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Ab initio elastic properties of talc from 0 to 12 GPa: Interpretation of seismicvelocities at mantle pressures and prediction of auxetic behaviour at low pressure

David Mainprice a,⁎, Yvon Le Page b, John Rodgers c, Paul Jouanna a

a Géosciences Montpellier UMR CNRS 5243, Université Montpellier II, 34095 Montpellier, Franceb ICPET, National Research Council of Canada, Ottawa, ON, Canada K1A 0R6c Toth Information Systems Inc., Ottawa, Canada

a b s t r a c ta r t i c l e i n f o

Article history:Received 29 March 2008Received in revised form 25 June 2008Accepted 23 July 2008Available online 9 September 2008

Editor: L. Stixrude

Keywords:talcab initioelastic propertiesseismic anisotropyupper mantlesubductionPoisson's ratio

Talc is a hydrousmagnesium rich layered silicate that iswidely disseminated in the Earth fromthe seafloor to over100 km depth, in ultra-high pressure metamorphism of oceanic crust. In this paper we determine the singlecrystal elastic constants at pressures from 0 to 12 GPa of talc triclinic (C 1

P) andmonoclinic (C2/c) polytypes using

ab initio methods. We find that talc has an extraordinarily high elastic anisotropy at zero pressure that reduceswith increasing pressure. The exceptional anisotropy is complemented by a negative Poisson's ratio for manydirections in crystal space. Calculations show that talc is not only one of very few common minerals to exhibitauxetic behaviour, but the magnitude of this effect may be the largest reported so far for a mineral. Thecompression (Vp) and shear (Vs) wave velocity anisotropy is 80% and 85% for the triclinic polytype. At pressureswhere talc is known be stable in the Earth (up to 5 GPa) the Vp and Vs anisotropy is reduced to about 40% for bothvelocities,which is still a very high value. Vp is slowparallel to the c-axis and fast perpendicular to it. This remainsunchanged with increasing pressure and is observed in both polytypes. The shear wave splitting (differencebetween fast and slow S-wave velocities) at low pressure has high values in the plane normal to the c-axis, with amaximumnear the a⁎-axis in the triclinic and the b-axis in the monoclinic polytype. The c-axis is the direction ofminimum splitting. The pattern of shear wave splitting does not change significantly with pressure.The volume fraction of talc varies between 11 and 41% for hydrated mantle rocks, but the lack of data on thecrystallographic preferred orientation (CPO) precludes a detailed analysis of the impact of talc on seismic anisotropyin subduction zones. However, it is highly likely that CPO can easily develop in zones of deformation due to the platyhabit of talc crystals. For random aggregates of talc, the isotropic Vp, Vs and Vp/Vs ratio have significantly lowervalues than those of antigorite and may explain low-velocity regions in the mantle wedge. Vp/Vs ratios are morecomplex in anisotropic media because there are fast and slow S-waves, resulting in Vp/Vs1 and Vp/Vs2 ratios forevery propagation direction,making interpretation difficult in deformed polycrystalline talc with a CPO. Talc on thesubduction plate boundary can strongly influence guided wave velocity as CPO would develop in this region ofintense shearing. The very lowcoefficient of friction (b0.1) of talc above 100 °C could also explain silent earthquakesat shallow depths (ca 30 km) along the subduction plate boundaries, frequently responsible for tsunami.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

Talc (Mg3Si4O10(OH)2) is a hydrous magnesium silicate mineralwith 2:1 trioctrahedral layered silicate structure with slabsmade of anMg-octrahedral (brucite) layer sandwiched between two Si-tetrahe-dral (silicate) layers (Fig.1). It is of great geological importance having awidespread occurrence from the surface of the seafloor to depths of150 km in subduction zones. Talc typically has a water content of 4.7%by weight, and hence plays an important role in recycling water intothe deep Earth. It has recently been observed to form by depositionfrom hydrothermal vents on the seafloor (Dekov et al., 2008). Perhaps,

it is best known to form during the serpentinization of abyssal peri-dotites at mid-ocean ridges (e.g. Cannat, 1993; Mével, 2003), beingassociatedwith alteration of orthopyroxene to serpentine+talc, wheretalc can represent up to 41% by volume of the rock (e.g. Hacker et al.,2003). In subduction zones, it forms in the mantle wedge from thebreakdown of antigorite serpentine to give talc+forsterite+water (e.g.Evans et al., 1976). This reaction has recently been found to explain theglobal pattern of seismicity that displays double Benioff zones(Brudzinski et al., 2007). At greater depths, talc releases water in thedehydration reaction to enstatite at ca 1.7 GPa 660 °C (b60 km depth)(e.g. Ulmer and Trommsdorff, 1995; Wunder and Schreyer, 1997;Pawley, 1998; Melekhova et al., 2006). Talc should accordingly bepresent from shallowdepths to 60 km. At 100 kmormore, talc forms inoceanic crust at ultra-high pressure metamorphic (UHPM) conditionsduring the hydration of mafic eclogites, even for H2O undersaturated

Earth and Planetary Science Letters 274 (2008) 327–338

⁎ Corresponding author. Tel.: +33 467143283; fax: +33 467143603.E-mail address: [email protected] (D. Mainprice).

0012-821X/$ – see front matter © 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.epsl.2008.07.047

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conditions lacking a free fluid phase (Poli and Schmidt, 1997). Talc is awidespread metamorphic mineral in many high pressure blueschistand whiteschist terrains. It is associated there with kyanite and evencoesite involved in volatile recycling in subduction zones (Chopin,1984; Schreyer, 1977; Liou and Zhang, 1995; Spandler et al., 2007) atpressures up to 4 GPa in mafic and pelitic rocks.

Talc is also an important structural elementof layered silicates, suchas phlogopite, clinochlore and 10 Å phase (e.g. Welch and Marshall,2001; Fumagalli and Strixrude, 2007; Parry et al., 2007). As such, itsstudy has fundamental significance for the study of layered silicates ingeneral. At 710 °C and 5 GPa (ca 150 kmdepth), talc breaks down to theclosely related 10 Å phase Mg3Si4O10(OH)2.xH2O (Bose and Ganguly,1995; Pawley and Wood, 1995). Talc is made of a central trioctahedralbrucite layer composed of Mg-octahedra sandwiched in-between twohexagonal silicate tetrahedral rings (Fig. 1). The oxygen atoms of thetwo remaining OH groups belong to the brucite layer, and are thereforelocated inside the talc slab. Valence sums in the Pauling sense (Pauling,1960) are perfectly obeyed, making the talc slab a neutral inorganicpolymer devoid of a dipole moment. Adhesion between talc slabs isthen due solely to van der Waals forces. It is then not surprising thattalc corresponds to the bottom of Mohs' scale for mineral hardness,with aMohshardness of 1 bydefinition. Fivemainpropertiesmake talca unique mineral of industrial importance: low coefficient of friction,platyness, chemical inertness, affinity for organic chemicals andhydrophobicity.

Talc is tectonically important because its very low coefficient offriction of about 0.2 at room temperature reduces to less than 0.1 attemperatures above 100 °C (Moore and Lockner, 2008; Escartín et al.,2008). Talc is also inherently mechanically stable with velocity-strengthening behaviour. Its strengthening with strain magnitudemakes it an ideal mineral to allow stable sliding along fault surfaces.

In addition, talc is often formed by hydrothermal alteration by silica-containingfluids that permeate rocks and faults surfaces, creating talc atmechanically active sites. For example, talc has recently been identifiedon the stable fast creeping section (28mm/yr) of the SanAndreas fault inthe San Juan Bautista area (Moore and Rymer, 2007), as well as atdetachment faults along slow spreading mid-oceanic ridges (Boschiet al., 2006). It has also been proposed that talc forms in forearc mantlealong the plate interface and that this limits the depth of subductionthrust earthquakes (Peacock and Hyndman, 1999). We suggest that thepresence of talc in shallow subduction (less than 60 km) may facilitateslip on the subduction interface that causes ‘silent’ earthquakes oftenassociated with tsunami (e.g. Kawasaki, 2004).

