A Nanoscale Simulation Study of Elastic Properties of Gaspeite

Abstract The study of structural and mechanical properties of carbonate rock is an interesting subject in engineering and its different applications. In this paper, the crystal structure of gaspeite (NiCO3) is investigated by carrying out molecular dynamics simulations based on energy minimization technique using an interatomic interaction potential. At first, we focus on the structural properties of gaspeite mineral. And then, the elastic properties are calculated, including the elastic constants, bulk modulus, shear modulus, the S- and P-wave velocities. In the next part of this paper, the pressure effect will be studied on the structural and elastic properties of NiCO3 at high pressure.


INTRODUCTION
Carbonate rocks are common minerals found in sedimentary environments.They have a general formula of XCO 3 where X represents one or more metal ions (+2), such as Ba, Ca, Cd, Co, Cu, Fe, Mg, Mn, Ni, Pb, Sr, Zn and others.The study of the elastic properties of carbonate rocks involves several engineering areas (civil engineering, petroleum engineering, etc.), i.e., the elastic properties of carbonate rock can be used as parameters in different geological and geomechanical models.These properties depend on several factors, including chemical composition, crystal orientation, pressure and temperature.
To study the gaspeite (or nickel carbonate mineral), molecular dynamics (MD) simulations based on energy minimization technique have been used.The DM simulation is a theoretical method used in the last two decades to study and evaluate a wide range of quantities (structural, physical, mechanical, thermodynamics, etc.) of materials.
In this paper, we present the elastic properties of gaspeite under normal conditions and under the effect of pressure.The paper is organized as follows.In Section 2, we describe the calculation details related to this study.In Section 3, we present the results and discussion.The last section contains conclusions.

THEORETICAL BACKGROUND
Many theoretical studies based on the atomic scale simulations have been made to develop interatomic potentials describing the carbonate rocks (Pavese et al. [20], Dove et al. [10], Jackson and Price [15], Catti et al. [7], Parker et al. [19], Fisler et al. [11], Archer et al. [2].The interatomic potential based on the interaction of Rohl et al. [23] was used to describe the gaspeite. In this study, a modified Born model for description of solids was used to perform simulations, in which the ionic material is treated as a collection of point ions with electrostatic and short-range forces acting between them (Born and Huang [5]).The contribution of the electrostatic interactions (longrange) is determined by using the Ewald sum whereas the short-range terms contribution for all ion-ion interaction of the crystal are summed within a predetermined cutoff.The procedure is described in detail by Catlow and Mackrodt [6].The oxygen atom is represented by a core and a shell (Dick and Overhauser [9]).The harmonic spring potential is used for the core-shell.The interatomic potential model employed here is The potential forms are presented in Table 1.The forms are described in detail by Rohl et al. [23].For the potential acting on the CO 3 group, i.e., interactions with the outside of the Oxygen shell, there are intermolecular interactions of O-O and Buckingham interaction of Ni-O.The interaction potentials between atoms of the same group CO 3 (i.e., interactions with the oxygen core) are grouped O-O intramolecular interactions and C-O Morse interaction.A three-body term (O-C-O) has been incorporated in the model, to maintain the 120° bond angle in the CO 3 group.A four-body potential (O-C-O-O) associated with the out-of-plane displacement of C within the CO 3 group was added to the model.Buckingham potential is integrated between Ni-C (Rohl et al. [23], Austen et al. [3]).All potential parameters used in this work were taken initially from the work of Rohl et al. [23].
Before we begin our calculation, some potential parameters are adjusted by using the GULP code, so as to minimize the sum of squared differences between the experimental and calculated properties (Gale [12], [13]).The experimental crystal structure (lattice parameters as a, b, c, α, β and γ and fractional coordinates) and bulk modulus are the only parameters used during the fit; elastic constants are not taken into account.The adjusted parameters of the interatomic potentials are listed in Table 1.
where f calc and f obs are the calculated and observed quantities and ω is a weighting factor.
In this study, the energy minimization technique is performed at constant pressure, which allows the atomic positions and lattice parameters to vary.The Newton-Raphson procedure is used to find the minima, with the Broyden-Fletcher-Goldfarb-Shanno (BFGS) scheme to update the Hessian (Gale [12], [13]).Table 2 shows a comparison between experimental and calculated structural parameters of the gaspeite.
We present a comparison of the initial (a, b, c, α, β and γ from experiments) and the final (after fitting) configuration with the corresponding variation.The percentage difference between the lattice parameters of experimental and adjusted is acceptable.The report parameter c/a obtained is 3.19 which is in good agreement with the experimental values of 3.13 and 3.2 found by Pertlik [22], and Zhang and Reeder [28], respectively.

