The Astrophysical Journal, 776:24 (17pp), 2013 October 10

doi:10.1088/0004-637X/776/1/24


C 2013. The American Astronomical Society. All rights reserved. Printed in the U.S.A.

FORMULATION OF NON-STEADY-STATE DUST FORMATIONPROCESS IN ASTROPHYSICAL ENVIRONMENTS
Takaya Nozawa1 and Takashi Kozasa2

1 Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa, Chiba 277-8583, Japan; takaya.nozawa@ipmu.jp
2 Department of Cosmosciences, Graduate School of Science, Hokkaido University, Sapporo 060-0810, Japan

Received 2013 June 7; accepted 2013 August 7; published 2013 September 23

ABSTRACTThe non-steady-state formation of small clusters and the growth of grains accompanied by chemical reactions areformulated under the consideration that the collision of key gas species (key molecule) controls the kinetics of dust
formation process. The formula allows us to evaluate the size distribution and condensation efficiency of dust formedin astrophysical environments. We apply the formulation to the formation of C and MgSiO3 grains in the ejecta ofsupernovae, as an example, to investigate how the non-steady effect influences the formation process, condensation
efficiency fcon,∞ , and average radius aave,∞ of newly formed grains in comparison with the results calculatedwith the steady-state nucleation rate. We show that the steady-state nucleation rate is a good approximation if thecollision timescale of key molecule τcoll is much smaller than the timescale τsat with which the supersaturation ratio
increases; otherwise the effect of the non-steady state becomes remarkable, leading to a lower fcon,∞ and a larger
aave,∞ . Examining the results of calculations, we reveal that the steady-state nucleation rate is applicable if thecooling gas satisfies Λ ≡ τsat /τcoll  30 during the formation of dust, and find that fcon,∞ and aave,∞ are uniquelydetermined by Λon at the onset time ton of dust formation. The approximation formulae for fcon,∞ and aave,∞ as afunction of Λon could be useful in estimating the mass and typical size of newly formed grains from observed ormodel-predicted physical properties not only in supernova ejecta but also in mass-loss winds from evolved stars.
Key words: dust, extinction – stars: winds, outflows – supernovae: general – supernovae: individual (SN 2010jl)
Online-only material: color figure

understand not only the total mass but also the size distributionof dust produced in the ejecta of SNe.
The formation process of dust in the SN ejecta has beenstudied mainly with the classical nucleation theory and its
extension (Kozasa et al. 1989, 1991; Todini & Ferrara 2001;
Nozawa et al. 2003, 2008, 2010, 2011; Bianchi & Schneider2007). In the nucleation theory, the condensation of dust is
described by the formation of stable seed nuclei (called criticalclusters) and their growth, where the formation rate of criticalclusters is derived by assuming the nucleation current to be
in a steady state. This theory enables us to predict the size
distribution and mass of condensing grain species, and theresults of the dust formation calculations have nicely explained
the mass of dust formed in SN 1987A (Kozasa et al. 1991) andthe formation and evolution processes of dust in Cas A (Nozawa
et al. 2010).
However, it has been argued that the validity for the application of classical nucleation theory could not be justified in therarefied gas typical of dust-forming region of astrophysical interest (Donn & Nuth 1985 and references therein); in much lessdense systems, the timescale on which the nucleation current
achieves a steady state must be longer than the timescales ofthe evolutions of the gas density and temperature, which would
render the application of the steady-state nucleation rate questionable. On the other hand, Paquette & Nuth (2011) suggested
that the lack of the steady-state condition is unlikely to change
the radius and number density of newly formed grains significantly. They showed that the resulting size distribution of dust islittle affected even if the steady-state nucleation rate is reduced
by a few orders of magnitude, although they did not clarify theeffect of a non-steady state on the formation process of dust.
In this paper, we develop a method managing the dust
formation process without postulating a steady state, which is

1. INTRODUCTION
In astrophysical environments, dust grains condense in themetal-rich cooling gas such as the stellar winds from evolvedstars and the expanding ejecta of novae and supernovae (SNe).
Once the newly formed dust grains are injected into theinterstellar medium (ISM), they cause interstellar extinction
and diffuse infrared emission, and also serve as catalysts forH2 formation and as building materials of such planets as welive on. Hence, the investigation of the formation and evolutionof dust is indispensable in disclosing the nature of objects athigh redshifts, the radiative process and energy balance in the
ISM, and the formation history of stars and planetary systems.
In particular, the origin of dust has been hotly debated since
the discoveries of a huge amount of dust grains at redshifts
higher than z = 5 (Gall et al. 2011 and references therein).
In an early epoch of the universe, core-collapse SNe arisingfrom short-lived massive stars are likely to be dominant sourcesof dust (e.g., Dwek et al. 2007). In fact, recent far-infrared tosubmillimeter observations of young supernova (SN) remnants,
SN 1987A (Matsuura et al. 2011; Lakićević et al. 2012), Cas A(Sibthorpe et al. 2010; Barlow et al. 2010), and Crab (Gomez
et al. 2012), have reported the presence of a subsolar massof cool dust formed in the ejecta, which seems to be highenough to account for the observed amount of dust at highredshifts. However, these cool grains have not yet undergone thedestruction in the hot gas swept up by the reverse and forward
shocks, and thus their mass does not necessarily represent theamount of dust finally ejected by SNe. What fraction of newly
formed grains can survive on their journeys to and in the ISM
heavily depends on their sizes at the time of formation (e.g.,
Nozawa et al. 2006, 2007). Thus, in order to reveal the rolesof SNe as sources of dust in the universe, it is essential to1

The Astrophysical Journal, 776:24 (17pp), 2013 October 10

Nozawa & Kozasa

expected to be more appropriate in astrophysical application.
In Section 2, we formulate the non-steady-state dust formation
process, including chemical reactions at the time of formationof clusters and grains. After describing a simple model of thetime evolutions of the gas temperature and density in the ejectaof SNe in Section 3, we present, in Section 4, the results ofthe calculations for the formation of C and MgSiO3 grains anddiscuss the effect of the non-steady state and its dependence
on the physical conditions in the ejecta. In Section 5, wedemonstrate that the average radius and condensation efficiency
can be uniquely determined by the ratio of the supersaturation
timescale to the collision timescale at the time when thecondensation efficiency rises to 10−10 . Our conclusions arepresented in Section 6. We also present the detailed derivation of
the steady-state nucleation rate for the formation of compoundgrains, such as silicates, in Appendix A.

where a is the grain radius, s is the sticking probability of the key
molecule onto grains, Ω0 is the volume of the condensate per
key molecule, k is the Boltzmann constant, T is the temperatureof the gas, and S is the supersaturation ratio.
The successive formation of clusters and the growth of grainscause the depletion of the key molecules. The time variation of
the concentration c1 of the key molecule is determined from theequation of the mass conservation:c10 − c1 =

kT
2π m1

 12



1
,
c1 1 −
Sa 3 (t, t  ) 
dt ,
a03

(3)

2.1. Case for a Single-element GrainIn this subsection, as a reference, we consider the formationof clusters whose chemical composition is the same as that of
the key molecule (hereafter we refer to such a grain as a “singleelement grain”) in order to clarify how the formation process ofclusters is formulated by means of the kinetic approach. In thiscase, the formation of clusters proceeds through attachment anddetachment of a key molecule as follows:
X + X  X2 ,Xn−1 + X  Xn

(4)
for 3  n  n∗ ,(5)where X and Xn represent the key molecule and the n-mer
cluster, respectively. Then, the current density Jn (t) is given byJn (t) = αn−1 cn−1 c1 − βn cn

for 2  n  n∗ ,(6)where αn is the attachment rate coefficient of a monomer to an nmer, and βn is the detachment rate coefficient of a monomer froman n-mer. In general, αn has not been measured for the materialsof astrophysical interest. Thus, considering that collisions ofmonomers control the kinetics of attachment, we evaluate αn as
follows:

 12
kT
sn
2
4π a02 n 3
,
(7)
αn =
1 + δ1n
2π mn,1
where sn is the sticking probability of a monomer onto an
n-mer, δ1n is the Kronecker’s delta, and mn,1 = nm1 /(n + 1) is
the reduced mass of a monomer and an n-mer. The detachment
rate coefficient βn (n  2) can be related to αn−1 through theprinciple of detailed balance:(1)βn = αn−1where Jn (t) is the net current density from (n − 1)-mer to n-mer.
Here we consider that a cluster containing more key molecules
than n = n∗ can be treated as a macroscopic grain (hereafter
simply called a “grain”). Given the concentration c1 and the mass
m1 of the key molecule, the growth rate of grains is given byt0

Jn∗ (t  )where a0 = (3Ω0 /4π )1/3 is the hypothetical radius of thecondensate per key molecule, and a(t, t  ) is the radius of agrain that reaches n = n∗ at t  and is measured at t.
In the following subsections, we describe how the current
density Jn and the supersaturation ratio S can be presented
according to the chemical reaction at the time of formation.

