O2 Diffusion in Amorphous SiO2 Nanoparticles Probed by Outgassing

Article pubs.acs.org/JPCC

O2 Diffusion in Amorphous SiO2 Nanoparticles Probed by Outgassing G. Iovino, S. Agnello,* F. M. Gelardi, and R. Boscaino Dipartimento di Fisica, Università di Palermo, Via Archirafi 36, I-90123 Palermo, Italy ABSTRACT: An experimental study of the O2 diffusion process in nanoparticles of amorphous SiO2 in the temperature range from 98 to 157 °C was carried out by Raman and photoluminescence techniques. We studied O2 diffusion in high purity silica nanoparticles with a mean diameter of 14, 20, and 40 nm detecting the outgassing of molecules trapped during the manufacturing. The kinetics of diffusion is well described for all the investigated nanoparticles by the Fick’s equation proving its applicability to nanoscale systems. The diffusion coefficient features an Arrhenius law temperature dependence in the explored temperature range, and the diffusion coefficient values are in good agreement with extrapolation of Arrhenius law from higher temperature studies.

■

that higher material density inhibits molecule diffusion.17,24 Furthermore, it has been shown that diffusion of the O2 can occur without interaction with the network oxygen or by contrast with a molecule−network exchange of atoms.25,26 This latter process is negligible below 900 °C where the exchangefree diffusion length reaches values of ∼1 μm. As a final concern, it has been evidenced that network modifiers could have minor effects on the O2 migration, and the more relevant species could be SiOH through a dehydroxylation process, causing an interstitial water driven exchange of oxygen with the SiO2 network at high temperatures.26,27 The precedent studies have been typically carried out at temperatures above 500 °C due to a limitation in the detectability of the O2 in bulk systems caused by the low penetration. Studies at lower temperature and in nanodimensional silica are limited and deserve to be done in view of the above-mentioned applications at the nanoscale. In addition, the typical dimension of nanomaterials, reduced below the exchange-free diffusion length, introduces a relevant interest at the fundamental level to establish a deeper comprehension of the features of the diffusion process. In this work, the O2 diffusion in high purity silica nanoparticles of mean diameter between 14 and 40 nm has been studied by thermal treatments at temperatures below 160 °C. The used silica materials have interstitial O2 by production, and the diffusion process has been investigated through the outgassing at temperatures not investigatable in laboratory time for bulk systems. The used materials are also characterized by different structural features, enabling one to elucidate the role these latter have on the diffusion.

INTRODUCTION Amorphous SiO2, or silica, is a prototype material in the investigation of solid state physics and glass systems.1,2 This material, apart from basic physics, is relevant in many optics and electronic devices of today technology. In particular, it finds frontier applications in nanoscale optical systems,3−5 and also in the electronics field in which the increasing requirement of high miniaturization pushes toward nanosize layers of dielectrics.6 In these contexts, the dimensions reduced to the nanometer scale impose a comparison of the physical properties of the nanomaterial with analogous properties of the bulk system, having a typical size of the order of μm or larger. Particular interest is linked to the diffusion of small particles present in the environment, as they could affect the functionality and realization of devices. For example, the thermal oxidation of silicon in metal oxide semiconductor devices usually involves the migration of oxygen through a layer of SiO2 on the top of a Si one, this process being the prominent one in limiting the oxidation of Si.7−10 Furthermore, as the role of oxygen diffusion has been shown to be relevant in affecting the properties of nanoscale devices as the core shell systems, large interest is devoted to this topic in connection with nanoscale systems.9−14 In addition, evidence has been put forward that some differences with respect to bulk materials occur due also to structural constraints arising in connection to the confined size.15 From a general point of view, the diffusion of small molecules has been widely investigated for bulk silica materials.16−21 At variance, a few studies have been focused on the direct experimental investigation of the diffusion process of small molecules in nanoscale systems.15,22,23 The study of oxygen diffusion in bulk amorphous SiO2 systems has evidenced that different features of the material could affect the kinetics of the molecule. A prominent role of amorphous structure and density of the material has been put forward both by experimental and simulative works, suggesting © 2012 American Chemical Society

