Journal of Crystallization Process and Technology, 2013, 3, 119-122 Published Online October 2013 (
Copyright © 2013 SciRes. JCPT
Thermodinamic Interpretaion of the Morphology
Individuality of Natural and Synthesized Apatite Single
Takaomi Suzuki, Haruka Takemae, Mika Yoshida
Department of Environmental Science and Technol ogy, Fac ulty of E ngineering, Shinshu University , Nagano, Japan.
Received July 24th, 2013; revised August 23rd, 2013; accepted September 5th, 2013
Copyright © 2013 Takaomi Suzuki et al. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
Specific surface free energy (SSFE) of natural calcium fluorapatite from the same mother rock and synthesized barium
chlorapatite from the same lot was determined using contact angle of water and formamide droplets, compared with
grown length of crystal face (h). The experimentally obtained SSFEs have different values even for the same index
faces of the different crystals. The SSFEs also have wide distribution for each face of crystals. Observed SSFE is con-
sidered to be not only the SSFE of ideally flat terrace face, but also includes the contribution of strep free energy.
Though the crystals we experimentally obtained were growth form, the relationship between SSFE and h was almost
proportional, which looks like satisfying Wulff’s relationship qualitatively. The slope of SSFE versus h line shows the
driving force of crystal growth, and the line for larger crystal has less steep slope. The driving force of crystal growth
for larger crystal is smaller, which also means that the chemical potential is larger for larger crystal. Th e ind ividu ality of
crystals for the same lot can be explained by the difference of the chemical potential of each crystal.
Keywords: Crystal Morphology; Single Crystal Growth; Apatite Crystal
1. Introduction
Measurement of specific surface free energy (SSFE) of
solid using contact angle of liquid is very popular and
well accepted in the field of polymer science. Especially,
automatic contact angle meter with software for the
analysis of SSFE can be obtained commercially (for ex-
ample, CA-V Series, Kyowa Interface Science). How-
ever, in the field of crystal science, the measurement of
SSFE of crystals is believed to be difficult. In fact, ex-
perimental determination of SSFE of crystal is very few,
even though the significance of SSFE for crystal growth
is well discussed. Only a few trials to determine the sur-
face energy of crystals were reported. For example, bond-
ing energy of quartz was measured by introducing a
crack into the crystal, and the energy of breaking was
determined as the surface energy of the quartz crystal [1].
Determination of SSFE of polymer surface is not diffi-
cult because we can easily determine the contact angle
with a good reproducibility. However, the con tact angles
of liquid on inorganic crystal face have very wide distri-
bution, and we need to take a lot of photographs to de-
termine the average of contact angle. We determined
SSFE of barium chlorapatite single crystals from contact
angle of liquids [2] for the first time. At that time we
used more than 4000 photographs in order to determine
the average contact angle of liquids. We also determined
the SSFE of ruby crystals from the contact angle of liq-
uid droplet and discussed the relationship between the
grown length of the crystal face [3]. Although the ruby
crystals were growth shape, the relationship between the
SSFE and h were almost proportional, which looks satis-
fying Wulff’s relationship. However, Wulff’s relationship
should be satisfied for ideal equilibrium system, and the
crystal face should be ideally flat. At this time, we ex-
tended our experimental technique for natural mineral
crystals in order to study the relationship between SSFE
and h for the less ideal but more real system.
2. Experimental Procedure
Natural calcium fluorapatite crystals (Cerro del Mercado
Cd. Durango, Mexico) were produced from a cluster of
crystals with mother rock using a hammer and a chisel.
Thermodinamic Interpretaion of the Morphology
Individuality of Natural and Synthesized Apatite Single Crystals
Copyright © 2013 SciRes. JCPT
The mother rock is fragile and the interface between the
crystal and the mother rock can be easily separated. The
advantage of this sample is that a clean and as-grown flat
crystal face can be obtained. Ideal form of the fluorapa-
tite crystal is hexagonal prism with pyramidal end faces.
The indices of the prismatic faces is
1010 and that of
the pyramidal faces is
1011 .
Four clean crystals without serious damage were ob-
tained from a mother rock and named crystal (A), (B),
(C), and (D) (Figure 1). Because not all the faces of the
crystal look clean and flat, the cleanest face in the
1011 faces is determined as
1011 face and used
for the experiment. Also the adjacent
1010 face of
each crystal was used. Crystals of barium chlorapatite
were synthesized using NaCl flux method [4]. Three well
formed crystals were named crystal (a), (b), and (c) (Fig-
ure 2).
The crystals were sonicated with ethanol and thor-
oughly dried. Each single crystal was fixed on a glass
plate using a small piece of clay for contact angle meas-
urement. Water and formamide was dropped onto the
crystal using a micropipette. The droplets with a volume
of 0.1 mm3 were observed using a digital camera
Figure 1. Photographs of natural fluorapatite crystals (A),
(B), (C), and (D). The scale of the paper under the crystal is
1 mm. The larger crystal is more opaque.
1mm 1mm 1mm
Figure 2. Photographs of synthesized chlorapatite crystals
(a), (b), and (c).
with a magnifying lens. The details of measurement of
the contact angles of the droplets are described elsewhere
3. Results
The values of SSFE, S
, can be calculated from the ob-
tained contact angle of the liquid,
, using the Fowkes
approximation [5] and Wu’s harmonic mean equations
The polar and dispersed components of the liquid were
= 22.1 mN/m,
= 50.7 mN/m for water and
= 39.5 mN/m,
= 18.7 mN/m for formamide,
which were taken from reported data [7].
The length of the normal line from the center of the
crystal to each face, hi, was obtained from the photograph
and the hi was considered to be the grown length of the
ith face. The SSFE of natural calcium fluorapatite was
compared with hi and shown in Figure 3. The line join-
ing the points of
1010 and
1011 faces of crystal
(A) in Figure 3 almost passes through the origin. On the
other hand, the slope of the line is less steep for larger
crystals. The relationship between calculated SSFE of
synthesized barium chlorapatite (a), (b), and (c) are shown
in Figure 4 with the p lots of natural calcium fluorapatite
(A)-(D). They are found between the plots of crystal (A)
and (B).
4. Theoretical Background
Surface free energy of a crystal is a sum of the free en-
ergy of each face as,
urfi i
where i
is the SSFE and i
is the surface area of ith
h: length of normal line (mm)
Specific Surface Free Energy (mN/m)
Figure 3. Relationship between the specific surface free
energy and the length of the normal line to each face of
natural fluorapatite crystals (A), (B), (C), and (D).
Thermodinamic Interpretaion of the Morphology
Individuality of Natural and Synthesized Apatite Single Crystals
Copyright © 2013 SciRes. JCPT
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,
G, and
well known Wulff’s relationship [8] is obtained as,
ii S
  (2)
is the driving force of crystal growth, and
v is atomic volume of the crystal. If the crystals were
ideally equilibrium, the relationship between the SSFE,
, 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,
, can be described as,
obs terr
where terr
is SSFE of ideally flat face, L is step density,
is step free energy. In the real system, the sur-
face free energy of crystal can be described as,
real obs
urfi i
and the minimization of real
G requires the relationship
as ,
obs terr
ii ii
  (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
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
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),
, and that of crystal (D),
, is described as,
 
