1 Introduction

Particle size reduction to the nanometer scale significantly affects magnetic properties of magnetic nanoparticles (MNPs) due to deleterious impact of surface atoms on the effective magnetization of the MNPs [1, 2]. However, when optimal performance is sought in bio-applications such as magnetic hyperthermia therapy [3, 4], drug delivery [5, 6], and magnetic resonance imaging [7, 8], it is important to retain the magnetization values M at room temperature as close as possible to the corresponding bulk ones [9, 10]. Hence, it is important to address size-dependent changes of magnetic properties in nanometer-scaled particles.

As the size of the MNPs decreases below the micrometer-size range several new phenomena appear, including superparamagnetism [11, 12], reduced saturation magnetization [13, 14], and non-saturated and open hysteresis loop at high magnetic fields [15, 16]. For a given material with effective magnetic anisotropy \(\mathrm{K}_{\mathrm{eff}}\), the superparamagnetic behavior appears when the particle volume \(V\) is small enough that the thermal energy can overcome the anisotropy energy barrier \(\mathrm{E}={\mathrm{K}}_{\mathrm{eff}} V\) separating magnetization easy axes [17, 18]. On the other hand, the reduced and non-saturating magnetization phenomena originate from spin-disordered configuration at the MNP surface, that can be explained by the core–shell model.

This core–shell configuration on MNPs consists of a spin-disordered shell, known as magnetic dead layer (due to its zero net magnetization), surrounding a core with ferro/ferri magnetic-ordered spins [19, 20]. Deterioration of magnetic order in the dead layer is originated from the surface effects in this region. Structural distortions at the MNPs boundaries result in breaking atomic bonds and consequently frustrating exchange interactions between surface and core spins, which in turn lead to the orientation deviation of surface spins with respect to the core ones [19, 21,22,23,24]. In ferrites, like magnetite, exchange interactions occur through intermediation of oxygen ions (called super-exchange interactions), and therefore the presence of defects and impurities in surface sites or missing of oxygen ions can spread the spin-disordered region into the core [21,22,23].

Consequently, the decrease in \(\mathrm{M}_{\mathrm{s}}\) with the size reduction, which is one consequence of surface effects, has well been described by considering a model in which the MNPs are composed with a core having bulk-like magnetic properties and a surrounding shell composed of a magnetically disordered layer. As the particle size decreases the impaired magnetic order of the surface layer increasingly determines the magnetic properties of a given MNP. For iron oxide NPs with 0.9 nm magnetic dead layer thickness, Kim et al. reported that 61.4% of spins in 12 nm-sized MNPs are magnetically disordered, while the figure increasingly reaches 99.4% in 2.2 nm-sized MNPs [25]. In addition, the decreasing trend of \(\mathrm{M}_{\mathrm{s}}\) with the size reduction has been reported in many articles [26,27,28,29,30,31,32,33]. Nevertheless, to the best of our knowledge, there are not any experimentally estimated values for the magnetic dead layer thickness of the magnetite NPs with considering the coating ligand-related effect on the measured magnetization.

Several methods have been introduced to synthesize different types of nano-scaled particles, including co-precipitation, hydrothermal, solid state, microwave, microemulsion, and thermal decomposition [34,35,36,37,38,39,40,41]. Among them, co-precipitation method in combination with hydrothermal process is one of the low-cost and most common techniques for preparation of size-controlled MNPs. In this work, the effect of size reduction on the magnetization behavior of variable-sized DM NPs (from 3.1 to 18.9 nm), synthesized by hydrothermal-assisted co-precipitation method, were investigated. Specifically, the magnetic dead layer thickness of DM NPs using real values of \(\mathrm{M}_{\mathrm{s}}\) at high magnetic fields for the magnetic part of DM NPs were estimated by eliminating the weight contribution of non-magnetic coating layer to the whole magnetization. The \(\mathrm{K}_{\mathrm{eff}}\) values were obtained using a law of approach to saturation were, for the first time, modeled by including an extra contribution of dipolar interactions due to the formation of particle clusters with chain-like structure. The obtained results can provide new insight into the modification of magnetic properties of MNPs, especially for applications in which an accurate determination of \(\mathrm{M}_{\mathrm{s}}\) is required.

