Granular systems are composed of many particles with sizes ranging from few nanometers to more than 100 nanometers. Among these materials, magnetic nanoparticles (MNPs) are made of materials that at bulk scale show an intrinsic magnetization properties (FexOy, Co), although MNPs made of gold (Au) have been also reported . However, because of their nanometric scale, the magnetic properties they exhibit are strongly temperature and size dependent, differing notably from bulk properties .
One important size dependent feature is the multidomain or monodomain magnetic structure of the particle. Large nanoparticles present a multidomain structure (see figure 2a): magnetostatic energy is too high to maintain a monodomain structure. However, below a certain critical size nanoparticles become singledomain entities (see figure 2b). Complex micromagnetic calculations yield a critical size of 29.7 nm for magnetite nanoparticles .
Large multi-domain or single-domain MNPs exhibit open hysteresis loops (see figure 1a), that is, they exhibit magnetization in the absence of an externally applied magnetic field and a coercive field that is required to reverse the magnetization direction. The presence of magnetic domains reduces the coercive field, so single-domain particles tend to have wider hysteresis loop .
However, as the particle size decreases, the thermal agitation of particles becomes important. This agitation causes the magnetic dipole to reverse spontaneously from one direction to another. Nevertheless, when the magnetization of a certain sample is measured, the averaged value of a huge amount of particles (~103 ) is obtained. It can be shown that at equilibrium, the averaged magnetization follows approximately the Langevin curve (see figure 1b), with null coercive field and zero magnetization in the absence of external magnetic field . This regime is called superparamagnetic regime.
Figure 1: a) Open hysteresis loop under an externally applied magnetic field ( Happ) with coercive field (Hcb) Langevin curve of a superparamagnetic MNP sample. Dashed line shows the nonlinearities of the curve. c) Relaxation through Néel and Brown relaxation mechanisms.
If the system is cooled down, the thermal agitation becomes less important until the dipoles do not switch spontaneously and thus, the MNP sample has a transition from superparamagnetic to open-loop regime. The critical temperature at which the MNP have this transition is called the blocking temperature (Tb ) which depends on the material and shape of the nanoparticle .
As it has mentioned, the macroscopic magnetization in superparamagnetic regime follows the Langevin curve when the sample is at equilibrium. However, the magnetization needs a certain time, called relaxation time (T ), to reach the equilibrium (see figure 1c). The before mentioned thermal agitation is the responsible of the so called Néel relaxation . Néel relaxation mechanism is present in every MNP system. However, in stable MNP colloids there is a second relaxation mechanism called Brown relaxation, which is caused by the Brownian motion of the particle inside the fluid . The effective relaxation time results from the combination of both mechanism.
The relaxation time is an important parameter that characterizes the dynamic magnetization of the MNPs. When the measurement time is much larger than the relaxation time of the system, the magnetization will have enough time to reach the equilibrium state and it will follow the Langevin curve, that is, a null loop with null coercive field. On the contrary, when the measurement time is shorter than the relaxation time, the magnetization will not be able to reach the equilibrium state, resulting in an open hysteresis loop (see figure 1a). In other words, the MNPs present superparamagnetic features when low frequency or static magnetic fields are applied, whereas they present open hysteresis loops when higher frequency fields are applied (> 50 kHz).
Medical applications. The aforementioned properties of the MNP system make them suitable for several medical applications. First, in order to introduce the particles into a biological system, agglomerations must be avoided. Fortunately, superparamagnetic particles do not present magnetization in the absence of external magnetic fields. Hence, they are less likely to agglomerate inside veins or arteries due to the absence of magnetic attraction. A remarkably stable colloid can be obtained surrounding the particles with an appropriate polymer or lipid protecting layer. Iron oxides (FexOy) nanoparticles are the best candidates for medical applications thanks to their highest biocompatibility and to their natural presence in the body (e.g. ferritin, hemosiderin, transferrin, hemoglobin).
In this scope, the correct positioning of MNPs into the body is crucial for any successful medical application. This can be achieved by means of active and passive targeting. As soon as particles are injected into blood circulatory system, they are subjected mainly to the action of the macrophages . The principal goal of the macrophage cells is to clear any exogenous element in the body. To do so, they capture any invading entity and concentrate them in areas with high macrophage activity. This phenomenon gives rise to a passive targeting of MNPs to areas with high macrophage activity (i.e. liver or infection areas). The probability to redirect particles to the desired target tissues can be enhanced labelling their surface with ligands that specifically bind to surface receptors on target cells. This kind of targeting is called active targeting (see figure 2c). In addition, the magnetic nanoparticles can be directed to desired areas by the usage of magnetic gradients .
Regarding the therapeutic applications, the magnetic particles can be used as a heat sources in magnetic hyperthermia, a novel cancer therapy . When applying an external high frequency magnetic field, the magnetization of MNPs follows an open loop hysteresis cycle (see figure 1a) and therefore, they absorb energy from the magnetic field. This energy is converted into heat, heating the tissues surrounding the nanoparticles and eventually killing tumor cells. What is more, drugs can be attached to the MNPs by several linkers. Then, once magnetic particles are located in the correct place, the medicaments are releases directly into the affected cells (i.e. cancer cells) .
Regarding the diagnostic applications, there are several technics to detect and map the position of MNPs inside the body. Therefore, since MNPs can be targeted to specific cells, they can be used as tracers for specific diseases. According to Langevin curve, MNPs tends to saturate faster than other materials present in biological systems. This rapid saturation causes nonlinearities in the magnetic response (see figure 1b) that can be used to detect the presence of MNPs . What is more, based on this principle, the spatial distribution of magnetic nanoparticles is obtained by means of magnetic particle imaging (MPI) technique  (see figure 1d). Finally, MNPs can be used as a contrast agents in magnetic resonance imaging due to the distortion in T1 and T2 relaxation times caused by the magnetic moment of the particles .
Figure 2: a) Schematic view of a multidomain nanoparticle. b) Schematic view of a monodomain nanoparticle. τ is the relaxation time of the sample whereas is the measurement time. e) Schematic explanation of the active targeting. f) An MPI image of a similar ‘‘Cal’’ phantom showing millimetre-scale resolution .
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