In this section, we formulate the non-steady-state formationof clusters and dust grains accompanied by chemical reactions,by means of a kinetic approach. Most of the grain species ofastrophysical interest, like silicate, have no monomer moleculewith the same chemical composition as the condensate. Thisimplies that the formation of such compound grains proceeds
via the chemical reactions involving the relevant gas species,
although the reaction pathways and their rate constants are
not well known. One of the methods of evading the difficultyin treating the formation of compound grains without thedetailed knowledge of the chemical pathways and reaction
constants is to employ the concept of a key species (keymolecule), which is defined as a gaseous reactant with theleast collisional frequency among the gaseous reactants, as
proposed by Kozasa & Hasegawa (1987). In this method, twobody reactions between a cluster and the key molecule areconsidered to control the kinetics of the chemical reaction. The
concept of the key molecule has also been applied to the growth
process of compound grains in circumstellar envelopes, as well
as molecular clouds (e.g., Ferrarotti & Gail 2001; Zhukovska
et al. 2008). In what follows, we refer to the cluster composed ofn-key molecules as n-mer, and assume that clusters are spherical.
We also assume the temperature of clusters to be the same as
the gas temperature.
First, for simplicity, we shall consider a cooling gas in aclosed box with the initial concentration of the key moleculec10 at a time t = t0 . As the gas cools down, the condensationof dust grains proceeds through formation and growth of n-mer
clusters via the attachment of the key molecules. In principle,
the time evolution of concentration c(n, t) = cn of n-mers canbe described by a set of differential equations:

da
= sΩ0
dt

 t
ncn +

n=2

2. FORMULATION OF NON-STEADY-STATEFORMATION PROCESS OF CLUSTERS
AND DUST GRAINS

dcn= Jn (t) − Jn+1 (t) for 2  n  n∗ ,
dt

n
∗ −1

c̊n−1
c̊1 ,
c̊n

(8)

where c̊n is the concentration of the n-mer in the gas inthermodynamic equilibrium at a temperature T. Then, thecurrent density Jn (t) is reduced toc̊n−1 c̊1.
(9)Jn (t) = αn−1 c1 cn−1 − cnc̊n c1(2)
2

The Astrophysical Journal, 776:24 (17pp), 2013 October 10

Nozawa & Kozasa

2.2.1. Formation of Dimer (n = 2)

From the law of mass action stemming from the condition thatthe sum of the chemical potentials of the reactants is equal to thatof the products in chemical equilibrium (see Landau & Lifshitz1976),

p̊n−1 p̊11(g̊n − g̊n−1 − g̊1 ) ,
(10)
= exp
p̊n pskT

We consider that the dimer formation proceeds through thereaction:2(X + ν1 A1 + · · · + νi Ai )  Z2 + 2(η1 B1 + · · · + ηj Bj ).
(17)
Under the consideration that the collisions of key molecules
controls the kinetics of the chemical reaction, the current density
J2 (t) is written as

where p̊n = c̊n kT and g̊n are, respectively, the partial pressure
and the chemical potential at a standard pressure ps of the n-mer.
The factor c̊n−1 c̊1 /c̊n c1 of the second term in the parenthesis on
the right-hand side of Equation (9) is written as c̊n−1 c̊1p1
1(g̊n − g̊n−1 − g̊1 ) − ln
.
(11)
= exp
c̊n c1kT
ps
By introducing the supersaturation ratio S defined as


 
p1
1
p1
(g̊c − g̊1 ) + ln
ln S = ln
=−
,
p1v
kT
ps



1
(g̊n − g̊n−1 − g̊c ) − ln S.
kT

(12)

(13)

b=
(14)

c1
c̊1

i
k=1
j
k=1

ckA /c̊kA
ckB /c̊kB

νk
ηk =

ω =1+

(15)

i

k=1

supposing that cn∗  cn∗ +1 for n∗  1.

In order to derive the formula as generally as possible, we
consider that an n-mer cluster Zn containing n of the key
molecule X is generated from the following chemical reactions:for 3  n  n∗ ,

i
k=1
j
k=1

pkA /p̊kA
pkB /p̊kB

νk
ηk ,

(20)

νk −

j


ηk ,

(21)

k=1

the factor c1 /c̊2 b2 of the second term in the parenthesis on theright-hand side of Equation (19) can be rewritten as follows;
because2.2. Case for a Multi-Element Grain

Zn−1 + (X + ν1 A1 + · · · + νi Ai )
 Zn + (η1 B1 + · · · + ηj Bj )p1
p̊1

where c̊kA and c̊kB (p̊kA and p̊kB ) are the concentrations (partial
gas pressures) of the kth gaseous reactants and products, respectively, in the gas in thermodynamic equilibrium at a temperature
T. As is the case for a single-element grain, by applying the lawof mass action and introducing ω (=0)3 defined asfor 2  n  n∗ .
Note that, as seen from Equations (1) and (14), the current
density Jn∗ cannot be evaluated without any relation betweencn∗ and Jn∗ +1 . Thus, in order to close Equations (1) to (3), we
introduce a closure relation that the current density from n∗ -merto (n∗ + 1)-mer, Jn∗ +1 , is approximated by
Jn∗ +1 (t)  αn∗ c1 cn∗ [1 − exp(γn∗ +1 )],

(18)

,

with

Then, the current density Jn (t) is expressed as
Jn (t) = αn−1 c1 [cn−1 − cn exp(γn )]2where the forward reaction coefficient α1 is the same as thatgiven in Equation (7). The form of the second term on theright-hand side of Equation (18) is based on the principleof detailed balance that the ratio of the forward reaction
coefficient to the backward reaction coefficient K is expressed asj
K = c̊2 [ k=1 (c̊kB )ηk /c̊1 ik=1 (c̊kA )νk ]2 in chemical equilibrium.
Then, the current density J2 (t) is represented as


c1 1
(19)
J2 (t) = α1 c1 c1 − c2c̊2 b2where g̊c and p1v are, respectively, the chemical potential ata standard pressure ps and the vapor pressure of the bulk
condensate, the exponent in Equation (11) is represented as
γn =

j
B ηk
k=1 (ck )iA νk
k=1 (ck )J2 (t) = α1 c12 − β2 c2

1
 i

νk ω
νk 2
i
A
c1
p1 p̊1 k=1 p̊kA /pkA
k=1 ck /c1=ηkjBc̊2 b2p̊2 p1 jk=1 p̊kB /pkB ηk
k=1 ck /c1
1
 
νk ω
i
ps ω−1 k=1 pkA /ps
×
ηk
j
B
p1
k=1 pk /ps


 
 i
νk 2 
νk −2
i
A
A
p̊
/p
p1
ps p̊1 2s
k=1
k=1 pk /ps
k
=
ηk
ηk
j
j
B
B
p̊2 psps
k=1 p̊k /ps
k=1 pk /ps1
  iνk ωAp1
k=1 pk /ps
×
ηk
j
B
ps
k=1 pk /ps(16)

where Ak (k = 1–i) and Bk (k = 1–j) denote the gaseous
reactants and products, respectively, and the stoichiometriccoefficients of the gaseous reactants (νk ) and products (ηk )are normalized to the key molecule. In the following, we
designate the physical quantities of the gaseous reactants Ak andproducts Bk by attaching the superscript A and B, respectively;
for example, the concentrations (partial pressures) of the gas
species Ak are denoted as ckA (pkA ). Below we first formulate theformation of dimer and then describe the formation of n-mer. In
the formulation, we assume that clusters with n  2 have thesame stoichiometric composition as the condensate.

Note that the formulation is applicable except for the case of ω = 0 such as
the endothermic dissociative reaction C2 H2 + C2 H2  (C2 )2 + 2H2 for theformation of the dimer (C2 )2 .

3

3

The Astrophysical Journal, 776:24 (17pp), 2013 October 10 
ps p̊1 2p̊2 ps
 
p1
×
ps

=

i
k=1
j
k=1
i
k=1
j
k=1

p̊kA /psνk

p̊kB /ps

ηk

2.2.2. Formation of n-mer (n  3)

2

For the chemical reaction (Equation (16)) for the formationof n-mers (3  n  n∗ ), the current density Jn (t) is given by

1
ω −2

νk

pkA /ps

Nozawa & Kozasa

(22)

,

η
pkB /ps k

Jn (t) = αn−1 cn−1 c1 − βn cn

we have
ps
p̊2



p̊1
ps

2 

i
k=1
j
k=1

νk

p̊kA /psηp̊kB /ps k

2

=

c1 Π
c̊2 b2



,
(23)


Π=

Ξ=

i
k=1
j
k=1

ckA /c1

νk

ckB /c1

ηk

i
k=1
j
k=1

ν
pkA /ps k
η .
pkB /ps kp̊n−1 p̊1
p̊n ps

1
ω

(24)

p̊1
ps

νk ,

(32)

2 

i
k=1
j
k=1

p̊kA /psνk

p̊kB /ps

ηk

2

(25)


1
(g̊2 − 2Δg̊gas ) ,
= exp
kT

k=1

νk g̊kA −

j




νk

With ln S defined by Equation (30), the exponent in Equation (35) is rewritten as



γn =

ηk g̊kB ,(27)

where g̊kA and g̊kB are the chemical potentials of kth gaseous reactants and products at a standard pressure ps , respectively, thefactor in the second term in the parenthesis on the right-handside of Equation (19) is reduced to(28)

The exponent in Equation (28) is written as
 




1
1
1
1
γ2 =
g̊2 − 2 −g̊c − Δg̊gas − 2 −
ln S,
kT
ω
ω
ω
(29)
where the supersaturation ratio S is defined as 1p1
ln S = − (g̊c − Δg̊gas ) + ln
+ ln Ξ,kT
psand Equation (19) is finally reduced to1J2 (t) = α1 c1 c1 − c2 exp(γ2 ) .
Π(36)

