Elasticity

The mechanical or elastic properties of a material constitute a very important group of properties determining the behavior of the material under an applied pressure. Clear expositions of the physical magnitudes involved, their significance and determination may by found in several sources (Nye, 1985; Ortega-Castro, 2007).

The mechanical stability of a given structure, its hardness and other related mechanical properties are determined by its elastic behavior and determine to a great extend the material durability and behavior under a large range of environmental conditions. For example, in terms of the elastic properties of minerals we can study the variation of their structures under different pressure conditions and to predict their behavior in the Earth’s interior.

The determination of mechanical properties amounts to evaluate the deformation produced in the structure by a series of applied stresses. From these deformations the elasticity tensor may be determined and, from it, the mechanical stability of the corresponding structure may be studied and all the mechanical properties may be evaluated. Due to the complexity of their experimental

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measurement and its great importance, theoretical methods aimed to their determination have been developed in the last decades.

Therefore, the elasticity tensor of a given compound provides a complete description of the response of the material to external stresses in the elastic limit (Nye, 1985). The analysis of this tensor, which is usually correlated with many mechanical properties, also helps to understand the nature of the bonding in the material.

3.2.9.1 Mechanical Properties and stability

At the optimized structures relaxed with respect to Hellmann-Feynman forces, elastic constants can be calculated at T = 0 K as the second derivatives of the energy with respect to the strain, i.e.:

𝐶 =1 𝑉

𝜕 𝑈

𝜕𝜀 𝜕𝜀 (3.156) where 𝑉 is the volume, 𝑈 is the total energy of the system, and 𝜀 is the infinitesimal displacement (or infinitesimal strain). They can also be calculated from stress-strain relations. With this purpose, finite deformation technique is employed in CASTEP. In this technique, finite programmed symmetry-adapted strains (Nye, 1985) may be used to extract individual elastic constants from the stress tensor obtained as response of the system to the applied strains. For the calculation of elastic tensor, this stress-based method appears to be more efficient than the energy- based methods and the use of DFPT (Yu et al., 2010).

The Voigt notation, which relates the elastic stiffness coefficients 𝐶 (𝑖, 𝑗, 𝑘, 𝑙 = 𝑥, 𝑦, 𝑧) in the different directions in the crystal to the 𝐶 (𝑖, 𝑗 = 1, . . ,6) elastic constants, was adopted in this study, i.e. 𝑥𝑥 → 1, 1, 𝑦𝑦 → 2, 𝑧𝑧 → 3, 𝑦𝑧 → 4, 𝑧𝑥 → 5, 𝑥𝑦 → 6.

The 6×6 elasticity tensor was determined by performing finite distortions of the equilibrium lattice and deriving the elastic constants from the strain-stress relationship (Le Page and Saxe, 2002). The elastic behaviour of the systems is determined to a great extent by the crystal symmetry of the system and, for example, for orthorhombic and monoclinic crystals the symmetric elasticity tensor is determined by 9 and 13 elastic constants, respectively (Nye, 1985; Weck et al., 2015):

𝐶 = ⎝ ⎜ ⎜ ⎛ 𝐶 𝐶 𝐶 0 0 0 𝐶 𝐶 𝐶 0 0 0 𝐶 𝐶 𝐶 0 0 0 0 0 0 𝐶 0 0 0 0 0 0 𝐶 0 0 0 0 0 0 𝐶 ⎠ ⎟ ⎟ ⎞ (3.157. 𝑎) and, 𝐶 = ⎝ ⎜ ⎜ ⎛ 𝐶 𝐶 𝐶 0 𝐶 0 𝐶 𝐶 𝐶 0 𝐶 0 𝐶 𝐶 𝐶 0 𝐶 0 0 0 0 𝐶 0 𝐶 𝐶 𝐶 𝐶 0 𝐶 0 0 0 0 𝐶 0 𝐶 ⎠ ⎟ ⎟ ⎞ (3.157. 𝑏)