2 Material and methods

2.1 Materials

Ferrous and ferric salts (FeCl2·4H2O and FeCl3·6H2O), sodium hydroxide, and dextran (Mw \(\approx 10 \mathrm{KDa}\)) were purchased from Sigma Aldrich company. All chemicals were in the analytical grade, and were used without any further purification.

2.2 Synthesis of DM NPs

The DM NPs were synthesized by combination of co-precipitation and hydrothermal methods. The detail of experimental procedure is given elsewhere [42]. Briefly, co-precipitation of iron salts (FeCl2·4H2O and FeCl3·6H2O with molar ratio of 1:2) in the presence of dextran was performed under nitrogen atmosphere and vigorous stirring at 80 \(^\circ{\rm C}\) by raising pH to 12 and then stabilizing the conditions for 1 h. The hydrothermal process was also implemented on the resultant black suspension at different aging temperatures from 120 to 220 \(^\circ{\rm C}\). The synthesis conditions of seven prepared samples are summarized in Table 1.

Table 1 Synthesis conditions, TEM particle size, and some characteristics of the DM NPs
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2.3 Characterization

The X-ray diffraction (XRD) patterns of samples were obtained by the powder X-ray diffraction (XRD, Philips, X-pert) with the Cu-\(\mathrm{K}_{\alpha }\) radiation. The morphology and particle size of the samples were investigated by the transmission electron microscopy (TEM, Phylips- CM120) and field emission scanning electron microscopy (FESEM, ZEISS, SIGMA VP). The changes of residual mass with temperature were investigated under nitrogen atmosphere (30–900 \(^\circ{\rm C}\)) by TG analysis and using Perkin Elmer instrument. The magnetic properties of the samples were evaluated at room temperature by a vibrating sample magnetometer (VSM, Meghnatis Kavir Kashan Co., Iran) instrument.

3 Results and discussion

The TEM images of the DM NPs are shown in Fig. 1. The corresponding size distribution histograms, obtained by measuring over more than 500 NPs from different pictures, and the lognormal fits are also shown in the inset of each image. The obtained average particle sizes are listed in the Table 1. As can be seen, the average sizes of the DM NPs increase by raising the aging temperature in the hydrothermal process. To provide a better characterization, the morphology of DM19 NPs was also investigated by the FESEM analysis and the result is shown in Fig. 2.

Fig. 1
figure 1

TEM images of the DM NPs. The corresponding size distribution histograms and the lognormal fits are shown in the inset of the TEM images

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Fig. 2
figure 2

FESEM images of the DM19 NPs with two different resolutions

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The crystal structure of the samples was determined by XRD analysis and the results are shown in Fig. 3. All reflection peaks in the XRD patterns are in accordance with characteristic peaks (220), (311), (400), (422), (511), (440), (620), and (533) of the cubic spinel structure of Fe3O4 (JCPDS Card No. 75-0449), confirming the formation of the pure magnetite phase in our samples. Moreover, the diffraction peaks steadily become sharper and more intense from the DM3 to DM19 sample as a result of the particle size enhancement, in agreement with TEM results.

Fig. 3
figure 3

XRD spectra of the DM NPs, indicating their characteristic peaks

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The hysteresis loops of DM NPs were obtained by VSM analysis and the saturation magnetization values (\({\mathrm{M}}_{\mathrm{s}}\)) are summarized in Table 1. We note that our largest applied field in VSM measurements could be insufficient for complete saturation of samples, and thus the \(\mathrm{M}_{\mathrm{s}}^{*}\) values were estimated using extrapolation of magnetization (M) versus the inverse of magnetic field strength (\(\frac{1}{H})\) curves. To this end, initial magnetization curves of DM NPs (shown in Fig. 4) were utilized to make M vs. \(\frac{1}{\mathrm{H}}\) curves using their data near saturation. Both the \({\mathrm{M}}_{\mathrm{s}}^{*}\) and \({\mathrm{M}}_{\mathrm{s}}\) values are presented per unit of total mass (\({g}_{\mathrm{Dex}-\mathrm{M}}\)) which comprises the mass of magnetite NPs (\({g}_{\mathrm{Mag}}\)), dextran layer, and absorbed water on NPs surface. Therefore, \(\mathrm{M}_{\mathrm{s}}^{**}\) values (real saturation magnetizations at high magnetic filed strengths for pure magnetite NPs) were estimated using effective magnetic material mass obtained by remanence weight (\(\mathrm{m}_{\mathrm{r}}\)) in TG analyses. The values of \(\mathrm{M}_{\mathrm{s}}^{*}\), and \(\mathrm{M}_{\mathrm{s}}^{**}\) are also summarized in the Table 1. As it can be seen, \(\mathrm{M}_{\mathrm{s}}^{**}\) values are smaller than those of the bulk material in the range \(92-100\,\mathrm{emu}/\mathrm{g}\) [17, 43,44,45]. Moreover, they dramatically decrease with the particle size reduction.

