Biomedical Engineering Reference
In-Depth Information
Considering the theory of chemical reaction kinetics, the HA crystallization should fol-
low the Arrhenius equation [167,170,182] as represented in Equation 6.19:
d
d
x
t
(
)
x
n
r
=
=
k
1
−
(6.19)
where
x
represents the conversion ratio of HA from amorphous calcium phosphate to crys-
talline HA. This can be recognized as the IOC value for the various degrees of crystal-
lization of the HT-HACs displayed in Figure 6.15. The symbol
r
is the crystallization rate
and
k
is the rate constant. The rate constant (
k
) represents the reaction rate for the kinetics
of Arrhenius equation, and it can be recognized as the crystallization rate during heat
treatments. Next, the crystallization rate as well as the activation energy of hydrothermal
crystallization can be quantitatively evaluated by the IOC data. Equation 6.20 is obtained
from the natural logarithmic (ln) of 6.19, and it should be noted that the conversion ratio
(i.e., the IOC) determines the overall crystallization kinetics.
dIOC
d
ln
=
n
ln(
1
−
IOC
)
+
ln
k
(6.20)
t
According to the derivation in a previous study [120], the reaction order
n
, which is cal-
culated from the slope by the least squares fitting method of Equation 6.20 at a maximum
coefficient of determination (
R
2
), is approximated from 1.8 to 2.1 for the atmospheric and
vacuum crystallization treatments of plasma-sprayed HACs at 500°C to 800°C. Since the
total reaction order should be an integer, it can be deduced that HA crystallization should
follow the second-order reaction kinetics of the Arrhenius equation, and Equation 6.19 can
be converted into Equation 6.21. Table 6.5 lists the rate constant
k
, which can be calculated
from the slope of the linear plots of (1 − IOC)
−1
versus heat-treatment time (
t
) from the
integration of Equation 6.21 for the atmospheric and vacuum crystallization treatments.
Based on the reaction kinetics of Arrhenius equation, the rate constant (
k
) can be thought
as the reaction rate, and it represents the crystallization rate during heat treatments. Except
for considering the oxidation problem of Ti-6Al-4V during high-temperature heating, the
experimental results, as shown in Table 6.5, demonstrate that heat treatments in an ambient
TABLE 6.5
Results of Crystallization Kinetics of Various Heat-Treated HACs
RateConstant,
k
k
=
A
exp(−
E
a
/
RT
)
k
500
k
600
k
800
ActivationEnergy,
E
a
(kJ/mol)
a
Vacuum heating
b
0.197
0.365
0.586
24.6
Atmospheric heating
b
0.272
0.379
0.612
19.1
k
100
k
150
k
200
Hydrothermal treatment
0.294
0.591
0.905
16.6
Source:
Yang C.W. and Lui, T.S.,
Mater. Trans
., 48(2), 211-218, 120. 2007. With permission.
a
Data calculated from the slope of ln
k
vs. heating temperatures (1/
T
) graph.
b
The parameters
k
500
,
k
600
, and
k
800
represent the crystallization rate constant for 500°C, 600°C, and 800°C heat
treatments in the air and in vacuum, respectively.
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