Slip systems and plastic shear anisotropy in Mg SiO ... · Slip systems and plastic shear...

Slip systems and plastic shear anisotropy in Mg2SiO4 ringwooditeinsights from numerical modelling

PHILIPPE CARREZ1 PATRICK CORDIER1 DAVID MAINPRICE2 and ANDREA TOMMASI2

1Laboratoire de Structure et Proprietes de lrsquoEtat Solide ndash UMR CNRS 8008 Universite des Sciences et Technologiesde Lille Cite Scientifique Bat C6 F-59655 Villeneuve drsquoAscq France

Corresponding author e-mail philippecarrezuniv-lille1fr2Laboratoire de Tectonophysique UMR CNRS 5568 Universite de Montpellier II F-34095 Montpellier cedex 5 France

Abstract Knowledge on the deformation mechanisms of Mg2SiO4 ringwoodite is important for the understanding of flow andseismic anisotropy in the Earthrsquos mantle transition zone We report here the first numerical modelling of dislocation structures inringwoodite The dislocation properties are calculated through the Peierls-Nabarro model using the generalized stacking fault (GSF)results as a starting model The GSF are determined from first-principle calculations using the code VASP They enable us todetermine the relative ease of slip for dislocation glide systems in ringwoodite The dislocation properties such as core spreading andPeierls stresses were determined for the easy dislocation glide systems Our results show that frac12 lsaquo 110 8 110 and frac12 lsaquo 110 8 111 arethe easiest slip systems in ringwoodite at 20 GPa and 0 K These results are used as input of a viscoplastic model to predict thedeformation of a ringwoodite rich aggregate Calculated crystal preferred orientation (CPO) accounts satisfactorily for experimentaldata available from either diamond anvil cell or D-DIA experiments

Key-words ringwoodite deformation mechanisms dislocations slip systems first-principle calculations Peierls-Nabarro modelseismic anisotropy Earth mantle transition zone

1 Introduction

(MgFe)2SiO4 ringwoodite is a high-pressure polymorph ofolivine with a spinel structure It is widely considered to bethe most abundant phase in the lower half of the mantle tran-sition zone in the depth range 550ndash670 km (Irifune amp Ring-wood 1987) Understanding crystal defects and plasticity ofringwoodite is thus critical for modelling the dynamics ofthe interior of the Earth Information on defects in ringwoo-dite have first been obtained from the microstructural studyof shocked chondrite meteorites Madon amp Poirier (1983)have reported evidence for frac12 lsaquo 110 8 111 dislocation slipfrom observations of the Tenham meteorite frac14 lsaquo 110 8 110stacking faults have also been reported in numerous cases(Putnis amp Price 1979 Vaughan amp Kohlstedt 1981 Madonamp Poirier 1983 Rubie amp Brearley 1990 1994) but they areusually interpreted as resulting from phase transformationsand do not seem to be related to plasticity

Conducting plastic deformation experiments in P T con-ditions of the deep Earth is still one of the most importantchallenges for mineral physics The newly developed defor-mation experiments dedicated to high-pressure (Deforma-tion-DIA Wang et al 2003 and Rotational Drickamer Ap-paratus Yamasaki amp Karato 2001) cannot reach the P Tconditions prevailing in the lower part of the transition zoneAbove 15 GPa deformation experiments must be undertak-

en with a multianvil apparatus (eg Cordier amp Rubie 2001Cordier et al 2004) or with a diamond anvil cell (eg Wenket al 2004) The first deformation experiment on(MgFe)2SiO4 ringwoodite has been performed by Karato etal (1998) at 16 GPa and 1600 K using a shear deformationassembly in a multianvil apparatus Two slip systems havebeen identified by transmission electron microscopyfrac12 lsaquo 110 8 111 and frac12 lsaquo 110 8 100 In contrast a more re-cent study of Mg2SiO4 ringwoodite samples deformed at22 GPa between 1400 and 1500degC (Thurel 2001) has re-vealed evidence for slip on frac12 lsaquo 110 8 111 and onfrac12 lsaquo 110 8 110 Flow laws for ringwoodite at subductionzone conditions (20 GPa T lsaquo 1350degC) have been obtainedrecently by Xu et al (2003) using a T-cup multianvil appa-ratus with in situ measurements based on synchrotron-gen-erated X-rays but no microstructural investigations wereconducted on these samples Room-temperature deforma-tion of ringwoodite has been achieved using diamond anvilcells (Kavner amp Duffy 2001 Wenk et al 2004) Ring-woodite transformed in situ from San Carlos olivine isshown to develop weak preferred orientations with 110lattice planes perpendicular to the compression direction(Wenk et al 2004) Recently room-temperature deforma-tion experiments have been performed on Mg2SiO4 ring-woodite using the D-DIA (Nishiyama et al 2005 Wenk etal 2005) where stress-strain curves could be measured in

Eur J Mineral2006 18 149ndash160

DOI 1011270935-122120060018-01490935-1221060018-0149 $ 540

ˇ 2006 E Schweizerbartrsquosche Verlagsbuchhandlung D-70176 Stuttgart

situ at high pressures These studies based on in situ crystalpreferred orientation measurements emphasize that slip onfrac12 lsaquo 110 8 111 dominates deformation under those condi-tions

Computational materials science is an alternative ap-proach to study plastic deformation The Peierls-Nabarro(PN) model provides a conceptual framework for address-ing the issue of dislocation structures (Hirth amp Lothe 1982)The model is essentially based on continuum mechanics (farfrom the defect) but inelastic displacements in the disloca-tion core are given a specific description A recent approachfor describing forces in the dislocation core is based on theconcept of generalized stacking fault (GSF) surfaces (Vitek1968 Christian amp Vitek 1970) The value of the GSF is ob-tained by shearing half of the infinite crystal over the otherhalf The fault is called bdquogeneralizedrdquo because it is unstablefor most values of shear and must be balanced by a restoringforce (which is introduced in the PN model) The GSF cannowadays be calculated accurately from first-principlesThe GSF gives access to the ideal shear strength which is theupper bound on the mechanical strength of a solid This ap-proach has been applied to many metallic systems Al (Pax-ton et al 1991 Hartford et al 1998 Roundy et al 1999Ogata et al 2002) Cu (Roundy et al 1999 Ogata et al2002) Pd (Hartford et al 1998) Fe (Clatterbuck et al2002) Mo (Xu amp Moriarty 1996 Luo et al 2002) Nb (Luoet al 2002) W (Krenn et al 2001 Roundy et al 2001) Ta(Soumlderlind amp Moriarty 1998) and Zr (Domain et al 2004)Recently we have calculated GSF for forsterite (Durinck etal 2005) and shown that despite a complex crystal chemis-try this approach could account for plastic anisotropy in thismineral It is thus possible to build more realistic models ofdislocations using a PN model in which the inelastic dis-placements in the dislocation core are described using theGSF approach Several examples are available in the recentliterature Al (Sun amp Kaxiras 1997 Hartford et al 1998)Pd (Hartford et al 1998) NiAl and FeAl (Medvedeva et al1996) TiAl and CuAu (Mryasov et al 1998) Si (Kaxiras ampDuesbery 1993 Joos et al 1994 de Koning et al 1998)and MgO (Miranda amp Scandolo 2005)

In the present study we use the GSF framework to assessplastic strain anisotropy of ringwoodite at the atomic scalefrom first-principle calculations Dislocation properties arederived from GSF through the Peierls-Nabarro (PN) modelThis approach is combined with elasticity to infer the possi-ble slip systems in ringwoodite Self-consistent viscoplasticmodels are finally used to evaluate the crystal-preferred ori-entation that will result from the activation of these slip sys-tems and compare our results with ringwoodite crystal pre-ferred orientations produced in recent low-temperature de-formation experiments

2 Crystallography of slip in the spinel structure

Ringwoodite exhibits a spinel structure based on a face-cen-tred-cubic packing of the oxygen sublattice Silicon atomsare located in the tetrahedral sites whereas octahedral sitesare occupied by magnesium and iron atoms The spacegroup of Mg2SiO4 is Fd3m and the lattice parameter is

Fig 1 Ringwoodite structure viewed along lsaquo 110 8 with potentialslip planes The unit cells are shown in dashed lines

