A Theoretical Analysis of Magnetic Particle Alignment in External Magnetic Fields Affected by Viscosity and Brownian Motion

Featured Application

Iron oxide nanoparticles with highly nonlinear magnetic behavior are attractive for biomedical applications, including biosensing using the rotational freedom of particles for detection of biomarkers for cancer cells and for contrast enhancement in magnetic resonance imaging (MRI). Hyperthermia therapy has been used for cancer therapy, and magnetic particle imaging (MPI) is a promising new imaging modality that can spatially resolve the concentration of nanoparticles. For the success of the technology, understanding the nanoparticle rotation mechanism is necessary. The presented computational model can be used in the study of magnetic particle alignment phenomena as the non-Markovian process with memory in the external force field as part of generalized Langevin theory. It can elucidate the significance of each kind of torque in this phenomenon, or can serve as the estimator of the characteristic time of magnetic particle alignment in a wide range of magnitudes of the external magnetic flux density field. Our results, therefore, have far-reaching implications for understanding and advancement of these emerging biomedical technologies.

Abstract

The interaction of an external magnetic field with magnetic objects affects their response and is a fundamental property for many biomedical applications, including magnetic resonance and particle imaging, electromagnetic hyperthermia, and magnetic targeting and separation. Magnetic alignment and relaxation are widely studied in the context of these applications. In this study, we theoretically investigate the alignment dynamics of a rotational magnetic particle as an inverse process to Brownian relaxation. The selected external magnetic flux density ranges from $5\mathsf{\mu }\mathrm{T}$ to $5\mathrm{T}$. We found that the viscous torque for arbitrary rotating particles with a history term due to the inertia and friction of the surrounding ambient water has a significant effect in strong magnetic fields (range 1–$5\mathrm{T}$). In this range, oscillatory behavior due to the inertial torque of the particle also occurs, and the stochastic Brownian torque diminishes. In contrast, for weak fields (range 5–$50\mathsf{\mu }\mathrm{T}$), the history term of the viscous torque and the inertial torque can be neglected, and the stochastic Brownian torque induced by random collisions of the surrounding fluid molecules becomes dominant. These results contribute to a better understanding of the molecular mechanisms of magnetic particle alignment in external magnetic fields and have important implications in a variety of biomedical applications.

1. Introduction

When subjected to an external magnetic field, a magnetic particle will respond with translational motion in the gradient magnetic field and its self-rotation, or rotation of its 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 Néel 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], magnetorelaxometry [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 ${\overrightarrow{T}}_{\mathrm{v}}\left(t\right)$, 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 ${\overrightarrow{T}}_{\mathrm{B}}\left(t\right)$, 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]. Particularly, the borderline from ballistic to diffusion behavior should be considered [40].

Therefore, in this study, our aim was to develop a complex model of the magnetic alignment of a single magnetic particle in an external homogeneous magnetic field. Originally, a quiescent ambient viscous fluid, together with consideration of the arbitrary viscous torque with the history acceleration term, as well as the stochastic Brownian torque for a defined temperature. The rotational movement of the sphere can then be modeled by a Langevin-like equation with a stochastic fluctuating torque due to random impulses from the many neighboring fluid molecules (similar to the Langevin equation in [41,42]), with a modified viscous torque through the use of the history term (see, for example, [43,44,45]).

The description of such a system is far from trivial. The stochastic term brings white noise to the model, which is mostly discontinuous and has infinite variations [46]. We consider not only a simple system of ordinary differential equations (ODEs) but also stochastic integro-differential equations (SIDEs), which do not possess a simple analytic or numerical solution. Recently, we showed in [5,47,48] how to approach a similar system of integro-differential equations (IDEs) containing a non-local Basset acceleration history term numerically, with a variable timestep order of one, quadrature scheme. However, the current situation is more complex because the stochastic term brings the additional difficulty of solving the current system of SIDEs with a similar method. Therefore, we use an approach arising from the simple finite difference method of Euler [49].

The computer source codes developed in this research are available as the Supplementary Materials.

