We investigated the field-dependent Néel relaxation time of magnetic nanoparticles (MNPs) in an AC excitation field. Specifically, a fundamental component of the magnetization for immobilized MNPs was studied by numerical simulation while changing the frequency f and amplitude Hac of the field. From the simulation results, we clarified the Hac dependence of the effective Néel relaxation time τN,e and obtained an empirical expression for τN,e(Hac) for the first time. The expression was obtained for the cases when the angle of the easy axis of magnetization in MNPs is fixed and randomly distributed. Using the Hac dependencies of τN,e and the previously reported Brownian relaxation time τB,e, we showed that the behavior of suspended MNPs changes from Brownian-dominant to Néel-dominant when Hac increases, even when the MNP parameters are fixed, and we obtained an expression for the boundary field between them. Furthermore, we classified several types of responses for the suspended MNPs in the AC field using the magnitude relationship among τN,e(Hac), τB,e(Hac), and 1/(2πf). Finally, we experimentally verified the classification, and reasonable agreement was observed between the experiment and analysis. The results are useful for determining suitable MNP parameters and excitation conditions for various biomedical applications.

Magnetic nanoparticles (MNPs) have been widely studied for biomedical applications, such as magnetic particle imaging,1,2 bio-sensing,3,4 and magnetic hyperthermia.5,6 These applications generally rely on the magnetization (M–H) curve of MNPs under an applied AC field H(t) = Hacsin(2πft), where Hac and f are the amplitude and frequency of the field, respectively. The AC M–H curve is determined by the Néel relaxation mechanism for immobilized MNPs. However, both the Brownian and Néel relaxation mechanisms affect the AC M–H curve for suspended MNPs, and the MNP parameters and excitation conditions determine which relaxation type occurs faster and becomes dominant. As a result, suspended MNPs exhibit various characteristics depending on the MNP parameters and excitation conditions. The effects of these factors on the AC M–H curve for suspended MNPs have been studied both theoretically and experimentally.7–20 

The dominant relaxation behavior for suspended MNPs is determined by the magnitude relationship between τN,e(Hac) and τB,e(Hac) when an AC field is applied, where τN,e(Hac) and τB,e(Hac) are the field-dependent Néel and Brownian relaxation times, respectively. These dependencies must be clarified to determine the dominant relaxation. For Brownian relaxation, an expression for τB,e(Hac) has been empirically obtained based on numerical simulations.17–19 For Néel relaxation, however, an expression for τN (Hdc) has only been obtained for the case when a DC (or step) field Hdc is applied to MNPs.21,22 This expression for τN(Hdc) has been tentatively used to approximate τN,e(Hac). Field-dependent relaxation times were also measured, and the experimental results were compared with analytical ones.23–25 

In our previous studies,26,27 we investigated the conditions under which the Brownian or Néel relaxation becomes dominant for suspended MNPs by approximating τN,e(Hac) using τN(Hdc). We showed that the MNP behavior changes from Brownian-dominant to Néel-dominant when Hac increases, even when the MNP parameters are fixed. This occurs because the reduction in τN,e with increasing Hac is much more significant than the reduction in τB,e. However, for a more quantitative analysis, it is necessary to obtain an expression for τN,e(Hac) that applies to a practical case (i.e., when the AC field is applied to an ensemble of immobilized MNPs with randomly oriented easy axes). By obtaining the expression for τN,e(Hac), we can classify the responses of suspended MNPs to an AC field into several types using the magnitude relationship among τN,e(Hac), τB,e(Hac), and 1/(2πf). This classification is necessary to determine the suitable MNP parameters and excitation conditions for various biomedical applications.

In this study, we first obtained an expression for τN,e(Hac) under an applied AC field. For this purpose, we studied the AC M–H curves of immobilized MNPs using numerical simulations. Specifically, the frequency dependence of the imaginary part of the fundamental component of magnetization, Im[M1], was investigated. Analyzing the Im[M1]–f curve, we obtained the τN,e–Hac curve and an empirical expression for τN,e(Hac). We obtained τN,e(Hac) for both fixed and randomly distributed angles of the easy axis in MNPs. Next, we obtained a boundary field that separates Néel- and Brownian-dominant regions for suspended MNPs, where the field satisfies the condition τN,e(Hac) = τB,e(Hac). We also classified several types of responses for suspended MNPs in an AC field. Characteristic fields were obtained to determine the boundaries of these response types. Finally, we performed an experiment to verify the classification, and reasonable agreement was obtained between the experiment and analysis.

We obtained the Im[M1]–f curve for immobilized MNPs by solving the Fokker–Planck equation by numerical simulation,13 where we neglected magnetic dipole–dipole interactions between MNPs. Because the dynamic behavior of MNPs is determined by the values of σ = KVc/(kBT) and ξac = μ0Hacm/(kBT), the simulation was performed for a wide range of σ and ξac, where dc and Vc = (π/6)dc3 are the diameter and volume of the magnetic core, respectively; m = MsVc is the magnetic moment; K and Ms are the effective anisotropy constant and saturation magnetization, respectively; kB is the Boltzmann constant; and T is the absolute temperature. Note that ξac can be expressed as ξac = 2σhac, where hac = Hac/Hk is the field normalized by the characteristic field Hk = 2K/(μ0Ms). Numerical simulation was performed for σ = 10–30, hac = 0.05–0.8 (or ξac = 0.1σ–1.6σ), f = 1–107 Hz, and T = 300 K.

