Synthesis-Dependent Structural and Magnetic Properties of Monodomain Cobalt Ferrite Nanoparticles

Abstract

This research examines the structural and magnetic properties of monodomain cobalt ferrite nanoparticles with the formula (Co_1−xFe_x)^A[Fe_2−xCo_x]^BO₄. The particles were synthesized using various methods, including coprecipitation (with and without ultrasonic assistance), coprecipitation followed by mechanochemical treatment, microemulsion, and microwave-assisted hydrothermal techniques. The resulting materials were extensively analyzed using X-ray diffraction (XRD) and magnetic measurements to investigate how different synthesis methods affect the structure and cation distribution in nanoscale CoFe₂O₄. For particles ranging from 15.8 to 19.0 nm in size, the coercivity showed a near-linear increase from 302 Oe to 1195 Oe as particle size increased. Saturation magnetization values fell between 62.6 emu g⁻¹ and 74.3 emu g⁻¹, primarily influenced by the inversion coefficient x (0.58–0.85). XRD analysis revealed that as the larger Co²⁺ cations migrate from B- to A-sites (decreasing x), the lattice constants and inter-cation hopping distances increase, while the average strength of super-exchange interactions decreases. This study establishes a connection between the magnetic properties of the synthesized samples and their structural features. Importantly, this research demonstrates that careful selection of the synthesis method can be used to control the magnetic properties of these nanoparticles.

1. Introduction

Cobalt ferrite belongs to the class of spinel oxides, a large group of cubic symmetry crystals with the general formula AB₂X₄ whose name originates from the MgAl₂O₄ mineral of the same crystal structure. A and B denote to two- and tri-valent cations, respectively, and X is a two-valent anion (O, S, Se, Te). Nanoparticles of transition-metal spinel ferrites have been the subject of increasing interest due to their magnetic and catalytic properties, different from those of bulk materials [1,2,3]. Normally, cobalt ferrite has an inverse spinel structure [4]. Transition-metal ferrite nanoparticles find applications in magnetic storage systems [5,6], magnetic nanocomposite systems [7], photocatalysis [8], signal processing [9], technology as a part of transformer cores [10], high-definition TV deflection yokes [11], telecommunication systems [12], and medicine for drug delivery [1,13], hyperthermia [14,15,16], and magnetic resonance imaging [17]. The properties of nanoferrites, and thus their inherent application, largely depend on the size of the nanoparticles and the cation distribution between the A- and B-sites. As the particle size decreases, the surface area increases, improving the catalytic properties. In such a scenario, each nanoparticle becomes a single magnetic domain. The magnetic properties of nanoferrites are directly related to cation distribution. However, when speaking of a single-domain range of sizes, magnetization also depends on size [18]. The structure and size of nanoparticles are primarily influenced by the conditions of the synthesis procedure [19,20,21,22,23].

CoFe₂O₄, a well-known hard magnetic material, possesses moderate saturation bulk magnetization (M_s) at room temperature, high chemical stability, good electrical insulation, and significant mechanical hardness [24]. The saturation magnetization of CoFe₂O₄ nanoparticles mainly depends on the cation distribution; i.e., it decreases with the increase in cation inversion. On the other hand, the saturation magnetization value increases with increasing nanoparticle size. Annealing treatment has significant influence on the magnetic properties of ferrite nanoparticles since their M_s values rise, although the cation inversion effect is more pronounced [25]. Nanoparticles sintered at ~1300 K have magnetization that reaches a value such as that in the bulk (single crystal) material [26,27]. Otherwise, the saturation magnetization of CoFe₂O₄ nanoparticles is commonly lower than the one of the bulk material, mainly due to the existence of a “dead” layer on the nanoparticle surfaces (a layer with a depleted magnetic contribution due to disordered surface structure). The effect of the disordered surface layer is more pronounced if the nanoparticles are smaller. Cobalt ferrite has been studied in detail due to its high coercivity connected with high magnetic anisotropy which is almost ten times greater than that of NiFe₂O₄ and Fe₃O₄ [28,29]. In the case of bigger, multidomain nanoparticles, coercivity is relatively low. When the size of the nanoparticles decreases and reaches the characteristic limit value, i.e., when the whole particle becomes one magnetic domain, the coercive field reaches its maximum [30]. In the case of smaller, monodomain nanoparticles, coercivity and anisotropy increase with particle size but also depend on the synthesis method and the regularity of the nanoparticle structure.

Cobalt ferrite has an intriguing relationship between size, structural, and magnetic properties. To investigate the influence of the synthesis route on the structural features and cation distribution in nanosized monodomain cobalt ferrite, the samples were prepared with several commonly used synthesis techniques, namely coprecipitation, ultrasonically assisted coprecipitation, coprecipitation followed by mechanochemical treatment, microemulsion, and microwave-assisted hydrothermal synthesis. This paper contributes to a better understanding of the relationship between structural and magnetic properties, hopefully allowing the control of magnetic properties through the management of the synthesis method.

2. Experimental Procedures

All chemicals (iron (III) chloride hexahydrate (FeCl₃·6H₂O, 98%), cobalt (II) chloride hexahydrate (CoCl₂·6H₂O, 98%), sodium hydroxide (NaOH, >97%), cetyltrimethylammonium bromide (CTAB, >98%), n-butanol (99.8%), n-hexanol (>99%), iron(III) nitrate nonahydrate (Fe(NO₃)₃·9H₂O, >99.95), cobalt(II) nitrate hexahydrate (Co(NO₃)₂·6H₂O, >99.99), ammonium hydroxide solution (28% NH₃ in water), and absolute ethanol) were obtained from Sigma-Aldrich (St. Louis, MO, USA) (p.a. quality) and used without additional purification.

Stoichiometric cobalt ferrite nanomaterials were synthesized with five different methods, i.e., coprecipitation (CO), ultrasonically assisted coprecipitation (US-CO), coprecipitation followed by mechanochemical treatment (MC-CO), microwave-assisted hydrothermal (MW-HT), and microemulsion (ME) synthesis, as presented in previous research [31].

The water solution (50 mL) of 0.02 mol FeCl₃∙6H₂O and 0.01 mol CoCl₂∙6H₂O was heated until boiling point was reached and then an excess of 1 M NaOH solution was added. The pH value of the suspension was approximately 11. The reaction mixture was heated for 1 h. After cooling at room temperature, the precipitate was collected and rinsed with deionized water to remove the excess NaOH (neutral pH). One part of the dried product was pulverized in an agate mortar and annealed in an electrical furnace at 450 °C for 1 h (sample CO). The second part was mechanochemically treated in a planetary ball mill (PM100CM, Retsch GmbH, Düsseldorf, Germany). A hardened-steel vial (500 cm³ volume) filled with 10 hardened-steel balls (8 mm in diameter) was used as the milling medium. The powder was ball-milled for 10 h at 500 rpm with a ball-to-sample mass ratio of 20:1 (sample MC-CO).

Sample US-CO was obtained in the same manner as sample CO, with the aid of ultrasound.

MW-HT synthesis was conducted using stoichiometric amounts of chloride salts of Fe³⁺ and Co²⁺ dissolved in deionized water. A small excess of ammonia was added (pH value ≈ 10). The mixture vessels were placed in an HPR-1000/10S high-pressure segmented rotor and heated in the microwave digester (ETHOS 1, Advanced Microwave Digestion System, MILESTONE, Milan, Italy). The power of the microwave irradiation was set in the range of 0–1000 W, with linear heating of the mixture at 20 °C/min. The mixture was then heated at 200 °C for 20 min at a maximum pressure of 100 bars. The prepared particles were separated from the solution using vacuum filtration. The precipitate was then washed several times with deionized water to remove the excess chlorides. The synthesized nanoparticles were dried at 70 °C for one day. The powder was pulverized in an agate mortar (sample MW-HT).

The ME sample was obtained using CTAB as the surfactant, n-butanol as the cosurfactant, n-hexanol as the oil phase, and an aqueous solution as the water phase. The microemulsion was composed of 15 wt% hexanol and 45 wt% aqueous solution. The surfactant-to-cosurfactant ratio was 60:40. Identical microemulsion systems with different aqueous phases were prepared. The first microemulsion contained aqueous solutions of a stoichiometric amount of Fe³⁺ and Co²⁺ nitrates, while the second one contained an aqueous solution of ammonia. The microemulsions were heated at 90 °C for 1 h. The pH value was about 11. Then, the water/ethanol mixture was added to the final microemulsion and precipitated by centrifugation. To remove residues of oil and surfactant, the precipitate was washed with absolute ethanol several times, separated from the solution by vacuum filtration, and dried at 70 °C. The precipitate was pulverized in an agate mortar and annealed in an electrical furnace at 450 °C for 1 h.

XRD spectra were recorded at 300 K with a Rigaku SmartLab automated powder X-ray diffractometer with Cu K_α1 (λ = 1.54059 Å) radiation equipped with a D/teX Ultra 250 stripped 1D detector in the XRF reduction mode. The measurement range was 15–90° 2θ with a step of 0.01° and a scan speed of 2° min⁻¹. Obtained spectra were analyzed with the FullProf Suite program [32].

Magnetic measurements were performed using a Quantum Design Physical Property Measurement System (PPMS) equipped with a 9 T superconducting magnet and a vibrating sample magnetometer (VSM). Temperature dependences of the magnetization, M(T), were measured upon heating in the zero-field cooled (ZFC) and the field-cooled (FC) regime at 100 Oe from 5 to 300 K. The hysteresis loop of magnetization, M(H), was measured in the field range ±90 kOe at 300 K.

3. Results and Discussion

3.1. Structure of Cubic Spinels

In the spinel structure, with the space group Fd$\overline{3}$m (No. 227), divalent anions X²⁻ (X = O, S, Se, Te) form a close-packed face-centered cubic lattice (f.c.c.). One elementary unit cell contains 56 atoms (8 formula units) and a primitive tetrahedral cell contains 14 atoms—two formula units. Between the 32 anions in the elementary cell, there are 64 tetrahedrally coordinated interstitial positions and 32 octahedrally coordinated interstitial sites. Divalent A²⁺ cations are in one-eighth of the possible tetrahedral A-sites, and the trivalent B³⁺ cations occupy one-half of the possible octahedrally coordinated B-sites.

