Biomedical Engineering Reference
In-Depth Information
in ( 3.12 ) is a physical quantity known as the step kinetic coefficient. It
corresponds to the rate coefficient when a growth unit is incorporated into a step
and is expressed as
The
ˇ
using activation energy E for growth unit
incorporation. The C e is the equilibrium concentration. The
ˇ
exp
. E=k B T/
˛
in ( 3.13 ) is the surface
D ˛.h$/ 1=2 .The
energy and is related to edge free energy
on the step front:
$
is the molecular volume of the crystal. Factor B is given by
B D Z.2NC $ =/
v exp
. E=k B T/:
(3.14)
where Z
is the Zeldovich factor, and N
and C are constants. Since
Z
D
./ 3=2 =4 .k B T/ 1=2 ,
.1 C / g 1=2
B D 0:5N $ C
v exp
. E=k B T /.1 C / f ln
exp f 2 =.k B T/ 2 ln
.1 C / g ;
(3.15)
and ultimately the growth rate is given as
R D K 2=3 .1 C / 1=3 f ln
.1 C / g 1=6 exp
Π2 = f 3.k B T/ 2 ln
.1 C / g :
(3.16)
Here, K is a constant that includes E . Taking the logarithms on both sides of ( 3.16 )
gives
ŒR=Œ 2=3 .1 C / 1=3 f ln
.1 C / g 1=6 D ln
K f 2 =3.k B T/ 2 g 1=
ln
ln
.1 C /:
(3.17)
Calculating the left side from the measured growth rate and plotting it against
1=
ln
.1 C /
gives the most important parameter for two-dimensional nucleation,
.Thevalueof
for the c -face of the HAP crystal was determined to be
3:3 k B T
.
When surface energy
, calculated from growth rate measurements based on the
CC method, is converted to
˛
, its value is estimated as
D 1:1 2:0 k B T
,which
is considerably smaller than
. In the CC method, the growth crystal face is
not specified; hence, there are problems with direct comparisons between the two
values. However, assuming that the results of the CC method are representative of
the
3:3 k B T
value of the c -face makes it possible to explain the differences between the
two in terms of the nucleation mode.
The growth rate has been measured using the phase-shift method for supersatura-
tion ranging from 9.8 to 22.0. Although measurements have been performed below
this range, the results were deemed unusable because of the large distribution of
the data. In contrast, the supersaturation range used for measurements with the CC
method was set to 0.4-7.0, which is considerably lower than for the phase-shift
method. As seen in the two-dimensional nucleation of soluble inorganic crystals,
heterogeneous nucleation with impurities as active sites is more likely to occur at
low supersaturation, and the apparent
markedly decreases. With HAP as well, for
which heterogeneous nucleation occurs at low supersaturation and homogeneous
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