For low-symmetry monoclinic phases, a number of possible formulations of the generic Born elastic stability conditions (Born, 1940; Born and Huang, 1954) for an unstressed single crystal have been proposed. Popular criteria for mechanical stability are given by (Wu et al., 2007; Weck et al., 2015):

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Cii > 0 (𝑖 = 1, …, 6) (3.158. 𝑎) C33 C55 − C35 C35 > 0 (3.158. 𝑏) C44 C66 – C46 C46 > 0 (3.158. 𝑐) C22 C33 – 2 C23 > 0 (3.158. 𝑑) C11 + C22 + C33 + 2 (C12 + C13 + C23)> 0 (3.158. 𝑒) C22 (C33 C55 – C35 C35) + 2 C23 C25 C35 – C23 C23 C55 – C25 C25 C33 > 0 (3.158. 𝑓) 2 (h-i) + g C55 > 0 (3.158. 𝑔) with g = C11 C22 C33 – C11 C23 C23 – C22 C13 C13 – C33 C12 C12 + 2 C12 C13 C23 h = C15 C25 (C33 C12 – C13 C23) + C15 C35 (C22 C13 – C12 C23) + C25 C35 (C11 C23 – C12 C13) i = C15 C15 (C22 C33 – C23 C23) + C25 C25 (C11 C33 – C13 C13) + C35 C35 (C11 C22 – C12 C12) (3.159) While these are necessary (Watt, 1980), they are not sufficient criteria for mechanical stability, as recently noted by Mouhat and Coudert (2014). In particular, the generic necessary and sufficient Born criterion is that all eigenvalues of the 𝐶 matrix be positive. The insatisfaction of the mechanical stability conditions for a given phase is a strong indication of phase metastability (Weck et al., 2015).

For the orthorhombic single crystals, the necessary and sufficient Born criteria for mechanical stability are (Mouhat and Coudert, 2014; Weck et al., 2015):

Cii > 0 (𝑖 = 1,4,5,6) (3.160. 𝑎) C11 C22 − C12 > 0 (3.160. 𝑏) C11 C22 C33 + 2C12 C13 C23 – C11 C23 – C22 C13 – C33 C12 > 0 (3.160. 𝑐) The bulk and shear moduli in the Voigt (Voigt, 1962) approximation, respectively, were calculated for both monoclinic and orthorhombic polycrystalline aggregates using the formulas:

𝐵 =1

9(𝐶 + 𝐶 + 𝐶 + 2 [𝐶 + 𝐶 + 𝐶 ]) (3.161. 𝑎) and,

𝐺 = 1

15(𝐶 + 𝐶 + 𝐶 − 𝐶 − 𝐶 − 𝐶 + 3[𝐶 + 𝐶 + 𝐶 ]) (3.161. 𝑏) While the strain is assumed to be uniform throughout the aggregate of crystals in Voigt’s method, the approximation formulated by Reuss (1929) considers the stress to be uniform and averaging of the relations expressing the strain is carried out. The Reuss methodology was also used to compute the isotropic elastic properties of and polycrystalline aggregates.

The bulk and shear moduli within the Reuss approximation were obtained for monoclinic phases using the expressions:

151 𝐵 = Ω (𝑎 [ 𝐶 + 𝐶 − 2 𝐶 ] + 𝑏 [2 𝐶 − 2 𝐶 − 𝐶 ] + 𝑐 [ 𝐶 − 2 𝐶 ] + 𝑑[2 𝐶 + 2 𝐶 − 𝐶 − 2 𝐶 ] + 2 𝑒 [ 𝐶 − 𝐶 ] + 𝑓) (3.162. 𝑎) 𝐺 = 15 {4 (𝑎 [ 𝐶 + 𝐶 + 𝐶 ] + 𝑏 [𝐶 − 𝐶 − 𝐶 ] + 𝑐 [ 𝐶 + 𝐶 ] + 𝑑 [ 𝐶 − 𝐶 − 𝐶 − 𝐶 ] + 𝑒 [ 𝐶 − 𝐶 ] + 𝑓)/𝛺 + 3 (𝑔/𝛺 + [𝐶 + 𝐶 ]/[ 𝐶 𝐶 − 𝐶 𝐶 ])} (3.162. 𝑏) where, 𝛺 = 2 𝐶 𝐶 ( 𝐶 𝐶 − 𝐶 𝐶 ) + 𝐶 𝐶 ( 𝐶 𝐶 − 𝐶 𝐶 ) + 𝐶 𝐶 ( 𝐶 𝐶 − 𝐶 𝐶 ) − [𝐶 𝐶 ( 𝐶 𝐶 − 𝐶 𝐶 ) + 𝐶 𝐶 ( 𝐶 𝐶 − 𝐶 𝐶 ) + 𝐶 𝐶 ( 𝐶 𝐶 − 𝐶 𝐶 )] + 𝑔𝐶 (3.163. 𝑎) 𝑎 = 𝐶 𝐶 − 𝐶 𝐶 (3.163. 𝑏) 𝑏 = 𝐶 𝐶 − 𝐶 𝐶 (3.163. 𝑐) 𝑐 = 𝐶 𝐶 − 𝐶 𝐶 (3.163. 𝑑) 𝑑 = 𝐶 𝐶 − 𝐶 𝐶 (3.163. 𝑒) 𝑒 = 𝐶 𝐶 − 𝐶 𝐶 (3.163. 𝑓) 𝑓 = 𝐶 (𝐶 𝐶 − 𝐶 𝐶 ) − 𝐶 (𝐶 𝐶 − 𝐶 𝐶 ) + 𝐶 (𝐶 𝐶 − 𝐶 𝐶 ) + 𝐶 (𝐶 𝐶 − 𝐶 𝐶 ) (3.163. 𝑔) and 𝑔 was defined previously.

For polycrystalline aggregates of orthorhombic symmetry, the Reuss bulk and shear moduli were calculated using the expressions:

𝐵 = ∆ ( 𝐶 [ 𝐶 + 𝐶 − 2 𝐶 ] + 𝐶 [ 𝐶 − 2 𝐶 ] − 2 𝐶 𝐶 + 𝐶 [2 𝐶 − 𝐶 ] + 𝐶 [2 𝐶 − 𝐶 ] + 𝐶 [ 2 𝐶 − 𝐶 ]) (3.164. 𝑎) and 𝐺 = 15 {4 (𝐶 [ 𝐶 + 𝐶 + 𝐶 ] + 𝐶 [𝐶 + 𝐶 ] + 𝐶 𝐶 − 𝐶 [ 𝐶 + 𝐶 ] − 𝐶 [ 𝐶 + 𝐶 ] − 𝐶 [ 𝐶 + 𝐶 ])/∆ + 3 (𝐶 + 𝐶 + 𝐶 )} (3.164. 𝑏) where, ∆= 𝐶 ( 𝐶 𝐶 − 𝐶 𝐶 ) + 𝐶 ( 𝐶 𝐶 − 𝐶 𝐶 ) + 𝐶 ( 𝐶 𝐶 − 𝐶 𝐶 ) (3.165) The Voigt and Reuss methods employed above for bulk and shear modulus calculations yield differences which are generally small. However, they may lead to differences seem larger than usually obtained for crystalline systems with strong anisotropy, such as studtite, metastudtite and rutherfordine (Weck et al., 2015; Colmenero et al., 2017a) with structures featuring large differences between elastic constants along different directions.

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As shown by Hill (Hill, 1952). the Reuss and Voigt approximations result in lower and upper limits, respectively, of polycrystalline constants and practical estimates of the polycrystalline bulk and shear moduli in the Hill approximation were computed using the formulas:

𝐵 = 1/2(𝐵 + 𝐵 ) (3.166. 𝑎) 𝐺 = 1/2(𝐺 + 𝐺 ) (3.166. 𝑏) In an effort to assess the uncertainty of the elastic parameters and the derived elastic properties, the bulk modulus may be calculated using fits to an equation of state and comparing the resulting value with the bulk modulus calculated from the elastic constants in the Voigt, Reuss, and Hill approximations.

