Microscopic mechanism of sandstone hydration in Yungang Grottoes, China

2.2.1 Establishment and optimization of models

The physical and mechanical properties of minerals in the sandstone in the macroscopic scale are mainly determined by those in the microscopic scale. To investigate the microscopic physical and mechanical properties of minerals, the unit cell and super cell models should be established according to the Crystallography Open Database [13]. The size of a super cell may affect the simulation result [14]. To reduce errors, for each mineral, the super cell is constructed by 64 unit cells in a style of 4A × 4B × 4C.

In nature, molecules usually exist with low energy in a stable state. Therefore, the unit cells and super cells need to take geometric optimization so as to minimize the energy. To minimize the energy, it is necessary to compute the atomic interactions between the unit cells and super cells. The interaction forces exerted by other atoms are associated with the positions. The universal force field (UFF) [15] was used in the study to describe the potential energy, demonstrating in the function of atom coordinates. The potential energy of UFF is given by equation (1):

where E is the potential energy, E_b is the bond’s stretching energy, E_a is the bending potential energy of bond angles, E_t is the distortion potential energy of normal dihedral angle, E_i is the distortion potential energy of abnormal dihedral angle, E_e is the potential energy of Coulomb action and E_v is the potential energy of van der Waals action. Bonding interaction may be described by the first four terms, while non-bonding interaction may be described by the last two terms.

The Coulomb energy between the two point charges may be obtained by equation (2):

where k_e is the Coulomb constant, q₁ and q₂ are the volumes of two point charges and R is the distance between two point charges.

The potential energy of the van der Waals may be obtained by the Lennard–Jones potential (12-6) [16] (see equation (3)):

where r is the distance between atoms, ε is the minimum value of potential function and σ is the value of r when the potential function equals to 0.

To optimize unit cell, the MM simulation configurations were set up as follows in the software Material Studio. The smart method was adopted to improve the precision. Electrostatic forces between the point charges were computed by Ewald Sum method [17], with the precision of 10⁻⁵ kcal/mol. van der Waals energy was obtained based on the atom method. To speedup the computation of van der Waals energy, it is necessary to set the cutoff distance. The reason is that both the attractive forces and the repulsive forces weaken rapidly with an increase in distance r (see equation (3)), although an atom interacts with all atoms in the molecule. Then the forces of the two molecules are computed only within the cutoff distance. According to the nearest image convention, the cutoff distance should be less than the half of the shortest cell length. Therefore, the unit cell cutoff distances of quartz, K-feldspar, Na-feldspar and kaolinite were, respectively, 2.45, 3.55, 3.55 and 2.55 Å. Super cell cutoff distances of those minerals were all 12.5 Å, with spline width = 1 Å and buffer width = 0.5 Å. The external pressure was set at 0 GPa. The number of iteration steps was set at 300. After optimization, the stable conformations of unit cells and super cells were then obtained for the later computations.

2.2.2 Computations of physical properties of minerals

Microscopic physical properties of minerals were represented by volumes and densities in the study. Known the cell lengths, the angles between edges and masses (M) of the optimized unit cells and super cells, the volume (V) and the densities (ρ) are obtained, respectively, by equations (4) and (5):

where a, b and c are the axes lengths of cells, respectively; α, β and γ are the included angles of the axes, respectively.

2.2.3 Computations of mechanical properties of minerals

Microscopic mechanical properties of minerals were represented by Young’s modulus and Poisson’s ratio in the study. Computations were based on Theodorou and Suter’s method [18], which assumes the stress–strain relation satisfies the generalized Hooke’s law. Set the time of strain at nine and the maximum strain amplitude at 0.4%. The elastic properties related to stress and strain may be obtained by equations (6) and (7):

where T = initial temperature, V₀ = the volume of initial mineral cell, A = Helmholtz free energy, defined as the deformation potential energy, σlm and σnk (l, m, n, k = 1, 2, 3, 4, 5, 6) = the stresses applied on cells and εlm and εlm = the corresponding strains, −1 = the inverse matrix.

Thus, the stiffness matrix C_lmnk and the flexibility matrix S_lmnk were obtained. The minerals are usually polycrystalline, the Voigt–Reuss–Hill (VRH) [19,20] approximation method was then used to compute the elastic constants, namely, the bulk moduli K and the shear modui G. The VRH approximation method may be described as equations (8) and (9):

where V and R are the moduli of cells in Voigt and Reuss obtained by equations (10)–(13). The second-order tensors C_ij and S_ij in equations (10)–(13) are, respectively, reduced from the fourth-order tensors C_lmnk and S_lmnk in equations (6) and (7), considering the symmetry of the stress and strains.