2. Materials and Methods

2.1. Physical Model

If we consider a spherical magnetic particle with radius R, mass density ${\rho }_{\mathrm{p}}$, and norm of magnetic moment ${\mu }_{\mathrm{p}}$, initially perpendicularly rotated to an external homogeneous magnetic field with flux density ${\overrightarrow{B}}_0\equiv B_0\overrightarrow{i}$, i.e., ${\overrightarrow{\mu }}_{\mathrm{p}}\left(0\right)\equiv {\mu }_{\mathrm{p}}\overrightarrow{j}$, located in a viscous ambient fluid with mass density $\rho$, dynamic viscosity $\eta$, and kinematic viscosity $\nu \equiv \eta /\rho$, then the rotational movement of the particle (alignment to the direction of the external magnetic field) can be described with a system of differential equations (DEs):

$$\begin{array}{cc}\hfill {\overrightarrow{\overrightarrow{I}}}_{\mathrm{p}}·\frac{\mathrm{d}\overrightarrow{\omega }\left(t\right)}{\mathrm{d}t}& ={\overrightarrow{\mu }}_{\mathrm{p}}\times {\overrightarrow{B}}_0+{\overrightarrow{T}}_{\mathrm{v}}\left(t\right)+{\overrightarrow{T}}_{\mathrm{B}}\left(t\right),\hfill \end{array}$$

(1)

where ${\overrightarrow{\overrightarrow{I}}}_{\mathrm{p}}$ is the moment of inertia tensor of the spherical particle, which for simple rotation about the particle axis reduces to the scalar $I_{\mathrm{p}}=\frac{2}{5}{m}_{\mathrm{p}}{R}^2$ and $\overrightarrow{\omega }\equiv \omega \overrightarrow{k}$ is the angular velocity of the rotational movement of the spherical particle. Unit vectors $\overrightarrow{i}$, $\overrightarrow{j}$, and $\overrightarrow{k}$ form the basis of a Cartesian coordinate system. $m_{\mathrm{p}}=V_{\mathrm{p}}{\rho }_{\mathrm{p}}$ is the weight of the magnetic particle and $V_{\mathrm{p}}$ is its volume. The first term on the right-hand side of Equation (1) is the magnetic torque exerted by the external homogeneous magnetic field ${\overrightarrow{B}}_0$ on the particle with magnetic moment ${\overrightarrow{\mu }}_{\mathrm{p}}={\mu }_{\mathrm{p}}(cos\phi \overrightarrow{i}-sin\phi \overrightarrow{j})$. The second term in Equation (1) is the viscous torque of the viscous ambient fluid on the rotating spherical particle. The final term in Equation (1) is the random Brownian torque exerted on the spherical particle.

Due to the axial symmetry of the rotational movement of the spherical particle ($\overrightarrow{\omega }\Vert \overrightarrow{k}$), the system of DEs (1) becomes a scalar differential equation:

$$\begin{array}{cc}\hfill \underset{\mathrm{inertial}}{\underbrace{I_{\mathrm{p}}\frac{\mathrm{d}\omega \left(t\right)}{\mathrm{d}t}}}& =\underset{\mathrm{magnetic}}{\underbrace{{\mu }_{\mathrm{p}}{B}_{0}sin\phi }}+\underset{\mathrm{viscous}}{\underbrace{T_{\mathrm{v}}\left(t\right)}}+\underset{\mathrm{Brownian}}{\underbrace{T_{\mathrm{B}}\left(t\right)}},\hfill \end{array}$$

(2)

which together with:

$$\begin{array}{cc}\hfill \frac{\mathrm{d}\phi \left(t\right)}{\mathrm{d}t}& =-\omega \left(t\right),\hfill \end{array}$$

(3)

and initial conditions:

$$\begin{array}{cccc}\hfill \phi \left(0\right)& =\frac{\pi }{2}\mathrm{rad},\hfill & \hfill \omega \left(0\right)& =0\mathrm{rad}/\mathrm{s},\hfill \end{array}$$

(4)

define the rotational alignment of the spherical particle with magnetic moment ${\mu }_{\mathrm{p}}$ in the external homogeneous magnetic field with flux density $B_0$ and viscous fluid with thermodynamic temperature T.

