from the laser focus. Wheneverthis happens, the MLB is repositioned by manipulating themanual stick connected to the sample assembly as quickly
as possible. The pyrometer records the temperature of
the MLB at 0.01 s intervals. Unfortunately, however, thepyrometer is too heavy for manual handling and its focal
position is adjusted with a two-axis tilt stage, which slows
down its realignment. This disadvantage has resulted in
the temperature measurement in many experiments to beincomplete.

Page 4 of 37

The moment the laser irradiation commences, a smokelike haze develops around the sample. The smoke is asheet of clusters of ultra-fine solid particles that condensed from the vapor by collision with the cold ambientgas. The mean free path of Ar gas at 0.3 bar and 300 K is∼300 nm. The ambient gas, being secondarily heated by
the sample assembly, starts to make a convective motion
and carries away the condensed particles. The convection
continues until the laser power is shut off, and condensates remain suspended throughout the experiments. To
collect the condensates, a valve to the vacuum pot is
opened and the condensates are vacuum-suctioned onto afilter paper.
We apply two laser powers, 200 and 50 W, in expectationof some temperature effects on the chemical fractionations. Thus, we have four sets of combinations, viz.,two initial compositions (Lsil and Allende) and two laser
powers. For convenience, we distinguish four groups ofexperiments with IDs: L200 (Lsil and 200 W), L50, A200
(Allende and 200 W) and A50.
After each run, glass droplets are collected from the Cuplates with a needle under a microscope. Glass droplets
are also collected from the chamber wall with a brush.
Most of the condensed particles are collected on a filter
paper by vacuum suction; those attached to the Cu-plates
and the chamber wall are also collected with a brush but
not used for analysis.
Chemical analysis

A collection of the recovered glass droplets from each run
was handled as follows. Because the number of droplets
is different from run to run, <200-5000, we randomly
picked 100-500 droplets from each collection, dependingon the size of collection. The selected droplets for eachrun were then divided into five bins in accordance with
their diameters, mounted on separate graphite discs (10mm in diameter) with epoxy resin, polished down so as
to expose cross-sections of all the mounted droplets, andcarbon-coated. The condensed particles on a filter paper
are easy to handle since they are compressed into a sheetlike layer, which is removed from the paper and mounted
on a carbon conductive adhesive tape and carbon-coatedon its surface.
Instrumental chemical analyses were made on thosesamples described above, with a FE-SEM-EDS (model
JEOL JSM-7001F) at the Pheasant Memorial Laboratory for Geochemistry and Cosmochemistry, Institute forStudy of the Earth’s Interior, Misasa, Japan. The analytical conditions, 15 kV for accelerating voltage and 3 nA
for probe current, were chosen for probing glassy andamorphous samples without evaporating volatile Na andK. The glass droplets were found to be uniform in composition; therefore, a single beam-rastering analysis overa large sample area was made on each droplet. The sheet

Nakano and Hashimoto Progress in Earth and Planetary Science(2020) 7:47

of condensed particles was relatively inhomogeneous, sothat three beam-rastering analyses covering nearly anentire sheet area were applied.

Experimental results
Table 2 shows the experimental conditions and someresults including maximum temperature recorded withpyrometer, and mass loss (%) relative to the initial mass,for four groups of experiments, discriminated with theirIDs. The recorded Tmax needs caution as indicated in“Experimental procedure” section and is explained indetail in “Time-temperature profiles” section. The massloss (%) is calculated by a simple subtraction of the post
weight of the assembly (sample + Re + graphite) from its
initial weight; it includes weight losses not only of sample
but also of Re wire and graphite disk. This is because theassembly parts are difficult to separate from one anotherafter the experiments. In “Chemical evolution of MLB
driven by evaporation” section, we adopt an alternative
and unambiguous definition of ‘vaporization degree’.
We have carried out many preliminary experiments(∼150 runs) before the present work, using similar initial materials and methods. Droplets and condensates
produced in these experiments were not chemically analyzed, but they back up the present work, especially inthe temperature measurements. Instead of presenting thewhole data of those experiments, we point out the use of
preliminary data wherever necessary.
Characteristics of the droplets and the condensed particles

Figures 2 and 16 in Appendix A show video frames ofthe experiments. Bright and straight rays are the trajectories of silicate droplets that were ejected radially from thesample (Fig. 2a). Silicate vapor condenses immediately bycolliding with a cold inert gas, making clusters of ultrafine particles that circulate in the convective motion of thegas inside the chamber (Fig. 2b).
The halos are the effect of light-scattering by thecondensed particles, and obstruct the view of thelaser-irradiated sample in the video images. Observation through a filtered-scope attached to the pyrometer
revealed a restless and vibrating sample with many dotted lights and shades changing incessantly. They are vapor
bubbles that grow and collapse on the sample surface(“Jet-droplets as origin of chondrules” section).
Figure 3a shows an example of silicate droplets
quenched as glass, which were collected from inside thechamber. The smallest is several μm in diameter, while
the largest is ∼400 μm in diameter. Most droplets are in
the range 10-100 μm. Figure 3b is a SEM image of thecondensed particles. Individual particles are 50-100 nm
in diameter and are connected 3-dimensionally to one
another. They are probably amorphous judging from theirround shapes.

Page 5 of 37

Time-temperature profiles

As described in “Experimental procedure” section, thepyrometer scope was not always pointed at the hot spot of
the charge. As a result, the Tmax in Table 2 does not necessarily represent the true maximum temperature reachedby the sample. This is particularly so in the 200 W experiments due to a short time for adjusting the scope position.
In the 50 W experiments, however, it was mostly successful because more time was available to reposition thepyrometer.
Figure 4 shows a time-temperature profile of a successfully recorded run with a laser power of 200 W (run# 3
from preliminary experiments). The temperature reaches
2200 °C in one second, then it rises gradually to its maximum, ∼2600 °C in 14 s. The inflection in temperature at∼2200 °C, referred to as TB (and tB for its time), is similarin all experiment groups, L200, L50, A200 and A50. For
the 200 W experiments, the average (tB , TB ) is (0.6 s, 2217
°C) from eight selected runs, and for 50 W it is (15 s, 2202
°C) from twenty-two selected runs.
The maximum temperature reached for each experiment group is obtained by averaging several highest Tmax
values in the same group of experiments, including thepresent and preliminary experiments. We selected runs#
138, 115, 60-2, and 36 for L200; runs# 91, 59 and 57 for
A200; runs# 123, 121, 119, 118, 117, 113 and 24 for L50;
and runs# 109 and 104 for A50. Their average Tmax values
are 2865 °C, 2673 °C, 2693 °C and 2576 °C, respectively,
as shown in Table 3. These values would change a little byadding or subtracting runs in their selected lists, but theirdifferences can be clearly indicated that Tmax for 200 W ishigher than for 50 W, and Tmax for Lsil is higher than forAllende.
The recorded temperatures with pyrometer need corrections to account for absorption and scattering by thequartz window and the prism. The combined effective
transmittance measured in advance was 95%. The pyrometer’s output assumes a unit emissivity, viz., ε = 1, for any
measured object, but emissivities of both Lsil and Allende
are unknown. Table 3 shows Tmax and TB values for four
experiment groups after corrections of the transmittanceand four emissivity values (0.9, 0.8, 0.7 and 0.6).
TB is most likely a boiling point at 0.3 bar, because it
is constant regardless of the laser powers applied. (It alsomeans that the boiling points of Lsil and Allende are similar). Silicate droplets were almost immediately (withina second) ejected from the samples when using a laser
power of 200 W, while ejection began 10-20 s after the
onset for 50 W. In “Jet-droplets as origin of chondrules”
section, it is shown that those droplets are jet-droplets
released by bursting of vapor bubbles during boiling.
The reason why the time-temperature derivative
becomes small after TB is reached is that the boiling regulates the sample temperature due to the large latent heat(2020) 7:47

Nakano and Hashimoto Progress in Earth and Planetary Science

Page 6 of 37

Table 2 Experimental conditions and results
Run#

InitialMaterial

Laser
Power
(W)

Ambient
Gas
(bar)

Tmax
(°C)

Heating
Duration(s)

SampleShape

Holder
SystemSampleInitialWeight (g)

Mass
Loss
(wt%)

Recov.
Drop.∗1

Recov.
Vap.∗2

Group ID: L200
114

Lsil

200

0.3Ar + 0.01H2

2603

14

m.c.

G+Re

0.01480

9.3

n

Y

115

Lsil

200

0.3Ar + 0.01H2

2928

20

m.c.

G+Re

0.01407

?

Y

Y

126

Lsil

200

0.3Ar + 0.01H2

1987

2

m.c.

G+Re

0.01520

13.7

Y

Y

127

Lsil

200

0.3Ar + 0.01H2

2123

2

m.c.

G+Re

0.01510

27.3

Y

Y

128

Lsil

200

0.3Ar + 0.01H2

1811

4

m.c.

G+Re

0.01481

4.0

Y

Y

129

Lsil

200

0.3Ar + 0.01H2

2212

8

m.c.

G+Re

0.01489

48.0

Y

Y

130

Lsil

200

0.3Ar + 0.01H2

1855

6

m.c.

G+Re

0.01493

29.2

Y

Y

131

Lsil

200

0.3Ar + 0.01H2

2345

9

m.c.

G+Re

0.01454

81.8

Y

Y

132

Lsil

200

0.3Ar + 0.01H2

2183

8

m.c.

G+Re

0.01497

97.4

Y

Y

133

Lsil

200

0.3Ar + 0.01H2

2077

1

m.c.

G+Re

0.01460

77.7

Y

Y

137-1

Lsil

200

0.3Ar + 0.01H2

1940

1

m.c.

G+Re

0.01437

4.5

Y

Y

137-2

re

200

0.3Ar + 0.01H2

1870

2

r.r.

G+Re

—

5.6

Y

y

137-3

re

200

0.3Ar + 0.01H2

1910

2

r.r.

G+Re

—

16.5

Y

y

137-4

re

200

0.3Ar + 0.01H2

1608

2

r.r.

G+Re

—

1.5

n

y

137-5

re

200

0.3Ar + 0.01H2

2182

5

r.r.

G+Re

—

22.6

Y

y

137-6

re

200

0.3Ar + 0.01H2

1693

2

r.r.

G+Re

—

4.0

Y

y

138

Lsil

200

0.3Ar

2799

24

m.c.

G+Re

0.01501

104.0

Y

Y

113

Lsil

50

0.3Ar + 0.01H2

2672

245

m.c.

G+Re

0.01498

>100

Y

Y

117

Lsil

50

0.3Ar + 0.01H2

2613

31

m.c.

G+Re

0.01508

32.7

Y

Y

118

Lsil

50

0.3Ar + 0.01H2

2776

61

m.c.

G+Re

0.01499

47.2

Y

Y

119

Lsil

50

0.3Ar + 0.01H2

2702

87

m.c.

G+Re

0.01485

49.8

Y

Y

120

Lsil

50

0.3Ar + 0.01H2

2450

128

m.c.

G+Re

0.01443

81.6

Y

Y

121

Lsil

50

0.3Ar + 0.01H2

2733

132

m.c.

G+Re

0.01490

94.5

Y

Y

122

Lsil

50

0.3Ar + 0.01H2

2387

81

m.c.

G+Re

0.01379

35.1

Y

Y

123

Lsil

50

0.3Ar + 0.01H2

2695

140

m.c.

G+Re

0.01481

41.9

Y

Y

124

Lsil

50

0.3Ar + 0.01H2

2298

464

m.c.

G+Re

0.01507

60.3

n

Y

125-1

Lsil

50

0.3Ar + 0.01H2

1600

17

m.c.

G+Re

0.01468

3.5

n

Y

125-2

re

50

0.3Ar + 0.01H2

1617

16

r.r.

G+Re

—

3.6

n

y

125-3

re

50

0.3Ar + 0.01H2

1723

16

r.r.

G+Re

—

9.7

n

y

125-4

re

50

0.3Ar + 0.01H2

1450

17

r.r.

G+Re

—

17.3

n

y

125-5

re

50

0.3Ar + 0.01H2

2123

17

r.r.

G+Re

—

2.0

n

y

134

Lsil

50

0.3Ar + 0.01H2

1770

22

m.c.

G+Re

0.01488

2.4

n

Y

135

Lsil

50

0.3Ar + 0.01H2

1484

2

m.c.

G+Re

0.01500

1.3

n

Y

136

Lsil

50

0.3Ar + 0.01H2

1505

19

m.c.

G+Re

0.01415

10.9

n

Y

91

Allende200

0.3Ar + 0.01H2

2608

8

m.c.

G+Re

0.01437

117.0

Y

Y

93-1

Allende200

0.3Ar + 0.01H2

1010

4

m.c.

G+Re

0.01500

6.5

Y

Y

93-2

re

200

0.3Ar + 0.01H2

2084

3

r.r.

G+Re

—

17.2

Y

Y

93-3

re

200

0.3Ar + 0.01H2

1532

3

r.r.

G+Re

—

52.4

Y

Y

93-4

re

200

0.3Ar + 0.01H2

2189

2

r.r.

G+Re

—

24.9

Y

Y

Group ID: L50

Group ID: A200

(2020) 7:47

Nakano and Hashimoto Progress in Earth and Planetary Science

Page 7 of 37

Table 2 Experimental conditions and results (Continued)
Run#

InitialMaterial

Laser
Power
(W)

Ambient
Gas
(bar)

Tmax

SampleShape

Holder
SystemSampleInitialWeight (g)

Mass
Loss
(wt%)

Recov.
Drop.∗1

Recov.
Vap.∗2(°C)

Heating
Duration
(s)

93-5

re

200

0.3Ar + 0.01H2

2372

1

r.r.

G+Re

—

30.2

Y

Y

93-6

re

200

0.3Ar + 0.01H2

2215

2

r.r.

G+Re

—

6.9

Y

Y

93-7

re

200

0.3Ar + 0.01H2

2139

9

r.r.

G+Re

—

7.2

Y

Y

94

Allende200

0.3Ar + 0.01H2

902

2

m.c.

G+Re

0.01491

44.0

Y

Y

95

Allende200

0.3Ar + 0.01H2

2280

2

m.c.

G+Re

0.01482

?

n

Y

96

Allende200

0.3Ar + 0.01H2

2187

3

m.c.

G+Re

0.01467

61.0

Y

Y

97

Allende200

0.3Ar + 0.01H2

2052

4

m.c.

G+Re

0.01446

46.0

Y

Y

98

Allende200

0.3Ar + 0.01H2

2148

2

m.c.

G+Re

0.01443

45.0

Y

Y

99

Allende200

0.3Ar + 0.01H2

1701

10

m.c.

G+Re

0.01446

61.0

Y

Y

100

Allende200

0.3Ar + 0.01H2

2031

12

m.c.

G+Re

0.01477

59.0

Y

Y

101

Allende200

0.3Ar + 0.01H2

2122

14

m.c.

G+Re

0.01397

104.0

Y

Y

112

Allende200

0.3Ar + 0.01H2

2495

6

m.c.

G+Re

0.01491

>100

Y

Y

Group ID: A50
90

Allende50

0.3Ar + 0.01H2

2313

218

m.c.

G+Re

0.01428

97.0

Y

Y

102-1

Allende50

0.3Ar + 0.01H2

1436

3

m.c.

G+Re

0.01469

3.4

n

Y

102-2

re

50

0.3Ar + 0.01H2

1723

27

r.r.

G+Re

—

8.9

n

y

102-3

re

50

0.3Ar + 0.01H2

1471

13

r.r.

G+Re

—

23.3

n

y

102-4

re

50

0.3Ar + 0.01H2

1075

7

r.r.

G+Re

—

10.5

n

y

102-5

re

50

0.3Ar + 0.01H2

806

6

r.r.

G+Re

—

3.1

n

y

102-6

re

50

0.3Ar + 0.01H2

1329

7

r.r.

G+Re

—

2.2

n

y

102-7

re

50

0.3Ar + 0.01H2

1391

13

r.r.

G+Re

—

7.7

n

y

102-8

re

50

0.3Ar + 0.01H2

809

13

r.r.

G+Re

—

7.9

n

y

102-9

re

50

0.3Ar + 0.01H2

1195

34

r.r.

G+Re

—

16.3

n

y

102-10re50

0.3Ar + 0.01H2

2258

112

r.r.

G+Re

—

14.4

y

y

103

Allende50

0.3Ar + 0.01H2

2177

34

m.c.

G+Re

0.01429

23.1

Y

Y

104

Allende50

0.3Ar + 0.01H2

2591

186

m.c.

G+Re

0.01474

45.0

Y

Y

105

Allende50

0.3Ar + 0.01H2

1855

26

m.c.

G+Re

0.01395

24.9

Y

Y

106

Allende50

0.3Ar + 0.01H2

2364

85

m.c.

