an ab initio study of the structure and properties of aluminum hydroxide: gibbsite and bayerite

An ab Initio Study of the Structure and Properties of Aluminum Hydroxide: Gibbsite andBayerite

Julian D. Gale,*,† Andrew L. Rohl, ‡ Victor Milman, § and Michele C. Warren|

Department of Chemistry, Imperial College of Science, Technology and Medicine, South Kensington,SW7 2AY, U.K., School of Applied Chemistry, Curtin UniVersity of Technology, P.O. Box U1987,Perth, Western Australia 6845, Molecular Simulations, Inc., The Quorum, Barnwell Road,Cambridge, CB5 8RE, U.K., and Department of Earth Sciences, UniVersity of Manchester, Oxford Road,Manchester, M13 9PL, U.K.

ReceiVed: May 10, 2001; In Final Form: June 28, 2001

The two most important polymorphs of aluminum hydroxide, namely gibbsite and bayerite, have been studiedfor the first time using solid state ab initio quantum mechanical methods, both using plane wave and localizedbasis set methodologies, within the framework of nonlocal density functional theory. The fully optimizedstructures have been determined for both phases, yielding improved hydrogen positions in the case of gibbsitefor which the only previous information is from X-ray data. Mechanical properties have been calculated forgibbsite, including the full elastic constants tensor and the bulk modulus. The latter is found to be 55 GPa,which is significantly lower than a recent experimental estimate. Vibrational spectra have been calculated forboth phases and assignments of the hydroxyl stretching modes are proposed.

Introduction

The study of gibbsite has significance from both a mineral-ogical and commercial point of view. Gibbsite is an importantore of aluminum and is one of three minerals that make upbauxite, the raw material for the Bayer process for the extractionof aluminum. Bauxite is composed of aluminum oxide andhydroxide minerals such as gibbsite, boehmite, AlO(OH) anddiaspore, AlO(OH), as well as clays, silt, and iron oxides, andhydroxides. Because the aluminum minerals are soluble in strongcaustic solutions, they are separated from the rest of the ore byaddition of sodium hydroxide. In the rate-limiting step of theBayer process, they are precipitated out from the caustic solutionas gibbsite. Aluminum hydroxide (Al(OH)3) has three otherpolymorphs; bayerite, nordstrandite, and doyleite,1-3 althoughbayerite is the only one with any industrial significance, sinceit is an undesirable product in various stages of the Bayerprocess. In addition, bayerite incorporates less sodium duringits precipitation than gibbsite and thus is used commercially toprepare aluminum oxide catalyst supports.4

All of the polymorphs of aluminum hydroxide are composedof layers of aluminum octahedra with hydroxyl groups on eitherside that hydrogen bond the layers together. The differencebetween the polymorphs is the stacking sequence of the layers.Gibbsite and bayerite represent the two extremes of the stackingsequence, with nordstrandite and doyleite being intermediatestructures. The structures of gibbsite and bayerite, both of whichadopt the space groupP21/n, are illustrated in Figure 1. If welabel the stacking of two oxygen layers that contain thealuminum ions as AB, the figure shows that the bayeritestructure consists of these double layers stacked directly on top

of each other resulting in an ABABABAB... structure. In theother extreme, gibbsite, each double layer is a reflection of theprevious one, resulting in an ABBAABBA... stacking motif.Also note that the figure clearly shows that only two-thirds ofthe octahedral sites are occupied by aluminum ions in eachdouble layer.

Although the basic structures of all the polymorphs areknown, there is still much to be characterized. From a structuralperspective, the most widely used crystal structure for gibbsiteis based on X-ray diffraction,5 thus leading to poor characteriza-tion of the hydroxyl groups. Beyond the structure, there is stillmuch to be determined concerning the physical properties andrelative stabilities of the polymorphs where there are either gapsin the experimental information or uncertainties concerning someof the values. Experimental determinations are, in general,hindered by difficulties in obtaining pure samples with largecrystal dimensions.

There have been a small number of previous theoreticalattempts to study gibbsite using both interatomic potentials6,7

and quantum mechanics.8 However, the quantum mechanicalwork to date has been based on the cluster approach which limits

* To whom correspondence should be addressed. Email: [email protected]: +44 207 594 5804.