Figure 1(a) shows an example of the Im[M1]–f curves for immobilized MNPs with randomly oriented easy axes (i.e., when the angle of the easy axis of magnetization with respect to the direction of the applied field, β, is randomly distributed). The symbols represent the results obtained from the numerical simulation for σ = 20 and hac = 0.1–0.5. Im[M1] exhibited a peak value at a specific frequency, which is denoted as fNp, and fNp increased with increasing hac.

FIG. 1.

(a) The Im[M1]–f curves for immobilized MNPs with randomly oriented easy axes. The symbols represent results obtained by numerical simulation for σ  = 20 and hac = 0.1–0.5. (b) The fNphac curves obtained for σ  = 20 and different values of β. The dotted lines in (a) and (b) are included as guides for visualization.

FIG. 1.

(a) The Im[M1]–f curves for immobilized MNPs with randomly oriented easy axes. The symbols represent results obtained by numerical simulation for σ  = 20 and hac = 0.1–0.5. (b) The fNphac curves obtained for σ  = 20 and different values of β. The dotted lines in (a) and (b) are included as guides for visualization.

Close modal

Figure 1(b) shows the fNphac curves obtained for σ = 20. Closed circles were obtained for MNPs with randomly oriented easy axes, from the results shown in Fig. 1(a). For comparison, the fNphac curves are also shown for β  = 0°, 15°, and 45°. For hac < 0.2, the fNphac curve was almost independent of β. However, the fNphac curve was significantly affected by β for hac > 0.2, and fNp became higher with an increase in β. The value of fNp for β = 15° was close to that for MNPs with a random distribution of β.

As in the case of the previously reported Brownian relaxation time τB,e(Hac),17 we used fNp, to define the effective Néel relaxation time τN,e(hac) as
τ N , e ( h a c ) = 1 2 π f Np .
(1)

1. MNPs with a fixed β value

First, we studied the case for β = 0. The τN,e–hac curve was obtained by substituting the fNphac curve [Fig. 1(b)] into Eq. (1). The symbols in Fig. 2(a) represent the simulation results obtained for σ = 10–30. As shown, τN,e rapidly decreased with increasing hac, and the decrease was more significant for larger σ values.

FIG. 2.

(a) The τN,e–hac curves for β  = 0. The symbols represent the simulation results obtained for σ = 10–30, whereas the lines were calculated using Eqs. (2)–(4). (b) The Δɛhac curves for σ  = 14, 22, and 30. The symbols represent the simulation results, whereas the lines were calculated using Eq. (4).

FIG. 2.

(a) The τN,e–hac curves for β  = 0. The symbols represent the simulation results obtained for σ = 10–30, whereas the lines were calculated using Eqs. (2)–(4). (b) The Δɛhac curves for σ  = 14, 22, and 30. The symbols represent the simulation results, whereas the lines were calculated using Eq. (4).

Close modal
The hac dependencies of τN,e in Fig. 2(a) were fitted using the following equation:
τ N , e ( h a c ) = τ 0 π σ 1 1 h a c exp [ σ Δ ε ( h a c ) ] .
(2)

We note that the functional form of Eq. (2) was decided based on the function τN (hdc) obtained in Ref. 21 for the DC field hdc and β = 0. The functional form of Eq. (2) is an approximate expression for τN(hdc) and hdc = 0.05–0.7, where Δɛ represents the field dependence of the energy barrier, and τ0 = 10−9 s is the characteristic time.

From Eq. (2), we obtain
σ Δ ε ( h a c ) = ln { τ N , e τ 0 σ π ( 1 h a c ) } .
(3)

Substituting the simulation results for the τN,e–hac curve [symbols in Fig. 2(a)] into Eq. (3), we can obtain the Δɛhac curve. The symbols in Fig. 2(b) show the Δɛhac curves obtained for σ = 14, 22, and 30. As shown, Δɛ decreased with increasing hac, and the Δɛhac curve slightly depended on σ.

An empirical expression for the Δɛhac curve in Fig. 2(b) was obtained as
Δ ε ( h a c ) = ( 1 k h a c ) 2 for β = 0 ,
(4)
with
k = 0.66 + 0.0056 × σ .
(5)

The lines in Fig. 2(b) were calculated using Eq. (4) and agreed well with the simulation results. The lines in Fig. 2(a) represent the field-dependent Néel relaxation time calculated using Eqs. (2) and (4) and agree well with the simulation results.

We note that Δɛ is given by Δ ε ( h d c ) = ( 1 h d c ) 2 when a DC field hdc is applied and β = 0.21 Therefore, khac in Eq. (4) represents an effective field when an AC field hac is applied. This effective field slightly depended on σ, as shown in Eq. (5), and k changed from 0.72 to 0.83 when σ changed from 10 to 30.

Next, we studied the Néel relaxation time for an arbitrary value of β. Figure 3(a) shows the τN,e–hac curves obtained for β = 0°, 15°, 25°, and 45° when σ = 20: These curves were obtained using the same procedure applied in the case of β = 0. As shown, τN,e became shorter with increasing β.

FIG. 3.

(a) The τN,e–hac curves for different β values. The symbols represent the simulation results for σ = 20, whereas the lines were calculated using Eqs. (2) and (7). (b) The ɛ/σθ curve calculated from Eq. (6) for β = 45°, σ = 20, and hdc = 0.2. The inset represents the magnetic moment vectors m1 and m2 at θ1 and θ2, respectively. Solid and dotted lines in the inset represent the directions of the applied field and easy axis, respectively.

FIG. 3.

(a) The τN,e–hac curves for different β values. The symbols represent the simulation results for σ = 20, whereas the lines were calculated using Eqs. (2) and (7). (b) The ɛ/σθ curve calculated from Eq. (6) for β = 45°, σ = 20, and hdc = 0.2. The inset represents the magnetic moment vectors m1 and m2 at θ1 and θ2, respectively. Solid and dotted lines in the inset represent the directions of the applied field and easy axis, respectively.

