static lattice simulation of feldspar solid solutions: ferroelastic instabilities and order/disorder

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Static lattice simulationof feldspar solid solutions:Ferroelastic instabilities andorder/disorderSimon A.T. Redfern a , Martin T. Dove a & DavidR.R. Wood aa Department of Earth Sciences , University ofCambridge , Downing Street, Cambridge CB23EQ, UKPublished online: 19 Aug 2006.

To cite this article: Simon A.T. Redfern , Martin T. Dove & David R.R. Wood (1997)Static lattice simulation of feldspar solid solutions: Ferroelastic instabilities andorder/disorder, Phase Transitions: A Multinational Journal, 61:1-4, 173-194, DOI:10.1080/01411599708223736

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STATIC LATTICE SIMULATION OF FELDSPAR SOLID SOLUTIONS:

FERROELASTIC INSTABILITIES AND ORDER/DISORDER

SIMON A.T. REDFERN *, MARTIN T. DOVE and DAVID R.R. WOOD

Department of Earth Sciences, University of Cambridge, Downing Street, Cambridge CB2 3EQ. UK

(Received in final form 12 December 1996)

Lattice energy minimisation calculations with empirical shell model potentials have been performed on feldspar solid solutions as a function of composition and tetrahedral AI/B order. The monoclinic-tnclinic phase transitions have been simulated across the K,Na,_,AISi~O~ and SrxCal-xA12Si208 solid solutions. In both cases it is found that the transitions are driven by an elastic softening without critical softening of an optic phonon. The results for the ferroelastic spontaneous strains are consistent with experimental observation, and explain the nature of the coupling between the ferroelastic instability and Al/Si ordering.

Keyworh: Feldspar; Albite; Anorthite; Elastic moduli; Order-disorder; Ferroelastic

INTRODUCTION

The structural family of feldspars (MT408) accommodates a range of alkali and alkaline-earth cations on the M-sites, which comprise large cavities within the tetrahedral framework (T-cations being A1 and Si). This structure type shows phase transitions from an aristotype C2/m structure to lower-symmetry IT, CT and Pi polymorphs. Two processes are crucial to comprehend if the structural stability of feldspars is to be understood. First, the processes controlling displacive phase

* Corresponding author.

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174 S.A.T. REDFERN er al.

transitions which arise from changes in M-site content, temperature ( T ) or pressure ( P ) , and secondly, changes in the degree of order of the A13+ and Si4+ cations on the T-sites. Here, we describe the contribution that is made in understanding these processes by employing simple transferable empirical potential models to simulate feldspar solid solutions, including modelling the effects of varying the degree of T-site order on these solid solutions. These calculations have enabled us to elucidate the fundamental driving mechanisms for some of the composition-dependent structural instabilities, as well as to probe for the coupling between these structural instabilities and the varying degrees of T-site disorder in feldspars.

The solid solutions of interest to us in this study are indicated in Fig. 1. We have simulated the C2/m-CT monoclinic-triclinic phase transition in the alkali feldspar solid solution (K,Na, _,)AlSi3O8 as well as the 12/c- I i monoclinic-triclinic transition in (Sr,Cal -JA12Si208 feldspars. In each case, the transition occurs as a function of temperature and composition. We have chosen to model the transition behaviour at 0 K using static lattice and lattice dynamics simulations, and have therefore restricted ourselves to the study of the transfor- mations as a function of composition alone. The changes in space group are 12/c-I1 and C2Im-CT in the (Sr,Ca1-.r)A12Si208 and (K,Nal-,)A1Si3O8 solid solutions, respectively. Recent theoretical work with the rigid unit mode model (Hammonds et al., 1996), applied to the feldspar structure, suggests that there is not an optic instability, but that there is considerable softening of the acoustic modes, leading to the possibility that the observed structural phase transitions in these feldspars are due to intrinsic elastic instabilities. In both cases, if the transition is proper ferroelastic the stability condition that is broken at the symmetry change is c44c66 - C& > 0 (Cowley, 1976). Clearly, we would like to be able to measure the individual elastic constants across the solid solutions to test whether these transitions are indeed proper ferroelastic, since we would then be able to model the driving free energy for the phase transitions in terms of an appropriate expansion of the spontaneous strains. While these strains may easily be measured from the observed composition- dependent behaviour of the cell parameters, the experimental measurement of the composition dependence of the elastic constants of these materials is very difficult indeed. We would also like to be able