Received: January 19, 2012 Revised: April 24, 2012 Published: April 30, 2012 11351

dx.doi.org/10.1021/jp3006734 | J. Phys. Chem. C 2012, 116, 11351−11356

The Journal of Physical Chemistry C

■

Article

EXPERIMENTAL PROCEDURE Sample and Measurements. Commercially available hydrophilic silica nanoparticle powders produced by the pyrogenic technique, known as Aerosil fumed silica, have been used as starting materials.28,29 In particular, the study was carried out on Aerosil 150, Aerosil 90, and Aerosil OX50, that are characterized by a specific surface of 150, 90, and 50 m2/g, and average particle diameters of 14, 20, and 40 nm, respectively. For shortness, in the following, the nicknames AE150, AE90, and AEOX50 will be used. These materials are characterized by the presence of interstitial O2 introduced during their manufacturing. In order to handle the powders, they were pressed by a uniaxial mechanical press at ∼0.3 GPa to obtain tablets. The experiments reported in this work were carried out on pieces of typical size 4 × 4 × 2 mm3, derived from these tablets. In particular, different pieces for each sample were thermally treated in air in a furnace at fixed temperatures in the range between 98 and 157 °C to make interstitial oxygen molecules outflow. The temperature was stabilized within ±1 °C by the temperature control of the furnace. The heating and cooling times from room temperature to the thermal treatment temperature and vice versa were about 2 min. For each sample type and temperature, different pieces were subjected to thermal treatments of incremental duration until the equilibrium state was achieved and no further change of interstitial O2 was detected. The interstitial O2 concentration after each thermal treatment was determined by measurements carried out with a RAMII Bruker FT-Raman spectrometer equipped with a Nd:YAG laser source at 1064 nm that enabled both the intrinsic vibrational modes of silica and the photoluminescence (PL) emission from singlet oxygen to be detected.30,31 It was previously demonstrated that in these measurements the amplitude of the PL emission is proportional to the O2 concentration, and this latter can be determined by comparison of the PL amplitude with the amplitude of intrinsic Raman bands of silica.30,32 In our setup, the source power was fixed at 500 mW and the detection system spectral resolution at 15 cm−1 to increase the sensitivity. Due to the simultaneous occurrence of Raman scattering and photoluminescence, these measurements are reported in the following as Raman/PL. Data Analysis. Treatment of experimental data of the O2 content time dependence has been based on diffusion theory assuming a spherical geometry. In particular, Fick’s diffusion equation has been used to analyze the outgas process, fitting its solution to the experimental data.19 The analytical solution has been determined by supposing that the powder is composed by identical spherical nanoparticles with a diameter equal to the average value reported by the producer.28,29 Furthermore, it was assumed that O2 molecules were initially uniformly distributed in each sphere and the approximation was used that the oxygen concentration at the surface of each nanoparticles instantaneously reached its equilibrium value.27,33 This latter is determined by the partial pressure of O2 in the air atmosphere and by the explored temperature. The obtained solution averaged over the nanoparticle spherical volume can be put in the following form:34 C(t ) − C i 6 =1− 2 Cf − C i π

∞

∑ n=1

1 −π 2n2Dt / r 2 e n2

coefficient, and r is the radius of the nanoparticles. It is worth noting that the Cf value should be determined by the equilibrium condition with the content of O2 in the air atmosphere at each temperature. By this fitting procedure, the diffusion coefficient has been found for each temperature and investigated sample type, and is reported in the following. Finally, the use of the relative change of concentrations, defined in eq 1, to describe the kinetics of O2 avoids that results could be affected by incorrect estimation of the absolute concentration of the interstitial molecules, as well as that incorrect comparison between different samples could occur due to changes in the emission quantum yield, as recently suggested.8

■

RESULTS Preliminary to each treatment, the samples were characterized by Raman/PL measurements to evidence the presence of structural differences between the different materials and to establish the initial content of O2. Figure 1 shows the acquired

Figure 1. Raman/PL spectra of the as received samples AEOX50, AE90, and AE150. The spectra are scaled to have equal amplitude of the silica Raman band at ∼440 cm−1.

spectra for the as received AEOX50, AE90, and AE150. In all the spectra, the typical bands associated with vibrational modes of silica can be recognized below 1200 cm−1.35 It can be observed that the amplitude of the bands at about 490 and 600 cm−1 associated with four- and three-membered rings, respectively,36 changes among the samples, suggesting that some structural differences occur for the nanoparticles of different diameter, with a larger effect for the AE150 sample.35,37 Furthermore, a band, associated with OH groups,38 of increasing amplitude on decreasing the nanoparticle mean diameter is observed at about 980 cm−1. All the spectra present also a band peaked at 1538 cm−1, that corresponds to an absolute wavelength of 1272 nm (7862 cm−1) and is attributed to the photoluminescence of interstitial O2.21,30−32 The amplitude of the observed signal relative to the main Raman scattering band at about 450 cm−1 depends on the nanoparticles size, increasing with increasing average diameter. This interstitial O2 is trapped in the nanoparticles during their manufacturing procedure and, as a consequence, is assumed to be uniformly distributed inside them. Figure 2 reports the effects of thermal treatments at 127 °C in air for the AEOX50 sample. It is observed that the amplitude

