Biomedical Engineering Reference
In-Depth Information
finding that the edge free energy required for nucleation is discontinuously smaller
at low supersaturation. As will be discussed later, even for HAP growth, there are
cases in which discrepancies between edge-free-energy data sets can be explained
by considering a heterogeneous nucleation model.
3.4
HAP Crystal Growth Mechanisms
3.4.1
Driving Force for Growth
In a two-component simple aqueous solution system of soluble salts,
supersaturation—the thermodynamic driving force for growth—is determined by
the solubility curve of the solid phase. This method is sufficiently useful as an
approximation; there are no significant problems even when interpreting data on
growth rate versus supersaturation and in calculating physical constants related
to growth. However, in the case of insoluble calcium phosphate salts, including
HAP, there are three problems in particular: the possibility of multistep dissociation
equilibrium of the phosphate group in solution, the possibility of ion pair formation
between calcium and phosphate ions, and the possibility that the solutions in
which the crystals are grown contains cations and anions other than calcium and
phosphate. For these reasons, the driving force is normally defined using the ion
product
I
P
—defined as the ionic strength
I
calculated from the concentration and
the total charge of each ion in the solution—and the solubility product of the solid
phase
K
sp
(solely a function of temperature).
Let
a
i
and
c
i
be the activity and concentration, respectively, of the
i
th ion
contained in the solution.
a
i
D
f
˙
c
i
;
(3.2)
where
f
˙
is the mean activity coefficient, which, from the Debye-Huckel approxi-
mation, is expressed by the following equations using ionic strength
I
:
z
i
2
p
A
I
log
f
˙
D
p
(3.3)
1
C
Br
i
I
X
c
i
z
i
2
:
I
D
0:5
(3.4)
The
z
i
r
i
is the effective radius of each ion, and
A
and
B
are theoretical coefficients determined by the specific permittivity of solvent
is the charge of the
i
th ion,
"
r
and
absolute temperature
T
, commonly using
1:8246
10
6
."
r
T/
3=2
A
D
(3.5)
