Erika J. Palin

Computational methods for the study of energies of cation distributions: applications to cation-ordering phase transitions and solid solutions

Abstract The structural and thermodynamic properties of minerals are strongly affected by cation site-ordering processes. We describe methods to determine the main interatomic interactions that drive the ordering process, which are based on parameterizing model Hamiltonians using empirical interatomic potentials and/or ab initio quantum mechanics methods. The methods are illustrated by a number of case study examples, including Al/Si ordering in aluminosilicates, Mg/Ca ordering in garnets, simultaneous Al/Si and Mg/Al ordering in pyroxenes, micas and amphiboles, and Mg/Al non-convergent ordering in spinel using only quantum mechanical methods.

Monte Carlo methods for the study of cation ordering in minerals

Abstract This paper reviews recent applications of Monte Carlo methods for the study of cation ordering in minerals. We describe the program Ossia99, designed for the simulation of complex ordering processes and for use on parallel computers. A number of applications for the study of long-range and short-range order are described, including the use of the Monte Carlo methods to compute quantities measured in an NMR experiment. The method of thermodynamic integration for the determination of the free energy is described in some detail, and several applications of the method to determine the thermodynamics of disordered systems are outlined.

Monte Carlo simulations of ordering of Al, Fe, and Mg cations in the octahedral sheet of smectites and illites

Abstract The ordering of Al3+, Fe3+, and Mg2+ cations along the octahedral sheet in dioctahedral 2:1 phyllosilicates was studied theoretically. The distribution of Fe3+/Mg2+ was studied in the octahedral sheet and is compared with the Al3+/Fe3+ and Al3+/Mg2+ distributions. The cation exchange interaction parameters Jn, as first, second, third, and fourth nearest neighbors were calculated by means of empirical interatomic potentials. Several compositions with different interlayer cations, and tetrahedral charge close to ferric smectites, illites, and nontronites were studied. From these Jn values, a trend to form FeMg pairs was observed in the Fe/Mg system. Monte Carlo (MC) simulations based on the previously calculated cation exchange potentials Jn of these systems showed ordering phase transitions in the distribution of the octahedral cations, with different ordering patterns in each case. The two-species model was extended to a three-species ordering MC simulation model. A new procedure to study the ordering of three species is presented in this paper. We present for the first time a theoretical study of the ordering of three octahedral cations Al3+, Fe3+, and Mg2+ in clays, describing compositions more realistic for dioctahedral clay minerals, by means of Monte Carlo simulations based only on atomistic models. Short-range ordering of Fe was found in compositions of smectites and illites reproducing experimental cation distribution patterns.

A computational investigation of the Al/Fe/Mg order-disorder behavior in the dioctahedral sheet of phyllosilicates

Abstract In previous papers, we investigated via Monte Carlo simulation the order-disorder behavior of an individual octahedral phyllosilicate sheet, with respect to two-species systems Al/Fe, Al/Mg, and Fe/Mg, and some three-species systems Al/Fe/Mg that were relevant to clay compositions found in nature. We have extended the work on Al/Fe/Mg systems to include a wide range of different octahedral compositions that can represent different natural and synthetic clay minerals, by means of Monte Carlo simulations based only on atomistic models. In many cases, phase transitions do not occur, in that long-range order is not attained, but most systems exhibit short-range order at low temperatures. The ordering of the octahedral cations is highly dependent on the cation composition.

A Monte Carlo investigation of the thermodynamics of cation ordering in 2-3 spinels

