Structural Characteristics and Mechanical and Thermodynamic

Review pubs.acs.org/CR

Structural Characteristics and Mechanical and Thermodynamic Properties of Nanocrystalline TiO2 Hengzhong Zhang* and Jillian F. Banfield Department of Earth and Planetary Science, University of California, Berkeley, California 94720, United States Corresponding Author Notes Biographies Acknowledgments References

1. INTRODUCTION Titania (TiO2) is one of the most chemically stable, environmentally compatible, and functionally versatile oxide materials. Due to its modified material properties and chemical reactivity at small sizes, nanocrystalline titania (nano TiO2) finds applications in many emerging areas, including solar energy harvesting and hydrogen generation from water through photochemical reactions1,2 and biomedical implants using devices coated with titania nanofilms,3,4 in addition to traditional applications for pigment, metallurgy, catalysis, etc. The modified material properties and functions are determined by the structures of nano TiO2, which are influenced by many external factors, predominantly temperature, pressure, and the surrounding environment. As titania may exist in one of several TiO2 phases, the phase stability of nano TiO2 at different conditions is key to its applications. Also, the mechanical properties of nano TiO2, such as the elasticity and deformation behavior, determine whether it can be used in places where mechanical compatibility is essential (e.g., for biomedical applications). In view of these, needed is a comprehensive understanding of the structures and mechanical and thermodynamic properties of nano TiO2 at small sizes and in different environments. Research over more than a decade has advanced understanding of many aspects of nano TiO2, including its synthesis, processing, properties, and applications.1 Within the many publications describing the structures and mechanical and thermodynamic properties of nano TiO2, inconsistences and even contradictory results are not uncommon. This hinders our understanding of fundamental structure−property relationships, impeding development of TiO2 nanomaterials. Here, we review literature reports on structural characteristics of TiO2 nanoparticles, particle size effects on the elasticity of nano TiO2, and both theoretical and experimental investigations of phase stability of TiO2 at small particle sizes and under different conditions. Thermodynamic results were analyzed using rigorous thermodynamic principles and some corrections are provided where thermodynamic treatments were incomplete. We also try to resolve a literature debate on the phase stability

CONTENTS 1. Introduction 2. Structures of Nanocrystalline TiO2 2.1. Structures of 11 TiO2 Crystalline Phases 2.2. TiO2 Phase Identification Using X-ray Diffraction and Pair-Distribution Function Analysis 2.3. Core−Shell Structure in TiO2 Nanoparticles 2.4. Lattice Contraction or Expansion in TiO2 Nanoparticles 2.5. Dislocations in TiO2 Nanoparticles 3. Mechanical Properties of Nanocrystalline TiO2 3.1. Bulk Modulus of Nano TiO2 by High-Pressure X-ray Diffraction 3.2. Elasticity of Nanostructured TiO2 by Nanoindentation and Compression/Bending Tests 4. Thermodynamic Properties of Nanocrystalline TiO2 4.1. General Thermodynamic Formulation 4.2. Phase Stability of Nano TiO2 Based on Computational Modeling 4.2.1. Phase Stability of Nano TiO2 Based on Atomistic and/or Molecular Dynamics Simulations 4.2.2. Phase Stability of Nano TiO2 Based on First Principle Calculations 4.3. Phase Stability of Nano TiO2 by Experimental Investigations 4.3.1. Surface Energies from Calorimetry Measurements 4.3.2. Surface Free Energies from Electrochemical Measurements 4.3.3. Surface (Free) Energies from X-ray Diffraction Determinations 4.3.4. Phase Stability Inferred from Nano TiO2 Syntheses and Post Treatments 5. Summary Author Information © 2014 American Chemical Society

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Special Issue: 2014 Titanium Dioxide Nanomaterials

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Received: February 12, 2014 Published: July 15, 2014 9613

dx.doi.org/10.1021/cr500072j | Chem. Rev. 2014, 114, 9613−9644

9614

ICSD #15409

3 6 18.53

8 0.03108

orthorhombic Pbca 61 9.184 5.447 5.145 90 90 90 4.123 octahedron

brookite

ICSD #41056

6 49.16

8 0.02815

monoclinic C2/m 12 12.1787 3.7412 6.5249 90 107.054 90 3.734 octahedron

TiO2(B)

CODd #1008514 + ref 9

6 73.05

8 0.02609

tetragonal I4/m 87 10.161 10.161 2.970 90 90 90 3.461 octahedron

TiO2(H)

ICSD #75179

6 68.49

4 0.02916

orthorhombic Pbnm 62 4.9022 9.4590 2.9585 90 90 90 3.868 octahedron

TiO2(R)

6 8.86 high P phase ICSD #15328

4 0.03263

orthorhombic Pbcn 60 4.515 5.497 4.939 90 90 90 4.329 octahedron

TiO2(II)

7 155.55 high P phase COD #9015355

monoclinic P21/c 14 4.589 4.849 4.736 90 98.6 90 5.092 augmented triangular prism 4 0.03839

baddeleyite

7 141.07 high P phase ICSDd #173960 + ref 13

8 0.03958

orthorhombic Pbca 61 9.052 4.836 4.617 90 90 90 5.251 distorted augmented triangular prism

OI


4 0.04345

orthorhombic Pnma 62 5.163 2.989 5.966 90 90 90 5.763 triaugmented triangular prism

OII


4 0.04343

cubic Fm3m ̅ 225 4.516 4.516 4.516 90 90 90 5.761 cube

cubic

a

Calculated. bPer polyhedron. cRelative to that of rutile; calculated at 0 K and 0 GPa. dModified from the referred structure file in the database ICSD 19 or COD 20 using structure parameters from the cited literature.

COD #9008213

4 6 24.75

2 6 0

ICSD #9161

4 0.02936

2 0.03203

#polyhedra per unit cell #polyhedra per unit cell volume (1/Å3) #edge sharingb CN(Ti) lattice energyc (kJ/mol) note reference

tetragonal I41/amd 141 3.7842 3.7842 9.5146 90 90 90 3.895 octahedron

anatase

tetragonal P42/mnm 136 4.5941 4.5941 2.9589 90 90 90 4.248 octahedron

rutile

crystal system space group group # a (Å) b (Å) c (Å) α (°) β (°) γ (°) densitya (g/cm3) polyhedron type

phase

Table 1. Structural Characteristics of 11 TiO2 Crystalline Phases (Structure Data Taken from Cited References)

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Figure 1. continued

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Figure 1. Structures of 11 TiO2 phases rendered (using the Vesta program21) in the ball-and-stick models and the polyhedron models. (a) rutile, (b) anatase, (c) brookite, (d) TiO2(B), (e) hollandite-like TiO2(H), (f) ramsdellite-like TiO2(R), (g) columbite-like TiO2(II), (h) baddeleyite-like phase, (i) OI phase, (j) cotunnite-like OII phase, and (k) fluorite-like cubic phase. Red balls: O atoms; blue balls: Ti atoms. Unit cells are outlined using thin lines.

Figure 2. X-ray diffraction patterns of 11 TiO2 bulk phases (Co Kα1 radiation, X-ray wavelength 1.7890 Å) calculated using the Visualize 29 program.

of very small and isolated TiO2 nanoparticles. Ultimately, we

2. STRUCTURES OF NANOCRYSTALLINE TIO2

hope that this provides a more correct and comprehensive

2.1. Structures of 11 TiO2 Crystalline Phases

understanding of the properties of nano TiO2.

There are at least 11 reported bulk and/or nanocrystalline phases of TiO2. These are rutile, anatase, brookite, the 9616


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Figure 3. Pair distribution functions of ∼4 nm particles of 11 TiO2 phases calculated using the Debyer program.32 The nanoparticle models were constructed from the coordinates of bulk TiO2 phases (Table 1).

insight into solid-state phase transformation mechanisms.23,24 For instance, twin planes in anatase have the brookite structure and can nucleate the anatase to brookite phase transformation (e.g., ref 23). Similarly, common structural slabs are present in anatase and TiO2(B). The surface of the slab in TiO2(B) defines the plane upon which steps anatase develop during the thermally driven phase transformation.25 Three phases of TiO2, rutile, brookite, and anatase, are commonly recognized to occur naturally. TiO2 (B) has been reported in natural samples a few instances, based on electron microscopic observation.22,25 For rutile, brookite and anatase, the numbers of shared edges are, respectively, 2, 3, and 4 out of the 12 edges per octahedron.26 Increased edge sharing decreases the Ti−Ti distances and hence increases the structure energy. This deduction is consistent with the results of lattice energy calculation based on the force field description of the TiO2 crystals,27 calculated using the program GULP 28 (Table 1). The calculated results show that for the six ambient/lowpressure bulk phases, the relative phase stability is rutile > brookite > anatase > TiO2(B) > TiO2(R) > TiO2(H), which scales roughly with the number of polyhedra per unit cell volume of a phase (Table 1).

columbite (α-PbO2)-like phase TiO2(II), the baddeleyite (ZrO 2 )-like phase, TiO 2 (B), the hollandite-like phase TiO 2 (H), the ramsdellite (MnO 2 or VO 2 )-like phase TiO2(R), the fluorite (CaF2)-like cubic phase, TiO2-OI, and the cotunnite(PbCl2)-like phase TiO2-OII (refs 5−12, 14, and 15.). In addition, there are at least 3 reported noncrystalline TiO2 phases: a low density amorphous TiO2 and two high density amorphous TiO2 types.16−18 The structural characteristics of the 11 crystalline phases are summarized in Table 1. The ball-and-stick and polyhedral renderings of these structures are illustrated in Figure 1. In Table 1, the first 6 phases (rutile to TiO2(R)) are stable at ambient or low-pressure. Their densities range from ∼3.5 (TiO2(H)) to ∼4.2 (rutile) g/cm3. The last 5 phases in Table 1 (TiO2(II) to cubic) are high pressure phases. Their densities range from ∼4.3 (TiO2(II)) to ∼5.8 (cubic) g/cm3. All TiO2 phases can be viewed as constructed by Ti-O polyhedra connected by a variable number of shared corners, edges, and/or faces. The Ti-O polyhedra in the low-pressure phases are all octahedra, whereas those of the high-pressure phases are octahedra, augmented triangular prisms, triaugmented triangular prisms, or cubes. The change in the shape of the polyhedron is a consequence of the change in the coordination number of Ti from 6 to 7, 8, and 9 as the pressure increases. Some of the phases share extended structural blocks, for example anatase and TiO2(B) contain chains of edge sharing octahedra in a single orientation whereas these chains occur in two orientations in rutile, brookite and TiO2(H) (see Figure 1). In fact, anatase, brookite and TiO2(II) are polytypes, i.e., they differ in the stacking arrangement of similar structural layers.22 These considerations can provide

2.2. TiO2 Phase Identification Using X-ray Diffraction and Pair-Distribution Function Analysis

In many cases, powder X-ray diffraction (XRD) is used to identify the phases of polycrystalline TiO2 materials (i.e., the bulk phases of TiO2). The calculated XRD patterns of the 11 TiO2 bulk phases are shown in Figure 2. In the given 2θ ranges, the XRD patterns are sufficiently distinct, making identification of a bulk TiO2 phase routine. However, at very small sizes, the broadened peaks of one or more TiO2 phases may overlap, 9617


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reduce the surface distortion and the shell thickness, increasing the crystallinity of the whole particle. The nano TiO2 core−shell structure was proved by synchrotron WAXS measurements and X-ray absorption spectroscopy of ∼2−3 nm TiO2 nanoparticles (Figure 5)

making it difficult to identify TiO2 phases when the particle size is small. Thus, full profile Rietveld analyses (see, e.g., ref 30) are needed for identification of very small (e.g., < 3 nm) nanophases, especially when multiple TiO2 nanophases are present in a same sample. An alternative method, atomic pair-distribution function (PDF) analysis31 that directly reflects the atomic pair correlations in a structure, may be used for identification of nano TiO2 phases. PDF data are generally derived from synchrotron wide-angle X-ray scattering (WAXS) of the samples.31 Due to the high brightness of the synchrotron Xray beams, there are very high signal-to-noise ratios in the X-ray scattering images and hence the derived PDF data can be of very high quality. Figure 3 shows calculated PDFs of ∼4 nm particles (from models without consideration of surface and lattice relaxations) of the 11 TiO2 phases. The positions of the first strongest peaks in these PDFs represent the average Ti-O bond lengths, ranging from ∼1.88 to 1.98 Å in the TiO2 phases. The PDF peaks damp drastically with increasing radial distance, mainly due to the finite sizes of the nanoparticles (in experimental PDFs, the instrumental resolution also contributes to this). Other indirect structure determination methods, such as Raman spectroscopy, can also be used to identify TiO2 nanophases, as reviewed in refs 1 and 33. 2.3. Core−Shell Structure in TiO2 Nanoparticles

