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 *H*_{ac} of the field. From the simulation results, we clarified the *H*_{ac} dependence of the effective Néel relaxation time *τ*_{N,e} and obtained an empirical expression for *τ*_{N,e}(*H*_{ac}) 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 *H*_{ac} dependencies of *τ*_{N,e} and the previously reported Brownian relaxation time *τ*_{B,e}, we showed that the behavior of suspended MNPs changes from Brownian-dominant to Néel-dominant when *H*_{ac} 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 *τ*_{N,e}(*H*_{ac}), *τ*_{B,e}(*H*_{ac}), and 1/(2π*f*). 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.

## I. INTRODUCTION

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*) = *H*_{ac}sin(2π*ft*), where *H*_{ac} 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 *τ*_{N,e}(*H*_{ac}) and *τ*_{B,e}(*H*_{ac}) when an AC field is applied, where *τ*_{N,e}(*H*_{ac}) and *τ*_{B,e}(*H*_{ac}) 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 *τ*_{B,e}(*H*_{ac}) has been empirically obtained based on numerical simulations.^{17–19} For Néel relaxation, however, an expression for *τ*_{N} (*H*_{dc}) has only been obtained for the case when a DC (or step) field *H*_{dc} is applied to MNPs.^{21,22} This expression for *τ*_{N}(*H*_{dc}) has been tentatively used to approximate *τ*_{N,e}(*H*_{ac}). 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 *τ*_{N,e}(*H*_{ac}) using *τ*_{N}(*H*_{dc}). We showed that the MNP behavior changes from Brownian-dominant to Néel-dominant when *H*_{ac} increases, even when the MNP parameters are fixed. This occurs because the reduction in *τ*_{N,e} with increasing *H*_{ac} is much more significant than the reduction in *τ*_{B,e}. However, for a more quantitative analysis, it is necessary to obtain an expression for *τ*_{N,e}(*H*_{ac}) 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 *τ*_{N,e}(*H*_{ac}), we can classify the responses of suspended MNPs to an AC field into several types using the magnitude relationship among *τ*_{N,e}(*H*_{ac}), *τ*_{B,e}(*H*_{ac}), and 1/(2π*f*). 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 *τ*_{N,e}(*H*_{ac}) 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[*M*_{1}], was investigated. Analyzing the Im[*M*_{1}]–*f* curve, we obtained the *τ*_{N,e}*–H*_{ac} curve and an empirical expression for *τ*_{N,e}(*H*_{ac}). We obtained *τ*_{N,e}(*H*_{ac}) 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 *τ*_{N,e}(*H*_{ac}) = *τ*_{B,e}(*H*_{ac}). 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.

## II. NÉEL RELAXATION TIME

### A. Definition of effective Néel relaxation time

We obtained the Im[*M*_{1}]–*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* = *KV*_{c}/(*k _{B}T*) and

*ξ*

_{ac}=

*μ*

_{0}

*H*

_{ac}

*m/*(k

_{B}

*T*), the simulation was performed for a wide range of

*σ*and

*ξ*

_{ac}, where

*d*

_{c}and

*V*

_{c}= (π/6)

*d*

_{c}

^{3}are the diameter and volume of the magnetic core, respectively;

*m*=

*M*

_{s}

*V*

_{c}is the magnetic moment;

*K*and

*M*

_{s}are the effective anisotropy constant and saturation magnetization, respectively;

*k*is the Boltzmann constant; and

_{B}*T*is the absolute temperature. Note that

*ξ*

_{ac}can be expressed as

*ξ*

_{ac}= 2

*σh*

_{ac}, where

*h*

_{ac}=

*H*

_{ac}/

*H*

_{k}is the field normalized by the characteristic field

*H*

_{k}= 2

*K*/(

*μ*

_{0}

*M*

_{s}). Numerical simulation was performed for

*σ*= 10–30,

*h*

_{ac}= 0.05–0.8 (or

*ξ*

_{ac}= 0.1

*σ*–1.6

*σ*),

*f*= 1–10

^{7}Hz, and

*T*= 300 K.

