# Origin of Low-${}^{26}{\rm Al}/{}^{27}{\rm Al}$ Corundum/Hibonite Inclusions
in Meteorites

Steven J. Desch School of Earth and Space Exploration, Arizona State
University,
PO Box 871404, Tempe, AZ 85287-1404  Emilie T. Dunham Department of Earth,
Planetary, and Space Sciences, University of California, Los Angeles, Los
Angeles, CA 90095, and Department of Earth and Planetary Sciences, University
of California, Santa Cruz, Santa Cruz CA 95064  Ashley K. Herbst School of
Earth and Space Exploration, Arizona State University,
PO Box 871404, Tempe, AZ 85287-1404  Cayman T. Unterborn Southwest Research
Institute, 6220 Culebra Road,
San Antonio, TX 78238  Thomas Sharp School of Earth and Space Exploration,
Arizona State University,
PO Box 871404, Tempe, AZ 85287-1404  Maitrayee Bose School of Earth and Space
Exploration, Arizona State University,
PO Box 871404, Tempe, AZ 85287-1404  Prajkta Mane Lunar and Planetary
Institute,
3600 Bay Area Blvd., Houston TX 77058  Curtis D. Williams Arctic Slope
Regional Corporation Federal, 11091 Sunset Hills Road #800, Reston VA 20190

###### Abstract

Most meteoritic calcium-rich, aluminum-rich inclusions (CAIs) formed from a
reservoir with $\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$}\approx 5\times
10^{-5}$, but some record lower $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm
Al}$})_{0}$, demanding they sampled a reservoir without live ${}^{26}{\rm
Al}$. This has been interpreted as evidence for “late injection” of supernova
material into our protoplanetary disk. We instead interpret the heterogeneity
as chemical, demonstrating that these inclusions are strongly associated with
the refractory phases corundum or hibonite. We name them “Low-${}^{26}{\rm
Al}/{}^{27}{\rm Al}$ Corundum/Hibonite Inclusions” (LAACHIs). We present a
detailed astrophysical model for LAACHI formation in which they derive their
Al from presolar corundum, spinel or hibonite grains $0.5-2\,\mu{\rm m}$ in
size with no live ${}^{26}{\rm Al}$; live ${}^{26}{\rm Al}$ is carried on
smaller ($<$50 nm) presolar chromium spinel grains from recent nearby Wolf-
Rayet stars or supernovae. In hot ($\approx$ 1350-1425 K) regions of the disk
these grains, and perovskite grains, would be the only survivors. These
negatively charged grains would grow to sizes $1-10^{3}\,\mu{\rm m}$, even
incorporating positively charged perovskite grains, but not the small,
negatively charged ${}^{26}{\rm Al}$-bearing grains. Chemical and isotopic
fractionations due to grain charging was a significant process in hot regions
of the disk. Our model explains the sizes, compositions, oxygen isotopic
signatures, and the large, correlated ${}^{48}{\rm Ca}$ and ${}^{50}{\rm Ti}$
anomalies (if carried by presolar perovskite) of LAACHIs, and especially how
they incorporated no ${}^{26}{\rm Al}$ in a solar nebula with uniform,
canonical ${}^{26}{\rm Al}/{}^{27}{\rm Al}$. A late injection of supernova
material is obviated, although formation of the Sun in a high-mass star-
forming region is demanded.

meteorites (1038) — protoplanetary disks (1300) — solar system formation
(1530) — supernovae (1668) — dust physics (2229)

##  1 Introduction

###  1.1 Low ${}^{26}{\rm Al}/{}^{27}{\rm Al}$ in Some Inclusions

One of the enduring mysteries of Solar System formation is the origin of the
short-lived radionuclides (SLRs), in particular the important isotope
${}^{26}{\rm Al}$, which decays to ${}^{26}{\rm Mg}$ with a half-life of 0.72
Myr (Auer et al., 2009; Kondev, 2021). When this isotope was discovered, it
was immediately recognized for its potential to be the cause of asteroid
melting (Urey, 1955). Its existence in the solar nebula at a level
$\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$}\approx 5\times 10^{-5}$ was
established later (Lee et al., 1976), by measurements of Al and Mg isotopes in
calcium-rich, aluminum-rich inclusions (CAIs). A linear correlation, or
“isochron”, between ${}^{26}{\rm Mg}/{}^{24}{\rm Mg}$ and ${}^{27}{\rm
Al}/{}^{24}{\rm Mg}$ measurements in minerals in an inclusion is a signature
that ${}^{26}{\rm Mg}$ formed by decay of ${}^{26}{\rm Al}$, and the slope of
the linear correlation is the initial $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm
Al}$})_{0}$ ratio in the solid when it formed, or when it last achieved
isotopic closure. The differences in $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm
Al}$})_{0}$ between two inclusions (e.g., CAIs and chondrules) can be used to
determine the time difference between their formations, assuming they formed
from the same reservoir with common initial $(\mbox{${}^{26}{\rm
Al}/{}^{27}{\rm Al}$})_{\rm SS}$. After measurements of thousands of CAIs, a
“canonical” Solar System value has been established (MacPherson et al., 1995),
and refined today to be $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{\rm
SS}=(5.23\pm 0.13)\times 10^{-5}$ (Jacobsen et al., 2008). Although debate
continues whether some CAIs record slightly higher values (Simon & Young,
2011), the vast majority of CAIs formed with very similar ratios, apparently
recording a spatially uniform ratio $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm
Al}$})_{\rm SS}$ in the solar nebula, with the slight variations (among those
not melted later) interpreted as small differences in formation time, within
about $5\times 10^{5}$ years (e.g., MacPherson et al., 2012; Liu et al., 2019;
Kawasaki et al., 2020).

The canonical $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{\rm SS}$ ratio has
been inferred from the majority of CAIs, which primarily contain such minerals
as melilite, anorthite, Ti-rich clinopyroxene, and spinel. Depending on their
exact mineralogy and thermal histories, these may be categorized as fluffy
type A (FTA), compact type A (CTA), type B, or type C CAIs (e.g., Grossman,
1975; Grossman et al., 2000; MacPherson et al., 2022). These can be considered
“normal” CAIs, and the community’s intuition about CAIs and ${}^{26}{\rm Al}$
in the solar nebula was initially shaped by measurements in large (up to cm-
sized) type B CAIs from the Allende (CV3) carbonaceous chondrite in
particular.

Some types of CAIs, not falling into the categories above, record much lower
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$ ratios (see the review by
Krot et al., 2012). In particular, PLAty Crystals of hibonite (PLACs) are
$10-100\,\mu{\rm m}$-sized fragments or aggregates of hibonite laths that tend
to record low values $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}<3\times
10^{-6}$ (Ireland, 1988, 1990; Liu et al., 2009; Kööp et al., 2016a). Numerous
corundum grains tens of microns in size exhibit similarly small
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$ ratios (Virag et al., 1991;
Makide et al., 2011, 2013). A large fraction of grossite-dominated CAIs from
CH chondrites have been shown to record $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm
Al}$})_{0}\approx 4\times 10^{-7}$ (Krot, 2019). There are also a few examples
of large, corundum- and/or hibonite-dominated inclusions: M98-8 (from the CM
chondrite Murchison), DOM 31-2 (from the CO chondrite Dominion Range 08006),
and A-COR-01 (from the CO chondrite Allan Hills 77307) record values from
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}<1.6\times 10^{-6}$, down to
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}\approx(1.63\pm 0.31)\times
10^{-7}$ (Simon et al., 2002, 2019b; Bodénan et al., 2020). The most difficult
to explain of these may be the hibonite-dominated inclusion HAL, which
apparently records $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}\approx
5\times 10^{-8}$ (Fahey et al., 1987b).

These low values of $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}\sim
10^{-7}$ cannot be explained as a late formation or thermal resetting of the
Al-Mg chronometer of materials that sampled the reservoir with canonical
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{\rm SS}$. Most of these
inclusions would have been incorporated into their parent body asteroids
within 3 Myr, when $\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$}>3\times 10^{-6}$
still, and remained too cold to alter the Al-Mg isochron. The cause of the
heterogeneity in $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$ ratios has
been an ongoing mystery for decades.

###  1.2 Causes of the ${}^{26}{\rm Al}/{}^{27}{\rm Al}$ Heterogeneities?

While many papers have identified this clear heterogeneity of ${}^{26}{\rm
Al}$, there is no consensus in the field for its origin. The term
“heterogeneity” is agnostic about cause, but often it is implied that the
heterogeneities are erased by physical mixing between materials at different
parts of the disk (e.g., Krot et al., 2012; Mishra & Chaussidon, 2014), by
processes such as those described by Boss (2013). Such spatial heterogeneities
might be expected if ${}^{26}{\rm Al}$ were produced by irradiation by solar
energetic particles in the solar nebula (e.g., Gounelle et al., 2001; Gaches
et al., 2020; Jacquet, 2019), but there are severe limitations to production
of ${}^{26}{\rm Al}$ in ${\rm H}_{2}$ gas (Clayton & Jin, 1995), or in
material devoid of ${\rm H}_{2}$ (Desch et al., 2010). Wood (1998) suggested
that the molecular cloud was heterogeneous in its ${}^{26}{\rm Al}$
distribution, leading to ${}^{26}{\rm Al}$-free gas during “clumpy infall”;
likewise, spatial variations in isotopic compositions of disk materials,
because of variations in isotopic compositions of gas infalling from the
molecular cloud, are the underlying assumptions of several contemporary models
(Nanne et al., 2019; Pignatale et al., 2019; Lichtenberg et al., 2021; Liu et
al., 2022a). However, these assumptions are untested and such models are
counter to the expectation that the molecular cloud was spatially homogeneous
(Pan et al., 2012) and well mixed during infall (Kuffmeier et al., 2017), even
before mixing within the disk.

The evidence for large-scale spatial heterogeneities is weak in any case.
Bollard et al. (2019) argued that chondrules formed in a region with lower (by
a factor of 2) ${}^{26}{\rm Al}/{}^{27}{\rm Al}$ than CAIs did, although this
interpretation was based on the untested assumption that CAIs achieved
isotopic closure in the Pb-Pb system at the same time they closed in the Al-Mg
system (Desch et al., 2023a). Similarly, Larsen et al. (2011) argued for the
solar nebula generally having lower (factor of 2) ${}^{26}{\rm Al}/{}^{27}{\rm
Al}$ ratios than the CAI-forming region, based on measured ${}^{26}{\rm Mg}$
deficits in CAIs; but recently Gregory et al. (2020) measured greater deficits
in forsteritic inclusions, again favoring a uniform distribution of
${}^{26}{\rm Al}$. These variations, if they existed, would not explain
${}^{26}{\rm Al}/{}^{27}{\rm Al}$ ratios as low as $\sim 10^{-7}$, anyway.

Currently, the interpretation of a temporal heterogeneity, and the “late
injection” model (Sahijpal & Goswami, 1998; Sahijpal et al., 2000), is most
commonly accepted (e.g., Liu et al., 2009; Makide et al., 2011; Krot et al.,
2012; Van Kooten et al., 2016; Krot, 2019; Ku et al., 2022). In this model,
the solar nebula formed with little ${}^{26}{\rm Al}$, which was instead
acquired later, just prior to the formation of most CAIs, either due to some
late input to the Sun’s molecular cloud (Foster & Boss, 1997), or injected
into the Sun’s protoplanetary disk (Ouellette et al., 2007) by a nearby core
collapse supernova injecting gas or, more likely, dust grains. While this
scenario cannot be ruled out, the likelihood of a supernova exploding at the
right distance from the Sun’s protoplanetary disk, during its first $<10^{5}$
years, is very small $\sim 0.1-1\%$ (Ouellette et al., 2010).

Although spatial and temporal heterogeneities are popular explanations, there
is a high likelihood that a forming solar system would simply acquire
${}^{26}{\rm Al}$ from its molecular cloud at near-canonical values (Jura et
al., 2013; Desch et al., 2023c). Any ${}^{26}{\rm Al}$ inherited from the
molecular cloud would have been present and well-mixed throughout the solar
nebula, from the earliest times, which would argue against any spatial or
temporal heterogeneities.

Therefore, we instead favor the interpretation of a chemical heterogeneity. We
note that all of the inclusions that sampled a subcanonical ${}^{26}{\rm
Al}/{}^{27}{\rm Al}$ reservoir are dominated by the refractory minerals of
corundum [${\rm Al}_{2}{\rm O}_{3}$], the calcium aluminate hibonite [${\rm
CaAl}_{12}{\rm O}_{19}$], or more rarely the calcium aluminate grossite [${\rm
CaAl}_{4}{\rm O}_{7}$]. These Al-dominated minerals are some of the most
refractory minerals found in meteorites, and hibonite and grossite are both
formed by reaction of Ca vapor with corundum (e.g., Ebel & Grossman, 2000;
Unterborn & Panero, 2017). The association of low-${}^{26}{\rm Al}/{}^{27}{\rm
Al}$ ratios with these minerals, especially corundum and hibonite, is so
strong that we collectively call such meteoritic components Low-${}^{26}{\rm
Al}/{}^{27}{\rm Al}$ Corundum/Hibonite Inclusions (LAACHIs).

Ireland (1990) and Larsen et al. (2020) offered a chemical heterogeneity
interpretation of ${}^{26}{\rm Al}$ distributions, suggesting that inclusions
low in ${}^{26}{\rm Al}$ may have formed from material that was predominantly
from presolar grains that had spent enough time in the interstellar medium
(ISM) that effectively all their ${}^{26}{\rm Al}$ had decayed. In the model
of Larsen et al. (2020), these grains would be preferentially destroyed at
high temperatures, and their ${}^{26}{\rm Al}$ somehow not incorporated into
inclusions. Given the wide spread of isotopic ratios in presolar grains, this
would be consistent with the observation that low-${}^{26}{\rm Al}/{}^{27}{\rm
Al}$ inclusions often show large anomalies in $\epsilon^{50}{\rm Ti}$, whereas
inclusions with canonical ${}^{26}{\rm Al}/{}^{27}{\rm Al}$ ratios show a
small spread in $\epsilon^{50}{\rm Ti}$ values (Fahey et al., 1987a; Kööp et
al., 2016a). Indeed, there are known examples of aluminum oxide presolar
grains, including: corundum (e.g., Hutcheon et al., 1994; Nguyen et al.,
2007); spinel (e.g., Nguyen et al., 2003); and hibonite (e.g., Choi et al.,
1999; Krestina et al., 2002; Nittler et al., 2008; Zega et al., 2011) grains.
These grains likely formed in the outflows of asymptotic giant branch (AGB)
stars, supernova explosions, or other stellar nucleosynthetic events,
throughout Galactic history, likely surviving in the interstellar medium (ISM)
for $10^{8}-10^{9}$ yr (Jones et al., 1994; Heck et al., 2020). Even if they
formed with such a high ratio as $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm
Al}$})_{0}\sim 1$, they would have $\mbox{${}^{26}{\rm Al}/{}^{27}{\rm
Al}$}<4\times 10^{-9}$ after just 20 Myr, so that the vast majority of
presolar grains would contain essentially no live ${}^{26}{\rm Al}$.

Objections to the idea that LAACHIs are predominantly formed from presolar
grains include the fact that they have oxygen isotopic signatures that exactly
match other known solar nebula reservoirs, not the wide range of values seen
in presolar grains, and despite the observed heterogeneity in ${}^{50}{\rm
Ti}$. Most importantly, there has not been an astrophysical model for how
these Al-rich presolar grains—and only these Al-rich presolar grains—could
have coagulated with each other into $10-100\,\mu{\rm m}$ objects like PLACs,
and yet avoided accreting ${}^{26}{\rm Al}$ from the gas, or any carriers of
${}^{26}{\rm Al}$.

###  1.3 Outline of this Paper

The goal of this paper is to present a quantitative astrophysical model for
how inclusions, in particular those dominated by corundum and hibonite (or
grossite) could have formed without significant ${}^{26}{\rm Al}$, in a solar
nebula with uniform $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{\rm
SS}\approx 5\times 10^{-5}$. Once such inclusions formed, isotopic exchange
with other materials in the solar nebula would allow these inclusions to
acquire more canonical ${}^{26}{\rm Al}/{}^{27}{\rm Al}$ ratios. This paper
addresses the greater challenge, which is to explain how any inclusions could
have formed without ${}^{26}{\rm Al}$, in a solar nebula with ${}^{26}{\rm
Al}$ uniformly distributed even in the molecular cloud.

The paper is organized as follows. In §2 we review the meteoritic data that
shows that some inclusions record $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm
Al}$})_{0}<3\times 10^{-6}$, which cannot be explained by late resetting. We
show that these comprise only inclusions dominated by corundum or hibonite (or
grossite), which we call LAACHIs; and that conversely, inclusions dominated by
these phases mostly formed with low $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm
Al}$})_{0}$. In §3 we present an astrophysical model for formation of LAACHIs.
We argue these formed from partially processed presolar grains of corundum and
spinel and hibonite, in a very early stage of nebular evolution. A central,
novel aspect of our model is that chemical and isotopic fractionations occur
because of electrical charging of coagulating dust grains in hot regions of
the solar nebula. Our model explains the growth of LAACHIs up to tens to
hundreds of microns in size, devoid of ${}^{26}{\rm Al}$, but with correlated
excesses/deficits in ${}^{48}{\rm Ca}$ and ${}^{50}{\rm Ti}$, as well as their
solar oxygen isotopic compositions and other features. In §4 we discuss the
implications for observable features of these inclusions, including refractory
metal abundances, stable isotope anomalies, short-lived radionuclide
abundances, and mineralogy and microstructures. In §5 we draw conclusions and
discuss implications for Solar System chronometry and formation environment.

