Mon. Not. R. Astron. Soc. 418, 571–582 (2011)

doi:10.1111/j.1365-2966.2011.19506.x

On the effects of microphysical grain properties on the yields
of carbonaceous dust from Type II supernovae
David W. Fallest,1 Takaya Nozawa,2 Ken’ichi Nomoto,2 Hideyuki Umeda,3Keiichi Maeda,2 Takashi Kozasa4 and Davide Lazzati1
1 Department of Physics, NC State University, 2401 Stinson Drive, Raleigh, NC 27695-8202, USA
2 Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa 277-8583, Japan
3 Department of Astronomy, School of Science, University of Tokyo, Tokyo 113-0033, Japan
4 Department of Cosmosciences, Graduate School of Science, Hokkaido University, Sapporo 060-0810, Japan

Accepted 2011 July 25. Received 2011 July 21; in original form 2011 May 14

ABSTRACT

We study the role of the unknown microphysical properties of carbonaceous dust particles
in determining the amount and size distribution of carbonaceous dust condensed in Type IIsupernova (SN) explosions. We parametrize the microphysical properties in terms of the shapefactor of the grain and the sticking coefficient of gas-phase carbon atoms on to the grainsurfaces. We find that the amount of dust formed is fairly independent of these properties,
within the parameter range considered, though limited by the available amount of carbon
atoms not locked in CO molecules. However, we find that the condensation times and size
distributions of dust grains depend sensitively on the microphysical parameters, with the mass
distributions being weighted towards larger effective radii for conditions considering grains
with higher sticking coefficients and/or more aspherical shapes. We discuss that this leads toimportant consequences on the predicted extinction law of SN dust and on the survival rate of
the formed grains as they pass through the reverse shock of the SN. We conclude that a moredetailed understanding of the dust formation process and of the microphysical properties ofeach dust species needs to be achieved before robust prediction on the SN dust yields can be
performed.
Key words: supernovae: general – dust, extinction.

1 I N T RO D U C T I O NInterstellar dust, once considered to be little more than a nuisanceto astronomical observations, is one of the most interesting areasof astrophysical research today (Li & Greenberg 2003). One aspect of particular interest is the origin of dust at high redshift (z >5). Possible sources of interstellar dust that have been considered
include outflows from asymptotic giant branch stars (Morgan & Edmunds 2003; Zhukovska, Gail & Trieloff 2008; Valiante et al. 2009),
Wolf–Rayet systems (Cherchneff 2010), quasars (Elvis, Marengo &Karovska 2002) and supernova (SN) explosions (Kozasa, Hasegawa& Nomoto 1989, 1991; Todini & Ferrara 2001; Nozawa et al. 2003;
Bianchi & Schneider 2007; Kozasa et al. 2009; Cherchneff 2010).
How much dust can be attributed to each of these possible sources
at such high redshift remains unclear. Supernovae (SNe) are considered by some to be likely contributors of much of the dust in
the early Universe because their progenitors are quite massive andconsequently have short lifetimes (Bromm, Coppi & Larson 2002;
 E-mail: dwfalles@unity.ncsu.edu
C 2011 The Authors
C 2011 RASMonthly Notices of the Royal Astronomical Society Morgan & Edmunds 2003; Maiolino et al. 2004; Dwek, Galliano& Jones 2007). However, such early SNe have not been observed,
making determinations of their dust contributions difficult. Instead,
we need to consider dust yield predictions of more recent SNe,which have been observed, and then extrapolate the dust yields toearlier SNe. The theoretical predictions of dust yields for recent SNe(Kozasa et al. 1991; Todini & Ferrara 2001; Nozawa et al. 2008),
however, are too large compared to observations, with discrepancies that can be as high as 3–4 orders of magnitude in some cases(Hoyle & Wickramasinghe 1970; Lucy et al. 1989; Wooden et al.
1993; Elmhamdi et al. 2003; Meikle et al. 2007; Kotak et al. 2009).
Observations of young supernova remnants (SNRs), however, haveconfirmed dust masses 1–2 orders of magnitude greater than SNe.
Some of these SNRs have observed dust masses of 0.02–0.054 M
by Spitzer (Rho et al. 2008), 0.06 M by AKARI (Sibthorpe et al.
2010) and 0.075 M by Hershel (Barlow et al. 2010), in Cassiopeia A, and 0.04–0.1 M by Spitzer in the pulsar wind nebulaG54.1+0.3 (Temim et al. 2010). Additionally, Nozawa et al. (2010)
have demonstrated that the observed spectral energy distribution of
Cassiopeia A can be well reproduced by the calculations of dust
formation in Type IIb SNR, and the mass in the SNR is 0.07 M .

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D. W. Fallest et al.

