Biomedical Engineering Reference
In-Depth Information
Where
σ
is the surface tension of the liquid,
r
is the capillary radius,
θ
the contact
angle,
g
is gravity's acceleration,
the liquid's density and
h
max
is the maximum
height of the liquid inside the capillary. To describe the dynamical aspects of the
liquid ' s infi ltration in the tubular capillary a laminar fl ow is assumed, so Poi-
seuille's law can be applied, giving:
ρ
8
2
η
h
hh
h
⎡
⎢
⎡
⎢
⎤
⎥
−
⎤
⎥
max
t
=
rg
h
ln
(11.4)
max
ρ
−
max
Where
is the liquid's viscosity,
t
is the time and
h
the height of the liquid inside
the capillary. With these results, an estimation of the maximum infi ltration height
can be done using the physical properties of molten Si [Gern, 1997; Gerwien,
1986; Grabmaier, 1981]. A carbon preform with a mean pore diameter of 5
η
m can
be infi ltrated with liquid Si up to approximately 25 m high. Clearly the infi ltration
process is very fast. For instance, a carbon preform with pore diameters ranging
from 5
μ
m will be infi ltrated in less than two seconds. Although the
large channels are infi ltrated faster than small ones (due to viscosity effects), the
capillary pressure is much larger in the small channels, so because a large channel
is connected to some small channels (see Figure 11.9), these will extract liquid Si
from the large one until a pressure equilibrium is reached.
Most authors agree in the existence of an initial phase where the surface
carbon is dissolved in the liquid Si and then is nucleated as SiC in the carbon
surface. The latter growth of SiC can then be controlled by two different mecha-
nisms. One of them is the diffusion of C and/or Si through the previously formed
SiC, as described by Hon et al. [Hon, 1979; Hon, 1980], Fitzer and Gadow [Fitzer,
1985; Fitzer, 1986] and Zhou and Singh [Zhou, 1995]. The other possible process
is the solution-precipitation of all the carbon present in the silicon, as described
by Pampuch et al. [Pampuch, 1986; Pampuch, 1987]. Whether one mechanism or
the other dominates will depend on the morphology of the carbon preform. On
bulk, poreless carbon once the initial SiC layer is formed Si and/or C will have to
diffuse through the SiC and this is a slow process, due to the low C and Si diffu-
sion coeffi cients in SiC [Hon, 1979; Hon, 1979; Hong, 1980; Hong, 1981]. Typically,
the formation of a 10
μ
m to 100
μ
μ
m thick SiC layer takes about one hour at 1500 °C.
-SiC formation reaction follows fi rst order kinetics, so the concentration
of each phase can be modelled using:
The
β
[]
=
[
(
−
[]
)
[
CC CC
f
+
0
f
exp
()
t
(11.5)
i
i
i
i
Where
k
is the reaction constant of the SiC formation reaction and
C
i
is the
concentration of phase
i
.
C
i
f
is the concentration of each phase for an infi nite
reaction time, which has to be introduced since Si is introduced in excess in the
process, and
C
i
0
is the initial concentration of phase
i
. Pampuch et al. studied the
SiC formation process by reaction of bundles of C fi bres 4 - 6
m in diameter with
liquid Si at 1422 ° C and 1439 ° C [Pampuch, 1986 ; Pampuch, 1987 ], and determined
the reaction constant
k
using DTA analysis. They also estimated the reaction
μ
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