##  2 Meteoritic Evidence for Low ${}^{26}{\rm Al}/{}^{27}{\rm Al}$

###  2.1 Background

The definition of “CAIs” as objects rich in Ca and Al encompasses many
different types of inclusions with different mineralogies, sizes and
morphologies, and $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$ ratios. Our
goal in this section is to review these different types and show that the
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})$ ratio in the solar nebula
reservoir sampled by all CAIs is uniform and canonical, except for some CAIs
dominated by the refractory phases corundum and/or hibonite or grossite.
Conversely, we show that among the types of CAIs dominated by corundum and/or
hibonite or grossite, the majority sample a reservoir with very subcanonical
${}^{26}{\rm Al}/{}^{27}{\rm Al}$. It is because the association between low
${}^{26}{\rm Al}/{}^{27}{\rm Al}$ ratios and corundum and/or hibonite (and
less commonly grossite) mineralogies is so strong that we use the term
“LAACHIs” to collectively describe these inclusions. It is likely LAACHIs have
a common origin, separate from most CAIs.

Most CAIs are characterized by mineralogies rich in melilite [a solid solution
of ${\rm Ca}_{2}{\rm Al}_{2}{\rm SiO}_{7}$ and ${\rm Ca}_{2}{\rm Mg}{\rm
Si}_{2}{\rm O}_{7}$], anorthite [${\rm CaAl}_{2}{\rm Si}_{2}{\rm O}_{8}$],
Ti,Al-rich pyroxene [${\rm Ca}({\rm Mg,Al,Ti})({\rm Si,Al})_{2}{\rm O}_{6}$],
and spinel [${\rm MgAl}_{2}{\rm O}_{4}$], with most (types CTA, B, C) showing
igneous textures, and FTA showing signatures of having grown from aggregates
of condensates, plus reaction with gas (Krot et al., 2009; MacPherson, 2014).
Most of these CAIs record nearly identical ratios $(\mbox{${}^{26}{\rm
Al}/{}^{27}{\rm Al}$})_{0}\approx 5.23\times 10^{-5}$ (Jacobsen et al., 2008),
indicating formation from a common reservoir at a similar time, and defining
the canonical $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{\rm SS}$ ratio.

Other, rarer types of CAIs have been identified as recording
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$ too low to have formed from
the same reservoir (Krot et al., 2012). Although it is always possible to
thermally reset the Al-Mg system after ${}^{26}{\rm Al}$ has decayed, and
record a lower $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$, the
temperatures in the parent bodies of carbonaceous chondrites generally were
too low for that, and any resetting had to take place in the solar nebula.
Because carbonaceous chondrites generally, and the CV3 chondrite Allende in
particular, assembled between 2 and 3 Myr after most CAIs (e.g., Desch et al.,
2018), CAIs would have to record $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm
Al}$})_{0}>3\times 10^{-6}$. The CAI H 030/L from an R chondrite of petrologic
type 4 (Rout et al., 2009) is the rare exception where low
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$ might be explained by
resetting on the parent body. In general, CAIs with $(\mbox{${}^{26}{\rm
Al}/{}^{27}{\rm Al}$})_{0}<3\times 10^{-6}$ must have sampled a different
reservoir, with $\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$}<(\mbox{${}^{26}{\rm
Al}/{}^{27}{\rm Al}$})_{\rm SS}$.

In the following, we compare many properties of LAACHIs and non-LAACHI CAIs,
including: their sizes, morphologies, mineralogies, MgO and TiO2 contents,
initial $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$ ratios, oxygen
isotopic compositions ($\Delta^{17}{\rm O}$), and titanium isotopic
compositions ($\epsilon^{50}{\rm Ti}$). Here $\Delta^{17}{\rm
O}=\delta^{17}{\rm O}-0.517\,\delta^{18}{\rm O}$, where $\delta^{17}{\rm
O}=\left[({}^{17}{\rm O}/{}^{16}{\rm O})_{\rm CAI}/({}^{17}{\rm O}/{}^{16}{\rm
O})_{\rm std}-1\right]$ and $\delta^{18}{\rm O}=\left[({}^{18}{\rm
O}/{}^{16}{\rm O})_{\rm CAI}/({}^{18}{\rm O}/{}^{16}{\rm O})_{\rm
std}-1\right]$ are the deviations in molar ratios of oxygen isotopes from a
terrestrial standard, expressed in parts per thousand (per mil, or ‰). Whereas
both $\delta^{17}{\rm O}$ and $\delta^{18}{\rm O}$ can be altered during
chemical reactions or evaporation, changes in $\Delta^{17}{\rm O}$ generally
require mixing with another isotopic reservoir. Likewise, $\epsilon^{50}{\rm
Ti}=\left[({}^{50}{\rm Ti}/{}^{47}{\rm Ti})_{\rm CAI}/({}^{50}{\rm
Ti}/{}^{47}{\rm Ti})_{\rm std}-1\right]$, measured in parts per ten thousand,
or epsilon units. In the following, the ratios have been corrected for mass-
dependent fractionation fixing the ${}^{49}{\rm Ti}/{}^{47}{\rm Ti}$ ratio. In
§2.2 we review the properties of ‘normal’ CAIs and the intermediate objects
known as Spinel-HIBonite inclusions (SHIBs). In §2.3 we review the same
properties for LAACHIs, types of CAIs that commonly record $\mbox{${}^{26}{\rm
Al}/{}^{27}{\rm Al}$}<3\times 10^{-6}$. These include: hibonite-dominated
Fractionation and Unknown Nuclear Effects (FUN) CAIs; corundum grains; PLAty
Crystals of hibonite (PLACs), and other PLAC-like CAIs; Blue AGgregates of
hibonites (BAGs); grossite-dominated CAIs; and other unusual inclusions
dominated by corundum and hibonite. In §2.4 we summarize the data to show that
the dichotomy in $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$}_{0}$ values is
clearly associated with corundum or hibonite, justifying the classification of
some CAIs as LAACHIs.

###  2.2 Non-LAACHI CAIS

####  2.2.1 Normal CAIs

As a baseline for comparison, we review ‘normal’ CAIs, which we define as the
multi-minerallic CAIs of types A, B and C. While normal CAIs are found in all
chondrite types, most descriptions are of CAIs from CV chondrites (e.g.,
MacPherson & Krot, 2014). Type A CAIs are primarily melilite, with some Ti-
rich clinopyroxene and spinel. CTA CAIs have igneous textures, while FTA CAIs
appear to be aggregates of grains that condensed from the gas or at least
reacted with the gas. Type B and C CAIs have igneous textures, and contain
melilite, Ti-rich clinopyroxene, anorthite, and spinel. Normal CAIs have
distinct populations in each carbonaceous chondrite type and show a wide range
of mineralogies, textures, shapes, and sizes. For example, the most common
type of CAIs in CM, Ordinary, and Enstatite chondrites are spinel-rich and
hibonite-rich; CK CAIs are commonly rich in secondary minerals; CO CAIs
commonly contain hibonite and grossite; while CR CAIs tend to be melilite-rich
(Russell et al., 1998; Aléon et al., 2002; Chaumard et al., 2014; Fendrich &
Ebel, 2021; Lin et al., 2006). While many CAIs contain hibonite or grossite,
the vast majority of them are not dominated by these minerals. CAIs can be
small (tens of $\mu{\rm m}$), or up to centimeter-sized in the case of type B
CAIs from CV chondrites like Allende.

Oxygen isotopic compositions of normal CAIs can extend from $\Delta^{17}{\rm
O}\approx-26$ to $-22\mbox{\text{\textperthousand}}$, with an average of
$\approx-24\mbox{\text{\textperthousand}}$ (Krot et al., 2010). Titanium
isotopic compositions cluster very tightly around $\epsilon^{50}{\rm
Ti}\approx+9$, varying from about $+2$ to $+15$ epsilon units (Williams et
al., 2016; Davis et al., 2018; Torrano et al., 2019).

The first measurements of $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$
ratios were undertaken in large type B CAIs from Allende. MacPherson et al.
(1995) and MacPherson et al. (2005) created histograms of $(\mbox{${}^{26}{\rm
Al}/{}^{27}{\rm Al}$}_{0}$ ratios in all CAIs, including many CAIs we do not
consider ‘normal’. These distributions appear to be bimodal, with the majority
of CAIs near the canonical value $\mbox{${}^{26}{\rm Al}/{}^{27}{\rm
Al}$}\approx 5\times 10^{-5}$ and a tail of inclusions extending to
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}\approx 0$ (Davis, 2022).

####  2.2.2 SHIBs

Spinel-HIBonite inclusions (SHIBs) are a type of CAI we consider intermediate
between normal CAIs and LAACHIs. They are composed predominantly of spinel and
hibonite, with accessory perovskite. Voids and cracks are commonly observed in
both untreated and acid-treated SHIBs, and the former generally contains
alteration phases while untreated SHIBs sustain partial rims of secondary, or
sometimes primary silicates. Grain sizes of SHIBs are quite variable, ranging
from 30 $\mu{\rm m}$ to 160 $\mu{\rm m}$ in their longest dimension (Kööp et
al., 2016b). The MgO content for the hibonite in SHIBs ranges from 0.4 to 3.1
wt% while TiO2 ranges from 0.2 to 6.2 wt%. Kööp et al. (2016a) reported that
hibonite in SHIBs often is very zoned in MgO and TiO2.

SHIBs generally vary in their oxygen isotopic ratios from $\Delta^{17}{\rm
O}\approx-25$ to $-16\mbox{\text{\textperthousand}}$, with an average
$\approx-23\mbox{\text{\textperthousand}}$, almost identical to normal CAIs
(Kööp et al., 2016b). Their titanium isotopic ratios vary anywhere from
$\epsilon^{50}{\rm Ti}\approx-100$ to $+100$ epsilon units (Kööp et al.,
2016b).

The distribution of initial $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$
values for SHIBs tends to skew strongly to the canonical value $\approx
5\times 10^{-5}$, with none found with $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm
Al}$})_{0}<3\times 10^{-6}$ (Liu et al., 2009, 2012, 2019; Kööp et al.,
2016b). There is a preponderence of nearly canonical ${}^{26}{\rm
Al}/{}^{27}{\rm Al}$ ratios in both ‘normal’ CAIs (defined by their mineralogy
and lacking in corundum or hibonite) and SHIBs (containing hibonite but also
significant spinel) These are not examples of classes of CAIs that are
generally ${}^{26}{\rm Al}$-poor.

###  2.3 LAACHIs

In contrast to normal CAIs and SHIBs, other classes of CAIs as a rule exhibit
low $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$ ratios and appear to
sample a different reservoir. These include FUN CAIs, corundum grains, PLACs,
grossite-bearing CAIs, and other hibonite- or corundum-bearing inclusions. We
review these here.

####  2.3.1 FUN CAIs

FUN CAIs are Ca-rich, Al-rich inclusions with both Unknown Nucleosynthetic
(UN) mass-independent isotopic anomalies and mass-dependent Fractionation (F)
of isotopes due presumably to evaporation (Mendybaev et al., 2013, 2017). This
definition permits a variety of mineralogies, and many types of CAIs can be
classified as FUN. To date, about 21 inclusions have been identified as FUN
CAIs (Dunham et al., 2021).

The $\Delta^{17}{\rm O}$ values of FUN CAIs can range from
$-25\mbox{\text{\textperthousand}}$ to $-16\mbox{\text{\textperthousand}}$
(Krot et al., 2014), but are generally $\Delta^{17}{\rm
O}\approx-23\mbox{\text{\textperthousand}}$, indistinguishable from normal
CAIs, although they show strong mass-dependent fractionation in borth
$\delta^{17}{\rm O}$ and $\delta^{18}{\rm O}$ (Krot et al., 2010). Their
titanium isotopic ratios vary, but appear correlated with whether or not they
are dominated by hibonite: hibonite-rich FUN CAIs can see $\epsilon^{50}{\rm
Ti}$ vary from about -200 to +200; FUN CAIs not dominated by hibonite tend to
cluster strongly around a single value $\approx-40$ (Torrano et al., 2021;
Dunham et al., 2021).

FUN CAIs have been traditionally thought of as forming from an ${}^{26}{\rm
Al}$-poor reservoir (e.g., Krot et al., 2012), but this is true only of the
subset of FUN CAIs containing significant hibonite. As reviewed by Dunham et
al. (2021), only the examples of H030/L, HAL, 2-6-6, and 1-9-1 definitely
formed with $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}<3\times 10^{-6}$,
and these all contain significant hibonite, or are dominated by it. In
contrast, only seven of the 21 known FUN CAIs (CMS-1, EK1-4-1, GG3, CG14,
STP-1, C1, and Vig 1623-5) do not contain hibonite. These are all large,
multi-minerallic CAIs identifiable as type A or B, and all record
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$ ratios $>3\times 10^{-6}$
(Dunham et al., 2021; Marin-Carbonne et al., 2012). These FUN CAIs without
hibonite could have formed from the same canonical reservoir as other CAIs,
later in the lifetime of the solar nebula, but before parent body accretion.
FUN CAIs strongly suggest a correlation between hibonite and
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$.

####  2.3.2 Corundum Grains

The corundum grains that have been analyzed are single, isolated grains with
exceptionally small sizes, ranging from 0.5 $\mu{\rm m}$ to 15 $\mu{\rm m}$
(Virag et al., 1991; Makide et al., 2011; Nakamura et al., 2007). The corundum
is very low in TiO2 and generally MgO is below detection limits, while any
hibonite included in these CAIs tends to be low in both MgO and TiO2 (Makide
et al., 2013; Virag et al., 1991). Corundum grains often have sharp features,
suggesting they are fragments of larger corundum grains (Virag et al., 1991).
Some of the corundum grains within the matrices of their host samples analyzed
by Nakamura et al. (2007) were found as aggregates of two to six similarly-
sized grains. Other corundum inclusions are seen rimmed in hibonite (Needham
et al., 2017).

The oxygen isotopic compositions of corundum grains range from
$\Delta^{17}{\rm O}\approx-31$ to $-14\mbox{\text{\textperthousand}}$, with an
average near that of other CAIs, $\Delta^{17}{\rm
O}\approx-23\mbox{\text{\textperthousand}}$ (Makide et al., 2011). Corundum
grains have not, to our knowledge, been analyzed for $\epsilon^{50}{\rm Ti}$.

Over 110 corundum grains have been analyzed for $(\mbox{${}^{26}{\rm
Al}/{}^{27}{\rm Al}$})_{0}$ (Virag et al., 1991; Makide et al., 2011; Nakamura
et al., 2007). Because the corundum grains are so small, it is difficult to
measure more than one spot on most grains, so most of the $(\mbox{${}^{26}{\rm
Al}/{}^{27}{\rm Al}$})_{0}$ ratios are inferred from “model” isochrons in
which the intercept is fixed to the value of bulk chondrites. Given that many
PLACs show deficits in ${}^{26}{\rm Mg}$ (§4.2.2), model isochrons may
underestimate the slope. Despite this, it is clear that these grains were
generally deficient in ${}^{26}{\rm Al}$. Although almost half of all analyzed
corundum grains exhibit $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$
values approaching the canonical value, we calculate a modal value for the
corundum grains of $<1\times 10^{-6}$.

####  2.3.3 PLACs and PLAC-like CAIs

PLAty hibonite Crystals (PLACs) typically range in size from 30 $\mu{\rm m}$
to 150 $\mu{\rm m}$ (Kööp et al., 2016a). They are traditionally characterized
as having large, platy or plate-like hibonite crystals, but may range in
morphology from platy to equant or stubby, and sometimes appear to be
aggregates (Kööp et al., 2016a; Liu et al., 2012). Many PLACs appear to have
fragmented from larger objects or aggregates with sizes $>150\,\mu{\rm m}$
(Kööp et al., 2016a).

PLACs exhibit a wider range of oxygen isotopic ratios than normal CAIs,
$\Delta^{17}{\rm O}\approx-32$ to $-19\mbox{\text{\textperthousand}}$, with
average $\Delta^{17}{\rm O}\approx-22\mbox{\text{\textperthousand}}$, but
similar to the range seen in corundum grains (Kööp et al., 2016a). PLACs are
known for their large positive and negative anomalies in stable isotope ratios
such as ${}^{50}{\rm Ti}$, which ranges from -700 to +2700 epsilon units.

PLACs generally exhibit low, subcanonical $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm
Al}$})_{0}$ ratios (Ireland, 1988, 1990; Kööp et al., 2016a; Liu et al., 2012;
Liu, 2008). Although a minority (about a third) of PLACs record the canonical
value of $5\times 10^{-5}$, we calculate a modal $(\mbox{${}^{26}{\rm
Al}/{}^{27}{\rm Al}$})_{0}$ value for PLACs of $2\times 10^{-6}$, more than an
order of magnitude lower than the canonical value.

Blue AGregates (BAGs) are another type of hibonite-rich inclusion, similar to
but not as common as PLACs. Ireland (1988) analyzed three BAGs and described
them as less than 80 $\mu{\rm m}$ in diameter and and as a conglomerate of
hibonite plates and fragments ranging from 2 $\mu{\rm m}$ to 20 $\mu{\rm m}$.
Perovskite grains are common, but they are no larger than 2 $\mu{\rm m}$ in
diameter Ireland (1988). Determination of $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm
Al}$})_{0}$ has been attempted for five BAGs, and is generally low. Ireland
(1988) and Ireland (1990) found upper limits $(\mbox{${}^{26}{\rm
Al}/{}^{27}{\rm Al}$})_{0}<1\times 10^{-5}$ in some; Liu et al. (2009)
detected no excesses of ${}^{26}{\rm Mg}$ in others.