G+Re

0.01423

38.3

Y

Y

107

Allende50

0.3Ar + 0.01H2

2472

188

m.c.

G+Re

0.01456

42.7

Y

Y

108

Allende50

0.3Ar + 0.01H2

2469

230

m.c.

G+Re

0.01470

46.4

Y

Y

109

Allende50

0.3Ar + 0.01H2

2559

417

m.c.

G+Re

0.01419

53.1

Y

Y

110

Allende50

0.3Ar + 0.01H2

2420

749

m.c.

G+Re

0.01426

70.2

Y

Y

111

Allende50

0.3Ar + 0.01H2

2288

598

m.c.

G+Re

0.01462

90.8

Y

Y

Note. ∗1: Recovery of droplets. ∗2: Recovery of vapor. Allende: Allende meteorite.
Lsil : synthetic L silicate. re: residue of the preceding run. m.c.: miniature cylinder.
r.r.: rounded residue. G+Re: Graphite disk and rhenium wire. Y: recovered and analyzed.
y: recovered but not analyzed. n: not recovered. Tmax : lower limits for Tmax (see the text)of vaporization. The steady increase in temperature afterwards may result from two effects: the evaporation rate,
hence latent heat loss rate, becomes gradually smaller due
to the selective loss of volatiles with time, and the radiativeloss of energy becomes smaller due to gradual shrinkageof the sample by evaporation. The latent heat control also
explains why the difference in Tmax between L200 and
L50, or between A200 and A50, is only 100-200 °C, while

the laser power differs by a factor of four. The lower Tmax
for Allende as compared to Lsil is explained by a higher
concentration of FeO, which is more volatile than MgO, inthe former.
Accepting TB as the boiling point at 0.3 bar, TB ∼2300°C (which corresponds to ε = 0.9 in Table 3) is too low forboiling (Hashimoto 1983). Therefore, ε ≤ 0.8 should be
more realistic.

Nakano and Hashimoto Progress in Earth and Planetary Science(2020) 7:47

Fig. 2 Video frames (30 frames s−1 and 1/60 s for exposure) of thescenes in the experiments for a run# 84 and b run# 32. The molten
sample (MLB) sits in the center of the bright halo, and the laser beam
comes from the 10 o’clock direction. a A dozen droplets, seen as
bright and straight rays, are radially ejected from the MLB almost
simultaneously. (One droplet is actually seen at the tip of the ray inthe 10 o’clock position). b A smoke composed of condensed particles
rises from the evaporating MLB, while clusters of previously
condensed particles form sheet-like structures in the convective
motions of the ambient gas. This scene was taken within a fraction of
second after the laser power was shut off, when the MLB temperature
dropped below the boiling point

Page 8 of 37

Fig. 3 a Glass-quenched droplets collected from inside the chamber.
Scale bar = 100 μm. They are generally spherical in shape, with a flatface on one side that results from collision with a Cu-plate or thechamber wall. Necking between the spherules is an optical artifact. b
SEM image of the condensed particles collected on a filter paper.
Scale bar = 100 nm. Globules of 100 nm or less in diameter are
3-dimensionally connected to one anotherSize-frequency distribution of silicate droplets

We collected all droplets (677 in total) in run# SZ6 (runconditions similar to L200) and nearly all droplets inrun# 138 to determine the size-frequency distributionof the ejected droplets. All droplets were measured fortheir diameters under a microscope. In run# SZ6, theirrange in diameter is 9-390 μm, and were sorted into sizebins of every 5 μm width for a normal-scale histogram(Fig. 5a). The same droplets were size-sorted into common log-scale bins of every 0.065 width (Fig. 5b). The
result is a log-normal-like distribution, which is generally recognized as a typical size-frequency distribution ofchondrules (Friedrich et al. 2015). In a similar manner,
the size-frequency distribution of the droplets in run#138 (3067 droplets) was obtained, which also resembles alog-normal distribution.

Fig. 4 Time-temperature profile of the MLB surface in run# 3 (200 W).
The temperature is recorded automatically once it exceeds 700 °C.
The laser power was shut off after 14 s, at which point radiative
cooling started

(2020) 7:47

Nakano and Hashimoto Progress in Earth and Planetary Science

Page 9 of 37

Table 3 Measured and corrected temperatures of MLB surfaceID

tm 0.95, ε 0.9

Meas (ε=1)

tm 0.95, ε 0.8

tm 0.95, ε 0.7

tm 0.95, ε 0.6

TB

Tmax

TB

Tmax

TB

Tmax

TB

Tmax

TB

Tmax

L200

2217

2865

2317

2991

2395

3088

2488

3206

2597

3343

A200

2217

2673

2317

2792

2395

2883

2488

2994

2597

3123

L50

2202

2693

2302

2812

2379

2904

2472

3016

2580

3146

A50

2202

2576

2302

2690

2379

2778

2472

2886

2580

3011

Note. Temperature is shown in °C. Meas: Measured temperature.
tm: Transmittance. ε: Emissivity. See text for TB and Tmax

The total mass of ejected droplets are calculated fromtheir size and frequency, assuming a complete sphere anddensity of 3 g cm−1 . The mass of ejection thus obtained
is divided by the total mass lost by the sample to givea mass fraction of droplets to the total mass loss. The
mass fractions for runs# SZ6 and 138 are 11.9 and 11.0%,
respectively.
Chemical evolution of MLB driven by evaporation

The silicate droplets are sprayed pieces of the boilingMLB. Therefore, their chemical compositions represent
that of the MLB at the time of their ejection. Since thedroplets are ejected all the time as long as boiling continues, they are good tracers of any compositional changes in
the MLB that is driven by continuous evaporation.

80(a) Normal scale

Frequency

60

.
MXAl = M XA1

40

0
0

50

100

150

200

250

300

350

400

Droplet size in micrometer
70(b) Log scale

60
50
40
30
20
10
0
0.03

0.39

0.78

1.17

1.56

1.95

2.34

2.79

3.18

Droplet log size/micrometer

Fig. 5 Size-frequency distribution of silicate droplets recovered fromrun# SZ6. a Normal scale. b Common logarithmic scale(1)

Here M is a total number of moles of the ten elements in the MLB, and XAl is a molar fraction of Al in
the MLB; superscript ( ) denotes post-evaporation and nosuperscript pre-evaporation.
Wedefine the vaporizationdegree (VD) of the MLB as follows,
VD(%) = 100(

20

Frequency

The relative timing of ejection of a particular droplet
can be known from the Al concentration in the droplet,
for the reasons described below. An aluminum oxide(AlO1.5 ) is one of the super-refractory oxides amongst theconstituents of the chondritic meteorites. It is by far themost refractory of the ten elements (in their oxide forms)
included in the present experiments, and does not evaporate
until the other nine elements have completely vaporized(Hashimoto et al. 1979; Hashimoto 1983). The unchanging
mass of Al in the MLBs guarantees the following relationship, as long as mass loss is caused solely by evaporation.

XAl,0
M0 − M) = 100(1 −  ).
M0
XAl

(2)

Here M0 denotes the initial total number of moles in theMLB, and XAl,0 denotes the initial mole fraction of Al in
the MLB. Because each silicate droplet is a piece of theMLB, it registers its moment of ejection as MLB’s VD%
and also represents the chemical composition of the MLB
at that moment.
Figure 6 shows the concentrations of the nine elementsin the ejected droplets as functions of their Al concentrations for the L200 experiment group (856 droplets altogether for chemical analyses) and for the A200 experimentgroup (851 droplets). The Al concentration increases fromits initial value 2.88 mol% for Lsil and 3.40 mol% forAllende to the highest 39.42 mol% for L200 and 71.46
mol% for A200, corresponding to a VD range from 0 to
92.7% for L200 and from 0 to 95.2% for A200. The highest
Al mol% (or VD%) might be larger since not all dropletsfrom the same group of experiments are used for chemical analyses. The results for the groups, L50 and A50 show
trends that are similar to L200 and A200, respectively.
We may divide the ten elements into five groups according to their difference in evaporative fractionation. 1)
Alkaline elements, Na and K, are very volatile and distillout of MLB in a very early phase of evaporation. In theNakano and Hashimoto Progress in Earth and Planetary Science(2020) 7:47

Page 10 of 37

Fig. 6 Element concentration in mol% in the droplets as a function of Al mol% for the experiment groups L200 and A200. ‘All droplets’ indicate all of
the droplets analyzed for each group. The ‘Initial’ composition is either Lsil for L200 or Allende for A200. For ‘Ensemble average’, refer to “Derivationof relative volatilities from the experimental results” section and Table 4. ‘Calculation’ denotes the calculated composition of the MLB that evolves by
evaporation, starting with the initial composition, either of Lsil or of Allende. See “Application of relative volatilities to an arbitrary MLB: self-consistency
test” section for detail. The dotted lines in the Ca and Ti panels indicate extension of the lines that connect the origin and their initial concentration

(2020) 7:47

Nakano and Hashimoto Progress in Earth and Planetary Science

present experiments, 80-90% of these elements were lostby the time boiling started. 2) Si, Fe, Cr and Mn are moderately volatile and their concentrations in the MLB steadilydecrease as VD increases. 3) Mg is more refractory than
the second group of elements. Thus, at first it increases inconcentration in the MLB as the second group are preferentially lost, but soon comes its turn for depletion. 4) Ca
and Ti are more refractory than Mg, and become steadilyenriched in the MLB up until Al reaches ∼60 mol% (or VD∼95%), when they start to decrease in concentration (as isseen in A200). 5) Al is more refractory than Ca and Ti, anddoes not evaporate before Ca and Ti distill out.
Relative volatilities of the ten elements
Definition of relative volatility and mathematical method of
its derivation

Hashimoto (1983) defined ‘volatility’ (ζ ) of an oxide component in a silicate liquid to quantify its ease for escape tovacuum.
Ji(3)
ζi = ,
Ci
where Ji is the vacuum evaporation rate of component i
per unit area and unit time (mole cm−2 s−1 ) at a given
temperature, and Ci is its molar concentration (molecm−3 ) in the silicate liquid. Generally, evaporation rate
is proportional to the concentration of the component.
Hence, volatility is expected to be roughly constant. He
applied it to the three components, SiO2 , MgO and FeO
in the five component silicate liquid, and found that theirvolatilities are nearly constant for a wide range of theirconcentrations at a constant temperature.
We propose that the relative magnitudes in volatilities ofthe oxide components are enough to predict the chemical
fractionations caused by evaporation, and extend this idea
by introducing ‘relative volatility’ of j-th to i-th component
in the liquid,
Zj/i =

ζj
.
ζi

(4)

Unlike volatilities, the relative volatilities are insensitive totemperature variations as long as the volatilities of the jth and i-th components vary proportional to the inverse
temperature with similar slopes. We take advantage of thisquality in the analyses to follow, i.e., data analyses from
“Derivation of relative volatilities from the experimental
results” section to “Application of relative volatilities toan arbitrary MLB: self-consistency test” section, and alsomodel analyses in fitting chondrule chemistry in “Modelcalculation of chondrule chemistry” section.
We consider an infinitesimal evaporation of a multicomponent silicate liquid which include oxide components iand j as well as Al-oxide (AlO1.5 ), for an infinitesimal time
dt. Then the mass of evaporation of the i-th component(in moles) in dt is,Page 11 of 37

dmi = Ji Adt.

(5)

Here A is the surface area of the silicate liquid. Ci is written
using the molar volume of the liquid (VM ) and the molar
fraction of i (Xi ) in the liquid,
Ci =

Xi
.
VM

(6)

Substitution of Eqs. 3 and 6 into Eq. 5 makesdmi =ζi Xi Adt.
VM

Hence it follows
dmj
ζj Xj
=
.
dmi
ζi Xi

(7)

(8)

Considering the Al-oxide as a non-evaporating substance as explained in “Chemical evolution of MLB driven
by evaporation” section and using Eq. 1, dmi is expressedasdmi = mi − mi= MXi − M Xi
=M − XX
Xi XAl
i Al
.

XAl

(9)

Here again, superscript ( ) denotes post-evaporation andno superscript pre-evaporation. Hence it follows
 − XX
Xj XAl
dmj
j Al
=
.

dmi
Xi XAl − Xi XAl(10)

 /X by q and equating Eq. 8 with
Finally, substituting XAl
Al
Eq. 10,qXj − Xj Xi
ζj
=
.
ζi
qXi − Xi Xj(11)Thus, relative volatility is a function of molar fractions
only. We take SiO2 for the i-th component unless otherwise noted.
If evaporation is not infinitesimal, the term Xi /Xj in
Eq. 11 should be replaced with (Xi + Xi )/(Xj + Xj ), andthe obtained relative volatility is an average value.
Derivation of relative volatilities from the experimental
results

We obtain relative volatilities for the ten oxide components for the two groups, L200 and A200, from the chemical compositions of the ejected droplets. Theoreticallyspeaking, a set of two droplet compositions with different VD values is enough to calculate relative volatilities
for all elements relative to Si by using Eq. 11. The actual
compositional data, however, indicate that the droplet
compositions do not define smooth functions of VD or Al
mol%, but more or less scatter about the general trends.
Since (relative) volatility is basically a differential quantity,
it is sensitive to data scattering. There is a means to getNakano and Hashimoto Progress in Earth and Planetary Science(2020) 7:47

away from this difficulty, but beforehand let us specify apossible cause for the scattering of the data.
Although analytical errors contribute to data dispersion,
it is not considered as the main cause because of the relatively high precision of the chemical analyses. We foundthat the FeO contents in the droplets are too variablefrom one droplet to another even if their Al mol% or VD
are nearly identical. This was more prominent for A200
droplets than L200 droplets. We identified that chemical reactions of the MLB with the sample holder, either
the graphite disk or Re-wire, caused FeO to be partiallyreduced into metal, leaving the MLB depleted in FeO inplaces where the liquid silicate was in direct contact withthose holder materials. Allende contains twice as much
FeO as that of Lsil in its initial composition, making itvery susceptible to the reduction reactions. If local heterogeneity in FeO content is not effectively averaged out bydiffusion (slow) and stirring (fast) driven by boiling, it will
persist. Silicate droplets were ejected from many places in
the MLB and should sample local heterogeneity in FeO
concentration. The FeO heterogeneity extends its effectto other oxide components in MLB because the chemicalcomposition is always normalized to 100%.
We took 856 droplets from the L200 experiments (851droplets from A200), arranged them in the increasingorder of the Al content, and divided them into thirteen
ensembles in the following manner. Since droplets with
low Al concentrations are large in number, the sizes ofensembles (viz., the number of droplets in them) are configured so that each ensemble is spaced as equally as
possible in terms of its average Al concentration. In addition, to obtain a smooth change in chemical composition,
each ensemble overlaps its quarter or half of constituent
droplets with the preceding as well as following ensemble (a method of ensemble moving average). The averagechemical compositions of thirteen ensembles for L200 are
shown in Table 4 and Fig. 6 (upper nine panels). Those for
A200 are shown in Table 4 and Fig. 6 (lower nine panels).
The lower sections in Table 4 show relative volatilitiesof the ten elements calculated with Eq. 11 from adjacentpairs of ensemble averages of their concentrations. Notethat the term Xi /Xj is replaced with (Xi + Xi )/(Xj + Xj )in Eq. 11, because it is a finite evaporation that proceeded
between successive ensembles.
Relative volatilities of Mg and Fe

Table 4 shows that the relative volatility of Mg to Si(ZMg/Si ) increases as VD or Al mol% increases. This apparent trend is not related to Al, but ZMg/Si is clearly related
to the Mg/Si ratio as indicated by their linear relationshipin Fig. 7a. In liquid silicates, the type and concentration of network-modifying cations that coordinate to theanionic SiO4 tetrahedra dictate their physical and thermodynamic properties (Toop and Somis 1962; Yokokawa

