applied sciences  
A Theoretical Analysis of Magnetic Particle Alignment in External Magnetic Fields Affected by Viscosity and Brownian Motion  
[PERSON],*, [PERSON], [PERSON] and [PERSON]  
Department of Imaging Methods, Institute of Measurement Science, Slovak Academy of Sciences, Dubravska Cesta 9, 8410 Bratislava, Slovakia; [EMAIL_ADDRESS] Department of Nuclear Physics and Biophysics, Faculty of Mathematics, Physics and Informatics, Comenius University, Mlynska Dolina Fl1, 842 48 Bratislava, Slovakia; [EMAIL_ADDRESS] Biomedical Center Martin, Jessenius Faculty of Medicine in Martin, Comenius University in Bratislava, Mala Hora 4, 036 01 Martin, Slovakia; [EMAIL_ADDRESS] Correspondence: [EMAIL_ADDRESS]  
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magnetic moment, to the direction of the external magnetic field as it is applied in various cases of biomedical applications widely studied experimentally and theoretically [1; 2; 3; 4; 5; 6; 7; 8; 9; 10]. These latter effects are simply denoted as magnetic alignment and are inverse processes to Brownian and Neel relaxation [11; 12]. Both magnetic alignment and relaxation are important in, for example, magnetic particle contrast imaging and quantification in magnetic resonance and particle imaging [8; 9; 13; 14; 15], electromagnetic hyperthermia [6; 7; 16; 17], theranostics [18; 19; 20], magnetore-laxometry [10; 21; 22], and navigation in the geomagnetic field [23; 24; 25; 26; 27]. Magnetic particle alignment at the nano- and micro-scale requires a complex approach for simulating a wide range of possible strengths of an acting external magnetic field. However, to our knowledge, such a study does not currently exist. If we restrict only the particle rotating (as a whole with a magnetic moment) to the direction of an external homogeneous magnetic field, the particle experiences a magnetic torque that can be easily described analytically [28]. An object moving or rotating in a viscous ambient fluid experiences an action against its movement due to the internal friction of the ambient fluid. Such a fluid on the surface of the object moves with the object with the same velocity as its surface, i.e., it fulfills the no-slip boundary condition. The velocity of the layers of the fluid parallel to the object surface decreases with increasing distance between the layer and particle surface due to the internal friction of the fluid [29]. If the object is a sphere rotating about its axis with an arbitrary angular velocity, the torque felt by the sphere due to the internal friction of the fluid and its inertia adds an integral term for the whole history and development of angular acceleration to the quasi-steady viscous torque, as shown previously [30]. This integral history term is analogous to the Basset history force [31; 32] (such a non-local Basset force was studied by our group recently as the correction of the magnetic particle separation dynamics in [5]). This term has its origin in the vortices of an ambient fluid around an arbitrary rotating sphere. The viscous torque is denoted as \(\vec{T}_{\rm v}(t)\), i.e., an \"arbitrary viscous torque\". Furthermore, when a spherical particle is sufficiently small, it experiences the stochastic effect of the Brownian motion of ambient fluid molecules [33], which depends on its temperature. This contribution to the overall torque is denoted as the \(\vec{T}_{\rm B}(t)\), i.e., a stochastic Brownian torque. In most studies [34; 35; 36], the history acceleration term of the arbitrary viscous torque, inertia, and especially stochastic Brownian torques, are assumed to be negligible, and a simple viscous torque (a quasi-steady term, \(\gamma\omega=8\pi\eta R^{3}\omega\), for a sphere of radius \(R\) rotating with angular velocity \(\omega\) in an ambient viscous fluid with dynamic viscosity \(\eta\)[37]) are considered, as in [17; 25; 33]. Alternatively, the thermal disturbance of magnetic alignment through the rotational diffusion model can be analyzed [38; 39].