Formation of carbon nanoparticles by the condensation of supersaturated atomic vapor obtained by the laser photolysis of C3O2

Article abstract of DOI:10.1134/S0023158407020036

A new technique is suggested for obtaining nanoparticles from highly supersaturated vapor resulting from the laser photolysis of volatile compounds. The growth of carbon nanoparticles resulting from C₃O₂ photolysis has been studied in detail. Absorbing UV quanta (from an Ar-F excimer laser), C₃O₂ molecules decompose to yield atomic carbon vapor with precisely known and readily controllable parameters. This is followed by the condensation of supersaturated carbon vapor and the formation of carbon nanoparticles. These processes have been investigated by the laser extinction and laser-induced incandescence (LII) methods in wide ranges of experimental conditions (carbon vapor concentration, nature of the diluent gas, and gas pressure). The current and ultimate particle sizes and the kinetic parameters of particle growth have been determined. The characteristic time of particle growth ranges between 20 and 1000 μs, depending on photolysis conditions. The ultimate particle size determined by electron microscopy is 5-12 nm for all experimental conditions. It increases with increasing total gas pressure and carbon vapor partial pressure and depends on the diluent gas. The translational energy accommodation coefficients for the Ar, He, CO, and C₃O ₂ molecules interacting with the carbon particle surface have been determined by comparing the LII and electron microscopic particle sizes. A simple model has been constructed to describe the condensation of carbon nanoparticles from supersaturated atomic vapor. According to this model, the main process in nanoparticle formation is surface growth through the addition of separate atoms to the nucleation cluster. The nucleus concentrations for various condensation parameters have been determined by comparing experimental and calculated data.

Full text of DOI:10.1134/S0023158407020036

ISSN 0023-1584, Kinetics and Catalysis, 2007, Vol. 48, No. 2, pp. 194–203. © MAIK “Nauka /Interperiodica” (Russia), 2007.
Original Russian Text © E.V. Gurentsov, A.V. Eremin, C. Schulz, 2007, published in Kinetika i Kataliz, 2007, Vol. 48, No. 2, pp. 210–219.
Formation of Carbon Nanoparticles by the Condensation
of Supersaturated Atomic Vapor Obtained
by the Laser Photolysis of C3O2
E. V. Gurentsova, A. V. Eremina, and C. Schulzb
a Institute of Thermal Physics of Extreme States, Russian Academy of Sciences, Moscow, 125412 Russia
b Institut für Verbrennung und Gasdynamik, 47048 Duisburg, Germany
e-mail: gurentsov@ihed.ras.ru
Received November 11, 2005
Abstract—A new technique is suggested for obtaining nanoparticles from highly supersaturated vapor result-
ing from the laser photolysis of volatile compounds. The growth of carbon nanoparticles resulting from C3O2
photolysis has been studied in detail. Absorbing UV quanta (from an Ar–F excimer laser), C3O2 molecules
decompose to yield atomic carbon vapor with precisely known and readily controllable parameters. This is fol-
lowed by the condensation of supersaturated carbon vapor and the formation of carbon nanoparticles. These
processes have been investigated by the laser extinction and laser-induced incandescence (LII) methods in wide
ranges of experimental conditions (carbon vapor concentration, nature of the diluent gas, and gas pressure). The
current and ultimate particle sizes and the kinetic parameters of particle growth have been determined. The
characteristic time of particle growth ranges between 20 and 1000 µs, depending on photolysis conditions. The
ultimate particle size determined by electron microscopy is 5–12 nm for all experimental conditions. It
increases with increasing total gas pressure and carbon vapor partial pressure and depends on the diluent gas.
The translational energy accommodation coefficients for the Ar, He, CO, and C3O2 molecules interacting with
the carbon particle surface have been determined by comparing the LII and electron microscopic particle sizes.
A simple model has been constructed to describe the condensation of carbon nanoparticles from supersaturated
atomic vapor. According to this model, the main process in nanoparticle formation is surface growth through
the addition of separate atoms to the nucleation cluster. The nucleus concentrations for various condensation
parameters have been determined by comparing experimental and calculated data.
DOI: 10.1134/S0023158407020036
INTRODUCTION
ties and developing methods for diagnostics of growing
nanoparticles during their formation.
The problem of quantitatively describing the forma-
tion of carbon nanoparticles is important for a wide
variety of applications ranging from combustion to the
manufacture of new materials. Although this problem
has been the subject of numerous studies, both experi-
mental and theoretical, it is still far from being solved.
