Structural and elastic properties of CaCO3 hydrated phases: A dispersion-corrected density functional theory study

Journal of Physics and Chemistry of Solids 138 (2020) 109295

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Structural and elastic properties of CaCO3 hydrated phases: A dispersion-corrected density functional theory study G. Chahi a, D. Bradai b, I. Belabbas a, * a b

Equipe de Cristallographie et Simulation des Mat�eriaux, Laboratoire de Physico-Chimie des Mat�eriaux et Catalyse, Universit�e de B�ejaia, 06000, B�ejaia, Algeria Facult�e de Physique, Universit�e des Sciences et de la Technologie Houari Boumediene, BP32, El-Alia, Bab Ezzouar, 16111, Algiers, Algeria

A R T I C L E I N F O

A B S T R A C T

Keywords: Hydrated CaCO3 Ikaite Monohydrocalcite Elastic properties Density functional theory DFT-D2 Density functional perturbation theory

Structural and elastic properties of the two known hydrated phases of calcium carbonate (i.e., ikaite and mon­ ohydrocalcite) were investigated. A comparative study was conducted where computer atomistic simulations based on standard density functional theory (DFT-PBE) and dispersion-corrected density functional theory (DFTD2) were performed. Properties such as the elastic constants, the bulk modulus, the Young modulus, the shear modulus, the Poisson ratio, the velocities of acoustic waves, and the Debye temperature were evaluated for the first time at the DFT level of theory. As most of the properties investigated have not been measured experi­ mentally yet, the DFT-PBE and DFT-D2 values provide limits that allow bracketing of the unknown experimental values. The evolution with pressure of the structural and elastic properties of ikaite and monohydrocalcite was investigated in the range from 0 to 5 GPa. In monohydrocalcite, a brittle-ductile transition is predicted to occur between 1.3 and 2.2 GPa.

1. Introduction Calcium carbonate (CaCO3) is a very common natural mineral [1,2]. It has attracted much interest in a variety of fields, such as geology, biology, and climatology [3,4]. Because it is one of the most abundant minerals on Earth, knowledge of its physical properties, among those of other minerals, is essential in rationalizing seismological data and un­ derstanding the dynamics of Earth’s crust [5]. CaCO3 is also important in biology; namely, in the context of biomineralization [4]. It is synthe­ sized by various living organisms, such as bivalves and gastropods, to produce solid shells that serve to protect their soft bodies [4]. Calcium carbonate has also been recognized as a major reservoir of inorganic carbon, which has an impact on global climate warming [3]. Calcium carbonate exhibits several allotropic forms among hydrous and anhydrous ones. Calcite, aragonite, and vaterite are the most com­ mon anhydrous polymorphs of CaCO3 [6]. Calcite is the stablest poly­ morph in ambient thermodynamic conditions and aragonite is stable at higher pressures. Although vaterite is the least thermodynamically favorable anhydrous polymorph, it is usually encountered in living or­ ganisms, where it is stabilized via the biomineralization process [4]. CalciteII, III, IIIb, IV, V, and VI are other anhydrous polymorphs which occur at high pressures and temperatures [7,8].

Calcium carbonate has two known hydrous phases:hexahydrocalcite and monohydrocalcite [9]. Hexahydrocalcite, also called ikaite, is a hexahydrated phase of calcium carbonate that forms in carbonate-rich cold marine water. Its name is derived from that of the Ikka fjord in southwestern Greenland, where it was discovered in 1963 as submarine columns [10]. Ikaite has also been found in marine sediments in the Arctic, the Antarctic, and Alaska, as well as Japan and the Republic of the Congo [11–14]. Monohydrocalcite was synthesized in the laboratory first by Brooks et al. [15] in 1950 and later by Van Tassel [16] in 1962. The first report of monohydrocalcite in nature was by Sapozhnikov and Zvetkov [17] in 1959. They found it to occur as an impure sediment in Issyk-Kul, a lake in Kyrgyzstan. Monohydrocalcite was also found in vertebrate otoliths [18] and was found to be produced in air-conditioning systems [19]. Although the hydrated phases of calcium carbonate are rare in nature, under geological conditions, they are frequently formed in the intermediate stages of crystallization of stabler calcium carbonate forms under biogenic and abiogenic conditions [20, 21]. At variance with the anhydrous polymorphs of CaCO3, for which the physical properties are well documented, few reports are available in the literature concerning hydrous phases [6]. However, the properties of ikaite have been relatively more investigated than those of

* Corresponding author. URL: http://[email protected] (I. Belabbas). https://doi.org/10.1016/j.jpcs.2019.109295 Received 17 June 2019; Received in revised form 30 November 2019; Accepted 2 December 2019 Available online 5 December 2019 0022-3697/© 2019 Published by Elsevier Ltd.

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Journal of Physics and Chemistry of Solids 138 (2020) 109295

Fig. 1. The crystal structures of the two hydrated phases of calcium carbonate: (a) primitive unitcell of ikaite; (b) primitive unitcell of monohydrocalcite. Calcium atoms are presented in blue, oxygen atoms in red, hydrogen atoms in white, and carbon atoms in brown. Hydrogen bonds are shown as fine dashed lines.

monohydrocalcite. The thermodynamic stability of synthetic ikaite was investigated experimentally by Marland [22] in 1975. Marland attempted to provide the phase diagram of ikaite with respect to water and anhydrous phases. Marland [22] demonstrated that although ikaite is thermodynamically unstable at atmospheric pressure, it is stable at room temperature above 0.5 GPa. Dickens and Brown [23] first determined the crystal structure of synthetic ikaite, at 120 � C, by using oscillation photographs. How­ ever, this technique did not allow them to obtain the hydrogen positions directly from the data. The crystal structure of ikaite was subsequently refined by Swainson and Hammond [24] by using the neutron powder diffraction technique. This allowed them to identify and characterize the hydrogen bond network of ikaite. By using synchrotron X-ray powder diffraction, Lennie et al. [25] studied the thermal expansion behavior of ikaite between 114 and 293K. They found that ikaite exhibits a volume expansion coefficient that is intermediate between that of ice and that of deuterated gypsum. By using the same technique, Lennie [26] investi­ gated the compressibility of ikaite for pressures up to 4 GPa and found it to exhibit anisotropy along the crystallographic axes. An experimental thermochemical investigation of natural mono­ hydrocalcite was performed by Hull and Turnbull [27] in 1973. They identified the parameters that impact the stabilization of mono­ hydrocalcite. The crystal structure of monohydrocalcite was first determined, in 1981, by Effenberger [28]. This structure was further refined, in 2008, by Swainson [29] by using both neutron and X-ray powder diffraction. Few theoretical investigations of the physical properties of ikaite and monohydrocalcite are available in the literature. Demichelis et al. [9] performed calculations based on density functional theory (DFT) to investigate the structure of the two hydrous phases while focusing on the characterization of their hydrogen bond networks. Demichelis et al. [30] subsequently conducted a comparative study where thermochemical properties of the two phases were evaluated. Recently, Costa et al. [31] performed DFT calculations on ikaite and monohydrocalcite to investi­ gate their electronic, vibrational, and thermodynamic properties. They also simulated infrared and Raman spectra of the two phases to identify their optical fingerprints. By using empirical potentials, Sekkal and Zaoui [32] investigated some elastic properties of the two hydrous phases of CaCO3. To the best of our knowledge, this is the only report on the elastic properties of ikaite and monohydrocalcite. Many physical properties of the two hydrous phases of calcium carbonate are missing from databases as they have not been measured experimentally or evaluated theoretically. This is particularly the case for the elastic properties. Because the two hydrous phases occur at high pressure and low temperature, knowing the evolution of their properties with pressure and temperature is of great interest. This is strong moti­ vation for our current research. In the present article, an atomistic investigation of the structural and

