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Finite strain Landau theory of high pressure phase 0 transformations 1 0 2 W. Schranz1, A. Tröster1, J. Koppensteiner1 and n R. Miletich2 a J 1FacultyofPhysics,UniversityofVienna,Boltzmanngasse5,A-1090Wien, 8 Austria 1 2 MineralogischesInstitut,UniversitätHeidelberg,ImNeuenheimerFeld236, 69120Heidelberg,Germany ] E-mail: [email protected] i c s - Abstract. The properties of materials near structural phase transitions are l often successfully described in the framework of Landau theory. While usually r t the focus is on phase transitions, which are induced by temperature changes m approaching a critical temperature Tc, here we will discuss structural phase transformations driven by high hydrostatic pressure, as they are of major . t importance for understanding processes in the interior of our earth. Since at a very high pressures the deformations of a material are generally very large, m one needs to apply a fully nonlinear description taking physical as well as - geometricalnonlinearities(finitestrains)intoaccount. Inparticularitisnecessary d to retune conventional Landau theory to describe such phase transitions. In n [A. Tröster, W. Schranz and R. Miletich, Phys. Rev. Lett.88, 55503 (2002)] we o haveconstructedaLandau-typefreeenergybasedonanorderparameterpart,an c orderparameter-(finite)strain coupling andanonlinear elasticterm. Thismodel [ provides an excellent and efficient framework for the systematic study of phase transformationsforawiderangeofmaterialsuptoultrahighpressures. 1 WeillustratethemodelonthespecificexampleofBaCr(Si4O10),showingthatit v fullyaccountsfortheelasticsofteningwhichisobservednearthepressureinduced 6 phasetransformation. 9 9 2 . 1 0 0 1 : v i X r a Finite strain Landau theory of high pressure phase transformations 2 1. Introduction A major achievementof EkhardSalje andcoworkerswasto show thatLandau theory canbeveryefficientlyappliedforthedesriptionofexperimentaldataneartemperature induced structural phase transitions in minerals (Salje, 1990; Salje, 1992; Carpenter, Salje and Graeme-Barber,1998). There are a couple of reasons for this success. First of all, as coupling to strains induce long range forces and enhance anisotropy, strain- inducedinteractionsalwaystendtosuppressfluctuations,leadingtoameanfieldtype of transition (Bratkovsky, 1995, Tsatskis, 1994). Depending on the nature of the system(i.e.onthetypeofcouplingbetweenstrainandorderparameteritsanisotropy and the boundary conditions imposed on its surface) strain effects also may change the character of the transition from second to first order (Bergman and Halperin, 1976). Now in principle, a Landau potential can be obtained from a low order Taylor expansion of the mean field free energy. The problem of how to determine the range of validity of this expansion naturally arises. To answer this question one observes that in many mineral systems the phase transitions turn out to be of the displacive type (Dove, 1997). For such systems, the above authors were able to show that Landau theory indeed allows to reproduce experimental data in an extremely broad temperaturerange,inparticulardowntoverylowtemperaturesbytakingintoaccount the effect of quantum saturation (Salje, Wruck and Thomas, 1991). The situation is not so simple for the other extreme, i.e. order-disorder phase transitions, since it is no longer possible to approximate their free energy by a low order polynomial in the orderparameter(Giddy,DoveandHeine,1989). Inpassing,wenotethatrealsystems are often of mixed, i.e. displacive/order-disordercharacter (Meyer, 2000; Sondergeld, 