The spinel is called normal (A)[B₂]O₄ when A²⁺ occupies only tetrahedral positions and B³⁺ are positioned in octahedral sites. The anions are not in the ideal f.c.c. positions, depending on the difference in the effective ionic radii of the A- and B-cations, except that the spinel structures can be partially or completely inverse, with the general formula (A_1−xB_x)[B_2−xA_x]O₄, where x is a degree of inversion; i.e., a part of the cations from the octahedral positions migrated to the tetrahedral sites. An alternative way to view the spinel structure is to observe the overall cation sublattice that possesses the same structure as the cubic MgCu₂ C15 Laves phase and the same length of unit cell as the f.c.c. anion lattice [33]. For a precise definition of the spinel unit cell, except the length of the unit cell, a, it is also necessary to know the anion (or deformation) parameter u.

Description of the atomic positions in the spinel structure is dependent on the choice of setting for the origin in the Fd$\overline{3}$m space group. Two different Wyckoff positions with point symmetries $\overline{4}$3m and $\overline{3}$m are possible choices for the unit-cell origin. Moreover, the origin can be assigned to either a vacant site or an occupied lattice site. If the origin is an occupied A-site, i.e., A²⁺ cation (setting 1), its Wyckoff position will be 8a (0, 0, 0), and then the cation in the B-site has Wyckoff position 16d (5/8, 5/8, 5/8) and the Wyckoff position of the anions is 32e (point symmetry 3m), with fractional coordinates (u, u, u). For an ideal close-packed face-centered cubic anion arrangement, u = 3⁄8 (=0.375) for the $\overline{4}$3m origin. The given fractional coordinates are along the body diagonal of the unit cell. On the basis of crystal symmetry, programs for Rietveld refinement generate coordinates of all other atoms in the unit cell and corresponding diffraction pattern. If the origin is on inversion center at the octahedral vacancy (setting 2) with point symmetry $\overline{3}$m, all the lattice sites are translated by the vector [−1⁄8a,−1⁄8a,−1⁄8a] and the fractional coordinates of the symmetry points will be changed: the Wyckoff position of the A²⁺ cation will be 8a (1/8, 1/8, 1/8), the cation in the B-site has Wyckoff position 16d (1/2, 1/2, 1/2), and the fractional coordinates of the anions in 32e are u = 1/4 (=0.250) [27]. Programs for the crystallographic calculation PowderCell 2.4 [34] and MAUD (Material Analysis Using Diffraction) [35,36] support the first configuration, while FullProf Suite [32,37,38] supports the second one.

The VESTA (Visualization for Electronic and STructural Analysis) program [39] can display both variants of the spinel unit cell (knowing the corresponding “cif”-files) with ions as spheres or with cations in the tetrahedral and octahedral voids, for example. Nevertheless, the schematic representation of the unit cell is often used in papers because it allows for easier observation of the geometrical relationships within the cell.

A conventional schematic representation of the elementary unit cell of the cubic spinel structure with the origin in an occupied A-site (Fd$\overline{3}$m: setting 1) is presented in Figure 1, left. This simple sketch of the atomic arrangement (with no fully drawn octahedral bonds) makes it easy to see the connection between the lattice constant and the octahedral and tetrahedral bond lengths. The tetrahedrally coordinated A-cations form a diamond cubic sublattice with a repeat motif equal to the lattice parameter a. Two different “building blocks” with side length a/4 are placed alternately in the octants of the A-cation unit cell. The contours of the primitive cell are plotted within the elementary cell in the middle of Figure 1. Along the body diagonal of the primitive and elementary cell, there are the following sites: the A-cation in Wyckoff position 8a (0, 0, 0), the octahedral vacancy at 16c (1/8, 1/8, 1/8), the A-cation in position 8a (1/4, 1/4, 1/4), the anion in 32e (3/8, 3/8, 3/8), the tetrahedral vacancy at 8b (1/2, 1/2, 1/2), the B-cation in Wyckoff position 16d (5/8, 5/8, 5/8), the tetrahedral vacancy at 8b (3/4, 3/4, 3/4), and the anion in 32e (7/8, 7/8, 7/8).

Cobalt ferrite has a predominantly inverse spinel structure with the general formula (Co_1−xFe_x)[Fe_2−xCo_x]O₄. The mean ionic radii of the tetrahedral and octahedral sites for the partially inverse cobalt ferrite are:

<r_A> = (1 − x)·r(Co)_A + x·r(Fe)_A

(1)

<r_B> = 1/2·[(2 − x)·r(Fe)_B + x·r(Co)_B]

(2)

The cation-to-anion distance in the tetrahedron or octahedron, i.e., bond length, is the sum of the corresponding mean ionic radius and oxygen anion radius: R_tet,oct = $\langle r_{A,B}\rangle +r_O$.

Consider the bottom row of the “building elements” with side lengths a/4 (Figure 1). The space between them is a/4 and corresponds to the bond in the octahedron (plotted only in the bottom row in the added sketch of the octahedron). The value of the lattice parameter can be calculated using the bond lengths of the adjacent octahedron and tetrahedron as 3·a/4 = 2·R_oct + 2·R_tet/$\sqrt{3}$. From this relation follows:

$${\mathrm{a}}_{\mathrm{calc}}=8/3\left[{\displaystyle \frac{\left(\langle r_A\rangle +r_O\right)}{\sqrt{3}}}+\left(\langle r_B\rangle +r_O\right)\right]$$

(3)

The magnitude of the ionic radius in a crystal depends on the chemical surroundings in which the ion is located. According to the magnetic measurements and the determination of the cation distribution (by Rietveld analysis, neutron diffraction, XPS measurements, or Mössbauer spectroscopy), the Co²⁺ cations and Fe³⁺ cations are in the high-spin state in cobalt ferrite [40]. The effective ionic radii in the tetrahedral and octahedral coordination and oxygen anion radius are given in the Shannon comprehensive review [41]. The effective radius of the Co²⁺ cation with the high-spin (HS) electron arrangement in the tetrahedral position is r(Co²⁺)_A = 0.58 Å and in the octahedral site radius is r(Co²⁺)_B = 0.745 Å. For Fe³⁺ in the tetrahedral position, the effective radius is r(Fe³⁺)_A = 0.485 Å (HS), and in the octahedral position, the site radius is r(Fe³⁺)_B = 0.645 Å. The oxygen radius is r_O = 1.38 Å (the oxygen anions have a four-fold coordination, formed by three B-cations and one A-cation (32e)). The mean ionic radii of the tetrahedral and octahedral sites can be calculated as follows:

a_calc = 8.4175 − 0.0129·x

(4)

Relation (4) shows that the cobalt ferrite with the maximal degree of the inversion will have the smallest lattice constant.

On the other side, the cation-to-anion distances, R_tet,oct = <r_A,B> + <r_O>, for the case in Figure 1 when the unit cell origin is the A-cation and the ideal $u^{\overline{4}3m}$ = 3/8 are given by:

$$R_{\mathrm{tet}}=a \sqrt{3} \left(u-1/4\right)$$

(5)

$$R_{\mathrm{oct}}=a \sqrt{2{\left(u-3/8\right)}^2+{\left(5/8-u\right)}^2} \approx a\left(5/8-u\right)$$

(6)

When the unit cell origin is the B—vacancy and the ideal $u^{\overline{3}m }$ = 1/4, the cation-to-anion distances are given by:

$$R_{\mathrm{tet}}=a \sqrt{3} \left(u-1/8\right)$$

(7)

$$R_{\mathrm{oct}}=a \sqrt{3u^2-2u+3/8)}.$$

(8)

$u^{\overline{4}3m}$ is converted to $u^{\overline{3}m }$ and Equations (5) and (6) in (7) and (8), respectively, when the value of u from Equations (5) and (6) is replaced by u + 1⁄8 (because $u^{\overline{4}3m}=u^{\overline{3}m }+1/8$) [33]. The degree of inversion x can be calculated using the equations for the tetrahedral and octahedral bond lengths expressed over experimentally determined a and u together with the relations R_tet,oct = $\langle r_{A,B}\rangle +r_O$ and the equations for the mean ionic radii of the tetrahedral and octahedral site, Equations (1) and (2).

The cation-to-cation distances, the hopping lengths, in the spinel elementary cell can be easily deduced from Figure 1. The smallest distance is between the cations in the B-sites:

$$R_{\mathrm{BB}}={\mathrm{B}}_{16\mathrm{d}}-{\mathrm{B}}_{16\mathrm{d}}=a \sqrt{2}/4=0.353553·a$$

(9)

Tetrahedral A-cations are the furthest from each other:

$$R_{\mathrm{AA}}={\mathrm{A}}_{8\mathrm{a}}-{\mathrm{A}}_{8\mathrm{a}}=a \sqrt{3}/4=0.433013·a$$

(10)

The mutual distance of the cations at the adjacent A and B sites is:

$$R_{\mathrm{AB}}={\mathrm{A}}_{8\mathrm{a}}-{\mathrm{B}}_{16\mathrm{d}}=a \sqrt{11}/8=0.414578·a$$

(11)

The distances between the cations depend only on the lattice constant a. The cation–ation distances, cation–anion bonds, and interbond angles are usually denoted as in Figure 2.

All formulas for calculating cation–cation and cation–anion distances and angles between bonds (including formulas already given by Equations (5), (6), (9)–(11)) are shown in

Table 1. Formulas for calculating cation–cation (M-M) and cation–anion distances (M-O) and angles between bonds (including formulas given by Equations (5), (6), (9)–(11)).

