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Crystallization kinetics study of cerium titanate CeTi2O6
Journal of Physics and Chemistry of Solids 75 (2014) 265–270
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Crystallization kinetics study of cerium titanate CeTi2O6 Václav Valeš a,n, Lenka Matějová b, Zdeněk Matěj a, Tereza Brunátová a, Václav Holý a a
Charles University in Prague, Faculty of Mathematics and Physics, Department of Condensed Matter Physics, Ke Karlovu 5, 121 16 Prague 2, Czech Republic Institute of Chemical Process Fundamentals of the ASCR, v. v. i., Department of Catalysis and Reaction Engineering, Rozvojová 135, 165 02 Prague 6, Czech Republic
b
art ic l e i nf o
a b s t r a c t
Article history: Received 25 July 2013 Received in revised form 1 October 2013 Accepted 11 October 2013 Available online 18 October 2013
Cerium titanate CeTi2O6 has been investigated recently for its photocatalytic activity and as a safe analogue to actinide-containing brannerite-like titanates (UTi2O6, PuTi2O6, e.g.) which are intensively studied because of their use for storing nuclear waste. In this paper we report on the monoclinic phase CeTi2O6 obtained from the Ti–Ce oxide mixture prepared by a reverse micelles directed sol–gel method and subsequently annealed. The kinetics of the isothermal crystallization process is investigated by means of Johnson–Mehl–Avrami–Kolmogorov equation. The effective activation energy of the formation of CeTi2O6 particles, which is an important parameter for its synthesis, is estimated. & 2013 Elsevier Ltd. All rights reserved.
Keywords: A. Ceramics B. Sol–gel growth C. x-ray diffraction D. Thermodynamic properties
1. Introduction Titania–ceria mixed oxides have been studied recently because of their application potential in sensing films of gas sensors [1], as an electrode material in electrochromic devices [2], coatings for chemical–mechanical polishing [3], self-cleaning surfaces [4], photocatalysts with visible light induced activity [5,6] and catalysts support [7]. Various preparation methods based on sol–gel processes have been reported [1,2,4,5,7,8]. In a certain range of Ti:Ce content in the mixture, ternary cerium titanates [8–10] crystallize. These cerium titanates can form different phases with different oxidation states of Ce ions. In a reducing atmosphere during annealing process phases with Ce3 þ ions are formed, such as Ce2TiO5, Ce2Ti2O7, and Ce4Ti9O24, while annealing in the air gives rise to Ce4 þ ions containing phases, CeTiO4, and CeTi2O6 [11]. Phases involving Ce4 þ ions exhibit higher photocatalytic activity than those involving Ce3 þ ions [11]. The structure of the monoclinic CeTi2O6 phase (space group C2/m, unique axis b) consists of zigzag layers of Ti atoms that are coordinated in distorted octahedra and layers of distorted Ce octahedra [12,13], see inset of the Fig. 4. The Ce atom is located in the Wyckoff position 2a, Ti and all oxygen atoms O1,2,3 are in the positions 4i. A detailed structural study [14] has shown recently that the structure of CeTi2O6 is rather non-stoichiometric (should be expressed as Ce1 xTi2O6 2x, with x¼ 0.025). However our study was not so sensitive to the presence of Ce and O vacancies, hence we assumed a stoichiometric form here (x ¼0). The crystal phase is
n
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isomorphous to the structure of UTi2O6 (brannerite), which is a part of synroc—a material developed to bind nuclear waste elements [12,15]. Since the study of actinide-containing materials is associated with difficulties, CeTi2O6 has been studied as their analogue—it has similar ionic radius and coordination environment [16,17]. In this paper we investigate the kinetics of crystallization of CeTi2O6 phase prepared by a reverse micelles directed sol–gel method. Earlier we studied four samples prepared by the same method with different ratios of Ti and Ce in the initial mixture (from 90% of Ti to 30% of Ti in molar units, see [18]), however in this paper we present the detailed results achieved for the mixture of nominal molar composition of 70% Ti and 30% Ce, since this particular sample after annealing consisted of almost pure monoclinic CeTi2O6. Understanding the crystallization process of cerium titanate is essential for optimizing the conditions of its preparation.
2. Experimental 2.1. Synthesis We considered a series of four Ti–Ce oxide samples varying in the nominal molar ratio of Ti and Ce (Ti90–Ce10, Ti70–Ce30, Ti50– Ce50, and Ti30–Ce70) prepared in the free powder form and calcined for 4 h at 350 1C. Samples Ti90–Ce10 and Ti30–Ce70 crystallized into binary oxides of Ti (anatase and rutile) and Ce (cerianite) [18], sample Ti50–Ce50 contained crystalline cubic cerium oxide (cerianite), anatase TiO2 and ternary phase of CeTi2O6. Sample Ti70–Ce30 crystallized to almost pure CeTi2O6 monoclinic phase with minor amount of anatase-TiO2 (Fig. 1).
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In this paper we study in detail the kinetics of the formation of this ternary phase in the sample Ti70–Ce30 by means of x-ray diffraction. For the synthesis of Ti70–Ce30 mixed oxide, the following chemicals were used: cyclohexane (Aldrich, HPLC grade), nonionic surfactant Triton X-114 ((1,1,3,3-tetramethylbutyl)phenyl– polyethylene glycol, C29H52O8.5, Aldrich), absolute ethanol (water content max. 0.2 vol%), titanium(IV) isopropoxide (Ti(OCH(CH3)2)4, 99.999%, Aldrich), cerium(III) nitrate hexahydrate (Ce(NO3)3 6 H2O, Aldrich) and distilled water. Ti70–Ce30 mixed oxide as a powder was prepared via the sol– gel processing controlled within reverse micelles of nonionic surfactant Triton X-114 in cyclohexane and the thermal treatment [18]. The first step of synthesis consisted in the preparation of ceria sol: cerium(III) nitrate hexahydrate (1.1982 g) was dissolved in absolute ethanol (3 ml) under intense stirring. In the second step cyclohexane (16.4 ml) was mixed with Triton X-114 (14.4 ml) and distilled water (0.2 ml) followed by the addition of ceria sol. Prepared slightly yellow-coloured sol was stirred for 20 min.