(1)

where Ci, Cf, and C(t) are the initial, final, and at time t O2 volume averaged concentrations, respectively, D is the diffusion 11352

dx.doi.org/10.1021/jp3006734 | J. Phys. Chem. C 2012, 116, 11351−11356

The Journal of Physical Chemistry C

Article

Figure 2. Raman/PL spectra of the O2 emission in the AEOX50 sample thermally treated in air atmosphere at 127 °C for increasing time.

Figure 3. Evolution of the relative O2 content (C(t) − Ci)/(Cf − Ci) in the sample AEOX50 at temperatures between 98 and 157 °C. Ci and Cf are the experimental O2 concentrations before the treatments and at the equilibrium state. The x-axis r value coincides with the AEOX50 average radius of 20 nm, the time t is the duration of each thermal treatment, and the first point for the lowest t/r2 value refers to the not treated sample.

of the O2 emission band at 1538 cm−1 decreases on increasing the duration of the treatment without detectable changes in shape. As already reported in previous experiments,31 we observe that no relevant effects are induced on the other bands detected by the Raman/PL, proving that the observed changes of the O2 PL band amplitude are related to changes in the population of these molecules only. A decrease of the O2 emission is found also for the treatments at the other temperatures here considered. It is worth noting that a possible reduction of OH content due to thermally induced dehydration processes39 should result in an increase of the O2 PL quantum yield, as recently suggested,8 and, as a consequence, an increase of the PL amplitude. This effect is contrary to our experimental findings, and if present, it can be neglected. Since the amplitude of the Raman/PL emission is proportional to the volume average of the concentration, C(t), of O2 in the nanoparticles, we used the PL amplitude to evaluate the relative concentration change with time: (C(t) − Ci)/(Cf − Ci). As reported in Figure 3 for the sample AEOX50 treated at temperatures between 98 and 157 °C, on increasing the temperature, the time necessary to reach the equilibrium content of O2 decreases. The O2 concentration as a function of the thermal treatment duration in the AE90 and AE150 samples has been investigated at the representative temperatures of 98, 127, and 157 °C. Also for these samples, the concentration decreases on increasing treatment duration time. Comparison among the O2 concentration kinetics in all three employed samples is easily reported plotting (C(t) − Ci)/(Cf − Ci) as a function of the rescaled time t/r2, as shown in Figure 4. This representation, as shown by eq 1, singles out the dependence on the diffusion coefficient and removes any dependence on geometrical factors of the different nanoparticles. For each sample, we used its starting and its final O2 concentration values as estimated from Raman/ PL measurements, and the average radius of its nanoparticles. From these data, it can be observed that for each temperature the three different samples feature very similar rescaled kinetics. Furthermore, it can be observed that, for a fixed sample, that means for fixed r value, the time needed to reach a stationary value decreases on increasing temperature.

■

DISCUSSION The reported experiments are carried out at temperatures much lower than those usually employed to investigate the diffusion process of O2 and have enabled us to characterize the diffusion kinetics in an unexplored temperature range below 500 °C. It is worth noting that the possibility to explore a low temperature range as compared to usual bulk samples is mainly due to the small dimension of the nanoparticles used and the high surface/ volume ratio. This aspect allows detecting diffusion depth not traceable in bulk systems. The data for the sample AEOX50, reported in Figure 2, evidence that a monotonous decrease of the O2 associated PL band is induced by thermal treatments at 127 °C. The investigated process can be specifically attributed to the diffusion of O2 molecules out of the nanoparticles. In fact, on increasing the temperature, the O2 solubility in silica is expected to decrease, thus causing a reduction of the concentration of these molecules in nanoparticles.8 A decrease of the embedded molecules content is observed for the other temperatures in the AEOX50 and also for the other samples AE90 and AE150. On this basis, it can be asserted that in the temperature range below 157 °C the diffusion process is activated in all the investigated nanoparticles, their specific structure or impurity content not inhibiting it. In this respect, it is worth noting that the Raman measurements have shown that some structural differences occur among the nanoparticles of the different samples, as evidenced by the variation of amplitudes of the bands at about 490 and 600 cm−1. Since the thermal treatments induce no detectable modification of the silica vibrational modes, it is guessed that the network structures of the nanoparticles are unmodified in the explored temperature range.31 On the basis of this observation, it can be concluded that the reported results on the diffusion process are characteristic of the unique network structure characterizing both the as grown and treated nanoparticles. The data for the AEOX50 sample at the different temperatures reported in Figure 3 evidence that the kinetics 11353

