On the theory of Al,Si ordering in albite

Calculations of the equilibrium distribution of Al, Si in the albite framework based on quasi-chemical theories of order, disorder transformations (Yang 1945; Yang and Li 1947; Li 1949) were made for a two-dimensional framework model. The ordering is caused by the energy of Al, Si interchange between sites of different crystal-chemical types and the energy of nearest neighbour interaction. By taking into account the decrease in the energy of interchange between sites with increasing disordering and with increasing temperature, and by examining different relationships for site-to-site interchange energy and the nearest neighbour interaction, it is possible to understand the basic characteristics of the transformation from low (essentially ordered) to high (essentially disordered) albite as revealed by experiment. These characteristics are: (1) abrupt variation of the equilibrium degree of order within a narrow temperature range and possible first order phase transformation for the transition from low-albite to high-albite, (2) hysteresis of the synthetic high albite transformation path and of the low albite hydrothermal “annealing” path, (3) presence of a temperature range where high albite is stable and has a continually changing equilibrium degree of order.     

Kinetic rate laws derived from order parameter theory III: Al,Si ordering in sanidine

The time dependence of the ordering and disordering of Al and Si in sanidine is described within the framework of the kinetic rate laws developed in papers I and II of this series. It was found that the relevant order parameter Q _t is homogeneous and non-conserved. The rate law is:

Al/Si interdiffusion in albite: effect of pressure and the role of hydrogen

The effect of pressure on the rate of Al/Si disorder in albite has been determined at temperatures from 800° C to 1050° C and at pressures up to 24 kbar, using dried samples in welded Pt containers, in piston-cylinder devices and internally-heated gas apparatus. In the piston-cylinder device with NaCl medium, the effect of pressure is profound. A pure low albite from Clear Creek, California reaches the equilibrium state of disorder at 850° C and 22 kbar in 10 h, whereas at 6 kbar it has not equilibrated in three weeks, and at one bar it probably cannot be disordered at 850° C in the laboratory. The enhancement of Al/Si interdiffusion takes place under dry conditions: any H₂O penetrating the samples would have produced melting, and none was observed. Hydrogen, however, is produced by dissociation of moisture in the pressure medium and can penetrate the Pt sample capsules. If the samples are deprived of hydrogen by replacing NaCl with glass or by embedding the samples in a hydrogen getter such as Fe₂O₃ or ZnO, the order-disorder reaction is greatly inhibited.A mechanism is suggested in which OH^– groups are formed by hydrogen hopping in albite. The smaller charge on the tetrahedral complex induces transient coordination of Al greater than four at elevated pressures, facilitating Al/Si interchange during the bond-breaking process. This results in an exponential pressure dependence of diffusion. It is also suggested that the presence of hydrogen at high pressures can greatly effect the mechanical properties and reactivity of deep crustal and mantle rocks. In nature, as well as in the laboratory, equilibration at elevated pressures may be as much dependent on the availability of hydrogen as on time or temperature.     

Mg-cordierite: Si/Al ordering,optical properties,and distortion

Flux-grown crystals of Mg-cordierite, Mg_1.93 Al_3.95 Si_5.07 O₁₈ synthesized by Lee and Pentecost (1976) appear biaxial (2V _x=10°–25°) under the polarizing microscope whereas their distortion index =0°. Between crossed polars, (001) sections display lamellar and cyclic twinning on {110} and, less frequently, {310}. As duration of annealing at 1,300° C, increased, 2V _x increased. Simultaneously, undulatory extinction and intragrain variations in 2V _x increased slightly up to 4 h annealing, then steadily decreased. For this Mg-cordierite, which lacks significant channel H₂O or CO₂, 2V _x and reach maxima of 88° (=589 nm) and 0.25° after 42 h of annealing but other sectors still display lesser values for 2V _x. Presumably, to the extent 2V _x is less than 88°, these sectors represent intergrown submicros copically twinned orthorhombic domains and thus possess shortrange but not long-range order. Annealing at 1,300° C likely increased long-range order by promoting growth of larger domains at the expense of smaller ones. Ultimately, two differently oriented domains, growing toward each other by annexation (and re-orientation) of smaller domains, meet in a twin boundary that, with time, tends to become straight.The cause of intermediate values for , whether compositional or from submicroscopic {110} or {310} twinning, may be revealed by single crystal X-ray photographs. Streaking of diffraction spots along a^* or b^* (but not c^*) will indicate such twinning as the sole or major cause.     