Figure 1(a) shows an example of the Im[*M*_{1}]–*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 *h*_{ac} = 0.1–0.5. Im[*M*_{1}] exhibited a peak value at a specific frequency, which is denoted as *f*_{Np}, and *f*_{Np} increased with increasing *h*_{ac}.

Figure 1(b) shows the *f*_{Np}–*h*_{ac} 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 *f*_{Np}–*h*_{ac} curves are also shown for *β* = 0°, 15°, and 45°. For *h*_{ac} < 0.2, the *f*_{Np}–*h*_{ac} curve was almost independent of *β*. However, the *f*_{Np}–*h*_{ac} curve was significantly affected by *β* for *h*_{ac} > 0.2, and *f*_{Np} became higher with an increase in *β*. The value of *f*_{Np} for *β* = 15° was close to that for MNPs with a random distribution of *β*.

*τ*

_{B,e}(

*H*

_{ac}),

^{17}we used

*f*

_{Np}, to define the effective Néel relaxation time

*τ*

_{N,e}(

*h*

_{ac}) as

### B. Dependence of *τ*_{N,e} on *h*_{ac}

#### 1. MNPs with a fixed *β* value

First, we studied the case for *β* = 0. The *τ*_{N,e}*–h*_{ac} curve was obtained by substituting the *f*_{Np}–*h*_{ac} curve [Fig. 1(b)] into Eq. (1). The symbols in Fig. 2(a) represent the simulation results obtained for *σ* = 10–30. As shown, *τ*_{N,e} rapidly decreased with increasing *h*_{ac}, and the decrease was more significant for larger *σ* values.

*h*

_{ac}dependencies of

*τ*

_{N,e}in Fig. 2(a) were fitted using the following equation:

We note that the functional form of Eq. (2) was decided based on the function *τ*_{N} (*h*_{dc}) obtained in Ref. 21 for the DC field *h*_{dc} and *β* = 0. The functional form of Eq. (2) is an approximate expression for *τ*_{N}(*h*_{dc}) and *h*_{dc} = 0.05–0.7, where Δ*ɛ* represents the field dependence of the energy barrier, and *τ*_{0} = 10^{−9} s is the characteristic time.

Substituting the simulation results for the *τ*_{N,e}*–h*_{ac} curve [symbols in Fig. 2(a)] into Eq. (3), we can obtain the Δ*ɛ*–*h*_{ac} curve. The symbols in Fig. 2(b) show the Δ*ɛ*–*h*_{ac} curves obtained for *σ* = 14, 22, and 30. As shown, Δ*ɛ* decreased with increasing *h*_{ac}, and the Δ*ɛ*–*h*_{ac} curve slightly depended on *σ*.

*ɛ*–

*h*

_{ac}curve in Fig. 2(b) was obtained as

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 Δ*ɛ* is given by $\Delta \epsilon ( h d c)= ( 1 \u2212 h d c ) 2$when a DC field *h*_{dc} is applied and *β* = 0.^{21} Therefore, *kh*_{ac} in Eq. (4) represents an effective field when an AC field *h*_{ac} 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 *τ*_{N,e}*–h*_{ac} 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, *τ*_{N,e} became shorter with increasing *β*.

*β*dependence of

*τ*

_{N,e}can be considered as follows. For a finite value of

*β*, the energy of the MNPs system normalized by k

_{B}

*T*, namely,

*ɛ*, is given by

^{28}

^{,}

*θ*is the angle of the magnetic moment vector

*m**,*and

*ξ*

_{dc}=

*μ*

_{0}

*H*

_{dc}

*m/*(k

_{B}

*T*). Figure 3(b) shows an example of the

*ɛ*/

*σ*–

*θ*curve calculated using Eq. (6) for

*β*= 45°,

*σ*= 20, and

*h*

_{dc}= 0.2. As shown,

*ɛ*exhibited two local minima at

*θ*=

*θ*

_{1}and

*θ*

_{2}. The magnetic moment vectors

*m*_{1}and

*m*_{2}corresponding to

*θ*

_{1}and

*θ*

_{2}, respectively, are schematically shown in the inset in Fig. 3(b). The energy barrier for the transition from

*m*_{2}to

*m*_{1}is represented by Δ

*ɛ*. In our previous paper, we obtained Δ

*ɛ*(

*h*

_{dc},

*β*) for a finite

*β*value under an applied DC field.