† Department of Chemistry.‡ School of Applied Chemistry.§ Molecular Simulations, Inc..| Department of Earth Sciences.

Figure 1. The structure of the two main polymorphs of aluminumhydroxide, (a) bayerite and (b) gibbsite, illustrating the difference inthe stacking of the layers.

10236 J. Phys. Chem. B2001,105,10236-10242

10.1021/jp011795e CCC: $20.00 © 2001 American Chemical SocietyPublished on Web 09/28/2001

its ability to distinguish between different polymorphs ofaluminum hydroxide and to determine the structure in a mannerthat properly acccounts for the intrinsic periodicity.

The aim of the present work is to perform the first study, tothe best of our knowledge, of the energetics, structure, andproperties of gibbsite and bayerite, as the most significantpolymorphs of aluminum hydroxide, using ab initio methodswithin periodic boundary conditions.

Methods

In the present study, two different approaches have been usedfor the calculation of the properties of gibbsite and bayerite.Both are based on the use of density functional theory,9 whichrepresents, at the present time and for systems of this complex-ity, the only practical quantum mechanical method for studyingmaterials properties with the inclusion of electron correlation.All calculations have been performed using the same gradient-corrected functional, namely, that of Perdew, Burke, andErnzerhof (PBE).10

The two approaches taken differ only in the basis sets andalgorithms used in order to evaluate the electronic structure.On one hand, we have used plane waves to expand the wavefunctions in the solid state,11 whereas in the second case alocalized real space basis was employed. Plane waves have thedistinct advantage of being able to systematically converge thebasis set by purely increasing the kinetic energy cutoff, whilethe real space basis set has the advantage of being able to tacklelarger system sizes. Hence one of the aims of this study is tovalidate the real space approach for future studies of morecomplex systems, including surfaces and defects.

Plane wave calculations were performed using the programCASTEP12 with a kinetic energy cutoff of 340 eV. The nucleiand core electrons were represented by ultrasoft pseudopotentialsof the form described by Vanderbilt13 on the basis of valenceelectron configurations of 3s23p1, 2s22p6, and 1s1, for Al, O,and H, respectively. Pseudopotentials were generated based onall electron calculations utilizing both the LDA and PBEfunctionals for the exchange-correlation potential. The Brillouinzone was sampled using a Monkhorst-Pack grid14 with shrink-ing factors of 2× 4 × 2 for gibbsite and 4× 2 × 2 for bayerite.

As a complement to the plane wave calculations, we havealso utilized a methodology which is suitable for the study ofmuch larger systems, as embodied in the program SIESTA.15-17

In particular, the approach taken is designed to yield improvedscaling with system size which can be made to be linear, via asuitable choice of algorithm, when the prefactor makes thisfavorable.18 For the size of the unit cell being considered here,the use of linear scaling techniques for solving the SCFprocedure is not beneficial, relative to conventional diagonal-ization. However, for future studies on more complex environ-ments, this situation will be reversed.

In the approach taken within SIESTA, calculations areperformed using a localized basis set which consists of numericaltabulations of the exact solutions to the pseudo-atomic prob-lem,19 again with a pseudopotential representation of the coreelectrons and nucleus. Furthermore, the orbitals can be radiallyconfined in a balanced manner across all species by specifyingthe energy increase that is to be allowed. By adjusting thisenergy shift, an appropriate compromise between convergenceto the unconfined results and computational speed can beachieved. Here an energy shift of 0.005 Ry was found to be asuitable choice. This spatial confinement also leads to theimproved scaling with respect to system size. All atoms arerepresented by the Troullier-Martins norm-conserving pseudo-

potentials.20 The basis set was extended for the calculations tobe of the double-ú form, with a norm of 0.15 for the outerexponent, and polarization functions were utilized for all atoms,including hydrogens. For the evaluation of the Hartree andexchange-correlation potentials a real space mesh is used withan associated kinetic energy cutoff. For the calculations in thepresent work, a mesh cutoff of 250 Ry was found to offer ahigh level of convergence. Identical Monkhorst-Packk-pointgrids to those employed in the plane wave calculation were usedin the real space basis case to approximate the integration acrossthe Brillouin zone.

