We conducted rigorous studies by X-ray diffraction of the MCFO nanoparticles in the entire range of composition to understand the structure-property relationship, which in turn affected the efficiency of hyperthermia of the nanoparticles. Nanoparticles’ size, shape, and magnetic characteristics affect specific loss power (SLP) and maximum attainable temperature (T _max ) of hyperthermia, the ultimate engineering parameters for hyperthermia efficiency. The relaxation phenomena that incur self-heating properties are the Néel and Brownian relaxations and hysteresis loss. Néel relaxation is directly proportional to the volume and anisotropy of the nanoparticles. On the other hand, Brownian motion is directly proportional to the size of the nanoparticles. Nanoparticle size, cation distribution, and magnetic state (ferro/superparamagnetic) affect hysteresis loss. The structure of Co-Mg mixed-spinel ferrites undergoes considerable change throughout the whole spectrum of composition of MCFO due to the nano-magnetism of MgFe₂O₄ and CoFe₂O₄ at an ultrasmall scale, which again changes with particle size. Hence, acquiring an in-depth understanding of the structure of MCFO nanoparticles at different compositions would contribute to perceiving structure-property relationship, which would affect SLP and T _max of hyperthermia.

At an annealing temperature of 200°C and when x = 0 to 0.3, the XRD pattern exhibits a broad diffuse hump at the 35° position. We delineated in the TEM section that the existence of lattice fringes within the x = 0 to 0.3 range indicates ultra-small particle size. The peaks corresponding to the spinel structure are observable at the annealing temperature of 200°C for the composition, x ≥ 0.4. Peak shift occurs as x increases due to the difference in the ionic radii of Co²⁺ (0.072 nm) and Mg²⁺ (0.065 nm) (Islam et al., 2022; Anjum et al., 2017). At an annealing temperature of 400°C, similar variations occur, although all six peaks are noticeable for x ≥ 0.3. At the annealing temperatures of 600°C and 800°C, all the peaks are clearly visible with narrower peak widths, indicating grain growth and higher crystallinity right from x = 0. The peaks have shifted towards the lower angle side, indicating an increase in lattice spacing due to the grain growth. The nanoparticles possess a high ratio of surface area to volume. As the particle size increases, the core of the particles increases, and the surface atoms reduce. Consequently, the ordered phase increases, replacing an incomplete coordination of disordered surface atoms. This results in a peak shift as the crystallinity and grain growth increase (Fouad et al., 2019).

The peak width of the diffraction peaks estimates the crystallite size. The kinematical theory of x-ray diffraction reveals that the total intensity of each diffraction peak from a single crystal, which relates to the scattering angle, 2θ, is directly proportional to the volume of the crystal (Valério and Morelhao 2019). As the volume of the crystallites increases, the peak area also increases while the peak width (FWHM) β decreases. The width of the peak is inversely proportional to the size of the crystallite. The width of each peak ascertains the mean particle size of the crystallites. It is common practice to utilize the width of the maximum intensity peak, which has the highest multiplicity factor. In the current study, the peak of Miller indices (311) exhibits the highest multiplicity factor, resulting in the maximum intensity we used to determine the crystallite size. Figure 2 demonstrates that the peak width decreases as the crystallinity increases due to a concurrent rise in Co concentration and annealing temperature. Crystallinity and grain growth also result in a displacement of the peaks. This is because smaller particle size results in larger surface areas. When the crystallinity increases, the surface area decreases due to the coalescing of smaller grains and the increased coordination number, while the lattice strain decreases. By altering the lattice spacing, d, the position of the peak changes following Bragg’s Law.