^{29}Using an effective field

*kh*

_{ac}for the AC field, as in the case of

*β*= 0, we express Δ

*ɛ*(

*h*

_{ac},

*β*) as

Substituting Eq. (7) into Eq. (2), we can obtain the *τ*_{N,e}*–h*_{ac} curve. Here, *c*_{1} in Eq. (7) was taken as an adjustable parameter and was determined as *c*_{1} = 0.62 to obtain the best fit between the simulation and analysis for the *τ*_{N,e}*–h*_{ac} 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 *τ*_{N,e}*–h*_{ac} curve can only be correctly calculated from Eq. (7) for *β* < 45°, where the *M*–*H* curve is mainly determined by the reversal of magnetic moment ** m** over the barrier Δ

*ɛ*. For

*β*> 45°, however,

*θ*

_{1}and

*θ*

_{2}change significantly with

*h*

_{dc}. Because magnetization in the direction of the applied field is given by $M\u221d m 1cos \theta 1+ m 2cos \theta 2$, changes in

*θ*

_{1}and

*θ*

_{2}cause changes in the

*M*–

*H*curve. This effect becomes larger for larger

*β*values.

^{28}As a result, the deviation between the

*τ*

_{N,e}

*–h*

_{ac}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[*M*_{1}]–*f* curves for this case are shown in Fig. 1(a), from which the *τ*_{N,e}*–h*_{ac} 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.

*τ*

_{N,e}

*–h*

_{ac}curve using Eq. (2) and the field-dependent Δ

*ɛ*given by

*a*

_{1}and

*a*

_{2}were determined to obtain the best fit between Eq. (8) and the Δ

*ɛ*−

*h*

_{ac}curve obtained by simulation for

*h*

_{ac}< 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 *τ*_{N,e}(*h*_{ac}) for practical MNPs with randomly oriented easy axes.

Figure 4(b) shows the comparison of the *τ*_{N,e}*–h*_{ac} 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 *h*_{ac} values. However, for high *h*_{ac} values, *τ*_{N,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.

### C. Characteristic field *h*_{NT}

*h*

_{NT}by determining the field at which the field-dependent Néel relaxation time satisfies the condition 2π

*fτ*

_{N,e}(

*h*

_{NT}) = 1. For MNPs with a random distribution of

*β*, the equation for

*h*

_{NT}is obtained by substituting Eqs. (2) and (8) into this condition, and

*h*

_{NT}is obtained by numerically solving the resulting equation. An explicit equation for

*h*

_{NT}can be obtained by approximating the term 1/(1 −

*h*

_{ac}) in Eq. (2) with a constant value

*c*

_{1}, and

*c*

_{1}= 1.5 was chosen to obtain the best fit between the numerical calculation of

*h*

_{NT}and Eq. (10). In this case, we obtained

*h*

_{NT}as

*a*

_{1}and

*a*

_{2}are given in Eq. (9). Note that

*h*

_{NT}= 0 for

*f*<

*f*

_{Nc}, where

*f*

_{Nc}is the frequency that satisfies the condition 1 + (1/

*σ*)ln(

*C*) = 0 and is given by 1/

_{N}*f*

_{Nc}= 3π

*τ*

_{0}(π/

*σ*)

^{1/2}exp(

*σ*). Using the frequency

*f*

_{N,0}= 1/(2π

*τ*

_{N,0}),

*f*

_{Nc}can be expressed as

*f*

_{Nc}= (1/3)

*f*

_{N,0}, where

*τ*

_{N,0}(

*σ*) = (

*τ*

_{0}/2)(π/

*σ*)

^{1/2}exp(

*σ*) is the Néel relaxation time at

*h*

_{ac}= 0.