For isolated TiO2 nanoparticles in vacuum, the surface structures, characterized by many unsatisfied (dangling) bonds,34 may be significantly distorted (relative to the bulk counterparts) as a result of the energy minimization of the whole particle as one system. Figure 4 illustrates the structure

Figure 5. Structure of a ∼2 nm amorphous TiO2 nanoparticle obtained from reverse Monte Carlo simulation, which fits best both the pair distribution function (PDF) derived from the wide-angle Xray scattering (WAXS) and the X-ray absorption (XAS) spectra of the amorphous samples. (a) Whole particle, (b) Cross section. Reprinted figure with permission from Zhang, H.; Chen, B.; Banfield, J. F. and Waychunas, G. A., Phys. Rev. B, 78, 214106, 2008 (Reference 38; http://dx.doi.org/10.1103/PhysRevB.78.214106). Copyright 2008 American Physical Society. Figure 4. Molecular dynamics simulation prediction of the structure (cross section) of an ∼5 nm anatase particle in vacuum. The surface layer (A) and the near-surface layer (B) are highly distorted, but the interior of the nanoparticle (C) retains typical anatase bulk structure. Region C disappears as particle diameter decreases toward ∼2 nm. Reprinted with permission from ref 35. Copyright 2001 Mineralogical Society of America and Geochemical Society.

synthesized from hydrolysis of titanium ethoxide at 0 °C.37,38 Reverse Monte Carlo simulations based on the PDF data (derived from the WAXS) confirmed the core−shell structure 38 (Figure 5). Due to the surface truncation and distortion in the particle shell and the lattice contraction in the core in small (3−5 nm) anatase particles, the average coordination number of Ti is reduced from 6 (in bulk anatase) to ∼5, and the average Ti-O bond length is shortened from 1.96 (in bulk anatase) to 1.92 Å, based on extended X-ray absorption fine structure analysis of these particles.39 In other systems, such as ∼3 nm ZnS nanoparticles, the surface distortion (or the crystallinity of the whole particle) can be easily tuned via changing the nanoparticle surface environment (see above), such as water,40,41 ionic salts and organic molecules,41 and the nanoparticle aggregation state.42 Although the chemical bonding is different in ZnS and TiO2 crystals (more covalent vs more ionic), the presence of the core−shell structures in their small nanoparticles arises from the same

of a ∼5 nm anatase particle, predicted from molecular dynamics (MD) simulations.35 Although the surface layer appears amorphous, the particle interior remains basically the bulk crystalline structure of anatase. Such a particle structure is often called the core−shell structure, a structure with a crystalline core and an amorphous shell. Now, this view of the nanoparticle structure is general in the literature, such as in ref 36. If nanoparticles are in an environment other than vacuum, such as in water, many dangling bonds may be partially satisfied via surface complexation with surrounding molecules. This can 9618


4−35 nm 2−7 nm anatase

3−12 nm anatase and 12−22 nm rutile in HCl fluid 5−40 nm anatase 3−25 nm anatase

5−26 nm rutile 5−23 nm rutile

no

no no

no no

hydrolysis of Ti isopropoxide in supercritical fluid with HCl or H2SO4 (in situ)

hydrolysis of Ti isopropoxide in supercritical water fluid (in situ) hydrolysis of titanium butoxide and calcination; gel from titanium isopropoxide and calcination hydrothermal hydrolysis of TiCl4 hydrothermal

5−40 anatase

no yes no

10−400 nm anatase 9−45 nm anatase 2−14 nm anatase 2−66 nm anatase

no yes no no

particle size and phase

sol−gel using Ti ethoxide/ isopropoxide solvothermal hydrolysis of TiCl4 in benzyl alcohol

sol−gel using Ti ethoxide sol−gel using TiCl4 sol−gel using Ti isopropoxide sol−gel using TiCl4 and tetrabutyltitanate; commercial samples chemical vapor synthesis

synthesis method

w/ int stnd

contraction for anatase and rutile in HCl fluid; expansion for anatase in H2SO4 fluid contraction contraction

negative in a, c negative in a, c (ex-situ expt.); negative in a, positive in c (in situ expt.) negative in a, c for anatase and rutile in HCl fluid; positive in a, c for anatase in H2SO4 fluid negative in a, c positive in a, negative in c

expansion expansion

contraction contraction (ex-situ); contraction then expansion (in situ)

positive in a, negative c

positive in a negative in c

expansion

positive in a, negative in c positive in a, negative in c

unit cell volume change contraction contraction expansion then contraction contraction then expansion

negative in a, c

strain in a, c

Table 2. Observed Lattice Contraction or Expansion of Nanocrystalline TiO2 as the Particle Size Decreases

negative negative

positive

positive for anatase and rutile in HCl fluid; negative for anatase in H2SO4 fluid positive

positive positive (ex-situ); positive then negative (in situ)

negative

positive positive either negative or positive positive then negative

surface stress

surface dipole Ti4+ vacancies

surface binding of benzyl alcohol in in situ expt. strong surface binding by SO42‑

surface adsorbed water

Ti vacancy

cause of negative surface stress

54 55

53

52

51

49 50

48

44 45 46 47

ref

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source, the incomplete satisfaction of the surface bonds due to the truncation of the crystal structure at the surfaces, which results in the surface distortion. 2.4. Lattice Contraction or Expansion in TiO2 Nanoparticles

The internal structure of small nanoparticles experiences high pressure due to surface stress.35 This introduces lattice contraction (i.e., negative strain) or expansion (i.e., positive strain) within the nanoparticles relative to the bulk phase, depending on whether the surface stress is tensile (positive value) or compressive (negative value). 35 The lattice contraction or expansion is usually calculated from the change in the unit cell volume at small sizes. XRD is commonly used to determine the lattice parameters of nanocrystalline TiO2. The accuracy of the measured 2θ angles is critical to the accuracy of the obtained lattice parameters. For obtaining high accuracy unit cell volumes, internal standards (such as NIST Si43) are needed for calibration of the measured 2θ angles. However, in many reports (see Table 2), internal standards were not used; instead, Rietveld analysis was commonly used. In Rietveld analysis, the 2θ shift can be optimized, together with many other adjustable parameters, in the global minimization of the intensity residuals. As the 2θ shift is adjustable in the minimization rather than fixed from calibrations using internal standards, Rietveld analysis cannot guarantee the accuracy of the derived lattice parameters. Many factors affect the surface stress of nanoparticles, including crystal structure, particle morphology and size, and nanoparticle surface state and surface environment (e.g., surface coating by organic molecules, adsorption of water molecules, and immersion in organic solvents, or solutions with different compositions). TiO2 nanoparticles synthesized from chemical routes such as hydrolysis of titanium alkoxide inevitably have surface-associated impurities due to adsorption of organic molecules and/or ions from the solution. This affects the nature and magnitude of the surface stress. Hence, both lattice expansion and lattice contraction in TiO2 nanomaterials have been observed experimentally (see Table 2). In anatase samples prepared by the sol−gel method using either titanium ethoxide 44 or TiCl4 45 as the Ti source, negative strain in the lattice parameters a and c and lattice contraction with decreasing particle size was observed. Djerdj and Tonejc determined the lattice parameters of several anatase samples (∼2−14 nm) synthesized using sol−gel method (using titanium isopropoxide as the Ti source).46 Rietveld analyses were used to derive the lattice parameters from XRD or selected area electron diffraction (SAED) data. Results showed that the lattice parameter c decreased by up to ∼0.7% and a increased by up to ∼0.3% when the particle sizes decreased from ∼14 to ∼4 nm. However, these authors reported nonmonotonic variation of the unit cell volume with the particle size (Figure 6). As no internal standard was used in the determinations and minor brookite phase was present together with the nano anatase, which could affect the surface stress of anatase nanoparticles, the observed nonmonotonic variation of the unit cell volume with the particle size may be apparent than inherent. In a report by Swamy et al.,47 2−66 nm anatase particles, some of which were prepared from TiCl4 and tetrabutyltitanate and some were obtained from commercial sources, also showed positive and negative strains in a and c, respectively. The unit cell contracted and then expanded as the particle size

Figure 6. Particle size dependence of the unit cell volume of anatase nanoparticles refined from Rietveld analysis using (a) XRD data and (b) SAED data. V0 is the unit cell volume of bulk anatase. Lit. value [15] is the value cited by Reference 46. Reprinted from (Reference 46) J. Alloys Compd., 413, Djerdj, I. and Tonejc, A. M., Structural investigations of nanocrystalline TiO2 samples, 159−174, Copyright 2006, with permission from Elsevier.

decreased. They ascribed this to the increasing Ti vacancy concentration in TiO2 as the particle size decreased. Ahmad et al. synthesized ∼5−40 nm anatase nanoparticles using a chemical vapor route.48 They observed size-dependence of the lattice parameters, similar to that reported by Djerdj and Tonejc.46 However, the unit cell volume appeared to increase monotonically with decreasing particle size. They attributed this to the adsorption of water molecules by nanoparticle surfaces, which caused parallel surface defect dipoles,54 leading to a negative surface stress and hence lattice expansion at small particle sizes. We synthesized 4−35 nm anatase particles using hydrolysis of Ti ethoxide (or Ti isopropoxide) at low temperatures (0−4 °C) and subsequently annealed the samples at different temperatures.37,49 Anatase nanoparticles are nearly spherical. NIST Si was used as an internal standard for 2θ calibration. XRD-based measurements of lattice parameters showed negative strains in both a and c in the samples, with higher strain in c than in a. The anatase lattice contracted as the particle size decreased (see Figure 7). These data were used to derive the surface free energy of nanocrystalline anatase as a function of particle size (section 4.3.3.2). Jensen at al. used both in situ synchrotron XRD and ex situ XRD to determine the size dependence of the lattice parameters of nanocrystalline anatase (2−7 nm) formed from 9620


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and rutile (∼12−22 nm), while lattice expansion was observed in the fluid with H2SO4 for nano anatase. The different structural response to different solution chemistry could be explained by the different binding strengths of Cl− and SO42‑ on the titania surface. Stronger binding of the latter than the former could result in a negative surface stress and hence a lattice expansion at small sizes, similar to the behavior of nano anatase bound by benzyl alcohol.50 A recent in situ synchrotron XRD study of pulsed supercritical synthesis of anatase nanoparticles in water− isopropanol mixture showed lattice contraction as the particle size decreased.52 The results are similar to those reported in our ex situ XRD study of the anatase samples synthesized using similar chemical reagents.49 In 3−25 nm anatase derived from titanium alkoxide and calcination, the lattice contraction was also observed at decreasing particle size.53 In these cases, the negative lattice strains at small sizes are from predominantly the “structure strains”. Li et al. synthesized several 5−26 nm nano rutile samples via hydrothermal hydrolysis of TiCl4 and determined the size dependence of the unit cell volume.54 It was found that the unit cell volume increased linearly with the reciprocal particle size in these samples. This is in contrast to those observed in some anatase samples (see e.g., ref 44, 45, and 49) and was explained as the result of repulsion of surface defect dipoles generated due to the distortion of the Ti-O octahedra on the nanoparticle surfaces.54 As rutile is in the shape of nanorods, the anisotropy in the particle morphology may also play a role in the lattice parameter changes. Kuznetsov et al. prepared 5−23 nm nanocrystalline rutile using hydrothermal hydrolysis of TiCl4.55 The size-dependence of the lattice parameters derived from synchrotron XRD showed increase in a and decrease in c with decreasing particle size.55 The unit cell volume increased by ∼1% when the size decreased to ∼5 nm. Following Swamy et al.,47 they attributed this to the increase in Ti4+ vacancies at small sizes. In nano CeO2 prepared by mixing cerium nitrate solution with an ammonium reagent, the lattice expansion was also explained by increased concentrations of point defects with decreasing particle size.56 However, Diehm et al. showed that point defects of various charge states are not the leading causes for the lattice expansion.57 Comparing the results from above studies, it is clear that both the particle size and the particle surface environment (that determines the particle surface state) have a significant impact on the structural state (assessed by measurement of lattice parameters) of TiO2 nanoparticles.

