Investigation of the Structure and Properties of Magnetic Nanopowders of Magnetite–Maghemite Solid Solutions by SAPNS

Abstract

Nanopowders of the magnetite–maghemite series have been synthesized by coprecipitation from aqueous solutions and by the sol–gel method, and a comparative comprehensive study of their structure has been carried out using X-ray diffraction analysis, scanning electron microscopy, low-temperature nitrogen adsorption, and small-angle polarized neutron scattering. It has been established that the resulting iron oxide nanopowders are porous systems that, depending on the synthesis method, have a one-level, two-level (for powders obtained by aqueous synthesis), or three-level (for powders obtained by the sol–gel method) hierarchical structure organization with different scales and different types of aggregation for each of the structural levels, and the characteristic size for the larger level in both cases is >45 nm. It has been revealed that the magnetic structure of the obtained iron oxide powders, regardless of the synthesis method, consists of superparamagnetic particles with a characteristic radius of magnetic R_M ⁓ 4 nm and magnetic–nuclear cross-correlations R_MN ⁓ 3 nm for powders obtained by the sol–gel method, and R_M ⁓ 5–11 nm and R_MN ⁓ 4–8 nm for powders obtained via aqueous route, depending on the synthesis conditions.

The method of fabricating magnetic nanoparticles by coprecipitation from aqueous salt solutions has long been widely used in practice [1–6]. Its advantages are associated primarily with the simplicity of the technological operations used, the availability of starting materials, and a small environmental impact [1]. The resulting nanoparticles do not need to be washed from organic solvents; they do not contain harmful toxic impurities. This is especially important when using powders in medicine and agriculture. The process can also be scaled up, for example, using microreactors with impinging swirling flows [7]. The main disadvantage of this method, as well as other liquid-phase methods for the synthesis of nanopowders, is a significant dependence of their structure and properties on the synthesis conditions at all stages of this process (selection and ratio of starting components, coprecipitation mode, extraction from the mother liquor, washing, drying, heat treatment). Despite the large number of studies on the production of magnetic nanopowders of magnetite and maghemite, there are no clear instructions in the scientific literature for reliable control over their shape, size, phase composition, superatomic structure, and magnetic properties. It should be noted that during the synthesis by coprecipitation of iron(II, III) salts from aqueous solutions in air without the addition of oxidizing or reducing reagents, it is difficult to obtain nanopowders of magnetic nanoparticles with the composition corresponding to the pure phase of maghemite or magnetite. As our long-term studies have shown, under these conditions, the resulting nanoparticles, as a rule, have the phase composition of magnetite–maghemite solid solutions [8, 9].

Using classical research methods (X-ray powder diffraction analysis, scanning and transmission electron microscopy, IR spectroscopy, low-temperature nitrogen adsorption), it is almost impossible to quantitatively characterize the superatomic structure of nanopowders and the type of aggregation, which is crucial for their reproducible synthesis and practical use. At the same time, such an opportunity is provided by the small-angle X-ray scattering (SAXS) and small-angle neutron scattering (SANS) methods [10–14]. In particular, Danks et al. [15] note the breakthrough role of the SANS method for sol–gel synthesis and study of composites with specified morphological features.

Even greater difficulties arise when it is necessary to characterize the magnetic superatomic structure of nanopowders. To study the structure of nanosized powders, including iron oxides, scanning and transmission electron microscopy, X-ray diffraction analysis, and Mössbauer spectroscopy are usually used. However, these techniques do not provide information about the spatial distribution and nature of spin correlations in the material under study, although this is important for characterizing the structure. At the same time, this information can be obtained by using the SANS method, primarily the small-angle polarized neutron scattering (SAPNS) method [16–18], in particular, by determining the contribution of magnetic-nuclear interference in a nanopowder sample [19–24].

For magnetic nanopowders having the composition of magnetite–maghemite solid solutions with different ratios of Fe²⁺/Fe³⁺ cations, we were able to find only a few similar studies [25–27] devoted to the study of their nuclear and magnetic mesostructure, including the assessment of the magnetic-nuclear component. At the same time, it is the structural features of the magnetic superatomic structure and its relationship with the phase composition and morphology of nanoparticles, their texture and magnetic properties that are of interest for the targeted use of nanoparticles, for example, in medicine and agricultural technologies [28–36].

This study focuses on quantitative characterization of the nuclear and magnetic superatomic structures of magnetite–maghemite nanopowders by the SAPNS method, correlation of these data with the characteristics obtained previously by classical methods, analysis of the influence of synthesis conditions (coprecipitation from aqueous solutions, sol–gel method) of nanopowders, and comparison of them with the characteristics of natural and commercial magnetite nanopowders.

To solve this problem, we relied on the data that we had previously obtained when studying the structure and properties of nanopowders of iron oxides of the maghemite–magnetite series synthesized by coprecipitation from aqueous solutions and the sol–gel method, as well as reference samples, commercial magnetite nanopowder and dispersed natural mineral magnetite [8, 13, 23, 31].

