AB INITIO THERMODYNAMIC AND STRUCTURAL STUDIES OFCATIONIC DISORDERED MgAl204 SPINEL

S.Da Rocha1,2 and P.Thibaud eau2

'Lab oraroir e d'Electrod ynamiqu e des Mat eriaux AvancesUMR 6157 CNRS CEAFaculte des Sciences et TechniquesUniversite Francois Rab elaisPare de Grandmont, F-37000 Tours, France2Commissariat a I'Energie AtomiqueLe Rip ault , BP 16, F-37260 Monts, France

INTRODUCTION

Spinels belong to the ceramic oxide family and have a wide range of applicationsin geophysics [16], magnetism [26] and irradiat ed environments [44, 45]. The gen­era l formula of a 2-3 spinel is AB204, where A is a divalent cation (such as Mg2+,Zn2+, Cd2+, etc ) and B a tr ivalent cat ion (such as AI3+, Ga3+, In3+, etc). Amongthe spinels fam ily, MgAl20 4 is often considered as a model of spinel structure wherethe oxygen atoms form a close-packed pseudo face cente red-cubic sublattice. Amongall the 96 possible interst ices of the anions lat ti ce, 64 are tet ra hedra l (IV) and 32 areoctahedra l (VI). In a so-called normal arrangement , one over 8 tetrahedra l sites is occu­pied by divalent cations, whereas tri valent cat ions occupy half of octahedral sites [30].Barth and Posnjak [3] have outl ined that the normal arrangement cannot reproducethe intensity of X-ray diffract ion pattern of some spinels. So, they have suggested anarrangement where equivalent positi ons are occupied by different atoms according the

following genera l formula [BIV(AB)VI 04 ]' However , it is now well known that spinels

can accommoda te cationic disorder [27, 11]. An inversion parameter, usually lab elledx, has been introduced to quantify the numb er of t rivalent B ions (here AI3+) locat edin tetrahedral int ersti ces.

The MgAl204 spinel can then be described by the following formula

(Mg(l_xlAlx)IV [MgxAl(2_x)tI

0 4,

21

where x ranges from °(normal spinel), to 1 (fully inverse spinel) . It has been shownthat cat ionic disorder is act ivated by either pressure [13] or part icle irra diat ions suchas neutrons, elect rons or Ne+ ions [29, 31]. However, the usual and probably easiestway to act ivate disorder by cat ionic exchange in both natural or synthetic spinel, is toheat the st udied specimen [7]. Successive heat pulses can also be app lied on samplesto synthesize directly disordered spinels [28] .

It is now commonly admit ted that most of spinels belong to Fd3m (227) space group .However , experimental observations report ed by many groups [8, 9, 14, 28, 35] suggest alower symm et ry space group for some of these st ructures. In particular , Sclunocker andWaldner [28] have out lined that in an inverse spinel, domains of reduced symmet ry arerespons ible for observed extra Bragg reflections. Thi s suggests a possible connect ionbetween the inversion parameter x and the space group of the st ructure, whereas Haashas claimed that cat ion disorder does not induce natural change in st ructure symmet ry[10].

Up to now, most of the reported experimental studies on spinels deal with th edeterminati on of x with respect to temperat ure. Elect ron spin resonance (ESR) on anatural MgAI204 spinel for temperat ures lower than 1278 K [27] and high-resolution27Al nuclear magnetic resonance (NMR) on quenched natural or synthet ic samples[41, 19, 17] have been int ensively developed to investigate such behavior. Several othertechniques have been reported such as Raman spectroscopy [4], neutron diffraction [23,25] which allow in situ measurements and X-ray diffraction [42, 2]. Because of variousdifferent experimenta l techniques and samples preparations, discrepancies appear inthe reported data.

