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  • [object Object] Surendra K. Saxena

This monograph is written for advanced students and research workers in the fields of mineralogy, petrology and physical geochemistry. It introduces the reader to the popular solution theories and the special problems connected with the treatment of heterogeneous phase equilibria involving complex crystalline solutions. The results and discussions in the recent publications of J. B. Thompson, D. Waldbaum, R. F. Mueller, R. Kretz, E. J. Green, S. Ghose and from my work during the last eight years, have been used extensively in the book.

  • [object Object] Surendra K. Saxena

Thermodynamic relations between the concentration of a component in a solution and its chemical potential and other thermodynamic functions of mixing are presented here. The details of the simplifying assumptions and the methods of statistical thermodynamics have been given by Denbigh (1965), Guggenheim (1952, 1967), and Prigogine and Defay (1954), among others. Recently Thompson (1967) also considered the properties of simple solutions. Besides a summary of thermodynamic relations in binary, ternary, and multicomponent solutions, the difficulties encountered in their application to silicate minerals will be considered. Some of these problems, such as the choice of a component and definition of its chemical potential in a silicate, have been discussed by Ramberg (1952a, 1963), Kretz (1961), and Thompson (1969).

  • [object Object] Surendra K. Saxena

Composition of coexisting minerals occurring in rocks or in experiments are the main source of data to be used in obtaining the information on the thermodynamic behaviour of silicate crystalline solutions. It is, therefore, necessary to use certain solution models to relate the observed compositional variables to the thermodynamic functions of mixing. Although such models are based on specific statistical theory and employ certain assumptions regarding the molecular forces in the crystalline lattice, the equations obtained for the thermodynamic functions of mixing are mathematically equivalent to those of other mathematical models which are not bound to any special physical interpretation. The regular solution model of Guggenheim (1952) is considered in detail in this chapter.

  • [object Object] Surendra K. Saxena

For binary or multi-component crystalline solutions, the stability is determined by the diffusion processes. Above a certain temperature two end member crystals may form a complete crystalline solution series. At lower temperatures, the solution may become unstable in a certain range of the composition. Figure 1 shows a miscibility gap in a binary crystalline solution (A, B)M. Above Tc , the critical temperature of unmixing, the solution is continuous between AM and BM and irrespective of whether AM and BM are structurally similar or dissimilar, one can pass from an AM rich phase to a BM rich phase without any observable break or transition point. Below Tc there are two coexisting phases both crystalline solutions one rich in AM and the other rich in BM. At the critical solution point C the two phases become identical.

  • [object Object] Surendra K. Saxena

Experimental data on the distribution of a component between two coexisting crystalline solutions at a fixed P and T for systems such as olivine and pyroxene have been collected by Nafziger and Muan (1967), Larimer (1968), and Medaris (1969). Distribution data are also available for natural assemblages, but the P and T of their formation are indefinite. The distribution data from natural assemblages in many cases may be found to represent ion-exchange equilibrium closely. If precise P and T are not important, such data may be used to obtain useful information on the thermodynamic nature of mixing in the minerals. For this purpose, the thermodynamic equations according to various solution models for binary solutions presented in this chapter may be used.

  • [object Object] Surendra K. Saxena

An experimental measurement of activities of components in a crystalline solution, particularly the silicates, is beset with difficulties and the measured values are subject to large errors. Therefore, obtaining such activity-composition relations from phase diagrams would be very convenient.

  • [object Object] Surendra K. Saxena

Long range order-disorder phenomena in silicates differ from those in alloys in several important respects. First, as opposed to alloys, silicates contain structural sites with definite polyhedral geometries and order-disorder is noticed whenever an ion occupies two or more sites which differ in their polyhedral shape and size. Second, usually only the ions which occupy the nonequivalent sites, take part in the order-disorder and the remaining silicate framework remains more or less inert. Third, when the ions are sufficiently alike in their charge and size, the site preference energies (corresponding to the difference in binding energy of the ion between the nonequivalent sites) are not strongly dependent on the degree of order as is usual in many binary alloys. Fourth, complete order or disorder in silicates is generally not possible because the variation in composition in binary solution would require that some of the ions of one species must inevitable occupy some of the sites belonging to the other species and also because there may be potential barrier for the ordering process to continue below a certain temperature. At the high temperature side there is usually a phase transformation or melting before complete disorder can be attained.