####  2.3.4 Grossite-bearing CAIs

Although not as common as corundum or hibonite, grossite is a closely related
refractory calcium aluminate. Examples of CAIs dominated by grossite have been
identified. These CAIs have a large range in sizes, from 60 $\mu{\rm m}$ to
200 $\mu{\rm m}$ and have been found mostly in CO (Simon et al., 2019a) and CH
chondrites (e.g., Krot, 2019). Aside from grossite, these CAIs can exhibit a
wide mineralogical range containing perovskite, spinel, and/or hibonite
enclosed within the grossite cores. The hibonite is lath-like in some
grossite-enriched CAIs, and layers of melilite have been observed enclosing
the cores. Rims, or partial rims, of gehlenite or spinel have been observed.
These grossite-rich CAIs also have been observed as aggregates with gehlenite
rims and pore space and matrix between the nodules (Simon et al., 2019a).

Like corundum grains and PLACs, grossite-bearing CAIs show a wide range in
oxygen isotopic ratios, from $\Delta^{17}{\rm O}\approx-34$ to
$-9\mbox{\text{\textperthousand}}$, with an average $\Delta^{17}{\rm
O}\approx-19\mbox{\text{\textperthousand}}$, slightly more ${}^{16}{\rm
O}$-poor than other CAIs (Krot et al., 2019). We are not aware of titanium
isotopic measurements of grossite-rich CAIs.

Simon et al. (2019a) reported a bimodal distribution of initial
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$ values, with 38% near the
canonical value of $5\times 10^{-5}$; but the majority are low, and we
calculate the modal value of $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$
for grossite-rich CAIs is $<1\times 10^{-6}$. Complicating the issue are two
facts: the strongest evidence for low $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm
Al}$})_{0}$ ratios among grossite CAIs comes from those in CH chondrites, and
although some grossite-dominated CAIs from CH chondrites show low
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$, as low as $(3.7\pm
3.1)\times 10^{-7}$ (Krot et al., 2017), these may have been subject to a late
resetting. Many CAIs in CH chondrites appear to have been reprocessed, or at
least thermally reset, in the impact plume created by the collision between
asteroids that led to the CB/CH chondrites (Krot et al., 2005). This event
occurred at 5.8 Myr after CAIs (Desch et al., 2023b), when ${}^{26}{\rm
Al}/{}^{27}{\rm Al}$ would have been $\approx 2\times 10^{-7}$. It is expected
that the oxygen isotopic composition of the CAIs also would have been reset in
the impact plume. The fact that $\Delta^{17}{\rm O}$ in grossite-rich CAIs
extends to very ${}^{16}{\rm O}$-poor compositions suggests this may have
occurred to some CAIs, but perhaps not to others with more typical
${}^{16}{\rm O}$-rich compositions. At any rate, at least some grossite-
bearing CAIs in general appear to have formed from a low-${}^{26}{\rm
Al}/{}^{27}{\rm Al}$ reservoir.

####  2.3.5 Other Inclusions

In addition to the examples above, there are unusual, larger, multi-minerallic
inclusions dominated by corundum and/or hibonite.

M98-8 has been described by Simon et al. (2002). It is a rounded, $90\times
75\,\mu{\rm m}$ refractory inclusion from the Murchison (CM) carbonaceous
chondrite. It consists of rounded and anhedral 5 $\mu{\rm m}$ to 15 $\mu{\rm
m}$ corundum grains enveloped in hibonite laths that are 10 $\mu{\rm m}$ wide
and 30 $\mu{\rm m}$ to 40 $\mu{\rm m}$ long. Perovskite grains, a few microns
in size, are observed towards the inclusion’s edge and at the edge of the
hibonite laths. The inclusion has a bit of a fluffy texture due to the
triangular- or trapezoidal-shaped void spaces between the hibonite crystals.
Oxygen isotopic ratios within M98-8 appear consistent with a single value
$\Delta^{17}{\rm O}\approx-24\mbox{\text{\textperthousand}}$. We are not aware
of titanium isotopic measurements of this inclusion. This corundum- and
hibonite-bearing inclusion records a low $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm
Al}$})_{0}$ value of $(-1.9\pm 3.5)\times 10^{-6}$.

DOM 31-2 is a $230\times 220\,\mu{\rm m}$ inclusion found in the CO3.00
carbonaceous chondrite Dominion Range (DOM) 08006, described by Simon et al.
(2019b). It consists of a core of hibonite laths contained in a grossite and
perovskite mantle with anhedral spinel in between, which is then contained in
melilite approximately 10 $\mu{\rm m}$ to 15 $\mu{\rm m}$ wide. Void space is
observed within the grossite and perovskite mantle as well as in between the
hibonite laths in the core. Outside the melilite layer is a 10 $\mu{\rm m}$\-
to 15 $\mu{\rm m}$-sized thick layer of diopside enclosing FeNi metal within.
Lastly, a region of unstructured enstatite and forsterite grains and metal
occur. Oxygen isotopic ratios in all minerals, including the hibonite and
grossite, cluster tightly around $\Delta^{17}{\rm
O}\approx-23\mbox{\text{\textperthousand}}$. Its titanium isotopic composition
is $\epsilon^{50}{\rm Ti}=+88\pm 20$ (Simon et al., 2019b). Hibonite in this
UN inclusion records an ${}^{26}{\rm Al}/{}^{27}{\rm Al}$ initial value of
$(-3\pm 8)\times 10^{-7}$.

A-COR-01, from ALHA 77307 (CO3.0), has been described by Bodénan et al.
(2020). It is a sub-rounded, $175\times 140\,\mu{\rm m}$ -sized inclusion with
lath-like hibonite crystals with 5 $\mu{\rm m}$\- to 10 $\mu{\rm m}$-sized,
anhedral corundum grains in the core. This core is layered with spinel, which
is then layered with a 5 $\mu{\rm m}$\- to 10 $\mu{\rm m}$-sized rim of
diopside. There is a 1 to $4\,\mu{\rm m}$-sized crack that separated the
spinel from the diopside layer, which Bodénan et al. (2020) attribute to a
mechanical separation during an energetic event, e.g., an impact. Oxygen
isotopic ratios in the core minerals of corundum and hibonite are
$\Delta^{17}{\rm O}\approx-31.2\pm 3.0\mbox{\text{\textperthousand}}$ and
$-33.5\pm 2.3\mbox{\text{\textperthousand}}$, in contrast to the ratio in the
diopside rim, $\Delta^{17}{\rm O}\approx-24.8\pm
0.5\mbox{\text{\textperthousand}}$, which suggests the ratios in the hibonite
and corundum are primary. We are not aware of titanium isotopic measurements
for this inclusion. This hibonite- and corundum-bearing inclusion records an
${}^{26}{\rm Al}/{}^{27}{\rm Al}$ initial value $(1.67\pm 0.31)\times
10^{-7}$.

HAL (Hibonite from ALlende) is a $100\times 200\,\mu{\rm m}$-sized inclusion
from Allende composed of a hibonite core with exsolved fine needles of a Ti-
rich phase, and a $120\times 220\,\mu{\rm m}$-sized black rim (Fahey et al.,
1987b; Allen et al., 1980). This black, compact rim looks similar to
devitrified glass and is composed of an anisotropic Al-Fe oxide with a
feathery-like texture. The outer portion is composed of a fine-grained, very
friable rim with thickness up to $\approx 2000\,\mu{\rm m}$. This friable rim
is made up of five distinct layers with an aggregate thickness of 90 $\mu{\rm
m}$ to 450 $\mu{\rm m}$ and composed of a variable mineralogy that probably
formed on the parent body. The oxygen isotopic ratios have been altered by
mass-dependent fractionation due to evaporation, but are mostly consistent
with $\Delta^{17}{\rm O}\approx-25\mbox{\text{\textperthousand}}$ (Lee et al.,
1980). HAL is isotopically anomalous in titanium, with $\epsilon^{50}{\rm
Ti}\approx+160$ epsilon units (Fahey et al., 1987b). Fahey et al. (1987b)
reported $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}=(5.2\pm 1.7)\times
10^{-8}$ for this inclusion.

SHAL (Son of HAL) is a HAL-like Allende inclusion made up of a $500\times
200\,\mu{\rm m}$-sized single hibonite grain along with blocky-like perovskite
roughly 200 $\mu{\rm m}$ in size (Liu & Keller, 2017). SHAL perovskite and
hibonite appear devoid of ${}^{26}{\rm Mg}$ excesses attributed to the
${}^{26}{\rm Al}$ decay (Liu et al., 2016).

HIDALGO (HIbonite in Dar AL Gani cO3) is a HAL-like, nearly stoichiometrically
pure single hibonite inclusion with a size of $300\times 300\,\mu{\rm m}$. It
contains 0.2 wt% TiO2 and MgO below the detection limit, making it the most
Mg-depleted hibonite found so far (Liu et al., 2021). This inclusion records
an $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$ value of $(1.5\pm
0.02)\times 10^{-5}$ (Liu et al., 2021).

###  2.4 Summary

While normal CAIs and SHIBs attest to a widespread uniform and canonical value
$\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$}\approx 5\times 10^{-5}$ in the solar
nebula, some inclusions in meteorites formed with $(\mbox{${}^{26}{\rm
Al}/{}^{27}{\rm Al}$})_{0}$ as low as $\sim 10^{-7}$, which cannot be
explained by late resetting, and must have formed from a reservoir with
subcanonical ${}^{26}{\rm Al}/{}^{27}{\rm Al}$. As outlined above, these
“LAACHIs” include: some hibonite-bearing FUN CAIs, most corundum grains, PLACs
and PLAC-like CAIs, BAGs, grossite-bearing CAIs, as well as some other unusual
inclusions dominated by hibonite and/or corundum. The properties of LAACHIs,
SHIBs, and normal CAIs are summarized in Table 1. We list their typical sizes;
their mineralogies and morphologies, which suggest origins as aggregates or
fragments; their MgO and TiO2 contents; their typical oxygen isotopic
compositions; their typical ${}^{50}{\rm Ti}$ values; and their typical
initial $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$ ratios.

In Figure 1 we present compiled literature values of the $(\mbox{${}^{26}{\rm
Al}/{}^{27}{\rm Al}$})_{0}$ ratios of hibonite-rich FUN CAIs, corundum grains,
PLACs, and other hibonite-bearing inclusions. These are displayed as
histograms and as kernel density estimations we have calculated. Each
population, especially corundum grains and spinel-bearing inclusions, has a
minority of grains that show canonical ratios $(\mbox{${}^{26}{\rm
Al}/{}^{27}{\rm Al}$})_{0}\approx 5\times 10^{-5}$; but all types of LAACHIs
show modal values $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}\leq 2\times
10^{-6}$, meaning the majority of them must have formed from a reservoir with
subcanonical ${}^{26}{\rm Al}/{}^{27}{\rm Al}$. The distribution of
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$ ratios in SHIBs is shown for
comparison. Figure 1 is to be compared to similar plots of normal CAIs, which
show a strong preponderance to have values near the canonical ratio (e.g.,
MacPherson et al., 2005; Davis, 2022).

These probability density functions make clear that among inclusions dominated
by corundum or hibonite or grossite, $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm
Al}$})_{0}$ ratios are indeed low, with the modal value $\approx 2\times
10^{-6}$. Conversely, all types of inclusions with low modal
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$ are ones dominated by
corundum or hibonite or grossite. CAIs not dominated by corundum or hibonite
or grossite tend to have $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$
ratios consistent with formation from a reservoir with canonical ${}^{26}{\rm
Al}/{}^{27}{\rm Al}$. It is this evident dichotomy that leads us to categorize
many CAIs as LAACHIs. LAACHIs are typically tens to hundreds of microns in
size, with many having morphologies suggestive of aggregation, and many
resembling fragments of larger objects.

Table 1: Properties of LAACHIs and other CAIs

Objects | Sizes ($\mu{\rm m}$) | Morphology | MgO, TiO2 |  $\Delta^{17}{\rm O}$ (‰) |  $\epsilon^{50}{\rm Ti}$ †† |  $({}^{26}{\rm Al}/{}^{27}{\rm Al})_{0}$ †
---|---|---|---|---|---|---
Hibonite-rich F(UN) and HAL-like CAIsa | $80-300$ |  Large lath-like or needle-like hib. grains with interstitial sp. grains and layers of mel. and diop. Void spaces in hib. |  0.02-0.6 wt%, 0.03-6.4 wt% |  $\approx-24$ |  $\pm 200$ |  $<2\times 10^{-6}$ 50% Low
Corundum grainsb | $0.5-15$ |  Single cor. grains, sometimes aggregates. |  $\sim 0$ wt%, $\sim 0.09$ wt% |  -31 to -14 Avg. -23 |  - |  $<1\times 10^{-6}$ 43% Low
PLACs and PLAC-like CAIsc | 30 - 150 |  Large hib. grains (may be aggregates), ranging from platy to stubby and sp.-rimmed to sp.-free. Pv. found as inclusions within hib. and between grains in aggregates. Voids within aggregates. |  0.2 - 1.4 wt%, 0.7 - 3.1 wt% |  $-32$ to $-19$ Avg. -22 |  -700 to +2700 |  $\approx 2\times 10^{-6}$ 72% Low
Grossite-rich CAIsd | $60-200$ |  Gros.-rich, can contain hib., pv., mel., sp. within grossite or in rims. |  0.03 - 1.5 wt%, 0.05 - 1.6 wt% |  $-34$ to $-9$ Avg. -19 |  - |  $<1\times 10^{-6}$ 50% Low
Other Corundum $\pm$ Hibonite $\pm$ Spinel Inclusionsd | $30-175$ |  Lath-shaped or platy hib. with cor. and/or sp. grains within or surrounding hib. Void spaces within and/or between hib. or cor. grains. |  0.02- 13 wt%, 0.05 - 11 wt% |  $\approx-24$ |  $\pm$ tens? |  $<2\times 10^{-6}$ 27% Low
SHIBsf | $30-160$ |  Abundant hib. and sp., with pv., FeO-poor silicates, refractory metal blebs. Voids, cracks common. |  0.4 - 3.1 wt%, 0.2 - 6.2 wt% |  $-24$ to $-22$ Avg. -23 |  $\approx\pm 100$ |  $\approx 4\times 10^{-5}$ 0% Low
‘Normal’ CAIsg | $5-20,000$ |  Variable, with mel.-rich type A common. |  |  $-26$ to $-22$, Avg. -24 |  +2 to +15 ($\approx+9$) |  $\approx 5\times 10^{-5}$ rare low
|  |  |  |  |  |

Note. — Hib. $=$ hibonite, Cor. $=$ corundum, Gros. $=$ grossite, Sp. $=$
spinel, Pv. $=$ perovskite, Mel. $=$ melilite, Diop. $=$ diopside.
${}^{\dagger}{\rm Modal}$ value (see Fig. 1), and ‘Low’ fraction with
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}<3\times 10^{-6}$.
${}^{\dagger\dagger}{\rm Using}$ 49/47 normalization. References: $a$
(Hibonite-rich F(UN) CAIs): Russell et al. (1998), Ushikubo et al. (2007),
Rout et al. (2009), Kööp et al. (2018), Simon et al. (2019b); (HAL-like) Fahey
et al. (1987b), Ireland et al. (1992), Liu et al. (2016), Liu et al. (2021);
Lee et al. (1980); $b$ (corundum grains): Virag et al. (1991), Nakamura et al.
(2007), Makide et al. (2011); $c$ (PLACs and PLAC-like CAIs): Fahey et al.
(1987b); Trinquier et al. (2009), Liu et al. (2009), Liu et al. (2012), Kööp
et al. (2016a), Liu et al. (2019), Shollenberger et al. (2022); $d$ (Grossite-
rich CAIs): Simon et al. (2019a), Krot et al. (2019) and references therein,
Weber et al. (1995); $e$ (Other Corundum $\pm$ Hibonite $\pm$ Spinel): Hinton
& Bischoff (1984), Bar-Matthews et al. (1982), Fahey (1988), Simon et al.
(2002), Sugiura & Krot (2007), Rout et al. (2009), Makide et al. (2013), Liu
et al. (2019), Bodénan et al. (2020); $f$ (SHIBs): Liu et al. (2009), Liu et
al. (2012), Kööp et al. (2016b), Ireland (1990); $g$ (Normal CAIs): MacPherson
et al. (1995), McKeegan & Davis (2003), MacPherson et al. (2005), Krot et al.
(2010), Williams et al. (2016), Davis et al. (2018).

Figure 1: Histograms (a) and kernel density estimation (KDE) (b) of compiled
literature values of initial $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$
ratios in types of inclusions that are considered low in ${}^{26}{\rm Al}$,
including: corundum grains; PLACs; hibonite-rich FUN CAIs; and other hibonite-
or corundum-bearing inclusions. the distributions are strongly bimodal: a
minority have more canonical initial ratios $(\mbox{${}^{26}{\rm
Al}/{}^{27}{\rm Al}$})_{0}\approx 5\times 10^{-5}$, but the majority of these
inclusions with low $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$ record
modal $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}<2\times 10^{-6}$ (black
dotted line in (b)). All these types of inclusions with low modal
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$ are dominated by corundum or
hibonite (or sometimes grossite), and we categorize such inclusions as
Low-${}^{26}{\rm Al}/{}^{27}{\rm Al}$ Corundum/Hibonite Inclusions (LAACHIs).
Other non-LAACHI CAIs, not dominated by corundum or hibonite, record canonical
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$ ratios, including: SHIBs,
whose KDE is shown (dashed orange curve); and ‘normal’ type A/B/C CAIs (black
dotted line ”canonical”) dominated by melilite or anorthite, too numerous to
show, which cluster strongly around $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm
Al}$})_{0}\approx 5\times 10^{-5}$.

