Polymorphic Transformations and Particle Coarsening in

Apr 18, 2007 - Carina Dahlberg , Sergey V. Dvinskikh , Michael Schuleit , and István Furó .... Hengzhong Zhang , Michael P. Finnegan , Jillian F. Ba...
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J. Phys. Chem. C 2007, 111, 6621-6629

6621

Polymorphic Transformations and Particle Coarsening in Nanocrystalline Titania Ceramic Powders and Membranes Hengzhong Zhang* and Jillian F. Banfield Department of Earth and Planetary Science, UniVersity of California Berkeley, 307 McCone Hall, Berkeley, California 94720 ReceiVed: NoVember 17, 2006; In Final Form: March 12, 2007

X-ray diffraction (XRD), transmission electron microscopy (TEM), and nitrogen gas adsorption were used to characterize the phase composition, particle size, and aggregation state of nanocrystalline titania ceramic powders and membranes. Powders have more free surfaces, less compact packing of nanoparticles, and fewer nanoparticle-nanoparticle contacts than membranes and contain many large nanopores. The kinetics of the phase transformations among the three nanophases of titania (anatase, brookite, and rutile) in both types of samples were determined by XRD over the temperature range 500-600 °C. Kinetic modeling shows 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 frequency factors and the activation energies for phase transformations in powders are lower than those in membranes. Both the fusion of nanoparticles by recrystallization of one upon another and their phase transformation contribute to nanoparticle coarsening. Coarsening equations were derived by taking into account the particle size dependence of the activation energy of particle fusion. Surface energies of anatase, brookite, and rutile derived by fitting the experimental data to the coarsening equations are higher in loosely packed powders compared to more densely packed titania membranes. The results indicate that the nanoparticle aggregation state has a marked influence on the kinetics of phase transformation and particle coarsening, probably due to the lowering of nanoparticle surface energy by interparticle interactions.

Introduction Anatase, rutile, and brookite are the most common polymorphs of titania (TiO2). In sol-gel synthesis of nanocrystalline titania, anatase is the major product. However, samples often contain some brookite (see, e.g., ref 1). In fact, by varying the synthesis conditions, the relative amount of anatase and brookite can be changed.2 Upon heating, both anatase and brookite ultimately transform to rutile. Properties of titania materials depend upon the titania polymorph present. For example, films of sintered anatase nanoparticles are promising electrode materials for lithium batteries.3 Anatase is the essential phase, while rutile and brookite are non-effective.3 Understanding the rates and mechanisms of polymorphic transformations in nanocrystalline materials is critical for fabrication of titania products with desired properties. Results for titania are also important in that they elucidate the general ways in which particle size can affect the behavior of materials. At low and intermediate temperatures (465-700 °C), transformation of nanocrystalline anatase to rutile is known to occur by interface and/or surface nucleation.4,5 The aggregation structure of nanoparticles influences the reaction rate constant via effects on the frequency factor and the activation energy. Moreover, the number of surface sites and the availability of interfaces between particles are directly related to the particle packing. Thus, aggregate structure of a sample can significantly affect the kinetics of phase transformations. At the same rate * Address correspondence to this author. E-mail: [email protected]. edu.

constant, in a fully dispersed sample where the availability of surface sites is maximized, the transformation should be faster in powders than in membranes if the transformation occurs predominantly via surface nucleation. Conversely, the number of particle-particle contacts per unit volume is much higher in membranes than in powders, and the transformation should be faster in membranes than in powders if the transformation occurs predominantly via interface nucleation. The influence of the aggregation structure on the transformation kinetics can be complex when both modes of nucleation coexist. Therefore, aggregate structure is an important material property that determines the kinetics and the mechanisms of the transformation in nanostructured materials. In previous work, kinetics of phase transformation in nanocrystalline titania powders4-12 and membranes13,14 has been studied separately. In most cases, presence of the brookite phase was not considered. Ye et al.8 and Zhang and Banfield12have studied transformation sequences between nanocrystalline titania in powder samples when brookite phase was present. Kumar et al.13 and Lin et al.14 have studied phase transformation in nanocrystalline titania membranes. However, the existence, and thus the participation of the brookite phase in the polymorphic transformation, was neglected in their works [the X-ray diffraction patterns (Cu KR radiation) of Lin et al.14 and Kumar et al.13 show the (121) peak of brookite (ref JCPDS Card No. 29-1360) at around 2θ ) 30.8°]. In the present work, we used X-ray diffraction to measure the transformation kinetics between nanocrystalline brookite, anatase, and rutile and their particle coarsening in membrane and powder samples. By modeling and comparing the kinetics in both powders and membranes, we

