Cryst. Res. Technol. 42, No. 12, 1217 – 1221 (2007) / DOI 10.1002/crat.200711008
Measurement of specific surface free energy of ruby and quartz
single crystals using contact angle of liquids
Takaomi Suzuki*1, Naoki Sugihara1, Eiichi Iguchi1, Katsuya Teshima1, Shuji Oishi1, and
Masayuki Kawasaki2
1
2
Faculty of Engineering, Shinshu University, 4-17-1 Wakasato, Nagano 380-8553, Japan
Nihon Dempa Kogyo Co. Ltd. (NDK), 1275-2, Kamihirose, Sayama 350-1321, Japan
Received 20 May 2007, accepted 7 August 2007
Published online 10 November 2007
Key words surface, free energy, crystals.
PACS 68.35.Md
The specific surface free energy of ruby and quartz single crystal was experimentally obtained using contact
angle of water and formamide droplets on the crystal surfaces, and compared with the morphology of each
crystal. The ruby crystals satisfied Wulff’s relationship even though their shape were not equilibrium form.
The specific surface free energies of the growing faces of synthetic quartz crystal, -X, +X, Z, and S faces
were obtained as 51.9, 55.6, 57.4, and 58.9 mN/m, respectively. The growth rates of these faces were 0.09,
0.23, 0.28, and 0.33 mm/day, respectively. The growth rate of each face of the quartz crystal can be regarded
as a function of the experimentally obtained specific surface free energy.
© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1
Introduction
The specific surface free energy (SSFE) is significant value to determine the morphology of a single crystal.
Especially, Wulff’s relationship [1] is well known as,
γi
= constant
(1)
hi
where γ i is the specific surface free energy of the ith (arbitrary) face of the crystal and hi is the length of the
normal line to the ith face from Wulff’s point in the crystal. This relationship was theoretically introduced, and
some theoretical approach for studying the relationship between the SSFE and the morphology were performed
[2,3], but experimental study for confirmation of this theory is very few [4], because the measurement of SSFE
of crystal surface was believed to be difficult. Recently, we performed an experimental trial to investigate the
SSFE of chlorapatite and ruby single crystal by measurement of contact angles of liquids and compared with
the morphology of crystals [5-10]. Even though, Wulff’s relationship should be satisfied only for the crystals of
equilibrium form, the arbitral chosen crystals satisfied eq. (1). The sizes of crystals we synthesized were 1~2
mm, which were too big to be regarded equilibrium form. In this paper we are discussing the relationship
between our experimentally obtained SSFE and morphology of crystals in growth form.
2
Experimental
Ruby crystals were synthesized using a MoO3 flux. The details of the synthesis of the sample crystals are
described elsewhere [11,12]. The form of ruby crystal was basically a hexagonal bipyramidal with (1123) faces
____________________
* Corresponding author: e-mail: [email protected]
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Takaomi Suzuki et al.: Measurement of specific surface free energy
and small (0001) end faces. Some crystals have (1013) faces. The sizes of the crystals were <3 mm in length
and < 2 mm in width. We chose tow well formed ruby crystals and named ruby(A) and ruby(B). A single
crystal of synthetic quartz was grown at NDK Co. Ltd. by the industrial hydrothermal method using a NaOH
solution [13-15]. The schematic picture of the synthesized quartz crystal and its faces are shown in figure 1.
The sample crystal was cut in the plane of perpendicular to Y ax of the crystal.
Fig. 1 Synthesized quartz crystal and its faces.
Fig. 2 A vertical sectional view taken on X-Z plane.
The droplet of water or formamide was dropped on the (0001), (1123) and (1013) faces of ruby crystals, and on
S, Z, +X, and -X faces of the quartz crystal using a micropipet. The droplets sized ~ 0.1 mm3 were observed
using a digital camera (Nikon COOLPIX 910) with a magnifying lens (Nikon 8×20D). The details of the
measurement are described elsewhere [6,8-10]. We took more than 200 photographs for each crystal and used
the photographs in which the boundary between the liquid and solid was clearly recognized. The contact angles
of the droplets were measured manually using a graduator and printed photographs.
Table 1 Contact angles of water and formamide on the faces of two ruby crystals.