8071 A (Ringwood amp Major1970) The slip direction inspinel is always observed to be parallel to the shortest latticevector of the fcc oxygen lattice frac12 lsaquo 110 8 The observed slipplane is variable The most commonly observed slip planesare 111 and 110 although 100 have been reported inmagnetite nickel ferrite and chromite (see a detailed reviewin Mitchell 1999) Figure 1 shows the structure of ringwoo-dite viewed along lsaquo 110 8 The planes usually reported asslip planes in the spinel structure are indeed potential slipplanes for ringwoodite as they do not require breaking thestrong Si-O bonds In oxides with spinel structures disloca-tions are generally observed to be dissociated into collinearpartials following the reaction frac12 lsaquo 110 8 rarr frac14 lsaquo 110 8 +frac14 lsaquo 110 8 The dissociation width increase with increasingdeviation from stoichiometry implying that the stackingfault energy decrease (between 180 and 20 mJm2 Mitchell1999) A variety of dissociation planes have been reported111 110 100 and 311

3 Computational procedures

3a Ab initio calculations

Calculations were performed using the ab initio total-ener-gy calculation package VASP (Vienna Ab Initio Package)developed by Kresse and Hafner (Kresse amp Hafner 19931994 Kresse amp Furthmuumlller 1996a) This code is based onthe first-principles density functional theory and solves theeffective one-electron Hamiltonian involving a functionalof the electron density to describe the exchange-correlationpotential It gives access to the total energy of a periodic sys-tem with as a single input the atomic numbers of atomsComputational efficiency is achieved using a plane wavebasis set for the expansion of the single electron wave func-tions and fast numerical algorithms to perform self-consis-tent calculations (Kresse amp Furthmuumlller 1996b) Within thisscheme we used the Generalised Gradient Approximation(GGA) derived by Perdrew amp Wang (1992) and ultrasoft

150 P Carrez P Cordier D Mainprice A Tommasi

pseudopotentials (eg Vanderbilt 1990 or Kresse amp Hafner1994) Using this assumption the outmost core radius forthe Mg Si and O atoms are 1058 0953 and 0820 A re-spectively Computation convergence better than 410ndash5 eVatom is achieved in all simulations by using a single energycut-off value of 600 eV for the plane wave expansion Thefirst Brillouin zone is sampled using a Monkhorst-Pack grid(Monkhorst amp Pack 1976) adapted for each supercell ge-ometry in order to achieve the full energy convergence Asan example ringwoodite unit cell calculations were per-formed using a 4x4x4 grid with a convergence energy lessthan 025 meV for all external pressure conditions

The ringwoodite crystallographic structure was optimi-sed (full relaxation of the cell parameters and of the atomicpositions within the cell) for pressure conditions rangingfrom 0 to 32 GPa To determine the athermal elastic con-stants of the cubic cell we strained the equilibrium cell us-ing adapted deformations which were kept lower than 2To account for the pressure effect the calculations of theelastic constants were performed along the lines describedby Barron amp Klein (1965) As an illustration c11 can be ob-tained by applying a strain e11 = e to the pressurized cell Theassociated elastic energy variation is then

2 EV

= ndash pe + 12c11 e2 (1)

where p is the confining pressureFor c44 determination we can apply a strain e12 = e and

deduce c44 from

2 EV

= 2c44 e2 (2)

c12 can be determined by applying e11 = e22 = e which leadsto

2 EV

= ndash 2pe + (c11 + c12 ndash p)e2 (3)

The elastic constants are then determined from the fit of thetotal energy versus strain by a second order polynomial

Calculating a GSF for a given slip system requires a su-percell with an adapted geometry The supercell is built on aCartesian reference frame defined by the normal to thestacking fault plane (located in the middle of the supercell)and by the shear direction All our supercells contain a Bur-gers vector of the spinel structure frac12 lsaquo 110 8 which is theshortest lattice repeat along that direction The last directionis then defined as the cross product of the two previous De-spite the fact that ringwoodite has a cubic structure we de-cided to build the supercell with only one stacking fault andit was necessary to add a vacuum buffer parallel to the stack-ing plane In this way the number of atoms in the supercellwas reduced compared to the symmetric construction withtwo stacking fault It is shown that a thickness of 6 A of thebuffer vacuum and five atomic layers on both sides of thestacking fault are sufficient to guarantee energy accuracybetter than 001 with a 2x2x2 Monkhorst-Pack grid TheGSF are then calculated by imposing a given shear displace-ment value to the upper part of the supercell

For calculations of relaxed GSF we tried to allow themaximum number of degrees of freedoms for the atoms

without performing full relaxations calculations The super-cell vectors are kept fixed at the values obtained for a bulksystem submitted to the pressure of interest Atoms presenton the two surfaces are maintained fixed to mimic the actionof the surrounding bulk atoms in the direction normal to theshear plane Using these conditions we preserve pressureand shear mostly by imposing conditions on silicon atomswhich are only allowed to accommodate shear perpendicu-larly to the slip plane Finally the remaining atoms are fullyrelaxed in the three directions to account for structural de-grees of freedom associated to the crystal chemistry

3b Polycrystalline plasticity models

Development of crystal preferred orientations (CPO) inringwoodite polycrystals deformed in simple shear underhigh-pressure conditions is simulated using a viscoplasticself-consistent (VPSC) model (Molinari et al 1987 Le-bensohn amp Tome 1993) In this model as in all polycrystalplasticity approaches CPO evolution is essentially con-trolled by the imposed deformation the initial texture andthe active slip systems The latter depend on the mineralstructure but also on the temperature and pressure condi-tions which control their relative strength or critical re-solved shear stress (CRSS) Extensive testing on metallicalloys (Lebensohn amp Tome 1993 Loge et al 2000) halite(Lebensohn et al 2003) and garnet (Mainprice et al2004) as well as on highly anisotropic minerals such as cal-cite (Tome et al 1991) olivine (Wenk et al 1991 Tom-masi et al 2000) clinopyroxene (Bascou et al 2002) andwadsleyite (Tommasi et al 2004) show that this modelproduces robust CPO predictions Detailed informationabout the calculation procedure can be found in Tome ampCanova (2000)

In the present study we investigate the evolution of ring-woodite CPO for two end-member deformation regimessimple shear and axial shortening Actual flow in the transi-tion zone is most likely three-dimensional but regions sub-mitted to large deformations probably display a strong shearcomponent whose orientation (horizontal or steeply dip-ping) will depend on the large-scale convection patternMoreover CPO obtained in axial shortening simulationscan be compared to ringwoodite CPO measured duringcompression experiments at high pressure (Wenk et al2004 Wenk et al 2005) The only tuning parameters are theactive slip systems for ringwoodite their CRSS and stressexponent The CRSS for the various slip systems in ring-woodite are inferred from the Peierls stress analysis (see be-low) Hardening is not considered in the present models be-cause there are no constraints on the hardening behavior ofringwoodite Moreover the high temperatures prevailing attransition zone conditions should allow for easy recoveryThe actual stress exponent for ringwoodite is unknownhowever oxide spinels are usually characterized by n = 4(whatever the stoichiometry see Mitchell 1999) Simula-tions were thus run for n = 3 to 5 Indeed VPSC simulationsare not very sensitive to variations of n values within thisrange increasing n enhances the plastic anisotropy andhence slightly accelerates the CPO evolution

Slip systems in Mg2SiO4 ringwoodite 151

4 Elasticity4a Elastic constants

We have calculated the anisotropic athermal elastic con-stants cij up to 32 GPa The results are presented in Fig 2 Ingeneral the elastic constants increase linearly with increas-ing pressure The slope is more pronounced for c11 com-pared to c12 and c44 The anisotropy factor for cubic symme-try crystals A = 2c44(c11 ndash c12) decreases from 124 to 111when pressure increases from 0 to 20 GPa The elastic an-isotropy of ringwoodite is thus small with nearly no anisot-ropy effect at typical pressure conditions of the transitionzone (around 20 GPa) The Cauchy relation c12 ndash c44 = 2P isvalid only when all interatomic forces are central As pres-sure increases the calculated values of c12 ndash c44 ndash 2P de-crease to reach ndash13 GPa at 20 GPa and ndash3 GPa at 30 GPaThis indicates that bonding in ringwoodite exhibits a moreionic character as pressure increases

Figure 2 shows that our calculated elastic constants com-pare well with experimental data from Sinogeikin et al(2001) and with Local Density Approximation (LDA) cal-culations from Kiefer et al (1997) Our results are systemat-ically below Sinogeikin et alrsquos as expected from GGA cal-culations which are known to generally overestimate theunit cell volume and induce under binding The unit cell pa-rameter determined in this study is 813 A at 0 GPa which isindeed higher than the experimental value of 807 A mea-sured at 300 K by Ringwood amp Major (1970) or Meng et al(1994) Conversely elastic constants calculated using LDAapproach by Kiefer et al (1997) are higher than the experi-mental results