For the polymorphs of aluminum hydroxide considered here,the structures were fully relaxed, including the cell parameters,in the athermal limit. The gamma point phonons have beendetermined using SIESTA by central finite differencing of theanalytical first derivatives with a displacement of 0.04 Bohrabout the optimized configuration. Given the size of the unitcell for these materials, the phonons at the gamma point areexpected to be representative of the full spectra. Furthermore,the principal interest is in comparison of results with infraredand Raman spectra, which primarily explore lowQ values. Inthe future, the application of linear response techniques21 willmake it more facile to access phonons in other regions of theBrillouin zone.

In addition, the mechanical properties of gibbsite have beenpredicted with the plane wave methodology. Finite differencesare again used, in this case in the strain, and from calculationof the resulting stress, the full elastic constants tensor may bedetermined for the relaxed structure.22,23 Since this tensordescribes the full set of second derivatives of the energy withrespect to the strain, other properties such as the Young’s moduliand Poisson ratios can then easily be determined. Importantly,it is necessary to relax the internal degrees of freedom for eachpoint of the finite differences with respect to applied strain. Thisapproach has already been successfully applied to low symmetrysystems: the elastic constants of two orthorhombic polymorphswere accurately reproduced and the values for a third, triclinic,polymorph predicted ahead of experiment.24 For gibbsite, wehave used four different strain patterns, each with four ampli-tudes between-0.3 and+0.3%. For each strain pattern, thesymmetry andk-points were adjusted to reflect the reducedsymmetry of the distorted structure. For each of the stresscomponents generated as a result of each strain pattern, a least-squares fit of stress against strain yielded one of the elasticconstants. The bulk modulus of gibbsite is also available directlyfrom the elastic constants tensor. However, for comparison wehave also determined the value by fitting the pressure-volumecurve to a third-order Birch-Murnaghan equation of state overthe pressure range 0-10 GPa.

Prior to the present work, much of the data known concerningthe physical properties of gibbsite had only been determinedtheoretically, using interatomic potential models. However,given many of these were parametrized largely on the basis ofstructural data, rather than with the benefit of curvature relatedinformation as would normally be desirable, it is interesting toevaluate how such models fare against the ab initio data. Hencewe present comparisons for the mechanical properties asdetermined according to shell model interatomic potentialswithin the framework of an ionic description. Here all calcula-tions were performed using the program GULP25 whichdetermines such properties analytically. The particular param-eters used are those fitted by Fleming et al7 specifically foraluminum hydroxide.

Properties of Aluminum Hydroxide J. Phys. Chem. B, Vol. 105, No. 42, 200110237

Results

Bulk Structure. First of all, we consider the performance ofthe two density functional methodologies for the bulk structures,commencing with the plane wave technique, which will act asa reference due to the greater ease of systematic convergencewith respect to the basis set. For both gibbsite and bayerite,their structures have been determined by diffraction methods,including the positions of the hydrogen atoms. However, in thecase of gibbsite, the main structural determination was per-formed with X-rays5 and thus the O-H bond lengths aresystematically underestimated, while for bayerite a deutratedsample was used in neutron diffraction.26

Using the pseudopotential plane wave method we have fullyoptimized the structures of gibbsite and bayerite, the final unitcell dimensions being compared against experiment in Table1. Although we are principally concerned with the valuesdetermined using the GGA of Perdew, Burke and Ernzerhof, anumber of other results are also given for gibbsite. Includedare results for the cases where ultrasoft pseudopotentialsgenerated based on all electron LDA reference calculations areused in a bulk calculation with the LDA (LDA/LDA) and PBE(LDA/PBE) functionals. These results serve to illustrate theconsequences of utilizing pseudopotentials which are inconsis-tent with the final functional used for the solid-state calculation.