Fig. 4
figure 4

Initial magnetization curves of the DM NPs as a function of applied magnetic field, obtained from the saturated hysteresis loops

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Decline in the saturation magnetization with decrease in the particle size, which has already been observed in other experimental works [46,47,48,49,50], can be originated both from redistribution of cations between two sub-lattices of spinel structure and spin disorder on the particle surface. It has been reported that the distribution of cations between tetrahedral and octahedral sites of spinel structure can significantly affect the magnetization of ferrite NPs [76]. Magnetite (\({\mathrm{Fe}}_{3}{\mathrm{O}}_{4}\)) is an interesting member of the spinel ferrite family with inverse structure as \(\left({\mathrm{Fe}}^{3+}\right) ({\mathrm{Fe}}^{2+}{\mathrm{Fe}}^{3+}){\mathrm{O}}_{4}\) in which the parentheses indicate the tetrahedral and octahedral sites, respectively [4, 76]. Considering the magnetic moments of \({\mathrm{Fe}}^{3+}\) and \({\mathrm{Fe}}^{2+}\) ions as 5 and 4 μB, respectively, the net magnetic moment of each magnetite molecule is simply calculated as 4 μB. Assuming the probable redistribution of a fraction (x) of cations, a partially deviated inverse spinel structure as (\({{\mathrm{Fe}}^{2+}}_{\mathrm{x}}{{\mathrm{Fe}}^{3+}}_{1-\mathrm{x}}\)) (\({{\mathrm{Fe}}^{2+}}_{1-\mathrm{x}}{{\mathrm{Fe}}^{3+}}_{1+\mathrm{x}}\))\({\mathrm{O}}_{4}\) with the net magnetic moment (4 + 2x) μB is formed [51]. Accordingly, even if the size reduction causes a change in the arrangement of cations in the spinel structure of the magnetite NPs, magnetization is expected to increase. In other words, the magnetization reduction can exclusively be attributed to the existence of the magnetically inert layer on the surface of the MNPs in the core–shell model.

Assuming that the magnetic dead layer has (a) a negligible net magnetization and (b) a thickness, t, independent of particle size, D, the saturation magnetization \(\mathrm{M}_{\mathrm{s}}\) is given by Eq. (1)

$$\mathrm{M}_{\mathrm{s}} = {\mathrm{M}}_{\mathrm{s}0} {(1-2\mathrm{t}/{\mathrm{D}})}^{3}$$
(1)

where \(\mathrm{M}_{\mathrm{s}0}\) is the saturation magnetization of bulk material. Equation (1) indicates that the decrease of our experimental values of \(\mathrm{M}_{\mathrm{s}}^{**}\) (i.e., the magnetization corrected for the dextran mass) should be most relevant for the smallest MNPs. The plot of \({\mathrm{M}_{\mathrm{s}}^{**}}^{1/3}\) vs. \(\frac{1}{\mathrm{D}}\) data (Fig. 5) could be well fitted by a linear function as expected from Eq. (1), obtaining the values of \(\mathrm{M}_{\mathrm{s}0}=\) 98.8 \(\mathrm{emu}/\mathrm{g}\) and \(t=\) 6.8 Å for the bulk saturation magnetization and dead layer thickness, respectively (\(\mathrm{R}^{2}=\) 0.9662).

Fig. 5
figure 5

The plot of the cube root of the saturation magnetization (\({\mathrm{M}_{\mathrm{s}}^{**}}^{1/3}\)) versus the inverse of the average diameter (\(1/\mathrm{D}\)) of the DM NPs. The circles and the solid line show the experimental data and a linear fit to them, respectively