We used Eq. 1 to determine the grain size where D is the average crystallite size, λ is the X-ray wavelength, β is the full width at half maximum value of the highest intensity (311) peak in radians, and θ is the Bragg angle (Kumar et al., 2013). D = 0.94 λ β ⁡ cos ⁡ θ (1)

Figure 3 presents the variations of (a) the particle size D, (b) the observed lattice parameter a, (c) the X-ray density, (d) the specific surface area of the particles S, (e) the ionic radius of the tetrahedral site r _A , (f) the ionic radius of octahedral site r _B , (g) the theoretical lattice parameter ath, (h) the hopping length for tetrahedral site d _A , (i) the hopping length for octahedral site d _B , (j) the bond length of tetrahedral site d _AX , (k) the bond length of octahedral site d _BX , (l) the tetrahedral edge d _AXE , (m) the shared octahedral edge d _BXE , and (n) the unshared octahedral edge d _BXEU of MCFO nanoparticles at different values of x and annealed samples. Supplementary Table S1–S4 present the relevant data.

Figure 3A represents the variation of particle size with Co²⁺ content, x of MCFO ferrite nanoparticles annealed at different temperatures. Particle size increases with an increase in x and annealing temperature. The observed lattice parameter for each plane was calculated using Bragg’s law presented in Eq. 2 Kumar et al. (2013). d h k l = a h 2 + k 2 + l 2 (2)

Figure 3A presents the precise values of the lattice parameters determined using the Nelson-Riley function, F (θ). The intercept of the linear extrapolation of a vs. F (θ) for each plane gives the precise value of the lattice parameter (Cullity 2014; Chintala et al., 2021).

Figure 3B shows that the lattice parameter increases linearly with the increase in x and annealing temperature, which indicates that the dependence of the lattice parameter with composition follows Vegard’s Law (Denton and Ashcroft, 1991; Yadav et al., 2017). The lattice parameter increases with an increase in Co²⁺ content because the ionic radius of the Mg²⁺ (0.065 nm) ion is smaller than that of the Co²⁺ (0.072 nm) ion (Islam et al., 2022; Abraham et al., 2018). The lattice parameter also increases with an increase in annealing temperature because the core of the grains increases, replacing the disordered surface atoms with an increase in annealing temperature, and surface tension decreases (Li et al., 2020).

Figure 3C represents the variation of X-ray density with Co²⁺ content, x of MCFO nanoparticles annealed at different annealing temperatures. The X-ray density increases with an increase in x and annealing temperature. We calculated the specific surface area of the particles (S), the radius of the tetrahedral site (r _A ), the radius of the octahedral site (r _B ), the theoretical lattice parameter (ath), the hopping length for the tetrahedral site (d _A ), hopping length for the octahedral site (d _B ), the tetrahedral and the octahedral bond length(d _AX and d _BX ), the tetrahedral edge (d _AXE ), and the shared and unshared octahedral edge (d _BXE and d _BXEU ) for cubic spinel ferrite nanoparticles using the equation given in the literature (Satalkar and Kane, 2016). We used the equations in the supplementary section and Supplementary Table S1–S4 present all data at different compositions and annealing temperatures. Figures 3D–N shows the variation of S, r _A , r _B , d _A , d _B , d _AX , d _BX ,d _AXE , d _BXE , and d _BXEU with Co²⁺ content, x of MCFO nanoparticles annealed at different annealing temperatures. The value of specific surface area, S, decreases with Co²⁺ content, x, and annealing temperature. This is expected because, when particle size increases with x and annealing temperatures, the specific surface area decreases since the core of the nanoparticles grows at the expense of disordered surface atoms. The value of r _A , r _B , d _A , d _B , d _AX , d _BX , d _AXE , d _BXE , and d _BXEU increases with an increase in Co²⁺ content and annealing temperature because particle size increases with an increase in Co²⁺ content and annealing temperature, which is associated with cation redistribution with grain growth. Co²⁺ has a tendency to occupy B-site, while Mg²⁺ ion has the tendency to occupy A-site. Changes in composition and annealing temperatures cause changes in cation distribution, which lead to the change in tetrahedral and octahedral radii, hopping length at the tetrahedral and octahedral sites, the bond length, and shared and unshared tetrahedral and octahedral edges.

