Source: https://www.groundai.com/project/temperature-effects-on-drift-of-suspended-single-domain-particles-induced-by-the-magnus-force/
Timestamp: 2019-04-25 09:43:52+00:00

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We study the temperature dependence of the drift velocity of single-domain ferromagnetic particles induced by the Magnus force in a dilute suspension. A set of stochastic equations describing the translational and rotational dynamics of particles is derived, and the particle drift velocity that depends on components of the average particle magnetization is introduced. The Fokker-Planck equation for the probability density of magnetization orientations is solved analytically in the limit of strong thermal fluctuations for both the planar rotor and general models. Using these solutions, we calculate the drift velocity and show that the out-of-plane fluctuations of magnetization, which are not accounted for in the planar rotor model, play an important role. In the general case of arbitrary fluctuations, we investigate the temperature dependence of the drift velocity by numerically simulating a set of effective stochastic differential equations for the magnetization dynamics.
Recently, the phenomenon of drift motion of single-domain ferromagnetic particles suspended in a viscous fluid has been analytically predicted and numerically confirmed ;  . This phenomenon arises from the Magnus effect, and drift of particles occurs if the external driving force, which induces their oscillatory motion, is properly synchronized with the rotating magnetic field, which induces their non-uniform rotation. The main characteristic of drift motion, the particle drift velocity, has been calculated both analytically and numerically. Because the magnitude and direction of this velocity are easy to operate, the drift phenomenon has been proposed to separate ferromagnetic particles in suspensions.
The approach used in ;  was purely deterministic, i.e., thermal fluctuations in the translational and rotational dynamics of particles were completely ignored. According to  , this is approximately true if the particle size exceeds a few tens or even hundreds of nanometers. At the same time, to be single-domain, the particle size must be less than a critical one, which usually does not exceed hundreds of nanometers (see, e.g., Refs. ;  ). As a consequence, the applicability of the deterministic approach is restricted only to relatively large single-domain particles. Therefore, in order to describe the drift motion of smaller particles, thermal fluctuations must be taken into account.
One of the most powerful tools to study the influence of thermal fluctuations on the behavior of dynamical systems is a set of stochastic differential equations (e.g., Langevin equations) in which the action of these fluctuations is modelled by Gaussian white noises ; ;  . The main advantage of these equations is that their solution is a Markov process, whose probability density function obeys the Fokker-Planck equation, see also Refs. ;  . Within this framework, the stochastic magnetization dynamics in fixed single-domain particles was first investigated by Brown  . To describe this dynamics, he introduced the stochastic Landau-Lifshitz-Gilbert equation, in which thermal fluctuations are accounted for by a random magnetic field with Gaussian white noise components, and derived the corresponding Fokker-Planck equation. This approach (based on the stochastic Landau-Lifshitz-Gilbert or Landau-Lifshitz equations) has become a standard tool for studying the magnetization dynamics in nanosystems  . In particular, it was employed to study the influence of uniformly rotating magnetic field on the nanoparticle magnetization ;  , magnetic relaxation ;  , and precessional states of the magnetization  .
A set of stochastic differential equations for the rotational motion of ferromagnetic particles in a viscous fluid depends on whether the magnetization vector is frozen into the particle body (see Ref.  and references therein) or not ; ; ;  . In the former case, which holds if the anisotropy magnetic field noticeably exceeds the external one, the rotational dynamics of particles satisfies the Newton’s second law for rotation with total torque containing Gaussian white noise contributions. This approach is widely used in studying the role of thermal fluctuations in the rotational dynamics of suspended particles  (about the constructive role of these fluctuations see, e.g., Refs. ; ;  ). But the role of temperature in the drift motion of single-domain ferromagnetic particles induced by the Magnus force has never been investigated before. Therefore, to fill this gap, in this work we develop a statistical theory of drift of ferromagnetic particles and determine the dependence of the particle drift velocity on temperature.
The outline of the paper is as follows. In Sec. II, we derive a set of stochastic equations that describes the translational and rotational dynamics of single-domain ferromagnetic particles in a viscous fluid and accounts for the influence of thermal fluctuations. The particle drift velocity, which is induced by the Magnus force and depends on the average particle magnetization, is introduced in Sec. III. To find the average magnetization, in Sec. IV we obtain the Fokker-Planck equation for the probability density function of magnetization orientations. In the same section, using this equation we propose a set of effective stochastic differential equations describing the magnetization dynamics, which is much simpler than the original one. In Sec. V, in the case of strong thermal fluctuations we find analytically the steady-state solutions of the Fokker-Planck equations, which correspond to the planar rotor and general models, and on this basis calculate the drift velocity as a function of temperature for these models. In Sec. VI, we examine our theoretical predictions numerically and present numerical results on the temperature dependence of the drift velocity of Co nanoparticles in water, obtained by solving a set of effective stochastic equations. Finally, our main findings are summarized in Sec. VII.
