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Another important aspect to consider, when a magnetic material is reduced to the nanoscale, is the saturation magnetization of the material, which is influenced by such reduction.

In the case of bulk magnetic material, the saturation magnetization (M_sat) (theoretically obtained in intense magnetic field and at low temperatures) is equal to the spontaneous magnetization (M_s), being a known parameter of material. Example, in the case of Fe, the saturation magnetization at room temperature is 1714 kA m⁻¹ (Cullity and Graham 2009), and in the case of Fe₃O₄, it is 477.5 kA m⁻¹ (Smit and Wijin 1961). The spontaneous magnetization of the bulk magnetic material decreases with temperature, having the maximum value at 0 K (M_s(0)) and zero at a temperature, generally high (hundreds of degrees), which is Curie temperature in the case of ferromagnetics and Nèel temperature in the case of ferrimagnetic materials. The temperature variation of the spontaneous magnetization of massive ferromagnetic material, such as Fe, Co, and Ni (Figure 1.6a), is a universal function that does not involve indeterminate constants:

(1.11)

Figure 1.6 (a) Relative saturation magnetization of iron, cobalt, and nickel as a function of relative temperature. Calculated curves are shown for three values of J.

Source: Cullity and Graham (2009). Reproduced with permission from John Wiley & Sons;

(b) Curves that represent the dependence of the saturation magnetization on temperature according to Eq. (1.12) (curve α) and Eqs. (1.13) and (1.14) (curve β); experimental curve (□).

Source: Caizer (2005a). Reproduced with permission from Springer Nature.

At low temperatures, this variation is well described by Bloch's law (law in T^3/2 (Bloch 1930), deduced from the spin wave model,

(1.12)

which is the exact solution of the Hamiltonian Heisemberg at low temperatures. In Eq. (1.12), B is a constant whose value depends on the exchange integral.

This dependence is well verified experimentally both for bulk ferromagnetic materials (Fe, Ni) (Aldred and Frohle 1972; Aldred 1975) and for some bulk spinel ferrites, such as Mn_xFe_{3 − x}O₄ ferrite for 0.2 < x < 1.0 (Dillon 1962).

However, when the magnetic material has nanometric sizes, such as nanoparticles, some theoretical calculations and some experimental results have shown that the temperature exponent is greater than the value 3/2, corresponding to the bulk material (Hendriksen et al. 1992, 1993; Linderoth et al. 1993). Thus, Morais et al. (2000) showed that, depending on the temperature of the saturation magnetization in the range 4.2–293 K, in the case of a ferrofluid with NiFe₂O₄ nanoparticles having an average diameter of 11.1 nm, it deviates from the law corresponding to the bulk material. In the case of magnetite (Fe₃O₄) nanoparticles coated with oleic acid, Caizer (2003b) shows that the law is well verified for T² instead of T^3/2, and the constant of M_s(0) in Eq. (1.12) becomes in this case a function of temperature:

(1.13)

However, in the case of Mn_0.6Fe_2.4O₄ nanoparticles coated with oleic acid (Caizer 2005a), it is found that is verified in the law of Eq. (1.12), but M_s(0) is no longer a constant but is temperature‐dependent according to the formula:

(1.14)

In this equation, n is the concentration of the nanoparticles, and c_i are constants whose value is known, resulting from the fitting of the experimental data. In Figure 1.6b is shown the dependence on M_sat − T in this case, where the deviation is highlighted: (α) is the curve for bulk and (β) is the curve for nanoparticles. These results as well as others (Caizer 2002) show that T^3/2 valid in the case of bulk magnetic material is no longer verified in the case of magnetic nanoparticles, and there are different interpretations for this.

Another important aspect in the case of nanoparticles is the fact that the saturation magnetization measured experimentally at room temperature is lower than that corresponding to the same bulk magnetic material (Berkowitz et al. 1975; Zhang et al. 1997; Kodama 1999; Caizer and Stefanescu 2002; Caizer 2003b). It has also been found experimentally that the decrease in saturation magnetization is all the more pronounced the smaller the nanoparticles, a few nm. In Figure 1.7 is shown the dependency of saturation magnetization as a function of the average nanoparticle diameter in the case of Ni_0.35Zn_0.65Fe₂O₄ nanoparticles (Caizer and Stefanescu 2002). Also, a similar dependency is shown in Figure 1.12b for the magnetic moment of the nanoparticles (Wu et al. 2017).

Figure 1.7 (a) Specific saturation magnetization as a function of the mean diameter of nanocrystallites.

Source: Caizer and Stefanescu (2002). Reprinted by permission from IOP Publishing.

(b) Core‐shell pattern of the spherical nanoparticle.

Source: Caizer (2016). Springer Nature.

In Ref. (Caizer 2016) is presented in more detail the aspects that lead to the decrease of the saturation magnetization of the nanoparticles, considering the core‐shell model in which there is a layer on the surface of the nanoparticles where the magnetic moments are no longer aligned as in the ordered magnetic core. As a result, in the case of nanoparticles, a magnetic diameter (D_m) determined by the nanoparticle core in which the spins are magnetically aligned must be considered, which is generally smaller than the physical diameter (D) of the nanoparticles: D_m < D. Difference (D − D_m) becomes even more pronounced in the case of ferrimagnetic nanoparticles (Berkowitz et al. 1975, 1980; Kodama et al. 1996), surfactated (Caizer 2002) or in SiO₂ matrix, this reaching even up to 2–3 nm (Caizer et al. 2003; Caizer 2008).

To conclude, if in the case of bulk magnetic material, the saturation magnetization is a well‐defined value, being a material parameter, characteristic of the substance type, in the case of nanoparticles it is generally smaller, and decreases with the decrease in diameter to nanometers size. This is a very important aspect that must be taken into account in biomedical applications. Thus, in order not to introduce errors in the application of magnetic nanoparticles, it is recommended, before conducting any experiment, to determine/measure the saturation magnetization of the nanoparticles, and also the variation of saturation magnetization with temperature, if there is such a dependency in the targeted application.