Dipolar interaction effects in core–shell magnetic nanoparticles

Abstract Dipolar interaction between magnetic nanoparticles (MNPs) affects their magnetization properties, and the effect is particularly significant for multi-core MNPs and high-concentration MNP samples. In this work, we performed numerical simulations using the Landau–Lifshitz–Gilbert equation to investigate the effect of dipolar interaction on magnetization properties. In these simulations, we introduced a cubic model with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mn>10</mml:mn> <mml:mn>00</mml:mn> </mml:math> MNPs whose core size <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">c</mml:mi> </mml:mrow> </mml:msub> </mml:math> ranges from 10 to 20 nm and whose easy axes are randomly oriented. The distance <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">g</mml:mi> </mml:mrow> </mml:msub> </mml:math> between two adjacent MNPs in the cubic model was varied from 10–70 nm. We quantified the critical interparticle distance <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">crit</mml:mi> </mml:mrow> </mml:msub> </mml:math> — defined as the minimum surface-to-surface distance between magnetic nanoparticles (MNPs) at which dipolar interaction becomes negligible. Assuming a uniform spherical coating layer, this corresponds to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">crit</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:msub> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">m</mml:mi> </mml:mrow> </mml:msub> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">m</mml:mi> </mml:mrow> </mml:msub> </mml:math> is the minimum coating thickness required to eliminate dipolar effects for a given <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">c</mml:mi> </mml:mrow> </mml:msub> </mml:math> . Our results reveal that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">crit</mml:mi> </mml:mrow> </mml:msub> </mml:math> varies with harmonic order: the values of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">crit</mml:mi> </mml:mrow> </mml:msub> </mml:math> for the third <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> </mml:msub> </mml:math> and fifth <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>5</mml:mn> </mml:mrow> </mml:msub> </mml:math> harmonic components of magnetization are larger than that for the fundamental component <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:math> , indicating that higher-order harmonics commonly used in MPI are more sensitive to dipolar interaction. Moreover, we found <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> <mml:mrow>