In our model, the ferromagnetic particles moving in a viscous fluid are considered to be spherical and smooth. Their radius a is assumed to be so small that the single-domain state is realized, and thus the particle magnetization M is a function of time only: M=M(t). If the anisotropy magnetic field is large enough, then the vector M is approximately parallel to the anisotropy axis. In this approximation the magnetization is frozen into the particle body, and hence its dynamics is governed by the kinematic differential equation dM/dt=ω×M, where ω=ω(t) is the angular particle velocity. We also assume that the translational and rotational Reynolds numbers, Ret and Rer, are small compared to 1 (since maxa usually does not exceed a few hundreds of nanometers ;  , this condition is not very restrictive for suspended particles). In this case, the inertial effects in the particle dynamics can be neglected and, as a consequence, the equations for the translational and rotational motions of particles are reduced to the force and torque balance equations, F=0 and T=0, respectively.
Thus, in our model, the translational and rotational motions of suspended ferromagnetic particles are described by the stochastic equations (2), (4), and (5). They show that, while the particle rotation influences (due to the Magnus force) the translational motion, the rotational motion does not depend on the translational one. Therefore, to find statistical characteristics of the translational motion of these particles, including their drift velocity, we must first determine the rotational properties that are described by Eqs. (4) and (5).
Here, the angle ψ is a given function of the dimensionless time τ, which, similar to the driving force, is assumed to satisfy the antisymmetry condition ψ(τ+1/2)=−ψ(τ). It should also be stressed that this condition implies that ψ(τ) is a periodic function with period 1.
It is these stochastic differential equations that we will use to study the effect of thermal fluctuations on the drift of particles.
It is important to note that expression (17) is valid only if the particle angular velocity ω does not contain a white-noise contribution. The reason is that the multiplication of generalized functions, including Gaussian white noises, is not mathematically defined (for more details see, e.g., Ref.  ). Therefore, if ω is given by (9), then the particle velocity (17) is not appropriate for studying the stochastic translational motion of suspended particles (since u contains the term proportional to ξ×ν). However, expression (17) can be used to find the drift velocity of these particles, see just below.
(γ≪1) characterizes the contribution of the Magnus force to ⟨u⟩. Expression (19) shows that thermal fluctuations influence the mean particle velocity only through the dimensionless mean magnetization ⟨m⟩ of the particle.
Thus, the drift motion of particles induced by the Magnus force occurs along the y axis with the drift velocity (22). In general, this velocity can be calculated by numerically solving a set of coupled stochastic equations (13) describing the magnetization dynamics. But to determine the drift velocity analytically, it is convenient to introduce the Fokker-Planck equation associated with these stochastic equations.
As usually, P should satisfy the normalization condition ∫π0∫2π0Pd\thetaupd\chiup=1 and the initial condition P(\thetaup,\chiup,0)=δ(\thetaup−θ0)δ(\chiup−χ0), where θ0=θ(0) and χ0=χ(0).
Then, using the symmetry condition Pst(\thetaup,\chiup,ξ)=Pst(π−\thetaup,\chiup,ξ), which follows from the Fokker-Planck equation (23), one can easily make sure that ∫π0cos\thetaupPst(\thetaup,\chiup,ξ)d\thetaup=0 and, hence, the condition ⟨cosθst(ξ)⟩=0 indeed holds.
This means that solutions of Eqs. (13) and (27) with respect to the polar and lag angles θ(τ) and χ(τ) have the same probability density P(\thetaup,\chiup,τ). The word ‘effective’ is used here to emphasize that a set of Eqs. (27) (i) is not a set of ordinary stochastic differential equations (because the noiseless term β2cotθ arises, nevertheless, from thermal fluctuations) and (ii) is much simpler than a set of basic equations (13).
An interesting property of Eqs. (27) that follows from the results of Ref.  is that, in spite of the multiplicative character of the noise ζ2, the statistical characteristics of their solution do not depend on interpretation of noises ζi  . Note also that sets of effective stochastic differential equations describing the rotational dynamics of ferromagnetic particles in a viscous fluid and the magnetization dynamics in ferromagnetic particles embedded into a solid matrix have been proposed in Refs.  and  , respectively.
