Anatase and Rutile Surfaces with Adsorbates

Representative of Acidic and Basic Conditions

A. S. Barnard a,b,∗ P. Zapol a,b,c L. A. Curtiss b,c

aCenter for Nanoscale Materials, Argonne National Laboratory,9700 South Cass Ave, Argonne, IL, 60439

bMaterials Science Division, Argonne National Laboratory,9700 South Cass Ave, Argonne, IL, 60439

cChemistry Division, Argonne National Laboratory,9700 South Cass Ave, Argonne, IL, 60439

Abstract

Presented here are density functional theory results of the structure and energet-ics of selected low index stoichiometric surfaces of the anatase and rutile titaniumdioxide polymorphs, passivated with complete monolayers of adsorbates chosen torepresent acidic and basic conditions. The adsorbates differ in each case by varyingthe hydrogen to oxygen ratio with respect to a neutral, water-terminated surface.The results are compared for each of the thirty surfaces examined here, to iden-tify relationships between surface chemistry, surface free energy, surface stress and(upper-most) surface tri-layer reconstructions. Within our model, the results showthat termination with water consistently results in the lowest values of surface freeenergy, but not necessarily the lowest surface stress.

Key words: density functional calculations, low index single crystal surfaces,chemisorption, surface relaxation and reconstruction, surface free energy, surfacestress, titanium dioxidePACS: 68.35.-p, 68.35.Bs, 68.35.Md, 68.43.Bc, 82.65.+r

1 Introduction

The current interest in titanium dioxide nanoparticles for advanced photo-chemical applications[1] has prompted a number of studies to analyze theproperties of titanium dioxide surfaces under various conditions, such as acids

∗ amanda.barnard@anl.gov

Preprint submitted to Elsevier Science 10 March 2005

and bases[2,3]. TiO2 nanoparticles are typically generated via sol-gel synthe-sis, and it has been known for some time that the pH value of the sol-gel isa decisive factor for controlling the final particle size[4], shape[5], phase[4,6,7]and agglomeration[8].

In the case of anatase (for example), it has been reported that synthesis un-der basic conditions results in small cubic-like nanocrystals with {112} and{103} facets[9], hexagonal nanocrystals [10], or short rod-like nanocrystalswith {010}, {101} and {001} facets[2,10]. In contrast, acidic conditions havebeen reported to result almost exclusively in truncated tetragonal bipyramidalnanocrystals with {101}, {001} and {010} facets[9,11,12]. Further, it has beenfound that the growth rate of anatase is dependent on pH[13]; and that anexcess dilution of the particle density during synthesis causes partial disso-lution of TiO2 nanocrystals[9] (or melting[8]), possibly resulting in sphericalnanoparticles. In each case however, the final nano-morphology is dependentupon the value of the pH, and hence the properties of the nanocrystals willbe sensitive to the chemistry at the surfaces[13–15]. This sensitivity to pHis especially important when designing nanoparticles suitable for interfacingwith organic molecules [16].

In general, the surface science of the rutile titanium dioxide polymorph, andto a lesser extent the anatase polymorph, has been widely investigated andan excellent review of the experimental and theoretical results of many papersare given in reference [17]. Although there have been numerous studies inves-tigating the structure and energetics of H2O on stoichiometric anatase[18–22]and rutile[18,19,22–24] surfaces, a rigorous and systematic comparison of thestructure and energetics depending upon various surface adsorbates, using thesame computational technique (and convergence criteria) applied to multiplesurfaces, has not previously been undertaken.

Presented here are results of highly accurate first principles calculations ofthe surface structure and energetics of the selected low index surfaces of sto-ichiometric rutile and anatase, with adsorbates representing acidic, neutraland base conditions. From experiment, it is known that the structure of sur-faces are determined by acid–base equilibria involving TiOH surface hydroxylgroups[17,25–29]. Under neutral pH conditions, the surfaces are found to beterminated with water adsorbates (either as molecular H2O or as dissociatedOH−+H+) capping the under-coordinated surface sites[17,25]. When in acidicsolutions protonation occurs, with the (lowest pH) limiting case occuring whenall under-coordinated surface sites are protonated. Similarly, when in basic so-lution, the surfaces are deprotonated[27,30], and the (upper pH) limiting caseinvolves total deprotonation and all under-coordinated surface sites termi-nated with O−.

The surface conditions have been modelled here via the application of a com-

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plete monolayer of suitable adsorbates, and by varying the ratio of hydrogento oxygen in the adsorbates to represent different degrees of surface acidity.Hydrogen–rich (or H–terminated) surfaces were used to represent moderately(or highly) acidic conditions, whereas hydrogen–poor (or O–terminated) sur-faces were used to represent moderately (or highly) basic conditions. Therespective chemical potentials were then constructed with respect to the neu-tral (water terminated) surface. In our model, the surfaces are not changedsince we do not consider any charge balance ions that will be present in realsolutions.

Although there is only limited experimental evidence for the more “extreme”conditions, namely the fully hydrogenated and fully oxygenated surfaces, itis important to point out that these are the limiting cases for the presentstudy. There is a reasonable amount of evidence to support the terminationof TiO2 surfaces with oxygen under basic conditions[25,27,30,31]. However,although it has been found that TiO2 surfaces may be readily hydrogenated,it is unclear if the under-coordinated surface Ti atoms (as well as the surfacebridging oxygen atoms) will be capped[17]. Previously we have reported[32] oncalculations of hydrogen on selected surfaces of anatase and rutile, comparingresults of fully hydrogenated surfaces (where both under-coordinated Ti and Oatoms on the surface are capped) with partially hydrogenated surface (whereonly the under-coordinated O atoms are capped) and clean surfaces (whereno surface atoms are capped)[32]. Although we use the fully hydrogenatedversion as the (high pH) limiting case herein, we therefore direct the readerto reference [32] for results pertaining to the capping of Ti with H on TiO2

surfaces.

In the present study, section 3.1 begins by examining the relaxed structureof all thirty surfaces, by analyzing the displacement of the atoms in the up-per (surface) tri-layer perpendicular to the surface, and comparing the bondlengths between the surface atoms and the adsorbates. Details of the selectedadsorbates will be outlined in each subsection. In section 3.2, the chemicalpotentials are constructed and the Gibbs surface free energies and surfacestresses calculated and compared as a function of surface chemistry.