Fig 2 Elastic constants evolution in function of pressure (Dia-monds = C11 triangles = C12 and circles = C44) Results of our GGAcalculations (filled symbols) are compared to experimental data(open symbols) from Sinogeikin et al (2001) and to LDA results(grey symbols) from Kieffer et al (1997)

4b Dislocation line energy

The elastic energy Eel per unit length of a straight dislocationis given by

Eel = K( rsquo )b2

4 lsquoln ( R

r0) (4)

where K( rsquo ) the energy coefficient is function of the disloca-tion character which is defined by the angle ( rsquo ) between theline direction and the Burgers vector b is the Burgers vectorand r0 and R are the integration boundary limits (Hirth amp Lo-the 1982) For an isotropic crystal K( rsquo ) is

K( rsquo ) = micro (sin 2 rsquo1 ndash r

+ cos 2 rsquo ) (5)

where micro is the shear modulus and r the Poisson ratio Themain effect of elastic anisotropy on the dislocation line energyis found in the energy coefficient K which can be calculatedwithin the frame of the Stroh theory (Hirth amp Lothe 1982)The DisDi software (Douin 1987 Douin et al 1986) is basedon this theory It has been used here to calculate the energy co-efficients K( rsquo ) and the dislocation line energies The disloca-tion line energies predicted using elastic constants calculatedat 20 GPa are presented on Fig 3 for several slip systemsthose corresponding to frac12 lsaquo 110 8 slip in the three planes100 110 and 111 as well as two slip systems corre-sponding to lsaquo 100 8 slip in 011 and 001 planes Using ex-perimental values for the elastic constants or changing thepressure has no significant influence on the result Slip alonglsaquo 100 8 appears clearly less favourable from the energetic

point of view with a line energy twice as large as those corre-sponding to frac12 lsaquo 110 8 slip Figure 3 confirms that the effectsof elastic anisotropy are very smallndash the ratio between the line energies of edge and screw dis-

locations varies from 137 and 140 for frac12 lsaquo 110 8 depend-ing on the glide plane which is very close from the valuederived from the isotropic case 1(1 ndash r ) raquo 14

ndash the line energy of a frac12 lsaquo 110 8 dislocation is almost inde-pendent of the choice of slip plane

Fig 3 Dislocation line energies for several potential slip systems(elastic constants from our calculations at 20 GPa)

5 Plastic shear5a Generalized Stacking Faults (GSF)

The GSF excess energies have been calculated for severalshear configurations Three of them correspond to the slip


a) b) c)Fig 4 Unsheared supercells of various GSF calculations The shear planes are presented in light grey SiO4 tetrahedra are displayed in darkgrey white atoms corresponding to Mg Atoms in dark at surfaces are either O or Mg fixed The supercells are used to calculate the GSF asso-ciated with the following slip systemsa) frac12 lsaquo 110 8 001 b) frac12 lsaquo 110 8 011 lsaquo 100 8 011 c) frac12 lsaquo 110 8 111

systems commonly reported in the spinel structurefrac12 lsaquo 110 8 001 frac12 lsaquo 110 8 110 and frac12 lsaquo 110 8 111 as dis-cussed in section 2 Possible shear along lsaquo 100 8 on 001and 001 has also been considered The supercell con-structed for the GSF calculations are presented on Fig 4Following the results of Durinck et al (2005) the fault level(in the middle of the supercell) is chosen so as not to cut thestrong SiO4 structural units In case of shear on 001 twocutting levels could be considered one in the Mg layer andone between the SiO4 tetrahedra We have chosen to shearthe structure along the Mg layer which corresponds to thelargest interplanar distance

The unrelaxed GSF are presented on Fig 5 The energy isdetermined for several shear values S between 0 and b with-out any relaxation of the atomic position It is shown thatlsaquo 100 8 slip is very unfavourable due to strong impingement

of magnesium and oxygen atoms (Fig 5) The three barriersassociated with frac12 lsaquo 110 8 slip are very similar with a camel-hump shape The only significant difference lies in the 50 shear excess energy which is higher for slip on 001

More realistic values can be obtained by allowing atomicrelaxations at each shear step We tried to leave the atoms as

Fig 5 Unrelaxed generalized stacking fault excess energies as afunction of displacement shear S (expressed as a fraction of the mod-ulus of the Burgers vector frac12 lsaquo 110 8 ) Pressure 0 GPa

free as possible in their displacements Shear has to be im-posed however This is achieved by using the boundary con-ditions and the atomic relaxation scheme defined in sec-tion 3 The excess energy barriers at 0 GPa after relaxationare presented on Fig 6 As expected atomic relaxations re-sult in a significant decrease of the GSF energies Howeverthe barriers keep their shapes This is important as a camel-hump barrier suggest the possibility of a core extensionwithin the plane with potential implications for dislocationmobility as discussed later It is interesting to note that thethree barriers shown on Fig 6 are clearly distinct after relax-ation This shows that the relaxation mechanisms are verysensitive to the actual atomic environments at the shearplane The highest energy barrier is associated withfrac12 lsaquo 110 8 001 whereas frac12 lsaquo 110 8 110 corresponds to thelowest (see Table 1) It is difficult however to infer the rela-tive resistance to plastic shear simply by looking at the ener-gy barriers The simplest approach consists in calculatingthe shear resistance from the derivative of the GSF (Vıtek1974 Medvedeva et al 1996 Sun amp Kaxiras 1997 Hart-ford et al 1998) which gives access to the restoring force

Frarr

(S) = ndashrarrgrad (S) (6)

Fig 6 Influence of atomic relaxations on the GSF at 0 GPa


Table 1 Parameters issued from the GSF calculations max is themaximum value of the excess energy barrier S max is the ideal shearstress In the last column the ISS are normalised by the anisotropicshear moduli calculated from anisotropic elasticity All values arepresented at 0 and 20 GPa

Pressure(GPa)

Slipplane

b (A) micro lsaquo uvw 8 hkl

(GPa) max

(Jm2)S max

(GPa)S max

micro lsaquo uvw 8 hkl

0001

575131 24 24 019

110 105 12 17 016111 112 14 15 013

20001

557142 33 44 031

110 128 21 26 020111 132 23 25 019

The ideal shear strength (ISS) which is defined as the bdquomaxi-mum resolved shear stress that an ideal perfect crystal cansuffer without plastically deformingrdquo (Paxton et al 1991) isidentified to the maximum stress S max along the shear dis-placement The values of S max are presented in Table 1 Theresistance to plastic shear is very comparable on 110 andon 111 (the maximum of the slope is found at the begin-ning of the curve which governs the value of the ISS) Nor-malizing S max by the shear modulus enables comparisonwith other structures Most metals yield values in the range009ndash017 micro (Paxton et al 1991 Soumlderlind et al 1998Roundy et al 1999 Krenn et al 2001) which comparevery well with our values for the easiest slip planes 111and 110

The influence of pressure on plastic shear of ringwooditeis presented on Fig 7 which shows the relaxed GSF at20 GPa a pressure representative of the lower part of thetransition zone where ringwoodite is stable For this pur-pose a new reference state is obtained under pressurewhich is then sheared Atoms present at the interface withthe vacuum layer (shown in black on Fig 4) must obviouslybe fixed As expected pressure makes ringwoodite strongerThe influence of pressure on plastic deformation can havetwo origins however One is due to the evolution with pres-sure of the elastic constants The increase with pressure ofthe normalized ISS presented in Table 1 demonstrates thatthe elastic contribution is not the only one The atomic con-figuration in the shear plane also plays an important roleThe energy barriers of Fig 7 and the ISS presented in Table1 suggest that homogeneous plastic shear at 20 GPa is easierin 110 and 111 The two planes exhibit very compara-ble behaviours

5b From GSF to dislocations using the Peierls-Nabarro model

The Peierls-Nabarro model assumes that the misfit region ofinelastic displacement is restricted to the glide plane where-as linear elasticity applies far from it The dislocation corre-sponds to a continuous distribution of shear S(x) along theglide plane (x is the coordinate normal to the dislocation linein the glide plane) S(x) represents the disregistry across theglide plane The stress generated by the shear displacement

Fig 7 Relaxed GSF at 20 GPa

S(x) can be represented by a continuous distribution of in-finitesimal dislocations with density ρ(x)

ρ(x) = dS(x)dx

(7)

The Burgers vector b is the sum of those of all these infini-tesimal dislocations leading to the following normalisationcondition

⌠⌡ndash rsaquo

+ rsaquoρ(x)dx = ⌠

⌡ndash rsaquo

+ rsaquo dS(x)dx

dx = b (8)

The restoring force F acting between atoms on either sidesof the interface is balanced by the resultant stress of the dis-tribution of infinitesimal dislocations This leads to the so-called PN equation