Page 12 of 37

and Niwa 1969; Masson 1972; Hess 1977; Richet and Bottinga 1986; Stebbins et al. 1995). Comparing Fe and Mg,
Mg is a stronger base and selectively coordinates to SiO4
(Yokokawa and Niwa 1969), making Mg more refractory
than Fe. As the Mg concentration in the liquid increases,
however, the bond energy per Mg atom to the SiO4 tetrahedron should be reduced, thus making it less refractory.
Si should act in opposite way: with more Mg the totalbond energy per SiO4 should become larger, making it
more refractory as the Mg concentration increases. This
explains why the relative volatility of Mg increases withincreasing Mg/Si ratio.
The relative volatility of Mg for A200 is systematicallyhigher than for L200. This may be explained by either (1)the effect of FeO concentration, or (2) the effect of temperature. Allende has twice as much FeO compared to
Lsil . Although the basicity of Fe is far smaller than that ofMg (Yokokawa and Niwa 1969), its overabundance would
have an obstructing effect on Mg in monopolizing SiO4
tetrahedra. However, the difference in relative volatility
in Fig. 7a does not reduce even if the FeO concentration in MLB decreases to a level of 5 mol% for A200. So
this effect alone is not enough. Hashimoto (1983) indicated that the volatility of MgO is relatively increased atlower temperatures compared to that of SiO2 . As shownin “Time-temperature profiles” section, the recorded Tmax
for A200 experiments is lower than for L200 by 200 K. The
temperature effect, however, explains only 12% increase
in relative volatility. Therefore, we presume both effects
work together.
ZMg/Si rises abruptly at high Mg/Si ratios, 2.7 for L200
and 2.3 for A200. The sudden increases are caused by
the high concentration of Ca in the MLB (Hashimoto1983) because Ca is a stronger network-modifier than Mg(Yokokawa and Niwa 1969). As long as Ca concentration
is low, its impeding effect on Mg is limited. As evaporation
proceeds, however, Ca is steadily increased in expenseof more volatile elements. When Ca/Si ratio exceeds 0.6,
where more than halves of SiO4 tetrahedra are preferentially coordinated to Ca, Mg cations start to be freed
from bondage with the SiO4 tetrahedra. Therefore, it isthe effect not of high Mg/Si ratio but of concentrated Cathat raises ZMg/Si higher than unity.
Figure 7b shows a linear anti-correlation between ZMg/Si
and ZFe/Mg that obtain by dividing ZFe/Si with ZMg/Si ,for both L200 and A200. This anti-correlation may be
explained by the competitive relationship between Mg and
Fe in coordinating to SiO4 tetrahedra. The linearity fails at
the highest values of ZMg/Si for both L200 and A200; thisis again explained as the effect of high Ca concentration.
The failure at the low end for L200 can be explained by alarge structural change in the silicate liquid which should
happen when Mg/Si ratio crosses 1. The data point of
L200 with the lowest ZMg/Si , viz., 0.09, corresponds to the(2020) 7:47

Nakano and Hashimoto Progress in Earth and Planetary Science

Page 13 of 37

Table 4 Average chemical compositions of 13 ensembles (upper sections) and relative volatilities of elements (lower sections)
L200

Ensemble Average, mol% (Total =100)

E#

data#

Na

Mg

Al

Si

K

Ca

Ti

Cr

Mn

Fe

0

initial

1.824

37.2852.882

40.982

0.680

2.220

0.783

0.405

0.456

12.484

0.910

0.054

1

1-400

0.198

44.1213.506

36.707

0.075

2.718

0.935

0.274

0.411

11.053

1.202

0.074

2

301-600

0.041

48.5524.083

33.189

0.023

3.157

1.079

0.198

0.366

9.311

1.463

0.095

3

401-700

0.022

50.5594.415

31.655

0.016

3.421

1.162

0.170

0.334

8.246

1.597

0.108

4

501-7500.015

52.2744.787

30.341

0.014

3.731

1.261

0.148

0.296

7.132

1.723

0.123

5

650-800

0.006

55.383

5.904

27.578

0.012

4.619

1.533

0.095

0.206

4.665

2.008

0.167

6

716-8150.004

56.593

6.911

25.797

0.011

5.360

1.763

0.068

0.148

3.344

2.194

0.208

7

751-830

0.003

56.6808.308

24.019

0.011

6.400

2.075

0.047

0.103

2.352

2.360

0.266

8

781-8400.003

55.867

10.14922.013

0.013

7.738

2.475

0.032

0.066

1.643

2.538

0.352

9

801-8500.003

53.83812.40020.161

0.015

9.386

2.966

0.026

0.046

1.159

2.670

0.466

10

826-853

0.005

49.51315.99117.9080.015

12.046

3.744

0.015

0.037

0.726

2.765

0.673

11

840-8550.011

44.02219.96715.9060.017

14.901

4.548

0.019

0.029

0.579

2.768

0.937

12

852-8560.048

27.23931.031

11.6190.023

22.929

6.708

0.020

0.028

0.354

2.344

1.973

13

856

0.073

14.92039.418

8.515

0.049

28.994

7.953

0.038

0.000

0.040

1.752

3.405

E-pairζFe /ζMg

Na

Mg

Al

Si

K

Ca

Ti

Cr

Mn

Fe

Mg/Si

Ca/Si

0-1

11.487

5.899

0.090

0.000

1.000

5.878

(0.021)

0.060

1.908

0.977

1.038

1.048

0.064

1-2

5.728

5.789

0.223

0.000

1.000

4.828

0.010

0.038

1.860

1.059

1.279

1.326

0.084

2-3

5.310

5.564

0.300

0.000

1.000

3.467

(0.018)

0.030

1.840

1.358

1.591

1.528

0.101

3-4

4.745

3.446

0.386

0.000

1.000

1.552

(0.046)(0.007)

1.797

1.637

1.831

1.659

0.115

4-5

4.149

3.772

0.497

0.000

1.000

1.354

(0.012)

0.046

2.120

1.866

2.061

1.859

0.144

5-6

3.593

2.148

0.605

0.000

1.000

0.826

0.038

0.079

2.191

2.173

2.175

2.098

0.187

6-7

2.922

1.282

0.714

0.000

1.000

0.690

0.026

0.082

2.153

2.108

2.086

2.274

0.236

7-8

2.593

1.423

0.746

0.000

1.000

0.159

0.036

0.083

2.009

2.206

1.935

2.445

0.307

8-9

2.301

0.753

0.823

0.000

1.000

0.377

0.025

0.067

1.412

1.966

1.894

2.601

0.406

9-10

2.116

(0.689)

0.907

0.000

1.000

0.650

0.013

0.056

2.095

1.298

1.918

2.715

0.563

10-11

1.318

(1.818)

0.997

0.000

1.000

0.285

0.027

0.080

(0.039)

1.290

1.314

2.766

0.797

11-12

1.013

(1.072)

1.214

0.000

1.000

0.150

0.013

0.067

0.509

0.658

1.230

2.589

1.374

2.222

(0.332)

1.505

0.000

1.000

(0.877)

0.008

0.125

(0.671)

4.096

3.345

2.094

2.579

ζj /ζi (i = Si)

12-13

A200

Mg/Si

Ca/Si

pair average

Ensemble Average, mol% (Total =100)

E#

data#

Na

Mg

Al

Si

K

Ca

Ti

Cr

Mn

Fe

Mg/Si

Ca/Si

0

initial

0.836635.1933.400

32.8220.0367

2.467

0.100

0.394

0.146

24.605

1.072

0.075

1

31∼2000.066540.933

4.246

31.1790.0131

3.181

0.122

0.310

0.140

19.808

1.313

0.102

2

101∼300

0.063142.4954.521

30.6760.0145

3.399

0.130

0.292

0.137

18.272

1.385

0.111

3

301∼650

0.039746.381

5.317

29.4930.0119

3.996

0.154

0.241

0.121

14.245

1.573

0.135

4

501∼730

0.026850.342

6.420

28.5000.0115

4.815

0.184

0.191

0.089

9.419

1.766

0.169

5

601∼7700.017452.9437.639

27.0700.0115

5.736

0.216

0.142

0.061

6.164

1.956

0.212

6

681∼805

0.011054.3629.385

25.0840.0107

7.088

0.263

0.090

0.037

3.669

2.167

0.283

7

721∼8150.009454.19910.48624.1660.0108

7.909

0.287

0.065

0.029

2.839

2.243

0.327

8

801∼830

0.004750.51815.29021.1320.0115

11.414

0.407

0.038

0.014

1.171

2.391

0.540

9

825∼834

0.003243.66520.97718.8460.013815.236

0.539

0.031

0.007

0.681

2.317

0.808

10

834∼835

0.026320.92640.2118.816

0.010728.658

0.960

0.031

0.004

0.358

2.374

3.251

(2020) 7:47

Nakano and Hashimoto Progress in Earth and Planetary Science

Page 14 of 37

Table 4 Average chemical compositions of 13 ensembles (upper sections) and relative volatilities of elements (lower sections)(Continued)
11835∼8480.05370.268

58.8640.465

0.016639.792

0.482

0.010

0.010

0.038

0.576

85.57

12

848∼851

0.08160.013

66.7870.061

0.022931.995

0.813

0.021

0.006

0.199

0.218

527.9

13

851

0.10290.000

71.4600.036

0.034427.090

0.938

0.035

0.000

0.302

0.000

752.3

E-pairζFe /ζMg

Na

Mg

Al

Si

K

Ca

Ti

Cr

Mn

Fe

Mg/Si

0-1

6.200

7.066

0.259

0.000

1.000

4.279

(0.116)

0.065

1.683

0.966

1.604

1.189

0.088

1-2

5.666

1.448

0.320

0.000

1.000

(0.439)(0.046)

0.005

1.582

1.087

1.815

1.349

0.106

2-3

5.512

3.076

0.369

0.000

1.000

1.769

0.002

(0.043)

1.755

1.399

2.036

1.477

0.123

3-4

5.616

2.585

0.477

0.000

1.000

1.013

0.010

0.042

1.885

2.217

2.681

1.668

0.152

4-5

4.808

2.659

0.547

0.000

1.000

0.770

(0.006)

0.059

2.072

2.455

2.630

1.859

0.190

5-6

3.987

2.337

0.635

0.000

1.000

0.998

(0.020)

0.037

2.329

2.451

2.533

2.057

0.246

6-7

3.213

1.774

0.769

0.000

1.000

0.685

0.009

0.161

2.933

2.489

2.470

2.204

0.304

7-8

2.720

2.054

0.874

0.000

1.000

0.595

0.020

0.053

1.774

2.071

2.378

2.312

0.427

8-9

1.835

1.661

1.073

0.000

1.000

0.318

0.063

0.083

1.174

2.327

1.968

2.356

0.667

9-10

0.939

(0.697)

0.984

0.000

1.000

0.653

0.013

0.050

0.479

0.883

0.924

2.335

1.587

10-11

0.858

(0.141)

1.069

0.000

1.000

(0.027)

0.024

0.477

0.630

(0.236)

0.917

2.284

7.376

11-12

(0.638)(0.172)

1.164

0.000

1.000

(0.117)0.206(0.231)(0.342)0.412(0.743)

0.535

136.6

12-13

(0.167)(0.283)

3.585

0.000

1.000

(0.575)0.405(0.131)(0.749)3.585(0.599)

0.137

611.5

ζj /ζi (i = Si)pair average
Ca/Si

Note. E# denotes ensemble number. E-pair denotes ensemble pair. ‘Mg/Si’ and ‘Ca/Si’ denote molar ratios of elements in each ensemble.
‘pair average’ is average of ensemble pair. Subscript j denotes elements. Bracketed values denote negative values

ensemble pair 0-1 in Table 4, which encompasses Mg/Si
ratio from 0.9 to 1.2. A200, on the other hand, does notcross 1 in Mg/Si because the initial Mg/Si for Allende is
1.072.
The systematic shift in ZFe/Mg of A200 from L200
in Fig. 7b may be a temperature effect. According toHashimoto (1983), the volatility of FeO is relatively
increased at lower temperatures compared to that of
MgO. A temperature difference of 200 K in the temperature range near 3000 K changes ZFe/Mg by a factor of 1.3,
explaining the offset of A200 from L200.
Relative volatilities of other elements

The relative volatility of Cr, ZCr/Si , for L200 is nearly
constant from ensemble pair 0-1 through 9-10, and its
average is 1.94. For A200, ZCr/Si correlates well with and isroughly proportional to ZFe/Si with a proportionality factor of 0.9. ZMn/Si is roughly proportional to ZFe/Si with aproportionality factor of 0.9 for L200 and 0.8 for A200.
The values of ZNa/Si and ZK/Si for L200 are valid only
for the ensemble pairs 0-1, 1-2, 2-3 and 3-4 for Na, and
0-1 and 1-2 for K. This is because the Na concentration beyond ensemble 4 and the K concentration beyond
ensemble 2 are below their detection limits (0.01 mol%for Na; 0.02 mol% for K) in chemical analyses. Hencethe average of four ensemble pairs for Na and that oftwo pairs for K give 5.2 for ZNa/Si and 5.4 for ZK/Si ,respectively. These values may, however, represent theirrelative volatilities only appropriate to low concentrations
in Na and K. The shortcoming comes from the way oforganizing the ensembles in “Derivation of relative volatilities from the experimental results” section, where moderately volatile elements are optimized so as to derive
their relative volatilities over concentration ranges. Veryvolatile Na and K evaporate in the earliest phase and theirconcentration decrease quickly down to their detection
limits.
To try to overcome these shortcomings, we havearranged another set of thirteen ensembles, using thesame 856 droplets data, which allocates more numbers ofensembles for the early phase of evaporation. The derived
average values for ZNa/Si and ZK/Si are 5.41 from six
ensemble pairs (1-2, 2-3, 3-4, 4-5, 5-6 and 6-7) and 5.05
from four pairs (1-2, 2-3, 3-4 and 4-5), respectively. They
are the values to be used in the following sections.
The first pair 0-1 of the new set, viz., the tie between
the initial composition and the earliest sixty droplets,
however, reveals ZNa/Si and ZK/Si to be ∼10, muchlarger than the other pairs. If the number of the earliest droplets is reduced to as small as five, the pair 0-1
reveals ZNa/Si and ZK/Si to be ∼30. This means that
Na and K at the earliest evaporation phase of MLB
are much more volatile, relative to Si, than in later
phases. This is explained as a temperature effect because
highly volatile elements such as Na and K have considerably smaller heats of vaporization compared to that

Nakano and Hashimoto Progress in Earth and Planetary Science(2020) 7:47

Page 15 of 37

ZTi/Si . Al does not evaporate or is assumed not to, hence
ZAl/Si = 0. By definition ZSi/Si = 1.
Application of relative volatilities to an arbitrary MLB:
self-consistency test

Fig. 7 a ZMg/Si as a function of the Mg/Si molar ratio for L200 andA200. b ZFe/Mg as a function of ZMg/Si for L200 and A200. Each data
point represents an ensemble pair in Table 4. Linear or quadratic
regressions to the data are shown as solid lines, with their equations.
X: abscissa. Y: ordinateof SiO2 . As described in “Time-temperature profiles”
section, it took the MLB ∼0.5 s to reach its boiling point,
by which time significant amounts of Na and K havealready evaporated at relatively low temperatures. The
first few droplets must have stemmed from MLB justwhen it reached a full boiling point.
The ZNa/Si derived for A200 gives lower values than for
L200, 2.42 on average of five ensemble pairs. The firstpair gives ∼7 for ZNa/Si , again much larger than the otherpairs. A further analysis has indicated that ZNa/Si is ∼10at the earliest evaporation phase. The initial concentrationof K in Allende is already too close to its detection limit,
making it impossible to derive a reliable ZK/Si .
Ca and Ti are too refractory to evaporate until latestages, making it difficult to accurately determine theirrelative volatilities for early ensemble pairs. Late ensemble pairs are too rich in Ca (viz., Ca/Si >0.6) and Al,
representing Ca, Al-rich liquids rather than a silicate liquid. From these considerations, the ensemble pair 9-10 for
L200 is only suitable for determination of ZCa/Si and ZTi/Si
for a chondritic silicate liquid. We adopt 0.01 for ZCa/Si
and 0.06 for ZTi/Si for L200. For A200, the ensemble pair
7-8 is suitable, and we adopt 0.02 for ZCa/Si and 0.05 forIntroduction of the new quantitative measure, relative
volatility, has made clear the following important facts. 1)
The Mg/Si ratio in MLB is a dominant factor that determines volatilities of the major elements, Mg, Si and Fe
as well as Mn. The result is quite essential in evaluating evaporative fractionation of chondritic materials andalso of planetary materials, which are basically Mg, Fe-richsilicate bodies. 2) The relative volatilities of all elements
estimated here are similar both for L200 and for A200,
implying that the relative volatilities are insensitive to theinitial composition of the MLB. This fact provides support
for our attempt to predict the evaporative fractionation
trends of silicate liquids having arbitrary compositions.
We wish to predict evaporative fractionation of a silicate
liquid having an arbitrary initial composition by adopting
the relative volatility: Zj/i (i = SiO2 ). We assume that a silicate MLB is composed of oxides of the ten elements, Na,
Mg, Al, Si, K, Ca, Ti, Cr, Mn and Fe, and is homogeneousin composition throughout evaporation. After infinitesimal evaporation of the MLB, the molar fraction of j-thoxide is written by using Eqs. 4 and 11,
Xj = qXj − Zj/i Xj (q −

Xi
).
Xi

(12)