EXPERIMENTAL

Description of research objects. Magnetic iron oxide nanopowders were synthesized by two methods: coprecipitation from aqueous solutions and the sol–gel method. During the synthesis process, both by coprecipitation from aqueous solutions of iron(II, III) chlorides with aqueous ammonia, and by sol–gel route from a solution of iron(III) nitrate in ethylene glycol, various technological techniques were used to shift the synthesis process towards obtaining magnetic nanopowder of one of the phases, magnetite or maghemite (Table 1) [8, 9, 23, 30, 31]. During coprecipitation from aqueous solutions, the reaction mixture was subjected to ultrasonic homogenization or bubbling with argon on slight heating, the precipitate was modified with oleic acid or kept for a long time in the mother solution. In two cases, the synthesis of nanopowders by the sol–gel method was carried out under the same conditions, but heat treatment at high temperatures was carried out both in vacuum and in air.

Table 1.   Specific features of the different synthetic routes of magnetic nanopowders of iron oxides

A comprehensive study of the physicochemical properties of these nanopowders was carried out. Key characteristics of nanopowders are presented in Table 2.

Table 2.   Characteristics of iron oxide nanopowders synthesized by coprecipitation from aqueous solutions of iron salts and by the sol–gel route in comparison with the commercial and natural magnetite and literature data

X-ray powder diffraction analysis of the crystal lattice parameters of the oxides showed that all synthesized iron oxide nanopowders had a phase composition of the magnetite–maghemite series. The powders fabricated by coprecipitation from aqueous solutions had the composition of magnetite–maghemite solid solutions with different Fe²⁺-to-Fe³⁺ ratios in them, while iron oxide nanopowders obtained by the sol–gel method had a composition closest to magnetite or maghemite.

A similar conclusion could be drawn from the IR spectroscopy data. The spectra revealed bands characteristic of both magnetite (580 cm^–1) and maghemite (559, 632 cm^–1) [35]. For a nanopowder with a surface modified by oleic acid, additional bands were found indicating its presence: 2927 (CH₂, asymmetric vibrations), 2852 (CH₂, symmetric vibrations), 1706 (C=O), and 1409 (CH₃) cm^–1 [36].

A commercial powder and a pre-ground natural mineral powder were also studied for comparison and interpretation of data. Both powders corresponded to the phase composition of magnetite (Table 2).

Nanopowders obtained by a sol–gel route consisted of smaller nanoparticles (D_CSD ~ 8–12 nm) compared to powders obtained by coprecipitation (D_CSD ~ 12–19 nm) and to reference samples (D_CSD ~ 61–63 nm).

Powders synthesized by both methods were magnetically soft materials, and their specific residual magnetization increased with an increase in particle size (Table 2).

All synthesized iron oxide nanopowders, regardless of the synthesis method, had a developed surface (S_BET ⁓ 52–88 m²/g) and a large specific mesopore volume (${{V}_{{P/{{P}_{0}}}}}$_→0.99 = 0.26–0.43 cm³/g) compared to commercial and natural magnetite powders (S_BET ⁓ 12 and 2 m²/g, ${{V}_{{P/{{P}_{0}}}}}$_→0.99 = 0.03 and 0.005 cm³/g, respectively). At the same time, differences in synthesis methods have different effects on the pores shape and size, which indirectly indicates a difference in the superatomic structure, morphology, and type of aggregation of nanoparticles in the powders under study.

Experimental techniques. SAPNS measurements were carried out on the KWS-1 small-angle diffractometer (FRM-II reactor, Garching, Germany) operating in a mode close to point geometry. The experiment used a beam of polarized neutrons with an initial polarization P₀ ~ 0.95 and a wavelength λ = 0.5 nm with Δλ/λ = 0.1. The sample–detector distance SD = 8 m made it possible to measure the neutron scattering intensity in the momentum transfer range 0.08 < q < 1 nm⁻¹. Scattered neutrons were recorded with a two-dimensional scintillation position-sensitive detector based on ⁶Li (128 × 128 cells with a spatial resolution of 5 × 5 mm²).