Despite the importance of talc for many domains of Earth andMaterials Sciences, the complete single crystal elastic tensor has neverbeen measured because gem-quality crystals of suitable size for expe-rimental measurement are extremely rare. They would in any case beextremely fragile because of talc's easy cleavage on the basal plane. Inthis study, we have determined the elastic constants of the triclinicand monoclinic C2/c polytypes as a function of pressure using ab initiomethods. The seismic properties that we can calculate from theseelastic constants will provide constraints on seismological explorationfor talc at depth in the Earth. We will show that the elastic propertiesof talc are exceptionally anisotropic, probably the most anisotropic fora naturally occurring mineral at room pressure.

2. Computation details

2.1. Ab initio computations

VASP (Kresse, 1993; Kresse and Hafner, 1993, 1994) was used for allcomputations. All VASP input files were generated and interpreted by

Fig. 1. The structure of the talc triclinic polymorph at ambient pressure composed of layers of Si–O tetrahedra and Mg–O octahedra. In the basal plane the Si–O tetrahedra form 6-membered rings with a hydrogen atom bonded to a non-silicate oxygen at their center.

328 D. Mainprice et al. / Earth and Planetary Science Letters 274 (2008) 327–338


Materials Toolkit (Le Page and Rodgers, 2005). PW91 (Perdew et al.,1996) GGA PAW potentials (Kresse and Joubert, 1999) were used for allcalculations. Wave functions were iteratively optimized with theDavidson blocked scheme combined with reciprocal space projectors(Davidson, 1983) down to an electronic convergence of 1×10−7 eV.Reciprocal space integration was performed using a Monkhorst–Packscheme (Monkhorst and Pack, 1976). Energy corrections wereimplemented as Methfessel–Paxton smearing of order 1 and width0.2 eV (Methfessel and Paxton, 1989). The convergence criterion forforces was 1×10−4 eV/Å and the cutoff energy was 400 eV. No spinpolarization corrections were performed. Unless noted otherwise, a6×6×6k-mesh was used for all calculations.

2.2. Model selection and implementation of calculations

2.2.1. Triclinic polytype, Z=2We used the structure results of Perdikatsis and Burzlaff, 1981

reported in space group C 1P

as our initial structure model. The choiceof this non-standard space group setting by those authors stems fromthe nearly orthogonal directions [100] and [120] in the primitive meshof the talc layer. We first optimized its atom coordinates, retaining itszero-pressure experimental cell data. The corresponding calculatedpressure was −2.5 GPa. This value constitutes the offset betweenexperimental and calculated pressures. Such an offset is expected andis due to slight imperfections in the representation of the core electrondensity for calculated ab initio pseudo-potentials. A prior discussion inLe Page and Saxe (2002) concluded that both GGA PAW and LDApotentials systematically underestimate pressures, but the corres-ponding offset is usually smaller with GGA PAW than LDA. This is themain reason for our preference for those pseudo-potentials. Allcalculated pressures reported in this paper, for example in Supple-mentary data (Tables 1, 2 and 3) are corrected for this offset which isprinted in the write-up.

We performed eight optimizations of cell and structure from 0 GPaup to 12 GPa by implementing a Pulay stress (Pulay, 1980) that wasthe sum of the desired pressure and the printed offset. This pressurerange allows comparison with existing experimental results for talccompression at ambient temperature. Elastic coefficients werecalculated according to the stress–strain approach as developed inLe Page and Saxe (2002) for each of the resulting eight pressures,using for this the implementation in Materials Toolkit (Le Page andRodgers, 2005). The orthogonal axes were selected according to theInstitute of Radio Engineers (IRE) scheme (Brainerd et al., 1949) forthe triclinic cell. The Cartesian X3 axis was accordingly selected alongthe crystallographic c-axis, and X1 along the reciprocal axis a⁎,ensuring that b is in the X1, X2 plane. The mutually perpendicular X2axis is then chosen to complete the right-handed Cartesian system(X1, X2, X3). Two strain magnitudes of 0.5 and 0.75% were used forelastic calculations. As a consequence, least-squares extraction of theset of 21 independent elastic tensor coefficients for triclinic symmetryrequired creation of 25 cell distortions and the optimization ofcorresponding atom coordinates. Each elastic calculation accordinglytook approximately 5 cpu×days on Intel Xeon PCs with serialVASP.4.6.3 under Linux.

2.2.2. Monoclinic polytype, Z=4We used the cell data from Pawley et al., 2002 and the fractional

atom coordinates from the X-ray crystal structure by Gruner, 1934 as astarting point, with addition of a H atom at 0.97 Å from Gruner's OH1position. We processed it exactly as the triclinic polytype above,resulting in a −3.0 GPa pressure offset that we subtracted from allsubsequently calculated pressures. The IRE axes were X3 alongcrystallographic axis c and X1 along a⁎, giving X2 along both b andb⁎ for monoclinic symmetry. In view of the very long c-axis, the k-mesh was selected as 6×6×2. Each elastic calculation took about10 cpu×days on the same cluster as above.

3. Results

Supplementary data (Tables 1a and b) respectively report structureoptimization results for the triclinic and the monoclinic polytypes for aset of eight offset-corrected pressures. Calculated elastic tensorcoefficients (Cij in 2-index Voigt notation, see e.g. Nye, 1985) aresimilarly printed in Supplementary data (Tables 2a and b) with theisotropic bulk modulus K and the shear modulus G. Least-squaresstandard uncertainties for individual tensor coefficients in those Tablesare typically about 2 GPa. Supplementary data (Tables 3a and b) listvalues for the corresponding cell volumes, density, and the isotropicVoigt–Reuss–Hill average velocities for compressional and shear waves.

3.1. Compression of the crystal structure

Our predicted cell axes and volumes from Supplementary data(Table 1a and b) as a function of pressure are shown with relativecompression values in Fig. 2. As expected, the cell axes and volumesdecrease with increasing pressure for both polytypes. However, there

Fig. 2. Talc cell parameters for polytypes C 1P

and C2/c predicted by density functionaltheory (DFT) using VASP compared with the experimental results of Pawley et al.(2002), presented as relative compression X/X0 where X0 is the value at zero GPa.Notice that the good agreement between theory and experiment, well withinexperimental scatter and error for both polytypes. A. Relative compression of cellaxes a/a0, b/b0 and c/c0. B. Relative compression of cell volume V/V0.

329D. Mainprice et al. / Earth and Planetary Science Letters 274 (2008) 327–338


is a considerable anisotropy in the compressibility, the c-axis beingmuch more compressible than the axes a and b.

3.2. Elastic constants and linear compressibility

The elastic constants show a very strong anisotropy of stiffnesscomponents between the directions normal to the c-axis (C11, C22) andthe c-axis direction (C33) (Fig. 3). A similar anisotropy is observed forthe shear components associated with the c-axis (C66) and normal tothis direction (C44, C55). The two polytypes have similar values alongthe c-axis (C33, C66), whereas the components normal to c-axis (C11,C22, C44, C55) have similar values in the C 1

Ppolytype. They show larger

differences for the C2/c polytype, for compression (C11, C22) and shearcomponents (C44, C55). The effect of pressure is very important for C33

as might be expected for a layer silicate, whereas C66 is virtuallypressure independent. Pressure has a larger effect on componentsnormal to the c-axis for shear (C44, C55) than compression (C11, C22).

From the single crystal elastic compliance tensor Sij (=C ij−1), it is

possible to calculate the linear compressibility in any crystallographicdirection (e.g. Nye, 1985). The linear compressibility thus calculated ina, b and c directions from the Sij at any givenpressure is shown in Fig. 4.Note that with this way of proceeding, the linear compressibility iscalculated at a single pressure value, and is not determined as the ratioof a finite cell dimensional change due to a finite pressure change. Atzero pressure the calculated linear compressibility along the c-axis ismuch higher for the polytype than C2/cwhereas, at all other pressures,the linear compressibility is very similar for both polytypes. The de-crease in linear compressibility is most marked for the c-axis, whereasa- and b-axes change very little above 2 GPa.