10.1021/jp067665t CCC: $37.00 © 2007 American Chemical Society Published on Web 04/18/2007

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Zhang and Banfield 2θ ) 22° and 34° at a scanning rate of 0.04 deg/min. An XRD pattern of a sample heated at 500 °C for 11.7 h indicates the coexistence of anatase, brookite, and rutile (Figure 1). A numerical peak decomposition method was applied to separate the overlapping of the anatase (101) peak with brookite (120) and (111) peaks.12 The integrated intensities (I) and the fullwidths at half-maximum (fwhm) of the anatase (101), brookite (121), and rutile (110) peaks obtained from the numerical treatment were used to calculate respectively the weight fractions (x) and the average particle sizes (D-diameters) of the three phases by using the following relationships12

xAna ) Figure 1. A XRD pattern collected for a nanocrystalline titania powder sample heated at 500 °C for 11.7 h. The overlap of anatase (101) peak with brookite (120) and (111) peaks can be separated numerically by using a peak decomposition method described in ref 12.

provide insights into the influence of the aggregation state on the transformation kinetics. Analytical equations describing titania nanoparticle coarsening accompanying phase transformation yielded surface energies of the three titania polymorphs and reveal a link between aggregation state and average surface energy.

xRut )

xBro )

0.886IA(101) 0.886IA(101) + IR(110) + 2.721IB(121) IR(110) 0.886IA(101) + IR(110) + 2.721IB(121) 2.721IB(121) 0.886IA(101) + IR(110) + 2.721IB(121)

Membrane samples of titania were synthesized by the solgel method.7 A mixture of water, titanium isopropoxide, and concentrated nitric acid (catalyst) in the ratio of 150:15:1 (volume ratio) was boiled and refluxed in a round-bottom flask connected with a coil condenser. Gel formed after reaction for ∼72 h. Drying a thin layer of the gel in a weighing boat produced titania membrane. Powder samples of titania were obtained by grinding the same batch of membranes in an agate mortar with a pestle for at least 30 min. After pretreatments at 325 °C for 2 h, the membrane and powder samples were used as starting materials for kinetic experiments. Kinetic experiments were conducted isothermally at temperatures ranging from 500 to 600 °C with an interval of 20 °C. Samples of ∼40 mg each were placed in alumina crucibles and heated in an electric furnace held at the chosen temperatures for different lengths of time. Each experiment was run at least two times. The heated samples were then quenched in air and examined by powder X-ray diffraction (XRD) with use of a Scintag PADV diffractometer with Cu KR radiation (35 kV, 40 mA). In cases of membrane samples, they were ground for the XRD characterization. XRD patterns were collected between

(1b)

(1c)

and the Scherrer equation15

D) Experimental Section

(1a)

0.90λ fwhm cosθ

(2)