Figure 7. Size-dependent unit cell dimensions of nanocrystalline anatase. (a) Changes in lattice parameters a (closed circles) and c (open circles). (b) Change in unit cell volume. Reproduced from ref 49 with permission of the PCCP Owner Societies.

solvothermal hydrolysis of TiCl4 in benzyl alcohol at different temperatures (40−150 °C).50 In ex situ XRD experiments, where samples were cleaned beforehand by washing, Rietveld analysis showed negative strain in both a and c as the particle size decreased, with a larger strain in c than in a. However, in situ XRD experiments showed a different trend; the a displayed ∼1% negative strain while the c ∼2.5% positive strain at the onset of particle detection. The discrepancy in the two cases was interpreted as the consequence of the strong ligand (benzyl alcohol) binding on the growing anatase particles in the in situ experiments, which resulted in the preferred growth along c and hence its positive strain at small sizes. They infer that the strains can be decomposed into two components, the in-plane “structure strain” arising from the surface free energy and the positive “chemical strain” due to the surface ligand binding. The former accounted for the observed negative strain (due to positive surface stress) in the ex situ experiments; the latter for the positive strain (due to negative surface stress) in the in situ experiments. When the latter outweighs the former, a net positive strain and hence an expansion of the lattice parameter was generated, as observed in c at small sizes in the in situ experiments. In a relevant study, using in situ synchrotron XRD, Mi et al. studied the crystal growth and phase transformation of nano TiO2 (anatase, brookite and rutile) in supercritical fluid (with HCl or H2SO4 as a component) at 300 °C and 25 MPa.51 Rietveld analysis revealed substantially different lattice changes as a function of particle size in the fluid with HCl vs that with H2SO4. Lattice contractions (in a, c and unit cell volume) were observed in the fluid with HCl for nano anatase (∼3−12 nm)

2.5. Dislocations in TiO2 Nanoparticles

Conventionally, small nanoparticles were considered dislocation free because strong repulsive interactions between closely spaced dislocations should drive the dislocations from the nanoparticle structure.58,59 However, studies using highresolution transmission electron microscopy (HRTEM) showed presence of dislocations in hydrothermally coarsened anatase nanoparticles 59 (see Figure 8) and in anatase particles formed by annealing of dry nano powders 46 (Figure 9). Dislocations in anatase particles have been attributed to nanoparticle growth via oriented attachment (OA),23,59,60 i.e., the attachment of two or more smaller nanoparticles in crystallographically appropriate directions, forming a larger single crystal when the interface is eliminated. In OA, steps on a particle surface can be trapped at the particle−particle interface, 9621


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Particle size, morphology, aggregation state, and surface impurities (inherited from the synthesis) can all affect the bulk modulus of a nano TiO2 sample, as seen from data compiled in Table 3. Pressure medium (with/without its application) and the type of medium, not only affects the pressure transmission (i.e., hydrostatic vs nonhydrostatic conditions) but also offers a specific surface environment around nanoparticles. Thus, pressure medium may also influence the measured bulk modulus values. In order to study the size dependence of the bulk modulus of a nano TiO2 phase, samples should be consistent in all specifications except for the required variation in the particle size. This requires that the used samples be prepared using a same or very similar synthesis route so as to minimize effects from other sources, such as surface impurities, particle morphology and aggregation state. However, many data sets from the literature (see Table 3) are from samples of highly discrete sizes and/or of different sources. This makes it difficult to investigate the size-dependence of the bulk modulus in a consistent way. Given these considerations, we prepared a series of singlephase nanocrystalline anatase samples via heat treatment of amorphous titania produced from hydrolysis of titanium ethoxide (or isopropoxide) at low-temperatures (0−4 °C).37,78 Samples of equidimensional (near spherical) anatase with average particle sizes of ∼4−45 nm were prepared. Synchrotron HPXRD was used to determine the unit cell volumes of each sample during compression and decompression.66 The unit cell volumes of nano anatase at different pressures were obtained from Rietveld analyses and then fitted to the Birch−Murnaghan equation of state67 for derivation of the bulk modulus (see Table 3). Results showed that the variation of the bulk modulus with the particle size is nonmonotonic (Figure 10). 66 The bulk modulus exhibits a maximum value of ∼240 GPa at ∼15 nm. The spectacular variation of the bulk modulus of nano anatase with particle size was interpreted using a core−shell structure model (see section 2.3). As the particle size decreases, there are two opposing size effects influencing the bulk modulus, which are related to the generation of dislocations under high pressure 79 and their interactions at different particle sizes.66 First, the bulk modulus is increased due to the enhanced overlap of dislocation strain fields in smaller nanoparticles. Second, at very small sizes, interlocking dislocation networks are absent due to the inability to retain the dislocations inside small particles at high pressure. Further, the more compressible surface layers (i.e., particle shells with lower bulk modulus) represent a larger fraction of the particle at smaller particle sizes. The first and the second size effects dominate at particle sizes above and below ∼15 nm, respectively, explaining the nonmonotonic variation in bulk modulus with the particle size. Based on the core−shell structure model, an analytical equation was developed.66 This described the observed data quite well (see Figure 10). In comparison, an analytical model 80 based on approximate description of the size dependence of the average bond length and bond energy in a crystal failed to describe the observed data in ref 66, opposing to its claimed success. Al-Khatatbeh et al. measured the bulk modulus of 20 and 40 nm anatase particles.71 The samples were from two different commercial suppliers, and specifications of the synthesis methods and particle morphology were not provided. Their obtained bulk moduli differ significantly from those by Chen et

Figure 8. (a) HRTEM image of two attached anatasae particles showing the presence of an edge dislocation at the attaching interface. (b) HRTEM image (recorded down [100] anatase) of a portion of a anatase crystal formed by oriented attachment of at least four primary anatase particles. Arrowheads and lines (spaced 0.48 nm apart) indicate edge dislocations. From (ref 59) Penn, R. L.; Banfield, J. F. Science 1998, 281, 969. Reprinted with permission from American Association for the Advancement of Science.

Figure 9. (a) High-resolution image of an anatase sample prepared by sol−gel method and annealing at 500 °C. (b) Filtered image, obtained with anatase 101 reflection. The dislocations are denoted by black arrows. Reprinted from (ref 46) J. Alloys Compd., 413, Djerdj, I. and Tonejc, A. M., Structural investigations of nanocrystalline TiO2 samples, 159−174, Copyright 2006, with permission from Elsevier.

giving rise to dislocations. 59 Dislocations generated via oriented attachment or coalescence of nanoparticles were also observed in TiO2(B) nanowires synthesized by ion exchangingthermal treatment of titanate nanowires 61 and nanoparticles of rutile and anatase produced by laser ablation of titanium in oxygen flow.62−64 Recent in situ fluid-cell TEM observations revealed the dynamics of OA of FeOOH nanoparticles.65 This directly confirmed the OA process and provided insight into the dynamics of formation and migration of dislocations. Under high pressure, dislocations can also be generated and distributed in vicinity of grain boundaries of TiO2 nanoparticle.66 This has significant effects on the mechanical properties of nano TiO2 at different particle sizes (see section 3.1)

3. MECHANICAL PROPERTIES OF NANOCRYSTALLINE TIO2 3.1. Bulk Modulus of Nano TiO2 by High-Pressure X-ray Diffraction

Elasticity, represented by the bulk modulus, is an important mechanical property of materials. The bulk moduli of several nanocrystalline TiO2 phases have been studied rather extensively using synchrotron high-pressure X-ray diffraction and diamond anvil cells (DAC) techniques.67 Table 3 lists bulk moduli of several nano and bulk phases of TiO2. 9622


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Table 3. Bulk Modulus of Several Nanocrystalline and Bulk TiO2 Phases (Bulk Modulus Data Taken from Cited References) phase anatase anatase anatase

anatase

synthesis method commercial commercial nano samples: high-temp colloidal method

bulk sample: commercial nanosample: crystallization from amorphous nanopowders

nanoparticle shape NAa NA [100] elongated rod rice-shaped

near spherical

bulk: commercial

a

anatase

commercial

NA

anatase

hydrothermal

[100] elongated rod

baddeleyite baddeleyite rutile rutile

via compression of commercial anatase via heated compression of commercial 32 nm anatase NIST standard milling of bulk rutile

rutile

hydrothermal

rutile

ethylene glycol-mediated synthesis

elongated particle

elongated particle near spherical

particle size (nm)

bulk modulus (GPa)

bulk 30−40 3.5 × 21.0

178 243 204

3.8 × 5.0

319

bulk 4.0 6.5 7.2 9.0 13.5 15.0 21.3 24.0 30.1 45.0 bulk 20 40 50−200

241 185 199 211 230 237 239, 245 234 228 226 215 179 169 198 176

bulk 25−35

pressure range (GPa)

pressure medium

ref & note

0−8 0−16 0−10

NaCl none 16:3:1 M-E-Wb

68 69 nonhydrostatic 70 rather scattered data

1−12

4:1 M-E

66

0−11

16:3:1 M-E-W

71

0−10

4:1 M-E

72

303 298

17−40 15−46

Ar Ar

73 12

bulk 10 bulk 15

230 211 210 234

0−8 0−20

4:1 M-E M-E-W

74 75

0−18

NA

25

204

0−9

4:1 M-E

76 possibly nonhydrostatic 77

b

NA = not available. M = methanol, E = ethanol, W = water; number in volume ratio.

equation of state by fixing its zero pressure value to that of one of their commercial samples.71 This yielded a bulk modulus value much less than that reported previously.66 However, such a retreatment may not produce a value more accurate and reliable than that given originally,66 as the particle characteristics of the 15 nm sample 66 may differ appreciably (e.g., in surface impurity levels and particle morphology) from the commercial sample.71 Moreover, the constrained fitting may not guarantee simultaneous optimization of all model parameters. The particle shape effect on the bulk modulus of nano anatase can be seen from the work of Park et al.70 However, their unit cell volume data are rather sparse and hence the accuracy of the bulk modulus may be limited (Table 3). Their obtained bulk modulus of bulk anatase is 241 GPa, much higher than those (∼179 GPa) obtained in refs 68 and 66. Nonetheless, it was shown that the bulk modulus of the short “rice-shaped” nanoparticles (319 GPa) is much higher than that of the long “rod-shaped” ones (204 GPa), possibly due to the higher microstrain already contained in the former.70 This demonstrates the importance of understanding particle morphology when studying the size-dependence of the bulk modulus of nano TiO2. We now briefly discuss the bulk modulus of bulk TiO2-OII for reference in future studies of the bulk modules of nano OII. Using lattice dynamic and ab initio simulations, Dubrovinsky et

Figure 10. Size dependence of the bulk modulus of titania nanoparticles. In theoretical analysis, a core−shell model was applied to the nanoparticles. The solid curve represents the effective bulk modulus of the whole particle (core and shell); the dashed curve represents the bulk modulus of the particle core. Reprinted figure with permission from Chen, B., et al., Phys. Rev. B, 79,125406, 2009 (ref 66; http://dx.doi.org/10.1103/PhysRevB.79.125406). Copyright 2009 American Physical Society.

al. for particles of similar sizes.66 This discrepancy may come from the inconsistences in the surface and aggregation states and/or particle morphology of the nanoparticles in their samples. In a retreatment, they fitted the unit cell volume of the 15 nm sample of Chen at al.66 to the Birch−Murnaghan 9623


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al. calculated the bulk moduli of several known TiO2 phases.14 They found that the OII phase has the highest bulk modulus of ∼380−386 GPa among all the TiO2 phases. Then, using anatase or rutile as the starting materials contained in a laserheated diamond anvil cell, they produced TiO2 OII phase via heating at 1300−1500 °C and compression at 60−65 GPa. The experimentally determined bulk modulus at ambient temperature was 431 GPa, and the hardness (measured at 155−160 K) was 38 GPa. Based on these, they claimed the OII phase as the hardest known oxide material. However, later work showed that the bulk modulus of the OII phase has a much lower value than that reported in ref 14: 261 and 300 GPa by density functional theory (DFT) calculations using GGA and LDA, respectively, and 312 GPa by experimental determinations.81 Another experimental study resulted in a bulk modulus of 294 and 306 GPa for the bulk OII phase when the equation of state is described using the second and third Birch−Murnaghan equations, respectively.82 The previously reported very high value (431 GPa) was ascribed 82 to the nonhydrostatic condition used in the determinations by Dubrovinsky et al.