Figures 4A–E presents the variation of the interionic distances between the cations, (f–i) the cation-anion distance, and (j–n) the bond angle of Mg_1−x Co _x Fe₂O₄ nanoparticles annealed at 200°C, 400°C, 600°C, and 800°C. Figures 4A–E presents the interionic distances between cations b, c, d, e, and f obtained using the equation presented in the literature by (Satalkar and Kane, 2016). We presented all the equations in the supplementary section and the deduced data in Supplementary Table S5–S8. Figures 4F–I present the spaces between cations and anions p, q, r, and s using the equations in the supplementary section and the deduced data in Supplementary Table S5–S8. The cation-cation distances and cation-anion distances increase with an increase in Co²⁺ content and annealing temperature because the ionic radius of Co²⁺ (0.072 nm) is higher than that of Mg²⁺ (0.065 nm). Figures 4J–N show the variations in bond angles θ ₁, θ ₂, θ ₃, θ ₄, and θ ₅ at different compositions and annealing temperatures. Change in cation distribution with composition and annealing temperature also leads to change in the cation-cation, cation-anion distances, and the bond angles.

Figures 5A,B displays transmission electron microscopy (TEM) images of Mg_1−xCo_xFe₂O₄ nanoparticles, where x ranges from 0 to 1 with an increment of 0.1. The nanoparticles were annealed at temperatures of 200, 400C, 600, and 800°C. The inset displays selected area diffraction (SAED) patterns, exhibiting the most prominent peak (311) as well as additional planes such as (220), (400), (420), (511), and (440). The d_hkl values in the SAED patterns were determined using Velox software, and the corresponding diffractograms were indexed accordingly. In addition, SAED patterns also indicate that the Debye rings are wide for smaller particle sizes. The sharpness of the rings increases as the crystallinity and particle size increase. The TEM images exhibit semi-spherical particles that exhibit grain growth as the cobalt concentration x and annealing temperatures increase. The particle size distributions for all annealing temperatures and compositions were determined by analyzing the histogram of the size distribution, which exhibits a log-normal distribution. Figures 6A–D displays the log-normal distribution of MCFO nanoparticles and Figure 6E presents the relationship between particle size and composition, as obtained from the histogram. The data also indicates that the particle size tends to rise as the cobalt content and annealing temperature increase. Figure 3A depicts the relationship between the size of crystallite and x, obtained by X-ray diffraction, at various annealing temperatures. The particle sizes measured by X-ray diffraction (XRD) in Figure 3A and transmission electron microscopy (TEM) in Figure 6E are in good agreement.

Figure 7 displays high-resolution transmission electron microscopy (HRTEM) images of MCFO nanoparticles subjected to different annealing temperatures: 200, 400, 600, and 800°C. The HRTEM images indicate that all the samples under investigation possess a nanocrystalline structure, with the degree of crystallinity being directly proportional to both the x and the annealing temperatures. MCFO with higher cobalt concentration and higher annealing temperatures exhibited distinctive lattice fringes in the HRTEM image, which result in a significant enhancement of crystallinity.

Figure 8 displays chitosan-MCFO nanohybrids with varying compositions (0 ≤ x ≤ 1; ∆x = 0.1) that have been subjected to annealing at a temperature of 200°C. The illustration demonstrates that the particles exhibit a higher degree of dispersion when they are coated with chitosan. These figures can be compared with the images depicted in Figure 5 of uncoated samples annealing at the same temperature of 200°C. The samples annealed at other temperatures are anticipated to exhibit a more comparable dispersion of chitosan-MCFO nanohybrid than the MCFO nanoparticles.

To conduct a more detailed structure analysis, we employed Raman spectroscopy to examine MCFO nanoparticles subjected to annealing at 200°C–800°C to find vibrational modes. Figure 9 displays the Raman spectra of MCFO nanoparticles of annealed samples. The spectra were acquired in the range 190–1,000 cm⁻¹ at room temperature. It was well-established in the literature that MCFO ferrites possess a partially inverse spinel structure belonging to the Fd−3m space group. The vibrational modes associated with this space group are A_1g (R), E_g (R), F_1g, 3F_2g (R), 2A_2u, 2E_u, 4F_1u (IR), and 2F_2u. The notation “R" indicates Raman active vibrational modes, “IR” represents infrared–active vibrational modes, and the other modes are silent. The Raman spectrum would exhibit the Raman active modes A_1g, E_g, F_2g(3), F_2g(2), and F_2g(1) (Soler et al., 2004; Galinetto et al., 2018).