Contact angles
Water
Formamide
θ (1123)
52 ± 6°
41 ± 5°
Ruby (A)
θ (1013)
53 ± 4°
39 ± 4°
θ (0001)
28 ± 6°
12 ± 5°
Ruby (B)
θ (1123)
θ (1013)
44 ± 6°
50 ± 6°
32 ± 5°
37 ± 5°
Table 2 Contact angles of water and formamide droplets on S, Z, +X, -X faces of quartz crystal.
Liquid
Water
Formamide
3
Contact angles of liquids on each face
S
Z
+X
41.0 ± 4.0°
44.2 ± 3.6°
46.7 ± 4.0°
27.0 ± 3.1°
27.4 ± 2.7°
30.4 ± 4.7°
-X
53.8 ± 3.4°
33.8 ± 3.8°
Results
The obtained contact angles of water and formamide on ruby(A) and ruby(B) are summarized in table 1, and
those on S, Z, +X, and -X faces of the quartz crystal are summarized in table 2. The contact angle for the water
and formamide droplets had a wide distribution on each face of the crystal. The difference between the
maximum and minimum angles for each individual angle was almost 20°. The estimated average values and
standard deviations are shown in tables 1 and 2. A schematic picture of the cross section of the quartz crystal is
shown in figure 2. The cross section of the crystal was well polished, and the boundary between the seed
crystal and grown crystal became visible. The lengths of perpendicular lines from seed surface to each grown
crystal face were measured. The growth term of this crystal was 50 days, and the growth rate was calculated as
0.09, 0.23, 0.28, and 0.33 mm/day for -X, +X, Z, and S faces, respectively. The surfaces of these faces are not
flat, the details are described in other literature [13-15]. Especially the edge part of Z face, which is bounded to
-X face has swelling as shown in figure 2.
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Cryst. Res. Technol. 42, No. 12 (2007)
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Discussion
4.1 Calculation of SSFE
The values of SSFE, γ S , can be obtained from the contact angle of the liquid, θ , using the Fowkes
approximation [16] and Wu’s harmonic and geometric mean equations [17] as
⎛ γd γd
γp γp ⎞
γ LV (1 + cos θ ) = 4 ⎜ d LV S d + p LV S p ⎟
⎝ γ LV + γ S γ LV + γ S ⎠
(
d
γ LV (1 + cos θ ) = 2 ( γ LV
γ Sd )
1/ 2
p
+ ( γ LV
γ Sp )
1/ 2
(2)
)
(3)
where γ LV is the surface tension of the liquid, γ Sd and γ Sp are the dispersion and polar components of the SSFE
d
p
of the solid, respectively, and γ LV
and γ LV
are those of the surface tension of the liquid, respectively. These
equations are widely accepted for use in studies on the surface free energy of polymer surfaces [18,19] or
inorganic systems such as mica [20,21] and silica [22]. We have also used the equations for calculating the
specific surface free energies of chlorapatite crystals [6,8-10] and discussed the relationship between the SSFE
and the morphology of the crystals. We used Eq. (2) in this study, because this equation gave more
reproducible and reasonable surface free energy values than Eq. (3) in our previous studies [6,8-10]. The polar
and dispersed components of the liquid were taken from reported data [18].
Fig. 3 Relationship between the specific
surface free energy and the length of the
normal line to each face of ruby crystal (A)
and (B).
4.2 SSFE of ruby crystals and interpretation of Wulff’s relationship
The length from the center of the crystal to the ith face, hi , was geometrically calculated from the photo of the
crystals. The relationship between γ i and hi are shown in figure 3. Because γ i is proportional to hi , Wulff’s
relationship is satisfied for each crystal. If the relationship between the crystal surface and the environmental is
ideally equilibrium condition, the shape of crystal should be a unique Wulff shape. The constant in Eq. (1) is
known to be ∆µ / 2vS , where ∆µ is chemical potential difference between liquid and solid and vS is an atomic
volume of the crystal. This ∆µ is determined at the crystal growth condition: chemical potential difference
between flux liquid and the growing crystal at high temperature. On the other hand, we observed contact angles
of water and formamide in atmosphere at room temperature. The SSFE we obtained by this experiment should
be distinguished from γ i in Eq. (1), and we propose an experimental SSFE, γ iex , instead of γ i . Because Eq.
