Size Distribution of Fluidized Nanoparticle Agglomeratesfrom Agglomeration and Fragmentation: A Population Balance Study

Gas fluidization is emerging as a promising means for functionalizing nanoparticles in a scalable manner [1,2]. Yet, the practical use of fluidization in nanoparticle technology is currently hampered by a lack of fundamental understanding of the mechanisms behind nanoparticle fluidization. It is well known that nanoparticles are not fluidized as individual particles but rather in the form of highly porous fractal agglomerates that can be as large as hundreds of microns. Nanoparticle fluidization is in fact determined by the properties of such agglomerates (i.e., size and density) rather than by those of the individual particles [3]. For this reason, nanoparticle-agglomerate or nanopowder fluidization [4] are considered to be more accurate terms. Such large agglomerates are thought to be the result of a dynamic equilibrium between agglomeration and fragmentation events resulting from collisions induced by the fluidization process. However, a detailed understanding of how such dynamic equilibrium determines the size distribution of the fluidized agglomerates is still missing. In particular, the interplay between the inter-particle forces acting at the nanoparticle-scale, the multi-level structure of the agglomerates, and the collision events has not yet been fully described.

In this contribution, we present a study of the size distribution of fluidized nanoparticle agglomerates based on a population balance model that incorporates at the same time (1) a multi-scale description of the agglomerates [5] and (2) a detailed account of the outcomes of collision events. In particular, our model builds on the works of Brilliantov et al. [6] on agglomeration and fragmentation and of Liu [7] on the kinetic theory of granular flow applied to fluidization.

By analysing data from literature, we find that the experimental size distribution of fluidized nanoparticle agglomerates at dynamic equilibrium is independent of the nanoparticle properties and operating conditions when the distribution is rescaled with respect to the average agglomerate size. This suggests a universal mechanism for breakage and agglomeration under standard operation conditions. The population balance proposed here captures this feature when the size of the fragments resulting from a collision event follows an exponential law. Our model also sheds light into why the agglomerate size distribution of some nanoparticles, such as TiO₂ P25 (d_p = 21 nm), is nearly independent of the type of surface, i.e. hydrophilic vs. hydrophobic, whereas the size distribution for other nanoparticles, e.g. Al₂O₃ AluC (d_p = 13 nm) and SiO₂ A130 (d_p = 16 nm), is highly sensitive to the type of surface. We find that this is due to two combined factors: the high density of TiO₂ nanoparticles, which results in a large granular energy, and the relatively large nanoparticle size for this material, which results in a smaller tensile strength of the agglomerates. The way these two variables affect the outcome of a collision event in the balance explains the observed experimental results. TiO₂ fluidization is expected to be improved by using nanoparticles with smaller particle size and a hydrophobic coating. Such insights can be used to optimize and intensify nanopowder fluidization.

[1] Salameh, S., Gomez-Hernandez, J., Goulas, A., Van Bui, H., & van Ommen, J. R. (2017). Advances in scalable gas-phase manufacturing and processing of nanostructured solids: A review. Particuology, 30, 15-39.

[2] Van Bui, H., Grillo, F., & van Ommen, J. R. (2017). Atomic and molecular layer deposition: off the beaten track. Chemical Communications, 53(1), 45-71.

[3] Yao, W., Guangsheng, G., Fei, W., & Jun, W. (2002). Fluidization and agglomerate structure of SiO₂ nanoparticles. Powder Technology, 124(1-2), 152-159.

[4] van Ommen, J. R., Valverde, J. M., & Pfeffer, R. (2012). Fluidization of nanopowders: a review. Journal of Nanoparticle Research, 14(3), 737.

[5] de Martín, L., Bouwman, W. G., & van Ommen, J. R. (2014). Multidimensional nature of fluidized nanoparticle agglomerates. Langmuir, 30(42), 12696-12702.

[6] Brilliantov, N., Krapivsky, P. L., Bodrova, A., Spahn, F., Hayakawa, H., Stadnichuk, V., & Schmidt, J. (2015). Size distribution of particles in Saturn’s rings from aggregation and fragmentation. Proceedings of the National Academy of Sciences, 112(31), 9536-9541.

[7] Liu, L. (2011). Kinetic theory of aggregation in granular flow. AIChE Journal, 57(12), 3331-3343.