4. THERMODYNAMIC PROPERTIES OF NANOCRYSTALLINE TIO2 Bulk rutile is well-known to be more stable than bulk anatase at ambient conditions (see section 2.1). This has been proved once again by recent calorimetry determinations at temperatures up to 1300 K.87 However, production of titania using wet chemical routes such as a hydrolysis method (e.g., ref 88) often results in formation of the anatase phase. In fact, titania pigments produced via application of sulfuric acid in the 1920s to 1930s were mostly anatase.89 Over the past few decades, numerous titania syntheses via hydrolysis of titanium alkoxide also resulted in the formation of nanocrystalline anatase.1 These facts suggest that anatase upon its nucleation from the solution may be thermodynamically more stable than rutile. Anatase particles produced from wet chemical routes are usually in fine crystallites, with average diameters ranging from a few to tens of nanometers. Upon heating, the fine anatase particles coarsen and transform to rutile. In the 1960s, Yoganarasimhan and Yao 88 studied the influence of the grain size (not the particle size as described in ref 88) and the surface area of a few samples of anatase particles on the kinetics of transformation to rutile. They found that the smaller the grain size (and the higher the surface area), the faster the transformation. Subsequently, Banfield et al. studied the transformation kinetics systematically using a series of nanocrystalline anatase samples synthesized by sol−gel method.90 These authors attributed the more rapid transformation of nano compared to bulk anatase to rutile to the much lower measured transformation activation energy. They also proposed that anatase must reach a critical size before it transforms to rutile. Later, Gribb and Banfield 91 found that the average particle size of anatase upon first detection of the rutile phase appeared to be constant (∼15 nm; see Figure 11), with relatively little

3.2. Elasticity of Nanostructured TiO2 by Nanoindentation and Compression/Bending Tests

The hardness and modulus of elasticity of TiO2 nanofilms or nanowires can be measured using nanoindentation and compression/bending tests under microscopic operations.83,84 Kern et al. produced anatase films on AISI 316L stainless steel and Ti6Al4V alloy substrates using electrolytical depositions and subsequent annealing. 3 The anatase films were ∼140−190 nm thick, with slight Fe and Cr contamination due to thermal diffusion during annealing. Using scratch tests and nanoindentation, the hardness of the films was found to be 5−7 GPa and the modulus of elasticity (Young’s modulus) was 228 and 116 GPa for the two kinds of films. The Young’s modulus is close to that of the underlying substrate, providing good conditions for mechanical deformation of the film and the substrate as an integrated system. Conventional and nanostructured titania coatings on AISI 1018 low-carbon steel were produced by air plasma spray and high velocity oxy-fuel processes.85 The elasticity of various specimens (the film and the substrate as a whole system) was measured using cantilever bending experiments. The elastic modulus of the nano coated, conventionally coated and uncoated specimens is, respectively, 240, 215, and 195 GPa. This shows the enhanced elastic modulus of the specimen due to the adoption of the TiO2 nanostructures. Chang et al. synthesized anatase nanowires tens of microns in length and 100−350 nm in diameter by transforming titanate nanowires formed at high alkalinity and post-thermal treatment. 86 Using nano compression and bending tests manipulated within a scanning electronic microscopy (SEM), the modulus of elasticity was found to be 10−12 GPa. This value is significantly lower than the bulk modulus of anatase nanoparticles (see Table 3). Anisotropy in mechanical properties of anatase nanowires is expected based on the anisotropy in its crystal structure along different crystallographic orientations. However, their low measured value for the rather large nanowires cannot be explained by their suggested sources (e.g., the surface stress effect).

Figure 11. Average particle size of anatase in samples for kinetic experiments containing the first-detected rutile. Reprinted with permission from ref 91. Copyright 1997 Mineralogical Society of America.

dependence on heating temperature (350−525 °C) and reaction time (∼0.5−500 h). Based on these results, they suggested a phase stability reversal of rutile and anatase when the particle size is smaller than ∼15 nm (the so-called stability crossover size, or critical size). They attributed this effect to the higher surface energy of rutile than anatase. This research inspired later extensive research of the phase stability of nano TiO2, both experimentally and theoretically (as described below). 9624


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4.1. General Thermodynamic Formulation

particle dimension and morphology, and solution composition. In the following, we review research reports on thermodynamic phase stability of nanocrystalline TiO2 in accordance with the general thermodynamic principles.

A general thermodynamic description of a nanoparticle system was given previously.35 Suppose the nanoparticle system is comprised by nanoparticles (solid) immersed in an environmental fluid (liquid or gas). There are NS components in the nanoparticles. The nanoparticles may be present as phase α and/or phase β, depending on their phase stabilities at the equilibrium condition. In the fluid phase (solution), there are NF species. At thermodynamic equilibrium, the electrochemical potential of the same species in different phases becomes equal, and the total free energy of the nanoparticle system reaches its minimum. Temperature, external pressure, composition, morphology, and size of each phase all affect the free energy of the system. When these variables change, the total free energy G also changes in a way determined by the second law of thermodynamics: NS + NF

dG = −(

∑ ∑

Computational methods, including molecular dynamics (MD) simulations and density functional theory (DFT) calculations, have been used to study the structures and energies of TiO2 crystal surfaces or the whole TiO2 nanoparticles. The surface energies or energies of the whole nanoparticles were obtained from the simulations, which then were used for analysis of phase stability of TiO2 nanophases. 4.2.1. Phase Stability of Nano TiO2 Based on Atomistic and/or Molecular Dynamics Simulations. 4.2.1.1. Average Surface Energies of Anatase and Rutile with Wulff Morphologies. Oliver et al.92 calculated the surface energies of several surfaces of rutile and anatase using atomistic simulations based on the force-field description of TiO2 crystals developed by Matsui and Akoagi.27 The surface energies of rutile {011},{110},{100} and {221} are, respectively, 1.85, 1.78, 2.08 and 2.02 J/m2 ; those of anatase {011} and {001}, are, respectively 1.40 and 1.28 J/m2. These surfaces are expressed in the crystal morphologies constructed from the Wulff’s theorem,93,94 which are quite consistent with the observed experimental crystal shapes.92 The average surface energies for these surfaces are, respectively, 1.93 and 1.34 J/m2 for rutile and anatase. The higher average surface energy of rutile than anatase obtained by Oliver et al. was supported by calculations by Lazzeri et al.95,96 DFT calculations with LDA and PBE exchange-correlation functional were used to study the rutile (110) and anatase (101), (100), (001), (110), and (103) surfaces.95,96 The results showed that the average surface energy of a rutile crystal is higher than that of an anatase crystal, and this trend is independent of the form of the exchangecorrelation functional used. Both the atomistic simulation92 and DFT95,96 results support the deduction by Gribb and Banfield.91 4.2.1.2. Phase Stability of Nano Anatase and Rutile in Dry Conditions. For simplification, anatase and rutile nanoparticles were treated spherical in the thermodynamic analysis in ref 97. Suppose the average radius of anatase particles is r (average particle size D = 2r). Considering the phase transformation from one anatase nanoparticle to the rutile phase:

NS + NF

Si̅ ·ni)dT + (

i=1 NS + NF

+

4.2. Phase Stability of Nano TiO2 Based on Computational Modeling

∑

Vi̅ ·ni)d(P + Pexc)

i=1 J

μi̅ dni +

i=1

K

L

∑ γjdAj + ∑ σkdlk + ∑ λl j=1

k=1

l=1

(1)

where T, P, Pexc, and ni represent temperature, external pressure, excess pressure induced by surface stress in fine particles (for fluid, this term is zero), as well as the molar number of species i, respectively; S̅i, V̅ i, and μ̅i are the partial molar entropy, partial molar volume and electrochemical potential (which equals the chemical potential in the case of neutral species) of species i, respectively; γj is the interfacial free energy of interface j with area Aj formed by the fluid phase and one face of the nanoparticle; σk is the line energy of edge k with length lk formed by two interfaces; and λl is the point energy of vertex l formed by two edges or three faces. If there are grain boundaries inside nanoparticles, contributions to the total free energy from the grain boundaries, junctions of grain boundaries, and vertices of junctions can also be summarized in the last three terms of equation 1, respectively. In equilibrium at a certain condition (T, P, and parameters determining the geometry of each phase), the partial differential of G with respect to ni equals zero, producing NS+NF coupled differential equations. These NS+NF equations can be solved, giving the equilibrium concentrations of the NS+NF species in the system. At constant temperature and external pressure, equation 1 becomes NS + NF

dG = (

∑

NS + NF

Vi̅ ·ni)dPexc +

i=1 K

+

i=1

J

μi̅ dni +

∑ γjdAj j=1

L

∑ σkdlk + ∑ λl k=1

∑

l=1

(2)

Anatase nanoparticle (radius r )

As the particle size, particle morphology and particle surface environment can all affect the extensive quantities (such as μ̅i and γj) and/or the intensive quantities (such as ni and Aj) of the system, they all have some influence on the total free energy of the system and thus the phase stability of each nanophase. Phase transformation, particle coarsening and redistribution of species in different phases may occur to approach the minimum free energy state (equilibrium state) of the system. Then, the thermodynamic stability of each phase is determined by the system state variables, such as temperature, pressure,

= Rutile nanoparticle (radius ∼ r )

Upon full conversion from anatase to rutile, the radius of rutile = (ρA/ρR)1/3r = (3.9/4.2)1/3r = 0.98r ≅ r, where ρ is the density of titania (see Table 1) and subscripts R and A represent, respectively, rutile and anatase. Then, the free energy change (ΔG0) of the phase transformation is constituted by the contributions from the bulk phases, the surface free energy terms and the surface pressure terms 97 9625


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ΔG 0 = {Δf G 0(T , rutile) − Δf G 0(T , anatase)} ⎧ ⎛γ γ ⎞⎫ ⎪ 3M ⎪ ⎜⎜ R − A ⎟⎟⎬ + {ΔPR Vm,R − ΔPAVm,A } +⎨ ⎪ ρA ⎠⎪ ⎩ r ⎝ ρR ⎭ = Δf G 0(T , rutile) − Δf G 0(T , anatase) γ ⎞ M⎛ γ + (2t + 3) ⎜⎜ R − A ⎟⎟ ρA ⎠ r ⎝ ρR (3)

where ΔfG0 is the standard free energy of formation (available from the literature; see ref 97), γ the surface free energy of nanoparticles of titania, Vm the molar volume, and M the molecular weight. ΔP = Pexc is the excess pressure acting on the nanoparticles caused by the surface stress f described by the Young −Laplace equation:

Pexc = ΔP =

2f r

(4)

The surface stress is related to the surface free energy via the Shuttleworth equation98−100 f=γ+

∂γ ∂ε

(5)

where ε is the surface strain. Explicit relationship between surface free energy and surface stress is unknown for titania. In equation 3, surface stress was assumed t times the surface free energy, i.e., f = tγ. Using the average surface energies (as the surface enthalpies HS) derived from the atomistic simulations 92 (see section 4.2.1.1) and the estimated surface entropies (SS; see ref 97) to calculate the surface free energies (HS − TSS) of anatase and rutile, we analyzed the phase stability of nano anatase and rutile.97 According to equation 3, ΔG0 (T, r) = 0 defines the phase boundary between nano anatase and rutile (where both phases have the same thermodynamic stability). The calculated phase diagram is shown in Figure 12a. It is seen that, with the consideration of the surface stress in addition to the surface free energy, the calculated particle size of anatase (∼14−15 nm) for the phase stability crossover (between anatase and rutile) agrees with the experimental observations of Gribb and Banfield 91 quite well. This is the first thermodynamic analysis of the phase stability of the nanocrystalline TiO2 system. Note that, although the analysis assumed spherical shapes in the nanoparticles, the Wulff shapes were actually implied since the average surface energies were calculated from those of the surfaces exposed in the Wulff-constructed titania crystals. Using a thermodynamic formulation similar to eq 3 but with estimated surface stresses that are higher than the surface free energies of anatase and rutile, i.e., fA = 2.15 J/m2 > γA = 1.32 J/ m2 and f R = 2.75 J/m2 > γR = 1.91 J/m2, Lu et al. obtained a particle size similar to the one in ref 97 for the rutile to anatase phase stability reversal.101 However, the estimated surface stress of anatase is much higher than the value determined recently by experiments 49 (see section 4.3.3.2). 4.2.1.3. Phase Stability of Nano Anatase and Rutile under Hydrothermal Conditions. Under hydrothermal conditions, the interfacial free energies between titania surfaces and water are lower than the surface energies of the same surfaces in air (or vacuum) due to the surface hydration by water molecules. This makes the average interfacial free energy of anatase closer to that of rutile,35 favoring the transformation from nano

Figure 12. (a) Phase boundary between nanocrystalline anatase and rutile in dry conditions. Line A: calculated without consideration of surface stress (t = 0 in eq 3); line B: calculated with consideration of surface stress (t = 1 in eq 3). Points: size of anatase upon first detection of rutile.91 Reproduced from ref 97 with permission of The Royal Society of Chemistry. (b) Calculated phase diagram for the nano-TiO2 system under hydrothermal conditions (t = 1). Reprinted with permission from ref 35. Copyright 2001 Mineralogical Society of America and Geochemical Society.