Close modal
The β dependence of τN,e can be considered as follows. For a finite value of β, the energy of the MNPs system normalized by kBT, namely, ɛ, is given by28,
ε = σ si n 2 ( θ β ) ξ d c cos ( θ ) ,
(6)
where θ is the angle of the magnetic moment vector m, and ξdc = μ0Hdcm/(kBT). Figure 3(b) shows an example of the ɛ/σθ curve calculated using Eq. (6) for β = 45°, σ = 20, and hdc = 0.2. As shown, ɛ exhibited two local minima at θ  = θ1 and θ2. The magnetic moment vectors m1 and m2 corresponding to θ1 and θ2, respectively, are schematically shown in the inset in Fig. 3(b). The energy barrier for the transition from m2 to m1 is represented by Δɛ. In our previous paper, we obtained Δɛ(hdc, β) for a finite β value under an applied DC field.29 Using an effective field khac for the AC field, as in the case of β = 0, we express Δɛ(hac, β) as
Δ ε ( h ac , β ) = ( 1 k h a c ) 2 c 1 [ k h a c 0.78 ( k h a c ) 2 ] sin ( 2 π β / 180 ) .
(7)

Substituting Eq. (7) into Eq. (2), we can obtain the τN,e–hac curve. Here, c1 in Eq. (7) was taken as an adjustable parameter and was determined as c1 = 0.62 to obtain the best fit between the simulation and analysis for the τN,e–hac curve. The lines in Fig. 3(a) represent the analytical results calculated using Eqs. (2) and (7), which agreed reasonably well with the simulation.

We note that the τN,e–hac curve can only be correctly calculated from Eq. (7) for β < 45°, where the MH curve is mainly determined by the reversal of magnetic moment m over the barrier Δɛ. For β > 45°, however, θ1 and θ2 change significantly with hdc. Because magnetization in the direction of the applied field is given by M m 1 cos θ 1 + m 2 cos θ 2, changes in θ1 and θ2 cause changes in the MH curve. This effect becomes larger for larger β values.28 As a result, the deviation between the τN,e–hac curve calculated from Eq. (7) and simulation becomes larger with increasing β for β > 45°.

2. MNPs with a random distribution of β

Next, we considered practical MNPs with randomly oriented easy axes (i.e., for the case when β is uniformly distributed in an ensemble of MNPs). The Im[M1]–f curves for this case are shown in Fig. 1(a), from which the τN,e–hac curves were obtained using the same procedure as before. Figure 4(a) shows the results, where symbols represent the simulation results for σ = 10–30.

FIG. 4.

(a) The τN,e–hac curves for MNPs with randomly oriented easy axes. The symbols represent the simulation results, whereas the lines were calculated using Eqs. (2) and (8). (b) Comparison of the τN,ehac curves for β  = 0 (closed symbols) and randomly distributed β (open symbols). Simulation results for σ  = 14 and 30 are shown, and the lines are included for visualization.

FIG. 4.

(a) The τN,e–hac curves for MNPs with randomly oriented easy axes. The symbols represent the simulation results, whereas the lines were calculated using Eqs. (2) and (8). (b) Comparison of the τN,ehac curves for β  = 0 (closed symbols) and randomly distributed β (open symbols). Simulation results for σ  = 14 and 30 are shown, and the lines are included for visualization.

Close modal
We fitted the τN,e–hac curve using Eq. (2) and the field-dependent Δɛ given by
Δ ε ( h a c ) = 1 a 1 h a c + a 2 h a c 2 for random β ,
(8)
with
a 1 = 0.92 + 0.034 σ and a 2 = 0.45 + 0.045 σ ,
(9)
where a1 and a2 were determined to obtain the best fit between Eq. (8) and the Δɛhac curve obtained by simulation for hac < 0.8 and σ < 30

The lines in Fig. 4(a) were calculated using Eqs. (2) and (8) and agreed well with the simulation results. Therefore, Eqs. (2) and (8) can be used to calculate τN,e(hac) for practical MNPs with randomly oriented easy axes.

Figure 4(b) shows the comparison of the τN,e–hac curves obtained for β  = 0 and randomly distributed β. The simulation results for σ  = 14 and 30 are shown, where the closed and open symbols represent the results obtained for β = 0 and randomly distributed β, respectively. The difference between the two cases was small for low hac values. However, for high hac values, τN,e for the random distribution of β (open symbols) became considerably smaller compared with that for β = 0 (closed symbols). The difference was more significant for σ = 30 than σ  = 14.

We then defined the characteristic field hNT by determining the field at which the field-dependent Néel relaxation time satisfies the condition 2πN,e(hNT) = 1. For MNPs with a random distribution of β, the equation for hNT is obtained by substituting Eqs. (2) and (8) into this condition, and hNT is obtained by numerically solving the resulting equation. An explicit equation for hNT can be obtained by approximating the term 1/(1 − hac) in Eq. (2) with a constant value c1, and c1 = 1.5 was chosen to obtain the best fit between the numerical calculation of hNT and Eq. (10). In this case, we obtained hNT as
h NT = 1 2 a 2 { a 1 a 1 2 4 a 2 [ 1 + ( 1 / σ ) ln ( C N ) ] } for f > f N c ,
(10)
with
C N = 3 π f τ 0 π / σ ,
(11)
where a1 and a2 are given in Eq. (9). Note that hNT = 0 for f < fNc, where fNc is the frequency that satisfies the condition 1 + (1/σ)ln(CN) = 0 and is given by 1/fNc = 3πτ0(π/σ)1/2 exp(σ). Using the frequency fN,0 = 1/(2πτN,0), fNc can be expressed as fNc = (1/3)fN,0, where τN,0(σ) = (τ0/2)(π/σ)1/2exp(σ) is the Néel relaxation time at hac = 0.21 

We note that hNT is also related to the field that gives the peak value of the Im[M1]–hac curve. Figure 5(a) shows the Im[M1]–hac curves simulated for σ = 20 and different f values. For small hac values, Im[M1] rapidly increased with increasing hac. Then, Im[M1] had a peak value at a specific value of hac, denoted as hNp, and gradually decreased with increasing hac. The field hNp increased with increasing f.