2 Methodology

As is traditionally done when investigating the structure and energetics ofmaterials surfaces using computational methods, two-dimensional rutile andanatase lattice slabs were generated by taking a bulk lattice and adding a 10 Avacuum layer in the crystallographic plane of interest. The ‘cleaved’ surfaceswere then terminated with a complete monolayer (θ = 1) of adsorbates selectedto describe the particular surface chemistry in each case. These terminations

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will be described for each type of surface chemistry in the next section. All ofthe surface slabs were relaxed prior to calculation of the total energies usingfirst principles methods.

All calculations herein have been carried out using density functional the-ory (DFT) within the generalized-gradient approximation (GGA), with theexchange-correlation functional of Perdew and Wang (PW91)[33]. This hasbeen implemented via the Vienna Ab initio Simulation Package (VASP)[34,35],which spans reciprocal space with a plane-wave basis, in this case up to a ki-netic energy cutoff of 270 eV. We have used the Linear Tetrahedron Method(LTM) with a 4 × 4 × 4 Monkhorst-Pack k-point mesh, for both the initialrelaxations of the TiO2 slabs, and the final calculation of surface energies andsurface stresses. Although this choice of k-mesh results in some superfluousk-points in the non-periodic direction of the surface slabs, inclusion of thesek-points is more consistent with the LTM.

The electronic relaxation technique used here is an efficient matrix diago-nalization routine based on a sequential band-by-band residual minimizationmethod of single–electron energies[36,37], with direct inversion in the iterativesubspace, whereas the ionic relaxation involves minimization of the Hellmann-Feynman forces. During the relaxations we have used ultra–soft, gradient–corrected Vanderbilt–type pseudopotentials (US-PP)[38,39] and real–spaceprojected wavefunctions (to decrease the computational cost), and have re-laxed to an energy convergence of 10−4 eV (equating to a force convergence of10−2 eV/A). The following (final) static single point energy calculations werethen performed using the Projected Augmented Wave (PAW) potentials[40],with a basis set increased to a cutoff of 350 eV and reciprocal–space projectedwave function (to improve accuracy), also to an energy convergence of 10−4

eV. PAW potentials are generally considered to be more accurate than theultra–soft pseudopotentials[41], since the radial cutoffs (core radii) are smallerthan the radii used for the US pseudopotentials, and the fact that the PAWpotentials reconstruct the exact valence wave function with all nodes in thecore region (all electron).

3 Discussion of Results

3.1 Surface Structure

In the following subsections, the relaxed surface structures are presented forthe (100), (001) and (101) surfaces of anatase; and the (100), (110) and (011)surfaces of rutile. These surfaces have been chosen for consideration as theyhave previously been shown to dominate the morphology of anatase and rutile

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nanocrystals[32]. In the nomenclature used herein each atom in the upper-mostTiO2 tri-layer has been labelled according to species, with the subscript de-noting the atomic layer. The oxygen associated with the adsorbates is denotedsimply as O (with no subscript).

The following sections are arranged according to the type of surface chemistry.In each case stoichiometric lattices have been used for the surface slabs; con-sisting of 12 atomic layers and a total of 48 atoms in the case of the anatase(001) surface, 18 atomic layers and a total of 72 atoms for the anatase (100)and (101) surfaces, 18 atomic layers and 72 atoms for the rutile (100) surface,12 atomic layers and 72 atoms for the rutile (110) surface, and 27 atomic layersand 108 atoms for the rutile (011) surface. The specific adsorbates coveragewill be described for each surface in the following sections 3.1.1 to 3.1.5.

In terms of results, the displacements of the surface atoms (perpendicular tothe surface) are listed in Tables 1 and 2 for anatase and rutile, respectively.This will be accompanied (in section 3.2) by results for the surface free energyγ and surface stress σ for each of the fully relaxed surface.

3.1.1 Fully Hydrogenated Surfaces

To represent “highly acidic” conditions, where an excess of hydrogen is presenton the surfaces, all under-coordinated surface sites have been terminatedwith a hydrogen atom. This equates (structurally) to fully hydrogenated (H–terminated) surfaces[32]. In the following description of the surfaces underthese conditions, each atom has been labelled in Figure 1 according to species.All of the anatase surfaces, along with the rutile (100) and (011) surfaces, areterminated with a total of 16 hydrogen adsorbates (8 per surface of the slab),and the rutile (110) surface is terminated with a total of 12 adsorbates (6 persurface of the slab). As described later in section 3.2, these are values for thenumber of adsorbates (referred to as Nad) for each surface. The anatase andrutile surfaces are shown in the left and right images of Figure 1, respectively.

In the case of the H–terminated anatase (001) surface (shown in 1a), the entireupper tri-layer was found to undergo an outward displacement. This was mostpronounced for the H–terminated O(1) bridging oxygens, and least pronouncedfor the O(3) sub-surface oxygens. The outward displacement of the Ti(2) atomsof 0.12 A (see Table 1) was also found to be three times the displacementpreviously reported for the clean (001) surface[32].

The H–terminated anatase (100) surface (shown in Figure 1c) also exhibiteda net outward relaxation. The H–terminated O(1) bridging oxygens exhibiteda considerable outward relaxation of 0.13 A (see Table 1), whereas the O(3)

atoms were found to maintain position. The net outward relaxation of theupper tri-layer is further enhanced by the outward displacement of the Ti(2)

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atoms, which have been previously found to relax inwards on the clean anatase(100) surface[32].

In contrast to the (001) and (100) surfaces, when the H–terminated anatase(101) surface is relaxed (Figure 1e) the results show that the outward dis-placement is dominated by the sub-surface O(3) atoms and the Ti(2) surfacetitanium atoms, rather than the bridging oxygens[32].

Upon relaxation, the Ti(2) of the H–terminated rutile (100) surface was foundto contract inward (perpendicular to the surface) as shown in Figure 1b. Also,the hydrogen atoms terminating the O(1) bridging oxygens were found to re-orient so that the O–H bonds were directed inward. The resulting Ti(2)-O(1)-Hbond angles were approximately 100.2◦. This, combined with the threefoldcoordinated O(3) atoms relaxing outwards, cause the formation of hydrogenbonds between the H atoms and the O(3) with a bond length found to be 1.49A (as indicated by the dashed lines in Figure 1b)[32].