The iron oxide powders were placed in a 1-mm thick quartz cell. The measurements were carried out in a “zero” field (H ⁓ 0) and an external magnetic field H = 1 T applied in the horizontal direction perpendicular to the incident neutron beam. In the experiment, we measured the dependence of the neutron scattering intensity on q for neutron polarization P₀ directed parallel to I⁺(q, $P_{0}^{ + }$) and antiparallel I⁻(q, $P_{0}^{ - }$) to the external magnetic field H. The initial spectra were corrected using a standard procedure, considering scattering by the installation fittings and quartz cell, as well as the laboratory hall background. To separate the isotropic and anisotropic scattering components, radial averaging of the intensity was carried out in the vicinity of the angles α = 0 and π/2 in the detector plane (averaging sector ±2°), which corresponded to the directions along and across the applied magnetic field H. This averaging led to the system of equations:

$$\left\{ {\begin{array}{*{20}{c}} {{{I}^{{ + ~}}}\left( {q,0} \right) = I_{\parallel }^{ + }\left( {\text{q}} \right) = \left\langle {F_{{\text{N}}}^{2}\left( q \right)} \right\rangle } \\ {{{I}^{ + }}\left( {q,\frac{\pi }{2}} \right) = I_{ \bot }^{ + }\left( q \right) = \left\langle {F_{{\text{N}}}^{2}\left( q \right)} \right\rangle + \left\langle {F_{{\text{M}}}^{2}\left( q \right)} \right\rangle - ~2P\left\langle {{{F}_{{\text{N}}}}\left( q \right){{F}_{{\text{M}}}}\left( q \right)} \right\rangle } \\ {{{I}^{{ - ~}}}\left( {q,0} \right) = I_{\parallel }^{ - }\left( {\text{q}} \right) = \left\langle {F_{N}^{2}\left( q \right)} \right\rangle } \\ {{{I}^{ - }}\left( {q,\frac{\pi }{2}} \right) = I_{ \bot }^{ - }\left( q \right) = \left\langle {F_{{\text{N}}}^{2}\left( q \right)} \right\rangle + \left\langle {F_{{\text{M}}}^{2}\left( q \right)} \right\rangle + 2P\varepsilon \left\langle {{{F}_{{\text{N}}}}\left( q \right){{F}_{{\text{M}}}}\left( q \right)} \right\rangle } \end{array}.~} \right.$$

(1)

This system was used to determine the nuclear ⟨$F_{{\text{N}}}^{2}$ (q)⟩, magnetic ⟨$F_{{\text{M}}}^{2}$(q)⟩, and interference ⟨F_N(q)F_M(q)⟩ contributions to the overall scattering intensity I(q) = (I⁺(q, $P_{0}^{ + }$) + I⁻(q, $P_{0}^{ - }$))/2.

Assuming that nuclear scattering is independent of magnetic field, the magnetic contribution to the scattering intensity at Н ⁓ 0 was determined as:

$${{\left\langle {F_{{\text{M}}}^{2}\left( q \right)} \right\rangle }_{{H\sim 0}}} = \frac{3}{2}\left( {{{I}_{{H\sim 0}}}(q) - {{{\left\langle {F_{{\text{N}}}^{2}} \right\rangle }}_{{H\,\, = \,\,1\,\,{\text{T}}}}}} \right).$$

(2)

The obtained scattering intensities were reduced to absolute values by normalizing to the incoherent scattering cross section of plexiglass, considering the detector efficiency and bulk density ρ_b for each powder. The QtiKWS software program was used for data preprocessing [40].

The magnetic properties of iron oxide powders were measured using an experimental setup based on the nuclear magnetic resonance (NMR) method, as described in detail in [41]. The magnetization was determined as the difference between the measured values of induction and strength of a constant and uniform magnetic field in accordance with the classical equation:

$$M = \frac{B}{{{{\mu }_{0}}}} - H,$$

(3)

where В is the magnetic field induction, μ₀ is the magnetic constant, and H is the magnetic field strength.

RESULTS AND DISCUSSION

Figure 1 shows two-dimensional scattering intensities on iron oxide samples obtained for two polarization states of the neutron beam $({{I}^{ - }}$(q, α) and I⁺(q, α)) during measurements in “zero” (H ⁓ 0) and horizontal magnetic fields ( Н = 1 Т), respectively, as well as their difference ΔI_MN(q, α) = ${{I}^{ - }}$(q, α) − I⁺(q, α) (magnetic-nuclear interference term) during measurements in an external magnetic field. For the spectra measured in a magnetic field H = 1 T, the observed scattering pattern is anisotropic with a significant change in the aspect ratios for the two polarization states (Fig. 1). The difference signal ΔI_MN(q, α), where all background contributions are subtracted, shows an angular dependence on α with negligible intensity along the direction of the applied magnetic field H. Separated contributions to scattering (nuclear $\left\langle {F_{{\text{N}}}^{2}\left( q \right)} \right\rangle$, magnetic $~{{\left\langle {F_{{\text{M}}}^{2}\left( q \right)} \right\rangle }_{{H\,\,~ = \,\,1{\text{ T}}}}},$ and interference ⟨F_N(q)F_M(q)⟩_{H = 1 T}) are presented in Figs. 2a–2e. From the above data, it is clear that nuclear scattering $\left\langle {F_{{\text{N}}}^{2}\left( q \right)} \right\rangle$ for all iron oxide nanopowders, with the exception of natural and commercial Fe₃O₄ samples, significantly (almost by an order of magnitude) exceeds magnetic scattering $~{{\left\langle {F_{{\text{M}}}^{2}\left( q \right)} \right\rangle }_{{H~\,\, = \,\,1{\text{ T}}}}}$.