  • [object Object] Surendra K. Saxena

Orthopyroxene is one of the few important rock-forming minerals that can be considered as quasi-binary without significant loss of accuracy. Usually more than 95% of the mineral is a crystalline solution of the end members enstatite (MgSiO3) and ferrosilite (FeSiO3). Fe2+ and Mg2+ are distributed between two nonequivalent sites M 1 and M 2. With the use of X-ray or Mössbauer spectroscopic technique, it is possible to determine the proportion of Fe2+ in the two nonequivalent sites (Evans, Ghose and Hafner, 1967). These data can be used with the help of suitable solution models to determine the thermodynamic properties of the solution (Virgo and Hafner, 1969; Saxena and Ghose, 1971).

  • [object Object] Surendra K. Saxena

Olivines (Fe, Mg, Ca)2Si2O4 are important constituents of many igneous and metamorphic rocks. The thermodynamic properties of olivines have been studied recently by several workers (Kitayama and Katsura, 1968; Nafziger and Muan, 1967; Olsen and Bunch, 1970; Warner, 1971). Bowen and Schairer (1935) calculated the molal heats of fusion for forsterite (Mg2SiO4) and fayalite (Fe2SiO4). Sahama and Torgesson (1949) determined the heat of solution in HF for the crystalline solution series. The thermodynamic nature of the (Mg-Fe) olivine crystalline solution appears to be closely ideal at high temperatures from the data of Bowen and Schairer and from the cation distribution data (see for example Olsen and Bunch, 1970; Medaris, 1969). However, Nafziger and Muan (1967) and Kitayama and Katsura (1968) note a definite deviation from ideality in olivines at 1200° C. The results of the various workers are reviewed and some new calculations on activity-composition relations are presented in this chapter.

  • [object Object] Surendra K. Saxena

Although important information on crystal structure and experimental phase relationship has been accumulating for decades, it is only recently that attempts have been made to use such information in understanding the thermodynamic properties of the pure end member feldspar or their crystalline solutions. Allmann and Hellner (1962), Perchuk (1965), Perchuk and Ryabchikov (1968), Thompson and Waldbaum (1968a, 1968 b, 1969), and Thompson (1969) have discussed and calculated the energy functions of feldspar solutions. Waldbaum (1968) has calculated the thermodynamic properties of alkali feldspars while Holm and Kleppa (1968) and Brown (1971) have calculated the ideal configurational entropies for plagioclase. Most of the energy calculations for the feldspar crystalline solution have been based on the experimental data on phase relationships as collected by Orville (1963), Bowen and Tuttle (1950) and Luth and Tuttle (1966). Recently Orville (1972), Bachinski and Muller (1971) and Seck (1971a) have published more experimental data on binary and ternary feldspars.

  • [object Object] Surendra K. Saxena

The concept of metamorphic facies evolved through the attempts of the petrologists to distinguish the mineral assemblages formed at different P and T in the field. In several cases, the physical and chemical conditions of the formation of rocks have been simulated in the laboratory. However, the meagerness of relevant thermodynamic data on rock-forming minerals makes it difficult to interpret meaningfully and check the validity of most experiments. From existing knowledge, it is possible to obtain only certain qualitative to semiquantitative estimates of P and T of the formation of a mineral assemblage. Such methods are based on the knowledge of the chemical reactions that occur as a result of changing P and T within and in between the crystalline solutions.

... This is not surprising given the large difference in the ionic radius between F (or OH) and Cl (i.e., F À : 1.33 Å , OH À : 1.34 Å , Cl À : 1.81 Å ; Shannon and Prewitt, 1969;White et al., 2005), which can result in very different interactions between these anions and their neighboring atoms (e.g., Ca and O; Hughes and Rakovan, 2015;Hughes et al., 2014;Hughes et al., 2016;Hughes et al., 2018). Non-ideal mixing has been found in several multicomponent minerals (Saxena, 1973), such as feldspars (e.g., Thompson and Hovis, 1979;Ghiorso, 1984;Holland and Powell, 2003), and garnet (e.g., Ganguly and Saxena, 1984;Ganguly et al., 1996), but has been poorly constrained for the binary/ternary apatite at magmatic temperatures (e.g., >700°C). ...