3.3. Anisotropic Poisson's ratio

The anisotropic Poisson's ratio (νijkl) varies in a complexmannerwithdirection. Poisson's ratio is defined by the elastic strain in twoorthogonal directions, the longitudinal (or axial) direction (xi) andtransverse (or lateral) direction (yi). The lateral strain isdefinedby −εijyiyjalong y and the longitudinal strain by εijxixj along x, where εij is theinfinitesimal strain tensor. The anisotropic Poisson's ratio (νijkl) is givenas the ratio of lateral to longitudinal strain (Sirotin and Shakolskaya,1982) as

mijkl ¼ −eijyiyj=eklxkx1 ¼ −Sijklxixjyky1=Smnopxmxnxoxp ð1Þ

where Sijkl is the elastic compliance 4th rank tensor in 4-index notation.To compare νijkl to seismic wave propagationwe could choose the axialand transverse strain directions to be parallel to the orthogonal P- and S-wave polarization directions respectively. However, this analogy isclearly not exact, as the strains associated with seismic waves in anelastic medium require shear strain components not present in thedefinition of Poisson's ratio. For an orthogonal reference frame, x, y, z,extension along x direction typically results in contraction strains alongy and z for positive Poisson's ratio, which is commonly the case. IfPoisson's ratio is negative for a particular choice of x and orthogonaldirection, such as z, then the strain along z will be extensional. Fig. 5shows the strains associated with Poisson's ratio for various axialextension directions for c 1

Pthe talc polytype at pressures of 0.0 and

3.9 GPa. For many directions in talc at zero pressure, the anisotropicPoisson's ratio is negative; meaning the extension in the axial directionwill result in extension in the transverse direction, for that pair of axialand transverse directions. At a pressure of 3.9 GPa only two directionshave negative anisotropic Poisson's ratio.

3.4. Wave velocities and anisotropy

3.4.1. Isotropic random aggregatesFor aggregates of talc with random crystallographic preferred

orientation (CPO), the seismicwave velocity can be estimated from theisotropic Voigt–Reuss–Hill (VRH) average calculated from the singlecrystal elastic tensor and the density. VRH average is the arithmeticmean of the upper bound Voigt and lower bound Reuss averages. Astalc is very anisotropic, the Voigt and Reuss bounds are widely sepa-rated as can been seen in Supplementary data (Tables 2a and b). TheVp and Vs VRH average velocities for the C2/c polytype has highervelocity than C 1

Pand both rise rapidly with pressure up to 2 GPa and

more slowly up to 12 GPa (Fig. 6). The Vp/Vs ratio behaves in a similarmanner with increasing pressure, but difference between thedifference between C 1

Pand C2/c polytypes is very small.

3.4.2. Anisotropic single crystal and aggregates with CPOTo understand the evolution of the seismic anisotropy of talc as a

function of pressure, the seismic velocities for a given propagationdirection are calculated from the Cij and the density at each pressureusing Christoffel's equation (e.g. Mainprice, 2007). The percentageanisotropy (A) is defined here as A=200 (Vmaximum−Vminimum)/

Fig. 3. Single crystal elastic constants of the leading diagonal of the elastic tensor (Cij; i= j) oftalc as a function of pressure. C11, C22 andC33 are directly related toVp,whereas C44, C55 andC66 are related to Vs, propagation respectively along X1 (a⁎-axis),X2 and X3 (c-axis) elasticreference axes, where X2=c×a⁎ for C 1

Pand X2=b for C2/c.

Fig. 4. The volume and linear compressibility along a-, b- and c-axes at each pressurecalculated from the elastic constants for C 1

Pand C2/c polymorphs of talc. The predicted

values of linear compressibility (βa,βb, andβc) are similar around2GPa to those quotedbyPawley et al., 2002 from a linear fit of cell axes between 10−4 and 5 GPawithβa=3.9×10−3,βb=2.7×10−3 andβc=11.8×10−3 GPa−1 (shown by black line over pressure range 0–5 GPa).



Fig. 5. Anisotropic Poisson's ratio of the C 1P

polytype of Talc at low and high pressure. On the 3D upper hemisphere stereogram the black lines radiating out from the centre are axialextension directions. For each extension direction there is a circle, which represents the circumference of an undeformed cylinder. The thick black line represents the circumferenceof a elastically deformed cylinder after 1% elastic extension, multiply by 200. The coloured lines represent the max (blue) and min (green) direction of positive Poisson's ratio. Linesare indicated in red where Poisson's ratio is negative. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)



(Vmaximum+Vminimum), where the maximum and minimum arefound by exploring a hemisphere of all possible propagation directions.Note that for P-wave velocities Vp, propagation directions of maximumandminimumVp are of course different, and the percentage anisotropyis calculated globally on the whole hemisphere. In contrast to theisotropic case, there are in general two orthogonally polarized S-wavesfor each propagation directionwith different velocities in an anisotropicmedium. The anisotropy AVs can then be defined for each direction.Contoured upper hemisphere stereograms of P-wave velocity (Vp),percentage shearwave anisotropy (AVs) called also shearwave splitting,as well as polarization (Vs1) of the fastest S-wave are shown in Fig. 7 forthe zero andhigh pressures for the c1 andC2/cpolytypes. Firstly, at zeropressure, the pattern of the velocity for Vp is similar in both polytypeswith high velocities normal to the c-axis and lower velocities parallelto it. The pattern for AVs is differentwith high values neara⁎ and b in theC 1P

and C2/c polytypes respectively. The low values of AVs are near thec-axis in both polytypes. The fastest S-wave (s1) has a polarization inthe basal plane for both polytypes,which is typical for layer silicates. Theanisotropyof talc is the highest ever reported for amineralwithAVp andAVs for the C 1

Ppolytype of 79.0 and 85.6% respectively. The anisotropy

of the C2/c polytype is slightly lower with 75.4 and 32.3% for Vp and Vsrespectively. At the higher pressure (3.9 GPa forC 1

Pand 2.8 GPa for C2/c,

representative of stability range of talc in rocks) the anisotropy is greatlyreduced, but remains high. The patterns remain similar if not identicalfor Vp. However the AVs pattern changes to include a small region ofhigher value around the c-axis. The fastest S-wave (s1) has a polarizationin the basal plane for both polytypes. In anisotropic elastic media thereare three velocities (Vp, Vs1, Vs2) and hence two ratios Vp/Vs1 and Vp/Vs2, and of course these ratios vary with direction. The distribution ofVp/Vs1 and Vp/Vs2 ratios for both polytypes is illustrated in Fig. 8. Atzero pressure the Vp/Vs1 ratio has high values (maximum 1.72 and 1.86for c 1

Pand C2/c polytypes respectively) normal to the c-axis and low

values parallel to the c-axis (minimum 1.06 and 1.04 for c 1P

and C2/cpolytypes respectively). For Vp/Vs2 ratios, the high values (maximum4.08 and 2.56 for c 1

Pand C2/c polytypes respectively) are also normal to

the c-axis. At the higher pressures the distribution of theVp/Vs1 andVp/Vs2 ratios do not change dramatically. The anisotropy of Vp/Vs1 ratio isbetween 35.5 to 56.4%, whereas the Vp/Vs2 ratio has much higheranisotropies between 52.5 and 91.4%. The anisotropy of both velocityratios slightly reduces with increasing pressure.

The anisotropy as a function of pressure is illustrated in Fig. 9A forP- and S-waves. The anisotropy is very high at low pressure (about 80%

for P and S) and decreases to half these values with increasingpressure to 5 GPa (40%), a pressure typical of the stability range forbearing metamorphic rocks at in situ PT conditions. Further pressureincrease to about 12 GPa reduces the anisotropy of P and S toapproximately 16%. For isotropic media, there is only one Vp/Vs ratioas the velocity is the same for all propagation directions with only oneP-wave and one S-wave velocity. For the anisotropic case themaximum and minimum ratios of Vp/Vs1 and Vp/Vs2 are shown asa function of pressure for the C 1

Ppolytype in Fig. 9B. Vp/Vs1 and Vp/

Vs2 correspond to ratios using the first and second arriving S-waves atseismic station respectively.

4. Discussion

4.1. Comparison with published results

4.1.1. Cell compression and elastic moduliThe experimental results of Pawleyet al. (2002)have shown that talc,

like other layered silicates is most compressible perpendicular to layers.Note that Pawley et al. (2002) reduced their data for monoclinic sym-metry. We have accordingly adjusted our results for the C 1

Ppolytype

with triclinic symmetry by doubling the length of the c-axis whenmaking the comparisons in the Fig. 2. The agreement is excellent for thea- and b-axes for both polytypes. The agreement of the c-axis is good,but better for C2/c than C 1

Ppolytype in Fig. 2. In Fig. 2 we can also see

that there is considerable scatter in the experimental data for the a- andb-axes, but for example the agreement between experiment and theoryis excellent for the volume compressibility for both polytypes.