^{21}

We note that *h*_{NT} is also related to the field that gives the peak value of the Im[*M*_{1}]–*h*_{ac} curve. Figure 5(a) shows the Im[*M*_{1}]–*h*_{ac} curves simulated for *σ* = 20 and different *f* values. For small *h*_{ac} values, Im[*M*_{1}] rapidly increased with increasing *h*_{ac}. Then, Im[*M*_{1}] had a peak value at a specific value of *h*_{ac}, denoted as *h*_{Np}, and gradually decreased with increasing *h*_{ac}. The field *h*_{Np} increased with increasing *f*.

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

In our previous paper,^{30} the field represented by *h*_{ac,th} was used as a threshold field for the reversal of the magnetic moment against the energy barrier. It can be shown that *h*_{NT} is slightly larger than *h*_{ac,th} and can be approximately expressed as $ h N T\u2248 h a c , t h+0.05.$

## III. NÉEL- AND BROWNIAN-DOMINANT REGIONS

### A. Characteristic fields *h*_{BT} and *h*_{BY}

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 *τ*_{N,e}(*H*_{ac}),*τ*_{B,e}(*H*_{ac}), and 1/(2π*f*). To classify the responses of suspended MNPs, we define two fields in addition to *h*_{NT} in Eq. (10), as follows.

*M*

_{1}]–

*f*curve for the Brownian-dominant case and obtained the field-dependent Brownian relaxation time,

*τ*

_{B,e}, as

^{17}

^{,}

*h*

_{ac}= 0,

*η*is the liquid viscosity, and

*V*

_{H}= (π/6)

*d*

_{H}

^{3}and

*d*

_{H}are the hydrodynamic volume and diameter of an MNP, respectively. Notably, the term 0.07

*ξ*

_{ac}

^{2}in Eq. (12) should be refined to 0.126

*ξ*

_{ac}

^{1.72}when we consider very large

*ξ*

_{ac}values up to 300.

^{19}We used Eq. (12) because the simulation described in Sec. II was performed for

*ξ*

_{ac}< 50 in the present study.

*h*

_{BT}by considering the field that satisfies the condition 2π

*fτ*

_{B,e}(

*h*

_{BT}) = 1. The field

*h*

_{BT}can be obtained by solving the condition with Eq. (12). Because

*ξ*

_{ac}= 2σ

*h*

_{ac}, we obtained

Note that *h*_{BT} = 0 for *f* < *f*_{B,0}, where *f*_{B,0} is given by *τ*_{B,0} as *f*_{B,0} = 1/(2π*τ*_{B,0}).

The *h*_{NT}*–f* and *h*_{BT}*–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 *h*_{NT}*–f* curves obtained by numerically solving the equation for *h*_{NT} (i.e., 2π*fτ*_{N,e}(*h*_{NT}) = 1). The red lines in Fig. 6 represent the *h*_{NT}*–f* curves calculated using Eq. (10) and agree well with the numerical results. The blue lines in Fig. 6 present the *h*_{BT}*–f* curves calculated using Eq. (13). In the calculation, we set *d*_{H} = 50 nm and *η* = 0.86 mPa s for water, which gave *τ*_{B,0} = 41 *μ*s and *f*_{B,0} = 3.9 kHz.