FIG. 5.

(a) The Im[M1]–hac curves for σ = 20 and different f values. The symbols represent simulation results, and dotted lines are included for visualization. (b) Dependence of hNp on f obtained for σ = 20. The solid line represents hNT calculated using Eq. (10).

FIG. 5.

(a) The Im[M1]–hac curves for σ = 20 and different f values. The symbols represent simulation results, and dotted lines are included for visualization. (b) Dependence of hNp on f obtained for σ = 20. The solid line represents hNT calculated using Eq. (10).

Close modal

The circles in Fig. 5(b) show the simulated frequency dependence of hNp obtained for σ = 20. The solid line represents hNT calculated using Eq. (10). As shown, hNT is approximately equal to hNp, although strictly speaking, it is not the same as hNp.

In our previous paper,30 the field represented by hac,th was used as a threshold field for the reversal of the magnetic moment against the energy barrier. It can be shown that hNT is slightly larger than hac,th and can be approximately expressed as h N T h a c , t h + 0.05.

For MNPs suspended in liquid, both Brownian and Néel relaxations affect the AC M–H curve. Their response to the AC field depends on the magnitude relationship among τN,e(Hac),τB,e(Hac), and 1/(2πf). To classify the responses of suspended MNPs, we define two fields in addition to hNT in Eq. (10), as follows.

We consider the case when MNP behavior is dominated by Brownian relaxation. Previously, we studied the Im[M1]–f curve for the Brownian-dominant case and obtained the field-dependent Brownian relaxation time, τB,e, as17,
τ B , e ( h a c ) = τ B , 0 1 + 0.07 ξ a c 2 ,
(12)
where τ B , 0 = 3 η V H / ( k B T ) is the Brownian relaxation time for hac = 0, η is the liquid viscosity, and VH = (π/6)dH3 and dH are the hydrodynamic volume and diameter of an MNP, respectively. Notably, the term 0.07ξac2 in Eq. (12) should be refined to 0.126ξac1.72 when we consider very large ξac values up to 300.19 We used Eq. (12) because the simulation described in Sec. II was performed for ξac < 50 in the present study.
Similar to the Néel-dominant case, we define the field hBT by considering the field that satisfies the condition 2πB,e(hBT) = 1. The field hBT can be obtained by solving the condition with Eq. (12). Because ξac = 2σhac, we obtained
h BT = 1.91 σ ( 2 π f τ B , 0 ) 2 1 for f > f B , 0 .
(13)

Note that hBT = 0 for f < fB,0, where fB,0 is given by τB,0 as fB,0 = 1/(2πτB,0).

The hNT–f and hBT–f curves are shown in Fig. 6, and the results obtained for σ = 13 and 30 are shown in Figs. 6(a) and 6(b), respectively. The red circles in Fig. 6 show the hNT–f curves obtained by numerically solving the equation for hNT (i.e., 2πN,e(hNT) = 1). The red lines in Fig. 6 represent the hNT–f curves calculated using Eq. (10) and agree well with the numerical results. The blue lines in Fig. 6 present the hBT–f curves calculated using Eq. (13). In the calculation, we set dH = 50 nm and η = 0.86 mPa s for water, which gave τB,0 = 41 μs and fB,0 = 3.9 kHz.

FIG. 6.

The hBT–f and hNT–f curves: (a) σ = 13 and dH = 50 nm, (b) σ = 30 and dH = 50 nm. The blue lines are the hBT–f curves calculated using Eq. (13). The red lines are the hNT–f curves calculated using Eq. (10). The red circles are obtained by solving for hNT numerically, τN,e(hNT) = 1/(2πf).

FIG. 6.

The hBT–f and hNT–f curves: (a) σ = 13 and dH = 50 nm, (b) σ = 30 and dH = 50 nm. The blue lines are the hBT–f curves calculated using Eq. (13). The red lines are the hNT–f curves calculated using Eq. (10). The red circles are obtained by solving for hNT numerically, τN,e(hNT) = 1/(2πf).