... The activity coefficient of one component in a binary/ ternary solution can be expressed using the Gibbs freeenergy interaction parameter W G (Hildebrand, 1929;Wohl, 1946). The simplest model is to assume a simple symmetric mixture, where for a A-B binary solution at such condition, the W G s of A-B and B-A are equal to each other, and both expressed as W G A;B (Saxena, 1973;Anderson and Crerar, 1993). Similarly, for a simple symmetric ternary solution of A-B-C, the W G s along the three binaries can be expressed as W G A;B , W G A;C and W G B;C , respectively. ...

... A;B in a simple symmetric mixture can be expressed using the excess mixing Gibbs free energy (D m G ex Guggenheim, 1967), whose value is usually derived from calorimetric measurements of the excess mixing enthalpy and entropy (Haselton et al., 1983). For an ideal solution, D m G ex A;B = 0 so thatW G A;B = 0, whereas for non-ideal solutions, both D m G ex A;B andW G A;B differ from zero (Saxena, 1973 ...

The abundance and composition of volatiles in subvolcanic melts play a key role in controlling eruptive styles of volcanoes, but they are difficult to determine directly due to volatile loss during magma transport to the surface and eruption. Most constraints on volatile abundances are obtained by studying melt inclusions in minerals, but not all samples contain suitable inclusions, and they can be modified by a range of post-entrapment and re-equilibration processes. Apatite incorporates several volatile elements such as F, Cl, H, C and S into its structure, and thus has been proposed as an alternative tool for tracking the melt volatile contents. However, application of the apatite approach replies on the partitioning of volatiles between apatite and silicate melts, which has been found to show non-Nernstian behaviour but yet to be quantified. Here we propose a thermodynamic model that considers the non-ideal mixing in apatite solution, and includes the interaction parameters (WG) and Gibbs free energy properties calculated by regressing experimental data from the literature. We find that WG for the Cl-F binary join is larger than those for the Cl-OH, and F-OH joins, indicating a stronger non-ideality. We propose two equations for calculating the exchange coefficients (KD) between apatite (Ap) and silicate melts as: lnKDOH-FAp-melt=-1RT×{94,600±5600-40(±0.1)×T-1000×7±4×XFAp-XOHAp-11±7×XClAp} and lnKDOH-ClAp-melt=-1RT×{72,900±2900-34(±0.3)×T-1000×[5±2×XClAp-XOHAp-10(±8)×XFAp]} where temperature (T) is in kelvins, apatite compositions are expressed in mole fractions (XiAp), and R is the universal gas constant. With the two equations above, we established a calculation procedure for estimating the water concentrations in the melt, and have developed it into an online calculator (https://apthermo.wovodat.org/). Application of this method to volcanic apatite from the literature (e.g. from Pinatubo, Campi Flegrei, Santiaguito and Augustine) gives melt water concentrations that are equivalent or higher than those measured from melt inclusions. Our new calibrations of the exchange coefficients allow us to obtain more robust estimates of melt volatile budgets, which provide more insights into the effects of volatiles on a variety of volcanic and plutonic phenomena.

... Glass forming studies have tackled immiscibility from experimental studies, mainly within a thermodynamic approach [73]. In such terms, liquid immiscibility, under constant pressure and temperature, depends on the variation of the Gibbs free energy with the melt composition [72]. It is the enthalpy of mixing minus the entropy of mixing. ...

  • jean louis Vigneresse jean louis Vigneresse

The notion of fluids immiscibility is examined from the field observations (i.e. rock samples or measurements in drill holes) and experimental point of view, as well as from usual thermodynamics and theoretical hard-soft and acid–base and density functional theory concepts. Basically, fluids immiscibility relates to a difference in chemical potentials between the elements in presence, i.e. some differences in the first derivative of their free energy. The latter usually plots as a quartic curve with upward concave curvature. However, it can develop a local maximum, inducing immiscibility. The theoretical chemistry concepts consider in addition to chemical potential, the second derivatives of the energy, as hardness and electrophilicity, or polarizability and magnetizability. Those are due to external field components, electric or magnetic, whereas the two former derivatives depend directly on the quantitative changes in the electronic cloud. Those descriptors give insights to reactions evolution during the reaction path. Their introduction in the definition of fluids immiscibility is examined, documenting chemical reactions and path reflecting an open system. Mapping the values of molecule polarizability shows a linear trend from silica to fayalite and leucite, i.e. magnesian and alkali-alumino silicates. A local positive anomaly in polarizability, due to the near-by influence of phosphorous controls the extent of the immiscibility area. The technique could easily be applied in other fields to control hydrophilicity or hydrophobicity.