Because small thermal fluctuations do not affect strongly the deterministic drift of particles, which was studied in detail in Refs. ;  , below we analytically study the role of large thermal fluctuations only. For this purpose, we consider the stochastic particle dynamics within the framework of the planar rotor model and general model.
is the ratio of magnetic to thermal energy and I0(ϵ) is the modified Bessel function of the first kind and order zero. Note, since the von Mises distribution is the circular analog of the normal distribution, it is widely used in various applications, see, e.g., Refs. ;  .
with Pst(\chiup,ξ)=limn→∞P(\chiup,n+ξ). Our aim here is to determine the drift velocity (34) in the limit of large thermal fluctuations, when β→∞.
According to this result, strong thermal fluctuations essentially decrease the drift velocity (⟨sy⟩ tends to zero quadratically as ϵ→0). In addition, it shows that ⟨sy⟩ depends linearly on the amplitude ψm of the magnetic field azimuthal angle and periodically on the initial phase ϕ of the external force.
Because the deterministic dynamics of the particle magnetization occurs in the plane of magnetic field rotation ;  , the planar rotor model is quite appropriate to describe the drift motion of ferromagnetic particles in the absence of thermal fluctuations. However, the presence of such fluctuations, especially strong ones, casts doubt on the applicability of this model. The reason is that the out-of-plane fluctuations of the magnetization vector, which always exist, are excluded from consideration in the planar rotor model. Therefore, to investigate the influence of these fluctuations on the drift velocity, we will use a general Fokker-Planck equation (23).
if ψ(τ) is defined by (40).
By comparing formulas (39) and (48), we see that the drift velocity predicted by the planar rotor model in the case of large thermal fluctuations is three times larger than that predicted by the general model. Because the only difference between these models is the out-of-plane fluctuations of the magnetization vector, we may conclude that these fluctuations are responsible for this decrease of the drift velocity.
where N(≫1) is the total number of runs (these formulas become exact if N,n→∞). Finally, using (50) and (51), we determined the drift velocity (34) within the planar rotor model and the drift velocity (22) within the general model.
Figure 1: The reduced drift velocity as a function of the parameter ϵ for ϵ≪1. The inverted and upright triangles represent the simulation data for the planar rotor and general models, and the solid lines 1 and 2 represent the theoretical results from (42) and (49), respectively.
agreement with analytical predictions, if the parameter ϵ is small enough. (The difference between the numerical and analytical results grows with increasing ϵ because of limitation of the second-order approximation.) This validates both our analytical results and simulation procedure. Moreover, the obtained numerical results confirm a strong influence of the out-of-plane fluctuations of the particle magnetization on the drift velocity. Therefore, to account for these fluctuations, in our further analysis we use a general set of effective stochastic differential equations (27).
which well reproduces the experimental data for the dynamic viscosity of water in the reference interval ΔT (see Ref.  , Tab. 4.9). Here, A=−3.5318, B=220.57, C=149.39, T is in Kelvin, and η is in poise (P).
Figure 2: The temperature dependence of the reduced drift velocity for Co nanoparticles of different radii suspended in water. The dependence at β=0 corresponds to the large radius limit.
a=2nm (line with inverted triangles), and a=3nm (line with upright triangles). With increasing the particle radius a, the temperature dependence of the drift velocity very rapidly approaches the limiting function (represented by the line with diamonds), which corresponds to the noiseless case β=0. In particular, the difference between these functions can safely be neglected already at a>5nm (this radius slowly increases with decreasing Hm). This means that thermal fluctuations reduce the drift velocity of rather small particles (when a<5nm), while in the case of larger particles their drift velocity depends on temperature solely due to the temperature dependence of the dynamic viscosity of water.
Figure 3: The reduced drift velocity of Co nanoparticle as a function of temperature for different values of the initial phase ϕ. The particle radius is a=3nm and the other parameters are the same as in Fig. 2.
may expect that temperature change can also change the sign of ⟨sy⟩, if the initial phase is properly chosen. This expectation is confirmed by our simulation results presented in Fig. 3. From a physical point of view, the reversal of the drift direction occurs when the Magnus force averaged over the time period and thermal fluctuations changes sign, which in turn happens if the translational and rotational oscillations of particles are appropriately synchronized. This synchronization can be realized by changing both the initial phase (that is quite obvious) and temperature. It seems that the observed temperature-induced reversal of the drift direction can be used to improve the method of separation of suspended ferromagnetic particles based on the Magnus effect  .