It should be mentioned here that the chemical reaction at
the time of formation is included in the current density only
through the factor Π given by Equation (24), ω in Equation (21),and the supersaturation ratio S defined by Equation (30).
The formation process of multi-element grains formulated byintroducing the key molecule can be treated as the naturalextension of the formation process of single-element grains;in fact, Equations (31) and (37) can be reduced to Equation (14)by substituting Π = ω = 1 calculated for νkA = νkB = 0.
Note that in principle the current density Jn can be evaluatedonce the chemical potentials of n-mers are given. However, thechemical potential is currently available only for tiny clusters(n  5) of very few materials of astrophysical interest (e.g.,
Goumans & Bromley 2012). Therefore, the so-called capillary
approximation, which is the practice for estimating the chemical
potential of an n-mer in terms of the chemical potential ofa monomer in the bulk condensate and the surface energy(Abraham 1974; Blander & Katz 1972), is generally adopted
for evaluating the current density as well as the steady-state
nucleation rate. For example, g̊n is expressed as

k=1

c1
1
1
(g̊2 − 2Δg̊gas )= exp
c̊2 b2ΠkT
  
p1
1ln+ ln Ξ .
− 2−
ω
ps

1
(g̊n − g̊n−1 − g̊c ) − ln S,
kT

and consequently, the current density Jn (t) for 3  n  n∗ isgiven byJn (t) = αn−1 c1 cn−1 − cn exp(γn ) .
(37)

with
i



1
(g̊n − g̊n−1 − Δg̊gas ) ,η = expkT
p̊kB /ps kp̊kA /ps

the factor c̊n−1 /c̊n b in Equation (33) can be reduced to


 
1
c̊n−1 1p1
− ln Ξ .
= exp(g̊n − g̊n−1 − Δg̊gas ) − ln
c̊n b
kT
ps
(35)

(26)

Δg̊gas = g̊1 +

i
k=1
j
k=1

(34)

Then, by applying the law of mass action,


ckA

ηk

From the law of mass action,

where

ps
p̊2

ckB

with αn defined by Equation (7), and the principle of detailedbalance leads to the equationc̊n−1 1.
(33)
Jn (t) = αn−1 c1 cn−1 − cnc̊n b

2− ω1

p1
Ξ
ps

j
k=1
i
k=1

2

g̊n = 4π a02 σ (n − 1) 3 + (n − 1)g̊c + g̊1

(30)

(38)

for a single-element grain (e.g., Yasuda & Kozasa 2012), where
σ is the surface tension of bulk condensate. By substituting g̊ninto Equation (13), γn for a single-element grain is written as22γn = μ (n − 1) 3 − (n − 2) 3 − ln S,
(39)

(31)
4

The Astrophysical Journal, 776:24 (17pp), 2013 October 10

Nozawa & Kozasa

where μ = 4π a02 σ/kT . Inspecting the change of the chemical potential γ2 for the formation of a dimer from the reactants given by Equation (29) for a multi-element grain and γn(Equation (A5)) for the formulation of the steady-state nucleation rate in comparison with the corresponding one given by
Equation (13) for a single-element grain, we can see that thefactor 1/ω in Equation (29) represents the contribution of the
key molecule to the change of chemical potential. Thus,
the chemical potential of an n-mer with n  2 for a
multi-element grain can be defined as

2 

1 3
1
1
2
g̊n = 4π a0 σ n −g̊c + Δg̊gas ,
+ n−
ω
ω
ω

The equation of the mass conservation for the key molecule
given by
 t
n
∗ −1
a 3 (t, t  ) 
ncn V +V (t  )Jn∗ (t  )dt ,c̃1 V − c1 V =
a03
t0
n=2
(46)
being divided by c̃1 V , is also rewritten as
1 − Y1 =γ2 = μ 2 −

1
ω

 23



1
ln S
− 2−
ω

for n = 2,

(40)

By introducing
Ki (t) =γn = μ1n−ω− ln S 23

1− n−1−
ω

(41)

(42)

for 2  n  n∗ ,

(43)

(48)

YkA =

ckA
c̃A
= k − νkA (1 − Y1 ),
c̃1
c̃1

(50)

YkB =

ckB
c̃B
= k + ηkB (1 − Y1 ),
c̃1
c̃1

(51)

respectively. In addition, since the grains nucleated in a timeinterval between t and t + dt have the radii between a anda + da, the size distribution function f (a) of newly formed
grains is calculated by(44)

f (a)da = c̃1 (t)I∗ (t)dt.

where Yn = cn /c̃1 represents the normalized concentration ofn-mers, and the normalized current density from the (n−1)-merto the n-mer In = Jn /c̃1 is given by⎧−1⎨[Yn−1 − Yn Π exp (γn )] for n = 2−1
In = τn−1
× [Yn−1 − Yn exp (γn )] for 3  n  n∗ ,⎩Yn−1 [1 − exp (γn )] for n = n∗ + 1

for i = 0−3

where c̃kA and c̃kB are the nominal concentrations of kth gaseous
reactant and product, respectively. These equations can besolved, given the initial abundances of gaseous reactants andproducts as well as clusters (Yn ) at t = t0 , together with the time
evolutions of gas temperature and density.
Note that Y1 = c1 /c̃1 represents the number fraction of the key
molecules left in the gas phase (so-called depletion efficiency),
K0 represents the number density of dust grains (K0 = ndust /c̃1 ),and K3 represents the number fraction of the key molecules
locked in dust grains. Hence, the condensation efficiency fcon (t)and volume-equivalent average radius aave (t) are calculated by1K3 (t) 3,(52)
fcon (t) = K3 (t) and aave = a0
K0 (t)

Then, Equation (43), being divided by c̃1 V , is reduced todYn= In − In+1dt

a i (t, t  ) 
dt ,
a0i

and

Equations (1) and (3) describe, respectively, the time evolution of concentrations of n-mer clusters and the conservationof the key molecule in a fixed volume. However, the formationof dust generally takes place in a cooling gas flow, such as thestellar winds from evolved stars and the expanding ejecta of
SNe and novae. Therefore, it is more useful to formulate theformation process of dust in a frame comoving with the gas.
Let us consider a specific volume V (t) comoving with the gas.
The nominal concentration of the key molecule c̃1 (t) is definedas the concentration without the depletion due to the formationof clusters and dust grains. Thus, c̃1 (t) at a time t holds therelation c10 V (t0 ) = c̃1 (t)V (t). The current from the (n − 1)mer to the n-mer in V (t) is given as Jn (t)V (t), and the time
variation of the number of n-mers in V (t), Nn (t) = cn (t)V (t),is expressed asfor 2  n  n∗ .

I∗ (t  )and Y1 is calculated from Equation (47). The concentrationsof gaseous reactants and products except for the key molecule,taking into account the depletion due to the formation of clusters
and the growth of grains, are evaluated, respectively, by

2.3. Formulation of Cluster and Dust Formation
in a Cooling Gas FlowdNn
= V (t) (Jn − Jn+1 )
dt

(47)

and I∗ = In∗ , K3 is calculated by solving the following
simultaneous differential equations:
 
⎧
i
i da
⎨
3
dKi
Ki−1 for i = 1−3I∗ (t)n∗ +
(49)
=
a0 dt
⎩
dt
for i = 0,I∗ (t)

 23

for 3  n  n∗ .

 t
t0

and


nYn + K3 .

n=2

so as to be consistent with the formula for a single-element
grain. Then, γn is evaluated by


n
∗ −1

(53)

3. APPLICATION TO DUST FORMATION
IN THE EJECTA OF SNe
We apply the derived formula to the formation of dust grains in
the ejecta of SNe. The aim is to reveal how the non-steady-state
effect and the physical conditions of the dust-forming regions
affect the condensation efficiency, average grain radius, andsize distribution. Also, in the following sections, we discuss theapplicability of the steady-state nucleation rate.

(45)

−1
= αn−1 c1 .
with τn−1

5The Astrophysical Journal, 776:24 (17pp), 2013 October 10

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3.1. Grain Species and Elemental Composition of the Gas
In the ejecta of SNe, the macroscopic mixing of elements islikely to be caused by the Rayleigh–Taylor instability. Although
Cherchneff & Dwek (2009) have claimed that hydrogen atoms
mixed with heavy elements play a critical role in the formationof precursor molecules of dust grains, the microscopic mixing of hydrogen penetrating into the inner layer is absolutely
impossible within the timescale of a few years, because molecular diffusion length is much smaller than the typical sizes of
the gas shell and clumps in the ejecta (Deneault et al. 2003;
Clayton 2013). Thus, we suppose the onion-like composition as
an elemental composition in the inner ejecta of SNe, and individually consider the formation of C and MgSiO3 grains, which
are expected to form in the carbon-rich layer and the oxygenrich layer of SNe, respectively. The chemical reactions at thetime of the formation of C and MgSiO3 grains and clusters,
along with the physical constants necessary for the calculations,
are taken from Table 2 of Nozawa et al. (2003).
It has been believed that carbon reacts with oxygen to produce
CO molecules, and that carbon atoms tied up in CO cannot be
available for the formation of C grains because CO is stableagainst dissociation. However, in the ejecta of SNe, some of CO
molecules might be destroyed by the collisions with energetic
electrons and charge transfer reactions with the ionized inert gas(Liu & Dalgarno 1994, 1996; Clayton et al. 1999, 2001; Clayton
2013). Here, we do not consider the formation and destruction
processes of CO molecules, since we treat the initial abundanceof carbon atoms available for dust formation as a parameter.
For MgSiO3 grains, we assume that the key molecule is the
SiO molecules, which have been considered to be a precursor for
the formation of silicate grains in SNe (e.g., Kozasa et al. 1989;
Kotak et al. 2009). The initial abundance of SiO is also treatedas a parameter. The number ratios of Mg and O atoms to SiO
molecules are taken to be cMg,0 /cSiO,0 = 2 and cO,0 /cSiO,0 = 20.
These abundance ratios are typical in the oxygen-rich layer ofsolar-metallicity SNe if almost all Si atoms are bound to SiO
molecules (see, e.g., Figure 1 of Nozawa et al. 2010).