##  3 Astrophysical Model for Formation of LAACHIs

We develop here a simple but quantitative model to demonstrate that
$10-100\,\mu{\rm m}$ LAACHIs could have formed in the solar nebula, with low
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$, despite the widespread
existence of ${}^{26}{\rm Al}$. The key elements of this model are depicted in
Figure 2. As presolar grains find their way into regions $>1350$ K, pyroxene,
olivine and other grains vaporize; corundum, spinel and hibonite are
essentially the only solids in these regions. Spinel is likely transformed to
corundum by loss of Mg, without gain or loss of Al atoms. Corundum may
partially or completely convert to hibonite by gaining Ca, without gain or
loss of Al. Hibonite could convert to corundum by loss of Ca (or even gain Ca
to convert to grossite), or may remain as hibonite, all depending on what
phases are thermodynamically favored. As almost the only solids in the hot
midplane region, micron-sized grains of corundum or hibonite quickly aggregate
and grow into objects tens of microns in size, within hundreds of years. Only
large grains (a few to tens of $\mu{\rm m}$ in size) can coagulate, and these
large grains derive from AGB stars and contain no live ${}^{26}{\rm Al}$. We
assume that all live ${}^{26}{\rm Al}$ is carried on small (tens of nm) grains
that cannot be accreted into the aggregates because they are too negatively
charged, and repelled from the negatively charged corundum/hibonite
aggregates. These grains include ${}^{26}{\rm Al}$-bearing refractory grains
condensed in the ejecta of a recent, nearby supernova or Wolf-Rayet star.
Because ejecta densities drop so quickly in these environments, dust grains do
not grow as large as in AGB outflows. These small ${}^{26}{\rm Al}$-bearing
grains also include refractory grains condensed in the environment from Al
vapor from vaporized, non-refractory grains. We assume that ${}^{48}{\rm Ca}$
and ${}^{50}{\rm Ti}$ anomalies are carried on similarly small grains of
perovskite [${\rm CaTiO}_{3}$] that, we argue, are positively charged and can
be accreted. As such, LAACHIs would be devoid of live ${}^{26}{\rm Al}$, but
would show correlated anomalies in ${}^{48}{\rm Ca}$ and ${}^{50}{\rm Ti}$.
LAACHIs are likely to diffuse out of the hot midplane region in thousands of
years, before they grow larger than $100\,\mu{\rm m}$. Once outside of this
region, they may combine with other material to form objects like hibonite-
bearing CAIs and SHIBs. We further hypothesize that spinel in SHIBs formed in
other, cooler regions where it was possible to accrete the small grains
containing live ${}^{26}{\rm Al}$, so that although SHIBs carry hibonite, they
would show more canonical $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$
ratios.

Figure 2:  Illustration of our model for formation of Low-${}^{26}{\rm
Al}/{}^{27}{\rm Al}$ Corundum/Hibonite Inclusions (LAACHIs). Very early in
solar nebula evolution ($<10^{5}$ yr), most grains were presolar. These grains
diffused into hot ($T>1350$ K; light brown), turbulent ($\alpha\sim 10^{-3}$)
regions $\sim 0.6$ AU from the Sun, within a scale height of the disk
midplane. 1.) Corundum, spinel, hibonite, and perovskite were practically the
only surviving grains; silicates vaporized, releasing Ca and other vapor,
recondensing as corundum. Spinel rapidly converted to corundum by loss of Mg.
Corundum reacted with Ca vapor to form hibonite. 2.) Grains in this region
were charged according to their size and material work function, with most
grains negatively charged, except for positively charged perovskite. 3.) Their
relative velocities were sufficient for the largest grains to coagulate, but
small ${}^{26}{\rm Al}$-bearing grains generally were repelled. Aggregates of
${}^{26}{\rm Al}$-free corundum and hibonite grew to $10-10^{2}\,\mu{\rm m}$
in hundreds of years. 4.) Growth beyond this size was inhibited by
fragmentation. 5.) LAACHIs escaped this region after thousands of years.
(Credit: Sue Selkirk, ASU.)

In this section we provide quantitative assessments of the validity of each
element of our hypothesis. We begin in §3.1 by describing the temperature
structure and other aspects of the hot region of the protoplanetary disk in
which LAACHIs form. In §3.2 we discuss what phases are thermodynamically
favored and how minerals transform from one phase to another. Almost all Al is
in the solid phase.  In §3.3 we discuss the likely presolar carriers of Al and
${}^{26}{\rm Al}$, and their sizes and abundances, both in the solar nebula
and in the LAACHI-forming region. In §3.4 we demonstrate that these presolar
grains could exchange oxygen isotopes with the disk gas. In §3.5 we quantify
the electrical charging of dust grains in the hot environment and show that
micron-sized grains typically would have hundreds of electrons per grain; as a
result, we show, the nanospinels would be repulsed by the corundum/hibonite
grains (and each other), but that the corundum/hibonite grains could collide
with and stick to each other. Next, in §3.6 we quantify the growth rate of
LAACHIs and show that objects in the typical size range $10-100\,\mu{\rm m}$
form before they leave this hot region of the disk. In §3.7 we calculate the
incorporation of ${}^{26}{\rm Al}$ into these growing particles, and
demonstrate that they would generally form with $(\mbox{${}^{26}{\rm
Al}/{}^{27}{\rm Al}$})_{0}<10^{-7}$. These various stages are depicted in
Figure 2. In §3.8 we discuss their fate after leaving this region, and their
observable characteristics.

###  3.1 The LAACHI-Forming Environment

We hypothesize that LAACHIs uniquely formed without ${}^{26}{\rm Al}$, early,
and in a region hot enough to vaporize almost but not quite all solids. The
production of PLACs and other low-${}^{26}{\rm Al}/{}^{27}{\rm Al}$ inclusions
is widely accepted to have taken place early in the evolution of the solar
nebula, before most of its (presolar) material had been thermally processed
(e.g., Liu et al., 2009; Makide et al., 2011; Krot et al., 2012). This stage
could have occurred even before the Sun had mostly formed. One of the more
widely used treatments of the solar nebula at this early time is that of Yang
& Ciesla (2012), and we adopt their canonical model here.

We assume formation of LAACHIs at about $r=0.6\,{\rm AU}$, at $\approx 0.1$
Myr, when the Sun had a mass of only $0.36\,M_{\odot}$, yielding an orbital
frequency $\Omega=2.6\times 10^{-7}\,{\rm s}^{-1}$. At this point, the mass
accretion rate through the disk was $\dot{M}\approx 5.0\times
10^{-7}\,M_{\odot}\,{\rm yr}^{-1}$ and its surface density $\Sigma\approx
1.0\times 10^{4}\,{\rm g}\,{\rm cm}^{-2}$. The midplane temperature was
$\approx 1425\,{\rm K}$, so that the isothermal sound speed was $C=2.25\,{\rm
km}\,{\rm s}^{-1}$. The scale height $H=C/\Omega=8.8\times 10^{11}\,{\rm
cm}=0.06\,{\rm AU}$, the midplane density $\rho(0)=5.5\times 10^{-9}\,{\rm
g}\,{\rm cm}^{-3}$, and the midplane pressure $P=2.8\times 10^{-4}$ bar. The
assumed turbulence parameter was $\alpha=1\times 10^{-3}$, and the turbulent
viscosity was $\nu=\alpha CH=2.0\times 10^{14}\,{\rm s}^{-1}$.

Under these conditions, the disk would have had a vertical temperature
gradient, with temperatures of a few hundred K near the surface, increasing to
$\approx 1425\,{\rm K}$ at the midplane. The surface temperature was dominated
by accretional energy, so that $\sigma T_{\rm
eff}^{4}=(9/8)\Sigma\,\nu\,\Omega^{2}$, or $T_{\rm eff}=235\,{\rm K}$. With
increasing optical depth $\tau$ into the disk, the temperature increased, as
$T\approx T_{\rm eff}\,(3\tau/4)^{1/4}$ (Hubeny, 1990), and so reached 1350 K,
the temperature at which most silicates evaporate at the relevant pressures,
at optical depth $\tau\approx 1430$. For the Rosseland mean opacity assumed by
Yang & Ciesla (2012), $\kappa=1\,{\rm cm}^{2}\,{\rm g}^{-1}$, this entailed a
surface density of $1430\,{\rm g}\,{\rm cm}^{-2}$ between the 1350 K layer and
the surface, a small fraction of the total surface density of the disk. The
$T=1350\,{\rm K}$ surface would have been at a height $z=1.18\,H$ above the
midplane. We assume an opacity $\kappa\approx 0.1\,{\rm cm}^{2}\,{\rm g}^{-1}$
in the hotter regions (see calculation in §3.6), so that the total optical
depth to the midplane was 1790, and the temperature at the midplane 1425 K.
The conditions at the 1350 K point at $z=1.18\,H$, below which silicates were
vaporized, were $\rho=2.7\times 10^{-9}\,{\rm g}\,{\rm cm}^{-3}$, and
$P=1.3\times 10^{-4}$ bar.

We note that this hot midplane region would have been sufficiently ionized to
support the magnetorotational instability (MRI) and therefore a turbulence
parameter $1\times 10^{-3}$. Desch & Turner (2015) calculated ionization
fractions $\sim 10^{-7}$, and Elsasser numbers $\gg 1$ (for both Ohmic
dissipation and ambipolar diffusion) at temperaures 1350 - 1500 K in a similar
region. Simulations by Sano et al. (2004) show that under a variety of
assumptions (zero or non-zero net magnetic flux, isothermal or adiabatic
conditions), the MRI saturates at levels equivalent to $\alpha\sim 10^{-3}$,
justifying the assumption of Yang & Ciesla (2012), and indicating a self-
consistent disk structure.

An important quantity is how much time grains would have spent in such a
portion of the disk before diffusing out of it. The average time taken to
vertically diffuse a distance $L\approx 1.18H$ is $L^{2}/D$, where $D=\nu/{\rm
Sc}$, ${\rm Sc}\approx 0.7$ being the appropriate Schmidt number (Desch et
al., 2017). This yields a median time spent in the $T>1350\,{\rm K}$ region of
$\approx 120$ years (if remaining at $r=0.6$ AU). Only $\sim 10\%$ of grains
would have spent less than 20 years in the region, and only $\sim 10\%$ would
have stayed longer than 3700 years. Escape is also possible by radial
diffusion. The radial temperature gradient in the disk was likely 100 K per
0.1 AU, and a typical time to diffuse radially outward $L\sim 0.2\,{\rm AU}$
(200 K) would be $\sim 1000$ years. Particles typically would not have spent
more than hundreds of years in this region, but those particles that escaped
and found their way into chondrites could retain a record of the unique
conditions in this environment.

###  3.2 Stable Phases in the LAACHI-Forming Environment

Regions like the LAACHI-forming environment are hot enough ($>1350$ K) to
vaporize silicates. Almost the only minerals that could survive in such a
region are the Ca- and Al-bearing minerals that characterize CAIs. We have
calculated the thermodynamically favored phases using the ArCCoS condensation
code (Unterborn & Panero, 2017), benchmarked against the condensation
calculations of Ebel & Grossman (2000). At a pressure $P=1\times 10^{-4}$ bar
(useful for comparisons to other calculations, and close to the pressure in
the LAACHI-forming region; §3.1), assuming solar compositions (Lodders, 2003),
we find that olivine [$({\rm Mg},{\rm Fe})_{2}{\rm SiO}_{4}$] vaporizes fully
at 1354 K, and pyroxene [$({\rm Mg},{\rm Fe}){\rm SiO}_{3}$] at 1290 K. We
adopt 1350 K as the temperature at which silicates vaporize and the opacity
drops. Above those temperatures, only a handful of minerals are
thermodynamically favored: spinel [${\rm MgAl}_{2}{\rm O}_{4}$], at
$\approx\\!\\!1375-1387\,{\rm K}$; hibonite [${\rm CaAl}_{12}{\rm O}_{19}$],
at $\approx\\!\\!1386-1395\,{\rm K}$; grossite [${\rm CaAl}_{4}{\rm O}_{7}$],
at $\approx\\!\\!1394-1463\,{\rm K}$; krotite [${\rm CaAl}_{2}{\rm O}_{4}$],
at $\approx\\!\\!1461-1516\,{\rm K}$; grossite, at
$\approx\\!\\!1493-1588\,{\rm K}$; hibonite, at $\approx\\!\\!1579-1609\,{\rm
K}$; and corundum [${\rm Al}_{2}{\rm O}_{3}$], at
$\approx\\!\\!1606-1665\,{\rm K}$. Gehlenite [${\rm Ca}_{2}{\rm Al}_{2}{\rm
SiO}_{7}$] (an end-member of melilite) is also stable from 1364 K to 1478 K.
Among phases without Al, diopside [${\rm CaMgSi}_{2}{\rm O}_{6}$] is unstable
at temperatures $>1360$ K although solid solutions with Ti-rich pyroxene may
be more stable. Perovskite [${\rm CaTiO}_{3}$] is favored from 1316 - 1583 K,
but anatase [${\rm TiO}_{2}$] is unstable above 1132 K. The temperatures of
these phase transformations all increase by the same small amount ($<10$ K) at
pressures of $3\times 10^{-4}$ bar. Figure 3 shows the phases into which Al
partitions, as a function of temperature, at $P=1\times 10^{-4}$ bar.

These stability fields determine the fate of the Al in silicate grains that
vaporize upon entry into the hot midplane region. Depending on the
temperature, this Al could condense as any of the phases in Figure 3. For
reasons discussed below, we consider it most likely that the Al vapor will
condense by self-nucleation into numerous small ($<$ tens of nm) grains.These
grains are likely corundum that then transforms into other relevant phases
(e.g., hibonite). Very little Al would be in the gas phase: the vapor pressure
of Al over corundum is $2.5\times 10^{-15}$ atm at 1425 K, and $1.5\times
10^{-16}$ atm at 1350 K (Takigawa et al., 2015). Equivalently, a fraction
$1.6\times 10^{-6}$ of all Al atoms are in the gas phase at 1425 K, and only
$2.0\times 10^{-7}$ at 1350 K.

In contrast to Al, a significant fraction of Ca should be in vapor form at
$T>1360\,{\rm K}$ above which diopside vaporizes, although we note that more
Ca might be found to condense if the ArCCoS code had included solid solutions
of Ca-rich pyroxene, Ti-rich pyroxene, and kushiroite [${\rm CaAlAlSiO}_{6}$].
The major solids in this temperature regime—presolar spinel and corundum—do
not contain Ca. Although not yet discovered among presolar grains, Dauphas et
al. (2014) suggested the existence of presolar perovskite [${\rm CaTiO}_{3}$]
grains. These refractory grains would have survived in the hot midplane
region, but even if all Ti condensed as perovskite, fewer than 0.03 Ca atoms
per Al atom would be sequestered in these solids ($<4$% of all Ca atoms), far
short of the 0.75 Ca atoms per Al atom in a solar abundance (Lodders, 2003).
Even if all Al were in the form of grossite, this would consume only another
33% of all Ca. Throughout much of the hot midplane region, most Ca would have
been in vapor form, unless it could have fully reacted with solids, forming
gehlenite (melilite), which is unlikely.

These calculations reveal the equilibrium, thermodynamically favored phases,
but kinetic effects matter as well. For example, a spinel grain entering the
hot midplane region would quickly find itself in regions $>1387\,{\rm K}$
where it was unstable. This does not mean it would be destroyed; as discussed
by Zega et al. (2021), spinel grains can remain intact by transforming to a
different solid phase quickly. Because Ca and Si vapor existed in the hot
midplane region, it would have been possible for corundum and spinel (and
hibonite) to transform to more thermodynamically favored minerals such as
gehlenite and grossite; but these transformations to thermodynamically favored
phases likely were kinetically inhibited, due to the slow diffusion of the
tetravalent cation ${\rm Si}^{4+}$ into the minerals. Instead, as a spinel
grain passed through regions with temperatures high enough that spinel was no
longer stable, it would have lost Mg (as ${\rm Mg}^{+}$) and O, creating a
residue of corundum, as is experimentally observed (Rutman et al., 1968). This
process would have been limited only by diffusion of Mg through spinel, which
is rapid. The diffusion coefficient for Mg in spinel, at about 1600 K, is
$D_{\rm Mg}\sim 10^{-16}\,{\rm m}^{2}\,{\rm s}^{-1}$ (Sheng et al., 1992;
Liermann & Ganguly, 2002; Vogt et al., 2015), about three orders of magnitude
faster than self-diffusion of O in spinel (Van Orman & Crispin, 2010). We
estimate $D_{\rm Mg}\sim 10^{-19}\,{\rm m}^{2}\,{\rm s}^{-1}$ at 1350 K, in
which case a 500 nm-sized spinel grain would have transformed to corundum
within $\sim 10$ days. Nanometer-scale Cr-rich spinel grains would transform
even faster, by loss of both Mg and Cr. Significantly, the transformation of
spinel to corundum would not have required gain or loss of Al atoms or
${}^{26}{\rm Al}$, and indeed very few Al atoms would have existed in the gas
phase.

