3.4 Anisotropic elastic properties
The elastic stiffness tensor elements Cij of the
calcium carbonate hydrates are calculated via first-principles
calculations with the stress-strain method based on the general Hooke’s
law.[31] These
elastic constants and modulus results are listed in Table 3.Cij is the vital mechanical property, which is
very beneficial to understanding the mechanical properties of calcium
carbonate hydrates. In this work, we also calculate all elastic
constants of the calcium carbonate hydrates which are mechanically
stable on the basis of the Born-Huang’s criterions[32,
33]and the specific criteria can be
found in reference [26].
CaCO3·1/2H2O with monoclinic structure
is not easy to be compressed under the external uniaxial stress along
the [001] and [010] directions because it has the largestC 11 and C 22 with 109.28
GPa and129.65 GPa, respectively. While
CaCO3·6H2O is easy to be compressed
along a-axis and b-axis compared with other calcium carbonate hydrates.
The deformation resistance of these calcium carbonate hydrates is
determined by many factors such different crystal structures and
orientation of bonds. C 12 characterizes resist
shear deformation at (100) crystal plane along the direction. In Table
3, the C 12 value of
CaCO3·1/2H2O is 36.33 GPa which is
larger than other calcium carbonate hydrates, indicating that it is hard
to be shear deformation along direction.
When elastic constants (Cij ) if calcium carbonate
hydrates are obtained, the polycrystalline modulus including bulk and
shear modulus iscalculated according to the elastic constants
matrix[33], which
can be see the attachment for details. In order to further apprehend the
mechanical anisotropy of
CaCO3·x H2O (x= 1/2, 1 and
6), the three-dimensional (3D) surface of Young’s modulus for them are
plotted in Fig. 6. The 3D representation of Young’s modulus is given by
the following equations:
For the hexagonal structure (CaCO3·1H2O)[34]
(11)
For the monoclinic structure
(CaCO3·1/2H2Oand
CaCO3·6H2O)
(12)
Where represents the elastic
compliance constants,, and are the directional cosines in spherical
coordinates with respect to θ and φ (). As shown in Fig.
6, CaCO3·6H2O displays strong anisotropy
because the shapes are deviated from the perfect sphere.
CaCO3·1/2H2O also represents strong
anisotropy, while CaCO3·H2O shows weak
anisotropy because the graph is close to a sphere. These results are in
agreement with the universal anisotropic index
(A U) and the specific formula
forA U and percent anisotropic
index(AB and AG ) as
following [35].
(13)
(14)
(15)
where BV, BR, GVand GR represent the bulk modulus and share
modulus estimation within Voigt and Reuss approximations, respectively.
The anisotropy index of
CaCO3·x H2O (x= 1/2, 1 and
6) can be determined by the value zero, if the value is very close to
zero, it indicates the less anisotropy, vice versa. The results are
listed in Table 3. CaCO3·6H2O has the
highest values of the AU , indicating that the
elastic properties of CaCO3·6H2O havethe
strongest anisotropy. Similarly, the value of AG ,AB also confirmed this results. From Table 3, the
universal anisotropic index of
CaCO3·x H2O (x= 1/2, 1 and
6) formedthe following sequence:
CaCO3·6H2O
(0.742) > CaCO3·1/2H2O
(0.454) > CaCO3·H2O
(0.155). We can clearly see that the Young’s modulus of these three
calcium carbonate hydrates hasdifferent surface constructions due to the
different crystal structures. Furthermore, planar projections of the
Young’s modulus of the calcium carbonate hydrates on the (001) and (110)
crystallographic planes are shown in Fig. 6. The anisotropy of Young’s
modulus for all the calcium carbonate hydrates on the (110) plane is
stronger than (001) plane. What’s more,
CaCO3·6H2O has the strongest anisotropy
of Young’s modulus among them due to the most remarkable anisotropic
geometry of the surface contour.
Intrinsic
hardness (HV ) is also an important index for the
calcium carbonate hydrates because their application is not indirectly
related to the hardness. In our work, we choose Chen’s
model[36] and
Tian’s model[37] to
calculate the hardness of the calcium carbonate hydrates, which can be
expressed as follows:
(16)
(17)
wherek is the Pugh ratio as B/G. It is clearly seen that the
values of hardness are excellent close through Chen’s model and Tian’s
model. CaCO3·6H2O has the largest
hardness with 6.386 GPa, while CaCO3·H2O
has the smallest with 4.274 GPa due to different atomic constitute and
disparate hydrates. In Tian’s model, the hardness of calcium carbonate
hydrates comply with following sequence:
CaCO3·1/2H2O >
CaCO3·6H2O >
CaCO3·H2O.