Reconciling the dust yield prediction for local SNe can be done
via one of two channels: improving our understanding of the dust
formation process, leading to a substantially decreased prediction,or revising the observational constraints to account for a higher
dust yield than so far implied. It is possible that some amountof dust has avoided observation. Dust particles absorb light andre-emit the energy at infrared wavelengths. However, cold dust at
temperatures of a few tens of kelvin could escape detection at midinfrared wavelengths. Additionally, areas where the dust is opticallythick could obscure some amount of dust, again allowing some dust
to not be detected. Dust clumping may also affect the estimates ofdust mass from absorption (Sugerman et al. 2006), since it is usuallyassumed that the surface filling factor of dust is close to unity,
while substantial clumping could be present due to the intrinsic
inhomogeneity of the ejecta and Rayleigh–Taylor instabilities inthe expanding ejecta (Wooden 1997; Douvion, Lagage & Cesarsky1999).
On the theoretical side, all the estimates for dust productionin SNe are based on the so-called classical theory of nucleation(e.g. Becker & Döring 1935). It has been argued, however, thatthe use of classical nucleation theory in astrophysical environments
is questionable (Donn & Nuth 1985; Lazzati 2008). In addition,
almost all SN dust nucleation models thus far have considered theformation of spherical grains, and assumed any atoms/moleculesthat contact the grain will adhere to the grain; conditions that reflect
maximally efficient nucleation. It is therefore not entirely surprisingthat the theoretical estimates of SN dust yields, based on upper limitof nucleation efficiency, are in excess of those from observations. It
should be noted that Bianchi & Schneider (2007) have consideredless than maximally efficient nucleation by assuming the probabilityof atoms/molecules adhering to the grain is less than unity, resulting
in smaller dust yields.
Since dust formation is a highly non-linear phenomenon, understanding the effects of different nucleation rates on the final dust
yields is difficult. To check the effects of different nucleation conditions we have performed a parametric study, in which we consider
carbonaceous dust production in a SN explosion by varying theshape of the forming grains as well as the sticking coefficient, i.e.
the probability that an incoming monomer will stick to the grainrather than bounce off and remain in the gas phase. Our study is
phenomenological and aims at understanding which conclusionsof previous nucleation studies are robust to changing the parameters, and which may need to be investigated more thoroughly. A
self-consistent nucleation model in astrophysical conditions will
be achieved by involving a kinetic approach (Donn & Nuth 1985;
Lazzati 2008; Keith & Lazzati 2011) and detailed chemistry ofprecursor molecules (Cherchneff & Dwek 2009, 2010). Such a detailed approach, however, is still under development and is not yet
applicable to large-scale simulations like the one that we use here,
and that have been used in previous investigations of e.g. Todini &Ferrara (2001); Nozawa et al. (2003, 2010).
This paper is organized as follows. In Section 2 we detail the
nucleation theory that we adopted. In Section 3 we describe thenumerical code used for the computations, and in Section 4 we
describe our results. In Section 5 we finally discuss the implicationand limitations of our results and lay out future perspectives for SN
dust studies.

atoms to solid clusters of amorphous carbon dust. There is a phaseequilibrium pressure where both the gas and solid phases are stablewithin a given volume. The phase equilibrium pressure depends
on the temperature of the materials in the gas phase within thevolume. Nucleation of clusters of atoms/molecules of the new phaseis favoured when the gas phase is supersaturated (i.e. the pressure of
the gas is higher than the phase equilibrium pressure). The higher thesupersaturation (i.e. the ratio of the pressure to the phase equilibriumpressure), the smaller the size of the stable clusters that are able
to form. The size of the smallest stable cluster able to form at
a given temperature and density is called the critical cluster size.
Clusters that are smaller than the critical size will tend to evaporate,
while larger clusters will tend to continue to grow. The goal of any
nucleation theory is to calculate how many critical clusters form perunit volume per unit time.
The classical theory of nucleation considers nucleation as a thermodynamical process in quasi-equilibrium (Becker & Döring 1935;
Feder et al. 1966; Kashchiev 2000). Besides the supersaturationlevel, the physical properties of the nucleating material affect thesize of a critical cluster. The classical nucleation theory assumes that
all clusters share the same properties, such as surface tension andshape, independent of their size. The theory also assumes that those
properties are equal to those of a macroscopic sample. Moreover,the clusters are assumed to have a uniform equilibrium temperaturethat is equal to the temperature of the surrounding gas. These basic
assumptions are problematic because macroscopic thermodynamic
properties are not expected to be applicable to clusters of only a few
atoms.
A different approach is provided by the kinetic theory of nucleation, which is applicable to very small cluster sizes. The kinetic
theory relies on calculating the attachment and detachment frequencies of monomers to a cluster (Kashchiev 2000). Furthermore, incontrast to the classical nucleation theory, the kinetic theory follows
the formation of clusters smaller than the critical cluster size. In theframework of the kinetic theory, the critical cluster is the cluster
whose attachment and detachment frequencies are equal, and thus
it is stable. The downside of the kinetic theory is that attachment anddetachment frequencies for all cluster sizes need to be calculated
in order to determine the overall nucleation rate. The main aim of
this paper is to reveal the dependence of dust formation processeson the microphysical properties of grains. In order to achieve this,we adopt the simpler classical nucleation theory, rather than using
the kinetic theory that demands a more complicated treatment.
When the supersaturation level is S > 1, nucleation can take placebecause the free energy of the new phase is lower than that of theold phase (Kashchiev 2000). The change in the free energy is dueto the work necessary to form the critical size clusters. Energy isreleased in the formation of the volume of the cluster, but energy
is required in order to form the surface of the cluster (Kashchiev
2000). The nucleation rate (equation 1) is given basically by two
factors: the number density of critical clusters and a kinetic factor
that describes the rate at which clusters become large enough to bestable (i.e. critical size or larger).
The general equation of stationary nucleation is given by(Kashchiev 2000, their equation 13.39):

