Detecting deeply subducted crust from the elasticity of hollandite

Earth and Planetary Science Letters 288 (2009) 349–358

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Earth and Planetary Science Letters j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e p s l

Detecting deeply subducted crust from the elasticity of hollandite Mainak Mookherjee ⁎, Gerd Steinle-Neumann Bayrisches Geoinstitut, Universitat Bayreuth, Universitatstrasse, 30, Bayreuth D95440, Germany

a r t i c l e

i n f o

Article history: Received 2 June 2009 Received in revised form 10 September 2009 Accepted 20 September 2009 Available online 14 October 2009 Editor: R.D. van der Hilst Keywords: hollandite elasticity mantle discontinuity subduction crustal materials

a b s t r a c t Subduction of differentiated continental and oceanic crusts through sediments and basalt to the deep mantle has been shown to be a likely source for the geochemical signature of ocean island basalts that are enriched in large ion lithophile elements such as K, Na, Rb, and Sr. At high pressure such a lithology will consist of stishovite, majorite and hollandite, where hollandite (KAlSi3O8) can readily host the large ion lithophile elements, and is hence a geochemically important phase. Here we study the elasticity of hollandite up to lower mantle pressure by electronic structure simulations and attempt to constrain the volume percent of hollandite in a subduction zone environment. In agreement with experiments we predict a phase transition from a low pressure tetragonal phase to a high pressure monoclinic phase at 33 GPa. The phase transition has significant effects on the elastic properties of hollandite, with an increase in shear modulus of 10%. Based on the computed reflection coefficient across the transition and observed reflectance for mid-mantle seismic scatterers (920 km discontinuity) we constrain the maximum volume of hollandite to be around 5% in a subduction zone environment. © 2009 Elsevier B.V. All rights reserved.

1. Introduction Potassium (40K) is an important heat producing radionuclide in the Earth and other terrestrial planets (Wasserburg et al., 1964). However, potassium (K) in the bulk silicate Earth (∼240 ppm) is depleted compared to carbonaceous chondrite (CI, ∼ 550 ppm) (Mcdonough and Sun, 1995), and this depletion has been investigated for many decades. K could have been volatalized during accretion and differentiation and been lost to space (Mcdonough and Sun, 1995) or exhibit siderophile behavior at high pressure, with K being removed from the silicate Earth to the core (Bukowinski, 1976; Lee and Jeanloz, 2003; Murthy et al., 2003; Lee et al., 2004). Although the concentration of potassium is low in the bulk silicate earth, it is an important geochemical tracer. Owing to its large ionic size potassium readily fractionates to basaltic magma during mantle melting. Basaltic volcanism samples the deep interior of the Earth from a depth of 60–300 km beneath mid oceanic ridges (Dasgupta and Hirschmann, 2006), to the core mantle boundary region where partial melts might be present at the base of the plumes (Zou et al., 2007) and subsequently be transported to the surface (Montelli et al., 2004). Based on trace elements, basalts are broadly categorized into mid oceanic ridge basalts (MORB) which are depleted with respect to large ion lithophile (LIL) and associated trace elements and ocean island basalts (OIB) which are enriched in LIL elements (Zindler and Hart, 1986; Hofmann, 1997). As a consequence solid Earth geochemists

⁎ Corresponding author. Tel.: +49 921 553746; fax: +49 921 553769. E-mail address: [email protected] (M. Mookherjee). 0012-821X/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.epsl.2009.09.037

have proposed deep reservoirs of missing trace elements in the Earth's transition zone and lower mantle. Armstrong (1968) proposed that subduction of sediments derived from continental crust to the deep mantle and the ultimate return of sediment in plumes could explain the chemical signatures in OIBs. Recently Rapp et al. (2008) demonstrated that ∼ 10% continental sediment could match the trace element signatures for Pitcairn OIBs, and around 40–50% of continental sediment could explain trace element signatures of Gaussberg Lamprotites, Antartica (Murphy et al., 2002). This notion is supported by isotopic signatures (enriched 87Sr/86Sr and 143Nd/ 144 Nd) in Samoan lavas that are indicative of recycled sediments with compositions similar to continental crust (Jackson et al., 2007). Phase equilibria studies on sediments of average continental crust composition between 6 and 24 GPa revealed majorite, stishovite and hollandite mineral phases in equal proportions (Irifune et al., 1994), showing the importance of the hollandite phase for subducted sediments. Prior to the recent geochemical work by Jackson et al. (2007), Hofmann (1997) argued that the OIB signature is mainly due to subducted oceanic crust and not by continental sediments. Even in this case Wang and Takahashi (1999) demonstrated that basalts enriched with potassium such as OIB, will stabilize hollandite as the liquidus phase at transition zone conditions. This finds support in the discovery of hollandite with a 92% K-end member component in Kankan diamond inclusions, which are likely to be derived from the lower mantle (Stachel et al., 2000). Similarly, natural KAlSi3O8 hollandite has been reported in melt veins of shocked meteorite with crystallization pressures of ∼ 23 GPa (Langenhorst and Poirier, 2000).