Abstract The Monte Carlo (MC) simulation technique is a powerful tool for the investigation of thermodynamic and kinetic phenomena in minerals, and is especially well suited to the study of cation ordering. We have performed MC simulations of eight end-member 2-3 spinels (X2+ = Mg, Fe, Zn, Ni; X3+ = Al, Fe) using pair interaction parameters, Ji, and chemical potentials, μ, derived from atomistic simulations. The Ji values for all but one of these spinels are remarkably similar, despite their different character (normal vs. inverse). The sign of μ, and hence the tendency to form a normal or inverse spinel, was correctly predicted in all cases. Agreement between the simulated and observed cation distributions as a function of temperature is good for the normal spinels and poor for the inverse spinels. Agreement could be greatly improved for the inverse spinels through relatively modest adjustments to the simulation parameters (usually increasing the strength of the tetrahedral-octahedral, T-O, interactions, and decreasing the magnitude of μ). We have developed an atomistic random-mixing model for cation ordering in spinels and compared it with the macroscopic O’Neill-Navrotsky model. In so doing, we have determined the relative contributions of μ, tetrahedral-tetrahedral (T-T), octahedral-octahedral (O-O), and T-O interactions to the O’Neill-Navrotsky coefficients α and β. We found that the value of β depends on the relative enthalpy contributions of (T-T + O-O) vs. T-O interactions, a useful insight considering the large spread of values found experimentally to be taken by β. We used the thermodynamic integration technique to quantify the entropy, and hence the amount of short-range order, present in the spinels studied. We found that there is virtually no short-range order in the normal spinels. There is significant short-range order in the inverse spinels, though in the experimentally accessible temperature range, the contribution of this short-range order to the entropy is comparatively small. At very low temperatures, we find that the octahedral cations in the inverse spinels become ordered, reducing the symmetry to P4122, in agreement with other simulated findings for 2-3 spinels and experimental findings for 4-2 spinels

A computational study of order-disorder phenomena in Mg2TiO4 spinel (qandilite)

Abstract We have used a combination of classical and quantum-mechanical atomistic calculations, together with Monte Carlo simulations, to study order-disorder phenomena in the spinel mineral qandilite, Mg2TiO4. Using an interatomic potential model akin to those previously used for 2-3 spinels yielded a general increase in energy E as a function of inversion parameter x, and thus incorrectly predicted a normal-spinel ground state, whereas the E(x) behavior as modeled by density-functional theory exhibited a maximum at an intermediate degree of inversion and correctly predicted an inverse-spinel ground state. We therefore used the quantum-mechanical simulations to derive pair interaction parameters (for nearest-neighbor tetrahedral-tetrahedral, octahedral-octahedral, and tetrahedral-octahedral interactions) and chemical potential to use in Monte Carlo simulations of order-disorder in qandilite. The simulated cation distributions compared favorably with those obtained experimentally, although the long-range ordering transition to the tetragonal P4122 phase was not observed when using only nearest-neighbor interactions. However, this transition could be observed following the addition of two extra parameters to the model. The simulations were used to calculate the effect of short- and long-range cation order on the configurational entropy of qandilite as a function of temperature. The calculated entropy of the hightemperature cubic phase was in very good agreement with the experimental value recently determined, supporting the suggestion that there is considerable short-range order in qandilite.

Effect of the tetrahedral charge on the order-disorder of the cation distribution in the octahedral sheet of smectites and illites by computational methods

The order-disorder behavior of the isomorphous cation substitution of the octahedral sheet of phyllosilicates was investigated by Monte Carlo simulations based only on atomistic models in some three-species systems Al/Fe/Mg including a wide range of different octahedral compositions that can be relevant to clay compositions found in nature, especially for smectites and illites. In many cases, phase transitions do not occur, in that long-range order is not attained, but most systems exhibit short-range order at low temperature. The ordering of the octahedral cations is highly dependent on the cation composition. Variations in the tetrahedral charge (smectite vs. illite) produce slight differences in the cation distribution and the short-range and long-range order of octahedral cations do not change drastically. The average size of Fe clusters and the long-range order of Fe are not larger in illites than in smectites as previous reports concluded, but the proportion of Fe3+ cations non-clustered is higher in smectites than in illites. This behavior supports the experimental behavior of the Fe effect on the Al-NMR signal, which is lower in illites than in smectites.