To complete st ructura l analysis and avoid experimental difficulties, some theoret icalinvest igat ions have been carried out to st udy disorder effect on spinel stability. Wei andZhang [40] have selected 18 members among 2-3 and 4-2 spinels to evaluate the mostst ab le form between normal and inverse configura tions. They have also calculated at abinitio level, st ructural parameters such as the lattice constant (a) , the average internaloxygen posit ion (u) and band gaps in bot h configurations. Most of these spinels arefound to be more stable in normal structure except for MgGa20 4, MgIn204 in the2-3 family and GeMg204 and all Si spinels in the 4-2 group. Moreover , band gaps arefound to be smaller in the inverse configurat ion. To explore cationic disorder cont inuousvariat ions, Warren et al. [37, 38] have invest igated disordered spinel thermodynamicpropert ies with tempe rature. In their approach, short ranged one-site cluster potent ialshave been parameterized from a small numb er of disordered configurat ions within theDensity Funct ional Th eory (DFT ) in both Local Density (LDA) and General GradientApproximati on (GGA). Th ese potentials have been applied in Monte Carlo simulat ionsto predict disordering thermodynamics state variables.

Alternatively, the evaluat ion of volume thermal expansion and specific heat ofalumina-magnesium systems with temperat ure have been evalua ted using both molec­ular dynamics and experimentally derived interat omic potent ials [20]. In spite of awell reproduced heat capac ity curve, the predicted thermal expansion appears to besignificantly lower at high temperat ure than experimentally observed.

It is import ant to observe that most of experimenta l and theoretical investigationsare concentrated on spinels cations behavior with temperature. But to our knowledge,

22

it seems that no complete study has been previously carried out to evaluate th e varia­tion of both struct ura l and thermodynamical spinel properties with cationic disorder.

In th is study, ab ini tio calculations have been performed to evalua te firstly, the in­fluence of cationic disorder on internal energy variat ion (U) and excess of heat capacityevolution and secondly, the relative density behavior. The cationic disorder parameterversus tempera ture curve is deduced as a by product , according to an effective ther­modynamic model using both a regular solution and a quadr atic form of the internalenergy [22]. Results are compared to available experimental dat a.

CALCULATION METHODOLOGY

To approximate the cont inuous variat ion of x , a 56 atoms supercell is generated.Among all the cations , N Mg and Al ions are randomly exchanged, with N rangingfrom 0 to 8, so the inversion parameter x, is defined as Jif . Because it is computert ime demanding, only 5 different supercells for each cationic arr angement are selectedto average cell energy and structura l behavior of the disordered cryst al.

Simulati ons are performed using th e ABINIT [1] code within Density FunctionalTheory (DFT ) and Local Density Approxim ation (LDA) . The valence electrons aredescribed by pseudopotent ials developed on a plane waves basis set. The genera t ion ofth e cat ions pseudopotenti als follows th e scheme prop osed by Haman [12], whereas theoxygen follows the Troullier and Martins scheme [36]. The particular choice of thesepseudopot enti al schemes is det ailed elsewhere [34]. During calculat ions, symmetriesare turned off to impose P I space group .

In a previous study, Warr en et al. [37] have selected a finite 2x2 x2 Monkhorst-Packset of irreducible k-points for all their configurat ions. However , because the variat ionof geomet ry indu ced by disorder has an import ant influence on th e genera t ion of anoptimum k-points grid, different configurat ions must be tr eat ed with the same accuracy.Thus, only the first Brillouin zone cent re point was considered in this study to integrateth e DFT derived prop erti es.

For each disorder rate x and each configurat ion, cell vector s and atomic positions arerelaxed after severa l self-consistent cycles. The simulation stops when all the forces andresulting stresses are close to a desired precision. As DFT calculat ions are performed atOK, an effective thermodynamic model , firstly introduced by Necl [21] and popu larizedby O'Neill and Navrotsky [22] is used to determine the equilibrium inversion parameterfor a given temperature. The main quantity in this model is th e Gibbs free entha lpyexcess induced by disorder (boG) . boG usually includes a quadr ati c internal energy(boU ), a well defined form of configurat ion entropy (boSe), and a work of externalpressure via PboV product are found . As calculat ions have been carried out withoutexternal applied pressure, no ext ra work induced by pressure is necessary. Then, th eGibbs free enth alpy per molecule can be expressed by equa tion 1.

with

boU

boG = boU - T boSe (1)

(2)