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The \(\mathrm{M}_{\mathrm{s}0}\) obtained from the fit is consistent with reported values of bulk magnetite \(92-100\,\mathrm{emu}/\mathrm{g}\) [17, 43,44,45]. The value of the thickness \(t=\) 6.8 Å obtained from the fit is somewhat smaller than the Fe3O4 lattice constant (8.39 Å) [52] but comparable to previous findings t = 6 Å found by Chen et al. [53], and Zheng et al. [54] for \({\mathrm{MnFe}}_{2}{\mathrm{O}}_{4}\) NPs at 300 K. Other \(t\) values consistent with our findings have been reported in different systems, including values of t = 5 Å (at 10 K) [53], and 4.5 Å (at 20 K) [55] for \({\mathrm{MnFe}}_{2}{\mathrm{O}}_{4}\) MNPs, and 10 Å (at 5 K) for \({\mathrm{CoFe}}_{2}{\mathrm{O}}_{4}\) NPs [56]. Interestingly, \({\mathrm{CuFe}}_{2}{\mathrm{O}}_{4}\) NPs with sizes from 10 to 60 nm produced by mechanical milling were reported to have a much larger t \(\approx\) 2.5 nm spin-disordered layer thickness, consistent with the high-energy collisions during mechanosynthesis [57]. The only measured values for the bulk saturation magnetization \(\mathrm{M}_{\mathrm{s}0}\) and thickness \(t\) in \({\mathrm{Fe}}_{3}{\mathrm{O}}_{4}\) MNPs, to the best of our knowledge, have been reported by Caruntu et al. to be \(\mathrm{M}_{\mathrm{s}0}=\) 88.65 \(\mathrm{emu}/\mathrm{g}\) and \(t=\) 2.26 Å at 300 K [58]. These values are smaller than the estimation values presented here because they have obtained from the plot of \(\mathrm{M}_{\mathrm{s}}\) vs. \(\frac{1}{\mathrm{D}}\) data using the equation \(\mathrm{M}_{\mathrm{s}} ={\mathrm{M}}_{\mathrm{s}0} \left(1-6\mathrm{t}/{\mathrm{D}}\right)\) which is the expansion of Eq. 1. However, comparison between samples obtained from different synthesis routes can be misleading, since the thickness of the disordered surface is expected to depend on the details of the energy landscape involved in each synthesis route. We also note that the thickness t obtained from the extrapolation method could be affected by the \(\mathrm{M}_{\mathrm{s}}\) values assumed for the magnetic core at high magnetic filed. Our estimations are based on considering the \(\mathrm{M}_{\mathrm{s}}^{**}\) values obtained from the extrapolation of M vs. \(\frac{1}{H}\) curves and the TG results.

Using the obtained t value and assuming the spherical shape for the DM NPs, the magnetic-disordered content for the DM3 NPs was calculated to be 82.3%. It means that the magnetic order only exists in 17.7% of the particle volume. By increasing the particle size, the magnetic-ordered content increases to 79.9% for the DM19 NPs. Therefore, the effect of surface magnetic disorder on the magnetic behavior of MNPs significantly diminishes with increasing the particle size, consistent with the decrease of the surface/volume ratio as the particle size increases.

The magnetization data as a function of the applied field H can be analyzed in terms of the law of approach to saturation, according to Eq. (2) [59]:

$$\mathrm{M} = \mathrm{M}_\mathrm{S} \left( {1 - \frac{\mathrm{b}}{{\mathrm{H}^{2} }}} \right)$$
(2)

where the parameter b is related to the magnetocrystalline anisotropy, which can be obtained from the M (H) data near saturation by a linear fitting of the \(\frac{\mathrm{M}}{\mathrm{M}_{s}}\) (or) \(\frac{{\mathrm{M}}^{**}}{{\mathrm{M}}_{\mathrm{S}}^{**}}\) in our assumption) versus \(1/{\mathrm{H}}^{2}\) curve. Following the procedure reported in [60], we calculated \(\mathrm{K}_{\mathrm{eff}}\) for a uniaxial magnetic anisotropy by Eq. (3) [61, 62].

$$\mathrm{K}_{\mathrm{eff}}={\upmu }_{0}\mathrm {M}_{\mathrm{S}}{\left(\frac{15}{4}{\mathrm{b}}\right)}^{1/2}$$
(3)

