# An Improved Method for Estimating Core Size Distributions of Magnetic Nanoparticles via Magnetization Harmonics

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## Abstract

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## 1. Introduction

## 2. Models and Methods

_{ac}= Hsin(ωt), can be described by the Langevin function [12,17,18,19]. Here, H is the amplitude of AC excitation field, w = 2πf is the angular frequency, and f is the frequency of the AC excitation field. The ensemble magnetization M can be described as follows:

_{T}is the total volume of the MNPs, D

_{min}and D

_{max}are the minimum and maximum, k

_{B}is Boltzmann’s constant, T is the absolute temperature, M

_{s}is the saturation magnetization, m is the magnetic moment, and µ

_{0}is the permeability of the vacuum.

_{2j-1}and A

_{2j-1}can be expressed as:

_{2j−1}and C

_{2j−1}represent the (2j−1)-th harmonic amplitude of magnetization from single MNP with D

_{k}and ensemble of MNPs, respectively. As shown in Equation (6), the contribution of the MNP with D

_{k}to the amplitude of the harmonics is given by the factor n(D

_{k})V(D

_{k}).

_{2j-1}was measured for N different amplitudes of the AC excitation field, i.e., H

_{i}(i = 1, 2, …, N). By introducing the N × 1 vector

**C**

_{2j-1}with component C

_{2j-1}(H

_{i}), the K × 1 vector

**X**with component n(D

_{k})V(D

_{k}), and the N × K matrix

**B**

_{2j-1}with component $\frac{1}{{V}_{T}}{A}_{2j-1}\left({H}_{i},{D}_{k}\right)\mathsf{\Delta}D$, the following equation can be obtained:

_{k}can be given as:

_{i}, the component C

_{2j-1}(H

_{i}) of vector

**Y**can be obtained via a digital phase sensitive detection (DPSD) algorithm. The component A

_{ik}of vector

**B**can be determined based on Equations (9)–(11), and thus the vector

**X**with component n(D

_{k})V(D

_{k}) can be obtained by solving the inverse problem given by Equation (8).

## 3. Simulation Results

_{k}were calculated via Equations (9)–(11), respectively, and the core size distribution was estimated by the SVD algorithm based on Equation (8). Figure 2d illustrates the estimated core size distributions for each MNP sample with different parameters. The solid lines indicate the original core size distributions, while symbols indicate the distributions estimated with harmonic amplitudes. There is excellent agreement between the estimated distributions and the original ones. Moreover, we used the estimated distributions to reconstruct the harmonic amplitudes of magnetization. As shown in Figure 2a–c, the first, third, and fifth harmonics of magnetization reconstructed by estimated distributions agree well with those of the original magnetization.

## 4. Experimental Results and Discussion

_{2}O with 0.03% NaN

_{3}. The AC susceptibility was first evaluated within the frequency range 10–100 kHz to choose an available frequency without the effect of Brownian relaxation. Then we measured the magnetization harmonics of the MNP sample at different AC excitation fields and estimated the core size distribution via Equation (8). Finally, the distribution was estimated via the M–H curve method to compare the results with that based on magnetization harmonics.

#### 4.1. Excitation Frequency without the Effect of Brownian Relaxation

_{0}with a frequency ranging over 10–100 kHz. The AC susceptibility of the MNP sample is plotted in Figure 3. The imaginary part had a peak at frequency f = 1/(2πτ

_{B}), where τ

_{B}is the Brownian relaxation time. When the excitation frequency was less than 1 kHz, the real part of the AC susceptibility approached a constant value (DC susceptibility), which indicated that the effect of Brownian relaxation on the MNP magnetization response could be ignored at low frequencies. Note that the magnetization of noninteracting and identical MNPs under AC excitation fields with low frequencies can be described by the Langevin function.

#### 4.2. Estimation of Core Size Distribution

_{i}(H = 0) = 0.