Figure 1. Gas temperature T0 vs. the concentration of the key molecule c10
at ln S = 0 for C (solid) and MgSiO3 grains (dashed). The key moleculefor MgSiO3 grains are assumed to be SiO molecules, and the number ratiosof Mg and O atoms to SiO molecules are taken to be cMg,0 /cSiO,0 = 2 andcO,0 /cSiO,0 = 20, respectively.
(A color version of this figure is available in the online journal.)

As the gas cools down, it shifts from unsaturated states(ln S < 0) to supersaturated ones (ln S > 0) in which theformation of dust grains occurs. Thus, we take t0 as a time
when ln S = 0, and determine the equilibrium temperature T0
from the equation
c10 kT0A+ ln Ξ = 0(56)− B + ln
ln S =
T0
ps
for a given initial concentration of the key molecule c10 (andgiven abundance ratios of reactants and products). Throughout
this paper, the chemical potential for the formation of a bulk
condensate from reactants per key molecule is approximated as(g̊c − Δg̊gas )/kT = −A/T + B with the numerical values A
and B from Table 2 of Nozawa et al. (2003). Figure 1 showsthe equilibrium temperature T0 for C and MgSiO3 grains as afunction of c10 . For MgSiO3 , cMg /cSiO = 2 and cO /cSiO = 20
are adopted as mentioned above, but T0 is insensitive to thechanges in the abundance ratios. For both the grain species, T0
is higher for higher c10 , and T0  2000 K for C grains andT0  1500 K for MgSiO3 grains with c10 = 108 cm−3 .
As mentioned in Section 2, the condensation of dust isachieved through the formation of clusters and their growth.
Thus, it is convenient to define the timescales characterizingthese processes along with given temporal evolutions of the gas
density and temperature. The time evolution of current density,and thus the formation of clusters, are regulated mainly by thetime evolution of ln S in γn (see Equations (41), (42), and (45)).
Then, assuming that the depletion of the key molecule isnegligible at the earlier stage of dust formation, we introduce thetimescale of supersaturation τsat with which the supersaturation
ratio S increases as follows:
3(γ − 1) AγωA −1d ln S−1=−∼ τcool
τsat
≡
,
dt
t
T
γ −1
T
(57)

3.2. Evolution of the Gas Density and Temperature
The number density and size distribution of newly formed
dust depend on the time evolution of the density and temperatureof the gas. In the ejecta of SNe, the gas expands homologously
after ∼1 day, and the nominal concentration of a gas species
decreases as −3tc̃(t) = c0
,
(54)
t0
where c0 is the concentration at a time t = t0 .
On the other hand, the temperature of the gas in the ejecta
is determined by the balance between the energy input due tothe decay of radioactive elements and the energy output due
to expansion and radiative cooling. In this study, as in someprevious works (e.g., Kozasa et al. 1989), we assume the time
evolution of the gas temperature as −3(γ −1)t
T (t) = T0
,
(55)
t0

where τcool = t/3(γ − 1) is the timescale of gas cooling forthe current model. The second term on the right-hand side in
Equation (57) is generated from the time differentiation of theterm ln[(p1 /ps )Ξ] in Equation (30) for the time evolutions of
the gas density and temperature in Equations (54) and (55). On

where T0 is the gas temperature at t0 , and γ is a constantparameter. In the calculations of dust formation, we employ
the capillary approximation expressed by Equations (38) to (42)for evaluating the chemical potentials g̊n and γn .
6The Astrophysical Journal, 776:24 (17pp), 2013 October 10

Nozawa & Kozasa

the other hand, the growth of dust grains proceeds through thecollision (attachment) of the key species onto their surfaces, andthe collision timescale is defined as

 12
kT
−1
τcoll
≡ s4π a02 c̃1
.
(58)
2π m1

4.1. High-density Case
Figure 2 illustrates the formation process of C grains as afunction of time (x = t/t0 ) for c10 = 108 cm−3 , t0 = 300 days,and γ = 1.25; Figure 2(a) depicts the evolutions of abundancesof n-mer clusters Yn (n  2) as well as of the key molecules Y1 .
Figure 2(b) depicts the time evolutions of the current density ofn∗ -mer I∗ , the average grain radius aave , and the condensation
efficiency fcon . Note that, in the figure, the time evolution of theaverage radius is plotted after the time at which the condensation
efficiency reaches 10−10 , and hereafter the time is referred to as
the onset time xon of dust formation.5 Figure 2(c) presents thetime evolutions of the supersaturation ratio S and the critical size
ncrit that is defined as the size satisfying the condition γn = 0.
Figure 2(d) presents the time evolutions of current densities In
for the formation of given n-mers. The results for MgSiO3 grains
are provided in Figure 3.
For both grain species, the non-steady-state formation processof dust is described as follows. As can be seen from panels (c)and (d) of Figures 2 and 3, the increase in ln S, induced by thedecrease in gas temperature with time, leads to the progressive
formation of clusters with larger n. Once ln S reaches 2, at
which ncrit  100, I∗ becomes high enough that some amountof grains with n  100 start to form. A further increase in ln S
enhances In , producing a much greater number of clusters andgrains. Note that the onset time of dust formation is later for
C grains (xon  1.08) than for MgSiO3 grains (xon  1.03),
which stems partly from the longer timescale of supersaturationand partly from the larger surface tension: τsat  10 days andσ = 1400 erg cm−3 for C grains, whereas τsat  2.5 days andσ = 400 erg cm−3 for MgSiO3 grains. Since newly formed
grains grow efficiently to cause the consumption of the key
molecule, the supersaturation ratio S reaches a maximum andthen decreases. The critical size nsmax
crit at S = Smax is ∼20 for C
grains (∼10 for MgSiO3 grains). The current densities In for theformation of clusters with n < nsmax
crit drop almost abruptly justbefore S becomes Smax , whereas those for n > nsmax
crit -mer reachalmost the same maximum value at S  Smax and then decrease
quickly. Accordingly, I∗ has a sharp peak; the gas temperature
at the peak of I∗ is 1850 K (1490 K) for C (MgSiO3 ) grains,
being lower than its equilibrium temperature T0 = 1990 K(1530 K). After that, dust grains continue to grow until fcon  1
by consuming almost all of the key molecules.
It should be noted here that the current densities for theformation of clusters with n  nsmax
crit around S = Smax , being
almost independent of n, reach a steady-state value in this highdensity case for which the condition of τcoll
τsat is satisfied(see Section 5). In addition, the steady-state value excellently
matches the steady-state current density Is , as seen from panels(b) and (d) in Figures 2 and 3, where we overplot the results
obtained from the steady model (dotted lines).
The behavior of the formation process of clusters and grainsin this high-density case can be qualitatively understood byinspecting the time evolution of eγn regulating the backward
reactions in Equation (45). The factor eγn is a decreasing
function of n, and in a supersaturated gas, it becomes below
unity for n larger than a critical size ncrit approximately given as


2μ 3
1
(60)
ncrit − ω
3 ln S3.3. Calculations of Dust Formation
In what follows, we adopt n∗ = 100 as the minimum numberof the key molecule that is regarded as a grain. This number
corresponds to the minimum radius of grains: a∗ = 5.9 Å
for C grains and a∗ = 10.8 Å for MgSiO3 grains. The effectof changing n∗ on the results of calculations is examined inAppendix B. We take a sticking probability of sn = 1 onto all
sizes of clusters and grains. Among the three free parameters
c10 , γ , and t0 in the models described above, we adopt γ = 1.25
and t0 = 300 days as the standard values representing the gas
cooling rate and the equilibrium time at which ln S = 0 in the SN
ejecta, although the cases for different γ and t0 are examined aswell. The calculations are performed up to the time long enoughso that the current density I∗ and the growth rate of grains canbecome negligibly small.
The formation of dust grains from the gas phase takes placewhen ln S > 0.4 On the other hand, our non-steady-statecalculations with the initial conditions of ln S < 0 and cn = 0(n  2) have confirmed the following: the formation of small(n  10) clusters is possible even if ln S < 0, although theirabundances are very small. The abundances of small clusters at
ln S  0 are the same as the steady-state values as a result of toohigh backward reaction rates (exp(γn )  1; see Equation (45)).
Therefore, the steady-state abundances of small clusters at
ln S = 0 are taken as the initial values for the simulations
starting from t = t0 .
We also compare the results obtained from a set of formulae
as described in Section 2 (hereafter referred to as the nonsteady model) with those calculated from the revised steadystate nucleation rate (hereafter the steady model) given by
Js = scrit Ω0

2σ
π m1

 12



4 μ3
,
c12 Π exp −27 (ln S)2(59)

for which the detailed derivation is presented in Appendix A.
In the steady model which does not involve the formation ofclusters, the calculations of dust formation are performed byreplacing I∗ with Is = Js /c̃1 in Equation (47), without the firstterm on the right-hand side, and by replacing n∗ with ncrit given
in Equation (A9). In what follows, we refer to In and Is as
current densities and steady-state current density, respectively,
for convenience.
4. RESULTS OF DUST FORMATION CALCULATIONSGiven the parameters γ and t0 representing the cooling anddynamical times of the gas, the concentration c10 controls the
behavior of the formation processes of clusters and grains. We
first present the results of the calculations in the case that theinitial concentration of the key molecule is high enough thatthe assumption of a steady state is considered to be a good
approximation. Then, we demonstrate the results for the lowdensity case. We also explore the dependence of the results on
t0 and γ in Section 4.3.