K2 lsquo

⌠⌡ndash rsaquo

+ rsaquo 1x ndash xrsquo[dS(xrsquo)

dxrsquo ]dxrsquo = K2 lsquo

⌠⌡ndash rsaquo

+ rsaquo ρ(xrsquo)x ndash xrsquo

dxrsquo = F (S(x)) (9)

where K is the energy coefficient introduced above whichdepends on the dislocation character rsquo (equation 5) In theoriginal model the PN equation was solved by introducinga sinusoidal restoring force yielding a well known analyticalsolution However Joos et al (1994) pointed out that in areal crystal the restoring force could be quite different fromsinusoidal Christian amp Vitek (1970) have suggested that theGSF could be used to calculate the restoring force Indeedthe GSF is usually not stable and must be balanced by a re-

storing force F(S) = ndash part partS

(see equation (6)) which can be

used to solve the PN equationIn this paper we follow the methodology proposed by

Joos et al (1994) and already applied with success in sever-al cases (eg Hartford et al 1998) The disregistry distribu-tion in the dislocation core S(x) was obtained by searchingfor a solution in the form

S(x) = b2

+ blsquo 7

N

i = 1[ i middot arctan x ndash xi

ci(10)

where [ i xi and ci are variational constants Using the previ-ous form of S(x) the infinitesimal dislocation density ρ(x) is

ρ(x) = dS(x)dx

= blsquo 7

N

i = 1[ i

ci(x ndash xi)

2 + ci2 (11)


a) b) c)Fig 8 Example of a determination of the dislocation density ρ and disregistry S for a frac12 lsaquo 110 8 111 screw dislocations calculated at 0 GPa(a) GSF and derived restoring force F (dashed line) plotted as a function of Sb (b) Restoring force F (from ab initio calculations) and fittedsolution FPN as a function of disregistry S (equation 10) The restoring force F from ab initio calculations are represented by the open circlesand the curve represents the solution of the fit FPN (equation 12) (c) Disregistry S (dashed line) and dislocation density ρ plotted against thedistance of the dislocation core

As the disregistry S(x) and the density ρ(x) must be solutionof the normalisation condition the [ i are constrained by

7N

i = 1[ i = 1 Then the previous disregistry function is used to

solve the PN equation Substituting the disregistry into theleft-hand side of the PN equation gives the restoring force

FPN (x) = Kb2 lsquo 7

N

i = 1[ i middot

x ndash xi(x ndash xi)

2 + ci2 (12)

The variational constants [ i xi and ci are obtained from aleast square minimisation of the difference between FPN andthe restoring force F from our ab initio calculations Practi-

Fig 9 frac12 lsaquo 110 8 100 slip system Infinitesimal dislocation densitydistributions ρ corresponding to screw and edge dislocations at 0(dashed line) and 20 GPa

cally the restoring force is obtained by derivating the GSFcalculated ab initio It is then fitted by a sine series N = 3(equation 10) is sufficient to describe the disregistry S(x) ofa simple (ie with one maximum only) energy barrier where-as barriers with a camel hump requires more terms we use N= 6 An example of calculation for a screw dislocation of thesystem frac12 lsaquo 110 8 111 is shown on Fig 8 Figure 8b showsthe adjustment of FPN on the restoring force F derived fromthe GSF (Fig 8a) The disregistry S(x) and the correspond-ing dislocation density ρ(x) are presented on Fig 8c

Following this method the Peierls model has been ap-plied to calculate the edge and screw dislocations belongingto the three slip systems frac12 lsaquo 110 8 001 frac12 lsaquo 110 8 110

Fig 10 Same as Fig 9 for frac12 lsaquo 110 8 110 slip system



and frac12 lsaquo 110 8 111 The results are presented in Fig 9 10and 11 with the numerical values being given in Tables 2and 3 The dislocation cores are always found to be signifi-cantly spread in the glide plane The dislocation densityfunctions exhibit two maxima Camel-hump GSF corre-spond therefore to a tendency of the dislocation core to dis-sociate The dislocation core width ˇ is defined as the dis-tance between the two peaks of the density function At0 GPa core spreading is maximum in the 110 planes andminimum in the 100 planes Pressure is found to have asignificant effect on the dislocation core geometries Appli-cation of a 20 GPa pressure results in a more compact dislo-cation core Dislocations in 110 and 111 exhibit verycomparable dislocation cores at 20 GPa

Finally the PN model can be used to calculate the Peierlsenergy and the stress (called the Peierls stress) required toovercome this energy barrier Lattice discretisation must bere-introduced at this point as previously the crystal was con-sidered as an elastic continuum medium Let us considerthat in the perfect crystal the distance between planes in thex direction is arsquo The misfit energy can be considered as thesum of misfit energies between pairs of atomic planes andcan be written as

W(u) = 7+ rsaquo

m=ndash rsaquo (S(marsquo ndash u)) middot arsquo (13)

The Peierls stress is given by

c P = max 1b

dW(u)du (14)

Table 2 Parameters for the PN model for frac12 lsaquo 110 8 screw disloca-tions The energy coefficient K (equation 5) and arsquo (periodicity of thePeierls energy W) are input of the PN model The width of disloca-tion distribution ˇ is obtained as the distance separating the two max-imum of the dislocation density function ρ 2 W and c p are respec-tively the height of the Peierls energy barrier and the Peierls stressneeded to overcome this barrier

Pressure(GPa)

Slipplane

K(0)(GPa)

arsquo (A) ˇ (A) 2 W(eVA)

c p (GPa)

0001

117a radic22 10 029 6

110 a 37 067 11111 a radic64 20 021 4

20001

127a radic22 6 102 34

110 a 15 065 10111 a radic64 14 061 14

Table 3 Parameters for the PN model for frac12 lsaquo 110 8 edge disloca-tions Parameters are defined as in Table 2

Pressure(GPa)

Slipplane

K(90)(GPa)


c p (GPa)

0001 158

a radic2213 032 6

110 150 55 021 4111 154 26 009 15

20001 177

a radic229 037 12

110 174 20 018 35111 176 19 019 55

The results of Peierls stresses calculations are presented inTables 2 and 3 for both screw and edge dislocations Pres-sure is found to increase the Peierls stress of dislocationsgliding in 100 and 111 Dislocations gliding in 110do not present a strong sensitivity to pressure At 20 GPaPeierls stresses determined for edge or screw dislocationsconfirm the results of ISS that suggest glide on 110 and111 planes is easier than on 100 planes

6 Prediction of easy slip systems in ringwoodite

A few criteria are usually proposed to predict or account forthe choice of the slip systems in crystals (Cordier 2002)ndash Burgers vectors are chosen among the shortest lattice re-

peatsndash Glide planes are chosen among close-packed planesndash The slip plane corresponds to the dissociation plane of

the dislocationsndash Easy slip systems correspond to minimum values of laquob

dhklraquo (Chalmers-Martius criterion (Chalmers amp Martius1952) where b is the Burgers vector magnitude and dhklthe interplanar spacing of the slip plane)Selecting the smallest Burgers vectors is a criterion of

minimum dislocation energy It is based on the idea that aneasy slip system is a slip system where the energy increaseyielded by the multiplication of dislocations required formaintaining plastic deformation is smallest We have calcu-lated the elastic energies associated with several slip sys-tems and shown that in agreement with the first rule listed


above frac12 lsaquo 110 8 slip is more likely than lsaquo 100 8 slip Wehave seen that following crystal chemistry arguments andrecent calculations of Durinck et al (2005) the criterion ofbdquoclose-packed planesrdquo must be replaced in silicates by thechoice of planes that do not cut strong Si-O bonds The weakelastic anisotropy of ringwoodite does not allow us to distin-guish between the possible slip planes which satisfy this re-quirement 001 110 and 111

Our calculations of GSF bring further information on thepossibility of shearing the ringwoodite structure The occur-rence of lsaquo 100 8 slip is ruled out by strong atomic impinge-ments Rigid body plastic shear is intrinsically easier in110 and 111 Plastic shear on 001 seems alreadymost difficult at this stage The Chalmers-Martius criterionassesses the easiness of slip from the Peierls-Nabarro modelin its most simple form (sine profile of the potential) Thesearch for possible dissociation planes can be assisted by theGSF calculations The existence of a minimum of the GSFcurves suggests a possible dissociation This has been con-firmed by the Peierls models of the dislocation cores pre-sented in section 5 Once again glide is suggested to be easi-er on 110 and 111 by this approach as a tendency forcore spreading is usually related to a greater mobility Itmust be underlined that the calculations reported here repre-sent a first approach and do not pretend to give a definiteview of the dislocation core structure in ringwoodite In par-ticular it must be remembered that the Peierls model con-siders planar cores only which represent a strong assump-tion The strongest tendency for core spreading in 110 and