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LATTICE SIMULATION OF FELDSPAR SOLID SOLUTIONS 175

FIGURE 1 Compositions of interest in this study. The feldspar ternary is shown with the extents of high-temperature solid solution indicated schematically by stippled regions. This paper describes calculations done on the KA1Si308-NaAISi308 solid solution in which a transition from C2/m to C1 occurs in temperature-composition space, for disordered samples. We also consider the 12/c to I1 transition in the CaAl2SizO8-SrAl2Si2O8 solid solution.

to determine how the stability condition c44c66 - Cj6 > 0 is broken, if the transition is indeed proper ferroelastic: does it come about due to changes in one of the three elastic moduli involved, or by the combination of them being naturally soft and hence sensitive to varia- tions in temperature, pressure or composition? The phase transitions themselves induce considerable transformation twinning, and single untwinned crystals of triclinic feldspars of a quality suitable for elastic moduli measurement are rare. This means that the off-diagonal components of the elastic constants are particularly susceptible to measurement errors arising from the sample itself. The experimental difficulty of such measurement explains why there are no good data available for the solid solutions of interest to us, and the only route to testing the stability condition CuC, - c& > 0 for these solid solutions is to conduct static lattice calculations, where the elastic constants are determined within the calculation as part of the derivative properties of the structure. We should say at this point that static

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176 S.A.T. REDFERN pt ul.

lattice calculations with empirical potentials are a particularly robust method of determining elastic and structural properties of tetrahedral framework silicates, since the potentials (incorporating three-body terms to model the tetrahedral interactions) were in part developed by fitting to observed structural parameters for a range of aluminosilicate minerals. In addition to the information on elastic constants, the equilibrium crystal structure given by the static lattice calculations can also be used to probe details of the distortion of the structure, principally through the monoclinic-triclinic spontaneous ferroelastic strains, as given for feldspars by Redfern and Salje (1987).

The background to the lattice energy minimisation and lattice dynamics calculation techniques we have employed has been described extensively in the literature (Price ef al., 1987; Catlow, 1988; Dove, 1989; Winkler et al., 1991; Patel er al., 1991). We used the THBREL, THBPHON and GULP programs, which are particularly suited for the simulation of silicates. The pair interactions between atoms were modelled using the standard Coulomb and Buckingham potentials:

where the parameters C, B and p depend only on the atom pairs, and the charges Q were assumed to have formal values. The polarisation of the oxygen ions was treated within the shell model: the anion is separated into a massless outer shell and an inner massive core, the charge is partitioned between the core and shell, and the core and shell interact via a simple harmonic energy that depends on only the separation d of the positions of the core and shell:

The pair interactions described above are assumed to operate only with the shell component of the 0' anion. Additionally, we used three-body terms to simulate the bond-bending interactions at the T- sites. These energies depend on the angle O two O atoms subtend at a common bonded silicon or aluminium atom and simulates part of the

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covalency inherent in the Si04 and A104 tetrahedra:

The potential parameters used in our calculations are listed in Dove and Redfern (1997). A measure of the transferability of these potentials is given by a comparison of the results obtained from them for the static lattice minimisation of unit cells of the end-member feldspars of interest to us, and the observed unit-cell parameters of these structures. From Table I we see that the three-body potential model enables the accurate calculation of the structural properties of these feldspars. The agreement between modelled and observed structures that we obtain is superior to the early studies of Post and Burnham (1987) and Pate1 et al. (1991) and comparable with the accuracy suggested by Purton and Catlow (1990) using the same methods.