The viscous torque ${\overrightarrow{T}}_{\mathrm{v}}\left(t\right)=T_{\mathrm{v}}\left(t\right)\overrightarrow{k}$ exerted on the spherical particle for arbitrary angular velocity $\omega \left(t\right)$ can be expressed according to [30]:

$$\begin{array}{cc}\hfill T_{\mathrm{v}}\left(t\right)& \equiv \underset{\mathrm{arbitrary}\mathrm{viscous}\mathrm{torque}}{\underbrace{\underset{\mathrm{quasi}-\mathrm{steady}}{\underbrace{-\gamma \omega \left(t\right)}}\underset{\mathrm{history}\mathrm{acceleration}\mathrm{torque}}{\underbrace{-\frac{\gamma }{3\sqrt{\pi }}\underset{-\infty }{\overset{t}{\int }}K(t-\tau )\frac{\mathrm{d}\omega \left(\tau \right)}{\mathrm{d}\tau }\mathrm{d}\tau }}}},\hfill \end{array}$$

(5)

where $\gamma \equiv 8\pi \eta R^3$ is the Stokes coefficient of the viscous torque and:

$$\begin{array}{cc}\hfill K(t-\tau )& \equiv \frac{R}{\sqrt{\nu (t-\tau )}}-\sqrt{\pi }exp\left(\frac{\nu (t-\tau )}{R^2}\right)erfc\left(\frac{\sqrt{\nu (t-\tau )}}{R}\right)\hfill \end{array}$$

(6)

is a kernel function. This brings a challenge to the solution of the system of DEs (2)–(3) with initial conditions (4), due to the history integral term in equation (5) over the whole evolution of the angular acceleration of the particle in time, known as the history acceleration torque.

Moreover, the fluctuating Brownian torque is a stochastic term generated with random impulses from the neighboring fluid molecules and its mathematical representation yields the properties of the Gaussian white noise phenomenon $W\left(t\right)$, i.e., in the first moment $\langle W\left(t\right)\rangle =0$ (zero mean) and in the second $\langle W\left(t\right)W(t+\tau )\rangle =\delta \left(\tau \right)$ (uncorrelation) [41], where $\langle \cdots \rangle$ represents an ensemble average and $\delta \left(\tau \right)$ is the Dirac delta function. The stochastic Brownian torque can then be expressed as:

$$\begin{array}{cc}\hfill T_{\mathrm{B}}\left(t\right)& \equiv \sqrt{2k_{\mathrm{B}}T\gamma }W\left(t\right),\hfill \end{array}$$

(7)

where $k_{\mathrm{B}}$ and T are the Boltzmann constant and thermodynamic temperature, respectively.

The studied problem is not only the system of ODEs but also the system of SIDEs.

2.2. Solution

To solve the system of SIDEs (2)–(7), we used the first-order integration method, generalizing the Euler method for stochastic differential equations (finite difference approach).

For memory integral integration in the arbitrary viscous torque evaluation of the acceleration torque, an order one quadrature scheme, similar to those presented in [50], is generally used. However, in contrast, we have used a different kernel function (6) arising from the definition of viscous torque equation (5). The integral occurring in this equation can be transcribed using integration by parts:

$$\begin{array}{cc}\hfill \underset{t_0}{\overset{t}{\int }}K(t-\tau )\frac{\mathrm{d}\omega \left(\tau \right)}{\mathrm{d}\tau }\mathrm{d}\tau +K(t-t_0)\omega \left(t_0\right)& =\frac{\mathrm{d}}{\mathrm{d}t}\underset{t_0}{\overset{t}{\int }}K(t-\tau )\omega \left(\tau \right)\mathrm{d}\tau .\hfill \end{array}$$

(8)