2 N U C L E AT I O N

where A is the kinetic factor, W ∗ is the work needed to form thecritical size cluster from the gaseous state, k is the Boltzmann constant, and T is the gas temperature. The kinetic factor, A, is given

Nucleation is the first step of a first-order phase transition. In thecase we consider here, the phase transition is from a gas of carbon

JS = A exp (−W ∗ /kT ),(1)

C 2011 The Authors, MNRAS 418, 571–582
C 2011 RASMonthly Notices of the Royal Astronomical Society SN dust microphysics
by A = A exp (μ/kT), and A is 3
1/2 
c σ
pe v0 C0 ,A = γ ∗2
18π m0kT
where γ ∗ is the size-dependent sticking coefficient, c is the shapefactor of the cluster, σ is the surface tension, m0 and v 0 are, respectively, the mass and volume of the monomer of nucleating material,
pe is the phase-equilibrium pressure, and C0 is the concentrationof sites where nucleation can occur [equations (13.41) and (13.44),
respectively, in Kashchiev (2000)]. The concentration of gaseousmonomers, C1 , is related to the concentration of nucleation sites
by C1 = C0 exp (−W 1 /kT), where W 1 is the work needed to forma cluster consisting of one monomer [equation (7.5) in Kashchiev(2000)]. We consider the monomer in the gaseous state to be indistinguishable from the monomer in the condensed state, so thatW 1 = 0 and C1 = C0 . Using the supersaturation ratio, S = C1 /C1,e =
p/pe and μ = kT ln S, where C1,e = pe /kT is the gaseous monomer
concentration at the phase-equilibrium pressure and p is the partialpressure of gaseous monomers, we obtain 3 2 1/2
c v0 σC12 .
A = γ∗
18π2 m0
While the sticking coefficient may depend on the size of the cluster,
we consider it to be constant, so that γ ∗ = γ = constant. Finally,using the work to form a critical sized cluster W ∗ = 4c3 v 20 σ 3 /27μ2[Kashchiev 2000, equation (4.8)], we find our stationary nucleation
rate equation to be 3 2 1/2


c v0 σ−4c3 v02 σ 3
2
C
exp
.
(2)
JS = γ1
18π2 m027(kT )3 (ln S)2After nucleation the clusters grow through impingement ofmonomers upon the cluster. To find how much the clusters grow,we begin by finding the volume of the newly nucleated cluster.
The clusters nucleate with some critical number of monomers, n∗[equation (4.7) in Kashchiev 2000]:n∗ =8c3 v02 σ 3
.
27μ3

(3)

The volume of the critical cluster is then v ∗ = n∗ v 0 . The change inthe cluster’s volume over time depends on the sticking coefficientγ , the surface area of the cluster , the concentration of monomers
C1 , the volume of the monomer v 0 and the average relative speed ofthe monomers with respect to the cluster. In this paper, we consider
clusters that can nucleate with aspherical shapes through the useof the shape factor c. The shape factor is a dimensionless quantitythat relates the surface area  of an object to its volume V by c =/V 2/3 (Kashchiev 2000). Thus we find the change in volume overtime to be

1/2
kT
dV
2
.
(4)
= γ cV 3 C1 v0
dt
2πm0

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so that the total number of gas-phase atoms remains constant, andV(t, t ) is the volume of a cluster formed at time t and measured attime t. Rather than computing the integral on the right-hand side,
we instead calculate C1 (t) as described in step (iii) in the next section. From equations (2) and (4), we see that J S and dV /dt are
simply proportional to the sticking coefficient γ . Thus, we expectthat reduced sticking coefficients will suppress both nucleation andgrain growth. The dependence of J S on the shape factor c is more
complicated, since it appears in both the kinetic factor and the exponential term. The shape factor in the exponential term, however, will
dominate the nucleation rate equation and we expect that increased
shape factors will suppress nucleation. On the other hand, grain
growth (dV/dt) is simply proportional to c, and thus an increase in
the shape factor will increase the cluster growth rate.

