1. Introduction

Clay minerals are the most abundant materials in sedimentary basin. The most common like kaolinite is found in various mudstones[1].

Kaolin minerals are dioctahedral clays of 1:1 layer type with chemical composition Al₂Si₂O₅(OH)₄. Kaolinite, Dickite, and Nacrite are polytypes.

Nacrite is the rarest of the kaolin polymorph, it is characterized by Al₄Si₄O₁₈H₈ structural formula, monoclinic structure of space group of C_c , a= 8.906(2), b = 5.146(1), c = 15.664(3)Å,β= 113.58(3)° and V = 657.9(3) ų[2]. The basic structural element of kaolin minerals consists in a tetrahedral[SiO₄] and an octahedral[AlO₂(OH)₄] sheet linked together by a common plane of oxygens and hydroxyls (figure 1). Only 2/3 octahedral sites are occupied by Al³⁺, the remaining 1/3 being vacant[3]. Nacrite is structurally distinct from other members of kaolin group, being the most closely stacked in the c-axis direction.

However, elastic properties of clay minerals are almost unknown[4, 5] and there is few available experimental data. This is mainly due to the difficulty presented by their intrinsic properties. The small grain size makes it difficult to isolate an individual crystal of clay large enough to measure acoustic properties[6, 7].

Despite their importance, the single-crystal elastic moduliof clay minerals have not been measured experimentally, because of technical difficulties associated with the small grain size. There is also considerable uncertainty in atomic force microscopy[6].

Recently, several group attempts to predict, theoretically, elastic constant and mechanical properties of kaolin family; For kaolinite, values from first-principles calculations exist for the ideal structure[8, 9], or from molecular dynamics simulation[4, 10]. For Nacrite, only a first principle calculations exists[11] to predict elastic behaviour.

In this work, a prediction of elastic constant, shear modulus, bulk modulus, Young modulus along a, b and c directions, the S- and P-wave velocities as well as Poisson ratio were also evaluated using molecular dynamics simulation based on GULP program.

2. Materials and Methods

2.1. Simulation Principle

The structural and mechanical properties of Nacrite were simulated using GULP program[12].

The GULP program offers added advantage that it permits the inclusion of atom polarisability through the use of the core-shell model[13].

The potential model describing the effective forces acting between the atoms in the structure has the following components[14]:

*A two-body short-range term describes repulsions from electron cloud overlap and attractions due to dispersion and covalence. In this study we describe cation-O and the O-O interactions using a Buckingham function: , where the exponential term describes the repulsive energy and the r⁶ term the longer range attraction (table 1).

*A three-body short-range term describes angular dependent covalent forces. A simple approach is to include bond-bending terms about the tetrahedral cation of the type: U_thb = 1/2K_thb(θ − θ₀)². where K_thb is the harmonic three-body force constant, and θ and θ₀ are the observed and ideal tetrahedral O-T-O bond angles, respectively (table 1).

Table 1. Potential used in simulation Two body short range interactiona Atom 1 Atom 2 Potential A/eV ρ/Å C/eVÅ6 Al1 c O1 s Buckingham 1460 0.299 0.00 O2 c Al1 c Buckingham 1140 0.299 0.00 Si1 c O1 s Buckingham 1280 0.321 10.7 O2 c Si1 c Buckingham 984 0.321 10.7 O1 s O1 s Buckingham 22800 0.149 27.9 O2 c O1 s Buckingham 22800 0.149 27.9 O2 c O2 c Buckingham 22800 0.149 27.9 H1 c O1 s Buckingham 312 0.250 0.00 Morse potential Atom 1 Atom 2 Potential De/eV β/Å−1 re/Å H1 c O2 c Morse-Coulomb 7.05 2.20 0.949 Shell model interaction Atom 1 Atom 2 Potential Ks/eVÅ−2 O1 c O1 s Spring 74.90 Three body interaction Atom 1 Atom 2 Atom3 Kthb/eV rad−2 θ/° Si1 c O1 s O1 s 2.097 109.470 Al1 c O2 c O1 s 2.097 109.470

Figure 1. Projection onto (001) of basal oxygens in the tetrahedral sheet of the second layer onto the octahedral sheet of the first layer. Small solid circles are H+ positions found in this study. Small open circles are centers of the vacant octahedral[17]

* A term to describe electronic polarizability is required if dielectric and dynamic properties are to be modelled accurately. In this study the shell model was used, which provides a simple mechanical model of electronic polarizability. The core-shell self energy is given by Us = 1/2K_sr² where K_s is the harmonic spring constant and r is the core-shell separation (Table 1).

