3.1. Effect on Crystal Structure of Kaolinite
The microscopic mechanisms of the Mg(II) and Na(I) doping in kaolinite were studied first before considering the influence of doping on the elastic properties. The atomic structure for the relaxed kaolinite was calculated and numbered (Figure 1
), during which the high-symmetry doping site for Mg(II) and Na(I) to substitute for the Al2
atom in kaolinite was considered.
According to the previous experimental data [25
] and calculated results [28
], the kaolinite consisted of dioctahedral 1:1 layer structures with an ideal composition of Al4
]. The unit crystal layer was composed of a silicon oxygen tetrahedron (SiO4
) and an aluminum oxygen octahedron (AlO4
]. All the top oxygen atoms of the silicon oxygen tetrahedron layer pointed in the same direction and were connected with the Al in the aluminum oxygen octahedron. Only two-thirds of the center positions of the aluminum oxygen octahedron were occupied by the Al, and the remaining one-third formed the vacancy; the occupied octahedron and the vacancy were regularly intersected. This could be arranged for other instances. Four of the six oxygen atoms linked to Al were hydroxyl oxygen (OH
), and the remaining two were top oxygen (Oa
). According to the position of the hydroxyl, it could be divided into interlayer hydroxyl and internal hydroxyl. The three oxygen atoms of the silicon ring system were called bottom oxygen (Ob
). The crystal cell of kaolinite used to calculate the electronic structure included 34 atoms, namely four Al atoms, four Si atoms, eight H atoms, and 18 O atoms. Table 1
lists the optimal structural parameters of the kaolinite model. The calculated structural parameters were a
= 5.155 Å, b
= 5.155 Å, c
= 7.405 Å, α
= 75.14°, β
= 84.12°, and γ
= 60.18°, as shown in Table 1
. The DFT calculation results of kaolinite crystal were in good agreement with the experimental data [19
], indicating that the DFT method adopted in this paper was reliable, laying a foundation for the subsequent reliable doping results.
The average calculated bond length of the kaolinite model is listed in Table 2
, and the calculated value of DTF matched well with the experimental value [34
]. The Al–O bond length in kaolinite was longer than the Si–O bond length, which means that the strength of the Al–O bond was weaker than the Si–O bond, which can partly explain why the Al ion in an octahedral position in kaolinite was more easily replaced by some cations than in a tetrahedral position. The Si–Oa
connected to the alumina octahedron was shorter than the Si–Ob
bond not connected to the alumina octahedron; similarly, the Al–Oa
connected to the silica tetrahedron was shorter than the Al–Ob
not connected to the silica tetrahedron. The calculation results exhibited that the ionic bond of kaolinite along the c
axis was weaker than that along the a
plane. The results can partly explain why the large deformation of lattice in kaolinite is always observed along the c
also shows the average calculated bond length of the Mg-doped and Na-doped kaolinite crystal. The doping cations had the greatest effect on the Al2
–O bond length, but had little effect on the Si–O and O–H bond. Further, we compared the influence of Mg(II) and Na(I) on the average bond length, which was in the following order: Na–OH
and the change trend of the Si–O bond was opposite. Because Na had the largest ion radius, the bond length of the Na–O bond was also the largest. At the same time, the larger ion radius of Na “pushed” the top oxygen (Oa
) outward, making the top oxygen (Oa
) closer to the tetrahedral Si, so the Si–O bond had a small degree. From this, we could see that the average bond length of the doped atom and the O atom was positively related to the atomic radius of the doped atom.
From the change of average bond length, it could be seen that the type of replacement cation was an important factor affecting bond length. Therefore, it was necessary to study the change rate of the atomic layer spacing of the Mg-doped and Na-doped kaolinite crystal relative to that of the undoped kaolinite crystal. This is listed in Table 3
, where Δdij
represents the atomic layer spacing of layer i
and layer j
. From the table, we can see that the doping cations had a great influence on the related Δd23
, but had a little influence on the distant Δd12
, and Δd56
. The influence of Na(I) on the atomic layer spacing was greater than that of Mg(II); that is, the change of the atomic layer spacing was also related to the atomic radius of the doped atom.
3.2. Effect on the Electronic Structure of Kaolinite
To better understand the influence of substitutional impurities on the interaction of various elements in kaolinite crystals, the band structure, density of states, and charge density of different doped kaolinite crystals were analyzed.