5 The threshold value 10−10 is arbitrary, but, given that f
con rises up quickly
with time, it does not affect the conclusion of this paper as long as the value isless than ∼10−5 .

4The steady-state nucleation rate in Equation (59) can be applied only for
S > 1; see Appendix A.

7

The Astrophysical Journal, 776:24 (17pp), 2013 October 10

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Figure 2. Formation process of C grains as a function of time (x = t/t0 ) and the resulting size distribution for c10 = 108 cm−3 , t0 = 300 days, and γ = 1.25;(a) abundances of the key molecule (n = 1) and n-mer clusters with n = 2–100; (b) current density of n∗ -mer (I∗ ), average grain radii (aave ) in units of μm, andcondensation efficiency (fcon ); (c) supersaturation ratio (S) and critical size (ncrit ); (d) current densities of n-mers (In ) and the steady-state current densities (Is , dottedline); (e) final size distribution spectra of clusters (blue) and grains (red). For comparison, the results from the steady model are shown by dotted lines in (b) and (e).
In (d), I20 , I100 , and Is are overlapped.

for ncrit  1. Note that this expression for ncrit is equivalent
to that for the steady-state current density as defined by
Equation (A9). At the initial phase of ln S  2, ncrit is verylarge (ncrit > 100; see Figures 2(c) and 3(c)), so the backward
reactions are dominant (eγn  1) for any size of n-mers. In thiscase, the abundances of n-mers at each time approximately takethe values Yn  Yn−1 e−γn , and the current to larger n-mers is
considerably smaller (In /In−1
1). On the other hand, as ln S
increases with time, ncrit falls below n∗ = 100, reaching down
to 10–20. In this phase, the backward reactions are suppressed(eγn < 1) for n  ncrit , so the n-mers can grow exclusively
through the collisions with the key molecules. Since the collisiontimescale is extremely small (τsat /τcoll  1) in this high-density
case, the reactions for n  ncrit proceed instantaneously, whichmakes the current densities In a steady state (i.e., In  In−1
for n  ncrit ). Furthermore, the agreement of I∗ with Is can beinterpreted as follows: even if ln S is high enough, the backward
reactions remain predominant for n  ncrit , where the relation
Yn  Yn−1 e−γn holds. Thus, the abundance of ncrit -mers can beapproximately estimated asYncrit  Y1 exp −


ncrit



γn

n=2



1
 Y1 exp −μ ncrit −
ω

 23



1
ln S .
+ ncrit −
ω
(61)

8

The Astrophysical Journal, 776:24 (17pp), 2013 October 10

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Figure 3. Same as Figure 2 but for MgSiO3 grains.

Then, the current of n∗ -mers is found to be on the orders of


4μ3
2
−1
3τ
.
I∗  Incrit ∼ τn−1
Y
∼
(n
−
1)
Y
exp
−
n
crit
1
crit
coll
crit −127(ln S)2(62)

S to decrease in the course of time evolution of gas density andtemperature, as is demonstrated in the case of a high initial gas
density.
Figures 2(e) and 3(e) show the final size distributions ofgrains and clusters. The size distribution is lognormal-like withaave,∞ = 0.07 μm for C grains and aave,∞ = 0.08 μm for
MgSiO3 grains. Because the size distributions of grains follow
the time evolution of I∗ , and Is is equal to I∗ , we can see thatthe size distribution in the non-steady model is the same asthat of the steady model. In this high-density case with thefinal condensation efficiency fcon,∞  1, the abundance ofthe key molecules locked in the clusters is extremely small( nYn < 10−7 ). Our calculations show that, when the initialconcentration is as high as c10  107 cm−3 , all carbon atoms

The exponential term in Equation (62), which dominates thetime evolution of the current density, has the same form as theterm in the steady-state current density Is . Hence, the current
density for the formation of n∗ -mers I∗ is essentially equal to thesteady-state current density Is . This allows us to conclude thatthe application of the steady-state current density could be valid
as long as the consumption of the key molecules due to formationand growth of clusters and grains causes the supersaturation ratio
9The Astrophysical Journal, 776:24 (17pp), 2013 October 10

Nozawa & Kozasa

Figure 4. Same as Figure 2 but for c10 = 105 cm−3 , t0 = 300 days, and γ = 1.25.

and SiO molecules are ultimately locked in grains with the size
distributions identical to those in the steady models.

increase (Figures 4(c) and 5(c)), and I∗ gradually decreases
after passing the peak, in contrast to the high-density case
(Figures 4(d) and 5(d)), which results in the formation of manysmall grains with n  1000. Finally, the depletion due toformation of clusters and grains and/or the dilution due tothe expansion of the gas makes the concentration of the key
molecules too low to advance the further growth, and theabundances of clusters approach to the constant values (seeFigures 4(a) and 5(a)).
Figures 4(b) and 5(b) compare the time evolutions of I∗ (andIs ), aave , and fcon between the non-steady and the steady models.
It can be seen that the non-steady current density I∗ rises up at atime later than the steady-state current density Is , corresponding
to the later onset time of dust formation, and that its peak value

4.2. Low-density Case
Figures 4 and 5 show the formation processes of C andMgSiO3 grains, respectively, for c10 = 105 cm−3 , t0 = 300 days,and γ = 1.25. Even for such a low initial gas density, theformation of n-mers progresses as ln S increases, as is thesame as the high-density case. However, even if grains withn  100 are produced, they cannot grow efficiently throughattachment of the key molecules because the collision timescale
is considerably longer (τsat /τcoll ∼ 1–10). Thus, despite thefact that the formation and growth of clusters and grains
significantly consume the key molecules, ln S continues to
10

The Astrophysical Journal, 776:24 (17pp), 2013 October 10

Nozawa & Kozasa

Figure 5. Same as Figure 4 but for MgSiO3 grains.

is much smaller than that in the steady model. This is at odds
with the high-density case, indicating that the steady model is nolonger appropriate for this low-density case with τsat /τcoll  10
during the evolution. For clusters with n  ncrit , as mentioned in
the previous subsection, Yn evolves as Yn  Yn−1 e−γn , and Incrit
at each time is on the order of Is (cf. Equation (62)). However,the collision timescale is too long for the current densities In toestablish a steady state at n  ncrit , so I∗ remains much lower
than Incrit  Is (Figures 4(d) and 5(d)).
There also appear to be differences in the final average radiusand condensation efficiency of dust grains between the nonsteady and steady models. The final condensation efficiency inthe non-steady model is fcon,∞  0.3 for C grains (fcon,∞ 
0.01 for MgSiO3 grains), which is lower than fcon,∞ = 1 in thesteady model for both the grain species. In both models, few
key molecules are finally left in the gas phase (fdep,∞ < 10−5 ),indicating that 70% (99%) of key molecules are bound toclusters for the non-steady model. On the other hand, the final
average radius of dust grains is larger for the non-steady model(aave,∞ = 0.0007μm for C grains and 0.0011 μm for MgSiO3
grains) than for the steady model (aave,∞ = 0.0004 μm for C
grains and 0.0005 μm for MgSiO3 grains).
The discrepancies in the average grain radius and condensation efficiency between the two models simply reflect the difference in the minimum size considered as bulk grains. As can beseen from Figures 4(e) and 5(e), the final size distribution in thesteady model is quite similar to the combined size distributionof grains and clusters in the non-steady model. This is because
11The Astrophysical Journal, 776:24 (17pp), 2013 October 10

Nozawa & Kozasa

ncrit and τcoll are the same in both the models, so the numberof ncrit -mer formed at a given time and the growth rate are essentially identical. However, in the steady model, clusters thatmeet n  ncrit are taken as bulk grains, and ncrit is normally lessthan 10, even down to 1–2 in the low-density case as shownin Figures 4 and 5. Thus, the steady model, which regards small
n-mers as grains, leads to a smaller average grain radius anda higher condensation efficiency. Although we cannot have aclear number of constituent atoms to distinguish between small
clusters and grains, it must be unreasonable to consider clusters
with n  10 to hold the properties of bulk grains.
The results of calculations for this low initial density clearlyattest that the application of the steady-state current density
overestimates the condensation efficiency and underestimates
the average grain radius for dust formation in less dense and/or
rapidly cooling environments. Also, it may be useful to point out
here that the application of the steady-state current density witha given cutoff value of critical size (e.g., Bianchi & Schneider2007) in low-density/rapidly cooling environments, arguing theinadequacy of the extension to smaller critical sizes, leads tothe considerable depression of the condensation efficiency andenhancement of the average grain radius, and cannot reproduce
the combined size distribution of clusters and grains.
4.3. Dependence on t0 and γ
In Sections 4.1 and 4.2 we have demonstrated how the initialconcentration of the key molecule affects the formation process
and properties of newly formed grains. In this subsection, weinvestigate the dependence of the other free parameters t0 and γ
on the formation process, average radius, and size distributionof dust grains.
Figure 6(a) plots the formation process of C grains for t0 =100, 300, and 600 days with c10 = 107 cm−3 and γ = 1.25.
The results of the calculations show that a larger t0 leads toa smaller peak of I∗ as well as a little earlier onset time ofdust formation xon . This is explained as follows: as seen from
Equation (57), the timescale of supersaturation in terms of x,(d ln S/dx)−1 , is independent of t0 , while the timescale of grain
growth, (d ln a/dx)−1 ∝ τcoll /t0 , is inversely proportional to t0 .
This means that the increase in t0 makes grain growth more
active but has little impact on the number of clusters at a given
time x. Therefore, for a larger t0 , dust grains capture the key
molecules more efficiently through their growth, which causesa faster rise of fcon and a faster drop of I∗ (a smaller peak ofI∗ ) and results in the grain size distribution weighted toward alarger radius (see Figure 6(b)). Thus, the increase in t0 enhances
the effect of grain growth relative to formation of clusters andacts to produce dust grains with large average radii.
Figure 7 gives the results of the calculations for the formationof MgSiO3 , adopting γ = 1.25, 1.4, and 1.6 for c10 = 107
cm−3 and t0 = 300 days. For a larger γ , which corresponds
to a more rapid cooling of the gas, the onset time of dust
formation is earlier, and I∗ has a higher peak. Again considering
the timescale in terms of x, the timescale of supersaturation,(d ln S/dx)−1 ∝ x −3γ +4 /(γ − 1) decreases with increasing
γ , whereas the timescale of grain growth, (d ln a/dx)−1 ∝x 3(γ +1)/2 , increases. Hence, for a larger γ , a more rapid increasein ln S, as well as a more rapid decrease in ncrit , leads to theformation of a larger number of clusters with n  ncrit at an
earlier time before grain growth efficiently consumes the key
molecules. It should be noted that the collision timescale τcollin the models considered here is still short enough so that theconsumption of the key molecules due to grain growth makes