Fig. 1.

Experimental two-dimensional scattering intensities for two polarization states of the incident neutron beam and their difference ΔI_MN(q, α) = ${{I}^{ - }}$(q, α) ^– I⁺(q, α) obtained for iron oxide nanopowders during measurements in the external magnetic field H = 1 T. The square in the center of the detector is the trace from the beam absorber (beamstop).

Fig. 2.

Plots of nuclear dΣ_N(q)/dΩ ($\bigcirc$), magnetic dΣ_M(q)/dΩ ($\square$), and magnetic-nuclear interference dΣ_NM(q)/dΩ ($\diamondsuit$) contributions to the SAPNS cross sections vs. q obtained from the two-dimensional spectra (Fig. 1) for iron oxides (a) γ-Fe₂O₃ (no. 1), (b) γ-Fe₂O₃^–Fe₃O₄ (no. 2), (c) γ-Fe₂O₃–Fe₃O₄@OleicAcid (no. 3), (d) Fe₃O₄–γ-Fe₂O₃ (no. 4), (e) natural Fe₃O₄, and (f) commercial Fe₃O₄. Solid lines show the results of experimental data fitting by Fqs. (4)–(7) and (9).

SAPNS Nuclear Cross Section dΣ_N(q)/dΩ (Н = 1 Т)

The scattering pattern observed for the nuclear component of the scattering cross section dΣ_N(q)/dΩ SAPNS (Fig. 2) is characteristic of porous systems (solid phase–pore) with a disordered structure [10–14, 42–44]. At the same time, the behavior of dΣ_N(q)/dΩ depends on the conditions of synthesis of iron oxide nanopowders.

In particular, for nanopowders with a solid solution composition from the middle of the magnetite–maghemite series γ-Fe₂O₃–Fe₃O₄ (C/P 2, Fig. 2b), including nanopowders after surface modification with oleic acid γ-Fe₂O₃–Fe₃O₄@OleicAcid (C/P 3) (Fig. 2c), the common feature is the presence of two q ranges on the scattering curves, where the behavior of dΣ_N(q)/dΩ obeys the power laws q^–Δ with different values of the exponents Δ = n₁ and n₂, respectively. Near the crossover point q_c (the transition point from one scattering mode to the other), the behavior of dΣ_N(q)/dΩ is satisfactorily described by an exponential dependence (Guinier mode [45]). The observed SANS pattern is typical of scattering on two-level hierarchical structures with different characteristic scales and different types of aggregation for each level [14, 46, 47]. The convex shape of the dΣ_N(q)/dΩ (n₁ > n₂) curves clearly indicates that the inhomogeneities of the subsequent (larger in characteristic size R_c) structural level are formed from smaller inhomogeneities of the previous structural level, i.e. R_c2 > R_c1.

It should be noted that the absence of deviation of the scattering curves dΣ_N(q)/dΩ from the power-law dependence q^–n2 at small q values indicates that the characteristic size of second-level inhomogeneities R_с2 exceeds the maximum size of inhomogeneities R_max scattering from which can be recorded in an experiment with the given resolution of the device. In this case, R_с2 > R_max ⁓ 3.5/q_min ⁓ 45 nm [48].

Based on the above, to analyze the SANS data, we used a unified exponential-power expression that considers the presence of two structural levels in the scattering system [49]:

$$\begin{gathered} \frac{{d{{\Sigma }}\left( q \right)}}{{d{{\Omega }}}} = \mathop \sum \limits_{i = 0}^1 ({{G}_{i}}{\kern 1pt} \exp \left( { - \frac{{{{q}^{2}}R_{{gi}}^{2}}}{3}} \right) + ~{{B}_{i}}{\text{exp}}\left( { - \frac{{{{q}^{2}}R_{{g\left( {i - 1} \right)}}^{2}}}{3}} \right) \\ \times \,\,{{\left[ {\frac{{{{{\left( {{\text{erf}}\left( {\frac{{q{{R}_{{gi}}}}}{{\sqrt 6 }}} \right)} \right)}}^{3}}}}{q}} \right]}^{{{{n}_{i}}}}}. \\ \end{gathered} $$

(4)

The summation is carried out over the number of structural levels. In general, this expression determines the presence of four free parameters for each structural level, such as the Guinier prefactor G_i, the gyration radius R_gi, the power prefactor B_i, and the exponent n_i.