Close modal
As shown in Figs. 6(a) and 6(b), the hBTf and hNTf curves intersect at a field represented by hBY. We note that the condition τB,e(hac) = τN,e(hac) is satisfied at hac = hBY. Substituting Eq. (12) for τB,e(hac) and Eqs. (2) and (8) for τN,e(hac) into this condition, we obtain the equation for hBY, and hBY can be obtained by solving the equation numerically. An explicit equation for hBY can be obtained by approximating the term 1 + 0.07ξac2 in Eq. (12) as 1 + 0.07ξac2 = 1 + 0.28 hac2σ2 = 1 + c2σ2, where c2 is a parameter that is assumed to be independent of hac. The value of c2 was determined to obtain good agreement between the numerical calculation of hBY and Eq. (14). Empirical expression for c2 was obtained as c 2 = 0.004 ( σ 1.5 σ B Y ) . Here, σBY is given by the condition τN,e(h = 0) = τB,0. Therefore, we obtained hBY as
h B Y = 1 2 a 2 { a 1 a 1 2 4 a 2 [ 1 + ( 1 / σ ) ln ( C B Y ) ] } ,
(14)
with
C B Y = 1.5 τ 0 τ B , 0 π σ 1 + 0.004 ( σ 1.5 σ B Y ) σ 2 .
(15)
Because τN,e(h = 0) is given by τN,e(h = 0) = τ0(π/σ)1/2exp(σ) from Eq. (2), σBY satisfies the condition τ0(π/σBY)1/2exp(σBY) = τB,0. An approximate expression for σBY can be obtained as
σ B Y = 0.86 1.05 × ln ( 2 τ 0 τ B , 0 ) .
(16)

Figure 7 presents the dependence of hBY on σ for different values of dH. The results for dH = 30, 50, and 100 nm are shown, corresponding to τB,0 = 8.8, 41, and 326 μs. The circles show hBY obtained by numerically solving the equation τN,e(hBY) = τB,e(hBY), while the lines were calculated using Eqs. (14)–(16). As shown, hBY became larger with increasing σ. The value of hBY also increases with decreasing dH. These dependencies quantitatively agreed between the numerical result and Eqs. (14)–(16).

FIG. 7.

Dependence of hBY on σ. Circles present the numerical results for hBY, and the solid lines are calculated using Eqs. (14)–(16). Results for dH = 30, 50, and 100 nm are shown.

FIG. 7.

Dependence of hBY on σ. Circles present the numerical results for hBY, and the solid lines are calculated using Eqs. (14)–(16). Results for dH = 30, 50, and 100 nm are shown.

Close modal

We note that, in our previous study,27 the field represented by hac,BR was used for the boundary between Brownian- and Néel-dominant regions, where τN,e(hac) was approximated using τN(hdc). The field hBY in Eq. (14) provides a more accurate boundary between the two regions.

Using the fields hNT, hBT, and hBY, we can classify the responses of suspended MNPs to the AC field into several types, as listed in Table I. In Fig. 6, the region for each response type is shown in the hac-f plane. First, the field hBY gives the boundary between the Néel- and Brownian-dominant regions. For hac > hBY, Néel relaxation becomes dominant, but Brownian relaxation becomes dominant for hac < hBY. In Fig. 6, the region for hac > hBY is represented by N, and the region for hac < hBY is represented by B.

TABLE I.

Classification by the type of operating mechanism.

NéelConditionBrownianCondition
N0 f < fNc and τN,e < τB,e B0 f < fNc and τN,e > τB,e 
N1 τN,e < τB,e < 1/(2πfB1 τB,e < τN,e < 1/(2πf
N2 τN,e < 1/(2πf) < τB,e B2 τB,e < 1/(2πf) < τN,e 
N3 1/(2πf) < τN,e < τB,e B3 1/(2πf) < τB,e < τN,e 
NéelConditionBrownianCondition
N0 f < fNc and τN,e < τB,e B0 f < fNc and τN,e > τB,e 
N1 τN,e < τB,e < 1/(2πfB1 τB,e < τN,e < 1/(2πf
N2 τN,e < 1/(2πf) < τB,e B2 τB,e < 1/(2πf) < τN,e 
N3 1/(2πf) < τN,e < τB,e B3 1/(2πf) < τB,e < τN,e 

In the Néel-dominant region, the response of MNPs to the AC field also depends on the magnitude relationship between 1/(2πf) and τN,e. Therefore, region N is subdivided into four regions, as listed in Table I (i.e., regions N0, N1, N2, and N3). Similarly, the Brownian-dominant region is subdivided into four regions, namely B0, B1, B2, and B3, depending on the magnitude relationship between 1/(2πf) and τB,e. In Fig. 6, these regions are shown in the hac-f plane for σ  = 13 and 30.

Region N0 in Fig. 6(a) is determined for f  < fNc and hBY < hac. In this region, both the Néel and Brownian relaxation times at hac = 0 satisfy the conditions τN,0 < 1/(2πf) and τB,0 < 1/(2πf). Thus, the magnetic moment vector m can respond to the AC field with negligible delay time.

Region N1 is determined by the field with hBY < hac and hBT < hac. In this region, the condition τN,e < τB,e < 1/(2πf) is satisfied. Therefore, MNPs mainly respond to the AC field via Néel relaxation, although Brownian relaxation slightly affects MNP behavior.

Region N2 is determined by the field with hNT < hac < hBT. In this region, the condition τN,e < 1/(2πf) <τB,e is satisfied. Therefore, the effect of Brownian relaxation can be neglected, and the AC M–H curve is mostly determined by the Néel relaxation mechanism.

Region N3 is determined by the field with hBY<hac < hNT. In this region, the condition 1/(2πf) < τN,e < τB,e is satisfied, and the response of MNPs via Néel relaxation becomes small.

In regions N1 and N2, the alignment of the easy axes in suspended MNPs is caused by the AC field.7–10 Namely, the angle β for suspended MNPs converges around a specific value, unlike immobilized MNPs with a random distribution of β. In these regions, it was also shown that the AC M–H curve for suspended MNPs can be approximated using that for immobilized MNPs with partially aligned easy axes.12,26,29 Strictly speaking, it is necessary to obtain the AC M–H curve for the suspended MNPs by simultaneously solving the dynamic behavior of the magnetic moment vector m and unit vector along the easy axis n.31 

In region B0, the magnetic moment vector m can respond to the AC field with negligible delay time, as in region N0.