Figure 10 illustrates the rendition of the Raman spectra of annealed nanoparticles achieved by applying the Gaussian function after subtracting the background and performing deconvolution. Gaussian function yielded the validation between the experimental and theoretical data. The A_1g modes correspond to the symmetric stretching of MeO₄ (where Me represents Co and Mg) and FeO₄ at the A-site. The E_g modes correspond to the bending of oxygen atoms with respect to the iron atom at the B-site. The F_2g(3) modes correspond to the antisymmetric bending of oxygen in the presence of Fe while the F_2g(2) modes depict the asymmetric stretching of Fe in conjunction with O. The F_2g(1) modes exhibit the translational movement of both Fe and O. The Raman peaks within the range of 660–720 cm⁻¹ correspond to the vibrational modes of the A-site, while the Raman peaks within the range of 460–660 cm⁻¹ correspond to the vibrational modes of the B-site in the ferrites (Soler et al., 2004; Galinetto et al., 2018; Baig et al., 2021).

Figures 11A–D illustrate the relationship between the area integral and composition at the annealing temperatures ranging from 200 to 800°C. The largest A-site occupancy for both Me and Fe in Figure 11A is observed for MgFe₂O₄ at an annealing temperature of 200°C, corresponding to a particle size range of 4.3–9.6 nm. The rise in Co content leads to a decrease in the occupancy of both Me (Mg and Co) and Fe on the A-site, while E_g and F_2g(2) increase due to the increase in B-site occupancy. These findings indicate that for 4.3–9.6 nm particle size, compositions with a higher concentration of magnesium tend to favor the normal spinel structure, while compositions with a higher concentration of cobalt tend to favor the inverse spinel structure. The area integral of additional small intensity peaks, F_2g(1) and F_2g(3), remain constant regardless of the composition.

At 400°C, for particle size range 5.5–15.8 nm in Figure 11B, the relationship between the area integral A_1g and F_2g(2) becomes relatively random with x. Nevertheless, there is a noticeable tendency of decreased A_1g and increased F_2g(2) with x. This reflects a transformation of the spinel structure from normal to inverse spinel structure. In Figure 11C, the scattered nature of the area integral with composition is more pronounced for the particle size range of 10.1–24.9 nm at an annealing temperature of 600°C. Nevertheless, it was seen that the area integral of A_1g for Fe ions is dropping (A-site occupancy), while the area integral of F_2g(2) is growing (B-site occupancy). This indicates a decrease in A-site occupancy and an increase in B-site occupancy with the increase of Co, resulting in a transition from a normal to an inverse spinel structure with x. At 800°C, specifically for particle sizes ranging from 15.9 to 30.3 nm, the occupancy of A-sites declines while B-sites increase with the increase of Co. This indicates a transition in the spinel structure from normal to inverse spinel with x. The Raman shift values for the Raman-active modes A_1g, E_g, F_2g(1), F_2g(2), and F_2g(3) of the Mg_1−xCo_xFe₂O₄ composition annealed at temperatures ranging from 200°C to 800°C are provided in Tables 1, Table 2, Table 3, and Table 4. These values were obtained using Gaussian fitting and deconvolution of the spectra shown in Figure 10. The frequencies of A_1g vibrational modes decrease when the cobalt content, x, increases in all samples. This is due to the substitution of lighter magnesium (24.3050 emu) ions with heavier cobalt (58.9332 amu) ions with the increase of Co²⁺. The Raman shift of F_2g(2) also exhibits a reduction as the Co²⁺ increases. In this case also, magnesium (24.3050 amu) ions are replaced with the heavier cobalt (58.9332 amu) ions, which have a greater atomic mass. To summarize, particles with higher magnesium content have a more pronounced normal spinel structure. Conversely, particles containing higher amounts of cobalt display a more pronounced inverse spinel structure. Raman shift basically decreases at all values of x for both A and B sites because of the greater atomic mass Co²⁺ than Mg²⁺.