Now, we can divide the time span by the sequence of $n-1$ constant timestep intervals $h={\tau }_{i+1}-{\tau }_i$ for $i=1$ to $n-1$, where ${\tau }_1=t_0$ and ${\tau }_n=t$, which gives:

$$\begin{array}{cc}\hfill \frac{\mathrm{d}}{\mathrm{d}t}\underset{t_0}{\overset{t}{\int }}K(t-\tau )\omega \left(\tau \right)\mathrm{d}\tau =\frac{\mathrm{d}}{\mathrm{d}t}\sum _{i=1}^{n-1}\underset{{\tau }_i}{\overset{{\tau }_{i+1}}{\int }}K(t-\tau )\omega \left(\tau \right)\mathrm{d}\tau & \equiv \frac{\mathrm{d}}{\mathrm{d}t}\sum _{i=1}^{n-1}{I}_i\left(t\right).\hfill \end{array}$$

(9)

If we now examine the simplest case, a linear approximation, the calculation leads to an order one quadrature scheme. By approximating $\omega \left(\tau \right)$ linearly in the interval $\tau \in \left[{\tau }_i,{\tau }_{i+1}\right]$:

$$\begin{array}{cc}\hfill \omega \left(\tau \right)& =\omega \left({\tau }_i\right)+\frac{\omega \left({\tau }_{i+1}\right)-\omega \left({\tau }_i\right)}{h}(\tau -{\tau }_i)+\mathcal{O}\left(h^2\right),\hfill \end{array}$$

(10)

we obtain:

$$\begin{array}{cc}\hfill I_i\left(t\right)& \equiv \underset{{\tau }_i}{\overset{{\tau }_{i+1}}{\int }}K(t-\tau )\omega \left(\tau \right)\mathrm{d}\tau =\left[\omega \left({\tau }_i\right)+\mathcal{O}\left(h^2\right)\right]\underset{0}{\overset{h}{\int }}K(t-{\tau }_i-\tau )\mathrm{d}\tau +\frac{\omega \left({\tau }_{i+1}\right)-\omega \left({\tau }_i\right)}{h}\underset{0}{\overset{h}{\int }}\tau K(t-{\tau }_i-\tau )\mathrm{d}\tau .\hfill \end{array}$$

(11)

The integrals in Equation (11), with kernel function (6), can be computed analytically to yield:

$$\begin{array}{cc}\hfill I_i\left(t\right)& \approx \omega \left({\tau }_i\right)\sqrt{\pi }\frac{R^2}{\nu }{\left[exp\frac{\nu \left(t-{\tau }_i-\tau \right)}{R^2}erfc\frac{\sqrt{\nu (t-{\tau }_i-\tau )}}{R}\right]}_{\tau =0}^h+\hfill \\ \hfill & +\frac{\omega \left({\tau }_{i+1}\right)-\omega \left({\tau }_i\right)}{h}{\left[\frac{2R^3}{{\nu }^{3/2}}\sqrt{t-{\tau }_i-\tau }+\sqrt{\pi }\left(\frac{R^4}{{\nu }^2}+\frac{R^2}{\nu }\tau \right)exp\frac{\nu \left(t-{\tau }_i-\tau \right)}{R^2}erfc\frac{\sqrt{\nu (t-{\tau }_i-\tau )}}{R}\right]}_{\tau =0}^h,\hfill \end{array}$$

(12)

if higher orders of h, than the first, are omitted.

