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Review
Computer simulations of cations order-disorder in 2:1 dioctahedral phyllosilicates using cation-exchange potentials and monte carlo methods
Article first published online: 23 MAY 2014
DOI: 10.1002/qua.24703
Copyright © 2014 Wiley Periodicals, Inc.
Issue
International Journal of Quantum Chemistry
Special Issue: Highlights of QUITEL 2013
Volume 114, Issue 19, pages 1257–1286, October 5, 2014
Additional Information
How to Cite
How to cite this article: Int. J. Quantum Chem. 2014, 114, 1257–1286. DOI: 10.1002/qua.24703
, , , , , ,Publication History
- Issue published online: 18 AUG 2014
- Article first published online: 23 MAY 2014
- Manuscript Accepted: 11 APR 2014
- Manuscript Revised: 10 APR 2014
- Manuscript Received: 21 JAN 2014
Funded by
- EPSRC
- Royal Society
- MCYT. Grant Number: BTE2002-03838
- Spanish MEC and European FEDER funds. Grant Numbers: CGL2005-02681, CGL2008-02850, PR2008-0288 projects
- Junta de Andalucía. Grant Numbers: RNM-363, RNM-264, RNM-3581 project
- Abstract
- Article
- References
- Cited By
Keywords:
- cation ordering;
- cation-exchange potentials;
- Monte Carlo simulations;
- phyllosilicates;
- empirical potential;
- DFT
Abstract
- Top of page
- Abstract
- Introduction
- Structure of 2:1 Dioctahedral Phyllosilicates
- Methodology
- Case Studies
- Conclusions
- Acknowledgments
- Appendix: A: The Ordering Energy Expression for the Case of Two Species Ordering Across One Type of Site (example: Al–Si in Muscovite)
- Appendix: B: Force Field of Empirical Interatomic Potential Used to Study Phyllosilicates
- Appendix: C: The Ordering Energy Expression for the Case of Three Species Ordering Across One Type of Site (Example: Al–Fe–Mg in Illite/Smectite)
- Appendix: D: Full List of Al–Fe–Mg Compositions Simulated
- Appendix: E: The Ordering Energy Expression for the Case of Three Cations Ordering Across Three Types of Site, in Two Distinct A–B and A–C Networks (Example: Al–Si and Al–Mg in Phengite)
- Biographies
This article reviews the use of Monte Carlo methods with cation-exchange potentials and effective Hamiltonians, based on empirical potentials and quantum-mechanical calculations, for the study of cation ordering in phyllosilicates. The basic methodology is described, and the application of the methods is illustrated with a number of key example case studies. These include Al–Si ordering in muscovite, Al–Fe–Mg ordering (both binary and ternary compositions) in the octahedral illite/smectite sheet, examination of the ordering behavior of phengite, in which the octahedral sites are occupied by Al and Mg and the tetrahedral sites by Al and Si, and Al–Si ordering in the tetrahedral phyllosilicate sheet with variable Al:Si ratio. In several cases, complex ordering processes were found. The essential conclusion from this work is that computer simulation studies of this nature can be a valuable tool in ordering studies of many nanomaterials. © 2014 Wiley Periodicals, Inc.
Introduction
- Top of page
- Abstract
- Introduction
- Structure of 2:1 Dioctahedral Phyllosilicates
- Methodology
- Case Studies
- Conclusions
- Acknowledgments
- Appendix: A: The Ordering Energy Expression for the Case of Two Species Ordering Across One Type of Site (example: Al–Si in Muscovite)
- Appendix: B: Force Field of Empirical Interatomic Potential Used to Study Phyllosilicates
- Appendix: C: The Ordering Energy Expression for the Case of Three Species Ordering Across One Type of Site (Example: Al–Fe–Mg in Illite/Smectite)
- Appendix: D: Full List of Al–Fe–Mg Compositions Simulated
- Appendix: E: The Ordering Energy Expression for the Case of Three Cations Ordering Across Three Types of Site, in Two Distinct A–B and A–C Networks (Example: Al–Si and Al–Mg in Phengite)
- Biographies
The ordering and disordering of cations within minerals and ceramics has been a prominent feature of research into these materials for several years. In particular, there is a lot of interest in ordering phase transitions and in the role of short-range order in determining the thermodynamic properties of solid solutions. The development of methods and techniques to study atomic ordering are of great interest for the characterization, preparation-monitoring, and design of the new amorphous nanomaterials, doped nanomaterials, metal alloys, synthetic metallic oxides, synthetic phyllosilicates, and zeolites.
More effort is required in the understanding of the physical aspects of cation order-disorder for helping in the interpretation of experimental work. This aspect is more significant in phyllosilicates because they are structurally flexible minerals, able to accommodate many chemical substitutions, and existing in a variety of different structural polytypes. From the point of view of cation ordering, phyllosilicates are interesting in view of their structural inhomogeneity. Besides, there are some disadvantages to the study of order-disorder in phyllosilicates by certain traditional techniques, such as X-ray diffraction and infrared (IR) spectroscopy. In X-ray diffraction, the scattering factors of Mg and Al are similar, making it difficult to distinguish the octahedral site occupancies, such that average bond distances have to be used.[1] In IR spectroscopy, the OH vibrational frequencies mainly depend on the first neighbor octahedral cations; these frequencies can yield information only about pairs of atoms that are linked to OH groups in the octahedral sheet, and so only short-range ordering information can be obtained.[2] Furthermore, the small crystal size of the clay minerals has always made it difficult to study them experimentally. For these reasons, in many cases, to obtain good results in cation ordering studies, a combination of experimental methods with molecular modeling, force field, first principles, and/or simulation methods is necessary.[3]
In micas, Herrero et al.[4] and Palin et al.,[5] by means of experimental and theoretical methods, found a high dispersion of ^{IV}Al^{3+} (prefix superindex in Roman number on the cation means the symmetry of the sheet: tetrahedral or octahedral sheet), which is in agreement with the Loewenstein rule of AlAl pair avoidance. Conversely, some variation was found in the cation distribution for the octahedral sheet of phyllosilicates. A nonrandom distribution of cations was found in the octahedral sheet of celadonites, where the Al^{3+} and Fe^{3+} show a tendency to segregate.[6] Drits et al.[7] studied cation ordering in celadonites, glauconites, and Fe-illites by diverse experimental techniques (IR, Mössbauer, and EXAFS spectroscopies) and simulations by probabilistic methods; they found a certain degree of short-range ordering. By means of ^{27}Al NMR, Schroeder[8] found Fe^{3+} mixes with Al^{3+} in illite-smectite of shales with low Fe^{3+} content, but that Fe segregates from Al in Fe-rich samples. Grauby et al.[9] found that Al^{3+} and Fe^{3+} tend to mix rather than to segregate in some samples of synthesized smectites. These results were opposed to the Muller et al. observations,[10] where, in montmorillonites, Mg^{2+} and Fe^{3+} form clusters. On the contrary, in the same minerals, a random distribution of Fe^{3+} in samples with high Fe content was found by Vantelon et al.[11]
Combining experimental results (FTIR and ^{27}Al NMR spectroscopy) with Reverse Monte Carlo (RMC) simulations on interstratified illite-smectite samples, a short-range ordering in the octahedral cations was found with a tendency of Fe^{3+} segregation and Mg^{2+} dispersion.[12, 13] Quantum-mechanical calculations of periodic crystal lattices for a series of dioctahedral 2:1 phyllosilicates in the trans-vacant polymorph (see next section), with tetrahedral and octahedral cation substitutions in different configurations, showed that the most stable configurations had two Fe^{3+} in two nearest-neighbor octahedra, while Mg^{2+} and ^{IV}Al^{3+} cations were found to be kept apart and dispersed, respectively.[14-16] Dainyak et al.[17] studied the redistribution of octahedral cations (Al^{3+}, Mg^{2+}, Fe^{3+} and Fe^{2+}) in the transformation of illite-smectite (or -vermiculite) to illite-tobelite-smectite (or –vermiculite) by means of Mössbauer and IR spectroscopies and simulations of both spectroscopical data; they found ordered clusters of mixed cations in which divalent and trivalent cations were alternated, plus dispersed clusters of Fe^{3+}; in the transformation process, these last clusters degenerate to shorter clusters. By means of neutron powder diffraction, Redfern[1] studied the ^{VI}Mg^{2+} and ^{VI}Al^{3+} ordering in phengite, finding ordering. For dioctahedral smectites, Drits et al.[18] proposed that octahedral cation order-disorder probably had important consequences in the formation of cis-vacant and trans-vacant sheets, and that the pairs MgOHMg could favor the formation of cis-vacant montmorillonites. Tunega et al.[19] reported quantum-mechanical studies, using plane-wave periodic calculations for different cationic arrangement in supercells, and found the short-range ordering is the usual ordering in dioctahedral phyllosilicates, but the long-range ordering would be difficult in these minerals. Marchel and Stanjek[20] used a combination of X-ray fluorescence and IR spectroscopy to study short-range ordering in smectites, in both cis-vacant and trans-vacant polymorphs; they found homonuclear pairs are randomly distributed but aligned along the OH-bonded directions; however, heteronuclear pairs are increased in cis-vacant smectites but decreased in size in nontronites. Gailhanou et al.[21] determined the configurational entropy terms for the octahedral sheet cations in illite, smectite, and beidellite, by distributing the Al^{3+}, Fe^{3+}, Fe^{2+}, and Mg^{2+} cations following computational results. Dainyak et al.,[22] used quasicontinuous model-independent quadrupole splitting distribution spectral analysis combined with crystal-chemical modeling fitted to Mössbauer spectra to study glauconites. They found ^{VI}Fe-clustering, domain structure, and distinct tendencies in the homogeneous charge dispersion. Quantum-mechanical calculations of different compositions and cation configurations of 2:1 dioctahedral phyllosilicates in the cis-vacant polymorph showed that the ^{VI}Mg^{2+}, ^{VI}Fe^{3+}, and ^{IV}Al^{3+} isomorphous substitutions follow the same trends than in the trans-vacant polymorphs.[23, 24] Considering all these experimental and computational results, it is difficult to draw definitive conclusions for the cation ordering in these minerals.
Long-range order is amenable to experimental study using diffraction methods, and short-range order can be measured using magnetic resonance methods. Several computational and theoretical methods for the study of cation ordering have been developed in recent years. These include the cluster variation method[25, 26] and approaches based on configurational averaging.[27]
This review is concerned with developing models of the energetics of ordering, and using Monte Carlo (MC) methods to study the variation of equilibrium ordered states (both long- and short-range) with temperature. These methods have been successfully applied to a wide range of minerals, as reviewed by Bosenick et al.[28] and Warren et al.,[29] and more recently to a range of examples within the family of 2:1 phyllosilicate minerals.
The rationale behind using computer simulations to study cation ordering in phyllosilicates is that, due to their inhomogeneous structures, cation ordering can be strongly inhibited by kinetics. Considerable effort has been directed toward determining cation distributions in phyllosilicates by experimental means, as discussed above, but the results of such studies can be ambiguous or inconclusive because of the kinetics. Using a thermodynamics-based computational approach, such as that of MC simulation, tackles the issue from a different perspective—one simply selects the input parameters (e.g., temperature) and is then able to obtain (and visualize directly using appropriate software) the cation distributions as controlled by thermodynamics, rather than by kinetics.
Our first purpose in this article is to give a review of the cation-exchange potentials and MC methods as applied to the study of cation ordering in anhydrous 2:1 dioctahedral phyllosilicate minerals, with examples covering ordering in both the tetrahedral and octahedral layers. This review draws together a number of results to enable some general conclusions to be drawn. Our second purpose is to use this review to illustrate the potential of this approach for studies of other members of the phyllosilicate family and other materials.
In the first part of this article, we discuss the methods used to study cation ordering. The second part reviews in more detail the results that have been obtained by application of these methods, beginning by considering the simplest case, that of two ordering species in a single sheet of the phyllosilicate structure. The first example concerns Al and Si ordering in the tetrahedral sheet, such as would be appropriate to phyllosilicates with Al:Si ratios from 1:1 to 1:7. This encompasses minerals such as margarite (Al:Si = 1:1), muscovite (Al:Si = 1:3), and phengite (Al:Si = 1:7). This is extended by a similar example, namely the octahedral ordering of Al and Fe, Al and Mg, and Fe and Mg, such as would be appropriate to the illite/smectite minerals. We will show that even these compositionally simple systems can demonstrate considerable complexity in their ordering behavior, the study of which has enabled us to clarify certain previous experimental and theoretical investigations.
We then extend this to the case of three ordering species, Al, Fe, and Mg, in a single octahedral sheet, as would once again be relevant to illite/smectites. The behavior now observed is remarkably diverse, despite its dependence only on the ratios of cations.