The Young’s modulus E and Poisson’s ratio υ were obtained by equations (14) and (15) [21]:

2.2.4 Water and temperature influences

Considering the interlayer water and adsorbed water may influence the mechanical properties of minerals, the sorption module in the Material Studio was then used for simulations. The water molecules were geometrically optimized before simulations. Because the water and mineral molecules were small, the Metropolis method was adopted. The interlayer water molecules were inserted into the minerals. For the adsorbed water, considering that the hydration radius is 3.0 to 4.5 Å, new unit cells were established with b_w = b + 4.5 Å, which is the length of unit cell with adsorbed water molecules. Other original unit cell parameters remain unchanged. In the simulations, the mineral unit cells may adsorb up to five water molecules. Unit cells of minerals with 1, 3 and 5 adsorbed water molecules were thereafter established.

In the study, the MD simulation was used to investigate the influence of temperature on the microscopic physical and mechanical properties of minerals, the NPT (number of particles N, pressure P and temperature T) ensemble was used to simulate the microscopic physical and mechanical properties of minerals, and the Nose–Hoover and Berendsen methods [22,23] were adopted to control the temperature and pressure, respectively. The time step and total time were set at 1.0 fs and 200 ps, respectively. The initial velocity of system was then obtained by the Maxwell–Boltzmann distribution randomly. Because the weathering occurs on the surface layer of Yungang Grottoes sandstone as the temperature varies from −30°C to 30°C, the simulations were carried out at 243.15, 273.15 and 303.15 K under one atmospheric pressure.

3.2.1 Microscopic physical properties of minerals

The microscopic physical properties of the unit and super cells of minerals are listed in Table 2. From Table 2 it can be seen that the volumes of quartz, K-feldspar, Na-feldspar and kaolinite unit-cell are 110.53, 683.21, 672.98 and 299.41 ų, respectively; the volumes of quartz, K-feldspar, Na-feldspar and kaolinite super cell are 7072.00, 43024.36, 43694.59 and 19146.75 ų, respectively; the densities of unit cells and super cells of quartz, K-feldspar, Na-feldspar and kaolinite are 2.71, 2.71, 2.59 and 2.86 g cm⁻³, respectively.

3.2.2 Verification of microscopic physical properties of minerals

The microscopic densities of quartz, K-feldspar, Na-feldspar and kaolinite cells and super cells obtained by MM simulation were compared with those from the published data set in the macroscopic and microscopic scales (see Table 3).

From Table 3 it can be seen that the densities of quartz, K-feldspar, Na-feldspar and kaolinite in MM simulation are 2.71, 2.71, 2.59 and 2.86 g cm⁻³, respectively. According to the published data set, the macroscopic density of those are, respectively, 2.65, 2.57, 2.61 and 2.68 g cm⁻³ or lower 2.26%, lower 5.45%, higher 0.77% and lower 6.71% than those, respectively, from the MM simulations; the densities of quartz, Na-feldspar and kaolinite are, respectively, 2.70 g cm⁻³ [24], 2.62 g cm⁻³ [25] and 2.62 g cm⁻³ [26], or lower 0.37%, greater 1.15% and lower 9.16% than those, respectively, from the MM simulation.

In summary, the microscopic densities of unit and super cells of quartz, K-feldspar, Na-feldspar and kaolinite obtained by the MM simulation agree well with the published data set. The differences between the simulated results and the published data set may be caused by the following factors. (1) In a natural state, it is difficult to obtain the density of the pure minerals, and the densities of other minerals may affect the result. (2) The parameters (such as cell lengths and angles) used in the microscopic models of the minerals in the simulation may be different from those in the published data set.

3.2.3 Microscopic mechanical properties of minerals

The microscopic mechanical properties of the unit cells and super cells of minerals were obtained by MM simulation and are shown in Table 4. From Table 4 it can be seen that the Young’s moduli of the unit cells of quartz, K-feldspar, Na-feldspar and kaolinite are 215.6, 126.54, 102.56 and 162.12 GPa, respectively; the Young’s moduli of the super cells of quartz, K-feldspar, Na-feldspar and kaolinite are 214.16, 126.48, 102.87 and 162.88 GPa, respectively; the Poisson’s ratio of unit cells and super cells of quartz, K-feldspar, Na-feldspar and kaolinite are 0.17, 0.23, 0.21 and 0.26, respectively.