2000),andthecharacterizationofcrossoversystemsinbetweenadisplaciveandorder- disorder type is still a matter of active research (Perez-Mato, 2000; Rubtsov, Hlinka and Janssen, 2000; Tröster, Dellago and Schranz, 2005). In “traditional” Landau theory it is difficult to include pressure effects beyond the infinitesimal strain approximation. Therefore one frequently resorts to replacing the elastic background energy by its harmonic truncation and consequently treats the strains as infinitesimal. Since the total strains appearing in temperature induced phase transitions are as a rule quite small this is well justified. However, when we discuss the role of extremely high pressure in driving the transition, the validity of this assumption must be seriously doubted. At high pressure a phase transition is usually detected by anomalies e. g. in the systemvolumeV(P), whichinturnresultfromanomaliesinthe pressuredependence of the lattice parameters a (P), i=1,2,3, near a critical pressure P . In many cases i c thisleadstoanadditional(withrespecttothe”background”)nonlinearP-dependence of lattice parameters and/or volume. As a function of pressure, the ”stiffness” of any solid is characterized by the isothermal compressibility κ(P)= −dlogV(P)/dP. Varioustheoreticalconceptsareusuallyemployedtoderiveso-calledequationsofstate (EOS) (Anderson, 1995; Angel, 2000a; Angel, 2000b), which – in absence of phase transitions – describe the hydrostatic pressure dependence of the crystal’s volume V(P). In presence of a high pressure phase transition (HPPT) a commonly adopted practice is to merely fit the corresponding V(P)-behavior to a number of differently parametrizedEsOSforeachphase(SchulteandHolzapfel,1995;Kru¨gerandHolzapfel, 1992;ChesnutandVohra,2000),and,apartfromcomputersimulations,inthecaseof a reconstructive phase transition there is little more one can do at present. However, Finite strain Landau theory of high pressure phase transformations 3 from a theoretical point of view, although this purely phenomenological procedure does fit experimental volume data in many cases, it neither describes the pressure behaviour of individual strain components nor provides any possible further insight intothemechanismsofaHPPT.Amoreprofoundtheoreticalapproachisthereforeof vital interest to a broadaudience reachingform physicists studying the high pressure behaviorofmaterials(crystals,liquidcrystals,complexliquids,biologicalmembranes, etc.) to geologists investigating the earth’s minerals and bulk properties. Indeed, for HPPT’s of the group-subgroup type, it is natural to expect that one could do much better by applying the concepts of Landau theory. In particular, we should be able to connect the high and low pressure phase by a uniform thermodynamic description. However, here one faces a marked difference compared tothetemperaturedrivencase. Athighpressures,astheinter-atomicforcesopposing further compression, a crystal’s volume and lattice parameters develop large strains in a pronounced non-linear way, which corrupts any serious attempt to treat the elastic energy in the framework of the infinitesimal strain approximation. For these reasons it is necessary to reconsider the construction of Landau theory at extremely high pressures and large deformations (Tröster, Schranz and Miletich, 2002). Here we summarize the main ideas of this new theoretical approach and discuss some consequences for the elastic behaviour of materials. 