M-M Hopping Lengths	M-O Bond Lengths	θ Interbond Angles
b = RBB = a $\sqrt{2}$/4	p = Roct = a (5/8 − u)	θ1 = cos−1 [${\displaystyle \frac{{\mathrm{p}}^2+{\mathrm{q}}^2-{\mathrm{c}}^2}{2\mathrm{pq}}}$]
c = RAB = a $\sqrt{11}$/8	q = Rtet = a $\sqrt{3}$(u − 1/4)	θ2 = cos−1 [${\displaystyle \frac{{\mathrm{p}}^2+{\mathrm{r}}^2-{\mathrm{e}}^2}{2\mathrm{pr}}}$]
d = RAA = a $\sqrt{3}$/4	r = a $\sqrt{11}$(u − 1/8)/2	θ3 = cos−1 [${\displaystyle \frac{2{\mathrm{p}}^2-{\mathrm{b}}^2}{2{\mathrm{p}}^2}}$]
e = a $3\sqrt{3}$/8	s = a $\sqrt{3}$(1/3u + 1/8)	θ4 = cos−1 [${\displaystyle \frac{{\mathrm{p}}^2+{\mathrm{s}}^2-{\mathrm{f}}^2}{2\mathrm{ps}}}$]
f = a $\sqrt{6}$/4		θ5 = cos−1 [${\displaystyle \frac{{\mathrm{r}}^2+{\mathrm{q}}^2-{\mathrm{d}}^2}{2\mathrm{rq}}}$]

.

As parameter u increases, the oxygen anions in the tetrahedrons move along the 〈111〉 direction, changing the volumes of the tetrahedrons but not affecting the $\overline{4}$3m symmetry. Due to the increase in the tetrahedral interstices, the octahedral B-sites become correspondingly smaller and more deformed. Variation in u is reflected in the bond-angle variations. These structural variations have important consequences on the properties of the material, especially the magnetic properties of ferrites. For instance, the angles θ₁ = <AOB, θ₃ = <BOB, and θ₅ = <AOA are the coupling angles for the most important super-exchange interactions. The values of the angles in the ideal f.c.c. spinel lattice are θ₁ = 126° (θ₁ = 125.264° ≈ 125.3° for oxide ferrimagnetic lattice [33]), θ₂ = 154°, θ₃ = 90°, θ₄ = 125°, and θ₅ = 79° [42]. The hopping lengths e and f are too large and together with the corresponding interbond angles θ₂ and θ₄, respectively, do not allow the creation of significant super-exchange interactions. Later, in the discussion of magnetic properties, we will calculate the values of these distances and angles for various coefficients of the cation inversion in the examined cobalt ferrite samples and evaluate their influence on the super-exchange interactions.

3.2. Structure Investigation of Cobalt Ferrite Nanoparticles by XRD

The X-ray diffractograms of five cobalt ferrite samples obtained with the various synthesis methods are presented in Figure 3, along with the Bragg positions and differences in calculated and experimental intensities. All investigated samples are well crystalized in the single-phase spinel structure with characteristic Bragg reflections, corroborating JCPDS PDF 22-1086. The diffractograms were thoroughly analyzed by means of the FullProf Suite program [32]. To make it easier to see the details of the XRD patterns, a small interval of 2θ (from 29° to 38°), with the reflections (220), (311), and (222) for all patterns, is shown on the left side of Figure 3. The peaks were practically Lorentzian. There was a noticeable difference in the 2θ positions and broadening of the corresponding reflections in the different samples. The auxiliary vertical lines coincided with the Bragg reflections of the sample with a maximal value of 2θ, i.e., the smallest lattice constant. The sample obtained by US-CO had the smallest lattice constant, followed by the samples prepared by ME, CO, MW-HT, and MC-CO, respectively. The sample ME had the smallest peak width (Figure 3).

Although a broadening of reflections increases with 2θ, the crystallite sizes can be well evaluated from the strongest peak (311) with Debye–Scherrer’s formula: D = K λ_Cu/(w₃₁₁·cosθ₃₁₁), where K is a shape factor equal to 0.94 for spherical particles of cubic symmetry, λ_Cu is a wavelength of the Cu_kα radiation used, and w_hkl is the FWHM (full width at half maximum) of a diffraction peak at the 2θ_hkl position. The instrumental broadening of the peak is neglected because the measured width of the Bragg reflections for the nanoparticles is about ten times greater. Structural parameters, a—lattice constant, u—oxygen positional parameter, B_eff—effective temperature factor, and x—coefficient of inversion derived by Rietveld profile refinement [43], are listed in

Table 2. Structural parameters of CoFe₂O₄ samples calculated from diffractograms: a—lattice constant, u—oxygen positional parameter, x—coefficient of inversion, B_eff—effective temperature factor, D—crystallite size, and e_str—internal strain obtained by Rietveld refinement and W-H analysis. ρ_XRD is the volume density of the samples and A_m is the specific surface area. In the last column are two agreement factors for Rietveld profile fitting: R_wp—weighted profile and goodness of fit χ².

Sample	aexp [Å]	$u^{\overline{3}m}$	x	Beff [Å2]	D [nm] (D311)	estr·10−4	ρXRD [g/cm3]	Am [m2/g]	Rwp χ2
US-CO	8.3407	0.2540	0.80	0.543	15.80 (15.21)	21.00	5.373	70.68	19.5 3.22
ME	8.3503	0.2545	0.80	0.678	19.00 (19.59)	5.78	5.367	58.84	16.4 4.17
CO	8.3619	0.2548	0.78	0.584	16.47 (16.19)	9.09	5.331	68.34	21.79 4.11
MW-HT	8.3732	0.2554	0.67	0.336	17.34 (15.73)	16.8	5.309	65.18	16.77 2.47
MC-CO	8.3825	0.2558	0.59	0.627	16.46 (15.47)	20.0	5.272	69.14	16.3 2.75

, together with the average crystallite size (D) and internal strain (e_str) obtained with the Williamson–Hall (W-H) plot:

w_hkl·cosθ_hkl = K·λ_Cu/D + 4·e_str·sinθ_hkl

(12)

The strain was assumed to be uniform in all crystallographic directions. From the linear fit to the data (where 4·sinθ_hkl is on the abscissa and the ordinate is w_hkl·cosθ_hkl), the crystalline size was estimated from the ordinate intercept and the strain e_str from the slope of the fit.

The crystallite sizes estimated from the strongest (311) peak with Debye–Scherrer’s formula are mostly smaller than the size obtained with the W-H plot (Table 2, D₃₁₁ in brackets). In Debye–Scherrer’s formula, a microstrain broadening is neglected. On the other hand, the Williamson–Hall analysis is a method where both main causes of broadening—size-induced and strain-induced broadening—are deconvoluted by considering the peak width as a function of θ [44,45].

The expected decrease in microstrain with an increase in the size of the nanoparticles is observed. The microstrain parameter of sample CO was slightly lower in comparison to the parameters obtained for the other investigated samples. In the case of the MC-CO sample, the microstrain value increased noticeably due to the internal stresses.

It can be noticed that the obtained lattice parameters of the studied nanocrystals were smaller than the calculated ones. This is common for nanomaterials since a lot of structural imperfections exist on the surface of nanoparticles.

The volume density obtained by Rietveld refinement, ρ_XRD, is a theoretical density for an obtained lattice constant, ρ_XRD = 8·M/(N_A·a_exp³), where M is the molecular mass and N_A is Avogadro’s number. The ρ_XRD value is higher than that of the bulk material (ρ_b = 3.24 g cm⁻³). The specific surface area, A_m, is calculated under the approximation that the nanoparticles are spherical, A_m = 6/(ρ_XRD·D). The A_m value decreases with the increase in the nanoparticle size. Although the Rietveld profile-refined diffractograms look satisfactorily good, the quality of fit, χ², remains insufficiently sensitive to the changes in the cation distribution x. Therefore, the obtained values for x should be taken with caution. Generally, this is a problem with cations whose atomic factors are of similar magnitudes. The difference in diffraction intensities between the normal and inverse CoFe₂O₄ is less than the line thickness, so the diffractograms look identical (see Supplementary Materials).

As already mentioned, the experimentally obtained lattice constants for the nanocrystalline CoFe₂O₄ samples were significantly smaller than those calculated via Shannon effective ion radii [41] (Equation (4)) (Table 2). As the crystal dimension decreases, the surface/volume ratio increases and the amount of disturbed surface structures with broken bonds and vacancies increases. The expected difference in the amount of internal stress between the largest and smallest nanoparticles is noticeable (Table 2). The magnitude of the internal stress and the lattice constant of the nanocrystals depends mostly on the synthesis conditions. In the tested samples, there is no connection between the nanoparticles’ size and the size of the lattice constant. It is obvious that in the examined range of nanoparticle sizes, cationic inversion has an incomparably more significant influence on the lattice constant value.

At the same time, the contraction of the crystal lattice is accompanied by the decrease in the oxygen parameter value. The value of the oxygen positional parameter depends on the difference in the size of the cations and their redistribution by the tetrahedral and octahedral sites. The huge specific surface area of the nanoparticles with incomplete bonds, vacancies, and other imperfections significantly contributed to the effective value of the lattice constant and oxygen parameter values.

It is shown that the intensity of the Bragg diffractions does not change sufficiently with variation in the inversion coefficient of the cations with close atomic scattering factors (see Supplementary Materials). Unlike cobalt ferrite, where changes in cation inversion do not significantly affect the Bragg reflection intensities (making it difficult to accurately determine the inversion parameter x through profile spectra refinement), the lattice constant strongly influences the positions of the Bragg reflections and can be measured with high precision. Starting from the values of the lattice constant, a_exp, and the oxygen parameter, u, obtained by XRD refinement and assuming that the cation radii were not changed significantly in the nanoparticles, the cation–anion distances in the tetrahedrons and octahedrons, R_tet and R_oct, were calculated using Equations (7) and (8). R_tet,oct = <r_A,B> + r_o, via Equations (1) and (2), is the estimated coefficient of the cation inversion, x, and average oxygen radius, <r_o>, in the cobalt ferrite nanocrystals produced with the various synthesis methods. The obtained values are given in

Table 3. Calculated values of average cation-to-anion distances in tetrahedrons (R_tet) and octahedrons (R_oct), effective oxygen radius <r_o>, and the degree of the inversion (x) as functions of a_exp and u, relations (7), (8), (1), and (2). In the last column is the vacancy parameter, β. Samples are in order of ascending lattice constant.