Finally, we turn to the case of three ordering species, Al, Si, and Mg, distributed across interacting tetrahedral and octahedral sheets, where Al and Si can occupy tetrahedral sites and Al and Mg can occupy octahedral sites. This study illustrates the behavior of phengite mica, with a tetrahedral Al:Si ratio of 1:7 and an octahedral Al:Mg ratio of 3:1. The presence of Al in both tetrahedral and octahedral sites results in the possibility of coupling between the octahedral and tetrahedral ordering processes, and indeed this is what is observed in our simulations.
From all these examples, we show that many useful insights can be gained into the complex ordering behavior of phyllosilicate minerals by using the MC simulation technique. The general conclusion that can be drawn from this work is that simple systems show relatively well-defined ordering behavior, but as the system complexity increases, so does the ordering behavior: competing ordering schemes may arise, and different processes such as exsolution and short-range ordering can be observed as a function of composition. The main point is that the ordering behavior of any complex system is not inherently predictable, and the sometimes inconclusive nature of experimental examinations of complex phyllosilicates means that simulation studies provide an excellent approach to studying cation behavior.
Structure of 2:1 Dioctahedral Phyllosilicates
- Top of page
- Abstract
- Introduction
- Structure of 2:1 Dioctahedral Phyllosilicates
- Methodology
- Case Studies
- Conclusions
- Acknowledgments
- Appendix: A: The Ordering Energy Expression for the Case of Two Species Ordering Across One Type of Site (example: Al–Si in Muscovite)
- Appendix: B: Force Field of Empirical Interatomic Potential Used to Study Phyllosilicates
- Appendix: C: The Ordering Energy Expression for the Case of Three Species Ordering Across One Type of Site (Example: Al–Fe–Mg in Illite/Smectite)
- Appendix: D: Full List of Al–Fe–Mg Compositions Simulated
- Appendix: E: The Ordering Energy Expression for the Case of Three Cations Ordering Across Three Types of Site, in Two Distinct A–B and A–C Networks (Example: Al–Si and Al–Mg in Phengite)
- Biographies
The structure of phyllosilicates, also called layer silicates, shows a diversity of chemical compositions, different types of bonding (covalent, coordination, ionic, and van der Waals), and structural and cation ordering.
The structure is formed by stacking sheets of different structure and composition (see example in Fig. 1). The tetrahedral sheet (T) is formed by tetrahedral units of SiO_{4} bonded with three other SiO_{4} units by the oxygens of one triangular base of the tetrahedron. These oxygens are called basal oxygens. Six SiO_{4} units form a ring of quasihexagonal symmetry (or ditrigonal symmetry, Fig. 2a). This ditrigonal ring bonds with other ones, forming a plane of effectively infinite extension, known as the basal plane; these bonded tetrahedra form the T sheet. In 2:1 phyllosilicates, two T sheets sandwich another sheet of octahedral polyhedra (O) (Fig. 1). This octahedral sheet is formed by the four oxygens coming from the T sheets, but not part of the basal plane—the so-called apical oxygens—whose silicon-oxygen bonds are approximately perpendicular to the basal plane, two of them coming from one T sheet and other two coming from the other T sheet. The other two vertices of the octahedra are hydroxyl groups, which are placed approximately at the level of the apical oxygens in the centre of the ditrigonal rings. A TOT stack of sheets forms a “layer.” Half of the unit cell contains four tetrahedra and three octahedra. These TOT layers are stacked along the c crystallographic axis, and the space between two TOT layers is known as the interlayer space. This stacking is known as 2:1. If all octahedra of the O sheets are occupied by cations (mainly Mg^{2+}), this gives rise to the trioctahedral phyllosilicate series; if only two-thirds of the octahedra are occupied by cations (generally Al^{3+}), and one per half-unit cell is empty, this gives rise to the dioctahedral phyllosilicate series. Different compositions are possible by substituting T and O cations with others; the most frequent substitutions are Al^{3+} in the T sheet, and Mg^{2+}, Fe^{3+}, and Fe^{2+} in the O sheet. When substitutions have less charge than three in the O sheet or less than four in the T sheet, the TOT layer has a net negative charge, which is compensated by other cations hosted in the interlayer space. These cations are generally Na^{+}, K^{+}, Mg^{2+}, and Ca^{2+}. The end member of the 2:1 dioctahedral phyllosilicates, with no cation substitution, is pyrophyllite, which has the structural formula [Al_{2}(Si_{4})O_{10}(OH)_{2}] (per half unit-cell), where the Si in the T sheet are written in parenthesis. These minerals, with only two substitution cations (X and Y as octahedral and tetrahedral cations, respectively) show the general formula [I_{x+y}Al_{2-x} (Si_{4-y}Y_{y})O_{10}(OH)_{2}] (x = 0 − 1, y =1 − 0), where I is the interlayer cation species, if X or Y has a charge of less than three or four, respectively. Nonetheless, a tetrahedral charge exceeding 1 per formula unit can occur. More than one, two, or three cations can be substituted in the O and T sheets, and a large diversity of mineral series can be generated. In some series of minerals, the interlayer space can also contain H_{2}O and other polar molecules. In some cases, hydroxyl groups can be substituted by F.
In 2:1 dioctahedral phyllosilicates, the OH groups can be differently located in the vacant octahedral position, forming two polymorphs: (i) if two OH groups are located in trans position in the octahedral vacancy, the trans-vacant polymorph is found; (ii) if two OH group are located in cis position in the vacancy, the cis-vacant polymorph is found.[30, 31]
The bonds found in the T sheet can be considered of covalent nature; in the O sheet, coordination bonds are found. If the layers are charge-neutral, they are bonded by van der Waals forces; if the layers are negatively charged and cations are present in the interlayer space, a mix of these last forces and ionic bonds occur in the interlayer space. So, the nature of bonds is diverse in these minerals.
Taking into account the charge of the layers, the common 2:1 dioctahedral phyllosilicates are usually classified as follows: (i) if x + y = 0, the mineral is called pyrophyllite; (ii) x + y ≈ 0.2–0.6, and the octahedral charge is larger than the tetrahedral charge, the series of smectites (montmorillonite, beidellite, nontronite, …) is found; (iii) if x + y ≈ 0.6–0.85, we find the so-called vermiculite; (iv) if x + y ≈ 0.85–1, and the tetrahedral charge is larger than the octahedral charge, we find the so-called “true” micas, for example, the muscovite-paragonite series, phengite, celadonite, illite, and so forth; and (v) if x + y ≈ 1.8 – 2.0, and the charge is mainly tetrahedral, and interlayer cations are generally divalent, the so-called brittle micas are found–for example, margarite.
Other important structural features of the crystal structure of phyllosilicates are related to the different sequences of stacking of the TOT layers, in such a way that different polytypes are produced. The nomenclature of these polytypes is related to the sequence of stacking and symmetry of the crystal. As significant examples, we can mention the 2M and 3T polytypes, of which nomenclature comes from the monoclinic (M) or triclinic (T) symmetry of the polytypes, respectively, and the prefix number from the number of layers in the stacking sequence. Much more detailed description about the structure of phyllosilicates can be found in previous works of Moore et. al.[32] and Guggenheim.[33]
Although T and O sheets are linked by means of the apical oxygens, the fit between these two sheets is not perfect. As such, the symmetry of the tetrahedral rings is reduced to ditrigonal symmetry (Fig. 2a). The basal oxygens do not form a perfect plane either—some corrugation is seen—and the O_{basal} – Si – O_{apical} angles deviate from the perfect tetrahedral angle. So structural distortions appears in the crystal structure, which can undergo order-disorder phenomena.[34]
Interacting species | A (eV) | ρ (Å) | C (eV.Å^{−6}) | Reference | Used in |
---|---|---|---|---|---|
| |||||
Si c – O1 c | 999.9 | 0.3012 | 0 | 1 | Mu Ph IS |
Si c – O2 s | 1283.907 | 0.3205 | 10.66 | 1 | Mu Ph IS |
Al c – O1 c | 1142.677 | 0.2991 | 0 | 2 | Mu IS |
Al c – O2 s | 1460.3 | 0.2991 | 0 | 1 | Mu Ph IS |
Al c – O1 c | 1460.3 | 0.2991 | 0 | 1* | Ph |
O2 s – O2 s | 22764 | 0.149 | 27.88 | 1 | Mu Ph IS |
H c – O2 s | 325 | 0.25 | 0 | 3 | Mu Ph IS |
Mg c – O1 c | 1428.5 | 0.2945 | 0 | 1 | Ph |
Mg c – O1 c | 1142.677 | 0.2945 | 0 | 3 | IS |
Mg c – O2 s | 1428.5 | 0.2945 | 0 | 1 | Ph IS |
Na c – O2 s | 1271.504 | 0.3 | 0 | 4 | IS |
K c – O2 s | 65269.7 | 0.213 | 0 | 1 | Mu Ph IS |
Fe c – O2 s | 3219.335 | 0.2641 | 0 | 4 | IS |
Potential type and interacting species | Parameters and values | Used in | |||
---|---|---|---|---|---|
| |||||
Morse | D (eV) | α (Å^{−1}) | r_{0} (Å) | r_{max} (Å) | |
O1c – Hc [43] | 7.0525 | 2.1986 | 0.9485 | 1.4 | Mu Ph IS |
Three-body | k (eV rad^{−2}) | θ_{0} (°) | r_{max}(M–O) (Å) | r_{max}(O–O) (Å) | |
O – Si c – O [43] | 2.0974 | 109.47 | 1.8 | 3.2 | Mu Ph IS |
O – Al1 c – O [43] | 2.0974 | 109.47 | 1.8 | 3.2 | Mu IS |
O – Al1 c – O [47] | 2.0974 | 109.47 | 1.95 | 3.4 | Ph |
O – Al2 c – O [43] | 2.0974 | 90 | 2.2 | 3.2 | Mu Ph IS |
O – Mg c – O [45] | 2.0974 | 90 | 2.2 | 3.2 | Ph IS |
O – Fe c – O [45] | 2.0974 | 90 | 2.2 | 3.2 | IS |
Spring (core-shell) | K (eV.Å^{−2}) | ||||
O2 c – O2 s [45] | 74.92 | Mu Ph IS |
Parameter | Type | Distance (Å) | Value (eV) |
---|---|---|---|
| |||
J_{1} | Intrasheet | 2.97–3.08 | 1.0(1) |
J_{2} | Intrasheet | 5.22 | 0.23(5) |
J_{3} | Intrasheet | 5.96–6.05 | 0.00(5)* |
J_{4} | Intrasheet | 7.92–8.04 | 0.13(4) |
J_{5} | Intralayer | 5.56 | 0.38(14) |
J_{6} | Intralayer | 5.69 | −0.07(10)* |
J_{7} | Intralayer | 6.24 | −0.05(12)* |
J_{8} | Intralayer | 6.47 | 0.11(13) |
J_{9} | Intralayer | 6.96 | 0.15(13) |
J_{10} | Intralayer | 7.02 | 0.07(9) |
J_{11} | Interlayer | 4.51 | 0.0(2)* |
J_{12} | Interlayer | 5.41–5.46 | −0.16(16) |
J_{13} | Interlayer | 6.82–6.98 | −0.1(1) |
J_{14} | Interlayer | 7.48–7.55 | −0.08(9) |
If translational symmetry is exhibited, a perfect and ideal crystallinity would be shown in these minerals; however, crystallinity is diverse and depends of the environment of the crystal, chemical composition, thermodynamics, and kinetics of the crystal growth. Imperfect crystallinity is highlighted by a degree of disorder of the molecular groups, and the cation distributions. [33, 35, 36] Furthermore, distinct cation substitutions in the two sheets produce changes in the bond lengths, bond angles, polyhedral surfaces, and volumes of the molecular groups, and consequently, in the thickness of the sheets, and ditrigonal symmetry. All these distortions can be emphasized by a disordered distribution of the cation substitutions.[37] Furthermore, if charge-generating cation substitutions occur in the tetrahedral sheet, the charge in the layer is more localized around the substitutions; however, if charge-generating substitutions occur in the octahedral sheet, the charge is much more delocalized in the layer. These charge distributions in the layer can also affect the ordering of the interlayer cations, and consequently the OH group bond lengths and angles, and their hydrogen bonds with the apical and basal oxygens, are also affected.[38]
Structural distortions and cation order-disorder in the crystal structure are dependent on the temperature and pressure history of the material; structural distortions and cation ordering phase transitions can be produced by both temperature and pressure.[35, 39] Sufficient time at sufficiently low temperatures can yield long-range-ordered structural distortion and/or cation distributions, with precise transition temperature; however, short-range ordering can also appear in these minerals.[37] The ordering and the phase transition temperature will depend on different factors, including chemical composition, concentration, structure, and environment. Therefore, determining all these structural distortions and cation distributions, ordering and phase transition temperatures can yield important information about the original thermodynamic and kinetic conditions, stability, and crystal growth of the minerals and, therefore, important mineralogical, petrological, geochemical, and geophysical consequences can be extracted.