In turn, for the nanopowder that is closest in composition to maghemite γ-Fe₂O (C/P 1) (Fig. 2a) and, conversely, the one closest to magnetite, among the nanopowders obtained by coprecipitation of Fe₃O₄–γ‑Fe₂O₃ (C/P 4) (Fig. 2d), two q ranges are also observed on the scattering curves, where the dΣ_N(q)/dΩ behavior is described by a power-law dependence q^–Δ with different exponents Δ = n₁ and n₂. However, the closeness of the exponent n₂ to 1 indicates that the observed SANS occurs in systems consisting of randomly oriented, highly elongated anisodiametric (nonspherical) inhomogeneities of radius R_c and length L [50–52]. Therefore, their corresponding Guinier region should include two ranges of q. The absence of the second Guinier region in this case indicates that the length L > R_max = 45 nm. The range q > q_c, where the behavior of the scattering cross section dΣ_N(q)/dΩ is described by the power-law dependence q^–n1, corresponds to the Porod regime [53].

Thus, when analyzing the dΣ_N(q)/dΩ curves for γ‑Fe₂O₃ (C/P 1) (Fig. 2a) and Fe₃O₄–γ-Fe₂O₃ (C/P 4) (Fig. 2d), a generalized empirical Guinier–Porod model was used [54]:

$$\left\{ {\begin{array}{*{20}{c}} {\frac{{d{{\Sigma }}\left( q \right)}}{{d{{\Omega }}}} = \frac{G}{{{{q}^{{{{n}_{2}}}}}}}\exp \left( { - \frac{{{{q}^{2}}R_{g}^{2}}}{{3 - {{n}_{2}}}}} \right)~\,\,{\text{at}}\,\,q < {{q}_{{\text{c}}}}} \\ {\frac{{d{{\Sigma }}\left( q \right)}}{{d{{\Omega }}}} = \frac{B}{{{{q}^{{{{n}_{1}}}}}}}{\text{ at }}~q > {{q}_{{\text{c}}}}~} \end{array}} \right.,$$

(5)

where (3 – n₂) is the dimensional factor; G is the Guinier prefactor; R_g is the gyration radius, which is R_g = R_с^/$\sqrt 2$ for highly elongated objects; B is the power prefactor; n₁ is the exponent.

The behavior of the SANS cross section dΣ_N(q)/dΩ for natural Fe₃O₄ (Fig. 2e) is satisfactorily described by two power-law dependences:

$$\frac{{d{{\Sigma }}\left( q \right)}}{{d{{\Omega }}}} = \frac{{{{B}_{2}}}}{{{{q}^{{{{n}_{2}}}}}}} + \frac{B}{{{{q}^{{{{n}_{1}}}}}}} + {{I}_{{{\text{inc}}}}},$$

(6)

which also corresponds to scattering on a disordered structure consisting of two types of inhomogeneities with different characteristic scales and different types of aggregation. At the same time, it is not possible to estimate the characteristic size R_с1 of inhomogeneities of the first type from the available data due to the overlap in the corresponding q range of scattering from large-scale inhomogeneities of the second type, the characteristic size of which is R_с2 > R_max = 45 nm.

For commercial Fe₃O₄, the observed SANS (Fig. 2f) in the entire q range is described only by a power-law dependence q^–n, which corresponds to scattering on magnetite particles with R_с > R_max = 45 nm.

The results obtained by convolution of Eqs. (4)–(6) with the installation resolution function and their least squares processing (LSM) are presented in Fig. 2 and Table 3.

Table 3.   Superatomic structure parameters of synthesized iron oxide nanopowders and natural and commercial magnetite determined from analysis of the nuclear component SAPNS dΣ_N(q)/dΩ

According to the data obtained (Table 3), the γ‑Fe₂O₃–Fe₃O₄ (C/P 2) nanopowder consists of almost smooth particles with a characteristic size R_с1 ⁓ 7 nm, from which mass-fractal clusters with a dimension D_M = 2.33 are formed at the second structural level. At the same time, for the γ-Fe₂O₃–Fe₃O₄@Oleic-Acid (C/P 3) nanopowder modified with oleic acid, from particles of the first structural level with characteristic dimensions R_с1 ⁓ 6 nm, having a developed fractal surface with a dimension D_S = 2.55, mass-fractal clusters with dimension D_M = 2.41 are formed at the second structural level. A similar picture of structure formation in the type of hierarchical fractal structures was previously observed for nanopowders of iron oxides Fe₃O₄ (S/G 5) and γ-Fe₂O₃ (S/G 6), synthesized by the sol–gel method [23].

In turn, nanopowders of composition, both practically corresponding to the composition of maghemite γ-Fe₂O₃ (C/P 1), and closer to magnetite Fe₃O₄–γ‑Fe₂O₃ (C/P 4) (Table 3), consist of randomly oriented strongly elongated nonspherical (anisodiametric) particles with a gyration radius R_c ⁓ 5 and 6 nm, respectively. In the iron oxide γ-Fe₂O₃ nanopowder (C/P 1) these are practically smooth particles, while in the Fe₃O₄–γ-Fe₂O₃ (C/P 4) nanopowder, the particles have a developed fractal surface with the dimension D_S = 2.30. It should be noted that the characteristic particle sizes of the first structural level of all synthesized iron oxide nanopowders, obtained from the analysis of SANS data, generally correlate with the average crystallite sizes (D_CSD) obtained by XRD (Table 2).