#### 4.3. Comparison via Reconstruction of Magnetization and AC Harmonics

#### 4.4. Discussion

_{s}ξ/3 = M

_{s}xsin(ωt)/3; i.e., its magnetization is in the linear regime and only the first harmonic magnetization can be observed for the most part, as shown in Equations (9)–(11). In this case, information on the core size distribution, which is included in C

_{2j-1}in Equation (6), is lost. Consequently, the MNP distribution that satisfies x < 1 cannot be accurately estimated. The core sizes of MNPs with x = 1 are 18.5, 16.8, and 15.6 nm for maximum AC excitation field amplitudes µ

_{0}H

_{max}= 6, 8, and 10 mT, respectively. As shown in Figure 8, there is a trough at 12–15 nm. Therefore, distributions of MNPs with core sizes larger than 15 nm can be accurately obtained for µ

_{0}H

_{max}= 10 mT. As shown in Figure 8, estimated distributions of core sizes larger than 15 nm gradually deviate from that obtained for µ

_{0}H

_{max}= 10 mT with decreasing µ

_{0}H

_{max}. Moreover, the densities at 4.8 nm are strongly affected and deviate from that obtained for µ

_{0}H

_{max}= 10 mT.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Magnetization of MNPs depends on the core size distribution. (

**a**) Waveforms of MNP magnetization for different core size distributions. (

**b**) Frequency spectrum. The sampling frequency was 1 MHz and the sampling cycle was 50. Waveforms of the magnetization at equilibrium for each excitation field amplitude are plotted in (

**a**).

**Figure 2.**Simulation results of the (

**a**) 1st, (

**b**) 3rd, (

**c**) 5th harmonics of MNP magnetization, and (

**d**) the estimated MNP core size distributions using the harmonic amplitudes.

**Figure 4.**The (

**a**) magnetization of MNP samples under different magnetic field excitations, and the (

**b**) 1st, (

**c**) 3rd, (

**d**) 5th harmonics of MNP magnetization. The AC magnetic excitation field was 1.5–10 mT, with a step size of 0.5 mT at a frequency of 200 Hz. The experimental data at different magnetic field excitations was shown in one period.

**Figure 5.**The core size distribution of SHP-20 estimated by harmonics amplitudes via the singular value decomposition (SVD) method (red) and M–H curves via the NNLS method (blue).

**Figure 6.**Experimental and reconstructed static M–H curves of SHP20. The illustration is the error between the experimental data and the reconstructed results.

**Figure 7.**Experimental and reconstructed results of harmonic amplitudes. The points are experimental data, the red lines indicate analytical results reconstructed with the core size distribution estimated via harmonics amplitudes, and the blue lines represent that estimated by M–H curves. (

**a**–

**c**) are the 1st, 3rd, and 5th harmonics, respectively.

**Figure 8.**Core size distribution of SHP-20 estimated by magnetization harmonics with different excited magnetic field strengths.

k = 1 | k = 2 | |
---|---|---|

Weight ω_{k} | 0.9 | 0.1 |

Geometric mean μ_{k} (nm) | 21 | 34 |

Geometric standard deviation σ_{k} | 0.16 | 0.06 |

k = 1 | k = 2 | |
---|---|---|

Weight ω_{k} | 0.3 | 0.7 |

Geometric mean μ_{k} (nm) | 16 | 30 |

Geometric standard deviation σ_{k} | 0.3 | 0.1 |

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**MDPI and ACS Style**

Sun, Y.; Ye, N.; Wang, D.; Du, Z.; Bai, S.; Yoshida, T.
An Improved Method for Estimating Core Size Distributions of Magnetic Nanoparticles via Magnetization Harmonics. *Nanomaterials* **2020**, *10*, 1623.
https://doi.org/10.3390/nano10091623

**AMA Style**

Sun Y, Ye N, Wang D, Du Z, Bai S, Yoshida T.
An Improved Method for Estimating Core Size Distributions of Magnetic Nanoparticles via Magnetization Harmonics. *Nanomaterials*. 2020; 10(9):1623.
https://doi.org/10.3390/nano10091623

**Chicago/Turabian Style**

Sun, Yi, Na Ye, Dandan Wang, Zhongzhou Du, Shi Bai, and Takashi Yoshida.
2020. "An Improved Method for Estimating Core Size Distributions of Magnetic Nanoparticles via Magnetization Harmonics" *Nanomaterials* 10, no. 9: 1623.
https://doi.org/10.3390/nano10091623