Fourier-transform infrared spectroscopy (FTIR) of MCFO nanoparticles annealed at 200°C, 400°C, 600°C, and 800°C are presented in Figure 12. Absorption peaks at 3,430–3,493 cm⁻¹ were observed due to the stretching vibration of the O-H group (Zeeshan et al., 2018). Two characteristic peaks of spinel ferrite of the cubic structure at the lower frequency region were observed. The higher frequency band is related to the stretching vibration of a metal-oxide bond at the tetrahedral (A) site and the lower frequency band is related to the stretching vibration of a metal-oxide bond at the octahedral (B) site because the bond length of the A site is shorter than the bond length of the B site (Zeeshan et al., 2018; Naseri et al., 2014; Gadkari et al., 2009; Pawlak and Mucha, 2003; Josyulu and Sobhanadri, 1981; Mund and Ahuja, 2016; Sharma et al., 2016). The frequency of both the higher frequency band and the lower frequency band shifted towards the higher frequency region with an increase in Co²⁺ content because Co²⁺ (0.072 nm) replaces Mg²⁺ (0.065 nm) at the A site and Fe²+ (0.0645 nm) at the B site which extends the covalent bond. Both the frequency bands shifted toward the higher frequency region with increasing annealing temperatures because particle size increases with an increase in annealing temperature.

Magnesium-rich compositions prefer normal spinel structures in their Raman spectroscopy, whereas cobalt-rich compositions demonstrate a preference for mixed spinel structures. Raman and FTIR spectroscopy produce similar outcomes. The reason is that the vibrational modes of the inverse spinel structure exhibit more clarity compared to those of the normal spinel structure, making them more readily detectable using Raman spectroscopy. FTIR spectroscopy can detect a wider variety of vibrational modes, which allows it to accurately identify both inverse and normal spinel structures. Consequently, these two methods can be employed in conjunction to discern and distinguish various spinel configurations. Additionally, they aid in the characterization of the vibrational modes of A-site and B-site at different compositions and annealing temperatures. At a temperature of 200°C, an absorption band with a lower frequency (v₁) was detected in the range of 403–418 cm⁻¹, while an absorption band with a higher frequency (ν₂) was found in the range of 603–622 cm⁻¹ for various compositions. The lower frequency absorption band (ν₁) was detected at a temperature of 400°C, with a range of 398–415 cm⁻¹. Similarly, the higher frequency absorption band (ν₂) was identified within the range of 597–618 cm⁻¹. The absorption band (ν₁) with lower frequency, ranging from 395 to 412 cm⁻¹, was seen at a temperature of 600°C. Additionally, the absorption band (ν₂) with higher frequency, ranging from 593 to 610 cm⁻¹, was also observed at the same temperature. Finally, at a temperature of 800°C, the absorption band with a lower frequency (ν₁) was detected in the range of 391–408 cm⁻¹, while the absorption band with a higher frequency (ν₂) was seen in the range of 580–600 cm⁻¹. The frequency band changed towards the lower frequency region as the particle size (annealing temperature) increased, due to the contraction of the covalent bond with larger particle sizes (annealing temperatures). Smaller particles exhibit a greater surface-to-volume ratio, which leads to incomplete coordination. In contrast, larger particles have a smaller surface area, resulting in more complete coordination and causing covalent bonds to decrease.

The force constants (FC) for the A site (F_CT) and B site (F_CO) are obtained using the following relation: F C = 4 π 2 c 2 ν 2 m (4) where c is the speed of light, v is the vibration frequency of the A site and B site, and m is the reduced mass for the Fe²+and O²⁻ ions (Yadav et al., 2017). Variations of F _CT and F _CO of Mg_1−x Co _x Fe₂O₄ ferrites annealed at 200°C, 400°C, 600°C, and 800°C with Co²⁺ content, x are presented in Figure 12B,C. The data are presented in Supplementary Table S9. The F _CT and F _CO increases with increasing Co²⁺ content x because the bond length of the A site and the B site decreases with an increase in Co²⁺content (Rana et al., 2010). In our previous study, Islam et al. (2022), FTIR spectra of uncoated and chitosan-coated MCFO in the as-dried condition are presented. The good bond of chitosan and MCFO is manifested from the peak shift of A and B-site bonding. FTIR spectra of uncoated and chitosan-coated MCFO in the as-dried condition are presented. The chitosan and MCFO bonded well, which was manifested in the peak shift of A and B-site bonding before and after coating. We expect a similar situation in the present study also.

