The nonlinear elasticity and related properties of anhydrous and hydrous γ-Mg2SiO4: a theoretical study

Abstract

The second-order elastic constants up to 30 GPa, which encompass the stability field of the spinel forms, their pressure derivatives and the third-order elastic constants of both hydrous and anhydrous γ-Mg2SiO4 have been obtained theoretically. A combination of deformation theory and finite strain elasticity theory has been employed to arrive at the expressions for second-order and third-order elastic constants from the strain energy of the lattice. The strain energy is calculated by taking into account the interactions up to second nearest neighbours in the γ-Mg2SiO4 lattice. This is then compared with the strain-dependent lattice energy from continuum model approximation to obtain the expression of elastic constants. The second-order elastic constants C ij compare favourably with the measurements in the case of anhydrous as well as hydrous γ-Mg2SiO4 and with other calculations on the anhydrous phase. All the third-order elastic constants of both the compounds are negative. The third-order elastic constant C144(−52.41 and −45.07 GPa for anhydrous and hydrous γ-Mg2SiO4, respectively) representing the anisotropy of shear mode has a smaller value than C111 (−2443.94 and −2101.25 GPa for anhydrous and hydrous phases, respectively), which corresponds to the longitudinal mode. The pressure-induced variations in the longitudinal elastic constants (i.e.,dC11/dp) are relatively large (4.08 and 4.09 for dry and hydrous ringwoodite, respectively) compared with those for the shear (0.22 and 0.32 for dry and hydrous ringwoodite, respectively) and off-diagonal constants (1.40 and 1.41 for dry and hydrous ringwoodite, respectively). The variation of the shear moduli Cs and anisotropy factor A with pressure have also been studied. The average value of elastic anisotropy is 0.835 in the case of anhydrous γ-Mg2SiO4 and 0.830 in the hydrous phase. The reversal of sign of the Cauchy pressure C12 – C44, which describes the angular character of atomic bonding in metals and other compounds, at around 21 GPa for both the compounds may be a precursor to the phase transition from ringwoodite to periclase and perovskite at an elevated temperature. The aggregate elastic properties like the adiabatic bulk modulus K (175.4 and 150.2 GPa for anhydrous and hydrous phases, respectively), and the isotropic compressional (P) and shear (S) wave velocities were calculated and the mode Gruneisen Parameters (GPs) of the acoustic waves were determined based on the quasi-harmonic approximation. The low temperature limit % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqyaaaaaaaaWqVb % xEG8MzGieBHn2Aa8qabaGafq4SdCMbaebacaGGOaGaaGimaiaacMca % aaa!3E73! $$\bar \gamma (0)$$ of both hydrous and anhydrous phases of γ-Mg2SiO4 are positive (1.69 and 1.78, respectively, for hydrous and anhydrous phases) and hence we expect the thermal expansion to be positive down to absolute zero. The Anderson–Gruneisen parameter δ obtained for hydrous as well as anhydrous phases of γ-Mg2SiO4 from the second-order and third-order elastic constants are 2.30 and 2.29, respectively.