The linear compressibility calculated from our elastic constantsat individual pressures reveals more detail about the variation incompressibility with pressure than the approach used by Pawley et al.(2002) of fitting a Murnaghan equation of state (EOS) over the pressurerange 10−4 to 5 GPa, which gives an average value for these conditions.Pawley et al. (2002) report linear compressibilities for the a-, b- and c-axes of βa=3.9×10−3, βb=2.7×10−3 and βc=11.8×10−3 GPa−1 which arevery similar to our values for the a- and b-axes, although our values haveβbNβa, with the difference being small above 3 GPa. At zero pressureour values forβc aremuchhigher than the results of Pawleyet al. (2002),being much higher for the C 1

Pthan C2/c polytype, whereas the agree-

ment with our predictions is excellent at higher pressures. There are anumber of physical properties that have significantly different valuesbetween C 1

Pand C2/c polytypes at zero pressure, begging the question

of which polytype best corresponds to experimentally measured pro-perties of talc.

Cautionary comments by Stixrude (2002) in his study of C 1P

polytype about the large difference between Voigt and Reussestimates for the bulk modulus K show that he probably calculatedthe elastic tensor, but does not report it. He fitted a 4th order Birch–Murnaghan EOS to LDA-calculated cell volumes as a function ofpressure, and so determined K0 the zero-pressure bulk modulus andits first and second derivatives (K0′, K0″). We have fitted the 4th orderBirch–Murnaghan EOS using the program EOS-FIT 5.2 (Angel, 2000) toour P–V results and the experimental results of Pawley et al. (2002)and the values of K0, K0′, K0″ are given in Supplementary data Table 4.The 4th order EOS determined K0 is in reasonable agreement betweenall studies (44.5, 37.8, 32.1 and 34.3 GPa for Pawley et al., Strixrude, c 1

P

and C2/c polytypes respectively). The agreement between Stixrude(2002) values and our calculations for c 1

Pis good, especially for K0′

(see Supplementary data Table 5). The agreement between the bulkmodulus determined by Pawley et al. (2002) by fitting a Murnaghanequation of state to their data and our calculations is also very good,whereas the discrepancy for K0′ is much larger (13.5 in our calcu-lations, 13.6 in Stixrude and 6.0 from Pawley et al.). The better agree-ment of our Reuss estimate of shear modulus G of 18.89 GPa for c 1

P

and 42.24 GPa for C2/c polytype with the 0.5 GPa experimental valueof Bailey and Holloway (2000) of 22.6 GPa is indicative of an

Fig. 6. Voigt–Reuss–Hill (VRH) average isotropic compressional (Vp) and shear (Vs)velocities and their ratio (Vp/Vs) as a function of pressure calculated from the elasticconstants.



anisotropy problem in their samples prepared by uniaxial coldpressing 5 μm grain size talc powder successively three times toattain 100% density (2793 kg/m3). We will address this point belowwhen considering the Vp velocities.

No full elastic tensor for talc, either experimental or calculated,seems to have been published so far, maybe with the partial exceptionof Gales and Mahanti (1989). In a lattice-dynamical study using two-body and three-body bond-bending phenomenological potentials atambient conditions, they derived values for C11, C13, C33, C44 and C66that they expected to be general for the whole family of 2:1 trio-

ctahedral layered silicates towhich the talc belongs. The generic valuesthey published, reproduced here in Supplementary data Table 5, are inrough agreement with those we predict here using ab initio methods.

Weworried that our 0GPa elastic tensor coefficients forC 1P

polytypemight be poor due to highly non-linear behaviour of some calculatedstress values under the two 0.5% and 0.75% strain magnitudes that weimposed. We accordingly performed an additional least-squarescalculation using five strains: 0.25, 0.30, 0.35, 0.40 and 0.45%. Theresidual between the 366 stress components calculated by VASP andthose recalculated from least-squares tensor coefficients was quite

Fig. 7. Evolution of P-wave velocity (Vp) distribution, anisotropy of S-waves (AVs) and fastest S-wave (Vs1) polarization, at pressures 0 and 3.9 GPa, for the Talc C 1P

(top) and 0 and3.2 GPa for C2/c (bottom) polytypes. The short black lines on the right-hand figures are the fastest S-wave (Vs1) polarizations, the slowest S-wave (Vs2) polarization are orthogonal toVs1, but not marked for clarity. Lambert equal-area upper hemisphere projections with elastic tensor orthogonal axes X1=a⁎ (north) and X3=c-axis (centre), X2=c×a⁎ for C 1

Pand

X2=b for C2/c (east) (see text for discussion).



acceptable at 0.91%. The numerical values of all 21 extracted coefficientswere in statistical agreement with those obtained previously from thetwo larger strains and reported in Supplementary data (Table 2a). VRHvalues for the additional calculation with five strains for K and G were43.38 and 39.09 GPa respectively, and therefore not very different fromthose calculatedpreviously (Table 2a).We interpret the result of this testas disproving the existence of an overwhelming non-linearity thatwould invalidate our calculated elastic constants.We feel thatour resultsin Supplementary data (Table 2) then correspond to the state-of-the-artfor this type of calculations.

Talc and many other layered silicates have very high elastic aniso-tropy. We have calculated the anisotropic Poisson's ratio from ourelastic tensors at low and high pressure, which are illustrated in Fig. 5.At low pressure, talc exhibits negative Poisson's ratio inmany differentcrystallographic directions. Negative Poisson's ratio or lateral exten-sion in response to stretching is a characteristic of auxetic materials(Yang et al., 2004) that were first described in 1987. Hence talc is anauxetic material, and to our knowledge is the only naturally occurringmineral to show this property to such a degree. At a pressure of 3.9 GPatalc has the more classical positive Poisson's ratio in most directions.

Fig. 8. Evolution of Vp/Vs1 and Vp/Vs2 distribution at pressures 0 and 3.9 GPa, for the Talc C 1P

(top) and 0 and 3.2 GPa for C2/c (bottom) polytypes. Lambert equal-area upperhemisphere projections with elastic tensor orthogonal axes X1=a⁎ (north) and X3=c-axis (centre), X2=c×a⁎ for C 1

Pand X2=b for C2/c (east) (see text for discussion).



Negative Poisson's ratio also occurs in α-cristobalite, where it isattributed to the rotation of the Si–O tetrahedral (Yeganeh-Haeri et al.,1992). Layered structures are also known to favour auxetic behaviourbyallowing local bending (Milton,1992). Stixrude (2002) reported thatcompression in the basal plane is accommodated primarily by theMg-octahedra and that the octahedral have unusual tendency to flatten onexpansion. Probably some combination of the layered structure,tetrahedral rotation and octahedral distortion are needed to explainthe macroscopic auxetic behaviour of talc in 3 dimensions.

4.1.2. Acoustic velocitiesThe experimental acoustic Vp velocities are compared with cal-

culated values in Fig. 10. The excellent agreement at low pressurebetween the calculated P-wave velocities and experiments of Alexan-drov and Ryhova (1961) in the [010] and [001] directions suggests thatthe zero-pressure elasticmoduli that showexceptional anisotropy for P-and S-wave are realistic. For Vp parallel to the b-axis both polytypes givealmost identical results, whereas for Vp parallel to the C-axis the agree-mentwith experiments ismuch closer for the c1

Ppolytype. TheVpvalue

measured by Bailey and Holloway (2000) at 0.5 GPa is well below theVRH average, being very close to the predicted velocity along the c-axisfor C 1

Ppolytype. Asmentioned before, this is similar to the case of their

shear modulus derived from their Vs measurement, which is closer toourReuss average. Given their report that the samples have100%density

(i.e. no porosity), then the most likely explanation of their low P-wavevelocity is that their sample has a crystallographic preferred orientationdue to the three successive uniaxial compressions used to obtain fulldensity. Given that all layered silicates tend to have platy habits, their c-axis tends to rotate towards the compaction direction, which is also thepropagation direction used for velocity measurement in the axialcylindrical samples of Bailey and Holloway (2000). Hence even a smalldegree of alignment of very anisotropic talc platelets will cause thevelocity to be lower than the VRH average of a random aggregate.

