Distribution functions of magnetic nanoparticles determined by a numerical inversion method

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.


Introduction
As evident from recent review articles [1][2][3][4], biomedical application of magnetic iron oxide nanoparticles (IONPs) is presently an intensely investigated, multi-disciplinary field of research. However, to this day there exists no standardized way of characterizing IONPs. In this sense, regulatory work is necessary prior to clinical application to guarantee a safe and effective implementation of IONPs [5,6]. To disclose unambiguously the interrelations between structural and magnetic properties of ensembles of IONPs usually a combination of several techniques is applied, such as transmission electron microscopy (TEM), dynamic light scattering (DLS), small-angle x-ray scattering (SAXS), DC magnetization (DCM) and AC susceptibility (ACS) measurements. The standardization process then should include a precise control of the measurement procedures itself as well as of the data analysis.
The theoretical framework defining the interrelations between particle structure and magnetic properties is well established in the ideal case of homogeneously magnetized, single-domain and non-interacting particles [7][8][9]. The same applies to the relaxation dynamics of colloidally dispersed magnetic nanoparticles. These relations are commonly used to extract for example the intrinsic core or hydrodynamic size distributions from DCM [10][11][12] or ACS measurements [13,14]. As model functions for DCM and ACS measurements, the Langevin function [15] (DCM) and Debye model [16] (ACS) are used, respectively. The classical approach to extract the characteristic distributions is to fit the experimental data with the respective model function under assumptions regarding the line shape. Usually a log-normal shape is a priori assumed for the intrinsic distributions [10][11][12][13][14]. But several factors can result in deviations of the macroscopically determined, apparent distribution functions from the intrinsic ones. In particular dipolar interactions can lead to particle agglomeration [17][18][19][20], which can significantly modify the macroscopically detected magnetization behavior of the particle ensemble. To infer the intrinsic moment or relaxation time distribution from data fits, several mean field approaches exist to analytically model the influence of dipolar interactions on isothermal [21][22][23] or dynamic magnetization measurements [24,25], respectively.
However, an alternative approach to analytical model fits can be found in the literature. In [26][27][28][29] for example, the discrete, apparent moment distributions of IONP ensembles were extracted from M(H) data numerically, using simply the Langevin equation as a model function. In these cases no a priori assumptions regarding the line shape of the extracted distribution have to be made and data analysis is ultimately performed by interpreting the obtained apparent distributions. Similar numerical approaches are also commonly used to infer the hydrodynamic size distribution of nanoparticles from DLS measurements [30,31] or to analyze the small-angle scattering data of nanoparticle ensembles [32][33][34][35][36].
In the current work we use the same numerical approach as applied in [36] for the analysis of magnetic multi-core nanoparticles to systematically evaluate SAXS, DCM and ACS data of a dilute, colloidal dispersion of single-domain IONPs. Initially, we extracted the discrete particle size distribution from the SAXS data, the moment distribution from the DCM data and the relaxation time distribution from the ACS data. Afterwards, we further analyzed the distributions to disclose the relations between the structural and the magnetic properties of the nanoparticles. This approach, in combination with classical TEM and DLS analysis, as well as magnetization measurements of the immobilized particles, enabled us to perform a very detailed characterization of the IONP ensemble. Additionally, we highlight the influence of particle agglomeration and/ or uncorrelated surface spins on the measurement signals by comparing the extracted distributions to the results of standard model fits, where a log-normal shape of the distributions was a priori assumed.