 /X . Replacing j with a dummy k andHere q = XAl
Alsumming both sides of Eq. 12 for all ten oxides,X 
Xk = qXk − (q − i )(Zk/i Xk ).
(13)
Xi
k

k

k

Appreciating that the sum of molar fractions of alloxides is unity,X (Zk/i Xk ).
(14)
1 = q − (q − i )Xik
Replacing the term k (Zk/i Xk ) with b, it follows(bq − q + 1).
(15)b
Accordingly, Xi , the molar fraction of i-th oxide in the
MLB after infinitesimal evaporation is obtained from its
previous value Xi . By inserting Xi into Eq. 12, Xj for allother oxides are calculated. With q taken to be close tounity (e.g., 1.005), Eqs. 12 and 15 are iteratively solved togive a set of Xi and Xj for any given Al mol% or VD% of
MLB starting from an arbitrary initial set of Xi and Xj .
Expressing the composition of the vapor with molar
fractions Yl (l = i or j), the vapor composition at every iterdif
ation step (Yl , called differential vapor composition) is
given byXi = Xi

Nakano and Hashimoto Progress in Earth and Planetary ScienceXXl − ql
dif.
Yl = 
X
(Xk − qk )k(2020) 7:47

(16)

Here k takes all ten oxides. The composition of the total
int orvapor accumulated in N iterations from the start, Yl,N
the integrated vapor composition, is given by summing
the differential vapor over all of the iteration steps,
 dif
Yl,n
N
int
=   dif .
(17)
Yl,N
Yk,n
k N

dif
dif
Here Yl,n and Yk,n are the differential vapor compositionsat iteration step# n.
In the above derivation, we have not included a possible contribution of the ejection of jet-droplets during thecourse of evaporative fractionation of MLB. Even if thejet-droplet mass is taken into account, the chemical compositions of the MLB and the differential vapor are notaffected and are given by Eqs. 12, 15 and 16, since only themass of the MLB is reduced. The integrated vapor composition, however, is affected and Eq. 17 is replaced by,


 
dif
(1 − εn ) Yl,n
n
N
int


=
.
(18)
Yl,N
 
dif
(1 − εn ) Yk,nk N

nHere εn is the mass proportion of jet-droplets ejected atstep# n, relative to the mass of the MLB at step# n. In thecase where the ejected mass as jet-droplets is negligibly
small, Eq. 17 is strictly valid.
We have applied Eqs. 12, 15, 16 and 17 to the initial
MLB compositions of Lsil and Allende, to check for selfconsistency of the set of relative volatilities determined for
all ten elements, and whether they reproduce the chemical composition of the MLB (viz., droplets) and vapor(viz., condensates) obtained in the experiments. The setof relative volatilities and their applicable conditions are
summarized in Table 5. The iteration was executed with aconstant q (either 1.005 or 1.002, depending on a required
precision) from VD = 0 to 95%. The result is shown in
Figs. 6 (upper nine panels) for L200. As shown in Fig. 6,
the calculation reproduces the composition trajectories ofthe MLB defined by droplets quite well. Figure 8 compares the calculated trajectories of the integrated vapor
with the bulk compositions of condensed particles inthe experiments. Although the bulk condensates (experiments) scatter in composition due to their own heterogeneity (“Chemical analysis” section), their general trends
are reproduced by the calculated trajectories rather well.
The convincing result in the self-consistency test justifies
both the L200 experiment and the use of relative volatilities derived from it. It also indicates that the effect ofPage 16 of 37

mass loss by ejection of jet-droplets on the composition of
the integrated vapor is small, since the total mass loss inour experiments as jet-droplets was less than ∼10% of theinitial mass of the MLBs.
The same calculations were made for A200. The set of
relative volatilities for A200 used is shown in Table 5.
The compositional trajectories of the MLB defined by the
droplets are well reproduced by the calculation, as shownin Fig. 6 (lower nine panels). As shown in Fig. 8 (lower
six panels), however, the integrated vapor does not follow the general trends of the bulk condensates, except for
Na. The Fe content in the bulk condensates is systematically low as compared to that of the calculated vapor,being explained by the loss of FeO from the MLB as aresult of reduction reactions that proceeded in the Allendeexperiments (“Derivation of relative volatilities from theexperimental results” section). Its effect is two fold: 1) theMLB and droplets from it become depleted in FeO, indicating superficially a high evaporation rate of FeO andhence its large relative volatility that is used in the calculation; 2) the condensates from the vapor are depleted
in FeO, indicating superficially its low evaporation rate in
the experiment. The high Si (and Mg) content in the bulk
condensate, as compared to that in the calculated integrated vapor, is considered a compensating effect of the
Fe depletion as a result of normalization to 100 mol%. It
should not largely affect minor elements such as Mn andCr, but they also considerably deviate from the calculated
trajectories. As a result, the discrepancies in composition
between the experimental condensates and the calculated
integrated vapor disqualifies A200 for further use of its
derived relative volatilities.

Discussion
It is required for a proper model of chondrule formation that the 3.5 fundamental features of chondrules must
be explained. 1) Why is their maximum size 1-5 mm? In
addition, why do they distribute in log-normal-like size
distribution? 2) Why are their chemical compositions sovariable? 3) Why are their petrographic textures variable?
We attempt to answer all these questions in the following
discussion.
First we identify the silicate droplets in our experiments
as splash droplets that eject from collapsing cavities ofvapor bubbles, known as ‘jet-droplets’ in the field of atmospheric and oceanic science. If jet-droplets are origin ofchondrules, the characteristic size of bubbles, approximately ten times the size of the droplets, requires a dense
and large silicate body as a chondrule source, named ‘duston’. Accordingly it is postulated that chondrules are jetdroplets from large boiling silicate bodies (“Jet-droplets asorigin of chondrules” section).
A boiling duston should accompany a large vapor
cloud expanding adiabatically, where jet-droplets must

Nakano and Hashimoto Progress in Earth and Planetary Science(2020) 7:47

Page 17 of 37

Table 5 The estimated relative volatilities and their applicable conditions
Conditions

ZMg/Si
L200

C < 1 and X < 1.052

0.093

C < 1 and 1.052 < X < 2.924

0.4859X − 0.4208

C < 1 and X > 2.924

1

C>1

1
A200

C < 0.6 and X < 0.8937

0.1

C < 0.6 and 0.8937 < X < 2.758

0.4826X − 0.3313

C < 0.6 and X > 2.758

1

C > 0.6

1
Conditions

ZFe/MgL200

C < 1 and Y < 0.2232

284.003Y 2 − 132.367Y + 21.124

C < 1 and 0.2232 < Y < 1.0631−5.589Y + 6.941

C < 1 and Y > 1.0631

1

C>1

1
A200

C < 0.667 and Y < 1.205−5.534Y + 7.669

C < 0.667 and Y > 1.205

10.667 < C < 1.587−0.9733C + 2.4834

C > 1.587

0.9242
Conditions

ZNa/Si

ZK/Si

ZCa/SiZTi/SiZCr/SiZMn/SiZFe/Si

L200
VD < 8.5%

f(VD)

f(VD)

0.01

0.06

1.94

0.9ZFe/Si

ZFe/Mg ×ZMg/Si

VD > 8.5%

5.41

5.05

0.01

0.06

1.94

0.9ZFe/Si

ZFe/Mg ×ZMg/Si

VD < 9%f1 (VD)

f2 (VD)

0.02

0.05

0.9ZFe/Si

0.8ZFe/Si

ZFe/Mg ×ZMg/Si

VD > 9%

2.42

2.42

0.02

0.05

0.9ZFe/Si

0.8ZFe/Si

ZFe/Mg ×ZMg/Si

A200

Note. C: Ca/Si molar ratio. X: Mg/Si molar ratio. Y: ZMg/Si .
f (VD) = 0.2854(VD)2 − 5.194VD + 29.27.
f1 (VD) = −0.8867VD + 10.35.
f2 (VD) = −0.2568VD + 5.09

react with vapor and with condensed dust, thereby
re-accommodate volatiles lost during evaporation. Thismechanism should produce a wide spectrum in chondrule chemistry. A model calculation with use of experimentally obtained relative volatilities demonstrates thatvarious chondrule compositions are nearly completely
reproduced from the initial CI composition by considering the vaporization degree and redox state of Fe (“Originof chondrule chemistry” section). The model also naturally explains other features of chondrules (“Origin ofother features of chondrules” section).
A possible mechanism required for heating dustonsmust be energetic enough to cause their surface to boil.
We will explore such a heat source in the companion paper(II). Here we take it as a prerequisite because boiling is a
key to the formation of chondrules in our model.
The main bodies of dustons with their surfaces heatscorched, are decoupled from chondrules and fine dust
due to the aerodynamic effect, and are assembled intoplanetesimals by their self-gravities. The fate of dustons
as well as chondrite-forming materials after the chondrule
formation will be explored in detail in the companion
paper (II).
Jet-droplets as origin of chondrules
Evidence for the ejected droplets being jet-droplets

We first point out preceding works on ejection of dropletsfrom bursting air bubbles on the surfaces of fresh and sea

Nakano and Hashimoto Progress in Earth and Planetary Science(2020) 7:47

Page 18 of 37

Fig. 8 Numerical reproduction of evaporation and condensation trajectories for the experiment groups L200 and A200. Element concentration in
mol% is shown in each panel as a function of Mg mol%. ‘Calc. droplets’, ‘Calc. differential vapor’ and ‘Calc. integrated vapor’ are numerical results(see “Application of relative volatilities to an arbitrary MLB: self-consistency test” section). ‘Condensates in L200’ are the bulk compositions of thecondensed particles in the experiments that belong to L200, and are compared to ‘Calc. integrated vapor’. The L50 data are added for reinforcement
since they exhibit similar trends to the L200 data. The same approach was applied when plotting and modeling the A200 data. The ‘Initial’
composition is either Lsil for L200 or Allende for A200

water (e.g., Kientzler et al. 1954; Hayami and Toba 1958;
Blanchard 1989). Two distinct mechanisms, jet-dropletsand film-droplets, have been studied as a possible major
source of sea-salt aerosols in terrestrial atmosphere (e.g.,
O’Dowd et al. 1997). (Some details of these mechanisms
are described in Appendix A).

Jet-droplets are produced in a series of steps (Georgescu
et al. 2002): a bubble bursts on a liquid surface; the gaping
bubble cavity collapses; a surface capillary wave converges
axially, generating an upward high-speed liquid jet; and a
successive fragmentation of the jet produces one to several jet-droplets. These are visually shown in successive

Nakano and Hashimoto Progress in Earth and Planetary Science(2020) 7:47

frames in Fig. 14d in Appendix A (MacIntyre 1972). One
noticeable characteristics of jet-droplets is their verticalejection from a bulk liquid surface. In our experiments silicate droplets are ejected radially from the spherical MLB
surface (Figs. 2a and 16 in Appendix A), indicating that
their ejection angle is normal to the spherical surface.
Other features of jet-droplets known from the freshwater and seawater experiments can be compared with ourexperiments. Spile (1994, 1995, 1997), in his high-speed
camera analyses, observed 1-6 droplets in each bubble
burst event, with ejection speeds of 0.1-10 m s−1 . In our
experiments, 1-3 droplets were observed per bubble burst,
with velocities of 0.3-4 m s−1 (see Fig. 16 in Appendix
A). Our measurements, however, were biased by the limitations of the available video images: the viewing areaof ∼90 mm across and the shutter speed 30 frames s−1
meant that the maximum number and velocity of droplets
were limited to 3 and 4 m s−1 , respectively.
In the water jet-droplet experiments, the size ratio of
jet-droplet to its original air bubble is known to be 1/101/5. In our preliminary experiment (run# V2; see Fig. 15 inAppendix A), which used a high-speed video camera, thesize ratio of droplet/bubble was observed to be ∼1/6.
There is another line of evidence for jet-droplets, whichused CO bubbles in iron liquid and N2 bubbles in liquid mercury (Distin et al. 1968). Figure 3 in their paper
shows that a single bubble burst produced more than threeiron droplets, and it is also calculated from the figurethat the droplet/bubble size ratio was ∼1/10 and thatthe droplets were ejected at 4 m s−1 . The size ratio of
mercury droplet/bubble is estimated 1/11-1/7 from theirdescription (Distin et al. 1968).
Admittedly, our experiments lack high-speed images ofthe moment of ejection of jet-droplets, but all lines of evidence acquired so far indicate that jet-droplet phenomena
are common to any kind of liquid with any kind of bubble
gas in it.
Evidence for chondrules being jet-droplets

If jet-droplets are the origin of chondrules, the maximum size of chondrules, 1-5 mm depending on chemical
types of chondrites, must be explained by the maximum
size possible for jet-droplets. Our experiments produceda droplet with the largest diameter ∼0.4 mm, while thediameter of the MLB at its earliest stage before significant evaporation was 2.1 mm. Distin et al. (1968) observed
the largest iron droplet ∼1 mm in diameter, while thediameter of molten iron body (2 g) is calculated to be
8.2 mm. King (1982, 1983) reported a ∼0.7 mm silicate
drop (in King 1982, figure 8), while the initial diameter of molten bulk Allende sample (∼2 g) was ∼10 mm.
All three experiments indicate that the maximum droplet
sizes observed in them are 1/5-1/15 times those of theMLBs. This fact implies that their observed maximum

Page 19 of 37

droplet sizes are controlled by the maximum possible bubble sizes, or strictly saying the maximum cavity diameters
after they burst, which should be no larger than the sizesof MLBs. This may lead to a tentative suggestion that the
maximum chondrule diameter was determined by the sizeof their precursory melt body and is 1/10-1/5 of the latter.
Yet another plausible constraint as to the maximum
chondrule size comes from the study of water jet-droplets.
It is known that there are upper size limits for bubbles thatcan produce jet-droplets upon bursting at the surface ofwater and other solvents (Resch and Afeti 1991a; Suzuki
and Mitachi 1995; Georgescu et al. 2002). For a distilledwater, the upper size limit is 5.4 mm in diameter (Suzuki
and Mitachi 1995).
According to Blantachette and Bigioni (2006) the capillary wave must travel a quarter of the circumferencealong the cavity surface from edge to bottom before being
damped by viscous dissipation (Fig. 14d in Appendix A).
It travels at the wave velocity given by vw = (σ k/ρ)1/2(Landau and Lifshitz 1987) and reaches the bottom in timet = (πRb /2)/(σ k/ρ)1/2 , where Rb is radius of the cavity,
σ surface tension, k wave number, ρ density of the liquid. The attenuation efficiency f is given by f = 2μk 2 /ρ(Landau and Lifshitz 1987), where μ is dynamic viscosity.
The condition for the wave to reach the bottom without
its complete dissipation is ft < 1. Thus the critical bubble
radius for ejection of jet-droplet is given by11(ρσ )1/2 .
(19)
Rcr =
μ πk 3/2
Georgescu et al. (2002) obtained numerically critical bubble radii for twenty kinds of liquids (Table 7 inAppendix B), including freshwater and seawater, in good
agreement with experimentally determined critical bubble radii. We have calculated wave numbers k for thetwenty solvents using Eq. 19 and the values for Rcr , ρ, σ
and μ given by Georgescu et al. (2002), and found that
there is a good linear correlation between log μ and log
k as shown in Fig. 17 in Appendix B. Using Eq. 25, we
can calculate wave numbers for various silicate liquids
from their known dynamic viscosity, and apply its result
to obtain their critical bubble radii using Eq. 19.
Table 6 shows Rcr for peridotite, dunite and enstatitecompositions at 2400, 2700 and 3000 °C. The surface
tension was assumed constant, 0.4 N m−1 , for all temperatures since its temperature dependence is small. We
Table 6 Critical bubble radiusT

Rcr (m)(°C)