In this study, the libraries of potential used were the Catlow 1992[15]. The minimum energy configurations were derived by minimizing the potential energy as a function of the atomic coordinates and unit cell parameters.

3. Results and Discussion

3.1. Structural Simulation

The Nacrite structure used in the present work is initially based on crystallographic data[2] (figure 1), which gives, after optimization, a unit cell of C1 symmetry with lattice parameters a = 5.233 A˚ , b = 5.14 A˚ , c = 15.664 A˚ , and angles: α= 11.0933°, β= 111.0933°, γ = 59.2107° (table 2). The computed energy is equal to -848.784 eV.

Table 2. Bibliographic and simulated unit cell Nacrite CGA[11] Nacrite LDA[11] Nacrite[17] This Work ρ (g/cm3) 2.494 2.607 - 2.68 a (Å) 9.020 8.910 8.906 5.233 b (Å) 5.221 5.165 5.146 5.14 c (Å) 14.881 14.571 15.664 15.6640 α (°) 90 90 90 111.0933 β (°) 101.11 101.24 113.58 111.0933 γ (°) 90 90 90 59.2107

Table 3. Simulated atomic coordinates of Nacrite Atom x/a y/b z/c Al1 -0.164600 0.475000 0.217700 Al1 0.165600 0.814600 0.217300 Si1 -0.268800 0.671200 0.027700 Si1 -0.611600 1.354600 0.027000 O1 -0.508800 0.991000 -0.004100 O1 0.025300 0.519300 -0.018800 O1 -0.438800 0.444600 -0.019300 O1 -0.168900 0.708100 0.138500 O1 0.450800 0.472800 0.138500 O2 0.067700 0.091300 0.143200 O2 -0.076200 1.192200 0.283800 O2 -0.451600 0.807600 0.283800 O2 0.162700 0.575900 0.282600 OH1 -0.508800 0.991000 -0.004100 OH1 0.025300 0.519300 -0.018800 OH1 -0.438800 0.444600 -0.019300 OH1 -0.168900 0.708100 0.138500 OH1 0.450800 0.472800 0.138100 H1 0.183600 1.085600 0.127700 H1 0.080140 1.014860 0.327900 H1 -0.366400 0.852800 0.322800 H1 0.293100 0.570300 0.322800

Table 3 shows the structure, cell parameter, and unique atomic coordinates of the simulated structure determined using GULP program. In fact, in Nacrite, only one of the basal oxygens is located so it can pair up ideally with a directed hydrogen bond of this type. The other two basal oxygens (0₁ and 0₂) are located so that the actual bonds must be nearly at right angles to the ideal directed bonds, and this must affect the nacrite stability adversely[2].

3.2. Mechanical Properties

3.2.1. Elastic Constant

The elastic constant represent the second derivatives of the energy density with respect to strain:

(1)

where Cij is a component of the stiffness matrix, C, E is the energy expression, V is the volume of the unit cell, and εi and εj are strain components.

There by describing the mechanical hardness of the material with respect to deformation. Since there are 6 possible strains within the notation scheme employed here, the elastic constant tensor is a 6 x 6 symmetric matrix. The 21 potentially independent matrix elements are usually reduced considerably by symmetry.

In table 4, elastic constant tensor is presented as well as previous results. The obtained elastic constant shows that C₁₁≈C₂₂ and higher than C₃₃, indicating that Nacrite is more flexible in a and b direction than c one. These results present some different values from bibliographic data. Differences in elastic constants between various structures may, be partially due to difference in orientation of hydroxyl group[11]. Contrary to kaolinite which present more flexibility along a direction than b one[10].