The rutile (110) surface (see Figure 1d) contains inequivalent (normal to thesurface) Ti atoms lying in a centered rectangular arrangement. The Ti(2) atomsbeneath the bridging oxygens are sixfold coordinated, while the remaining Ti(2)atoms are fivefold coordinated (on the clean surface). The latter Ti atomswere found to displace quite differently when H–terminated. The outwarddisplacement of the fivefold coordinated H–terminated Ti(2) atoms was foundto be an order of magnitude larger than that of the sixfold coordinated Ti(2).It was also found that the outward displacement of the O(3) exceeded thatof the H–terminated O(1) bridging oxygens. A partially hydrogenated rutile(110) surface (denoted as the ‘R Model’ surface) was examined by Leconteet al [42], using DFT GGA and Vanderbilt–type pseudopotentials. Althoughthese authors did not report the magnitude of the relaxation of the surfacetri-layer with respect the bulk, they did list the O–H bond length as a functionof slab thickness. For a slab of nine atomic layers (as used herein), their O–Hbond length of 0.968 A is slightly shorter than the 0.99 A listed in Table 2[32].

The H–terminated rutile (011) surface was also examined, as shown in Figure1f. The bonds between the uppermost O(1) atoms and the Ti(2) atoms formtwo symmetrical pairs, creating zig-zag chains of twofold coordinated O(1)

atoms, and threefold coordinated surface O(3) atoms in the first and thirdatomic layers, respectively. Hydrogen termination was found to induce rathersignificant outward displacement of the O(1) and Ti(2), but a moderate outwarddisplacement of the O(3) atoms (see Table 2)[32].

As mentioned above, previously we have reported calculations[32] of hydro-gen on these anatase and rutile surface, comparing results of fully hydro-genated surfaces (where both under-coordinated Ti and O atoms on the sur-face are capped) with partially hydrogenated surface (where only the under-

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coordinated O atoms are capped) and clean surfaces (where no surface atomsare capped). We therefore direct the reader to reference [32] to compare theeffects of capping Ti with H on TiO2 surfaces.

3.1.2 Hydrogen–rich Surfaces

To represent “moderately acidic” conditions, where an abundance of hydro-gen is present on the surfaces, the under-coordinated titanium sites have beencapped with a water molecule and the under-coordinated oxygen sites havebeen capped with a hydrogen atom. This equates to hydrogen–rich (with re-spect to the H2O terminated) surface mono-layer on anatase and rutile, and(once again) each atom has been labelled in Figures 2, showing the relaxedstructure of the anatase and rutile surfaces to the left and right, respectively.In these cases, the anatase surfaces and rutile (100) and (011) surfaces areterminated with a total of 8 (H2O+H) adsorbates (4 per surface of the slab),and the rutile (110) surface is terminated with a total of 6 adsorbates (3 persurface of the slab). These values are referred to as Nad for each surface insection 3.2.

The hydrogen–rich surface tri-layers of the anatase (001) surface are shown inFigure 2a. Compared with the H–terminated (highly acidic) surface describedabove, the outward displacement of the O(1) and O(3)oxygen atoms was foundto increase, but decrease for the Ti(2) atoms. This is most significant in thecase of the O(3) sub-surface oxygen. Interestingly, the water molecules boundto the Ti(2) atoms are oriented perfectly perpendicular to the surface, with theO–H bonds aligned in the [010] direction. The final Ti(2)–OH2 bond length of2.64 A (see Table 1) is quite long, whereas the O(1)–H bonds length was foundto be almost equivalent to that of O–H bonds in the water adsorbates.

When under hydrogen–rich conditions, the anatase (100) surface (shown inFigure 2c) was found to undergo a small outward relaxation very similar tothe H–terminated surface described above. The displacement of the oxygensslightly exceeded that observed on the H–terminated surface, but the displace-ment of the Ti(2) atoms was found to be the same. The Ti(2)–OH2 bonds werevertically aligned, perpendicular to the surface, with a bond length of 2.33 A(see Table 1) . This is less than the hydrogen–rich anatase (001) surface, butstill significantly longer than the Ti–O bonds in bulk region.

Under these conditions, the anatase (101) surface (Figure 2e) exhibited almostno outward displacement of the O(1) and Ti(2) atoms (0.04 A), but was againdominated by the relatively large outward displacement of the O(3) atoms(0.24 A). The Ti(2)–OH2 bond lengths were found to be 2.30 A (see Table 1);shorter than both the (001) and (100) hydrogen–rich surfaces, but in goodagreement with the Ti(2)–OH2 bond length of 2.28 A reported by Vittadini et

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al[20] for this bond on a H2O terminated surface.

The hydrogen–rich rutile (100) surface (shown in Figure 2b) was found tobe unstable. Although an outward displacement of all surface tri-layer atomswas observed, the desorption of the water molecules (leaving only the extrahydrogen atoms bound to the bridging oxygens) results in a surface that is nolonger terminated in the appropriate way, and is therefore not representativeof a moderately acidic rutile (100) surface. Due to this instability, this surfacehas been omitted from the remainder of the study.

In contrast to the anatase surfaces, the hydrogen–rich rutile (110) surface(Figure 2d) showed a significantly more pronounced outward displacement ofthe O(1) atoms, than the Ti(2) and O(3) atoms. It is also interesting to note thatthe displacement of the inequivalent Ti(2) atoms is almost identical under theseconditions, whereas they were quite different on the H–terminated (highlyacidic) surface. This is attributed to the termination of the under-coordinatedTi(2) atoms with the O atoms of the water molecules, even though (once again)the Ti(2)–OH2 bond lengths of 2.30 A (see Table 2) are longer than bulk Ti–O bonds. Figure 2d also shows that the Ti(2)–OH2 bonds were found to tiltupon relaxation, to accommodate the formation of hydrogen bonds betweenhydrogen atoms of the water molecules and the O(1) atoms. The length ofthese bonds was found to vary between 1.63 to 1.65 A (see Table 2) .

A most interesting result was observed on the hydrogen–rich rutile (011) sur-face (Figure 2f), where the O(1) and Ti(2) atoms were found to relax inwards.This is especially significant in the case of the Ti(2) atoms, which is a uniqueresult among the rutile (011) surfaces examined herein. In all other instancesthese atoms exhibit an outward relaxation. In addition to this, the 2.42 ATi(2)–OH2 bonds (see Table 2) are significantly longer than the rutile apicaland equatorial Ti–O bond lengths (and longer than any other terminatingbond length on this surface, irrespective of passivation conditions), but werefound to adopt angular positions very similar to the bulk.