2. New approach for high pressure phase transformations As stated above, the conventional infinitesimal strain approach is clearly insufficient foraimingatanaccuratedescriptionofHPPT’s. Instead,athighpressurestheelastic response of a system must be characterizedin terms of a appropriatenonlinearstrain measure like the Lagrangian or Eulerian strain tensor. Let X and X′ denote an undeformed reference system and a deformed system, respectively. Then, from the ′ deformation gradient tensor α = ∂X /∂X one constructs the Lagrangian strain ik i k tensor e = 1( α α −δ ). In the present work, we will consider three pairs ik 2 n ni nk ik of systems X, X′ at hydrostatic external pressures P (Fig.1). For measuring the P “backgroundstrain”e(P), which is defined as the Lagrangianstraindisplayedby the system at the constraint of zero order parameter Q¯ (see below), X(P) denotes the corresponding state and X(P = 0) is used as a reference system. For measuring the spontaneous strain ǫ resulting from the emergence of Q¯ 6= 0, X(P) plays the role of a (hypothetical) “floating” background reference system. On the other hand, abovethecriticalpressurbePc thetotalstrainη(P),whichisobtainedasthenonlinear superposition η = e+α+.ǫ.α. of the spontaneous strain ǫ appearing “on top of” the backgroundstraine(P), is the strainmeasuredin the fully deformedstate Y(P), which is the one actuallydisbplayedby the system, the systembX(0)againplayingthe role of the reference system. In the spirit of Landau’s theory of phase transitions the Landau free energy F(Q,ǫ) is built from an order parameter part F (Q), an order parameter-strain Q couplingFQǫ(Q,ǫ;X(P))andapurelyelasticfreeenergytermFǫ(ǫ;X(P))(Toledano and Toledano, 1988). The order parameter part F (Q) is usually written as a Q polynomial in the variables Q which is constructed using the theory of invariants of irreducible representations of the space groups (Kovalev, 1993). The construction of the couplingtermFQǫ(Q,ǫ;X(P))is moretricky. Measuringthestrainswithrespect to the floating background reference system X(P) and assuming linear-quadratic couplingofstrainandorderparameterwithcouplingcoefficientsdIJ(X(P)),wewrite: ij Finite strain Landau theory of high pressure phase transformations 4 e ) ij s t ni X u . b r a ( V Y h e ij ij X(0) X(P) Y(P) h ij 0 P (arb.units) Figure 1. Sketch ofdeformationstates andsuperpositionofstrains. F(Q,ǫ;X(P))=V(P)Φ(Q;X(P))+V(P) dIJ(X(P))Q Q ǫ ij I J ij IJij X b +F0(ǫ;X(P)) b(1) HPPT’sarecharacterizedbythefactthatthetotalstrainηisnotsmall. However, b for phase transitionscloseto secondorderthere is atleasta nonzeropressureinterval starting at P where the spontaneous strain ǫ induced by the nonzero equilibrium c order parameter Q¯ can still be treated as infinitesimal. In this approximation the pure elastic free energy part of the Landau potbential with respect to X(P) reads: 1 F0(ǫ;X(P))≈V(P) τijǫij Cijkl(X(P))ǫijǫkl (2)  2  ij ijkl X X where C (X(bP)) arethe crystal’sthermbodynamic elasticconbstbantsatpressure ijkl P. Inserting Eq.2 into Eq.1 and minimizing with respect to ǫ yields: ij dKL(X(P))Q¯ Q¯ + C (X(P))ǫ b(Q¯)≡0 (3) ij K L ijkl kl KL kl X X Solving Eq.3 for ǫ (P) and inserting into the equilibrbium equation for the order ij parameter b Finite strain Landau theory of high pressure phase transformations 5 ∂Φ(Q¯;X(P)) 0≡ +2 dKL(X(P))Q¯ ǫ (Q¯) (4) ∂Q¯ ij L ij K L X one obtains the renormalized order parameter potentbial written with respect to the deformed state X(P): Φ (Q;X(P)):=Φ(Q;X(P))− R − 1 Q Q Q Q dIJ(X(P))C−1 (P)dKL(X(P)) (5) 2 I J K L ij ijkl kl  IJKL ijkl X X   In the spirit of Landau theory we now make the simple but crucial assumption that the Landau and coupling coefficients of the potential depend on P only in a “geometrical” way, i.e. through volume ratios and geometrical transformation rules between the reference states X(P) and X(0), such that the P dependence of the equlibrium order pararameter