To install StudyMoose App tap and then “Add to Home Screen”
Save to my list
Remove from my list
Grossular garnet is a mineral of significant importance in the Earth's upper mantle and mantle change zone. This paper explores the thermodynamic properties of grossular garnet using equations of state. Various equations of state, including the Murnaghan and Birch-Murnaghan models, are discussed to understand the behavior of grossular garnet under high-pressure and high-temperature conditions. Additionally, the compositional dependence of thermal expansion and elastic moduli of grossular garnet is investigated. The results provide valuable insights into the properties of this mineral, which is essential in petrogenetic processes and subduction zones.
Garnets are essential constituents of the Earth's upper mantle and mantle change zone, playing a vital role in high-pressure and high-temperature petrogenetic processes.
They are also significant components of subducted oceanic crust. Among the various types of garnets, grossular garnet (Ca3Al2Si3O12) is known for its diverse coloration. Grossular garnet is often found in contact-transformed limestones, associated with minerals like vesuvianite, diopside, wollastonite, and wernerite. Additionally, a derivative of grossular, known as hydrogrossular, is used as a carving gemstone.
The equation of state (EOS) relates the volume (V), temperature (T), and pressure (P) in a state of thermodynamic equilibrium, given by the equation f(P,V,T) = 0. An EOS is a fundamental thermodynamic relationship describing the state of matter under specific physical conditions.
Verified writer
Proficient in: Physics
4.7 (657)
“ Really polite, and a great writer! Task done as described and better, responded to all my questions promptly too! ”
It is typically developed for pure substances and can be extended to mixtures using appropriate mixing rules.
A well-defined EOS allows for the calculation of various thermodynamic properties using limited experimental data.
Historical developments in EOS include Boyle's law (1662) and the more recent Mie-Gruniesen and Birch-Murnaghan models, which are widely used for solids. The Mie-Gruniesen model, developed by Gustav Mie in 1903, provided an intermolecular potential for high-temperature equations of state in solids. In 1912, Eduard Gruniesen extended this model to temperatures below the Debye temperature, giving rise to the Mie-Gruniesen EOS.
Murnaghan's EOS, introduced in 1944, is based on the conservation of mass, Hooke's law for small stress variations, and the assumption that the bulk modulus is linear with respect to pressure. This assumption has been validated at low pressure through comparisons with experimental data. The Murnaghan EOS has been further refined, including higher-order Taylor expansions with pressure, referred to as the modified Murnaghan equation. However, due to the complexity of higher-order derivatives and experimental limitations, the most common form remains the linear term truncation.
Murnaghan noted that the bulk modulus (K) is linear in pressure (P) for high compressions:
K(P) = K0 + K0'P
The Birch-Murnaghan equation relates volume (V) and pressure (P) and can be expressed as:
PV = B0 + (B0'/2)(V - V0)2
Gustav Mie's intermolecular potential, extended by Eduard Grüneisen, introduced the Grüneisen model, which relates pressure (p), volume (V), the Grüneisen parameter (Γ), and internal energy (e):
pV = -Γe
The integrated Grüneisen model incorporates internal energy and pressure at absolute zero (0K):
pV = -Γ(e - e0)
The Mie-Grüneisen relationship describes the vibrational effect of changes in the crystal lattice volume:
ρ0/ρ = [1 - Co2(1 - Γo)](Up/Us)s
Donald G. Isaak, Orson L. Anderson, and Hitoshi Oda (1992) presented new high temperature elasticity data on two grossular garnet specimens. One specimen was single-crystal, of nearly endmember grossular, the other is polycrystalline with about 22% molar andradite. Their data extended the high temperature regime for which any garnet elasticity data was available from 1000 to 1350 K and the compositional range of temperature data to near endmember grossular. They also present new data on the thermal expansivity of calcium-rich garnet. No differences were found in the temperature T derivatives at ambient conditions of the isotropic bulk Ks and shear moduli when comparing results between these two specimens. Small, but measurable, nonlinear temperature dependences of most of the elastic moduli were observed. Several dimensionless parameters were computed with the new data and used to illustrate the effects of different assumptions on elastic equations of state extrapolated to high temperatures. Their results had bearing on the amount of diopside required to explain the shear velocity gradients in Earth's transition zone.