In region B1, the condition τB,e < τN,e < 1/(2πf) is satisfied. Therefore, MNPs mainly respond to the AC field via Brownian relaxation, although the Néel relaxation slightly affects MNP behavior.

In region B2, the condition τB,e < 1/(2πf) < τN,e is satisfied. Therefore, the effect of the Néel relaxation can be neglected, and the AC M–H curve is mostly determined by Brownian relaxation.

In region B3, the condition 1/(2πf) < τB,e < τN,e is satisfied, and the response of MNPs via Brownian relaxation becomes small.

When we compare the results for σ = 13 [Fig. 6(a)] and 30 [Fig. 6(b)], regions N0 and B0 disappear for σ = 30. For σ = 13, the hBY value is small, and regions N1 and N2 (Néel-dominant regions) are wide. For σ = 30, however, hBY becomes large, and region B2 becomes wide, corresponding to the increase in the Brownian-dominant region for suspended MNPs.

As shown in Fig. 6, the behavior of suspended MNPs becomes Brownian- or Néel-dominant depending on the hac and f of the excitation field. We experimentally verified this property using an MNP sample called MS1(Meito Sangyo), which was magnetically fractionated from Ferucarbotran (Meito Sangyo) to obtain MNPs with large magnetic moments.32 In the experiment, we used MNPs dispersed in water and MNPs dispersed in glycerol. Specifically, 150 μg of MNPs was dispersed in 150 μl of water. The glycerol sample, in which 150 μg of MNPs was included, was prepared by mixing 144 μl of glycerol and 6 μl of suspended MS1. The viscosity of the glycerol sample was calculated to be as large as η = 411 mPa s,33 which was much larger than that of the water sample, η = 0.86 mPa s. Therefore, a glycerol sample was used to study the case when Brownian relaxation can be neglected.

The parameters of the MS1 sample were estimated in our previous paper.26 We obtained Ms = 360 kA/m from the DC M–H curve in solution. The distribution of dc in the sample was also evaluated from the analysis of the DC M–H curve, ranging from 15 to 32 nm, with a typical value of 24 nm. If we use the estimated value of K = 7 kJ/m3,26 σ widely ranged from 3 to 29. The distribution of dH was evaluated by dynamic light scattering, ranging from 30 to 100 nm, with a typical value of 50 nm. When we used η = 0.86 mPa s for water, τΒ,0 ranged from 8.8 to 326 μs (or fΒ,0 ranged from 18 to 0.5 kHz).

As shown in Eqs. (10) and (13), HNT and HBT are determined by σ and τB,0. Although σ and τB,0 were distributed in the sample, we assumed representative MNPs with typical values and made a semi-quantitative discussion in the present study. Figure 8(a) presents the HNT–f and HBT–f curves calculated for the water sample with σ = 19.4 and τΒ,0 = 90 μs. The procedure for determining these values is shown in Figs. 8(b) and 9(a). As shown in Fig. 8(a), the HNT–f and HBT–f curves intersect at μ0Hac = μ0HBY = 12 mT for the water sample. We note that fB,0 and HBY become almost zero for the glycerol sample, and regions N1, B1, B2, and B3 disappear, as shown in Fig. 8(a). Therefore, for the glycerol sample, only the regions N2 and N3 existed for Hac > HNT and Hac < HNT, respectively.

FIG. 8.

(a) The HNT–f and HBT–f curves for the MS1 sample dispersed in water. (b) The Im[M1]–Hac curves for the MS1 sample dispersed in glycerol, using f = 3, 10, and 20 kHz. The vertical bars represent HNT for each frequency.

FIG. 8.

(a) The HNT–f and HBT–f curves for the MS1 sample dispersed in water. (b) The Im[M1]–Hac curves for the MS1 sample dispersed in glycerol, using f = 3, 10, and 20 kHz. The vertical bars represent HNT for each frequency.

Close modal
FIG. 9.

The Im[M1]–μ0Hac curves for water and glycerol samples: (a) f = 3 kHz and (b) f = 20 kHz. (c) The ratio of Im[M1] for the water sample to that for the glycerol sample as a function of Hac. The symbols represent experimental results, and lines are included for visualization.

FIG. 9.

The Im[M1]–μ0Hac curves for water and glycerol samples: (a) f = 3 kHz and (b) f = 20 kHz. (c) The ratio of Im[M1] for the water sample to that for the glycerol sample as a function of Hac. The symbols represent experimental results, and lines are included for visualization.

Close modal

We first measured the Im[M1]–Hac curve for the glycerol sample. The results are shown in Fig. 8(b) for f = 3, 10, and 20 kHz. The Im[M1]–Hac curve had a peak value at a specific value of Hac, which increased with increasing f. This was consistent with the simulation results shown in Fig. 5(a). The measured Im[M1]–Hac curve, however, had a broad peak owing to the distribution of dc in the sample. As shown in Fig. 5(b), the field that gives the peak of the Im[M1]–Hac curve is approximately equal to HNT for the Néel-dominant case. Therefore, for simplicity, we estimated HNT from the field that gave the peak of the Im[M1]–Hac curve. We obtained μ0HNT = 11, 13, and 15 mT for f = 3, 10, and 20 kHz, respectively, as shown by the vertical bars in Fig. 8(b). In Fig. 8(a), circles represent these experimental values of μ0HNT. When we calculated the HNT–f curve shown in Fig. 8(a), we took σ in Eq. (10) as an adjustable parameter and determined σ = 19.4 to obtain the best fit between the experiment and Eq. (10).