3.2.4 Anisotropy

The anisotropy are important mechanical properties of K-feldspar, Na-feldspar and kaolinite because the unit cells of these minerals are, respectively, monoclinic, triclinic and triclinic systems which are highly asymmetric. To investigate anisotropy, the Young’s moduli and Poisson’s ratios of mineral unit cells were obtained by simulations and are shown in Table 5.

From Table 5 it can be seen that the Young’s moduli of K-feldspar in the Y and Z directions are 3.161 and 5.455 times, respectively, greater than those in the X direction; the Young’s moduli of the Na-feldspar in the Y and Z directions are 5.566 and 8.832 times, respectively, greater than those in the X direction; the Young’s moduli of kaolinite in the X and Y directions are 1.972 and 6.442 times, respectively, greater than those in the Z direction. Minerals have positive Poisson’s ratios in all directions, indicating that their volumes do not expand under external load.

The Poisson’s ratios of K-feldspar and Na-feldspar in the yx direction are 0.90 and 1.28, respectively, which are obviously greater than the macroscopic Poisson’s ratio with the maximum of 0.5. That is partly because the optimized unit cell models used in the calculation have structural defects in some particular directions.

3.2.5 Water and temperature influences

Kaolinite is an octahedron 1:1 layered silicate with a unit structural layer. Halloysite structure is a layer of water molecules sandwiched by two layers of kaolinite. The chemical formula of halloysite is Al₄[Si₄O₁₀](OH)₈·4H₂O, which has a unit structural layer height of 10.1 Å and a water molecular layer thickness of 2.9 Å. The lattice parameters of halloysite unit cell are α = 90°, β = 100°, γ = 90°, a = 5.20 Å, b = 8.92 Å and c = 10.25 Å. Considering the influence of the interlayer water on the mechanical properties of kaolinite, a unit cell of halloysite was established and optimized.

The mechanical properties of K-feldspar, Na-feldspar, kaolinite and halloysite’s unit cells with various absorbed water molecules were obtained (see Table 6). From Table 6 it can be seen that, as adsorbed water molecules increases from 0 to 5, the Young’s moduli of K-feldspar, Na-feldspar, kaolinite and halloysite decrease by 73.77%, 72.80%, 42.60% and 47.46%, respectively; the Poisson’s ratios of K-feldspar and halloysite are almost unchanged; the Poisson’s ratio of Na-feldspar increase by 14.25% and that of kaolinite decrease by 11.54%.

Microscopic physical and mechanical properties of the unit cells of minerals at various temperatures were also obtained (see Table 7). From Table 7 it can be seen that as temperature rises from 243.15 to 303.15 K, the volume of the unit cell of quartz, K-feldspar, Na-feldspar and kaolinite increases by 0.25%, 0.23%, 0.17% and 0.31%, respectively; the density of these minerals decreases by 0.26%, 0.22%, 0.15% and 0.28%, respectively; the Young’s moduli of these minerals decrease by 0.12%, unchange, increase by 0.17% and decrease by 7.53%, respectively; the Poisson’s ratio of these minerals decreases by 1.18%, unchanges, decreases by 0.48% and increases by 3.85%, respectively.

3.4.1 Microscopic mechanism of macroscopic physical and mechanical properties of minerals

The microscopic physical and mechanical properties of minerals have great implications for the minerals in the macroscopic scale. In this scale, the Young’s modulus and Poisson’s ratio of the quartz–feldspar sandstone in the horizontal direction is significantly greater than those in the vertical direction [27]. From Table 5 it can be seen that the Young’s moduli and Poisson’s ratios of K-feldspar and Na-feldspar in the horizontal direction were significantly greater than those of feldspars in the vertical direction. This is because the K-feldspar and Na-feldspar have the strongest bonds, or the Al–O bond and the Si–O bond in their silicate structures increase the intrinsic lattice resistance of dislocation. Therefore, feldspar–quartz sandstone composed of K-feldspar and Na-feldspar has an initial anisotropy in the microscopic scale.