3 S I M U L AT I O N S
We concentrate on the formation of carbonaceous grains (clusters)from carbon atoms in the expanding material of a core-collapse SN(CCSN). Our simulations are based on the hydrodynamic results
and elemental composition for the unmixed ejecta of a CCSN of
a 20 M progenitor star with metallicity Z = 0 and an explosion
energy of 1051 erg by Umeda & Nomoto (2002) (see also Nomoto
et al. 2006).
Table 1 describes the data necessary for the calculation of carbon
grain formation. To compute the supersaturation of the expandinggas, we find the phase equilibrium pressure by pe = p0 e−A/T +B ,where p0 is the standard pressure, T is the temperature, and thevalues of A and B, listed here in Table 1, are taken from table 2 of
Nozawa et al. (2003). As the material ejected by the CCSN expands
it also cools. Figs 1 and 2 show the evolution of the density andtemperature, respectively, of the expanding and cooling material for
two of the enclosed mass subshells that we refer to in the rest of
this work.
Table 1. Carbon properties.
A/104
(K)

B

σ
(erg cm−2 )v0(10−24 cm−3 )m0(10−23 g)

8.647 26

19.0422

1400

8.805

1.995

In the same manner as Nozawa et al. (2003), we compute thedepletion of the available nucleation material through mass conservation:
 t
J (t  ) V (t, t  ) C1 (t)= 1 − Y1 =
dt ,
(5)
1−

v0
C̃1 (t)te C̃1 (t )where C̃1 is the nominal concentration of monomers – the concentration expected should nucleation not occur, which in an expanding
shell of volume V shell can be found usingC̃1 (t = tn )Vshell (t = tn ) = C̃1 (t = t0 )Vshell (t = t0 ),
C 2011 The Authors, MNRAS 418, 571–582
C 2011 RASMonthly Notices of the Royal Astronomical Society 

Figure 1. Density evolution for 4.96 (black) and 6.0 M (red) enclosed
mass subshells up to 1000 d after the SN explosion.

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D. W. Fallest et al.

Figure 2. Temperature evolution for 4.96 (black) and 6.0 M (red) enclosed
mass subshells up to 1000 d after the SN explosion.

We study four different values of the unknown sticking coefficient, γ = 1.0, 0.1, 0.01 and 0.001, neglecting any dependence ofγ on the temperature of the gas and the size of the cluster. For eachvalue of the sticking coefficient, we study six different values forthe shape factor, c = (36π)1/3 , 5.4, 6.0, 7.0, 9.0 and 12.0, corresponding to shapes ranging from a sphere to a flattened cylinder
similar to a coin. We therefore performed 24 simulations in total.
A sticking coefficient of γ = 1.0 and a shape factor for a sphere ofc = (36π)1/3 are the usual parameters used in previous nucleation
works.
As in Nozawa et al. (2003), we assume that the stable formation of
CO molecules occurs prior to grain nucleation and that carbon grains
will form only where the initial number fraction of carbon is higher
than that of oxygen. Under this assumption, the number fraction offree carbon atoms available for dust formation is obtained simply
from the initial number fraction of carbon minus the number fraction
oxygen. We divide the expanding gases into a series of enclosed
mass shells beginning at ∼4.93 M and ending at ∼6.21 M . In
this range of enclosed masses, the number fraction of carbon atoms,
after the formation of CO, is highest; ranging between 2 × 10−1 and8 × 10−9 . Fig. 3 shows the number fraction of carbon and oxygen
atoms in the expanding ejecta from the hydrodynamic results of
Umeda & Nomoto (2002). The solid (blue) line indicates the number
fraction of carbon left over after the formation of CO molecules. It
should be noted for completeness that grain nucleation can occur
at enclosed masses larger than we consider here, but is extremely
inefficient due to low carbon number fractions, ∼10−14 and below.
For each mass subshell, our code starts by following the evolutionof the density and temperature (see Figs 1 and 2, respectively,
for examples) of the gas until the condition of supersaturation issatisfied. From that point on, at each time-step the code performs
three operations.
(i) First, the code computes the number of critical clusters thatare formed given the saturation, temperature and partial pressure of
the carbon atoms [according to equation (2)].
(ii) Secondly, the code grows any pre-existing grain formed atearlier times according equation (4).
(iii) Finally, the code subtracts from the carbon in the gas phase
the amount of carbon that has been locked in the solid phase by
the processes in steps (i) and (ii). The concentration of gas-phase

Figure 3. Number fraction of helium (green dot–dashed), carbon (black
dashed), oxygen (red dotted) and silicon (magenta dot–dashed) atoms for
enclosed masses from 2.45 to 8.7 M . The solid (blue) line is the carbon
number fraction after the formation of CO molecules.

carbon is evolved according toVgrains,nv0 Vshell,n
C1 (tn−1 ) −C̃1 (tn−1 ) − C̃1 (tn ) ,C̃1 (tn )C1 (tn ) = C1 (tn−1 ) −where V grains,n is the total change in volume of grains from theprevious time-step to the current time-step. Then, the first termon the right-hand side is the concentration of gas-phase carbon
from the previous time-step, the second term is the change in theconcentration of solid-phase carbon (it is a summation of all grainchanges, including formation of new critical clusters and the growthof existing grains), and the final term accounts for the decrease inconcentration of gas-phase carbon due to the expansion of the shell.
This process is repeated for each mass shell until the concentration of gas-phase carbon is reduced to 1 per cent of its original value.
The dust distributions from all shells are then summed together toproduce the final dust yield of each particular set (γ , c).
4 R E S U LT S
4.1 Spherical grains
After the SN explosion, the hot gases expand and cool. The coolingallows the gases to reach supersaturation conditions. Once the supersaturation level becomes greater than unity, nucleation can occur.
The supersaturation level continues to rise and the nucleation rateincreases over time until depletion of available material becomessignificant and the supersaturation level begins to drop, after which
the nucleation rate falls off quickly. Eventually the gas is no longer
supersaturated and nucleation ceases. However, grain growth is still
possible.
Fig. 4 shows the nucleation rate for spherically shaped grains at
enclosed mass coordinate 4.96 M for the four sticking coefficients.
We chose this particular enclosed mass shell because it contains
the highest abundance of carbon atoms, after CO formation, of all
our mass shells. The solid curve corresponds to c = (36π)1/3 andγ = 1.0, the parameters generally used for nucleation studies. The
dashed, dot–dashed and dotted curves correspond to γ = 0.1, 0.01
C 2011 The Authors, MNRAS 418, 571–582
C 2011 RASMonthly Notices of the Royal Astronomical Society SN dust microphysics