Table 4. Elastic constant of Nacrite Nacrite CGA Nacrite LDA This work C11 147.6 131.8 215.24 C22 160.8 157.9 215.64 C33 64.8 75 67.92 C44 7.8 13.2 17.09 C55 8.8 17.7 10.92 C66 62.4 62.2 47.26 C12 45.4 41.5 117.14 C13 5.7 10.2 18.33 C14 0 0.6 0.42 C15 4.4 -16 1.54 C16 0 0.1 -1.7 C23 12 14 24.57 C24 0 0.4 1.93 C25 2.8 -3.8 2.53 C26 0 0.1 1.45 C34 0 0.5 -0.26 C35 0.2 9.3 -0.46 C36 0 0.1 -5.41 C45 0 -0.5 -5.34 C46 0.6 -9.4 -0.49 C56 0 -0.1 -1.32

3.2.2. Bulk and Shear Moduli

Like the elastic constant tensor, the bulk (K) and shear (G) moduli contain information regarding the hardness of a material with respect to various types of deformation. The bulk modulus is much more facile to determine experimentally than the elastic constant tensor. If the structure of a material is studied as a function of applied isotropic pressure, then a plot of pressure versus volume can be fitted to an equation of state where the bulk modulus is one of the curve parameters. Typically, a third- or fourth-order Birch-Murnaghan equation of state is utilized. Alternatively, the bulk and shear moduli are also clearly related to the elements of the elastic constant.

In this study, predicted bulk equal to 56.41 GPa. Experimentally, bulk modulus values for kaolinite have been found between 21 and 55 GPa[15]. The theoretical models based on measured quantities on other sheet silicates[6] for the bulk modulus give a value of 55.5 GPa.

Calculated shear modulus gives a 20.28 GPa value, it is comparable to the one reported in literature using molecular dynamics simulation[10].

3.2.3. Poisson’s Ratios

When materials are compressed along a particular axis they are most commonly observed to expand in directions orthogonal to the applied load. The property that characterizes this behavior is the Poisson’s ratio, which is defined as the ratio between the negative transverse and longitudinal strains. The majority of materials are characterized by a positive Poisson’s ratio (ν)[16].

The poison’s ratios ν_ij in the Ox_i-Ox_j planes (i,j=1,2,3) for loading in the Ox_i direction are then given by:

(2)

Table 5 shows the calculated Poisson’s ratios values of Nacrite. The Poisson’s ratios exhibit prominent anisotropy, with values ranging from 0.03 to 0.53.

Table 5. Young moduli and Poisson’s ratio of Nacrite Stress axis x y z Young moduli (GPa) 151.11037 147.53536 64.17514 Poissons Ratio (x) 0.52302 0.03067 Poissons Ratio (y) 0.53569 0.10011 Poissons Ratio (z) 0.07221 0.23015

3.2.4. Young’s and Shear Mouduli

The young’s moduli E_i for loading in the Ox_i (i=1,2,3) directions are given by:

(3)

The applicability of this definition gives E₁=151.11 GPa, E₂=147.53 GPa and E₃=64.17 GPa. Those value are lower compared with those found for Kaolinite (E₁=187.22 GPa, E₂=192.57 GPa and E₃=49.53 GPa). The elasticity of Nacrite is highly anisotropic; the Young’s modulus in the[100] and[010] directions (E₁ and E₂) is larger than E₃ (Young moduli in[001] direction). It is notable that Nacrite is less compressible compared to kaolinite.

3.2.5. Acoustic Velocities

Acoustic velocities are key quantities in the interpretation of seismic data. The polycrystalline averages of these acoustic velocities in a solid can be derived from the bulk and shear moduli of the material, as well as the density, ρ. There are two values, that for a transverse wave, Vs and that for a longitudinal wave, Vp, which are given by the following equations:

(4)

(5)

In this study Vp=17.59 Km/s, Vs=8.68 Km/s and giving Vp/Vs ratio equal to 2.02; this ratios is relatively high by comparing to those found for kaolinite 1.82[17].

4. Conclusions

This paper reports on prediction of elastic constant of Nacrite. The complete Cij elastic moduli were determined using molecular dynamics simulation. The elasticity of Nacrite is highly anisotropic. The calculated elastic constant tensor along c is much lower than the constants calculated along a and b. Nacrite deviates slightly by exhibiting a softer response, smaller elastic constants, which is a result of the weaker interlayer hydrogen bonding.