Methodology
- Top of page
- Abstract
- Introduction
- Structure of 2:1 Dioctahedral Phyllosilicates
- Methodology
- Case Studies
- Conclusions
- Acknowledgments
- Appendix: A: The Ordering Energy Expression for the Case of Two Species Ordering Across One Type of Site (example: Al–Si in Muscovite)
- Appendix: B: Force Field of Empirical Interatomic Potential Used to Study Phyllosilicates
- Appendix: C: The Ordering Energy Expression for the Case of Three Species Ordering Across One Type of Site (Example: Al–Fe–Mg in Illite/Smectite)
- Appendix: D: Full List of Al–Fe–Mg Compositions Simulated
- Appendix: E: The Ordering Energy Expression for the Case of Three Cations Ordering Across Three Types of Site, in Two Distinct A–B and A–C Networks (Example: Al–Si and Al–Mg in Phengite)
- Biographies
Previous atomistic calculations of cation ordering in phyllosilicates have compared relative energies of several cation configurations. This approach yields only a partial view of cation ordering and is highly dependent on the particular cases studied. The approach reviewed in this work offers a more general scenario where many cation configurations can be investigated for different compositions, being extensible to any nanomaterial with natural origin (long geological formation or reaction times) or synthetically obtained (short formation times).
Basic approach to studying cation ordering
The procedure for studying cation ordering comprises two stages. The first stage involves building a model of the crystal structure to be investigated, and optimizing its lattice energy through an empirical or quantum-mechanical computational method. The aim of this stage is to calculate interaction parameters between certain atoms across certain distances. The second stage of the process then involves the use of the calculated interaction parameters in MC simulations of ordering as a function of temperature. We have described the method in detail elsewhere,[28, 29] and therefore restrict ourselves here to a brief treatment.
The energy of a network of crystallographic sites across which cations can order can be written in terms of the numbers of bonds of particular types occurring in the network, and the energies of these bonds. Thus, for example, in a single network of symmetrically-equivalent sites containing A and B atoms, the three types of bonds are A–A, B–B, and A–B, and the energy would be written as
- (1)
It turns out that, although this equation is written in terms of the numbers of three different types of bond, it is possible to rewrite it in terms of only one bond variable, as N_{AA}, N_{BB}, and N_{AB} are interdependent. This makes Eq. (1) much easier to manipulate with reference to cation ordering studies. If we choose N_{AA} as the fundamental bond variable, Eq. (1) becomes
- (2)
In this expression, the first term is a constant component of the energy which is not involved in the ordering process. Equation (1) considers only nearest-neighbor interactions between atoms; in Eq. (2), we have extended the representation, through a summation, to interactions over all distances j. J denotes the interaction parameters across particular distances, otherwise known as cation exchange potentials; these parameters are central to the study of cation ordering.
The full derivation of Eq. (2) from Eq. (1) is given in Appendix A. The case of two cations ordering across one type of site is a relatively simple one, and as such further similar derivations are necessary for more complex cases—for example, that of more than two types of cation, or cation interactions between two networks of sites. These derivations will also be given in subsequent Appendices.
Cation-exchange potential models
To calculate the energy of any periodic crystalline lattice, two complementary methodologies have been used: (i) the empirical potential force-field method; and (ii) the quantum-mechanical method. The first has the advantage of a low computational cost; for this reason, we can realize a large set of calculations with a variety of compositions. With the second methodology, within the DFT but with increased computational cost, similar calculations to the first methodology were performed.
The lattice energy calculations can be realized with the programs GULP[40] and SIESTA,[41, 42] both use periodic boundary conditions to calculate crystal lattices. Within these programs, the crystal structure to be investigated is specified in terms of a set of fractional coordinates and lattice parameters. GULP has a set of empirical interatomic potentials which give an approximate mathematical form to the interactions between atoms. There are specific formulae for these potentials, and tuneable parameters within the formulae describing the behavior for interactions between different atoms. The Ewald sum can be used to account for the Coulombic contribution to the energy. The interatomic potential models are discussed in Appendix B, and the parameter values used for modeling phyllosilicates are tabulated in Tables 1 and 2. Extensive testing of these potentials is discussed elsewhere.[43, 45]
SIESTA uses DFT[41, 42] with Troullier–Martins norm-conserving pseudopotentials.[48] It uses linear combination of numerical atomic orbitals and it has been shown to describe the crystal lattice structure of phyllosilicates quite well.[14, 49] Furthermore, it yields efficient calculations for large systems. In the approach of SIESTA, calculations are performed using a localized basis set that consists of numerical tabulations of the exact solutions to the pseudoatomic problem. The basis sets should be extended in the double-ζ form with additional polarization functions to ensure a high-quality calculation. The basis sets and pseudopotentials can be previously optimized, since previous calculations showed a better convergence and yielded results in geometry and vibration frequencies closer to experimental data.[50, 51] The generalized gradient approximation and the Perdew–Burke–Ernzerhof[52] parameterization of the exchange-correlation functional are commonly used, since they have described quite well the crystal structure of phyllosilicates. For the evaluation of the Hartree and exchange-correlation potentials, a real space mesh and a mesh cutoff of 300 Ry offer a high level of convergence for all compositions.[53] The calculations exploring only the Γ-point of the Brillouin zone is enough for these solids. Spin polarization should be included in the calculation of all configurations with Fe^{3+} with all the spins of the Fe^{3+} cations having the same orientation to avoid any additional effect of the spin ordering in the cation ordering study.
Model construction and determination of cation-exchange potentials
The phyllosilicate models generated are based on the structural parameters, namely the cell parameters and the atomic coordinates. Initial values for these quantities are usually obtained from experimental data. Owing to the high diversity of cation substitutions in phyllosilicates, lower symmetry than the unit cell or supercells were used to describe the most possible substitutions. When the number of variables in the cation substitution is too high, some simplifications can be used.
For empirical potentials-based calculations, a model is produced which employs the virtual crystal approximation (VCA). This is a partial occupancy model in which the sites across which ordering can occur are each occupied by a virtual atom consisting of a statistical average of the ordering cations. For example, a 1:3 ratio of A and B species would be represented in the VCA as an atom consisting of 0.25 A and 0.75 B on each site available for ordering. This means that these cations are randomly distributed depending only on the relative proportion of cations, whereas the ordering of the other cations can be considered more easily and with low computational effort. The VCA approach can be applied within the empirical force-fields methodology, but cannot be applied in DFT calculations and limited cation substitutions have to be used. For the study of octahedral ordering with DFT, at least 2 × 2 × 1 supercells should be used with periodic boundary conditions, containing 160–168 atoms in total, and with 16 octahedral cation sites, to describe properly the cation ordering.
Once the structural model is built, the cation sites for ordering are filled in randomly depending on the relative cation proportion. Using a bespoke code, like MCCLAY,[13] a large number of cation configurations can be generated randomly. To ensure that the whole space of possible configurations has been explored, a large number (60–100) of cation configurations should be generated. A few highly ordered cation configurations should also be included to ensure appropriate sampling of the configuration space.
All configurations of these supercells with different compositions and cation distributions are optimized, allowing the relaxation of the atomic positions and the cell parameters simultaneously. The structures resulting from this optimization are verified against the experimental structure (cell parameters, atomic positions, average bond lengths, etc.) to ensure good agreement. The optimized parameters are then used as the basis for the construction of a data set consisting of many configurations, each of which has the correct proportion of ordering cations randomly located on the sites available for ordering. The purpose of producing this data set is to solve the energy equation, Eq. (2), for the J values. This equation is simply a multilinear regression problem, and as such can be solved using standard least-squares-based methods, as follows. All configurations in the data set are optimized, using GULP or SIESTA, to produce a set of E values. The number of linkages of a particular type (A–A in the general example) for each configuration can be computed through a spreadsheet method[28] or by a bespoke code like MCCLAY.[13] It is then possible to use a computer algorithm to determine values for both E_{0} and J. The quality of the calculated parameters can be assessed by plotting the observed energy (obtained from GULP or SIESTA) against the calculated energy from Eq. (2); there are several statistical tests which can be performed to evaluate the goodness of the fit. One of these is the correlation coefficient, R^{2}, which in this work is given by
- (3)
E is the energy of each particular configuration, ΔE the difference between the calculated (with GULP or SIESTA) lattice energy and the Hamiltonian lattice energy, and the angle brackets denote averages over all configurations. The closer the value of R^{2} to 1, the better is the fit. Most methods for performing multilinear regression also yield values for the errors on the calculated quantities; these can sometimes appear to be rather high, but this is because the errors can be correlated with one another, rather than being due to any fault with the analysis.
MC simulations
Once a set of J values has been obtained through the procedure above, these values can be used in MC simulations of ordering. The MC algorithm follows the standard Metropolis procedure: a particular configuration of atoms is set up at the beginning of the simulation, and two atoms are selected at random and swapped. The energy change associated with the swap is evaluated; if it is negative, the swap is retained and the next MC step follows with the new configuration as a starting point. If the energy change is positive, however, the swap is retained subject to the probability test (in which k_{B} is the Boltzmann constant):
- (4)
and the next MC step follows. This procedure gives efficient sampling of the allowed phase space for a particular system. The random nature of the sampling allows the system to evolve rapidly and with a sufficiently high number of iterations, high statistical accuracy can be obtained.
The MC simulations undertaken in this work are all performed with the program OSSIA.[29] As it happens, the energy as it is given in Eq. (2) is not especially suitable for manipulation within a MC algorithm, since it does not allow one to consider the ordering site by site. Instead, it is more straightforward to define a site variable S which characterizes the occupancy of a given site. Considering the two-species example discussed in the introduction above, we define the variable S_{j} for a site j, such that S_{j} has a value of 1 if the site is occupied by an A atom and zero if occupied by a B atom. Then, it is possible to consider the product S_{i}S_{j} for two sites i and j, so that the product is zero unless both sites are occupied by an A atom. This means that the number of A–A interactions can be given as
- (5)
where the bracket notation signifies that the summation occurs over all interactions with no interaction being counted twice.
The Hamiltonian of the system is then given by
- (6)
which is identical to the formulation given in Eq. (2), but neglecting the constant term E_{0}. It is also possible to include a chemical potential term to model the preference of a particular type of cation for a particular site, but this has not been employed in our phyllosilicate work.
The standard output from the MC simulations consists simply of the expectation value of the energy, , and the expectation value of the square of the energy, , as a function of temperature. However, other useful quantities can also be computed; the heat capacity of the system can be calculated from and :
- (7)
This is a useful quantity since it diverges at a continuous phase transition, and the heat capacity profile often allows a more accurate determination of the phase transition temperature T_{c} than can be obtained from the energy evolution.
In addition to the heat capacity, it is also possible to compute one or more order parameters. In OSSIA, order parameters are calculated site by site using the formula
- (8)
where Q_{i} denotes the order parameter for a particular site i, s_{i} denotes the occupancy with respect to one cation of site i averaged over all unit cells, s_{i,∞} indicates the occupancy at T = ∞ and s_{i,0} indicates the occupancy at T = 0. The overall order parameter is then given by
- (9)
where N is the number of sites. This generally gives the desired behavior of the order parameter, namely that disorder should correspond to Q = 0 and order to Q = 1, although in complex systems further normalization may be necessary.
This order parameter formulation, of course, requires the user to state explicitly the ordered state, that is, the state corresponding to Q = 1. This may not be known initially, and hence must be postulated somehow. The usual strategy for achieving this is to begin by performing a “hot start” MC simulation, whereby the system is set up with atoms randomly located on the sites and ordering is monitored with decreasing temperature. This usually reveals the ordered structure, and then a “cold start” simulation can be performed, where the system is initially constructed in an ordered state and disordering is monitored with increasing temperature. A further simulation mode has been employed, which we have called the “warm start”; this allows the simulation of systems in which the chemical composition is not commensurate with ordering at a unit-cell level.
From the order parameter and its mean-squared value, the susceptibility of the system can be determined:
- (10)
Like the heat capacity, the evolution of the susceptibility also exhibits a divergence at a continuous phase transition, being therefore another useful measure of the value of T_{c}.
The critical points to note about all these methods are that (a) the empirical interatomic potentials we use are well-tested, (b) the formalism underlying the method is shown to provide a good description of the lattice energy of the system, with observed and calculated lattice energies being in good agreement (correlation coefficients between the two showing an excellent fit), (c) DFT is a method of choice for most molecules and solids, which yields very valuable and reliable results, and (d) the use of the MC method to model cation ordering as a function of temperature works well.
Case Studies
- Top of page
- Abstract
- Introduction
- Structure of 2:1 Dioctahedral Phyllosilicates
- Methodology
- Case Studies
- Conclusions
- Acknowledgments
- Appendix: A: The Ordering Energy Expression for the Case of Two Species Ordering Across One Type of Site (example: Al–Si in Muscovite)
- Appendix: B: Force Field of Empirical Interatomic Potential Used to Study Phyllosilicates
- Appendix: C: The Ordering Energy Expression for the Case of Three Species Ordering Across One Type of Site (Example: Al–Fe–Mg in Illite/Smectite)
- Appendix: D: Full List of Al–Fe–Mg Compositions Simulated
- Appendix: E: The Ordering Energy Expression for the Case of Three Cations Ordering Across Three Types of Site, in Two Distinct A–B and A–C Networks (Example: Al–Si and Al–Mg in Phengite)
- Biographies
This section describes the results on studies of various phyllosilicate minerals, and is arranged in order of increasing complexity of the systems, in terms of the number of ordering species and the number of types of site across which ordering occurs. We begin by considering two ordering species in one type of site, then three ordering species in one type of site, and finally three ordering species across two different site types. Subsequently, we will bring together the results from these investigations and discuss the insights into 2:1 dioctahedral phyllosilicates' behavior.