(1) is effective only for equilibrium shape of crystal, the ratio of γ iex to hi is not ∆µ / 2vS . However, the ratio
of γ iex to hi is constant for each crystal in growth shape. In order to interpret the physical meaning of the ratio
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Takaomi Suzuki et al.: Measurement of specific surface free energy
of γ iex to hi , we propose the concept of steric energy density, 3 γ iex / hi , as following. The definition of SSFE is
the surface free energy of the face per unit area as
γi =
Gi
Si
(4)
where Gi and Si are the surface free energy and surface area of the ith face, respectively. On the other hand,
the pyramid with the base of the ith face and height of hi has a volume as
Vi =
Si hi
3
(5)
Therefore, when γ iex is substituted in Eq. (1) and the ratio of SSFE and hi was multiplied three, it gives the
steric energy density of the crystal as
3γ iex Gi G
=
=
h
Vi V
(6)
where G and V are the total surface free energy and volume of the crystal. It is possible to mention that the
steric energy density of each pyramid which radially grown from the center of the crystal is equal for the single
crystal. The constant values, 3 γ iex / hi , are 150 Jm-3 and 262 Jm-3 for ruby crystals (A) and (B), respectively.
The difference of this constant might originate from the density of dislocations in each crystal. Such
dislocations cause the steps on the crystal surfaces and the density of steps influence observed SSFE, because
the observed SSFE contains the step free energy [7-10]. Also the fact that the value 3 γ iex / hi of crystal (B) is
larger than crystal (A), indicates that the lager energy by dislocation is involved in the crystal (B). This value
of steric energy density should restrict the morphology of the crystal, even including the size of the crystal.
Fig. 4 Specific surface free energy of S, Z, +X, and -X face
of quartz crystal as a function of growth rate.
Fig. 5 Schematic picture of crystal growth of small
or large crystal.
4.3 SSFE of quartz crystal
Calculated SSFE of each face of quartz crystal are shown in figure 4 as a function of growth rate. The
relationship between SSFE and growth rate is almost linear, and the surface with larger SSFE has larger growth
rate. That is, the surface with smaller SSFE has smaller growth rate, but such surface with smaller growth rate
has larger ability to extend its surface area. During the crystal growth process, the total surface free energy is
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Cryst. Res. Technol. 42, No. 12 (2007)
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considered to become minimum. Therefore, the surface area of the face with larger SSFE should become
smaller, on the other hand, the surface area of the face with smaller SSFE extends. An energetic advantage
should be reflected on the morphology of small crystals as chlorapatite or ruby, which we studied previously.
figure 5 shows a schematic picture of crystal growth, when the SSFE of face Z, γ Z , is larger than that of face X, γ − X : γ Z > γ − X . The quartz crystal we used in this study is too large to cause the morphology change, but
local deformation of the crystal face is produced. For example, the swelling of Z face at the bounded edge to
+X face can be explained as following. The SSFE of Z and -X faces is estimated to be 57.4 and 51.9 mN/m,
respectively. When the crystal was growing, the total surface free energy could be kept smaller by extending X face as shown in figure 5, because the crystal is too large to change the whole shape of the crystal. The
growth rate of crystal faces in growth form is considered to proportional to the driving force on each growing
crystal face, which is also regarded as difference of chemical potential, ∆µ , between the solution and the
crystal. Our experimental SSFE may involve in the value of ∆µ for the crystal grow process.
5
Conclusions
The experimentally obtained SSFE, γ iex , was proportional to the normal length from center to each face of the
ruby crystals, and it looks as if Wulff’s equation is satisfied even for growth shape crystals. Therefore, we
proposed the value of steric energy density, 3 γ iex / hi , which represents the character of real crystal in growth
shape. The experimentally obtained SSFE of quartz crystal is a linear function of the growth rate, which
suggests that the SSFE reflects the driving force of crystal growth.
Acknowledgments This research was supported by the CLUSTER of the Ministry of Education, Culture, Sports, Science
and Technology and the Grant-in-Aid for Scientific Research (C) from the Japanese Government (Project No. 19560674).
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