The composition-dependent structural behaviour in these solid solutions was modelled using effective potentials for the M-sites with an occupancy x of one cation and (1-x) of the other. Winkler et al. (1991) have shown that such potentials can be formulated using the condition that their first and second differentials should be equal to the weighted mean of the differentials of the pure end-member potentials at the observed atomic separation ro:

The same method was used to obtain effective potentials for the tetrahedral cations corresponding to various degrees of Al/Si order. For example, x for the T-sites of the totally disordered alkali feldspars corresponds to an occupancy of 0.25 A1 and 0.75 Si. This procedure is equivalent to making the standard “mean-field” approximation, since it neglects local fluctuations in the ordering. Our calculations, therefore, will give a mean-field picture of feldspar behaviour. It has previously been noted that long-range elastic interactions within these tectosilicates tend to induce near-mean-field behaviour in most instances (Redfern and Salje, 1987), and hence the approximation is validated.

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178 S.A.T. REDFERN et al.

TABLE I member feldspars

Comparison of calculated and observed unit-cell parameters for end-

STA12Si208 CaAI2Si2O8 NaA1Si308 KAISi308

calc.' obxt calc.' obs.t calc.' 0bs.l calc.' o b d

a 8.316 8.395 8.181 8.139 8.180 8.154 8.584 8.539 b 12.879 12.977 12.874 12.815 12.850 12.869 12.989 13.015 C 14.037 14.270 14.174 13.890 7.056 7.107 7.134 7.179 a 91.422 90 93.150 93.592 93.205 93.521 90 90 B 115.878 115.440 115.810 117.119 116.985 116.458 115.99 115.99 Y 90.728 90 91.260 90.488 90.310 90.257 90 90

' Completely ordered AI/Si distribution. From McGuinn and Redfern (1994a).

'Completely disordered Al/Si distribution. Results for high albite from Prewitt e ta / . (1976).

'I Results for sanidine from Phillips and Ribbe (1973).

COMPOSITION-DEPENDENT BEHAVIOUR OF ALKALI FELDSPARS

(i) The Disordered Solid Solution

A number of experimental studies of the displacive monoclinic- triclinic transition in disordered alkali feldspars have been under- taken, including those of Grundy and Brown (1969), Henderson (1979), Kroll et al. (1980), Harrison and Salje (1994), Hayward and Salje (1996), Wood (1997), Hovis (1977), and Kroll et al. (1986) (all by powder X-ray diffraction), and those of Zang et al. (1996) and Salje (1 986) by vibrational spectroscopy. Heat capacity measurements through the transition were also made on an alkali feldspar (Or31) by Salje et al. (1985), the only other direct thermodynamic data coming from Hemingway et al. (1981) who measure C, for disordered albite away from the phase transition.

In a monoclinic to triclinic zone centred transition, the E~ strain scales as the order parameter Q. The data from Grundy and Brown (1969), Kroll et al. (1980), Harrison and Salje (1994), Hayward and Salje (1996) and Wood (1997) all demonstrate that E; 0: T below T,, according to the behaviour expected for a classical continuous second-order transition, with T, x 1250 K in pure disordered NaA1Si308. Similar behaviour is noted for the composition dependence of strain

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measured at room temperature across the disordered alkali feldspar solid solution. Hayward and Salje (1996) show that at compositions away from a plateau region near the NaAlSi@8 end-member, E: 0: X (where X is composition). The composition at which the transition occurs at 0 K, using a classical extrapolation, is approximately Na0.561(0.&1Si308. Calorimetric data from Salje et al. (1985) also confirms the continuous thermodynamic character of the transition.

