(the magnetic moment of the nanoparticle) is no longer stable, reversing at 180°, due to thermal activation (at a temperature). Thus, the magnetic moment of the nanoparticle, below the threshold diameter Vth, fluctuates along the direction of easy magnetization (Figure 1.16b) (Néel 1949). The inversion or not of the magnetic moment of the nanoparticle in the absence of the external magnetic field will depend on the barrier energy given by the uniaxial magnetic anisotropy energy (applied in the case of nanoparticles) (Caizer 2016):
(1.28)
where Ku is the uniaxial magnetic anisotropy, V is the magnetic volume of the nanoparticle, and φ is the angle between the magnetic moment and the easy magnetization axis. In this case, the barrier energy (KuVm) is maximum at φ = 90° (Figure 1.17a). Figure 1.17a shows the case of the Co ferrite nanoparticle with mean magnetic diameter of 10 nm and Ku = 1.6 × 105 J m−3, the energy being calculated in eV (electron‐volt) (Caizer and Tura 2006).
Figure 1.17 Nanoparticle energy as a function of ψ and φ angles for (a) H = 0 and (b) H = 100 kA m−1; (c) Orientation of spontaneous magnetization Ms of a nanoparticle with respect to external field H direction (q) and the easy magnetization axis (z).
Source: Reprinted from Caizer and Tura (2006), with permission of Elsevier.
In the presence of an external magnetic field, the situation changes radically, depending on the orientation of the magnetic field and the magnetic anisotropy axes of the nanoparticle (Caizer and Tura 2006) (Figure 1.17b). The situation becomes more complex in a nanoparticles system when the uniaxial magnetic anisotropy axes are oriented in all directions (Caizer 2004b).
In the absence of the magnetic field (H = 0) and for a type of material, there will be a statistical probability of the magnetic moments of the nanoparticles passing over the potential barrier, this being higher as the volume Vm of the nanoparticles will be lower and the temperature (T) higher. This process is characterized by a time called magnetic relaxation time (Nèel 1949),
where τ0 is a time constant that is generally of 10−9 (Back et al. 1998).
In biomedical applications, the dispersions of magnetic nanoparticles (nanoparticles dispersed in pharmaceutical liquids) are often used for various purposes (Caizer 2010; Caizer 2017). Thus, in these cases, the magnetic nanoparticles can move freely in the liquid (due to thermal agitation) and/or under the action of an external magnetic field. Thus, in this case, besides the rotation of the magnetic moments inside the nanoparticles (Nèel magnetic relaxation) (Figure 1.18a), there will also be a rotation of the nanoparticles with its fixed magnetic moment along the easy magnetization (Figure 1.18b) (Laurent et al. 2008; Reeves and Weaver 2014). This process is Brown magnetic relaxation (Brown 1963) and is quantitatively characterized by Brown relaxation time,
where η is the viscosity coefficient and Vh is the hydrodynamic volume of nanoparticle.
Figure 1.18 Illustration of the two components of the magnetic relaxation of a magnetic fluid: (a) Nèel and (b) Brown relaxation.
Source: Reprinted with permission from Laurent et al. (2008). Copyright 2008 American Chemical Society.
Of course, in reality, in pharmaceutical suspensions both relaxation processes can take place, a situation in which a relaxation time given by the formula will be taken into account (…),
(1.31)
In practice, on given applications, first, it will be necessary to analyze the contribution of each relaxation process (Nèel–Brown) to the total magnetic relaxation time (tau), as there may be situations in which one of the processes can be neglected. For example, in the case of highly viscous dispersion media, or in the case of injection of nanoparticles into tissues/tumors, in which small magnetically soft nanoparticles are dispersed, in general, the Brown relaxation time may be neglected.
Similar situations can also occur in the case of biocompatible nanoparticles with core‐shell structure for biomedical applications (coated with organic and/or biofunctionalized with large molecules), where the shell can be thick or really thick (even thicker than the diameter of the magnetic core of nanoparticles) (Figure 1.19) (Wells et al. 2017).
Figure 1.19 Schematic diagram of a single‐core magnetic nanoparticle. Note that the magnetic core is a single magnetic object that may be either a monocrystalline or a polycrystalline single magnetic domain, which responds to an applied magnetic field in a single, net, coherent manner.
Source: Wells et al. (2017). CC BY 3.0.
In these cases, in the formula (1.30), the hydrodynamic diameter (Dh) must be taken into account:
(1.32)
where d is the thickness of the organic layer from the surface of the nanoparticle, which will significantly increase the Brawn relaxation time (τB) compared to the Nèel relaxation time (τN). In many situations, tB can be higher or much higher than τN (τB>/>>τN), so only τN will be used in the calculations.
1.1.8 Dynamic Magnetic Behavior
1.1.8.1 Relaxation Time, Measurement Time, and Blocking Temperature
Below are some important aspects regarding the magnetic relaxation in nanoparticles and the determination of the threshold volume (Vth) from Eq.