MECHANICAL PROPERTIES
In this work, we have used the mathematical formulation of the mechanical properties as given in the paper of Gale and Rohl [14].The elastic constants are represented by the second derivative of the energy density with respect to strain where V is the volume, U is the energy and ε is the strain.Therefore, the elastic constant tensor is a 6 × 6 symmetric matrix.The 21 potentially independent matrix elements are usually reduced considerably by symmetry (Nye [18]).For a material rhombohedral, the matrix elements are C 11 , C 33 , C 44 , C 12 , C 13 , C 14 .
The elastic compliances S can be readily calculated from the above expression by inverting the matrix (i.e., S = C -1 ).The elastic compliances, S ij , can be easily obtained from the above expression by inverting the matrix.The C ij and S ij obtained are given in Table 3.We note that C 33 is greater in value compared to other constants, which indicates that the gaspeite is stiffer in this direction.
In isotropic, polycrystalline aggregate, there are only two independent elastic constants, such as the bulk (B) and shear (G) moduli.Other constants are expressed in terms of B and G.We are interested in calculating the value of B ang G from the values of C ij and S ij using the relationships given in the work of Gale and Rohl [14].
The bulk modulus represents the ratio of the variation in pressure to the volume compression of a material.
B is the bulk modulus, P is the pressure, V is the volume, and T is the temperature.Experimentally, the bulk modulus value for gaspeite of 131 GPa was evaluated by Zhang and Reeder [28] using energy-dispersive X-ray, EDX.From energy minimization calculations, Fisler et al. [11], Austen et al. [3] and Wang et al. [26] reported the value of bulk modulus.The proposed values are a little larger than the experimental value.The bulk modulus, B, of gaspeite can be evaluated from the elastic compliances or elastic constants.The bulk modulus is given in equation ( 5) by two definitions, the lower (Reuss) bounds and the upper (Voigt) bounds (Gale and Rohl [14]).Using Reuss definition, the value obtained of B = 129.12GPa is in good trend with the only experimental values found in literature (Zhang and Reeder [28]).The bulk modulus (Voigt) obtained is 136.68 GPa.
The shear modulus describes the response of materials to deformation in shear.It can also be calculated employing the Reuss and Voigt definitions (see equations (6)).The values obtained for shear modulus are 73.89GPa and 82.13 GPa using the Reuss and Voigt definitions, respectively.
where S ij and C ij are the elastic compliances and elastic constants, respectively, given in Table 4.
The next property determined is Young's modulus.Young's modulus is defined by the ratio of stress to strain.It can be calculated from the elastic compliances in three Cartesian directions a, b and c (Gale and Rohl [14]).The values obtained for gaspeite crystal are  The obtained V p , V s and υ are 7.233, 4.121 kms −1 and 0.26, respectively.
The calculated mechanical quantities for gaspeite, bulk modulus, shear modulus, Young's moduli, elastic wave velocities (V s and V p ) and the Poisson ratio, are summarized in Table 4 compared to experimental findings and theoretical works.