There is a small deviation from the linear relationship between the transition temperature and composition close to albite end member in the alkali feldspar solid solution series. In a region between 0 and 2 mol% KAlSi308, the transition temperature appears to be independent of composition (Hayward and Salje, 1996). This region is, therefore, referred to as a plateau. This result indicates that the strain interaction length in the dilute solid solution is approximately on the scale of 2-3 unit cells, and it demontrates that while the transformation can be approximated by the mean-field theories, the interaction lengths within the crystal are not infinite.

The experimental observations do not in themselves answer the question as to what the driving mechanism behind the transition is. The monoclinic-triclinic transition in alkali feldspars could, for example, be driven either by an acoustic soft mode and or by an optical soft mode. We do know that Raman spectroscopic analysis by Salje (1 986) found no evidence of an optical soft mode, however. Salje (1 985) pursued this fundamental question concerning the driving mechanism in his discussion of a Landau model for the behaviour of Na-feldspar. He wrote the excess free energy for the transition in the form of a Landau expansion:

where Q represents an order parameter, referring to a sublattice distortion, E the spontaneous strain, c the coupling between the strain and the order parameter, and C refers to the bare elastic constants. The coupling is written in a bilinear form since Q has the same symmetry as the strain.

Salje (1 985) highlighted three possible independent mechanisms which might be responsible for driving the transition. The transition may be driven by a pure elastic instability, or by the behaviour of the

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180 S.A.T. REDFERN cf a/

order parameter or by the behaviour of the coupling between the order parameter and strain. Each mechanism has subtle implications for the transition as discussed below:

(a) If the transition is a pure elastic instability it will be driven by the softening of the critical combination of elastic constants c44c66- c& = 0 and the strains E~ and €6 correspond to the order parameters, as described above. This elastic softening precludes the need for optical softening at the transition, as there is no contribution to the transition from sublattice distortions. For a simple second-order transition, in which the bare elastic constants do not vary with temperature or composition, the effective elastic constants which soften will vary linearly with temperature or composition. The transition is then referred to as true proper ferroelastic, as described for NaA1Si308 by Carpenter ( 1997).

(b) The second possibility, favoured by Salje et al. (1985), is that the transition is driven by Q, where Q is a sublattice distortion (some form of spontaneous order of the structure). In this case the ratio &4/&

is independent of temperature, implying the rhombic section is constant with T. Thompson and Hovis's (1978) observations of the rhombic section of analbite provide some evidence in support of this hypothesis.

The driving order parameter, Q, is coupled linearly to the strain E .

As a result, both the optic and acoustic phonons may soften close to T,. They do not, however, have to both soften at the same temperature, as discussed by Wadhawan (1982), and the coupling between them alters the observed phase transition temperature to a value T,*. If they soften at different temperatures, the mode which softens at the highest temperature can be considered to be the driving mode. If there is an instability in the elastic constants at a temperature T:, while an optic instability would, independently, have softened at a lower temperature, it can be demonstrated that the difference in temperatures of the soft modes is related to the strength of the coupling between the order parameter, Q, and the strain, E. The distinction between T, and T,' begins to disappear as C-0, and the pseudo- proper-ferroelastic case becomes indistinguishable form the proper ferroelastic case, i.e. a transition described by one order parameter without coupling.

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The result of a pseudo-proper transition mechanism is that both the critical optic and acoustic modes soften as the transition is approached, although only one mode has to go completely soft. At a second-order phase transition, the renormalised elastic constants will be non-linear functions of temperature, even when bare elastic constants and the coupling are independent of temperature.

(c) The third possibility is that the transition is driven by the temperature dependence of the coupling constant, c, between the strain and an order parameter. Again, considering the second-order case and assuming the coupling constant is linearly dependent upon temperature, the elastic constants will be a non-linear function of temperature or composition across the solid solution.