4.2. Geophysical implications

Talc is probably themost elastically anisotropicmineral reported todate. Talc is also widely disseminated in the oceanic crust and uppermantle, both as hydrothermal alteration product and from thebreakdown of antigorite, commonly located on actively deformingthrust faults. Talc is certainly a mineral present in subduction zones, inparticular in the mantle wedge and probably on the plate interfacewhere talc may form in ultramafic rocks by the reactionwith silica richfluids rising from the dehydration zone below. Talc is also present inUHPM rocks and more widely in continental orogenic belts, such asblueschists and whiteschists. Talc also occurs in sufficient volumes tobe mined commercially in many places in the world. Hence talc must,at least locally, have an impact on seismic anisotropy. The impact oftalc on that anisotropy will depend on its volume fraction and thedegree of crystal preferred orientation (CPO). Estimates from Hackeret al. (2003) place the volume fraction of talc in subduction zones at16% in low-grade prehnite–pumpellyite facies, 11–15% in hydratedlherzolite and harzburgite respectively, and finally up to 41% in com-pletely transformed antigorite serpentine in mantle wedge.

In the case of subduction zones it has become standard practice touse the isotropic velocities andPoisson's ratio to characterize the seismicresponse related to hydrated minerals (e.g. serpentine group minerals,Christensen, 1996, 2004) or the presence of dehydration fluids (e.g. Ito,1990; Kono et al., 2007). If talc is not deformed, it is likely to be randomlyoriented, and hence have isotropic physical properties. In this case thevolume fraction of talc may be deduced from isotropic values Vp, Vs andVp/Vs (see Fig. 6) by assuming simple linear proportionality betweenvelocity and volume fractions of candidateminerals. For example, in themantle wedge, if talc is present in high volumes, this is mainly due to

Fig.10. Theoretical predictions of P-wave velocities in [010] and [001] directions, with VRHisotropic averagevelocity as a function of pressure. The experimental P-wave velocities in asingle crystal of talc P-wave along [010] and [001], with values of 9.00 and 3.73 km/srespectively (open large squares), at room pressure (10−4 GPa) and temperature reportedby Alexandrov and Ryzhova (1961). The experimental P-wave velocity of 5.27 km/s (greyfilled large square) in cold pressed polycrystalline talc reported by Bailey and Holloway(2000) at a pressure of 0.5 GPa pressure and temperature of 25 °C.

Fig. 9. Seismic anisotropy of Talc. A. C 1P

and C2/c polytypes P-waves (AVp) and S-waves(AVs) anisotropy as a function of pressure. B. Vp/Vs anisotropy of the C 1

Ppolytype of talc

(see text for definition).



antigorite breakdown. According to the experimental measurements ofChristensen (2004), antigorite at 1.0 GPa has Vp=7.96 km/s,Vs=4.59 km/s and Vp/Vs=1.73. Our calculations predict that the c 1

P

polytype of talc at 0.9 GPa has Vp=6.59, Vs=4.15 km/s and Vp/Vs=1.59.Those values are respectively lower than for antigorite by 1.37, 0.44 km/sin Vp, Vs, and 0.14 in Vp/Vs. Hence the observed low velocities and lowVp/Vs in some mantle wedges could be due to the presence of talc asfound in theflat slab segment of the Andes (Wagner et al., 2005), centralAlaska wedge (Rossi et al., 2005), and in the Kurile back arc (Zheng andLay, 2006) where Vp/Vs values lower than 1.7 have been reported. Inaddition the breakdown of antigorite produces talc together withforsterite, the pure Mg end-member of the olivine series, which has thelowest Vp/Vs ratio of the olivine series at 1.74. An aggregate of forsteriteand talc should then have a low Vp/Vs ratio.

Given the highly anisotropic nature of talc and the volume fractionspresent, a talc polycrystal with CPO will influence the seismic aniso-tropy of subduction zones. Experimental deformation does producekinking and some alignment of the talc platelets (Escartín et al., 2008),but no quantitative study of CPOwasmade. Talc-amphibolite and talc-serpentine schists recovered from core-complexes near Atlantic mid-ocean ridge show heterogeneous high strain crystal–plastic textures(Boschi et al., 2006; Escartín et al., 2003). This probably causes CPO, butno CPOmeasurements have beenmade on these samples. Although noCPO measurements are available for talc, the seismic anisotropy ofcarbonated schist (38% talc) from a depth of 4673 m in the Kolaborehole has been measured in the laboratory. The Vp anisotropy ofthis sample is 26.5% with highest velocities in the foliation and lowestnormal to the foliation (Kern et al., 2001). The seismic anisotropyof thissample is compatible with talc having a CPO with the c-axis normal tothe foliation. The presence of anisotropy will make the interpretationof velocity ratios very complex due to the presence of Vp, Vs1 and Vs2,hence two ratios Vp/Vs1 and Vp/Vs2 that vary with the propagationdirection as illustrated in Fig. 8. The highest ratios of Vp/Vs1 and Vp/Vs2 occur approximately in the basal plane of talc and lowest ratiosoccur near the c-axis. There can be considerable variation of the Vp/Vs2 ratio in the basal plane with values differing by amountscomprised between 0.3 and 1.0. The anisotropic velocity ratios canvary between 1.1 and 4.1 depending on the propagation direction. Weconsider two ray paths of the seismic wave sampling the maximumvalues of Vp/Vs1 or Vp/Vs2. In spite of the difference in ray paths, wecan nevertheless state that: (i) the second arriving S-wave correspondsto the highest ratio at all pressures and decreases with increasingpressure, (ii) the first arriving S-wave corresponds to the next highestratio at all pressures. Vp/Vs1 is almost independent of pressure, andnearly equal to the isotropic VRH Vp/Vs ratio (Fig. 9B). We assume thatthe ray path of the seismic wave samples the minimum values of Vp/Vs1 or Vp/Vs2. It then follows that their values are very low andincrease slightly with pressure, but they remain lower than the iso-tropic VRH Vp/Vs ratio with essentially the same values for first andsecond arriving S-waves. The ray paths associated with minimumvalues of Vp/Vs1 or Vp/Vs2 could be in similar directions, near the c-axis. Clearly in an anisotropicmedium, Vp/Vs ratios are very difficult tointerpret, but great attention should be paid to the ray path.

A low-velocity layered structure 2–8 km thick at the top interfaceof subducting slabs has been observed down to depths greater than150 km using guided waves by Abers (2005). In the layered structure,Vp is 14% slower than in the cold slab interior at 50 km depth, reducingto less than 4% slower below 150 km depth (Abers et al., 2003; Abers,2005; Helffrich and Abers, 1997; Hori et al., 1985; Matsuzawa et al.,1986). We can make a rough estimate of the degree of preferredorientation necessary to account for the 10% change in Vp of from 50to 150 km depth range. Firstly, the degree of CPO can vary betweenzero for a random aggregate and one for a perfect alignment (i.e. asingle crystal). Secondly, the depth range we want to consider cor-responds to pressures of 1.4 to 4.7 GPa, for which Vp of a randomaggregate is given by the VRH values shown in Fig. 10, resulting in a

8.6% increase for Vp. For a perfectly aligned aggregate we can use thevariation of Vp with pressure shown in Fig. 10 for a single crystal,where we have the two extreme choices, either propagation along[010] or [001]. For propagation along [010] there is almost no variationwith pressure, hence we must use [001] to illustrate the effect of CPO.For propagation along the c-axis, Vp would increase by 21.1% from 1.4to 4.7 GPa. Assuming linear relationship between CPO and velocitychange, an alignment of 11% percent of the c-axis of the talc crystalswith the propagation direction could produce this observed 10%velocity change provided temperature effects and other phasechanges are ignored. It is likely that the deformation in the layeredlow-velocity structure on the top plate interface is intense shearing,causing development of a strong CPO. The c-axis of talc would mainlyalign normal to the interface whereas the a and b-axes would beparallel to the propagation direction of guided waves. The implicationof this analysis is that even a weak CPO would explain the velocitychange with depth as VRH Vp of a random aggregate already explains86% of the observed velocity increase. Obviously this calculation isidealized, as we have considered a rock composed of 100% talc. Giventhat talc is a structural element associated with chlorite and 10 Åphase, we could consider this argument as indicative of changes withpressure in other hydrated phases likely to be present in this depthrange. Abers (2005) has shown that the properties of this layer cannotbe explained by typical upper mantle mineralogy. We suggest herethat hydrated phases with pressure-dependent seismic propertiesmay explain the change of P-wave velocity with depth.