Si,Al ordering in leucite by high-resolution 27Al MAS NMR spectroscopy

High-resolution ²⁷Al MAS NMR spectra of natural leucite recorded at H ₀=11.7T contain three resolvable resonances at ²⁷Al δ _i = 69.2, 64.7, and 61.0±0.5 ppm. These three resonances are assigned to the three inequivalent framework positions of leucite: T3, T2, and T1, respectively. Fitting the observed spectra yields a Si,Al distribution for leucite in which approximately one-half of the Al is in T1 and one-quarter in each of T2 and T3. This Si,Al distribution differs substantially from those obtained by previous workers using ²⁹Si NMR spectroscopy and X-ray diffraction. New ²⁹Si NMR spectra and revision of previously reported ²⁹Si NMR peak assignments, however, make the ²⁷Al and ²⁹Si NMR results consistent. The ²⁷Al δ _i correlate linearly with the mean T-O-T′ bond angles of the average structure, which allows the peak assignments to be made. However, this correlation lies distinctly toward higher frequency and larger bond angles than correlations for Si,Al ordered aluminosilicates, suggesting that the mean T(Al)-O-T′(Si) bond angle for each site in leucite is smaller than the mean bond angle of the average structure, which is averaged over T(Al)-O-T′(Si) and T(Si)-OT′(Si,Al) angles.     

Equilibrium thermodynamics of Al/Si ordering in anorthite

Variations in the equilibrium degree of Al/Si order in anorthite have been investigated experimentally over the temperature range 800-1535° C. Spontaneous strain measurements give the temperature dependence of the macroscopic order parameter, Q, defined with respect to the \(C\bar 1 \rightleftharpoons I\bar 1\) phase transition, while high temperature solution calorimetric data allow the relationship between Q and excess enthalpy, H, to be determined. The thermodynamic behaviour can be described by a Landau expansion in one order parameter if the transition is first order in character, with an equilibrium transition temperature, T _tr, of ～2595 K and a jump in Q from 0 to ～0.65 at T_tr. The coefficients in this Landau expansion have been allowed to vary with composition, using Q=1 at 0 K for pure anorthite as a reference point for the order parameter. Published data for H and Q at different compositions allow the calibration of the additional parameters such that the free energy due to the \(C\bar 1 \rightleftharpoons I\bar 1\) transition in anorthite-rich plagioclase feldspars may be expressed (in cal. mole^-1) as: \(\begin{gathered}G = \tfrac{1}{2} \cdot 9(T - 2283 + 2525X_{Ab} )Q^2 \\ {\text{ + }}\tfrac{1}{4}( - 26642 + 121100X_{Ab} )Q^4 \\ {\text{ + }}\tfrac{1}{6}(47395 - 98663X_{Ab} )Q^6 \\ \end{gathered}\) where X _Ab is the mole fraction of albite component. The nature of the transition changes from first order in pure anorthite through tricritical at ～An₇₈ to second order, with increasing albite content. The magnitude of the free energy of \()\) ordering reduces markedly as X _Ab increases. At ～700° C incommensurate ordering in crystals with compositions ～An₅₀–An₇₀ needs to have an associated free energy reduction of only a few hundred calories to provide a more stable structure. These results, together with a simple mixing model for the disordered ( \()\) ) solid solution, an assumed tricritical model for the incommensurate ordering and published data for ordering in albite have been used to calculate a set of possible free energy relations for the plagioclase system. The incommensurate structure should appear on the equilibrium phase diagram, but its apparent stability with respect to the assemblage albite plus anorthite at low temperatures depends on the values assigned to the mixing parameters of the $$$$ solid solution.     

Computer modelling of Al/Si ordering in sillimanite

Computer simulation is used to investigate the effect of Al/Si disordering over the tetrahedral sites on the lattice energy and the lattice constants of the mineral sillimanite Al₂SiO₅. A methodology for an atomistic assessment of the energy of the reaction 2(Si-O-Al)→(Si-O-Si)+(Al-O-Al) and its various contributions is established. This ordering energy is 0.97 eV for nearest neighbour sites in the ab-plane and 0.56 eV for those separated in the c-direction. The large difference is due to a greater constraint on the atomic relaxation in the ab-plane and shows the structural dependence of the ordering energy. Its magnitude appears to be determined by a complicated balance between Coulomb and short-range repulsive energy involving strain over many bonds, both in the ordered and disordered structures. There is also a significant interaction between second neighbour sites whereas the contribution of more distant neighbours is negligible. The lattice energies of most of the 154 configurations studied show a linear behaviour as a function of short-range order, specified by the number of Al-Al pairs. The ordering temperature T_c, estimated on the basis of a statistical mechanical model of disordering, and the calculated ordering energies are in semi-quantitative agreement with experimental values.     