Scenario 1: Two cation species ordering across one type of site
Al–Si ordering in muscovite
Muscovite, [Si_{6}Al_{2}]O_{20}(OH)_{4}, is a relatively simple mineral with respect to cation ordering.[5] It can be seen from the chemical formula that there is scope for ordering across the tetrahedral sites only, which contain Al and Si in a 1:3 ratio.
Model of the tetrahedral sheet
The tetrahedral sheet can be considered as a network of sites across which are distributed two cation species, Al and Si. The ordering equation for this case is Eq. (2), here written with specific reference to the cations involved:
- (11)
The derivation of this equation from the general formula for the energy in Eq. (1) is contained in Appendix A. Here, we simply note that with a set of values for E and N_{AlAl} it is possible to compute the j values of J, as detailed above.
Determination of cation-exchange potentials (J) for the tetrahedral sheet
The structural parameters for the initial muscovite model were taken from an X-ray diffraction study of the 2M_{1} polytype.[54] Fourteen interactions (J values) were defined to form the muscovite model. Some of these occur over distances between tetrahedral sites in the same sheet (“intrasheet”), some over distances between tetrahedral sites in different sheets in the same TOT layer (“intralayer”), and some over distances between tetrahedral sites in tetrahedral sheets in adjacent TOT layers (“interlayer”). The three different types of interaction are illustrated in Figure 1. Examples of the distances corresponding to intrasheet, intralayer, and interlayer J parameters are illustrated in Figures 2a and 2b.
The J values determined by multilinear regression are given in Table 3. The coefficient R^{2} was 0.92. The starred parameter values in Table 3 are very small, such that setting them to zero had only a negligible effect on the quality of the fit. Conversely, the largest parameter of the set is J_{1}, indicating that nearest-neighbor Al–Al interactions are unfavorable. This value of J_{1} compares favorably with J_{1} values determined previously for other aluminosilicate minerals.[28]
MC simulations of muscovite
The first muscovite simulations were of one tetrahedral sheet, with composition Al:Si = 1:3, the aim being to investigate the ordering behavior in two dimensions. The first four J values given in Table 3, that is, the intrasheet parameters, were used in the simulation. Initially, a hot start simulation was performed; this revealed attempts at long-range ordering at low temperature, so order parameters describing this ordering were constructed. Four-order parameters were required; all describe the same ordering scheme, but with different unit cell origins. These four configurations, labeled A, C, E, G, are illustrated in Figure 3, and were used to perform a cold start simulation. Also illustrated in Figure 3 are the four corresponding order parameters, labeled B, D, F, H, for the other possible orientation of the tetrahedral sheet, which is rotated by 180° with respect to the first.
The results obtained from the cold start simulation for the order parameter are shown in Figure 4, and a low-temperature snapshot is shown in Figure 5. These figures show that at low temperatures, long-range order occurs, with a transition temperature T_{c} of ∼1900 K. In the ordered configuration, all Al–Al interactions are J_{3}, forming a “superhexagon” structure, since the Al–Al linkages form larger hexagons around a central hexagonal ring of tetrahedral sites.[5]
The next simulations were of interacting tetrahedral sheets, that is, using all parameters in Table 3. In these simulations, all the sheets ordered, but there was no long-range ordering of sheets with respect to one another—many different combinations of the configurations A–H were observed, with no apparent preference for any combination over any other. A long-range-ordered scheme was proposed by building and optimizing GULP models of all the possible configuration combinations, and selecting the lowest-energy structure of these. This ordered state was used as the starting configuration for a cold start simulation. Comparison of these results with those for two-dimensional (2D) ordering shows that the two simulations have very similar behavior (results not illustrated, for this reason), except that the transition temperature is higher (∼2140 K) for the 3D simulation than for the 2D simulation, and the changes at the phase transition are faster for the 3D system than the 2D.
These simulations allow us to make some suggestions as to the kinetics of the process of ordering in the tetrahedral sheets. MC simulations do not incorporate a kinetic component, being dependent only on thermodynamic parameters, but in spite of this we can draw conclusions based on the differences between the 2D- and 3D-ordering behavior. Specifically, the very fact that no 3D-ordering was observed when we performed a 3D hot start simulation suggests kinetic constraints on the ordering process. The 2D ordering will set in first, such that when the temperature is reached at which 3D ordering should occur, the kinetic barrier to rearrangement of the ordered sheets of cations with respect to one another will be too great.
The similarity between the order parameter profiles in the 2D and 3D simulations also indicates this fact. For the 3D cold start simulation, when the temperature is reached at which the system should disorder, the individual sheets will still be highly ordered, and the sheets are not able to reorder to remove the correlation of neighboring sheets. Thus, the cold start result for 3D ordering resembles the 2D ordering profile.
Extensive previous studies of tetrahedral Al–Si ordering in phyllosilicates have been undertaken by Herrero, Sanz, and coworkers.[56-58] For the muscovite composition, they proposed an ordering pattern, according to the homogeneous dispersion of charges model, which differs from the one we observe in our simulations. Both patterns are based on a chain structure in which the Al line up with respect to one another, interacting across J_{3} distances, forming zig-zag chains. There are then two ways in which these zig-zag chains can line up: either in phase or out of phase. In our (out-of-phase, superhexagon) structure, there are no Al–Al J_{4} linkages, whereas in the other (in-phase) structure, Al–Al J_{4} linkages do exist. The J parameters that we have determined include a positive J_{4}, which indicates that Al–Al linkages across J_{4} distances should not occur if the energy is to be at its lowest. This therefore indicates that, of the two possible ordering schemes proposed, the superhexagon structure is favorable.
Further discussion of tetrahedral ordering in muscovite is given below in the study of the general Al–Si ordering behavior of the tetrahedral sheet.
Al–Fe, Al–Mg, and Fe–Mg ordering in the octahedral phyllosilicate sheet
We next considered the case of the octahedral phyllosilicate sheet, and the ordering behavior of the binary systems Al–Fe, Al–Mg, and Fe–Mg.[37, 59, 60] We will show later in this article that these systems must be characterized as groundwork for studying the case of Al–Fe–Mg ordering in the octahedral sheet.
Determination of cation-exchange potentials (J) for the octahedral sheet
Four interactions, again labeled J_{1–4}, were considered for the octahedral sheet model. The layout of the octahedral sheet, in terms of cationic positions, is virtually identical to that of the tetrahedral sheet. This means that the distances corresponding to the interactions in the octahedral sheet are the same as those for the tetrahedral sheet (approximately 3, 5.2, 6, and 8 Å respectively); therefore, we have not illustrated the octahedral sheet separately.
The equation modeling ordering behavior is still that of two species in one site type, that is, Eq. (11), albeit with different cations. However, it is still true that the numbers of bonds of each type are interdependent, so that only one variable (e.g., N_{AlAl}) is required in the equation.
In this case, ordering was studied via an empirical potentials or DFT calculations. Several initial two-species models were produced. Details of these models are given in Table 4; they each have different octahedral compositions. Other variations in composition are also incorporated, namely different interlayer cation species (Na vs. K) and different tetrahedral charge (TC). A low TC of 0.28/unit cell corresponds to smectites, and a high TC of 0.8/unit cell is more typical of illites.
GULP | SIESTA | ||||||||
---|---|---|---|---|---|---|---|---|---|
| |||||||||
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
Si | 7.72 | 7.72 | 7.20 | 7.72 | 7.20 | 7.72 | 8 | 8 | 8 |
^{IV}Al | 0.28 | 0.28 | 0.80 | 0.28 | 0.80 | 0.28 | – | – | – |
^{VI}Al | 3 | 3 | 3 | 3 | 3 | – | 3 | 3 | |
Mg | 1 | 1 | 1 | – | – | 2 | 1 | 2 | |
Fe^{3+} | – | – | – | 1 | 1 | 2 | – | 1 | 2 |
IC | K_{1.28} | Na_{1.28} | K_{1.8} | K_{0.28} | K_{0.8} | K_{2.28} | Na_{1} | – | Na_{2} |
a (Å) | 5.22(1) | 5.22(1) | 5.22(1) | 5.24(1) | 5.22(0) | 5.20(1) | 5.18 | 5.16 | 5.18 |
b (Å) | 8.91(2) | 8.91(1) | 8.93(1) | 8.90(2) | 8.93(1) | 8.96(1) | 8.98 | 8.96 | 8.98 |
c (Å) | 10.14(5) | 10.18(2) | 10.03(2) | 9.54(3) | 10.24(1) | 9.87(1) | 10.05 | 9.35 | 10.05 |
β (°) | 102.5(3) | 105.1(3) | 102.4(2) | 96.0(2) | 102.5(1) | 101.6(1) | 101.4 | 100.4 | 101.4 |
In the empirical potentials approach, an initial VCA model was created, using the cell parameters quoted by Tsipursky and Drits[61] for an illite/smectite sample and from X-ray diffraction data of pyrophyllite[62] for model (8) where there is no interlayer cation. The cell was optimized with GULP and the resulting cell parameters and atomic positions were then used to generate a 2×2×1 supercell, containing 16 octahedral sites. Ninety configurations for using in empirical potential calculations, and a maximum of 53 configurations for the DFT method, were generated with the MCCLAY program,[13] adding several ordered configurations, to sample the parameter space appropriately. The compositions are included in Table 4 with cation ratios Al:Mg = 1:3, Al:Fe = 1:3, and Fe:Mg = 1:1. Notice that models used in the DFT calculation do not have TC, and the cation compositions are slightly different from those calculated with the empirical potential method. Note that for layer silicates and other systems, we found that the lattice cell parameters do not change significantly with the cation distribution and thus do not contribute significantly to the resultant ordering energies.[14, 28] All configurations of these supercells with different compositions and cation distributions were optimized.
The two-species Js (for Al–Mg, Al–Fe and Fe–Mg systems) were then determined using Eq. (11). These parameters are presented in Table 5.
GULP | SIESTA | ||||||||
---|---|---|---|---|---|---|---|---|---|
Parameter and distance | 1 (AlMg) | 2 (AlMg) | 3 (AlMg) | 4 (AlFe) | 5 (AlFe) | 6 (FeMg) | 7(AlMg) | 8(AlFe) | 9(FeMg) |
| |||||||||
J_{1}, < 3.3 Å | 0.656(16) | 0.652(14) | 0.620(18) | 0.025(2) | 0.015(3) | 0.456(31) | 0.598(7) | 0.122(8) | 0.497(4) |
J_{2}, 5.1–5.3 Å | 0.168(10) | 0.162(10) | 0.151(11) | 0.007(1) | 0.005(2) | 0.101(23) | 0.154(1) | −0.007(3) | 0.112(9) |
J_{3}, 5.8–6.2 Å | 0.089(11) | 0.088(10) | 0.066(12) | 0.003(1) | 0.008(3) | –0.003(27) | 0.096(3) | −0.006(9) | 0.051(5) |
J_{4}, 7.7–8.2 Å | 0.025(8) | 0.015(9) | 0.030(9) | 0.003(1) | 0.001(2) | 0.075(20) | 0.011(3) | 0.003(2) | −0.019(7) |
The values of the exchange potentials obtained from the SIESTA calculations yield to the cationic mixture of Al and Mg in the Al_{3}Mg composition due to the highly positive value of J_{1} (0.598 eV) and the smaller values of the rest of the potentials (J_{2} = 0.154 eV, J_{3} = 0.096 eV, J_{4} = 0.011 eV).[37] These values are close to those found based on empirical interatomic potentials (J_{1} = 0.652 eV, J_{2} = 0.162 eV, J_{3} = 0.088eV, and J_{4} = 0.015 eV).[59] The high values of J_{1} indicate that Mg^{2+} will tend to be dispersed all along the octahedral sheet. Similar differences between the J values obtained with GULP and SIESTA calculations were found recently in other phyllosilicates.[63]
The exchange potentials for the Al_{3}Fe series obtained with DFT calculations (J_{1} = 0.122 eV, J_{2} = –0.007 eV, J_{3} = –0.006 eV, and J_{4} = 0.003 eV) are significantly smaller than those of the Al_{3}Mg series, and J_{1} for Al/Fe is larger than that obtained by using empirical interatomic potentials[59] (J_{1} = 0.025 eV, J_{2} = 0.007eV, J_{3} = 0.003 eV, and J_{4} = 0.003 eV). The low value of J_{1} indicates that the aggregation tendency of Fe^{3+} is greater than the other two cations. Finally, the cation-exchange potentials of the Fe/Mg configurations present the same trend as in Al/Mg compositions. In this case, the potentials are higher than those from the Al/Fe composition and lower than for the Al/Mg series. Therefore, in the case of the composition Fe/Mg, the Mg will tend to form scattered configurations. These values agree with the results from empirical interatomic potentials.
