III.3 Silica reduction kinetics in silicon powder compacts .1 Experimental approach .1Experimental approach
III.3.3 Model for silicon compact oxidation
a) Basics
According to the previous section III.2, oxidation of silicon is controlled by reaction (R5), assuming that the silica layer does not preclude the release of SiO(g) from the silicon-silica interface.
Si(s) + SiO2(s) = 2SiO(g) (R5)
This reaction is a combination of two others, (R1) and (R2), that describes the oxidation state of silicon depending on the transition temperature, T*m, or corresponding water vapor pressure surrounding the sample at the transition, HO
P 2 (T*m) (Figure III.1).
For temperatures lower than T*m, a mass gain is observed and is related to passive oxidation of silicon according to the following reaction:
Si(s) + 2H2O(g) = SiO2(s) + 2H2(g) (R1)
For temperatures higher than T*m, a mass loss is observed and is related to the dissociation of silica (R5) and active oxidation of silicon according to reaction:
Si(s) + H2O(g) = SiO(g) + H2(g) (R2)
The mass transport kinetics can be entirely described in three steps as sketched on Figure III.8 for a compact in horizontal position:
(Step 1) The diffusion through a silica layer of nanometric thickness on the silicon particles in the non-reduced area. This step is assumed to not limit the departure of SiO(g) after the passive to active transition as the layer is assumed porous or cracked.
But it certainly limits passive oxidation kinetics of silicon at low temperature.
However, the silica growth at the sample surface is small under the water vapor pressure considered here. Diffusion kinetics of water through the silica layer is then not taken into account and passive oxidation kinetics is overestimated.
(Step 2) The molecular/Knudsen diffusion of SiO(g) and H2O(g) in the porosity of the reduced area. A reduction front, at the radial position rr, moves from the edge (rr = rc) to the center (rr = 0) of the cylindrical compact. rr can be related to the position rg of the grain coarsening area, as observed on Figure III.7, since surface diffusion and then grain coarsening would become dominant after the silica removal [Cob90].
(Step 3) The molecular diffusion of H2O(g) and SiO(g) outside the compact over a height zf, along the furnace tube. (Appendix B.3.3).
As a first approximation, the shrinkage is neglected and the radius of the cylinder compact, rc, is assumed constant.
Figure III.8 : Schematic representation of the model derived. The fluxes of gases involved during the thermal oxidation of silicon compacts are represented. At the particle surface (Step 1), the diffusion of silicon monoxide occurs through holes or solid state diffusion. Inside the cylindrical compact, between 0 and rr, silicon particles are covered with silica and the equilibrium (R5) controls the rate of SiO(g)
release (non-reduced area). Between rr and rc, silicon particles are free from oxide (reduced area) and gaseous species diffuse inside pores (Step 2). Out of the compact, gaseous species diffuse in the furnace tube in a stagnant mixture of He-4 mol.% H2
(Step 3).
b) Model equations
A kinetic model for diffusion is derived inside (Step 2) and outside (Step 3) the porous compact (Figure III.8), assuming steady-state conditions as verified in Appendix B.3.2.
h
out
jSiO z
2 out
jH O z r rr
rc
2 in
jH O r Reduced area Porous Si without SiO2
Non reduced area Si particles with SiO2
in
jSiO r
z zf
0
Wall of the furnace
SiO tube
j
Step 1 Si particle SiO2
Step 2
Step 3
The diffusion flux density of gaseous species j, jj, is given by Equation (III.11), where Pj is the pressure gradient of the species j, Dj, their diffusion coefficient, R the gas constant and T the temperature.
j j
j R P
T
j D (III.11)
The diffusion flux densities HinO
j 2 and jSiOin of H2O(g) and SiO(g) inside the porosity of the reduced area (Step 2) are derived in cylindrical coordinates in Equations (III.12) and (III.13), r being the radial position in the cylindrical compact, HinO
D 2 and DSiOin the diffusion coefficients inside the porosity of the reduced area.
r r r P P T j D
r
r 1
R ln
r c
O H O H in
O in H
O H
r 2 c
2 2
2 (III.12)
r r r P P T j D
r
r 1
R ln
r c
SiO SiO in in SiO
SiO
r c
(III.13)
The diffusion flux densities HoutO
j 2 and jSiOout of H2O(g) and SiO(g) out of the compact (Step 3) are derived in linear coordinates in Equations (III.14) and (III.15), z being the vertical position in the furnace tube, HoutO
D 2 and DSiOout the diffusion coefficients outside the compact.
z P P T j D
r
z c
2 2 2 2
O H O H out
O out H
O
H R (III.14)
z P P T j D
r
z c
SiO SiO out out SiO
SiO R (III.15)
c) Boundary conditions
At the reaction front position, rr, there is competition between the formation of silica and silicon monoxide according to the monovariant equilibrium (R5). Thus, fixing the temperature, the pressure of silicon monoxide at the reaction front position, PSiOrr , is given by the equilibrium pressure of reaction (R5), in Equation (III.16), where P° is the standard pressure.
