Thermodinamic Interpretaion of the Morphology
Individuality of Natural and Synthesized Apatite Single Crystals
Copyright © 2013 SciRes. JCPT
121
0
10
20
30
40
50
60
70
01234567
(b)
(a)
(c)
(1010)
(1011)
●
○
Specific Surface Free Energy(mN/m)
h: length of normal line (mm)
Figure 4. Relationship between the specific surface free
energy and the length of the normal line to each face of
chlorapatite crystals (a), (b), and (c). Plots of natural fluo-
rapatite crystals are also shown by small marks.
face. The equilibrium shape of the crystal is determined
by minimization of the surface free energy,
urf
G, and
well known Wulff’s relationship [8] is obtained as,
2constant
ii S
hv
(2)
where
is the driving force of crystal growth, and
S
v is atomic volume of the crystal. If the crystals were
ideally equilibrium, the relationship between the SSFE,
i
, and the grown length of the crystal, i
h, satisfies
Equation (2). Therefore, all crystals in the same system
should be the similar. Also, if the crystal face is ideally
flat, like terrace, the SSFE of the same index face should
be the same for different crystals. Therefore, the crystal
habit must be the same for the equilibrium system.
5. Discussion
Though ideal equilibrium crystal shape is uniform, real
crystals from the same lot have different shape and they
are called growth shape. Our experimental result shows
that the SSFE of the same index face for the different
crystals are not the same, and the contact angles of liquid
for one individual crystal face have wide distribution,
which corresponds to the wide distribution of SSFE of a
crystal face. Such wide distribution of observed SSFE
can be caused by step free energy. The real crystals have
not only ideal flat terrace face, but also steps are distrib-
uted on the surface of crystal face. The observed SSFE,
obs
i
, can be described as,
obs terr
ii
L
(3)
where terr
i
is SSFE of ideally flat face, L is step density,
and
is step free energy. In the real system, the sur-
face free energy of crystal can be described as,
real obs
urfi i
GA
(4)
and the minimization of real
urf
G requires the relationship
as ,
constant
obs terr
ii ii
hLh
(5)
The distribution of the step density is considered to
cause the wide distribution of the observed SSFE of the
crystal. Also, if each crystal satisfies the thermodynamic
equilibrium and stable state, and the ind ividuality of each
crystal can be explained by the difference of the step
density distribution. Th e line binding th e plot of
1010
and
1011 of crystal (A) passes a point near origin.
Crystal (A) is most close to equilibrium condition, be-
cause crystal (A) is the smallest, and it satisfied Equation
(5). On the other hand, the lines binding the plot of
1010 and
1011 of larger crystals are apart from
origin, because they are apart from equilibrium, and
hardly satisfy Equation (5). However, the slope of lines
drawn from origin to the center of points between
1010 and
1011 which is shown by broken lines in
Figure 3 roughly correspo nd to the constant of 2S
v
of Equation (2). For example, the slope of line for crystal
(D) is less steep than that of crystal (A), indicating the
relationship between the driving force of crystal (A),
A
, and that of crystal (D),
D
, is described as,
AD
(6)
Because both crystals are produced from the same
mother rock, the chemical potential of liquid phase
should be th e same. The driving for ce of both crystal can
be described as,
AA
liq sol
(7)
and
DD
liq sol
(8)
where liq
, (A)
ol
, and
D
ol
is the chemical potential
of liquid phase, that of crystal (A) and (D), respectively.
The relationship of Equation (6) requires the relationship
as,
AD
ol sol
(9)
The chemical potential for larger crystal should be
larger than that for smaller crystal. The larger chemical
potential is considered to be resulted from the defects or
contamination in the crystal. In fact, the photographs of
crystals in Figure 1 show that the larger crystals look
dirtier than the smaller crystals. This relationship is
schematically described in Figure 5, where dots and
lines in the picture indicate contamination and defects.
Such contamination and defects make the crystal unsta-