Considering first of all the results for gibbsite, we see theusual trend for the LDA/LDA calculation to underestimate theunit cell volume, in this case dramatically by 10%. Becausethe structure consists of layers in theabplane, the largest singleerror arises through an overcompression of thec axis byapproximately 0.5 Å. This is due to the tendency of LDA toyield hydrogen bonds, which determine the interlayer spacing,that are too short,27 though the intralayer cell parameters arealso contracted with respect to experiment. Performing acalculation using the PBE functional with LDA generatedpseudopotentials leads to a cell volume which is too small by3%, while the correct consistent PBE calculation gives anoverestimation by only 1%. This is broadly consistent with whatmight be expected based on previous calculations, in that GGAfunctionals tend to overestimate cell volumes by a few percent,though the error here is a little lower than typical. Often theLDA/GGA combination is found to give fortuitously goodresults, since the LDA pseudopotentials counteract the overes-timation of volume due to the GGA bulk calculation. However,for gibbsite this is not found to be the case. Bayerite, in contrast,demonstrates just under a 4% overestimation of the cell volumedespite showing much better reproduction of the monoclinicangle than for gibbsite. This larger error can be attributed tothe fact that in gibbsite the interlayerc axis is underestimated,which largely negates the fact that the unit cell is too large in

the ab plane for both polymorphs, while in bayerite alldimensions are overestimated.

Given that the hydrogen positions are poorly characterizedfor gibbsite, a further calculation has been performed in whichthe unit cell and heavy atom positions were held fixed at thosefrom Saalfeld and Wedde5 and just the hydrogen positions wereoptimized. The numbering of the hydrogen/deuterium atomswithin the structures of bayerite and gibbsite are illustrated inFigure 2. The calculated hydrogen fractional coordinates forgibbsite are given in Table 2 along with the hydroxyl bondlengths. As expected, the optimized hydrogen positions are muchmore consistent with typical hydroxyl bond lengths. Hence webelieve that these fractional coordinates represent a more reliableset than those from the original experimental data and wouldbe suitable for future structural refinement. In comparison,Kubicki and Apitz8 obtain O-H distances in the range 0.94-0.95 Å from their cluster calculation at the HF/3-21G(d,p) level.This systematic difference is to be expected given the tendencyof Hartree-Fock calculations to underestimate bond lengths,while nonlocal density functional theory tends to lead tooverestimation.

Having determined precise reference data for gibbsite andbayerite with converged plane wave calculations, the localizedbasis set approach was used to calculate structures for gibbsiteand bayerite. The results are given in Table 3 again comparedto experimental data. As can be seen, all individual cell lengthsare reproduced to better than 3%, while the volumes areoverestimated by 4%. Generally the real space approach showssimilar trends to the PBE/PBE plane wave calculation exceptthat the overestimation of the volume is more uniform andslightly larger. Perhaps the most significant discrepancy is themonoclinic angle for gibbsite which is too small, although theagreement for bayerite is much better. In terms of the hydroxylgroups, the localized basis approach yields O-H bond lengths

TABLE 1: Comparison of Experimental and CalculatedCell Parameters for Gibbsite and Bayerite According toPlane Wave Calculationsa

gibbsite bayerite

parameter exptl5 LDA/LDA LDA/PBE PBE/PBE exptl26 PBE/PBE

a(Å) 8.684 8.504 8.623 8.798 5.063 5.107b(Å) 5.078 4.867 5.017 5.110 8.672 8.852c(Å) 9.736 9.224 9.598 9.674 9.425 9.517â(°) 94.54 93.06 92.76 92.54 90.26 90.47vol (Å3) 428.0 381.2 414.7 434.4 413.8 430.2

a In the case of gibbsite, results are included for the cases LDA/LDA, LDA/PBE, and PBE/PBE, where the first functional was usedin the generation of the pseudopotentials and the second for the actualoptimization of the polymorph.

Figure 2. The labeling of the hydrogen/deuterium atoms within theunit cell of (a) bayerite and (b) gibbsite.

TABLE 2: Optimized Fractional Coordinates for theAsymmetric Unit Hydrogens in Gibbsite and TheirCorresponding O-H Bond Lengths (in Angstroms)

site x y z calcdr(O-H) exptl r(O-H)5

H1 0.0757 0.1393 0.8741 0.974 0.749H2 0.5729 0.5548 0.8977 0.981 0.773H3 0.4951 0.1105 0.7949 0.984 0.838H4 0.9497 0.8156 0.8866 0.981 0.875H5 0.2941 0.7181 0.7933 0.986 0.883H6 0.8058 0.1624 0.7970 0.988 0.861

10238 J. Phys. Chem. B, Vol. 105, No. 42, 2001 Gale et al.

that are systematically of the order of 0.01 Å longer than thosefrom the plane wave calculations, though this discrepancy couldbe reduced by adding further basis functions. For calibration,the present computational conditions in SIESTA yield an O-Hbond length of 0.972 Å (0.957 Å) and an angle of 104.3°(104.5°) for an isolated water molecule, where the experimentalvalues are given in parentheses.