A method that considers the stochastic term (7) in the solution of stochastic differential equations (SDEs), using a finite difference approach was shown in [42] and also used in [33]. The approach utilizes a discrete sequence of random numbers $W_i$ that mimics the properties of $W\left(t\right)$ and is stationary with zero mean, as well as having to fulfill $\langle W{\left(t\right)}^2\rangle =1$ for each value of t, which in a discrete sequence sense means that $\langle {(W_i\Delta t)}^2\rangle /\Delta t=1$, i.e., the $W_i$ has variance $1/\Delta t$. Furthermore, because $W\left(t\right)$ is uncorrelated, we assume $W_i$ and $W_j$ to be independent for $i\ne j$, i.e., we use a sequence of uncorrelated random numbers with zero mean and variance $1/\Delta t$. The realization of a such sequence is as simple as:

$$\begin{array}{c}\hfill W_i=\frac{w_i}{\sqrt{\Delta t}}\equiv \frac{w_i}{\sqrt{h}},\end{array}$$

(13)

where $w_i$ is a Gaussian random number with zero mean and unit variance. For the timestep $\Delta t$, we used the standard notation h, as used above. The discrete stochastic Brownian torque value in the i-th timestep is then given by:

$$\begin{array}{c}\hfill T_{\mathrm{B},i}\equiv \sqrt{\frac{2k_{\mathrm{B}}T\gamma }{h}}{w}_i.\end{array}$$

(14)

In a discrete sense, the solution for the n-th step of the angle and angular velocity can be expressed using the finite difference method as:

$$\begin{array}{cc}\hfill {\phi }_n& =\frac{2I_{\mathrm{p}}+\gamma h}{I_{\mathrm{p}}+\gamma h}{\phi }_{n-1}-\frac{I_{\mathrm{p}}}{I_{\mathrm{p}}+\gamma h}{\phi }_{n-2}-\frac{{\mu }_{\mathrm{p}}{B}_0{h}^2}{I_{\mathrm{p}}+\gamma h}sin{\phi }_{n-1}-\frac{h^2}{I_{\mathrm{p}}+\gamma h}{T}_{\mathrm{acc},n}-\frac{\sqrt{2k_{\mathrm{B}}T\gamma }{h}^{\frac{3}{2}}}{I_{\mathrm{p}}+\gamma h}{w}_n,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill {\omega }_n& =-\frac{{\phi }_n-{\phi }_{n-1}}{h},\hfill \end{array}$$

(16)

where the history acceleration torque fulfills in the n-th step:

$$\begin{array}{cc}\hfill T_{\mathrm{acc},n}& =-\frac{\gamma }{3\sqrt{\pi }}\left[-K(t-t_0)\omega \left(t_0\right)+\frac{S_n-S_{n-1}}{h}\right]\hfill \end{array}$$

(17)

with $S_n\equiv {\displaystyle \sum _{i=1}^{n-1}}{I}_i\left(t\right)$ and $S_{n-1}$ is the $S_n$ from the previous step.

3. Results and Discussion

We simulated the rotational alignment of a spherical MyOne $1.0\mathsf{\mu }\mathrm{m}$ microparticle [ThermoFischer Scientific, Waltham, MA USA, Dynabeads™, MyOne™, available online: https://www.thermofisher.com/order/catalog/product/65012?SID=srch-srp-65012 (accessed on 31 May 2018)] in an external homogeneous magnetic field and water as an ambient viscous fluid, with its parameters shown in

Table 1. Values of parameters used in simulations.