Experimental procedure
The synthesis of the IONPs was carried out by thermal decomposition of an iron oleate complex mixed with oleic acid in 1-octadecene [37]. In a round-bottomed flask equipped with a mechanical stirrer and a reflux condenser under a nitrogen flow, the iron(III) oleate (4.5 g, 5 mmol, 7% iron) was mixed with oleic acid (0.7 g) in 1-octadecene (50 ml). The mixture was stirred and heated up until reflux (320 C • ) at 3 C min 1 -• with a heating mantle. Stirring and nitrogen flow were stopped once the temperature reached 100 C • . The resulting suspension was washed with ethanol, centrifuged at 3944 g and redispersed by agitation (ten times). The precipitated particles were finally dried under a nitrogen flow, redispersed in toluene and transferred to water by ligand exchange with dimercaptosuccinic acid (DMSA) [38]. The pH value of the final water-based colloidal dispersion of the nanoparticles was adjusted to 7. This colloidal dispersion was directly measured by SAXS, DCM and ACS without any sample modification, although for ACS the sample was diluted ten-fold with distilled water.
To determine the amount of iron within the colloidal dispersion and hence estimate the particle concentration, inductively coupled plasma optical emission spectrometry (ICP-OES) was carried with an apparatus from Perkin Elmer, model OPTIME 2100DV. The wavelengths used for iron determination were 238.204 and 239.562 nm. For an appropriate determination it is necessary to digest the nanoparticle dispersion prior to analysis. An aliquot of sample (50 l m ) was mixed with 2 ml of HCl until complete digestion was obtained.
The main structural properties of the sample were determined by TEM, DLS and finally SAXS.
To obtain the core-size distribution, images of the particles were taken using a 200 keV JEOL-2000 FXII TEM and the largest dimension of 500 particles was measured.
The hydrodynamic size of the particles dispersed in water was measured with a ZETASIZER NANO-ZS device (Malvern Instruments) using the DLS mode. The measurements of the 100-fold diluted dispersion were done at 25 C • with an equilibration time of 60 s. A measurement consisted of 24 runs, with acquisition times of 10 s. The hydrodynamic diameter, or z-average, as well as the polydispersity index PDI, was derived from the cumulant analysis, wherein a single particle size was assumed and a single exponential fit was applied to the autocorrelation function.
The SAXS measurement of the colloidal dispersion of IONPs was carried out on a Kratky system with slit focus, SAXSess by Anton Paar, Graz, Austria. The measurement was performed as an absolute intensity measurement by also measuring the scattering curves of the empty capillary and water. These were subtracted from the measured scattering curve of the sample during the data reduction procedure using the implemented SAXSquant software. The resultant curve was deconvoluted with the beam profile curve to correct for the slit focus smearing.
The magnetic properties of the particles were determined by measuring the isothermal magnetization curve of both the immobilized as well as colloidally dispersed particles.
To immobilize the particles and hence suppress a rotation of the particles in the field direction, a droplet of 5 l m of the colloidal dispersion was put on cotton wool. The isothermal magnetization measurement was recorded with a Quantum design SQUID VSM 7T with Quick Switch and Evercool at 300 K in a field range of The isothermal DCM measurement of the colloidal dispersion of IONPs was performed at 300 K in a magnetic property measurement system (MPMS XL, Quantum Design, USA). The magnetic field was varied in the field range H 0 4.9 T 0 m =  and the size of the consecutive field step was changed logarithmically to ensure a sufficient number of measurements at low fields. The time between each field step was about 3 min, providing quasi-static measurement conditions. A volume of 30 l m of the suspension was measured a total of five times with the same measurement conditions and the data points were averaged. From the data the diamagnetic contributions of the empty sample holder and the water were subtracted. The corrected magnetic moment in A m 2 was normalized to the mass of iron determined by ICP-OES to obtain the magnetization in A m 2 kg −1 Fe . To investigate the relaxation dynamics of the particles dispersed in water the colloidal dispersion was measured using ACS. The measurement was performed at 300 K utilizing two custom-built susceptometers [13]. The frequency was swept from 10 Hz to 10 kHz and from 200 Hz to 1 MHz in logarithmical steps. The amplitudes of the excitation field amounted to 567 T m and 90 T m , respectively, to ensure that the magnetization response was in the linear regime. To suppress hydrodynamic interactions between the particles, the dispersion was diluted ten-fold. A total volume of 150 L m of the resulting colloid was measured. To convert the measured signal to volume susceptibility c w ( ), both systems were calibrated with Dy O 2 3 powder samples with known susceptibility. Since the measurement was performed only once, for each data point a noise of 0.01 max c ¢ was artificially added.

Theoretical framework
In this section we present the theoretical background regarding the magnetic properties of colloidally dispersed IONPs. On this basis, the approaches to analyze the experimental data of the SAXS, DCM and ACS measurements of the colloidal dispersion are introduced.