Sample	Rtet [Å]	Roct [Å]	<ro> [Å]	x	β %
US-CO	1.8636	2.05235	1.36468	0.85	2.34
ME	1.87298	2.05069	1.36685	0.79	2.03
CO	1.87992	2.05112	1.3695	0.73	1.41
MW-HT	1.89117	2.04908	1.37204	0.64	1.24
MC-CO	1.89907	2.04816	1.37412	0.58	0.93

.

The results listed in Table 3 are graphically presented in Figure 4a–c. Figure 4c also shows the inversion coefficients obtained from the Rietveld profile refinement, x-Rietv, together with x-struct, obtained by subsequent structural analysis. It is seen that the x-Rietv deviates within the limits of ±0.05 related to the x-struct, determined with greater accuracy.

At the same time, the structural analysis was used to determine the average oxygen radius <r_o> in the cobalt ferrite nanocrystals. The nanoparticles have a high proportion of surface defects. Such defects include incomplete tetrahedrons and octahedrons, which cause the contraction of cation–anion bonds. This contraction directly leads to a decrease in the average radius of the oxygen anion (<ro>). The lowest values of <r_o> were observed for the samples with the smallest lattice constants, i.e., for the powders obtained by US-CO and ME. Both samples had a high cation inversion coefficient x, confirming that for the nanoparticles in the investigated range, the size of x had a decisive influence on the lattice constant value.

In that case, the vacancy parameter β, defined as the relative volume of the missing ions at the nodal points of the spinel structure, is not a reliable indicator of the number of vacancies in the sample. The vacancy parameter can be evaluated as β = [(a_calc³ − a_exp³)/a_calc³] ·100%, where a_calc is the theoretical value of the crystal cell parameter calculated by Equation (4) for the supposed coefficient of the inversion x. For larger nanoparticles, the influence of the cation inversion coefficient on the lattice constant is not dominant, and β has the expected trend; i.e., it is a good indicator of the number of vacancies in the sample. A previous study clearly showed that β decreased with an increase in the size of the nanoparticles from 27 to 53 nm [46].

The calculated vacancy parameters for the studied samples are given in Table 3. The vacancy parameters decreased with the increasing value of the oxygen radius and the increase in the lattice constant value, as expected. However, it does not show any dependence on the size of the nanoparticles. Larger nanoparticles must have a less disordered crystal structure due to a smaller surface-to-volume ratio and therefore a lower β value.

The structural XRD analysis based on the measured values of the lattice constant (a_exp) and oxygen parameter (u) of the nanoparticles gives a more reliable value of the cation inversion. However, it should be emphasized that it is the average cation inversion of the material, which represents a “mixture” of crystalline cores with the undisturbed structure and shells of the structurally altered material since a noticeable difference in the size of a_exp and a_calc, as well as an almost ten times greater slope of the function a_exp = f(x), occurred. In the case of the relatively small nanoparticles which were examined in this work (<20 nm), not only the cation inversion affects the value of the lattice constant. Therefore, it is necessary to compare the data obtained by XRD analysis with the magnetic measurements, which can separate the part of the predominantly regular structure and the contribution of the incomplete tetrahedrons and octahedrons in the surface layers.

3.3. Magnetic Properties of Cobalt Ferrite Nanoparticles

The magnetizations of the cobalt ferrite samples obtained with the various synthesis methods are shown in Figure 5 as a function of temperature and magnetic field. The temperature dependencies of the magnetizations are measured by zero-field-cooled (ZFC) and field-cooled (FC) protocols at the applied field of 100 Oe. An irreversible magnetic behavior is shown by the splitting between the ZFC and FC curves. The irreversibility arises from the competition between the energy required for the magnetic moment reorientation in field direction vs. the energy barrier associated with the magnetoelectricity and the crystalline anisotropy.

The FC magnetization shows less temperature dependence than the ZFC magnetization. This indicates that the particle assemblies retain their magnetic history. The CO and MC-CO samples exhibit slight magnetization decreases above 50 K, becoming nearly constant at 300 K. In contrast, the samples obtained with the ME and MW-HT methods maintain almost constant FC magnetization up to about 200 K and then slightly increase near 300 K. At 5 K, the FC magnetization values are notably higher for US-CO, CO, and MC-CO samples (M_US-CO = 4.1 emu g⁻¹, M_CO = 4 emu g⁻¹, M_MC-CO = 3.5 emu g⁻¹) compared to the MW-HT and ME samples (1.75 emu g⁻¹ and 1.2 emu g⁻¹, respectively), suggesting easier magnetic moment rotation in the field direction for the former group. This aligns with their lower coercivities.

The shape of the FC curves in the irreversibility region indicates the interparticle interaction intensity. The FC magnetization increases with the rising of temperature (for T < T_B) when the dipolar interaction is significant [46]. Vargas et al. showed that an increase in the dipolar interaction simultaneously affects a decrease in the relative height of the FC magnetization with respect to the maximum of the ZFC curve [47]. Increasing dipolar interaction affects the relative height of the FC magnetization compared to the ZFC curve maximum. The MW-HT and ME samples show signs of greater dipolar interaction than the US-CO, CO, and MC-CO samples. Initial FC magnetization values gradually decrease and the slope of the FC curves changes as the nanoparticles’ sizes increase (Figure 5). A reduction in the initial FC magnetization value for the US-CO sample and a slight increase in the magnetization for the ME sample as temperature rises was observed.

The characteristics of the FC/ZFC curves indicate moderately wide distributions of particle sizes and therefore similar distributions of blocking temperatures in the particle assemblies. In addition, it is clearly seen that the temperature of the maximum of the ZFC curves, T_max, is over 300 K. The blocking temperatures can be determined, T_B (T_B ≈ T_x crossover temperature) according to Livsey et al. in the inflection point of the ZFC magnetization curves, i.e., in the maximum of $-{\displaystyle \frac{d\left(\mathsf{\Delta }M\right)}{dT}}$, where ΔM = M_FC − M_ZFC, [48]. Other authors imply that T_B corresponds to the maximum of the $-{\displaystyle \frac{1}{T}} {\displaystyle \frac{d\left(\mathsf{\Delta }M\right)}{dT}}$ [49,50]. Micha et al. state that “the blocking temperature, T_B, is determined by the FC and ZFC junction” [51]. In the following text, the term “blocking temperature” will mean the maximum of $-{\displaystyle \frac{d\left(\mathsf{\Delta }M\right)}{dT}}$. To estimate the T_B of the nanocrystalline CoFe₂O₄ materials obtained with the different synthesis methods, a modified Stoner–Wohlfarth two-stage model can be used [48,52]. This model implies the existence of non-interacting, spherical, uniaxial nanoparticles.

As can be seen with SEM and TEM [31], the CoFe₂O₄ nanoparticles obtained with the investigated methods are agglomerated and have different shapes and sizes. Defined crystal planes are seen on the largest particles, while smaller particles are generally spherical. The nanoparticles with sizes up to 20 nm usually grow spherical due to the large surface tension originating from the large surface-to-volume ratio [27] in the investigated nanoparticles.

Therefore, the first condition for applying the modified Stoner–Wohlfarth model was not met, but the nanoparticles are supposed to be mostly spherical. The presence of a single magnetic domain (uniaxial structure) was evaluated by observing a consistent rise in coercivity as the nanoparticle size increased. This steady increase is a clear sign of a single-domain configuration. In Figure 5, it is clearly visible that the investigated nanoparticles have almost linear growth of coercivity. At a characteristic critical nanoparticle size, a multidomain structure appears in order to lower the internal energy of the crystallites, and with a further increase in the size of the nanoparticles, the coercivity begins to decrease, reaching the coercivity of the bulk material (~230 Oe) [30]. According to Maaz and Kim, the single-domain limit for CoFe₂O₄ is about 28 nm [30].

One of the significant features in the M(H) loops of the nanoparticles is that the magnetization does not saturate, even at high applied field. This is generally attributed to the canted or disordered spins at the surface of the nanoparticles that are difficult to align along the field direction. The numerical results of measuring the dependence of the magnetization of the cobalt ferrite samples on the applied magnetic field (Figure 5) are given in

Table 4. Measured saturation magnetization M_s at 90 kOe (with absolute error Δ ≈ ±0.1 emu/g), calculated saturation magnetization M_s(calc), slope of the magnetization in the near-zero field region (dM/dH)_H=0, slope of the magnetization in the high-field region (dM/dH)_H>>Hc, coercivity H_c (Δ_max = ±5 Oe) magnetic remanence M_r (Δ ≈ ±0.1 emu/g), and “squareness” M_r/M_s (Δ ≈ ±0.01) for the samples obtained with various synthesis methods. Samples are in order of ascending lattice constant.

	Ms (90 kOe) [emu/g] (n [μB]) ♥	Ms (calc) [emu/g] (n [μB]) ♥	(dM/dH) for H = 0 [emu/g/kOe]	(dM/dH) for H>>Hc [emu/g/kOe]	Hc [Oe]	Mr [emu/g]	Mr/Ms
US-CO	61.0 (2.56)	62.6 (2.63)	43.7	0.037	291	13.2	0.21
ME	69.3 (2.91)	71.1 (2.99)	28.4	0.046	1206	26.8	0.38
CO	66.3 (2.78)	68.0 (2.86)	46.7	0.048	330	14.7	0.22
MW-HT	70.0 (2.94)	72.4 (3.04)	29.7	0.045	723	18.1	0.24
MC-CO	72.4 (3.04)	74.3 (3.12)	36.6	0.035	407	15.8	0.22

n ^♥ = 234.623·M_s/5585.

.

Due to the asymptotic increase in magnetization for high fields, the maximal saturation magnetization M_s(calc) can be obtained from extrapolating the magnetization function M = f(1/H) to 1/H = 0. The measured saturation magnetization M_s at 90 kOe and extrapolated saturation magnetization M_s(calc) are shown in Table 4. In the following text, M_s instead of M_s(calc) will denote the maximum possible magnetization in the sample.

The values of the slope of the magnetization in the near-zero field region (dM/dH)_H=0 show that smaller nanoparticles, obtained with different coprecipitation methods, follow changes in the magnetic field faster. The slope of the magnetization in the high-field region (dM/dH)_H>>Hc implies that samples with a smaller slope (US-CO and MC-CO) have nanoparticles with a relatively small amount of disturbed spins and probably thin surface layers.