Next, we compared the Im[M1]–Hac curves obtained for MNPs suspended in glycerol and water. Figure 9(a) shows the result for f = 3 kHz. Circles and rectangles represent the results for the glycerol and water samples, respectively. The difference in the Im[M1]–Hac curve between the water and glycerol samples became small for μ0Hac > 12 mT. Therefore, we estimated μ0HBY = 12 mT for this sample. Substituting this value into Eq. (14), we obtained τΒ,0 = 90 μs, and, thus, the HBT–f curve in Fig. 8(a) was calculated using τΒ,0 = 90 μs. For the water sample, the Im[M1]–Hac curve for f = 3 kHz had a peak value at a specific field μ0Hac =  6 mT, as shown in Fig. 9(a). In Fig. 8(a), the rectangles represent this specific field for f = 3 kHz, which was close to the HBT–f curve.

In Fig. 9(c), the ratio of Im[M1] for the water sample to that for the glycerol sample is shown as a function of Hac. For f = 3 kHz, the ratio was close to 1 for Hac > HBY, which means that Im[M1] was almost independent of viscosity in this region. This occurred because MNPs in the water sample operate in region N1 in Fig. 8(a) for Hac > HBY and f = 3 kHz, and the MNP behavior is dominated by Néel relaxation, as in the case of the glycerol sample. For f = 3 kHz and Hac < HBY, however, the ratio significantly increased with decreasing Hac, as shown in Fig. 9(c). This result indicates that the Im[M1]–Hac curve for the water sample is strongly affected by Brownian relaxation in this case. This is because MNPs operated in regions B1, B2, and B3 for f = 3 kHz and Hac < HBY, and Hac existed near HBT for these regions, as shown in Fig. 8(a). In this case, Brownian relaxation strongly affects the AC M–H curve, as discussed in Sec. III.

Figure 9(b) shows the Im[M1]–Hac curves for f = 20 kHz. The difference in the Im[M1]–Hac curve between the water and glycerol samples was small for all Hac values. As shown in Fig. 9(c), the ratio of Im[M1] for the water sample to that for the glycerol sample was approximately 1.1 for Hac > HBY. The reason for this small difference remains unclear. The ratio only slightly increased with decreasing Hac, even for Hac < HBY. Furthermore, MNPs in the water sample operated in region B3 for Hac < HBY, as shown in Fig. 8(a). Because μ0HBT is calculated as 46 mT for f = 20 kHz using Eq. (13), the field Hac for region B3 existed far below HBT. Therefore, the effect of Brownian relaxation became small, as discussed in Sec. III.

In Fig. 9(c), the results for f = 10 kHz are also shown. In this case, the increase in the ratio for Hac < HBY became larger than that for f = 20 kHz. This is because HBT for f = 10 kHz becomes smaller than that for f = 20 kHz, where μ0HBT = 23 and 46 mT for f = 10 and 20 kHz, respectively. As a result, the field Hac for region B3 became closer to HBT for f = 10 kHz compared with the case of f = 20 kHz. Therefore, the effect of Brownian relaxation for f = 10 kHz is larger than that for f = 20 kHz.

The agreements between experiment and analysis shown in Figs. 8 and 9 partly validate the classification of the MNP responses using Eqs. (10), (13), and (14). However, for a more quantitative comparison of Im[M1], we must account for the distributions of dc and dH in the sample.

We investigated the field-dependent Néel relaxation time of MNPs, τN,e(Hac), under an applied AC excitation field and obtained an empirical expression for τN,e(Hac) for the first time. The expression was obtained for cases when the easy-axis angle β is fixed and randomly distributed, which is necessary to discuss the dynamic behavior of MNPs. Next, we classified the response of suspended MNPs to the AC field into several types using the magnitude relationship among τN,e(Hac), τB,e(Hac), and 1/(2πf). We also obtained characteristic fields for the boundaries between these types (i.e., HBY, HNT, and HBT). Finally, we aimed to experimentally verify the classification, and reasonable agreement was obtained between the experiment and analysis. This classification of the response type is useful for identifying the suitable MNP parameters and excitation conditions for specific biomedical applications.

This work was supported, in part, by the Japan Society for the Promotion of Science (JSPS) KAKENHI [Grant Nos. JP20H05652, JP21H01343, and JP23K17750]. We thank Robert Ireland, PhD, from Edanz (https://jp.edanz.com/ac) for editing a draft of this manuscript.

The authors have no conflicts to disclose.

Takashi Yoshida: Conceptualization (equal); Data curation (lead); Formal analysis (supporting); Funding acquisition (lead); Investigation (equal); Writing – original draft (supporting). Keiji Enpuku: Conceptualization (equal); Data curation (supporting); Formal analysis (lead); Investigation (equal); Writing – original draft (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