Figure 6. (a) Time evolutions of I∗ , aave , and fcon and (b) the final size
distribution spectra of newly formed C grains for t0 = 100 days (dotted),
300 days (solid), and 600 days (dashed). The other parameters are set to be
c10 = 107 cm−3 and γ = 1.25.

ln S decrease during the evolution. Consequently, in the modelwith larger γ , I∗ increases more rapidly and has a higher andnarrower peak at an earlier time, and the average grain radius
as well as the peak radius of size distribution becomes smaller,
as seen in Figures 7(a) and (b). In conclusion, the increase inγ makes the formation of clusters more active relative to grain
growth and induces the formation of dust grains with small
average radii even if τsat /τcoll is not much larger than unity.
5. THE SCALING RELATIONS FOR AVERAGE GRAINRADIUS AND CONDENSATION EFFICIENCYThe objects to be clarified in the study of dust formation inastrophysical environments are not only the chemical composition of dust grains but also their amount and size distribution. In
the ejecta of SNe, the knowledge of size distribution of newly
formed dust is crucial for unraveling what amount and size ofdust grains are finally ejected from SNe to the ISM, because
the destruction efficiency of dust by the reverse shock heavily
depends on the size distribution (e.g., Nozawa et al. 2007). Our
results show that the size distribution of dust formed for given
time evolutions of gas density and temperature is lognormal-likeas long as τsat /τcoll  1 during the formation of dust. Therefore,
the average grain radius can be taken as a representative measurement of the size distribution. The condensation efficiency
is also a fundamental quantity in estimating the mass of newly
formed dust. In this section, we explore how the average grainradius and condensation efficiency can be constrained from aphysical condition at the time of dust formation, and derive the12The Astrophysical Journal, 776:24 (17pp), 2013 October 10

Nozawa & Kozasa

Figure 7. (a) Time evolutions of I∗ , aave , and fcon , and (b) the final size
distribution spectra of newly formed MgSiO3 grains for γ = 1.25 (solid),
1.4 (dashed), and 1.7 (dotted). The other parameters are set to be c10 = 107
cm−3 and t0 = 300 days.

Figure 8. Final condensation efficiencies fcon,∞ and average grain radii aave,∞as a function of Λon for γ = 1.1, 1.3, 1.5, and 1.7; (a) for C grains and (b) for
MgSiO3 grains. The thin gray lines are the results from the steady model with
γ = 1.3.

scaling relations for the average radius and condensation efficiency, referring to the results of the calculations presented in theprevious section. In addition, we clarify in what conditions thesteady-state nucleation rate is applicable. Then, we shall present
some examples of the application of the scaling relations for theformation of dust in SNe.

order to get the information on physical conditions in the ejecta
from the observations and vice versa. One of the best indicators
for the formation time of dust would be the condensation time, tc ,defined as the time when I∗ has a peak. However, the depletionof the key molecules due to the formation and growth of clusters
and grains at tc is considerably large (Y1  0.1–0.2 at tc ), which
has significant effects on the relevant physical quantities. Thus,
we adopt as the time of dust formation the onset time of dust
formation, ton , defined as the time at which fcon reaches 10−10 .
The non-dimensional physical quantity Λon characterizing theformation process of dust grains is given as

5.1. The Physical Quantity Characterizing Dust Formation
Process and the Scaling Relations
The results in Section 4 demonstrate that the formation
process of dust is determined by the competition between theformation of clusters and the growth of grains. Although theonset time of dust formation (and the condensation time atwhich the current density I∗ reaches the maximum), averagegrain radius, and condensation efficiency depend on c10 , γ , andt0 in a complicated manner, we have shown that the behavior
of formation processes of dust grains can be qualitatively
interpreted in terms of τsat and τcoll during the formation of
dust. It has been shown in the studies based on the steady-state
nucleation rate that the average radius and number density of
dust grains formed in a cooling gas undergoing macroscopic
motion can be scaled by a non-dimensional quantity Λ =τsat /τcoll at the condensation time when the nucleation rate
reaches the maximum (e.g., Hasegawa & Kozasa 1988).
The formation time of dust is the most direct information thatcan be obtained from observations of SNe. Hence, it is usefulto relate the average grain radius and condensation efficiency to
the gas density and temperature at the time of dust formation, in

Λon ≡


1
kTon 2ton
Tonτsat (ton )∼× s4π a02 c̃onτcoll (ton )3(γ − 1) A
2π m1
3




2
Ton
ton
c̃on
C  s ,∼γ − 1 1.0 108 cm−3
2000 K
300 days(63)

where c̃on = c̃1 (ton ) and Ton = T (ton ), and C = 1.94 × 103(1.15 × 103 ) for C grains (MgSiO3 grains). In Equation (63),−1−1
we employ the approximation τsat
 (A/T )τcool(see
Equation (57)), although we have calculated Λon without
using this approximation in the following figures.
Figure 8 presents the final average grain radius aave,∞ andcondensation efficiency fcon,∞ as a function of Λon calculatedfor γ = 1.1, 1.3, 1.5, and 1.7 by covering a wide range of c10
and t0 . The figures show that, as Λon increases, aave,∞ and fcon,∞13

The Astrophysical Journal, 776:24 (17pp), 2013 October 10

Nozawa & Kozasa

Table 1Fitting Parameters for Equations (64) and (65)C

MgSiO3

5.946 × 10−4

2

−1.21
0.854

1.076 × 10−3
−1.19
0.871

χ1
χ2
χ3

4.15
1.91
0.088

6.38
2.17
0.049

a∗ (μm)1

Note. The minimal radius of bulk grain a∗ is for the number of constituent
monomer n∗ = 100.

increase, and fcon,∞ = 1 at Λon  20–30 for both C and MgSiO3
grains; for a larger Λon , grain growth becomes more dominant
over the formation of clusters, and, as a result, larger grains
are formed to lock up all of the key molecules. The remarkableconsequence of Figure 8 is that aave,∞ and fcon,∞ for different
γ (especially for γ  1.2) are, respectively, plotted almost
completely as a single curve for both C grains (Figure 8(a)) andMgSiO3 grains (Figure 8(b)). This means that the average grainradius and condensation efficiency can be uniquely determinedby one parameter Λon , except for C grains formed in extremely
slowly cooling gas with low densities corresponding to the caseof γ = 1.1 with Λon  10. In the figures, we also plot aave,∞and fcon,∞ from the steady model for γ = 1.3. They deviate
from those from the non-steady model at Λon  30, where thesteady model predicts too small aave,∞ to be regarded as bulk
grains with keeping fcon,∞ = 1. In other words, the steady-state
nucleation rate is applicable only if Λon  30.
We derive the approximation formulae describing the dependence of aave,∞ and fcon,∞ on Λon for the non-steady model,
which are, respectively, given by
aave,∞− 1 = 1 + 2 log Λon
(64)
log
a∗

Figure 9. Final average grain radii aave,∞ and condensation efficiencies fcon,∞calculated by the fitting formulae, respectively, Equations (64) and (65) as afunction of Λon (solid lines), and the results of the simulations for γ = 1.25
(dashed lines); (a) for C grains and (b) for MgSiO3 grains. The hatched regions
display the typical ranges of Λon for dust formation (a) in the carbon-rich He
layer and (b) in the oxygen-rich layer, for Type II-P SNe (cyan; Nozawa et al.
2003) and Type IIb SNe (magenta; Nozawa et al. 2010).