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These natural samples suggest that hollandite can host other alkali and alkaline earth cations such as Na and Ca. However, in laboratory experiments the amount of sodium in hollandite seems to be limited to 40 mol% of NaAlSi3O8 (Yagi et al., 1994). Gillet et al. (2000), on the other hand, found evidence of NaAlSi3O8 end member in shocked meteorites and suggested that the solubility of sodium is likely to be enhanced by pressure. Tutti (2007) demonstrated that the Na-endmember of hollandite is stable only in a narrow range of pressure between 19 and 23 GPa (at 2273 K) and decomposes to the calcium ferrite structure beyond that pressures. Under ambient conditions, hollandite has tetragonal symmetry I4/m (Bystrom and Bystrom, 1949; Ringwood et al., 1967). Based on in-situ Xray diffraction in diamond anvil cell experiments (Sueda et al., 2004) showed that hollandite undergoes a phase transition from the tetragonal (hollandite-I) to a monoclinic (hollandite-II) phase with I2/m symmetry (Cadee and Verschoor, 1978) at 20 GPa at room temperature. The tetragonal to monoclinic transition is a second order displacive transition involving rotation of the Si(Al)O6 octahedral units, reducing the tunnel size at high pressures (Fig. 1). (Nishiyama et al., 2005) determined a positive Clapeyron slope P[GPa]= 16.6 + 0.007 ×T[K] for the tetragonal to monoclinic transformation. A lower transition pressure of 15 GPa has been reported based on Raman spectroscopy (Liu et al., 2009), and might be related to non-hydrostatic stresses; the largely unconstrained chemistry of the hollandite investigated may as well influence the transition pressure. Although much is known about the high pressure phase equilibria and equation of state for hollandite, no data exist for high pressure

elasticity. Elasticity data of hollandite is crucial to relate petrology and elasticity of subducted sediments and could hence be used to provide geophysical constraints on the amount of sediment subducted. In the present study, we explore the high pressure behavior of K-hollandite (hollandite-I and hollandite-II) as a function of pressure, both in terms of crystal chemistry and elastic properties. From the full elastic constant tensor for the tetragonal and monoclinic phases we attempt to constrain the amount of hollandite that is likely to be present in subduction zone environment the Earth's lower mantle. 2. Methods Density functional theory (DFT) (Kohn and Sham, 1965) has proved to be a powerful tool for studying the structure and thermodynamics of Earth materials at high pressure. While the theory is exact in principle, the form of exchange and correlation, accounting for electronic many-body interactions, is unknown and must be approximated. We use the local density approximation (LDA) which assumes that exchange and correlation are the same as in the uniform electron gas with the corresponding electron density. Although by construction the LDA approximation is expected to work only for systems with slowly varying electron density, LDA has been successfully applied to a wide range of silicate materials (Stixrude et al., 1998). In order to further facilitate solving the DFT equations, we use the pseudopotential method (Heine, 1970): in a pseudopotential the core (i.e. non-valence) electrons of an atom are replaced by an effective potential, so that a smaller number of

Fig. 1. (a) Outline of the tetragonal hollandite-I unit cell (blue) and the monoclinic hollandite-II unit cell. Coordinate axes for I4/m (blue), C2/m (green) and I2/m (blue) are also shown. Potassium ions (blue circles) are shown at the (0,0,0) position and (1/2,1/2,1/2) position in tetragonal and monoclinic I-type cell or (1/2,1/2,0) position in monoclinic C-type cell. The blue (b and c) and green arrows (d and e) refer to the projection direction for the structures in panels b through e. (b) Converged I4/m type structure at 21 GPa with Al octahedral units in light blue and Si octahedral units in dark blue, potassium ion are shown by the purple spheres. The structure is projected down the [001]tet direction, and the 4/m symmetry is generated by placing the aluminium atoms in different locations along the [001]tet direction which can be better seen along the [100]tet direction in (c). (d) shows the converged monoclinic structure C2/m at 21 GPa, outlining the monoclinic C-type unit cell and the monoclinic I-type unit cell. Note that at this pressure the monoclinic distortion is negligible; (e) converged monoclinic structure at 80 GPa, showing the monoclinic distortion in the tunnel shape.

M. Mookherjee, G. Steinle-Neumann / Earth and Planetary Science Letters 288 (2009) 349–358

electrons need to be considered in the computations, and a smaller set of basis functions suffice to accurately represent the charge density. Energetics of K-hollandite is evaluated with the Vienna ab-initio simulation package (VASP) (Kresse and Hafner, 1993; Kresse and Furthmuller, 1996a,b) with a plane wave basis set and ultrasoft Vanderbilt pseudopotentials (Vanderbilt, 1990; Kresse et al., 1992). Hollandite has a general chemical formula AB4O8, with BO6 octahedral units and AO8 distorted tetrahedral prism. Its crystal structure has tetragonal I4/m symmetry (space-group no. 87) (Fig. 1a) (Ringwood et al., 1967), and is related to the rutile structure: The (Si, Al)O6 octahedral units form double chains parallel to the [001] direction. These chains share corners with neighboring double chains to form a framework structure (Fig. 1). The average octahedral bond length dictates the size of unit cell, and the radius ratio RA/RB (tunnel cation A and octahedral cation B) determines the symmetry of the structure (Cheary, 1986). In K-hollandite (KAlSi3O8) K occupies the distorted tetrahedral prisms (tunnel), Al and Si are located on the octahedral site (Ringwood et al., 1967). At high pressures hollandite has a monoclinic space-group symmetry I2/m. We have used C2/m (space-group no. 12) (Fig. 1c, d) and transformed the lattice parameters back to I2/m for comparison with the tetragonal symmetry I4/m, since there is no formal description of I2/m in the crystallographic literature. All computations are performed in a 104 atom cell consisting of a 1 × 1 × 4 super cell based on a 26 atom body centered unit cell. Since our calculations are static, we have ordered the Al and Si to generate the tetragonal symmetry (Fig. 1a, b). We use an energy cutoff of Ecut = 400 eV and reciprocal space sampling is restricted to the Γ-point. A series of convergence test demonstrated that these computational parameters yield pressures and total energies that are converged to within 0.01 GPa and 10 meV/atom respectively. We analyze bulk compression behavior as well as that of structural units using a third order Birch–Murnaghan equation of state (Birch, 1978). The linear compressibility of the axes is similarly evaluated using a linear formulation of Eulerian finite strain theory (Davies,