We have attempted to distinguish between these three possible ori- gins of the transition in the alkali feldspar solid solution using computer modelling. The completely disordered (K,Na, -,)AISi308 solid solution was simulated by static lattice minimisation using the Prewitt et al. (1976) triclinic disordered structure for NaA1Si03 as the starting point for triclinic structures and the disordered sanidine of Phillips and Ribbe (1973) for the monoclinic structures. For compositions greater than around 22.5 mol% KA1Si03 all structures (using either set of starting coordinates) minimised to a stable monoclinic (C2/m) structure. For the more Na-rich compositions the triclinic minimised structures were found to have a lower lattice energy than the isochemical monoclinic structures. Metastable monoclinic structures could be minimised as local minima in the potential energy surface, allowing the comparison of paraphase and ferrophase structural characteristics directly. Hence properties such as spontaneous strain can be computed directly without recourse to extrapolating the paraphase values from K-rich compositions to the Na-rich stability field.

The composition-dependent behaviour of the computed principal symmetry breaking spontaneous strains ( E ~ and E ~ ) is shown in Fig. 2. The results are in broad general agreement with the experimentally observed structural response, although they differ in the details. For example, the computed transition occurs at a somewhat more albitic composition than that found in natural samples (22.5 mol% KAISi03 compared with 34.4molYo found by Kroll et al., 1986). In addition, the critical exponent, p, suggested by the behaviour of ~4 is greater

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182 S.A.T. REDFERN e / a/.

0 10 20

mol% KAISi308 0

FIGURE 2 Variation of the symmetry-breaking strains in the completely disordered KAlSi3O8-NaAISi3O8 solid solution, computed from the behaviour of the ferrophase and paraphase cell parameters obtained from static lattice energy minimisation. The transition from C2/m to C1 appears continuous as a function of composition.

than the value of 0.5 found experimentally. The fact that the simple transferable potential we have employed replicates the phase transition at all is noteworthy, since the potentials are not optimised to the par- ticular structures that we are looking at, and these structures are in themselves very complex. The energy surface of the potential energy function changes from a single shallow minimum to a double well potential across the composition range of interest, and this must arise from a very delicate balance of all the interatomic forces involved which would be susceptible to small errors in the model. The fact that the critical composition does not exactly match observation does not in any way detract from the results, since it is simply a reflection of this very delicate balance associated with the bifurcation of the energy surface in composition space. The discrepancies may, to some

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extent, be accounted for by the manner in which we have approximated the chemical mixing, since the approach we have adopted does not take into account the local structural relaxations around the sub- stituted cations. We therefore expect properties such as excess volume to be overestimated. The maximum calculated excess volume in the solid solution is in the order of 12A3: twice that observed experimentally for disordered alkali feldspars by Kroll et al. (1987). As the elastic constants are related to the molar volume of the crystal, this overestimation of the excess volume may also tend to push the transition towards a lower volume, that is towards the albite end of the solid solution.

Clearly, our assumption of a mean field model is a reasonable first approximation, as the correlation lengths of strain interactions in feldspars are reasonably large. This model will obviously not repli- cate the more subtle effects such as the plateau effect (Hayward and Salje, 1996) or low-temperature quantum saturation, although it does seem to accurately portray the gross elastic response of the aluminosilicate framework to changing M-site occupancy.

Lattice dynamic calculations for the same compositions were carried out, and the composition dependence of the frequency of the lowest-lying optic phonon at k = 0 is shown in Fig. 3. This demon- strates that the computed phase transition at around 22.5mol% KA1SiO3 is not associated with optic phonon softening. On the other hand, the elastic moduli demonstrate that the transition is accom- panied by an instability in the quantity c44c66 - C& with none of the individual elastic moduli showing independent softening (Figs. 4 and 5). In other words, the transition arises from the fine balance between c426 (which is small across the entire composition range) and the product C4.C66, which is very sensitive to K+ content. c44c66 - C& falls to zero at the transition, which may therefore be described as proper ferroelastic driven by an acoustic instability.

