Hysteresis properties are often the most common parameters that are used for assessing composition, domain state, or particles size of ferromagnetic minerals in geological materials (Day et al., 1977; Dekkers, 1988; Heider, 1988; Geiss et al., 1996; Heider et al., 1996; Dunlop, 2002; Roberts et al., 2018), biological samples (Moskowitz et al., 1993; Brem et al., 2006), and synthesized magnetic particles (Sotiriou et al., 2011; Widdrat et al., 2014; Starsich et al., 2016; Crippa et al., 2017). Geological materials, in particular, can contain varying proportions of different ferromagnetic minerals, or in the case of only a single magnetic phase, a spectrum of particle sizes. Further aggregation of synthetic nanoparticles can also produce a spectrum of effective or apparent particle sizes (Hirt et al., 2017).

In this study we are interested specifically in mixtures of stable single domain (SSD) and superparamagnetic (SP) magnetite. Mixtures of SSD and SP magnetite are of particular interest in environmental, biological and material studies. The ability to estimate the size distribution of magnetite can be important in environmental studies, because it is indicative of geochemical processes that can lead to authigenic growth of new magnetite from a precursor phase or chemical reduction of an existing magnetite. Biological magnetite in bacteria often consists of a range of particle sizes in the SP to SSD range (Faivre and Schuler, 2008; Katzmann et al., 2013; Jacob and Suthindhiran, 2016), and understanding variations in size distribution can provide insight into the provenance or fate after the bacteria die and magnetosomes are deposited in a sediment. A further biological application is associated with magnetite nanoparticles that are either introduced into human or animal tissue through inhalation (Maher et al., 2016; Calderon-Garciduenas et al., 2019), or through biochemical processes associated with an imbalance of iron in the body (Dobson, 2004; Beyhum et al., 2005; Hosking et al., 2018). In these cases the particle size of iron oxides may be important in assessing potential health hazards. Information on particle size spectrum is particularly important in the application of magnetic nanoparticles for medical therapeutics, because the magnetic properties need to be tailored to provide the best response for a specific application. For example, in hyperthermia treatment of cancer cells, the heating response of magnetic nanoparticles will be related to their ability to respond to an alternating field to generate heat (Crippa et al., 2017; Starsich et al., 2018).

Different magnetic methods are available to assess whether a material contains SP ferromagnetic particles. The most definitive methods involve measuring induced magnetization, particularly AC susceptibility, as a function of temperature, which indicates at what temperature the particles undergo blocking. If, however, one is limited to instruments that only measure at room temperature, then one of the most common methods, which is used to indicate the presence of SP ferromagnetic particles, is frequency dependent susceptibility. The method became popular in environmental magnetic studies with the introduction of the Bartington Instruments MS2 susceptibility meter, which offers application of two different AC fields, i.e., 470 and 4,700 Hz (Dearing et al., 1996). Later AGICO introduced a susceptibility bridge that offered measurement in three different frequencies, 976, 3,904, and 15,606 Hz. Hrouda (2011) has provided a detailed discussion of the many factors that will influence whether a significant frequency dependence in susceptibility can be measured. This includes the strength of the contribution from diamagnetic, paramagnetic and even multi-domain (MD) ferrimagnetic fractions to the bulk susceptibility, and the particle size distribution of the SP fraction. Anisotropy factors and saturation magnetization, however, can also play a role in modeling the degree of frequency dependence as a function of particle size. Hrouda (2011) demonstrates that in the case of magnetite or maghaemite only a relatively narrow range of particle sizes contribute to the frequency dependent susceptibility, i.e., only magnetite particles with a diameter between 16.05–16.73 nm when using F1 (976 Hz) and F3 (15,616 Hz) on an Agico susceptibility bridge, or 16.64–17.68 nm when using the Bartington susceptibility meter with frequencies of 470–4,700 Hz. This means that frequency dependent susceptibility may not detect a significant portion of SP particles, if the particle diameters lie outside these ranges.