The potential of numerical simulation of the growth of
carbon nanoparticles is limited by the uncertainty of the
structure and thermodynamic properties of small car-
bon clusters. The main difficulties in experimental stud-
ies (including experiments dealing with flames, electric
arcs, laser ablation of carbon, and pyrolysis of carbon-
containing compounds) arise from uncertainties in
experimental conditions (temperature, pressure, and
carbon concentration) and from the effects of active
impurities [1–4].
Here, we report a radically new approach to produc-
ing supersaturated vapor, specifically, the photolysis of
volatile compounds under the action of UV radiation.
This technique was for the first time suggested for the
preparation and examination of nanoparticles [5].
Supersaturated carbon vapor was obtained from carbon
suboxide (C3O2), a volatile carbon-containing com-
pound. Under the action of UV radiation with λ <
207 nm, the C3O2 molecule decomposes into a carbon
atom and two inert CO molecules [6]. This provides a
means of obtaining supersaturated carbon atom vapor
with a precisely known and readily controllable carbon
concentration.
Let us evaluate the efficiency of this process in a
simple way. Suppose that the volume containing origi-
nal molecules is exposed to a photon flux of density F
(photon/cm2). The absorbed part of the flux will be
expressed as
Thus, the main prospects for the detailed investiga-
tion of the formation of condensed carbon particles are
establishing experimental conditions under which it
∆F = F[1 – exp(–CC O l )]
(1)
3
2
will be possible to obtain carbon vapor with precisely
known, homogeneous, and readily controllable proper- and, for an optically thin layer,
194

FORMATION OF CARBON NANOPARTICLES BY THE CONDENSATION
195
Projection 1
∆F ≈ FσCC O l ,
(2)
3
2
6
where CC O is the concentration of absorbing mole-
cules and 3σ2 is the absorption cross section, which is
estimated at 10–17–10–18 cm2 for C3O2 and λ = 190–
200 nm [6]. Each absorbed photon produces one car-
bon atom [6] (i.e., the quantum yield of photolysis is 1).
It is interesting that the total photolysis yield Y, defined
as the total number of atoms formed (which is equal to
the number of absorbed photons) divided by the total
number of molecules encountered by the photon beam,
3
1
2
4
Projection 2
∆F/M (photon/molecule), where M = CC O l , is inde-
3
5
pendent of the concentration of original mo2lecules (C)
14
2
and is expressed as
∆F
9
7
-------
Y =
≈ Fσ.
(3)
8
M
10
Thus, for complete ë3é2 dissociation followed by
nanoparticle formation, the quantum flux density
should be of the order 1018 photon/cm2. For a photon
energy of about 6 eV, the radiation energy density
should be of the order 1 J/cm2.
However, carbon vapor condensable into nanoparti-
cles will still form at a lower energy density, with the
difference that the environment will contain undissoci-
ated ë3é2 molecules, which do not exert any significant
effect on condensation at room temperature [7], and the
total yield of nanoparticles will be lower. By varying
the initial conditions (the concentration of the carbon-
containing compound, the diluent gas, and pressure), it
is possible to study the kinetics of condensation and of
the formation of carbon nanoparticles.
13
11
12
Fig. 1. Experimental setup: (1) excimer laser, (2) reactor
cell, (3–5) energy meters, (6) dielectric mirror, (7) photo-
diode, (8) He–Ne laser, (9) quartz plate, (10) converging
lens, (11) Nd:YAG laser, (12) photomultiplier, (13) filter
(λ = 694 nm), and (14) (λ = 633 nm).
between 10 and 1000 mbar. The source of photons for
ë3é2 photodissociation was an argon–fluorine excimer
laser (Lambda Physik EMG 150TMSC) generating λ =
193 nm UV light with a pulse duration of ~20 ns and a
pulse energy of 100 mJ. The laser beam, with a rectan-
gular cross section, was focused so that its cross section
coincided with one of the cell walls. The optical path
The new method suggested here for obtaining nano-
particles has the advantages of low energy demand, a
wide variety of means of controlling the synthesis out-
comes, and readily variable and reliably controllable
photosynthesis conditions. These features make the
process very promising for fundamental research in length was 10 mm. In order to determine the energy
nanoparticle condensation and growth and offer the
prospect of producing nanomaterials with controllable
properties.