Thermodynamics of Al/Al avoidance in the ordering of Al/Si tetrahedral framework structures

The main driving force behind Al/Si ordering in tetrahedral framework aluminosilicates is nearest-neighbour Al/Al avoidance. Computer simulation is used to explore the direct consequences of such Al/Al avoidance. The main result is that the order-disorder transition temperature T _c falls dramatically as the concentration x of Al in the structure is reduced, and if the only interactions are those associated with nearest-neighbour Al/Al avoidance, T _c becomes zero for x less than some critical value x _c , where x _c =0.31 for the feldspar framework and x _c =0.34 for cordierite. Also a large degree of short range order is found above T _c . Both results differ radically from the standard Bragg-Williams model. Plots of entropy and enthalpy of ordering are given as functions of x and T, which may be used to interpret experimental data or for extrapolation into ranges of x and T inaccessible to experiment. Received: 14 May 1997 / Revised, accepted: 2 June 1997     

Computational study of tetrahedral Al–Si ordering in muscovite

 The nature of Al–Si ordering across the tetrahedral sites in muscovite, K₂Al₄(Si₆Al₂O₂₀)(OH)₄, was investigated using various computational techniques. Values of the atomic exchange interaction parameters J _l were obtained. From these parameters, a two-dimensional Al–Si ordering scheme was deduced. The transition temperature T _c for this two-dimensional ordering is 1900 K. There are several possible ordering schemes in three dimensions, based on different stacking sequences of ordered sheets of tetrahedral sites. Monte Carlo simulations of both two-dimensional and three-dimensional ordering were performed, but in the three-dimensional simulation only the two-dimensional ordering is seen, implying that three-dimensional ordering is too slow to be attained during the timescale of the simulation. The effect of the three-dimensional interactions is to raise the two-dimensional ordering temperature to 2140 K. From the three-dimensional Monte Carlo simulation, the frequency of occurrence of 4Si0Al, 3Si1Al, 2Si2Al and 1Si3Al clusters was determined, which match those inferred by ²⁹Si MAS–NMR measurements reasonably well. In fact, the match suggests that the cation ordering seen in experiments corresponds to a configuration with considerable short-range order but no long-range order, similar to a state that is at a temperature just above an ordering phase transition. Received: 28 August 2000 / Accepted: 12 March 2001     

A computational study of Al/Si ordering in cordierite

The ordering of Al and Si in Mg cordierite Mg₂Al₄Si₅O₁₈ is considered using computer simulation. First the enthalpy of interaction J _ij between sites is derived by computer modelling 101 different Al/Si configurations and analysing their energies. They are compared with similar results for three other minerals and with ab initio calculations to assess the whole approach. Secondly the ordering process is studied using Monte Carlo simulation applied to the J _ij . The ordering phase transition temperature T _c is found as 1800°C in reasonable agreement with the experimental estimate of 1450° C. These are much lower than the estimate T _c(ABW)≈7600°C obtained from Bragg-Williams theory. Strong short-range order sets in below T _c(ABW), and the reasons for much lower temperature T _c of long-range ordering are discussed. Strong short-range also sets in very rapidly in a simulated anneal, in agreement with experiment. Thirdly an attempt is made to compare our calculated enthalpies directly with the results of NMR and calorimetry experiments, not completely successfully. A free energy ΔG≈4.6 eV for the activation barrier for ordering is suggested.     

Al,Si ordering in cordierite using “magic angle spinning” NMR

A method of determining the number of Al-O-Al bonds per unit cell from ²⁹Si nuclear magnetic resonance (NMR) data of synthetic cordierites with increasing Si, Al order is described. The number of Al-O-Al bonds is found to vary linearly with the logarithm of the annealing time. This may be correlated with previously published heat of solution data on similar samples (Carpenter et al. 1983) to determine the enthalpy change Δh, associated with a single Al?Si interchange in cordierite. Δh is found to be 8.1 kcal/mole. The NMR data show that the short range Al, Si order cannot be described in terms of twin domains of ordered orthorhombic cordierite. An ordering model derived from group theoretical constraints on possible Al, Si distributions within the hexagonal symmetry of the cordierite is found to provide a better fit to the NMR data.     

CBED and ALCHEMI investigation of local symmetry and Al,Si ordering in K-feldspars

CBED (convergent beam electron diffraction) and ALCHEMI (atom location by channelling enhanced microanalysis) have been used to determine the symmetry and Al-order parameter 2t ₁ within homogeneous regions (of the order of 100 nm in diameter) of high sanidine and low microcline. The results are in excellent agreement with those obtained on the same crystals from X-ray measurements of average T-O bond lengths. In view of this agreement, CBED and ALCHEMI were then applied to an orthoclase with a well-developed tweed microstructure. The results indicate that the specimen is partially ordered and that the average symmetry is triclinic. The lattice modulations involve values of γ* in the range 90°<γ*<90.5°, with correspondingly small variations of 2t ₁ about an average value of 0.67.     

