Distribution functions of magnetic nanoparticles determined by a numerical inversion method

Abstract

In the present study, we applied a regularized inversion method to extract the particle size, magnetic moment and relaxation-time distribution of magnetic nanoparticles from small-angle x-ray scattering (SAXS), DC magnetization (DCM) and AC susceptibility (ACS) measurements. For the measurements the particles were colloidally dispersed in water. At first approximation the particles could be assumed to be spherically shaped and homogeneously magnetized single-domain particles. As model functions for the inversion, we used the particle form factor of a sphere (SAXS), the Langevin function (DCM) and the Debye model (ACS). The extracted distributions exhibited features/peaks that could be distinctly attributed to the individually dispersed and non-interacting nanoparticles. Further analysis of these peaks enabled, in combination with a prior characterization of the particle ensemble by electron microscopy and dynamic light scattering, a detailed structural and magnetic characterization of the particles. Additionally, all three extracted distributions featured peaks, which indicated deviations of the scattering (SAXS), magnetization (DCM) or relaxation (ACS) behavior from the one expected for individually dispersed, homogeneously magnetized nanoparticles. These deviations could be mainly attributed to partial agglomeration (SAXS, DCM, ACS), uncorrelated surface spins (DCM) and/or intra-well relaxation processes (ACS). The main advantage of the numerical inversion method is that no ad hoc assumptions regarding the line shape of the extracted distribution functions are required, which enabled the detection of these contributions. We highlighted this by comparing the results with the results obtained by standard model fits, where the functional form of the distributions was a priori assumed to be log-normal shaped.

Internal Structure and Magnetic Properties in Cobalt Ferrite Nanoparticles: Influence of the Synthesis Method

“…To compare the particle size evaluated by TEM and the crystallographic size estimated by XRD it is important to point out that, while the TEM analysis provides a number‐averaged size distribution, XRD (as well as the magnetic measurements that will be presented later) reflects a volume‐averaged median size. Nevertheless, it is possible to calculate a volume‐weighted size distribution from TEM results, whose median diameter is shifted to larger values and presents the same deviation (more details on this point can be found in the Supporting Information). Following this approach, the volume‐weighted TEM median diameter ( D TEM ) gives , as reported in Table together with its associated standard deviation given by .…”

Influence of clustering on the magnetic properties and hyperthermia performance of iron oxide nanoparticles

“…The corrected magnetic moment in Am 2 was normalized to the iron mass, which was determined by inductively coupled plasma optical emission spectrometry (ICP-OES), to obtain the magnetization M in units of Am 2 /kg Fe . From the resulting M (H) curves we derived the underlying apparent moment distributions P (µ) by numerical inversion [28,31].…”

“…The complex volume susceptibility χ(ω) = χ (ω) + iχ (ω) was measured with a custom-built setup described in [34], which uses an field amplitude of 95 µT, following the protocol described in [20]. By a numerical inversion of the ACS spectra we derived P (ω c ) [31,35]. The uncertainty was not known and we assumed for each data point a reasonable value of σ = 0.05 • χ max .…”

Dynamics of Magnetic Fluids in Crossed DC and AC Magnetic Fields

“…In Figure 6, the vertical dashed line is used to denote the maximum diameter xmax ≈ 20-25 nm of magnetite particles, which were still observable in an electron microscope in highly stable magnetic fluids and powders obtained by the chemical precipitation method [9,10,[29][30][31][32]. In real solutions, particles with magnetic cores of large diameter are absent, since the concentration of iron salts in Let us consider the problem of particle size distribution in magnetic fluids in more detail.…”