Coercivity, H_c, increases remarkably with the size of the magnetic domains. The magnetic remanence M_r and “squareness” M_r/M_s for the obtained samples are also given in Table 4. The presence of super-paramagnetic nanoparticles in each sample, smaller than average size measured by XRD, reduces the value of “squareness” and the magnetization curves become less steep. It can be seen that “squareness” increases with the average nanoparticle size (see Table 2 and Table 4), which is an indication of a smaller influence of the disturbed surface layer in bigger nanoparticles.

The maximal difference in the average sizes of the CoFe₂O₄ nanoparticles of the different samples is less than ±1.6 nm (<19%), but some of the physical quantities, shown in Table 4, vary considerably more. That is a direct consequence of the synthesis method. For small nanoparticles (<28 nm), the value of the saturation magnetization depends primarily on the coefficient of the cation inversion, but structural factors also play a significant role. The synthesis conditions, average nanoparticle size, distribution, and morphology of the nanoparticles crucially affect the magnetic remanence and coercivity. The maximum difference in M_r between the investigated samples is 79%, and for coercivity, the maximal difference is even 150%.

3.4. Estimation of Blocking Temperature from ZFC/FC Curves by Fitting Procedure

A modified Stoner–Wohlfarth model with certain limitations, as previously seen, can be used to estimate the blocking temperature in the tested CoFe₂O₄ samples. We have mostly spherical, uniaxial monodomain nanoparticles, but their interaction is not negligible. In the Stoner–Wohlfarth model, the particles are considered to be completely blocked as long as the temperature is below their blocking temperature T_B, with the magnetization in the ZFC (heating) regime equal to the initial magnetization M_ZFC (t = 0) = M_ZFC (T→0) = M_b. The blocked magnetic moment is M_b = μ₀μ²H/(3K), where μ is the magnetic dipole moment (μ = N·μ_at = M_s·v). N is the density and number of atomic moments μ_at per unit volume, and v is the volume of the nanoparticle. M_s is a saturation magnetization (per unit volume)—the maximum magnetization value that the system can reach, which corresponds to the perfect aligning of all magnetic moments to the external magnetic field H used in measurement. K is the magnetic anisotropy energy (MAE), K = K_eff·v, and K_eff is the effective anisotropy constant, i.e., the density of anisotropic energy. At T > T_B, the nanoparticles become perfectly superparamagnetic (at thermodynamic equilibrium) with magnetization M_eq = μ₀·μ²·H/(3·k_B·T). The crossover between the two regimes is supposed to be abrupt (T_B = T_x) and occurs at a transition temperature related to the magnetic anisotropic energy. The Stoner–Wohlfarth two-stage model is modified by introducing the size distribution of the nanoparticles. This approximation is quite satisfactory for assemblies of magnetic nanoparticles with a wide size distribution [48].

A range of particle sizes implies a range of blocking temperatures in the particle assemblies. The average Néel time, τ, for a particle to flip from one well to another is given by the Néel–Arrhenius law [53]:

τ = τ_o exp (K_eff·v/(k_B·T)),

(13)

where τ_o is a characteristic relaxation time, typically taken as 10⁻⁹–10⁻¹² s [54,55]. If τ is the measurement time, τ_m, we can define the blocking temperature for a given experimental conditions as:

T_B = K_eff·v/[k_B·ln(τ_m/τ_o)]

(14)

The value of α = ln(τ_m/τ_o) for usual conditions in magnetic measurement, with τ_m = 100 s, is 25–32. If τ_m = 10⁻⁸ s, like in Mössbauer measurements, α is 2–9 [56]. Changes in the Mössbauer spectra related to the transition to the superparamagnetic state appear at higher temperatures compared to the magnetic measurements [57]. T_B depends on the measurement conditions, so the relevant time frame (τ_m), temperature, and size of the nanoparticles (that affect τ_o) must be specified. The blocking temperature T_B is equal to the crossover temperature T_x in the approximation that α = ln(τ_m/τ_o) is not dependent on v [50].

The distribution of the particle volumes gives rise to a distribution of blocking temperatures T_B(v). So, the ZFC magnetization and the FC magnetization become, respectively [54]:

$$M_{\mathrm{ZFC}}\left(T\right)={\displaystyle \frac{{\mu }_o·\mathrm{H}·{\mathrm{M}}_{\mathrm{s}}^2}{3·k_B·\mathrm{T}}}·{\displaystyle \frac{1}{v_a}}{{\displaystyle \int }}_0^{v_{m\left(T\right)}}{v}^2·f\left(v\right)dv+{\displaystyle \frac{{\mu }_o·\mathrm{H}·{\mathrm{M}}_{\mathrm{s}}^2}{3·{\mathrm{K}}_{eff}}}·{\displaystyle \frac{1}{v_a}}{{\displaystyle \int }}_{v_{m\left(T\right)}}^{\infty }v·f\left(v\right)dv$$

(15)

$$M_{\mathrm{FC}}\left(T\right)={\displaystyle \frac{{\mu }_o·\mathrm{H}·{\mathrm{M}}_{\mathrm{s}}^2}{3·k_B·\mathrm{T}}}·{\displaystyle \frac{1}{v_a}}{{\displaystyle \int }}_0^{v_{m\left(T\right)}}{v}^2·f\left(v\right)dv+{\displaystyle \frac{\mathsf{\alpha }·{\mu }_o·\mathrm{H}·{\mathrm{M}}_{\mathrm{s}}^2}{3·{\mathrm{K}}_{eff}}}·{\displaystyle \frac{1}{v_a}}{{\displaystyle \int }}_{v_{m\left(T\right)}}^{\infty }v·f\left(v\right)dv$$

(16)

The first integral in Equations (15) and (16) represents the contribution of the superparamagnetic particles, while the second corresponds to the blocked ones. The expected value (mean, average) of the particle volume is v_a, and v_m(T) is the limit (maximum) volume of the nanoparticles in the superparamagnetic state at a given temperature, v_m(T) = α·k_B·T/K_eff. The probability density function f(v) can be the volume log-normal or normal Gaussian distribution with mean value ν_a and “width”, i.e., standard deviation, σ.

The contribution of the superparamagnetic particles is the same for the M_FC and M_ZFC curves and is canceled in the difference between M_FC and M_ZFC magnetization. The difference ΔM = M_FC − M_ZFC can be used to simplify the estimation of different quantities such as T_B, K_eff, v_a, and the standard deviation σ.

$$\mathsf{\Delta }M={\displaystyle \frac{\left(\mathsf{\alpha }-1\right)·{\mu }_o·\mathrm{H}·M_s^2}{3·{\mathrm{K}}_{eff}}}·{\displaystyle \frac{1}{v_a}}{{\displaystyle \int }}_{v_{m\left(T\right)}}^{\infty }v·f\left(v\right)dv$$

(17)

Instead of a distribution of particle volumes, the distribution of the reduced volumes, y, which is the same as the distribution of the T/T_B temperature, y = v/v_a = T/T_B, due to the linear relation α·k_B·T_B = K_eff·v_a can be used. In that case, the contribution of the blocked particles becomes [58,59,60,61]:

$$\mathsf{\Delta }M={\displaystyle \frac{\left(\mathsf{\alpha }-1\right)·{\mu }_o·\mathrm{H}·M_s^2}{3·{\mathrm{K}}_{eff}}}·{{\displaystyle \int }}_{y_m}^{\infty }f\left(y\right)dy$$

(18)

If the distribution of particle volumes f(v) is log-normal, the same is f(y) $={\displaystyle \frac{1}{\sigma \sqrt{2\pi }}}·{\displaystyle \frac{1}{y}}·e^{-{\displaystyle \frac{l{n}^2\left(y\right)}{2·{\sigma }^2}}}$.

The probability that y has values higher than y_m is:

$${{\displaystyle \int }}_{y_m}^{\infty }{\displaystyle \frac{1}{\sigma \sqrt{2\pi }}}·{\displaystyle \frac{1}{y}}{e}^{-{\displaystyle \frac{l{n}^2\left(y\right)}{2·{\sigma }^2}}}dy={\displaystyle \frac{1}{2}}\left[1-\mathrm{Erf}\left({\displaystyle \frac{\ln \left(y_m\right)}{\sigma ·\sqrt{2}}}\right)\right]$$

(19)

A similar result is obtained for the normal distribution.

The contribution from the blocked particles in ΔM = M_FC − M_ZFC can be written as:

$$\mathsf{\Delta }M\left(\mathrm{T}\right)={\displaystyle \frac{\left(\alpha -1\right)·{\mu }_o·\mathrm{H}·{\mathrm{M}}_{\mathrm{s}}^2}{3·{\mathrm{K}}_{eff}}}·{\displaystyle \frac{1}{2}}\left[1-\mathrm{Erf}\left(\mathrm{x},\sigma \right)\right]$$

(20)

Now, we can introduce the distribution of the blocking temperatures and directly fit the function ΔM(T) = M_FC − M_ZFC as a function of temperature. For the normal distribution, it is x = ${\displaystyle \frac{1}{\sigma ·\sqrt{2}}}$ (T/T_B − 1), and for the log-normal distribution x = ${\displaystyle \frac{1}{\sigma ·\sqrt{2}}}$·ln(T/T_Bn). Here, T_Bn is the median on the natural scale, T_Bn = exp(μ), and the mode, the global maximum of the T-distribution, is T_B = exp(μ − σ²).

The program for the fitting of the blocking temperature was tested on the data for the Fe₃O₄ samples with all T_B-s in the measurement interval. The data were used from [47]. Vargas et al. made samples from the same nanomaterial diluted in paraffin, with Fe₃O₄ concentrations of 0.05%, 0.5%, 5%, 45%, and 100% [47]. Obviously, all the nanoparticles had the same size distribution and the same anisotropy, but with increasing nanoparticle concentration, the dipole interaction increased in the samples, leading to an apparent increase in measured T_B and a change in the appearance of the FC and ZFC curves.