Figure 4. Nucleation rates for spherical grains [c = (36π)1/3 ], for the four
considered sticking coefficients of γ = 1.0, 0.1, 0.01 and 0.001. Filled circles
indicate the time and rate of maximum nucleation. Rates are calculated at
an enclosed mass coordinate 4.96 M .

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Figure 6. Nucleation rates for spherical carbon grains at an enclosed coordinate of 6.00 M for four sticking coefficients. Condensation times areindicated by filled circles.

and 0.001, respectively. To be consistent with the works of Kozasa& Hasegawa (1987), Kozasa et al. (1989), Kozasa et al. (1991) andNozawa et al. (2003), we consider the time at which the nucleation
rate is at its maximum to be the condensation time of the grains. We
show these condensation times as filled circles at the maxima of thenucleation rates in the figure.
At early times, the reduced sticking coefficient makes the formation of critically sized grains more difficult and results in a suppressed nucleation rate. In the absence of strong nucleation, carbon
atoms are not depleted from the gas and the saturation continues toincrease. The reduced sticking coefficient thus causes the time atwhich nucleation is at its maximum to be delayed, and the nucleation to take place at higher saturation levels. As a consequence, alarger number of critical clusters can form with much smaller size
(equation 3).
Fig. 5 shows the maximum nucleation rates for spherical grains as
a function of enclosed mass for our four sticking coefficients. For enclosed masses up to M r ∼ 5.87 M , the nucleation rate maxima for
reduced sticking coefficients exhibit similar behaviour as in Fig. 4.
At greater enclosed masses, however, the behaviour is inverted anda reduced sticking coefficient results in depressed maximum nucle-

ation rates. This difference in behaviour is due to the reduction in
the number fraction of carbon available, from ∼10−2.5 to ∼10−5(see Fig. 3), which corresponds to a reduction in the concentrationof carbon monomers. Since the nucleation rate (equation 2) is proportional to the concentration of monomers squared, the drop in the
carbon concentration consequently drops the nucleation rate. Thus,
the nucleation rates do not peak as strongly (Fig. 6), drawing out
nucleation to later times for all sticking coefficients, so that a catastrophic reduction in the available material does not occur. Thus, thenucleation rates for reduced sticking coefficients remain depressed
throughout the simulation time for higher enclosed mass shells.
Fig. 7 shows the condensation times corresponding to the maximum nucleation rates shown in Fig. 5. The condensation timesof the solid curve (c = (36π)1/3 , γ = 1.0) are in good agreementwith the condensation times for carbon grains reported in Nozawa
et al. (2003). Here the lower sticking coefficients result in delayedcondensation times up to M r ∼ 5.87 M , outside which the condensation time is only slightly delayed, or no longer delayed, whencompared to the case of γ = 1.0. The more noticeably delayed condensation times, as well as the similarity of condensation times forall our sticking coefficients, at higher enclosed mass shells are also
due to the drawn out nucleation process already discussed above.

Figure 5. Maximum nucleation rates for spherical carbon grains at enclosed
masses <6.2 M for four sticking coefficients.

Figure 7. Condensation times of spherical carbon grain as a function of
enclosed mass for four sticking coefficients.

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D. W. Fallest et al.
Table 2. Total mass of carbon grains formed.
Mass (10−2 M )c(36π)1/3
5.4
6.0
7.0
9.0
12.0

γ = 1.0

γ = 0.1

γ = 0.01

γ = 0.001

4.95
4.94
4.92
4.89
4.82
4.64

4.93
4.91
4.89
4.86
4.76
4.52

4.89
4.87
4.84
4.79
4.65
4.27

4.79
4.75
4.70
4.62
4.34
3.32

4.2 Non-spherical grains

Figure 8. Size distribution of carbon grains for γ = 1.0, 0.1, 0.01, 0.001,
with c = (36π)1/3 . For reference, the solid (red) line represents the powerlaw distribution with the form of N r ∝ r−3.5 , which has been suggested asthat of interstellar grains (e.g. Mathis, Rumpl & Nordsieck 1977).

Figure 9. Mass distribution of carbon grains for γ = 1.0, 0.1, 0.01, 0.001,
with c = (36π)1/3 .