Here, we report a detailed experimental study of the
kinetics of carbon nanoparticle growth by condensation
from highly supersaturated vapor and describe the
observed kinetics in terms of a simple cluster growth
model.
absorbed by the ë3é2 molecules, we measured, in each
experiment, the radiation energy before and after the
pumped cell, using a calibrated photodiode and a cali-
brated calorimeter. From these measurements, the
absorption cross section of ë3é2 at λ = 193 nm was
determined to be 9 × 10–19 cm2. This value is in good
agreement with previous data [6]. At this absorption
cross section and a cell length of 1 cm, the optically thin
layer conditions were stringently fulfilled only at C3O2
pressures of ~10 mbar. It is at these pressures that most
of our experiments were conducted. At higher ë3é2
pressures, the concentration of the resulting vapor was
somewhat nonuniformly distributed along the cell
length. The average carbon atom concentration result-
ing from ë3é2 photolysis was derived from energy
absorption data under the assumption that the absorp-
tion of a photon with λ < 207 nm yields one C atom and
two CO molecules [6]. The carbon atom concentrations
EXPERIMENTAL PROCEDURES
Experiments were carried out at room temperature
in a rectangular quartz cell with a volume of 0.5 cm3
(10 × 10 × 5 mm). The cell was pumped and was then
filled with C3O2 premixed with an inert gas. The forma-
tion of nanoparticles was studied by the laser extinction
and the laser-induced incandescence (LII) methods
using the setup shown in Fig. 1. The ë3é2 concentra-
tion was varied between 1 and 100%; the diluent gas
was Ar, He, Co, or Kr; and the total pressure was varied and the partial pressures of carbon vapor resulting from
KINETICS AND CATALYSIS Vol. 48 No. 2 2007

196
GURENTSOV et al.
Table 1. Carbon vapor concentration and partial pressure
The source of radiation was a pulsed Nd:YAG laser
(Quanta Ray DCR11) generating λ = 1064 nm light
with a pulse duration of about 10 ns. The profile of the
laser beam was a disc 1 mm in diameter, and the energy
density in a pulse was set below 0.4 W/m2 to avoid par-
ticle evaporation [10]. The time profiles of incandes-
cence were recorded using a system consisting of a
photomultiplier (Hamamatsu R7400 U-4), a narrow-
bandpass filter with a transmission peak at 694 nm
placed before the photomultiplier, and a digital oscillo-
scope (LeCroy LT 342) with a frequency bandwidth of
500 MHz. The time-resolving power of this system was
2 ns or better.
as a function of the C3O2 pressure
C3O2 partial
Concentration of
Partial pressure
pressure, mbar C atoms × 10–16, cm–3 of C atoms, mbar
10
30
1.15
1.85
3.40
0.50
0.85
1.50
100
ë3é2 photolysis are listed in Table 1. The error in car-
bon atom concentration measurements was 25%.
The average size of growing nanoparticles was
determined by the LII method [8, 9]. This method is
based on measuring radiation from nanoparticles
heated with a laser pulse as short as a few nanoseconds.
The radiation from the particles peaks in a short time
owing to heating and then declines because of cooling.
A typical profile of the incandescence signal is shown
in Fig. 2. The determination of the particle sizes from
the time profiles of incandescence intensity is based on
the energy and mass conservation equations for a spher-
ical particle. The absorbed energy of laser radiation is
dissipated by radiative and conductive heat transfer to
the environment and is spent on the evaporation of the
material of the particles. The heating and cooling peri-
ods do not overlap. The radiation heat flux and evapo-
ration were not taken into account in our experiments
because the particle were heated to relatively low tem-
peratures (≤3500 K) and, therefore, their cooling was
mainly due to collisions with surrounding gas mole-
cules. Under conditions of free molecular heat transfer
between the particles and the gas (Kn ꢀ 1), considering
that the laser-heated volume was small (0.03 of the cell
volume), the temperature of the gas around the particle
was invariable and was equal to the ambient tempera-
ture.
The absorbance of the gas medium containing a
condensed phase was measured using the laser extinc-
tion method [1, 4]. The source of radiation was a con-
tinuous-wave He–Ne laser. The laser beam was passed
through the cell, and the transmitted beam intensity was
recorded using a system consisting of an active photo-
diode (E2VUV, Spindler & Hoyer) with a time-resolv-
ing power of 1 µs or better, a narrow-bandpass filter
with a transmission peak at 633 nm, and a digital oscil-
loscope (LeCroy 9314 AM) with a frequency band-
width of 400 MHz. A typical extinction profile is pre-
sented in Fig. 3.
EXPERIMENTAL DATA
Optical measurements. In extinction measure-
ments, we used the absorbance of the medium (D)
determined as
D = –ln(I/I0),
(4)
where I is the beam intensity transmitted by the
medium and I0 is the beam intensity in vacuo (Fig. 3).