Energetic Stability

Perhaps the principal question in this area concerns the orderof stability of the polymorphs. The general predominance ofgibbsite in nature suggests that this may be the more stable form.Furthermore, bayerite slowly transforms into gibbsite on stand-ing in contact with the appropriate solution at 300 K.28

Experimental calorimetric measurement of the difference in theheats of formation at 298 K also suggests that gibbsite is morestable by approximately 5 kJ/mol.4

In our calculations gibbsite is found to be lower in energyby 7.7 kJ/mol than bayerite at 0 K based on the plane wavecalculations. A similar difference of 6.3 kJ/mol is obtained fromthe localized basis set approach, again confirming that bothapproaches are consistent with each other. The magnitude ofthe energy difference is small and near the limit of the likelyaccuracy of present density functionals, making it hard to becertain that the ordering is correct based on this alone. However,both measurements agree well with the experimental estimatethus giving added confidence. Furthermore, the similarity ofboth structures should lead to cancellation of systematic errorsin the density functional used. It has been argued that bayeriteshould be the more stable phase due to the fact that it is moredense than gibbsite at ambient pressure. However, this logic isnot conclusive, because there are counter examples of hydrogenbonded molecular crystalline systems in which the most densepolymorph is less stable.29 In such cases, the key criterion isoptimization of the directionality of the interactions, rather thanthe straightforward number density.

Beyond the difference in the internal energies of the materials,the influence of the thermodynamics of vibration should alsobe allowed for. As will be discussed later, the gamma pointphonons have been determined for both gibbsite and bayeriteusing the real space approach. Hence we can readily calculatethe contribution of the zero point energy to the stabilitydifference. This term is found to be quite small at 1.2 kJ/moland again favors gibbsite over bayerite. This additional prefer-ence for gibbsite can be readily understood, because the higherdensity of bayerite leads to a blue shift in the hydroxyl stretchingmodes that dominate the zero point energy term.

Mechanical Properties

There have been previous theoretical studies of aluminumhydroxide polymorphs that have attempted to empiricallyparametrize interatomic potentials to reproduce the structures

of such phases.7 Potential models are particularly important forthe prediction of morphology and surface-related properties,given the interest in the growth of gibbsite, where calculationson many surfaces have to be performed.30 This would representa computationally challenging task for ab initio methods andtherefore reliable force fields are crucial in this area. Inclusionof information concerning the curvature of the energy surfaceabout the experimental structure is of paramount importance inthe derivation of good empirical potentials.31 This requires datasuch as the elastic constants tensor, the bulk modulus, dielectricconstants, and other physical properties which are related tothe second derivatives of the energy with respect to eitherinternal or external strain. In the case of aluminum hydroxidethere is a paucity of data of this nature.

Recently an X-ray diffraction study of the structure of gibbsiteunder applied pressure has been published32 which suggests thatit ultimately undergoes a pressure induced phase transition tonordstrandite. From the data the authors also managed to extractthe bulk modulus as being 85 GPa, though there was consider-able noise in the curve of volume vs pressure. This representsthe only experimental information concerning the mechanicalstiffness of aluminum hydroxide of which we are aware.

To better characterize pure gibbsite as a material, we haveused ab initio plane wave calculations to calculate both the bulkmodulus and to predict the elastic constants tensor. The latterinformation will prove particularly valuable in the futurerefinement of empirical potentials for this class of material, asexplained previously. Although it is unlikely that the elasticconstants of gibbsite will be measured experimentally for theforeseeable future, due to the small size of crystals that can beobtained, there are new methods for their determination thatcan use much smaller samples than other approaches,33 butwhich rely on refining initial estimates. Hence, the present valueswould prove useful for providing starting values for suchexperiments.

The symmetry-unique elements of the elastic constants tensorare reported in Table 4 along with limits on the numericaluncertainties arising from the finite difference procedure. Alsogiven for comparison are the same values as calculated using arecent shell model interatomic potential derived for the studyof gibbsite.7 Although there are some significant differencesbetween the ab initio and potential-based values, most particu-larly for the value ofC33, the general trends in values and theabsolute magnitude of the on-diagonal terms for strain withinthe layers are generally quite reasonable, especially consideringthere was no information included in the fit concerning themechanical properties. However, there is clearly scope forrefinement of the potential in the light of these results.