Titanium(IV) isopropoxide (1.887 ml) was injected fluently into the mixture in the final step. The micellar intensely yellowcoloured titania–ceria sol was stirred for other 20 min. After that the homogeneous transparent sol was poured into Petri's dish in a thin layer ( 3 mm) and the dish was left standing on air up to the conversion of the sol into the gel. The gelation period was max. 60 h. Generally, the Ti70–Ce30 sol was prepared keeping the molar ratio of cyclohexane:Triton X-114:H2O: Ti(OCH(CH3)2)4 þCe (NO3)3 6H2O at 16.5:3:3:1 with the used amount of absolute ethanol 3 ml. The Ti70–Ce30 gel was scraped out from the Petri's dish to a calcination cup and thermally treated in air at 350 1C for 4 h with the heating rate 1 1C/min in order to produce ‘asprepared‘ Ti70–Ce30 brightly yellow-coloured powder for in-situ x-ray diffraction (XRD) studies. The calcination has been performed in order to convert the gel into the powder form. This conversion is accompanied by a removal of all organic part of the gel i.e. solvents, surfactant, organic rest of metal oxide precursors which were used in the preparation. 2.2. Characterization by non-diffraction methods We performed x-ray fluorescence (XRF), Raman spectroscopy, and nitrogen desorption experiments on the whole series of samples with different Ti:Ce content in our previous work. Details can be found in [18]. For the study here information on the actual molar composition of the samples is desirable. From the XRF analysis [18] for the sample studied in this work (nominally 70% of Ti atoms) the molar ratio of Ti atoms was found to be (72.2 7 1.4)%. 2.3. X-ray diffraction characterization
Fig. 1. Diffraction curves of samples with various content of Ti and Ce after annealing at 900 1C. The grey lines in the bottom correspond to diffraction positions of all phases found in the samples (C—cerianite CeO2, T—ternary CeTi2O6, A—anatase TiO2, R—rutile TiO2). In the sample Ti30–Ce70 cerianite, rutile and anatase phases were found, in the sample Ti50–Ce50 cerianite, CeTi2O6 and anatase, in the sample Ti70–Ce30 CeTi2O6, and in the sample Ti90–Ce10 anatase, rutile, cerianite. The most intensive peaks corresponding to the major phases in the sample are indicated by the appropriate symbols.
X-ray diffraction curves were measured by the PANalyticalMPD diffractometer in the conventional focusing Bragg-Brentano geometry with variable slits using Ni-filtered CuKα characteristic radiation. PIXcel linear detector was used for a fast collection of the scattered intensity in the diffraction angle range 101–1201. The heating (up to 700 1C, in the air, heating rate 60 1C/min) was performed with a radiant heater placed around the sample. The sample was placed in an alumina crucible with a Pt–Rh thermocouple immersed into the sample inside the crucible. The annealing conditions were the same for all four temperatures, so
Fig. 2. The Williamson-Hall plot of selected diffraction peaks (black stars) of the ex-situ sample annealed at 650 1C. The anisotropy in strain can be seen in the data. For the fit of the anisotropy parameters, only clearly non-overlapping peaks were chosen (those with grey background). From the fit we calculated the integral breaths of the remaining diffractions (grey line). The dotted lines represent the linear fits of {h00} and {00l} diffractions with a common intercept at 1/d ¼0. The plotted breadths were corrected for the instrumental broadening.
V. Valeš et al. / Journal of Physics and Chemistry of Solids 75 (2014) 265–270
the relative uncertainty of the temperature determination is 710 1C. Each individual in-situ scan took 10 min. The measured data were simulated using the whole powder pattern modellingbased [19] software MStruct [20]. Although the main subject of this study is the crystallization kinetics of CeTi2O6, i.e. the analysis of time evolution of diffracted intensity, some attention was also given to the proper choice of a diffraction line profile model as it was essential to achieve a good pattern fit. Analysing the Williamson-Hall plot of the ex-situ data of the sample annealed at 650 1C a strong anisotropy of strain has been found (Fig. 2). There are several models [21–23] describing this effect. We used the model introduced by Popa. Within this model the relative change of the interplanar distance in the direction of the diffraction vector (perpendicular to diffracting lattice planes) defines the hkl-dependent strain εhkl ¼ Δdhkl/dhkl. In a stress free powder due to local variations of the lattice parameters its variance 〈ε2hkl〉 can be nonzero and the square root of the variance defines a phenomenological parameter—microstrain (ehkl). For the polycrystalline material of Laue group 2/m symmetry the model of Popa gives [21]: 4
4
4
2 2
2 2
2 2
Table 2 Reduced sample weight fraction, calculated as the weight fraction of the sample from the weighted amount of individual materials (studied sample and reference silicon) reduced by the weight of non-stoichiometric excess Ti atoms that cannot crystallize into CeTi2O6, is compared to the weight fraction of CeTi2O6 phase obtained from the diffraction fit. Sample
650 670 700
Sample weight (g) 0.0432 0.0427 0.0639
3
Parameter E1 E2 E3 E4 E5 E6 E7 E8 E9
ð1Þ
where a is the lattice parameter, Hhkl ¼ 2sinΘhkl/λ is the diffraction vector length and E1–E9 are nine independent phenomenological parameters. An origin of the local variations of the lattice parameters can be of different type. For the material under study we can consider: e.g. local variation of the chemical composition or dislocations. Faults in the layers stacking would induce even more complex peak broadening effects [19]. For dislocations the model in [23] could even assign to E1-E9 parameters physically relevant values and the dislocation density could be determined. Since dislocation Burgers vectors types and elastic constants of the material are necessary, it is out of the scope of this work. For modelling of the line profiles from the microstrain effect the simple Pseudo–Voigt function was used. It was convoluted with other broadening effects (instrumental, size) [20]. For the ex-situ data the additional refined parameters were lattice parameters, isotropic Debye–Waller temperature factors for individual atom types (Ce, Ti, O), atomic positions and the parameters of the crystallite size distribution. The crystallites were assumed to be spherical with log-normal size distribution described by median M and s-parameter, where es is the geometric standard deviation. The diffraction is a volume sensitive method. Hence the sizes reported in this paper are the volume weighted diameters which are considered to be the natural presentation of the information embedded in the diffraction data [24,25]. The refined parameters can be found in Tables 1 and 3. For the fitting of the in-situ data, atomic positions (see the Table 1) and the geometric standard deviation of the particle size distribution (s ¼0.65) were fixed to the values obtained from the ex-situ measurement. In order to assess the growth kinetic, the diffraction scans were rather rapid so that the quality of the data collected did not allow fitting these parameters. Table 1 Atomic positions of CeTi2O6 phase obtained from the post annealing fit. All zeros arise from the symmetry of the individual Wyckoff positions and therefore have not been refined. Atom
x
Y
z
Ce Ti O1 O2 O3
0 0.824(1) 0.968(2) 0.652(2) 0.273(2)
0 0 0 0 0
0 0.388(1) 0.297(2) 0.112(2) 0.396(2)
Reduced sample weight fraction (%)
CeTi2O6 weight fraction from diffration (%)
597 5 61 75 567 5
63.0 7 0.5 56.2 7 0.5 51.7 7 0.5
Table 3 The anisotropic strain parameters obtained from the post-annealing (ex-situ) data.
