There has been a limited number of papers on the crystallite size dependence of the bulk modulus for the spinel structure oxides, and no clear trend has been observed for various materials. From the reported papers, some materials exhibit a decrease of bulk modulus with decreasing particle size, while some nanomaterials show compressibility similar to their bulk counterparts [4]. Materials with opposite tendency are also known. The mineral maghemite, γ-Fe₂O₃ has a cubic spinel type structure. Nanocrystalline maghemite, have been reported as much less compressible than bulk material (K₀=305 ± 15 GPa and K₀=203 ± 10 GPa for nanocrystalline and bulk material, respectively) [5]. Another sample with a structural formula Fe₈[Fe_13.33 _2.67]O₃₂, (where  denotes vacancy) had an exceptionally low bulk modulus of bulk and nanocrystalline form of K₀=124±11 GPa with K'₀=30±7 and K₀=143±14 GPa with K'₀=43±9 respectively. Here the first derivative values K'₀ are exceptionally high and both deviations have been ascribed to the presence of vacancies on octahedral sites [6].

Another factor affecting the bulk modulus is the distribution of cations, characterized by inversion parameter. This parameter is defined as the fraction of divalent metal cations that moves to octahedral sites from tetrahedral sites. Theoretical calculations, using the DFT formalism, have indicated sensitivity of the bulk modulus to the inversion parameter in AlCo₂O₄ of K₀=235.7 GPa and K₀=221.8 GPa in the normal and fully inversed spinel, respectively [7]. A small change in chemical composition of spinel can lead to significant changes in its compression behavior. The bulk modulus of Mg_0.4Al_0.6[Al_1.8_0.2]O₄ ('defect sample') is only 171(2) GPa while for stoichiometric disordered spinel MgAl₂O₄ is of 193 GPa [8]. However, in another paper, concerning the equation of state, Nestola et al. [9] have indicated that there are no differences in the bulk modulus for the ordered and disordered spinel samples.

In addition to the above factors, also the measurement conditions affect the results, especially, the pressure hydrostaticity level (dependent on the applied pressure transmitting medium) seems to be crucial for proper interpretation of compression results for oxides with the spinel structure [10,11]. Synchrotron radiation measurements revealed that LiMn₂O₄ displays an extremely sensitive structural response to deviatoric stress [3], however, nanostructured materials can accommodate more stress compared to their bulk counterparts [12].

Acknowledgements: The research leading to these results has received funding from the European Community’s Seventh Framework Program (FP7/2007-2013) under Grant agreement no 226716.

References

[1] A.M. Pendás, A. Costales, M.A. Blanco, J.M. Recio, V. Luaña, "Local compressibilities in crystals", Phys. Rev. B, 62 (2000) 13970-0-8.
[2] L. Gerward, J.Z. Jiang, J. Staun Olsen, J.M. Recio, A. Waśkowska, "X-ray diffraction at high pressure and high or low temperature using synchrotron radiation: Selected applications in studies of spinel structures", J. Alloys Compd., 401 (2005) 11-17.
[3] P. Piszora, W. Nowicki, J. Darul, "High-pressure metaelastic properties of Li_xMn_3-xO₄ (x=0.87; 0.94; 1.00)", J. Mater. Chem., 18 (2008) 2447-2452.
[4] B. Chen, D. Penwell, L.R. Benedetti, R. Jeanloz, M.B. Kruger, “Particle-size effect on the compressibility of nanocrystalline alumina”, Phys. Rev. B, 66 (2002) 144101-1-4.
[5] J. Z. Jiang, "Phase transformations in nanocrystals", J. Mater. Sci., 39 (2004) 5103-5110.
[6] H. Zhu, Y. Maa, H. Yang, C. Ji, D. Hou, L. Guo, "Pressure induced phase transition of nanocrystalline and bulk maghemite (γ-Fe₂O₃) to hematite (α-Fe₂O₃)", J. Phys. Chem. Solids, 71 (2010) 1183-1186.
[7] R. Franco, F. Tielens, M. Calatayud, J. M. Recio, "Cation distributions on CoAl₂O₄ and Co₂AlO₄ spinels: pressure and temperature effects", Int. J. High Pressure Res., 28 (2008) 521-524.
[8] F. Nestola, J.R. Smyth, M. Parisatto, L. Secco, F. Princivalle, M. Bruno, M. Prencipe, A. Dal Negro, "Effects of non-stoichiometry on the spinel structure at high pressure: Implications for Earth’s mantle mineralogy", Geochim. Cosmochim. Acta, 73 (2009) 489-492.
[9] F. Nestola, T. Boffa Ballaran, T. Balic-Zunic, F. Princivalle, L. Secco, A. Dal Negro, "Comparative compressibility and structural behavior of spinel MgAl₂O₄ at high pressures: The independency on the degree of cation order", Am. Miner., 92 (2007) 1838-1843.
[10] P. Piszora, "In-situ investigations of LiMn₂O₄ at high pressure", Z. Kristallogr. Suppl., 26 (2007) 387-392.
[11] G.D. Gata, I. Kantor, T.B. Ballaran, L. Dubrovinsky, C. McCammon, "Effect of non-hydrostatic conditions on the elastic behavior of magnetite: An in situ single-crystal X-ray diffraction study", Phys. Chem. Miner., 34 (2007) 627-635.
[12] Y. Lin, Y. Yang, H. Ma, Y. Cui, W.L. Mao, “Compressional behavior of bulk and nanorod LiMn₂O₄ under nonhydrostatic stress”, J. Phys. Chem. C, 115 (2011) 9844-9849.

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