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1973a,b). To compute the full elastic constant tensor we strain the lattice and let the internal degree of freedom of the crystal structure relax consistent with the symmetry: elastic constants are obtained through the changes in stress tensor (σ̳) with respect to applied strain (ε̳). We apply positive and negative ε̳ of magnitude 1% in order to accurately determine σ̳ in the appropriate limit of zero strain. The details of the method are outlined in (Karki and Stixrude, 1999). This formulation of elastic constants governs elastic wave propagation, and stability criteria at general (finite) pressure (Steinle-Neumann and Cohen, 2004). The tetragonal hollandite-I has seven independent elastic constants c11(=c22), c12, c13(=c23), c33, c44(=c55), c16(=−c26) and c66 (Nye, 1985), monoclinic hollandite-II (with [010] axis aligned to 2-fold symmetry) has thirteen independent elastic constants c11, c12, c13, c15, c22, c23, c25, c33, c35, c44, c46, c55 and c66 (Nye, 1985). 3. Results 3.1. Compressibility and structure at high pressure Calculated pressure–volume relations for hollandite-I and hollandite-II are shown in Fig. 2. We find that the Birch– Murnaghan equation of state (Birch, 1978) describes the compression behavior over the entire pressure range well. Based on our static calculations we find that the enthalpy of hollandite-I is smaller at low pressures, with a transition to hollandite-II around 33 GPa (Fig. 2, inset). This is in reasonable agreement with experimental observation where the structural phase transformation from hollandite-I to hollandite-II occurs at 22 GPa (Sueda et al., 2004; Ferroir et al., 2006). The discrepancy may be partly due to the ordering scheme of Al and Si that we have adopted in our simulation. Although our calculation is static, in order to generate the I4/m symmetry of hollandite-I, we have ordered the Al and Si atoms in different layers (Fig. 1) such that it mimics Al–Si disorder which is likely to occur at higher temperature. This might explain why our static transition

Fig. 2. Equation of state of hollandite: LDA theory for hollandite-I (tetragonal, blue open circles) and hollandite-II (monoclinic, green open circles), with black solid lines for finite strain fits to the computational results. The red line shows the LDA equation of state for hollandite-I with zero point and thermal corrections to 300 K. Smaller symbols show experimental data (red open squares (Zhang et al., 1993); black open circles (Ferroir et al., 2006); blue open diamonds (Hirao et al., 2008)).The inset on the upper right shows ΔHmon–tet for the tetragonal to monoclinic transformation.

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M. Mookherjee, G. Steinle-Neumann / Earth and Planetary Science Letters 288 (2009) 349–358

pressure is around 33 GPa while experiments at room temperature find the transition at 22 GPa. However, at high temperatures (∼1800 K), transition pressures are around 30 GPa based on a positive Clapeyron slope of P[GPa] = 16.6 + 0.007 × T[K] (Nishiyama et al., 2005). In addition, LDA often overestimates transition pressures (Yong et al., 2008). In terms of equation of state (Fig. 2) and lattice parameter evolution under compression (Fig. 3) we find good agreement with experimental data. The static zero pressure volume,V0 for hollandite-I within LDA is 227.7 Å3. This is 3.6% smaller than V0 determined by single crystal X-ray diffraction (Table 1). The static zero pressure value of the bulk modulus, K0, within LDA is 225 GPa. K0 from single crystal X-ray diffraction is 180 GPa (Zhang et al., 1993), whereas more recent insitu synchroton powder diffraction yield K0 = 201 GPa (Ferroir et al., 2006), more consistent with the computational results. The discrepancy between these studies is likely due to the limited pressure range explored by (Zhang et al., 1993) (up to 4.5 GPa). Moreover, (Zhang et al., 1993) noted that when the zero pressure volume (V0) is unconstrained, K0 varies between 187 and 193 GPa for corresponding K0′ values of 2.8 to 6.