3.1.3 Hydrated Surfaces

To represent “neutral” conditions two configurations of water terminated sur-faces were investigated, corresponding to molecular adsorption and dissocia-tive adsorption. In the case of the molecular adsorption configuration all of theunder-coordinated titanium sites have been capped with a water molecule andthe under-coordinated bridging oxygen sites have been left vacant; whereasfor the dissociative adsorption configuration the under-coordinated titaniumatoms were terminated with OH groups, and the under-coordinated oxygenswere terminated with a hydrogen atom. Once again, the anatase surfaces andrutile (100) and (011) surfaces are terminated with a total of 8 (H2O or OH+H)

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adsorbates (4 per surface of the slab), and the rutile (110) surface is termi-nated with a total of 6 adsorbates (3 per surface of the slab). These valueswill be referred to as Nad for each surface in section 3.2.

Previously we have undertaken a detailed study of the adsorption configura-tions and energetics of water adsorbed on the anatase and rutile surfaces con-sidered in the present study[22]. Although both configurations were examinedfor each anatase and rutile surface, to avoid repetition only the lowest energyconfiguration has been chosen for presentation here. The hydrated anataseand rutile surfaces are shown in the left and right images of Figure 3, andthe displacements perpendicular to the surface are listed in Tables 1 and 2,respectively.

The hydrated surface tri-layers of the anatase (001) surface are shown in Figure3a. Dissociative adsorption was found to be energetically favorable on thissurface[18,22], with the H and OH terminations oriented perpendicular to thesurface after relaxation. Compared with the acidic surfaces described above,the outward displacement of the upper tri-layer was significantly less, and wasdominated by the displacement of the Ti(2) atoms. The final Ti(2)–OH bondlength of 1.93 A (Table 1) was just under the equatorial Ti–O bonds lengthof anatase, and was significantly less than the Ti(2)–OH2 bond lengths onthe hydrogen–rich (001) surface. The displacement of selected atoms on thehydrated anatase (001) surface was examined by Bredow and Jug[19] using thesemi-empirical SINDO1 method and model clusters. The authors found thatthe bridging O(1) atoms relaxed outward by 0.12 A, the Ti(2) atom relaxedoutward by 0.16 A and the Ti(2)–OH bond length was 1.85 A, when a 4×4×3cluster was used.

When covered with a monolayer of dissociated water[22], the anatase (100)surface (Figure 3c) was found to undergo an outward relaxation of the O(3)

and Ti(2) atoms, and a small inward relaxation of the O(1) atoms. The Ti(2)–OH bond length was found to be 1.87 A (Table 1), which is smaller than theapical and equatorial Ti–O bonds in the bulk. The most interesting aspect ofthis surface was the bending of the O–H bonds of the OH groups toward eachother, resulting in a H–H distance of 2.6 A.

In the case of the anatase (101) surface, molecular adsorption was found tobe energetically preferred[22] as indicated in Figure 3e. The binding energyfor the H2O molecule adsorbed onto the Ti(2) site of -0.56 eV (as opposed tothe binding energy for the dissociated OH adsorbed onto the Ti(2) site andH adsorbed on the O(1) site of -0.48 eV) is in agreement with the resultsof Vittadini et al[20] and Tilocca and Selloni[43,44], who also concluded infavour of molecular adsorption. In contrast to the anatase (101) surfaces witha high fraction of H in the adsorbates, both O(1) and Ti(2) atoms were found tocontract inward following relaxation, and the O(3) was found to relax outward.

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The inward relaxation of the upper atomic layers under neutral conditions isa unique result for the anatase (101) surface. The Ti(2)–OH2 bond length of2.28 A (Table 1) matches the length of 2.28 A reported by Vittadini et al[20]for this bond. This surface is also equivalent to the 2D commensurate watermonolayer examined by Tilocca and Selloni[44], who calculated a O−→O(1)

distance of 3.04 A, slightly shorter than the 3.2 A calculated here.

Like the hydrated anatase (100) surface, the hydrated rutile (100) surface wasfound to prefer dissociative adsorption[22] (as shown in Figure 3b), whichis generally considered to be the case in other experimental and theoreticalstudies[17]. In general, the (outward) relaxation of the hydrated surface wasfound to be similar to the clean surface[32], and was significantly less than inthe acidic surfaces. The Ti(2)–OH bond length of 1.90 A (Table 2) was foundto be less than the Ti(2)–OH2 bond length on the hydrogen-rich rutile (100)surface, and just slight less than the equatorial Ti–O bond length of bulkrutile.

The hydrated rutile (110) surface (see Figure 3d) was found to prefer molecularadsorption[22]. There is some disagreement in the literature as to whether wa-ter adsorbs on this surface molecularly or dissociatively, but in most cases ex-perimental investigations indicate molecular adsorption[17]. Other theoreticalstudies have concluded that molecular adsorption is most probable[47,48,45],and it has also been suggested that molecular and dissociative water may co-exist on the rutile (110) surface[23]. The binding energy for the H2O moleculeadsorbed onto the Ti(2) site of -0.82 eV is in agreement with the results Lindanet al.[23] of -0.99 eV, and Bandura et al[45] of -0.95 eV; although their bindingenergies for the dissociated OH adsorbed onto the Ti(2) site and H adsorbed onthe O(1) site of -0.91 eV and -0.79 eV (respectively) are lower in energy thanthe value of -0.27 eV that we calculated for the present surface[22]. The effectof charge on this surface has been recently examined by Predota et al[46].

Even though our results indicate molecular adsorption, the structure of thesurface is somewhat distorted by the presence of water, as shown in Figure 3d.The Ti(2)–O bond length of 2.30A measured here is in good agreement with thelength of 2.22A measured my Bandura et al[45] using DFT GGA (PW91) andplane waves. The sixfold coordinated Ti(2) atoms relaxing outward, whereasthe fivefold coordinated Ti(2) atoms bound to the water molecules relaxinginward. This layer distortion has been observed before[24,48], and is in partdue to the accommodation of H-bonding between the bridging O(1) atoms andhydrogens in the water molecules. The length of these H-bonds of 1.66–1.67 Ain good agreement with the result 1.6-1.8 A calculated by Langel[48] using CarParinello Molecular Dynamics (CPMD), and the value of 1.61 A calculatedby Ferris and Wang using the self consistent field (SCF) method and 3-21Gbasis set; but is considerably less than the 2.22 A calculated by Menetrey etal[24] using PW91-USPP.