elastic properties of the two hydrous phases of calcium carbonate is presented. Our study was conducted at the DFT level, where standard and dispersion-corrected approaches were adopted and compared. Be­ sides our evaluating the different properties (structural and elastic) at zero pressure, their dependence on pressure up to 5 GPa was investi­ gated to fill the existing gap. This constitutes a first main step that may be followed by an investigation of the temperature dependence of the previously mentioned properties. This article is organized as follows: After an introduction where current knowledge of CaCO3 polymorphism is summarized, the crys­ tallographic models used and the calculation settings adopted are described in Section 2. Our results on the structural and elastic prop­ erties of ikaite and monohydrocalcite as well as their directional and pressure dependences are presented in Section 3. Section 4 is dedicated to the presentation of our conclusions. 2. Models and computational details The two hydrated phases of calcium carbonate, which are hexahy­ drocalcite (CaCO3⋅6H2O), also called ikaite, and monohydrocalcite (CaCO3⋅H2O), are considered. The crystal structures of these phases were determined experimentally by Swainson and Hammond [24,29], who used the neutron powder diffraction technique. It was found that ikaite crystallizes in the monoclinic system and belongs to the C2/c space group (Fig. 1a). It has a centered conventional unit cell that contains four formulas units, resulting in 92 atoms. For the primitive unitcell, the number of atoms is reduced to46 (Fig. 1a). The lattice pa­ rameters of the conventional unitcell determined by Swainson and Hammond [24] are a ¼ 8.7316 Å,b ¼ 8.2830 Å, c ¼ 10.9629 Å, and β ¼ 110.4� . Monohydrocalcite crystallizes in the hexagonal system and be­ longs to the P31 space group (Fig. 1b). It has a primitive unitcell that contains nine formulas units, resulting in 72 atoms. The lattice param­ eters given by Swainson [29] are a ¼ 10.5547 Å and c ¼ 7.5644 Å. The DFT calculations [33,34] were performed with the ABINIT software suite [35] on the primitive unitcells of the two hydrated phases of CaCO3. In this code, pseudopotentials are used to account for the effect of the nuclei and the core electrons and a basis set of plane waves is used to expand the valence one-electron Kohn-Sham orbitals. Norm-conserving. Troullier-Matrins pseudopotentials [36] were used to describe the effect of core electrons. The pseudopotentials used were created and optimized by the Rappe group at Pennsylvania University [37]. In our calculations, the 3s, 3p, and 4p orbitals of calcium, the 2s and 2p orbitals of carbon, the 2s and 2p orbitals of oxygen, and the 1s orbital of hydrogen were explicitly included in the valence. Each of the previously mentioned orbitals was expanded in a plane-wave basis set with an energy cutoff of 70Ry. Integrals over the Brillouin zonewere evaluated through uniform Monkhorst-Pack grids [38]. The Brillouin zone 2

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Journal of Physics and Chemistry of Solids 138 (2020) 109295

Table 1 Lattice parameters and unitcell volume of ikaite and monohydrocalcite. Experimental values reported by Swainson and Hammond [24,29], present calculated values (DFT-PBE, DFT-D2), and calculated values (DFT-D2) reported by Dimichelis et al. [9] and Costa et al. [31]. Ikaite

Monohydrocalcite

Experiment [24,29]

Present work (DFT-PBE)

Present work (DFT-D2)

Previous work (DFT-D2) [9]

Previous work (DFT-D2) [31]

a ¼ 8.7316 Å b ¼ 8.2830 Å c ¼ 10.9629 Å β ¼ 110.36� V ¼ 743.34 Å3 a ¼ 10.5547 Å c ¼ 7.5644 Å V ¼ 729.79 Å3

a ¼ 8.8806 Å b ¼ 8.3169 Å c ¼ 10.9665 Å β ¼ 108.50� V ¼ 768.05 Å3 a ¼ 10.6348 Å c ¼ 7.6355 Å V ¼ 747.88 Å3

a ¼ 8.6283 Å b ¼ 8.2408 Å c ¼ 10.8023 Å β ¼ 109.49� V ¼ 724.07 Å3 a ¼ 10.5041 Å c ¼ 7.5569 Å V ¼ 722.10 Å3

a ¼ 8.6268 Å b ¼ 8.2499 Å c ¼ 10.7875 Å β ¼ 108.71� V ¼ 727.19 Å3 a ¼ 10.5653 Å c ¼ 7.5871 Å V ¼ 725.41 Å3

a ¼ 8.6560 Å b ¼ 8.2510 Å c ¼ 10.8660 Å β ¼ 109.09� V ¼ 733.36 Å3 a ¼ 10.4950 Å c ¼ 7.5460 Å V ¼ 719.80 Å3

sampling was performed over an unshifted 4 � 4 � 2 k-point grid (14 points) and a 2 � 2 � 4 k-point grid (6 points) for ikaite and mono­ hydrocalcite, respectively. The exchange and correlation effects were treated in the framework of the generalized gradient approximation (GGA) with use of the Perdew-Burke-Ernzerhof (PBE) functional [39]. Besides standard DFT calculations, dispersion-corrected DFT calcu­ lations were also performed. The latter calculations aim at better describing the van der Waals interactions involved by the hydrogen bonds in the hydrated CaCO3 phases. We then used the DFT-D2 approach of Grimme [40], where the long-range van der Waals in­ teractions are described through an analytical pairwise potential, which exhibits an asymptotic 1/R6 dependence. This approach has the great advantage in including van der Waals interactions in an effective way. In the DFT-D2 method, the total energy of the system is written as [40]. EDFT

D2

where Cij are the clamped-ion elastic constants,V is the volume of the unit cell at equilibrium. ðK 1 Þmn represent the components of the inverse force constant matrix, and ^nk are the components of the force-response internal-strain tensor. 3. Results and discussion In the following, the results concerning structural and elastic prop­ erties of the two hydrated phases of CaCO3 (i.e., ikaite and mono­ hydrocalcite) are presented. These properties were evaluated at zero pressure and their evolution was also determined from 0 to 5 GPa. Although the two hydrated phases were experimentally observed at high pressures, the calculations at 0 GPa served as test calculations, which are necessary to validate the computational schemes we used. The derived reference values of the properties of interest were compared with values available in the literature.

(1)

¼ EDFT þ Edisp

where EDFT represents the total energy of the system as obtained by DFT calculations and Edisp is a semiempirical correction expressed as [40]. Edisp ¼

S6

XX Cij i

j6¼i

6 fdmp R6ij

Rij

�

3.1. Structural properties

(2)

Table 1 lists the values of the calculated DFT-PBE and DFT-D2 lattice parameters and the unitcell volumes of ikaite and monohydrocalcite at 0 GPa. For comparison, experimental and theoretical DFT-D2 values, obtained by Demichelis et al. [9,30] and Costa et al. [31], are reported as well. The calculated values of the lattice parameters of both ikaite and monohydrocalcite are in very good agreement with the experimental values reported by Swainson et al. [24,29]. By following the general trend of GGA-type functionals, our DFT-PBE values are overestimated with respect to the experimental values of the lattice constants of ikaite (þ1.7% for a, þ0.4% for b, and þ0.03% for c) and monohydrocalcite (þ0.7% for a or b and þ0.9% for c) (Table 1). The lattice constants of both ikaite and monohydrocalcite calculated by DFT-D2 are shorter than the DFT-PBE ones (Table 1). This is a consequence of including attractive van der Waals interactions in DFT-D2 calculations, which leads normally to a reduction of the lattice constants. Otherwise, our calculated DFT-D2 lattice constants of both ikaite and monohydrocalcite are in excellent agreement with those evaluated at the same level of theory by Dimi­ chelis et al. [9] and Costa et al. [31](Table 1). Our calculated DFT-D2 lattice constants of ikaite are underestimated with respect to the experimental ones ( 1.1% for a, 0.5% for b, and 1.4% for c). The same trend was observed for monohydrocalcite ( 0.4% for a or b and 0.09% for c). This is in accordance with the results obtained by Costa et al. [31] and Demichelis et al. [9]. The experimental unitcell volumes of both ikaite and monohydrocalcite are thus well bracketed by the calculated DFT-D2 and DFT-PBE values, which constitute lower and upper limits.