Using the b values obtained from Eq. (2) and the \(\mathrm{M}_{\mathrm{S}}^{**}\) values from Table 1, Eq. (3) yielded the \(\mathrm{K}_{\mathrm{eff}}\) values for DM NPs listed in Table 2. These calculated values are higher than the bulk anisotropy constant (\({\mathrm{K}}_{\mathrm{bulk \,magnetite}}=\) 1.35 \(\times {10}^{4}\mathrm{J}/{\mathrm{m}}^{3}\) [63]), reflecting the surface effects as particle size decreases. This size dependence is consistent with previous findings in iron oxide NPs [58, 60, 63, 64] that reported a rising trend for \(\mathrm{K}_{\mathrm{eff}}\) with size reduction. The size dependency of the \(\mathrm{K}_{\mathrm{eff}}\) is in good agreement with previous results on spherical Fe3O4 MNPs showing a decrease in \(\mathrm{K}_{\mathrm{eff}}\) from 4.74 \(\times {10}^{5} \mathrm{J}/{\mathrm{m}}^{3}\) to 1.11 \(\times {10}^{5}\mathrm{ J}/{\mathrm{m}}^{3}\) with increasing particle size from 6 to 11 nm [58]. Similar changes in the \(\mathrm{K}_{\mathrm{eff}}\) for cubic magnetite NPs have been observed, with a reduction from 77 \(\times {10}^{3} \mathrm{J}/{\mathrm{m}}^{3}\) (20 nm diameter) to 42 \(\times {10}^{3} \mathrm{J}/{\mathrm{m}}^{3}\) (40 nm diameter) [60]. Sarkar and Mandal also reported a decreasing trend in the \(\mathrm{K}_{\mathrm{eff}}\) from 1.84 \(\times {10}^{5} \mathrm{J}/{\mathrm{m}}^{3}\) (7.23 nm diameter) to 1.25 \(\times {10}^{5}\mathrm{ J}/{\mathrm{m}}^{3}\) (11 nm diameter) for chain-like magnetite NPs [63].

Table 2 The calculated \({\mathrm{K}}_{\mathrm{eff}}\) values of the DM NPs
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Assuming that the magnetocrystalline anisotropy Kbulk of the magnetic cores is constant along our series of samples with different particle sizes, additional contributions to the effective magnetic anisotropy \(\mathrm{K}_{\mathrm{eff}}\) come from shape and/or surface anisotropies as well as magnetic dipolar interactions among MNPs [65,66,67]. Since the overall shape of our MNPs does not change significantly along the series of samples as observed from TEM images [42], the contributions from shape anisotropy to \(\mathrm{K}_{\mathrm{eff}}\) can be ignored. The interactions between particles include exchange and dipole–dipole interactions. Exchange interactions act only between particles in contact. Since the dextran-coating makes the MNPs to be separated by at least a distance twice the coating layer thickness, such interactions are no considerable in our system [58, 68, 69]. On the other hand, our experimental determination of the evolution in both \(\mathrm{K}_{\mathrm{eff}}\) and \(\mathrm{K}_{\mathrm{S}}\) was made on non-diluted samples, so dipolar magnetic interparticle interactions could not be negligible in the analysis of the single-particle magnetic anisotropy. Additionally, since the \(\mathrm{M}_{\mathrm{s}}\) values in our MNPs decrease with decreasing particle size (see Table 1) the same trend is to be expected for the strength of dipole–dipole interactions.

The phenomenological expression for the anisotropy Keff originally proposed by Bødker et al. [70] as Eq. (4),

$$\mathrm{K}_{{\text{eff}}} V \cong {\mathrm{K}}_{{{\text{bulk}}}} {\mathrm{V}} + {\mathrm{K}}_{\mathrm{s}} {\mathrm{S}}$$
(4)

where \(\mathrm{K}_{\mathrm{bulk}}\) is the bulk anisotropy energy per unit volume, and \(\mathrm{K}_{\mathrm{s}}\) is the surface density of anisotropy energy. Assuming that the particles are spherical with diameter D, Bødker et al. simplified the Eq. (4) as Eq. (5) which has experimentally been found on many different systems [71, 72].

$$\mathrm{K}_{{{\text{eff}}}} \cong {\mathrm{K}}_{{{\text{bulk}}}} + \frac{{6{\mathrm{K}}_{\mathrm{s}} }}{\left\langle {\mathrm{D}} \right\rangle }$$
(5)

Using symmetry arguments and assuming that surface anisotropy is normal to the particle surface, Bødker et al. [70] showed that for a perfectly spherical particle a zero contribution from surface anisotropy should be expected. We note here that this is an empirical expression, and the hypothesis that the surface contribution to the effective anisotropy is simply additive has yet to be demonstrated.