PRESSURE EFFECT
Understanding the rocks mineral behavior depending on earth depth can be combined with knowl-edge of the physical quantities.We then focused on the evaluation of these properties and we studied their variation under the effect of hydrostatic pressure using molecular dynamics simulations.The behavior under pressure of gaspeite is presented in a range from 0 to 500 MPa corresponding to depths greater than 10 km.BENAZZOUZ 14 volume is 1.77, which is almost the same slope (equal to 1.83) that represents the volume variation in the work of Zhang and Reeder [28].Zhang and Reeder studied the pressure effect on the NiCO 3 in the interval from 0 to 7.29 GPa.We notice that the density increases with pressure up to 4.37g/cm 3 .
The decrease of the lattice parameters a, b and c is observed with pressure and stability of α, β and γ angles.The pressure effect on a (= b) and c parameters of gaspeite can be compared to the experimental work of Zhang and Reeder [28].The variation of a and c axes with pressure is given by: a = 4.6463-2E-05*P and c = 14.536-2E-05*P,where P is the pressure in GPa, and the axis length is in Ǻ.

EFFECT ON THE MECHANICAL PROPERTIES
In this section, the effect of hydrostatic pressure on the mechanical properties of gaspeite is presented.Pressure dependence of the elastic constants of gaspeite is shown in Fig. 5.The elastic constants are almost stable with a slight increase for C 11 and C 33 .From 0 to 500 MPa, the C 33 (corresponding to the c-axis perpendicular to the atomic planes) is greater than C 11 corresponding to the a-axis, which shows that this structure is more compressive in the direction perpendicular to the layers parallel to (100) plane.The potential energy has contributed to reduction of the volume of the unit cell and increase of the stiffness constants.Figure 6 illustrates a pressure dependence of the isotropic bulk (B) and shear (G) moduli.A linear increase presents that the gaspeite becomes more rigid in this pressure range.The bulk modulus increases from 129 GPa up to 132 GPa.The increase of the bulk modulus is more significant than that for shear modulus.
The variation in Young's modulus with pressure is shown in Fig. 7. Young's modulus increases linearly in the three directions up to the pressure of 500 MPa.This can be explained from the fact that its crystallographic structure is composed of planes of nickel ions along the c direction alternately with the carbonate groups CO 3 .
Figure 8 illustrates a linear variation of the transverse wave velocity, V s , and the longitudinal wave velocity, V p .Under pressure, gaspeite became more rigid and this is represented by the slight increase in the value of the elastic wave velocities.Finally, Fig. 7 shows the evolution of azimuthal anisotropy A s and A p as a function of pressure.The A s and A p waves are defined as

CONCLUSION
In this work, the molecular dynamics based on energy minimization approach was employed to study the gaspeite (nickel carbonate).Potential fitting has been used in order to make calculation reliable.The potential fitting was made considering the experimental value of the bulk modulus of gaspeite 131 GPa (Zhang and Reeder [28]).
The structural and mechanical properties were performed at atmospheric pressure.The values of bulk modulus and shear modulus were evaluated as 129.12 GPa and 73.89 GPa, respectively.In the next step, the pressure effect was introduced to predict the behavior of these properties.The results are in good agreement with available experimental measurements and some theoretical work.

Fig. 1 .
Fig. 1.Structure of gaspeite.O atoms are in red, C atoms are in black and Ni atoms in blue 2 .

S
= 227.86GPa.Concerning the acoustic wave velocities (acoustic transverse wave velocity, V s , and longitudinal wave velocity, V p ) and the Poisson ratio can be calculated from B and G as follows Figures 3 and 4 depict the variation of volume, density, the cell parameters a, b, c and angles α, β and γ with pressure.The slope value of decrease in

Fig. 3 .
Fig. 3. Volume and density variation of gaspeite with pressure

Fig. 7 .Fig. 8 .
Fig. 7. Young's modulus of gaspeite as a function of pressure along a, b, and c directions

Table 2 .
Comparison between the simulated and experimental structural properties of gaspeite a Pertlik 1986, b Zhang 1999, c Wang 2011.