andlog fcon,∞ = χ1 [tanh (χ2 log Λon + χ3 ) − 1],(2003), the concentration of carbon atoms in the carbon-rich
He layer of Type II-P SNe is found to be c̃on  108 –109 cm−3
at ton = 330 days. For the reasonable values of γ (1.5–1.7)
and Ton (2000 K), Equation (64) presents aave,∞ =0.03–0.3
μm (Λon = (0.3–8) × 104 ; see Figure 9(a)), which is consistent
with the average grain radii given in Figure 7 of Nozawa et al.
(2003). In Type IIb SNe with much less massive hydrogenenvelopes, the density of the gas at ton = 330 days is by afactor of 100–300 lower than in Type II-P (see Figure 2 of
Nozawa et al. (2010)), and Equation (64) leads to aave,∞ ∼0.001 μm (Λon ∼ 100). This radius also agrees with thatobtained in Nozawa et al. (2010). These simple analyses suggestthat our previous calculations by a theory of the non-steady-state
nucleation and grain growth applying the steady-state nucleation
rate with a relaxation time toward the steady-state rate (seeNozawa et al. 2003) were performed under the condition thatthe steady-state approximation is appropriate.
Observationally, there have been few studies that reported atypical size of dust formed in SNe. Recently, Maeda et al. (2013)
clearly detected the formation of C grains in the luminous TypeIIn SN 2010jl around day 550 after the explosion by the opticalthrough the near-infrared observation. They suggested that the
typical radius of the dust grains is less than 0.1 μm (moreprobably 0.01 μm) to account for the wavelength dependenceof obscuration of hydrogen emission lines. These C grains(65)

where the fitting parameters 1 , 2 , and χk (k = 1–3) are given in
Table 1. In Figure 9, aave,∞ and fcon,∞ calculated by the above
fitting formulae are compared with the results of simulationsfor γ = 1.25. Equation (64) reproduces the calculated averageradii with the error less than 5% for Λon  106 .
It should be emphasized here that the scaling relations
given above are independent of the initial conditions of thecalculations and the time evolution of the gas density. In
fact, we have performed the dust formation calculations for
expanding gas flows with a constant velocity by changing γ , andconfirmed that the resulting aave,∞ and fcon,∞ entirely coincidewith those shown in Figure 8. Therefore, the average grainradius and condensation efficiency of a given grain species canbe universally described by the corresponding non-dimensional
physical quantity Λon .
5.2. Application of the Scaling Relations
to Dust Formation in SNe
Equation (64) allows us to estimate the typical size of newly
formed grains once we know the density of the gas in the ejecta
and the onset time of dust formation (or the formation timeof dust). For example, from Figures 1 and 2 in Nozawa et al.
14

The Astrophysical Journal, 776:24 (17pp), 2013 October 10

Nozawa & Kozasa

are likely to have formed not in the ejecta but in relatively
dense clumps in the shocked circumstellar shell, however, it
would be interesting to compare this with our results. Basedon the simple argument of optical depth, Maeda et al. (2013)
showed that the gas density in the interclump medium mustbe cgas  109 cm−3 . Then, assuming the abundance of carbon
atoms to be c1 /cgas = 10−4 , and with typical values of γ and Ton ,Λon  8 × 102 (D/100), where D is the density contrast between
the dense clumps and the interclump medium. Adopting D =100–1000, Equation (64) yields aave,∞  0.01–0.08 μm, whichis consistent with the grain size estimated from the observation.
This emphasizes that Equations (64) and (65) could provide
powerful constraints on the properties of newly formed grains
through the gas density and condensation time extracted fromobservations as well as those predicted from theoretical models.

that the steady-state nucleation rate is applicable under thecondition of Λon  30, irrespective of grain species; otherwise
the application of the steady-state nucleation rate results inthe formation of a large number of unreasonably small grains
with the condensation efficiency considerably higher than that
calculated by the non-steady rate. Furthermore, we have derived
the scaling relations for the average radius and condensation
efficiency of C and MgSiO3 grains as a function of Λon . The
approximation formulae depend neither on the time evolutionof the gas density and temperature, nor on the initial condition,and thus could serve as a universal relation to predict the mass
and average size of newly formed grains from the observations
and/or the model calculations of explosions of SNe and novae
as well as mass-loss winds from stars.
We are grateful to the anonymous referee for critical comments that improved the manuscript. This research has been
supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan, and by the Grant-in-Aid
for Scientific Research of the Japan Society for the Promotionof Science (20340038, 22684004, and 23224004).

6. SUMMARY
We have developed a new formulation describing the nonsteady-state formation of small clusters and grains in a selfconsistent manner, taking into account chemical reactions at
the time of dust formation and assuming that the temperaturesof small clusters are the same as that of the gas. Given
the chemical potentials of small clusters, the formula can beapplied to investigate the formation process of dust grains
in rarefied astrophysical environments, where the steady-state
nucleation rate is not applicable. It should be pointed out herethat the formation process of dust is formulated in the present
study under the assumption that the smallest cluster (dimer)
has the same chemical composition as the grains. However,the formulation can be extended for the case that chemicalcompositions of small clusters are different from the grains,given the chemical reaction paths and chemical potentials. Also,
the formula can be extended and applied to explore the effectsof the difference in temperatures of small clusters and the gas(Kozasa et al. 1996; Yasuda & Kozasa 2012) as well as thetemperature fluctuation (Keith & Lazzati 2011) and the shapeof small clusters (Fallest et al. 2011) on the formation process ofdust grains. These subjects will be studied in the future works.
Applying the new formulation with the capillary approximation for evaluating the chemical potentials of small grains, wehave investigated the formation processes of C and MgSiO3
grains over a wide range of physical conditions expected in theejecta of SNe. The results of the calculations have shown that the
behavior of non-steady-state formation process of small clusters and grains can be qualitatively interpreted in terms of thetemporal evolutions of the collision timescale of key molecule
τcoll and the supersaturation timescale of the gas τsat during theformation of dust. In the condition that τcoll
τsat , the formation process of dust grains can be completely reproduced by thesteady-state nucleation rate, and grains form with the condensation efficiency fcon  1, otherwise the formation of clusters
and grains proceeds in a non-steady state, and the resulting condensation efficiency is fcon < 1, with the efficiency of growthof clusters and grains being depressed considerably.
Analyzing the results of the model calculations, we foundthat the condensation efficiency and average radius of newly
formed grains can be fully described by one non-dimensional
quantity Λon , the ratio of the supersaturation timescale to thecollision timescale at the onset time of dust formation, although
the time evolutions of gas temperature and density considerably
influence the formation process as well as the average grainradius and condensation efficiency. Also, we have revealed

APPENDIX AA REVISED FORMULA OF STEADY-STATENUCLEATION RATE
The rate of the steady-state homogeneous nucleation accompanied by chemical reactions of relevant gaseous molecules has
been developed by introducing the concept of the key species(Kozasa & Hasegawa 1987) and has been applied for the formation of dust not only in the ejecta of SNe but also in thesolar nebula (Kozasa & Hasegawa 1988) and in circumstellar
envelopes of asymptotic giant branch stars (e.g., Kozasa & Sogawa 1997, 1998; Chigai et al. 1999). Yamamoto et al. (2001)
tried to generalize the formulation for the steady-state nucleation rate and derived an analytic formula in which the effectof chemical reactions at the time of formation is included as acorrection factor in the form of functions of partial pressures ofreactants (except for the key species) and products, as well as
the standard gas pressure. However, the formula is incomplete
because the nucleation rate depends on the standard gas pressure
explicitly in the case that the number of product molecules is different from that of reactant molecules except for the key species.
Thus, in this appendix, we reformulate the analytic formula forthe rate of the steady-state nucleation proceeding through thechemical reactions at the time of formation.
Here we consider the formation of dust grains throughthe chemical reactions (16) and (17). In a steady state, thecurrent density Jn from (n − 1)-mer to n-mer is independentof n, being identical to the steady-state nucleation rate Js .
Thus, Equations (19) and (33) lead to the following relations,
respectively:
Js
c2 1
=1−
2
c̊2 b2
α1 c1

(A1)

and
Js
cn−1
cn 1
=
−
c̊n−1
c̊n b
αn−1 c1 c̊n−1

for n  3,

(A2)

where b is defined by Equation (20). By summing up
Equation (A1) and Equation (A2) multiplied by 1/bn−1
15

The Astrophysical Journal, 776:24 (17pp), 2013 October 10

Nozawa & Kozasa

successively, the nucleation rate Js can be derived from theequation


n

1
1
cn 1
Js
+
.
(A3)
=1−
2
i
c̊n bnα1 c1 i=2 αi c1 c̊i bThis revised formula of the steady-state nucleation rate includes
a correction factor Π defined as

By applying Equations (28) and (35), the factor 1/c̊n bn on thesecond term in the right-hand side of Equation (A3) is written as

(A12)


Π=

γn =

1
kT

 




1
1
1
g̊n − n −g̊c − Δg̊gas − n −
ln S,
ω
ω
ω
(A5)

where S is the supersaturation ratio defined by Equation (30).
Since γn  −n ln S and c̊n bn ∝ S n for n  1, the second termon the right-hand side of Equation (A3) approaches to zero for
S > 1 as n → ∞. Then, the steady-state nucleation rate Js isgiven by
∞
∞


1
1
1
1
1
=
+
=
+
exp(γi ).
Js
α1 c12 i=2 αi c1 c̊i bi
α1 c12 i=2 αi c12 Π

νk

ckB

ηk



1
ω

=

i
k=1
j
k=1

ckA /c1

νk

ckB /c1

ηk

1
ω

.