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Like the rutile (100) surface, the hydrated rutile (011) surface (Figure 3f)was also found to prefer dissociative adsorption[22]. The upper tri-layer wasfound to relax outward approximately 0.1 A more than the clean surface. TheTi(2)–OH2 bonds (of 2.14 to 2.16 A, as listed in Table 2) were longer thanthe rutile apical and equatorial Ti–O bond lengths. Hydrogen bonds were alsoobserved between the hydrogen atoms connected to the bridging oxygens andthe oxygen atoms of the OH groups, but with a considerably reduced lengthof 1.44–1.47 A.

More information on the structure and energetics of the hydrated surface maybe found in reference [22].

3.1.4 Hydrogen–poor Surfaces

To represent “moderately basic conditions”, where a deficiency of hydrogenexists on the surfaces, the under-coordinated titanium sites have been cappedwith an OH molecule and the under-coordinated oxygen sites left vacant(as has been established by other researchers[25,27,30,31]). This equates tohydrogen–poor (with respect to the H2O terminated) surface mono-layers,and once again each atom has been labelled in Figures 4, showing the relaxedstructure of the anatase and rutile surfaces to the left and right, respectively.The anatase surfaces and rutile (100) and (011) surfaces are terminated witha total of 8 OH adsorbates, and the rutile (110) surface is terminated with atotal of 6 adsorbates, referred to as Nad for each surface in section 3.2.

The hydrogen–poor anatase (001) surface is shown in Figure 4a. In contrast tothe hydrogen–rich and hydrated (001) surfaces described above, the hydrogen–poor surface exhibited a considerable inward displacement of the O(1) bridgingoxygens. There was however, still an outward displacement of the Ti(2) andO(3) atoms, resulting in the Ti(2) becoming the outer-most atomic layer ofthe surface tri-layer. Also, the final Ti(2)–OH bond length of 1.80 A (Table1) was shorter than the equatorial Ti–O bonds length of anatase. This bondlength is however, in good agreement with the results of Vittadini et al[20].The structure of the hydrated anatase (001) surface was examined, and theirresults showed that for a coverage of θ = 0.5 (where the O(1)–H terminationswere removed, leaving an effective OH–terminated surface) the Ti(2)–OH bondlengths varied between 1.74–1.93 A[20].

Like the hydrogen–poor anatase (001) surface, the (100) surface (shown inFigure 4c) was found to undergo an outward displacement of the Ti(2) atoms,and a small inward displacement of the O(3) and O(1) atoms upon relaxation.This resulted in a net inward contraction of the surface tri-layer. The Ti(2)–OHbonds were found to be vertically aligned (perpendicular to the surface) witha bond length of 2.23 A (Table 1).

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However, although inward displacements were observed in the hydrogen–pooranatase (001) and (100) surfaces, the hydrogen–poor (101) surface was foundto undergo only outward displacements (Figure 4e). The Ti(2)–OH bond lengthof 1.85 A (Table 1) was found to be considerably less than the Ti(2)–OH2

bond length of the neutral and moderately acidic surfaces and, as shown inFigure 4e, the O–H bonds of the terminations were found to tilt of the surface,adopting angles akin to the bulk.

In the case of the hydrogen–poor rutile (100) surface (Figure 4b) the O(1) werefound to undergo an inward displacement and, in the case of the Ti(2) and O(3)

atoms, an outward displacement. This was greatest for the O(3) atoms. TheTi(2)–OH bond length showed a considerably degree of variability, rangingfrom 1.88 to 1.94 A, as indicated in Table 2.

The relaxation of the hydrogen–poor rutile (110) surface (Figure 4d) wasclearly dominated by the outward displacement of the O(1) atoms, and toa lesser degree the outward displacements of the fivefold coordinated and six-fold coordinated Ti(2) atoms, respectively; whereas the O(3) atoms exhibitedno net displacement (perpendicular to the surface) at all. The Ti(2)–OH bondsremained vertically aligned, with an average bond length of 1.80 A (Table 2).

Another interesting relaxation was observed on the hydrogen–poor rutile (011)surface. Here, the Ti(2) atoms exhibited strong outward displacements (al-though with some degree of variation) and the the O(1) atoms were found torelax inwards. This combined with the tilting of the short (1.80 to 1.84 A)Ti(2)–OH bonds caused the formation of an unusual surface structure, bestdescribed by referring to Figure 4f, where the O(1) and Ti(2) atoms lie almostin the same plane. Unfortunately, Figure 4f fails to capture the exact arrange-ment of the O–H bonds of the terminations, that have aligned in the (011)direction (out of the page) toward the neighboring O atoms. The distancebetween the H atoms and the neighboring O atoms of 1.91 A however, doesnot suggest the formation of hydrogen bonds.

3.1.5 Oxygenated Surfaces

To represent “highly basic” conditions, where there is an absence of hydrogenon the surfaces, all under-coordinated titanium atoms have been terminatedwith an oxygen atom and the under-coordinated bridging oxygen sites areleft vacant. This equates (structurally) to a fully oxygenated (O–terminated)surfaces, as shown in Figure 5 where the anatase and rutile surfaces are shownin the left and right images, respectively. Like the surfaces above, the anatasesurfaces and rutile (100) and (011) surfaces are terminated with a total of 8oxygen adsorbates, and the rutile (110) surface is terminated with a total of6 oxygen adsorbates, referred to as Nad in section 3.2.

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The O–terminated, highly basic anatase (001) surface shown in Figure 5a,like the moderately basic surface, exhibited a strong inward displacement ofthe O(1) bridging oxygens, and an outward displacement of the Ti(2) and O(3)

atoms. The O(1) and Ti(2) atoms in the final structure lie almost in the sameplane, with the 1.78 A Ti(2)–O bonds (Table 1) oriented perpendicular to thesurface.

When O–terminated, the anatase (100) surface (shown in Figure 5c) also ex-hibited a strong inward displacement of the O(1) atoms, together with a minorinward relaxation of the Ti(2) atoms, and outward displacement of the O(3)

atoms. In this case, the O(3) and Ti(2) atoms in the final structure lie almostin the same plane, with the O(1) atoms lying beneath. The terminating oxy-gen atoms also shifted position laterally during the relaxation (in the [101]direction) becoming centered over the O(1) oxygens. The final Ti(2)–O bondswere found to be long, at 2.25 A; whereas the resulting O(1)–O bonds wereextremely short, at only 1.51 A (Table 1).