Table 1 Equation of state parameters for low-pressure hollandite-I (tetragonal) and highpressure hollandite-II (monoclinic) from experimental and theoretical studies. V0 (Å3)

K0 (GPa)

K′0

Theory (this study) Hollandite-I (static) Hollandite-I (300 K)

227.65 232.23

225.0 212.0

4.3 4.3

Hollandite-II Hollandite-II (300 K)

228.02 234.95

220.5 207.7

3.9 3.9

241.06a 236.73b 236.26c 237.60d 237.01e

180.0 183.0 201.4

4.0 4.0 4.0

232.30f 237.01g

232.0 181.0

4.0 4.9

Experiments

Hollandite-I

Hollandite-II a

Powder X-ray diffraction: Ringwood et al. (1967). Powder X-ray diffraction: Yamada et al. (1984). c Single-crystal X-ray diffraction: Zhang et al. (1993). d In situ synchroton X-ray diffraction: Nishiyama et al. (2005). e In situ synchroton powder X-ray diffraction (Ferroir et al., 2006). f In situ synchroton X-ray diffraction with laser heated diamond anvil cell, K′0 fixed at 4.0. g In situ synchroton X-ray diffraction with laser heated diamond anvil cell, parameters obtained by combining data of hollandite-II phase from Hirao et al. (2008) and Ferroir et al. (2006). b

The difference between theory (LDA) and experiment (Ferroir et al., 2006) is similar to that found in other silicates and oxides relevant to the Earth's mantle (Mookherjee and Stixrude, 2006, 2009; Mookherjee and Steinle-Neumann, 2009) and can be attributed to the influence of phonon excitation, and the approximations to the exchange-correlation functional. We correct for the thermal and zero point motion effects by a pressure correction, estimating the phonon excitation with a Debye–Grüneisen model following Stixrude (2002). The zero point pressure is given by Pzp =9nγkBθD/8V, where θD is Debye temperature, γ is the Grüneisen parameter, V is the volume of the unit cell, n is the number of atoms in the unit cell and kB is the Boltzmann constant. In the Debye approximation, the thermal energy is given by 3

θ =T

Eth = 9nkB TðT =θD Þ ∫0D

Fig. 3. Evolution of lattice parameters under compression: (a) lattice parameters a and b; (b) lattice parameter c. Computational results are shown in large open circles for hollandite-I (blue) and hollandite-II (green). The green line is shown as a guide to the eye for the computed monoclinic lattice parameters, and the black line is a finite strain fit to the hollandite-I results. Experimental data are shown in smaller symbols (black open circles (Ferroir et al., 2006); blue open diamonds (Hirao et al., 2008)).

3

x

ðx = ðe  1ÞÞdx

ð1Þ

and the thermal pressure by Pth = γEth/V. We estimate the elastic Debye temperature based on the formulation θD = 251.2(ρ/M̅)1/3 v̅, where v ̅ is related to the average seismic velocity, ρ is density and M̅ is pffiffiffiffiffiffiffiffiffiffiffiffi mean atomic mass (Poirier, 2000). We assume that  v ∼vB = K0 = ρ, the bulk sound velocity, with K0 ∼ 201 GPa and density, ρ ∼ 3.91 gm/ cm3 from the experiment (Ferroir et al., 2006). These values yield v ̅ ∼ 7.17 km/s and θD ∼ 1000 K. θD ∼ 1000 K is typical for octahedrally coordinated silicate phases relevant to the Earth's mantle (Poirier, 2000). We calculate a thermodynamic Grüneisen parameter of γ0 = αVK0/CV ∼ 2, with the thermal expansivity α ∼ 3.3 × 10− 5 K− 1 (Akaogi et al., 2004). The heat capacity CV ∼ 195.4 Jmol− 1 K− 1 is obtained from CP ∼ 200.3 Jmol− 1 K− 1 (Yong et al., 2006) using the relation CV =CP − α2 TK0/ρ. We assume a volume dependence of the Grüneisen parameter q = ∂ln(γ)∂ln(V) ∼ 1 which has proved to be a reasonable approximation for mantle silicates (Kiefer et al., 2001). Applying the corrections for phonon excitations yields equation of state parameters at 300 K, which are in better agreement with the experiment (Table 1). The linear compressibility, Kα where α refers to a and c-axes is defined as Fα = Kα + mα fα

ð2Þ

M. Mookherjee, G. Steinle-Neumann / Earth and Planetary Science Letters 288 (2009) 349–358

where fα is the linear Eulerian finite strain, and mα is function of third order elastic constants fα =

1 2



α α0

2

 1

ð3Þ

and Fα is the normalized pressure Fα =

P fα ð1 + 2fα Þð1 + 2fV Þ

ð4Þ

and fV is the volume Eulerian strain (Davies, 1974). The linear compressibilities exhibit strong anisotropy with the [001] direction being almost 1.5 times stiffer than the [100] direction (Ka = 601 GPa and Kc = 892 GPa). The anisotropy is in good agreement with (Zhang et al., 1993) who report Ka = 549 GPa and Kc = 961 GPa. The stiffness along [001] is related to the Si(Al)–Si(Al) repulsion (Fig. 1). By

