Thermodinamic Interpretaion of the Morphology Individuality of Natural and Synthesized Apatite Single Crystals

doi:10.4236/jcpt.2013.34019

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Journal of Crystallization Process and Technology, 2013, 3, 119-122

http://dx.doi.org/10.4236/jcpt.2013.34019 Published Online October 2013 (http://www.scirp.org/journal/jcpt)

119

Thermodinamic Interpretaion of the Morphology

Individuality of Natural and Synthesized Apatite Single

Crystals

Takaomi Suzuki, Haruka Takemae, Mika Yoshida

Department of Environmental Science and Technol ogy, Fac ulty of E ngineering, Shinshu University , Nagano, Japan.

Email: [email protected]

Received July 24th, 2013; revised August 23rd, 2013; accepted September 5th, 2013

License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly

cited.

ABSTRACT

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

120

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

[2].

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

[6].

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



 (1)

where i



is the SSFE and i

is the surface area of ith

01234567

h: length of normal line (mm)

Specific Surface Free Energy (mN/m)

(A)

(B)

(C)

(D)

●

○

)0110(

)1110(

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

121

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





  (2)

where



 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,

obs



, can be described as,

obs terr







(3)

where terr



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







(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





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)

and

 

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

as,

 

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-

Thermodinamic Interpretaion of the Morphology

Individuality of Natural and Synthesized Apatite Single Crystals

122

sol

A)(



sol

D)(



liq



)(A



)(D





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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