In eq 2, λ is the wavelength of Cu KR radiation (1.5418 Å) and 0.90 is the Scherrer constant. The precision and accuracy of the determination method were checked by using a standard sample consisting of nanocrystalline anatase and bulk rutile. The determined particle size of anatase was 10.7 ( 0.8 nm (standard deviation). The determined anatase content (wt %) was 38.2 ( 2.4 (standard deviation), in good agreement with that of the prepared portion (39.6%). The relative standard deviations are all close to 7% (size: 0.8/10.7 = 7%; content: 2.4/38.2 = 6%). Nitrogen (N2) adsorption onto titania samples was carried out at 77 K to determine the surface areas, using the BrunauerEmmett-Teller (BET) equation, and to calculate the pore size distribution, using the Barrett-Johner-Halenda (BJH) method.16 The N2 adsorption measurements were performed with an Accelerated Surface Area and Porosimetry System (Micrometrics ASAP2010). Transmission electron microscopy (TEM) characterization of sample microstructures was done with a Philips CM200 highresolution transmission electron microscope operated at 200 kV. A TEM specimen was prepared by dripping titania nanoparticles

Figure 2. (a) Isotherms for adsorptions of nitrogen on nanocrystalline titania powders (upper solid curve) and membranes (lower solid curve) at 77 K. (b) Pore size distributions measured in powders (solid line) and membranes (dashed line).

Nanocrystalline Titania Ceramic Powders and Membranes

J. Phys. Chem. C, Vol. 111, No. 18, 2007 6623

Figure 3. Phase contents of nanocrystalline anatase (a, d), brookite (b, e), and rutile (c, f) in powders (a-c) and membranes (d-f) heated at 500 (diamond), 520 (square), 540 (triangle), 560 (×), 580 (/), and 600 °C (circle).

dispersed in water, using an ultrasonic method, onto a carboncoated copper TEM grid, followed by drying in air. Experimental Results and Discussion (a) Characterization of Starting Materials. XRD measurements showed that the starting powder sample consisted of 46.3% (weight percentage) anatase and 53.7% brookite. The membrane sample consisted of 53.4% anatase and 46.6% brookite. The particle sizes of anatase and brookite in both samples are all ∼7 nm (in powder: anatase 7.6 nm, brookite 7.1 nm; in membrane: anatase 7.0 nm, brookite 7.5 nm). This average particle size is consistent with that measured from the TEM study of the samples prepared in the same way as in our previous work.7 Figure 2 shows the isotherms of adsorption of N2 on the samples and the calculated pore size distributions. The N2

adsorption on the powder sample exhibits a type-III shape, and that on the membranes a type-V shape.17 Most pores present within the powders and the membranes are 2-5 nm in diameter (Figure 2b), suggesting that the powders consist of a number of membrane clusters produced by grinding. In the powders there are also many pores bigger than 10 nm that are not seen in the membranes, indicating that the aggregation state of the powder sample was highly reduced by crushing. Due to the lower concentration of bigger pores in the membrane sample, adsorption of N2 on the sample was saturated when P/P0 g 0.65 (Figure 2a). In contrast, in the powder sample the adsorption of N2 increases dramatically as P/P0 f 1 due to unlimited capillary condensation of N2 on available bigger pores (Figure 2a). By using the BET equation and the data points between P/P0 ) 0.1 and 0.35, the surface areas of the powder

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Figure 4. Average particle sizes of nanocrystalline anatase (a, d), brookite (b, e), and rutile (c, f) in powders (a-c) and membranes (d-f) heated at 500 (diamond), 520 (square), 540 (triangle), 560 (×), 580 (/), and 600 °C (circle).

and membrane samples were calculated to be 140.5 and 121.7 m2/g, respectively. Given that the nanoparticles are near spherical and assuming that they are isolated from each other, the surface area of a titania sample is 214 m2/g, using the nanoparticle diameter ) 7.0 nm (approximated based on XRD determinations) and the density of titania ) 4.01 g/cm3 (i.e., the average of 3.89 g/cm3 for anatase and 4.13 g/cm3 for brookite). The BET surface areas are smaller than the calculated value because nanoparticles are aggregated and interfaces are inaccessible to N2 molecules. The amount of contact area in the membrane and powder samples is respectively 214-121.7 ) 92 m2/g and 214-140.5 ) 74 m2/ g. This confirms that the degree of aggregation in the membranes is higher than that in the powders. In summary, XRD, TEM, and N2 adsorption measurements reveal that both the powder and membrane samples are composed of nanocrystalline titania