The time dependence of D for one experimental mode
is plotted in Fig. 4 (curve 1). Particle growth can
Amplitude, V
1.0
Amplitude, V
0
–0.08
0.8
0.6
0.4
0.2
0
I(t)
I0
–0.32
–0.36
0
100
200
300
400
500
t, µs
0
100
200 t, ns
Fig. 2. Observed profile of the LII signal.
Fig. 3. Observed extinction profile.
KINETICS AND CATALYSIS Vol. 48 No. 2 2007

FORMATION OF CARBON NANOPARTICLES BY THE CONDENSATION
197
Particle size, nm
30
D
0.02
2
1
20
10
0
0.01
0
0
200
400
600
800
0
200
400
600
800
t, µs
t, µs
Fig. 4. Time variation of the absorbance of the carbon nano-
particles for the 10 mbar C O + 1 bar Ar mixture: (1)
Fig. 5. Time variation of the carbon nanoparticle size for the
10 mbar C O + 1 bar Ar mixture. The points are experi-
3
2
3 2
experimental data and (2) fitted curve.
mental data, and the solid line is the fitted curve.
roughly be described by the following relaxation equa- and pressure for argon-diluted mixtures (Figs. 6, 7).
tion:
Approximation of the experimental data indicated that
keff is a quadratic function of the initial concentration of
carbon atoms (C0),
dD
dt
------- = keff(Dmax – D(t)),
(5)
keff = BC02 ,
(7)
where keff is the effective particle growth rate and Dmax
is the maximum absorbance. The experimental absor-
bance data were fitted to the exponential function
(Fig. 4, curve 2)
where B = 2 × 10–28 cm6/s (Fig. 6), and that the particle
growth rate increases progressively less rapidly
(Fig. 7). Furthermore, keff is nearly pressure-indepen-
dent starting at P = 200–300 mbar. The pressure depen-
dence of keff (Fig. 7) can be fitted to the exponential
equation
D = Dmax[1 – exp(–kefft)].
(6)
In order to derive the nanoparticle size from the
observed LII signal, it is necessary to know the physical
properties of the nanoparticles, namely, thermal con-
ductivity, density, and the radiation absorption coeffi-
cient, as well as thermal energy accommodation coeffi-
cient (α) data for collisions of the particles with mole-
cules of various diluent gases. In this study, we used the
same physical data for soot particles and the same
accommodation coefficient (α = 1) as in previous stud-
ies of C3O2 pyrolysis [11, 12], although it has recently
been demonstrated that α can differ significantly from
1, depending on the diluent gas [13, 14]. (The variation
of α will be discussed below.) Figure 5 shows the time
profile of the size of the growing nanoparticles accord-
ing to LII data (the points represent experimental data,
and the solid curve is an exponential fit similar to the fit
used in absorbance data processing (Eq. (6)). A simul-
taneous analysis of the absorbance and particle size
(Figs. 4, 5) time profiles demonstrated that, under fixed
conditions, both can be fitted to one exponential func-
tion (like that defined by Eq. (6)) with the same coeffi-
cient keff. In other words, the absorbance of the con-
densed phase and the particle size increase at the same
rate.
P
80
⎛
⎛
⎞⎞
⎠⎠
keff = 0.018 1 – exp –-----
,
(8)
⎝
⎝
where P is expressed in millibars.
keff × 10–5, s–1
2.5
2.0
1.5
1.0
0.5
0
1
2
3
4
Concentration of C atoms × 10–16, cm–3
From the LII time profiles of the particle size
obtained at different C3O2 pressures or at different car-
bon vapor concentrations in the mixture, we derived keff
as a function of the initial concentration of carbon atoms
Fig. 6. Growth rate of the carbon nanoparticles as a function
of the initial concentration of carbon atoms in the C O –Ar
mixture. P = 100 mbar. The points are experimental data,
and the solid line is the fitted curve.
3
2
KINETICS AND CATALYSIS Vol. 48 No. 2 2007

198
GURENTSOV et al.
keff × 10–4, s–1
2.0
1.5
1.0
0.5
0
200
400
600
800 1000
P, mbar
Fig. 7. Growth rate of the carbon nanoparticles as a function
of the C O –Ar mixture pressure. P = 100 mbar. The points
3
2
are experimental data, and the solid line is the fitted curve.
20 nm
Electron microscopy. In some experiments of each
series, we deposited nanoparticles onto a microscope
grid placed on the bottom of the quartz cell. The parti-
cles were examined to determine their size distribution
and structure. Figure 8 shows a micrograph of nanopar-
ticles produced in one experiment. Clearly, the particles
are near-spherical. From the micrographs of the nano-
particles obtained in all experiments, we derived the
final particle size of the nanoparticles. The observed
particle size distribution is close to the lognormal dis-
tribution with a geometric deviation from the mean of
δ = 1.2–1.4 (Table 2).