Parameter	Symbol	Value	Unit
Finite timestep	h	${10}^{-8\phantom{1}}$	$\mathrm{s}$
Boltzmann constant	$k_{\mathrm{B}}$	$1.3807\times {10}^{-23}$	$\mathrm{J}{\mathrm{K}}^{-1}$
Thermodynamic temperature	T	$293.15$	$\mathrm{K}$
Fluid ${}^{\mathrm{a}}$ dynamic viscosity	$\eta$	${10}^{-3\phantom{1}}$	$\mathrm{Pa}\mathrm{s}$
Fluid mass density	$\rho$	1000	$\mathrm{kg}{\mathrm{m}}^{-3}$
Fluid kinematic viscosity	$\nu$	$\eta /\rho$	${\mathrm{m}}^2{\mathrm{s}}^{-1}$
Magnetic flux density norm	$B_0$	$5\times {10}^{-6},\cdots ,5$	$\mathrm{T}$
Particle ${}^{\mathrm{b}}$ diameter	$2R$	${10}^{-6\phantom{1}}$	$\mathrm{m}$
Particle mass density	${\rho }_{\mathrm{p}}$	$1792$	$\mathrm{kg}{\mathrm{m}}^{-3}$
Particle volume	$V_{\mathrm{p}}$	$\frac{4}{3}\pi R^3$	${\mathrm{m}}^3$
Particle weight	$m_{\mathrm{p}}$	$V_{\mathrm{p}}{\rho }_{\mathrm{p}}$	$\mathrm{kg}$
Particle moment of inertia	$I_{\mathrm{p}}$	$\frac{2}{5}{m}_{\mathrm{p}}{R}^2$	$\mathrm{kg}{\mathrm{m}}^2$
Particle saturation magnetization	$M_{\mathrm{sp}}$	$43.0\times {10}^{3\phantom{-1}}$	$\mathrm{A}{\mathrm{m}}^{-1}$
Particle magnetic moment ${}^{\mathrm{c}}$	${\mu }_{\mathrm{p}}$	$2.25\times {10}^{-14}$	$\mathrm{A}{\mathrm{m}}^2$
Stokes coefficient	$\gamma$	$8\pi \eta R^3$	$\mathrm{kg}{\mathrm{m}}^2{\mathrm{s}}^{-1}$

${}^{\mathrm{a}}$ Water as an ambient fluid. ${}^{\mathrm{b}}$ Parameters of commercially available magnetic particle MyOne $1 \mathsf{\mu }\mathrm{m}$ (ThermoFischer Scientific, Waltham, MA USA, Dynabeads™, MyOne™). ${}^{\mathrm{c}}$ Magnetic moment calculated as ${\mu }_{\mathrm{p}}=M_{\mathrm{sp}}{V}_{\mathrm{p}}$.

.

3.1. Comparison of Models

Simulations were performed for four different combinations of the considered effects and these were denoted as four different models, as shown in

Table 2. Model notation with specified torques involved in simulations.

No.	Torques Involved	Notation
(i)	inertial, magnetic, quasi-steady viscous, stochastic Brownian	SDE ${}^{\mathrm{a}}$
(ii)	inertial, magnetic, arbitrary viscous, stochastic Brownian	SIDE ${}^{\mathrm{b}}$
(iii)	inertial, magnetic, quasi-steady viscous, no stochastic Brownian	ODE ${}^{\mathrm{c}}$
(iv)	inertial, magnetic, arbitrary viscous, no stochastic Brownian	IDE ${}^{\mathrm{d}}$

${}^{\mathrm{a}}$ Stochastic differential equations. ${}^{\mathrm{b}}$ Stochastic integro-differential equations. ${}^{\mathrm{c}}$ Ordinary differential equations. ${}^{\mathrm{d}}$ Integro-differential equations.

, where the abbreviated notation of each model is explicitly defined. The magnitude of the external homogeneous magnetic flux density field $B_0$ was used for each model with a scale of $5\mathsf{\mu }\mathrm{T}$ to $5\mathrm{T}$. The obtained results are shown in Figure 1.

From the solutions shown in Figure 1, a strong dependence for the speed of magnetic particle alignment on the magnitude of the magnetic flux density acting on the particle can be seen. For strong fields, the process of magnetic alignment is rapid and slowed down with decreasing $B_0$.

Strong magnetic fields acting on the magnetic particles in water as the ambient fluid at a temperature of $293.15\mathrm{K}$ cause rapid magnetic particle alignment with a characteristic timescale of the order of microseconds. In weaker fields, this time rises significantly by approximately one order of magnitude with each decreasing order of magnitude of the external magnetic flux density field.

It can be seen that the time evolution of the angle and angular velocity for the strongest $B_0$ field and for all modeled combinations of considered effects exhibit characteristic oscillations and nearly exponential decaying behavior. With a decrease in the magnetic flux density $B_0$ field, the oscillations disappear even though the almost exponential decaying behavior persists.