Peridotite

DuniteEnstatite

3000

0.01275

0.013290.010362700

0.011960.012480.009712400

0.01104

0.011540.00901

Nakano and Hashimoto Progress in Earth and Planetary Science(2020) 7:47

have used data by Bottinga and Weill (1972) for viscosityand data by Nelson and Carmichael (1979) for densities,
and extrapolated those to high temperatures. Chondritic
composition is more or less close to dunite among thethree compositions. Accordingly 2Rcr , viz., the maximum
bubble diameter for jet-droplet ejection, is 23-27 mm for
2400-3000 °C, and the corresponding jet-droplet size is
4.6-5.4 mm in diameter by assuming the 1/5 size ratioof droplet to bubble (the size ratio slightly increases withbubble size). This size range of jet-droplets is very similar
to the maximum chondrule size of ordinary chondrites,
hence explains the upper limit of the chondrule size as wellas the lower abundances for larger chondrules. An use of
non-Arrhenian T-dependence of viscosity (e.g., Giordano
et al. 2008) in the calculation, instead of the Bottinga andWeill (1972) data extraporated to higher temperatures,
reduces Rcr by only 10%. Thus, this conclusion does notchange.
The maximum chondrule sizes for carbonaceous chondrites are generally smaller than those for ordinary chondrites (Friedrich et al. 2015). It could be explained if
the boiling temperatures of dustons were below 2400 °C
because (1) the higher viscosity for lower-temperatureliquids should dampen the capillary wave more effectively, (2) smaller bubbles obey the ∼1/10 rule for thejet-droplet/bubble size ratio, rather than 1/5 for large bubbles, and (3) carbonaceous chondrites, hence their precursors should contain more volatiles (e.g., H2 O and organic
compounds), that would lower their boiling temperatures.
If the size of a MLB is larger than the Rcr of bubbles,
Rcr determines the chondrule maximum size; if a MLB
is smaller than Rcr , the size of the MLB determines theupper limit for chondrule size. For ordinary chondrites
whose maximum chondrule diameter ranges in 2-6 mm(Friedrich et al. 2015) it is required that their MLBs mustbe larger than 10-30 mm in diameter. For other types ofchondrites whose chondrule sizes tend to be smaller thanthose in ordinary chondrites, the sizes of their MLBs could
be smaller than 10-30 mm. This, however, is not decisive
because the lower-temperature boiling could also explain
their smaller size ranges. It is also plausible that only limited areas of their MLBs were heated above their boilingpoint; in that case jet-droplets were ejected from bubbles
much smaller than the MLB size.
Another feature that characterizes chondrules is theirsize range and size-frequency distribution. The sizefrequency distributions of chondrules have been measured for nearly all types of chondrites. Most data, however, are biased because the chondrule size is usually
determined on polished sections of meteorites, whichmeans that a measured size is nothing but a crosssectional diameter. Statistically hundreds or thousands ofmeasurements on a single chondrite would produce asize-frequency distribution somewhat similar to its real

Page 20 of 37

distribution, but it tends to be shifted towards smaller
sizes and also creates an artifact in that there is no cutoff for the smallest size. The measurements using thedisaggregation method is only trusted, in which statistically meaningful analyses (e.g., at least two hundreds
measurement for each chondrite) show log-normal-likesize-frequency distributions with a size range (maximum/minimum size ratio) of 10-15 (Friedrich et al. 2015).
As shown in “Size-frequency distribution of silicate
droplets” section, the size-frequency distribution of jetdroplets in two experimental runs exhibited log-normallike distributions. This makes a strong argument that jetdroplets from bursting bubbles are origin of chondrules.
In his solar-furnace evaporation experiments using theMurchison meteorite, King (1982) observed droplet ejections similar to our experiments. Although he described
them as a sputtering product, we suppose that they were
jet-droplets judging from their size distribution shown in
Rosin’s size-frequency diagram. King (1982) is the first
who proposed that no particular size sorting mechanism(e.g., Liffman and Brown 1995) is necessary after chondrule formation to explain their log-normal distribution.
Because of the straightforward size relationshipbetween the jet-droplet and its natal bubble, it appears
that the size-frequency distribution of bubbles mustobey a log-normal size distribution. We presume thatthe product of the bubble size frequency and its bursting
probability determines the size-frequency distribution ofjet-droplets.
Duston as a source of chondrules

If chondrules were jet-droplets ejected from a molten silicate body, there should have existed a solid precursory
body before its melting. The body is presumed to be composed of primordial dust in PPD, that is dense enoughfor holding a molten surface layer and larger than 20
mm in diameter from the consideration made in previous sections. We name such a body ‘duston1 ’ in order
to distinguish it from the conventional image of fluffy
dust-balls.
The collisional dust growth has been studied experimentally and its size limit known as a bouncing barrierhas been suggested, several mm in its final diameter(e.g., Blum and Wurm 2008). However, the dust-collisionexperiments were performed mostly using aggregates ofmonomer grains (usually amorphous silica) with a single
grain size, ∼1 μm. Ingredient particles of primordial dust,
as inferred from interstellar dust, are extremely variablein their physical and chemical properties, including their
1 ‘Duston’ is defined as a dense and large body (>20 mm in diameter) consistingof primordial dust in PPD, that is postulated to have existed as a source ofchondrules. This new term is a hybrid of English and Japanese in a shortenedform of dust-ton, which means a large, rounded unified mass of dust. As an
example ‘Futon’, a Japanese mattress, is a combination of ‘fu’ and ‘ton’, whichmeant historically a large, round bundle (ton) of clothes (fu) to sit on.

Nakano and Hashimoto Progress in Earth and Planetary Science(2020) 7:47

grain sizes (e.g., Kim et al. 1994; Draine 2003; Natta et al.
2007; Waters 2015). The behavior of dust aggregates incollision: either stick, bounce, fragment or erode is hardto predict if various materials and their combinations withvarious grain sizes, especially in nanometer ranges, areinvolved (Blum 2018). Therefore, we still do not know if
there is a barrier for the dust growth and how large thesize limit would be if there was a barrier.
The existence of large dust aggregates is suggested
from theoretical studies of giant planet formation, whichhave shown that pebble-sized materials are a key to thefast growth of giant planetary cores (Lambrechts andJohansen 2014; Levison et al. 2015). In-situ observationsof the comet 67P/Churyumov-Gerasimenko in Rosetta
mission revealed a very wide range in size of dust aggregates, from 10−5 to 10 meters (e.g., Fulle et al. 2016;
Poulet et al. 2016; Ott et al. 2017; Pajola et al. 2017;
Blum et al. 2017). The size range differs depending on thevarious dust-sensitive instruments onboard Rosetta andPhilae, but it appears that most of the mass resides in theforms of decimeter particles (Blum et al. 2017). Blum etal. (2017) concluded that 67P/Churyumov-Gerasimenko
likely formed through a gentle gravitational collapse ofa bound clump of several-mm-sized pebbles, intermixed
with microscopic ice particles. Recently, a multiwavelength analysis of (sub-)mm and cm continuum interferometric observations of three PPDs has shown that the
maximum dust grain size decreases with distance from thecentral star and it tends towards similar values ∼2 cm inthe innermost spatially resolved region 10 au < R < 20 au(Tazzari et al. 2016).
We suggest that the grain growth of primordial dust
inevitably led to the formation of dustons, and that dustons dominated the dust mass function by the time chondrules started to form.
Origin of chondrule chemistry
Vapor cloud model

A boiling duston, as proposed here for the formation ofchondrules, should be accompanied by a high density silicate vapor cloud around it. In a thin gas environment
such as PPD, the vapor cloud would expand adiabatically with a subsonic velocity and subsequently cool toa supersaturated state. Jet-droplets, on the other hand,
rapidly cool by radiative emission and possibly fall into asupercooled state. Naturally, they would condense vapor
molecules upon collision en-route to the outside of thecloud and also incorporate the homogeneously-nucleatedsilicate dust at the vapor cloud-disk gas boundary. Toward
a quantitative estimate of such processes, we first consider
a simple steady state of the vapor cloud.
Figure 9 depicts a steady state vapor cloud with a boiling
duston (surface temperature ∼3000 K) of radius r0 in thecenter. We assume that an adiabatically expanding vapor

Page 21 of 37

Fig. 9 Schematic diagram of a steady-state vapor cloud with a boiling
duston surface (r = r0 ) at constant T0 = 3000 K and P0 = 1 bar.
Adiabatic expansion of the vapor cloud ends at r ∼ 100r0 , where its
pressure becomes equivalent to that of the disk gas (assumed 10−5bar) and a whole condensation occurs as shown in gray belt. The
abscissa is in log scale except r < r0 . Jet-droplets, shown in filled
circle, eject from the duston surface and may grow in size byrecondensation of the vapor and by incorporation of condensed
particles on their way out

condenses as its pressure becomes equivalent to that of
the surrounding disk gas (taken as 10−5 bar in Fig. 9),for two reasons. (1) Condensation of vapor needs a large
heat sink for its latent heat to be absorbed, which the disk
gas should provide. (2) Without preexisting solid nuclei,
vapor must condense homogeneously. Homogeneously
condensed particles in general are extremely fine-grained,
typically one to tens of nm (e.g., Kato 1976; Rietmeijer etal. 1999, 2002, 2009). Particles much smaller than 1 μm
cannot radiate at infrared wavelengths, thus cannot cool
effectively. Even if they condensed, they should immediately reevaporate due to the latent heat deposited on
them.
Nucleation and condensation of the vapor may happen
well inside the vapor cloud, at least partially, if the maximum supersaturation at which a significant number ofcondensation nuclei form is reached before the vapor isdepressurized to 10−5 bar. The maximum supersaturation, in turn, is determined by the interplay between two
opposite influences: cooling of the vapor due to the workof expansion, and heating of the vapor due to the latent
heat liberated during condensation (Raizer 1960). Here, adetailed examination on the nucleation is difficult and outof scope. However, even if the homogeneous nucleation
occurred before the vapor was depressurized to 10−5 bar,
the condensed particles would be accumulated near theboundary with the disk gas, because they should keep thesame momentum as they had in the vapor state before
condensation.

Nakano and Hashimoto Progress in Earth and Planetary Science(2020) 7:47

For simplicity, we choose the situation where the vapor
homogeneously condenses as it reaches the boundary.
Then, due to condensation, the silicate vapor pressure at
the boundary becomes negligible, which ensures a free
expansion of the following vapor that is continuously produced by the boiling duston. Hence, a steady state pressure
profile is established throughout the cloud in less than 0.1
s for dustons smaller than 1 m. The vapor flux from theduston surface (as denoted by subscript ‘0’) is equal to
the flux passing through a sphere of arbitrary radius (asdenoted by subscript ‘1’).
 
 
2 1
2 1
n0 c0 = 4πr1
n1 c1 .
(20)
4πr0
4
4
Here c = (8RT/πM)1/2 is the mean thermal velocity, r:
radius, n: vapor density (mole m−3 ), R: gas constant, T:
absolute temperature, and M: the mean molecular weightof the vapor. A free expansion (into a vacuum space)proceeds adiabatically, hence from Eq. 20
P1
=
P0



r1
r0

 −4γ

γ +1

T1
;
=
T0



r1
r0

 4(γ −1)
γ +1

.

(21)

Here γ is the adiabatic index, and P0 is the vapor pressureof silicate liquid at the duston surface T0 . Substituting γ= 1.6 for the silicate vapor and P0 = 1 bar at 3000 K fora ∼CI-chondritic composition, estimated from the dataof Hashimoto (1983), the pressure and temperature profiles are given as functions of normalized radius, r1 /r0 , asshown in Fig. 9.

Page 22 of 37

Now we turn to the jet-droplets that are ejected fromthe boiling duston. They travel through the vapor cloudin radial direction. We first make the simplest assumptionthat the jet-droplets do not interact with vapor at all, butthat they cool independently by their own radiation andevaporation. The radiation from the duston’s surface isalso considered. Then the droplet’s temperature T is givenby−4πa3 ρcp
dT = 4πa2 Q(T)dt;3
ro 2
1
ho Pe.
+√ v
Q(T) = σ T 4 − σ T04
4
r
2πMRT

(22)

Here a: radius of the droplet (which is assumed constant because it changes little), ρ: its density (assumed
constant), cp : its specific heat (assumed constant), hov :its latent heat of vaporization (assumed constant), T: its
temperature (assumed uniform from surface to inside),
Pe : its vapor pressure, M: mean molecular weight of thevapor (assumed constant), R: gas constant, and σ : StefanBoltzmann constant.
Integration of Eq. 22 and consideration of travel timet = (r − r0 )/Ve (Ve : ejection speed) giveT
T0

Q−1 (T)dT = −(

3
r0
r
)(
)( − 1).
ρcp aVe r0(23)

Equation 23 indicates that the term, r0 /aVe determines
the temperature profile of the ejected droplet. Basedon the present experiments and some considerations

Fig. 10 Upper panel: Temperature profile of the expanding vapor (Texp,vap : solid line), and those of ejected jet-droplets (Tdrop ) with a constantproduct of aVe = 2 × 10−3 m2 s−1 , starting from 3000 K as functions of the normalized radius, r/r0 . Four r0 values (0.03, 0.1, 0.3, and 1 m) areselected for the radius of duston. The temperature profile of the expanding vapor is independent of the duston’s normalized radius. Teqv is thecondensation temperature of a saturated silicate vapor whose pressure is equivalent to that of the expanding vapor, shown in the lower panel.
Lower panel: Pressure profile of the adiabatically expanding vapor (Pexp,vap ). Teqv is higher than the droplet temperatures, meaning that the
expanding vapor is always supersaturated for the droplets

Nakano and Hashimoto Progress in Earth and Planetary Science(2020) 7:47

(Appendix C), there is a good relationship between Rb(bubble radius) and Ve : Ve = A exp(−αRb ), (A = 14.6
m s−1 , α = 330.3 m−1 ) and between a and Rb : log a =1.1267 log Rb − 0.4579. The product, aVe ranges from
1×10−3 to 2.75×10−3 m2 s−1 , corresponding to the radiusfrom a = 8.5 × 10−5 to 2.0 × 10−3 m for chondrules inordinary chondrites.
Figure 10 shows temperature profiles of jet-droplet withthe duston’s radius as a parameter, r0 = 0.03, 0.1, 0.3 and 1
m. Here we used ρ = 3 × 103 kg m−3 , σ = 5.6705 × 10−8
J m−2 K−4 , cp = 1.458 × 103 J kg−1 K−1 , and hov =1.936 × 107 J kg−1 . We have used enstatite liquid for values of cp and hov (Chase et al. 1985). We have also taken
aVe = 2 × 10−3 m2 s−1 as average. A most notable featureis that the vapor is always super-saturated for jet-droplets,
as Teqv (condensation temperature of a saturated silicate
vapor whose pressure is equivalent to that of the expanding vapor) is always higher than the droplet temperatures.
This means that the vapor molecules that collide with jetdroplets effectively condense on them. Indeed, it points
out that a quantitative analysis including the effect ofdroplet-vapor interaction is necessary.
Smaller dustons are covered with smaller vapor clouds,
while ejection speeds of jet-droplets do not depend on
duston size. It implies that the jet-droplets from small dustons suffer less from condensation of the vapor. With large
dustons, on the other hand, droplets would cool down
below their glass transition temperatures, ∼1000 K, before
their reaching the vapor condensation zone. In this case,the droplets would not be able to dissolve the condensed
particles into their liquids, which are rich in volatiles suchas Na and K, although they would condense more massof moderately-volatile vapor by collision on the way. We
therefore expect that we may be able to identify the size
range of dustons that satisfies highly variable chemical
and textural characteristics observed for chondrules. Fora satisfactory argument, however, we need a quantitative
estimate that considers exchange processes between jetdroplets and the vapor cloud in terms of mass, energy andmomentum.
Model calculation of chondrule chemistry(i) Semi-quantitative Approach We next evaluate thechemistry of jet-droplets, vapor and condensate thatwould evolve as a result of continuous evaporation of theMLB, viz., the molten surface of a duston. Our model
predicts that chondrules consist of these three components because on their ways out from the vapor cloud,jet-droplets inevitably take vapor and condensate into
their liquid masses. The question is when in the evaporative evolution of MLB they are ejected and how muchuptake of vapor and condensate is possible. Another question is about characterization of dustons that should be a
primary source of chondrules. We assume dustons to be

Page 23 of 37

dense aggregates of primordial dust which has the sameelemental composition as that of the rocky componentof the Sun or of CI chondrite. The elemental composition alone, however, is not enough to characterize dustonssince their chemical or redox states must be specified.
The initial materials in our experiments, Lsil andAllende, included the ten oxide elements for analyses: Na,
Mg, Al, Si, K, Ca, Ti, Cr, Mn and Fe, which are mostabundant in CI (except for extreme volatiles and organic
elements). We have found that the relative volatilities of
the ten elements determined from the L200 experiments
are most suitable for further use (“Application of relative volatilities to an arbitrary MLB: self-consistency test”
section). All the nine elements except Fe are stable as oxide
in normal redox environments assumed in PPD, but Fe
would take various redox states such as oxides, sulfides,
and metal. When melted, the oxides including Fe-oxide
form one silicate liquid, but Fe metal and sulfides do not.
Hence we take FeO content in the silicate as a parameter.
In the following analyses, we adopt the abovementioned ten elements having CI abundance ratios onmolar basis (Lodders 2003) and normalize the composition to 100 mol% in total. This is called our CI composition. The FeO content in our CI composition is adjusted asfollows. We define fs (≤1) as the fraction of Fe that exists asoxide in the total Fe in our CI composition. This operation
makes the total oxide mol% becomes less than 100, thenit is normalized to 100 mol%. The normalized composition is different from our CI composition in Fe content butotherwise keeps CI abundance ratios among other nineelements. We call it CIsil , as it represents a silicate composition when melted. Naturally CIsil with fs = 1 is identicalto our CI composition.
We investigate the evaporative chemical fractionation ofprimordial dust with CIsil as initial composition for fs values 0.01, 0.1, 0.3, 0.5 and 1, assuming that it occurred in
PPD environment at temperatures high enough to cause
boiling. We use the relative volatilities for the ten elements
determined from the L200 experiments and apply themto CIsil composition, as is done in “Application of relative volatilities to an arbitrary MLB: self-consistency test”
section for Lsil . We also note that the relative volatilities are properties that are insensitive to temperature
variations as explained in “Definition of relative volatility and mathematical method of its derivation” section.
This makes it logically sound for them to be applied tochondrules as their exact formation temperatures are notknown but for being in the boiling range. Given the initial composition of MLB (Xi for Si and Xj for other nineelements), viz., CIsil , a combination of Eqs. 12 and 15
gives a next set of composition (Xi and Xj ) for MLB after
an infinitesimal evaporation, which in turn is used for anext step of evaporation and so on. In reality, a small but
finite evaporation is imposed by setting the step index