TABLE 3: Comparison of Experimental and CalculatedCell Parameters for Gibbsite and Bayerite Using theLocalized Basis Set Methodology Embodied within SIESTA

gibbsite bayerite

parameter exptl5 calcd exptl26 calcd

a (Å) 8.684 8.887 5.063 5.149b (Å) 5.078 5.171 8.672 8.915c (Å) 9.736 9.696 9.425 9.422R (deg) 90.00 90.00 90.00 90.00â (deg) 94.54 91.47 90.26 90.49γ (deg) 90.00 90.00 90.00 90.00

TABLE 4: Elastic Constants of Gibbsite Calculated UsingBoth ab Initio and Ionic force field Methodsa

elastic constant ab initio force field

C11 130.9( 2.3 122.1C22 144.7( 6.6 141.8C33 120.0( 3.8 76.5C44 23.0( 2.2 10.4C55 28.7( 2.4 13.7C66 55.1( 0.7 34.0C12 56.0( 1.7 71.8C13 4.0( 2.5 3.4C15 -4.6( 1.1 -8.8C23 6.3( 2.6 8.5C25 -1.2( 0.7 -0.6C35 -4.6( 0.5 -1.5C46 -0.6( 0.2 -3.0

a For the ab initio values, the numerical uncertainties are also quoted.All values are in GPa.


Perhaps the most significant aspect of the elastic constantstensor is that it reveals information concerning the anisotropyin the mechanical stiffness of gibbsite that is not discerniblefrom the bulk modulus. As expected, the crystal is most readilydeformed along the direction corresponding toC33 (excludingshearing). However, what is more surprising is the low degreeof anisotropy, which suggests that the force constants fordistorting the interlayer hydrogen bonding are only marginallyweaker than those for perturbing the octahedra within the layers.Conversely, from considering C44, C55, andC66, the distortionof the octahedra is almost twice as stiff as the interactionbetween the layers with respect to shearing.

The bulk modulus of gibbsite has been determined in twoways, both with the plane wave methodology. Both the fittingof the pressure vs volume curve to an equation of state and thedirect calculation from the elastic constants tensor yielded thesame value, within numerical precision, of 55 GPa. Thiscompares to the experimental estimate of 85 GPa and the valuefrom the interatomic potentials of 45 GPa. While the overes-timation of the cell volume is partly responsible for thediscrepancy between theory and experiment, this is insufficientto explain such a large error. Indeed, the accumulated evidencefrom previous calculations of bulk moduli on a variety ofmaterials is that GGA estimated values are rarely wrong by morethan a few percent, except in pathological cases, such as vander Waals solids. Certainly an underestimation by 35% wouldbe very surprising. Given the difficulties of accurately measuringthe equation of state under pressure experimentally for gibbsite,primarily due to issues of sample quality, it would be wrong tosuggest that the density functional approach had failed in thiscase. Recently, the bulk modulus of diaspore,R-AlOOH, hasbeen calculated using the same approach as in the presentwork.34 The calculated value of 148 GPa agreed well withexperimental values obtained from the determination of theelastic constants, but differs from values extracted from high-pressure X-ray measurements, just as in the case of gibbsite.

There is also a further possible explanation for the disagree-ment. Because of the conditions under which gibbsite is usuallyprepared, it is difficult to obtain a pure sample of the material.Very commonly there will be sodium, or other cation, impuritiespresent in the experimental sample. If such cations act to bindthe layers together more strongly than the hydrogen bonds thenthis would raise the bulk modulus. Further calculations are inprogress to determine the influence of sodium doping on theproperties of gibbsite which may resolve this issue.