Fig 12 Ringwoodite crystal preferred orientations predicted by VPSC models at an equivalent strain of 1 (shear strain of 173) Critical re-solved shear stresses (CRSS) for the 3 slips systems used in the models are displayed to the left of each pole figure Lower hemisphere equal-area projections contours at intervals of 05 multiple of a uniform distribution 1000 grains Dextral shear SD = shear direction NSP = nor-mal to the shear plane which is marked by the horizontal line Black line marks the foliation (main flattening plane which normal is Z) X= lineation (main stretching direction)

111 is however an important information The calculationof the Peierls stresses not only gives more realistic valuesthan ISS it also allows us to account for the influence of thecore geometry (spreading) As both ISS and core spreadinglead to the same conclusion it is not surprising that the Pei-erls stresses provide additional support All together ourcalculations suggest that at 20 GPa the easy slip systems ofringwoodite are frac12 lsaquo 110 8 110 and frac12 lsaquo 110 8 111 Itmust be noted that those slip systems exhibit a very differentevolution with pressure frac12 lsaquo 110 8 110 softens slightlywhile frac12 lsaquo 110 8 111 hardens significantly with pressureOur calculations compare very well with recent in situ stressmeasurements from Nishiyama et al (2005) consideringthat their experiments are dominated by frac12 lsaquo 110 8 111 slipas shown by Wenk et al (2005) Compared to previous ex-periments in diamond cells the measurements of Nishiya-ma et al (2005) allows the distinction of the actual influenceof pressure on plastic shear by removing the strain-harden-ing contribution Their stress values compare well with ourPeierls stresses and moreover the pressure evolution ofstress reported by Nishiyama et al (2005) is in good agree-ment with their conclusions of dominant frac12 lsaquo 110 8 111

7 From single crystals to polycrystals

In Fig 12 we show CPO developed in an aggregate con-taining one thousand initially spherical and randomly ori-ented ringwoodite grains deformed in simple shear after an


Fig 13 Slip systems activity at an equivalent strain of 1 as a functionof relative strength of 100 slip relative to 110 and 111 in thesimple shear VPSC simulations The activity of a slip system is cal-culated by averaging over all grains the contribution of this slip sys-tem to the local strain rate

Fig 14 Inverse pole figure showing the orientation of the shorteningdirection relative to the ringwoodite crystal axes predicted by VPSCmodels with critical resolved shear stresses (CRSS) for slip on110 111 and 100 defined as CRSS110 = CRSS111 = 12CRSS100 after an axial shortening of 25 Contours at intervals of02 multiple of a uniform distribution 1000 grains

equivalent strain of 1 which corresponds to a dextral simpleshear strain of 173 The models which are presented in-volve easy slip along frac12 lsaquo 110 8 on 110 and 111 ieCRSS110 = CRSS111 = 1 and increasingly harder slip on100 ie CRSS100 = 1 2 or 10 and a rate sensitivity n =3 Ringwoodite CPO are well developed but weak (maxi-mum concentrations are always below 2 multiples of an uni-form distribution) The CPO are almost insensitive to thevariation of CRSS for 100 slip lsaquo 100 8 axes tend to aligneither within or normal to the foliation (main flatteningplane) with a weak maximum close to the lineation (mainstretching direction) in those models in which slip on 100planes are hindered lsaquo 111 8 and lsaquo 110 8 axes display both arough 6-fold distribution with a symmetry axis normal to theshear plane However lsaquo 110 8 displays a maximum close tothe shear direction and some dispersion normal to the linea-tion or within the shear plane whereas lsaquo 111 8 shows twosymmetric maxima at ca 30deg to the shear direction and agirdle distribution in a plane at high angle to the shear direc-tion Once formed these CPO evolve slowly with increas-

Fig 15 Slip systems activity as a function of strain in an axial com-pression VPSC simulation The activity of a slip system is calculatedby averaging over all grains the contribution of this slip system to thelocal strain rate

ing strain CPO modelled for an equivalent strain of 5 showpatterns and intensities (maximum concentrations oflsaquo 100 8 lsaquo 110 8 and lsaquo 111 8 are 17 195 and 18 multiples

of an uniform density respectively) similar to those ob-served after an equivalent strain of 1 Results for modelswith n = 5 show similar patterns but stronger concentrations(maximum concentrations of lsaquo 100 8 lsaquo 110 8 and lsaquo 111 8are 197 269 and 273 multiples of an uniform density re-spectively for a model with CRSS110 = CRSS111 = 12CRSS100 at an equivalent strain of 1) In all cases slip on111 dominates the plasticity of the aggregate (Fig 13)The higher activity of 111 systems results from geometri-cal constraints Indeed in a cubic mineral this mode is com-posed by 12 systems whereas 110 and 100 modes arecomposed by 6 systems each

CPO formed in axial compression models in whichCRSS110 = CRSS111 = CRSS100 or CRSS110 =CRSS111 = 12 CRSS100 reproduce the pattern observedin room temperature compression experiments in a diamondanvil cell (Wenk et al 2004) and in a D-DIA (Wenk et al2005) 110 planes tend to orient normal to the compres-sion direction (Fig 14) because deformation in these mod-els is essentially accommodated by slip on 111 planes(Fig 15) The orientation of 110 normal to compressiondirection in these models results therefore from a geometri-cal effect associated with the simultaneous activation of var-ious frac12 lsaquo 110 8 111 systems Similar geometrical effectshave also been observed to occur in other minerals like cli-nopyroxenes (Bascou et al 2002) These results highlightthat the interpretation of slip system activity from the analy-sis of measured CPO is not straightforward when deforma-tion is essentially accommodated by prismatic or pyramidalslip systems emphasising the importance of TEM data andof forward simulations like the ones presented here in theevaluation of the slip systemsrsquo activity

8 Concluding remarks

It is now possible to use first-principle calculations to assessthe resistance of minerals to plastic shear from GSF calcula-


tions We show in this study that GSF can be used as input datato infer dislocations properties from the Peierls model Thisapproach has been used here for the first time to predict theplastic properties of a mineral under high-pressure conditionsOur calculations suggest that frac12 lsaquo 110 8 110 andfrac12 lsaquo 110 8 111 are the easiest slip systems in ringwoodite at20 GPa VPSC calculations of CPO using these data areshown to account satisfactorily for experimental data obtainedat room temperature in diamond anvil cell and D-DIA experi-ments The approach illustrated in the present work opens anew field in the study of plasticity of minerals under extremepressure conditions The next step will be of course to incorpo-rate the influence of temperature on plastic slip Our approachcannot predict if a change of slip system occurs with tempera-ture We observe that our predictions are in agreement with theHP-HT deformation experiments of Thurel (2001) but notwith those of Karato et al (1998) who reported slip on 100It is not clear whether the discrepancy between those results isdue to a sampling problem or if other parameters (differencesin the iron content for instance) are involved In any case thedeformation of ringwoodite seems to produce relatively weakcrystal preferred orientations Given the modest elastic anisot-ropy of ringwoodite at pressure and temperature of the transi-tion zone (Mainprice et al 2000) and the weak crystal pre-ferred orientations it develops it is unlikely that plastic defor-mation of ringwoodite generates a marked seismic anisotropyin the transition zone between 520 and 670 km depth

Acknowledgements We acknowledge the contribution ofC Terrier and F Vasconcelos for the calculations and thankJ Douin (CEMES Toulouse) for helpful discussionsThoughtful comments from the associate editor H-RWenk and an anonymous referee helped us in improving thepaper This work was supported by CNRS-INSU under theDyETI programme Computational resources were provid-ed by IDRIS (project 031685) and CRI-USTL supportedby the Fonds Europeens de Developpement Regional

ReferencesBarron THK amp Klein ML (1965) Second-order elastic con-

stants of a solid under stress Proc Phys Soc 85 523ndash531Bascou J Tommasi A Mainprice D (2002) Plastic deformation

and development of clinopyroxene lattice preferred orientationsin eclogites J Struct Geol 24 1357ndash1368

Chalmers B amp Martius UM (1952) Slip planes and the energy ofdislocations Proc R Soc London A 213 175ndash185

Christian JW amp Vitek V (1970) Dislocations and StackingFaults Rep Prog Phys 33

Clatterbuck DM Krenn CR Cohen ML Morris Jr JW(2003) Phonon instabilities and the ideal strength of aluminumPhys Rev B Condensed Matter 91 5501ndash5504