Nakano and Hashimoto Progress in Earth and Planetary Science(2020) 7:47

q = 1.005-1.001 to be applied to each step of evaporation. A small (q − 1) is required at large VD% or at highAl concentration in MLB for keeping a good precision in
the approximation. At any step of evaporation jet-droplets
may be produced from MLB, thus they have the same
composition (Xi and Xj ) as that of MLB at their ejection
moment. Equations 16 and 17 give compositions of differential vapor and integrated vapor, respectively, in each
iteration step. The differential vapor corresponds to thevapor of the vapor cloud at every moment of evaporation,
which jet-droplets may travel through. The integrated
vapor is the total vapor that accumulates from the start
until a current evaporation step is reached, and corresponds to a bulk of condensed particles distributing near
the border of the vapor cloud, which jet-droplets may go
across.
Figure 11 shows concentrations (Na, Si and Fe) in
jet-droplet, differential vapor and integrated vapor as afunction of Mg concentration, in each of iteration step.
In the same figures, chemical compositions of all chondrules in all types of chondrites (in so far as we surveyed) are plotted (see Appendix D for the sources ofchondrule data). The composition of MLB evolves from
CIsil to Mg-rich one till Mg reaches 55.8, 57.5, 58.3,
59.2 and 59.7% (corresponding to fs = 1, 0.5, 0.3, 0.1
and 0.01, respectively), where it turns to Mg-poorer
one. The overturn is the effect of Ca enrichment as
explained in “Relative volatilities of Mg and Fe” section.
The Na, Si and Fe concentrations in MLB decrease continuously as evaporation proceeds. The Na concentration of differential as well as integrated vapor starts with∼58% (which is scaled out in the figures) and decreases

Page 24 of 37

monotonously as evaporation proceeds. The Fe and Si
concentrations in the vapor (both differential and integrated) increase to their maximum values then decrease
monotonously.
The triangle in each of Fig. 11 consists of the three tielines between the compositions of the jet-droplet, differential vapor and integrated vapor at a specified evaporation step, in order to show an example that the chondrulecompositions inside the triangle can be expressed by combination of the three components. It is, however, only abinary expression (e.g., Mg-Na) and does not guaranteethat the same chondrules are enclosed in the triangle atthe same evaporation step in another binary system (e.g.,
Mg-Si). This problem is completely removed in a quantitative treatment described in “(ii) Quantitative Approach”
section.
Yet we emphasize the two points. 1) The composition
space which combinations of the three components withvariable evaporation degrees (VD%) and fs values define
appears to cover the entire space in which chondrules distribute. 2) A wide fs range, 0.0-0.7 is required to cover all
chondrules. The first point supports our hypothesis thatchondrules were jet-droplets which subsequently reactedwith vapor and condensate in the vapor cloud. The second
point implies that the primary CI-composition dust which
made dustons was quite variable in redox state, imposing arather strong constraint on characterization of primordialdust. These points are reinforced in the following section.

(ii) Quantitative Approach We wish to calculate thechemical compositions of the three components and theirmixing ratio that reproduce chemical compositions ofFig. 11 Binary diagrams showing that a majority of real chondrules (crosses) is covered by the composition space defined by combinations of thethree components, jet-dropet (in red), differential vapor (in light blue) and integrated vapor (in deep blue), that evolve by evaporation of theprimordial silicates. Trajectories of chemical compositions (mol%) calculated for the three components are shown separately for five different initial
CIsil compositions with different fs values, 0.01, 0.1, 0.3, 0.5 and 1. For the sake of visual clarity the parameter fs is not shown in the figures; in the twoleft-most figures, a smaller fs corresponds to higher concentrations in Mg, Na and Si for all three components; in the right-most figure, a smaller fs
corresponds to lower Fe concentration. Open triangle in each figure connects the three components that coexist at a specified evaporation step (or
VD%) for the same fs value, in order to show the composition space that can be expressed by combination of the three components. Suchexamples are shown: in the left, middle, and right figure, the triangle corresponds to fs & VD = 0.1 & 20.1%, 0.1 & 63.3%, and 0.5 & 63.3%, respectively.
‘Chondrules’ are compositions of real chondrules from all types of chondrites analyzed so far (see Appendix D for their data source). ’Initials’ areinitial compositions, CIsil corresponding to the five selected fs values

Nakano and Hashimoto Progress in Earth and Planetary Science(2020) 7:47

silicate chondrules, by solving five simultaneous linear
equations,
Am,n Fn = Bm (m = 1−5

and n = 1−5).

(24)Here Bm represents a molar fraction of element m = 1-4(Na, Mg, Al and Si in ascending order) in a silicate chondrule. The basis of molar fraction is the ten elements as
described before. B5 = 1. Fn , an unknown to be determined, represents a mixing proportion (in molar basis) of
the five components n = 1-5 (1: jet-droplet, 2: differential
vapor, 3: integrated vapor,4: final integrated vapor, and
5: our CI). By definition n Fn = 1. Am,n is a matrix todescribe molar fractions of Na, Mg, Al and Si (m = 1-4)
among the ten elements in each of the five components(n = 1-5). A5,n = 1 (n = 1-5).
Am,n (m = 1-4; n = 1-3) can be obtained by calculating the evaporative chemical fractionation starting with
CIsil , similar to the one described in “(i) Semi-quantitative
Approach” section, at any specified evaporation step (or
VD%) and for any specified fs as a parameter. Am,4 (m =1-4) is a chemical composition of the ‘final’ integrated
vapor, viz., that of the total condensates that accumulate
by the end of evaporation process from the MLB. It is distinguished from the composition of an integrated vapor
that accumulates as condensates by a specified evaporation step, while more evaporation from the MLB is still to
go. Am,5 (m = 1-4) is the chemical composition of our CI
as described in “(i) Semi-quantitative Approach” section.
Among the five components, only the first three components (n = 1-3) are needed to make chondrules. The
two extra components (n = 4 and 5) are ones for making a matrix of chondrite. Equation 24 is originally devised
for reproducing bulk composition of chondrites. The useof Eq. 24, however, is quite beneficial for reproducing
chondrule compositions because the four elemental concentrations (Na, Mg, Al and Si) in chondrule composition
instead of two can be used as constraints for solving theequation by finding a solution that imposes F4 = F5 = 0.
It extremely enhances precision since the fifth element,
Fe is already parameterized with fs , leaving the five minor
elements unconstrained. Although Ca is relatively abundant among the five minor elements, it is too refractory
to fractionate from Al for a moderate evaporation degree(VD<95%) which appears to be appropriate for mostchondrules. Therefore, the Ca/Al ratios of chondrules
should be constant, viz., equal to that of the CI chondrite,
if the original composition before their evaporative fractionations is CIsil . It means that the Ca concentrations inchondrules cannot be a constraint to determine the VD%of chondrules. The remaining four elements, K, Ti, Cr and
Mn, if combined, amount to less than 1 mol%, thus hardly
affect the concentrations of other six elements when the
total mol% of the ten oxides is normalized to 100. Besides,
these minor elements had better not be included in the

Page 25 of 37

constraints because their concentrations are usually close
to and sometimes below their detection limits in ordinary
EPMA analyses on chondrules. For these reasons, we use
the concentrations of the four elements, Na, Mg, Al and
Si in chondrules as the constraints, and the parameter fs
for the initial FeO concentration in CIsil in the following
calculations.
Calculation proceeds with a step size q = 1.001 starting with CIsil for a specified fs , and obtains a set of Am,nat every step number. Am,4 is temporarily assigned to Am,3
that would obtain at twice the step number. This procedure inherits from the bulk chondrite fitting, as will
be described in the companion paper (II). Here such an
assignment has no influence on a final solution because F4
and F5 are eventually brought to zero.
Now Eq. 24 is solved at every step number to give themolar fractions Fn . Computation is repeated with fs andstep number as parameters so as to satisfy the following
requirements: Fn (n = 1-3) to be all positive; F4 and F5 tobe zero, or smaller than 0.005. We find a single solution
which satisfies these conditions and identify fs and step
number (or VD%) for a given chondrule.
Chondrule chemical compositions used for the present
numerical investigation are as follows: 88 chondrules in
H3 chondrite (ALH-77299 analyzed by Ikeda 1983), 88
chondrules in E3 chondrite (Y-691 by Ikeda 1983), and 394
chondrules in total in L chondrites (Y-74191 by Kimura
et al. 1979, ALH-77015 by Fujimaki et al. 1981; Nagahara
1981a; Ikeda 1983, and ALH-77249 by Kimura 1983).
We were able to determine the formation condition(fs , VD%, F1 , F2 and F3 ) for majority of chondrules. For
some chondrules, however, the calculation did not satisfy the aforementioned requirements and failed to give
answers. The probability of giving right answers is 86%(76/88) for H3 chondrules, 75% (66/88) for E3 chondrules, and ∼60% for L3 chondrules (50-80% depending
on authors of EPMA analysis). A reasonably high probability confirms validity of our model.
Figure 12 (upper panel) shows frequencies of the chondrules as a function of VD%. The VD% ranges from 5 to
85 for H chondrules and from 10 to 90 for L chondrules.
The mode is 25% for H and 35% for L. E chondrules distribute in a narrower range, 10-65%, with its mode 35%.
These results implies the followings. 1) MLB, the sourceof chondrules, started to boil in early stages of heating andejected jet-droplets. 2) Ejection of jet-droplets was mostactive in the middle stage of heating. 3) The ejection continued until evaporation of MLB reached 65-90 VD%. In
our experiments (L200 and A200), ejection of jet-droplets
started with VD ∼5%. Therefore, it is suggested that there
was a time lag for the MLBs before they ejected the firstjet-droplets, during which temperature kept rising to theirboiling points while evaporation proceeded up to 5-10%of their initial masses.

Nakano and Hashimoto Progress in Earth and Planetary Science(2020) 7:47

Figure 12 (lower panel) shows frequencies of the chondrules as a function of fs . H and L chondrules distribute in
the range from 0.05 to 0.5. Compared with L chondrules,
H chondrules tend to have smaller fs values. E chondrules have smaller fs values, ranging from 0.025 to 0.15,
with one chondrule with 0.4. Variable fs values indicate
various redox states in the initial materials, viz., dustons. A straightforward interpretation is that each duston
had a different redox state or that duston was heterogeneous in distribution of the Fe oxide. Another possibility
is that C-rich, refractory organic materials (CROMs) similar to CHON in comets (e.g., Kissel and Krueger 1987)
and to IOM (e.g., Alexander et al. 2007) and organic
nano-globules (e.g., Nakamura-Messenger et al. 2006) incarbonaceous chondrites, were incorporated into the duston and reacted with Fe oxides when it was heated tohigh temperatures. If CROMs were distributed heterogeneously inside the duston, they would have contributed to
its internal heterogeneity in redox state. A big difference inthe fs range between E chondrules and H & L chondrules
may be attributed either to a difference in the initial redox
state of dustons or to a different abundance of CROMs industons.
Figure 13 (upper panels) shows the mixing proportion of the three components Fn (n = 1: jet-droplets,
2: differential vapor and 3: integrated vapor) determined
for the same sets of chondrules as in Fig. 12. H and

Fig. 12 Frequency of chondrules as functions of VD% and fs

Page 26 of 37

L chondrules show similar trends. The proportion ofjet-droplets that contributes to the mass of chondrules
decrease with increasing VD, while that of the integrated
vapor increases. This trend is expected because morevapor should deposit near the border of the vapor cloudas VD increases. The proportion of the differential vapor
shows no correlation with VD, and varies in wide range forboth H and L. This indicates that the jet-droplets sufferedfrom various degrees of condensation while traveling in
the vapor cloud. Its cause is most likely a size difference
among the jet-droplets as the efficiency of condensation
is directly related to the mass/surface-area ratio. Anothercause may be a size difference among dustons; a large
duston should allow more time for a jet-droplet to travel
through and also provide a denser vapor. E chondrules, on
the other hand, have small proportions of integrated vapor
compared to H and L. Instead, the proportion of differential vapor increases as VD increases, compensating thedecrease in jet-droplet mass proportion.
Figure 13 (lower panels) shows concentration ratios ofelements (Mg/Si and Na/Si) as a function of VD% for
the same sets of chondrules. As VD increases, the Mg/Si
ratio increases. The Na/Si ratio, however, has no correlation with VD, contrary to the common sense that volatileelements should deplete as evaporation proceeds. The
observed trend can be explained by recondensation of
the differential vapor and incorporation of the integrated
vapor (as condensates) onto the jet-droplets.
The retention of alkaline elements by chondrules has
been an issue of debate since chondrules experienced hightemperatures at least above their melting points while theykeep a level of alkaline concentrations similar to that ofbulk chondrites which contain them. In the framework of
the condensation model of liquid substances in PPD (e.g.,
Ebel and Grossman 2000), it has been shown that a dust
enrichment (as high as 1000×) alone was not sufficientenough to retain alkaline elements unless the gas pressurein PPD was assumed as high as 10−3 bar. Hewins et al.
(2005) have shown in their experiments that in a low density gas (< 10−4 bar) such as in PPD, silicate liquids could
not be held at their melting points, 1700-1900 K even forminutes without losing a significant amount of alkaline
elements.
We, therefore, contend that ejection of jet-droplets into
the vapor cloud that forms around a boiling duston could
only explain the retention of both moderately-volatileelements such as Si, Fe, Cr and Mn and volatile elements such as Na and K in chondrules because thesupercooled jet-droplets will immediately condense thesupersaturated vapor, which is enriched in moderatelyvolatile-element, during their flights in the vapor cloud,and also will incorporate the fine condensates that areenriched in once evaporated volatiles and distribute near
the vapor-cloud/disk-gas boundary. We also predict that

Nakano and Hashimoto Progress in Earth and Planetary Science(2020) 7:47

Page 27 of 37

Fig. 13 Upper panels: Calculated proportions of the three components that constitute chondrules as a function of calculated VD%. Lower panels:
Element/Si molar ratio in the chondrules as a function of calculated VD%

most chondrules were cold enough and completed theirchemical makeup when they got out of the vapor cloud
into the cold and thin environment in PPD.
We have focused our investigation on the chondrulesthat can be explained with our model calculations above,
but a significant number of chondrules could not be fitted
for their chemical compositions under the imposed constraints. The 14% in H chondrules, 25% in E chondrules,
and 50% in L chondrules for the worst case have been left
unfitted.
In the above numerical estimate, we have tacitly
assumed that the chondrule composition data which wereobtained with EPMA are reliable. But we have some doubt
about this assumption. We discuss this issue in detail in
Appendix E.
Origin of other features of chondrules

Our model naturally explains other recognized features ofchondrules.
Petrographic textures of chondrules

Chondrules have many different petrographic textures,e.g., porphyritic, granular, barred, radial, cryptcrystalline,
and glassy (e.g., Gooding and Keil 1981). Laboratory
experiments have been performed to reproduce thesevarious textures as a function of cooling rate of molten
silicate charges in electric furnaces (e.g., Lofgren and
Russell 1986; Hewins et al. 2005). The obtained coolingrates which correspond to unique textures have stimulated many theoretical scientists to evaluate the formation

condition of chondrules and possible heat sources for
melting (e.g., Boss 1996; Desch and Connolly 2002).
We disagree with the conventional belief about the cooling rates of chondrules. There is a critical problem in thoseexperiments that negates the meaning of the derived cooling rates. The experiments almost unequivocally used a
Pt-wire loop to hold a molten silicate charge with a helpof surface tension, while in space there needs no support
to hold chondrules. The holder provides nucleation centers for liquid to solidify and thus controls nucleation andcrystallization process that follows (e.g., Hollomon andTurnbull 1953; Huisman et al. 1997).
Free-floating, laser-heating experiments using drop
towers, high-speed gas flows or ultrasonic sound waves
that levitate molten silicate charges remove the problem
of preexisting solid nuclei (Nelson et al. 1972; Blander etal. 1976; Nagashima et al. 2008; Srivastava et al. 2010).
These experiments have clearly shown that the supercooling degree of a melt charge is a crucial factor that
determines petrographic textures upon its solidification.
The fact indicates not only that previous electric-furnace,Pt-suspending, cooling experiments no longer apply tochondrules, but that precursory silicate droplets had been
in a state of supercooling just before their solidification.
From their petrographic insight, Kimura et al. (1979) havealready concluded that chondrules are products of supercooled liquid silicates.
Our chondrule formation model postulates that thesupercooling of jet-droplets was inevitable due to theirfast cooling in the adiabatically expanding vapor with