Further transformation of corundum, to hibonite or even grossite, would have
occurred upon reaction with Ca vapor, plus diffusion of Ca into the corundum
grains. Ceramicists produce hibonite [${\rm CaO}\cdot 6({\rm Al}_{2}{\rm
O}_{3})$, or “CA6”] by reacting corundum grains with Ca delivered via solid-
state diffusion from adjacent grains of grossite [${\rm CaO}\cdot 2({\rm
Al}_{2}{\rm O}_{3})$, or “CA2”] (e.g., Pieta et al., 2015). Single hibonite
crystals tens of microns in size formed in these experiments, larger than the
grain sizes in the powders. Hibonite forms by Ca and O diffusing along the
crystal surface perpendicular to the $c$ axis, which is why hibonite forms
plates (Chen et al., 2016). Reaction of Ca vapor with corundum is likely to
have created at least rims of hibonite on corundum surfaces, such as is
observed in many hibonite-rimmed corundum grains (e.g., Nakamura et al.,
2007).

Whether corundum could have completely transformed to hibonite or grossite
depends on whether Ca could have diffused into grain interiors over relevant
timescales. Transformation of a $\sim\\!\\!1\,\mu{\rm m}$-radius grain in,
say, 3 years requires a diffusion coefficient $D_{\rm Ca}>10^{-21}\,{\rm
m}^{2}\,{\rm s}^{-1}$ or so at $\approx 1350-1425\,{\rm K}$ in the relevant
minerals. Unfortunately, the values of $D_{\rm Ca}$ in most of these minerals
are not well known. In general, the diffusion of ${\rm Ca}^{2+}$ in oxides is
presumed to be slow, because of its larger ionic radius compared to ${\rm
Fe}^{2+}$ or ${\rm Mg}^{2+}$. Nakamura et al. (2007) estimated the diffusion
of Ca in corundum to be no more than 10 times as fast as the diffusion of Al,
for which they took $D_{\rm Al}(T)=1.6\times 10^{-5}\,\exp(-(510\,{\rm
kJ}\,{\rm mol}^{-1})/RT)\,{\rm m}^{2}\,{\rm s}^{-1}$, following Gall et al.
(1994). At $T=1400\,{\rm K}$, this would yield $D_{\rm Ca}=1.5\times
10^{-23}\,{\rm m}^{2}\,{\rm s}^{-1}$, and a corundum grain would take about
1400 years to transform. However, we would expect diffusion of the divalent
cation ${\rm Ca}^{2+}$ to more than an order of magnitude faster than the
diffusion of trivalent ${\rm Al}^{3+}$. A more direct estimate of $D_{\rm Ca}$
can be made from the experiments by Pieta et al. (2015), in which CA6 formed
via diffusion of Ca through corundum grains of diameter $10\,\mu{\rm m}$, in
only 3 hours at $1600^{\circ}{\rm C}$, implying $D_{\rm Ca}\sim 2\times
10^{-15}\,{\rm m}^{2}\,{\rm s}^{-1}$ at 1873 K. As a divalent cation, Ca
probably has an activation energy similar to O, which would imply $D_{\rm
Ca}\sim 1.5\times 10^{-21}\,{\rm m}^{2}\,{\rm s}^{-1}$ at 1350 K. At these
rates, a $1\,\mu{\rm m}$-radius corundum grain could transform to hibonite in
about 20 years. We view it as likely but not definite that a corundum grain
would have fully transformed to hibonite during the hundreds of years it spent
in the hot midplane region.

While spinel would have transformed rapidly to corundum, and corundum could
have transformed to hibonite, it is not clear that these minerals would have
transformed to the other thermodynamically favored phases, including gehlenite
and grossite. Production of gehlenite requires inward diffusion of Si. The
diffusion coefficients of Si into corundum or spinel are not known, but as a
tetravalent cation, it could be expected that $D_{\rm Si}\ll D_{\rm O}$ in
these minerals, strongly suggesting that gehlenite production might have been
kinetically inhibited. Production of grossite probably could have proceeded by
Ca diffusion into hibonite, but diffusion coefficients in hibonite are not
known. However, Russell et al. (1998) noted that Mg diffuses more slowly in
hibonite than in other CAI minerals.

Given that hibonite is thermodynamically favored over a temperature range seen
in our modeled hot midplane region, and easy to produce by reaction with
corundum, we consider corundum and hibonite to be the most likely phases
present. We emphasize that the preponderance of large corundum and hibonite
crystals is not because these condensed in this region from a solar nebula gas
that cooled from above 1665 K, and then cooled below 1605 K to allow
transformation to hibonite. Corundum and hibonite were likely the common
phases because presolar grains were corundum, spinel, or hibonite, and at the
relevant temperatures ($\approx 1350-1425\,{\rm K}$), spinel would have
converted to corundum, and corundum would have converted to hibonite, within
the $\sim 10^{3}$ residence time of grains in the hot midplane region, and
other phases would have vaporized.

As spinel loses Mg and O atoms and converts to corundum, it loses 28% of its
mass, and its density increases from $3.6\,{\rm g}\,{\rm cm}^{-3}$ to
$4.0\,{\rm g}\,{\rm cm}^{-3}$. A 500 nm-sized spinel grain becomes a 430 nm-
sized corundum grain. A 50 nm-sized nanospinel grain becomes a 43 nm-sized
corundum grain. Conversion of corundum to hibonite by addition of Ca and O
atoms will increase its mass by 9% and decrease its density to $3.8\,{\rm
g}\,{\rm cm}^{-3}$. We assume corundum grains do not change in size or density
as they convert to hibonite.

Figure 3: Fraction of all Al atoms existing in various Al-bearing phases at
different temperatures in a solar composition gas. Below 1500 K, essentially
all Al exists in solid phases. Above 1650 K it is entirely in gas form.
Calculated using ArCCoS code.

###  3.3 Presolar Materials

A foundational aspect of our model is the interpretation that some of the most
primitive “Solar System” solids are not condensates from a solar-composition
gas that cooled from temperatures $>1600$ K, but are instead partially or
completely altered presolar grains that were heated from molecular cloud
temperatures $\sim 10$ K. If it is assumed from the start that PLACs and
similar objects derived from condensates from a completely vaporized solar-
composition gas, then their lack of ${}^{26}{\rm Al}$ necessarily would be
because the solar nebula lacked ${}^{26}{\rm Al}$. The only way not to
implicitly assume the solar nebula lacked ${}^{26}{\rm Al}$ early on is to
consider the possibility that these objects formed from material that retained
some memory of their presolar nature. In particular, one cannot explain the
existence of Al-bearing minerals lacking in ${}^{26}{\rm Al}$ unless one
considers the presolar carriers of Al and ${}^{26}{\rm Al}$ in the solar
nebula. The hypothesis we present here requires that a significant fraction of
Al resided in presolar micron-sized corundum or hibonite grains (or in
minerals that converted easily to corundum and hibonite), but that all live
${}^{26}{\rm Al}$ resided in a small fraction of tiny (tens of nm in size)
grains, of similar mineralogy.

From studies of the least altered chondrites, a wide variety of presolar
grains are known to have existed in the solar nebula, as reviewed by Zinner
(2014): diamonds and graphitic carbon; silicon carbide; silicates like olivine
and pyroxene; perovskite; aluminates like corundum, hibonite and spinel;
${}^{54}{\rm Cr}$-rich nanospinels; and many others. Most are present among
chondritic matrix grains at 1-100 ppm levels, but that is just the tiny
fraction of materials that have not been thermally or chemically processed in
the solar nebula or on the parent bodies. Originally, of course, 100% of dust
in the solar system was presolar. The exact abundances in the solar nebula of
different types of presolar grains depend on their primordial abundances in
the ISM, but also on their ability to survive disk processes, parent body
processes, and the process of extracting them from their host chondrites.
These efficiencies are difficult to quantify, although attempts are being made
(Hoppe et al., 2017).

The vast majority of grains are expected to derive from asymptotic giant
branch (AGB) stars, from progenitors with initial masses $0.5$ to
$8\,M_{\odot}$. Integration of the stellar initial mass function suggests
their outflows should produce more dust than from stars $\geq 8\,M_{\odot}$ in
mass, and this seems to be the case. The sources of aluminum oxide presolar
grains can be determined based on their oxygen isotope ratios (Nittler et al.,
1997)—at least for those isolated grains found in chondrites, that started
with distinctly non-solar isotopic ratios and that never exchanged oxygen
isotopes with the solar nebula. Presolar grains with oxygen isotopes in Groups
1, 2 or 3 probably derive from AGB stars, while grains from Group 4—with
${}^{18}{\rm O}/{}^{17}{\rm O}>5.2$—probably derive from core-collapse
supernovae (Nittler et al., 2008, 2020b). At first only $\approx 10\%$ of
(Nguyen et al., 2007; Hynes & Gyngard, 2009) presolar oxide grains were
classified as coming from core-collapse supernovas, but as the spatial
resolution of mass spectrometers has improved and allowed detection of smaller
grains, and as magnesium isotope ratios have been measured more precisely,
mass fractions $\approx 20\%$ have been inferred (Hoppe et al., 2015; Liu et
al., 2022b). Similarly, Zega et al. (2011) analyzed five presolar hibonite
grains, and concluded that four had Group 1 or 2 oxygen isotopic compositions,
but one (KH2) was Group 4. Thus, about 80% of Solar System Al came from grains
condensed in AGB outflows. The remaining grains probably came from supernova
explosions (or Wolf-Rayet winds, also from massive stars $>20\,M_{\odot}$)
over Galactic history.

####  3.3.1 Al-bearing Presolar Grains, their Abundances, and their Sizes

We first consider the presolar carriers of Al and their sizes. As reviewed by
Zinner (2014), most extant presolar Al-bearing oxide grains are corundum
[${\rm Al}_{2}{\rm O}_{3}$] or spinel [${\rm Mg}{\rm Al}_{2}{\rm O}_{4}$],
with some also being hibonite [${\rm Ca}{\rm Al}_{12}{\rm O}_{19}$] and
chromium spinel [${\rm Mg}({\rm Cr,Al})_{2}{\rm O}_{4}$]. In the last decade,
presolar silicate grains have increasingly been recognized as the most
abundant type of presolar grain (Floss & Haenecour, 2016). Presolar silicate
grains themselves often contain Al, with the few measured to date showing
Al/Mg $\sim 0.01$ to 0.1 (Hoppe et al., 2021). Compared to the solar ratio
Al/Mg $=0.08$ (Lodders, 2003), this suggests that a non-negligible but poorly
known fraction of all Al is carried on presolar silicates. Taking an average
value 0.03 and comparing to the solar ratio Al/Mg $=0.08$, we estimate that
roughly 60% of Al in presolar grains resides in aluminum oxides, and 40%
resides in silicates. We assume similar proportions among grains from AGB
stars and massive stars.

Among presolar aluminum oxide grains, roughly half appear to be spinel, and
the other half corundum and hibonite, which are not easily distinguished
(Nguyen et al., 2007, 2010; Vollmer et al., 2009; Floss & Stadermann, 2009,
2012; Bose et al., 2010, 2012; Haenecour et al., 2018; Nittler et al., 2020a,
2008; Zega et al., 2011). The proportion that are hibonite is not well known,
but appears to be small: they make up one third of Al-oxide presolar grains in
the LL3.2 chondrite Krymka, but a much smaller fraction in other
unequilibrated ordinary chondrites (UOCs), and are absent in the H/L3.6
chondrite Tieschitz (Nittler et al., 2008). Because corundum readily
transforms to hibonite in the hot midplane region (§3.4), and the two minerals
have similar sizes and properties, we do not distinguish between them, and
refer to ‘corundum’ to mean a mix of mostly presolar corundum and some
hibonite. Again we assume the proportions of spinel and corundum/hibonite are
similar whether grains derive from AGB stars or supernovae/Wolf-Rayet stars.

The typical sizes of presolar corundum grains are a few microns (Makide et
al., 2011), and the typical sizes of presolar hibonite grains are several
hundred nm to several microns (Nittler et al., 2008; Zega et al., 2011). The
typical sizes of presolar spinel grains are $\approx 0.5\,\mu{\rm m}$ (e.g.,
Zinner et al., 2005). These sizes are characteristic of the $\approx 80\%$ of
grains that derive from AGB stars, but dust grains from the much faster
outflows of supernovae / Wolf-Rayet stars will be smaller. Theoretical
calculations (e.g., Nozawa et al., 2003) predict that corundum grains in
supernova ejects should grow no larger than about 10 nm in radius, and even
the largest silicate grains will not exceed 100 nm in radius ($0.2\,\mu{\rm
m}$ in size). Crystal domains also tend to be smaller (10-460 nm) in supernova
SiC grains than the $>500$ nm domain sizes in AGB SiC grains (Stroud et al.,
2004; Hynes, 2010), strongly indicating faster cooling and condensation in
supernova ejecta (Hynes, 2010). From meteoritic studies, higher proportions of
Group 4 (supernova) presolar oxides are found among grain size fractions below
a few hundred nm (Nguyen et al., 2007).

“Nanospinels” are presolar grains composed of Cr-rich spinel [$({\rm Mg},{\rm
Fe})({\rm Al},{\rm Cr})_{2}{\rm O}_{4}$] that are $<200$ nm in size (Liu et
al., 2022b). They are remarkable for having ${}^{54}{\rm Cr}/{}^{52}{\rm Cr}$
ratios $3.6\times$ solar ($\epsilon^{54}{\rm Cr}=+26,000$), and may be the
sole carriers of the $\epsilon^{54}{\rm Cr}$ positive anomalies in the Solar
System. Their isotopic anomalies are consistent with formation in a core-
collapse type II supernova, probably in its O/Ne or O/C zones (Dauphas et al.,
2010; Nittler et al., 2018). Dauphas et al. (2010) suggested they were
typically $<100$ nm in size but also noted they are concentrated into the
colloidal fractions of Murchison and Orgueil separates, indicating sizes
$<30-34$ nm. If the mass of nanospinels correlates with the $\epsilon^{54}{\rm
Cr}$ anomaly, the typical sizes of nanospinels would be in these smallest
particles, i.e., $<50$ nm.

From the considerations above, including comparing the sizes of these
nanospinels with most spinels, formed in AGB outflows, we make the
approximation that presolar grains formed in supernova ejecta or Wolf-Rayet
winds are a factor of 10 smaller than the similar minerals formed in AGB
outflows.

Once in the hot midplane region, presolar silicate and SiC grains will
vaporize, and the 75% or so of Al atoms in them will enter the gas phase.
These atoms will recondense as corundum or other grains, as described above
(§3.2). The sizes of these condensates will be likely be small; in other
contexts, condensation nuclei can be as small as tens of nm (Nozawa et al.,
2003). We will assume these condensates have typical radii $a_{\rm c}\sim 22$
nm, like the smallest presolar grains. Together these grains are a major
reservoir of Al and dominate the surface area of solids in the LAACHI-forming
region.

####  3.3.2 Live ${}^{26}{\rm Al}$ in Presolar Grains

Almost none of the grains carrying Al discussed above should have carried live
${}^{26}{\rm Al}$. Because the maximum plausible ratio in a grain formed in a
stellar outflow or explosion is $\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$}\sim
1$, decay of just 20 Myr is sufficient to reduce the ratio to
$\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$}<10^{-8}$ within the grain. The
majority ($\approx 80\%$) of presolar corundum and spinel grains come from AGB
stars, which are not at all associated with star-forming regions. The
probability of a molecular cloud being contaminated by a passing AGB star is
$\sim 10^{-6}-10^{-9}$ (Kastner & Myers, 1994; Ouellette et al., 2010), and it
can be safely assumed this did not occur. Those presolar grains in the Sun’s
molecular cloud from AGB stars therefore would have been from much earlier
generations of star formation, after having spent $10^{8}-10^{9}$ years in the
ISM (Jones et al., 1996; Hirashita et al., 2016; Heck et al., 2020). AGB
grains should contain essentially zero ${}^{26}{\rm Al}$. Similar arguments
would apply to grains from novas, or any stellar source taking more than tens
of Myr to evolve.

Live ${}^{26}{\rm Al}$ would be carried exclusively by grains from massive
stars. Unlike other stellar sources, core-collapse supernovae (and Wolf-Rayet
winds from their progenitors) are associated with star-forming regions, and
some may have promptly delivered grains to the Sun’s molecular cloud (Hester &
Desch, 2005; Pan et al., 2012; Desch et al., 2023c). Only a small fraction of
supernova grains would have been of recent origin, though: like grains from
AGB stars, most supernova grains would have been produced over Galactic
history, with most recording random ages also $\sim 10^{8}-10^{9}$ years. The
fraction of grains formed in supernovae or Wolf-Rayet stars that contained
live ${}^{26}{\rm Al}$ was probably $\approx(20\,{\rm Myr})/(10^{9}\,{\rm
yr})\sim 2\%$.

The initial ratios of those grains from recent, nearby massive stars was
probably $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}\sim 10^{-2}$. Zinner
(2014) compiled initial $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$
ratios for Group 4 oxide grains made of corundum, hibonite or spinel (Figure
15 of that paper). Corundum grains formed with $(\mbox{${}^{26}{\rm
Al}/{}^{27}{\rm Al}$})_{0}\approx 3\times 10^{-3}$ to $7\times 10^{-2}$, a
hibonite grain formed with $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm
Al}$})_{0}\approx 1\times 10^{-2}$, and spinel grains formed with
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}\approx 3\times 10^{-3}$ to
$5\times 10^{-2}$. There is also the example of an extremely large
($25\,\mu{\rm m}$) presolar SiC grain Bonanza of supernova origin, with Al-
bearing phases within it that apparently record $(\mbox{${}^{26}{\rm
Al}/{}^{27}{\rm Al}$})_{0}\sim 0.9$ (Gyngard et al., 2018). Our model does not
depend strongly on the exact value, so based on the above literature, we adopt
$\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$}\approx 0.0125$ ($250\times$
canonical) in the ${}^{26}{\rm Al}$-bearing presolar grains in the solar
nebula. This could reflect the value in the supernova ejecta, plus immediate
delivery to the Sun’s molecular cloud right before its collapse; or this could
represent an initial $\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$}\approx 0.9$ in
the ejecta of supernovae about 5 Myr before Solar System formation.