Analysis of SANS data (Table 3) showed that commercial Fe₃O₄ powder consists of large-scale particles (R_с > 45 nm) having a diffuse interface surface (n > 4) [55], while natural Fe₃O₄ contains inhomogeneities of two types: large-scale inhomogeneities (R_c > 45 nm) with an almost smooth interface (D_S ⁓ 2) and smaller scale inhomogeneities with a developed fractal surface (D_S = 2.46).

SAPNS Magnetic Cross Section dΣ_M(q)/dΩ (H = 1 T)

For superparamagnetic nanoparticles upon magnetization saturation, magnetic scattering becomes completely anisotropic, while nuclear scattering remains isotropic. As can be seen from Fig. 2, the magnetic scattering cross section dΣ_M(q)/dΩ is statistically resolvable for natural and commercial Fe₃O₄, as well as for the γ-Fe₂O₃–Fe₃O₄@OleicAcid (C/P 3) solid solution. In this case, the behavior of magnetic scattering dΣ_M(q)/dΩ is satisfactorily described by power-law dependences:

$$\frac{{d{{\Sigma }}\left( q \right)}}{{d{{\Omega }}}} = \frac{{{{B}_{2}}}}{{{{q}^{4}}}} + \frac{B}{{{{q}^{2}}}} + {{I}_{{inc}}},$$

(7)

which corresponds to scattering on two types of spin correlations. The term q^–4 corresponds to scattering on large-scale magnetic fluctuations, and q^–2 is characteristic of scattering on spin correlations of the type of critical fluctuations [17].

For the remaining synthesized iron oxide nanopowders, the magnetic scattering cross section dΣ_M(q)/dΩ is small (no more than 10%) compared to nuclear scattering dΣ_M(q)/dΩ and is statistically resolvable only at small q values, q < 0.15 nm^–1, which corresponds to scattering on large-scale magnetic fluctuations that appear when saturation magnetization of the material is achielved. In this regard, a quantitative analysis of magnetic scattering dΣ_M(q)/dΩ for synthesized iron oxide samples is practically impossible. At the same time, this problem can be solved by analyzing the interference contribution dΣ_MN(q)/dΩ to the total SAPNS, which is determined by the product of the magnetic and nuclear scattering amplitudes, i.e. by the first, rather than the second, as in the case of intensity measurements, degree of magnetic scattering amplitude, which determines the higher sensitivity of the method [17].

SAPNS Magnetic–Nuclear Cross Section dΣ_MN(q)/dΩ (Н = 1Т)

Analysis of the magnetic–nuclear interference scattering contribution to the overall SAPNS in the α = π/2 direction perpendicular to the applied magnetic field H = 1 T (Fig. 2) has demonstrated that the SAPNS dΣ_MN(q)/dΩ curves for nanopowders nearly corresponding to the maghemite γ-Fe₂O₃ composition (C/P 1) (Fig. 2a) and for solid solutions from the middle of the γ-Fe₂O₃–Fe₃O₄ series (C/P 2) (Fig. 2b), and shifted to magnetite Fe₃O₄–γ-Fe₂O₃ (C/P 4) (Fig. 2d) are satisfactorily fitted by the squared Lorentzian:

$$\frac{{d{{{{\Sigma }}}_{{{\text{MN}}}}}\left( q \right)}}{{d{{\Omega }}}}\left( q \right) = \frac{A}{{{{{\left( {{{q}^{2}} + {{\kappa }^{2}}} \right)}}^{2}}}},$$

(8)

where A is a free parameter, and κ = 1/R_MN is the inverse correlation radius of the magnetic-nuclear contrasting and, accordingly, scattering region. In coordinate representation, this expression corresponds to scattering on a spin correlator $\left\langle {{{S}_{i}},{{S}_{j}}} \right\rangle$ that decays exponentially with distance r:

$$\left\langle {{{S}_{i}}{{S}_{j}}~} \right\rangle \propto {\text{exp}}\left( { - \frac{r}{{{{R}_{{{\text{MN}}}}}}}} \right).$$

(9)

In the case of a nanopowder fabricated by the same method as the C/P 2 nanopowder, but with the surface modified with oleic acid γ-Fe₂O₃–Fe₃O₄@Oleic Acid (C/P 3) (Fig. 2c), the observed magnetic-nuclear interference scattering is described by the sum of two terms:

$$\frac{{d{{\Sigma }_{{{\text{MN}}}}}\left( q \right)}}{{d\Omega }}\left( q \right) = \frac{{{{A}_{2}}}}{{{{q}^{2}}}} + \frac{{{{A}_{1}}}}{{{{{\left( {{{q}^{2}} + {{\kappa }^{2}}} \right)}}^{2}}}},$$

(10)

where the first term ~q^–2 corresponds to the scattering on large-scale spin density fluctuations [17].