Nakano and Hashimoto Progress in Earth and Planetary Science(2020) 7:47

no preexisting solid nuclei. Depending on the size andvelocity of jet-droplets, the volume of the vapor clouds,
and the temperature and density profile of the cloud, thejet-droplets would have experienced a various degree ofsupercooling. This might have created a variety of petrographic textures as seen in chondrules.
In their high-speed gas flow, laser-heating experiments,
Nagashima et al. (2008) have pointed out that silicate
liquids with high SiO2 contents do not crystallize but
becomes glasses irrespective of supercooling rates andthat solid seeds were required to impact with the supercooled liquids to initiate their crystallization. In ourmodel, we can predict a situation where supercooled jetdroplets were thrust into a zone with abundant condensed
particles which would have grown in size large enough toact as seeds.
Relict grains in porphyritic chondrules

Porphyritic chondrules often contain relict grains (mostly
olivine), which have been interpreted as evidence forincompleting melting during the chondrule formation(e.g., Nagahara 1981b ; Rambaldi 1981). We consider thatthe feature of relict grains observed in many of porphyritic
chondrules is a by-product of our chondrule formation
mechanism. We have two alternative explanations for it.
(i) The crystallization of a supercooled liquid droplet isaccompanied by a release of large latent heat, a recalescence. It would raise the temperature of crystallizing
chondrules above their melting temperatures momentarily, causing a remelting of just crystallized grains inside
the chondrules. Whether it happens or not depends on
the indicator Tn /Thyp , where Tn is the supercooling degree and Thyp , defined by Hf /CpL (Hf : heatof fusion, CpL : specific heat of liquid), is the hypercooling
limit. If Tn /Thyp is smaller than 1, a remelting occurs.
Accoring to the free-floating-crystallization experiment of
a forsterite liquid by Srivastava et al. (2010), a porphyritic
texture was obtained with the value ∼0.7. Thus, a porphyritic chondrule could remelt momentarily by recalescence. If so, the omnipresent occurrence of relict grains(mostly olivine) is a natural consequence of the crystallization from the supercooled state. For a conclusive
argument, however, we need further free-floating crystallization experiments that would use supercooled liquids
with more realistic chemical compositions.
(ii) Alternatively, the relict grains are reminiscent of
the incomplete remelting of the preexisting chondrulesthat solidified earlier. In our model, solidified chondrules
emerge from the vapor cloud with a velocity of 1-20 m s−1 .
Depending on the number density of dustons in space,which are simultaneously producing chondrules as jet
droplets, chondrules may cross a neighboring hot vapor
cloud and get partially remelted. In the companion paper(II), we investigate possibilities of various mechanisms

Page 28 of 37

for heating dustons. The probability of chondrules suffering from remelting would be surely dependent on themechanisms.
Compound chondrules

In ordinary chondrites some fraction of chondrules,
approximately 3%, are recognized as compound chondrules: a pair of chondrules that stick to each other (e.g.,
Wasson et al. 1995; Ciesla et al. 2004). To produce compound chondrules, either of the paired chondrules haveto be in the liquid state (or at sub-solidus temperatures
where some plastic deformation is possible) (Gooding and
Keil 1981). Arakawa and Nakamoto (2016) argued thatcompound chondrules are likely to be formed throughcollisions among supercooled droplets. The presence ofcompound chondrules as high as 3% requires a high number density of chondrule precursory droplets for themto encounter frequently and also demands coexistence ofboth liquid and solid droplets in the same environment.
Our chondrule formation model predicts numerous jetdroplets, in the order of ten to thousand (cm−2 s−1 ) being
ejected from a duston surface, depending on conditions.
The ejection direction is generally normal to the liquidsurface but can be tilted up to 20° from the normal inwater experiments (Rossodivita and Andreussi 1999). If
the duston surface was uneven, jet-droplets from various
crests, flanks and pits would cross each other’s trajectories to enhance the chance of collision. If a number densityof dustons was high, there might be also a chance that
jet-droplets having already solidified pass through nearby
vapor clouds and meet supercooled droplets.
The mutual collisional frequency would be calculated if
proper populations of jet-droplets and dustons are specified, since the speed of jet-droplets is known as a functionof their size from the present experiments (Appendix C).

Conclusions
This paper consists of two parts. The first part describes
our newly designed experiment and its links to the chondrules in three ways:(i) Boiling is a key to the formation of chondrules.
Bursting of vapor bobbles on a surface of molten silicate creates a swarm of jet-droplets that are ejected fromcollapsed bubble cavities. These droplets exhibit a lognormal-like size frequency distribution, one of the major
characteristics of chondrules. This discovery makes anextra size-sorting process redundant.
(ii) The formation mechanism of jet-droplets requires adense and large (>3 cm across) silicate body as a sourceof chondrules. This conclusion is led by combination of
the 1/10-1/5th law as to the size ratio of jet-droplet/bubble
diameter and the maximum size of silicate vapor bubbles
that can produce jet-droplets from their collapsed cavities.
Many studies on water jet-droplets in the atmosphericNakano and Hashimoto Progress in Earth and Planetary Science(2020) 7:47

and oceanic science reinforce this conclusion. The silicate
body is presumably an aggregate grown from primordialdust in PPD, which we name ‘duston’.
(iii) The most important information that the present
experiment provides is the full range of evaporation andcondensation pathways for chondritic compositions. They
are reduced to a newly defined quantity, ‘relative volatilities’ of elements, which can be used to predict chemical
fractionations caused by evaporation and condensation
for a silicate material having an arbitrary initial composition, including the CI composition.
The second part constructs a new formation model ofchondrules with a particular emphasis on their chemistry,and highlights the following four aspects.
(iv) The source material for chondrules, viz., duston, hasa CI-like chemical composition with its Fe oxide content
fs , viz., redox state, as a parameter.
(v) In PPD, a duston surface is exposed to a high flux ofheat which makes its surface boil. We expect that a large,
adiabatically-expanding vapor cloud develops around the
duston and that supercooled jet-droplets travel though it.
This creates a situation that jet-droplets, viz., chondrule
nuclei, incorporate the supersaturated vapor and the finecondensates on their way out from the cloud.
(vi) By assuming that the three components (jetdroplets, differential vapor and integrated vapor) makea bulk of chondrule, we are able to predict chondrulecompositions using relative volatilities of the ten elements determined by the experiment. First, we draw awhole range of trajectories in two-component composition spaces with fs as a parameter, swept by each of
the three components. This semi-quantitative approachshows that the three sets of composition trajectories cover

Page 29 of 37

the whole composition field occupied by real chondrules
analyzed to today. The quantitative approach solves fivesimultaneous linear equations to give the exact chondrulecomposition, VD (vaporization degree) and fs . Nearly 600
actual chondrules are fitted with this method, and ∼75%of them give answers. They give the ranges in VD and fs , aswell as the mixing ratios of the three components, leading
to the conclusion that chondrule chemistry is consistent
with a single source material having CI-like composition,with a wide range of redox state (fs = 0.02-0.55), with awide range of vaporization degree (VD = 5-90%), and witha wide range in the mixing ratio of the three components.
(vii) Other features of chondrules, i.e., textures and compound chondrules, are explained as a natural consequenceof the proposed model.
We have left an issue of a possible heat source in PPD,
that had brought dustons to temperatures high enough toboil. This issue is handled with in the companion paper(II).

Appendix(A) Ejection of silicate droplets by bursting of vapor
bubbles

Bursting of sub-surface air bubbles in water generates
two distinct types of water droplets (Lhuissier and Villermaux 2012). The bursting commences the moment the
bubble’s film cap ruptures in some mechanically weakplace (Fig. 14a), usually along the pinching ring aroundthe bubble. Due to the surface tension the film cap rolls
up along its original curvature (Fig. 14b) to the other sideof its perimeter (in ∼100 μs after the rupture; Resch etal. 1986). In the meantime, the rolled film mass is fragmented to pieces of strings, then to multiple droplets by

Fig. 14 a, b and c: schematic representation of bursting of an air bubble in succession (modified after Lhuissier and Villermaux 2012, figure 18). Only
the film cap of the bubble is shown. a A pin-hole rupture in the film expands. b The rolled-up film starts to disintegrate into pieces of strings byRayleigh-Taylor instability. c The instability continues to break up strings into multiple droplets, which is detached and ejected in tangential
direction from the original film curvature, called film droplets. d: schematic representation of jet-droplet formation (modified after MacIntyre 1972,
figure 1). The succession of profiles are ∼1/6000 s apart, from label 1 through 10. The pseudo-circle in red indicates a shape of bubble just before
the film rupturing. The profile 1 represents the moment the film cap is stowed away into its base perimeter by surface tension. The perimeter (nowthe edge of the gaping cavity) starts to collapse due to a continuous action of surface tension, generating a capillary wave that travels along thecavity wall to the bottom (profiles 2 through 5). The wave converges in the center to form an upward-moving jet (profiles 6 through 9). The jet starts
to deform into nodes (profile 10) due to Savart-Plateau-Rayleigh instability (Duchemin et al. 2002)

Nakano and Hashimoto Progress in Earth and Planetary Science(2020) 7:47

Rayleigh-Taylor instability, and eventually ejected in tangential direction (Fig. 14c). They are called film droplets.
Because the film rolling is very fast, a gaping cavity isleft and its thin edge starts to collapse inward by surface
tension, initiating a capillary wave (Fig. 14d: profiles 1-5).
The wave converges at the bottom of the cavity and turns
into a jet that protrudes in normal (viz., vertical) direction(Fig. 14d: profiles 6-10). The jet subsequently disintegrates
into one to several droplets, depending on its momentum, by Savart-Plateau-Rayleigh instability (Duchemin etal. 2002). It takes 1-100 ms after the film cap rupturing forjet droplets to emerge (Resch et al. 1986). There are many
works on the relationships between the bubble size andthe droplet (both film and jet) properties (size, number
and velocity) (Virij 1966; Resch et al. 1986; Blanchard andSyzdek 1988; Resch and Afeti 1991b, 1992; Spiel 1998; Wu
1994; Lhuissier and Villermaux 2012; Kientzler et al. 1954;
Hayami and Toba 1958; Spiel 1995, 1997; Rossodivita andAndreussi 1999; Georgescu et al. 2002; Lee et al. 2011).
The diameters of jet-droplets are known experimentallyand theoretically to be 1/10-1/5 of the diameters of the
bubbles from which they are ejected (e.g., Kientzler et al.
1954; Hayami and Toba 1958; Duchemin et al. 2002). The
number of droplets that produce from a single jet is one
to six; it reduces to one when the bubble diameter is large,
and increases up to six when it gets smaller. If the bubble diameter exceeds 5.4 mm for fresh water and 6.0 mm
for seawater, no jet-droplet produces (Georgescu et al.
2002). This is related to the viscous dissipation of capillary
wave when it travels along the cavity wall (Blanchette andBigioni 2006). The number of film-droplets, on the other
hand, is large, 1-110 (Speil 1998) and increases with the
bubble size. The diameter of film-droplets is <1/30 of the
bubble diameter. Since jet- and film-droplets are mutuallydifferent in their characters, it is rather easy to distinguishthem.
In the early phase of our experimental research, we
carried out several boiling experiments using a tungsten
resistance heater in vacuum. By doping 3 wt.% carbon
into the experimental charge (∼CI-chondritic composition), the liquid silicate boiled at 1500 °C due to the CO gas
that formed by reaction of carbon with iron oxide. A highspeed camera was available by courtesy of Dr. Arakawa,
Institute of Low Temperature Science, Hokkaido University.
In run# V1, we observed ∼500 bubble bursts in 10469
frames of image (shutter speed 1/5000 s; 2 s in total) andidentified only 6 events in which a single droplet each wasejected. In run# V2, we observed ∼400 bubble bursts in
25124 frames (1/5000 s; 5 s) and identified 3 events two of
which are a single shot of droplet, but one event produced
three jet-droplets as well as six film-droplets (Fig. 15).
The six film-droplets ejected at the same time withvelocity ∼3 m s−1 in a fan-like direction. The thicknessof their trajectories is translated to their diameters, ∼40

Page 30 of 37

μm and uniform. The three jet-droplets were various in
size, and ejected in different directions due to gravity. The
fastest of the three was ∼1 m s−1 . The slowest one was thelargest in size, ∼180 μm in diameter. The bubble diameter just before its bursting is 1080 μm. Hence the largest
jet-droplet is ∼1/6 of and the film-droplets are ∼1/27 of
the bubble size, which conforms to the droplet/bubble
ratios for water. The reason why the ejection frequencyof droplets is so small compared to the number of bubble
bursts can be explained by the high viscosity of liquid silicate at such a low temperature (1500 °C), which dampened
capillary waves and also reduced film rolling speeds.
A high speed camera was unavailable for the present
experiments that used laser beam heating. Instead weused a conventional video camera (frame rate = 30 fm
s−1 ; exposure time = 1/60 s). Experiments with a low
laser power are suitable for observing jets owing to a low
degree of halation, but jetting is limited to vicinity ofthe direct laser spot (∼10 o’clock direction). Figure 16a-c(run# 2; 65 W) shows elongated jets (due to the longexposure time) along with earlier split droplets (seen as
elongated trajectories) on their extensions. Figure 16d-f
shows jet-droplets in run# 39-2, 39-1 and 38 (all 100 W),
respectively. Thin trajectories tend to elongate longer,
indicating that smaller droplets have higher ejection
velocities. A bunch of trajectories in very close proximity appears to originate in the same jet source. There
are many such instances observed in these frames. It is
possible that the source jet shifts its position while spewing droplets one after another. It is also possible thatdroplets with different masses and velocities take different
paths due to gravity. Experiments with a high laser power(200 W) produce jet-droplets in all directions because theentire sample is kept above the boiling temperature, as
shown in Fig. 16g&h (run# 83). A strong halation, however, shrouds their source jets. Rarely a number of droplets
eject in a fan-like, synchronized fashion, suggesting them
being film droplets from a single bubble burst, as shownin Fig. 16i (run# 83). Figure 16j (run# 32) shows a lightscattering cloud of condensed, ultra-fine particles that
accumulates in a late phase of evaporation.
(B) Critical bubble radius for ejection of jet-droplet

Georgescu et al. (2002) obtained numerically critical bubble radii for twenty kinds of liquids (table 1 in theirpaper), including freshwater and seawater. Table 7 is anextended version of their table 1, where we have included
three kinds of rock melts at three temperatures (# 22-29)
with their physical properties. We have used data by
Bottinga and Weill (1972) for viscosity and data by Nelson and Carmichael (1979) for densities, and extrapolated those to high temperatures. The surface tension,
0.4 N m−1 , is taken constant for all melts at temperatures for the following reasons. There are no pertinent

Nakano and Hashimoto Progress in Earth and Planetary Science(2020) 7:47

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Fig. 15 High speed camera visions (1/5000 s apart) from left to right. Between the 4th and 5th frames, four frames are omitted. The sample charge(∼CI-chondritic composition doped with 3 wt.% carbon) is suspended with a PtRh wire and is in its total melting at 1500 °C. The horizontal width ofthe charge is ∼5 mm. A bulged bubble on the 1st frame is already ruptured on the 2nd frame, where multiple film-droplets are ejected while a jet isstill developing from the center of the collapsed cavity. On the 3rd and 4th frames, the six film-droplets are spreading out in concert, while the firstjet-droplet is detached from tip of the jet. Skipping four frames, the 5th frame shows three jet-droplets moving away from the mother liquid body.
Their direction of motion is up and far side of the picture. The film-droplets are out of the 5th frame because they move fast

data for multicomponent rock melts. The surface tensions in binary silicate melts (e.g., MgO-SiO2 , CaO-SiO2 ,MnO-SiO2 and FeO-SiO2 ) are similar in values and theirtemperature derivative is small (Turkdogan 1983). So wetake that of MgSiO3 (enstatite melt) ∼0.4 N m−1 at ∼1600°C for all melts and temperatures.
The computed Rcr values by Georgescu et al. (2002)
combined with Eq. 19 give wave numbers for critical bubbles. They are shown in Table 7 for twenty liquids. We

have looked into the relationship between viscosity and(critical) wave numbers for these liquids and found a good
linear correlation between log μ and log k as shown in
Fig. 17.
log (k/m−1 ) = −0.6003 log (μ/Pa s) + 2.1516.