A dipolar field reduces the ordering of the magnetic moments and has a role similar to the effect of the temperature. The corresponding temperature Td = Ed/k_B, where Ed is dipolar energy, can be introduced in the presented model by writing T_B + Td instead of T_B (=T_Beff) and fitting Td as a new parameter (the Vogel–Fulcher law). Indeed, this is provided that the true value of T_B is known. The excellent fits of the diluted and powder Fe₃O₄ data (fits that give the same T_B as the maximum of the experimental $-{\displaystyle \frac{d\left(\mathsf{\Delta }M\right)}{dT}}$) with the same temperature distributions were obtained, as it was expected.

Due to the lack of samples of diluted nanoparticles in which the dipole interaction is negligible, the fitting results could only be the type and width of the distribution of the effective blocking temperatures. Figure 6 shows the result of the ΔM(T) = M_FC − M_ZFC fitting, as well as the obtained distribution functions whose maxima correspond to the blocking temperature. The “blocking temperatures” obtained by the fitting were surprisingly low. This means that the actual blocking temperatures of the prepared nanomaterials could be even lower. Accurate determination of the real T_B would be important for the application of nanomaterials in the form of ferrofluids, particularly for biomedical application.

From the papers concerning the T_B of the CoFe₂O₄ nanomaterials diluted in organic [62] or inorganic [23,58] matrices, one cannot determine unambiguously the impact of the dipolar interaction on T_B and estimate the real T_B (not effective) for a given size of nanoparticles. We did not find articles in which the authors systematically dealt with the dipole interaction in CoFe₂O₄ nanoparticle ensembles.

Babić-Stojić at al. determined the temperature of the maximum of the ZFC magnetization, T_max = 156 K, for the organic liquid suspension of 0.84% vol. fraction CoFe₂O₄ nanoparticles with a size of about 5–6 nm [62]. The ZFC-FC magnetization was recorded under H = 100 Oe. Unfortunately, no ZFC-FC measurement of the powder sample was performed under the same field. Only under the field of 1000 Oe were both measurements performed. The maximal temperatures of the corresponding ZFC magnetizations for the suspension and for the nanopowder are observed to be matched at approximately 164 K. Probably, a sufficiently strong magnetic field partially compensates for the effects of the dipolar interaction. However, we used the ZFC/FC magnetization data for the suspension measured at 100 Oe to determine the exact T_B (and test the program again). By the fitting of the ΔM(T) = M_FC − M_ZFC data, we obtained T_B = 105 K and σ = 0.33. Almost the same blocking temperature and the same Gaussian distribution were obtained by direct differentiation of the experimental ΔM(T) in the Origin Lab program.

To the best of our knowledge, there are no relevant data in the literature to determine the relationship between the dimensions of the CoFe₂O₄ nanoparticles and their exact T_B-s. In this study, the effective blocking temperatures T_Beff and the standard deviation (“width”) of their distributions were estimated as is usual for solid materials.

In all cases, including the presented CoFe₂O₄ samples and the data utilized from Babić-Stojić et al. [62] and Vargas et al. [47], the distributions of the blocking temperatures are found to conform to a normal (Gaussian) distribution. The effective blocking temperatures and relative widths of the distributions shown in Figure 6 are also given in

Table 5. Effective blocking temperatures T_Beff and standard deviations σ obtained by fitting of ΔM, initial magnetic moment of ZFC magnetization, M_b, anisotropy constant K_eff (5 K) calculated from the initial M_b measured at 5 K (values obtained with residual magnetization are omitted), and anisotropy constant K_eff (300 K) estimated from the high field magnetization by fitting “the law of approach to saturation”. The magnetic core diameter D_m = $\sqrt[3]{\left(6/\mathsf{\pi }\right)·30k_B{T}_B/K_{eff}}$, calculated via anisotropy characteristics T_B and K_eff, is compared with the magnetic core diameter D_c calculated by core–“dead” shell model. Estimated volume percentages of disturbed, magnetically “dead” layers, V_shell, are given in the last column. Samples are in order of ascending nanoparticle sizes.

Sample/ (D [nm])	TBeff [K]	σ	Mb [emu/g]	Keff * (5 K) [105 A/m3]	Keff (300 K) [105 A/m3]	Dm [nm]	Dc [nm]	Vshell %
US-CO/ (15.8)	-	-	-	-	4.04	-	13.79	34
MC-CO/ (16.46)	269	0.33	0.096	18.0 *	4.25	7.94	14.18	36
CO/ (16.47)	-	-	-	-	4.35	-	14.29	35
MW-HT/ (17.34)	271	0.30	0.10	16.5 *	5.21	7.44	15.00	35
ME/ (19)	280	0.35	0.09	18.8 *	5.53	7.37	17.08	27

* K_eff (5 K) calculated with M_s values 40% higher than M_s measured at 300K.

.

According to the Stoner–Wohlfarth model, blocking temperature is directly dependent on magnetic anisotropic energy K_eff·v. As can be seen in Figure 6, a fairly uniform increase in T_B with augmentation in the nanoparticle size was achieved, consistent with the model [48,59].

3.5. The Effective Anisotropy Constant, K_eff

The effective anisotropy constant K_eff can be calculated from the blocking magnetization M_ZFC(T→0) = M_b as K_eff = μ_o·H·M_sv²/(3·M_bv). The label “v” emphasizes that the magnetization must be calculated per unit volume. The saturation magnetization at 300 K is known from the measurement of magnetization as a function of the applied magnetic field (Table 4). However, in the equation for the blocking magnetization at T→0, M_b, it should be the M_s-value obtained at T→0. Chatterjee et al. [63] and Ananthramaiah and Joy [64] obtained magnetizations at 5 K which were 1.25 to 1.7 times larger than at 300 K. Based on the aforementioned literature, it could be assumed that the magnetization of smaller nanoparticles increases more with decreasing temperature. For nanoparticles with the size 15–20 nm, the magnetization at T→0 should be about 40% higher than the M_s measured at 300 K. The calculated values of the anisotropy constants K_eff with the adjusted value of M_s are rather close to the value of the anisotropy constant for single crystals at 5 K obtained by Shenker [65]. According to Shenker, the temperature dependence of the effective anisotropy (precisely the principal anisotropy constant K₁) for bulk CoFe₂O₄ with cubic symmetry follows the empirical relationship:

K₁ = 19.6·10⁵·exp(−1.9·10⁻⁵·T²) J m⁻³

(21)

It can easily be calculated that the anisotropy constant K₁ at 5 K for the bulk is 19.6·10⁵ [J m⁻³] and at 300 K is 3.5·10⁵ [J m⁻³].

In Table 5 are given the values of K_eff at 5 K (marked by “*”) obtained from the measured initial blocking magnetizations M_b and with M_s values 40% higher than the values measured at 300 K, which is the expected increase in M_s at T→0 K [56,60]. The low-temperature values for K_eff obtained with the specified M_s correction are somewhat lower than Shenker’s value obtained on a single crystal [60].

Considering that we do not have the measured values of the magnetization of the examined nanopowders at 5 K, we will determine K_eff at 300 K using the “law of approach to saturation” on high magnetic fields. The values of the effective anisotropy constants estimated on the basis of the “law of approach to saturation” on high magnetic fields are more reliable because all the quantities required for the calculation are already determined at the same temperature.

According to Brown’s theory [66], internal forces lead to laws of the form M/M_s = 1 − 1/H^n/2, where n = 1, 2, 3, for the point, line, and plane concentrations of forces, respectively, or n = 4 for forces uniform throughout an extended volume. A lattice distortion concentrated in a small region will produce a deviation from perfect saturation over a much larger region. When two or more of these types of force distributions are present and there is no correlation between them, their contributions to M(H) must be added. The empirical relation for H>>H_c, known as the “law of approach to saturation”, is consistent with the following theory:

$$M\left(\mathrm{H}\right)=M\mathrm{s}·\left(1-{\displaystyle \frac{A}{\mathrm{H}}}-{\displaystyle \frac{B}{{\mathrm{H}}^2}}+\dots \right)+{\mathsf{\chi }}_{\mathrm{p}}·\mathrm{H}$$

(22)

Brown showed that the coefficient A is proportional to the number of dislocations and therefore to the plastic strain. As the investigated nanoparticles are monodomain, the term A/H can be neglected. Coefficient B is related to the effective anisotropy constant as:

$$B={\displaystyle \frac{4·K_{eff}^2}{15·{\mathsf{\mu }}_0^2·{\mathrm{M}}_{\mathrm{s}}^2}}={\displaystyle \frac{4\left(K_{sh}^2+K_{st }^2+K_{su}^2+K_{cry}^2\right)}{15·{\mathsf{\mu }}_0^2·{\mathrm{M}}_{\mathrm{s}}^2}}$$

(23)

where K²_eff is the sum of the squares of all the contributions to the anisotropy–shape anisotropy K_sh, stress anisotropy K_st, surface anisotropy K_su, and magnetocrystalline anisotropy K_cry [63].

In the case of uniform anisotropy, the effective anisotropy constant is:

$$K_{\mathrm{eff}}={\mu }_{\mathrm{o}}·M_{\mathrm{s}}·\sqrt{{\displaystyle \frac{15·B}{4}}}$$

(24)

For randomly oriented spherical polycrystalline nanomaterials with cubic symmetry and dominant magnetocrystalline anisotropy, the effective anisotropy constant will be approximately equal to the principal anisotropy constant K₁ [63,67]:

$$K_{\mathrm{eff}}=K_1={\mu }_{\mathrm{o}}·M_{\mathrm{s}}·\sqrt{{\displaystyle \frac{105·B}{8}}}$$

(25)

In the case of nanomaterials, the linear term in the equation for the “law of approach to saturation” indicates the effect of canted spins on the surface of nanoparticles. These spins begin to contribute to the total magnetization only at very strong magnetic fields (χ_p·H—“forced magnetization term”).