In order to gain more insight into the behaviour of the reduced drift velocity of Co nanoparticles, we calculated also its dependence on the magnetic field characteristics, namely, on the magnitude Hm, frequency Ω, and angle ψm. The dependence of ⟨sy⟩/γ on Hm for a nanoparticle of radius a=5nm determined at room temperature T=295K, Ω=5×105rad/s, ψm=1rad, and different values of ϕ is shown in Fig. 4. Since the change of Hm is equivalent to changing the parameter α, the obtained results are similar to those derived in Ref.  in the deterministic approach. Figure 5 illustrates the dependence of ⟨sy⟩/γ on Ω calculated for T=295K, Hm=10Oe, ψm=1rad, ϕ=0rad, and different particle radii. As seen, decreasing the particle size decreases the absolute value of the reduced drift velocity and shifts the minimum of this velocity to the right. This occurs because the magnitude of thermal fluctuations, which is characterized by the parameter β, increases as the particle size decreases, see (8). Finally, in Fig. 6 we show the dependence of ⟨sy⟩/γ on ψm calculated for a=5nm, Hm=10Oe, Ω=5×105rad/s, ϕ=0rad, and different temperatures. In this case, the influence of temperature on the reduced drift velocity is caused by both the temperature dependence of the dynamic viscosity of water and thermal fluctuations. It should also be kept in mind that the dependencies of the dimensional drift velocity vdr=vm⟨sy⟩ on temperature differ from that obtained for the reduced drift velocity ⟨sy⟩/γ. The reason is that the quantities vm and γ, according to their definitions, depend on T through the dynamic viscosity of water.
Figure 4: The reduced drift velocity of Co nanoparticle as a function of the magnetic field magnitude Hm for different values of the initial phase ϕ. Line 1 corresponds to ϕ=0.4rad, line 2 to ϕ=2.0rad, and line 3 to ϕ=(π+0.4)rad.
Figure 5: The frequency dependence of the reduced drift velocity for different radii of Co nanoparticles. Line 1 corresponds to the large radius limit (β=0), line 2 to a=5nm, and line 3 to a=4nm. The Ω-axis is in logarithmic scale.
Figure 6: The reduced drift velocity of Co nanoparticle as a function of the angle ψm for different temperatures.
We have studied analytically and numerically the temperature dependence of the drift velocity of suspended single-domain ferromagnetic particles induced by the Magnus force. Our approach is based on a set of stochastic differential equations that describes the translational and rotational motions of particles with frozen magnetization in the limit of small Reynolds numbers. In this approximation, the rotational motion, which is caused by a rotating magnetic field, affects (due to the Magnus effect) the translational motion, which is caused by an external harmonic force. In contrast, the translational motion does not influence the rotational one. These features of particle dynamics were used to derive a general expression for the particle drift velocity in the steady state. It shows that, as in the deterministic case ;  , drift occurs when the translational and rotational motions of particles are properly synchronized, and thermal fluctuations do not destroy the drift motion; they only decrease the drift velocity. Thus, the statistical theory developed here confirms the existence of the drift phenomenon in suspensions of ferromagnetic particles and has a much wider range of applicability compared to the deterministic one.
Using the planar rotor model and general model, we have solved the corresponding Fokker-Planck equations in the limit of strong thermal fluctuations and, on this basis, calculated the particle drift velocity for these models. It turned out that the out-of-plane fluctuations, which are not accounted for by the planar rotor model, play an important role, resulting in decreasing the drift velocity by three times.
To verify our analytical results obtained in the case of large thermal fluctuation and to analyze the dependence of the drift velocity on temperature and other parameters, we have solved numerically a set of effective stochastic differential equations describing the magnetization dynamics. In this way, it has been shown that the analytical and numerical results are in good agreement with each other, if the magnetic energy of a particle is very small compared to the thermal energy. Next, using the validated numerical procedure, we have calculated the temperature dependence of the drift velocity of Co nanoparticles suspended in water. An important feature of our approach is that the temperature dependence of the dynamic viscosity of water is explicitly taken into account. This permitted us to determine the drift velocity in a wide temperature interval and clarify the role of thermal fluctuations. It has been shown, in particular, that the temperature dependence of the drift velocity of rather large particles is almost completely determined by the temperature dependence of the dynamic viscosity of water, and thus thermal fluctuations do not noticeably affect the drift velocity of these particles. In contrast, thermal fluctuations strongly influence (decrease) the drift velocity of smaller particles whose size is of the order of a few nanometers or less. But the most remarkable feature of the drift velocity is that its sign (i.e., the drift direction) can be changed by changing temperature, if the initial phase of the external driving force is chosen appropriately. This opens new perspectives in the development of innovative methods for separation of ferromagnetic particles in suspensions.
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