5. Conclusions

Exploring the pressure dependence of triclinic and monoclinicpolytypes of talc using ab initiomethods, we have shown that talc hasexceptional anisotropic elastic properties, and that this anisotropyreduces with increasing pressure. We have calculated the completeelastic single crystal tensors to 12 GPa for both polytypes. Comparisonbetween theory and experiment of the few elastic parameters avail-able in the literature illustrate that the theoretical calculations matchthe unit cell compression and experimental ultrasonic velocity data toa high degree of fidelity. Comparison between experiment and theorypredictions indicates that triclinic polytype matches the experimentaldata better than monoclinic, particularly for measurements madealong the c-axis. The elastic anisotropy of talc is exceptionally high atzero pressure, perhaps the highest of any naturally occurring mineral.For example, the exceptional anisotropy results in negative Poisson'sratio for many extension (or compression) directions in crystal coor-dinates. We predict talc to be an auxetic mineral at low pressures, andto even be the most auxetic mineral reported so far. The seismicanisotropy is also exceptional for Vp and Vs anisotropies of 80% and85% for the triclinic polytype. The seismic anisotropy reduces to about40% at 5 GPa, the limit of the observed natural occurrence of talcUHPM rocks in subduction zones.

The volume fraction of talc in subduction zones is thought to bebetween11 and 41% (e.g. Hacker et al., 2003). Given the high anisotropyof talc, it is expected to influence seismic anisotropy in the presence ofCPO. At present there seems to be no data on the CPO of talc, but givenits platy habit, it is likely to develop CPO via a variety of mechanisms(crystal plasticity, kinking, anisotropic growth..) in a way similar toother metamorphic layer silicates (e.g. biotite). If CPO is present, thenthe traditional use of Vp/Vs ratios for seismic interpretation in terms ofrock composition could be seriously complicated by the presence oftwo ratios in an anisotropic talc-rich rock, namely Vp/Vs1 and Vp/Vs2.Anisotropic talc layers will also influence guided wave velocity at thesubductionplate upper boundary and is a possible cause of the velocitychanges with depth (and pressure) in the low-velocity zone at slabupper interface. For a random aggregate of talc, the isotropic velocitieswill be lower than those of antigorite. Indeed the breakdown ofantigorite results in the production of talc+forsterite+water. Mg-rich



forsterite has the lowest Vp/Vs in the olivine series, hence the mixtureof talc plus forsterite will produce very low Vp/Vs values below 1.7, asobserved in some subduction zones.

Acknowledgements

DM thanks Benoit Ildefonse for discussions about the origin of talcin oceanic rocks. We thank the editor Lars Stixrude for a helpfulsuggestion concerning the role of Mg-octahedra during compressionand the two reviewers for pertinent remarks, in particular about the k-mesh, which lead to significant improvements for C2/c polytype. D.M.thanks Ross Angle for a copy of his program EOS-FIT 5.2 used for the4th order Birch–Murnaghan finite strain equation of state. This con-tributionwas made possible thanks to the support for DM from INSU-CNRS SEDIT (France) and the European Science Foundation (ESF)under the EUROCORES Programme EuroMinScI, through contract No.ERAS-CT-2003-980409 of the European Commission, DG Research,FP6.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at doi:10.1016/j.epsl.2008.07.047.

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Ab initio elastic properties of talc from 0 to 12 GPa: interpretation of seismic

velocities at mantle pressures and prediction of auxetic behaviour at low

pressure.

David Mainprice*a, Yvon Le Page b, John Rodgers c and Paul Jouanna a

aGéosciences Montpellier UMR CNRS 5243, Université Montpellier II, 34095

Montpellier, France. bICPET, National Research Council of Canada, Ottawa, ON, Canada, K1A 0R6.

cToth Information Systems Inc., Ottawa, Canada.

* corresponding author tel: +33-467143283; fax: +33-467143603

email: [email protected]

Submitted to Earth and Planetary Sciences Letters 21st March 2008

Revised 25th June 2008

Supplementary Material

Supplementary Material

Legends for Tables Table 1a. Optimized crystal structures for the triclinic polytype of talc at various pressures. The basic structure model for talc is: Space group: (number 2); atoms with z coordinate in the interval -1/2≤z<1/2 belong to a same talc layer. Table 1b. Optimized crystal structures for the monoclinic polytype of talc, space group C2/c (number 15); atoms with z coordinate in the interval -1/2≤z<1/2 belong to a same talc layer. Table 2a. Calculated elastic stiffness coefficients, isotropic Bulk and Shear moduli averages in GPa from 0 to 11.6 GPa pressure for the triclinic ( ) polytype. Table 2b. Calculated elastic stiffness coefficients, isotropic Bulk and Shear moduli averages in GPa in GPa from 0 to 12.5 GPa pressure for the monoclinic (C2/c) polytype. Table 3a. Cell volume, density and isotropic VRH average Vp and Vs for the triclinic ( ) polytype. Table 3b. Cell volume, density and isotropic VRH average Vp and Vs for the monoclinic (C2/c) polytype. Table 4. Equation of State : Birch-Murnaghan 4th order equation Table 5. Comparison with other zero-pressure results, except Bailey and Holloway (2000) at 0.5 GPa.

Table 1a. Optimized crystal structures for the triclinic polytype of talc at various pressures. The basic structure model for talc is: Space group: (number 2); atoms with z coordinate in the interval -1/2≤z<1/2 belong to a same talc layer. pressure(GPa) 0 0.87 1.96 3.89 5.90 7.89 9.71 11.60

a (Å) 5.2957 5.2760 5.2561 5.2227 5.1869 5.1561 5.1307 5.1052 b 9.1810 9.1496 9.1194 9.0612 9.0050 8.9541 8.9074 8.8629 c 9.4228 9.2574 9.1633 9.0216 8.9149 8.8262 8.7578 8.6948 α 90.372 90.322 90.359 90.299 90.340 90.290 90.225 90.171 β 98.880 99.504 99.941 100.110 100.611 100.826 100.941 101.029

cell

γ 90.110 90.183 90.241 90.286 90.330 90.343 90.331 90.313 x 0.247557881 0.253704817 0.257219263 0.259497733 0.263527067 0.265466906 0.266595981 0.267475796 y 0.502070336 0.502265900 0.502496480 0.502306516 0.502682539 0.502275407 0.501761415 0.501387096

Si 1 4i

z 0.293402008 0.298472426 0.301242222 0.305013099 0.308274088 0.310781901 0.312647691 0.314376972 x 0.248158070 0.254692614 0.258407254 0.261257923 0.265816183 0.268334053 0.269909242 0.271169805 y 0.835505871 0.835859161 0.836236097 0.836325053 0.836948562 0.836765931 0.836455526 0.836203594

Si 2 4i

z 0.293534319 0.298597256 0.301341765 0.305102192 0.308270210 0.310757081 0.312590442 0.314299500 x 0 0 0 0 0 0 0 0 y 0 0 0 0 0 0 0 0

Mg 3 2a -1 z 0 0 0 0 0 0 0 0

x 0.500057974 0.500329394 0.500570462 0.500865508 0.501163585 0.501338559 0.501537718 0.501753008 y 0.833246947 0.833129151 0.833024296 0.832862755 0.832714629 0.832590852 0.832484984 0.832386610

Mg 4 4i

z 0.999804648 0.999636304 0.999523317 0.999205556 0.998936572 0.998640798 0.998421339 0.998249794 x 0.200274153 0.203223371 0.204761256 0.206198009 0.208221477 0.209498920 0.210286433 0.210916146 y 0.834342407 0.834598820 0.834855613 0.834962082 0.835346427 0.835362927 0.835263733 0.835143502

O 5 4i

z 0.117860981 0.119803523 0.120748817 0.121995830 0.123091073 0.123961965 0.124601736 0.125226379 x 0.697540778 0.699913673 0.701512968 0.702515224 0.704435510 0.705260434 0.705888670 0.706351620 y 0.667491081 0.667563886 0.667787248 0.667871252 0.668174117 0.668194979 0.668142587 0.668063513