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contrast, the softness along [100] is related to the deformation of the tunnels in the hollandite structure. The linear .compressibilities . . are related to bulk compressibility as 1 K Volume = 2 K a + 1 K c and in 0 the present study the values are consistent with the relationship. The linear compressibilities from single crystal diffraction (Zhang et al., 1993) yield K0 = 213 GPa, considerably greater than the K0 (180 GPa) reported from the P–V data in the same study (Table 1). The apparent discrepancy might be related to their treating the linear compressibility data with a linear fit in pressure and the volume-compression data with a finite strain fit. In our study we have consistently used finite strain fits to determine bulk and linear compressibilities. The octahedral units are rigid (have high bulk moduli) compared to the tunnels. In particular, the silicon octahedron is stiffer with a 6 6 bulk modulus KSiO of 342 GPa (with VSiO of 7.39 Å3 and K0′SiO6 ∼ 5.8). 0 0 This is in excellent agreement with compressibility of SiO6 octahedron 6 6 in stishovite (KSiO = 303–346 GPa (Andrault et al., 1998) and KSiO = 0 0 342 GPa (Ross et al., 1990)). The aluminium octahedra are signifi6 6 cantly softer with a bulk modulus of KAlO = 259 GPa (VAlO = 8.42 Å3 0 0 and pressure derivative of bulk modulus of AlO6 units, K0′AlO6 ∼ 4.9), consistent with mineral phases relevant for the mantle (Hazen et al., 6 2000): KAlO = 211–260 GPa. However, Zhang et al. (1993) reported an 0 6 anomalously low incompressibility for Si(Al)O6 (KSi(Al)O = 153 GPa) 0 and attributed such low bulk modulus to the substitution of silicon with aluminium. Since the stiffness along the [001] direction is greater than along [100], we might expect a relationship between the octahedral compression and linear compressibility of the crystal along the [001] direction (Kc). The average polyhedral bulk modulus Koct 0 should be one third of the linear modulus Kc. We define the average polyhedral SiO6 compression Koct + 0 (for a hollandite with Al:Si ∼ 1:3) as (3/4) × K0 AlO 6 (1/4)K0 ∼ 321 GPa which is greater than K c/3 ∼ 297 GPa > K a/ 3 ∼ 200 GPa. This indicates that the octahedral sites are not compressed isotropically. This is further documented in the quadratic 6

elongation Q = ∑ ðli = l0 Þ2 = 6, where we have assumed l0 to be the i=1

average octahedral bond length (Robinson et al., 1971) in hollandite-I. QSi decreases from 0.976 at 0 GPa to 0.926 at 30 GPa and QAl increases from 1.003 at 0 GPa to 1.004 at 30 GPa. However, note that the average octahedral stiffness could, to a large extent, account for the stiffness along the [001] direction. The relatively softer [100] direction

Fig. 4. (a) Correlation between unit cell volume and average octahedral bond distance; blue open circles and green open diamonds show LDA results for tetragonal hollandite-I and monoclinic hollandite-II. The octahedral bond length, on the x-axis refers to ¼average + ¾average. Black open circles show experimental data on a number of hollandite structured compounds (Ringwood et al., 1967) (b) Pressure evolution of hinge angle formed between adjacent octahedral units.

Fig. 5. Evolution of angle γ as a function of pressure; green open circles are for monoclinic hollandite-II. Note that at low pressure, i.e. below 22 GPa (in the stability field of hollandite-I) the γ values of hollandite-II (monoclinic) phase is almost constant and is to within 1.20 of the expected value of 900 for the tetragonal hollandite-I. Experimental data are shown by black open circles (Ferroir et al., 2006) and blue open diamonds (Hirao et al., 2008).

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M. Mookherjee, G. Steinle-Neumann / Earth and Planetary Science Letters 288 (2009) 349–358

can be explained by bending of Si(Al)–O–Si(Al) angles, with the octahedra behaving as rigid units. The unit cell size of hollandite is linearly dependent on the average octahedral bond distance (Fig. 4). This also implies that the size of the tunnel is related to the octahedral framework and average Si(Al)–O distances (Ringwood et al., 1967; Zhang et al., 1993). The KO8 polyhedra are very compressible with a bulk modulus of 8 KKO 0 = 157 GPa. (Zhang et al., 1993) reported an unusually large bulk modulus of 181 GPa for the KO8 polyhedra, which is inconsistent with other studies. Using the bulk modulus of the polyhedron and bond distance relationship 1/K = 37 × (d3/z) × 10− 5 Gpa− 1, with z the formal charge of the cation (z ∼ 1 for potassium) and 1/K the com8 pressibility (Hazen and Prewitt, 1977), we estimate KKO 0 = 132 GPa for the powder diffraction data by Hirao et al. (2008) (with the average K–O distance d = 2.733 Å at 128 GPa), more consistent with our results than the single crystal data. (Hazen and Prewitt, 1977) demonstrated that the K–O bond is one of the most compressible

bonds among the metal–oxygen pairs in oxide minerals. As shown above, the (Si,Al)O6 octahedra are the rigid units in hollandite and the size and shape of the KO8 polyhedra are determined by the hinge angle between adjacent tetrahedral units (Fig. 1). Under compression the hinge angle between the neighboring octahedral chains changes to reduce the tunnel size as the symmetry changes from tetragonal to monoclinic (Fig. 4). In addition the angle between [100] and [010] (90° in hollandite-I) changes under compression as the structure transforms to hollandite-II (Fig. 5).

3.2. Elasticity Elastic constants of the low pressure tetragonal phase are found to increase monotonically and slightly sub-linearly with P, up to 20 GPa tet (Fig. 6). Upon further compression, ctet 11 and c66 soften. This behavior suggests that hollandite-I is elastically unstable at higher pressure.