(25)

Applying Eq. 25 we obtain (critical) wave numbers for thethree kinds of rock melts as shown in the 2nd rightmost
column of # 21-29 in Table 7. Then by applying Eq. 19,

Fig. 16 Snapshots of jet-droplet ejection from the boiling sample. The video camera speed is 30 fm s−1 , and its exposure time is 1/60 s. a, b & c: run#2 (65 W). The central bright ball is an effect of halation. The actual sample diameter is about one half of the ball. The field of view is ∼35 mm across.
In this low laser power (65 W) experiment only the laser spot and its vicinity is above boiling temperature, and hence jet-droplets are ejected in ∼10o’clock direction where laser beam comes from. a and b are successive video frames. In a, the first jet-droplet is ejected while the jet is still
extending for a second droplet. In b, the first and second jet-droplets are seen. The length of their trajectories and the exposure time give theirvelocities, ∼0.6 m s−1 . d, e & f: run #39-2, #39-1, and #38 (all 100 W), respectively. Thin trajectories tend to elongate longer, indicating that smaller
droplets have higher ejection speeds. g, h & i: run #83 (200 W). Experiments with high laser power (200 W) produce jet droplets in all directions
because the entire sample are kept above boiling temperature. In g, thick and bent trajectories of several jet-droplets indicate their large size. In h,thin, straight and elongated trajectories stand out, indicating smaller and faster droplets than in g. Rarely a number of droplets are ejected in a
fan-like, synchronized fashion, suggesting them being film droplets from a single bubble burst, as shown in i. j: run #32 (200 W). In a late phase ofevaporation as shown in j, a thick cloud of condensed, ultra-fine particles, a source of light-scattering, develops and circulates in convective motion

Nakano and Hashimoto Progress in Earth and Planetary Science(2020) 7:47

Page 32 of 37

Table 7 Critical bubble radii for various liquids (modified after Georgescu et al. 2002, table 1)
Density#Surface

ViscosityρTensionμ(kg m−3 )σ (N m−1 )(Pa s)

Liquid type

Wave number (m−1 )Cr.bub.rad.
log μCr.bub.rad.

exp.∗1comput.∗2

calc.∗3graph∗4calc.∗5

Rcr (m)Rcr (m)

log k

log k

Rcr (m)1

acetic acid1046

0.02761.220E-03-2.914

2.06E-03

2.32E-03

3.854

–

–

2

acetone791

0.02333.240E-04-3.4892.37E-03

2.46E-03

4.156

–

–

3

aniline1022

0.042674.466E-03-2.350–2.04E-03

3.575

–

–

4

benzene885

0.02888

6.470E-04-3.189

–2.42E-03

4.008

–

–

5

carbon tetrachloride

1594

0.02629.750E-04-3.011–

1.29E-03

4.142

–

–

6

CFC-1131617

0.01999.600E-04-3.018

1.34E-03

1.19E-03

4.133

–

–

7

distilled water

998.2

0.07274

1.002E-03-2.9992.70E-03

2.87E-03

3.983

–

–

8

ethanol798

0.02281.194E-03-2.923

1.84E-03

2.16E-03

3.814

–

–

9

e-w (2-98 wt%)

993

0.06551.090E-03-2.963

2.29E-03

2.54E-03

3.978

–

–

10

g-w (85-15 wt%)

1220

0.0657

1.108E-01-0.955–

1.98E-03

2.742

–

–

11

g-w (79-21 wt%)

1210

0.06644.970E-02-1.304–

2.15E-03

2.951

–

–

12

g-w (70.5-29.5 wt%)

1185

0.06762.350E-02-1.629

–2.34E-03

3.143

–

–

13

g-w (65-35 wt%)

1170

0.0683

1.540E-02-1.812–2.60E-03

3.234

–

–

14

g-w (50-50 wt%)

1130

0.06945.980E-03-2.223–

2.72E-03

3.493

–

–

15

i-propanol

791

0.02142.420E-03-2.616

1.40E-03

1.42E-03

3.721

–

–

16

methanol

804

0.02265.600E-04-3.252

2.22E-03

2.40E-03

4.003

–

–

17

mineral oil

840

0.029

1.700E-02-1.770–2.18E-03

3.085

–

–

18

n-butanol

816

0.02472.980E-03-2.526

1.25E-03

1.28E-03

3.716

–

–

19

n-propanol

805

0.02372.190E-03-2.6601.50E-03

1.87E-03

3.687

–

–

20

seawater (NaCl 3.4%)1024

0.08023

1.126E-03-2.9483.00E-03

2.94E-03

3.960

–

–

21

peridotite 3000 °C

2462

0.4

4.490E-04-3.348

–

–

–

4.161

1.275E-02

22

dunite 3000 °C

2577

0.4

3.700E-04-3.432

–

–

–

4.212

1.329E-02

23

enstatite 3000 °C

2503

0.4

3.910E-03-2.407

–

–

–

3.597

1.036E-02

24

peridotite 2700 °C

2508

0.4

9.350E-04-3.029

–

–

–

3.970

1.196E-02

25

dunite 2700 °C

2612

0.4

7.470E-04-3.127

–

–

–

4.029

1.248E-02

26

enstatite 2700 °C

2475

0.4

7.070E-03-2.151

–

–

–

3.443

9.714E-03

27

peridotite 2400 °C

2556

0.4

2.290E-03-2.640

–

–

–

3.736

1.104E-02

28

dunite 2400 °C

2652

0.4

1.770E-03-2.752

–

–

–

3.804

1.154E-02

29

enstatite 2400 °C

2458

0.4

1.460E-02-1.837

–

–

–

3.254

9.007E-03

Note. g-w: glycerol-water solution. e-w: ethanol-water solution. Cr.bub.rad.: Critical bubble radius.
exp.*1: experimentally obtained values, see references in Georgescu et al. (2002). comput.*2: numerically obtained by Georgescu et al. (2002)
calc.*3: calculated from Rcr with Eq. 19. graph*4: obtained from μ with Eq. 25. calc.*5: calculated from k with Eq. 19

we obtain Rcr for these rock melts at three temperatures
as shown in the rightmost column in Table 7 and also in
Table 6.
(C) Relationship between jet-droplet size and its ejection
speed

We attempt to estimate ejection speed of jet-droplets fromthe silicate liquid, with the known relationship between
droplet size and its ejection speed determined for waterjet-droplets as a guide. The source of energy for giving

rise to the formation of jet-droplets is the sum of the surface energy and the potential energy stored in the gaping
cavity formed just after bursting of a bubble. In PPD, thepotential energy is ignored due to its micro-gravity environment, and on earth it can be ignored for a small bubble
because the surface tension dominates. In the following,
the cavity is assumed to be a hemisphere with its radius
Rb , then the surface energy stored in the cavity is 2πR2b σ ,where σ is the surface tension of the liquid. Subtracting
the surface energy needed to mend the liquid surface toNakano and Hashimoto Progress in Earth and Planetary Science(2020) 7:47

Fig. 17 log μ versus log k of various liquids in Table 7

a flat plane πR2b σ , the available energy for jet-droplet formation is πR2b σ . Assuming n droplets of equal radius a tobe produced in a single bursting event, a total 4πa2 σ n isrequired to form droplets. The remaining energy is available for the kinetic energy of droplet ejection. Neglecting
viscous dissipation during the whole process, the ejection
speed of the droplets is given by1/2

3 σ 1 φ2
(
− 4)
.
(26)
Ve =
2ρa n
Here ρ is density of the liquid and φ = Rb /a is the size
ratio of bubble/droplet.
According to the experiments by Spiel (1997), five jetdroplets of 20 μm in radius eject in bursting of an air
bubble of radius 200 μm. Substituting a = 2 × 10−5 m,
φ = 10, n = 5, σ = 0.07274 N m−1 and ρ = 1000kg m−3 , into Eq. 26, Ve is calculated to be 9.3 m s−1 .
This is comparable to the value 9.2 m−1 that is obtainedby extrapolation of the droplet radius vs. ejection speed
relation of Spiel (1997), indicating that Eq. 26 is a good
approximation. Obviously gravity and viscous dissipation
has little effect on ejection of jet-droplets from a small
bubble.
The same procedure is applied to the silicate liquid andjet-droplets therefrom. Substituting σ = 0.4 N m−1 andρ = 2550 kg m−3 (at 3000 K) and taking the same value
for a, φ and n, we obtain Ve = 13.7 m s−1 .
For large bubbles, it is known experimentally (Resch andAfeti 1991b; Suzuki and Mitachi 1995) and theoretically(Georgescu et al. 2002) that jet-droplets do not form due
to viscous dissipation for the bubble radius > 3 × 10−3 m
for water. A droplet that marginally form at Rb ∼ 3 × 10−3
m has a speed ∼0.1 m s−1 (Hayami and Toba 1958; Spiel
1995). Spile (1997) obtained the relationship between bubble radius and ejection velocity of jet-droplets as Ve =12.12 exp(−1.390 × 103 Rb ). Application of this equation

Page 33 of 37

to Rb = 3 × 10−3 m and a = 6 × 10−4 m, gives Ve = 0.2
m s−1 , in good agreement with the observation. Here weused φ = Rb /a = 5 for large bubbles.
In “Evidence for chondrules being jet-droplets” section,
we derived the critical bubble radius Rcr = 1.25 × 10−2
m for production of jet-droplets from the silicate liquid.
We assume that a speed of jet-droplet that may eject by
bursting of the critical bubble is 0.2 m s−1 just as fromwater. By fitting an attenuation relation of the form Ve =A exp(−αRb ) to the two sets of (Rb , Ve ), viz., (2 × 10−4 ,13.7) and (1.25 × 10−2 , 0.2), we obtain A = 14.6 m s−1
and α = 330.3 m−1 . We further assume that φ = 10 for
a = 2 × 10−4 m and φ = 5 for a = 1.25 × 10−2 m, andobtain the relation, log a = 1.1267 log Rb − 0.4579. These
two equations are used in “Vapor cloud model” section.
We have found that the value α is very close to 4/Rcr for
yet unknown reason. In case of water, the computer simulation by Georgescu et al. (2002) gave Rcr = 2.87 × 10−3m, which result in 4/Rcr = 1.393 × 103 m−1 , being very
close to the value α = 1.390 × 103 m−1 by Spiel (1997)
experiments. For our silicate liquid (dunite or peridotite),
4/Rcr = 4/0.0125 = 320 m−1 , which is close to α = 330.3
m−1 as obtained above. If this relation holds for any liquid, we may write Ve = A exp(−Rb /R∗b ); R∗b = Rcr /4(R∗b : bubble scale radius for jetting). It also indicates thatthe ejection speed has an upper limit: A = 12.1 m s−1 forwater and A = 14.6 m s−1 for silicate liquid.
(D) Data sources of chondrule compositions

Table 8 lists references of chondrule chemical compositions used for the numerical investigation in “Modelcalculation of chondrule chemistry” section.
(E) Assessment of chondrule analyses

A bulk elemental, X-ray analysis with Electron Probe
Micro-Analyzer on chondrules is made on a cross-sectionof chondrules as they expose by cutting their parent meteorites. Two fundamental problems are attached to thismethod for determining the bulk composition of chondrules.
1) A mineral proportion that reveals on a cross-sectional
area does not necessarily represent its real proportion in
an entire chondrule volume. Chondrules generally consistof several mineral species including glass. They have theirown unique chemical compositions. In many cases they
appear as relatively large crystals and distribute rather
heterogeneously inside chondrules. Therefore, a chancecross-section may not represent their average proportionsin the bulk of a chondrule. On the other hand, chondrules with a finer petrographic texture also exist in large
number. In that case a cross-section is not a bad manifestation of bulk chondrules. Whether or not a cross-section
represents a true bulk mineral proportion is hard to beevaluated especially after cross-sectioning.

Nakano and Hashimoto Progress in Earth and Planetary Science(2020) 7:47

Page 34 of 37

Table 8 Data source of chondrule compositions
Chondrite type

Chondrite name

Reference

H3

ALH 77299

Ikeda (1983)

H3

Y 790986

Ikeda (1983)

L3

ALH 77015

Fujimaki et al. (1981), Nagahara (1981a), Ikeda (1983)

L3

ALH 77249

Kimura (1983)

L3

Y 74191

Ikeda and Takeda (1979), Kimura et al. (1979)

LL3

ALH 764

Ikeda (1980)

LL3

Semarkona

Jones and Scott (1988), Jones (1990), Jones (1996), Hewins et al. (2012)

EH3

Y 691

Ikeda (1983)

CM2

EL-Quss Abu Said

Hezel and Palme (2010)

CO3

ALH-77003

Ikeda (1982)

CO3

Y 790992

Ikeda (1983)

CO3

Kainsaz

Hezel and Palme (2010)

CV3

Efremovka

Hezel and Palme (2010)

CV3

AllendeIsobe et al. (1986)

CR2

Renazzo

Hezel and Palme (2010)

2) ZAF and Bence & Albee correction methods for X-ray
intensities (e.g., Reed 2005) are optimized for a small electron beam (micro-probe) diameter because they assume
a uniform composition. A bulk chondrule analysis with
EPMA, however, uses in almost all cases a large (viz., defocused) electron beam, 30-100 μm in diameter, dependingon analysts, to cover a large area as possible in a single analysis. Usually many analyses are made to cover awhole cross-sectional area and averaged for a bulk composition. A large electron beam excites X-rays from alarge area of sample which may encompass more than
two compositionally different phases or minerals, which
violates normal matrix corrections in the built-in computation program and may become a source of large errors.
In order to overcome this problem, various compensation
procedures have been developed (e.g., Albee et al. 1977;
Bastin et al. 1983). Ikeda (1980) determined calibration
coefficients for correcting broad-beam effects element by
element, by synthesizing artificial mineral assemblages ofknown chemical compositions similar to chondrules, andcalibrating apparent concentrations of elements determined with EPMA with known values. Without suchbroad-beam corrections and depending on elements, anerror of a few tens of percent is not uncommon (Albee etal. 1977; Ikeda 1980).
Our numerical investigation on chondrules outlined
above had originally included Y-74191 (L3) chondrules
analyzed by Ikeda and Takeda (1979). Their analyses,
however, had given unusually high concentrations in Na
and Al compared to other L3 chondrules, obviously due
to no correction for broad-beam analysis. Our calculation
was capable of determining the formation conditions only
for 6 out of 43 chondrules (viz., 14%) whose compositions

had been provided by Ikeda and Takeda (1979). This
unusually low percentage reconfirms the importance ofthe broad (defocused) beam corrections. These six chondrules are not included in Figs. 12 and 13.
Although it is not an analytical problem, there is an
important factor that alters primary chemical compositions of chondrules: alteration or reactions of chondruleswith matrix materials that are usually postulated to haveoccurred in meteorite parent bodies. The reason why wechose petrographic type 3 chondrites for our research isthat they are least-altered among those which belong to
the same chemical type (Van Schums and Wood 1967).
Abbreviations
MLB: Mother liquid body; VD: Vaporization degree; CROMs: C-rich, refractoryorganic materials, CAI: Calcium-aluminum-rich inclusion, IOM: Insolubleorganic materials
Acknowledgements
We are greatly indebted to the members of the Planetary and Space Sciencegroup, Hokkaido University, for their valuable comments throughout this workand to Dr. A. Kouchi, Institute of Low Temperature Science, Hokkaido University
for useful discussion. The chemical analysis on the experimental products inthis work were made at Pheasant Memorial Laboratory for Geochemistry andCosmochemistry, Institute for Study of the Earth’s Interior, Misasa. We greatly
appreciate Dr. E. Nakamura and his staffs in Misasa for providing such an
opportunity. We wish to thank Mr. Z. Nakamura and his men of Tanaka Ltd. fortheir continuous and sincere support. Machine shop in Hokkaido University is
acknowledged for their technical helps. We thank the anonymous reviewers fortheir careful reading of our manuscript and their comments and suggestions.
Authors’ contributions
YN designed the experimental set-up, conducted the experiments, performed
the analysis, and wrote the paper. AH supervised the study, conceived themodel, made the numerical calculations, and wrote the paper with YN. The
author(s) read and approved the final manuscript.
Funding
This research received no specific grant from any funding agency in the
public, commercial, or not-for-profit sectors.

Nakano and Hashimoto Progress in Earth and Planetary Science(2020) 7:47

Availability of data and materials
Experimental data is available from the corresponding author upon
reasonable request.
Competing interests
The authors declare that they have no competing interest.
Author details
1 Department of Cosmosciences, Hokkaido University, N10 W8, Sapporo
060-0810, Japan. 2 Suns & Fishermen Society, Sapporo, Japan.
Received: 11 November 2019 Accepted: 17 May 2020

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