MC simulations of two-species behavior in the octahedral sheet
Simulations were performed on the following binary octahedral sheet compositions: Al:Mg = 1:1, Al:Mg = 3:1, Al:Fe = 1:1, Al:Fe = 3:1, Fe:Mg = 1:1, Fe:Mg = 3:1. All of these systems exhibited an order-disorder phase transition. The ordered structures are shown in Figure 6. For the 1:1 systems, an “ABABAB” pattern forms, so-called because of the alternating A and B cations around a hexagonal ring (Fig. 6a). No other long-range-ordered pattern is possible if there are to be no like-atom nearest-neighbor pairs, which is of course what is required by the large J_{1} values in Table 5. For the 3:1 systems, two ordered patterns are possible; these correspond to the in-phase and out-of-phase patterns discussed above for muscovite (albeit with different cations). The out-of-phase (superhexagon) structure is seen for the Al:Fe = 3:1 and Fe:Mg = 3:1 compositions (Fig. 6c), whereas the in-phase configuration is seen for Al:Mg = 3:1 (Fig. 6b). As was the case for muscovite, it is the relative values of the Js that dictate which of these patterns should form.
In simulating the compositions above, all nine sets of Js were used—sets 1 to 3 and 7(DFT) for Al–Mg systems, sets 4, 5 and 8(DFT) for Al–Fe systems, and set 6 and 9(DFT) for Fe–Mg systems. The different J sets for Al–Mg systems did not give different behavior, for either the 1:1 or 3:1 composition and DFT and empirical potential calculations; the only slight variation was in the value of T_{c}. The different sets for Al–Fe, however, gave different results for the 3:1 composition. The behavior of this composition when simulated with J set 5 and 8 are illustrated in Figures 7a and 7b, respectively. When the J set 5 is used (Fig. 7a), the superhexagon pattern is different to that obtained with J set 4 (Fig. 6c), in the first one the pattern is commanded by J_{2} and in the latter the behavior is commanded by the J_{3}. No long-range order is observed in Figure 7a. Similarly, when the J set 8 (DFT) is used, no superhexagon pattern is obtained, instead an exsolution of Al-Fe pairs is generated, where the ordered pattern is controlled by J_{2} (Fig. 7b). This different ordering behavior is due to the fact that, in J set 5 and 8 (DFT), J_{2} is relatively small in comparison to J_{3}. These differences could be due from no presence of interlayer cations or differences in the tetrahedral charge in the J set 8 (DFT), and the small differences between J_{i} values for Al-Fe systems.
Tetrahedral Al–Si ordering as a function of Al:Si ratio in phyllosilicates
Having examined the tetrahedral behavior of a specific phyllosilicate, muscovite, we then extended our studies to the case of the general tetrahedral sheet.[55] So far in this discussion we have seen that, even for a simple case, ordering behavior can be a function both of cation ratios and of J parameters and their relative magnitudes. For example, (a) in muscovite, the ordering behavior was ultimately determined by the value of J_{3}; (b) for Al–Fe ordering, different behavior can occur with different J values; and (c) for 1:1 ratios of octahedral cations, the ordering scheme observed is always different from that for 3:1 ratios. These facts led us to consider the effect of fixing the values for the tetrahedral Js and changing the Al:Si ratio. By this approach, we can investigate the effect of composition on both the ordering behavior (long-range vs. short-range) and the ordering scheme which operates.
The model parameters of the tetrahedral sheet were discussed above for muscovite. The ordered 1:3 scheme determined for that composition, however, is only one possible ordering scheme which may operate in the tetrahedral sheet; different compositions may be expected to order according to different ordering schemes, as was the case in the octahedral sheet above. Initially, two ordering schemes were considered: the 1:3 (muscovite) scheme, and a 1:1 “ABABAB” ordering scheme with Al and Si cations alternating around the six-membered rings of tetrahedral sites, as was seen for the 1:1 ratios in the Al–Fe–Mg systems above. These two ordering schemes are shown in Figures 8a and 8b. We will refer to order parameters corresponding to these schemes (i.e., for which Q = 1 for the perfect order shown in Fig. 8) as Q_{1:3} and Q_{1:1}. On performing simulations of a variety of compositions, however, another ordering scheme became evident, which corresponds to a composition of Al:Si = 1:2 and is shown in Figure 8c. Further simulations were therefore performed with an additional order parameter corresponding to Al:Si = 1:2, which we have labeled Q_{1:2}.
To differentiate between the compositions studied, we will use a variable x, which indicates the fraction of Al in the tetrahedral sheet. This fraction will be presented in 64ths; although many compositions can be simplified (e.g., x = 16/64 = ¼, i.e., 1:3 Al:Si ratio), we will leave such fractions unsimplified for ease of comparison.
Neither of the two simulation methods previously discussed (hot and cold start) is suitable for modeling the systems under investigation here, because the majority of compositions studied are not identical to the ordered states corresponding to the order parameters discussed above. Instead, we need to employ a method which can deal with partial ordering, and we have termed this a “warm start.” In this method, the structure is initially set up in a particular ordered state. Then, cations of one type are selected at random and changed to cations of the other type until a specified desired composition is reached, whereupon the simulation proceeds as would a cold start simulation. For example, a simulation discussed as “the x = 20/64 simulation started in the 1:2 scheme” would be set up with the perfectly ordered 1:2 pattern (composition corresponding to x = 21⅓/64), then atoms of Al would be changed to Si until the composition corresponded to x = 20/64, and then a cold start simulation would begin.
Simulations were grouped according to composition. The muscovite composition, corresponds to x = 16/64, the phengite composition to x = 8/64, and the margarite composition to x = 32/64. Compositions with x ≤ 16/64 form the “Al-poor” group, Compositions with x ≥ 26/64 form the “Al-rich” group, and compositions falling between these two limits form the “intermediate” group.
MC simulations of the tetrahedral sheet
Figure 9 shows the order parameter evolution as a function of temperature for each composition, divided according to the groups discussed above. The simulations for the Al-poor group were started in the 1:3 ordered scheme, whereas those for the Al-rich group were started in the 1:1 ordered scheme. For the intermediate systems, simulation results are presented for simulations started in all three ordering schemes (1:1, 1:2, 1:3). Figure 10 shows low-temperature snapshots for some of the compositions studied.
The behavior of the Al-poor and Al-rich systems is quite similar. In each, there is one composition for which long-range order occurs (16/64 and 32/64 respectively, by definition of the order parameters!), and the more the composition deviates from this, the less clear is the ordering. For the Al-poor systems, this corresponds to a complete lack of ordering at dilute compositions—the form of the order parameter curves for x ≤ 11/64 is such that one cannot determine a T_{c} value. For the Al-rich systems, the form of the curves remains fairly clear, since the site occupancies corresponding to the 1:1 ordering are still partly fulfilled by the 1:2 ordering pattern.
In the intermediate systems, the behavior of the 1:1 and 1:3 order parameters is complementary. That is, the “best” order parameter curve for simulations started in the 1:3 scheme is for x = 18/64 and the “best” order parameter curve for those started in the 1:1 scheme is for x = 24/64. This is as would be expected, as these compositions are the closest in their groups to the long-range-ordered compositions. The behavior of the simulations started in the 1:2 scheme is similar to that of those started in the 1:1 scheme, since (as stated above) the two ordering schemes are closely related.
As was the case above for the octahedral sheet, varying only the composition of the tetrahedral sheet (and not the interaction parameters) a variety of cation behavior can be observed.
Metastability and dilution in the tetrahedral sheet
The study of a range of compositions with respect to different ordering schemes facilitates the identification of metastability at certain compositions. That is, for a particular composition, one can plot the order parameter results for simulations started in different ordering schemes and note the prevalence of one order parameter at the expense of another. A specific example is that of x = 18/64 (Fig. 11) in which the Q_{1:1} is higher in value at low T than Q_{1:3}, but then decreases with a corresponding increase in Q_{1:3} at higher T, before disorder occurs. In fact, it is easier to identify fields of metastability in a plot of transition temperature T_{c} as a function of composition (Fig. 12); these fields have been hatched on that Figure. Although, as previously discussed, MC simulations do not incorporate a time component, it can easily be envisaged that this metastability might manifest itself in natural samples of the relevant compositions, if we assume that there is no kinetic effect which greatly favors one ordering scheme over another.
Figure 12 also illustrates the dilution phenomenon as a sharp decrease in T_{c} values (for the Q_{1:3} order parameter) as x decreases. A critical value of x (x_{c}, i.e., the value at which T_{c} = 0) cannot be determined accurately, as the order parameter graphs are not clear enough, but an approximate value of x_{c} ≈ 8/64–10/64 can be ascertained from the form of the Q_{1:3} curve; thus, compositions with Al/Si ratios lower than this cannot show ordering behavior. We can suggest that the same dilution behavior could occur for the Q_{1:1} order parameter, since it too shows a steep drop in T_{c} with decreasing x, but this is complicated on the graph by Q_{1:2}, which is related to Q_{1:1}.
Comparison of simulation results with ^{29}Si MAS-NMR experiments on phyllosilicates
We are able to use our MC code to record the number of nearest-neighbors to each Si atom in a configuration, and whether these neighbors are Al or Si. From these, we can compute the proportions of different clusters of atoms around Si, which is what is recorded by a ^{29}Si MAS-NMR experiment. Therefore, comparisons between our simulations and existing NMR data for phyllosilicates can be performed. The existing data are summarized in Table 6 for a variety of compositions of the tetrahedral sheet.
x/64 (E) | Origin | x/64 (S) | Si–0Al | Si–1Al | Si–2Al | Si–3Al |
---|---|---|---|---|---|---|
| ||||||
7.36 | Synthetic [64] | 8 | 61 | 39 | 0 | 0 |
9.6 | Synthetic [64] | 10 | 52 | 44 | 4 | 0 |
12.8 | Synthetic [64] | 13 | 35 | 55 | 10 | 0 |
16.64 | Synthetic [64] | 16 | 19 | 57 | 24 | 0 |
17.92 | Natural [57] | 18 | 15.5 | 54 | 30.5 | 0 |
27.52 | Synthetic [57] | 28 | 1 | 13.8 | 38 | 47.2 |
32 | Natural [57] | 32 | 0 | 0 | 0 | 100 |
Figures 13a–d show the simulated normalized NMR peak intensities for the compositions x = 8/64, 10/64, 13/64 and 16/64. Horizontal lines on these figures show the normalized NMR peak intensities obtained from experiments on compositionally-similar samples (the composition comparisons are given in Table 6).[5, 65]
It is clear from Figures 13a–d that there is no agreement between the experimental and simulated NMR data except at high temperatures (i.e., in the disordered regime).
At low temperatures, none of our simulations of Al-poor systems (except the 16/64 composition) exhibit long-range order, though one can note that occasional Al atoms line up at J_{3} positions (compare 8/64 and 16/64 snapshots in Fig. 10). Indeed, the most dilute compositions are not able to order, for reasons discussed earlier in this article. It is thus implied that the experimental samples are likely to be disordered, or at most short-range-ordered. The interpretation for this has been previously discussed for muscovite:[5] since the mineral samples were unlikely to have formed at the temperatures at which the experimental and simulation intensities matched (> 2500 K), it was more likely that during their formation, they had not reached their equilibrium cation distributions on cooling, but instead had traveled along a kinetic pathway that passes through short-range-ordered states similar to those found just above the simulated phase transition temperature (i.e., in the short-range-ordered regime). It is reasonable to make the same interpretation here, since the 16/64 simulation corresponds to that of muscovite and the other samples show sufficiently similar behavior.
In Figures 13e–g, data are shown for the composition x = 18/64; this is the only composition within the intermediate group for which experimental data exist. The three graphs are for simulations started in each of the three ordering schemes. All three simulations show rather similar results; the best agreement with experimental data is at high temperatures. Additionally, the metastable, low-temperature regions on the 1:1 scheme and 1:2 scheme plots are also in reasonable agreement with the experimental results; it is, however, less likely that these regions represent the configuration in the sample, since this is a natural sample and is more likely to have equilibrated than the synthetic samples in the Al-poor category.
The x = 28/64 and x = 32/64 simulation results are shown in Figures 13h–k. In the x = 28/64 case, there is agreement between the simulation and experiment at high temperature, and at low temperature in the simulation started in the 1:3 scheme. Again, the latter agreement is for the part of the simulation where the behavior is metastable. The 28/64 experimental sample is synthetic, so it is possible here that the sample is not well-equilibrated and hence could correspond to the metastable configuration. However, the metastable simulation configurations occur only up to ∼250 K in this system, and the sample synthesis took place between 1120 and 1350 K,[66] so the metastable configurations are unlikely to occur.