Figure 6 (a) shows fitting the \(\mathrm{K}_{\mathrm{eff}}\) vs. 1/D data using Eq. (5), from which a value \(\mathrm{K}_{\mathrm{s}}=\) 2.11 \(\times {10}^{5}\mathrm{J}/{\mathrm{m}}^{3}\) was obtained for our DM NPs samples. However, as clearly seen in Fig. 6a, there is a large deviation from the linear behaviour for large 1/D values. These deviations could be explained by deviations from spherical shape that are not included in Eq. (5). Indeed, for different particle morphologies, Eq. (4) should include an additional contribution with a different \(\mathrm{K}_{\mathrm{eff}}\) vs. 1/D slope. However, as previously mentioned no major change in MNPs overall morphology can be observed in our series of size-increased samples, despite using the hydrothermal route [42].

Fig. 6
figure 6

The plot of the effective anisotropy (\(\mathrm{K}_{\mathrm{eff}}\)) vs. 1/D data of the DM NPs. The circles and the solid lines respectively show the calculated values and the linear fits to them using a Eq. (5) and b Eq. (6)

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It should be noted that the original work by Bødker et al., did not consider any contributions from dipolar interactions to the collective behavior of nanoparticles. It is well known that dipolar interactions between MNPs favor the formation of the chain-like clusters in large enough MNPs for which the magnetic dipole–dipole interaction energy can surpass the thermal energy even at room temperature [60, 73,74,75,76,77]. To include these interactions, we modified Eqs. 4 and 5 including a parabolic term as Eq. (6):

$$\mathrm{K}_{{{\text{eff}}}} \cong {\mathrm{K}}_{{{\text{bulk}}}} + \frac{{6{\mathrm{K}}_{s} }}{\left\langle {\mathrm{D}} \right\rangle } + \frac{{{\mathrm{E}}_{l} }}{{\left\langle {\mathrm{D}^{2} } \right\rangle }}$$
(6)

in which \(\mathrm{E}_{\mathrm{l}}\) has the unit of a linear density of anisotropy energy. Figure 6(b) clearly shows that using Eq. (6) the fit of experimental data can be extended to the full range of particle sizes. The last term in Eq. (6) could be understood as originated from the formation of one-dimensional chains of MNPs (i.e., head-to-tail orientation) due to dipolar interactions. Consistent with Eq. (6) the last term is more relevant for larger 1/D values, reflecting the fact that clustering and chain formation is favoured for larger MNPs due to their larger dipolar moment. These interactions originate the extra contribution to the anisotropy in the Eq. (6) and, with this assumption and using Eq. (6) to fit the \(\mathrm{K}_{\mathrm{eff}}\) vs. 1/D data, a value of \(\mathrm{K}_{\mathrm{s}}=\) 1.04 \(\times {10}^{5}\mathrm{J}/{\mathrm{m}}^{3}\) was obtained for our DM NPs samples. The obtained value shows that surface anisotropy gives an important contribution to the effective anisotropy of small MNPs. Surface anisotropy originates from the lack of long-range crystalline order in surface layer where breaking the crystal structure symmetry due to the lower and more variable coordination of cations results in perturbation in crystal field and consequently modification of magnetocrystalline anisotropy [21, 23].

4 Conclusion

We have successfully used a series of magnetite (Fe3O4) nanoparticles increasing sizes from 3.1 to 18.9 nm to investigate size-dependent changes in their magnetic properties. Our results revealed that the decrease in the saturation magnetization \(\mathrm{M}_{\mathrm{s}}\) with decreasing size can be explained by a magnetically disordered surface layer, and fitting the experimental data the values of the magnetic dead layer thickness and \(\mathrm{M}_{\mathrm{s}}\) were estimated as \(\mathrm{t}=\) 6.8 Å and \(\mathrm{M}_{\mathrm{s}}=\) 98.8 \(\mathrm{emu}/\mathrm{g}\), respectively. We used a modified relation for calculating the contribution of the surface anisotropy \(\mathrm{K}_{\mathrm{S}}\) to the effective anisotropy \(\mathrm{K}_{\mathrm{eff}}\) by adding the contributions from dipolar interactions to the original model proposed by Bødker et al., obtaining a good fit for the whole range of MNPs sizes. Our analysis provides a clearer picture of the effects of the spin-disordered surface configuration on the magnetic properties in MNPs of diverse sizes.