and that the powder samples have more free surfaces and less dense packing of nanoparticles. (b) Results of Kinetic Experiments. The variations of the weight percentages of anatase, brookite, and rutile with time are depicted in Figure 3. In both powders and membranes, the amount of anatase increases initially, passes through a maximum, and then decreases (Figure 3a,d). In contrast, the amount of brookite decreases constantly (Figure 3b,e) and the amount of rutile increases monotonically (Figure 3c,f). These results indicate that brookite transforms to anatase early in the experiment. Rutile, however, may form from brookite and/or anatase. Whether anatase transforms to brookite (as observed in ref 12) or brookite transforms to anatase (as observed in this work) depends on the relative phase stabilities of the two nanophases, which in turn depends on their relative sizes. The enthalpy (to

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J. Phys. Chem. C, Vol. 111, No. 18, 2007 6625

Figure 5. TEM images of nanoparticles in a powder sample heated at 600 °C for 1 h (without pretreatment at 325 °C): (a) anatase nanoparticles ∼25 nm in size and (b) a rutile nanoparticle ∼75 nm in size.

Figure 6. Arrhenius plots of the kinetic constants for the transformations B f A (square), B f R (diamond), and A f R (triangle) in powders (a) and membranes (b). The lines are from linear least-square fits to eq 10. (c) Comparisons between the kinetic constants of powders (solid lines) and membranes (dotted lines) at different reciprocal temperatures for the transformations B f A (line pair 1), A f R (line pair 2), and B f R (line pair 3). Note that lines in pair 3 were shifted 1 unit down for clarity.

approximate free energy) vs particle size diagram in ref 12 shows that anatase is more stable than brookite when they are equal in size and both are smaller than ∼11 nm in diameter. The initial particle sizes of the two phases are all ∼7 nm in the powder and membrane samples. Accordingly brookite transforms to anatase early in the experiment because anatase is more stable. As seen from parts b and e of Figure 3, the reaction time at which brookite disappears is shorter in membranes than in powders at T g 540 °C. This indicates that the conversion from brookite to anatase and/or rutile is faster in membranes than in powders, possibly due to the higher degree of aggregation in the membranes. The correlation between reaction rate and

packing density is expected if, as suggested previously,4,5 the reaction initiates at interfaces. The crystal growth rate of anatase is faster than that of brookite, and that of rutile is faster than that of anatase (Figure 4). It is important to note that many of the size vs time curves show a “∼”-like (sigmoidal) shape, especially at higher temperatures (e.g., g540 °C for brookite and g560 °C for anatase and rutile). Such a phenomenon was not noticed previously for coarsening of titania nanoparticles in air (for example, see Figures 2 and 3 of ref 7), and it is related to the multiple sources of particles (see below). TEM images of titania nanoparticles in a powder sample heated at 600 °C for 1 h indicate that each titania nanoparticle

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is composed of a single phase (Figure 5). Lattice fringe spacings for nanoparticles in parts a and b of Figure 5 are 0.36 and 0.43 nm, respectively, close to the anatase (101) d-value of 0.352 nm and the rutile (100) d-value of 0.459 nm, respectively. Thus, it is likely that the nanoparticle in Figure 5a is anatase and that in Figure 5b is rutile. The fact that the rutile nanoparticle (∼75 nm, Figure 5b) is much bigger than the anatase particle (∼25 nm, Figure 5a) is consistent with the particle size determination from XRD (Figure 4a,c).

since the brookite structure can be converted to rutile structural elements by simple shearing.22 Thus, reaction 5 may be first order with respect to the number of brookite nanoparticles. In fact, a first-order kinetics was observed for brookite to rutile transformation in bulk materials.23 Since the weight fraction (x) and the average particle size (D) of each phase are known (Figures 3 and 4), the number of particles (N) in a sample with a mass m can be calculated for each phase

