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.