anatase to rutile. In addition, the elevated water vapor pressure also favors the transformation, as rutile has a higher density than anatase. These factors decrease the rutile and anatase stability crossover from ∼14 nm (Figure 12a) at standard pressure and in air to ∼11 nm (Figure 12b), implying that under hydrothermal conditions nanocrystalline anatase is more easily converted to rutile. This calculated critical size (∼11 nm) is quite close to experimentally observed sizes of anatase (∼8− 11 nm) upon first detection of formed rutile in nano anatase samples treated hydrothermally at 250 °C.102 4.2.1.4. Phase Stability of Nano Anatase, Brookite and Rutile in Dry Conditions. In wet-chemistry (e.g., sol−gel) synthesis of nanocrystalline TiO2, brookite often forms together with anatase.44,103,104 Upon heating, both nano anatase and brookite transform to rutile.104 The relative stability of the three nanophases at a same particle size was analyzed using the relative enthalpy of the three phases, as shown in Figure 13.104 In the calculation, the employed surface enthalpies of rutile (1.93 J/m2), brookite (1.66 J/m2) and anatase (1.34 J/m2) were from the atomistic simulations 92 (see above) or estimations.104 Results show that anatase is more stable at < 11 nm, brookite is more stable between 11 and 35 nm, and rutile is more stable at > 35 nm. If brookite is not involved into 9626


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Figure 13. Variation of enthalpies of anatase (A), brookite (B), and rutile (R) with particle size. Reprinted with permission from ref 104. Copyright 2000 American Chemical Society.

the phase transformation, the anatase to rutile transformation occurs at 16 nm, similar to the size (14 −15 nm) from the more rigorous analysis.97 The closeness of the phase stability of brookite and anatase at small sizes may explain the concurrence of brookite with anatase at many sol−gel syntheses. 4.2.1.5. Phase Stability of Very Small Anatase, Brookite, and Rutile Nanoparticles in Dry Conditions. Naicker et al. used MD simulations to study the structures and energies of 2− 6 nm anatase, brookite and rutile nanoparticles at different temperatures.105 The employed force field was from Matsui and Akaogi.27 The surface energies of the particles were derived from the excess energies of the whole nanoparticles relative to the bulk phases, rather than from averaging those of the individually exposed surfaces as in ref 97. Results showed that the surface energies increase with increasing particle size (see Figure 14), which is in good agreement with our recent

Figure 15. Energies of anatase, rutile and brookite nanoparticles relative to bulk rutile as a function of surface area at 300 K based on molecular dynamics simulations. Reprinted with permission from ref 105. Copyright 2005 American Chemical Society.

constructed particles should have lower average surface energies (i.e., 1.93 and 1.34 J/m2 for rutile and anatase, respectively97) than the small and isolated spherical particles (e.g., 2.12 and 1.82 J/m2 for ∼2 nm rutile and anatase, respectively105) since some high energy surfaces can be exposed in the latter. The difference in the surface energies of rutile and anatase is larger in the former case (0.59 J/m2) than in the latter case (0.30 J/ m2). Thus, the larger Δγ = 0.59 J/m2 and smaller Δγ= 0.30 J/ m2 for the two cases result in a rutile to anatase stability crossover at a larger size (∼14−25 nm) for the Wullf-shaped nanoparticles and at a smaller size (∼2.5 nm) for the isolated and spherical nanoparticles. Using the same force field for TiO227 and MD simulations, Zhou and Fichthorn106 studied the energetics of small faceted anatase and rutile particles (in contrast to the spherical nanoparticles by Naicker et al.105). Anatase particles were enclosed by {101} and {001} surfaces, and rutile particles by {110} and {101} surfaces. The energies as a function of TiO2 units are illustrated in Figure 16. Figure 16a shows that the anatase to rutile transition occurs at several TiO2 units (and hence at several particle sizes), depending on the length ratio of {110} to {101} for anatase and the ratio of {110} to {101} for rutile. This shows clearly the influence of the particle morphology on the phase stability crossover of nano anatase and rutile. The phase stability reversal for the energy-minimized shapes of anatase and rutile occurs at ∼267 TiO2 units, corresponding to an anatase particle with 2.1 nm thickness along normal of {101} and 3.6 nm length in the [001] direction. The average of the two dimensions (∼2.9 nm) is close to that (∼2.5 nm) for spherical particles in ref 105. It is interesting to note, however, that in explicit MD calculations using spherical shapes, the phase stability crossover for anatase and rutile is at ∼4.1 nm by Zhou and Fichthorn,106 slightly larger than the size predicted by Naicker et al. 105 4.2.2. Phase Stability of Nano TiO2 Based on First Principle Calculations. First principle calculations, predominantly based on the density functional theory, were used to study the structures and energies of specific surfaces of bulk TiO2 crystals. The obtained surface energies (and surface stress data in some cases) were then used to construct the equilibrium

Figure 14. Surface energies as a function of particle diameter at 300 K by molecular dynamics simulations. Reprinted with permission from ref 105. Copyright 2005 American Chemical Society.

experimental determinations for anatase particles with sizes < 15 nm.49 The MD predicted nano anatase to rutile phase stability crossover is at ∼2.5 nm (see Figure 15), far less than those in refs 97 and 101. The difference in the calculated crossover sizes in the studies in refs 97 and 105 can be ascribed to the difference in the nanoparticle morphology. In ref 97, considered were expressed surfaces with relatively low surface energies in the Wulffconstructed crystals (see above). This is more typical for coarsened (and thus often aggregated) nanoparticles with rather developed morphologies. In contrast, in ref 105 the MD relaxed 2 - 6 nm nanoparticles are isolated and rather spherical. Many crystal surfaces can be exposed in such small and spherical particles. Comparing these two cases, the Wulff9627


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respectively. This complicated formulation was used for consideration of the shape effect on nanoparticle phase stability in the works of Barnard et al.107 Note that, in eqs 1−5 and 6−11 (below), different symbols may have been used to represent the same physical quantity (e.g., for the surface stress, f was used in the former and σ in the latter). In case of spherical or near spherical particles, the fraction of volume contraction (e) under surface pressure ΔP (caused by surface stress σ; also obsoletely termed surface tension in ref 107) can be estimated using the compressibility β (=inverse of bulk modulus) of the particle:107 e=

ΔV 2σ = ΔPβ = β V r

(7)

In case of faceted crystals, the more complicated Stoneham formulation 108 can be used. However, this may introduce other quantities that are hard to obtain. Barnard and Zapol used another way to calculate e:109 e=

2 ∑i fi σi 2σ β= β r r

(8)

where σi is the surface stress of face i and r the average radius of the particle; f i bears the same meaning as in equation 6. If an external pressure (Pext) is applied to the nanoparticles, eq 8 becomes110,111 Figure 16. Potential energy of (a) series of quasi-Wulff shapes of anatase (A, red symbols) and rutile (R, green symbols) with various length-to-width ratios and (b) minimum-energy shapes as a function of the number of TiO2 units. The size of the particle (in nm) along the normal of anatase {101} plane and rutile ⟨110⟩ direction is given in parentheses. O ions are red and Ti ions are gray. Reprinted with permission from ref 106. Copyright 2012 American Chemical Society.

e=

⎛ 2 ∑ f σi ⎞ ⎛ 2σ ⎞ i i ⎜ + Pext⎟β = ⎜⎜ + Pext⎟⎟β ⎝ r ⎠ ⎝ r ⎠

Neglecting the free energy contributions from the edges and corners for nanoparticles with large enough sizes, eq 6 is simplified to109 Gx0 = Δf Gx0 +

crystal morphology and to analyze the phase stability in nanocrystalline TiO2 systems. 4.2.2.1. Free Energy of Nanoparticles with Certain Morphology. Barnard et al. have done a series of DFT calculations for analyses of the phase stability of anatase and rutile nanocrystals exhibiting certain morphologies (see below). In their thermodynamic model (equation 6), the total free energy of a nanoparticle is constituted by the bulk energy, the surface energies from all expressed surfaces (faces), the edge energies of all exposed edges and corner energies of all exhibited corners in the crystal.107 The crystal volume change under surface pressure at small sizes was considered. Gx0

=

Δf Gx0

k

M (1 − e)[q ∑ fi γxi] ρx i

(10)

However, in eqs 6 and 10, the correction of the particle molar volume under surface pressure by application of a scale factor (1 − e) does not really account for the increased free energy due to the acquisition of the excess surface pressure. This additional free energy should be calculated as ∫ V dP ≈ VPexc = VΔP (see eqs 1 and 4). As such, for a more accurate thermodynamic description, eq 10 should be modified as Gx0 = Δf Gx0 + = Δf Gx0 +

M + (1 − e)[q ∑ fi γxi(T ) + p ∑ gjλxj(T ) ρx i j

+ w ∑ hk εxk(T )]

(9)

M (1 − e)[q ∑ fi γxi + Pexc] ρx i ⎡ 2 ∑i fi σxi ⎤ M ⎥ (1 − e)⎢q ∑ fi γxi + ⎥⎦ ρx r ⎣⎢ i

(11)

A similar correction term (last term in eq 11) should also be applied to eq 6. If an external pressure (Pext) is applied to the nanoparticles, Pext should also be added to the square brackets. In eqs 6−10, if the density (ρx) and compressibility (β) of the bulk phase are used in the total free energy calculations, i.e., without consideration of the expected size effect (see Table 3) on both quantities, calculation of the molar volume using (M)/ (ρx)(1 − e) may introduce significant errors on Gx0 at small nanoparticle sizes (e.g., < 5 nm). 4.2.2.2. Phase Stability Crossover in Anatase and Rutile Nanoparticles with Clean and Hydrogenated Surfaces. Structures, surface energies and surface stresses (“surface tensions”) of clean and hydrogenated (i.e., hydrogen-terminated) low index surfaces of anatase and rutile were studied

(6)

0

In the equation, Gx is the total molar free energy of the nanoparticle at temperature T; Δf Gx0 is the molar free energy of formation of the bulk counterpart; M is the molecular weight; ρx is the density; e is the fraction of volume contraction under surface pressure (called volume dilation in ref 107); q, p and w are, respectively, the ratios of the total surface area, edge length and number of corners to the particle molar volume; γxi, λxj and εxk are, respectively, the surface energy of face i, edge energy of edge j and corner energy of corner k; f i, gj, and hk are, respectively, the fractions of face i, edge j and corner k in the total surface area, total edge length and total number of corners, 9628


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using DFT calculations by Barnard and Zapol.112 The simulated hydrogenated surfaces were used to represent protonated TiO2 surfaces at highly acidic aqueous conditions.112 Considered surfaces were anatase (001), (100), (110), and (101) and rutile (001), (100), (110), and (011). Three methods were used to treat the in-plane lattice parameters of a considered surface and the internal parameters of a titania crystal in the calculations. The derived surface energy and surface stress data using the third method were more reliable, because the method adopted a uniform dilation in the two surface dimensions and an optimization of the internal lattice parameters as well. On average, the surface energies of rutile are higher than those of anatase, consistent with previous calculations.92,95,96 The Wulff-constructed morphologies of anatase and rutile are shown in Figure 17. The hydrogenation affected the

Figure 18. Free energy as a function of number of TiO2 units for anatase and rutile crystals (in shapes in Figure 17) with clean (a), partially hydrogenated (b), and fully hydrogenated (c) surfaces. The phase stability crossover is indicated by the arrow at the given anatase sizes. Reprinted figure with permission from Barnard, A. S. and Zapol, P., Phys. Rev. B, 70, 235403, 2004 (ref 112; http://dx.doi.org/10.1103/ PhysRevB.70.235403). Copyright 2004 American Physical Society.