The O–terminated anatase (101) surface (Figure 5e) exhibited no outwarddisplacement of the O(1) atoms, and only minor outward displacements of theand Ti(2) and O(3) atoms (0.04 and 0.14 A, respectively). The average Ti(2)–Obond length was found to be 1.80 A (Table 1).

The O–terminated rutile (100) surface (Figure 5b) was found to be quitesimilar to the hydrogen–poor (OH–terminated) surface, characterized by aminor inward displacement of the O(1) atoms and an outward displacement ofthe Ti(2) and O(3) atoms. In this case the average Ti(2)–O bond lengths wasfound to be 1.83 A (Table 2).

The O–terminated rutile (110) surface (see Figure 5d) underwent a net out-ward displacement upon relaxation, most prominently in the O(1) atoms andfivefold coordinated Ti(2) atoms. The displacement of the sixfold coordinatedTi(2) atoms and the O(3) is comparatively minor. A unique result for the rutile(110) surface is the inequivalent displacements of the O(3) atoms, ranging from0.06 A to 0.15 A (Table 2). This variation is evident in Figure 5d. Some vari-ation was also observed in the final length of the terminating Ti(2)–O bonds,ranging from 1.78 to 1.95 A.

Finally, another interesting relaxation was observed on the O–terminated ru-tile (011) surface (Figure 5f), with the Ti(2) atoms undergoing strong outwarddisplacements and the O(1) undergoing inward displacements. Again, the tilt-ing of the Ti(2)–O bonds in the (011) direction formed a surface structurewhere the O(1) and Ti(2) atoms lie in the same plane. Some variation was alsoobserved in the final length of the terminating Ti(2)–O bonds. It is interestingto note that although the adsorption of O in the surface of TiO2 does not(intuitively) seem as “extreme” as the adsorption of H, the degree of restruc-

13

turing is significantly greater for the oxygenated (011) surface than for thehydrogenated (011) surface described in section 3.1.1.

All of the displacements of the surface atoms (perpendicular to the surface)for the anatase and rutile surfaces under each set of conditions are listed inTables 1 and 2.

3.2 Surface Energetics

As mentioned above, the Gibbs surface free energy per unit area γ and surfacestress σ for the fully relaxed surfaces have also been calculated for each ofthe anatase and rutile surfaces passivated with adsorbates representative ofacidic and basic conditions. The value of γ was calculated from the energyper stoichiometric unit of the bulk (Ebulk) and the total energy of the surface(Esurface

N ) slabs using the expression,

γ =G

A=

1

2A

(Esurface

N −NEbulk −Nadµad

)(1)

where G is the free energy of the surfaces, A is the area of the surface andN is the number of TiO2 units in the (stoichiometric) cell. Here, µad is thechemical potential of the adsorbates, and Nad is the total number of adsorbateunits present. In principle, the quantity of surface adsorbates may be alteredby varying Nad, which is indicative of the coverage not composition of theadsorbates. Hence, Nad=Nmax

ad describes a complete monolayer coverage witheach under–coordinated surface site terminated (as used herein), and Nad=0will describes a clean surface[32]. In this context, the values of Nad have beenlisted at the beginning of sections 3.1.1 to 3.1.5. The composition of theseadsorbates (as described in the sections above) is defined chemically withinµad.

In order to determine µad, it is necessary to have a reliable description ofhydrogen–rich and hydrogen–poor solutions. Brief tests were undertaken todetermine how well the PW91–PAW method describes the dissociation energyof water molecules by calculating the dissociation energy from the followingendothermic reactions. For the reaction,

H2O −→ 2H + O (2)

the calculated energy was 10.34 eV, which is in reasonable agreement with theexperimental value of 9.51 eV. However, for the reaction,

H2O −→ OH + H (3)

14

the calculated energy of 9.44 eV is considerably higher than the experimentalvalue of 5.12 eV. This is attributed to the inaccuracy of the OH energy (ratherthan that calculated for H2O or H) since the result for the reaction,

OH −→ O + H (4)

of 0.90 is much lower than the experimental value of 4.39 eV. These resultssuggest that while the PW91–PAW method is capable of describing waterand hydrogen systems well, it does not give a good description of the OHradical. Therefore the chemical potential µad is constructed from the chemicalpotentials of water and hydrogen,

µad = nH2OµH2O + nHµH + nOµO. (5)

This may be calculated using for each particular type of adsorbate with theappropriate values of nH2O, nO and nH (which are indicative of compositionof the adsorbates, not coverage) and,

µH2O = EH2O +hνH2O

2+ kBT

[ln( PV

kBT

)], (6)

µH =1

2

(EH2 +

hνH2

2+ kBT

[ln( PV

kBT

)]), (7)

µO = µH2O − 2µH . (8)

Here kB is Boltzmann’s constant, T , P and ν are the temperature, pressureand sum of the vibrational frequencies in the reservoir, and V is the quantumvolume[49],

V =( h2

2πmkBT

)3/2. (9)

In this particular approximation, the assumption has been made that no freeatoms or ions exist in the solution, and each chemical potential is formedwith respect to the hydrated (water covered) surface. The construction ofµad in this way ensures that chemical potentials for the hydrogen–rich andhydrogen–poor conditions are obtained using reliable molecular energies, withthe required oxygen to hydrogen ratio. The hydrogen to oxygen ratio peradsorbate is easily set by inserting integer values for nH2O, nO and nH . For theH–terminated surfaces nH2O=nO=0 and nH=1; for the hydrogen–rich surfacesnH2O=nH=1 and nO=0; for the hydrated surface nH2O=1 and nH=nO=0;for the hydrogen–poor surfaces nH2O=1, nH2O=-1 and nO=0; and for the O–terminated surfaces nH2O=nH=0 and nO=1. Using these values the form of

15

µad per adsorbate is determined, and then multiplied by Nad in equation 1 toobtain the desired degree of surface coverage. For the purposes of this study,experimental values were used for ν, the chemical potential were calculatedat ambient temperature and pressure (298.15 K and 101.33 kPa), and thevalues of EH2O and EH2 were calculated explicitly for free molecules (in thegas phase).

The surface energies have been calculated as given in equation 1 for anataseand rutile, with the corresponding values of γ listed in Table 3.