Background
Within this work we assume that the IONPs are spherically shaped with a core diameter d c (figure 1) and are single-domain particles. Experimental studies and simulations [39][40][41][42][43][44] indicate that the atomic spins at the surface of magnetic nanoparticles are usually disordered due to lack of coordination, vacancies and/or surface anisotropy. This can be depicted by a core-shell structure of the particle with d d m c < , where d m is the diameter of the homogeneously magnetized core (single-domain, figure 1). Therefore, in the following we will differentiate between the total core volume V d gat T 300 K = are listed in table 1, as well as other intrinsic properties relevant for the current study [7,9].
In > dispersed in a viscous matrix on the other hand, can only align in the field direction by a rotation of the entire particle, the so-called Brownian rotation. In the case of DCM measurements, the magnetization behavior of one macrospin dispersed in a purely viscous matrix is independent of the alignment process and can be described by the Langevin function [15]:  gat T 300 K = [7,9]. Here ρ is the volumetric mass density, w Fe the weight percentage of iron, M S the saturation magnetization, K 1 the first order magnetocrystalline anisotropy constant and s r the scattering length density. The scattering length density of water is 9.5 10 nm The time constant 0 t in equation (3) is generally taken as 10 s 9 - [13]. In equation (4), η is the viscosity of the surrounding medium and d h the hydrodynamic diameter of the particle (figure 1). The hydrodynamic diameter is usually larger than the core diameter d c due to a surfactant shell and additional friction contributions, such as surface charges [46]. The effective relaxation time )is dominated by the faster process and as a result the measurement signal of the ACS experiments are sensitive to the respective relaxation process [47].
As depicted above, the quasi-static and dynamic magnetic properties critically depend on the structural properties of the particles. Within this work we determined the core size d c of the particles by TEM and SAXS, the hydrodynamic size d h by DLS, the magnetic moment μ and magnetic core size d m by DCM, and the relaxation time τ by ACS. The main objective was to correlate the various parameters. When characterizing nanoparticle ensembles it has to be considered that the core and hydrodynamic sizes usually have a distribution width, and thus also magnetic moments as well as relaxation times [48]. In the following we explain how the various distributions were determined in this study.

Data analysis
With TEM the core-size distribution was directly obtained by measuring N = 500 particles and the DLS setup provided the detected hydrodynamic size distribution. Both techniques were used for the pre-characterization of the sample.
In the case of SAXS, DCM and ACS the relevant distribution p(z) of parameter z had to be extracted from the data D(x). For SAXS D(x) is the scattering intensity I(q), with q being the scattering vector and p z The distribution p(z) describes the continuous probability density function of the paramater z and it can be written Here c is a characteristic prefactor and f x z , ( )the corresponding model function. To determine p(z) from the experimental data we applied two approaches in this study. (1) By a standard least square fit of the data under the assumption of a log-normal distribution p(z). (2) By numerical inversion without a priori assumptions regarding the line shape.

Log-normal fit
With respect to magnetic nanoparticle ensembles it is often assumed that the particle size is logarithmically distributed, resulting in log-normal distributions of the magnetic moments [10][11][12] and relaxation times [13]. The probability density of a log-normally distributed variable z is where σ is the standard deviation of the log of the distribution, z¯the median and z z exp the mean value. For comparison purposes between the different distributions, it is important to consider that they vary with respect to their weighting, depending on the nature of the applied technique. In this work it has to be differentiated between number-weighted distributions p N and volume-weighted distributions p V [49]. The characteristic values of these distributions-such as broadness, median and arithmetic mean-are subscripted in the following always with either N or V to indicate whether they correspond to the number or volumeweighted distribution, respectively. For number and volume-weighted log-normal distributions the standard deviations are in theory identical ( V N s s s = = ) but the medians z V and z N are shifted. In the case of homogeneous spheres, e.g. z z exp 3 [49]. In this work we determined the number-weighted core-size distribution p d Here p f is the volume fraction of particles, s r D the scattering length density contrast between the particles and the surrounding medium (table 1) and bkg the background level due to incoherent scattering. As a model function we used the form factor F q d , c ( )of a homogeneous sphere [50]: From the magnetization data M(H) the volume-weighted moment distribution p V m ( )can be extracted by fitting with [11] To determine the number-weighted magnetic core diameter distribution p d N m ( )we modeled the data in this work with [11]: From the ACS measurements the relaxation time distribution can be extracted by fitting the ACS spectra with In this case the prefactor 0 * c as well as the weighting of the relaxation time distribution p t ( ) critically depends on the correlation between the particle moments μ and τ [14]. For the simple case that they are not correlated ñ is the number-weighted mean magnetic core volume, and p t ( ) is number-weighted. Normally, however, a correlation between μ and τ has to be considered, as discussed for example in [14,51,52].