Determination of the degree of Al,Si order Q od in kinetically disordered albite using hard mode infrared spectroscopy

Hard Mode Infrared Spectroscopy (HMIS) is used to correlate the line shifts Δω, the intensity changes ΔA and the variations of spectral line widths Γ of infrared absorption bands with the degree of Al, Si ordering, Q _od, in kinetically disordered Na-feldspar. A simple relationship Δω∝ΔA∝ΔΓ∝Q _od ² was found with the phonon band around 650 cm^?1 being particularly sensitive to small changes of the degree of Al, Si ordering. It is shown that the average degree of Al, Si order can be determined from HMIS with an accuracy of ca. 8 percent using 50 mg of Na-feldspar. The experimental results agree well with recent X-ray determinations using identical samples. The significance of HMIS for the study of kinetic processes in minerals is explained.     

Enthalpy effects associated with Al/Si ordering in anhydrous Mg-cordierite

Measurements of the heats of solution (ΔH_soln) in molten Pb₂B₂O₅ at 708°C of anhydrous magnesian cordierites, prepared with a range of structural states, show that the enthalpy effect associated with Al/Si ordering is substantial (? 9.76 ± 1.56 kcal mole^?1). Differences in the state of order between synthetic cordierites used in phase equilibrium studies and cordierites in the natural environment could lead to significant errors in the estimation of palaeo-pressures and temperatures. A continuous change of ΔH_soln with annealing time supports the suggestion of putnis (1980) that the hexagonal → orthorhombic transformation in cordierite, which can occur via a modulated structure, is truly continuous under metastable conditions. In addition, a linear relation between ΔH_soln and the logarithm of annealing time has been found, which provides some insight into the nature of the ordering mechanisms at an atomic level. Al and Si exchanges occur continuously between neighbouring tetrahedral sites with a net drift towards increasing order. No kinetic or thermochemical distinction can be made between the development of long range and short range order.The enthalpy of vitrification (~ 12 kcal mole^?1) for a metastable stuffed β-quartz polymorph of cordierite composition is similar to that for pure quartz (on a per two oxygen basis), while the heat of vitrification for even the most disordered cordierite seen in this study is more than a factor of three greater (~40 kcal mole^?1). This is consistent with the view that cordierite glass resembles the quartz structure more closely than the crystalline cordierite structure, and that crystallisation of the glass below ~900°C is controlled by a tetrahedral framework.     

Computational study of tetrahedral Al–Si and octahedral Al–Mg ordering in phengite

 As part of a wider study of the nature and origins of cation order–disorder in micas, a variety of computational techniques have been used to investigate the nature of tetrahedral and octahedral ordering in phengite, K₂ ^[6](Al₃Mg)^[4](Si₇Al)O₂₀(OH)₄. Values of the atomic exchange interaction parameters J _n used to model the energies of order–disorder were calculated. Both tetrahedral Al–Si and octahedral Al–Mg ordering were studied and hence three types of interaction parameter were necessary: for T–T, O–O and T–O interactions (where T denotes tetrahedral sites and O denotes octahedral sites). Values for the T–T and O–O interactions were taken from results on other systems, whilst we calculated new values for the T–O interactions. We have demonstrated that modelling the octahedral and tetrahedral sheets alone and independently produces different results from modelling a whole T–O–T layer, hence justifying the inclusion of the T–O interactions. Simulations of a whole T–O–T layer of phengite indicated the presence of short-range order, but no long-range order was observed. Received: 8 August 2002 / Accepted: 14 February 2003 Acknowledgements The authors are grateful to EPSRC (EJP) and the Royal Society (CIS) for financial support. Monte Carlo simulations were performed on the Mineral Physics Group's Beowulf cluster and the University of Cambridge's High Performance Computing Facility.     

Al,Si ordering on cordierite using “magic angle spinning” NMR

Silicon-29 “magic angle spinning” nuclear magnetic resonance (NMR) spectroscopy has been used to study the changes in local Si environment during Al, Si ordering in synthetic cordierite, Mg₂Al₄Si₅O₁₈. In the most disordered form, crystallized from a glass, eight distinct tetrahedral sites for silicon can be identified and assigned, while there are only two distinguishable Si sites in the well-annealed ordered form. This allows the changes in the Si site environments to be determined as a function of annealing time for the transformation from the disordered to the ordered form. The first crystallized state has a considerable degree of partitioning between T₁ and T₂ sites with the following site occupancies: T₁ ? Al:Si=0.80:0.20, T₂?Al:Si=0.27:0.73 The changes in Si environment are approximately linear with log time. The measured values of ²⁹Si isotropic chemical shift do not fit well to previously determined correlations of shift with various structural parameters.     