dx.doi.org/10.1021/jp3006734 | J. Phys. Chem. C 2012, 116, 11351−11356

The Journal of Physical Chemistry C

Article

measure of time. The experimental data can be fitted using eq 1, valid under the approximation of diffusion in a spherical geometry. By this fitting procedure, the value of D can be determined at the different temperatures investigated. The best fit curves are reported in Figure 3 and show a good agreement with the data. The obtained D values are collected in Table 1, and show a monotonic increase of D with temperature. As shown in Figure 5, an Arrhenius plot of the determined D values evidences the dependence on temperature as D0e−ΔEa/kT,

Figure 5. Arrhenius plot of the diffusion coefficient in the samples AEOX50 (triangles), AE90 (squares), and AE150 (circles). The linear best fit of the AEOX50 data is reported as a full line.

where D0 is the pre-exponential factor, ΔEa the activation energy, k the Boltzmann constant, and T the absolute temperature. The best fit parameters are D0 = (1.4 ± 1.3) × 1010 nm2/min and ΔEa = (0.82 ± 0.06) eV. The orders of magnitude are compatible with literature data for bulk systems, and the absolute values are a little bit lower than those previously reported.8 In detail, the diffusion coefficient extrapolated from literature data in the low temperature limit of our experiments evidences that D is in the range 0.008−0.06 nm2/min for 98 °C and 0.6−3 nm2/min for 157 °C. These intervals partially overlap with the here obtained values reported in Table 1. This agreement permits one to validate the model of diffusion based on Fick’s equation also for nanometer size silica particles. The small differences in the absolute values of D0 and ΔEa with respect to bulk silica could be attributed to the differences in the amorphous structure of nanoparticles that is evidenced by the Raman bands below 1200 cm−1; see Figure 1.40 The comparison of diffusion kinetics among the nanoparticles with different average radius and somewhat different

Figure 4. Evolution of the relative O2 content (C(t) − Ci)/(Cf − Ci) in the samples AEOX50, AE90, and AE150 at different temperatures between 98 and 157 °C. Ci and Cf are the experimental O2 concentrations before the treatments and at the equilibrium state in each sample. The x-axis r values are the average radii of the different nanoparticles: 20 nm for AEOX50, 10 nm for AE90, and 7 nm for AE150; the time t is the duration of each thermal treatment, and the first points for the lowest t/r2 value refer to the not treated samples. Lines represent best fit curves obtained by fitting the experimental data using eq 1.

of outgassing is completed at shorter time on increasing temperature. In fact, having a fixed average value of the radius, r, of the nanoparticles, the x-axis of Figure 3 is essentially a

Table 1. Diffusion Coefficients in nm2/min Obtained by Fitting the Experimental Data Obtained for Thermal Treatments at Temperatures from 98 up to 157 °C with eq 1a

a

sample

98 °C

AE150 AE90 AEOX50

0.07 ± 0.05 0.08 ± 0.04 0.04 ± 0.02

113 °C

127 °C

0.11 ± 0.04

0.7 ± 0.5 0.4 ± 0.2 0.18 ± 0.06

143 °C

157 °C

0.6 ± 0.2

3±2 2±1 1.3 ± 0.4

The best fit curves are shown in Figures 3 and 4. 11354

dx.doi.org/10.1021/jp3006734 | J. Phys. Chem. C 2012, 116, 11351−11356

The Journal of Physical Chemistry C

Article

effects of reduced size and modified amorphous structure are guessed, giving minor changes with respect to bulk diffusion dynamics.