... The mathematical expression for the configurational entropy for the hollandite system is complicated due to the coupled nature of the solid solution, which includes two-site mixing (i.e., mixing on both A and B sites) and an interdependency of the A-site and B-site composition. Following the derivation for the configurational entropy for the plagioclase [(Ca 1−x Na x )(Al 2−x Si 2+x )O 8 ] coupled-substitution solid solution 25 , the configurational entropy for the 1 × 1 × 1 unit cell of hollandite, with the number of mixing components N = 4, is defined in Equation 2. . The configurational entropy is divided into two sets of terms, the first describing the A-site mixing and the second describing the B-site mixing, where the total number of occupied A-site cations is 1.33 and the total number of B-site cations is 8. ...

The titanate-based hollandite structure is proposed as an effective ceramic waste form for Cs-immobilization. In this study, quantum-mechanical calculations were used to quantify the impact of A-site and B-site ordering on the structural stability of hollandite with compositions BaxCsy(MzTi8-z)O16, where M = Zn2+, Ga3+, and Al3+. The calculated enthalpy of formation agrees with experimental measurements of related hollandite phases from melt solution calorimetry. Ground state geometry optimizations show that, for intermediate compositions (e.g., CsBaGa6Ti18O48), the presence of both Cs and Ba in the A-site tunnels is not energetically favored. However, the decay heat generated during storage of the Cs-containing waste form may overcome the energetics of Ba and Cs mixing in the tunnel structure of hollandite. The ability of the hollandite structure to accommodate the radioparagenesis of Cs to Ba is critical for long term performance of the waste. For the first time, B-site ordering was observed along the tunnel direction ([001] zone axis) for the Ga-hollandite compositions, as well as the intermediate Al-hollandite composition. These compositionally dependent structural features, and associated formation enthalpies, are of importance to the stability and radiation damage tolerance of ceramic waste forms.

The fractionation of Ni isotopes during Ni coprecipitation with calcite was measured at pH = 6.2 and pCO2 = 1 atm as a function of calcite growth rate. Light Ni isotopes are preferentially incorporated into calcite during its coprecipitation, which is likely due to a longer Ni-O bond length in calcite compared to that of the Ni aquo complex. The extent of Ni isotope fractionation between Ni in the solid and the aqueous fluid phase increases from -0.3 to -0.9 ‰ as the calcite growth rate slows from 10-7.3 to 10-8..3 mol m⁻² s⁻¹. This behaviour can be attributed to the strong hydration of the Ni²⁺ aqueous ion. As mineral growth rates depend strongly on the degree of supersaturation of the fluid relative to the mineral, the results of this study suggest that the Ni isotopic composition of natural calcite can potentially provide insight into the saturation state of seawater with respect to calcite at the time that this mineral formed. In addition, calculations based on our results suggest that the incorporation of Ni into calcite could be a significant sink of light Ni in the ocean.

  • Gautam Sen Gautam Sen

This chapter is concerned with metamorphic reactions that produce new minerals at the expense of reactant minerals in the protolith. Many of these reactions are useful thermometers and barometers. It is possible for the metamorphic petrologist to deduce the evolutionary history of a metamorphic complex in terms of pressure–temperature–time (or P–T–t) path. These topics and a brief presentation on metamorphic rocks are made in this chapter.

  • Vadim S. Urusov Vadim S. Urusov

The golden age of classic crystal chemistry (1920s–30s) yielded many well-known empirical rules and generalizations concerning the formation of solid solutions or isomorphous mixtures (mixed crystals). Among them are Vegard's rule of additive dependence of lattice spacings on composition, Goldschmidt and Hume-Roseri's rules of maximal 15% difference of ionic or atomic radii for the existence of wide miscibility, Goldschmidt and Fersman's rules of substitution polarity (in relation to sizes and charges of the ions replacing each other), and the criteria of proximity of polarizabilities or electronegativities of substituents, etc. In the sections that follow we will return to an analysis of these rules from a more sophisticated and modern point of view.

  • Charles R. Ross II

A phase transition has been defined as "the phenomenon that a system exhibits a qualitative change at one sharply defined parameter value [the critical point], if that value is changed continuously" (Stauffer, 1979). What the "parameter" and "change" are depends upon the system that is examined; a classic example is the spontaneous magnetization of magnetite (Fe3O4) as a function of temperature. Here the critical point (the Curie temperature) is the temperature at which (with increasing temperature) spontaneous magnetization is lost.

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