Fig. 8. Micrograph of the carbon nanoparticles obtained by
photolysis in the 30 mbar C O + 1 bar He mixture
3
2
(×100000).
DISCUSSION
Analyzing the particle size data, we found that the
particle sizes determined by LII under the assumption
that α = 1 far exceed the electron microscopic particle
sizes (Table 2). The unlikely assumption that, late in the
process, the particles change their structure so that their
effective size decreases severalfold was disproved by
LII measurements at long particle formation times
(up to 0.1 s), which demonstrated that the particle size
was almost invariable.An analysis of the dependence of
the LII particle size data on the properties of the parti-
cles demonstrated that the LII and electron microscopic
data cannot be reconciled by varying the thermal con-
ductivity, density, or the radiation absorption coeffi-
cient of the particle within any reasonable range. There-
fore, the only plausible cause of this discrepancy is that
the accommodation coefficient α is not equal to unity
and depends on the diluent gas. By equating the LII par-
ticle sizes measured at a long nanoparticle formation
time to the corresponding electron microscopic particle
sizes, we arrived at the following accommodation coef-
ficients: for Ar and CO, α = 0.44; for C3O2, α = 0.51;
for He, α = 0.1. These results are illustrated in Fig. 9,
which shows how the true particle size changes with
time in various diluent gases and presents the ultimate
particle sizes measured by electron microscopy. The
ultimate particle size is almost independent of the ini-
tial concentration of carbon vapor (note, however, that
the initial concentration was varied only in a rather nar-
row range of 1.15 × 1016 to 3.4 × 1016 cm–3), but it
Table 2. Average size of carbon particles (d) obtained under
various conditions and the corresponding geometric devia-
tions (δ)
Mixture composition
10 mbar C3O2
d, nm
δ
5.7
5.5
5.5
4.6
12
1.2
1.2
1.2
1.2
1.4
1.2
1.4
1.3
1.4
1.2
1.2
1.2
10 mbar C3O2 + 100 mbar Ar
10 mbar C3O2 + 100 mbar CO
10 mbar C3O2 + 100 mbar He
10 mbar C3O2 + 1 bar Ar
30 mbar C3O2
5.5
12
30 mbar C3O2 + 1 bar Ar
100 mbar C3O2
9
100 mbar C3O2 + 1 bar Ar
10 mbar C3O2 + 300 mbar Ar
10 mbar C3O2 + 1 bar He
40 mbar C3O2 + 400 mbar Kr
11
5.7
5.5
5.5
KINETICS AND CATALYSIS Vol. 48 No. 2 2007

FORMATION OF CARBON NANOPARTICLES BY THE CONDENSATION
199
Particle size, nm
depends on pressure in a way depending on the diluent
gas.
15
1
(‡)
This result allows some important assumptions to be
made as to the role of the diluent gas in the formation
of carbon nanoparticles. We will proceed from the con-
cept that the carbon vapor condensation process con-
sists of two steps, namely, the formation of nucleation
clusters and the surface growth of the nucleation clus-
ters by the addition of separate atoms to their surfaces
[7, 15]. (This is followed by the nanoparticle coagula-
tion (agglomeration) stage, which proceeds on a much
longer time scale and, therefore, can be ignored here;
furthermore, the LII method allows one to determine
only the size of a separate particle, not the size of an
agglomerate.)
2
3
4
5
6
10
5
0
10–5
10–4
10–3
10–2
10–1
20
In the framework of these concepts, it can be
deduced that not only the particle growth rate at the late
stages of the process (Fig. 7) but also the concentration
of nucleation clusters depend on pressure and the
nature of the diluent gas. Indeed, the surface growth of
particles does not change the number density of these
particles and comes to an end as the carbon atoms are
consumed. Therefore, the higher the concentration of
nucleation clusters, the smaller the ultimate size of the
particles. This assumption seems quite plausible con-
sidering the significant role of deactivating collisions in
the recombination and coagulation of small carbon
clusters consisting of n < ncr atoms, where ncr is the
number of atoms in a nucleation cluster. The kinetics of
these processes is beyond the scope of this study. Here,
we will attempt only to develop the numerical model
describing the evolution of the size and number density
of the clusters as a function of the carbon atom concen-
tration in the gas phase and the nature and pressure of
the diluent gas.
