**2. Theoretical Background of Magnetism**

In order to basically understand the potential which controlling magnetic forces might offer for novel treatment options for liver fibrosis, it is important to comprehensively understand nanoscale magnetism. Nano-sized particles are often used as tools in this regard since they can specifically target cells or structures in the body. Magnetic nanoparticles (MNPs) usually refer to ferro- or ferrimagnetic crystals sized below 100 nm [43–45]. At the nanoscale, many physical properties, including the short- and long-range magnetic interactions, contribute to the overall functionality of these nanoparticles [46]. The magnetic material is characterized by a strong response under an action of the magnetic field. The investigation of a family of 3D nanocrystals has been growing with impressive speed over the past decades. Among them, the MNPs are an exciting class of material for biomedical applications [43,47–50]. Magnetism in matter is related to spins − the smallest magnetic units referred to as atoms are composed of ferro(i-)magnets, which interact via a quantum phenomenon called the exchange interaction that leads to the formation of long-range ordered areas (magnetic domains) [44,46]. Owing to the superposition character of the magnetic field, the total magnetic moment of this area is equal to the sum of individual moments (μ) of each atom. If a magnetic field with the strength H is applied, the magnetic moments prefer to align along the direction of the magnetic field to reduce the total energy (by the domain walls movement in the macroscopic body). Thus, when the magnetic field is strong enough to align all individual magnetic moments, a ferromagnetic body is saturated, and the magnetic moment of this system equals N·μ, where N is the total number on the individual magnetic moments of the atoms in the system. For characterization of material mass or volume-weighted parameters in magnetization saturation, MS = N·μ/m or N·μ/V (Am2/kg and A/m

in SI, or emu/g and emu/cm3 in CGS). The behavior of magnetization has a hysteretic character for ferro(i-)magnetic materials because of specialties of the magnetization processes such as domain wall pinning on defects. Those irreversible processes lead to nonzero magnetization (MR) at the remnant state when the magnetic field is off. Intrinsic energy, which keeps the spins in a certain direction in the absence of a magnetic field, is called the magnetic anisotropy energy and it determines the hysteresis loop width or coercivity field μ0HC (T in SI and Oe in CGS).

Macroscopic magnets tend to reduce their magnetic moments (or, in other words, to reduce their magnetostatic energy) and split into randomly oriented domains [44,45]. Magnetic domains are separated by domain walls − intermediate states are required to rotate spins in differently magnetized domains to reduce exchange interaction with the interface. When the size of MNPs is comparable with the size of domain walls, the split into domains is no more energetically favorable, and MNPs transform to the single-domain state. A single particle presents a saturated magnet with magnetization equal to the saturation magnetization value. In the ideal case, all spins below Curie temperature are oriented in one direction, and because of exchange interaction, their behavior can be described by the superposition of all spins. The orientation of this macrospin in the absence of a magnetic field is defined by easy axis−positions where total energy is minimal. The energy which separates the macrospin at a certain position is magnetic anisotropic energy (Ea). That means that to change orientation along the axis, the system needs to overcome the energy barrier equal to Ea = K × V. Thus, particles of bigger volume (V) have higher anisotropy energy and the second term, K, is an anisotropic constant defined by material and structural-morphological properties of MNPs. Several sources contribute to anisotropy with constant K. Among them the more pronounced are the shape anisotropy arising from magnetostatic interactions of poles and the magnetocrystalline anisotropy coming from spin-orbital coupling. The behavior in the magnetic field of assembly of randomly oriented single-domain MNPs with one easy axis was described by Stoner and Wohlfarth in 1948 [51]. Magnetic properties of single-domain MNPs are different compared to bulk analogs – they may have higher values of remnant magnetization and coercivity field because the coherent rotation magnetization reversal mechanism is more difficult than the domain wall moving and additional contributions to anisotropy coming from the surface.

The range of diameters when MNPs pass in the single-domain state is 20–800 nm, for magnetic iron oxides (magnetite Fe3O4 or magnetite γ-Fe2O3) it is 80–90 nm [44,52,53]. For MNPs of smaller size, the magnetic anisotropy energy becomes comparable to the thermal energy K × V ≈ kB × T, where kB is the Boltzmann constant. In this regime, called superparamagnetic (SPM), the temperature induces random switching of magnetization of single MNPs, and thus the time-averaged MR and HC of MNPs assembly are zero [54]. The probability of this fluctuation is described by the Néel relaxation time:

$$
\pi\_N = \pi\_0 \mathbf{e}^{\frac{K\mathbf{V}}{k\_B T}} \tag{1}
$$

where τ<sup>0</sup> is an attempt time ~10−<sup>9</sup> s [44]. The temperature at which MNPs act in the SPM regime at a fixed observation time is the so-called blocking temperature TB. For quasi-static measurements the typical measuring time is 100 s and the blocking temperature can be evaluated as TB = K × V/25kB. The fluctuating magnetic moment of MNPs with TB less than the ambient temperature is favorable for their colloidal stability since it reduces dipolar interactions among them. Moreover, SPM MNPs do not change their magnetic properties when the viscosity of the medium is changed, for example, if MNPs are internalized by cells [55].