Cordier P (2002) Dislocation and slip systems of mantle mineralsin bdquoPlastic deformation of minerals and rocksrdquo Reviews in Min-eralogy 41 Mineralogical Society of America ed

Cordier P amp Rubie DC (2001) Plastic deformation of mineralsunder extreme pressure using a multi-anvil apparatus Mater SciEng A Struct Mater 309 38ndash43

Cordier P Ungar T Zsoldos L Tichy G (2004) Dislocationcreep in MgSiO3 Perovskite at conditions of the Earthrsquos upper-most lower mantle Nature 428 837ndash840

de Koning M Antonelli A Bazant MZ Kaxiras E Justo JF(1998) Finite-temperature molecular-dynamics study of unsta-ble stacking fault free energies in silicon Phys Rev B CondensedMatter 58 12555ndash12558

Domain C Besson R Legris A (2004) Atomic-scale ab initiostudy of the Zr-H system II Interaction of H with plane defectsand mechanical properties Acta Materialia 52 1495ndash1502

Douin J (1987) Structure fine des dislocations et plasticite dansNi3Al PhD Thesis University of Poitiers France

Douin J Veyssiere P Beauchamp P (1986) Dislocation line sta-bility in Ni3Al Philos Mag A 54 375ndash393

Durinck J Legris A Cordier P (2005) Influence of crystal chem-istry on ideal plastic shear anisotropy in forsterite first principlecalculations Am Mineral 90 1072ndash1077

Hartford J von Sydow B Wahnstroumlm G Lundqvist BI (1998)Peierls barrier and stresses for edge dislocations in Pd and Al cal-culated from first principles Phys Rev B 58 2487ndash2496

Hirth JP amp Lothe J (1982) Theory of dislocations Wiley-Inter-science Publication

Irifune T amp Ringwood AE (1987) Phase transformations in aharzburgite composition to 26 GPa implications for dynamicalbehavior of the subducting slab Earth Planet Sci Lett 86 365ndash376

Joos B Ren Q Duesbery MS (1994) Peierls-Nabarro model ofdislocations in silicon with generalized stacking-fault restoringforces Phys Rev B 50 5890ndash5898

Karato SI Dupas-Bruzek C Rubie DC (1998) Plastic defor-mation of silicate spinel under the transition-zone conditions ofthe Earthrsquos mantle Nature 395 266ndash269

Kavner A amp Duffy TS (2001) Strength and elasticity of ringwoo-dite at upper mantle pressures Geophys Res Lett 28 2691ndash2694

Kaxiras E amp Duesbery MS (1993) Free energies of generalyzedstacking faults in Si and implications for the brittle-ductile transi-tion Phys Rev Lett 70 3752ndash3755

Kiefer B Stixrude L Wentzcovitch RM (1997) Calculatedelastic constants and anisotropy of Mg2SiO4 spinel at high pres-sure Geophys Res Lett 24 2841ndash2844

Krenn CR Roundy D Morris Jr JW Cohen ML (2001) thenonlinear elastic behavior and ideal shear strength of Al and CuMaterials Sci Eng A 317 44ndash48

Kresse G amp Furthmuumlller J (1996a) Efficiency of ab-initio total en-ergy calculations for metals and semiconductors using a plane-wave basis set Comput Mat Sci 6 15ndash50

ndashndash (1996b) Efficient iterative schemes for ab initio total-energy cal-culations using a plane-wave basis set Phys Rev B 54 11169

Kresse G amp Hafner J (1993) Ab initio molecular dynamics for liq-uid metals Phys Rev B 47 558

ndashndash (1994) Norm-conserving and ultrasoft pseudopotentials forfirst-row and transition elements J Phys Condens Mat 68245

Lebensohn RA amp Tome CN (1993) A self-consistent anisotrop-ic approach for the simulation of plastic deformation and texturedevelopment of polycrystal application to zirconium alloys Ac-ta Metall Mater 41 2611ndash2624

Lebensohn RA Dawson PR Kern HM Wenk HR (2003)Heterogeneous deformation and texture development in halitepolycrystals comparison of different modeling approaches andexperimental data Tectonophysics 370 287ndash311

Loge RE Signorelli JW Chastel YB Perrin MY LebensohnRA (2000) Sensitivity of alpha-Zy4 high-temperature deforma-tion textures to beta-quenched precipitate structure and to recrystal-lization application to hot extrusion Acta Mater 48 3917ndash3930


Luo W Roundy D Cohen ML Morris Jr JW (2002) Idealstrength of bcc molybdenum and niobium Phys Rev B B66 1ndash7

Madon M amp Poirier JP (1983) Transmission electron microscopeobservation of a b and g (Mg Fe)2SiO4 in shocked meteoritesplanar defects and polymorphic transitions Phys Earth PlanetInt 33 31ndash44

Mainprice D Barruol G Ben Ismaıl W (2000) The seismic an-isotropy of the Earthrsquos mantle from single crystal to polycrystalin bdquoEarth deep interior mineral physics and tomography from theatomic scale to the global scaleldquo SI Karato Ed American Geo-physical Union Washington DC 237ndash264

Mainprice D Bascou J Cordier P Tommasi A (2004) Crystalpreferred orientations of garnet comparisons between numericalsimulations and electron back-scattered diffraction (EBSD) mea-surements in naturally deformed eclogites J Struct Geol 262089ndash2102

Medvedeva NI Mryasov ON Gornostyrev YN Novikov DLFreeman AJ (1996) First-principles total-energy calculationsfor planar shear and cleavage decohesion processes in B2-or-dered NiAl and FeAl Phys Rev B 54 13506ndash13514

Miranda CR amp Scandolo S (2005) Computational materials sci-ence meets geophysics dislocations and slip planes of MgOComp Phys Comm 169 24ndash27

Mitchell TE (1999) Dislocations and mechanical properties ofMgO-Al2O3 spinel single crystals J Amer Ceram Soc 823305ndash3316

Molinari A Canova GR Azhy S (1987) A self-consistent ap-proach of the large deformation crystal polycrystal viscoplastici-ty Acta Metall 35 2983ndash2994

Monkhorst HJ amp Pack JD (1976) Special points for Brillouin-zone integrations Phys Rev B 23 5048ndash5192

Mryasov ON Gornostyrev YN Freeman AJ (1998) General-ized stacking-fault energetics and dislocation properties com-pact versus spread unit-dislocation structures in TiAl and CuAuPhys Rev B Condensed Matter 58 11927ndash11932

Nishiyama N Wang Y Uchida T Rivers ML Sutton SR(2005) Pressure and strain dependence of the strength of sinteredpolycrystalline Mg2SiO4 ringwoodite Geophys Res Lett 32L04307

Ogata S Li J Yip S (2002) Ideal pure shear strength of alumi-num and copper Science 298 807ndash811

Paxton AT Gumbsch P Methfessel M (1991) A quantum me-chanical calculation of the theoretical strength of metals PhilosMag Lett 63 267ndash274

Perdew JP amp Wang Y (1992) Accurate and simple analytic repre-sentation of the electron-gas correlation energy Phys Rev B 4513244ndash13249

Putnis A amp Price GD (1979) High pressure (MgFe)2SiO4 phasesin the Tenham chondritic meteorite Nature 280 217ndash218

Roundy D Krenn CR Cohen ML Morris Jr JW (1999) Idealshear strengths of fcc aluminum and copper Phys Rev Lett 822713ndash2716

Roundy D Krenn CR Cohen ML Morris Jr JW (2001) Theideal strength of tungsten Philos Mag A 81 1725ndash1747

Rubie DC amp Brearley AJ (1990) Mechanism of the g-b-phasetransformation of Mg2SiO4 at high pressure and temperature Na-ture 348 628ndash631

ndashndash (1994) Phase transitions between beta and gamma (MgFe)2SiO4

in the Earthrsquos mantle Mechanisms ans rheological implicationsScience 264 1445ndash1448

Sinogeikin SV Bass JD Katsura T (2001) Single-crystal elas-ticity of g-(Mg091Fe009)2Si04 to high pressures and to high tem-peratures Geophys Res Lett 28 4335ndash4338

Soumlderlind P amp Moriarty JA (1998) First-principles theory of Taup to 10 Mbar pressure Structural and mechanical propertiesPhys Rev B 57 10340ndash10350

Sun Y amp Kaxiras E (1997) Slip energy barrier in aluminum andimplications for ductile-brittle behaviour Philos Mag A 751117ndash1127

Thurel E (2001) Etude par microscopie electronique en transmis-sion des mecanismes de deformation de la wadleyite et de la ring-woodite PhD Thesis University of Lille France