Reducing the sticking coefficient makes both nucleation and grain
growth more difficult. A reduced sticking coefficient results in alarger number of smaller grains, fewer larger grains and a smaller
maximum grain radius. In Figs 8 and 9, respectively, we show thesize and mass distributions for spherical grains. For the case of γ =1.0, the maximum grain size achieved is between 2 and 3 μm. Whenthe sticking coefficient is reduced to 0.001, the maximum grain size
is decreased to less than 0.01 μm.
Since the reduced sticking coefficient causes larger numbers ofsmall grains to form, the small grains contain more relative mass
than the larger grains, as can be seen in Fig. 9. However, the total
mass of the grains is relatively robust. The total masses of dust grains
for the spherical case are shown in Table 2, along with the asphericalcases which are discussed in the next section. Even though the onset
of nucleation is delayed due to the reduced sticking coefficient at
enclosed masses less than 5.87 M , where the majority of available
carbon is contained, the subsequent grain growth consumes almost
all of the carbon atoms for γ > 0.001. Thus, the total mass of carbon
dust is principally determined by the mass of pre-existing carbon
atoms.

We also calculated nucleation rates for aspherical grains. We choosea range of shape factors from c = 5.4 to 12.0. While c = 6.0 is theshape factor of a cube, each shape factor can correspond to a numberof different grain shapes. The shape factor can be thought of as adeviation from the spherical case; the bigger the shape factor, thelarger the deviation from a sphere and the larger the surface for agiven volume.
In Fig. 10, we show the nucleation rates for the same enclosed
mass shell as shown in Fig. 4, for increasing shape factors with γ =1.0. Since the shape factor appears in the exponential term of thenucleation rate equation as J S ∝ exp (−c3 ) (equation 2), the higher
shape factors reduce the nucleation rate. Therefore, the higher supersaturation levels at later times need to be attained so that thenucleation rate becomes high enough for significant depletion of thegas due to the growth of newly formed grains. However, the maximum nucleation rates for increasing shape factors are decreased.
This is due to the fact that with a larger surface-to-volume ratio
aspherical grains tend to grow faster, producing a sizeable depletion of the gas-phase carbon even for moderate nucleation rates.
For all mass coordinates, increasing the shape factor leads to lower
nucleation rates and delayed condensation times (Figs 11a and 12a,
respectively).
In Figs 11(b–d), we show the nucleation rate maxima for all shapefactors for each of the other three sticking coefficients. For any sticking coefficient, the shape factor has the same effect of suppressing
the nucleation rate. All condensation times, shown in Figs 12(b–d),
increase with the increased shape factor. Furthermore, for reduced

Figure 10. Nucleation rates as a function of time for six shape factors with
γ = 1.0. Filled circles indicate maximum nucleation rate. Rates shown arefor nucleation within a shell of enclosed mass of ∼4.96–4.97 M .
C 2011 The Authors, MNRAS 418, 571–582
C 2011 RASMonthly Notices of the Royal Astronomical Society SN dust microphysics

Figure 11. Maximum nucleation rates for all six shape factors and four sticking coefficients.

Figure 12. Condensation times for six shape factors and four sticking coefficients.
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D. W. Fallest et al.

sticking coefficients, greater shape factors lead to greater delays in
the condensation time.
The shape factor c = (36π)1/3 is the only shape factor that has
just one associated shape, the sphere. The other shape factors weconsider do not necessarily have a unique grain shape. To find asize distribution that is easily comparable to the spherical case, we
define a volume equivalent radius for the aspherical grains as
3V 1/3,(6)
reff =
4π
where V is the grain volume.
Figs 13(a–d) show the size distributions for all shape factors
and sticking coefficients. As the shape factor is increased, the size
and number of larger grains also increase. We find that the largest
maximum grain radii are formed with c = 12.0 and γ = 1.0, andhave a volume equivalent radius of almost 18 μm. On the other hand,the smallest maximum grain radii (∼0.008 μm) are formed withc = (36π)1/3 and γ = 0.001. As the sticking coefficient is reduced,
the maximum grain radii for each shape factor are also reduced, andthe number of smaller grains is increased. Even though there is amuch larger number of small grains than large grains, the majorityof the dust mass is contained within intermediate sized grains. In
Fig. 14(a) (γ = 1.0), most of the dust mass is contained in grains
with volume equivalent radii between 0.01 and 0.5 μm.
The masses of carbon grains formed (Figs 14a–d) are dominatedby the relatively small numbers of larger sized grains as the shapefactor is increased. However, as the sticking coefficient is reduced(Figs 14b–d), even the masses of the grains formed with the largest
shape factor become dominated by the smaller sized grains. The
total mass of carbon grains formed for increased shape factors isgiven in Table 2 and shown in Fig. 15. Again, we note that despite