(b)
1
2
3
4
15
10
5
0
10–5
10–4
10–3
(c)
10–2
10–1
10
1
2
3
4
8
6
4
2
Numerical Simulation
Let us construct a simple model to qualitatively
describe the experimental data that refer to the late
stages of the growth of the particles that are detectable
by LII (i.e., particles with d > 1 nm). Consider the con-
densation of homogeneous, highly supersaturated
atomic vapor diluted with a nonreactive gas at various
pressures. Suppose that some concentration C0 of car-
bon atoms (spheres of diameter d and volume V = πd3/6,
whose temperature is T and thermal velocity is uÒ) has
formed instantaneously in some volume containing a
diluent gas, whose concentration is Cg. Assume that
nucleation clusters containing ncr atoms form rather
rapidly in the atomic vapor, resulting in the cluster
number density Np.
10–5
10–4
10–3
t, s
10–2
10–1
Fig. 9. Comparison between LII and electron microscopic
particle size data for nanoparticles obtained from (a) pure
C O and (b, c) C O diluted with (b)Ar and (c) He. (a): (1–
3) LII and (4–6) electron microscopic data for samples
obtained at C O pressures of (2, 5) 10, (1, 4) 30, and (3, 6)
3
2
3 2
2
3
The last assumption was analyzed by numerical cal-
culations for the early stages of carbon vapor condensa-
tion. The kinetic scheme for the recombination and
coagulation of small carbon clusters containing up to
30 atoms was taken from an earlier publication [7]. The
thermodynamic properties of the small clusters were
100 mbar. (b): (1, 2) LII and (3, 4) electron microscopic data
for samples 1, 2 obtained from the (1, 3) 10 mbar C O +
3
2
1 bar Ar and (2, 4) 30 mbar C O + 1 bar Ar mixtures.
3
2
(c): (1, 2) LII and (3, 4) electron microscopic data for sam-
ples 1, 2 obtained from the (1, 3) 10 mbar C O + 1 bar He
3
2
and (2, 4) 10 mbar C O + 100 bar He mixtures.
3
2
KINETICS AND CATALYSIS Vol. 48 No. 2 2007

200
GURENTSOV et al.
estimated by extrapolating available data, as was done
by Krestinin et al. [15]. The calculations were carried
out using the program package Chemkin II. The calcu-
lated data indicated that carbon molecules containing
up to six atoms appear as early as 0.2 µs after the
appearance of carbon vapor in the gas phase consisting
of an inert gas (1 bar) and a C3O2 admixture.
1
6
3
dd
dt
--
----- = 2uc C0V – Npπd (t)
(15)
× [1 – exp(–ugπd2(t)Cgτ)].
In order to solve this equation, it is necessary to specify
both initial conditions (C0 and Cg) and the initial size of
the nucleation cluster, which is defined as
Since the equilibrium state of carbon under the
given conditions is a condensed phase, we will assume
that a recombination event (or the surface growth of a
particle) takes place each time a carbon atom collides
against a nucleation cluster (or, at later stages, against a
particle) and that the collision cross section is equal to
the particle surface area S. The frequency of collisions
between a carbon atom and a particle (Zc) will be
expressed as
6ncrV1/3
------------------
d0 =
.
(16)
π
The value of d0 was varied over the ncr range of 3 to 30,
and these variations did not modify the calculated data
to any significant extent.
Equation (15) contains two unknown parameters,
namely, the number density of nucleation clusters (Np)
and the lifetime of the atom–particle complex (τ). The
latter can be estimated from the experimental pressure
dependence of the particle growth rate (Eq. (8)). Since
all experiments were conducted at room temperature,
Eq. (8) can be represented in terms of the diluent gas
concentration:
Zc = ucSC,
(9)
where C is the current concentration of carbon atoms.
However, it will be assumed that the complex resulting
from a collision between an atom and a particle is
unstable and will break down after some time τ unless
a stabilizing collision between this complex and a mol-
ecule of the surrounding gas (which consists largely of
a buffer gas) takes place within this period of time. The
characteristic lifetime of the unstable complex, τ, may
Cg
⎛
⎛
⎞⎞
keff = 0.018 1 – exp –-------------------------- .
(17)
2.16 × 1018
⎝
⎝
⎠⎠
be of the same order of magnitude as the characteristic With this representation, the exponential term in empir-
time of the vibrations of the atom–particle bond. The ical equation (17) is very similar to Eq. (11). Therefore,
particle–gas molecule collision frequency Zg is the lifetime of the particles can be estimated from the
expressed as
condition
Zgτ = ugπd2(t)Cgτ = Cg/2.16 × 1018.