Computer simulation of Al-Mg ordering in glaucophane and a comparison with infrared spectroscopy

TheorderingofMgandAlovertheoctahedralsitesinglaucophane, (A) (8) Na2 (6) (Mg3Al2) (4) Si8O22(OH)2,hasbeenstudied by Monte Carlo simulation using a model Hamiltonian parameterised using both empirical interatomic interactions and ab initio calculations. It is found that Al is fully ordered at the M(2) site, with disorder beginning to appear for temperatures above ~1000 K. Infrared spectra of threesynthetichigh-PT glaucophane-nybo ite amphiboles were also collected and theOH-stretching frequencies usedtoinferthestateofAl-Mgordering.Thespectraofallthreeamphibolescompriseonlytwopeaksat ~3662cm -1 and~3720cm -1 , correspondingtoMgMgMg- OH- (A) andMgMgMg- OH- (A) Na,respectively.TheseinfraredspectrashowunequivocallythatM(1) andM(3) sites are fully occupied by Mg, and, therefore, (6) Al isfully orderedat M(2),in agreement withthebehaviour predicted by the computational studies and bond-valence considerations. Such a highly (6) Al-ordered state for alkali amphiboles contrasts starkly with calcic amphiboles synthesized under similar pressure-temperatureconditions,whichhaveahighdegreeof (6) AldisorderoverM(2)andM(3)sites.Thisdifferencebetweenalkali and calcic amphibolesshows the major influence that the M(4) cation (monovalent versus divalent) has, via its bonding relations to O(4), in controlling the ordering of trivalent cations over the octahedral sites in amphiboles.

A computational model of cation ordering in the magnesioferrite-qandilite (MgFe2O4-Mg2TiO4) solid solution and its potential application to titanomagnetite (Fe3O4-Fe2TiO4)

Abstract Cation ordering in the magnesioferrite-qandilite (MgFe2O4-Mg2TiO4) solid solution has been investigated using an interatomic potential model combined with Monte Carlo simulations. The dominant chemical interaction controlling the thermodynamic mixing behavior of the solid solution is a positive nearest-neighbor pairwise interaction between tetrahedrally coordinated Fe3+ and octahedrally coordinated Ti4+ (JTOFeTi). The predicted cation distribution evolves gradually from the Néel-Chevalier model to the Akimoto model as a function of increasing JTOFeTi, with JTOFeTi = 1000 ± 100 K providing an adequate description of both the temperature and composition dependence of the cation distribution and the presence of a miscibility gap. Although Mg is a good analog of Fe2+ in end-member spinels, a comparison of model predictions for MgFe2O4-Mg2TiO4 with observed cation ordering behavior in titanomagnetite (Fe3O4-Fe2TiO4) demonstrates that the analog breaks down for Fe3O4-rich compositions, where a value of JTOFeTi closer to zero is needed to explain the observed cation distribution. It is proposed that screening of Ti4+ by mobile charge carriers on the octahedral sublattice is responsible for the dramatic reduction in JTOFeTi. If confirmed, this conclusion will have significant implications for attempts to create a realistic thermodynamic model of titanomagnetite.

Investigation of Al/Si ordering in tetrahedral phyllosilicate sheets by Monte Carlo simulation

Abstract We have investigated by Monte Carlo simulation the Al/Si ordering behavior of the tetrahedral phyllosilicate sheet, with a variety of compositions from Al1Si1 to Al1Si7, using atomic interaction parameters determined for the tetrahedral sheet in muscovite. Three different ordering schemes operate, depending on composition, with relatively Al-poor systems ordering in a muscovite-like (Al:Si = 1:3) pattern and relatively Al-rich compositions ordering in an “ABABAB” or margarite-like (Al:Si = 1:1) pattern, where ABABAB indicates the arrangement of atoms around a hexagonal ring of tetrahedral cation sites. The pattern corresponding to Al:Si = 1:2 occurs in intermediate compositions, but always in conjunction with another ordering pattern, except for one composition close to Al:Si = 1:2. Simulations of the same composition but with different ordering schemes can show different behavior, and this is evidence for metastability fields. The transition temperature for order-disorder Tc is strongly dependent on composition, and the dilution effect can be observed at low Al concentrations, with a critical concentration xc between 0.12 and 0.15.