23

where ke is the Boltzmann constant. The constants a and (3 are deduced from ab initiocalculat ions and the equilibrium condition &~xG = 0 is numerically solved to determinethe value of the mean equilibrium cat ionic distribution for a given temperature. Fur­thermore, using the derived equilibrium curve into the configurat ional entropy, the heatcapacity excess induced by disorder is given as

(4)

RESULTS AND DI SCUSSION

Variation of internal Energy

The averaged energy differences ,6,U(x ) = U(x '" 0) - U(x = 0), expressed ineV/ molecule, are calculated as shown in Figure 1 and compared with result s fromWarr en et al. [37]. Discrepan cies between the two calculations could have their originin different spannings of the configurational space. Our calculated mean-squared erroron ,6,U clearly shows tha t a single calculation per inversion parameter is not probablysufficient to take into account the configurational space variety. As the internal energydifferences are small, the convergence of forces on atoms and t he resulting stress onthe cell have been carefully monitored during calculat ions.

• Averaged internal energyvariation evaluated in DFT-LOA (this study)- - Quadratic fit of the averaged b,U ab initiocalculated (this study)

0040 x Internal energy variation evaluatedfromWarren et at. within DFT-LOA

x

~O.30

~;5 0.20<]

0.10

0.4 0.6 0.8

x

1.0

Figure 1: Averaged internal energy variat ion with cat ionic disorder expressed ineV/ molecule calculated with in LDA (0) and compared to LDA Warren et al. results(x) [37].

24

A quadr ati c fit of fl.U(x) provides both averaged values a and (3 and mean-squarederrors fl.a and fI.(3 . These values are compared with available experimental dat a asshown in Ta ble 1. In their analysis of cat ionic distribution , O'Neill and Navrotsky[22] have shown that a and (3 parameters should be approximately of equal magnitudeand opposite in sign. The main suggested reason of this sign difference was inducedby th e ionic character of th e bindings in spinels. As shown by Thibaudeau et at.[34] in the normal spinel, the Born effect ive charges on atoms are very close to thefull ionic charges. So, at least for low disorder rate, there is no obvious reason th atthe ionic char acter is deeply altere d with in the exchange of ions. As shown in Table1, this st udy and some of the reported dat a [37, 17,23] fulfill the opposite sign criterion.

D er ivat ion of equilibrium curve and hea t capacity

Reference a fl.a (3 fI.(3 MethodTh eory Thi s study 0.48 0.08 -0 .25 0.06 DFT-LDA

Warr en et at. [37] 0.60 -0.22 DFT-LDAMillard et at. [19] 0.26 0.06 0.06 0.10 RMN 27AlMaekawa et at. [17] 0.36 0.06 -0 .33 0.06 RMN 27Al

Exp . Peterson et at. [23] 0.32 0.01 -0.10 0.0 3 neutron diffractionRedfern et at. [25] 0.34 0.01 0.05 0.02 neutron diffractionAndreozzi et at. [2] 0.24 0.02 0.14 0.05 X-ray diffraction

Table 1: Comparison of experiment al and ab initio calculated a and (3 parameters andmean-squared erro rs .

Variati ons of th e disorder rate with tempera ture are deduced from the equilibriumcondition on th e Gibbs free entha lpy. The derived curve is compared to most of avail­able experimental dat a as shown in Figure 2. It seems th at the mean-squared errorsinduced by fl.a and fI.(3 on cationic disorder curve was never rep ort ed previously. How­ever Figure 2. shows that significant differences between internal energy parametersare not sufficient to predict accurate variations of x versus temperature. This curvealso shows that the uncert ainties fl.a and fI.(3 can give an insight into conclusions be­tween different experimental and theoretic al works. Because the experimental exte ntof tempera tures is limited , an accurate determination of the internal energy variationappears to be more difficult for the quadratic coefficient (3 than for a.

