Study of the phase composition of Fe₂O₃ nanoparticles

Václav Valeš¹, Jana Poltierová-Vejpravová¹, Petr Brázda², Alice Mantlíková¹, and Václav Holý¹

¹Department of Condensed Matter Physics, Faculty of Mathematics and Physics, Charles University in Prague, Czech Republic

²Department of Inorganic Chemistry, Faculty of Science, Charles University in Prague, Czech Republic

Introduction

Important physical properties of nanoparticles are determined mainly by their atomic structure, especially by their phase composition and the presence of structure defects. X-ray diffraction is a good tool for studying the structure of the nanoparticles, its application for very small particles is however limited by very small intensity of the scattered wave. For this reason special experimental setups, like e.g. diffraction with small incidence angle, are used and many experiments have to be done at synchrotrons. Standard methods of the measured data analysis based on the description of the diffraction using instrumental functions and functions of physical broadening of the lines fail in the case of very small particles. An ab-initio calculation method (based on the Debye formula [1, 2]) has to be used instead.

In this work the Debye formula is used for the description of the diffraction of iron oxide samples measured at ANKA synchrotron in Karlsruhe. Using this approach we determine basic parameters of the particles such as lattice parameters and the size of the particles, as well as the presence of different phases. During annealing, subsequent phase transitions from γ-Fe₂O₃ to ε-Fe₂O₃ and to α-Fe₂O₃ take place. New phases nucleate probably at the surface of the nanoparticles and the phase transformation proceeds towards the particle center ([3, 4]), so that the structure of the nanoparticles can be described by a core-shell model; this model was used in the Debye-formula based simulation. From the analysis of the experimental data we determined the kinetic parameters of the phase transitions and their dependence on the nanoparticle sizes.

Measured samples

The great interest in Fe₂O₃ nanoparticles is caused mainly by magnetic properties of these particles, namely extremely high room temperature coercivity of epsilon phase of these iron oxide nanoparticles. The samples were prepared by ex-situ annealing of organic precursors and then measured at ANKA synchrotron in Karlsruhe with incidence angle 5º and the wavelength of 0.95007 Å. The primary beam was monochromatized by a 2x111Si monochromator, the diffracted radiation was measured by a point detector equipped with a narrow entrance slit and a filter suppressing the Fe-fluorescence. The series of samples was prepared with the final annealing temperature from 900 ºC to 1150 ºC with the step of 50 ºC. To the temperature of 900 ºC all the samples were heated at the speed 1 ºC per minute and stayed at this temperature for 4 hours. As for the sample with the final temperature 900 ºC this was the whole procedure. The other samples were then with the same speed heated to their final annealing temperature with the 4-hour waiting each 50 ºC up to their final heating temperature. This procedure causes the creation of Fe₂O₃ nanoparticles in the amorphous SiO₂ matrix. From the literature [5] it is known that the particles created at the lowest final temperature should be in the form of maghemite and with increasing final temperature the phase of Fe₂O₃ particles should change to ε and hematite.

Theoretical description

Debye formula in Eq. (1), which has been used for the x-ray data analysis describes the intensity distribution of the samples consisting of the same randomly oriented particles, knowing the positions of the atoms in one such a particle.

(1)

where the double sum goes over all atoms in the particle, Q is the length of the scattering vector, f_i is the atom form factor of the i-th atom and r_ij is the distance between i-th and j-th atom. The formula is valid for any arrangement of atoms in any particle; no lattice is needed; only exact positions of atoms in the particle are important. The only technical limit of using of this equation is the number of terms in the double sum. For instance, a particle of Fe₂O₃ of diameter of 13 nm contains about 10⁵ atoms, which means that there are 10¹⁰ interatomic distances that have to be taken into account for every Q. For this reason, the distribution function of atomic pairs was calculated and a histogram of all interatomic distances was created; an example of such histogram is shown in Fig. 1 corresponding to a spherical particle with the radius of 40 Å, the histogram has been constructed using the step width of 0.01 Å.