structure is reported in Figure 4. The overall dependence (C(t) − Ci)/(Cf − Ci) on t/r2 is comparable for all the samples, suggesting the applicability of Fick’s equation also for nanoparticles with diameter smaller than 40 nm. The fit of data in Figure 4 with eq 1 enabled us to determine the diffusion coefficient also for the samples AE90 and AE150. The obtained results are reported in the inset of Figure 5 and are summarized in Table 1. It can be observed that a systematic dependence of D on the average size of the particles is present, suggesting that the somewhat different structure affects the diffusion process. In particular, the smallest nanoparticles of the sample AE150 have the largest diffusion coefficient. On the basis of Raman spectra, this sample has also the most modified structure, with a larger population of small membered rings denoting the presence of stressed regions, as recently evidenced.40 This result could be compared with indirect observation of O2 diffusion in bulk samples mechanically densified, where it was found that more densified and stressed structures inhibit molecular oxygen diffusion.17 These findings disagree with ours, showing that the nanodimension and the modified structure favor somehow the diffusion process. It is worth noting that in ref 17 a structural modification much larger than that of our samples and extended to the overall volume of the system was involved, and we cannot rule out that other structural and size effects, due to the nanometric dimension of the particles, could be present in our case. In general, a dependence of the diffusion process on the size and structure of silica has been suggested by simulative works, evidencing that the reduced size overall could enhance the diffusion.24,41 Thus, the effect of nanodimensions in our samples could overcome the inhibition caused by the weakly stressed structure. The tendency to increase the diffusion coefficient in connection to the tendency of the particles to have a more stressed structure on decreasing their size could then be commented in the framework of the suggested model of not uniform particle structure.40,42 In detail, it has been predicted that the nanoparticles are formed by an interior region and an exterior shell, comprising their surface, more stressed than the interior one, that at variance is similar to the regular amorphous structure of silica. It can be expected that overall this binary system could feature a different diffusion coefficient with respect to a uniform matrix, either densified or not, especially on decreasing the nanoparticle size. The higher relevance of surface shell with respect to the interior region on decreasing the nanoparticle size indeed correlates with the tendency to increase the effect on the diffusion coefficient, as shown in the inset of Figure 5. These aspects need to be further investigated both experimentally and theoretically. Finally, it is worth emphasizing that the dynamics in the temperature range explored in the present work is essentially due to particle motion within interstices of amorphous matrix without exchange with matrix constituents. This is a relevant point in view of the possible oxidation mechanisms occurring through small size layers of silica in ambient conditions.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +39 091 23891703. Fax: +39 091 6162461. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors would like to thank the people of the LAMP group (http://www.fisica.unipa.it/amorphous/) for useful discussions. Technical assistance by G. Napoli and G. Tricomi is acknowledged.

■

REFERENCES

(1) Defects in SiO2 and Related Dielectrics: Science and Technology; Pacchioni, G., Skuja, L., Griscom, D. L., Eds.; Kuwer Academic Publishers: Dordrecht, The Netherlands, 2000. (2) Silicon Based Materials and Devices; Nalwa, H. S., Ed.; Academic Press: San Diego, CA, 2001. (3) Burns, A.; Ow, H.; Weisner, U. Chem. Soc. Rev. 2006, 35, 1028− 1042. (4) Lu, Y.; Yin, Y.; Li, Z.-Y.; Xia, Y. Nano Lett. 2002, 2, 785−788. (5) Noginov, M. A.; Zhu, G.; Belgrave, A. M.; Bakker, R.; Shalaev, V. M.; Narimanov, E. E.; Stout, S.; Herz, E.; Suteewong, T.; Wiesner, U. Nature 2009, 460, 1110−1113. (6) Defects in Microelectronic Materials and Devices; Fleetwood, D. M., Pantelides, S. T., Schrimpf, R. D., Eds.; CRC Press: Boca Raton, FL, 2008. (7) Deal, B. E.; Grove, A. S. J. Appl. Phys. 1965, 36, 3770−3776. (8) Kajihara, K.; Kamioka, H.; Hirano, M.; Miura, T.; Skuja, L.; Hosono, H. J. Appl. Phys. 2005, 98, 013529-1−013529-7. (9) Watanabe, T.; Tatsumura, K.; Ohdomari, I. Phys. Rev. Lett. 2006, 96, 196102-1−196102-4. (10) Ohta, H.; Watanabe, T.; Ohdomari, I. Phys. Rev. B 2008, 78, 155326-1−155326-7. (11) Correa-Duarte, M. A.; Giersig, M.; Liz-Marzán, L. M. Chem. Phys. Lett. 1998, 286, 497−501. (12) Yang, Y.; Jing, L.; Yu, X.; Yan, D.; Gao, M. Chem. Mater. 2007, 19, 4123−4128. (13) Orlandini, S.; Meloni, S.; Ippolito, M.; Colombo, L. Phys. Rev. B 2010, 81, 014203-1−014203-8. (14) Syutkin, V.; Korolev, V. V. J. Non-Cryst. Solids 2011, 357, 3781− 3784. (15) Okada, R.; Iijima, S. Appl. Phys. Lett. 1991, 58, 1662−1663. (16) Norton, F. Nature 1961, 191, 701. (17) Devine, R. A. B.; Capponi, J. J.; Arndt, J. Phys. Rev. B 1987, 35, 770−773. (18) Hamann, D. R. Phys. Rev. Lett. 1998, 81, 3447−3450. (19) Diffusion of reactive molecules in solids and melts; Doremus, R. H., Ed.; John Wiley and Sons: New York, 2002. (20) Tournou, C.; Shelby, J. E. Phys. Chem. Glasses 2005, 46, 559− 563. (21) Kajihara, K.; Miura, T.; Kamioka, H.; Aiba, A.; Uramoto, M.; Morimoto, Y.; Hirano, M.; Skuja, L.; Hosono, H. J. Non-Cryst. Solids 2008, 354, 224−232. (22) Carraway, E. R.; Demas, J. N.; DeGraff, B. A. Langmuir 1991, 7, 2991−2998. (23) Kim, S.; Yin, Y.; Paul Alivisatos, A.; Somorjai, G. A.; Yates, J. T. J. J. Am. Chem. Soc. 2007, 129, 9510−9513. (24) Bongiorno, A.; Pasquarello, A. Phys. Rev. Lett. 2002, 88, 1259011−125901-4. (25) Kajihara, K.; Miura, T.; Kamioka, H.; Hirano, M.; Skuja, L.; Hosono, H. Phys. Rev. Lett. 2009, 102, 175502-1−175502-4.