Other methods to detect the presence of SP particles include measurement of hysteresis loops or first-order reversal curves. Tauxe et al. (1996) simulated hysteresis loops for mixtures of SSD and SP magnetite, using a Langevin function. They showed that models predict wasp-waisted hysteresis curves and pot-bellied loops in such mixtures depending on their relative contributions to magnetization. Dunlop (2002) illustrated how hysteresis parameters will be influenced by mixtures of SP magnetite of different sizes and SSD magnetite. Later studies demonstrated that the thermal relaxation of the SP particles was not considered in this study and that SSD-SP mixtures would lead to mixing curves that have lower than predicted coercivity ratios (Lanci and Kent, 2003; Heslop, 2005). Two studies on artificial mixtures of SSD and SP magnetite illustrated that the ratio between saturation remanent magnetization and saturation magnetization (M_RS/M_S) decreases significantly with the addition of SP material and coercivity ratios lower than predicted by Dunlop (2002), which would be expected because SP particles do not carry a remanent magnetization (Dunlop and Carter-Stiglitz, 2006; Kumari et al., 2015). In the past several years, Lascu et al. (2015) and later Harrison et al. (2018) introduced a method to perform principal component analysis (PCA) on first order reversal curves (FORC) data (FORCem) to unravel mixtures of phases with contrasting magnetic properties, including domain state, and/or composition. The method was applied to different combinations of synthetic binary mixtures of multi-domain (MD), vortex state (V), SSD and SP magnetite, or SSD and SP greigite. Liu et al. (2019) demonstrated how PCA of FORC data could be used to unravel mixtures of SSD magnetite and hematite.

We evaluate methods that can be used to obtain a qualitative estimate for the concentration of SP and SSD particles of magnetite in a mixed system at room temperature. For this purpose, we focus on the Day-Dunlop plot, FORC-PCA, and the relative contributions of the reversible and irreversible magnetizations (Kumari et al., 2014). Two SSD-SP mixing series are compared, in which the SP component of one series has 20 nm average particle size and the second series has ca. 11 nm. Results from three further data sets are compared to these mixtures series. They include geological samples, brain tissue samples, and synthetic SP magnetite. This evaluation should provide insight into which methods are most suitable for discerning relative concentration of SP in mixtures with SSD ferrimagnetic phases. We further highlight the limitations of the methods, particularly when analyzing natural samples that may not meet the assumptions of non-interacting, pure magnetite.

The two sets of SSD-SP mixtures show similarities in their change in magnetic properties that would be expected with an addition of a second phase that is ferromagnetic, but which carries “no” remanent magnetization. The difference between the two series arises largely from the difference in the core size of the SP end member, with 20 nm in Series 1 and ca. 11 nm in Series 2. Considering that the hysteresis and FORC measurements were made using an averaging time of 100 ms, this would mean that the boundary between SP and SSD behaviour for non-interacting magnetite is around 23 nm. Note the longest averaging time that was used in the FORC measurements was 300 ms, which would shift the boundary slightly higher to around 24 nm.

The two SP end members do not show any frequency dependency with respect to the applied frequencies; therefore, χ_FD cannot be used to provide information on the concentration of SP particles. Our results highlight the limitations of using the methods due to the frequencies that are available in commercial instruments.

The hysteresis loops for Series 1 show wasp-waisting with the addition of the SP end member starting between 9 and 19% SP-content. Series 2, however, only shows a weak wasp-waisting by sample 10; what is notable, however, is a narrowing of the hysteresis loop. This difference reflects probably the more rapid relaxation of particles in Series 2. The Day-Dunlop plot is one of the most commonly employed methods to assess particle size or domain state of magnetite, based on hysteresis parameters. There is a major concern, which is associated with using hysteresis properties, to estimate particles size. Day et al. (1977) noted that if magnetite particles are interacting, this will lower M_RS/M_S, B_C and B_CR. Muxworthy et al. (2003) examined the effect of magnetostatic interactions of hysteresis parameters using micromagnetic models, and demonstrated that the degree to which the parameters will be affected, will be dependent on particle anisotropies, degree of alignment, and particle configuration. For magnetite in the SSD range, magnetostatic interactions can lead to significantly lower values of M_RS/M_S, B_C and B_CR.