Fig. 6. Elastic constants as a function of pressure. (a)–(c) show longitudinal, off-diagonal, and shear elastic constants for tetragonal hollandite-I with symmetry I4/m, respectively. mon (d)–(f) show the same groups for monoclinic hollandite-II with symmetry C2/m. Black lines are finite strain fits to the results. Note that the ctet 33 (along [001]tet , equivalent to c22 ) is mon consistently stiffer than ctet 11 (and c11 ). For relation between coordinate systems refer to Fig. 1.

M. Mookherjee, G. Steinle-Neumann / Earth and Planetary Science Letters 288 (2009) 349–358 Table 2 Elastic constants (cij), bulk (K) and shear (G) moduli of monoclinic and tetragonal hollandite at zero pressure (subscript 0) and pressure dependence (superscript ′). Monoclinic

c11 c22 c33 c44 c55 c66 c12 c13 c23 c15 c25 c35 c46 KVoigt KHill KReuss GVoigt GHill GReuss

Tetragonal

M0

M′0

M0

M′0

395 575 382 182 48 177 132 140 107 − 16 4 38 13 226 223 220 140 137 134

5.9 6.3 5.6 1.2 4.0 1.5 2.8 3.3 2.7 0.1 0.4 0.2 0.3 4.1 4.1 4.2 2.3 2.2 2.0

342

5.2

568 165

5.7 1.1

129 186 118

0.5 2.8 2.0

234 232 230 145 134 124

4.1 4.1 4.2 1.5 1.0 0.6

The calculated elastic constants are in agreement with the linear tet moduli, since ctet 33 is stiffer than c11 . mon mon In hollandite-II c22 is stiffer than cmon 11 and c33 , since the [010]mon direction in hollandite-II with monoclinic symmetry (C2/m) is equivalent to the [001]tet direction in the tetragonal hollandite-I (I4/m). In general, the relation between the tetragonal and monoclinic elastic constants is as follows: for longitudinal elastic constants ctet 33 cormon responds to cmon (ctet 22 33 ∼ c22 ), for the off-diagonal elastic constants

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mon tet mon tet mon tet ctet 12 ∼ c13 c16 ∼ c15 and c13 ∼ c12 . For the shear elastic constants c66 tet mon corresponds to cmon 55 (c66 ∼ c55 ). tet As hollandite-I is compressed, the elastic constants ctet 11 and c66 soften tet above 50 GPa (Fig. 6), and the Born stability criterion ctet −c −P > 0 is 11 12 violated around 50 GPa. This indicates that hollandite-I is mechanically unstable above this pressure. The elastic constants in the hollandite-II phase do not show any sign of softening under compression. The isotropic bulk (K) and shear (G) moduli (Table 2) are determined using the Voigt–Reuss–Hill averaging scheme. We also calculate the variation of P-and S-wave velocities with pressure (Fig. 7). At low pressures, the S-wave velocity initially increases and then decreases. At the transition, it increases by 10% and upon further compression up to 100 GPa it shows sub-linear behavior. By contrast, the P-wave velocity increases only by 2% across the transition. This is related to the fact that the bulk modulus across the transition remains largely unaffected, whereas the shear modulus softens in the stability field of hollandite-I and shows a discontinuous increase across the hollandite-I to hollandite-II transition at around 30 GPa. To asses anisotropy in hollandite we calculate the pressure variations of the single crystal azimuthal anisotropy for P- and the two polarizations of the S-waves, defined as (Mainprice, 1990)

AX ð%Þ =

VX max VX min × 200; VX max + VX min

ð5Þ

where VX refers to VP,VS1 or VS2 respectively, and maximum polarization anisotropy (for S-waves) ASp ð%Þ =

VS1 max  VS2 min × 200: VS1 max + VS2 min

ð8Þ

Fig. 7. (a) P- and S-wave velocities for hollandite as a function of pressure, indicating the discontinuity at 33 GPa. Velocities at pressures lower than 33 GPa are calculated from the tetragonal hollandite-I and velocities at pressure higher than 33 GPa are calculated from monoclinic hollandite-II. The S-wave velocity shows a 10% discontinuity at the transition. The upper dotted line is the Voigt limit and the lower dashed line represent the Reuss limit for the P- and S- wave velocities, with the Voigt–Reuss–Hill average in the solid line. Pressures of the transition zone and at 920 km depth are indicated. (b) Seismic anisotropy calculated for tetragonal hollandite-I and monoclinic hollandite-II, exhibiting a significant S-wave polarization anisotropy in hollandite-I near the transition pressure, and a discontinuous behavior for S-wave azimuthal anisotropy across the transition. The P-wave anisotropy is only slightly affected by the transition.