The elastic constants of both the monoclinic paraphase and the relaxed triclinic ferrophase are non-linear functions of composition. However, removing the contribution of the bare elastic constants reveals that the renormalised triclinic elastic constants are linearly dependent on composition (Fig. 6). This, therefore, further confirms that the transition is proper ferroelastic rather than pseudo-proper ferroelastic, and that the first of the three explanations for the origin

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184 S.A.T. REDFERN et a/.

mol% KAISi308

FIGURE 3 Composition dependence of the frequency of the lowest-lying optic phonon (at k = 0) across the disordered KAISi30e-NaAISi308 solid solution, determined from lattice dynamics calculation. The C2/m to C1 transition does not result from optic phonon softening.

T 6 -

6-

3 . 5 4-.\

c44

w

E O - c46

-4 . . I I '

0 20 40 60 80 100 -4! . . I I ' I

0 20 40 60 80 100

mol% KAISi308

FIGURE 4 Composition dependence of individual elastic constants across the disordered KAISi3O8-NaAISi3O8 solid solution. The C2/m to C I transition does not result from softening of any one individual elastic constant.

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mol% KAISi308

FIGURE 5 Composition dependence of the combination of elastic constants CuC, - C& across the disordered KA1Si308-NaAISi30s solid solution. This combination goes to zero in the C 2 / m phase at around 22.5 mol% KAISi308, demonstrating that the transition is driven by an instability in this combination and is hence proper ferroelastic in character.

2.0

0.5 -

0 1 I I I

0 5 10 1 5 20 d

mol% KAISi308

5

FIGURE 6 Composition dependence of the renormalised elastic constants C,, C,, and C- (equal to the triclinic relaxed elastic constants minus the bare elastic constants) across the disordered KAISi3O8-NaAISi3O8 solid solution below the monoclinic- triclinic transition.

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186 S.A.T. REDFERN cf al.

of the transformation, listed above, correctly describes the critical behaviour.

(ii) The Effect of Al/Si Order

Similar calculations have been carried out for this composition range, but with varying degrees of order on the tetrahedral sites. This can easily be accomplished by varying the mean Al/Si occupancy of each T-site. It should be noted that both convergent and non-convergent ordering in alkali feldspars can been investigated by this technique, but here we simply focus on the computed structural and elastic properties across the completely ordered solid solution. A fuller discussion of this computational approach to investigation of varying degrees of tetrahedral order in plagioclase and alkali feldspar solid solutions appears elsewhere (Wood, 1997).

The behaviour of the critical cell angles of the completely ordered alkali feldspars is shown in Fig. 7, together with those of the disordered solid solution, shown for comparison. Also shown are the experimentally observed cell angles for ordered and disordered synthetic samples. The computations accurately describe the observed behaviour of these cell angles. The transformation to monoclinic cannot occur in ordered alkali feldspars, but we see that the cell parameters nonetheless vary non-linearly across the solid solution associated with the remaining elastic non-linearity. This elastic behaviour is seen in Fig. 8, where the behaviour of the elastic constants determined from our calculations of these ordered alkali feldspars is illustrated. In the presence of Al/Si order the critical combination c44c66 - Ci6 does not fall to zero across the composition range and the transition to monoclinic cannot arise. We see that the model not only identifies the mechanism of the phase transition in the disordered alkali feldspars, it also correctly predicts the behaviour of the elastic and structural properties as a function of Al/Si order, as is seen from the behaviour of the cell angle for these ordered alkali feldspars. We intend to exploit this predictive power of the computational model we have used here in planned studies which will aim to examine the bulk and anisotropic elastic properties of feldspars as a function of their degree of mean-field tetrahedral order, a problem that is extremely dificult to address experimentally.

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n v

P P a 3 - -

94.

93.

92.

91 a

90.

89,

88.

87.

94.

93.

92.