In the case of anatase, the hydrated (and H–terminated) surfaces are lowest(and next lowest) in energy. This indicates that these surfaces prefers condi-tions with a high fraction of H on the surface, as opposed to hydrogen–pooror O–terminated conditions. Also, although the (101) surface is consistentlylowest in energy with a greater fraction of H on the surface, the (100) surfacegains stability as the fraction of hydrogen in the adsorbates decreases. Theanatase (100) surface becomes the lowest in energy when O–terminated.

In the case of rutile, the (110) surface is consistently the lowest energy surface,irrespective of surface chemistry. With the exception of the H–terminatedsurface, the (011) surface is consistently the least (energetically) favorable.Recall that the hydrogen–rich rutile (100) surface was found to be the leaststable over all (as discussed above).

In addition to the surface free energy, the isotropic and anisotropic componentsof the surface stress σ[50] has also been investigated. It can be shown usingsimple thermodynamic arguments that the value of the σ may be obtainedusing the expression,

σ =∂G

∂A≈ ∆G

∆A. (10)

It is important to point out that while γ is defined as the energy per unit area,σ is the rate of change of the energy with respect to area. Using equation 10the calculation of σ is quite straight forward. By applying a two-dimensionaluniform dilation in the plane of the surface (including optimization of allinternal parameters) and calculating the free energy G as shown in Equation1 for each area, the change in free energy (∆G) was found for a set of areadilations (∆A). After plotting these results and fitting to a polynomial, anestimate of the isotropic surface stress may be obtained from the coefficient ofthe linear term. In the case of titanium dioxide, an estimate of the anisotropiccomponent of the surface stress may be obtained from the coefficient of thequadratic term divided by A.

For each of the surfaces, the isotropic and anisotropic components of the sur-face stress have been calculated. The isotropic components have been listed

16

in Table 4, and the anisotropic components in Table 5. In the majority ofcases, the anisotropic component of the surface stress is negative (indicatingan expansion, that would reduce the net compression of the anatase (101) sur-faces, for example), with the exception of the neutral anatase (100) surface,the highly basic rutile (110) surface and the moderately acidic rutile (011)surface. It is interesting to note that each of these three exceptions also ex-hibited unusual displacements or bond lengths upon relaxation, as describedin section 3.1.

By comparing values of γ and σ (corresponding values in Tables 3 and 4) foreach of the thirty surfaces it may be seen that although the hydrated surfacesresult in the lowest surface free energy, these conditions do not necessarily giverise to the lowest surface stress. This is true for the anatase (101) surface wherethe water terminations minimize the surface free energy but a H–terminationsminimize the surface stress; and all of the rutile surfaces (considered here).The results for the rutile (011) surface also indicate how sensitive the surfacestress is to the chemistry of the surface, being completely uncorrelated to thetrends observed for surface energies. This highlights the importance of makingexplicit calculations of surface stress in the presence of adsorbates, rather thanassumptions based on the surface energies.

4 Conclusions

Presented here are density functional theory results of the surface relaxation,surface energies and surface stress of selected low index surfaces of anatasean rutile, under acidic and basic conditions. Each particular degree of acid-ity has been represented by varying the ratio of hydrogen to oxygen atomsconstituents in suitable mono-layers of adsorbates; to simulate “highly acidic”(H–terminated), “moderately acidic” (hydrogen–rich), “neutral” (hydrated),“moderately basic” (hydrogen–poor) and “highly basic” (O–terminated) con-ditions.

The displacements of the upper surface tri-layers (with respect to the un-relaxed, bulk-terminated surface) are presented for all thirty surfaces inves-tigated, along with the surface–adsorbate bond lengths. The results show avariety of unique surface reconstructions, dependent upon the surface chem-istry, that are in good agreement with the results of other comparable studies(where available). We would however like to point out that the effects of tem-perature upon the structure of the surface is unknown, and it is likely thatfurther (more severe) restructuring of the hydrogenated and oxygenated sur-face may be expected when heated.

Trends in the surface free energy have been observed, as a function of surface

17

chemistry. In general the results here indicate that the surface free energyis lowest when hydrated (representing neutral conditions) or with a higherfraction of H on the surface, on both anatase and rutile surfaces. Conversely,the surfaces with a equal ratio of H and O in the adsorbates, or when O–terminated, generally have the a higher surface free energy. However, trendsin the surface free energy appear uncorrelated with the surface stress. These re-sults highlight, the importance of explicit calculation of the surface stress whenexamining surfaces under conditions such as those considered here, rather thanmaking implicit estimates from the surface free energy.

It is anticipated that these results will offer a guide as to possible charac-teristics of TiO2 surfaces under acidic and basic conditions, and hence theproperties of anatase and rutile nanocrystals that are sensitive to the chem-istry of the surfaces.

Acknowledgments

This work has been supported by the U.S. Department of Energy BES-ChemicalSciences, under Contract W-31-109-ENG-38. Computational resources for thisproject have been supplied by Argonne National Laboratory - LaboratoryComputing Resource Center, Pacific Northwest National Laboratory Molecu-lar Science Computing Facility and the U.S. Department of Energy NationalEnergy Research Scientific Computing Center. The authors would like tothank Tijana Rajh for advice regarding the suitability of adsorbates usedin this study.

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20

Figure 1: Relaxed low index H–terminated stoichiometric surfaces of titaniumdioxide, with adsorbates chosen to represent “highly acidic” conditions. Thesurfaces selected include the (a) anatase (001) surface viewed from the [010]direction, (b) rutile (100) surface viewed from the [001] direction, (c) anatase(100) surface viewed from the [010] direction, (d) rutile (110) surface viewedfrom the [001] direction, (e) anatase (101) surface viewed from the [010]direction, and (f) rutile (011) surface viewed from the [011] direction.

Figure 2: Relaxed low index hydrogen–rich stoichiometric surfaces oftitanium dioxide, with adsorbates chosen to represent “moderately acidic”conditions. The surfaces selected include the (a) anatase (001), (b) rutile(100), (c) anatase (100), (d) rutile (110), (e) anatase (101) and (f) rutile(011) surfaces. The orientations are as described for Figure 1.

Figure 3: Relaxed low index hydrated stoichiometric surfaces of titaniumdioxide, with adsorbates chosen to represent “neutral” conditions. The H2Omolecules may show either molecular or dissociative adsorbtion[22], howeveronly the lowest energy configuration is shown here. The surfaces selectedinclude the (a) anatase (001), (b) rutile (100), (c) anatase (100), (d) rutile(110), (e) anatase (101) and (f) rutile (011) surfaces. The orientations are asdescribed for Figure 1.