Kinetic Modeling

N)

(a) Kinetic Modeling of Phase Transformation. On the basis of the experimental results, we propose the following phase transformation sequences: k1

B 98 A k2

(4)

k3

B 98 R

( )

(6)

where K is a coefficient relevant to the mass of the sample, and F is the density of the phase (3.89, 4.13, and 4.25 g/cm3 respectively for anatase, brookite, and rutile).24 Figure A1 in the Supporting Information shows the variation of the number of titania nanoparticles with time at various temperatures (K was set arbitrarily to 10). Modeling of the transformation kinetics by using the number of particles (Figure A1, Supporting Information) as the substance quantity was unsuccessful due to significant errors introduced by the much smaller number of particles of rutile compared to anatase and brookite. Therefore, we use the weight fraction as the substance quantity in our subsequent kinetic modeling. With the consideration of the reaction orders above, we write the following kinetic equations for transformation reactions 3-5:

(3)

A 98 R

6m x x xm )K 3 ) 3 3 π πD DF DF F 6

(5)

where B, A, and R stand for brookite, anatase, and rutile, respectively, and k1, k2, and k3 are the kinetic constants for the corresponding transformations 3, 4, and 5, respectively. The kinetics of the phase transformation from nanocrystalline anatase to rutile, and that from nanometer-sized amorphous titania to nanocrystalline anatase were studied previously.4-7,12,18-20 The following summarizes major results from these studies. (a) A titania nanoparticle consists of a single phase, as also was observed in this study (Figure 5). (b) In the transformation from nanocrystalline A f R, rutile nucleates at interfaces between anatase nanoparticles (e.g., twining interfaces of anatase {112}19). (c) Once rutile nuclei are formed, they grow through the whole nanoparticles rapidly. (d) At lower temperatures (∼600 °C), surface nucleation becomes important. The rate of the transformation via surface nucleation is proportional to the number of anatase nanoparticles. The temperatures in the present work are e600 °C. On the basis of the results of previous studies, the rate of reaction 4, A f R, is limited by interface nucleation of rutile on anataseanatase nanoparticle contacts. Thus, reaction 4 is second order with respect to the number of anatase nanoparticles. A TEM study showed that brookite structure occurs in {112} anatase twin interfaces.21 In fact, brookite is a polytype of anatase.21 Similarly, anatase structural elements may be produced at interfaces between appropriately oriented brookite nanoparticles. Hence, reaction 3, B f A, is likely to occur via interface nucleation at contacts of anatase nanoparticles. So reaction 3 may be second order with respect to the number of brookite nanoparticles. On the other hand, transformation of brookite nanoparticles to rutile may occur via surface nucleation,

dxA ) k1xB2 - k2xA2 dt

(7)

dxB ) -k1xB2 - k3xB dt

(8)

dxR ) k2xA2 + k3xB dt

(9)

The initial conditions are xA ) xA0, xB ) xB0, and xR ) 0 at time t ) 0. Full analytical solutions to the above differential equations are not available. However, they can be solved numerically. In this work, we used GEPASI25 (a program for simulations of kinetics of chemical reactions that can also be used to optimize kinetic parameters by fitting experimental data to designed kinetic models) to optimize the kinetic constants k1, k2, and k3 by minimizing the differences between the calculated phase contents and the experimental data. Figures A2 and A3 in the Supporting Information show the results of the kinetic modeling of the experimental data (Figure 3) based on reactions 3-5 and eqs 7-9. It is seen that the kinetic model reproduces the experimental observations quite well. The optimized kinetic parameters are shown in Figure 6. We also tested kinetic models for other probable combinations of the reaction orders for the conversion of brookite to anatase (reactions 3) and brookite to rutile (reaction 5): 1-1 (both are first order), 1-2 (reaction 3 is first order, reaction 5 is second

TABLE 1: Frequency Factors and Activation Energies for Phase Transformation in Nanocrystalline Titania Powders and Membranes BfA sample powder membrane