The results were obtained by convolution of Eqs. (7) and (9) with the setup resolution function and their processing using the LSM. The results obtained are presented in Fig. 2 and in Table 4.

Table 4.   Characteristic sizes of iron oxide particles determined from analysis of XRD and SAPNS data as compared with remanent magnetization values

According to Table 4, the characteristic sizes R_MN of magnetic-nuclear correlations obtained from the analysis of SAPNS are smaller than the characteristic sizes R_с of nuclear inhomogeneities (Table 3). However, the R_MN size correspond to the average sizes of magnetic-nuclear correlations, rather than to its upper limit, as in the case of the characteristic size R_с of nuclear correlations in the expressions used in the analysis of dΣ_N(q)/dΩ.

SANS Magnetic Cross Section dΣ_M(q)/dΩ (H ⁓ 0 T)

Assuming that nuclear scattering is isotropic and independent of the applied magnetic field, we determined the magnetic contribution ${{\left\langle {F_{{\text{M}}}^{2}\left( q \right)} \right\rangle }_{{H\sim 0}}}$ to the SANS intensity in the case of H ⁓ 0 by Eq. (2). The corresponding cross sections dΣ_M(q)/dΩ of the magnetic scattering for the synthesized iron oxide nanopowders are shown in Fig. 3. As can be seen, a statistically resolvable magnetic scattering dΣ_M(q)/dΩ is observed for all synthesized iron oxide nanopowders. In particular, for nanopowders γ-Fe₂O₃ (C/P 1) (Fig. 3a), γ-Fe₂O₃–Fe₃O₄ (C/P 2) (Fig. 3b), and Fe₃O₄–γ‑Fe₂O₃ (C/P 4) (Fig. 3d), the SAPNS dΣ_M(q)/dΩ behavior, as in the case of the analysis of magnetic-nuclear interference for these samples (Figs. 2a, 2b, 2d), is satisfactorily described by a squared Lorentzian.

Fig. 3.

SAPNS magnetic cross sections dΣ_M(q)/dΩ at H ⁓ 0 vs. q for iron oxides (a) γ-Fe₂O₃ (no. 1), (b) γ-Fe₂O₃^–Fe₃O₄ (no. 2), (c) γ-Fe₂O₃–Fe₃O₄@OleicAcid (no. 3), and (d) Fe₃O₄–γ-Fe₂O₃ (no. 4). Solid lines show the results of experimental data fitting by Eqs. (7) and (9).

In the case of γ-Fe₂O₃–Fe₃O₄@OleicAcid (C/P 3) (Fig. 3c) nanopowder modified with oleic acid, an equation proportional to ~q^–4 was used, which indicates the presence of large-scale spin density fluctuations in the scattering system.

The results presented in Fig. 3 and in Table 4 were obtained using the procedure described earlier.

Previously [23], we have comprehensively studied the supermolecular structure with an assessment of the nuclear and magnetic components for nanopowders synthesized by a sol–gel route. Features of the synthesis of these nanopowders and their properties, studied by classical methods, are given in Tables 1 and 2, and the parameters of the superatomic structure obtained from the analysis of the nuclear and magnetic-nuclear components are presented in Tables 3 and 4.

The comparative analysis by the SAPNS method of the nuclear superatomic structure of the synthesized iron oxide powders carried out in this work has demonstrated that they are porous systems, which, depending on the synthesis method, have a one-level (for C/P 1 and C/P 4 powders), two-level (for C/P 2 and C/P 3 powders), or three-level (for powders obtained by the sol–gel method) hierarchical structure organization with a different characteristic scale and type of aggregation for each of the structural levels, and the characteristic size R_c for the larger level in both cases exceeds 45 nm.

In particular, powders prepared by coprecipitation from aqueous solutions, both practically corresponding to the composition of maghemite γ-Fe₂O₃ and those closer to magnetite Fe₃O₄–γ-Fe₂O₃, consist of randomly oriented, highly elongated nonspherical (anisodiametric) particles with a characteristic size R_с1 ⁓ 5 and 6 nm, respectively. The γ-Fe₂O₃ powder consists of practically smooth particles, while in the powder closer in composition to magnetite Fe₃O₄–γ‑Fe₂O₃, the particles have a developed fractal surface with the dimension D_S1 = 2.30. The γ-Fe₂O₃–Fe₃O₄ solid solution powder consists of almost smooth particles with a characteristic size of R_с1 ⁓ 7 nm, which form mass-fractal clusters with a dimension of D_M2 = 2.33 at the second structural level. At the same time, for the γ-Fe₂O₃–Fe₃O₄ solid solution powder modified with oleic acid, particles of the first structural level with characteristic dimensions R_с1 ⁓ 6 nm have a developed fractal surface with dimension D_S1 = 2.55, which form mass-fractal clusters with dimension D_M2 = 2.41 at the second structural level.