Numerical inversion
Here j = N is the number of bins of the reconstructed histogram p(z) with width z D , where p(z) represents the probability density function of the variable z so that p z 1 has to be minimized, with x s s = ( ) being the measurement uncertainty or standard deviation at each data point. In equation (13) (8)) and the number-weighted core-size distribution p d N c ( ) was extracted with (equation (12)): Comparison with equation (7) shows that c V c (1)) and from M(H) the volume-weighted moment distribution p V m ( )was extracted according to: In this case c D = M S (equation (9)). (2)) and from the measurement the relaxation time distribution p t ( ) was determined by simultaneously adjusting the real and imaginary part of the complex volume susceptibility c w ( ): (11)). As discussed above, the prefactor 0 * c and the weighting of the relaxation time distribution depends on the correlation between μ and τ. For further details regarding this issue see [14,51,52].
Solving of equation (13), however, is an ill-conditioned problem. Therefore, we implemented a nonnegative constraint and additionally applied a regularized inversion. To determine the probability for a given regularization parameter we performed a Bayesian analysis, as introduced in [34,35]. The computational details can be found in the appendix.
For a detailed analysis and interpretation of the extracted distributions a pre-characterization of the sample was necessary. A structural pre-characterization included the measurement of the iron concentration in the colloidal dispersion by ICP-OES as well as the determination of the core-size distribution p d c ( )by TEM and the hydrodynamic size distribution p d h ( )of the particles in dispersion by DLS. Additionally, we characterized the magnetic properties of the immobilized particles to estimate the number of thermally blocked particles at 300 K.  , which can be primarily attributed to an increased hydrodynamic volume due to the surfactant layer and surface charges (figure 1). Additionally, it has to be considered that with DLS the determined hydrodynamic size is intensity-weighted and hence the signal is dominated by large particles.

Pre-characterization of the nanoparticles
Considering the quite large size of the particles it was important to monitor the stability of the colloidal dispersion. With a core diameter of about 19 nm and assuming a homogeneous magnetization with M 0.48 MAm S = (table 1), the dipolar energy between two neighboring particles with parallel aligned moments can be estimated [22] to be in the range of k T 10 B . Thus, dipolar interactions could lead to particle agglomeration and ultimately coagulation. However, the measured z-average and PDI basically did not change over a time span of 15 months, and no sedimentation was observed. In addition, the measured zeta potential remained constant over time and was determined to be about 30 mV -. The same applied for the pH value, which remained at 7, and hence we can conclude that the sample remained stable over time across all metrics. We attributed the colloidal stability to the comparatively low volume concentration of nanoparticles, and their surrounding surfactant layer and surface charge.
Later on, the SAXS, DCM and ACS measurements of the colloidally dispersed IONPs are analyzed. In a colloidal dispersion, the static magnetization behavior of thermally blocked and non-blocked (superparamagnetic) particles is identical, provided the particles are individually dispersed in the viscous matrix. Hence, to estimate the amount of thermally blocked particles a physical rotation (Brownian rotation) of the particles has to be suppressed. Additionally, large average distances between the particles have to be assured to avoid dipolar interactions, which can significantly alter the magnetization behavior of such ensembles [53,54]. For this purpose the particles were immobilized and measured as discussed in section 2.