A

(1014

/min)

3.99 98.6

AfR Ea (kJ/mol)

12

A (10 /min)

252 276

1.86 207

BfR Ea (kJ/mol)

14

A (10 /min)

Ea (kJ/mol)

231 264

1.36 4.65

263 269

Nanocrystalline Titania Ceramic Powders and Membranes

Figure 7. Schematic diagram showing nanoparticle packing in powders (a) and membranes (b). Compared with membranes, powders have more bigger pores and fewer nanoparticle-nanoparticle contacts inside the aggregates, but have more free surfaces on the nanoparticle edges.

order), and 2-2 (both are second order), for comparison with 2-1 (reaction 3 is second order, reaction 5 is first order). Only the 2-1 combination fit the data well, consist with structural considerations of phase transformation mechanisms for B f A and B f R. In parts a and b of Figure 6, the optimized kinetic constants are plotted as functions of the temperature in an Arrhenius relationship (eq 10)

ln k ) ln A -

Ea RT

(10)

where A is the frequency factor, Ea the activation energy, T the absolute temperature, and R the gas constant (8.314 J/(mol‚K)). Table 1 summarizes the frequency factors and the activation energies for transformations 3-5, obtained from linear [ln k vs 1/T] regression fit of the data points to eq 10. In the regression, the correlation coefficients (r2) are ∼0.98 for B f A and A f R and ∼0.94 for B f R. The very closeness of every r2 to 1 indicates that the determined kinetics satisfies the Arrhenius equation quite well and that the difference between the derived kinetic parameters (Table 1) for powders and membranes is intrinsic rather than due to scattering in the data. The Ea for the nanopowder A f R (231 kJ/mol) is in good agreement with that (227 kJ/mol) from a previous study.26 The Ea for the nanopowder B f R (263 kJ/mol) is higher than that (240 kJ/mol) for bulk B f R.23 The increase of Ea at decreased sizes may be a consequence of a stronger binding between atoms at smaller sizes.26 Compared with membrane samples, powder samples have lower frequency factors and lower activation energies for all reactions. Figure 6c compares the rate constants calculated from eq 10 of powder and membrane samples. It is seen that at T < 600 °C (i.e., 1000/T >1.15) for B f A and at T < 560 °C (i.e., 1000/T >1.20) for A f R, the transformations are faster in powders than in membranes. At higher temperatures, the transformation is faster in membranes than in powders. For B f R, the transformation is faster in membranes than in powders. (b) Significance of Nanoparticle Packing on Kinetic Parameters. Surface area and pore size distribution determined by N2 adsorption revealed that the powder sample has more free surfaces but less compact nanoparticle packing than the membrane sample, as illustrated by the schematics in Figure 7. In other words, nanoparticle powders have fewer nanoparticlenanoparticle interfaces than membranes, and hence the average surface/interface energy of titania is higher in powders than in membranes. This is confirmed by the kinetic modeling of the coarsening data (see below). Consequently, nanoparticles in powders have higher molar free energy than those in membranes, i.e., the driving force for a phase transformation is higher in powders than in membranes. Accordingly, for the same