DEPENDENCE ON n∗
Here we describe how the change in n∗ in Equation (1) affects
the formation process of dust as well as the average radius andcondensation efficiency of newly formed grains. First of all,as demonstrated in Section 4.1 and discussed in Section 5, thechange of n∗ does not influence the results of calculations as longas the formation of dust grains proceeds under the conditionof Λ = τsat /τcoll  1 and n∗  nsmax
crit , because the current
densities In for n > nsmax
crit achieve a steady-state value, being
independent of n. Thus, here we examine the effect of change
in n∗ for the model in which the non-steady-state effect for theformation process of grains is significant.
Figure 10(a) presents the time evolutions of the current I∗ ,average grain radius aave , and condensation efficiency fcon of
C grains for n∗ = 100, 300, and 1000 with c10 = 106 cm−3 ,t0 = 300 days, and γ = 1.25. The figure shows that, as n∗increases, the onset time of dust formation is slightly delayed,
and the peak value of I∗ decreases. In addition, a larger n∗ leads
to a smaller fcon and a larger aave : fcon = 1.0 and aave = 0.0016
μm for n∗ = 100, whereas fcon = 0.9 and aave = 0.0019 μm forn∗ = 1000. Thus, the formation process as well as the resulting
properties of dust grains seems to depend on the adopted valueof n∗ .
However, we recognize that the dependence of n∗ is superficial. As is seen from the final size distributions displayed in
Figure 10(b), where the values for n∗ = 300 and 1000 are
shifted down, respectively, by factors of 0.1 and 0.01 for clarity, the combined size distributions of clusters and grains are
identical to each other, regardless of n∗ . It should be notedthat this demonstrates that the change in n∗ does not influence the formation processes of clusters and grains themselves.
The independence of the combined size distribution on n∗ canbe understood as follows: the evolutions of In and Yn are thesame for n  n∗,min = 100 in all the three cases. On the otherAs long as 1/αc12
1, the summation in Equation (A6) canbe replaced with the integration, and the nucleation rate Js canbe approximately calculated as
 ∞
1
1
1
 2
exp(γi )di.
(A7)
Js
c1 Π 2 αiUnder the capillary approximation given by Equation (40), γnis expressed as
2 


1 3
1
ln S,
(A8)
− n−
γn = μ n −
ω
ω

(A9)

Thus, the integration of Equation (A7) with the saddle-point
method results in the steady-state nucleation rate given by


2
 12
3
2σ
ncrit
Js = sncrit Ω02
π mncrit ,1(ncrit − 1/ω) 3


4 μ3
,
× c12 Π exp −27 (ln S)2(A10)
and, for ncrit  1,1
2σ 2 24 μ3.
c1 Π exp −Js = sncrit Ω0
π m1
27 (ln S)2

ckA

APPENDIX B(A6)and have a maximum at ncrit given by

1
1 3
2 μ
ncrit −=.
ω
3 ln S

c1ω−1

i
k=1
j
k=1

It should be emphasized that the factor Π is expressed asfunctions of concentration ratios of gaseous reactants/products
to the key molecule and does not include the standard pressure
ps explicitly.
In most of the past works for the formation of dust in theejecta of SNe (e.g., Nozawa et al. 2003; Todini & Ferrara 2001),
the factor Π has been set to be unity. Since Π > 1 for theformation of a multi-element grain with no gaseous productsuch as in the ejecta of SNe (see Nozawa et al. 2003), the revised
formula provides a nucleation rate larger than the previous one;for the elemental abundances of the gas (cMg,0 /cSiO,0 = 2 andcO,0 /cSiO,0 = 20) for the formation of MgSiO3 used in thetext, the revised formula presents the nucleation rate only by afactor of Π = 5.3 higher than the old one. This enhancement
cannot be significant because Paquette & Nuth (2011) showedthat the resulting size distribution of dust is little affected even
if the steady-state nucleation rate is changed by a few orders ofmagnitude.

1
1
1
exp
(g̊n − nΔg̊gas )=c̊n bnc1 ΠkT  
1
p1
1+ ln Ξ ≡ln− n−exp(γ  )
ω
ps
c1 Π
(A4)
with

1

(A11)
16

The Astrophysical Journal, 776:24 (17pp), 2013 October 10

Nozawa & Kozasa

size of grains is only increased by a factor of 101/3 = 2.15. We
also note that small grains formed in the SN ejecta would bedestroyed quickly by the passage of the reverse shock before
being injected into the ISM (Nozawa et al. 2007, 2010; Silvia
et al. 2010, 2012). Thus, when we consider the final mass of dust
ejected from SNe, the uncertainty in the mass of small grains,resulting from the uncertainty in n∗ , could not be important.
REFERENCES
Abraham, F. F. 1974, Homogeneous Nucleation Theory (New York: Academic)
Barlow, M. J., Krause, O., Swinyard, B. M., et al. 2010, A&A, 518, L138
Bianchi, S., & Schneider, R. 2007, MNRAS, 378, 973
Blander, M., & Katz, J. 1972, JSP, 4, 55
Cherchneff, I., & Dwek, E. 2009, ApJ, 703, 642
Chigai, T., Yamamoto, T., & Kozasa, T. 1999, ApJ, 510, 999
Clayton, D. D. 2013, ApJ, 762, 5
Clayton, D. D., Deneault, E. A.-N., & Meyer, B. S. 2001, ApJ, 562, 480
Clayton, D. D., Liu, W., & Dalgarno, A. 1999, Sci, 283, 1290
Deneault, E. A.-N., Clayton, D. D., & Heger, A. 2003, ApJ, 594, 312
Donn, B., & Nuth, J. A. 1985, ApJ, 288, 187
Dwek, E., Galliano, F., & Jones, A. P. 2007, ApJ, 662, 927
Fallest, D. W., Nozawa, T., Nomoto, K., et al. 2011, MNRAS, 418, 571
Ferrarotti, A. S., & Gail, H.-P. 2001, A&A, 371, 133
Gall, C., Hjorth, J., & Andersen, A. C. 2011, A&ARv, 19, 43
Gomez, H. L., Krause, O., Barlow, M. J., et al. 2012, ApJ, 760, 96
Goumans, T. P. M., & Bromley, S. T. 2012, MNRAS, 420, 3344
Hasegawa, H., & Kozasa, T. 1988, PThPS, 96, 107
Keith, A. C., & Lazzati, D. 2011, MNRAS, 410, 685
Kotak, R., Meikle, W. P. S., Farrah, D., et al. 2009, ApJ, 704, 306
Kozasa, T., Dorschner, J., Henning, Th., & Stognienko, R. 1996, A&A,
307, 551
Kozasa, T., & Hasegawa, H. 1987, PThPh, 77, 1402
Kozasa, T., & Hasegawa, H. 1988, Icar, 73, 180
Kozasa, T., Hasegawa, H., & Nomoto, K. 1989, ApJ, 344, 325
Kozasa, T., Hasegawa, H., & Nomoto, K. 1991, A&A, 249, 474
Kozasa, T., & Sogawa, H. 1997, Ap&SS, 251, 165
Kozasa, T., & Sogawa, H. 1998, Ap&SS, 255, 437
Lakićević, M., van Loon, J. Th., Stanke, T., De Breck, C., & Patat, F. 2012, A&A,
541, L1
Landau, L. D., & Lifshitz, E. M. 1976, Statistical Physics (3rd ed., Part 1;
Oxford: Pergamon)
Liu, W., & Dalgarno, A. 1994, ApJ, 428, 769
Liu, W., & Dalgarno, A. 1996, ApJ, 471, 780
Maeda, K., Nozawa, T., Sahu, D. K., et al. 2013, ApJ, 776, 5
Matsuura, M., Dwek, E., Meixner, M., et al. 2011, Sci, 333, 1258
Nozawa, T., Kozasa, T., & Habe, A. 2006, ApJ, 648, 435
Nozawa, T., Kozasa, T., Habe, A., et al. 2007, ApJ, 666, 955
Nozawa, T., Kozasa, T., Tominaga, N., et al. 2008, ApJ, 684, 1343
Nozawa, T., Kozasa, T., Tominaga, N., et al. 2010, ApJ, 713, 356
Nozawa, T., Kozasa, T., Umeda, H., Maeda, K., & Nomoto, K. 2003, ApJ,
598, 785
Nozawa, T., Maeda, K., Kozasa, T., et al. 2011, ApJ, 736, 45
Paquette, J. A., & Nuth, J. A. 2011, ApJL, 737, L6
Sibthorpe, B., Ade, P. A. R., Bock, J. J., et al. 2010, ApJ, 719, 1553
Silvia, D. W., Smith, B. D., & Shull, J. M. 2010, ApJ, 715, 1575
Silvia, D. W., Smith, B. D., & Shull, J. M. 2012, ApJ, 748, 12
Todini, P., & Ferrara, A. 2001, MNRAS, 325, 726
Yamamoto, T., Chigai, T., Watanabe, S., & Kozasa, T. 2001, A&A, 380, 373
Yasuda, Y., & Kozasa, T. 2012, ApJ, 745, 159
Zhukovska, S., Gail, H.-P., & Trieloff, M. 2008, A&A, 479, 453

Figure 10. (a) Time evolutions of I∗ , aave , and fcon , and (b) the final size
distribution spectra of newly formed C clusters (blue) and grains (red) forn∗ = 100 (solid), 300 (dashed), and 1000 (dotted). The other parameters are
set to be c10 = 106 cm−3 , t0 = 300 days, and γ = 1.25. For clarity, the size
distributions for n∗ = 300 and 1000 are reduced by a factor of 0.1 and 0.01,
respectively.

hand, for n > n∗,min , the current density is In  αn−1 c1 Yn−1 under the condition that ln S  (2μ)/(3n1/3 ), as can be seen from
Equations (42) and (45). Thus, the combined size distribution
can be expected to be independent of n∗,min since, in the model
considered here, ln S is not depressed very much by the depletion of the key molecules due to grain growth. The increase inn∗ only enhances the minimal size of grains, which results in
the reduced number of atoms finally locked in grains (i.e., condensation efficiency) and the increased average radius of grains
for the given combined size distribution of clusters and grains.
In conclusion, the change in n∗ does not vary the formation
process of dust grains but affects the resulting condensation
efficiency and average radius of grains. The value of n∗ isimportant only for dust formation at low gas densities where
the average radius of newly formed grains is very small(aave  0.005 μm), as is shown above. It should be kept in mind
that, even if n∗ is enhanced from n∗ = 100 to 1000, the minimal

17