is due to the presence of the order parameter-strain coupling. TransformingEq.5tothe undeformedreferencestateX(0),the equilibrium order parameter Q¯ can then be calculated as the minimum of Φ (Q;X(0))=Φ(Q;X(0))+ dIJ(X(0))e (P)Q Q − R ij ij I J IJij X T (P) − dIJ(X(0)) ijkl dKL(X(0))Q Q Q Q (6) ij 2 kl I J K L IJKL Xijkl where V(0) T (X(P))≡ α α C−1 (X(P))α α (7a) rost V(P) ir jo ijkl ks lt ijkl X V(0) dIJ (X(P)) ≡ α dIJ(X(0))α (7b) mn V(P) mi ij nj ij X and the total (Lagrange) strain η turns out to be ij η =e − Q¯ Q¯ dKL(X(0))T (P) (8) ij ij K L kl ijkl KL kl X X Applying the above procedure and employing the usual assumptions concerning the temperature behaviour of the Landau parameters, one is able to calculate pressure and temperature dependencies of thermodynamic quantities like specific heat, strains and elastic constants, soft mode behaviour, etc. near HPPTs. A concrete example of the above theory at work will be given below. However, the above constructions obviously rely on the knowledge of the pressure dependence of the elastic constants C (X(P)) ≡ C (P) (or equivalently, the Birch coefficients ijkl ijkl B (P)=P(δ δ −δ δ −δ δ )+C (P))andthe resultingbackgroundstrains ijkl ij kl il jk ik jl ijkl e (P). Notethatthese quantities appearalsointhe renormalizedfreeenergydensity ij Eq.6. To calculate it we proceed in the following way. Finite strain Landau theory of high pressure phase transformations 6 Forcrystalclassesthatdevelopnoshearstrainsunderhydrostaticpressure(cubic, tetragonal, hexagonal, orthorhombic) the deformation tensor α (P) = α (P)δ is ij i ij diagonal and the axial compressibilities satisfy the equations 1 dα (P) − i = S (P)=:κ (P) (i=1,2,3) (9) ik i α (P) dP i k X with the initial conditions α (0) = 1, where S (P) := B−1(P) denote the i ij ij compliance tensor elements and we switched to Voigt notation. Therefore the deformation tensor components α (P) as well as the finite strain tensor components i ǫ (P) or lattice parameters a (P) can be easily calulated by integrating Eq.9, once ij i the functions S (P), i,k =1,2,3 are known. ik In our recent work (Tröster, Schranz and Miletich, 2002) we proposed the expansion ∞ κ(P) S (P)= S0 + κnPn (10) ij κ0 ij ij ! n=1 X where κ(P) is the bulk compressibility, κ0 =κ(P =0) and Si0j denotes the zero- pressure compliance. The expansion coefficients κn obey the sum rule κn = ij ij ij 0 ∀ n ∈ N. Eq.10, which is exact at infinite n, yields an elegant parametrization P of the pressure dependence of the compliance tensor, since it essentially factorizes out the main nonlinearity of the compliance through the bulk compressibility κ(P). The elastic anisotropyis taken into accountby expanding the remainder in powers of P, as we anticipate that the very nature of the expansion allows it to be truncated at low orders with good accuracy. Indeed we have shown in a very recent work (Koppensteiner, Tröster and Schranz, 2006) that even a low order truncation of our ansatz Eq.10 excellently fits the experimental high pressure data of olivine, fluorite, garnet, magnesium oxide and stishovite. In fact, for all these examples a truncation at order n = 1 turned out to be sufficient to produce accurate results, the only exception being magnesium oxide, where the expansion Eq.10 had to be taken to n = 2. This very encouraging result implies that our new thermodynamic theory indeedallowsforadramaticreductionofthenumberoffitparametersascomparedto conventional nonlinear approaches. Table1 shows the numbers of relevant (nonshear) elasticconstantsofsecond,thirdandfourthorderforvariouspointgroupsymmetries. Ourapproachrequiresonlyq2−1additionalunknownparametersκ1ij (atordern=1), whereasafitofanexpansionoftheelasticenergyincludingnonlinearelasticconstants up to fourth order introduces q3+q4 additional fit parameters. For example, in the case of cubic I symmetry, our new approach requires q2 −1 = 1 fit parameters as comparedtoq2+q3 =7unknownparametersofaconventionalfourthordernonlinear theory. For lower symmetry the advantages of our approach are even more obvious. For instance, in the case of orthorhombic symmetry our new parametrization needs 5 parameters, which should be compared to a total of 25 unknowns for conventional nonlinear elasticity! The combination of high pressure Landau theory and the use of the expansion (10) for the background system X(P) is illustrated by an analysis of measurements oftetragonalBaCr(Si4O10) single crystals. Using x-raydiffraction,the P-dependence of lattice parameters a1(P) = a2(P), a3(P), and the unit cell volumes V(P) were Finite strain Landau theory of high pressure phase transformations 7 Table 1. Numbers q2.q3,q4 of nonshear elastic constants of second, third and fourthorder(RoyandDasgupta,1988; Brendel,1979). cubic I cubic II hexagonal I hexagonal II tetragonal I tetragonal II Orthorhombic (432,¯43m (23,m3) (622,6mm (6,¯6 (422,4mm (4,¯4 (222,mm2 m3m) ¯6m2,6/mmm) 6/m) ¯42m,4/mmm) 4/m) mmm) q2 2 2 4 4 4 4 6 q3 3 4 6 6 6 7 10 q4 4 5 13 13 9 9 15 16.0 ) Å ( 15.5 3 a 15.0 920 900 880 860 7.5 3V(Å) 840 Å) 820 ( 800 a1 7.4 780 760 0 2 4 6 8 10 P(GPa) 7.3 0 2 4 6 8 10 P(GPa) Figure 2. Pressure dependence of lattice parameters and unit cell volume of BaCr(Si4O10). Pointsaremeasureddata,linesarefitsusingthepresenttheory. measuredverydetailedatroomtemperatureinadiamondanvilcell(Tröster,Schranz and Miletich, 2002). One finds a tetragonal-tetragonal HPPT at approximately 2.24 GPa, characterized by a discontinuitiy in a1(P), a3(P) and V(P). Consistent with the observed pressure hysteresis behavior, the transition can be classified as being of (weakly) first order. The order parameter part is constructed in the following standard way (Ko- Finite strain Landau theory of high pressure phase transformations 8 Table 2. FitparametersforthehighpressureLandautheoryofBaCr(Si4O10). BaCr(Si4O10) A 0.45GPa B -0.2GPa C 20GPa a1(0)=a2(0) 7.535˚A a3(0) 16.09˚A κ01 0.0035GPa−1 κ03 0.001GPa−1 S101 0.0035GPa−1 S30 0.0089GPa−1 K0′ 4.1 κ111 1.9×10−4 GPa−2 κ112 -1.7×10−4 GPa−2 κ113 2.3×10−4 GPa−2 d1 13.75GPa−1 d3 -0.33GPa−1 valev, 1993): The symmetry reduction P4/ncc to P4212 is driven by the onedimen- sionalirreduciblerepresentationτ2 atthewavevectork=0,yieldingaonecomponent order parameter Q, which is zero in the paraphase (P < P ) and nonzero in the dis- c torted phase (P >P ). For Φ(Q,X) we assume c A B C Φ(Q,X)= Q2+ Q4+ Q6 (11) 2 4 6 where A,C > 0. The tetragonal symmetry also dictates d1(X) = d2(X) 6= d3(X). LetK0 :=κ−1(0)denotetheisothermalbulkmodulusatP =0. TheMurnaghan equation of state (MEOS) (Anderson, 1995) v(P):= V(P) =(1+K0′P/K0)−1/K0′ (12) V(0) which is based on the simple ansatz κ−1(P)=:K(P)=K0+K0′P, is frequently used to describe (P,V)-data and is known to usually reproduce the values of K(P) correctly up to volume changes somewhat larger than v(P) > 0.9 while being algebraically much simpler than other approaches such as the ”Vinet” or the ”Birch- Murnaghan” EsOS (Anderson, 1995; Angel, 2000b) used for higher compression ranges. Fig.2 shows corresponding fits of unit cell volume and axes of BaCr(Si4O10) using the parametervalues ofTable2. Note, however,thatourtheory canin principle be usedincombinationwithanyoftheseEsOS,orevenafunctionV(P)derivedfrom anexperimentalmeasurementoracomputersimulation. Withthesevaluesourmodel confines possible pressure ranges for