Tome CN amp Canova G (2000) Self-consistent modelling of het-erogeneous plasticity in bdquoTexture and Anisotropy ndash PreferredOrientations in Polycrystals and their effects on Materials Proper-tiesldquo UF Kocks CN Tome HR Wenk Eds Cambridge Uni-versity Press Cambridge 466ndash509

Tome CN Wenk H-R Canova G Kocks UF (1991) Simula-tions of texture development in calcite comparison of polycrys-tal plasticity theories J Geophys Res 96 11865ndash11875

Tommasi A Mainprice D Canova G Chastel Y (2000) Visco-plastic self-consistent and equilibrium-based modeling of olivinelattice preferred orientations Implications for the upper mantle seis-mic anisotropy J Geophys Res Solid Earth 105 7893ndash7908

Tommasi A Cordier P Mainprice D Couvy H Thoraval C(2004) Strain-induced seismic anisotropy of wadsleyite poly-crystal constraints on flow patterns in the mantle transition zoneJ Geophys Res 109 B12405

Vanderbilt D (1990) Soft self-consistent pseudopotentials in a gen-eralized eigenvalue formalism Phys Rev B 41 7892ndash7895

Vaughan PJ amp Kohlstedt DL (1981) Cation stacking faults in mag-nesium germanate spinel Phys Chem Minerals 7 241ndash245

Vıtek V (1968) Intrinsic stacking faults in body-centered cubiccrystals Phil Mag A 18 773ndash786

ndash (1974) Theory of the core structures of dislocations in body-cen-tered-cubic metals Crystal Lattice Defects 5 1ndash34

Walker AM (2004) Computational studies of point defects anddislocations in forsterite (Mg2SiO4) and some implications forthe rheology of mantle olivine PhD Thesis University CollegeLondon UK

Wang Y Durham WB Getting I Weidner DJ (2003) The de-formation-DIA a new apparatus for high-temperature triaxial de-formation to pressures up to 15 GPa Rev Sci Instrum 74 3002ndash3011

Wenk HR Lonardelli I Pehl J Devine J Prakapenka V ShenG Mao H-K (2004) In situ observation of texture develop-ment in olivine ringwoodite magnesiowuumlstite and silicate pe-rovskite Earth Planet Sci Lett 226 507ndash519

Wenk HR Bennet K Canova GR Molinari A (1991) Model-ling plastic deformation of peridotite with the self-consistent the-ory J Geophys Res 96 8337ndash8349

Wenk HR Ischia G Nishiyama N Wang Y Uchida T (2005)Texture development and deformation mechanisms in ringwoo-dite Phys Earth Planet Int 152 191ndash199

Xu W amp Moriarty JA (1996) Atomistic simulation of ideal shearstrength point defects and screw dislocations in bcc transitionmetals Mo as a prototype Phys Rev B 54 6941ndash6951

Xu YQ Weidner DJ Chen JH Vaughan MT Wang YBUchida T (2003) Flow-law for ringwoodite at subduction zoneconditions Phys Earth Planet Interiors 136 3ndash9

Yamazaki D amp Karato S (2001) High-pressure rotational defor-mation apparatus to 15 GPa Rev Sci Instr 72 4207ndash4211

Received 20 July 2005Modified version received 17 October 2005Accepted 14 November 2005


situ at high pressures These studies based on in situ crystalpreferred orientation measurements emphasize that slip onfrac12 lsaquo 110 8 111 dominates deformation under those condi-tions

Computational materials science is an alternative ap-proach to study plastic deformation The Peierls-Nabarro(PN) model provides a conceptual framework for address-ing the issue of dislocation structures (Hirth amp Lothe 1982)The model is essentially based on continuum mechanics (farfrom the defect) but inelastic displacements in the disloca-tion core are given a specific description A recent approachfor describing forces in the dislocation core is based on theconcept of generalized stacking fault (GSF) surfaces (Vitek1968 Christian amp Vitek 1970) The value of the GSF is ob-tained by shearing half of the infinite crystal over the otherhalf The fault is called bdquogeneralizedrdquo because it is unstablefor most values of shear and must be balanced by a restoringforce (which is introduced in the PN model) The GSF cannowadays be calculated accurately from first-principlesThe GSF gives access to the ideal shear strength which is theupper bound on the mechanical strength of a solid This ap-proach has been applied to many metallic systems Al (Pax-ton et al 1991 Hartford et al 1998 Roundy et al 1999Ogata et al 2002) Cu (Roundy et al 1999 Ogata et al2002) Pd (Hartford et al 1998) Fe (Clatterbuck et al2002) Mo (Xu amp Moriarty 1996 Luo et al 2002) Nb (Luoet al 2002) W (Krenn et al 2001 Roundy et al 2001) Ta(Soumlderlind amp Moriarty 1998) and Zr (Domain et al 2004)Recently we have calculated GSF for forsterite (Durinck etal 2005) and shown that despite a complex crystal chemis-try this approach could account for plastic anisotropy in thismineral It is thus possible to build more realistic models ofdislocations using a PN model in which the inelastic dis-placements in the dislocation core are described using theGSF approach Several examples are available in the recentliterature Al (Sun amp Kaxiras 1997 Hartford et al 1998)Pd (Hartford et al 1998) NiAl and FeAl (Medvedeva et al1996) TiAl and CuAu (Mryasov et al 1998) Si (Kaxiras ampDuesbery 1993 Joos et al 1994 de Koning et al 1998)and MgO (Miranda amp Scandolo 2005)

In the present study we use the GSF framework to assessplastic strain anisotropy of ringwoodite at the atomic scalefrom first-principle calculations Dislocation properties arederived from GSF through the Peierls-Nabarro (PN) modelThis approach is combined with elasticity to infer the possi-ble slip systems in ringwoodite Self-consistent viscoplasticmodels are finally used to evaluate the crystal-preferred ori-entation that will result from the activation of these slip sys-tems and compare our results with ringwoodite crystal pre-ferred orientations produced in recent low-temperature de-formation experiments

2 Crystallography of slip in the spinel structure

Ringwoodite exhibits a spinel structure based on a face-cen-tred-cubic packing of the oxygen sublattice Silicon atomsare located in the tetrahedral sites whereas octahedral sitesare occupied by magnesium and iron atoms The spacegroup of Mg2SiO4 is Fd3m and the lattice parameter is

Fig 1 Ringwoodite structure viewed along lsaquo 110 8 with potentialslip planes The unit cells are shown in dashed lines

8071 A (Ringwood amp Major1970) The slip direction inspinel is always observed to be parallel to the shortest latticevector of the fcc oxygen lattice frac12 lsaquo 110 8 The observed slipplane is variable The most commonly observed slip planesare 111 and 110 although 100 have been reported inmagnetite nickel ferrite and chromite (see a detailed reviewin Mitchell 1999) Figure 1 shows the structure of ringwoo-dite viewed along lsaquo 110 8 The planes usually reported asslip planes in the spinel structure are indeed potential slipplanes for ringwoodite as they do not require breaking thestrong Si-O bonds In oxides with spinel structures disloca-tions are generally observed to be dissociated into collinearpartials following the reaction frac12 lsaquo 110 8 rarr frac14 lsaquo 110 8 +frac14 lsaquo 110 8 The dissociation width increase with increasingdeviation from stoichiometry implying that the stackingfault energy decrease (between 180 and 20 mJm2 Mitchell1999) A variety of dissociation planes have been reported111 110 100 and 311

3 Computational procedures

3a Ab initio calculations

Calculations were performed using the ab initio total-ener-gy calculation package VASP (Vienna Ab Initio Package)developed by Kresse and Hafner (Kresse amp Hafner 19931994 Kresse amp Furthmuumlller 1996a) This code is based onthe first-principles density functional theory and solves theeffective one-electron Hamiltonian involving a functionalof the electron density to describe the exchange-correlationpotential It gives access to the total energy of a periodic sys-tem with as a single input the atomic numbers of atomsComputational efficiency is achieved using a plane wavebasis set for the expansion of the single electron wave func-tions and fast numerical algorithms to perform self-consis-tent calculations (Kresse amp Furthmuumlller 1996b) Within thisscheme we used the Generalised Gradient Approximation(GGA) derived by Perdrew amp Wang (1992) and ultrasoft




2 EV

= ndash pe + 12c11 e2 (1)


deduce c44 from

2 EV

= 2c44 e2 (2)


2 EV
















Eel = K( rsquo )b2

4 lsquoln ( R

r0) (4)



+ cos 2 rsquo ) (5)
















Frarr





Pressure(GPa)

Slipplane


(GPa) max

(Jm2)S max

(GPa)S max


0001

575131 24 24 019

110 105 12 17 016111 112 14 15 013

20001

557142 33 44 031

110 128 21 26 020111 132 23 25 019







ρ(x) = dS(x)dx

(7)