Figure 1. Calculated histogram of interatomic distances in a spherical Fe2O3 particle of radius of 40 Å. The histogram step is 0.01 Å.	Figure 2. Calculation of diffraction curves for different phases of Fe2O3. The full line corresponds to the pure maghemite particle of radius of 50 Å; the dotted one to the pure ε-Fe2O3 particle of the same radius; and the dashed line represents the diffraction from the particles of radius 50 Å, which consist of the core (radius 40 Å) of maghemite and the shell of ε-Fe2O3.

Since we introduced the histogram of interatomic distances, we can rewrite the Eq. (1) using calculated data from the histogram, i.e. we know the multiplicity of each of interval of distances. The rewritten form of Debye formula in equation (2) enables us to perform calculations for much larger samples. For the intensity we can write

(2)

where m_i is the multiplicity factor for the i-th interval of distances. The expression in Eq. (2) is valid only for one type of atoms in the particle, which is not our case (because of different atom form factors). This fact requires only some technical changes, which do not affect the fundamental meaning of Eq. (2).

The phase transition from one phase to the other is supposed to take place from the particle surface to its center. For this reason the core-shell model of the particle (particle consisting of two different phases) has been introduced to the Debye formula program. In order to have a brief look in to the behavior of the simulated data calculated by our model, diffraction curves for different phases were calculated (Fig. 2). From this picture the difference between the maghemite and ε phases of Fe₂O₃ can be seen as well as the effect of the core-shell structure of these two phases, which causes some “mixture” of the diffraction pattern of both phases.

Data analysis

Several samples from the series described above were analysed by the Debye-formula approach using the core-shell model, assuming that the interface of the two phases moves from the surface to the center of the particle. The data from the Figs. 3 – 5 (samples A – C) were fitted by hand and the results are summarized in Table 1; the errors were estimated from this fit too. The background was approximated ad-hoc by a polynomial of the third power. The broad peak around 13º is caused by the amorphous SiO₂ matrix and for our fitting is not important. The fits describe the measured data well and the parameters of the core-shell model were obtained. The sample D (Fig. 6) could not have been fitted because of a too large size of the particles that made the simulation extremely time-consuming.

Figure 3. Sample A. Measured data and fit of the sample annealed at the 900 ºC as the highest temperature.	Figure 4. Sample B. Measured data and fit of the sample annealed at the 950 ºC as the highest temperature.

Figure 5. Sample C. Measured data and fit of the sample annealed at the 1000 ºC as the highest temperature.	Figure 6. Sample D. Measured data of the sample annealed at the 1100 ºC as the highest temperature.

Table 1. Results obtained from the measured data fitting

Sample	Total radius (Å)	Maghemite (%)	ε-Fe2O3 (%)
A	40 ± 4	34 ± 5	66 ± 5
B	50 ± 5	26 ± 3	74 ± 3
C	58 ± 5	0 ± 8	100 ± 8

From the fitting of the measured data we obtained the total size of analyzed samples (A – C) and the fraction of the maghemite and ε phase assuming the core-shell model with maghemite as a core. It can be seen that the size of the particles increase with increasing annealing temperature and that the fraction of maghemite decreases and it completely vanishes at the temperature of 1000 ºC. This corresponds to the assumption presented above. As for the sample D, which has not been analysed, the hematite diffraction peaks appear.

Both he core-shell model of the nanoparticles and the Debye formula are suitable tools for the analysis of our samples. In the future we have to investigate, whether it is possible to distinguish between the core and the shell, i.e., whether we can determine which phase is in the core and which one is in the shell. A method, which would enable us to analyze larger particles, has to be implemented as well.

References

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2. A. Cervellino, C. Giannini and A. Guagliardi, J. Appl. Cryst. 36 (2003), 1148.

3. Chang-Woo Lee, Sung-Soo Jung and Jai-Sung Lee, Materials Letters 62 (2008), 561.

4. M. Gich, C. Frontera, A. Roig et al, Chemistry of Materials 18 (2006), 3889.

5. P. Brázda, D. Nižňanský, J.-L. Rehspringer, J. Poltierová Vejpravová, J. Sol-Gel Sci. Technol., 51 (2009), 78-83.