hysteresis effects to 2.2 GPa–2.4 GPa. One also calculates that the geometricalerrorintroducedin assuming the spontaneous strainǫ to be infinitesimal is smaller than 0.9%, yielding an error < 0.1% in the total strain η. b Figure 3 shows the pressure dependence of the bulk modulus of BaCr(Si4O10) calculated from the experimental data of V(P) as compared to the predictions of Finite strain Landau theory of high pressure phase transformations 9 110 100 BaCrSi O 4 10 90 80 ) 70 a P G 60 ( K 50 40 30 present theory Murnaghan EOS 20 10 0 0 2 4 6 8 10 P (GPa) Figure 3. Pressure dependence of bulk modulus K(P) calculated from the experimental data of V(P) (points). The lines show calculations based on piecewise EOS fitting (thin line) and the present high pressure Landau theory (thickline). the different approaches discussed above. It is clearly evident that the conventional fitting procedure, which is based on a piecewise fitting of EOS in the high and low pressurephase,respectively,underestimatestheelasticanomalybyatleastafactorof two, whereas the presenthigh pressureadapted Landau theory reproduces the elastic anomaly very well. The same behaviour is found for the axial imcompressibilities (Fig.4). It is worth noting that a very similar softening was also observed in other examples ofpressure-inducedphase transitions, notably for the bulk modulus of solid C60 near its fcc-sc transition (Pintschovius, 1999) and for the longitudinal acoustic modes near the cubic-tetragonal transition of BaTiO3 (Ishidate, 1989). 3. Summary Summarizing, our theory allows to compute a wealth of experimental observables (e.g. lattice parameters, elastic constants, specific heat, soft modes, etc.) from a quite transparent thermodynamic model based on coupling finite strain elasticity to Landautheory. ThereareseveraladvantagesofthisnewapproachtoHPPTs: Atfirst there is a drastic reduction of fit parameters as compared to an expansion in terms of nonlinear elastic constants (Table1). Moreover the present approach allows for a direct connection to EOS fitting procedures. Compared to the method of piecwise EOSfittingthemainadvantageofourmethodisthatitallowsforaunifiedconsistent Finite strain Landau theory of high pressure phase transformations 10 BaCrSi O 4 10 300 ) a K P a G ( 200 uli high d low symmetry o symmetry m c K 100 sti c a el K 0 0 2 P 4 6 8 10 c Pressure (GPa) Figure 4. Pressure dependence of axial incompressibilites Ka,Kc calculated from the lattice parameters a1,a3 (points) and the bulk modulus K compared withtheresultsofthepresentLandautheory(lines). description of high and low symmetry phases with a single set of parameters. Atpresent,ourtheoryiscapableofdealingwithgroup-subgrouptransitionsinvolving symmetries of cubic, tetragonal, orthorhombic or hexagonal type. However, it is possible to generalize our approach to arbitrary symmetries once the pressure dependence of the shear elastic constants (in addition to that of the longitudinal ones) is known. Further work in this direction is in progress. Acknowledgments Support by the Austrian FWF (P19284-N20) and the University of Vienna (Initiativkolleg IK 1022-N) is gratefully acknowledged. References Anderson,O.L.(1995).Equations of State of Solids for Geophysics and Ceramic Sciences, Oxford UniversityPress,Oxford. Angel,R.J.(2000a).High-Pressure Structural Phase Transitions, in ”Transformation Processes in Minerals”, Reviews in Mineralogy and Geochemistry, Vol. 39, p. 85, ed. S.A.T. Redfern and M.A.Carpenter. Angel,R.J.(2000b).Equationsofstate,InR.M.Hazen(ed),High-Temperature-High-PressureCrystal Chemistry,ReviewsinMineralogy40. Bergman,D.J.andHalperin,B.I.,(1976).Phys.Rev.B13,2145-75.