(18)
Zg = ugSCg,
(10)
where ug is the thermal velocity of the buffer gas mole-
cules. The probability of a cluster colliding with a diluent
gas molecule (X) within the time τ will then be equal to
For particle sizes between 1 and 10 nm, the lifetime is
estimated at τ ≅ 5 × 10–10–5 × 10–12 s. These values
somewhat exceed the characteristic times of molecular
vibrations. However, since the probability of energy-
exchange collisions is finite, this result seems quite
plausible. Note that the real probability of an active
atom–particle complex breaking down must decrease
as the size (number of degrees of freedom) of this com-
plex increases. However, for the particle sizes consid-
ered (n > 1000), this trend is weak. At the same time,
the low probability of energy-exchange collisions
seems to reflect the difference between the real mecha-
nism of the deactivating collisions and the mechanism
of strong collisions.
For this reason, in further calculations, we used the
mean value of τ ≅ 5 × 10–11 s, the initial size of a nucle-
ation cluster was taken to be ncr = 6, and the only vari-
able parameter was the number density of nucleation
clusters (Np).
The time profile calculations using this equation
were carried out by the Runge–Kutta method employ-
ing the Maple 9.5 program.
X = (1 – exp(–Zgτ)),
(11)
and the kinetic equation for the increasing number of
atoms in the particle (n) will appear as
dn
dt
-----
= XZc = ucSC[1 – exp(–ugSCgτ)].
(12)
The current carbon atom concentration balance is writ-
ten as
C = C0 – Npn,
(13)
and, under the assumption that the particles are spheri-
cal, n is related to the particle size as
πd3
--------
n =
.
(14)
6V
Hence, we arrive at the following kinetic equation for
the current particle diameter d at various initial concen-
Figure 10 presents calculated and experimental data
trations of carbon vapor (C0) and diluent gas concentra- for the growth of carbon nanoparticles at a constant
tions (Cg):
pressure (b) and at a constant concentration of carbon
KINETICS AND CATALYSIS Vol. 48 No. 2 2007

FORMATION OF CARBON NANOPARTICLES BY THE CONDENSATION
201
Particle size, nm
atoms (a) and compares the data obtained for two dilu-
ent gases, namely, argon and helium (c).
(‡)
The calculated data indicate that the model ade-
quately describes both the particle growth time and the
ultimate particle size data at Np = 1010–1011 cm–3. For
the mixtures diluted with argon at P = 1 bar, the best fit
between the experimental and calculated data is
observed for Np = 1010 cm–3. For a lower argon or C3O2
15
3
10
5
pressure, Np = 1011 cm–3; for helium-diluted mixtures,
2
1
Np = 3 × 1011 cm–3. Thus, according to our calculated
data, the number of nucleation clusters decreases with
increasing pressure for a given buffer gas and is
30 times larger in helium than in argon under the same
conditions. Qualitatively, this result is in good agree-
ment with the fact that the efficiency of the energy-
exchange collisions between small clusters and the sur-
rounding gas molecules (which are necessary for nucle-
ation) decreases with decreasing pressure or on passing
from argon to helium, a lighter diluent. A quantitative
analysis of this result should be made using a kinetic
model of the growth and coagulation of small carbon
clusters and is beyond the scope of this study.
0
0
200
400
600
(b)
800 1000 1200
20
15
10
5
2
1
Number Density of Particles
Proceeding from the results of the above simulation,
it is pertinent to attempt an analysis of the experimental
Np data. Using the results of the simultaneous measure-
ments of the particle size and absorbance, it is possible
to relate the number density of particles in a given vol-
ume to the particle formation time. The expression for
Np can be written as follows [16]:
0
0
100
200
(c)
300
400
14
12
10
8
2
6 f v
--------
Np =
,
(19)
πd3
D
H
6
----
f v =
.
(20)
1
4
Here, fv is the volume fraction of the particles, D is the
observed absorbance, d is the particle diameter, and H
is a function characterizing the optical properties of the
particles [16]:
2
0
0
100
200
t, µs
300
400
6πIm[(m2 – 1)/(m2 + 2)]
--------------------------------------------------------------
H =
,
(21)
λ
Fig. 10. Effects of (a) pressure, (b) the C O concentration,
3
2
and (c) the nature of the diluent gas on the growth of carbon
nanoparticles. (a): (1) 10 mbar C O , (2) 10 mbar C O +
where λ is the wavelength of absorbed light and m = k –
iz is the refractive index, for which
3
2
3 2
100 mbar Ar, and (3) 10 mbar C O + 1 bar Ar. (b):
3
2
6zk
Im[(m2 – 1)/(m2 + 2)] =
. (22)
(1) 10 mbar C O + 1 bar Ar and (2) 30 mbar C O + 1 bar Ar.