The ratio 𝐷 = 𝐵/𝐺 of the bulk modulus divided by the shear modulus was proposed by Pugh (1954) as a simple indicator of the correlation between the ductile/brittle properties of crystals and their elastic constants. A material is considered ductile if 𝐷 > 1.75, otherwise it is brittle. As discussed by Pugh (1954) and Frantsevich et al. (1983), the Poisson’s ratio, 𝜈, can also be utilized to measure the malleability of crystalline compounds and is related to the Pugh’s ratio given above by the relation 𝐷 = (3 − 6𝜈)/(8 + 2𝜈). The Poisson’s ratio is close to 1/3 for ductile materials, while it is generally much less than 1/3 for brittle materials. Using the bulk and shear moduli, Poisson’s ratio, 𝜈, can be obtained using the expression:

𝜈 = (3𝐵 − 2𝐺)/[2(2𝐵 + 𝐺)] (3.167) The hardness of these systems is computed according to a recently introduced empirical scheme (Niu et al., 2011; Chen et al., 2011) that correlates the Vickers hardness and Pugh ratio (𝐷 = 𝐵/𝐺). The Vickers hardness, 𝐻, for systems characterized by large 𝐷 ratios, have smaller values. Young’s modulus, corresponding to the ratio of the stress to strain (𝐸 = 𝜎/𝜀), was computed using the formula:

𝐸 = 9𝐵𝐺/(3𝐵 + 𝐺) (3.168) Alternatively, the axial components of Young’s modulus, were derived from the elastic compliances, with its components along the 𝑎, 𝑏, 𝑐 directions expressed as 𝐸 = 𝑆 , 𝐸 = 𝑆 and 𝐸 = 𝑆 . The elastic compliances, 𝑆 , can be readily obtained by inverting the elastic constant tensor, i.e. 𝑆 = 𝐶 .

In order to assess the elastic anisotropy, the shear anisotropic factors for the {100} (𝐴 ), {010} (𝐴 ) and {001} (𝐴 ) crystallographic planes may be computed using the formulas:

𝐴 = 4𝐶 𝐶 + 𝐶 − 2𝐶 (3.169. 𝑎) 𝐴 = 4𝐶 𝐶 + 𝐶 − 2𝐶 (3.169. 𝑏) 𝐴 = 4𝐶 𝐶 + 𝐶 − 2𝐶 (3.169. 𝑐) Alternatively, the percentage of anisotropy in compression and shear was obtained using:

𝐴 =𝐵 − 𝐵

𝐵 + 𝐵 ×100 (3.170. 𝑎)

𝐴 =𝐺 − 𝐺

𝐺 + 𝐺 ×100 (3.170. 𝑏) In terms of the recently introduced universal anisotropy index (Ranganathan and Ostoja- Starzewski, 2008):

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𝐴 = 5 𝐺

𝐺 +

𝐵

𝐵 − 6 (3.171) The acoustic transverse wave velocity, 𝑣 , and longitudinal wave velocity, 𝑣 , can also be derived from the bulk and shear moduli using the formulas:

𝑣 = 𝐺

𝜌 (3.172. 𝑎) and

𝑣 = 3𝐵 + 4𝐺

3𝜌 (3.172. 𝑏) Using the computed acoustic transverse and longitudinal wave velocities, the mean sound velocity, 𝑣 , can be obtained using the expression:

𝑣 = 1 3 2 𝑣 + 1 𝑣 (3.173) and the Debye temperature from:

𝜃 = ℎ 𝑘 3𝑛 4𝜋 𝑁 𝜌 𝑀 𝑣 (3.174) where ℎ is Planck’s constant, 𝑘 is Boltzmann’s constant, 𝑁 is Avogadro’s number, 𝑛 and 𝑀 are the number of atoms and molar mass per formula unit, and 𝜌 is the crystal density. The Debye temperature is related to important properties such as the specific heat or melting temperature of crystalline structures; in particular, at relatively low temperature, 𝜃 calculated from elastic constants or obtained from calorimetric measurements are similar.

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