91,

90

89

88

- - - - a

I I I I

L - - - - - - - - - 7- - - analblte - high sanidine

Tow albite - low microclin

1 I I I

0 20 40 60 80 100

mol% KAISi308

FIGURE 7 Composition dependence of Q and y cell angles across the completely ordered and disordered KAISi308-NaAISi30s solid solution (calculated from static lattice energy minimisation and shown in (a). The transition to C2/m cannot occur in the presence of any degree of AI/Si order, but the cell angles still behave non-linearly across the solid solution due to coupling to the incipient elastic instability. The behaviour of the cell angles agrees well with that observed experimentally, shown in (b) (after Kroll et al., 1986). validating the use of this simple transferable potential model in the study of these complex aluminosilicate structures.

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FIGURE 8 Computed composition dependence of the critical combination of elastic constants C-C, - C$ across the completely ordered and disordered KAISi308- NaAISi3O8 solid solutions. The ordered solid solution displays elastic softening at compositions somewhat more K-rich than the transition in the disordered solid solution, but softening is not complete and the transition from the triclinic structure to the C2/m phase cannot occur.

COMPOSITION-DEPENDENT BEHAVIOUR OF Ca, Sr FELDSPARS

The monoclinic-triclinic phase transition also occurs in Sr,CaI -,A12SizOs solid solutions. Here, the transition may occur for any degree of Al/Si order/disorder, since it is associated with an instability at the Brillouin zone centre while the order/disorder transition in these "2 : 2" feldspars (2 A1 to 2 Si) is a zone-boundary process. We have carried out simulations of the monoclinic- triclinic displacive phase transitions in this solid solution as a function of degree of Al/Si order/disorder. Although these feldspars remain highly ordered up to their melting points, there is a discernible decrease in

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Al/Si order with temperature as the structure approaches the transition from an I-body centred lattice to a C-face centred lattice (Carpenter et al., 1990; Phillips et al., 1997). The coupling between t h s zone- boundary ordering process and the zone centre monoclinic-triclinic transition observed as a function of composition (McGuinn and Redfern, 1994a,b; Tribaudino et al., 1993) has been the subject of some considerable speculation by experimentalists. Here we have attempted to elucidate the potential coupling effects, and to identify the driving mechanism of the phase transition.

We used the known structures of the 12/c SrA12Si208 end-member (Chiari er al., 1975) and 11 anorthite (Kempster et al., 1962) for our starting coordinates and successively computed the properties of triclinic and monoclinic members of the intermediate solid solution in an analogous manner to that described for the disordered alkali feldspars above. In the (Ca,Srl -JA12Si208 solid solution, however, the results indicate that the triclinic phase is stable across the whole composition range at OK, which is in agreement with the anticipated form of the temperature-composition phase diagram described by Tribaudino (1994). We find that a transition to 12/c would occur for completely ordered (Ca, Sr)A12Si208 feldspars beyond the end-member at around 102.5mol% SrAI2Si2O8, as is shown by Fig. 9. The nature of the elastic instability is revealed by the behaviour of the C,, c66, and c46 elastic constants. Lattice dynamic calculations across the solid solution demonstrate that the transition occurs as a result of an elastic instability alone, with no significant softening of optic modes (Fig. 10). Similar to the alkali feldspar solid solution, the results indicate that the transition is driven by the softening of the combination C, . c66 - c&6, which goes to zero at the transition. The discontinuities in the computed cell parameters (and derived spontaneous strains) and elastic constants indicate that the transition is first-order in character, although not strongly so.