3
2
3 2
--------------------------------------------------
(k2 – z2 + 2)2 + 4z2k2
(c): (1) 10 mbar C O + 1 bar He and (2) 10 mbar C O +
3
2
3 2
1≤ bar Ar.
As is clear from these relationships, Np is proportional
to the function that accounts for the optical properties
of the nanoparticles and is inversely proportional to the
cubed particle size. The calculations were performed
for two different refractive indexes. In the first case, the
refractive index of soot particles was taken to be m1 =
particle sizes. In the second case, we used the refractive
index suggested for carbon particles smaller than 1 nm
that are transparent to visible light: m2 = 1.4 – i0.08
[17]. Figure 11 illustrates the time dependence of the
1.57 – i0.56, the value used in the LII determination of number densities thus calculated (Np1 and Np2) for one
KINETICS AND CATALYSIS Vol. 48 No. 2 2007

202
Np, cm–3
GURENTSOV et al.
(2) The ultimate yield of nanoparticles depends on
the energy spent on the photodissociation of the starting
molecules. The highest yield of nanoparticles is
observed at an energy density of about 1 J/cm2.
1012
1011
1010
109
(3) The current size of growing nanoparticles was
measured by the laser-induced incandescence method
in wide ranges of total pressures and C3O2 concentra-
tions and for a variety of diluent gases. The growth time
of the carbon nanoparticles does not exceed 1 ms and
depends strongly on the total gas pressure at P = 10–
1000 mbar.
1
3
2
(4) The ultimate size of the nanoparticles was mea-
sured using an electron microscope. The average size of
the particles resulting from the photolysis of carbon
suboxide is 4–12 nm. The translational energy accom-
modation coefficients for the Ar, He, CO, and C3O2
molecules colliding against a carbon particle were
determined by comparing the particle sizes measured
by LII and electron microscopy.
108
0
200
400
600
800
t, µs
Fig. 11. Comparison between the experimental and the cal-
culated nanoparticle number densities as a function of the
particle formation time for different optical properties
assigned to the nanoparticles: (1, 2) experimental data pro-
cessed using the refractive index of (1) soot particles (m =
1.56 – i0.57) and (2) ultrafine carbon particles (m = 1.4 –
(5) The growth rate of the nanoparticles resulting
from photolysis was determined. The growth rate is
proportional to the squared initial concentration of car-
bon atoms and grows at a decreasing rate as the total
gas pressure in the system is raised to 300 mbar. As the
total pressure is further increased, the particle growth
rate remains invariable.
1
2
i0.08) [17] and (3) numerical simulation data. The starting
mixture is 10 mbar C O + 1 bar Ar.
3
2
set of experimental conditions (curves 1, 2). Figure 11
also shows the Np value that is used in the model as the
initial concentration of nucleation clusters (line 3).
(6) A simple model has been suggested to describe
the condensation of carbon nanoparticles from super-
saturated atomic vapor. According to this model, the
main process in nanoparticle condensation from super-
saturated vapor is surface growth through the addition
of separate atoms to the nucleation cluster. The concen-
tration of nucleation clusters for various pressures and
diluent gases was determined by comparing experi-
mental and calculated data.
The initial portion of the experimental number den-
sity curve indicates that Np (determined under the
assumption that the refractive index is constant) falls
from its maximum to its steady-state value within
~150 µs, while the theoretical number density is set to
be equal. This difference is due to the fact that the par-
ticles radically change their optical properties as they
grow from a few atoms to their ultimate size. The ulti-
mate Np value determined using the optical properties
of soot (Fig. 11, curve 1) is closer to the model value,
and the Np curve calculated using the optical properties
of ultrafine particles [17] (Fig. 11, curve 2) intersects
the model Np line at ~15 µs. Similar results were
obtained for the other experimental conditions. These
results seem to be significant for independent, laser
extinction–based analysis of the growth kinetics of the
particles and of the evolution of their optical properties.
(7) The number density of particles estimated from
the experimental particle size and absorbance data is in
satisfactory agreement with the model calculations.
ACKNOWLEDGMENTS
We are grateful to Dr. H. Jander (University of Got-
tingen) and N. Schlosser (University of Duisburg) for
the electron microscopic examination of nanoparticles
and to M. Hoffman (University of Heidelberg) for
assistance in our preparations for the experimental
study. This work was supported by the Russian Founda-
tion for Basic Research, DFG, and INTAS.
CONCLUSIONS
The experimental data obtained in this study and the
numerical model developed to describe these data sug-
gest the following inferences:
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