For the SDE and SIDE models, weakening the external $B_0$ fields stochastic Brownian torque results in a random pattern in the evolution of the angle and angular velocity with the appearance of stochastic jumps, firstly for $\omega \left(t\right)$ and also for $\phi \left(t\right)$ for the weakest $B_0$ fields (see Figure 1e–g).

3.2. Strong $B_0$ Field Limit Case

For the strong magnetic flux density field limit case, the time evolution of $\phi \left(t\right)$ and $\omega \left(t\right)$ in Figure 1a,b is shown, and it is clear that the viscous history acceleration torque as part of the arbitrary viscous torque starts to have a significant effect. The involvement of the history acceleration torque term in the SIDE and IDE models results in a reduction in the amplitude of the angle and angular velocity at the early stage of alignment in comparison with the SDE and ODE models. For $B_0=5\mathrm{T}$ (Figure 1a), this correction of the $\phi \left(t\right)$ amplitude caused by the history acceleration torque is a few percent. The correction of the $\omega \left(t\right)$ amplitude is >10%. Furthermore, the stochastic Brownian torque effect is minimal and, for the temperature considered, can be neglected.

For a better illustration of the significance of each torque in the strong $B_0$ field limit for the complex SIDE model (i.e., inertial, magnetic, arbitrary viscous, and stochastic Brownian torques) during the whole time evolution, see Figure 2.

The observed oscillatory character of the $\phi \left(t\right)$ and $\omega \left(t\right)$ time evolution in the strongest $B_0$ field has its origin in the involvement of the inertial torque in the description of the models. In contrast, its diminishment, for instance, at a low Reynolds number limit, results in a loss of this oscillatory behavior.

As much as the usual inertia contribution, the non-local Basset force is usually negligible at the macroscopic observation time scales considered in standard tracking experiments, but its effects have been shown to be prominent at short time-scales in [51,52]. This finding is an analogue to the behavior of our studied system with the non-Basset kernel of the history acceleration torque for arbitrary rotating sphere.

3.3. Weak $B_0$ Field Limit Case

The simulations for the SIDE and IDE models and for weaker $B_0$ fields were prematurely stopped due to excessively high values of exponents in the expression for the kernel function for the arbitrary viscous torque (history acceleration torque). Therefore, in the weakest $B_0$ fields, we only focused on the models without the history acceleration torque. In this weak $B_0$ field limit case, Figure 1f,g, the stochastic Brownian torque effect rises and starts to dominate. In contrast, even the viscous history acceleration torque (part of the arbitrary viscous torque) is numerically unreachable; its effect is minimal and can therefore be neglected.

The manifestation of stochastic behavior for the time evolution of angle $\phi \left(t\right)$ is only visible for the weakest magnetic flux density fields. Even for the time evolution of the angular velocity $\omega \left(t\right)$, it occurs for the medium magnitudes of the magnetic field. The stochasticity of $\phi \left(t\right)$ will also be visible for the same values of $B_0$ as for the stochastic behavior of $\omega \left(t\right)$ in the case of omitting the inertial torque from the models (low Reynolds number limit, not shown here). This discrepancy only disappears in the case of times significantly longer than the inertial time ${\tau }_{\mathrm{inertial}}\equiv I_{\mathrm{p}}/\gamma =0.03\mathsf{\mu }\mathrm{s}$ when both the time evolution of the angle for models with the inertial torque and without it are jagged because the microscopic details are not resolvable. Otherwise, the time evolution of the angle for the model with the inertial torque is smoothed [42].

The zero mean and uncorrelation of stochastic Brownian impulses to the magnetic particle are clearly visible in the averaged time evolution of $\phi \left(t\right)$ for an ensemble of $N=1000$ samples of rotational magnetic particle alignment, as shown in Figure 3a for the SDE model, meaning that the evolution replicates the time dependence of $\phi \left(t\right)$ of the single ODE model. Furthermore, its region of variance, defined with lines of $\mathrm{mean}\pm \mathrm{std}$, does not change with the size of the finite timestep h (see Figure 3b).