Dunlop (2002) presented a theoretical study in which he demonstrated how hysteresis parameters will change for SSD-SP mixtures, for SP particles between 10 and 30 nm. Dunlop and Carter-Stiglitz (2006) carried out an experimental study on the hysteresis properties of a mixture of SSD magnetite and a ferrofluid of SP magnetite, and found that B_CR/B_C was lower than predicted, which they attributed to interactions. Roberts et al. (2018), provides a thorough discussion of the weakness of using this method, where they note, among other points, that Dunlop (2002) does not consider thermal relaxation in his mixtures of SSD-SP magnetite, which will have an effect on the magnetization ratio. When thermal relaxation of the viscous particle is considered, there is a more rapid drop in M_RS/M_S (Lanci and Kent, 2003; Kumari et al., 2015; Roberts et al., 2018). This is supported by experimental results (Dunlop-Cater-Stiglitz, 2006; Kumari et al., 2015).

Figure 2 shows the progression of M_RS/M_S and B_CR/B_C for the two SSD-SP series. Series 1 has higher coercivity ratios relative to Series 2, which would be expected due to the larger volume of the SP particles. How this is reflected in the natural samples or synthetic magnetic nanoparticles is seen in Figure 5. The samples from the three sets, which show coercivity ratios that would require the presence of very fine particles (<10 nm), according to Dunlop’s (2002) theoretical results. The coercivity ratios, however, are consistent for a broad particle size spectra in the SP range when accounting for thermal relaxation (Lanci and Kent, 2003). It should also be noted that none of these three sets of samples show wasp-waisting (Supplementary Figures S6–S8). Samples AF23 and XL are synthetic nanoparticles with better constrained particle sizes that are close to the SP-SSD boundary. These two samples show the lowest magnetization ratios, similar to what was found in the SP end member of the mixing series. In summary there is a progression to lower values of M_RS/M_S and higher values of B_CR/B_C with increasing SP concentration, but all sample sets lie below the theoretical progression that is expected by only assuming Langevin behaviour.

The FORC diagrams for Series 1 shows a gradual progression of a FORC distribution from a diagram typical for SSD magnetite, which lies away from the origin. The first indication of an SP contribution at the origin of the FORC diagram is for the sample with 19% SP content, and this contribution becomes larger successively with each addition (Supplementary Figure S3). The SP end-member is confined to the origin with a slight upward shift. The second series only has a FORC distribution that is typical for magnetotactic bacteria with a broader coercivity spectrum that extends to the origin of the plot. There is no noticeable change in the FORC diagram, other than a gradual decrease in the intensity of the distribution, until the until 10th addition, when the hysteresis loop shows a weak wasp-waisting. The SP end member has only a weak expression at the origin, which suggests that the FORCs will be less influenced due to the faster relaxation of the smaller particle diameter in comparison to the SP end member of Series 1.

PCA analysis of the FORC data provides a more quantitative way to observe changes. Series 1 shows a regular change in the PC1 score with each increment (Figure 3; Supplementary Table S6), and the calculated FORC diagrams of the end members are in agreement with the FORC diagrams in their major features. If we compare Series 2 with Series 1, the starting SSD end member has a notable SP-contribution (86% SSD-concentration). The sample with the highest SP concentration (sample 17) compares with the sample with ca. 50% SSD--concentration, but the SP end-member of Series 2 falls around the 68% concentration of Series 1. The fact that a higher SSD concentration is predicted for the SP end member of Series 2 indicates that the PCA cannot properly assess the SP end member with its fast relaxation. This point is discussed below.