O6 4i

z 0.112917382 0.115395841 0.117187853 0.119675439 0.121941918 0.123605302 0.125044005 0.126331574 x 0.198641900 0.200775017 0.201999020 0.202652344 0.204108129 0.204709240 0.205171227 0.205552150 O7

4i y 0.500966128 0.500853439 0.500783325 0.500433105 0.500280295 0.499879248 0.499546945 0.499291513

z 0.117791265 0.119735917 0.120733586 0.121986434 0.123198708 0.124096122 0.124757475 0.125398455 x 0.031019607 0.042923450 0.050452493 0.058534588 0.068489289 0.074747779 0.079094021 0.083018814 y 0.936374296 0.941183200 0.944760985 0.950159430 0.955764755 0.959405453 0.962046813 0.964670625

O8 4i

z 0.351782035 0.357189117 0.360073461 0.363556402 0.366397069 0.368409585 0.369893051 0.371234645 x 0.531545475 0.543423498 0.551000378 0.559068480 0.568946524 0.575285792 0.579773547 0.583723689 y 0.902156484 0.898284935 0.895647183 0.890575275 0.886750096 0.882887184 0.879542711 0.876491328

O9 4i

z 0.353669600 0.360516598 0.364326716 0.369540672 0.374033780 0.377464039 0.379835710 0.381850414 x 0.229429911 0.228170061 0.225790424 0.217834713 0.212413285 0.206606069 0.201722379 0.196776733 y 0.669180181 0.669424071 0.669771066 0.669692283 0.670262283 0.669963531 0.669548905 0.669240576

O10 4i

z 0.352218612 0.357779112 0.360475738 0.364067858 0.366506352 0.368288520 0.369693959 0.370940996 x 0.725670238 0.730147193 0.733530974 0.735361734 0.739519596 0.741442215 0.742656135 0.743636965 y 0.667614777 0.667417290 0.667598485 0.667473300 0.667625176 0.667423585 0.667217582 0.667046704

H11 4i

z 0.217101208 0.221614871 0.224621478 0.228823948 0.232548776 0.235370779 0.237699572 0.239800157

Table 1b. Optimized crystal structures for the monoclinic polytype of talc, space group C2/c (number 15); atoms with z coordinate in the interval -1/2≤z<1/2 belong to a same talc layer. pressure(GPa) 0 0.96 1.82 3.22 5.63 7.61 9.13 10.62

a (Å) 5.2976 5.2869 5.2673 5.2423 5.2082 5.1773 5.1561 5.1349 b 9.1447 9.1211 9.0824 9.0364 8.9639 8.9115 8.8752 8.8400 c 18.8083 18.6430 18.4229 18.2126 17.9006 17.7112 17.5921 17.4850

cell

β 101.471 101.777 101.710 101.605 101.732 101.711 101.697 101.680 x 0.772481036 0.774853873 0.774984635 0.775103366 0.775921126 0.776020964 0.776084796 0.775956486 y 0.999485056 0.999418051 0.999328924 0.999206320 0.998943988 0.998685750 0.998546505 0.998413798

Si 1 8f

z 0.148033648 0.149298318 0.150737423 0.152091989 0.154131390 0.155374996 0.156137798 0.156830417 x 0.272568995 0.274900631 0.274996268 0.275069871 0.275776534 0.275820817 0.275851879 0.275694249 y 0.166647749 0.166724847 0.166847419 0.167032521 0.167223864 0.167413189 0.167518107 0.167576095

Si 2 8f

z 0.148045168 0.149308176 0.150746494 0.152102495 0.154144734 0.155388753 0.156145938 0.156827638 x 0 0 0 0 0 0 0 0 y 0 0 0 0 0 0 0 0

Mg 3 4a -1 z 0 0 0 0 0 0 0 0

x 0.999542226 0.999278931 0.998976282 0.998639188 0.997961976 0.997491844 0.997226389 0.996961080 y 0.333251756 0.333187663 0.333112394 0.333044722 0.332901528 0.332813259 0.332754564 0.332699697

Mg 4 8f

z 0.999884769 0.999813367 0.999758068 0.999706975 0.999494689 0.999331901 0.999239778 0.999160079 x 0.208951850 0.209886213 0.209902921 0.209849415 0.210091593 0.210045280 0.210022392 0.209891597 y 0.499134032 0.498987181 0.498832361 0.498643041 0.498279100 0.497974313 0.497833278 0.497749770

O 5 8f

z 0.059389470 0.059873106 0.060395799 0.060881819 0.061550179 0.061998829 0.062269879 0.062523462 x 0.209342211 0.210135844 0.209990170 0.209776973 0.209574407 0.209290847 0.209143805 0.208947254 y 0.167065478 0.167211632 0.167401235 0.167627892 0.167945647 0.168197565 0.168296684 0.168328359

O6 8f

z 0.059419525 0.059906507 0.060439655 0.060933995 0.061611310 0.062068342 0.062335152 0.062576785 x 0.207299882 0.208614103 0.208967421 0.209195882 0.210548773 0.211198758 0.211616169 0.211949980 y 0.833467182 0.833490259 0.833496652 0.833501764 0.833546336 0.833607734 0.833650323 0.833687256

O7 8f

z 0.057085245 0.057786208 0.058676866 0.059561807 0.060970598 0.061855936 0.062406832 0.062889017 O8 x 0.063315399 0.068399112 0.071784352 0.075639200 0.081915802 0.085726761 0.088157807 0.090324403

y 0.062977614 0.060711039 0.057244592 0.053198270 0.047475923 0.043527811 0.041035880 0.038618848 8f z 0.177319295 0.178703424 0.180209298 0.181486726 0.183265274 0.184247833 0.184845431 0.185389580 x 0.563307028 0.568394088 0.571744606 0.575617000 0.581825315 0.585593105 0.587988071 0.590119455 y 0.103069786 0.105358184 0.108872846 0.113003166 0.118655783 0.122532077 0.124997183 0.127341859

O9 8f

z 0.177382522 0.178741939 0.180212829 0.181462465 0.183167689 0.184118679 0.184689063 0.185218860 x 0.255140154 0.253813046 0.247794062 0.240609850 0.231437280 0.224811778 0.220483841 0.215889394 y 0.333066949 0.333072545 0.333096980 0.333129933 0.333106795 0.333079206 0.333061038 0.333025490

O10 8f

z 0.178695178 0.180397202 0.182368176 0.184235289 0.187025319 0.188756351 0.189746649 0.190551886 x 0.243381871 0.245830670 0.245964748 0.245726333 0.247302239 0.247878216 0.248286695 0.248559009 y 0.834148452 0.834134920 0.834112126 0.833985364 0.834012650 0.834131773 0.834167497 0.834215537

H11 8f

z 0.109703353 0.110924065 0.112434494 0.113904052 0.116272948 0.117727729 0.118638379 0.119447017

Table 2a. Calculated elastic stiffness coefficients, isotropic Bulk and Shear moduli averages in GPa from 0 to 11.60 GPa pressure for the triclinic ( ) polytype. Pressure (GPa) 0 0.87 1.96 3.89 5.90 7.89 9.71 11.60

C11 219.83 206.22 211.08 207.57 211.00 219.91 227.45 229.97 C12 59.66 53.08 55.45 48.59 52.77 61.45 66.21 71.42 C13 -4.82 6.79 11.98 21.63 27.39 38.42 49.30 55.90 C14 -0.82 -1.55 -1.67 -0.95 -0.58 0.22 -0.42 -1.56 C15 -33.87 -33.16 -29.98 -26.29 -22.22 -19.58 -16.11 -13.03 C16 -1.04 -1.02 0.51 2.24 3.52 4.87 5.61 5.89 C22 216.38 223.48 217.63 214.28 212.71 220.55 221.95 226.78 C23 -3.67 10.66 13.92 25.42 34.22 46.45 50.30 61.36 C24 1.79 2.12 3.79 5.32 6.84 7.56 8.53 6.44 C25 -16.51 -14.85 -10.85 -8.12 -1.72 -3.73 -2.72 -2.27 C26 -0.62 0.15 1.98 5.56 5.29 6.85 7.68 8.92 C33 48.89 78.00 99.43 120.06 136.17 162.38 170.98 193.66 C34 4.12 2.62 3.49 7.39 7.75 9.92 8.83 7.21 C35 -15.52 -9.52 -5.95 -5.12 -1.15 -5.58 -7.41 -7.42 C36 -3.59 -5.23 -5.99 -8.08 -5.65 -4.09 -3.61 -2.63 C44 26.54 32.52 41.45 48.68 57.80 66.42 71.62 76.75 C45 -3.60 -5.80 -9.08 -9.89 -10.65 -9.97 -9.87 -8.62 C46 -6.41 -7.36 -6.75 -6.78 -4.29 -3.85 -2.20 -1.36 C55 22.85 31.22 32.97 40.68 47.77 56.92 63.62 69.77 C56 -1.67 -3.11 -3.25 -4.07 -2.56 -1.49 -0.81 -1.25 C66 78.29 79.12 80.68 79.12 80.69 80.80 80.68 78.82