3
2
Si weight (g) 0.0220 0.0253 0.0294
e2hkl ¼ ðE1 h þ E2 l þ E3 k þ 2E4 h l þ 2E5 l k þ 2E6 h k þ 4E7 h l þ 4E8 l h þ 4E9 k lhÞ=ðH hkl aÞ4 ;
267
Value (1e-4) 0.050(7) 1.20(7) 9.8(4) 0.29(4) 4.2(3) 0.12(8) 0.022(7) 0.01(3) 0.1(1)
The other refined parameters were the lattice parameters, isotropic Debye–Waller factors, median of the particle size distribution and the anisotropic strain parameters.
3. Kinetics We investigated the kinetics of the formation of CeTi2O6 phase. The time-dependent amount of the crystallized material from amorphous state at constant temperature can be described by the Johnson–Mehl–Avrami–Kolmogorov (JMAK) equation for homogeneous crystallization [26]: n
YðtÞ ¼ 1 e Kðt t0 Þ ;
ð2Þ
where Y is the volume fraction of crystallized matter, t is the time (variable), t0 is the time constant of the beginning of the crystallization, n is the Avrami exponent describing the nature of the crystallization and K is a temperature-dependent factor, following the Arhenius equation: K ¼ K 0 e Ea =kB T ;
ð3Þ
where K0 is a constant, Ea is the effective activation energy of the whole crystallization process involving both the creation of the nuclei and the growth of the particle, kB is the Boltzmann constant and T is the absolute temperature. Possible values of the exponent n will be discussed later.
4. Results and discussion From the diffraction data the formation of CeTi2O6 phase as a function of time can be clearly observed. The sample annealed at 650 1C starts to crystallize about 45 min after reaching the annealing temperature (Fig. 3(a)). On the other hand, the sample annealed at the highest temperature, 700 1C, exhibit the CeTi2O6-related peaks immediately after reaching the annealing temperature (Fig. 3(b)). The analysis of the CeTi2O6 phase structure was performed at the sample annealed at 650 1C for 6 h (until the crystallization was fully saturated) from the data measured ex-situ after cooling down to the
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room temperature (Fig. 4). From the Rietveld refinement we determined the volume–weighted mean value of diameter of the particles DV ¼(125710) nm and the lattice parameters: a¼9.8146(5) Å, b¼ 3.7022(2) Å, c¼6.7720(4) Å, β ¼ 118.69(1)1, which are slightly smaller than the values from literature [8,16]. The atomic positions are summarized in the Table 1 and they are in a good agreement with the published data—(ICDD, 2012, PDF 01-014-5813). In the high statistics ex-situ scan a 101 diffraction peak from anatase has been observed. Due to the non-stoichiometry of the initial mixture (72.2%Ti, 27.8%Ce) an excess of Ti atoms can be expected. The weight fraction of Ti atoms that do not contribute to the CeTi2O6 phase is nominally 10.8% of both Ce and Ti atoms together. From the fit of the measured data (Fig. 4) the relative weight content of anatase phase is (7.270.5)% which corresponds to the weight fraction of Ti atoms (6.1 70.5)% in the ensemble of Ti and Ce atoms in anatase and CeTi2O6. This value suggests that at
Fig. 3. In the panel (a), the time evolution of the sample Ti70–Ce30 annealed at 650 1C expressed by x-ray diffraction curves taken after 0.75, 1.75, 2.75, 3.75 h from the reaching the annealing temperature is shown. The formation of new phase from amorphous state is clearly visible. The curves are vertically shifted for clarity. The grey lines in the bottom depict the positions of the CeTi2O6 diffraction maxima. Panel (b) shows the diffraction pattern for the samples annealed at all four temperatures (650, 670, 685, and 700 1C) 30 min after reaching the annealing temperature.
least a half of excess Ti atoms that cannot be incorporated in CeTi2O6 phase forms crystalline anatase-TiO2. The form of the amorphous state, which can be observed before crystallization of CeTi2O6 when annealing at 650 1C, was qualitatively studied by modelling the measured broad peaks with very small (diameter 1 nm) particles. From these rough simulations (Fig. 5) it can be seen, that the positions of the maxima may correspond to cerianite particles. It indicates that in this “amorphous state” Ce tends to form very small particles (with just several atoms) while titanium oxide remains in genuine amorphous state. This is similar to what was observed for mixtures with different Ti–Ce molar ratio also by Raman spectroscopy [18]. For the study of the kinetics of the crystallization, we fitted the in-situ xrd data (as a function of time, at four different temperatures, 650 1C, 670 1C, 685 1C, and 700 1C). In this temperature range the crystallization proceeds in a reasonable speed and the data from individual temperatures differ enough. From the fits we obtained scale factors which correspond to the amount of crystallized CeTi2O6 phase. The time dependence of this amount has been fitted by the JMAK Eq. (1). The asymptotic value lim Y ðt Þ of the t-1 fitted JMAK curve was set to 100%. Therefore the 100% value corresponds to the saturated amount of the CeTi2O6 phase and does not need to be directly related to the total sample weight. The degree of crystallinity is discussed later. It can be seen (Fig. 6) that for the sample annealed nominally at 685 1C the crystallization proceeds slower than for the sample annealed nominally at 670 1C. This discrepancy is induced by the
Fig. 5. Measured diffraction data (grey dots) of the sample just after reaching 650 1C. The broad maxima are fitted with small particles (diameter 1 nm) of cerianite (full line), CeTi2O6 (dash-dotted line) and rutile (dash line). The fit for anatase particles is not displayed for clarity; its peaks are shifted to even lower angles compared to rutile.