In spinels, several thermodynamic observables exhibit rapid variat ions with tem­perature which suggest possible second-order phase tr ansit ion. A discontinuity in celledge and thermal expansion coefficient has been observed by several groups [42, 33],but Kashii et at. [15] using NMR technique have reported, on the basis of kinetic con­siderations , that the cat ionic equilibrium disorder reaction is a first-order transition. Inthis st udy, the second derivati ve calculation of the free energy fl.G with respect to thetemperature was performed. Once the equilibrium variat ion of x versus tempera tureis known, the determination of configurational entro py and its first derivative dropnaturally according to Eq.(4). As shown in Figure 3, fl.Cp exhibits a peak which ischarac terist ic of a continuous phase transit ion. The harm onic par t of heat capacity at

25

0.6

0 5

0.4

0.3

0.2

0.1

Evolution of disorder rate with temperature- simulated with DFf-LDA (this study)

Andreozzi et al.Millard et al.Peterson et al.

x Redfernet al.Wood et al.Yamanakaet at.

T

(5)

T(K)

Figure 2: Evoluti on of disorder rate x obtained by resolution of equa tion 1 compared toneutron scat tering experiments [25](x), [23](+ ), X-ray measurements [42](6 ), [2](.),and 27Al NMR experiments [19](0 ), [41](*).

constant pressure is the main contribution to the tot al capacity in spinels. However,the increase at maximum of heat capacity induced by disorder is about 15 J/ mol.Kand should be experimentally observed.

This maximum value suggest s a definition of a crit ical temp erature for this sys­tem. However , an another crit ical temperature is usually introduced to describe order­disorder transitions [24]. It s determination is st rongly connected to th e Bragg-Williamsmodel. In this model, the solut ion of

(0;1:>; ) = 0,Q Q=O.T=T,

where Q is an order parameter, gives th e required temp erature Te . In the case of2-3 spinels, it is possible to define an order parameter Q = 1 - 1x connected tothe cat ionic occupation, which varies from 1 (maximum order ) to 0 (random disor­der ). Solving equa t ion 5 with the Gibbs free energy form int roduced previously, givesT; = -~ where kB stands for the Boltzmann constant . Using our derived ab 'initiovalues, T; :::::: 860/(. This temperature is in fair agreement with the previously reportedexperimental values of 870 K < t: :::;970 K [42], r; :::::: 950 K [39], t:» 930 K [33]. AsT; is positive defined , the sign of (3 is expected to be negati ve or zero.

Density with disorder behavior

Discrepancies have been report ed on the density behavior with the cat ionic disorder.Indeed, it is difficult to determine th e volume variat ion induced only by disorder onquenched samples. Wood et al. [41] have carried out 27Al NMR experiments on

26

18

Variation of heat capacity due to disorder calc ulated- in DFf-LDA (this study)

15

_- 12'",

'0E 9

'0:u<l

6

00 250 500 750 1000 1250 1500 1750 2000 2250 2500T (K )

Figur e 3: heat capac ity indu ced by disorder curve as a funct ion of temperature

quenched synthetic spinels and found a cell volume increase of 0.03 %for an increase ofx from 0.21 to 0.39. An X-ray st udy performed on a quenched single-crystal has showna decrease of cell para meters when the disorder increases [42]. Using a descriptio nbased on ionic radii, O'Neill and Navrotsky [22] have predicted a decrease of volumewith disord er for all the 2-3 spinels with cation s of similar size to Mg2+ and Fe3+.

Ab initio calculations performed in thi s study valida te density increases with cationicexchange. T hey are compared with recent X-ray measurements on quenched single-­crystal [2] on Figure 4. Conclusive agreement is found for the range of temperatureexperimentally observed .

Infrared vibrational spectra of disordered spinel

Maradudin and coworkers [18] have demonst rated that the macroscopic low-frequen­cy static dielectric permi t t ivity tensor Eij(W ) which gives the infrared spectra maincontribution , is a sum of both an ionic par t and a limit value of a pure electro niccont ribut ion. According to Gonze and Lee [6], one has

where 1 ::::: i, j ::::: 3 and 0 0 is the cell volume, ZK ,ii' are the Born effective tensors,E'0 is the electronic cont ributio n to the dielectric permi ttivity te nsor. T his tensor isgiven by the knowledge of the mixed derivat ive of the total electronic energy over th emacroscopic elect ric field. Here w~ represent the dynamical matri x eigenvalues and

27

Relative density simulated in DFf· LDA (this study)y X-ray measurementrealized by Andreozzi et al. [26)

1.02

06 0.8

1.0 15

ofa.