####  3.3.3 Summary

In Table 2 we list the types of Al-bearing presolar grains that would exist in
the solar nebula overall, their stellar sources, the percentage of Al atoms in
each phase, their typical sizes (diameters), and the typical initial
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$ values. It is assumed that
80% of Al atoms are in the large grains from AGB stars, and 20% from
supernovae and Wolf-Rayet stars. It is assumed that 40% of Al atoms are in
silicates / SiC grains, and 60% in aluminum oxides. of aluminum oxides, it is
assumed that equal amounts are in corundum / hibonite grains and in spinel
grains. In Table 3 we list the Al-bearing grains that would exist in the
LAACHI-forming region, their sources, the percentage of Al atoms in each
phase, their typical sizes, their midplane densities (scaled to gas density
$5.5\times 10^{-9}\,{\rm g}\,{\rm cm}^{-3}$), and their typical initial
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$ values. It is assumed that
the non-refractory grains vaporize in the LAACHI-forming region and recondense
as small particles; for simplicity we assume they have the same sizes as the
smallest grains, the nanospinels. We assume spinels grains are transformed to
corundum by loss of Mg with some reduction in volume.

Table 2:  Al-bearing presolar grains in the solar nebula.

Mineral | Source | % of all Al | Size | ${}^{26}{\rm Al}/{}^{27}{\rm Al}$
---|---|---|---|---
Silicate, SiC, etc. | AGBa | 32% | microns? | $\approx 0$
Cor / Hib | AGB | 24% | $\sim 2\,\mu{\rm m}$ | $\approx 0$
Spinel | AGB | 24% | $\sim 0.5\,\mu{\rm m}$ | $\approx 0$
Silicate, SiC, etc. | SN/WRb | 8% |  $\sim 0.1\,\mu{\rm m}$? | 98% $\approx 0$
|  |  |  | 2% $\approx 0.0125$
Cor / Hib | SN/WR | 6% | $\sim 0.2\,\mu{\rm m}$ | 98% $\approx 0$
|  |  |  | 2% $\sim 0.0125$
Spinel | SN/WR | 6% |  $\approx 50$ nm | 98% $\approx 0$
|  |  |  | 2% $\sim 0.0125$

Note. — $a$. Asymptotic Giant Branch star. $b$ Core-collapse supernova or
Wolf-Rayet star.

Table 3:  Al-bearing presolar grains in the LAACHI-forming region.

Mineral | Source | % of all Al | Size |  $n({\rm cm}^{-3})$ a | ${}^{26}{\rm Al}/{}^{27}{\rm Al}$
---|---|---|---|---|---
Cor / Hib | Condensates | 40% | 44 nm | 1540 | $5\times 10^{-5}$
Cor / Hib | AGBb | 24% | $2\,\mu{\rm m}$ | 0.0098 | $\approx 0$
Cor / Hib | AGB | 24% | $0.44\,\mu{\rm m}$ | 0.92 | $\approx 0$
Cor / Hib | SN/WRc | 6% | $0.2\,\mu{\rm m}$ | 2.5 | 98% $\approx 0$
|  |  |  |  | 2% $\approx 0.0125$
Cor / Hib | SN/WR | 6% | 44 nm | 231 | 98% $\approx 0$
|  |  |  |  | 2% $\approx 0.0125$

Note. — $a$ Particle density, assuming $\rho_{\rm g}=5.5\times 10^{-9}\,{\rm
g}\,{\rm cm}^{-3}$. $b$. Asymptotic Giant Branch star. $c$ Core-collapse
supernova or Wolf-Rayet star.

###  3.4 Diffusive Exchange of Oxygen Atoms

One potential objection to the idea that LAACHIs derive from presolar grains
is that they have oxygen isotopic ratios matching solar nebula reservoirs
(§2). For our hypothesis to be viable, presolar grains must have exchanged
most of their oxygen atoms with the gas, presumably while in the hot midplane
region. Given the typical excursions of oxygen isotopic ratios in presolar
grains ($\sim 10^{3}\,\mbox{\text{\textperthousand}}$) and the precision of
such measurements ($\sim 1\mbox{\text{\textperthousand}}$), roughly 99.9% of
all oxygen atoms in a grain must be exchanged.

The fraction of atoms exchanged by a spherical grain of radius $a$ in a time
$t$ is determined by the ratio $t/t_{\rm diff}$, where $t_{\rm
diff}=a^{2}/D(T)$, and $D(T)$ is the temperature-dependent diffusion
coefficient. Direct numerical integration of the diffusion equation (or
comparison with Equation 6.20 of Crank (1979)) shows that 99.9% of atoms will
exchange after a time $t>0.69\,t_{\rm diff}$. At a given temperature $T$, we
assume a diffusion coefficient governed by an Arrhenius relationship:
$D(T)=D_{0}\,\exp(-Q/RT)$. For lattice diffusion of O atoms through spinel, we
adopt $D_{0}=2.2\times 10^{-7}\,{\rm m}^{2}\,{\rm s}^{-1}$ and $Q=404\,{\rm
kJ}\,{\rm mol}^{-1}$ (Ryerson & McKeegan, 1994). For the gas constant
$R=8.3145\,{\rm kJ}\,{\rm mol}^{-1}$, $Q/R=48,600$ K. Diffusion of O atoms
through corundum has been reviewed by Heuer (2008) and Doremus (2006). We
adopt the value of Reddy (1979), as conveyed by Heuer (2008): $D_{0}=2.5\,{\rm
m}^{2}\,{\rm s}^{-1}$ and $Q=625\,{\rm kJ}\,{\rm mol}^{-1}$, and $Q/R=75,200$
K. To our knowledge, oxygen diffusion data do not exist for hibonites or other
Al-bearing phases.

While small, these diffusion rates are high enough to allow nearly complete
exchange of oxygen atoms with the gas in the hot midplane region. For
$a=0.25\,\mu{\rm m}$-radius spinel grains at 1350 K, $D=5.1\times
10^{-23}\,{\rm m}^{2}\,{\rm s}^{-1}$, $t_{\rm diff}=39$ years, and exchange
would have been 99.9% complete after 27 years. Nanospinel grains $\sim 25$ nm
in radius would have equilibrated much faster. Likewise, for corundum grains
at 1350 K, $D=1.6\times 10^{-24}\,{\rm m}^{2}\,{\rm s}^{-1}$, and at 1425 K,
$D=3.1\times 10^{-23}\,{\rm m}^{2}\,{\rm s}^{-1}$. For grains with radius
$a=1\,\mu{\rm m}$, the values of $t_{\rm diff}$ are $2.0\times 10^{4}$ yr and
$1.0\times 10^{3}$ yr, respectively. Isotopic exchange at these temperatures
would be 99.9% complete in 13,000 yr to 710 yr, and would be 90% complete in
5900 yr to 200 yr. Almost all grains at 0.6 AU would have spent hundreds of
years near the midplane before diffusing out of the region or being accreted,
and therefore would be largely ($>90\%$) equilibrated with the nebular gas in
their oxygen isotopes with the ${\rm H}_{2}{\rm O}$, CO, and silicate vapor in
the gas. These would have the oxygen isotopic signature near either that of
the Sun, at $\Delta^{17}{\rm O}\approx-29\mbox{\text{\textperthousand}}$
(McKeegan et al., 2011); or combined H2O vapor and CO gas, at $\Delta^{17}{\rm
O}\approx-35\mbox{\text{\textperthousand}}$ (Krot et al., 2010); or the
reservoir sampled by the majority of CAIs, $\Delta^{17}{\rm
O}\approx-23\mbox{\text{\textperthousand}}$ (Kööp et al., 2020).

###  3.5 Charging of Particles

By virtue of the fact that the region where PLACs and other LAACHIs formed
must have been very hot ($T>1350$ K), they inevitably would have been charged,
as quantified by Desch & Turner (2015). First, all potassium atoms (the most
easily ionized abundant atom) would have been in the gas phase, and several
percent of them ionized, close to as predicted by the Saha equation,
generating free electrons. Dust grains would have been very negatively charged
by the gas, due to adsorption of free electrons on their surfaces, balanced by
thermionic emission of electrons. Thermionic emission is a process whose rate
depends on the effective work function $W_{\rm eff}$ of the solid, as
$\exp(-W_{\rm eff}/kT)$, where $W_{\rm eff}=W+Ze^{2}/a$, $W$ is the work
function of the material, and $Ze$ the charge of the grain. The work function
is the energy required to remove an electron from the solid, and larger work
functions mean fewer electrons are ejected. The work function is formally
defined for conductors, but thermionic emission from insulators is also
governed by a work function-like term, via the Richardson equation. Because of
this dependence, grains made of materials with high work functions will be
negatively charged, and those with low work functions will be positively
charged.

Desch & Turner (2015) investigated cases nearly identical to the situation
considered here: a single population of dust grains with radius $a=1\,\mu{\rm
m}$, in a gas of density $n_{\rm H2}=1.0\times 10^{14}\,{\rm cm}^{-3}$ and
temperatures up to 1500 K. The solids-to-gas ratio was taken to be a canonical
$0.01$, but a ratio $10^{-4}$ was also considered (Figure 11 of Desch and
Turner 2015), and found not to affect grain charge or electron density above a
temperature of 1100 K. At high densities, gas-phase electrons come mostly from
thermionic emission, and are lost mostly to adsorption to grain surfaces, both
effects being proportional to the dust surface area, which therefore is not
something grain charges are sensitive to. In practice, Desch & Turner (2015)
found that grains made of materials with work functions $<4$ eV will be
positively charged, and those with work functions $>4$ eV will be negatively
charged.

The LAACHI-forming region considered here differs from the case considered by
Desch & Turner (2015) in that there are multiple populations of
corundum/hibonite grains with different radii. It is straightforward to show
that when exposed to the same hot gas and electron density, each population
will charge to the same voltage, and therefore the same $W_{\rm eff}$. In
practice this means the charge on each grain is proportional to its radius,
$a$.

Calculation of their exact charges depends on the work functions of corundum
and hibonite. As discussed by Desch & Turner (2015), a wide variety of
astrophysically relevant materials have work functions near 5 eV, including
quartz [${\rm SiO}_{2}$] and many other oxides, with $W\approx 5.0\,{\rm eV}$
(Fomenko, 1966), and lunar regolith simulant (mostly silicates), also near
$W=5.0\,{\rm eV}$ (Feuerbacher et al., 1972). Corundum’s work function is
estimated as 4.7 eV (Fomenko, 1966). Data do not exist for the work functions
of hibonite and spinel. We therefore adopt $W=4.7\,{\rm eV}$ for all these
minerals. As long as their work functions are at least 4.5 eV or so, the
grains are negatively charged and repel each other, and none of our results
are modified.

At typical temperatures in the hot midplane region of $T=1350\,{\rm K}$ and
$1425\,{\rm K}$, $a=1\,\mu{\rm m}$ grains would have been charged to $-280e$
and $-240e$, and voltages of about -0.35 V (Desch & Turner, 2015). The charges
on the $0.25\,\mu{\rm m}$ radius grains at these temperatures would be $-70e$
and $-60e$. The charges on the $0.025\,\mu{\rm m}$ former nanospinels would
have been $-7e$, and $-6e$. The results are summarized in Table 4.

Table 4: Work functions and charges of refractory grains possibly pertinent to
the LAACHI-forming region.

Material | Work Functiona | Chargeb
---|---|---
Perovskite (${\rm CaTiO}_{3}$) | 3.0 eV | $+400e$
Titanates (e.g., ${\rm SrTiO}_{3}$) | 3.1 eV | $+360e$
Hafnia (${\rm HfO}_{2}$) |  $\sim$ 3.6 eV | $+160e$
Zirconia (${\rm ZrO}_{2}$) |  $\sim$ 4.0 eV | $0e$
Refractory Metal (Mo,Ru) |  $\sim 4.5$ eV | $-200e$
Silicon Carbide (SiC) | 4.5 eV | $-200e$
Graphite (C) | 4.7 eV | $-240e$
Corundum (${\rm Al}_{2}{\rm O}_{3}$) | 4.7 eV | $-240e$
Hibonite (${\rm CaAl}_{12}{\rm O}_{19}$) |  $\sim 4.7$ eV | $-240e$

Note. — $a$. See text for details and references. $b$. For $T=1425$ K and
radius $a=1\,\mu{\rm m}$. For grains with different radii, charge is
multiplied by $(a/1\,\mu{\rm m})$.

###  3.6 Coagulation and Growth

In the hot, LAACHI-forming environment, grains would have collided and stuck
together, forming larger aggregates. For practical purposes, all the grains
were corundum or hibonite (with presumed identical properties), including
former spinel grains, with radii as listed in Table 2: $1.0\,\mu{\rm m}$,
$0.25\,\mu{\rm m}$, $0.1\,\mu{\rm m}$, $0.025\,\mu{\rm m}$, or $a_{\rm c}\sim
10$ nm. A ‘large’ corundum grain of radius $a_{1}$ would sweep up other grains
of radius $a_{2}$ at a rate $n_{2}\,\pi(a_{1}+a_{2})^{2}\,V_{12}\,S_{12}$,
where $n_{2}$ is the number density of other grains, $V_{12}$ the relative
velocity between the large grain and the other grains, and $S_{12}$ a factor
to account for sticking efficiency and electrical effects on the cross
sections. Here we quantify these terms, using the densities and properties
listed in Table 2. The densities are estimated assuming a midplane density
$\rho=5.5\times 10^{-9}\,{\rm g}\,{\rm cm}^{-3}$, so that $n_{\rm
H2}=\rho/(1.4m_{\rm H2})=1.2\times 10^{15}\,{\rm cm}^{-3}$, and the abundance
ratios of Lodders (2003), yielding a density of Al atoms $8.1\times
10^{9}\,{\rm cm}^{-3}$. These densities of course would scale linearly with
the gas density $\rho_{\rm g}$.

For dust surviving in the LAACHI-forming region, the opacity $\kappa$ at
$\lambda=2\,\mu{\rm m}$ (the Wien peak at 1500 K) associated with this
assemblage is

$\kappa\approx\left(\frac{\rho_{\rm d}}{\rho_{\rm
g}}\right)\,\frac{3}{4\rho_{\rm cor}}\,\frac{Q_{\rm ext}}{a}\approx 0.1\,{\rm
cm}^{2}\,{\rm g}^{-1},$

where the dust-to-gas ratio $\rho_{\rm d}/\rho_{\rm g}\approx 1.26\times
10^{-4}$ if all the Al is in corundum grains, assuming solar abundances
(Lodders, 2003), and $Q_{\rm ext}/a\approx 4000\,{\rm cm}^{-1}$ for alumina
particles formed by combustion (Koike et al., 1995). This opacity is
consistent with the vertical thermal structure described in §3.1.

The relative velocities between two grains 1 and 2 are on the order of tens of
cm/s, and obey the relation

$V_{12}^{2}=V_{\rm B,1}^{2}+V_{\rm B,2}^{2}+V_{\rm T,12}^{2},$ (1)

where $V_{\rm B}=(kT/m)^{1/2}$ is the Brownian velocity of grains with mass
$m$, and $V_{\rm T}$ is a velocity difference due to turbulence (Xiang et al.,
2020). The velocities of former nanospinel grains are dominated by Brownian
velocities $V_{\rm B}\approx 27\,\left(T/1400\,{\rm
K}\right)^{1/2}\,\left(a/0.025\,\mu{\rm m}\right)^{-3/2}\,{\rm cm}\,{\rm
s}^{-1}$, but Brownian motion is negligible for larger particles. The velocity
difference between two particles 1 and 2 ($a_{1}>a_{2}$) is given by Ormel &
Cuzzi (2007) as

$V_{\rm T,12}=V_{\rm g}\,{\rm Re}^{1/4}\,\left({\rm St}_{1}-{\rm
St}_{2}\right)$ (2)

in the limit that ${\rm St}_{1}<{\rm Re}^{-1/2}$, where the Stokes number of
particle 1 is ${\rm St}_{1}=\Omega\,t_{\rm stop,1}$ and its stopping time is
$t_{\rm stop,1}=\rho_{\rm cor}a_{1}/(\rho_{\rm g}C)$, where
$C=(kT/\bar{m})^{1/2}$. The velocity of the largest-scale eddies is $V_{\rm
g}=(3\alpha/2)^{1/2}C$, which would have been $8.7\times 10^{3}\,{\rm
cm}\,{\rm s}^{-1}$ in our model. The Reynolds number is ${\rm Re}=\alpha
C^{2}\Omega^{-1}/\nu_{\rm m}$, where $\nu_{\rm m}\approx 4\times 10^{3}\,{\rm
cm}^{2}\,{\rm s}^{-1}$ is the molecular viscosity of an ${\rm H}_{2}$-He gas
at 1350-1425 K (see Desch, 2007), yielding ${\rm Re}=4-7\times 10^{10}$. For
conditions at the midplane, this formula for the relative velocity is obeyed
by particles up to about 100 microns in size, and the relative velocity tends
to approach $0.38\,(a_{1}-a_{2})\,{\rm cm}\,{\rm s}^{-1}$ if $a_{1}$ and
$a_{2}$ are measured in microns. For larger particles, with ${\rm St}>{\rm
Re}^{-1/2}$ (and $a_{1}\gg a_{2}$), the relative velocity is

$V_{\rm T,12}=1.73\,V_{\rm g}\,{\rm St}_{1}^{1/2}.$ (3)

(Ormel & Cuzzi, 2007). Typical relative velocities would have been tens to
hundreds of cm/s, and would have scaled with density $\rho$ and turbulence
level $\alpha$ as $\alpha^{3/4}\,\rho^{-3/4}$.