The variation of magnetization (M) with an applied magnetic field (H) of MCFO ferrite nanoparticles annealed at 200°C–800°C are presented in Figure 13. The magnetization increases with an increase in Co²⁺ content because Co²⁺ replaces Mg²⁺, and the magnetic moment of Co²⁺ (3.88µ _B ) is higher than Mg²⁺ (0µ _B ) (Mund and Ahuja 2016; Anis-Ur-Rehman et al., 2011). The magnetization also increases with an increase in annealing temperature because of the increase in the ordered core replacing disordered surface atoms. The saturation magnetizations (M _s ) were determined by using Law of Approach to Saturation Jr. and Silva (2011). The anisotropy constants (K) were obtained by using the relation K = H c × M s 0.96 (5) where H _c is the coercive field (Yadav et al., 2017). Figure 14 shows the variation of (M _s ), anisotropy constant (K), coercivity (H_c), and remanence ratio (M _r /M _s ) with particle size of MCFO ferrite nanoparticles with compositions and annealing temperatures. The data are presented in Supplementary Table S10. The values of M _s and M _r /M _s increase with the particle size because of the increase of ferrimagnetic core at the expense of the disordered surface. Further, we observed from Raman and FTIR spectroscopy that B-site occupancy of Fe tends to increase rather than the A-site occupancy, which increases the A-B exchange interaction. The values of K and H _c increase because, with the increase of particle size, the particles become multidomain and there is an increased amount of Co. However, with the increase of Co²⁺, all the values of M _s , M _r /M _s , K, and H _c increase because of the higher magnetic moment of Co²⁺ (3.88µ _B ) than Mg²⁺ (0µ _B ). The values of K and H _c increase with an increase in Co²⁺ content because of the higher anisotropy of CFO (Mund and Ahuja, 2016).

Figure 15 shows the time-dependence temperature evolution of chitosan-MCFO nanohybrids with different particle sizes for the entire range of composition for the concentration of 4 mg/mL. Similar experiments were also carried out for three more concentrations of 0.5, 1, and 2 mg/mL, which show similar heating profile. The temperature increases with time linearly first and then it reaches a plateau value. The temperature evolution of chitosan-MCFO nanohybrid increase with the particle size for all compositions. The temperature at the plateau region also increases with the Co²⁺ content because of the increase in magnetic moment with Co²⁺. Habib et al. (2008) reported similar behavior for Fe-Co alloy, magnetite, and maghemite nanoparticles.

Figures 16, 17 represent the variation of T _max and specific loss power (SLP) with the particle size of chitosan-MCFO nanohybrid of different concentrations. The data are presented in Supplementary Table S11, S12. Initially, the value of T _max and SLP increases with particle size for each sample, reaches a maximum, and then decreases. It is intriguing to note that both T _max and SLP exhibit similar behaviors with x and particle size. However, with the increase of nanoparticle concentration in water from 0.5 to 4 mg/mL, specific loss power decreases while T _max increases. This might be because, with the increase of concentration, Brownian relaxation will be impaired because of agglomeration of the particles at the initial stage of particle heating. This is because the initial slope of time-dependence temperature curve determines SLP. With time, when the particles get sufficient energy, this hindrance of Brownian relaxation is overcome. By the time the temperature reaches the plateau value, particles gain sufficient energy to increase T _max with concentration.