The ordering behaviour of albite in aqueous solutions at 1 kbar

Albite crystallized hydrothermally from gels of Ab₁₀₀ and Ab₉₀ (SiO₂)₁₀ (wt %) composition changes in degree of Al, Si order with time to reach a steady value. The steady values are influenced by fluid and gel composition as well as temperature. Albites grown in long experiments in the presence of NaOH and excess SiO₂ are more ordered than albites grown in similar experiments with pure water and excess SiO₂. This ‘catalytic’ effect declines as temperature is increased above 600 ° C. Partially ordered albites heated hydrothermally at 700 ° and 725 ° C decrease in degree of order. When heated at 600 °, 650 ° or 675 ° C the degree of order either increases very slightly, or remains the same depending on the degree of order initially present.     

Al,Ge cation ordering in BaAl2Ge2O8-feldspar: monodomain ordering kinetics

The kinetics of monodomain order-disorder processes in monoclinic (I2/c) BaAl₂Ge₂O₈-feldspar have been investigated by X-ray powder diffraction, Hard Mode IR Spectroscopy, and TEM darkfield imaging on quenched samples. Compared to the behaviour predicted by the TDGL approximation ordering kinetics observed at low temperatures slow down significantly when equilibrium is approached. Such a delay is not observed in disordering experiments starting from essentially ordered cation distributions. The deviation from TDGL behaviour is interpreted in terms of partial order parameter conservation in a non-uniformly ordered phase. Modifications to the uniform TDGL rate equation are tested against the available data. An activation energy of 352 ± 28 kJ/mol is obtained for Al,Ge ordering. The mixing coefficient ξ_c ²/ξ², which describes the degree of order parameter conservation, is obtained as a function of temperature. While this coefficient vanishes in the vicinity of the transformation temperature T _tr, it saturates towards a level of ξ_c ²/ξ²≈0.4 for T→0. ξ _c ²/ξ² determines the kinetic stability of ordered clusters quenched from T≈T _tr. Received: 21 April 1999 / Revised, accepted: 19 July 1999     

Kinetic rate laws derived from order parameter theory IV: Kinetics of Al,Si disordering in Na-feldspars

The kinetic rate laws of Al-Si disordering under dry conditions (T = 1353K, 1253 K, 1223 K, 1183 K) and in the presence of water (p = 1 kbar, T = 1023 K, 1073 K, 1103 K) were studied both experimentally and theoretically. A gradual change of the degree of order was found under dry conditions. For intermediate degrees of order broad distributions of the order parameter Q _od occur. The variations of Q _od are correlated with structural modulations as observed in the transmission electron microscope. The time evolution of the mean value of Q _od can be well described by the rate law: $$\frac{{dQ_{od} }}{{dt}} = - \frac{\gamma }{{RT}}\exp \sum\limits_{i = 1}^n {X_i^2 } \left[ {\frac{{ - (G_a^0 + \varepsilon (\Delta Q_{od} )^2 )}}{{RT}}} \right]\frac{{dG}}{{dQ_{od} }}$$ with the excess Gibbs energy G and G _a ⁰ = 433.8 kJ/mol, ?= -27.4 kJ/mol, γ = 1.687 · 10¹⁴ h ^?1. Under wet conditions, two processes were found which occur simultaneously. Firstly, some material renucleated with the equilibrium degree of order. Secondly, the bulk of the material transformed following the same rate law as under dry conditions but with the reduced activation energy G _a ⁰ = 332.0 kJ/mol and ? = -43.0 kJ/ mol, γ = 1.047 · 10¹³ h^?1. The applicability of the kinetic theory is discussed and some ideas for the analysis of geological observations are evolved.     

A computer simulation study of Al/Si ordering in gehlenite and the paradox of the low transition temperature

The ordering of Al and Si in gehlenite is considered using computer simulation. The enthalpy of ordering ΔH per 2Al+2Si atoms is found to be 0.52 eV. It is dominated by the nearest neighbour interaction, but the analysis is carried out to fifth neighbours. The nearest neighbour interaction differs significantly from that for other materials. The structure does not have a connected network of ordering sites, which mainly accounts for the unobservably low transition temperature for Al/Si ordering. Two alternatives are given for the likely ordering pattern.