Steeve Gréaux et al. (2010) examined the thermoelastic parameters of synthetic Ca3Al2Si3O12 (Grossular Garnet) in situ at high-pressure and high-temperature by energy dispersive X-ray diffraction, using a Kawai-type multi-anvil press apparatus coupled with synchrotron radiation. Measurements have been conducted at pressures up to 20 GPa and temperatures up to 1,650 K: this P, T range covered the entire high-P, T stability field of grossular garnet. By fitting their P–V–T data by means of the high-temperature third order Birch–Murnaghan or the Mie–Grüneisen–Debye thermal equations of state, gives the thermoelastic parameters. From the comparison of the two different approaches, they proposed to constrain the bulk modulus of grossular garnet.
Yoshio Kono et al. (2010) conducted a study in which simultaneous ultrasonic elastic wave velocity and in situ synchrotron X-ray measurements on grossular garnet were carried out up to 17 GPa and 1650 K. P- and . These data yielded a pressure derivative of the bulk modulus and shear modulus were in good agreement with those of garnets with variable chemical compositions. Temperature dependence of the bulk modulus of grossular is similar to that of other garnets, while the temperature dependence of the shear modulus of grossular is higher than those of magnesium end-member garnets and pyrolitic garnet.
Kenji Kawai and Taku Tsuchiya (2012) studied the effect of temperature and pressure on grossular garnet in 2012 using the first principles computation method, it was found to dissociate into an assemblage of CaSiO3 Ca-perovskite (Pv) and Al2O3 corundum (Cor) at about 23.4 GPa, accompanied by remarkable jumps of compressional wave (8.0%), shear wave (11.6%), bulk sound (5.7%) velocities, and density (12.1%). The equation of state parameters (zero-pressure bulk modulus B0 and its pressure derivative B0′) were determined by fitting the data to the thirdorder Birch-Murnaghan equation. The parameters B0 and B0′ were in good agreement with experimental data. [Pavese et al., 2001; Zhang et al., 1999; Olijnyk et al., 1991; Kono et al., 2010].
Wei Du et al. (2015) conducted a study in which unit cell parameters of a series of synthetic garnets with the pyrope, grossular, and four intermediate compositions were measured up to about 900K and to 10 GPa using synchrotron X-ray powder diffraction. Coefficients of thermal expansion of pyrope-grossular garnets uniformly increase with temperature. Values for the two end members pyrope and grossular are identical. Bulk modulus of grossular Κ0 (with fixed Κ0′, the pressure derivative of the bulk modulus) and bulk modulus of pyrope Κ0(with fixed Κ0′) using a third order Birch-Murnaghan equation of state, was found to be consistent with previously reported values. The compositional dependence of bulk modulus resembles the compositional dependence of thermal expansion. It was found that excess volumes in the pyrope-grossular series remained relatively large even at high pressure (~6GPa) and temperature (~800K), supporting the observation of crystal exsolution on this garnet join.
S. Milani et al. (2017) conducted a study on thermoelastic behavior of grossular garnet at high temperature and pressure and the results were divided into four sets of data: Temperature-Volume data, Temperature-Volume-Bulk Modulus data, Pressure-Volume data and P-V-T-K and PVT equation of state. This data was analyzed and the results were obtained. A continous increase in unit cell volume is observed as a function of temperature with no evidence of any irreversible change. T-V data and T-Ks data from Isaak et al.(1992) of grossular yielded identical results as those obtained from T-V-K data. This implies that the assumption γ is a constant is reasonably valid. The unit cell volume of grossular decreases smoothly with increasing pressure upto a maximum hydrostatic pressure of 7.5 Gpa. Such pressure covers the stability pressure range of upper mantle.
Jay D. Bass et al. (1989) A garnet structure is most likely to bear aluminium in its phases. Thus the elastic properties of the garnet have stood interesting to constrain the composition of earth’s mantle. The grossular (Gr) was simply labelled 'pastel grossular' and is of unknown origin. The appearance and the infrared (IR) spectrum of the grossular are similar to those of African specimens from Kenya and Tanzania (G. Rossman, personal communication, 1988). The elastic moduli of grossular garnet estimates the properties like lattice parameter, refractive index and density from several recent studies. A hydrogrossular component would decrease the refractive index and increase the lattice parameter of grossular relative to the pure grossular end-member; no evidence was found for an appreciable hydrous component on the basis of the measurements. The single crystal elastic properties of the grossular at 1 atm pressure and 220C temperature by Brillouin spectroscopy are accurate within better than ±1% except for mixed modulli.