An interesting outcome of our computations is the observation that the strains &6 and which are symmetrically equivalent as far as the phase transition is concerned, do not scale linearly to one another. This appears to be inherent, and we note that ~4 (which is related to the a angle) behaves as the primary order parameter for the phase transition with &6 coupled non-linearly. This is the first time that the importance of higher order coupling terms in controlling

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190

. 15--

10- cu co 0"

3 0" 5:

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S.A.T. REDFERN el (I/

- % - - - -

\ \ \ \ \ \ \ \ \ I I

I I I I I

FIGURE 9 Computed composition dependence of the critical combination of elastic constants CMC, - C& across-the completely ordered CaA12SizOs-SrAl2Si2Os solid solution. A transition from I1 to f 2 / c occurs at around 100mol% SrAI2Si2O8, in agreement with experimental observation of this system. These calculations further demonstrate that the transition is driven by softening of this combination of elastic constants and is ferroelastic in character.

the cell parameter evolution has been recognised in these materials. For some time now it has been noted that ferroelastic transitions most strongly influence the cy cell angle in feldspars (see, for example, Salje et al., 1985). The y cell angle has previously been recognised to vary non-uniformly with o in triclinic feldspars. It has also been demonstrated, however, that Al/Si order/disorder (described by an order parameter Qod) most strongly affects y. Hence, non-linearity between cx and y at displacive transitions in feldspars has previously been attributed to coupled changes in Al/Si order which induce vari- ations in y. These results show, on the other hand, that non-linearity between the two triclinic cell angles occurs inherently in the absence of changes in Al/Si order, and variation in Qod need not be invoked to explain the apparently strange composition dependence of y. We

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0.4 0 20 40 60 80 100

mol% SrAI2Si208

FIGURE 10 Computed composition dependence of the frequency of the lowest k = 0 optic phonon across the completely ordered CaA12Si2Os-SrAI2Si2O8 solid solution. The transition is not driven by optic softening, although coupling of the optic hard modes to the acoustic instability in the triclinic phase (shown by the dashed line) is clear.

notice that y increases from zero at the transition from 12/c to IT, and then decreases on further increase of Ca content. Since we know that Qod couples strongly and positively to y, we might expect these marked changes in y to favour an increase in Qod across the phase transition with a subsequent gradual reduction in Qod with increasing Ca content. Such a change in Al/Si order associated with increasing Ca content across the Z2/c to Zi phase transition has indeed been observed experimentally in a recent NMR study (Phillips et al., 1997).

The influence of Al/Si order/disorder on the I2 /c to IT transition has been investigated by conducting static lattice calculations of the (CaxSr1-x)A1$3i208 solid solution with varying degrees of tetrahedral order, mixing A1 and Si on the T-sites as we did for the alkali feldspars. The results are present fully in Dove and Redfern (1997), but in summary we have found that decreasing the tetrahedral order shifts the phase transition boundary towards the Ca end-member, in

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192 S.A.T. REDFERN et a1

agreement with the experimental findings of Tribaudino et al. (1993, 1995). Our results also indicate that Al/Si disorder also reduces the slope, dT/dX of the phase boundary between the I2/c and 11 structures in T-X space.

CONCLUSIONS

Calculations on both the (Ca, Sr)A12Si208 and disordered- (Na, K)A1Si3O8 feldspar solid solutions as functions of composition show that both the Z2/c-IT and the C2/m-Ci phase transitions arise from the vanishing of the quantity C M C ~ - C42, and may be described as proper ferroelastic. These results indicate the usefulness of the mixed potential approach in investigating phase transition and order-disorder phenomena in aluminosilicates. The empirical potential models we have employed successfully simulate the essential elastic and structural features of these complex structures, and allow the investigation of mean-field effects across a large number of structural and chemical states. These calculations were performed on modest workstations at low cost, but provide significant new physical insights into the systems of interest. Much has been written on the transferability of the model we have employed, the results we obtain suggest that it is indeed rather general for these structures and has strong predictive power. While phase transition boundaries were not predicted at exactly the correct critical compositions, it should be realised that the fact that phase transitions have been successfully computed at all with this model indicates that the rather simple atomic interactions that it incorporates are sufficient to provide the shallow potential wells and easily distorted energy surfaces needed to provide a bifurcation into a double well at the low symmetry compositions, and hence are useful for the study of subtle distortive phase transitions such as these.

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static lattice simulation of feldspar solid solutions: ferroelastic instabilities and order/disorder

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