Fig. 4. Measured (grey dots) and fitted (black line) diffraction curves of the sample annealed at 650 1C for 6 h (until the crystallization was fully saturated) after cooling down to the room temperature. The grey line is the difference curve. The inset displays the structure of CeTi2O6 in polyhedral representation.
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269
Fig. 6. Measured data (points) and their fit (lines) of the CeTi2O6-crystallized fraction as a function of time for various temperatures. The depicted error bar shows the estimated error for all points. The inset displays the plot of ln(K) vs. reciprocal of the annealing temperature. From the linear fit of these data the value of effective activation energy of CeTi2O6 particle formation has been determined. The vertical error is caused by the fit of the JMAK equation, the horizontal one is due to the uncertainty of temperature (7 10 1C).
uncertainty of the temperature determination of 710 1C. A potential uncertainty of the determination of the annealing time does not play role in the analysis since (1) the annealing processes were as similar to each other as possible (inducing maximal error of 1 min) and (2) the time difference would affect the t0 parameter and does not influence essentially the parameter K used in estimation of the activation energy. The calcination of the samples during their preparation should not cause any effect on particle crystallization since the calcination temperature was much lower than crystallization temperatures and all the samples were calcined in the exactly same way. From the JMAK fit (Fig. 6) the temperature-dependent parameters K were obtained, and by their fitting to Eq. (2) the effective activation energy of the whole crystallization process Ea ¼ 1.5(4) eV was calculated (Fig. 6, inset). This result is of the same order as the total activation energies of pure Ti (from 1.3 to 2.5 eV, depending on the specific phase and the preparation procedure; the activation energy of the growth of anatase particles is in the range 0.35– 0.9 eV and for Ce oxides (from 0.3 to 1.7 eV) [27–30]. The total activation energy of CeTi2O6 obtained from the heavy ion irradiation amorphization was reported to be 0.95 eV [31], which is slightly smaller value than we obtained. The Avrami exponent n from Eq. (1) describes the nature of the crystallization. It depends on the dimensionality of the growth and on the rate of formation of new nuclei [26,32]. We assumed n to have a constant value for all four temperatures and fitted the crystallization curves for all temperatures together. From the fit we found the value of n ¼1.7(2), which does not agree with standard theoretical values of Avrami exponent. The Avrami exponent n in the JMAK theory is a sum of two terms n ¼ a þb c: The parameter a is the exponent in the time dependence of the number of nuclei N per unit volume of untransformed material: N t a : The parameter b describes the number of dimensions of the growth (b¼ 3 for three-dimensional (3D) growth), and c¼1, 0.5 for the interfacecontrolled and diffusion-controlled growth, respectively. For the interface-controlled 3D growth (which is expected for free powder particles) the Avrami coefficient should reach the value of 3–4.
This disagreement is probably caused by the fact that the JMAK theory supposes a homogeneous nucleation but in our case of a free powder, consisting of small particles, heterogeneous nucleation on the surface of the particles is more likely to happen. In the case of heterogeneous nucleation the decrease of Avrami exponent n has been predicted and observed [33,34]. The volume–weighted mean value of the diameter of CeTi2O6 particles is not changing systematically with time and for all annealing temperatures for all samples it is in the range (90– 130) nm which is in an agreement with the ex-situ data after cooling. This can be seen already from the Fig. 3(a) demonstrating that already narrow peaks start to appear during the crystallization. From this fact it follows, that we do not see the particle growth but only the creation of new nuclei of CeTi2O6 phase which then rapidly grow up to the observed size. Thus, the activation energy of the particle growth is much smaller than the activation energy of the nucleation. In addition the degree of the sample crystallinity after annealing has been studied by mixing a certain amount of fully crystalline Si standard into the samples. The measured data were not of the quality to distinguish the anatase peak as in the high-quality scan, therefore the obtained results have to be compared to the values reduced by the non-stoichiometry excess Ti atoms that crystallize to anatase or remain amorphous. From the results (Table 2) it can be seen that the entire stoichiometric part of the sample (within the errors) has crystallized to CeTi2O6. The errors of the reduced weight fractions arise from the error of the x-ray fluorescence measurements and from the uncertainties of the sample weighting and corresponding manipulation. Finally to illustrate the main trends in the anisotropic diffraction line broadening, the integral breadths of several hkl-reflections were used to construct the Williamson-Hall plot (Fig. 2). There are two orders of reflections from the {h00} and {00l} planes. The common intersect of the linear fits through them gives the hkl-independent (constant) broadening level which could be related to the size-effect. The rest of the broadening for any hkl-reflection can be well explained by the anisotropic
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microstrain effect described earlier. This is illustrated in Fig. 2 by a model line which reproduces the characteristic hkl-anisotropy in the data. To have an idea of the effect strength the microstrain was evaluated in the directions perpendicular to the {h00} planes to be eh00 E0.3%. It is evident from Fig. 2 that in some directions it is even higher. On the contrary it is negligible (e00l E0%) in the directions perpendicular to the {00l} planes. The CeTi2O6 is not such intensively studied material as its two counterparts (CeO2 [28], TiO2 [24]) and hence it is difficult to conclude if the physical origin of the microstrain effect is in the presence of dislocations, or it is related to the local variation of the lattice parameters e.g. due to clustering of vacancies, which were reported in [14]. This would require a thorough study. 5. Conclusion We measured the diffraction patterns of the Ti–Ce mixed oxide sample with nominal molar content of Ti atoms of 70% at four temperatures (650 1C, 670 1C, 685 1C, 700 1C) depending on time. The crystallization of cerium titanate CeTi2O6 has been observed. We fitted the measured data with a model including log-normal distribution of particle sizes and an anisotropic microstrain in the particles. From the fit of the measured data the lattice parameters, the parameters of the particle distribution, anisotropic strain parameters and the atomic positions were determined. The main scope of this work, however, was to investigate the kinetics of the cerium titanate crystallization. The effective activation energy was found to be Ea ¼1.5(4) eV and it is mostly composed of the activation energy of the particle nucleation. The growth of the particles has not been seen; a direct formation of particles with sizes in the range (90–130) nm has been observed. The study of the crystalinity of the sample after annealing revealed that the whole part of the sample that corresponds to the stoichiometry of CeTi2O6 crystallizes into this phase indeed. The excess Ti atoms form crystalline anatase-TiO2 or remain in an amorphous state. Acknowledgements This project has been supported by the Grant Agency of the Czech Republic (project numbers P204-11-0785, P108-11-1539 and 104/09/P290).