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At low pressures, ASp is greater than the AS2 azimuthal anisotropy (Fig. 7b). In the vicinity of the transition, ASp increases to around 80% and then drops to 30%. At low pressures, the azimuthal S-wave anisotropy (AS2) is the lowest, it gradually increases with P-wave anisotropy (AP) and across the transition it changes discontinuously to 30%. At high pressures the S-wave (AS1 and AS2) anisotropies are very similar and upon further compression they both reduce to around 20% at 100 GPa. At low pressures P-wave anisotropy (AP) is around 30% and decreases gradually upon compression to 10% at around 100 GPa, almost unaffected by the transition. 4. Geophysical significance In a global geochemical context it is becoming increasingly important to incorporate deeply subducted sediments in the recycled basaltic crust to explain the distinct isotopic signatures expected for OIBs (Chauvel et al., 2008; Jackson et al., 2007; Plank and van Keken, 2008). Similarly, increasing the basalt fraction in the lower mantle can better explain its seismic structure (Xu et al., 2008). Since hollandite is stabilized in potassium rich basalt (Wang and Takahashi, 1999), sediments (Irifune et al., 1994) and deeply subducted continental crust (Wu et al., 2009) it is important to incorporate the thermoelastic parameters of hollandite when interpreting mantle structure in subduction zone environments. As discussed above, K-hollandite undergoes a structural transformation at ∼20 GPa at room temperatures (Ferroir et al., 2006). In the Earth's mantle, however, the transition from hollandite-I to hollandite-II is expected to be affected by different parameters, including temperature and composition (e.g., Na–K substitution). The temperature dependence of the hollandite-I to hollandite-II transition has been reported with a positive Clapeyron slope, (Nishiyama et al., 2005). This would imply that the transition is likely to occur at ∼ 30 GPa for a typical mantle adiabat (Piazzoni et al., 2007). The effect of cation substitution (e.g., Na–K) is likely to have the opposite effect on the transition pressure. However, as outlined above, sodium content in hollandite appears to be limited (Yagi et al., 1994). Independent of the details of the chemistry of hollandite and temperature in the mantle hollandite-II will be stabilized at the base of the transition zone or the upper part of lower mantle in subduction related lithologies. Despite our calculations yielding a higher transition pressure under static conditions, the computed elastic constants can shed light on the interpretation of seismic structure in the presence of K-enriched lithologies. Velocity–density systematics have been successfully applied to decipher the composition of the earth's interior (Birch, 1960; Mcqueen et al., 1964). Fig. 8 shows the correlation between the bulk sound velocity (VΦ) and density (ρ) of various potassium bearing mineral phases and glasses. They tend to obey a linear relationship between velocity and density (Birch's law). It becomes evident that the normal mantle phases also have very similar velocity–density characteristics. Unlike hydrous phases, which are seismically slow compared to mantle phases and occur separately in velocity–density plots (Mookherjee and Stixrude, 2009) K-bearing phases are dense and have velocities similar to the mantle phases, making it difficult to map composition heterogeneity based on velocity–density relation. On the other hand, the hollandite-I to hollandite-II phase transition involves 2% and 10% changes in P- and S-wave velocities, respectively (Fig. 7), with the VS contrast strong enough to cause seismic reflections. Quantitatively, we estimate the normal incidence frequency-dependent reflection coefficient, Rvertical = (ρIIvII − ρIvI) / (ρIIvII + ρIvI), where I and II refer to hollandite-I and hollandite-II, ρ is the density and v is the velocity associated with this transition. The magnitude of Rvertical for the step discontinuity is 10% for S-wave velocity, for a pure hollandite assemblage. Such a transition could account for the occurrence of mid-mantle scatterers as mapped in seismology (Kaneshima and Helffrich, 1999; Kaneshima and Helffrich,

Fig. 8. Velocity–density systematics for several potassium bearing and typical mantle phases. Data on potassium bearing phases (green open circles) are compiled from the literature, Hol: hollandite (KAlSi3 O 8 ), (Ferroir et al., 2006); ph-X: phase-X (K2Mg2Si2O7), (Mookherjee and Steinle-Neumann, 2009); K-pv: potassium neighborite with perovskite structure (KMgF3) (Vaitheeswaran et al., 2007); potassium bearing layered hydrous silicate, Mus: muscovite (Comodi and Zanazzi 1995) and phlog: phlogopite (Hazen and Finger 1978); K-Cym: Cymrite (KAlSi3O8.H2O) (Fasshauer et al., 1997); Or: potassium feldspar, orthoclase (KAlSi3O8) (Mcqueen et al., 1964); Wdg: wadeite glass (K2Si4O9) (Dickinson et al., 1990). The rock types and mineral phases (blue open diamonds) are from Mcqueen et al, (1964) and the thick black line represents average mantle based on compilation by Poirier (2000). Our results on hollandite-I (blue line) and hollandite-II (green line) (including pressure dependence) are shown for comparison. The parentheses next to the abbreviations show the mean atomic mass for the respective phases.