The anisotropy at 300 K calculated by Shenker’s formula [65] for a cobalt ferrite single crystal is 3.5 × 10⁵ J m⁻³. Fitting the “law of approach to saturation” M(H) of sintered polycrystalline cobalt ferrite gives roughly similar values: (2–4) × 10⁵ J m⁻³, depending on the synthesis method [64]. The effective anisotropy constants for the investigated CoFe₂O₄ nanopowders also obtained by fitting the “law of approach” are shown in Table 5 as K_eff (300 K). The obtained values for the nanopowders (4–5.5) × 10⁵ J m⁻³ are slightly higher than for sintered polycrystalline cobalt ferrite. The observed difference can be attributed to the surface anisotropy of K_su. It is noticeable that the anisotropy constants K_eff(300K) for various samples have similar a trend as the corresponding T_B.

The fitting of the magnetization M(H) in the range of H>>H_c is illustrated in Figure 7. It can be seen that the values of the effective anisotropy constants in the case of ME and MW-HT samples are higher than for nanomaterials obtained by the coprecipitation methods. (The shapes of the ZFC and FC magnetization curves also confirm this.) The obtained almost linear dependence of K_eff on H_c verifies the direct proportionality of these two physical quantities. Both quantities increase expectedly with the size of the nanoparticles.

Values of anisotropy constants in CoFe₂O₄ are almost ten times higher than in other ferrites [24] due to significant spin–orbit interactions of the Co²⁺ ions. It has been observed that the single-ion magnetocrystalline anisotropy for Co²⁺ located in the octahedral site is much larger when related to the tetrahedral site [68]. This implies that the higher degree of the inversion in bulk (when the influence of nanoparticle’s size has no effect) leads to a higher value of anisotropy. For small nanoparticles, the influence of the size is dominant.

The slope of the M(H) curves and the difference in the magnitude of the slope between the samples obtained with the different synthesis routes can be better observed in Figure 7 than in Figure 5. Achieving magnetic saturation in nanoparticle systems is complicated by several factors. The surface of the nanoparticles often exhibits disordered structures and magnetic moments, which interferes with uniform magnetic alignment. Additionally, nanoparticle samples typically contain a distribution of particle sizes, including both larger particles and smaller superparamagnetic ones. This size variation leads to different magnetic behaviors within the same sample. The combination of surface disorder and size distribution makes it challenging for the entire nanoparticle system to reach a state of complete magnetic saturation. Nevertheless, the fact that samples MC-CO and US-CO, with the smallest average size of nanoparticles, at the same time have the smallest slopes of the linear part of the function M(H), for H>>H_c clearly shows that the methods of their synthesis yielded nanoparticles with the narrowest size distribution and probably the thinnest disordered surface layers.

The size of the magnetic cores, i.e., the central part of the nanoparticles with undisturbed magnetic moments, were estimated from Equation (14): T_B = K_eff·v/(k_B·α) as D_m = $\sqrt[3]{\left(6/\mathsf{\pi }\right)·30·k_B·T_B/K_{eff}}$, with K_eff (300 K). The obtained values of the magnetic cores show that the thicknesses of the “dead” layers, t = (D_XRD − D_m)/2, are from 4.3 nm for MC-CO to 5.8 nm for ME. This trend corresponds to the difference in slopes, ΔM/ΔH at H>>H_c (Table 4), but the obtained values seem to be too high. In the next section, a different estimation of the size of the magnetic cores (D_c) will be presented. With the approximation used, more realistic values for the “dead” layers are obtained at t ≈ 1 nm.

3.6. Average Magnetic Moment of Co and Fe Cations at 300 K

The magnetic properties of the cobalt ferrite originate from the spin magnetic moment of the unpaired 3d electrons of the transition metals. Metal cations in the tetrahedral and octahedral coordination of oxygen anions have a relatively small crystal-field splitting Δ, which leads to a high-spin (HS) arrangement.

Theoretically, the contribution to the magnetic moment from the free spins in the Fe³⁺ ion (with n = 5 unpaired spins) or in the Co²⁺ ion (with n = 3) is $\sqrt{n\left(n+2\right)}$, which gives 5.92 μ_B and 3.87 μ_B, respectively. Although it is theoretically possible that Fe³⁺ and Co²⁺ have the stated values of the magnetic moments, such high values indicate perfect ordering and are not realistic in the case of nanoparticles. Commonly used values for low-temperature magnetic moments in nanomaterials are 5 μ_B for Fe³⁺ magnetic cations and 3 μ_B (or 3.5 μ_B) for Co²⁺ magnetic cations [62,69,70].

Neutron diffraction measurements on CoFe₂O₄ confirm that at low temperatures, the magnetic moments of the A- and B-cations are antiparallel [71]. In that case, Néel’s model of collinear ferrimagnetism (which stands for many inverse spinel ferrites) can be applied [72]. Therefore, the resulting magnetic moment per formula unit of cobalt ferrite (Co_1−xFe_x)^A[Fe_2−xCo_x]^BO₄ will be:

M_s = M^B − M^A = [(2 − x)·μ_Fe + x·μ_Co]^B − [(1 − x)·μ_Co + x·μ_Fe]^A

(26)

where μ_Fe and μ_Co are the magnetic moments of the Fe and Co cations, respectively, expressed by the number of Bohr magnetons (μ_B = 9.27 × 10⁻²⁴ J T⁻¹).

In the inverse cobalt ferrite (x = 1), the value of the magnetization is equal to the magnetic moment of Co²⁺ (the magnetic moment of the Fe³⁺ from the tetrahedral site compensates the same antiparallel moment from the octahedral site). Guillot et al. observed a spontaneous magnetization of 3.95 μ_B in the inverse single crystal. The obtained value is close to the saturation magnetization of 4 μ_B. The experimental result is higher than the theoretical prediction for Co²⁺ ions. The discrepancy suggests that there is an additional orbital contribution to the magnetic moment of cobalt [26].

Based on the measured values of magnetization M_s at T = 300 K (Table 4) and the inversion coefficients obtained by XRD structural analysis (Table 3), it can be estimated that the average magnetic moments μ_Fe and μ_Co are equal to 3.25 μ_B and 2.5 μ_B, respectively. Both values are about 1.7 times smaller compared to the theoretical value or 1.4 times smaller compared to the commonly used magnetic moments in the CoFe₂O₄ nanomaterials for T→0 K, which is in agreement with the experimental data for nanoparticles with a size of 15–20 nm. As already mentioned, in the case of nanoparticles, the experimental magnetic moments of the cations in nanostructures are smaller than the theoretical ones [62,73].

Different interplanar distances, characteristic of the FCC crystal structure, were previously observed and identified by high-resolution transmission microscopy (HR-TEM) [74,75,76,77], thus confirming that the central part of the nanoparticles has an undisturbed structure such as that in the bulk. This magnetic core, with undisturbed magnetic moments (M_c), is surrounded by a layer with a disturbed structure and much lower magnetic moments (unoriented, canted, or both). If we assume for the sake of simplicity that this layer is “magnetically dead”, then the total magnetization of the nanoparticle would originate only from the core. With this approximation, we can easily estimate the possible diameter of the core.

If we calculate the magnetic moment using the Néel model, Equation (26), with the inversion coefficient obtained by XRD analysis and the values of μ_Fe = 5 μ_B and μ_Co = 3.36 μ_B as they are in single crystals at room temperature [26], we should obtain an enlarged value (M_s*):

M_s* = 2 × 5μ_B (1 − x) + 3.36μ_B (2x − 1)

(27)

For the MC-CO sample, for instance, with the degree of inversion x = 0.58 and M_s = 3.12 μ_B (74.2 emu g⁻¹), the enlarged magnetization would be M_s* = 4.74 μ_B (113 emu g⁻¹) at room temperature. To recall, the saturation magnetization is defined as the maximum possible magnetization of the sample per unit volume. Common values of M_s in units of emu g⁻¹ must be divided by the average volume of the nanoparticles (π/6)·D³. Then, by comparing M_s and M_s*, the ratio of the respective nanoparticle diameters is obtained. This means that the diameter of the core, D_c, is:

D_c = (M_s/M_s*)^1/3 · D

(28)

Diameter D is obtained by Rietveld analysis of the X-diffractograms and Williamson–Hall plot (Table 2). In the case of the MC-CO sample, the magnetic core diameter should be D_c = 14.32 nm and the thickness of the “dead” layer t = (D − D_c)/2 = 1.07 nm.

The calculated values of the magnetic core diameters D_c for all the investigated CoFe₂O₄ samples are given in Table 5. The thickness of the disturbed surface layer obtained in this way is in better accordance with the literature data. Along with D_c, the volume percentages (V_shell) of the disturbed structures in the corresponding samples are given. This evaluation shows that the investigated nanomaterials produced with different synthesis methods have a thickness of “dead” layers (t) of about 1 nm, independent of the average particle size. Indeed, the total amount of disturbed surface structures and superparamagnetic particles is higher in the samples with smaller nanoparticles. The exception is the “US-CO” sample, obtained by ultrasonically assisted coprecipitation, which has a narrow distribution of nanoparticle sizes (characteristic for this synthesis method [78]) and therefore a small number of superparamagnetic particles.