Figure 4: Relaxed low index hydrogen–poor stoichiometric surfaces oftitanium dioxide, with adsorbates chosen to represent “moderately basic”conditions. The surfaces selected include the (a) anatase (001), (b) rutile(100), (c) anatase (100), (d) rutile (110), (e) anatase (101) and (f) rutile(011) surfaces. The orientations are as described for Figure 1.

Figure 5: Relaxed low index O–terminated stoichiometric surfaces oftitanium dioxide, with adsorbates chosen to represent “highly basic”conditions. The surfaces selected include the (a) anatase (001), (b) rutile(100), (c) anatase (100), (d) rutile (110), (e) anatase (101) and (f) rutile(011) surfaces. The orientations are as described for Figure 1.

21

Table 1Comparison of average displacements (in A), normal to the surface, of atoms in theuppermost tri-layer of the anatase surfaces, along with the Ti–O and O–H bondslengths.

Surface Label Hydrogenated[32] Hydrogen–rich Hydrated[22] Hydrogen–poor Oxygenated

(001) O(1) 0.23 0.34 0.03 -0.40 -0.30

Ti(2) 0.12 0.08 0.09 0.17 0.06

O(3) 0.08 0.23 0.02 0.11 0.12

O(1)–H 1.00 1.00 1.01 — —

Ti(2)–H 1.79 — — — —

Ti(2)–O — 2.64 1.93 1.80 1.78

O–H — 1.01 0.99 0.98 —

(100) O(1) 0.13 0.17 -0.02 -0.08 -0.21

Ti(2) 0.07 0.07 0.10 0.10 -0.07

O(3) 0.00 0.11 0.01 -0.05 0.08

O(1)–H 0.99 0.99 0.99 — —

Ti(2)–H 1.72 — — — —

Ti(2)–O — 2.30 1.87 2.23 2.25

O–H — 1.00 0.99 0.98 —

(101) O(1) 0.02 0.04 -0.01 0.01 0.00

Ti(2) 0.16 0.04 -0.05 0.26 0.04

O(3) 0.24 0.24 0.10 0.09 0.14

O(1)–H 0.99 0.99 — — —

Ti(2)–H 1.74 — — — —

Ti(2)–O — 2.30 2.28 1.85 1.80

O–H — 1.00 0.99 0.99 —

22

Table 2Comparison of average displacements (in A), normal to the surface, of atoms inthe uppermost tri-layer of the rutile surfaces, along with the Ti–O and O–H bondslengths. ∗ sixfold coordinated, † fivefold coordinated. ‡ Unstable due to the desorp-tion of water molecules during the structural relaxation.

Surface Label Hydrogenated[32] Hydrogen–rich Hydrated[22] Hydrogen–poor Oxygenated

(100) O(1) 0.18 0.23‡ 0.13 -0.11 -0.05

Ti(2) -0.15 0.42‡ 0.02 0.17 0.12

O(3) 0.62 0.16‡ 0.16 0.19 0.19

O(1)–H 1.07 1.01 –1.02‡ 1.01 — —

Ti(2)–H 1.80 — — — —

Ti(2)–O — — 1.90 1.88 – 1.94 1.83

O–H — 0.99 – 1.01‡ 0.99 1.00 —

(110) O(1) 0.11 0.30 0.08 0.42 0.25

Ti(2) 0.02∗, 0.23† 0.03∗, 0.06† 0.35∗, 0.36† 0.17∗, 0.21† 0.05∗, 0.49†

O(3) 0.15 0.15 -0.05 0.00 -0.06 – 0.15

O(1)–H 0.99 0.99 — — —

Ti(2)–H 1.73 — — — —

Ti(2)–O — 2.35 – 2.38 2.30 1.80 1.78 – 1.95

O–H — 1.00 – 1.03 0.99 0.98 —

(011) O(1) 0.32 -0.01 0.17 -0.14 -0.18

Ti(2) 0.20 -0.15 – -0.20 0.19 0.41 – 0.54 0.66

O(3) 0.06 0.09 0.25 0.18 0.14

O(1)–H 1.02 1.01 – 1.03 — — —

Ti(2)–H 1.80 — — — —

Ti(2)–O — 2.42 2.14–2.16 1.80 – 1.84 1.94 – 2.04

O–H — 1.00 – 1.03 0.99 0.99 – 1.01 —

23

Table 3The surface free energy γ (in J/m2) for the low index stoichiometric anatase andrutile surfaces, calculated from equation 1.

Hydrogenated[32] Hydrogen–rich Hydrated[22] Hydrogen–poor Oxygenated

Anatase

(001) 1.88 2.28 1.55 1.89 2.55

(100) 1.41 1.67 1.13 1.58 1.53

(101) 1.14 1.41 1.03 1.50 2.07

Rutile

(100) 3.07 — 1.57 1.91 2.55

(110) 1.72 1.60 1.08 1.30 1.60

(011) 2.55 2.94 1.79 3.58 4.02

24

Table 4The isotropic component of the surface stress σ (in J/m2) for the low index stoi-chiometric anatase and rutile surfaces, calculated from equation 10.

Hydrogenated[32] Hydrogen–rich Hydrated[22] Hydrogen–poor Oxygenated

Anatase

(001) 0.91 0.41 -0.37 0.83 1.28

(100) -0.19 -0.08 -0.59 0.04 0.35

(101) 0.09 0.21 0.45 0.54 0.28

Rutile

(100) 0.80 — 0.61 0.49 0.63

(110) 1.27 1.65 0.92 1.69 0.67

(011) 1.38 -2.63 1.36 -1.33 -0.55

25

Table 5The anisotropic component of the surface stress (in mJ/m2) for the low index stoi-chiometric anatase and rutile surfaces.

Hydrogenated Hydrogen–rich Hydrated Hydrogen–poor Oxygenated

Anatase

(001) -4.00 -5.94 -3.85 -3.24 -7.72

(100) -1.37 -1.60 4.16 -4.03 -1.17

(101) -1.28 -1.55 -2.26 -0.90 -0.34

Rutile

(100) -15.02 — -5.36 -6.11 -6.90

(110) -2.50 -3.82 -2.55 -12.89 3.55

(011) -6.16 8.00 -3.13 -23.82 -3.91

26

Fig. 1.

27

Fig. 2.

28

Fig. 3.

29

Fig. 4.

30

Fig. 5.

31