The stochastic Brownian torque term contribution to the overall dynamics of magnetic particle alignment increases with the weakening external magnetic flux density field $B_0$, as discussed. This contribution will be even more visible for an increased thermodynamic temperature of the ambient fluid, while the stochastic Brownian torque depends on the square root of the thermodynamic temperature, Equation (7).

3.4. Characteristic Time of Magnetic Particle Alignment

The time evolution of $\phi \left(t\right)$ for the simplest model, the ODE model, involving the inertial, magnetic, quasi-steady viscous, and no stochastic Brownian torques, was used as the input data for the least-square minimization fit using a fitting function:

$$\begin{array}{cc}\hfill \phi \left(t\right)& =\frac{\pi }{2}exp\left(-\frac{t}{{\tau }_{\mathrm{char}}}\right)\hfill \end{array}$$

(18)

with the founded parameter ${\tau }_{\mathrm{char}}$ for each $B_0$. The obtained fits with estimated values of parameter ${\tau }_{\mathrm{char}}$ are shown in Figure 4a–g. This parameter ${\tau }_{\mathrm{char}}$ has the meaning of time t when the angle $\phi \left(t\right)$ reduces to the value of $\frac{\pi }{2\mathrm{e}}$ and, therefore, can be denoted as the characteristic time of magnetic particle alignment. The obtained fits from simulations are in good agreement for the whole range of concerned $B_0$, except for its strong limit case ($B_0=5000$ $\mathrm{mT}$ case in Figure 4g).

The estimated values of ${\tau }_{\mathrm{char}}$ for each $B_0$ were further used to find the expression of their dependence. The following fitting function was used:

$$\begin{array}{cc}\hfill {\tau }_{\mathrm{char}}\left(B_0\right)& =\frac{C}{B_0}.\hfill \end{array}$$

(19)

Parameter C was found from the least-square minimization fit with the value $C=0.169\mathsf{\mu }\mathrm{s}\mathrm{T}$ (see Figure 4h).

3.5. Limitations

To summarize, a complex theoretical model of rotational magnetic particle alignment for external homogeneous magnetic flux density magnitudes $B_0$ from $5\mathsf{\mu }\mathrm{T}$ to $5\mathrm{T}$ in water as an ambient viscous fluid at room temperature has been presented. It has been shown that the significance of the arbitrary viscous torque history term increases in the strong $B_0$ limit. In contrast, in the weak $B_0$ field limit, it diminishes, and the stochastic Brownian torque effect starts to manifest. In addition, the characteristic time of magnetic particle alignment for the entire scale of $B_0$ magnitudes has been estimated.

The presented models bring several simplifications and limitations to the studied system, as discussed below.

The weakening of the $B_0$ field causes prolongation of the time needed for rotational magnetic particle alignment. This implies increasing the argument of the exponential function in the kernel function definition with the lowering of $B_0$, which is limited in the numerical realization of the $f\left(x\right)=exp\left(x\right)$ function. The solution of this problem is simple. The effect of the history term in the weak field is minimal and diminishes, so we do not need to solve the whole problem (SIDE model) in the weak $B_0$ limit and can neglect it and focus only on the solution of the SDE model.

Due to the non-locality of the history acceleration torque, the stochastic Brownian torque should be not mathematically represented as a Gaussian white random process in Langevin theory [53], but instead as a “colored” one [45]. However, the correlation in the time of the random process, due to the neglection of the history term in the weak $B_0$ limit, when the stochastic Brownian torque effect manifests, diminishes. Therefore, the stochastic Brownian torque can be considered as a zero-mean uncorrelated random process. In reality, the system memory can affect the “color” of the Brownian random process [45,51]. The external forces can also affect the memory of the system and thermal force as it was discussed, for example in [54,55,56,57,58]. A solution of the studied problem in this way is still missing.

The particle magnetic moment of the considered models comes from the assumption of its constant value, which in reality is not fulfilled, while the used particle is paramagnetic. Therefore, the involvement of a magnetic moment saturation process in future models will be convenient.