Biomedical Engineering Reference
In-Depth Information
50:29
10
8
."
r
T/
1=2
B
D
:
(3.6)
The driving force for growth, i.e., the chemical potential difference between the
solid phase and liquid phase
,is
kT
D
ln
a
i
a
ei
D
ln
I
P
K
sp
;
(3.7)
where
a
ei
is the ionic activity at equilibrium. Between
and supersaturation
,
the relationship
=kT
D
ln
.1
C
/
holds. Therefore,
I
P
K
sp
1:
D
(3.8)
However, to enable comparisons under a consistent supersaturation, even be-
tween solid phases that have naturally different stoichiometric ratios, an equation
standardized by
N
, the sum of the valences for each ion comprising the solid phase,
is used [
42
]:
1=N
I
P
K
sp
D
1:
(3.9)
For Ca
10
(PO
4
)
6
(OH)
2
, which is the stoichiometric composition of HAP,
N
is 18. In the form of (
3.9
), a constant supersaturation is applied even if the
descriptive formula for the chemical composition of the target solid phase differs.
For HAP,
K
sp
is occasionally defined on the basis of its half-unit cell Ca
5
(PO
4
)
3
OH.
However, this value turns out to be approximately the square root of the
K
sp
value
for Ca
10
(PO
4
)
6
(OH)
2
, the corresponding full-unit cell. Therefore, supersaturation
calculated using (
3.9
) is the same value independent of the definition of the unit cell.
Equation (
3.9
) includes an ionic product term. For this reason, even if the
calcium or phosphate concentrations are constant, the supersaturation greatly varies
depending on the background salt concentration, and this affects the conditions for
growth and phase transformation. For example, the structural reconstruction-type
phase transformation of ACP to HAP, which is described elsewhere [
24
], occurs
when only calcium and phosphate ions (excluding hydrogen and hydroxide ions)
are contained in the solution. When the supersaturation is raised by adding 10 mM
of KCl to this solution, the phase transformation switches to a heterogeneous-type
nucleation in which HAP forms around ACP as the nucleus (or template).
3.4.2
Biological HAP Formation Model
The HAP growth mechanisms, including nucleation, differ considerably depending
on whether they are viewed from a biological or a physicochemical perspective.