12 5 R
SiO SiO
r P 5 P K
Pr (III.16)
In the reduced area, particles are no longer covered with silica. Reaction (R2) of equilibrium constant K2 controls the pressures of silicon monoxide and water inside the sample.
Si(s) + H2O(g) = SiO(g) + H2(g) (R2)
From Equation (III.17), the vapor pressure of water is found negligible as regards the pressure of silicon monoxide, in the temperature range of active oxidation (T > 1000 °C).
r r
r
r P
K P
P
P P 3 SiO
2 SiO H O
H2 2 2.10 then PHrO PSiOr
2 (III.17)
Since HinO
D 2 and DSiOin are of the same order of magnitude, HinO
j 2 can be neglected with respect to jSiOin (Equations (III.12) and (III.13)).
Since the water vapor pressure in the compact is much lower than the silicon monoxide one (Equation (III.17)), water is entirely consumed at the surface of the compact through the silicon oxidation. The water vapor pressure at the surface of the compact, c
2O H
Pr , can be neglected with respect to the value far from the compact, f
2O H
Pz . Indeed, PSiOrc cannot be superior to PSiOR5 and Equation (III.18) can be written for temperatures below 1400 °C.
Pa 10 .
6 2
2 R SiO H 2 SiO H O H
5 c
2 c c c 2
2 P K
P P K P
P P P
r r r
r then f 1 1.8Pa
2 c
2O HO
H
z
r P
P (III.18)
Far from the sample, the temperature is low, silicon monoxide condensate in the form of silicon and silica and the partial pressure, PSiOzf , becomes negligible as regards PSiOrc , while the partial pressure of H2O(g), f
2O H
Pz , is at least the value in the experimental gas mixture (see Appendix B.3.4).
d) Model derivation
Under steady-state conditions, the net transport of oxygen atoms reaching and leaving the sample vanishes (Equation (III.19)) and the silicon monoxide pressure at the surface of the cylinder, PSiOrc (Equation (III.20)), is deduced from Equations (III.12) to (III.19), Sin being the lateral cylindrical sample surface. Sout is twice the furnace tube section to take into account for the diffusion in the lower part as well as in the upper part of the tube.
in in SiO out out
O H out
SiO j 2 S j S
j (III.19)
in out
f c r out c SiO in
SiO
in out
f c r out c
O H O H in out
f c r out c SiO in
SiO
in R SiO
SiO SiO
ln ln ln
2 f
2 5
c
S S z r r D r
D
S S z r r D r
P S S z r r D r
D P D
Pr z (III.20)
The mass loss rate is derived in Equation (III.21) considering that H2O arrival is responsible for a mass gain and SiO departure for a mass loss. Mj is the molar mass of a specie j.
SiO in out
f c r out c SiO in
SiO
in out
f c r out c
O H out SiO O H
SiO in out
f c r out c SiO in
SiO
in SiO out R SiO
SiO O out
O H O H
f out
out SiO c
out SiO O c out
O H
ln ln
ln
R 2
f 2
5 2
f 2 2
M S S z r r D r
D
S S z r r D r
D P
M S S z r r D r
D
D P D
M D P
z T
S
S M r j M r t j
m
z z
(III.21)
In the same way, the silica mass variation in the compact is assessed in Equation (III.22), by considering that the molar variation of silica is half the oxygen one.
in out
f c r out c SiO in
SiO
in out
f c r out c
O H out SiO O H
in out
f c r out c SiO in
SiO
in SiO out R SiO
SiO out
O H O H
SiO f
out SiO
ln ln
ln 2
1
R 2
f 2
5 2
f 2
2 2
S S z r r D r
D
S S z r r D r
D P
S S z r r D r
D
D P D
D P
z M T
S t
m
z z
(III.22)
e) Application to the description of silicon oxidation sequences
During the passive oxidation of the compact, the model is reduced to a more simple expression where the influxes, jjin, do not exist. Equilibrium (R5) is always realized at the edge of the sample (rr = rc). Equations (III.20), (III.21) and (III.22) can be written in a more simple way in Equations (III.23), (III.24) and (III.25), respectively.
5
c R
SiO r
SiO P
P (III.23)
SiO out SiO R SiO O out
O H O H f out
5 2
f
R P2 D M P D M
z T S t
m z
(III.24)
out SiO R SiO out
O H O H SiO f
out
SiO 5
2 f 2 2 2
2 1
R M P D P D
z T
S t
m z
(III.25)
The rate of mass loss (Equation (III.24)) is nil at a temperature T*m, where the water vapor pressure far from the sample, f
2O H
Pz , can be deduced from Equation (III.26) and corresponds to the water partial pressure, 5
2
WR O
PH , found in section III.1 (Equation (III.4) and Figure III.1).
m R * out SiO
O H
out SiO O SiO O
H
5
2 f
2 P T
D D M
Pz M (III.26)
For the 1.25 °C min-1 cycle, the rate of mass loss is nil at 1020 °C, thus giving a water pressure of 1.8 Pa far from the sample.
As the molar flux of silicon monoxide overpass the water vapor flux arriving at the surface of the compact, the silica starts to dissociate and the passive to active transition occurs as defined by Wagner [Wag58]. The reaction front position is located at rc as long as the silica produced during the passive oxidation is not entirely reduced. Then, the reaction front starts to move to the center of the compact and the model is described by the general Equations (III.20), (III.21) and (III.22). The variation of the reaction front position (rr) must be assessed in order to solve these equations. Assuming that the silica is initially uniformly distributed at the particle surfaces throughout the sample, the reaction front position as function of time, t, rr (t), is given by Equation (III.27).
c
r ( 0)
) ) (
(
2
2 r
t m
t t m
r
SiO
SiO (III.27)
An iterative calculation is performed, rr (t) is sequentially uploaded in Equations (III.20), (III.21) and (III.22) thanks to Equation (III.27). The silica mass gradually decreases, causing the reaction front to move forward, which tends to reduce the rate of mass loss.
Finally, the silica mass and rr position become nil, Equations (III.20), (III.21) and (III.22) simplify respectively in Equations (III.28), (III.29) and (III.30) during active oxidation (R2).
Equation (III.29) is simply the rate of mass loss as estimated by Wagner (Appendix A).
out SiO out
O H O H SiO
f 2 2 c
D P D
Pr z (III.28)
Si out
O H O H f out
2 f
R P 2 D M
z T
S t
m z
(III.29)
0
SiO2
t
m (III.30)
The rate of mass loss no longer depends on the equilibrium constant of reaction (R5). It is then monitored by the water vapor pressure in the gas mixture, f
2O H
Pz , and the diffusion length of the species out of the compact, zf. f
2O H
Pz being equal to 1.8 Pa, zf is found to be 40 mm for the 1.25 °C min-1 cycle, a value slightly larger than the uniform temperature area in the furnace tube (30 mm) beyond which a condensate film of silicon and silica is observed (Figure III.4).
f) Assessment of model parameters
Diffusion coefficients out of the compact, Djout, are the molecular diffusion coefficients (Equation (III.31)), estimated as a function of molecular characteristics and temperature from the semi-empiric approach of Chapman-Enskog [BSL01, AS62] in Appendix B.4.1.
mol j out
j D
D (III.31)
Diffusion coefficients inside the compact (Equation (III.32)) are function of the molecular diffusion coefficient, Djmol, and the Knudsen diffusion coefficient, DKnudsenj , accounting for the elastic collisions between gas molecules and the pore surface. DKnudsenj is calculated from the gas kinetic theory [BSL01] in Appendix B.4.2, replacing the molecular mean free path by the pore size estimated between 1 and 10 µm from SEM observations.
1
Knudsen j mol j in
j
1 1
D D
D p (III.32)
The volume fraction of pores, p, is fixed as 40 %, a value comprised between the initial (47 %) and final (36 %) experimental values (density curve in Figure III.6). The pore tortuosity, , is at least equal to the minimal molecular path around spherical particles, i.e.
2. The initial amount of silica in the compact, ( 0)
2 t
mSiO (Equation (III.27)), is related to the thickness of the oxide covering the particles and to the sample mass. This value is adjusted in the model to fit the experimental global mass loss and is consistent with the IGA
measurements. It corresponds to an initial thickness of 0.52 nm for the native oxide (0.43 ± 0.10 nm from IGA).