The efficiency of accretion must also be multiplied by a factor
$S_{12}=(1-{\rm PE}/{\rm KE})$, where KE is the kinetic energy and PE the
electrical potential energy, to account for the electrical repulsion or
attraction between two particles (Xiang et al., 2020). This term is exactly
analogous to the gravitational focusing factor $[1+V_{\rm esc}^{2}/V_{\rm
rel}^{2}]$ that increases the collision cross section for two masses; except
that unlike gravity, for which PE is always negative and the term $1-{\rm
PE}/{\rm KE}>1$, enhancing the cross section, here PE is negative (enhancing
the cross section) only for oppositely charged particles, and PE is positive
(reducing the cross section) for similarly charged particles. It is understood
that this term vanishes if ${\rm KE}<{\rm PE}$, i.e., if particles lack
sufficient kinetic energy to overcome their mutual repulsion. As in the case
of gravitational focusing, we can write this term as $(1-V_{\rm
crit}^{2}/V_{12}^{2})$, where $V_{\rm crit}$ plays the role of escape
velocity. For particles of identical corundum composition,

$V_{\rm crit}=\left(\frac{3Z_{1}Z_{2}\,e^{2}}{2\pi\rho_{\rm
cor}}\right)^{1/2}\,\left(\frac{a_{1}^{3}+a_{2}^{3}}{a_{1}+a_{2}}\right)^{1/2}\,\frac{1}{a_{1}^{3/2}a_{2}^{3/2}},$
(4)

or $V_{\rm crit}=8.3\,$
$\left[(a_{1}^{3}+a_{2}^{3})/(a_{1}+a_{2})\right]^{1/2}/(a_{1}a_{2})\,\times$
$\left[\left|\bar{Z}\right|/500\right]\,$ ${\rm cm}\,{\rm s}^{-1}$, where
$a_{1}$ and $a_{2}$ are measured in microns, and $\bar{Z}e$ is the charge on a
$1\,\mu{\rm m}$-radius grain. If $a_{1}\gg a_{2}$, then this expression
reduces to $V_{\rm crit}=(8.3/a_{2})\,(\bar{Z}/500)\,{\rm cm}\,{\rm s}^{-1}$,
independent of the size of the larger particle. Large particles must have
velocities exceeding tens of cm/s in order to sweep up ${}^{26}{\rm
Al}$-bearing nanospinels. It is because this critical relative velocity is
comparable to the expected relative velocities that accretion may be inhibited
by electric charge.

Table 5: Expected ($V_{12}$) and critical ($V_{\rm crit}$) relative velocities (cm/s) of pairs of corundum particles, at $z=1.18H$, where $T=1350$ K. Bold type denotes cases where $V_{12}>V_{\rm crit}$ and grains can accrete. | $a_{2}=1\,\mu{\rm m}$ | $a_{2}=0.22\,\mu{\rm m}$ | $a_{2}=22\,{\rm nm}$
---|---|---|---
$a_{1}=1000\,\mu{\rm m}$ | $V_{12}={\bf 193}$ | $V_{12}={\bf 193}$ | $V_{12}=195$
| $V_{\rm crit}={\bf 4.7}$ | $V_{\rm crit}={\bf 21.1}$ | $V_{\rm crit}=211$
$a_{1}=250\,\mu{\rm m}$ | $V_{12}={\bf 96.3}$ | $V_{12}={\bf 96.3}$ | $V_{12}=102$
| $V_{\rm crit}={\bf 4.6}$ | $V_{\rm crit}={\bf 21.1}$ | $V_{\rm crit}=211$
$a_{1}=100\,\mu{\rm m}$ | $V_{12}={\bf 60.9}$ | $V_{12}={\bf 60.9}$ | $V_{12}=68.9$
| $V_{\rm crit}={\bf 4.6}$ | $V_{\rm crit}={\bf 21.1}$ | $V_{\rm crit}=211$
$a_{1}=25\,\mu{\rm m}$ | $V_{12}={\bf 15.3}$ | $V_{12}=15.8$ | $V_{12}=36.0$
| $V_{\rm crit}={\bf 4.6}$ | $V_{\rm crit}=21.0$ | $V_{\rm crit}={211}$
$a_{1}=10\,\mu{\rm m}$ | $V_{12}={\bf 5.7}$ | $V_{12}=6.3$ | $V_{12}=32.9$
| $V_{\rm crit}={\bf 4.4}$ | $V_{\rm crit}=20.9$ | $V_{\rm crit}=211$
$a_{1}=2.5\,\mu{\rm m}$ | $V_{12}=1.0$ | $V_{12}=1.8$ | $V_{12}=32.4$
| $V_{\rm crit}=4.1$ | $V_{\rm crit}=20.3$ | $V_{\rm crit}=210$
$a_{1}=1\,\mu{\rm m}$ | $V_{12}=0.15$ | $V_{12}=1.1$ | $V_{12}=32.3$
| $V_{\rm crit}=4.7$ | $V_{\rm crit}=19.2$ | $V_{\rm crit}=209$
$a_{1}=0.22\,\mu{\rm m}$ | $V_{12}=1.1$ | $V_{12}=1.5$ | $V_{12}=32.3$
| $V_{\rm crit}=19.2$ | $V_{\rm crit}=21.1$ | $V_{\rm crit}=202$
$a_{1}=22\,{\rm nm}$ | $V_{12}=32.3$ | $V_{12}=32.3$ | $V_{12}=45.7$
| $V_{\rm crit}=209$ | $V_{\rm crit}=202$ | $V_{\rm crit}=211$
Table 6: Expected ($V_{12}$) and critical ($V_{\rm crit}$) relative velocities (cm/s) of pairs of corundum particles, at $z=0H$, where $T=1425$ K. Bold type denotes cases where $V_{12}>V_{\rm crit}$ and grains can accrete. | $a_{2}=1\,\mu{\rm m}$ | $a_{2}=0.25\,\mu{\rm m}$ | $a_{2}=25\,{\rm nm}$
---|---|---|---
$a_{1}=1000\,\mu{\rm m}$ | $V_{12}={\bf 138}$ | $V_{12}={\bf 138}$ | $V_{12}=142$
| $V_{\rm crit}={\bf 4.0}$ | $V_{\rm crit}={\bf 18.1}$ | $V_{\rm crit}=181$
$a_{1}=250\,\mu{\rm m}$ | $V_{12}={\bf 68.9}$ | $V_{12}={\bf 68.9}$ | $V_{12}=76.5$
| $V_{\rm crit}={\bf 4.0}$ | $V_{\rm crit}={\bf 18.1}$ | $V_{\rm crit}=181$
$a_{1}=100\,\mu{\rm m}$ | $V_{12}={\bf 37.6}$ | $V_{12}={\bf 37.9}$ | $V_{12}=50.4$
| $V_{\rm crit}={\bf 4.0}$ | $V_{\rm crit}={\bf 18.1}$ | $V_{\rm crit}=181$
$a_{1}=25\,\mu{\rm m}$ | $V_{12}={\bf 9.1}$ | $V_{12}=9.5$ | $V_{12}=34.5$
| $V_{\rm crit}={\bf 3.9}$ | $V_{\rm crit}=18.0$ | $V_{\rm crit}=181$
$a_{1}=10\,\mu{\rm m}$ | $V_{12}=3.4$ | $V_{12}=3.9$ | $V_{12}=33.4$
| $V_{\rm crit}=3.8$ | $V_{\rm crit}=17.9$ | $V_{\rm crit}=181$
$a_{1}=2.5\,\mu{\rm m}$ | $V_{12}=0.6$ | $V_{12}=1.4$ | $V_{12}=33.2$
| $V_{\rm crit}=3.5$ | $V_{\rm crit}=17.4$ | $V_{\rm crit}=180$
$a_{1}=1\,\mu{\rm m}$ | $V_{12}=0.2$ | $V_{12}=1.1$ | $V_{12}=33.2$
| $V_{\rm crit}=4.0$ | $V_{\rm crit}=16.5$ | $V_{\rm crit}=179$
$a_{1}=0.22\,\mu{\rm m}$ | $V_{12}=1.1$ | $V_{12}=1.5$ | $V_{12}=33.2$
| $V_{\rm crit}=16.5$ | $V_{\rm crit}=18.1$ | $V_{\rm crit}=173$
$a_{1}=22\,{\rm nm}$ | $V_{12}=33.2$ | $V_{12}=33.2$ | $V_{12}=47.0$
| $V_{\rm crit}=179$ | $V_{\rm crit}=173$ | $V_{\rm crit}=181$

In Table 5 we list the relative velocities, $V_{12}$ and critical velocities
for collision, $V_{\rm crit}$, between pairs of particles in our model, based
on conditions at the $z=1.18\,H$ surface ($\rho_{\rm g}=2.7\times
10^{-9}\,{\rm g}\,{\rm cm}^{-3}$, $T=1350\,{\rm K}$). In Table 6 we list the
same information for conditions at the $z=0$ midplane ($\rho_{\rm g}=5.5\times
10^{-9}\,{\rm g}\,{\rm cm}^{-3}$, $T=1425\,{\rm K}$). Particles are assumed to
be corundum, with work functions 4.7 eV. Several trends are evident.

First, throughout the hot midplane region, even particles up to several
microns in radius would have been incapable of overcoming their mutual
electrical repulsion to collide with each other. In contrast, ‘seed’ particles
at least $10\,\mu{\rm m}$ in radius could sweep up other micron-sized
particles (but not $0.25\,\mu{\rm m}$ particles, unless the seed particles
were hundreds of microns in size). This is because in turbulent eddies, larger
particles have greater velocities relative to other particles. Exceptionally
large presolar grains might have served as seed particles: some of the largest
presolar hibonite grains studied are several microns in size (Zega et al.,
2011), and other presolar grains approach $20\,\mu{\rm m}$ in size (Zinner,
2014). A more likely source of seed particles is fragments of previously
formed, large, PLAC-like LAACHIs. One object that had previously grown to
radius $1\,{\rm mm}$ and then fragmented could produce up to $10^{6}$
fragments with radius $10\,\mu{\rm m}$, which could then each accrete
$1\,\mu{\rm m}$-radius grains and grow up to $1\,{\rm mm}$ in radius
themselves, before either fragmenting or diffusing out of the region.

Another trend evident from Tables 5 and 6 is that not even large particles
could have accreted the small, ${}^{26}{\rm Al}$-bearing former nanospinels,
unless they were $>1200\,\mu{\rm m}$ in radius (at $z=1.18H$) or
$>1700\,\mu{\rm m}$ in radius (at $z\\!\\!=\\!\\!0$). No LAACHIs less than
about 2 or 3 mm in size would have accreted any ${}^{26}{\rm Al}$, at least
for the model parameters explored here.

Assuming perfect sticking efficiencies for accreted grains, the rate at which
a large (seed) grain with radius $a$ gained mass by accreting $1\,\mu{\rm m}$
particles at $z=0H$ would have been

$\frac{d}{dt}\left(\frac{4\pi}{3}\rho_{\rm
cor}a^{3}\right)=\frac{4\pi}{3}\rho_{\rm cor}(1\,\mu{\rm m})^{3}\,\times$
$n_{2}\,\pi\left(a+1\,\mu{\rm m}\right)^{2}\,\left(0.38\,{\rm cm}\,{\rm
s}^{-1}\right)\,\left(\frac{a}{1\mu{\rm m}}-1\right),$ (5)

where $n_{2}=1.0\times 10^{-2}\,{\rm cm}^{-3}$ refers to the density of the
$1\,\mu{\rm m}$-radius corundum grains only. Defining $y=a/(1\,\mu{\rm m})$,
we can rewrite this as

$\frac{dy}{dt}=\frac{1}{t_{\rm
grow}}\,\left(y+1\right)\,\left(1-\frac{1}{y^{2}}\right)\approx\frac{y}{t_{\rm
grow}},$ (6)

where

$t_{\rm grow}=810\,\left(\frac{\alpha}{1\times
10^{-3}}\right)^{-3/4}\,\left(\frac{\rho}{5.5\times 10^{-9}\,{\rm g}\,{\rm
cm}^{-3}}\right)^{-1/4}\,{\rm yr}.$ (7)

The growth timescale would be only $\sim 10\%$ longer at $z=1.18\,H$. The
solution to Equation 6 approaches $y(t)=y(0)\,\exp(+t/t_{\rm grow})$ in the
limit that $y(0)\gg 1$. Because of the exponential nature of the growth, the
time to grow from a starting radius $a=10\,\mu{\rm m}$, to a particle 1000
times more massive, with $a=100\,\mu{\rm m}$, would have been about 1860
years; and the time taken to grow to $a=1000\,\mu{\rm m}$ would have been no
more than an additional 1860 years, and possibly no more than an additional
930 years, as larger particles can sweep up the smaller grains in the LAACHI-
forming region.

These growth times of thousands of years are comparable to the time spent by
particles in the hot midplane region. A distribution of particles all starting
at $z\\!\\!=\\!\\!0$, would have density proportional to $\exp(-z^{2}/(4Dt))$
after a time $t$; the fraction still at $z<1.18\,H$ and remaining in the hot
midplane region would be ${\rm erf}((1.18H)/\sqrt{(}4Dt))$ $={\rm
erf}(\sqrt{30.0\,{\rm yr}/t})$. Above we defined the typical time in the hot
region to be $(1.18H)^{2}/D\approx 120\,{\rm yr}$, and indeed the fraction of
particles that would have remained after 120 years would have been ${\rm
erf}(0.50)=52\%$. The fraction of particles that escape to cooler regions of
the disk, where they might escape further reprocessing, is ${\rm erfc}(\Delta
r/\sqrt{4Dt})/2$ $={\rm erfc}(\sqrt{255\,{\rm yr}/t})/2$, where $\Delta
r\approx 0.2$ AU (§3.1). After 1000 years, about 25% of all inclusions in the
LAACHI-forming region will have diffused out beyond 0.8 AU. The fraction of
particles that diffuse to cooler regions, while remaining in the hot midplane
region, is the product of these, which peaks at $\approx 5\%$ between 1000 and
2000 years. In this timespan, particles can grow by factors of 3 to 10 in
radius (i.e., from $10\,\mu{\rm m}$ to $30-100\,\mu{\rm m}$) and factors of 30
to $10^{3}$ in mass. These sizes and proportions appear consistent with
observations of PLACs and corundum grains.

In principle, some LAACHIs could have remained in the hot midplane region for
longer and grown to much larger sizes; but several mechanisms work against
growth to scales larger than about 1 mm. The most important one is
fragmentation: as particles grow to sizes larger than about 1 mm in radius,
their velocities relative to other particles exceed $140\,{\rm cm}\,{\rm
s}^{-1}$. This begins to exceed the fragmentation threshold velocity,
estimated from laboratory experiments to be about $1\,{\rm m}\,{\rm s}^{-1}$
(Güttler et al., 2010). Indeed, many PLACs appear to be fragments of larger
objects hundreds of microns in size (§2.3.3; Kööp et al., 2016a). The survival
of the singularly large inclusion HAL may owe to its apparently melted nature.
To form its three large hibonite laths, it would seem necessary to melt and
recrystallize this aggregate somehow; this would be consistent with the known
isotopic fractionation of HAL and the presence of glass (Fahey et al., 1987b).
Other inclusions, not quite as large as HAL (e.g., M98-8, 31-2) do not appear
melted; we conjecture that had they been larger they would have fragmented.

Large objects the size of HAL also may be removed effectively from the LAACHI-
forming region. Objects will settle to the midplane if ${\rm St}>0.2\alpha$
(Desch et al., 2018); for this environment that means radius $>2$ mm. Once
there, they might be removed efficiently by meridional flow (Desch et al.,
2018).