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Journal of Physics and Chemistry of Solids journal homepage: www.elsevier.com/locate/jpcs
Crystallization kinetics study of cerium titanate CeTi2O6 Václav Valeš a,n, Lenka Matějová b, Zdeněk Matěj a, Tereza Brunátová a, Václav Holý a a
Charles University in Prague, Faculty of Mathematics and Physics, Department of Condensed Matter Physics, Ke Karlovu 5, 121 16 Prague 2, Czech Republic Institute of Chemical Process Fundamentals of the ASCR, v. v. i., Department of Catalysis and Reaction Engineering, Rozvojová 135, 165 02 Prague 6, Czech Republic
b
art ic l e i nf o
a b s t r a c t
Article history: Received 25 July 2013 Received in revised form 1 October 2013 Accepted 11 October 2013 Available online 18 October 2013
Cerium titanate CeTi2O6 has been investigated recently for its photocatalytic activity and as a safe analogue to actinide-containing brannerite-like titanates (UTi2O6, PuTi2O6, e.g.) which are intensively studied because of their use for storing nuclear waste. In this paper we report on the monoclinic phase CeTi2O6 obtained from the Ti–Ce oxide mixture prepared by a reverse micelles directed sol–gel method and subsequently annealed. The kinetics of the isothermal crystallization process is investigated by means of Johnson–Mehl–Avrami–Kolmogorov equation. The effective activation energy of the formation of CeTi2O6 particles, which is an important parameter for its synthesis, is estimated. & 2013 Elsevier Ltd. All rights reserved.
Keywords: A. Ceramics B. Sol–gel growth C. x-ray diffraction D. Thermodynamic properties
1. Introduction Titania–ceria mixed oxides have been studied recently because of their application potential in sensing films of gas sensors [1], as an electrode material in electrochromic devices [2], coatings for chemical–mechanical polishing [3], self-cleaning surfaces [4], photocatalysts with visible light induced activity [5,6] and catalysts support [7]. Various preparation methods based on sol–gel processes have been reported [1,2,4,5,7,8]. In a certain range of Ti:Ce content in the mixture, ternary cerium titanates [8–10] crystallize. These cerium titanates can form different phases with different oxidation states of Ce ions. In a reducing atmosphere during annealing process phases with Ce3 þ ions are formed, such as Ce2TiO5, Ce2Ti2O7, and Ce4Ti9O24, while annealing in the air gives rise to Ce4 þ ions containing phases, CeTiO4, and CeTi2O6 [11]. Phases involving Ce4 þ ions exhibit higher photocatalytic activity than those involving Ce3 þ ions [11]. The structure of the monoclinic CeTi2O6 phase (space group C2/m, unique axis b) consists of zigzag layers of Ti atoms that are coordinated in distorted octahedra and layers of distorted Ce octahedra [12,13], see inset of the Fig. 4. The Ce atom is located in the Wyckoff position 2a, Ti and all oxygen atoms O1,2,3 are in the positions 4i. A detailed structural study [14] has shown recently that the structure of CeTi2O6 is rather non-stoichiometric (should be expressed as Ce1 xTi2O6 2x, with x¼ 0.025). However our study was not so sensitive to the presence of Ce and O vacancies, hence we assumed a stoichiometric form here (x ¼0). The crystal phase is
n
Corresponding author. Tel.: þ 420 221 911 482. E-mail address: [email protected] (V. Valeš).
0022-3697/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jpcs.2013.10.001
isomorphous to the structure of UTi2O6 (brannerite), which is a part of synroc—a material developed to bind nuclear waste elements [12,15]. Since the study of actinide-containing materials is associated with difficulties, CeTi2O6 has been studied as their analogue—it has similar ionic radius and coordination environment [16,17]. In this paper we investigate the kinetics of crystallization of CeTi2O6 phase prepared by a reverse micelles directed sol–gel method. Earlier we studied four samples prepared by the same method with different ratios of Ti and Ce in the initial mixture (from 90% of Ti to 30% of Ti in molar units, see [18]), however in this paper we present the detailed results achieved for the mixture of nominal molar composition of 70% Ti and 30% Ce, since this particular sample after annealing consisted of almost pure monoclinic CeTi2O6. Understanding the crystallization process of cerium titanate is essential for optimizing the conditions of its preparation.
2. Experimental 2.1. Synthesis We considered a series of four Ti–Ce oxide samples varying in the nominal molar ratio of Ti and Ce (Ti90–Ce10, Ti70–Ce30, Ti50– Ce50, and Ti30–Ce70) prepared in the free powder form and calcined for 4 h at 350 1C. Samples Ti90–Ce10 and Ti30–Ce70 crystallized into binary oxides of Ti (anatase and rutile) and Ce (cerianite) [18], sample Ti50–Ce50 contained crystalline cubic cerium oxide (cerianite), anatase TiO2 and ternary phase of CeTi2O6. Sample Ti70–Ce30 crystallized to almost pure CeTi2O6 monoclinic phase with minor amount of anatase-TiO2 (Fig. 1).
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In this paper we study in detail the kinetics of the formation of this ternary phase in the sample Ti70–Ce30 by means of x-ray diffraction. For the synthesis of Ti70–Ce30 mixed oxide, the following chemicals were used: cyclohexane (Aldrich, HPLC grade), nonionic surfactant Triton X-114 ((1,1,3,3-tetramethylbutyl)phenyl– polyethylene glycol, C29H52O8.5, Aldrich), absolute ethanol (water content max. 0.2 vol%), titanium(IV) isopropoxide (Ti(OCH(CH3)2)4, 99.999%, Aldrich), cerium(III) nitrate hexahydrate (Ce(NO3)3 6 H2O, Aldrich) and distilled water. Ti70–Ce30 mixed oxide as a powder was prepared via the sol– gel processing controlled within reverse micelles of nonionic surfactant Triton X-114 in cyclohexane and the thermal treatment [18]. The first step of synthesis consisted in the preparation of ceria sol: cerium(III) nitrate hexahydrate (1.1982 g) was dissolved in absolute ethanol (3 ml) under intense stirring. In the second step cyclohexane (16.4 ml) was mixed with Triton X-114 (14.4 ml) and distilled water (0.2 ml) followed by the addition of ceria sol. Prepared slightly yellow-coloured sol was stirred for 20 min.
