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. 2021 Jan 12;11:733. doi: 10.1038/s41598-020-79871-1

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© The Author(s) 2021

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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Dipole–dipole interaction model and nanofluid characterization. (a) A schematic model of dipole–dipole interaction between two SPNPs. d_c–c, m, and θ represent center-to-center interparticle distance, the magnetic moment of each MNP, and angle between tow dipole induced by the SPNPs, respectively. (b) A TEM image of 25 nm Mg_x–γFe₂O₃ SPNP (inset, the size distribution of Mg_x-γFe₂O₃ SPNP measured from TEM image). (c) A liquid VSM measurement of magnetic properties of Mg_x-γFe₂O₃ nanofluids. (inset, a picture of Mg_x–γFe₂O₃ nanofluid sample mounted on the VSM holder). (d) M–H major loop of Mg_x–γFe₂O₃ nanofluids measured at the sweeping field of ± 5 kOe (inset, minor M–H loop). (e) The hydrodynamic sizes before (top) and after (bottom) applied H_DC,appl (5 min@0.5 T). (f,g) Measured hydrodynamic size (f) and polydispersity index (PDI, g) of the nanofluids before and after the applied H_DC,appl.