*h*

_{BT}–

*f*and

*h*

_{NT}–

*f*curves intersect at a field represented by

*h*

_{BY}. We note that the condition

*τ*

_{B,e}(

*h*

_{ac}) =

*τ*

_{N,e}(

*h*

_{ac}) is satisfied at

*h*

_{ac}=

*h*

_{BY}. Substituting Eq. (12) for

*τ*

_{B,e}(

*h*

_{ac}) and Eqs. (2) and (8) for

*τ*

_{N,e}(

*h*

_{ac}) into this condition, we obtain the equation for

*h*

_{BY}, and

*h*

_{BY}can be obtained by solving the equation numerically. An explicit equation for

*h*

_{BY}can be obtained by approximating the term 1 + 0.07

*ξ*

_{ac}

^{2}in Eq. (12) as 1 + 0.07

*ξ*

_{ac}

^{2}= 1 + 0.28

*h*

_{ac}

^{2}

*σ*

^{2}= 1 +

*c*

_{2}

*σ*

^{2}, where

*c*

_{2}is a parameter that is assumed to be independent of

*h*

_{ac}. The value of

*c*

_{2}was determined to obtain good agreement between the numerical calculation of

*h*

_{BY}and Eq. (14). Empirical expression for

*c*

_{2}was obtained as $ c 2=0.004( \sigma \u2212 1.5 \u2212 \sigma B Y).$ Here,

*σ*

_{BY}is given by the condition

*τ*

_{N,e}(

*h*= 0) =

*τ*

_{B,0}. Therefore, we obtained

*h*

_{BY}as

*τ*

_{N,e}(

*h*= 0) is given by

*τ*

_{N,e}(

*h*= 0) =

*τ*

_{0}(π/

*σ*)

^{1/2}exp(

*σ*) from Eq. (2),

*σ*

_{BY}satisfies the condition

*τ*

_{0}(π/

*σ*

_{BY})

^{1/2}exp(

*σ*

_{BY}) =

*τ*

_{B,0}. An approximate expression for

*σ*

_{BY}can be obtained as

Figure 7 presents the dependence of *h*_{BY} on *σ* for different values of d* _{H}*. The results for

*d*

_{H}= 30, 50, and 100 nm are shown, corresponding to

*τ*

_{B,0}= 8.8, 41, and 326

*μ*s. The circles show

*h*

_{BY}obtained by numerically solving the equation

*τ*

_{N,e}(

*h*

_{BY}) =

*τ*

_{B,e}(

*h*

_{BY}), while the lines were calculated using Eqs. (14)–(16). As shown,

*h*

_{BY}became larger with increasing

*σ*. The value of

*h*

_{BY}also increases with decreasing

*d*

_{H}. These dependencies quantitatively agreed between the numerical result and Eqs. (14)–(16).

### B. Classification by type of operating mechanism

Using the fields *h*_{NT}, *h*_{BT}, and *h*_{BY}, 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 *h*_{ac}*-f* plane. First, the field *h*_{BY} gives the boundary between the Néel- and Brownian-dominant regions. For *h*_{ac} > *h*_{BY}, Néel relaxation becomes dominant, but Brownian relaxation becomes dominant for *h*_{ac} < *h*_{BY}. In Fig. 6, the region for *h*_{ac} > *h*_{BY} is represented by N, and the region for *h*_{ac} < *h*_{BY} is represented by B.