j>i

where dumping function fdmp ðRij Þ has the form [40]. � fdmp Rij ¼

�

1 þ exp

1 � d Rij Rt

�� 1

(3)

The summation in Eq. (2) runs over all ij atom pairs forming the system, where Rij represents the distance between atom i and atom j. S6 is a global scaling factor that depends on the exchange and correlation functional used in the DFT calculations. Rt denotes the sum of the van

der Waals radii of atoms i and j. Cij6 represents the dispersion coefficient of the atom pair (ij) and is related to the dispersion coefficients of in­ dividual atoms through [40]. qffiffiffiffiffiffiffiffiffiffi Cij6 ¼ Ci6 Cj6 (4)

In the present calculations, the DFT-D2 parameters used are those derived by Grimme [40] while performing DFT calculations and using the PBE exchange and correlation functional. In both DFT-PBE and DFT-D2 calculations, the geometry relaxations of the unitcells were performed by use of the Broyden-Fletcher-GoldfarbShanno algorithm [41]. The convergence criterion of forces adopted was 10 5 hartree/bohr, whereas that of stresses was 10 7 hartree/bohr3. Density functional perturbation theory–based calculations [42,43] were performed to evaluate the elastic constants of the two hydrated polymorphs of CaCO3 at equilibrium or at a given pressure. For that purpose, the approach of Hamann et al. [44] was used. In the latter, the elastic constants Cij are given by Ref. [45]. Cij ¼ Cij

� 1 ^mi K 1 mn ^nj V

3.2. Elastic properties The monoclinic and the hexagonal symmetries impose the following forms on the tensors of the elastic constants {Cij} of ikaite and mono­ hydrocalcite, respectively [46]:

(5)

3

G. Chahi et al.

2

Journal of Physics and Chemistry of Solids 138 (2020) 109295

C11 C12 C13 0 6 C12 C22 C23 0 6 6 C13 C23 C33 0 6 0 0 0 C44 6 4 C15 C25 C35 0 0 0 0 C46 and 2 C11 C12 C13 6 C12 C11 C13 6 6 C13 C13 C33 6 0 0 0 C 6 44 4 0 0 0 0 0 0 0 0

C15 C25 C35 0 C55 0

0 0 0 C46 0 C66

3 7 7 7 7 7 5

Table 2 Elastic constants (GPa), elastic moduli (GPa), Pugh ratio, Poisson ratio, veloc­ ities of transverse and longitudinal acoustic waves (km s 1), and Debye tem­ perature (K) of ikaite, monohydrocalcite, and calcite. Present DFT-PBE and DFTD2 calculated values as well as the values calculated by Sekkal and Zaoui [32] using the potentials of Raiteri et al. [49] and Xiao et al. [50] are reported.

(6)

Property

3 0 0 0 0 0 07 7 0 0 07 0 0 7 7 5 C44 0 0 C66

(7)

Ikaite C11 C12 C13 C15 C22 C23 C25 C33 C35 C44 C46 C55 C66 B G B/G E

BV þ BR 2

(8)

G¼

GV þ GR 2

(9)

where BV and BR represent the Voigt and Reuss averages of the bulk modulus, respectively, and GV and GR represent the Voigt and Reuss averages of the shear modulus, respectively. The Voigt and Reuss averages of the bulk modulus are given, respectively, by Ref. [47]. 1 BV ¼ ½C11 þ C22 þ C33 þ 2ðC12 þ C13 þ C23 Þ� 9 BR ¼ ½S11 þ S22 þ S33 þ 2ðS12 þ S13 þ S23 Þ�

(10) (11)

1

The Voigt and the Reuss averages of the shear modulus are given, respectively, by Ref. [47]. GV ¼

1 ½C11 þ C22 þ C33 15

GR ¼ 15½4ðS11 þ S22 þ S33 Þ

(12)

ðC12 þ C13 þ C23 Þ þ 3ðC44 þ C55 þ C66 Þ� 4ðS12 þ S13 þ S23 Þ þ 3ðS44 þ S55 þ S66 Þ�

1

(13)

The Young modulus and the Poisson ratio can be evaluated from the bulk and shear moduliby the relations [48]. E¼

9BG G þ 3B

(14)

� � 1 3B 2G 2 G þ 3B

(15)

and

ν¼

Present work

(DFTPBE)

(DFT-D2)

46.19 18.05 14.36 1.20 75.57 18.93 0.52 64.73 5.79 12.97 1.43 11.38 17.24 31.08 16.40 1.895 41.85 ν 0.276 Vt 2.91 Vl 5.23 3.24 Vm TD 475.72 Monohydrocalcite C11 83.33 22.54 C12 C13 27.54 C33 76.63 25.67 C44 C66 30.39 B 44.28 G 27.28 B/G 1.623 E 67.90 ν 0.244 3.40 Vt Vl 5.85 Vm 3.77 514.68 TD Calcite C11 146.35 C12 57.24 C13 52.08 16.99 C14 C33 84.01 32.25 C44 C66 44.55 B 74.62 G 29.25 B/G 2.551 E 77.60 ν 0.327 3.33 Vt Vl 5.32 4.20 Vm TD 536.81

By inverting the tensor of the elastic constants {Cij}, one can obtain the tensor of the elastic compliances {Sij}. Different elastic moduli, such as the bulk modulus (B), the shear modulus (G), and the Young modulus (E), and the Poisson ratio (ν) can be readily evaluated from the elastic constants or the elastic compliances. The adopted values for the bulk and shear moduli are those given by the Voigt-Reuss-Hill arithmetic average [47]: B¼

Present work

The velocities of the acoustic waves and the Debye temperature are related to the bulk and shear moduli. The average speed of the acoustic waves is expressed as [48]. � � �� 1=3 1 2 1 Vm ¼ þ 3 (16) 3 3 Vt Vl where Vl and Vt stand for the velocities of the longitudinal and transverse acoustic waves, respectively, which are related to the bulk and shear moduli through the following relations [48]:

4

Previous work [32] (potential of Raiteri et al.)

Previous work [32] (potential of Xiao et al.)