Figure 17. Calculated morphologies of anatase (left) and rutile (right) with clean (top), partially hydrogenated (center) and fully hydrogenated (bottom) surfaces. Reprinted figure with permission from Barnard, A. S. and Zapol, P., Phys. Rev. B, 70, 235403, 2004 (ref 112; http://dx.doi.org/10.1103/PhysRevB.70.235403). Copyright 2004 American Physical Society.

equilibrium morphologies to some extent, as expected, due to its influence on the surface energies of the exposed surfaces. The calculated phase stability crossover for nano anatase and rutile (based on equation 10) depends on the surface states of the crystals showing the Wulff morphologies.109,112 With clean surfaces, the crossover average sizes are 9.3 nm for anatase and 9.0 for rutile; with partially hydrogenated surfaces, the sizes are 8.9 nm for anatase and 8.6 for rutile; and with fully hydrogenated surfaces, the sizes are 23.1 nm for anatase and 22.1 for rutile (see Figure 18). Interestingly, if anatase and rutile particles are spherical rather than in the Wulff shapes, the crossover size is ∼2.6 nm,112 close to that by MD105,106 and other DFT calculations.113 This can also be attributed to the

higher surface energies in the spherical particles yet smaller differences in them for anatase and rutile at the small sizes (see section 4.2.1.5). The results from these works demonstrated the importance of the nanoparticle surface state (determined by the surface environment) on the phase stability of nano TiO2, because it can modify the particle morphology and hence the average surface energies and surface stresses. The surface state effect is manifested when considering many possible equilibria between anatase and rutile crystals with different shapes (such as anatase with bipyramid or truncated bipyramid shapes in equilibrium 9629


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Table 4. Surface Energy (γ, J/m2) and Surface Stress ( f, N/m) of Anatase Crystals from DFT Calculationsa,b surface state clean (fully) hydrogenated hydrogen-rich (partially hydrogenated) hydrated hydrogen-poor oxygenated

γ (001)

f (001)

γ (100)

f (100)

γ (101)

f (101)

ref

0.51 0.84 1.88 0.86 2.28 1.55 1.89 2.55

2.07 0.91 0.91 0.50 0.41 −0.37 0.83 1.28

0.39 0.65 1.41 0.55 1.67 1.13 1.58 1.53

0.60 −0.19 −0.19 0.23 −0.08 −0.59 0.04 0.35

0.35 0.63 1.14 0.51 1.41 1.03 1.50 2.07

0.51 0.09 0.09 0.71 0.21 0.45 0.54 0.28

112 112 114 112 114 114 114 114

a

Table adapted partially from (ref 114) Surf. Sci., 582, Barnard, A. S.; Zapol, P., Curtiss, L. A., Anatase and rutile surfaces with adsorbates representative of acidic and basic conditions, 173−188, Copyright 2005, with permission from Elsevier. bTable adapted partially with permission from Barnard, A. S. and Zapol, P., Phys. Rev. B 2004, 70, 235403 (ref 112; http://dx.doi.org/10.1103/PhysRevB.70.235403). Copyright 2004 by the American Physical Society.

Table 5. Surface Energy (γ, J/m2) and Surface Stress ( f, N/m) of Rutile Crystals from DFT Calculationsa,b γ (100)

f (100)

γ (110)

f (110)

γ (011)

f (011)

ref

hydrogen-rich (partially hydrogenated)

0.60 1.82 3.07 0.71

0.95 0.80 0.80 0.66

hydrated hydrogen-poor oxygenated

1.57 1.91 2.55

0.61 0.49 0.63

0.47 0.84 1.72 0.56 1.60 1.08 1.30 1.60

1.25 1.27 1.27 1.96 1.65 0.92 1.69 0.67

0.95 1.19 2.55 1.02 2.94 1.79 3.58 4.02

1.50 1.38 1.38 0.39 −2.63 1.36 −1.33 −0.55

112 112 114 112 114 114 114 114

surface state clean (fully) hydrogenated

a

Table adapted partially from (ref 114) Surf. Sci., 582, Barnard, A. S.; Zapol, P., Curtiss, L. A., Anatase and rutile surfaces with adsorbates representative of acidic and basic conditions, 173−188, Copyright 2005, with permission from Elsevier. bTable adapted partially with permission from Barnard, A. S. and Zapol, P., Phys. Rev. B 2004, 70, 235403 (ref 112; http://dx.doi.org/10.1103/PhysRevB.70.235403). Copyright 2004 by the American Physical Society.

J/m2 (oxygenated). The surface stresses of rutile are also generally greater than those of anatase. However, the surface energies of hydrated surfaces (i.e., with adsorption of H2O) are all higher than those in vacuum (i.e., clean surface) (Tables 4 and 5). This is inconsistent with the general trend of decrease of surface energy upon adsorption of water molecules, as seen from calorimetry determinations (section 4.3.1.4). With the data in Tables 4 and 5, Barnard and Curtiss optimized the shapes of anatase and rutile nanoparticles with different surface states by making use of both the surface energy and surface stress of each surface using a variant of the Wulffconstruction,115 as shown in Figure 19. Again, the surface state dependence of the morphologies is clearly shown. Then, the free energies of nano TiO2 were calculated from equation 10 as a function of the number of TiO2 units for anatase and rutile particles in the optimized shapes, as shown in Figure 20. The phase stability crossover for anatase and rutile occurs at sizes of approximately 22.7, 18.4, 15.1, 13.2, and 6.9 nm, respectively, for titania nanocrystals with fully hydrogenated surface, hydrogen-rich surface, hydrated surface, hydrogen-poor surface and oxygenated surface.115,116 Note again that the surface pressure contribution (i.e., the ∫ V dP ≈ VPexc = VΔPterm) to the free energy was not considered in these calculations. Due to this, the stability crossover sizes (transition sizes) might have been underestimated, possibly by ∼6 nm each (see section 4.2.2.2). When the particle surfaces are clean (i.e., in vacuum or air), the transition size is 9.6 nm,117 similar to the one (9.3 nm) reported previously by Barnard et al. (Figure 18a, ref 112.). In comparison, the transition size in water (i.e., for hydrated surfaces) is 15.1 nm (Figure 20c).115 This size is even larger than that (9.3 or 9.6 nm) in vacuum (or air), contradicting our

with rutile in the Wulff shapes possessing different height-towidth ratios), as seen from several energy plots in ref 109. However, as indicated above, these calculations (based on equation 10) did not consider the energy contribution by the surface pressure (i.e., the ∫ V dP ≈ VPexc = VΔP term). It is known that without consideration of this term, the phase stability crossover can be underestimated by ∼6 nm (see Figure 12a). Adding this size to the anatase size for clean surfaces (9.3 nm, Figure 18a), the rutile to anatase phase stability reversal is at ∼15 nm, close to that (∼14−15 nm) by previous analysis based on average surface energies from MD simulations.97 4.2.2.3. Phase Stability Crossover in Anatase and Rutile Nanoparticles with Hydrogenated, Hydrated, and Oxygenated Surfaces. To represent the surface states of anatase and rutile crystals in aqueous solutions that are highly acidic, moderately acidic, neutral, moderately basic and highly basic, respectively, fully hydrogenated, hydrogen-rich, hydrated, hydrogen-poor and oxygenated TiO2 crystal surfaces corresponding to above conditions, respectively, were constructed for DFT calculations.114 Anatase (001), (100), and (101) and rutile (100), (110), and (011) surfaces were considered. In each surface state, the structure of a considered surface was relaxed, leading to changes in the atomic bond lengths and/or atomic rearrangements (i.e., the surface reconstruction) with respect to the as-constructed surface structure. Surface energies and surface stresses were calculated (see Tables 4 and 5). The general trend of higher average surface energy of rutile than anatase was apparent for all surface states. The average surface energy of anatase vs that of rutile at above five surface states, are, respectively, 1.48 vs 2.45 J/m2 (fully hydrogenated), 1.78 vs 2.27 J/m 2 (hydrogen-rich), 1.24 vs 1.48 J/m 2 (hydrated), 1.66 vs 2.26 (hydrogen-poor), and 2.05 vs 2.72 9630


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alkalinity of the solution in this case, the phase boundary also changes with the solution conditions. The anatase = rutile phase boundaries are shown in blue and red lines, respectively, for the neutral and basic conditions (Figure 22). The stability region of anatase at a neutral condition (wheat color area) is narrower than that at a basic condition (wheat color area + green yellow area), indicating anatase is more favored in basic solutions, in agreement with experimental observations (see section 4.3.4.1). In ref 110, the region between the blue and red boundary lines (i.e., the green yellow area) was interpreted as a region for coexistence of nano anatase and rutile with a surface state between fully hydrated and fully hydroxylated states. However, this violates the phase rule. For this system, C = 1 (TiO2), adjustable state variables = 2 (temperature and size; external pressure is fixed). Then the number of degree of freedom F = C − P (number of phases) + 2. If two phases (anatase and rutile coexist), F = 1 − 2 + 2 = 1, which means only one variable is free to change. Hence, equilibrium of two phases can only occur on a phase boundary (such as the blue curve in case of hydrated surfaces), not in a “coexistence” area where F = 2. The “coexistence” area should be considered as a collection of all possible phase boundaries (such as the dashed curve) as the surface state changes from fully hydrated to fully hydroxylated. This corresponds to partially H2O-terminated and partially OH-terminated surfaces in different proportions. Again, the surface state is determined by the solution chemistry. If one considers H2O as a component of the system, one more liquid phase is added. In this case, the above analysis of F remains unchanged (F = 2 − 3 + 2 = 1). The reported experimental observation of nano anatase together with rutile in the synthesis of nano TiO2 in 0.2 M HCl at 180 °C 110 does not affirm true thermodynamic equilibrium between the two phases. Moreover, the phase boundary at acidic conditions was not considered in Figure 22. Bear in mind also that the thermodynamic calculations110 using equation 10 did not consider the surface pressure contribution to the total free energy (i.e., the ∫ V dP ≈ VPexc = VΔP term).

Figure 19. Morphology predicted for anatase (top) with (a) hydrogenated surfaces (b) with hydrogen-rich surface adsorbates, (c) hydrated surfaces, (d) hydrogen-poor adsorbates, and (e) oxygenated surfaces, and rutile (bottom) with (f) hydrogenated surfaces, (g) with hydrogen-rich surface adsorbates, (h) hydrated surfaces, (i) hydrogenpoor adsorbates, and (j) oxygenated surfaces. Reprinted with permission from ref 115. Copyright 2005 American Chemical Society.

previous analyses (Figure 12) supported by experimental observations (see section 4.2.1.3). This indicates that the theoretical prediction of higher surface energies of TiO2 in water117 than in vacuum (air)112 (see Tables 4 and 5) is unreasonable, as proved by calorimetry measurements (see section 4.3.1.4). Based on the calculated results (Figure 20), a phase stability diagram was constructed (Figure 21). According to this diagram, the stability region of anatase decreases with increasing solution pH. However, this trend is inconsistent with many experimental observations that show that nano rutile is favored at highly acidic conditions and nano anatase in highly basic conditions (see section 4.3.4.1). The contradiction may have two explanations. First, the representation of the TiO2 surfaces at basic conditions using the oxygenated terminations may be far from realistic, as hydroxylated (−OH) rather than oxygenated (-O) surfaces are usually expected, especially at high alkalinity. Second, at pH values different from the point of zero charge of titania (pHPZC = ∼5.9−6.3, ref 78), titania nanoparticle surfaces are charged. The electrostatic potential also contributes to the potential energy of the particles (i.e., it needs to be included into the electrochemical potential term; see eq 1). This was not included in the thermodynamic formulation by equation 10. 4.2.2.4. Phase Stability Crossover in Anatase and Rutile Nanoparticles with Hydrated and Hydroxylated Surfaces. Having recognized oxygenated TiO2 surfaces cannot accurately represent the surface states of titania crystals in basic aqueous solutions (section 4.2.2.3), Barnard and Xu used hydroxyl (OH)-terminated surfaces to represent the surface state in basic conditions.110 Using DFT derived surface energy and surface stress data, they optimized the nanocrystal morphologies of anatase and rutile as a function of both the temperature and particle size for both the hydrated (terminated by H2O) and hydroxylated (OH terminated) surfaces. Then, the free energies of anatase and rutile as a function of temperature and number of TiO2 units (from eq 10) were used to calculate the anatase = rutile phase boundary, as shown in the temperature-particle size phase diagram in Figure 22. Figure 22 shows that the phase boundary between nano anatase and rutile depends on the surface state. As the surface state is determined by the solution chemistry, the acidity or the

4.3. Phase Stability of Nano TiO2 by Experimental Investigations

4.3.1. Surface Energies from Calorimetry Measurements. 4.3.1.1. Differential Scanning Calorimetry. An early attempt by Terwilliger and Chiang to measure the grain boundary energies of nano TiO2 (rutile) employed a differential scanning calorimetry (DSC) technique.118 However, as there were many sources contributing to the measured enthalpy change, including phase transformation of residual anatase phase to rutile, oxidation due to initial nonstoichiometry in TiO2, relief of lattice strain upon heating and the heat content in temperature scanning, many corrections were needed. This made the measured grain boundary energies inaccurate. The obtained specific excess enthalpy at relatively low temperatures (600−780 °C) and fine grain sizes (30−200 nm) is 0.5−1.0 J/ m2, whereas the value averaged over a larger temperature (600−1300 °C) and size (30 nm−2 μm) range is 1.3−1.7 J/m2. 4.3.1.2. Solution Calorimetry with Enthalpy Corrected for Physically Adsorbed Water (Bulk Water). Measurements of the surface enthalpies (surface energies) of titania nanoparticles 36 have been made using microcalorimeters.119 The first calorimetry determinations of nano rutile, anatase and brookite were conducted by Ranade et al. in 2002 using high9631


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Figure 20. Free energy as a function of the number of TiO2 units for anatase and rutile with (a) hydrogenated surfaces, (b) hydrogen-rich surface adsorbates, (c) hydrated surfaces, (d) hydrogen-poor adsorbates, and (e) oxygenated surfaces, calculated using the shapes given in Figure 19 and the surface energy and surface stress data in Tables 4 and 5. The intersection points indicate the phase transition at the designated average sizes of anatase. Reprinted with permission from ref 115. Copyright 2005 American Chemical Society.