The agreement of the x = 32/64 case with experiment is the reverse of that seen in the other systems—the NMR intensities agree with the simulated intensities at low temperatures, that is, in the ordered regime. The explanation for this is that there is only one feasible ordered configuration for a 1:1 Al:Si ratio, and hence real samples will order in this pattern, since the introduction of even a few like-atom nearest-neighbor interactions will raise the energy of the system considerably. An alternative way of stating this is that at this composition, short-range order drives the formation of long-range order (the opposite of what is seen in dilute compositions, where short-range order can exist without long-range order). The strict alternation of Al and Si is indeed what has been observed by X-ray diffraction[67, 68] and neutron diffraction studies.[69]
Scenario 2: Three cation species ordering across one type of site
Al–Fe–Mg ordering in the octahedral phyllosilicate sheet
Having observed the complexity of the behavior of binary systems in the octahedral sheet, the case of the ordering of Al, Fe, and Mg in the octahedral phyllosilicate sheet is a natural extension, allowing us to probe the ordering behavior of three cation types.[37, 60, 70] This is an important advance in terms of cation ordering of solids, since the octahedral composition of most phyllosilicates comprises significant amounts of at least three species. The three-species octahedral ordering focuses on the octahedral sheet of illite/smectite minerals, and we constrain ourselves to the dioctahedral case.
Model of the octahedral sheet with three cation species
The derivation of the energy model corresponding to the case of three cation types in one network is given in Appendix C. The form of the model is similar to that of two cation types in one network, but the Js are more complex. The essential point to note is that it is easier to calculate two-species models for the three combinations of pairs of cations (Al–Fe, Al–Mg, Fe–Mg), by the approach already discussed in Scenario 1 above, and then compute three-species Js from the two-species ones through the equations
- (12)
where the two-species parameters appear on the left-hand side of these equations and the three-species parameters on the right-hand side.
Determination of cation-exchange potentials for three-species ordering in the octahedral sheet
The three-species J values were obtained (Table 7) from the two-species ones (Table 5), according to Eq. (12). Initially, the average values of two species J can be used. However, the samples in Table 4 can also be considered according to their tetrahedral charge: samples 1, 4, 6 which are typical of smectites and 3, 5, 6 which are typical of illites. This classification enables us to compute a further two sets of three-species Js, for smectitic and illitic compositions, respectively, which in turn allows us to investigate the possible differences in octahedral ordering behavior for illites and smectites. (It should be noted at this point that sample 2 is excluded from the classification as it contains Na as the interlayer cation instead of K, and sample 6 is considered to be a member of both sets as it is the only Fe–Mg composition.) In fact, four sets of three-species Js (Table 7) were calculated. The first and second sets of values were computed from illitic and smectitic samples, respectively, the third set was calculated from the average of the two species J, and the fourth set comes from the DFT two species J. This enables a broad study of three-species behavior to be undertaken with two different methodologies.
Interaction type | J_{1} | J_{2} | J_{3} | J_{4} |
---|---|---|---|---|
| ||||
Al–Al | 0.090a | 0.028a | 0.039a | −0.022a |
0.113b | 0.037b | 0.048b | –0.024b | |
0.105c | 0.033c | 0.049c | –0.029c | |
Al-Al (DFT) | 0.112d | 0.017d | 0.019d | 0.017d |
Fe–Fe | –0.075a | –0.023a | –0.031a | 0.023a |
–0.088b | –0.030b | –0.045b | 0.027b | |
–0.085c | –0.027c | –0.044c | 0.031c | |
Fe-Fe (DFT) | 0.011d | −0.024d | −0.026d | −0.014d |
Mg–Mg | 0.531a | 0.124a | 0.028a | 0.052a |
0.544b | 0.131b | 0.042b | 0.049b | |
0.545c | 0.127c | 0.041c | 0.049c | |
Mg-Mg (DFT) | 0.487d | 0.137d | 0.077d | −0.006d |
The values of J_{1} and J_{2} of Mg-Mg are drastically different that the rest, being the highest values, which means the Mg tend to be dispersed. DFT and empirical potential values show the same trend. The J_{1}, J_{2}, and J_{3} of Fe-Fe system are the lowest values and even they show negatives values, indicating an aggregation trend. In this system, the J_{1} value obtained with DFT calculations is positive and higher that the rest of J, this difference can come from the different compositions of the samples calculated with DFT.
MC simulations of three-species behavior in the octahedral sheet
We performed simulations of many different Al–Fe–Mg ratios, across the whole range of possible compositions, using the average J values in Table 7. The aim of performing many simulations of diverse compositions was to investigate the behavior with reference to both natural and synthetic phyllosilicates. We are unable to present results for all compositions here; however, a list of all compositions studied is given in Appendix D.
Figures 14a and 14b show heat capacity profiles and low-temperature snapshots for the compositions Al:Fe:Mg = 1:1:1, 1:3:2, 2:1:1. These compositions are representative examples of long-range order, exsolution (phase separation), and short-range order, respectively. Figure 14c shows only the Fe atoms in these three compositions, with nearest-neighbor Fe atoms having been linked together to elucidate the patterns in the structure. The 1:1:1 corresponds compositionally to a celadonite; it exhibits a striking pattern, in which the Fe atoms cluster together to fill all the octahedral sites in a hexagonal ring. The ordering in this composition is almost perfect. In the 1:3:2 composition, distinct regions of different behavior can be observed; at the top of the Figure, the behavior is similar to that of the 1:1:1 composition, whereas at the bottom right a two-species region of the superhexagon pattern, with composition corresponding to Fe:Mg = 3:1, is seen (compare Fig. 6c). This composition is rich in both Fe and Mg, and is unlikely to correspond to a natural composition for a dioctahedral phyllosilicate (although it could be considered as an Mg-rich nontronite). Finally, the 2:1:1 composition could represent a natural or synthetic smectite, and has Fe atoms mostly arranged in pairs or small chains—no long-range order is seen. However, short-range order is evident: Fe atoms cluster together in pairs or occasional larger clusters, and Mg atoms segregate from one another. In fact, this Fe and Mg behavior is observed as a general trend throughout our simulations of the octahedral sheet.[70, 71] The segregation of Mg is consistent with experimentally observed behavior in smectites, illites[12] and nontronites. The clustering of Fe is in agreement with experimental results for natural illites and smectites from IR and NMR experiments and from RMC simulations based thereon.[13] Experiments on nontronites have also found small domains of Fe separated by Al and Mg,[72] which is consistent with the lack of magnetic ordering found in these minerals at low temperature.[73]
The differences between the heat capacity profiles for 1:1:1 and 2:1:1 are typical: a phase transition is typified by divergent behavior, which is modified in a sample of finite size to a sharp, distinct peak. Conversely, the 2:1:1 composition displays a “Schottky anomaly,” which is a broad, asymmetric hump in the heat capacity which corresponds to a lack of ordering at the long-range scale.
These three samples are discussed here to illustrate the wide variety of possible behavior that can be obtained with only one set of J parameters, that is, by varying only the composition of the system. The three compositions discussed above represent both natural and synthetic clays, and are therefore interesting with respect to both of these aspects of clay mineralogy. We now consider these two different aspects with some specific examples and applications.
Natural clay minerals: illites and smectites
Among the most common naturally occurring clay minerals, are the illites and smectites which are typically Al-rich and relatively Mg- and Fe-poor. As previously stated, the main difference between the two mineral groups is the tetrahedral charge, TC, with higher TC being more typical of illites, and lower TC of smectites.
A long-standing puzzle in the field of clay mineralogy is the effect of the octahedral Fe distribution on the ^{27}Al NMR signal obtained from clay samples. The overall effect of Fe on this signal is inhibition; the paramagnetic character of Fe causes it to interact with the applied field. This inhibitor effect is stronger in smectites than in illites, and previous explanations of this phenomenon included the possibility of long-range order in illites, or a difference in the Fe cluster size in illites from that in smectites.[13]
An MC investigation of the behavior of illites and smectites should permit the resolution of this issue. The most suitable composition to study for this investigation is Al:Fe:Mg = 4:1:1, as it is a common composition for both illites and smectites. MC simulations of this composition with all four sets of Js were performed (Table 7).
Low-T snapshots of this composition, for each of the average empirical, illitic and smectitic parameter simulations, are shown in Figure 15. Also presented in Figure 15 are plots showing Al cations only with Al nearest-neighbors linked, Fe only with the Fe nearest-neighbors linked, and Mg only with, Mg atoms linked up to J_{3} distances. One can observe from Figure 15 that long-range order does not occur. For Al cations, some local linear patterns appear disordered along the octahedral sheet, and once more clustering of Fe and dispersal of Mg are evident. Fe cations form small clusters and Mg cations are separated with more than three neighbors distances. These phenomena are seen in all these sets of results. The conundrum of differing Fe inhibition of Al-NMR signal is solved by comparing our results for illitic and smectitic systems (Figs. 15b and 15c): namely, the cluster sizes in the illitic and smectitic samples are similar, but the number of isolated Fe atoms is much larger in the smectite simulation than in the illite. Then, the inhibition effect of Fe is higher in smectites than in illites according to previous experimental results.
Synthetic 2:1 dioctahedral phyllosilicates: the magnetic behavior of ^{VI}Fe-bearing compositions
A further function of our simulation approach is to offer insights in the field of 2:1 dioctahedral phyllosilicates synthesis. In this field, the philosophy is not to explain the behavior of existing samples, but to generate particular physical and/or chemical properties in a clay sample by focusing on a particular chemical composition and/or synthesis approach (kinetic history, thermal history etc.).
One particularly interesting issue is the magnetic behavior of Fe-bearing clay minerals. Experiments on natural samples suggest that there is no magnetic ordering;[73] and this is supported by the observation of small domains of Fe with intervening Al and Mg in some nontronites.[72] However, if magnetic ordering were to occur in these layer silicates, it could give rise to interesting magnetic properties, in view of the quasi-2D nature of the clay mineral structure. To illustrate this point, we present here a selection of simulation results in which novel Fe patterns are observed, and relate this to the compositions of the samples.
In illite/smectite samples with low Al but high Fe and Mg, the iron forms small clusters which are not interconnected (e.g., 1:3:2, already shown in Fig. 14c). This behavior is also observed in the nontronites (e.g., 1:8:3, Fig. 16a), although one nontronite sample (1:4:1, Fig. 16b) did exhibit interconnected Fe regions. As stated above, experimental studies of nontronite[72] revealed no magnetic ordering at low temperature, which would be consistent with a lack of connections between clusters. Furthermore, Vantelon et al. (2003)[11] used EXAFS and IR spectroscopy to study several ferruginous smectites, finding Fe clustering and Mg dispersion in most of the samples, and also concluding that samples from only one locality exhibited an ordered Fe distribution, with the rest either being disordered or short-range-ordered. The 2:3:1 sample (Fig. 16c) is another in which Fe atoms form chains; this is iron-rich but outside the common compositional range for nontronites. If the amount of Al is increased to 50%, with Mg > Fe, long-range ordering decreases, and the Fe clusters take different shapes (e.g., 3:1:2, Fig. 16d). If on the other hand, Fe > Mg, the clusters become longer (e.g., 7:4:1, Fig. 16e) and can begin to form chains (e.g., 3:2:1, Fig. 16f). In the DFT average set, for the last composition (3:2:1), the Fe cations are homogeneously distributed along the lattice, without no AlAl, FeFe, MgMg, and FeMg first-neighbor pairs are observed and all first-neighbor pairs are AlFe and AlMg.
The magnetic (or otherwise) behavior of a sample does of course depend on the behavior of adjacent octahedral sheets—it is possible that effects in a single sheet could be cancelled out by polytypic rotations of adjacent sheets, for example, and this is a factor which is not addressed by these simulations of only one octahedral sheet. The point is that our simulations have shown the effect of composition on the Fe distribution, which could facilitate the synthesis of compositions which are most likely to exhibit magnetic behavior, thus saving experimental time and resources.
Scenario 3: Three cation species ordering across two types of site
Al–Si and Al–Mg ordering in phengite
We now turn to phengite, [Al_{3}Mg]^{IV}[Si_{7}Al]O_{20}(OH)_{4}, which is related to muscovite by the Tschermak substitution, ^{IV}Si^{VI}Mg ^{IV}Al^{VI}Al.[47] This is a more complex case than the case of muscovite discussed above. There is now a mixture of Al and Mg on the octahedral sites, and a mixture of Al and Si on the tetrahedral sites, and this leads to the possibility of coupling between the ordering processes in the tetrahedra and octahedra.
Model of phengite: two interacting networks of sites, one containing A and B cations, the other A and C cations
The derivation of the energy expression governing this system is given in Appendix 5. The bond energy expression is actually quite similar to that for muscovite, Eq. (11):
- (13)
except that the parameters are now defined differently, being for the description of interactions between the two networks. In practice, this does not affect our analysis; however, we can use the single-network formalism for the interactions within the octahedral and tetrahedral sheets.