62.71 23.64 22.20 2.52 93.13 24.45 0.77 76.66 8.10 13.74 0.78 12.31 19.83 40.25 18.64 2.160 48.43 0.299 3.10 5.81 3.46 518.99

41.70 18.80 27.80 – 96.10 18.7 – 82.00 – 8.00 – 8.90 16.20 36.30 31.1 1.167 – – – – – –

59.20 20.40 26.30 – 75.40 16.8 – 62.70 – 4.20 – 2.70 12.90 15.6 13.7 1.139 – – – – – –

89.25 24.35 30.72 78.53 27.41 32.46 47.61 28.71 1.659 71.70 0.249 3.43 5.93 3.80 525.04

99.30 25.60 34.80 91.90 52.80 36.80 53.40 36.90 1.447 – – – – – –

98.80 28.90 42.20 88.10 32.10 34.90 44.4 27.6 1.609 – – – – – –

G. Chahi et al.

Vt ¼

� �1=2 G

� Vl ¼

ρ �1=2 4G þ 3B 3ρ

Journal of Physics and Chemistry of Solids 138 (2020) 109295

values. This is likely due to a transferability-related problem of the CaCO3 parameterizations of the two potentials. Our calculations demonstrate that the elastic moduli (B, G, E) as well as the average velocity of acoustic waves (Vm) and the Debye tempera­ ture (TD) of the two hydrous polymorphs of CaCO3 (i.e. ikaite and monohydrocalcite) are lower than those of calcite. Calcite and monohydrocalcite have similar values of the shear modulus, indicating their similar resistance to plastic deformation. This is justified in the framework of the sliding model of the plastic defor­ mation of a prefect crystal where a sinusoidal restoring force is assumed [54]. In this model, the theoretical critical shear stress is demonstrated to be proportional to the shear modulus [54]. Comparison of the elastic properties of the two hydrous CaCO3 polymorphs shows that mono­ hydrocalcite has higher values of the elastic moduli (B, G, E) than ikaite. The same trend was observed for the average velocity of acoustic waves (Vm) and the Debye temperature (TD). By evaluating the B/G ratio, one can analyze the ductility of a given material following the Pugh criterion [55]. According to the latter, a material with a B/Gratio higher than 1.75 is considered as ductile, while one with a B/G ratio lower than 1.75 is considered as brittle. As a reference, we calculated the B/G ratio of calcite, which was found to be 2.55, thus indicating a ductile behavior. The B/G ratio obtained for ikaite is 2.16, indicating that it is ductile, but less so than calcite. For monohydrocalcite, the calculated B/G ratio of 1.62 suggests that it is brittle. From our results it appears that incorporation of water molecules into CaCO3 leads to a decrease of the B/G ratio and thus to a reduction of its ductility. Monohydrocalcite, with a single water molecule per unit formula, was revealed to be brittle, while ikaite, which contains six water molecules per unit formula, was found to be ductile. This suggests that incorporating more water molecules into a CaCO3-based crystal leads to an increase of its ductility. This is at variance with the report of Sekkal and Zaoui [32] based on empirical potentials. The values of the elastic moduli (bulk modulus, shear modulus, Young modulus, and Poisson ratio) presented in Table 2 can be considered as averages over all directions in space. Ikaite and mono­ hydrocalcite belong to monoclinic and hexagonal crystal systems, which are known to be structurally anisotropic, and so their elastic properties should exhibit a directional dependence. Inspecting the directional dependence of the different elastic properties, by revealing the di­ rections along which these properties are maximal or minimal, is very relevant for understanding the mechanical behavior of the two hydrous phases of CaCO3. The directional dependence of the Young modulus as a function of elastic compliances and direction cosines for the monoclinic system is given by Ref. [46].

(17) (18)

where ρ represents the density of the crystal. The Debye temperature is related to the average sound velocity (Vm) through the following relation [48]: � �1=3 h 3N TD ¼ Vm (19) kB 4πV where V and N are the volume and number of atoms in the unitcell, respectively, kB is the Boltzmann constant, and h is the Planck constant. The elastic constants, the elastic moduli, the velocities of acoustic waves, and the Debye temperature of both ikaite and monohydrocalcite have not been fully determined experimentally yet. To the best of our knowledge, they have not been evaluated at the DFT level of theory either. The only existing theoretical report is that of Sekkal and Zaoui [32] and it is based on empirical potentials. Sekkal and Zaoui used two different potentials developed and parameterized for CaCO3-based sys­ tems by Raiteri et al. [49] and Xiao et al. [50]. The list of elastic con­ stants of ikaite provided by Sekkal and Zaoui [32] is incomplete. Our calculated DFT-PBE and DFT-D2 values of the elastic constants, elastic moduli (bulk modulus (B), shear modulus (G), Young modulus (E), and Poisson ratio (ν)), velocities of acoustic waves (Vt, Vl, Vm), and Debye temperature (TD) of ikaite, monohydrocalcite, and calcite are listed in Table 2. These values were obtained at 0 GPa. For comparison, the theoretical values provided by Sekkal and Zaoui [32] are also re­ ported. The calculations on calcite were performed with the PBE ex­ change and correlation functional with an energy cutoff of 70Ry and use of an unshifted 6 � 6 � 6 k-point grid, with 32 reduced k points, to sample the Brillouin zone. As mentioned previously, the elastic constants and the elastic moduli of both ikaite and monohydrocalcite, quantities which have been shown to be important for instance for the estimation of the point defect for­ mation (and migration) parameters [51], are missing from databases. Our calculations revealed that among the 13 independent elastic con­ stants of ikaite, four of them are negative (C15, C25, C35, C46). This is contrasted with monohydrocalcite, where all six independent elastic constants were found to be positive. Among the elastic constants of ikaite reported by Sekkal and Zaoui [32], those with negative values were missing. Both ikaite and monohydrocalcite were found to be stable at 0 GPa, as their elastic constants fulfill the stability criteria of Born and Huang [52] for monoclinic and hexagonal crystals. However, the pres­ ence of negative elastic constants indicates the possible occurrence in ikaite of either a structural transformation or a phase transition, which may be induced by pressure or temperature [53]. For both ikaite and monohydrocalcite, the elastic constants calcu­ lated by DFT-D2 are greater than those calculated by DFT-PBE (Table 2). This is a direct consequence of the overbinding resulting from the reduction of the lattice constants when attractive van der Waals in­ teractions are included in DFT-D2. This also explains why the DFT-D2 values of the elastic moduli (B, G, E), the acoustic wave velocities (Vt, Vl, Vm), and the Debye temperature (TD) are all greater than the DFT-PBE ones. As concluded for the lattice constants, the unknown experimental values of the elastic moduli (B, G, E) and those of derived properties (Vt, Vl, Vm, TD) of ikaite and monohydrocalcite are likely to be bracketed by the calculated DFT-PBE and DFT-D2 values. From Table 2, it appears that the potential with the parameterization of Xiao et al. [50] leads to values of the bulk and shear moduli of monohydrocalcite that agree better with our values than does the po­ tential with the parameterization of Raiteri et al. [49]. For ikaite, the two parameterizations lead to discrepancies and thus fail to produce values of the bulk and shear moduli that simultaneously agree with our

1 4 ¼ l S11 þ 2 l21 l22 S12 þ 2 l21 l23 S13 þ 2 l31 l3 S15 þ l42 S22 þ 2 l22 l23 S23 þ 2 l1 l22 l3 S25 E 3 þ l43 S33 þ 2 l1 l33 S35 þ l22 l22 S44 þ 2 l1 l22 l3 S46 þ l21 l23 S55 þ l21 l22 S66

(20)

and for the hexagonal system is given by Ref. [46]. 1 ¼ 1 E

� l3 2 S11 þ l43 S33 þ l23 1

� l23 ð2S13 þ S44 Þ

(21)

wherel1 ; l2 ; and l3 are the direction cosines of a unit vector associated with a given orientation with respect to the OX, OY, and OZ axes, respectively. The so-called linear compressibility (K) expresses the relative decrease in the length of a line when the crystal is subjected to unit hydrostatic pressure. Its directional dependence for the monoclinic system is given by Ref. [46]. K ¼ ðS11 þ S12 þ S13 Þl21 þ ðS12 þ S22 þ S23 Þl22 þ ðS13 þ S23 þ S33 Þl23 þ ðS15 þ S25 þ S35 Þl1 l3

5

(22)

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Journal of Physics and Chemistry of Solids 138 (2020) 109295