###  3.7 Incorporation of ${}^{26}{\rm Al}$ from the Gas

With the growth of the particles quantitatively described, it is possible to
estimate the amount of ${}^{26}{\rm Al}$ they would incorporate from the gas.
In the region where $T=1425\,{\rm K}$ and $\rho_{\rm gas}=5.5\times
10^{-9}\,{\rm g}\,{\rm cm}^{-3}$, only a fraction $\sim 1.6\times 10^{-6}$ of
all Al atoms will be in the gas phase in equilibrium. The number density of
gas-phase Al atoms will be $n_{\rm Al}=1.3\times 10^{4}\,{\rm cm}^{-3}$. The
${}^{26}{\rm Al}/{}^{27}{\rm Al}$ ratio of this gas is likely to be close to
the solar system average, ${\cal R}=(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm
Al}$})\approx$ ${\cal R}_{\rm SS}=5\times 10^{-5}$, as it comprises Al from a
variety of presolar grains, including perhaps silicate and SiC grains from a
recent, nearby supernova or Wolf-Rayet star, but mostly other presolar grains
from AGB stars that have no live ${}^{26}{\rm Al}$. Particles will acquire
${}^{26}{\rm Al}$ atoms from the gas at a rate proportional to their surface
area:

$\frac{d}{dt}\left[{}^{26}{\rm Al}\right]=\,4\pi a^{2}\,n_{\rm Al}\,{\cal
R}\,\frac{1}{4}\left(\frac{8kT}{\pi m_{\rm Al}}\right)^{1/2}\,S,$ (8)

where $S\approx 0.05$ is the sticking coefficient (Takigawa et al., 2015).
Assuming $a(t)=a_{0}\exp\left(+t/t_{\rm grow}\right)$, the solution to
Equation 6, we find the final number of ${}^{26}{\rm Al}$ atoms in a particle
of final radius $a$ is

${}^{26}{\rm Al}=\frac{1}{2}\left(a^{2}-a_{0}^{2}\right)\,n_{\rm Al}\,{\cal
R}\,\frac{1}{4}\left(\frac{8kT}{\pi m_{\rm Al}}\right)^{1/2}\,S\,t_{\rm
grow}.$ (9)

The total number of ${}^{27}{\rm Al}$ atoms in the particle is

${}^{27}{\rm Al}=\frac{4\pi}{3}a^{3}\,\frac{\rho_{\rm cor}}{51m_{\rm p}},$
(10)

where $51m_{\rm p}$ is the mass of ${\rm Al}_{2}{\rm O}_{3}$ per Al atom).
After growing to a size $a$, the average $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm
Al}$})$ in a particle is

$\frac{{}^{26}{\rm Al}}{{}^{27}{\rm
Al}}=\frac{3}{2}\,\left(\frac{a^{2}-a_{0}^{2}}{a^{3}}\right)\,\frac{51m_{\rm
p}}{\rho_{\rm cor}}\,n_{\rm Al}\,{\cal R}\,\frac{1}{4}\left(\frac{8kT}{\pi
m_{\rm Al}}\right)^{1/2}\,S\,t_{\rm grow}.$ (11)

The quantity in parentheses has a maximum of $0.4/a_{0}$ at a radius
$a=1.7a_{0}$, and decreases as $1/a$ at large $a$. Because no particle spends
all its time at the midplane, the average value of $n_{\rm Al}$ is lower than
the midplane value; integrating over $z$, we find the average value is lower
by a factor of 0.45. Assuming this, and adopting $t_{\rm grow}=810\,{\rm yr}$
and $S\approx 0.05$ from above, yields

$\frac{{}^{26}{\rm Al}}{{}^{27}{\rm Al}}\approx 0.06\,{\cal
R}\,\left(\frac{a}{1\,\mu{\rm m}}\right)^{-1}.$ (12)

Particles like HAL with an average radius $\approx 70\,\mu{\rm m}$, would have
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}\approx 0.002\,{\cal R}$, or
about $4\times 10^{-8}$.

More sophisticated modeling should be undertaken to explore the diversity of
particle histories, but we conclude from this simple treatment that growth of
aggregates of corundum / hibonite, to tens or hundreds of microns, is
predicted to take place during the thousands of years or so particles spent in
this environment, and that growth to these sizes would not generally have
included live ${}^{26}{\rm Al}$. Electrical charge has been considered in
previous models of coagulation and growth (e.g., Okuzumi, 2009; Xiang et al.,
2020; Akimkin et al., 2020), but to our knowledge, it has not been
contemplated how this would lead to chemical or isotopic fractionations.

###  3.8 Subsequent Fate

After diffusing out of the hot midplane region, newly-formed LAACHIs (PLACs,
corundum grains, etc.) would find themselves in cooler environments where
other phases are thermodynamically favored, as described in §3.4.

One possible outcome is that reaction with the gas could transform hibonite or
corundum to the thermodynamically favored phases spinel, melilite, anorthite
or diopside, by addition of Mg, Si, and Ca. This is probably limited, though,
as diffusion of these cations into the minerals could have been kinetically
inhibited at lower temperatures. The inclusion A-COR-01 has a rim of diopside
about $5-10\,\mu{\rm m}$ thick (Bodénan et al., 2020). Assuming the rate-
limiting step is diffusion of Ca into corundum, and estimating $D_{\rm
Ca}\approx 1.5\times 10^{-21}\,{\rm m}^{2}\,{\rm s}^{-1}$ at 1350 K (§3.4),
implies a rim of thickness $\approx 6\,\mu{\rm m}$ would grow in $\approx
700\,{\rm yr}$, roughly the time it would take to diffuse radially to
conditions 50 K cooler.

Another possible outcome is the accretion of spinel produced at lower
temperatures elsewhere in the disk. In adjacent regions of the disk with
slightly lower temperatures (e.g., $\approx 1200\,{\rm K}$, beyond about 0.8
AU), such that olivine and pyroxene would have been stable, the density of
solids would be higher by a factor $\sim 10^{2}$ relative to the LAACHI-
forming region. While grains would have had similar charges, and the relative
velocities would have been similar, a key difference is that silicate
particles could have condensed and quickly ($\sim 10^{4}$ years) grown to
sizes $>1\,{\rm mm}$ capable of accreting small, ${}^{26}{\rm Al}$-bearing
grains. We do not model these regions, but consider it likely that spinel or
other minerals formed in these regions would have contained live ${}^{26}{\rm
Al}$. Agglomerations of spinel and hibonite would resemble SHIBs, with near-
canonical ${}^{26}{\rm Al}/{}^{27}{\rm Al}$ ratios. In general, as LAACHIs
interact with material outside the hot midplane region where they formed,
there would be opportunities for them to incorporate ${}^{26}{\rm Al}$-bearing
material.

##  4 Discussion

The detailed model presented here provides a framework for interpreting the
observations made of PLACs and other LAACHIs, to infer their origins and to
make predictions about other potentially observable features. One potential
application is the understanding of Rare Earth Element (REE) patterns. Many
PLACs are depleted in Yb and Eu (Ireland, 1990), with condensation
temperatures of 1475 K and 1347 K (Lodders, 2003). Eu, at least, would not
fully condense in the LAACHI-forming region. However, an understanding of the
REE patterns in PLACs and other LAACHIs lays beyond the scope of this paper.
Here we discuss the implications for refractory metal abundances, stable
isotope anomalies, short-lived radionuclides, and detailed mineralogy of
LAACHIs. These can be used to test the hypothesis presented here.

###  4.1 Refractory Metal

Almost as important to the question of what PLACs and other LAACHIs grew from
is what they did not include during their growth. Although corundum and
hibonite are among the most refractory mineral phases to condense from a solar
composition gas, there are other, rarer, minerals that could survive at even
higher temperatures. Besides perovskite, zirconia [${\rm ZrO}_{2}$] and hafnia
[${\rm HfO}_{2}$] are stable (at $10^{-4}$ bar) up to 1758 K and 1694 K
(Lodders, 2003). Refractory metal alloys of Mo, Ru, Rh, W, Re, Os, Ir and Pt
condense at temperatures between 1403 K for Pt, and 1817 K for Re (Lodders,
2003). Refractory metal nuggets (RMNs) are a phase rich in these elements;
although it has been argued that these primarily form by precipitation from
CAI melts (Schwander et al., 2015), a subset of RMNs appear more consistent
with condensation, and certainly a small fraction of RMNs appear to be of
presolar origin (Schwander et al., 2015; Daly et al., 2017). Therefore it is
reasonable to assume that at least some refractory metal grains would have
existed in the LAACHI-forming region. Despite this, and although refractory
metal is commonly found in SHIBs, it is relatively rare in PLACs (Schwander et
al., 2015).

In the context of the model presented here, refractory metal grains would not
have been incorporated into PLACs. The work functions of the refractory metals
above range from 4.3 eV for Mo and 4.5 eV for W, and 4.6 eV for Ru, to 5.3 eV
for Ir and Pt (Fomenko, 1966). Grains made of refractory metals would have
been micron-sized and negatively charged in the LAACHI-forming environment,
although not quite as much as corundum and hibonite grains (being dominated by
Mo and Ru, with typical work function $\sim 4.5$ eV). Their electrical
repulsion would have excluded them from the growing PLACs, same as the
${}^{26}{\rm Al}$-bearing corundum grains. Interestingly, the work functions
of ${\rm HfO}_{2}$ and ${\rm ZrO}_{2}$ are roughly 3.6 eV and 3.8 eV,
respectively (Fomenko, 1966). If bare grains of these materials existed, they
would be neutral to positively charged, and effectively accreted, although
they would be exceedingly rare.

###  4.2 Stable Isotope Anomalies

####  4.2.1 Oxygen

A potential argument against a mostly presolar origin for the Al in PLACs and
other LAACHIs is that their oxygen isotopes do not reflect the wide variety
exhibited by presolar grains (e.g., Zinner, 2014). As outlined in §3.3,
diffusive exchange of oxygen isotopes with the nebula is rapid in regions that
reached 1350 K: roughly 25 years for $0.25\,\mu{\rm m}$-radius spinel grains
to exchange 99.9% of O atoms; and 3500 years for $1\,\mu{\rm m}$-radius
corundum grains, to exchange 90% of O atoms. At 1425 K, a $1\,\mu{\rm m}$
radius corundum grain exchanges 90% of oxygen in 190 years. These are
comparable to or shorter than the typical time a PLAC would have spent in the
hot midplane region, roughly hundreds of years. There is a strong sensitivity
to temperature, however, and corundum grains would take thousands of years to
equilibrate at 1325 K. Any PLACs that formed mostly from grains that didn’t
heat above 1325 K might not have fully exchanged with the solar nebula gas at
$\Delta^{17}{\rm O}\approx-35\mbox{\text{\textperthousand}}$ to
$-23\mbox{\text{\textperthousand}}$, and might appear mixed with a presolar
composition that presumably was near $\Delta^{17}{\rm O}\approx
0\mbox{\text{\textperthousand}}$, with variable $\delta^{18}{\rm O}$ (Krot et
al., 2010). In fact, many PLACs have $\Delta^{17}{\rm O}$ extending up to
$\approx-17\mbox{\text{\textperthousand}}$, and those that do show a slightly
greater spread in $\delta^{18}{\rm O}$ (Kööp et al., 2016a). We conclude that
in many PLACs ($\Delta^{17}{\rm O}<-23\mbox{\text{\textperthousand}}$), oxygen
isotope exchange with the solar nebula gas was very ($>99.9\%$) complete, but
that in some ($\Delta^{17}{\rm O}>-23\mbox{\text{\textperthousand}}$),
isotopic exchange perhaps was not quite as complete. This is consistent with
the predicted timescales for oxygen isotopic equilibration in the LAACHI-
forming environment.

####  4.2.2 Magnesium

For the most part, the Mg isotopic compositions of PLACs and related
inclusions do not appear different from solar (Liu, 2008). In some inclusions,
analysis spots show excesses in ${}^{26}{\rm Mg}/{}^{24}{\rm Mg}$, and
isochrons can be constructed, revealing the decay of ${}^{26}{\rm Al}$. This
is not true in most of the inclusions, and in fact analysis spots in many
PLACs show deficits of $3-4\mbox{\text{\textperthousand}}$ in the ${}^{26}{\rm
Mg}/{}^{24}{\rm Mg}$ ratio relative to the chondritic value (Ireland, 1988;
Liu, 2008, and references therein). This observation has been difficult to
explain, but our model predicts such a nucleosynthetic effect.

In the context of our model, PLACs and other LAACHIs mostly would have
exchanged Mg isotopes with the abundant Mg vapor in the gas, with a solar
composition; but some might have retained Mg from the minor amounts in the Al-
bearing phases. These would not have been spinel or corundum: presolar
corundum grains contain very little Mg, and most Mg would have diffused out of
spinel as it converted to corundum. Any Mg remaining in these grains would
have equilibrated with solar Mg. In contrast, Mg can exist as a minor element
in hibonite, which contains structural Mg in the form of coupled substitution
of ${\rm Mg}^{2+}$ and ${\rm Ti}^{4+}$ for two ${\rm Al}^{3+}$ atoms. As a
tetravalent cation, ${\rm Ti}^{4+}$ may not diffuse out of hibonite easily,
and this may allow hibonite to retain the paired ${\rm Mg}$. PLACs would have
mostly exchanged Mg with the gas and isotopically equilibrated; but to the
extent they retain a presolar signature, it would be from the Mg in presolar
hibonite grains.

Among presolar oxide grains of spinel and corundum, deficits in ${}^{26}{\rm
Mg}/{}^{24}{\rm Mg}$ are found almost exclusively on Group 1 and Group 3
grains. Similarly, the only measured Group 3 presolar hibonite grain exhibited
a deficit of $-28\mbox{\text{\textperthousand}}$ (Nittler et al., 2008). The
majority of presolar aluminum oxide grains, those from AGB stars, appear to
have deficits of ${}^{26}{\rm Mg}$. In contrast, supernova-derived Group 4
grains of spinel, hibonite, and corundum show $\delta^{26}{\rm Mg}$ excesses
(from the presolar grain database, http://presolar.physics.wustl.edu; Hynes &
Gyngard, 2009). As only a small fraction of Mg existed in solid form in the
LAACHI-forming environment, it is difficult to make quantitative predictions
of the deviation of Mg isotopic compositions relative to bulk Solar System or
terrestrial standards; but of the Mg in the region, LAACHIs would have
preferentially accreted Mg from the low-$\delta^{26}{\rm Mg}$ Group 1 and 3
spinel and corundum grains and Group 3 hibonite grains, and excluded the
high-$\delta^{26}{\rm Mg}$ Group 4 nanospinels. This potentially explains the
deficits in ${}^{26}{\rm Mg}$ of $\approx-3\mbox{\text{\textperthousand}}$, if
roughly 10% of their Mg retained the signature of presolar hibonites.

####  4.2.3 Calcium and Aluminum

LAACHIs would not be expected to show any major deviations from bulk Solar
System (“chondritic”) in most Ca or Al isotopic ratios. Al definitely would
have recorded a presolar isotopic signature: it is a major element in
corundum, spinel, and hibonite, and Al in these minerals would not be
exchanged with the gas. However, Al is monoisotopic.

Ca is a refractory element with multiple isotopes. While a minor phase in
presolar corundum and spinel (CaO $\ll 1$wt%), presolar hibonite contains 8wt%
CaO. However, this Ca would have mostly exchanged with solar-composition Ca
vapor in the gas (§3.4) In contrast to ${\rm Mg}^{2+}$, which is coupled to
${\rm Ti}^{4+}$ in hibonite, ${\rm Ca}^{2+}$ could have diffused through
hibonite within hundreds of years or less, although the exact diffusion
coefficients are unknown.

The only means of acquiring a Ca isotopic composition recognizable as presolar
is through inheritance of isotopic anomalies by accretion of presolar
perovskite grains. Such grains formed in supernovae are the likely source of
${}^{48}{\rm Ca}$ excesses (Dauphas et al., 2014). Being small grains, one
might expect them to also exchange Ca atoms with the gas: the diffusion
coefficients of Ca and Ti through perovskite are unknown, but the diffusion of
similar Ba through ${\rm BaTiO}_{3}$ is $3\times 10^{-18}\,{\rm m}^{2}\,{\rm
s}^{-1}$ at 1350 K (Sažinas et al., 2017), implying exchange with the gas in
about 4 days, assuming a grain radius of $1\,\mu{\rm m}$ (e.g., Han et al.,
2015). As short as this is, it is longer than the average time for a
perovskite grain to collide with and stick to a corundum or hibonite grain,
about 0.5 days (after accounting for the focusing due to their charge;
§4.2.4). Perovskite grains liberated from evaporating aggregates of silicates
therefore might not spend enough time as bare grains to exchange with the gas
and lose their signature of ${}^{48}{\rm Ca}$ excess, before being
incorporated into LAACHIs.

Without pursuing detailed modeling, our expectation is that with the exception
of ${}^{48}{\rm Ca}$ and other anomalies acquired from perovskite grains,
LAACHIs should appear solar in their Ca isotopic compositions. Measurements of
$\delta^{42}{\rm Ca}$ and $\delta^{43}{\rm Ca}$ in PLACs indeed are not
resolved from a solar composition (Kööp et al., 2016a).

####  4.2.4 Titanium

Another well-known feature of PLACs is that they exhibit large and correlated
excursions in both $\epsilon^{50}{\rm Ti}$ and $\epsilon^{48}{\rm Ca}$
anomalies (Zinner et al., 1986, 1987; Fahey et al., 1987b, a; Ireland, 1990;
Sahijpal et al., 2000; Meyer & Zinner, 2006; Liu et al., 2009; Kööp et al.,
2016a). Dauphas et al. (2014) have argued compellingly that the carrier of
positive anomalies is presolar perovskite [${\rm CaTiO}_{3}$] grains formed in
supernova ejecta containing both the neutron-rich isotopes ${}^{48}{\rm Ca}$
and ${}^{50}{\rm Ti}$. Acquisition of a greater-than-average number of such
carriers would lead to ${}^{50}{\rm Ti}$ excesses, while incorporation of a
lower-than-average number would lead to deficits, relative to terrestrial
(near Solar System average). These carriers have not been discovered, but
having formed in supernova ejecta they are likely to be small, $<$ tens of nm
in size.

Like the small nanospinel grains excluded from growing LAACHIs, presolar
perovskite grains would have been small enough to be excluded from PLACs if
they were negatively charged, which would mean PLACs could not carry
${}^{48}{\rm Ca}$ or ${}^{50}{\rm Ti}$ excesses; remarkably, though,
perovskite in the LAACHI-forming region is likely to have been positively