Determination of cation-exchange potentials for phengite
For the investigation of phengite, we used the J values determined for muscovite for interactions within the tetrahedral sheet (T–T), and those determined for a smectite (sample 3 from Table 5; this composition is very close to phengite) for interactions within the octahedral sheet (O–O). The initial structural parameters were those obtained for a 2M_{1} phengite by Gueven.[74] The usual set of configurations can be generated, but instead of calculating a complete set of interactions, the TT and OO parameters can be fixed, and new values can be calculated only for the interactions between the tetrahedra and octahedra (TO), according to Eq. (13). In the set of configurations, we used a different Al–O1 Buckingham potential from the one used in other studies (see Table 1), and we modified the Al–O three-body potential to allow larger AlO_{4} tetrahedra (see Table 2). These two factors gave us better results, that is, more of the configurations were successfully optimized with GULP, thereby providing a larger data set from which to compute the J values.
The T–O interaction parameters are labeled J_{a–d} and are shown in Table 8. Figure 17 also shows examples of the distances in the structure over which these interactions operate.
Parameter | Distance (Å) | Value (eV) |
---|---|---|
| ||
J_{a} | 3.1–3.23 | 0.75(8) |
J_{b} | 4.37–4.6 | 0.06(10) |
J_{c} | 5.27–5.42 | 0.02(8) |
J_{d} | 5.13–5.24 | 0.24(10) |
MC simulations of phengite
Simulations on three systems relevant to phengite were performed: a single tetrahedral sheet, a single octahedral sheet and a T–O–T layer.
The simulation of the tetrahedral sheet used the first four muscovite T–T Js from Table 3, and had a cation composition of Al:Si = 1:7. This simulation is thus equivalent to that shown in Figure 10a; there is no long-range order at low temperature. There is a certain degree of short-range order, with Al atoms lining up with respect to each other across J_{3} linkages. The lack of long-range order is not surprising in view of the composition of the sheet. A cation ratio of 1:7 means that it is possible to have complete avoidance of Al–Al nearest-neighbor pairs without the need for long-range order. This phenomenon has been termed “dilution” by workers in Cambridge, and has been discussed previously with respect to other minerals, for example, leucite.[75, 76]
The octahedral sheet simulation was for the composition Al:Mg = 3:1, and used the illite/smectite O–O Js from sample 3 in Table 5 (this composition being very close to that of phengite). The results of the simulation hence correspond to those in Figure 6b. The pattern consists of chains of Mg atoms linked through J_{3} interactions, with adjacent chains being linked by J_{4} interactions; the so-called “in-phase” arrangement (compare the out-of-phase, or superhexagon, arrangement seen in the 3:1 cation ratio of the muscovite tetrahedral sheet, Figure 5, and the 3:1 Al:Fe composition of the octahedral sheet, Fig. 6c).
The T–O–T layer simulation used the intrasheet muscovite T–T Js (first four Js in Table 3), the illite/smectite O–O Js (Table 5, sample 3), and the T–O Js calculated for phengite (Table 8). In addition to the various J parameters, an artificial energy term was included; the idea of this term is to avoid the placing of Mg on tetrahedral sites and Si on octahedral sites (neither of these is known in naturally-occurring phyllosilicate minerals). The artificial energy term has the effect of causing a prohibitively large increase in the energy if tetrahedral Mg or octahedral Si is generated in a MC simulation step.
The simulation cell size and shape are different from those used for the T sheet and O sheet, but this does not affect the results, which are shown in Figure 18. It can be noted immediately from Figure 18 that the behavior of the whole T–O–T layer is different from that of the O sheet and T sheet alone (compare the T sheets in Figs. 18b and 18d with the T sheet in Fig. 10a, and the O sheet in Fig. 18c with the O sheet in Fig. 6b). In the whole-layer simulation, the tetrahedral Al atoms are even more dispersed than before, across distances of ∼9 Å. The octahedral Mg atoms are arranged mostly in nearest-neighbor pairs.
At this point, it is worth noting that no simulation of interacting T–O–T layers was performed. The reason for this is that in the simulations of muscovite, the intralayer, and interlayer parameters were found mostly to be much smaller than the intrasheet parameters, and the intrasheet ordering was thus found to be more important than the ordering of interacting layers. This argument can be extended to phengite, using the muscovite T–T Js in the phengite simulations.[47]
Discussion and comparison with experiment
The results from the T-only and O-only simulations for phengite show markedly different behavior from the T–O–T layer simulations. This proves unequivocally that the T–O interactions are important if one is to create a realistic model for ordering in phengite, which is an important result with respect to simulations of coupled and/or competing ordering processes.
The simulation results can be compared with experimental results for phengite. Pavese et al. [77-79] have conducted many experimental studies of the 2M_{1} and 3T polytypes of phengite. Much of their work suggests a low degree of octahedral cation order in both polytypes, but by contrast, their most recent investigation of a different natural phengite 3T sample from that previously studied, concluded that the sample exhibited both octahedral and tetrahedral order.[80] The authors of those works are, however, somewhat cautious in their assertion of octahedral order in the first set of samples, having suggested that their results may be due to structural defects such as stacking faults.[79] The computational results we have obtained here do not indicate long-range order; instead, they indicate ordered regions of moderate size (∼25–50 Å). The earlier set of results obtained by Pavese et al. may well reflect such medium-range order. It is likely that any possible kinetic impediment to the evolution of long-range order in natural samples would affect the measured values of long-range order from diffraction experiments, as would the presence of stacking faults.
The most interesting experimental technique with which can be compared these simulation results is ^{29}Si MAS-NMR studies of Al–Si ordering in phyllosilicates, discussed in more detail above. Referring back to Table 8, one can note that the experimental study[64] of a sample with Al:Si = 0.12:1 had normalized peak intensities of 39% Si–2Si1Al and 61% Si–3Si. The composition of this sample compares favorably with that of phengite (Al:Si = 1:7). However, interestingly, the tetrahedral sheet configurations from both the T-only and the TOT layer simulations give similar calculated NMR intensities, 42% Si–2Si1Al and 58% Si–3Si, in spite of their very different configurations (compare Figs. 6b and 18b/d). From this, the NMR cannot be used to determine the cation configuration in the tetrahedral phengite sheet, as more than one configuration gives a similar NMR signal. On the other hand, the observed signal most likely indicates a really dispersed pattern of Al atoms, as would be expected for such a dilute composition.
Conclusions
- Top of page
- Abstract
- Introduction
- Structure of 2:1 Dioctahedral Phyllosilicates
- Methodology
- Case Studies
- Conclusions
- Acknowledgments
- Appendix: A: The Ordering Energy Expression for the Case of Two Species Ordering Across One Type of Site (example: Al–Si in Muscovite)
- Appendix: B: Force Field of Empirical Interatomic Potential Used to Study Phyllosilicates
- Appendix: C: The Ordering Energy Expression for the Case of Three Species Ordering Across One Type of Site (Example: Al–Fe–Mg in Illite/Smectite)
- Appendix: D: Full List of Al–Fe–Mg Compositions Simulated
- Appendix: E: The Ordering Energy Expression for the Case of Three Cations Ordering Across Three Types of Site, in Two Distinct A–B and A–C Networks (Example: Al–Si and Al–Mg in Phengite)
- Biographies
The aim of this article has been to demonstrate the ways in which MC simulation results can provide the phyllosilicates community with useful insights into cation ordering behavior in layer silicates. Here, we will flag the main conclusions which we believe are of particular interest to fellow researchers in phyllosilicates.
At this point, we remind the reader that our simulations use a technique whose scope is purely thermodynamic—the concept of swapping a pair of randomly selected atoms which may be a long distance from one another does not equate to a realistic process in kinetic terms. Consequently, the configurations that are obtained from MC simulations will correspond to representative equilibrium states, and may not reproduce configurations found in natural materials where ordering is hindered kinetically. That said, the work in the first example, representing the Al/Si ordering in muscovite, suggests that short-range ordering may correspond to the equilibrium state at higher temperatures. On the other hand, there is a lot of advantage in using a method in which kinetic constraints are “turned off,” in that it is able to provide detailed information about the fundamental ordering processes in a way that may be impossible from experiment.
With this in mind, we can turn to specific findings from the research reviewed in this article.
First, we have illustrated that the muscovite tetrahedral sheet exhibits long-range order, and we have shown that, from two ordered states proposed on the basis of homogeneous charge distribution (“in-phase” and “out-of-phase”/“superhexagon”) in the muscovite tetrahedral sheet, the latter ordering is exhibited. Simulations show that 2D ordering sets in before 3D ordering, and thereby effectively precludes the latter. This is because the interactions within one tetrahedral sheet are very much stronger than those between tetrahedral sheets, and this itself is an important result that we might have expected for these very anisotropic mineral structures.
Second, the extension of the muscovite study to the case of the general Al–Si ordering behavior in the tetrahedral sheet has shown a wide range of behavior therein. By changing only the composition, three possible ordering schemes exist across the compositional range from Al:Si = 1:7 to Al:Si = 1:1. Evidence for metastability is also displayed by certain compositions, which is of interest in terms of the effects of kinetic inhibition on the observed atomic configurations in 2:1 dioctahedral phyllosilicates.
Third, we have shown simulated NMR peak intensities for tetrahedral sheets with various Al:Si ratios and with respect to several possible ordering schemes, which has enabled us to conclude that the only experimental system in which long-range order is attained is the Al:Si = 1:1 composition. All other experiment-simulation comparisons were in the best agreement either in the vicinity of, or above, the ordering transition temperature. This means that natural and synthetic samples previously studied by experiment are most likely disordered or partially ordered.
Fourth, a large number of simulations of three-species ordering has been performed in the octahedral phyllosilicate sheet. Using only one set of interaction parameters, very different behaviors can be observed depending on Al:Fe:Mg ratio. Few compositions exhibit long-range order, some exhibit exsolution, but most exhibit only short-range order. A general tendency of Fe to form clusters and of Mg to distance itself from other Mg atoms has been observed, which is in agreement with experimental findings and the results of RMC simulations based on NMR and IR experiments.
Fifth, by using modified values for the octahedral sheet interaction parameters, it is possible to clarify the phenomenon of the effect of Fe on the ^{27}Al NMR signal from the octahedral sites of phyllosilicates. Previous studies had suggested that the different behavior of illites and smectites (namely that the effect of Fe on the NMR signal from smectites is greater than the effect on the signal from illites) was probably due to different Fe cluster sizes in these minerals. We have shown that it is not due to differing cluster sizes, as these are broadly similar in the two groups. Instead, there are fewer isolated Fe cations in illites than in smectites, and this is what causes the difference.
Sixth, these simulations have shown that the phengite is not expected to exhibit long-range order. This is a useful result in the context of the many and varied experimental results on both 2M_{1} and 3T polytypes. Although neither the effects of polytypic variation, nor of stacking faults has been investigated, the presence of medium-range-ordered domains of approximate size 25–50 Å is demonstrated, the existence of which is in accord with a significant proportion of the experimental conclusions. The ^{29}Si MAS NMR may not be a particularly useful technique for determining the tetrahedral ordering behavior of phyllosilicates with low Al:Si ratios, since for a given Al:Si ratio there is more than one possible atomic configuration which will yield similar NMR peak intensities, because the Al atoms are spread out, and NMR is a probe for short-range structure.
Finally, it is well known that DFT calculations are a valuable tool for modeling chemical compounds, minerals, and materials. The cation-exchange potentials calculated by the two different methods (DFT and empirical potentials) are similar in most cases. Although the computational effort for the DFT method is large in comparison with the empirical approach, the similarity between results from each method does not suggest that employing the dual approach is without value: rather, it gives confidence in the reliability of the results we have obtained. The ab initio nature of the DFT method highlights its power as a general tool for studying cation order-disorder phenomena.
What these results taken together show is that the methodologies reviewed in this article are capable of giving insights into the ordering processes and mechanisms that may be hidden from experiment. The methodologies themselves are flexible and robust, particularly since the interatomic potentials for aluminosilicates are reasonably well defined for the empirical potential method. Here, we have used our research on 2:1 dioctahedral phyllosilicates as a case study, but the great compositional variety in this mineral system effectively illustrates the extensibility of the exchange potential methodology: the approach should work with equal success for the study of other layer silicates, including cation ordering with the tetrahedral and octahedral layers of trioctahedral phyllosilicates, and ordering of the interlayer cations. More widely, it should also be applicable to the study of other natural or synthetic nanomaterials; indeed, we hope that this article will encourage this.