Néel . | Condition . | Brownian . | Condition . |
---|---|---|---|

N0 | f < f_{Nc} and τ_{N,e} < τ_{B,e} | B0 | f < f_{Nc} and τ_{N,e} > τ_{B,e} |

N1 | τ_{N,e} < τ_{B,e} < 1/(2πf) | B1 | τ_{B,e} < τ_{N,e} < 1/(2πf) |

N2 | τ_{N,e} < 1/(2πf) < τ_{B,e} | B2 | τ_{B,e} < 1/(2πf) < τ_{N,e} |

N3 | 1/(2πf) < τ_{N,e} < τ_{B,e} | B3 | 1/(2πf) < τ_{B,e} < τ_{N,e} |

Néel . | Condition . | Brownian . | Condition . |
---|---|---|---|

N0 | f < f_{Nc} and τ_{N,e} < τ_{B,e} | B0 | f < f_{Nc} and τ_{N,e} > τ_{B,e} |

N1 | τ_{N,e} < τ_{B,e} < 1/(2πf) | B1 | τ_{B,e} < τ_{N,e} < 1/(2πf) |

N2 | τ_{N,e} < 1/(2πf) < τ_{B,e} | B2 | τ_{B,e} < 1/(2πf) < τ_{N,e} |

N3 | 1/(2πf) < τ_{N,e} < τ_{B,e} | B3 | 1/(2πf) < τ_{B,e} < τ_{N,e} |

In the Néel-dominant region, the response of MNPs to the AC field also depends on the magnitude relationship between 1/(2π*f*) and *τ*_{N,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/(2π*f*) and *τ*_{B,e}. In Fig. 6, these regions are shown in the *h*_{ac}*-f* plane for *σ* = 13 and 30.

Region N0 in Fig. 6(a) is determined for *f* < *f*_{Nc} and *h*_{BY} < *h*_{ac}. In this region, both the Néel and Brownian relaxation times at *h*_{ac} = 0 satisfy the conditions *τ*_{N},_{0} < 1/(2π*f*) and *τ*_{B},_{0} < 1/(2π*f*). Thus, the magnetic moment vector ** m** can respond to the AC field with negligible delay time.

Region N1 is determined by the field with *h*_{BY} < *h*_{ac} and *h*_{BT} < *h*_{ac}. In this region, the condition *τ*_{N,e} < *τ*_{B,e} < 1/(2π*f*) 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 *h*_{NT} < *h*_{ac} < *h*_{BT}. In this region, the condition *τ*_{N,e} < 1/(2π*f*) <*τ*_{B,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 *h*_{BY} *<* *h*_{ac} < *h*_{NT}. In this region, the condition 1/(2π*f*) < *τ*_{N,e} < *τ*_{B,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 *τ*_{B,e} < *τ*_{N,e} < 1/(2π*f*) 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 *τ*_{B,e} < 1/(2π*f*) < *τ*_{N,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/(2π*f*) < *τ*_{B,e} < *τ*_{N,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 *h*_{BY} value is small, and regions N1 and N2 (Néel-dominant regions) are wide. For *σ* = 30, however, *h*_{BY} becomes large, and region B2 becomes wide, corresponding to the increase in the Brownian-dominant region for suspended MNPs.

## IV. EXPERIMENT

As shown in Fig. 6, the behavior of suspended MNPs becomes Brownian- or Néel-dominant depending on the *h*_{ac} 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 *μ*g of MNPs was dispersed in 150 *μ*l of water. The glycerol sample, in which 150 *μ*g of MNPs was included, was prepared by mixing 144 *μ*l of glycerol and 6 *μ*l 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 *M*_{s} = 360 kA/m from the DC *M–H* curve in solution. The distribution of *d*_{c} 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/m^{3,26} *σ* widely ranged from 3 to 29. The distribution of *d*_{H} 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, *τ*_{Β,0} ranged from 8.8 to 326 *μ*s (or *f*_{Β,0} ranged from 18 to 0.5 kHz).

As shown in Eqs. (10) and (13), *H*_{NT} and *H*_{BT} are determined by *σ* and *τ*_{B,0}. Although *σ* and *τ*_{B,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 *H*_{NT}*–f* and *H*_{BT}*–f* curves calculated for the water sample with *σ* = 19.4 and *τ*_{Β,0} = 90 *μ*s. The procedure for determining these values is shown in Figs. 8(b) and 9(a). As shown in Fig. 8(a), the *H*_{NT}*–f* and *H*_{BT}*–f* curves intersect at *μ*_{0}*H*_{ac} = *μ*_{0}*H*_{BY} = 12 mT for the water sample. We note that *f*_{B,0} and *H*_{BY} 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 *H*_{ac} > *H*_{NT} and *H*_{ac} < *H*_{NT}, respectively.

We first measured the Im[*M*_{1}]–*H*_{ac} curve for the glycerol sample. The results are shown in Fig. 8(b) for *f* = 3, 10, and 20 kHz. The Im[*M*_{1}]–*H*_{ac} curve had a peak value at a specific value of *H*_{ac}, which increased with increasing *f*. This was consistent with the simulation results shown in Fig. 5(a). The measured Im[*M*_{1}]–*H*_{ac} curve, however, had a broad peak owing to the distribution of *d*_{c} in the sample. As shown in Fig. 5(b), the field that gives the peak of the Im[*M*_{1}]–*H*_{ac} curve is approximately equal to *H*_{NT} for the Néel-dominant case. Therefore, for simplicity, we estimated *H*_{NT} from the field that gave the peak of the Im[*M*_{1}]–*H*_{ac} curve. We obtained *μ*_{0}*H*_{NT} = 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 *μ*_{0}*H*_{NT}. When we calculated the *H*_{NT}*–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[*M*_{1}]–*H*_{ac} 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[*M*_{1}]–*H*_{ac} curve between the water and glycerol samples became small for *μ*_{0}*H*_{ac} > 12 mT. Therefore, we estimated *μ*_{0}*H*_{BY} = 12 mT for this sample. Substituting this value into Eq. (14), we obtained *τ*_{Β,0} = 90 *μ*s, and, thus, the *H*_{BT}*–f* curve in Fig. 8(a) was calculated using *τ*_{Β,0} = 90 *μ*s. For the water sample, the Im[*M*_{1}]–*H*_{ac} curve for *f* = 3 kHz had a peak value at a specific field *μ*_{0}*H*_{ac} = 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 *H*_{BT}*–f* curve.

In Fig. 9(c), the ratio of Im[*M*_{1}] for the water sample to that for the glycerol sample is shown as a function of *H*_{ac}. For *f* = 3 kHz, the ratio was close to 1 for *H*_{ac} > *H*_{BY}, which means that Im[*M*_{1}] 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 *H*_{ac} > *H*_{BY} 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 *H*_{ac} < *H*_{BY}, however, the ratio significantly increased with decreasing *H*_{ac}, as shown in Fig. 9(c). This result indicates that the Im[*M*_{1}]–*H*_{ac} 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 *H*_{ac} < *H*_{BY}, and *H*_{ac} existed near *H*_{BT} 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[*M*_{1}]–*H*_{ac} curves for *f* = 20 kHz. The difference in the Im[*M*_{1}]–*H*_{ac} curve between the water and glycerol samples was small for all *H*_{ac} values. As shown in Fig. 9(c), the ratio of Im[*M*_{1}] for the water sample to that for the glycerol sample was approximately 1.1 for *H*_{ac} > *H*_{BY}. The reason for this small difference remains unclear. The ratio only slightly increased with decreasing *H*_{ac}, even for *H*_{ac} < *H*_{BY}. Furthermore, MNPs in the water sample operated in region B3 for *H*_{ac} < *H*_{BY}, as shown in Fig. 8(a). Because *μ*_{0}*H*_{BT} is calculated as 46 mT for *f* = 20 kHz using Eq. (13), the field *H*_{ac} for region B3 existed far below *H*_{BT}. 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 *H*_{ac} < *H*_{BY} became larger than that for *f* = 20 kHz. This is because *H*_{BT} for *f* = 10 kHz becomes smaller than that for *f* = 20 kHz, where *μ*_{0}*H*_{BT} = 23 and 46 mT for *f* = 10 and 20 kHz, respectively. As a result, the field *H*_{ac} for region B3 became closer to *H*_{BT} 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.

## V. CONCLUSION

We investigated the field-dependent Néel relaxation time of MNPs, *τ*_{N,e}(*H*_{ac}), under an applied AC excitation field and obtained an empirical expression for *τ*_{N,e}(*H*_{ac}) 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 *τ*_{N,e}(*H*_{ac}), *τ*_{B,e}(*H*_{ac}), and 1/(2π*f*). We also obtained characteristic fields for the boundaries between these types (i.e., *H*_{BY}, *H*_{NT}, and *H*_{BT}). 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.

## ACKNOWLEDGMENTS

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.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**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).

## DATA AVAILABILITY

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