Titanium(IV) isopropoxide (1.887 ml) was injected fluently into the mixture in the final step. The micellar intensely yellowcoloured titania–ceria sol was stirred for other 20 min. After that the homogeneous transparent sol was poured into Petri's dish in a thin layer ( 3 mm) and the dish was left standing on air up to the conversion of the sol into the gel. The gelation period was max. 60 h. Generally, the Ti70–Ce30 sol was prepared keeping the molar ratio of cyclohexane:Triton X-114:H2O: Ti(OCH(CH3)2)4 þCe (NO3)3 6H2O at 16.5:3:3:1 with the used amount of absolute ethanol 3 ml. The Ti70–Ce30 gel was scraped out from the Petri's dish to a calcination cup and thermally treated in air at 350 1C for 4 h with the heating rate 1 1C/min in order to produce ‘asprepared‘ Ti70–Ce30 brightly yellow-coloured powder for in-situ x-ray diffraction (XRD) studies. The calcination has been performed in order to convert the gel into the powder form. This conversion is accompanied by a removal of all organic part of the gel i.e. solvents, surfactant, organic rest of metal oxide precursors which were used in the preparation. 2.2. Characterization by non-diffraction methods We performed x-ray fluorescence (XRF), Raman spectroscopy, and nitrogen desorption experiments on the whole series of samples with different Ti:Ce content in our previous work. Details can be found in [18]. For the study here information on the actual molar composition of the samples is desirable. From the XRF analysis [18] for the sample studied in this work (nominally 70% of Ti atoms) the molar ratio of Ti atoms was found to be (72.2 7 1.4)%. 2.3. X-ray diffraction characterization
Fig. 1. Diffraction curves of samples with various content of Ti and Ce after annealing at 900 1C. The grey lines in the bottom correspond to diffraction positions of all phases found in the samples (C—cerianite CeO2, T—ternary CeTi2O6, A—anatase TiO2, R—rutile TiO2). In the sample Ti30–Ce70 cerianite, rutile and anatase phases were found, in the sample Ti50–Ce50 cerianite, CeTi2O6 and anatase, in the sample Ti70–Ce30 CeTi2O6, and in the sample Ti90–Ce10 anatase, rutile, cerianite. The most intensive peaks corresponding to the major phases in the sample are indicated by the appropriate symbols.
X-ray diffraction curves were measured by the PANalyticalMPD diffractometer in the conventional focusing Bragg-Brentano geometry with variable slits using Ni-filtered CuKα characteristic radiation. PIXcel linear detector was used for a fast collection of the scattered intensity in the diffraction angle range 101–1201. The heating (up to 700 1C, in the air, heating rate 60 1C/min) was performed with a radiant heater placed around the sample. The sample was placed in an alumina crucible with a Pt–Rh thermocouple immersed into the sample inside the crucible. The annealing conditions were the same for all four temperatures, so
Fig. 2. The Williamson-Hall plot of selected diffraction peaks (black stars) of the ex-situ sample annealed at 650 1C. The anisotropy in strain can be seen in the data. For the fit of the anisotropy parameters, only clearly non-overlapping peaks were chosen (those with grey background). From the fit we calculated the integral breaths of the remaining diffractions (grey line). The dotted lines represent the linear fits of {h00} and {00l} diffractions with a common intercept at 1/d ¼0. The plotted breadths were corrected for the instrumental broadening.
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the relative uncertainty of the temperature determination is 710 1C. Each individual in-situ scan took 10 min. The measured data were simulated using the whole powder pattern modellingbased [19] software MStruct [20]. Although the main subject of this study is the crystallization kinetics of CeTi2O6, i.e. the analysis of time evolution of diffracted intensity, some attention was also given to the proper choice of a diffraction line profile model as it was essential to achieve a good pattern fit. Analysing the Williamson-Hall plot of the ex-situ data of the sample annealed at 650 1C a strong anisotropy of strain has been found (Fig. 2). There are several models [21–23] describing this effect. We used the model introduced by Popa. Within this model the relative change of the interplanar distance in the direction of the diffraction vector (perpendicular to diffracting lattice planes) defines the hkl-dependent strain εhkl ¼ Δdhkl/dhkl. In a stress free powder due to local variations of the lattice parameters its variance 〈ε2hkl〉 can be nonzero and the square root of the variance defines a phenomenological parameter—microstrain (ehkl). For the polycrystalline material of Laue group 2/m symmetry the model of Popa gives [21]: 4
4
4
2 2
2 2
2 2
Table 2 Reduced sample weight fraction, calculated as the weight fraction of the sample from the weighted amount of individual materials (studied sample and reference silicon) reduced by the weight of non-stoichiometric excess Ti atoms that cannot crystallize into CeTi2O6, is compared to the weight fraction of CeTi2O6 phase obtained from the diffraction fit. Sample
650 670 700
Sample weight (g) 0.0432 0.0427 0.0639
3
Parameter E1 E2 E3 E4 E5 E6 E7 E8 E9
ð1Þ
where a is the lattice parameter, Hhkl ¼ 2sinΘhkl/λ is the diffraction vector length and E1–E9 are nine independent phenomenological parameters. An origin of the local variations of the lattice parameters can be of different type. For the material under study we can consider: e.g. local variation of the chemical composition or dislocations. Faults in the layers stacking would induce even more complex peak broadening effects [19]. For dislocations the model in [23] could even assign to E1-E9 parameters physically relevant values and the dislocation density could be determined. Since dislocation Burgers vectors types and elastic constants of the material are necessary, it is out of the scope of this work. For modelling of the line profiles from the microstrain effect the simple Pseudo–Voigt function was used. It was convoluted with other broadening effects (instrumental, size) [20]. For the ex-situ data the additional refined parameters were lattice parameters, isotropic Debye–Waller temperature factors for individual atom types (Ce, Ti, O), atomic positions and the parameters of the crystallite size distribution. The crystallites were assumed to be spherical with log-normal size distribution described by median M and s-parameter, where es is the geometric standard deviation. The diffraction is a volume sensitive method. Hence the sizes reported in this paper are the volume weighted diameters which are considered to be the natural presentation of the information embedded in the diffraction data [24,25]. The refined parameters can be found in Tables 1 and 3. For the fitting of the in-situ data, atomic positions (see the Table 1) and the geometric standard deviation of the particle size distribution (s ¼0.65) were fixed to the values obtained from the ex-situ measurement. In order to assess the growth kinetic, the diffraction scans were rather rapid so that the quality of the data collected did not allow fitting these parameters. Table 1 Atomic positions of CeTi2O6 phase obtained from the post annealing fit. All zeros arise from the symmetry of the individual Wyckoff positions and therefore have not been refined. Atom
x
Y
z
Ce Ti O1 O2 O3
0 0.824(1) 0.968(2) 0.652(2) 0.273(2)
0 0 0 0 0
0 0.388(1) 0.297(2) 0.112(2) 0.396(2)
Reduced sample weight fraction (%)
CeTi2O6 weight fraction from diffration (%)
597 5 61 75 567 5
63.0 7 0.5 56.2 7 0.5 51.7 7 0.5
Table 3 The anisotropic strain parameters obtained from the post-annealing (ex-situ) data.