 (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,
 
liq sol
 (7)
 
liq sol
 (8)
where liq
, (A)
, and
is the chemical potential
of liquid phase, that of crystal (A) and (D), respectively.
The relationship of Equation (6) requires the relationship
 
ol sol
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-
Thermodinamic Interpretaion of the Morphology
Individuality of Natural and Synthesized Apatite Single Crystals
Copyright © 2013 SciRes. JCPT
Figure 5. Schematic pict ure of relationship between driving
force and the chemical potential of crystals. The dots and
lines in crystal (D) indicate contamination and defects. Be-
cause of such contamination and defects, the cry stal (D) has
larger chemical potential than crystal (A).
ble, and the energy of the bulk crystal is higher. There-
fore, the chemical potential of the crystal (D),
, is
larger than that of (A),
, as Equation (9). The SSFE
of synthesized barium chlorapatite (a)-(c) were also
qualitatively satisfied relationship of Equation (5). The
SSFE for
1010 is larger than that for
1011 . The
relationship between SSFE and h is close to natural crys-
tals (A) and (B).
6. Conclusion
Though the crystals from the same mother rock or the
same crucible have a variety of shapes, their shapes are
resulted from the thermodynamic equilibrium. The ob-
served SSFE was almost proportional to the grown
length of the face, and the individuality of the crystal
shape is explained by the minimization of surface free
energy including step free energy.
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