■

CONCLUSION The reported experimental investigation permitted direct characterization of the diffusion process at temperatures much lower than those usually explored for bulk samples and determination of the diffusion coefficient for nanometric silica samples. The obtained results support the applicability of Fick’s diffusion model in the nanometer size range and show that extrapolation of high temperature predictions is valid. Some 11355

dx.doi.org/10.1021/jp3006734 | J. Phys. Chem. C 2012, 116, 11351−11356

The Journal of Physical Chemistry C

Article

(26) Kajihara, K.; Miura, T.; Kamioka, H.; Hirano, M.; Skuja, L.; Hosono, H. Phys. Rev. B 2011, 83, 064202-1−064202-12. (27) Doremus, R. H. J. Non-Cryst. Solids 2004, 349, 242−247. (28) Basic Characteristics of Aerosil, 4th ed.; Degussa: Frankfurt, 2001. (29) Evonik industries online catalog, http://www.aerosil.com/ product/aerosil/en/products/hydrophilic-fumed-silica/pages/default. aspx, 2010. (30) Skuja, L.; Guttler, B. Phys. Rev. Lett. 1996, 77, 2093−2096. (31) Agnello, S.; Cannas, M.; Vaccaro, L.; Vaccaro, G.; Gelardi, F. M.; Leone, M.; Militello, V.; Boscaino, R. J. Phys. Chem. C 2011, 115, 12831−12835. (32) Skuja, L.; Guttler, B.; Schiel, D.; Silin, A. R. J. Appl. Phys. 1998, 83, 6106−6110. (33) Kajihara, K.; Kamioka, H.; Hirano, M.; Miura, T.; Skuja, L.; Hosono, H. J. Appl. Phys. 2005, 98, 013528-1−013528-5. (34) The mathematics of diffusion, 2nd ed.; Crank, J., Ed.; Clarendon Press: Oxford, U.K., 1979. (35) Geissberger, A. E.; Galeener, F. L. Phys. Rev. B 1983, 28, 3266− 3271. (36) Galeener, F. L. Solid State Commun. 1982, 44, 1037−1040. (37) Hehlen, B. J. Phys.: Condens. Matter 2010, 22, 025401-1− 025401-6. (38) Morrow, B. A.; McFarlain, A. J. J. Phys. Chem. 1992, 96, 1395− 1400. (39) Young, G. J. Colloid Sci. 1958, 13, 67−85. (40) Vaccaro, G.; Agnello, S.; Buscarino, G.; Gelardi, F. M. J. Phys. Chem. C 2010, 114, 13991−13997. (41) Bongiorno, A.; Pasquarello, A. Phys. Rev. B 2004, 70, 195312-1− 195312-14. (42) Vaccaro, G.; Buscarino, G.; Agnello, S.; Sporea, A.; Oproiu, C.; Sporea, D. G.; Gelardi, F. M. J. Phys. Chem. C 2012, 116, 144−149.

11356

dx.doi.org/10.1021/jp3006734 | J. Phys. Chem. C 2012, 116, 11351−11356