The first structural level of the γ-Fe₂O₃ powder obtained by the sol–gel method consists of almost smooth particles with a characteristic size of R_с1 ⁓ 6 nm, which at the second structural level aggregate into mass-fractal clusters with the dimension D_M2 = 2.74 and the upper limit of self-similarity R_с2 ⁓ 16 nm, which form mass-fractal aggregates with the dimension D_M3 = 2.35 at the third structural level. At the same time, the Fe₃O₄ powder obtained by the sol–gel method at the first structural level also consists of small particles (according to SAXS data, R_с1 ⁓ 2 nm) with an almost smooth surface, which at the second structural level aggregate into surface-fractal clusters with the dimension D_S2 = 2.46 and the upper limit of self-similarity R_с2 ⁓ 5 nm, forming surface-fractal aggregates with the dimension D_S3 = 2.82 at the third structural level.

A detailed analysis of the SAPNS data made it possible to establish that the magnetic structure of the obtained iron oxide powders, regardless of the synthesis method, consists of superparamagnetic particles with a characteristic magnetic R_М ⁓ 4 nm and magnetic-nuclear cross-correlation radius R_MN ⁓ 3 nm for powders obtained by the sol–gel method, and R_M ⁓ 5–11 nm and R_MN ⁓ 4–8 nm for powders obtained by coprecipitation from aqueous synthesis, depending on the production conditions. In iron oxide powders synthesized by the sol–gel method, there are also spin correlations of the short-range order type between these superparamagnetic particles with interparticle magnetic correlation radii ζ_M ⁓ 16 and 25 nm for Fe₃O₄ and γ-Fe₂O₃, respectively [23].

CONCLUSIONS

The structures of iron oxide powders synthesized by coprecipitation from aqueous solutions and by the sol–gel route have been comprehensively studied by XRD, SEM, low-temperature nitrogen adsorption, and small-angle polarized neutron scattering methods.

It has been established that the synthesized iron oxides are porous systems; depending on the synthesis method, they differ in the hierarchical organization of the structure with a characteristic scale and type of aggregation different for each of the structural levels. Powders obtained by coprecipitation from aqueous solutions are characterized by a one-level or two-level hierarchical organization of the structure, and powders obtained by the sol–gel method are characterized by a three-level structure. The characteristic size R_c for the larger level in both cases exceeds 45 nm. It has been found that the first structural level of γ-Fe₂O₃, Fe₃O₄ powders obtained by the sol–gel method, as well as powders that practically correspond to the composition of maghemite γ-Fe₂O₃ and the γ-Fe₂O₃–Fe₃O₄ solid solution obtained by coprecipitation, consists of almost smooth particles with a characteristic size R_с1 ⁓ 6, 2, 5, and 6 nm, respectively. At the same time, the first structural level of powders closer in composition to magnetite Fe₃O₄–γ-Fe₂O₃, and of the γ-Fe₂O₃–Fe₃O₄ solid solution modified with oleic acid obtained by coprecipitation consists of particles with a developed fractal surface with the dimension D_S1 = 2.30 and 2.55, respectively.

The second structural level was determined only for γ-Fe₂O₃, Fe₃O₄ powders obtained by the sol–gel method, as well as for γ-Fe₂O₃–Fe₃O₄ and γ-Fe₂O₃–Fe₃O₄ solid solution powders modified with oleic acid obtained by coprecipitation. It has been established that the primary particles of γ-Fe₂O₃ powders obtained by the sol–gel method, as well of the powders of the γ-Fe₂O₃–Fe₃O₄ solid solution and the γ-Fe₂O₃–Fe₃O₄ solid solution modified with oleic acid synthesized by coprecipitation, form at the second structural level mass-fractal clusters with dimensions D_M2 = 2.74, 2.33, and 2.41, respectively, while primary particles of the Fe₃O₄ powder obtained by the sol–gel method form surface-fractal clusters with the dimension D_S2 = 2.46.

The third structural level is observed only for γ‑Fe₂O₃ and Fe₃O₄ powders obtained by the sol–gel method; at these level, mass-fractal aggregates with the dimension D_M3 = 2.35 and surface-fractal aggregates with the dimension D_S3 = 2.82, respectively, are formed.

Analysis of SAPNS data has demonstrated that the magnetic structure of the fabricated iron oxide powders, regardless of the synthesis method, consists of superparamagnetic particles with the characteristic magnetic R_M ~ 4 nm and magnetic-nuclear cross-correlation radius R_MN ~ 3 nm for powders obtained by the sol–gel method, and R_M ~ 5–11 nm and R_MN ~ 4–8 nm for powders obtained by coprecipitation from aqueous solutions, depending on the synthesis conditions.