3
2
Si weight (g) 0.0220 0.0253 0.0294
e2hkl ¼ ðE1 h þ E2 l þ E3 k þ 2E4 h l þ 2E5 l k þ 2E6 h k þ 4E7 h l þ 4E8 l h þ 4E9 k lhÞ=ðH hkl aÞ4 ;
267
Value (1e-4) 0.050(7) 1.20(7) 9.8(4) 0.29(4) 4.2(3) 0.12(8) 0.022(7) 0.01(3) 0.1(1)
The other refined parameters were the lattice parameters, isotropic Debye–Waller factors, median of the particle size distribution and the anisotropic strain parameters.
3. Kinetics We investigated the kinetics of the formation of CeTi2O6 phase. The time-dependent amount of the crystallized material from amorphous state at constant temperature can be described by the Johnson–Mehl–Avrami–Kolmogorov (JMAK) equation for homogeneous crystallization [26]: n
YðtÞ ¼ 1 e Kðt t0 Þ ;
ð2Þ
where Y is the volume fraction of crystallized matter, t is the time (variable), t0 is the time constant of the beginning of the crystallization, n is the Avrami exponent describing the nature of the crystallization and K is a temperature-dependent factor, following the Arhenius equation: K ¼ K 0 e Ea =kB T ;
ð3Þ
where K0 is a constant, Ea is the effective activation energy of the whole crystallization process involving both the creation of the nuclei and the growth of the particle, kB is the Boltzmann constant and T is the absolute temperature. Possible values of the exponent n will be discussed later.
4. Results and discussion From the diffraction data the formation of CeTi2O6 phase as a function of time can be clearly observed. The sample annealed at 650 1C starts to crystallize about 45 min after reaching the annealing temperature (Fig. 3(a)). On the other hand, the sample annealed at the highest temperature, 700 1C, exhibit the CeTi2O6-related peaks immediately after reaching the annealing temperature (Fig. 3(b)). The analysis of the CeTi2O6 phase structure was performed at the sample annealed at 650 1C for 6 h (until the crystallization was fully saturated) from the data measured ex-situ after cooling down to the
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room temperature (Fig. 4). From the Rietveld refinement we determined the volume–weighted mean value of diameter of the particles DV ¼(125710) nm and the lattice parameters: a¼9.8146(5) Å, b¼ 3.7022(2) Å, c¼6.7720(4) Å, β ¼ 118.69(1)1, which are slightly smaller than the values from literature [8,16]. The atomic positions are summarized in the Table 1 and they are in a good agreement with the published data—(ICDD, 2012, PDF 01-014-5813). In the high statistics ex-situ scan a 101 diffraction peak from anatase has been observed. Due to the non-stoichiometry of the initial mixture (72.2%Ti, 27.8%Ce) an excess of Ti atoms can be expected. The weight fraction of Ti atoms that do not contribute to the CeTi2O6 phase is nominally 10.8% of both Ce and Ti atoms together. From the fit of the measured data (Fig. 4) the relative weight content of anatase phase is (7.270.5)% which corresponds to the weight fraction of Ti atoms (6.1 70.5)% in the ensemble of Ti and Ce atoms in anatase and CeTi2O6. This value suggests that at
Fig. 3. In the panel (a), the time evolution of the sample Ti70–Ce30 annealed at 650 1C expressed by x-ray diffraction curves taken after 0.75, 1.75, 2.75, 3.75 h from the reaching the annealing temperature is shown. The formation of new phase from amorphous state is clearly visible. The curves are vertically shifted for clarity. The grey lines in the bottom depict the positions of the CeTi2O6 diffraction maxima. Panel (b) shows the diffraction pattern for the samples annealed at all four temperatures (650, 670, 685, and 700 1C) 30 min after reaching the annealing temperature.
least a half of excess Ti atoms that cannot be incorporated in CeTi2O6 phase forms crystalline anatase-TiO2. The form of the amorphous state, which can be observed before crystallization of CeTi2O6 when annealing at 650 1C, was qualitatively studied by modelling the measured broad peaks with very small (diameter 1 nm) particles. From these rough simulations (Fig. 5) it can be seen, that the positions of the maxima may correspond to cerianite particles. It indicates that in this “amorphous state” Ce tends to form very small particles (with just several atoms) while titanium oxide remains in genuine amorphous state. This is similar to what was observed for mixtures with different Ti–Ce molar ratio also by Raman spectroscopy [18]. For the study of the kinetics of the crystallization, we fitted the in-situ xrd data (as a function of time, at four different temperatures, 650 1C, 670 1C, 685 1C, and 700 1C). In this temperature range the crystallization proceeds in a reasonable speed and the data from individual temperatures differ enough. From the fits we obtained scale factors which correspond to the amount of crystallized CeTi2O6 phase. The time dependence of this amount has been fitted by the JMAK Eq. (1). The asymptotic value lim Y ðt Þ of the t-1 fitted JMAK curve was set to 100%. Therefore the 100% value corresponds to the saturated amount of the CeTi2O6 phase and does not need to be directly related to the total sample weight. The degree of crystallinity is discussed later. It can be seen (Fig. 6) that for the sample annealed nominally at 685 1C the crystallization proceeds slower than for the sample annealed nominally at 670 1C. This discrepancy is induced by the
Fig. 5. Measured diffraction data (grey dots) of the sample just after reaching 650 1C. The broad maxima are fitted with small particles (diameter 1 nm) of cerianite (full line), CeTi2O6 (dash-dotted line) and rutile (dash line). The fit for anatase particles is not displayed for clarity; its peaks are shifted to even lower angles compared to rutile.