A comparison of the samples from geological samples projected onto Series 1 shows that the lacustrine sediments fall toward an SSD concentration of around 76–83% of Series 1 in PCA analysis (Figure 6). Bacterial magnetite is thought to be the main source of magnetite in the samples and this comparable to the SD end member of Series 2, which is bacterial magnetite. The Tiva Canyon Tuff sample lies by ca. 50% contribution of SP particles from Series 1 similar to the higher SP concentration in Series 2. Note that a largely SP content that was found by Till et al. (2011) from low temperature blocking in field-cooled and zero-field-cooled experiments. The biological samples also lie around an SSD concentration of around 74–80%. Note that all of the biological samples show an increase in isothermal remanent magnetization when measured at 77 K (Brem, 2006; Blaser, 2008). This confirms that all samples contain a certain amount of SP magnetite, but it is not possible to determine the concentration. The synthetic samples show a larger spread in particle size, which is in agreement to physical measurements. Samples VR4 and VR8 are mesocrystals of magnetite made up of ca. 9–11 nm crystallites that agglomerate into a single crystal with 45 nm diameter. The SP content reflects the remainder of crystallites or smaller aggregates in the colloidal solution (Reichel et al., 2017). Sample FCA35 is a synthetic magnetite with ca. 16 ± 1 nm core (pers. comm. F. Crippa). It is SP at room temperature but undergoes partial blocking at 250 K, where the PCA analysis suggest that ca. 50% of the magnetization is blocked. At 50 K, ca. 83% of the magnetization is blocked in comparison to Series 1. AF23 has a 23 ± 1 nm magnetite core (pers. comm. A. Fink), which is close to the SP-SSD boundary, and the same is found for XL, whose core diameter is close to being magnetically ordered at room temperature (Hirt et al., 2017). The two samples, however, have markedly different PC1 scores. Although it is difficult to reconcile this difference, it should be noted that AF23 consists of a single magnetite crystal, whereas XL is a mesocrystal, i.e., an assemblage of smaller subunits, in which all have the same crystallographic orientation.

The last method to qualitatively assess the SP contribution to the magnetization is a comparison of the reversible to irreversible magnetization. This method assumes that magnetite is non-interacting as in the case of the Day-Dunlop plot. Both mixing series show an increase in M_rev/M_irrev with increasing SP concentration. This parameter can be plotted as a function of B_C, which will decrease as the SP-content increases. We use a log plot, because the increase in the ratio is small for low concentrations for the SP end member. There is a progressive increase in M_rev/M_irrev accompanied by the decrease in B_C, which would be expected. The track that the progression follows will be sensitive to the coercivity of the SSD end member, i.e., where the progression starts on the plot. This effect is seen for the lake sediments that have a higher coercivity than the SSD end member of Series 1, and therefore lie to the higher coercivity end of Series 1. The size of the SP end member affects how rapidly M_rev/M_irrev will increase; therefore, higher M_rev/M_irrev will be reached for lower SP content. Although one would expect M_rev/M_irrev < 0.5 for magnetite that is controlled by magnetocrystalline anisotropy, it is also found in the samples for which there is independent evidence for interaction between particles (Wohlfarth ratios).

Plots of ∂M_rev/∂B and ∂M_irrev/∂B are also of value (cf., Supplementary Figures S3, S4, S6–S8). For Series 1 M_irrev is characterized by a Gaussian curve. With the first addition of an SP component, little change is seen in the general shape of the hysteresis curve, or the FORC diagram, but the Gaussian curve shows a weak deviation on its flank around ∂B = 0. This feature becomes stronger with each addition of the SP material until two separate peaks are seen. It is these two distinct coercivity contributions that lead to wasp-waisting in the hysteresis curve. Note, in the case of Series 2, however, this secondary peak is not seen in M_irrev.