J. Phys. Chem. C, Vol. 111, No. 18, 2007 6627 transformation, the activation energy is lower in powders than in membranes (Table 1). In a powder sample, there are more surface O and Ti atoms in a unit volume than in a membrane sample because there are more free surfaces and more large pores (Figure 7). This means on average the binding of the surface atoms with the interior atoms of the nanoparticles in powders is not as strong as that in membranes. As a result, the force constants and hence the vibration frequencies of the surface atoms of the nanoparticles in powders are lower than those in membranes. Since a frequency factor scales with the vibration frequency along a reaction coordinate,27 the frequency factors for the phase transformations in powders are lower than those in membranes (see Table 1). (c) Kinetics of Nanoparticle Coarsening That Accompanies Phase Transformations. When there is no phase transformation, particle coarsening proceeds via pure (or classical) crystal growth of nanoparticles, i.e., merging (or fusion) of particles by recrystallization of one upon another. However, if a phase transformation occurs, new nanoparticles form or nanoparticles are consumed. Both processes affect the particle size distribution, and hence nanoparticle coarsening. Let us consider the coarsening of anatase nanoparticles. At lower temperatures (e.g., at 500 °C), most brookite transforms to anatase (Figure 3a) rather than to rutile (Figure 3c). Thus, the number of anatase particles equals those from pure anatase crystal growth plus those formed by B f A via second-order reaction 3. Both processes contribute to an increase in size over time (Figure 4a), yielding a consistent trend that resembles that for pure crystal growth. At a higher temperature (e.g., at 600 °C), however, while brookite transforms to anatase (Figure 3b), anatase transforms to rutile (Figure 3a). Thus, the number of anatase particles equals those from pure anatase crystal growth plus those formed by B f A via second-order reaction 3 minus those consumed by A f R via second-order reaction 4. At shorter reaction times (e.g., 10 h, 600 °C), more and larger anatase particles are created by B f A (because two brookite particles convert to one anatase particle), while fewer anatase particles are converted to rutile. Thus, in effect, the size of anatase increases with time (Figure 4a). At longer reaction times (e.g., 120 h, 600 °C), no brookite remains and anatase particles are transforming to rutile, with preferential conversion of larger anatase particles (because bigger anatase particles are less stable; see ref 28). This causes the average particle size of anatase to decrease (Figure 4a). At even longer reaction times (e.g., 240 h, 600 °C) the transformation from anatase to rutile slows down (Figure 3a,c) and crystal growth of anatase predominates again. Over the whole time range the size vs time curve has a sigmoidal (“∼”) shape (at 600 °C). In the intermediate temperature range (520-580 °C), the size vs time curve changes from a regular shape to a “∼” shape. The “∼” shapes observed for particle coarsening of brookite and rutile (Figure 4b,c) can also be explained based on similar considerations. Due to the multiple sources for coarsening of titania nanoparticles, it is inappropriate to apply the conventional powder growth law (eq A11 in the Supporting Information) alone to describe the net rate of nanoparticle coarsening that accompanies phase transformations. Furthermore, in the recrystallization of nanoparticles, smaller nanoparticles need lower activation energy than larger ones because the driving force for coarsening is greater in small particles than in big ones. Mathematical treatments of the multiple sources for coarsening lead to the following coarsening equations for nanocrys-

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TABLE 2: Surface Energies (J/m2) of Anatase, Brookite, and Rutile Derived from Nanoparticle Coarsening Data powder sample

membrane sample

temp (°C)

anatase

brookite

rutile

anatase

brookite

rutile

500 520 540 560 580 600 average refs 12 and 28 ref 29 ref 30

0.74 0.81 0.90 0.99 1.05 1.12 0.93 [0.19]b 1.34 0.4 0.74

1.48 0.68 0.69 1.61 1.41 1.69 1.26 [0.58] 1.66 1.0

1.16 3.82 2.17 3.88 (9.81)a (20.9) 2.76 [1.60] 1.93 2.2 2.22

0.59 0.71 0.85 0.92 0.87 0.77 0.79 [0.20]

∼0 0.50 0.90 0.69 2.61 1.69 1.07 [1.54]

0.45 1.98 1.64 (9.95) (9.55) (36.6) 1.35 [0.90]

a Values in parentheses were not included in the calculation of the averages. b Values in brackets are the maximum deviations of data points from the averages.

talline anatase (eq 11), brookite (eq 12), and rutile (eq 13) (see the Supporting Information for the derivation):

(

Mγ

)

∫0t 0.26xA2LA exp 10FADAART

(

dt )

[1.29DB - DA]k1xB2 dt (11) Mγ

∫0t 0.26xB2LB exp 10FBDBBRT x

(

Mγ

∫t0t DRRLR exp 10FRDRRRT

∫t)0t xA dDA - ∫0t

)

dt )