3.7. Exchange Interaction in Investigated Nanoparticles of CoFe₂O₄

According to spin–wave theory, the unpaired 3d electrons in transition metals are coupled by the super-exchange interaction via oxygen ions separating the magnetic ions. Electrons are shared between the 3d orbitals of the metal ions and the 2p orbitals of the oxygen ions. The interaction strength is proportional to the overlap between these orbitals and directly depends on the bond length and angle between the metal ions and oxygen ion [79,80,81]. A strong antiferromagnetic structure is made when the magnetic ion (with partially filled d-shells), ligand, and neighboring magnetic ion tend to form an angle of ~180° and a weak ferromagnetism or mediate antiferromagnetism when the angle is ~90°. In spinel ferrites, there are three types of significant super-exchange interactions. (The remaining two super-exchange interactions are negligible due to the large distance between the cations.) The strongest is J_AB (A-O-B), with angle θ₁; then J_BB (B-O-B), with angle θ₃; and weak J_AA (A-O-A), with angle θ₅ (see Figure 2). A and B refer to the ions on the tetrahedral and octahedral sites, respectively. The ideal spinel value of A-O-B angle θ₁ is 125.3°. Since there can be two types of ions in the A and B positions, the actual number of interactions is 10 for partially inverse ferrite or 6 for inverse ferrite. The J_AA interaction could be expected to be the weakest due to the largest distance between the tetrahedral A–A-cations in the crystal lattice of the spinel ferrite; see Equations (9)–(11). At the same time, the angle for the overlap of the A-O-A orbitals is much less than 90° and unfavorable compared to the angle between the B-cations. Any alteration in cation distribution changes the lattice constant and the oxygen parameter, thereby altering the spin interactions that determine the super-exchange interactions. Ferrimagnetic ordering occurs because the negative (antiferromagnetic) exchange interaction J_AB between the magnetic cations occupying the tetrahedral A-sites and the octahedral B-sites dominates the also antiferromagnetic intra-sublattice exchange interactions J_AA and J_BB, causing each of the two magnetic sublattices on the A-sites and on the B-sites to be aligned. Apparently, there is a complex magnetic structure within the sublattices. The substructure of the A-sublattice was studied on the example of Co₃O₄, whose crystal field determines the low-spin configuration of the Co³⁺ cations in the octahedral sublattice (S = 0), so the antiferromagnetic arrangement that exists up to 40 K originates only from the magnetic moments of the Co²⁺ cations from the tetrahedral A-sub-sublattices [82]. The substructure of the B-sublattice was investigated in the normal spinels ZnFe₂O₄ and CdFe₂O₄ with nonmagnetic cations in the tetrahedral A-sublattice. Antiferromagnetic ordering in these ferrites up to 9 K originates from the ordering of the Fe³⁺ cations with S = 5/2 on the B-sub-sublattices [83]. Relatively high T_N in Co₃O₄ related to ZnFe₂O₄ and CdFe₂O₄ (although weaker exchange interaction J_AA would be expected due to the greater distance between tetrahedral cations) originates from the additional spin–orbit coupling in the Co²⁺ cations of Co₃O₄.

The exchange constants are determined experimentally based on the temperature dependence of spontaneous magnetization and magnetic susceptibility. The main difficulty is to determine a set of exchange constants which simultaneously satisfy the thermal variation in the spontaneous magnetization and the high-temperature change of the magnetic susceptibility. Various experimentally obtained values for interaction constants can be found in the literature, which the authors show to be in agreement with Anderson spin–wave theory [84,85]. The reliable values given by Srivastava et al. are listed as J_AA = −15 K (Fe_tet−Fe_tet), J_AB′ = −22.7 K (Fe_tet−Co_oct), J_AB’’ = − 26 K (Fe_tet−Fe_oct), J_B′B′ = +46.9 K (Co_oct−Co_oct), J_B′B″ = − 18.5 K (Co_oct−Fe_oct), and J_B″B″ = −7.5 K (Fe_oct−Fe_oct) [85]. All interactions are antiferromagnetic, except Co−Co at octahedral sites. Sawatzky et al. used the molecular-field theory and the analysis of Mössbauer spectra to show that the interaction constants are slightly smaller, but the difference in J_AB′ and J_AB″ is greater in favor of the Fe_tet−Fe_oct interaction [86]. The values of J_AB″ and J_AB′ were found to be −20.07 K and −13.7 K, respectively.

The change in the strength of the super-exchange interactions in cobalt ferrite with the change in the inversion coefficient can only be roughly illustrated here using the structural data obtained by XRD analysis. X-ray diffraction provides “average data” for the investigated material, so the mean distances between cations and the mean angles between different pairs of cations represent a measure of the strength of the “average” super-exchange interactions.

As the lattice constant increases and the inversion coefficient decreases, all hopping distances (b, c, d, e, and f) increase because they depend only on a_exp (see Table 1). We have already seen that the octahedral bonds R_oct = p decrease, while the tetrahedral bonds R_tet = q increase (Table 3). Other cation–anion distances, r and s, also grow. It can be seen that with a decrease in the degree of inversion x (increased number of Co cations in tetrahedral sites), deviations from the ideal values for spinel ferrites increase (

Table 6. The mean angles between different pairs of cations; see Figure 2. Ideal values for spinel oxides are in brackets. The angles corresponding to the most important super-exchange interactions are underlined. The remaining two super-exchange interactions are negligible due to the large distance between the cations.

	x	θ1 (A-O-B) (125.3°)	θ2 (A-O-B) (154°)	θ3 (B-O-B) (90°)	θ4 (B-O-B) (125°)	θ5 (A-O-A) (79°)
US-CO	0.85	123.86	152.21	91.88	125.89	77.69
ME	0.79	123.70	151.95	92.12	125.95	77.43
CO	0.73	123.61	151.79	92.27	125.98	77.27
MW-HT	0.64	123.42	151.49	92.56	126.05	76.96
MC-CO	0.58	123.29	151.28	92.75	126.09	76.75

). Enlarging the distances between the magnetic cations and increasing the deviation of the value of the interbond angles with decreasing x lead to a decrease in the “average” strength of the super-exchange interactions. This also leads to a decrease in the magnetic ordering temperature [87].

In the ideal case, the completely inverse structure of cobalt ferrite, with x = 1, (Fe)^A[FeCo]^BO₄, each Fe³⁺ cation in the A-site would be surrounded by 12 cations from the B-site: 6 Fe³⁺ cations and 6 Co²⁺ cations. At the same time, J(Fe_tet − O − Fe_oct) > J(Fe_tet − O − Co_oct). The cation Fe³⁺ in the B-site would be surrounded by 6 Fe³⁺ cations from the A-site [86,88]. This is the highest possible exchange energy that the Fe³⁺ in the B-site can have (and the smallest that the Fe³⁺ in the A-site can have). Obviously, when the inversion coefficient x decreases, when Co²⁺ cations migrate from the B- to the A-site, the total super-exchange energy in the B-site will decrease and, in the A-site, energy will increase by the same amount. Since the tetrahedral A-cation has a larger number of B-neighbors, it will have a higher total exchange energy and will therefore be less affected by the changes in cation inversion. In contrast, the B-cation undergoes much more significant changes. At first glance, it seems that the total energy does not change; however, as we have already shown by XRD structural analysis (Table 6), the migration of the larger Co²⁺ cations from the B-sites to the A-sites, i.e., decreasing of the inversion x, leads to an increase in the lattice constants and all hopping distances and a decrease in the average strength of the super-exchange interactions (and also to a decrease in the magnetic ordering temperature).

3.8. Influence of Surface Layer, Internal Stress, and Magnetic Anisotropy on the Value of Magnetization

At the surface of the nanoparticles, some exchange bonds are removed and the perturbations of the crystal symmetry at the surfaces should affect the magnetocrystalline anisotropy. The so-called “dead” layer is not completely inactive. The term “magnetically depleted” layer would be more appropriate. At high magnetic fields, there is a partial ordering of the magnetic moments from the surface layers of the nanoparticles. Kodama developed a model for describing the combined effect of reduced coordination and surface anisotropy in ionic materials [89]. He assumed that the ion-to-ion exchange interactions have the same magnitude for bulk and surface atoms. The total exchange interaction is considerably less for surface atoms due to their lower coordination and broken exchange bonds owing to oxygen vacancies or bonding with ligands other than oxygen. The high annealing temperature favors particle growth, improves the crystal structure, and decreases the dead or inert layer responsible for the low magnetization [90].

The surface of the nanoparticles and the internal stress caused by the synthesis conditions greatly affect the values of the magnetization. Maaz et al. showed that the magnetization of the samples obtained with the wet synthesis method increases uniformly with the increasing annealing temperature [18]. Annealing leads to an increase in the size of the nanoparticles. The crystal structure of the nanoparticles is improved; i.e., the volume of the regular crystal structure increases due to the reduction in the disordered surface layer of the nanoparticles. Annealing at temperatures <1000 °C apparently did not lead to a significant change in the inversion coefficient. Among these samples obtained by different synthesis methods but at similar temperatures, the relationship between the saturation magnetization and the nanoparticle size is more complex than that obtained in Ref. [18].

Figure 8 shows the dominant influence of the inversion coefficient on the achieved saturation magnetization (stars). It is obvious that the sample “ME” has a higher saturation magnetization than expected. This sample has the highest magnetic anisotropy constant and effective blocking temperature also (Table 5), which is a sign of a very regular crystal structure in a large part of the sample.

A linear fit (solid red line) to the magnetizations M_s(x) has the parameters μ_Fe = 3.25 μ_B and μ_Co = 2.5 μ_B, which correspond to the mean values of the cation magnetic moments at 300 K. The obtained values are expected for the magnetic moments of the cations in nanoparticles of similar sizes (about 40% lower than the magnetizations at low temperatures).

4. Conclusions

This study examined stoichiometric single-domain cobalt ferrite nanoparticles produced by five different synthesis methods. XRD analysis confirmed the formation of pure CoFe₂O₄ with Fd$\overline{3}$m symmetry in all samples. The lattice constant was found to increase more significantly with decreasing cation inversion in the smaller nanoparticles compared to the larger ones. Internal stress was more influenced by synthesis conditions than by particle size within the 15.8 to 19.0 nm range studied.

Hysteresis loops show magnetization asymptotic increase for high fields and do not achieve saturation at ±90 kOe. This is generally attributed to the canted or disordered spins at the surface of the nanoparticles. Magnetic measurements showed that saturation magnetization was primarily influenced by the inversion coefficient, with M_s decreasing as cation inversion increased. Particle size had the dominant effect on magnetic remanence and coercivity, both of which increased linearly with particle size, confirming the single-domain nature of the nanoparticles. The surface layer characteristics, influenced by the synthesis method, affected the magnetization behavior in the high-field region. The slope of the magnetization in the high-field region is the smallest in the case of the samples synthesized with ultrasonically assisted coprecipitation and coprecipitation followed by mechanochemical treatment. At a first glance, this was unexpected due to the small average size of these nanoparticles. However, the synthesis had a decisive influence on the thickness of the surface layer. Smaller particles showed easier magnetic moment rotation and lower anisotropy, while slightly larger particles exhibited higher dipolar interaction and anisotropy. The blocking temperatures and effective anisotropy constants increased with particle size.

By combining XRD analysis and magnetic measurements, this study provides a comprehensive understanding of how synthesis procedures and nanoparticle size variations influence the magnetic properties of cobalt ferrite nanoparticles.