Analysis of distribution functions
In this section we present and discuss the particle size, magnetic moment and relaxation time distributions determined from the SAXS, DCM and ACS measurements of the colloidal dispersion. First, we estimated the distributions in each case by fitting the data under the usual assumption of log-normal shape. Afterwards, we determined the discrete, apparent distributions by numerical inversion without a priori assumptions regarding the line shape. We analyzed the discrete distributions further to extract the core size (SAXS), magnetic core size (DCM) and hydrodynamic size (ACS) distributions.
5.1. Small-angle x-ray scattering (SAXS) 5.1.1. Log-normal fit Figure 4 shows the measured SAXS intensity I(q). The data was fitted with equation (7) using either the scattering length density contrast between particle and water 31.  . Comparison with the distribution determined by TEM ( figure 5, table 2) shows that the log-normal distribution according to SAXS is broader and shifted to larger values. It can be observed that in the low q-range of figure 4 the forward scattering of the sample is significantly enhanced compared to the fit, which indicates the presence of larger structures, i.e. agglomerates. To verify this we determined the discrete core-size distribution numerically.

Numerical inversion
We numerically inversed the data using the form factor of a sphere as model function (equation (14)) to extract the particle size distribution p d N c ( ). In doing so, the incoherent scattering background bkg can be, in principal, implemented as a fitting parameter [35], however, here the result bkg 3.4 10 cm    To get a deeper physical understanding of the results, we investigated what a core-shell model [50] predicts regarding the extracted size distribution. For this purpose we simulated the scattering intensity I q sim ( ) of an ensemble of IONPs surrounded by a surfactant layer with the core-shell model [50], assuming a log-normal distribution of the core sizes. The free parameters in the core-shell model were (i) the shell (surfactant) thickness s (figure 1), (ii) the scattering length density s shell r of the shell and (iii) the broadness σ and median value d c,N of the number-weighted core-size distribution. Afterwards, I q sim ( ) was numerically inversed the same way as the experimental data and the free parameters adjusted until good agreement between the extracted distribution and peaks 2 and 3 of p d N sum c ( )was obtained ( figure 7). This approach is comparable to a core-shell model fit of the reciprocal scattering data I(q), which is the standard approach to analyzing the structural properties of nanoparticles surrounded by a surfactant layer [56]. The difference, however, is that we can analyze separately the scattering behavior of the individual particles, which is achieved by focusing on the adjustment of peaks 2 and 3. By contrast, a fit of I(q) would include all scattering contributions (peaks 1-5).
The best agreement between p d   ( )is plotted in figure 5 and, as can be seen, is in accordance with the core-size distribution determined by TEM (table 2), just slightly broader. In the following we will regard the distribution p d N core c ( )as the real core-size distribution of the individual IONPs according to SAXS.
Furthermore, numerical inversion offers the opportunity of observing and hence characterizing larger structures present in the dispersion by analyzing the peaks in the large d c -range (peaks 4 and 5 in figure 7). We surmise that these larger structures are particle agglomerates. With sizes in the range 37-47 nm the agglomerates apparently consist of only a few particles (   (table 1). That the magnetization curve is not completely saturated, even at 4.9 T, can be attributed to a paramagnetic-like contribution of uncorrelated surface spins [39,57]. To correct the magnetization curve from all linear contributions in the high field range, the last three data points were fitted linearly to obtain A m kg T  8 and 9). Two possible contributing factors to the deviations are (i) the unknown magnetization behavior of the surface spins and (ii) particle agglomerates. From these discrepancies it can be concluded that the isothermal magnetization behavior of the colloidal particle ensemble cannot be modeled ad hoc by a single log-normal distribution, which has also been observed for other systems [57,58]. To verify this, we determined the apparent moment distribution of the particles by numerical inversion.