In summary, three methods were evaluated for detecting SP magnetite in mixtures with SSD magnetite. Both the Day-Dunlop plot and M_rev/M_irrev will see a progression that is related to the amount of SP particles in a material. Critical for the progression in the Day-Dunlop plot is the SP particles size, whereas it is the coercivity of the SSD end member that will be important for M_rev/M_irrev plotted as a function of B_C. Both methods, however, assume that the magnetite is non-interacting. If interactions occur between the magnetite particles, then there will be a reduction in M_RS/M_S, B_C, and B_CR, which would predict falsely a higher SP concentration. This may not be serious for weak interactions, as seen in the example of OP 18 compared to SOSP1-059, but would not be suitable for an assembly of interacting particles. A critical point is that both approaches are also not applicable if there are particles that are larger than SSD, which would also contribute to the reversible magnetization.

FORC analysis and PCA of the FORC measurements is suitable in detecting SP content, particularly when the SP particles are near to the SP-SSD boundary. The method loses its sensitivity when the SP particles are very fine and have short relaxation times. Fine SP particles will still contribute to the induced magnetization, which is why they affect hysteresis and M_rev/M_irrev, but will have less of a contribution to the FORC distribution. The SP concentration, which the PCA analysis calculates, will be dependent on knowing what the correct end members are so that they can be fitted. This can often be a limiting factor, particularly in natural samples that can show variability within a given locality, and the number of end members may not be clear. The smoothing factor, which is used to process the FORCs, will also play a role in any comparison of FORC results, therefore all data sets should be processed using the same smoothing factor.

In addition to problems that will arise if there are particles that are larger than SSD, or are interacting magnetically, we extend further caveats with respect to all three approaches. Firstly, the presence of other mineral phases, e.g., hematite, will greatly affect hysteresis properties. Mixtures of different ferromagnetic minerals with different particle sizes will lead to competing factors, which will not be possible to separate. Secondly, performing a slope correction could remove part of the contribution from the superparamagnetic contribution. Thirdly, averaging time of measurements is another factor that needs consideration and plays an important role in any comparison. Samples should only be evaluated if their measurement has been made using the same averaging time. Longer averaging time will lead to more relaxation during a measurement, thus influencing the reversible magnetization.

We can ask how often one deals with ideal mixtures of SSD-SP particles that are only made up of non-interacting magnetite. They are most likely to be found in synthetic particles, and the above approaches can be very useful in the characterization of these systems. Depending on the source of the ferromagnetic particles in biological systems, these are also often limited to finer particles. The problem in tissue samples, however can be clustering, and therefore meeting the criterion of non-interaction. For example, Brem et al. (2006) found clustering between magnetite/maghaemite particles in the brain tissues that were studied, but measurements had to be made on freeze-dried tissue that was compressed. They could not conclude if the particles were clustered naturally or due to sample preparation. In any case these presented methods can provide useful information in a comparison of different tissues, although some overestimation of the SP concentration may occur. The same can be said for natural materials. These can have more variety in the composition of ferromagnetic (s.l.) phases, and in particle size, which may not include only SP and SSD particle size. The degree of magnetic interaction may also be more common. Critical is that magnetite is the main ferromagnetic phase, and there is independent evidence that the effect due to interactions is not large.

We have evaluated three methods to estimate the relative contribution of SP magnetite to mixtures with SSD magnetite, using two mixing series of SSD magnetite with SP magnetite, which have two different particle sizes. Both the Day-Dunlop plot or M_rev/M_irrev plotted as a function of B_C will detect changes in the SP contribution to the magnetization. PCA analysis of FORC measurements will also detect changes in the relative amount of SP particles, but is less sensitive in distinguishing small differences. The particle size or size distribution will affect the relatively change in magnetization with increasing SP concentration in all three methods. For this reason, the methods provide a qualitative method for assessing the concentration of SP magnetite, although comparisons within a selected group of samples may provide semi-quantitative information.