From Table 5 it can be also seen that the mechanical anisotropies of K-feldspar and Na-feldspar are similar, agreeing with the isomorphism in the macroscopic scale. From Figure 1(b and c) it can be seen that both K-feldspar and Na-feldspar have tetrahedrons of [Al–O₄] and [Si–O₄] connected with each other to generate a chain connected with four tetrahedrons. The structural differences between K-feldspar and Na-feldspar are mainly due to the radii of ions. The greater radius of potassium ion results in more expansion of the tetrahedron frames of K-feldspar. On the contrary, the tetrahedron frames of Na-feldspar may collapse in this condition.

From Table 5 it can be seen that kaolinite has the lowest strength in the Z direction due to the existence of OH⋯O hydrogen bond that weakens the interlayer force in that direction. Therefore, kaolinite easily fails in the Z direction. When existing in the interlayer of kaolinite, the water molecules may furthermore weaken the interlayer hydrogen bond. Thus, from Table 6 it can be seen that the Young’s modulus of halloysite decreases by 75.75%, compared with that of kaolinite. In the macroscopic scale, the crystal layer of halloysite will be displayed as a tubular or curled scale shape [28].

3.4.2 Microscopic mechanism of sandstone hydration

Now we provide an explanation of the sandstone hydration. As the water molecules permeate into the crystal of minerals and bring out the cations to generate new minerals, changes in the sandstone structure may occur both in the macroscopic scale and in the microscopic scale. The average humidity in the Yungang Grottoes are usually 40%–50% and may rise to 60%–80% after rain [1]. Moreover, the Yungang Grottoes has been polluted by the coal industry [29], which will enhance the precipitation acidity. In the acidic environment, the feldspar of the Yungang Grottoes sandstone may be hydrated as follows [30]:

In this chemical formula, the feldspar converts into kaolinite by the actions water molecules and hydrogen ions. Therefore, the replaced metal cations will be taken away by the acid fluid.

In hydration, the mineral strengths in the sandstone of Yungang Grottoes become weak. Four main reasons account for that in a microscope scale:

1. From Table 6 it can be seen that as the adsorbed water molecules increase from 0 to 1, huge decreases are observed in the Young’s moduli of K-feldspar and Na-feldspar. The reason is that the adsorbed polarized water molecules may disturb the cations on the crystal surface and lead to structural damage when K-feldspar and Na-feldspar start to adsorb these water molecules.

2. From Table 6 it can be seen that as the numbers of adsorbed water molecules increase, the Young’s moduli of minerals greatly decrease to 73.77%, 72.80%, 42.60% and 47.46%. Tan [31] suggested that the osmotic pressure between the inside and outside crystals caused by the concentration gradient of ions in solution transfers the water molecules from low concentration zone to high concentration zone, which may apply the repulsive forces inside the crystal and then the crystal expands. These changes may damage the original structure of the mineral crystal and lead to further declination in the mechanical properties of the sandstone.

3. From Table 4, it can be seen that the Young’s modulus of kaolinite is 28.18% and 58.07%, respectively, greater than that of K-feldspar and Na-feldspar. Nevertheless, the kaolin formed in hydration [32] is subjected to combine with water molecules and therefore has a good plasticity.

4. The varying temperature will also cause the deterioration of the mechanical properties. From Table 7 it can be understood that as the temperature rises, the volumes of minerals increase. In the macroscopic scale, this increase may lead to the expansion of the mineral particles and to furthermore generate or expand the fractures in the sandstone.

The weakening in the strength by hydration has a significant impact on the weathering of the Yungang Grottoes sandstone. In Table 8 it can be observed that during the moderately weathering stage which is less affected less hydration, the main minerals of the sandstone are quartz and feldspar and a few calcium minerals such as calcite and ankerite. As shown in Figure 4(a), at Grotto 14, discoloration was observed on the surface of sandstone with a few cracks and light loss of material and the shapes of the sandstone carving may be clearly identified. Nevertheless, as presented in Table 8, the highly weathering stage was heavily affected by hydration, wherein the feldspar has been mostly hydrated into kaolinite; the main minerals of the sandstone are quartz and kaolinite; and calcite and ankerite decomposed. As shown in Figure 4(b) at Grotto 29 where the sandstone was restored, the surface layer has been highly weathered, leading to the strip near the surface layer; the sandstone carvings were blurred and may hardly be recognized.