the difference in the size distribution due to different microphysicalparameters, the total mass of dust that condenses is fairly robust(within a factor of 1.5) and constitutes almost the total amountof carbon that was left in the gas phase after the creation of CO
molecules.
5 DISCUSSION
We have shown that varying the microphysical properties of dust
grains has important effects on the condensation times, nucleation
rates and size distributions of carbon dust grains from Type II SN
explosions. However, the total mass of dust is only modestly affectedby the changes in the grain properties. An inadequate choice of the
shape or sticking coefficient is not therefore a viable explanation
for the discrepancy between the mass of dust grains predicted in SN
explosions and the observed dust mass in local Type II SNe (seeSection 1 for a more thorough discussion and references).
We find that a larger saturation is necessary to achieve efficient
nucleation with either a small sticking coefficient (γ < 1) or for
aspherical grains [c > (36π)1/3 ]. For that reason, all our simulations
show that the condensation time grows when sticking coefficients
less than unity or grain shapes that are different from spherical are
adopted. However, differences can be found in the nucleation rates
and final size distribution of the grains. Simulations with a low valueof the sticking coefficient show a delayed nucleation but very highnucleation rates, thereby producing large quantities of small grains.
This is due to the fact that low sticking coefficients inhibit both
nucleation and grain growth and, therefore, all the carbon remains
in the gas phase until a high saturation level is reached. At that point,
many small grains are nucleated and the atomic carbon is quickly
depleted. Bianchi & Schneider (2007) found similar effects when

Figure 13. Size distribution of carbon grains for six shape factors and four sticking coefficients.
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Figure 14. Mass distribution of carbon grain for six shape factors and four sticking coefficients.

Figure 15. Total mass of carbon dust formed for all simulations.

calculating dust nucleation with a sticking coefficient of γ = 0.1.
Asphericity of the grains, on the other hand, inhibits nucleation but
enhances grain growth. As a consequence, even if fewer grains are
nucleated, they grow fast and the result is a grain size distribution
characterized by less numerous, larger grains.
We find that the total mass of carbonaceous dust formed remains
relatively stable even with sticking coefficients as low as 0.001. For
the spherical case only, we explored the possibility of even smaller
sticking coefficients, down to γ = 10−9 . We find that the mass ofcarbon dust formed becomes significantly reduced for sticking coefficients of γ = 10−7 and below (see Fig. 16). With sufficiently low
values of sticking coefficient (below 10−8 ) there is virtually no dustC 2011 The Authors, MNRAS 418, 571–582
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Figure 16. Total mass of dust formed for spherical carbonaceous grains
with sticking coefficients down to γ = 10−9 .

formation, but the required sticking coefficients seem unphysicallylow.
In terms of the observable properties of the SN-condensed dust,
we find that the quantity that is most affected is the extinction curve(see Fig. 17). Not surprisingly, simulations with small sticking coefficient (which, as explained above, produce large amounts of small
grains), result in a very steep extinction curve at far-UV wavelengths, with RV values between 3 and 3.5, shown in Fig. 18. We
make special note that the RV values for γ = 1.0 are relatively high,even though the size distribution of the grains is consistent withthat of interstellar grains (see Fig. 8), because we account for only

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Figure 17. Extinction curves for each sticking coefficient and shape factor.

Figure 18. RV values for 23 simulations. Not shown is RV = 41 for c =12.0 and γ = 1.0.

carbon grains and that including other grains, such as silicates, could
decrease the RV values. On the other hand, simulations with very
aspherical grains and relatively high sticking coefficients produce
larger grains and, consequently, grey extinction curves. Without
the knowledge of the values of the sticking coefficient and of the
shape factor it is therefore impossible to predict the extinction curveof SN-condensed dust. This is a particularly worrying conclusion
since the extinction curve is relatively easy to measure, even at highredshift, and could be used as an observational constraint for theorigin of dust in the various environments. For example, Maiolino
et al. (2004) compared the extinction curve measured in a quasar atz = 6.2 to the extinction curve calculated using the dust model by