1.01,,i,,

1.005 1T..~.10 0.2 0.4

T,

j.c

T

i.c

i

Figure 4: Density evolut ion with x ab initio calculate d (0), compared with X-raymeasur ements (6) [2]

Um q the corresponding eigenvectors with the following normalization

(7)

where M k are the ato mic masses.Powerful techniques have been developed a few years ago to calculate vibrat ional

properties of exte nded systems through density functional per turbation theory (DFPT).Thi s technique has been applied successfully to evaluate all vibrational modes at th ezone centre for normal cubic MgAl204 spinel [34]. The Born effect ive charges havealso been evaluated within DFPT and compared to a set of new exper iments on syn­th et ic spinels [34]. The agreement was found sat isfactory , demonstr atin g that the ioniccharacter was preserved in thi s spinel even on synthet ic and probably small disorderedcompound.

Many groups have repor ted extra infrared modes for synthet ic MgAh04 spinels[43, 34] and highly non-symmetric peaks in the Raman spectra for quenched samples[5]. The precise origin of these ext ra modes is unclear [43] but ext ra Ram an mode near727 em" ! was clear ly assigned to th e symmetric AI-O stretching vibration of Al04groups created by the redistribution of some aluminum ions from oct ahedral to tetra­hedral sites . However , to explore the possibility that infrared modes originate fromcat ionic disorder , dynamical matrix eigenvectors and eigenvalues have been calculatedwithin DFPT for several inversions parameters and averaged on different configurations.The imaginary parts of the low-frequency dielectri c tens or components are reported inFigure 5. Lifetime is requir ed for each mode such as its corresponding damping overits frequency is a constant for a given temperature [32]. As room tempera ture is highenough to neglect anharmonic residual quantum effect for low temperatures , assum-

28

50

\122

x=O.l25 3340 \ 2

1323. expoPalik

30

w

:§

20

10

0100 200 800 900

Wavenumber (em -1)

50

x=O.250

40

30

j20

10

0100

Wavenumber (em 0

1)

700

II2233121323

800

Figure 5: Evolut ion of infrared spectra for different disorder rate

29

ing dampings linear behavior wit h temperature is a natural approximat ion. Assumingconstant int eracti ons of phonons is a crude approximation. However an examination ofboth LO and TO dampings and frequencies measured for two sets of synthet ic spinelsprovides a linear regression coefficient of 0.8. This value is high enough to consider theprop ortion between dampings and frequencies constant . The corres pon ding value ofth is const ant was fitted using previously measured dampings and frequencies [34] . Forsimplicity, we assume that anharmonicity is sufficient ly small for low inversion parame­ter such as this constant rema ins unchanged. Using these approximatio ns, the inte nsityof all the T 1u norm al spinel infrare d modes decreases when the inversion parameter in­creases as shown in Figur e 5. The resultin g widths become broader and severa l modesnear 800 cm- 1 and 180 ern" ! become intense. These extra modes have been previouslyobserved in different synt het ic spinels and then could have their origin in the defectivenature of t he spinel lat tice.

CONCLUSION

Structural and thermody namic properties of disordered MgAl204 spinel are calcu­lated by DF T with plane-wave basis set and pseudopotentials. Several configurationsare considered to average each property to take account of configurational variety.The evolution of int ern al energy is calculated as a funct ion of cationic disorder rat e,variat ions of disorder rate with tempera ture are determined using an effective th ermo­dynamic model. By taking into account the mean-squared errors obtained on a and (3parameters, an original approach of the cat ionic equilibrium curve is introduced. Fur­thermore , the excess of heat capacity shows a typical behavior which is the sign of acontinuous transition and a critical temperature is also evalua ted for a perfect randomcrystal. In additio n, it has been shown that th e density of the ma teri al increases withincreas ing disord er. Finally, infrared spect ra calculation of selecte d inversion param­eters have been carried out within DFPT and have shown extra modes which couldoriginate in thi s par ticular defective nature of the spinel lattice.

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