1.
L. R.
Croft
,
P. W.
Goodwill
,
J. J.
Konkle
,
H.
Arami
,
D. A.
Price
,
A. X.
Li
,
E. U.
Saritas
, and
S. M.
Conolly
,
Med. Phys.
43
,
424
(
2016
).
2.
M.
Graeser
,
F.
Thieben
,
P.
Szwargulski
,
F.
Werner
,
N.
Gdaniec
,
M.
Boberg
,
F.
Griese
,
M.
Möddel
,
P.
Ludewig
,
D.
van de Ven
,
O. M.
Weber
,
O.
Woywode
,
B.
Gleich
, and
T.
Knopp
,
Nat. Commun.
10
,
1936
(
2019
).
3.
J.-J.
Chieh
,
W.-C.
Wei
,
S.-H.
Liao
,
H.-H.
Chen
,
Y.-F.
Lee
,
F.-C.
Lin
,
M.-H.
Chiang
,
M.-J.
Chiu
,
H.-E.
Horng
, and
S.-Y.
Yang
,
Sensors
18
,
1043
(
2018
).
4.
K.
Wu
,
D.
Su
,
R.
Saha
,
J.
Liu
,
V. K.
Chugh
, and
J.-P.
Wang
,
ACS Appl. Nano Mater.
3
,
4972
4989
(
2020
).
5.
E. A.
Périgo
,
G.
Hemery
,
O.
Sandre
,
D.
Ortega
,
E.
Garaio
,
F.
Plazaola
, and
F. J.
Teran
,
Appl. Phys. Rev.
2
,
041302
(
2015
).
6.
D.
Chang
,
M.
Lim
,
J. A. C. M.
Goos
,
R.
Qiao
,
Y. Y.
Ng
,
F. M.
Mansfeld
,
M.
Jackson
,
T. P.
Davis
, and
M.
Kavallaris
,
Front. Pharmacol.
9
,
831
(
2018
).
7.
H.
Mamiya
and
B.
Jeyadevan
,
Sci. Rep.
1
,
157
(
2011
).
8.
T.
Yoshida
,
S.
Bai
,
A.
Hirokawa
,
K.
Tanabe
, and
K.
Enpuku
,
J. Magn. Magn. Mater.
380
,
105
110
(
2015
).
9.
S. B.
Trisnanto
,
S.
Ota
, and
Y.
Takemura
,
Appl. Phys. Express
11
,
075001
(
2018
).
10.
M.
Suwa
,
A.
Uotani
, and
S.
Tsukahara
,
Appl. Phys. Lett.
116
,
262403
(
2020
).
11.
S.
Ruta
,
R.
Chantrell
, and
O.
Hovorka
,
Sci. Rep.
5
,
9090
(
2015
).
12.
S. A.
Shah
,
D. B.
Reeves
,
R. M.
Ferguson
,
J. B.
Weaver
, and
K. M.
Krishnan
,
Phys. Rev. B
92
,
094438
(
2015
).
13.
A. L.
Elrefai
,
K.
Enpuku
, and
T.
Yoshida
,
J. Appl. Phys.
129
,
093905
(
2021
).
14.
K.
Enpuku
,
A. L.
Elrefai
,
J.
Gotou
,
S.
Yamamura
,
T.
Sasayama
, and
T.
Yoshida
,
J. Appl. Phys.
130
,
113903
(
2021
).
15.
C.
Shasha
and
K. M.
Krishnan
,
Adv. Mater.
33
,
1904131
(
2021
).
16.
H.
Albers
,
T.
Kluth
, and
T.
Knopp
,
J. Magn. Magn. Mater.
541
,
168508
(
2022
).
17.
T.
Yoshida
and
K.
Enpuku
,
Jpn. J. Appl. Phys.
48
,
127002
(
2009
).
18.
R. J.
Deissler
,
Y.
Wu
, and
M. A.
Martens
,
Med. Phys.
41
,
012301
(
2014
).
19.
H.
Remmer
,
M.
Gratz
,
A.
Tschöpe
, and
F.
Ludwig
,
IEEE Trans. Magn.
53
,
1
(
2017
).
20.
P.
Ilg
and
M.
Kröger
,
Phys. Chem. Chem. Phys.
22
,
22244
22259
(
2020
).
21.
W. T.
Coffey
,
P. J.
Cregg
, and
Y. U. P.
Kalmykov
, “
On the theory of Debye and Néel relaxation of single domain ferromagnetic particles
,”
Adv. Chem. Phys.
83
,
263
464
(
1998
).
22.
A. R.
Chalifour
,
J. C.
Davidson
,
N. R.
Anderson
,
T. M.
Crawford
, and
K. L.
Livesey
,
Phys. Rev. B
104
,
094433
(
2021
).
23.
T.
Yoshida
,
K.
Ogawa
,
K.
Enpuku
,
N.
Usuki
, and
H.
Kanzaki
,
Jpn. J. Appl. Phys.
49
,
053001
(
2010
).
24.
J.
Dieckhoff
,
D.
Eberbeck
,
M.
Schilling
, and
F.
Ludwig
,
J. Appl. Phys.
119
,
043903
(
2016
).
25.
S.
Draack
,
T.
Viereck
,
F.
Nording
,
K.-J.
Janssen
,
M.
Schilling
, and
F.
Ludwig
,
J. Magn. Magn. Mater.
474
,
570
573
(
2019
).
26.
K.
Enpuku
,
S.
Yamamura
, and
T.
Yoshida
,
J. Magn. Magn. Mater.
564
,
170089
(
2022
).
27.
K.
Enpuku
,
Y.
Sun
,
H.
Zhang
, and
T.
Yoshida
,
J. Magn. Magn. Mater.
579
,
170878
(
2023
).
28.
N. A.
Usov
and
Y. B.
Grebenshchikov
,
J. Appl. Phys.
106
,
023917
(
2009
).
29.
K.
Enpuku
and
T.
Yoshida
,
AIP Adv.
12
,
055211
(
2022
).
30.
K.
Enpuku
and
T.
Yoshida
,
AIP Adv.
11
,
125123
(
2021
).
31.
N. A.
Usov
and
B. Y.
Liubimov
,
J. Appl. Phys.
112
,
023901
(
2012
).
32.
T.
Yoshida
,
N. M.
Othman
, and
K.
Enpuku
,
J. Appl. Phys.
114
,
173908
(
2013
).
33.
N.
Cheng
,
Ind. Eng. Chem. Res.
47
,
3285
3288
(
2008
).