Fig. 4. Measured (grey dots) and fitted (black line) diffraction curves of the sample annealed at 650 1C for 6 h (until the crystallization was fully saturated) after cooling down to the room temperature. The grey line is the difference curve. The inset displays the structure of CeTi2O6 in polyhedral representation.
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269
Fig. 6. Measured data (points) and their fit (lines) of the CeTi2O6-crystallized fraction as a function of time for various temperatures. The depicted error bar shows the estimated error for all points. The inset displays the plot of ln(K) vs. reciprocal of the annealing temperature. From the linear fit of these data the value of effective activation energy of CeTi2O6 particle formation has been determined. The vertical error is caused by the fit of the JMAK equation, the horizontal one is due to the uncertainty of temperature (7 10 1C).
uncertainty of the temperature determination of 710 1C. A potential uncertainty of the determination of the annealing time does not play role in the analysis since (1) the annealing processes were as similar to each other as possible (inducing maximal error of 1 min) and (2) the time difference would affect the t0 parameter and does not influence essentially the parameter K used in estimation of the activation energy. The calcination of the samples during their preparation should not cause any effect on particle crystallization since the calcination temperature was much lower than crystallization temperatures and all the samples were calcined in the exactly same way. From the JMAK fit (Fig. 6) the temperature-dependent parameters K were obtained, and by their fitting to Eq. (2) the effective activation energy of the whole crystallization process Ea ¼ 1.5(4) eV was calculated (Fig. 6, inset). This result is of the same order as the total activation energies of pure Ti (from 1.3 to 2.5 eV, depending on the specific phase and the preparation procedure; the activation energy of the growth of anatase particles is in the range 0.35– 0.9 eV and for Ce oxides (from 0.3 to 1.7 eV) [27–30]. The total activation energy of CeTi2O6 obtained from the heavy ion irradiation amorphization was reported to be 0.95 eV [31], which is slightly smaller value than we obtained. The Avrami exponent n from Eq. (1) describes the nature of the crystallization. It depends on the dimensionality of the growth and on the rate of formation of new nuclei [26,32]. We assumed n to have a constant value for all four temperatures and fitted the crystallization curves for all temperatures together. From the fit we found the value of n ¼1.7(2), which does not agree with standard theoretical values of Avrami exponent. The Avrami exponent n in the JMAK theory is a sum of two terms n ¼ a þb c: The parameter a is the exponent in the time dependence of the number of nuclei N per unit volume of untransformed material: N t a : The parameter b describes the number of dimensions of the growth (b¼ 3 for three-dimensional (3D) growth), and c¼1, 0.5 for the interfacecontrolled and diffusion-controlled growth, respectively. For the interface-controlled 3D growth (which is expected for free powder particles) the Avrami coefficient should reach the value of 3–4.
This disagreement is probably caused by the fact that the JMAK theory supposes a homogeneous nucleation but in our case of a free powder, consisting of small particles, heterogeneous nucleation on the surface of the particles is more likely to happen. In the case of heterogeneous nucleation the decrease of Avrami exponent n has been predicted and observed [33,34]. The volume–weighted mean value of the diameter of CeTi2O6 particles is not changing systematically with time and for all annealing temperatures for all samples it is in the range (90– 130) nm which is in an agreement with the ex-situ data after cooling. This can be seen already from the Fig. 3(a) demonstrating that already narrow peaks start to appear during the crystallization. From this fact it follows, that we do not see the particle growth but only the creation of new nuclei of CeTi2O6 phase which then rapidly grow up to the observed size. Thus, the activation energy of the particle growth is much smaller than the activation energy of the nucleation. In addition the degree of the sample crystallinity after annealing has been studied by mixing a certain amount of fully crystalline Si standard into the samples. The measured data were not of the quality to distinguish the anatase peak as in the high-quality scan, therefore the obtained results have to be compared to the values reduced by the non-stoichiometry excess Ti atoms that crystallize to anatase or remain amorphous. From the results (Table 2) it can be seen that the entire stoichiometric part of the sample (within the errors) has crystallized to CeTi2O6. The errors of the reduced weight fractions arise from the error of the x-ray fluorescence measurements and from the uncertainties of the sample weighting and corresponding manipulation. Finally to illustrate the main trends in the anisotropic diffraction line broadening, the integral breadths of several hkl-reflections were used to construct the Williamson-Hall plot (Fig. 2). There are two orders of reflections from the {h00} and {00l} planes. The common intersect of the linear fits through them gives the hkl-independent (constant) broadening level which could be related to the size-effect. The rest of the broadening for any hkl-reflection can be well explained by the anisotropic
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microstrain effect described earlier. This is illustrated in Fig. 2 by a model line which reproduces the characteristic hkl-anisotropy in the data. To have an idea of the effect strength the microstrain was evaluated in the directions perpendicular to the {h00} planes to be eh00 E0.3%. It is evident from Fig. 2 that in some directions it is even higher. On the contrary it is negligible (e00l E0%) in the directions perpendicular to the {00l} planes. The CeTi2O6 is not such intensively studied material as its two counterparts (CeO2 [28], TiO2 [24]) and hence it is difficult to conclude if the physical origin of the microstrain effect is in the presence of dislocations, or it is related to the local variation of the lattice parameters e.g. due to clustering of vacancies, which were reported in [14]. This would require a thorough study. 5. Conclusion We measured the diffraction patterns of the Ti–Ce mixed oxide sample with nominal molar content of Ti atoms of 70% at four temperatures (650 1C, 670 1C, 685 1C, 700 1C) depending on time. The crystallization of cerium titanate CeTi2O6 has been observed. We fitted the measured data with a model including log-normal distribution of particle sizes and an anisotropic microstrain in the particles. From the fit of the measured data the lattice parameters, the parameters of the particle distribution, anisotropic strain parameters and the atomic positions were determined. The main scope of this work, however, was to investigate the kinetics of the cerium titanate crystallization. The effective activation energy was found to be Ea ¼1.5(4) eV and it is mostly composed of the activation energy of the particle nucleation. The growth of the particles has not been seen; a direct formation of particles with sizes in the range (90–130) nm has been observed. The study of the crystalinity of the sample after annealing revealed that the whole part of the sample that corresponds to the stoichiometry of CeTi2O6 crystallizes into this phase indeed. The excess Ti atoms form crystalline anatase-TiO2 or remain in an amorphous state. Acknowledgements This project has been supported by the Grant Agency of the Czech Republic (project numbers P204-11-0785, P108-11-1539 and 104/09/P290).
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