⌠⌡ndash rsaquo


⌡ndash rsaquo

+ rsaquo dS(x)dx

dx = b (8)


K2 lsquo

⌠⌡ndash rsaquo



⌠⌡ndash rsaquo








S(x) = b2

+ blsquo 7

N


ci(10)


ρ(x) = dS(x)dx

= blsquo 7

N

i = 1[ i

ci(x ndash xi)

2 + ci2 (11)




7N




N

i = 1[ i middot


2 + ci2 (12)










W(u) = 7+ rsaquo



c P = max 1b

dW(u)du (14)


Pressure(GPa)

Slipplane

K(0)(GPa)


c p (GPa)

0001

117a radic22 10 029 6

110 a 37 067 11111 a radic64 20 021 4

20001

127a radic22 6 102 34

110 a 15 065 10111 a radic64 14 061 14


Pressure(GPa)

Slipplane

K(90)(GPa)


c p (GPa)

0001 158

a radic2213 032 6

110 150 55 021 4111 154 26 009 15

20001 177

a radic229 037 12

110 174 20 018 35111 176 19 019 55







































































































2 EV

= ndash pe + 12c11 e2 (1)


deduce c44 from

2 EV

= 2c44 e2 (2)


2 EV
















Eel = K( rsquo )b2

4 lsquoln ( R

r0) (4)



+ cos 2 rsquo ) (5)
















Frarr





Pressure(GPa)

Slipplane


(GPa) max

(Jm2)S max

(GPa)S max


0001

575131 24 24 019

110 105 12 17 016111 112 14 15 013

20001

557142 33 44 031

110 128 21 26 020111 132 23 25 019







ρ(x) = dS(x)dx

(7)


⌠⌡ndash rsaquo


⌡ndash rsaquo

+ rsaquo dS(x)dx

dx = b (8)


K2 lsquo

⌠⌡ndash rsaquo



⌠⌡ndash rsaquo








S(x) = b2

+ blsquo 7

N


ci(10)


ρ(x) = dS(x)dx

= blsquo 7

N

i = 1[ i

ci(x ndash xi)

2 + ci2 (11)




7N




N

i = 1[ i middot


2 + ci2 (12)










W(u) = 7+ rsaquo



c P = max 1b

dW(u)du (14)


Pressure(GPa)

Slipplane

K(0)(GPa)


c p (GPa)

0001

117a radic22 10 029 6

110 a 37 067 11111 a radic64 20 021 4

20001

127a radic22 6 102 34

110 a 15 065 10111 a radic64 14 061 14


Pressure(GPa)

Slipplane

K(90)(GPa)


c p (GPa)

0001 158

a radic2213 032 6

110 150 55 021 4111 154 26 009 15

20001 177

a radic229 037 12

110 174 20 018 35111 176 19 019 55











































































































Eel = K( rsquo )b2

4 lsquoln ( R

r0) (4)



+ cos 2 rsquo ) (5)
















Frarr





Pressure(GPa)

Slipplane


(GPa) max

(Jm2)S max

(GPa)S max


0001

575131 24 24 019

110 105 12 17 016111 112 14 15 013

20001

557142 33 44 031

110 128 21 26 020111 132 23 25 019







ρ(x) = dS(x)dx

(7)


⌠⌡ndash rsaquo


⌡ndash rsaquo

+ rsaquo dS(x)dx

dx = b (8)


K2 lsquo

⌠⌡ndash rsaquo



⌠⌡ndash rsaquo








S(x) = b2

+ blsquo 7

N


ci(10)


ρ(x) = dS(x)dx

= blsquo 7

N

i = 1[ i

ci(x ndash xi)

2 + ci2 (11)




7N




N

i = 1[ i middot


2 + ci2 (12)










W(u) = 7+ rsaquo



c P = max 1b

dW(u)du (14)


Pressure(GPa)

Slipplane

K(0)(GPa)


c p (GPa)

0001

117a radic22 10 029 6

110 a 37 067 11111 a radic64 20 021 4

20001

127a radic22 6 102 34

110 a 15 065 10111 a radic64 14 061 14


Pressure(GPa)

Slipplane

K(90)(GPa)


c p (GPa)

0001 158

a radic2213 032 6

110 150 55 021 4111 154 26 009 15

20001 177

a radic229 037 12

110 174 20 018 35111 176 19 019 55












































































































Frarr





Pressure(GPa)

Slipplane


(GPa) max

(Jm2)S max

(GPa)S max


0001

575131 24 24 019

110 105 12 17 016111 112 14 15 013

20001

557142 33 44 031

110 128 21 26 020111 132 23 25 019







ρ(x) = dS(x)dx

(7)


⌠⌡ndash rsaquo


⌡ndash rsaquo

+ rsaquo dS(x)dx

dx = b (8)


K2 lsquo

⌠⌡ndash rsaquo



⌠⌡ndash rsaquo








S(x) = b2

+ blsquo 7

N


ci(10)


ρ(x) = dS(x)dx

= blsquo 7

N

i = 1[ i

ci(x ndash xi)

2 + ci2 (11)




7N




N

i = 1[ i middot


2 + ci2 (12)










W(u) = 7+ rsaquo



c P = max 1b

dW(u)du (14)


Pressure(GPa)

Slipplane

K(0)(GPa)


c p (GPa)

0001

117a radic22 10 029 6

110 a 37 067 11111 a radic64 20 021 4

20001

127a radic22 6 102 34

110 a 15 065 10111 a radic64 14 061 14


Pressure(GPa)

Slipplane

K(90)(GPa)


c p (GPa)

0001 158

a radic2213 032 6

110 150 55 021 4111 154 26 009 15

20001 177

a radic229 037 12

110 174 20 018 35111 176 19 019 55






































































































Pressure(GPa)

Slipplane


(GPa) max

(Jm2)S max

(GPa)S max


0001

575131 24 24 019

110 105 12 17 016111 112 14 15 013

20001

557142 33 44 031

110 128 21 26 020111 132 23 25 019







ρ(x) = dS(x)dx

(7)


⌠⌡ndash rsaquo


⌡ndash rsaquo

+ rsaquo dS(x)dx

dx = b (8)


K2 lsquo

⌠⌡ndash rsaquo



⌠⌡ndash rsaquo








S(x) = b2

+ blsquo 7

N


ci(10)


ρ(x) = dS(x)dx

= blsquo 7

N

i = 1[ i

ci(x ndash xi)

2 + ci2 (11)




7N




N

i = 1[ i middot


2 + ci2 (12)










W(u) = 7+ rsaquo



c P = max 1b

dW(u)du (14)


Pressure(GPa)

Slipplane

K(0)(GPa)


c p (GPa)

0001

117a radic22 10 029 6

110 a 37 067 11111 a radic64 20 021 4

20001

127a radic22 6 102 34

110 a 15 065 10111 a radic64 14 061 14


Pressure(GPa)

Slipplane

K(90)(GPa)


c p (GPa)

0001 158

a radic2213 032 6

110 150 55 021 4111 154 26 009 15

20001 177

a radic229 037 12

110 174 20 018 35111 176 19 019 55







































































































7N




N

i = 1[ i middot


2 + ci2 (12)










W(u) = 7+ rsaquo



c P = max 1b

dW(u)du (14)


Pressure(GPa)

Slipplane

K(0)(GPa)


c p (GPa)

0001

117a radic22 10 029 6

110 a 37 067 11111 a radic64 20 021 4

20001

127a radic22 6 102 34

110 a 15 065 10111 a radic64 14 061 14


Pressure(GPa)

Slipplane

K(90)(GPa)


c p (GPa)

0001 158

a radic2213 032 6

110 150 55 021 4111 154 26 009 15

20001 177

a radic229 037 12

110 174 20 018 35111 176 19 019 55








































































































W(u) = 7+ rsaquo



c P = max 1b

dW(u)du (14)


Pressure(GPa)

Slipplane

K(0)(GPa)


c p (GPa)

0001

117a radic22 10 029 6

110 a 37 067 11111 a radic64 20 021 4

20001

127a radic22 6 102 34

110 a 15 065 10111 a radic64 14 061 14


Pressure(GPa)

Slipplane

K(90)(GPa)


c p (GPa)

0001 158

a radic2213 032 6

110 150 55 021 4111 154 26 009 15

20001 177

a radic229 037 12

110 174 20 018 35111 176 19 019 55





































































































Slip systems and plastic shear anisotropy in Mg SiO ... · Slip systems and plastic shear...

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Transcript of Slip systems and plastic shear anisotropy in Mg SiO ... · Slip systems and plastic shear...