The band theory of solids simplifies the motion of electrons in crystals into independent motion in an equivalent potential field, i.e., single electron approximation. The separate energy levels of each atom in the crystal became very dense and split into a series of quasi-continuous energy bands. Each band consisted of N atoms. Each energy level could be occupied with two electrons of opposite spin according to the Pauli exclusion principle, so each band could hold up to 2N electrons. At absolute zero, the electrons in an insulator crystal would fill the energy levels in turn from low to high. In the band structure, one band was completely filled with electrons, had the highest energy, and was termed the valence band, and the next higher energy band was termed the conduction band. The energy separation between the highest valence band state (valence band maximum) and the lowest conduction band state (conduction band minimum) was called the band gap, and their energy difference was termed the band gap width.
The band gap width determined the conductivity of the material. The band gap width of the metal conductor was very small or zero; however, the band gap width of the insulator was large (about 3–6 eV). The band gap width of the semiconductor was between the conductor and the insulator, at about 0.1–2 eV. The energy band structure diagrams of the undoped, Mg-doped, and Na-doped kaolinite crystal are shown in Figure 2
. In the energy band structure diagram, the transverse axis is the high symmetry point in the simple Brillouin region, which is Γ(0,0,0), B(0.5,0,0), F(0,0.5,0), and G(0,0,0.5), and the vertical axis is the energy. The above results exhibited that the energy band structure of the kaolinite was relatively flat and its Fermi energy level was located at the valence band maximum (VBM). The VBM values of the undoped, Mg-doped, and Na-doped kaolinite were at the G(0,0,0.5) in the Brillouin zone, and the conduction band minimum (CBM) values were at the Γ(0,0,0) in the Brillouin zone. The indirect band gap width of the undoped kaolinite was 4.99 eV, that of the Mg-doped kaolinite was 5.03 eV, and that of the Na-doped kaolinite was 5.12 eV. Therefore, kaolinite doped with different elements had typical insulator characteristics. The order of the band gap width of each crystal was Al4
. That is, the doping of Mg(II) and Na(I) made it more difficult for electrons to obtain enough energy to move from the valence to conduction band state.
In solid-state physics, the density of state, as a visualization result of band structure, can reflect the distribution of electrons in each orbit, as well as the interaction between atoms, and reveal the information of chemical bonds. In Figure 3
, we plot the total and partial density of states of the undoped, Mg-doped, and Na-doped kaolinite crystal to understand the effect of Mg(II) and Na(I) doping on the bonding and electronic structure of kaolinite. We set the Fermi energy EF
to 0 eV. Due to the different symmetry and positions, the partial density of state values of different types of oxygen atoms are plotted in Figure 3
. The partial density of state values of different oxygen atoms had similar change trends, which may be due to the high ionic degree of oxygen atoms. As shown in Figure 3
, the valence band of 10 eV < E
was mainly composed of the O 2p
state, which implied a large amount of charge transfer from Al 3p
, Si 3p
and H 1s
states to O 2p
states. The 2s
states of O bonded with Mg and O adjacent to Al had higher peaks, indicating that Mg is more metallic than Al and had more electron transfer to O. In the same way, the 2s
states of the Na-bound O and the O adjacent to Mg had higher peaks, indicating that Na was more metallic than Mg and had more electrons transferred to the O adjacent to it.
On the other hand, it can be seen from Figure 3
that some residual charges existed in the 3s
states of Al and Si atom of kaolinite crystal, which means that there were obvious covalent components in the chemical bonds of Al–O and Si–O in kaolinite. At the same time, under the action of a local electric field, the overlap degree of the 3p
orbit of Mg and 2p
orbit of adjacent O was lower than that of Al, which made the covalent bond composition of Mg–O lower than that of Al–O. There was almost no overlap between the 3p
orbit of Na and 2p
orbit of the adjacent O, and the covalent bond composition of Na–O was very small. In other words, the doping of Na(I) and Mg(II) reduced the covalent bond composition of the crystal.
To better understand the bonding characteristics between atoms in the kaolinite system, a charge density analysis of kaolinite crystal was carried out. Figure 4
a shows the charge density diagram of the plane of O2
in the undoped kaolinite crystal. Figure 4
b,c shows the charge density diagram of the plane of O2
in the Mg-doped kaolinite and that of O2
in the Na-doped kaolinite, to aid in understanding the influence of doping atoms on the bonding and electronic structure of kaolinite more intuitively and vividly. It can be seen from the Figure 4
that the charge density of Na is smaller than that of Mg and Al. The localization of the charge density of O is stronger, and the charge density around O is more and more disorientated towards the direction of the doped atom (Mg/Na). This shows that the covalent bond properties of the Mg–O bond and Na–O bond are weaker than that of the Al–O bond, and the ionic bond properties are stronger.