Numerical inversion
We inversed the data with equation (15) to extract p V m ( ). The range of the extracted moment distribution was set to be 10 10 Am     agreement with the experimental data ( figure 9). As shown in figure 9, the residuals are nearly zero over the whole field range, indicating the high quality of the fit by numerical inversion. Therefore, the extracted moment distribution c p D V sum m m D · ( ) seems to be a very good representation of the real intrinsic moment distribution. Over the whole μ-range several peaks in c p D V sum m m D · ( ) can be observed. In the following we assume that the main peak at 10 Am 18 2 -corresponds to the magnetic cores of the individually dispersed nanoparticles, whose magnetization behavior can be modeled over the whole field range, each by a single Langevin function. The other peaks describe magnetization contributions that lead to deviations from the simple Langevin-type behavior, e.g. the uncorrelated surface spins or agglomerates [59,60].
The main peak in figure 10 starts at the index j = 54 and ends at j = 78. The volume-weighted mean moment of the individual cores is determined from the main peak to can be attributed to the reduced magnetic core volume compared to the total particle core volume due to the surface layer with uncorrelated spins. This can be shown by transforming the main peak of the extracted moment distribution to a number-weighted magnetic core-size distribution p d   figure 10). Accordingly, the agglomerates can significantly contribute to the ACS spectra of the colloidal dispersion, as shown in the following.
The peaks in the low moment range on the other hand correlate to non-linear magnetization processes in the high field range. We surmise that they can be either attributed to dipolar interactions within the agglomerates [59] or to uncorrelated surface spins.  , table 2). But with DLS we determined the intensity-weighted mean of the hydrodynamic diameter, and the expected volume-or number-weighted means would, in theory, be significantly smaller. Thus we surmise that the peak at 10 rad s 5 / is mostly a result of Brownian relaxation processes. To fit measurements of polydisperse nanoparticle ensembles, however, normally Néel contributions also have to be considered, which is done for example in the generalized Debye model [14,51,52]. Qualitatively, this model is comparable to the superposition of two log-normal functions p a p a p 1 1 2 (11) in the case that Néel and Brownian contributions do not overlap. Fitting c¢ and c under the assumption of a bimodal distribution of relaxation times resulted in the high frequency part ( 10 Hz 4 w > ) in good agreement with the data ( figure 12). From the extracted relaxation time distributions structural parameters such as the hydrodynamic size or the magnetic core size can be determined [14,51,52]. However, as can be seen in figure 12, in the low frequency range significant deviations between data and fit are observed, which indicates the presence of slow relaxation processes. It is safe to assume that these can be attributed to the contributions from particle agglomerates. To extract the complete spectra of relaxation times we numerically inversed the experimental data using equation (16) as a model function.

Numerical inversion
To adjust the experimental data shown in figure 12 over the whole ω range, the range of the extracted relaxation time distribution had to be set to 10 10 s  and a smaller peak (shoulder) at 3 10 s 7´-. As mentioned before, the main peak seems to correspond to the Brownian relaxation of the individually dispersed and thermally blocked particles, whereas the smaller peak can be attributed to small particles, which respond by Néel relaxation. The remaining very small relaxation times can be probably attributed to intra-well relaxation processes [63,64] (transversal relaxation). Additionally, contributions in the very large relaxation time range are observed (10 10 s As mentioned before, it is safe to assume that these slow processes correspond to the relaxation dynamics of the agglomerated particles, as discussed for example in [14]. Hence, by numeric inversion we could isolate the different contributions and subsequently analyze them separately. First, we estimated the hydrodynamic diameter distribution of the particles from the main peak of p sum t t D ( ) . Considering that now the individually dispersed particles could be analyzed separately, we could assume a log-normal shape for the hydrodynamic size distribution.
For this purpose, we calculated the Brownian relaxation time distribution P b t ( )with equation (4) for a given log-normal distribution of the hydrodynamic diameter d h until visually good agreement was found between    figure 14 was obtained when the anisotropy constant was set to K 9 kJ m 3 = -, which is inbetween the magnetocrystalline anisotropy constant of bulk magnetite and maghemite (table 1). The quite good agreement with p sum t t D ( ) seems to verify that this peak corresponds to the Néel relaxation of small particles. With K 9 kJ m 3 = the estimated effective anisotropy constant is in good agreement with other studies, where K was derived from ACS measurements and found to be in the range of K 10 kJ m 3 = for similar iron oxide nanoparticles [14,65]. However, it has to be taken into account that in the case of cubic anisotropy the expected value for the effective anisotropy constant relevant for equation (3) would be K 1 12 1 ( )| | [66] and hence much lower than K 9 kJ m 3 = -. A possible explanation for this discrepancy is that the effective anisotropy constant of the IONPs is dominated by surface anisotropy, as theorized in [67].