2003). Such mid-mantle scatterers have been traditionally explained by reduction of S-wave velocity due to transition in stishovite (Karki et al., 1997), which could occur in basalt fractions. Pure SiO2 stishovite exhibits the transformation at around 47 GPa; however, substitution of H2O and Al2O3 could reduce the transition pressures (Lakshtanov et al., 2007) and match the seismological observation. Kawakatsu and Niu (1994) estimated the reflectivity of the 920 km feature in a multicorridor stack of ScS reverberation to be 0.5%. For this discontinuity to be caused by hollandite it would require 5 vol% of this phase. If we consider a sediment package consisting of equal proportion of hollandite, majorite and stishovite (Irifune et al., 1994) then the maximum amount of sediment that could cause the 920 km discontinuity would be 15 vol%. The uncertainty in this prediction is large, because the bounds on aggregate elastic properties of the material in the vicinity of the transition based on continuous stress (Reuss) and strain (Voigt) show significant differences (Fig. 7a). As discussed earlier, hollandite exhibits strong seismic anisotropy and shows remarkable changes in anisotropy across the displacive transition (Fig. 7). Slabs penetrating the lower mantle can develop significant anisotropy through deviatoric stresses (Nippress et al., 2004) as the slab encounters higher viscosity in the lower mantle (Steinberger, 2000). While globally no significant anisotropy in the uppermost lower mantle has been observed, there is ongoing discussion of anisotropy (or the absence thereof) in the Tonga– Kermadec and New Hebrides subduction zones (Wookey et al., 2002;

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Wookey and Kendall, 2004; Heintz, 2006). If such anisotropy exists the highly anisotropic hollandite might be an important candidate to explain such properties. Further work on its plastic deformation would be crucial to test this hypothesis. If indeed hollandite is subducted down to the lower mantle, it is likely to be a phase which would host most of the available potassium, and hence the mass balance and partitioning of potassium among major lower mantle phases such as perovskite, post -perovskite and core need to be revisited (Lee and Jeanloz, 2003; Lee et al., 2004, 2009). 5. Conclusions Using first-principle calculations we studied the high pressure behavior of KAlSi3O8 hollandite. Hollandite consists of silicon and aluminium ions in octahedral coordination that are arranged to form tunnels where alkali cation such as potassium and other large ion lithophile or incompatible elements are accommodated. At high pressure the octachedra behave as rigid units and compression occurs by rotation of the Si(Al)–O–Si(Al) angles. Hollandite is a very dense mineral, and based on velocity–density systematics, it is very difficult to map out the presence of crustal components from the rest of the mantle. Based on the enthalpy differences, at 30 GPa hollandite undergoes a transformation from a tetragonal (hollandite-I) to a monoclinic phase (hollandite-II). Associated with this transition is a significant increase in the shear wave velocity. Consequently, this transition might be related to mid-mantle elastic scatterers that have been observed seismologically in subduction zone settings. Acknowledgements Computations were performed at the Leibniz Computer Center Munich. Constructive criticism by three anonymous reviewers is greatly appreciated. We greatly appreciate helpful discussion with Prof. Lars Stixrude, Prof. Falko Langenhorst, Prof. David Mainprice, Dr. Zurab Chemia and Dr. Tiziana Boffa-Ballaran. This work was supported by the European Commission through the Marie-Curie Research Training Network ‘c2c’ (contract no. MRTN-CT-2006-035957). References Akaogi, M., Kamii, N., Kishi, A., Kojitani, H., 2004. Calorimetric study on high-pressure transitions in KAlSi3O8. Phys. Chem. Mineral. 31 (2), 85–91. Andrault, D., Fiquet, G., Guyot, F., Hanfland, M., 1998. Pressure-induced Landau-type transition in stishovite. Science 282 (5389), 720–724. Armstrong, R.l., 1968. A model for evolution of strontium and lead isotopes in a dynamic Earth. Rev. Geophys. 6 (2), 179–199. Birch, F., 1960. The velocity of compressional waves in rocks to 10-kilobars.1. J. Geophys. Res. 65 (4), 1083–1102. Birch, F., 1978. Finite strain isotherm and velocities for single-crystal and polycrystalline NaCl at high-pressures and 300-degree-K. J. Geophys. Res. 83 (Nb3), 1257–1268. Bukowinski, M.S.T., 1976. Effect of pressure on physics and chemistry of potassium. Geophys. Res. Lett. 3 (8), 491–494. Bystrom, A., Bystrom, A.M., 1949. Crystal structure of hollandite. Nature 164 (4183), 1128. Cadee, M.C., Verschoor, G.C., 1978. Barium tin chromium-oxide, a new hollandite phase. Acta Crystallogr. B 34, 3554–3558 (Dec). Chauvel, C., Lewin, E., Carpentier, M., Arndt, N.T., Marini, J.C., 2008. Role of recycled oceanic basalt and sediment in generating the Hf–Nd mantle array. Nat. Geosci. 1 (1), 64–67. Cheary, R.W., 1986. An analysis of the structural characteristics of hollandite compounds. Acta Crystallogr. B 42, 229–236. Comodi, P., Zanazzi, P.F., 1995. High-pressure structural study of muscovite. Phys. Chem. Mineral. 22 (3), 170–177. Dasgupta, R., Hirschmann, M.M., 2006. Melting in the Earth's deep upper mantle caused by carbon dioxide. Nature 440 (7084), 659–662. Davies, G.F., 1973a. Invariant finite strain measures in elasticity and lattice-dynamics. J. Phys. Chem. Solids 34 (5), 841–845. Davies, G.F., 1973b. Quasi-harmonic finite strain equations of state of solids. J. Phys. Chem. Solids 34 (8), 1417–1429. Davies, G.F., 1974. Effective elastic-moduli under hydrostatic stress. 1. Quasi-harmonic theory. J. Phys. Chem. Solids 35 (11), 1513–1520.

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