It will be interesting to revisit the expressions responsible for the self-heating properties of the nanoparticles and the dependence on the particle size and magnetic anisotropy with an alternating magnetic field. The self-heating properties of the nanoparticles are affected by the Néel and Brownian relaxations and hysteresis loss (Torres et al., 2019; Barati et al., 2014; Reeves and Weaver, 2014; Ferguson et al., 2013). By definition, specific loss power is the energy absorbed by the nanoparticles per unit volume and time and can be expressed as, S L P   W / g = P / ρ (6) Where, P = π µ o χ o H 2 f   2 π f τ e f f   / 1 + 2 π f τ e f f 2 (7)

P is the mean volumetric dissipation power, ρ is the density of the magnetic material, χ _o is the initial magnetic susceptibility, H is the magnetic field amplitude, f is the frequency, and µ _o is the permeability of the vacuum (Ota and Takemura, 2019). The τ _eff is the weighted average of the Néel and Brownian relaxations and related to the Néel and Brownian relaxations by the following formula τ e f f = τ N + τ B / τ N   τ B (8)

The Néel and Brownian relaxations are very effective for the self-heating properties of nanoparticles up to small anisotropy and particle size, which can be expressed as, τ N = τ o ⁡ exp K V / k B T   (9) and τ B = 4 π η r H 3 / k B T (10) where K is the anisotropy of the magnetic nanoparticle, V is the volume of the magnetic core, η is the viscosity of the solvent, r _H is the hydrodynamic radius of the magnetic nanoparticle, k _B is the Boltzmann’s constant, and T is the temperature. The above equations show that the Néel relaxation is directly proportional to the anisotropy and the particle size, while the Brownian relaxation is directly proportional to the hydrodynamic diameter, which is again related to the particle size. There exists a critical diameter above which hysteresis loss comes into play for the self-heating properties of the nanoparticles (Barati et al., 2014; Islam et al., 2020),

The critical diameter can be expressed using the following relation: D c r = 6 ⁡ ln t m f o k B T / π K 1 / 3 (11) Where, K is the anisotropy constant of MCFO, f _o is the frequency, k _B the Boltzmann’s constant, T the temperature, and t _m the measurement time (Barati et al., 2014). The given expression shows an inverse relationship between magnetic anisotropy and critical diameter. The anisotropy constant causes the critical diameter to decrease as the Co²⁺ concentration increases. Beyond the critical diameter, anisotropy exceeds the threshold for Néel relaxation. For lower Co²⁺ content and smaller particle sizes, Néel relaxation dominates the effective relaxation time, τ_eff. However, Brownian relaxation dominates the relaxation mechanism beyond the critical diameter. Figure 18 depicts the particle size dependence of Néel and Brownian relaxation time, as well as the effective relaxation time. It is observed in the figure that the Néel relaxation, τ _N, is faster while Brownian relaxation, τ _B, is slower. Up to x = 0.5, the Néel relaxation time, τ _N , and the effective relaxation time, τ _eff , coincide fully. For x = 0.6, τ _N and τ _eff starts separating beyond 16 nm particle size. For x = 0.7 and above there is crossover at a certain particle diameter below which τ _N is faster and in this range τ _eff coincides with τ _N . Above the crossover, τ _B is faster and in this range τ _eff coincides with τ _B . The magnetic domain’s influence on anisotropy is widely recognized. Particles are monocrystalline and monodomain, having minimal anisotropy at small sizes. However, as particle size grows, they become multidomain, resulting in the presence of anisotropy. Anisotropy rises with particle size. As the Co²⁺ content increases, particle size and critical diameters reduce further because of the intrinsic increase of anisotropy. In Figures 16, 17, the SLP and T_max increase up to the critical diameter, reach a maximum, and then decrease with particle size. The foregoing rationale illustrates how anisotropy and average particle size affect these parameters.

The range of T_max and SLP obtained in this study can be efficiently used for the annihilation of cancer cells as we observed in our previous studies (Hoque et al., 2016; Hoque et al., 2021; Hyder and Hoque., 2017). We observed above 98% annihilation of 9L Gliosarcoma cancer cells by using CoFe₂O₄ and Fe_xCo_1-xFe₂O₄ and chitosan nanohybrid as the thermotherapeutic agent for 15 min exposure to an rf magnetic field. We expect similar annihilation of cancer cells by the chitosan-MCFO nanohybrids for comparable specific loss power and T_max.