)

dt )

∫t)0t xB dDB

(12)

∫t0t xR dDR - ∫t0t [k2xA2

(1.22DA - DR) + k3xB(0.99DB - DR)] dt (13)

In the above equations, M is the molecular weight of titania (79.9 g/mol), γ the surface energy of a titania phase, k a kinetic constant for a phase transformation [reactions 3, 4, or 5] obtained from the kinetic modeling of the phase transformations (Figure 6), and L a coarsening rate constant. The time t0 in eq 13 is the time when rutile was first detected in a time-series experiment at a fixed temperature. The right-hand sides of eqs 11-13 can be calculated numerically with experimental data (Figures 3 and 4) and the kinetic parameters for phase transformation reactions 3-5 (Figure 6). The left-hand sides can also be calculated, given that γ and L are known. Thus, the unknown surface energy γ (and L as well) can be obtained by minimizing the sum of the differences between the left-hand side and the right-hand side of an equation (eqs 11, 12, or 13) for all data points (see Figure A4 of the Supporting Information for an example). Table 2 lists the surface energies of anatase, brookite, and rutile thus obtained. The scattering in these data may be due to that in their numerical derivations there are only limited numbers of data points in the numerical integrations of eqs 11-13. It is seen that in general the surface energies of titania increase with the temperature. The derivative of the surface energy with respect to the temperature gives the excess heat capacity possessed by surface atoms. At T > 580 °C for powders and T > 560 °C for membranes, the derived surface energies of rutile are higher than expected. The rutile coarsening mechanism considered here (see the Supporting Information) may not be able to fully account for growth at higher temperatures. Furthermore, the particle sizes of rutile calculated by using the Scherrer equation (eq 2) are less accurate at high temperatures, because the broadening of XRD peaks from large rutile particles is minimal. These factors may contribute to the anomaly in the derived surface energies. Thus, in the calculation of the average surface energies of rutile, the values at higher temperature were excluded.

The average surface energies of anatase, brookite, and rutile obtained from the coarsening data (Table 2) are in fairly good agreement with those calculated from results of atomistic lattice calculations and empirical estimations,12,28 and those determined by high-temperature calorimetry.29,30 The average surface energy of rutile is greater than that of brookite, and that of brookite is greater than that of anatase. Table 2 also shows that the surface energy of a titania phase in membranes is lower than that in powders. This is because more and stronger interactions between atoms on interfaces of nanoparticles in membranes reduce the energy difference between the surfaces and the interior of the nanoparticles. Conclusions Compared with the membranes, powders are looser aggregations of nanoparticles and contain a significant number of big nanopores (>10 nm). In the temperature range of 500-600 °C, brookite transforms to anatase and anatase transforms to rutile via interface nucleation, and brookite transforms to rutile via surface nucleation. The transformation sequence reflects the relative phase stabilities of the nanophases present at the reaction conditions. The activation energies and the frequency factors for the phase transformations are lower in the powders than in the membranes. Recrystallization of a single phase and transformations from one phase to another all contribute to the nanoparticle coarsening. The surface energies of titania derived from the coarsening data are in good agreement with the literature values. The above results emphasize the importance of physical status of nanostructured materials on the phase transformation kinetics and the influence of the kinetics on the nanoparticle coarsening as well. Acknowledgment. The authors thank Dr. P. Mendes for providing the GEPASI package. Financial support for this work was provided by the U.S. Department of Energy (Grant No. DE-FG03-01ER15218) and the National Science Foundation (Grant No. EAR-0123967). Supporting Information Available: Graphs showing the variation of the number of titania nanoparticles with time and kinetic modeling of experimental data for phase transformations in nanocrystalline thtania powders and membranes, as well as details of the derivation of the time evolution of the average particle size of nanocrystalline titania. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Bokhimi; Morales, A.; Novaro, O.; Lopez, T.; Gomez, R. J. Mater. Res. 1995, 10, 2788.

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