Table 3 Fitting parameters for the lattice constants (a, b, c; Å), unitcell volume (V;Å3), bulk modulus (B; GPa), shear modulus (G; GPa), Young modulus (E; GPa), and linear compressibilites (Ka, Kb, Kc; TPa 1) of ikaite and monohydrocalcite in the pressure range from 0 to 5 GPa. Property Ikaite a b c V B G E Ka Kb

and for the hexagonal system is given by Ref. [46]. S13

S33 Þl23

C (DFT-PBE, DFTD2)

0.0075772, 0.00567637 0.00249951, 0.00249318 0.00576663, 0.00349673 1.19805, 0.633606 0.176856, 0.159526 0.0754875, 0.0927276 0.218498, 0.187755 0.202517, 0.0902484 0.043126, 0.0334218

0.130895, 0.0985381 0.054306, 0.0480779 0.0914683, 0.0558395 23.3701, 18.418 4.78427, 4.21081

8.88373, 8.62512

0.984481, 0.514077

16.4415, 18.714

3.06027, 1.78638 2.52255, 1.44045 0.680535, 0.514305 1.34469, 0.941971

3.06027, 1.78638 15.771, 10.9252 6.81164, 5.66071

0.0772211, 0.0710884 0.0551066, 0.0504673 16.1521, 14.503 4.56211, 4.87989

10.633, 10.5015

1.74686, 1.6921

27.3123, 28.7332

4.79399, 4.73948

67.9779, 71.7827

0.702964, 0.657122 0.684831, 0.625031

8.31463, 8.23564 10.9618, 10.7934 767.99, 723.641 30.9524, 40.3962

8.86442, 6.50112

7.63684, 7.55762 747.721, 721.778 44.3308, 47.6757

7.4122, 6.76821 7.674, 7.38588

ikaite and monohydrocalcite. The surfaces are represented in an orthonormal frame OXYZ for which the axes are oriented according to the usual conventions adopted for monoclinic (ikaite) or hexagonal (monohydrocalcite) crystals. The 3D graphical representations were generated at 0 GPa by use of an angular increment of about 4� . As for a given property (E or K) both standard DFT-PBE and DFT-D2 calculations led to the same directional dependence, we chose to display only DFTD2 results. For an isotropic property one would expect a spherical 3D graphical representation of its directional dependence. The degree of elastic anisotropy is then directly reflected by the degree of deviation of the shape of the 3D surface from a sphere. Fig. 2 displays the directional dependence of the Young modulus and the linear compressibility of ikaite. For a monoclinic structure, the OX and OY axes are oriented, respectively, along the unitcell a-axis and baxis, with the b-axis being the two fold symmetry axis. The unitcell c-axis is normally inclined with respect to OZ. The Young modulus and the linear compressibility exhibit 3D surfaces of different shapes (Fig. 2). The Young modulus along the a-axis is 53.04 GPa and along the b-axis is 80.0 GPa; however, it is 64.64 GPa along the c-axis. The maximum value of the Young modulus over all directions is along the unitcell b-axis, while the minimum value of 29.41 GPa was found along the direction defined by the vector 0:84! a þ 0:55! c . The hierarchy of linear com­ pressibilies along the three unit-cell axes was found to be opposite that of the Young moduli. The linear compressibility along the a-axis is 10.98 TPa 1 and along the b-axis is 5.67 TPa 1. The latter is the lowest value of the linear compressibility over all directions. The value along the c-axis was found to be 6.51 TPa 1. The maximum value of the linear compressibility of 14.33 TPa 1 over all directions was found along the direction defined by the vector 0:94! a þ 0:35! c. Fig. 3 displays the directional dependence of the Young modulus and

Fig. 3. Directional dependence of (a) the Young’s modulus (GPa) and (b) the linear compressibility (TPa 1) of monohydrocalcite.

ðS11 þ S12

B (DFT-PBE, DFT-D2)

Kc 0.0918238, 0.101373 Monohydrocalcite a 0.00312248, 0.0031423 c 0.00202276, 0.00142206 V 0.69475, 0.615651 B 0.107848, 0.0948713 G 0.0835724, 0.069589 E 0.211874, 0.181544 Ka, Kb 0.0471751, 0.0418418 Kc 0.0470451, 0.0436153

Fig. 2. Directional dependence of (a) the Young modulus (GPa) and (b) the linear compressibility (TPa 1) of ikaite.

K ¼ ðS11 þ S12 þ S13 Þ

A (DFT-PBE, DFT-D2)

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Figs. 2 and 3 display 3D graphical representations of the directional dependence of the Young modulus and the linear compressibility of 6

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Journal of Physics and Chemistry of Solids 138 (2020) 109295

Fig. 4. Pressure dependence of lattice constants a (a), b (b), and c (c) and the unitcell volume of ikaite (d). DFT-PBE values are in black, DFT-D2 values are in blue, and the experimental data from Swainson and Hammond [24] are in red.

the linear compressibility of monohydrocalcite. For a hexagonal struc­ ture, the OZ axis is parallel to the unitcell c-axis, which is the six fold symmetry axis, and the OX axis is oriented along the unitcell a-axis. The Young modulus and the linear compressibility exhibit 3D surfaces with similar shapes (Fig. 3). The Young modulus was found to have a maximum value in directions corresponding into the (0001) basal plane and a minimum value along the c-axis (Fig. 3a). The opposite was found for the linear compressibility (Fig. 3b). In the basal plane the two pre­ viously mentioned properties are isotropic as the projection of the 3D surface is circular in this plane. In the basal plane, the Young modulus and the linear compressibility are. 75.3 GPa and 6.80 TPa 1, respectively. Along the c-axis, the Young modulus and the linear compressibility are 61.9 GPa and 7.42 TPa 1, respectively.

Where A, B, and C are fitting parameters and X refers to the property of interest (i.e., one of the lattice constants, the unitcell volume, or one of the elastic moduli). The values of the fitting parameters for the prop­ erties investigated are summarized in Table 3. Fig. 4 displays the pressure dependence of the lattice constants and the unitcell volume of ikaite in the pressure range from 0 to 5 GPa. DFTPBE and DFT-D2 values are reported as well as the experimental results obtained by Lennie [26]. The pressure curves for the three lattice con­ stants as well as that for the unitcell volume exhibit a normal behavior (i. e., a decrease on increase of pressure). The experimental values of lattice constant a were found to lie between the values calculated by DFT-PBE and DFT-D2 up to 4 GPa. While the experimental values of lattice con­ stant parameter are close to the DFT-D2 values, those of lattice constant c are larger than the DFT-PBE and DFT-D2 values. For the unitcell vol­ ume, the experimental values are bracketed by the DFT-PBE and DFT-D2 values for pressures lower than 4 GPa. Fig. 5 displays the evolution with pressure, up to 5 GPa, of the bulk modulus, the shear modulus, and the Young modulus of ikaite. Fig. 5a shows that the bulk modulus of ikaite increases with increase of the pressure, where the DFT-PBE and DFT-D2 values vary from 31.0 to 50.45 GPa and from 40.25 to 57.60 GPa, respectively, in the pressure range investigated. From the values of the fitting parameters (Table 3) it appears that the fitting curves for both the DFT-PBE data and the DFTD2 data are concave down parabolas. Fig. 5b shows that the shear modulus increases with increase of the pressure, where the DFT-PBE and DFT-D2 values vary from 16.40 to 19.6 GPa and from 18.64 to 23.58

3.3. Pressure dependence After we had investigated the structural and elastic properties of both ikaite and monohydrocalcite at 0 GPa, we investigated the evolution of these properties with pressure up to 5 GPa. Pressure increments of 0.1 and 0.5 GPa were used for ikaite and monohydrocalcite, respectively. The pressure dependence of the previously mentioned properties was fitted according to the following quadratic function: XðPÞ ¼ AP2 þ BP þ C

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Journal of Physics and Chemistry of Solids 138 (2020) 109295

Fig. 5. Pressure dependence of (a) the bulk modulus B, (b) the shear modulus G, and (c) the Young modulus E of ikaite. DFT-PBE values are in black and DFT-D2 values are in blue.