3.3. Effect on the Mechanical Properties of Kaolinite
The elastic constant matrix of the undoped kaolinite was obtained by density functional theory and the VASP software package. So as to evaluate the effectiveness and rationality of the calculation method and parameter setting, the elastic matrix constant of the undoped kaolinite was calculated first and compared with the existing experimental data [35
] (see Table 4
). The DFT calculation results of the elastic matrix constant of kaolinite crystal were in good agreement with the experimental data. According to the calculation results in Table 4
, the elastic constants of the undoped kaolinite crystals met the mechanical stability conditions. The C33
of the undoped kaolinite crystal was obviously smaller than C11
; that is, the elastic constant related to the elastic property in the direction of vertical crystal surface was obviously smaller than that in the direction of the parallel crystal surface, showing that the van der Waals force effect between crystal layers of the undoped kaolinite crystal was far smaller than the bond effect in the crystal layer.
The elastic constant matrix of the Mg-doped and Na-doped kaolinite is also listed in Table 4
. Comparing the elastic constants of different models, it was obvious that the change of C11
in the Mg-doped and Na-doped kaolinite was greater than that of C33
; that is, the influence of doping on the stiffness of parallel crystal plane was greater.
The elastic modulus reflected the ability of the material to resist deformation on the macro scale and reflected the bonding strength between atoms, ions, or molecules on the micro scale. The greater the modulus of elasticity, the greater was the stress required for the material to resist elastic deformation, that is, the greater the stiffness of the material. The bulk modulus (B
) is a relatively stable constant, indicating the ability of the material to resist to volume changes. The shear modulus (G
) indicates the ability of material to resist shear strain. The greater the shear modulus was, the stronger the material’s rigidity was. Young’s modulus (E
) is the elastic modulus along the longitudinal direction, which is a physical quantity describing the deformation resistance of solid materials. It can be seen from Table 5
that the elastic modulus of the undoped kaolinite crystal calculated by the VASP software package matched the experimental value [37
], which provided a guarantee for calculating the mechanical and electronic properties of the Mg-doped and Na-doped kaolinite. By calculation, the bulk modulus (B
) of kaolinite crystal was 44.001 GPa, and the bulk modulus (B
) of the Mg-doped and Na-doped kaolinite were smaller than that of the undoped kaolinite crystal, which proved that the incompressibility of the octahedral replacement model was weaker. The Young’s modulus (E
) of the undoped kaolinite crystal was 57.783 GPa, and the values of the Mg-doped and Na-doped kaolinite were smaller than that of the undoped kaolinite crystal, which also proved that the ability of octahedral displacement model materials to resist deformation was reduced. The shear modulus (G
) of the kaolinite crystal was 22.552 GPa. Similarly, the shear modulus (G
) of the two octahedral displacement models decreased, that is, the rigidity of the material decreased. All the above calculations showed that the doping of Mg(II) and Na(I) reduced the stress required for the material to resist further elastic deformation. The reason for this phenomenon is that the doping of Mg(II) and Na(I) increased the distance between atoms, thus reducing the van der Waals force between the layers and the bonding in the crystal.
The brittle toughness of the material was related to the Poisson’s ratio (μ), which represented the shear stability of the material. In general, a large Poisson’s ratio was more ductile than the resulting material. The calculated value of the Poisson’s ratio (μ) of the undoped kaolinite crystal was 0.281, and the Poisson’s ratio (μ) of the octahedral displacement models was larger than that of the kaolinite crystal, which proved that the ductility of the Mg-doped and Na-doped kaolinite increased, which means that the plasticity of the material was improved. We often use the Pugh’s modulus ratio G/B to predict the brittle ductile behavior of materials. When the ratio of G/B of the material is less than 0.57, it is ductile; otherwise, the material is brittle. The G/B ratio of the kaolinite crystal was 0.513, so the kaolinite material had ductility. The G/B ratio of the octahedral replacement model was significantly smaller than that of the kaolinite crystal, so the doping of Mg(II) and Na(I) made the material show better ductility; that is, the plasticity of the material was greatly improved. The Cauchy pressure Pc (C12–C44) can also reflect the brittle ductile behavior of the material. The calculated value of Pc of the kaolinite crystal was 42.805 GPa. The values of Pc of the octahedral displacement model were higher than those of the kaolinite crystal, the bonding direction was weaker, the mobility was lower, and the toughness of the material was improved. Finally, the Vickers hardness (Hv) could be used to express the hardness of materials. The calculation formula of Hv is Hv = 2(K2G)0.583 − 3, where K is the Pugh’s modulus ratio G/B. The calculated value of Hv of the kaolinite crystal was 2.643 GPa. The Vickers hardness of the octahedron replacement model decreased, implying that the hardening influence of the material was weakened. The doping of Mg(II) and Na(I) made the elastic wave velocity decrease.