Vibrational Spectra

One of the best techniques for in situ identification of differentpolymorphs of aluminum hydroxide is via characterization ofthe vibrational modes,35 particularly using Raman spectro-scopy.36 Here there are distinct differences in the hydroxylstretching bands between gibbsite and bayerite, with the peaksfor the latter being systematically shifted to higher wavenumber.As discussed in the methodology section, the vibrations of bothminerals have been determined using finite differences. Cur-rently this yields only the mode frequencies and not theirintensities, thus making full comparison against experimentdifficult. Furthermore, it is well-known that there are smallsystematic errors in vibrational frequencies associated with theparticular quantum mechanical Hamiltonian and computationalconditions, such as the basis set used, partially as a consequenceof deviations from experimental geometries, as well as differ-ences due to anharmonicity. Hence, it is common to introducescaling factors to correct, to a first approximation, for some of

these differences, which are typically of the order of a fewpercent.37 In the present work, the reported frequencies will beunscaled and consequently direct quantitative agreement is notexpected, nor should it be, since we would be comparingcalculated harmonic frequencies with anharmonic experimentalvalues. The focus will therefore not be on the absolute positions,but the trends within the modes and the assignment of thespectra.

There is one further complication when calculating thevibrational frequencies at the gamma point with this method,in that the LO-TO splitting that should occur will be absent.Consequently, some of the calculated frequencies will be slightlyin error. However, we can readily assess the extent of theproblem using the interatomic potential methodology where itis trivial to run calculations exactly at the center of the Brillouinzone and fractionally away from it. Such calculations indicatethat the largest shift of any mode is less than 30 cm-1, and thisonly occurs for a single hydroxyl shift. Most frequencies areonly perturbed by a few wavenumbers, which is of the order ofthe numerical uncertainty and therefore negligible. Hence wewill neglect the influence of the LO-TO splitting in the followingresults and discussion.

The most readily identified peaks in the vibrational spectrumare the hydroxyl stretching modes because they are completelyseparated from all other frequencies. Calculated mode frequen-cies for gibbsite and bayerite are given in Table 5 along withassignments to particular hydrogens based on the dominantcontributors to the eigenvectors. The nomenclature for thehydrogens refers to the labeling given in Figure 2. It should benoted that while all the stretching modes have been apportionedto particular hydrogens, in many cases there is mixing betweenthe modes of different hydrogens. For example, in the case ofgibbsite H3, H4 and H5 tend to contaminate each other’seigenvectors significantly, while for bayerite there are combina-tions involving H1 with H2, and H4 with H5. However, somehydrogens give rise to quite distinct modes. For gibbsite, thepeaks due to H1 stretches are separated from other modes byover a 100 cm-1 at the high-frequency end of the spectrum,while those due to H6 are confined to the low-frequency extremeof the stretching region. In the case of bayerite, there is nodistinct gap at the low-frequency end of the hydroxyl stretches,but there is again a clear splitting at the other extreme due toH6.

When the structure of both gibbsite and bayerite are consid-ered, the hydrogens are oriented primarily either in the planeof the layers (xy plane) or perpendicular to this (z axis). Thus,the vibrational modes can be seen to be polarized accordinglybased on the dominant hydrogen contributors. What is found isthat of the 24 hydroxyl stretching modes, the 12 with the highestfrequencies are those that lie mainly in thexy layers. From thisit is possible to infer that the hydrogen bonding between layers

TABLE 5: Hydroxyl Stretching Frequencies (inWavenumbers) and Their Assignments for Gibbsite andBayeritea

atom gibbsite bayerite

H1 3569, 3570, 3575, 3578 3306, 3308, 3309, 3317H2 3423, 3425, 3426, 3426 3324, 3326, 3335, 3355H3 3302, 3311, 3322, 3416 3387, 3390, 3397, 3441H4 3437, 3438, 3440, 3440 3464, 3465, 3468, 3469H5 3272, 3274, 3285, 3297 3471, 3473, 3480, 3481H6 3189, 3192, 3207, 3212 3610, 3612, 3615, 3618

a Note for the assignments, the specified atom represents thedominant hydrogen contributing to the eigenvector, but not necessarilythe only one.


is greater than that within the layers, since stronger interactionsof this kind lead to an increased red shift of the frequencies.This is what would be expected based on the fact that there ismore freedom to achieve the optimum hydrogen bondingdistance between the layers since it is only a matter of findingthe best compromise between the three symmetry-distincthydrogens oriented in this direction.