GPa, respectively. The values of the fitting parameters (Table 3) reveal that fitting curve for the DFT-PBE data is a concave down parabola, while that for the DFT-D2 data is a concave up parabola. Fig. 5c shows that the Young modulus increases with increase of the pressure, where the DFT-PBE and DFT-D2 values vary from 41.85 to 52.05 GPa and from 48.43 to 62.25 GPa, respectively. The values of the fitting parameters (Table 3) reveal that fitting curve for the DFT-PBE data is a concave down parabola, while that for the DFT-D2 data is a concave up parabola. Fig. 6 shows the pressure dependence of the lattice constants and the unitcell volume of monohydrocalcite in the pressure range from 0 to 5 GPa. Similarly to ikaite, the curves for the two lattice constants as well as that for the unitcell volume exhibit a normal behavior on increase of the pressure. No experimental reports on the structural properties of mon­ ohydrocalcite are available, but at 0 GPa we expect the experimental curve for the unitcell volume to be delimited by the DFT-PBE and DFTD2 ones, just like in the case of ikaite. Fig. 7 shows the evolution with pressure of the bulk modulus, the shear modulus, and the Young modulus of monohydrocalcite up to 5 GPa. The three elastic moduli investigated increase with increase of the pressure. The values of the fitting parameters (Table 3) reveal that the fitting curves for both the DFT-PBE data and the DFT-D2 data are concave down parabolas. From 0 to 5 GPa, the DFT-PBE and DFT-D2 values of the bulk modulus vary from 44.28 to 64.47 GPa and from 47.61 to 69.73 GPa, respectively, the DFT-PBE and DFT-D2 values of the shear modulus vary from 27.28 to 33.98 GPa and from 28.71 to 35.47

GPa, respectively, and the DFT-PBE and DFT-D2 values of the Young modulus vary from 67.90 to 86.71 GPa and from 71.70 to 91 GPa, respectively. Fig. 8 shows the evolution with pressure of the B/G ratio of ikaite and monohydrocalcite up to 5GaP. As pointed out previously, the B/G ratio allows the ductility of the two hydrated CaCO3 phases to be analyzed. As shown in Fig. 8a, the DFT-PBE and DFT-D2 curves exhibit B/G ratios higher than 1.75, thus indicating that ikaite remains ductile in the whole pressure range considered. The DFT-PBE curve is linear with a positive slope, while the DFT-D2 curve exhibits a maximum at a pressure of about 3.5 GPa before decreasing. Fig. 8b shows that the mono­ hydrocalcite DFT-PBE and DFT-D2 curves of the B/G ratio increase monotonically from 0 to 5 GPa. More importantly, it indicates the occurrence of a brittle-ductile transition in the previously mentioned pressure range. According to DFT-PBE, the transition occurs at about 2.2 GPa, while DFT-D2 locates it at about 1.3 GPa. The previous values likely give low and high estimates of the experimental brittle-ductile transition pressure. The linear compressibility is one of the few elastic properties that have been determined experimentally, along with its pressure depen­ dence, for ikaite. However, no experimental report is available for monohydrocalcite. Figs. 9 and 10 show the evolution with pressure of the relative linear compressibilities (Ka, Kb, and Kc) of ikaite and mon­ ohydrocalcite along the three crystallographic axes a,b, and c. For ikaite, the DFT-PBE and DFT-D2 curves (Fig. 9) demonstrate a general decrease of the linear compressibilities on increase of the 8

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Journal of Physics and Chemistry of Solids 138 (2020) 109295

Fig. 6. Pressure dependence of lattice constants a (a) and c (b) and the unitcell volume of monohydrocalcite (c). DFT-PBE values are in red and DFT-D2 values are in blue.

pressure. The values of the fitting parameters (Table 3) reveal that the fitting curves for both the DFT-PBE data and the DFT-D2 data are concave up parabolas. Fig. 9 shows that the linear compressibilities along the three crystallographic axes are in the order Kb < Kc < Ka. Such a hierarchy agrees with the experimental findings of Lennie [26]. From 0 to 5 GPa, the DFT-PBE and DFT-D2 values of Ka decrease from 15.87 to 8.08 TPa 1 and from 10.98 to 5.95 TPa 1, respectively, the DFT-PBE and DFT-D2 values of Kb decrease from 6.84 to 4.46 TPa 1 and from 5.67 to 3.92 TPa 1, respectively, and the DFT-PBE and DFT-D2 values of Kc decrease from 8.9 to 4.40 TPa 1 and from 6.51 to 4.28 TPa 1, respectively. The hexagonal symmetry of monohydrocalcite causes the linear compressibilities along the a-axis (Ka) and along the b-axis (Kb) to be equal as the corresponding directions belong to the basal plane. The DFT-PBE and DFT-D2 curves (Fig. 10) show that the linear compress­ ibility along the c-axis is greater than along the a-axis and the b-axis (Kc > Ka,Kb). Similarly to the case of ikaite, the values of the fitting pa­ rameters (Table 3) reveal that fitting curves for both the DFT-PBE data and the DFT-D2 data are concave up parabolas. The DFT-PBE and DFTD2 values of Kc vary from 7.7 to 5.4 TPa 1 and from 7.41 to 5.33 TPa 1, respectively, andt he DFT-PBE and DFT-D2 values of Ka and Kb vary from 7.44 to 5.05 TPa 1 and from 6.8 to 4.51 TPa 1, respectively.

that is, ikaite (CaCO3⋅6H2O) and monohydrocalcite. (CaCO3⋅H2O). A comparative study was conducted and calculations based on standard DFT (DFT-PBE) and dispersion-corrected DFT (DFTD2) were performed. Structural properties (lattice parameters, unitcell volumes), elastic properties (elastic constants, elastic moduli), and some derived properties (acoustic wave velocities, Debye temperature) of the two phases were evaluated and their pressure dependence was investi­ gated between 0 and 5 GPa. To the best of our knowledge, the present work constitutes the first investigation of the elastic properties of the two hydrated phases of calcium carbonate at the DFT level of theory. As most of the properties investigated here have not been measured experimentally yet, the DFTPBE and DFT-D2 values provide limits that allow bracketing of the un­ known experimental values. For both ikaite and monohydrocalcite, the DFT-PBE lattice constants were larger than the DFT-D2 ones. This is a direct consequence of including attractive van der Waals interactions in DFT-D2, which leads normally to a reduction of the lattice constants compared with the DFTPBE case. Consequently, the elastic constants calculated by DFT-D2 are greater than those calculated by DFT-PBE. This is also the case for the elastic moduli (bulk modulus, shear modulus, Young modulus), the acoustic wave velocities, and the Debye temperature, which are all greater than the DFT-PBE values. Our calculations demonstrate that the elastic moduli of ikaite and monohydrocalcite have lower values than those of calcite. This is due to the difference in the strength of the ionic bonds involved in calcite and the hydrogen bonds involved in the hydrated phases. Otherwise,

4. Summary and conclusion Computer atomistic simulations were performed to investigate some physical properties of the two hydrated phases of calcium carbonate; 9

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Journal of Physics and Chemistry of Solids 138 (2020) 109295

Fig. 7. Pressure dependence of (a) the bulk modulus B, (b) the shear modulus G, and (c) the Young modulus E of monohydrocalcite. DFT-PBE values are in black and DFT-D2 values are in blue.