Voigt 65.27 72.09 76.76 81.47 87.63 99.50 105.78 114.20 Reuss 18.89 45.05 58.48 70.71 81.18 94.58 101.72 111.97

K Bulk (GPa) VRH 42.08 58.57 67.62 76.09 84.41 97.04 103.75 113.08

Voigt 54.46 57.72 60.80 63.45 66.95 71.26 73.49 75.85 Reuss 26.42 40.70 46.12 53.54 60.33 67.53 71.01 74.43

G Shear (GPa) VRH 40.44 49.21 53.46 58.49 63.64 69.40 72.25 75.14

Table 2b. Calculated elastic stiffness coefficients, isotropic Bulk and Shear moduli averages in GPa from 0 to 12.52 GPa pressure for the monoclinic (C2/c) polytype. Pressure (GPa) 0 0.15 0.35 0.64 0.93 1.72 2.81 4.60 6.77 8.18 10.66 12.52

C11 243.23 241.89 241.12 240.54 238.34 238.4 239.28 230.92 227.97 234.72 234.92 241.48 C12 96.35 93.76 94.27 93.66 91.39 87.18 85.62 77.98 80.04 86.18 89.14 96.64 C13 23.56 17.38 14.78 16.76 19.12 21.92 27.22 31.48 34.24 43.22 51.76 54.98 C15 -29.94 -27.65 -28.17 -28.42 -27.37 -22.79 -18.98 -11.92 -4.69 -0.29 4.67 8.33 C22 256.62 252.34 250.71 250.8 251.26 245.75 248.02 239.37 241.85 246.29 249.86 257.94 C23 10.25 9.65 9.45 7.88 9.05 7.89 17.54 18.59 28.58 38.36 48.94 57.47 C25 -15.24 -13.01 -13.62 -12.72 -13.71 -12.75 -11.48 -11.22 -9.82 -9.12 -7.46 -7.28 C33 53.72 67.22 70.58 72.71 76.45 90.89 117 122.98 145.15 159.19 177.75 190.06 C35 -19.05 -22.31 -23.39 -19.62 -19.88 -16.88 -18.04 -15.18 -14.55 -11.69 -9.43 -10.16 C44 40.12 39.05 42.09 48.58 46.58 41.42 45.35 48.31 54.34 61.54 68.98 71.51 C46 -5.48 -6.25 -4.37 -0.25 -3.92 -5.02 -4.63 -4.06 -2.32 -1.42 0.55 -0.08 C55 54.92 52.68 51.68 52.37 52.75 59.28 63.37 69.06 74.52 78.15 83.31 86.52 C66 74.49 75.72 76.09 76.8 77.05 77.08 78.98 78.92 79.69 79.82 78.55 78.55

Voigt 90.43 89.23 88.82 88.96 89.46 89.89 96.12 94.38 100.08 108.41 115.80 123.07 Reuss 39.55 44.49 44.75 48.41 50.83 60.24 74.67 78.38 89.55 100.13 110.57 118.33

K Bulk (GPa) Hill 64.99 66.86 66.79 68.69 70.15 75.07 85.39 86.38 94.81 104.27 113.19 120.7

Voigt 62.13 62.87 63.57 65.27 65.04 66.09 69.14 70.27 73.18 75.40 77.68 79.34 Reuss 47.07 49.20 50.40 54.04 54.23 57.08 62.44 65.17 69.88 73.25 76.33 77.92

G Shear (GPa) Hill 54.6 56.03 56.98 59.65 59.64 61.59 65.79 67.72 71.53 74.32 77.01 78.63

Table 3a. Cell volume, density and isotropic VRH average Vp and Vs for the triclinic ( ) polytype Pressure (GPa) Cell

volume (Å3) Density (kg/m3)

Vp (km/s) Vs (km/s)

0.00 452.637 2782.76 5.87 3.81 0.87 440.736 2857.90 6.59 4.15 1.96 432.609 2911.58 6.91 4.28 3.89 420.292 2996.91 7.17 4.42 5.90 409.255 3077.73 7.41 4.55 7.89 400.219 3147.22 7.76 4.69 9.71 392.954 3205.41 7.90 4.75 11.60 386.138 3261.99 8.08 4.80

Table 3b. Cell volume, density and isotropic VRH average Vp and Vs for the monoclinic (C2/c) polytype. Pressure (GPa) Cell

volume (Å3) Density (kg/m3) Vp (km/s) Vp (km/s)

0.00 912.263 2761.44 7.06 4.45 0.15 907.442 2776.11 7.14 4.49 0.35 905.344 2782.54 7.16 4.52 0.64 899.436 2800.82 7.27 4.61 0.93 892.570 2822.36 7.28 4.60 1.72 876.270 2874.86 7.39 4.63 2.81 859.697 2930.28 7.69 4.74 4.60 837.192 3009.04 7.66 4.74 6.77 814.414 3093.20 7.84 4.81 8.18 801.729 3142.15 8.04 4.86 10.66 782.284 3220.26 8.19 4.89 12.52 769.012 3275.83 8.30 4.90

Table 4. Equation of State : Birch-Murnaghan 4th order equation Pawley et al.

2002 Strixrude 2002

Triclinic Monoclinic C2/c

Vo (Å3) 911.9 217.5 452.0 913.2 Ko (GPa) 44.5 37.8 32.1 34.3 K0' 3.0 13.6 17.4 11.5 K0'' (GPa-1) +1.0 -4.0 -9.1 -3.0 Maximum Pressure (GPa) 6.2 26.0 11.6 12.5 Number of points 10 13 8 12 Weighted χ2 3.36 - 0.63 0.23 Goodness of fit R % 3.80 - 0.88 0.68 Max dP (GPa) 0.22 - 0.11 0.08 The data of Pawley et al. 2003, plus our results for the triclinic and monoclinic polytypes were fitted using the program EOS-FIT 5.2 (available at http://www.crystal.vt.edu/crystal/software.html) by Ross Angel (Virginia Tech, Blacksburg, USA). Strixrude 2002 is the EOS for LDA static given in his table 1. Max dP is the maximum difference in pressure at a given volume between the observations and the fitted EOS. See Angel (2000) for discussion of EOS. Table 5. Comparison with other zero-pressure results, except Bailey and Holloway (2000) at 0.5 GPa.

Experimental Calculated Pawley et

al., 2002 EOS

Babeyko et al., 1994

Bailey and

Holloway, 2000

ultrasonic

This study

VRH average

This study C2/c VRH

average

Stixrude 2002 EOS

Gales and Mahanti,1989

Lattice dynamics

K0 (GPa) 41.0 59.2 ─ 42.9 64.99 37.8 ─ K0' 6.0 ─ ─ 13.5 ─ 13.6 ─

K0'' (GPa-1) ─ ─ ─ -2.2 ─ -4.0 ─ G (GPa) ─ 31.4 22.6 40.4 54.6 ─ ─

C11 (GPa) ─ ─ ─ 219.8 243.2 ─ 161.1 C13 (GPa) ─ ─ ─ -4.8 23.6 ─ 8.4

C33 (GPa) ─ ─ ─ 48.9 53.7 ─ 50.1 C44 (GPa) ─ ─ ─ 26.5 40.2 ─ 14.4 C66 (GPa) ─ ─ ─ 78.6 74.5 ─ 55.0

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