Todini & Ferrara (2001). They find that the data and the theoreticalprediction are in good agreement and conclude that the dust observed in SDSS J104845.05+463718.3 is indeed condensed in SN
explosions (see also Stratta et al. 2007). In light of our results, suchconclusions need confirmation once a complete theory of SN dust
nucleation is obtained.
The dust that condenses, however, is not the dust that is ejected
into the interstellar medium (ISM). Dust produced in a CCSN has
to travel through the reverse shock before being released into theISM. The reverse shock can destroy most of the dust, in particular the smaller dust grains (Draine & Salpeter 1979; Nozawa et al.
2007; Nath, Laskar & Shull 2008; Silvia, Smith & Shull 2010). If
astrophysical dust formation is indeed characterized by small values of the sticking coefficient, it is likely that the amount of dust
formed is reduced significantly by the reverse shock. Conversely, increased shape factors allow for the formation of larger grains which
would survive shock processing. The microphysical properties ofdust grains can therefore affect the mass of dust that is injected in
the ISM, even though they affect only marginally the dust that iscondensed during the early stages of the explosion.
Grain nucleation with non-spherical shapes may be more complicated than we considered here. We have assumed that the shapefactors of the grains do not change as the grains grow. Since we assume here that grains grow by the addition of monomers, the shapeof small clusters can change as monomers attach, in turn altering
the shape factor. Another route to take could be to nucleate grains
at an arbitrary shape factor and then allow the grains to grow into
spheres. For example, one may turn grains into spheres when thenumber of monomers is larger than a given value. In this case, increased shape factors may not lead to such large grains as we showin our size distributions (Fig. 13).
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This work is purely a parametric study that aims to show that the
microphysical properties of the grains are important, but not to point
to any specific values of γ and c to be used in nucleation calculations.
Therefore, we have chosen to neglect important factors that need beconsidered in a complete SN dust nucleation model. These include
the choice of progenitor model, the presence of other dust species(see Nozawa et al. 2003) as well as charged molecules that may
interfere with carbonaceous dust condensation, and the destructionof CO molecules, due to photodissociation or collisions with fastelectrons and charged particles (Petuchowski et al. 1989; Lepp,
Dalgarno & McCray 1990; Liu, Dalgarno & Lepp 1992; Clayton,
Deneault & Meyer 2001) that can inject additional carbon atoms
into the available monomer concentration (see Todini & Ferrara
2001; Bianchi & Schneider 2007).
A zero-metallicity 20 M CCSN progenitor model was chosenbecause SNe at high redshift are expected to have zero metallicity.
In general, the relative abundances of major elements in the He core
are not significantly different among the SN models with different
metallicities. Thus, the species of dust formed do not depend on
the metallicity of the SN progenitor star (Nozawa et al. 2010).
Additionally, the gas density and temperature in the He core arealmost independent of the progenitor mass and metallicity as longas the kinetic energy of the explosion is the same (Nozawa et al.
2003). Therefore, SN models with non-zero metallicities, or withdifferent progenitor masses, are expected to show similar effects on
carbon grains formation as we see here. This means that the mass ofcarbon dust formed in the SN ejecta is rather insensitive to changes
in the sticking coefficient and shape factor and is purely determinedby the mass of carbon atoms available for dust formation in the Helayer. However, to confirm such expectations, additional progenitor
models would need to be investigated.
Perhaps more important than the choice in progenitor model isthe dissociation process of CO molecules. In this paper we assumed
the formation of CO molecules to be complete and considered only
the condensation process of C grains in the He layer where C/O > 1.
In the expanding ejecta, CO molecules could be destroyed throughinteractions with fast electrons from radioactively decaying 56 Co
and charged particles such as He+ and Ne+ (Petuchowski et al.
1989; Lepp et al. 1990; Liu et al. 1992; Clayton, Liu & Dalgarno
1999; Kwong, Chen & Fang 2000; Clayton, Deneault & Meyer
2001; Deneault, Clayton & Meyer 2006), allowing for more freecarbon (and oxygen) atoms to be available for grain formation thanwe consider here. However, the number abundance of silicon atoms
is too small for most of the enclosed mass regions (M = 4.93–
6.21 M ) in this work (see Fig. 3), so that the formation of SiC andsilicate grains cannot be expected. Therefore, the dissociation of
CO molecules due to interactions with He+ only results in a slightenhancement of the final mass of carbon grains and never changes
our conclusion on the dependence of formation process of C grainson the microphysical properties.
On the other hand, Clayton et al. (1999) and Deneault et al.
(2006) show that CO dissociation enables carbon dust grains toform even in O-rich layer where C/O < 1. Given that the abundanceof silicon atoms in the O-rich layer is higher than in the He layer,
the formation of SiC grains could be expected there. However, as
discussed in Nozawa et al. (2003), even if free carbon and silicon
atoms coexist abundantly, the nucleation theory does not predict theformation of SiC grains. The formation process of large SiC grains
as appeared in pre-solar grains, as well as formation efficiency ofmolecules, is to be pursued in more sophisticated studies of dust
formation. Furthermore, in the O-rich layer, the formation of silicate
grains is also feasible. Bianchi & Schneider (2007) show that theC 2011 The Authors, MNRAS 418, 571–582
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formation of silicate grains is more sensitive to changes in sticking
coefficient than carbon grains, and the mass of silicate grains formed
can be reduced for even γ = 0.1. The inclusion of silicate grains
could affect the resulting extinction curves.
We have adopted the thermodynamic approach for this study because it involves the simplest nucleation equations; however, use of
the kinetic theory of nucleation should be considered in the future.
The kinetic theory still needs to take the sticking coefficient into
account, but makes consideration of an evolving shape factor unnecessary, because the shape of the grain from a complex solid (fora few molecules) to a sphere (for ∼100 molecules) is intrinsicallytaken into account.

AC K N OW L E D G M E N T S
We would like to thank the reviewer for insightful comments andsuggestions for the improvement of this work. We would also like tothank Raffaella Schneider and Andrea Ferrara for their comments.
Thanks go to the Institute for the Physics and Mathematics of the
Universe (IPMU), University of Tokyo, Kashiwa, Japan for theirgenerous hospitality, and to the NSF East Asia and Pacific Summer
Institute (EAPSI) Japan 2010 program (award #1015575) and theJapanese Society for Promotion of Science for their generous support. This work has also been supported in part by World PremierInternational Research Center Initiative, MEXT, Japan.
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C 2011 RASMonthly Notices of the Royal Astronomical Society 