Fig. 8. Pressure dependence of the B/G ratio of (a) ikaite and (b) monohydrocalcite. DFT-PBE values are in red and DFT-D2 values are in blue. The green line indicates B/G ¼ 1.75.

monohydrocalcite was found to possess greater values of the elastic moduli, acoustic wave velocities, and Debye temperature than ikaite. In accordance with the experimental report of Lennie [26] for ikaite, we found that the highest linear compressibility is along the a-axis, followed by that along the c-axis and the b-axis. In monohydrocalcite, our cal­ culations led to the prediction that the linear compressibility is higher along the c-axis than along the directions belonging to the basal plane.

The difference in elastic properties of the two hydrated CaCO3 phases is attributed to their different crystal structures as well as to the different features of their complex networks of hydrogen bonds. Besides our determining the structural and elastic properties of ikaite and monohydrocalcite at zero pressure, we also investigated their evo­ lution between 0 and 5 GPa. The different properties were found to exhibit a quadratic dependence on pressure. In the pressure range 10

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Journal of Physics and Chemistry of Solids 138 (2020) 109295

Fig. 9. Pressure dependence of linear compressibilities (Ka, Kb, and Kc) of ikaite: (a) DFT-PBE values and (b) DFT-D2 values.

Fig. 10. Pressure dependence of linear compressibilities (Ka and Kc) of monohydrocalcite: (a) DFT-PBE values and (b) DFT-D2 values.

investigated, ikaite was found to be ductile; however, in mono­ hydrocalcite, a brittle-ductile transition is predicted to occur between 1.3 and 2.2 GPa.

Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.jpcs.2019.109295.

Data availability

Author contributions

The raw data required to reproduce these findings can be obtained on request from the corresponding author.

I.Belabbas designed the research, analyzed the results, and wrote the article. G. Chahi performed the calculations and analyzed the results. D. Bradai participated to the discussions and in the analysis of the results.

Declaration of competing interest Ghiles CHAHI, Djamel BRADAI and Imad BELABBAS declare that they have no conflict of interest.

References [1] W.A. Deer, R.A. Howie, J. Zussmann, Introduction to the Rock Forming Minerals, second ed., Harlow Longman, 1992. [2] H. Becker, R. Altherr, Evidence from ultra-high-pressure marbles for recycling of sediments into the mantle, Nature 358 (1992) 745–748, https://doi.org/10.1038/ 358745a0. [3] R. Dasgupta, M.M. Hirschmann, The deep carbon cycle and melting in Earth’s interior, Earth Planet. Sci. Lett. 298 (2010) 1–13, https://doi.org/10.1016/j. epsl.2010.06.039. [4] H. Lowenstam, S. Weiner, On Biomineralization, Oxford University Press, New York, 1989. [5] K. Bullen, Introduction to the Theory of Seismology, fourth ed., Cambridge University Press, 1985. [6] S.A.T. Redfern, Structural variations in carbonate, Rev. Mineral. Geochem. 41 (2000) 289–308.

Acknowledgments I. Belabbas and G. Chahi acknowledge the University of Bejaia and the Algerian Ministry of HigherEducation and Scientific Research for funding. This work was achieved in the framework of the CNEPRU project B00L02UN060120150003. The Centre R�egional Informatique et d’Applications Num�eriques de Normandie (CRIANN; http://www. criann.fr) is acknowledged for providing computational time.

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Journal of Physics and Chemistry of Solids 138 (2020) 109295

[7] R. Belkofsi, O. Adjaoud, I. Belabbas, Pressure induced phase transitions and elastic properties of CaCO3 polymorphs: a density functional theory study, Model. Simul. Mater. Sci. Eng. 26 (2018), 065004, https://doi.org/10.1088/1361-651X/aacbed. [8] N. Ishizawa, H. Setoguchi, K. Yanagisawa, Structural evolution of calcite at high temperatures: phase v unveiled, Sci. Rep. 3 (2013) 2832, https://doi.org/10.1038/ srep02832. [9] R. Demichelis, P. Raiteri, J.D. Gale, Structure of hydrated calcium carbonates: a first-principles study, J. Cryst. Growth 401 (2014) 33–37, https://doi.org/ 10.1016/j.jcrysgro.2013.10.064. [10] M. Fleischer, New mineral names, Am. Mineral. 49 (1964) 439–448. [11] C. Stein, A.J. Smith, Authigenic carbonate nodules in the nankai trough, site 583, Initial Rep. Deep Sea Drill. Proj. 87 (1985) 659–668. [12] J.H.F. Jansen, G.F. Woensdregt, M.J. Kooistra, S.J. Van Der Gaast, Ikaite pseudomorphs in the Zaire deep-sea fan: an intermediate between calcite and porous calcite, Geology 15 (1987) 245–248, https://doi.org/10.1130/0091-7613 (1987)15<245:IPITZD>2.0.CO;2. [13] G.L. Kennedy, D.M. Hopkins, W.J. Pickthorn, Geological Survey of Alaska Annual Meeting, Abstract, programme. 9 (1987) 725. [14] C.R. Omelon, W.H. Pollard, G.M. Marion, Seasonal formation of ikaite (CaCO3.6H2O) in saline spring discharge at expedition fiord, Canadian high arctic: assessing conditional constraints for natural cyrstal growth, Geochem. Cosmochim. Acta 65 (2001) 1429–1437, https://doi.org/10.1016/S0016-7037(00)00620-7. [15] R. Brooks, L.M. Clark, E.F. Thurston, Calcium carbonate and its hydrates, Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 243 (2006) 145–167, https://doi.org/ 10.1098/rsta.1950.0016. [16] R. Van Tassel, Carbonatniederschl€ age aus gemischten calcium-magnesiumchloridl€ osungen, Z. Anorg. Allg. Chem. 319 (1962) 107–112. [17] D.G. Sapozhnikov, A.I. Zvetkov, Precipitation of hydrous calcium carbonate on the bottom of Lake Issyk-Kul, Dokl. Akad. Nauk SSSR 124 (1959) 402–404. [18] C. Carlstrom, A crystallographic study of vertebrate otoliths, Biol. Bull. 125 (1963) 441–463. [19] H. Marschner, Hydrocalcite (CaCO3⋅H2O) and nesquehonite (MgCO3⋅3H2O) in carbonate scales, Science 165 (1969) 1119–1121, https://doi.org/10.1126/ science.165.3898.1119. [20] C.C. Tang, S.P. Thompson, J.E. Parker, A.R. Lennie, F. Azough, K. Kato, The ikaiteto-vaterite transformation: new evidence from diffraction and imaging, J. Appl. Crystallogr. 42 (2009) 225–233, https://doi.org/10.1107/S0021889809005810. [21] M. Saharay, A.O. Yazaydin, R.J. Kirkpatrick, Dehydration-induced amorphous phases of calcium carbonate, J. Phys. Chem. B 117 (2013) 3328–3336, https://doi. org/10.1021/jp308353t. [22] G. Marland, The stability of CaCO3⋅6H2O (ikaite), Geochem. Cosmochim. Acta 39 (1975) 83–91, https://doi.org/10.1016/0016-7037(75)90186-6. [23] B. Dickens, W.E. Brown, The crystal structure of calcium carbonate hexahydrate at ~ –120� , Inorg. Chem. 9 (1970) 480–486, https://doi.org/10.1021/ic50085a010. [24] I.P. Swainson, R.P. Hammond, Hydrogen bonding in ikaite, CaCO3.6H2O, Mineral. Mag. 67 (2003) 555–562, https://doi.org/10.1180/0026461036730117. [25] A.R. Lennie, C.C. Tang, S.P. Thompson, The structure and thermal expansion behaviour of ikaite, CaCO3. 6H2O, from T ¼ 114 to T ¼ 293 K, Mineral. Mag. 68 (2004) 135–146, https://doi.org/10.1180/0026461046810176. [26] A.R. Lennie, Ikaite (CaCO3.6H2O) compressibility at high water pressure: a synchrotron X-ray diffraction study, Mineral. Mag. 69 (2005) 325–335, https://doi. org/10.1180/0026461056930254. [27] H. Hull, A.G. Turnbull, A thermochemical study of monohydrocalcite, Geochem. Cosmochim. Acta 37 (1973) 685–694, https://doi.org/10.1016/0016-7037(73) 90227-5. [28] H. Effenberger, Kristallstruktur und Infrarot-Absorptionsspektrum von synthetischem Monohydrocalcit, CaCO3H2OCrystal structure and infrared absorption spectrum of synthetic monohydrocalcite, CaCO3H2O, Monatshefte F r Chemie. 112 (2004) 899–909, https://doi.org/10.1007/bf00905061. [29] I.P. Swainson, The structure of monohydrocalcite and the phase composition of the beachrock deposits of Lake Butler and Lake Fellmongery, South Australia, Am. Mineral. 93 (2008) 1014–1018, https://doi.org/10.2138/am.2008.2825. [30] R. Demichelis, P. Raiteri, J.D. Gale, R. Dovesi, Examining the accuracy of density functional theory for predicting the thermodynamics of water incorporation into minerals: the hydrates of calcium carbonate, J. Phys. Chem. C 117 (2013) 17814–17823, https://doi.org/10.1021/jp4048105. [31] S.N. Costa, V.N. Freire, E.W.S. Caetano, F.F. Maia, C.A. Barboza, U.L. Fulco, E. L. Albuquerque, DFT calculations with van der Waals interactions of hydrated calcium carbonate crystals CaCO3⋅(H2O, 6H2O): structural, electronic, optical, and vibrational properties, J. Phys. Chem. 120 (2016) 5752–5765, https://doi.org/ 10.1021/acs.jpca.6b05436.