B.J. Hensen et al (1975) The properties of mixed garnet solid solutions have been studied by many authors and some have experimental evidences . The composition activity relationships of garnets is of major interest to understand the activity coefficient and interaction coefficient of grossular garnet. Values of activity coefficient for garnets with 10-12 mole % grossular have been obtained at 10000 ,1100o,12000 and 13000 C at pressures between 15 and 21 Kb. This data is consistent with the regular solid model for Grossular-pyrope solid solutions. The interaction parameter (W) is inversely proportional to temperature and is given by W= 7460-4.3 T cals (T in K). Pyrope-grossular solid solutions shows a significant positive deviation from ideality.
Mu chi and J. Michael Brown (1997) et.al Laser induced phonon spectroscopy has been used for determining the elastic constants and equation of state at 250C temperature of a natural garnet to a pressure of 20 GPa. The moduli of garnet linearly changes with pressure. The adiabatic bulk modulus in GPa is given by K = 170.8 + 4.09 P and the shear modulus is = 94.7 + 1.76P. The derivative of the shear modulus with respect to temperature at constant density is derived from the temperature and pressure dependencies of the elastic constants. Murnaghan equation with po = 3.810 g/cm3 , KT =169.5 GPa, K’ = 4.09, i.e. p = 3.810 (l +0.024 described the pressure dependence of density. All elastic constants of a pyrope rich garnet are linear in accordance with pressure within experimental limits of 20Gpa. Since the bulk modulus is only a function of density, the pressure dependence of thermal expansion is estimated; a is reduced by 32% at 20 GPa. The shear modulus has a small constant-density temperature derivative at 1 bar.
Suzuki and Anderson (1983) et.al reported the constant pressure derivatives and the coefficients of thermal expansion at 1 bar and 298 K, a = 2.55x10-5 K-1 and at 993 K, 3.31x10-5 K-1. At 20 GPa thermal expansion is reduced 32% from its 1 bar value. It is the most striking feature of the data which shows the small pressure and temperature dependence of the tabulated derivatives.
Robert M. Hazen and Larry W. Finger (2008) et.al studied that variations in the structure and physical properties of garnet with pressure helps in framing the model of earth’s interior. On the basis of empirical compression equation, Hazen suggested a silicon tetrahedral modulus of 8000 kbar. Pyrope shows inverse relationship between structural changes with temperature and pressure. The crystal structures and compressibilities of the end-member garnets pyrope (synthetic Mg3A12Si30112 at 1 bar and 16, 31, 43, and 56 kbar) and grossular (natural Ca3A12Si3012 at 1 bar and 19, 35, and 61 kbar) have been determined from three-dimensional, single-crystal X-ray data. Cubic unit-cell dimensions for pyrope a = 11.456 at 1 bar and 11.332A at 50 kbar; for grossular a 11.846A at 1 bar and 11.720A at 50 kbar. Bulk moduli of pyrope and grossular are both approximately 1350 kbar, which agrees with previous static compression data but is significantly lower than elastic constant data.
Table 1. Comparison of lattice parameters of grossular garnet at ambient P, T conditions
| S.No. | Grossular Composition | V0(Å3) | References |
|---|---|---|---|
| 1 | Synthetic Ca3Al2Si3O12 | 1661.6(1) | Steeve Gréaux et al. (2010) |
| 2 | (Ca2.96Mn0.04)(Al10.95Fe0.05)Si3O12 | 1661.9(4) | Novak and Gibbs (1971) |
| 3 | (Ca2.96Mn0.04)(Al10.95Fe0.05)Si3O12 | 1662.3(8) | Meagher (1975) |
| 4 | Natural Ca3Al2Si3O12 | 1664.2(2) | Sawada (1999) |
| 5 | Synthetic Ca3Al2Si3O12 | 1664.1(2) | Rodehorst et al. (2002) |
| 6 | Synthetic Ca3Al2Si3O12 | 1664 (2) | Yoshio Kono et al. (2010) |
| 7 | Ca3Al2Si3O12 + 0.5 wt% FeO | 1662.3(4) | Hazen and Finger (1978) |
| 8 | Natural Ca3Al2Si3O12 | 1660.2(3) | Zhang et al. (1999) |
| 9 | (Ca2.95Mg0.04Fe0.01)(Al1.97Ti0.02Mn0.01)Si2.99O12 | 1663.6 | Conrad et al. (1999) |
| 10 | Ca3Al2Si3O12 | 1666.9 | Nobes et al. (2000) |
| 11 | (Ca2.90Fe0.10)(Al1.95Ti0.04Mn0.01)Si2.99O12 | 1666.4(4) | Pavese et al. (2000) |
