13096 J. Phys. Chem. B, Vol. 110, No. 26, 2006
Trunov et al.
limited number of experiments, did not result in any further
weight increase. Two heating rates of 1 and 5 °C/min were used
in the study.
by two different processes: a mass change m˘ mox due to the
tr
direct aluminum oxidation and a mass change m˘ due to
polymorphic phase transformations. Because of the accepted
sequence of the polymorphic phase transitions: amorphous f
14
Oxidation Modeling
γ f R, the specific masses of the amorphous, γ-, and R-oxides,
referred to by the subscripts am, γ, and R, respectively, are
determined for each instant of time (or for each temperature)
from the following equations:
Recently, oxidation of micron-sized aluminum powders was
described considering formation of different polymorphs of
1
4-16
Al2O3.
At different oxidation stages, the oxidation rates
are limited by the diffusion resistance of different crystal-
lographic modifications of alumina. The oxidation model
described in detail elsewhere16 was modified in this work to
account for the effect of particle size distribution on the diffusion
and oxidation rates. After this modification, the model was used
to predict the TGA curves expected for each powder sample
while the PSD for each powder was obtained from SAXS
measurements. All particles were assumed to be spherical.
Major Processes during Aluminum Oxidation. Initially,
aluminum particles are covered by a thin layer of “natural”
amorphous oxide, and it should be noted that recent research
ox
am
tr
F V˙
)
∫
[ m˘ - m˘
]P(D) dD
(2.1)
(2.2)
(2.3)
am am
amfγ
ox
tr
tr
γfR
F V˙ )
∫
[ m˘ γ + m˘
- m˘ ]P(D) dD
γ
γ
amfγ
ox
tr
γfR
F V˙ )
∫
[ m˘ R + m˘ ]P(D) dD
R
R
2
µAl
ox
ox
ox
F V˙ ) -
∫
[ m˘ + m˘ γ + m˘ R ]P(D) dD
(2.4)
Al Al
am
µ
Al O
2
3
where V˙ is the rate of volume change, F is density, and all the
variables in the square brackets are functions of particle
diameter. The temperature of aluminum particles, T, affecting
the rates of mass change, m˘ i, increases from room temperature
at a constant rate used in the oxidation TGA experiments. The
modeled TGA weight change was calculated by integrating eqs
21
using prompt gamma neutron activation analysis and high-
resolution transmission electron microscopy22 showed that the
amorphous oxide coatings on aluminum nanopowders include
impurities of hydroxide, trapped water molecules, and boron.
In addition, single crystalline layers were observed on the surface
and in localized areas of the otherwise amorphous oxide. Such
impurities as hydroxide are expected to become more significant
for aged particles. However, addressing the effect of such
impurities on the aluminum oxidation was beyond the scope of
this paper. These effects, together with effects of specific surface
morphology (e.g., nonspherical and agglomerated particles) and
bulk impurities (e.g., iron, copper, and potassium detected by
2
.1-2.4 over time numerically.
The rates of mass change due to direct oxidation were
described by the model of diffusion for the rate-limiting species.
To simplify the boundary conditions necessary to compute the
diffusion mass flows, the model allows the direct oxidative
growth of only one alumina polymorph at any given time.
Specifically, at any given time only the alumina polymorph with
the highest diffusion resistance, which is tracked during the
calculations, is allowed to grow because of the direct oxidation.
The phase transformations can occur simultaneously and may
contribute to the growth of any coexisting alumina polymorph.
Thus, if several alumina polymorphs coexist, each particle is
surrounded by concentric shells of the respective oxide poly-
morphs. The spherical symmetry allows one to recalculate
volumes of alumina and aluminum core for each specific particle
size using equations similar to 2.1-2.4, but written for an
individual particle, without the integration over PSD.
22
X-ray photoelectron spectroscopy ) will certainly reduce the
accuracy of the aluminum oxidation model16 that is also used
in the present paper.
Neglecting the impurities present in the initial amorphous
oxide layer it is simply assumed that at low temperatures its
thickness increases. At elevated temperatures, the amorphous
alumina transforms into a denser γ-Al2O3 polymorph. This phase
transformation can reduce the thickness and the diffusion
resistance of the oxide layer. For thin oxide layers observed at
high heating rates, this phase change can also result in local
discontinuities in the oxide coverage.1
4,16
Thus, the rate of
Diffusion-Limited Oxidation. For each polymorph, the rate
of direct, diffusion-limited oxidation for a spherical oxide layer
oxidation accelerates rapidly until a continuous, polycrystalline
layer of the γ-Al2O3 is produced. Note that the decomposition
of any hydroxide phase would lead to a similar volume
reduction, accelerating oxidation as well. As the temperature
continues increasing, the layer of the γ-Al2O3 grows until
crystallites of an even denser R-oxide start forming. Formation
of a higher density oxide polymorph once again results in a
reduction in the overall oxide thickness and the diffusion
resistance of the oxide layer. An accelerated oxidation continues
until the oxide layer transforms into a continuous, polycrystalline
film. According to this model,14,16 five processes need to be
analyzed to describe the oxidation quantitatively. In the order
of their occurrence at the increasing temperatures, these
processes are (1) growth of amorphous oxide, (2) the amorphous
to γ-alumina phase change, (3) growth of γ-alumina, (4) the γ
to R-alumina phase change, and (5) growth of R-alumina.
General Model Formulation. The mass of each alumina
polymorph and the mass of metallic aluminum are computed
as a function of time and temperature. To account for the effect
of specific PSD, the computations are made for each particle
size and then integrated over the PSD. According to the model,
changes in the mass of each oxide polymorph can be caused
of outer diameter, D , is described using an Arrhenius type
expression
i
-
1
1
2
Ei
1
1
Di
ox
m˘ i ) C exp -
-
(3)
i
(
)(
)
RT D
i-1
where the subscript i indicates the specific oxide polymorph,
that is, amorphous, γ, or R alumina, subscript i-1 indicates
the underlying substrate or “parent” material, which could be
aluminum, amorphous, or γ alumina, respectively, C and Ei
i
are the parameters describing the diffusion of rate-limiting
species in different alumina polymorphs, and R is the ideal gas
constant. The values of C , and E were the same as determined
i
i
recently for the oxidation of micron-sized aluminum particles.16
Diffusion in Thin Oxide Layers. Additional assumptions
were necessary to describe the oxidation kinetics when the oxide
thickness was comparable to the size of individual γ- and
R-oxide crystallites. When such crystallites are initially formed,
the discontinuities in the oxide coverage are produced.1 The
initial diffusion resistance of the newly formed γ- and R-oxide
crystallites was neglected. A critical thickness for each of the
4,16