[32] W. Sekkal, A. Zaoui, Nanoscale analysis of the morphology and surface stability of calcium carbonate polymorphs, Sci. Rep. 3 (2013) 1587, https://doi.org/10.1038/ srep01587. [33] P. Hohenberg, W. Kohn, Inhomogeneous electron gas, Phys. Rev. 136 (1964) 864, https://doi.org/10.1103/PhysRev.136.B864. [34] W. Kohn, L.J. Sham, Self-consistent equations including exchange and correlation effects, Phys. Rev. 140 (1965) 1133, https://doi.org/10.1103/PhysRev.140. A1133. [35] X. Gonze, F. Jollet, F. Abreu Araujo, D. Adams, B. Amadon, T. Applencourt, C. Audouze, J.M. Beuken, J. Bieder, A. Bokhanchuk, E. Bousquet, F. Bruneval, D. Caliste, M. C^ ot� e, F. Dahm, F. Da Pieve, M. Delaveau, M. Di Gennaro, B. Dorado, C. Espejo, G. Geneste, L. Genovese, A. Gerossier, M. Giantomassi, Y. Gillet, D. R. Hamann, L. He, G. Jomard, J. Laflamme Janssen, S. Le Roux, A. Levitt, A. Lherbier, F. Liu, I. Luka�cevi�c, A. Martin, C. Martins, M.J.T. Oliveira, S. Ponc� e, Y. Pouillon, T. Rangel, G.M. Rignanese, A.H. Romero, B. Rousseau, O. Rubel, A. A. Shukri, M. Stankovski, M. Torrent, M.J. Van Setten, B. Van Troeye, M. J. Verstraete, D. Waroquiers, J. Wiktor, B. Xu, A. Zhou, J.W. Zwanziger, Recent developments in the ABINIT software package, Comput. Phys. Commun. 205 (2016) 106–131, https://doi.org/10.1016/j.cpc.2016.04.003. [36] N. Troullier, J.L. Martins, Efficient pseudopotentials for plane-wave calculations, Phys. Rev. B 43 (1991) 1993–2006, https://doi.org/10.1103/PhysRevB.43.1993. [37] http://sas.upenn.edu/rappegroup/research/pseudo-potential-gga.html. [38] J.P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77 (1996) 3865–3868, https://doi.org/10.1103/ PhysRevLett.77.3865. [39] H.J. Monkhorst, J.D. Pack, Special points for Brillouin-zone integrations, Phys. Rev. B 13 (1976) 5188–5192, https://doi.org/10.1103/PhysRevB.13.5188. [40] S. Grimme, Semiempirical GGA-type density functional constructed with a longrange dispersion correction, J. Comput. Chem. 27 (2006) 1787–1799, https://doi. org/10.1002/jcc.20495. [41] T.H. Fischer, J. Alml€ of, General methods for geometry and wave function optimization, J. Phys. Chem. 96 (1992) 9768–9774, https://doi.org/10.1021/ j100203a036. [42] S. Baroni, S. De Gironcoli, A. Dal Corso, P. Giannozzi, Phonons and related crystal properties from density-functional perturbation theory, Rev. Mod. Phys. (2001) 515–562, https://doi.org/10.1103/RevModPhys.73.515. [43] X. Gonze, Perturbation expansion of variational principles at arbitrary order, Phys. Rev. A 52 (1995) 1086–1095, https://doi.org/10.1103/PhysRevA.52.1086. [44] D.R. Hamann, X. Wu, K.M. Rabe, D. Vanderbilt, Metric tensor formulation of strain in density-functional perturbation theory, Phys. Rev. B Condens. Matter Mater. Phys. 71 (2005), 035117, https://doi.org/10.1103/PhysRevB.71.035117. [45] X. Wu, D. Vanderbilt, D.R. Hamann, Systematic treatment of displacements, strains, and electric fields in density-functional perturbation theory, Phys. Rev. B Condens. Matter Mater. Phys. 72 (2005), 035105, https://doi.org/10.1103/ PhysRevB.72.035105. [46] J.F. Nye, Physical Properties of Crystal, Oxford University Press, London, 1957. [47] A.R. Oganov, Computer Simulation Studies of Minerals, PhD Thesis University College London, 2002. [48] Q.J. Liu, N.C. Zhang, F.S. Liu, Z.T. Liu, Structural, electronic, optical, elastic properties and Born effective charges of monoclinic HfO2 from first-principles calculations, Chin. Phys. B 23 (2014), 047101, https://doi.org/10.1088/16741056/23/4/047101. [49] P. Raiteri, J.D. Gale, D. Quigley, P.M. Rodger, Derivation of an accurate force-field for simulating the growth of calcium carbonate from aqueous solution: a new model for the calcite?water interface, J. Phys. Chem. C 114 (2010) 5997–6010, https://doi.org/10.1021/jp910977a. [50] S. Xiao, S.A. Edwards, F. Gr€ ater, A new transferable forcefield for simulating the mechanics of CaCO 3 crystals, J. Phys. Chem. C 115 (2011), https://doi.org/ 10.1021/jp202743v, 20067–20075. [51] P. Varotsos, Point defect parameters in b-PbF2 revisited, Solid State Ionics179 (2008) 438. [52] M. Born, K. Huang, Dynamical Theory of Crystal Lattices, London Oxford University Press, 1954. [53] R.S. Lakes, W.J. Drugan, Dramatically stiffer elastic composite materials due to a negative stiffness phase? J. Mech. Phys. Solids 50 (2002) 979. [54] D. Hull, Introduction to Dislocations, second ed., Pergamon Press, 1975. [55] S.F. Pugh, XCII. Relations between the elastic moduli and the plastic properties of polycrystalline pure metals, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 45 (2014) 823–843, https://doi.org/10.1080/ 14786440808520496.

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