Acknowledgments
- Top of page
- Abstract
- Introduction
- Structure of 2:1 Dioctahedral Phyllosilicates
- Methodology
- Case Studies
- Conclusions
- Acknowledgments
- Appendix: A: The Ordering Energy Expression for the Case of Two Species Ordering Across One Type of Site (example: Al–Si in Muscovite)
- Appendix: B: Force Field of Empirical Interatomic Potential Used to Study Phyllosilicates
- Appendix: C: The Ordering Energy Expression for the Case of Three Species Ordering Across One Type of Site (Example: Al–Fe–Mg in Illite/Smectite)
- Appendix: D: Full List of Al–Fe–Mg Compositions Simulated
- Appendix: E: The Ordering Energy Expression for the Case of Three Cations Ordering Across Three Types of Site, in Two Distinct A–B and A–C Networks (Example: Al–Si and Al–Mg in Phengite)
- Biographies
The Monte Carlo simulations were performed on the Cambridge Mineral Physics Group's Beowulf cluster at the Department of Earth Sciences, and on the Cambridge-Cranfield High Performance Computing Facility (CCHPCF). Authors are grateful to the “Centro de Cálculo de la Universidad de Granada” (UGRGRID) and “Centro de Computación de Galicia (CESGA) for allowing the use of their computational facilities”.
Appendix: A: The Ordering Energy Expression for the Case of Two Species Ordering Across One Type of Site (example: Al–Si in Muscovite)
- Top of page
- Abstract
- Introduction
- Structure of 2:1 Dioctahedral Phyllosilicates
- Methodology
- Case Studies
- Conclusions
- Acknowledgments
- Appendix: A: The Ordering Energy Expression for the Case of Two Species Ordering Across One Type of Site (example: Al–Si in Muscovite)
- Appendix: B: Force Field of Empirical Interatomic Potential Used to Study Phyllosilicates
- Appendix: C: The Ordering Energy Expression for the Case of Three Species Ordering Across One Type of Site (Example: Al–Fe–Mg in Illite/Smectite)
- Appendix: D: Full List of Al–Fe–Mg Compositions Simulated
- Appendix: E: The Ordering Energy Expression for the Case of Three Cations Ordering Across Three Types of Site, in Two Distinct A–B and A–C Networks (Example: Al–Si and Al–Mg in Phengite)
- Biographies
We begin from Eq. (1), which we have now written explicitly in terms of Al and Si cations:
- (A1)
We represent the fraction of Al atoms as x and the fraction of Si atoms as 1 – x. The probability of encountering an Al–Al interaction is given as P_{AlAl} and so the number of Al–Al interactions, N_{AlAl}, can be denoted as
- (A2)
where N is the total number of sites in the network and z is the number of neighboring sites. The numbers of Al–Si and Si–Si interactions can be shown to depend only on the number of Al–Al interactions, as follows. The probability of encountering an Al–Si interaction is 1 – P_{AlAl} –— if a given Al atom does not have an Al neighbor, then the neighbour must be Si. Hence
- (A3)
and finally, the number of Si–Si interactions is just the total number of interactions less the number of Al–Al and Al–Si interactions:
- (A4)
As stated in Part I of this article, we have written the formula for each N parameter in terms of N_{AlAl} only. This means that, finally, we can write Eq. (1) explicitly in terms of N_{AlAl} only:
- (A5)
with —the energy involved in forming an Al–Al linkage and an Si–Si linkage at the expense of two Al–Si linkages. This represents an exchange of cations, hence we named J as cation-exchange potential. All the terms which are not dependent on any N parameters have been grouped into the constant E_{0}. Equation (A5) is trivially extensible to more than one J parameter (i.e., ), with this latter equation being equivalent to Eq. (2), and thus to Eq. (10) with the omission of the constant term.
Appendix: B: Force Field of Empirical Interatomic Potential Used to Study Phyllosilicates
- Top of page
- Abstract
- Introduction
- Structure of 2:1 Dioctahedral Phyllosilicates
- Methodology
- Case Studies
- Conclusions
- Acknowledgments
- Appendix: A: The Ordering Energy Expression for the Case of Two Species Ordering Across One Type of Site (example: Al–Si in Muscovite)
- Appendix: B: Force Field of Empirical Interatomic Potential Used to Study Phyllosilicates
- Appendix: C: The Ordering Energy Expression for the Case of Three Species Ordering Across One Type of Site (Example: Al–Fe–Mg in Illite/Smectite)
- Appendix: D: Full List of Al–Fe–Mg Compositions Simulated
- Appendix: E: The Ordering Energy Expression for the Case of Three Cations Ordering Across Three Types of Site, in Two Distinct A–B and A–C Networks (Example: Al–Si and Al–Mg in Phengite)
- Biographies
In the equations which follow, r indicates an atomic separation and θ a bond angle, with an equilibrium value being denoted by a zero subscript.
Short-range interactions (e.g., K–O, Al–O) are modeled using the Buckingham potential form:
- (B1)
in which A, ρ and C represent tunable parameters. The second term in Eq. (B1) describes the dispersive interaction, and is usually zero, except in the case of Si–O and O–O parameters.
The O–H interactions of the hydroxyl group are modeled using the Morse potential:
- (B2)
in which a, D and r_{0} are tunable parameters. This description is considered to be sufficient for the O–H group; the constituent atoms are not coupled via any other interactions.
Interactions for atoms comprising polyhedral units such as the AlO_{6} octahedron or SiO_{4} tetrahedron are described by the three-body potential:
- (B3)
where the equilibrium bond angle, θ_{0}, is usually taken to be that subtended at the centre of an ideal polyhedron (i.e., 90° for an octahedron and 109.47° for a tetrahedron).
Finally, the polarizability of the oxygen atom is handled by use of the shell model for O–O interactions, such that the oxygen atom is decomposed into two parts, the “core,” consisting of the nucleus and tightly-bound inner electrons, and the “shell”, comprising the outer electrons and considered to be massless. The core and shell are coupled together by a harmonic interaction of the form
- (B4)
where K is a tunable parameter and d is the separation of the centres of the core and shell.
Appendix: C: The Ordering Energy Expression for the Case of Three Species Ordering Across One Type of Site (Example: Al–Fe–Mg in Illite/Smectite)
- Top of page
- Abstract
- Introduction
- Structure of 2:1 Dioctahedral Phyllosilicates
- Methodology
- Case Studies
- Conclusions
- Acknowledgments
- Appendix: A: The Ordering Energy Expression for the Case of Two Species Ordering Across One Type of Site (example: Al–Si in Muscovite)
- Appendix: B: Force Field of Empirical Interatomic Potential Used to Study Phyllosilicates
- Appendix: C: The Ordering Energy Expression for the Case of Three Species Ordering Across One Type of Site (Example: Al–Fe–Mg in Illite/Smectite)
- Appendix: D: Full List of Al–Fe–Mg Compositions Simulated
- Appendix: E: The Ordering Energy Expression for the Case of Three Cations Ordering Across Three Types of Site, in Two Distinct A–B and A–C Networks (Example: Al–Si and Al–Mg in Phengite)
- Biographies
For the case of three ordering cations in one network, the formalism becomes a little more complicated than that in Appendix A. By analogy with Eq. (A), the energy of a system with Al, Fe, and Mg cations can be written as
- (C1)
As was the case for two ordering cations in one network, the number of Al–Al interactions can be expressed in terms of the probability of encountering an Al–Al interaction:
- (C2)
The number of interactions of Al with other atoms is given by the expression
- (C3)
with analogous expressions following for the other pairs of cations. It is hence easier to express the equations as a matrix:
- (C4)
the solutions of which are
- (C5)
Placing these values into Eq. (C1) gives
- (C6)
which can be rearranged in terms of N_{AlAl}, N_{FeFe} and N_{MgMg} as follows:
- (C7)
with
- (C8)
To determine the J parameters for a three-species system of A, B and C cations, we use the fact that the parameters for three-species systems are related to those for the three two-species systems which can occur (A and B, A and C, B and C). For Al, Fe, and Mg cations, the relevant two-species equations are analogous to Eq. (A5) above, and are as follows:
- (C9)
and these can be combined to give the three-species parameters:
- (C10)
In practice, then, simulating three-species systems is achieved by first computing the values for the two-species systems, and then using these to generate those for the three-species systems.
Appendix: D: Full List of Al–Fe–Mg Compositions Simulated
- Top of page
- Abstract
- Introduction
- Structure of 2:1 Dioctahedral Phyllosilicates
- Methodology
- Case Studies
- Conclusions
- Acknowledgments
- Appendix: A: The Ordering Energy Expression for the Case of Two Species Ordering Across One Type of Site (example: Al–Si in Muscovite)
- Appendix: B: Force Field of Empirical Interatomic Potential Used to Study Phyllosilicates
- Appendix: C: The Ordering Energy Expression for the Case of Three Species Ordering Across One Type of Site (Example: Al–Fe–Mg in Illite/Smectite)
- Appendix: D: Full List of Al–Fe–Mg Compositions Simulated
- Appendix: E: The Ordering Energy Expression for the Case of Three Cations Ordering Across Three Types of Site, in Two Distinct A–B and A–C Networks (Example: Al–Si and Al–Mg in Phengite)
- Biographies
The following is a list of three-species compositions studied, and further details are available from the authors on request. The compositions are listed according to their labels in this article (if applicable), and then as percentages (to 1 d.p.) of total dioctahedral occupancy, for easy comparison.
Composition | % ^{VI}Al | % ^{VI}Fe | % ^{VI}Mg |
---|---|---|---|
1:1:1 | 33.3 | 33.3 | 33.3 |
1:1:4 | 16.7 | 16.7 | 66.7 |
1:2:3 | 16.7 | 33.3 | 50.0 |
12:7:5 | 50.0 | 29.2 | 20.8 |
1:3:2 | 16.7 | 50.0 | 33.3 |
1:4:1 | 16.7 | 66.7 | 16.7 |
1:8:3 | 8.3 | 66.7 | 25.0 |
2:1:1 | 50.0 | 25.0 | 25.0 |
2:1:3 | 33.3 | 16.7 | 50.0 |
2:3:1 | 33.3 | 50.0 | 16.7 |
2:7:3 | 16.7 | 58.3 | 25.0 |
3:1:2 | 50.0 | 16.7 | 33.3 |
3:2:1 | 50.0 | 33.3 | 16.7 |
4:1:1 | 66.7 | 16.7 | 16.7 |
4:5:3 | 33.3 | 41.7 | 25.0 |
7:2:3 | 58.3 | 16.7 | 25.0 |
7:4:1 | 58.3 | 33.3 | 8.3 |
8:1:3 | 66.7 | 8.3 | 25.0 |
9:1:2 | 75.0 | 8.3 | 16.7 |
Appendix: E: The Ordering Energy Expression for the Case of Three Cations Ordering Across Three Types of Site, in Two Distinct A–B and A–C Networks (Example: Al–Si and Al–Mg in Phengite)
- Top of page
- Abstract
- Introduction
- Structure of 2:1 Dioctahedral Phyllosilicates
- Methodology
- Case Studies
- Conclusions
- Acknowledgments
- Appendix: A: The Ordering Energy Expression for the Case of Two Species Ordering Across One Type of Site (example: Al–Si in Muscovite)
- Appendix: B: Force Field of Empirical Interatomic Potential Used to Study Phyllosilicates
- Appendix: C: The Ordering Energy Expression for the Case of Three Species Ordering Across One Type of Site (Example: Al–Fe–Mg in Illite/Smectite)
- Appendix: D: Full List of Al–Fe–Mg Compositions Simulated
- Appendix: E: The Ordering Energy Expression for the Case of Three Cations Ordering Across Three Types of Site, in Two Distinct A–B and A–C Networks (Example: Al–Si and Al–Mg in Phengite)
- Biographies
For the case of phengite, the interaction parameters between two networks can be calculated—the parameters within each of the individual networks can be computed by a method analogous to that for muscovite above.
There are N atoms in each network, and xN Al and (1 – x)N Si cations in one network and yN Al and (1 – y)N Mg cations in the other network. Here, we will consider the simplest case where each site in the first network interacts with only one site in the second network. The parameter N_{AlAl} now represents the number of Al–Al interactions between the networks. Thus, the number of Al–Mg bonds between the networks is
- (E1)
and the number of SiAl bonds between the networks is
- (E2)
so that the number of SiMg bonds between the networks is
- (E3)
The bond energy expression thus becomes
- (E4)
The interaction parameter (cation-exchange potential) here has been explicitly labeled J_{Al–Al} instead of just J here, as it is no longer symmetric with respect to the different atom types.
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Biographies
- Top of page
- Abstract
- Introduction
- Structure of 2:1 Dioctahedral Phyllosilicates
- Methodology
- Case Studies
- Conclusions
- Acknowledgments
- Appendix: A: The Ordering Energy Expression for the Case of Two Species Ordering Across One Type of Site (example: Al–Si in Muscovite)
- Appendix: B: Force Field of Empirical Interatomic Potential Used to Study Phyllosilicates
- Appendix: C: The Ordering Energy Expression for the Case of Three Species Ordering Across One Type of Site (Example: Al–Fe–Mg in Illite/Smectite)
- Appendix: D: Full List of Al–Fe–Mg Compositions Simulated
- Appendix: E: The Ordering Energy Expression for the Case of Three Cations Ordering Across Three Types of Site, in Two Distinct A–B and A–C Networks (Example: Al–Si and Al–Mg in Phengite)
- Biographies