Recently Wang and Johnston38 have made an experimentalassignment of the hydroxyl stretching modes of gibbsite basedon the polarization direction of each band observed whencompared to the orientation of the hydroxyl groups in the crystalstructure. They identified six bands at 3623, 3526, 3519, 3433,3370, and 3363 cm-1 which were assigned to H1, H2, H4, H6,H3, and H5, respectively. Their results are broadly consistentwith the present findings. In particular, both studies find thatthe three highest modes are associated with the stretching modesperpendicular to the 001 direction and that H1 has the greatestfrequency with a significant separation. The order of H2 andH4 differs, though given the closeness of the modes it isprobably beyond the accuracy of the calculation to make anunambiguous assignment here. Similarly the ordering within thetriad of vibrations parallel to 001 differs. Once prediction ofthe intensities for the modes becomes routine, for both infraredand Raman spectra, then it may be feasible to determine thevalidity of the present assignments.

The quantitative description of the gibbsite vibrationalspectrum, with respect to the mode positions, is somewhatlimited in comparison to the experimental data. The highestfrequency band is predicted to be approximately 50 cm-1 toolow. This tendency to underestimate the stretching frequencies,in contrast to most ab initio calculations that typically tend tooverestimate them, is largely related to the fact that the hydroxylbond lengths are too long. Hence a determination of thevibrational frequencies using the plane wave approach wouldbe more accurate. More discouraging is that the range offrequencies spanned by the hydroxyl stretching modes is toogreat in the calculation, with the lowest mode disagreeing bythe order of 170 cm-1. Again some allowance must be madefor the fact that not all modes in the calculated phonon spectrumwill have nonzero intensity and for the width of the experimentalbands.

Although the quantitative positions of the hydroxyl stretchingmodes are not neccessarily accurate, it is still possible toconsider the shift in modes that occurs between the two phases.Experimentally a shift of approximately+36 cm-1 is found inthe highest peak when going from gibbsite to bayerite, whichis reproduced by the calculations with a corresponding displace-ment of+40 cm-1. This shift can be understood on the basisof the smaller unit cell volume of bayerite in comparison togibbsite. Similarly the hydroxyl bending modes demonstrate thesame tendency with highest frequencies of 1138 and 1152 cm-1

for gibbsite and bayerite, respectively.

Conclusions

The energetic and structural properties of the two dominantpolymorphs of aluminum hydroxide have been determined forthe first time using ab initio techniques. Both plane wave andlocalized basis set approaches have been employed in order toevaluate their relative performance, as well as characterizingthe intrinsic uncertainties associated with the choice of com-putational implementation. The present results are in accord withthe available experimental calorimetric data by finding thatgibbsite is the thermodynamically favored material at lowtemperatures and ambient pressure.

Structurally, both the ab initio techniques employed demon-strated the systematic deviations that would be expected: theGGA calculations tended to overestimate the unit cell volume,while using the LDA leads to a more dramatic overcompressionof the cell due to shrinking of the interlayer spacing. Whencontrasting the plane wave and localized basis set results, theformer are found to be consistently superior for the structuredue to the greater degree of convergence of the basis set. Thiscan of course be rectified by increasing the localized basis fromdouble-ú plus polarization to include more flexibility. However,the present basis set represents an acceptable compromisebetween precision and computational speed that would betractable for calculations on larger systems.

The full elastic constants tensor has been determined in thecase of gibbsite by finite differences. In turn, this has been usedto yield the bulk modulus for this phase that agrees well withthe value determined from fitting an equation of state to thecalculated variation of the unit cell volume with applied isotropicexternal pressure. The calculated value of 55 GPa is much lowerthan a recent experiment determination where it was found tobe 85 GPa. It is proposed that one possible cause of thedifference in values, beyond experimental and calculationuncertainties, is the presence of impurities within the sampleof gibbsite. Calculations are currently under way to study thelocation of sodium within gibbsite and the influence it wouldhave on the bulk modulus.

Acknowledgment. J.D.G. would like to thank the RoyalSociety for a University Research Fellowship and the EPSRCfor provision of computing facilities. A.L.R. and J.D.G. alsothank the ARC for the award of an IREX grant. The financialsupport of the Australian Federal Government through itsCooperative Research Centres Program is also gratefullyacknowledged.

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an ab initio study of the structure and properties of aluminum hydroxide: gibbsite and bayerite

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