temperature oxide melt drop solution calorimetry.120 Limited sets of nano samples were synthesized and characterized (by XRD and surface area measurements) for the calorimetric determinations. A nano sample was dropped into a 3Na2O· 4MoO3 melt (kept at 975 K) contained in the measurement cell of a Calvet twin microcalorimeter. The enthalpy change associated with the dissolution of the sample in the molten solvent was measured by the calorimeter. Through designed thermochemical cycles, the enthalpy change with respect to bulk rutile at 298 K was obtained after correction for the adsorbed water by assuming water molecules were adsorbed physically. The transformation enthalpy of a sample was then defined as the enthalpy change (which has a positive value). This way, the transformation enthalpy of a nano sample equals the transformation enthalpy of a bulk sample plus the total surface enthalpy of the nano sample. For instance, for a nano anatase sample:

ΔHtrans(bulk rutile = nano anatase) = ΔHtrans(bulk rutile = bulk anatase) + γ(anatase)· A(anatase) (12)

where γ is the surface enthalpy and A is the surface area. According to eq 12, a plot of the transformation enthalpy of a nano sample versus the sample surface area should produce a straight line, the intercept of which gives the transformation enthalpy of the corresponding bulk phase and the slope gives the surface enthalpy of the nano sample. Figure 23a−c shows such plots for samples of nano rutile, anatase and brookite. The surface enthalpies derived from linear least-squares fittings are 2.2, 0.4, and 1.0 J/m2 respectively for rutile, anatase, and brookite.120 The values derived from atomistic simulations and estimations (1.93, 1.34, and 1.66 J/m2 respectively for rutile, anatase, and brookite104) differ from these values significantly, especially for anatase and brookite. This may be due to the differences between the surfaces of the experimental samples120 and those expressed in the Wulff shapes from the simulation.92 Other factors contributing to the deviations may include 9632


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However, the stability crossover sizes from the calorimetry experiments differ evidently from those in ref 104 (Figure 13). These deviations come from the differences in calorimetric surface energies and those from the atomistic simulations (see above). 4.3.1.3. Solution Calorimetry with Enthalpy Corrected For Chemically and/or Physically Adsorbed Water. An improved calorimetry determination of the surface enthalpies of nano rutile and anatase was done by Levchenko et al. in 2006 by using a number of phase-pure anatase and rutile samples with sizes ranging from 6 to 40 nm and surface areas 5−270 m2/g.121 Rutile particles were nanorods while anatase particle were approximately spherical. As titania samples synthesized and/or processed in air commonly adsorb (both chemically and physically) water molecules, the enthalpy of solution of a titania sample was carefully corrected for the enthalpy of desorption of water upon dissolution of titania using the enthalpy of adsorption of water measured independently. The formation (transformation) enthalpies of rutile and anatase samples are shown in Figure 24a,b. From these data, the derived surface enthalpies of anhydrous rutile and anatase are, respectively, 2.22 and 0.74 J/m2. For hydrated rutile and anatase, these are, respectively, 1.89 and 0.51 J/m2. These values are comparable to the corresponding values (2.2 and 0.4 J/m2) from the previous work.120 The updated values should be considered more accurate. A revised enthalpy vs surface areas diagram for rutile, anatase and brookite based on the updated surface enthalpy data is shown in Figure 24c. From this plot, anatase is more stable at < 30 nm, brookite is more stable in 34−200 nm, and rutile is more stable at > 200 nm. The rutile to anatase stability crossover is at 1100 m2/mol (or ∼104 nm).121 A recalculation showed this size is ∼67 nm.122 This updated critical size is closer to the value (∼14−15 nm) from the thermodynamic analysis 97 than the value (∼79 nm) predicted based on the previous calorimetric determination 120 (see above). Note that this size (∼67 nm) is for transition between rutile nanorods (not spherical particles) and near spherical anatase particles. Later work showed nanoparticle morphology has a significant influence on the surface enthalpy (next section). The surface enthalpy of a nanocrystalline rutile-structured Sn0.586Ti0.414O2 solid solution was determined similarly.123 It was found that the value (1.68 and 2.02 J/m2, respectively, for hydrated and anhydrous surfaces) is well close to the molar fraction-weighted sum of the corresponding values of the two end-members (SnO2 and TiO2). This suggests that Sn and Ti atoms are indistinguishably distributed in the cation sites both in the nanoparticle interior and on the particle surfaces. 4.3.1.4. Solution Calorimetry with Consideration of Influence of Nanoparticle Morphology on Enthalpy. Using a method similar to that by Levchenko et al.,121 Park et al. investigated the influence of particle morphology on the surface energy (or surface enthalpy) of nano anatase.124 One sample consisted of anatase nanowires with diameters from 20 to 50 nm and lengths up to several microns, another of sea-urchinlike anatase assemblies (3D assemblies) with diameters 0.4 to 1.6 μm and a hollow core with a diameter of 100−200 nm. The 3D assemblies were composed of many one-dimensional nanostructures ∼10 nm in diameter. Surfaces of both kinds of samples were dominated by {101} surfaces. However, other crystal surfaces were also exhibited in both samples. Combined results (Table 6) from refs 121 and 124 show that, the surface enthalpy of anatase 3D assemblies is higher

Figure 21. Phase stability diagram of nano anatase and rutile at different solution conditions constructed based on the results from ref 115. The variations of the phase stability regions of nano anatase and rutile with pH are inconsistent with a majority of experimental observations.

Figure 22. Temperature−particle size phase diagram for nano TiO2 with different surface terminations (hydrated, hydroxylated, or a mixture of the two). RT line indicates room temperature; “coexistence” was supposed to be a region for coexistence of both anatase and rutile (violating the phase rule). Adapted with permission from ref 110. Copyright 2008 American Chemical Society.

insufficient corrections in the calorimetry data (e.g., for the chemically adsorbed water), limited phase purity in some of the samples (mixtures of anatase and rutile), and unaccounted uncertainty in the derived data due to availability of very limited number of samples (only one sample for brookite). Figure 23d shows the plots of the transformation enthalpy for all the three titania phases calculated using the calorimetrymeasured surface enthalpy values.120 It is seen that, for bulk TiO2 phases (corresponding to zero surface area), the phase stability order is rutile > brookite > anatase. For the three nano phases, anatase is more stable at small sizes (surface area > 3174 m2/mol (∼39 nm)); rutile is more stable at large sizes (surface area < 592 m2/mol (∼193 nm)), and brookite is more stable in intermediate sizes (surface area between 592 m2/mol (∼198 nm) and 3174 m2/mol (∼37 nm)). Enthalpies of rutile and anatase cross at 1452 m2/mol (∼79 nm). These experimental results confirmed our previous thermodynamic prediction concerning the size-dependent stability sequence based on atomistic simulation derived surface energies.97,104 9633


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Figure 23. Enthalpy of nanocrystalline samples with respect to bulk rutile vs surface area for (a) nano rutile samples; (b) nano anatase samples and normalized nano anatase−rutile mixture samples (dashed curves represent 95% confidence limits for the mean); and (c) one nano brookite sample. In (a) to (c), straight lines are from linear regressions. (d) shows the phase stability crossover of titania. The darker line segments indicate the energetically stable phases. Reprinted with permission from (ref 120) Ranade, M. R.; Navrotsky, A.; Zhang, H. Z.; Banfield, J. F.; Elder, S. H.; Zaban, A.; Borse, P. H.; Kulkarni, S. K.; Doran, G. S.; Whitfield, H. J. Energetics of nanocrystalline TiO2, Proc. Natl. Acad. Sci. U.S.A. 2002, 99, 6476. Copyright 2002 National Academy of Sciences, U.S.A.

than that of anatase nanorods, and the surface enthalpy of anatase nanorods is higher than that of spherical nanoparticles. This is true for both hydrated and anhydrous surfaces. The results (Table 6) also show that the surface enthalpy of particles with anhydrous surfaces is higher than that with hydrated surfaces. Thus, in phase stability analysis using these measured data, the particle morphology and surface states should be taken into account. For instance, if one considers the phase stability change in approximately spherical titania nanoparticles in heat treatment above 400 °C, the surface enthalpies of anhydrous anatase nanoparticles (0.74 J/m2) and rutile nanoparticles (1.3 or 1.7 J/m2 by two estimation methods; Table 6), not those of nanowire/nanorods, should be used. This example would produce a stability crossover at anatase sizes 41 and 69 nm, respectively, for the two different methods for estimation of the surface enthalpy of anhydrous rutile nanoparticles (see Table 6). 4.3.1.5. Treatment of Calorimetry Data with Consideration of Nanoparticle Aggregation State. Considering that nanoparticles in powder samples are more or less aggregated, i.e., nanoparticles have solid−solid interfaces (grain boundaries) in addition to solid−vapor surfaces, Castro and Wang122 retreated the calorimetry data of Levchenko et al. 121 by decomposing the measured excess enthalpy (ΔHexc) into the total grain-boundary energy and total surface energy of the particles. Using nano anatase as an example, eq 12 should be rewritten as

ΔHexc = ΔHtrans(bulk rutile = nano anatase) − ΔHtrans(bulk rutile = bulk anatase) = γSS·A SS + γSV ·A SV (13)

where γSS and γSV are respectively the grain boundary energy and surface energy of the nanoparticles, and ASS and ASV are, respectively, their grain boundary area and surface area. ASS values were calculated from the average particle sizes from XRD measurements with consideration of the particle morphology; ASV values were obtained from gas adsorption measurements. Using a regression method, the following results were obtained for nano TiO2 with anhydrous surfaces: for anatase nanoparticles, γSV = 0.67 J/m2 and γSS = 0.20 J/m2; for rutile nanorods, γSV = 2.06 J/m2 and γSS = 0.87 J/m2.122 The surface energies thus obtained are generally lower than those obtained without consideration of nanoparticle aggregation (Table 6). With above derived γ data, the enthalpies of nano anatase (particles) and rutile (nanorods) were plotted in 3D as a function of both the surface area and grain boundary area,122,125 as shown in Figure 25. This diagram shows that the rutile to anatase stability crossover is a function of both the surface area and grain boundary area, or equivalently, a function of both the particle size and aggregation state at given particle morphologies. 4.3.2. Surface Free Energies from Electrochemical Measurements. Balaya and Maier designed a solid state cell for derivation of the free energy of anatase nanoparticles (with average sizes 5, 15, and 100 nm in three samples) via measurements of the electromotive force (emf), i.e., the open circuit cell voltage of the following electrochemical cell.126 9634


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Table 6. Surface Enthalpy (γ, J/m2) of Nano Anatase and Rutile from Calorimetric Determinations (Experimental Data Taken from refs 121 and 124) phase

anatase

rutile

γ (hydrated)

γ (anhydrous)

nanoparticles

0.51a

0.74a

nanowires/ nanorods 3D assemblies

1.07b

1.24b

1.29b

1.41b

morphology

a

γ (hydrated)

γ (anhydrous)

(1.3)c, (0.9)d 1.89a

(1.7)c, (1.3)d 2.22a

b

Reference 121. Reference 124. cEstimated under the assumption that the surface enthalpy of nanoparticles is lower than that of nanowires /nanorods by the same amount as in anatase, i.e. Δγ = γ(nanowires/nanorods) - γ(nanoparticle) = 1.07 − 0. 51 = 0.6 J/m2 for hydrated surface, or 1.24 − 0.74 = 0.5 J/m2 for anhydrous surface. d Estimated under the assumption that the ratio of the surface enthalpy of nanoparticles to that of nanowires /nanorods is the same as that for anatase, i.e., ratio = γ(nanoparticle)/γ(nanowires/nanorods) = 0.51/ 1.07 = 0.5 for hydrated surface, or 0.74/1.24 = 0.6 for anhydrous surface.

The anode reaction is Na 2Ti6O13 = 6TiO2 (bulk rutile) + 2Na + +

1 O2 + 2e− 2 (14)

The cathode reaction is 6TiO2 (nano anatase) + 2Na + +

1 O2 + 2e− 2

= Na 2Ti6O13

(15)

The total cell reaction is (14) + (15): 6TiO2 (nano anatase) = 6TiO2 (bulk rutile)

(16)

Thus, the emf E is E=−

6μ b0− rut − 6μn0− ana 2F

=

3μn0− ana − 3μ b0− rut F

Or, μn0− ana − μ b0− rut =

Figure 24. Formation enthalpies vs surface area for (a) rutile and (b) anatase. Solid lines are fits to the data (with adsorbed water correction for anhydrous surfaces) using equation 12. Dotted lines are 95% confidence bands. Dashed lines are the fits to data with bulk water correction (for hydrated surfaces). (c) Energetics of nano titania. Coarse line shows energetic stability regions of different phases. Reprinted with permission from ref 121. Copyright 2006 American Chemical Society.