Final remarks
Prior to an application of magnetic nanoparticles, for example in biomedicine, the main physical properties of the ensembles (structural as well as magnetic) must be evaluated, ideally in a standardized way [5,6]. Amongst other properties, of outmost importance is knowledge on the characteristic particle sizes, such as core, magnetic and hydrodynamic size. The usual approach to determining the relevant size distributions from measurements (scattering, magnetization, susceptibility) is to assume a priori a log-normal shape and fit the data accordingly [10][11][12][13][14]. However, subtle details relevant for the application of magnetic nanoparticles can be missed, as we have revealed in our results described above. Our numerical inversion procedure, derived from a method used for the analysis of small-angle scattering data [34,35], is a finer approach to revealing the distribution functions. This method does not imply a complication in deriving the relevant physical parameters compared to classical model fits but in fact our aim is to reach an easily applicable numerical approach. For this purpose the applied model functions are intended to be kept as simple as possible.
In this study we initially extracted the particle size distribution of an ensemble of iron oxide nanoparticles by numerical inversion from SAXS data of the colloidal particle dispersion. Then we determined the moment distribution of the particle ensemble from the isothermal magnetization (DCM) measurement and its relaxation time distribution from the ACS measurement of the colloidal dispersion, using the same numerical approach. As model functions we simply used the particle form factor of a sphere (SAXS), the Langevin function (DCM) and the Debye model (ACS). In figures 6, 10 and 13 the extracted size, moment and relaxation time distributions are shown. In all three cases the distributions deviate from the commonly expected log-normal shape.
These deviations are well interpreted as stemming partly from particle agglomerates (SAXS, DCM, ACS). It also appears, unambiguously, that uncorrelated surface spins modify the expected magnetization behavior (DCM). Such surface spins are a common finding in magnetic nanoparticles as described, e.g., in [39][40][41][42][43][44]. Finally, a good description of the relaxation time spectra may be connected to the presence of intra-well relaxation processes [63,64]. Accordingly, the distributions determined by numerical inversion exhibit characteristic peaks, which we attributed to these contributions. However, we additionally observed peaks in the discrete distributions, which we could distinctly ascribe to the individually dispersed and non-interacting particles. Ultimately, evaluation of these peaks allowed us to estimate their core, magnetic core and hydrodynamic sizes as well as relevant ensemble parameters, summarized in table. 3.
According to these results the mean core size of the particles was 19 nm and the mean magnetic core size 18 nm. This indicated the presence of a 0.5 nm thick surface layer of uncorrelated surface spins. Furthermore, from an analysis of the as determined size and moment distributions we calculated the volume fraction of agglomerated particles to be 0.08. From the extracted relaxation time distribution we estimated the hydrodynamic size distribution of the particles to have a mean value of 32.2 nm. Additionally, we could evaluate the effective anisotropy constant to be 9 kJ m 3 -, which is in good agreement with other studies where K was derived from ACS measurements [14,65]. Taking all these comments together it can be concluded that the analysis of the discrete distribution functions extracted by numerical inversion enables a detailed analysis of the structural and magnetic properties of the particle ensemble. This is fundamental for technical application and related standardization criteria. Naturally, the inversion method cannot eliminate the errors caused by an inadequacy of the applied model functions, which need to be carefully chosen. The main strength of the numerical inversion method is then the possible separation of contributions of the model-like contributions of the individually dispersed particles from the total measurement signal. The numerical approach to determining such distribution functions without a priori assumptions regarding the line shape is universally applicable, as we have shown in this study by means of scattering, magnetization and ACS data. The code for the numerical inversion of the SAXS, DCM and ACS data used for this study was written in Python and is available from the authors.

Appendix: numerical inversion
Solving equation (13) is in general an ill-conditioned problem, in particular due to measurement uncertainties. This can give rise to large, unphysical oscillations in the extracted distribution. To overcome this problem two approaches were combined. First, a non-negative constraint [68] was implemented as done in [26,28] to ensure positive values P z 0 j > ( ) in the extracted distributions. Additionally, a Tikhonov regularization was applied to force smooth distributions. In this case, instead of equation (13) is minimized, here additionally with the constraint P z 0  ( ) . The N × N matrix L is a regularization matrix which is generally chosen to force smooth solutions, weighted by the regularization parameter α. To additionally force the extracted distribution to approach zero at the start and end points the following non-singular approximation of the discrete second-order derivative operator was used: , with 0 N,1 being a zero vector of length N. To find the optimal value for the regularization parameter α the a posteriori probability or evidence P a ( ) according to [35] was determined: