Biomedical Engineering Reference
In-Depth Information
constant using diffusion data from Hon et al. [Hon, 1979; Hon, 1980] and assum-
ing the reaction was controlled by diffusion through the SiC layer, concluding
that the reaction had to be controlled by solution-precipitation of the C in the
molten Si, as diffusion estimations gave much lower reaction rates than observed
[Varela, 2007 ].
For porous carbon preforms with wall thickness smaller than 10
m, the dif-
fusion mechanisms play no signifi cant role. Instead, the carbon is dissolved in the
molten silicon and precipitated as silicon carbide near the carbon walls when the
solution supersaturates. In the wood based carbon precursors used for bioSiC
the carbon wall thickness is about 2
μ
m, so the SiC formation should be exclu-
sively controlled by solution-precipitation. However, as some of the channels in
the preform have a diameter of the same range as the wall thickness, they can be
closed as the SiC precipitates and thus coalesce with surrounding small channels,
yielding an effective SiC layer that is signifi cantly thicker, forcing a scenario
where the SiC growth can only happen by a diffusion mechanism. This is also sug-
gested by the presence of unreacted C in bioSiC (Figures 11.5 and 11.10).
In this scheme, the liquid Si must dissolve the cell walls of the carbon preform
to form a solution which will then precipitate as micron-sized SiC grains. An esti-
mation of the time required for the solution of the preform into the molten Si can
be made using the Nerst-Noves-Whitney equation:
±
1
μ
dm
dt
DA c
(
−
c
)
(11.6)
c
s
0
−
=
ζ
()
t
In Equation 11.6
dm
c
/ dt
is the carbon mass dissolved into the molten Si per unit
time,
D
is the diffusion coeffi cient of carbon in liquid Si for a given temperature,
A
is the C-Si(L) contact surface,
−
(t)
is the thickness of the solution layer,
c
s
is the
carbon concentration and
c
0
is the initial carbon concentration, which is zero as
initially the molten Si contains no C. For this example's considerations and based
on microstructural evidence, assume an initial wall thickness of one
ζ
m and
suppose that the temperature is constant through the C/Si interface and its sur-
roundings, due to the high thermal conductivity of molten Si [Bartlett, 1967].
Further, suppose that the carbon concentration at the interface is that of equilib-
rium at the reaction's temperature, that is,
c
s
= c
e
for all
t
μ
≥
0, being
c
e
the equilib-
rium concentration. Also, approximate
ζ
(t)
∼
4(
Dt
)
½
[Crank, 1975 ]. Substituting
into equation (4), one obtains:
−
dm
dt
DT Ac
Dt
()
DT Ac
t
()
(11.7)
c
e
e
=
=
4
4
Equation 11.7 make it possible to estimate the time needed to dissolve
a
with
one
m-thick carbon wall as a function of temperature
T
. For these calculations
c
e
has been taken from the works of Scace [Scace, 1959] and
D
has been taken
from [Gnesin, 1973]. It has been shown that the heat generated by the reaction
can raise the temperature in the reaction zone by about 500 °C. This has been
μ
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