μ0n−ana

FE 3

(17)

μ0b−rut

where and are, respectively, the chemical potentials of nano anatase and bulk rutile at standard conditions; F is the Faraday constant (96485 C/mol). Note that a “−” sign is missing in the left side of the equivalent equation in ref 126 (its eq 5). For nano anatase, Balaya and Maier attributed the increased free energy (or chemical potential) solely due to the ∫ V dP ≈ VPexc = VΔP term at small sizes (see eq 1):

Here, following the electrochemistry conventions, we have rewritten the cell notation and the corresponding electrode reactions (to replace those given in ref 126.). - terminal) Au, O2, Na2Ti6O13, TiO2 (bulk rutile) |β″Al2O3(NaO) |TiO2(nano anatase), Na2Ti6O13, O2, Au (+ terminal The anode (“−” terminal) was a composite of Au, Na2Ti6O13 and bulk rutile powders. The cathode (“+” terminal) was a composite of Au, Na2Ti6O13 and nano anatase powders. The cell was operated at several temperatures between 250 and 450 °C.

⎛ 2f ⎞ μn0− ana = μ b0− ana + PexcVm = μ b0− ana + ⎜ ⎟Vm ⎝r⎠ = μ b0− ana +

⎛ 2γ ⎞ ⎟V ⎝ r ⎠ m

⎜

(18)

Although not stated in ref 126, the surface free energy γ (called surface/interface tension there) was implied to be the same as the surface stress f, as seen from above equation. A combination of eqs 18 and 17 gives 9635


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Figure 25. Enthalpy of nanocrystalline titania as a function of both the grain boundary area and surface area. The circles represent hypothetical microstructures at that interface area condition. To the left there are nanograined samples, while to the right nanoparticles free of strong agglomerates. Reprinted from ref 125. Mater. Lett., 96, Castro, R. H. R., On the thermodynamic stability of nanocrystalline ceramics, 45−56, Copyright 2013, with permission from Elsevier.

μ b0− ana − μ b0− rut +

⎛ 2γ ⎞ FE ⎜ ⎟V = ⎝ r ⎠ m 3

⎛ 2f ⎞ μn0− ana = μ b0− ana + PexcVm + γA m = μ b0− ana + ⎜ ⎟Vm ⎝r⎠ 3V ⎛γ ⎞ + γ m = μ b0− ana + (2t + 3)⎜ ⎟Vm ⎝r ⎠ (21) r

i.e., γ=

⎞ r ⎛ FE ⎜ + μ b0− rut − μ b0− ana ⎟ ⎝ ⎠ 2Vm 3

where the surface stress is assumed t times the surface free energy in magnitude (i.e., f = tγ).97 If t = 1, eq 21 becomes

(19)

Based on the above equation, the surface/interfacial free energy γ was calculated for nano anatase at different sizes and temperatures. Then, the surface/interface enthalpy (ΔHS) and entropy (ΔSS) were obtained from least-squares fitting of γ against the temperature T based on γ = ΔH S − T ΔS S

μn0− ana = μ b0− ana +

⎛ 5γ ⎞ ⎜ ⎟V ⎝ r ⎠ m

(22)

Equation 19 becomes

(20)

γ=

The obtained results are shown in Figure 26: ΔHS = 1.1 J/m2 and ΔSS = 0.043 J/m2·K.126 They regarded the obtained surface/interface enthalpy comparable to the one (1.32 J/m2) obtained from molecular simulations.97 Bear in mind, however, that eq 18 does not consider the real surface/interface free energy contribution (i.e., the γAm term; see eq 1). With this consideration, equation 18 should be modified to

⎞ r ⎛ FE ⎜ + μ b0− rut − μ b0− ana ⎟ ⎝ ⎠ 5Vm 3

(23)

Thus, the corrected surface/interface enthalpy and entropy are ΔHS = 2/5 × 1.1 = 0.44 J/m2 and ΔSS = 2/5 × 0.043 = 0.017 J/m2·K for nano anatase. Given that anatase nanoparticles in the electrode were dispersed in the composite materials after compression, the nanoparticle surfaces should be somewhat between free surfaces and dense grain boundaries. Thus, a corrected ΔHS = 0.44 J/m2 is reasonable, since it is between γSV = 0.67 J/m2 and γSS = 0.20 J/m2 based on calorimetry measurements 122 (section 4.3.1.5). 4.3.3. Surface (Free) Energies from X-ray Diffraction Determinations. 4.3.3.1. Surface Energies Derived From Coarsening Kinetics of Nano Titania. Nanoparticle coarsening is driven by reduction of the total surface energy of all the particles (γ·A) by decreasing the total surface area (as A(large sizes) < A(small sizes)). The surface energy weighs in the coarsening kinetics by entering into the activation energy term. Via kinetic modeling of the coarsening data, it is possible to derive the surface energy of the nanoparticles. Using nano titania powder and membrane samples consisting initially of anatase and brookite phases, we studied the kinetics of phase transformation and particle coarsening at temperatures 500−600 °C.127 The difference between the powder samples and the membrane samples lies in the aggregation status of the nanoparticles, as illustrated in Figure 27a,d. Compared to membranes, powders have more bigger pores, less nanoparticle−nanoparticle contacts inside the aggregates, and more free surfaces on the nanoparticle edges. Thus, the average

Figure 26. Plot of surface/interface tension versus temperature for the anatase nanoparticles. Note ΔHS and ΔSS shown in the diagram need corrections (see text). Reproduced from ref 126 with permission from the PCCP Owner Societies. 9636


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Figure 27. Schematic diagrams showing nanoparticle packing in powders (a) and membranes (d). Phase contents (b, e) and average particle size (c, f) of nano anatase in powders (b, c) and membranes (e, f) heated at 500 °C (diamond), 520 °C (square), 540 °C (triangle), 560 °C (×), 580 °C (*), and 600 °C (circle). Reprinted with permission from ref 127. Copyright 2007 American Chemical Society.

Table 7. Surface Energy (J/m2) of Anatase, Brookite, and Rutile Nanoparticles in Powder and Membrane Samples Derived from Coarsening Kinetics (Data Taken from ref 127)

surface energies of the powders should be higher than those of the membranes. The variations with the time and temperature of the phase contents and average particle sizes were determined from XRD.127 Figure 27 exemplifies variations of both the phase content (b, e) and average particle size (c, f) of nano anatase with time at different temperatures. Kinetic modeling of the phase transformation showed that smaller brookite nanoparticles preferentially transform to anatase and anatase transforms to rutile via interface nucleation and growth. Larger brookite nanoparticles transform preferentially to rutile via surface nucleation and growth. The coarsening kinetics (size vs time data) was modeled with our derived coarsening equations,127 from which the surface energies of nano titania were derived (Table 7). For powder samples, γ(anatase) = 0.93 J/m2, γ(brookite) = 1.26 J/m2 and γ(rutile) = 2.76 J/m2. For membrane samples, γ(anatase) = 0.79 J/m2, γ(brookite) = 1.07 J/m2 and γ(rutile) = 1.35 J/m2.

sample form

anatase

brookite

rutile

powders membranes

0.93 0.79

1.26 1.07

2.76 1.35

The relative magnitudes of the surface energies of the three phases, and those of the more open surfaces (in powders) and denser aggregates (in membranes), are all consistent with calorimetry determinations (sections 4.3.1.2 and 4.3.1.5). 4.3.3.2. Surface Free Energies Derived From Size-Dependent Lattice Contractions. The surface free energy (γ) and the surface stress ( f) are related via the Shuttleworth equation 5. For near spherical nanoparticles, if the size dependence of the 9637


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surface stress is known, solution of the Shuttleworth equation generates the following relationships:49 γ=

γ=

2 D2

∫D

fD dD

(D0 ≤ D ≤ D1)

(24a)

0

D12γ1 D

D

2

+

2 D2

∫D

D

Df dD

(D ≥ D1)

1

(24b)

where D is the average particle size (diameter) of the nanoparticles. D0 is the smallest size of a nanoparticle retaining its crystallinity (∼one unit cell, e.g. ∼1 nm for anatase). D1 is the smallest particle size experimentally accessible, which should be as close to D0 as possible. Inserting equation 24a at D = D1 into equation 24b results in γ=

2 [ D2

∫D

D1

fD dD +

0

∫D

D

fD dD]

(D ≥ D1)

1

(24c)

Equations 24a and 24c can be unified as γ=

2 D2

∫D

D

fD dD

(D ≥ D0 ≈ 1 nm)

0

(24d)

eq 24 shows that the surface free energy of nanoparticles equals in magnitude the size-weighted average of the surface stress. Note that, in derivation of above equations,49 the surface free energy (γ0) at D0 was assumed zero, an adequate approximation for nanoparticles with clean surfaces. If the surface free energy at D0 is not zero, as would be for nanoparticles with appreciable and strong adsorption or binding of surface molecules, the above equation should be modified to: 2 γ = γ0 + 2 D

∫D

Figure 28. Size-dependent surface stress (a) and surface free energy (b) of nanocrystalline anatase. Circles and squares were calculated, respectively, with size-dependent bulk modulus of nano anatase and bulk modulus of bulk anatase. In (b), triangles are surface enthalpies of nano anatase, calculated using calorimetry data reported in ref 121, where the surface enthalpy was reported as an average value (dashed line). Reproduced from ref 49 with permission of the PCCP Owner Societies.

D

fD dD

(D ≥ D0 ≈ 1 nm)

0

(24e)

Using eqs 24a−24e, the size dependent surface free energy can be derived. The size-dependent surface stress can be obtained from the nanoparticle lattice contraction (V0 − V) determined from XRD at different particle sizes 49 (a combination of eqs 4 and 7): V − V0 4f ΔV ΔP = =− B=− B D V0 V0

atoms in the nanoparticles will increase due to an increasing proportion of high-energy sites (i.e., atoms have more dangling bonds). Second, as the particle size becomes very small, the nanoparticle surface and interior regions become closer and closer in structure and energy due to pervasive internal distortion, causing a decrease in the surface free energy. The overall effect is that as the particle size decreases, the surface free energy first increases and then decreases. The decrease of the surface energy with decreasing particle size at small size ( D0 and the stability region of rutile is expanded (shifted to the right). If we include brookite in the analysis, one case (when anatase size D1 < D0) is illustrated in Figure 30d. Here, we assume the pHPZC of brookite is between those of rutile and anatase because the surface energy and the bulk free energy of brookite are intermediate between those of the two phases.120,127 A stability region of brookite between those of rutile and anatase would explain the formation of nano brookite at an intermediate acidity in ref 130. 4.3.4.2. Debate on Metastability of Isolated QuantumSized Anatase Particles. Recently, Satoh et al. studied the phase transformation in annealing isolated and quantum-sized (Q-size) titania particles (1−2 nm) distributed on quartz substrates based on change in the optical bandgap for the particles determined from optical waveguide (OWG) spectra.134 The Q-size titania particles (containing 14TiO2, 30TiO2 and 60TiO2 units, respectively) were prepared using the phenylazomethine dendrimers (DPA) template method.135 The needed quantity (e.g., 60) of Ti(acac)Cl3 units (acac = acetylacetonate) was assembled into the DPA template. Then the Q-size TiO2 particles were fabricated on the substrates using hydrolysis and thermolysis methods that transformed the Ti(acac)Cl 3 within the DPA into rutile and anatase, respectively. Under “bulk” conditions, i.e., without application of the substrates, the hydrolysis method (catalyzed by HCl) produced rutile particles ∼8.2 nm in diameter and the 9640


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Figure 30. Revised diagram (to Figure 29c) for the total free energy versus pH for nano anatase and rutile as the particle size changes (a−c). In (d), nano brookite is included. See text for details.

shape, and other factors. Thus, use of consistent samples for investigation of all material properties and size effects is very important. Various thermodynamic analyses of the phase stability in nano TiO2 systems based on both the molecular simulations and experimental determinations were reviewed and some deficiencies in certain thermodynamic treatments were identified. Corrections and revisions were given whenever possible, producing better agreement with experimental results. In particular, the analyses of the phase stability of nano rutile, brookite and anatase in aqueous solutions with different degrees of acidity or alkalinity are revised. Finally, a new perspective was provided on the debate related to the stability or metastability of anatase nanoparticles. We hope that the information presented here will guide future fundamental and applied research on titania nanomaterials.

isolated titania particles (thought to be rutile converted from anatase) in ref 134 are right within or near this size range (2.5− 4 nm) for the anatase to rutile stability crossover. This would just support the conclusion that the very small and isolated anatase particles (

Structural Characteristics and Mechanical and Thermodynamic

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